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TITLE: Connection between splitting field and quotient ring of polynomials
QUESTION [0 upvotes]: Suppose that $f(x) \in \mathbb{F}_p[x]$ is an irreducible polynomial. We will call a field $L$ a splitting field, if $f(x) = (x-a_1) \ldots (x-a_n)$ in $L$. Also i know how to construct a field from $f(x)$ and ring of polynomials $\mathbb{F}_p[x]$. Just by taking a quotient ring $\mathbb{F}_p[x] / (f)$, and doing this we add a root of $f(x)$. Is it true that $\mathbb{F}_p[x] / (f)$ is a splitting field of $f(x)$? If no, what is a counterexample?And what is if we look at an arbitrary field $F$ instead of $\mathbb{F}_p$?
REPLY [0 votes]: $$K=\mathbb F_p[x]/(f(x))$$ is a field of $p^d$ elements, with $d=\deg f.$ The field has, as automorphisms:
$$\phi_k:\alpha\mapsto \alpha^{p^k}, k=0,1,\dots,d-1$$ which fix $\mathbb F_p.$
This means $\phi_k(\langle x\rangle)$ are all roots of $f.$ If $f$ doesn’t split, then you must have $$\phi_i(\langle x\rangle)=\phi_j(\langle x\rangle),$$ for some $0\leq i<j<d.$
Then $$\langle x\rangle^{p^i}=\langle x\rangle ^{p^j}$$ or $x^{p^j}-x^{p^i}$ is divisible by $f(x)$ in $\mathbb F_p[x].$ But $$x^{p^j}-x^{p^i}=(x^{p^{j-i}}-x)^{p^i}$$ in $\mathbb F_p[x].$
So this means $f$ must divide $x^{p^k}-x$ for $1\leq k=j-i <d-1.$
But $\phi_k(\langle x\rangle)=\langle x\rangle$ means, for any polynomial $p(x)\in\mathbb F_p[x],$ $$\phi_k(\langle p( x)\rangle)=\langle p(x)\rangle.$$ But that means $x^{p^k}-x$ has every element of $K$ as a root, and $|K|=p^d>p^k.$
The finite fields are special. This is true for irreducible $f\in \mathbb F_q[x]$ over any finite field $\mathbb F_q,$ but is not in general true over other fields, even fields of finite characteristic.
For example, in the field $F=\mathbb F_p(x),$ the field of fractions of $\mathbb F_p[x],$ the polynomial $y^p-x$ does not split in $F[y]/(y^p-x).$
| 130,150
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By Uran Botobekov
May 2, 2019, the CACI Analyst
In January 2019, the Central Asian terrorist group Katibat Tawhid wal Jihad (KTJ) publicly renewed its Bayat (oath of allegiance) to Ayman al-Zawahiri, al Qaeda’s global chief. More than seven years after the killing of Osama bin Laden, al Qaeda continues to attract Central Asian Salafi groups. This trend has intensified since the start of the Syrian civil war, where several thousand radical Islamists from Central Asia went to wage Jihad. The conflict has allowed al Qaeda to claim Syria as its newest and most important safe haven for a global Jihad ideology. The ideological assimilation of the Central Asian groups with al Qaeda took place precisely in Syria..
| 37,489
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Melanie is a passionate ambassador for New Zealand Riding for the Disabled (NZRDA) and a volunteer at North Shore RDA in Stillwater.
Riding develops and improves balance, coordination, and muscle tone. Interaction with horses improves concentration, self-discipline and self-esteem. Not to mention the improvement in communication and social skills. A horse it seems is a far better listener that you or I could ever be!
RDA receives no government funding, relying on sponsorship, grants and donations.
Reaching more Riders. Changing more lives.
If you can help please visit or call 04 234 6090.
Melanie's.
| 319,851
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:
Guys,
The last year has been a rather difficult one for me and I have spent a lot of time soul searching and trying to figure out which end is up. One thing in particular has taken me quite a long time to come to terms with, and I want to address it here and now because it's way past due.
Within the past year, I have fallen into a state of depression, which has deeply affected my life. I have had a hard time sleeping at night, eating, and have lost a lot of interest in swimming. For the first time in my career, I was ready to quit swimming at the end of the summer. Not many people have noticed something has been wrong. The ones that did, I just blamed all the stress on my schoolwork, which has been my scapegoat. But, I miss sleeping at night; being happy and being the person I used to be when it didn't really matter. learnt that this is normal and its. I simply couldn't continue to hide this from you anymore because it was slowly killing me. I want you to know me for who I am. I pray that this will not change anything, but I know for some of you this is uneasy. I want you all to know that I am here to have an open dialogue. If you have questions or concerns please be honest with me as I am being with you. Do not hesitate to ask me questions if you have any (appropriate ones). I know this email has been a bit heavy and I apologize for that. All I can do is hope you will accept me for who I am and realize that I really haven't changed. I am still the same me.
I love you guys & thank you for listening.
Best,
Matt
Congratulations, Matt!.
| 171,545
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Great Idea for a toy, BUT...... worthless for chewers!
Verified Buyer Reviewed by dramy from ny on Monday, April 11, 2016
My dog loved the noise that this toy made. However,TRULY in 5 minutes, he had it torn apart! He is a 90 lb german shorthaired pointer mix...
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| 406,771
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As Mother's Day has passed Boots have slashed the price of some of their Gift Sets down to just £16 each, when they're worth as much as £49! Choose from Ted Baker, Sanctuary, Soap & Glory and more, but you'll need to move fast as these will fly out at this price!
Here's what you can get:
- Ted Baker Blush Bouquet Cosmetic Bag Gift* worth £31 now £16
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- Sanctuary Spa With Love Gift* worth £35 now £16
- Champneys Reward & Restore Gift Set* worth £49 now £16
- Joules Wonderful Weekend Bag* was £24 now £16
- Jack Wills Bathing Gift Set* was £18 now £16
I have my eye on the Champneys to tuck away as a pressie as it's such a huge saving, and a bit of Sopa & Glory is always welcome in my house, so I may have to bag a treat for myself too!
Click and collect is free, home delivery costs from £3.50 or is free on orders of £45 or more.
I want the ted baker set
Il order them all then
thanks !
Sanctuary one is lovely xxx
... quite like the ted one!
Ooo I swear you’ll fuelling my shopping addiction lol xx
these r amazingive ordered some Christmas pressies
I like the Ted Baker one xx
What’s stopping you get on it
I am gonna I was showing you as well
Got a few bits fab bargains
What do you think?
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\begin{document}
\title{Pointwise estimates for the derivative of algebraic polynomials}
\author{Adrian Savchuk}
\address{Taras Shevchenko National University of Kyiv, Kyiv, Ukraine}
\curraddr{Faculty of Mathematics and Mechanics, Academician Glushkov Avenue 4,
02000, Kyiv, Ukraine}
\address{University of Toulon, Toulon, France}
\curraddr{UFR Sciences et Techniques, 83130, La Garde, France}
\email{adrian.savchuk.v@gmail.com, adrian-savchuk@etud.univ-tln.fr}
\subjclass[2020]{Primary 30A08, 30E10; Secondary 41A20}
\keywords{Algebraic polynomial, Logarithmic derivative, Bernstein inequality}
\begin{abstract}
We give the sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum_{k=0}^{n}a_kz^k$, $a_n\not=0,$ for the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$
to be true for all $z\in\overline{\mathbb D}:=\{w\in\mathbb C : |w|\le 1\}$.
\end{abstract}
\maketitle
\section{Introduction and main result}
Let $P$ be an algebraic polynomial with complex coefficients and let $z_1, z_2,\ldots, z_m$ be a distinct zeros of $P$ with multiplicities $r_1, r_2,\ldots, r_m$ respectively, $\sum_{k=1}^{m}r_k=\deg P$. Further we assume that $z_k$ are numerated in arbitrary manner so that $|z_1|\le|z_2|\le\cdots\le|z_m|$.
Consider the real part of the logarithmic derivative of $P$. We have
\begin{eqnarray}\label{fraction}
\mathop{\rm Re}\frac{zP'(z)}{P(z)}&=&\mathop{\rm Re}\sum_{k=1}^{m}\frac{r_kz}{z-z_k}\\
&=&\frac{n}{2}+\frac{1}{2}\sum_{k=1}^{m}r_k\frac{|z|^2-|z_k|^2}{|z-z_k|^2}\label{fraction2},
\end{eqnarray}
where $n=\deg P$. It follows that for all $z\in\mathbb C\setminus\{z_1,\ldots, z_m\}$,
\begin{equation}\label{lower est}
\left|\frac{n}{2}+\frac{1}{2}\sum_{k=1}^{m}r_k\frac{|z|^2-|z_k|^2}{|z-z_k|^2}\right|\le
\left|\frac{zP'(z)}{P(z)}\right|.
\end{equation}
Denote $\mathbb D:=\{z\in\mathbb D : |z|<1\},$ $\mathbb T:=\{z\in\mathbb C : |z|=1\}$ and we let by $d\sigma$ denote the normalized Lebesgue measure on $\mathbb T$. Assume $z_k\not\in\mathbb T$, $k=1,\ldots,m,$ then by integrating the last inequality along $\mathbb T$ we get
\begin{equation}\label{lower est int}
\sum_{k=1}^{j}r_k\le\int_{\mathbb T}\left|\frac{P'}{P}\right|d\sigma\le\max_{z\in\mathbb T}\left|\frac{P'(z)}{P(z)}\right|,
\end{equation}
where $j$ is positive integer $\le m$ such that $|z_j|<1<|z_{j+1}|$. Here and further we put $\sum_{k=1}^{0}=0$.
From (\ref{lower est}), (\ref{lower est int}) and from the well-known Bernstein inequality, that say that
\begin{equation}\label{B1}
\max_{z\in\mathbb T}|P'(z)|\le n\max_{z\in\mathbb T}|P(z)|,
\end{equation}
we readily conclude that for any algebraic polynomial $P$, $\deg P=n,$ having all its zeros in $\mathbb D$, the following inequalities holds
\begin{equation}\label{ineq}
\frac{n}{1+|z_m|}\le\min_{t\in\mathbb T}\left|\frac{P'(z)}{P(z)}\right|\le n\le \max_{z\in\mathbb T}\left|\frac{P'(z)}{P(z)}\right|.
\end{equation}
The first inequality was observed by Govil \cite{Gov}, the second one is the consequence of (\ref{B1}) and the third one is the consequence of (\ref{lower est int}). All these results are sharp. The equalities occurs for the polynomial $P(z)=a_n(z-c)^n$ for suitable $c\in\mathbb D$.
Assume now that all zeros $z_1,\ldots, z_m$ of $P$ lies in the domain $\mathbb U:=\{z\in\mathbb C : |z|\ge1\}$. Then it follows from (\ref{fraction2}) that
\[
\mathrm{Re}\frac{zP'(z)}{P(z)}\le\frac{n}{2}
\]
for all $z\in\overline{\mathbb D}\setminus\{z_1,\ldots,z_m\}$. This gives, as was noted by Aziz \cite{Aziz} (see also Lemma \ref{prop1} below),
\begin{equation}\label{Aziz ineq}
|zP'(z)|\le|nP(z)-zP'(z)|
\end{equation}
for all $z\in\overline{\mathbb D}$.
On the other side, it is easy to see that if $P$ is a polynomial of degree $n$ having all its zeros in $\mathbb U_2:=\{z\in\mathbb C : |z|\ge 2\}$, then
\begin{eqnarray*}
\max_{z\in\overline{\mathbb D}}\left|\frac{zP'(z)}{P(z)}\right|&\le&\sum_{k=1}^{m}\frac{r_k}{|z_k|-1}\\
&\le&n.
\end{eqnarray*}
This is equivalent to
\begin{equation}\label{B2}
|zP'(z)|\le n|P(z)|
\end{equation}
for all $z\in\overline{\mathbb D}$. We will call the last relation a pointwise Bernstein inequality.
Combining (\ref{Aziz ineq}) and (\ref{B2}), we obtain, for all $z\in\overline{\mathbb D}$,
\begin{equation}\label{B4}
|zP'(z)|\le\min\left(|nP(z)-zP'(z)|, n|P(z)|\right),
\end{equation}
provided $\{z_1,\ldots,z_m\}\in\mathbb U_2$. For example, if $P(z)=(2+z)^n$, then (\ref{B4}) gives
\[
|z|\le\min\left(2,|2+z|\right)=
\begin{cases}
2,\hfill&\mbox{if}~z\in\overline{\mathbb D}\setminus\{w\in\mathbb C: |2+w|\le 2\},\cr
|2+z|,\hfill&\mbox{otherwise},
\end{cases}
\]
for all $z\in\overline{\mathbb D}$. Equality occurs here only in the point $z=-1$.
In this note we give the sufficient condition on coefficients of a polynomial $P$ for the pointwise Bernstein inequality
to be true for all $z\in\overline{\mathbb D}$. As we will see, this condition implies (\ref{B4}) and does not require that all zeros of $P$ must be in $\mathbb U_2$.
For further information about the estimates of derivative and the logarithmic derivative of polynomials we refer to \cite{Mil}, \cite{Rah}, \cite{Sheil-Small}, \cite{Danch} and references therein.
Our main result is the following theorem.
\begin{theorem}\label{main thm} Let $n\in\mathbb Z_+$ and $\{k_\nu\}_{\nu=0}^n$, $0\le k_0<k_1<\ldots<k_n$, be positive integers and let $P(z)=\sum_{\nu=0}^{n}a_\nu z^{k_\nu}$ be an algebraic polynomial of degree $k_n$ with coefficients $\{a_\nu\}_{\nu=0}^n\in\mathbb C\setminus\{0\}$. If
\begin{equation}\label{main condition}
\mathop{\rm \min_{t\in\overline{\mathbb D}}Re}\sum_{j=0}^{n-\nu}\frac{a_{j+\nu}}{a_{\nu}}t^{k_{j+\nu}-k_\nu}\ge\frac{1}{2},\quad\nu=0,1,\ldots,n,
\end{equation}
then the following assertions are holds true:
(i) the polynomial $P$ have no zeros in $\overline{\mathbb D}$, provided $k_0=0$, and have no zeros in $\overline{\mathbb D}\setminus\{0\}$ for $k_0>0$;
(ii) for all $z\in\overline{\mathbb D}$
\begin{equation}\label{B3}
|zP'(z)|\le k_n|P(z)|.
\end{equation}
If $z\in\mathbb D$ the equality occurs here only in case $n=0$, that is for $P(z)=a_0z^{k_0}$, $k_0>0$;
(iii) if $k_0=0$ and $n\ge 1$, then we have
\begin{equation}\label{strong B}
|P'(z)|< k_n|P(z)|
\end{equation}
for all $z\in\mathbb D$.
\end{theorem}
\begin{remark}
Let $P$ be as in Theorem \ref{main thm}. Then we have the implication $(ii)\Rightarrow(i)$.
\end{remark}
This is a consequence of the Riemann's theorem on removable singularities applied to the function
\[
z\mapsto\frac{zP'(z)}{P(z)}=\sum_{k=1}^{m}\frac{r_kz}{z-z_k}.
\]
\begin{corollary}
Let $P$ be as in Theorem \ref{main thm} with $a_0\ge a_1\ge\ldots\ge a_{n}>0$, $n\in\mathbb N$ and $k_0=0$. If
\[
0\le\Delta^2 (a_\nu):=
\begin{cases}
a_{\nu+2}-2a_{\nu+1}+a_\nu,\hfill&\mbox{if}~\nu=0,1,\ldots,n-2,\cr
a_{n-1}-2a_n,\hfill&\mbox{if}~\nu=n-1,\cr
a_n,\hfill&\mbox{if}~\nu=n,
\end{cases}
\]
then there holds
\[
|zP'(z)|\le\min\left(|k_nP(z)-zP'(z)|, k_n|P(z)|\right)
\]
for all $z\in\overline{\mathbb D}$.
\end{corollary}
Indeed, for each $\nu=0,1,\ldots, n$ the sequence $\{\lambda_{k,\nu}\}_{k=0}^{n-\nu+1},$ where
\[
\lambda_{k,\nu}=
\begin{cases}
\displaystyle\frac{a_{k+\nu}}{a_\nu},\hfill&\mbox{if}~k=0,1,\ldots,n-\nu,\cr
0,\hfill&\mbox{if}~k=n-\nu+1,
\end{cases}
\]
is non-negative, monotonically non-increasing and convex, i. e. $\lambda_{0,\nu}\ge\lambda_{1,\nu}\ge\ldots\ge\lambda_{n-\nu,\nu}>\lambda_{n-\nu+1,\nu}=0$ and $\Delta^2(\lambda_{k,\nu})\ge0$ for $k=0,1,\ldots,n-\nu+1$. Thus by Fej\'er theorem (see \cite{Mil}, p. 310) the trigonometric polynomials
\[
\frac{\lambda_{0,\nu}}{2}+\sum_{k=1}^{n-\nu}\lambda_{k,\nu}\cos kx,\quad\nu=0,1,\ldots,n,
\]
are non-negative for all $x\in\mathbb R$. This is equivalent to the condition (\ref{main condition}).
\begin{example}
Let $n\in\mathbb N\setminus\{1\}$ and let
\[
P(z)=\sum_{k=0}^{n}(n+1-k)z^k.
\]
Then for $t=\mathrm e^{\mathrm ix},$ $x\in\mathbb R$, we have
\begin{eqnarray*}
\frac{1}{2}+\mathop{\rm Re}\sum_{k=1}^{n-\nu}\frac{n+1-(k+\nu)}{n+1-\nu}t^k&=&\frac{1}{2}+\sum_{k=1}^{n-\nu}\left(1-\frac{k}{n+1-\nu}\right)\cos kx\\
&=&F_{n-\nu+1}(x)\ge 0,
\end{eqnarray*}
for all $x\in\mathbb R,~\nu=0,1,\ldots,n$, where $F_k$ is the Fej\'er kernel (see \cite{Mil}, pp. 311, 313).
Therefore, combining (\ref{Aziz ineq}) and (\ref{B3}), we get
\[
\left|\sum_{k=1}^{n}(n+1-k)kz^k\right|\le\min\left(\left|\sum_{k=0}^{n-1}(n+1-k)(n-k)z^k\right|,n\left|\sum_{k=0}^{n}(n+1-k)z^k\right|\right).
\]
By Enestr\"om--Kakeya theorem (see \cite{Rah}, p.255) with refinement given by Anderson, Saff and Varga \cite{And} (see Corollary 2), zeros of $P$ satisfy $|z_k|<2$, $k=1,\ldots,n$.
\end{example}
\section{Lemmas}
For the proof of Theorem 1.1 we require the following lemmas.
\begin{lemma}\label{prop1}
Let $P$ and $Q$ be a functions defined on a compact set $K\subset\mathbb C$, $\mathcal Z(Q):=\{z\in\mathbb C : Q(z)=0\}$ and $K\setminus\mathcal Z(Q)\not=\emptyset$. In order that
\[
|P(z)-Q(z)|\le|P(z)|
\]
for all $z\in K$ it is necessary and sufficient that
\[
\inf_{z\in K\setminus\mathcal Z(Q)}\mathop{\rm Re}\frac{P(z)}{Q(z)}\ge\frac{1}{2}.
\]
\end{lemma}
\begin{proof}The assertion readily follows from the obvious identity
\[
\left|\frac{P(z)}{Q(z)}\right|^2-\left|\frac{P(z)}{Q(z)}-1\right|^2=2\mathbb{\rm Re}\frac{P(z)}{Q(z)}-1
\]
for $z\in K\setminus\mathcal Z(Q)$.
\end{proof}
\begin{lemma}
\label{lemma2}
Let $P(z)=\sum_{j=0}^{n}a_jz^j$, $n\in\mathbb N,$ and $a_n\not=0$. Then for all $z\in\mathbb C\setminus\{0\}$ and $w\in\mathbb C$ we have
\[
\left|z\frac{P(z)-P(w)}{z-w}\right|\le A(z,w)\max_{k=0,\ldots,n-1}\left|P(z)-\sum_{j=0}^{k}a_jz^j\right|,
\]
where
\[
A(z,w)=
\begin{cases}
\displaystyle \frac{|z|^n-|w|^n}{|z|^{n-1}(|z|-|w|)},\hfill&\mbox{if}~|z|\not=|w|,\cr
n,\hfill&\mbox{if}~|z|=|w|.
\end{cases}
\]
The result is best possible and the equality holds for the polynomial $P(z)=a_0+a_nz^n$ in case $\arg z=\arg w$.
\end{lemma}
\begin{proof} Fix $z\in\mathbb C\setminus\{0\}$. Summation by parts yields
\[
P(w)=
P(z)\left(\frac{w}{z}\right)^n+\left(1-\frac{w}{z}\right)\sum_{k=1}^{n}\left(\sum_{j=0}^{k-1}a_jz^j\right)\left(\frac{w}{z}\right)^{k-1}.
\]
This gives
\begin{equation}\label{main formula}
z\frac{P(z)-P(w)}{z-w}=\sum_{k=0}^{n-1}\left(P(z)-\sum_{j=0}^{k}a_jz^j\right)\left(\frac{w}{z}\right)^k.
\end{equation}
The result follows.
\end{proof}
\section{Proor of Theorem \ref{main thm}}
Denote
\[
\rho_k(P)(z):=\sum_{j=k}^{k_n}c_jz^j,~k=0,1,\ldots,k_n,
\]
where
\[
c_j=
\begin{cases}
0,\hfill&\mbox{if}~j\not\in\{k_\nu\}_{\nu=0}^n,\cr
a_j,\hfill&\mbox{if}~j\in\{k_\nu\}_{\nu=0}^n.
\end{cases}
\]
(i) By Lemma \ref{prop1} the conditions (\ref{main condition}) are equivalent to
\begin{equation}\label{decrease ineq}
|P(z)|\ge|\rho_{k_0}(P)(z)|\ge\ldots\ge|\rho_{k_n}(P)(z)|=\left|a_{n}z^{k_n}\right|\quad\forall z\in\overline{\mathbb D}.
\end{equation}
This gives that $P(z)\not=0$ for $z\in\overline{\mathbb D}\setminus\{0\}$. If $k_0=0$ then in addition $P(0)=a_{k_0}\not=0$.
(ii) It follows from (\ref{decrease ineq}) that the sequence $\{|\rho_{k_\nu}(P)(z)|\}_{\nu=0}^n$ is non-increasing. Since $\rho_{j}(P)=\rho_{k_\nu}(P)$
for $k_{\nu-1}<j\le k_{\nu}$, $\nu=0,1,\ldots,n,$ where $k_{-1}=-1$, we conclude that the sequence $\{|\rho_j(P)(z)|\}_{j=0}^{k_n}$ also is non-increasing. Therefore by Lemma \ref{lemma2} we get
\begin{eqnarray*}
\left|z\frac{P(z)-P(zt)}{1-t}\right|&\le& k_n|\rho_{n_0}(P)(z)|\\
&\le&k_n|P(z)|
\end{eqnarray*}
for all $t\in\mathbb T$. In particularly, for $t=1$ we obtain (\ref{B3}).
Now assume that for some $z\in\mathbb D$ in (\ref{B3}) occurs equality. Then according to the assertion (i) proved above, the function
\[
F(t):=\frac{tP'(t)}{k_nP(t)}=\frac{k_0}{k_n}+\frac{(k_1-k_0)a_1}{k_na_0}t^{k_1-k_0}+\ldots
\]
is holomorphic in $\mathbb D$, $|F(t)|\le 1$ for all $t\in\mathbb D$ and $|F(z)|=1$. Therefore by maximum modulus principle $F(t)=c$ for all $t\in\mathbb D$ with $|c|=1$. But $F(0)=k_0/k_n$. So $c=k_0/k_n=1$. These are equivalent to $n=0$ and $P(t)=\mathrm{e^M}t^{k_0}$ for some $M\in\mathbb C$.
(iii) Let $k_0=0$. Then according to the above conclusions about the function $F$, we have that $F(0)=0$. Therefore by Schwarz lemma we get $|F(t)|\le|t|$ for all $t\in\mathbb D$. Moreover, if $|F(z)|=|z|$ for some $z\in\mathbb D\setminus\{0\}$, then $F(t)=ct$ for some $c\in\mathbb C$ with $|c|=1$. Once this is done, it follows that
\[
c=F'(t)=\frac{k_1^2a_1}{k_na_0}t^{k_1-1}+\cdots,\quad t\in\mathbb D.
\]
Hence, necessarily $k_1=1$ and $|a_1|=k_n|a_0|$. However, under condition (\ref{main condition}),
\begin{eqnarray*}
\left|\frac{a_1}{a_0}\right|&=&\left|\int_\mathbb Tt^{k_1-k_0}\mathrm{Re}\left(1+2\sum_{j=1}^n\frac{a_j}{a_0}t^{k_j-k_0}\right)d\sigma(t)\right|\\
&\le&\int_\mathbb T\mathrm{Re}\left(1+2\sum_{j=1}^n\frac{a_j}{a_0}t^{k_j-k_0}\right)d\sigma(t)\\
&=&1.
\end{eqnarray*}
Thus $k_n=1$ or equivalently, $n=1$. On the other side, for $n=1$ the condition (\ref{main condition}) implies $|a_0|\ge 2|a_1|$. This is a contradiction. Hence, $|F(t)|<|t|$ for all $t\in\mathbb D$.
The proof is complete.
\bibliographystyle{amsplain}
| 85,562
|
Abstract
For transient symmetric Levy processes we determine a uniform lower bound for the Hausdorff dimension of the range of a process on various time sets. This complements earlier work which provided a uniform upper bound. An example is provided in which both bounds are attained.
Citation
W. J. Hendricks. "A Uniform Lower Bound for Hausdorff Dimension for Transient Symmetric Levy Processes." Ann. Probab. 11 (3) 589 - 592, August, 1983.
Information
| 46,670
|
\begin{document}
\title{ Polynomial equations
and rank of matrices over $\mathbb{F}_{2} $ related to persymmetric matrices}
\author{Jorgen~Cherly}
\address{D\'epartement de Math\'ematiques, Universit\'e de
Brest, 29238 Brest cedex~3, France}
\email{Jorgen.Cherly@univ-brest.fr}
\email{andersen69@wanadoo.fr}
\maketitle
\begin{abstract}
Dans cet article nous illustrons par quelques exemples la connexion entre le nombre de solutions
d'\' equations polyn\^omiales satisfaisant des conditions de degr\' es et le nombre de rang i matrices rattach\' ees aux
matrices persym\' etriques
\end{abstract}
\selectlanguage{english}
\begin{abstract}
In this paper we illustrate by some examples the connection between the number of solutions of polynomial
equations satisfying degree conditions and the number of rank i matrices related to persymmetric matrices.
\end{abstract}
\newpage
\maketitle
\newpage
\tableofcontents
\newpage
\section{Notations}
\label{sec 1}
A matrix $[\alpha_{i,j}] $ is persymmetric if $\alpha _{i,j} = \alpha_{r,s}$ for i+j=r+s.\\
\vspace{0.1 cm}
We denote by $ \Gamma_{i}^{s\times k} $ the number of rank i persymmetric $s\times k $ matrices over $\mathbb{F}_{2}$
of the form :
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \alpha _{4}& \ldots & \alpha _{k} & \alpha _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\alpha _{s-1} & \alpha _{s} & \alpha _{s+1} & \ldots & \alpha _{k+s-3} & \alpha _{k+s-2} \\
\alpha _{s} & \alpha _{s+1} & \alpha _{s+2} & \ldots & \alpha _{k+s-2} & \alpha _{k+s-1} \\
\end{array} \right). $$ \vspace{0.1 cm}\\
We denote by $ \Gamma_{i}^{\left[s\atop s+m \right]\times k} $
the number of rank i double persymmetric $(2s+m) \times k $ matrices over $\mathbb{F}_{2}$
of the form:\vspace{0.1 cm}
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \alpha _{4}& \ldots & \alpha _{k} & \alpha _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\alpha _{s-1} & \alpha _{s} & \alpha _{s +1} & \ldots & \alpha _{s+k-3} & \alpha _{s+k-2} \\
\alpha _{s} & \alpha _{s+1} & \alpha _{s +2} & \ldots & \alpha _{s+k-2} & \alpha _{s+k-1} \\
\hline \\
\beta _{1} & \beta _{2} & \beta _{3} & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \beta _{4} & \ldots & \beta_{k} & \beta _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\beta _{m+1} & \beta _{m+2} & \beta _{m+3} & \ldots & \beta_{k+m-1} & \beta _{k+m} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\beta _{s+m-1} & \beta _{s+m} & \beta _{s+m+1} & \ldots & \beta_{s+m+k-3} & \beta _{s+m+k-2} \\
\beta _{s+m} & \beta _{s+m+1} & \beta _{s+m+2} & \ldots & \beta_{s+m+k-2} & \beta _{s+m+k-1}
\end{array} \right). $$ \vspace{0.1 cm}\\
We denote by $ \Gamma_{i}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k} $
the number of rank i double persymmetric $(3s+2m+l) \times k $ matrices over $\mathbb{F}_{2}$
of the form:\vspace{0.1 cm}
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \alpha _{k} & \alpha _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\alpha _{s-1} & \alpha _{s} & \ldots & \alpha _{s+k-3} & \alpha _{s+k-2} \\
\alpha _{s } & \alpha _{s +1} & \ldots & \alpha _{s +k-2}& \alpha _{s +k-1}\\
\hline \\
\beta _{1} & \beta _{2} & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \beta_{k} & \beta _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\beta _{m+1} & \beta _{m+2} & \ldots & \beta_{k+m-1} & \beta _{k+m} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\beta _{s+m-1} & \beta _{s+m} & \ldots & \beta_{s+m+k-3} & \beta _{s+m+k-2} \\
\beta _{s+m} & \beta _{s+m+1} & \ldots & \beta _{s+m+k-2} & \beta _{s+m+k-1}\\
\hline \\
\gamma _{1} & \gamma _{2} & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3} & \ldots & \gamma _{k} & \gamma _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{m+1} & \gamma _{m+2} & \ldots & \gamma _{k+m-1} & \gamma _{k+m} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{s+m-1} & \gamma _{s+m} & \ldots & \gamma _{s+m+k-3} & \gamma _{s+m+k-2} \\
\gamma _{s+m} & \gamma _{s+m+1} & \ldots & \gamma _{s+m+k-2} & \gamma _{s+m+k-1}\\
\gamma _{s+m+1} & \gamma _{s+m+2} & \ldots & \gamma _{s+m+k-1} & \gamma _{s+m+k}\\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{s+m+l} & \gamma _{s+m+l+1} & \ldots & \gamma _{s+m+l+k-2} & \gamma _{s+m+l+k-1}
\end{array} \right). $$
\newpage
\section{Polynomial equations related to persymmetric matrices}
\label{sec 2 }
\begin{example}
The number $ \Gamma_{i}^{5\times 5} $ of persymmetric $5\times 5$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {ccccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} & \alpha _{5} \\
\alpha _{2} & \alpha _{3} & \alpha_{4} & \alpha _{5} & \alpha _{6} \\
\alpha _{3} & \alpha _{4} & \alpha_{5} & \alpha _{6} & \alpha _{7} \\
\alpha _{4} & \alpha _{5} & \alpha_{6} & \alpha _{7} & \alpha _{8} \\
\alpha _{5} & \alpha _{6} & \alpha_{7} & \alpha _{8} & \alpha _{9}
\end{array} \right). $$
is equal to \\
\begin{equation*}
\left\{\begin{array}{ccc}
1 & if & i= 0, \\
3 & if & i= 1, \\
12 & if & i= 2, \\
48 & if & i= 3, \\
192 & if & i= 4, \\
256 & if & i= 5.
\end{array}\right.\
\end{equation*}
The general formula is given by D.E. Daykin [1] and in [2] in the case q = 2.\\
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1, \ldots,Y_q,Z_q) \in \big(\mathbb{F}_{2}[T] \big)^{2q} $ of the polynomial equation
$$ Y_1Z_1 + Y_2Z_2 + \ldots + Y_qZ_q = 0 $$ \\
satisfying the degree conditions \\
$$ degY_j \leq 4 , \quad degZ_j \leq 4 \quad for 1\leq j \leq q, $$ \\
are connected with the numbers $ \Gamma_{i} $ of persymmetric
$5\times 5$ matrices over $\mathbb{F}_{2} $
of rank i, in the way that the number $ R_{q} $ can be written as a linear combination of the $\Gamma_{i}.$\\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{10q-9}\sum_{i=0}^{5}\Gamma_{i}2^{-qi} \\
& = \left\{\begin{array}{ccc}
63 & if & q=1,\\
8704 & if & q=2, \\
2^{10q-9}\left[1+ 3\cdot2^{-q} +12\cdot2^{-2q} + 48\cdot2^{-3q} +192\cdot2^{-4q} +256\cdot2^{-5q} \right]
& if & 3\leq q.
\end{array}\right.
\end{align*}
\end{example}
\begin{example}
The number $ \Gamma_{i} = \Gamma_{i}^{2\times 2} $ of persymmetric $2\times 2$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {ccccc}
\alpha _{1} & \alpha _{2} \\
\alpha _{2} & \alpha _{3}
\end{array} \right). $$
is equal to \\
\begin{equation*}
\left\{\begin{array}{ccc}
1 & if & i= 0, \\
3 & if & i= 1, \\
4& if & i= 2, \\
\end{array}\right.\
\end{equation*}
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1, \ldots,Y_q,Z_q) \in \big(\mathbb{F}_{2}[T] \big)^{2q} $ of the polynomial equation
$$ Y_1Z_1 + Y_2Z_2 + \ldots + Y_qZ_q = 0 $$ \\
satisfying the degree conditions \\
$$ degY_j \leq 1 , \quad degZ_j \leq 1 \quad for 1\leq j \leq q, $$ \\
is connected with the numbers $ \Gamma_{i} $ of persymmetric
$2\times 2$ matrices over $\mathbb{F}_{2} $
of rank i, in the way that the number $ R_{q} $ can be written as a linear combination of the $\Gamma_{i}.$\\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{4q-3}\sum_{i=0}^{2}\Gamma_{i}2^{-qi} \\
& = \left\{\begin{array}{ccc}
7 & if & q=1,\\
64 & if & q=2, \\
2^{4q-3}\left[1+ 3\cdot2^{-q} +4\cdot2^{-2q} \right]
& if & 3\leq q.
\end{array}\right.
\end{align*}
\end{example}
\begin{example}
The fraction of square persymmetric matrices which are invertible is equal to $ {1\over 2} $
\end{example}
\begin{example}
The number $ \Gamma _{i}^{\Big[\substack{5 \\ 1 }\Big] \times 5}$ of rank i matrices of the form \\
$$ \left ( \begin{array} {ccccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} & \alpha _{5} \\
\alpha _{2} & \alpha _{3} & \alpha_{4} & \alpha _{5} & \alpha _{6} \\
\alpha _{3} & \alpha _{4} & \alpha_{5} & \alpha _{6} & \alpha _{7} \\
\alpha _{4} & \alpha _{5} & \alpha_{6} & \alpha _{7} & \alpha _{8} \\
\alpha _{5} & \alpha _{6} & \alpha_{7} & \alpha _{8} & \alpha _{9}\\
\hline
\beta _{1} & \beta _{2} & \beta _{3} & \beta_{4} & \beta_{5}
\end{array} \right) $$
is equal to \vspace{0.1 cm} \\
$$ 2^{i}\Gamma _{i}^{5 \times 5}+
(2^{5}-2^{i-1})\cdot\Gamma _{i-1}^{5\times 5}\quad for\quad 0\leq i\leq 5, $$\vspace{0.1 cm}\\
\begin{equation*}
= \begin{cases}
1 & \text{if } i = 0, \\
37 & \text{if } i=1, \\
138 & \text{if } i = 2, \\
720 & \text{if } i = 3, \\
4224 & \text{if } i = 4. \\
11264 & \text{if } i = 5. \\
\end{cases}
\end{equation*}
\vspace{0.1 cm}
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1,U_{1}, Y_2,Z_2,U_{2},\ldots , Y_q,Z_q,U_{q} ) \in \big(\mathbb{F}_{2}[T] \big)^{3q} $ \vspace{0.5 cm}\\
of the polynomial equations\\
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0 \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 4 , \quad degZ_j \leq 4 ,
\quad degU_{j}\leq 0 , \quad \text{for} \quad 1\leq j \leq q $$ \\
is connected with the numbers $\Gamma _{i}^{\Big[\substack{5 \\ 1 }\Big] \times 5}$
, in the way that the number $ R_{q} $ can be written as a linear combination of the $\Gamma _{i}^{\Big[\substack{5 \\ 1 }\Big] \times 5}$\\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{11q-14}\sum_{i = 0}^{5}
\Gamma _{i}^{\left[\stackrel{5}{1}\right]\times 5}2^{-iq}\\
& = \left\{\begin{array}{ccc}
95 & if & q=1,\\
14752 & if & q=2, \\
2^{11q-14}\left[1+ 37\cdot2^{-q} +138\cdot2^{-2q} + 720\cdot2^{-3q} +4224\cdot2^{-4q} +11264\cdot2^{-5q} \right]
& if & 3\leq q.
\end{array}\right.
\end{align*}
\end{example}
\newpage
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop (2) \right]\times 4} $ of rank i matrices of the form \\
\begin{displaymath}
\left ( \begin{array} {cccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} \\
\alpha _{4} & \alpha _{5} & \alpha_{6} & \alpha _{7} \\
\alpha _{7} & \alpha _{8} & \alpha_{9} & \alpha _{10} \\
\alpha _{10} & \alpha _{11} & \alpha_{12} & \alpha _{13}
\end{array} \right ) \; \overset{\text{rank}}{\sim} \;
\left ( \begin{array} {cccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} \\
\alpha _{2} & \alpha _{3} & \alpha_{4} & \alpha _{5} \\
\hline
\beta _{11} & \beta _{12} & \beta _{13} & \beta_{14} \\
\beta _{21} & \beta _{22} & \beta _{23} & \beta_{24}
\end{array} \right )
\end{displaymath} \\
is equal to \vspace{0.1 cm} \\
$2^{2i}\cdot\Gamma_{i}^{2\times 4}+3\cdot2^{i-1}\cdot(2^{4}-2^{i-1})\cdot \Gamma_{i-1}^{2\times 4} +(2^{4}-2^{i-1})\cdot(2^{4}-2^{i-2})\cdot\Gamma_{i-2}^{2\times 4}\quad \text{for} \; 0\leqslant i \leqslant 4$
\begin{equation*}
= \begin{cases}
1 & \text{if } i = 0, \\
57 & \text{if } i=1, \\
910 & \text{if } i = 2, \\
4536 & \text{if } i = 3, \\
2688 & \text{if } i = 4. \\
\end{cases}
\end{equation*}
\vspace{0.1 cm}
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1,U_{1},V_{1}, Y_2,Z_2,U_{2},V_{2},\ldots , Y_q,Z_q,U_{q},V_{q} ) \in \big(\mathbb{F}_{2}[T] \big)^{4q} $ \vspace{0.5 cm}\\
of the polynomial equations\\
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0 \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0 \\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0 \\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 3 , \quad degZ_j \leq 1 ,
\quad degU_{j}\leq 0 , \quad degV_{j}\leq 0 \quad \text{for} \quad 1\leq j \leq q $$ \\
is equal to\\
$ 2^{8q-13} \sum_{i=0}^{4} \Gamma_{i}^{\left[2\atop (2) \right]\times 4} \cdot2^{-iq} =
2^{8q-13}\left[1+ 57\cdot2^{-q} +910\cdot2^{-2q} + 4536\cdot2^{-3q} +2688\cdot2^{-4q} \right] $
\end{example}
For more details see [2]
\section{Polynomial equations related to double persymmetric matrices}
\label{sec 3 }
\begin{example}
The most simple problem concerning double persymmetric matrices with entries in $\mathbb{F}_{2}$:\\
Compute the number $ \Gamma_{i}^{\left[2\atop 2 \right]\times k} $ of double persymmetric matrices in $\mathbb{F}_{2}$
of rank $0\leq i\leq \inf(4,k) $ of the form:
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \alpha _{k} & \alpha _{k+1} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \beta_{k} & \beta _{k+1}
\end{array} \right). $$
We get
\begin{equation*}
\Gamma_{i}^{\left[2\atop 2 \right]\times k} =
\begin{cases}
1 & \text{if } i = 0,\; k \geq 1, \\
9 & \text{if } i=1,\; k > 1, \\
3\cdot 2^{k+1} + 30 & \text{if } i = 2 ,\; k > 2, \\
21\cdot 2^{k+1} -168 & \text{if } i=3 ,\; k > 3,\\
2^{2k+2} -3\cdot2^{k+4} + 128 & \text{if } i=4 ,\; k \geq 4
\end{cases}
\end{equation*}
\begin{equation*}
\Gamma_{i}^{\left[2\atop 2 \right]\times i} =
\begin{cases}
15 & \text{if } i = 1, \\
54 & \text{if } i=2, \\
168 & \text{if } i=3 ,\\
384 & \text{if } i=4
\end{cases}
\end{equation*}
\vspace{0.1 cm}
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1,U_{1}, Y_2,Z_2,U_{2},\ldots , Y_q,Z_q,U_{q} ) \in \big(\mathbb{F}_{2}[T] \big)^{3q} $ \vspace{0.5 cm}\\
of the polynomial equations\\
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0 \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq k-1 , \quad degZ_j \leq 1 ,
\quad degU_{j}\leq 1 , \quad \text{for} \quad 1\leq j \leq q $$ \\
is connected with the numbers $\Gamma _{i}^{\Big[\substack{2 \\ 2 }\Big] \times k}$
, in the way that the number $ R_{q} $ can be written as a linear combination of the $\Gamma _{i}^{\Big[\substack{2 \\ 2 }\Big] \times k}$\\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{(k+4)q-2k-2}\sum_{i = 0}^{inf(4,k)}
\Gamma _{i}^{\left[\stackrel{2}{2}\right]\times k}2^{-iq}\\
& = \left\{\begin{array}{ccc}
2^{k} +15 & if & q=1,k\geqslant 4\\
2^{2k+8} +27\cdot2^{k+1}+192 & if & q=2,k\geqslant 4 \\
2^{kq +4q -2k -2}\cdot \left[1+ 9\cdot2^{-q} +(3\cdot2^{k+1} +30)\cdot2^{-2q} + (21\cdot2^{k+1} -168)\cdot2^{-3q} \right] \\
+ 2^{kq +4q -2k -2}\cdot \left[ (2^{2k+2} -3\cdot2^{k+4} +128) \cdot 2^{-4q} \right]
& if & 3\leq q, k\geqslant 4
\end{array}\right.
\end{align*}
\textbf{The case k = 1}
\begin{equation*}
R_{q}
= 2^{5q-4}\sum_{i = 0}^{1}
\Gamma _{i}^{\left[\stackrel{2}{2}\right]\times 1}2^{-iq} = 2^{5q-4} +15\cdot2^{4q-4}
\end{equation*}
\textbf{The case k = 2}
\begin{equation*}
R_{q}
= 2^{6q-6}\sum_{i = 0}^{2}
\Gamma _{i}^{\left[\stackrel{2}{2}\right]\times 2}2^{-iq} = 2^{6q-6} +9\cdot2^{5q-6} + 54\cdot2^{4q-6}
\end{equation*}
\textbf{The case k = 3}
\begin{equation*}
R_{q}
= 2^{7q-8}\sum_{i = 0}^{3}
\Gamma _{i}^{\left[\stackrel{2}{2}\right]\times 3}2^{-iq} = 2^{7q-8} +9\cdot2^{6q-8} + 78\cdot2^{5q-8} + 168\cdot2^{4q-8}
\end{equation*}
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[5\atop 5 \right]\times k} $ of double persymmetric $10\times k$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3} & \alpha _{4} & \ldots & \alpha_{k+1} & \alpha _{k+2} \\
\alpha _{4} & \alpha _{5} & \ldots & \alpha_{k+2} & \alpha _{k+3} \\
\alpha _{5} & \alpha _{6} & \ldots & \alpha_{k+3} & \alpha _{k+4} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \beta_{k+1} & \beta _{k+2} \\
\beta _{4} & \beta _{5} & \ldots & \beta_{k+2} & \beta _{k+3} \\
\beta _{5} & \beta _{6} & \ldots & \beta_{k+3} & \beta _{k+4}
\end{array} \right). $$
is equal to \\
\begin{equation*}
\begin{cases}
1 & \text{if } i = 0,\; k \geq 1, \\
9 & \text{if } i=1,\; k > 1, \\
78 & \text{if } i = 2 ,\; k > 2, \\
648 & \text{if } i=3 ,\; k > 3,\\
5280 & \text{if } i=4 ,\; k > 4,\\
3\cdot2^{k+4} + 39552 & \text{if } i=5 ,\; k > 5,\\
21\cdot2^{k+4} + 290304& \text{if } i=6 ,\; k > 6,\\
21\cdot2^{k+7} +1892352 & \text{if } i=7 ,\; k > 7,\\
21\cdot2^{k+10} + 825753 6& \text{if } i=8 ,\; k > 8,\\
21\cdot2^{k+13} - 44040192& \text{if } i=9 ,\; k > 9,\\
2^{2k + 8} -3\cdot 2^{k+16} + 2^{25} & \text{if } i = 10 ,\; k \geq 10.
\end{cases}
\end{equation*}
\textbf{The case k=i}
\begin{equation*}
\Gamma_{i}^{\left[5\atop 5 \right]\times i} =
\begin{cases}
2^{10} -1& \text{if } i=1, \\
2^{12} -10& \text{if } i=2, \\
2^{14} -88 & \text{if } i=3, \\
2^{16} -736 & \text{if } i=4, \\
2^{18} - 6016 & \text{if } i=5, \\
2^{20} - 48640 & \text{if } i=6, \\
2^{22} - 385024& \text{if } i=7, \\
2^{24} - 3014656 & \text{if } i=8, \\
2^{26} - 23068672 & \text{if } i=9, \\
2^{28} - 167772160 & \text{if } i=10,
\end{cases}
\end{equation*}
\textbf{The case k=4}
The number $ \Gamma_{i}^{\left[5\atop 5 \right]\times 4} $ of double persymmetric $10\times 4$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} & \alpha _{4} \\
\alpha _{2 } & \alpha _{3} & \alpha _{4} & \alpha _{5} \\
\alpha _{3} & \alpha _{4} & \alpha_{5} & \alpha _{6} \\
\alpha _{4} & \alpha _{5} & \alpha_{6} & \alpha _{7} \\
\alpha _{5} & \alpha _{6} & \alpha_{7} & \alpha _{8} \\
\hline
\beta _{1} & \beta _{2} & \beta_{3} & \beta _{4} \\
\beta _{2} & \beta _{3} & \beta_{4} & \beta _{5} \\
\beta _{3} & \beta _{4} & \beta_{5} & \beta _{6} \\
\beta _{4} & \beta _{5} & \beta_{6} & \beta _{7} \\
\beta _{5} & \beta _{6} & \beta_{7} & \beta _{8}
\end{array} \right). $$
is equal to \\
\begin{equation*}
\begin{cases}
1 & \text{if } i = 0, \\
9 & \text{if } i=1 \\
78 & \text{if } i = 2 , \\
648 & \text{if } i=3 \\
64800 & \text{if } i=4
\end{cases}
\end{equation*}
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1,U_{1}, Y_2,Z_2,U_{2},\ldots , Y_q,Z_q,U_{q} ) \in \big(\mathbb{F}_{2}[T] \big)^{3q} $ \vspace{0.5 cm}\\
of the polynomial equations\\
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0 \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 3 , \quad degZ_j \leq 4 ,
\quad degU_{j}\leq 4 , \quad \text{for} \quad 1\leq j \leq q $$ \\
is connected with the numbers $\Gamma _{i}^{\Big[\substack{5 \\ 5 }\Big] \times 4}$
, in the way that the number $ R_{q} $ can be written as a linear combination of the $\Gamma _{i}^{\Big[\substack{5 \\ 5 }\Big] \times 4}$\\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{14q-16}\sum_{i = 0}^{4}
\Gamma _{i}^{\left[\stackrel{5}{5}\right]\times 4}2^{-iq}\\
& = 2^{14q-16}\cdot[1+9\cdot2^{-q} + 78\cdot2^{-2q} +648\cdot2^{-3q} + 64800\cdot2^{-4q} ]
\end{align*}
\end{example}
\begin{example}
The fraction of square double persymmetric matrices which are invertible is equal to $ {3\over 8} $
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop 2+3 \right]\times 4} $ of double persymmetric $7\times 4$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} & \alpha _{4} \\
\alpha _{2 } & \alpha _{3} & \alpha _{4} & \alpha _{5} \\
\hline
\beta _{1} & \beta _{2} & \beta_{3} & \beta _{4} \\
\beta _{2} & \beta _{3} & \beta_{4} & \beta _{5} \\
\beta _{3} & \beta _{4} & \beta_{5} & \beta _{6} \\
\beta _{4} & \beta _{5} & \beta_{6} & \beta _{7} \\
\beta _{5} & \beta _{6} & \beta_{7} & \beta _{8}
\end{array} \right). $$
is given by\\
\begin{equation*}
\begin{cases}
1 & \text{if } i = 0, \\
9 & \text{if } i=1,\\
94 & \text{if } i = 2, \\
600 & \text{if } i=3 ,\\
7488 & \text{if } i=4
\end{cases}
\end{equation*}
\textbf{Application:}\\
The number $ R_{q} $ of solutions
$(Y_1,Z_1,U_{1}, Y_2,Z_2,U_{2},\ldots , Y_q,Z_q,U_{q} ) \in \big(\mathbb{F}_{2}[T] \big)^{3q} $ \vspace{0.5 cm}\\
of the polynomial equations\\
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0 \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 3 , \quad degZ_j \leq 1,
\quad degU_{j}\leq 4 , \quad \text{for} \quad 1\leq j \leq q $$ \\
is connected with the numbers $\Gamma _{i}^{\Big[\substack{2 \\ 2+3 }\Big] \times 4}$
, in the way that the number $ R_{q} $ can be written as a linear combination of the $\Gamma _{i}^{\Big[\substack{2 \\ 2+3 }\Big] \times 4}$\\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{11q-13}\sum_{i = 0}^{4}
\Gamma _{i}^{\left[\stackrel{2}{2+3}\right]\times 4}2^{-iq}\\
& = 2^{11q-13}\cdot[1+9\cdot2^{-q} + 94\cdot2^{-2q} +600\cdot2^{-3q} + 7488\cdot2^{-4q} ]
\end{align*}
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop {2+3\atop (1)} \right]\times 4} $ of rank i $8\times 4$ matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} & \alpha _{4} \\
\alpha _{2 } & \alpha _{3} & \alpha _{4} & \alpha _{5} \\
\hline
\beta _{1} & \beta _{2} & \beta_{3} & \beta _{4} \\
\beta _{2} & \beta _{3} & \beta_{4} & \beta _{5} \\
\beta _{3} & \beta _{4} & \beta_{5} & \beta _{6} \\
\beta _{4} & \beta _{5} & \beta_{6} & \beta _{7} \\
\beta _{5} & \beta _{6} & \beta_{7} & \beta _{8} \\
\hline
\gamma_{11} & \gamma_{12} & \gamma_{13} & \gamma_{4}
\end{array} \right). $$
is equal to \\
$$ 2^{i}\cdot \Gamma _{i}^{\Big[\substack{2 \\ 2+3 }\Big] \times 4}+(2^{4} -2^{i-1})\cdot \Gamma _{i-1}^{\Big[\substack{2 \\ 2+3 }\Big] \times 4}
\text{for} \quad 1\leqslant i \leqslant 4 $$
\begin{equation*}
= \begin{cases}
1 & \text{if } i = 0, \\
33 & \text{if } i=1,\\
502 & \text{if } i = 2, \\
5928 & \text{if } i=3 ,\\
124608 & \text{if } i=4
\end{cases}
\end{equation*}
\textbf{Application:}
The number $ R_{q} $ of solutions
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q})\in \big( \mathbb{F}_{2}[T ] \big)^{4q} $
of the polynomial equations\\
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 3 , \quad degZ_j \leq 1,
\quad degU_{j}\leq 4 , \quad degV_{j}\leq 0 \quad \text{for} \quad 1\leq j \leq q $$ \\
is connected with the numbers $ \Gamma_{i}^{\left[2\atop {2+3\atop (1)} \right]\times 4} $
, in the way that the number $ R_{q} $ can be written as a linear combination of the $ \Gamma_{i}^{\left[2\atop {2+3\atop( 1)} \right]\times 4} $
More precisely:\\
\begin{align*}
& R_{q}
= 2^{12q-17}\sum_{i = 0}^{4}
\Gamma_{i}^{\left[2\atop {2+3\atop (1)} \right]\times 4} 2^{-iq}\\
& = 2^{12q-17}\cdot[1+33\cdot2^{-q} + 502\cdot2^{-2q} +5928\cdot2^{-3q} + 124608\cdot2^{-4q} ]
\end{align*}
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop {2\atop (4)} \right]\times 4} $ of rank i matrices of the form \\
\begin{displaymath}
\left ( \begin{array} {cccccccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} & \alpha _{5} & \alpha _{6} & \alpha_{7} & \alpha _{8} \\
\alpha _{7} & \alpha _{8} & \alpha_{9} & \alpha _{10} & \alpha _{11} & \alpha _{12} & \alpha_{13} & \alpha _{14} \\
\alpha _{13} & \alpha _{14} & \alpha_{15} & \alpha _{16} & \alpha _{17} & \alpha _{18} & \alpha_{19} & \alpha _{20} \\
\alpha _{19} & \alpha _{20} & \alpha_{21} & \alpha _{22} & \alpha _{23} & \alpha _{24} & \alpha_{25} & \alpha _{26}
\end{array} \right ) \; \overset{\text{rank}}{\sim} \;
\left ( \begin{array} {cccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} \\
\alpha _{2} & \alpha _{3} & \alpha_{4} & \alpha _{5} \\
\hline \\
\beta _{1} & \beta _{2} & \beta_{3} & \beta _{4} \\
\beta _{2} & \beta _{3} & \beta_{4} & \beta _{5} \\
\hline\\
\gamma _{11} & \gamma _{12} & \gamma _{13} & \gamma_{14} \\
\gamma _{21} & \gamma _{22} & \gamma _{23} & \gamma_{24} \\
\gamma _{31} & \gamma _{32} & \gamma _{33} & \gamma_{34} \\
\gamma _{41} & \gamma _{42} & \gamma _{43} & \gamma_{44}
\end{array} \right )
\end{displaymath} \\
is equal to \vspace{0.1 cm} \\
$ 2^{4i}\cdot\Gamma _{i}^{\Big[\substack{2 \\ 2 }\Big] \times 4}
+15\cdot2^{(i-1)3}(2^{k}-2^{i-1})\cdot\Gamma _{i-1}^{\Big[\substack{2 \\ 2 }\Big] \times 4}\\
+ 35\cdot2^{2i-4}(2^{k}-2^{i-1})(2^{k}-2^{i-2})\cdot \Gamma _{i-2}^{\Big[\substack{2 \\ 2 }\Big] \times 4} \\
+ 15\cdot2^{i-3}(2^{k}-2^{i-1})(2^{k}-2^{i-2})(2^{k}-2^{i-3})\Gamma _{i-3}^{\Big[\substack{2 \\ 2 }\Big] \times 4} \\
+ (2^{k}-2^{i-1})(2^{k}-2^{i-2})(2^{k}-2^{i-3})(2^{k}-2^{i-4})\Gamma _{i-4}^{\Big[\substack{2 \\ 2 }\Big] \times 4}
for\quad 0\leq i\leq inf(4,8), $\vspace{0.5 cm} \\
\begin{equation*}
= \begin{cases}
1 & \text{if } i = 0, \\
369 & \text{if } i=1, \\
54726 & \text{if } i = 2, \\
3765384 & \text{if } i = 3, \\
63288384 & \text{if } i = 4. \\
\end{cases}
\end{equation*}
\vspace{0.1 cm}
Then the number of solutions \\
$(Y_1,Z_1,U_{1},V_{1}^{(1)},V_{2}^{(1)}, V_{3}^{(1)},V_{4}^{(1)}, Y_2,Z_2,U_{2},V_{1}^{(2)},V_{2}^{(2)},
V_{3}^{(2)},V_{4}^{(2)},\ldots Y_q,Z_q,U_{q},V_{1}^{(q)},V_{2}^{(q)}, V_{3}^{(q)},V_{4}^{(q)} )\in \big( \mathbb{F}_{2}[T ] \big)^{7q} $ \vspace{0.5 cm}\\
of the polynomial equations \vspace{0.5 cm}
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0 \\
Y_{1}U_{1} +Y_{2}U_{2}+ \ldots + Y_{q}U_{q} = 0 \\
Y_{1}V_{1}^{(1)} + Y_{2}V_{1}^{(2)} + \ldots + Y_{q}V_{1}^{(q)} = 0 \\
Y_{1}V_{2}^{(1)} + Y_{2}V_{2}^{(2)} + \ldots + Y_{q}V_{2}^{(q)} = 0\\
Y_{1}V_{3}^{(1)} + Y_{2}V_{3}^{(2)} + \ldots + Y_{q}V_{3}^{(q)} = 0\\
Y_{1}V_{4}^{(1)} + Y_{2}V_{4}^{(2)} + \ldots + Y_{q}V_{4}^{(q)} = 0
\end{array}\right.\]
satisfying the degree conditions \\
$$ \deg Y_i \leq 3 , \quad \deg Z_i \leq 1 ,\quad \deg U_i \leq 1 ,
\quad \deg V_{j}^{i} \leq 0 , \quad for \quad 1\leq j\leq 4 \quad 1\leq i \leq q $$ \\
is equal to
$$ 2^{12q - 26}\sum_{i = 0}^{4}
\Gamma_{i}^{\left[ 2\atop{ 2\atop (4)} \right]\times k} 2^{-iq}= 2^{12q-26}\left[1+ 369\cdot2^{-q} +54726\cdot2^{-2q} + 3765384\cdot2^{-3q} +
63288384 \cdot2^{-4q} \right] $$
\end{example}
For more details see [3], [4]
\newpage
\section{Polynomial equations related to triple persymmetric matrices}
\label{sec 4}
\begin{example}
The number $ \Gamma_{i}^{\left[1\atop {1\atop 1} \right]\times 1} $ of triple persymmetric $3\times 1$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} \\
\hline \\
\beta _{1} \\
\hline \\
\gamma _{1} \\
\end{array} \right). $$
is equal to
\[ \begin{cases}
1 &\text{if } i = 0 \\
7 & \text{if } i = 2 \\
\end{cases}
\]
\textbf{Application:}
The number of solutions $(y_1,z_1,u_{1},v_{1}, \ldots,y_q,z_q,u_{q},v_{q}) \in \mathbb{F}_{2}^{4q} $ of the following system
of quadratic equations : \\
\[\left\{\begin{array}{c}
y_{1}z_{1} +y_{2}z_{2}+ \ldots + y_{q}z_{q} = 0, \\
y_{1}u_{1} + y_{2}u_{2} + \ldots + y_{q}u_{q} = 0,\\
y_{1}v_{1} + y_{2}v_{2} + \ldots + y_{q}v_{q} = 0,\\
\end{array}\right.\]
is equal to
\begin{align*}
& 2^{4q-3}\cdot \sum_{i = 0}^{1} \Gamma_{i}^{\left[1\atop{ 1\atop 1} \right]\times 1} \cdot2^{- qi} \\
& = 2^{4q-3}\cdot\big(1 + 7\cdot2^{-q} + \big ) \\
& = 2^{4q-3} + 7\cdot2^{3q-3}
\end{align*}
\textbf{Generalization}
The number $ \Gamma_{i}^{\left[1\atop{\vdots \atop 1}\right]\times 3} $ of n-times persymmetric $n\times 1$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} \\
\hline \\
\alpha _{2} \\
\hline \\
\vdots \\
\hline \\
\alpha _{n} \\
\end{array} \right). $$
is equal to
\[ \begin{cases}
1 &\text{if } i = 0 \\
2^n -1 & \text{if } i = 2 \\
\end{cases}
\]
\textbf{Application:}
The number of solutions $(y_1,z_1^{(1)},z_{1}^{(2)}, \ldots, z_{1}^{(n)}, \ldots, y_q,z_{q}^{(1)},z_{q}^{(2)}, \ldots, z_{q}^{(n)} ) \in \mathbb{F}_{2}^{(n+1)q} $ of the following system
of quadratic equations : \\
\[\left\{\begin{array}{c}
y_{1}z_{1}^{(1)} +y_{2}z_{2}^{(1)}+ \ldots + y_{q}z_{q}^{(1)} = 0, \\
y_{1}z_{1}^{(2)} +y_{2}z_{2}^{(2)}+ \ldots + y_{q}z_{q}^{(2)} = 0, \\
y_{1}z_{1}^{(3)} +y_{2}z_{2}^{(3)}+ \ldots + y_{q}z_{q}^{(3)} = 0, \\
\vdots \\
y_{1}z_{1}^{(n)} +y_{2}z_{2}^{(n)}+ \ldots + y_{q}z_{q}^{(n)} = 0
\end{array}\right.\]
is equal to
\begin{align*}
& 2^{(n+1)q-n}\cdot \sum_{i = 0}^{1} \Gamma_{i}^{\left[1\atop{\vdots \atop 1}\right]\times 3}\cdot2^{- qi} \\
& = 2^{(n+1)q-n}\cdot\big(1 + (2^n -1)\cdot2^{-q}\big ) = 2^{nq-n}\cdot[2^q+2^n-1]
\end{align*}
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop {2\atop 2} \right]\times 6} $ of triple persymmetric $6\times 6$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} & \alpha _{4} & \alpha _{5} & \alpha _{6} \\
\alpha _{2 } & \alpha _{3} & \alpha _{4} & \alpha _{5} & \alpha _{6} & \alpha _{7} \\
\hline
\beta _{1} & \beta _{2} & \beta _{3}& \beta _{4} & \beta_{5} & \beta _{6} \\
\beta _{2} & \beta _{3} & \beta _{3}& \beta _{5} & \beta_{6} & \beta _{7} \\
\hline
\gamma _{1} & \gamma _{2} & \gamma _{3}& \gamma _{4} & \gamma _{5} & \gamma _{6} \\
\gamma _{2} & \gamma _{3} & \gamma _{4} & \gamma _{5} & \gamma _{6} & \gamma _{7} \\
\end{array} \right). $$
is equal to
\[ \begin{cases}
1 &\text{if } i = 0 \\
21 &\text{if } i = 1 \\
1162 & \text{if } i = 2 \\
20160 & \text{if } i = 3 \\
258720 & \text{if } i = 4 \\
1128960 & \text{if } i = 5 \\
688128 & \text{if } i = 6
\end{cases}
\]
\textbf{Application:}
The number $ R_{q} $ of solutions \\
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q}) \in \big( \mathbb{F}_{2}[T ] \big)^{4q} $ of the polynomial equations
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 5 , \quad degZ_j \leq 1 ,\quad degU_{j}\leq 1,\quad degV_{j}\leq 1 \quad for \quad 1\leq j \leq q. $$ \\
is equal to
\begin{align*}
& 2^{12q-21}\cdot \sum_{i = 0}^{6} \Gamma_{i}^{\left[2\atop{ 2\atop 2} \right]\times 6} \cdot2^{- qi} \\
& = 2^{12q-21}\cdot\big(1 + 21\cdot2^{-q} + 1162\cdot2^{-2q} + 20160\cdot2^{-3q} + 258720\cdot2^{-4q} + 1128960 \cdot2^{-5q}
+ 688128 \cdot2^{-6q}\big ) \\
& = 2^{6q-21}\cdot\big(2^{6q} + 21\cdot2^{5q} + 1162\cdot2^{4q} + 20160\cdot2^{3q} + 258720\cdot2^{2q} + 1128960 \cdot2^{q}
+ 688128 \big )
\end{align*}
\end{example}
\begin{example}
The fraction of square triple persymmetric matrices which are invertible is equal to $ {21\over 64} $
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop {2\atop 2} \right]\times 2} $ of triple persymmetric $6\times 2$ matrices over $\mathbb{F}_{2}$ of rank i of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} \\
\alpha _{2 } & \alpha _{3} \\
\hline
\beta _{1} & \beta _{2} \\
\beta _{2} & \beta _{3} \\
\hline
\gamma _{1} & \gamma _{2} \\
\gamma _{2} & \gamma _{3} \\
\end{array} \right). $$
is equal to
\[ \begin{cases}
1 &\text{if } i = 0 \\
21 &\text{if } i = 1 \\
490 & \text{if } i = 2 \\
\end{cases}
\]
\textbf{Application:}
The number $ R_{q} $ of solutions \\
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q}) \in \big( \mathbb{F}_{2}[T ] \big)^{4q} $ of the polynomial equations
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq 1 , \quad degZ_j \leq 1 ,\quad degU_{j}\leq 1,\quad degV_{j}\leq 1 \quad for \quad 1\leq j \leq q. $$ \\
is equal to
\begin{align*}
& 2^{8q-9}\cdot \sum_{i = 0}^{2} \Gamma_{i}^{\left[2\atop{ 2\atop 2} \right]\times 2} \cdot2^{- qi} \\
& = 2^{8q-9}\cdot\big(1 + 21\cdot2^{-q} + 490\cdot2^{-2q} \big ) \\
& = 2^{8q-9} + 21\cdot2^{7q-9} + 490\cdot2^{6q-9}
\end{align*}
\textbf{Application :}
The number of solutions $ (x_1,x_2,\ldots, x_{15},x_{16}) \in \mathbb{F}_{2}^{16} $ of the following system
of quadratic equations : \\
\[\left\{\begin{array}{c}
x_{1}x_{5} +x_{3}x_{7} = 0, \\
x_{1}x_{9} +x_{3}x_{11} = 0,\\
x_{1}x_{13} +x_{3}x_{15} = 0,\\
x_{1}x_{6} +x_{2}x_{5} + x_{3}x_{8} +x_{4}x_{7} = 0\\
x_{1}x_{10} +x_{2}x_{9} + x_{3}x_{12} +x_{4}x_{11} = 0\\
x_{1}x_{14} +x_{2}x_{13} + x_{3}x_{16} +x_{4}x_{15} = 0\\
x_{2}x_{6} +x_{4}x_{8} = 0, \\
x_{2}x_{10} +x_{4}x_{12} = 0,\\
x_{2}x_{14} +x_{4}x_{16} = 0,\\
\end{array}\right.\]
is equal to $R_{2} = 2^{7} + 21\cdot2^{5} + 490\cdot2^{3} = 4720 $\\
\begin{proof}
Set
\[\left\{\begin{array}{c}
Y_{1} = x_{1} +x_{2} \cdot T \\
Y_{2} = x_{3} +x_{4} \cdot T \\
Z_{1} = x_{5} +x_{6} \cdot T \\
Z_{2} = x_{7} +x_{8} \cdot T \\
U_{1} = x_{9} +x_{10} \cdot T \\
U_{2} = x_{11} +x_{12} \cdot T \\
V_{1} = x_{13} +x_{14} \cdot T \\
V_{2} = x_{15} +x_{16} \cdot T
\end{array}\right.\]
Then we obtain
\begin{equation*}
\begin{cases}
Y_{1}Z_{1} +Y_{2}Z_{2}= 0, \\
Y_{1}U_{1} + Y_{2}U_{2} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} = 0
\end{cases} \; \Leftrightarrow \; \begin{cases}
x_{1}x_{5} +x_{3}x_{7} = 0, \\
x_{1}x_{9} +x_{3}x_{11} = 0,\\
x_{1}x_{13} +x_{3}x_{15} = 0,\\
x_{1}x_{6} +x_{2}x_{5} + x_{3}x_{8} +x_{4}x_{7} = 0\\
x_{1}x_{10} +x_{2}x_{9} + x_{3}x_{12} +x_{4}x_{11} = 0\\
x_{1}x_{14} +x_{2}x_{13} + x_{3}x_{16} +x_{4}x_{15} = 0\\
x_{2}x_{6} +x_{4}x_{8} = 0, \\
x_{2}x_{10} +x_{4}x_{12} = 0,\\
x_{2}x_{14} +x_{4}x_{16} = 0,\\
\end{cases}
\end{equation*}
\end{proof}
\end{example}
\begin{example}
The number $ \Gamma_{i}^{\left[2\atop {2\atop {2 \atop (3)}} \right]\times 4} $ of rank i matrices of the form \\
\begin{displaymath}
\left ( \begin{array} {ccccccccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} & \alpha _{5} & \alpha _{6} & \alpha_{7} & \alpha _{8} & \alpha _{9} \\
\alpha _{7} & \alpha _{8} & \alpha_{9} & \alpha _{10} & \alpha _{11} & \alpha _{12} & \alpha_{13} & \alpha _{14} & \alpha _{15} \\
\alpha _{13} & \alpha _{14} & \alpha_{15} & \alpha _{16} & \alpha _{17} & \alpha _{18} & \alpha_{19} & \alpha _{20} & \alpha _{21} \\
\alpha _{19} & \alpha _{20} & \alpha_{21} & \alpha _{22} & \alpha _{23} & \alpha _{24} & \alpha_{25} & \alpha _{26} & \alpha _{27}
\end{array} \right ) \; \overset{\text{rank}}{\sim} \;
\left ( \begin{array} {cccc}
\alpha _{1} & \alpha _{2} & \alpha_{3} & \alpha _{4} \\
\alpha _{2} & \alpha _{3} & \alpha_{4} & \alpha _{5} \\
\hline \\
\beta _{1} & \beta _{2} & \beta_{3} & \beta _{4} \\
\beta _{2} & \beta _{3} & \beta_{4} & \beta _{5} \\
\hline \\
\gamma _{1} & \gamma _{2} & \gamma _{3} & \gamma_{4} \\
\gamma _{2} & \gamma _{3} & \gamma _{4} & \gamma_{5} \\
\hline \\
\delta_{11} & \delta_{12} & \delta _{13} & \delta_{14} \\
\delta_{21} & \delta_{22} & \delta _{23} & \delta_{24} \\
\delta_{31} & \delta_{32} & \delta _{33} & \delta_{34}
\end{array} \right )
\end{displaymath} \\
is equal to \vspace{0.1 cm} \\
$ 2^{3i}\cdot \Gamma_{i}^{\left[2\atop {2\atop 2} \right]\times 4}\\
+7\cdot2^{(i-1)2}(2^{4}-2^{i-1})\cdot \Gamma_{i-1}^{\left[2\atop {2\atop 2} \right]\times 4}\\
+ 7\cdot2^{i-2}(2^{4}-2^{i-1})(2^{4}-2^{i-2})\cdot \Gamma_{i-2}^{\left[2\atop {2\atop 2} \right]\times 4} \\
+ (2^{4}-2^{i-1})(2^{4}-2^{i-2})(2^{4}-2^{i-3})\cdot \Gamma_{i-3}^{\left[2\atop {2\atop 2} \right]\times 4} for\quad 0\leq i\leq inf(4,9), $\vspace{0.5 cm} \\
Hence :\\
\begin{equation*}
\Gamma_{i}^{\left[2\atop {2\atop {2 \atop (3)}} \right]\times 4} = \begin{cases}
1 & \text{if } i = 0, \\
273 & \text{if } i=1, \\
41062 & \text{if } i = 2, \\
3807048 & \text{if } i = 3, \\
130369344 & \text{if } i = 4. \\
\end{cases}
\end{equation*}
\vspace{0.1 cm}
\end{example}
For more details see [4]
\section{The inverse problem}
\label{sec 5}
How to compute the number $ R_{q} $ of solutions \\
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q}) \in \big( \mathbb{F}_{2}[T ] \big)^{4q} $ of the polynomial equations
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq k-1 , \quad degZ_j \leq s-1 ,\quad degU_{j}\leq s+m-1,\quad degV_{j}\leq s+m+l-1 \quad for \quad 1\leq j\leq q. ?$$ \\
\textbf{Response: } \vspace{0,1 cm} \\
We need only to compute
the number $\Gamma_{i}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k} $ of triple persymmetric $(3s+2m+l)\times k $
rank i matrices
over $\mathbb{F}_{2}$ for $ 0\leqslant i \leqslant \inf (k, 3s+2m+l) $ of the form
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \alpha _{k} & \alpha _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\alpha _{s-1} & \alpha _{s} & \ldots & \alpha _{s+k-3} & \alpha _{s+k-2} \\
\alpha _{s } & \alpha _{s +1} & \ldots & \alpha _{s +k-2}& \alpha _{s +k-1}\\
\hline \\
\beta _{1} & \beta _{2} & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \beta_{k} & \beta _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\beta _{m+1} & \beta _{m+2} & \ldots & \beta_{k+m-1} & \beta _{k+m} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\beta _{s+m-1} & \beta _{s+m} & \ldots & \beta_{s+m+k-3} & \beta _{s+m+k-2} \\
\beta _{s+m} & \beta _{s+m+1} & \ldots & \beta _{s+m+k-2} & \beta _{s+m+k-1}\\
\hline \\
\gamma _{1} & \gamma _{2} & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3} & \ldots & \gamma _{k} & \gamma _{k+1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{m+1} & \gamma _{m+2} & \ldots & \gamma _{k+m-1} & \gamma _{k+m} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{s+m-1} & \gamma _{s+m} & \ldots & \gamma _{s+m+k-3} & \gamma _{s+m+k-2} \\
\gamma _{s+m} & \gamma _{s+m+1} & \ldots & \gamma _{s+m+k-2} & \gamma _{s+m+k-1}\\
\gamma _{s+m+1} & \gamma _{s+m+2} & \ldots & \gamma _{s+m+k-1} & \gamma _{s+m+k}\\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{s+m+l} & \gamma _{s+m+l+1} & \ldots & \gamma _{s+m+l+k-2} & \gamma _{s+m+l+k-1}
\end{array} \right). $$
To compute those numbers we use the following reductions formulas:\\
\begin{align*}
\Gamma _{2s+1+m+j}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k} & = 16^{j}\cdot \Gamma _{2s+1+m}^{\left[s\atop{ s+m\atop s+m+l-j} \right]\times (k-j)}
& \text{if } 0 \leq j \leq l,\; k \geqslant 2s+1+m+j \\
\Gamma _{2s+1+m+l+j}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k} & = 16^{2j+l}\cdot \Gamma _{2s+1+m-j}^{\left[s\atop{ s+m-j\atop s+m-j} \right]\times (k-2j-l)}
& \text{if } 0 \leq j \leq m, \; k \geqslant 2s+1+m+l+j \\
\Gamma _{2s+1+2m+l+j}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k} & = 16^{2m+l+3j}\cdot \Gamma _{2(s-j)+1}^{\left[s-j\atop{ s-j\atop s-j} \right]\times (k-2m-l-3j)}
& \text{if } 0 \leq j \leq s-1, \; k \geqslant 2s+1+2m+l+j
\end{align*}
Then $R_{q}$ is equal to a linear combination of the $\Gamma_{i}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k} \; \text{for} \; 0\leqslant i\leqslant \inf(k, 3s+2m+l)$ \\
More precisely:\\
\begin{align*}
& R_{q}
= 2^{(k+3s+2m+l)q -(3k+3s+2m+l-3)} \sum_{i = 0}^{\inf(k, 3s+2m+l)}
\Gamma_{i}^{\left[s\atop {s+m\atop s+m+l} \right]\times k} 2^{-iq}
\end{align*}
\textbf{Remark : } \vspace{0.1 cm} \\
We have computed the $\Gamma_{i}^{\left[s\atop{ s+m\atop s+m+l} \right]\times k}$ for $ 0\leqslant i\leqslant \inf(k, 3s+2m+l)$
in the case $l=0.$ [see (4)]
\subsection{The inverse problem in the case $s=2,\;m =3,\;l\geqslant 4$ }
\label{subsec 1}
The results in this subsection are new.
\begin{example}
To compute the number $ R_{q} $ of solutions \\
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q}) \in \big( \mathbb{F}_{2}[T ] \big)^{4q} $ of the polynomial equations
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq k-1 , \quad degZ_j \leq 1 ,\quad degU_{j}\leq 4,\quad degV_{j}\leq 4+l \quad for \quad 1\leq j \leq q, $$ \\
we need only to compute
the number $\Gamma_{i}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k} $ of triple persymmetric $(12+l)\times k $ rank i matrices
over $\mathbb{F}_{2}$ for $ 0\leqslant i\leqslant \inf(k, 12+l) $ of the form
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \alpha _{k} & \alpha _{k+1} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \beta_{k+1} & \beta _{k+2} \\
\beta _{4} & \beta _{5} & \ldots & \beta_{k+2} & \beta _{k+3} \\
\beta _{5} & \beta _{6} & \ldots & \beta_{k+3} & \beta _{k+4} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3} & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{5+l} & \gamma _{6+l} & \ldots & \gamma _{k+l+3} & \gamma _{k+l+4} \\
\end{array} \right). $$
Proceeding as in (4), using the following reductions formulas:\\
\begin{align*}
\Gamma _{8+j}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k} & = 16^{j}\cdot \Gamma _{8}^{\left[2\atop{ 2+3\atop 2+3+l-j} \right]\times (k-j)}
& \text{if } 0 \leq j \leq l, \; k \geqslant 8+j \\
\Gamma _{8+l+j}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k} & = 16^{2j+l}\cdot \Gamma _{8-j}^{\left[2\atop{ 2+3-j\atop 2+3-j} \right]\times (k-2j-l)}
& \text{if } 0 \leq j \leq 3, \; k \geqslant 8+l+j \\
\Gamma _{11+l+j}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k} & = 16^{6+l+3j}\cdot \Gamma _{2(2-j)+1}^{\left[2-j\atop{ 2-j\atop 2-j} \right]\times (k-6-l-3j)}
& \text{if } 0 \leq j \leq 1, \; k \geqslant 11+l+j
\end{align*}
we obtain:\\
\begin{equation*}
\Gamma_{i}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \; k > 0\\
21 & \text{if } i = 1, \; k > 1\\
2^{k+1} + r_{2} (=362 ) & \text{if } i = 2, \; k > 2 \\
9\cdot2^{k+1} + r_{3} (=6048) & \text{if } i = 3, \; k > 3 \\
39\cdot2^{k+2} + r_{4} (=98784) & \text{if } i = 4, \; k > 4\\
97\cdot2^{k+4} + r_{5}(= 1580288) & \text{if } i = 5, \; k > 5 \\
225\cdot2^{k+6} + r_{6}(= 25135104) & \text{if } i = 6, \; k > 6 \\
2^{2k+5} +417\cdot2^{k+8} +r_{7}(= 402571264=24571\cdot2^{14} ) & \text{if } i = 7, \; k > 7
\end{cases}
\end{equation*}
\begin{equation*}
\Gamma_{8+i}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k}
= \begin{cases}
3\cdot2^{2k+5+2i} + 105\cdot2^{k+13+3i} + r_{8+i}(= 12285\cdot2^{19+4i}) & \text{if } 0\leqslant i \leqslant l-4, \; k > 8+i \\
3\cdot2^{2k+2l-1} + 137\cdot2^{k+3l+4} + r_{8+l-3}(=12029\cdot2^{23+4\cdot(l-4)}) & \text{if } i = l-3, \; k > 8+i \\
3\cdot2^{2k+2l+1} + 213\cdot2^{k+3l+7} + r_{8+l-2}(= 11373\cdot2^{27+4\cdot(l-4)}) & \text{if } i = l-2, \; k > 8+i \\
11\cdot2^{2k+2l+3} + 333\cdot2^{k+3l+9} + r_{8+l-1} (= 10159\cdot2^{31+4\cdot(l-4)}) & \text{if } i = l-1, \; k > 8+i \\
\end{cases}
\end{equation*}
\begin{equation*}
\Gamma_{8+l+i}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k}
= \begin{cases}
21\cdot2^{2k+2l+5} + 2331\cdot2^{k+3l+11} + r_{8+l} (= 15435\cdot2^{34+4\cdot(l-4)}) & \text{if } i = 0, \; k > 8+l+i \\
21\cdot2^{2k+2l+8} + 2163\cdot2^{k+3l+15} + r_{9+l}(= 2835\cdot2^{39+4\cdot(l-4)}) & \text{if } i = 1, \; k > 8+l+i \\
53\cdot2^{2k+2l+11} + 1311\cdot2^{k+3l+19} + r_{10+l} (= -1417\cdot2^{44+4\cdot(l-4)} ) & \text{if } i = 2, \; k > 8+l+i \\
\end{cases}
\end{equation*}
\begin{equation*}
\Gamma_{11+l+i}^{\left[2\atop{ 2+3\atop 2+3+l} \right]\times k}
= \begin{cases}
105\cdot2^{2k+2l+14} -315\cdot2^{k+3l+23} + r_{11+l} (= 105\cdot2^{49+4\cdot(l-4)}) & \text{if } i = 0, \; k > 11+l+i \\
2^{3k+l+9} - 7\cdot2^{2k+2l+18} + 7\cdot2^{k+3l+28} + r_{12+l} (= -2^{55+4\cdot(l-4)} ) & \text{if } i =1,\; k\geqslant 11+l+i
\end{cases}
\end{equation*}
Then we get:\\
\begin{align*}
& R_{q}
= 2^{(k+12+l)q -(3k+l+9)} \sum_{i = 0}^{\inf(k, 12+l)}
\Gamma_{i}^{\left[2\atop {2+3\atop 2+3+l} \right]\times k} 2^{-iq}
\end{align*}
\end{example}
\subsection{The inverse problem in the case $s=2,\;m =3,\; l=4,\; k=6$}
\label{subsec 2}
The results in this subsection are new.
\begin{example}
We get:\\
\begin{equation*}
\Gamma_{i}^{\left[2\atop{ 2+3\atop 2+3+4} \right]\times 6}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
490 & \text{if } i = 2, \\
7200 & \text{if } i = 3, \\
108768 & \text{if } i = 4, \\
1679616 & \text{if } i = 5, \\
2145687552 & \text{if } i = 6,
\end{cases}
\end{equation*}
The number $ R_{q} $ of solutions \\
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q}) \in \big( \mathbb{F}_{2}[T ] \big)^{4q} $ of the polynomial equations
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_i \leq 5 , \quad degZ_i \leq 1 ,\quad degU_{i}\leq 4,\quad degV_{i}\leq 8 \quad for \quad 1\leq i \leq q. $$ \\
is equal to\\
$ 2^{22q - 31} \sum_{i = 0}^{6}
\Gamma_{i}^{\left[2\atop {2+3\atop 2+3+4} \right]\times 6} 2^{-iq} $
\end{example}
\newpage
\subsection{The inverse problem in the case $s=3,\;m =0,\; l\geqslant 0$}
\label{subsec 3}
The results in this subsection are new.
\begin{example}
\textbf{$s=3,\;m=0,\;l=0.$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3} \right]\times k} $ of triple persymmetric $9\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\end{array} \right). $$
is equal to
\[ \Gamma_{i}^{\left[3\atop{ 3\atop 3} \right]\times k}=
\begin{cases}
1 &\text{if } i = 0 \\
21 &\text{if } i = 1 \\
378 &\text{if } i = 2 \\
7\cdot2^{k+ 2} + 5936 & \text{if } i = 3 \\
147\cdot2^{k+ 2} + 84672 & \text{if } i = 4 \\
147\cdot9\cdot2^{k+ 3} + 959616 & \text{if } i = 5 \\
7\cdot2^{2k+4} + 2121\cdot2^{k+ 6} + 5863424 & \text{if } i = 6\\
105\cdot2^{2k+4} + 2625\cdot2^{k+ 9} - 92897280 & \text{if } i = 7 \\
105\cdot2^{2k+8} - 315\cdot2^{k+ 14} + 220200960 & \text{if } i = 8 \\
2^{3k+6} - 7\cdot2^{2k+12} + 7\cdot2^{k+ 19} - 134217728 & \text{if } i = 9,\;k\geq 9
\end{cases}
\]
\end{example}
\begin{example}
\textbf{$s=3,\;m=0,\;l=1.$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3+1} \right]\times k} $ of triple persymmetric $10\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\gamma _{4} & \gamma _{5} & \ldots & \ldots & \gamma _{k+2} & \gamma _{k+3}
\end{array} \right). $$
is equal to
\begin{equation*}
\Gamma_{i}^{\left[3\atop {3\atop 3+1} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
378 & \text{if } i = 2, \\
3\cdot2^{k+2} +6192 & \text{if } i = 3, \\
71\cdot2^{k+2} +124864 & \text{if } i = 4, \\
651\cdot2^{k+3} +1246848 & \text{if } i = 5, \\
2^{2k+4} + 645\cdot2^{k+7} +15464448 & \text{if } i = 6, \\
27\cdot2^{2k+4} + 531\cdot2^{k+11} +93782016 & \text{if } i = 7, \\
105\cdot2^{2k+6} + 2625\cdot2^{k+12} -1486356480 & \text{if } i = 8, \\
105\cdot2^{2k+10} -315\cdot2^{k+17} +3523215360 & \text{if } i = 9, \\
2^{3k+7} - 7\cdot2^{2k+14} + 7\cdot2^{k+22} -2147483648 & \text{if } \; i =10
\end{cases}
\end{equation*}
\end{example}
\begin{example}
\textbf{$s=3,\;m=0,\;l=2.$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3+2} \right]\times k} $ of triple persymmetric $11\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\gamma _{4} & \gamma _{5} & \ldots & \ldots & \gamma _{k+2} & \gamma _{k+3} \\
\gamma _{5} & \gamma _{6} & \ldots & \ldots & \gamma _{k+3} & \gamma _{k+4} \\
\end{array} \right). $$
is equal to
\begin{equation*}
\Gamma_{i}^{\left[3\atop {3\atop 3+2} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
378 & \text{if } i = 2, \\
3\cdot2^{k+2} +6192 & \text{if } i = 3, \\
39\cdot2^{k+2} +99264 & \text{if } i = 4, \\
347\cdot2^{k+3} + 1503872 & \text{if } i = 5, \\
2^{2k+4} + 618\cdot2^{k+6} +21426176 & \text{if } i = 6, \\
3\cdot2^{2k+4} + 1293\cdot2^{k+9} +246448128 & \text{if } i = 7, \\
27\cdot2^{2k+6} + 531\cdot2^{k+14} +1500512256 & \text{if } i = 8, \\
105\cdot2^{2k+8} + 2625\cdot2^{k+15} - 92897280\cdot2^{8} & \text{if } i = 9, \\
105\cdot2^{2k+12} -315\cdot2^{k+20} +53760\cdot2^{20} & \text{if } i = 10, \\
2^{3k+8} - 7\cdot2^{2k+16} + 7\cdot2^{k+25} - 2^{35} & \text{if } i =11
\end{cases}
\end{equation*}
\end{example}
\begin{example}
\textbf{$s=3,\;m=0,\;l=3.$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3+3} \right]\times k} $ of triple persymmetric $12\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\gamma _{4} & \gamma _{5} & \ldots & \ldots & \gamma _{k+2} & \gamma _{k+3} \\
\gamma _{5} & \gamma _{6} & \ldots & \ldots & \gamma _{k+3} & \gamma _{k+4} \\
\gamma _{6} & \gamma _{7} & \ldots & \ldots & \gamma _{k+4} & \gamma _{k+5}
\end{array} \right). $$
is equal to
\begin{equation*}
\Gamma_{i}^{\left[3\atop {3\atop 3+3} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
378 & \text{if } i = 2, \\
3\cdot2^{k+2} +6192 & \text{if } i = 3, \\
39\cdot2^{k+2} +99264 & \text{if } i = 4, \\
219\cdot2^{k+3} + 1569408 & \text{if } i = 5, \\
2^{2k+4} + 314\cdot2^{k+6} +24113152 & \text{if } i = 6, \\
3\cdot2^{2k+4} + 621\cdot2^{k+9} +342786048 & \text{if } i = 7, \\
3\cdot2^{2k+6} + 1293\cdot2^{k+12} +3943170048 & \text{if } i = 8, \\
27\cdot2^{2k+8} + 531\cdot2^{k+17} + 93782016\cdot2^{8} & \text{if } i = 9, \\
105\cdot2^{2k+10} + 2625\cdot2^{k+18} - 92897280\cdot2^{12} & \text{if } i = 10, \\
105\cdot2^{2k+14} -315\cdot2^{k+23} +53760\cdot2^{24} & \text{if } i = 11, \\
2^{3k+9} - 7\cdot2^{2k+18} + 7\cdot2^{k+28} - 2^{39} & \text{if } i =12
\end{cases}
\end{equation*}
\end{example}
\begin{example}
\textbf{$s=3,\;m=0,\;l=4.$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3+4} \right]\times k} $ of triple persymmetric $13\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\gamma _{4} & \gamma _{5} & \ldots & \ldots & \gamma _{k+2} & \gamma _{k+3} \\
\gamma _{5} & \gamma _{6} & \ldots & \ldots & \gamma _{k+3} & \gamma _{k+4} \\
\gamma _{6} & \gamma _{7} & \ldots & \ldots & \gamma _{k+4} & \gamma _{k+5} \\
\gamma _{7} & \gamma _{8} & \ldots & \ldots & \gamma _{k+5} & \gamma _{k+6}
\end{array} \right). $$
is equal to
\begin{equation*}
\Gamma_{i}^{\left[3\atop {3\atop 3+4} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
378 & \text{if } i = 2, \\
3\cdot2^{k+2} +6192 & \text{if } i = 3, \\
39\cdot2^{k+2} +99264 & \text{if } i = 4, \\
219\cdot2^{k+3} + 1569408 & \text{if } i = 5, \\
2^{2k+4} + 186\cdot2^{k+6} +25161728 & \text{if } i = 6, \\
3\cdot2^{2k+4} + 317\cdot2^{k+9} +38577664 & \text{if } i = 7, \\
3\cdot2^{2k+6} + 621\cdot2^{k+12} +5356032\cdot2^{10} & \text{if } i = 8, \\
3\cdot2^{2k+8} + 1293\cdot2^{k+15} + 240672\cdot 2^{18} & \text{if } i = 9, \\
27\cdot2^{2k+10} + 531\cdot2^{k+20} + 93782016\cdot 2^{12} & \text{if } i = 10, \\
105\cdot2^{2k+12} + 2625\cdot2^{k+21} - 92897280\cdot 2^{16} & \text{if } i = 11, \\
105\cdot2^{2k+16} -315\cdot2^{k+26} +53760\cdot 2^{28} & \text{if } i = 12, \\
2^{3k+10} - 7\cdot2^{2k+20} + 7\cdot2^{k+31} - 2^{43} & \text{if } i =13
\end{cases}
\end{equation*}
\end{example}
\begin{example}
\textbf{$s=3,\;m=0,\;l=5.$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3+5} \right]\times k} $ of triple persymmetric $14\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\gamma _{4} & \gamma _{5} & \ldots & \ldots & \gamma _{k+2} & \gamma _{k+3} \\
\gamma _{5} & \gamma _{6} & \ldots & \ldots & \gamma _{k+3} & \gamma _{k+4} \\
\gamma _{6} & \gamma _{7} & \ldots & \ldots & \gamma _{k+4} & \gamma _{k+5} \\
\gamma _{7} & \gamma _{8} & \ldots & \ldots & \gamma _{k+5} & \gamma _{k+6} \\
\gamma _{8} & \gamma _{9} & \ldots & \ldots & \gamma _{k+6} & \gamma _{k+7}
\end{array} \right). $$
is equal to
\begin{equation*}
\Gamma_{i}^{\left[3\atop{ 3\atop 3+5} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
378 & \text{if } i = 2, \\
3\cdot2^{k+2} +6192 & \text{if } i = 3, \\
39\cdot2^{k+2} +99264 & \text{if } i = 4, \\
219\cdot2^{k+3} + 1569408 & \text{if } i = 5, \\
2^{2k+4} + 186\cdot2^{k+6} +25161728 & \text{if } i = 6, \\
3\cdot2^{2k+4} + 189\cdot2^{k+9} +402554880 & \text{if } i = 7, \\
3\cdot2^{2k+6} + 317\cdot2^{k+12} +385777664\cdot2^{4} & \text{if } i = 8, \\
3\cdot2^{2k+8} + 621\cdot2^{k+15} +342786048\cdot2^{8} & \text{if } i = 9, \\
3\cdot2^{2k+10} + 1293\cdot2^{k+18} + 246448128\cdot 2^{12} & \text{if } i = 10, \\
27\cdot2^{2k+12} + 531\cdot2^{k+23} + 93782016\cdot 2^{16} & \text{if } i = 11, \\
105\cdot2^{2k+14} + 2625\cdot2^{k+24} - 92897280\cdot 2^{20} & \text{if } i = 12, \\
105\cdot2^{2k+18} -315\cdot2^{k+29} +53760\cdot 2^{32} & \text{if } i = 13, \\
2^{3k+11} - 7\cdot2^{2k+22} + 7\cdot2^{k+34} - 2^{47} & \text{if } i =14
\end{cases}
\end{equation*}
\end{example}
\begin{example}
\textbf{$s=3,\;m=0,\; l\geqslant 5$}
The number $ \Gamma_{i}^{\left[3\atop {3\atop 3+l} \right]\times k} $ of triple persymmetric $(9+l)\times k$ rank i matrices over $\mathbb{F}_{2}$ of the form \\
$$ \left ( \begin{array} {cccccc}
\alpha _{1} & \alpha _{2} & \ldots & \ldots & \alpha _{k-1} & \alpha _{k} \\
\alpha _{2 } & \alpha _{3} & \ldots & \ldots & \alpha _{k} & \alpha _{k+1} \\
\alpha _{3 } & \alpha _{4} & \ldots & \ldots & \alpha _{k+1} & \alpha _{k+2} \\
\hline
\beta _{1} & \beta _{2} & \ldots & \ldots & \beta_{k-1} & \beta _{k} \\
\beta _{2} & \beta _{3} & \ldots & \ldots & \beta_{k} & \beta _{k+1} \\
\beta _{3} & \beta _{4} & \ldots & \ldots & \beta_{k+1} & \beta _{k+2} \\
\hline
\gamma _{1} & \gamma _{2} & \ldots & \ldots & \gamma _{k-1} & \gamma _{k} \\
\gamma _{2} & \gamma _{3}& \ldots & \ldots & \gamma _{k} & \gamma _{k+1} \\
\gamma _{3} & \gamma _{4} & \ldots & \ldots & \gamma _{k+1} & \gamma _{k+2} \\
\gamma _{4} & \gamma _{5} & \ldots & \ldots & \gamma _{k+2} & \gamma _{k+3} \\
\gamma _{5} & \gamma _{6} & \ldots & \ldots & \gamma _{k+3} & \gamma _{k+4} \\
\gamma _{6} & \gamma _{7} & \ldots & \ldots & \gamma _{k+4} & \gamma _{k+5} \\
\gamma _{7} & \gamma _{8} & \ldots & \ldots & \gamma _{k+5} & \gamma _{k+6} \\
\gamma _{8} & \gamma _{9} & \ldots & \ldots & \gamma _{k+6} & \gamma _{k+7} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{3+l} & \gamma _{4+l} & \ldots & \ldots & \gamma _{k+1+l} & \gamma _{k+2+l} \\
\end{array} \right). $$
is equal to
\begin{equation*}
\Gamma_{i}^{\left[3\atop{ 3\atop 3+l} \right]\times k}
= \begin{cases}
1 & \text{if } i = 0, \\
21 & \text{if } i = 1, \\
378 & \text{if } i = 2, \\
3\cdot2^{k+2} +6192 & \text{if } i = 3, \\
39\cdot2^{k+2} +99264 & \text{if } i = 4, \\
219\cdot2^{k+3} + 1569408 & \text{if } i = 5, \\
2^{2k+4} + 186\cdot2^{k+6} +25161728 & \text{if } i = 6, \\
3\cdot2^{2k+2i-10} + 189\cdot2^{k+3i-12} +402554880\cdot 2^{4(i-7)} & \text{if } 7\leqslant i\leqslant l+2 \\
3\cdot2^{2k+2l-4} + 317\cdot2^{k+3l-3} +38577664\cdot2^{4l-16} & \text{if } i = l+3, \\
3\cdot2^{2k+2l-2} + 621\cdot2^{k+3l} +342786048\cdot2^{4l-12} & \text{if } i=l+4, \\
3\cdot2^{2k+2l} + 1293\cdot2^{k+3l+3} +246448128\cdot2^{4l-8} & \text{if } i=l+5, \\
27\cdot2^{2k+2l+2} + 531\cdot2^{k+3l+8} +93782016\cdot2^{4l-4} & \text{if } i=l+6, \\
105\cdot2^{2k+2l+4} + 2625\cdot2^{k+3l+9} -92897280\cdot2^{4l} & \text{if } i=l+7, \\
105\cdot2^{2k+2l+8} -315\cdot2^{k+3l+14} +53760\cdot2^{4l+12} & \text{if } i=l+8 \\
2^{3k+l+6} -7\cdot2^{2k+2l+12} +7\cdot2^{k+3l+19}-2^{4l+27} & \text{if } i = l+9
\end{cases}
\end{equation*}
\begin{proof}
We compute $ \Gamma_{i}^{\left[3\atop{ 3\atop 3+l} \right]\times k} $ for $0\leqslant i\leqslant l+9$ by using the following two reduction formulas:\\
\begin{align*}
\Gamma_{7+j}^{\left[3\atop{ 3\atop 3+l} \right]\times k}
& =2^{4j}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3+l-j} \right]\times (k-j)} & \text{if } 0\leq j\leq l, \; k \geqslant 7+j \\
\Gamma_{7+l+j}^{\left[3\atop{ 3\atop 3+l} \right]\times k}
& =2^{4l+12j}\cdot\Gamma_{2(3-j)+1}^{\left[3-j\atop{ 3-j\atop 3-j} \right]\times (k-l-3j)} & \text{if } 0\leq j\leq 2, \; k \geqslant 7+l+j
\end{align*}
combined with the formula:\\
\begin{equation*}
\Gamma_{7}^{\left[3\atop{ 3\atop 3+l} \right]\times k} = \Gamma_{7}^{\left[3\atop{ 3\atop 3+5} \right]\times k} \quad \text{if }\; l\geqslant 5, \; k > 7
\end{equation*}
More precisely : (with $ k > 7+j $)\\
\begin{equation*}
\Gamma_{7+j}^{\left[3\atop{ 3\atop 3+l} \right]\times k}
= \begin{cases}
2^{4j}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3+5} \right]\times (k-j)} =
3\cdot2^{2k+2j+4} + 189\cdot2^{k+3j+9} +402554880\cdot2^{4j} & \text{if } 0\leqslant j \leqslant l-5, \\
2^{4l-16}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3+4} \right]\times (k-l+4)} =
3\cdot2^{2k+2l-4} + 317\cdot2^{k+3l-3} +38577664\cdot2^{4l-16} & \text{if } j = l-4, \\
2^{4l-12}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3+3} \right]\times (k-l+3)} =
3\cdot2^{2k+2l-2} + 621\cdot2^{k+3l} +342786048\cdot2^{4l-12} & \text{if } j = l-3, \\
2^{4l-8}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3+2} \right]\times (k-l+2)} =
3\cdot2^{2k+2l} + 1293\cdot2^{k+3l+3} +246448128\cdot2^{4l-8} & \text{if } j = l-2, \\
2^{4l-4}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3+1} \right]\times (k-l+1)} =
27\cdot2^{2k+2l+2} + 531\cdot2^{k+3l+8} +93782016\cdot2^{4l-4} & \text{if } j = l-1, \\
2^{4l}\cdot\Gamma_{7}^{\left[3\atop{ 3\atop 3} \right]\times (k-l)} =
105\cdot2^{2k+2l+4} + 2625\cdot2^{k+3l+9} -92897280\cdot2^{4l} & \text{if } j = l, \\
2^{4l+12}\cdot\Gamma_{5}^{\left[2\atop{ 2\atop 2} \right]\times (k-l-3)} =
105\cdot2^{2k+2l+8} -315\cdot2^{k+3l+14} +53760\cdot2^{4l+12} & \text{if } j = l+1, \\
2^{4l+24}\cdot\Gamma_{3}^{\left[1\atop{ 1\atop 1} \right]\times (k-l-6)} =
2^{3k+l+6} -7\cdot2^{2k+2l+12} +7\cdot2^{k+3l+19}-2^{4l+27} & \text{if } j = l+2
\end{cases}
\end{equation*}
\end{proof}
The number $ R_{q} $ of solutions \\
$(Y_1,Z_1,U_{1},V_{1}, \ldots,Y_q,Z_q,U_{q},V_{q}) \in \big( \mathbb{F}_{2}[T ] \big)^{4q} $ of the polynomial equations
\[\left\{\begin{array}{c}
Y_{1}Z_{1} +Y_{2}Z_{2}+ \ldots + Y_{q}Z_{q} = 0, \\
Y_{1}U_{1} + Y_{2}U_{2} + \ldots + Y_{q}U_{q} = 0,\\
Y_{1}V_{1} + Y_{2}V_{2} + \ldots + Y_{q}V_{q} = 0,\\
\end{array}\right.\]
satisfying the degree conditions \\
$$ degY_j \leq k-1 , \quad degZ_j \leq 2 ,\quad degU_{j}\leq 2,\quad degV_{j}\leq 2+l \quad for \quad 1\leq j \leq q, $$ \\
is equal to
\begin{align*}
2^{(k+9+l)q -(3k+l+6)} \sum_{i = 0}^{\inf(k, 9+l)}
\Gamma_{i}^{\left[3\atop {3\atop 3+l} \right]\times k} 2^{-iq}
\end{align*}
In particular: \\
\begin{align*}
R_{1}= 2^{-2k+3} \sum_{i = 0}^{\inf(k, 9+l)}
\Gamma_{i}^{\left[3\atop {3\atop 3+l} \right]\times k} 2^{-i} = 2^{9+l} +2^{k} -1
\end{align*}
\end{example}
\selectlanguage{english}
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TITLE: Does category Cat encode all it's properties in itself?
QUESTION [0 upvotes]: I want specifically know if one can tell that the morphisms between objects in a category are functors or not without being told that the studies category is $Cat$? That means does the objects and morphisms in between them in $Cat$ contain all the information needed to recognize it's morphisms as functors, or such recognition is something external to this category?
REPLY [3 votes]: The question as it is right know seems to be the following:
Given a category $\mathcal C$, how can I know if $\operatorname{Mor}(\mathcal C) = \operatorname{Mor}(\mathsf{Cat})$?
Then the question does not make much sense. For example, you can probably find a bijection $\phi : \operatorname{Mor}(\mathsf{Cat}) \to \operatorname{Mor}(\mathsf{Set})$ and then craft a new category $\mathcal C$ whose objects are the small sets of and whose morphism class is $\operatorname{Mor}(\mathsf{Cat})$ by setting the sets $\mathcal C(S,T)$ to be $\phi^{-1}(\mathsf{Set}(S,T))$ for each sets $S,T$. So $\mathcal C$ is now a category whose morphism are functors. But what you have done is just renaming each set-function by the name of a functor. It does not tell you anything about $\mathcal C$ or its properties. (And actually, depending on your foundations, this construction might be simply wrong/ill-typed.)
A more sensible question is:
Given a category $\mathcal C$, how can I know if $\mathcal C$ is equivalent to $\mathsf{Cat}$?
Then the question has been tackle by Lawvere under the name Elementary Theory of the Category of Categories (see ETCC).
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INFJ personality type is one of the sixteen personality types according to the MBTI typology. INFJ is referred to as “Artist”.
The basic characteristics of this personality type are:
- The rarest type, whimsical dreamer doubts about himself
- Perfectionist, often controls things
- Excellent observer, very empathetic
- Practically oriented, likes order
- Prefers solitude and tranquility
- Rich inner world
- Prefers deep relationships and unconditional loyalty
- Gives great emphasis on justice
Kroeger, Thuesen and Rutledge indicate following characteristics and recommendations:
- At work: provides inspiration for others and well-intentioned commitment to the organization
- Professional development: must understand that his enthusiasm for future projects may seem unrealistic to others
- Leadership position: uses imagination, inspires by moral context towards change and development
INFJ personality type in practice: Every type of person is suited for different work. People with INFJ personality type usually excel in organizations in the following professions:
- Corporate Psychologist
- HR specialist
- Lecturer
- Marketing Manager
- Specialist for Development and Education
Following important people had the INFJ personality type:
- Aristophanes - Greek dramatist
- Goethe - poet
- Nelson Mandela - President of South Africa
- Oprah Winfrey - TV show host
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\begin{document}
\title{The Classification of Generalized Riemann Derivatives}
\author{J. Marshall Ash}
\address{Department of Mathematics, DePaul University\\
Chicago, IL 60614}
\email{mash@condor.depaul.edu}
\urladdr{http://www.depaul.edu/\symbol{126}mash/}
\author{Stefan Catoiu}
\email{scatoiu@condor.depaul.edu}
\urladdr{http://www.depaul.edu/\symbol{126}scatoiu/}
\author{William Chin}
\email{wchin@condor.depaul.edu}
\urladdr{http://www.depaul.edu/\symbol{126}wchin/}
\thanks{This paper is in final form and no
version of it will be submitted for publication elsewhere.}
\date{v22: December 16, 2014}
\subjclass{Primary 26A24; Secondary 26A27, 16S34}
\keywords{derivatives, generalized derivatives, Riemann
derivatives, $\mathcal{A}$-derivatives, generalized Riemann derivatives, even and odd differences, group algebras}
\begin{abstract}
We characterize all pairs $(\Delta_{\mathcal{A}},\Delta_{\mathcal{B}})$ of generalized Riemann differences for which $\mathcal{A}$-differentiability implies $\mathcal{B}$-differentiability.
Two generalized Riemann derivatives $\mathcal{A}$ and $\mathcal{B}$ are equivalent if a function has a derivative in the sense of $\mathcal{A}$ at a real number $x$ if and only if it has a derivative in the sense of $\mathcal{B}$ at $x$. We determine the equivalence classes for this equivalence relation.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{Motivation}
The $n$th Riemann difference of a function $f$ is the difference
\begin{equation*}
\Delta_{n}f(x,h)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}f(x+\left( n-k\right) h),
\end{equation*}
and the $n$th symmetric Riemann difference of $f$ is
\begin{equation*}
\Delta_{n}^{s}f(x,h)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}f(x+(\frac{n}{2}-k)h).
\end{equation*}
The function $f$ is $n$ times Riemann (resp. symmetric Riemann)
differentiable at $x$ if the limit $\displaystyle R_{n}f(x)=\lim_{h
\rightarrow 0}\frac{\Delta_{n}f(x,h)}{h^{n}}\text{ (resp. $R_{n}^{s}f(x)=\lim_{h
\rightarrow 0}\frac{\Delta_{n}^{s}f(x,h)}{h^{n}}$)}$ exists as a finite number.
If $f$ is $n$ times differentiable at $x$, one can use its $n$th Taylor polynomial about $x$ to see that $f$ is $n$
times Riemann and symmetric Riemann differentiable at $x$ and $
R_{n}f(x)=R_{n}^{s}f(x)=f^{\left( n\right) }(x)$. With the exception of the $n=1$ forward Riemann case, where the definition of Riemann differentiation is the same as the one for ordinary differentiation, the converse of this is in general false. If $n\geq 2$, the function
\[ f(x)=\begin{cases}
0, &\text{if }x\in \mathbb{Q}\\
x, &\text{if }x\notin \mathbb{Q}
\end{cases}
\]
is $n$ times Riemann differentiable at $x=0$, without being first order differentiable at zero.
In the symmetric Riemann case, every discontinuous at zero odd function is symmetric Riemann differentiable at zero of all even orders without necessarily being differentiable at zero of any order, and every discontinuous at zero even function is symmetric Riemann differentiable at zero of all odd orders without necessarily being differentiable at zero of any order. Moreover, symmetric Riemann differentiability of a certain order does not imply symmetric Riemann differentiability of a different order. For example, at $x=0$ the even function
\[
f(x)=\begin{cases}
0, & \text{if } x\notin \mathbf{Q}\\
1, & \text{if } x\in \mathbf{Q}
\end{cases}
\]
is symmetric Riemann differentiable of all odd orders, and it is not symmetric Riemann differentiable of any even order. This example might lead one to guess that,
in the symmetric Riemann case, higher order differentiability implies lower order differentiability when the two orders have the same parity. This is disproved in greater generality in Section 6.1, Corollary 2.
Riemann derivatives were generalized in \cite{As}. A generalized $n$th
Riemann difference of a function $f$ is a difference of the form
\begin{equation}
\Delta _{\mathcal{A}}f(x,h)=\sum_{i=1}^{m}A_{i}f(x+a_{i}h),
\label{GRD}
\end{equation}
where $\mathcal{A}=\{A_{1},\ldots ,A_{m};a_{1},\ldots ,a_{m}\}$ is a set of $2m$
parameters, with $a_{1},\ldots ,a_{m}$ distinct, whose elements satisfy the
Vandermonde conditions $\sum_{i=1}^{m}A_{i}a_{i}^{j}=n!\cdot \delta _{jn}$,
for $j=0,1,\ldots ,n$. By linear algebra, the excess number $e=m-\left( n+1\right) $ is non-negative. Some
interesting examples of $\mathcal{A}$-derivatives with positive excess appear in
numerical analysis; see \cite{AJ} and \cite{AJJ}. The \textit{$\mathcal{A}$-derivative} of $f$
is defined by the limit
\begin{equation*}
D_{\mathcal{A}}f(x)=\lim_{h\rightarrow 0}\frac{\Delta _{\mathcal{A}}f(x,h)}{
h^{n}}.
\end{equation*}
For any $\mathcal{A}$, the derivative associated with $\mathcal{A}$ is a generalized Riemann derivative. Conversely, any generalized Riemann derivative is an $\mathcal{A}$-derivative for some $\mathcal{A}$. For simplicity, we will use the notation $\mathcal{A}$-derivative to mean both a particular and a generalized Riemann derivative. Context will always make clear which is meant. We have found this abuse of notation convenient. (E.g., the $\mathcal{A}$-derivatives $\mathcal{B}=\{1,-1;1,0\}$ and $\mathcal{C}=\{1,-1;1/2,-1/2\}$ are inequivalent, since $|x|$ has a $\mathcal{C}$-derivative at $x=0$, but not a $\mathcal{B}$-derivative there.)
Most of the results for classical Riemann derivatives hold true for $
\mathcal{A}$-derivatives of differentiable functions $f$. For example, it is true that
\begin{equation}
\text{ordinary $n$th derivative exists at $x\Longrightarrow $ every $n$th $\mathcal{A}
$-derivative exists at $x$} \label{imp}
\end{equation}
and, as seen before, the converse of this is in general false for each $n$.
Our main motivation is the the following theorem of \cite{ACC} that, in the particular case of $n=1$, classifies all $\mathcal{A}$-derivatives for which the converse of (\ref{imp}) is true.
\begin{theorem}
\begin{description}
\item[A] The first order $\mathcal{A}$-derivatives which are dilates ($
h\rightarrow sh$, for some $s\neq 0$) of
\begin{equation}
\lim_{h\rightarrow 0}\frac{Af(x+rh)+Af(x-rh)-2Af(x)+f(x+h)-f(x-h)}{2h},
\label{Ar der}
\end{equation}
where $Ar\neq 0$ are equivalent to ordinary differentiation.
\item[B] Given any other $\mathcal{A}$-derivative of any order $n=1,2,\dots $
, there is a (Lebesgue) measurable function $f\left( x\right) $ such that $D_{\mathcal{A
}}f\left( 0\right) $ exists, but the $n$th (Peano) derivative $f_{n}\left( 0\right) $
does not.
\end{description}
\end{theorem}
\begin{remark}
Part A of the above theorem displays some $\mathcal{A}$-derivatives
equivalent to the ordinary first order derivative. Part B asserts that no
first order $\mathcal{A}$-derivative that is not mentioned in Part A is equivalent to
ordinary first order differentiation and also that no higher order $
\mathcal{A}$-derivative is equivalent to ordinary differentiation of the
same order. This makes Theorem 1 the best possible result with regard to
reversing the implication in (\ref{imp}). On the other hand, since the first
forward Riemann derivative is the same as the first ordinary derivative,
Theorem 1 classifies all $\mathcal{A}$-derivatives of order $1$ that are
equivalent to the Riemann derivative $R_{1}f(x)=\lim_{h\rightarrow 0}\frac{
f(x+h)-f(x)}{h}$. This leads to the following more general problem: given any generalized Riemann derivative
$\mathcal{B}$, determine all generalized Riemann derivatives $\mathcal{A}$
such that
\begin{equation}
\text{$\mathcal{B}$-derivative exists at $x\Longleftrightarrow $
$\mathcal{A}$-derivative exists at $x$.} \label{A-B}
\end{equation}
This is the main goal of the present work.
\end{remark}
\begin{remark} The terms of the numerator in (\ref{Ar der}) fall into two categories: the sum of the
first three terms is a scalar multiple of an $r$-dilate of the even
difference $f(x+h)+f(x-h)-2f(x)$, and the remaining two terms add up to the odd
difference $f(x+h)-f(x-h)$. This motivates us to expect that a
classification of $\mathcal{A}$-derivatives given by the equivalence (\ref
{A-B}) will be stated in terms of dilates, even differences, and odd
differences.
\end{remark}
\subsection{Even and odd differences} A (not necessarily generalized Riemann) difference
\[
\Delta_{\mathcal{A}}f(x,h)=\sum A_if(x+a_ih)
\]
(where $\mathcal{A}=\{A_1,\ldots ,A_m;a_1,\ldots ,a_m\}$ is a set of $2m$ parameters with $a_1,\ldots ,a_m$ distinct)
is \textit{even} if $\Delta_{\mathcal{A}}f(x,-h)=\Delta_{\mathcal{A}}f(x,h)$ and it is \textit{odd} if $\Delta_{\mathcal{A}}f(x,-h)=-\Delta_{\mathcal{A}}f(x,h)$. For example, the symmetric Riemann difference $\Delta_nf(x,h)$ is even when $n$ is even and it is odd when $n$ is odd, while $\Delta_{\mathcal{A}}f(x,h)=2f(x+h)+f(x-h)-3f(x)$ is neither even nor odd.
Each difference $\Delta_{\mathcal{A}}$ gives rise to an even difference $\Delta_{\mathcal{A}}^{ev}$ and an odd difference $\Delta_{\mathcal{A}}^{odd}$, defined as
\begin{equation}
\begin{aligned}
\Delta_{\mathcal{A}}^{ev}f(x,h)&=\frac {\Delta_{\mathcal{A}}f(x,h)+\Delta_{\mathcal{A}}f(x,-h)}2=\sum_iA_i\frac { f(x+a_ih)+f(x-a_ih)}2 ,\\
\Delta_{\mathcal{A}}^{odd}f(x,h)&=\frac {\Delta_{\mathcal{A}}f(x,h)-\Delta_{\mathcal{A}}f(x,-h)}2=\sum_iA_i\frac { f(x+a_ih)-f(x-a_ih)}2 .
\end{aligned}
\label{evenalignedodd}
\end{equation}
The difference $\Delta_{\mathcal{A}}$ is even if and only if $\Delta_{\mathcal{A}}^{ev}=\Delta_{\mathcal{A}}$, and it is odd if and only if $\Delta_{\mathcal{A}}^{odd}=\Delta_{\mathcal{A}}$.
In addition, we have
\begin{equation}
\Delta_{\mathcal{A}}f(x,h)=\Delta_{\mathcal{A}}^{ev}f(x,h)+\Delta_{\mathcal{A}}^{odd}f(x,h).
\label{even-odd}
\end{equation}
Conversely, whenever $\Delta_{\mathcal{A}}$ is written as a sum $\Delta_{\mathcal{A}}=\Delta_{\mathcal{B}}+\Delta_{\mathcal{C}}$
of an even difference $\Delta_{\mathcal{B}}$ and an odd difference $\Delta_{\mathcal{C}}$, we must have
$\Delta_{\mathcal{B}}=\Delta_{\mathcal{A}}^{ev}$ and $\Delta_{\mathcal{C}}=\Delta_{\mathcal{A}}^{odd}$.
Relation (\ref{even-odd}) is therefore the unique writing of $\Delta_{\mathcal{A}}$ as a sum of an even difference and an odd difference, the \textit{components} of $\Delta_{\mathcal{A}}$.
\subsection{Results}
Our main result is the classification of all $\mathcal{A}$-derivatives given by the equivalence of generalized Riemann differentiation of (\ref{A-B}). Its statement can be written in a compact form by correlating parity of the order of the derivative and the parity of the component differences. For this, we define two maps
\[
\epsilon =\epsilon (n)=\begin{cases}
ev,& \text{if $n$ even}\\
odd, & \text{if $n$ odd}
\end{cases}
\text{ and }
\epsilon '=\epsilon '(n)=\begin{cases}
odd,& \text{if $n$ even}\\
ev, & \text{if $n$ odd}.
\end{cases}
\]
Given an $n$th generalized Riemann difference $\Delta _{\mathcal{B}}f(x,h)$, by the Vandermonde conditions, the difference
${\Delta _{\mathcal{B}}f(x,sh)}/{s^n}$ is the only scalar multiple of its $s$-dilate that is also an $n$th generalized Riemann difference. We call this a \textit{scaling} of $\Delta _{\mathcal{B}}f(x,h)$.
We have found the following complete classification of $\mathcal{A}$-derivatives.
\begin{theorem}
\label{2}Let $\mathcal{A}$ and $\mathcal{B}$ be generalized Riemann derivatives of orders $m$ and $n$. Then $
\mathcal{B}$ is equivalent to $\mathcal{A}$ if and only if $m=n$ and there are non-zero constants $A,r$, and $s$ so that
\begin{equation}
\frac{\Delta _{\mathcal{B}}f\left( x,h\right) }{h^{n}}=\frac{\Delta _{
\mathcal{A}}^{\epsilon }f(x,sh)+A\Delta _{\mathcal{A}}^{\epsilon ^{\prime
}}f(x, rh )}{\left( sh\right) ^{n}}. \label{Ar der1}
\end{equation}
This means $\Delta _{\mathcal{B}}^{\epsilon }f(x,h)$ is a scaling of $\Delta _{\mathcal{A}}^{\epsilon }f(x,h)$ and $\Delta _{\mathcal{B}
}^{\epsilon ^{\prime }}f(x,h)$ is a non-zero scalar multiple of a dilate of $
\Delta _{\mathcal{A}}^{\epsilon ^{\prime }}f(x,h)$.
\end{theorem}
\begin{example}
\label{e:2}(i) Let $\Delta _{\mathcal{A}}=\Delta _{n}^{s}$ be the $n$th
symmetric Riemann difference. Theorem 2 says that the only generalized
derivatives that are equivalent to the $n$th symmetric Riemann derivative $
R_{n}^{s}$ are the non-zero scalings of it.
(ii) By taking $\Delta _{\mathcal{A}}f(x,h)=\Delta _{1}f(x,h)=f(x+h)-f(x)$,
the first (forward) Riemann difference, the computation
\begin{equation*}
\begin{aligned} \Delta_{\mathcal{A}}^{\epsilon
}f(x,h)=\Delta_{\mathcal{A}}^{odd }f(x,h)&=\frac
{[f(x+h)-f(x)]-[f(x-h)-f(x)]}2\\ &=\frac {f(x+h)-f(x-h)}2\\
\Delta_{\mathcal{A}}^{\epsilon '}f(x,h)=\Delta_{\mathcal{A}}^{ev
}f(x,h)&=\frac {[f(x+h)-f(x)]+[f(x-h)-f(x)]}2\\ &=\frac
{f(x+h)-2f(x)+f(x-h)}2 \end{aligned}
\end{equation*}
shows that the classification in Theorem 2 is a generalization of the
classification in Theorem 1.
(iii) The first order $\mathcal{A}$-derivative with excess $e=1$ given by
\begin{equation*}
f_{\ast }\left( x\right) =\lim_{h\rightarrow 0}\frac{\left( \frac{1}{2}-\tau
\right) f\left( x+\left( \tau +1\right) h\right) +2\tau f\left( x+\tau
h\right) -\left( \frac{1}{2}+\tau \right) f\left( x+\left( \tau -1\right)
h\right) }{h},
\end{equation*}
where $\tau =1/\sqrt{3}$, arises in numerical analysis; see \cite{AJ}. A very
simple calculation done at the end of this paper using a convenient group algebra notation
developed in Section 5 shows that Theorem \ref{2} asserts that the most general
first order $A$-derivative equivalent to $f_{\ast }\left( x\right) $ is
\begin{equation*}
\lim_{h\rightarrow 0}\frac{\Delta ^{odd}f\left(x, sh\right) }{sh}+A\frac{\Delta
^{ev}f\left(x, rh\right) }{h}
\end{equation*}
where
\begin{eqnarray*}
\Delta ^{odd}f\left(x, h\right) &=\left( \frac{1}{2}-\tau \right) \frac{f\left( x+\left( \tau +1\right) h\right) -f\left( x-\left( \tau +1\right)h\right) }{2}+2\tau \frac{f\left( x+\tau h\right) -f\left( x-\tau h\right) }{2}\\
&-\left( \frac{1}{2}+\tau \right) \frac{f\left( x+\left( \tau -1\right)
h\right) -f\left( x-\left( \tau -1\right) h\right) }{2},
\end{eqnarray*}
\begin{eqnarray*}
\Delta ^{ev}f\left(x, h\right) &=\left( \frac{1}{2}-\tau \right) \frac{f\left( x+\left( \tau +1\right) h\right) +f\left( x-\left( \tau +1\right)h\right) }{2}+2\tau \frac{f\left( x+\tau h\right) +f\left( x-\tau h\right) }{2} \\
&+\left( \frac{1}{2}+\tau \right) \frac{f\left( x+\left( \tau -1\right)
h\right) +f\left( x-\left( \tau -1\right) h\right) }{2},
\end{eqnarray*}
and $s$, $r$, and $A$ are nonzero constants.
\end{example}
We actually have a much more general theorem. For a given generalized Riemann derivative $\mathcal{B}$, it classifies all generalized Riemann derivatives $\mathcal{A}$ such that
\begin{equation}
\text{$\mathcal{B}$-derivative exists at $x\Longrightarrow $ $\mathcal{A}$-derivative exists at $x$.} \label{A->B}
\end{equation}
The result is as follows:
\begin{theorem}
Let $\mathcal{A}$ and $\mathcal{B}$ be generalized Riemann derivatives of orders $m$ and $n$, respectively. Then $
\mathcal{B}$-differentiation implies $\mathcal{A}$-differentiation if and only if $m=n$ and, for every function $f$,
$\Delta_{\mathcal{A}}^{\epsilon }f(x,h)$ and $\Delta_{\mathcal{A}}^{\epsilon '}f(x,h)$ are finite linear combinations
\[\begin{aligned}
\Delta_{\mathcal{A}}^{\epsilon }f(x,h)&=\sum_iU_i\Delta_{\mathcal{B}}^{\epsilon }f(x,u_ih)\\
\Delta_{\mathcal{A}}^{\epsilon '}f(x,h)&=\sum_iV_i\Delta_{\mathcal{B}}^{\epsilon '}f(x,v_ih)
\end{aligned}
\]
of non-zero
$u_i$-dilates of $\Delta_{\mathcal{B}}^{\epsilon }f(x,h)$ and $v_i$-dilates of $\Delta_{\mathcal{B}}^{\epsilon '}f(x,h)$.
\end{theorem}
A basic fact about ordinary derivatives is that the the existence of a lower order derivative does not imply the existence of a higher order derivative. The same is true about generalized Riemann derivatives. Specifically, the next example shows that the existence of a generalized Riemann derivative of a certain order does not imply the existence of any generalized Riemann derivative of higher order. This rules out the case $m>n$ in the above theorem.
\begin{example}
Fix $\mathcal{A}=\{A_{1},\ldots ;a_{1},\ldots \}$ of order $m$ and $\mathcal{B}=\{B_{1},\ldots ;b_{1},\ldots \}$ of order $n$, with $m>n$. We construct a function $f$ which is $\mathcal{B}$-differentiable and not $\mathcal{A}$-differentiable. Let $K$ be the the subfield of $\mathbb{R}$ generated over the rationals by all $a_i$'s and $b_i$'s, and define
\[
f(x)=\begin{cases}
x^m & \text{, if }x\in K\\
0 & \text{, if }x\notin K.
\end{cases}
\]
Note that by the definitions of $K$ and $f$, the operators $\lim_{h\rightarrow 0,h\in K}$ and $\lim_{h\rightarrow 0,h\notin K}$ act independently. They are easy to compute when applied to the quotients $\frac {\Delta_{\mathcal{A}}f(0,h)}{h^m}$ and $ \frac {\Delta_{\mathcal{B}}f(0,h)}{h^n}$. In the first case, these are the different numbers
\[\lim_{h\rightarrow 0,h\in K }\frac {\Delta_{\mathcal{A}}f(0,h)}{h^m}=\frac {d^m(x^m)}{dx^m}(0)=m!\;\text{ and }\;
\lim_{h\rightarrow 0,h\notin K}\frac {\Delta_{\mathcal{A}}f(0,h)}{h^m}=0,\]
so $D_{\mathcal{A}}f(0)$ does not exist. In the second case, the limits
\[\lim_{h\rightarrow 0,h\in K}\frac {\Delta_{\mathcal{B}}f(0,h)}{h^n}=\frac {d^n(x^m)}{dx^n}(0)=0\;\text{ and }\;
\lim_{h\rightarrow 0,h\notin K}\frac {\Delta_{\mathcal{B}}f(0,h)}{h^n}=0\]
are equal, so $D_{\mathcal{B}}f(0)$ exists. In both cases we used the fact that if the $k$th ordinany derivative of a function exists, then any $k$th generalized Riemann derivative exists and they are equal.
Since $K$ is countable, $f$ is measurable.
\end{example}
\begin{example} (i) By taking $\Delta_{\mathcal{B}}f(x,h)=\Delta_1f(x,h)=f(x+h)-f(x)$, the first Riemann difference we studied before, by Theorem 3 the first $\mathcal{A}$-derivatives implied by $R_1$ look like a first order linear combination of non-zero dilates of $R_1^s$ plus a linear combination of dilates of $R_2^s$. It is not hard to see that all these form the class of all first order $\mathcal{A}$-derivatives. Indeed, we already know that
``first order ordinary differentiable implies $\mathcal{A}$-differentiable, for every first order $\mathcal{A}$-derivative.''
(ii) By Theorem 2, there are no even first order and no odd second order $\mathcal{A}$-derivatives. More generally, for each $n$, there are no $n$th $\mathcal{A}$-derivatives of opposite parity.
By Theorem 3, the first symmetric Riemann derivative $R_1^s$ implies all odd first order $\mathcal{A}$-derivatives, and the second symmetric Riemann derivative $R_2^s$ implies all even second order $\mathcal{A}$-derivatives.
As we shall see next, these two implications do not extend to higher order symmetric Riemann derivatives.
(iii) Consider the difference
\[
\Delta_{\mathcal{A}}=\frac{f(2h)-2f(h)+2f(-h)-f(-2h)}{2}
\]
(the reader can check that this satisfies the Vandermonde conditions for a third order generalized Riemann difference) and let $\Delta_{\mathcal{B}}=\Delta_3^sf(0,h)$ be the third symmetric Riemann difference.
We claim that $\Delta_{\mathcal{A}}$ cannot be written as a linear combination of dilates of $\Delta_{\mathcal{B}}$ to deduce, by Theorem 3, that the generalized Riemann derivative corresponding to $\mathcal{A}$ is not implied by the symmetric Riemann derivative $R_3^s$.
To prove the claim, suppose $\Delta_{\mathcal{A}}$ is a linear combination of dilates of $\Delta_{\mathcal{B}}$. We write
\[
\Delta_{\mathcal{A}}=\sum_{i=1}^k\lambda_i\Delta_3^sf(0,r_ih)
=\sum_{i=1}^k\lambda_i\left[ f(\frac 32r_ih)-3f(\frac 12r_ih)+3f(-\frac 12r_ih)-f(-\frac 32r_ih)\right],
\]
where $\lambda_i\neq 0$, for all $i$. Since $\Delta_3^s$ is odd, all $r_i$'s may be taken to be positive, say $0<r_1<r_2<\cdots <r_k$. Note that the terms of the form $Af(rh)$ for the largest (resp. smallest) positive $r$ that appear in the expansions of both sides must be identical.
The largest of the dilates on each side appears only once, so $2=\frac 32r_k$; similarly, each smallest positive dilate appears only once, so $1=\frac 12r_1$.
In particular, $r_k=\frac 43$ is smaller than $r_1=2$, a contradiction.
\end{example}
More elegant equivalent formulations of the statements of Theorems 2 and 3 are given in Theorem 5, using the language of ideals of the group algebra $kG$, where the ground field $k=\mathbb{R}$ is the field of real numbers, and the group $G=\mathbb{R}^{\times }$ is the multiplicative group of non-zero real numbers. Then both theorems are proved in their reformulated forms.
A second kind of equivalence for generalized Riemann derivatives is a.e. equivalence. Say that two $n$th order generalized Riemann derivatives are \textit{a.e. equivalent} if for every measurable function, the set of all real $x$ where exactly one of them exists has measure $0$. In the 1930s, J. Marcinkiewicz and A. Zygmund proved that the Riemann derivative and the symmetric Riemann derivative are a.e. equivalent; see \cite{MZ}. In 1967 it was shown that \textit{all} $n$th order generalized Riemann derivatives are a.e. equivalent; see \cite{As}. This equivalence relation is as coarse as possible, having only one class for each degree. Until recently, the authors believed that the equivalence relation discussed in this paper was as fine as possible, each class consisting only of the scalings of a single $\mathcal{A}$-derivative. Theorem 1 corrected that false notion. Theorem 2 gives the exact answer.
\section{Orders of even and odd differences}
Let $\Delta _{\mathcal{A}}f(x,h)=\sum_{i=1}^{m}A_{i}f(x+a_{i}h)$ be an $n$th generalized Riemann difference as defined in (\ref{GRD}). By (\ref{evenalignedodd}), the decomposition (\ref{even-odd}) can be written as
\begin{equation}
\begin{aligned}
\Delta_{\mathcal{A}}f(x,h)&=\sum_i B_i[f(x+b_ih)+f(x-b_ih)-2f(x)]\\
&+\sum_i C_i[f(x+c_ih)-f(x-c_ih)],
\label{2.1}
\end{aligned}
\end{equation}
where the $b_i$'s and $c_i$'s are all distinct and positive. The architecture of the above brackets automatically implies the $n$th Vandermonde condition $\sum_iA_i=0$ corresponding to $j=0$. The $n$th Vandermonde system then translates into the following:
\begin{equation}
\sum_iB_i[b_i^j+(-b_i)^j]+\sum_iC_i[c_i^j-(-c_i)^j]=n!\delta_{nj},
\label{BC}
\end{equation}
for $j=1,2,\ldots ,n$. If $S_1(j)$ and $S_2(j)$ are the above left and right sums, an immediate consequence of (\ref{BC}) is that
\begin{equation}
\begin{cases}
S_1(j)=0,&\text{if $j$ odd,}\\
S_2(j)=0,&\text{if $j$ even.}
\end{cases}
\label{S1S2}
\end{equation}
Assume that $n$ is an even number. Then (\ref{BC}) and (\ref{S1S2}) imply that
\begin{equation}
\begin{cases}
S_1(j)=n!\delta_{nj},&\text{if $j=2,4,...,n$,}\\
S_2(j)=0,&\text{if $j=1,3,...,n-1$.}
\end{cases}
\label{S1S2'}
\end{equation}
The systems (\ref{S1S2}) and (\ref{S1S2'}) together say that $\Delta_{\mathcal{A}}^{ev}$ satisfies the Vandermonde equations of order $n$, for $j=1,2,\ldots ,n$, and $\Delta_{\mathcal{A}}^{odd}$ satisfies Vandermonde equations of order higher than $n$, for $j=1,2,\ldots ,n$. With the Vandermonde equation for $j=0$ trivially satisfied for both component differences, we deduce that $\Delta_{\mathcal{A}}^{ev}$ is of order $n$ and $\Delta_{\mathcal{A}}^{odd}$ is either zero or of order $>n$. A similar discussion is conducted when $n$ is an odd number.
Let $\Delta _{\mathcal{A}}f(x,h)=\sum_{i=1}^{m}A_{i}f(x+a_{i}h)$ be a difference of a function $f$. By (\ref{even-odd}) and the notation of Section 1, we write $\Delta_{\mathcal{A}}f(x,h)=\Delta_{\mathcal{A}}^{\epsilon }f(x,h)+\Delta_{\mathcal{A}}^{\epsilon '}f(x,h)$ as a sum of its even and odd parts, where $\epsilon $ respects the parity of $n$ and $\epsilon '$ the opposite parity.
\begin{theorem}
The difference $\Delta_{\mathcal{A}}f(x,h)=\sum_{i=1}^{m}A_{i}f(x+a_{i}h)$ is a generalized $n$th Riemann difference if and only if the following two conditions hold:
\begin{enumerate}
\item[(i)] $\Delta _{\mathcal{A}}^{\epsilon }f(x,h)$ is a generalized Riemann difference of order $n$;
\item[(ii)] $\Delta _{\mathcal{A}}^{\epsilon '}f(x,h)$ is a scalar multiple of a generalized Riemann difference of order $>n$.
\end{enumerate}
\end{theorem}
\begin{proof}
One implication is already proved above. Conversely, suppose conditions (i) and (ii) are satisfied and that $n$ is an even number. By (\ref{S1S2'}), condition (i) translates into the Vandermonde system
\[
\sum_iA_i\frac {a_i^k+(-a_i)^k}2=n!\delta_{nk},
\]
for $k=0,1,\ldots ,n$, and condition (ii) translates into
\[
\sum_iA_i\frac {a_i^k-(-a_i)^k}2=0,
\]
for $k=0,1,\ldots ,n$.
We have
\[
\sum_iA_ia_i^k=\sum_iA_i\frac {a_i^k+(-a_i)^k}2+\sum_iA_i\frac {a_i^k-(-a_i)^k}2=n!\delta_{nk}+0=n!\delta_{nk},
\]
for $k=0,1,\ldots ,n$, so $\Delta _{\mathcal{A}}f(x,h)$ is an $n$th Riemann difference. The case when $n$ is odd is treated in a similar manner.
\end{proof}
\section{Dilations and the group algebra of $\mathbb{R}^{\times }$} The dilation by a non-zero real number $r$ of the difference $\Delta_{\mathcal{A}}f(x,h)$ is the difference
\[
\Delta_{\mathcal{B}}f(x,h)=\sum A_if(x+a_irh).
\]
We write this as $x_r\cdot \Delta_{\mathcal{A}}=\Delta_{\mathcal{B}}$. Since
$x_1\cdot \Delta_{\mathcal{A}}=\Delta_{\mathcal{A}}$ and
$x_{rs}\cdot \Delta_{\mathcal{A}}=x_r\cdot (x_s\cdot \Delta_{\mathcal{A}})$, dilation is a group action of the multiplicative group of non-zero real numbers $\mathbb{R}^{\times }$ on the real vector space of all differences of $f$.
On the other hand, since a linear combination with real coefficients of dilates of a difference is also a difference, this gives an action of the real vector space $\text{span}_{\mathbb{R}}\{x_r\;|\;r\in \mathbb{R}^{\times }\}$ on the same space of differences. For example,
\[
(3x_2-ex_{\sqrt{3}})\cdot \Delta_{\mathcal{A}}=3(x_2\cdot \Delta_{\mathcal{A}})-e(x_{\sqrt{3}}\cdot \Delta_{\mathcal{A}}).
\]
The group action and the vector space action together give an action of the group algebra $k\mathbb{R}^{\times }$ of the multiplicative group of the real numbers over the real field $k=\mathbb{R}$. This is the vector space
\[
\mathbf{A}=k\mathbb{R}^{\times }=\text{span}_{k}\{x_r\;|\;r\in \mathbb{R}^{\times }\},
\]
with the multiplication of basis elements given by
\[
x_{r}x_{s}=x_{rs}\text{, for all $r,s\in \mathbb{R}^{\times }$.}
\]
For example, in $\mathbf{A}$ we have \[(2x_1-7x_{-5})(\sqrt{2}x_3-x_{\pi })=2\sqrt{2}x_3-7\sqrt{2} x_{-15}-2x_{\pi }+7x_{-5\pi }.\]
The above discussion can be formalized by saying that the space $D=D(f,x,h)$ of all differences of a function $f$ at $x$ and $h$ without $h$-constant term becomes an $\mathbf{A}$-module via the map $\mathbf{A}\times D\rightarrow D$, given by
\[(x_r,f(x+ah))\mapsto x_r\cdot f(x+ah):=f(x+rah)\text{, for all }r,a\in \mathbb{R}^{\times }.\]
We observe that the unique linear map $\theta :\mathbf{A}\rightarrow D$, defined by $\theta (x_a)=f(x+ah)$, for all $a\in \mathbb{R}^{\times }$, is actually an onto $\mathbf{A}$-module map. Indeed, $\theta (x_rx_a)=\theta (x_{ra})=f(x+rah)=x_r\cdot f(x+ah)=x_r\cdot \theta (x_{a})$. An easy application of this is that
\[D=\theta (\mathbf{A})=\theta (\mathbf{A}x_1)=\mathbf{A}\cdot \theta (x_1)=\mathbf{A}\cdot f(x+h),\]
so $D$ is a cyclic $\mathbf{A}$-module generated by $f(x+h)$.
Indeed, each difference $\Delta_{\mathcal{A}}f(x,h)=\sum_iA_if(x+a_ih)$ of $f$ without $h$-constant term $Af(x)$ can be written as the action
\begin{equation}
\Delta_{\mathcal{A}}f(x,h)=(\sum_iA_ix_{a_i})\cdot f(x+h).
\label{orbit}
\end{equation}
For more properties of group algebras see \cite{P}.
We also define the algebra $\mathbf{B}$ to be the $k$-semigroup algebra of the multiplicative semigroup (actually monoid) of all real numbers, by adjoining the absorbing basis element $x_0$. Its multiplication is given by $x_0x_r=x_0$, for all real numbers $r$. Since
\[ x_0\cdot f(x+h)=f(x),\]
in light of (\ref{orbit}), the space of all differences of $f$ at $x$ and $h$ is a cyclic $\mathbf{B}$-module generated by $f(x+h)$. Being more inclusive, the action of $\mathbf{B}$ is more general than the action of its subalgebra $\mathbf{A}$. On the other hand, the difference between $\mathbf{B}$ and $\mathbf{A}$ is just the basis element $x_0$. This translates into the space of all differences of $f$ at $x$ and $h$ splitting into differences that do and those that do not contain $f(x)$ as a term. Our concern being with $n$th $\mathcal{A}$-differences of $f$, the only adjustment we make between the actions of algebras $\mathbf{A}$ and $\mathbf{B}$ on $f(x+h)$ is at the Vandermonde conditions for $j=0$. See the sentence following equation (\ref{2.1}) to understand how the $j=0$ condition naturally fits into the theory.
We shall see that questions about generalized derivatives translate into questions about principal ideals of $\mathbf{A}$.
\section{Properties of the groups $\mathbb{R}^{+}$ and $\mathbb{R}^{\times }$}
The multiplicative group $\mathbb{R}^{\times }$ of non-zero real numbers has a single torsion element $-1$ of order two. \textit{Torsion elements} in a group are its elements of finite order. A \textit{torsion group} is a group whose elements are all of finite order.
The group isomorphism $r\mapsto (\text{sign}(r),|r|)$ gives a decomposition of $\mathbb{R}^{\times }$ as a direct product $\langle -1\rangle \times \mathbb{R}^+$ of its torsion subgroup $\langle -1\rangle $ and its torsion-free subgroup $\mathbb{R}^+$.
We digress for a paragraph and recall coproducts and direct sums. The \textit{direct product} of a family of groups $\{G_i\}_{i\in I}$ is their Cartesian product
\[\prod_{i\in I}G_i=\{(g_i)_{i}\;|\;g_i\in G_i,i\in I\},\]
with the componentwise multiplication. Their \textit{coproduct} or \textit{direct sum} is the subgroup \[\coprod G_i=\bigoplus_{i\in I}G_i=\{(g_i)_{i}\;|\;g_i=1_{G_i}\text{, for all but finitely many }i\in I\}.\]
The above two definitions coincide precisely when the indexing set $I$ is finite. When dealing with multiplicative groups, we prefer the coproduct $\coprod $ because the direct sum notation $\oplus $ connotes additivity. The direct product enjoys the uniqueness of expression of elements by their components, while the coproduct additionally requires the finiteness of expressions. The usual notion of a direct sum of vector spaces is an abelian group direct sum with an additional scalar structure. The finiteness condition comes from the property that every element of a vector space is a finite linear combination of basis elements.
We show that the multiplicative group $\mathbb{R}^{+}$ is the coproduct of $2^{\aleph_0}$ copies of $\mathbb{Q}$. A way to see this is to first notice that $\log :\mathbb{R}^{+}\rightarrow \mathbb{R}$ is a group isomorphism from the multiplicative group $\mathbb{R}^{+}$ to the additive group $\mathbb{R}$ (with inverse map $\exp :\mathbb{R}\rightarrow \mathbb{R}^{+}$), and observe that $\mathbb{R}$ is a direct sum of copies of $\mathbb{Q}$. Indeed, the field $\mathbb{R}$ is a vector space over its subfield $\mathbb{Q}$, hence it has a basis, say $\{ \log \lambda_i\}_{i\in I}$ (known as a Hamel basis). So we see that $\mathbb{R}$ is a direct sum of the $2^{\aleph_0}$ one-dimensional subspaces generated by the basis elements. The subspaces are each isomorphic to $\mathbb{Q}$ as abelian groups.
In fact, it is apparent that $\log :\mathbb{R}^{+}\rightarrow \mathbb{R}$ is also a $\mathbb{Q}$-vector space isomorphism, where the scalar multiplication on the multiplicative abelian group $\mathbb{R}^{+}$ is defined by exponentiation. So now $\{\lambda_i\}_{i\in I}$ is a basis of $\mathbb{R}^{+}$ over $\mathbb{Q}$, and the span of each $\lambda_i$ is written $\lambda_i^{\mathbb{Q}}$. Thus we have the $\mathbb{Q}$-vector space decomposition
\[
\mathbb{R}^{+}=\exp (\sum \mathbb{Q} \log \lambda_i)=\coprod \lambda_i^{\mathbb{Q}}.
\]
Equivalently, every element of $\mathbb{R}^{+}$ is uniquely expressible as a finite product $\prod \lambda_i^{q_i}$, with $q_i\in \mathbb{Q}$.
\begin{remark}
The decomposition of $\mathbb{R}^{+}$ also follows from the structure theory for abelian torsion-free divisible groups; see \cite{F} or \cite{K}. This involves the basic fact that abelian divisible groups are injective $\mathbb{Z}$-modules, a core concept of homological algebra; see \cite{AF} or \cite{CE}.
\end{remark}
We consider the algebra of generalized polynomials $k[x]_{\mathbb{Q}}$ where rational exponents are allowed. One can define $k[x]_{\mathbb{Q}}$ as the union
\[
\bigcup_{n>0}k[x^{\pm \frac 1n} ]
\]
of Laurent polynomial rings using formal $n$th roots of the indeterminate $x$. Algebras of generalized polynomials $k[x_i\;|\;i\in I]_{\mathbb{Q}}$ in many variables $x_i$ are defined similarly. These algebras have bases of generalized monomials given by finite products $\prod x_i^{q_i}$ with $q_i\in \mathbb{Q}$ where the multiplication is the obvious one, agreeing with the ones in the polynomial subrings $k[x_i^{\pm \frac 1n} \;|\;i\in I]$.
\begin{lemma}
The group algebra $k\mathbb{R}^{+}$ is a generalized polynomial algebra over $k$ in continuum-many variables.
\end{lemma}
\begin{proof}
Define a map $\nu :k[x_i\;|\;i\in I]_{\mathbb{Q}}\rightarrow k\mathbb{R}^{+}$ on monomials by
\[
\nu (\prod x_i^{q_i})=x_{\prod\lambda_i^{q_i}}
\]
and extend $k$-linearly. By the discussion above concerning the uniqueness for the direct sum decomposition, $\nu $ is a $k$-linear bijection. It also follows directly from the definition of group algebra $k\mathbb{R}^{+}$ that $\nu $ is an algebra homomorphism. This completes the proof of the lemma.
\end{proof}
\section{The algebras $\mathbf{A}$, $\mathbf{B}$ and their associated differences}
This section is useful in getting familiar with the basic properties of the algebras $\mathbf{A}$ and $\mathbf{B}$ and their interpretation with differences. It is essential to understanding the rest of the paper, though non-algebraists may wish to skim it on the first pass.
\subsection{Properties of the algebras $\mathbf{A}$ and $\mathbf{B}$.} The group algebra $\mathbf{A}=k\mathbb{R}^{\times }$ and the semigroup algebra $\mathbf{B}=k\mathbb{R}$ have many basic properties some of which translating into properties of differences of a function $f$. Here are some of these properties.
\begin{enumerate}
\item[(AB1)] Both $\mathbf{A}$ and $\mathbf{B}$ have the same identity element $1=x_1$.
\item[(AB2)] Let $\sigma =x_{-1}$. The elements $\pm \sigma $ are the only elements of $\mathbf{A}$ and $\mathbf{B}$ with multiplicative order $2$.
\item[(AB3)] $\mathbf{A}$ contains two orthogonal idempotents $e=\frac 12(1+\sigma )$ and $d=\frac 12(1-\sigma )$ such that $e+d=1=x_1$. These are elements that satisfy $e^2=e$, $d^2=d$, and $de=0$. Pairwise orthogonal idempotents that add up to one are responsible for the decomposition of an algebra as a direct sum of ideals that are also unital subalgebras. In our case,
\begin{equation}
\mathbf{A}=e\mathbf{A}\oplus d\mathbf{A},
\label{eA+dA}
\end{equation}
where $e$ is the identity of $e\mathbf{A}$ and $d$ is the identity of $d\mathbf{A}$.
Algebra $\mathbf{B}$ contains the absorbing idempotent $x_0$, and $\mathbf{B}=kx_0\oplus \mathbf{A}$, a direct sum of unital subalgebras, not ideals. Note that $ex_0=x_0$ and $dx_0=0$. Therefore
\begin{equation}
\mathbf{B}=(kx_0\oplus e\mathbf{A})\oplus d\mathbf{A}=e\mathbf{B}\oplus d\mathbf{B}.
\label{eB+dB}
\end{equation}
\item[(AB4)] Let $e_r=\frac 12(x_r+x_{-r})=ex_r\in e\mathbf{A}$ and $d_r=\frac 12(x_r-x_{-r})=dx_r\in d\mathbf{A}$. \textit{E.g.} we have $d_1=d$ and $e_1=e$. An element of $\mathbf{A}$ looks like $\alpha =\sum_r A_rx_{r}$ and a generic element of $e\mathbf{A}$ is $e\alpha =\sum_r A_re_{r}$. Since $e_{-r}=e_r$, we can write $e\alpha =\sum_{r>0} (A_r+A_{-r})e_{r}$, so
\[e\mathbf{A}=\sum_{r>0} ke_{r}\text{ and similarly }d\mathbf{A}=\sum_{r>0} kd_{r}.\]
\item[(AB5)] After relabeling of coefficients, the above generic element of $e\mathbf{A}$ has the form of a finite sum indexed by positive real numbers
\[
e\alpha =\sum_{r>0} A_re_{r}=\sum_{r>0} A_r\frac {x_r+x_{-r}}2.
\]
Its expression is uniquely determined by twice its positive part $\sum_{r>0} A_rx_r$.
The structure of $e\mathbf{A}$ is then revealed by the mapping $k\mathbb{R}^+\rightarrow e\mathbf{A} $ given by $x_r\mapsto e_r$, for $r>0$. One easily checks that this map is an algebra isomorphism. The result of this is that $e\mathbf{A}$ and (similarly) $d\mathbf{A}$ are isomorphic to the group algebra $k\mathbb{R}^+$. By Lemma 1, we see that both $e\mathbf{A}$ and $d\mathbf{A}$ are generalized polynomial algebras in $2^{\aleph_0}$ variables. This structure will be used in Example 4.
\item[(AB6)] Equation (\ref{eA+dA}) is the decomposition of $\mathbf{A}$ as a direct sum of its eigenspaces given by multiplication by $\sigma $. Specifically, $e\mathbf{A}$ is the $\sigma $-eigenspace with eigenvalue $+1$ and $d\mathbf{A}$ is the $\sigma $-eigenspace with eigenvalue $-1$. We call $e\mathbf{A}$ the even part of $\mathbf{A}$, and $d\mathbf{A}$ is the odd part of $\mathbf{A}$.
\item[(AB7)] The elements of $\mathbf{A}$ of the form $cx_r$, where $c,r\in\mathbb{R}^{\times }$ are invertible in $\mathbf{A}$. These are the \textit{trivial units} of $\mathbf{A}$; their inverses are $c^{-1}x_{r^{-1}}$. We shall see next that the trivial units are not all of the units of $\mathbf{A}$.
It is well-known (see \cite{P}, Chapter 13) that the group algebra of a torsion-free abelian group has only trivial units. In particular, the same is true for the group algebra $k\mathbb{R}^+$ and by (AB5) for its isomorphic copies $e\mathbf{A}$, and $d\mathbf{A}$. The (trivial) units in $e\mathbf{A}$ are $e$-multiples of trivial units in $\mathbf{A}$. This means they are of the form $Aex_r=Ae_r$, for $0\neq A\in k$ and $r>0$. By the same token, the units of $d\mathbf{A}$ are of the form $Bdx_s=Bd_s$, for $0\neq B\in k$ and $s>0$. Since the units of a direct sum of algebras are sums of the units of the summands, by (\ref{eA+dA}) we conclude that the units of $\mathbf{A}$ are the elements of the form
\[ Ae_r+Bd_s,\]
for $0\neq A,B\in k$ and $r,s>0$.
\item[(AB8)] Let $V=\sum_{r\in k}kx_r$. Then the $k$-space of all real functions
\[\mathcal{F}(k)=\{f\;|\;f:k\rightarrow k\}\] is isomorphic to the dual $k$-space
\[V^{*}=\text{Hom}_k(V,k)=\{\varphi \;|\;\varphi :V\rightarrow k\}\]
via the mapping $\mathcal{F}(k)\ni f\mapsto \varphi \in V^{*}$, where $\varphi $ is the linear map defined on basis elements of $V$ by $\varphi (x_r)=f(r)$, for all $r\in k$.
\item[(AB9)] This property, which looks more technical than it really is, is only needed in the proof of Corollary 1.
\newline Let $k'$ be a subfield of $k$ and $V'=\sum_{r\in k'}k'x_r$. Each function $f:k'\rightarrow k'$ can be viewed as a function $f:k\rightarrow k$ by setting $f(t)=0$, for $t\in k\setminus k'$. In this way, the function space $\mathcal{F}(k')$ embeds naturally in $\mathcal{F}(k)$ as the $k'$-subspace
\[\mathcal{F}'(k)=\{f\in \mathcal{F}(k)\;|\;R(f)\subseteq k'\text{ and }k\setminus k'\subseteq N(f)\},\]
where $R(f)$ and $N(f)$ are the range and the nullset of $f$.
By (AB8) applied to $k'$ and $V'$ in place of $k$ and $V$, we have $\mathcal{F}(k')$ is $k'$-linear isomorphic to ${V'}^{*}=\text{Hom}_{k'}(V',k')$.
Let $\mathcal{S}=\{1\}\cup \mathcal{T}$ be a basis of $k$ over $k'$. Then $\bigcup_{r\in k}\mathcal{S}x_r$ is a $k'$-basis of $V$.
Each $k'$-linear map $\varphi :V'\rightarrow k'$ naturally extends to a $k'$-linear map $\varphi :V\rightarrow k$ by setting $\varphi(t)=0$, for $t\in \left(\bigcup_{r\in k}\mathcal{S}x_r\right)\setminus \{x_r|r\in k'\}$, which is also an element of $V^{*}$. In this way, ${V'}^{*}$ is $k'$-linearly isomorphic to
\[V^{*'}=\{\varphi \in V^{*}\;|\;R(\varphi )\subseteq k'\text{ and }(\bigcup_{r\in k}\mathcal{S}x_r)\setminus \{x_r|r\in k'\}\subseteq N(\varphi )\}\]
which is a $k'$-subspace of $V^{*}$. Moreover, $\mathcal{F}'(k)$ is isomorphic to $V^{*'}$ as a $k'$-space via the mapping $\mathcal{F}'(k)\ni f\mapsto \varphi \in V^{*'}$, where $\varphi $ is defined on $k'$-basis elements of $V$ by $\varphi (x_r)=f(r)$ and $\varphi (\mathcal{T}x_r)=0$, for all $r\in k$.
\end{enumerate}
\[
\begin{array}{ccccc}
\mathcal{F}(k') &\rightarrow &\mathcal{F'}(k)&\hookrightarrow &\mathcal{F}(k)\\
\downarrow & & \downarrow & & \downarrow \\
{V'}^{*}&\rightarrow &V^{*'}&\hookrightarrow & V^{*}
\end{array}
\]
The above diagram illustrates the relationship between the vector spaces and linear maps considered in (AB8) and (AB9). The right vertical arrow is the $k$-linear isomorphism of (AB8). The four arrows to the left are the $k'$-linear isomorphisms of (AB9). The two hook-arrows are inclusions of $k'$-subspaces.
\subsection{And their interpretation with differences.} The above properties translate into the language of even and odd differences of Section 1.2 or group algebra actions of Section 3. For example, we have
\[
\begin{aligned}
e\cdot f(x+h)&=\frac {f(x+h)+f(x-h)}2,\\
d\cdot f(x+h)&=\frac {f(x+h)-f(x-h)}2.
\end{aligned}
\]
More generally, if $\alpha =\sum_iA_ix_{a_i}$ is the group algebra element determined by $\Delta_{\mathcal{A}} $ in (\ref{orbit}), then
\[
\begin{aligned}
(e\alpha )\cdot f(x+h)=e\cdot \Delta_{\mathcal{A}}f(x,h)&=\Delta_{\mathcal{A}}^{ev}f(x,h),\\
(d\alpha )\cdot f(x+h)=d\cdot \Delta_{\mathcal{A}}f(x,h)&=\Delta_{\mathcal{A}}^{odd}f(x,h).
\end{aligned}
\]
The second equalities say that the actions of $e$ and $d$ on the space $\mathcal{D}$ of all differences of $f$ at $x$ and $h$ map these differences onto their even and odd parts, respectively. For the first equalities, we notice that $\alpha \in \mathbf{A}$ (resp. $\alpha \in \mathbf{B}$) is equivalent to $\Delta_{\mathcal{A}}f(x,h)$ is a difference without $f(x)$-term (resp. $\Delta_{\mathcal{A}}f(x,h)$ is any difference of $f$). We deduce that the submodules of $\mathcal{D}$ generated by $f(x+h)$ under the actions of $e\mathbf{A}$ and $d\mathbf{A}$ (resp. $e\mathbf{B}$ and $d\mathbf{B}$) are exactly all even and odd differences without $f(x)$-term (resp. all even and odd differences of $f$). Note that odd differences do not have $f(x)$-terms, so the actions of $d\mathbf{A}$ and $d\mathbf{B}$ on $f(x+h)$ coincide. The natural conclusion from this discussion is that the actions of $\mathbf{A}$ and $\mathbf{B}$ on $f(x+h)$, which are sums of the actions of their components given in (\ref{eA+dA}) and (\ref{eB+dB}), amount to splitting their corresponding differences into even and odd components given in (\ref{even-odd}).
\section{Translation Theorem and Proof of Main Results}
The major result of this section, Theorem 5, translates the implication and the equivalence of generalized derivatives into the inclusion and equality of principal ideals of algebras $\mathbf{A}$ or $\mathbf{B}$. We then prove the main classification results of Theorems 2 and 3 stated in the Introduction by means of classifying principal ideals of $\mathbf{A}$ or $\mathbf{B}$ by inclusion or equality. We prove this easily by reference to the basic group algebra properties (AB1)-(AB9) of Section 5.
\subsection{Translation Theorem.} Let $\alpha =\sum_iA_ix_{a_i}$ and $\beta =\sum_iB_ix_{b_i}$ be the elements of $\mathbf{B}$ that correspond to the differences $\Delta_{\mathcal{A}}f(x,h)=\sum_iA_if(x+a_ih)$ and $\Delta_{\mathcal{B}}f(x,h)=\sum_iB_if(x+b_ih)$, as defined in Section 5.2. Call $\{ x_{b_1},x_{b_2},\ldots \}$ the \textit{support} of $\beta $.
As examples, $\alpha =d_1$ corresponds to the first symmetric Riemann difference $\Delta_1^sf(x,h)$, $\alpha =x_1-x_0$ corresponds to the first (forward) Riemann difference $\Delta_1f(x,h)$, and $\alpha =A(e_r-x_0)+d_1$ corresponds to the first difference $\Delta_{\mathcal{A}}f(x,h)$ of Theorem 1.
The ideal of $\mathbf{B}$ generated by $\alpha $ is denoted by $(\alpha )$. Let $\alpha_r =\alpha x_r=\sum A_ix_{a_ir}$ be the \textit{dilate} of $\alpha $ by $r\in k^{\times }$. Note especially that $x_r$ invertible implies $(\alpha )=(\alpha_r)$, and since $(\alpha )$ is the set of all $\mathbf{B}$-multiples of $\alpha $, the ideal $(\alpha )$ is the linear span of all dilates of $\alpha $.
Moreover, since in arguments about $\mathcal{A}$-differentiability of a general function $f$ at $x$ we can always assume without loss that $x=0$ and $f(0)=0$, this amounts to factoring out the ideal $(x_0)$ of $\mathbf{B}$, that is to projecting down to $\mathbf{A}$. In this process, the ideals of $\mathbf{B}$ may be assumed to be ideals of $\mathbf{A}$.
\begin{theorem} With the above notation, if $\alpha $ and $\beta $ correspond to order $n$ generalized Riemann differences $\Delta_{\mathcal{A}}$ and $\Delta_{\mathcal{B}}$, then
\begin{enumerate}
\item[(i)] $(\alpha )\supseteq (\beta )$ iff ``$\mathcal{A}$-differentiable at $x$ $\Longrightarrow $ $\mathcal{B}$-differentiable at $x$''.
\item[(ii)] $(\alpha )= (\beta )$ iff ``$\mathcal{A}$-differentiable at $x$ $\Longleftrightarrow $ $\mathcal{B}$-differentiable at $x$''.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) Assume $(\alpha )\supseteq (\beta )$. We write $\beta $ as a finite sum $\beta=\sum_rc_r\alpha_r$ with $r,c_r\in k$. Let $f$ be $\alpha $-differentiable at $x$ and denote $D_{\alpha }f(x)=d\in k$. As an abuse of notation, we write $\alpha $-differentiable, $D_{\alpha }f(x)$ and $\alpha (x,h)$ to respectively denote $\mathcal{A}$-differentiable, $D_{\mathcal{A}}f(x)$ and $\Delta_{\mathcal{A}}f(x,h)$. Since
\[
D_{\alpha_r }f(x)=\lim_{h\rightarrow 0}\frac {\alpha_r(x,h)}{h^n}=\lim_{h\rightarrow 0}\frac {\alpha(x,rh)}{(rh)^n}\cdot r^n=r^nD_{\alpha }f(x),
\]
linearity of the limit operator makes $f$ a $\beta$-differentiable function at $x$ and $D_{\beta }f(x)=\sum_rc_rr^nd$. This sum actually equals $d$, since $\beta $ is a generalized Riemann derivative.
It remains to prove the converse. Let $G$ be the subgroup of $k^{\times }$ generated by all non-zero $a_i$'s and $b_i$'s. Then $G$ is countable while the set of its cosets in $k^{\times }$ is not. Consider $\{s_n\}_{n>0}$ be a sequence of representatives of cosets of $G$ in $k^{\times }$ such that $\lim_{n\rightarrow \infty }s_n=0$. We prove the contrapositive statement: assuming $(\alpha )\nsupseteq (\beta )$, we show that there exists a function $f$ such that $D_{\mathcal{A}}f(0)$ exists but $D_{\mathcal{B}}f(0)$ does not exist.
Observe that, by assumption and the obvious fact that $(\beta_{s_n})=(\beta )$, we have $\beta_{s_n}\notin (\alpha )$ for all $n$. Moreover, the $\beta_{s_n}$'s are linearly independent modulo the ideal $(\alpha )$. To see this, we note that the support of $\beta $ is included in $G$ and this makes the supports of the $\beta_{s_n}$'s included in the $Gs_n$'s, hence they are pairwise disjoint. Suppose that $\sum \lambda_n\beta_{s_n}\in (\alpha )$, for $\lambda_n\in k$. We write this as $\sum \lambda_n\beta_{s_n}=\alpha \sum \mu_ix_{r_i}$, for $\mu_i\in k$ and $r_i\in k^{\times}$. For each $n$, let $t_{s_n}$ be the sum of all terms $\mu_ix_{r_i}$ of the last sum for which $r_i\in G{s_n}$, and let $t'$ be the sum of the remaining terms. The last equation becomes $\sum \lambda_n\beta_{s_n}=\alpha (t'+\sum t_{s_n})=\alpha t'+\sum \alpha t_{s_n}$. The expression of an element of a group algebra as a sum of elements with supports in distinct cosets of a subgroup is unique.
Thus $\lambda_n\beta_{s_n}=\alpha t_{s_n}$, for all $n$, and $0=\alpha t'$. If $\lambda_n\neq 0$, for some $n$, then
$\beta_{s_n}\in (\alpha )$, a contradiction.
Let $V$ be the vector space defined in (AB8) of Section 5.1. The axiom of choice implies that every linearly independent subset of $V$ can be completed to a basis, so let $W$ be a complement of the subspace $(\alpha )\oplus \sum k\beta_{s_n}$ in $V$. This means that
\[
V=(\alpha )\oplus \sum k\beta_{s_n}\oplus W.
\]
Define a functional $\varphi \in V^{*}$ by setting $\varphi $ identically equal to zero on both $(\alpha )$ and $W$, and $\varphi (\beta_{s_n})=1$, for all $n$. Then the corresponding function $f$ has
\[ \Delta_{\mathcal{A}}f(0,h)=\sum A_if(a_ih)=\sum A_i\varphi (x_{a_ih})=\varphi \left(\sum A_ix_{a_ih}\right)=\varphi (\alpha_h)=0,\]
since $\alpha_h\in (\alpha )$, for all $h$. Thus $D_{\mathcal{A}}f(0)=\lim_{h\rightarrow 0}\frac {\Delta_{\mathcal{A}}f(0,h)}{h^n} =0$. On the other hand,
\[ \Delta_{\mathcal{B}}f(0,s_m)=\sum B_if(b_is_m)=\sum B_i\varphi (x_{b_is_m})=\varphi \left(\sum B_ix_{b_is_m}\right)=\varphi (\beta_{s_m})=1,\]
for all $m$, implies that $D_{\mathcal{B}}f(0)=\lim_{m\rightarrow \infty }\frac {\Delta_{\mathcal{B}}f(0,s_m)}{s_m^n} =\lim_{m\rightarrow \infty }\frac {1}{s_m^n} $ does not exist as a finite number.
Part (ii) is an easy consequence of part (i).
\end{proof}
\begin{corollary}
The counterexample $f$ constructed in the proof of Theorem 5 may not be a measurable function. Nevertheless, the proof can be adapted to make $f$ measurable.
\end{corollary}
\begin{proof} We follow closely the proof of Theorem 5 and stress only the differences. Let $k'$ be the subfield of $k$ generated over the rationals by all $A_i,a_i,B_i,b_i$ and $\{s_n\}_{n\geq 1}$. This is a countable field. Recall from earlier in the section that the ideal $(\alpha )$ is the $k$-span of all shifts $\alpha_r=\alpha x_r$, for $r\in k$. We define $[\alpha ]$ to be the $k'$-span of all shifts $\alpha_r$, for $r\in k'$. Then $(\alpha )=([\alpha ])$ and also $(\beta )=([\beta ])$. Note that the $\beta_{s_n}$'s are $k'$-linearly independent modulo $[\alpha ]$, since they are $k$-linearly independent modulo $(\alpha )$. Let $W$ be a $k'$-complement of $[\alpha ]\oplus \sum k'\beta_{s_n}$ in $V$. This means that
\[
V=[\alpha ]\oplus \sum k'\beta_{s_n}\oplus W.
\]
Since the first two terms above are part of $V'$, using the $k'$-basis of $V$ of (AB9), we may assume that $W$ contains all basis elements of $V$ that are not basis elements of $V'$.
Define a functional $\varphi \in \text{Hom}_{k'}(V,k)$ by setting $\varphi $ identically equal to zero on both $[\alpha ]$ and $W$, and $\varphi (\beta_{s_n})=1$, for all $n$. Then $\varphi $ is an element of $V^{*'}$. Let $f\in \mathcal{F}'(k)$ be the function that corresponds to $\varphi $ via the last isomorphism in (AB9). Let $h$ be any real number. If $h\in k'$, then $\alpha_h\in [\alpha ]$, so
\[ \Delta_{\mathcal{A}}f(0,h)=\sum A_if(a_ih)=\sum A_i\varphi (x_{a_ih})=\varphi \left(\sum A_ix_{a_ih}\right)=\varphi (\alpha_h)=0.\]
If $h\in k\setminus k'$, then
\[ \Delta_{\mathcal{A}}f(0,h)=\sum A_if(a_ih)=\sum A_i\varphi (x_{a_ih})=\sum A_i\cdot 0=0,\]
by the definition of $\varphi $, since $a_ih\in k\setminus k'$ makes $x_{a_ih}\in W$. The rest follows the last part of the proof of Theorem 5. The function $f$ is measurable, since it is non-zero on a subset of the countable set $\sum k'\beta_{s_n}$.
\end{proof}
Let $n\geq 2$. If the ordinary $n$th derivative exists at $x$, so does the
ordinary $m$th derivative for each $m$, $1\leq m<n$. This property fails for
all $\mathcal{A}$-derivatives.
\begin{corollary}
Let $\mathcal{A}$ be a generalized Riemann derivative of order $n$, $n\geq 2$
. If $\mathcal{B}$ is any generalized Riemann derivative of order $m$ where $
1\leq m<n$, then there is a measurable function $f$ so that $D_{\mathcal{A}
}f\left( 0\right) $ exists, but $D_{\mathcal{B}}f\left( 0\right) $ does not.
\end{corollary}
\begin{proof}
Let $\mathcal{A=}\left\{ A_{1},\dots ;a_{1},\dots \right\} $ and $\mathcal{B=
}\left\{ B_{1},\dots ;b_{1},\dots \right\} $. Since $m<n$, the $m$th Vandermonde conditions
force $\sum A_{i}a_{i}^{m}=0$ and $\sum B_{i}b_{i}^{m}=m!$. Consequently $
\Delta _{\mathcal{B}}$ cannot be a linear combination of dilates of $\Delta
_{\mathcal{A}}$. Just as in the proof of Theorem 5 above, this leads to the
existence of the desired function $f$.
\end{proof}
\subsection{Proof of the main results.} We are now ready to prove Theorems 2 and 3 announced in the Introduction.
\begin{proof}[Proof of Theorem 3] Example 2 rules out the case $m>n$, and Corollary 2 rules out the case $m<n$, so $m=n$. By Theorem 5(i), $\mathcal{A}$-differentiation implies $\mathcal{B}$-differentiation is equivalent to $(\alpha )\supseteq (\beta )$. We write $\mathbf{A}=e\mathbf{A}\oplus d\mathbf{A} =\mathbf{A}^{\epsilon }\oplus \mathbf{A}^{\epsilon '}$, by (\ref{eA+dA}). The same relation yields $(\alpha )=\alpha \mathbf{A}=\alpha (e+d)\mathbf{A}=\alpha e\mathbf{A}\oplus \alpha d\mathbf{A}=(\alpha e)\oplus (\alpha d) =(\alpha^{\epsilon })\oplus (\alpha^{\epsilon '})$. A similar expression holds for $ (\beta )$. Basic ideal theory in direct sums of algebras makes the inclusion $(\alpha )\supseteq (\beta )$ equivalent to
both $(\alpha^{\epsilon })\supseteq (\beta^{\epsilon })$ and $(\alpha^{\epsilon '})\supseteq (\beta^{\epsilon '})$. These are clearly equivalent to the two desired equations.
\end{proof}
\begin{proof}[Proof of Theorem 2] The equality $m=n$ follows from Theorem 3. By Theorem 5, ``$\mathcal{A}$-differentiable $\Longleftrightarrow $ $\mathcal{B}$-differentiable'' is equivalent to ``$(\alpha )=(\beta )$'', that is to $\beta =u\alpha $, for some invertible element $u\in \mathbf{A}$.
By (AB7), we write $u=Ae_r+Bd_s$,
for $0\neq A,B\in k$ and $r,s>0$. Consequently, $\beta =Ae_r\alpha +Bd_s\alpha =Ax_re\alpha +Bx_sd\alpha $. Uniqueness of writing of $\beta $ and $\alpha $ as sums of components makes $\beta^{\epsilon }$ and $\beta^{\epsilon '}$
scalar multiples of dilates of $\alpha^{\epsilon }$ and $\alpha^{\epsilon '}$, respectively. Moreover, the equation $m=n$ makes both $\alpha $ and $\beta $ correspond to $n$th generalized Riemann differences and, by Theorem 4(i), the same is true for $\beta^{\epsilon }$ and $\alpha^{\epsilon }$. We conclude that $\Delta _{\mathcal{B}}^{\epsilon }f(x,h)$ is a scaling of $\Delta _{\mathcal{A}}^{\epsilon }f(x,h)$ and $\Delta _{\mathcal{B}
}^{\epsilon ^{\prime }}f(x,h)$ is a non-zero scalar multiple of a dilate of $
\Delta _{\mathcal{A}}^{\epsilon ^{\prime }}f(x,h)$.
\end{proof}
\subsection{Two examples}
Recall from Section 1 that Example 3(iii) contained an application of Theorem 3. It provided two third generalized Riemann differences $\Delta_{\mathcal{A}}$ and $\Delta_{\mathcal{B}}$ such that $\mathcal{B}$-differentiation does not imply $\mathcal{A}$-differentiation. The next example is an application of Theorem 5. It obtains the same result in a different way and additionally shows that for the same two differences $\mathcal{A}$-differentiation does not imply $\mathcal{B}$-differentiation. Similar ring- theoretic arguments can be used for many other examples.
\begin{example} Let $\Delta_{\mathcal{A}}$ and $\Delta_{\mathcal{B}}$ be the third differences of Example 3(iii). The group algebra elements corresponding to them are
\[\alpha =d_2-2d_1\text{ and }\beta =2d_{\frac 32}-6d_{\frac 12}=2x_{\frac 12}\beta',\]
where $\beta'=d_3-3d_1$. We observe that both $\alpha $ and $\beta $ are elements of $d\mathbf{A}$ since they correspond to odd differences, and $(\beta )=(\beta' )$ since the factor $2x_{\frac 12}$ is a unit in $\mathbf{A}$.
The integers $2$ and $3$ are not rational powers of each other, so they are multiplicatively $\mathbb{Q}$-linearly independent. Take the basis elements ${\lambda_1}=\log 2$ and ${\lambda_2}=\log 3$ for $\mathbb{R}$ over $\mathbb{Q}$ as in Section 4. By (AB5) we can pass from $k\mathbb{R}^+$ to $d\mathbf{A}$, and see that the elements $d_2$ and $d_3$ are algebraically independent over $k$. Since $d_1=d$ is the identity element of $d\mathbf{A}$, so too are $\alpha $ and $\beta' $. Since $\alpha $ and $\beta' $ are not multiples of each other, the ideals
\[\alpha k[\alpha, \beta' ]=(\alpha )\cap k[\alpha, \beta' ]\text{ and }\beta' k[\alpha, \beta' ]=(\beta' )\cap k[\alpha, \beta' ]\]
of $k[\alpha, \beta' ]=\mathbf{A}\cap k[\alpha, \beta' ]$ are incomparable (not included in each other), and hence the same is true about the ideals $(\alpha )$ and $(\beta' )=(\beta )$ of $\mathbf{A}$. By Theorem 5(i), neither of $\mathcal{A}$-differentiability and $\mathcal{B}$-differentiability implies the other.
\end{example}
\begin{example}
Part (iii) of Example \ref{e:2} discussed $f_{\ast }\left( x\right) $, the
first order $\mathcal{A}$-derivative with excess $e=1$ associated with the
difference quotient
\begin{equation*}
\frac{\left( \frac{1}{2}-\tau \right) f\left( x+\left( \tau +1\right)
h\right) +2\tau f\left( x+\tau h\right) -\left( \frac{1}{2}+\tau \right)
f\left( x+\left( \tau -1\right) h\right) }{h},
\end{equation*}
where $\tau =1/\sqrt{3}$. We will now derive the result shown there.
The derivative $f_{\ast }$ is not equivalent to ordinary differentiation. To
see this, either invoke Theorem 1 or directly test $f_{\ast }$ at $x=0$ on
the characteristic function of $\left\{ 0\right\} $. Theorem 2 allows us to
identify all $\mathcal{A}$-derivatives that are equivalent to this one.
Decompose the quotient into even and odd parts as follows.
\begin{eqnarray*}
&&\frac{\left( \frac{1}{2}-\tau \right) x_{\left( \tau +1\right) h}+2\tau
x_{\tau h}-\left( \frac{1}{2}+\tau \right) x_{\left( \tau -1\right) h}}{h}=
\\
&&\frac{\left( \frac{1}{2}-\tau \right) \left( e_{\left( \tau +1\right)
h}+d_{\left( \tau +1\right) h}\right) +2\tau \left( e_{\tau h}+d_{\tau
h}\right) -\left( \frac{1}{2}+\tau \right) \left( e_{\left( \tau -1\right)
h}+d_{\left( \tau -1\right) h}\right) }{h}= \\
&&\frac{\Delta ^{ev}f\left(x, h\right) }{h}+\frac{\Delta ^{odd}f\left(x, h\right)
}{h}
\end{eqnarray*}
where
\begin{equation*}
\Delta ^{ev}f\left(x, h\right) =\left( \frac{1}{2}-\tau \right) e_{\left( \tau
+1\right) h}+2\tau e_{\tau h}-\left( \frac{1}{2}+\tau \right) e_{\left( \tau
-1\right) h}
\end{equation*}
and
\begin{equation*}
\Delta ^{odd}f\left(x, h\right) =\left( \frac{1}{2}-\tau \right) d_{\left( \tau
+1\right) h}+2\tau d_{\tau h}-\left( \frac{1}{2}+\tau \right) d_{\left( \tau
-1\right) h}.
\end{equation*}
Now apply Theorem \ref{2} to find that the most general first order $
\mathcal{A}$-derivative equivalent to this one is associated with
\begin{equation*}
\frac{\Delta ^{odd}f\left(x, sh\right) }{sh}+A\frac{\Delta ^{ev}f\left(
x,rh\right) }{h}
\end{equation*}
where $s$, $r$ and $A$ are nonzero constants.
\end{example}
| 184,878
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\section{Introduction}
\label{sec:intro}
Methods based on global Radial Basis Functions (RBFs) have become quite popular for the numerical solution of the partial differential equations (PDEs) due to their ability to handle scattered node layouts, their simplicity of implementation and their spectral accuracy and convergence on smooth problems. While these methods have been successfully applied to the solution of PDEs on planar regions~\cite{Fasshauer:2007}, they have also been applied to PDEs on the two-sphere $\mathbb{S}^2$ (\emph{e.g.}~\cite{Gia:2005, FlyerWright:2007, FlyerWright:2009}).
Many methods have been developed for the solution of the class of PDEs known as diffusion (or reaction-diffusion) equations on more general surfaces. Of these, the so-called \emph{intrinsic methods} attempt to solve PDEs using surface-based meshes and coordinates intrinsic to the surface under consideration; this approach can be efficient since the dimension of the discretization is restricted to the dimension of the surface under consideration (\emph{e.g.}~\cite{Calhoun2009,Dziuk2007}). However, such intrinsic coordinates can contain singularities or distortions which are difficult to accomodate. A popular alernative is the class of so-called \emph{embedded, narrow-band methods} that extend the PDE to the embedding space, construct differential operators in extrinsic coordinates, and then restrict them to a narrow band around the surface (\emph{e.g.}~\cite{MacDonaldRuuth2009,Piret2012}). Such methods incur the additional expense of solving equations in the dimension of the embedding space; the curse of dimensionality will ensure these costs will grow rapidly depending on the order of accuracy of the method.
RBFs have recently been used to compute an approximation to the surface Laplacian in the context of a pseudospectral method for reaction-diffusion equations on manifolds~\cite{FuselierWright2012}. In that study, global RBF interpolants were used to approximate the surface Laplacian at a set of ``scattered'' nodes on a given surface, combining the advantages of intrinsic methods with those of the embedded methods. This method showed very high rates of convergence on smooth problems on parametrically and implicitly defined manifolds. However, for $N$ points on the surface, the cost of that method scales as $O(N^3)$. Furthermore, the dense nature of the resulting differentiation matrices means that the cost of applying those matrices to solution vectors is $O(N^2)$, assuming the manifold is static. Our goal is to develop a method that is less costly to apply than the global RBF method while still retaining the ability to use scattered nodes on the surface to approximate derivatives, thereby combining the benefits of the intrinsic and narrow-band approaches. Our motivation is to eventually apply this method for the simulation of chemical reactions on evolving surfaces of platelets and red blood cells. For this, we turn to RBF-generated Finite Differences (RBF-FD).
First discussed by Tolstykh~\cite{Tolstykh2003}, RBF-FD formulas are generated from RBF interpolation over \emph{local} sets of nodes on the surface. This type of method is conceptually similar to the standard FD method with the exception that the differentiation weights enforce the exact reproduction of derivatives of shifts of RBFs (rather than derivatives of polynomials as is the case with the
standard FD method) on each local set of nodes being considered. This results in sparse matrices like in the standard FD method, but with the added advantage that the RBF-FD method can naturally handle
irregular geometries and scattered node layouts. We note that the RBF-FD method has proven successful for a number of other applications in planar domains in two and higher dimensions (\emph{e.g.}~\cite{ShuDing2003,CecilQian2004,Wright200699,Chandhini2007,SPLM}). The RBF-FD method has also been shown to be successful on the surface of a sphere~\cite{FoL11,FlyerLehto2012} for convective flows by stabilization with hyperviscosity.
An RBF-FD method for the solution of diffusion and reaction-diffusion equations on general 1D surfaces embedded in 2D domains was recently developed~\cite{ShankarWrightEtAlIJNMF2013}. In our experiments, a straightforward extension of that approach to 2D surfaces proved to be unstable, requiring hyperviscosity-based stabilization as in the case of the RBF-FD method for purely convective flows. In this work, we modify the RBF-FD formulation presented in \cite{ShankarWrightEtAlIJNMF2013}, and present numerical and algorithmic strategies for generating RBF-FD operators on general surfaces. Our approach appears to do away with the need for hyperviscosity-based stabilization.
The remainder of the paper is organized as follows. In Section 2, we briefly review RBF interpolation of both scalar and vector data on scattered node sets in $\mathbb{R}^d$. Section 3 discusses the formulation of surface differential operators in Cartesian coordinates. Section 4 then goes on to describe how these differential operators are discretized in the form of \emph{sparse} differentiation matrices and presents a method-of-lines formulation for the solution of diffusion and reaction-diffusion equations on surfaces; this section also presents important implementation details and comments on the computational complexity of our RBF-FD method. In Section 5, we detail our shape parameter optimization approach and illustrate how it can be used to stabilize the RBF-FD discretization of the surface Laplacian without the need for hyperviscosity-based stabilization. In Section 6, we numerically demonstrate the convergence of our method for different stencil sizes (on two different surfaces) for the forced scalar diffusion equation using two different approaches to selecting the shape parameter $\ep$. Section 7 demonstrates applications of the method to simulations of Turing Patterns on two classes of surfaces: implicit and parametric surfaces, and more general surfaces represented only by point clouds. We conclude our paper with a summary and discussion of future research directions in Section 8.
\textbf{Note}: Throughout this paper, we will use the terms surface or manifold to refer to smooth embedded submanifolds of codimension one in $\mathbb{R}^d$ with no boundary, with the specific case of $d = 3$. Although not pursued here, straightforward extensions are possible for manifolds of higher codimension, or manifolds of codimension $1$ embedded in higher or lower dimensional spaces.
| 191,948
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As many of us gather with family on Dies Natalis Solis Invicti and the year draws to a close, now seems as good a time as any to touch on just a few of the highlights and trends of the past year. There will be plenty that I don’t cover here, but feel free to add in your thoughts:
Unicorns heading for the cull
There’s no end of people elaborating on the end of the ride for Unicorns (or for some, the bursting of a bubble entirely), but one thing seems clear; Unicorns are having challenges going to IPO with their enormous valuations, which when coupled with the end of near-zero interest rates means that valuations will fall and VC funding is likely to diminish. The resulting shortened runways will lead to reductions of workforce, increased pressure on sustainability and potentially many hasty acquisitions. Funding will be more sparse for new or smaller companies. It’s not clear if this will be RIP Good Times, or just a small hump.
Me-Too Messaging
Messaging is becoming an extremely crowded space. It’s likely that the shift to messaging is a combination of 1) people initially looking to avoid text message fees and 2) communities shifting back to the more “natural” state of communicating amongst small groups of close friends, as opposed to status updates to large groups. Indeed, in mobile, messaging has overtaken social. This shift has resulted in many people jumping into the space, hoping to grab a significant user base. Unfortunately, general messaging requires massive numbers of users to be useful, and faces all of the fragmentation issues that Instant Messaging faced before it. To win in this space, entrants will need to be either a) extremely niche (with a focus on community), b) massively adopted (for example, by being part of a large social network like Facebook), or c) provide the infrastructure for others to implement in their services. Everyone else will fade away.
Move to Secure
The introduction of App Transport Security by Apple is just the latest move by the industry towards the use of secure encryption for everything. The days of plain HTTP are numbered — All new protocols such as SPDY, HTTP/2 and QUIC are by default encrypted, providing assurances to end users that nobody has intercepted or manipulated their cat videos. Expect everything to be secure by the end of 2017 if not sooner.
Seamless Experiences
The increasing number of devices that make up our digital footprint (think Laptop, Phone, Watch, Fitness Tracker, Cameras, Thermostats etc etc) require that we move beyond the web, apps and IOT to “Seamless Experiences”. These maintain your state and experience across multiple mediums, screen sizes and capabilities. They pull in data from disparate sources, present it in the context of the device you are using, and correlate data over time to infer context. For examples of this in action over the last year, look to the Apple Watch, Google Cards and Amazon Echo.
Net Neutrality
This was a big one earlier in the year, as US legislators sought to ensure that all sites played on a level field, with no option to give an advantage to any particular one in return for money. In the early days of the web, this was pretty much a given (although large infrastructure vendors absolutely proposed this model in order to sell DiffServ capable routers and switches). Today, as cable companies seek new revenue models, it was clear that this “gentlemen’s agreement” of the early web needed to be established in law in order to preserve a fair competitive environment. The landmark ruling in Feb achieved this, but cable companies are not yet done trying to fight it. Look for more on this next year.
Final Thoughts
As we move into 2016, we are clearly looking at a world that is full of distributed data sources, from a multitude of device types, all-HTTPS, all the time. The infrastructure required to process this data is not insignificant, and we will see individual companies increase their server infrastructure build outs as well as looking to cloud services to help scale this out. Content will be increasingly structured, and presented through more “natural” interfaces such as intelligent assistants or simplified card-style results. We’re just on the precipice of where this will take us, and I’m positive we will need to change the way we build and deploy services as a result. It’s going to be an interesting 2016 for sure.
| 119,045
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Once, the answer is both “yes” and “no”. Sometimes it truly is plug-and-play with models working well across multiple simulators. But more often than not, moving VHDL-AMS models between simulators from different vendors isn’t as easy as you might think. The problem is not in the language itself, but in how the vendor interprets and implements language features.
Standard languages are typically supported by a standards body which, with the help of its members, defines language functionality and documents required features in a Language Reference Manual (LRM). LRMs are weighty documents, usually several hundred pages long, containing everything there is to know about a language. The latest version of the VHDL-AMS LRM (IEEE STD 1076.1-2009) stretches on for 340+ pages. For the average reader, even someone with a background in modeling and simulation, it is not a light read.
To support a language standard, tool vendors must first interpret the LRM as accurately as possible, then implement their interpretation in a simulation program. Problems relating to model portability start with LRM research and generally take one or two forms. First, the tool vendor needs to determine which language features to support. It shouldn’t come as a surprise that developing simulation support for a new language is not an overnight task. Supporting a new language often takes several man years of development work, especially if language support first requires a new simulator. A simulator won’t be viable in the market until it first supports a core set of language features, but opinions about what constitutes a core feature set vary a bit between tool vendors. This leads to language features supported in one simulator that are not supported in another.
Once a tool vendor determines which language features to support, the next model portability stumbling block results from language feature implementation. In other words, once a tool vendor decides which features to support, the next step is determining exactly how to support each feature. Hopefully, the LRM is written well enough to limit implementation errors. But errors do creep in. Tool vendors exacerbate this problem when they are lax in strict adherence to LRM requirements. As an example, some simulators allow non-standard syntax that might cause errors in tools that more strictly interpret the LRM. Once again, this leads to models that work in the parent simulator but possibly not in others.
Language standards are not new to the simulation tool industry. Nor is the ability to move models between tools from different vendors. Users have successfully done so with standards-based digital models for years (think simulation models based on the popular VHDL and Verilog logic modeling languages). Mixed-technology language standards, however, lag logic modeling languages in model portability, but are gaining popularity. As mixed-technology languages like VHDL-AMS grow in popularity, so will their broader acceptance. And as acceptance grows, so will multi-vendor simulation support until seamless model portability is a reality.
VHDL-AMS Stress Modeling – Part 1 « Mike Jensen's Blog
8:11 PM Jan 7, 2013
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\begin{document}
\title[Sums of CR functions from competing CR structures]
{Sums of CR functions from competing CR structures}
\author[David E. Barrett]{David E. Barrett}
\address{Department of Mathematics\\University of Michigan
\\Ann Arbor, MI 48109-1043 USA }
\email{barrett@umich.edu}
\author[Dusty Grundmeier]{Dusty Grundmeier}
\address{Department of Mathematics \\
Harvard University \\
Cambridge, MA 02138-2901 USA }
\email{deg@math.harvard.edu}
\thanks{{\em 2010 Mathematics Subject Classification:} 32V10}
\thanks{The first author was supported in part by NSF grants number DMS-1161735 and DMS-1500142.}
\date{\today}
\begin{abstract}
In this paper we characterize sums of CR functions from competing CR structures in two scenarios.
In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions.
\end{abstract}
\maketitle
\section{Introduction}\label{S:Intro}
The Dirichlet problem for pluriharmonic functions is a natural problem in several complex variables with a long history going back at least to Amoroso \cite{Amo}, Severi \cite{Sev}, Wirtinger \cite{Wir}, and others. It was known early on that the problem is not solvable for general boundary data, so we may try to characterize the admissible boundary values with a system of tangential partial differential operators. This was first done for the ball by Bedford in \cite{Bed1}; see \S \ref{SS:ball} for details. More precisely, given a bounded domain $\Omega$ with smooth boundary $S$, we seek a system $\mathcal{L}$ of partial differential operators tangential to $S$ such that a function $u\in \mathcal{C}^{\infty}(S,\C)$ satisfies $\mathcal{L}u=0$ if and only if there exists $U\in \mathcal{C}^{\infty}(\overline{\Omega})$ such that $U|_S = u$ and $\partial \bar{\partial} U =0$. The problem may also be considered locally.
While natural in its own right, this problem also arises in less direct fashion in many areas of
complex analysis and geometry. For instance, this problem plays a fundamental role in Graham's work on Bergman Laplacian \cite{Gra}, Lee's work on pseudo-Einstein structures \cite{Lee}, and Case, Chanillo, and Yang's work on CR Paneitz operators (see \cite{CCY} and the references therein).
From another point of view, the existence of non-trivial restrictions on pluriharmonic boundary values points to the need to look elsewhere (such as to the Monge-Amp\`ere equations studied in \cite{BeTa}) for Dirichlet problems solvable for general boundary data.
The pluriharmonic boundary value problem is closely related to the problem of characterizing sums of CR functions from different, competing CR structures; indeed, when the competing CR structures are conjugate then these problems coincide (in simply-connected settings); see Theorems \ref{T:glo-prob} and \ref{T:loc-prob} below. Another natural construction leading to competing CR structures arises from the study of projective duality (see \S \ref{S:proj} or \cite{Bar} for precise definitions)
In each of these two scenarios we precisely characterize sums of CR functions from the two competing CR structures in the setting of two-dimensional circular domains satisfying appropriate convexity conditions.
For conjugate structures we assume strong pseudoconvexity; our result appears as Theorem \ref{T:cir-plh} below. In the projective duality scenario we assume strong $\C$-convexity (see \S \ref{S:pdh}) and the main result appears as Theorem \ref{T:cir-proj}.
Our techniques for these two related problems are interconnected to a surprising extent, and the reader will notice that the projective dual scenario actually turns out to have more structure and symmetry.
This paper is organized as follows. In \S \ref{S:conj} we focus on the case of conjugate CR structures (the pluriharmonic case). In \S \ref{S:proj} we study the competing CR structures coming from projective duality. In \S \ref{S:pfs} we prove the main result in the projective setting. In \S \ref{S:pfs2} we prove the main result in the case of conjugate structures. Finally in \S \ref{S:furth} we consider uniqueness questions and further connections with projective duality results in \cite{Bar}.
\section{Conjugate structures}\label{S:conj}
\subsection{Results on the ball} \label{SS:ball}
Early work focused on the case of the ball $B^n$ in $\mathbb{C}^n$. In particular, Nirenberg observed that there is no second-order system of differential operators tangent to $S^3$ that exactly characterize pluriharmonic functions (see \S \ref{S:Nir} for more details). Third-order characterizations were developed by Bedford in the global case and Audibert in the local case (which requires stronger conditions).
To state these results we define the tangential operators
\begin{align}
L_{kl} = z_k \vf{\bar{z}_l}-z_l \vf{\bar{z}_k} && \bar{L_{kl}} = \bar{z}_k \vf{z_l}-\bar{z}_l \vf{z_k}
\end{align}
for $1\leq k,l \leq n$.
\begin{Thm}[{[Bed1]}]\label{T:Bglobal}
Let $u$ be smooth on $S^{2n-1}$, then $$\bar{L_{kl}} \bar{L_{kl}} L_{kl} u=0$$ for $1\leq k,l \leq n$ if and only if $u$ extends to a pluriharmonic function on $B^n$. \end{Thm}
\begin{Thm}[{[Aud]}]\label{T:Alocal} Let $S$ be a relatively open subset of $S^{2n-1}$.
and let $u$ be smooth on $S$. Then $$L_{jk} L_{lm} \bar{L_{rs}} u=0=\bar{L_{jk}} \bar{L_{lm}} L_{rs} u$$ for $1\leq j,k,l,m,r,s \leq n$ if and only if $u$ extends to a pluriharmonic function on a one-sided neighborhood of $S$. \end{Thm}
For a treatment of both of these results along with further details and examples, see \S 18.3 of \cite{Rud}.
\subsection{Other results}
Laville \cite{Lav1, Lav2} also gave a fourth order operator to solve the global problem. In \cite{BeFe} Bedford and Federbush solved the local problem in the more general setting where $b\Omega$ has non-zero Levi form at some point. Later in \cite{Bed2}, Bedford used the induced boundary complex $(\partial \bar{\partial})_b$ to solve the local problem in certain settings. In Lee's work \cite{Lee} on pseudo-Einstein structures, he gives a characterization for abstract CR manifolds using third order pseudohermitian covariant derivatives. Case, Chanillo, and Yang study when the kernel of the CR Paneitz operator characterizes CR-pluriharmonic functions (see \cite{CCY} and the references therein).
\subsection{Relation to decomposition on the boundary}\label{SS:conj-bg}
\begin{Remark}
Outside of the proof of Theorem \ref{T:pair} below, all forms, functions, and submanifolds will be assumed $\mathcal{C}^\infty$-smooth.
\end{Remark}
\begin{Thm} \label{T:glo-prob} Let $S\subset\C^n$ be a compact connected and simply-connected real hypersurface and let $\Omega$ be the bounded domain with boundary $S$. Then for $u\colon S\to \C$
the following conditions are equivalent.
\refstepcounter{equation}\label{N:plh-CR-glob}
\begin{enum}
\item $u$ extends to a (smooth) function $U$ on $\bar\Omega$ that is pluriharmonic on $\Omega$.
\label{I:gloext}
\item $u$ is the sum of a CR function and a conjugate-CR-function. \label{I:glodecomp}
\end{enum}
\end{Thm}
\begin{proof}
In the proof that \itemref{N:plh-CR-glob}{I:gloext} implies \itemref{N:plh-CR-glob}{I:glodecomp}, the CR term is the restriction to $S$ of an anti-derivative for $\dee U$ on a simply-connected one-sided neighborhood of $S$, and the conjugate-CR term is the restriction to $S$ of an anti-derivative for $\deebar U$ on a one-sided neighborhood of $S$ (adjusting one term by a constant as needed).
To see that \itemref{N:plh-CR-glob}{I:glodecomp} implies \itemref{N:plh-CR-glob}{I:gloext} we use the global CR extension result \cite[Thm.\,2.3.2]{Hor} to extend the terms to holomorphic and conjugate-holomorphic functions, respectively; $U$ is then the sum of the extensions.
\end{proof}
\begin{Thm} \label{T:loc-prob}
Let $S\subset\C^n$ be a simply-connected strongly pseudoconvex real hypersurface.
Then for $u\colon S\to \C$ the following conditions are equivalent.
\refstepcounter{equation}\label{N:plh-CR-loc}
\begin{enum}
\item \label{I:locext} there is an open subset $W$ of $\C^n$ with $S\subset\bndry W$ (with $W$ lying locally on the pseudoconvex side of $S$) so that $u$ extends to a (smooth) function $U$ on $W\cup S$ that is pluriharmonic on $W$.
\item \label{I:locdecomp} $u$ is the sum of a CR function and a conjugate-CR-function.
\end{enum}
\end{Thm}
\begin{proof} The proof follows the proof of Theorem \ref{T:glo-prob} above, replacing the global CR extension result by the Hans Lewy local CR extension result as stated in \cite[Sec.\,14.1, Thm.\,1]{Bog}.
\end{proof}
\subsection{Circular hypersurfaces in $\C^2$}
\begin{Thm}\label{T:cir-plh}
Let $S\subset\C^2$ be a strongly pseudoconvex circular hypersurface. Then there exist
nowhere-vanishing tangential vector fields $X,Y$ on $S$ satisfying the following conditions.
\refstepcounter{equation}\label{N:XY-cond}
\begin{enum}
\item If $u$ is a function on a relatively open subset of $S$ then $u$ is CR if and only if $Xu=0$.
\label{I:aaa}
\item If $u$ is a function on a relatively open subset of $S$ then $u$ is CR if and only if
$Y\bar u=0$.\label{I:bbb}
\item If $S$ is compact then a function $u$ on $S$ is a pluriharmonic boundary value (in the sense of Theorem \ref{T:glo-prob}) if and only if $XXYu=0$. \label{I:cir-plh-glo}
\item A function $u$ on a relatively open subset of $S$ is a pluriharmonic boundary value (in the sense of Theorem \ref{T:loc-prob}) if and only if $XXYu=0=\bar{XXY}u$. \label{I:cir-plh-loc}
\end{enum}
\end{Thm}
It is not possible in general to have $Y=\bar X$.
Theorem \ref{T:cir-plh} is proved in \S\ref{S:pfs} below.
\section{Projective dual structures}\label{S:proj}
\subsection{Projective dual hypersurfaces} \label{S:pdh}
Let $S\subset\C^n$ be an oriented real hypersurface with defining function $\rho$. $S$ is said to be {\em strongly $\C$-convex} if $S$ locally equivalent via a projective transformation (that is, via an automorphism of projective space) to a strongly convex hypersurface; this condition is equivalent to either of the following two equivalent conditions:
\refstepcounter{equation}\label{N:strCconv-cond}
\begin{enum}
\item the second fundamental form for $S$ is positive definite on the maximal complex subspace $H_zS$ of each $T_zS$;
\medskip
\item the complex tangent (affine) hyperplanes for $S$ lie to one side (the ``concave side") of $S$ near the point of tangency with minimal order of contact.
\end{enum}
\begin{Prop}\label{P:gl-Cconv}
When $S$ is compact and strongly $\C$-convex the complex tangent hyperplanes for $S$ are in fact disjoint from the domain bounded by $S$.
\end{Prop}
\begin{proof}\,
\cite[\S2.5]{APS}.
\end{proof}
We note that strongly $\C$-convex hypersurfaces are also strongly pseudoconvex.
A circular hypersurface (that is, a hypersurface invariant under rotations $z\mapsto e^{i\theta}z$)
is strongly $\C$-convex if and only if it is strongly convex \cite[Prop.\,3.7]{Cer}.
The proper general context for the notion of strong $\C$-convexity is in the study of real hypersurfaces in complex projective space $\CP^n$ (see for example \cite{Bar} and \cite{APS}).
We specialize now to the two-dimensional case.
\begin{Lem}\label{L:w-def} Let $S\subset\C^2$ be a compact strongly $\C$-convex hypersurface enclosing the origin.
Then there is a uniquely-determined map
\begin{align*}
\mapdef{\dual\colon S}{\C^2\setminus\{0\}}{z}{w(z)=(w_1(z),w_2(z))}
\end{align*}
satisfying
\refstepcounter{equation}\label{N:w-def}
\begin{enum}
\item $z_1w_1+z_2w_2=1$ on $S$; \label{I:key-rel}
\item the vector field
\begin{equation*}\label{E:Ydef}
Y\eqdef w_2\vf{z_1}-w_1\vf{z_2}
\end{equation*}
is tangent
to $S$.
Moreover, $Y$ annihilates conjugate-CR functions on any
relatively open subset of $S$. \label{I:tang(1,0)}
\end{enum}
\end{Lem}
\begin{proof} It is easy to check that \itemref{N:w-def}{I:key-rel} and \itemref{N:w-def}{I:tang(1,0)} force
\begin{align*}
w_1(z)&=\frac{\frac{\dee\rho}{\dee z_1}}{z_1\frac{\dee\rho}{\dee z_1}+z_2\frac{\dee\rho}{\dee z_2}}\\
w_2(z)&=\frac{\frac{\dee\rho}{\dee z_2}}{z_1\frac{\dee\rho}{\dee z_1}+z_2\frac{\dee\rho}{\dee z_2}}.
\end{align*}
establishing uniqueness. Existence follows provided that the denominators do not vanish; but the vanishing of the denominators occurs precisely when the complex tangent line for $S$ at $z$ passes through the origin, and Proposition \ref{P:gl-Cconv} above guarantees that this does not occur under the given hypotheses.
\end{proof}
\begin{Remark}
It is clear from the proof that the conclusions of Lemma \ref{L:w-def} also hold under the assumption that $S$ is a (not necessariy compact) hypersurface satisfying
\begin{equation}\label{E:0spec}
\text{no complex tangent line for $S$ passes through the origin.}
\end{equation}
\end{Remark}
\begin{Remark}
Any tangential vector field annihilating conjugate-CR functions will be a scalar multiple of $Y$.
\end{Remark}
\begin{Remark}
The complex line tangent to $S$ at $z$ is given by
\begin{equation}\label{E:Ctan}
\{\zeta\in\C^2\colon w_1(z)\zeta_1+w_2(z)\zeta_2=1\}.
\end{equation}
\end{Remark}
\begin{Remark}\label{R:ann}
The maximal complex subspace $H_zS$ of each $T_zS$ is annihilated by the form $w_1\,dz_1+w_2\,dz_2$.
\end{Remark}
\begin{Prop}\label{P:loc-diff} For $S$ strongly $\C$-convex satisfying \eqref{E:0spec} the map
$\dual$ is a local diffeomorphism onto an immersed strongly $\C$-convex hypersurface $S^*$, with each maximal complex subspace $H_zS$ of $T_zS$ mapped (non-$\C$-linearly) by $\dual_z'$ onto the corresponding maximal complex subspace of $H_{w(z)}S^*$. For $S$ strongly $\C$-convex and compact $S^*$ is an embedded strongly $\C$-convex hypersurface and $\dual$ is a diffeomorphism.
\end{Prop}
\begin{proof}\,
\cite[\S 6]{Bar}, \cite[\S2.5]{APS}.
\end{proof}
For $S$ strongly $\C$-convex satisfying \eqref{E:0spec} we may extend $\dual$ to a smooth map on an open set in $\C^2$; the extended map $\dual^\star$ will be a local diffeomorphism in some neighborhood $U$ of $S$. We may then define vector fields $\vf{w_1}, \vf{w_2}, \vf{\bar{w_1}}, \vf{\bar{w_2}}$ on $U$ by applying $\left(\left(\dualx\right)\inv\right)'$ to the corresponding vector fields on $\dualx(U)$; these newly-defined vector fields will depend on the choice of the extension
$\dualx$.
\begin{Lem}\label{L:Vdef}
The non-vanishing vector field
\begin{equation*}
V\eqdef z_2\vf{w_1}-z_1\vf{w_2}\label{E:V-def}
\end{equation*}
is tangent to $S$ and is independent of the choice of the extension $\dualx$.
\end{Lem}
\begin{proof} From \itemref{N:w-def}{I:key-rel} we have
\begin{align*}
0 &= d(z_1w_1+z_2w_2)\\
&= z_1\,dw_1+z_2\,dw_2 + w_1\,dz_1 + w_2\,dz_2
\end{align*}
on $T_zS$.
From Remark \ref{R:ann} we deduce that the null space in $T_z\C^2$ of $z_1\,dw_1+z_2\,dw_2$ is precisely the maximal complex subspace $H_z S$ of $T_z S$
(and moreover the null space in $\left(T_z\C^2\right)\otimes\C$ of of $z_1\,dw_1+z_2\,dw_2$ is precisely $\left(H_z S\right)\otimes\C$).
If we apply $z_1\,dw_1+z_2\,dw_2$ to $V$ we obtain
\begin{equation*}
z_1\cdot Vw_1+ z_2\cdot Vw_2 = z_1 \cdot z_2 - z_2 \cdot z_1=0
\end{equation*}
showing that $V$ takes values in $\left(H_z S\right)\otimes\C$ and is thus tangential.
If an alternate tangential vector field $\tilde V$ is constructed with the use of an alternate extension $\widetilde{\dualx}$ of $\dual$ then
\begin{align*}
\tilde Vw_j&=\pm z_{3-j}=Vw_j\\
\tilde V\bar w_j &= 0 = V\bar w_j
\end{align*}
along $S$, so $\tilde V = V$ along $S$.
\end{proof}
\begin{Def}
A function $u$ on relatively open subset of $S$ will be called {\em dual-CR} if $\bar Vu=0$.
\end{Def}
\begin{Ex}
If $S$ is the unit sphere in $\C^2$ then $w(z)=\bar z$ and the set of dual-CR functions on $S$ coincides with the set of conjugate-CR functions on $S$.
\end{Ex}
The set of dual-CR functions will only rarely coincide with the set of conjugate-CR functions as we see from the following two related results.
\begin{Thm} If $S$ is a compact strongly $\C$-convex hypersurface in $\C^2$ then the
set of dual-CR functions on $S$ will coincide with the the set of conjugate-CR functions on $S$ if and only if $S$ is a complex-affine image of the unit sphere.
\end{Thm}
\begin{Thm} If $S$ is a strongly $\C$-convex hypersurface in $\C^2$ then the
set of dual-CR functions on $S$ will coincide with the the set of conjugate-CR functions on $S$ if and only if $S$ is locally the image of a relatively open subset of the unit sphere by a projective transformation.
\end{Thm}
For proofs of these results see \cite{Jen}, \cite{DeTr}, and \cite{Bol}.
\begin{Remark}
The constructions of the vector fields $Y$ and $V$ transform naturally under complex-affine mapping of $S$. The construction of the dual-CR structure transforms naturally under projective transformation of $S$. (See for example \cite[\S 6]{Bar}.)
\end{Remark}
\begin{Lem}\label{L:YV-rel}
Relations of the form
\begin{align*}
V&= \chi Y + \sigma \bar Y\\
Y&= \kappa V + \xi \bar V
\end{align*}
hold along $S$ with $\sigma$ and $\xi$ nowhere vanishing.
\end{Lem}
\begin{proof}
This follows from the following facts:
\begin{itemize}
\item $V, \bar V, Y$ and $\bar Y$ all take values in the two-dimensional space $\left(H_z S\right)\otimes\C$;
\item $V$ and $\bar V$ are $\C$-linearly independent, as are $Y$ and $\bar Y$;
\item the non-$\C$-linearity of the map $\dual'_z\colon \left(H_z S\right)\otimes\C\to \left(H_z S*\right)\otimes\C$ (see Proposition \ref{P:loc-diff}).
\end{itemize}
\end{proof}
\begin{Lem}
If $f_1, f_2$ are CR functions and $g_1, g_2$ are dual-CR functions on a connected relatively open subset $W$ of $S$ with $f_1+g_1=f_2+g_2$ then $g_2-g_1=f_1-f_2$ is constant.
\end{Lem}
\begin{proof}
From Lemma \ref{L:YV-rel} we deduce that the directional derivatives of $g_2-g_1=f_1-f_2$ vanish in every direction belong to the maximal complex subspace of $TS$. Applying one Lie bracket we find that in fact all directional derivatives along $S$ of $g_2-g_1=f_1-f_2$ vanish.
\end{proof}
\begin{Cor}\label{C:sc}
If $W$ is a simply-connected relatively open subset of $S$ and $u$ is a function on $W$ that is locally decomposable as the sum of a CR function and a dual-CR function then $u$ is decomposable on all of $W$ as the sum of a CR function and a dual-CR function.
\end{Cor}
\subsection{Circular hypersurfaces in $\C^2$}
\begin{Thm}\label{T:cir-proj}
Let $S\subset\C^2$ be a strongly ($\C$-)convex circular hypersurface. Then there exist
tangential vector fields $X,T$ on $S$ satisfying the following conditions.
\refstepcounter{equation}\label{N:XT-cond}
\begin{enum}
\item There are scalar functions $\phi$ and $\psi$ so that
\begin{align*}
X&= V+\phi \bar{V}\\
T&= Y+\psi \bar{Y}.
\end{align*}
\label{I:aa}
\item If $u$ is a function on a relatively open subset of $S$ then $f$ is CR if and only if $Xu=0$; equivalently, $X$ is a non-vanishing scalar multiple $\alpha \bar Y$ of $\bar Y$.\label{I:bb}
\item If $u$ is a function on a relatively open subset of $S$ then $g$ is dual-CR if and only if $Tu=0$; equivalently, $T$ is a non-vanishing scalar multiple $\beta \bar V$ of $\bar V$.\label{I:cc}
\item If $S$ is compact then a function $u$ on $S$ is the sum of a CR function and a dual-CR function if and only if $XXTu=0$. \label{I:cir-proj-glo}
\item If $S$ is simply-connected (but not necessarily compact) then a function $u$ on $S$ is the sum of a CR function and a dual-CR function if and only if $XXTu=0=TTXu$. \label{I:cir-proj-loc}
\end{enum}
\end{Thm}
As we shall see the vector field $X$ in Theorem \ref{T:cir-proj} will also work as the vector field $X$ in Theorem \ref{T:cir-plh}.
\begin{Ex} (Compare \cite{Aud}.)
The function $\frac{z_1}{w_2}$ satisfies $XXT\frac{z_1}{w_2}=0$ but is not globally defined. Since $TTX\frac{z_1}{w_2}=2\ne 0$ this function is not locally the sum of a CR function and a dual-CR function.
\end{Ex}
Conditions \itemref{N:XT-cond}{I:aa}, \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} uniquely determine $X$ and $T$. See \S \ref{S:uniq} for some discussion of what can happen without condition \itemref{N:XT-cond}{I:aa}.
\section{Proof of Theorem \ref{T:cir-proj}}\label{S:pfs}
To prove Theorem \ref{T:cir-proj} we start by consulting Lemma \ref{L:YV-rel}
and note that \itemref{N:XT-cond}{I:aa}, \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} will hold if we set
\begin{align*}
\alpha&=1/\bar\xi,&
\beta&=1/\bar\sigma,\\
\phi&=\bar\kappa/\bar\xi,&
\psi&=\bar\chi/\bar\sigma;
\end{align*}
it remains to check \itemref{N:XT-cond}{I:cir-proj-glo} and \itemref{N:XT-cond}{I:cir-proj-loc}.
We note for future reference and the reader's convenience that
\begin{align}
Xw_1&=z_2&
Xw_2&=-z_1\notag\\
\bar Yw_1&=\bar\xi z_2&
\bar Yw_2&=-\bar\xi z_1\notag\\
X\bar{w}_1 &= \phi \bar{z}_2 & X \bar{w}_2&=-\phi \bar{z}_1\notag\\
Xz_1&=\bar Yz_1=0&
Xz_2&=\bar Yz_2=0\notag\\
X\bar z_1 &= \alpha \bar w_2&
X\bar z_2 &= -\alpha \bar w_1\notag\\%needed?
Tz_1&=w_2&
Tz_2 &= -w_1\label{E:diff-rules}\\
\bar V z_1 &= \bar\sigma w_2&
\bar V z_2 &= -\bar\sigma w_1\notag\\
T \bar{z}_1 &= \psi \bar{w}_2 & T\bar{z}_2 &= -\psi \bar{w}_1 \notag\\
Tw_1&=\bar Vw_1=0&
Tw_2&=\bar Vw_2=0\notag\\
T\bar w_1 &= \beta z_2&
T\bar w_2 &= -\beta z_1.\notag
\end{align}
\begin{Lem}\label{L:YVbrack}
\begin{align*}
[Y,\bar Y] &= \bar\xi \left(z_1 \vf{ z_1}+ z_2 \vf{ z_2}\right)-\xi \left(\bar z_1 \vf{\bar z_1} + \bar z_2 \vf{\bar z_2}\right)
\\
[V,\bar V] &= \bar\sigma \left(w_1 \vf{ w_1}+ w_2 \vf{ w_2}\right)
-\sigma \left(\bar w_1 \vf{\bar w_1} + \bar w_2 \vf{\bar w_2}\right).
\end{align*}
\end{Lem}
\begin{proof}
The first statement follows from
\begin{equation*}
[Y,\bar Y] = \left(Y\bar w_2\right)\vf{\bar z_1}-\left(Y\bar w_1\right)\vf{\bar z_2}
-\left(\bar Y w_2\right)\vf{ z_1}+\left(\bar Y w_1\right)\vf{ z_2}
\end{equation*}
along with \eqref{E:diff-rules}.
The proof of the second statement is similar.
\end{proof}
We note that the assumption that $S$ is circular has not been used so far in this section. We now bring it into play by introducing the real tangential vector field
\begin{equation*}
R \eqdef i \left( z_1 \vf{z_1}+ z_2 \vf{z_2} -
\bar z_1 \vf{\bar z_1}- \bar z_2 \vf{\bar z_2}\right)
\end{equation*}
generating the rotations of $z\mapsto e^{i\theta}z$ of $S$.
\begin{Lem} The following hold.
\refstepcounter{equation}\label{N:rot-lem1}
\begin{enum}
\item $\bar\xi=\xi$ \label{I:xir}
\item $\bar\sigma=\sigma$ \label{I:sigmar}
\item $\bar\alpha=\alpha$ \label{I:alphar}
\item $\bar\beta=\beta$ \label{I:betar}
\item $R=-i \left( w_1 \vf{w_1}+ w_2 \vf{w_2} -
\bar w_1 \vf{\bar w_1}- \bar w_2 \vf{\bar w_2}\right)$ \label{I:wrot}
\item $[Y,\bar Y]=-i\xi R$ \label{I:YbarY}
\item $[V,\bar V]=i\sigma R$ \label{I:VbarV}
\item $[X,Y]=iR-(Y\alpha)\bar Y$ \label{I:XY*}
\end{enum}
\end{Lem}
\begin{proof}
We start by considering the tangential vector field
\begin{equation*}
[Y,\bar Y]+i\xi R = (\bar \xi - \xi)\left( z_1 \vf{z_1}+ z_2 \vf{z_2}\right);
\end{equation*}
if \itemref{N:rot-lem1}{I:xir} fails then $z_1 \vf{z_1}+ z_2 \vf{z_2}$ is a non-vanishing holomorphic tangential vector field on some non-empty relatively open subset of $S$, contradicting the strong pseudoconvexity of $S$.
To prove \itemref{N:rot-lem1}{I:wrot} we first note from Lemma \ref{L:w-def} that $w\left(e^{i\theta}z\right)=e^{-i\theta} w(z)$; differentiation with respect to $\theta$ yields \itemref{N:rot-lem1}{I:wrot}.
The proof of \itemref{N:rot-lem1}{I:xir} now may be adapted to prove \itemref{N:rot-lem1}{I:sigmar}.
\itemref{N:rot-lem1}{I:alphar} and \itemref{N:rot-lem1}{I:betar} follow immediately.
Using Lemma \ref{L:YVbrack} in combination with \itemref{N:rot-lem1}{I:xir} and \itemref{N:rot-lem1}{I:sigmar} we obtain \itemref{N:rot-lem1}{I:YbarY} and \itemref{N:rot-lem1}{I:VbarV}.
From \itemref{N:XT-cond}{I:bb} and \itemref{N:rot-lem1}{I:YbarY} we obtain \itemref{N:rot-lem1}{I:XY}.
\end{proof}
\begin{Lem}\label{L:XTbrack}
$[X,T]=iR$.
\end{Lem}
\begin{proof}
On the one hand,
\begin{align*}
[X,T] &= [V+\phi\bar V,\beta\bar V]\\
&=\left( (V+\phi\bar V)\beta-\beta(\bar V\phi)
\right)\bar V + i \beta\sigma R\\
&= \left( (V+\phi\bar V)\beta-\beta(\bar V\phi)
\right)\bar V + i R.
\end{align*}
On the other hand,
\begin{align*}
[X,T] &= [\alpha \bar Y,Y + \psi\bar Y]\\
&=\left( \alpha(\bar Y\psi)-(Y+\psi\bar Y)\alpha
\right)\bar Y + i \alpha \xi R\\
&=\left( \alpha(\bar Y\psi)-(Y+\psi\bar Y)\alpha
\right)\bar Y + i R.
\end{align*}
Since $\bar V$ and $\bar Y$ are linearly independent, it follows that $[X,T]= i R$.
\end{proof}
\begin{Lem} The following hold.
\refstepcounter{equation}\label{N:Rbrack}
\begin{enum}
\item $[R,Y]=-2iY$ \label{I:RY}
\item $[R,\bar Y]=2i\bar Y$ \label{I:RbY}
\item $[R,V]=2iV$ \label{I:RV}
\item $[R,\bar V]=-2i\bar V$ \label{I:RbV}
\item $[R,X]=2iX$ \label{I:RX}
\item $[R,\bar X]=-2i\bar X$ \label{I:RbX}
\item $[R,T]=-2iT$ \label{I:RT}
\item $[R,\bar T]=2i\bar T$ \label{I:RbT}
\item $R\alpha=0$ \label{I:Ralpha}
\item $R\beta=0$ \label{I:Rbeta}
\end{enum}
\end{Lem}
\begin{proof}
\itemref{N:Rbrack}{I:RY}, \itemref{N:Rbrack}{I:RbY}, \itemref{N:Rbrack}{I:RV} and \itemref{N:Rbrack}{I:RbV} follow from direct calculation.
For \itemref{N:Rbrack}{I:RT} first note that writing $T=\beta\bar V$ and using \itemref{N:Rbrack}{I:RbV} we see that $[R,T]$ is a scalar multiple of $T$. Then writing
\begin{equation*}
[R,T] = [R,Y+\psi\bar Y]= -2iY + (\text{multiple of }\bar Y)
\end{equation*}
we conclude using \itemref{N:XT-cond}{I:aa} that $[R,T]=-2iT$. The proof of \itemref{N:Rbrack}{I:RX} is similar, and \itemref{N:Rbrack}{I:RbX} and \itemref{N:Rbrack}{I:RbT} follow by conjugation.
Using \itemref{N:XT-cond}{I:bb} along with \itemref{N:Rbrack}{I:RbY} and \itemref{N:Rbrack}{I:RX} we obtain \itemref{N:Rbrack}{I:Ralpha}; \itemref{N:Rbrack}{I:Rbeta} is proved similarly.
\end{proof}
\begin{Lem}\label{L:XXker}
$XXf=0$ if and only if $f=f_1w_1+f_2w_2$ with $f_1, f_2$ CR.
\end{Lem}
\begin{proof}
From \itemref{N:XT-cond}{I:bb} and \eqref{E:diff-rules} it is clear that $XX\left(f_1w_1+f_2w_2\right)=0$ if $f_1$ and $ f_2$ are CR.
For the other direction, suppose that $XXf=0$. Then setting
\begin{align*}
f_1&\eqdef z_1f+w_2Xf\\
f_2&\eqdef z_2f-w_1Xf.
\end{align*}
it is clear that $f=f_1w_1+f_2w_2$; with the use of \itemref{N:XT-cond}{I:bb} and \eqref{E:diff-rules} it is also easy to check that $f_1$ and $f_2$ are CR.
\end{proof}
\begin{Lem}\label{L:proj-1eq}
Suppose that $XXTu=0$ so that by Lemma \ref{L:XXker} we may write $Tu=f_1w_1+f_2w_2$ with $f_1, f_2$ CR. Then
\begin{equation}\label{E:proj-2op}
TTXu = \frac{\dee f_1}{\dee z_1} + \frac{\dee f_2}{\dee z_2}.
\end{equation}
In particular, $TTXu$ is CR.
\end{Lem}
The non-tangential derivatives appearing in \eqref{E:proj-2op} may be interpreted using the Hans Lewy local CR extension result previously mentioned in the proof of Theorem \ref{T:loc-prob}, or else by rewriting them in terms of tangential derivatives (as in the last step of the proof below).
\begin{proof} We have
\begin{align*}
TTXu
&= TXTu + T[T,X]u \\
&= TX\left(f_1w_1+f_2w_2\right)-iTRu && \text{(Lemma \ref{L:XTbrack})}\\
&= T\left(f_1z_2-f_2z_1\right) - iRTu - i[T,R]u && \text{\itemref{N:XT-cond}{I:bb}, \eqref{E:diff-rules} }\\
&= T\left(f_1z_2-f_2z_1\right) - iR\left(f_1w_1+f_2w_2\right) + 2Tu
&& \text{ \itemref{N:Rbrack}{I:RT} }\\
&= \left(Tf_1\right)z_2 - f_1w_1 - \left(Tf_2\right)z_1 - f_2w_2 \\
&\qquad
-i\left(Rf_1\right)w_1 - f_2w_2 - i\left(Rf_2\right)w_2 - f_2 w_2 \\
&\qquad + 2\left(f_1w_1+f_2w_2\right)
&& \text{\eqref{E:diff-rules}, \itemref{N:rot-lem1}{I:wrot} }\\
&= \left(z_2 T - i w_1 R\right) f_2 - \left( z_1 T + iw_2 R\right) f_2\\
&= \left(z_2 Y - i w_1 R\right) f_2 - \left( z_1 Y + iw_2 R\right) f_2\\
&= \frac{\dee f_1}{\dee z_1} + \frac{\dee f_2}{\dee z_2}.
\end{align*}
\end{proof}
\begin{Lem}\label{L:spec-2ord} The following hold.
\refstepcounter{equation}\label{N:spec-2ord}
\begin{enum}
\item The operator $XT$ maps CR functions to CR functions. \label{I:XT}
\item The operator $XY$ maps CR functions to CR functions. \label{I:XY}
\item The operator $TX$ maps dual-CR functions to dual-CR functions. \label{I:TX}
\item The operator $\bar{XY}$ maps conjugate-CR functions to conjugate-CR functions. \label{I:bar-XY}
\end{enum}
\end{Lem}
\begin{proof}
To prove \itemref{N:spec-2ord}{I:XT} and \itemref{N:spec-2ord}{I:XY} note that for $u$ CR we have $XTu=XYu=-z_1\frac{\dee u}{\dee z_1}-z_2\frac{\dee u}{\dee z_2}$ which is also CR. The other proofs are similar.
\end{proof}
\begin{proof}[Proof of \itemref{N:XT-cond}{I:cir-proj-loc}] To get the required lower bound on the null spaces, it will suffice to show that $XXT$ and $TTX$ annihilate CR functions and dual-CR functions. This follows from \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} along with \itemref{N:spec-2ord}{I:XT} and \itemref{N:spec-2ord}{I:TX}.
For the other direction, if $XXTu=0=TTXu$ then from Lemma \ref{L:proj-1eq} we have a
closed 1-form $\omega\eqdef f_2\,dz_1 - f_1\,dz_2$ on $S$ where $f_1$ and $f_2$ are CR functions satisfying $Tu=f_1w_1+f_2w_2$. Since $S$ is simply-connected we may write $\omega=df$ with $f$ CR. Then from \itemref{N:XT-cond}{I:aa} we have
\begin{align*}
Tf &= Yf\\
&=w_2f_2+w_1f_1\\
&= Tu.
\end{align*}
Thus $u$ is the sum of the CR function $f$ and the dual-CR function $u-f$.
\end{proof}
To set up the proof of the global result \itemref{N:XT-cond}{I:cir-proj-glo} we introduce
the form
\begin{equation}\label{E:nu-def}
\nu\eqdef (z_2\,dz_1-z_1\,dz_2)\w dw_1\w dw_2
\end{equation}
and the $\C$-bilinear pairing
\begin{equation}\label{E:pair-def}
\lma \mu,\eta\rma \eqdef \intl_S \mu\eta\cdot\nu
\end{equation}
between functions on $S$ (but see Technical Remark \ref{R:tech} below).
\begin{Lem}\label{L:parts}
$\lma T\gamma, \eta \rma=-\lma \gamma, T\eta \rma$.
\end{Lem}
\begin{proof}
\allowdisplaybreaks
\begin{align*}
\lma T\gamma, \eta \rma+\lma \gamma, T\eta \rma
&= \intl_S T(\gamma\eta)\cdot\nu\\
&= \intl_S \iota_T d(\gamma\eta)\cdot\nu\\
&= \intl_S d(\gamma\eta)\cdot \iota_T \nu\\
&= \intl_S d(\gamma\eta \cdot \iota_T \nu)- \intl_S \gamma\eta\cdot d(\iota_T \nu)\\
&= 0- \intl_S \gamma\eta\cdot d(\iota_T ((z_2\,dz_1-z_1\,dz_2)\w dw_1\w dw_2)\\
&= - \intl_S \gamma\eta\cdot d((z_2\cdot Tz_1-z_1\cdot Tz_2)\cdot dw_1\w dw_2)\\
&\qquad\qquad +\intl_S \gamma\eta\cdot d( (z_2\,dz_1-z_1\,dz_2)\cdot Tw_1\w dw_2)\\
&\qquad\qquad -\intl_S \gamma\eta\cdot d( (z_2\,dz_1-z_1\,dz_2)\w dw_1\cdot Tw_2)\\
&= - \intl_S \gamma\eta\cdot d((z_2w_2+z_1w_1) \,dw_1\w dw_2)+0-0\\
&= - \intl_S \gamma\eta\cdot d(dw_1\w dw_2)\\
&=0.
\end{align*}
Here we have quoted
\begin{itemize}
\item the definition \eqref{E:pair-def} of the pairing $\lma\cdot\rma$;
\item the Leibniz rule $\iota_T(\varphi_1\w\varphi_2)=(\iota_T\varphi_1)\w\varphi_2+(-1)^{\deg\varphi_1}\varphi_1\w(\iota_T\varphi_2)$ for the interior product $\iota_T$;
\item the fact that $S$ is integral for 4-forms;
\item Stokes' theorem;
\item the rules \eqref{E:diff-rules};
\item the relation \itemref{N:w-def}{I:key-rel}.
\end{itemize}
\end{proof}
\begin{Thm}\label{T:pair}
A function $\mu$ on a compact strongly $\C$-convex hypersurface $S$ is CR if and only if $\lma \mu, \eta \rma =0$ for all (smooth) dual-CR $\eta$ on $S$.
\end{Thm}
\begin{proof}\,
\cite[(4.3d) from Theorem 3]{Bar}. (Note also definition enclosing \cite[(4.2)]{Bar}.)
\end{proof}
\begin{proof}[Proof of \itemref{N:XT-cond}{I:cir-proj-glo}] Assume that
$XXTu=0$. Noting that $S$ is simply-connected, from \itemref{N:XT-cond}{I:cir-proj-loc} it suffices to prove that $TTXu=0$. From Lemma \ref{L:proj-1eq} we know that
$TTXu$ is CR. By Theorem \ref{T:pair} it will suffice to show that
\begin{equation*}
\lma TTXu, \eta \rma=0
\end{equation*}
for dual-CR $\eta$. But from Lemma \ref{L:parts} we have
\begin{align*}
\lma TTXu, \eta \rma&=- \lma TXu, T\eta \rma\\
&=0
\end{align*}
as required.
\end{proof}
\begin{Remark}\label{R:pd} From symmetry of formulas in Lemma \ref{L:w-def} and \ref{L:Vdef} we have that $X_{S^*}=\dual_* T_{S}, T_{S^*}=\dual_* X_{S}$ and $S^{**}=S$. These facts serve to explain why the formulas throughout this section appear in dual pairs.
\end{Remark}
\begin{Tech}\label{R:tech}
In \cite{Bar} the pairing \eqref{E:pair-def} applies not to functions $\mu,\nu$ but rather to forms
$\mu(z)\,(dz_1\w dz_2)^{2/3}$, $\mu(w)\,(dw_1\w dw_2)^{2/3}$; the additional notation is important in \cite{Bar} for keeping track of invariance properties under projective transformation but is not needed here.
Note also that \eqref{E:pair-def} coincides (up to a constant) with the pairing (3.1.8) in \cite{APS} with $s=w_1\,dz_1+w_2\,dz_2$.
\end{Tech}
\section{Proof of Theorem \ref{T:cir-plh}}\label{S:pfs2}
\begin{Lem}\label{L:prlh-1eq}
Suppose that $XXYu=0$ so that by Lemma \ref{L:XXker} we may write $Yu=f_1w_1+f_2w_2$ with $f_1, f_2$ CR. Then
\begin{equation}\label{E:plh-2op}
\bar{XXY}u = \alpha\left(\frac{\dee f_1}{\dee z_1} + \frac{\dee f_2}{\dee z_2}\right).
\end{equation}
In particular, $\alpha\inv\bar{XXY}u$ is CR.
\end{Lem}
\begin{proof}
We have
\begin{align*}
\allowdisplaybreaks
\bar{XXY}u
&= \bar{XYX}u
+ \bar{X} [\bar X,\bar Y] u\\
&=\bar{XY} \left( \alpha \left(f_1w_1+f_2 w_2\right) \right)
+\bar{X}\left( -iR-(\bar Y\alpha)Y\right) u && \text{\itemref{N:XT-cond}{I:bb},\itemref{N:rot-lem1}{I:alphar},\itemref{N:rot-lem1}{I:XY}}\\
&=\bar X\left( \alpha\bar Y \left( f_1w_1+f_2 w_2 \right)\right)
-i\bar X R u \\
&=\bar X\left(f_1z_2-f_2 z_1 \right)
-iR \bar X u
-i[\bar X, R] u && \text{\itemref{N:XT-cond}{I:bb},\eqref{E:diff-rules}}\\
&= \bar X\left(f_1z_2-f_2 z_1 \right)
-iR \left( \alpha \left( f_1w_1+f_2 w_2 \right) \right)
+2 \bar X u && \text{\itemref{N:XT-cond}{I:bb},\itemref{N:Rbrack}{I:RbX}}\\
&= (\bar X f_1)\cdot z_2-f_1\cdot\alpha w_1-(\bar X f_2) \cdot z_1
- f_2 \cdot\alpha w_2\\
&\qquad - i \alpha \left( (Rf_1)\cdot w_1-f_1\cdot(iw_1)+(Rf_2)\cdot w_2-f_2\cdot(iw_2) \right)
\\
&\qquad
+ 2 \alpha \left(f_1w_1+f_2 w_2\right) && \text{\eqref{E:diff-rules},\itemref{N:Rbrack}{I:Ralpha},\itemref{N:rot-lem1}{I:wrot},\itemref{N:XT-cond}{I:bb}}\\
&= (\bar X f_1)\cdot z_2-(\bar X f_2) \cdot z_1
- i \alpha \left( (Rf_1)\cdot w_1+(Rf_2)\cdot w_2 \right)
\\
&= \alpha\left(z_2Y-iw_1R)f_1-(z_1Y+iw_2R)f_2\right)\\
&= \alpha \left( \frac{\dee f_1}{\dee z_1}+ \frac{\dee f_2}{\dee z_2}\right).\\
\end{align*}
\end{proof}
\begin{proof}[Proof of \itemref{N:XY-cond}{I:cir-plh-loc}]
To get the required lower bound on the null spaces, it will suffice to show that $XXY$ and $\bar{XXY}$ annihilate CR functions and conjugate-CR functions. This follows from \itemref{N:XY-cond}{I:aaa} along with \itemref{N:spec-2ord}{I:XY} and \itemref{N:spec-2ord}{I:bar-XY}.
For the other direction, if $XXYu=0=\bar{XXY}u$ then from Lemma \ref{L:proj-1eq} we have a
closed 1-form $\tilde\omega\eqdef f_2\,dz_1 - f_1\,dz_2$ on the open subset of $S$ where $f_1$ and $f_2$ are CR functions satisfying $Yu=f_1w_1+f_2w_2$. Restricting our attention to a simply-connected subset, we may write $\omega=d f$ with $ f$ CR. Then we have
\begin{align*}
Y f
&=w_2f_2+w_1f_1\\
&= Yu.
\end{align*}
Thus $u$ is the sum of the CR function $f$ and the conjugate-CR function $u-f$.
The general case follows by localization.
\end{proof}
\begin{Lem}\label{L:Ydiv}
$\operatorname{div} Y\eqdef \frac{\dee w_2}{\dee z_1} - \frac{\dee w_1}{\dee z_2}$ and $\operatorname{div} \bar Y\eqdef {\frac{\dee \bar w_2}{\dee \bar z_1}} - \bar{\frac{\dee \bar w_1}{\dee \bar z_2}}$ vanish on $S$.
\end{Lem}
\begin{proof}
Since $S$ is circular, any defining function $\rho$ for $S$ will satisfy
$\Im \left( z_1\frac{\dee\rho}{\dee z_1}+z_2\frac{\dee\rho}{\dee z_2} \right)
=-\frac{R\rho}{2} = 0$. Adjusting our choice of defining function we may arrange that
$z_1\frac{\dee\rho}{\dee z_1}+z_2\frac{\dee\rho}{\dee z_2}\equiv 1$ in some neighborhood of
$S$. Then from the proof of Lemma \ref{L:w-def} we have $\frac{\dee w_2}{\dee z_1} - \frac{\dee w_1}{\dee z_2}=\frac{\dee^2 \rho}{\dee z_1 \dee z_2}-
\frac{\dee^2 \rho}{\dee z_2 \dee z_1}=0$.
The remaining statement follows by conjugation.
\end{proof}
\begin{Lem}\label{L:parts-plh}
$\displaystyle{\intl_S \left(X\gamma\right)\eta\,\frac{dS}{\alpha} = - \intl_S \gamma\left(X\eta\right)\,\frac{dS}{\alpha}}$
\end{Lem}
\begin{proof}
\begin{align*}
\intl_S \left(X\gamma\right)\eta\,\frac{dS}{\alpha}
&= \intl_S \left(\bar Y\gamma\right)\eta\,dS && \text{\itemref{N:XT-cond}{I:bb}}\\
&= -\intl_S \gamma\left(\bar Y\eta\right)\,dS && \text{(Lemma \ref{L:Ydiv})}\\
&= - \intl_S \gamma\left(X\eta\right)\,\frac{dS}{\alpha} && \text{\itemref{N:XT-cond}{I:bb}}
\end{align*}
(The integration by parts above may be justified by applying the divergence theorem on a tubular neighborhood of $S$ and passing to a limit.)
\end{proof}
\begin{proof}[Proof of \itemref{N:XY-cond}{I:cir-plh-glo}]
Assume that
$XXYu=0$. Noting that $S$ is simply-connected, from \itemref{N:XY-cond}{I:cir-plh-loc} it suffices to prove that $\bar{XXY}u=0$. From Lemma \ref{L:proj-1eq} we know that
$\alpha\inv\bar{XXY}u$ is CR. The desired conclusion now follows from
\begin{align*}
\intl_S \left|\bar{XXY}u\right|^2\,\frac{dS}{\alpha^2}
&=\intl_S \alpha\inv \bar{XXY}u\cdot {XXY}\bar u\,\frac{dS}{\alpha}\\
&= - \intl_S X\left( \alpha\inv\bar{XXY}u \right) \cdot {XY}\bar u\,\frac{dS}{\alpha}
&&\text{(Lemma \ref{L:parts-plh})}\\
&= - \intl_S 0 \cdot {XY}\bar u\,\frac{dS}{\alpha}\\
&=0.
\end{align*}
\end{proof}
\section{Further comments}\label{S:furth}
\subsection{Remarks on uniqueness}\label{S:uniq}
\begin{Thm}\label{T:plh-un}
Suppose that in the setting of Theorem \ref{T:cir-proj} we have vector fields $\tilde X,\tilde T$ satisfying (suitably-modified) \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} with the property that $\tilde X\tilde X\tilde T$ annihilates CR functions and dual-CR functions. Then there are CR functions $f_1, f_2$ and $f_3$ so that $f_1w_1+f_2w_2$ and $f_3$ are non-vanishing and
\begin{align*}
\tilde X&= f_3(f_1w_1+f_2w_2)^2 X\\
\tilde T&= \frac{1}{f_1w_1+f_2w_2} T.
\end{align*}
\end{Thm}
\begin{proof}
From \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} we have $\tilde X=\gamma X$, $\tilde T=\eta T$ with non-vanishing scalar functions $\gamma$ and $\eta$.
By routine computation we have
\begin{equation*}
\tilde X\tilde X\tilde T
= \gamma^2\eta XXT + \gamma\Big( \left(2\gamma(X\eta)+\eta(X\gamma)\right) XT
+ (X(\gamma(X\eta))T\Big).
\end{equation*}
The operator $\left(2\gamma(X\eta)+\eta(X\gamma)\right) XT
+ (X(\gamma(X\eta))T$ must in particular annihilate CR functions.
But if $f$ is CR then using Lemma \ref{L:XTbrack} we have
\begin{equation*}
\Big( \left(2\gamma(X\eta)+\eta(X\gamma)\right) XT
+ (X(\gamma(X\eta))T\Big)f = \Big( i\left(2\gamma(X\eta)+\eta(X\gamma)\right) R
+ (X(\gamma(X\eta))T\Big)f
\end{equation*}
Since $R$ and $T$ are $\C$-linearly independent and $f$ is arbitrary it follows that we must have
\begin{align*}
X(\gamma\eta^2)=2\gamma(X\eta)+\eta(X\gamma) &= 0\\
X(\gamma(X\eta) &= 0.
\end{align*}
We set $f_3=\gamma\eta^2$ which is CR and non-vanishing. Then the second equation above
yields
\begin{align*}
-f_3 \cdot XX(\eta\inv)
&= X\left(f_3 \,\eta^{-2}(X\eta)\right)\\
&=X(\gamma(X\eta)\\
&=0
\end{align*}
and hence $XX(\eta\inv)=0$. From Lemma \ref{L:XXker} we have
$\eta=\frac{1}{f_1w_1+f_2w_2}$ with $f_1$ and $f_2$ CR. The result now follows.
\end{proof}
\begin{Thm}
Suppose that in the setting of Theorem \ref{T:cir-plh} we have vector fields $\tilde X,\tilde T$ satisfying (suitably-modified) \itemref{N:XY-cond}{I:aaa} and \itemref{N:XY-cond}{I:bbb} with the property that $\tilde X\tilde X\tilde Y$ annihilates CR functions and conjugate-CR functions. Then there are CR functions $f_1, f_2$ and $f_3$ so that $f_1w_1+f_2w_2$ and $f_3$ are non-vanishing and
\begin{align*}
\tilde X&= f_3(f_1w_1+f_2w_2)^2 X\\
\tilde Y&= \frac{1}{f_1w_1+f_2w_2} Y.
\end{align*}
\end{Thm}
The proof is similar to that of Theorem \ref{T:plh-un}, using \itemref{N:rot-lem1}{I:XY*} in place of Lemma \ref{L:XXker}.
\subsection{Nirenberg-type result}\label{S:Nir}
\begin{Prop}\label{P:Nir-plh}
Given a point $p$ on a strongly pseudoconvex hypersurface $S\subset\C^2$, any 2-jet at $p$ of a $\C$-valued function on $S$ is the 2-jet of the restriction to $S$ of a pluriharmonic function on $\C^2$.
\end{Prop}
\begin{proof}
After performing a standard local biholomorphic change of coordinates we may reduce to the case where $p=0$ and $S$ is described near 0 by an equation of the form
\[y_2=z_1\bar z_1+O(\|(z_1,x_2)\|)^3.\] The projection $(z_1,x_2+iy_2)\mapsto (z_1,x_2)$ induces a bijection between 2-jets at 0 along $S$ and 2-jets at 0 along $\C\times\R$. It suffices now to note that the 2-jet
\begin{equation*}
A+Bz_1+C\bar{z}_1+Dx_2+E z_1^2+ F \bar{z}_1^2 + G z_1\bar{z}_1 + H z_1 x_2 + I \bar{z}_1 x_2+Jx_2^2
\end{equation*}
is induced by the pluriharmonic polynomial
\begin{equation*}
A+Bz_1+C\bar{z}_1+\frac{D-iG}{2} z_2 + \frac{D+iG}{2} \bar{z}_2+ E z_1^2+ F \bar{z}_1^2 +
Hz_1z_2 + I \bar{z}_1\bar{z}_2 + J\bar{z}_2^2.
\end{equation*}
\end{proof}
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\begin{document}
\title{Downlink Cellular Network Analysis with Multi-slope Path Loss Models}
\author{Xinchen~Zhang and Jeffrey G. Andrews\vspace{-0.8cm}
\thanks{Manuscript date \today.
Xinchen Zhang ({\tt xzhang7@alumni.nd.edu}) is now with Qualcomm Inc., San Diego, CA, USA. This work is completed when he was with the University of Texas at Austin, Austin, Texas, USA.
Jeffrey G. Andrews ({\tt jandrews@ece.utexas.edu}) is the corresponding author of this paper and is with the Wireless Networking and Communications Group (WNCG), the Univeristy of Texas at Austin, Austin, Texas, USA.
This work was supported by the Simons Foundation and the National Science Foundation CIF-1016649.}}
\markboth{}{Updated \today}
\maketitle
\begin{abstract}
Existing cellular network analyses, and even simulations, typically use the standard path loss model where received power decays like $\|x\|^{-\alpha}$ over a distance $\|x\|$. This standard path loss model is quite idealized, and in most scenarios the path loss exponent $\alpha$ is itself a function of $\|x\|$, typically an increasing one. Enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values. In this paper we study \emph{multi-slope} path loss models, where different distance ranges are subject to different path loss exponents. We focus on the dual-slope path loss function, which is a piece-wise power law and continuous and accurately approximates many practical scenarios. We derive the distributions of SIR, SNR, and finally SINR before finding the potential throughput scaling, which provides insight on the observed cell-splitting rate gain. The exact mathematical results show that the SIR monotonically decreases with network density, while the converse is true for SNR, and thus the network coverage probability in terms of SINR is maximized at some finite density. With ultra-densification (network density goes to infinity), there exists a \emph{phase transition} in the near-field path loss exponent $\alpha_0$: if $\alpha_0 >1$ unbounded potential throughput can be achieved asymptotically; if $\alpha_0 <1$, ultra-densification leads in the extreme case to zero throughput.
\end{abstract}
\section{Introduction\label{sec:Intro}}
A fundamental property of wireless signal transmissions is that their power rapidly decays over distance. In particular, in free space we know from the Friis equation that over a distance $\| x \|$ the received signal power $P_r$ is given in terms of the transmit power $P_t$ as
\begin{equation}
P_r = P_t G_t G_r \left( \frac{\lambda_c}{4 \pi \| x \|} \right)^2,
\end{equation}
for a wavelength $\lambda_c$ and antenna transmit and receive gains $G_t$ and $G_r$. In terrestrial environments, propagation is much more complex to characterize due to ground reflections, scattering, blocking/shadowing, and other physical features. Since the number of possible realizations of propagation environments is infinite, simplified models stemming from the Friis equation are typically adopted that have at least some measure of empirical support. A nearly universal characteristic of such models is that the distance dependence is generalized to $\| x \|^{-\alpha}$, with the path loss exponent $\alpha$ being a parameter that can be roughly fit to the environment. In decibel terms, this gives a form
\begin{equation}
\label{eq:PLintro}
P_r = P_t + K_0 - 10 \alpha \log_{10} \| x \|,
\end{equation}
where $K_0$ is a catch-all constant that gives the path loss (in dB) at a distance $\| x \| = 1$. For example, in the Friis equation, $K_0 = G_t G_r (\lambda_c/ 4\pi)^2$. Hence the slope of the path loss (in dB) is constant and is determined only by $\alpha$ in such a model, which we will term a \emph{standard path loss model}.
\subsection{The Case for Multi-Slope Path Loss Models}
Although the standard path loss model has a great deal of history, and is the basis for most existing cellular network theory, analysis, simulation and design,
it is also known to lead to unrealistic results in some special cases \cite{net:Inaltekin09jsac,net:Dousse04}.
Besides, the standard path loss model does not accurately capture the dependence of the path loss exponent $\alpha$ on the link distance in many important situations. We now enumerate a few of these as examples, along with the possible consequences to future cellular network optimization and design.
\textbf{The two-ray model.} Even a very simple two-ray model with one direct path and one ground-reflected path results in a pronounced dual slope path loss behavior \cite{GoldsmithBook,SchillingMilsteinPickholtzBrunoKanterakisKullbackErcegBiedermanFishmanSalerno1991,FeuersteinBlackardRappaportEtAl1994}. In particular, with transmit and receive antenna heights of $h_t$ and $h_r$, below a critical distance $R_c \approx 4 h_t h_r / \lambda_c$ the path loss exponent is $\alpha = 2$, while above this distance it changes to $\alpha = 4$. For a plausible values $h_t = 10$~m, $h_r = 2$~m, and a carrier frequency $f_c = c/\lambda_c = 1$ GHz, we have $R_c=267$~m as a rough threshold. It is worth emphasizing that there is a massive difference between $\| x \|^{-2}$ and $\| x \|^{-4}$ for most reasonable values of $\| x \|$, and that splitting the difference by using $\alpha \approx 3$ results in large errors in both regimes.
\textbf{Dense or clustered networks.} Wireless networks are rapidly increasing in density, and in doing so are becoming ever more irregular \cite{And13}. This causes increasing variations in the link distances and the number of appreciable interferers, and makes a ``one size fits all'' path loss model ever more dubious. For example, a cellular user equipment (UE) might connect with its closest macrocell (or microcell) that is beyond the critical distance, while experiencing interference from nearby closed access femtocells that are within the critical distance but cannot be connected to \cite{XiaCha10}. In such a case the SINR would be greatly over-estimated with a standard path loss model, and the gain from interference avoidance or cancellation techniques greatly underestimated. Or if a nearby picocell was connected to, while interference mostly originated from more distant BSs, the SINR would be greatly underestimated. In general, standard path loss models may not paint an accurate picture of what happens as networks densify, which is a key theme that we explore in this paper.
\textbf{Millimeter wave cellular networks.}
The intriguing possibility of using millimeter waves -- $\lambda \in (1,10)$ mm, \ie carrier frequencies of $30$ to $300$ GHz -- for cellular communication makes the revisitation of propagation models particularly urgent \cite{PiKha11, Rap2013ItWillWork}. A key feature of millimeter wave systems is their sensitivity to blocking \cite{RanRapErk14, BaiHeath2015}. One recently proposed model with considerable empirical support is to use one path loss exponent $\alpha_{0} \approx 2$ for line of sight (LOS) links and another $\alpha_{1} \approx 3.5$ for non line of sight (NLOS) links \cite{GhoshJSAC}. Statistically, LOS links are shorter than NLOS links, and a critical distance $R_c$ can be used to approximate the two regimes \cite{SinAnd14b,BaiHeath2015}. Here, $R_c$ is an environmentally dependent random variable, but it could be approximated by the mean LOS distance. For example, in urban parts of New York City and Chicago, this method will lead to $R_c \approx 70$~m \cite{SinAnd14b}, whereas in environments with less blocking $R_c$ would increase. Although such approximations require considerable further investigation, generalizing to at least a dual-slope model appears essential for millimeter wave cellular systems.
The above examples make clear that a dual (or more) slope path loss model is highly desirable for analysis. And indeed, such a model is very close to many scenarios in the WINNER II path loss model adopted for 3GPP-LTE standardization \cite{LTERelease9,KyostiMeinilaHentilaEtAl2007} and is well-supported by many measurements see, \cite{ErcegGhassemzadehTaylorEtAl1992,FeuersteinBlackardRappaportEtAl1994a,HamptonMerhebLainEtAl2006} and the references therein.
A more detailed comparison between the dual (or more) slope path loss models
and the models in the standardization activities can be found in \cite{LiuXiaoSoong2014}.
\subsection{Contributions}
The overall contribution of this paper is to analyze the coverage probability (SINR distribution) and potential throughput of a downlink cellular network under multi-slope path loss models, with a focus on the dual-slope model. This can largely be viewed as a generalization of \cite{net:Andrews12tcom} which used the standard path loss model and derived fairly simple closed form solutions particularly for the case of $\alpha = 4$. One notable observation from \cite{net:Andrews12tcom} was that the coverage probability (SINR) can only increase with BS density, and does not depend on the BS density once it is sufficiently large (thus rendering noise negligible compared to interference). We term this property \emph{SINR invariance}, and will see that it does not even approximately hold with a dual-slope model.
Although the results for a multi-slope path loss model are predictably more complicated than with the standard path loss model, in various limiting cases several crisp statements can be made. Below, we summarize the main contributions of this paper:
\begin{itemize}
\item
We derive numerically tractable integral-form expressions as well as tight closed-form estimates for the coverage probability (\SINR distribution) in cellular networks with multi-slope path loss functions and Poisson distributed BSs.
\item
Focusing on the dual-slope case, we prove that network SIR decreases with increasing network density. Since SNR always increases with network densification, the spectral efficiency is a non-monotonic function of network density, rendering an (finite) optimal density that maximizes the coverage probability, \ie the probability of meeting a particular $\SINR$ target. These results stand in sharp contrast to the SINR invariance observed under the standard path loss model \cite{net:Andrews12tcom, net:Dhillon12jsac}.
\item
However, the network potential throughput still (asymptotically) \emph{linearly} scales with network density $\lambda$ if the near-field path loss exponent $\alpha_0$ is larger than 2. The scaling rate becomes $\Omega(\lambda^{2-\frac{2}{\alpha_0}})$, \ie sublinear, for $1<\alpha_0<2$.
\item
On the other hand, with ultra-densification, \ie the network density $\lambda$ goes to infinity, the potential throughput always scales unboundedly if $\alpha_0>1$, despite the fact that the coverage probability goes to zero in the limit when $\alpha_0<2$. A \emph{phase-transition} happens at $\alpha_0 = 1$: if $\alpha_0 < 1$, ultra-densification ($\lambda \to 0$) always leads ultimately to zero throughput.
\item
The above results are shown to generalize to multi-slope path loss functions, with an arbitrary number of increasing path loss exponents.
\end{itemize}
\subsection{Paper Organization}
In the rest of the paper, we will first introduce the path loss and network models in Section~\ref{sec:sys}. General coverage probability expressions under the dual-slope path loss model are derived in Section~\ref{sec:GenCovProbE}. Section~\ref{sec:SIR} specializes in the interference-limited case, derives key differential properties of the network performance and contrasts them with those found under the standard path loss function. The case with noise is analyzed in Section~\ref{sec:SINRnTPScaling}. Section~\ref{sec:Mslope} generalizes the conclusions drawn in the dual-slope case to the multi-slope case. Concluding remarks are given in Section~\ref{sec:conclu}.
\section{System Model\label{sec:sys}}
\subsection{Network and Path Loss Models}
Consider a typical downlink UE located at the origin $o$.
We assume the BS distribution is governed by a marked Poisson point process (PPP) $\hat\Phi = \{(x_i, h_{x_i})\}\subset \R^2\times \R^+$,
where the ground process $\Phi = \{x_i\}\subset \R^2$ is a homogeneous PPP
with intensity $\lambda$ and $h_{x_i}$ is the (power) fading gain from the BS at $x_i$ to the typical user $o$.\footnote{
The PPP-based cellular network model is well accepted for cellular network analysis, see, \eg \cite{SinAnd14b,Zhang14twc,net:Dhillon12jsac,Bar13Lattice}, and is supported both empirically \cite{net:Andrews12tcom} and theoretically \cite{Bar13Lattice,KeelerRossXia2014}.}
For simplicity but without loss of generality, we assume that all BSs transmit with unit power
and let $l:\R^+\to\R^+$ denote the path loss function.
Then, at the origin $o$, the received power from the BS $x$ is $h_x l(\|x\|)$,
where $\|x-y\|$ is the Euclidean distance between $x$ and $y$.
While some of the results in this paper hold irrespective of the fading distribution,
we will focus on Rayleigh fading, \ie $h_x$ are iid exponentially distributed with unit mean.
With slight abuse of notation, we may write $l(\|x\|)$ as $l(x)$ for simplicity.
In the following, we (formally) define the few path loss functions of interest.
\begin{definition}[Standard Path Loss Function]
The standard (power-law) path loss function is
\begin{equation}
l_\textnormal{1}(\alpha;x) = \|x\|^{-\alpha}.
\label{equ:stdplfunc}
\end{equation}
\end{definition}
This simple version is a suitable simplification of \eqref{eq:PLintro} since $K_0$ is assumed to be the same for all links and can simply be folded into the noise power. Thus, we can write \eqref{equ:stdplfunc} in dB as $- 10 \alpha \log_{10} \| x \|$ and the standard path loss function can be also interpreted as the \emph{single-slope} model
with slope $-\alpha$. The subscript $1$ indicates this single-slope property.
Many efforts have been made to identify the ``right'' path loss exponent $\alpha$.
It is empirically observed that $\alpha$ is generally best approximated as a constant between $2$ to $5$ which depends on the carrier frequency as well as the physical environment (indoor/outdoor). However, as we noted at the outset, this model has severe limitations, and several motivating examples lead us to consider a dual-slope path loss function.
\begin{definition}[Dual-slope Path Loss Function]
The dual-slope (power-law) path loss function \cite{SarkarJiKimMedouriSalazar-Palma2003} is
\begin{equation}
l_2(\alpha_0,\alpha_1;x) = \left\{
\begin{array}{lr}
\|x\|^{-\alpha_0}, & \|x\|\leq R_c\\
\eta \|x\|^{-\alpha_1}, & \|x\|> R_c,
\end{array}
\right.
\label{equ:biexpPL}
\end{equation}
where $\eta \triangleq R_c^{\alpha_1-\alpha_0}$, $R_c>0$ is the \emph{critical distance}, and
$\alpha_0$ and $\alpha_1$ are the \emph{near-} and \emph{far-field path loss exponents} with $0\leq \alpha_0\leq\alpha_1$.
\end{definition}
Clearly, the dual-slope path loss model has two slopes in a dB scale,
which we stress with the subscript $2$. The constant $\eta$ is introduced to maintain continuity and complies with the definitions in \cite{SarkarJiKimMedouriSalazar-Palma2003,GoldsmithBook,SchillingMilsteinPickholtzBrunoKanterakisKullbackErcegBiedermanFishmanSalerno1991}.
\begin{remark}
The dual-slope path loss function as defined above is a more general version of the standard path loss function
with the following three important special cases.
\begin{itemize}
\item
The standard path loss function can be retrieved (from the dual-slope path loss function) by setting $\alpha_0=\alpha_1 = \alpha$ in \eqref{equ:biexpPL}.
\item
Letting $R_c \to \infty$, we have $l_2(\alpha_0,\alpha_1;x) = \|x\|^{-\alpha_0}= l_1(\alpha_0;x) $.
Analogously, when $R_c\to 0$, $l_2(\alpha_0,\alpha_1;x) = \eta \|x\|^{-\alpha_1} = \eta l_1(\alpha_1;x) $.
These two special limiting instances of the dual-slope path loss
function will become important in our later coverage analyses.
\item
If $\alpha_0 = 0,~ \alpha_1>2$,
the dual-slope path loss model can be rewritten as
\begin{equation}
l_2(0,\alpha_1;x) = \min\{1, \eta \|x\|^{-\alpha_1}\},
\label{equ:bdedPL}
\end{equation}
where $\eta = R_c^{\alpha_1}$.
Here, \eqref{equ:bdedPL} can be interpreted as the \emph{bounded} single-slope path loss function,
\ie a fixed path loss up to $R_c$ and a single path loss exponent afterwards.
Indeed, this path loss function is also often used in the literature, see \eg \cite{HeathKountourisBai2013,net:Haenggi11cl,net:Dousse04,FranceschettiDousseTseEtAl2007},
and with experimental support seen in \cite{net:Haenggi05commag}.
\end{itemize}
\end{remark}
More generally, one can consider a finite number of path loss exponents (and critical distances) and
obtain a continuous, multi-slope path loss function, which we now define.
\begin{definition}[Multi-slope ($N$-slope) Path Loss Model]
For $N\in\mathbb{N}^+$, the $N$-slope path loss model
\begin{equation}
l_N(\{\alpha_i\}_{i=1}^{N-1};x) = K_n \|x\|^{-\alpha_n}
\end{equation}
for $\|x\|\in [R_{n}, R_{n+1}), n\in[N-1]\cup\{0\}$,\footnote{We use $[n]$ to denote the set $\{1,2,\cdots,n\}$.}
where $K_0 = 1$ and $K_n = \prod_{i=1}^n R_i^{\alpha_i-\alpha_{i-1}},~\forall n\in [N-1]$, $0=R_0<R_1<\cdots<R_N=\infty$,
$0\leq \alpha_0\leq\alpha_1\leq\cdots\leq\alpha_{N-1}$, $\alpha_{N-1}>2$.
\label{def:Multi-slopePLM}
\end{definition}
Def.~\ref{def:Multi-slopePLM} is consistent with the piece-wise linear model in \cite[Sect. 2.5.4]{GoldsmithBook}.
Clearly, when $N=2$, the multi-slope path loss function becomes dual-slope,
and $N=1$ gives the standard path loss model.
More importantly, the $N$-slope path loss function provides a means
to study more general path loss functions which decay faster
than power law functions.
For notational simplicity, the path loss exponents parameterizing
the path loss functions may be omitted when they are obvious from the context,
\ie we may write $l_1(\alpha; \cdot)$, $l_2(\alpha_0,\alpha_1; \cdot)$, $l_N(\{\alpha_i\}_{i=1}^{N-1};\cdot)$
as $l_1(\cdot)$, $l_2(\cdot)$, $l_N(\cdot)$, respectively.
\subsection{SINR-based Coverage\label{subsect:SINRCov}}
The main metric of this paper is the coverage probability of the typical user at $o$,\footnote{
By the stationarity of $\Phi$, the result will not change if an \emph{arbitrary} location (independent of $\Phi$) is chosen instead of $o$.}
defined as the probability that the received SIR or SINR at the user is larger than
a target $ T$.
When the user is always associated with the nearest BS,
\ie one with the least path loss and highest average received power,
the SINR can be written as
\begin{equation*}
\SINR_l = \frac{h_{x^*}l(x^*)}{\sum_{y\in\Phi\setminus\{x^*\}} h_y l(y) + \sigma^2},
\end{equation*}
where the subscript $l$ in SINR is to emphasize that the SINR is defined
under any path loss function $l$, $x_{*}\triangleq \argmax_{x} l(x)$ and $ \sigma^2$ can be considered as the receiver-side noise
power normalized by the transmit power and other propagation constants, \eg loss at $\|x\|=1$.
Then, the SINR coverage probability can be formalized as
\begin{equation}
\Pc^\SINR_{l} (\lambda, T) \triangleq \P(\SINR_l> T).
\label{equ:PcSINRdef}
\end{equation}
where the parameters $(\lambda, T)$ may be omitted if they are obvious in the context.
It is clear from \eqref{equ:PcSINRdef} that
$\Pc^\SINR_{l} (\lambda, \cdot)$ is the ccdf of the SINR at the typical user.
In addition to the coverage probability, we further define the coverage density and the potential throughput as
our primary metrics for the area spectral efficiency under network scaling.
\begin{definition}[Coverage Density and Potential (Single-rate) Throughput]
\label{def:PT}
The \emph{coverage density} of a cellular network under path loss function $l(\cdot)$ is
\begin{equation*}
\mu_{l}(\lambda, T) \triangleq \lambda \Pc_l^\SINR (\lambda, T)
\end{equation*}
where $\lambda$ is the network (infrastructure) density and $T$ is the $\SINR$ target. It has units of BSs/Area.
The \emph{potential throughput} is
\begin{equation*}
\tau_l(\lambda, T) \triangleq \log_2(1+T) \mu_{l}(\lambda, T),
\end{equation*}
which has units of bps/Hz/m$^2$, the same as area spectral efficiency.
\end{definition}
Whereas the coverage probability can be used to capture the spectral efficiency distribution (since they have a 1:1 relation),
the potential throughput gives an indication of the (maximum) cell splitting gain, which would occur if all BSs remain fully loaded as the network densifies.
To see this, consider the case of \emph{simultaneous densification},
\ie the densities of network infrastructure (BSs) and users scale at the same rate.
Assuming the user process is stationary and independent of $\Phi$,
it is not difficult to observe that
under simultaneous densification, the scaling of the area spectral efficiency (ASE), defined as the number
of bits received per unit time, frequency and area, is
\emph{the same} as the scaling of the potential throughput\footnote{This argument could be made rigorous by introducing further assumptions on the scheduling procedure and traffic statistics but is beyond the content of this paper.},
which by the Def.~\ref{def:PT} also equals that of the coverage density.
For example, using the standard path loss function,
\cite{net:Andrews12tcom} shows that the network density does not change the SINR distribution
in the interference-limited case, which leads to the potential throughput growing
linearly with the network density with simultaneous densification and implies a linear scaling for the cellular network area spectral efficiency.
\section{The General Coverage Probability Expressions\label{sec:GenCovProbE}}
The stochastic geometry framework provides a tractable way to characterize the coverage probability for cellular networks. Generally speaking, an integral form of the coverage probability can be derived under arbitrary fading regardless of the path loss function \cite{net:Sousa90,net:Haenggi09jsac}. However, Rayleigh fading (having an exponential power pdf) is nearly always used due to its outstanding tractability, and as seen in \cite{net:Andrews12tcom}, it yields similar results to other fading/shadowing distributions (as long as they have the same mean) due to the spatial averaging inherent to stochastic geometry. Admittedly, the standard path loss function does give some extra tractability which cannot be duplicated with more general path loss functions.
In this section, we give an explicit expression for the coverage probability
for general path loss function and demonstrate that it can be simplified in terms
of Gauss hypergeometric functions under the dual-slope power law path loss function.
\begin{lemma}
The coverage probability under a nearest BS association
and general path loss function $l(\cdot)$ is $\Pc^\SINR_l(\lambda, T) =$
\begin{multline}
\lambda\pi \int_0^\infty \exp\left({-\lambda\pi y \Big(1+\int_1^\infty
\frac{ T}{ T+\frac{l(\sqrt{y})}{l(\sqrt{t y})}}\d t\Big)}\right) \\
\times e^{- T \sigma^2/l(\sqrt{y})} \d y.
\label{equ:NBSACvrgP}
\end{multline}
\label{lem:CovPGenPL}
\end{lemma}
The proof of Lemma~\ref{lem:CovPGenPL} is analogous to that of \cite[Theorem 1]{net:Andrews12tcom}.
It is a result of the probability generating functional (PGFL) and the nearest neighbor distribution of the PPP,
and changes of variables.
We omit the proof for brevity.
Note that the tractability exposed in \cite{net:Andrews12tcom} hinges on the fact that
$\frac{l(\sqrt{y})}{l(\sqrt{ty})}$ is independent of $y$ under the standard path loss function
which does not apply for general path loss functions.
However, \eqref{equ:NBSACvrgP} does allow numerical computation for the coverage
probability for general path loss functions.
For the dual-slope path loss function,
\eqref{equ:NBSACvrgP} can be further simplified as in the following theorem.
\begin{theorem}
The coverage probability under the dual-slope path loss function
is
\begin{multline}
\Pc^\SINR_{l_2}(\lambda, T) =
\lambda\pi R_c^2 \int_0^1 e^{-\lambda\pi R_c^2 I(\delta_0,\delta_1, T; x)
- T \sigma^2 x^{\frac{\alpha_0}{2}} R_c^{\alpha_0}} \d x \\
+ \lambda\pi R_c^2 \int_1^{\infty} e^{ -\lambda\pi R_c^2 x C_{-\delta_1}( T)
- T \sigma^2 x^{\frac{\alpha_1}{2}} R_c^{\alpha_0}} \d x ,
\label{equ:2expCP}
\end{multline}
where $I(\delta_0,\delta_1, T; x) =$
\[ C_{\delta_0}\left(\frac{1}{ T x^{\frac{1}{\delta_0}}}\right)
+C_{-\delta_1}({ T x^{\frac{1}{\delta_0}}})
+ x \left(1-C_{\delta_0}\left(\frac{1}{ T}\right)\right) -1,
\]
$C_\beta(x) = {_2F}_1(1,\beta;1+\beta;-x)$, where
${_2F}_1 (a,b;c;z)$ is the \emph{Gauss hypergeometric function}, $\delta_0=2/\alpha_0$,\footnote{If $\alpha_0=0$, we interpret $\delta_0=\infty$.} $\delta_1=2/\alpha_1$.\label{thm:2exp}
\end{theorem}
\begin{IEEEproof}
The proof follows directly from Lemma~\ref{lem:CovPGenPL} and changes of variables.
\end{IEEEproof}
The first term in \eqref{equ:2expCP} represents the coverage probability when the distance to the serving BS is less than the critical distance $R_c$, and the second term is the coverage probability when it is farther than $R_c$. The intervals of integral $(0,1)$ and $(1,\infty)$ result from a change of variables.
In most reasonably dense (e.g. urban) existing cellular networks, interference dominates the noise power,
making the signal-to-interference ratio (SIR), $\SIR_l \triangleq \SINR_l |_{ \sigma^2=0}$, an accurate approximation to SINR.
Such an approximation has been adopted in many cellular network analyses, see \eg \cite{Zhang14twc}.
If we define the SIR coverage probability
$\Pc^\SIR_{l} (\lambda, T) \triangleq \P(\SIR_l> T)$ as the probability that the received SIR
at the typical user is above the threshold $ T$,
Theorem~\ref{thm:2exp} yields the following important observation.
\begin{fact}[Near-field-BS Invariance]
For two dual-slope path loss function $l_2(\cdot)$ and $l'_2(\cdot)$ with
the same path loss exponents but different critical distances $R_c$ and $R'_c$,
the effect of density and the critical distance
on the SIR coverage probability is \emph{equivalent}
in the sense that $\Pc_{l_2}^\SIR(\lambda, T) = \Pc_{l_2'}^\SIR(\lambda', T)$
as long as $\lambda R_c^2 = \lambda' (R'_c)^2$, \ie
the mean numbers of the near-field BSs are the same.
\label{fact:lambdaR2}
\end{fact}
\begin{remark}[Loss of SIR-invariance]
Under the standard path loss model,
the $\SIR$ coverage probability is independent of the network density \cite{net:Andrews12tcom}, i.e. \SIR-invariance holds.
Fact~\ref{fact:lambdaR2} looks similar but is much weaker than the \SIR-invariance property.
Under the dual-slope path loss function, $\Pc_{l_2}^\SIR(\lambda,R_c)$ is held constant only if
$R_c$ scales with $1/\sqrt{\lambda}$ as the network densifies.
Since empirically $R_c\propto f_\textnormal{c}$,
the ambition of maintaining the same spectral efficiency with higher network density is equivalent to asking for more bandwidth at the lower end of the spectrum, an unrealistic request.
\end{remark}
\begin{remark}[Requirements for Finite Interference]
Unlike for the standard path loss function, where $\alpha>2$ is typically required to
guarantee (almost surely) bounded interference,
Theorem~\ref{thm:2exp} and \eqref{equ:2expCP} only requires $\alpha_1>2$.
Intuitively, the interfering region under $\alpha_0$ is always finite and thus does not contribute
infinite interference (at finite network density) and $\alpha_1>2$ guarantees the interference from beyond the critical distance is
bounded.
\end{remark}
\begin{remark}[Simplifying Special Cases]
For some particular choices of $\alpha_0$ and $\alpha_1$,
the need for hypergeometric functions in \eqref{equ:2expCP}
can be eliminated.
In particular, we have
$C_1(x) = \frac{\log(1+x)}{x}$, $C_{-\frac{1}{2}}(x)=1+\sqrt{x}\arctan \sqrt{x}$, $C_\frac{1}{2}(x)=\frac{\arctan\sqrt{x}}{\sqrt{x}}$, $C_2(x) = \frac{2(x-\log(1+x))}{x^2}$ and $C_\infty(x) = \frac{1}{1+x}$.
Consequently, many important special cases can be expressed without special functions
including $[\alpha_0\; \alpha_1] = [2\; 4],[1\; 4],[0\; 4]$.
\end{remark}
Among these several special cases, the most interesting one is probably $[\alpha_0\; \alpha_1] = [2\; 4]$
which coincides with the well-known two-ray model. We thus conclude this section with a corollary highlighting this case.
\begin{corollary}
The \SINR coverage probability under a dual-slope path loss function with $\alpha_0 = 2$, $\alpha_1 = 4$ is
\begin{multline}
\Pc^\SINR_{l_2}(\lambda, T) =
\lambda\pi R_c^2 \int_0^1 e^{-\lambda\pi R_c^2 I(\delta_0,\delta_1, T; x)
- T \sigma^2 x R_c^2} \d x \\
+ \lambda\pi R_c^2 \int_1^{\infty} e^{ -\lambda\pi R_c^2 x (1+\sqrt{T}\arctan\sqrt{T})
- T \sigma^2 x^{2} R_c^2} \d x ,
\end{multline}
where
\begin{multline*}
I(\delta_0,\delta_1, T; x) =xT\log\left(1+\frac{1}{xT}\right) \\
+ \sqrt{xT}\arctan\sqrt{xT} + x\left(1-T\log\big(1+\frac{1}{T}\big)\right).
\end{multline*}
\end{corollary}
\section{The Interference-limited Case\label{sec:SIR}}
Theorem~\ref{thm:2exp} gives an exact expression of the coverage probability in the cellular
network modeled by a PPP.
In this section, we refine our understanding about the dual-slope path loss function by
comparing against the standard path loss function, and highlighting the differences.
\begin{lemma}
For an arbitrary marked point pattern (including fading) $\hat\Phi (\omega) \subset \R^2\times \R^+$
associated with sample $\omega\in\Omega$
and any $ T>0$,
\begin{itemize}
\item
$\SIR_{l_1(\alpha_0;\cdot)} (\omega) > T$ implies $\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) \geq T$,
\item
$\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) > T$ implies $\SIR_{l_1(\alpha_1;\cdot)} (\omega) \geq T$,
\end{itemize}
where $\SIR_l(\omega)$ is the $\SIR$ at the typical user under path loss function $l(\cdot)$ for a sample $\omega$.
\label{lem:SIRbounds}
\end{lemma}
\begin{IEEEproof}
See Appendix~\ref{app:SIRbounds}.
\end{IEEEproof}
\begin{remark}[Generality of SIR Bounds]
Lemma~\ref{lem:SIRbounds} is stated for an arbitrary realization of the network topology
and fading, and does not depend on any statistical assumptions.
It is purely based on the nature of the path loss functions in question.
\end{remark}
An immediate consequence of Lemma~\ref{lem:SIRbounds} is the SIR coverage ordering
of cellular networks with general fading and BS location statistics.
\begin{theorem}\label{thm:PcSIRordering}
For random wireless networks modeled by arbitrary point process and fading, under the nearest BS
association policy, the following $\SIR$ coverage probability ordering holds for arbitrary $0\leq\alpha_0\leq \alpha_1$:
\begin{equation*}
\Pc^{\SIR}_{l_1(\alpha_1;\cdot)}(\cdot, T)\geq \Pc^{\SIR}_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T) \geq \Pc^{\SIR}_{l_1(\alpha_0;\cdot)}(\cdot, T).
\end{equation*}
\end{theorem}
The proof of Theorem~\ref{thm:PcSIRordering} follows directly from Lemma~\ref{lem:SIRbounds}
and is omitted from the paper.
In addition to being an important characterization of the dual-slope
path loss function, Theorem~\ref{thm:PcSIRordering} leads to the following interesting
fact that has been observed in many special cases.
\begin{corollary}
For random wireless networks modeled by arbitrary point process and fading, under the nearest BS
association policy, the $\SIR$ coverage probability
is a monotonically increasing function of the path loss exponent for the standard path loss function.
\label{cor:SICcovSinglePLEmonotonicity}
\end{corollary}
Cor.~\ref{cor:SICcovSinglePLEmonotonicity} follows directly from the observation
that $\Pc^{\SIR}_{l_1(\alpha_1;\cdot)} (\cdot, T) \geq \Pc^{\SIR}_{l_1(\alpha_0;\cdot)} (\cdot, T)$ is true for all $0\leq\alpha_0\leq \alpha_1$
(including the case of $\alpha_1\leq 2$).
Although given Rayleigh fading and the PPP model,
the fact that SIR coverage monotonically increases
with $\alpha$ is known before, \eg easily inferred from the expressions in \cite{net:Andrews12tcom}.
Theorem~\ref{thm:PcSIRordering} shows that such monotonicity is a nature of the (standard) path loss function
and is independent of any network and fading statistics.
Furthermore, the theorem includes the case $\alpha\leq 2$, which was often excluded in conventional analyses.
Since the coverage probability under the standard path loss function is well-known for $\alpha>2$ \cite{net:Andrews12tcom}, Theorem~\ref{thm:PcSIRordering} leads to computable bounds on the SIR coverage probability with the dual-slope path loss function.
A natural question follows: what if the dual-slope model is applied but with $\alpha_0\leq 2$?
Although $\alpha\leq 2$ is not particularly interesting under the standard path loss function since it is both intractable and not empirically supported, a
small ($\leq 2$) near-field path loss exponent is relevant under the dual-slope model since
both early reports in the traditional cellular frequency bands \cite{SchillingMilsteinPickholtzBrunoKanterakisKullbackErcegBiedermanFishmanSalerno1991}
and recent measurements at the millimeter wave bands \cite{AzarRap2013ICC28Goutdoor} suggest that small near-field path loss exponents are definitely plausible. Intuitively, a small $\alpha_0 < 2$ simply means that for short distances, the path loss effects are fairly negligible versus for example the positive impact of reflections or directionality. The following proposition highlights an interesting feature of this small $\alpha_0$ case.
\begin{proposition}
Under the dual-slope pathloss model, when $\alpha_0\leq 2$,
the $\SIR$ and $\SINR$ coverage probabilities $\Pc^\SIR_{l_2}$ and $\Pc^\SINR_{l_2}$
go to zero as $\lambda\to\infty$.
\label{prop:densezerocoverage}
\end{proposition}
\begin{IEEEproof}
See Appendix~\ref{app:densezerocoverage}.
\end{IEEEproof}
The most important implication from Prop.~\ref{prop:densezerocoverage} is that
ultra-densification could eventually lead to near-universal outage if $\alpha_0\leq 2$.
It is worth stressing that this asymptotically zero coverage probability happens if and only if $\alpha_0\leq 2$ and for any $\alpha_0>2$,
(still) $\Pc^\SINR_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T)>0$ for all $ T,\lambda>0$.
Combining Prop.~\ref{prop:densezerocoverage} with Theorem~\ref{thm:PcSIRordering} leads to the following corollary.
\begin{corollary}
Under the standard path loss model, the typical user has an SINR and SIR coverage probability of zero almost surely
if the path loss exponent is no larger than $2$.
\label{cor:0CovP}
\end{corollary}
The following lemma strengthens Theorem~\ref{thm:PcSIRordering} by showing that
the upper and lower bounds on $\Pc^{\SIR}_{l_2(\alpha_0,\alpha_1;\cdot)}$
are achievable by varying the network density.
\begin{lemma}
The following is true for any $ T\geq 0$:
\begin{itemize}
\item
$\lim_{\lambda\to\infty} \Pc^\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\lambda, T) = \lim_{\lambda\to\infty} \Pc^\SINR_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T) = \Pc^\SIR_{l_1(\alpha_0;\cdot)} (\cdot, T)$.
\item
$\lim_{\lambda\to 0} \Pc^\SIR_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T) = \Pc^\SIR_{l_1(\alpha_1;\cdot)} (\cdot, T)$
\end{itemize}
\label{lem:BoundAchi}
\end{lemma}
\begin{IEEEproof}
First, we realize that both $\Pc^\SIR_{l_1(\alpha_0;\cdot)}(\lambda, T)$ and $\Pc^\SIR_{l_1(\alpha_1;\cdot)}(\lambda, T)$
are independent from $\lambda$.
This fact is most well known for the case $\alpha_0,\alpha_1 > 2$, see, \eg \cite{net:Andrews12tcom}, but Cor.~\ref{cor:0CovP}
confirms that it is true for all $\alpha_0,\alpha_1>0$.
By Theorem~\ref{thm:PcSIRordering}, we have $\Pc^{\SIR}_{l_1(\alpha_1;\cdot)}(\cdot, T)\geq \Pc^{\SIR}_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T) \geq \Pc^{\SIR}_{l_1(\alpha_0;\cdot)}(\cdot, T)$ for all $\lambda>0$.
To show the convergence, we make use of Fact~\ref{fact:lambdaR2}.
Instead of letting $\lambda\to\infty$ (and $\lambda\to 0$ resp.), we consider equivalently
$R_c\to\infty$ ($R_c\to 0$ resp.).
But by the definition of the dual-slope path loss function, such
scaling results in $l_1(\alpha_0;\cdot)$ ( $\eta l_1(\alpha_1;\cdot)$ resp.).
The lemma is completed by observing that $ \Pc^\SIR_{l_2} \to \Pc^\SINR_{l_2}$ as $\lambda\to\infty$.
\end{IEEEproof}
Theorem~\ref{thm:PcSIRordering} and Lemma~\ref{lem:BoundAchi} point to the perhaps counter-intuitive
conclusion that $\SIR$ coverage probability decays with network densification.
This is formalized in the following lemma.
\begin{lemma}[SIR monotonicity]
Under the dual-slope path loss function and arbitrary fading distribution, $ \Pc^\SIR_{l_2} (\lambda_1, T) \geq \Pc^\SIR_{l_2}(\lambda_2, T)$, for all $\lambda_1\leq\lambda_2$, $ T \geq 0$ and $0\leq\alpha_0\leq\alpha_1$.
\label{lem:SIRmono}
\end{lemma}
\begin{IEEEproof}
See Appendix~\ref{app:SIRmono}.
\end{IEEEproof}
Fig.~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_3_alpha1_4} plots the SIR coverage probability as a function of $\lambda$
for $ T = -10,-5,0,5,10$ dB (top to bottom).
Consistent with Lemma~\ref{lem:SIRmono}, we see SIR coverage decreases with increasing density.
The convergence of $\Pc^\SIR_{l_2} (\lambda, T)$ as $\lambda\to\infty$
and $\lambda\to 0$ is also verified in the figure.
In Fig.~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_3_alpha1_4}, we use $\alpha_0 = 3>2$ and thus positive
coverage probability is expected as $\lambda\to\infty$.
In contrast, Fig.~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_2_alpha1_4}
demonstrates the coverage probability scaling predicted by Prop.~\ref{prop:densezerocoverage} and Lemma~\ref{lem:SIRmono},
\ie $\Pc^\SIR_{l_2} (\lambda, T)$
keeps decreasing (to zero) regardless of $ T$ as $\lambda$ increases.
The sharp visual difference between Figs.~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_3_alpha1_4}~and~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_2_alpha1_4}
highlights the \emph{phase transition} on $\alpha_0$ with 2 being the critical exponent.
\begin{figure}[th]
\centering
\psfrag{lambda}[c][c]{$\lambda$}
\psfrag{Pc}[c][t]{$\Pc^\SIR_{l_2(\alpha_0,\alpha_1;\cdot)}$}
\begin{overpic}[width=\linewidth]{SIRcvrgProbxlambda-10-10_alpha0_3_alpha1_4.eps}
\put(25,75){\vector(0,-1){60}}
\put(27,11){$ T=-10,-5,0,5,10$ dB}
\end{overpic}
\caption{SIR coverage scaling with network density when $\alpha_0 = 3$, $\alpha_1 = 4$, $R_c=1$. \label{fig:SIRcvrgProbxlambda-10-10_alpha0_3_alpha1_4}}
\end{figure}
\begin{figure}[tbh]
\centering
\psfrag{lambda}[c][c]{$\lambda$}
\psfrag{Pc}[c][t]{$\Pc^\SIR_{l_2(\alpha_0,\alpha_1;\cdot)}$}
\begin{overpic}[width=\linewidth]{SIRcvrgProbxlambda-10-10_alpha0_2_alpha1_4.eps}
\put(65,65){\vector(-1,-1){50}}
\put(52,68){$ T=-10,-5,0,5,10$ dB}
\end{overpic}
\caption{SIR coverage scaling with network density when $\alpha_0 = 2$, $\alpha_1 = 4$, $R_c=1$. \label{fig:SIRcvrgProbxlambda-10-10_alpha0_2_alpha1_4}}
\end{figure}
Intuitively, one can understand this result by considering the best-case scenario for SIR which would occur at a low density, where the UE is connected to the nearest BS which is in the near-field and all the interfering BSs are located in the far-field and thus more rapidly attenuate. Increasing the density in such a case could only reduce SIR, since interfering BSs would soon be added in the near-field. This density regime is where we observe the the transition from higher to lower SIR in
Figs.~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_3_alpha1_4}~and~\ref{fig:SIRcvrgProbxlambda-10-10_alpha0_2_alpha1_4}. Asymptotically, an infinite number of BSs will be present in the near-field, and we are back to SIR-invariance as observed for the standard path loss model since we have only a single relevant path loss exponent, $\alpha_0$.
A hasty conclusion from this discussion is that to optimize the SIR coverage probability, one
can simply let the density of the network go to zero. While this statement is true, it is not of much practical relevance
since as the network density goes to zero the received signal power goes to zero as well,
and the network is no longer interference-limited. Thus, unlike the standard path loss case, where the interference-limited
assumption is often justifiable in the coverage analysis,
the dual-slope path loss function increases the importance of including noise.
\section{SINR Coverage and Throughput Scaling\label{sec:SINRnTPScaling}}
\subsection{The Tension between SIR and SNR\label{sec:SIRnSNR}}
As shown in Sect.~\ref{sec:SIR}, BS densification generally reduces
the SIR coverage under the dual-slope path loss model.
Yet, bringing the BSs closer to the users clearly increases SNR.
Thus, the optimal density of the network introduces a tradeoff between SIR and SNR.
While a closed-form expression of the SINR coverage does not exist in general,
there are multiple ways to characterize the coverage probabilty
as a function of the network density in addition to directly applying the integral
expression in Theorem~\ref{thm:2exp}.
\subsubsection{SNR Coverage Analysis\label{subsec:SNR}}
To complement the SIR coverage analysis in Sect.~\ref{sec:SIR},
it is natural to focus on the SNR coverage probability, defined as the probability
that $\SNR_l \triangleq h_{x^*} l(x^*) / \sigma^2 > T$.
Such analysis has not attracted much attention under the standard path loss model due to the SINR monotonicity
under the standard path loss function (\ie $\SINR$ increases monotonically with network density),
but becomes relevant for the dual-slope path loss model.
The analysis is also important for noise-limited systems including the emerging mmWave networks \cite{RanRapErk14}.
\begin{lemma}[The SNR Coverage Probability]\label{lem:SNRCovP}
The SNR coverage probability under the dual-slope path loss model is
\begin{multline}
\Pc^\SNR_{l_2}(\lambda, T) = \lambda\pi R_c^2 \int^1_0 e^{-\lambda \pi R_c^2 y
- T \sigma^2 R_c^{\alpha_0} y^{\frac{\alpha_0}{2}}} \d y \\
+ \lambda \pi R_c^2 \int_1^\infty e^{-\lambda\pi R_c^2 y - T \sigma^2 R_c^{\alpha_0} y^{ \frac{\alpha_1}{2}}}\d y.
\label{equ:SNRCovP}
\end{multline}
\end{lemma}
The proof of Lemma~\ref{lem:SNRCovP} is strightforward, using the well-known distance distribution
in Poisson networks \cite{net:Haenggi05tit} and so is omitted from the paper.
The first term in \eqref{equ:SNRCovP} corresponds to the case where the serving BS station
is within distance $R_c$ to the typical user and the second term to the case where the serving BS
is farther than $R_c$ away from the user.
In general, the SNR coverage probability \eqref{equ:SNRCovP}
cannot be written in closed-form. But for the special case $\alpha_0=2$
and $\alpha_1 = 4$, it can be simplified as in the following corollary.
\begin{corollary}
For $\alpha_0=2$, $\alpha_1 = 4$, the $\SNR$ coverage probability is
\begin{multline}
\Pc^\SNR_{l_2(2,4;\cdot)}(\lambda, T) = \frac{\lambda\pi}{\lambda\pi+ T \sigma^2} (1-e^{-(\lambda\pi+ T \sigma^2) R_c^2}) \\
+\frac{\lambda \pi^{\frac{3}{2}}R_c}{\sqrt{ T \sigma^2}} e^{\frac{\lambda^2\pi^2 R_c^2}{4 T \sigma^2}}
\Q\left(\frac{\lambda\pi+2 T \sigma^2}{\sqrt{2 T \sigma^2}}R_c\right),
\end{multline}
where $\Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-{t^2}/{2}}\d t$.
\end{corollary}
Naturally, both $\Pc^\SNR_{l_2}(\lambda, T)$ and $\Pc^\SIR_{l_2}(\lambda, T)$ are upper bounds on $\Pc^\SINR_{l_2}(\lambda, T)$ and the former is asymptotically tight for $\lambda\to 0$ and the latter for $\lambda\to\infty$. Taking the minimum of them could result in an informative characterization of the interplay between interference and noise as the network densifies.
Fig.~\ref{fig:SINRcvrgProbxalpha02_alpha14_lambda1e-05_10000000000}
and Fig.~\ref{fig:SINRcvrgProbxalpha03_alpha14_lambda0point0001_100}
compare the coverage probability
for the case $\alpha_0=2,3$ and $\alpha_1 = 4$.
As expected, we observe that the SINR coverage probability is maximized for some finite $\lambda$
which effectively strikes a balance between SIR coverage and SNR coverage.
The former decreases with $\lambda$; the latter increases with $\lambda$;
both of them are upper bounds on the SINR coverage probability.
Fig.~\ref{fig:SINRcvrgProbxalpha02_alpha14_lambda1e-05_10000000000} also verifies Prop.~\ref{prop:densezerocoverage}
as the coverage probability goes to zero as $\lambda\to\infty$.
Fig.~\ref{fig:SINRcvrgProbxalpha03_alpha14_lambda0point0001_100} is consistent with Lemma~\ref{lem:BoundAchi} and shows that ultra-densification will lead to constant positive coverage probability if $\alpha_0>2$.
The numerical example also suggests that in this case, the decay of coverage probability with
densification is smaller for low and high SINR but larger for medium SINR.
\begin{figure}[tbh]
\centering
\psfrag{lambda}[c][c]{$\lambda$}
\psfrag{Pc}[c][t]{$\Pc^\SINR_{l_2}$, $\Pc^\SIR_{l_2}$, $\Pc^\SNR_{l_2}$}
\begin{overpic}[width=\linewidth]{SINRcvrgProbxalpha02_alpha14_lambda0point001_100nogrid_c.eps}
\put(74,65){\vector(-1,0){7}}
\put(76,64){$ T=-10$ dB}
\put(74,35){\vector(-1,0){10}}
\put(76,34){$ T=0$ dB}
\put(74,13){\vector(-1,0){10}}
\put(76,12){$ T=10$ dB}
\end{overpic}
\caption{SINR, SIR, and SNR coverage scaling vs. network density, with $\alpha_0 = 2$, $\alpha_1 = 4$, $R_c=1$, $ \sigma^2 = 1$.\label{fig:SINRcvrgProbxalpha02_alpha14_lambda1e-05_10000000000}}
\end{figure}
\begin{figure}[tbh]
\centering
\psfrag{lambda}[c][c]{$\lambda$}
\psfrag{Pc}[c][t]{$\Pc^\SINR_{l_2(\alpha_0,\alpha_1;\cdot)}$, $\Pc^\SIR_{l_2(\alpha_0,\alpha_1;\cdot)}$, $\Pc^\SNR_{l_2(\alpha_0,\alpha_1;\cdot)}$}
\begin{overpic}[width=\linewidth]{SINRcvrgProbxalpha03_alpha14_lambda0point001_100nogrid_c.eps}
\put(73,55){\vector(-1,1){5}}
\put(75,54){$ T=-10$ dB}
\put(75,39){\vector(-1,0){5}}
\put(77,38){$ T=0$ dB}
\put(73,17){\vector(-1,0){5}}
\put(75,16){$ T=10$ dB}
\end{overpic}
\caption{SINR, SIR, and SNR coverage scaling vs. network density, with $\alpha_0 = 3$, $\alpha_1 = 4$, $R_c=1$, $ \sigma^2 = 1$.\label{fig:SINRcvrgProbxalpha03_alpha14_lambda0point0001_100}}
\end{figure}
\subsubsection{\SINR Distribution for the Two-ray Model}
For the special case of $\alpha_0 = 2$ and $\alpha_1 = 4$, it is possible to derive
a tight lower bound on the coverage probability and thus the ccdf of $\SINR$ as
an alternative to the numerical integral in Theorem~\ref{thm:2exp}.
\begin{proposition}
For $\alpha_0=2$ and $\alpha_1 =4$, we have the (closed-form) lower bound in \eqref{equ:SINR24LB} (at the top of the next page),
where $\gtrsim$ denotes larger than and asymptotically equal to (with respect to both $\lambda\to 0$ and $T \to 0$),
$C_{-\frac{1}{2}}(x)=1+\sqrt{x}\arctan \sqrt{x}$,
$\rho_0(\lambda, T, \sigma^2) = {\lambda\pi(1+ T)+ T \sigma^2}$,
$\rho_1(\lambda, T, \sigma^2) = {\lambda\pi R_c}/{\sqrt{ T \sigma^2}}$,
$\gamma_\textnormal{E} \approx 0.577$ is the Euler-Mascheroni constant, $\Ei(x) = \gamma(0,x) = \int_x^\infty \frac{e^{-t}}{t} \d t$
is the exponential integral function.
\label{prop:SINRcovLB}
\end{proposition}
\begin{figure*}[!t]
\normalsize
\begin{multline}
\Pc^\SINR_{l_2(2,4;\cdot)}(\lambda, T) \gtrsim
\frac{\lambda\pi}{\rho_0(\lambda, T, \sigma^2)}
\left( 1- e^{ -{\rho_0(\lambda, T, \sigma^2) R_c^2}} \right)
e^{\frac{\lambda\pi T}{\rho_0(\lambda, T, \sigma^2)}
\left( 1- \frac{\gamma_\textnormal{E} + \log\left(\rho_0(\lambda, T, \sigma^2)\right) + \Ei(\rho_0(\lambda, T, \sigma^2))}{1-\exp(\rho_0(\lambda, T, \sigma^2) R_c^2)} \right)} \\
+ \sqrt{\pi}\rho_1(\lambda, T, \sigma^2) e^{\frac{1}{4}\left({C_{-\frac{1}{2}}( T)}\rho_1(\lambda, T, \sigma^2)\right)^2}
\Q\left(\frac{1}{\sqrt{2}}\rho_1(\lambda, T, \sigma^2) C_{-\frac{1}{2}}( T) + \sqrt{2 T \sigma^2} R_c\right)
\label{equ:SINR24LB}
\end{multline}
\hrulefill
\vspace*{4pt}
\end{figure*}
\begin{IEEEproof}
See Appendix~\ref{app:SINRcovLB}.
\end{IEEEproof}
\begin{figure}[tbh]
\centering
\psfrag{theta}[c][b]{$ T$}
\psfrag{Pc}[c][t]{$\Pc^\SINR_{l_2(\alpha_0,\alpha_1;\cdot)}$}
\psfrag{ccc}{$\lambda = 0.1$}
\psfrag{bbb}{$\lambda = 10$}
\psfrag{aaa}{$\lambda = 1$}
\begin{overpic}[width=\linewidth]{SINRccdfalpha02_alpha14_lambda0point1_10_c.eps}
\end{overpic}
\caption{SINR ccdf from simulation, Theorem~\ref{thm:2exp} and Prop.~\ref{prop:SINRcovLB}.
Here, $\alpha_0 = 2$, $\alpha_1 = 4$, $R_c=1$, $ \sigma^2 = 1$.\label{fig:SINRccdfalpha02_alpha14_lambda0point1_10_c}}
\end{figure}
Prop.~\ref{prop:SINRcovLB} does not involve numerical integral and is instead based
on two well-known special functions: the Q-function and the exponential integral function.
Fig.~\ref{fig:SINRccdfalpha02_alpha14_lambda0point1_10_c} compares the lower bound in Prop.~\ref{prop:SINRcovLB}
with simulation results and the integral expression in Theorem~\ref{thm:2exp}.
The figure numerically verifies the asymptotic tightness of the bound for $ T\to 0$ and/or $\lambda\to 0$.
With the $\lambda = 1$ curve on top,
Fig.~\ref{fig:SINRccdfalpha02_alpha14_lambda0point1_10_c} also confirms that \SINR
does not increase with network density\footnote{More precisely, the \SINR first increases with the network density
(in the noise-limited regime) and then decreases with the network density (in the interference-limited regime).}, as expected.
\subsection{Throughput Scaling}
The coverage probability analysis alone does not provide a complete characterization of the network
performance scaling under densification.
To understand how the area spectral efficiency scales, we further study the potential throughput
defined in Sect.~\ref{subsect:SINRCov}.
By the definition of the potential throughput (Def.~\ref{def:PT}) and Theorem~\ref{thm:PcSIRordering},
we immediately obtain following lemma.
\begin{lemma}
Under the dual-slope path loss model and full-load assumption, the potential network throughput grows \emph{linearly} with BS
density $\lambda$ (as $\lambda\to\infty$) if $\alpha_0>2$.
\label{lem:capacityLinearScaling}
\end{lemma}
\begin{IEEEproof}
If $\alpha_0>2$, $\Pc^\SIR_{l_1(\alpha_1;\cdot)}$ and $\Pc^{\SIR}_{l_1(\alpha_0;\cdot)}$ are positive and invariant
with network density $\lambda$ for any $ T\geq 0$ \cite{net:Andrews12tcom}.
Under the full load assumption, the coverage density $\lambda \Pc^\SIR_{l_1(\alpha_0;\cdot)} \leq \mu_{l_2(\alpha_0,\alpha_1;\cdot)} \leq \lambda \Pc^\SIR_{l_1(\alpha_1;\cdot)}$.
Consequently, $ \mu_{l_2}(\lambda,T)= \Theta(\lambda)$ and in interference-limited network.
By the definition of potential throughput in Def.~\ref{def:PT}, we further have $ \tau_{l_2}(\lambda,T)= \Theta(\lambda)$.
When $\lambda\to\infty$, interference dominates noise and thus the same throughput scaling holds
in noisy networks.
\end{IEEEproof}
While Lemma~\ref{lem:capacityLinearScaling} is encouraging,
it is only for the case $\alpha_0 > 2$.
On the other hand, Prop.~\ref{prop:densezerocoverage} shows that if $\alpha_0\leq 2$, the coverage probability decays to zero as the network densifies.
This may lead to the pessimistic conjecture that the potential throughput would decrease with the network density.
Fortunately, this is not necessarily true.
A complete characterization of the potential throughput scaling is given in the following theorem.
\begin{theorem}[Throughput Scaling under the Dual-slope Model]
Under the dual-slope path loss model, as $\lambda\to\infty$, the potential throughput $\tau_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T)$
\begin{enumerate}
\item
grows linearly with $\lambda$ if $\alpha_0>2 $,
\label{itm:Tscaling1}
\item
scales sublinearly with rate $\lambda^{2-\frac{2}{\alpha_0}}$ if $1<\alpha_0 < 2 $,
\label{itm:Tscaling2}
\item
decays to zero if $\alpha_0< 1$.
\label{itm:Tscaling3}
\end{enumerate}
\label{thm:TScaling}
\end{theorem}
\begin{IEEEproof}
See Appendix~\ref{app:TScaling}.
\end{IEEEproof}
Due to the technical subtlety, Theorem~\ref{thm:TScaling} does not include the cases of $\alpha_0 =1, 2$
(a slightly different proof technique needs to be tailored exclusively for these points).
Yet, by continuity, we conjecture that the potential throughput
scales linearly at $\alpha_0 = 2$ and converges to some finite value at $\alpha_0=1$.
Theorem~\ref{thm:TScaling} provides theoretical justification to the potential of cell densification
despite the slightly pessimistic results given in Prop.~\ref{prop:densezerocoverage}.
Under the dual slope model, even if $\alpha_0 < 2$ and the coverage probability
goes to zero as the network densifies, the cell splitting gain can still scale up
the potential throughput of the network as long as $\alpha_0 > 1$
which practically holds in most of the cases of interest.
\begin{figure}[tbh]
\centering
\psfrag{lambda}[c][c]{$\lambda$}
\psfrag{T}[c][t]{$\tau_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda, T)$}
\includegraphics[width=\linewidth]{PTScalingtheta1_R1_lambda0point01_1000_c.eps}
\caption{Potential throughput scaling with network density.
Here, $\alpha_0 = 0.9,1,1.8,3$, $\alpha_1 = 4$, $R_c=1$, $ \sigma^2 = 0$, $ T = 1$.\label{fig:PTScalingtheta1_R1_lambda0point01_1000_c}}
\end{figure}
Fig.~\ref{fig:PTScalingtheta1_R1_lambda0point01_1000_c} verifies the scaling
results given in Theorem~\ref{thm:TScaling}.
As expected, we observe a \emph{phase transition} at $\alpha_0 = 1$:
if $\alpha_0<1$, the asymptotic potential throughput goes to zero;
if $\alpha_0>1$, it goes to infinity.
For $\alpha_0=1$, numerical results suggest that asymptotic potential throughput
converge to a \emph{positive} finite value.
\section{Multi-slope Path Loss Model\label{sec:Mslope}}
The previous sections have focused on the dual-slope path loss function.
Since Lemma~\ref{lem:CovPGenPL} applies for arbitrary path loss functions (whenever the integral exists),
explicit (integral) expression for the coverage probability of the multi-slope path loss function (Def.~\ref{def:Multi-slopePLM})
can be derived analogous to Theorem~\ref{thm:2exp}.
\begin{theorem}
The coverage probability under the $N$-slope path loss model ($N\geq 3$) is
in \eqref{equ:PcMslp},
where $I_i(\{\alpha_l\},\{R_l\}, T; x)$ is given in \eqref{equ:IiMslp} (both equations are at the top of the next page),
$\delta_i = 2/\alpha_i,~i \in [N-1]\cup\{0\}$ and $C_\beta(x) = {_2F}_1(1,\beta;1+\beta;-x)$.
\label{thm:Nexp}
\end{theorem}
\begin{figure*}[!t]
\normalsize
\begin{equation}
\Pc^\SINR_{l_N}(\lambda, T) =
\lambda\pi \left( \sum_{i=0}^{N-2}\int_{R_i^2}^{R_{i+1}^2} e^{-\lambda\pi I_i(\{\alpha_l\},\{R_l\}, T; x)
}
e^{- T \sigma^2 x^{\frac{\alpha_i}{2}}/K_i } \d x
+ \int_{R_{N-1}^2}^{\infty} e^{ -\lambda\pi x C_{-\delta_{N-1}}\left( T \right) } e^{- T \sigma^2 x^{\frac{\alpha_{N-1}}{2}}/K_{N-1}} \d x \right)
\label{equ:PcMslp}
\end{equation}
\begin{multline}
I_i(\{\alpha_l\},\{R_l\}, T; x) =
x \left(1-C_{\delta_i}\Big(\frac{1}{ T}\Big)\right)
+ R_{i+1}^2 C_{\delta_i}\left( \frac{R_{i+1}^{\alpha_i}}{ T x^{\frac{\alpha_i}{2}}} \right) \\
+ \sum_{j=i+1}^{N-2} \left(
R_{j+1}^2 C_{\delta_j}\bigg(\frac{K_i}{K_j} \frac{R_{j+1}^{\alpha_j}}{T x^{\frac{\alpha_i}{2}}}\bigg)
-R_{j}^2 C_{\delta_j}\bigg(\frac{K_i}{K_j} \frac{R_{j}^{\alpha_j}}{T x^{\frac{\alpha_i}{2}}}\bigg)
\right)
+ R_{N-1}^2 C_{-\delta_{N-1}}\left( \frac{K_{N-1}}{K_i}\frac{T x^{\frac{\alpha_i}{2}}}{R_{N-1}^{\alpha_{N-1}}} \right)
- R_{N-1}^2
\label{equ:IiMslp}
\end{multline}
\hrulefill
\vspace*{4pt}
\end{figure*}
While Def.~\ref{def:Multi-slopePLM} requires the path loss exponents for the multi-slope path loss function to be increasing,
the proof of Theorem~\ref{thm:Nexp} does not depend on the ordering.
Thus, Theorem~\ref{thm:Nexp} is true even when $\{\alpha_l\}$ are arbitrarily ordered (so is Theorem~\ref{thm:2exp}).
In the practically important case of ordered path loss exponents (Def.~\ref{def:Multi-slopePLM}), all conclusions drawn in Sect.~\ref{sec:SIR} extend to the multi-slope case.
In the following theorem, we summarize these main conclusions.
\begin{theorem}
The coverage probability with the multi-slope path loss function given by Def.~\ref{def:Multi-slopePLM} satisfies the following properties:
\begin{itemize}
\item
$\Pc^\SIR_{l_1(\alpha_0;\cdot)} (\cdot, T) \lesssim \lim_{\lambda\to\infty} \Pc^\SIR_{l_N} (\lambda, T) $ (as $\lambda \to \infty$).
\item
$\lim_{\lambda\to 0} \Pc^\SIR_{l_N}(\lambda, T) \lesssim \Pc^\SIR_{l_1(\alpha_{N-1};\cdot)} (\cdot, T)$ (as $\lambda \to 0$).
\item
$ \Pc^\SIR_{l_N} (\lambda_1, T) \geq \Pc^\SIR_{l_N}(\lambda_2, T)$, for all $\lambda_1\leq\lambda_2$ and $ T \geq 0$
\end{itemize}
where $\lesssim$ denotes less than equal to and asymptotically equal to, $l_i(\|x\|) = K_i \|x\|^{-\alpha_i}$, for $i\in\{0,1,2,\cdots,N\}$.
\end{theorem}
In Fig.~\ref{fig:SINRccdfalpha_v_024_Rb_v_1267_lambda1e-07_0.001_W_1e-08_c}, we validate Theorem~\ref{thm:Nexp} with simulations.
We combine the classic two-ray model with a bounded path loss model to create a 3-slope path loss model with $[\alpha_0\; \alpha_1\; \alpha_2] = [0\; 2\; 4]$
and $[R_1\; R_2] = [1\; 267]$.\footnote{Here, we use standard units and $R_2=267$ m comes from the two-ray example mentioned in Sect.~\ref{sec:Intro}.}
The noise variance is set to $10^{-8}$, corresponding to an $80$ dB \SNR at unit distance. An exact match between analysis and simulation is observed in the figure. Despite the more refined model, similar trends can be observed as in the case with the dual slope model (Fig.~\ref{fig:SINRccdfalpha02_alpha14_lambda0point1_10_c}).
\begin{figure}[tbh]
\centering
\psfrag{theta}[c][b]{$ T$}
\psfrag{Pc}[c][t]{$\Pc^\SINR_{l_3}$}
\begin{overpic}[width=\linewidth]{SINRccdfalpha_v_024_Rb_v_1267_lambda1e-07_0point001_W_1e-08_c.eps}
\put(55,60){\vector(-4,-3){20}}
\put(57,60){$ \lambda=10^{-5}, 10^{-7}, 10^{-3}$}
\end{overpic}
\caption{SINR ccdf from simulation and Theorem~\ref{thm:Nexp}.
Here, the number of slopes $N=3$, $[\alpha_0\; \alpha_1\; \alpha_2] = [0\; 2\; 4]$
and $[R_1\; R_2] = [1\; 267]$, $ \sigma^2 = 10^{-8}$.\label{fig:SINRccdfalpha_v_024_Rb_v_1267_lambda1e-07_0.001_W_1e-08_c}}
\end{figure}
Similarly, following the same proof techniques of those of Theorem~\ref{thm:TScaling},
it is straightforward to generalize the throughput scaling results from the dual-slope
path loss model to the multi-slope path loss model, resulting in the following theorem.
\begin{theorem}[Throughput Scaling under Multi-slope Path Loss Model]
Under the multi-slope path loss model, as $\lambda\to\infty$, the potential throughput $\tau_{l_N}(\lambda, T)$
\begin{enumerate}
\item
grows linearly with $\lambda$ if $\alpha_0>2 $,
\label{itm:MTscaling1}
\item
scales sublinearly with rate $\lambda^{2-\frac{2}{\alpha_0}}$ if $1<\alpha_0 < 2 $,
\label{itm:MTscaling2}
\item
decays to zero if $\alpha_0< 1$.
\label{itm:MTscaling3}
\end{enumerate}
\label{thm:MTScaling}
\end{theorem}
Theorem~\ref{thm:MTScaling} shows that there (still) exists
a phase transition for the asymptotic scaling of network throughput under the multi-slope
path loss model,
and the phase transition happens at the same critical values of $\alpha_0$.
Intuitively, in the ultra-dense regime ($\lambda\to\infty$),
infinitely number of BSs are in the nearest field (subject to path loss exponent $\alpha_0$),
making the scaling independent of $\alpha_n, n\geq 1$.
Nevertheless, the values of $\alpha_n, n\geq 1$ as well as $R_n, n\in [N-1]$
affect the $\SINR$ distribution in the non-asymptotic regime.
\section{Conclusions\label{sec:conclu}}
This paper analyzes cellular network coverage probability and potential throughput under the dual-slope path loss model. We show that despite being a seemingly minor generalization, the dual-slope path loss model produces many surprising observations that stand in sharp contrast to results derived under standard path loss models. In particular, we show the monotonic decrease of $\SIR$ with infrastructure density and the existence of a coverage-maximizing density. Both results are consistent with recent findings based on
other non-standard path loss functions \cite{RamasamyGantiMadhow2013,BaiHeath2015,BaiVazeHeath2014,BaccelliZhang2015}.
By studying the potential throughput, we show that there exists a phase transition on the asymptotic potential throughput of the network. If the near-field path loss exponent $\alpha_0$ is less than one, the potential throughput goes to zero as the network densifies.
If $\alpha_0>1$, the potential throughput grows (unboundedly) with denser network deployment,
but the growth rate may be sublinear depending on the path loss exponent. Since in most practical cases, we have $\alpha_0>1$, this implies network scalability even without intelligent scheduling.
We believe this paper should lead to further scrutiny of the idealized standard path loss model. The dual-slope and multi-slope path loss functions are important potential substitutes with much more precision and seemingly adequate tractability.
As the cellular network densifies and new technologies are introduced, existing knowledge need to be refined in view of these models. For example, (i) local cell coordination and coordinated multipoint processing (CoMP) may be much more powerful than previously predicted since near-field interferers can produce much stronger interference than far-field ones; (ii) successive interference cancellation (SIC) may be less useful or more dependent on power control since near-by transmitters may produce less differentiable received powers; (iii) in HetNets, closed subscriber groups may be more harmful to nearby users, and the benefit of load balancing may be less than expected due to higher received power from nearby small cells and lower received power from far-away macrocells; and (iv) device-to-device (D2D) communication may be (even) more power-efficient than foreseen due to smaller near-field path loss, but demanding more careful scheduling to mitigate near-field interference.
\section{Acknowledgments}
The authors wish to thank Anthony Soong (Huawei) for the suggestion to investigate dual-slope path loss models and for providing some empirical data supporting their accuracy. The authors also wish to thank Sarabjot Singh (Nokia) for comments on early drafts of the paper
and sharing his insights on recent millimeter wave research.
\appendices
\section{Proof of Lemma~\ref{lem:SIRbounds}\label{app:SIRbounds}}
\begin{IEEEproof}
Since the lemma states for arbitrary realization $\omega$,
the statistics of the marked point process is not relevant.
Instead of carrying $\omega$ for the rest of the proof,
we will use $h_x$, $x\in\Phi$ to denote $h_x(\omega)$, $x(\omega)\in\Phi(\omega)$ for simplicity.
First, we focus on the first part (first bullet) of the lemma and assume $\SIR_{l_1(\alpha_0;\cdot)}(\omega) > T$.
The proof proceeds in two cases separately: $\|x^*\|\leq R_c$ and $\|x^*\|>R_c$.
For $\|x^*\|\leq R_c$, we have $l_2(\alpha_0,\alpha_1;x^*) = l_1(\alpha_0;x^*)$.
Since $l_2(\alpha_0,\alpha_1;x)\leq l_1(\alpha_0;x),\; \forall x\neq o$,
we obtain $h_{x^*} l_2(\alpha_0,\alpha_1;x^*) = h_{x^*} l_1(\alpha_0;x^*) > T \sum_{y\in\Phi\setminus\{x^*\}} h_y l_1(\alpha_0;y) \geq T \sum_{y\in\Phi\setminus\{x^*\}} h_y l_2(\alpha_0,\alpha_1;y)$,
\ie $\SIR_{l_1(\alpha_0;\cdot)} (\omega) > T$ implies $\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) > T$.
For $\|x^*\|> R_c$, given $\SIR_{l_1(\alpha_0;\cdot)} (\omega) > T$, we have
\begin{align*}
h_{x^*} l_2(\alpha_0,\alpha_1;x^*) &= h_{x^*} \eta \|x^*\|^{-\alpha_1} \\
&= h_{x^*} \eta\|x^*\|^{-\alpha_0} \frac{\|x^*\|^{\alpha_0}}{\|x^*\|^{\alpha_1}} \\
&\stackrel{\textnormal{(a)}}{>} T \eta \frac{\|x^*\|^{\alpha_0}}{\|x^*\|^{\alpha_1}} \sum^{\|y\|>\|x^*\|}_{y\in\Phi} h_y l_1(\alpha_0;y) \\\displaybreak[0]
&= T \sum^{\|y\|>\|x^*\|}_{y\in\Phi} h_y \eta\|y\|^{-\alpha_1} \left(\frac{\|y\|}{\|x^*\|}\right)^{\vartriangle\alpha} \\
&\stackrel{\textnormal{(b)}}{>} T \sum_{y\in\Phi\setminus\{x^*\}} h_y \eta\|y\|^{-\alpha_1} \\
&= T \sum_{y\in\Phi\setminus\{x^*\}} h_y l_2(\alpha_0,\alpha_1;y),
\end{align*}
where $\eta = R_c^{\vartriangle\alpha}$, $\vartriangle\alpha = \alpha_1 - \alpha_0$, (a) is due to $\SIR_{l_1(\alpha_0;\cdot)} (\omega) > T$ and (b) comes from the fact that $\|x^*\|<\|y\|,\;\forall y\in\Phi\setminus\{x^*\}$.
The same idea applies to the proof of the second part of the lemma.
To make the proof more strightforward, we first prove that $\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) > T$ implies $\SIR_{\eta l_1(\alpha_1;\cdot)} (\omega) \geq T$ as follows:
If $\|x^*\|> R_c$, $l_2(\alpha_0,\alpha_1;x) = \eta l_1(\alpha_1;x)$ for all $x_i\in\Phi\cap(x^*,\infty)$
and thus $\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) > T \Longleftrightarrow \SIR_{\eta l_1(\alpha_2;\cdot)} (\omega) \geq T$.
If $\|x^*\| \leq R_c$, given, $\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) > T$, we have
\begin{align*}
&h_{x^*} \eta l_1(\alpha_1;x^*)=h_{x^*} \eta \|x^*\|^{-\alpha_1} \\
&\stackrel{\textnormal{(c)}}{>} T \eta \|x^*\|^{\vartriangle\alpha}
\left(\sum^{\|y\|\leq R_c}_{y\in\Phi\setminus\{x^*\}} h_y \|y\|^{-\alpha_0}
+ \sum^{\|y\|>R_c}_{y\in\Phi} h_y \eta\|y\|^{-\alpha_1} \right) \\
&= T \left(\sum^{\|y\|\leq R_c}_{y\in\Phi\setminus\{x^*\}} h_y \eta \|y\|^{-\alpha_1} \left(\frac{\|y\|}{\|x^*\|}\right)^{\vartriangle\alpha} \right. \\
&\phantom{=}~ +\left. \sum^{\|y\|>R_c}_{y\in\Phi} h_y \eta\|y\|^{-\alpha_1} \left(\frac{R_c}{\|x^*\|}\right)^{\vartriangle\alpha}\right) \\
&\stackrel{\textnormal{(d)}}{>} T \sum_{y\in\Phi\setminus\{x^*\}} h_y \eta \|y\|^{-\alpha_1}
= T \sum_{y\in\Phi\setminus\{x^*\}} h_y \eta l_1(\alpha_1;y),
\end{align*}
where (c) is due to the assumption $\SIR_{l_2(\alpha_0,\alpha_1;\cdot)} (\omega) > T$ and (d) takes into account the fact that
$\|y\|>\|x^*\|,\;\forall y\in\Phi\setminus\{x^*\}$ and $\|x^*\|\leq R_c$.
Realizing that $\SIR_{ l_1(\alpha_1;\cdot)} (\omega) = \SIR_{k l_1(\alpha_1;\cdot)} (\omega)$ for all $k>0$,
we complete the proof for the second part of the lemma.
\end{IEEEproof}
\section{Proof of Prop.~\ref{prop:densezerocoverage} \label{app:densezerocoverage}}
\begin{IEEEproof}
The following proof is to show $\Pc^\SIR_{l_2}\to 0$ as $\lambda\to\infty$.
Since $\Pc^\SINR_{l_2}\leq\Pc^\SIR_{l_2}$ the same result for SINR coverage follows naturally.
We will first focus on the case of $\alpha_0<2$.
The result of $\alpha=2$ then follows from the continuity of the coverage probability expression \eqref{equ:2expCP}.
Using Lemma ~\ref{lem:CovPGenPL} and setting $W=0$, we can upper bound the coverage probability as follows,
$\Pc^\SIR_{l_2} $
\begin{align*}
&\pleq{a} \lambda\pi \int_0^\infty \exp\left({{-\lambda\pi y \Bigg(1+\int_1^{1 \vee \frac{R_c^2}{y}}
\frac{ T}{ T+\frac{l_2(\sqrt{y})}{l_2(\sqrt{t y})}}\d t\Bigg)}} \right) \d y \\
&= \lambda\pi \int_0^{R_c^2} \exp\left({-\lambda\pi y \Big(1+\int_1^{\frac{R_c^2}{y}}
\frac{ T}{ T+t^{\frac{\alpha_0}{2}}}\d t \Big)}\right) \d y \\
&\phantom{=}~ + \lambda\pi \int_{R_c^2}^\infty e^{-\lambda\pi R_c^2} \d y \\
&\peq{b} {A(\lambda, R_c, \alpha_0, T)}
+ e^{-\lambda\pi R_c^2},\numberthis\label{equ:alpha0<2CPUB}
\end{align*}
where $a\vee b = \max\{a,b\}$, ${A(\lambda, R_c, \alpha_0, T)} \triangleq $
\begin{equation*}
{\lambda\pi R_c^2 \int_0^{1} \exp\left({-\lambda\pi u R_c^2 \Big(1+\int_1^{\frac{1}{u}}
\frac{ T}{ T+t^{\frac{\alpha_0}{2}}}\d t \Big)}\right) \d u},
\end{equation*}
(b) is based on the change of variable $y\to u R_c^2$, and (a) is based on
truncating the interval of integration in the exponent with the intuition of ignoring the interference
coming from BSs farther than $R_c$.
Since the second term of \eqref{equ:alpha0<2CPUB} converges to zero with $\lambda\to\infty$,
to prove the lemma we only need to show that $A(\lambda,R_c,\alpha_0, T)$
goes to zero.
For an increasing sequence of $\lambda_n$, let
\begin{equation}
f_n(x) = \lambda_n \pi R_c^2 \exp\left({-\lambda_n\pi R_c^2 x \left(1+\int_1^{\frac{1}{x}}
\frac{ T}{ T+t^{\frac{\alpha_0}{2}}}\d t\right)}\right).
\label{equ:fn}
\end{equation}
It is clear that $f_n(x) \to 0$ almost everywhere on $(0,1)$.
Also, \[0\leq f_n (x) \leq g(x) \triangleq \frac{1}{x e \left(1+\int_1^{\frac{1}{x}}
\frac{ T}{ T+t^{\frac{\alpha_0}{2}}}\d t\right)}\]
and it is straightforward to check that $g(x)$ is integrable on $(0,1)$ for $0 \leq \alpha_0<2$.
By the dominated convergence theorem, we have $\lim_{\lambda\to\infty} A(\lambda, R_c, \alpha_0, T) = 0$ and thus complete the proof.
\end{IEEEproof}
\section{Proof of Lemma~\ref{lem:SIRmono}\label{app:SIRmono}}
\begin{IEEEproof}
Consider a linear mapping $f:\R^2\to\R^2$ such that $f(x) = ax$ for $a>1$.
By the mapping theorem\cite{net:mh12}, it is easy to show that for any homogeneous PPP $\Phi\subset\R^2$
with intensity $\lambda$, $f(\Phi)$ is also a homogeneous PPP on $\R^2$ with intensity $\lambda/a$.
With slight abuse of notation, we let the same mapping operate on the space of the marked PPP (but take effect only on the ground process), \ie $f(\hat\Phi) = f(\{(x_i, h_{x_i})\}) = \{(a x_i, h_{x_i})\}$.
By the same argument, $f(\hat\Phi)$ is a marked PPP with intensity $\lambda/a$
and marked by the same iid fading marks.
If we define the indicator function $\chi_{ T,l} : \R^2\times\R^+ \to \{0,1\}$ as follows:
\begin{equation*}
\chi_{ T,l}(\hat\Phi) = \left\{
\begin{array}{ll}
1,& \textnormal{if } h_{x^*}l(x^*)> T \sum_{x\in\Phi\setminus\{x^*\}} h_xl(x)
\\
0,& \textnormal{otherwise,}
\end{array}
\right.
\end{equation*}
we have $\Pc_l^\SIR(\lambda, T) = \E \left[ \chi_{ T,l}\left(\hat\Phi(\lambda)\right)\right]$,
where we use $\hat\Phi(\lambda)$ to emphasize that the density of the ground process is $\lambda$.
The key of the proof then comes from the observation that
$\chi_{ T,l_2}(\hat\phi) \leq \chi_{ T,l_2}\left(f(\hat\phi)\right)$
for all marked point pattern $\hat\phi = \{(x_i, h_{x_i})\}\subset\R^2\times\R^+$ and $a>1$.
More specifically, if $\|x^*\|>R_c$, then $\chi_{ T,l_2}(\hat\phi) = 1$ implies
$ h_{x^*} \|x^*\|^{-\alpha_1} > T\sum_{x\in\phi\setminus\{x^*\}} h_x \|x\|^{-\alpha_1}$.
Multiplying both sides of the inequality by $a^{-\alpha_1},\; a>1$ leads to the conclusion
that $\chi_{ T,l}\left(f(\hat\phi)\right) = 1$.
If $\|x^*\|<R_c$, we need to separate two cases: $a\|x^*\| \leq R_c$ and $a\|x^*\| > R_c$.
In the former case, $l_2(a x^*) = a^{-\alpha_0} l_2(x^*)$, thus $\chi_{ T,l_2}(\hat\phi) = 1$ implies
\begin{multline*}
h_{x^*}l_2(ax^*) = h_{x^*} a^{-\alpha_0}l_2(x^*) \\
> T\left(\sum_{x\in\phi\setminus\{x^*\}}^{\|x\|\leq\frac{R_c}{a}} h_x a^{-\alpha_0} l_2(x)
+ \sum_{x\in\phi}^{\|x\|\in(\frac{R_c}{a},R_c]} h_x a^{-\alpha_0} l_2(x) \right. \\ \left.
+ \sum_{x\in\phi}^{\|x\|>R_c} h_x a^{-\alpha_0} l_2(x)\right),
\end{multline*}
where for $\|x\|\in(\|x^*\|,\frac{R_c}{a}]$, we have $a^{-\alpha_0}l_2(x) = l_2(x)$;
for $\|x\|\in(\frac{R_c}{a},R_c]$,
we have $a^{-\alpha_0}l_2(x) = \left(\frac{x}{R_c/a}\right)^{\vartriangle\alpha} l_2(ax) \geq l_2(ax)$,
where $\vartriangle\alpha = \alpha_1-\alpha_0$;
for $\|x\|>R_c$, we have $a^{-\alpha_0}l_2(x) = a^{-\alpha_0} \eta x^{-\alpha_1} \geq \eta (ax)^{-\alpha_1} = l_2(ax)$
since $a>1$ and $\vartriangle\alpha>0$.
These observations lead to the conclusion that $h_{x^*} l_2(ax^*)> T\sum_{x\in\phi\setminus\{x^*\}} h_x l_2(ax)$,
\ie $\chi_{ T,l_2}\left(f(\hat\phi)\right) = 1$.
In the latter case where $a\|x^*\|>R_c$, if $\chi_{ T,l_2}(\hat\phi) = 1$, we have
\begin{align*}
&h_{x^*}l_2(ax^*) = h_{x^*} \eta (ax^*)^{-\alpha_1} = h_{x^*} l_2(x) \frac{\eta a^{-\alpha_1}}{\|x^*\|^{\vartriangle\alpha}} \\
&\stackrel{\textnormal{(a)}}{>} T \frac{\eta a^{-\alpha_1}}{\|x^*\|^{\vartriangle\alpha}}
\left(\sum_{x\in\phi\setminus\{x^*\}}^{\|x\|\leq R_c} h_x l_2(x) + \sum_{x\in\phi}^{\|x\|> R_c} h_x l_2(x) \right) \\
&= T \left(\sum_{x\in\phi\setminus\{x^*\}}^{\|x\|\leq R_c} h_x \eta \|a x\|^{-\alpha_1}
\left(\frac{\|x\|}{\|x^*\|}\right)^{\vartriangle\alpha} \right. \\
&\phantom{=}~\left. + \sum_{x\in\phi}^{\|x\|> R_c} h_x \eta \|a x\|^{-\alpha_1}
\left(\frac{R_c}{\|x^*\|}\right)^{\vartriangle\alpha} \right) \\
&\stackrel{\textnormal{(b)}}{>} T
\left(\sum_{x\in\phi\setminus\{x^*\}}^{\|x\|\leq R_c} h_x \eta \|a x\|^{-\alpha_1}
+ \sum_{x\in\phi}^{\|x\|> R_c} h_x \eta \|a x\|^{-\alpha_1}
\right) \\
&= T \sum_{x\in\phi\setminus\{x^*\}} h_x l_2(ax) \numberthis\label{equ:lax>laxC3}
\end{align*}
where (a) is the due to the assumption $\chi_{ T,l_2}(\hat\phi) = 1$,
and (b) takes into account the assumption $ax^*>R_c$ and $a>1$.
\eqref{equ:lax>laxC3} again leads to $\chi_{ T,l_2}\left(f(\hat\phi)\right) = 1$.
Therefore,
\begin{multline*}
\Pc_{l_2}^\SIR(\lambda, T) = \E \left[\chi_{ T,l_2}\left(\hat\Phi(\lambda)\right)\right]
\\
\leq \E \left[ \chi_{ T,l_2}\left(f(\hat\Phi(\lambda))\right)\right] \peq{c} \E \left[\chi_{ T,l_2}\left(\hat\Phi(\lambda/a)\right)\right],
\end{multline*}
where (c) comes from the fact that $f(\hat\Phi(\lambda))$ is
a marked homogeneous PPP with intensity $\lambda/a$ and with the same iid mark distribution as that of $\hat\Phi(\lambda)$.
\end{IEEEproof}
\section{Proof of Prop.~\ref{prop:SINRcovLB}\label{app:SINRcovLB}}
\begin{IEEEproof}
We start from Lemma~\ref{lem:CovPGenPL}. The $\SINR$
coverage probability can be written as $\Pc^\SINR_{l_2(2,4;\cdot)}(\lambda,T) =$
\begin{multline}
\underbrace{\lambda \pi \int_0^{R_c^2} e^{{-\lambda\pi y \left(1+\int_1^\infty
\frac{T}{T+{y^{-1}}/{l_2(2,4;\sqrt{t y})}}\d t\right)} - \sigma^2 T y} \d y}_{A} \\
+ \underbrace{\lambda \pi \int_{R_c^2}^\infty e^{{-\lambda\pi y \left(1+\int_1^\infty
\frac{T}{T+ R_c^2 y^{-1}/{l_2(2,4;\sqrt{t y})}}\d t\right)-\sigma^2 T y^2/ R_c^2}} \d y}_{B},
\label{equ:SINRCov}
\end{multline}
where $A$ ($B$, resp.) is the probability that the user being covered by an BS closer (farther, resp.) than $R_c$.
$B$ can be simplified (with the change of variable $y\to x R_c^2$, as in Thm.~\ref{thm:2exp}) into
\begin{multline*}
\lambda\pi R_c^2 \int_1^{\infty} \exp\left( -\lambda\pi R_c^2 x C_{-\frac{1}{2}}(T)\right) e^{-T \sigma^2 x^2 R_c^2} \d x
= \\
\sqrt{\pi}\rho_1(\lambda,T,\sigma^2) e^{\frac{1}{4}\left({C_{-\frac{1}{2}}(T)}\rho_1(\lambda,T,\sigma^2)\right)^2} \\
\times \Q\left(\frac{1}{\sqrt{2}}\rho_1(\lambda,T,\sigma^2) C_{-\frac{1}{2}}(T) + \sqrt{2T \sigma^2} R_c\right).
\end{multline*}
Thus, to prove the proposition, it is just to lower bound $A$ of \eqref{equ:SINRCov}.
We first focus on the exponent inside the integral and observe that
\begin{align*}
&\int_1^\infty \frac{T}{T+{y^{-1}}/{l_2(2,4;\sqrt{t y})}}\d t \\
&= \int_1^{\frac{R_c^2}{y}} \frac{T}{T+{t}}\d t
+ \int_{\frac{R_c^2}{y}}^\infty \frac{T}{T+ t^2 y/R_c^2}\d t \\
&\pleq{a} T \int_1^{\frac{R_c^2}{y}} \frac{1}{t} \d t
+ T R_c^2 \frac{1}{y} \int_{\frac{R_c^2}{y}}^\infty \frac{1}{t^2} \d t \\
&= T \log\left(\frac{y}{R_c^2}\right) + T. \numberthis\label{equ:intUB24}
\end{align*}
Applying \eqref{equ:intUB24} and a change of variables $y \to x R_c^2$,
we obtain
\begin{equation}
A \geq \lambda \pi R_c^2 \int_0^1 e^{\lambda\pi R_c^2 T x\log(x)} e^{ - \rho_1(\lambda,T,\sigma^2) R_c^2 x} \d x,
\end{equation}
which can be viewed as $K_1 \E[e^{\lambda\pi R_c^2 T X\log(X)}]$
for random variable $X$ with pdf $f_X(x) = K_2 e^{ - \rho_1(\lambda,T,\sigma^2) R_c^2 x}$
($K_1,K_2\in\R^+$ are normalization factors).
Since $e^{x}$ is convex, we apply Jensen's inequality
\begin{equation}
K_1 \E[e^{\lambda\pi R_c^2 T X\log(X)}]\geq K_1 e^{\lambda\pi R_c^2 T \E[X\log(X)]}
\label{equ:Jensenex}
\end{equation}
and obtain the desired bound.
To see the asymptotic tightness as $\lambda\to 0$,
we can examine the alternative representation of \eqref{equ:SINRCov} in \eqref{equ:2expCP}
and make the following observation which essentially generalizes Fact~\ref{fact:lambdaR2} to
the noisy case: letting $\lambda\to 0 $ (but keeping $R_c$ and $\sigma^2$ fixed) produces the same
effect on $P^\SINR_{l_2}$ as letting $R_c\to 0$ but keeping $\sigma^2 R_c^{\alpha_0}$ and $\lambda$ fixed.
Due to the physical meaning of $A$ and $B$, this implies $B$ dominates $A$ in \eqref{equ:SINRCov} as $\lambda\to 0$.
Since $B$ is exact in the lower bound (in \eqref{equ:SINRCov}, we only lower bounded $A$.), the bound is tight as $\lambda\to 0$.
The asymptotic tightness as $T \to 0$ is observed by examining the (only)
two inequality applied in the derivation: (a) in \eqref{equ:intUB24} and the Jensen's inequality in \eqref{equ:Jensenex}.
Both are tight as $T\to 0$.
\end{IEEEproof}
\section{Proof of Theorem~\ref{thm:TScaling}\label{app:TScaling}}
\begin{IEEEproof}
Since as network density goes to infinity the network becomes interference limited,
it suffices to consider only the case where $W=0$ and the result holds even with noise.
\ref{itm:Tscaling1}) comes directly from Lemma~\ref{lem:capacityLinearScaling}.
To show \ref{itm:Tscaling3}), one could use the same techinques in the proof
of Prop.~\ref{prop:densezerocoverage} thanks to the simple relation between
coverage probability and the potential throughput.
Basically multiplying both sides of \eqref{equ:alpha0<2CPUB} by $\lambda$ gives an upper bound
on the coverage density, \ie $\mu_{l_2}(\lambda, T) \leq \lambda A(\lambda, R_c, \alpha_0, T) + \lambda\exp(-\lambda\pi R_c^2)$,
where the second term goes to zero as $\lambda\to\infty$.
The first term can also be shown to converge to zero by the dominated convergence theorem.
In particular, using similar construction to \eqref{equ:fn}, $\lambda_n f_n(\cdot)$ goes to
zero almost everywhere and is upperbounded by
\[
g'(x) = 4/\left( \pi R_c^2 x^2 \left(1+\int_1^{\frac{1}{x}}
\frac{ T}{ T+t^{\frac{\alpha_0}{2}}}\d t\right)^2\right),
\]
which is integrable on $(0,1)$ if $\alpha_0 <1$.
To prove \ref{itm:Tscaling2}), we focus on showing that $\Pc^\SIR_{l_2}(\lambda, T) = \Omega (\lambda^{1-\frac{2}{\alpha_0}})$
as $\lambda\to\infty$ given $1\leq \alpha_0 <2$.
We start from Lemma~\ref{lem:CovPGenPL}.
By truncating the (outer) infinite integral to only $(0,R_c^2)$,
we have a lower bound on the coverage probability $\Pc^\SIR_{l_2}(\lambda, T) \geq$
\begin{equation}
\lambda \pi \int_0^{R_c^2} \exp\left({-\lambda\pi y \Big(1+\int_1^\infty
\frac{ T}{ T+{y^{-\frac{\alpha_0}{2}}}/{l_2(\sqrt{t y})}}\d t\Big)}\right) \d y
\label{equ:PcLB}
\end{equation}
which is essentially the probability
that the typical user being covered by a BS within distance $R_c$.
Further,
\begin{align*}
&\int_1^\infty \frac{ T}{ T+{y^{-\frac{\alpha_0}{2}}}/{l_2(\sqrt{t y})}}\d t \\
&= \int_1^{\frac{R_c^2}{y}} \frac{ T}{ T+{t^{\frac{\alpha_0}{2}}}}\d t
+ \int_{\frac{R_c^2}{y}}^\infty \frac{ T}{ T+ t^{\frac{\alpha_1}{2}} y^{\frac{\alpha_1-\alpha_0}{2}}/\eta}\d t \\
&\leq T \int_1^{\frac{R_c^2}{y}} t^{-\frac{\alpha_0}{2}}\d t
+ T \eta y^{-\frac{\alpha_1-\alpha_0}{2}} \int_{\frac{R_c^2}{y}}^\infty t^{-\frac{\alpha_1}{2}} \d t \\
&= -\frac{2 T}{2-\alpha_0} + \frac{2 T R_c^{2-\alpha_0} (\alpha_1-\alpha_0)}{(2-\alpha_0)(\alpha_1-2)} y^{\frac{\alpha_0}{2}-1}.
\end{align*}
This leads to a simplification of the lower bound in \eqref{equ:PcLB}.
After a change of variable $y\to x R_c^2$, we obtain $\Pc^\SIR_{l_2}(\lambda, T) \geq$
\begin{align*}
\lambda\pi R_c^2 \int_0^1 e^{-\lambda\pi R_c^2 \left(1-\frac{2 T}{2-\alpha_0}\right)x}
e^{-\lambda\pi R_c^2 \frac{2 T(\alpha_1-\alpha_0)}{(2-\alpha_0)(\alpha_1-2)}x^\frac{\alpha_0}{2}}\d x,
\end{align*}
which can be lower bounded for $ T\in(0,1-\frac{\alpha_0}{2})$ and $ T\in[1-\frac{\alpha_0}{2},\infty)$ separately.
If $ T\in(0,1-\frac{\alpha_0}{2})$, we have $1-\frac{2 T}{2-\alpha_0}>0$ and
\begin{align*}
\Pc^\SIR_{l_2}(\lambda, T)
&\pgeq{a} \lambda\pi R_c^2 \int_0^1 e^{-\lambda\pi R_c^2 \left(1+\frac{2 T}{\alpha_1-2}\right) x^\frac{\alpha_0}{2}} \d x \\
& = \delta_0 \frac{(\lambda\pi R_c^2)^{1-\delta_0}}{\left(1+\frac{2 T}{\alpha_1-2}\right)^{\delta_0}}
\gamma\left(\delta_0, \lambda \pi R_c^2 \Big(1+\frac{2 T}{\alpha_1-2}\Big)\right),
\end{align*}
where $\delta_0=2/\alpha_0$, (a) is due to the fact that $x \leq x^{\alpha_0/2}$ for $0<x<1$ and $\alpha_0\leq 2$
and $\gamma(t,z) = \int_0^z x^{t-1} e^{-x} \d x$ is the \emph{lower} incomplete gamma function.
If $ T\in[1-\frac{\alpha_0}{2},\infty)$, we have
\begin{multline*}
\Pc^\SIR_{l_2}(\lambda, T)
\geq \lambda\pi R_c^2 \int_0^1
e^{-\lambda\pi R_c^2 \frac{2 T(\alpha_1-\alpha_0)}{(2-\alpha_0)(\alpha_1-2)}x^\frac{\alpha_0}{2}}\d x \\
= \delta_0 \frac{(\lambda\pi R_c^2)^{1-\delta_0}}{\left(\frac{2 T(\alpha_1-\alpha_0)}{(2-\alpha_0)(\alpha_1-2)}\right)^{\delta_0}}
\gamma\left(\delta_0, \lambda \pi R_c^2 \Big(\frac{2 T(\alpha_1-\alpha_0)}{(2-\alpha_0)(\alpha_1-2)}\Big)\right).
\end{multline*}
Since
\begin{multline}
\lim_{\lambda\to\infty}\gamma\left(\delta_0, \lambda \pi R_c^2\big(1+\frac{2 T}{\alpha_1-2}\big)\right) \\
= \lim_{\lambda\to\infty} \gamma\left(\delta_0, \lambda \pi R_c^2 \big(\frac{2 T(\alpha_1-\alpha_0)}{(2-\alpha_0)(\alpha_1-2)}\big)\right) = \gamma(\delta_0),
\end{multline}
we have $\Pc^\SIR_{l_2} (\lambda, T) = \Omega(\lambda^{1-\delta_0})$ for all $ T>0$,
and thus $\tau_{l_2}(\lambda, T) = \Omega(\lambda^{2-\delta_0}) = \mu_{l_2}(\lambda, T)$.
\end{IEEEproof}
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(*
File: Fishburn_Impossibility.thy
Author: Manuel Eberl, TU München
Author: Christian Saile, TU München
Incompatibility of Pareto Optimality and Fishburn-Strategy-Proofness
*)
section \<open>Main impossibility result\<close>
theory Fishburn_Impossibility
imports
Social_Choice_Functions
begin
subsection \<open>Setting of the base case\<close>
text \<open>
Suppose we have an anonymous, Fishburn-strategyproof, and Pareto-efficient SCF
for three agents $A1$ to $A3$ and three alternatives $a$, $b$, and $c$. We will derive
a contradiction from this.
\<close>
locale fb_impossibility_3_3 =
strategyproof_anonymous_scf agents alts scf Fishb +
pareto_efficient_scf agents alts scf
for agents :: "'agent set" and alts :: "'alt set" and scf A1 A2 A3 a b c +
assumes agents_eq: "agents = {A1, A2, A3}"
assumes alts_eq: "alts = {a, b, c}"
assumes distinct_agents: "distinct [A1, A2, A3]"
assumes distinct_alts: "distinct [a, b, c]"
begin
text \<open>
We first give some simple rules that will allow us to break down the strategyproofness
and support conditions more easily later.
\<close>
lemma agents_neq [simp]: "A1 \<noteq> A2" "A2 \<noteq> A1" "A1 \<noteq> A3" "A3 \<noteq> A1" "A2 \<noteq> A3" "A3 \<noteq> A2"
using distinct_agents by auto
lemma alts_neq [simp]: "a \<noteq> b" "a \<noteq> c" "b \<noteq> c" "b \<noteq> a" "c \<noteq> a" "c \<noteq> b"
using distinct_alts by auto
lemma agent_in_agents [simp]: "A1 \<in> agents" "A2 \<in> agents" "A3 \<in> agents"
by (simp_all add: agents_eq)
lemma alt_in_alts [simp]: "a \<in> alts" "b \<in> alts" "c \<in> alts"
by (simp_all add: alts_eq)
lemma Bex_alts: "(\<exists>x\<in>alts. P x) \<longleftrightarrow> P a \<or> P b \<or> P c"
by (simp add: alts_eq)
lemma eval_pareto_loser_aux:
assumes "is_pref_profile R"
shows "x \<in> pareto_losers R \<longleftrightarrow> (\<exists>y\<in>{a,b,c}. x \<prec>[Pareto(R)] y)"
proof -
interpret pref_profile_wf agents alts R by fact
have *: "y \<in> {a,b,c}" if "x \<prec>[Pareto(R)] y" for y
using Pareto.strict_not_outside[of x y] that by (simp add: alts_eq)
show ?thesis by (auto simp: pareto_losers_def dest: *)
qed
lemma eval_Pareto:
assumes "is_pref_profile R"
shows "x \<prec>[Pareto(R)] y \<longleftrightarrow> (\<forall>i\<in>{A1,A2,A3}. x \<preceq>[R i] y) \<and> (\<exists>i\<in>{A1,A2,A3}. \<not>x \<succeq>[R i] y)"
proof -
interpret R: pref_profile_wf agents alts by fact
show ?thesis unfolding R.Pareto_strict_iff by (auto simp: strongly_preferred_def agents_eq)
qed
lemmas eval_pareto = eval_pareto_loser_aux eval_Pareto
lemma pareto_efficiency: "is_pref_profile R \<Longrightarrow> x \<in> pareto_losers R \<Longrightarrow> x \<notin> scf R"
using pareto_efficient[of R] by blast
lemma Ball_scf:
assumes "is_pref_profile R"
shows "(\<forall>x\<in>scf R. P x) \<longleftrightarrow> (a \<notin> scf R \<or> P a) \<and> (b \<notin> scf R \<or> P b) \<and> (c \<notin> scf R \<or> P c)"
using scf_alts[OF assms] unfolding alts_eq by blast
lemma Ball_scf_diff:
assumes "is_pref_profile R1" "is_pref_profile R2"
shows "(\<forall>x\<in>scf R1 - scf R2. P x) \<longleftrightarrow>
(a \<in> scf R2 \<or> a \<notin> scf R1 \<or> P a) \<and> (b \<in> scf R2 \<or> b \<notin> scf R1 \<or> P b) \<and>
(c \<in> scf R2 \<or> c \<notin> scf R1 \<or> P c)"
using assms[THEN scf_alts] unfolding alts_eq by blast
lemma scf_nonempty':
assumes "is_pref_profile R"
shows "\<exists>x\<in>alts. x \<in> scf R"
using scf_nonempty[OF assms] scf_alts[OF assms] by blast
subsection \<open>Definition of Preference Profiles and Fact Gathering\<close>
text \<open>
We now define the 21 preference profile that will lead to the impossibility result.
\<close>
preference_profile
agents: agents
alts: alts
where R1 = A1: [a, c], b A2: [a, c], b A3: b, c, a
and R2 = A1: c, [a, b] A2: b, c, a A3: c, b, a
and R3 = A1: [a, c], b A2: b, c, a A3: c, b, a
and R4 = A1: [a, c], b A2: a, b, c A3: b, c, a
and R5 = A1: c, [a, b] A2: a, b, c A3: b, c, a
and R6 = A1: b, [a, c] A2: c, [a, b] A3: b, c, a
and R7 = A1: [a, c], b A2: b, [a, c] A3: b, c, a
and R8 = A1: [b, c], a A2: a, [b, c] A3: a, c, b
and R9 = A1: [b, c], a A2: b, [a, c] A3: a, b, c
and R10 = A1: c, [a, b] A2: a, b, c A3: c, b, a
and R11 = A1: [a, c], b A2: a, b, c A3: c, b, a
and R12 = A1: c, [a, b] A2: b, a, c A3: c, b, a
and R13 = A1: [a, c], b A2: b, a, c A3: c, b, a
and R14 = A1: a, [b, c] A2: c, [a, b] A3: a, c, b
and R15 = A1: [b, c], a A2: a, [b, c] A3: a, b, c
and R16 = A1: [a, b], c A2: c, [a, b] A3: a, b, c
and R17 = A1: a, [b, c] A2: a, b, c A3: b, c, a
and R18 = A1: [a, c], b A2: b, [a, c] A3: b, a, c
and R19 = A1: a, [b, c] A2: c, [a, b] A3: a, b, c
and R20 = A1: b, [a, c] A2: a, b, c A3: b, a, c
and R21 = A1: [b, c], a A2: a, b, c A3: b, c, a
by (simp_all add: agents_eq alts_eq)
lemmas R_wfs =
R1.wf R2.wf R3.wf R4.wf R5.wf R6.wf R7.wf R8.wf R9.wf R10.wf R11.wf R12.wf R13.wf R14.wf R15.wf
R16.wf R17.wf R18.wf R19.wf R20.wf R21.wf
lemmas R_evals =
R1.eval R2.eval R3.eval R4.eval R5.eval R6.eval R7.eval R8.eval R9.eval R10.eval R11.eval R12.eval R13.eval
R14.eval R15.eval R16.eval R17.eval R18.eval R19.eval R20.eval R21.eval
lemmas nonemptiness = R_wfs [THEN scf_nonempty', unfolded Bex_alts]
text \<open>
We show the support conditions from Pareto efficiency
\<close>
lemma [simp]: "a \<notin> scf R1" by (rule pareto_efficiency) (simp_all add: eval_pareto R1.eval)
lemma [simp]: "a \<notin> scf R2" by (rule pareto_efficiency) (simp_all add: eval_pareto R2.eval)
lemma [simp]: "a \<notin> scf R3" by (rule pareto_efficiency) (simp_all add: eval_pareto R3.eval)
lemma [simp]: "a \<notin> scf R6" by (rule pareto_efficiency) (simp_all add: eval_pareto R6.eval)
lemma [simp]: "a \<notin> scf R7" by (rule pareto_efficiency) (simp_all add: eval_pareto R7.eval)
lemma [simp]: "b \<notin> scf R8" by (rule pareto_efficiency) (simp_all add: eval_pareto R8.eval)
lemma [simp]: "c \<notin> scf R9" by (rule pareto_efficiency) (simp_all add: eval_pareto R9.eval)
lemma [simp]: "a \<notin> scf R12" by (rule pareto_efficiency) (simp_all add: eval_pareto R12.eval)
lemma [simp]: "b \<notin> scf R14" by (rule pareto_efficiency) (simp_all add: eval_pareto R14.eval)
lemma [simp]: "c \<notin> scf R15" by (rule pareto_efficiency) (simp_all add: eval_pareto R15.eval)
lemma [simp]: "b \<notin> scf R16" by (rule pareto_efficiency) (simp_all add: eval_pareto R16.eval)
lemma [simp]: "c \<notin> scf R17" by (rule pareto_efficiency) (simp_all add: eval_pareto R17.eval)
lemma [simp]: "c \<notin> scf R18" by (rule pareto_efficiency) (simp_all add: eval_pareto R18.eval)
lemma [simp]: "b \<notin> scf R19" by (rule pareto_efficiency) (simp_all add: eval_pareto R19.eval)
lemma [simp]: "c \<notin> scf R20" by (rule pareto_efficiency) (simp_all add: eval_pareto R20.eval)
lemma [simp]: "c \<notin> scf R21" by (rule pareto_efficiency) (simp_all add: eval_pareto R21.eval)
text \<open>
We derive the strategyproofness conditions:
\<close>
lemma s41: "\<not> scf R4 \<succ>[Fishb(R1 A2)] scf R1"
by (intro strategyproof'[where j = A2]) (simp_all add: R4.eval R1.eval)
lemma s32: "\<not> scf R3 \<succ>[Fishb(R2 A1)] scf R2"
by (intro strategyproof'[where j = A1]) (simp_all add: R3.eval R2.eval)
lemma s122: "\<not> scf R12 \<succ>[Fishb(R2 A2)] scf R2"
by (intro strategyproof'[where j = A2]) (simp_all add: R12.eval R2.eval)
lemma s133: "\<not> scf R13 \<succ>[Fishb(R3 A2)] scf R3"
by (intro strategyproof'[where j = A2]) (simp_all add: R13.eval R3.eval)
lemma s102: "\<not> scf R10 \<succ>[Fishb(R2 A2)] scf R2"
by (intro strategyproof'[where j = A2]) (simp_all add: R10.eval R2.eval)
lemma s13: "\<not> scf R1 \<succ>[Fishb(R3 A3)] scf R3"
by (intro strategyproof'[where j = A2]) (simp_all add: R1.eval R3.eval)
lemma s54: "\<not> scf R5 \<succ>[Fishb(R4 A1)] scf R4"
by (intro strategyproof'[where j = A1]) (simp_all add: R5.eval R4.eval)
lemma s174: "\<not> scf R17 \<succ>[Fishb(R4 A1)] scf R4"
by (intro strategyproof'[where j = A1]) (simp_all add: R17.eval R4.eval)
lemma s74: "\<not> scf R7 \<succ>[Fishb(R4 A2)] scf R4"
by (intro strategyproof'[where j = A2]) (simp_all add: R7.eval R4.eval)
lemma s114: "\<not> scf R11 \<succ>[Fishb(R4 A3)] scf R4"
by (intro strategyproof'[where j = A3]) (simp_all add: R11.eval R4.eval)
lemma s45: "\<not> scf R4 \<succ>[Fishb(R5 A1)] scf R5"
by (intro strategyproof'[where j = A1]) (simp_all add: R4.eval R5.eval)
lemma s65: "\<not> scf R6 \<succ>[Fishb(R5 A2)] scf R5"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R5.eval R6.eval)
lemma s105: "\<not> scf R10 \<succ>[Fishb(R5 A3)] scf R5"
by (intro strategyproof'[where j = A3]) (simp_all add: R10.eval R5.eval)
lemma s67: "\<not> scf R6 \<succ>[Fishb(R7 A1)] scf R7"
by (intro strategyproof'[where j = A2]) (simp_all add: insert_commute R6.eval R7.eval)
lemma s187: "\<not> scf R18 \<succ>[Fishb(R7 A3)] scf R7"
by (intro strategyproof'[where j = A3]) (simp_all add: insert_commute R7.eval R18.eval)
lemma s219: "\<not> scf R21 \<succ>[Fishb(R9 A2)] scf R9"
by (intro strategyproof'[where j = A3]) (simp_all add: insert_commute R9.eval R21.eval)
lemma s1011: "\<not> scf R10 \<succ>[Fishb(R11 A1)] scf R11"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R10.eval R11.eval)
lemma s1012: "\<not> scf R10 \<succ>[Fishb(R12 A2)] scf R12"
by (intro strategyproof'[where j = A2]) (simp_all add: insert_commute R10.eval R12.eval)
lemma s1213: "\<not> scf R12 \<succ>[Fishb(R13 A1)] scf R13"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R12.eval R13.eval)
lemma s1113: "\<not> scf R11 \<succ>[Fishb(R13 A2)] scf R13"
by (intro strategyproof'[where j = A2]) (simp_all add: insert_commute R11.eval R13.eval)
lemma s1813: "\<not> scf R18 \<succ>[Fishb(R13 A3)] scf R13"
by (intro strategyproof'[where j = A2]) (simp_all add: insert_commute R18.eval R13.eval)
lemma s814: "\<not> scf R8 \<succ>[Fishb(R14 A2)] scf R14"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R8.eval R14.eval)
lemma s1914: "\<not> scf R19 \<succ>[Fishb(R14 A3)] scf R14"
by (intro strategyproof'[where j = A3]) (simp_all add: insert_commute R19.eval R14.eval)
lemma s1715: "\<not> scf R17 \<succ>[Fishb(R15 A1)] scf R15"
by (intro strategyproof'[where j = A3]) (simp_all add: insert_commute R17.eval R15.eval)
lemma s815: "\<not> scf R8 \<succ>[Fishb(R15 A3)] scf R15"
by (intro strategyproof'[where j = A3]) (simp_all add: insert_commute R8.eval R15.eval)
lemma s516: "\<not> scf R5 \<succ>[Fishb(R16 A1)] scf R16"
by (intro strategyproof'[where j = A3]) (simp_all add: insert_commute R5.eval R16.eval)
lemma s517: "\<not> scf R5 \<succ>[Fishb(R17 A1)] scf R17"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R5.eval R17.eval)
lemma s1619: "\<not> scf R16 \<succ>[Fishb(R19 A1)] scf R19"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R16.eval R19.eval)
lemma s1820: "\<not> scf R18 \<succ>[Fishb(R20 A2)] scf R20"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R18.eval R20.eval)
lemma s920: "\<not> scf R9 \<succ>[Fishb(R20 A3)] scf R20"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R20.eval R9.eval)
lemma s521: "\<not> scf R5 \<succ>[Fishb(R21 A1)] scf R21"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R21.eval R5.eval)
lemma s421: "\<not> scf R4 \<succ>[Fishb(R21 A1)] scf R21"
by (intro strategyproof'[where j = A1]) (simp_all add: insert_commute R21.eval R4.eval)
lemmas sp = s41 s32 s122 s102 s133 s13 s54 s174 s54 s74 s114 s45 s65 s105 s67
s187 s219 s1011 s1012 s1213 s1113 s1813 s814 s1914 s1715 s815 s516
s517 s1619 s1820 s920 s521 s421
text \<open>
We now use the simplifier to break down the strategyproofness conditions into SAT formulae.
This takes a few seconds, so we use some low-level ML code to at least do the simplification
in parallel.
\<close>
local_setup \<open>fn lthy =>
let
val lthy' = lthy addsimps @{thms Fishb_strict_iff Ball_scf Ball_scf_diff R_evals}
val thms = Par_List.map (Simplifier.asm_full_simplify lthy') @{thms sp}
in
Local_Theory.notes [((@{binding sp'}, []), [(thms, [])])] lthy |> snd
end
\<close>
text \<open>
We show that the strategyproofness conditions, the non-emptiness conditions (i.\,e.\ every SCF
must return at least one winner), and the efficiency conditions are not satisfiable together,
which means that the SCF whose existence we assumed simply cannot exist.
\<close>
theorem absurd: False
using sp' and nonemptiness [simplified] by satx
end
subsection \<open>Lifting to more than 3 agents and alternatives\<close>
text \<open>
We now employ the standard lifting argument outlined before to lift this impossibility
from 3 agents and alternatives to any setting with at least 3 agents and alternatives.
\<close>
locale fb_impossibility =
strategyproof_anonymous_scf agents alts scf Fishb +
pareto_efficient_scf agents alts scf
for agents :: "'agent set" and alts :: "'alt set" and scf +
assumes card_agents_ge: "card agents \<ge> 3"
and card_alts_ge: "card alts \<ge> 3"
begin
(* TODO: Move? *)
lemma finite_list':
assumes "finite A"
obtains xs where "A = set xs" "distinct xs" "length xs = card A"
proof -
from assms obtain xs where "set xs = A" using finite_list by blast
thus ?thesis using distinct_card[of "remdups xs"]
by (intro that[of "remdups xs"]) simp_all
qed
lemma finite_list_subset:
assumes "finite A" "card A \<ge> n"
obtains xs where "set xs \<subseteq> A" "distinct xs" "length xs = n"
proof -
obtain xs where "A = set xs" "distinct xs" "length xs = card A"
using finite_list'[OF assms(1)] by blast
with assms show ?thesis
by (intro that[of "take n xs"]) (simp_all add: set_take_subset)
qed
lemma card_ge_3E:
assumes "finite A" "card A \<ge> 3"
obtains a b c where "distinct [a,b,c]" "{a,b,c} \<subseteq> A"
proof -
from finite_list_subset[OF assms]
obtain xs where xs: "set xs \<subseteq> A" "distinct xs" "length xs = 3" by auto
then obtain a b c where "xs = [a, b, c]"
by (auto simp: eval_nat_numeral length_Suc_conv)
with xs show ?thesis by (intro that[of a b c]) simp_all
qed
theorem absurd: False
proof -
from card_ge_3E[OF finite_agents card_agents_ge] obtain A1 A2 A3
where agents: "distinct [A1, A2, A3]" "{A1, A2, A3} \<subseteq> agents" .
let ?agents' = "{A1, A2, A3}"
from card_ge_3E[OF finite_alts card_alts_ge]
obtain a b c where alts: "distinct [a, b, c]" "{a, b, c} \<subseteq> alts" .
let ?alts' = "{a, b, c}"
interpret scf_lowering_anonymous agents alts scf ?agents' ?alts'
by standard (use agents alts in auto)
interpret liftable_set_extension ?alts' alts Fishb
by (intro Fishburn_liftable alts)
interpret scf_lowering_strategyproof agents alts scf ?agents' ?alts' Fishb ..
interpret fb_impossibility_3_3 ?agents' ?alts' lowered A1 A2 A3 a b c
by standard (use agents alts in simp_all)
from absurd show False .
qed
end
end
| 211,463
|
\begin{document}
\maketitle
\begin{abstract}
In this article, we prove that for a finite quiver $Q$ the equivalence class of a potential up to formal change of variables of the complete path algebra $\wh{\CC Q}$, is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming it is finite dimensional. This result can be viewed as a noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra is finite dimensional. These results have several interesting applications. First, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential. The contraction algebras associated to three dimensional flopping contractions provide a class of examples of finite dimensional Jacobi algebras. As the second application, we prove that the contraction algebra together with the class in its 0-th Hochschild homology represented by the potential determines the relative singularity category defined by Kalck and Yang and the classical singularity category of Buchweitz and Orlov. This result provides an evidence for Donovan-Wemyss conjecture, which asserts that the contraction algebra is a complete invariant for the underlying singularities of three dimensional flopping contractions.
\end{abstract}
\section{Introduction}
\newtheorem{maintheorem}{\bf{Theorem}}
\renewcommand{\themaintheorem}{\Alph{maintheorem}}
\newtheorem{mainconjecture}[maintheorem]{\bf{Conjecture}}
\renewcommand{\themainconjecture}{}
\subsection{Jacobi algebras and Calabi-Yau algebras}
The systematic study of Calabi-Yau algebras (see definitions in Section \ref{sec:CY-Ginzdg}) was initiated by Ginzburg in his milestone paper \cite{Ginz}. In the last decade, it finds enormous applications in algebraic geometry, representation theory and topology. Fix a field $k$. Ginzburg conjectured that all 3-dimensional Calabi-Yau algebras should arise as Jacobi algebras, where a Jacobi algebra $\Lm(Q,\Phi)$ is defined to be the quotient algebra of the path algebra $kQ$ of a finite quiver $Q$ by a two-sided ideal generated by the cyclic derivatives of a potential $\Phi\in kQ_{\cy}:=kQ/[kQ,kQ]$ (see Definition \ref{superpotential-free}). This conjecture has been confirmed by Van den Bergh for all complete Calabi-Yau algebras \cite{VdB15}, while the general case has been disproved by Ben Davison \cite{Da}.
In this article, we mostly work on Jacobi algebras $\wh{\Lm}(Q,\Phi)$ of a complete path algebra $\wh{kQ}$.
On the other hand, not all Jacobi algebras satisfy the (smooth) Calabi-Yau property (see definition in Section \ref{sec:CY-Ginzdg}). The contraction algebras associated to 3-dimensional flopping contractions, defined by Donovan and Wemyss \cite{DW13}, give such examples. For every finite quiver $Q$ and potential $\Phi$, Ginzburg defined a natural complete dg-algebra $\hD(Q,\Phi)$ associating to it, where $H^0(\hD(Q,\Phi))\cong \wh{\Lm}(Q,\Phi)$. Keller proved that $\hD(Q,\Phi)$ always satisfies the Calabi-Yau property, regardless of whether $\wh{\Lm}(Q,\Phi)$ is smooth or not \cite{KV09}. A geometric proof for this statement was given by Van den Bergh in the Appendix of \cite{KV09}. The result of Keller and Van den Bergh suggests that we should view $\hD(Q,\Phi)$ as a ``derived thickening" of the the Jacobi algebra $\wh{\Lm}(Q,\Phi)$. This point of view, already contained in Ginzburg's original paper, is motivated by Donaldson-Thomas theory. In DT theory, the moduli space of stable sheaves on Calabi-Yau threefold is locally isomorphic to a critical set of a function. Indeed, the moduli space admits an enhancement as a derived scheme (stack) of virtual dimension zero in the sense of \cite{BCFHR}. Although the moduli space itself is highly singular, the derived moduli space is a better behaved object. It is well-known that the derived structure is determined by the moduli space together with its obstruction theory.
\subsection{Main results}
If we make a heuristic comparison between the Jacobi algebra and the moduli space of stable sheaves in DT theory, then it is natural to ask the following question:{\bf{ to what extent the complete Ginzburg dg-algebra is determined by the algebra structure on the Jacobi algebra?}} In this paper, we give an answer to this question.
\begin{maintheorem}(Theorem \ref{rigidity-Ginzburg})\label{mainthm-rigidity}
Fix a finite quiver $Q$. Let $\Phi,\Psi\in \wh{\CC Q}_{\ol{\cy}}$ be two potentials of order $\geq 3$, such that the Jacobi algebras
$\wh{\Lm}(Q,\Phi)$ and $\wh{\Lm}(Q,\Psi)$ are both finite dimensional. Assume there is an $\CC Q_0$-algebra isomorphism $\gamma: \wh{\Lm}(Q,\Phi)\to\wh{\Lm}(Q,\Psi)$ so that
$\gamma_*([\Phi])=[\Psi]$. Then there exists a dg-$\CC Q_0$-algebra isomorphism
\[
\xymatrix{
\Gamma: \hD(Q,\Phi)\ar[r]^{\cong} &\hD(Q,\Psi)
}
\]
such that $\Gamma(t_i)=t_i$ for all nodes $i$ of $Q$.
\end{maintheorem}
As an immediate corollary, we show that the (topological) \emph{generalized cluster category} is determined by the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential (Corollary \ref{clustercat}).
The above theorem is a consequence of a Mather-Yau type theorem for finite dimensional Jacobi algebras. In singularity theory, Mather and Yau proved that the germ of a holomorphic function $f$ on $(\CC^n,0)$ is determined by its Tjurina algebra $\cO_{\CC^n,0}/(f, \frac{\partial f}{\partial x_1},\ldots, \frac{\partial f}{\partial x_1})$(see Section 1 of \cite{MY}). In \cite{GH85}, Gaffney and Hauser proved a vast generalization of the Mather-Yau theorem, where in particular the isolatedness assumption is removed. Recently, Greuel and Pham generalizes Mather-Yau theorem to the case of formal power series \cite{GP}. They also prove a weaker version of the theorem for the positive characteristic case. Our main theorem is a Mather-Yau type result on complete path algebra.
\begin{maintheorem}(Theorem \ref{ncMY})\label{mainthm-MY}
Fix a finite quiver $Q$. Let $\Phi,\Psi\in \wh{\CC Q}_{\ol{\cy}}$ be two potentials of order $\geq 3$, such that the Jacobi algebras $\wh{\Lm}(Q,\Phi)$ and $\wh{\Lm}(Q,\Psi)$ are both finite dimensional. Then the following two statements are equivalent:
\begin{enumerate}
\item[(1)] There is an $\CC Q_0$-algebra isomorphism $\gamma: \wh{\Lm}(Q,\Phi)\cong\wh{\Lm}(Q,\Psi)$ so that $\gamma_*([\Phi])=[\Psi]$.
\item[(2)] $\Phi$ and $\Psi$ are right equivalent.
\end{enumerate}
\end{maintheorem}
In a series of classical papers \cite{Ma68}\cite{Ma69}, Mather studied various equivalence relations on germ of smooth and analytic functions. He has observed that the tangent spaces of these equivalence classes can be interpreted as Jacobi ideal and its variations. Moreover, Mather has found infinitesimal criteria of checking whether two germs lie in the same equivalence class. The classical Mather-Yau theorem can be proved using it.
In noncommutative differential calculus, the notion of cyclic derivative was first defined by Rota, Sagan and Stein \cite{RRS}. Ginzburg used cyclic derivatives of the potential to define the Jacobi algebra for noncommutative algebras.
The main difficulty to implement Mather's technique on noncommutative functions is that the tangent space of a right equivalence class of functions, which is a subspace of the push forward of the space of derivations, admits no module structure. So in noncommutative world, there is no a priori relationship between the tangent space of right equivalence class and the Jacobi ideal. We overcome this problem by observing that both space of derivations and space of cyclic derivations are quotients of the space of double derivations, which does admit a bimodule structure. Moreover, the actions of derivations and cyclic derivations on the space of potentials have the identical orbits. On the other hand, the group of automorphisms is infinite dimensional. To reduce it to a finite dimensional problem, we establish a finite determinacy result for noncommutative formal series, which is of independent interest.
\begin{maintheorem}(Theorem 3.16)
Let $Q$ be a finite quiver and $\Phi\in \wh{\CC Q}_{\ol{\cy}}$. If the Jacobi algebra $\wh{\Lm}(Q,\Phi)$ is finite dimensional then $\Phi$ is finitely determined. More precisely, if
$\wh{\rJ}(Q,\Phi) \supseteq \wh{\fm}^r$ for some integer $r\geq 0$ then $\Phi$ is $(r+1)$-determined. In particular, any potential with finite dimensional Jacobi algebra is right equivalent to a potential in $\CC Q_{\cy}$.
\end{maintheorem}
The second difficulty is that in the formal world not all derivations come from tangents of automorphisms. To be more precise, we are not allowed to take translation of origin as in the smooth or analytic world. We resolve this problem by proving a bootstrapping lemma on Jacobi ideals (see Proposition \ref{Bootstrapping}).
In Section \ref{sec:Pre}, we collect basic notations and results on noncommutative differential calculus. In particular, the definition of Jacobi algebra for path algebra (resp. complete path algebra) are presented in Definition \ref{superpotential-free} (resp. in Definition \ref{superpotential-complete}). Section \ref{sec:Proof} is devoted to the proof of Theorem B. In Section \ref{sec:app}, we recall the definitions of complete Ginzburg dg-algebras and Calabi-Yau algebras, and prove Theorem A.
\subsection{Applications to 3-dimensional birational geometry}
In \cite{DW13}, Donovan and Wemyss associated to every minimal model of complete local Gorenstein 3-fold a finite dimensional Jacobi algebra, called the \emph{contraction algebra}. In \cite{Wem14}, Wemyss proposed an algorithm to study minimal model program of 3-folds via contraction algebras and their mutations, called \emph{homological MMP}. A fundamental question in homological MMP is the following conjecture of Donovan and Wemyss.
\begin{mainconjecture}(\emph{\cite[Conjecture~1.4]{DW13}})\label{conj-dw}
Suppose that $Y\to X$ and $Y^\prime\to X^\prime$ are 3-dimensional flopping contractions of smooth quasi-projective 3-folds $Y$ and $Y^\prime$, to threefolds $X$ and $X^\prime$ with isolated singular points $p$ and $p^\prime$ respectively. To these, associate the contraction algebras $\Lm$ and $\Lm^\p$. Then the completions of stalks at $p$ and $p^\prime$ are isomorphic if and
only if $\Lm\cong \Lm^\p$ as algebras.
\end{mainconjecture}
It is known that contraction algebra recovers many numerical invariants for flopping contraction, including the length, the width and the Gopkakumar-Vafa invariants (see \cite{DW13} \cite{HT16}).
In the work of Toda and the first author \cite{HT16}, a natural $\ZZ/2$-graded $A_\infty$-enhancement has been constructed for the contraction algebra. In \cite{Hua17}, the first author show that the completion of the singularity is determined by this $A_\infty$-enhancement. Although the $\ZZ/2$-graded $A_\infty$-enhancement contains sufficient information, it can only be constructed in the $\ZZ/2$-graded dg-category of matrix factorizations. This is against the philosophy of homological MMP. First, for flips such $\ZZ/2$-graded category does not exist while contraction algebra can still be defined. Second, to take care of the mutations it is better to work on the derived category of the resolutions instead of the derived category of singularities.
Using the noncommutative Mather-Yau theorem, we prove that the relative and classical singularity categories can be recovered from the contraction algebra together with the class represented by the potential, which is an evidence of the validity of Conjecture D.
\begin{maintheorem}(Theorem \ref{contractionDsg})\label{mainthm-contraction}
Let $f: Y\to X=\Spec R$ and $f^\p: Y^\p\to X^\p=\Spec R^\p$ be two 3-dimensional formal flopping contractions such that the exceptional fibers of $f$ and $f^\p$ have the same number of irreducible components, with the associated contraction algebras $\Lm(Y,\Phi)$ and $\Lm(Y^\p,\Psi)$, respectively. Assume there is a $l$-algebra isomorphism $\gamma: \Lm(Y,\Phi)\to\Lm(Y^\p,\Psi)$ so that $\gamma_*([\Phi])=[\Psi]$. Then there are triangle equivalences
\[
\Delta_R(Y)\cong \Delta_{R^\p}(Y^\p) \quad \text{and} \quad \D_{sg}(R) \cong \D_{sg}(R^\p)
\]
between the relative singularity categories and the classical singularity categories respectively.
\end{maintheorem}
\paragraph{Acknowledgments.} The first author wishes to thank Michael Wemyss for generously sharing his insights on contraction algebras and in particular for explaining the approach to Conjecture \ref{conj-dw} via relative singularity categories during a visit to Glasgow in May 2017. Both authors want to thank Joe Karmazyn for inspiring discussion on relative singularity categories and the relations to Ginzburg dg-algebras. We also benefit from the valuable communication with Bernhard Keller, Gert-Martin Greuel, Martin Kalck and Sasha Polishchuk. The research of the first author was supported by RGC General Research Fund no. 17330316, no. 17308017 and Early Career grant no. 27300214. The research of the second author was supported by NSFC grant no. 11601480 and RGC Early Career grant no. 27300214.
\section{Preliminaries}\label{sec:Pre}
In this section, we collect basic notations and terminologies that are of concern. In particular, we recall the definition of Jacobi algebras of path algebras and of complete path algebras of finte quivers. In addition, for completeness and reader's convenience, some relevant well-known facts on complete path algebras are presented in full details in our own notations.
Throughout, we fix a base commutative ring $k$ with unit and a finite dimensional separable $k$-algebra $l$. Unadorned tensor products are over $k$.
\subsection{Basic terminologies}
A $l$-algebra $A$ is a $k$-algebra $A$ equipped with a $k$-algebra monomorphism $\eta: l\to A$. Note that the image of $l$ is in general not central even when $l$ is commutative. We are mainly interested in the case when $l=k e_1+\ldots +k e_n$ for central orthogonal idempotents $e_i$.
Let $U$ be a right $l$-module and $V$ be left $l$-module. The tensor product $U\ot_l V$ is the quotient of $U\ot V$ by the $k$-submodule generated $xe_i\ot y-x\ot e_i y$ for any $x\in U$, $y\in V$ and $i=1,\ldots, n$. If $U$ and $V$ are $l$-bimodules, we call a map $f:M\to N$ \emph{$l$-linear} if it is a morphism of $l$-bimodules.
Let $A$ be an associative ($l$-)algebra. Set $A_{ij}:=e_i A e_j$. The multiplication map $\mu:A\ot A\to A$ is equivalent with its restriction
\[
\mu: A_{ij}\ot A_{jk}\to A_{ik}~~~~\text{for all } i,j,k=1,\ldots,n.
\]
In other words, it factors through the map $A\ot_l A\to A$, which will be denoted by the same symbol $\mu$ by an abuse of notations.
The tensor product space $A\ot A$ carries two commuting $A$-bimodule structures, called the \emph{outer} (resp. \emph{inner}) bimodule structure, defined by
\[
a(b^\p\ot b^{\p\p})c:=ab^\p\ot b^{\p\p}c~~~~~(~\text{resp.} \quad a*(b^\p\ot b^{\p\p})*c:=b^\p c\ot ab^{\p\p} ~).
\]
We denote by $A\stackrel{out}{\ot}A$ (resp $A\stackrel{in}{\ot} A$) the bimodule with respect to the outer (resp. inner) structure. Because $l$ is a sub algebra of $A$, they are in particular $l$-bimodules.
The flip map $\tau:A\stackrel{out}{\ot}A \to A\stackrel{in}{\ot}A$, defined by $a^\p\ot a^\pp$ to $a^\pp\ot a^\p$, is an isomorphism of $A$-bimodules, $\mu:A\stackrel{out}{\ot}A \to A$ is a homomorphism of $A$-bimodules but in general $\mu: A\stackrel{in}{\ot}A \to A$ is not. Unless otherwise stated, we simply view $A\ot A$ as the bimodule $A\stackrel{out}{\ot}A$. Also, the category of $A$-bimodules is denoted by $A\Bimod$. The multiplication map $\mu$ is $l$-linear.
A \emph{(relative) $l$-derivation of $A$ in an $A$-bimodule $M$} is defined to be a $l$-linear map $\delta:A\to M$ satisfies the Leibniz rule, that is $\delta(ab) =\delta(a)b+ a\delta(b)$ for all $a,b\in A$. If $e$ is an idempotent, it follows that
\[
\delta(e)=\delta(e^2)=\delta(e)e+e\delta(e)=2\delta(e^2)=2\delta(e).
\] Therefore $\delta(l)=0$ and $\delta(A_{ij})\subset M_{ij}$ with $M_{ij}:=e_i M e_j$.
Denote by $\der_l(A,M)$ the set of all $l$-derivations of $A$ in $M$, which naturally carries a $k$-module structure (but not a $l$-module structure). The elements of
\[
\der_l ( A): =\der_l(A,A) ~~~~~ (~\text{resp.} \quad \dder_l (A):= \der_l(A,A\ot A)~ )
\]
are called the \emph{$l$-derivations of $A$} (resp. \emph{double $l$-derivations of $A$}). For a general double derivation $\delta\in \dder_l A$ and $f\in A$, we shall write in Sweedler's notation that
\begin{align}\label{Sweedler}
\delta(f)= \delta(f)^\p\ot\delta(f)^\pp.
\end{align}
The inner bimodule structure of $A\ot A$ naturally yields a bimodule structure on $\dder_l(A)$. In contrast, $\der_l (A)$ doesn't have canonical left nor right $A$-module structures in general (not even a $l$-module structure). The multiplication map $\mu$ induces a $k$-linear map $\mu\circ-: \dder_l(A)\to \der_l(A)$ given by $\delta\mapsto \mu\circ \delta$. We refer to $\mu\circ \delta$ the $l$-derivation corresponding to the double $l$-derivation $\delta$.
Let us put on the space of $k$-module endomorphisms $\Hom_k(A,A)$ the $A$-bimodule structure defined by
\[
a_1fa_2: b\mapsto a_1 f(b) a_2, ~~~~~ f\in \Hom_k(A,A), ~ a_1, a_2, b\in A.
\]
Though the map $\dder_l(A) \xrightarrow{\mu\circ- } \Hom_k(A,A)$ doesn't preserves bimodule structures, the map
\[
\mu\circ \tau\circ- : \dder_l (A) \to \Hom_k(A,A)
\]
is clearly a homomorphism of $A$-bimodules. Denote by $\cder_l (A)$ the image of this map and call its elements \emph{cyclic $l$-derivations of $A$}. For a double $l$-derivation $\delta\in \dder_l(A)$, we shall refer to $\mu\circ \tau\circ \delta$ the cyclic $l$-derivation corresponding to $\delta$. Note that by definition $\cder_l(A)$ is an $A$-sub-bimodule of $\Hom_k(A,A)$, and hence is itself an $A$-bimodule.
The $A$-bimodule $\Omega_{A|l}$, of \emph{noncommutative (relative) K\"ahler $l$-differentials of $A$}, is defined to be the kernel of the multiplication map $\mu: A \ot_l A\to A$. The exterior differentiation map
\[
d: A\to \Omega_{A|l}, \quad a\mapsto d a:=1\ot a-a\ot 1
\]
is then a $l$-derivation of $A$ in the bimodule $\Omega_{A|l}$. There is a canonical isomorphism of $k$-modules
\[
\Hom_{A-{\rm Bimod}}(\Omega_{A|l},M) \xrightarrow{\cong} \der_l(A,M), \quad f\mapsto f\circ d.
\]
In another word, the exterior map $d:A\to \Omega_{A|l}$ is a universal $l$-derivation of $A$.
We collect some trivial properties on (cyclic) derivations in the following lemma.
\begin{lemma}\label{cycprop}
Let $A$ be a $l$-algebra and fix an element $\Phi\in A_\cy:=A/[A,A]$. Let $\pi:A\to A_\cy$ be the canonical projection and $\phi\in A$ a representative of $\Phi$.
\begin{enumerate}
\item[$(1)$] $\xi([A,A]) \subseteq [A,A]$ for every $\xi\in \der_l(A)$. Consequently, the assignment $\der_l(A) \ni \xi \mapsto \pi(\xi (\phi))$ only depends on $\Phi$ and defines a $k$-linear map $\Phi_{\#}: \der_l (A) \to A_\cy $.
\item[$(2)$] $D([A,A])=0$ for every $D\in \cder_l (A)$. Consequently, the assignment $\cder_l(A)\ni D \mapsto D(\phi)$ only depends on $\Phi$ and defines an $A$-bimodule morphism $\Phi_*: \cder_l (A)\to A$.
\item[$(3)$] We have the following commutative diagram:
\begin{align}
\xymatrix{
\dder_l (A) \ar@{->>}[r]^-{\mu\circ \tau\circ-} \ar[d]^{\mu\circ-} & \cder_l( A)\ar[r]^-{\Phi_*} & A\ar[d]^{\pi} \\
\der_l ( A) \ar[rr]^{\Phi_{\#}} & & A_\cy .
} \label{derivation}
\end{align}
Consequently, if $\dder_l(A) \xrightarrow{\mu\circ-} \der_l(A)$ is surjective then $\im (\Phi_{\#}) = \im (\pi\circ \Phi_*)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that $A_\cy$ admits only a $k$-module structure, but not a $l$-module structure.
Part (1) of the lemma and the second statement of part (2) are clear. To see the first statement of part (2), let $\delta\in \dder_l(A)$ be a double $l$-derivation and $f,g\in A$. Then
\begin{align*}
(\mu\circ \tau\circ \delta)([f,g])=\delta(f)^\pp g \delta(f)^\p+\delta(g)^\pp f\delta(g)^\p-\delta(g)^\pp f\delta(g)^\p -\delta(f)^\pp g\delta(f)^\p=0.
\end{align*}
From this we know $D([A,A])=0$ for every $D\in \cder_l(A)$. Part (3) of the lemma is a direct consequence of the fact that $(\mu\delta)(f)-(\mu\tau\delta)(f)\in [A,A]$ for every $\delta\in \dder_l(A)$ and $f\in A$.
\end{proof}
\subsection{Jacobi algebras of quivers with potentials}
Let $Q$ be an arbitrary finite quiver. Denote by $Q_0$ and $Q_1$ the sets of nodes and arrows of the quiver. Define $s,t: Q_1\to Q_0$ to be the \emph{source} and \emph{target} maps. Denote by $kQ$ the path algebra of $Q$ with respect to the path concatenation defined as follows. Let $a$ be a path from $i$ to $j$ and $b$ be a path from $j$ to $k$. Then $ab$ is a path from $i$ to $k$ defined by $a$ followed by $b$. Denote by $e_i$ the empty path at the node $i$. Denote by $Q_1^{(ij)}$ the set of arrows with source $i$ and target $j$.
Denote by $\fm$ the two sided ideal generated by all arrows and by $l$ the subalgebra $kQ_0$ which is canonically isomorphic to the quotient algebra $kQ/\fm$. As a consequence, $kQ$ is an \emph{augmented} $l$-algebra. An element of $kQ$ is a finite $k$-linear combination of paths. The \emph{length} or \emph{degree} $|p|$ of a path $p$ is defined in the obvious way. The ideal $\fm^r$ consists of finite sum of paths of length $\geq r$.
It is easy to check that $l$-derivations of $kQ$ in an $kQ$-bimodule are uniquely determined by their value at all $a\in Q_1$. Thus $kQ$ has double derivations $\frac{\partial}{\partial a}:kQ\to kQ\ot kQ$ given by
\begin{align}\label{Di}
\frac{\partial}{\partial a}( b) = \delta_{a, b} ~e_{s(a)}\ot e_{t(a)}.
\end{align}
The $kQ$-bimodule $\dder_l (kQ)$ is generated by $\frac{\partial}{\partial a}$, $a\in Q_1$, and the map $\mu\circ-: \dder_l (kQ)\to \der_l (kQ)$ is surjective.
By \eqref{Di}, $\mu\circ \frac{\partial}{\partial a}=0$ for any arrow $a$ such that $s(a)\neq t(a)$.
Denote by $D_{a}$ the cyclic $l$-derivation corresponding to $\frac{\partial~}{\partial a}$, and so it take a path $p$ to
\[
D_{a}(p)=\sum\nolimits_{p=uav}vu
\] where $u,v$ are paths. By (2) of Lemma \ref{cycprop}, $D_a([kQ,kQ])=0$ for any $a\in Q_1$. In particular, if $p$ is a path such that $s(p)\neq t(p)$ then $D_a(p)=0$ for any $a$.
This cyclic derivation in the special case of $n$-loop quiver was first discovered by Rota, Sagan and Stein \cite{RRS}. Clearly, the $kQ$-bimodule $\cder_l (kQ)$ is generated by $D_a$, $a\in Q_1$. In addition, the $kQ$-bimodule $\Omega_{kQ|l}$ is generated by $\d a$, $a\in Q_1$.
\begin{definition}\label{superpotential-free}
Elements of $kQ_\cy:=kQ/[kQ,kQ]$ are called \emph{potentials} of $kQ$. Given a potential $\Phi\in kQ_\cy$, the two sided ideal
\[
\rJ(Q,\Phi):=\im(\Phi_*)
\]
is called the \emph{Jacobi ideal} of $\Phi$, where $\Phi_*:\cder_l(kQ)\to kQ$ is the $kQ$-bimodule homomorphism constructed in Lemma \ref{cycprop} (2). The associative algebra
\[
\Lm(Q, \Phi) := kQ/\rJ(Q,\Phi)
\]
is called the \emph{Jacobi algebra} of $\Phi$. Like $kQ$, it is an augmented $l$-algebra.
\end{definition}
The above definition of Jacobi algebras coincides with the conventional one
because $\rJ(Q,\Phi)$ is generated by $\Phi_*(D_a)$, $a\in Q_1$, as an ideal of $kQ$. Jacobi algebras of path algebras of quivers are key objects of interest in literatures.
\subsection{Complete Jacobi algebras}
Given a finite quiver $Q$, denote by $\wh{kQ}$ the completion of $kQ$ with respect to the two sided ideal $\fm\subset kQ$.
Elements of $\wh{kQ}$ are (infinite) formal series $\sum_{w} a_w w$, where $w$ runs over all paths (of finite length) and $a_w\in k$. Note that $\wh{kQ}$ contains the $kQ$ as a $l$-subalgebra. Let $\wh{\fm} \subseteq \wh{kQ}$ be the ideal generated by all arrows. For $r\geq0$, $\wh{\fm}^r$ consists of formal series with no terms of degree $< r$. For any subspace $U$ of $\wh{kQ}$, let $U^{cl}$ be the closure of $U$ with respect to the $\wh{\fm}$-adic topology on $\wh{kQ}$. Note that $U^{cl}= \cap_{r\geq 0 } (U+\wh{\fm}^r)$.
Clearly, $l$-derivations of $\wh{kQ}$ are uniquely determined by their value at all $a\in Q_1$. However, it is generally not true for $l$-derivations of $\wh{kQ}$ in an arbitrary $\wh{kQ}$-bimodule. In particular,
the assignment (\ref{Di}) does not extend to a double derivation on $\wh{kQ} $ since its value on a general formal series will not lie in the algebraic tensor product $\wh{kQ}\ot\wh{kQ}$ which admits only finite sums. Thus we need an alternative definition of double derivations of $\wh{kQ}$ to deal with noncommutative calculus on $\wh{kQ}$.
\begin{remark}
Different from that of $kQ$, the $\wh{kQ}$-bimodule $\Omega_{\wh{kQ}|l}$ is not generated by $d a$ for $a\in Q_1$. For example, take $Q$ to be a quiver with one node and one loop. Then $\wh{kQ}\cong k[[x]]$ and consider a formal series $\sum_{n=0}^\infty a_n x^n$ with generic coefficients. Then
\begin{align*}
d\Big(\sum_{n=0}^\infty a_n x^n\Big)&=dx\sum_{n=0}^\infty a_{n+1} x^n+xdx\sum_{n=0}^\infty a_{n+2} x^n+\ldots
\end{align*}
which can not be expressed as a finite sum of $f\cdot dx \cdot g$ for some formal series $f,g$. It is an easy exercise to show that this can be done only when $\sum_{n=0}^\infty a_n x^n$ is a geometric series.
\end{remark}
Let $\wh{kQ}\wh{\ot} \wh{kQ}$ be the $k$-module whose elements are formal series of the form
$\sum\nolimits_{u,v} A_{u,v}~ u\ot v$, where $u,v$ runs over all paths. This is nothing but the adic completion of $\wh{kQ}\ot\wh{kQ}$ with respect to the ideal $\wh{\fm}\ot\wh{kQ}+\wh{kQ}\ot \wh{\fm}$. It contains $\wh{kQ}\ot \wh{kQ}$ as a subspace under the identification
\[(\sum_{u} a'_u~u) \ot (\sum_{v} a''_v~v) \mapsto \sum_{u,v} a'_ua''_v~ u\ot v.\]
There are two obvious bimodule structures on $\wh{kQ} \wh{\ot} \wh{kQ}$, which we call the outer and the inner bimodule structures respectively, extends those on the subspace $\wh{kQ} \ot \wh{kQ}$.
Unless otherwise stated, we view $\wh{kQ}\wh{\ot} \wh{kQ}$ as a $\wh{kQ}$-bimodule with respect to the outer bimodule structure. In addition, there are linear maps $\wh{\mu}: \wh{kQ}\wh{\ot} \wh{kQ}\to \wh{kQ}$ and $\wh{\tau}: \wh{kQ}\wh{\ot} \wh{kQ} \to \wh{kQ}\wh{\ot} \wh{kQ}$ given respectively by
\[
\wh{\mu}(\sum_{u,v} a_{u,v} u\ot v) = \sum_{w} (\sum_{w=uv} a_{u,v})~w \quad \text{and} \quad \wh{\tau} (\sum_{u,v} a_{u,v} u\ot v) = \sum_{u,v} a_{v,u} u\ot v.
\]
Clearly, $\wh{\mu}$ is a bimodule homomorphism extends $\mu$, and $\wh{\tau}$ extends $\tau$.
We call derivations of $\wh{kQ}$ in the $\wh{kQ}$-bimodule $\wh{kQ}\wh{\ot}\wh{kQ}$ \emph{double $l$-derivations} of $\wh{kQ}$. The inner bimodule structure on $\wh{kQ}\wh{\ot}\wh{kQ}$ naturally yields a bimodule structure on the space
\[
\wh{\dder}_l(\wh{kQ}):= \der_l(\wh{kQ}, \wh{kQ}\wh{\ot}\wh{kQ}).
\]
For any $\delta \in \wh{\dder}_l(\wh{kQ})$ and any $f\in \wh{kQ}$, we also write $\delta(f)$ in Sweedler's notation as (\ref{Sweedler}), but one shall bear in mind that this notation is an infinite sum. Clearly, double derivations of $\wh{kQ}$ are uniquely determined by their values on all $a\in Q_1$. Thus, by abuse of notation, we have double derivations
\[\frac{\partial~}{\partial a}: \wh{kQ} \to \wh{kQ}\wh{\ot} \wh{kQ}, ~~~~ a\mapsto \delta_{a,b}~e_{s(a)}\ot e_{t(a)},~~\text{for}~a\in Q_1\]
extending the double derivations $\frac{\partial~}{\partial a}:kQ\to kQ\ot kQ$ constructed in (\ref{Di}). Moreover, every double derivation of $\wh{kQ}$ has a unique representation of the form
\begin{align}\label{representation-doub}
\sum_{a\in Q_1} \sum_{u,v} A_{u,v}^{(a)}~ u* \frac{\partial~}{\partial a} * v,~~~~~A_{u,v}^{(a)}\in k,
\end{align}
where $t(u)=t(a)$ and $s(v)=s(a)$, and $*$ denotes the scalar multiplication of the bimodule structure of $\wh{\dder}_l(\wh{kQ})$. The infinite sum (\ref{representation-doub}) makes sense in the obvious way. Further, note that the map $\wh{\mu}\circ-: \wh{\dder}_l(\wh{kQ}) \to \der_l(\wh{kQ})$ is surjective and the map \[\wh{\mu} \circ \wh{\tau} \circ-: \wh{\dder}_l(\wh{kQ}) \to \Hom(\wh{kQ},\wh{kQ})\] is a bimodule homomorphism. The image of $\wh{\mu} \circ \wh{\tau} \circ-$ is denoted by $\wh{\cder}_l(\wh{kQ})$ and its elements are called \emph{cyclic $l$-derivations} of $\wh{kQ}$. By abuse of notation, we have cyclic $l$-derivations
\[
D_{a}:= \wh{\mu} \circ \wh{\tau} \circ \frac{\partial~}{\partial a}: \wh{kQ}\to \wh{kQ}
\]
extending the cyclic derivations $D_{a}: kQ\to kQ$. By (\ref{representation-doub}), every cyclic $l$-derivation of $\wh{kQ}$ has a decomposition (not necessary unique) of the form
\begin{align}\label{representation-cyc}
\sum_{a\in Q_1} \sum_{u,v} A_{u,v}^{(a)}~ u\cdot D_a\cdot v,~~~~~A_{u,v}^{(a)}\in k,
\end{align}
where $t(u)=t(a)$ and $s(v)=s(a)$.
Note that all derivations and cyclic derivations of $\wh{kQ}$ are continuous with respect to the $\wh{\fm}$-adic topology on $\wh{kQ}$. Consequently, $\xi([\wh{kQ},\wh{kQ}]^{cl}) \subseteq [\wh{kQ},\wh{kQ}]^{cl}$ for each derivation $\xi\in \der_l(\wh{kQ})$, and $D([\wh{kQ},\wh{kQ}]^{cl})=0$ for each cyclic derivation $D\in \wh{\cder}_l(\wh{kQ})$. In addition, note that \[\wh{\mu}(\delta(\phi)) - \wh{\mu}(\wh{\tau} (\delta(\phi))) \in [\wh{kQ},\wh{kQ}]^{cl}\] for all double derivations $\delta\in \wh{\dder}_l(\wh{kQ})$ and all formal series $\phi\in \wh{kQ}$.
Given an element $\Phi$ of \[\wh{kQ}_{\ol{\cy}}:=\wh{kQ}/[\wh{kQ}, \wh{kQ}]^{cl},\]
the previous discussion implies that all its representatives $\phi$ yields the same commutative diagram
\begin{align}
\xymatrix{
\wh{\dder}_l (\wh{kQ}) \ar@{->>}[r]^-{\wh{\mu}\circ \wh{\tau}\circ-} \ar@{->>}[d]^-{\wh{\mu}\circ-} & \wh{\cder}_l (\wh{kQ}) \ar[r]^-{\Phi_*} & \wh{kQ}\ar[d]^{\pi} \\
\der_l (\wh{kQ}) \ar[rr]^-{\Phi_{\#}} & & \wh{kQ}_{\ol{\cy}},
} \label{derivation+}
\end{align}
where $\pi$ is the projection, $\Phi_{\#}: \xi \mapsto \pi(\xi(\phi))$ and $\Phi_*: D\mapsto D(\phi)$. Clearly, $\Phi_*$ is a bimodule homomorphism, so $\im (\Phi_*)$ is a two sided ideal of $\wh{kQ}$. Moreover, one has
\[
(\{~ \Phi_*(D_a)~ |~ a\in Q_1~ \}) \subseteq \im(\Phi_*) \subseteq (\{~\Phi_*(D_a) ~|~a\in Q_1~ \})^{cl}.
\]
Note that both equalities hold when $(\{~ \Phi_*(D_a)~ |~ a\in Q_1~ \})\supseteq \wh{\fm}^r$ for some $r\geq0$.
\begin{definition}\label{superpotential-complete}
Elements of $\wh{kQ}_{\ol{\cy}}:= \wh{kQ}/[\wh{kQ},\wh{kQ}]^{cl}$ are called \emph{potentials} of $\wh{kQ}$. Given a potential $\Phi\in \wh{kQ}_{\ol{\cy}}$, the two sided ideal
\[
\wh{\rJ}(Q,\Phi):= \im(\Phi_*)
\]
is called the \emph{Jacobi ideal} of $\Phi$, where $\Phi_*: \wh{\cder}_l(\wh{kQ}) \to \wh{kQ}$ is the $\wh{kQ}$-bimodule homomorphism occurs in Diagram (\ref{derivation+}). The associative algebra
\[
\wh{\Lm}(Q, \Phi) := \wh{kQ}/\wh{\rJ}(Q,\Phi),
\]
is called the \emph{Jacobi algebra} of $\Phi$. The smallest integer $r\geq 0$ such that $\Phi\in \pi(\wh{\fm}^r)$ is called the \emph{order} of $\Phi$.
\end{definition}
\begin{remark}We fix a linear order on $Q_1$. Recall that two cycles $u,~v$ are {\em conjugate} if there are paths $w_1,w_2$ such that $u=w_1w_2$ and $v=w_2w_1$. Equivalent classes under this equivalence relation are called \emph{necklaces} or \emph{conjugacy classes}. Also recall that a path $u$ is \emph{lexicographically smaller} than another word $v$ if there exist factorizations $u=waw'$ and $v=wbw''$ with $a<b$. Obviously, this order relation restricts to a total order on all necklaces. Let us call a cycle \emph{standard} if it is maximal in its necklace. Then every potential of $kQ$ (resp. $\wh{kQ}$) has a unique representative which is a finite linear combination (resp. formal linear combination) of standard cycles. We shall refer to such unique representatives of potentials the \emph{canonical representative}.\end{remark}
\begin{lemma}\label{Jacobi-closed}
Fix a potential $\Phi\in \wh{kQ}_{\ol{\cy}}$. Suppose that $k$ is noetherian. Then the canonical map
\begin{equation}\label{canonical-map}
\wh{\Lm}(Q,\Phi) \to \lim_{r\to \infty} \wh{kQ}/(\wh{\rJ}(Q,\Phi)+\wh{\fm}^r)
\end{equation}
is an isomorphism of $l$-algebras. Consequently, $\wh{\rJ}(Q,\Phi)= (\{~ \Phi_*(D_a)~ |~ a\in Q_1~ \})^{cl}$.
\end{lemma}
\begin{proof}
Since $k$ is noetherian, every inverse system of finitely generated $k$-modules over the direct set $(\mathbb{N},\leq)$ satisfies the Mittag-Leffler condition (See \cite[Tag 0595]{Stack} for the definition), and moreover all such inverse systems form an abelian category. Then by \cite[Tag, 0598]{Stack}, taking inverse limit is an exact functor from this category to the category of $k$-modules.
Let $\fd:\wh{\dder}_l(\wh{kQ}) \to \wh{kQ}$ be the map given by $\delta\mapsto \Phi_*(\wh{\mu}\circ \wh{\tau}\circ \delta)$. By definition, $\fd$ is an $\wh{kQ}$-bimodule homomorphism and $\im(\fd) =\wh{\rJ}(Q,\Phi)$. Clearly, the exact sequence
\[\wh{\dder}_l(\wh{kQ})\xrightarrow{\fd} \wh{kQ} \to \wh{\Lm}(Q,\Phi)\to 0\]
induces an exact sequence of inverse system of finitely generated $k$-modules over $(\mathbb{N}, \leq)$ as follows:
\begin{equation}\label{inverse-system}
\{~\wh{\dder}_l(\wh{kQ})/ \wh{\dder}_l(\wh{kQ})_{\geq r}~\}_{r\in \mathbb{N}} \to \{~\wh{kQ}/\wh{\fm}^r~\}_{r\in \mathbb{N}} \to \{~\wh{kQ}/(\wh{\rJ}(Q,\Phi)+\wh{\fm}^r)~\}_{r\in \mathbb{N}} \to 0,
\end{equation}
where $\wh{\dder}_l(\wh{kQ})_{\geq r}:= \sum_{i=0}^r \wh{\fm}^i * \wh{\dder}_l(\wh{kQ}) * \wh{\fm}^{r-i}$. Clearly, all the morphism are $l$-linear.
Consider the following commutative diagram
\[
\xymatrix{
\wh{\dder}_l(\wh{kQ}) \ar[r]^-{\fd} \ar[d]^-{\eta_1}& \wh{kQ}\ar[r] \ar[d]^-{\eta_2} & \wh{kQ}/ \wh{\rJ}(Q,\Phi) \ar[r]\ar[d]^-{\eta_3} &0\\
\lim_{r\to \infty}\wh{\dder}_l(\wh{kQ})/ \wh{\dder}_l(\wh{kQ})_{\geq r} \ar[r] & \lim_{r\to\infty} \wh{kQ}/ \wh{\fm}^r \ar[r] & \lim_{r\to \infty}\wh{kQ}/(\wh{\rJ}(Q,\Phi)+\wh{\fm}^r) \ar[r] & 0,
}
\]
where the bottom is the limit of the sequence (\ref{inverse-system}) and $\eta_1,\eta_2,\eta_3$ are the canonical maps. By the general result mentioned in the last paragraph, the bottom sequence is also exact. It is easy to check that $\eta_1$ and $\eta_2$ is an isomorphism, so is $\eta_3$. This prove the first statement.
To see the second statement, it suffice to show $\wh{\rJ}(Q,\Phi)$ is closed with respect to the $\wh{\fm}$-adic topology. But we have $\wh{\rJ}(Q,\Phi) = \cap_{r\geq0} (\wh{\rJ}(Q,\Phi)+\wh{\fm}^r) = \wh{\rJ}(Q,\Phi)^{cl}$.
\end{proof}
\begin{remark}
In the above lemma, the statement that $\wh{\rJ}(Q,\Phi)=(\{~ \Phi_*(D_a)~ |~ a\in Q_1~ \})^{cl}$ has been claimed in \cite[Lemma 2.8]{KY11} without a proof; and the statement that the canonical map (\ref{canonical-map}) is an isomorphism is equivalent to say that the Jacobi algebra $\wh{\Lm}(Q,\Phi)$ is pseudocompact, which is actually a special case of the general result \cite[Lemma A.12]{KY11}. Nevertheless, to avoid involve too much, we give here a direct demonstration for reader's convenience.
\end{remark}
One may view potentials of $kQ$ as potentials of $\wh{kQ}$ under the canonical injection $kQ_\cy \to \wh{kQ}_{\ol{\cy}}$. Given a potential $\Phi\in kQ_{\cy}$, we have a compare homomorphism of $l$-algebras
\begin{equation*}\label{compare-map}
\Lm(Q,\Phi) \to \wh{\Lm}(Q, \Phi).
\end{equation*}
\begin{lemma}\label{superpotential-compare}
Fix a potential $\Phi\in kQ_\cy$ of order $\geq2$. Let $\mathfrak{a}:= \fm / \rJ(Q,\Phi)$. Suppose that $k$ is noetherian. Then the compare map $\Lm(Q,\Phi) \to \wh{\Lm}(Q,\Phi)$ factors into a composition as follows
\[
\Lm(Q, \Phi) \to \lim_{r\to \infty} \Lm(Q, \Phi)/\mathfrak{a}^r \xrightarrow{\cong} \wh{\Lm}(Q, \Phi),
\]
where the first map is the canonical one and the second map is an $l$-algebra isomorphism. Consequently, if $\mathfrak{a}$ is nilpotent, then the compare map $\Lm(Q,\Phi) \to \wh{\Lm}(Q, \Phi)$ is an $l$-algebra isomorphism.
\end{lemma}
\begin{proof}
Let $\mathfrak{b}=\wh{\fm}/ \wh{\rJ}(Q,\Phi)$. It is easy to check that the compare map $\Lm(Q,\Phi)\to \wh{\Lm}(Q,\Phi)$ induces an isomorphism of inverse systems over $(\mathbb{N},\leq)$ of the form
\[
\{~\Lm(Q,\Phi)/\mathfrak{a}^r~\}_{r\in \mathbb{N}} \to \{~\wh{\Lm}(Q,\Phi)/\mathfrak{b}^r~\}_{r\in \mathbb{N}}.
\]
Then we have a commutative diagram of $l$-algebra homomorphisms
\[
\xymatrix{
\Lm(Q,\Phi) \ar[r] \ar[d] & \wh{\Lm}(Q,\Phi) \ar[d] \\
\lim_{r\to \infty} \Lm(Q,\Phi)/\mathfrak{a}^r \ar[r]^{\cong} & \lim_{r\to \infty} \wh{\Lm}(Q,\Phi)/\mathfrak{b}^r,
}
\]
where all maps are the natural one. By Lemma \ref{Jacobi-closed}, the map $\wh{\Lm}(Q,\Phi)\to \lim_{r\to \infty} \wh{\Lm}(Q,\Phi)/\mathfrak{b}^r$ is an isomorphism. Thereof the first statement follows. The second statement is clear.
\end{proof}
\begin{remark}
In general, the compare map $\Lm(Q,\Phi) \to \wh{\Lm}(Q, \Phi)$ is neither injective nor surjective.
It may happens that $\Lm(Q,\Phi)$ is not finitely generated but $\wh{\Lm}(Q,\Phi)$ is as $k$-modules. Also, it may happens that $\Lm(Q,\Phi)$ and $\wh{\Lm}(Q,\Phi)$ are finitely generated $k$-modules of different rank. We give three toy examples below, one for each of the considerations.
\begin{enumerate}
\item[$(1)$] Take $k$ to be a field and $Q$ to be the quiver with one node and two loops, i.e. $kQ\cong k\lg x, y\rg$, and $\Phi= \pi(\frac{1}{2}x^2 + \frac{1}{3} x^3)$. Then $\Phi_*(D_{x}) = x+x^2$ and $\Phi_*(D_{y})=0$. In this case, $\Lm(Q,\Phi)= k\lg x,y\rg/(x+x^2)$ and $\wh{\Lm}(Q,\Phi)= k\lgg x,y\rgg/(x)^{cl} \cong k[[y]]$. The canonical homomorphism $k\lg x,y\rg/(x+x^2) \to k[[y]]$, which is given by $x\mapsto 0$ and $y\mapsto y$, has nonzero kernel contains the class of $x$ and has $k[y]$ as its image.
\item[$(2)$] Take $k$ and $Q$ to be defined as the previous example, and $\Phi= \pi(\frac{1}{2}x^2 + \frac{1}{3} x^3 + \frac{1}{2}y^2 + \frac{1}{3} y^3)$. Then $\Phi_*(D_{x}) = x+ x^2$ and $\Phi_*(D_{y}) = y+y^2$. It is easy to check that $\Lm(Q,\Phi) = k\lg x,y\rg/(x+x^2, y+y^2)$ is not finite dimensional but $\wh{\Lm}(Q,\Phi) = k\lgg x,y\rgg/ (x,y)^{cl}$ is of dimension one.
\item[$(3)$] Take $k$ to be a field and $Q$ to be the quiver with one node and one loop, i.e. $kQ=k\lg x\rg$ and $\Phi=\pi(\frac{1}{2}x^2+\frac{1}{3}x^3)$. Again, $\Phi_*(D_{x}) = x(1+x)$. So $\Lm(Q,\Phi)$ is two diemsnional but $\wh{\Lm}(Q,\Phi)$ is one dimensional. Note that $\Lm(Q, \Phi)$ is not local.
\end{enumerate}
\end{remark}
There are some relations between Jacobi algebras of complete path algebras and that of power series algebras. Fix $i\in Q_0$. Let $k[[Q_1^{(ii)}]]$ be the power series algebra generated by $Q_1^{(ii)}$. For any
$f\in k[[Q_1^{(ii)}]]$, we denote by $f_a$ the formal partial derivative of $f$ with respect to $a\in Q_1^{(ii)}$. The \emph{Jacobi algebra} of
$k[[Q_1^{(ii)}]]$ associated to $f$ is defined to be the commutative algebra
\[
\Lm(k[[Q_1^{(ii)}]], f):=k[[Q_1^{(ii)}]] / (f_a|a\in Q^{(ii)}_1).
\]
Let $\iota^{(i)}: \wh{kQ} \to k[Q_1^{(ii)}]$ be the $k$-algebra homomorphism defined by $e_j\mapsto \delta_{ij}$ for nodes $j\in Q_0$, $a\mapsto a$ for arrows $a\in Q_1^{(ii)}$ and $a\mapsto 0$ for arrows $a\in Q_1\backslash Q_1^{(ii)}$. Clearly, $\iota^{(i)}$ factors as
\[
\wh{kQ}\xrightarrow{\pi} \wh{kQ}_{\ol{\cy}}\xrightarrow{\ol{\iota}^{(i)}} k[[Q_1^{(ii)}]].
\]
\begin{lemma}\label{abelianization}
Fix a node $i\in Q_0$ and a potential $\Phi\in \wh{kQ}_{\ol{\cy}}$. We have
\[
\iota^{(i)}(\Phi_*(D_a))=\Big(\ol{\iota}^{(i)}(\Phi)\Big)_a, \quad a\in Q_1.
\]
Consequently, $\iota^{(i)}$ induces a surjective $k$-algebra homomorphism
$\wh{\Lm}(Q,\Phi)\to \Lm(k[[Q_1^{(ii)}]], \iota^{(i)}(\Phi))$.
\end{lemma}
\begin{proof}
If a cycle $w$ contains an arrow $a\notin Q_1^{(ii)}$, then both sides of the equation (with $\Phi$ replaced by $w$) vanish by the definition of $\iota^{(i)}$.
For an cycle $w=b_1\ldots b_r$ with $b_s\in Q_1^{(ii)}$, one can readily check that
\[
\iota^{(i)}\big(D_a(w)\big)=\sum_{\{s|b_s=a\}}\iota^{(i)}\big(b_{s+1}\ldots b_rb_1\ldots b_{s-1}\big)=\iota^{(i)}(w)_{a}.
\]
Since the maps $\iota^{(i)}$, $D_a$ and $(-)_{a}$ all commute with taking formal sums, the result follows.
\end{proof}
\subsection{Some technical results on complete path algebras}
\begin{lemma}[Chain rule]\label{chain-rule}
Let $Q$ and $Q'$ be two finite quivers. Let $H: \wh{kQ}\to \wh{kQ'}$ be an algebra homomorphism such that $H(\{ e_i ~ |~ i \in Q_0 \} \subseteq \{ e_j ~|~ j\in Q_0' \}$.
Let $h_a:= H(a)$ for $a\in Q_1$. Then for any $\phi\in \wh{kQ}$ and $\beta\in Q'_1$ we have
\[
\frac{\partial}{\partial \beta}\big(H(\phi)\big)=\sum_{a\in Q_1} \bigg(\frac{\partial h_a}{\partial \beta}\bigg)^\pp * (H\wh{\ot}H) (\frac{\partial \phi}{\partial a})* \bigg(\frac{\partial h_a}{\partial \beta}\bigg)^\p
\]
with $*$ means the scalar multiplication with respect to the inner bimodule structure on $\wh{kQ'}\wh{\ot} \wh{kQ'}$ and $H\wh{\ot}H: \wh{kQ}\wh{\ot} \wh{kQ} \to \wh{kQ'}\wh{\ot} \wh{kQ'}$ is given by $u\ot v\mapsto H(u)\ot H(v)$. Consequently,
\begin{equation}\label{chainrule2}
D_{\beta}(H(\phi)) = \sum_{a\in Q_1} (\frac{\partial h_a}{\partial \beta})^\pp \cdot H (D_{a} \phi) \cdot (\frac{\partial h_a}{\partial \beta})^\p.
\end{equation}
Here we used the Sweedler's notation for the double derivation $\frac{\partial~}{\partial \beta}$ acting on $h_a$. Note that because a double derivation takes value in $\wh{kQ'}\cot\wh{kQ'}$, the Sweedler's notation is an infinite sum.\end{lemma}
\begin{proof}
Suppose $\phi=\sum_w A_ww$, where $w$ runs over all paths of $Q$. Then we have
\begin{eqnarray*}
\frac{\partial}{\partial \beta}\big(H(\phi)\big) &=& \sum\nolimits_w A_w~ \frac{\partial}{\partial \beta}\big( H(w)\big) \\
&=& \sum_w A_w~ \sum_{a\in Q_1} (\frac{\partial h_a}{\partial \beta})^\pp * (H\wh{\ot}H)\big(\frac{\partial w}{\partial a}\big) * (\frac{\partial h_a}{\partial \beta})^\p\\
&=& \sum_{a\in Q_1} (\frac{\partial h_a}{\partial \beta})^\pp * (H\wh{\ot}H) \big (\sum_{w} A_w~ \frac{\partial w}{\partial a} \big )* (\frac{\partial h_a}{\partial \beta})^\p\\
&=& \sum_{a\in Q_1} (\frac{\partial h_a}{\partial \beta})^\pp * (H\wh{\ot}H) \big ( \frac{\partial \phi}{\partial a}\big) * (\frac{\partial h_a}{\partial \beta})^\p,
\end{eqnarray*}
as required. Since $\wh{\mu}\circ \wh{\tau}\circ (H\wh{\ot}H) = H\circ \wh{\mu} \circ \wh{\tau}$, the desired formula for cyclic derivations follows immediately from the justified formula for double derivations.
\end{proof}
The next result (for free algebras) was stated in \cite[Proposition 1.5.13]{Ginz} without a proof.
\begin{lemma}[Poincare lemma]\label{Poincare}
Let $Q$ be a finite quiver.
Fix $f_a \in e_{s(a)}\cdot \wh{kQ}\cdot e_{t(a)}$
for each $a\in Q_1$. Suppose $k$ contains a subring which is a field of characteristic $0$. Then
\[
\sum_{a\in Q_1} [a,f_a]=0 \Longleftrightarrow \exists~ \phi\in \wh{kQ} \text{ such that } f_a=D_a(\phi), ~\text{for all}~ a\in Q_1.
\]
\end{lemma}
\begin{proof}
To prove the if part, it suffices to check the case when $\phi$ is a cycle $w=b_1\ldots b_r$. Then
\begin{align*}
\sum_{a\in Q_1} [a, D_a(w)]&=\sum_{a\in Q_1}\sum_{\{s|b_s=a\}} \big (ab_{s+1}\ldots b_r\ldots b_{s-1}-b_{s+1}\ldots b_r\ldots b_{s-1}a \big )\\
&=\sum_{s=1}^r b_sb_{s+1}\ldots b_r\ldots b_{s-1}-\sum_{s=1}^r b_{s+1}\ldots b_r\ldots b_{s-1} b_s\\
&=0.
\end{align*}
Now we prove the only if part. For a collection of formal series $\{f_a|a\in Q_1\}$,
we define its antiderivative to be an element $\phi\in \wh{kQ}$ defined by
\[
\phi:=\sum_{a\in Q_1} \sum_{r\geq 1} \frac{1}{r}~a \cdot f_a[r-1]
\] where $f_a[r-1]$ is the sum of paths of degree $r-1$ that occur in $f_a$. It suffices to verify that $D_a(\sum_b bf_b[r-1])=rf_a[r-1]$. Without loss of generality, we may simply assume that $f_a$ are homogeneous of degree $r$ for all $a\in Q_1$. Then by the assumption $\sum_{a\in Q_1} f_a a=\sum_{a\in Q_1} af_a$, $\phi$ is invariant under the cyclic permutations. Therefore $D_a\phi=f_a$ by the definition of $D_a$.
\end{proof}
Let $Q$ be a finite quiver. A $l$-algebra endomorphism of $\wh{kQ}$ is a $k$-algebra endomorphism $H$ such that $H(e_i)=e_i$ for all $i\in Q_0$.
Let $H$ be a $l$-algebra endomorphism of $kQ$. For any $a\in Q_1$
\[
H(a)\in e_{s(a)}\cdot \wh{kQ}\cdot e_{t(a)}.
\]
Moreover, $H$ preserves the decreasing filtration
\[
\wh{\fm}\supset \wh{\fm}^2\supset \wh{\fm}^3\supset \ldots
\]
A $l$-algebra endomorphism is called a $l$-algebra automorphism if it is invertible.
\begin{lemma}[Inverse function theorem]\label{inverse}
Suppose $H$ is an $l$-endomorphism of $\wh{kQ}$ that induces an isomorphism $\wh{\fm}/\wh{\fm}^2 \xrightarrow{\cong} \wh{\fm}/\wh{\fm}^2$. Then $H$ is invertible.
\end{lemma}
\begin{proof}
By assumption we can choose an $l$-automorphism $T:\wh{kQ} \to \wh{kQ}$ induced by a collection of invertible linear transformations on the spaces of arrows with fixed source and target such that the composition $G:=H\circ T$ satisfying
\[
G(a) \equiv a \mod \wh{\fm}^2, ~~~~~a\in Q_1.
\]
For a formal series $f$, let us denote $f[r]$ to be the sum of all terms of degree $r$ that occur in $f$. Then
\[
G(f[r]) \equiv f[r] \mod \wh{\fm}^{r+1}.
\]
For $a\in Q_1$, we let $g_{a,1}=a$ and then inductively set
\[
g_{a,r}:= - G(g_{a,1}+\ldots +g_{a,r-1})[r], ~~~ r\geq 2.
\]
Note that $g_{a,r}$ consists of terms of degree $r$. Let $g_a:=\sum_{r\geq 1} g_{a,r}$. Then for $r\geq 1$ we have
\[
\begin{array}{llll}
G(g_a) & \equiv& G(g_{a,1}+\ldots +g_{a,r}) & \mod \wh{\fm}^{r+1}\\
& \equiv& G(g_{a,1}+\ldots +g_{a,r-1}) +G(g_{a,r}) & \mod \wh{\fm}^{r+1} \\
& \equiv& a -g_{a,r} + G(g_{a,r}) & \mod \wh{\fm}^{r+1} \\
& \equiv& a & \mod \wh{\fm}^{r+1}.
\end{array}
\]
Here, the third ``$\equiv$'' can be easily obtained by induction of $r$.
Consequently, $G(g_a) =a$ and therefore $G$ is surjective. Now let us consider the $r$-jet space $J^r:= \wh{kQ}/ \wh{\fm}^{r+1}$. They are all free $k$-modules with a finite basis. The map $G$ then induces an inverse system of ($k$-)linear maps
\[
G_{r}: J^r\to J^r.
\]
Clearly, all $G_{r}$ are surjective and thereof they are all invertible. This can be seen by consider these maps as square matrices with entries in $k$ and then the determinant trick applies. Since $G$ is just the inverse limit of $G_{r}$, it is invertible. Therefore $H$ is also invertible.
\end{proof}
\begin{lemma}[Nakayama Lemma]\label{Nakayama}
Let $Q$ be a finite quiver. Suppose $N$ is an ideal of $\wh{kQ}$ with $N+ \wh{\fm}^{r} \supseteq \wh{\fm}^{r-1}$ for some $r\geq1$. Then $N\supseteq \wh{\fm}^{r-1}$.
\end{lemma}
\begin{proof}
Note that $\wh{\fm}^{r-1}$ is a finitely generated left $\wh{kQ}$-module. Also note that elements of $1+ \wh{\fm}$ are all invertible in $\wh{kQ}$, so $\wh{\fm}$ contains in the Jacobson radical of $\wh{F}$. By modularity,
\[
N\cap \wh{\fm}^{r-1}+ \wh{\fm}^{r} = (N+ \wh{\fm}^{r}) \cap \wh{\fm}^{r-1}= \wh{\fm}^{r-1}.
\]
Then by \cite[Lemma 4.22]{Lam}, the Nakayama Lemma, we have $N\cap \wh{\fm}^{r-1} = \wh{\fm}^{r-1}$ and so $N\supseteq \wh{\fm}^{r-1}$.
\end{proof}
\section{Noncommutative Mather-Yau theorem}\label{sec:Proof}
This section is devoted to establish a noncommutative analogue of the well-known Mather-Yau type theorem in the hypersurface singularity theory \cite{MY, BY90, Yau84}.
Throughout, $k$ stands for a commutative ring with unit, $Q$ stands for a finite quiver and $l=kQ_0$, which is a subalgebra of $\wh{kQ}$. Let $\wh{\fm} \subseteq \wh{kQ}$ be the ideal generated by arrows.
We continue to use the notations appear in Diagram (\ref{derivation+}). So $\pi: \wh{kQ}\to \wh{kQ}_{\ol{\cy}}$ is the projection map, and for each potential $\Phi\in \wh{kQ}_{\ol{\cy}}$ there are maps $\Phi_*: \wh{\cder}_l(\wh{kQ}) \to \wh{kQ}$ and $\Phi_\#: \der_l(\wh{kQ}) \to \wh{kQ}_{\ol{\cy}}$.
Also, let $\wh{\dder}_l^+ (\wh{kQ})$ be the space of double derivations of $\wh{kQ}$ that map $\wh{\fm}$ to $\wh{\fm} \wh{\ot} \wh{kQ}+ \wh{kQ}\wh{\ot} \wh{\fm}$, and $\der_l^+ (\wh{kQ})$ (resp. $\wh{\cder}_l^+ (\wh{kQ})$) the space of derivations (resp. cyclic derivations) of $\wh{kQ}$ that preserves $\wh{\fm}$. It is easy to check that $\wh{\mu}(\wh{\dder}_l^+(\wh{kQ})) = \der_l^+(\wh{kQ})$ and $\wh{\mu}(\wh{\tau}(\wh{\dder}_l^+(\wh{kQ}))) = \wh{\cder}_l^+(\wh{kQ})$.
\subsection{Right equivalence of potentials}
We denote by $\mathcal{G}:=\Aut_l(\wh{kQ},\wh{\fm})$ the group of $l$-algebra automorphisms of $\wh{kQ}$ that preserves $\wh{\fm}$. It is a subgroup of $\Aut_l(\wh{kQ})$, the group of all $l$-algebra automorphisms of $\wh{kQ}$. In the case when $k$ is a field, $\mathcal{G}=\Aut_l(\wh{F})$. Note that $\mathcal{G}$ acts on $\wh{kQ}_{\ol{\cy}}$ in the obvious way.
\begin{definition}\label{right-equivalent}
For potentials $\Phi,\Psi\in \wh{kQ}_{\ol{\cy}}$, we say $\Phi$ is \emph{(formally) right equivalent} to $\Psi$ and write $\Phi \sim \Psi$, if $\Phi$ and $\Psi$ lie in the same $\cG$-orbit.
\end{definition}
For potentials of the non complete path algebra $kQ$, one may similarly define the \emph{algebraically right equivalence} in terms of the action of the group of $l$-algebra automorphisms of $kQ$ on $kQ_\cy$. It turns out that two potentials of $kQ$ can be algebraically right equivalent but not formally, and vice versa. This subtle difference can be checked by the following example.
\begin{example}
Let $k=\CC$ and $Q$ the quiver with one node and one loop. Consider the potentials $\Phi: = \frac{1}{2}x^2+\frac{1}{3}x^3$ and $\Psi:= \frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3$ of $F$. We have $H(\Phi) =\Psi$ for $H: kQ\to kQ$ the $\CC$-algebra automorphism given by $x\mapsto x-1$, so $\Phi$ and $\Psi$ are algebraically right equivalent. Since $\Phi$ is not a unit of $\wh{kQ}$ but $\Psi$ is, $\Phi$ and $\Psi$ are not formally right equivalent. Now consider the potentials $\Phi':= x^2$ and $\Psi':= x^2+ x^3$ of $kQ$. It is straightforward to show that $\Phi'$ and $\Psi'$ are not algebraically right equivalent. But we have $H'(\Phi') = \Psi'$ for $H': \wh{kQ}\to \wh{kQ}$ the $\CC$-algebra automorphism given by $x\mapsto x\cdot u$, where $u$ is the power series expansion of $(1+x)^{1/2}$ at $0$ (so $u^2=1+x$).
\end{example}
The main concern of this section is to explore relations between the isomorphism of Jacobi algebras of $\wh{kQ}$ and the right equivalence relation of potentials of $\wh{kQ}$.
The next result was already obtained in \cite[Proposition 3.7]{DWZ} in a more general form. For completeness and reader's convenience, we give a demonstration in our own notations.
\begin{prop}\label{Jacobi-transform}
Let $\Phi \in \wh{kQ}_{\ol{\cy}}$ and $H\in \mathcal{G}$. Then
\[
H(\wh{\rJ}(Q,\Phi)) =\wh{\rJ}(Q, H(\Phi)).
\]
Consequently, $H$ induces an isomorphism of $l$-algebras $\wh{\Lm}(Q, \Phi) \cong \wh{\Lm}(Q, H(\Phi))$.
\end{prop}
\begin{proof}
By the chain rule (Lemma \ref{chain-rule}), we have $H(\Phi)_*(D_a) \in H(\wh{\rJ}(Q,\Phi))$ for all $a\in Q_1$. Then by Lemma \ref{Jacobi-closed}, $\wh{\rJ}(Q, H(\Phi)) =\{ ~H(\Phi)_*(D_a) ~|~ a\in Q_1~\}^{cl} \subseteq H(\wh{\rJ}(Q,\Phi))$.
Symmetrically, we have $\wh{\rJ}(Q,\Phi) \subseteq H^{-1}(\wh{\rJ}(Q,H(\Phi)))$. Thus $H(\wh{\rJ}(Q,\Phi)) = \wh{\rJ}(Q,H(\Phi))$.
\end{proof}
\begin{remark}
Proposition \ref{Jacobi-transform} tells us that two right equivalent potentials of $\wh{kQ}$ have isomorphic Jacobi algebras. The converse is not true, even for $k=\CC$ under the additional assumption that the two isomorphic Jacobi algebras are finite dimensional. For example, take $Q$ to be the quiver with one node and one loop, $\Phi: = \frac{1}{2}x^2+\frac{1}{3}x^3$ and $\Psi:= \frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3$. Then $\Phi_*(D_x) = x+x^2$ and $\Psi_*(D_x)=(x-1)+ (x-1)^2$. So $\Lm(\wh{Q}, \Phi) $ and $\Lm(\wh{Q}, \Psi)$ are both isomorphic to the algebra $\CC$. However, $\Phi$ and $\Psi$ are not right equivalent simply because $\Phi$ is not a unit of $\wh{kQ}$ but $\Psi$ is.
\end{remark}
Given an integer $r\geq0$, the \emph{$r$-th jet space} of $\wh{kQ}$ is defined to be the quotient $l$-algebra $J^r:=\wh{kQ}/\wh{\fm}^{r+1}$. Clearly, the projection map $\wh{kQ}\to J^r$ induces a canonical surjective map
\[q_r: \wh{kQ}_{\ol{\cy}} \to J^r_\cy:=J^r/[J^r,J^r]\]
with kernel $\pi(\wh{\fm}^{r+1})$. The image of a potential $\Phi \in \wh{kQ}_{\ol{\cy}}$ under this map is denoted by $\Phi^{(r)}$. For two potentials $\Psi_1,\Psi_2\in \wh{kQ}_{\ol{\cy}}$, $\Psi_1^{(r)} = \Psi_2^{(r)}$ if and only if their canonical representatives share the same coefficients for standard cycles of length $\leq r$.
\begin{prop}\label{isomorphism-criterior-0}
Let $\Phi, \Psi\in \wh{kQ}_{\ol{\cy}}$ be potentials satisfy that $\Phi^{(r)} = \Psi^{(r)}$ in $J_\cy^r$ for some $r>0$. Suppose $\wh{\rJ}(Q,\Phi) \supseteq \wh{\fm}^{r-1}$. Then
\[\wh{\rJ}(Q,\Phi)=\wh{\rJ}(Q,\Psi).\] Consequently, $\wh{\Lm}(Q, \Phi) = \wh{\Lm}(Q, \Psi)$ as $l$-algebras.
\end{prop}
\begin{proof}
Note that $D(\wh{\fm}^s)\subseteq \wh{\fm}^{s-1}$ for any $D\in \wh{\cder}_l (\wh{kQ}) $ and any $s\geq 1$. It follows that
\begin{align*}
\wh{\rJ}(Q,\Psi)+\wh{\fm}^{r}= \wh{\rJ}(Q,\Phi)+ \wh{\fm}^{r} = \wh{\rJ}(Q,\Phi) \supseteq \wh{\fm}^{r-1}.
\end{align*}
By Lemma \ref{Nakayama}, we also have $\wh{\rJ}(Q,\Psi) \supseteq \wh{\fm}^{r-1}$ and consequently
\[
\wh{\rJ}(Q,\Psi) = \wh{\rJ}(Q,\Psi)+\wh{\fm}^{r}= \wh{\rJ}(Q,\Phi)+ \wh{\fm}^{r} = \wh{\rJ}(Q,\Phi).
\]
The equality of Jacobi algebras is just by definition.
\end{proof}
\subsection{Statement of the nc Mather-Yau theorem}
Given a potential $\Phi\in \wh{kQ}_{\ol{\cy}}$, let $\wh{\fm}_\Phi:= \wh{\fm}/ \wh{\rJ}(Q, \Phi)$, which is an ideal of $\wh{\Lm}(Q, \Phi)$. By Lemma \ref{Jacobi-closed}, the $\wh{\fm}_\Phi$-adic topology of $\wh{\Lm}(Q, \Phi)$ is complete. Let
\[
\wh{\Lm}(Q, \Phi)_{\ol{\cy}}:=\wh{\Lm}(Q, \Phi)/[\wh{\Lm}(Q, \Phi), \wh{\Lm}(Q, \Phi)]^{cl}.
\]
Clearly, if $\wh{\Lm}(Q, \Phi)$ is a finitely generated $k$-module, then
\[\wh{\Lm}(Q, \Phi)_{\ol{\cy}}=\wh{\Lm}(Q, \Phi)_\cy=HH_0(\wh{\Lm}(Q, \Phi)).\]
The projection map $\wh{kQ} \to \wh{\Lm}(Q, \Phi)$ induces a natural map
\[p_\Phi: \wh{kQ}_{\ol{\cy}} \to \wh{\Lm}(Q, \Phi)_{\ol{\cy}}\]
with kernel $\pi(\wh{\rJ}(Q,\Phi))$. For any $\Theta\in \wh{kQ}_{\ol{\cy}}$, we write $[\Theta]= [\Theta]_\Phi:= p_\Phi(\Theta).$
Given a homomorphism $\gamma: A\to B$ of $k$-algebras, we denote by $\gamma_*: A_\cy\to B_\cy$ the induced map. Our main result in this section is the next theorem, which is a noncommutative analogue of the Mather-Yau theorem for isolated hypersurface singularities.
\begin{theorem}[nc Mather-Yau Theorem]\label{ncMY}
Let $Q$ be a finite quiver. Let $\Phi, \Psi\in \wh{\CC Q}_{\ol{\cy}}$ two potentials of order $\geq 3$ such that the Jacobi algebras $\wh{\Lm}(Q, \Phi)$ and $\wh{\Lm}(Q, \Psi)$ are both finite dimensional. Then the following two statements are equivalent:
\begin{enumerate}
\item[$(1)$] There is an $\CC Q_0$-algebra isomorphism $\gamma: \wh{\Lm}(Q, \Phi)\cong\wh{\Lm}(Q, \Psi)$ so that $\gamma_*([\Phi]_\Phi)=[\Psi]_\Psi$.
\item[$(2)$] $\Phi$ and $\Psi$ are right equivalent.
\end{enumerate}
\end{theorem}
\begin{proof}
We postpone the proof to the end of this section. It needs some technical results which have interest in their own right and will be presented in the following two subsections. Our discussion is modified from that in \cite{MY}, which deals with isolated hypersurface singularities.
\end{proof}
\begin{remark}
Instead of considering the $\cG$-orbits in $\wh{kQ}_{\ol{\cy}}$, we may rephrase an enhanced version of the nc Mather-Yau theorem for $\cG$-orbits in $\wh{kQ}$. Unfortunately, this cannot hold. Take $Q$ to be the quiver with one node and two arrows. Choose any formal series $\phi \in \wh{\fm}^4$ with finite dimensional Jacobi algebra. Then $\psi:=\phi+ x^2y-yx^2 \not\in \wh{\fm}^4$. Clearly, there is an isomorphism of Jacobi algebras for $\phi$ and $\psi$. However, $\phi$ and $\psi$ are not in the same $\cG$-orbit because any automorphism of $\wh{kQ}$ should preserve powers of $\wh{\fm}$.
\end{remark}
\begin{definition}\label{quasi-homogeneous}
We call a potential $\Phi \in \wh{kQ}_{\ol{\cy}}$ \emph{quasi-homogeneous} if $[\Phi]_\Phi$ is zero in $\wh{\Lm}(Q, \Phi)_{\ol{\cy}}$, or equivalently, if $\Phi$ is contained in $\pi(\wh{\rJ}(Q,\Phi))$.
\end{definition}
\begin{corollary}\label{ncMY-QH}
Let $Q$ be a finite quiver. Let $\Phi, \Psi\in \wh{\CC Q}_{\ol{\cy}}$ two quasi-homogeneous potentials of order $\geq 3$ such that the Jacobi algebras $\wh{\Lm}(Q,\Phi)$ and $\wh{\Lm}(Q,\Psi)$ are both finite dimensional. Then the following two statements are equivalent:
\begin{enumerate}
\item[$(1)$] There is an $\CC Q_0$-algebra isomorphism $\wh{\Lm}(Q,\Phi)\cong\wh{\Lm}(Q,\Psi)$.
\item[$(2)$] $\Phi$ and $\Psi$ are right equivalent.
\end{enumerate}
\end{corollary}
\begin{proof}
Since $[\Phi]_\Phi$ and $[\Psi]_\Psi$ are both zero, the result is an immediate consequence of Theorem \ref{ncMY}.
\end{proof}
Quasi-homogeneous potentials are abundant. Let us call a potential of $\wh{kQ}$ \emph{weighted-homogeneous} if it has a representative with all terms have the same degree with respect to some choice of positive weight of the arrows of $Q$. For example, take $Q$ to be the quiver with one node and two arrows $x,y$. Then $\Phi=\pi(x^2y-\frac{1}{4}y^4)\in \wh{kQ}_{\ol{\cy}}$ is weighted-homogeneous. Note that all weighted-homogeneous potentials must lie in $kQ_\cy$ because there are only finitely many paths of the same degree with respect to any choice of weights on arrows.
\begin{lemma}
Weighted-homogeneous potentials of $\wh{kQ}$ are quasi-homogeneous.
\end{lemma}
\begin{proof}
Suppose that all terms of the potential $\Phi\in \wh{kQ}_{\ol{\cy}}$ have the same degree $d$ with respect to the choice of weight $|a| =r_a>0$. Then $\Phi = \pi ( \sum_{a\in Q_1}\frac{r_a}{d} a \cdot \Phi_*(D_a) )$, and so $\Phi$ is quasi-homogeneous.
\end{proof}
\begin{remark}
It is easy to check that the set of quasi-homogeneous potentials of $\wh{kQ}$ are closed under the action of the group $\mathcal{G}$. So for any weighted-homogeneous $\Phi$, the potential $H(\Phi)$ will be quasi-homogeneous but not weighted-homogeneous for some obvious choices of $H\in \mathcal{G}$.
\end{remark}
\begin{remark}
Recall from the hypersurface singularity theory that a power series $f\in k[[x_1,\ldots, x_n]]$ is quasi-homogeneous if $f$ lies in the ideal generated by the derivatives $f_{x_1},\ldots, f_{x_n}$. By Lemma \ref{abelianization}, the abelianization of a quasi-homogeneous potential of $\wh{kQ}$ (at any node) is quasi-homogeneous. So we may construct many non-quasi-homogeneous potentials from this perspective.
We refer to Example \ref{non-quasi-homo} for an example of non-quasi-homogeneous superpotentials but whose abelianization is a quasi-homogeneous power series.
\end{remark}
\subsection{Bootstrapping on Jacobi ideals}
The remaining of this section aims to prove the nc Mather-Yau theorem (Theorem \ref{ncMY}). This subsection is devoted to establish a bootstrap relation between the Jacobi ideal $\wh{\rJ}(Q,\Phi)$ and the higher Jacobi ideal $\wh{\fm}\cdot \wh{\rJ}(Q,\Phi)+ \wh{\rJ}(Q,\Phi) \cdot \wh{\fm}$ for potentials $\Phi\in \wh{kQ}$. The prototype of this argument in the complex analytic case traces back to \cite{MY}.
For our end, we need the following separation lemma for power series rings.
\begin{lemma}[Separation lemma]\label{separation}
Let $\mathcal{P}:= k[[x_1,\ldots, x_n]]$ be the power series ring over $k$ and let $\mathfrak{a}$ be the ideal generated by $x_1,\ldots, x_n$. Let $f_1,\ldots, f_n$ be a sequence of $n$ elements in $\mathfrak{a}$. Suppose that $k$ is a field and the quotient $k$-algebra $\mathcal{P}/(f_1,\ldots, f_n)$ is finite dimensional over $k$. Then there exists a homomorphism $\eta: \mathcal{P}\to \ol{k}[[T]]$ of $k$-algebras, where $\ol{k}$ is the algebraic closure of $k$, satisfying that $\eta(\mathfrak{a}) \subseteq (T)$, $\eta(f_1)\neq 0$ and $\eta(f_i)=0$ for $i>1$.
\end{lemma}
\begin{proof}
Let $I$ be the ideal of $\mathcal{P}$ generated by $f_2,\ldots, f_n$. By an easy (Krull) dimension argument, one obtain that $\dim \mathcal{P}/I = 1$ because $\dim \mathcal{P} =n$ and $\dim \mathcal{P}/(f_1,\ldots, f_n)=0$. Fix a minimal prime ideal $\mathfrak{p}$ over $I$ and let $A:= \mathcal{P}/\mathfrak{p}$. Clearly, $f_1\not \in \mathfrak{p}$ and $A$ is a noetherian local domain with residue field $k$ and of dimension $1$. Moreover, $A$ is complete by \cite[Tag 0325]{Stack}. Let $B$ be the integral closure of $A$ in the fraction field of $A$. By standard commutative ring theory, $B$ is a domain of dimension $1$ and some maximal ideal of $B$ contains $\mathfrak{a}$. By \cite[Tag 032W]{Stack}, $B$ is a finite module over $A$. Then, by \cite[Tag 0325]{Stack} again, $B$ is also a noetherian complete local ring. It follows that the inclusion map $A\to B$ is a finite local homomorphism. So the residue field $\mathbb{K}$ of $B$ is of finite dimensional over $k$ and thereof we may identify $\mathbb{K}$ as a $k$-subalgebra of $\ol{k}$. It is well-known that a Noetherian normal local domain of dimension $1$ is regular. By \cite[Tag 0C0S]{Stack}, $B\cong \mathbb{K}[[T]]$ as $\mathbb{K}$-algebras. Let $\eta: \mathcal{P}\to \ol{k}[[T]]$ be the composition of the following sequence of homomorphisms of $k$-algebras
\[
\mathcal{P} \xrightarrow{} A \xrightarrow{\subseteq } B \xrightarrow{\cong} \mathbb{K}[[T]] \xrightarrow{\subseteq} \ol{k}[[T]],
\]
where the first one is the projection map. Clearly, $\eta$ satisfies the requirements.
\end{proof}
\begin{prop}[Bootstrapping]\label{Bootstrapping}
Let $\Phi\in\wh{kQ}_{\ol{\cy}}$ be a potential of order $\geq 2$. Suppose that $k$ is a field and the Jacobi algebra $\Lm(\wh{Q},\Phi)$ is finite dimensional. Then
\begin{enumerate}
\item[$(1)$] $\Phi \in \pi(\wh{\rJ}(Q,\Phi))$ (i.e. $\Phi$ is quasi-homogeneous) if and only if $\Phi\in \pi(\wh{\fm}\cdot \wh{\rJ}(Q,\Phi)+\wh{\rJ}(Q,\Phi)\cdot \wh{\fm})$.
\item[$(2)$] For any potential $\Psi\in \wh{F}_{\ol{\cy}}$ of order $\geq 2$ with $\wh{\rJ}(Q,\Psi) = \wh{\rJ}(Q,\Phi)$, it follows that $\Phi-\Psi\in \pi(\wh{\rJ}(Q,\Phi))$ if and only if $\Phi-\Psi\in \pi(\wh{\fm}\cdot\wh{\rJ}(Q,\Phi)+\wh{\rJ}(Q,\Phi)\cdot \wh{\fm})$.
\end{enumerate}
\end{prop}
\begin{proof}
The if part of both statements are obvious. For the only if part, we prove by contradiction. To simplify the notation, let $\phi_a=\Phi_*(D_a)$ and $\psi_a= \Psi_*(D_a)$ for $a\in Q_1$. Write $Q_1=A\sqcup L$, where $A$ consists of arrows between distinct nodes and $L$ consists of loops. For a power series $f\in \ol{k}[[T]]$, let $o(f)$ denote the order of $f$. It is defined to be the minimal degree of terms that occurs in $f$.
For every node $i$, let $\iota^{(i)}: \wh{kQ}\to k[[Q_1^{(ii)}]]$ be the abelianization map constructed before Lemma \ref{abelianization}. For every $a\in Q^{(ii)}_1$, let $f_a = \iota^{(i)}(\phi_a)$. Clearly, $k[[Q_1^{(ii)}]]/ (f_a: a\in Q^{(ii)}_1)$ is finite dimensional over $k$. If the set $Q^{(ii)}_1\neq \emptyset$, then by Lemma \ref{separation}, for any arrow $b\in Q^{(ii)}_1$ one may choose a local homomorphism $\eta_b^{(i)}: k[[Q_1^{(ii)}]] \to \ol{k}[[T]]$ such that $\eta_b^{(i)}(f_b) \neq 0$ but $\eta_b^{(i)}(f_a) =0$ for $a\neq b$. Define a $k$-algebra homomorphism $\omega_b^{(i)}: \wh{kQ} \to \ol{k}[[T]]$ by $$\omega_b^{(i)} = \eta_b^{(i)} \circ \iota^{(i)}.$$ Clearly, $\omega_b^{(i)}(\wh{\fm}) \subseteq (T)$, $\omega_b^{(i)}(e_j) = \delta_{ij}$, $\omega_b^{(i)}(\phi_b)\neq 0$ and $\omega_b^{(i)} (\phi_a) =0$ for all $a\neq b$. Moreover, $\omega_b^{(i)}$ factors through $\pi: \wh{kQ}\to \wh{kQ}_{\ol{\cy}}$ by a linear map $\overline{\omega}_b^{(i)}: \wh{kQ}_{\ol{\cy}} \to \ol{k}[[T]]$. By the chain rule,
\[
\frac{d~ \ol{\omega}_b^{(i)}(\Phi)}{d~T} = \sum_{s(a)=t(a)=i} \omega_b^{(i)}(\phi_a)\cdot \frac{d~ \omega_b^{(i)}(a)}{d~ T} = \omega_b^{(i)}(\phi_b)\cdot \frac{d~ \omega_b^{(i)}(b)}{d~ T}.
\]
Consequently, $o(\omega_b^{(i)}(\phi_b))< o(\ol{\omega}_b^{(i)}(\Phi))$.
To see the only if part of (1), assume that $\Phi\in \pi(\wh{\rJ}(Q,\Phi))$, i.e.
\[
\Phi =\pi(\sum_{a\in Q_1} h_a \cdot \phi_a )=\pi(\sum_{a\in A} h_a \cdot \phi_a+\sum_{a\in L} h_a \cdot \phi_a ), \quad \text{where }~ h_a \in e_{s(a)} \cdot \wh{kQ}\cdot e_{t(a)}.
\]
Clearly, $h_a\in \wh{\fm}$ for $a\in A$. Suppose that for some $b \in L$ such that $s(b)=t(b)=i$,
\[
h_{b}=\sum_w \lambda_{b,w}\cdot w\in e_i\cdot \wh{kQ}\cdot e_i,
\]
where $w$ runs over cycles based at $i$ and $\lambda_{b,w}\in k$, having $\lambda_{b,e_i}\neq 0$. Since
\[
\ol{\omega}_b^{(i)}(\Phi) = \sum_{a\in Q_1} \omega_b^{(i)}(h_a)\cdot \omega_b^{(i)}(\phi_a) = \omega_b^{(i)}(h_b)\cdot \omega_b^{(i)}(\phi_b)
\]
and $\omega_b^{(i)}(h_b)$ is a unit of $\ol{k}[[T]]$, we then obtain $o(\omega_b^{(i)}(\phi_b))=o(\ol{\omega}_b^{(i)}(\Phi))$, which is absurd.
To see the only if part of (2), let
\[
\Psi-\Phi=\pi(\sum_{a\in Q_1} g_a \cdot \phi_a )=\pi(\sum_{a\in A} g_a \cdot \phi_a+\sum_{a\in L} g_a \cdot \phi_a ), \quad \text{where }~ g_a \in e_{s(a)} \cdot \wh{kQ}\cdot e_{t(a)}
\]
Again, it is clear that $g_a\in \wh{\fm}$ for $a\in A$. Suppose that for some $b \in L$ such that $s(b)=t(b)=i$,
\[
g_{b}=\sum_w \lambda'_{b,w}\cdot w\in e_i\cdot \wh{kQ}\cdot e_i,
\]
where $w$ runs over cycles based at $i$ and $\lambda'_{b,w}\in k$, having $\lambda'_{b,e_i}\neq 0$.
Since
\[\ol{\omega}_b^{(i)} (\Psi) = \ol{\omega}_b^{(i)}(\Phi) + \sum_{a\in Q_1}\omega_b^{(i)}(g_a)\cdot \omega_b^{(i)}(\phi_a) = \ol{\omega}_b^{(i)}(\Phi) + \omega_b^{(i)}(g_b)\cdot \omega_b^{(i)}(\phi_b)\] and $\omega_b^{(i)}(g_b)$ is a unit of $\ol{k}[[T]]$, we then obtain
$o(\ol{\omega}_b^{(i)}(\Psi)) = o(\omega_b^{(i)}(\phi_b))$. Note that every ideal $J$ of $\ol{k}[[T]]$ is principal; write $o(J)$ for the order of its generator. Since
\[
\frac{d~ \ol{\omega}_b^{(i)}(\Psi)}{d~T} = \sum_{s(a)=t(a)=i} \omega_b^{(i)}(\psi_a)\cdot \frac{d~ \omega_b^{(i)}(a)}{d~ T},
\]
we obtain $o(\omega_b^{(i)}(\psi_a))< o(\ol{\omega}_b^{(i)}(\Psi))$ for at least one loop $a$ such that $s(a)=t(a)=i$. Hence
\[
o\bigg(\big(\omega_b^{(i)}(\psi_a): a\in Q^{(ii)}_1\big)\bigg)< o(\ol{\omega}_b^{(i)}(\Psi)) = o(\omega_b^{(i)}(\phi_b)) = o\bigg( \big(\omega_b^{(i)}(\phi_a): a\in Q^{(ii)}_1\big) \bigg).
\]
But one has $$\big(\omega_b^{(i)}(\psi_a): a\in Q^{(ii)}_1\big) = \omega_b^{(i)} (\wh{\rJ}(Q,\Psi)) = \omega_b^{(i)} (\wh{\rJ}(Q,\Phi)) = \big(\omega_b^{(i)}(\phi_a): a\in Q^{(ii)}_1\big),$$
which yields a contradiction.
\end{proof}
\subsection{Finite determinacy}
Given an integer $r\geq0$, let $\mathcal{G}^r$ be the group of all $l$-algebra automorphisms of $J^r=\wh{kQ}/\wh{\fm}^{r+1}$. Clearly, the canonical map $\mathcal{G}\to \mathcal{G}^r$ is surjective. A potential $\Phi\in \wh{kQ}_{\ol{\cy}}$ is called \emph{$r$-determined} (with respect to $\mathcal{G}$) if $\Phi^{(r)}\in \cG^r\cdot \Psi^{(r)}$ implies $\Phi \sim \Psi $ for all $\Psi\in \wh{kQ}_{\ol{\cy}}$. Clearly, it is equivalent to the condition that $\Phi^{(r)}=\Psi^{(r)}$ implies $\Phi\sim\Psi$ for all $\Psi \in \wh{kQ}_{\ol{\cy}}$. If $\Phi$ is $r$-determined for some $r\geq 0$ then it is called \emph{finitely determined} (with respect to $\mathcal{G}$).
\begin{remark}
If $\Phi\in \wh{kQ}_{\ol{\cy}}$ is $r$-determined for some integer $r\geq0$ then $\Phi \sim \Phi^{(r)}$.
\end{remark}
This subsection is devoted to prove the following theorem, which can be viewed as a noncommutative analogue of Mather's infinitesimal criterion \cite[Theorem 1.2]{Wall}.
\begin{theorem}[Finite determinacy] \label{finite-determinacy}
Let $Q$ be a finite quiver and $\Phi\in \wh{\CC Q}_{\ol{\cy}}$ a potential. If the Jacobi algebra $\wh{\Lm}(Q,\Phi)$ is finite dimensional then $\Phi$ is finitely determined. More precisely, if
$\wh{\rJ}(Q,\Phi) \supseteq \wh{\fm}^r$ for some integer $r\geq 0$ then $\Phi$ is $(r+1)$-determined.
\end{theorem}
The proof of this theorem will be addressed after several auxiliary results.
Let us fix some notations once for all. We denote $K$ for the algebra of entire functions on the complex plane $\CC$. The base ring $k$ that we need below are $\CC$ and $K$. Let $l=\CC Q_0$ and $\wh{\fm}$ the ideal of $\wh{\CC Q}$ generated by arrows. Let $L= K Q_0$ and $\wh{\fn}$ the ideal of $\wh{KQ}$ generated by arrows.
We identify $\wh{\CC Q}$ (resp. $\wh{\CC Q}_{\ol{\cy}}$) as a subspace of $\wh{KQ}$ (resp. $\wh{KQ}_{\ol{\cy}}$) in the natural way. Since $l$-algebra automorphisms of $\wh{\CC Q}$ and $L$-algebra automorphisms of $\wh{KQ}$ are both uniquely determined by their values on the arrows, one may naturally identify the group $\Aut_l(\wh{\CC Q}) = \Aut_l(\wh{\CC Q}, \wh{\fm})$ as a subgroup of $\Aut_L(\wh{KQ},\wh{\fn})$. For every $t\in \CC$, let
\[
(-)_t: \wh{KQ} \to \wh{\CC Q}, ~~~~~f\mapsto f_t
\]
be the map given by evaluating coefficients at $t$. Furthermore, there is a map
\[
(-)_t: \Aut_L(\wh{KQ},\wh{\fn}) \to \Aut_l(\wh{\CC Q},\wh{\fm}), ~~~~~ H\mapsto H_t: a \mapsto H(a)_t, \quad a\in Q_1.
\]
One may consider $H\in \Aut_L(\wh{KQ},\wh{\fn})$ as an analytic curve $t\mapsto H_t$ of $l$-automorphisms of $\wh{\CC Q}$.
Given a formal series $f=\sum_{w} a_w(t) w\in \wh{KQ}$, the derivative $\frac{d\, f}{d\, t}$ is defined to be the formal series $\sum_{w} a^\p_w(t) w$. It is easy to check that take derivation preserves cyclic equivalence relation on $\wh{KQ}$. Consequently, one may naturally define $\frac{d\, \Phi}{d\, t}$ for any superpotential $\Phi\in \wh{KQ}_{\ol{\cy}}$.
\begin{lemma}\label{integrate-derivation}
Fix an element $f_a\in e_{s(a)}\cdot \wh{\fn}\cdot e_{t(a)}$ for every arrow $a\in Q_1$. There exists an automorphism $H\in {\rm Aut}_{L} (\wh{KQ}, \mathfrak{n})$ such that
\[
H_0=\id \quad \text{and} \quad\frac{~d\, H(a)}{d~ t} = -H(f_a)\text{ for } a\in Q_1.
\]
\end{lemma}
\begin{proof}
Write $f_a = \sum_{w} \lambda_{a,w}(t) ~w$,
where $w$ runs over all paths and $\lambda_{a,w}(t) \in K$. Note that $\lambda_{a,e_i} = 0$ for all arrows $a$ and nodes $i$, and $\lambda_{a,w} =0$ for all arrows $a$ and paths $w$ that don't have the same source and end.
For each pair of nodes $i$ and $j$, let
\[
M_{ij}(t):=[\lambda_{a,b}(t)]_{a,b\in Q_1^{(ij)}},
\]
which is an $(Q_1^{(ij)} \times Q_1^{(ij)})$-matrix with entries in $K$.
If desired $H$ exists, write
\[
H(a) = \sum_w \gamma_{a,w}(t)~ w, \quad a\in Q_1.
\]
Then we would have
\begin{enumerate}
\item[(1)] $\gamma_{a,e_i} (t)= 0$ for all arrows $a$ and nodes $i$, and $\gamma_{a,w} (t)=0$ for all arrows $a$ and paths $w$ that don't have the same source and end.
\end{enumerate}
Moreover, we would also have:
\[
\sum_{|w|\geq 1}\gamma_{a,w}'(t)~w = - \sum_{|w|\geq1} \lambda_{a,w}(t) ~H(w), \quad a\in Q_1.
\]
Compare the coefficients we then would have for each pair of nodes $i$ and $j$ that
\begin{enumerate}
\item[(2)] $(\gamma_{a,b}(0))_{a\in Q_1^{(ij)}}=(\delta_{a,b})_{a\in Q_1^{(ij)}}$ for all arrows from $i$ to $j$, and $(\gamma_{a,w}(0))_{a\in Q_1^{(ij)}}=0$ for all paths $w$ from $i$ to $j$ of length
$\geq 2$.
\item[(3)] For paths $w$ from $i$ to $j$ of length $\geq 1$,
\[
(\gamma_{a,w}'(t))_{a\in Q_1^{(ij)}} = -M_{ij}(t)\cdot (\gamma_{a,w}(t))_{a\in Q_1^{(ij)}}+(F_{a,w}(t))_{a\in Q_1^{(ij)}},
\] where $F_{a,w}(t) \in e_i \cdot \wh{KQ}\cdot e_j$ is a finite linear sum of finite products of elements in
\[
\{ \lambda_{a,u}(t)~ |~ 2 \leq |u| \leq |w|, ~ s(u)=i, ~ t(u)=j \} \bigcup \{\gamma_{b,v}(t)~|~ b\in Q_1, ~ 1 \leq |v| < |w| \}.
\]
Here the operation of $(Q_1^{(ij)} \times Q_1^{(ij)})$-matrices on $Q_1^{(ij)}$-tuples is defined in the natural way.
\end{enumerate}
By the stand theory of analytic ODE (see \cite{CLbook}), we can construct all these coefficients $\gamma_{a,w}(t)\in K$ by induction on the length of $w$
from the above three conditions (1), (2) and (3). Then of course the induced $L$-algebra endomorphism $H$ of $\wh{KQ}$ satisfies the ODE, and also
\[
[\gamma_{a,b}(t)]_{a,b\in Q_1^{(ij)}} = e^{N_{ij}(t)}, \quad i,j\in Q_0,
\]
where $N_{ij}(t)$ is the primitive of $M_{ij}(t)$ with $N_{ij}(0)$ the identity $(Q_1^{(ij)} \times Q_1^{(ij)})$-matrix.
In particular, $[\gamma_{a,b}(t)]_{a,b\in Q_1^{(ij)}}$
is invertible with inverse $e^{-N_{ij}(t)}$ for each pair of nodes $i$ and $j$. By Lemma \ref{inverse}, $H$ is an $L$-algebra automorphism of $\wh{KQ}$ preserves $\wh{\fn}$.
\end{proof}
\begin{lemma}[Local triviality]\label{thm-localtri}
Let $\Theta\in \wh{KQ}_{\ol{\cy}}$ be a potential. Suppose
$\frac{d \Theta}{d t} \in \Theta_\#(\der_L^+(\wh{KQ}))$. Then
there exists an automorphism $H\in \Aut_L(\wh{KQ},\wh{\fn})$ such that
\[
H_0=\id \quad \text{and} \quad H(\Theta)=\Theta_0 \text{ in } \wh{KQ}_{\ol{\cy}}.
\]
\end{lemma}
\begin{proof}
Choose $\xi\in \der_L^+(\wh{KQ})$ such that $\frac{d \Theta}{d t} = \Theta_\#(\xi)$. Then there exists a double $L$-derivation
\[
\delta=\sum_{a\in Q_1}^n \sum_{u,v} \lambda_{u,v}^{(a)}(t)~ u* \frac{\partial~}{\partial a} * v \in \wh{\dder}_L^+(\wh{KQ}),
\]
where $u$ and $v$ runs over paths with $t(u) = t(a)$ and $s(v) =s(a)$ respectively, such that $\xi=\wh{\mu}\circ \delta$. In $\wh{KQ}_{\ol{\cy}}$, we have
\begin{eqnarray*}
\Theta_\#(\xi)
&=& \pi\bigg ( \Theta_*\big (\wh{\mu}\circ \wh{\tau} \circ \delta \big ) \bigg )\\
&=& \pi \big ( \sum_{a\in Q_1} \sum_{u,v} \lambda_{u,v}^{(a)}(t)~ u\cdot \Theta_*(D_a) \cdot v \big )\\
&=& \pi \big ( \sum_{a\in Q_1} \sum_{u,v} \lambda_{u,v}^{(a)}(t)~ vu\cdot \Theta_*(D_a) \big )\\
&=& \pi \big ( \sum_{a\in Q_1} \xi(a)\cdot \Theta_*(D_a) \big )
\end{eqnarray*}
Note that $\xi(a)\in e_{s(a)} \cdot \wh{\fn}\cdot e_{t(a)}$. Let $H\in {\rm Aut}_{L} (\wh{KQ}, \mathfrak{n})$ be chosen as in Lemma \ref{integrate-derivation}. So
\[
H_0=\id \quad \text{and} \quad\frac{~d\, H(a)}{d~ t} = -H(\xi(a))\text{ for } a\in Q_1.
\]
Then, in $\wh{KQ}_{\ol{\cy}}$ we have
\begin{eqnarray*}
\frac{~~d\, H(\Theta)}{d~ t} &=& H(\frac{~d\, \Theta}{d~ t}) + \pi\bigg(\sum_{a\in Q_1}\frac{~d\, H(a)}{d~ t}\cdot H \big (\Theta_*(D_a)\big) \bigg) \\
&=&H \bigg ( \Theta_\#(\xi)- \pi\big(\sum_{a\in Q_1} \xi(a) \cdot \Theta_*(D_a) \big )\bigg)\\
&=&0.
\end{eqnarray*}
Thus all coefficients of the canonical representative of $H(\Theta) $ are constants and hence $H(\Theta)= H(\Theta)_0= H_0(\Theta_0) = \Theta_0$.
\end{proof}
\begin{lemma}\label{tangent-compare-1}
For any potential $\Phi\in \wh{\CC Q}_{\ol{\cy}} \subseteq \wh{KQ}_{\ol{\cy}} $, we have
\[
\Phi_* \big (\wh{\cder}_l^+ (\wh{\CC Q}) \big ) \supseteq \wh{\fm}^r \Longleftrightarrow \Phi_* \big (\wh{\cder}_{L}^+ (\wh{KQ}) \big ) \supseteq \wh{\fn}^r, ~~~ r>0.
\]
\end{lemma}
\begin{proof}
We may identify $\wh{\CC Q}\wh{\ot}_\CC \wh{\CC Q}$ with a subspace of $\wh{KQ}\wh{\ot}_K\wh{KQ}$ in the obvious way. Since double $l$-derivations of $\wh{\CC Q}$ and double $L$-derivations of $\wh{KQ}$ are uniquely determined by their values at arrows, we may naturally identify $\wh{\dder}_l(\wh{\CC Q})$ with a subspace of $\wh{\dder}_L(\wh{KQ})$. By the definition of cyclic derivations, the forward implication is clear.
To see the backward implication, fix an arbitrary element $f\in \wh{\fm}^r\subseteq \wh{\fn}^r$.
Choose a double $L$-derivation $\delta\in \wh{\dder}_L^+(\wh{KQ})$ with $\Phi_*(\wh{\mu}_{\wh{KQ}} \circ \wh{\tau}_{\wh{KQ}}\circ\delta) = f$. It is easy to see that every element $g\in \wh{KQ}\wh{\ot}_K\wh{KQ}$ can be uniquely decomposed into the following form:
\[
g=g_1+ t\cdot g_2, \quad g_1\in \wh{\CC Q}\wh{\ot}_\CC \wh{\CC Q}, ~~g_2\in \wh{KQ}\wh{\ot}_K\wh{KQ}.
\]
Using the decompositions
\[
\delta(a) = \delta(a)_1+ t\cdot \delta(a)_2, \quad a\in Q_1
\]
as above, we get a decomposition
\[
\delta= \delta_1 +t\delta_2, ~~~\delta_1\in \wh{\dder}_l^+ (\wh{\CC Q}) \subseteq \wh{\dder}_L^+(\wh{KQ}), ~~\delta_2\in \wh{\dder}_L^+(\wh{KQ}).
\]
In fact, $\delta_1$ (resp. $\delta_2$) is given by $\delta_1(a) = \delta(a)_1$ (resp. $\delta_2(a) = \delta(a)_2$).
We have
\[
f = \Phi_*(\wh{\mu}_{\wh{KQ}}\circ \wh{\tau}_{\wh{KQ}}\circ \delta)= \Phi_*(\wh{\mu}_{\wh{\CC Q}}\circ \wh{\tau}_{\wh{\CC Q}}\circ \delta_1) +t \cdot \Phi_*(\wh{\mu}_{\wh{KQ}}\circ \wh{\tau}_{\wh{KQ}}\circ \delta_2).
\]
It follows that $f= \Phi_*(\wh{\mu}_{\wh{\CC Q}}\circ \wh{\tau}_{\wh{\CC Q}}\circ \delta_1) $ and hence $f \in \Phi_*\big(\wh{\cder}_l^+( \wh{F}) \big)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{finite-determinacy}]
It suffices to show the second statement. Suppose $\wh{\rJ}(Q,\Phi) \supseteq \wh{\fm}^r$. We proceed to show $\Phi$ is $(r+1)$-determined. Suppose $\Psi\in \wh{\CC Q}_{\ol{\cy}}$ such that $\Psi^{(r+1)}=\Phi^{(r+1)}$. Let
\[
\Theta:= \Phi+ t(\Psi-\Phi) \in \wh{KQ}_{\ol{\cy}}.
\]
Clearly, we have
\[
\Phi_* \big (\wh{\cder}_l^+ (\wh{\CC Q}) \big ) \supseteq \wh{\fm}^{r+1}.
\]
Then Lemma \ref{tangent-compare-1} tells us
\[
\Phi_* \big (\wh{\cder}_L^+ (\wh{KQ}) \big ) \supseteq \wh{\fn}^{r+1}.
\]
Since $\Theta$ and $\Phi$ has the same $(r+1)$-jet in $\wh{KQ}_{\ol{\cy}}$, it follows readily that
\[
\Theta_*\big (\wh{\cder}_{L} ^+(\wh{KQ}) \big) +\wh{\fn}^{r+2} = \Phi_* \big (\wh{\cder}_{L} ^+ (\wh{KQ}) \big) +\wh{\fn}^{r+2}\supseteq \wh{\fn}^{r+1}.
\]
Then Lemma \ref{Nakayama}, the Nakayama lemma, tells us
\[
\Theta_*\big (\wh{\cder}_{L} ^+(\wh{KQ}) \big) \supseteq \wh{\fn}^{r+1}.
\]
Consequently,
\[
\Theta_{\#} \big (\der_{L} ^+ (\wh{KQ}) \big ) = \pi \big(\Theta_* \big (\wh{\cder}_{L} ^+(\wh{KQ})\big )\big )\supseteq \pi(\wh{\fn}^{r+1}) \ni \Psi-\Phi = \frac{~d\, \Theta}{d~ t}.
\]
Apply Lemma \ref{thm-localtri}, there is an automorphism $H\in \Aut_L(\wh{KQ},\wh{\fn})$ such that $H(\Theta) = \Theta_0 = \Phi$. In particular,
$H_1(\Psi)= H_1(\Theta_1) = H(\Theta)_1 = \Phi $
and so $\Psi$ is right equivalent to $\Phi$.
\end{proof}
\subsection{Proof of the nc Mather-Yau theorem (Theorem \ref{ncMY})}
\begin{proof}[Proof of Theorem \ref{ncMY}]
By Proposition \ref{Jacobi-transform}, it remains to show (1) implies (2).
Let $\gamma: \wh{\Lm}(Q,\Phi)\to \wh{\Lm}(Q,\Psi)$ be an $l$-algebra isomorphism such that $\gamma_*([\Phi]_\Phi)=[\Psi]_\Psi$. We claim that $\gamma$ can be lifted to an $l$-algebra automorphism of $\wh{\CC Q}$.
Denote the image of arrows $a\in Q_1$ in $ \wh{\Lm}(Q,\Phi)$ and $\wh{\Lm}(Q,\Psi)$ both by $\ol{a}$. Clearly, any lifting $h_a\in e_{s(a)} \cdot \CC Q \cdot e_{t(a)}$ of $\gamma(\ol{a})$, $a\in Q_1$, lifts $\gamma$ to an $l$-algebra endomorphism $H: a\mapsto h_a$ of $\wh{\CC Q}$. In other words, we have a commutative diagram:
\[
\xymatrix{
\wh{\CC Q}\ar[d]\ar[r]^H & \wh{\CC Q}\ar[d]\\
\wh{\Lm}(Q,\Phi)\ar[r]^\gamma & \wh{\Lm}(Q,\Psi).
}
\]
Recall that $\wh{\fm}_\Phi \subset \wh{\Lm}(Q,\Phi)$ is defined to be $\wh{\fm}/ \wh{\rJ}(Q, \Phi)$ and similarly for $\wh{\fm}_\Psi\subset \wh{\Lm}(Q,\Psi)$. Because $\Phi$ and $\Psi$ are of order $\geq3$, there is a canonical isomorphism of $l$-bimodules $\wh{\fm}/\wh{\fm}^2\cong \wh{\fm}_\Phi/\wh{\fm}_\Phi^2\cong \wh{\fm}_\Psi/\wh{\fm}_\Psi^2$. Because $\gamma$ induces an isomorphism on $\wh{\fm}_\Phi/\wh{\fm}_\Phi^2\cong \wh{\fm}_\Psi/\wh{\fm}_\Psi^2$, $H$ induces an isomorphism on $\wh{\fm}/\wh{\fm}^2$. By Lemma \ref{inverse}, $H$ is invertible.
By the assumption $\gamma_*([\Phi]_\Phi) = [\Psi]_\Psi$ we have $[H(\Phi)]_\Psi=[\Psi]_\Psi$, and by Lemma \ref{Jacobi-transform} we have
\[
\wh{\rJ}(Q,\Psi) = H(\wh{\rJ}(Q,\Phi)) = \wh{\rJ}(Q,H(\Phi)).
\]
Thus, without lost of generality, we may replace $\Phi$ by $H(\Phi)$ and assume in priori that
\[
\wh{\rJ}(Q,\Phi) = \wh{\rJ}(Q,\Psi) \quad \text{ and }\quad [\Phi]_\Phi=[\Psi]_\Psi.
\]
Let $r$ be the minimal integer so that $\wh{\rJ}(Q,\Phi) \supseteq \wh{\fm}^r$.
By finite determinacy (Theorem \ref{finite-determinacy}), it suffice to show that $\Phi^{(s)}$ and $\Psi^{(s)}$ lie in the same orbit of $\cG^s=\Aut_l(J^s)$ for $s=r+1$. If $\Phi^{(s)}=\Psi^{(s)}$ then there is nothing to proof. So we may assume further that $\Phi^{(s)}\neq \Psi^{(s)}$.
Since $J^s_{\cy}$ is a finite dimensional vector space, it has a natural complex manifold structure. Also, it is not hard to check that $\mathcal{G}^s$ is a complex Lie group acts analytically on $J^s_\cy$. So the orbit $\mathcal{G}^s\cdot \Xi^{(s)}$ is an immersed submanifold of $J^s_\cy$ for any potential $\Xi\in \wh{\CC Q}_{\ol{\cy}}$. We proceed to calculate $T_{\Xi^{(s)}}(\mathcal{G}^s\cdot \Xi^{(s)})$, the tangent space of $\mathcal{G}^s\cdot \Xi^{(s)}$ at $\Xi^{(s)}$. It is easy to check that $\delta(\wh{\fm}/ \wh{\fm}^{s+1}) \subseteq \wh{\fm}/ \wh{\fm}^{s+1}$ for every $\delta\in \der_l(J^s)$, so the canonical map $\rho_s: \der_l^+ (\wh{\CC Q}) \to \der_l(J^s)$ is surjective. Consequently, we have a commutative diagram of vector spaces over $\CC$ as following:
\[
\xymatrix{
\wh{\dder}_l^+ (\wh{\CC Q}) \ar@{->>}[r]^-{\wh{\mu}\circ \wh{\tau}\circ-} \ar@{->>}[d]^-{\wh{\mu}\circ-} & \wh{\cder}_l^+ (\wh{\CC Q}) \ar[r]^-{\Xi_*} & \wh{\CC Q}\ar[d]^{\pi} \\
\der_l^+ (\wh{\CC Q}) \ar@{->>}[d]^{\rho_s} \ar[rr]^-{\Xi_{\#}} & & \wh{\CC Q}_{\ol{\cy}} \ar[d]^{q_s} \\
\der_l(J^s) \ar[rr]^{(\Xi^{(s)})_{\#}} && J^s_\cy,
}
\]
where $(\Xi^{(s)})_{\#}$ is constructed in Lemma \ref{cycprop} (1). Recall that $\der_l(J^s)$ is the tangent space of $\mathcal{G}^s$ at the identity map, we have
\[
T_{\Xi^{(s)}}(\mathcal{G}^s\cdot \Xi^{(s)}) = \im ((\Xi^{(s)})_{\#}) =q_s \bigg(\pi \big(\wh{\fm}\cdot \wh{\rJ}(Q,\Xi) + \wh{\rJ}(Q,\Xi)\cdot \wh{\fm} \big)\bigg).
\]
Now consider the complex line $\cL:=\{~ \Theta_t^{(s)}=t\Psi^{(s)}+(1-t)\Phi^{(s)}~|~t\in\CC ~ \}$ contained in $J^s_\cy$. By the assumption that $\wh{\rJ}(Q,\Phi) = \wh{\rJ}(Q,\Psi)$, we have
\[T_{\Psi^{(s)}} (\mathcal{G}^s \cdot \Psi^{(s)}) = T_{\Phi^{(l)}} (\mathcal{G}^s\cdot \Phi^{(s)})=q_s \bigg(\pi \big(\wh{\fm}\cdot \wh{\rJ}(Q,\Phi) + \wh{\rJ}(Q,\Phi) \cdot \wh{\fm} \big)\bigg),\]
as subspaces of $J^s_\cy$. It follows that for any $t$ the tangent space $T_{\Theta_t^{(s)}} (\mathcal{G}^s\cdot \Theta_t^{(s)})$ is a subspace of $q_s \bigg(\pi \big(\wh{\fm}\cdot\wh{\rJ}(Q,\Phi) + \wh{\rJ}(Q,\Phi) \cdot \wh{\fm} \big)\bigg)$. Let $\cL_0$ be the subset of $\cL$ consisting of those $\Theta_t^{(s)}$ such that
\[
T_{\Theta_t^{(s)}} (\mathcal{G}\cdot \Theta_t^{(s)}) = q_s \bigg(\pi \big(\wh{\fm}\cdot \wh{\rJ}(Q,\Phi) + \wh{\rJ}(Q,\Phi)\cdot \wh{\fm} \big)\bigg).
\]
Then $\Phi$ and $\Psi$ are both in $\cL_0$. It remains to show that $\cL_0$ lies in the orbit $\cG^s \cdot\Phi^{(s)}$. By a standard lemma in the theory of Lie group (c.f. Lemma 1.1 \cite{Wall}), it suffices to check that
\begin{enumerate}
\item[(1)] The complement $\cL\backslash \cL_0$ is a finite set (so $\cL_0$ is a connected smooth submanifold of $J^s_\cy$);
\item[(2)] For all $\Theta_t^{(s)}\in \cL_0$, the dimension of $T_{\Theta_t^{(s)}}(\cG^s \cdot \Theta_t^{(s)})$ are the same;
\item[(3)] For all $\Theta_t^{(s)}\in \cL_0$, the tangent space $T_{\Theta_t^{(s)}}(\cL_0)$ is contained in $T_{\Theta_t^{(s)}}(\cG^s \cdot \Theta_t^{(s)})$.
\end{enumerate}
Condition (1) holds because $\cL\backslash \cL_0$ corresponds to the locus of the continuous family
\[\{(\Theta_{t}^{(s)})_{\#}: \der_l(J^s) \to J^s_\cy \}_{t\in \CC}\]
of linear maps between two finite dimensional spaces that have lower rank. Condition $(2)$ follows from the construction of $\cL_0$. Note that the tangent space of $\cL_0$ at each of its point is spanned by $\Phi^{(s)}-\Psi^{(s)} = q_s(\Phi-\Psi)$ in $J^s_\cy$. By Proposition \ref{Bootstrapping} (2), condition (3) holds if $\Phi-\Psi \in \pi(\wh{\rJ}(Q,\Phi))$, which is equivalent to the assumption that $[\Phi]_\Phi=[\Psi]_\Psi$.
\end{proof}
\section{Applications}\label{sec:app}
We give several applications of the noncommutative Mather-Yau theorem. In Section \ref{sec:CY-Ginzdg}, we recall the definitions of complete Ginzburg dg-algebra, topological Calabi-Yau algebra and topological generalized cluster category following \cite{KY11}, and prove Theorem \ref{mainthm-rigidity}. In Section \ref{sec:contraction1}, we define the relative and classical singularity category following \cite{KY12}. The latter is an example of generalized cluster category. After recalling the geometric context of 3-dimensional flopping contractions in \cite{DW13,HT16}, we define the contraction algebra. They provide examples of finite dimensional Jacobi algebras, whose derived category of finite dimensional modules are not (smooth) Calabi-Yau categories. Then we prove Theorem \ref{mainthm-contraction} and show several examples. In Section \ref{sec:conj}, we speculate about possible research directions on noncommutative approach to 3-dimensional birational geometry.
\subsection{A rigidity theorem on complete Ginzburg dg-algebra}\label{sec:CY-Ginzdg}
First we recall the definition of the \emph{complete Ginzburg dg-algebra} $\hD(Q,\Phi)$ associated to a finite quiver $Q$ and a potential $\Phi\in \wh{kQ}_{\ol{\cy}}$, where $k$ is a fixed field.
\begin{definition}(Ginzburg)
Let $Q$ be a finite quiver and $\Phi$ a potential on $Q$. Let $\ol{Q}$ be the graded quiver with the same vertices as $Q$ and whose arrows are
\begin{enumerate}
\item[$\bullet$] the arrows of $Q$ (of degree 0);
\item[$\bullet$] an arrow $\theta_a:j\to i$ of degree $-1$ for each arrow $a:i\to j$ of $Q$;
\item[$\bullet$] a loop $t_i:i\to i$ of degree $-2$ for each vertex $i$ of $Q$.
\end{enumerate}
The (complete) Ginzburg (dg)-algebra $\hD(Q,\Phi)$ is the dg $k$-algebra whose underlying graded algebra is the completion (in the category of graded vector spaces) of the graded path algebra $k\ol{Q}$ with respect to the two-sided ideal generated by the arrows of $\ol{Q}$. Its differential is the unique linear endomorphism homogeneous of degree 1 satisfying the Leibniz rule, and which takes the following values on the arrows of $\ol{Q}$:
\begin{enumerate}
\item[$\bullet$] $\fd e_i=0$ for $i\in Q_0$ where $e_i$ is the idempotent associated to $i$;
\item[$\bullet$] $\fd a=0$ for $a\in Q_1$;
\item[$\bullet$] $\fd(\theta_a)=\Phi_*(D_a)$ for $a\in Q_1$;
\item[$\bullet$] $\fd(t_i)=e_i(\sum_{a\in Q_1}[a,\theta_a])e_i$ for each $i\in Q_0$.
\end{enumerate}
It follows from Lemma \ref{Poincare} that $\hD(Q,\Phi)$ is a dg-$l$-algebra with $l=kQ_0$. Taking the topology into consideration, $\hD(Q,\Phi)$ is a pseudocompact dg-$l$-algebra in the sense of \cite[Appendix]{KY11}.
\end{definition}
The degree zero component of $\hD(Q,\Phi)$ is exactly the complete path algebra $\wh{kQ}$, which is itself a pseudocompact algebra. The $(-1)$- component $\hD^{-1}(Q,\Phi)$ consists of formal series of the form
\[
\sum_{a\in Q_1} \sum_{u,v} A_{u,v}^{(i)}~ u\cdot \theta_a \cdot v,~~~~~A_{u,v}^{(i)}\in k,
\]
where $t(u)=t(a)$ and $s(v)=s(a)$. Note that both $\hD^{-1}(Q,\Phi)$ and $\wh{\dder}_l(\wh{kQ})$ are pseudocompact $\wh{kQ}$-bimodules in the sense of \cite[Appendix]{KY11}.
Moreover, we have a commutative diagram
\begin{equation}\label{Ginzburg-superpotential}
\xymatrix{
\hD^{-1}(Q,\Phi) \ar[rr]^-{\fd} \ar[d]^{\cong} & & \wh{kQ} \ar[d]^-{=} \\
\wh{\dder}_l(\wh{kQ}) \ar[r]^-{\wh{\mu} \circ \wh{\tau} \circ- } & \wh{\cder}_l(\wh{kQ}) \ar[r]^-{\Phi_*} & \wh{kQ}.
}
\end{equation}
The map $\hD^{-1}(Q,\Phi) \xrightarrow{\cong} \wh{\dder}_l(\wh{kQ})$ appears above is the $\wh{kQ}$-bimodule isomorphism given by
\[
\sum_{a\in Q_1} \sum_{u,v} A_{u,v}^{(i)}~ u\cdot \theta_a \cdot v \mapsto \sum_{a\in Q_1} \sum_{u,v} A_{u,v}^{(i)}~ u * \frac{\partial}{\partial a} * v.
\]
Recall that the notation $*$ denotes the scalar multiplication of the bimodule structure of $\wh{\dder}_l(\wh{kQ})$ induced from the inner bimodule structure of $\wh{kQ}\wh{\ot}\wh{kQ}$.
\begin{lemma}
Let $Q$ be a finite quiver. For any potential $\Phi\in \wh{kQ}_{\ol{\cy}}$, one has
\[\wh{\Lm}(Q, \Phi) = H^0(\hD(Q,\Phi)).\]
\end{lemma}
\begin{proof}
This is a direct consequence of the commutative diagram (\ref{Ginzburg-superpotential}).
\end{proof}
The main theorem of this subsection is as follows.
\begin{theorem}\label{rigidity-Ginzburg}
Fix a finite quiver $Q$. Let $\Phi,\Psi\in \wh{\CC Q}_{\ol{\cy}}$ be two potentials of order $\geq 3$, such that the Jacobi algebras
$\wh{\Lm}(Q,\Phi)$ and $\wh{\Lm}(Q,\Psi)$ are both finite dimensional. Assume there is a $\CC Q_0$-algebra isomorphism $\gamma: \wh{\Lm}(Q,\Phi)\to\wh{\Lm}(Q,\Psi)$ so that
$\gamma_*([\Phi])=[\Psi]$. Then there exists a dg-$\CC Q_0$-algebra isomorphism
\[
\xymatrix{
\Gamma: \hD(Q,\Phi)\ar[r]^{\cong} &\hD(Q,\Psi)
}
\]
such that $\Gamma(t_i)=t_i$ for all $i\in Q_0$.
\end{theorem}
\begin{proof}
Let $l=\CC Q_0$. By the nc Mather-Yau theorem (Theorem \ref{ncMY}), there exists a $l$-algebra automorphism $H$ of $\wh{\CC Q}$ such that $H(\Phi)=\Psi$ in $\wh{\CC Q}_{\ol{\cy}}$. Choose such an automorphism $H$ so that $a\mapsto h_a$, and let $H^{-1}: a\mapsto h_a^{-1}$ be its inverse. {\bf{Warning}:} $h_a^{-1}$ refers to the component of the automorphism $H^{-1}$ that corresponds to $a$, instead of the inverse of $h_a$.
Define a dg-algebra homomorphism $\Gamma: \hD(Q,\Phi)\to \hD(Q,\Psi)$ by
\[
\Gamma: e_i\mapsto e_i,~~~ a\mapsto h_a, ~~~ \theta_a\mapsto \sum_{\beta\in Q_1} H\bigg(\big(\frac{\partial h_\beta^{-1}}{\partial a}\big)^\pp\bigg) \cdot \theta_\beta\cdot H\bigg(\big(\frac{\partial h_\beta^{-1}}{\partial a}\big)^\p\bigg),~~~ t_i\mapsto t_i.
\]
We need to check that $\Gamma$ is compatible with $\fd$. The above assignment defines a morphism of dg-algebras if and only if the following equalities hold:
\begin{eqnarray*}
\Gamma \big(\fd(\theta_a) \big ) &=&\fd \big(\Gamma(\theta_a) \big), \quad \quad a\in Q_1;\\
\Gamma\big( \fd(t_i) \big)&=&\fd \big( \Gamma(t_i) \big), \quad \quad i\in Q_0.
\end{eqnarray*}
We verify the first equality (using chain rule).
\begin{align*}
\Gamma \big(\fd(\theta_a) \big )&=\Gamma \big(\Phi_*(D_a)\big)=\Gamma \big( H^{-1}(\Psi)_*(D_a) \big)\\
&=H\bigg(\sum_{\beta \in Q_1}\big(\frac{\partial h_\beta^{-1}}{\partial a}\big)^\pp\cdot H^{-1}\big(\Psi_*(D_\beta)\big) \cdot \big(\frac{\partial h_\beta^{-1}}{\partial a}\big)^\p\bigg)\\
&=\sum_{\beta \in Q_1} H\bigg(\big(\frac{\partial h_\beta^{-1}}{\partial a}\big)^\pp\bigg) \cdot \Psi_*(D_\beta) \cdot H\bigg(\big(\frac{\partial h_\beta^{-1}}{\partial a}\big)^\p\bigg)\\
&=\fd \big(\Gamma(\theta_a) \big)
\end{align*}
By the canonical identification $\hD^{-1}(Q,\Phi) \cong \wh{\dder}_l(\wh{\CC Q})$, to verify the second equality, it suffices to show the following equality holds for any $a$.
\begin{align*}
a *\frac{\partial}{\partial a}-\frac{\partial}{\partial a}* a ~~=~~&\sum_{\beta\in Q_1} h_\beta *H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\pp\bigg)* \frac{\partial}{\partial a} *H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\p\bigg) \\ &- H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\pp\bigg)* \frac{\partial}{\partial a}*H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\p\bigg)* h_\beta.
\end{align*}
It is enough to check the application of both sides on $b\in Q_1$. For $b\neq a$, both sides equal to zero. The $b=a$ case reduces to the equality
\[
e_{s(a)}\ot a-a\ot e_{t(a)}=\sum_{\beta\in Q_1} H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\p\bigg)\ot h_\beta H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\pp\bigg) - H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\p\bigg)h_\beta \ot H\bigg(\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\pp\bigg)
\]
Applying $H^{-1}\wh{\ot} H^{-1}$ to both sides of the equation, it is equivalent to verify the identity
\[
e_{s(a)}\ot h^{-1}_a-h^{-1}_a\ot e_{t(a)}=\sum_{\beta\in Q_1} \big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\p\ot \beta\big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\pp- \big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\p \beta \ot \big(\frac{\partial h_a^{-1}}{\partial \beta}\big)^\pp.
\]
We claim this holds for arbitrary path $w = a_1\ldots a_r$ with $s(w)=s(a)$ and $t(w)=t(a)$, and therefore holds in general. Indeed, we have
\[
e_{s(a)}\ot w - w\ot e_{t(a)} =\sum_{\beta\in Q_1}\sum_{a_s=\beta} \big( a_1\ldots a_{s-1} \ot a_{s} a_{s+1} \ldots a_r -a_1\ldots a_{s-1} a_s \ot a_{s+1} \ldots a_r \big).
\]
Since
\[
\frac{\partial w}{\partial \beta}=\sum_{a_s=\beta}a_1\ldots a_{s-1} \ot a_{s+1} \ldots a_r,
\]
the desired identity follows.
Similarly, we may define a dg-morphism
\[
\Gamma^{-1}: e_i\mapsto e_i, ~~~ b\mapsto h^{-1}_b, ~~~ \theta_b\mapsto \sum_{\alpha\in Q_1} H^{-1}\bigg(\big(\frac{\partial h_\alpha}{\partial b}\big)^\pp\bigg)\cdot \theta_\alpha \cdot H^{-1}\bigg(\big(\frac{\partial h_\alpha}{\partial b}\big)^\p\bigg),~~~ t_i\mapsto t_i.
\]
Apply the canonical identification $\hD^{-1}(Q,\Phi) \cong \wh{\dder}_l(\wh{\CC Q})$ again, to prove $\Gamma^{-1}$ is the inverse of $\Gamma$, it suffices to check the identity
\[
\frac{\partial}{\partial b}=\sum_{\alpha\in Q_1}\sum_{\beta\in Q_1} \big(\frac{\partial h_\alpha^{-1}}{\partial b}\big)^\pp *H^{-1}\bigg(\big(\frac{\partial h_\beta}{\partial b}\big)^\pp\bigg) * \frac{\partial}{\partial \beta} *H^{-1}\bigg(\big(\frac{\partial h_\beta}{\partial b}\big)^\p\bigg)* \big(\frac{\partial h_\alpha^{-1}}{\partial b}\big)^\p
\]
It suffices to check the application of both sides on all $a$ such that $s(a)=s(b)$ and $t(a)=t(b)$. Then we get
\begin{align*}
\sum_{\alpha\in Q_1} H^{-1}\bigg(\big(\frac{\partial h_a}{\partial b}\big)^\p\bigg) \big(\frac{\partial h_\alpha^{-1}}{\partial b}\big)^\p\ot \big(\frac{\partial h_\alpha^{-1}}{\partial b}\big)^\pp H^{-1}\bigg(\big(\frac{\partial h_a}{\partial b}\big)^\pp\bigg) &=\frac{\partial\big(H^{-1}(h_a)\big)}{\partial b}\\
&=\delta_{a,b}e_{s(a)}\ot e_{t(a)}.
\end{align*}
The equality follows by the chain rule.
The desired identity follows.
\end{proof}
\begin{remark}
The condition $\Gamma(t_i)=t_i$ in Theorem \ref{rigidity-Ginzburg} can be interpreted as a volume-preserving condition in noncommutative geometry.
\end{remark}
\begin{remark}
When we finished our proof of Theorem \ref{rigidity-Ginzburg}, we learned that Keller and Yang has already got the fact that every $l$-algebra automorphism $H$ of $\wh{kQ}$, which transform $\Phi$ to $\Psi$, can be extended to a dg-$l$-algebra isomorphism $\hD(\wh{kQ}, \Phi) \xrightarrow{\cong} \hD(\wh{kQ}, \Psi)$, see \cite[Lemma 2.9]{KY11}. From this fact and the nc Mather-Yau theorem, one immediately obtain Theorem \ref{rigidity-Ginzburg} as well. However, we retain our proof in full detail for completeness and reader's convenience.
\end{remark}
The correct setup to discuss the complete Ginzburg dg-algebra is to use the language of pseudocompact dg-algebras and derived categories. We now state several definitions and results due to Keller and Van den Bergh. For simplicity, we will omit the definitions of pseudocompact dg-algebras and derived categories. The interested readers can find the details in the Appendix of \cite{KY11} and a generalization in Section 6 of \cite{VdB15}.
Let $l$ be a finite dimensional separable $k$-algebra over a field $k$.
Let $A$ be a pseudocompact dg-$l$-algebra. Denote the pseudocompact derived category of $A$ by $\D_{pc}(A)$. Define the \emph{perfect derived category} $\per_{pc}(A)$ to be the thick subcategory of $\D_{pc}(A)$ generated by the free $A$-module of rank 1. Define the \emph{finite-dimensional derived category} $\D_{fd,pc}(A)$ to be the full subcategory whose objects are the pseudocompact dg-modules $M$ such that
$\Hom_{D_{pc}(A)}(P,M)$ is finite dimensional for each perfect $P$. We say that $A$ is \emph{topologically homologically smooth} if the module $A$ considered as a pseudocompact dg-module over $A^e:=A \wh{\ot} A^{op}$ is quasi-isomorphic to a strictly perfect dg-module (See \cite[Appendix A.11]{KY11} for the definition).
Let $d$ be an integer. For an object $L$ of $\D_{pc}(A^e)$, define $L^\# =\RHom_{A^e}(L,A^e [d])$.
The dg-algebra $A$ is \emph{topologically bimodule $d$-Calabi-Yau} if there is an isomorphism
\[
A \xrightarrow{\cong} A^\#
\]
in $\D_{pc}(A^e)$. In this case, the category $\D_{fd,pc} (A)$
is $d$-Calabi-Yau as a triangulated category. If a pseudocompact dg-algebra $A$ is topologically homologically smooth and topological $d$-Calabi-Yau as a bimodule, we say $A$ is a \emph{topological $d$-Calabi-Yau algebra}. In this case, $\D_{fd,pc}(A)$ is a full subcategory of $\per_{pc}(A)$ (Proposition A.14 (c) \cite{KY11}).
Note that if a pseudocompact dg-algebra $A$ is topologically $d$-Calabi-Yau and has cohomology concentrating in degree $0$ then $H^0(A)$ is also topologically $d$-Calabi-Yau as a pseudocompact algebra (see \cite[Proposition A.14 (e) ]{KY11}). Here, the topological structure on $H^0(A)$ is inherited from $A$. If $A$ is algebraic (non-topological), then the notion of homologically smooth, bimodule Calabi-Yau and $d$-Calabi-Yau algebra can be defined similarly but with the pseudocompact derived category $\D_{pc}$ replaced by the algebraic derived category.
\begin{theorem}(\cite[Theorem A.17]{KY11})\label{GinzCY}
Complete Ginzburg dg-algebras are topologically $3$-Calabi-Yau.
\end{theorem}
Assume a pseudocompact dg-$l$-algebra $A$ satisfies the following additional conditions:
\begin{enumerate}
\item[$(1)$] A is topologically $3$-Calabi-Yau,
\item[$(2)$] for each $p>0$, the space $H^p(A)$ vanishes,
\item[$(3)$] the algebra $H^0(A)$ is finite-dimensional over $k$.
\end{enumerate}
The \emph{topological generalized cluster category} of $A$ is defined to be the Verdier quotient
$$\per_{pc}(A)/\D_{fd,pc}(A).$$
For a non-topological dg-$l$-algebra $A$, which is (non-topological) $3$-Calabi-Yau and satisfies conditions (2) and (3) above, the \emph{algebraic generalized cluster category} $\per(A)/\D_{fd}(A)$ was first studied by Amiot (see Theorem 2.1 \cite{Am}).
As an immediate consequence of Theorem \ref{rigidity-Ginzburg}, we have
\begin{corollary}\label{clustercat}
Fix a finite quiver $Q$. Let $\Phi,\Psi\in \wh{\CC Q}_{\ol{\cy}}$ be two potentials of order $\geq 3$, such that the Jacobi algebras
$\wh{\Lm}(Q,\Phi)$ and $\wh{\Lm}(Q,\Psi)$ are both finite dimensional. Assume there is an $\CC Q$-algebra isomorphism $\gamma: \wh{\Lm}(Q,\Phi)\to\wh{\Lm}(Q,\Psi)$ so that
$\gamma_*([\Phi])=[\Psi]$. Then the topological generalized cluster categories of $\hD(Q,\Phi)$ and of $\hD(Q,\Psi)$ are triangle equivalent. \hfill $\Box$
\end{corollary}
\subsection{Three dimensional flopping contractions}\label{sec:contraction1}
\begin{definition}\label{nccr}
Let $(R,\fm)$ be a complete commutative Noetherian local Gorenstein $k$-algebra of Krull dimension $n$ with isolated singularity and with residue field $k$. Denote the category of maximal Cohen--Macaulay (MCM) modules by $\CM_R$ and its stable category by $\ul{\CM}_R$. Let $N_0=R,N_1,N_2,\ldots,N_t$ be pairwise non-isomorphic indecomposables in $\CM_R$ and $A:=\End_R(\bigoplus_{i=0}^t N_i)$. We call $A$ a \emph{noncommutative resolution} (NCR) of $R$ if it has finite global dimension. A NCR is called a \emph{noncommutative crepant resolution} (NCCR) if $A$ further satisfies that
\begin{enumerate}
\item[$(a)$] $A\in \CM_R$;
\item[$(b)$] ${\rm{gldim}}(A)=n$.
\end{enumerate}
\end{definition}
If $A$ is a NCCR, we call $\op_{i=0}^t N_i$ a \emph{tilting module}.
Denote $\ol{l}$ for $A/\rad A $ and $e_i$ for the idempotent given by the projection $\bigoplus_{i=0}^t N_i\to N_i$. Let $S_0,S_1,\ldots,S_t$ be the simple $A$-modules with $S_i$ corresponding to the summand $N_i$ of $\bigoplus_{i=0}^t N_i$.
There is a fully faithful triangle functor \[\K^b(\proj-R)\to \D^b(\Mod-A),\] whose essential image equals the thick closure of $e_0A$ in $\D^b(\Mod-A)$.
\begin{definition}\cite[Definition 1.1, Definiton 1.2]{KY12}
The Verdier quotient category
\[
\Delta_R(A):=\frac{\D^b(\Mod-A)}{\K^b(\proj-R)}\cong\frac{\D^b(\Mod-A)}{{\rm{thick}}(e_0A)}
\]
is called the \emph{relative singularity category}, and
the Verdier quotient category
\[
\D_{sg}(R):=\frac{\D^b(\Mod-R)}{\K^b(\proj-R)}
\]
is called the \emph{classical singularity category}.
\end{definition}
Note that $\CM_R$ is a Frobenius category with $\proj (\CM_R)=\proj-R$, and there is a triangle equivalence $\D_{sg}(R) \cong \ul{\CM}_R:=\CM_R/\proj-R$ (see e.g. \cite{Bu87}).
De Thanhoffer de V\"olcsey and Van den Bergh prove that $\ul{\CM}_R$ admits an explicit
dg model in this case.
\begin{theorem}(\cite[Theorem 1.1]{ThV})\label{VdBTh}
There exists a finite dimensional $\ol{l}$-bimodule $V$ and a minimal model
$(\wh{T}_{\ol{l}}V,d)\iso A$ for $A$. Put $\Gamma=\wh{T}_{\ol{l}}V/\wh{T}_{\ol{l}}V e_0 \wh{T}_{\ol{l}} V$.
Then one has
\[
\ul{\CM}_R\cong \per(\Gamma)/\thick(S_1,\ldots, S_t)
\] and furthermore $\Gamma$ has the following properties
\begin{enumerate}
\item[$(1)$] $\Gamma$ has finite dimensional cohomology in each degree;
\item[$(2)$] $\Gamma$ is concentrated in negative degrees;
\item[$(3)$] $H^0\Gamma=A/Ae_0A$;
\item[$(4)$] As a graded algebra $\Gamma$ is of the form $\wh{T}_{l} V^0$ for $V^0=(1-e_0)V(1-e_0)$ and $l=\ol{l}/ke_0$.
\end{enumerate}
\end{theorem}
\begin{remark}
A related theorem was proved by Kalck and Yang (see \cite{KY12}).
\end{remark}
Now we recall the construction of the NCCR for the 3-dimensional flopping contractions. Let $k=\CC$ be the field of complex numbers.
\begin{definition}
Let $Y$ be a normal $\CC$-variety of dimension 3. A \emph{flopping contraction} is a proper birational morphism $f:Y\to X$ to a normal variety $X$ such that $f$ is an isomorphism in codimension one, and $K_Y$ is $f$-trivial. If $Y$ is smooth, then we call $f$ a smooth flopping contraction. In this case, the condition that $K_Y$ is $f$-trivial is redundant.
\end{definition}
If $f:Y\to X$ is a smooth flopping contraction, then $X$ has Gorenstein rational singularities. By passing to formal completion, we may assume $X=\Spec R$ for a complete local Gorenstein $\CC$-algebra $R$ of dimension 3. Denote the closed point of $X$ by $p$ and the reduced fiber of $f$ at $p$ by $C^\red$.
It is well known that $C^{\red}$ has a decomposition
\[
C^{\red}=\bigcup_{i=1}^t C^\red_i.
\]
where $C^{\red}_i\cong \PP^1$ (see Lemma 3.4.1 of \cite{VdB04}).
Let $\cL_i$ be a line bundle on $Y$ such that $\deg_{C_j}\cL_i=\delta_{ij}$. Define $\cN_i$ to be given by the maximal extension
\begin{equation}\label{extN}
\xymatrix{
0\ar[r] & \cL_i^{-1}\ar[r] & \cN_i\ar[r] &\cO_{Y}^{\oplus r_i}\ar[r] &0
}
\end{equation} associated to a minimal set of $r_i$ generators of $H^1(Y,\cL_i^{-1})$.
Set $N_i:=\bR f_*\cN_i=f_*\cN_i$. Then
\[
A:= \End_{R}(R\op N_1\op\ldots N_t)
\] is a NCCR.
Van den Bergh (\cite[Section~3.2.8]{VdB04}) showed that there is an equivalence of triangulated categories $\D^b(\rm{coh}(Y))\cong \D^b(\text{mod-}A)$. As a consequence, $A$ is homologically smooth and bimodule 3CY. In this case, we denote by $\Delta_R(Y)$ the relative singularity category associate to $A$.
\begin{definition}(\cite[Definition~2.8]{DW13}) \label{A_con}
Let $R$, $N_i$ and $A$ be those defined in \ref{extN}.
The \emph{contraction algebra} $A_{con}$ is defined to be $A/A e_0 A$, where $e_0$ is the idempotent that corresponds to the projection to $R$.
\end{definition}
By Theorem \ref{VdBTh}, the minimal model $(\wh{T}_{\ol{l}} V,d)$ of $A$ is (topologically) homologically smooth and bimodule 3CY. By Corollary 9.3 and Theorem 11.2.1 of \cite{VdB15}, it admits a structure of Ginzburg dg-algebra. The localization $\wh{T}_{\ol{l}}V/\wh{T}_{\ol{l}}V e_0 \wh{T}_{\ol{l}} V$ also admits a structure of Ginzburg dg algebra over the semisimple ring $l=ke_1+\ldots ke_t$. We denote it by $\hD(Y,\Phi)$, and its Jacobi algebra by $\wh{\Lm}(Y,\Phi)$. By $(3)$ of Theorem \ref{VdBTh}, the contraction algebra $A_{con}$ is isomorphic to $\wh{\Lm}(Y,\Phi)$.
\begin{remark}
For the construction of the minimal model $(\wh{T}_l V,d)$ and the potential $\Phi$, we refer the readers to \cite{VdB15}. We just remark that the underlying graded algebra $\hD(Y,\Phi)$ is determined by the graded vector spaces $\Ext^*_Y(\bigoplus_{i=1}^t\cO_{C^{\red}_i},\bigoplus_{i=1}^t\cO_{C^{\red}_i})$. It is proved in \cite{HuaKeller} that fixing the Calabi-Yau structure on $Y$, the potential $\Phi$ is unique up to right equivalences.
\end{remark}
\begin{theorem}\label{contractionDsg}
Let $f: Y\to X=\Spec R$ and $f^\p: Y^\p\to X^\p=\Spec R^\p$ be two 3-dimensional formal flopping contractions such that the exceptional fibers of $f$ and $f^\p$ have the same number of irreducible components, with the associated contraction algebras $\Lm(Y,\Phi)$ and $\Lm(Y^\p,\Psi)$, respectively. Assume there is a $l$-algebra isomorphism $\gamma: \Lm(Y,\Phi)\to\Lm(Y^\p,\Psi)$ so that $\gamma_*([\Phi])=[\Psi]$. Then there are triangle equivalences
\[
\Delta_R(Y)\cong \Delta_{R^\p}(Y^\p) \quad \text{and} \quad \D_{sg}(R) \cong \D_{sg}(R^\p).
\]
\end{theorem}
\begin{proof}
Let
$\hD(Y,\Phi)$ and $\hD(Y^\p,\Psi)$
be the Ginzburg dg-$l$-algebra associated to the two contractions.
Given a $l$-algebra isomorphism $\gamma: \Lm(Y,\Phi)\to\Lm(Y^\p,\Psi)$ such that $\gamma_*([\Phi])=[\Psi]$, by Theorem \ref{rigidity-Ginzburg} there exists an isomorphism of dg-$l$-algebras $\hD(Y,\Phi)\cong \hD(Y^\p,\Psi)$. From the definition of the relative singularity category, there is a triangle equivalence
$\per(\hD(Y,\Phi))\cong \Delta_R(Y)$. Therefore the isomorphism of Ginzburg dg algebras induces a triangle equivalence $\Delta_R(Y)\cong \Delta_{R^\p}(Y^\p)$. Because the thick closure of $S_1,\ldots,S_t$ coincides with the subcategory of $\per(\hD(Y,\Phi))$ consisting of modules with finite dimensional cohomology, we obtain a triangle equivalence $\ul{\CM}_R\cong \ul{\CM}_{R^\p}$, therefore a triangle equivalence $\D_{sg}(R)\cong \D_{sg}(R^\p)$.
\end{proof}
To finish this subsection, we demonstrate two examples of flopping contractions.
\begin{example}(Laufer's flop \cite[Example 1.3]{DW13})\label{LauferD4}
Consider the hypersurface $R= \CC[[u,v,x,y]]/(g)$, where
\[
g= u^2+v^2y-x(x^2+y^3).
\]
This is a weighted-homogeneous singularity with the weights of $(u,v,x,w)$ chosen to be $(3,2,2,2)$.
The Laufer flop contraction $Y\to \Spec R$ is obtained by blowing up the reflexive ideal $(x^2+y^3, vx+uy, ux-vy^2)$.
The noncommutative crepant resolution $A$ is the Jacobi algebra of the complete path algebra of the quiver
\begin{center}
\begin{tikzpicture}
\filldraw
(0,0) circle (1pt) node[align=left, left](1) {$e_1$}
(2,0) circle (1pt) node[align=left, right](2) {$e_2$} ;
\draw[->] (0,-0.1) to [bend right] node[midway,sloped] {$c$} (2,-0.1) ;
\draw[->] (2,0.1) to [bend right] node[midway,sloped] {$d$} (0,0.1) ;
\draw[->] (2) to [out=330,in=300,looseness=8] node[midway, below] {$a$} (2);
\draw[->] (2) to [out=-330,in=-300,looseness=8] node[midway, above] {$b$} (2);
\end{tikzpicture}
\end{center}
with the potential $W=a^2b-\frac{1}{4}b^4+cb^2d+cdcd$. The associated contraction algebra is
\[
\Lm(Y,\Phi) :=A/Ae_0A \cong \frac{\CC\lgg a,b\rgg}{(ab+ba,a^2-b^3)^{cl}}
\]
with $\Phi=a^2b-\frac{1}{4}b^4$. Note that $\Phi$ is a weighted-homogeneous potential.
\end{example}
The next example of flopping contraction is due to Brown and Wemyss (\cite[Example 2.1 ]{BW17}). It shares with the same Gopakumar-Vafa invariants (see definition in Section 4 of \cite{HT16}) with the Laufer flop in Example \ref{LauferD4}, but their associated contraction algebras are not isomorphic (see Theorem 1.1 of \cite{BW17}).
\begin{example}\label{non-quasi-homo}
Consider the hypersurface $R := \CC[[u, v, x, y]]/(g)$, where
\[
g :=u^2 +v^2(x+y)+x(x^2 +xy^2 +y^3).
\]
This singularity is not quasi-homogeneous.
Consider the flopping contraction $Y\to \Spec R$ obtained by blowing up the reflexive ideal $(vx-uy, xy^2+v^2, x^2y+uv)$. The associated contraction algebra is
\[
\Lm(Y,\Phi) \cong \frac{\CC\lgg a,b\rgg}{(ab+ba,a^2-b^3+a^2b)^{cl}} = \frac{\CC\lgg a,b\rgg}{(ab+ba, a^2+\sum_{r\geq 3} (-1)^rb^r)^{cl}}
\]
for $\Phi=a^2b-\sum_{n\geq 4}(-1)^r\frac{b^r}{r}.$ Now consider the algebra
\[
S= \frac{\CC\lg a,b\rg}{(ab+ba,a^2-b^3+a^2b)}.
\]
A direct computation shows that $a^3=0$ in $S$ and all ambiguities of the rewriting system
\[\{~ ba \mapsto -ab, \quad b^3\mapsto a^2+a^2b, \quad a^3\mapsto 0 ~\}\]
are resolvable. By the Diamond Lemma (see \cite[Theorem 1.2]{Berg}), $S$ is nine dimensional with basis
\[
1, ~ a, ~ b, ~ a^2, ~ ab,~ b^2, ~ a^2b, ~ ab^2, ~ a^2b^2.
\]
Moreover, $ab^3= b^6=0$ in $S$. In particular, $S$ is a local algebra. By a similar argument of the proof of Lemma \ref{superpotential-compare}, we have $\Lm\cong S$. By the division algorithm with respect to the above rewriting system, we also have
\[
\Phi = \frac{3}{4} a^2b -\frac{1}{20} a^2b^2 \neq0 \quad \text{in} ~~ \Lm(Y,\Phi).
\]
Note that the commutator space $[\Lm, \Lm]$ is spanned by $ab$, $a^2b$ and $ab^2$. Consequently,
\[
[\Phi] = -\frac{1}{20} a^2b^2 \neq0 \quad \text{in} ~~ \Lm(Y,\Phi)_\cy.
\]
Thus, $\Phi$ is not a quasi-homogeneous potential. However, it is easy to check that the abelianization of $\Phi$ is a quasi-homogeneous power series in $\CC[[a,b]]$.
Furthermore, by Diamond lemma, one can show that $\rJ(\CC\lg a, b\rg, \Phi^{(r)}) = (ab+ba,a^2-b^3+a^2b)$ and therefore $ \Lm(Y, \Phi^{(r)}) =\Lm(Y, \Phi) $ for $r\geq 5$. Then by the nc Mather-Yau theorem, $\Phi$ and $\Phi^{(r)}$ are right equivalent for $r\geq 5$. Also, we want to mention that the finite determinacy theorem also tells us that $\Phi$ is right equivalent to $\Phi^{(r)}$ for $r\geq 7$ because the Jacobi ideal $\rJ(Y, \Phi) \supseteq \wh{\fm}^6$.
\end{example}
\subsection{Some conjectures on contraction algebras}\label{sec:conj}
We finish this section by proposing several conjectures about contraction algebras.
\begin{definition}
Let $k$ be a field and $l=ke_1+\ldots+ke_t$. A \emph{polarized} $l$-algebra is a $l$-algebra $\Lm$ together with a class \[[\Phi]\in HH_0(\Lm,\Lm)=\Lm/[\Lm,\Lm],\] which is called a \emph{polarization}. A \emph{polarized morphism} between two polarized algebras $(A,[\Phi])$ and $(B,[\Psi])$ is a $l$-algebra morphism $\gamma: A\to B$ such that $\gamma_*([\Phi])=[\Psi]$.
\end{definition}
Theorem \ref{rigidity-Ginzburg} can be restated as two finite dimensional Jacobi algebras have isomorphic complete Ginzburg dg-algebras if they are isomorphic as polarized $l$-algebras.
It is clear that the contraction algebra defined as $H^0$ of the Ginzburg dg-algebra, is equipped with a natural polarization. We propose the following enhancement for the Donovan-Wemyss conjecture (\cite[Conjecture 1.4]{DW13}).
\begin{conjecture}
Suppose that $Y\to X=\Spec R$ and $Y^\p\to X^\p=\Spec R^\p$ are two 3-dimensional flopping contractions for complete local $\CC$-algebras $R$ and $R'$, with the associated contraction algebras $\Lm(Y,\Phi)$ and $\Lm(Y^\p,\Psi)$, respectively. Then $X$ and $X^\p$ are formally isomorphic at the singular points if and only if $\Lm(Y,\Phi)$ and $\Lm(Y^\p,\Psi)$ are isomorphic as polarized algebras.
\end{conjecture}
Let $f:Y\to X=\Spec R$ be a flopping contraction.
It was proved in Corollary 1.12 of \cite{Reid} that $X$ is indeed a hypersurface. The next conjecture relates the quasi-homogeneity of hypersurface singularities to the quasi-homogeneity of potentials.
\begin{conjecture}
Let $Y\to X$ be a 3-dimensional flopping contraction with contraction algebra $\Lm(Y,\Phi)$. Then $X$ is quasi-homogeneous if and only if $\Phi$ is quasi-homogeneous.
\end{conjecture}
| 101,763
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Oxfam urges you to support women cocoa farmers to celebrate International Women's Day.
Chocolate is a $100 billion industry – but most cocoa farmers live on less than $2 a day. And the women working on cocoa farms have it the hardest.
Women on cocoa farms get less training than men – and they get paid less (if they get paid at all!). Compared to men, women cocoa farmers struggle to get loans and rarely get to own the land they farm, even if they work the same plot their entire lives.
That's where you come in. The big chocolate companies employ millions of people in poor countries – and consumers like you and me have a powerful influence on their behavior.
Today, we're honoring women like Olga Adou, a cocoa farmer in Côte d’Ivoire who is beating the odds.
Olga understands the struggles women cocoa farmers face. She started a cocoa cooperative to support fellow farmers – helping them get training to improve their skills, access tools like fertilizers to strengthen their cocoa crop, and more. Olga is doing her part to change the way women cocoa farmers are treated – and she wants the big chocolate companies to start doing theirs.
Mars, Mondelez and Nestlé are the big companies that make Oreos, M&M's and Crunch. They can change their policies to help ensure that women cocoa farmers like Olga are treated equally – but they need to hear from consumers like you.
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| 48,397
|
Speaker: Dr. Henriette Elvang (Institute for Advanced Study, Princeton)
The unification of general relativity and quantum mechanics has long been an enticing, yet challenging, subject in theoretical physics. One difficulty is that graviton scattering amplitudes suffer from problematic divergences. However, it was recently proposed that all these divergences might cancel in a certain supersymmetric theory of gravity. Such surprising cancellations are currently being investigated through direct calculations of scattering amplitudes in supergravity. In my talk I will review these ideas and I will describe how new "on-shell methods" let us construct scattering amplitudes efficiently from a set of very simple building blocks. The techniques--which also apply to processes relevant for particle physics experiments --- reveal a structure much simpler than what is expected based on the standard Lagrangian formulations. This simplicity may guide us towards a new approach to quantum field theory.
| 92,928
|
» mkrlife - ttc after miscarriage
I was pregnant at about 4/5 weeks and started bleeding on Jaunuary 26th 2007. I did some Beta test to determine i was miscarriaging. The levels were 147. Since then I have not used protection and want to conceive right away. Througout this time my last beta test was 17 on 02/12/07. I think I ovulaed last week, lots and lots of cervical muscus. Can I still get pregnant even though my levels are showing 17? What are my chances of getting pregnant again and is it more dangerouse getting pregnant so soon? A few days ago I started feeling pregnant again, tired and nipples are tender. Would the beta testing pick that up yet? Thanks for your time and quick response.
Monica
» nubianqueen - ttc after miscarriageIn response to ttc after miscarriage posted by mkrlife:
Girlfriend, let go and let God,when he lets it happen your baby will live,be healthy and you will be happy.Stop all the things you are trying and just try praying." It really works ".
-- posted by nubianqueen
» mel206uk - Getting pregnant after miscarriage
Please follow the guidelines set forth in the Suite101 Posting Etiquette when adding to the discussion.
| 342,972
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| 337,891
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\begin{document}
\newcommand{\hgt}{\operatorname{ht}}
\newcommand{\Hht}{\operatorname{Hht}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Supp}{\operatorname{Supp}_R}
\newcommand{\Ass}{\operatorname{Ass}_R}
\newcommand{\Att}{\operatorname{Att}_R}
\newcommand{\Min}{\operatorname{Min}_R}
\newcommand{\depth}{\operatorname{depth}}
\newcommand{\pd}{\operatorname{pd}}
\newcommand{\ann}{\operatorname{ann}}
\newcommand{\grade}{\operatorname{grade}}
\newcommand{\Grade}{\operatorname{Grade}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\lc}{\operatorname{lc}}
\title[Finite Minimal Components]{On rings for which finitely generated ideals have only finitely many minimal components}
\author{Thomas Marley}
\address{Department of Mathematics\\
University of Nebraska-Lincoln\\
Lincoln, NE 68588-0130} \email{tmarley1@math.unl.edu}
\urladdr{http://www.math.unl.edu/\textasciitilde tmarley1}
\keywords{minimal primes of an ideal} \subjclass[2000]{13B24}
\date{December 20, 2005}
\begin{abstract} For a commutative ring $R$ we investigate the property
that the sets of minimal primes of finitely generated ideals of
$R$ are always finite. We prove this property passes to
polynomial ring extensions (in an arbitrary number of variables)
over $R$ as well as to $R$-algebras which are finitely presented
as $R$-modules.
\end{abstract}
\maketitle
\section{Introduction}
In \cite{OP} Ohm and Pendleton examine several topological
properties which may be possessed by the prime spectrum of a
commutative ring $R$. One of these properties, denoted in \cite{OP}
by {\it FC} for `finite components', is that every closed subset of
$\Spec R$ has a finite number of irreducible components, or
equivalently, that every quotient $R/I$ has a finite number of
minimal primes. In this paper, we will call a ring $R$ such that
$\Spec R$ has property FC an {\it FC-ring} or simply say {\it $R$ is
FC}. Such a condition on $R$ is useful when investigating questions
concerning heights of primes ideals. For example, using prime
avoidance one can show that if $P$ is a prime ideal of an FC-ring of
height at least $h$ then $P$ contains an ideal generated by $h$
elements which has height at least $h$. And if $R$ is a quasi-local
FC-ring of dimension $d$ then every radical ideal is the radical of
a $d$-generated ideal.
An obvious question is whether the FC property passes to finitely
generated algebras. Heinzer \cite{He} showed that if $R$ is an
FC-ring and $S$ is a finite $R$-algebra then $S$ is FC, while Ohm
and Pendleton prove that if $R$ is an FC-ring and $\dim R$ is finite
then $R[x]$ is FC. However, if $R$ is an infinite-dimensional
FC-ring $R[x]$ need not be FC. This is illustrated by the following
example:
\begin{ex} \label{OPex} {\rm (\cite[Example 2.9]{OP})
Let $V$ be a valuation domain of countably infinite rank and let
$\Spec R=\{P_i\mid i\in \mathbb N_0\}$ where $P_{i}\subset
P_{i+1}$ for all $i$. Clearly, V is an FC-ring. For each $i\in
\mathbb N_0$ let $a_i\in P_{i+1}\setminus P_i$. Let $x$ be an
indeterminate and set
$$f_i=a_i\prod_{j=0}^{i}(a_jx-1)\in V[x]$$
for each $i\in \mathbb N_0$. Then the ideal $I=(\{f_i\mid i\in
\mathbb N_0\})V[x]$ has infinitely many minimal primes (namely,
$(P_i,a_ix-1)V[x]$ for $i\in \mathbb N_0$). Hence, $V[x]$ is not
an FC-ring.}
\end{ex}
In this paper we examine the weaker property that every {\it
finitely generated} ideal of the ring has finitely many minimal
primes. In many applications of FC (such as the ones mentioned in
the first paragraph), one only needs the FC property for finitely
generated ideals. We show that this weaker property passes to
polynomial extensions (in any number of variables) as well as to
finitely presented algebras.
This paper arose in connection to the author's work on the
Cohen-Macaulay property for non-Noetherian rings \cite{HM}. It is an
open question whether the Cohen-Macaulay property (as defined in
\cite{HM}) passes to polynomial extensions. While the answer in
general is probably `no', there is evidence to support a positive
answer if finitely generated ideals of the base ring have only
finitely many minimal primes.
\section{Main Results}
All rings in this paper are assumed to be commutative with identity.
For a ring $R$ and an $R$-module $M$ we let $\Supp M$ denote the
support of $M$ and $\Min M$ denote the minimal elements of $\Supp
M$. We begin with the following definition:
\begin{df}{\rm Let $R$ be a ring. An $R$-module $M$ is said to be {\it FGFC} if for every
finitely generated $R$-submodule $N$ of $M$ it holds that $\Supp
M/N$ is the finite union of irreducible closed subsets of $\Spec R$;
equivalently, $\Min M/N$ is a finite set. A ring $R$ is an {\it
FGFC-ring} if it is FGFC as an $R$-module.}
\end{df}
We make some elementary observations:
\begin{prop} Let $R$ be an FGFC ring.
\begin{enumerate}[(a)]
\item Every finitely presented $R$-module is FGFC. \item If $I$ is
a finitely generated ideal then $R/I$ is FGFC. \item $R_S$ is an
FGFC ring for every multiplicatively closed set $S$ of $R$. \item
Every prime $p$ of finite height is the radical of a finitely
generated ideal. \item $R/p$ is an FGFC ring for every minimal
prime $p$ of $R$.
\end{enumerate}
\end{prop}
{\it Proof:} Note that if $M$ is finitely presented then $\Min M=
\Min R/F_0(M)$, where $F_0(M)$ is the zeroth Fitting ideal of $M$
(\cite[Proposition 20.7]{E}). Thus, $\Min M$ is finite. Since
every quotient of a finitely presented $R$-module by a finitely
generated submodule is also finitely presented, this proves (a).
Parts (b) and (c) are clear. Part (d) is proved by induction and
prime avoidance. Part (e) follows from (b) and (d). \qed
\medskip
Clearly, every FC-ring is FGFC. However, the converse is not
true. The ring $V[x]$ of Example \ref{OPex} is not FC but is FGFC
by the following theorem:
\begin{thm} Let $R$ be a ring. Then $R$ is an FGFC-ring if and only if
$R[x]$ is.
\end{thm}
{\it Proof:} Since $R\cong R[x]/(x)$ it is clear that $R$ is FGFC if
$R[x]$ is. Assume that $R$ is FGFC. For a finitely generated ideal
$I$ of $R[x]$ let $d(I):=\min \{\sum_{i=1}^n\deg f_i\mid
I=(f_1,\dots,f_n)\}$. If $d(I)=0$ then $I=JR[x]$ where $J$ is a
finitely generated ideal of $R$. Hence $\operatorname{Min}_{R[x]}
R[x]/I=\{pR[x]\mid p\in \Min R/J\}$, which is a finite set as $R$ is
FGFC. Let $I$ be a finitely generated ideal of $R[x]$ with $d(I)>0$
and assume the theorem holds for all finitely generated ideals $J$
of $R[x]$ such that $d(J)<d(I)$. Let $f_1,\dots,f_n$ be a set of
generators for $I$ such that $\sum_{i=1}^n\deg f_i=d(I)$. Without
loss of generality we may assume $\deg f_n=\min_{1\le i\le n}\{\deg
f_i\mid \deg f_i>0\}$. Let $c$ be the leading coefficient of $f_n$
and $Q$ a minimal prime containing $I$. If $c\in Q$ then $Q$ is
minimal over $I'=(f_1,\dots,f_{n-1}, f_n-cx^{\deg f_n}, c)\supset
I$. As $d(I')<d(I)$ we have that $\operatorname{Min}_{R[x]} R[x]/I'$
is a finite set. Suppose that $c\not\in Q$. Then $Q_c$ is minimal
over $I_c$. We claim that $\operatorname{Min}_{R_c[x]}R_c[x]/I_c$
is finite.
\medskip
{\it Case 1:} $\deg f_i\ge \deg f_n$ for some $i<n$.
In this case, since $f_n$ is monic in $R_c[x]$ we can replace the
generator $f_i$ by one of smaller degree (using $f_n$). Hence,
$d(I_c)<d(I)$ and $\operatorname{Min}_{R_c[x]}R_c[x]/I$ is a finite
set.
\medskip
{\it Case 2:} $\deg f_i<\deg f_n$ for all $i<n$.
By assumption on $\deg f_n$ we have $f_1,\cdots, f_{n-1}$ have
degree zero. By replacing $R$ with $R_c/(f_1,\dots,f_{n-1})R_c$
(which is still FGFC), we can reduce to the case $I=(f(x))$ where
$f(x)$ is a monic polynomial of positive degree. As $S=R[x]/(f(x))$
is a free $R$-module, the going-down theorem holds between $S$ and
$R$. Therefore, every minimal prime of $S$ contracts to a minimal
prime of $R$. Further, since $S$ is finite as an $R$-module, there
are only finitely many primes of $S$ contracting to a given prime of
$R$. Hence, $\operatorname{Min}_{R[x]} R[x]/(f(x))$ is finite. \qed
\medskip
As a consequence, we get the following:
\begin{cor} Let $R$ be an FGFC-ring and $X$
a (possibly infinite) set of indeterminates over $R$. Then $R[X]$
is an FGFC-ring.
\end{cor}
{\it Proof:} Let $I=(f_1,\dots,f_n)$ be a finitely generated ideal
of $R[X]$. Then there there exists $x_1,\dots,x_m\in X$ such that
$f_i\in R[x_1,\dots,x_m]$ for all $i$. Let $S=R[x_1,\dots,x_m]$,
$J=(f_1,\dots,f_n)S$ and $X'=X\setminus \{x_1,\dots,x_m\}$. By
the theorem (and induction) we have $\operatorname{Min}_SS/J$ is
finite. Furthermore, since $I=JS[X']$, every prime minimal over
$I$ is of the form $pS[X']$ for some $p\in
\operatorname{Min_S}S/J$. \qed
\medskip
It should be clear that, unlike FC, the FGFC property does not
pass to arbitrary finite ring extensions. For example, if $V[x]$
and $I$ are as in Example \ref{OPex}, then $V[x]$ is FGFC but
$V[x]/I$ is not. However, FGFC does pass to finitely presented
algebras:
\begin{cor} \label{fpalgebra} Let $R$ be an FGFC-ring. Then any $R$-algebra which is
finitely presented as an $R$-module is an FGFC-ring.
\end{cor}
{\it Proof:} Let $S$ be an $R$-algebra which is finitely presented
over $R$. Then certainly $S\cong R[x_1,\dots,x_n]/I$ for some
ideal $I$ of $R[x_1,\dots,x_n]$. By the theorem it is enough to
show that $I$ is finitely generated. Since $S$ is a finite
$R$-module, for each $i=1,\dots,n$ there exists a monic polynomial
$f_i(t)\in R[t]$ such that $f_i(x_i)\in I$. Let
$J=(f_1(x_1),\dots,f_n(x_n))R[x_1,\dots,x_n]$, $L=I/J$, and
$T=R[x_1,\dots,x_n]/J$. It suffices to show that $L$ is a
finitely generated ideal of $T$. In fact, since $T$ is a free
$R$-module of finite rank and $T/L$ is finitely presented over
$R$, it follows by the snake lemma that $L$ is finitely generated
as an $R$-module. \qed
\medskip
Given an integral ring extension $R\subset S$, it is clear that if
$S$ is FGFC then so is $R$. For, any prime minimal over a finitely
generated ideal $I$ of $R$ is contracted from a prime of $S$ minimal
over $IS$. However, FGFC does not in general ascend from $R$ to $S$
even if $R$ is coherent and $S$ is finite over $R$. To see this,
one can again modify the example of Ohm and Pendleton:
\begin{ex} {\rm Let $R=V[x]$ and $I$ be as in Example \ref{OPex}. Let
$S=R[y]/(yI,y^2-y)$. Then $R$ is a coherent FGFC-domain (cf.
\cite{GV}), $R\subset S$ and $S$ is a finite $R$-module. The set of
minimal primes of $S$ is $\{(P,y-1)S\mid P\in \Min R/I\}$, which is
infinite. Hence, $S$ is not FGFC.}
\end{ex}
As a final result, we prove that FGFC does ascend from coherent
domains to finite torsion-free algebras:
\begin{prop} Suppose $R$ is
a coherent domain and $S$ a finite $R$-algebra which is
torsion-free as an $R$-module. If $R$ is FGFC then so is $S$.
\end{prop}
{\it Proof:} The hypotheses imply that $S$ is isomorphic as an
$R$-module to a finitely generated submodule of $R^n$ for some
$n$. As $R$ is coherent, $S$ is finitely presented as an
$R$-module. By Corollary \ref{fpalgebra}, $S$ is an FGFC ring.
\qed
\medskip
| 99,483
|
TITLE: How to integrate $\frac{x}{e^x - 1}$ w.r.t. x?
QUESTION [2 upvotes]: A friend of mine and I wanted to solve the following indefinite integral but got stuck:
$$
\int \frac{x}{e^x - 1} dx.
$$
My approach:
Let
$$
I = \int \frac{x}{e^x - 1} dx.\\
\implies I = x\int \frac{dx}{e^x - 1} - \int \left ( \int \frac{dx}{e^x - 1} \right ) dx.
$$
Now, let $I_2 = \int \frac{dx}{e^x - 1}$. Also, let $z = e^x \implies dx = {dz \over z}$. Then,
$$
I_2 = \int \frac{dz}{z(z-1)} \\
\implies I_2 = \int {dz \over {z - 1}} - \int {dz \over z} \\
\implies I_2 = \ln (e^x - 1) - x.
$$
Substituting the value of $I_2$ in $I$, we get,
$$
I = x[\ln (e^x - 1) - x] + {x^2 \over 2} - \int \ln (e^x - 1) dx.
$$
I got stuck right here. Is it possible to proceed further?
REPLY [2 votes]: \begin{equation}
I = \int \frac{x}{\mathrm{e}^{x}-1} dx = -I_{1} = -\int \frac{x}{1-\mathrm{e}^{x}} dx
\end{equation}
Integrate by parts
\begin{align}
I_{1} &= \int \frac{x}{1-\mathrm{e}^{x}} dx \\
\tag{1}
&= x^{2} - x \ln(1-\mathrm{e}^{x}) - \int x dx + \int \ln(1-\mathrm{e}^{x}) dx \\
\tag{2}
&= x^{2} - x \ln(1-\mathrm{e}^{x}) - \frac{x^{2}}{2} - \mathrm{Li}_{2}(\mathrm{e}^{x})
\end{align}
and thus
\begin{equation}
\int \frac{x}{\mathrm{e}^{x}-1} dx = x \ln(1-\mathrm{e}^{x}) - \frac{x^{2}}{2} + \mathrm{Li}_{2}(\mathrm{e}^{x})
\end{equation}
Let $u=\mathrm{e}^{x}$
\begin{align}
\int \frac{1}{1-\mathrm{e}^{x}} dx &= \int \frac{1}{u(1-u)} du \\
&= \int \left( \frac{1}{u} + \frac{1}{1-u} \right) du \\
&= \ln \frac{u}{1-u} \\
&= x - \ln(1-\mathrm{e}^{x})
\end{align}
Let $u=\mathrm{e}^{x}$
\begin{align}
\int \ln(1-\mathrm{e}^{x}) dx &= \int \frac{\ln(1-u)}{u} du \\
&= -\mathrm{Li}_{2}(u) \\
&= -\mathrm{Li}_{2}(\mathrm{e}^{x})
\end{align}
where
\begin{equation}
\mathrm{Li}_{2}(z) = -\int\limits_{0}^{z} \frac{\ln(1-x)}{x} dx
\end{equation}
is the dilogarithm function.
| 28,219
|
\begin{document}
\title[Laguerre expansions for $C_0$-semigroups and resolvent operators]
{$C_0$-semigroups and resolvent operators approximated by Laguerre expansions}
\author{Luciano Abadias}
\address{Departamento de Matem\'aticas, Instituto Universitario de Matem\'aticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain.}
\email{labadias@unizar.es}
\author{Pedro J. Miana}
\address{Departamento de Matem\'aticas, Instituto Universitario de Matem\'aticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain.}
\email{pjmiana@unizar.es}
\thanks{Authors have been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Arag\'on; and Project UZCUD2014-CIE-09, Universidad de Zaragoza.}
\subjclass[2010]{Primary, 33C45, 47D06; Secondary, 41A35, 47A60.}
\dedicatory{To Laura and Pablo}
\keywords{Laguerre expansions, $C_0$-semigroups, resolvent operators, functional calculus.}
\begin{abstract} In this paper we introduce Laguerre expansions to approximate vector-valued functions expanding on the well-known scalar theorem.
We apply this result to approximate
$C_0$-semi\-groups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions, its Laplace transforms and the convergence of Laguerre series in Lebesgue spaces. The concluding section of this paper is devote to consider some examples of $C_0$-semigroups: shift, convolution and holomorphic semigroups where some of these results are improved.
\end{abstract}
\date{}
\maketitle
\section{Introduction}
Representations of functions through series of orthogonal polynomials such as Legendre, Hermite or Laguerre are well known in mathematical analysis and applied mathematics. They allow us to approximate functions by series of orthogonal polynomials on different types of convergence: a pointwise way, uniformly, or in Lebesgue norm. Two classical monographs where we can find this kind of results are \cite[Chapter 4]{Lebedev} and \cite[Chapter IX]{Szego}. In this paper we are concentrated on Laguerre expansions.
Rodrigues' formula gives a representation of generalized Laguerre polynomials, $$L_n^{(\alpha)}(t)=e^{t}\frac{t^{-\alpha}}{n!}\frac{d^n}{dt^n}(e^{-t}t^{n+\alpha}), \qquad t\in \RR,$$ for $\alpha\in \R$ and $n\in\N\cup\{ 0 \}.$ Here we present an interesting theorem which appears in \mbox{\cite[ Sec. 4.23]{Lebedev},} and whose statement was originally proved by J.V. Uspensky in \cite{Uspensky}. As we prove in Theorem \ref{upesvect}, this result also holds for vector-valued functions in abstract Banach spaces.
\begin{theorem} \label{upes} Let $\alpha>-1$ and $f:(0,\infty)\to\C$ be a differentiable function such that the integral $\int_{0}^{\infty}e^{-t}t^{\alpha}|f(t)|^2\,dt$ is finite, then the series $\displaystyle\sum_{n=0}^{\infty}c_n(f)\,L^{(\alpha)}_n(t),$ with $$c_n(f):=\frac{n!}{\Gamma(n+\alpha+1)}\int_{0}^{\infty}e^{-t}t^{\alpha}f(t)L^{(\alpha)}_n(t)\,dt,$$ converges pointwise to $f(t)$ for $t>0$.
\end{theorem}
There exists a large amount of results about Laguerre expansions: for example, Laguerre expansions of analytic functions are considered in \cite{Rusev}; the decay of coefficients is also studied to show properties of the function defined by the Laguerre expansions in \cite{Weniger}; and the algebraic structure related to the Laguerre expansions is treated in detail in \cite{Kanjin}.
In particular, Theorem \ref{upes} is applied to the function $e_a$ (where $e_a(t):=e^{-at})$ to obtain that \begin{equation}\label{expo2} e^{-a t}=\displaystyle\sum_{n=0}^{\infty}\frac{a^n}{(a +1)^{n+\alpha+1}}L_n^{(\alpha)}(t), \qquad a>0\end{equation} and the Laguerre expansion converges pointwise for $t>0$.
Through Laguerre polynomials, Laguerre functions are defined by $$\mathscr{L}_n^{(\alpha)}(t):=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}t^{\frac{\alpha}{2}}e^{-\frac{t}{2}}L_n^{(\alpha)}(t),\quad t>0,$$ for $\alpha>-1$. These functions form an orthonormal basis on the Hilbert space $L^2(\R_+).$ Furthermore, let $f$ be in $L^p(\R_+),$ $\frac{4}{3}<p<4,$ and $\displaystyle{a_k(f):=\int_0^{\infty}\mathscr{L}_k^{(\alpha)}(t)f(t)\,dt}$ for \mbox{$k\in \NN\cup\{0\}.$} Then $\lVert S_n(f)-f \rVert_p\to 0$ as $n\to\infty,$ with $$S_n(f)(t):=\displaystyle\sum_{k=0}^{n}a_k(f)\mathscr{L}_k^{(\alpha)}(t),\qquad t>0,$$ see \cite[Theorem 1]{Askey}.
On the other hand, a $C_0$-semigroup $(T(t))_{t\geq 0}$ is a one parameter family of linear and bounded operators on a Banach space $X$ which may be interpreted, roughly speaking, as $(e^{-tA})_{t\geq 0}$. The (densely defined) operator $-A,$ defined by $$-Ax:=\displaystyle\lim_{t\to 0^+}\frac{T(t)x-x}{t},\quad \hbox{when the limit exists} ,\, x\in D(A),$$ is called the infinitesimal generator of $C_0$-semigroup, see more details in the fourth section and in monographs \cite{ABHN, Nagel}. It seems natural to consider the formula (\ref{expo2}) in the context of $C_0$-semigroups; we prove that
$$
T(t)x=\displaystyle\sum_{n=0}^{\infty}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t), \qquad x\in D(A),
$$
in Theorem \ref{loc} (iii). The rate of convergence of $ \displaystyle\sum_{n=0}^{m}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t)$ to $T(t)x$ when $m\to\infty$ is also estimated in Theorem \ref{rate}.
In semigroup theory, there are different types approximations of $C_0$-semigroups as Trotter-Kato, Yosida, or Euler approximations. In a recent paper (\cite{Gomilko}), a new functional calculus is introduced to improve some rate of these approximations. Also Pad\'{e} approximations and the rate of convergence to the semigroup have been treated in \cite{BT, Egert, Neubrander}. Approximation of $C_0$-semigroups by resolvent series has been considered in \cite{Grimm} using some analytic functional calculus. In this work, authors also work on the Laguerre polynomials $(L^{(-1)}_n)_{n\ge 0}$, however their approach is completely different to ours because they consider other different approximations, compare \cite[Theorem 5.1]{Grimm} and Theorem \ref{loc} (iii).
The paper is organized as follows. In the second section, we consider the functions $t\mapsto\frac{n!}{\Gamma(n+\alpha+1)}e^{-t}t^{\alpha}L^{(\alpha)}_n(t)$ (for $\alpha>-1$ and $n\in \NN\cup\{0\}$) which have a key role in Theorem \ref{upes}. They satisfy interesting properties (similar to Laguerre polynomials, see Proposition \ref{alge} and \ref{propLag}). We estimate its $p$-norm in Theorem \ref{main2} and also the $p$-norm of its Laplace transform in Theorem \ref{phi}.
In the third section, we consider certain Laguerre expansions in Lebesgue space $L^p(\R_+)$ and abstract Banach spaces. In particular we give the Laguerre expansion for the fractional semigroup and for the exponential function $e_a$ in $L^1(\R_+)$ (Theorem \ref{fract}). To finish this section, we show a vector-valued version of Theorem \ref{upes} on an abstract Banach space $X$.
Main applications of Theorem \ref{upesvect} appear in the fourth and fifth section. In Theorem \ref{loc}, we express $C_0$-semigroup and resolvent operators though Laguerre expansions. In Theorem \ref{appr}, we apply this Laguerre expansion to express the resolvent semigroup subordinated to the original $C_0$-semigroup by a series representation. In Theorem \ref{rate}, we give the rate of the Laguerre expansion to the $C_0$-semigroup, which depends on the regularity of the initial data.
In the last section, we present some examples of $C_0$-semigroups and its Laguerre expansion: shift semigroup, convolution semigroups and in particular, Gauss and Poisson semigroups in Lebesgue spaces. For some differentiable and analytic semigroups, some previous results are improved.
Throughout the paper, we work on functions and operators defined in $\R_+,$ but there are also other results on expansions for functions defined in $\R$ using Hermite polynomials. Hermite polynomials are defined by Rodrigues' formula $$H_n(t)=(-1)^n e^{t^2}\frac{d^n}{dt^n}(e^{-t^2})\ \text{for }t\in\R\ \text{and}\ n\geq 0.$$ A similar result to Theorem \ref{upes} holds for functions defined on $\R$ and involves Hermite polynomials, see \cite[Sec. 4.15]{Lebedev}, in particular for $\lambda\in\C,$ $$e^{\lambda t}=e^{\frac{\lambda^2}{4}}\displaystyle\sum_{n=0}^{\infty}\frac{\lambda^n}{2^n n!}H_n(t), \qquad t\in \RR.$$ In the preprint \cite{AM}, we work on vector-valued approximations defined by Hermite polynomials, and its applications to $C_0$-groups and cosine operator families.
$\ $
\noindent {\bf Notation}. We write $\R_+:=[0,\infty).$ Given $1\leq p <\infty,$ let $L^p(\R_+)$ be the set of Lebesgue $p$-integrable functions, that is, $f$ is a measurable function and $$\Vert f \Vert_p:=\left(\int_0^{\infty}|f(t)|^p\,dt\right)^{\frac{1}{p}}<\infty;$$ for $p=2$, remind that $L^2(\R_+)$ is a Hilbert space with $\langle\,\,,\,\, \rangle$ the usual inner product, and $L^{\infty}(\R_+)$ the set of essential bounded Lebesgue functions with the norm $\lVert f\rVert_{\infty}:=\displaystyle\hbox{esssup}_{t\in\R_+}|f(t)|.$
We call $C_0(\R_+)$ the set of continuous functions defined in $[0,\infty)$ such that $\displaystyle\lim_{t\to \infty}f(t)=0,$ with the norm $\lVert \ \rVert_{\infty};$ and $\mathcal{H}_0(\C_+)$ the Banach space given by the set of bounded holomorphic functions defined in $\C_+=\{z\in\C\,:\,{\Re}z>0\}$ such that $\displaystyle\lim_{z\to \infty}f(z)=0,$ with the norm $||| f|||_{\infty}:=\displaystyle\sup_{z\in\C_+}|f(z)|.$ Furthermore, it is important to consider the following equivalence, which appears several times throughout the paper: for $\alpha>-1$ there exist $c_1 <c_2$ such that
\begin{equation} \label{main}
{c_1\over n^\alpha}\le {n!\over \Gamma(n+\alpha +1)}\le {c_2\over n^\alpha}, \qquad n\in \N.
\end{equation}
\section{ Laguerre functions and its Laplace transform}
\setcounter{theorem}{0}
\setcounter{equation}{0}
Generalized Laguerre polynomials $\{ L_n^{(\alpha)} \}_{n\ge 0}$ ($\alpha >-1$) are given by
$$
L_n^{(\alpha)}(t)=\sum_{k=0}^n(-1)^{k}{n+\alpha\choose n-k}{t^k\over k!}, \qquad t\ge 0;
$$
in particular $L_0^{(\alpha)}(t)=1$, $L_1^{(\alpha)}(t)=-t+\alpha+1$ and
$\displaystyle{L_2^{(\alpha)}(t)={t^2\over 2}-(\alpha+2)t+{(\alpha+2)(\alpha+1)\over 2}
}$. They satisfy the following condition of orthogonality: $${n!\over \Gamma(n+\alpha +1)}\int_0^{\infty}e^{-t}t^{\alpha}L_n^{(\alpha)}(t)L_m^{(\alpha)}(t)dt=\delta_{n,m},$$ for $n,m=0,1,2,\ldots,$ with $\delta_{n,m}$ is the Kronecker delta. The generalized Laguerre polynomials are solutions of second order differential equation $$\label{Laguerresoleq}ty''+(\alpha+1-t)y'+ny=0.$$
Now, we provide a list of well-known identities which these polynomials satisfy,
\begin{eqnarray*}
nL_n^{(\alpha)}(t)&=&(n+\alpha)L_{n-1}^{(\alpha)}(t)-tL_{n-1}^{(\alpha+1)}(t),\cr
tL_{n}^{(\alpha+1)}(t)&=&(n+\alpha)L_{n-1}^{(\alpha)}(t)-(n-t)L_n^{(\alpha)}(t),\cr
nL_n^{(\alpha)}(t)&=&(\alpha+1-t)L_{n-1}^{(\alpha+1)}(t)-tL_{n-2}^{(\alpha+2)}(t),\cr
0&=&(n+1)L_{n+1}^{(\alpha)}(t)+(t-\alpha-2n-2)L_{n}^{(\alpha)}(t)+(n+\alpha)L_{n-1}^{(\alpha)}(t),
\end{eqnarray*}
see for example \cite{Lebedev} and \cite{Szego}.
Muckenhoupt estimates for Laguerre functions $\mathscr{L}_n^{(\alpha)}$ are well known in the classical theory of orthogonal expansions, see \cite{Muckenhoupt} and \cite[Lemma 1.5.3]{Thangavelu}. As a consequence of these estimations,
\begin{eqnarray}\label{muck}
\Vert \mathscr{L}_n^{(0)} \Vert_1&\sim& \sqrt{n}, \qquad n\ge 1,\\
|L_n^{(\alpha)}(\lambda)|&\leq& C_{\lambda}n^{\frac{\alpha}{2}},\quad n\ge n_0,\label{mucklag}
\end{eqnarray}
hold for $\lambda>0,$ see \cite[Lemma 1]{Markett} and \cite[Lemma 1.5.4. (i)]{Thangavelu}. Similar results can be found in \cite[Lemma 5.1]{Eisner} and \cite[Theorem 8.1]{Gomilko}.
In this paper we are mainly interested in the following Laguerre functions.
\begin{definition}
For $\alpha\neq -1,-2,-3,\ldots,$ and $n\in \NN\cup\{0\}$, we denote by $\ell_n^{(\alpha)}$ the function defined in $(0,\infty)$ by
$$
\ell_n^{(\alpha)}(t):=\frac{n!}{\Gamma(n+\alpha+1)}t^{\alpha}e^{-t}L_n^{(\alpha)}(t).
$$
\end{definition}
Note that $\displaystyle\ell_n^{(\alpha)}(t)=\frac{1}{\Gamma(n+\alpha+1)}\frac{d^n}{dt^n}(e^{-t}t^{n+\alpha}),$ for $ t> 0,$ with $n=0,1,2,\ldots$ and $\ell_{0}^{(\alpha)}=I_{\alpha+1} $, where $(I_s)_{s\in\C_+}$ is the fractional semigroup defined by $$I_s(t):=\frac{1}{\Gamma(s)}e^{-t}t^{s-1}, \qquad t>0,\, s\in\C_+.$$ In \cite[Theorem 2.6]{Sinclair} the fractional semigroup $(I_s)_{z\in\C_+}$ is studied in detail; in particular $I_s\ast I_t=I_{t+s}$ for $s,\, t\in\C_+$ (i.e. algebraic semigroup) where
the convolution product of $f\ast g$ is defined by
$$
(f\ast g)(t):=\int_0^tf(t-s)g(s)ds, \qquad f,g\in L^1(\R_+),\quad t\ge 0.
$$
In the case of $\mu\in M(\R_+)$ is a non negative Borel measures on $\R_+$ of total variation, the convolution product $f\ast \mu$ is given by
$$
(f\ast \mu)(t):=\int_0^tf(t-s)d\mu(t), \qquad f \in L^1(\R_+),\quad t\ge 0.
$$
The Dirac delta $\delta_0$ verifies that $f\ast \delta_0 =f$ for any $f\in L^1(\R_+)$. We write by $f^{\ast n}=f\ast f^{\ast(n-1)}$ for $n\ge 2$ and $f^{\ast 1}=f$. Laguerre functions $\{ \ell_n^{(\alpha)}\}_{n\ge 0}$ satisfy the following algebraic property (for convolution product $\ast$).
\begin{proposition}\label{alge} For $\alpha>-1$, $n \in \NN$, and $e_1(t):=e^{-t}$ for $t>0$, we have that
\begin{equation}
\label{deco}
\ell_n^{(\alpha)}= (\delta_0-e_1)^{\ast n}\ast \ell_0^{(\alpha)}.
\end{equation}
Then the equality $\ell_{n}^{(\alpha)}*\ell_{m}^{(\beta)}=\ell_{n+m}^{(\alpha+\beta+1)}\ $ holds for all $\ \alpha ,\beta>-1 $ and $n,m\ge 0$.
\end{proposition}
\begin{proof} Note that $e_1^{\ast n}=I_n$, we write $I_0:=\delta_0$ and then
\begin{eqnarray*}
(\delta_0-e_1)^{\ast n}\ast\ell_n^{(\alpha)}&=&\sum_{k=0}^n(-1)^k{ n!\over k!(n-k)!}I_k\ast I_{\alpha+1}=\sum_{k=0}^n(-1)^k{ n!\over k!(n-k)!}I_{k+\alpha+1}\\
&=&{n!\over \Gamma(n+\alpha+1)}t^\alpha e^{-t} \sum_{k=0}^n(-1)^k{\Gamma(n+\alpha+1)\over (n-k)!\Gamma(k+\alpha +1)}{t^{k}\over k!}= \ell_n^{(\alpha)},
\end{eqnarray*}
for $\alpha>-1$, $n \in \NN$. The algebraic equality $\ell_{n}^{(\alpha)}*\ell_{m}^{(\beta)}=\ell_{n+m}^{(\alpha+\beta+1)}\ $ for $\ \alpha ,\beta>-1 $ and $n,m\ge 0$
is a straightforward consequence of the equality (\ref{deco}) and $\ell_{0}^{(\alpha)}=I_{\alpha+1} $.
\end{proof}
Functions $\{\ell_n^{(\alpha)}\}_{n\ge 0}$ satisfy recurrence relations, differential equations and some additional identities as the next proposition shows. The proof is left to the reader.
\begin{proposition}\label{propLag} For $\alpha\neq -1,-2,-3,\ldots,$ the family of functions $\{ \ell_n^{(\alpha)} \}_{n\ge 0}$ satisfies:
\begin{itemize}
\item[(i)] $$\ell_n^{(\alpha)}(t)=\ell_{n-1}^{(\alpha)}(t)-\ell_{n-1}^{(\alpha+1)}(t).$$
\item[(ii)] $$(n+\alpha+1)\ell_{n}^{(\alpha+1)}(t)=n \ell_{n-1}^{(\alpha)}(t)-(n-t )\ell_{n}^{(\alpha)}(t).$$
\item[(iii)] $$t \ell_{n}^{(\alpha)}(t)=(\alpha+1-t)\ell_{n-1}^{(\alpha+1)}(t)-(n-1)\ell_{n-2}^{(\alpha+2)}(t).$$
\item[(iv)] $$(n+\alpha+1)\ell_{n+1}^{(\alpha)}(t)+(t-\alpha-2n-1)\ell_{n}^{(\alpha)}(t)+n\ell_{n-1}^{(\alpha)}(t)=0.$$
\item[(v)] $$t\frac{d^2}{dt^2} \ell_{n}^{(\alpha)}(t)+(1-\alpha+t)\frac{d}{dt}\ell_{n}^{(\alpha)}(t)+(n+1)\ell_{n}^{(\alpha)}(t)=0.$$
\item[(vi)] For $k\geq 1$, $\displaystyle{\frac{d^k}{dt^k}\ell_{n}^{(\alpha)}(t)=\ell_{n+k}^{(\alpha-k)}(t).}$
\end{itemize}
\end{proposition}
In the next theorem, we present some estimates of $p$-Lebesgue norm of Laguerre functions $\{\ell^{(\alpha)}_n\}_{n\ge 0}$.
\begin{theorem} \label{main2} Take $p\ge 1$, $\alpha >-1 $ and the set of functions $\{\ell^{(\alpha)}_n\}_{n\ge 0}$. We denote by $\mathcal{Z}(L_n^{(\alpha)})$ the set of zeros of generalized Laguerre polynomial of degree $n$, $L_n^{(\alpha)}.$
\begin{enumerate}
\item[(i)] For $\alpha>0$, and $n\ge 0$, the inequality $\lVert \ell_{n}^{(\alpha)} \rVert_{\infty}\leq\lVert \ell_{n+1}^{(\alpha-1)} \rVert_1$ holds.
\item[(ii)] The set of functions $ \{\ell^{(\alpha)}_n\}_{n\ge 0} \subset L^1(\R_+)$ for $\alpha >{-1}$, and
$$
\displaystyle\max_{t\in\mathcal{Z}(L_n^{(\alpha)})}|\ell_{n-1}^{(\alpha)}(t)|\le\Vert \ell^{(\alpha)}_n\Vert_{1}\le {C_{\alpha}\over n^{\alpha \over 2}}, \qquad n\ge 1.
$$
\item[(iii)] The set of function $ \{\ell^{(\alpha)}_n\}_{n\ge 0} \subset L^p(\R_+)$ for $\alpha >-{1\over p}$ and
$
\displaystyle\lVert \ell^{(\alpha)}_n\rVert_{p}\leq C_{p,\alpha}\ n^{\frac{1}{2}} ,$ for $ n\ge 1.
$
\end{enumerate}
\end{theorem}
\begin{proof} To show (i), it is sufficient to use that $$|\ell_{n}^{(\alpha)}(t)|\leq \int_t^{\infty}|\frac{d}{ds}\ell_{n}^{(\alpha)}(s)|\,ds\leq\lVert \ell_{n+1}^{(\alpha-1)}\rVert_1,\qquad t>0,$$
where we have applied Proposition \ref{propLag} (vi). The second inequality in part (ii) is in \cite[(5.7.16)]{Szego}. By the part (i), and due to $\displaystyle\lim_{t\to 0+}\ell_{n-1}^{(\alpha+1)}(t)=0=\displaystyle\lim_{t\to \infty}\ell_{n-1}^{(\alpha+1)}(t)$, we obtain that
\begin{eqnarray*}\lVert \ell_n^{(\alpha)}\rVert_1&\geq& \lVert \ell_{n-1}^{(\alpha+1)} \rVert_{\infty} =\displaystyle\max_{\{ t\in\R_+\ |\ (\ell_{n-1}^{(\alpha+1)})'(t)=0 \}}\vert \ell_{n-1}^{(\alpha+1)}(t)\vert \\
&=&\displaystyle\max_{\{ t\in\R_+\ |\ \ell_{n}^{(\alpha)}(t)=0 \}}\vert\ell_{n-1}^{(\alpha+1)}(t)\vert =\displaystyle\max_{\{ t\in\R_+\ |\ L_{n}^{(\alpha)}(t)=0 \}}\vert \ell_{n-1}^{(\alpha)}(t)\vert,
\end{eqnarray*}
where we have used Proposition \ref{propLag} (vi) and (i) to get the result. The part (iii) is a direct consequence of Muckenhoupt estimates (\cite{Muckenhoupt} and \cite[Lemma 1.5.3]{Thangavelu}), see also \cite[Lemma 1]{Markett}.
\end{proof}
\begin{remark} {\rm To obtain other estimates for $\lVert \ell_n^{(\alpha)}\rVert_p,$ there exists the possibility of using different bounds of Laguerre polynomials. By \cite[Corollary 2.2]{Du}, we conclude that there is a constant $C_{\alpha}>0$ such that $$\lVert \ell_n^{(\alpha)}\rVert_p\leq C_{\alpha}\, n^{2+[\alpha]-\alpha},\qquad \alpha>-1,\quad p\ge 1,$$
where $[\alpha]$ is the integer part of the real number $\alpha.$ For $\alpha=0$, the bound $ \vert L_n^{(0)}(t)\vert \le e^{t\over 2}$ for $t\ge 0$ and $n\in \NN\cup\{0\}$ appears in \mbox{\cite[(7.21.3)]{Szego},} and then
$$
\Vert \ell^{(0)}_n\Vert_p\le\left({p\over 2}\right)^{1\over p}, \qquad p>1.
$$ In \cite[Theorems 7.6.2 and 7.6.4]{Szego}, \cite[Theorem 3]{Duran2} and \cite[Theorem 2]{Love}, other pointwise bounds for Laguerre polynomials are obtained. From those, bounds of $\Vert \ell^{(\alpha)}_n\Vert_p$ might be deduced.
}
\end{remark}
We denote with $\mathcal{L}$ the usual Laplace transform, $\mathcal{L}: L^1(\R_+)\to \mathcal{H}_0(\C_+)$ defined by
$$
\mathcal{L}(f)(z):=\int_0^\infty e^{-zt}f(t)dt, \qquad f\in L^1(\R_+),\quad z\in\C_+.
$$
Remind that the Laplace transform is a bounded linear operator, $||| \mathcal{L}(f)|||_\infty \le \Vert f\Vert_1$ such that $\mathcal{L}(f\ast g)=\mathcal{L}(f)\mathcal{L}(g)$ for $f,g\in L^1(\R_+)$. Observe that \begin{equation}\label{LaplaceLagFunc}
\mathcal{L}(\ell_{n}^{(\alpha)})(z)=\frac{z^n}{(z+1)^{n+\alpha+1}}, \qquad z\in\C_{+},
\end{equation}
see \cite[p.110]{Badii}. For convenience, we write by \begin{equation}\label{definit}\varphi_{n,\alpha}(z):=\frac{z^n}{(z+1)^{n+\alpha+1}}, \qquad z\in \C_+.\end{equation}
\begin{lemma} \label{222} Let $\varphi_{n,\alpha}$ defined by (\ref{definit}) for $n\in \N\cup\{0\}$ and $\alpha>-1$.
\begin{itemize}
\item[(i)] The equality $\varphi_{n,\alpha}'=n\varphi_{n-1,\alpha+2}-(\alpha+1)\varphi_{n,\alpha+1},$ holds $\text{ for }n\geq 1.$
\item[(ii)] For $j\ge 1$, we have that
$$\varphi_{n,\alpha}^{(j)}=\sum_{l=0}^{\min(j,n)}\frac{n!}{(n-l)!}b_{l,\alpha}\,\varphi_{n-l,\alpha+j+l}, $$
where $b_{l, \alpha}$ is a real number dependent on $l$ and $\alpha$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) Note that
$$\varphi_{n,\alpha}'(z)=\frac{nz^{n-1}}{(z+1)^{n+\alpha+1}}-\frac{(n+\alpha+1)z^n}{(z+1)^{n+\alpha+2}}=n\varphi_{n-1,\alpha+2}(z)-(\alpha+1)\varphi_{n,\alpha+1}(z),$$ for $z\in \C_+$ and $n\geq1$. To show the part (ii), first we consider $1\le j< n$. We prove the equality by the inductive method. For $j=1$, we just prove the equality in part (i). Now take $j+1\le n$ and again by part (i), we get the following:
\begin{eqnarray*}
&\quad&\varphi_{n,\alpha}^{(j+1)}=\left(\displaystyle\sum_{l=0}^{j}\frac{n!}{(n-l)!}b_{l,\alpha}\,\varphi_{n-l,\alpha+j+l}\right)'=\displaystyle\sum_{l=0}^{j}\frac{n!}{(n-l)!}b_{l,\alpha}\,\varphi_{n-l,\alpha+j+l}' \cr
&\quad&\quad=\displaystyle\sum_{l=1}^{j+1}\frac{n!}{(n-l)!}b_{l-1,\alpha}\varphi_{n-l,\alpha+j+l+1}-\displaystyle\sum_{l=0}^{j}\frac{n!}{(n-l)!}b_{l,\alpha}(\alpha+j+l+1)\varphi_{n-l,\alpha+j+l+1} \cr
&\quad&\quad=\displaystyle\sum_{l=0}^{j+1}\frac{n!}{(n-l)!}\widetilde{b_{l,\alpha}}\,\varphi_{n-l,\alpha+j+l+1},
\end{eqnarray*}
and the identity is obtained. In the case that $j\geq n$, note that $\varphi'_{0,\alpha}=-(\alpha+1)\varphi_{0,\alpha+1}$,
\begin{eqnarray*}
&\quad&\varphi_{n,\alpha}^{(j)}=\sum_{l=0}^{n}\frac{n!}{(n-l)!}b_{l,\alpha}\,\varphi_{n-l,\alpha+j+l},
\end{eqnarray*}
and then the equality is obtained.
\end{proof}
For $j\in \N,$ denote by $AC^{(j)}$ the Sobolev Banach space obtained as the completion of
the Schwartz class $\mathcal{S}(\R_+)$ (the set of restrictions to $[0, \infty)$ of
the Schwartz class $\mathcal{S}(\R)$, see more details in \cite[Definition 1.1]{Du} and \cite{Gale}) in the norm $$\lVert f \rVert_{(j)}:=\frac{1}{(j-1)!}\int_0^{\infty}|f^{(j)}(x)|x^{j-1}\,dx,\qquad f\in \mathcal{S}(\R_+).$$
Note that the following continuous embeddings hold: $(AC^{(j+1)},\lVert\ \rVert_{(j+1)})\hookrightarrow (AC^{(j)},\lVert\ \rVert_{(j)})$ $\hookrightarrow (C_0(\R_+),\lVert\ \rVert_{\infty})$ (see \cite[Proposition 3.1.(i)]{Gale}). These function spaces have been considered by several authors and extended considering Weyl fractional derivation, instead of the usual derivation in \cite{Gale}, see references and more details therein.
\begin{theorem}\label{phi} Let $\alpha>-1.$ \begin{itemize}
\item[(i)] For $1\leq p<\infty,$ $\varphi_{n,\alpha}\in L^p(\R_+),$ for each $n\geq 0$ whenever $\alpha>\frac{1}{p}-1, $ and $$\lVert \varphi_{n,\alpha} \rVert_p^p=\frac{\Gamma(np+1)\Gamma(p\alpha+p-1)}{\Gamma(p(n+\alpha+1))}. $$
\item[(ii)] For each $n\geq 1,$ there exists $N_\alpha> M_\alpha>0$ such that ${\frac{M_{\alpha}}{n^{\frac{\alpha+1}{2}}}\le |||\varphi_{n,\alpha}|||_{\infty}\le \frac{N_{\alpha}}{n^{\frac{\alpha+1}{2}}}.}$
\item[(iii)] For $j\in \N,$ $\varphi_{n,\alpha}\in AC^{(j)},$ for each $n\ge 0,$ and ${\lVert \varphi_{n,\alpha} \rVert_{(j)}\le\frac{C_{j,\alpha}}{n^{\alpha+1}}}$ for $ n\ge 1.$
\end{itemize}
\end{theorem}
\begin{proof}(i) Note that $\displaystyle{\int_0^{\infty}|\varphi_{n,\alpha}(t)|^p\,dt=\int_0^{\infty}\frac{t^{np}}{(t+1)^{p(n+\alpha+1)}}\,dt=\beta(np+1,p(\alpha+1)-1)},$ where $\beta$ is the Euler Beta function. (ii) By the formula \eqref{LaplaceLagFunc}, $\varphi_{n,\alpha}\in\mathcal{H}_0(\C_+), $ and $\displaystyle{|||\varphi_{n,\alpha}|||_{\infty}=\displaystyle\sup_{z\in \C_+}\frac{|z|^n}{|z+1|^{n+\alpha+1}}.}$ Furthermore, $|\varphi_{n,\alpha}(z)|\leq 1$ for all $z\in \C_+.$ We apply the Maximum Modulus Theorem to get that $$|||\varphi_{n,\alpha}|||_{\infty}=\displaystyle\max_{x\in\R}\frac{|ix|^n}{|ix+1|^{n+\alpha+1}}=\displaystyle\max_{x\in\R}\frac{|x|^n}{(x^2+1)^{\frac{n+\alpha+1}{2}}}.$$ We define $g(t):=\frac{t^n}{(t^2+1)^{\frac{n+\alpha+1}{2}}}$ for $t>0.$ We have that $$\displaystyle\max_{t\geq 0}g(t)=(\alpha+1)^{\frac{\alpha+1}{2}}\frac{n^\frac{n}{2}}{(n+\alpha+1)^{\frac{n+\alpha+1}{2}}}.$$ Then we conclude the result. (iii) It is enough to show for $n\ge j$. We apply the Lemma \ref{222} (ii) to get that
\begin{eqnarray*}
\lVert \varphi_{n,\alpha}\rVert_{(j)}&\leq &\frac{1}{(j-1)!}\displaystyle\sum_{l=0}^{j}\frac{n!}{(n-l)!}|b_{l,\alpha}|\int_0^{\infty}\frac{t^{n+j-l-1}}{(1+t)^{n+\alpha+j+1}}\,dt \\
&\leq & C_{j,\alpha}\frac{\Gamma(\alpha+j+1)}{(j-1)!\Gamma(n+\alpha+j+1)}\displaystyle\sum_{l=0}^{j}\frac{n!(n+j-l-1)!}{(n-l)!} \\
&\leq & (j+1) \tilde C_{j,\alpha}\frac{(n+j-1)!}{\Gamma(n+\alpha+j+1)}\le \frac{C_{j,\alpha}}{n^{\alpha+1}},
\end{eqnarray*}
where we use the inequality (\ref{main}).
\end{proof}
\begin{remark}\label{whittaker} {\rm Observe that if $\alpha>0,$ $\varphi_{n,\alpha}\in L^1(\R_+)$ for each $n\geq 0,$ and $$\mathcal{L}(\varphi_{n,\alpha})(\lambda)=n!\lambda^{\frac{\alpha-1}{2}}e^{\frac{\lambda}{2}}W_{-n-\frac{(\alpha+1)}{2},-\frac{\alpha}{2}}(\lambda)\,,\qquad \lambda\in \C_+$$ where $W_{k,\mu}$ is the Whittaker function, see \cite[p.24]{Badii}.
}
\end{remark}
\section{Laguerre expansions in Banach spaces}
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In this section we study Laguerre expansions on Lebesgue spaces $L^p(\R_+)$ and on abstract Banach spaces $X$. First we show that the span generated by the Laguerre functions $\{\ell_n^{(\alpha)}\}_{n\ge 0}$ are dense in $L^p(\R_+)$ for $\alpha>{-1\over p}$, Theorem \ref{densi} (ii). The fractional semigroup $\{I_\alpha\}_{ \alpha>0}$ and exponential functions $\{e_a\}_{a>0}$ may be expressed via Laguerre expansions, see Theorem \ref{fract}.
These expansions will be applied to obtain different representations of operator families related with semigroups theory in the next section. Following the original proof of J. V. Uspensky , we prove a vector-valued theorem of Laguerre expansions for continuous functions (Theorem \ref{upesvect}).
\begin{theorem} \label{densi} Take $1\leq p < \infty.$
\begin{itemize}
\item[(i)] For $\alpha>-\frac{1}{p}$, $\lambda>0$, and $\xi_{\alpha, \lambda}(t):= t^\alpha e^{-\lambda t}$ for $t>0$, we obtain that
$$\xi_{\alpha, \lambda+1}=\displaystyle\sum_{n=0}^{\infty}\frac{\lambda^n}{(\lambda +1)^{n+\alpha+1}}\frac{\Gamma(n+\alpha+1)}{n!}\ell_n^{(\alpha)} \hbox{ in $L^p(\R_+)$.}$$
\item[(ii)] The set span$\{\ell^{(\alpha)}_n \,\, \vert {n\ge 0}\}$ is dense in $L^p(\R_+)$ if $\alpha >-\frac{1}{p}.$
\end{itemize}
\end{theorem}
\begin{proof} (i) First note that for all $\lambda >0,$ $e^{-\lambda t}=\displaystyle\sum_{n=0}^{\infty}\frac{\lambda^n}{(\lambda +1)^{n+\alpha+1}}L_n^{(\alpha)}(t)$ pointwise, see formula (\ref{expo2}). Then $$t^{\alpha}e^{-(\lambda+1)t}=\displaystyle\sum_{n=0}^{\infty}\frac{\lambda^n}{(\lambda +1)^{n+\alpha+1}}\frac{\Gamma(n+\alpha+1)}{n!}\ell_n^{(\alpha)}(t), \qquad t>0.$$ Furthermore this convergence is in $L^p(\R_+):$
$$\lVert \displaystyle\sum_{n=0}^{\infty}\frac{\lambda^n}{(\lambda +1)^{n+\alpha+1}}\frac{\Gamma(n+\alpha+1)}{n!}\ell_n^{(\alpha)} \rVert_p\leq C_{p,\alpha}\displaystyle\sum_{n=0}^{\infty}\frac{\lambda^n}{(\lambda +1)^{n+\alpha+1}}n^{\alpha+\frac{1}{2}}<\infty, \qquad \lambda>0,$$
where we have applied Theorem \ref{main2} (iii). (ii) Using the part (i), it is enough to see that span$\{t^{\alpha}e^{-(\lambda+1)t} \,\, \vert {\lambda > 0}\}$ is dense in $L^p(\R_+)$ to get the result. Let $f\in L^q(\R_+)$ with $\frac{1}{p}+\frac{1}{q}=1$ such that $$\int_0^{\infty}f(t)t^{\alpha}e^{-(\lambda+1)t}\,dt=0, \qquad \lambda >0.$$ By H\"{o}lder inequality, the function $f\xi_{\alpha,1}\in L^1(\R_+)$ and then $$0=\int_0^{\infty}f(t)t^{\alpha}e^{-(\lambda+1)t}\,dt=\mathcal{L}(f\xi_{\alpha,1})(\lambda), \qquad \lambda >0.$$ Since the Laplace transform is injective in $L^1(\R_+)$, we conclude that $f= 0.$
\end{proof}
\begin{theorem}\label{fract}
\begin{itemize}
\item[(i)] For $2\beta>\alpha>-1,$ the equality $\displaystyle{I_{\beta+1}=\displaystyle\sum_{n=0}^{\infty}\binom{\alpha-\beta+n-1}{n}\ell_n^{(\alpha)}}$ holds in $L^1(\R_+).$
In particular, for $\alpha>2$, we have that $I_{\alpha}=\displaystyle\sum_{n=0}^{\infty}\ell_{n}^{(\alpha)}\ $ in $L^1(\R_+).$
\item[(ii)] For all $a>0,$ the equality $\displaystyle{e_a=\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}\mathscr{L}_n^{(0)} }$ holds in $L^1(\R_+).$
\end{itemize}
\end{theorem}
\begin{proof} (i) As $\displaystyle{\frac{1}{t^{\alpha-\beta}}=\sum_{n=0}^{\infty}\binom{\alpha-\beta+n-1}{n}\frac{n!\Gamma(\beta+1)}{\Gamma(n+\alpha+1)}L_n^{(\alpha)}(t)}$ for all $t>0$ with \mbox{$2\beta>\alpha-1,$} (see \cite[p.89]{Lebedev}) we obtain that $$\frac{t^{\beta}e^{-t}}{\Gamma(\beta+1)}=\displaystyle\sum_{n=0}^{\infty}\binom{\alpha-\beta+n-1}{n}\ell_n^{(\alpha)}(t), \qquad t>0.$$ This convergence is in $L^1(\R_+)$ when $2\beta>\alpha>-1:$ we apply Proposition \ref{main2} (ii) to get that $$\lVert \displaystyle\sum_{n=0}^{\infty}\binom{\alpha-\beta+n-1}{n}\ell_n^{(\alpha)} \rVert_1\leq \displaystyle\sum_{n=0}^{\infty}C_{\alpha}\frac{n^{\alpha-\beta-1}}{n^{\frac{\alpha}{2}}}.$$
(ii)
Since $\{\mathscr{L}_n^{(0)}\}$ is an orthonormal basis in $L^2(\R_+)$ and the function $e_a\in L^2(\R_+)$ for $a>0$, then the serie $\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}\mathscr{L}_n^{(0)}$ converges to $e_a$ in $L^2(\R_+)$ (in fact in $L^p(\R_+)$ for ${4\over 3}<p<4$, see Introduction and \cite{Askey}) for $a>0$:
$$
\langle e_a, \mathscr{L}_n^{(0)}\rangle=\int_0^\infty e^{-at} e^{-t\over 2}L_n^{(0)}(t)dt= \int_0^\infty e^{-\left(a-{1\over 2}\right)t}\ell_{n}^{(0)}(t)dt= \frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}},
$$
where we have applied the formula
(\ref{LaplaceLagFunc}). This convergence also holds in $L^1(\R_+)$:
$$
\lVert\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}\mathscr{L}_n^{(0)}\rVert_1\le
\sum_{n=0}^{\infty}\frac{b^n}{|a+\frac{1}{2}|}\Vert \mathscr{L}_n^{(0)}\Vert_1
\leq C\displaystyle\sum_{n=0}^{\infty}\frac{b^n}{|a+\frac{1}{2}|}\sqrt{n},
$$
with $b=|\frac{a-\frac{1}{2}}{a+\frac{1}{2}}|<1$ for $a>0,$ where we have applied the equivalence (\ref{muck}). Note that $e^{-at}=\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}\mathscr{L}_n^{(0)}(t)$ pointwise for $t>0$ by Theorem \ref{upes}
and we conclude the proof.
\end{proof}
Let $X$ be a Banach space, and $f:(0,\infty)\to X$ be a vector-valued continuous function. We say that this function $f$ is differentiable at $t$ if exists $$\displaystyle\lim_{h\to 0^+}\frac{1}{h}(f(t+h)-f(t))$$ in $X.$ Now, we give a vector-valued version of Theorem \ref{upes}. The proof is similar to the scalar case shown in \cite{Uspensky} and we include the proof to ease the reading.
\begin{theorem}\label{upesvect} Let $X$ be a Banach space and $f:(0,\infty)\to X$ a differentiable function such that the integral $\int_{0}^{\infty}e^{-t}t^{\alpha}\lVert f(t)\rVert^2\,dt$ is finite, then the series $\displaystyle\sum_{n=0}^{\infty}c_n(f)\,L^{(\alpha)}_n(t),$ with $$c_n(f)=\int_{0}^{\infty}\ell_n^{(\alpha)}(t)f(t)\,dt,$$ converges pointwise to $f(t).$
\end{theorem}
\begin{proof} By the Cauchy-Schwartz inequality, we get that $c_n(f)\in X,$ $$\lVert c_n(f) \rVert\leq (\lVert \mathscr{L}_n^{(\alpha)} \rVert_2)^{\frac{1}{2}}\left(\frac{n!}{\Gamma(n+\alpha+1)}\displaystyle\int_{0}^{\infty}e^{-t}t^\alpha\lVert f(t) \rVert^2dt\right)^{\frac{1}{2}}<\infty, $$ where we have applied that $\{\mathscr{L}_n^{(\alpha)}(t)\}_{n\ge 0}$ is a orthonormal basis in $L^2(\R_+)$ and $f$ satisfies the hypothesis.
Let $S_m(f)$ be the sum of the first $m+1$ terms of the series, $$S_m(f)(t):=\sum_{n=0}^m c_n(f)L_n^{(\alpha)}(t)=\displaystyle\int_{0}^{\infty}e^{-y}y^{\alpha}\varphi_{m}(t,y)f(y)\,dy,\qquad t>0,$$ where $\varphi_m(t,y)=\displaystyle{\sum_{n=0}^m\frac{n!}{\Gamma(n+\alpha+1)}L_n^{(\alpha)}(t)L_n^{(\alpha)}(y),}$ for $t,y>0.$ Note that $\varphi_m(t,y)=\varphi_m(y,t)$, for $t,y >0$, $\displaystyle\int_{0}^{\infty}e^{-y}y^\alpha L_m^{(\alpha)}(t,y)\,dy=1$ for $m\ge 0$ and $$\varphi_m(t,y)={(m+1)!\over \Gamma(m+\alpha+1)}\left({L_{m+1}^{(\alpha)}(t)L_m^{(\alpha)}(y)-L_{m+1}^{(\alpha)}(y)L_m^{(\alpha)}(t)\over y-t}\right), \qquad t\not=y >0,$$ see these properties in \cite[p. 611]{Uspensky}. Now, we write \begin{eqnarray*}
&\quad&S_m(f)(t)-f(t)=\int_{0}^{\infty}e^{-y}y^\alpha\varphi_m(t,y)(f(y)-f(t))\,dy, \qquad t>0, \\
\end{eqnarray*}
and we will conclude that $S_mf(t)-f(t)\to 0$ when $m \to \infty$ for $t>0$. Take $a<t<b$ and there exist $H,G$ such that $0<H<a<b<G$ and
$$
\int_0^H y^\alpha e^{-y}\varphi^2_m(y,t)dy <{C}, \qquad \int_G^\infty y^\alpha e^{-y}\varphi^2_m(y,t)dy <{C},
$$
for a constant $C>0$, see \cite[p. 614, formula (27)]{Uspensky}.
Take $\varepsilon>0$. There exist $H,G>0$ in the above conditions such that
\begin{eqnarray*}
&\quad&\Vert \int_{0}^{H}e^{-y}y^\alpha\varphi_m(t,y)(f(y)-f(t))\,dy\Vert \\&\quad& \qquad \le\left(\int_0^H y^\alpha e^{-y}\varphi^2_m(y,t)dy\right)^{1\over 2}\left(\int_0^H y^\alpha e^{-y}\Vert f(y)-f(t)\Vert^2dy\right)^{1\over 2}\le {\varepsilon\over 3}
\end{eqnarray*}
and similarly
\begin{eqnarray*}
&\quad&\Vert \int_{G}^{\infty}e^{-y}y^\alpha\varphi_m(t,y)(f(y)-f(t))\,dy\Vert \\&\quad& \qquad \le\left(\int_G^\infty y^\alpha e^{-y}\varphi^2_m(y,t)dy\right)^{1\over 2}\left(\int_G^\infty y^\alpha e^{-y}\Vert f(y)-f(t)\Vert^2dy\right)^{1\over 2}\le {\varepsilon\over 3}.
\end{eqnarray*}
Note that
\begin{eqnarray*}
\varphi_m(t,y)={\sqrt{(m+1)(m+\alpha+1)}\over \pi m}{(ty)^{{-\alpha \over 2}-{1\over 4}}e^{t+y\over 2}\over t-x}\left(T_m(t,y)+ {U_m(t,y)\over \sqrt{m}}\right),
\end{eqnarray*}
where
\begin{eqnarray*}
T_m(t,y)&=& \sqrt{y}\cos\left(2\sqrt{mt}-{2\alpha+1\over 4}\pi\right)\sin\left(2\sqrt{my}-{2\alpha+1\over 4}\pi\right)\\
&\quad&-\sqrt{t}\cos\left(2\sqrt{my}-{2\alpha+1\over 4}\pi\right)\sin\left(2\sqrt{mt}-{2\alpha+1\over 4}\pi\right)
\end{eqnarray*}
and $\vert U_n(t,y)\vert \le M$ for $y\in (H, G)$, see \cite[pp. 612-613]{Uspensky}. Now we define $$F(t,y)=\frac{f(y)-f(t)}{y-t}, \qquad t\not=y >0.$$ Observe that as function of $y,$ $F(t, \cdot)$ is continuous in $(0,\infty)$ for any $t>0$. Finally, we have that
\begin{eqnarray*}&\quad &\int_H^G e^{-y}y^\alpha\varphi_m(t,y)(f(y)-f(t))\,dy\\
&\quad&\qquad \qquad= C_m\, t^{{-\alpha \over 2}-{1\over 4}}e^{t\over 2}\int_H^G e^{-{y\over 2}}y^{{\alpha \over 2}-{1\over 4}}\left(T_m(t,y)+{U_m(t,y)\over \sqrt{m}}\right)F(t,y)\,dy,
\end{eqnarray*}
where $\displaystyle{\sup_{m\ge 1}C_m<\infty}$. Note that if $m\to \infty$, then
\begin{eqnarray*}\displaystyle{\int_H^G e^{-{y\over 2}}y^{{\alpha \over 2}-{1\over 4}}{U_m(t,y)\over \sqrt{m}}F(t,y)\,}dy\to 0,\qquad
\displaystyle{
\int_H^G e^{-{y\over 2}}y^{{\alpha \over 2}-{1\over 4}}T_m(t,y)F(t,y)\,dy} \to 0,
\end{eqnarray*} where we have applied the Riemann-Lebesgue Lemma in the second limit, see for example \cite[Theorem 1.8.1 c)]{ABHN}. We conclude that $$\displaystyle{\Vert \int_H^G e^{-y}y^\alpha\varphi_m(t,y)(f(y)-f(t))\,dy\Vert \le {\varepsilon \over 3}},$$ and $ \displaystyle\lim_{m\to\infty}\lVert S_m(f)(t)-f(t)\rVert=0,$ for $t>0$.\end{proof}
\begin{remark}\label{UMD} {\rm In fact, the original theorem of J. V. Uspensky is proved requiring less restrictive conditions on the function $f$, in particular, the existence of singular Dirichlet integral, see \cite[p. 617]{Uspensky}. In UMD Banach spaces, the convergence of the Laguerre expansion for continuous functions might be proved following the original proof given in \cite{Uspensky}.
On the other hand, a straightforward application of Theorem \ref{upes} allows to obtain the weak convergence of the partial series $S_m(f)(t)$ to the function $f(t)$ for $t>0$.}
\end{remark}
\section{$C_0$-semigroups and resolvent operators given by Laguerre expansions}
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In this section, we are interested in representing $C_0$-semigroups and resolvent operators on series of Laguerre polynomials. To do this, we will apply several results included in previous sections. However, first of all, we give some basic results from $C_0$-semigroup theory, for further details see monographies \cite{ABHN, Nagel}.
Given $-A$ a closed operator in a Banach space $X.$ The resolvent operator $\lambda\to (\lambda+A)^{-1}$ is analytic in the resolvent set $\rho(-A)$, and $$\frac{d^n}{d\lambda^n}(\lambda+A)^{-1}=(-1)^n n! (\lambda+A)^{-n-1}\ \text{for all }n\in\N,$$ see \cite[p.240]{Nagel}. We say that a $C_0$-semigroup $(T(t))_{t\geq 0}$ is uniformly bounded if $\lVert T(t)\rVert\leq M$ for all $t\geq 0,$ with $M$ a positive constant. Let $-A$ be the infinitesimal generator of a uniformly bounded $C_0$-semigroup $(T(t))_{t\geq 0}$. For $\alpha>0$ and $\lambda>0$ we define the fractional powers of the resolvent operator as below \begin{equation}\label{4}(\lambda+A)^{-\alpha}x:=\frac{1}{\Gamma(\alpha)}\int_0^{\infty}t^{\alpha-1}e^{-\lambda t}T(t)x\,dt,\qquad x\in X,\end{equation} see \cite[Proposition 11.1]{Komatsu}. The Cayley transform of $-A,$ i.e., $V:=(A-1)(A+1)^{-1},$ defines a bounded operator called the cogenerator of the $C_0$-semigroup, see for example \cite{Eisner, Gomilko}. Note that \begin{equation}\label{coge}A^n(A+1)^{-n-\alpha-1}x=\biggl(\frac{V+1}{2}\biggr)^n\biggl(\frac{1-V}{2}\biggr)^{\alpha+1}x, \qquad x\in X.
\end{equation}
Nice identities between the powers of the cogenerator and integral expressions which involve generalized Laguerre polynomials appear in \cite[Theorem 1]{Gomilko}, \cite[Lemma 4.4]{BZ} and \cite[Lemma 6.7]{Besselin}.
\begin{theorem}\label{loc} Let $(T(t))_{t\geq 0}$ be a uniformly bounded $C_0$-semigroup in a Banach space $X$ with infinitesimal generator $(-A, D(A)).$
\begin{itemize}
\item[(i)]For $n\in \NN\cup\{0\}$ and $\alpha >-1$,
$$
\int_{0}^{\infty}\ell_n^{(\alpha)}(t)T(t)x\,dt=A^n(A+1)^{-n-\alpha-1}x, \qquad x\in X.
$$
\item[(ii)] For $n\in \NN\cup\{0\}$ and $\alpha>0$,
$$
\int_{0}^{\infty}\ell_n^{(\alpha)}(t)(t+A)^{-1}x\,dt=\int_{0}^{\infty}\varphi_{n, \alpha}(t)T(t)x\,dt, \qquad x\in X.
$$
\item[(iii)] For $x\in D(A)$ and $\alpha>-1$, $$T(t)x=\displaystyle\sum_{n=0}^{\infty}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t), \qquad t>0.$$
\item[(iv)]For $x\in D(A)$ and $\alpha>-1$, $$T(t)x=\biggl(\frac{1-V}{2}\biggr)^{\alpha+1}\displaystyle\sum_{n=0}^{\infty}\biggl(\frac{V+1}{2}\biggr)^nx\,L^{(\alpha)}_n(t), \qquad t>0.$$
\item[(v)] For $x\in X$ and $\alpha >1$, $$(\lambda+A)^{-1}x=\displaystyle\sum_{n=0}^{\infty}\left(\int_{0}^{\infty}\varphi_{n, \alpha}(t)T(t)xdt\right)L^{(\alpha)}_n(\lambda), \qquad \lambda>0.$$
\end{itemize}
\end{theorem}
\begin{proof} (i) For $\alpha>-1 $ and $n\in \NN\cup\{0\}$, we have that
$$\int_{0}^{\infty}\ell^{(\alpha)}_n(t)T(t)x\,dt=\frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{\infty}\frac{d^n}{dt^n}(e^{-t}t^{n+\alpha})T(t)x\,dt, \qquad x\in X.$$ We integrate by parts to obtain that
$$\int_{0}^{\infty}\ell^{(\alpha)}_n(t)T(t)x\,dt= A^n\frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{\infty}e^{-t}t^{n+\alpha}T(t)x\,dt=A^n(A+1)^{-n-\alpha-1}x,
$$
where we use the formula \eqref{4}.
(ii) Take $x\in X$ and $\alpha>0$. The integral $\int_{0}^{\infty}\ell_n^{(\alpha)}(t)(t+A)^{-1}x\,dt$ converges due to $\Vert (t+A)^{-1}\Vert \le {M\over t}$ for $t>0$. Then
\begin{eqnarray*}
\int_{0}^{\infty}\ell_n^{(\alpha)}(t)(t+A)^{-1}x\,dt
&=&\frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{\infty}\frac{d^n}{dt^n}(e^{-t}t^{n+\alpha})(t+A)^{-1}x\,dt \\
&=&\frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{\infty}e^{-t}t^{n+\alpha}(\int_0^{\infty}s^ne^{-ts}T(s)x\,ds)\,dt \\
&=&\int_0^{\infty}\frac{s^n}{(s+1)^{n+\alpha+1}}T(s)x\,ds=\int_0^{\infty}\varphi_{n,\alpha}(s)T(s)x\,ds.
\end{eqnarray*}
(iii) For each $x\in D(A),$ the function $T(\cdot)x:\R_+\to X$ is differentiable at every point and ${d\over dt}T(t)x=-T(t)Ax$, see \cite[Definition 1.2, Chapter II]{Nagel}. In addition note that $$\int_{0}^{\infty}e^{-t}t^{\alpha}\lVert T(t)x\rVert^2\,dt\leq M^2\int_{0}^{\infty}e^{-t}t^{\alpha}\lVert x\rVert^2\,dt<\infty.$$ Then, we apply Theorem \ref{upesvect} to have that $\lVert T(t)x-\displaystyle\sum_{n=0}^{m}c_n(T(\cdot)x)\,L^{(\alpha)}_n(t) \rVert\to 0$ as $m\to\infty,$ for all $t> 0,$ with $$c_n(T(\cdot)x)=\int_{0}^{\infty}\ell^{(\alpha)}_n(t)T(t)x\,dt=A^n(A+1)^{-n-\alpha-1}x,$$ where we have applied the part (i). (iv) The proof of (iv) is a straightforward consequence of (iii) and (\ref{coge}). (v) Note that $\displaystyle\int_0^{\infty}e^{-t}t^{\alpha}\lVert (t+A)^{-1}\rVert^2 dt\leq M^2\int_0^{\infty}\frac{e^{-t}}{t^{2-\alpha}}dt<\infty$ if only if $\alpha>1.$ In this case $(\lambda+A)^{-1}x=\displaystyle\sum_{n=0}^{\infty}c_n((\cdot+A)^{-1}x)\,L^{(\alpha)}_n(\lambda)$ for $x\in X$ with $$
c_n((\cdot+A)^{-1}x)= \int_{0}^{\infty}\ell_n^{(\alpha)}(t)(t+A)^{-1}x\,dt=\int_{0}^{\infty}\varphi_{n, \alpha}(t)T(t)x\,dt,
$$
where we have applied the part (ii).
\end{proof}
For $C_0$-semigroups which are not uniformly bounded, we may perturb the infinitesimal generator to obtain uniformly bounded $C_0$-semigroups, and in this way, $-A-w$ generates a uniformly bounded $C_0$-semigroup, see for example \cite[p.60]{Nagel}.
\begin{corollary} Let $(T(t))_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $X$ with infinitesimal generator $(-A, D(A))$ and exponential bound $w,$ that is, $\lVert T(t)\rVert\leq Me^{wt}.$ Then for \mbox{$x\in D(A)$} and $\alpha>-1$, $$T(t)x=e^{wt}\displaystyle\sum_{n=0}^{\infty}(A+w)^n(A+w+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t), \qquad t>0.$$
\end{corollary}
Approximation theory of $C_0$-semigroups has a great importance in many mathematical areas, as PDEs, mathematical physics and probability
theory. There are a large number of results about approximation of $C_0$-semigroups using different approximations. A nice summary of these different approximations may be found in the recent paper \cite{Gomilko}. Subdiagonal Pad\'{e} approximations to $C_0$-semigroups are given in \cite{BT, Egert, Neubrander}.
Now we are interested in the approximation order of the $C_0$-semigroup by the $m-$th partial sum of the Laguerre series, where, $$T_{m,\alpha}(t)x:=\displaystyle\sum_{n=0}^{m}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t), \qquad x\in X, \quad t> 0.$$ Our approximation series are computationally better than the rational approximations (for example Euler approximation or Pad\'{e} approximation) since in each step one has to start to calculate all approximations over again (\mbox{$(1+t\frac{A}{m})^{-m}x,$ in the case of Euler approximation)} while for the Laguerre series one have already calculated until \mbox{$A^{m-1}(A+1)^{-(m-1)-\alpha-1}x,$} see \cite{Grimm}.
\begin{theorem}\label{rate} Let $(T(t))_{t\geq 0}$ be a uniformly bounded $C_0$-semigroup in a Banach space $X,$ $\lVert T(t)\rVert\leq M,$ with infinitesimal generator $-A,$ $\alpha>-1$.
\begin{itemize}
\item[(i)] For $x\in D(A^p)$ and $n\geq p,$ $$\lVert A^n(A+1)^{-n-\alpha-1}x\rVert\leq \frac{C} {n^{\frac{\alpha+p}{2}}}\lVert A^p x \rVert,$$ with $C$ a positive constant depending on $\alpha,$ $p$ and $M.$
\item[(ii)] For each $t>0$ there is a $m_0\in\N$ such that for all integer $2<p\leq m+1$ with $m\geq m_0,$ $$\lVert T(t)x-T_{m,\alpha}(t)x \rVert\leq \frac{ C_{t,p} \lVert A^p x\rVert}{m^{\frac{p}{2}-1}} \qquad x\in D(A^p),$$ where $C_{t,p}$ is a constant which depends only on $t>0$ and $p$.
\end{itemize}
\end{theorem}
\begin{proof}
(i) We write $B(x):=A^n(A+1)^{-n-\alpha-1}x=\int_0^{\infty}\ell_n^{(\alpha)}(t)T(t)x\,dt$ for $x\in X,$ see Theorem \ref{loc} (i). We apply Proposition \ref{propLag} (vi) and integrate it by parts to obtain that $$B(x)=\int_0^{\infty}\frac{d^p}{dt^p}(\ell_{n-p}^{(\alpha+p)}(t))T(t)x\,dt=\int_0^{\infty}\ell_{n-p}^{(\alpha+p)}(t)A^pT(t)x\,dt \qquad x\in D(A^p),$$
where we have used that $\frac{d^{p-j}}{dt^{p-j}}(\ell_{n-p}^{(\alpha+p)}(t))_{t=0}=0,$ and $\frac{d^{p-j}}{dt^{p-j}}(\ell_{n-p}^{(\alpha+p)}(t))_{t=\infty}=0,$ for $j=1,2,\ldots,p.$ By Theorem \ref{main2} (ii), $$\lVert B(x) \rVert\leq \frac{C} {n^{\frac{\alpha+p}{2}}}\displaystyle\sup_{ t\ge 0}\lVert A^p T(t)x \rVert < \frac{C} {n^{\frac{\alpha+p}{2}}}\lVert A^p x \rVert,$$ and we conclude the result.
(ii) By Muckenhoupt estimates,
we know that there is a $m_0\in\N$ such that $|L_m^{(\alpha)}(t)|\leq C_t m^{\frac{\alpha}{2}},$ for all $m\geq m_0,$ see (\ref{mucklag}). So, for $m\geq m_0$ and $x\in D(A^p)$ with $p\leq m+1,$ we apply Theorem \ref{loc} (iii) and (i) to get that $$
\lVert T(t)x-T_{m,\alpha}(t)x\rVert\leq\displaystyle\sum_{n=m+1}^{\infty}\lVert A^n(A+1)^{-n-\alpha-1}x \rVert |L_n^{(\alpha)}(t)|\leq\displaystyle\sum_{n=m+1}^{\infty}\frac{C_t}{n^{\frac{p}{2}}}\lVert A^p x\rVert\leq\frac{ C_{t,p} \lVert A^p x\rVert}{m^{\frac{p}{2}-1}},$$
where we have used the bound of part (i).
\end{proof}
The following result gives some representation formulae (via series of operators) of fractional powers of resolvent operator of $-A$. The Hille functional calculus is a linear and bounded map $f\mapsto f(A)$, $L^1(\R_+) \to {\mathcal B}(X)$, where
$$
f(A)x:=\int_0^\infty f(t)T(t)xdt, \qquad x\in X, \quad f\in L^1(\R_+),$$
and $(T(t))_{t\geq 0}$ is a uniformly bounded semigroup generated by $(-A, D(A))$ (see for example \cite[Chapter 3]{Sinclair}). Note that if $f= \sum_{n=1}^\infty f_n$ in $L^1(\R_+)$ then $f(A)= \sum_{n=1}^\infty f_n(A)$ in ${\mathcal B}(X)$.
\begin{theorem}\label{appr} Let $(-A, D(A))$ be the infinitesimal generator of a uniformly bounded $C_0$-semigroup $(T(t))_{t\geq 0}$ in a Banach space $X.$
\begin{itemize}
\item[(i)] For $2\beta>\alpha>-1,$ and $x\in X$,
$$(A+1)^{-\beta-1}=\displaystyle\sum_{n=0}^{\infty}\binom{\alpha-\beta+m-1}{m}A^n(A+1)^{-n-\alpha-1} \hbox{ in }{\mathcal B}(X).$$ In particular for $\alpha>2,$ we have that
$(A+1)^{-\alpha}=\displaystyle\sum_{n=0}^{\infty}A^n(A+1)^{-n-\alpha-1} \hbox{ in }{\mathcal B}(X).$
\item[(ii)] For all $a>0$, $(a+A)^{-1}=\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}(A-\frac{1}{2})^n(A+\frac{1}{2})^{-n-1}$ holds in ${\mathcal B}(X).$
\end{itemize}
\end{theorem}
\begin{proof}
(i) By Theorem \ref{fract} (i) and Theorem \ref{loc} (i), we have that \begin{eqnarray*}
(A+1)^{-\beta-1}x&=&\frac{1}{\Gamma(\beta+1)}\int_0^{\infty}t^{\beta}e^{-t}T(t)x\,dt=\displaystyle\sum_{n=0}^{\infty}\binom{\alpha-\beta+n-1}{n}\int_0^{\infty}\ell_n^{(\alpha)}(t)T(t)x\,dt \\
&=&\displaystyle\sum_{n=0}^{\infty}\binom{\alpha-\beta+n-1}{n}A^n(A+1)^{-n-\alpha-1}x.
\end{eqnarray*}
(ii) For $a>0$ and $x\in X,$ we have by Theorem \ref{fract} (ii),\begin{eqnarray*} (a+A)^{-1}x&=&\int_0^{\infty}e^{-at}T(t)x\,dt=\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}\int_0^{\infty}e^{-\frac{t}{2}}L_n^{(0)}(t)T(t)x\,dt \\
&=&\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}\int_0^{\infty}\ell_n^{(0)}(t)S(t)x\,dt,
\end{eqnarray*}
with $S(t):=e^{\frac{t}{2}}T(t)$is the $C_0$-semigroup generated by $\frac{1}{2}-A.$ By Theorem \ref{fract} (i), we conclude that $ \displaystyle{(a+A)^{-1}x=\displaystyle\sum_{n=0}^{\infty}\frac{(a-\frac{1}{2})^n}{(a+\frac{1}{2})^{n+1}}(A-\frac{1}{2})^n(A+\frac{1}{2})^{-n-1}x},$ for $x\in X.$
\end{proof}
\section{Examples and final comments}
In this section we apply our results to concrete examples: translation, convolution and multiplication semigroups in Lebesgue space. Holomorphic semigroups allow us to improve some previous results, compare Theorem \ref{Holomorphic} and Theorem \ref{loc}.
\setcounter{theorem}{0}
\setcounter{equation}{0}
\subsection{Translation semigroup}
Let $L^p(\R_+)$ be with $1\leq p<\infty.$ The translation semigroup (or shift semigroup) in $L^p(\R_+)$, $(T(t))_{t\geq 0},$ defined by \begin{displaymath}
T(t)f(x):=(\delta_t\ast f)(x)=\left\{\begin{array}{ll}
f(x-t),&x>t,\\
0,&x\leq t,
\end{array} \right.
\end{displaymath} is an isometry $C_0$-semigroup. The infinitesimal generator $-A$ is the usual derivation operator, $-A=-\frac{d}{dx}.$ Furthermore, $
(A+1)^{-n-\alpha-1}f=I_{n+\alpha +1}*f,$ and
$A^n(A+1)^{-n-\alpha-1}f=\ell^{(\alpha)}_n*f,$
for $n\in\NN\cup\{0\}$ and $\alpha >-1$. By Theorem \ref{loc} (iii), we obtain the formula
$$
\delta_t\ast f=\displaystyle\sum_{n=0}^{\infty}(\ell_n^{(\alpha)}\ast f)L_n^{(\alpha)}(t), \qquad f\in W^{(1),p}(\R_+),
$$
where $W^{(1),p}(\R_+)$ is the Sobolev space defined by \mbox{$W^{(1),p}(\R_+)=\{f\in L^p(\R_+)\,\,|\,\,f'\in L^p(\R_+)\}$.} In particular for $\alpha=0$ this formula has been considered in \cite[Section 3, Examples 3.1(4)]{Du} where Laguerre expansions of tempered distributions are studied.
\subsection{Convolution and multiplication semigroups}
Let $L^p(\R^m)$ with $m\geq 1$ and $1\leq p<\infty.$ For $t>0$, let \mbox{$k_t:\R^m\to\R$} be a convolution kernel on $L^p(\R^m)$ such that $\displaystyle\sup_{t>0}\Vert k_t\Vert_1<\infty$ and $(T(t))_{t\geq 0},$ defined by $$T(t)f(s):=(k_t*f)(s)= \int_{\R^m}k_t(s-u)f(u)du, \qquad f\in L^p(\R^m),$$ is a uniformly bounded $C_0$-semigroup, whose generator is denoted by $-A$.
Fixed $s\in\R^m,$ we suppose that the map $t \mapsto k_t(s)$ is differentiable with respect to $t$ and
\begin{equation}\label{formu}\int_{0}^{\infty}e^{-t}t^{\alpha}|k_t(s)|^2\,dt<\infty.
\end{equation}
By Theorem \ref{upes}, we have that $k_t(s)=\displaystyle\sum_{n=0}^{\infty}a_n(s)L_n^{(\alpha)}(t)$ for $s\in \R^m,$ and $t>0,$ where $$a_n(s)=\int_{0}^{\infty}\ell_n^{(\alpha)}(t)k_t(s)\,dt, \qquad s\in \R^m.$$ Furthermore, $a_n\in L^1(\R^n)$ and
$\Vert a_n\Vert_1\leq \displaystyle{\sup_{t>0}}\Vert k_t\Vert_1\, \Vert \ell_n^{(\alpha)}\Vert_1.$
By Fubini's Theorem and Theorem \ref{loc} (i), we obtain that $$A^n(A+1)^{-n-\alpha-1}f=a_n*f, \qquad f\in L^p(\R^m),$$ and therefore by Theorem \ref{loc} (iii), we have $\displaystyle{k_t*f=\displaystyle\sum_{n=0}^{\infty}(a_n\ast f)L_n^{(\alpha)}(t)}$ for $f\in D(A).$
Examples of convolutions semigroups are the Gaussian and Poisson kernels, $$g_t(s):=\frac{1}{(4\pi t)^{\frac{m}{2}}}e^{-\frac{\lVert s\rVert^2}{4t}}, \qquad p_t(s):=\frac{\Gamma(\frac{m+1}{2})}{\pi^{\frac{m+1}{2}}}\frac{t}{(t^2+\lVert s\rVert^2)^{\frac{m+1}{2}}}$$ see \cite[p.25]{Sinclair}. The condition (\ref{formu}) is verified by Gaussian and Poisson kernels if $s\neq 0$ for $\alpha>-1;$ in the case if $s=0,$ it is needed to consider $\alpha>m-1$ for Gaussian kernel and $\alpha>2m-1$ for Poisson kernel.
Now, let $q:\R^m\to\R_-$ be a continuous function, where $\R_{-}:=(-\infty,0]$. The family of operators in $C_0(\R^m)$, $(S_q(t))_{t\geq 0},$ defined by $S_q(t)f:=e^{tq}f,$ for $t>0,$ is a contraction $C_0$-semigroup whose infinitesimal generator $(-B, D(B))$ is given by $-Bf:=qf$ and $\ D(B)=\{f\in C_0(\R^m)\,\,|\,\, qf\in C_0(\R^m)\}$. In these cases, note that $$ B^n(B+1)^{-n-\alpha-1}f(s)=\frac{(-q)^n(s)}{(1-q(s))^{n+\alpha+1}}f(s), \qquad f\in C_0(\R^m), \quad s\in \R^m,$$ and $$S_q(t)f=\displaystyle\sum_{n=0}^{\infty}\frac{(-q)^n}{(1-q)^{n+\alpha+1}}f\,L_n^{(\alpha)}(t),\qquad f\in D(B),\quad t>0.$$
Some examples are the Fourier transforms of the Gaussian and Poisson semigroups, $q(s)=-\lVert s\rVert^2$ and $q(s)=-\lVert s\rVert$ (\cite[p.69]{Nagel}); or subordinated semigroups to them with $q(s)=-\text{log}(1+ \lVert s\rVert^2)$ and \mbox{$q(s)=-\text{log}(1+\lVert s\rVert),$} studied in details in \cite{Campos}.
To finish this subsection we are interested in identifying the functions $a_n$ in the cases of Poisson and Gaussian semigroup. We denote by \mbox{$\mathcal{F}: L^1(\R^m)\to C_0(\R^m)$} the usual Fourier transform, defined by
$\displaystyle \mathcal{F}(f)(s):=\int_{\R^m} e^{-i s\cdot u}f(u)\,du,$
with $s\cdot u$ the inner product in $\R^m.$ For the Poisson semigroup, we have $\mathcal{F}(a_n)(s)=\varphi_{n,\alpha}(\lVert s\rVert)$ for $s\in \R^m$ and \mbox{$\mathcal{F}(a_n)\in L^1(\R^m)$} for $\alpha>m-1$ and belongs to $L^2(\R^m)$ for $\alpha>\frac{m-2}{2}$, see functions $\varphi_{n,\alpha}$ defined in (\ref{definit}). For the Gaussian semigroup, $\mathcal{F}(a_n)(s)=\varphi_{n,\alpha}(\lVert s\rVert^2)$ for $s\in \R^m$ and \mbox{$\mathcal{F}(a_n) \in L^1(\R^m)$} for $\alpha>\frac{m-2}{2}$ and belongs to $L^2(\R^m)$ for $\alpha>\frac{m-4}{4}.$ To show both results, we use spherical coordinates and apply Theorem \ref{phi}\ (i). Note that in these cases $a_n$ is a radial function. Then $\mathcal{F}(a_n)=2\pi\mathcal{F}^{-1}(a_n)$ and $$a_n(s)=\frac{C_m}{(2\pi)^m}\int_0^{\infty}\mathcal{F}(a_n)(r)j_m(\lVert s\rVert r)\,r^{m-1}\,dr,\qquad s\in \R^m,$$ where $C_m$ is the area of the unit $(m-1)$-dimensional sphere and $j_m$
is the spherical Bessel function for $m\geq2$, see more details in \cite[p.133]{Dym}. In the particular case of $m=1,$ we have that $$a_n(s)=\frac{1}{2\pi}\int_0^{\infty}e^{i s r}\mathcal{F}(a_n)(r)\,dr+\frac{1}{2\pi}\int_0^{\infty}e^{- i s r}\mathcal{F}(a_n)(r)\,dr,$$ since $\mathcal{F}(a_n)$ is an even function. For $\alpha>0$ and the Poisson semigroup, we have that \begin{eqnarray*}
a_n(s)&=&\frac{1}{2\pi}\biggl(\displaystyle\lim_{\lambda\to i s}\mathcal{L}(\varphi_{n,\alpha})(\lambda)+\displaystyle\lim_{\lambda\to - i s}\mathcal{L}(\varphi_{n,\alpha})(\lambda)\biggr) \\
&=&\frac{n!}{2\pi}\biggl((-is)^{\frac{\alpha-1}{2}}e^{\frac{-i s}{2}}W_{-n-\frac{(1+\alpha)}{2},-\frac{\alpha}{2}}(-i s)+(is)^{\frac{\alpha-1}{2}}e^{\frac{i s}{2}}W_{-n-\frac{(1+\alpha)}{2},-\frac{\alpha}{2}}(i s)\biggr),
\end{eqnarray*}
for $s\in \R$ and $W_{k,\mu}$ is the Whittaker function; for the Gaussian semigroup, $$a_n(s)=\frac{1}{2\pi}\biggl(\displaystyle\lim_{\lambda\to i s}\mathcal{L}(\varphi_{n,\alpha}(t^2))(\lambda)+\displaystyle\lim_{\lambda\to -i s}\mathcal{L}(\varphi_{n,\alpha}(t^2))(\lambda)\biggr),\qquad s\in \R.$$
To calculate $\mathcal{L}(\varphi_{n,\alpha}(t^2))(\lambda)$ with $\lambda \in \C_+$, note that \begin{eqnarray*}
\mathcal{L}(\varphi_{n,\alpha}(t^2))(\lambda)=\frac{1}{\sqrt{\pi}}\int_0^{\infty}e^{-\frac{\lambda^2}{4 u^2}}\mathcal{L}(\varphi_{n,\alpha})(u^2)\,du
=\frac{n!}{\sqrt{\pi}}\int_0^{\infty}e^{-\frac{\lambda^2}{4 u^2}}u^{\alpha-1}e^{\frac{u^2}{2}}W_{-n-\frac{(1+\alpha)}{2},-\frac{\alpha}{2}}(u^2)\,du,
\end{eqnarray*}
where we have used Laplace transform properties and Remark \ref{whittaker}.
\subsection{Differentiable and holomorphic semigroups}
So far, we have considered $C_0$-semigroups defined in $[0,\infty).$ If we consider differentiable or holomorphic semigroups with certain growth assumptions, some previous results will be improved.
A $C_0$-semigroup $(T(t))_{t\ge 0}$ is called immediately differentiable if the orbit $t\mapsto T(t)x$ is differentiable for $t>0$, see \cite[Definition II.4.1]{Nagel}. A straightforward consequence of Theorem \ref{loc} (iii) is the following corollary. A similar result seems that it may hold in UMD spaces for every uniformly bounded $C_0$-semigroup, see Remark \ref{UMD}.
\begin{corollary}Let $-A$ be the infinitesimal generator of a immediately differentiable uniformly bounded $C_0$-semigroup $(T(t))_{t\geq 0}$ in a Banach space $X.$ Then
$$T(t)x=\sum_{n=0}^{\infty}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t),\qquad t>0, \quad x\in X.$$
\end{corollary}
A holomorphic semigroup $(T(z))_{z\in\C_+}$ is said of type $HG_{\beta}$ if $\displaystyle\lVert T(z) \rVert\leq C_{\nu}\biggl(\frac{|z|}{\Re z}\biggr)^{\nu},$ for $\Re z>0$ and every $\nu>\beta,$ see definition in \cite{Gale}. Classical holomorphic semigroup as the Gaussian and the Poisson semigroup satisfy the $HG_{\beta}$ condition for some $\beta>0$. Moreover, a large amount of different semigroups as those generated by $-\sqrt{\mathcal{L}},$ where $\mathcal{L}$ is the sub-Laplacian in $L^p(\mathbb{G})$ and $\mathbb{G}$ a Lie Group, or generated by $-(-\log(\lambda)+H),$ where $H$ is the strongly elliptic operator affiliated with a strongly continuous representation of a Lie Group into a Banach space, satisfy the $HG_{\beta}$ condition and can be found in \mbox{\cite[Section 5]{Gale2}.}
\begin{theorem}\label{Holomorphic} Let $-A$ be the infinitesimal generator of a uniformly bounded holomorphic $C_0$-semigroup $(T(z))_{z\in\C_+}$ of type $HG_{\beta}$ for some $\beta\ge 0$ in a Banach space $X.$
\begin{itemize}
\item[(i)] For $\alpha >-1$, there exists $C_{\alpha,\beta} >0$ such that
$$\lVert A^n(A+1)^{-n-\alpha-1}x \rVert\leq \frac{C_{\alpha,\beta}\Vert x\Vert}{n^{\alpha+1}}, \qquad x\in X,$$
\item[(ii)] For $\alpha>0$ and $t>0,$ $$\lVert T(t)x-\displaystyle\sum_{n=0}^{m}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t) \rVert\leq \frac{C_{\alpha,\beta,t}}{m^{\frac{\alpha}{2}}}\Vert x\Vert, \qquad x\in X.$$
\end{itemize}
\end{theorem}
\begin{proof} (i) By Theorem \ref{phi} (iii), $\varphi_{n,\alpha}\in AC^{(j)}$ for $j,n\in\N,$ and $\alpha>-1$. We apply holomorphic functional calculus $f\mapsto f(A)$ defined in \cite[Theorem 6.2.]{Gale} to get that
$$\lVert A^n(A+1)^{-n-\alpha-1}x \rVert= \lVert \varphi_{n,\alpha}(A)x \rVert\leq C\Vert x\Vert \,\Vert \varphi_{n,\alpha}\Vert_{(j)}\leq \frac{C_{\alpha,\beta}\Vert x\Vert}{n^{\alpha+1}},\qquad x\in X,
$$
for all $j>\beta+1$ and $x\in X$. (ii) For $t>0$ there is a $n_0\in\N$ such that the inequality $|L_n^{(\alpha)}(t)|\leq C_tn^{\frac{\alpha}{2}}$ holds for $n\geq n_0$, see formula (\ref{mucklag}).
We use part (i) to get that $$
\lVert T(t)x-\displaystyle\sum_{n=0}^{m}A^n(A+1)^{-n-\alpha-1}x\,L^{(\alpha)}_n(t)\rVert \leq \displaystyle\sum_{n=m+1}^{\infty}\frac{C_{\alpha,\beta,t}}{n^{1+\frac{\alpha}{2}}}\Vert x\Vert\leq\frac{C_{\alpha,\beta,t}}{m^{\frac{\alpha}{2}}}\lVert x\rVert, \quad x\in X,$$
and we conclude the result.
\end{proof}
\subsection*{Acknowledgements}
The authors wish to thank O. Blasco, M. L. Rezola and L. Roncal for the pieces of advice and assistance provided in order to obtain some previous results.
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\section{Proof of Theorem A}
Now we are ready for the proof of \hyperlink{theoremA}{Theorem A}.
\begin{proof}[Proof of \hyperlink{theoremA}{Theorem A}]
It is more convenient to work with the inverse of $L$, which will have two stable exponents, and in this case we have to show that $L^n\circ f_t$ decreases the strong stable exponent. Since the perturbations keeps the sum of the two stable exponents constant, this is equivalent to proving that the weak stable exponent increases.
Let us assume that $n>n_2(\geq n_1\geq n_0)$, $t>t_1(\geq t_0)$ and $t^{\alpha}>t_{\alpha}$.
We assume also that the conditions \hyperlink{condA}{(A)}, \hyperlink{condM}{(M)} and \hyperlink{condM}{(L)} are satisfied, so it holds
\begin{align*}
&t\lambda_{ws}^n<\frac 1{a_L},\\
&t\frac{\lambda_{ws}^n}{\lambda_u^n}<\frac{\gamma_M}{\gamma_L} \shortintertext{and}
&t^2\lambda_{ws}^{2n}<\frac 1{l_L}.
\end{align*}
We also assume that the variation of the $l$-Lipschitz unit vectorfields on the atoms of the partition $\xi$ is smaller that the separation of the preimages of the bad cone:
\begin{align*}\hypertarget{condSL}{}
&l_LD_Lt^2\lambda_{ws}^{2n}<s_Lt^{-1}, \shortintertext{or}
&t^3\lambda_{ws}^{2n}<\frac{s_L}{l_LD_L}\hspace{2cm} \text{(we call this {\bf Condition (SL)})}
\end{align*}
Let us remark that all the conditions above are satisfied if we take
\begin{equation}\label{eq:tn}
t=\lambda_{ws}^{-n\nu}
\end{equation}
for some $\nu\in\left(0,\frac 23\right)$ and for $n$ sufficiently large.
From Lemma \ref{le:markov} we have that
\begin{eqnarray*}
\mu_P^{\xi}\left(\cup_{P'\in\mathcal B(P)}P'\right)&<&\delta_L\left(t^{-\alpha}+\lambda_u^{-n}\right)\\
\mu_P^{\xi}\left(\cup_{P'\in\mathcal G^+(P)}P'\right)&>&\frac 13\\
\mu_P^{\xi}\left(\cup_{P'\in\mathcal G^-(P)}P'\right)&>&\frac 13.
\end{eqnarray*}
Since $t^{-\alpha}=\lambda_{ws}^{n\alpha\nu}>\lambda_u^{-n}$, the first relation can be rewriten as
\[
\mu_P^{\xi}\left(\cup_{P'\in\mathcal B(P)}P'\right)< 2\delta_Lt^{-\alpha}:=\delta.
\]
Let $\mathcal{X}$ be the family of unit vectorfields defined on atoms of $\xi$ and which are $l$-Lipschitz on the atom, $l=l_Lt^2\lambda_{ws}^{2n}<1$:
\[
\mathcal{X}=\{ X:P\rightarrow \mathbb PE^s:\ \ P\in\xi,\ X\hbox{ is $l$-Lipschitz}\}.
\]
Lemma \ref{le:lipschitz} shows that the family $\mathcal X$ is invariant under the pushed forward of the projectivization of $L^n\circ f_t$ (in the sense that the image of a $l$-Lipschitz unit vectorfield on $P\in\xi$ is several $l$-Lipschitz unit vectorfields defined on the atoms of $\xi$ from $(L^n\circ f_t)(P)$).
We consider the derivative cocycle of $L^n\circ f_t$ restricted to $E^s$, $D(L^n\circ f_t)|_{E^s}$.
We will check that the family of vector fields $\mathcal X$ is adapted as in Definition \ref{def:adapted}, for $\delta=2\delta_Lt^{-\alpha}$ and $\lambda=C_L\lambda_{ws}^{n-n\nu(1-\alpha)}$. We divide the vectorfields into ``good'' and ``bad'' in the following way:
\begin{eqnarray*}
\mathcal X^g&=&\{ X\in\mathcal X:\ X(p)\in C_g,\ \forall p\in P, \hbox{ where $P$ is the domain of $X$}\}\\
\mathcal X^b&=&\{ X\in\mathcal X:\ \exists p\in P \hbox{ with } X(p)\notin C_g, \hbox{ where $P$ is the domain of $X$}\}
\end{eqnarray*}
The good region is $G=G_t^{\alpha}$.
\begin{itemize}
\item
The condition \ref{item.condicion1} follows from the construction of the Markov partition in Lemma \ref{le:markov}.
\item
The condition \ref{item.condicion2} is obvious, we can take $X$ to be a constant vector in the good cone $C_g$.
\item
The condition \ref{item.condicion3} follows from Proposition \ref{prop:expansion}. The lower bound in \ref{eq:strong} is $C_L\lambda_{ws}^nt^{1-\alpha}$, but since we chose $t=\lambda_{ws}^{-n\nu}$ we have in fact the lower bound $\lambda=C_L\lambda_{ws}^{n-n\nu(1-\alpha)}$.
\item
The condition \ref{item.condicion4} follows also directly from Proposition \ref{prop:expansion}.
\item
The condition \ref{item.condicion5} is the most delicate. It follows from our choice of the Lipschitz constant of the vector fields in $\mathcal X$. The condition \hyperlink{condSL}{(SL)} implies that the variation of the vector field $X$ on an atom $P$ of the partition is smaller that the separation between the preimages of the bad cone given by Lemma \ref{le:separation}. This implies that the push forward of the vector field has to go inside the good cone at least on one of the sets $\mathcal G^+(P)$ or $\mathcal G^-(P)$. The choice of the Markov partition from Lemma \ref{le:markov} will give us the required condition.
\end{itemize}
Now we are in condition to apply Theorem \ref{teo.tecnico}. The cocycle is
\[
A=D(L^n\circ f_t)|_{E^s}=L^n\cdot Df_t|_{E^s}.
\]
The lower bound for the expansion in the good region is
\[
\lambda=C_L\lambda_{ws}^nt^{1-\alpha}=C_L\lambda_{ws}^{n-n\nu(1-\alpha)}.
\]
From Proposition \ref{prop:expansion} we see that the norm of the inverse of the cocycle is bounded by
\[
\| A^{-1}\|<C_L\lambda_{ss}^{-n}t=C_L\lambda_{ss}^{-n}\lambda_{ws}^{-n\nu}.
\]
Remember that also
\[
\delta=2\delta_Lt^{-\alpha}=2\delta_L\lambda_{ws}^{n\nu\alpha}.
\]
Since we have an adapted family of vector fields (for $\beta=\frac 13$), we obtain the inequality
\begin{eqnarray*}
\chi^+&>&\frac 1{1+3\delta}\log\frac {\lambda^{1-\delta}}{\| A^{-1}\|^{4\delta}}\\
&\geq&-C_L+\frac 1{1+3\delta}\log\frac{\lambda_{ws}^{[n-n\nu(1-\alpha)](1-\delta)}}{\lambda_{ss}^{-4n\delta}\lambda_{ws}^{-4n\nu\delta}}\\
&=&-C_L+\frac 1{1+3\delta}\log\left(\lambda_{ws}^{n-n\nu(1-\alpha)+n\delta(5\nu-\nu\alpha-1)}\lambda_{ss}^{4n\delta}\right)\\
&=&-C_L+\log\lambda_{ws}^n-\nu(1-\alpha)\log\lambda_{ws}^n+n\delta\left(\frac {-3\log\lambda_{ws}^{1-\nu(1-\alpha)}+\log\lambda_{ws}^{5\nu-\nu\alpha-1}+\log\lambda_{ss}^4}{1+3\delta}\right)\\
&=&-C_L+\log\lambda_{ws}^n-\nu(1-\alpha)\log\lambda_{ws}^n+\frac{n\delta C_{L,\alpha,\nu}}{1+3\delta}.
\end{eqnarray*}
Paring this with the facts $\delta=2\delta_L\lambda_{ws}^{n\nu\alpha}$ and $\lambda_{ws}<1$, we deduce
\[
\lim_{n\rightarrow\infty}\frac{n\delta C_{L,\alpha,\nu}}{1+3\delta}=\lim_{n\rightarrow\infty}\frac{2n\delta_L\lambda_{ws}^{n\nu\alpha} C_{L,\alpha,\nu}}{1+6\delta_L\lambda_{ws}^{n\nu\alpha}}=0,
\]
so the last term is negligible for large $n$. Then for such $n$'s we will obtain that the weak stable Lyapunov exponent of $L^n\circ f_t$ is greater than the weak stable Lyapunov exponent of $L^n$ ($=\log\lambda_{ws}^n$) by at least $-\nu(1-\alpha)\log_{ws}^n-C_L$, which is positive.
This finishes the proof.
\end{proof}
\begin{remark}
Let us comment that the maximal change that we can achieve for the weak stable Lyapunov exponent is close to two thirds of the weak stable exponent of the linear map $L^n$. This is done for $n$ large, $\nu$ close to $\frac 23$ and $\alpha$ close to $0$.
Then, in the context of \hyperlink{theoremA}{Theorem A}, we can increase the top Lyapunov exponent (strong unstable) by at most two thirds of the size of the second Lyapunov exponent (weak unstable).
\end{remark}
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Definition of pule
puled;
puling
Examples of pule in a sentence
a distressed baby puling in its crib
Origin and Etymology of pule
probably imitative
First Known Use: 14th century
Learn More about pule
See words that rhyme with pule Thesaurus: All synonyms and antonyms for pule
Writing? Check your grammar now!
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BOUNTIFUL - The ragged fence that tried to hide the large hole that once was Bountiful/Davis Art Center is gone.
The centrally located lot, still host to several large trees, is getting a new curb and will soon have sod installed.
City leaders plan on having the grass set in place and ready for action in time for residents to use during the Handcart Days Parade, July 23.
As part of the improvements, the road north of the area will be widened.
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Is it possible? My husband and I are planning to go on a trip to Europe with our two girls (3 and 1 y.o.) at the end of july 2014. We plan on spending one week in the bavaria area, possibly going to Prague, if there is time. Ideally, we would rather rent a space (airbnb) somewhat centrally located in bavaria, so we could go to salzburg, the castles, etc, on day trips. If there is time, we would love to visit prague as well, even if only for a couple of days. So, where should we stay and what are the must-do's during that time of the year? Thanks!'s possible but whether it's too much or not only you can answer.
I suggest basing in Munich as it is only 90 min from Salzburg by car (90-120 min by train) and there is an express bus that leaves several times a day from Munich that gets to Prague in 4.5 hours. Neuschwanstein Castle is about 2.5-3 hours by public transit from Munich (less than two by car). That is about as centrally located as you're going to get.
Just so you know, July is the height of tourist season and so the prices for lodging are accordingly high. It is also the summer school break for most of Germany (but not Bavaria).
We traveled to Bavaria and Salzburg with our (older) children in 2009, and split the time between Fussen and Salzburg. Our trip report is under my name if you're interested.
Fussen: Landhaus Kossel. Our favorite. A one-bedroom apartment with kitchen, plus a large living room with another bed. Terrific views, large yard for the children to play in; a very pleasant choice.
Salzburg. Haus Am Moos. We rented the family apartment and the space worked out beautifully. The family offers breakfast for their guests, as well. We now live in Vienna, and later this month we're spending a weekend again at Haus am Moos while we enjoy the Christmas markets.
In one week with 2 little ones IMHO this is way too much to do.
You can see parts of Bavaria and a day trip to Salzburg is doable - but long.
But Prague is 5 hours by car - so at least 10 hours of driving without stops. It would make sense only if you spend 3 nights (2 days) there - making seeing much of Bavaria and Salzburg at all impossible.
"Ideally, we would rather rent a space (airbnb) somewhat centrally located in bavaria, so we could go to salzburg, the castles, etc, on day trips. If there is time, we would love to visit prague as well..."
Suggest you base yourselves either in Salzburg itself, in nearby Berchtesgaden, or in Prien, Germany.
Prien is home to one of King Ludwig's "castles" - Herrenchiemsee, which like Neuschwanstein and Hohenschwnangau is a palace. Herrenchiemsee is situated on an island in Chiemsee Lake. Nice place.
Prien is 60 minutes by train from Salzburg, so you could daytrip there and back. It's about the same into Munich if you want to see Munich. You'll find a lot of rentals there as well (airbnb offers only the tiniest fraction of the total rentals available in Bavaria; check local tourist office sources instead.)
Berchtesgaden is close to Salzburg and would also be a possible daytrip from Prien - or vice versa.
Castles: Salzburg and nearby Werfen are your best options.
I can't see how you squeeze in Prague. See it another time.
Thanks everyone! Sounds like we can stay busy without going to Prague. Prague is one of those places I've been dreaming of going to visit, so it may be worth the wait.
Sparkchaser, thanks for the suggestion. I was actually thinking of flying in and out of Munich. So it may be a good place to call home base.
Fourfortravel, love the suggestions for stay outside of munich. Your post is so thorough! Super helpful. Thanks!
Nytraveler, totally agree. We'l skip Prague for now.
Russ, thanks for all the links and suggestions of places to stay.
I have another question:
How about the "Romantic Drive"? If we were to fly into Frankfurt, then out of Munich, would we have time to drive down and make it to Salzburg? This would mean a stop at a different place every night, which we are okay with. We've road the coast of california and oregon in recent years and had a blast! What are your thoughts on this drive? Worth it? Or is it best to stay in one spot and drive part of it?
Thanks again!
Driving the Romantic Road is not any more scenic or romantic than driving any other route through Bavaria, certainly nowhere near as gorgeous as the Big Sur Coastline...
There are some nice old-world towns on the RR with half-timbered buildings and old town walls that are worth seeing, but there are some nice towns on other routes south as well. Your time is already pretty short, and driving the RR would be a substantial detour. What you might do is research a few towns, see what interests you and, arrange to break up the trip south with one overnight stop in one of those towns.
You might check out Marktbreit, Ochsenfurt, and Iphofen, all pretty close to Würzburg:
Iphofen photos:
Weissenburg and Pappenheim are attractive little towns too.
Weissenburg:
Pappenheim:
You can take the train to all these places, if you prefer - it's often better than climbing into a car for a drive in strange territory after a long and possibly sleep-deprived international flight.
Hi Russ,
I'm starting to agree with you. It'll be stretch to arrive and drive this distance in such a short time. Also, as you point out, the Big Sur drive is very hard to beat! I was, however, looking for a scenic experience, as if I could feel an embrace from the Alps. On that note, is it better to approach the Alps from Germany, Switzerland, or Austria? I'm leaning towards staying stationary in one or two towns for the week (splitting the time between both places), then take day trips as suggested. Thanks again!
"I was, however, looking for a scenic experience, as if I could feel an embrace from the Alps."
The Berchtesgaden area has some spectacular Alps. If you stay or visit there, you'll want to visit the Königssee area.
You might want to ride the Jennerbahn there:
The Swiss Alps are even more spectacular. But they're pretty far off, and you aren't going there.
If mountains are for you, stay in or near Berchtesgaden for most or all of the week. Salt mines, Hitler's Eagle's Nest, amazing scenery... Do day trips to Salzburg, Prien and Herrenchiemsee, Werfen (Hohenwerfen Castle/Fortress; "Where Eagles Dare" was filmed there.)
Hallstatt Austria - picture perfect and an easy day trip:
Local tourist offices have hundreds of accommodations listings that you won't find elsewhere. Use the accommodations boxes at the top left of the Berchtesgaden area tourist office website to search for apartments (in English) which would no doubt be the most convenient for a couple with 2 small kids.
Thanks again Russ! Awesome tips!
Hi,
I agree with Russ's advice on the Berchtesgaden area as a base. We've stayed in the Berchtesgaden area in 10/06, 10/07, 9/09 and 11/10 and are heading back there soon for the Christmas Markets once again as we did in 11/10. From here we did easy day trips to Konigsee, Salzburg, St. Gilgen, Hallstatt, "Mad" King Ludwig's Herrenchiemsee, Bad Reichenhall, Hallein, Zell am See, etc.
If Neuschwanstein and Hohenschwangau are "must sees", consider splitting your time between the Berchtesgaden area and Fuessen. The Zugspitze, Germany's highest peak (which borders Austria) is easily accessed from Fuessen in nearby Ehrwald, Austria.
Except for 1 trip, we've always flown r/t from Munich. It's an easy 2 -2.5 hour drive to either Berchtesgaden or Fuessen.
If interested, we have some photo's (3 "pages" of them) at:
With 2 children, you may be interested in our favorite place to stay in the Berchtesgaden area (and probably anywhere we've been):
Just wanted to add that the Rossfeld Panorama Road and a summer luge ride are near Berchtesgaden.
Thanks pja1! Awesome tips!
The fact that you agree with Russ, who I was already agreeing with, helps me solidify my plans. Thank goodness I still have time (although buying airplane tickets would be the first move and the earlier, the better)!
I'm starting to feel more and more interested in Austria... Maybe even Switzerland. Although it's further out, I've read that the train rides are gorgeous, so it would facilitate getting from one country to the other. I would just have to pick the right place to stay in Switzerland.
Anyway... lots to think about. I sure do appreciate all of y'all's inputs!
Comment has been removed by Fodor's moderators
Although Airbnb ( and others such as vrbo ) are interesting ways to rent a place, Germany has a long tradition of renting rooms and apartments. As Russ said, the local tourist office will have listings of virtually all the places in the town and sometimes the neighboring areas. When you have decided on places you may want to stay, look up the town website, usually www.(town name).de
I try to stay in apartments. I would suggest staying in two places max. Many pride themselves on being kid friendly.
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Posted October 6, 2013
by Jerry Alatalo
Jeremy Gilley has invented a profound movement. He had an idea to name the date of September 21 as the “International Day of Peace“. His idea became a reality when the United Nations voted to do so in the General Assembly. Mr. Gilley recently created a documentary about his amazing experience -“Peace One Day: The Journey”- which should give any man or woman who views it a strong sense of potentials and
“Five great enemies to peace inhabit within us: viz., avarice, ambition, envy, anger, and pride. If those enemies were to be banished, we should infallibly enjoy perpetual peace.”
– Petrarch (1304-1374) Italian Poet
“No one is fool enough to choose war instead of peace. For in peace sons bury fathers, but war violates the order of nature, and fathers bury sons.”
– Herodotus (485-425 B.C.) Greek Historian
“There never was a good war or a bad peace.”
– Benjamin Franklin (1706-1790)
“Until you have become really, in actual fact, as brother to everyone, brotherhood will not come to pass.”
– Fyodor Dostoyevski (1821-1881) Russian Novelist
“War is an invention of the human mind. The human mind can invent peace.”
– Norman Cousins (1912-1990) American Writer
Jeremy Gilley invents peace.
Let all men, women, and children on this Earth strive to become inventors of peace.
Let today be the first day of World Peace Forever.
Related articles
- Russia celebrates World Peace Day with 24-hour charity broadcast (voiceofrussia.com)
- It’s World Peace Day! What will you do this Sept. 21? (ted.com)
- Peace Day (misbehavedwoman.wordpress.com)
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TITLE: Is parameterized complexity going to be the future of complexity theory?
QUESTION [10 upvotes]: I am a research scholar who works in Algorithms and Complexity theory, I use parameterized complexity to some extent. To me it appears that researchers in parameterized complexity are very active (I don't mean that other are not) in terms of number of research papers. I have seen that researchers from communication complexity, arithmetic complexity etc. are also using various parameters to greater extent.
Question : Is parameterized complexity going to be the future of complexity theory? Future just means number of research papers, number of researchers working in that area etc.
Please note that I am naive and may not be aware of many things.
REPLY [20 votes]: Predicting the future is nigh impossible, especially so for cutting-edge research. I don't think anyone predicted how much impact deep learning is now having or that cryptography would be taken over by indistinguishability obfuscation.
That said, I will say this much: I don't see any particular reason to expect parameterized complexity to take over. It's a mature field that has been active for something like 20 years. It doesn't really strike me as an up-and-coming area. To be clear, I think it's a successful area that will continue to thrive.
If you look on google trends, search interest in parameterized complexity has been declining. (Stick in some other terms for a comparison if you're interested.) If you look up the combined citations for the Downey-Fellows textbook Parameterized Complexity and their updated textbook, you see that they are pretty stable:
(Source: Google scholar. I added both books to my own profile, merged them, took a screenshot of the combined citations, and then deleted them from my profile.)
This is a healthy number of citations, but it is not the exponential growth that would make you think parameterized complexity is going to take over. Of course, this data is very flawed, but it's the best indication I can find of the global popularity of parameterized complexity.
Note that things can be very popular locally even if they aren't popular globally.
When I was an undergrad, I thought that I needed to learn about category theory because everyone around me was talking about it; I even bought a book. Then I moved on to grad school and never heard about it again; the book remains unread to this day.
Perhaps you are in a similar situation -- you are in a department where there is a lot of parameterized complexity going on, but, if you move somewhere else, the story will be completely different.
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\section{Theorems of Hall, Rado, Ore, and Perfect}
\PRFR{Jan 15th}
D.J.A.~Welsh gives the following very elegant generalization of the
theorems of Rado and Hall in \cite{We71}. From this generalization, the theorems of Hall, Rado, Ore, and Perfect follow as an easy corollary each. Before we present the theorem, we need some definitions.
\begin{definition}\PRFR{Jan 15th}
Let $I$ and $E$ be sets. A \deftext{family of subsets} of $E$ indexed by $I$ is
a map $A_{\bullet}\colon I\maparrow 2^{E}$ with domain $I$, such that for every
$i\in I$ the image $A_{i}$ is a subset of $E$.
We denote such a family by writing $(A_i)_{i\in I} \subseteq E$, or shorter\label{n:Afam}
$(A_i)_{i\in I}$ whenever $E$ is clear from the context. We call $(A_i)_{i\in I}$
\deftext{finite} if $I$ is finite. Further, we call $(A_i)_{i\in I}$ a \deftext{family of non-empty subsets}, if for all $i\in I$, $A_{i} \not= \emptyset$.
\end{definition}
\begin{definition}\PRFR{Jan 15th}
Let $I$, $E$ be sets, and let $\Acal = (A_i)_{i\in I} \subseteq E$ be a family of subsets of $E$. A \deftext{system of representatives} is a map $x_{\bullet}\colon I\maparrow E$ such that there is a bijection $\sigma \colon I\maparrow I$ with
$x_{i} \in A_{\sigma(i)}$ for all $i\in I$. We will denote such a family
by writing
\linebreak
$(x_i)_{i\in I} \in \Acal$. A system of representatives is called
\deftext{system of distinct representatives}, if $x_{\bullet}$ is an injective map.
A \deftext{transversal} of $\Acal$ is a subset $T\subseteq E$ such that
there is a bijection $\sigma\colon T\maparrow I$ with $t\in A_{\sigma(t)}$ for all $t\in T$. A \deftext{partial transversal} of $\Acal$ is a subset $P\subseteq E$ such that
there is an injection $\iota \colon P\maparrow I$ with $t\in A_{{\iota(t)}}$ for all $t\in P$.
If $P$ is a partial transversal of $\Acal$, we define the \deftext[defect of a partial transversal]{defect} of $P$ to be $\left| I \right| - \left| P \right|$,
i.e. the cardinality of those indices in $I$ that are not in the image of the corresponding $\iota$.
\end{definition}
\begin{theorem}\label{thm:radohall}\PRFR{Jan 15th}
Let $\Acal=(A_i)_{i\in I} \subseteq E$ be a finite family of non-empty subsets of $E$,
and let $\mu\colon 2^{E} \maparrow \N$ be a map with the properties that
\begin{enumerate}\ROMANENUM
\item for all $X \subseteq Y\subseteq E$, $\mu(X) \leq \mu(Y)$, and
\item for all $X,Y\subseteq E$, $\mu(X) + \mu(Y) \geq \mu(X\cap Y) + \mu(X\cup Y)$.
\end{enumerate}
Then there is a system of representatives $(x_i)_{i\in I} \in \Acal$ with the property that
\begin{enumerate}
\item[(1)] for all $J\subseteq I$, $\mu\left(\SET{x_i \mid i\in J}\right) \geq \left| J\right|$
\end{enumerate}
if and only if $\Acal$ has the property that
\begin{enumerate}
\item[(2)] for all $J\subseteq I$, $\mu \left(\bigcup_{i\in J} A_i\right) \geq \left| J\right|$.
\end{enumerate}
\end{theorem}
\noindent This proof of the theorem follows the course of \cite{We71} --- a very nice version of which can be found on p.100 of \cite{We76} --- and it focuses more on details than brevity.
\begin{proof}\PRFR{Jan 15th}
Let $(x_{i})_{i\in I} \in \Acal$ be such a system of representatives, that
{\em(1)} holds, and let $\sigma\colon I\maparrow I$ be a permutation
that has the property $x_{i}\in A_{\sigma(i)}$ for all $i\in I$. Let $J\subseteq I$, then
$\SET{x_{\sigma^{-1}(i)} ~\middle|~ i \in J} \subseteq \bigcup_{{i\in J}} A_{i}$.
By {\em (i)} $\mu$ is non-decreasing, therefore
\[ \left| J \right| = \left| \sigma^{{-1}}[J] \right| \leq
\mu\left(\SET{x_i ~\middle|~ i\in \sigma^{-1}[J]}\right) \leq \mu \left(\bigcup_{i\in J} A_i\right).\]
For the converse implication, we employ induction on the integer vector $v = \left( \left|A_{i}\right| \right)_{i\in I}$. The base case is $v_{i} = 1$ for all $i\in I$ where every $A_{i}$ is a singleton set, thus for any system of representatives $(x_i)_{i\in I} \in \Acal$, we have $A_{i} = \SET{x_{\sigma^{-1}(i)}}$ for all $i\in I$.
Therefore, $\SET{x_i ~\middle|~ i\in \sigma^{-1}[J]} = \bigcup_{i\in J} A_{i}$ and the equivalence is obvious. For the induction step, let $i'\in I$ such that $\left| A_{i'} \right| > 1$. In this case, we claim that there is some $x\in A_{i'}$, such that
the derived family
$\Acal' = (A'_i)_{i\in I}$ where $A'_{i} = A_{i}$ if $i\not= i'$, and $A'_{i'} = A_{i'}\BSET{x}$
still has the property {\em (2)}. Assume that this claim is false, then for any
$\dSET{x,y}\subseteq A_{i'}$ there are $J_{x},J_{y}\subseteq I\BSET{i'}$ such that
\begin{align*}
\mu\left( \left(A_{i'}\BSET{x}\right) \cup \bigcup_{i\in J_x} A_i \right) &
\leq \left|J_x \right| < \left|J_x\right| + 1 \text{, and}\\
\mu\left( \left(A_{i'}\BSET{y}\right) \cup \bigcup_{i\in J_y} A_i \right) &
\leq \left|J_y \right| < \left|J_y\right| + 1 .\\
\end{align*}
We use the submodularity {\em (ii)} of $\mu$ in order to obtain that
\begin{align*}
\mu\left( \left(A_{i'}\BSET{x}\right) \cup \bigcup_{i\in J_x} A_i \right) + \mu\left( \left(A_{i'}\BSET{y}\right) \cup \bigcup_{i\in J_y} A_i \right) & \geq \mu(B_{\cap}) +
\mu\left( A_{i'} \cup \bigcup_{i\in J_x\cup J_y} A_i \right)
\end{align*}
where \[ B_{\cap} = \left( \left(A_{i'}\BSET{x}\right) \cup \bigcup_{i\in J_x} A_i \right) \cap
\left( \left(A_{i'}\BSET{y}\right) \cup \bigcup_{i\in J_y} A_i \right).\]
Clearly, $\bigcup_{i\in J_x \cap J_y} A_{i} \subseteq B_{\cap}$, and since $\mu$ is non-decreasing due to property {\em (i)},
we obtain that
\begin{align*}
\mu(B_{\cap}) +
\mu\left( A_{i'} \cup \bigcup_{i\in J_x\cup J_y} A_i \right) & \geq
\mu\left(\bigcup_{i\in J_x\cap J_y} A_i\right) + \mu\left( A_{i'} \cup \bigcup_{i\in J_x\cup J_y} A_i \right).
\end{align*}
We now may use property {\em (2)} with $J = J_{x}\cup J_{y} \cup \SET{i'}$, and $J=J_{x}\cap J_{y}$, respectively. We add the respective inequalities and obtain
\begin{align*}
\mu\left( A_{i'} \cup \bigcup_{i\in J_x\cup J_y} A_i \right) + \mu\left( \bigcup_{i\in J_x\cap J_y} A_i \right) & \geq \left( \left| J_x \cup J_y \right| + 1\right) +
\left| J_x \cap J_y \right| = \left| J_x \right| + \left| J_y \right| + 1.
\end{align*}
Yet, this yields
\begin{align*}
\mu\left( \left(A_{i'}\BSET{x}\right) \cup \bigcup_{i\in J_x} A_i \right) + \mu\left( \left(A_{i'}\BSET{y}\right) \cup \bigcup_{i\in J_y} A_i \right) & \geq \left| J_x \right| + \left|J_y\right| + 1
\end{align*}
which contradicts
\begin{align*}
\mu\left( \left(A_{i'}\BSET{x}\right) \cup \bigcup_{i\in J_x} A_i \right) + \mu\left( \left(A_{i'}\BSET{y}\right) \cup \bigcup_{i\in J_y} A_i \right) & \leq \left| J_x \right| + \left|J_y\right|.
\end{align*}
Thus the claim holds, and since $\left|A'_{i'}\right| < v_{i'}$, we may use the induction hypothesis on $\Acal'$ which guarantuees the existence of a system of representatives $(x_i)_{i\in I}$ with
property {\em (1)}. Every such $(x_i)_{i\in I}$ is also a system of representatives of
$\Acal$, therefore $(x_i)_{i\in I}$ with {\em (1)} exists.
\end{proof}
\begin{corollary}[Hall]\label{cor:Hall}\PRFR{Jan 15th}
Let $\Acal = (A_i)_{i\in I}$ be a finite family of sets, then $\Acal$ has a transversal if and only if for all $J\subseteq I$,
$$\left| \bigcup_{{i\in J}} A_{i} \right| \geq \left| J \right|.$$
\end{corollary}
\begin{proof} \PRFR{Jan 15th}
Apply Theorem~\ref{thm:radohall} with $\mu(X) = \left| X \right|$ and $E = \bigcup_{{i\in I}} A_{i}$.
\end{proof}
\begin{corollary}[Rado]\label{cor:Rado}\PRFR{Jan 15th}
Let $M=(E,\Ical)$ be a matroid, and let $\Acal = (A_i)_{i\in I}$ be a finite family of subsets of $E$, then $\Acal$ has a transversal which is independent in $M$ if and only if for all $J\subseteq I$,
$$\rk_M \left( \bigcup_{{i\in J}} A_{i}\right) \geq \left| J \right|.$$
\end{corollary}
\begin{proof}\PRFR{Jan 15th}
Apply Theorem~\ref{thm:radohall} with $\mu(X) = \rk_M(X)$.
\end{proof}
\begin{corollary}[Ore]\PRFR{Jan 15th} Let $\Acal = (A_i)_{i\in I}$ be a finite family of sets, and $d\in \N$, then $\Acal$ has a partial transversal $T$ with defect $\leq d$ if and only if for all $J\subseteq I$,
$$\left| \bigcup_{{i\in J}} A_{i} \right| \geq \left| J \right| - d.$$
\end{corollary}
\begin{proof}\PRFR{Jan 15th}
Apply Theorem~\ref{thm:radohall} with $\mu(X) = \left| X \right| + d$ and $E = \bigcup_{{i\in I}} A_{i}$.
\end{proof}
\needspace{9\baselineskip}
\begin{corollary}[Perfect]\PRFR{Jan 15th} Let $M=(E,\Ical)$ be a matroid, $d\in \N$, and let $\Acal = (A_i)_{i\in I}$ be a finite family of subsets of $E$, then $\Acal$ has a partial transversal $T$ with defect $ \leq d$ which is independent in $M$ if and only if for all $J\subseteq I$,
$$\rk_M \left( \bigcup_{{i\in J}} A_{i}\right) \geq \left| J \right| - d.$$
\end{corollary}
\begin{proof}\PRFR{Jan 15th}
Apply Theorem~\ref{thm:radohall} with $\mu(X) = \rk_M(X) + d$.
\end{proof}
| 215,405
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TITLE: Curve in $\mathbb{A}^3$ that cannot be defined by 2 equations
QUESTION [12 upvotes]: According to a problem in Shafarevich 1.6, every curve in $\mathbb{A}^3$ can be cut out by 3 equations.
Can someone give me an example of a curve in $\mathbb{A}^3$ that is not cut out by 2 equations?
REPLY [15 votes]: This is mostly a long comment.
Consider the curve $C$ parametrized by $t\mapsto(t^3,t^4,t^5)$, as in Dylan's comment.
Let $A=k[x,y,z]$ be the polynomial ring in three variables, and let us consider it as graded ring with $x$, $y$ and $z$ of degrees $3$, $4$ and $5$, respectively.
Suppose $p=\sum_{i,j,k\geq0}\alpha_{i,j,k}x^iy^jz^k\in A$ is a polynomial such that $p(t^3,t^4,t^5)=0$, so that $p$ vanishes on the curve $C$. Then $$\sum_{\ell\geq0}\Bigl(\sum_{\substack{i,j,k\geq0\\3i+4j+5k=\ell}}\alpha_{i,j,k}\Bigr)t^\ell=0$$ so we see that for each $\ell\geq0$ we have $$\sum_{\substack{i,j,k\geq0\\3i+4j+5k=\ell}}\alpha_{i,j,k}=0.$$ Now, if for each $\ell\geq0$ we define the polynomial $$p_\ell=\sum_{\substack{i,j,k\geq0\\3i+4j+5k=\ell}}\alpha_{i,j,k}x^iy^jz^k,$$ we have first that $$p_\ell(t^3,t^4,t^5)=0,$$ so that $p_\ell$ vanishes on $C$, and $p_\ell$ is homogeneous of degree $\ell$ in our graded ring $A$. Our polynomial $p=\sum_{\ell\geq0}p_\ell$ is therefore a sum of homogeneous elements of $I$.
This means that the ideal $I\subseteq A$ of the polynomials which vanish on $C$ is homogeneous: it is generated by the homogeneous elements it contains.
Now, it is easy to see that the subspaces of $I$ of homogeneous elements of degrees $8$, $9$, and $10$ are $1$-dimensional, spanned respectively by the three polynomials $x z-y^2$, $y z-x^3$ and $z^2-x^2 y$. Moreover, it is also easy to see that all the homogeneous components of $I$ of degree less than $8$ are zero: in fact, the monomials $1$, $x$, $y$, $z$, $x^3$ and $xy$ span the direct sum of the homogeneous components of $A$ of degree less than $8$, and no non-zero linear combination of them vanishes on $C$, as a little computation will show.
Thinking a bit about what all this means, we see that the three polynomials $x z-y^2$, $y z-x^3$ and $z^2-x^2 y$ must belong to any generating set of $I$.
This shows that $C$ is not a complete intersection. This may or may not be what Shafarevich has in mind, as "cut" may also mean set-theoretically, as Dylan notes.
If he means the latter, then this example does not work: Ernst Kunz shows, in his Introduction to commutative algebra and algebraic geometry, that all monomial curves in affine $3$-space are set-theoretically complete intersections.
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\begin{document}
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\begin{flushright} \small
UUITP-08/10
\end{flushright}
\smallskip
\begin{center} \LARGE
{\bf Chiral de Rham complex on\\
special holonomy manifolds}
\\[12mm] \normalsize
{\bf Joel~Ekstrand$^a$, Reimundo~Heluani$^b$, \\
Johan~K\"all\'en$^a$ and Maxim Zabzine$^a$} \\[8mm]
{\small\it
$^a$Department of Physics and Astronomy,
Uppsala university,\\
Box 516,
SE-751\;20 Uppsala,
Sweden\\
~\\
$^b$Department of Mathematics, University of California,\\
Berkeley, CA 94720, USA
}
\end{center}
\vspace{7mm}
\begin{abstract}
\noindent
Interpreting the chiral de Rham complex (CDR) as a formal
Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for
the study of CDR on manifolds with special holonomy. We discuss classical and partial quantum
results. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the $N=2$ superconformal algebra)
on the space of global sections of CDR of a Calabi-Yau $3$-fold. This is the first example of such a vertex subalgebra which is non-linearly generated by a finite number of superfields.
\end{abstract}
\eject
\normalsize
\section{Introduction}
The chiral de Rham complex (CDR) was introduced by Malikov, Schechtman and Vaintrob in \cite{Malikov:1998dw}.
CDR is a sheaf of supersymmetric vertex algebras over a smooth manifold ${\cal M}$. It is defined by gluing free
chiral algebras on the overlaps of open sets of ${\cal M}$. Since the original work \cite{Malikov:1998dw}, there has been considerable
progress in understanding the mathematical aspects of CDR (cf. \cite{MR2038198, benzvi-2006, Malikov:2006rm} among others).
In physics literature, CDR appeared in the context of half-twisted sigma models
\cite{Kapustin:2005pt, Witten:2005px} and in the context
of infinite volume limits of sigma models \cite{Frenkel:2005ku, Frenkel:2006fy, Frenkel:2008vz}. The present work is the logical continuation of \cite{Ekstrand:2009zd}
where it was suggested to interpret CDR as a formal canonical quantization of the non-linear sigma model.
The question we would like to address is which vertex algebras can be attached to a manifold ${\cal M}$ within the CDR framework. It has been established that
one can attach different versions of super-Virasoro algebras as global sections of CDR. The details of
the construction depend on additional geometrical structures on ${\cal M}$. In this paper, we
construct extensions of the $N=2$ super-Virasoro algebra for Calabi-Yau $3$-folds. Moreover, we present
some partial results on the general possible extensions of the super-Virasoro algebra within the CDR framework.
In looking for possible vertex sub-algebras generated by global sections of CDR there are two possible ways to proceed, either
trying to guess the answer, or trying to find a systematic way to produce it. If we think of CDR as a formal quantization of the
non-linear sigma model, we can use the insight from classical sigma models in order to make an intelligent
guess about the answer in CDR. Indeed, any possible extension of the super-Virasoro algebra will be related
to the symmetries of the classical sigma model. Thus, the present result is inspired by the nearly 20 year old
observation by P.~Howe and G.~Papadopoulos in \cite{Howe:1991vs, Howe:1991ic},
where the relation between classical symmetries of non-linear sigma models and
special holonomy manifolds was observed. Our intention is to reformulate their result in terms of Poisson vertex algebras and
then try to quantize it within the framework of CDR.
In order to do so, we first describe a non-trivial way of associating global sections of CDR to differential forms on the target manifold ${\cal M}$. Locally, CDR is simply a $\beta\gamma$-$bc$ system with as many generators as the dimension of $\cal M$.
This gives natural embeddings of
the differential forms of $\cal M$ into CDR.
In this article, we consider an embedding inspired by the sigma model, that is different than the one introduced in \cite{Malikov:1998dw}.
In our embedding, one needs the Levi-Civita connection on the Riemannian manifold $\cal M$, to write the correct expressions. This is detailed in section \ref{lift} and Appendix \ref{inductionProof}.
On a special holonomy manifold there exist non-trivial and well-known covariantly constant
forms. The algebra generated by the corresponding superfields is a natural vertex algebra associated to any special-holonomy manifold. We show that this unified construction recovers all the known cases of \cite{Malikov:1998dw} \cite{benzvi-2006} \cite{heluani-2008}. Moreover, we show that for a Calabi-Yau threefold $\cal M$, the covariantly constant forms on $\cal M$ generate non-linearly a super-vertex algebra known as the Odake algebra. This is the first example of such a sub-super-vertex algebra of global sections of CDR which is
non-linearly generated by a finite number of superfields.
The paper is organized as follows: In Section \ref{s:classical} we review the Hamiltonian formalism for $N=(1,1)$
supersymmetric non-linear sigma model. We also set up the notations for the rest of the paper.
Section \ref{clAlgExt} deals with the classical symmetries of sigma models on special holonomy manifolds.
This section presents the Hamiltonian treatment of the results from \cite{Howe:1991vs, Howe:1991ic}, we also
give a list of Poisson vertex algebras associated to the different cases of special holonomy. In Section \ref{s:CDR} we briefly recall the formalism of SUSY vertex algebras and the definition of
CDR. We stress the physical interpretation of CDR as a formal canonical quantization of the non-linear sigma model.
In Section \ref{lift} we discuss how to promote the classical currents to well-defined sections of CDR.
Section \ref{CDRalgExt} presents our results at the quantum level. The main result is the construction of
two commuting copies of the Odake algebra on a Calabi-Yau $3$-fold. We also present some general remarks about other cases.
In Section \ref{s:summary} we present a summary and discuss the main complications in further possible calculations.
Many technicalities are collected in the appendices. In
Appendix \ref{crosscontractions} we collect some useful formulas on different special holonomy manifolds.
Appendix \ref{LBrules} contains a collection of properties of the quantum $\Lambda$-bracket.
In Appendix \ref{inductionProof} we present some explicit formulas accompanying those of Section \ref{lift}.
Appendix \ref{odakeCalc} presents the details of the calculation of the quantum Odake algebra.
\section{Classical sigma model}
\label{s:classical}
In this section we will review some basic facts about the classical sigma model. Especially, we will see the connection between covariantly constant forms on the target space and symmetries of the sigma model. We will also show how to write the model in the Hamiltonian framework.
\subsection{The sigma model in the Lagrangian formalism}
Consider the N=(1,1) supersymmetric sigma model defined on $\Sigma = S^{1}\times\mathbb{R}$. Its action is given by
\beq
S=\frac{1}{2}\int_{\Sigma}{d\sigma dt d\theta^{-}d\theta^{+} ~g_{ij}(\Phi)D_{+}\Phi^{i}D_{-}\Phi^{j}}.
\eeq{model}
We use N=(1,1) superfields $\Phi^{i}(\sigma,t,\theta^+,\theta^-)$. The circle $S^{1}$ is parametrized by $\sigma$, and $t$, the ``time'', is the coordinate on $\mathbb{R}$. The pair $\theta^\pm$ labels the spinor coordinates. The fields $\Phi^{i}$ are maps from $\Sigma^{2,2}$ into a target manifold $\cal{M}$, and $g_{ij}$ is the metric on $\cal{M}$. The odd derivatives $D_{\pm}$ and the even derivatives $\partial_{\pp}$ are defined by
\begin{align}
D_\pm &\equiv \frac{\d}{\d \theta^\pm} + \theta^\pm (\d_0 \pm \d_1)~,&
\partial_{\pp} \equiv D^2_\pm &= \d_0 \pm \d_1 ~,
\end{align}
where $\partial_0\equiv\frac{\partial}{\partial t}$ and $\partial_1\equiv\frac{\partial}{\partial \sigma}$. The equation of motion derived from this action is
\begin{equation} \label{eom}
D_{-}D_{+}\Phi^{i}+\G^{i}_{jk}D_{-}\Phi^{j}D_{+}\Phi^{k}=0 ~,
\end{equation}
where $\G^{i}_{jk}$ are the components of the Levi-Civita connection.
The model has N=(1,1) superconformal symmetry, with the corresponding current given by
\begin{equation}\label{eq:LagrTpmDef}
T_{\pm}=g_{ij}(\Phi)D_{\pm}\Phi^{i}\d_{\pp}\Phi^{j} ~.
\end{equation}
The equation of motion gives $D_{\mp} T_{\pm}=0$. This imply that $T_\pm$ are conserved, and also that $T_{\pm}=T_{\pm}(\sigma_{\pm},\theta_{\pm})$, \ie we have left and right moving currents. We can multiply $T_{\pm}$ by any function $f_{\pm}(\sigma_{\pm},\theta_{\pm})$ to form $\tilde{T}_{\pm}=f_{\pm}T_{\pm}$. $\tilde{T}_{\pm}$ still satisfy $D_{\mp}\tilde{T}_{\pm}=0$, and we therefore have infinitely many conserved currents. The components of the superfields $T_{\pm}$ are the Virasoro field and the Neveu-Schwarz supercurrent, respectively.
These are the only symmetries associated to a general Riemannian metric that we can find. However, as noticed
in \cite{Howe:1991vs, Howe:1991ic}, if $\cal{M}$ admits covariantly constant forms, the sigma model has additional symmetries. The argument goes as follows: consider a form $\omega=\omega_{i_{1}\ldots i_{n}} dx^{i_{1}}\wedge\ldots\wedge dx^{i_{n}}$ satisfying $\nabla \omega=0$, where $\nabla$ is the Levi-Civita connection. Then
\beq
J^{(n)}_{\pm}=\omega_{i_{1}\ldots i_{n}}(\Phi)D_{\pm}\Phi^{i_1}\ldots D_{\pm}\Phi^{i_n}
\eeq{LagrSymCurrDef}
satisfies $D_{\mp}J_{\pm}=0$ on-shell, \ie with the use of $\eqref{eom}$. This implies that $J^{(n)}_{\pm}=J^{(n)}_{\pm}(\sigma_{\pm},\theta_{\pm})$, and the components of $J^{(n)}_{\pm}$ will be left and right moving currents. By the same argument as above, we have infinitely many conserved currents.
The symmetries corresponding to the currents are
\beq
\delta_{\pm}\Phi^{i}=\epsilon_{\pm}g^{ii_{1}}\omega_{i_{1}\ldots i_{n}}D_{\pm}\Phi^{i_{2}}\ldots D_{\pm}\Phi^{i_{n}} ~,
\eeq{djdjk3393939}
where the parameter $\epsilon_{\pm}$ satifies $D_{\mp}\epsilon_{\pm}=0$. The action functional (\ref{model})
is invariant under (\ref{djdjk3393939}) if $\omega$ is covariantly constant with respect to Levi-Civita connection.
\subsection{The sigma model in the Hamiltonian formalism}
\label{classicalhamiltionian}
The sigma model $\eqref{model}$ can also be formulated in the Hamiltonian formalism \cite{Zabzine:2005qf, Bredthauer:2006hf, Zabzine:2006uz}. We integrate out one odd $\theta$, and identify the Hamiltonian and the phase space structure. In order to do so, we introduce new odd coordinates $\theta_0$ and $\theta_1$ by
\begin{align}
\theta_0 &= \frac{1}{\sqrt{2}} (\theta^+ + i \theta^-)~, & \theta_1 &=\frac{1}{\sqrt{2}}(\theta^+ - i \theta^-)~,
\end{align}
together with odd derivatives
\begin{align}\label{newD}
D_0 &= \frac{1}{\sqrt{2}} (D_+ - i D_-)~,&
D_1 &= \frac{1}{\sqrt{2}} (D_+ + i D_-)~,
\end{align}
which satisfy $D^2_0 = \d_1$, $D_1^2 = \d_1$ and
$D_1 D_0 + D_0 D_1 = 2 \d_0$. We also introduce new superfields
\begin{align} \label{newcoord}
\phi^i &\equiv \Phi^i |_{\theta_0=0}~, &
S_i &\equiv g_{i j} D_0 \Phi^j |_{\theta_0=0}~,
\end{align}
and new derivatives
\begin{align}
D_1 &\equiv D_1|_{\theta_0=0}~, &
\partial &\equiv \partial_1~.
\end{align}
After performing the $\theta_0$-integration, the action $\eqref{model}$ becomes
\beq
S= \int dt d\sigma d\theta_1 \left ( S_i \d_0 \phi^i - \frac{1}{2} {\cal H} \right )~,
\eeq{actionhamiltonian}
where
\beq
{\cal H} = \d \phi^i D_1 \phi^j g_{ij} + g^{ij} S_i D S_j
+ S_k D_1 \phi^i S_l g^{jl} \Gamma^k_{~ij}~.
\eeq{}
We see that the sigma model phase space
corresponds to a cotangent bundle $T^* {\cal LM}$, where ${\cal LM}= \{S^{1|1} \rightarrow \mathcal{M}\}$ is a superloop space. It is equipped with a natural symplectic structure
\beq
\int d\sigma\, d\theta_1 ~\delta S_i \wedge \delta \phi^i~.
\eeq{}
Thus, the space of functionals on $T^* {\cal LM}$ is equipped with
a (super) Poisson bracket $\{~,~\}$ generated
by the relation:
\beq
\{ \phi^i (\sigma, \theta_1), S_j (\sigma', \theta'_1) \} =
\delta^i_j \delta (\sigma - \sigma') \delta(\theta_1 -
\theta'_1)~.
\eeq{poisson}
From \eqref{actionhamiltonian}, the Hamiltonian is:
$$H = \frac{1}{2} \int d\sigma d\theta_1 ~{\cal H}.$$
This Hamiltonian, together with the Poisson bracket \eqref{poisson}, generates the same dynamics as we get from the action \eqref{model} and the variational principle. It is convenient to introduce new formal coordinates
on $S^{1|1}$: $\xi = e^{i\sigma}$ and $(i\xi)^{1/2} \theta = \theta_1$, which imply
\beq
(i\xi)^{1/2} D = D_1~,~~~~~~
(i\xi)^{1/2} d\theta = d\theta_1~,~~~~~
(i\xi)^{1/2} S_{i}(\xi, \theta) = S_i (\sigma, \theta_1)~.
\eeq{newvariables123}
Thus the Poisson bracket \eqref{poisson} becomes
\beq
\{ \phi^i (\xi, \theta), S_j (\xi', \theta') \} =
\delta^i_j~ \delta (\xi - \xi') \delta(\theta -
\theta)~.
\eeq{newposkdkeeee}
From now on, we will use the variables $(\xi, \theta)$ on $S^{1|1}$.
\subsubsection{Poisson vertex algebras and \texorpdfstring{$\Lambda$}{Λ}-brackets} \label{section:PVA}
Local functionals on $T^* {\cal LM}$ form a Poisson superalgebra.
This Poisson superalgebra can formally be described as a Poisson (super) vertex algebra.
These in turn can be understood as semi-classical limits of vertex algebras.
Alternatively, if we work locally on $\cal M$, we can construct a sheaf of Poisson (super) vertex algebras and consider its set of
global sections.
For an introduction to Poisson vertex algebras in the context of superfields and $\Lambda$-brackets we refer the reader to \cite{Heluani:2006pk}. For an extensive study
of sheaves of Poisson vertex algebras and their
relation to CDR see \cite{Malikov:2006rm}.
Below, we set the notational conventions used in the rest of the article.
The Poisson bracket between two local functionals has the following general form
\beq
\{A (\xi, \theta), B(\xi', \theta') \}= \sum_{\stackrel{j \geq 0}{J = 0,1}}
(-1)^J\d_{\xi'}^j D_{\xi'\theta'}^J
\delta(\xi-\xi')\delta(\theta-\theta') C_{(j|J)} (\xi', \theta') ~,
\eeq{}
where $C_{(j|J)} $ denotes the local functional multiplying the $(-1)^J\d_{\xi'}^j D_{\xi'\theta'}^J
\delta(\xi-\xi')\delta(\theta-\theta')$-term. This bracket
can be encoded as
\beq
\PLB{A}{B}= \sum_{\stackrel{j \geq 0}{J = 0,1}}
\Lambda^{j|J}C_{(j|J)} ~,
\eeq{}
where $\Lambda^{j|J} = \lambda^j \chi^J$, with formal even $\lambda$ and odd $\chi$ satisfying
$\chi^2 = - \lambda$. The $\L$'s encode derivatives of delta functions, and the translation between the two is
\begin{equation*}
\L^{j|J}\rightarrow(-1)^J\d_{\xi'}^j D_{\xi'\theta'}^J
\delta(\xi-\xi')\delta(\theta-\theta')~.
\end{equation*}
For example, we write $\eqref{poisson}$ as
\begin{equation*}
\PLB{\phi^i}{S_j}=\delta^i_j ~.
\end{equation*}
\subsubsection{Currents in phase space}
Next, we derive the currents \eqref{eq:LagrTpmDef} and \eqref{LagrSymCurrDef} in phase space coordinates. From \eqref{newD} and \eqref{newcoord} we note that
\begin{alignat}{2}
\label{ebase} D_{+}\Phi^{i}|_{\theta_{0}=0}&= \frac{\left(g^{ij}S_{j}+D\phi^{i}\right)}{\sqrt{2}} && \equiv e^{i}_{+} ~,\\
\label{ebase2} D_{-}\Phi^{i}|_{\theta_{0}=0}&=\frac{i\left(g^{ij}S_{j}-D\phi^{i}\right)}{\sqrt{2}} && \equiv i \; e^{i}_{-}~.
\end{alignat}
The factor of $i$ in the definition of $e^i_-$ is introduced for later computational convenience. The set of fields $\{ e^{i}_\pm \}$ is a suitable basis for the symmetry currents. For $J^{(n)}_{\pm}$ the rewriting is straightforward. Since $T_{\pm}$ have a term with a time derivative, it requires the use of the equations of motion. We get
\beq
\bs
T_{\pm}&=\pm\left(g_{ij} D e^{i}_{\pm} e_{\pm}^{j} +g_{ij} \G^{i}_{~kl} D\phi^{k} e_{\pm}^{l} e_{\pm}^{j} \right) ~, \\
J^{(n)}_{+}&=\frac{1}{n!}\omega_{i_1\ldots i_n}e_{+}^{i_{1}}\ldots e^{i_{n}}_{+} ~,\\
J^{(n)}_{-}&=\frac{i^{n}}{n!}\omega_{i_1\ldots i_n}e_{-}^{i_{1}}\ldots e^{i_{n}}_{-}~. \\
\end{split}
\eeq{currents}
It can be checked explicitly that $T_\pm$ generate the superconformal symmetries of the original theory
and the currents $J^{(n)}_\pm$ generate the symmetry \eqref{djdjk3393939} in the Hamiltonian formalism.
\section{Classical algebra extensions}
\label{clAlgExt}
We now investigate the classical algebra generated by the currents \eqref{currents}. A natural question to ask is whether the Poisson brackets between these currents can be expressed in terms of the same fields. The results in this section were obtained in \cite{Howe:1991vs, Howe:1991ic}, although in a different framework.
Between the fields \eqref{ebase}-\eqref{ebase2} we have the following brackets:
\begin{align}
\PLB{e^i_\pm }{ e^{j}_\pm }&=\pm \chi g^{ij} + \frac{1}{\sqrt{2}}\left( g^{kj}\Gamma^i_{mk}e_{\mp}^{m} - g^{ki}\Gamma^{j}_{mk} e_\pm^{m} \right) ~, \\
\PLB{e^i_+ }{ e^{j}_- }&=\frac{1}{\sqrt{2}}\left( g^{kj}\Gamma^i_{mk}e_+^{m}-g^{ki}\Gamma^{j}_{mk}e_-^{m}\right) ~,\\
\PLB{e^i_\pm }{ f(\phi) }&=\frac{1}{\sqrt{2}}g^{ij}f_{,j} ~,
\end{align}
where $f_{,j} = \partial_j f$.
If we only have $T_{\pm}$, we get the classical (super) Virasoro algebra.
In $\L$-bracket notation, the Virasoro algebra is written as
\beq
\bs
\PLB{T_{\pm}}{T_{\pm}}&=\left(2\partial+\chi D+3\lambda\right)T_{\pm} ~, \\
\PLB{T_{\mp}}{T_{\pm}}&=0 ~.
\end{split}
\eeq{eq:Wittalgebra}
The algebra between the currents corresponding to covariantly constant forms is straightforward to compute.
Using the above brackets, one can show that
\beq
\bs
\PLB{T_\pm}{J^{(n)}_\pm}&=\left(2\partial+\chi D +n\lambda\right)J^{(n)}_\pm ~,\\
\PLB{T_{\mp}}{J^{(n)}_{\pm}}&=0 ~,
\end{split}
\eeq{idontunderstandthisnumbering}
which shows that $J^{(n)}_{\pm}$ have conformal weight $\frac{n}{2}$ with respect to the left/right moving Virasoro field. Let us define
\beq
\bs
B^{i}_{+(n)}& = \frac{1}{n!}g^{ii_1}\omega_{i_1\ldots i_{n+1}}e^{i_{2}}_+\ldots e^{i_{n+1}}_+~,\\
B^{i}_{-(n)}& = \frac{i^{n}}{n!} g^{ii_1}\omega_{i_1\ldots i_{n+1}}e^{i_{2}}_- \ldots e^{i_{n+1}}_-~.
\end{split}
\eeq{}
We then find the following brackets between $J^{(n+1)}_\pm$ and $J^{(m+1)}_\pm$ :
\beq
\bs
\PLB{J^{(n+1)}_\pm}{J^{(m+1)}_\pm} &= (-1)^{n}\Bigl(\chi g_{ij}B^{i}_{\pm(n)}B^{j}_{\pm(m)} \\
&\qquad +g_{ij}DB^{i}_{\pm(n)}B^{j}_{\pm(m)}+g_{ij}\Gamma^{i}_{kl}D\phi^{l}B^{k}_{\pm(n)}B^{j}_{\pm(m)}\Bigr) ~, \\
\PLB{J_{\pm}^{(n+1)}}{J_{\mp}^{(m+1)}}&=0 ~.
\end{split}
\eeq{jnbjm}
\subsection{Currents from holonomy groups}
To further analyze the algebra \eqref{jnbjm}, we need to know more about the covariantly constants forms $\omega$ and the metric $g$.
The existence of covariantly constant forms is ultimately related to the holonomy of the Levi-Civita connection.
Fortunately, the possible holonomy groups for the Levi-Civita connection have been classified and their relation to
covariantly constant forms is known (see \cite{MR1787733} for a review of the subject). To use this classification, let us
assume that ${\cal M}$ is simply-connected and that the metric $g$ is irreducible (to avoid the holonomy group to be a product
of two groups of lower dimension). Finally, we assume that ${\cal M}$ is not locally a Riemannian symmetric space.
With these three assumptions, there are
seven different possible cases for the Riemannian holonomy group and in each one of them we understand the properties of the
corresponding covariantly constant forms. Below, we will give the details of the algebra \eqref{jnbjm} in each of these seven cases.
The relation between holonomy groups and the symmetries of non-linear sigma model has been noted in
\cite{Howe:1991vs, Howe:1991ic}.
In order to compute the structure of the corresponding algebras, we will need some algebraic properties of the invariant tensors on special holonomy manifolds.
We collect the relevant formulas in Appendix \ref{crosscontractions}. In particular, we use the formulas derived in \cite{karigiannis-2007,karigiannis-2007-2008}.
Below, we give explicit definitions
for these currents and compute their corresponding algebras.
\subsubsection{Orientable Riemannian manifold, \texorpdfstring{$SO(n)$}{SO(n)}}
On a general $n$-dimensional orientable Riemannian manifold the holonomy group is $SO(n)$, and we have the covariantly constant totally anti-symmetric tensor $ \epsilon_{i_1\ldots i_{n}}$. For $n > 2$ the Poisson bracket between the corresponding currents is zero. For $n=2$, since $SO(2)=U(1)$, and $\epsilon_{i_1 i_2}$ can be taken as the K\"ahler form, we get the $N=2$ supersymmetry algebra (see the next example).
\subsubsection{K\"ahler manifold, $U(n)$}
When the holonomy group is $U(n)$, $\dim\mathcal{M}=2n$, the manifold is K\"ahler and we have a covariantly constant $2$-form, the K\"ahler form $\omega$. Using $\omega = g I$, $I$ being the complex structure, the current is defined as
\beq
J_\pm^{(2)}= \pm \frac{1}{2} \omega_{ij} e^i_\pm e^j_\pm~,
\eeq{jwj39393sjjsj}
and we find that $\eqref{jnbjm}$ reduces to
\beq
\PLB{J^{(2)}_\pm}{J^{(2)}_\pm}=-T_\pm ~.\\
\eeq{jjdjdjjdjw2220}
We therefore get two commuting copies of the N=2 superconformal algebra when the target manifold is K\"ahler.
\subsubsection{Calabi-Yau, $SU(n)$}
When the holonomy group is $SU(n)$, $\dim\mathcal{M}=2n$, we are on a Calabi-Yau manifold. We then have, in addition to the K\"ahler form, a covariantly constant holomorphic $n$-form $\Omega$, and its complex conjugate $\bar{\Omega}$ at our disposal. Let us denote the corresponding currents $X^{(n)}_\pm$ and $\bar{X}^{(n)}_\pm$:
\begin{align}
X^{(n)}_+&=\frac{1}{n!}\Omega_{\alpha_{1}\ldots\alpha_{n}}e^{\alpha_1}_+\ldots e^{\alpha_n}_+ ~,
& X^{(n)}_- &=\frac{i^n}{n!}\Omega_{\alpha_{1}\ldots\alpha_{n}}e^{\alpha_1}_-\ldots e^{\alpha_n}_-~, \\
\bar{X}^{(n)}_+&=\frac{1}{n!}\bar{\Omega}_{\bar{\alpha}_1\ldots\bar{\alpha}_n}e^{\bar{\alpha}_1}_+\ldots e^{\bar{\alpha}_n}_+ ~,
& \bar{X}^{(n)}_-&=\frac{i^n}{n!}\bar{\Omega}_{\bar{\alpha}_1\ldots\bar{\alpha}_n}e^{\bar{\alpha}_1}_-\ldots e^{\bar{\alpha}_n}_- ~,
\end{align}
which are defined in addition to $J^{(2)}_\pm$ and $T_\pm$ on the Calabi-Yau manifold.
We here introduced complex coordinates, with indices $i= ({\alpha,\bar{\alpha}})$. Choosing an hermitian metric, we find, using the formulas in Appendix \ref{section:FormulasCalabiYau}, that $\eqref{jnbjm}$ reduces to
\beq
\bs
\PLB{J^{(2)}_\pm}{X^{(n)}_\pm}&= - i\left(n\chi X^{(n)}_\pm+DX^{(n)}_\pm \right)~, \\
\PLB{J^{(2)}_\pm}{\bar{X}^{(n)}_\pm}&= +i \left(n\chi \bar{X}^{(n)}_\pm+D\bar{X}^{(n)}_\pm \right)~, \\
\PLB{X^{(n)}_\pm}{X^{(n)}_\pm}&=0 ~,\\
\PLB{\bar{X}^{(n)}_\pm}{\bar{X}^{(n)}_\pm}&=0 ~,\\
\PLB{X^{(n)}_\pm}{\bar{X}^{(n)}_\pm}&= \frac{i^{n^2+1}}{(n-1)!} \Biggl(\frac{i}{2} (n-1) T\left(J^{(2)}_\pm\right)^{n-2} \\
& \qquad \qquad \qquad -\frac{1}{2}D\left(J^{(2)}_\pm\right)^{n-1}-\chi \left(J^{(2)}_\pm \right)^{n-1} \Biggr)~,
\end{split}
\eeq{clasis333939s9sjj}
where $J^{(2)}_\pm$ is defined in \eqref{jwj39393sjjsj}. Note that \eqref{idontunderstandthisnumbering} now reads
\begin{equation}
\begin{split}
\{ {T_\pm}_\Lambda X_{\pm}^{(n)} \} &= (2 \partial + \chi D + n \lambda) X_\pm^{(n)}, \\
\{ {T_\pm}_\Lambda \bar{X}_{\pm}^{(n)} \} &= (2 \partial + \chi D + n \lambda) \bar{X}_\pm^{(n)}, \\
\{ {T_\mp}_\Lambda {X}_{\pm}^{(n)} \} &= \{ {T_\mp}_\Lambda \bar{X}_{\pm}^{(n)} \} = 0.
\end{split}
\end{equation}
These relations, together with the remaining relations for $J^{(2)}_\pm$ and $T_\pm$, give
rise to the classical Odake algebra, a Poisson vertex algebra that one can attach to any Calabi-Yau manifold. Indeed, we have two commuting
copies of this algebra, \ie the plus-currents commute with the minus-currents.
Notice that the currents $(J^{(2)}_\pm, T_\pm, X^{(n)}_\pm, \bar{X}^{(n)}_\pm)$ satisfy extra constraints. For example, from $\omega \wedge \Omega= \omega \wedge \bar{\Omega}=0$ we would obtain the identities
\beq
J_\pm^{(2)} X^{(n)}_\pm=0~,~~~~~~~~~~~~~~~ J_\pm^{(2)} \bar{X}^{(n)}_\pm=0~,
\eeq{relations933is3}
which in turn are needed to check the Jacobi identity for \eqref{clasis333939s9sjj}.
\subsubsection{Hyperk\"ahler manifold, $Sp(n)$}
When the holonomy group is $Sp(n)$, $\dim\mathcal{M}=4n$, the manifold ${\cal M}$ is hyperk\"ahler. We have three complex
structures $I_A$, $A=1,2,3$, such that $I_A I_B = - \delta_{AB} + \epsilon_{ABC} I_C$.
The metric $g$ is Hermitian with respect to all $I_A$ and the forms $\omega_A = gI_A$ are covariantly constant.
For $\omega_A$ we denote the corresponding currents $J_{\pm A}^{(2)}$, $A=1,2,3$, where we use \eqref{jwj39393sjjsj}.
The algebra $\eqref{jnbjm}$ reduces to
\beq
\PLB{J^{(2)}_{\pm A}}{J^{(2)}_{\pm B}}=\epsilon_{ABC}\left(D+2\chi\right)J^{(2)}_{\pm C}-\delta_{AB}T_\pm~,
\eeq{hyperkahleralgebraJJ}
which, together with \eqref{eq:Wittalgebra}, generates the N=4 superconformal algebra.
\subsubsection{Quaternionic K\"ahler manifold, $Sp(n) \cdot Sp(1)$}
On a quaternionic K\"ahler manifold the holonomy group is $Sp(n) \cdot Sp(1)$. Locally, we have three almost
complex structures $J_A$ and three locally defined two-forms $\omega_A= gJ_A$.
Defining the covariantly constant form:
\beq
\Sigma=\sum_{i=A}^3 \omega_A \wedge \omega_A ~,
\eeq{}
and denoting the corresponding currents by $\Sigma_\pm$:
\beq
\Sigma_\pm = \frac{1}{4!}~ \Sigma_{ijkl} ~e^i_\pm e^j_\pm e^k_\pm e^l_\pm,
\eeq{ssskkk}
we obtain
\beq
\PLB{\Sigma_\pm }{\Sigma_\pm}= - 4 \Sigma_\pm T_\pm ~.
\eeq{}
\subsubsection{$G_{2}$-manifold}
$G_2$ is an example of an \emph{exceptional} holonomy group. A
$G_{2}$-manifold ${\cal M}$ is seven dimensional.
On such a manifold there are two covariantly constant forms, a 3-form $\Pi$ and its Hodge dual $\Psi$.
We denote the respective currents by the same letters $\Pi_\pm$ and $\Psi_\pm$,
\beq
\Pi_+ = \frac{1}{3!} \Pi_{ijk} e^i_+ e^j_+ e^k_+~,~~~~\Pi_- = \frac{i^3}{3!} \Pi_{ijk} e^i_- e^j_- e^k_-~,~~~~\Psi_\pm= \frac{1}{4!} \Psi_{ijkl} e^i_\pm e^j_\pm e^k_\pm e^l_\pm~.
\eeq{kskkskksk}
Using extensively the formulas presented in Appendix \ref{crosscontractions},
we find that $\eqref{jnbjm}$ reduces
to
\beq
\bs
\PLB{\Pi_\pm}{\Pi_\pm}&= -3D\Psi_\pm -6\chi\Psi_\pm ~,\\
\PLB{\Pi_\pm}{\Psi_\pm}&= 3T_\pm\Pi_\pm ~,\\
\PLB{\Psi_\pm}{\Psi_\pm}&= 10 T_\pm \Psi_\pm + 3 \Pi_\pm D \Pi_\pm ~.
\end{split}
\eeq{classicG2}
The right hand side in the last bracket can be written in many equivalent ways. The $4$-form
$\Psi$ is the Hodge dual of $\Pi$ and thus it is not independent data. At the level of currents
we can derive the relation
\beq
2T_\pm \Psi_\pm + \Pi_\pm D \Pi_\pm = 0 ~,
\eeq{classicalidealG2}
which follows from \eqref{PsiPsigPipi} and \eqref{kskkskksk}. Indeed, this relation would be needed if we study
the Jacobi identity for the above algebra without any reference to the definition of the corresponding currents.
\subsubsection{$Spin(7)$-manifold}
$Spin(7)$ is another example of an exceptional holonomy group.
$Spin(7)$-manifolds are eight dimensional and they admit a covariantly constant 4-form $\Theta$ which is self-dual with respect to the Hodge involution.
The corresponding currents $\Theta_\pm$ are defined as
\beq
\Theta_\pm = \frac{1}{4!} \Theta_{ijkl} e^i_\pm e^j_\pm e^k_\pm e^l_\pm~.
\eeq{odo3030kddjd}
We find that $\eqref{jnbjm}$ reduces
to
\beq
\PLB{\Theta_\pm}{\Theta_\pm}=6T_\pm\Theta_\pm ~.
\eeq{classicSpin7}
\section{The Chiral de Rham complex as a formal canonical quantization of the sigma model}
\label{s:CDR}
Above, we have considered classical algebra extensions. We now want to investigate the quantum counterpart of these algebras, in the CDR framework. Before doing so, in this section we give a short introduction to basic notions, such as vertex algebras, SUSY vertex algebras and CDR.
The latter is a sheaf of SUSY vertex algebras and it was introduced originally
in \cite{Malikov:1998dw}, here we follow the treatment presented in \cite{benzvi-2006}.
Also, we review the interpretation of CDR as a formal canonical quantization of the sigma model.
For further details the reader may consult \cite{Ekstrand:2009zd}.
\subsection{SUSY vertex algebras}
First, we introduce two formal variables $(\xi,\theta)$, where $\xi$ is even and $\theta$ is odd. Given a vector space V, we define an End(V)-valued superfield as
\beq
A(\xi,\theta)=\sum_{\stackrel{j\in\mathbb{Z}}{J=0,1}}{\xi^{-(j+1)}\theta^{1-J}A_{(j|J)}},\qquad A_{(j|J)}\in End(V)
\eeq{}
where for all $v\in V$, $A_{(j|J)}v=0$ for large enough j.
A SUSY vertex algebra \cite{Heluani:2006pk} consists of the data of a super vector space $V$,
an even vector $|0\rangle \in V$ (the vacuum vector), an odd endomorphism
$D$ (whose square is an even endomorphism which we denote $\d$),
and a parity preserving linear map $A \mapsto Y(A,\xi, \theta)$ from
$V$ to $\rm{End}(V)$-valued superfields (the state-superfield correspondence). This
data should satisfy the following set of axioms:
\begin{itemize}
\item Vacuum axioms:
\begin{equation*}
\begin{aligned}
Y(|0\rangle, \xi, \theta) &= \rm{Id}~, \\
Y(A, \xi, \theta) |0\rangle &= A + O(\xi, \theta)~, \\
D |0\rangle &= 0~.
\end{aligned}
\end{equation*}
\item Translation invariance:
\begin{equation*}
\begin{aligned}
{[} D, Y(A,\xi, \theta)] &= (\partial_\theta - \theta \partial_\xi)
Y(A,\xi, \theta)~,\\
{[}\d, Y(A,\xi, \theta)] &= \partial_\xi Y(A,\xi, \theta)~.
\end{aligned}
\end{equation*}
\item Locality:
\begin{equation*}
(\xi-\xi')^n [Y(A,\xi, \theta), Y(B,\xi', \theta')] = 0 ~,
\qquad n \gg 0~.
\end{equation*}
\end{itemize}
\label{defn:2.3}
(The notation $O(\xi, \theta)$ denotes a power series in $\xi$ and
$\theta$ without a constant
term in $\xi$.)
Given the vacuum axioms for a SUSY vertex algebra, we will use the state-field correspondence to identify a vector $A \in V$ with its corresponding
field $Y(A,\xi, \theta)$.
\label{rem:nosenosenose}
Given a SUSY vertex algebra $V$ and a vector $A \in V$, we expand the fields
\begin{equation*}
Y(A,\xi, \theta) = A(\xi, \theta)= \sum_{\substack{j \in \mathbb Z \\ J = 0,1}} \xi^{-1-j} \theta^{1-J}
A_{(j|J)}~,
\end{equation*}
and we call the endomorphisms $A_{(j|J)}$ the \emph{Fourier modes} of
$Y(A,\xi, \theta)$. Now, define the operations
\begin{equation}
\begin{aligned}
\LB{A}{B} &= \sum_{\substack{ j \geq 0 \\ J = 0,1} }
\frac{\Lambda^{j|J}}{j!} A_{(j|J)}B~, \\
A B &= A_{(-1|1)}B~,
\end{aligned}
\label{eq:2.4.2}
\end{equation}
where $\Lambda^{j|J} = \lambda^j \chi^J$, with $\lambda$ and $\chi$ being formal even and odd parameters, satisfying
$\chi^2 = - \lambda$.
The first operation is called the \emph{$\Lambda$-bracket} and the second is
called the \emph{normal ordered product}.
The $\L$-bracket is a practical way of writing an operator product expansion (OPE).
$\eqref{eq:2.4.2}$ corresponds to the commutator of superfields,
\begin{equation*}
[A(\xi, \theta), B(\xi', \theta')] = \! \sum_{\stackrel{j \geq 0}{J =
0,1}} \! \frac{(-1)^J \! \!}{j!} \Bigl(\d_{\xi'}^j D_{\xi'\theta'}^J
\delta(\xi-\xi')\delta(\theta-\theta')\Bigr)(A_{(j|J)}B) (\xi', \theta') ~,
\end{equation*}
we can therefore read the OPE from $\eqref{eq:2.4.2}$.
The $\L$'s encode derivatives of delta functions, and the translation between the two formalisms are given by
\begin{equation*}
\L^{j|J}\leftrightarrow(-1)^J\d_{\xi'}^j D_{\xi'\theta'}^J
\delta(\xi-\xi')\delta(\theta-\theta')~.
\end{equation*}
In particular, the $\L$-bracket contains the information of all the commutators of the Fourier modes of the respective fields.
The $\L$-bracket comes with a very efficient calculus, which allows us to compute the bracket between composite fields, once the bracket between the constituent fields are known. The rules of this calculus are listed in Appendix \ref{LBrules}.
For further details of the formalism the reader may consult \cite{Heluani:2006pk}.
It is worth emphasizing that the normal ordered product is not commutative nor associative. Neither does the $\L$-bracket fulfill the Leibniz rule. The deviation from these properties of the different operations can be seen from the rules in the appendix. These deviations have natural interpretations as ``quantum effects'', and the semiclassical limit
of a SUSY vertex algebra leads to a SUSY Poisson vertex algebra (which is the same as a Poisson superalgebra of local
functionals).
\subsection{The Chiral de Rham complex (CDR)}
As an example of a SUSY vertex algebra, consider the so called $\beta\gamma$-$bc$ system. It consists of $2n$ superfields $(\phi^{i},S_{j}), i,j=1,2\ldots n$. $\phi^{i}$ is an even field and $S_{i}$ is an odd field. The defining $\L$-brackets are given by
\beq
\bs
\LB{\phi^{i}}{S_{j}}&=\delta^{i}_{j} ~, \\
\LB{\phi^{i}}{\phi^{j}}&=0 ~,\\
\LB{S_{i}}{S_{j}}&=0 ~.
\end{split}
\eeq{betagamma}
If we expand these superfields
\beq
\phi^i = \gamma^i + \theta c^i~,~~~~~~~~~~~S_i = b_i + \theta \beta_i~,
\eeq{skskfe88kdk}
then we recover the standard $\beta\gamma$ and $bc$ systems.
Let $g^{a}(\phi)$ be an invertable function of $\phi$. The following is an automorphism of $\eqref{betagamma}$:
\begin{align} \label{gluing}
\tilde{\phi}^{a} &=g^{a}(\phi)~, & \tilde{S}_{a}&=\frac{\d f^{i}}{\d \tilde{\phi}^{a}}(g(\phi))S_{i} ~,
\end{align}
where $f$ is the inverse of $g$. The normal ordered product is used when defining the new fields. Recall that it is not associative. The statement $\LB{ \tilde{S}_a }{ \tilde{S}_b }=0$ is therefore non-trivial.
Using this fact we can make the following geometrical construction. Consider a smooth manifold ${\cal M}$, and attach
to each coordinate patch $\{x^i\}_{i=1}^n$ the $\beta\gamma$-$bc$ system $\{\phi^i, S_i\}_{i=1}^n$. We glue these systems on the intersections
using \eqref{gluing}. We
thus construct a sheaf $\Omega^{ch}(\cal{M})$ of $\beta\gamma$-$bc$ systems. This sheaf was first introduced in \cite{Malikov:1998dw}, and named the Chiral de Rham complex, which we abbreviate CDR. Although the formalism developed in \cite{Malikov:1998dw}
works in the analytic, algebraic and
smooth settings, most of the mathematical literature on CDR is dedicated to the algebraic setting. The setup relevant to us
was considered in \cite{MR2294219} and in superfield formalism in \cite{benzvi-2006}.
In the present
work, our treatment of CDR closely follow \cite{benzvi-2006}.
\subsection{Physical interpretation of CDR}
In the above construction of CDR, we can introduce a parameter $\hbar$ in the defining brackets $\eqref{betagamma}$ of the $\beta\gamma$-$bc$ system:
\beq
\bs
\LB{\phi^{i}}{S_{j}}&=\hbar\delta^{i}_{j} ~,\\
\LB{\phi^{i}}{\phi^{j}}&=0 ~,\\
\LB{S_{i}}{S_{j}}&=0 ~.
\end{split}
\eeq{betagammah}
Let us consider the bracket $\frac{1}{\hbar} \LB{a}{b}$.
From the rules of appendix \ref{LBrules}, we see that in the limit $\hbar\rightarrow 0$, the resulting bracket will fulfill the Leibniz rule, and the normal ordered product is both commutative and associative, since the deviations from these properties will be higher order in $\hbar$. The ``quasi-classical'' limit $\hbar\rightarrow 0$ is a well-defined operation on Vertex Algebras, and in this limit we get a Poisson Vertex Algebra, described in \ref{section:PVA} (see \cite{Ekstrand:2009zd} and references therein).
In section \ref{classicalhamiltionian} we showed that the phase space of the classical sigma model is the superloop space $T^{*}\cal{LM}$, equipped with the Poisson bracket $\eqref{poisson}$, which in $\L$-bracket notation can be written as
\beq
\PLB{\phi^{i}}{S_{j}}=\delta^{i}_{j} ~.
\eeq{poissonlambda}
This relation is invariant under changes of coordinates, when $\phi^{i}$ transforms as a coordinate and $S_{i}$ as a one-form.
We can therefore construct a global object, which can formally be regarded as a sheaf of Poisson Vertex Algebras.
We can interpret the structure $\eqref{betagammah}$ as a formal canonical quantization of the structure $\eqref{poissonlambda}$, and CDR as the formal canonical quantization of the sigma model. We interpret the $\L$-brackets for the $\beta\gamma$-$bc$ system as equal-time commutators, and the definition of normal ordering of composite fields is the standard one.
For a further explanation of the above interpretation, and more precise statements, we refer to \cite{Ekstrand:2009zd}.
\section{Constructing well-defined currents on CDR}
\label{lift}
Our goal is to extend the results of Section \ref{clAlgExt} from the classical setting to the quantum setting of CDR. Before calculating
any brackets, we have to understand how to define the appropriate global sections of CDR from the classical expressions
for $T_\pm$ and $J_\pm^{(n)}$. The main requirement is that these quantum fields coincide with their classical expressions when taking the limit $\hbar \rightarrow 0$.
We have seen above that we can glue the superfields $\phi^i$ and $S_i$ between patches using the normal ordered transformation rules
\eqref{gluing}. In this section, we will discuss whether composite operators, which are defined as normal ordered products of $\phi^{i},S_{j}$ and functions thereof, can be glued together to give rise to globally defined sections of CDR. Especially, we will investigate whether the symmetry currents of the classical sigma model can be ``quantized''. There are potential problems, since in the quantum setting
the expressions are not associative nor commutative anymore. Indeed, we will see that in general the classical currents are not invariant under a change of coordinates in the quantum setup. However, we can by hand add terms to the currents which will be of higher order in $\hbar$. These extra terms will by themselves not be invariant under a change of coordinates, but their anomalous parts will precisely cancel with the anomalous parts coming from the classical part. This resembles
of the introduction of a connection when constructing a covariant derivative in geometry.
Amusingly, we will see below that the structure we need in order to ``quantum covariantize'' the set of currents $J_{\pm}^{(n)}$ is the Levi-Civita connection. Below, we will give a general prescription on which terms one must add to a current constructed from an $n$-form, for any $n$, in order for it to be a well-defined section of the CDR. This construction is limited to currents constructed from forms on $\cal{M}$, the anti-symmetry of the indices is used extensively. Currents formed from other structures, for example the stress energy tensor formed from the metric, turns out to be much harder to handle, and in fact we can only give indirect proofs in special cases
that the stress energy tensor is a well-defined section of CDR.
The inhomogeneous transformation properties of the Levi-Civita connection was used in \cite{benzvi-2006} to construct global sections of CDR associated to K\"ahler forms. Our construction below generalizes their result to forms of arbitrary degree.
\subsection{Constructing well-defined sections from forms}
\label{welldefinedsections}
In this section, we will show how to modify the currents constructed from $n$-forms, which classically are of the form
\beq
\bs
J^{(n)}_{+c}&=\frac{1}{n!}\omega_{i_1\ldots i_n}e_{+}^{i_{1}}\ldots e^{i_{n}}_{+} ~,\\
J^{(n)}_{-c}&=\frac{i^{n}}{n!}\omega_{i_1\ldots i_n}e_{-}^{i_{1}}\ldots e^{i_{n}}_{-}~. \\
\end{split}
\eeq{}
We here have introduced a subscript ``c'', for ``classical''.
The statements in the rest of this subsection applies equally well to both the plus-sector and the minus-sector, and we skip the $\pm$-signs and factors of $i$.
In CDR, we in general have to choose in which order we multiply operators. For definiteness we choose the order of multiplication to be
\beq
(\omega_{j_1\ldots j_n}(\phi))\left(e^{j_{1}}(\ldots (e^{j_{n-1}}e^{j_{n}})\ldots )\right),
\eeq{}
although it can be shown that for this operator the order of multiplication does not matter.
Let us define new coordinates $\tilde{\phi}(\phi)^a$, and denote
\begin{align}
\tilde{f}^a_i &\equiv \frac{\partial \tilde{\phi}^a}{\partial \phi^i} ~, & f^i_a &\equiv \frac{\partial \phi^i}{\partial \tilde{\phi}^a}~.
\end{align}
The operator $e^i$ transforms as a vector, and we have $e^{i}=f^i_a e^{a}$. The $n$-form $\omega$ transforms as $\omega_{i_{1}\ldots i_{n}}=f^{a_{1}}_{i_{1}}\ldots f^{a_{n}}_{i_{n}}\omega_{a_{1}\ldots a_{n}}$. Because of the non-associativity of the normal order product, one can in general not simply pull out the factors of $f$, and for $n>1$, the operator $(\omega_{j_1\ldots j_n}(\phi))\left(e^{j_{1}}(\ldots (e^{j_{n-1}}e^{j_{n}})\ldots )\right)$ does not transform as a tensor. One has to add a non-tensorial object in order to get the right transformation properties. What we will do below is to introduce a new operator $E_{(n)}^{i_1 \ldots i_n}$ such that $\omega_{j_1\ldots j_n}(\phi)E_{(n)}^{i_1 \ldots i_n}$ does transform as a tensor, and reduces to $(\omega_{j_1\ldots j_n}(\phi))\left(e^{j_{1}}(\ldots (e^{j_{n-1}}e^{j_{n}})\ldots )\right)$ in the limit $\hbar\rightarrow 0$.
\subsubsection{The construction, plus-sector}
Let us first define
\beq
\bs
F_{\pm(0)} &\equiv 1~, \\
F_{+(k)}^{j_1\ldots j_k} &\equiv e_{+}^{j_1} ( e_{+}^{j_2}( \ldots ( e_{+}^{j_{k-1}} e_{+}^{j_k}) \ldots )) ~, \\
F_{-(k)}^{j_1\ldots j_k} &\equiv ie_{-}^{j_1} ( ie_{-}^{j_2}( \ldots ( ie_{-}^{j_{k-1}} ie_{-}^{j_k}) \ldots )) ~.
\end{split}
\eeq{}
From now on, we only work in the plus sector, and we suppress the ${+}$ subscript. Comments about the minus sector can be found in the next subsection. We want to construct operators $E_{(p)}^{i_1 \ldots i_p}$ that can be used as a base for $p$-forms, \ie an object that transforms as a $(p,0)$-tensor. We are going to assume that $E_{(p)}^{i_1 \ldots i_p}$ is anti-symmetrized in its indices.
Our construction is recursive. Given $E_{(k-1)}$, we want to construct $E_{(k)}$. First, we investigate how $e^{i_1} E_{(k-1)}^{i_2 \ldots i_{k}}$ transforms under the assumption that $e^b_{(n|1)} E_{(k-1)} = 0\quad \forall n > 0$, \ie that
the $\chi$-terms in the $\Lambda$-bracket between $e^b$ and $E_{(k-1)}$ only have a $\lambda^0$-part. We then have:
\beq
e^{i_1} E_{(k-1)}^{i_2 \ldots i_{k}} = \left(f^{i_1}_{a_1} \ldots f^{i_k}_{a_k}\right) \left(e^{a_1} E_{(k-1)}^{a_2 \ldots a_{k}} - \partial \tilde{f}^{a_1}_l f^l_b (e^b_{(0|1)} E_{(k-1)}^{a_2 \ldots a_{k}}) \right) ~,
\eeq{inhomoE}
where $(~_{(i|j)}~)$ is the $\lambda^i \chi^j$-part of the $\Lambda$-bracket. The above assumption is satisfied if $E_{(k)}$ is of the form $E_{(k)} = \sum_{i=0}^k h_{(i)}(\phi, \partial \phi) F_{(i)}$, where $\{ h_{(i)} \}$ are arbitrary functions of $\phi$ and $\partial \phi$. This claim will be justified below.
Because of the in-homogenous transformation of the connection, we can cancel the in-homogeneous part of \eqref{inhomoE} by adding a term proportional to $ \Gamma \partial \phi$. The sought $(p,0)$-tensorial object is then
\begin{align}
E_{(0)} & \equiv 1~, \\
E_{(k)}^{i_1 \ldots i_{k}} &\equiv e^{i_1} E_{(k-1)}^{i_2 \ldots i_{k}} + \Gamma_{k l}^{i_1} \partial \phi^k (e^l_{(0|1)} E_{(k-1)}^{i_2 \ldots i_{k}})~.
\label{Eptensor}
\end{align}
We now show that the form $E_{(k)} = \sum_{i=0}^k h_{(i)}(\phi, \partial \phi) F_{(i)}$ implies that there are no higher powers of $\lambda$ in the $\chi$-terms of the $\Lambda$-bracket between $e^b$ and $E_{(k-1)}$.
We have
\begin{align}
\LB{e^{i}}{e^{j}}&=\hbar\chi g^{ij} +\mathcal{O}(\chi^0 ) ~,
\end{align}
where we only keep the term proportional to $\chi$. From this it follows that
\beq
\LB{ e^{i}}{F_{(k)}^{j_1\ldots j_k} } = \hbar\chi \sum_{n=1}^{k} (-1)^{k+1} g^{i j_n} (e^{j_1} \ldots (\widehat{e^{j_n}} \ldots e^{j_k}) \ldots) +\mathcal{O}(\chi^0)~,
\eeq{}
where the term $\widehat{e^{j_n}}$ is omitted. When the indices are anti-symmetrized, this is
\beq
\LB{ e^{i}}{ F_{(k)}^{j_1\ldots j_k} } = \hbar\chi k g^{i j_1} F_{(k-1)}^{j_2\ldots j_k} +\mathcal{O}(\chi^0)~.
\eeq{lbracketF}
We also have
\beq
\begin{split}
\LB{h(\phi, \partial \phi) e^{i}}{F_{(k)}^{j_1\ldots j_k}} &= h(\phi,\partial \phi) \LB{ e^{i}}{ F_{(k)}^{j_1\ldots j_k}} + \mathcal{O}(\chi^0) ~, \\
\LB{ e^{i}}{h(\phi,\partial \phi) F_{(k)}^{j_1\ldots j_k}} &= h(\phi,\partial \phi) \LB{ e^{i}}{ F_{(k)}^{j_1\ldots j_k}} + \mathcal{O}(\chi^0) ~,
\end{split}
\eeq{lbracketfF}
where $h$ is an arbitrary function of $\phi$ and $\partial \phi$. Here we used the fact that the integral term does not have any $\chi$-factor.
From our formula \eqref{Eptensor}, we see by induction that $E_{(k)}$ is of the form $E_{(k)} = \sum_{i=0}^k h_{(i)}(\phi, \partial \phi) F_{(i)}$, and due to \eqref{lbracketfF}, there are no higher powers of $\lambda$ in the $\chi$-terms of the $\Lambda$-bracket between $e^b$ and $E_{(k-1)}$.
We now exemplify \eqref{Eptensor}.
For $p=1$, $E_{(1)}^i$ is just $e^i$. For $p=2$ \eqref{Eptensor} yields
\beq
E_{(2)}^{i j} = F_{(2)}^{i j} + \hbar\Gamma_{k l}^i g^{j k} \partial \phi^l~.
\eeq{}
$E_{(2)}^{i j}$ is antisymmetric on its own, and no anti-symmetrization is needed, since the symmetric part of $F_{(2)}$ and $\Gamma g \partial \phi$ cancel:
\beq
e^{(i} e^{j)} = - \hbar\Gamma_{k l}^{(i} {g\vphantom{\Gamma}}^{j) k} \partial \phi^l= \hbar\partial g^{i j}~.
\eeq{dsadsfs}
For $p=3$ and $p=4$ we get
\begin{equation}\label{eq:E3}
E_{(3)}^{i_1 i_2 i_3} =e^{i_1} E_{(2)}^{i_2 i_3} + 2\hbar\Gamma_{k l}^{i_1} g^{k i_2} \partial \phi^l e^{i_3} = F_{(3)}^{i_1 i_2 i_3} + 3\hbar \Gamma_{k l}^{i_1} g^{k i_2} \partial \phi^l e^{i_3} ~, \\
\end{equation}
and
\begin{align}
\begin{split}
E_{(4)}^{i_1 i_2 i_3 i_4} &=e^{i_1} E_{(3)}^{i_2 i_3 i_4} + 3 \hbar\Gamma_{k l}^{i_1} g^{k i_2} \partial \phi^l e^{i_3} e^{i_4} \\&\quad+ 3 \hbar^2\Gamma_{k_1 l_1}^{i_1} g^{k_1 i_2} \partial \phi^{l_1} \Gamma_{k_2 l_2}^{i_3} g^{k_2 i_4} \partial \phi^{l_2} \\
&= F_{(4)}^{i_1 i_2 i_3 i_4} + 6 \hbar \Gamma_{k l}^{i_1} g^{k i_2} \partial \phi^l e^{i_3} e^{i_4} \\&\quad+ 3 \hbar^2 \Gamma_{k_1 l_1}^{i_1} g^{k_1 i_2} \partial \phi^{l_1} \Gamma_{k_2 l_2}^{i_3} g^{k_2 i_4} \partial \phi^{l_2}~,
\end{split}
\end{align}
respectively.
Let us reintroduce the $\pm$ subscripts. The explicit expression for general $n$ is the following. Define the nested sum
\begin{equation*}
S_{r,s}=\sum_{k_{s}=0}^{r}{\sum_{k_{s-2}=0}^{k_{s}+1}{\ldots\sum_{k_1=0}^{k_3+1}{k_{1}\ldots k_{s-2}k_{s}}}}
\end{equation*}{}
and
\begin{equation*}
G^{j_{1}\ldots j_{n}}_{\pm(n-q-1)}=\Gamma_{k_1l_1}^{j_1}g^{j_2k_1}\partial\phi^{l_1}\ldots\Gamma_{k_ql_q}^{j_{q}}g^{j_{q+1}k_q}\d\phi^{l_{q}}F_{\pm(n-q-1)}^{j_{q+2}\ldots j_{n}}.
\end{equation*}
The subscript denotes how many $e$'s are present in the operator. Then the following transforms as a tensor
\beq
E^{j_{1}\ldots j_{n}}_{+(n)}=F^{j_{1}\ldots j_{n}}_{+(n)}+\!\!\!\! \sum_{q=1, q \text{ odd}}^{n-1-(n \text{ mod } 2)} \!\! \!\! {\hbar^{\frac{q+1}{2}}S_{n-q,q}G^{j_{1}\ldots j_{n}}_{+(n-q-1)}} ~.
\eeq{wellDefinedSections}
This can be proved by induction, see Appendix \ref{inductionProof}.
Using these operators, we can construct
\beq
J_{+q}^{(n)}=\frac{1}{n!}\omega_{j_{1}\ldots j_{n}}E^{j_{1}\ldots j_{n}} _{+(n)}.
\eeq{}
which is a well-defined section of CDR, and satisfies $J^{(n)}_{+q}\rightarrow J^{(n)}_{+c}$ when $\hbar \rightarrow 0$.
\subsubsection{Comments about the minus-sector}
Since the construction of well-defined tensors only uses the $\chi$-part of the $\Lambda$-bracket between the $e^{i}$'s, the construction in the minus-sector can be mapped to the construction in the plus-sector. Since we have
\beq
\bs
\LB{e_{+}^{i}}{e_{+}^{j}}&=\hbar\chi g^{ij}+\mathcal{O}(\chi^{0}) \\
F_{+(k)}^{j_1\ldots j_k}& \equiv e_{+}^{j_1} ( e_{+}^{j_2}( \ldots ( e_{+}^{j_{k-1}} e_{+}^{j_k}) \ldots )) ~
\end{split}
\eeq{}
and
\beq
\bs
\LB{ie_{-}^{i}}{ie_{-}^{j}}&=\hbar\chi g^{ij}+\mathcal{O}(\chi^{0}) \\
F_{-(k)}^{j_1\ldots j_k} &\equiv ie_{-}^{j_1} ( ie_{-}^{j_2}( \ldots ( ie_{-}^{j_{k-1}} ie_{-}^{j_k}) \ldots )) ~
\end{split}
\eeq{}
we can immediately draw the conclusion that
\begin{align}
E_{-(0)} & \equiv 1~, \\
E_{-(k)}^{i_1 \ldots i_{k}} &\equiv i e_-^{i_1} E_{-(k-1)}^{i_2 \ldots i_{k}} + \Gamma_{k l}^{i_1} \partial \phi^k ( i e^l_{-(0|1)} E_{-(k-1)}^{i_2 \ldots i_{k}})~.
\label{Eptensorminus}
\end{align}
or, explicitly,
\beq
E^{j_{1}\ldots j_{n}}_{-(n)}=F^{j_{1}\ldots j_{n}}_{-(n)}+\!\!\!\! \sum_{q=1, q \text{ odd}}^{n-1-(n \text{ mod } 2)} \!\! \!\! {\hbar^{\frac{q+1}{2}}S_{n-q,q}G^{j_{1}\ldots j_{n}}_{-(n-q-1)}} ~,
\eeq{}
transform as tensors. Using these operators, we can construct
\beq
J_{-q}^{(n)}=\frac{1}{n!}\omega_{j_{1}\ldots j_{n}}E^{j_{1}\ldots j_{n}} _{-(n)}.
\eeq{}
which is a well-defined section of CDR, and satisfies $J^{(n)}_{-q}\rightarrow J^{(n)}_{-c}$ when $\hbar \rightarrow 0$.
\subsection{Well-defined sections constructed from other tensors}
Above, we have constructed well-defined sections from given forms. We now investigate whether we can construct well-defined sections from other tensorial objects, in particular from the metric.
From equation $\eqref{currents}$, we see that classically, the energy momentum tensor for the sigma model is constructed using the metric. Also, we will later see that when the target space is a $G_2$ or a $Spin(7)$ manifold, in the quantum case but in the flat space limit, the operator $g_{ij}\d e^{i}e^{j}$ must be considered. It is therefore an interesting question whether these operators, constructed using the metric, can be modified into well-defined sections of CDR.
After covariantizing the derivative, we are asking whether
\beq
\bs
g_{ij}\d e^{i}e^{j}&+g_{ij}\G^{i}_{~kl} \d\phi^k e^l e^j ~,\\
g_{ij}De^{i}e^{j} &+ g_{ij}\G^{i}_{~kl} D\phi^k e^l e^j ~,
\end{split}
\eeq{metricCurrents}
are well-defined sections of CDR. In the above expressions we must also choose an order of multiplication.
It turns out that this question is very hard to answer. We cannot use the same construction as in section \ref{welldefinedsections}. That construction relied heavily on the anti-symmetry of the tensors we contract with, allowing us to move
around the $e^{i}$'s to a certain extent. Since $\LB{e^i}{e^{j}}= \hbar\chi g^{ij}+\mathcal{O}(\chi^{0})$, the $\chi$-part vanishes when the indices are anti-symmetrized. This luxury we do not have when we contract with symmetric tensors.
A further complication occurs in the construction of the energy momentum tensor since the derivative $D$ is involved.
The presence of $D$ adds $\chi$-terms in the relevant brackets, thus increasing the difficulties in the calculations.
Due to the above reasons we cannot say whether $\eqref{metricCurrents}$ are well-defined sections of CDR on a general Riemannian manifold. At present, we do not know in general how to extend them to well-defined sections of CDR.
However, for Calabi-Yau and hyperk\"ahler manifolds we know that the energy momentum tensor is well-defined, since we can generate it from well-defined sections due to supersymmetry, see the discussion in section \ref{N=2Algebra}.
\section{Algebra extensions on CDR}
\label{CDRalgExt}
\subsection{General setup}
We now would like to address the question to which extent the classical discussion about symmetry algebra extensions in section \ref{clAlgExt} can be taken over to the framework of CDR. In the last section, we found how to modify the classical currents into well-defined sections of CDR. We would now like to know whether these currents close under the $\Lambda$-bracket in the quantum setup. As a starting remark, let us mention that, although we can write down an explicit expression for $J_{q}^{(n)}$, \ie the quantum counterpart of the classical current formed from an $n$-form, it is quite complicated to do a general computation as in $\eqref{jnbjm}$ in the quantum case. From now on, we drop the superscript $(n)$ and the $q$-subscript from the currents $J_{q}^{(n)}$.
Two cases have already been worked out. In \cite{benzvi-2006,heluani-2008}, the authors considered K\"ahler and hyperk\"ahler manifolds. Interestingly, it was found that the $N=2$ algebra is anomalous unless we choose the metric to be Ricci-flat, hence CDR requires $\mathcal{M}$ to be a Calabi-Yau manifold. A nice feature of these two cases is that the issue of constructing a well-defined stress energy tensor by hand is circumvented, since it can be generated from the supersymmetry currents, which we know are well-defined sections. In summary: on a Calabi-Yau manifold $(T_{\pm},J_{\pm})$ are well-defined sections of CDR, and they generate two commuting copies of the superconformal N=(2,2) algebra, with central charge $\frac{3}{2}\dim\mathcal{M}$.
In this article, we will add the four currents constructed from the two additional covariantly constant forms present on a Calabi-Yau $3$-fold, namely the holomorphic volume form and its complex conjugate. We will show below that, together $(T_{\pm},J_{\pm}, X_{\pm},\bar{X}_{\pm})$ generate two commuting copies of the extension of the N=(2,2) algebra first investigated in \cite{Odake:1988bh}, henceforth called the Odake algebra.
The obvious next step is of course to treat the two exceptional cases, $G_{2}$ and $Spin(7)$. Here we are in much worse shape, for two reasons. Firstly, we do not know if the stress energy tensor is a well-defined section on these manifolds. Secondly, the ``quantum covariant'' form of the currents constructed on these two manifolds are more complicated than in the Calabi-Yau case. This, together with the lack of nice choices of coordinates (a feature used extensively in the Calabi-Yau calculation), makes this calculation very complicated. If we were able to solve the second problem, the first problem would be solved in the same way as for the Calabi-Yau case, since we again in principle can generate the stress energy tensor from well-defined currents. Below we compute these algebras with the assumption of a flat metric, and observe that it matches with the algebras calculated in the original work \cite{Shatashvili:1994zw}.
In the case when the $G_2$-manifold $\cal M$ is the product of a Calabi-Yau $3$-fold and $S^1$, we can attach CDR to each component in the product, and construct the $G_{2}$ currents from geometrical identities. This is much along the lines of \cite{FigueroaO'Farrill:1996hm}. Since we have reliably calculated the algebra on the Calabi-Yau factor, and the circle is a flat manifold, in this special case we are able to calculate the $G_2$ algebra within the CDR framework.
We have benefited from the Mathematica package Lambda \cite{Ekstrand2010}, when performing many of these calculations.
\subsection{N=2 algebra}
\label{N=2Algebra}
When $\cal{M}$ is a K\"ahler manifold, we can use the K\"ahler form $\omega$ to construct two currents, which, according to section \ref{welldefinedsections}, need to be modified with one extra term:
\beq
J_{\pm}= \pm \frac{1}{2} \omega_{ij}e_{\pm}^{i}e_{\pm}^{j}+\frac{1}{2}\hbar ~\G^{i}_{jk}g^{jl}\omega_{il}\d\phi^{k} ~.
\eeq{84848003dmmd}
Using \eqref{ebase}-\eqref{ebase2}, consider the following linear combinations of $J_\pm$:
\beq
\begin{split}
J_{1}&=- J_+ - J_- = I^{i}_{j}D\phi^{j}S_{i}+\hbar\Gamma_{jk}^{i}I^{j}_{i}\partial\phi^{k}~,\\
J_{2}&= - J_+ + J_- = \frac{1}{2}\left(\omega^{ij}S_{i}S_{j}-\omega_{ij}D\phi^{i}D\phi^{j}\right)~,
\end{split}
\eeq{}
where $I^{i}_{j}$ is the complex structure, and $\omega^{ij}\omega_{jk}=\delta^{i}_{k}$.
These global sections of CDR were studied in detail in \cite{benzvi-2006,heluani-2008} and they both
generate the $N=2$ algebra. In particular, in \cite{heluani-2008} it has been shown that $J_\pm$ generate
two commuting copies of $N=2$ algebra if the K\"ahler manifold is equipped with a Ricci-flat metric $g$,
\beq
\begin{split}
\LB{ T_\pm }{ T_\pm }&=\hbar\left(2\partial+\chi D+3\lambda\right)T_{\pm}+\hbar^{2}\frac{\dim\mathcal{M}}{2}\lambda^{2}\chi~, \\
\LB{T_{\pm}}{J_{\pm}}&=\hbar\left(2\partial+\chi D+2\lambda\right)J_{\pm}~, \\
\LB{J_{\pm}}{J_{\pm}}&=-\hbar T_{\pm}-\hbar^{2}\frac{\dim\mathcal{M}}{2}\lambda^{2}\chi ~,\\
\end{split}
\eeq{N2commd0383}
where $T_\pm$ are defined by the following expressions
\beq
\begin{split}
T_{+}+T_{-} &= D\phi^{i}DS_{i}+\partial\phi^{i}S_{i}-\hbar\partial D \log{\sqrt{det g_{ij}}} ~,\\
T_{+} - T_- &= g_{ij}D\phi^i \partial\phi^j+g^{ij}S_{i} DS_j+ \Gamma^{j}_{kl}g^{il}D\phi^{k}(S_j S_i)~.
\end{split}
\eeq{sskke030o3o3}
This calculations were performed in complex coordinates. It can be shown that $T_+ + T_-$ is well-defined,
and moreover $T_+ - T_-$ arises as
the brackets of well-defined sections of CDR. Therefore, we know that $T_\pm$ are well-defined sections.
We could rewrite $T_\pm$ in terms of $e_\pm$, however the expressions are messy and depend on
the particular choice of ordering.
The crucial fact is that in the semi-classical limit $\hbar \rightarrow 0$ the above quantum $T_\pm$
will collapse to the classical expressions \eqref{currents} for the super-Virasoro.
It is important to stress that, using the Jacobi identity for the $\L$-bracket, we can avoid $T_+-T_-$ in explicit calculations, and it is possible to prove that the ``quantum'' modified generators $(T_{\pm}, J_{\pm})$ generate two commuting copies of the $N=2$ superconformal algebra with central charge $\frac{3}{2}\dim\mathcal{M}$ if $\mathcal{M}$ is Calabi-Yau.
We will use similar tricks in the calculation of the extensions of this algebra on a Calabi-Yau $3$-fold.
\subsection{The Odake algebra}
\label{OdakeAlgebra}
When $\mathcal{M}$ is a Calabi-Yau $3$-fold, we can construct four additional symmetry currents
from the holomorphic volume form and its complex conjugate:
\begin{align}
X_{+}&= \frac{1}{3!} \Omega_{\alpha\beta\gamma}e_{+}^{\alpha}e_{+}^{\beta}e_{+}^{\gamma} ~,
&X_{-} &= \frac{i^3}{3!} \Omega_{\alpha\beta\gamma}e_{-}^{\alpha}e_{-}^{\beta}e_{-}^{\gamma}~,\\
\bar{X}_{+}&=\frac{1}{3!} \bar{\Omega}_{\bar{\alpha}\bar{\beta}\bar{\gamma}}e_{+}^{\bar{\alpha}}e_{+}^{\bar{\beta}}e_{+}^{\bar{\gamma}} ~,
&\bar{X}_{-}&=\frac{i^3}{3!} \bar{\Omega}_{\bar{\alpha}\bar{\beta}\bar{\gamma}}e_{-}^{\bar{\alpha}}e_{-}^{\bar{\beta}}e_{-}^{\bar{\gamma}} ~.
\end{align}
Here we have introduced complex coordinates $({\alpha,\bar{\alpha}})$. Since we have a K\"ahler metric $g$, the ``quantum'' corrections introduced in section \ref{welldefinedsections} vanish, see \eqref{eq:E3}. On a Calabi-Yau, we can choose coordinates in which the holomorphic volume forms are constant. This simplifies calculations considerably, although they are still quite lengthy. Below we state the main result of the present paper:
\begin{theorem}
On a Calabi-Yau $3$-fold, we can construct well-defined sections of CDR $(T_{\pm}, J_{\pm},X_{\pm},\bar{X}_{\pm})$, which
generates two commuting copies of the Odake algebra, given by \eqref{N2commd0383} together with the following brackets:
\begin{equation}
\begin{split}
\LB{X_\pm}{\bar X_\pm} &= - \frac{1}{2} \hbar \bigl( i T_\pm J_\pm - \uD J_\pm J_\pm - \chi J_\pm J_\pm \bigr ) \\
& + \hbar^2 \bigl( i\chi \partial J_\pm
+ \lambda T_\pm + i \lambda \uD J_\pm + 2 i \chi \lambda J_\pm \bigr ) + \hbar^3 \chi\lambda^{2} ~,\\
\LB{ J_\pm }{ X_\pm} & =-i\hbar\left(3\chi+D\right)X_\pm~,\\
\LB{ J_\pm }{ \bar X_\pm }& =+i\hbar \left(3\chi+D\right)\bar{X}_\pm~,\\
\LB{ X_\pm }{T_\pm }&= \hbar (3 \lambda + \chi \uD + 2 \partial)X_\pm ~.\\
\LB{ \bar X_\pm }{T_\pm }&= \hbar (3 \lambda + \chi \uD + 2 \partial)\bar{X}_\pm ~,\\
\LB{X_\pm}{X_\pm} &=0~,\\
\LB{\bar X_\pm}{\bar X_\pm}& =0~,\\
\end{split}
\label{eq:nueva}
\end{equation}
where the plus- and minus-sectors commute. The semi-classical limit $\hbar \rightarrow 0$ of this vertex algebra is the
Poisson vertex algebra given by \eqref{jjdjdjjdjw2220} and \eqref{clasis333939s9sjj} and discussed in section \ref{clAlgExt}.
\end{theorem}
\begin{proof}
The calculation is done in Appendix \ref{odakeCalc}. In this calculation, no further geometrical
constraints are found in order for the algebra to close.
\end{proof}
As in the classical case, there are non-trivial constraints these currents satisfy. The quantum analog of \eqref{relations933is3} is given by
\beq
J_\pm X_\pm = - i \hbar \partial X_\pm~,~~~~~~~~~~~ J_\pm \bar{X}_\pm = i \hbar \partial \bar{X}_\pm~.
\eeq{blsle00388e3}
These identities are needed to check the Jacobi conditions for \eqref{eq:nueva}.
\subsection{$G_2$ and $Spin(7)$}
As mentioned in the general discussion, we cannot choose coordinates in which the components of the covariantly constant forms, existing on $G_2$ and $Spin(7)$-manifolds, are constant. Moreover, we are not aware of a choice of coordinates in which the ``quantum corrections'' to the currents (see section \ref{lift}) vanishes. This makes the computations of the symmetry algebras quite challenging. Taking the ``flat space limit'', that is in practice choosing a constant metric and constant forms, we can compute a closed algebra, which will have several quantum terms compared to the ones computed in the classical setup. It is an open question if the algebras still hold in the general curved case, with the modifications of the currents in order to make them well-defined, since we are presently not able to do the calculations.
This subsection contains the $\beta\gamma$-$bc$ system realization of two commuting copies of the algebras found in
\cite{Shatashvili:1994zw,FigueroaO'Farrill:1996hm}. Moreover, we show that these algebras are quantizations of
the classical algebras from section \ref{clAlgExt}, thus showing that the Howe-Papadopoulos Poisson algebras \cite{Howe:1991vs, Howe:1991ic} for $G_2$ and $Spin(7)$ are the classical versions of the
Shatashvili-Vafa vertex algebras \cite{Shatashvili:1994zw}.
\subsubsection{$G_{2}$}
Let us choose a flat metric $g_{ij}$ and constant $\Pi_{ijk}$, with $*\Pi_{ijkl}=\Psi_{ijkl}$. Let us define
the currents
\beq
\bs
\Pi_+&= \frac{1}{3!} \Pi_{ijk}~e_+^ie_+^je_+^k ~,~~~~~~~~~~~~~~\Pi_-= \frac{i^3}{3!} \Pi_{ijk}~e_-^ie_-^je_-^k ~,\\
\Psi_\pm&= - \frac{1}{4!} \Psi_{ijkl}~e_\pm^ie_\pm^j e^k_\pm e^l_\pm \pm \hbar\frac{1}{2}g_{ij}\d e^{i}_\pm e^{j}_\pm ~.
\end{split}
\eeq{}
Then $\Pi_\pm$ are the primary fields with conformal weight $\frac{3}{2}$ with respect to the Virasoro
fields $T_\pm= \pm g_{ij}De_\pm^{i}e_\pm^{j}$ respectively. $\Psi_\pm$ have conformal weight $2$ but are not primary:
\beq
\LB{T_\pm}{\Psi_\pm}=\hbar\left(2\partial+4\lambda+\chi D\right)\Psi_\pm + \hbar^2 \frac{1}{2}\chi\lambda T_\pm + \hbar^3 \frac{7}{12}\lambda^{3} ~.
\eeq{}
We compute the following brackets between the currents:
\beq
\bs
\LB{\Pi_\pm}{\Pi_\pm}=& - 3\hbar D\Psi_\pm-\hbar^2\frac{3}{2}\partial T_\pm - 6\hbar\chi\Psi_\pm-
\hbar^{2}3\lambda T_\pm-\hbar^{3}\frac{7}{2}\lambda^{2}\chi ~,\\
\LB{\Pi_\pm}{\Psi_\pm}=&+3\hbar T_\pm \Pi_\pm + \hbar^2\frac{5}{2}\chi\partial\Pi_\pm+\hbar^2 3\lambda D\Pi_\pm+ \hbar^{3}\frac{15}{2}\chi\lambda \Pi_\pm ~,\\
\LB{\Psi_\pm}{\Psi_\pm}=& +\hbar^{3}\frac{9}{4}\partial^2T_\pm - \hbar^2\frac{9}{2}D\partial\Psi_\pm + 10 \hbar T_\pm
\Psi_\pm + 3\hbar \Pi_\pm D \Pi_\pm
\\
& + \hbar^{2} 5 \left( \chi \partial + \lambda D + 2 \lambda \chi\right) \Psi_\pm
\\&
+ \hbar^{3}\frac{9}{4} \lambda (\partial + \lambda) T_\pm
+\hbar^{4}\frac{35}{24}\chi\lambda^{3} ~.\\
\end{split}
\eeq{G_2Algebra}
We see that taking the limit $\hbar\rightarrow 0$ we get back the algebra computed with Poisson brackets in $\eqref{classicG2}$.
Using the flat realization of the tensors $\Psi$ and $\Pi$ given in \ref{localrepPiPsi}, we find the following
relation between currents
\beq
\hbar^{2}\frac{1}{4}\partial^2T_\pm - \hbar 2 D\partial\Psi_\pm + 2 T_\pm \Psi_\pm + \Pi_\pm D \Pi_\pm = 0 ~,
\eeq{G_2AlgebraIdeal}
which is the quantum version of the classical relation \eqref{classicalidealG2}.
With the formula \eqref{G_2AlgebraIdeal}, the constant part of $\LB{\Psi}{\Psi}$ can be written
\beq
\LB{\Psi_\pm}{\Psi_\pm}_1 = \hbar^{3}\frac{3}{4}\partial^2T_\pm + \hbar^{2} \frac{3}{2} D\partial\Psi_\pm +
4 \hbar T_\pm \Psi_\pm ~.
\eeq{G_2AlgebraConstPart}
Decomposing the currents as
\beq
\bs
T_\pm&=G_\pm+2\theta L_\pm ~,\\
\Pi_\pm=\phi_\pm+\theta K_\pm,& \qquad \Psi_\pm=- X_\pm- \theta M_\pm ~,
\end{split}
\eeq{}
we find two copies of the same algebra as in \cite{Shatashvili:1994zw}.
We notice from the structure of the algebra \eqref{G_2Algebra} that both the Virasoro field $T$ and the field $\Psi$ is generated from the field $\Pi$. Since we know from section \ref{welldefinedsections} how to make $\Pi$ well defined on a curved target manifold, in principle we can derive candidates for the Virasoro field T and the field $\Psi$ on a general $G_2$ manifold. As mentioned above, these calculations are technically complicated, and we are not able to perform them at the present stage.
\subsubsection{$Spin(7)$}
Let us again choose a flat metric $g_{ij}$ and a constant $\Theta_{ijkl}$. Define
\beq
\Theta_\pm= \frac{1}{4!} \Theta_{ijkl}~e_\pm^i e_\pm^je_\pm^ke_\pm^l \pm \hbar\frac{1}{2}g_{ij}\d e_\pm^{i}e_\pm^{j} ~.
\eeq{}
We compute the following brackets:
\beq
\begin{split}
\LB{T_\pm}{\Theta_\pm} =&\hbar \left( 2\partial + 4 \lambda+ \chi D \right) \Theta_\pm +
\hbar^{2}\frac{1}{2} \chi \lambda T_\pm + \hbar^{3}\frac{2}{3} \lambda^3,\\
\LB{\Theta_\pm}{\Theta_\pm} =& \hbar^{2}\frac{5}{2} \partial D \Theta_\pm +\hbar^{3} \frac{5}{4} \partial^2 T_\pm
+ 6 \hbar T_\pm \Theta_\pm
\\&+8 \hbar^{2} \left( \chi \partial + \lambda D + 2 \lambda \chi\right) \Theta_\pm
\\&+ \hbar^{3}\frac{15}{4} \lambda (\partial + \lambda) T_\pm
+ \hbar^{4}\frac{8}{3} \lambda^3 \chi ~.
\end{split}
\eeq{Spin7Algebra}
Taking the limit $\hbar\rightarrow 0$, we get back the classical result $\eqref{classicSpin7}$. Decomposing the superfields as:
\begin{align}
T_\pm &= G_\pm + 2 \theta L_\pm ~, & \Theta_\pm &= \tilde{X}_\pm + \theta \tilde{M}_\pm ~,
\end{align}
we find two copies of the same algebra as in \cite{Shatashvili:1994zw} (see also \cite{Heluani:2006pk}).
\subsubsection{$G_{2}=CY_{3}\times S^{1}$}
There is a special case of a $G_2$-manifold which we can address without taking the flat space limit. Consider a $G_2$-manifold $\cal M$ of the type ${\cal M}=CY_{3}\times S^{1}$, where $CY_{3}$ is a Calabi-Yau threefold and $S^{1}$ is a circle. This is an example of a compact $G_2$-manifold.
We can attach to the Calabi-Yau factor the sheaf of Vertex algebras considered in \ref{OdakeAlgebra}. We can add to these generators a free $\beta\gamma$-$bc$ system, generated by $e_\pm=\frac{1}{\sqrt{2}}\left(S \pm D\phi\right)$, where $\phi$ is identified with the coordinate on the circle. The super-Virasoro field is given by $T^{S^{1}}_\pm=\pm e_\pm De_\pm$.
On $\cal M$ the corresponding $3$-form $\Pi$ and its Hodge dual $\Psi$ are defined by
the following expressions
\begin{equation} \label{formsG2CYS1}
\begin{split}
\Pi&=Re(\Omega)+J\wedge dX^{7} ~,\\
\Psi&=Im(\Omega)\wedge dX^{7}+\frac{1}{2}J\wedge J ~,
\end{split}
\end{equation}
where $J$ is the K\"ahler form, $\Omega$ is the holomorphic volume form and $X^7$ is the coordinate along the circle. It is therefore natural to construct new operators from the operators in the Odake algebra by
\begin{equation}
\Pi_+=X_+ +\bar{X}_+ +J_+e_+ ~,
\end{equation}
where we temporarily only consider the plus-sector.
Let us denote
\begin{align}
R_+&=X_++\bar{X}_+ ~, &I_+&=-i (X_+-\bar{X}_+)~,
\end{align} (where $R$ is for real and $I$ for imaginary). We then have the following bracket between $\Pi_+$ and $\Pi_+$:
\begin{multline}
\LB{\Pi_+}{\Pi_+}=3\hbar D\left(\frac{J_+J_+}{2}-I_+e_+ +\frac{1}{2}\hbar e_+\partial e_+\right) \\
+6\hbar\chi\left(\frac{J_+J_+}{2}-I_+e_+ +\frac{1}{2}\hbar e_+\partial e_+\right) \\
-\hbar^2\frac{3}{2}\partial\left(T_++e_+De_+\right) -\hbar^2 3\lambda\left(T_++e_+De_+\right)-\hbar^3\frac{7}{2}\chi\lambda^{2} ~.
\end{multline}
We see that we reproduce the first part of the $G_2$-algebra, with $\Psi$ given by
\begin{equation}
\Psi_+=\frac{J_+J_+}{2}-I_+e_++\frac{1}{2}\hbar e_+\partial e_+ ~.
\end{equation}
This is what we expected from \eqref{formsG2CYS1}. When computing the rest of the algebra, we have to use the identities
\begin{align}
J_+I_+&=-\hbar\partial R_+ ~, &
J_+R_+&=\hbar\partial I_+ ~.
\end{align}
which hold on any Calabi-Yau manifold and are just a rewritten form of \eqref{blsle00388e3}. A similar analysis can be performed for the minus-sector.
Computing the rest of the brackets, using the above identities, we generate the full $G_2$ algebra \eqref{G_2Algebra}. Notice that we have proved the following proposition:
\begin{proposition} There exists an embedding of the vertex algebra \cite{Shatashvili:1994zw} (see \eqref{G_2Algebra}) associated to a manifold of special holonomy $G_2$ into the tensor product of the Odake vertex algebra \cite{Odake:1988bh} (see Section \ref{OdakeAlgebra}) associated to a Calabi-Yau $3$-fold and
a free boson-fermion system generated by one odd superfield $e$, such that $\LB{e}{e} = \chi$.
\end{proposition}
\section{Discussion}
\label{s:summary}
In this article we studied extensions of the super-Virasoro algebra within the framework of the chiral de Rham complex. The main result of
this work is the construction of two commuting copies of the Odake algebra associated to any Calabi-Yau $3$-fold. This is the first example of a non-linearly generated algebra arising in this manner. Another result of this article is the systematic construction of global sections of CDR from antisymmetric tensors on $\cal M$. Moreover,
we presented the full classical and partial quantum results for general extensions of the super-Virasoro
algebra. The central idea behind our consideration is the interpretation of CDR as a formal canonical
quantization of the non-linear sigma model.
The main unresolved questions are the full calculations of the algebras for $G_2$ and $Spin(7)$ manifolds. For $G_2$, we have a well defined generator in the curved case and in principle we can generate the other relevant operators through the brackets. By construction, these will be well-defined.
Unfortunately, at the present moment, these calculations appear to be too complicated to carry out.
One motivation for studying manifolds with special holonomies comes from string theory. After compactification, let $\mathcal{M}$ be the internal manifold. In order to have space-time supersymmetry, $\cal{M}$ must admit a covariantly constant spinor. For different dimensions of $\cal{M}$, the constraint of \textit{minimal} space-time supersymmetry leads to different choices of holonomy groups. For dimensions 6,7 and 8 the holonomy groups are $SU(3)$, $G_{2}$ and $Spin(7)$. In the quantum setup, the extensions of N=(2,2) symmetry algebra for $d=6$ were studied for the first time in \cite{Odake:1988bh}, whereas the cases $d=7$ and $d=8$ were first investigated in \cite{Shatashvili:1994zw}. A common feature for the calculations of these algebra extensions performed in the mentioned papers, is that they are performed in the large volume limit, which means that the metric is treated like a flat metric. With the construction of the CDR, and its interpretation as the canonical quantization of the sigma model, we can begin to compute the symmetry algebras in a reliable way without taking the large volume limit.
Finally, let us point out that our considerations give an embedding of the differential forms of $\cal{M}$ into CDR
which is different from the original work \cite{Malikov:1998dw}. Our embedding is motivated by sigma model
considerations, and it would be very interesting to further study the properties of this map. We hope to come
back to this issue elsewhere.
\section*{Acknowledgement}
The research of R.H. was supported by NSF grant DMS-0635607002.
The research of M.Z. was supported by VR-grant 621-2008-4273.
\pagebreak
\appendix
\appendixpage
\section{Special holonomy manifolds}
\label{crosscontractions}
In this appendix, we collect the relevant relations of the invariant tensors on special holonomy manifolds.
\subsection{K\"ahler manifolds}
On a K\"ahler manifold we have the K\"ahler form $\omega = g I$, where $g$ is a metric and $I$ is a complex structure.
We then have $\omega g^{-1} \omega = - g$. In components
\begin{equation}
\omega_{ij}g^{jk}\omega_{kl}=-g_{il} ~.
\end{equation}
\subsection{Calabi-Yau manifolds} \label{section:FormulasCalabiYau}
On a Calabi-Yau $n$-fold, we define the holomorphic volume form $\Omega$ and its complex
conjugate $\bar{\Omega}$. The following relation holds
\begin{equation} \label{holoformula}
\Omega_{\alpha_1\alpha_2\ldots\alpha_n}g^{\alpha_1\bar{\alpha}_1}\Omega_{\bar{\alpha}_1\bar{\alpha}_2\ldots \bar{\alpha}_n}= g_{\alpha_2\bar{\alpha}_2}\ldots g_{\alpha_n\bar{\alpha}_n} + ... ~,
\end{equation}
where dots stand for the terms required by antisymmetrization in $\alpha_1 .... \alpha_n$ and $\bar{\alpha}_1 ... \bar{\alpha}_n$.
In particular, for a Calabi-Yau $3$-fold we have
\beq
\Omega_{\alpha_1\alpha_2\alpha_3} g^{\alpha_1\bar{\alpha}_1} \Omega_{\bar{\alpha}_1\bar{\alpha}_2\bar{\alpha}_3}=
g_{\alpha_2\bar{\alpha}_2} g_{\alpha_3\bar{\alpha}_3} - g_{\alpha_2\bar{\alpha}_3} g_{\alpha_3\bar{\alpha}_2}~.
\eeq{ksks003383}
We also have the K\"ahler form $\omega$, and the following contraction between $\Omega$ and $\omega$:
\begin{equation}
\omega_{\alpha\bar{\alpha}}g^{\bar{\alpha}\beta_1}\Omega_{\beta_1 \ldots \beta_n}=i\Omega_{\alpha\beta_2 \ldots \beta_n} ~.
\end{equation}
\subsection{$G_2$-manifolds}
On a $G_2$ manifold we have couple of forms: a $3$-form $\Pi$ and its dual $*\Pi\equiv \Psi$. The following formulas are collected from \cite{karigiannis-2007}.
\subsubsection{Contracting $\Pi$ with itself}
\begin{eqnarray}
\Pi_{ijk} \Pi_{abc} g^{ia} g^{jb} g^{kc} &= & 42 \\
\Pi_{ijk} \Pi_{abc} g^{jb} g^{kc} & = & 6 g_{ia}
\\ \Pi_{ijk} \Pi_{abc} g^{kc} &= & g_{ia} g_{jb} - g_{ib} g_{ja} - \Psi_{ijab} \label{phiphi}
\end{eqnarray}
\subsubsection{Contracting $\Pi$ and $\Psi$}
\begin{eqnarray}
\Pi_{ijk} \Psi_{abcd} g^{ib} g^{jc} g^{kd} &= & 0 \\
\Pi_{ijk} \Psi_{abcd} g^{jc} g^{kd} &= & - 4 \Pi_{iab} \\ \Pi_{ijk}
\Psi_{abcd} g^{kd} & = & g_{ia} \Pi_{jbc} + g_{ib} \Pi_{ajc} +
g_{ic} \Pi_{abj} \\
&& {} - g_{aj} \Pi_{ibc} - g_{bj} \Pi_{aic} - g_{cj} \Pi_{abi} \nonumber
\label{phipsi}
\end{eqnarray}
\subsubsection{Contracting $\Psi$ with itself}
\begin{eqnarray}
\Psi_{ijkl} \Psi_{abcd} g^{ia} g^{jb} g^{kc} g^{ld} &= & 168 \\
\Psi_{ijkl} \Psi_{abcd} g^{jb} g^{kc} g^{ld} &=& 24 g_{ia} \\
\Psi_{ijkl} \Psi_{abcd} g^{kc} g^{ld} &=& 4 g_{ia} g_{jb} - 4
g_{ib} g_{ja} - 2 \Psi_{ijab} \\
\label{PsiPsigPipi}\Psi_{ijkl} \Psi_{abcd} g^{ld} &=& -\Pi_{ajk} \Pi_{ibc} - \Pi_{iak} \Pi_{jbc} - \Pi_{ija} \Pi_{kbc} \\
&& {} + g_{ia} g_{jb} g_{kc} + g_{ib} g_{jc} g_{ka} + g_{ic} g_{ja} g_{kb} \nonumber \\
&& {} - g_{ia} g_{jc} g_{kb} - g_{ib} g_{ja} g_{kc} - g_{ic} g_{jb} g_{ka} \nonumber \\
&& {} -g_{ia} \Psi_{jkbc} - g_{ja} \Psi_{kibc} - g_{ka} \Psi_{ijbc} \nonumber\\
&& {} + g_{ab} \Psi_{ijkc} - g_{ac} \Psi_{ijkb} \nonumber
\label{psipsi}
\end{eqnarray}
\subsubsection{Local representations of $\Pi$ and $\Psi$ } \label{localrepPiPsi}
In a local orthonormal and flat basis, where the metric is $\sum dx^i \otimes dx^i$, the forms $\Pi$ and $\Psi$ can be written \cite{Shatashvili:1994zw} as
\begin{align}
\Pi =& dx^1 dx^2 dx^5 +dx^1 dx^3 dx^6 +dx^1 dx^4 dx^7 -dx^2 dx^3 dx^7+ \\
&dx^2 dx^4 dx^6 -dx^3 dx^4 dx^5 +dx^5 dx^6 dx^7 ~, \nonumber \\
\Psi =& dx^1 dx^2 dx^3 dx^4 - dx^1 dx^2 dx^6 dx^7 + dx^1 dx^3 dx^5 dx^7 -\\
& dx^1 dx^4 dx^5 dx^6 + dx^2 dx^3 dx^5 dx^6 + \nonumber\\
& dx^2 dx^4 dx^5 dx^7 + dx^3 dx^4 dx^6 dx^7 ~, \nonumber
\end{align}
where the product is understood as the wedge product.
\subsection{$Spin(7)$-manifolds}
On a $Spin(7)$-manifold we have a self dual $4$-form $\Theta$. These formulas are collected from \cite{karigiannis-2007-2008}.
\begin{eqnarray}
\Theta_{ijkl} \Theta_{abcd} g^{ia} g^{jb} g^{kc} g^{ld} & = & 336 \\
\Theta_{ijkl} \Theta_{abcd} g^{jb} g^{kc} g^{ld} & = & 42 g_{ia} \\
\Theta_{ijkl} \Theta_{abcd} g^{kc} g^{ld} & = & 6 g_{ia} g_{jb} - 6
g_{ib} g_{ja} - 4 \Theta_{ijab} \\ \Theta_{ijkl} \Theta_{abcd} g^{ld} & = &
g_{ia} g_{jb} g_{kc} + g_{ib} g_{jc} g_{ka} + g_{ic}
g_{ja} g_{kb} \\
& & {} - g_{ia} g_{jc} g_{kb} - g_{ib} g_{ja}
g_{kc} - g_{ic} g_{jb} g_{ka} \nonumber \\
& & {} -g_{ia} \Theta_{jkbc} - g_{ja} \Theta_{kibc} - g_{ka} \Theta_{ijbc} \nonumber \\
& & {} -g_{ib} \Theta_{jkca} - g_{jb} \Theta_{kica} - g_{kb} \Theta_{ijca} \nonumber\\
& & {} -g_{ic} \Theta_{jkab} - g_{jc} \Theta_{kiab} - g_{kc} \Theta_{ijab} \nonumber
\end{eqnarray}
\section{\texorpdfstring{$\L$}{Lambda}-bracket calculus}\label{LBrules}
In this appendix we collect some properties of $\Lambda$-bracket calculus. For further explanations
and details, the reader may consult \cite{Heluani:2006pk}.
\begin{itemize}
\item Basic relations:
\begin{align}
\uD ^2&=\partial & [\uD,\partial]&=0 & [\uD,\lambda]&=0 & [\partial,\lambda]&=0 \\
\chi^2&=-\lambda & [\uD,\chi]&=2\lambda & [\partial,\chi]&=0
\end{align}
\item Sesquilinearity:
\begin{align}
\LB{ a}{ b} &= \chi \LB{a}{b}
& \LB{a}{ \uD b} &= -(-1)^{a} \left( \uD
+ \chi
\right) \LB{a}{b}
\\
\LB{\partial a }{ b} &= -\lambda \LB{a}{b}
&\LB{a}{\partial b} &= \left( \partial
+ \lambda
\right) \LB{a}{b}
\end{align}
\item Skew-symmetry:
\begin{equation}
\LB{a}{b} = (-1)^{a b} \LB[-\Lambda -
\nabla]{b}{ a}
\label{eq:k.skew.1}
\end{equation}
The bracket on the right hand side is computed as
follows: first compute $\LB[\Gamma]{b}{a}$, where $\Gamma =
(\gamma, \eta)$, then replace $\Gamma$ by $(-\lambda - \partial,
-\chi - \uD)$.
\item Jacobi identity:
\begin{equation}
\LB{a}{\LB[\Gamma]{b}{c}} = -(-1)^{a} \LB[\Gamma + \Lambda]{ \LB{a}{b} }{ c} +
(-1)^{(a+1)(b+1)} \LB[\Gamma]{b}{ \LB{a}{c} }
\end{equation}
where the first bracket on the right hand side is computed as in \eqref{eq:k.skew.1}.
\item Quasi-commutativity:
\begin{equation}
ab - (-1)^{ab} ba = \int_{-\nabla}^0 \LB{a}{b} \ud\Lambda
\end{equation}
where the integral $\int d\Lambda$ is $\partial_\chi \int d\lambda$.
\item Quasi-associativity:
\begin{equation} \label{eq:LambdaBrackeRulesQuasiAssociativity}
(ab)c - a(bc) = \sum_{j \geq 0}
a_{(-j-2|1)}b_{(j|1)}c + (-1)^{ab} \sum_{j \geq 0}
b_{(-j-2|1)} a_{(j|1)}c
\end{equation}
\item Quasi-Leibniz (non-commutative Wick formula):
\begin{equation}
\LB{a}{b c } = \LB{a}{b} c + (-1)^{(a+1)b}b
\LB{ a}{c} + \int_0^\Lambda \LB[\Gamma]{ \LB{a}{b} }{c} \ud \Gamma
\end{equation}
\end{itemize}
\section{Proof that \eqref{wellDefinedSections} solves \eqref{Eptensor}}
\label{inductionProof}
In this section, we will show that
\beq
E^{j_{1}\ldots j_{n}}_{+(n)}=F^{j_{1}\ldots j_{n}}_{+(n)}+\!\!\!\! \sum_{q=1, q \text{ odd}}^{n-1-(n \text{ mod } 2)} \!\! \!\! {\hbar^{\frac{q+1}{2}}S_{n-q,q}G^{j_{1}\ldots j_{n}}_{+(n-q-1)}} ~,
\eeq{generalFormula}
where
\beq
S_{r,s}=\sum_{k_{s}=0}^{r}{\sum_{k_{s-2}=0}^{k_{s}+1}{\ldots\sum_{k_1=0}^{k_3+1}{k_{1}\ldots k_{s-2}k_{s}}}}
\eeq{}
and
\beq
G^{j_{1}\ldots j_{n}}_{(n-q-1)}=\Gamma_{k_1l_1}^{j_1}g^{j_2k_1}\partial\phi^{l_1}\ldots\Gamma_{k_ql_q}^{j_{q}}g^{j_{q+1}k_q}\d\phi^{l_{q}}F_{(n-q-1)}^{j_{q+2}\ldots j_{n}} ~,
\eeq{}
is a solution to the recursive relation
\begin{align}
E_{(0)} & = 1~, \\
E_{(k)}^{j_1 \ldots j_{k}} &\equiv e^{j_1} E_{(k-1)}^{j_2 \ldots j_{k}} + \Gamma_{k l}^{j_1} \partial \phi^k (e^l_{(0|1)} E_{(k-1)}^{j_2 \ldots j_{k}})~.
\end{align}{}
We proof this for $n$ even, for notational convenience. The proof for $n$ odd is the same. We will always think of the free upper indices as anti-symmetrized, and we set $\hbar=1$. We will extensivly use the relations
\begin{align}
\LB{e^{i}}{e^{j}}_{\chi}&=g^{ij} ~,\\
\LB{ e^{i}}{ F_{(k)}^{j_1\ldots j_k} }_{\chi} &= k g^{i j_1} F_{(k-1)}^{j_2\ldots j_k} ~, \\
\LB{h(\phi, \partial \phi) e^{i}}{F_{(k)}^{j_1\ldots j_k}}_{\chi} &= \LB{ e^{i}}{h(\phi,\partial \phi) F_{(k)}^{j_1\ldots j_k}}_{\chi} \\
&= h(\phi,\partial \phi) \LB{ e^{i}}{ F_{(k)}^{j_1\ldots j_k}}_{\chi} \nonumber ~,
\end{align}
where $\LB{}{}_{\chi}$ denotes the $\chi$-part of the bracket, and $h$ is an arbitrary function of $\phi$ and $\partial \phi$ .
\begin{proof}
We will use induction. That it is true for $n=2$ follows without difficulty. From the recursion relation we find
\beq
E_{2}^{j_1 j_2}=F^{j_1 j_2}_{2}+ \Gamma_{k l}^{j_1} \partial \phi^kg^{lj_2} ~,
\eeq{}
and one easily convince oneself that we find the same result from $\eqref{generalFormula}$.
Let us now assume that it is true for $n=p-2$, \ie
\beq
E_{p-2}^{j_{3}\ldots j_{p}}=F^{j_{3}\ldots j_{p}}_{(p-2)}+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{S_{p-2-q,q}G^{j_{3}\ldots j_{p}}_{(p-q-3)}} ~.
\eeq{indass}
Using the recursion relation we find
\begin{multline}
E_{p}^{j_{1}\ldots j_{p}}=e^{j_1}\left\{e^{j_2} E_{p-2}^{j_{3}\ldots j_{p}}+\Gamma^{j_{2}}_{kl}\d\phi^{k}\left(e^{l}_{(0|1)}E_{p-2}^{j_{3}\ldots j_{p}}\right) \right\} \\
+\Gamma^{j_{1}}_{st}\d\phi^{s}\left\{e^{t}_{(0|1)}\left(e^{j_2}E_{p-2}^{j_{3}\ldots j_{p}}+\Gamma^{j_2}_{kl}\d\phi^{k}(e^{l}_{(0|1)}E_{p-2}^{j_{3}\ldots j_{p}})\right)\right\} ~.
\end{multline}
Using the induction assumption \eqref{indass}, the first term gives us
\beq
e^{j_1}e^{j_2} E_{p-2}^{j_{3}\ldots j_{p}}=F^{j_{1}\ldots j_{p}}_{(p)}+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-1)}}.
\eeq{1}
Since we anti-symmetrize the indices, we are allowed to move around the $e$'s freely.
Next, we need to find $(e^{l}_{(0|1)}E_{p-2}^{j_{3}\ldots j_{p}})$. Using the above relations we find that
\begin{align}
\LB{e^{l}}{F_{p-2}^{j_{3}\ldots j_{p}}}_{\chi}&=(p-2)g^{lj_3}F_{p-3}^{j_{4}\ldots j_{p}} ~,\\
\LB{e^{l}}{G^{j_{3}\ldots j_{p}}_{(p-q-3)}}_{\chi}&=(p-q-3)g^{lj_{3}}G^{j_{4}\ldots j_{p}}_{(p-q-4)} ~,
\end{align}
where we have used that $q$ is odd when moving the indices in the last line. These two relations give us
\beq
(e^{l}_{(0|1)}E_{p-2}^{j_{3}\ldots j_{p}})=(p-2)g^{lj_3}F_{p-3}^{j_{4}\ldots j_{p}}+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)S_{p-2-q,q}g^{lj_{3}}G^{j_{4}\ldots j_{p}}_{(p-q-4)}}.
\eeq{}
From this we deduce
\beq
\bs
\Gamma^{j_{2}}_{kl}\d\phi^{k}\left(e^{l}_{(0|1)}E_{p-2}^{j_{3}\ldots j_{p}}\right)&=(p-2)G_{p-3}^{j_{2}\ldots j_{p}}+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)S_{p-2-q,q}G^{j_{2}\ldots j_{p}}_{(p-q-4)}} \\
&\equiv H^{j_{2}\ldots j_{p}} ~.
\end{split}
\eeq{}
We find the second term to be
\beq
e^{j_{1}}H^{j_{2}\ldots j_{p}}=(p-2)G_{p-2}^{j_{1}\ldots j_{p}}+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-3)}} ~.
\eeq{2}
For the third term we need
\beq
\bs
&\LB{e^{t}}{e^{j_{2}}E_{p-2}^{j_{3}\ldots j_{p}}}_{\chi}=g^{tj_{2}}E_{p-2}^{j_{3}\ldots j_{p}} \\
&-e^{j_2} \left\{(p-2)g^{tj_{3}}F^{j_{4}\ldots j_{p}}_{(p-3)} +\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)S_{p-2-q,q}g^{tj_{3}}G^{j_{4}\ldots j_{p}}_{(p-q-4)}}\right\} \\
&=g^{tj_{2}}E_{p-2}^{j_{3}\ldots j_{p}}\\
&+\left\{(p-2)g^{tj_{2}}F^{j_{3}\ldots j_{p}}_{(p-2)} +\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)S_{p-2-q,q}g^{tj_{2}}G^{j_{3}\ldots j_{p}}_{(p-q-3)}}\right\} \\
&=\left(e^{t}_{(0|1)}(e^{j_2}E_{p-2}^{j_{3}\ldots j_{p}})\right).
\end{split}
\eeq{}
From this we find the third term to be
\begin{multline}\label{eq:longproof3}
\Gamma^{j_{1}}_{st}\d\phi^{s}\left(e^{t}_{(0|1)}(e^{j_2}E_{p-2}^{j_{3}\ldots j_{p}})\right)=
(p-1)G^{j_1\ldots j_{p}}_{p-2} \\
+ \sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-2)S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-3)}} ~.
\end{multline}
In the same way as for the other terms we find the fourth term to be
\begin{multline}
\Gamma^{j_{1}}_{st}\d\phi^{s}\left(e^{t}_{(0|1)}H^{j_2\ldots j_p}\right)=(p-2)(p-3)G^{j_{1}\ldots j_{p}}_{p-4}\\
+ \sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)(p-q-4)S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-5)}} ~.
\label{eq:longproof4}
\end{multline}
We now need to sum up $\eqref{1}$, $\eqref{2}$, $\eqref{eq:longproof3}$ and $\eqref{eq:longproof4}$, and we indeed get
\beq
F^{j_{1}\ldots j_{p}}_{(p)}+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-1}{S_{p-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-1)}}~.
\eeq{}
We see that the term $F^{j_{1}\ldots j_{p}}_{p}$ is present in $\eqref{1}$.
Next, we find the coefficients of $G^{j_{1}\ldots j_{p}}_{(p-q-1)}$ for $q=1,3, \ldots, p-1$. We treat $q=1$ and $q=3$ separately. Inspection of $\eqref{1}$, $\eqref{2}$, $\eqref{eq:longproof3}$ and $\eqref{eq:longproof4}$ gives us:
\begin{itemize}
\item $G_{p-2}$: \\
\begin{equation*}
S_{p-3,1}+(p-2)+(p-1)=S_{p-1,1}
\end{equation*}
\item $G_{p-4}$:
\begin{equation*}
\bs
S_{p-5,3}+(p-4)S_{p-3,1}&+(p-3)S_{p-3,1}+(p-2)(p-3)=\\
=S_{p-5,3}&+(p-4)S_{p-3,1}+(p-3)S_{p-2,1}= \\
&=S_{p-3,3}
\end{split}
\end{equation*}
\end{itemize}
The terms that we have not analyzed so far are
\begin{align*}
&\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-1)}}+\sum_{\stackrel{q=3}{\text{q odd}}}^{p-3}{(p-q-3)S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-3)}} \\
&+\sum_{\stackrel{q=3}{\text{q odd}}}^{p-3}{(p-q-2)S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-3)}}\\
&+\sum_{\stackrel{q=1}{\text{q odd}}}^{p-3}{(p-q-3)(p-q-4)S_{p-2-q,q}G^{j_{1}\ldots j_{p}}_{(p-q-5)}} \displaybreak[0] \\
&=\left\{\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{S_{p-2-q,q}}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q-1)S_{p-q,q-2}} \right\}G^{j_{1}\ldots j_{p}}_{(p-q-1)} \displaybreak[0] \\
&+\left\{\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q)S_{p-q,q-2}} +\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q+1)(p-q)S_{p+2-q,q-4}}\right\}G^{j_{1}\ldots j_{p}}_{(p-q-1)}
\end{align*}
where we in the second line have changed the summation indices and pulled out the common factor $G^{j_{1}\ldots j_{p}}_{(p-q-1)}$. Ignoring this factor, we find the coefficient to be
\begin{align*}
&\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{S_{p-2-q,q}}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q-1)S_{p-q,q-2}} \\
&\hphantom{=}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q)S_{p-q,q-2}} +\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q+1)(p-q)S_{p+2-q,q-4}} \displaybreak[0] \\
&=\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{S_{p-2-q,q}}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q-1)S_{p-q,q-2}}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{(p-q)S_{p+1-q,q-2}} \displaybreak[0] \\
&=\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{S_{p-2-q,q}}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{(p-q-1)S_{p-q,q-2}}+\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{(p-q)S_{p+1-q,q-2}} +S_{2,p-3} \displaybreak[0] \\
&=\sum_{\stackrel{q=5}{\text{q odd}}}^{p-3}{S_{p-q,q}}+S_{1,p-1} \\
&=\sum_{\stackrel{q=5}{\text{q odd}}}^{p-1}{S_{p-q,q}} ~,
\end{align*}
and we are done.
\end{proof}
\section{Calculation of the Odake algebra}
\label{odakeCalc}
First, we focus on the plus-sector and define the generators $X_+$, $T_+$ and $J_+$. We then compute the Odake algebra. After this, we show that the plus- and minus-sectors commute. The calculation of the minus-sector can be done in an analogous way, and we will therefore not give the explicit calculation. In this section we will suppress the factors of $\hbar$.
We choose coordinates where the holomorphic volume form is constant, which in particular implies $\Gamma^{\alpha}_{\alpha\beta}=\Gamma^{\bar\alpha}_{\bar\alpha\bar\beta}=g^{ i j} g_{ i j ,k} =0$ on the K\"ahler manifold.
On Calabi-Yau manifold we can always choose such coordinates locally.
\subsection{Setup}
Let us define
\begin{equation}
\begin{split}
e^i_+\equiv\frac{1}{\sqrt{2}}\left( g^{ij}S_{j} +D\phi^i\right) ~,\\
e^i_-\equiv\frac{1}{\sqrt{2}}\left(g^{ij}S_{j}-D\phi^i \right) ~.
\end{split}
\label{epem2SDphi}
\end{equation}
These fields satisfy:
\begin{equation}
\begin{split}
\LB{e^i_\pm }{ e^{j}_\pm }&=\pm \chi g^{ij} + \frac{1}{\sqrt{2}}\left( g^{kj}\Gamma^i_{mk}e_{\mp}^{m} - g^{ki}\Gamma^{j}_{mk} e_\pm^{m} \right) ~, \\
\LB{e^i_+ }{ e^{j}_- }&=\frac{1}{\sqrt{2}}\left( g^{kj}\Gamma^i_{mk}e_+^{m}-g^{ki}\Gamma^{j}_{mk}e_-^{m}\right)~,\\
\LB{e^i_\pm }{ f(\phi) }&=\frac{1}{\sqrt{2}}g^{ij}f_{,j} ~,\\
\LB{e^i_\pm }{ S_{j} }&=\pm \frac{1}{\sqrt{2}} \chi\delta^i_{j} - g^{ik}_{,j}S_k ~.
\end{split}
\end{equation}
On an Hermitian manifold, with the Hermitian connection, the brackets simplify:
\begin{equation} \label{kahlerbrackets}
\begin{split}
\LB{ e^{\alpha}_\pm }{ e^\beta _\pm }&=0 ~,\\
\LB{ e^{\alpha}_+ }{ e^\beta _- }&=0 ~,\\
\LB{ e^{\alpha}_\pm }{ e^{\bar\beta}_\pm }&=\pm \chi g^{\alpha\bar\beta} + \frac{1}{\sqrt{2}}\left( g^{\alpha\bar\beta}_{,\bar \nu} e_\pm^{\bar{\nu}} - g^{\alpha\bar\beta}_{,\nu} e_{\mp}^{\nu} \right) ~,\\
\LB{ e^{\alpha}_+ }{ e^{\bar\beta}_- }&=\frac{1}{\sqrt{2}}\left( g^{\alpha\bar{\beta}}_{,\bar \nu}e_-^{\bar{\nu}} - g^{\alpha \bar\beta}_{,\nu} e_+^{\nu} \right)~.
\end{split}
\end{equation}
\subsection{Defining the generators $X, T$ and $J$.}
We construct the generators $X_\pm$ and $\bar X_\pm$ from the invariant volume forms:
\begin{align} \label{xandxbargen}
X_+ &\equiv \frac{1}{3!} \Omega_{\alpha \beta \gamma} e_+^\alpha e_+^\beta e_+^\gamma ~, &
X_- &\equiv \frac{-i}{3!} \Omega_{\alpha \beta \gamma} e_-^\alpha e_-^\beta e_-^\gamma ~, \\
\bar X_+ &\equiv \frac{1}{3!} \bar\Omega_{\bar\alpha \bar\beta \bar\gamma} e_+^{\bar\alpha} e_+^{\bar\beta} e_+^{\bar\gamma} ~, &
\bar X_- &\equiv \frac{-i}{3!} \bar\Omega_{\bar\alpha \bar\beta \bar\gamma} e_-^{\bar\alpha} e_-^{\bar\beta} e_-^{\bar\gamma} ~.
\end{align}
Note that these are well-defined since we have associativity in the above expressions. This is consistent with the vanishing of the connection part in \eqref{eq:E3}.
From the Kähler form, $\omega= g I$, we construct the currents $J_\pm$:
\begin{equation} \label{eq:Jdef}
J_\pm =\pm\frac{1}{2}(\omega_{ij}e_\pm^i)e_\pm^{j} + \frac{1}{2} \omega_{ij} \Gamma_{k l}^i g^{j k} \partial \phi^l ~.
\end{equation}
In the coordinates where the volume form is constant the connection-part of \eqref{eq:Jdef} vanishes. From section \ref{N=2Algebra} we know that $J_\pm$ generates an $N=2$ superconformal algebra with central charge $c=9$, \ie
\begin{equation} \label{eq:JJalgebra}
\LB{J_\pm}{J_\pm}= - T_\pm - 3 \chi \lambda ~,
\end{equation}
where $T_\pm$ generates an $N=1$ superconformal algebra.
\subsection{Calculation of the plus-sector}
From now on we are going to focus on the plus-sector. In the rest of this subsection we only write out $\pm$ subscripts where necessary.
We now want to utilize \eqref{eq:JJalgebra} to find an expression for $T_+$. Going to complex coordinates, we can rewrite $J_+$ as
\begin{equation}
J_+ =- i (g_{\alpha \bar \beta} e_+^\alpha) e_+^{\bar \beta} ~.
\end{equation}
Let us define a shorthand notation:
\begin{equation} \label{eq:definingShortHandA}
A_{\bar \beta}\equiv g_{\alpha\bar \beta}e_+^ \alpha ~,
\end{equation}
so we have
\begin{equation} \label{eq:Aepb}
A_{\bar \beta} e_+^{\bar \beta} = i J_+ ~.
\end{equation}
From the bracket
\begin{equation}
\LB{A_{\bar \beta}}{ e_+^{\bar \alpha} }= \chi \; \delta_{\bar \beta}^{\bar \alpha} -\frac{1}{\sqrt{2}} \Gamma_{\bar \beta \bar \sigma}^{\bar \alpha } e_-^{\bar \sigma}
\end{equation}
we calculate
\begin{align}
\LB{A_{\bar \beta}}{ A_{\bar \alpha} e_+^{\bar \alpha} } &=- \chi \; A_{\bar \beta} - \frac{1}{\sqrt{2}} \Gamma_{\bar \beta \bar \sigma}^{\bar \alpha } A_{\bar \alpha} e_-^{\bar \sigma} \\
\LB{ e_+^{\bar \beta} }{ A_{\bar \alpha} e_+^{\bar \alpha} } &=+ \chi \; e_+^{\bar \beta} + \frac{1}{\sqrt{2}} \Gamma_{\bar \alpha \bar \sigma}^{\bar \beta } e_-^{\bar \sigma} e_+^{\bar \alpha}
\end{align}
From this we get
\begin{equation}
\begin{split}
\LB{ A_{\bar \alpha} e_+^{\bar \alpha} }{ A_{\bar \beta} e_+^{\bar \beta} } =&+ ( \chi A_{\bar \beta} + \uD A_{\bar \beta} - \frac{1}{\sqrt{2}} \Gamma_{\bar \beta \bar \sigma}^{\bar \alpha } A_{\bar \alpha} e_-^{\bar \sigma} )e_+^{\bar \beta} \\
&- A_{\bar \beta} ( - \chi \; e_+^{\bar \beta} - \uD e_+^{\bar \beta} + \frac{1}{\sqrt{2}} \Gamma_{\bar \alpha \bar \sigma}^{\bar \beta } e_-^{\bar \sigma} e_+^{\bar \alpha} ) \\
& + 3 \chi \lambda \\
=& \uD A_{\bar \beta} e_+^{\bar \beta} + A_{\bar \beta} \uD e_+^{\bar \beta} - \sqrt{2} ( \Gamma_{\bar \beta \bar \sigma}^{\bar \alpha } A_{\bar \alpha} e_-^{\bar \sigma} )e_+^{\bar \beta} + 3 \chi \lambda \\
=& 2 \uD A_{\bar \beta} e_+^{\bar \beta} + \sqrt{2} (g_{\alpha \bar\beta ,\bar\sigma} e_-^{\bar \sigma} e_+^{\alpha} )e_+^{\bar \beta} - \uD( A_{\bar \beta} e_+^{\bar \beta}) + 3 \chi \lambda ~.
\end{split}
\end{equation}
Let us again introduce some convenient notation:
\begin{equation} \label{eq:definingShortHandB}
B_{\bar \beta} \equiv \frac{1}{\sqrt{2}} g_{\alpha\bar\beta,\bar \sigma}e_-^{\bar \sigma}e_+^\alpha + \uD A_{\bar \beta} ~.
\end{equation}
So, we have
\begin{equation}
\LB{ J_+ }{ J_+ } = -2 B_{\bar \beta} e_+^{\bar \beta} + i \uD J - 3 \chi \lambda ~.
\end{equation}
Comparing with \eqref{eq:JJalgebra} we get that
\begin{equation} \label{eq:Bepb}
B_{\bar \beta} e_+^{\bar \beta} = \frac{1}{2} ( T_+ + i \uD J_+) ~.
\end{equation}
\subsubsection{$\LB{X}{\bar X}$}
We start with the most involved bracket, namely $\LB{X}{\bar{X}}$. We try to present the calculation step by step.
\subsubsection{$\LB{ X }{ \Omega_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} }$}
First, we want to calculate the bracket between $e^{\bar\alpha}_+$ and $X$. We start with $\LB{ e^{\bar\alpha}_+ }{ e^{[ \alpha}_+ e^{\beta ]}_+} $.
The field $e^{\alpha}_+ $ anti-commute with $\LB{ e^{\bar\alpha}_+ }{ e^\beta _+}$:
\begin{equation}
e^{\alpha}_+ \LB{ e^{\bar\alpha}_+ }{ e^\beta _+ } = - \LB{ e^{\bar\alpha}_+ }{ e^\beta _+} e^{\alpha}_+
\end{equation}
Also, the integral term in the Quasi-Leibniz formula \eqref{eq:LambdaBrackeRulesQuasiAssociativity} is zero. So, we have
\begin{align} \label{bracketepb2epep}
\LB{ e^{\bar\alpha}_+ }{ e^{[ \alpha}_+ e^{\beta ]}_+ }&= 2 \LB{ e^{\bar\alpha}_+ }{ e^{[\alpha}_+} \; e^{\beta ]}_+ ~,\\
\intertext{and, in general, }
\LB{ e^{\bar\alpha}_+ }{ e^{[ \alpha_1}_+ \ldots e^{\alpha_p]}_+ }&= p \LB{e^{\bar\alpha}_{+} }{ e^{[\alpha_1}_+ } \ldots e^{\alpha_p ] }_+ ~.
\end{align}
So
\begin{equation}
\LB{ e^{\bar\alpha}_+ }{ X } = \frac{1}{2} \chi \Omega_{\alpha\beta\gamma} g^{\alpha \bar\alpha} e^{\beta }_+e^{\gamma }_+ + \frac{1}{2} \Omega_{\alpha\beta\gamma} \frac{ \left( g^{\alpha \bar\alpha}_{, \nu} e_+^{\nu} - g^{\alpha \bar\alpha}_{,\bar\nu} e_-^{\bar\nu} \right) }{\sqrt{2}} e^{\beta }_+e^{\gamma }_+ ~.
\label{bracketepbX}
\end{equation}
No integral terms or ordering problems occurred so far.
From \eqref{bracketepbX} we find that
\begin{equation}
\LB{\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma} e^{\bar\alpha}_+ }{X} = \frac{1}{2}\Omega_{\alpha\beta\gamma}\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma} \left( \chi g^{\alpha \bar\alpha} + \frac{1}{\sqrt{2}}\left( g^{\alpha \bar\alpha}_{, \nu} e_+^{\nu} - g^{\alpha \bar \alpha}_{,\bar\nu} e_-^{\bar\nu} \right) \right) e^{\beta }_+e^{\gamma }_+ ~.
\end{equation}
Next, we rewrite the $\Omega \bar\Omega$-terms using
\begin{equation}
\Omega_{\alpha\beta\gamma}g^{\alpha\bar\alpha}\bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}=g_{\beta\bar\beta}g_{\gamma\bar\gamma}-g_{\beta\bar\gamma}g_{\gamma\bar\beta} ~.
\end{equation}
The term
\begin{equation}
(\Omega_{\alpha\beta\gamma}g^{\alpha \bar\alpha}_{, \nu} \bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma} ) e_+^{\nu} e^{\beta }_+e^{\gamma }_+ = \partial_\nu (g_{\beta\bar\beta}g_{\gamma\bar\gamma}-g_{\beta\bar\gamma}g_{\gamma\bar\beta} ) e_+^{\nu} e^{\beta }_+e^{\gamma }_+
\end{equation}
is zero because of the symmetries of the K\"ahler metric. We find
\begin{equation}
\LB{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_{+}^{\bar\alpha} }{ X }= \chi g_{\beta\bar\beta}g_{\gamma\bar\gamma} \; e_+^\beta e_+^\gamma - \frac{1}{ \sqrt{2}} e^{\bar\nu}_- \partial_{\bar\nu} (g_{\beta\bar\beta}g_{\gamma\bar\gamma}) \; e_+^\beta e_+^\gamma ~,
\end{equation}
so
\begin{equation} \label{eq:XOmegaepb}
\LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} }=(\chi+D) g_{\beta\bar\beta}g_{\gamma\bar\gamma} \; e_+^\beta e_+^\gamma +\frac{1}{\sqrt{2}} e^{\bar\nu}_- \partial_{\bar\nu}(g_{\beta\bar\beta}g_{\gamma\bar\gamma}) \; e_+^\beta e_+^\gamma ~.
\end{equation}
Using the shorthand notation \eqref{eq:definingShortHandA} and \eqref{eq:definingShortHandB}, this can be written as
\begin{equation} \label{eq:XOmegaepbAB}
\LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} }= \chi A_{\bar\beta} A_{\bar\gamma} + B_{[\bar\beta} A_{\bar\gamma]} ~,
\end{equation}
where the anti-symmetrization is without any factor and we used that
\begin{equation}
\LB{B_{\bar\gamma} }{ A_{\bar\beta}}_{\chi\lambda^\bullet} = 0 ~.
\end{equation}
The notation $\LB{~}{~}_{\chi\lambda^\bullet}$ stands for the $\chi\lambda^n$-part of the bracket $\LB{~}{~}$, $n\geq 0$.
\subsubsection{$\LB{ X }{ \bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha}e_+^{\bar\beta} }$}
We now want to calculate
\begin{equation}
\begin{split}
\LB{ X }{ \bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha}e_+^{\bar\beta} } =&
\LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } e_+^{\bar\beta} - e_+^{\bar\beta} \LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } \\
&+ \int^\Lambda_0 \LB[\Gamma]{ \LB{ X }{ \bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } }{ e_+^{\bar\beta} } \ud \Gamma \\
=& 2 \LB{ X }{ \bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } e_+^{\bar\beta} \\
&- \int^{0}_{-\Delta} \LB[\Gamma]{ e_+^{\bar\beta} }{ \LB{ X }{\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } } \ud \Gamma \\
& + \int^\Lambda_0 \LB[\Gamma]{ \LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } }{ e_+^{\bar\beta} } \ud \Gamma ~. \\
=& 2 \LB{ X }{ \bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } e_+^{\bar\beta} \\
& + \int^{\Lambda-\Delta}_0 \!\!\! \LB[\Gamma]{ \LB{ X }{\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } }{ e_+^{\bar\beta} } \ud \Gamma ~,
\end{split}
\end{equation}
where we have used that $ \LB[\Gamma]{ e_+^{\bar\beta} }{ \LB{ X }{\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } }$ has no $ \eta\gamma^n$-terms, with $n> 0$.
Using \eqref{eq:XOmegaepbAB}, and
the brackets
\begin{align} \label{eq:bracketsABepb}
\LB{A_{\bar\gamma} }{ e_+^{\bar\beta} }_{\chi\lambda^\bullet} &= \delta^{\bar\beta}_{\bar\gamma} ~, & \LB{B_{\bar\gamma} }{ e_+^{\bar\beta} }_{\chi\lambda^\bullet} &= 0 ~
\end{align}
the integral term is calculated to
\begin{equation}
\chi (\lambda - \partial)(-2 A_{\bar\gamma}) + (\lambda - \partial)(-2 B_{\bar\gamma})~,
\end{equation}
and we get
\begin{equation} \label{eq:XOmegaepbepbFinal}
\begin{split}
\LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha}e_+^{\bar\beta} } =& 2 \chi (A_{\bar\beta} A_{\bar\gamma} ) e_+^{\bar\beta} +2 ( B_{[\bar\beta} A_{\bar\gamma]}) e_+^{\bar\beta} \\
& +2 \partial B_{\bar\gamma} -2 \lambda B_{\bar\gamma}\\
& +2 \chi \partial A_{\bar\gamma} - 2 \chi \lambda A_{\bar\gamma} ~.
\end{split}
\end{equation}
\subsubsection{Calculation of $\LB{X}{\bar X}$}
We are now ready to compute $[X_{\Lambda}\bar{X}]$. We choose to do the calculation as follows:
\begin{multline} \label{eq:XXbfirst}
\LB{X}{\bar X} = \frac{1}{3!}( \LB{ X }{\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha}e_+^{\bar\beta} } e_+^{\bar\gamma} + (e_+^{\bar\beta}e_+^{\bar\gamma} ) \LB{ X }{ \bar{\Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha} } \\
+ \int_0^\Gamma \LB[\Gamma]{ \LB{ X }{\bar{ \Omega}_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha}e_+^{\bar\beta} } }{ e_+^{\bar\gamma} } \ud \Gamma ~).
\end{multline}
The two first terms are given by \eqref{eq:XOmegaepbAB} and \eqref{eq:XOmegaepbepbFinal}. The brackets relevant for the integral term are
\begin{align}
\LB{ (A_{\bar\beta} A_{\bar\gamma} ) e_+^{\bar\beta} }{ e_+^{\bar\gamma} } &= -2 \chi A_{\bar\beta} e_+^{\bar\beta} + 6 \chi \lambda + \ldots ~,\\
\LB{ ( B_{[\bar\beta} A_{\bar\gamma]}) e_+^{\bar\beta} }{ e_+^{\bar\gamma} } &= -2 \chi B_{\bar\beta} e_+^{\bar\beta} + \ldots~, \\
\LB{ \partial A_{\bar\gamma} }{ e_+^{\bar\gamma} } &= - 3 \chi \lambda + \ldots ~,\\
\LB{ A_{\bar\gamma} }{ e_+^{\bar\gamma} } &= + 3 \chi + \ldots ~.
\end{align}
The integral term in \eqref{eq:XXbfirst} therefore is
\begin{equation}
\begin{split}
\int_0^\Gamma \LB[\Gamma]{ \LB{ X }{ \Omega_{\bar\alpha\bar\beta\bar\gamma}e_+^{\bar\alpha}e_+^{\bar\beta} } }{ e_+^{\bar\gamma} } \ud \Gamma =&+ \chi ( -4 \lambda A_{\bar\beta} e_+^{\bar\beta} + 12 \frac{\lambda^2}{2} ) \\
&+ ( -4 \lambda B_{\bar\beta} e_+^{\bar\beta}) \\
&+ \chi ( - 6 \frac{\lambda^2}{2}) \\
&- \chi \lambda (+ 6 \lambda) \\
=&-4 \lambda B_{\bar\beta} e_+^{\bar\beta} -4 \chi \lambda A_{\bar\beta} e_+^{\bar\beta} -3 \chi \lambda^2 ~.
\end{split}
\end{equation}
Using this, \eqref{eq:XOmegaepbAB} and \eqref{eq:XOmegaepbepbFinal}, we can write \eqref{eq:XXbfirst} as
\begin{equation} \label{eq:XXbarTot}
\begin{split}
\LB{X}{\bar X} &= \frac{1}{3} \chi ((A_{\bar\beta} A_{\bar\alpha} ) e_+^{\bar\beta} )e_+^{\bar\alpha} + \frac{1}{3} (( B_{[\bar\beta} A_{\bar\alpha]}) e_+^{\bar\beta})e_+^{\bar\alpha} \\
& + \frac{1}{3} (\partial B_{\bar\alpha} )e_+^{\bar\alpha} - \frac{1}{3} \lambda B_{\bar\alpha} e_+^{\bar\alpha} \\
& + \frac{1}{3} \chi (\partial A_{\bar\alpha})e_+^{\bar\alpha} - \frac{1}{3} \chi \lambda A_{\bar\alpha} e_+^{\bar\alpha} \\
&+ \frac{1}{6} \chi (e_+^{\bar\beta}e_+^{\bar\alpha} ) ( A_{\bar\beta} A_{\bar\alpha} ) +\frac{1}{6} (e_+^{\bar\beta}e_+^{\bar\alpha} )( B_{[\bar\beta} A_{\bar\alpha]} ) \\
&-\frac{2}{3} \lambda B_{\bar\beta} e_+^{\bar\beta} -\frac{2}{3} \chi \lambda A_{\bar\beta} e_+^{\bar\beta} -\frac{1}{2} \chi \lambda^2 ~.
\end{split}
\end{equation}
We now go through the different terms occurring in this expression.
\subsubsection{$\LB{X}{\bar X}_1$}
The constant part of $\LB{X}{\bar X}$ is
\begin{equation}
\begin{split}
\LB{X}{\bar X}_1 &= \frac{1}{3} (e_+^{\bar \beta}e_+^{\bar \alpha}) \left( A_{\bar\alpha} B_{\bar\beta} \right) \\
&+\frac{1}{3} \left( ( B_{\bar\alpha} A_{\bar \beta}) e_+^{\bar \alpha} - ( B_{\bar\beta} A_{\bar\alpha}) e_+^{\bar \alpha} \right) e_+^{\bar \beta}\\
&+\frac{1}{3} \partial ( B_{\bar\beta} ) e_+^{\bar \beta} ~,
\end{split}\label{XXbarConstpartEA}
\end{equation}
where we have used \eqref{eq:bracketsABepb}. Using again \eqref{eq:bracketsABepb}, the first line of \eqref{XXbarConstpartEA} can be written as
\begin{equation}
\frac{1}{3} (e_+^{\bar \beta}e_+^{\bar \alpha}) \left( A_{\bar\alpha} B_{\bar\beta} \right) = -\frac{1}{3} \left( B_{\bar\beta} A_{\bar\alpha} \right) (e_+^{\bar \alpha}e_+^{\bar \beta}) + \frac{2}{3} \partial \left( e_+^{\bar \beta} B_{\bar\beta} \right) ~.
\label{XXbarConstpartEAL1}
\end{equation}
We can use quasi-associativity and move the parenthesis in the second line of \eqref{XXbarConstpartEA}. Once this is done, we can use anti-symmetry in $\bar\alpha$ and $\bar \beta$, and we get:
\begin{equation}
-\frac{1}{3} \left( ( B_{\bar\beta} A_{\bar\alpha} - B_{\bar\alpha} A_{\bar \beta}) e_+^{\bar \alpha} \right) e_+^{\bar \beta} = -\frac{2}{3} \left( B_{\bar\beta} A_{\bar\alpha} \right) (e_+^{\bar \alpha}e_+^{\bar \beta}) -\frac{2}{3} \partial ( e_+^{\bar \beta} ) B_{\bar\beta} ~.
\label{XXbarConstpartEAL2}
\end{equation}
With these rewritings, and noticing that $ \partial ( B_{\bar\beta} ) e_+^{\bar \beta} = e_+^{\bar \beta} \partial ( B_{\bar\beta} )$, eq.\ \eqref{XXbarConstpartEA} is
\begin{equation}
\begin{split}
\LB{X}{\bar X}_{1}=& - \left( B_{\bar\beta} A_{\bar\alpha} \right) (e_+^{\bar \alpha}e_+^{\bar \beta}) \\
&+ \frac{2}{3} \partial \left( e_+^{\bar \beta} B_{\bar\beta} \right) -\frac{2}{3} \partial ( e_+^{\bar \beta} ) B_{\bar\beta} +\frac{1}{3} e_+^{\bar \beta} \partial ( B_{\bar\beta} ) ~.
\end{split}
\label{XXbarConstpartEASimp}
\end{equation}
We now want to change $(B_{\bar\beta} A_{\bar\alpha} )(e_+^{\bar \alpha}e_+^{\bar \beta})$ to $(B_{\bar\beta}e_+^{\bar\beta})(A_{\bar \alpha}e_+^{\bar \alpha})$.
Going through the various steps needed to change the order of the fields, we find that
\begin{equation}
( B_{\bar\beta} A_{\bar\alpha}) (e_+^{\bar \alpha}e_+^{\bar \beta}) = (B_{\bar\beta}e_+^{\bar\beta})(A_{\bar \alpha}e_+^{\bar \alpha}) +e_+^{\bar \alpha} \partial(B_{\bar\alpha}) ~.
\label{XXbarConstpartEAL3}
\end{equation}
So, finally,
\begin{equation}
\begin{split}
\LB{X}{\bar X}_{1} =& - \left( B_{\bar\beta} e_+^{\bar \beta} \right) \left( A_{\bar\alpha} e_+^{\bar \alpha}\right) \\
&+ \frac{2}{3} \partial \left( e_+^{\bar \beta} B_{\bar\beta} \right) -\frac{2}{3} \partial ( e_+^{\bar \beta} ) B_{\bar\beta} +\frac{1}{3} e_+^{\bar \beta} \partial B_{\bar\beta} - e_+^{\bar \alpha} \partial B_{\bar\alpha} \\
=& - \left( B_{\bar\beta} e_+^{\bar \beta} \right) \left( A_{\bar\alpha} e_+^{\bar \alpha}\right) \\
=& -\frac{1}{2} ( T + i \uD J) i J ~,
\end{split}
\label{XXbarConstpartEAFinal}
\end{equation}
where we have used \eqref{eq:Aepb} and \eqref{eq:Bepb} to write the result in terms of the generators.
\subsubsection{$\LB{X}{\bar X}_\chi$}
We read of the $\chi$-terms from \eqref{eq:XXbarTot}:
\begin{equation}
\LB{X}{\bar X}_\chi = \frac{1}{3} ((A_{\bar\beta} A_{\bar\alpha} ) e_+^{\bar\beta} )e_+^{\bar\alpha} + \frac{1}{3} (\partial A_{\bar\alpha})e_+^{\bar\alpha} + \frac{1}{6} (e_+^{\bar\beta}e_+^{\bar\alpha} ) ( A_{\bar\beta} A_{\bar\alpha} ) ~.
\end{equation}
We need to do similar rearranging of these terms as we did for the constant part.
We calculate
\begin{align}
(e_+^{\bar\beta}e_+^{\bar\alpha} ) ( A_{\bar\beta} A_{\bar\alpha} ) &= - ( A_{\bar\beta} A_{\bar\alpha} ) (e_+^{\bar\alpha}e_+^{\bar \beta} ) + 4 \partial( A_{\bar\alpha} e_+^{\bar\alpha}) ~, \\
((A_{\bar\beta} A_{\bar\alpha} ) e_+^{\bar\beta} )e_+^{\bar\alpha} &= - ( A_{\bar\beta} A_{\bar\alpha} ) (e_+^{\bar\alpha}e_+^{\bar \beta} ) - 2 A_{\bar\alpha} \partial e_+^{\bar\alpha} ~, \\
( A_{\bar\beta} A_{\bar\alpha} ) (e_+^{\bar\alpha}e_+^{\bar \beta} ) &= ( A_{\bar\beta} e_+^{\bar \beta})( A_{\bar\alpha} e_+^{\bar\alpha}) + 3 \partial A_{\bar\alpha} e_+^{\bar\alpha} + A_{\bar\alpha} \partial e_+^{\bar\alpha} ~.
\end{align}
We then get
\begin{equation}
\LB{X}{\bar X}_\chi = - \frac{1}{2} ( A_{\bar\beta} e_+^{\bar \beta})( A_{\bar\alpha} e_+^{\bar\alpha}) -\frac{1}{2} \partial( A_{\bar\alpha} e_+^{\bar\alpha} ) = +\frac{1}{2} J J - \frac{1}{2} i \partial J ~.
\end{equation}
\subsubsection{$\LB{X}{\bar X}_{\lambda}$}
\begin{equation}
\LB{X}{\bar X}_\lambda = -B_{\bar\alpha} e_+^{\bar \alpha} = -\frac{1}{2} (T + i \uD J)~.
\end{equation}
\subsubsection{$\LB{X}{\bar X}_{\chi\lambda}$}
\begin{equation}
\LB{X}{\bar X}_{\chi \lambda} = -A_{\bar\alpha} e_+^{\bar \alpha} = - i J ~.
\end{equation}
\subsubsection{$\LB{X}{\bar X}_{\lambda^{2}}$}
For dimensional reasons, we do not want any contributions to this term. And, indeed, $\LB{X}{\bar X}_{\lambda^{2}} = 0$.
\subsubsection{$\LB{X}{\bar X}_{\chi\lambda^{2}}$}
This term, we can simply read from \eqref{eq:XXbarTot}:
\begin{equation}
\LB{X}{\bar X}_{\chi\lambda^{2}} = -\frac{1}{2} ~.
\end{equation}
\subsubsection{Collecting terms}
Collecting together the different terms we end up with
\begin{multline}
\LB{X}{\bar X} = - \frac{1}{2} \bigl( i T J - \uD J J - \chi J J + i\chi \partial J \\
+ \lambda T + i \lambda \uD J + 2 i \chi \lambda J + \chi\lambda^{2} \bigr )
\end{multline}
\subsubsection{\texorpdfstring{$\LB{X}{X}$}{[X,X]} and \texorpdfstring{$\LB{\bar X}{\bar X}$}{[X-bar,X-bar]} }
It follows immediatly from the basic brackets that $\LB{X}{X}=0$ since there are only holomorphic fields, $e_+^\alpha$, present. In the same way, $\LB{\bar X}{\bar X}=0$.
\subsubsection{\texorpdfstring{$\LB{J}{X}$}{[J,X]}}
As before, we work in coordinates where the volume form is constant. We compute
\begin{equation}
\begin{split}
\LB{ e^{\gamma}_+ }{ J } &= \LB{ e^{\gamma}_+ }{ \omega_{\alpha\bar{\beta}}e_+^{\alpha}e_+^{\bar{\beta}} } \\
&=\left( \frac{ g^{\gamma\bar\gamma}\omega_{\alpha\bar{\beta},\bar\gamma}e_+^\alpha }{\sqrt{2}} \right) e_+^{\bar\beta} + \omega_{\alpha\bar{\beta}} e_+^{\alpha} \left( \chi g^{\gamma\bar{\beta}}+
\frac{ g^{\gamma\bar\beta}_{,\bar \nu} e_+^{\bar{\nu}} - g^{\gamma\bar\beta}_{,\nu} e_{-}^{\nu} }{\sqrt{2}} \right) \\
&=i\chi e^{\gamma}_+ - i \frac{1}{\sqrt{2}}\Gamma^{\gamma}_{\alpha\beta}e^{\alpha}_+e^{\beta}_-.\\
\end{split}
\end{equation}
The integral term (not written out) is a derivative on the volume form and therefore zero.
Since the above bracket only has terms with $e^{\alpha}_\pm$, we do not have to worry about integral terms when computing the full bracket (remember that $\LB{e^{\alpha}_\pm}{e^{\beta}_\pm}=\LB{e^{\alpha}_+}{e^{\beta}_-}=0$). We get
\begin{equation}\label{eq:LBJX}
\LB{ J }{ X} =-i\left(3\chi+D\right)X+\frac{3i}{\sqrt{2}}\Gamma^{\alpha}_{\mu\nu}\Omega_{\alpha\beta\gamma}e^{\mu}_+e^{\nu}_-e_+^\beta e_+^\gamma =-i\left(3\chi+D\right)X.
\end{equation}
In the last step we have used that the term with the connection can be rewritten as a covariant derivative on $\Omega_{\alpha\beta\gamma}$, which is zero.
\subsubsection{\texorpdfstring{$\LB{J}{\bar X}$}{[J, X-bar]}}
The computation is similar as for $\LB{J}{X}$, and, up to a sign, the answer is the same:
\begin{equation}
\LB{ J }{ \bar X }=+i\left(3\chi+D\right)\bar{X}
\end{equation}
\subsubsection{\texorpdfstring{$\LB{ T }{ X }$}{[T,X]}}
We now want to calculate $\LB{ T }{ X }$.
In a direct approach, this bracket is harder then the previous ones to calculate. This is mainly because $T$ contains terms that are not linear in $e_+^{\bar \alpha}$. Also, terms with $\uD e_+^{\bar \alpha}$ makes the calculations more cumbersome. We therefore need to take a small detour to do the calculation, using the Jacobi identity.
We know that
\begin{equation}
\LB{J}{J} = - T - 3 \chi \lambda ~.
\end{equation}
We then have, using \eqref{eq:LBJX}:
\begin{equation}
\begin{split}
\LB{ X }{T }&=- \LB{X}{\LB[\Gamma]{J}{J}} = - \LB[\Gamma + \Lambda]{\LB{X}{J}}{J} -\LB[\Gamma]{J}{\LB{X}{J}} \\
&= - i \LB[\Gamma + \Lambda]{(3 \chi + 2 \uD)X}{J} - i \LB[\Gamma]{J}{(3\chi + 2\uD) X} \\
&= -i(-3\chi +2 (\chi +\eta) )\LB[\Gamma + \Lambda]{X}{J} -i(-3 \chi -2(\uD + \eta))\LB[\Gamma]{J}{X} \\
&= (3 \lambda + \chi \uD + 2 \partial)X ~.
\end{split}
\end{equation}
Since the bracket between $X$ and $J$ enters twice in the above calculation the different sign of $\LB{X}{J}$ and $\LB{\bar X}{J}$ does not matter, and we have
\begin{equation}
\LB{ \bar X }{T }= (3 \lambda + \chi \uD + 2 \partial)\bar{X} ~.
\end{equation}
To summarize, $(X_+,\bar{X}_+,J_+, T_+)$ generates the Odake algebra.
\subsection{Commuting sectors}
In \cite{heluani-2008} it is shown that $(J_+, T_+)$ and $(J_-, T_-)$ is two commuting copies of an $N=2$ superconformal algebra. Above, the algebra of $(X_+,\bar{X}_+,J_+, T_+)$ is computed. We now show that these generators commute with $(X_-,\bar{X}_-,J_-, T_-)$.
The bracket between $X_+$ and $X_-$ is zero since only holomorphic fields enter. Also $\LB{\bar X_+}{\bar X_-}=0$. For $X_+$ with $\bar{X}_-$ we compute
\begin{equation}
\LB{ \Omega_{\alpha\beta\gamma} e^\alpha_+ }{ \Omega_{\bar\alpha\bar\beta\bar\gamma}e_-^{\bar\alpha}e_-^{\bar\beta}e_-^{\bar\gamma} } = - \frac{3}{\sqrt{2}} \Omega_{\alpha\beta\gamma} g^{\alpha \bar\alpha}_{, \delta}\Omega_{\bar\alpha\bar\beta\bar\gamma} e^{\delta}_+e_-^{\bar\beta}e_-^{\bar\gamma}
\end{equation}
so
\begin{equation}
\begin{split}
\LB{ \bar X_- }{X_+} & \propto \Omega_{\alpha\beta\gamma} g^{\alpha \bar\alpha}_{, \delta} \Omega_{\bar\alpha\bar\beta\bar\gamma} e^{\delta}_+e_-^{\bar\beta}e_-^{\bar\gamma}e^\beta _+e^\gamma _+ \\
&= (g_{\beta \bar\beta} g_{\gamma\bar\gamma } - g_{\beta \bar\gamma } g_{\gamma \bar\beta})_{, \delta} e^{\delta}_+e^\beta _+e^\gamma _+ e_-^{\bar\beta}e_-^{\bar\gamma} \\
&=0 ~,
\end{split}
\end{equation}
due to the symmetries of the Kähler metric. In the same way $\LB{ X_- }{\bar X_+} =0$
The bracket between $X_+$ and $J_-$ are zero because
\begin{equation}
\LB{ J_- }{ e^\alpha_+} \propto \Gamma^{\alpha}_{\mu\nu}e^{\mu}_+e^{\nu}_- ~,
\end{equation}
so
\begin{equation}
\LB{ J_- }{ X_+ } \propto \Omega_{\alpha \beta \gamma} \Gamma^{\alpha}_{\mu\nu} e^{\mu}_+e^{\nu}_- e^{\beta}_+ e^{\gamma}_+ = 0
\end{equation}
since $\Omega$ is covariantly constant. The same argument goes for $\LB{J_- }{\bar X_+}$, $\LB{J_+ }{X_-}$, and $\LB{J_+ }{\bar X_-}$ . Using this, and \eqref{eq:JJalgebra} together with Jacobi identity, it follows that $\LB{X_+}{T_-}=\LB{\bar X_+}{T_-}=\LB{X_-}{T_+}=\LB{\bar X_-}{T_+}=0$.
Thus, $(X_+,\bar{X}_+,J_+, T_+)$ commutes with $(X_-,\bar{X}_-,J_-, T_-)$.
\subsection{Comments about the computation of the minus-sector}
As mentioned above, the calculation of the minus-sector is completely analagous to the computation of the plus-sector. However, due to the slight asymmetry between the two sectors in the defining brackets, \eqref{kahlerbrackets}, and the definition of the generators, \eqref{xandxbargen}, there will be some slight sign differences in the various steps. In the end, everything comes together to give us the same algebra in the minus-sector as well. Thus, $(X_-,\bar{X}_-,J_-, T_-)$ also gives us the Odake algebra.
\bibliographystyle{utphys}
\bibliography{algebraextensions}
\end{document}
| 167,387
|
Tedavi özeti
Hasta Bilgileri
- Age: 29
- Gender: male
- Invisalign Treatment Option: Invisalign Go
Total Treatment Time
- 3 months
Hizalayıcı sayısı
- Maxillary: 16
- Mandibulary: 16
Aligner Wear time
- 1 hafta
Alıkoyma
- Maxillary: Other clear Protocol: 20-22 hrs for 3 months, continue to wear during night time for 6-12 months
- Mandibular: Other clear
Results achieved
- Arklar hizalandı ve koordine edildi
- Functional overjet and overbite
- Midlines aligned
Yorumlar
- The patient consumed food with aligners to ensure that crossbite can be corrected. The case finished faster than expected with a fantastic outcome.
ClinCheck tedavi planı
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| 164,874
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{\bf Problem.} Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \sqrt{33},$ and
\[\overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} = 2.\]Find the cosine of the angle between vectors $\overrightarrow{EF}$ and $\overrightarrow{BC}.$
{\bf Level.} Level 5
{\bf Type.} Precalculus
{\bf Solution.} We can write
\begin{align*}
2 &= \overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} \\
&= \overrightarrow{AB} \cdot (\overrightarrow{AB} + \overrightarrow{BE}) + \overrightarrow{AC} \cdot (\overrightarrow{AB} + \overrightarrow{BF}) \\
&= \overrightarrow{AB} \cdot \overrightarrow{AB} + \overrightarrow{AB} \cdot \overrightarrow{BE} + \overrightarrow{AC} \cdot \overrightarrow{AB} + \overrightarrow{AC} \cdot \overrightarrow{BF}.
\end{align*}Since $AB = 1,$
\[\overrightarrow{AB} \cdot \overrightarrow{AB} = \|\overrightarrow{AB}\|^2 = 1.\]By the Law of Cosines,
\begin{align*}
\overrightarrow{AC} \cdot \overrightarrow{AB} &= AC \cdot AB \cdot \cos \angle BAC \\
&= \sqrt{33} \cdot 1 \cdot \frac{1^2 + (\sqrt{33})^2 - 6^2}{2 \cdot 1 \cdot \sqrt{33}} \\
&= -1.
\end{align*}Let $\theta$ be the angle between vectors $\overrightarrow{EF}$ and $\overrightarrow{BC}.$ Since $B$ is the midpoint of $\overline{EF},$ $\overrightarrow{BE} = -\overrightarrow{BF},$ so
\begin{align*}
\overrightarrow{AB} \cdot \overrightarrow{BE} + \overrightarrow{AC} \cdot \overrightarrow{BF} &= -\overrightarrow{AB} \cdot \overrightarrow{BF} + \overrightarrow{AC} \cdot \overrightarrow{BF} \\
&= (\overrightarrow{AC} - \overrightarrow{AB}) \cdot \overrightarrow{BF} \\
&= \overrightarrow{BC} \cdot \overrightarrow{BF} \\
&= BC \cdot BF \cdot \cos \theta \\
&= 3 \cos \theta.
\end{align*}Putting everything together, we get
\[1 - 1 + 3 \cos \theta = 2,\]so $\cos \theta = \boxed{\frac{2}{3}}.$
| 55,430
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\begin{document}
\setlength\abovedisplayskip{2pt}
\setlength\abovedisplayshortskip{0pt}
\setlength\belowdisplayskip{2pt}
\setlength\belowdisplayshortskip{0pt}
\begin{center}
{\bf \LARGE A Generalization of Exponential Class and Its
Applications\footnote{Corresponding author: GAO Hongya, E-mail:
ghy@hbu.cn, TEL: 863125079658, FAX: 863125079638.}}
\vspace{3mm}
{\small \textsc{GAO Hongya}\quad
\textsc{LIU Chao} \quad \textsc{TIAN Hong}\\}
{\small College of Mathematics and Computer Science, Hebei
University, Baoding, 071002, China\\}
\end{center}
\begin{center}
\begin{minipage}{135mm}
{\bf \small Abstract}.\hskip 2mm {\small A function space,
$L^{\theta, \infty)}(\Omega)$, $0\le \theta <\infty$, is defined. It
is proved that $L^{\theta, \infty)}(\Omega)$ is a Banach space which
is a generalization of exponential class. An alternative definition
of $L^{\theta, \infty)}(\Omega)$ space is given. As an application,
we obtain weak monotonicity property for very weak solutions of
$\cal A$-harmonic equation with variable coefficients under some
suitable conditions related to $L^{\theta, \infty)}(\Omega)$, which
provides a generalization of a known result due to Moscariello. A
weighted space $L^{\theta,\infty )}_w (\Omega)$ is also defined, and
the boundedness for the Hardy-Littlewood maximal operator $M_w$ and
a Calder\'on-Zygmund operator $T$ with respect to $L^{\theta,\infty
)}_w (\Omega)$ are obtained.}
{\bf AMS Subject Classification: } 46E30, 35J70.
{\bf Keywords:} Exponential class, weak monotonicity, very weak
solution, $\cal A$-harmonic equation, Hardy-Littlewood maximal
operator, Calder\'on-Zygmund operator.
\end{minipage}
\end{center}
\thispagestyle{fancyplain} \fancyhead{}
\fancyhead[L]{\textit{}\\
} \fancyfoot{} \vskip 6mm
\section{Introduction}
For $1<p<\infty$ and a bounded open subset $\Omega\subset \mbox
{R}^n$, the grand Lebesgue space $L^{p)}(\Omega)$ consists of all
functions $f(x)\in \bigcap _{0<\varepsilon \le p-1}
L^{p-\varepsilon} (\Omega)$ such that
$$
\|f\|_{p),\Omega} =\sup_{0<\varepsilon \le p-1} \left(\varepsilon
-\hspace{-4mm} \int_\Omega |f|^{p-\varepsilon} dx \right)^{\frac 1
{p-\varepsilon}} <\infty, \eqno(1.1)
$$
where $-\hspace{-3.5mm} \int_\Omega =\frac {1}{|\Omega|}
\int_\Omega$ stands for the integral mean over $\Omega$. The grand
Sobolev space $W_0^{1,p)} (\Omega)$ consists of all functions $u\in
\bigcap _{0<\varepsilon \le p-1} W_0^{1,p-\varepsilon} (\Omega)$
such that
$$
\|u\|_{W_0^{1,p)}} =\sup _{0<\varepsilon \le p-1} \left(\varepsilon
-\hspace {-4mm} \int_\Omega |\nabla f|^{p-\varepsilon} dx
\right)^{\frac 1 {p-\varepsilon}} <\infty. \eqno(1.2)
$$
These two spaces, slightly larger than $L^p(\Omega)$ and
$W_0^{1,p}(\Omega)$, respectively, were introduced in the paper [1]
by Iwaniec and Sbordone in 1992 where they studied the integrability
of the Jacobian under minimal hypotheses. For $p=n$ in [2] imbedding
theorems of Sobolev type were proved for functions $f\in
W_0^{1,n)}(\Omega)$. The small Lebesgue space $L^{(p}(\Omega)$ was
found by Fiorenza [3] in 2000 as the associate space of the grand
Lebesgue space $L^{p)}(\Omega)$. Fiorenza and Karadzhov gave in [4]
the following equivalent, explicit expressions for the norms of the
small and grand Lebesgue spaces, which depend only on the
non-decreasing rearrangement (provided that the underlying measure
space has measure 1):
$$
\|f\|_{L^{(p} } \approx \int_0^1 (1-\ln t)^{-\frac 1 p}
\left(\int_0^t [f^*(s)]^p ds \right)^{\frac 1 p}\frac {dt} t, \ \
1<p<\infty,
$$
$$
\|f\|_{L^{p)}} \approx \sup _{0<t<1} (1-\ln t)^{-\frac 1 p}
\left(\int_t^1 [f^*(s)]^p ds \right)^{\frac 1 p } , \ \ 1<p<\infty.
$$
In [5], Greco, Iwaniec and Sbordone gave two more general
definitions than (1.1) and (1.2) in order to derive existence and
uniqueness results for $p$-harmonic operators. For $1<p<\infty$ and
$0\le \theta <\infty$, the grand $L^p$ space, denoted by $L^{\theta,
p)}(\Omega)$, consists of functions $f\in \bigcap _{0<\varepsilon
\le p-1} L^{p-\varepsilon} (\Omega)$ such that
$$
\|f\|_{\theta, p)} =\sup_{0<\varepsilon\le p-1} \varepsilon^ {\frac
\theta p} \|f\|_{p-\varepsilon} <\infty, \eqno(1.3)
$$
where
$$
\|f\|_{p-\varepsilon} =\left(-\hspace {-4mm} \int_\Omega
|f|^{p-\varepsilon} dx \right)^{\frac 1 {p-\varepsilon}}.
\eqno(1.4)
$$
The grand Sobolev space $W^{\theta, p)}(\Omega)$ consists of all
functions $f$ belonging to $\bigcap _{0<\varepsilon \le p-1}
W^{1,p-\varepsilon} (\Omega)$ and such that $\nabla f \in L^{\theta,
p)}(\Omega)$. That is,
$$
W^{\theta, p)}(\Omega)=\left\{f\in \bigcap _{0<\varepsilon \le p-1}
W^{1,p-\varepsilon} (\Omega): \nabla f \in L^{\theta, p)}(\Omega)
\right\}. \eqno(1.5)
$$
Grand and small Lebesgue spaces are important tools in dealing with
regularity properties for very weak solutions of $\cal A$-harmonic
equation as well as weakly quasiregular mappings, see [6, 7].
The aim of the present paper is to provide a generalization
$L^{\theta, \infty)}(\Omega)$, $0\le \theta <\infty$, of exponential
calss $EXP(\Omega)$, and prove that it is a Banach space. An
alternative definition of $L^{\theta, \infty)}(\Omega)$ is given in
terms of weak Lebesgue spaces. As an application, we obtain weak
monotonicity property for very weak solutions of $\cal A$-harmonic
equation with variable coefficients under some suitable conditions
related to $L^{\theta, \infty)}(\Omega)$. This paper also consider a
weighted space $L_w^{\theta, \infty)}(\Omega)$, and some boundedness
result for classical operators with respect to this space.
In the sequel, the letter $C$ is used for various constants, and may
change from one occurrence to another.
\section{A Generalization of Exponential Class}
Recall that $EXP(\Omega)$, the exponential class, consists of all
measurable functions $f$ such that
$$
\int_\Omega e^{\lambda |f|}dx <\infty
$$
for some $\lambda >0$. It is a Banach space under the norm
$$
\|f\|_{EXP} =\inf \left\{\lambda >0: \int_\Omega e^{|f|/\lambda} dx
\le 2 \right\}.
$$
In this section, we define a space $L^{\theta, \infty)}(\Omega)$,
$0\le \theta <\infty$, which is a generalization of $EXP(\Omega)$,
and prove that it is a Banach space.
\begin{definition} For $ \theta\ge 0$, the space $L^{\theta, \infty)
}(\Omega)$ is defined by
$$
L^{\theta, \infty)}(\Omega) =\left\{f (x) \in \bigcap _{1\le
p<\infty}L^p(\Omega): \sup_{1\le p<\infty} \frac 1
{p^\theta}\left(-\hspace {-4mm} \int_\Omega|f (x)|^pdx
\right)^{\frac 1 p}<\infty \right\}. \eqno(2.1)
$$
\end{definition}
It is not difficult to see that
$$
L^{\theta, \infty)}(\Omega) =\left\{g(x) \in \bigcap _{1\le
p<\infty}L^p(\Omega): {\limsup}_{p\rightarrow \infty}\frac 1
{p^\theta}\left(-\hspace {-4mm} \int_\Omega|g(x)|^pdx \right)^{\frac
1 p}<\infty \right\}. \eqno(2.1)'
$$
There are two special cases of $L^{\theta,\infty)} (\Omega)$ that
are worth mentioning since they coincide with two known spaces.
\noindent {\bf Case 1}: $\theta =0$. In this case,
$$
L^{0, \infty)}(\Omega) =\left\{f (x) \in \bigcap _{1\le
p<\infty}L^p(\Omega): \sup_{1\le p<\infty} \left(-\hspace {-4mm}
\int_\Omega|f (x)|^pdx \right)^{\frac 1 p}<\infty \right\}.
$$
From the fact (see [8, P12])
$$
L^\infty (\Omega) =\left\{f\in \bigcap _{1\le p <\infty}
L^p(\Omega): \lim_{p\rightarrow \infty} \|f\|_p<\infty \right\},
$$
we get $L^{0,\infty)}(\Omega) =L^{\infty}(\Omega)$.
\noindent {\bf Case 2}: $\theta =1$. The following proposition shows
that $L^{\theta, \infty}(\Omega)$ can be regarded as a
generalization of $EXP(\Omega)$.
\noindent {\bf Proposition 2.1 } {$L^{1,
\infty)}(\Omega)=EXP(\Omega)$}.
\begin{proof}
In order to realize that a function in the $L^{1,\infty)}(\Omega)$
space is in $EXP(\Omega)$, it is sufficient to read the last lines
of [2]. The vice-versa is also true, see e.g. [9, Chap. VI, exercise
no. 17].
\end{proof}
It is clear that for any $0\le \theta <\theta'\le \infty$ and any
$q<\infty$, we have the inclusions
$$
L^\infty (\Omega) \subset L^{\theta, \infty)} (\Omega) \subset
L^{\theta', \infty)}(\Omega)\subset L^q(\Omega). \eqno(2.2)
$$
The following theorem shows that, if $\theta>0$, then $L^{\theta,
\infty )}(\Omega)$ is slightly larger than $L^\infty (\Omega)$.
\begin{theorem} For $\theta >0$, the space $L^\infty (\Omega)$ is a proper subspace of
$L^{\theta, \infty )}(\Omega)$.
\end{theorem}
\begin{proof} In the proof of Theorem 2.1 we always assume
$\theta>0$. Let $f (x) \in L^{\infty}(\Omega)$, then there exists a
constant $M<\infty$, such that $|f(x)|\le M$, a.e. $\Omega$. Thus,
$$
\sup_{1\le p<\infty} \frac 1 {p^\theta} \left(-\hspace {-4mm}
\int_\Omega|f(x)|^pdx \right)^{\frac 1 p} \le \sup _{1\le p<\infty}
\frac M {p^\theta} =M<\infty,
$$
which implies $f(x)\in L^{\theta, \infty )}(\Omega)$.
The following example shows that $L^\infty (\Omega) \subset
L^{\theta, \infty )}(\Omega)$ is a proper subset. Since we have the
inclusion (2.2), then it is no loss of generality to assume that
$\theta \le 1$. Consider the function $f(x)=(-\ln x)^\theta$
defined in the open interval $(0,1)$. It is obvious that $f(x)
\notin L^{\infty}(0,1)$. We now show that $f(x) \in L^{\theta,
\infty)}(0,1)$. In fact, for $m$ a positive integer, integration by
parts yields
$$
\begin{array}{llll}
\displaystyle \int_0^1 (-\ln x)^m dx
&=&\displaystyle \left. x(-\ln x )^m\right|_0^1 -\int_0^1 x d(-\ln x)^m \\
&=&\displaystyle -\lim_{x\rightarrow 0^+} x(-\ln x)^m +m\int_0^1
(-\ln x)^{m-1}dx.
\end{array} \eqno(2.3)
$$
By L'Hospital's Law, one has
$$
\lim_{x\rightarrow 0^+} x(-\ln x)^m =\lim_{x\rightarrow 0^+}\frac
{(-\ln x )^m}{\frac 1 x} =\lim_{x\rightarrow 0^+} \frac {m(-\ln
x)^{m-1}}{\frac 1 x } =\cdots =m!\lim_{x\rightarrow 0^+}x=0.
$$
This equality together with (2.3) yields
$$
\int_0^1 (-\ln x)^m dx=m\int_0^1 (-\ln x)^{m-1}dx.
$$
By induction,
$$
\int_0^1 f^m(x) dx =m\int_0^1 (-\ln x)^{m-1}dx=\cdots = m!\int_0^1dx
=m!. \eqno(2.4)
$$
Recall that the function
$$
p\mapsto \left(-\hspace{-4mm} \int_\Omega |f (x)|^p dx
\right)^{\frac 1 p}
$$
is non-decreasing, thus (2.4) yields
$$
\begin{array}{llll}
&\displaystyle \sup_{1\le p<\infty}\frac 1 {p^\theta}
\left(-\hspace{-4mm}\int_0^1 |f(x)|^pdx \right)^{\frac 1 p}\\
=&\displaystyle \sup_{1\le p<\infty}\left[\frac 1 {p} \left(\int_0^1 (-\ln x)
^{p\theta}dx \right)^{\frac 1 {p\theta}}\right]^\theta\\
\le&\displaystyle \sup_{1\le p<\infty}\left[ \frac 1 {p} \left(\int_0^1
(-\ln x)^{[p\theta]+1}dx \right)^{\frac 1 {[p\theta]+1}}\right]^\theta\\
=&\displaystyle \sup _{1\le p<\infty}\left[ \frac {([p\theta]+1)!
^{\frac 1 {[p\theta]+1}}}{p}\right]^\theta \le \sup _{1\le p<\infty}
\left[\frac {[p\theta]+1}{p} \right]^\theta\le 2,
\end{array}
$$
where we have used the assumption $\theta \le 1$, and $[p\theta]$ is
the integer part of $p\theta$. The proof of Theorem 2.1 has been
completed.
\end{proof}
For functions $f_1(x), f_2(x) \in L^{\theta, \infty )}(\Omega)$ and
$\alpha \in \mbox {R}$, the addition $f_1(x)+f_2(x)$ and the
multiplication $\alpha f_1(x)$ are defined as usual.
\begin{theorem}
$L^{\theta, \infty)}(\Omega)$ is a linear space on $\mbox {R}$.
\end{theorem}
\begin{proof}
This theorem is easy to prove, we omit the details.
\end{proof}
For $f(x)\in L^{\theta,\infty )}(\Omega)$, we define
$$
\|f\|_{\theta, \infty ),\Omega}=\sup_{1\le p<\infty} \frac 1
{p^\theta} \left(-\hspace {-4mm} \int_\Omega|f(x)|^pdx
\right)^{\frac 1 p}. \eqno(2.5)
$$
We drop the subscript $\Omega$ from $\|\cdot\|_{\theta, \infty ),
\Omega}$ when there is no possibility of confusion.
\begin{theorem}
$\|\cdot\|_{\theta, \infty)}$ is a norm.
\end{theorem}
\begin{proof}
(1) It is obvious that $\|f\|_{\theta,\infty)}\ge 0$ and
$\|f\|_{\theta,\infty )}=0$ if and only if $f =0$ a.e. $\Omega$;
(2) For any $f_1(x), f_2(x)\in L^{\theta, \infty)} (\Omega)$,
Minkowski inequality in $L^p (\Omega)$ yields
$$
\begin{array}{llll}
\|f_1 +f_2\|_{\theta,\infty)} &=&\displaystyle \sup_{1\le p <\infty}
\frac 1 {p^\theta} \left(-\hspace{-4mm} \int_\Omega |f_1+f_2|^pdx \right)^{\frac 1 p}\\
&\le & \displaystyle \sup_{1\le p <\infty} \frac 1 {p^\theta} \left[
\left(-\hspace{-4mm} \int_\Omega |f_1|^pdx \right)^{\frac 1 {p}}+
\left(-\hspace{-4mm} \int_\Omega |f_2|^pdx \right)^{\frac 1 {p}}\right]\\
&\le & \displaystyle \sup_{1\le p <\infty} \frac 1 {p^\theta}
\left(-\hspace{-4mm} \int_\Omega |f_1|^pdx \right)^{\frac 1 {p}}+
\sup_{1\le p <\infty} \frac 1 {p^\theta}
\left(-\hspace{-4mm} \int_\Omega |f_2|^pdx \right)^{\frac 1 {p}}\\
&= & \displaystyle \|f_1\|_{\theta, \infty)}
+\|f_2\|_{\theta,\infty)};
\end{array}
$$
(3) For all $\lambda \in \mbox {R}$ and all $f(x)\in L^{\theta,
\infty)}(\Omega)$, it is obvious that $\|\lambda f
\|_{\theta,\infty)}=|\lambda| \|f\|_{\theta,\infty)}$.
\end{proof}
\begin{theorem}
$\left(L^{\theta, \infty)}(\Omega), \|\cdot\|_{\theta,
\infty)}\right)$ is a Banach space.
\end{theorem}
\begin{proof}
Suppose that $\{f_n\}_{n=1}^\infty \subset L^{\theta, \infty)}
(\Omega)$, and for any positive integer $p$,
$$
\|f_{n+p} -f_n\|_{\theta, \infty)} \rightarrow 0, \ \ n \rightarrow
\infty. \eqno(2.6)
$$
Since $\Omega$ is $\sigma$-finite, then $\Omega =\bigcup
_{m=1}^\infty \Omega_m$ with $|\Omega_m|<\infty$. It is no loss of
generality to assume that the $\Omega_m$s are disjoint. (2.4)
implies that for any positive integer $p$,
$$
\int_{\Omega _m} |f _{n+p}(x) -f_n (x)|dx \rightarrow 0, \ \
n\rightarrow \infty.
$$
Thus, by the completeness of $L^1(\Omega _m)$, there exists
$f^{(m)}(x)\in L^1(\Omega_m)$, such that
$$
f_n(x) \rightarrow f^{(m)} (x), \ n\rightarrow \infty, \ \mbox { in
} L^1(\Omega_m). \eqno(2.7)
$$
Hence for any positive integer $m$, there exists a subsequence
$\{f_n^{(m)} (x)\}$ of $\{f^{m-1}_n(x)\}$,
$\{f_n^{(0)}(x)\}=\{f_n(x) \}$, such that
$$
f _n^{(m)} (x) \rightarrow f^{(m)} (x), \ \ n\rightarrow \infty, \
\mbox { a.e. } x\in \Omega _m.
$$
If we let
$$
f (x) =f^{(m)} (x), \ \ x\in \Omega_m, \ \ m=1,2,\cdots ,
$$
then
$$
f _n^{(n)}(x) \rightarrow f (x), \ \ n\rightarrow \infty, \mbox {
a.e. } x\in \Omega.
$$
It is no loss of generality to assume that the subsequence $\{f_n
^{(n)}(x)\}$ of $\{f _n(x)\}$ is itself, thus
$$
f_n(x) \rightarrow f (x), \ \ n\rightarrow \infty, \ \mbox { a.e. }
x\in \Omega.
$$
We now prove $f (x) \in L^{\theta, \infty)} (\Omega)$ and $\|f_n-f
\|_{\theta, \infty)}\rightarrow 0$, $(n\rightarrow \infty)$. In
fact, by (2.6), for any $\varepsilon >0$, there exists
$N=N(\varepsilon)$, such that if $n>N$, then
$$
\sup_{1\le q<\infty} \frac 1 {q^\theta} \left(-\hspace{-4mm}
\int_\Omega |f _{n+p}(x) -f_n(x)| ^q dx \right)^{\frac 1 q}
<\varepsilon.
$$
Let $p\rightarrow \infty$, one has
$$
\sup_{1\le q<\infty} \frac 1 {q^\theta} \left(-\hspace{-4mm}
\int_\Omega |f_{n}(x) -f(x)| ^q dx \right)^{\frac 1 q} <\varepsilon,
\ \ n>N.
$$
Hence $f (x) \in L^{\theta, \infty)}(\Omega)$, and $\|f _n(x) -f
(x)\|_{\theta, \infty)} \rightarrow 0$, $n\rightarrow \infty$. This
completes the proof of Theorem 2.4.
\end{proof}
\begin{definition} The grand Sobolev space $W^{\theta, \infty)}(\Omega)$
consists of all functions $f$ belonging to $\bigcap _{1\le
p<\infty}W^{1,\infty)}(\Omega)$ and such that $\nabla f \in
L^{\theta, \infty)}(\Omega)$. That is,
$$
W^{\theta, \infty)}(\Omega) =\left\{f\in \bigcap _{1\le
p<\infty}W^{1,\infty)}(\Omega): \nabla f \in L^{\theta,
\infty)}(\Omega) \right\}.
$$
\end{definition}
This definition will be used in Section 4.
\section{An Alternative Definition of $L^{\theta, \infty}(\Omega)$}
In this section, we give an alternative definition of $L^{\theta,
\infty)}(\Omega)$ in terms of weak Lebesgue spaces. Let us first
recall the definition of weak $L^p (0<p<\infty)$ spaces, or the
Marcinkiewicz spaces, $L_{weak}^p(\Omega)$, see [10, Chapter 1,
Section 2], [11, Chapter 2, Section 5] or [12, Chapter 2, Section
18].
\begin{definition}
Let $0<p<\infty$. We say that $f\in L_{weak}^{p}(\Omega)$ if and
only if there exists a positive constant $k=k(f)$ such that
$$
f_*(t)= |\{x\in \Omega: |f(x)|>t\}| \le \frac {k}{t^p} \eqno(3.1)
$$
for every $t>0$, where $|E|$ is the $n$-dimensional Lebesgue measure
of $E\subset \mbox {R}^n$, and $f_* (t)=|\{x\in \Omega: |f(x)|>t\}|$
denotes the distribution function of $f$.
\end{definition}
For $p>1$, we recall that if $f\in L_{weak}^p(\Omega)$, then $f\in
L^q (\Omega)$ for every $1\le q <p$, and $f\in L_{weak}^p(\Omega)$
if and only if for every measurable set $E\subset \Omega$, the
following inequality holds
$$
\int_E |f(x)| d x \le c|E|^{\frac {p-1}{p}}
$$
for some constant $c>0$.
(3.1) is equivalent to
$$
M_p(f)=\left[\frac 1 {|\Omega|} \sup_{t>0} t^p f_*(t) \right]^{\frac
1 p}<\infty. \eqno(3.2)
$$
Recall also that
$$
\int_\Omega |f(x)|^s dx =s\int_0^\infty t^{s-1} f_*(t)dt <\infty.
\eqno(3.3)
$$
\begin{definition} For $\theta\ge 0$, the weak space $L_{weak}^{\theta, \infty
}(\Omega)$ is defined by
$$
L_{weak}^{\theta, \infty} (\Omega) =\left\{f\in \bigcap _{1\le p
<\infty} L_{weak} ^p (\Omega): \sup _{1\le p <\infty}\frac {M_p(f)}
{p^\theta} <\infty\right\}. \eqno(3.5)
$$
\end{definition}
The following theorem shows that $L_{weak}^{\theta,
\infty}(\Omega)=L^{\theta, \infty)}(\Omega)$, thus
$L_{weak}^{\theta, \infty} (\Omega)$ can be regarded as an
alternative definition of the space $L^{\theta, \infty)} (\Omega)$.
\begin{theorem}
$$
L_{weak}^{\theta, \infty}(\Omega)=L^{\theta, \infty) }(\Omega).
$$
\end{theorem}
\begin{proof} We divided the proof into two steps.
{\bf Step 1 } $L_{weak}^{\theta, \infty}(\Omega)\subset L^{\theta,
\infty) }(\Omega)$.
If $1\le s <p$, for each $a>0$, one can split the integral in the
right-hand side of (3.3) to obtain
$$
\begin{array}{llll}
\displaystyle \int_\Omega |f|^s dx &=&\displaystyle s\int_0^a
t^{s-1} f_*(t)dt + s\int_a^\infty t^{s-1} f_*(t)dx \\
& \le &\displaystyle |\Omega |a^s +\frac {sa^{s-p}}{p-s} |\Omega|
M_p^p(f).
\end{array}
$$
The second integral has been estimated by the inequality $f_*(t) \le
|\Omega| t^{-p} M_p^p (f)$, which is a direct consequence of the
definition of the constant $M_p(f)$ (see (3.2)). Setting $a=M_p(f)$
we arrive at
$$
-\hspace{-4mm} \int_\Omega |f|^s dx \le M_p^s(f) +\frac s {p-s}
M_p^s(f) =\frac {p}{p-s} M_p(f).
$$
This implies
$$
\frac 1 {s^\theta} \left(-\hspace {-4mm} \int_\Omega |f|^s dx
\right)^{\frac 1 s } \le \frac 1 {s^\theta} \left(\frac
{p}{p-s}\right)^{\frac 1 s } M_p(f). \eqno(3.6)
$$
Therefore
$$
\begin{array}{llll}
&\displaystyle \sup_{1\le s <\infty} \frac 1 {s^\theta}
\left(-\hspace{-4mm}
\int_\Omega |f|^s dx \right)^{\frac 1 s } \\
=&\displaystyle \max \left\{ \sup_{1\le s <2} \frac 1 {s^\theta}
\left(-\hspace{-4mm} \int_\Omega |f|^s dx \right)^{\frac 1 s },
\sup_{2\le s <\infty } \frac 1 {s^\theta} \left(-\hspace{-4mm}
\int_\Omega |f|^s dx \right)^{\frac 1 s }\right\}\\
\le &\displaystyle \max \left\{\|f\|_2, \sup_{2\le s =p-1<\infty}
\frac 1 {s^\theta} \left(-\hspace{-4mm} \int_\Omega |f|^s dx
\right)^{\frac 1
s }\right\}\\
\le &\displaystyle \max \left\{\|f\|_2, \sup_{2\le s<\infty} \frac 1
{s^\theta} (s+1)^{\frac 1 s } M_{s+1} (f)\right\}\\
\le &\displaystyle \max \left\{\|f\|_2, 4\sup_{1\le s<\infty} \frac
{M_s(f)}{{s^\theta}}\right\}<\infty,
\end{array}
$$
here we have used (3.6) and the definition of $L_{weak}^\infty
(\Omega)$.
{\bf Step 2 } $L^{\theta, \infty) }(\Omega)\subset
L_{weak}^{\infty}(\Omega)$.
Since for any $t>0$,
$$
t^p f_*(t) =t^p \int_{\{x\in \Omega: |f(x)|>t\}} dx \le \int_{\{x\in
\Omega: |f(x)|>t\} }|f|^p dx \le \int_\Omega |f|^p dx,
$$
then
$$
\sup _{t>0} t^pf_*(t) \le \int_\Omega |f|^p dx .
$$
This implies
$$
M_p(f) =\left[\frac 1 {|\Omega|} \sup_{t>0} t^pf_*(t)\right]^{\frac
1 p} \le \left(-\hspace{-4mm} \int_\Omega |f|^p dx \right)^{\frac 1
p }.
$$
Thus
$$
\sup_{1\le p <\infty} \frac {M_p(f)}{p^\theta} \le \sup_{1\le p
<\infty}\frac 1 {p^\theta}\left(-\hspace{-4mm} \int_\Omega |f|^p dx
\right)^{\frac 1 p }<\infty.
$$
The proof of Theorem 3.1 has been completed.
\end{proof}
\section{An Application}
In this section, we give an application of the space $L^{\theta,
\infty} (\Omega)$ to monotonicity property of very weak solutions of
the $\cal A$-harmonic equation
$$
\mbox {div}{\cal A}(x,\nabla u(x)) =0, \eqno(4.1)
$$
where ${\cal A}: \Omega \times \mbox {R}^n \rightarrow \mbox {R}^n$
be a mapping satisfying the following assumptions:
(1) the mapping $x\mapsto {\cal A}(x,\xi)$ is measurable for all
$\xi \in \mbox {R}^n$,
(2) the mapping $\xi \mapsto {\cal A}(x,\xi)$ is continuous for a.e. $x\in \mbox {R}^n$,\\
for all $\xi \in \mbox {R}^n$, and a.e. $x\in \mbox {R}^n$,
(3)
$$
\langle {\cal A}(x,\xi), \xi \rangle \ge \gamma (x) |\xi|^p,
$$
(4)
$$
|{\cal A}(x,\xi)|\le \tau (x) |\xi|^{p-1},
$$
where $1<p<\infty$, $0<\gamma (x) \le \tau (x)<\infty$, a.e.
$\Omega$.
Conditions (1) and (2) insure that the composed mapping $x\mapsto
{\cal A}(x,g(x))$ is measurable whenever $g$ is measurable. The
degenerate ellipticity of the equation is described by condition
(3). Finally, condition (4) guarantees that, for any $0\le \theta
<\infty$ and any $\varepsilon>0$, ${\cal A}(x,\nabla u)$ can be
integrated for $u\in W^{\theta,p}(\Omega)$ against functions in
$W^{1,\frac {p-\varepsilon}{1-p\varepsilon}}(\Omega) $ with compact
support.
\begin{definition} A function
$u\in W_{loc}^{1,r} (\Omega)$, $\max \{1, p-1\} <r\le p$, is called
a very weak solution of (4.1), if
$$
\int_\Omega \langle {\cal A}(x,\nabla u(x)), \nabla \varphi (x)
\rangle dx =0
$$
for all $\varphi \in W_0^{1,\frac {r}{r-p+1}}(\Omega)$.
\end{definition}
A fruitful idea in dealing with the continuity properties of Sobolev
functions is the notion of monotonicity. In one dimension a function
$u:\Omega \rightarrow \mbox {R}$ is monotone if it satisfies both a
maximum and minimum principle on every subinterval. Equivalently, we
have the oscillation bounds $\mbox {osc}_I u \le \mbox
{osc}_{\partial I} u$ for every interval $I\subset \Omega$. The
definition of monotonicity in higher dimensions closely follows this
observation.
A continuous function $u:\Omega \rightarrow \mbox {R}^n$ defined in
a domain $\Omega \subset \mbox {R}^n$ is monotone if
$$
\mbox {osc}_B u \le \mbox {osc}_{\partial B} u
$$
for every ball $B\subset \mbox {R}^n$. This definition in fact goes
back to Lebesgue [13] in 1907 where he first showed the relevance of
the notion of monotonicity in the study of elliptic PDEs in the
plane. In order to handle very weak solutions of $\cal A$-harmonic
equation, we need to extend this concept, dropping the assumption of
continuity. The following definition can be found in [14], see also
[6, 7].
\begin{definition}
A real-valued function $u\in W_{loc}^{1,1}(\Omega)$ is said to be
weakly monotone if, for every ball $B\subset \Omega$ and all
constants $m\le M$ such that
$$
|M-u| -|u-m| +2u -m-M \in W_0^{1,1}(B), \eqno(4.2)
$$
we have
$$
m\le u(x) \le M \eqno(4.3)
$$
for almost every $x\in B$.
\end{definition}
For continuous functions (4.2) holds if and only if $m\le u(x)\le M$
on $\partial B$. Then (4.3) says we want the same condition in $B$,
that is the maximum and minimum principles.
Manfredi's paper [14] should be mentioned as the beginning of the
systematic study of weakly monotone functions. Koskela, Manfredi and
Villamor obtained in [15] that $\cal A$-harmonic functions are
weakly monotone. In [16], the first author obtained a result which
states that very weak solutions $u\in W_{loc}
^{1,p-\varepsilon}(\Omega)$ of the $\cal A$-harmonic equation are
weakly monotone provided $\varepsilon $ is small enough. The
objective of this section is to extend the operator $\cal A$ to
spaces slightly larger than $L^p(\Omega)$.
\begin{theorem}
Let $\gamma (x)>0$, a.e. $\Omega$, $\tau (x)\in L^{\theta_1,
\infty)} (\Omega)$. If $u\in W^{\theta_2,p)}(\Omega)$ is a very weak
solution to (4.1), then it is weakly monotone in $\Omega$ provided
that $\theta_1+\theta_2<1$.
\end{theorem}
\begin{proof}
For any ball $B\subset \Omega$ and $0<\varepsilon <1$, let
$$
\psi =(u-M)^+ -(m-u)^+ \in W_0^{1,p-\varepsilon} (B).
$$
It is obvious that
$$
\nabla \psi =\left\{
\begin{array}{llll}
0, & \mbox { for } m\le u(x) \le M,\\
\nabla u, & \mbox { otherwise, say, on a set } E\subset B.
\end{array}
\right.
$$
Consider the Hodge decomposition (see [6]),
$$
|\nabla \psi|^{-p\varepsilon} \nabla \psi =\nabla \varphi +h.
$$
The following estimate holds
$$
\|h\|_{\frac {p-\varepsilon}{1-p\varepsilon}} \le C \varepsilon
\|\nabla \psi\|_{p-\varepsilon}^{1-p\varepsilon}. \eqno(4.4)
$$
Definition 4.1 with $\varphi $ acting as a test function yields
$$
\int_E \langle {\cal A}(x,\nabla u), |\nabla u|^{-p\varepsilon}
\nabla u \rangle dx =\int_E \langle {\cal A}(x,\nabla u), h \rangle
dx. \eqno(4.5)
$$
H\"older's inequality together with the conditions (3), (4), (4.4)
and (4.5) yields
$$
\begin{array}{llll}
&\displaystyle \int_E \gamma (x) |\nabla u|^{p(1-\varepsilon)} dx\\
\le &\displaystyle \int_E \langle {\cal A}(x,\nabla u), |\nabla
u|^{-p\varepsilon}
\nabla u \rangle dx \\
=&\displaystyle \int_E \langle {\cal A}(x,\nabla u),h\rangle dx \\
\le &\displaystyle \int_E \tau (x) |\nabla u|^{p-1} |h|dx \\
\le &\displaystyle \|\tau\|_{\frac
{p-\varepsilon}{(p-1)\varepsilon}} \|\nabla
u\|_{p-\varepsilon}^{p-1} \|h\|_{\frac
{p-\varepsilon}{1-p\varepsilon}} \\
\le &\displaystyle C\varepsilon \|\tau\|_{\frac
{p-\varepsilon}{(p-1)\varepsilon}} \|\nabla
u\|_{p-\varepsilon}^{p(1-\varepsilon)}\\
=&\displaystyle C |E|\varepsilon \cdot \varepsilon ^{-\theta_2
(1-\varepsilon )} \left[\frac {p-\varepsilon }{(p-1)\varepsilon
}\right]^{\theta_1} \left[\frac {(p-1)\varepsilon }{p-\varepsilon
}\right]^{\theta_1} \left(-\hspace {-4mm} \int_E |\tau|^{\frac
{p-\varepsilon }{(p-1)\varepsilon }} dx \right)^{\frac
{(p-1)\varepsilon }{p-\varepsilon }}\times\\
&\displaystyle \times \varepsilon ^{\theta_2 (1-\varepsilon )}
\left(-\hspace {-4mm} \int_E |\nabla u|^{p-\varepsilon } \right)
^{\frac {p(1-\varepsilon )}{p-\varepsilon }}.
\end{array} \eqno(4.6)
$$
The condition $\tau \in L^{\theta_1, \infty) } (\Omega)$ implies
$$
\lim_{\varepsilon \rightarrow 0^+} \left[\frac {(p-1)\varepsilon
}{p-\varepsilon }\right]^{\theta_1} \left(-\hspace {-4mm} \int_E
|\tau|^{\frac {p-\varepsilon }{(p-1)\varepsilon }} dx \right)^{\frac
{(p-1)\varepsilon }{p-\varepsilon }}\le \|\tau \|_{\theta_1,
\infty)}<\infty. \eqno(4.7)
$$
Since $u\in W^{\theta_2,p)}(\Omega)$, then
$$
\lim_{\varepsilon \rightarrow 0^+}\varepsilon ^{\theta_2
(1-\varepsilon )} \left(-\hspace {-4mm} \int_E |\nabla
u|^{p-\varepsilon } \right) ^{\frac {p(1-\varepsilon
)}{p-\varepsilon }}\le \|\nabla u\|_{\theta_2, p)}^p <\infty,
\eqno(4.8)
$$
By $\theta_1 +\theta_2<1$, we have
$$
\lim_{\varepsilon \rightarrow 0^+} \varepsilon \cdot \varepsilon
^{-\theta_2 (1-\varepsilon )} \left[\frac {p-\varepsilon
}{(p-1)\varepsilon }\right]^{\theta_1}=\left(\frac p
{p-1}\right)^{\theta_1} \lim_{\varepsilon \rightarrow 0^+}
\varepsilon ^{1-\theta_2 (1-\varepsilon )-\theta_1} =0. \eqno(4.9)
$$
Combining (4.6)-(4.9), and taking into account the assumption
$\gamma (x)>0$, a.e. $\Omega$, we arrive at $\nabla u=0$, a.e. $E$.
This implies that $(u-M)^+ -(m-u)^+$ vanishes a.e. in $B$, and thus
$(u-M)^+ -(m-u)^+$ must be the zero function in $B$, completing the
proof of Theorem 4.1.
\end{proof}
\begin{rem}
We remark that the result in Theorem 4.1 is a generalization of a
result due to Moscariello, see [17, Corollary 4.1].
\end{rem}
\section{A Weighted Version}
A weight is a locally integrable function on $\mbox {R}^n$ which
takes values in $(0,\infty)$ almost everywhere. For a weight $w$ and
a measurable set $E$, we define $w(E)=\int_E w(x)dx$ and the
Lebesgue measure of $E$ by $|E|$. The weighted Lebesgue spaces with
respect to the measure $w(x)dx$ are denoted by $L^p_w$ with
$0<p<\infty$. Given a weight $w$, we say that $w$ satisfies the
doubling condition if there exists a constant $C>0$ such that for
any cube $Q$, we have $w(2Q)\le Cw(Q)$, where $2Q$ denotes the cube
with the same center as $Q$ whose side length is 2 times that of
$Q$. When $w$ satisfies this condition, we denote $w\in \Delta_2$,
for short.
A weight function $w$ is in the Muckenhoupt class $A_p$ with
$1<p<\infty$ if there exists $C>1$ such that for any cube $Q$
$$
\left(-\hspace {-4mm} \int_Q w (x)dx \right)\left(-\hspace {-4mm}
\int_Q w (x)^{1-p'}dx \right)^{p-1} \le C, \eqno(5.1)
$$
where $\frac 1 p +\frac 1 {p'}=1$. We define $A_\infty =\bigcup
_{1<p<\infty}A_p$.
Let $w$ be a weight. The Hardy-Littlewood maximal operator with
respect to the measure $w(x)dx$ is defined by
$$
M_wf(x) =\sup_{Q\ni x} \frac 1 {w(Q)} \int_Q |f(x)|w(x)dx.
$$
We say that $T$ is a Calder\'on-Zygmund operator if there exists a
function $K$ which satisfies the following conditions:
$$
Tf(x) =\mbox {p.v.} \int_{\mbox {R}^n} K(x-y) f(y) dy.
$$
$$
|K(x)| \le \frac {C_K}{|x|^n} \mbox { and } |\nabla K(x)|\le \frac
{C_K}{ |x|^{n+1}}, \ \ x\ne 0.
$$
For $w$ a weight and $0\le \theta <\infty$, we define the space
$L_w^{\theta, \infty)} (\Omega)$ as follows
$$
L^{\theta, \infty)}_w (\Omega)=\left\{f(x) \in \bigcap _{1<p<\infty}
L^p_w (\Omega): \|f\|_{L^{\theta, \infty)}_w (\Omega)}
<\infty\right\},
$$
where
$$
\|f\|_{L^{\theta, \infty)}_w(\Omega)} =\sup_{1<p<\infty} \frac 1
{p^\theta} \left( \frac 1 {w(\Omega)} \int_\Omega |f(x)|^p w(x) dx
\right)^{\frac 1 p}.
$$
The following lemma comes from [18].
\begin{lemma}
If $1<p<\infty$ and $w\in \Delta _2$, then the operator $M_w$ is
bounded on $L^p_w(\Omega)$.
\end{lemma}
\begin{theorem}
The operator $M_w$ is bounded on $L^{\theta, \infty)}_w (\Omega)$
for $0\le \theta <\infty$ and $w\in \Delta _2$.
\end{theorem}
\begin{proof}
By Lemma 5.1, since for $1<p<\infty$ and $w\in \Delta_2$, the
operator $M_w$ is bounded on $L^p_w (\Omega)$, then
$$
\left( \int_\Omega |M_w f(x)|^p w(x)dx \right)^{\frac 1 p} \le C
\left( \int_\Omega |f(x)|^p w(x)dx \right)^{\frac 1 p}.
$$
This implies
$$
\begin{array}{llll}
\displaystyle \|M_w f\|_{L^{\theta,\infty)}_w (\Omega)}
&\displaystyle =\sup _{1<p<\infty} \frac 1 {p^\theta} \left(\frac 1
{w(\Omega)} \int_\Omega |M_w f(x)|^p
w(x)dx \right)^{\frac 1 p } \\
&\displaystyle \le C\sup _{1<p<\infty} \frac 1 {p^\theta}
\left(\frac 1 {w(\Omega)} \int_\Omega |f(x)|^p w(x)dx \right)^{\frac
1 p } =\|f\|_{L^{\theta,\infty)}_w (\Omega)},
\end{array}
$$
completing the proof of Theorem 5.1
\end{proof}
The following lemma can be found in [19].
\begin{lemma}
If $w\in A_\infty$, then there exists $q\in (1,\infty)$ such that
$w\in A_q$.
\end{lemma}
The following lemma can be found in [20, 21].
\begin{lemma}
If $1<p<\infty$ and $w\in A_p$, then a Calder\'on-Zygmund operator
$T$ is bounded on $L^p_w (\Omega)$.
\end{lemma}
\begin{theorem}
A Calder\'on-Zygmund operator $T$ is bounded on $L^{\theta,
\infty)}_w (\Omega)$ for $0\le \theta <\infty$ and $w\in A_\infty$.
\end{theorem}
\begin{proof}
By $w\in A_\infty$ and Lemma 5.2, one has $w\in A_q$ for some $q\in
(1,\infty)$. For $1<p<q<\infty$, H\"older's inequality yields
$$
\begin{array}{llll}
&\displaystyle \int_\Omega |Tf(x)|^p w(x)dx =\int_\Omega |Tf(x)|^p
w(x)^{\frac p q} w(x)^{\frac {q-p}{p}}dx \\
\le &\displaystyle \left( \int_\Omega |Tf(x)|^q w(x)dx\right)^{\frac
p q} \left(\int_\Omega w(x)dx\right)^{\frac {q-p} q}.
\end{array}
$$
Thus
$$
\begin{array}{llll}
&\displaystyle \frac 1 {p^\theta} \left(\frac 1 {w(\Omega)}\int_
\Omega |Tf(x)|^p w(x)dx \right)^{\frac 1 p }\\
\le &\displaystyle \frac 1 {p^\theta} \left(\frac 1 {w(\Omega)}\int_
\Omega |Tf(x)|^q w(x)dx \right)^{\frac p q} \left(\frac 1
{w(\Omega)}\int_\Omega w(x)dx \right)^{\frac {q-p} q}\\
= &\displaystyle \frac 1 {p^\theta} \left(\frac 1 {w(\Omega)}\int_
\Omega |Tf(x)|^q w(x)dx \right)^{\frac p q} .
\end{array} \eqno(5.1)
$$
Lemma 5.3 yields
$$
\begin{array}{llll}
&\displaystyle \|Tf\|_{L^{\theta, \infty)}_w (\Omega)} \\
=&\displaystyle \max \left\{ \sup_{1<p<q} \frac {1}{p^\theta}
\left(\frac 1 {w(\Omega)} \int_\Omega |Tf(x)|^p w(x) dx
\right)^{\frac 1 p }, \sup_{q\le p<\infty} \frac {1}{p^\theta}
\left(\frac 1 {w(\Omega)} \int_\Omega
|Tf(x)|^p w(x) dx \right)^{\frac 1 p } \right\}\\
=&\displaystyle \max \left\{ \sup_{1<p<q} \frac {1}{p^\theta}
\left(\frac 1 {w(\Omega)} \int_\Omega |Tf(x)|^q w(x) dx
\right)^{\frac p q}, \sup_{q\le p<\infty} \frac {1}{p^\theta}
\left(\frac 1 {w(\Omega)} \int_\Omega
|Tf(x)|^p w(x) dx \right)^{\frac 1 p } \right\}\\
\le &\displaystyle \max \left\{ \sup_{1<p<q}\left(\frac q p
\right)^\theta, 1 \right\} \sup_{q\le p<\infty} \frac {1}{p^\theta}
\left(\frac 1 {w(\Omega)} \int_\Omega |Tf(x)|^p w(x) dx
\right)^{\frac 1 p }\\
\le &\displaystyle Cq^\theta \sup_{q\le p<\infty} \frac
{1}{p^\theta} \left(\frac 1 {w(\Omega)} \int_\Omega |Tf(x)|^p w(x)
dx\right)^{\frac 1 p }\\
\le &\displaystyle Cq^\theta \|f\|_{L^{\theta, \infty)}_w (\Omega)}.
\end{array}
$$
As desired.
\end{proof}
\vspace{4mm}
\noindent {\bf Acknowledgement } This study was funded by NSFC
(10971224) and NSF of Hebei Province (A2011201011).
\vspace{8mm}
\rm \footnotesize
\linespread{2}
| 95,290
|
The Affordable Care Act authorizes the Secretary to award grants to states for the purpose of improving their review of proposed rates in the individual and small group health insurance markets.1 The law appropriates $250 million for rate review grants for a five year period, fiscal years 2010 through 2014. Each state receiving a grant must submit data to HHS documenting all rate increases requested by issuers for major medical policies in both the individual and small group health insurance markets of that state. The Rate Review Grant Program awarded a total of $51 million to 45 states, 5 territories, and the District of Columbia through the first cycle of funding. Through the second cycle of funding, an additional $119 million was awarded to 30 states, three territories, and the District of Columbia.
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Research is a major activity at all levels within the College of Liberal Arts and Sciences, from undergraduate research experiences to thesis and dissertation work of graduate students to faculty scholarship. The research activities of this college are more diverse than those of any other college on campus and have gained the college a national and international reputation.
Point of Contact
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| 100,171
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TITLE: What is the probability of choosing a 4-digit number that starts with 1 or 2, and has at least 3 of the same digit?
QUESTION [0 upvotes]: This is a question that I haven’t been able to solve!
Some more information:
The four digit number must start with either 1 or 2.
I’ve assumed that combinations is required, as the order of the final three digits chosen doesn’t matter. For example, the number 2322 has three 2s, and so does 2223. Hence they are the same combination of 4 digits, (three 2s, one 3).
The digits can repeat as many times as needed. The only obvious restriction is that the 4-digit number must start with 1 or 2.
To find the probably, all the favourable outcomes (the number of 4-digit number combinations that start with 1 or 2, that have at least 3 of the same digit), must be divided by the total amount of 4-digit number combinations.
The range of 4-digit numbers is between 1000 and 9999, as a 4-digit number cannot start with 0. (0999 cannot occur, 0023 cannot either…)
To be considered a favourable outcome, the digit must repeat 3 or 4 times in the 4-digit number combination.
I’ve had a go:
By hand, I wrote every different 4-digit number combination that starts with 1 or 2, that has at least 3 of the same digit. There were 36 different combinations, although this may be incorrect due to technique used.
I assumed the answer would be 36 divided by the total number of different 4-digit number combinations.
I don’t know how to find these values, though.
How would you solve this?
REPLY [1 votes]: The way I interpret the question: We pick one of the integers from $1000,1001,1002,1003,\dots,9999$ uniformly at random. There are $9\times 10^3 = 9000$ of these numbers (seen by basic application of rule of product, nine options for the first digit, ten options for the second digit and so on... or by common knowledge of numbers). We ask what the probability is that the number we picked happens to have a starting digit of $1$ or $2$ and happens to have exactly three identical digits or exactly four identical digits.
Since we are dealing with uniform probability here, we may simply find the count of desired outcomes and divide by 9000 which is the total number of possible outcomes. (N.B. this approach of dividing good outcomes over total outcomes is only valid in such scenarios where we deal with a uniform distribution and is in general not going to give the correct answer).
Counting the desired outcomes, break into cases based on whether there are exactly three identical or exactly four identical numbers.
(3 identical numbers) Pick the first digit. It is either a $1$ or a $2$ for a total of two options. Given such a selection, pick an additional digit to appear different from the first for a total of nine options. Now, decide which of the four positions in the number is the "odd one out", being the digit that is not repeated for a total of four options.
Note, if the first digit was the odd one out that means that the alternate digit is the one that is repeated such as in the example of $2888$. Alternatively if the first digit was not selected to be the odd one out that means that the first digit was among those repeated such as the example of $2822$
(4 identical numbers). Pick the first digit, there are two options. All digits are the same.
$$2\cdot 9\cdot 4 + 2=74$$
The final probability then is $$\frac{74}{9000}$$
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Advantages fireplace that uses only outside air to sustain the fire. Air is drawn in from outside your home through a small vent into the firebox, where it supplies the fire, after which the new spent air leaves through another vent back outside your home. Due to the laws of convection, this process happens naturally as soon as the fire is started. These vents can be installed around obstacles and only require a small cut through your home’s exterior wall.
In addition to the radiant heat leaving the fireplace through the glass front, room air is circulated around the firebox to be heated and then pumped back into the room, providing all the warmth and comfort of a traditional fireplace.
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My Fireplace – Suppliers and professional installers of direct vent fireplaces
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Advice absorb each of the advice and you are sure to benefit from it.
To direct people toward your product, write articles on a site where everyone can contribute and build articles. Your article should be legitimate, related and well-informed to the product you are selling. Include a link to a page with more information on the merchandise as a resource at the end of your article.
When designing a website to advertise a product or service online, add a "resources" section. Adding a information section attracts clients to your site by upping your search engine ranking. It also means that viewers of your website are more likely to return to view those resources, and may wrap up buying your product afterwards.
If you are attempting to grow your business's web existence by using a blog, keep your updates short and efficient. Your regular updates ought to be about three hundred to five hundred words in length; for those who have a special characteristic you can extend this out to a thousand words. Any longer and you'll likely lose the reader's fascination before the end.
If you're looking for Online marketing advice, do not forget to check under a wide array of keywords and search conditions in order to find it. "Internet marketing" is not the only phrase you will discover advice under. Try SEO, Internet business tips, and other related words and phrases to find more information.
Making people aware of both positive aspects and disadvantaged of your advertised merchandise can make your visitors trust your claims more. While Use These On-line Advertising Tips to Grow Your enterprise shouldn't produce the negatives of your item sound too bad, placing them in with the positives (that ought to be exaggerated somewhat) is an excellent way to earn your audience's trust.
More than any one internet marketing strategy, it is trust between webpage users and site owners that brings about sales. Visitors who have constantly pleasant and useful interactions with a particular website are more inclined to hear that website's owner as it pertains time to produce a buying decision. Fostering trust is the key to converting faithful visitors into faithful customers.
Look at your website, you may love it because it is yours, but there is always room for improvement. If you are not open to the idea that your internet site can be improved, you will not have the website you desire and others want to go to. Not one website is ideal, and all websites can improve.
Your prices must be set in accord with the marketplace, and only the market. In the event that you attempt to go against the marketplace and set your own rates, you're finally going to lose money. Setting prices too low means you will not make practically enough money, and setting them too high means you may not make any money.
Know what your rivals is up to. Enhance Your Web Marketing Skills By Following These tips do not want to be out from the loop about what your competitor is doing. Be sure to review what they need to offer, and do it better than them then. This will give potential customers a reason to come quickly to your site and not your competitors.
Always make sure to put a description of the website right at the top of the page. There is no way that a user will be considering buying anything from your site if they cannot look at the page and find out what it is that you will be selling.
An important tip regarding Online marketing is to consider establishing a chat feature on your site. This is important since it provides an easy way for people to speak straight with a human and never have to wait for a contact or pick up the telephone and face a possible dialect barrier.
There are several ways for a organization to use internet marketing tools to promote their products. Options for transporting their marketing campaigns can be any sort of blend of the next: mass emails, company internet site(s), through banner, gadget, video and image advertisements and even through se's such as Google and Yahoo.
If you're obtaining bad reviews on your own site or blog for your product, usually do not take these down. Unless they're full of bad language and/or lies in what you're offering, keep people's thoughts up to ensure that others can easily see what they thought about the product. It shows a great deal of honesty and confidence on your part.
Design a logo design or a slogan for your business. It will make it easier for your customers to remember where it was that they got particular items and come back to order more. Memory is better damaged by visuals rather than words that you have read in the past.
A good online marketing tip is to up-sell your customers. If they're already buying something from your site, entice them with related goods that match what they are purchasing. Help to make it possible for them to easily add it with their original order and you will see your sales surge.
A good internet marketing strategy is to make your readers visualize that they have already bought your item. Make Helpful Advice For Any And All Marketing Online appear like your product is a necessity that they absolutely have to buy. This will cause them to become mounted on your product which will increase sales emotionally.
If you are using online marketing, tell the customers that they can earn a full rebate on the product that they purchase if they refer four people to your internet site. Massively Enhance Your Internet Marketing Enterprise With Guru-Authorized Ideas can help turn one sales into four. If you do not want to offer a full rebate, you can offer free e-books, coupons, program, and discounts on your products.
Today that you have browse the information and bits of advice in this article, you're sure to have realized that there have been some things which were not contained in your business plan which should have already been. Take the missing parts and fit in them into your plan for a profitable outcome.
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Away With the Out-of-date Tram !
Page 14
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DERIODICALLY we hear calls from various .11 quarters, not excluding well-known municipaltransport engineers, for the revival of interest in tramways, and even suggestions that they should be reinstituted or, where they already exist, extended.
It is our belief, however, that the tram, as a means for transport, is at least obsolescent, and in some cases obsolete. In these days of increasing road congestion, whichwill probably be greatly intensified after the war, the tram, with its tracks, must be looked upon as an anachronism, which forms one of the greatest obstructions to all other kinds of traffic. Although it may not actually itself be involved in many accidents, it certainly causes a considerable number by its inability to deviate from a fixed track.
It is possible that it may, in some instances, be capable of being operated at a lower cost than can the bus or trolleybus, and it does present the advantage (which, however, is shared by the trolleybus) that it operate g upon power developed from a national fuel. We have no objection to the tram as such. If it can be employed as a light railway and .removed from the road proper, then there is little, if anything, that can be said against it. It is its presence on the road which has proved to be its greatest disadvantage.
Its track alone is a source of danger and inconvenience to cyclists, motorists and other road -users. Even quite recently we have seen the wheels of horsed carts caught and broken in the " grooves, whilst many cyclists and motorcyclists have lost their lives or been gravely injured through skidding as the result of being caught in ,the lines or in the groove of the central conduit, 'where the latter is employed, and being thrown in the path of other traffic. The alteration in the surface between the ordinary road and that part carrying the lines may also be a source of danger.
With these factors in view, we cannot believe that there is any case for this type of passenger vehicle, and' the sooner it be removed from the road the better pleased will be the majority of the public. .
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TITLE: How to express the basis of a subspace?
QUESTION [2 upvotes]: Find a basis for this subspace of $\mathbb{R}^4$:
All vectors whose components add to zero.
I think I know what this is asking but I don't know how to express the final answer.
All vectors whose components add to zero means we have $3$ free variables in $\mathbb{R}^4$:
$$\begin{bmatrix} a\\ b\\ c\\ -(a+b+c)\\ \end{bmatrix}$$
Since we have three free variables we would need three vectors in a basis, but do you express the final answer as
$$a\begin{bmatrix} 1\\ 0\\ 0\\ -1\\ \end{bmatrix} +b\begin{bmatrix} 0\\ 1\\ 0\\ -1\\ \end{bmatrix} +c\begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$$
is a basis for all vectors whose components add to zero in $\mathbb{R}^4$?
REPLY [0 votes]: All vectors whose components add to zero$\iff$Nullspace of the matrix $A=\begin{bmatrix}1&1&1&1\end{bmatrix}$
Solutions to the equation
$$
Ax=0\implies A=\begin{bmatrix}1&1&1&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=\begin{bmatrix}0\\0\\0\\0\end{bmatrix}\\
x_1+x_2+x_3+x_4=0\\
\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=x_2\begin{bmatrix}-1\\1\\0\\0\end{bmatrix}+x_3\begin{bmatrix}-1\\0\\1\\0\end{bmatrix}+x4\begin{bmatrix}-1\\0\\0\\1\end{bmatrix}
$$
All vectors whose components add to zero=$span\bigg\{ \begin{bmatrix}-1\\1\\0\\0\end{bmatrix},\begin{bmatrix}-1\\0\\1\\0\end{bmatrix}, \begin{bmatrix}-1\\0\\0\\1\end{bmatrix}\bigg\}$
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Hi,
Tomorrow is the anniversary of Stalin’s last of three Trial of Twenty-One (1938), it’s Texas independence from Mexico (1836), the movie Sound of Music’s premier (1965), Mt Rainier National Park was established (1899), the Federal highway numbering system was introduced (1925) and the Battle of Bismark (1943). Birth anniversaries include actor Desi Arnaz (1917), Dr Seuss Theodor Geisel (1904), soldier & politician Sam Houston (1793), baseballer Mel Ott (1909), Soviet politician Mikhail Gorbachev (1931).
Well, Governor Corbett finally made the number one spot in a recent poll. Of the 15 governors up for reelection, he is the least likely to win his seat back. Maybe it was him gutting our educational system two years ago to balance the budget. Maybe it was his back handed way of trying to force a uniform code for his buddies in the fracking industry (I would be in favor of a uniform code, it’s just don’t do it unilaterally without input from the cities and counties effected). Maybe it was trying underhandedly give the operation of the cash cow lottery over to a third party out of England. Maybe it was his setting up the Commonwealth Financing Authority to award state contracts that has NO transparency. When the Trib was researching an article on this, ninety percent of the documents they were supplied were redacted. I wonder why he’s so unpopular.
Clairton Police Officer James Kuzak is going to be featured with two other officers in a documentary Heroes Behind the Badge series. If you recall, Officer Kuzak was shot when responding to a home invasion and left paralyzed from the waist down. The series follows officers that have given so much in the line of duty to keep us safe, others like him that have lost so much and the families of fallen officers that gave the ultimate sacrifice. The documentary will be aired at 7 pm on April 5 at the Hyundai West Club Lounge at Heinz Field. Tickets and more info at Ticket Fly or by calling 877-435-9849. Tickets are $10 and $15 in advance and all proceeds go to the National Law Enforcement Officers Memorial Fund.
Identity thieves and financial fraudsters are getting cleverer and cleverer. Users of social media frequently give the high school they attended, their mother’s maiden name, their pet’s name, (a frequent password used), and date of birth (not to be confused with their birthday which is less a security risk). Thieves with just this information can actually open credit cards and fraudulent bank accounts, shame on the banks for not being more strict on this, but the bottom line you can be saddled with the ramifications. Smart phone use to access you bank account in the open can be hacked, as well as your passwords if you keep them in the “Notes” section of your phone. Something you can easily do to help protect your phone is to add a password to use it. It only takes seconds to enter it and is just another layer of security you can add to your life.
Thinking about getting a new refrigerator, air conditioner or other high energy consuming product, check out Duquesne Light’s rebate program, or the Federal Energy Star for rebates first to see if what you are considering is covered or maybe another product may become more appealing.
The eagles that nested in Hays last year had one chick, this year Mrs Eagle just laid three eggs. The Trib in conjunction with Wild Earth has a live web cam you can access anytime to observe the nest. Just now, I’m watching mom keeping her eggs warm. It is sooo cool. Not only is it very cool to watch these magnificent creatures, it’s also very cool to realize that with as polluted as our planet is, we are making significant strides to clean up our mess that the eagles are able to eat the fish from our rivers. The eagles’ nest out in Harmar are in a very inconvenient location to see the nest (they actually stole the nest from some hawks last year) :). To make matters worse, the parking lot people used last year to view the nest and eagle activities is being used by PennDot as a staging area for the Hulton Bridge replacement project, so there’s no close parking.
The Pennsylvania Historical and Museum Commission is offering free admission to many of their historic sites on March 9 (Charter Day) on March in celebration of their 333rd birthday. Check out their site to see what peaks your interest.
Girl Scouts of America put a stop to the creativity of some young savvy girl scouts. The girls camped out outside several medical marijuana shops and was selling bunches of cookies. Can you say munchies?
It looks like some start ups are taking on the lawyers. There’s an app that’s been out since September called Shake, once you down load it, it will walk you through the process of creating a contract based on information you fill in. With a smart phone or tablet, you can create the document and have it signed with just the swipe of a finger. The best part is this app eliminates most of the legal jargon that so confuses everyone. What’s next, a divorce app?
Pascale Lemire set up the website Dog Shaming back in 2012 and has over 58 million page visits and 65,000 submissions from pet owners since inception. It’s pretty cute and changes all the time. Another cute site is Shame Your Pet.
The Croatian Church in Milvale has 22 murals by famed Maxo Vanka. St Nicholas parish was founded in 1894 and built the first church in 1900, but that was destroyed by fire in 1922. They built a new church and in the 1930′s the pastor got Maxo to create the murals adorning the church, covering 11,000 square feet. It costs about $17k to restore each mural, so St Nicholas is hosting a fund raiser next Friday, March 7 starting at 6 pm. General admission is $50 and for donations $100 and above you get the VIP treatment starting at 5:30. If you just want a tour sometime, there’s docent led tours on Saturdays at 11 am, noon and 1 pm. More info at their website or by calling 412-407-2570.
Coming to Carnegie in July is Apis Meadery on East Main Street. In case you don’t know, mead is brewed from fermenting honey and water, fruits, spices and grain. David Cerminara is a brewer at Penn Brewery and is looking to fill a niche. There’s a lot of great crafted beer breweries around, but no Meaderies. It should be a nice addition to the mix of locally crafted beverage choices.
Throughout the month of March, The Carnegie Museums of Art, Architecture and Natural History has free admission from 4 until 8 pm on Thursdays. A great way to see the 2013 Carnegie International before it closes March 16.
Well, bundle up and keep warm for hopefully out last big storm of the season, tomorrow it’s supposed to hit Pittsburgh starting in the afternoon,
ed
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Growing up a dancer, I’m accustomed to driving myself physically. I need to challenge myself physically in order to find be fulfilled.
But I’m also lazy. I fall into ruts of inactivity. I make excuses. I feel guilty. I crave that physical drive again and eventually I get myself moving. It’s a bitter and very unsatisfying cycle.
I want to be physically active and feel healthy EVERY DAY. I want to take care of my body. If I die young, I don’t want it to be because I didn’t take care of my physical health.
For me: physical health brings mental health. When I feel good….I feel happy. It’s a wonderful way to LIVE. Physical health includes feeling strong…feeling flexible…feeling mental peace. It’s about eating good food, staying active, enjoying the outdoors, and having fun.
I have been sedentary too long. I feel that drive pumping again. I want to challenge myself physically again. But this time..I don’t want to stop. I want it to become a way of life. I want it to be a bigger part of who I am. I want it to be important all the time- in action, not just thought. I want to stop thinking about it and begin doing it. TODAY.
How do you drive yourself physically and what motivates you to keep going?
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I start and stop way too often and that's one thing I struggle with! I don't have anyone around to get together and go workout with, which I think is a tremendous help since I'm not a self-motivator.
I'm not a self-motivator either. It helps me to have a set class to go to because then I feel like I shouldn't miss it. If I just plan to work out at home after work then I always find something else to do instead.
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Villa mh25309
- Villa
- 10 Sovepladser
- 5 Soveværelser
- 2 Badeværelser
A rare jewel: a spacious and comfortable house 100 metres from a great beach and with a private swimming pool. The inside of the house was completely renovated in 2019 and services are now fully included so more charges to be paid on site. This holiday home is in Cancale, a renowned seaside town. Sailing school and other surf sports are possible. The beaches are in walking distance. If you like oysters, here you will be catered for, because Cancale is renowned for its wild oysters! From Grouin Point where you can walk, you have a splendid view of the Bay of Mont-Saint-Michel. Mont-Saint-Michel itself is on the other side of the bay 50mins by road. Do not miss a visit to Saint-Malo (16km) and its fortified centre. Beautiful walks to do along the coast.
Heated (to be payed) enclosed private swimming pool (open from 01-06 till 30-09) with elec. cover. Tourism in Cancale. Cancale, jewel of the emerald coast Vis-a-vis the Mount Saint Michel, Cancale accomodates since always a number of visitors to the multiple and varied vocations. Also: That you are amateurs of nature and safeguarding of the environment, to come in Cancale will be for you a pilgrimage within a region where ecology and local development always could marry harmoniously. That you are amateurs of inheritance and traditions, to come in Cancale will be for you the discovery of proud people of his maritime past and its remote origins, which knew with the wire of times to maintain a certain art of living. That you are amateurs of gastronomy and good tables, to come in Cancale will be for you the meeting of famous chiefs whose passion made our city a remarkable site of the taste. Then do not hesitate any more, come to meet us in all seasons bus Cancale is discovered as well the summer as the winter thanks to the quality of its reception recognized of all. Town of Character to the image of its inhabitants, Cancale also knew to preserve what today does one of the jewels of Emerald Coast of it: its authenticity and richness of its traditions going back to the night of times. Ideas of visits to be made in the area? Initially a true council for a stay with Cancale: time can change quickly into better or less better. If the weather is nice: profit in forthwith, it is very risky to wait! If it rains, not of panic: the situation can be raised quickly. The variety of local curiosities contributes to this adaptation. The traditional ones St Malo: its ramparts, corsairs, Chateaubriant Dinard: its beach and the walk with the moonlight. Dinan and its medieval scenery. Do not forget Jerzual and its craftsmen. The Mount Saint Michel: to visit day as with fallen the night (visit in the enlightened Abbey) Attractions The large aquarium of Malo Saint is interesting and very ludic The castle of Bourbansais (between St Malo and Rennes) is a property including a zoo and a castle Walks On cancale, the site of the Beach of the Orchard (plage du Verger)(fine sand) is very sympathetic. The point of Grouin enables you to take a good bowl of air near the house! The chemin des douaniers (ways of the customs officers) in edge of coast are sportsmen and very pretty. You will find in the house (and with the tourist bureau) a complete documentation with descriptions of ballades. For the walkers, to start from the house, walk around the Pointe of Grouin is very pleasant. To take the street of Tintiaux (almost opposite the house) which joined the wild side of the Pointe To join the Pointe of the grouin either while skirting the road or by the chemin des Douaniers (way of the customs officers) To return to Port-Mer by the way of the customs officers. Internet sites to prepare your stay: Our good addresses For the courageous ones and the greedy ones: only one solution outward journey with the source: Fishing with foot in bay is the must but it is necessary to be equipped and attention with the tide - it really goes up very quickly. The gathering of the strawberries in St Méloir: you can go to gather them yourself (in season) or to go to buy them with the Farm which organizes, in season (to check the dates with the tourist bureau of Cancale) Monday evening a farm market with products of quality. FRUITS, FRESH VEGETABLES AND EARLY PRODUCTS (DETAIL) Small fruits of Bay Clossets 0299892198 For greedy the a little courageous ones: Moulds and oysters: There is much in Cancale but we were never disappointed at: MICHEL DANIEL 37, QUAI KENNEDY 35260 CANCALE Tel: 02 99 89 62 66 Alive shellfish : Les Viviers of Cancale Vauhariot 4 R Oyster 35260 CANCALE Tel: 02 23 15 17 00 Manche Crustacés zi Outre 35350 Gouesnière Tel .02 99 89 19 18 It should be noted that you will find in the kitchen a large stewpan adapted to the cooking of crabs and spiders! Poisson: a fisherman whose woman holds the fish shop what could be more normal! Cap Pilar Mr. Tachet Le tertre - street of Industry 35260 CANCALE Tel: 02 99 89 60 18 (Artisanal Zone opposite Super U behind the store of Breton specialities). A little courage however because the choice is that of the fishing of the day and thus the greatest choice is with the opening at 9H00. The service is typically cancalais! For the greedy absolutely not courageous, there remain the restaurants, the créperies, the port of La Houle is very famous: On the port our preferred are Pancake shops: we like the pancake shop Labordage and the créperie du port The Champlain Mother: brasserie of good quality The Troquet : a good bar 19, quai Gambetta Tél. : 02 99 89 99 42 Querrien for its environment marinates Quai Duguay Trouin Tel.: 02.99.89.64 .56 The Saint Cast for its sight and its kitchen - Road of the Cornice Tel.: 02 99 89 66 08 For the exceptional moments, two restaurant are famous: Tirel-Guerin house a family restaurant 1 star in Michelin guide to discover at the Station D 76 - 35350 Gouesnière Telephones: 02 99 89 10 46 Hotel with restaurant Houses of Bricourt - a not usurped world reputation - Jane & Olivier ROELLINGER Houses of Bricourt 1, rue Duguesclin - 35260 Cancale Tel.: 02 99 89 64 76 Visit at least the Internet site, it is a true treat: Return on earth for the day to day shopping: Super U is impossible to circumvent. You will find it on the road of St Malo (direct road) In the centre town of Cancale a broad Marché plus open Sunday morning and schedules in week There is also a new leader price. Sunday morning, behind the church, there is a sympathetic market.
Faciliteter - indendørs
Faciliteter - køkken
Faciliteter - udendørs
Tilgængelighed
Priser per uge
Pris fra Kr. 5444 Kr. 19146
Populære områder i Cancale
Feriebolig Frankrig | Feriebolig Bretagne | Feriebolig Ille-et-Vilaine | Feriebolig Cancale |
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Paulk: Johnson's past mastery bodes well at Brickyard
The Brickyard 400 may not be the biggest race of the Sprint Cup season, but it's likely to separate the championship contenders from the pretenders.
Five-time champion Jimmie Johnson has established he's the most dominant driver on the circuit this season.
He has been pushed some by Matt Kenseth, but unless he falters as he did in last year's season finale, his coronation as six-time champion seems inevitable.
In sweeping both races at Daytona and winning at Pocono, Johnson enters Sunday's race as the clear favorite and will start on the front row.
A victory at the storied Indianapolis Motor Speedway will bolster the already immeasurable confidence of Johnson and crew chief Chad Knaus.
If history is a barometer of success, then Johnson is in a perfect place. He is a four-time winner at the Brickyard — a track where his uncanny blend of patience and aggressiveness has yielded dividends.
“I think he will be fast, as usual,” Hendrick Motorsports teammate Dale Earnhardt Jr. said. “I'll be surprised if they're not one of the more competitive teams. It always seems like there's somebody else that ends up running well.”
Indeed, four-time Cup champion Jeff Gordon, Greg Biffle, Kyle Busch, Kasey Kahne and Tony Stewart have pressed Johnson at the Brickyard. And they'll need to be mistake-free if they are going to outhustle Johnson to the checkered flag and kiss the perfectly aligned bricks near the finish line.
“They certainly are in championship form, and I'd have to put them at the top of the list of teams to beat,” said Gordon, a four-time winner at the Brickyard. .”
In reality, only Kenseth can keep Johnson in check. Kenseth, tied with Johnson with four wins, can't afford an uneven performance like those that have left him sixth in points instead of second ahead of Clint Bowyer and Carl Edwards, who have only a single win between them.
For Bowyer, he needs a win this weekend to at least show Johnson he packs a big enough punch to tangle with him during the 10-race Chase.
“I'm certainly not right behind Jimmie, but I guess I'm the closest person to him,” said Bowyer, who trails Johnson by 56 points. “We've done a great job at being consistent, and that's what I like to focus on and then especially working into the Chase.
“I think these summer months have always sort of been difficult for me in my past, and what I've seen out of my team in these summer months I really like a lot, and it excites me about our chances for the Chase.”
Bowyer's Michael Waltrip Racing teammates — Martin Truex Jr. and Brian Vickers — both have captured checkered flags this season.
But Bowyer's consistency has positioned him to advance into the postseason. Truex could slide in as a wild card if he finishes outside the top 10.
“You can win the championship being consistently strong,” Bowyer said.
“We're second in points being consistently strong. I feel like — with an exception of one — we've covered our bases. We've done a good job of being there each and every week.”
Bowyer stumbled some at New Hampshire, but the No. 15 Toyota was as strong as Johnson's Hendrick Motorsports No. 48 Chevrolet and Kenseth's No. 20 Joe Gibbs Racing Toyota.
“It's a humbling sport,” Bowyer said. “Out of three races before (New Hampshire) were all solid top-fives. That's what I feel like where our team's at right now. We're a top-five team week in and week out. And if you do that, I know if you're going to get your wins.
“I'm really not concerned about that. Being consistent, staying in the hunt each and every weekend is what's going to give you a shot at a championship.”
Still, Bowyer has to flex his muscles at the Brickyard to get Johnson's attention.
Ralph N. Paulk is a staff writer for Trib Total Media. Reach him at rpaulk@tribweb.com or via Twitter @RalphPaulk_Trib. Listen to the Auto Racing Show with Ralph N. Paulk every Friday on TribLive Radio at 9 a.m. and 5 p.m.
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If you use Windows Media Center on your home theater PC you can configure it to run via voice commands using macros for Windows7's built-in Windows Speech Recognition software. The only additional cost is for a decent microphone if you don't already have one..
How-To: Voice control Windows Media Center | Inspect My Gadget
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Get simple communication tools, strategies, and tips to help you communicate with confidence.
- Hi, I'm Jeff Ansell, and welcome to Communicating with Confidence. In this course I'll show you ways you can get over your fear of public speaking and become an effective communicator. I'll start by going over techniques you can use to sound more confident. Then I'll go over ways to use your body language and gestures to look more confident. I'll cover a few ways to overcome the anxiety you might feel in public speaking situations. And finally, I'll show you how to bring it all together by going through some real coaching exercises.
Effective communication is a key skill in business and I'm here to show you ways to be your best. Let's get started with Communicating with Confidence.
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TITLE: What is internal space translation?
QUESTION [3 upvotes]: While I was reading the paper named "Classical time crystals," by A. Shapere and F. Wilczek, I found the following transformation.
$$f(x) \to f(x+e) - \frac{df}{dx}*e$$
It says the transformation is combined internal space-real space translation.
What is internal space translation (I guess it is $f(x) \to f(x+e) - \frac{df}{dx}*e$?) in both mathematical and physical sense?
The paper can be found here. The aforementioned formula is at the first paragragh of second column.
REPLY [0 votes]: The paper refers to a combined internal space-real space transformation.The first part, $\phi(x) \rightarrow \phi(x+\epsilon)$, is a real space traslation, as it shifts the position coordinate by $\epsilon$ as $x \rightarrow x+\epsilon$.
The second part, $\phi(x) \rightarrow \phi(x) -\epsilon\frac{d\phi}{dx}$, is the internal transformation. This naming comes from the fact that this transformation is acting on the angular variable itself ( or the field in the context of QFT or electromagnetism) and not on the spacetime as the traslation $x\rightarrow x+\epsilon$.
This is a kind of gauge transformations. The easiest to visualize is perhaps the electromagnetic gauge $^1$ where you write the vector potential $\textbf{A}$ as $\textbf{A} + \nabla f$ and the electric potential as $V \rightarrow V - \frac{df}{dt}$ which affect neither any of the "real" degrees of freedom of the system nor the fields ($\textbf{B}$ and $\textbf{E}$) themselves.
Classical Electrodynamics (1962). JD Jackson. Section 6.4 .
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\begin{document}
\begin{abstract}
The paper deals with existence and multiplicity of solutions
of the fractional Schr\"{o}dinger--Kirchhoff equation
involving an external magnetic potential. As a consequence,
the results can be applied to the special case
\begin{equation*}
(a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\,
\quad \text{in $\mathbb{R}^N$},
\end{equation*}
where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$,
$\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a
magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an
electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator.
In the super-- and sub--linear cases, the existence of least energy solutions
for the above problem is obtained by the mountain pass theorem, combined with the
Nehari method, and by the direct methods respectively. In the
superlinear--sublinear case, the existence of infinitely many solutions
is investigated by the symmetric mountain pass theorem.
\end{abstract}
\maketitle
\section{Introduction and main result}\label{sec1}
The paper deals with the existence of
solutions of the fractional {\em Schr\"{o}dinger--Kirchhoff} problem
\begin{equation}\label{eq1}
M([u]_{s,A}^2)(-\Delta)_A^su+V(x)u=f(x,|u|)u\quad \text{in $\mathbb{R}^N$},
\end{equation}
where hereafter $s\in(0,1)$, $N>2s$,
$$
[u]_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{1/2},
$$
$M:\mathbb{R}^+_0\rightarrow\mathbb{R}^+_0$ is a Kirchhoff function, $V:\mathbb{R}^N\rightarrow\mathbb{R}^+$ is a scalar
potential, $A:\mathbb{R}^N\rightarrow \mathbb{R}^N$ is a magnetic potential, and $(-\Delta )_A^s$ is the associated fractional magnetic operator which, up to a normalization constant, is defined as
\begin{equation*}
(-\Delta)_A^s\varphi(x)=2
\lim_{\varepsilon\rightarrow 0^+}\int_{\mathbb{R}^N\setminus B_\varepsilon(x)}\frac{\varphi(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}\varphi(y)}{|x-y|^{N+2s}}\,dy,\quad x\in\mathbb{R}^{N},
\end{equation*}
along functions $\varphi\in C_0^\infty(\mathbb{R}^N,\mathbb{C})$.
Henceforward $B_\varepsilon(x)$ denotes the ball of
$\mathbb{R}^N$ centered at $x\in\mathbb{R}^N$ and
radius $\varepsilon>0$. For details on fractional magnetic operators
we refer to \cite{PDMS} and to the references \cite{ichi1,ichi2,ichi3,purice} for the physical background.
The operator $(-\Delta)_A^s$ is consistent with the definition
of fractional Laplacian $(-\Delta )^s$ when $A\equiv0$. For
further details
on $(-\Delta )^s$, we refer the interested reader to \cite{r28}.
Nonlocal operators can be seen as the infinitesimal generators
of L\'{e}vy stable diffusion processes \cite{r8}. Moreover, they allow
us to develop a generalization of quantum mechanics and also
to describe the motion of a chain or an array of particles
that are connected by elastic springs as well as unusual diffusion
processes in turbulent fluid motions and material transports in
fractured media (for more details see for example
\cite{r8,r3,r7} and the references therein).
Indeed, the literature on nonlocal fractional operators and
on their applications is quite large, see for example the
recent monograph \cite{MBRS}, the extensive paper \cite{DMV} and
the references cited there.
The paper was motivated by some works appeared in recent years concerning the magnetic Schr\"{o}dinger equation
\begin{align}\label{eq1.01}
-(\nabla -{\rm i} A)^2u+V(x)u=f(x,|u|)u\quad\text{in $\mathbb{R}^N,$}
\end{align}
which has been extensively studied (see \cite{GAAS,SCSS,JDJV,KK,MS}). The magnetic Schr\"{o}dinger operator is defined as
\begin{align*}
-(\nabla -{\rm i} A)^2u=-\Delta u+2{\rm i}A(x)\cdot\nabla u+|A(x)|^2u+{\rm i}u\,{\rm div}A(x).
\end{align*}
As stated in \cite{MSBV}, up to correcting the operator
by the factor $(1-s)$, it follows that
$(-\Delta)^s_A u$ converges to $-(\nabla u-{\rm i} A)^2u$
as $s\uparrow1$. Thus, up to
normalization, the nonlocal case can be seen as an approximation of the local case (see Section~\ref{singularsect} for further details).
As $A=0$ and $M=1$, equation \eqref{eq1} becomes the fractional Schr\"{o}dinger equation
\begin{align*}
(-\Delta)^su+V(x)u=f(x,|u|)u\quad\text{in $\mathbb{R}^N,$}
\end{align*}
introduced by Laskin \cite{laskin1, laskin2}. Here
the nonlinearity $f$ satisfies general conditions.
We refer, for instance, to \cite{r15, r14, r17} and the
references therein for recent results.
Throughout the paper, without explicit mention, we also assume that
{\em $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$
and $V:\mathbb{R}^N\rightarrow\mathbb{R}^+$ are continuous
functions, and that $V$ satisfies},
\begin{itemize}
\item[$(V_1)$] {\em there exists $V_0>0$ such that $\inf_{\mathbb{R}^N} V \geq V_0$}.
\end{itemize}
The Kirchhoff function $M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0$
is assumed to be {\em continuous and to verify}
\begin{itemize}
\item[$(M_1)$] {\em for any $\tau>0$ there exists $\kappa=\kappa(\tau)>0$ such that $M(t)\geq\kappa$ for all $t\geq\tau$};
\item[$(M_2)$] {\em there exists $\theta\in[1,2_s^*/2)$ such that $tM(t)\leq \theta\mathscr{M}(t)$ for all $t\geq0$, $\mathscr{M}(t)=\int_0^t M(\tau)d\tau$}.
\end{itemize}
A simple typical example of $M$ is given by $M(t)=a+b\,t^{\theta-1}$
for $t\in\mathbb{R}^+_0$, where $a\in\mathbb{R}^+_0$, $b\in\mathbb{R}^+_0$
and $a+b>0$. When $M$ is of this type, problem \eqref{eq1}
is said to be {\em non--degenerate} if $a>0$, while it is
called {\em degenerate} if $a=0$.
Clearly, assumptions $(M_1)$ and $(M_2)$ cover the degenerate case.
It is worth pointing out that
the degenerate case is rather interesting and is treated in well--known
papers in Kirchhoff theory, see for example \cite{dAS}.
In the large literature on degenerate Kirchhoff problems,
the transverse oscillations of a stretched string, with nonlocal
flexural rigidity, depends continuously on the Sobolev deflection
norm of $u$ via $M(\|u\|^2)$. From a physical point of view, the fact
that $M(0)=0$ means that the base tension of the string is zero, a
very realistic model. We refer to \cite{AFP, capu, PXZ2, XZF2, XMTZ} and the
references therein for more details in bounded domains and in the whole space.
Recent existence results of solutions for fractional non--degenerate Kirchhoff problems are given, for example, in \cite{FV, XZF1, XZG, PS}.
Assumptions $(M_1)$ and $(M_2)$ on the Kirchhoff function $M$ are enough to assure the existence of solutions of~\eqref{eq1}.
However, to get the existence of ground states, we assume
also the further mild request
\begin{itemize}
\item[$(M_3)$] {\em there exists $m_0>0$ such that
$M(t)\geq m_0 t^{\theta-1}$ for all $t\in [0,1]$,}
\end{itemize}
{\em where $\theta$ is the number given in $(M_2)$ when $(M_2)$
is assumed, otherwise $\theta$ is any number greater
than or equal to 1}.
Of course, $(M_3)$ is satisfied also in the model case,
even when $M(0)=0$, that is in the degenerate case.
In \cite{PXZ2}, condition $(M_3)$ was also applied to
investigate the existence of entire solutions for the stationary
Kirchhoff type equations driven by the fractional $p$--Laplacian
operator in $\mathbb{R}^N$.
Superlinear nonlinearities $f$ satisfy
\begin{itemize}
\item[$(f_1)$] {\em $f\in\mathbb{R}^N\times\mathbb{R}^+\rightarrow\mathbb{R}$ is a Carath\'{e}odory function and there exist $C>0$ and $p\in (2\theta,2_s^*)$ such that}
\begin{align*}
|f(x,t)|\leq C(1+|t|^{p-2})\quad\mbox{\em for all }
(x,t)\in\mathbb{R}^N\times\mathbb{R}^+;
\end{align*}
\item[$(f_2)$] {\em There exists a constant $\mu>2\theta$ such that
$$0<\mu F(x,t)\leq f(x,t)t^2,\quad
F(x,t)=\int_0^tf(x,\tau)\tau d\tau,$$
whenever $x\in\mathbb{R}^N$ and $t\in \mathbb{R}^+$;}
\item[$(f_3)$] {\em $f(x,t)=o(1)$ as $t\rightarrow 0^+$, uniformly for $x\in\mathbb{R}^N$};
\item[$(f_4)$] $\displaystyle{\inf_{x\in\mathbb{R}^N}F(x,1)>0}$.
\end{itemize}
A typical example of $f$, verifying $(f_1)$--$(f_4)$, is given by $f(x, |u|)=|u|^{p-2}$, with $2\theta<p<2_{s}^{*}$.
The fractional solution spaces $\Ma(\mathbb{R}^N,\mathbb{C})$
and $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$
are introduced precisely in Section~\ref{sec3}.
\vskip2pt
\noindent
We say that $u\in \Ma(\mathbb{R}^N,\mathbb{C})$ (resp.\
$u\in H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$) is a (weak)
{\em solution} of \eqref{eq1}, if
\begin{align*}
\Re\bigg[M([u]_{s,A}^2)\iint_{\mathbb{R}^{2N}}\!\!&\frac{\big[u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y)\big]\cdot\big[\overline{\varphi(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}\varphi(y)}\big]}{|x-y|^{N+2s}} dxdy+\int_{\mathbb{R}^N}Vu\overline{\varphi} dx\bigg]\\
&\hspace{9truecm}=\Re\int_{\mathbb{R}^N}f(x,|u|)u\overline{\varphi} dx,
\end{align*}
for all $\varphi\in \Ma(\mathbb{R}^N,\mathbb{C})$
(resp.\ $\varphi\in H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$).
Now we are in a position to state the first existence result.
\begin{theorem}[Superlinear case]\label{th1}
Assume that $V$ satisfies $(V_1)$, $f$ satisfies $(f_1)$--$(f_4)$
and $M$ fulfills $(M_1)$--$(M_2)$. Then \eqref{eq1} admits
a nontrivial radial mountain pass solution
$u_0\in \Ma(\mathbb{R}^N,\mathbb{C})$.
Furthermore, if $M$ satisfies $(M_1)$--$(M_3)$, then
\eqref{eq1} has a ground state $u\in \Ma(\mathbb{R}^N,\mathbb{C})$
with positive energy.
\end{theorem}
\noindent
Sublinear nonlinearities $f$ verify
\begin{itemize}
\item[$(f_5)$] {\em There exist $q\in (1,2)$ and $a\in
L^\infty_{\rm loc}(\mathbb{R}^N)\cap L^{\frac{2}{2-q}}(\mathbb{R}^N)$
such that}
\begin{align*}
|f(x,t)|\leq a(x)t^{q-2}\quad\mbox{\em for all }
(x,t)\in\mathbb{R}^N\times\mathbb{R}^+.
\end{align*}
\item[$(f_6)$] {\em There exist $q\in (1,2)$, $\delta>0$, $a_0>0$ and
a nonempty open subset $\Omega$ of $\mathbb{R}^N$ such that}
\begin{align*}
|f(x,t)|\geq a_0t^{q-2}\quad\mbox{\em for all } (x,t)\in\Omega\times(0,\delta).
\end{align*}
\end{itemize}
A typical example of $f$, verifying $(f_5)$--$(f_6)$,
is $f(x, |u|)=(1+|x|^2)^{(q-2)/2}|u|^{q-2}$ with $1<q<2$.
The second result reads as follows.
\begin{theorem}[Sublinear case]\label{th2}
Assume that $V$ satisfies $(V_1)$, $f$ satisfies $(f_5)$--$(f_6)$
and $M$ is continuous in $\mathbb R^+_0$ and satisfies
$(M_1)$ and $(M_3)$, with $\vartheta\ge1$. Then \eqref{eq1}
admits a nontrivial solution $u\in H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$,
which is a ground sate of \eqref{eq1}.
\end{theorem}
\noindent
To get infinitely many solutions for equation
\eqref{eq1} in the local sublinear--superlinear case, we also assume
\begin{itemize}
\item[$(V_2)$] {\em There exists $h>0$ such that
\begin{align*}
\lim_{|y|\rightarrow\infty} {\mathscr L}^N\big(\{x\in B_h(y):V(x)\leq c\}\big)=0
\end{align*}
for all $c>0$.}
\item[$(f_7)$]
{\em $F(x,t)\geq 0$ for all $(x,t)\in\mathbb{R}^N\times\mathbb{R}^+_0$,
and there exist $q\in(1, 2)$, a nonempty open subset $\Omega$ of~$\mathbb{R}^N$
and $a_1>0$ such that}
\begin{align*}
F(x,t)\geq a_1t^{q}\quad\mbox{\em for all }
(x,t)\in\Omega\times\mathbb{R}^+.
\end{align*}
\end{itemize}
An example of $f$, which satisfies assumptions $(f_1)$ and $(f_7)$, is
\begin{align*}
f(x,t)=(1+|x|^2)^{(q-2)/2}t^{q-2}+t^{p-2}\quad\mbox{for all }
(x,t)\in\mathbb{R}^N\times\mathbb{R}^+_0,
\end{align*}
when $1<q<2\leq 2\theta<p<2_s^*$.
\begin{theorem}[Multiplicity -- local superlinear--sublinear case]\label{th3}
Assume that $V$ satisfies $(V_1)$--$(V_2)$, that
$f$ fulfills $(f_1)$ and $(f_7)$ and that $M$
is a continuous function in $\mathbb R^+_0$, verifying
$(M_1)$ and $(M_3)$, with $\vartheta\ge1$.
Then \eqref{eq1} admits a sequence $(u_k)_{k}$ of
nontrivial solutions.
\end{theorem}
\begin{remark}
{\rm $(i)$ Condition $(V_2)$, which is weaker than the
coercivity assumption: $V(x)\rightarrow \infty$ as $|x|\rightarrow \infty$,
was first proposed by {\em Bartsch} and {\em Wang} in~\cite{BW}
to overcome the lack of compactness.}
$(ii)$ {\rm To our best knowledge, Theorem~\ref{th3} is the first result
for the Schr\"{o}dinger--Kirchhoff equations involving
concave--convex nonlinearities in the fractional setting.
We also refer to \cite{XZF2} for some related multiplicity results.}
\end{remark}
The paper is organized as follows.
In Section~\ref{singularsect} we provide a few remarks about
the singular limit as $s\uparrow 1$.
In Section~\ref{sec3}, we recall some necessary definitions and
properties for the functional setting.
In Section~\ref{sec4}, we obtain some preliminary results.
In Section~\ref{sec5}, the existence of ground states of \eqref{eq1} is obtained by using the mountain pass theorem
together with the Nehari method, and by the direct
methods respectively.
In Section~\ref{sec6}, the existence of infinitely many solutions of \eqref{eq1} is obtained by using the symmetric mountain pass theorem.
\medskip
\noindent
{\bf Acknowledgements}.
Xiang Mingqi was supported by the Fundamental Research Funds for the Central Universities (No.\ 3122015L014). Patrizia Pucci and Marco Squassina are members of the {\em Gruppo Nazionale per l'Analisi Ma\-te\-ma\-ti\-ca, la Probabilit\`a e le loro Applicazioni} (GNAMPA) of the {\em Istituto Nazionale di Alta Matematica} (INdAM).
The manuscript was realized within the auspices of the INdAM -- GNAMPA Project
{\em Problemi variazionali su variet\`a Riemanniane e gruppi di Carnot} (Prot\_2016\_000421). P. Pucci was partly supported by the Italian MIUR project {\em Variational and perturbative aspects of nonlinear differential problems} (201274FYK7).
Binlin Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No.\ A201306).
\section{Remarks on the singular limit as $s\uparrow 1$}
\label{singularsect}
\renewcommand{\S}{{\mathbb S}}
\noindent
The functional framework investigated in the paper admits a very nice consistency
property with more familiar local problems, in the singular limit
as the fractional diffusion parameter $s$
approaches $1$.
Let $\Omega$ be a nonempty open subset of $\R^N$. We denote by $L^2(\Omega,
\mathbb C)$ the Lebesgue space of complex valued functions with
summable square, endowed with the norm $\|u\|_{L^2(\Omega,
\mathbb C)}$.\ We indicate by $H^s_A(\Omega)$ the space of functions
$u\in L^2(\Omega,\mathbb C)$ with finite magnetic Gagliardo semi--norm,
given by
$$
[u]_{H^s_A(\Omega)}=\left(\iint_{\Omega\times\Omega}\frac{|u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{1/2}.
$$
The space $H^s_A(\Omega)$ is equipped with the norm
$$
\|u\|_{H^s_A(\Omega)}=
\big(\|u\|_{L^2(\Omega,\mathbb C)}^2+[u]_{H^s_A(\Omega)}^2\big)^{1/2}.
$$
The space $H^s_{0,A}(\Omega)$ is the completion of $C^\infty_c(\Omega,
\mathbb C)$ in $H^s_A(\Omega)$.
Indeed, in the recent paper~\cite{MSBV}, the following theorem was proved, which is a Bourgain--Brezis--Mironescu
type result in the framework of magnetic Sobolev spaces.
\begin{proposition}[Theorems 1.1 and 1.2 of~\cite{MSBV}]\label{main}
Let $\Omega$ be an open bounded subset of $\R^N$, with Lipschitz
boundary and let $A$ be of class $C^2$ over $\overline{\Omega}$.
Then,
$$\lim_{s\uparrow 1}(1-s)
\iint_{\Omega\times\Omega}\frac{|u(x)-e^{\i (x-y)
\cdot A\left(\frac{x+y}{2}\right)}u(y)|^2}{|x-y|^{N+2s}}dxdy
=K_N\int_{\Omega}|\nabla u-\i A(x)u|^2dx
$$
for every $u\in H^1_{A}(\Omega)$, where
\begin{align*}
K_{N}=\frac{1}{2}\int_{\S^{N-1}}|\omega\cdot
{\bf e}|^{2}d\mathcal{H}^{N-1}(\omega),
\end{align*}
and $\S^{N-1}$ is the unit sphere of $\R^N$
and ${\bf e}$ any unit vector of $\R^{N}$. Furthermore,
$$\lim_{s\uparrow 1}(1-s)
\iint_{\R^{2N}}\frac{|u(x)-e^{\i (x-y)\cdot
A\left(\frac{x+y}{2}\right)}u(y)|^2}{|x-y|^{N+2s}}dxdy=
K_N\int_{\Omega}|\nabla u-\i A(x)u|^2dx
$$
for every $u\in H^1_{0,A}(\Omega)$.
\end{proposition}
Problem \eqref{eq1} could be treated in an arbitrary smooth
open bounded subset $\Omega$ of~$\R^N$, provided that
the solution space
is~$W$, which consists of all functions $u$ in $H^s_{A}(\R^N)$,
with $u=0$ in $\R^N\setminus\Omega$. More precisely, consider
the non--degenerate model case
$$
M(t)=a(s)+b(s)t,\qquad\mbox{where
$a(s)\approx 1-s\,\,$ and\,\, $b(s)\approx (1-s)^2b_0$\quad as $s\uparrow 1$.}
$$
Then the corresponding problem \eqref{eq1} in $\Omega$ writes as
\begin{equation*}
\begin{cases}
\left(1+(1-s)b_0\displaystyle\iint_{\R^{2N}}
\frac{|u(x)-e^{\i (x-y)\cdot A\left(\frac{x+y}{2}\right)}u(y)|^2}
{|x-y|^{N+2s}}dxdy\right)\widehat{(-\Delta)_A^s} u+V(x)u=f(x,|u|)u&\mbox{in }\Omega,\\
u=0 &\mbox{in }\mathbb{R}^N\setminus\Omega,
\end{cases}
\end{equation*}
where $u$ belongs to the solution space $W$ and
$$
\widehat{(-\Delta)_A^s} u=(1-s)(-\Delta)_A^su.
$$
This is natural since the Gagliardo semi--norms are typically multiplied by normalizing constants which vanish
at the rate of $1-s.$ Since by Proposition~\ref{main}
\begin{align*}
(1-s)\iint_{\R^{2N}}\frac{|u(x)-e^{\i (x-y)\cdot A\left(\frac{x+y}{2}\right)}u(y)|^2}{|x-y|^{N+2s}}dxdy & \approx
\int_{\Omega}|\nabla u-\i A(x)u|^2dx\quad\text{as }s\uparrow 1,\\
\widehat{(-\Delta)_A^s} u =(1-s)(-\Delta)_A^su &\approx -(\nabla u-{\rm i} A)^2u,\quad\text{as }s\uparrow 1,
\end{align*}
the above problem converges to the local problem
\begin{equation*}
\begin{cases}
-\left(1+ b_0\displaystyle\int_{\Omega}|\nabla u-\i A(x)u|^2dx\right)(\nabla u-{\rm i} A)^2u+V(x)u=f(x,|u|)u & \mbox{in $\Omega$},\\
u=0 & \mbox{on $\partial\Omega$},
\end{cases}
\end{equation*}
which as $A\to O$ reduces to
\begin{equation*}
\begin{cases}
-\left(1+ b_0\displaystyle\int_{\Omega}|\nabla u|^2dx\right)
\Delta u+V(x)u=f(x,|u|)u & \mbox{in }\Omega,\\
u=0 & \mbox{on }\partial\Omega.
\end{cases}
\end{equation*}
This is the classical model of a Schr\"{o}dinger--Kirchhoff
equation. When $b_0=0$, the last two problems become the
classical Schr\"odinger Dirichlet problems with or
without external magnetic potential~$A$.
\section{Functional setup}\label{sec3}
We first provide some basic functional setting that will be used in the next sections.
The critical exponent $2^*_s$ is defined as $2N/(N-2s).$
Let $L^2(\mathbb{R}^N,V)$ denote the Lebesgue space of real valued functions with
$V(x)|u|^2\in L^1({\mathbb{R}}^N),$ equipped with norm
$$
\|u\|_{2,V}=\left(\int_{\mathbb{R}^N}V(x)|u|^2 dx\right)^{{1}/{2}}\quad \text{for all }u\in L^2(\mathbb{R}^N,V).
$$
The fractional Sobolev space $H_{V}^{s}(\mathbb{R}^N)$ is then defined as
\begin{align*}
H^{s}_V(\mathbb{R}^N)=\big\{u\in L^2(\mathbb{R}^N,V):[u]_{s}<\infty\big\},
\end{align*}
where $[u]_{s}$ is the Gagliardo semi--norm
\begin{align*}
[u]_{s}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}
{|x-y|^{N+2s}}dxdy\right)^{{1}/{2}}.
\end{align*}
The space $H_{V}^{s}(\mathbb{R}^N)$ is endowed with the norm
\begin{align*}
\|u\|_s=\left(\|u\|_{2,V}^2+[u]_{s}^2\right)^{{1}/{2}}.
\end{align*}
The localized norm, on a compact subset $K$ of $\mathbb R^N$,
for the space $H^s_V(K)$, is denoted by
\begin{equation}
\label{localiz}
\|u\|_{s,K}=\left(\int_K V(x)|u|^2dx+\iint_{K\times
K}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{{1}/{2}}.
\end{equation}
The embedding $H^{s}_V(\mathbb{R}^N)\hookrightarrow L^{\nu}(\mathbb{R}^N)$ is continuous for any $\nu\in [2,2_s^*]$ by
\cite[Theorem 6.7]{r28}, namely there exists a positive constant $C$ such that
\begin{align*}
\|u\|_{L^{\nu}(\mathbb{R}^N)}\leq C \|u\|_s\quad\mbox{for all } u\in H^{s}_V(\mathbb{R}^N).
\end{align*}
Let us set
$$
H_{r,V}^{s}(\mathbb{R}^N)=\big\{u\in H^{s}_V(\mathbb{R}^N):u(x)=u(|x|)
\mbox{ for all }x\in\mathbb{R}^N\big\}.
$$
To prove the existence of radial weak solutions of \eqref{eq1}, we shall
use the following embedding theorem due to P.L. Lions.
\begin{theorem}[Compact embedding, I -- Th\'{e}or\`{e}me II.1
of \cite{Lions}]\label{th2.1} Let $N\ge2$.
For any $\alpha\in(2,2_s^*)$ the embedding
$
H_{r,V}^{s}(\mathbb{R}^N)\hookrightarrow\hookrightarrow L^{\alpha}(\mathbb{R}^N)
$
is compact.
\end{theorem}
\noindent
Furthermore, we also have
\begin{theorem}[Compact embedding, II -- Theorem 2.1 of \cite{PXZ}]
\label{th2.2}
Assume that conditions $(V_1)$--$(V_2)$ hold. Then, for any $\nu\in(2,2_s^*)$
the embedding
$
H_V^s(\mathbb{R}^N)\hookrightarrow\hookrightarrow L^\nu(\mathbb{R}^N)
$
is compact.
\end{theorem}
\noindent
Let $L_V^2(\mathbb{R}^N,\mathbb{C})$ be the Lebesgue space of functions $u:\mathbb{R}^N\to\mathbb{C}$
with $V|u|^2 \in L^1({\mathbb R}^N)$, endowed with the (real) scalar product
\begin{align*}
\langle u,v\rangle_{L^2,V}=\Re \int_{\mathbb{R}^N}V(x)u \overline{v}dx\quad \mbox{for all } u,v\in L^2(\mathbb{R}^N,\mathbb{C}),
\end{align*}
where $\bar z$ denotes complex conjugation of $z\in \mathbb{C}$.
Consider now, according to \cite{PDMS}, the magnetic Gagliardo
semi--norm given by
\begin{align*}
[u]_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{{\rm i}(x-y)
\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{1/2}.
\end{align*}
Define $H_{A,V}^{s}(\mathbb{R}^N)$ as the closure of $C_c^\infty(\mathbb{R}^N,\mathbb{C})$ with respect to the norm
\begin{align*}
\|u\|_{s,A}=\big(\|u\|_{L^2,V}^2+[u]_{s,A}^2\big)^{1/2}.
\end{align*}
A scalar product on $H_{A,V}^{s}(\mathbb{R}^N)$ is given by
\begin{align*}
\langle u,v\rangle_{s,A}=\langle u,v\rangle_{L^2,V}+\Re \iint_{\mathbb{R}^{2N}}\frac{\big[u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y)\big]\cdot\big[\overline{v(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}v(y)}\big]}{|x-y|^{N+2s}} dxdy.
\end{align*}
Arguing as in \cite[Proposition 2.1]{PDMS}, we see
that $\big(H_{A,V}^{s}(\mathbb{R}^N),\langle\cdot,\cdot\rangle_{s,A}\big)$
is a real Hilbert space.
\begin{lemma}\label{lemma2.1}
For each $u\in H^s_{A,V}(\mathbb{R}^N,\mathbb{C})$
\begin{align*}
|u|\in H^s_V(\mathbb{R}^N)\quad\mbox{and}\quad
\big\||u|\big\|_{s}\leq \|u\|_{s,A}.
\end{align*}
\end{lemma}
\begin{proof}
The assertion follows directly from the pointwise diamagnetic inequality
\begin{align*}
\big||u(x)|-|u(y)|\big|\leq \left|u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y)\right|,
\end{align*}
for a.e. $x,y\in\mathbb{R}^N$, see \cite[Lemma 3.1, Remark 3.2]{PDMS}.
\end{proof}
\noindent
Following Lemma~\ref{lemma2.1} and using the same discussion of~\cite[Lemma 3.5]{PDMS}, we have
\begin{lemma}\label{lemma2.2}
The embedding
$$
H_{A,V}^s(\mathbb{R}^N,\mathbb{C})\hookrightarrow L^p(\mathbb{R}^N,\mathbb{C})
$$
is continuous for all $p\in[2,2_s^*]$. Furthermore, for any compact
subset $K\subset \mathbb{R}^N$ and all $p\in[1,2_s^*)$ the embeddings
$$
H_{A,V}^s(\mathbb{R}^N,\mathbb{C})\hookrightarrow
H^s_V(K,\mathbb{C})\hookrightarrow \hookrightarrow L^p(K,\mathbb{C})
$$
are continuous and the latter is compact, where
$H^s_V(K,\mathbb{C})$ is endowed with \eqref{localiz}.
\end{lemma}
\noindent
Define now
\begin{align*}
\Ma(\mathbb{R}^N,\mathbb{C})=\big\{u\in H_{A,V}^s(\mathbb{R}^N,\mathbb{C}):u(x)=u(|x|),\,\, x\in\mathbb{R}^N\big\}.
\end{align*}
By Theorems~\ref{th2.1}--\ref{th2.2} and Lemma~\ref{lemma2.1}, we have the following lemma (cf. also \cite[Lemma 4.1]{PDMS}).
\begin{lemma}\label{lemma2.3}
Let $V$ satisfy $(V_1)$. Let $(u_n)_{n}$ be a bounded sequence
in $\Ma(\mathbb{R}^N,\mathbb{C})$. Then, up to a subsequence,
$(|u_n|)_{n}$ converges strongly to some function $u$
in $L^p(\mathbb{R}^N)$ for all $p\in(2,2_s^*)$.
Moreover, if $V$ satisfies $(V_1)$--$(V_2)$, then for all bounded
sequence $(u_n)_{n}$ in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$
the sequence $(|u_n|)_{n}$ admits a subsequence converging strongly
to some $u$ in $L^p(\mathbb{R}^N)$ for all $p\in[2,2_s^*)$.
\end{lemma}
\section{Preliminary results}\label{sec4}
The functional $\mathcal{I}:\Ma(\mathbb{R}^N,\mathbb{C})\to\mathbb{R}$, associated
with equation \eqref{eq1}, is defined by
\begin{align*}
\mathcal{I}(u)=\frac{1}{2}\mathscr{M}([u]_{s,A}^2)
+\frac{1}{2}\|u\|_{L^2,V}^2-\int_{\mathbb{R}^N}F(x,|u|)dx.
\end{align*}
It is easy to see that $\mathcal{I}$ is of class $C^1(\Ma(\mathbb{R}^N,\mathbb{C}),\mathbb{R})$ and
\begin{align*}
\langle \mathcal{I}^\prime(u),v\rangle=
&\Re\bigg[M([u]_{s,A}^2)\iint_{\mathbb{R}^{2N}}\frac{(u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y))(\overline{v(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}v(y)})}{|x-y|^{N+2s}} dxdy\\
&\hspace{6truecm}+\int_{\mathbb{R}^N}Vu\overline{v} dx\bigg]-\Re\int_{\mathbb{R}^N}f(x,|u|)u\overline{v}dx,
\end{align*}
for all $u,v\in \Ma(\mathbb{R}^N,\mathbb{C})$. Hereafter, $\langle\cdot,\cdot\rangle$ denotes the
duality pairing between $\big(\Ma(\mathbb{R}^N,\mathbb{C})\big)'$ and
$\Ma(\mathbb{R}^N,\mathbb{C})$.
Hence, the critical points of $\mathcal{I}$ are exactly the weak solutions of
\eqref{eq1}. Moreover, $\mathscr{M}([u]_{s,A}^2)$
is weakly lower semi--continuous in $\Ma(\mathbb{R}^N,\mathbb{C})$
by the weak lower semi--continuity of
$u\mapsto [u]_{s,A}^2$ jointly with the monotonicity
and continuity of $\mathscr{M}$. Hence, $\mathcal{I}$
is weakly lower semi--continuous in $\Ma(\mathbb{R}^N,\mathbb{C})$,
being $\int_{\mathbb{R}^N}F(x,|u|)dx$ weakly continuous
in $\Ma(\mathbb{R}^N,\mathbb{C})$.
\begin{definition} {\rm We say that $\mathcal{I}$ satisfies the $(PS)$ {\em condition} in $\Ma(\mathbb{R}^N,\mathbb{C})$, if any $(PS)$
sequence $(u_n)_{n}\subset \Ma(\mathbb{R}^N,\mathbb{C})$, namely
a sequence such that $(\mathcal{I}(u_n))_{n}$ is bounded and
$\mathcal{I}^\prime (u_n)\rightarrow 0$ as $n\rightarrow\infty$, admits a strongly convergent
subsequence in $\Ma(\mathbb{R}^N,\mathbb{C})$.}
\end{definition}
\begin{lemma}[Palais--Smale condition]
\label{lemma3.1}
Let $(M_1)$--$(M_2)$ and $(f_1)$--$(f_3)$ hold. Then $\mathcal{I}$
satisfies the $(PS)$ condition in
$\Ma(\mathbb{R}^N,\mathbb{C})$.
\end{lemma}
\begin{proof}
Let $(u_n)_{n}$ be a $(PS)$ sequence in $\Ma(\mathbb{R}^N,\mathbb{C})$. Then there
exists $C>0$ such that
$|\mathcal{I}(u_n)|\leq C$ and $|\langle \mathcal{I}^\prime (u_n),u_n\rangle|\leq C\|u_n\|_{s,A}$
for all $n$. As in Lemma~4.5 of~\cite{capu}, see
also \cite{colpuc},
we divide the proof into two parts.
\smallskip
\noindent
$\bullet$\,{\em Case} $\inf_{n\in\mathbb N}[u_n]_{s,A}=d>0$.
By $(M_1)$, there
exists $\kappa=\kappa(d)>0$ with $M(t)\geq \kappa>0$ for all $t\geq d$.
Thus, $(M_2)$ and $(f_2)$ yield
\begin{equation}\label{NEW}\begin{aligned}
C+C\|u_n\|_{s,A}&\geq \mathcal{I}(u_n)-\frac{1}{\mu}
\langle \mathcal{I}^\prime (u_n),u_n\rangle\\
&=\frac{1}{2}\mathscr{M}([u_n]_{s,A}^2)-\frac{1}{\mu}M([u_n]_{s,A}^2)[u_n]_{s,A}^2
+\left(\frac{1}{2}-\frac{1}{\mu}\right)\|u_n\|_{L^2,V}^2\\
&\ \ -\frac{1}{\mu}\int_{\mathbb{R}^N}(\mu F(x,|u_n|)-f(x,|u_n|)|u_n|^2)dx\\
&\geq \frac{1}{2}\mathscr{M}([u_n]_{s,A}^2)-\frac{1}{\mu}M([u_n]_{s,A}^2)[u_n]_{s,A}^2
+\left(\frac{1}{2}-\frac{1}{\mu}\right)\|u_n\|_{L^2,V}^2\\
&\geq \left(\frac{1}{2\theta}-\frac{1}{\mu}\right)M([u_n]_{s,A}^2)[u_n]_{s,A}^2
+\left(\frac{1}{2}-\frac{1}{\mu}\right)\|u_n\|_{L^2,V}^2\\
&\geq \kappa\left(\frac{1}{2\theta}-\frac{1}{\mu}\right)[u_n]_{s,A}^2+\left(\frac{1}{2}
-\frac{1}{\mu}\right)\|u_n\|_{L^2,V}^2.
\end{aligned}\end{equation}
This implies at once that $(u_n)_{n}$ is bounded in $\Ma(\mathbb{R}^N,\mathbb{C})$, being $\mu>2\theta$. Going if necessary to a subsequence,
thanks to Lemmas~\ref{lemma2.2} and~\ref{lemma2.3}, we have
\begin{align}\label{eq3.1}
&u_n\rightharpoonup u\ \ {\rm in}\ \Ma(\mathbb{R}^N,\mathbb{C}),\ \ u_n\rightarrow u\ \ {\rm a.e.\ in}\ \mathbb{R}^N,\nonumber\\
&|u_n|\rightarrow |u|\ \ {\rm in}\ L^p(\mathbb{R}^N), \\
&|u_n|\leq h\ \ {\rm a.e.\ in}\ \mathbb{R}^N,\ \ {\rm for\ some}\
h\in L^p(\mathbb{R}^N). \nonumber
\end{align}
To prove that $(u_n)_{n}$ converges strongly to $u$ in $\Ma(\mathbb{R}^N,\mathbb{C})$ as $n\to\infty$,
we first introduce a simple notation. Let $\varphi\in \Ma(\mathbb{R}^N,\mathbb{C})$ be fixed and
denote by $L(\varphi)$ the linear functional on $\Ma(\mathbb{R}^N,\mathbb{C})$ defined by
\begin{align}\label{Lu}
\langle L(\varphi), v\rangle
=\Re\iint_{\mathbb{R}^{2N}}
\frac{(\varphi(x)-e^{{\rm i}(x-y)\cdot
A(\frac{x+y}{2})}\varphi(y))}
{|x-y|^{N+2s}}(\overline{v(x)-e^{{\rm i}(x-y)\cdot
A(\frac{x+y}{2})}v(y)})dxdy,
\end{align}
for all $v\in\Ma(\mathbb{R}^N,\mathbb{C})$. Clearly, by the H\"{o}lder inequality, $L(\varphi)$ is continuous, being
\begin{align*}
|\langle L(\varphi),v\rangle|
\leq\|\varphi\|_{s,A}\|v\|_{s,A}.
\end{align*}
Hence the weak convergence in \eqref{eq3.1} gives
\begin{align*}
\lim_{n\rightarrow\infty}\langle L(u),u_n-u\rangle=0.
\end{align*}
Further, by the boundedness of $M([u_n]_{s,A}^2)$ we have
\begin{align}\label{eq3.2}
\lim_{n\rightarrow\infty}M([u_n]_{s,A}^2)\langle L(u),u_n-u\rangle=0.
\end{align}
By $(f_1)$ and $(f_3)$, for any $\varepsilon>0$ there exists $C_\varepsilon>0$ such that
\begin{equation}\label{g2}
|f(x,t)t|\leq \varepsilon|t|+C_\varepsilon|t|^{p-1}\ \ {\rm for \ all}\ x\in\mathbb{R}^N
\ {\rm and}\ t\in\mathbb{R}^+.
\end{equation}
Using the H\"{o}lder inequality, we obtain
\begin{equation}\label{eq3.3}\begin{aligned}
\int_{\mathbb{R}^N}\big|(f(x,&|u_n|)u_n-f(x,|u|)u)(\overline{u_n-u})\big|dx\\
&\leq \int_{\mathbb{R}^N}
[\varepsilon(|u_n|+|u|)+C_\varepsilon(|u_n|^{p-1}+|u|^{p-1})]|u_n-u|dx\\
&\leq \varepsilon(\|u_n\|_{L^2}+\|u\|_{L^2})\|u_n-u\|_{L^2}
+C_\varepsilon(\|u_n\|_{L^p}^{p-1}+\|u\|_{L^p}^{p-1})\|u_n-u\|_{L^p}\\
&\leq C\varepsilon+CC_\varepsilon\|u_n-u\|_{L^p}.
\end{aligned}\end{equation}
The Brezis--Lieb lemma and the fact that $|u_n|\rightarrow |u|$ in $L^p(\mathbb{R}^N)$ give
\begin{align*}
\lim_{n\rightarrow\infty}\int_{\mathbb{R}^N}|u_n-u|^pdx
=\lim_{n\rightarrow\infty}\int_{\mathbb{R}^N}\big(|u_n|^p-|u|^p
\big)dx=0.\end{align*}
Inserting this in \eqref{eq3.3}, we get
\begin{align}\label{eq3.4}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^N}(f(x,|u_n|)u_n-f(x,|u|)u)(\overline{u_n-u})dx=0,
\end{align}
since $\varepsilon$ is arbitrary.
Of course, $\langle\mathcal{I}'(u_n)-\mathcal{I}'(u),u_n-u\rangle\rightarrow0$ as $n\to\infty$, since $u_n\rightharpoonup u$ in $\Ma(\mathbb{R}^N,\mathbb{C})$ and $\mathcal{I}^\prime(u_n)\to 0$ in the dual space of $\Ma(\mathbb{R}^N,\mathbb{C})$. Thus,
\begin{align*}
o(1)&=\langle\mathcal{I}^\prime(u_n)-\mathcal{I}^\prime(u),u_n-u\rangle\\
&=M([u_n]_{s,A}^2)\langle L(u_n)-L(u),u_n-u\rangle+\|u_n-u\|_{L^2,V}^2\\
&+\left(M([u_n]_{s,A}^2)-M([u]_{s,A}^2)\right)\langle L(u),u_n-u\rangle-\Re\int_{\mathbb{R}^N}(f(x,|u_n|)u_n-f(x,|u|)u)(\overline{u_n-u})dx,
\end{align*}
this, together with \eqref{eq3.2} and \eqref{eq3.4}, implies that
\begin{align*}\lim_{n\rightarrow\infty}\left(
M([u_n]_{s,A}^2)\langle L(u_n)-L(u),u_n-u\rangle+\|u_n-u\|_{L^2,V}^2
\right)=0,
\end{align*}
which yields $u_n\rightarrow u$ in
$\Ma(\mathbb{R}^N,\mathbb{C})$, since $M([u_n]_{s,A}^2)\ge\kappa>0$
for all $n\geq 1$.
\smallskip
\noindent
$\bullet$\,\textit{Case} $\inf_{n\in\mathbb N}[u_n]_{s,A}=0$.
If $0$ is an isolated point for $([u_n]_{s,A})_n$, then there is a subsequence $([u_{n_k}]_{s,A})_k$ such that
$\inf_{k\in\mathbb N}[u_{n_k}]_{s,A}=d>0$ and one can proceed as before.\
If, instead, $0$ is an accumulation point for $([u_n]_{s,A})_n$, there is a subsequence, still labeled as $(u_n)_n$, such that
\begin{equation}\label{con5}
[u_n]_{s,A}\to 0,\,\,\,
\mbox{$u_n\to 0$ in }L^{2^*_s}(\mathbb R^N)\mbox{ and a.e. in }\mathbb R^N.
\end{equation}
We claim that $(u_n)_n$ converges strongly to $0$ in
$\Ma(\mathbb{R}^N,\mathbb{C})$. To this aim, we need only to show that
$\|u_n\|_{2,V}\to 0$ thanks to \eqref{con5}.
Now, \eqref{NEW} and \eqref{con5} yield that as $n\to\infty$
\begin{equation*}
C+C\|u_n\|_{2,V}+o(1)\ge \left(\frac{1}{2}
-\frac{1}{\mu}\right)\|u_n\|^2_{2,V}+o(1).
\end{equation*}
Hence, $(u_n)_n$ is bounded in $L^2(\mathbb R^N,V)$ and so
in $\Ma(\mathbb{R}^N,\mathbb{C})$. Thus, by \eqref{con5}
and Lemma~\ref{lemma2.2}
\begin{align}\label{con6}
u_n\rightharpoonup 0\mbox{ in $\Ma(\mathbb{R}^N,\mathbb{C})$
and $u_n\rightarrow 0$ in }L^{p}(\mathbb R^N),\end{align}
being $p\in(2,2^*_s)$.
Clearly, by \eqref{g2} and \eqref{con6}, for every $\varepsilon>0$
$$
\Bigg|\int_{\mathbb R^N}f(x,|u_n|)u_n^2dx\Bigg|\le\varepsilon\|u_n\|^{2}_{2} +C_\varepsilon\|u_n\|^p_p=\varepsilon C+o(1)
$$
as $n\to\infty$. Thus,
\begin{equation}\label{x9}
\lim_{n\to\infty}\int_{\mathbb R^N}f(x,|u_n|)u_n^2dx=0,
\end{equation}
being $\varepsilon>0$ arbitrary.
Obviously, $\langle\mathcal I'(u_n),u_n\rangle\to 0$ as $n\to\infty$, by \eqref{con6} and the fact that $\mathcal I'(u_n)\to 0$ in $\big(\Ma(\mathbb{R}^N,\mathbb{C})\big)'$. Hence, by the continuity of $M$ and \eqref{con5}--\eqref{x9}, we have
\begin{align*}
o(1)&=\langle \mathcal
I'(u_n),u_n\rangle=M([u_n]_{s,A}^2)[u_n]_{s,A}^2+\|u_n\|_{2,V}^2
-\int_{\mathbb R^N}f(x,|u_n|)u_n^2dx\\
&=\|u_n\|_{2,V}^2+o(1)
\end{align*}
as $n\to\infty$. This shows the claim.
Therefore, $\mathcal I$ satisfies the $(PS)$ condition
in $\Ma(\mathbb{R}^N,\mathbb{C})$ also in this second
case and this completes the proof.
\end{proof}
\noindent
Before going to the proof of Theorem \ref{th1}, we give some useful preliminary results.
\begin{lemma}[Mountain Pass Geometry I]\label{lemma3.2}
Assume that $(M_1)$--$(M_2)$, $(f_1)$ and $(f_3)$ hold. Then there exist constant $\varrho,\alpha>0$ such that
$\mathcal{I}(u)\geq \alpha$ for all $u\in \Ma(\mathbb{R}^N,\mathbb{C})$ with $\|u\|_{s,A}=\varrho$.
\end{lemma}
\begin{proof}
It follows from $(f_3)$ that for any $\varepsilon\in (0,1)$ there
exists $\delta=\delta(\varepsilon)>0$ such that
$|f(x,t)|\leq \varepsilon$ for all $x\in\mathbb{R}^N$
and $t\in[0,\delta]$.
On the other hand, $(f_1)$ yields that
$|f(x,t)|\leq C\big(1+\delta^{2-p}\big)|t|^{p-2}$
for all $x\in\mathbb{R}^N$ and $t>\delta$. In conclusion,
\begin{align}\label{eqbu}
|f(x,t)|\leq \varepsilon+C\big(1+\delta^{2-p}\big)|t|^{p-2}\quad
\mbox{for all } x\in\mathbb{R}^N \mbox{ and } t\in\mathbb R^+_0.
\end{align}
Whence, for some $C_\eps>0$, we get
\begin{align}\label{eq3.5}
|F(x,t)|\leq \int_0^t |f(x,\tau)\tau| d\tau
\leq \frac{\varepsilon}{2}t^2+C_\varepsilon t^{p},
\end{align}
for all $x\in\mathbb{R}^N$ and $t\geq0$.
Moreover, $(M_2)$ gives
\begin{align}\label{eqbu3.7}
\mathscr{M}(t)\geq \mathscr{M}(1)t^\theta\quad \mbox{for all }
t\in[0, 1],
\end{align}
while $(M_1)$ implies that $ \mathscr{M}(1)>0$.
Thus, using \eqref{eq3.5}, \eqref{eqbu3.7} and the H\"{o}lder
inequality, we obtain
for all $u\in \Ma(\mathbb{R}^N,\mathbb{C})$, with $\|u\|_{s,A}\leq 1$,
\begin{align*}
\mathcal{I}(u)&=\frac{1}{2}\mathscr{M}(\|u\|_{s,A}^2)+\frac{1}{2}\|u\|_{L^2,V}^2
-\int_{\mathbb{R}^N}F(x,|u|)dx\\
&\geq \frac{\mathscr{M}(1)}{2}[u]_{s,A}^{2\theta}+\frac{1}{2}\|u\|_{L^2,V}^2
-\frac{\varepsilon}{2}\int_{\mathbb{R}^N}|u|^2dx
-C_\varepsilon\int_{\mathbb{R^N}}|u|^pdx\\
&\geq \frac{\mathscr{M}(1)}{2}[u]_{s,A}^{2\theta}
+\Big(\frac{1}{2}-\frac{\varepsilon}{2V_0}\Big)\|u\|_{L^2,V}^2
-C_\varepsilon C_p^p\|u\|_{s,A}^p\\
&\geq \min\left\{\frac{\mathscr{M}(1)}{2},\frac{V_0-\varepsilon}{2V_0}\right\}([u]_{s,A}^{2\theta}
+\|u\|_{L^2,V}^2)-C_\varepsilon C_p^p\|u\|_{s,A}^p\\
&\geq \min\left\{\frac{\mathscr{M}(1)}{2},\frac{V_0-\varepsilon}{2V_0}\right\}([u]_{s,A}^{2\theta}
+\|u\|_{L^2,V}^{2\theta})-C_\varepsilon C_p^p\|u\|_{s,A}^p\\
&\geq 2^{1-\theta}\min\left\{\frac{\mathscr{M}(1)}{2},\frac{V_0-\varepsilon}{2V_0}\right\}([u]_{s,A}^{2}
+\|u\|_{L^2,V}^2)^{\theta}-C_\varepsilon C_p^p\|u\|_{s,A}^p\\
&=\left(2^{1-\theta}\min\left\{\frac{\mathscr{M}(1)}{2},
\frac{V_0-\varepsilon}{2V_0}\right\}-
C_\varepsilon C_p^p\|u\|_{s,A}^{p-2\vartheta}\right)
\|u\|_{s,A}^{2\vartheta},
\end{align*}
where $C_p$ is the embedding constant of $\Ma(\mathbb{R}^N,\mathbb{C})$ into
$L^p(\mathbb{R}^N,\mathbb{C})$ given by Lemma~\ref{lemma2.2}. Here we used that $\|u_n\|_{L^2,V}\leq \|u_n\|_{s,A}\leq 1$ and the inequality $(a+b)^\theta\leq 2^{\theta-1}(a^\theta+b^\theta)$ for all $a,b\geq0$.
Choosing $\varepsilon=V_0/2$ and taking $\|u\|_{s,A}=\varrho\in(0,1)$
so small that
$$
2^{1-\theta}\min\left\{\frac{\mathscr{M}(1)}{2},\frac{V_0}{4}\right\}-C_{V_0/2} C_p^p\varrho^{p-2\vartheta}>0,
$$
we have
\begin{align*}
\mathcal{I}(u)\geq \alpha=
\left(2^{1-\theta}\min\left\{\frac{\mathscr{M}(1)}{2},
\frac{V_0}{4}\right\}-C_{V_0/2} C_p^p\varrho^{p-2\vartheta}\right)
\varrho^{2\vartheta}>0,
\end{align*}
for all $u\in \Ma(\mathbb{R}^N,\mathbb{C})$, with $\|u\|_{s,A}=\varrho$.
\end{proof}
\begin{lemma}[Mountain Pass Geometry II]\label{lemma3.3}
Assume that $(M_1)$--$(M_2)$ and $(f_1)$--$(f_4)$ hold.
Then there exists
$e\in C_c^\infty(\mathbb{R}^N,\mathbb{C})$, with
$\|e\|_{s,A}\ge2$, such that
$\mathcal{I}(e)<0$. In particular, $\|e\|_{s,A}>\rho$,
where $\rho>0$ is the number introduced in Lemma~$\ref{lemma3.2}$
\end{lemma}
\begin{proof}
For any $x\in \mathbb{R}^N$, set $k(t)=F(x,t)t^{-\mu}$ for all $t\geq 1$.
Condition $(f_2)$ implies
that $k$ is nondecreasing on $[1,\infty)$. Therefore,
$k(t)\geq k(1)$ for any $t\geq1$, that is,
\begin{align}\label{eq3.6}
F(x,t)\geq F(x,1)t^\mu\geq c_F|t|^\mu\quad \mbox{for all }x\in\mathbb{R}^N
\mbox{and }t\geq1,
\end{align}
where $c_F=\inf_{x\in\mathbb{R}^N}F(x,1)>0$ by assumption $(f_4)$.
{From} $(f_3)$ there exists $\delta\in(0,1)$ such that
$|f(x,t)t|\leq t$ for all $x \in\mathbb{R}^N$
and $t\in[0,\delta]$.
Furthermore, $|f(x,t)|\leq 2C$ for all $x\in\mathbb{R}^N$ and all $t$, with $\delta<t\leq1$, thanks to $(f_1)$.
Hence, the above inequalities imply that
$f(x,t)t\geq -(1+2C)t$ for $x\in\mathbb{R}^N$ and $t\in[0,1]$.
Thus,
\begin{align}\label{eq3.7}
F(x,t)=\int_0^t f(x,\tau)\tau d\tau\geq -\frac{1+2C}{2}t^2\quad \mbox{for
all } x\in\mathbb{R}^N\mbox{ and }t\in[0,1].
\end{align}
Combining \eqref{eq3.6} with \eqref{eq3.7}, we obtain
\begin{align}\label{eq3.8}
F(x,t)\geq c_F|t|^\mu-C_F|t|^2\ \ {\rm for\ all}\ x\in\mathbb{R}^N\ {\rm and}\ t\geq0,
\end{align}
where $C_F=c_F+(1+2C)/2$.
Again $(M_2)$ gives
\begin{align}\label{eqbu3.8}
\mathscr{M}(t)\leq \mathscr{M}(1)t^\theta\quad \mbox{for all }t\geq1,
\end{align}
with $\mathscr{M}(1)>0$ by $(M_1)$.
Fix $u\in C_c^\infty(\mathbb{R}^N,\mathbb{C})$, with $[u]_{s,A}=1$.
By~\eqref{eq3.8} and~\eqref{eqbu3.8} as $t\rightarrow\infty$
\begin{align*}
\mathcal{I}(tu)
&=\frac{1}{2}\mathscr{M}([tu]_{s,A}^2)+\frac{1}{2}\|tu\|_{L^2,V}^2-
\int_{\mathbb{R}^N}F(x,t|u|)dx\\
&\leq \frac{\mathscr{M}(1)}{2}t^{2\theta}[u]_{s,A}^{2\theta}
+\frac{1}{2}\|tu\|_{L^2,V}^2
-c_F t^\mu \|u\|_{L^\mu(\mathbb{R}^N)}^\mu
+\frac{M_1}{V_0} t^2\|u\|_{L^2,V}^2\\
&\leq \frac{\mathscr{M}(1)}{2}t^{2\theta}-c_FC_\mu^\mu t^\mu\|u\|_{s,A}^\mu
+\left(\frac{M_1}{V_0} +\frac{1}{2}\right) t^2\|u\|_{L^2,V}^2\\
&\leq\frac{\mathscr{M}(1)}{2}t^{2\theta}-c_FC_\mu^\mu t^\mu[u]_{s,A}^\mu
+\left(\frac{M_1}{V_0} +\frac{1}{2}\right) t^2\|u\|_{L^2,V}^2\\
&=\frac{\mathscr{M}(1)}{2}t^{2\theta}-c_FC_\mu^\mu t^\mu
+\left(\frac{M_1}{V_0} +\frac{1}{2}\right) t^2\|u\|_{L^2,V}^2\to-\infty,
\end{align*}
since $2\le2\theta<\mu$.
The assertion follows at once, taking $e=T_0u$, with $T_0>0$ large
enough.
\end{proof}
\section{Proof of Theorems~\ref{th1} and~\ref{th2}}\label{sec5}
The following standard Mountain Pass Theorem will
be used to get our main result.
\begin{theorem}\label{th4.1}
Let $J$ be a functional on a real Banach space $E$ and
of class $C^1(E,\mathbb{R})$. Let us assume that there exists $\alpha$, $\rho>0$ such that\\
$(i)$ $J(u)\geq\alpha$ for all $u\in E$ with $\|u\|=\rho$,\\
$(ii)$ $J(0)=0$ and $J(e)<\alpha$ for some $e\in E$ with $\|e\|>\rho$.\\
Let us define $\Gamma=\{\gamma\in C([0,1];E):\gamma(0)=0,\gamma(1)=e\}$, and
\begin{align*}
c=\inf_{\gamma\in \Gamma}\max_{t\in[0,1]}J(\gamma(t)).
\end{align*}
Then there exists a sequence $(u_n)_n$ in $E$ such
that $J(u_n)\rightarrow c$ and $J^\prime(u_n)\rightarrow 0$
in $E^\prime$, the dual space of $E$, as $n\to\infty$.
\end{theorem}
\subsection{Proof of Theorem \ref{th1}}
Taking into account Lemmas \ref{lemma3.2} and \ref{lemma3.3}, by Theorem \ref{th4.1} there exists a sequence
$(u_n)_{n}\subset \Ma(\mathbb{R}^N,\mathbb{C})$ such that
$\mathcal{I}(u_n)\rightarrow c>0$ and $\mathcal{I}^\prime(u_n)\rightarrow 0$ as $n\to\infty$.
Then, in view of Lemma \ref{lemma3.1}, there exists a nontrivial
critical point $u_0\in \Ma(\mathbb{R}^N,\mathbb{C})$ of $\mathcal{I}$ with $\mathcal{I}(u_0)=c>0=\mathcal{I}(0)$.
Set $\mathscr{N}=\{u\in
\Ma(\mathbb{R}^N,\mathbb{C})\setminus\{0\}:\mathcal{I}^\prime(u)=0\}$.
Then $u_0\in \mathscr{N}\neq\emptyset$.
Next we show that $\mathcal{I}$ is coercive and bounded from below on $\mathscr{N}.$ Indeed, by $\mathcal{I}^\prime(u)=0$ and $(f_2)$, we get
\begin{align}\label{eq3.10}
\int_{\mathbb{R}^N}F(x,|u|)dx\leq\frac{1}{\mu}\int_{\mathbb{R}^N}f(x,|u|)|u|^2dx
=\frac{1}{\mu}\big(M([u]_{s,A}^2)[u]_{s,A}^2+\|u\|_{L^2,V}^2\big).
\end{align}
By using \eqref{eq3.10}, $(M_2)$ and the fact that $2\le2\theta<\mu$,
for all $u\in\mathscr{N}$, we have
\begin{align*}
\mathcal{I}(u)&
\geq \frac{1}{2}\mathscr{M}(\|u\|_{s,A}^2)+\frac{1}{2}\|u\|_{L^2,V}^2
-\frac{1}{\mu}(M([u]_{s,A}^2)[u]_{s,A}^2+\|u\|_{L^2,V}^2)\\
&=\left(\frac{1}{2\theta}
-\frac{1}{\mu}\right)M([u]_{s,A}^2)[u]_{s,A}^2+\left(\frac{1}{2}
-\frac{1}{\mu}\right)\|u\|_{L^2,V}^2\ge0.
\end{align*}
Hence, by $(M_1)$ and $(M_3)$ for $u\in\mathscr{N}$
\begin{align}\label{eq3.11}
\mathcal{I}(u)&\ge\left(\frac{1}{2\theta}-\frac{1}{\mu}\right)
\cdot\left(\|u\|_{L^2,V}^2
+\begin{cases}\kappa[u]_{s,A}^2,&\mbox{if }[u]_{s,A}\ge1\\
m_0[u]_{s,A}^{2\theta},&\mbox{if }[u]_{s,A}\le1\end{cases}\right),
\end{align}
where $\kappa=\kappa(1)>0$ by $(M_1)$.
Hence in all cases, for all $u\in\mathscr{N}$
$$\mathcal{I}(u)\ge\min\{\kappa,m_0\}
\left(\frac{1}{2\theta}-\frac{1}{\mu}\right)\|u\|_{s,A}^2-1,$$
by the elementary inequality $t^\theta\ge t-1$ for all
$t\in\mathbb R^+_0$.
In particular, $\mathcal{I}$ is coercive and bounded from
below on $\mathscr{N}$.
Define $c_{\rm min}=\inf\{\mathcal{I}(u):u\in\mathscr{N}\}$.
Clearly, $0\le c_{\rm min}\leq \mathcal{I}(u_0)=c$.
Let $(u_n)_{n}$ be a minimizing
for $c_{\rm min},$ namely $\mathcal{I}(u_n)\rightarrow c_{\rm min}$ and $\langle \mathcal{I}^\prime(u_n),u_n\rangle=0$. Then,
since $\mathscr{N}$ is a complete metric space, by Ekeland's variational principle we can find a new minimizing sequence,
still denoted by $(u_n)_{n}$, which is a $(PS)$ sequence for
$\mathcal{I}$ at the level $ c_{\rm min}$.
Moreover, Lemma~\ref{lemma3.1} implies that $(u_n)_{n}$ has a
convergence subsequence, which we
still denote by $(u_n)_{n}$, such that
$u_n\rightarrow u$ in $\Ma(\mathbb{R}^N,\mathbb{C})$. Thus
$c_{\rm min}=\mathcal{I}(u)$ and
$\langle\mathcal{I}'(u),u\rangle=0$.
We claim that $c_{\rm min}>0$.
Otherwise, there is $(u_n)_{n}\subset \Ma(\mathbb{R}^N,\mathbb{C})\setminus\{0\}$ with $\mathcal{I}^\prime(u_n)=0$
and $\mathcal{I}(u_n)\rightarrow 0$. This via \eqref{eq3.11} implies that $\|u_n\|_{s,A}\rightarrow0$.
On the other hand, by \eqref{eqbu}, we have for any $\varepsilon\in (0,V_0)$
\begin{equation*}
M([u_n]_{s,A}^2)[u_n]_{s,A}^2+\|u_n\|_{L^2,V}^2 =\int_{\mathbb{R}^N}f(x,|u_n|)|u_n|^2dx
\leq \frac{\varepsilon}{V_0}\|u_n\|^2_{L^2,V}+C_\varepsilon C_p^p\|u_n\|_{s,A}^p.
\end{equation*}
Thus, $M([u_n]_{s,A}^2)[u_n]_{s,A}^2+\Big(1-\varepsilon/V_0\Big)
\|u_n\|_{L^2,V}^2
\leq C_\varepsilon C_p^p\|u_n\|_{s,A}^p$.
Now take $N_1$ so large that $\|u_n\|_{s,A}\le1$ for all $n\ge N_1$.
Hence, $(M_3)$ implies that for all $n\ge N_1$
\begin{align*}
m_0[u_n]_{s,A}^{2\theta}+\big(1-\varepsilon/V_0\big)
\|u_n\|_{L^2,V}^{2\theta}
&\leq C_\varepsilon C_p^p\|u_n\|_{s,A}^p,
\end{align*}
that is
\begin{align*}
\min\big\{m_0,\big(1-\varepsilon/ V_0\big)\big\}\leq
C_\varepsilon C_p^p\|u_n\|_{s,A}^{p-2\theta}.
\end{align*}
This is a contradiction since $2\theta<p$ and proves the claim.
Thus, $u$ is a nontrivial critical point of $\mathcal{I}$,
with $\mathcal{I}(u)=c_{\rm min}>0$.
Therefore, $u$ is a ground state solution of \eqref{eq1}.\qed
\subsection{Proof of Theorem \ref{th2}}
By $(f_5)$, $(V_1)$ and the H\"{o}lder inequality, for all
$u\in H_{s,A}^s(\mathbb{R}^N,\mathbb{C})$ we have
\begin{align*}
\mathcal{I}(u) &\geq \frac{1}{2}\mathscr{M}([u]_{s,A}^2)+\frac{1}{2}\|u\|_{L^2,V}^2-
\int_{\mathbb{R}^N}a(x)|u|^{q}dx\\
&\geq \frac{1}{2}\mathscr{M}([u]_{s,A}^2)
+\frac{1}{2}\|u\|_{L^2,V}^2-\|a\|_{L^{\frac{2}{2-q}}}
\|u\|_{L^2}^q\\
&\geq \frac{1}{2}\mathscr{M}([u]_{s,A}^2)+\frac{1}{4}\|u\|_{L^2,V}^2
+\frac{V_0}{4}\|u\|_{L^2}^2-\|a\|_{L^{\frac{2}{2-q}}}\|u\|_{L^2}^q\\
&\geq \frac{1}{2}\mathscr{M}([u]_{s,A}^2)+\frac{1}{4}\|u\|_{L^2,V}^2-C_0,\\
&C_0=\frac{\|a\|_{L^{\frac{2}{2-q}}}}{2q}(2q-1)
\left(\frac{2\|a\|_{L^{\frac{2}{2-q}}}}
{qV_0}\right)^{q/(2-q)}.
\end{align*}
As shown in \eqref{eq3.11}, this, $(M_1)$ and $(M_3)$
imply at once that for all $u\in H_{s,A}^s(\mathbb{R}^N,\mathbb{C})$
$$\mathcal{I}(u) \ge\frac{\min\{\kappa,m_0\}}4\,
\|u\|_{s,A}^2-1-C_0,$$
$\kappa=\kappa(1)$.
Hence $\mathcal{I}$ is coercive and
bounded below on $H_{s,A}^s(\mathbb{R}^N,\mathbb{C})$. Set
\begin{align*}
J(u)=\frac{1}{2}\mathscr{M}([u]_{s,A}^2)+\frac{1}{2}\|u\|_{L^2,V}^2,\quad
H(u)=\int_{\mathbb{R}^N}F(x,|u|)dx
\end{align*}
for all $u\in H_{A,V}^s(\mathbb{R}^N)$.
Then $J$ is weakly lower semi--continuous in
$H_{A,V}^s(\mathbb{R}^N)$, since $\mathscr{M}$ is continuous and
monotone non--decreasing in $\mathbb R^+_0$. Moreover, by using a
similar discussion as \cite[Lemma~2.3]{PXZ2}, one can show that $H$ is weakly
continuous on $H_{A,V}^s(\mathbb{R}^N)$ under condition $(f_5)$.
Thus, $\mathcal{I}(u)=J(u)-H(u)$ is weakly lower semi--continuous
in $H_{A,V}^s(\mathbb{R}^N)$.
Then there exists $u_0\in H_{A,V}^s(\mathbb{R}^N)$ such that
$$\mathcal{I}(u_0)=\inf\{\mathcal{I}(u):u\in H_{A,V}^s(\mathbb{R}^N)\}.$$
Next we show $u_0\neq0$. Let $x_0\in \Omega$ and let $R>0$ such that $B_R(x_0)\subset \Omega$. Fix $\varphi\in C_0^\infty(B_{R}(x_0))$ with
$0\leq\varphi\leq 1$, $\| \varphi\|_{s,A}\leq C(R)$ and $\|\varphi\|_{L^q(B_R(x_0))}\neq 0$.
Then, by $(f_6)$ for all $t\in(0,\delta)$
\begin{align*}
\mathcal{I}(t\varphi)&\leq \frac{t^2}{2}\left(\sup_{0\leq \xi\leq (\delta C(R))^2}M(\xi)\right)[\varphi]_{s,A}^2+\frac{t^2}{2}\|\varphi\|_{L^2,V}^2
-t^{q}\int_{B_{R}(x_0)}a_0|\varphi|^qdx\\
&\leq \frac{t^2}{2}\left(\sup_{0\leq \xi\leq (\delta C(R))^2}M(\xi)+1\right)\|\varphi\|_{s,A}^2-
t^{q}a_0\|\varphi\|_{L^q(B_R(x_0))}.
\end{align*}
Since $1<q<2$, we get $\mathcal{I}(\bar t\varphi)<0$ by taking $\bar t>0$ small enough.
Hence $\mathcal{I}(u_0)\leq \mathcal{I}(\bar t\varphi)<0$,
and so $u_0$ is a nontrivial critical point.
In other words, $u_0$ is a nontrivial solution of \eqref{eq1}. \qed
\section{Proof of Theorem \ref{th3}}\label{sec6}
We first recall the following symmetric mountain pass theorem
in~\cite{RK}.
\begin{theorem}\label{th4}
Let $X$ be an infinite dimensional real Banach space.
Suppose that $J$ is in $C^1(X,\mathbb{R})$ and satisfies the following condition:\\
$(a)$ $J$ is even, bounded from below, $J(0)=0$ and $J$ satisfies
the $(PS)$ condition;\\
$(b)$ For each $k\in\mathbb{N}$ there exists
$E_k\subset \Gamma_k$ such that
$\sup_{u\in E_k}J(u)<0$, where
\begin{align*}
\Gamma_k=\{E: E\mbox{ is closed symmetric subset
of $X$ and } 0\notin E, \ \gamma(E)\geq k\}
\end{align*}
and $\gamma(E)$ is a genus of a closed symmetric set $E$.
Then $J$ admits a sequence of critical points $(u_k)_k$ such
that $J(u_k)\leq 0$, $u_k\neq0$ and $\|u_k\|\rightarrow0$
as $k\to\infty$.
\end{theorem}
Let $h\in C^1(\mathbb R^+_0,\mathbb{R})$ be a radial decreasing function such that $0\leq h(t)\leq 1$ for all
$t\in\mathbb R^+_0$, $h(t)=1$ for $0\leq t\leq 1$ and $h(t)=0$ for $t\geq 2$.
Let $\phi(u)=h(\|u\|^2_{s,A})$.
Following the idea of \cite{FV}, we consider the truncation functional
\begin{align*}
\mathcal{I}(u)=\frac{1}{2}\mathscr{M}([u]_{s,A}^2)
+\frac{1}{2}\|u\|_{L^2,V}^2-\phi(u)\int_{\mathbb{R}^N}F(x,|u|)dx.
\end{align*}
Clearly, $\mathcal{I}\in C^1(H_{A,V}^s(\mathbb{R}^N,\mathbb{C}),\mathbb{R})$ and
\begin{align*}
\langle \mathcal{I}^\prime(u),v\rangle&=M([u]_{s,A}^2)\langle L(u), v\rangle
+\Re\int_{\mathbb{R}^N}V(x)u\overline{v}dx\\
&\qquad-2\phi^\prime(u)\int_{\mathbb{R}^N}F(x,|u|)dx\cdot \langle L(u), v\rangle
-\phi(u)\Re\int_{\mathbb{R}^N}f(x,|u|)u\overline{v}dx
\end{align*}
for all $u,v\in H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$. Here $L(u)$ is the linear functional
on $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$, introduced in~\eqref{Lu}.
\subsection{Proof of Theorem \ref{th3}}
For all $u\in H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$, with $\|u\|_{s,A}\geq 2$, we get
$$\mathcal{I}(u)\geq \frac{1}{2}\mathscr{M}([u]_{s,A}^2)+\frac{1}{2}\|u\|_{L^2,V}^2\ge\frac{1}{2}\min\{\kappa,\,m_0\}\|u\|_{s,A}^2,$$
by $(M_1)$ and $(M_3)$, where $\kappa=\kappa(1)$, as in the proof of
Theorem~\ref{th2}. Hence
$\mathcal{I}(u)\rightarrow \infty$ as
$\|u\|_{s,A}\rightarrow \infty$ and $\mathcal{I}$ is coercive and bounded from below on $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$.
Let $(u_n)_n$ be a $(PS)$ sequence, i.e. $\mathcal{I}(u_n)$ is bounded and $\mathcal{I}^\prime(u_n)\rightarrow 0$ as $n\rightarrow\infty$. Then the coercivity of $\mathcal{I}$
implies that $(u_n)_n$ is bounded in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$. Without loss of generality, we assume that $u_n\rightharpoonup u$ in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$ and
$u_n\rightarrow u$ a.e. in $\mathbb{R}^N$.
We now claim that
\begin{align}\label{bueq4}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^N}(f(x,|u_n|)u_n-f(x,|u|)u)(\overline{u_n-u})dx=0.
\end{align}
Clearly, $|f(x,t)t|\leq C(|t|+|t|^{p-1})$ for all
$x\in\mathbb{R}^N$ and $t\in\mathbb{R}^+_0$ by $(f_1)$.
Using the H\"{o}lder inequality, we obtain
\begin{align}\label{eqbu3.3}
&\int_{\mathbb{R}^N}|(f(x,|u_n|)u_n-f(x,|u|)u)(\overline{u_n-u})|dx\nonumber\\
&\leq \int_{\mathbb{R}^N}
C[|u_n|+|u|+|u_n|^{p-1}+|u|^{p-1}]|u_n-u|dx\\
&\leq C(\|u_n\|_{L^2}+\|u\|_{L^2})\|u_n-u\|_{L^2}
+C(\|u_n\|_{L^p}^{p-1}+\|u\|_{L^p(\mathbb{R}^N)}^{p-1})\|u_n-u\|_{L^p(\mathbb{R}^N)}
\nonumber\\
&\leq C(\|u_n-u\|_{L^2}+\|u_n-u\|_{L^p}).\nonumber
\end{align}
Lemma \ref{lemma2.3} guarantees that $|u_n|\rightarrow |u|$ in $L^p(\mathbb{R}^N)$ and $|u_n|\rightarrow |u|$ in $L^2(\mathbb{R}^N)$.
Hence, $u_n\to u$ in $L^p(\mathbb{R}^N,\mathbb{C})$ and in $L^2(\mathbb{R}^N,\mathbb{C})$ by the Brezis--Lieb lemma.
Inserting these facts in \eqref{eqbu3.3}, we get the desired claim \eqref{bueq4}.
Now, $\langle \mathcal{I}^\prime(u_n)-\mathcal{I}^\prime(u),u_n-u\rangle\rightarrow0$,
since $\mathcal{I}^\prime(u_n)\rightarrow0$ and $u_n\rightharpoonup u$ in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$.
By \eqref{bueq4}, we have as $n\to\infty$
\begin{align*}
o(1)&=\langle \mathcal{I}^\prime(u_n)-\mathcal{I}^\prime(u),u_n-u\rangle=M([u_n]_{s,A}^2)\langle L(u_n), u_n-u\rangle-M([u]_{s,A}^2)\langle L(u), u_n-u\rangle\\
&+\Re\int_{\mathbb{R}^N}V(x)(u_n-u)\overline{(u_n-u)}dx
-2\phi^\prime(u_n)\int_{\mathbb{R}^N}F(x,|u_n|)dx\cdot \langle L(u_n), u_n-u\rangle\\
&-2\phi^\prime(u)\int_{\mathbb{R}^N}F(x,|u|)dx\cdot \langle L(u), u_n-u\rangle
-\phi(u_n)\Re\int_{\mathbb{R}^N}f(x,|u_n|)u_n(\overline{u_n-u})dx\\
&-\phi(u)\Re\int_{\mathbb{R}^N}f(x,|u|)u(\overline{u_n-u})dx.
\end{align*}
From $(f_7)$ and the facts that $u_n\rightharpoonup u$
in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$ and $\phi^\prime\leq 0$ it
follows that
\begin{align}\label{PPP}
0\le M([u_n]_{s,A}^2)\langle L(u_n)-L(u), u_n-u\rangle
+\Re\int_{\mathbb{R}^N}V(x)(u_n-u)\overline{(u_n-u)}dx\leq o(1).
\end{align}
We divide the proof into two parts.\smallskip
\noindent {\em Case $\inf_{n\in\mathbb N}[u_n]_{s,A}=d>0$.}
By $(M_1)$, there exists $\kappa=\kappa(d)>0$ with $M(t)\geq \kappa>0$ for all $t\geq d$.
This, together with \eqref{PPP}, implies that
\begin{align*}
\lim_{n\rightarrow\infty}\bigg[\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}(u_n(y)-u(y))|^2}{|x-y|^{N+2s}}dxdy
+\int_{\mathbb{R}^N}V(x)|u_n-u|^2dx\bigg]=0.
\end{align*}
Hence $u_n\rightarrow u$ in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$.
\smallskip
\noindent{\em Case $\inf_{n\in\mathbb N}[u_n]_{s,A}=0$}.
If $0$ is an isolated point for $([u_n]_{s,A})_n$, then there is a
subsequence $([u_{n_k}]_{s,A})_k$ such that
$\inf_{k\in\mathbb N}[u_{n_k}]_{s,A}=d>0$ and one can proceed as before.
If, instead, $0$ is an accumulation point for $([u_n]_{s,A})_n$, there is a subsequence,
still labeled as $(u_n)_n$, such that
$[u_n]_{s,A}\to 0$ and $u_n\to 0$ in $L^{2^*_s}(\mathbb{R}^N)$ as $n\to\infty$ and
again \eqref{PPP} implies at once that
$u_n\rightarrow 0$ in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$, since $\langle L(u_n)-L(u), u_n-u\rangle\to0$ and $M([u_n]_{s,A}^2)\to M(0)\ge0$.
In conclusion, $\mathcal{I}$ satisfies the $(PS)$ condition in $H_{A,V}^s(\mathbb{R}^N,\mathbb{C})$.
For each $k\in\mathbb N$, we take $k$ disjoint open sets $K_i$
such that $\bigcup_{i=1}^k K_i\subset\Omega$. For each
$i=1,\cdots,k$ let $u_i\in (H_{A,V}^s(\mathbb{R}^N,\mathbb{C})
\bigcap C_0^\infty(K_i,\mathbb{C}))\backslash\{0\}$, with $\|u_i\|_{s,A}=1$,
and $W_k={\rm span}\{u_1,u_2,\cdots,u_k\}.$
Therefore, for any $u\in W_k$, with
$\|u\|_{s,A}=\rho\leq 1$ small enough, we obtain by $(f_7)$,
being $q\in(1,2)$,
\begin{align*}
\mathcal{I}(u)&\leq \frac{1}{2}\left(\max_{0\leq t\leq 1}M(t)\right)[u]_{s,A}^2+\frac{1}{2}\|u\|_{L^2,V}^2
-\int_{\Omega}a_1|u|^qdx\\
&\leq \frac{1}{2}
\left(1+\max_{0\leq t\leq 1}M(t)\right)\|u\|_{s,A}^2
-C_k^qa_1\|u\|_{s,A}^q\\
&=\frac{1}{2}\left(1+\max_{0\leq t\leq 1}M(t)\right)\rho^2-
C_k^q a_1\rho^q<0,
\end{align*}
where $C_k>0$ is a constant such that $\|u\|_{L^q(\mathbb{R}^N,\mathbb{C})}
\le C_k\|u\|_{s,A}$ for all $u\in W_k$, since all norms on~$W_k$ are equivalent.
Therefore, we deduce
\begin{align*}
\{u\in W_k:\|u\|_{s,A}=\rho\}\subset \{u\in W_k:\mathcal{I}(u)<0\}.
\end{align*}
Obviously, $\gamma(\{u\in W_k:\|u\|_{s,A}=\rho\})=k$, see~\cite{chang}.
Hence by the monotonicity of the
genus $\gamma$, cf. \cite{krasnoselskii}, we obtain
\begin{align*}
\gamma(u\in W_k:\mathcal{I}(u)<0)\geq k.
\end{align*}
Choosing $E_k=\{u\in W_k:\mathcal{I}(u)<0\}$, we have $E_k\subset \Gamma_k$ and
$\sup_{u\in\Gamma_k}\mathcal{I}(u)<0$. Thus, all the assumptions of Theorem~\ref{th4} are satisfied,
Hence, there exists a sequence $(u_k)_k$ such that
$$\mathcal{I}(u_k)\leq 0,\quad \mathcal{I}^\prime(u_k)=0,\quad\mbox{and}\quad
\|u_k\|_{s,A}\rightarrow 0\mbox{ as }k\rightarrow\infty.$$
Therefore, we can take $k$ so large that
$\|u_k\|_{s,A}\leq 1$, and so these infinitely many functions $u_k$
are solutions of~\eqref{eq1}.\qed
\medskip
| 88,391
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TITLE: Problem related to continuous functions on real numbers
QUESTION [1 upvotes]: Let $X$ be a subset of $\mathbb R$ and let $f, g : X \to X$ be two continuous functions such that $f(X) \cap g(X) = \emptyset$ and $ f(X) \cup g(X) = X.$ Then which one of the following sets can't be equal to $X.$
$[0,1]$
$[0,1)$
$(0,1)$
$ \mathbb R.$
I am just feeling it too hard to start. Any suggestion will be highly appreciated. Thank you.
Update:
I was just giving a thought in the following way-
If $X = [0,1]$ then if $f(x)$ is open(or closed) then from continuity of $f$, $X$ becomes open(or closed) and then from continuity of $g$, $X$ becomes closed(or open), which in either case a contradiction. So $X \neq [0,1].$
(Here both $f(X)$ and $g(X)$ can't be open(or closed) as $[0,1]$ is connected.)
Now if $X = (0,1)$ then no contradiction arises in the above argument so it can be possible.
For $X=\mathbb R$ similarly it can be shown that it's a possible case.
In this way I am struck at the case where $ X =[0,1).$
Am I thinking in a right direction ?
REPLY [1 votes]: $X=[0,1]$ is indeed impossible, for otherwise $f[X], g[X]$ would both be compact (thus closed) and form a disconnection for $[0,1]$ which is connected. Contradiction.
| 5,006
|
TITLE: Plane from intersection line and point
QUESTION [1 upvotes]: The task: Determine the plane containing point $P( -5 , 2 , 3 )$ and going through the intersection line of the planes $2x + y + 5z = 31$ and $-4x + 5y + 4z = 50$
1.: Intersect the two given planes, resulting in a line in parameter form ( $X = P + t * V$ )
2.: Determine two arbitary points on the line
3.: Form the new plane using the two points on the line and the given point P
(by creating an equation system $ax + by + cz + d = 0$, with $3$ equations with the $x$, $y$ and $z$ values of the points inserted)
Is my approach for solving the problem right? (I don't ask for a solution!)
REPLY [3 votes]: For any two parameters $\lambda, \mu \in \mathbb{R}$, not both zero, the combination
$$
\lambda \cdot (2x+y+5z) + \mu\cdot(−4x+5y+4z)= \lambda \cdot 31 + \mu \cdot 50
$$
is the equation of a plane that has the same intersection line as the two given planes. (This is called the pencil of planes through that line.) Substitution of the point $(−5,2,3)$ gives:
$$
\lambda \cdot 7 + \mu \cdot 42 = \lambda \cdot 31 + \mu \cdot 50
$$
or $24 \lambda + 8 \mu = 0$. So you can take $\lambda = 1$ and $\mu=-3$ to find the requested plane.
| 20,505
|
Unknown
Tree damage and costs to vehicles
Reported via Android in the Trees category anonymously at 11:26, Sat 13 September 2014
Sent to Cherwell District Council 4 minutes later
Good morning.
The tree outside our house is damaging the paintwork, wipers and windows seals on my company vehicle and my family car.
I am a chauffeur and the company car has To be maintained in showroom condition. leaving for work at 5am in a car damaged by tree sediment and the occasional falling branch is unacceptable.
This car is a week old and cost over £70000. Any permanent damaged caused by this tree will be pursued through the necessary channels, if we cannot get a practical solution. This is a semi rural area surrounded by large gardens full of trees as well as parks and sports fields.
Please reply to acknowledge receipt. you should know that a copy of this is going to a director of my company.
Many thanks and regards Paul Stanley 07809 432020
Provide an update
Please note that updates are not sent to the council. Your information will only be used in accordance with our privacy policy
| 199,151
|
1/29/2017
I love weddings! They celebrate the love of a man and a woman as they promise to love each other forever. We will ignore the 50% divorce rate in this fantasy. Picking out the gift is one of the most fun parts of the occasion for me. There is a right way and a wrong way to go about getting a wedding gift for the bride and groom. We'll take a look at different options throughout this blog. Hopefully this will make your next wedding gift purchase a little easier to pick out.
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| 96,682
|
A man walking through the woods near a river hears desperate screams for help. He runs to the river to see someone struggling as the river pulls him downstream. He jumps in and pulls the person to safety.
As soon as he gets to the shore, he hears another person coming downstream, screaming for help. He jumps back in and rescues that person. Sure enough, just as he gets the second person to shore, another person comes down the river, screaming for help. He rescues that person, and another and still another. As more and more people come down river and he begins to tire, he stops jumping in and heads upriver.
When asked where he is going, he replies: "I'm going upstream to find out who is throwing these people in and stop them!"
This story came to mind after reading an article featuring a new phrase in the world of marriage and the family: So many 20- and 30-somethings are getting married and within only a few years divorced, the people who track their numbers call them "starter marriages."
I'm struggling for the right words to comment on "starter marriages." The words that keep coming to me are "no, no, wrong, no, no, hell no!"
I'm familiar with starter cars, starter homes, starter jobs. But starter marriages? What's next, starter kids? As in, these are the kids we practice parenting on, and later we raise some kids to adulthood.
It frustrates me when sociologists or some other "ologist" finds a creative label for a painful phenomenon, as if a clever name takes care of it.
The guy in the above story was doing a worthy thing, trying to help people who were drowning, just as I hope that whoever came up with the starter marriage label is trying to do a worthy thing. At some point, however, you have to go upstream and deal with what is causing the problem in the first place.
How to create a successful marriage is a crucial thing we need to learn but one rarely taught in school. So the question becomes, what do we need to know before we get married in order to have a successful marriage?
Here are some suggestions:
1) Choose well
It's often easier said than done. Love can make you blind. It can make you temporarily stupid, too. One way to choose well is to be aware of your own relationship radar - how you go about becoming attracted to certain people. If this radar is faulty, you likely will be attracted to someone who may not be good for you. In order to choose wisely, you may have to choose differently as well.
2) Pre-marital counseling
It's a great way to identify and work out some bugs early on. Whether you see a minister or therapist, you can discover areas that might be challenging for your relationship and learn skills and techniques for handling them. Believing problem areas will automatically get better after marriage is a cruel myth. Without learning methods for managing differences, they almost are guaranteed to get worse, not better.
3) Have a teachable spirit
Being teachable is a hallmark of success. Many people enter into marriage thinking they know how to do it right. I know I did. I even had a license and degree on my wall that said I was a marriage and family expert. Fortunately, I was blessed to have someone who was willing to hang around while I learned.
So, learn all you can about marriage, relationships, communication, etc.
Columnist Sydney J. Harris said "Almost no one is foolish enough to imagine that he automatically deserves great success in any field of activity; yet almost everyone believes that he automatically deserves success in marriage."
Read books, go to seminars, get good coaching when and even before you need it. You also need to learn from each other. Teach each other how to be each other's own unique partner.
4) Become a good heart-tender
When we get married, we become the caretaker of someone else's heart. We can break it, ignore it, or take great care of.
| 168,014
|
\begin{document}
\title{Quantum Monodromy in the Isotropic 3-Dimensional Harmonic Oscillator}
\author{Irina Chiscop$^1$, Holger R. Dullin$^2$, Konstantinos Efstathiou$^1$, and Holger Waalkens$^1$\\
\small $^1$Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen\\$^2$School of Mathematics and Statistics, University of Sydney}
\date{\today}
\begin{abstract}
The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems.
Separating in a particular coordinate system defines a system of three commuting operators,
one of which is the Hamiltonian.
We show that the joint spectrum of the Hamilton operator,
the $z$ component of the angular momentum, and a quartic integral
obtained from separation in prolate spheroidal coordinates
has quantum monodromy for sufficiently large energies.
This means that one cannot globally assign quantum numbers to the joint spectrum.
The effect can be classically explained by showing that the corresponding
Liouville integrable system has a non-degenerate focus-focus point, and hence Hamiltonian monodromy.
\end{abstract}
\maketitle
\section{Introduction}
The isotropic harmonic oscillator is at the same time the simplest and the most important system in physics.
The system is very special in both the classical and the quantum setting. All (nontrivial) solutions of the classical equations of motion are periodic and even have the same period. The quantum system is special in that it has an equidistant energy spectrum.
The best explanation of these special properties in both the classical and the quantum setting
are the symmetries of the system.
The energy spectrum is independent of the dimension, however, the degeneracy of the energy levels increases with dimension.
What we are going to show is that within a degenerate energy eigenspace
we can define a \emph{quantum integrable system} (QIS) whose joint spectrum is non-trivial in the sense that it does not allow for a global assignment of quantum numbers.
With a QIS for an $N-$dimensional isotropic harmonic oscillator we mean a set of
$N$ commuting operators $\mathcal{H} = (\hat H_1, \dots, \hat H_N)$ with say $\hat H_1$ being the Hamilton operator of the system.
Because the operators commute their spectra can be measured simultaneously:
$\hat H_i \psi = \lambda_{i} \psi$, $i=1, \dots, N$.
Together they define the joint spectrum which associates a point in $N-$dimensional space with coordinates $\lambda_{i}$ to each eigenfunction $\psi$.
It follows from the Bohr-Sommerfeld quantization of classical actions whose local existence in turn follows from the Liouville-Arnold Theorem \cite{Arnold78}
that the joint spectrum locally has the structure of a lattice $\Z^N$.
We show that for $N=3$ there is a QIS for which there is an obstruction to the global existence of action-angle variables due to monodromy \cite{Duistermaat80},
which manifests itself as a lattice defect in the joint spectrum that prevents the global assignment of quantum numbers \cite{CushDuist88,VuNgoc99,SadovskiiZhilinskii99,Zhilinskii2006}.
Monodromy and generalizations of monodromy \cite{Nekhoroshev2006,Sadovskii2007164} have been extensively studied in recent years and found for many different systems, see, e.g., \cite{Zhilinskii2011} and the references therein. Quantum monodromy explains, e.g., problems in assigning rovibrational spectra of molecules \cite{CDGHJLSZ04,Child2008,Assematetal2010}
or electronic spectra of atoms in external fields \cite{CushmanSadovskii99,EfstathiouSadovskiiZhilinskii2007}.
Moreover it provides a mechanism for excited-state quantum phase transitions \cite{Cejnaretal2006,Caprio20081106}.
The generalization of monodromy to scattering systems has been shown to lead to defects in the lattice of transparent states
in planar central scattering \cite{DullinWaalkens08}.
Monodromy can also play a role in spatiotemporal nonlinear wave systems \cite{Sugnyetal2009}, and
dynamical manifestations of monodromy have recently been studied in \cite{Delos14}.
Another way of thinking about our result is as follows. Due to the high degree of symmetry the
quantum harmonic oscillator is not only a QIS but it has additional independent operators
that commute with $\hat H$. Such a system is called super-integrable.
Important examples are systems that are separable in different coordinate systems.
Schwarzschild \cite{Schwarzschild1916}
was the first to point out that if the Hamilton-Jacobi equation of $H$ can be separated in more
than one coordinate system, the quantum energy eigenvalues of $\hat H$ are degenerate.
Such a Hamiltonian operator $\hat H$ is called multiseparable,
and is hence included in non-equivalent QIS's $\mathcal H$ and $\mathcal G$.
The simplest multiseparable systems are the free particle, the Kepler problem, and the harmonic oscillator.
A multiseparable system with $N$ degrees of freedom is superintegrable, because if both
$\mathcal{H}$ and $\mathcal{G}$ contain $\hat H$, then we have found
more than $N-1$ operators that commute with $\hat H$.
An important group of 3-dimensional superintegrable and multiseparable systems
is classified in \cite{Kress06}.
The classical geometry of superintegrable systems is well understood. Fixing the integrals defines tori of lower dimension than in the Liouville-Arnold Theorem and Nekhoroshev showed that one can construct lower dimensional action-angle coordinates in a kind of generalization of the Liouville-Arnold Theorem \cite{Nekhoroshev1972}.
More global aspects have been studied in \cite{FomenkoMishchenko78,Delzant87}.
The isotropic three-dimensional harmonic oscillator is maximally superintegrable which means that together with the Hamiltonian it has five independent integrals. The joint level sets are one-dimensional tori, i.e. periodic orbits, whose projection to configuration space are ellipses centered at the center of the force.
From the classical geometric point of view considering tori with half the dimension of phase space in
a super-integrable system appears somewhat arbitrary. However, from the quantum point of view it
is prudent to study all possible sets of commuting observables, because these tell us what can
be measured simultaneously as the uncertainty principle is trivial in this case.
Thus we are going to study a particular set of collections of ellipse shaped periodic orbits that form 3-tori in phase space,
and we will show that the joint quantum spectrum associated to these tori has quantum monodromy.
If a Hamiltonian $\hat H$ is super-integrable then there are distinct
QIS that share the given Hamiltonian $\hat H$, but form non-equivalent
QIS with in general different joint spectra. The eigenvalues of $\hat H$ and their degeneracy
are the same in each realisation, but the joint spectrum within a degenerate eigenspace
and the corresponding basis of eigenfunctions are different.
We are focusing on the case where the different QIS are obtained from
separation in different coordinate systems.
Separation in different coordinate systems gives different QIS with the same Hamiltonian $\hat H$.
A Hamiltonian that is multi-separable is also super-integrable, since there are more than $n$ integrals.
For the 3-dimensional harmonic oscillator this is well known.
On the one hand it separates in Cartesian coordinates into a sum of one-degree-of-freedom
harmonic oscillators, so that the wave function for the multi-dimensional case is simply a product
of wave functions for the one-dimensional case, which are given in terms of Hermite polynomials.
On the other hand it separates in spherical coordinates, which leads to wave functions that
are products of spherical harmonics and associated Laguerre polynomials.
The associated quantum numbers have different meaning, but the total number
of states of a three-dimensional harmonic oscillator with angular frequency $\omega$ and energy $E =\hbar \omega( n + 3/2)$ is $(n+1)(n+2)/2$ with ``principalÕÕ qauntum number $n=0,1,2,\ldots$.
In the first case we have a quantum number $n_i = 0, 1, 2,\dots$ for each 1D oscillator,
and the eigenvalues of $\hat H$ are $E = \hbar \omega (n + 3/2) = \hbar \omega (n_1 + n_2 + n_3 + 3/2)$.
In the second case (see, e.g., \cite{Griffiths2016})
we have $E = \hbar \omega (2k+l+3/2)$ for non-negative integer $k$ where $l$ is the total angular momentum
eigenvalue $l = n, n-2, n-4, \dots$ down to 0 or 1, depending on whether $n$ is even or odd, respectively.
In addition there is the usual ``magneticÕÕ quantum number $m = -l, \dots, l$.
In both cases the quantum states form a lattice in which lattice points can be uniquely labelled
by quantum numbers. The details of the two lattices are, however, different. In particular the actions are not even locally related by unimodular transformation.
Specifically, we are going to separate the isotropic harmonic oscillator in prolate spheroidal coordinates.
Prolate spheroidal coordinates are a family of coordinate systems where the family parameter $a$
is half the distance between the focus points of a family of confocal ellipses and hyperbolas, which in order to get corresponding coordinate surfaces are rotated about
the axis containing the focus points.
In the limit $a \to 0$ spherical coordinates are obtained, and in the limit $a \to \infty$ parabolic
coordinates are obtained.
Our main result is that when the energy $E > \frac12 \omega^2 a^2$ then the system has monodromy.
Our approach is similar to a recent analysis of the Kepler problem \cite{DullinWaalkens2018},
which through separation in prolate spheroidal coordinates leads to a quantum integrable system
that does not possess three global quantum numbers.
This paper is organized as follows. In Sec.~\ref{sec:separation} we introduce the classical three-dimensional isotropic harmonic oscillator, discuss its symmetries and its separation in prolate spheriodal coordinates.
In Sec.~\ref{sec:bifdiag} we compute the bifurcation diagram for the energy momentum map associated with separation in prolate spheroidal coordinates and prove the presence of monodromy.
The effect of monodromy on the quantum spectrum is studied in Sec.~\ref{sec:sepQM}.
We conclude with some comments in Sec.~\ref{sec:discussion}.
\section{Classical separation in prolate spheroidal coordinates}
\label{sec:separation}
The three-dimensional isotropic harmonic oscillator has Hamiltonian
\begin{equation}\label{eq:defH}
H = \frac12 |\mathbf{p}|^2 + \frac{\omega^2}{2} |\mathbf{r}|^2 \,,
\end{equation}
where $\mathbf{r} = (x,y,z)^T$
and $\mathbf{p} = (p_x, p_y, p_z)^T$ are the canonical variables on the phase space $T^*\mathbb{R}^3\cong \mathbb{R}^6$.
By choosing suitable units we can assume that the frequency $\omega$ has the value $1$.
But in order to identify terms arising from the potential we will keep $\omega$ in the equations below.
Not only are the three separated Hamiltonians
$$\mathbf{A} = ( \tfrac12 (p_x^2 + \omega^2 x^2), \tfrac12 (p_y^2 + \omega^2 y^2), \tfrac12 (p_z^2 + \omega^2 z^2) )^T $$
constants of motion, but so are the components of the angular momentum
$\mathbf{L} = \mathbf{r} \times \mathbf{p}$.
Not all these integrals are independent. But any
five of them are, so that $H$ is maximally superintegrable.
Define $$\mathbf{B} = ( \{L_x, A_y\} , \{L_y, A_z\} , \{L_z, A_x\} )^T,$$
where $\{ \cdot \,,\,\cdot \}$ is the Poisson bracket. The algebra of 9 quadratic integrals
$\mathbf{A, B, L}$ closes and defines a Lie-Poisson bracket, shown in Table~\ref{eq:poisson-cp2}, that is isomorphic to the Lie algebra $\mathfrak{su}(3)$ (see also \cite{Fradkin1965}). Fixing the relations between the integrals $\mathbf{A}$, $\mathbf{B}$, $\mathbf{L}$ defines an embedding of the reduced symplectic manifold $\mathbb{C}P^2$ into $\mathbb{R}^9$.
Here $\mathbb{C}P^2$ is the orbit space of the $S^1$ action induced on $\mathbb C^3 \simeq T^* \mathbb R^3$ by the Hamiltonian flow of $H$ \cite{Moser1970}.
The Hamiltonian $H = A_x + A_y + A_z $ is a Casimir. The algebra has two more Casimirs, the quadratic $C_2 = 2 \mathbf{A}^2 + \omega^2 \mathbf{L}^2 + \mathbf{B}^2$ and the cubic
\[
C_3 = 6 \mathrm{Re}(w_x w_y w_z) + \sum_{k=x,y,z} 2|w_k|^2(H-3A_k) - \frac{8}{27}(H-3A_k)^3,
\]
where $w_k = B_k + i \omega L_k$, $k=x,y,z$.
\begin{table}[htbp]
\renewcommand{\arraystretch}{1.2}
\begin{ruledtabular}
\begin{tabular}{C|CCCCCCCCC}
\{\downarrow,\rightarrow\} & A_x & A_y & A_z & L_x & L_y & L_z & B_x & B_y & B_z \\
\colrule
A_x & 0 & 0 & 0 & 0 & B_y & -B_z & 0 & -\omega ^2 L_y & \omega ^2 L_z \\
A_y & 0 & 0 & 0 & -B_x & 0 & B_z & \omega ^2 L_x & 0 & -\omega ^2 L_z \\
A_z & 0 & 0 & 0 & B_x & -B_y & 0 & -\omega ^2 L_x & \omega ^2 L_y & 0 \\
L_x & 0 & B_x & -B_x & 0 & L_z & -L_y & 2 A_z-2 A_y & -B_z & B_y \\
L_y & -B_y & 0 & B_y & -L_z & 0 & L_x & B_z & 2 A_x-2 A_z & -B_x \\
L_z & B_z & -B_z & 0 & L_y & -L_x & 0 & -B_y & B_x & 2 A_y-2 A_x \\
B_x & 0 & -\omega ^2 L_x & \omega ^2 L_x & 2 A_y-2 A_z & -B_z & B_y & 0 & -\omega ^2 L_z & \omega ^2 L_y \\
B_y & \omega ^2 L_y & 0 & -\omega ^2 L_y & B_z & 2 A_z-2 A_x & -B_x & \omega ^2 L_z & 0 & -\omega ^2 L_x \\
B_z & -\omega ^2 L_z & \omega ^2 L_z & 0 & -B_y & B_x & 2 A_x-2 A_y & -\omega ^2 L_y & \omega ^2 L_x & 0 \\
\end{tabular}
\end{ruledtabular}
\caption{Poisson structure on $\mathbb{C}P^2$.\label{eq:poisson-cp2}}
\end{table}
The huge symmetry of the isotropic harmonic oscillator is also reflected by its separability in different coordinate systems.
In fact, the three-dimensional oscillator separates in several different coordinate systems. The most well known are the systems of Cartesian coordinates and spherical coordinates (see, e.g., \cite{Fradkin1965}).
In this paper we will be studying the separation in prolate spheroidal coordinates. The separability in these coordinates is, e.g., mentioned in \cite{CoulsonJoseph1967}. The coordinates are defined with respect to
two focus points which we assume to be located on the $z$ axis at $\mathbf{a}=(0,0,a)$ and $-\mathbf{a}=(0,0,-a)$ where $a>0$.
The prolate spheroidal coordinates are then defined as
$$
(\xi,\eta,\varphi) = \Big(\frac{1}{2a} ( r_+ + r_-),\, \frac{1}{2a} (r_+ - r_-),\, \mathrm{arg}(x+\ui y) \Big) ,
$$
where $r_\pm = |\mathbf{r}\pm \mathbf{a}|$. They have ranges $\xi\ge1$, $-1\le \eta\le1$ and $0\le \varphi \le 2\pi$.
The surfaces of constant $\xi>1$ and $-1<\eta<1$ are confocal prolate ellipsoids and two-sheeted hyperboloids which are rotationally symmetric about the $z$ axis and have focus points at $\pm \mathbf{a}$.
For $\xi\to 1$, the ellipsoids collapse to the line segment connecting the focus points,
and for $\eta\to \pm 1$, the hyperboloids collapse to the half-lines consisting of the part of the $z$ axis above and below the focus points, respectively.
The Hamiltonian in prolate spheroidal coordinates becomes
$$
H = \frac12 \frac{1}{a^2(\xi^2-\eta^2)} (p_\xi^2(\xi^2-1) + p_\eta^2 (1-\eta^2)) + \frac{1}{2} \frac{p_\varphi^2}{a^2(\xi^2-1)(1-\eta^2)} + \frac12 a^2 \omega^2 (\xi^2 + \eta^2-1).
$$
The angle $\varphi$ is cyclic. So $p_\varphi$ which is the $z$ component of the angular momentum is a constant of motion.
Multiplying the energy equation $H=E$ by $2a^2(\xi^2-\eta^2)$ and reordering terms gives the separation constant
\begin{equation} \label{eq:G_xieta}
\begin{split}
G &:= - p_\xi^2(\xi^2-1) - \frac{l_z^2}{\xi^2-1} - a^4 \omega^2 \xi^2 (\xi^2 -1) + 2a^2 (\xi^2-1)E\\
&\phantom{:}=
\phantom{-} p_\eta^2 (1-\eta^2) + \frac{l_z^2}{1-\eta^2} + a^4 \omega^2 \eta^2(1-\eta^2) - 2a^2 (1-\eta^2)E,
\end{split}
\end{equation}
where we use $l_z$ to denote the value of $p_\varphi$.
Rewriting the separation constant in Cartesian coordinates gives
\begin{equation}\label{eq:G_Cart}
G = L_x^2 + L_y^2 + L_z^2 - 2 a^2 (A_x + A_y).
\end{equation}
The functions $\mathcal{G} = ( H, L_z, G)$ are independent and their mutual Poisson brackets vanish. They thus define a Liouville integrable system which as we will see has a singular foliation by Lagrangian tori with monodromy which we then also study quantum mechanically.
\section{Bifurcation diagram and reduction}
\label{sec:bifdiag}
Solving \eqref{eq:G_xieta} for the momenta $p_\eta$ and $p_\xi$ we get
\begin{equation} \label{eq:separated_momenta}
p^2_\xi = \frac{P(\xi)}{(\xi^2-1)^2} \quad \text{ and } \quad p^2_\eta = \frac{P(\eta)}{(\eta^2-1)^2} \,,
\end{equation}
where
\begin{equation} \label{eq:def_P(s)}
P(s) = -l_z^2 + 2 a^2 (1-s^2) \left[ \left(E - \frac12 a^2 \omega^2 s^2 \right) (1-s^2) + \frac{g}{2a^2} \right]
\end{equation}
with $g$ denoting the value of the separation constant $G$.
The roots of the polynomial $P(s)$ are turning points in the corresponding separated degree of freedom, i.e. roots in $[-1,1]$ correspond to turning points in the $(\eta , p_\eta)$ phase plane and roots in $[1,\infty)$ correspond to turning points in the $(\xi, p_\xi)$ phase plane. Critical motion occurs for values of the constants of motion where turning points collide, i.e. for double-roots of $P(s)$.
The bifurcation diagram, i.e. the set of critical values of the energy momentum map $\mathcal{G} = ( H, L_z, G): T^*\mathbb{R}^3 \to \mathbb{R}^3$, $(\mathbf{r} ,\mathbf{p} )\mapsto (E,l_z,g)$, can thus be found from
the vanishing of the discriminant of the polynomial $P(s)$. However, care has to be taken due to the singularities of the prolate spheroidal coordinates at the focus points.
In Sec.~\ref{sec:reduction} below we will therefore derive the bifurcation diagram more rigorously using the method of singular reduction \cite{CushBates}.
For $l_z=0$, the motion (in configuration space) takes place in invariant planes of constant angles about the $z$ axis.
We will consider this case first and study the case of
general $l_z$ afterwards.
\subsection{The two-dimensional harmonic oscillator ($l_z = 0$)}
\begin{figure}
\includegraphics[width=4.5cm]{bifdiag_planar_xfig} \hspace*{1cm}
\raisebox{3cm}{I}
\includegraphics[width=4cm]{poly_planar_I} \hspace*{1cm}
\raisebox{3cm}{II}
\includegraphics[width=4cm]{poly_planar_II} \\[1.5ex]
\raisebox{3cm}{III}
\includegraphics[width=4cm]{poly_planar_III}\hspace*{1cm}
\raisebox{3cm}{IV}
\includegraphics[width=4cm]{poly_planar_IV} \hspace*{1cm}
\raisebox{3cm}{V}
\includegraphics[width=4cm]{poly_planar_V}
\caption{Bifurcation diagram of the planar harmonic oscillator with energy momentum map $(H,G)$ (top left). The remaining panels show the graphs of the polynomial $P(s)$ for representative values of $(E,g)$ in the regions $I$ to $V$ marked in the $(g,h)$ plane.
In region I: all roots are real and satisfy $|s_{2\pm}|<|s_{3\pm}|<|s_{1\pm}|$.
In region II: $s_{2\pm}$ and $s_{3\pm}$ are complex.
In region III: all roots are real and satisfy $|s_{1\pm}|<|s_{2\pm}|<|s_{3\pm}|$.
In region IV: all roots are real and satisfy $|s_{2\pm}|<|s_{1\pm}|<|s_{3\pm}|$.
In region V: $s_{2\pm}$ are complex and $s_{3\pm}$ are real with $|s_{1\pm}|<|s_{3\pm}|$.
} \label{fig:bifdiag_planar}
\end{figure}
From the one-parameter family of two-dimensional harmonic oscillators with $l_z=0$ we will consider the one in the $(x,z)$ plane. This is an integrable system with the energy momentum map $(H,G)$ where $H$ and $G$ are the constants of motion defined in \eqref{eq:defH} and \eqref{eq:G_Cart} restricted to $y=p_y=0$.
For $l_z = 0$, the roots of $P(s)$ are
\begin{equation*}
\begin{split}
s_{1\pm} &=\pm 1, \\
s_{2\pm} &= \pm
{\frac{1}{2\omega a} {\sqrt { 2 {a}^{2}{\omega}^{2}+4\,h - 2\,
\sqrt{ (a^2\omega^2-2E)^2 - 4\, g {\omega}^{2}}
}}} ,\\
s_{3\pm} &= \pm
{\frac{1}{2\omega a} {\sqrt { 2 {a}^{2}{\omega}^{2}+4\,h + 2\,
\sqrt{ (a^2\omega^2-2E)^2 - 4\, g {\omega}^{2}}
}}} .
\end{split}
\end{equation*}
For values $(E,g)$ for which $s_{2\pm}$ and $s_{3\pm}$ are real, we have $|s_{2\pm}| \le |s_{3\pm}|$. If $s_{3\pm}$ are not real then $s_{2\pm}$ are also not real. But conversely $s_{3\pm}$ can be real even if $s_{2\pm}$ are not real.
The discriminant of $P(s)$ is
$$
\text{discrim} (P(s),s) = 64\, {a}^{12}{\omega}^{2}
\left( 2\,{a}^{2}E+g \right)
g^4
\left( (a^2\omega^2-2E)^2 - 4\, g { \omega}^{2} \right) ^{2} .
$$
Double roots occur for
$$
{\cal L}_1 := \{ g=-2\,a^2 E \}, \,
{\cal L}_2 := \{ g= 0 \}, \,
{\cal L}_3 := \{ g= {\frac {(a^2\omega^2-2\,E)^2 }{{
4\,\omega}^{2}}}
\}.
$$
The curves ${\cal L}_i$, $i=1,2,3$, divide the upper $(g,E)$ half plane into five region with different dispositions of roots as shown in Fig.~\ref{fig:bifdiag_planar}.
From the separated momenta in \eqref{eq:separated_momenta} we see that the values of the constants of motion facilitate
physical motion (i.e. real momenta) if the resulting $P(s)$ is positive somewhere in $[-1,1]$ and at the same time positive somewhere in $[1,\infty)$. From Fig.~\ref{fig:bifdiag_planar} we see that this is the case only for regions III and IV.
For a fixed energy $E\ge0$, the minimal value of $g$ is determined by the collision of the roots $s_{2\pm}$ at $0$.
Whereas for a fixed energy $E> \frac12 \omega^2a^2$, the maximal value of $g$ is determined by the collision of the pairs of roots $s_{2\pm}$ and $s_{3\pm}$, the maximal value of $g$
for a fixed energy $0< E < \frac12 \omega^2a^2$ is determined by the collision of the pairs of roots
$s_{3\pm}$ and $s_{1\pm}=\pm1$. At the boundary between regions III and IV, the pairs of roots $s_{2\pm}$ and $s_{1\pm}=\pm1$ collide.
For a value $(E,g)$ in region IV, the preimage under the energy momentum map $(H,G)$ is a two-torus consisting of a one-parameter family of periodic orbits whose projection to configuration space are ellipses which are enveloped by a caustic formed by the ellipse given by the coordinate line $\xi = s_{3+}$ and the two branches of the confocal hyperbola corresponding to the coordinate line $\eta= s_{2+}$ (see Fig.~\ref{fig:caustics}a).
For a value $(E,g)$ in region III, the preimage under the energy momentum map $(H,G)$ is a two-torus consisting of a one-parameter family of periodic orbits whose projection to configuration space are ellipses which are enveloped by a caustic formed by two confocal ellipses given by the coordinate lines $\xi = s_{2+}$ and $\xi = s_{3+}$, respectively (see Fig.~\ref{fig:caustics}c).
The boundary ${\cal L}_2 = \{ g=0 \}$ between regions III and IV is formed by critical values of the energy momentum map $(H,G)$ and the preimage consists of a one-parameter family of periodic orbits whose projection to the configuration space are ellipses which each contain the focus points $\pm \mathbf{a}$ (see Fig.~\ref{fig:caustics}b). The family in particular contains the periodic orbit oscillating along the $z$ axis with turning points $z_\pm = \pm \sqrt{2E} / \omega$, where $|z_\pm|>a$. The caustic is again formed by the ellipse $\xi = s_{3+}$.
For $(E,g)\in {\cal L}_2$ and $E<\frac12 \omega^2 a^2$, the preimage consists only of the periodic orbit oscillating along the $z$ axis between $z_\pm = \pm \sqrt{2E} / \omega$ where $z_\pm $ now has a modulus less than $a$.
For $(E,g)\in {\cal L}_3$, i.e. the maximal value of $g$ for fixed energy $E>\frac12 \omega^2 a^2$,
the preimage consists of two periodic orbits whose configuration space projections are the ellipse $\xi=s_{2+}=s_{3+}$.
For $(E,g)\in {\cal L}_1$, i.e. the minimal value of $g$ for fixed energy $E$, the preimage consists of the periodic orbit that is oscillating along the $x$ axis with turning points $x_\pm = \pm \sqrt{2E} / \omega$.
The tangental intersection of ${\cal L}_2$ and ${\cal L}_3$ at $(g,E)=(0,\frac12 \omega^2 a^2)$ corresponds to a pitchfork bifurcation where two ellipse shaped periodic orbits grow out of the periodic orbit along the $z$ axis.
\begin{figure}
\raisebox{4.5cm}{(a)} \includegraphics[height=4.3cm]{caustic_a_1_h_5_lz_0_g_plus_1}
\raisebox{4.5cm}{(b)} \includegraphics[height=4.3cm]{caustic_a_1_h_5_lz_0_g_0}
\raisebox{4.5cm}{(c)} \includegraphics[height=4.3cm]{caustic_a_1_h_5_lz_0_g_minus_1}
\raisebox{4.5cm}{(d)} \includegraphics[height=4.3cm]{caustic_h_5_lz_1_g_0_a_1}
\caption{Orbits and caustics for $h=5$, $l_z=0$ and $g=-1$ (region IV) in (a), $g=0$ (boundary III/IV) in (b) and $g=1$ (region III) in (c), and $h=5$, $l_z=1$ and $g=0$ in (d), where $\rho=\sqrt{x^2+y^2}$. In all panels $a=1$.
} \label{fig:caustics}
\end{figure}
\subsection{The three-dimensional harmonic oscillator (general $l_z$)}
\label{sec:bifdiag_gen}
Increasing the modulus of $l_z$ from zero we see from the definition of $P(s)$ in Eq.~\eqref{eq:def_P(s)}
that the graphs of the polynomial in Fig.~\ref{fig:bifdiag_planar} move downward. Even though we cannot easily give expressions for the roots of
$P(s)$ for $l_z\ne 0$ we see that
increasing $|l_z|$ from zero for fixed $E$ and $g$ the ranges of admissible $\eta$ and $\xi$ shrink. Moreover, as $P(\pm 1)=-l_z^2$,
the roots stay away from $\pm 1$ (the coordinate singularities of the prolate ellipsoidal coordinates) for $l_z \ne 0$.
For general $l_z$, the discriminant of $P(s)$ is
\begin{equation*}
\begin{split}
\text{discrim} (P(s),s) =
64\, {a}^{12}{\omega}^{2} & \left( 2\,{a}^{2}E+g-{ l_z^2} \right)
\left( 4\,{a}^{8}{ l_z^2}\,{\omega}^{6}-24\,{a}^{6}E{ l_z^2}\,{
\omega}^{4}-{a}^{4}{g}^{2}{\omega}^{4}-18\,{a}^{4}g{ l_z^2}\,{
\omega}^{4}+ \right. \\ & \left.
27\,{a}^{4}{{ l_z}}^4{\omega}^{4}+
48\,{a}^{4}{E}^{2
}{ l_z^2}\,{\omega}^{2}+4\,{a}^{2}{g}^{2}E{\omega}^{2}+36\,{a}^{2}g
E{ l_z^2}\,{\omega}^{2}-32\,{a}^{2}{E}^{3}{ l_z^2}+4\,{g}^{3}{
\omega}^{2}-4\,{g}^{2}{E}^{2} \right) ^{2}
\,.
\end{split}
\end{equation*}
The first (nonconstant) factor vanishes for
\begin{equation} \label{eq:min_g_general_lz}
g = {{\it l_z}}^{2} - 2\,{a}^{2}E.
\end{equation}
From $P(0)=g-l_z^2+2\, a^2 E$ we see that this is the condition for the local maximum of $P(s)$ at $s=0$ to have the value zero or equivalently the collision of roots at 0.
In order to see when the second nonconstant factor vanishes it is useful to write $P(s)$ as $(s-d)^2(a_4s^4 + a_3s^3 a_2s^2 + a_1s + a_0)$ where $d$ is the position of the double root. Comparing coefficients then gives
\begin{eqnarray}
g(d) &=& -a^2 \left( {d}^{2}-1 \right) \left({a}^{2}\omega^2 (3d^2-1) -4\,E \right) , \label{eq:g(d)} \\
l_z^2(d) &=& \phantom{-} a^2 \left( {d}^{2}-1 \right)^2 \left( {{a}^{2}\omega^2 (2d^2-1) -2\,E} \right) . \label{eq:lz(d)}
\end{eqnarray}
For fixed $E$ and $l_z$, the minimal value of $g$ is, similarly to the planar case ($l_z=0$),
determined by the occurrence of a double root of $P(s)$ at $0$, i.e. by Eq.~\eqref{eq:min_g_general_lz}.
The maximal value of $g$ for fixed $E$ and $l_z$ is similarly to the planar case determined by the collision of the two biggest roots of $P(s)$ and given by $g(d)$ in Eq.~ \eqref{eq:g(d)} for the corresponding $d>1$.
We present the bifurcation diagram as slices of constant energy for representative values of $E$.
We have to distinguish between the two cases $0<E<\frac12 \omega^2 a^2$ and $E>\frac12 \omega^2 a^2$ as shown in Fig.~\ref{fig:bifdiag_spatial}.
The upper branches of the bifurcation diagrams in Fig.~\ref{fig:bifdiag_spatial} result from $d>1$ in Eqs.~\eqref{eq:g(d)} and \eqref{eq:lz(d)}.
A kink at $l_z=0$ occurs when $E<\frac12 \omega^2 a^2$. This is because the second factor in $\eqref{eq:lz(d)}$ can be zero at a $d\ge 1$ only if $E>\frac12 \omega^2 a^2$ in which case there is no kink.
For $E<\frac12 \omega^2 a^2$, there is an isolated point at $(l_z,g)=(0,0)$.
This results from $d=\pm1$ in Eqs.~\eqref{eq:g(d)} and \eqref{eq:lz(d)}. The point is isolated because the second factor in $\eqref{eq:lz(d)}$ is negative for
$E>\frac12 \omega^2 a^2$ and $d=\pm 1$.
The preimage of a regular value of $(H,L_z,G)$ in the region enclosed by the outer lines bifurcation diagrams in Fig.~\ref{fig:bifdiag_spatial} corresponds to a three-torus formed by a two-parameter family of periodic orbits given by ellipses in configuration space which are enveloped by two-sheeted hyperboloids and two ellipsoids given by coordinate surfaces of the prolate spheroidal coordinates $\eta$ and $\xi$, respectively (see Fig.~\ref{fig:caustics}d).
The preimage of a critical value $(E,l_z,g)$ in the upper branches in Fig.~\ref{fig:bifdiag_spatial} is a two-dimensional torus consisting of periodic orbits that move on ellipsoids of constant $\xi$.
The preimage of a critical value $(E,l_z,g)$ in the lower branches consists of a two-dimensional torus formed by periodic orbits whose projections to configuration space are contained in the $(x,y)$ plane.
At the corners where $|l_z|$ reaches its maximal value $E/\omega$, the motion is along the circle of radius $(|l_z|/\omega)^{1/2}$ in the $(x,y)$ plane with the sense of rotation being determined by the sign of $l_z$.
\begin{figure}
\raisebox{4.5cm}{(a)}
\includegraphics[width=4.5cm]{momentum_map_small_h_xfig}
\raisebox{4.5cm}{(b)}
\includegraphics[width=4.5cm]{momentum_map_big_h_xfig}
\caption{Slices of constant energy through the spatial bifurcation diagram with $a=1$, $\omega=1$ and energies $E=1/4$ (a) and $E=4$ (b).
} \label{fig:bifdiag_spatial}
\end{figure}
For the planar case, we saw that the critical energy $E=\frac12 \omega^2 a^2$ corresponds to a pitchfork bifurcation. In the spatial case this becomes a Hamiltonian Hopf bifurcation which manifests itself as the vanishing of the kink and detachment of the isolated point in the bifurcation diagram when $E$ crosses the value $\frac12 \omega^2 a^2$. Note that the critical energy is the potential energy at the focus points of the prolate spheroidal coordinates.
\subsection{Reduction}
\label{sec:reduction}
The isolated point of the bifurcation diagram for energies $E>\frac12 \omega^2 a^2 $ leads to monodromy.
To see this more rigorously we proceed as follows.
For a classical maximally super-integrable Hamiltonian with compact energy surface, the flow of the Hamiltonian
is periodic. Therefore it is natural to consider symplectic reduction by the $S^1$ symmetry induced by
the Hamiltonian flow. This leads to a reduced system on a compact symplectic manifold.
On the reduced space which turns out to be $\mathbb{C}P^2$ we then have a two-degree-of-freedom Liouville integrable system $(L_z,G)$.
We will prove that for $E > \frac12 \omega^2 a^2$, this system has monodromy by showing the existence of a singular fibre with value $(l_z,g)=(0,0)$ (the isolated point discussed in the previous subsection) given by a 2-torus that is pinched at a focus-focus singular point.
To this end it is useful to also reduce the $S^1$ action
corresponding to the flow of $L_z$. As this $S^1$ action has isotropy, standard symplectic reduction is not applicable and we resort to singular reduction using the method of invariants instead. The result will be a one-degree-freedom system on a singular phase space. For a general introduction, we refer to \cite{CushBates}.
In order to reduce by the flows of $H$ and $L_z$ it is useful to rewrite $G$ as
\begin{equation}\label{eq:G_reduced}
G = L_z^2 - 2 R^2 - \frac{2}{\omega}(a^2 \omega^2 - H) R +\frac{1}{ \omega} X,
\end{equation}
where
\begin{eqnarray}
R &:=& \frac{1}{\omega}(A_x + A_y), \label{eq:def_R} \\
X &:=& \omega(L_x^2 + L_y^2) - 2A_z R. \label{eq:def_X}
\end{eqnarray}
The significance of this decomposition is that defining $Y$ by $$\{ R, X \} = -2 Y$$ we find
that the Poisson brackets between $R$, $X$, and $Y$ are closed. Specifically we have
$$\{ R, Y \} = 2 X \text{ and }\{ X, Y \} = 8 (H - \omega R) (\omega L_z^2 + H R - 2 \omega R^2)$$
and $(R$,X$, Y)$ form a closed Poisson algebra
with Casimir function
\begin{equation}\label{eq:DefCasimirFunction}
C=4 \omega^2 (H - \omega R)^2 ( R^2 - L_z^2) - \omega^2 ( X^2 + Y^2 ) = 0.
\end{equation}
Hence this achieves reduction to a single degree of freedom with phase space given by the zero level set of the Casimir function $C$.
A systematic way to achieve this reduction uses invariant polynomials.
This approach is moreover useful because it gives a classical analogue to creation and annihilation operators used in the quantization below.
The flows generated by $(H, L_z)$ define a $T^2$ action on the original phase space $T^*\mathbb{R}^3$.
Since both $H$ and $L_z$ are quadratic and they satisfy $\{ H, L_z \} = 0$ there is a linear symplectic transformation
that diagonalises both $H$ and $L_z$. It is given by
\[
x = \frac{1}{\sqrt{2\omega}} (p_1+p_2),\
y = \frac{1}{\sqrt{2\omega}} (q_1-q_2),\
z = \frac{1}{\sqrt{\omega}} q_3,\
p_x = -\sqrt{\frac{\omega}{2}} (q_1+q_2),\
p_y = \sqrt{\frac{\omega}{2}} (p_1-p_2),\
p_z = \sqrt{\omega} p_3.
\]
and in the new complex coordinates $z_k = p_k + i q_k$, $k=1,2,3$, we find
\[
H = \frac{\omega}{2} ( z_1 \bar z_1 + z_2 \bar z_2 + z_3 \bar z_3), \quad
L_z = \frac12( z_1 \bar z_1 - z_2 \bar z_2) \,.
\]
Additional invariant polynomials are
\[
R = \frac12( z_1 \bar z_1 + z_2 \bar z_2), \quad
X - i Y = \omega z_1 z_2 \bar z_3^2 \,.
\]
These invariants are related by the syzygy $C=0$ in Eq.~\eqref{eq:DefCasimirFunction}
and satisfy $|L_z| \le R \le H/\omega$.
The surface $C=0$ in the three-dimensional space $(X,Y,R)$ can be viewed as the reduced phase space. It is rotationally symmetric about the
$R$ axis. Due to a singularity at $R=E/\omega $ and another singularity at $R=0$ when $l_z=0$, the reduced space is
homeomorphic but not diffeomorphic to a two-dimensional sphere (see Figs.~\ref{fig:RedSpace}(a) and (b)).
The singularity at $R=0$ when $l_z=0$ results from nontrivial isotropy of the $S^1$ action of the flow of $L_z$.
$R=0$ implies that the full energy is contained in the $z$ degree of freedom and motion consists of oscillation along the $z$ axis. The corresponding phase space points are fixed points of the $S^1$ action of the flow of $L_z$. The value of $L_z$ is zero for this motion. For $R=E/\omega$, the energy is contained completely in the $x$ and $y$ degrees of freedom (see \eqref{eq:def_R}), i.e. the motion takes place in the $(x,y)$ plane. This includes also the motion along the circle of radius $(|l_z|/\omega)^{1/2}$ where the flows of $L_z$ and $G$ are parallel.
The dynamics on the reduced phase space is generated by $G$. As the system has only one degree of freedom
the solutions are given by the level sets of $G$ restricted to $C=0$.
As $G$ is independent of $Y$ the surfaces of constant $G$ are cylindrical in the space $(X,Y,R)$. Given the rotational symmetry of the reduced phase space the intersections of $G=g$ and $C=0$ can be studied in the slice $Y=0$ (see Fig.~\ref{fig:RedSpace}). Two intersection points in the slice result in a topological circle. Under variation of the value of the level $g$ the two intersection points collide at a tangency or the singular point where $R=E/\omega$ corresponding to the maximal and minimal values of $g$ for which there is an intersection, respectively. Both cases correspond to elliptic equilibrium points for the flow of $G$ on the reduced space. For $l_z=0$, one of the intersection points can be at the singular point where $R=0$. From Eq.~\eqref{eq:G_reduced} we see that the corresponding value of $g$ is $0$.
In this case the topological circle is not smooth. Away from the singular point $R=0$, the points on this curve correspond to circular orbits of the action of $L_z$ giving together with the fixed point of the action $L_z$ at $R=0$ a pinched 2-torus where the pinch is a focus-focus singular point in the space reduced by the flow of $H$. Reconstructing the reduction by the flow of $H$ results in the product of a pinched 2-torus and a circle in the original full phase space $T^*\mathbb{R}^3$.
The minimal value of $G$ attained at the singular point $R=E/\omega$ can be obtained from Eq.~\eqref{eq:G_reduced} and gives again \eqref{eq:min_g_general_lz}.
The maximal value of $G$ can be computed from the condition that $\nabla G$ and $\nabla C$ are dependent on $C=0$,
where $\nabla$ is with respect to the coordinates on the reduced space $(R, X, Y)$. Similarly to the computation of the maximal value of $g$ for fixed $E$ and $l_z$ in subsection~\ref{sec:bifdiag_gen} this leads to a cubic equation.
The critical energy at which the focus-focus singular point comes into existence corresponds to the collision of the tangency that gives the maximal value of $g$ with the singular point $R=0$. As mentioned in subsection~\ref{sec:bifdiag_gen} this corresponds to a Hamiltonian Hopf bifurcation. The critical energy can be computed from comparing the slope of the upper branch of the slice $Y=0$ of $C=0$ at $R=0$ which is $2E$ with the
slope of $G=0$ at $R=0$ which is $2a^2\omega^2-2E$. Equating the two gives the value $E=\frac12 \omega^2 a^2$ that we already found in
subsection~\ref{sec:bifdiag_gen}.
\begin{figure}
\raisebox{4cm}{(a)}\includegraphics[width=5cm]{ReducedSpace_E_0_25_lz_0_1_labelled}
\raisebox{4cm}{(b)}\includegraphics[width=5cm]{ReducedSpace_E_0_25_lz_0_labelled}
\raisebox{4cm}{(c)}\includegraphics[width=5cm]{ReducedSpace_E_4_lz_0_labelled} \\
\raisebox{5cm}{(d)}\includegraphics[width=5cm]{ReducedSpace_E_0_25_lz_0_1_slice}
\raisebox{5cm}{(e)}\includegraphics[width=5cm]{ReducedSpace_E_0_25_lz_0_slice}
\raisebox{5cm}{(f)}\includegraphics[width=5cm]{ReducedSpace_E_4_lz_0_slice}
\caption{Reduced space $C=0$ for $E=1/4$ and $l_z=0.1$ (a), $E=1/4$ and $l_z=0$ (b), and $E=4$ and $l_z=0$ (c).
The lower panels show the corresponding slices $Y=0$ (dashed) and contours $G=g$ with increments $\Delta g= 0.05$ in (d) and (e) and $\Delta g= 2$ in (f).
In all panels $a=\omega=1$.
} \label{fig:RedSpace}
\end{figure}
\subsection{Symplectic volume of the reduced phase space}
It follows from the Duistermaat-Heckman Theorem \cite{DuistermaatHeckman1982} that the symplectic volume (area) of the reduced phase space defined by $C = 0$ has a piecewise linear dependence on the global action $L_z$.
Indeed, introducing cylinder coordinates to parametrize the reduced phase space $C = 0$ as
$X = f(R) \sin\theta$ and $Y=f(R) \cos\theta$ we see from
$
\{ \theta, R \} = 2,
$
that the symplectic form on $C=0$ is
$
\frac12 d\theta \wedge dR.
$
Integrating the symplectic form over the reduced space $C=0$ gives
the symplectic volume
\begin{align*}
\mathrm{Vol}_{E,l_z} = \frac{\pi}{\omega} (E- \omega |l_z|),
\end{align*}
for fixed $E \ge \omega |l_z|$.
It follows from Weyl's law that $ \mathrm{Vol}_{E,l_z} /(2\pi \hbar) = (E-\omega |l_z|)/(2\hbar\omega)$
gives the mean number of quantum states for fixed $E$ and $l_z$ (see \cite{Ivrii2016} for a recent review). Indeed inserting $E=\hbar\omega (n+3/2)$ and $l_z=\hbar m$ we get
$ \mathrm{Vol}_{E,l_z} /(2\pi \hbar) = (n+3/2 -|m|)/2$. Counting the exact number of states for fixed $n$ and $m$
which is most easily done using the separation with respect to spherical coordinates (see the Introduction)
we get $ (n+2-|m|)/2$ if $n-|m|$ is even and $ (n+1-|m| )/2$ if $n-|m|$ is odd.
We see that Weyl's law is interpolating between the even and the odd case, see Fig.~\ref{fig:WeylLaw}(a).
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(170,60)(0,0)
\put(5,0){\includegraphics[width=5cm]{WeylLaw_n_10}}
\put(0, 50){(a)}
\put(15,43){$\mathrm{Vol}_{E,l_z}$}
\put(27, 0){$l_z/\hbar$}
\put(60,0){\includegraphics[width=5cm]{WeylLaw_n_11}}
\put(55, 50){(b)}
\put(70,43){$\mathrm{Vol}_{E,l_z}$}
\put(82, 0){$l_z/\hbar$}
\put(117,0){\includegraphics[width=5cm]{IntegratedWeylLaw_vs_E}}
\put(110, 50){(c)}
\put(112,43){$N(E)$}
\put(155, 0){$E/(\hbar \omega)$}
\end{picture}
\caption{$\mathrm{Vol}_{E,l_z} $ versus $m=l_z/\hbar$ for $E=\hbar \omega (n+3/2)$ and corresponding exact number of states (dots) for the even integer $n=10$ (a) and the odd integer $n=11$ (b).
(c) The area $N(E)$ under graphs of the form in (a) and (b) divided by $2\pi\hbar^2$ versus $E$ and the corresponding exact number of states (dots) at energies of $\hbar\omega(n+3/2)$.
}\label{fig:WeylLaw}
\end{figure}
The area under the graph of $\mathrm{Vol}_{E,l_z}$ as a function of $l_z$ for fixed $E=\hbar\omega(n+3/2)$ is $\pi \hbar^2 (n+3/2)^2 $. Dividing by the product of $2\pi \hbar$ and $\hbar$ (which is the distance between two consecutive quantum angular momenta eigenvalues $l_z$) gives $(n+3/2)^2/2$ which for $n\to\infty$ asymptotically agrees with the exact number of states
$(n+1)(n+2)/2$, see Fig.~\ref{fig:WeylLaw}(b).
\subsection{The limiting cases $a\to 0$ and $a\to \infty$}
\label{sec:LimitingCases}
From Eq.~\eqref{eq:G_Cart} we see that for $a\to 0$, the separation constant $G$ becomes the squared total angular momentum, $\mathbf{L}^2 = L_x^2 + L_y^2 + L_z^2 $. In the limit $a \to 0$ we thus obtain the Liouville integrable system given by $(H, L_z, |\mathbf{L}|^2)$ which corresponds to separation in spherical coordinates.
Note that the $a \to \infty$ limit of prolate spheroidal coordinates corresponds to parabolic coordinates, where the harmonic oscillator is not separable.
However, the scaled separation constant
\begin{equation}\label{eq:G_Cart_mod}
\tilde{G} = -\frac{1}{a^2} G = 2 (A_x + A_y) - \frac{1}{a^2} (L_x^2 + L_y^2 + L_z^2 ),
\end{equation}
has the well defined limit $2 (A_x + A_y) $ as $a\to \infty$.
The limit $a \to \infty$ then leads to the Liouville integrable system $(H, L_z, 2(A_x + A_y))$.
The standard separation in Cartesian coordinates leads to the integrable system $(H, A_x, A_y)$.
The reduction by the flow of $H$ gives as the reduced space the compact symplectic manifold $\mathbb{C}P^2$, see Sec.~\ref{sec:separation}.
Then the map $(A_x, A_y)$ associated with separation in Cartesian coordinates defines an effective toric action on $\mathbb{C}P^2$.
The image of $\mathbb{C}P^2$ under $(A_x, A_y)$ is therefore a Delzant polygon which is a convex polygon with special properties \cite{Delzant1988}, see Fig.~\ref{fig:polygon3}(a).
Similarly the map $(L_z,\frac{1}{\omega}(A_x + A_y)):\mathbb{C}P^2\to \mathbb{R}^2$ associated with separation in prolate spheroidal coordinates in the limit $a\to \infty$ also defines a toric, non-effective, action and its image is the convex, non-Delzant, polygon shown in Fig.~\ref{fig:polygon3}b. We here have scaled the separation constant in such a way that the $S^1$ actions associated with the flows of $L_z$ and $\frac{1}{\omega}(A_x + A_y)$ have the same period.
The image of the map $(L_z, |\mathbf{L}|):\mathbb{C}P^2\to \mathbb{R}^2$ associated with the limit $a\to 0$ and separation in spherical coordinates also gives the same polygon as in the previous case, see Fig.~\ref{fig:polygon3}c. However, whereas here $L_z$ is a global $S^1$ action this is not the case for $|\mathbf{L}|$ whose Hamiltonian vector field is singular at points with $\mathbf{L}=0$.
Because of this singularity $(L_z, |\mathbf{L}|)$ is not the moment map of a global toric action. Whereas the image is a convex polygon the singularity manifests itself when considering the joint quantum spectrum of the operators associated with the classical constants of motion. Whereas these form rectangular lattices in Figs.~\ref{fig:polygon3}(a) and (b) with lattice constants $\hbar$ this is not the case in Fig.~\ref{fig:polygon3}(c) where the distance between consecutive lattice layers is not constant in the vertical direction.
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(170,50)(0,0)
\put(5,11){\includegraphics[width=3.3cm]{Delzant_Ax_Ay}}
\put(0, 50){(a)}
\put(2,40){$a_y$}
\put(33, 10){$a_x$}
\put(45,0){\includegraphics[width=6cm]{Delzant_lz_Ax_plus_Ay}}
\put(39, 50){(b)}
\put(43,22){\rotatebox{90}{$(a_x+a_y)/\omega$}}
\put(77, 10){$l_z$}
\put(111,0){\includegraphics[width=6cm]{Delzant_L_lz}}
\put(108, 50){(c)}
\put(111,40){$l $}
\put(143, 10){$l_z$}
\end{picture}
\caption{The images of different maps of integrals $\mathbb{C}P^2\to \mathbb{R}$ where $\mathbb{C}P^2$ is the energy level set of the
harmonic oscillator reduced by the flow of the Hamiltonian $H$.
(a) The map of integrals $(A_x, A_y)$ associated with separation in Cartesian coordinates, where we denote the values of the functions $A_k$ by $a_k$, $k=x,y$.The image is enclosed by the triangle with corners $(0,0)$, $(0,E)$ and $(E,0)$.
(b) The map of integrals $(L_z, \frac{1}{\omega}(A_x+A_y))$ associated with the limit $(a \to \infty)$ when separating in prolate spheroidal coordinates. The image is enclosed by the triangle with corners $(0,0)$, $(E/\omega,E/\omega)$ and $(-E/\omega,E/\omega)$.
(c) The map of integrals $(L_z, |\mathbf{L}| )$ associated with separation in spherical coordinates and the limit $(a\to 0)$ in prolate spheroidal coordinates. Here $l$ denotes the value of the function $|\mathbf{L}| $. The image is enclosed by the triangle with corners $(0,0)$, $(E/\omega,E/\omega)$ and $(-E/\omega,E/\omega)$. The dots mark the joint spectrum of the corresponding quantum operators. The energy is chosen to be $E=\omega\hbar(n+3/2)$ with $n=11$. The values of $\hbar$ and $\omega$ are chosen as 1.
} \label{fig:polygon3}
\end{figure}
\rem{
\begin{figure}
\raisebox{5cm}{(a)} \raisebox{1cm}{\includegraphics[width=3.5cm]{Delzant_Ax_Ay}}
\raisebox{5cm}{(b)} \includegraphics[width=6cm]{Delzant_lz_Ax_plus_Ay}
\raisebox{5cm}{(c)} \includegraphics[width=6cm]{Delzant_L_lz}
\caption{The images of different maps of integrals $\mathbb{C}P^2\to \mathbb{R}$ where $\mathbb{C}P^2$ is the phase space of the
harmonic oscillator reduced by the flow of the Hamiltonian $H$.
(a) The map of integrals $(A_x, A_y)$ associated with separation in Cartesian coordinates. The image is enclosed by the triangle with corners $(0,0)$, $(0,E)$ and $(E,0)$.
(b) The map of integrals $(L_z, \frac{1}{\omega}(A_x+A_y))$ associated with the limit $(a \to \infty)$ when separating in prolate spheroidal coordinates. The image is enclosed by the triangle with corners $(0,0)$, $(E/\omega,E/\omega)$ and $(-E/\omega,E/\omega)$.
(c) The map of integrals $(L_z, |\mathbf{L}| )$ associated with separation in spherical coordinates and the limit $(a\to 0)$ in prolate spheroidal coordinates. The image is enclosed by the triangle with corners $(0,0)$, $(E/\omega,E/\omega)$ and $(-E/\omega,E/\omega)$. The dots mark the joint spectrum of the corresponding quantum operators. The energy is chosen to be $E=\omega\hbar(n+3/2)$ with $n=11$. The values of $\hbar$ and $\omega$ are chosen as 1.
} \label{fig:polygon3}
\end{figure}
}
\section{Quantum monodromy}
\label{sec:sepQM}
In this section we discuss the implications of the monodromy discussed in the previous section on the joint spectrum of the quantum mechanical version of the isotropic oscillator which is described by the operator
\[
\hat{H} = -\frac{\hbar^2}{2} \nabla^2 + \frac{\omega^2}{2} (x^2 + y^2 + z^2) \,.
\]
In prolate spherical coordinates the Schr\"odinger equation becomes
\[
-\frac{\hbar^2}{2} \left\{
\frac{1}{a^2(\xi^2-\eta^2)}
\left[
\frac{\partial}{\partial \xi} \left((\xi^2-1) \frac{\partial \Psi}{\partial \xi} \right)
+ \frac{\partial}{\partial \eta} \left((1-\eta^2) \frac{\partial \Psi}{\partial \eta} \right)
\right]
+ \frac{1}{a^2(\xi^2-1)(1-\eta^2)} \frac{\partial^2 \Psi}{\partial \phi^2}
\right\}
+ \frac{\omega^2}{2} a^2 (\xi^2 + \eta^2 -1 ) \Psi = E \Psi.
\]
Separating the Schr\"odinger equation in prolate spheroidal coordinates works similarly to the classical case discussed in Sec.~\ref{sec:separation}.
The separated equations for $\eta$ and $\xi$ are
\begin{equation} \label{eq:Ppsi}
- \hbar^2 \frac{1}{1 - s^2} \frac{d}{ds} \left[(1 - s^2) \frac{d\psi}{ds} \right] = \frac{P(s)}{(1-s^2)^2} \psi \,,
\end{equation}
where $P(s)$ is again the polynomial that we defined for the classical case in Eq.~\eqref{eq:def_P(s)},
with $l_z = \hbar m$.
This is the spheroidal wave equation with an additional term proportional to $\omega^2$ coming from the potential.
For $|s| < 1$ it describes the angular coordinate $\eta$, and for $s > 1$ the radial coordinate $\xi$
of spheroidal coordinates.
Analogously to the classical case the separation constant $g$ corresponds to the eigenvalue of the operator
\begin{equation}\label{eq:G_Cart_quantum}
\hat G = {\hat L}_x^2 + {\hat L}_y^2 + {\hat L}_z^2 - 2 a^2 ({\hat A}_x + {\hat A}_y)\, ,
\end{equation}
where for $k=x,y,z$, the ${\hat L}_k$, are the components of the standard angular momentum operator, and the ${\hat A}_{k}=-\frac12 \hbar^2 \partial^2_k + \frac12 \omega^2 k^2$ are the Hamilton operators of one-dimensional harmonic oscillators.
A WKB ansatz shows that the joint spectrum of the quantum integrable system $(\hat H, {\hat L}_z,\hat G)$ associated with the separation in prolate spheroidal coordinates can be computed semi-classically from a Bohr-Sommerfeld quantization of the actions according to
$I_\phi = \frac{1}{2\pi} \oint p_\phi \,\ud \phi = \hbar m $,
$I_\eta = \frac{1}{2\pi} \oint p_\eta \,\ud \eta = \hbar (n_\eta+\frac12) $ and $I_\xi = \frac{1}{2\pi} \oint p_\xi \,\ud \xi = \hbar(n_\xi + \frac12 )$ with $m \in \Z$
and non-negative quantum numbers $n_\eta$ and $n_\xi$.
Using the calculus of residues it is straightforward to show that $E = I_\eta + I_\xi + | I_\phi |$.
Taking the derivative with respect to $l_z$ using $I_\phi=l_z$ shows that
the actions $I_\eta$ and $I_\xi$ are not globally smooth functions of the constants of motion $(E,g,l_z)$.
This is an indication that the quantum numbers do not lead to a globally smooth labeling of the joint spectrum.
We will see this in more detail below.
\subsection{Confluent Heun equation}
It is well known that the spheroidal wave equation can be transformed into the confluent Heun equation
\cite{NIST:DLMF}. Adding the harmonic potential adds additional terms that dominate at infinity, and
so a different transformation needs to be used to transform \eqref{eq:Ppsi} into the Heun equation.
We change the independent variable $s$ in \eqref{eq:Ppsi} to $u$ by $s^2 = u$ and the dependent variable to $y$ by
$y(u) = \exp( a^2 \omega u/(2 \hbar) ) ( 1 - u)^{-m/2} \psi(s)$ which leads to
\[
y'' + \left( - \frac{a^2\omega}{\hbar} - \frac{m+1}{1-u} + \frac{1}{2 u}\right) y' + Q y = 0\,,
\]
where
\[
Q = \frac{ g - \hbar^2 m(m+1)}{4 \hbar^2 u ( 1 - u)} +
\frac{ a^2}{2\hbar^2} \left( \frac{\hbar (m+1)}{ 1-u} + \frac{E-\tfrac12 \hbar \omega}{ u} \right) \,.
\]
This is a particular case of the confluent Heun equation, with regular singular
points at 0 and 1, and an irregular singular point of rank 1 at infinity.
Each regular singular point has one root of the indicial equation equal to zero,
so we may look for a solution of the form $y(u) = \sum_k a_k u^k = \sum_k b_{2k} s^{2k}$.
This leads to the three-term recursion relation for the coefficients
\[
b_{k-2} A_{k-2} + b_k B_k - b_{k+2} \hbar^2 (k+1)(k+2) = b_k ( g + 2 a^2 E)\,,
\]
where $k$ is an even integer and
\begin{align*}
A_{k-2} & = 2 a^2 (E - \hbar \omega (m + k - \tfrac12 )), \\
B_k & = a^2 \hbar \omega (2k+1) + \hbar^2 (m+k)(m+k+1) \,.
\end{align*}
If we require that $y(u)$ is polynomial of degree $d$,
we need to require that for $k = 2 d + 2$ the coefficient $A_{k-2}$ vanishes,
and hence the quantisation condition
\[
E = \hbar \omega \left( m + 2 d + \tfrac 32\right) ,
\]
with principal quantum number $n = m + 2 d$ is found.
Fixing $E$ to some half-integer the spectrum of the tridiagonal matrix $M$ obtained
from the three-term recurrence relation
determines the spectrum of $ g + 2 a^2 E$.
In the limit $a \to 0$ the spectrum
becomes $n(n+1), ..., m(m+1)$. Note that fixing the energy and
allowing all possible degrees $d$ makes $m$ change in steps of 2.
Since $m$ in fact changes in steps of 1 there must be additional solutions.
The regular singular point at $u=0$ has another regular solution
with leading power $\sqrt{u} = s$, so that we make the Ansatz $y(u) = \sqrt{u} \sum_0 a_k u^k = \sum_0 b_{2k+1} s^{2k+1}$,
which leads to an odd function in $s$.
The same three-term recursion relation holds as above, except that now the index $k$ is odd.
For $a\to 0$ the spectrum is $n(n+1), \dots, (m+1)(m+2)$, as before in steps of 2 in $m$.
We note that in the spherical limit $a\to 0$ the Heun equation reduces to the associated Laguerre equation
with polynomial solutions $L_{(n-l)/2}^{l+1/2}(s^2)$ when $g = l(l+1)$.
\subsection{Algebraic computation of the joint spectrum}
Instead of starting from the spheroidal wave equation wave equation as illustrated in the previous subsection one can directly compute the joint spectrum algebraically by using creation and annihilation operators. As we will see this gives explicit expressions for the entries of a tri-diagonal matrix whose eigenvalues give the spectrum of $\hat G$ for fixed $E$ and $l$.
Instead of the usual creation and annihilation operators of the
harmonic oscillator we use operators that are written in the set of coordinates $(z_1, z_2, z_3)$ introduced in Sec.~\ref{sec:reduction}.
The transformation to the new coordinates diagonalises $\hat L_z$ and at the same time keeps $\hat H$ diagonal, so that
\[
\hat H = \hbar \omega ( a_1^\dagger a_1 + a_2^\dagger a_2 + a_3^\dagger a_3 + \tfrac32), \quad
\hat L_z = \hbar ( a_1^\dagger a_1 - a_2^\dagger a_2 ) \,.
\]
and the operator $\hat R$ corresponding to the classical $R$ in Eq.~\eqref{eq:def_R} reads
\[
\hat R = \hbar ( a_1^\dagger a_1 + a_2^\dagger a_2 + 1) \,.
\]
The operator $\hat X$ corresponding to $X$ in Eq.~\eqref{eq:def_X} is of higher degree, and thus care needs to be taken with the order of operators.
The classical $X$ can be written as $X = \frac{1}{2} \omega ( z_1 z_2 \bar z_3^2 + \bar z_1 \bar z_2 z_3^2)$.
We also need to preserve the relation (for operators!)
$\omega \hat L^2 = \omega \hat L_z^2 + 2 ( \hat H - \omega \hat R) \hat R + \hat X$, cf.~Eq.~\eqref{eq:def_X},
and this leads to
\[
\hat X = 2 \hbar^2 \omega \left( a_1^\dagger a_2^\dagger a_3^2 + a_1 a_2 ( a_3^\dagger)^2 - \frac12 \right) \,.
\]
With these expressions matrix elements can be computed. Denote a state
with three quantum numbers associated to the creation and annihilation operators $a_i$ and $a_i^\dagger$, $i=1,2,3$,
by $\ket{k_1, k_2, k_3}$, such that
\begin{align*}
a_1^\dagger \ket{k_1, k_2, k_3} & = \sqrt{ n_1 + 1} \ket{k_1+1, k_2, k_3}, \\
a_1 \ket{k_1, k_2, k_3} & = \sqrt{ n_1 } \ket{k_1-1, k_2, k_3}, \text{ for } k_1 \ge 1, \\
a_1 \ket{0, k_2, k_3} & = 0
\end{align*}
and similar relations for $a_2$ and $a_3$. This allows to verify
\begin{align*}
\hat H \ket{k_1, k_2, k_3} &= \hbar \omega (k_1 + k_2 + k_3 + \tfrac32) \ket{k_1, k_2, k_3}\,, \\
\hat L_z \ket{k_1, k_2, k_3} &= \hbar (k_ 1 - k_2) \ket{k_1, k_2, k_3} \,.
\end{align*}
In terms of the quantum numbers $(k_1,k_2,k_3)$ the principal and magnetic quantum numbers are $n = k_1 + k_2 +k_3$ and
$m = k_1 - k_2$, respectively. The space of states with fixed $n$ and fixed
$m$ is the span of the states of the form
\[
\ket{k} := \ket{k, k - m, n + m - 2k } , \quad \max(0, m) \le k \le \frac12 ( n + m) \,.
\]
Now the non-zero matrix elements of
\begin{align*}
\hat G
= \hat{\mathbf L}^2 - 2 a^2 \omega \hat R
= \hat{L}_z^2 - 2 \hat R^2 - \frac{2}{\omega} (a^2 \omega^2 - \hat H) \hat R + \frac{1}{\omega} \hat X,
\end{align*}
are given by
\begin{align*}
\bra{k} \hat G \ket{k} & = 2\hbar a^2 \omega ( m - 1 - 2k) - \hbar^2( 2+ 8 k(1+k) -4m -8k m + m^2 ) + \frac{\hbar^2}{\omega} (1+2k-m)(3+2n)\\
\bra{k} \hat G \ket{k+1} &= 2 \hbar^2 \sqrt{ (k+1)(k+1-m)(n-1+m-2k)(n+m-2k)}.
\end{align*}
The resulting joing spectrum of $({\hat L}_z,\hat G )$ for a fixed $n$ is shown in Fig.~\ref{fig:MoMap_mono} for a choice of parameters such that the energy $E$ is above the threshold value $\frac12 \omega^2 a^2$ for the occurrence of monodromy. As to be expected from the Bohr-Sommerfeld quantization of actions the spectrum locally has the structure of a regular grid.
Globally however the lattice has a defect as can be seen from transporting a lattice cell along a loop that encircles the isolated critical value of the energy momentum map at the origin.
In Fig.~\ref{fig:MoMap} the joint spectrum of $({\hat L}_z,\hat G)$ is shown for fixed $n$ and a small and large value of $a$, respectively.
As discussed in Sec.~\ref{sec:LimitingCases}, in the limits $a\to 0$ and $a\to\infty$ (and in the latter case changing to $\tilde{G}=-\frac{1}{a^2}G$) the images become the polygones shown in Fig.~\ref{fig:polygon3}.
\begin{figure}
\includegraphics[width=0.7\linewidth]{spectrum_a_2_5_mono_xfig}
\caption{Joint spectrum $(l_z,g)$ of $(\hat L_z, \hat G)$ (black dots) and
classical critical values (red),
for $n=20$, $\omega=1$, $\hbar=1$, and $a = 3/2$. There are $(n+1)(n+2)/2$ joint states.
The joint spectrum locally has a lattice structure which globally has a defect as can be seen from transporting a lattice cell around the isolated critical value at the origin.
} \label{fig:MoMap_mono}
\end{figure}
\begin{figure}
\raisebox{3cm}{(a)} \includegraphics[width=0.45\linewidth]{spectrum_a_1_xfig}
\raisebox{3cm}{(b)} \includegraphics[width=0.45\linewidth]{spectrum_a_10_xfig}
\caption{Joint spectrum $(l_z,g)$ of $(\hat L_z, \hat G)$ (black dots) and
classical critical values (red) for
$a = 1$ (a) and $a=10$ (b) and otherwise same parameters as in Fig.~\ref{fig:MoMap_mono}.
} \label{fig:MoMap}
\end{figure}
\section{Discussion}
\label{sec:discussion}
It is interesting to compare the two most important super-integrable systems, the Kepler problem and the harmonic oscillator, in the light of our analysis.
The Kepler problem has symmetry group $\mathrm{SO}(4)$ and reduction by the Hamiltonian flow leads to a system on $S^2 \times S^2$ \cite{Moser1970}.
The 3-dimensional harmonic oscillator has symmetry group $SU(3)$
and reduction by the Hamiltonian flow leads to a system on $\mathbb{C}P^2$.
Separation of both systems, the Kepler problem and the harmonic oscillator in 3 dimensions, in prolate spheroidal coordinates leads to Liouville integrable systems that are of toric type for sufficiently large $a$.
Here the technical meaning of toric type is that they are integrable systems with a global $T^n$ action for $n$ degrees of freedom, which implies that all singularities are of elliptic type.
To a toric system is associated the image of the momentum map of the $T^n$ action, and this is a Delzant polytope, a convex polytope with special properties \cite{Delzant1988}.
The Delzant polytope for the $T^2$ action of the reduced Kepler system on $S^2 \times S^2$ is a square (take the limit $a\to\infty$ in
Fig.~4 in \cite{DullinWaalkens2018})
while the Delzant polytope for the $T^2$ action of the reduced harmonic oscillator on $\mathbb{C}P^2$ is an isosceles right triangle, see Fig.~\ref{fig:polygon3}(a).
It is remarkable that the two simplest such polytopes appear as reductions from the Kepler problem and
from the harmonic oscillator. We note, however, that the harmonic oscillator as opposed to the Kepler problem does not separate in parabolic coordinates.
This is related to the fact that for the separation of the Kepler problem in prolate spheroidal coordinates, the origin is in a focus point, while for the oscillator the origin is the midpoint between the foci.
For decreasing family parameter $a$, both systems become semi-toric \cite{Pelayo2009,EFSTATHIOU2017104} through a supercritical Hamiltonian Hopf bifurcation.
It thus appears that the reduction of super-integrable systems by the flow of $H$ leads to natural and important examples of toric and semi-toric systems on compact symplectic manifolds.
\rem{
\section{Remarks / ToDo}
\begin{itemize}
\item What is the limit of $I_\xi$ and $I_\eta$ when $a \to 0$?
\item \RESOLVED{Introduce canonical variables on reduced phase space defined by Casimirs and compute the volume
of the reduced phase space, plot it as a function of $l$ (Duistermaat-Heckman). This will give a
piecewise linear function, and should account for Weyl's law. For the HO the slope must be
less, since the number of states is less. KE: This is already done.}
\item Mention the Sturm-Liouville weights so that we know what is orthogonal,
and check what they are for spheroidal coordinates, probably $=1$ in the algebraic form?
\item check details of $a\to 0$ limit of Heun equation; what about $a\to \infty$?
\end{itemize}
\hrule
\begin{itemize}
\item insert comments and references to work on superintegrable systems in the introduction (see the hydrogen paper)
\item \RESOLVED{HW: it would be nice to explicitly see somewhere in the classical part that the reduced phase space obtained from reducing the flow of $H$ is indeed $CP^2$. KE: It is a non-trivial exercise to start with the invariants and their syzygies (which, by the way, we don't include in the paper) and show that the space they describe is $CP^2$. The standard proof is indirect: invariants and the syzygies are known to give the orbit space of the $H$-induced $S^1$ action, and $CP^2$ is \emph{defined} to be the orbit space of exactly this $S^1$ action, so the result follows. I have added some text in Sec. II concerning this.}
\item \RESOLVED{HW: note that in the classical part there is a separating constant $G$ and $\tilde{G}= - G/a^2$ (note the sign). It seems that $\tilde{G}$ is used in the quantum part. But still there is a sign issue in the separated Schrodinger equation. In the classical part all signs and factors like $a$ and $\omega$ are checked very carefully (here only in the systematic way of deriving the invariants this still needs to be done; see comment in the text). KE: I believe that signs are OK now}\ISSUE{except perhaps for the Heun equation}
\item \ISSUE{a quantum monodromy picture where a lattice cell is transported around the singularity is still missing. KE: For HW?}
\item it would be nice to have a plot of the wave functions and caustics like in the hydrogen paper...
\item should the actions be discussed in more detail? At the moment their non smoothness is only mentioned in the WKB approximation in the quantum part. Should we compute the monodromy matrix from smoothing the actions like in Irina's thesis?
\end{itemize}
}
| 202,898
|
TITLE: Relation between Combinatorics and Sum of First $(n-1)$ Natural Numbers
QUESTION [0 upvotes]: Is it mere coincidence that ${n \choose 2}$ is equal to the sum of the first $(n-1)$ natural numbers both of which happen to be $\frac{n(n-1)}{2}$?
A Proof of the relation is appreciated. Thanks
REPLY [1 votes]: It is not.
The number of two-element subsets of the set $S = \{1, 2, 3, \ldots, n\}$ is
$$\binom{n}{2} = \frac{n!}{2!(n - 2)!} = \frac{n(n - 1)(n - 2)!}{2 \cdot 1 \cdot (n - 2)!} = \frac{n(n - 1)}{2}$$
We can also count the set by considering two-element subsets with largest element $i$. The number of such subsets is $i - 1$ since there are $i - 1$ ways to choose the smaller element. Since $i$ can vary from $1$ to $n$, the number of two-element subsets is
$$\sum_{i = 1}^{n} (i - 1) = \sum_{k = 0}^{n - 1} k = \frac{n(n - 1)}{2}$$
where we have made the substitution $k = i - 1$.
| 43,229
|
- Telephone:
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| 305,589
|
\begin{document}
\title[
Dynamical
systems with fast switching and slow diffusion]{
Dynamical systems with fast switching and slow diffusion: hyperbolic equilibria and stable limit cycles}
\author[N. H. Du]{Nguyen H. Du}
\address{Department of Mathematics, Mechanics and
Informatics\\
Hanoi National University\\
334 Nguyen Trai\\
Thanh Xuan\\
Hanoi\\
Vietnam}
\email{dunh@vnu.edu.vn}
\author[A. Hening]{Alexandru Hening }
\address{Department of Mathematics\\
Tufts University\\
Bromfield-Pearson Hall\\
503 Boston Avenue\\
Medford, MA 02155\\
United States
}
\email{alexandru.hening@tufts.edu}
\author[D. Nguyen]{Dang H. Nguyen }
\address{Department of Mathematics \\
University of Alabama\\
345 Gordon Palmer Hall\\
Box 870350 \\
Tuscaloosa, AL 35487-0350 \\
United States}
\email{dangnh.maths@gmail.com}
\author[G. Yin]{George Yin}
\address{Department of Mathematics\\
Wayne State University\\
Detroit, MI 48202\\
United States
}
\email{gyin@math.wayne.edu}
\maketitle
\begin{abstract}
We study the long-term qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of
certain
stochastic differential equations (SDE) with Markovian switching,
in which
the process switches at random times
among different systems of SDEs, when the switching is fast varying and the diffusion (white noise)
term is slowly changing.
The system is modeled by
\[
dX^{\eps,\delta}(t)=f(X^{\eps,\delta}(t), \alpha^\eps(t))dt+\sqrt{\delta}\sigma(X^{\eps,\delta}(t), \alpha^\eps(t))dW(t)
, \ X^\eps(0)=x,
\]
where $\alpha^\eps(t)$ is a
finite state space Markov chain with irreducible generator $Q=(q_{ij})$.
The relative changing rates of
the switching and the diffusion are
highlighted by the two small parameters $\eps$ and $\delta$.
We associate to the system the averaged ordinary differential equation (ODE)
\[
d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x,
\]
where $\bar f(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)\nu_i$ and $(\nu_1,\dots,\nu_{m_0})$ is the unique invariant probability measure of the Markov chain with generator $Q$.
Suppose that for each pair $(\eps,\delta)$ of parameters, the process has an invariant probability measure $\mu^{\eps,\delta}$,
and that the averaged ODE has a limit cycle in which there is an averaged occupation measure $\mu^0$ for the averaged equation. We are able to prove, under weak conditions, that if $\bar f$ has finitely many unstable or hyperbolic fixed points, then $\mu^{\eps,\delta}$ converges weakly to $\mu^0$ as $\eps \to 0$ and $\delta \to 0$. Our results generalize to the setting where the switching process $\alpha^\eps$ is state-dependent and is given by
\[
\PP\{\alpha^\eps(t+\Delta)=j~|~\alpha^\eps=i, X^{\eps,\delta}(s),\alpha^\eps(s), s\leq t\}=q_{ij}(X^{\eps,\delta}(t))\Delta+o(\Delta),~~ i\neq j
\]
as long as the generator $Q(\cdot)=(q_{ij}(\cdot))$ is bounded, Lipschitz, and irreducible for all $x\in\R^d$.
We conclude our analysis by studying in detail an example of a stochastic predator-prey model.
\end{abstract}
\tableofcontents
\section{Introduction}\label{sec:int}
Natural phenomena are almost always influenced by different types of random noise. In order to better understand the world around us, it
is important
to study random perturbations of dynamical systems.
In the continuous dynamical systems setup,
the focus then shifts from the study of the behavior of deterministic differential equations to that of
differential equations with switching
(piecewise deterministic Markov processes) or stochastic differential equations with switching.
The long-term behavior of these systems can be analyzed by a careful study of the ergodic properties of the induced Markov processes.
Quite often, the ``white noise'' in the system is small compared to the deterministic component. In
such cases, one is usually interested in knowing how well the deterministic system approximates the stochastic one. It is common to model continuous-time phenomena by stochastic differential equations of the type
\begin{equation}\label{eq1.1}
dx^\delta(t)=f(x^\delta(t))dt+\sqrt{\delta}\sigma(x^\delta(t))dW(t),
\end{equation}
where $f\cd$ and $\sigma\cd$ are sufficiently smooth
functions, $W(\cdot)$ is a standard $m$-dimensional Brownian motion, and $\delta>0$ is a small parameter.
If we let $\delta\downarrow 0$, one would expect that the solutions of \eqref{eq1.1} converge, in an appropriate sense, to
that
of a deterministic differential equation.
Versions of this problem have been studied extensively starting with Freidlin and Wentzell \cite{FW70, FW98}, Fleming \cite{WF}, Kifer \cite{K81} and Day \cite{D82}.
If the process $x^\delta(t)$ has a unique ergodic probability measure $\mu^\delta$ for each $\delta>0$ and the origin of the corresponding deterministic ODE
\begin{equation}\label{eq1.2}
dx=f(x)dt,
\end{equation}
is a globally asymptotic equilibrium point,
Holland established in \cite{CH1} asymptotic expansions of the expectation of
the underlying functionals with respect to the unique ergodic probability measures $\mu^\delta$.
In addition, in \cite{CH}, Holland considered the case when the ODE \eqref{eq1.2} has an asymptotically stable limit cycle and proved the weak convergence of the family $(\mu^\delta)_{\delta>0}$ to the unique stationary distribution that is concentrated on the limit cycle of the process from \eqref{eq1.2}.
Our interest in the current problem stems from applications
in ecology. Quite often, one models the dynamics of populations with continuous-time processes. This way we inherently assume that organisms can respond instantaneously to changes in the environment. However, in some cases the dynamics are better described by discrete-time models, in which demographic decisions are not made continuously. In order to model more complex systems, one has to analyze `hybrid' systems where both continuous and discrete dynamics coexist. Such systems arise naturally in ecology, engineering, operations research, and physics as well as in emerging applications in wireless communications, internet traffic modeling, and financial engineering; see \cite{YZ} for more references.
Recently, there has been renewed interest in studying piecewise deterministic
Markov
processes (PDMP) \cite{D84}. One may describe a PDMP by the use of a two component process. The first component is a continuous state process represented by the solution of a deterministic differential equation, whereas the second component is a discrete event process taking values in a finite set. This discrete event process is often modeled as a continuous-time Markov chain with a finite state space. At any given instance, the Markov chain takes a value (say $i$ in the state space), and the process sojourns in state $i$ for a random duration. During this period, the continuous state follows the flow given by a differential equation associated with $i$. Then at a random instance, the discrete event switches to another state $j\not =i$. The Markov chain sojourns in $j$ for a random duration, during which, the continuous state follows another flow associated with the discrete state $j$.
A careful study of such
processes has recently led to a
better understanding of
predator-prey communities where the predator evolves much faster than the prey \cite{C16} and for a possible explanation of how the competitive exclusion principle from ecology, which states that multiple species competing for the same number of small resources cannot coexist, can be violated because of switching \cite{BL16,HN18}.
It is natural to study the SDE counter-part of PDMP, that is, SDE with switching. Similarly to the piecewise deterministic Markov processes mentioned in the previous paragraph,
in this setting one follows a specific system of SDE for a random time after which the discrete event switches to another state, and the process is governed by a different system of SDE.
The resulting stochastic process has a discrete component (that switches among a finite number of discrete states) and a continuous component (the solution of SDE associate with
each fixed discrete event state).
We refer the reader to \cite{YZ} for an introduction to SDEs with switching.
Most of the work inspired by Freidlin and Wentzell has been concerned with local phenomena that involve the exit times and exit probabilities from neighbourhoods of equilibria.
One usually uses the theory of large deviations to analyze the exit problem from the domain of attraction of a stable equilibrium point. There are more complicated situations, as the one we treat in this paper, in which large deviation techniques are not sufficient, and one needs to analyze the distributional scaling limits for the exit distributions \cite{B11}.
There have been some previous important studies for multiscale systems with fast and slow scales \cite{DS12, DSW12}. These previous papers have looked at large deviations in the related setting where one has a slow diffusion and the coefficients are fastly oscillating. However, in contrast to our framework, the fast oscillations come from having periodic coefficients and introducing a factor $\frac{1}{\eps}$ into the periodic component of the coefficients. The way the fast oscillations are introduced in these previous papers is similar to how it is done when one does stochastic homogenization. In the present paper, the switching comes from a discrete random process $\alpha^\eps$.
In this paper, we consider dynamical systems represented by switching diffusions, in which
the switching is rapidly varying whereas the diffusion is slowly changing.
To be more precise, let $(\Omega, \F, \{\F_t\}, \PP)$ be a filtered probability space satisfying the usual conditions.
Consider the process $(X^{\eps,\delta})_{t\geq 0}$ defined by
\begin{equation}\label{eq2.1}
dX^{\eps,\delta}(t)=f(X^{\eps,\delta}(t), \alpha^\eps(t))dt+\sqrt{\delta}\sigma(X^{\eps,\delta}(t), \alpha^\eps(t))dW(t)
, \ X^{\eps,\delta}(0)=x,\end{equation}
where $W(t)$ is an $m$-dimensional
standard Brownian motion, $\alpha^\eps(t)$ is
a finite-state Markov chain that is independent of the Brownian motion and that has a
state space $\M=\{1,..., m_0\}$ and generator $Q/\eps=\big(q_{ij}/\eps\big)_{m_0\times m_0}$,
$X^{\eps,\delta}$ is an $\R^d$-valued process, $f: \R^d\times\M\to\R^d, \sigma: \R^d\times\M\to\R^{d\times m},$ and
$\eps, \delta>0$ are two small parameters.
We assume that the matrix $Q$ is irreducible. The irreducibility of $Q$ implies that the Markov chain associated
with $Q$, which will be denoted by $(\tilde \alpha(t))_{t\geq 0}$,
is ergodic thus has a unique stationary distribution $(\nu_1,\dots,\nu_{m_0})$.
We denote by $X^{\eps,\delta}_{x, i}(t)$
the solution of \eqref{eq2.1} at time $t\geq0$ when the initial value is $(x,i)$ and by $\alpha^\eps_i(t)$ the Markov chain started at $i$.
Let us explore, intuitively, what happens when $\eps$ and $\delta$
are very small.
In this setting, $\alpha^\eps(t)$ converges very fast to its stationary distribution $(\nu_1,\dots,\nu_{m_0})$ while the diffusion is asymptotically small.
As a result, on each finite time interval $[0,T]$ for $T>0$, a solution of equation \eqref{eq2.1} can be approximated by the solution $\bar X_x(t)$ to
\begin{equation}\label{eq2.2}
d\bar X(t)=\bar f(\bar X(t))dt, \ \bar X(0)=x,
\end{equation}
where $\bar f(x)=\sum_{i=1}^{m_0}f(x, i)\nu_i$.
However, if in lieu of a finite time horizon,
we look at the process on
the infinite time horizon
$[0,\infty)$,
it is not clear that $\bar X_x(t)$ is a good approximation. Suppose that equation \eqref{eq2.2} has a stable limit cycle. A natural question is whether the invariant measures $(\mu^{\eps,\delta})$ of the processes \eqref{eq2.1} converge weakly as $\eps\downarrow 0$ and $\delta\downarrow 0$, to the measure concentrated on the limit cycle. This is the main problem that we address in the current paper. In order to do this, we
substantially extend the results of \cite{CH} by considering the presence of both small diffusion and rapid switching.
Because of the presence of both the switching and the diffusion we need to develop new mathematical techniques.
In addition, even if there is no switching and we are in the SDE setting of \cite{CH},
our assumptions are weaker than those used in \cite{CH}.
\begin{rmk}
One might be interested in the following natural generalization of the setting presented above. The switching process $\alpha^\eps$ can be state dependent, that is
\[
\PP\{\alpha^\eps(t+\Delta)=j~|~\alpha^\eps=i, X^{\eps,\delta}(s),\alpha^\eps(s), s\leq t\}=q_{ij}(X^{\eps,\delta}(t))\Delta+o(\Delta).
\]
As long as the generator $Q(x)=(q_{ij}(x))$ is irreducible for each $x\in\R^d$ one can show that on each finite time interval $[0,T]$ one can approximate the process from \eqref{eq2.1} if $\eps,\delta$ are small by
\[
d\bar X = \bar f(\bar X(t))dt, \bar X(0)=x,
\]
where $\bar f(x)=\sum_{i=1}^{m_0} f(x,i)\nu_i(x)$ and $(\nu_1(x),\dots,\nu_{m_0}(x))$ is the stationary distribution of a Markov chain with generator $Q(x)=(q_{ij}(x))$. We will explain through a sequence of remarks that our results hold for this generalization.
\end{rmk}
The rest of the paper is organized as follows. The main assumptions and results appear in Section \ref{sec:2}. In Section \ref{sec:3}, we estimate the exit time of the solutions of \eqref{eq2.1} from neighborhoods around the stable manifolds of the critical points of $\bar f$. The proof of the main result is presented in Section \ref{sec:4}. In Section
\ref{sec:5}, we apply our results to a general predator-prey model. In addition to showcasing
our result in a specific setting, the proofs from Section \ref{sec:5} are interesting on their own right as they are quite technical and require the development of new tools. Finally, in Section \ref{sec:6}, we provide some numerical examples to illustrate our results from the predator-prey setting from Section \ref{sec:5}.
\subsection{Assumptions and main results}\label{sec:2}
We denote by $A'$ the transpose of a matrix $A$, by $|\cdot|$ the Euclidean norm of vectors in $\R^d$, and by $\|A\|:=\sup\{|Ax|: x\in\R^d, |x|=1\}$ the operator norm of a matrix $A\in \R^{d\times d}$. We also define $a\wedge b:=\min\{a, b\}$, $a\vee b:=\max\{a, b\}$, and the closed ball of radius $R>0$ centered at the origin $B_R:=\{x\in\R^d:|x|\leq R\}$.
We recall some definitions due to Conley \cite{C78}. Suppose we are given a flow $(\Phi_t(\cdot))_{t\in\R}$. A compact invariant set $K$ is called \textit{isolated} if there exists a neighborhood $V$ of $K$ such that $K$ is the maximal compact invariant set in $V$. A collection of nonempty sets $\{M_1,\dots,M_k\}$ is a \textit{Morse decomposition} for a compact invariant set $K$ if $M_1,\dots,M_k$ are pairwise disjoint, compact, isolated sets for the flow $\Phi$ restricted to $K$ and the following properties hold: 1) For each $x\in K$ there are integers $l=l(x)\leq m=m(x)$ for which the \textit{alpha limit set} of $x$, $\hat\alpha(x)=\bigcap_{t\leq 0} \overline{\{\Phi_s(x), s\in(-\infty,t] \}}$, satisfies $\hat\alpha(x)\subset M_l$ and the \textit{omega limit set} of $x$, $\hat\omega(x):=\bigcap_{t\geq 0} \overline{\{\Phi_s(x), s\in[t,\infty) \}}$, satisfies $\hat\omega(x)\subset M_m$ 2) If $l(x)=m(x)$ then $x\in M_l=M_m$.
\begin{asp}\label{asp1}
We impose the following assumptions for the processes modeled
by the systems \eqref{eq2.1} and \eqref{eq2.2}.
\begin{enumerate}[(i)]
\item
For each $i\in \M$,
$f(\cdot, i)$ and $\sigma(\cdot, i)$ are locally Lipschitz continuous.
\item There is an $a>0$ and a twice continuously differentiable
real-valued, nonnegative function $\Phi(\cdot)$ satisfying $\lim\limits_{R\to\infty}\inf\{\Phi(x): |x|\geq R\}=\infty$ and $(\nabla \Phi)'(x)f(x,i) \leq a(\Phi(x)+1)$,
for all $(x, i)\in\R^d\times\M$.
\item The vector field $\bar f(\cdot)$ has finitely many equilibrium points $\{x_1,\dots,x_{n_0-1}\}$ and a unique limit cycle $\Gamma$. The equilibrium points are either sources or hyperbolic points.
\item
There exists a Morse decomposition $\{M_1, M_2,\cdots, M_{n_0}\}$ of the flow associated with $\bar f$ such that
$M_{n_0}=\Gamma$ is the limit cycle and for any $i<n_0$ we have $M_i=\{x_i\}$ where $x_i$ is an equilibrium point.
\item There exists $\eps_0>0$ such that for all $0<\eps<\eps_0$, the system \eqref{eq2.1} has a unique solution.
Furthermore, for any $0<\eps<\eps_0$, the process $(X^{\eps,\delta}(t), \alpha^{\eps,\delta}(t))$ has the strong Markov property and has an invariant measure $\mu^{\eps,\delta}$.
The family $(\mu^{\eps,\delta})_{0<\eps<\eps_0}$ is tight, i.e., for any $\gamma>0$ there exists an $R=R_\eta>0$ such that $\mu^{\eps,\delta}(B_R)>1-\gamma$ for all $0<\eps<\eps_0$.
\end{enumerate}
\end{asp}
\begin{rmk}
We note that using Assumption (ii), we can work in a compact state space $\K\subset \R^n$ if the diffusion term from \eqref{eq2.1} is zero. Assumptions (i) and (ii) are needed in order to deduce the existence and boundedness of a unique solution to equation \eqref{eq2.1} in the absence of the diffusion term. Assumption (iv) is used to make sure that there exist no heteroclinic cycles.
Assumption (v) ensures that \eqref{eq2.1} has a unique solution that is strong Markov. Sufficient conditions that imply uniqueness and the strong Markov property exist in the literature \cite{MY, YZ}.
\end{rmk}
\begin{rmk}
In \cite{CH} the author studied \eqref{eq1.1} under the assumptions that
\begin{enumerate}
\item [(A1)] $f,\sigma\in C^2(\R^d)$.
\item [(A2)] The system \eqref{eq1.2} has a unique limit cycle.
\item [(A3)] There exists at most a finite number of critical points $x^*$ of $f$. At each critical point the Jacobian matrix has only positive real parts and the matrix $\sigma'\sigma$ is positive definite.
\item [(A4)] For any compact set $B$ not containing critical points and any $u>0$ there exists $T>0$ such that if $x\in B$, then
\[
d(x^0_x(t),\Gamma)<u,~\text{for}~t\geq T.
\]
\item [(A5)] There exists $\delta_0>0$ such that for $0<\delta<\delta_0$ the stochastic differential equation has a unique ergodic measure $\mu_\delta$. Furthermore, the family $(\mu_{\delta})_{0<\delta<\delta_0}$ is tight in $\R^d$.
\end{enumerate}
Our work generalizes \cite{CH} significantly in the following aspects.
First, we work with two types of randomness - one comes from the diffusion term and the other from the switching mechanism. Second, Assumption \ref{asp1} (i) is weaker than (A1). Third, we can have any hyperbolic fixed points whereas assumptions (A3)-(A4) imply that all fixed points are sources and the deterministic system converges uniformly to the limit cycle. In addition, we do not need $\sigma'\sigma$ to be positive definite at the critical points.
\end{rmk}
\begin{rmk}
There are several papers which look at the exit time asymptotics near hyperbolic fixed points of small perturbations of dynamical systems \cite{K81, B08,B11}. We do not assume like in these paper that the noise is uniformly elliptic and we have to deal with the additional complications of a stable limit cycle as well as the switching due to $\alpha^\eps$.
\end{rmk}
Let $T_\Gamma>0$ be the period of the limit cycle $\Gamma$. We can define a probability measure $\mu^0$, which is independent of the starting point $y\in \Gamma$, by
\begin{equation}\label{e:mu0}
\mu^0(\cdot)=\dfrac1{T_\Gamma}\int_0^{T_\Gamma} \1_{\{\bar X_y(s)\in \cdot\}}ds,
\end{equation}
where $\bar X_y(t)$ is the solution to equation \eqref{eq2.2} starting at $ \bar X(0)=y$ and $\1_{\{\cdot\}}$ is the indicator function.
The measure $\mu^0\cd$ is the averaged occupation measure of the process $\bar X$ restricted to the limit cycle $\Gamma$. Throughout the paper, we assume that $\delta$ depends on $\eps$, i.e. $\delta= \delta(\eps)$, and $\lim\limits_{\eps\downarrow 0}\delta (\eps)=0$. We will investigate the asymptotic behavior of the invariant probability measures $\mu^{\eps,\delta}$ as $\eps \downarrow 0$ in the following three cases:
\begin{equation}\label{eq:ep-dl}
\lim\limits_{\eps\downarrow 0}\dfrac{\delta}\eps=\left\{\begin{array}{ll}l\in(0,\infty), &\mbox{ case 1}\\0, &\mbox{ case 2}\\\infty, &\mbox{ case 3}.\end{array}\right.
\end{equation}
The multi-scale modeling approach we use is similar to the one from \cite{HY14}.
\begin{asp}\label{asp2}We
impose additional conditions corresponding to the cases from \eqref{eq:ep-dl}.
\begin{enumerate}
\item[1)] Suppose $\lim_{\epsilon\downarrow 0}\frac{\delta}{\eps}=l\in (0,\infty)$. For any critical point $x^*$ of $\bar f$ there exists $i^*\in \M$ such that $\beta'f(x^*, i^*)\ne 0$ or $\beta'\sigma(x^*, i^*)\ne0$ where $\beta$ is a normal vector of the stable manifold of \eqref{eq2.2} at $x^*$.
\item[2)] Suppose $\lim_{\epsilon\downarrow 0}\frac{\delta}{\eps}=0$. For any critical point $x^*$ of $\bar f$ there exists $i^*\in \M$ such that $\beta'f(x^*, i^*)\ne 0$ where $\beta$ is a normal vector of the stable manifold of \eqref{eq2.2} at $x^*$.
\item[3)] Suppose $\lim_{\epsilon\downarrow 0}\frac{\delta}{\eps}=\infty$. For any critical point $x^*$ of $\bar f$, there exists $i^*\in \M$ such that $\beta'\sigma(x^*, i^*)\ne 0$ where $\beta$ is a normal vector of the stable manifold of \eqref{eq2.2} at $x^*$.
\end{enumerate}
\end{asp}
The intuition for the conditions of Assumption \ref{asp2} is the following. In case 2, since $\delta$ tends to $0$ much faster than $\eps$,
for sufficiently small $\delta$,
the behavior of $X^{\eps,\delta}(t)$ will be close to the process $\xi^{\eps}(t)$ defined by
\begin{equation}\label{eq2.3}
d\xi^\eps(t)=f\big(\xi^\eps(t),\alpha^\eps(t)\big)dt.
\end{equation}
We denote from now on by $\xi^{\eps}_{x, i}(t)$ the solution of \eqref{eq2.3} at time $t\geq 0$ if the initial condition is $(x,i)$.
If for each $i\in \M$, $f(x^*, i)=0$ at a critical point $x^*$ of $\bar f$, the Dirac mass function at $x^*$, $\boldsymbol\delta_{x^*}$, will be an invariant
measure for $\xi^\eps(t)$. Because of this, the sequence of invariant probability measures $(\mu^{\eps,\delta})$ (or one of its subsequences) may converge to $\boldsymbol\delta_{x^*}$. In order to have the weak convergence of $(\mu^{\delta, \eps})_{\eps>0}$ to the measure $\mu^0$, we need to assume that there is an $i^*\in\M$ such that $\beta'f(x^*, i^*)\ne0$ where $\beta$ is a normal vector of the stable manifold of \eqref{eq2.2} at $x^*$. This guarantees that the process from \eqref{eq2.3} gets pushed away from the equilibrium $x^*$ and away from the stable manifold (where it could get pushed back towards the equilibrium).
In case 3, the switching is very fast compared to the diffusion term, so for small $\eps$ the process will behave like
\[
d \eta^\eps(t) = \bar f(\eta^\eps(t))dt + \sqrt{\delta}\sigma(\eta^\eps(t),i)dW(t).
\]
In order for the limit of $(\mu^{\eps,\delta})$ not to put mass on the critical point $x^*$ of $\bar f$, we need to suppose that there exists an $i^*\in\M$ such that $\beta'\sigma(x^*, i^*)\ne0$ where $\beta$ is a normal vector of the stable manifold of \eqref{eq2.2} at $x^*$
For case 1, since both the switching and the diffusion are on a similar scale, we need to assume that for each critical point $x^*$ of $\bar f$ there is $i^*\in\M$ satisfying either $\beta'\sigma(x^*, i^*)\ne0$ or $\beta'f(x^*, i^*)\ne0$.
The next theorem is the main result of this paper.
\begin{restatable}{thm}{main}\label{t:main}
Suppose Assumptions \ref{asp1} and \ref{asp2} hold. The family of invariant probability measures $(\mu^{\eps,\delta})_{\eps>0}$ converges weakly to the measure $\mu^0$ given by \eqref{e:mu0} in the sense that for every bounded and continuous function $g:\R^d\times\M\to \R$,
$$\lim\limits_{\eps\to0}\sum_{i=1}^m\int_{\R^d} g(x, i)\mu^{\eps,\delta}(dx, i)=\dfrac1{T_\Gamma}\int_{0}^{T_\Gamma}\bar g(\bar X_y(t))dt,$$
where $T_\Gamma$ is the period of the limit cycle, $y\in\Gamma$ and $\bar g(x)=\sum_{i\in\M}g(x, i)\nu_i$.
\end{restatable}
\begin{rmk}\label{r:state}
Theorem \ref{t:main} still holds if the switching component $\alpha^\eps$ is state-dependent with generator $Q(x)=(q_{ij}(x))_{\M\times\M}, x\in \R^d$ as long as $Q$ is bounded and satisfies the following conditions:
\begin{itemize}
\item For all $i, j$ the functions $q_{ii}(\cdot)$ and $\frac{q_{ij}(\cdot)}{q_{ii}(x\cdot)}$ are Lipschitz continuous.
\item If $q_{ij}(x)>0$ for some $x\in\R^d$ then $\inf_{x\in\R^d} \frac{q_{ij}(x)}{|q_{ii}(x)|}>0$.
\item For all $i$ we have $\inf_{x\in\R^d}|q_{ii}(x)|>0$.
\item There exists $k\in\N$ such that we have $\inf_{x\in\R^d} \hat q^{(k)}_{ij}(x)>0$ where $\hat Q(x)=(0\vee q_{ij}(x))_{\M\times\M}$, and $\hat Q^k(x)=(\hat q^{(k)}_{ij}(x))$.
\end{itemize}
We explain how one can do this in Remark \ref{r:dens}.
\end{rmk}
\subsection{An application of Theorem \ref{t:main}} We will exhibit an example where the result of Theorem \ref{t:main} applies. Recently there has been renewed
interest in stochastic population dynamics \cite{HN16, BL16, B18, HN18}.
Suppose we have a predator-prey system of the form
\begin{equation}\label{ex00}
\left\{\begin{array}{lll}
\disp {d \over dt}{x}(t)=x(t)\left[ a- bx(t)-y(t)h(x(t), y(t))\right]\\
\disp {d \over dt}
y(t)=y(t)\left[- c- d y(t)+x(t)fh(x(t), y(t))\right]
.\end{array}\right.
\end{equation}
Here $x(t), y(t)$ denote the densities of prey and predator
at time $t\geq 0$, respectively; $a, b, c, d, f>0$ describe the per-capita birth/death and competition rates, and $xh(x,y), yh(x,y)$ are the functional responses of the predator and the prey. For instance, if $h(x, y)$ is constant, the model is the classical Lotka-Volterra one \cite{L25, V28, GH79}.
If $$h(x, y)=\dfrac{m_1}{m_2(i)+m_3x+m_4y},$$ the functional response is of Beddington-DeAngelis type \cite{CC01}.
The setting of \eqref{ex00} is very general and encompasses many of the models used in the ecological literature.
We explore what happens in the fast-switching slow-noise limit for the following noisy extension of \eqref{ex00}
\begin{equation}\label{ex1}
\left\{\begin{array}{lll}d{X^{\eps,\delta}}(t)=X^{\eps,\delta}(t)\varphi\big(X^{\eps,\delta}(t), Y^{\eps,\delta}(t), \alpha^\eps(t))dt+\sqrt{\delta}\lambda(\alpha^\eps(t))X^{\eps,\delta}(t)dW_1(t)\\
d{Y^{\eps,\delta}}(t)=Y^{\eps,\delta}(t)\psi\big(X^{\eps,\delta}(t), Y^{\eps,\delta}(t), \alpha^\eps(t))dt+\sqrt{\delta}\rho(\alpha^\eps(t))Y^{\eps,\delta}(t)dW_2(t).\end{array}\right.
\end{equation}
Here
\bea \ad \varphi(x, y, i)=a(i)-b(i)x-yh(x, y, i) \ \hbox{
and }\\
\ad \psi(x, y, i)=-c(i)-d(i)y+f(i)xh(x, y, i), \eea
where
$a(\cdot), b(\cdot), c(\cdot), d(\cdot), f(\cdot), \lambda(\cdot), \rho(\cdot)$ are positive functions defined on $\M$,
$\delta=\delta(\eps)$ depends on $\eps$, $\lim\limits_{\eps\to0}\delta=0$,
$W_1(t)$ and $W_2(t)$ are independent Brownian motions, and
$\alpha^\eps$ is an independent Markov chain with generator $Q/\eps$. As before, the generator $Q$ is assumed to be irreducible so that the Markov chain has a unique stationary distribution given by $(\nu_1,\dots,\nu_{n_0})$.
The function $h(\cdot, \cdot, \cdot)$ is assumed to be positive, bounded, and continuous on $\M\times\R^2_+$.
For $g(\cdot)= a(\cdot), b(\cdot), c(\cdot), d(\cdot), f(\cdot), \varphi(\cdot), \psi(\cdot)$, define the \textit{averaged quantities} $\bar g:=\sum g(i)\nu_i$, $g_m=\min\{g(i):i\in\M\}, g_M=\max\{g(i):i\in\M\}$. Set
$h_1(x, y):=\sum h(x, y, i)\nu_i$ and $h_2(x, y):=\sum f(i)h(x, y, i)\nu_i.$
The existence and uniqueness of a global positive solution to \eqref{ex1} can be proved in the same manner as in \cite{JJ} or \cite{CDN} and is therefore omitted.
We denote by $Z^{\eps,\delta}_{z, i}(t)=(X^{\eps, \delta}_{z, i}(t), Y^{\eps, \delta}_{z, i}(t))$ the solution to \eqref{ex1} with initial value $\alpha^{\eps}(0)=i\in\M, Z^{\eps,\delta}_{z, i}(0)=z\in\R^{2}_+.$
Consider the averaged equation
\begin{equation}\label{ex2}
\left\{\begin{array}{lll}
\disp {d \over dt}{X}(t)=X(t)\bar\varphi(X(t), Y(t))
=X(t)\left[\bar a-\bar bX(t)-Y(t)h_1(X(t), Y(t))\right]\\
\disp {d \over dt}
Y(t)=Y(t)\bar\psi(X(t), Y(t)))=Y(t)\left[-\bar c-\bar d Y(t)+X(t)h_2(X(t), Y(t))\right]
.\end{array}\right.
\end{equation}
We denote by $\bar Z_z(t)=(\bar X_z(t), \bar Y_z(t))$, the solution to \eqref{ex2} with initial value $\bar Z_z(0)=z$.
\begin{asp}\label{asp5.1}
The following properties hold.
\begin{enumerate}
\item The system \eqref{ex2} has a finite number of positive
equilibria and a unique stable limit cycle $\Gamma$. In addition, any positive solution not starting at an equilibrium converges to the stable limit cycle.
\item The inequality $$\frac{\bar a}{\bar b}h_2\left(\frac{\bar a}{\bar b}, 0\right)>\bar c$$ is satisfied.
\end{enumerate}\end{asp}
\begin{rmk}
Note that the Jacobian of
$\Big(x\bar\phi(x,y), y\bar\psi(x,y)\Big)^\top$
at $\left(\frac{\bar a}{\bar b}, 0\right)$
has two eigenvalues: $-\bar c+\frac{\bar a}{\bar b}h_2(\frac{\bar a}{\bar b}, 0)$
and $-\frac{\bar b^2}{\bar a}<0$.
If $-\bar c+\frac{\bar a}{\bar b}h_2(\frac{\bar a}{\bar b}, 0)<0$,
then $\left(\frac{\bar a}{\bar b}, 0\right)$ is a stable equilibrium of \eqref{ex2}, which violates condition (i) of Assumption \ref{asp5.1}. This shows that condition (ii) is often contained in condition (i).
We note that the model \eqref{ex2} is quite general and as such conditions on the parameters for the existence and uniqueness of a limit cycle are in general complicated.
\end{rmk}
We can apply Theorem \ref{t:main} to this model if we can verify part (v) of Assumption \ref{asp1} since the other conditions are clearly satisfied.
Since the process $\alpha^\eps(t)$ is ergodic and the diffusion is nondegenerate, an invariant probability measure of the solution $Z^{\eps,\delta}(t)$ is unique if it exists.
It is unlikely that one could find a Lyapunov-type function satisfying the hypothesis of \cite[Theorem 3.26]{YZ} in order to prove the existence of an invariant probability measure.
In addition, the tightness of the family of invariant probability measures $(\mu^{\eps,\delta})_{\eps>0}$ cannot be proved using the methods from \cite{DDT, DDY}.
These difficulties can be overcome
with the help of a new technical tool. We partition the domain $(0,\infty)^2$ into several parts and then construct a truncated Lyapunov-type function. We then estimate the average probability that the solution belongs to a specific part of our partition.
This then allows us to prove that the family of invariant probability measures $(\mu^{\eps,\delta})_{\eps>0}$ is tight on the interior of $\R_+^2$, i.e. for any $\eta>0$, there are $0<\eps_0,\delta_0<1< L$ such that
for all $\eps<\eps_0,\delta<\delta_0$, the unique invariant measure $\mu^{\eps,\delta}$ of $(Z^{\eps,\delta}(t), \alpha^\eps(t))$ satisfies
$$\mu^{\eps,\delta}([ L^{-1}, L]^2)\geq 1-\eta.$$
We are able to prove the following result.
\begin{restatable}{thm}{mainnex}\label{thm5.1}
Suppose Assumption {\rm\ref{asp5.1}} holds. For
sufficiently small $\delta$ and $\eps$, the process given by \eqref{ex1} has a unique invariant probability measure $\mu^{\eps,\delta}$ with support in $\R_+^{2,\circ}$ $($where $\R^{2,\circ}_+$ denotes the interior of $\R^2_+)$. In addition:
\begin{enumerate}
\item[a)] If $\lim\limits_{\eps\to0}\dfrac\delta\eps=l\in(0,\infty]$, the family of invariant measures $(\mu^{\eps,\delta})_{\eps>0}$ converges weakly to $\mu^0$, the occupation measure of the limit cycle of \eqref{ex2}, as $\eps\to0$ (in the sense
of Theorem \ref{t:main}).
\item[b)] If $\lim\limits_{\eps\to0}\dfrac\delta\eps=0$ and at each critical point $(x^*, y^*)$ of $(\bar\varphi(x, y), \bar\psi(x, y))$, there is $i^*\in\M$ such that either $\varphi(x^*, y^*, i^*)\ne0$ or $\psi(x^*, y^*, i^*)\ne0$, then the family of invariant measures $(\mu^{\eps,\delta})_{\eps>0}$ converges weakly to $\mu^0$, the occupation measure of the limit cycle of \eqref{ex2}, as $\eps\to0$.
\end{enumerate}
\end{restatable}
\begin{rmk}
We note that on any finite time interval $[0,T]$ the solutions to \eqref{ex1} converge to the solutions of \eqref{ex2}. However, in ecology, people are interested in the long term behavior of ecosystems as $T\to \infty$. Therefore, the above result shows rigorously that \eqref{ex2} gives the correct long-term behavior.
\end{rmk}
\subsection{Sketch of proof of Theorem \ref{t:main}}
Because some parts of the proofs are very technical, in order to offer some intuition to the reader we present the main ideas in this subsection.
Condition (v) of Assumption \ref{asp1} is a tightness assumption for the family of invariant measures $(\mu^{\eps,\delta})_{0<\eps<\eps_0}$. This implies that any weak limit of $(\mu^{\eps,\delta})_{0<\eps<\eps_0}$ is an invariant measure of the limit system \eqref{eq2.2}. The main technical issue is to show that any subsequential limit of $(\mu^{\eps,\delta})_{0<\eps<\eps_0}$ does not assign any mass to any of the fixed points of $\bar f$. This is done by a careful analysis of the nature of the deterministic and stochastic systems near the attracting region $\chi_l:=\{y: \lim_{t\to\infty}\bar X_y(t)=x_l\}$, of an equilibrium $x_l$ of $\bar f$. Note that if $x_l$ is a source then $\chi_l=\{x_l\}$ while if $x_l$ is hyperbolic $\chi_l$ can be an unbounded set. This makes the problem hard.
In Section \ref{sec:3}, using large deviation techniques, we establish the following uniform estimate for the probability that the processes $X_{x,i}^{\eps,\delta}$ and $\bar X_x$ are close on a fixed time interval: For any $R$, $T$,
and $\gamma>0$, there is a $\kappa=\kappa(R,\gamma, T)>0$ such that
\begin{equation}\label{sk1}
\PP\left\{\left|X^{\eps,\delta}_{x,i}(t)-\bar X_x(t)\right|\geq\gamma ~\text{for some}~t\in [0,T] \right\}<\exp\left(-\frac{\kappa}{\eps+\delta}\right), x\in B_R.
\end{equation}
The main task is to estimate the time of exiting the attracting region, $\chi_l\cap B_R$, of an equilibrium $x_l$. To be precise, we show that $X_{x,i}^{\eps,\delta}$ leaves small neighborhoods of $\chi_l\cap B_R$ with strictly positive probability in finite time if we start close to $\chi_l\cap B_R$. We find uniform lower bounds for these probabilities.
In fact, for any
sufficiently small $\Delta>0$
and sufficiently large
$R>0$
to include all the sets $M_i, i=1,\dots,n_0$, we can find $\theta_1,\theta_3>0$, $H^\Delta_l>0$, and $ \eps_{l}(\Delta)$ such that for $\eps<\eps_{l}(\Delta)$,
\begin{equation}\label{sk2}
\PP\left\{\wdt \tau^{\eps,\delta}_{x, i}\leq H^\Delta_l\right\}\geq\psi^{\Delta,\eps}:=\exp\left(-\dfrac{\Delta}{\eps+\delta}\right),\,|x-x_l|<\theta_1,
\end{equation}
where
$$\wdt \tau^{\eps,\delta}_{x, i}:=\inf\{t\geq 0: X^{\eps,\delta}_{x,i}(t)\in B_R \text{ and } \dist(X^{\eps,\delta}_{x,i}(t),\chi_l)\geq \theta_3\}.$$
We prove the estimate \eqref{sk2} in the different cases as follows:
\begin{enumerate}
\item [1)] Suppose that there is an $i^*\in\M$ satisfying
$\beta'f(x_l,i^*)\neq 0$, where $\beta$ is a normal unit vector of the
stable manifold of \eqref{eq2.2} at $x_l$. Then we estimate the time $\alpha^\eps(t)$ stays in $i^*$ and consider the diffusion in this fixed state, that is
\[
dZ^\delta(t)=f(Z^\delta(t), i^*)dt+\sqrt{\delta}\sigma(Z^\delta(t), i^*)dW(t).
\]
Since the drift $f(x,i^*)$ is nonzero and pushes us away from the stable manifold of $x^*$, and the diffusion term is small, we can estimate the exit time $\wdt\tau^{\eps,\delta}_{x, i}$.
\item [2)] Suppose $\lim_{\eps\to0}\frac{\delta}{\eps}\in(0,\infty]$ and there is an $i^*$ such that $\beta'\sigma(x_l,i^*)\neq 0$. If $\lim_{\eps\to0}\frac{\delta}{\eps}<\infty$, suppose in addition that $\beta'f(x_l,i)=0, i\in \M$. We estimate the time $\alpha^\eps(t)$ to be
in $i^*$ and consider the diffusion component in the direction $\beta$ in this fixed state
\[
dZ^{\eps,\delta} = \sqrt{\delta}\beta'\sigma(Z^{\eps,\delta},\alpha^\eps)dW(t)
\]
Since the diffusion coefficient does not vanish close to $x_l$, we can do time change so that we get a Brownian motion. Then we can estimate the probability that the exit time exceeds a given number. Ultimately, we show that $Z^{\eps,\delta}$ and $\beta'X^{\eps,\delta}$ are close to each other.
\end{enumerate}
Comparing the rates in \eqref{sk1} with \eqref{sk2} is key to prove the main result in Section \ref{sec:4} (see e.g. \cite{CH, kifer12}). The idea is to estimate the time of exiting the attracting region, $\chi_l\cap B_R$, of an equilibrium $x_l$ as well as the time of coming back to this region. Then we prove that eventually, the probability of entering $\chi_l\cap B_R$ is very small compared to the probability of exiting the region.
If we start with $\bar X(0)$ close to $\chi_l\cap B_R$ then after a finite time $\bar X$ will be close to one of the equilibrium points or the limit cycle. Using this together with \eqref{sk1} and \eqref{sk2} we get that there exist neighborhoods $N_1, G_1$ of $\chi_l\cap B_R$ with $N_1\subset G_1$ such that
\[
\PP\{\tau^{\eps,\delta}_{x,i}<L\}>\dfrac1{8}\psi^{\Delta,\eps}, x\in N_1
\]
for some constant $L>0$ and
$$\tau^{\eps,\delta}_{x,i}=\inf\{t\geq0: X_{x,i}^{\eps,\delta}(t)\in B_R\setminus G_1\}.$$
This can be leveraged into showing that with high probability, if we start in $N_1$, we will leave the region $G_1\supset N_1$ in a finite, uniformly bounded, time:
\begin{equation}\label{sk3}
\PP\left\{\tau^{\eps,\delta}_{x,i}<T^{\eps,\delta}_{\Delta,1}\right\}>\frac12, x\in N_1
\end{equation}
where $T^{\eps,\delta}_{\Delta,1}:=C\exp\left(\dfrac{\Delta}{\eps+\delta}\right)$.
Using \eqref{sk1} we can find a constant $\hat T>0$, independent of $\eps$ such that
\begin{equation}\label{sk4}
\PP\left\{X^{\eps, \delta}_{x, i}(\hat T)\notin G_1 \right\}\geq1-\exp\Big(-\dfrac{\kappa}{\eps+\delta}\Big), x\in B_R\setminus N_1
\end{equation}
and that
\begin{equation}\label{sk5}
\PP\left\{X^{\eps, \delta}_{x, i}(t)\notin N_1, \ ~\text{for all}~ t\in[0,\hat T]\right\}\geq1-\exp\Big(-\dfrac{\kappa}{\eps+\delta}\Big), x\in B_R\setminus G_1.
\end{equation}
Note that $T^{\eps,\delta}_{\Delta,1}\to\infty$ as $\eps\to 0$. However, if we pick $\Delta<\kappa/2$, we have
\begin{equation}\label{sk6}
\lim_{\eps\to 0}T_{\Delta,1}^{\eps,\delta}\exp\left(-\dfrac{\kappa}{\eps+\delta}\right)= \lim_{\eps\to 0}\exp\left(\dfrac{\Delta}{\eps+\delta}\right)\exp\left(-\dfrac{\kappa}{\eps+\delta}\right)= 0.
\end{equation}
The estimate \eqref{sk6} shows the exit time is not long compared to the good rate of large deviations,
which will be used to show that invariant measures cannot put much mass on the equilibria.
Let $\widetilde X^{\eps, \delta}(t)$ be the stationary solution, whose distribution is $\mu^{\eps,\delta}$ for every time $t\geq 0$. Let $\tau^{\eps,\delta}$
be the first exit time of $\widetilde X^{\eps, \delta}(t)$ from $G_1$.
We can show that for any $\eta>0$ we can find $R>0$ such that $\mu^{\eps,\delta}(N_1)\leq 2\eta$ by using \eqref{sk4}, \eqref{sk5}, and \eqref{sk6} to find the probabilities of the events
\begin{align*}
K_1^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \tau^{\eps,\delta}\geq T_{\Delta,1}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_1\Big\}\\
K_2^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \tau^{\eps,\delta}< T_{\Delta,1}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_1\Big\}\\
K_3^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \widetilde X^{\eps,\delta}(0)\in B_R\setminus N_1\Big\}\\
K_4^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \widetilde X^{\eps,\delta}(0)\notin B_R\Big\}.
\end{align*}
Similar arguments show that for any $\eta>0$, we can find $R>0$ and neighborhoods $N_1,\dots, N_{n_0-1}$ of $\chi_1\cap B_R,\dots, \chi_{n_0-1}\cap B_R$ such that
$$\limsup_{\eps\to0} \mu^{\eps,\delta}(\cup_{j=1}^{n_0-1}N_j)\leq 2^{n_0}\eta.$$
Using this fact together with Assumption \ref{asp1} and Lemma \ref{lm2.2} we can establish, by a straightforward modification of the proof of \cite[Theorem 1]{CH}, that for any $\eta>0$ there is neighborhood $N$ of the limit cycle $\Gamma$ such that
$$\liminf_{\eps\downarrow 0} \mu^{\eps,\delta} (N)>1-2^{n_0}\eta.$$
\section{Estimates for the first exit times}\label{sec:3}
Define for any $i=1,\dots,n_0$ and $\theta>0$, the sets
$\chi_i:=\{y: \lim_{t\to\infty}\dist(\bar X_y(t),M_i)=0\}$ and
$M_{i,\theta}:=\{y: \dist(y, M_i)<\theta\}$.
Let $R_0>1$ be large enough such that $B_{R_0-1}$ contains all $M_i$, $i=1,\dots, n_0$.
Fix $\theta_0\in(0,1)$ such that $\{M_{i,2\theta_0}, i=1,\dots, n_0\}$ are mutually disjoint
and $M_{i,2\theta_0}\cap \chi_j=\emptyset$ for $j<i$. For any $\eta>0$, let $R=R_\eta>0$ such that $\mu^{\eps,\delta}(B_R)>1-\eta$ and $R<R_0$.
The following is a well-known exponential martingale inequality (see \cite[Theorem 1.7.4]{XM}).
\begin{lm}\label{l:exp}(Exponential martingale inequality)
Suppose $(g(t))$ is a real-valued $\F_t$-adapted process and $\int_0^Tg^2(t)dt<\infty$ almost surely. Then for any $a,b>0$ one has
$$\PP\left\{\sup_{t\in[0, T]}\left[\int_0^tg(s)dW(s)-\dfrac{a}{2}\int_0^tg^2(s)ds\right]>b\right\}\leq e^{-ab}.$$
\end{lm}
We will make use of this lemma repeatedly in the proofs to follow. The next result gives us estimates on how close the solutions to \eqref{eq2.1} and \eqref{eq2.3} are on a finite time interval if they have the same starting points.
The argument of the
proof is pretty standard. For completeness, it relegated
to Appendix \ref{a:1}.
\begin{lm}\label{lm2.2}
For any $R$, $T$, and $\gamma>0$, there is a $\kappa=\kappa(R,\gamma, T)>0$ such that
$$\PP\left\{\left|X^{\eps,\delta}_{x,i}(t)-\bar X_x(t)\right|\geq\gamma ~\text{for some}~t\in [0,T] \right\}<\exp\left(-\frac{\kappa}{\eps+\delta}\right), x\in B_R.$$
\end{lm}
\begin{lm}\label{lm3.1}
Let $N$ be an open set in $\R^d$ and let $\check\tau_{x, i}^{\eps, \delta}$ be any stopping time.
Suppose that there is an $\ell>0$ such that for all starting points
$(x, i)\in N\times\M$ one has $\PP\{\check\tau_{x, i}^{\eps, \delta}<\ell\}\geq a^{\eps,\delta}>0$, where $\lim\limits_{\eps\to0}a^{\eps,\delta}=0$.
Then $\PP\left\{\check\tau_{x, i}^{\eps, \delta}<\dfrac{\ell}{a^{\eps,\delta}}\right\}>1/2$ for $(x, i)\in N\times\M$ if $\eps$ is sufficiently small.
\end{lm}
\begin{lm}\label{lm2.5} The following properties hold:
\begin{enumerate}
\item For any $\theta>0, R>0$, there exists $\wdt T_1>0$ such that
for any $y\in B_R$,
$\bar X_y(t)\in M_{k,\theta}$ for some $t<\wdt T_1$, and some $k\in\{1,\dots, n_0\}$.
\item For any $y\in B_R\setminus\chi_1$ and any $\theta>0$, there exists $\wdt t_y>0$ such that $\bar X_y(t_y)\in\bigcup_{k=2}^{n_0} M_{k,\theta}$.
\item For any $\theta_1>0$, $R\geq R_0$, there exists $\theta_2>0$ such that
$\dist(\bar X_y(t), \chi_1)>\theta_2$ for any $t>0$ and $y\in B_R$ satisfying
$\dist(y,\chi_1)>\theta_1$.
\item Let $\beta$ be a normal unit vector of the stable manifold of \eqref{eq2.2} at an equilibrium $x_l$.
Then for any $m>0$, we can find $\wdt\theta_0>0$ such that
$\{y: |\beta' y|\geq\theta, |y|\leq m\theta\}\cap\chi_l=\emptyset$
for any $\theta\in(0,\wdt\theta_0]$
\end{enumerate}
\end{lm}
The following lemmas show that the process leaves small neighborhoods around the equilibrium points with strictly positive probability in finite time if we start close to the equilibrium points. Furthermore, this probability can be bounded below uniformly for all starting points close to the equilibrium. We need this because we want to show the convergence of the process to the limit cycle $\Gamma$.
\begin{lm}\label{lm3.2}
Consider an equilibrium $x_l$ and suppose there exists $i^*\in\M$ such that $\beta' f(x_l, i^*)\ne 0$ where $\beta$ is a normal unit vector
of the stable manifold of \eqref{eq2.2} at $x_l$.
Then for any $\Delta>0$ that is sufficiently small and any $R>R_0$, we can find $\theta_1,\theta_3>0$, $H^\Delta_l>0$, and $ \eps_{l}(\Delta)$ such that for $\eps<\eps_{l}(\Delta)$,
$$\PP\left\{\wdt \tau^{\eps,\delta}_{x, i}\leq H^\Delta_l\right\}\geq\psi^{\Delta,\eps}:=\exp\left(-\dfrac{\Delta}{\eps}\right),\,x\in M_{l,\theta_1},$$
where
$$\wdt \tau^{\eps,\delta}_{x, i}:=\inf\{t\geq 0: X^{\eps,\delta}_{x,i}(t)\in B_R \text{ and } \dist(X^{\eps,\delta}_{x,i}(t),\chi_l)\geq \theta_3\}.$$
\end{lm}
\begin{proof}
Suppose without loss of generality that $x_l=0$.
Let $\beta$ be a normal vector of the stable manifold at $0$ such that $|\beta|=1$ and $\beta'f(0, i^*)>0$. Since $f$ is locally Lipschitz we can find $a_1>0$ such that
\begin{equation}\label{e:a}
\beta'f(x, i^*)>a_1>0, |x|<\theta_0.
\end{equation}
Then $A_1:=\sup_{x<\theta_0}\left\{\frac{|f(x,i^*)|}{\beta'f(x,i^*)}\right\}<\infty$.
Since $\beta$ is perpendicular to the tangent of the stable manifold at $0$,
we can find $\theta_2\in\left(0, \frac1{2+3A_1}\left(\frac{a_1\Delta}{4|q_{i^*i^*}|}\wedge\theta_0\right)\right)$ such that
\begin{equation}\label{e:dist}
\dist(L_l^{\theta_2}, \chi_l):=\theta_3>0
\end{equation}
where
\begin{equation}\label{defL1}
L_l^{\theta_2}=\{x: |x|\leq(2+3A_1)\theta_2 \text{ and } |\beta'x|>\theta_2\}.
\end{equation}
The continuous dependence of the solutions of \eqref{eq2.2} on the starting point and the fact that $0$ is an equilibrium of \eqref{eq2.2} imply that $\bar X$ stays close to $0$ for a finite time if the starting point is close enough to $0$.
Using this, we can derive from Lemma \ref{lm2.2} that there exist numbers $\theta_1\in(0,\theta_2)$ and $k>0$ such that
\begin{equation}\label{lm3.2-e1}
\PP\left\{| X_{x,i}^{\eps,\delta}(t)|<\theta_2, 0<t<1+\frac{1}{|q_{i^*i^*}|}\right\}>1-\exp\left(-\frac{k}{\eps+\delta}\right)\, \text{ for all } x\in M_{l,\theta_1}, i\in\M.
\end{equation}
First, we consider the case $\alpha^\eps(0)=i^*$. Because of the independence
of $\alpha^\eps(\cdot)$ and $W(\cdot)$, if $\alpha^\eps(t)=i^*$ for all $t\in\Big[0, \frac{\Delta}{|q_{i^*i^*}|}\Big]$, the process $X^{\eps, \delta}_{x, i^*}(\cdot)$ has the same distribution on the time interval $\Big[0, \frac{\Delta}{|q_{i^*i^*}|}\Big]$ as that of
$Z^{\delta}_x$ given by
\begin{equation}\label{e:Z}
dZ^\delta(t)=f(Z^\delta(t), i^*)dt+\sqrt{\delta}\sigma(Z^\delta(t), i^*)dW(t).
\end{equation}
Define the bounded stopping time $$\rho^{\eps,\delta}_x:=\dfrac{\Delta}{|q_{i^*i^*}|}\wedge \inf\{t>0: |Z_x^\delta(t)|\geq \theta_0\}\wedge \inf\{t>0: \beta'Z_x^\delta(t)\geq \theta_2\}.$$
We have
\begin{equation}\label{e:betaZ}
\beta'Z_x^\delta(\rho_x^{\eps,\delta})=\beta'x+\int_0^{\rho^{\eps,\delta}_x}\beta'f(Z_x^\delta(s), i^*)ds+\int_0^{\rho^{\eps,\delta}_x}\sqrt{\delta}\beta'\sigma(Z_x^\delta(s), i^*)dW(s), |x|\leq \theta_0.
\end{equation}
By the exponential martingale inequality from Lemma \ref{l:exp}, there exists a constant $m_3>0$ independent of $\delta$ such that
$$\PP\left(\Omega^{\eps,\delta,1}_{x}\right)>\frac34 \text{ and }\PP\left(\Omega^{\eps,\delta,2}_{x,i}\right)>\frac34$$
where
\bea \ad
\Omega^{\eps,\delta,1}_{x}:=\Bigg\{-\int_0^t\sqrt{\delta}\beta'\sigma(Z_x^\delta(s), i^*)dW(s)
\\ \aad \quad\qquad\qquad
-\dfrac1{\sqrt{\delta}}\int_0^t\delta\beta'\sigma(Z_x^\delta(s), i^*)\sigma(Z_x^\delta(s), i^*)'\beta ds<m_3\sqrt{\delta}, t\in\left[0,{\rho^{\eps,\delta}_x}\right]\Bigg\}
\eea
and
\bea \ad
\Omega^{\eps,\delta,2}_{x}:=\Bigg\{\left|\int_0^t\sqrt{\delta}\sigma(Z_x^\delta(s), i^*)dW(s)\right|
\\ \aad \quad\qquad\qquad
-\dfrac1{\sqrt{\delta}}\int_0^t\delta\left|\sigma(Z_x^\delta(s), i^*)\sigma(Z_x^\delta(s), i^*)'\right| ds<m_3\sqrt{\delta}, t\in\left[0,{\rho^{\eps,\delta}_x}\right]\Bigg\}.
\eea
This implies that
\begin{equation}\label{e:12}
\PP\left(\Omega^{\eps,\delta,1}_{x}\cap \Omega^{\eps,\delta,2}_{x,i}\right)>\frac{1}{2}.
\end{equation}
Using \eqref{e:a} and \eqref{e:betaZ} we note that on the set $\Omega^{\eps,\delta,1}_{x}$
\begin{equation}\label{lm3.2-e7}
\begin{aligned}
\beta'Z_x^\delta(\rho_x^{\eps,\delta})>&\beta'x+\int_0^{\rho^{\eps,\delta}_x}\beta'f(Z_x^\delta(s), i^*)ds\\
& -\dfrac1{\sqrt{\delta}}\int_0^{\rho^{\eps,\delta}_x}\beta'\delta\sigma(Z_x^\delta(s), i^*)'\sigma(Z_x^\delta(s), i^*)\beta ds-m_3\sqrt{\delta}\\
\geq&-\theta_2+\int_0^{\rho^{\eps,\delta}_x}a_1ds-m_3\sqrt{\delta}
\end{aligned}
\end{equation}
Let $\delta$ be so small that $m_3\sqrt{\delta}<\dfrac{a_1}2\dfrac{\Delta}{|q_{i^*i^*}|}$.
If $\rho^{\eps,\delta}_x(\omega)=\dfrac{\Delta}{|q_{i^*i^*}|}$ for some $\omega\in\Omega^{\eps,\delta,1}_{x,i}$, using $\theta_2\leq\dfrac{a_1\Delta}{4|q_{i^*i^*}|}=\dfrac{a_1{\rho^{\eps,\delta}_x}}4$, we get
$$|\beta'Z_x^\delta({\rho^{\eps,\delta}_x}(\omega))|\leq \theta_2< -\theta_2+a_1{\rho^{\eps,\delta}_x}-m_3\sqrt{\delta},$$
which contradicts \eqref{lm3.2-e7}.
As a result, if $x\leq\theta_2, \omega\in\Omega^{\eps,\delta,1}_{x}$ and $\delta$ is sufficiently small, we have
\begin{equation}\label{e:O1rho}
\rho^{\eps,\delta}_x(\omega)<\dfrac{\Delta}{|q_{i^*i^*}|},
\end{equation}
and
by \eqref{e:betaZ} we have
\begin{equation}\label{lm3.2-e9}
\begin{aligned}
\int_0^{\rho^{\eps,\delta}_x}\beta'f(Z_x^\delta(s), i^*)ds
\leq& |\beta'Z_x^\delta(\rho_x^{\eps,\delta})|+|\beta'x|+\sqrt{\delta}\int_0^{\rho^{\eps,\delta}_x}\left|\sigma(Z_x^\delta(s), i^*)\sigma(Z_x^\delta(s), i^*)'\right| ds+m_3\sqrt{\delta} \\
<&3\theta_2.
\end{aligned}
\end{equation}
on $\Omega^{\eps,\delta,1}_{x}\cap \Omega^{\eps,\delta,2}_{x}$.
Using \eqref{e:Z} and \eqref{lm3.2-e9}, one sees
that if $\delta$ is sufficiently small and $|x|<\theta_2$ then for $\omega\in \Omega^{\eps,\delta,1}_{x}\cap \Omega^{\eps,\delta,2}_{x}$,
\begin{equation}\label{lm3.2-e8}
\begin{aligned}
|Z_x(\rho^{\eps,\delta}_x)|<&|x|+\int_0^{\rho^{\eps,\delta}_x}|f(Z_x(t),i^*)|dt+\sqrt{\delta}\int_0^{\rho^{\eps,\delta}_x}\left|\sigma(Z_x^\delta(s), i^*)\sigma(Z_x^\delta(s), i^*)'\right| ds+m_3 \sqrt{\delta}\\
<&2\theta_2+A_1 \int_0^{\rho^{\eps,\delta}_x}\beta'f(Z_x(t),i^*)dt\\
<&(2+3A_1)\theta_2<\theta_0,
\end{aligned}
\end{equation}
Combining \eqref{lm3.2-e8} with the definition of $\rho^{\eps,\delta}_x$ shows that
$\beta'Z_x(\rho^{\eps,\delta}_x)=\theta_2$ and
$|Z_x(\rho^{\eps,\delta}_x)|<(2+3A_1)\theta_2$ on $ \Omega^{\eps,\delta,1}_{x}\cap \Omega^{\eps,\delta,2}_{x}$.
As a result of this and \eqref{e:12},
if $|x|\leq\theta_2$,
$$\PP \left\{ \beta 'Z_x(t)\geq\theta_2, |Z_x(t)|\leq(2+3A_1)\theta_2 \text{ for some } t\in\left[0,\frac\Delta{|q_{i^*i^*}|}\right]\right\}\geq \PP\left(\Omega^{\eps,\delta,1}_{x}\cap \Omega^{\eps,\delta,2}_{x,i}\right)>\frac{1}{2}.$$
Let $$\zeta^{\eps, \delta}_{x, i}:=\inf\{t>0: \beta' X^{\eps,\delta}_{x,i}(t)\geq\theta_2, |X^{\eps,\delta}_{x,i}|\leq(2+3A_1)\theta_2\}=\inf\{t>0: X^{\eps,\delta}_{x,i}\in L_l^{\theta_2}\}.$$
Using the independence of $\alpha^\eps$, the paragraph before equation \eqref{e:Z}, and the last two equations, we obtain
\begin{equation}\label{lm3.2-e2}
\PP\left\{\zeta^{\eps, \delta}_{x, i^*}\leq \dfrac{\Delta}{|q_{i^*i^*}|}\right\}> \dfrac12\PP\left\{\alpha^\eps_{i^*}(t)=i^*, \ ~\text{for all}~ t\in\left[0, \dfrac{\Delta}{|q_{i^*i^*}|}\right]\right\}=\dfrac12\exp\left(-\dfrac{\Delta}{\eps}\right), \text{ if } |x|\leq\theta_1.
\end{equation}
Since $\alpha^\eps(t)$ is ergodic, for any sufficiently small $\eps$, i.e., small enough $\Delta$,
\begin{equation}\label{lm3.2-e3}
\PP\{\alpha^\eps_i(t)=i^* \mbox{ for some } t\in [0, 1]\}>{3 \over 4}, i\in\M.
\end{equation}
By the strong Markov property, we derive from \eqref{lm3.2-e1}, \eqref{lm3.2-e2}, and \eqref{lm3.2-e3} that for all $(x,i)\in M_{l,\theta_1}\times\M$ and for $\eps$ sufficiently small
\begin{equation}\label{lm3.2-e5}
\PP\left\{\zeta^{\eps, \delta}_{x, i}<1+\dfrac{\Delta}{|q_{i^*i^*}|}\right\}\geq {1\over 4}\exp\left(-\dfrac{\Delta}{\eps}\right).
\end{equation}
The proof is complete by combining this estimate with \eqref{e:dist}.
\end{proof}
\begin{lm}\label{lm3.3}
Suppose that $\lim\limits_{\eps\to0}{\delta\over\eps}=r>0$. Assume that at the equilibrium point $x_l$, one has $f(x_l, i)=0$ for all $i\in\M$, and there is $i^*\in\M$ for which
$\beta'\sigma(x_l,i)\ne 0$,
where $\beta$ is a normal unit vector of the stable manifold of \eqref{eq2.2} at $x_l$. Then for any sufficiently small $\Delta>0$ and any $R>R_0$, we can find $\theta_1,\theta_3>0$, $H^\Delta_l>0$, and $ \eps_{l}(\Delta)>0$ such that for $\eps<\eps_{l}(\Delta)$,
$$\PP\left\{\wdt \tau^{\eps,\delta}_{x, i}\leq H^\Delta_1\right\}\geq\psi^{\Delta,\eps}:=\exp\Big(-\dfrac{\Delta}{\delta}\Big),~\text{for all}~\,(x,i)\in M_{l,\theta_1}\times\M,$$
where
$$\wdt \tau^{\eps,\delta}_{x, i}=\inf\{t\geq 0: X^{\eps,\delta}_{x,i}(t)\in B_R \text{ and } \dist(X^{\eps,\delta}_{x,i}(t),\chi_l)\geq \theta_3\}.$$
\end{lm}
\begin{proof}
We can assume without loss of generality that $x_l=0$ and $\lim\limits_{\eps\to0}{\delta\over\eps}=1$. Since $\sigma$ is locally Lipschitz, we can find $a_2>0$ such that
\begin{equation}\label{e:a2}
a_2<\beta'(\sigma\sigma')(y, i^*)\beta, |y|< \theta_0.
\end{equation}
Let $K_l>0$ be such that $|f(x,i)|<K_l|x|$ and $|(\sigma'\sigma)(x,i)|<K_l$ if $|x|<\theta_0, i\in \M$. Fix $T>0$ such that $\dfrac{a_2\nu_{i^*}T}{2}>1$ and let $\theta_1>0$ be such that
\begin{equation}\label{e:theta1}
(2+K_lT)^2e^{K_lT}\theta_1<\theta_0
\end{equation}
and
$\dist(L_l^{\theta_1}, \chi_l):=\theta_3>0$ where
\begin{equation}\label{defL1}
L_l^{\theta_1}:=\{x: |x|\leq (2+K_lT)^2e^{K_lT}\theta_1 \text{ and } |\beta'x|>\theta_1\}.
\end{equation}
Define
$$\zeta_{t,x, i}:=\inf\left\{u>0:\int_0^u\beta'(\sigma\sigma')\left(\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\right)\beta ds\geq t\right\}.$$
For all $t\geq 0$, we have by \eqref{e:a2} and the ergodicity of the Markov chain $\alpha^\eps_i$ that $$\PP(\zeta_{t,x,i}<\infty)=1, |x|<\theta_0.$$ As a result the process $(M(t))_{t\geq 0}$ defined by
$$M(t)=\int_0^{\zeta_{t,x,i}}\beta'\sigma\left(\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\right)dW(s)$$ is a Brownian motion. This follows from the fact that $M(t)$ is a continuous martingale with quadratic variation $[M,M]_t=t, t\geq 0$.
Set $\theta_2:=(2+K_lT)\theta_1$. Since $M(1)$ has the distribution of a standard normal, for sufficiently small $\delta$, we have the estimate
\begin{equation}\label{e:Omega2}
\PP\{\sqrt{\delta}M(1)>\theta_2\}\geq\dfrac12\exp\left(-\dfrac{\theta_2^2}{\delta}\right), |x|<\theta_0.
\end{equation}
Using the large deviation principle (see \cite{HYZ}), we
can find $a_3=a_3(T)>0$ such that
\begin{equation}\label{ergodic-alpha}
\PP\left\{\dfrac{1}{T}\int_0^{T}\1_{\{\alpha_i^\eps(s)=i^*\}}ds>\dfrac{\nu_{i^*}}2\right\}\geq1-\exp\left(-\dfrac{a_3}{\eps}\right).
\end{equation}
Equation \eqref{e:a2}, the definition of $\zeta_{t,x,i}$, and $\dfrac{a_2\nu_{i^*}T}{2}>1$ yield
\bea \ad \PP\left\{\int_0^T\beta'(\sigma\sigma')\left(\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\right)\beta ds
\geq \dfrac{a_2\nu_{i^*}T}{2} \right\}\geq1-\exp\left(-\dfrac{a_3}{\eps}\right),~|x|<\theta_0,\eea
which leads to
\begin{equation}\label{e:zeta1}
\PP\{\zeta_{1, x, i}\leq T\}\geq1-\exp\left(-\dfrac{a_3}{\eps}\right), |x|<\theta_0.
\end{equation}
Define for $|x|<\theta_0$, $i\in\M$
$$
\begin{aligned}
\Omega_{x,i}^{\eps,\delta,3}:=\bigg\{&\left|\sqrt{\delta}\int_0^{t}\sigma\left(\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\right)dW(s)\right|\\
&\quad<\dfrac{\theta_2}\delta\int_0^t\delta\left|(\sigma'\sigma)\left(\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\right)\right|ds+\theta_2\leq (K_lT+1)\theta_2, t\in[0,T]\bigg\}
\end{aligned}
$$
and note that the last inequality holds by the definition of $K_l$.
By Lemma \ref{l:exp}
\begin{equation}\label{e:Omega3}
\PP(\Omega_{x,i}^{\eps,\delta,3})\geq 1-\exp\left(-\frac{2\theta_2^2}{\delta}\right), |x|<\theta_0.
\end{equation}
Define the stopping time
$$\zeta_{x,i}=\inf\{t>0: |\beta'X^{\eps,\delta}_{x,i}(t)|\geq\theta_1\}\wedge \inf\{t>0: |X^{\eps,\delta}_{x,i}(t)|\geq (K_l+2)\theta_2e^{K_lT} \}.$$
If $|x|\leq\theta_1$ and $\omega\in\{\sqrt{\delta}M(1)>\theta_2\}\cap \{\zeta_{1,x,i}\leq T\}\cap \Omega_{x,i}^{\eps,\delta,3}$,
we claim that we must have
\begin{equation}\label{ezT}
\zeta_{x,i}<T.
\end{equation}
We argue by contradiction.
Suppose the three events $\{\sqrt{\delta}M(1)>\theta_2\}$, $\{\zeta_{1,x,i}\leq T\}$, and $\{\zeta_{x, i}\geq\zeta_{1,x,i}\}$ happen simultaneously. Then we get the contradiction
\begin{align*}
\theta_2=(2+K_lT)\theta_1& <\sqrt{\delta}M(1)=\sqrt{\delta}\int_0^{\zeta_{1,x,i}}\beta'\sigma\left(\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\right)dW(s)\\
& \leq |\beta'X^{\eps, \delta}_{x, i}(\zeta_1)|+|\beta'x|+\Big|\int_0^{\zeta_{1,x,i}}\beta'f(X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s))ds\Big|\\
& \leq2\theta_1+\int_0^{\zeta_{1,x,i}}K_l|\beta'X^{\eps,\delta}_{x,i}(s)|ds< (2+K_lT)\theta_1=\theta_2,
\end{align*}
where we used that $\left(1\wedge\frac{\theta_0}{|X^{\eps, \delta}_{x, i}(s)|}\right)X^{\eps, \delta}_{x, i}(s)=X^{\eps, \delta}_{x, i}(s)$ if $s<\zeta_{x,i}$ by the definition of $\zeta_{x,i}$ and \eqref{e:theta1}.
For $|x|\leq\theta_1$ and $\omega\in\{\sqrt{\delta}M(1)>\theta_2\}
\cap \{\zeta_{x,i}\leq T\}\cap \Omega_{x,i}^{\eps,\delta,3}$,
for any $0\leq t\leq \zeta_{1,x,i}\leq T$,
$$
\begin{aligned}
|X^{\eps,\delta}_{x,i}(t)|\leq& |x|+ \sqrt{\delta}\left|\int_0^{t}\sigma\big(X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s)\big)dW(s)\right|
+\int_0^t|f(X^{\eps, \delta}_{x, i}(s),\alpha^\eps_i(s))|ds\\
<& (K_lT+2)\theta_2+K_l\int_0^t |X^{\eps, \delta}_{x, i}(s)|ds.
\end{aligned}
$$
This together with Gronwall's inequality implies that
$$|X^{\eps,\delta}_{x,i}(t)|< (K_lT+2)\theta_2 e^{K_l T}, t\in [0, \zeta_{x,i}]$$
Thus for $|x|\leq\theta_1$ and $\omega\in\{\sqrt{\delta}M(1)>\theta_2\}\cap \{\zeta_{x,i}\leq T\}\cap \Omega_{x,i}^{\eps,\delta,3}$,
we have that $\zeta_{x,i}<T$ and
$X^{\eps,\delta}_{x,i}(\zeta_{x,i})<(K_lT+2)\theta_2 e^{K_l T}$
and $\beta' X^{\eps,\delta}_{x,i}(\zeta_{x,i})\geq\theta_1$.
Since $\theta_2<a_3$ and $\lim_{\eps\to 0}\frac{\delta}{\eps}=1$ we have by \eqref{e:Omega2}, \eqref{e:zeta1}, \eqref{e:Omega3} and \eqref{ezT} that for all sufficiently small $\eps$
$$
\PP(\{\sqrt{\delta}M(1)>\theta_2\}\cap \{\zeta_{x,i}\leq T\}\cap \Omega_{x,i}^{\eps,\delta,3})\geq\dfrac14\exp\left(-\dfrac{\theta_2^2}\delta\right)
\geq \dfrac14\exp\left(-\dfrac{\Delta}\delta\right), |x|<\theta_1
$$
if $\Delta<\theta_2^2$,
which completes the proof.
\end{proof}
\begin{lm}\label{lm3.4}
Suppose that $\lim\limits_{\eps\to0}\dfrac\delta\eps=\infty$.
Assume that at the equilibrium point $x_l$ one can find $i^*\in\M$ such that $\beta'\sigma(x_l,i^*)\ne 0$
where $\beta$ is a normal unit vector of the stable manifold of \eqref{eq2.2} at $x_l$.
Then for any sufficiently small $\Delta>0$ and any $R<R_0$ we can find $\theta_1,\theta_3>0$, $H^\Delta_l>0$,and $ \eps_{1}(\Delta)$ such that for $\eps<\eps_{1}(\Delta)$,
$$\PP\left\{\wdt \tau^{\eps,\delta}_{x, i}\leq H^\Delta_l\right\}\geq\psi^{\Delta,\eps}:=\exp\Big(-\dfrac{\Delta}{\delta}\Big)~\text{for all}~\,(x,i)\in M_{l,\theta_1}\times\M ,$$
where
$$\wdt \tau^{\eps,\delta}_{x, i}=\inf\{t\geq 0: X^{\eps,\delta}_{x,i}(t)\in B_R \text{ and } \dist(X^{\eps,\delta}_{x,i}(t),\chi_l)\geq \theta_3\}.$$
\end{lm}
\begin{proof}
Assume, as in the previous lemmas, that $x_l=0$.
Pick a number $a_2>0$ for which $$a_2<\beta'(\sigma\sigma')(y, i^*)\beta, |y|<\theta_0.$$
Let $K_l>0$ be such that $|\bar f(x)|<K_l|x|$ and $|(\sigma'\sigma)(x,i)|<K_l$ whenever $|x|<\theta_0$,
and fix $T>0$ such that $\dfrac{a_2\nu_{i^*}T}{2}>1$.
Let $\theta_1>0$ be such that $(3+K_lT)^2e^{K_lT}\theta_1<\theta_0$
and
$\dist(L_l^{\theta_1}, \chi_l):=\theta_3>0$ where
\begin{equation}\label{defL11}
L_l^{\theta_1}=\{x: |x-x_l|\leq (3+K_lT)^2e^{K_lT}\theta_1 \text{ and } |\beta'(x-x_l)|>\theta_1\}.
\end{equation}
Define $\theta_2=(3+K_lT)\theta_1$ and let $a_2, M(t), T, \zeta_{1,x,i}$ be as in the proof of Lemma \ref{lm3.3}.
Arguing as in the proof of \eqref{e:zeta1}, we can find $a_3>0$ such that $$\PP\big\{\zeta_{1,x,i}\leq T\big\}\geq 1-\exp\left(-\dfrac{a_3}\eps\right), |x|<\theta_0.$$
Since $\bar f(0)=0$, we can apply the large deviation principle
(see \cite{HYZ}) to show that
there is $\kappa=\kappa(\Delta)>0$ such that
\begin{equation}\label{e:estA}
\PP(A)\geq 1-\exp\left(-\dfrac{\kappa}\eps\right),
\end{equation}
where $A:=\left\{\left|\int_0^{u}f(0,\alpha_i^\eps(s))ds\right|
<\theta_1, \text{ for all } u\in[0,T]\right\}$.
The estimates
\bea
M(1) \ad =\int_0^{\zeta_{1,x,i}}\beta'\sigma(X^{\eps,\delta}_{x, i}(s), \alpha_i^\eps(s))dW(s)\\
\ad \leq |\beta'X^{\eps, \delta}_{x, i}(\zeta_{1,x,i})|+|\beta'x|+\Big|\int_0^{\zeta_{1,x,i}}\beta' f(0,\alpha_i^\eps(s))ds\Big|\\
\aad \ +\int_0^{\zeta_{1,x,i}}\big|\beta'\big(f(X^{\eps, \delta}_{x, i}(s),\alpha_i^\eps(s))-f(0,\alpha_i^\eps(s))\big)\big|ds.
\eea
and
$$
\begin{aligned}
|X^{\eps,\delta}_{x,i}(t)|\leq& |x|+ \sqrt{\delta}\left|\int_0^{t}\sigma\big(X^{\eps, \delta}_{x, i}(s),\alpha_i^\eps(s)\big)dW(s)\right|
+\int_0^t|\bar f(X^{\eps, \delta}_{x, i}(s))|ds\\
&+\int_0^t |\bar f(X^{\eps, \delta}_{x, i}(s))-f(X^{\eps, \delta}_{x, i}(s),\alpha_i^\eps(s)|ds
\end{aligned}
$$
together with arguments similar to those from the proof of Lemma \ref{lm3.3} show that
$$
\PP\left\{ X^{\eps,\delta}_{x,i}(t)\in L_l^{\theta_1} \text{ for some } t\in[0,T]\right\}\geq\dfrac14\exp\left(-\frac{\Delta}\delta\right), (x,i)\in M_{l,\theta_1}\times\M
$$
if $\delta$ is sufficiently small.
\end{proof}
\begin{rmk}\label{r:dens}
The results in this section still hold true if one assumes the generator $Q(\cdot)$ of $\alpha(\cdot)$ is state dependent -- see an explanation of the exact setting in Remark \ref{r:state}. By the large deviation principle in \cite[Section 3]{budhiraja2018large} and the truncation arguments in Lemma \ref{lm2.1}, we can obtain Lemma \ref{lm2.2} for the case of state-dependent switching.
It should be noted that while \cite{budhiraja2018large} only considers Case 1 of \eqref{eq:ep-dl}, using the variational representation,
the arguments in \cite[Section 3]{budhiraja2018large} can be applied to obtain Lemma \ref{lm2.2} for the other cases.
We can also infer from the large deviation principle that \eqref{lm3.2-e3}, \eqref{ergodic-alpha} and \eqref{e:estA} hold in this setting.
As a result, Lemmas \ref{lm3.2}, \ref{lm3.3} and \ref{lm3.4} hold. These lemmas, in combination with the proofs from Section \ref{sec:4} imply that the main result, Theorem \ref{t:main}, remains unchanged if one has state-dependent switching.
\end{rmk}
\section{Proof of the main result}\label{sec:4}
This section provides the proofs of the convergence of $\mu^{\eps,\delta}$ for the three cases
given in \eqref{eq:ep-dl}.
\begin{prop}\label{p:nbhd}
For every $\eta>0$, there exists $R>R_0$ and neighborhoods $N_1,\dots, N_{n_0-1}$ of $\chi_1\cap B_R,\dots, \chi_{n_0-1}\cap B_R$ such that
$$\limsup_{\eps\to0} \mu^{\eps,\delta}(\cup_{j=1}^{n_0-1}N_j)\leq 2^{n_0}\eta.$$
\end{prop}
\begin{proof}
For any $\eta>0$, let $R>R_0$ be
such that $\mu^{\eps,\delta}(B_R)\geq 1-\eta.$ Define
$$S_1=\{y\in B_R: \dist(y,\chi_1\cap B_R)< \theta_0\}$$
In view of Lemma \ref{lm2.5}, there exists $c_2>0$ such that for all $t\geq 0$
\begin{equation}\label{extra-e3.1}
\dist(\bar X_y(t), \chi_1)\geq 2c_2\,\text{ for any }\,y\in B_R\setminus S_1.
\end{equation}
Define
$$G_1=\{y\in B_R: \dist(y,\chi_1\cap B_R)< c_2\}.$$
There exists $c_3>0$ such that
\begin{equation}\label{extra-e3.8}
\dist(\bar X_y(t), \chi_1)\geq 2c_3 \text{ for any }y\in B_R\setminus G_1,\,t\geq 0.
\end{equation}
Note that we have $2c_3\leq c_2$ and $2c_2\leq\theta_0$.
Define
$$N_1=\{y\in B_R: \dist(y,\chi_1\cap B_R)< c_3\}$$
In view of Lemma \ref{lm2.5},
for any $y\notin\chi_1$, there exists $\wdt t_y$ such that $\bar X_y(\wdt t_y)\in M_{i,\theta_0}\cap (B_R\setminus S_1)$
for some $i>1$.
This fact together with
the continuous dependence of solutions to initial values
and \eqref{extra-e3.1} implies that
there exists $\hat T>0$ such that
\begin{equation}\label{extra-e3.3}
\dist(\bar X_y(t), \chi_1)\geq 2c_2\,\text{ for any }\,t\geq \hat T, y\in B_R\setminus N_1.
\end{equation}
Let $\kappa=\kappa(R,c_3, \hat T)$ be as in Lemma \ref{lm2.2} and $\Delta<\frac{\kappa}2$ and $\theta_1$ and $\psi^\Delta_\eps$ be as in one of the Lemmas \ref{lm3.2}, \ref{lm3.3} and \ref{lm3.4} (depending on which case we are considering).
We have
\begin{equation}\label{extra-e3.6}
\PP(\wdt \tau_{x,i}^{\eps,\delta}<H^\Delta)\geq\psi^\Delta_\eps, x\in M_{1,\theta_1}
\end{equation}
where, as in Section \ref{sec:3}, the stopping time is
$$\wdt \tau^{\eps,\delta}_{x, i}=\inf\{t\geq 0: X^{\eps,\delta}_{x,i}(t)\in B_R \text{ and } \dist(X^{\eps,\delta}_{x,i}(t),\chi_1)\geq \theta_3\}.$$
Define
$$\tau^{\eps,\delta}_{x,i}=\inf\{t\geq0: X_{x,i}^{\eps,\delta}(t)\in B_R\setminus G_1\}.$$
It follows from part (1) of Lemma \ref{lm2.5} that for any $x\in N_1$, there exists a $\wdt T_1>0$ such that
$\bar X_x(t_x)\in \bigcup_{j=1}^{n_0} M_{j,\frac{\theta_1}2}\text{ for some } t_x\leq \wdt T_1$.
Suppose $\bar X_x(t_x)\in \bigcup_{j=2}^{n_0}M_{j,\frac{\theta_1}2}$.
Note that $\bigcup_{j=2}^{n_0} M_{j,\frac{\theta_1}2} \cap M_{1,c_3}=\emptyset$, $\theta_1<\theta_0$ and that by construction, $M_{1,2\theta_0}\cap \chi_j=\emptyset, j>1$. These facts imply that $\bigcup_{j=2}^{n_0} M_{j,\frac{\theta_1}2}\cap N_1=\emptyset$.
This together with Lemma \ref{lm2.2} and \eqref{extra-e3.3} implies
\begin{equation}\label{ex.e1}
\PP\{\tau^{\eps,\delta}_{x,i}<\tilde T_1+\hat T\}>\frac12
\end{equation}
for small $\eps>0$.
When $\eps$ is sufficiently small, we have by Lemma \ref{lm2.2} (applied with $\gamma=\frac{\theta_1}{2})$ that for any $x\in N_1$ satisfying
$\bar X_x(t_x)\in M_{1,\frac{\theta_1}2}$ that
\begin{equation}\label{extra-e3.4}
\PP\{X^{\eps,\delta}_{x,i}(t_x)\in M_{1,\theta_1}\}>\frac12.
\end{equation}
Similarly to \eqref{extra-e3.3},
there exists a $\wdt T_2>0$ such that
$$
\dist(\bar X_y(t), \chi_1)\geq 2c_2\,\text{ for any }\,t\geq \wdt T_2, y\in B_R, \dist(y,\chi_1)\geq\theta_3,
$$
which implies that by Lemma \ref{lm2.2}, for sufficiently small $\eps>0$,
\begin{equation}\label{extra-e3.5}
\PP\left\{\dist(X^{\eps,\delta}_{x,i}(\wdt T_2), \chi_1)\geq c_2\right\}>\frac{1}{2}\,\text{ for any }\, x\in B_R, \dist(x,\chi_1)\geq\theta_3, i\in\M.
\end{equation}
Putting together \eqref{extra-e3.6}, \eqref{extra-e3.4}, and \eqref{extra-e3.5}
we deduce that
\begin{equation}\label{ex.e2}
\PP\{\tau^{\eps,\delta}_{x,i}<\wdt T_1 +H^\Delta+\wdt T_2\}>\dfrac1{4}\psi^\Delta_{\eps}.
\end{equation}
for $\eps$ sufficiently small.
Combining \eqref{ex.e1} and \eqref{ex.e2},
we get that
\begin{equation}\label{ex.e22}
\PP\{\tau^{\eps,\delta}_{x,i}<H^\Delta+\wdt T_1+\wdt T_2+\hat T\}>\dfrac1{8}\psi^\Delta_{\eps}, x\in N_1.
\end{equation}
Define $T^{\eps,\delta}_{\Delta,1}:=4\dfrac{H^\Delta+\wdt T_1+\wdt T_2+\hat T}{\psi_{\eps,\delta}^\Delta}$. Applying Lemma \ref{lm3.1} to \eqref{ex.e22},
we have
\begin{equation}\label{ex.e3}
\PP\left\{\tau^{\eps,\delta}_{x,i}<T^{\eps,\delta}_{\Delta,1}\right\}>\frac12, x\in N_1.
\end{equation}
We will argue by contradiction that $\limsup\limits_{\eps\to0}\mu^{\eps,\delta}(N_1)\leq 2\eta$.
Assume that $\limsup\limits_{\eps\to0}\mu^{\eps,\delta}(N_1)>2\eta>0$.
Since $\Delta<\kappa/2$, we have
\begin{equation}\label{e:TD}
\lim_{\eps\to 0}T_{\Delta,1}^{\eps,\delta}\exp\left(-\dfrac{\kappa}{\eps+\delta}\right)= 0.
\end{equation}
Let $\widetilde X^{\eps, \delta}(t)$ be the stationary solution, whose
distribution is $\mu^{\eps,\delta}$
for every time $t\geq 0$. Let $\tau^{\eps,\delta}$
be the first exit time of $\widetilde X^{\eps, \delta}(t)$ from $G_1$.
Define the events
\begin{align*}
K_1^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \tau^{\eps,\delta}\geq T_{\Delta,1}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_1\Big\}\\
K_2^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \tau^{\eps,\delta}< T_{\Delta,1}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_1\Big\}\\
K_3^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \widetilde X^{\eps,\delta}(0)\in B_R\setminus N_1\Big\}\\
K_4^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \widetilde X^{\eps,\delta}(0)\notin B_R\Big\}.
\end{align*}
Note that the above events are disjoint and have union $N_1$. As such $$\mu^{\eps,\delta}(N_1)=\sum_{n=1}^4\PP\{K_n^{\eps, \delta}\}.$$
Using \eqref{ex.e3}, we get that
\begin{equation}\label{e:K1K4}
\PP(K_1^{\eps, \delta})\leq \dfrac12\mu^{\eps,\delta}(N_1) \ \hbox{ and }
\ \PP(K_4^{\eps, \delta})\leq 1-\mu^{\eps,\delta}(B_R)< \eta.
\end{equation}
Next, we estimate $\PP(K_3^{\eps, \delta}).$
It follows from Lemma \ref{lm2.2}, \eqref{extra-e3.8},
and \eqref{extra-e3.3} that if $\eps$ is sufficiently small
then
$$\PP\left\{X^{\eps, \delta}_{x, i}(\hat T)\notin G_1 \right\}\geq1-\exp\Big(-\dfrac{\kappa}{\eps+\delta}\Big), x\in B_R\setminus N_1$$
and
$$\PP\left\{X^{\eps, \delta}_{x, i}(t)\notin N_1, \ ~\text{for all}~ t\in[0,\hat T]\right\}\geq1-\exp\Big(-\dfrac{\kappa}{\eps+\delta}\Big), x\in B_R\setminus G_1.$$
Using the last two estimates together with the Markov property one sees that for any $x\in B_R\setminus G_1, i\in\M, s\in[0,T_{\Delta,1}^{\eps,\delta}]$,
\begin{equation}\label{e4.3}
\begin{aligned}
\PP\Big\{ X_{x,i}^{\eps, \delta}&(s )\in N_1\Big\}\\
=&\PP\Big\{ X_{x,i}^{\eps, \delta}(s )\in N_1, X_{x,i}^{\eps,\delta}(\hat T)\notin B_R\setminus G_1 \Big\}\\
&+\sum_{n=2}^{\lf s /\hat T\rf }\PP\Big\{ X_{x,i}^{\eps, \delta}(s )\in N_1, X_{x,i}^{\eps,\delta}(n\hat T)\notin B_R\setminus G_1 , X_{x,i}^{\eps,\delta}(\iota\hat T)\in B_R\setminus G_1 , \iota=1,...,n-1\Big\}\\
&+\PP\Big\{ X_{x,i}^{\eps, \delta}(s )\in N_1, X_{x,i}^{\eps,\delta}(\iota\hat T)\in B_R\setminus G_1 , \iota=1,...,[s /\hat T]\Big\}\\
\leq& \PP\Big\{ X_{x,i}^{\eps,\delta}(\hat T)\notin B_R\setminus G_1\Big\}+\sum_{n=2}^{\lf s /\hat T\rf}\PP\Big\{ X_{x,i}^{\eps,\delta}(n\hat T)\notin B_R\setminus G_1 , X_{x,i}^{\eps,\delta}((n-1)\hat T)\in B_R\setminus G_1 \}
\\
&+\PP\left\{X^{\eps, \delta}_{x, i}(t)\in N_1, \,\text{ for some } t\in \left[\left\lf s /\hat T\right\rf\hat T, \left\lf s /\hat T\right\rf\hat T+\hat T\right], X^{\eps, \delta}_{x, i}\left(\left\lf s /\hat T\right\rf\hat T\right)\in B_R\setminus G_1 \right\}\\
\leq&\left(\left\lf s /\hat T\right\rf+1\right)\exp\left(-\dfrac{\kappa}{\eps+\delta}\right)\\
\leq&\left(s /\hat T+1\right)\exp\left(-\dfrac{\kappa}{\eps+\delta}\right),
\end{aligned}
\end{equation}
where $\lf s /\hat T\rf $ denotes the integer part of $s /\hat T$.
Note that similar arguments show that \eqref{e4.3} also holds for all $s\in[\hat T,T^{\eps,\delta}_{x,i}]$ and $x\in B_R \setminus N_1$.
It follows from this with $s=T_{\Delta,1}^{\eps,\delta}$,
$$\PP(K_3^{\eps, \delta})=\PP\left\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \widetilde X^{\eps,\delta}(0)\in B_R\setminus N_1\right\}\leq\left(T_{\Delta,1}^{\eps,\delta}/\hat T+1\right)\exp\left(-\dfrac{\kappa}{\eps+\delta}\right).$$
This together with \eqref{e:TD} implies that
\begin{equation}\label{e:K3}
\lim_{\eps\downarrow 0}\PP(K_3^{\eps,\delta})=0.
\end{equation}
Using \eqref{e4.3} and the strong Markov property, we get
\begin{equation}\label{e:K2}
\begin{aligned}
\PP(K_2^{\eps, \delta})=&\PP\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta})\in N_1, \tau^{\eps,\delta}< T_{\Delta,1}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_1\Big\}\\
=&\int_0^{T_{\Delta,1}^{\eps,\delta}}\PP\{\tau^{\eps,\delta}\in dt\}\left[\sum_{i\in\M}\int_{\partial G_1}\PP\left\{ X_{x,i}^{\eps, \delta}(T_{\Delta,1}^{\eps,\delta}-t)\in N_1\right\}\PP\left\{\alpha^\eps(t)=i, \widetilde X^{\eps, \delta}(t)\in dx\right\}\right]\\
\leq& \left(T_{\Delta,1}^{\eps,\delta}/\hat T+1\right)\exp\left(-\dfrac{\kappa}{\eps+\delta}\right)\\
\to&0\text{ as } \eps\to0 \,\text{ due to }\,\eqref{e:TD}.
\end{aligned}
\end{equation}
Putting together the estimates \eqref{e:K1K4}, \eqref{e:K2}, and \eqref{e:K3}, we see that $$\limsup\limits_{\eps\to0}\mu^{\eps,\delta}(N_1)\leq \dfrac12\limsup\limits_{\eps\to0}\mu^{\eps,\delta}(N_1)+0+0+\eta,$$
which contradicts the assumption that $\limsup\limits_{\eps\to0}\mu^{\eps,\delta}(N_1)>2\eta$. We have therefore shown that
\[
\lim_{\eps\to 0} \mu^{\eps,\delta}(N_1)\leq 2\mu.
\]
Define
$$S_2=\{y\in B_R\setminus S_1: \dist(y,\chi_2\cap
B_R\setminus S_1)< \theta_0\}.$$
There exists $c_4>0$ such that
$\dist(\bar X_y(t), \chi_1)\geq 2c_4$ for any $y\in B_R\setminus S_1$.
Define
$$G_2=\{y\in B_R\setminus S_1: \dist(y,\chi_2\cap (B_R\setminus S_1))< c_4\}$$
There exists $c_5>0$ such that
$\dist(\bar X_y(t), \chi_1)\geq 2c_5$ for any $y\in B_R\setminus G_1$.
Define
$$N_2=\{y\in B_R\setminus S_1: \dist(y,\chi_1\cap B_R\setminus S_1)< c_5\}$$
Let $\hat T_2$ be such that
$\bar X_y(\hat T_2)\in B_R\setminus(S_1\cup S_2)$ given that $y\in B_R\setminus(S_1\cup N_2)$.
We can show, just as above, that
there exists a $T_{\Delta,2}^{\eps,\delta}$ such that $\lim_{\eps\to\infty} T_{\Delta,2}^{\eps,\delta}\exp\left(-\dfrac{\kappa}{\eps+\delta}\right)=0$ and
$$\PP\{\tau_{x,i}^{\eps,\delta}<T_{\Delta,2}^{\eps,\delta}\}>\frac12.$$
Define events
\begin{align*}
K_{1,2}^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,2}^{\eps,\delta})\in N_2, \tau_2^{\eps,\delta}\geq T_{\Delta,2}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_2\Big\}\\
K_{2,2}^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,2}^{\eps,\delta})\in N_2, \tau_2^{\eps,\delta}< T_{\Delta,2}^{\eps,\delta}, \widetilde X^{\eps,\delta}(0)\in N_2\Big\}\\
K_{3,2}^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,2}^{\eps,\delta})\in N_2, \widetilde X^{\eps,\delta}(0)\in B_R\setminus (S_1\cup N_2)\Big\}\\
K_{4,2}^{\eps, \delta}&=\Big\{\widetilde X^{\eps, \delta}(T_{\Delta,2}^{\eps,\delta})\in N_2, \widetilde X^{\eps,\delta}(0)\notin B_R\setminus S_1\Big\}.
\end{align*}
Applying the same arguments as in the previous part,
we can show that $\limsup_{\eps\to 0} \mu^{\eps,\delta}(N_2)\leq 4\eta$.
Continuing this process, we can construct neighborhoods $N_1,\dots, N_{n_0-1}$ of $\chi_1\cap B_R$, $\dots$,$\chi_{n_0-1}\cap B_R$ such that
$$\limsup_{\eps\to0} \mu^{\eps,\delta}(\cup_{j=1}^{n_0-1}N_j)\leq 2^{n_0}\eta.$$
\end{proof}
\main*
\begin{proof}
We have proved in Proposition \ref{p:nbhd} that for any $\eta>0$ we can find $R>0$ and neighborhoods $N_1,\dots, N_{n_0-1}$ of $\chi_1\cap B_R,\dots, \chi_{n_0-1}\cap B_R$ such that
$$\limsup_{\eps\to0} \mu^{\eps,\delta}(\cup_{j=1}^{n_0-1}N_j)\leq 2^{n_0+1}\eta.$$
Using this fact together with Assumption \ref{asp1} and Lemma \ref{lm2.2},
by a straightforward modification of the proof of \cite[Theorem 1]{CH},
we can establish
that for any $\vartheta>0$ there is neighborhood $N$ of the limit cycle $\Gamma$ such that
$$\liminf_{\eps\downarrow 0} \mu^{\eps,\delta} (N)>1-\vartheta.$$
\end{proof}
\section{Proof of Theorem \ref{thm5.1}}\label{sec:5}
To proceed, we first need some auxiliary results.
\begin{lm}\label{lm5.1}
There exist numbers $ K_1, K_2>0$ such that for any $0<\eps,\delta<1$ and any $(i_0, z_0)\in\M\times\R^{2,\circ}_+$,
we have
$$\dfrac1t\E\int_0^t|Z_{z_0, i_0}^{\eps,\delta}(s)|^2ds\leq
K_1(1+|z_0|), t\geq1,$$
and
$$\limsup\limits_{t\to\infty}\E|Z_{z_0, i_0}^{\eps,\delta}(t)|^2\leq K_2.$$
\end{lm}
\begin{proof}
Let
$\theta<\min\{f_Mb(i), d(i): i\in\M\}$.
Define $$\hat K_1=\sup\limits_{(x, y, i)\in\M\times\R^2_+}\{f_Mx(a(i)-b(i)x)-y(c(i)+d(i)y)+ \theta(x^2+y^2)\}<\infty.$$
Consider $\hat V(x, y, i)=f_M x+y.$
We can check that $\mathcal{L}^{\eps,\delta}\hat V(x, y, i)\leq\hat K_1-\theta(x^2+y^2),$
where $\mathcal{L}^{\eps,\delta}$ the generator associated with \eqref{ex1} (see \cite[p. 48]{MY} or \cite{YZ} for the formula of $\mathcal{L}^{\eps,\delta}$).
Similarly, we can verify that there is $\hat K_2>0$ such that for all $\eps<1,\delta<1$,
$\mathcal{L}^{\eps,\delta} (\hat V^2(x, y, i))\leq\hat K_2-\hat V^2(x, y, i)$.
For each $k >0$,
define the stopping time $\sigma_k=\inf\{t: x(t)+y(t)>k\}.$
By the generalized It\^o formula for $\hat V(x(t), y(t),\alpha^\eps(t))$
\begin{equation}\label{ex3}
\begin{aligned}
\E \hat V(Z_{z_0, i_0}^{\eps,\delta}(t\wedge\sigma_k), \alpha^\eps(t\wedge\sigma_k))
&= \hat V(z_0, i_0)+\E\int_0^{t\wedge\sigma_k}\mathcal{L}^{\eps,\delta}\hat V(Z_{z_0, i_0}^{\eps,\delta}(s), \alpha^\eps(s))ds\\
&\leq f_M x_0+y_0+\E\int_0^{t\wedge\sigma_k}\big[\hat K_1-\theta|Z_{z_0, i_0}^{\eps,\delta}(s)|^2\big]ds.
\end{aligned}
\end{equation}
Hence
$$\theta\E\int_0^{t\wedge\sigma_k}|Z_{z_0, i_0}^{\eps,\delta}(s)|^2ds\leq f_M x_0+y_0+\hat K_1t.$$
Letting $k\to\infty$ and dividing both sides by $\theta t$ we have
\begin{equation}\label{ex3b}
\dfrac1t\E\int_0^t|Z_{z_0, i_0}^{\eps,\delta}(s)|^2ds\leq \dfrac{f_M x_0+y_0}{\theta t}+\dfrac{\hat K_1}\theta.
\end{equation}
Applying the generalized It\^o formula to $e^{t}\hat V^2(Z_{z_0, i_0}^{\eps,\delta}(t), \alpha^{\eps}(t))$,
\begin{equation}\label{ex3a}
\begin{aligned}
\E e^{t\wedge\sigma_k}&\hat V^2(Z_{z_0, i_0}^{\eps,\delta}(t\wedge\sigma_k), \alpha^{\eps}(t\wedge\sigma_k))\\
&= \hat V^2(z_0, i_0)+\E\int_0^{t\wedge\sigma_k}e^s\big[(\hat V^2(Z_{z_0, i_0}^{\eps,\delta}(s), \alpha^\eps(s))+\mathcal{L}^{\eps,\delta}\hat V^2(Z_{z_0, i_0}^{\eps,\delta}(s), \alpha^\eps(s))\big]ds\\
&\leq(f_M x_0+y_0)^2+\hat K_2\E\int_0^{t\wedge\sigma_k}e^sds\leq (f_M x_0+y_0)^2+\hat K_2 e^t.
\end{aligned}
\end{equation}
Taking the limit as $k\to\infty$, and then dividing both sides by $e^t$, we have
\begin{equation}\label{ex3c}
\E \big[f_MX_{z_0, i_0}^{\eps,\delta}(t)+Y_{z_0, i_0}^{\eps,\delta}(t)\big]^2\leq (f_M x_0+y_0)^2e^{-t}+\hat K_2.
\end{equation}
The assertions of the lemma follow directly from \eqref{ex3b} and \eqref{ex3c}.
\end{proof}
\begin{lm}\label{lm5.1a}
There is a number $K_3>0$ such that
$$\dfrac1t\E\int_0^t\Big[ \varphi^2(Z_{z, i}^{\eps,\delta}(s), \alpha_i^\eps(s))+\psi^2(Z_{z, i}^{\eps,\delta}(s), \alpha_i^\eps(s))\Big]ds\leq K_3(1+|z|)$$
for all $\eps,\delta\in (0, 1], z\in\R^{2,\circ}_+, t\geq 1$.
\end{lm}
\begin{proof}
Since the function $h(\cdot, \cdot, i)$ is bounded, we can find $C>0$ such that
$$\varphi^2(z, i)+\psi^2(z, i)\leq C(1+|z|^2).$$
The claim follows by an application of Lemma \ref{lm5.1}.
\end{proof}
Recall that
the two equilibria of \eqref{ex2} on the boundary are both hyperbolic.
Note that the Jacobian of
$\Big(x\bar\phi(x,y), y\bar\psi(x,y)\Big)^\top$
at $\left(\frac{\bar a}{\bar b}, 0\right)$
has two eigenvalues: $-\bar c+\frac{\bar a}{\bar b}h_2\left(\frac{\bar a}{\bar b}, 0\right)>0$
and $-\frac{\bar b^2}{\bar a}<0$.
At $(0,0)$, the two eigenvalues are $\bar a>0$ and $-\bar c<0$, respectively.
If we consider the weighted average Lyapunov exponent,
we can see that the growth rate of $\dfrac{2\bar c}{\bar a} \frac{d\ln X(t)}{dt}+\frac{d\ln Y(t)}{dt}$
is positive both at $(0,0)$ and $\left(\frac{\bar a}{\bar b}, 0\right)$.
This suggests we should look at $\dfrac{2\bar c}{\bar a} \frac{d\ln X(t)}{dt}+\frac{d\ln Y(t)}{dt}$
in order to prove that the dynamics of \eqref{ex2} is pushed away from the boundary.
Then we can use approximation arguments to obtain the tightness of $(Z^{\eps,\delta})$ on $\R^{2,\circ}_+$.
Define
$$\Upsilon(z,i):=\frac{2\bar c}{\bar a}\varphi(z,i)+\psi(z,i)$$
and
$$\bar\Upsilon(z):=\frac{2\bar c}{\bar a}\bar\varphi(z)+\bar\psi(z).$$
We have the following lemma.
\begin{lm}\label{lm5.2}
Let $\gamma_0=\dfrac12 \Big(\bar c\wedge \big(-\bar c+\frac{\bar a}{\bar b}h_1(\frac{\bar a}{\bar b}, 0)\big)\Big)>0.$ For any $H>\frac{\bar a}{\bar b}+1$, there are numbers $T, \beta>0$ such that for all $z\in\{(x,y)\in\R^2_+\,|\, x\wedge y\leq\beta, x\vee y\leq H\}$
\begin{equation}\label{e0-lm5.2}
\bar X_z(T)\vee\bar Y_z(T)\leq H \text{ and } \dfrac1{T}\int_0^{T}\bar\Upsilon(\bar Z_z(t))dt\geq\gamma_0.
\end{equation}
\end{lm}
\begin{proof}
Since $\lim\limits_{t\to\infty}\bar Z_{(0, y)}(t)\to (0,0), \ \forall y\in\R_+$
and
\begin{equation}\label{e1-lm5.2}
\bar\Upsilon(0,0)=\frac{2\bar c}{\bar a}\bar\varphi(0,0)+\bar\psi(0,0)=\frac{2\bar c}{\bar a}\bar a- \bar c=\bar c\geq 2\gamma_0,
\end{equation}
there exists $T_1>0$ such that
\begin{equation}\label{e2-lm5.2}
\dfrac1t\int_0^t\bar\Upsilon(\bar Z_{(0, y)}(s))ds\geq\frac32\gamma_0\,\text{
for }\,t\geq T_1,\,y\in[0,H].
\end{equation}
By \eqref{e1-lm5.2} and the continuity of $\bar\Upsilon(\cdot)$, there exists $\beta_1\in (0, \frac{\bar a}{\bar b})$
such that
\begin{equation}\label{e3-lm5.2}
\bar\Upsilon(x,0)\geq \frac74\gamma_0,\text{ if } x\leq\beta_1.
\end{equation}
Since
$$\bar\Upsilon\left(\frac{\bar a}{\bar b},0\right)=\frac{2\bar c}{\bar a}\bar\varphi \left(\frac{\bar a}{\bar b},0\right)+\bar\psi \left(\frac{\bar a}{\bar b},0\right)=-\bar c+\frac{\bar a}{\bar b}h_1 \left(\frac{\bar a}{\bar b},0\right)\geq2\gamma_0$$
and
$$\lim\limits_{t\to\infty}\bar Z_{(x, 0)}(t)\to \left(\frac{\bar a}{\bar b},0\right), \ \forall x>0,$$
there exists a $T_2>0$ such that
\begin{equation}\label{e5-lm5.2}
\dfrac1t\int_0^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds\geq\frac74\gamma_0\,\text{
for }\,t\geq T_2,\,x\in[\beta_1,H].
\end{equation}
Let $\bar M_H=\sup_{x\in[0,H]}\left\{|\bar\Upsilon(x,0)|\right\}
$,
$\bar t_{x}=\inf\{t\geq0: X_{x,0}\geq\beta_1\}$
and $T_3=\left(4\frac{\bar M_H}{\gamma_0}+7\right)T_2$.
It can be seen from the equation of $\bar X(t)$ that
$\bar X_{(x,0)}(t)\in[\beta_1,H]$
if $t\geq \bar t_x, x\in(0,\beta_1]$.
For $t\geq T_3$, we can use \eqref{e3-lm5.2} and \eqref{e5-lm5.2} to estimate $\frac1t\int_0^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds$ in
the following three cases.
{\bf Case 1}. If $t-T_2\leq \bar t_x\leq t$
then
$$
\begin{aligned}
\int_0^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds
&=
\int_0^{\bar t_x}\bar\Upsilon(\bar Z_{(x, 0)}(s))ds
+\int_{\bar t_x}^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds\\
&\geq \frac74\gamma_0(t-T_2)-T_2\bar M_H\geq\frac32\gamma_0t,\,\,\bigg(\text{since }\, t\geq \Big(4\frac{\bar M_H}{\gamma_0}+7\Big)T_2\bigg).
\end{aligned}
$$
{\bf Case 2}. If $\bar t_x\leq t-T_2$,
then
$$
\begin{aligned}
\int_0^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds
&=
\int_0^{\bar t_x}\bar\Upsilon(\bar Z_{(x, 0)}(s))ds
+\int_{\bar t_x}^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds\\
&\geq \frac74\gamma_0(t-\bar t_x)+\frac74\gamma_0\bar t_x\geq\frac32\gamma_0t.
\end{aligned}
$$
{\bf Case 3}. If $\bar t_x\geq t$,
then
$$
\begin{aligned}
\int_0^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds
&=
\int_0^{\bar t_x}\bar\Upsilon(\bar Z_{(x, 0)}(s))ds
\geq \frac74\gamma_0\bar t_x\geq\frac32\gamma_0t.
\end{aligned}
$$
As a result,
\begin{equation}\label{e6-lm5.2}
\dfrac1t\int_0^t\bar\Upsilon(\bar Z_{(x, 0)}(s))ds\geq\frac32\gamma_0,\,\text{ if } t\geq T_3, x\in(0,H].
\end{equation}
Let $T=T_1\vee T_3$.
By the continuous dependence
of solutions on initial values,
there is $\beta>0$
such that
\begin{equation}\label{e7-lm5.2}
\bar X_z(T)\vee\bar Y_z(T)\leq H \text{ and } \dfrac1T\int_0^T\left|\bar\Upsilon(\bar Z_{z_1}(s))-\bar\Upsilon(\bar Z_{z_2}(s))\right|ds\leq \frac12\gamma_0
\end{equation}
given that $|z_1-z_2|\leq\beta, z_1,z_2\in[0,H]^2.$
Combining \eqref{e2-lm5.2}, \eqref{e6-lm5.2} and \eqref{e7-lm5.2}
we obtain the desired result.
\end{proof}
Generalizing the techniques in \cite{DY},
we divide the proof of the eventual tightness into two lemmas.
\begin{lm}\label{lm5.3}
For any $\Delta>0$, there exist $\eps_0, \delta_0, T>0$ and a compact set $\mathcal K\subset\R^{2,\circ}_+$ such that
$$\liminf\limits_{k\to\infty}\dfrac1k\sum_{n=0}^{k-1}\PP\left\{Z^{\eps,\delta}_{z_0, i_0}(nT)\in \mathcal K\right\}\geq 1-\dfrac\Delta3\, \text{ for any }\, \eps<\eps_0, \delta<\delta_0, z\in\R^{2,\circ}_+.$$
\end{lm}
\begin{proof}
For any $\Delta>0$, let $H=H(\Delta)>\frac{\bar a}{\bar b}+1$ be chosen later and define
$D=\{(x, y): 0<x, y\leq H\}$.
Let $T>0$ and $\beta>0$ such that \eqref{e0-lm5.2} is satisfied
and $D_1=\{(x, y): 0<x, y\leq H, x\wedge y<\beta\}\subset D$.
Define $V(x, y)=-\frac{2\bar c}{\bar a}\ln x-\ln y+ C$ where $C$ is a positive constant such that $V(z)\geq0\,\forall\, z\in D$. In view of the generalized It\^o formula,
$$
\begin{aligned}
V(Z_{z, i}^{\eps,\delta}(t))-V(z)=&\int_0^t\left[-\Upsilon\big(Z_{z, i}^{\eps,\delta}(s), \alpha^\eps(s)\big)+\dfrac{\delta}2\left(\frac{2\bar c}{\bar a}\lambda^2(\alpha^\eps(s))+\rho^2(\alpha^\eps(s))\right)\right]ds\\
&-\frac{2\bar c}{\bar a}\int_0^t\sqrt{\delta}\lambda(\alpha^\eps(s))dW_1(s)-\int_0^t\sqrt{\delta}\rho(\alpha^\eps(s))dW_2(s).
\end{aligned}
$$
For $A\in\F$, using Holder's inequality and It\^o's isometry, we have
\begin{equation}\label{ex11}
\begin{aligned}\E\Big(\1_{A}&\big|V(Z_{z, i}^{\eps,\delta}(T))-V(z)\big|\Big)\\
\leq&\left|\E\1_A\int_0^{T}\Upsilon\big(Z_{z, i}^{\eps,\delta}(t), \alpha^\eps(t)\big)dt\right|+\E\1_A\int_0^{T}\dfrac{\delta}2\left(\frac{2\bar c}{\bar a}\lambda^2(\alpha^\eps(t))+\rho^2(\alpha^\eps(t))\right)dt\\&+\frac{2\bar c}{\bar a}\E\1_A\left|\int_0^{T}\sqrt{\delta}\lambda(\alpha^\eps(t))dW_1(t)\right|+\E\1_A\left|\int_0^{T}\sqrt{\delta}\rho(\alpha^\eps(t))dW_2(t)\right|\\
\leq&T(\E\1_A)^{\frac12}\left(\E\int_0^{T}\left[\Upsilon\big(Z_{z, i}^{\eps,\delta}(t), \alpha^\eps(t)\big)+\dfrac\delta2\left(\frac{2\bar c}{\bar a}\lambda^2(\alpha^\eps(t))+\rho^2(\alpha^\eps(t))\right)\right]dt\right)^{\frac12}\\
&+\delta\sqrt{\PP(A)}\left(\E\int_0^T\left(\frac{2\bar c}{\bar a}\lambda^2(\alpha^\eps(t))+\rho^2(\alpha^\eps(t))\right)dt\right)^{\frac12}\\
\leq& K_4T(1+|z|)\sqrt{\PP(A)},
\end{aligned}
\end{equation}
where the last inequality follows from \eqref{lm5.1a} and the boundedness of $\rho(i)$ and $\lambda(i)$.
If $A=\Omega$, we have
\begin{equation}\label{ex11a}
\dfrac1{T}\E\Big(\big|V(Z_{z, i}^{\eps,\delta}(T))-V(z)\big|\Big)\leq K_4(1+|z|).
\end{equation}
Let $\hat H_{T}>H$ such that $\bar X_z(t)\vee \bar Y_z(t)\leq \hat H_{T}$ for all $z\in[0,H]^2,\, 0\leq t\leq {T}$ and
$$\bar{d}_H=\sup\left\{\left|\dfrac{\partial \bar\Upsilon}{\partial x}(x, y)\right|, \left|\dfrac{\partial \bar\Upsilon}{\partial y}(x, y)\right|: (x,y)\in\R^2_+, x\vee y\leq \hat H_{T}\right\}.$$
Let $\varsigma>0$. Lemma \ref{lm2.2} implies that there are $\delta_0,\eps_0$ such that if $\eps<\eps_0,\delta<\delta_0$,
\begin{equation}\label{ex5}
\PP\left\{|\bar X_z(t)-X_{z, i}^{\eps,\delta}(t)|+|\bar Y_z(t)-Y_{z, i}^{\eps,\delta}(t)|<1\wedge\dfrac{\gamma_0}{2\bar{d}_H}, ~\text{for all}~ t\in[0,{T}]\right\}>1-\dfrac\varsigma6, \,z\in \bar D.
\end{equation}
On the other hand, if $|\bar X_z(t)-X_{z, i}^{\eps,\delta}(t)|+|\bar Y_z(t)-Y_{z, i}^{\eps,\delta}(t)|<1\wedge\dfrac{\gamma_0}{2\bar d_H}$, we have
\begin{equation}\label{ex6}
\begin{aligned}
\bigg|\dfrac{1}{T}\int_0^{T}&\Upsilon(Z_{z, i}^{\eps,\delta}(t), \alpha^\eps(t))dt
- \dfrac{1}{T}\int_0^{T}\bar\Upsilon(\bar Z_{z, i}(t))dt\bigg|\\
\leq&\dfrac1{T}\left|\int_0^{T}\Big(\bar\Upsilon(Z_{z, i}^{\eps,\delta}(t))-\bar\Upsilon(\bar Z_{z, i}(t))\Big)dt\right|\\
& +\dfrac1{T}\left|\int_0^{T}\Big(\Upsilon(Z_{z, i}^{\eps,\delta}(t), \alpha^\eps(t))-\bar\Upsilon(Z_{z, i}^{\eps,\delta}(t))\Big)dt\right|\\
\leq&\dfrac{\gamma_0}2+\dfrac{F_H}{T}\int_0^{T}\sum_{j\in\M}\big|\1_{\{\alpha^\eps(t)=j\}}-v_j\big|dt
\end{aligned}
\end{equation}
where
$F_H:=\sup\{|\Upsilon(z, i)| i\in\M, z\in[0, K_{T}+1]^2\}.$
In view of \cite[Lemma 2.1]{HYZ},
\begin{equation}\label{ex7}
\E\bigg|\dfrac1T\int_0^{T}\sum_{j\in\M}\big|\1_{\{\alpha^\eps(t)=j\}}-v_j\big|dt\bigg|^2=\E\bigg|\dfrac\eps{T}\int_0^{T/\eps}\sum_{j\in\M}\big|\1_{\{\alpha(t)=j\}}-v_j\big|dt\bigg|^2\leq \dfrac{\kappa}{T}\eps
\end{equation}
for some constant $\kappa>0.$
On the one hand,
\begin{equation}\label{ex9}
\E\dfrac1{T}\left|\int_0^{T}\Big(-\frac{2\bar c}{\bar a}\lambda(\alpha^\eps(t))dW_1(t)-\rho(\alpha^\eps(t))dW_2(t)\Big)\right|^2\leq\frac{4\bar c^2}{\bar a^2}\lambda_M^2+\rho_M^2.
\end{equation}
Combining \eqref{e0-lm5.2}, \eqref{ex5}, \eqref{ex6}, \eqref{ex7}, and \eqref{ex9}, we can reselect $\eps_0$ and $\delta_0$ such that for $\eps<\eps_0,\delta<\delta_0$ we have
\begin{equation}\label{ex8}
\PP\left\{\dfrac{-1}{T}\int_0^{T}\Upsilon(\alpha^\eps(t), Z_{z, i}^{\eps,\delta}(t))dt\leq-0.5\gamma_0\right\}\geq 1-\dfrac\varsigma3,\,z\in D_1, i\in\M,
\end{equation}
\begin{equation}\label{ex8b}
\PP\Big\{X_{z, i}^{\eps,\delta}(T)\vee Y_{z, i}^{\eps,\delta}(T)\leq H \mbox{ (or equivalently } Z_{z, i}^{\eps,\delta}(T)\in D)\Big\}\geq1-\dfrac\varsigma3, z\in D_1
,\end{equation}
and
\begin{equation}\label{ex8a}
\PP\left\{\delta\vartheta+\dfrac{\sqrt{\delta}}{T}\left|\int_0^{T}\Big(\frac{2\bar c}{\bar a}\lambda(\alpha^\eps(t))dW_1(t)+\rho(\alpha^\eps(t))dW_2(t)\Big)dt\right|<0.25\gamma_0\right\}>1-\dfrac\varsigma3
\end{equation}
where
$\vartheta=\frac12\left(\frac{2\bar c}{\bar a}\lambda_M^2+\rho_M^2\right).$
Consequently, for any $(z, i)\in D_1\times\M$, there is a subset $\Omega_{z, i}^{\eps,\delta}\subset\Omega$ with $\PP(\Omega_{z, i}^{\eps,\delta})\geq1-\varsigma$
in which we have $Z_{z, i}^{\eps,\delta}(T)\in D$ and
\begin{equation}\label{ex10}
\begin{aligned}
\dfrac1{T}\big(V(Z_{z, i}^{\eps,\delta}({T}))-V(z)\big)\leq&\dfrac{-1}{T}\int_0^{T}\Upsilon(\alpha^\eps(t), Z_{z, i}^{\eps,\delta}(t))dt+\delta\vartheta\\&+\dfrac1{T}\Big|\int_0^{T}\sqrt{\delta}\Big(\frac{2\bar c}{\bar a}\lambda(\alpha^\eps(t))dW_1(t)+\rho(\alpha^\eps(t))dW_2(t)\Big)\Big|\\
\leq& -0.25\gamma_0
\end{aligned}
\end{equation}
On the other hand, we deduce from \eqref{ex11a} that for $z\in D$,
\begin{equation}\label{ex10a}
\PP\left\{\dfrac1{T}\big(V(Z_{z, i}^{\eps,\delta}({T}))-V(z)\big)\leq\Lambda\right\}\geq 1-\varsigma,
\end{equation}
where $\Lambda :=\frac{ K_4(1+2H)}\varsigma$.
Moreover, it also follows from \eqref{ex11a} that for $z\in D\setminus D_1$
$$\E V(Z_{z, i}^{\eps,\delta}({T})\leq \sup_{z\in D\setminus D_1}\big(V(z)+ K_4{T}|z|\big).$$
Define
\begin{equation}\label{ex10d}
L_1:=\sup_{z\in D\setminus D_1}V(z)+\Lambda T,~ L_2:=L_1+0.25\gamma_0,
\end{equation}
as well as $D_2:=\{(x, y)\in\R^{2,\circ}_+: (x, y)\in D, V(x, y)>L_2\}$ and $U(z)=V(z)\vee L_1.$ It is clear that
\begin{equation}\label{ex12a}
U(z_2)-U(z_1)\leq|V(z_2)-V(z_1)| \text{ for any }z_1, z_2\in\R^{2\circ}_+.
\end{equation}
It follows from \eqref{ex11}
that for any $\delta,\eps<1$, $A\in\F$, and $z\in D$, we have
\begin{equation}\label{ex12}
\dfrac1{T}\E\1_{A}\Big|V(Z_{z, i}^{\eps,\delta}(T))-V(z)\Big|\leq K_4(2H+1)\sqrt{\PP(A)}
.\end{equation}
Applying \eqref{ex12} and \eqref{ex12a} with $A=\Omega\setminus\Omega_{z, i}^{\eps,\delta}$,
we get
\begin{equation}\label{ex12b}
\dfrac1{T}\E\1_{\Omega\setminus\Omega_{z, i}^{\eps,\delta}}\Big[U(Z_{z, i}^{\eps,\delta}(T))-U(z)\Big]\leq K_4(2H+1)\sqrt{\varsigma},\text{ if } z\in D_1.
\end{equation}
In view of \eqref{ex10}, for
$z\in D_2\text{ we have }V\big(Z_{z, i}^{\eps,\delta}(T)\big)<V(z)-0.25\gamma_0T.$
By the definition of $D_2$,
we also have $L_1\leq V(z)-0.25\gamma_0T.$
Thus, for any $z\in D_2$ and $\omega\in\Omega^{\eps,\delta}_{z, i}$
$$U\big(Z_{z, i}^{\eps,\delta}(T)\big)=L_1\vee V\big(Z_{z, i}^{\eps,\delta}(T)\big)\leq V(z)-0.25\gamma_0T=U(z)-0.25\gamma_0T,$$
which implies
\begin{equation}\label{ex10c}
\dfrac1{T}\Big[\E\1_{\Omega_{z, i}^{\eps,\delta}}U(Z_{z, i}^{\eps,\delta}({T}))-\E\1_{\Omega_{z, i}^{\eps,\delta}}U(z)\Big]\leq-0.25\gamma_0\PP(\Omega_{z, i}^{\eps,\delta})\leq -0.25\gamma_0(1-\varsigma).
\end{equation}
Combining \eqref{ex12b} with \eqref{ex10c}
\begin{equation}\label{ex13}
\dfrac1{T}\Big[\E U(Z_{z, i}^{\eps,\delta}({T}))-U(z)\Big]\leq-0.25\gamma_0(1-\varsigma)+ K_4(2H+1)\sqrt{\varsigma}, \ \forall z\in D_2.
\end{equation}
For $z\in D_1\setminus D_2$,
and $\omega\in\Omega_{z, i}^{\eps,\delta}$, we have from \eqref{ex10} that $ V(Z_{z, i}^{\eps,\delta}({T}))\leq V(z)$. This shows that
$U(Z_{z, i}^{\eps,\delta}({T}))=L_1\vee V(Z_{z, i}^{\eps,\delta}({T}))\leq U(z)=V(z)\vee L_1.$
Hence, for $z\in D_1\setminus D_2$ and $\omega\in \Omega_{z, i}^{\eps,\delta}$ one has $$U(Z_{z, i}^{\eps,\delta}({T}))-U(z)\leq 0.$$
This and \eqref{ex12b} imply
\begin{equation}\label{ex14}
\dfrac1{T}\Big[\E U(Z_{z, i}^{\eps,\delta}({T}))-U(z)\Big]\leq K_4(2H+1)\sqrt{\varsigma}, \ \forall z\in D_1\setminus D_2.
\end{equation}
If $z\in D\setminus D_1$, $U(z)=L_1$ and we have from \eqref{ex10a} and \eqref{ex10d} that
$$\PP\big\{U(Z_{z, i}^{\eps,\delta}({T}))=L_1\big\}= \PP\big\{V(Z_{z, i}^{\eps,\delta}({T}))\leq L_1\big\}\geq 1-\varsigma.$$
Thus
$$\PP\{U(Z_{z, i}^{\eps,\delta}({T}))=U(z)\}\geq 1-\varsigma.$$
Use \eqref{ex12} and \eqref{ex12a} again to arrive at
\begin{equation}\label{ex16}
\dfrac1{T}\Big[\E U(Z_{z, i}^{\eps,\delta}({T}))-U(z)\Big]\leq K_4(2H+1)\sqrt{\varsigma}, \ \forall \,z\in D\setminus D_1.
\end{equation}
On the other hand, equations \eqref{ex11a} and \eqref{ex12a} imply
\begin{equation}\label{ex17}
\dfrac1{T}\Big[\E U(Z_{z, i}^{\eps,\delta}({T}))-U(z)\Big]\leq K_4(1+|z|), \ z\in\R^{2,\circ}_+.
\end{equation}
Pick an arbitrary $(z_0, i_0)\in\R^{2,\circ}_+\times\M$.
An application of the Markov property yields
$$
\begin{aligned}
\dfrac1{T}\Big[&\E U(Z_{z_0, i_0}^{\eps,\delta}((n+1){T}))-\E U(Z_{z_0, i_0}^{\eps,\delta}({nT}))\Big]\\
&=\sum_{i\in\M}\int_{\R^{2,\circ}_+}\dfrac1{T}\Big[\E U(Z_{z, i}^{\eps,\delta}({T}))-U(z)\Big]\PP\Big\{Z_{z_0, i_0}^{\eps,\delta}(n{T})\in dz, \alpha^\eps(t)=i\Big\}.
\end{aligned}
$$
Combining \eqref{ex13}, \eqref{ex14}, \eqref{ex16}, and \eqref{ex17}, we get
$$
\begin{aligned}
\dfrac1{T}\Big[\E& U(Z_{z_0, i_0}^{\eps,\delta}((n+1){T}))-\E U(Z_{z_0, i_0}^{\eps,\delta}({nT}))\Big]\\
\leq&-\big[0.25\gamma_0(1-\varsigma)- K_4(2H+1)\sqrt{\varsigma}\big]\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(n{T})\in D_2\big\}\\
& + K_4(2H+1)\sqrt{\varsigma}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(n{T})\in D\setminus D_2\big\}\\
&+ K_4 \E\1_{\{Z_{z_0, i_0}^{\eps,\delta}(n{T})\notin D\}}\big(1+|Z_{z_0, i_0}^{\eps,\delta}(n{T})|\big)\\
\leq&-0.25\gamma_0(1-\varsigma)\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(n{T})\in D_2\big\}+ K_4(2H+1)\sqrt{\varsigma}\\
&+ K_4\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(nT)\notin D\big\}\E\big(1+|Z_{z_0, i_0}^{\eps,\delta}(nT)|\big).
\end{aligned}
$$
Note that
\bea \ad \liminf\limits_{k\to\infty}\dfrac1k\sum_{n=0}^{k-1}\dfrac1{T}\Big[\E U(Z_{z_0, i_0}^{\eps,\delta}((n+1){T}))-\E U(Z_{z_0, i_0}^{\eps,\delta}({nT}))\Big]\\
\aad \
=\liminf\limits_{k\to\infty}\dfrac1{kT}\E U(Z_{z_0, i_0}^{\eps,\delta}(k{T}))\geq0.
\eea
This forces
\begin{equation}\label{ex17a}
\begin{aligned}
0.25\gamma_0(1-\varsigma)&\limsup\limits_{k\to\infty}\dfrac1k\sum_{n=0}^{k-1}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(n{T})\in D_2\big\}\\
\leq & K_4(2H+1)\sqrt{\varsigma}+ K_3\limsup\limits_{k\to\infty}\dfrac1k\sum_{n=1}^k\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(nT)\notin D\big\}\E\big(1+|Z_{z_0, i_0}^{\eps,\delta}(nT)|\big).
\end{aligned}
\end{equation}
In view of Lemma \ref{lm5.1}, we can choose $H=H(\Delta)$ independent of $(z_0, i_0)$ such that
\begin{equation}\label{ex18}
\limsup\limits_{t\to\infty}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(t)\notin D\big\}\leq \limsup\limits_{t\to\infty}\dfrac{\E(|Z_{z_0, i_0}^{\eps,\delta}(t)|)}
{H}\leq\dfrac{\Delta}{6}
,\end{equation}
and
\bea\ad K_4\limsup\limits_{t\to\infty}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(t)\notin D\big\}\E\big(1+|Z_{z_0, i_0}^{\eps,\delta}(t)|\big)\\
\aad \ \leq K_4\limsup\limits_{t\to\infty}\dfrac{\Big[\E\big(1+|Z_{z_0, i_0}^{\eps,\delta}(t)|\big)\Big]^2}{H}\leq\dfrac{0.1\gamma_0}{6}\Delta.
\eea
Hence, we have
\begin{equation}\label{ex19}
K_4\limsup\limits_{k\to\infty}\dfrac1k\sum_{n=1}^k\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(nT)\notin D\big\}\E\big(1+|Z_{z_0, i_0}^{\eps,\delta}(nT)|\big)\leq\dfrac{0.1\gamma_0}{6}\Delta.
\end{equation}
Choose $\varsigma=\varsigma(H)>0$ such that $0.25\gamma_0(1-\varsigma)\geq0.2\gamma_0$ and $ K_4(2H+1)\sqrt{\varsigma}\leq\dfrac{0.1\gamma_0}{6}\Delta$
and let $\eps_0=\eps_0(\varsigma, H),\delta_0(\varsigma, H)$ such that \eqref{ex8}, \eqref{ex8b}, and \eqref{ex8a} hold.
As a result, we get from \eqref{ex17a} and \eqref{ex19} that
\begin{equation}\label{ex20}
\limsup\limits_{k\to\infty}\dfrac1k\sum_{n=1}^k\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(nT)\in D_2\big\}\leq \dfrac{\Delta}{6}.
\end{equation}
This together with \eqref{ex18} and \eqref{ex20} shows that for any $\eps<\eps_0, \delta<\delta_0$, we have
\begin{equation}\label{ex21}
\liminf\limits_{k\to\infty}\dfrac1k\sum_{n=1}^k\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(nT)\in D\setminus D_2\big\}\geq1- \dfrac{\Delta}3.
\end{equation}
One can conclude the proof by noting that the set $D\setminus D_2$ is a compact subset of $\R^{2,\circ}_+$.
\end{proof}
\begin{lm}\label{lm5.4}
There are $ L>1$, $\eps_1=\eps(\Delta)>0$, and $\delta_1=\delta_1(\Delta)>0$ such that as long as $0<\eps<\eps_1, 0<\delta<\delta_1$, we have
$$\liminf\limits_{T\to\infty}\dfrac1T\int_0^T\PP\{Z^{\eps,\delta}_{z_0, i_0}(t)\in [ L^{-1}, L]^2\}\geq 1-\Delta, (z_0, i_0)\in\R^{2,\circ}_+\times\M.$$
\end{lm}
\begin{proof}
Let $D$ and $T$ as in Lemma \ref{lm5.3}.
Since $D\setminus D_2$ is a compact set in $\R^{2,\circ}_+$, by a modification of the proof of \cite[Theorem 2.1]{JJ}, we can show that there is a positive constant $ L>1$ such that
$\PP\{Z_{z, i}(t)\in [ L^{-1}, L]^2\}>1-\dfrac\Delta3, z\in D\setminus D_2, i\in M, 0\leq t\leq T$.
Hence, it follows from the Markov property of the solution that
\bea \ad\PP\{Z_{z_0, i_0}^{\eps, \delta}(t)\in[ L^{-1}, L]^2\}\\
\aad\quad \geq \Big(1-\dfrac{\Delta}3\Big)\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(jT)\in D\setminus D_2\big\}, t\in[jT, jT+T].\eea
Consequently,
\begin{align*}
\liminf\limits_{k\to\infty}\dfrac1{kT}&\int_{0}^{kT}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(t)\in [ L^{-1}, L]^2\big\}\,dt\\&\geq \Big(1-\dfrac{\Delta}3\Big)\liminf\limits_{k\to\infty}\dfrac1{k}\sum_{j=0}^{k-1}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(jT)\in D\setminus D_2)\big\}\geq 1-\Delta.
\end{align*}
It is readily seen from this estimate that
$$\liminf\limits_{T\to\infty}\dfrac1{T}\int_{0}^{T}\PP\big\{Z_{z_0, i_0}^{\eps,\delta}(t)\in [ L^{-1}, L]^2\big\}dt\geq 1-\Delta.$$
\end{proof}
\mainnex*
\begin{proof}
The conclusion of Lemma \ref{lm5.4}
is sufficient for the existence of a unique invariant probability measure
$\mu^{\eps,\delta}$ in $\R^{2,\circ}_+\times\M$ of $(Z^{\eps,\delta}(t), \alpha^\eps(t))$ (see \cite{LB} or \cite{MT}). Moreover, the empirical measures
$$\dfrac1t\int_0^t\PP\big\{Z^{\eps,\delta}_{z_0,i_0}(s)\in \cdot\big\}ds, t>0$$
converge weakly to the invariant measure $\mu^{\eps,\delta}$ as $t\to\infty$.
Applying Fatou's lemma to the above estimate yields
$$\mu^{\eps,\delta}([ L^{-1}, L]^2)\geq \Delta, \ \forall \,\eps<\eps_0, \delta<\delta_0.$$
This tightness implies Theorem \ref{thm5.1}.
\end{proof}
\subsection{An Example}
\label{sec:6}
In this section we provide a specific
example
under the setting
of Section \ref{sec:5}.
We consider the following stochastic predator-prey model with Holling functional response in a switching regime
\begin{equation}\label{nex1}
\left\{\begin{array}{ll} \disp d\xx (t)&\!\!\! \disp =\bigg[r(\alpha^\eps(t))\xx(t)\left(1-\dfrac{\xx(t)}{K(\alpha^\eps(t)}\right)-\dfrac{m(\alpha^\eps(t))\xx(t)\yy(t)}{a(\alpha^\eps(t))
+b(\alpha^\eps(t))\xx(t)}\bigg]dt\\
& \quad \ +\sqrt{\delta}\lambda(\alpha^\eps(t))\xx(t)dW_1(t)\\
\disp d\yy(t)&\!\!\! \disp =\yy(t)\bigg[-d(\alpha^\eps(t))+\dfrac{e(\alpha^\eps(t))
m(\alpha^\eps(t))\xx(t)}{a(\alpha^\eps(t))+b(\alpha^\eps(t))\xx(t)}-f(\alpha^\eps(t))\yy(t)\bigg]dt\\
&\quad \ +\sqrt{\delta}\rho(\alpha^\eps(t))\xx(t)dW_2(t),\end{array}\right.
\end{equation}
where $W_1$ and $W_2$ are two independent Brownian motions, $\alpha^\eps(t)$ is a Markov chain, that is
independent of the Brownian motions, with state space $\M=\{1, 2\}$ and generator $Q/\eps$ where
$$Q =
\left( \begin{array}{rr} -1 & 1 \\
1 & -1 \\ \end{array} \right),$$
and $r(1)=0.9, r(2)=1.1, K(1)=4.737, K(2)=5.238, m(1)=1.2, m(2)=0.8, a(1)=a(2)=1, b(1)=b(2)=1, d(1)=0.85, d(2)=1.15, e(1)=1, e(2)=2.5, f(1)=0.03, f(2)=0.01, \lambda(1)=1, \lambda(2)=2, \rho(1)=3, \rho(2)=1$.
As $\eps$ and $\delta$ tend to 0, solutions of equation \eqref{nex1} converge to the corresponding solutions of
\begin{equation}\label{nex2}
\left\{\begin{array}{lll}\disp {d \over dt} {x}(t)=x(t)\left(1-\dfrac{x(t)}{5}\right)-\dfrac{x(t)y(t)}{1+x(t)},
\\
\disp {d \over dt} {y}(t)=y(t)\left(-1+\dfrac{1.6x(t)}{1+x(t)}-0.02y(t)\right)
\end{array}\right.
\end{equation}
on any finite time interval $[0,T]$. The system \eqref{nex2} has the unique equilibrium $(x^*, y^*)=(1.836, 1.795)$.
Modifying \cite[Theorem 2.6]{SR} it can be seen that the solution of equation
\eqref{nex2} has a unique limit cycle $\Gamma$ that attracts all positive solutions except for
$(x^*, y^*)$.
Moreover, it is easy to check that the drift
\begin{equation}\label{nex4}
\left(\begin{array}{l}r(i)x(t)\left(1-\dfrac{x(t)}{K(i)}\right)-\dfrac{m(i)x(t)y(t)}{a(i)+b(i)x(t)}\\
y(t)\left(-d(i)-\dfrac{e(i)m(i)x(t)}{a(i)+b(i)x(t)}\right)-f(i)y(t)\end{array}\right)
\end{equation}
does not vanish at $(1.836, 1.795)$. The assumptions of Theorem \ref{thm5.1} hold in this example. As a result, the family $(\mu^{\eps,\delta})_{\eps>0}$ converges weakly as $\eps\downarrow 0$ to the stationary distribution of \eqref{nex2} that is concentrated on the limit cycle $\Gamma$.
We illustrate this convergence in Figures \ref{f1}, \ref{f2} and \ref{f3} below by graphing sample paths of \eqref{nex1} for different values of $(\eps,\delta)$.
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.2in,width=2.1in]{X_0_001.png}
\includegraphics[totalheight=2.2in,width=2.1in]{X_0_00005.png}
\includegraphics[totalheight=2.2in,width=2.1in]{X_limit.png}
\caption{From left to right: Graphs of the $x^{\eps,\delta}(t)$ component of \eqref{nex1} with $(\eps,\delta)=(0.001, 0.001)$, $(\eps,\delta)=(0.00005, 0.00005)$ and $x(t)$ of the averaged system \eqref{nex2} respectively.}
\label{f1}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.2in,width=2.1in]{Y_0_001.png}
\includegraphics[totalheight=2.2in,width=2.1in]{Y_0_00005.png}
\includegraphics[totalheight=2.2in,width=2.1in]{Y_limit.png}
\caption{From left to right: Graphs of the $y^{\eps,\delta}(t)$ component of \eqref{nex1} with $(\eps,\delta)=(0.001, 0.001)$, $(\eps,\delta)=(0.00005, 0.00005)$ and $y(t)$ of the averaged system \eqref{nex2} respectively.}
\label{f2}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.2in,width=2.1in]{figure1.png}
\includegraphics[totalheight=2.2in,width=2.1in]{figure2.png}
\includegraphics[totalheight=2.2in,width=2.1in]{figure3.png}
\caption{Phase portraits of \eqref{nex1} for different values of $\eps$ and $\delta$.}
\label{f3}
\end{figure}
{\bf Acknowledgments.} The research of
Nguyen H. Du was supported in part by
NAFOSTED n$_0$ 101.02 - 2011.21. Three of the authors were supported in part by the National Science Foundation under grants DMS-1207667 (George Yin), DMS-1853463 (Alexandru Hening) and DMS-1853467 (Dang Nguyen).
\clearpage
\bibliographystyle{amsalpha}
\bibliography{cycle}
\appendix
\section{Proofs of Lemmas from Section \ref{sec:3}}\label{a:1}
\begin{lm}\label{lm2.1}
For any $R, T, \gamma>0$, there exists a number $k_1=k_1(R, T, \gamma)>0$ such that for all sufficiently small $\delta$,
$$\PP\{|X^{\eps,\delta}_{x, i}(t)-\xi^{\eps}_{x, i}(t)|\geq\gamma, ~\text{for some}~t\in [0,T]\}<\exp\left(-\dfrac{k_1}{\delta}\right), x\in B_R,$$
where $X^{\eps,\delta}_{x, i}(t)$ and $\xi^{\eps}_{x, i}(t)$ are the solutions to
the systems \eqref{eq2.1} and \eqref{eq2.3} that have initial value $(x, i)$.
\end{lm}
\begin{proof}
By (i) and (ii) of Assumption \ref{asp1}, we can deduce the existence and boundedness of a unique solution to equation \eqref{eq2.3}
using
the
Lyapunov function method. Moreover, we can find $R_T>R>0$ such that almost surely
\begin{equation}\label{e:bound}
|\xi^{\eps}_{x, i}(t)|<R_T-\gamma, ~\text{for all}~t\in[0, T], x\in B_R.
\end{equation}
Let $h(\cdot)$ be a twice differentiable function with compact support such that $h(x)=1$ if $|x|\leq R_T$ and $h(x)=0$
if $|x|\geq R_T+1$.
Put $f_h(x, i)=h(x)f(x, i)$, $\sigma_h(x, i)=h(x)\sigma(x, i)$ and
let $Y^{\eps,\delta}_{x, i}(t)$ be the solution starting at $(x, i)$ of
\begin{equation}
d Y(t)= f_h(Y(t),\alpha^\eps(t))dt+\sqrt{\delta}\sigma_h(Y(t), \alpha^\eps(t)dW(t)
\end{equation}
Note that $Y^{\eps,\delta}_{x, i}(t)=X^{\eps,\delta}_{x, i}(t)$ up to the time $\zeta=\inf\{t>0: |X^{\eps,\delta}_{x, i}(t)|>R_T\}$.
Because of \eqref{e:bound},
the solution $\xi^{\eps}_{x, i}(t)$ to \eqref{eq2.3} coincides with the solution to
$$d Z(t)= f_h(Z(t),\alpha^\eps(t))dt$$ with
starting point $x\in B_R$ and $t\in [0,T]$.
We have from the generalized It\^o's formula that
for all $x\in B_R$ and $t\in [0,T]$,
\begin{equation}
\begin{aligned}
|Y&^{\eps,\delta}_{x, i}(t)-\xi^{\eps}_{x, i}(t)|^2\\
\leq&2\int_0^{t}|Y^{\eps,\delta}_{x, i}(s)-\xi^{\eps}_{x, i}(s)||f_h(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s))-f_h(\xi^{\eps}_{x, i}(s),\alpha^\eps(s))|ds\\
&+\int_0^{t}\delta\trace\big((\sigma_h\sigma_h')(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s))\big)ds\\
&+2\sqrt{\delta}\left|\int_0^{t}\big(Y^{\eps,\delta}_{x, i}(s)-\xi^{\eps}_{x, i}(s)\big)'\sigma_h(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s)\big)dW(s)\right|.
\end{aligned}
\end{equation}
Define
\begin{align*}
A=\bigg\{\omega\in \Omega ~:~ &\left|\int_0^{t}\sqrt{\delta}\left(Y^{\eps,\delta}_{x, i}(s)-\xi^{\eps}_{x, i}(s)\right)'\sigma_h(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s))dW(s)\right|\\
&\qquad-\dfrac1\delta\int_0^{t}\delta\left|Y^{\eps,\delta}_{x, i}(s)-\xi^{\eps}_{x, i}(s)\right|^2\left\|\sigma_h\sigma_h'(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s))\right\|ds\leq k_1\, \text{ for all }\, t\in[0,T]\bigg\}.
\end{align*}
By the exponential martingale inequality, we get that for any $\delta<k_1$
\begin{align*}
\PP(A)\geq 1-2\exp\left(-\dfrac{2k_1}\delta\right)\geq1-\exp\left(-\frac{k_1}\delta\right).
\end{align*}
Since $f_h$ is Lipschitz and $\sigma_h$ is bounded, there is
an $M_1>0$ such that for all $\omega\in A$,
\begin{equation}
\begin{aligned}
|Y^{\eps,\delta}_{x, i}&(t)-\xi^{\eps}_{x, i}(t)|^2\\
\leq & 2 \int_0^{t}|Y^{\eps,\delta}_{x, i}(s)-\xi^{\eps}_{x, i}(s)||f_h(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s))-f_h(\xi^{\eps}_{x, i}(s),\alpha^\eps(s))|ds\\
&\ +2\int_0^{t}\big|Y^{\eps,\delta}_{x, i}(s)-\xi^{\eps}_{x, i}(s)\big|^2\|\sigma_h\sigma_h'(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s)\big)\|ds\\
&\ +\int_0^{t}\delta\trace\big((\sigma_h\sigma_h')(Y^{\eps,\delta}_{x, i}(s),\alpha^\eps(s))\big)ds+2\int_0^tk_1ds\\
\leq & M_1\int_0^t|Y^{\eps,\delta}_{x, i}(t)-\xi^{\eps}_{x, i}(t)|^2ds+(2k_1+M_1\delta) t
.\end{aligned}
\end{equation}
For each $t\in[0, T]$, an application of Gronwall's inequality implies that on the set $A$,
$$|Y^{\eps,\delta}_{x, i}(t)-\xi^{\eps}_{x, i}(t)|^2\leq (2k_1+M_1\delta)T\exp(M_1T)<\gamma^2$$ for $0<\delta<k_1$ sufficiently small.
It also follows from this inequality that for $\omega\in A$ and $0<\delta<k_1$ sufficient small, we have $\zeta>T$, which implies $$|X^{\eps,\delta}_{x, i}(t)-\xi^{\eps}_{x, i}(t)|^2=|Y^{\eps,\delta}_{x, i}(t)-\xi^{\eps}_{x, i}(t)|^2<\gamma^2,$$ for all $ t\in[0,T]$.
\end{proof}
\begin{lm}\label{lm2.1b}
For each $x$ and $\gamma$, we can find $k_{\gamma,x}=k_{\gamma,x}(T)>0$ such that
$$\PP\left\{\left|\xi^{\eps}_{x, i}(t)-\bar X_x(t)\right|\geq\gamma ~\text{for some}~t\in [0,T]\right\}\leq\exp\left(-\frac{k_{\gamma,x}}\eps\right),$$
where $\bar X_x(t)$ is the solution to equation \eqref{eq2.2} with the initial value $x$.
\end{lm}
\begin{proof}
This follows from the large deviation principle shown in \cite{HYZ}. We note that the existence and boundedness of a unique solution to equation \eqref{eq2.3} follows from parts (i) and (ii) of Assumption \ref{asp1}.
\end{proof}
By combining the results of Lemmas \ref{lm2.1} and \ref{lm2.1b} we can
prove
Lemma \ref{lm2.2}.
\begin{proof}[Proof of Lemma \ref{lm2.2}]
By virtue of Lemma \ref{lm2.1b}, for each $x$ and $\gamma$, we have
$$\PP\left\{\left|\xi^{\eps}_{x, i}(t)-\bar X_x(t)\right|\geq\dfrac\gamma6 ~\text{for some}~t\in [0,T]\right\}\leq\exp\left(-\frac{k_{\gamma/6,x}}\eps\right).$$
By part $(ii)$ of Assumption \ref{asp1} together with the Lyapunov method for \eqref{eq2.3}, we can find $H_{R, T}>0$ such that $|\xi^{\eps}_{x, i}(t)|\leq H_{R, T}$ and $|\bar X_x(t)|\leq H_{R, T}$ for all $|x|\leq R$ and $0\leq t\leq T$. Since $f(\cdot, i)$ is locally Lipschitz for all $i\in\M$, there is a constant $M_2>0$ such that $|f(u, i)-f(v, i)|\leq M_2|u-v|$ for all $|u|\vee|v|\leq H_{R, T}$ and $i\in\M$. Using the Gronwall
inequality, we have for $|x|\vee|y|\leq R$, $i\in\M$ and any $t\in [0,T]$
\bea \ad |\xi^{\eps}_{x, i}(t)-\xi^{\eps}_{y,i}(t)|\leq|x-y|\exp(M_2T),\\
\ad
|\bar X_x(t)-\bar X_y(t)|\leq|x-y|\exp(M_2T).\eea
Let $\lambda=\dfrac{\gamma}6\exp(-M_2T)$. It is easy to see that for $|x-y|<\lambda$,
$$\PP\left\{\left|\xi^{\eps}_{y,i}(t)-\bar X_y(t)\right|\geq\dfrac\gamma2 ~\text{for some}~t\in [0,T] \right\}\leq\exp\Big(-\frac{k_{\gamma/6,x}}\eps\Big).$$
By the compactness of $B_R$, for $\gamma>0$, there is $k_2=k_2(R, T,\gamma)>0$ such that for all $x\in B_R$,
$$\PP\left\{\left|\xi^{\eps}_{x, i}(t)-\bar X_x(t)\right|\geq\dfrac\gamma2 ~\text{for some}~t\in [0,T]\right\}\leq \exp\left(-\frac{k_2}\eps\right).$$
Combining this with Lemma \ref{lm2.1}, we have
\bea \disp \PP\left\{\left|X^{\eps,\delta}_{x,i}(t)-\bar X_x(t)\right|\geq\gamma ~\text{for some}~t\in [0,T] \right\}\ad <\exp\left(-\frac{k_1(R, T,\gamma/2)}\delta\right)+\exp\left(-\frac{k_2}\eps\right)\\
\ad <\exp\left(-\frac{\kappa}{\eps+\delta}\right)\eea
for a suitable $\kappa=\kappa(R,T,\gamma)$ and for all sufficiently small $\eps$ and $\delta$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lm3.1}]
Let $n^{\eps,\delta}\in\N$ such that $n^{\eps,\delta}-1<\dfrac{1}{a^{\eps,\delta}}\leq n^{\eps,\delta}.$
We consider events $A_k=\{X_{x,i}^{\eps, \delta}(t)\in N, \ \forall (k-1)\ell <t\leq k\ell \}$.
We have $\PP(A_1)\leq1-a^{\eps,\delta}.$
By the Markov property,
\bea \disp\PP(A_k|A_1,...,A_{k-1})\ad =\int_{N}\PP\left\{\check\tau_y^{\eps, \delta}\leq \ell \right\}\PP\Big\{X_{x,i}^{\eps, \delta}((k-1)\ell )\in dy\Big|A_1,...,A_{k-1}\Big\}\\
\ad\leq1-a^{\eps,\delta}.\eea
As a result,
$$\PP(A_1A_2\cdots A_n)\leq (1-a^{\eps,\delta})^{n^{\eps,\delta}}$$
Since $\lim\limits_{\eps\to0}a^{\eps,\delta}=0$, we deduce that $\lim\limits_{\eps\to0}(1-a^{\eps,\delta})^{n^{\eps,\delta}}=e^{-1}$, which means that $(1-a^{\eps,\delta})^{n^{\eps,\delta}}<1/2$ for sufficiently small $\eps$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lm2.5}]
The proof is omitted because
it states some standard properties of dynamical systems.
Interested readers can refer to \cite{perko13}.
\end{proof}
\end{document}
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Fidel Castro described the destruction left by Category Four Hurricane Gustav as similar to the aftermath of a nuclear bombing. While the damage is extensive, and will take years to put right, the island’s renowned Civil Defense system evacuated hundreds of thousands of people from Gustav’s path, ensuring that whatever impact the hurricane might have, it would not cost human lives.
In co-operation with Cuba friendship groups across the nation, NSCUBA is calling on the Canadian government to act swiftly in the provision of emergency aid to Cuba and Haiti, also hard-hit by the storm.
Your financial contributions to our Hurricane Relief Fund (managed by the Canadian Network on Cuba (CNC)) are urgently needed to help Cuba obtain housing materials, rebuild the electrical grid and the communications infrastructure.
Please send cheques payable to Mackenzie-Papineau Memorial Fund and mail to:
Sharon Skup (treasurer)
56 Riverwood Terrace
Bolton, ON
L7E 1S4
And write Cuba Hurricane Fund on the memo line of the cheque. Charitable tax receipts will be issued.
The Mackenzie-Papineau Memorial Fund is a charitable organization and working with the CNC and members. [Revenue Canada registration # 88876 9197]
| 114,408
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Once again, I'm all apologies--this blog has been terribly neglected. This summer I've gotten a fairly decent start on my dissertation draft, written my first journal article, and planned a wedding, so this blog has slipped off my radar again. I've completely missed the boat on reacting to the London 2012 Olympics ceremonies...
Thoughts on the Olympics: Overall a successful event, partially because expectations weren't very high. I think people doubted London could compete with Beijing in terms of being 'spectacular' and overwhelming, so the organisers took a different approach. They didn't go for shock & awe--they went for a presentation of the Best of Britain, and relied heavily on British music to represent UK culture. I saw a mix of comments on Facebook, from foreigners and natives, and many of the positive comments came from people who had studied here and returned home. They expressed a sense of nostalgia for Britain--perfect evidence of educational exchange impact! For my part, I was surprisingly patriotic and sentimental about my adopted country--I got teary-eyed at the start, with the kid soloist singing 'Jerusalem'. I loved the references to Mary Poppins and Harry Potter, and the prominent use of so many Beatles songs in both the opening & closing ceremonies. I thought the celebration of the NHS was an interesting choice, and I wondered how Republicans back in the States reacted to that one, as they vow to repeal Obamacare (which doesn't go nearly as far as true socialized medicine, but they call him a 'socialist' anyway...sigh).
In terms of public diplomacy, I'm not sure that it was a success--but I also doubt that it was supposed to achieve much in the way of public diplomacy. This summer seems to be all about Britain, with the focus on the Jubilee, the Euros, and Team GB. In my 5 years here, I've never seen much in the way of patriotism--at least not in the way you see it in the States, with flags and t-shirts to express your pride--but now, Union Jack paraphernalia is everywhere. Looking at some of the references used, I think the ceremonies may have had more meaning for Brits than for the rest of the world. But I don't think that's necessarily a bad thing. The domestic political situation has been messy for 2 years now, the economy has been struggling for even longer--so maybe turning inwards and celebrating British culture is a way of coping. 'We might be divided by politics and money and class, but we all love the Beatles, right? Remember the Spice Girls, they were good, too? And British comedy is great, isn't it? Let's get Eric Idle in to cheer us all up!'
To sum it all up, the ceremonies changed my mind a bit about the use of the Olympics as a public diplomacy tool. They represented Britain well, of course, but I don't think they were overly concerned with the reaction of foreign publics--at least not as self-consciously as China and Canada have been in the past two Olympics.
| 83,090
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\subsection{Complex surfaces}
Throughout this paper, $X$ will denote a complex surface, by which we mean a
connected compact complex manifold of complex dimension two. Usually
$X$ will be rational. The book \cite{BHPV} is a good
general reference for complex surfaces. Here we recount only needed facts.
Given divisors,
$D,D'$ on $X$, we will write $D\sim D'$ to denote linear equivalence,
$D\leq D'$ if $D' = D + E$ where $E$ is an effective divisor and
$D\lesssim D'$ if $D' - D$ is linearly equivalent to an effective divisor.
By a \emph{curve} in $X$, we will mean a reduced effective divisor. We let
$\pic(X)$ denote the Picard group on $X$, i.e. divisors modulo linear
equivalence. We let $K_X\in\pic(X)$ denote the (class of) a canonical divisor
on $X$, which is to say, the divisor of a meromorphic two form on $X$.
Taking chern classes associates each element of $\pic(X)$ with a cohomology
class in $H^{1,1}(X)\cap H^2(X,\Z)$. We will have need of the larger group
$H^{1,1}_\R(X) \eqdef H^{1,1}(X)\cap H^2(X,\R)$. We call a class
$\theta\in H^{1,1}_\R(X)$ \emph{nef} if $\theta^2 \geq 0$ and
$\theta\cdot C\geq 0$ for any complex curve. We will repeatedly rely on the
following consequence of the Hodge index theorem.
\begin{thm}
\label{hodgethm}
If $\theta\in H_\R^{1,1}(X)$ is a non-trivial nef class, and $C$ is a curve,
then $\theta\cdot C = 0$ implies that either
\begin{itemize}
\item the intersection form is negative when restricted to divisors supported
on $C$; or
\item $\theta^2=0$ and there exists an effective divisor $D$ supported on $C$
such that $D \sim t\theta$ for some $t>0$.
\end{itemize}
In particular if $\theta$ has positive self-intersection, then the
intersection form is negative definite on $C$.
\end{thm}
\begin{proof}
The hypotheses imply that $\theta\cdot D = 0$ for every divisor $D$
supported on $C$. Suppose that the intersection form restricted to $C$
is not negative definite. That is,
there is a non-trivial divisor $D$ with $\supp D\subset C$
and $D^2 \geq 0$. Then we may write $D = D_+ - D_-$ as a difference
of effective divisors supported on $C$ with no irreducible components in
common. Since $D_+\cdot D_-\geq 0$, we have
$$
0\leq D^2 \leq D_+^2 + D_-^2,
$$
so replacing $D$ with $D_+$ or $D_-$ allows us to assume that $D$ is
effective. In particular, $D$ represents a non-trivial class in
$H^{1,1}_{\R}(X)$. Since $D\cdot \theta = 0$ and $\theta^2,D^2 \geq 0$,
we see that the intersection form is non-negative on the subspace of
$H^{1,1}_\R(X)$ generated by $D$ and $\theta$. By Corollary 2.15
in \cite[page 143]{BHPV}, such a subspace must be one-dimensional.
Thus $D = t\theta$ for some $t>0$.
\end{proof}
By the \emph{genus} $g(C)$ of a curve $C\subset X$, we will mean the quantity
$1-\chi(\mathcal{O}_C)$, or equivalently, $1+h^0(K_C)$ minus the number of
connected components of $C$. If $C$ is smooth and irreducible, then $g(C)$ is
just the usual genus of $C$ as a Riemann surface. If $C$ is merely
irreducible, then $g(C)$ is usually called the \emph{arithmetic genus} of $C$,
and in this case it dominates the genus of the Riemann surface obtained by
desingularizing $C$. If $C$ is connected then $g(C) \geq 0$, but our notion
of genus is a bit non-standard in that we do not generally require
connectedness of $C$ in what follows. For any curve $C$, connected or not, we
have the following \emph{genus formula}
\begin{equation}
\label{genusformula}
g(C) = \frac{C\cdot(C+K_X)}{2} + 1.
\end{equation}
\subsection{Birational maps}
Now suppose that $Y$ is a second complex surface and $f:X\to Y$ is a
birational map of $X$ onto $Y$. That is, $f$ maps some Zariski open subset of
$X$ biholomorphically onto its image in $Y$. In general the complement of
this subset will consist of a finite union of rational curves collapsed by
$f$ to points, and a finite set $I(f)$ of points on which $f$ cannot be
defined as a continuous map. We call the contracted curves \emph{exceptional}
and the points in $I(f)$ \emph{indeterminate} for $f$. The birational inverse
$f^{-1}:Y\to X$ of $f$ is obtained by inverting $f$ on the Zariski open set
where $f$ acts biholomorphically. Note that what we call a birational map is
perhaps more commonly called a birational correspondence, the former term
often being understood to mean that $I(f)=\emptyset$.
We adopt the following conventions concerning images of proper subvarieties
of $X$. If $C\subset X$ is an irreducible curve, then $f(C)$ is defined to be
$\overline{f(C-I(f))}$, which is a point if $C$ is exceptional for $f$ and a
curve otherwise. If $p\in X$ is a point of indeterminacy, then $f(p)$ will
denote the union of $f^{-1}$-exceptional curves that $f^{-1}$ maps to $p$. We
apply the same conventions to images under $f^{-1}$.
Our convention for the inverse image of an irreducible curve extends by
linearity to a \emph{proper transform} action $f^\sharp D$ of $f$ on divisors
$D$, provided we identify points with zero. We also have the
\emph{total transform} action $f^* D$ of $f$ on divisors obtained by pulling
back local defining functions for $D$ by $f$. Total transform has the
advantage that it preserves linear equivalence and therefore descends to a
linear map $f^*:\pic(Y)\to \pic(X)$. We denote the proper and
total transform under $f^{-1}$ by $f_*$ and $f_\sharp$, respectively.
In general, $f^* D - f^\sharp D$ is an effective divisor with support equal to
a union of exceptional curves mapped by $f$ to points in $\supp D$. It will
be important for us to be more precise about this point. To do so, we use the
`graph' $\Gamma(f)$ of $f$ obtained by minimally desingularizing the variety
$$
\overline{\{(x,f(x))\in X\times Y:x\notin I(f)\}}.
$$
We let $\pi_1:\Gamma(f)\to X$, $\pi_2:\Gamma(f)\to Y$ denote projections onto
first and second coordinates. Thus $\Gamma(f)$ is an irreducible complex
surface and $\pi_1,\pi_2$ are \emph{proper modifications} of their respective
targets, each holomorphic and birational and therefore each equal to a finite
composition of point blowups. One sees readily that
$f = \pi_2\circ \pi_1^{-1}$, and that the exceptional and indeterminacy sets
of $f$ are the images under $\pi_1$ of the exceptional sets of $\pi_2$ and
$\pi_1$, respectively. Given a decomposition $\sigma_n\circ\dots
\circ\sigma_1$ of $\pi_2$ into point blowups, we let $E(\sigma_j)$ denote the
center of the blowup $\sigma_j$ and
$$
\hat E_j(f) = \sigma_1^*\dots \sigma_{j-1}^* E(\sigma_j),\quad
E_j(f) = \pi_{1*}\hat E_j(f).
$$
In particular, $\bigcup \supp E_j(f)$ is the exceptional set. We call
the individual divisors $E_j(f)$ the \emph{exceptional components} of $f$
and call their sum $E(f)\eqdef \sum E_j(f)$ the \emph{exceptional divisor}
of $f$. It should be noted that, as we have defined them, the exceptional
components of $f$ are
connected, but in general they are neither reduced nor irreducible.
The following proposition assembles some further information about the
exceptional components. These can be readily deduced from well-known facts
about point blowups. We recall that the \emph{multiplicity}
of a curve $C$ at a point $p$ is just the minimal multiplicity of the
intersection of $C$ with an analytic disk meeting $C$ only at $p$.
\begin{prop}
\label{exceptional}
Let $\sigma_j$, $E_j(f)$, and $E(f)$ be as above,
and $C\subset X$ be a curve.
\begin{itemize}
\item $E(f) = (f^*\eta) - f^*(\eta)$ for any
meromorphic two form $\eta$ on $X$ (here $(\eta)$ denotes the divisor of
$\eta$). Less precisely, $E(f)\sim K_X - f^* K_X.$
\item $E_j(f)$ and $E_i(f)$ have irreducible components in common if and only
if $f(E_j(f)) = f(E_i(f))$. If this is the case, then $i \leq j$ implies that
$E_i(f)\leq E_j(f)$.
\item The multiplicity with which an irreducible curve $E$ occurs in
$E(f)$ is bounded above by a constant that depends only on the number
of exceptional components $E_j(f)$ that include $E$.
\item $f^* C - f^\sharp C = \sum c_j E_j(f)$ where $c_j$ is the multiplicity
of $(\sigma_n\circ \dots\circ\sigma_{j+1})^\sharp(C)$ at the point
$\sigma_j(E(\sigma_j))$.
\item In particular, $c_j$ vanishes if $p_j \eqdef f(E_j(f)) \notin C$,
$c_j\leq 1$ if $p_j$ is a smooth point of $C$, and $c_j>0$ if $p_j\in C$
and $E_j(f)$ is not dominated by any other exceptional component of $f$.
\item Hence (in light of the 2nd and 5th items),
$\supp f^* C - f^\sharp C = f^{-1}(C\cap I(f^{-1}))$.
\end{itemize}
\end{prop}
We will also need the following elementary fact.
\begin{lem}
\label{sing1}
Let $C\subset X$ be a curve such that no component of $C$ is exceptional
for $f$. If $p\in C-I(f)$, then multiplicity of $f(C)$ at $f(p)$ is
no smaller than that of $C$ at $p$. In particular, $f(p)$ is singular
for $f(C)$ if $p$ is singular for $C$.
\end{lem}
\ignore{\begin{proof}
If $f$ acts biholomorphically at $p$, the result is obvious. Otherwise
$f$ is holomorphic near $p$ and decomposes locally as a composition of point
blowups. Hence it suffices to verify the lemma for the case where $f$
is a point blowup. In this case, the result follows from the facts that
the multiplicity of the intersection of $C$ with a generic smooth disk will
be minimal and that the image of a generic smooth disk under a point blowup
will be smooth.
\end{proof}}
\subsection{Classification of birational self-maps}
Supposing that $f:X\self$ is a birational self-map, we now recall some
additional information from \cite{DiFa01}. First of all, there are pullback
and pushforward actions $f^*,f_*:H^{1,1}_\R (X)\self$ compatible with the
total transforms $f^*,f_*:\pic(X)\self$. The actions are adjoint with respect
to intersections, which is to say that
\begin{equation}
\label{adjoint}
f^*\alpha \cdot \beta = \alpha\cdot f_*\beta,
\end{equation}
for all $\alpha,\beta\in H^{1,1}_\R(X)$. Less obviously, $f^{n*}$
is `intersection increasing', meaning
$$
(f^{n*}\alpha)^2 \geq \alpha^2
$$
The \emph{first dynamical degree} of $f$ is the quantity
$$
\lambda(f) := \lim_{n\to\infty} \norm{f^{n*}}^{1/n} \geq 1.
$$
It is less clear than it might seem that $\lambda(f)$ is well-defined, as it
can happen that $(f^n)^* \neq (f^*)^n$ for $n$ large enough. However,
$\lambda(f)$ can be shown to be invariant under birational change of
coordinate and one can take advantage of this to choose a good surface on
which to work.
\begin{thm}
\label{asthm}
The following are equivalent for a birational map $f:X\self$ on a complex
surface.
\begin{itemize}
\item $(f^n)^* = (f^*)^n$ for all $n\in\Z$.
\item $I(f^n)\cap I(f^{-n}) = \emptyset$ for all $n\in\N$.
\item $f^n(C) \notin I(f)$ for any $f$-exceptional curve $C$.
\item $f^{-n}(C)\notin I(f^{-1})$ for any $f^{-1}$ exceptional curve $C$.
\end{itemize}
By blowing up finitely many points in $X$, one can always arrange that
these conditions are satisfied.
\end{thm}
We will call maps satisfying the equivalent conditions of this theorem
\as (for \emph{algebraically} or \emph{analytically stable}).
If $f$ is \as, then $\lambda = \lambda(f)$ is just the spectral
radius of $f^*$. If $X$ is K\"ahler, then there is a nef class $\theta^+$
satisfying
$$
f^* \theta^+ = \lambda\theta^+.
$$
From \eqref{adjoint}
we have that $\lambda(f^{-1}) = \lambda(f)$,
so we let $\theta^-$ denote the corresponding class for $f^{-1}$. The
following theorem summarizes many of the main results of \cite{DiFa01}, and
we will rely heavily on it here.
\begin{thm}
\label{classthm}
If $f:X\self$ is an \as birational map of a complex K\"ahler surface $X$
with $\lambda(f)=1$, then exactly one of the following is true (after
contracting curves in $\supp E(f^n)$, if necessary).
\begin{itemize}
\item $\norm{f^{n*}}$ is bounded independent of $n$, and $f$ is an
automorphism some iterate of which is isotopic to the identity.
\item $\norm{f^{n*}} \sim n$ and $f$ preserves a rational fibration. In
this case $\theta^+ = \theta^-$ is the class of a generic fiber.
\item $\norm{f^{n*}} \sim n^2$ and $f$ is an automorphism preserving an
elliptic fibration. Again $\theta^+=\theta^-$ is the class of a generic fiber.
\end{itemize}
If, on the other hand, $\lambda(f) > 1$, then $\theta^+\cdot \theta^- > 0$
and either $X$ is rational or $f$ is (up to contracting exceptional curves)
an automorphism of a torus, an Enriques surface, or a K3 surface.
\end{thm}
\noindent
We remark that the classes $\theta^\pm$ are unique up to positive multiples
whenever $\norm{f^{n*}}$ is unbounded, and indeed under the unboundedness
assumption, we have
$$
\lim_{n\to\infty} \frac{f^{n*}\theta}{\norm{f^{n*}}} = c\theta^+
$$
for any K\"ahler class $\theta$ and some constant $c = c(\theta) > 0$.
In what follows, we will largely ignore the case in which $\norm{f^{n*}}$ is
bounded. After all, if some iterate of $f$ is the identity map, then every
curve in $X$ will be $f$-invariant.
To close this section, we recall a result from \cite{BeDi05a}, which we
will use in section \ref{elliptic}.
\begin{thm}
\label{criticalthm}
If $f:X\self$ is an \as birational map of a complex K\"ahler surface $X$
with $\lambda(f)>1$, then after contracting curves in $\supp E(f^n)$, we can
arrange additionally that
$\theta^+\cdot f(p) > 0$ for every $p\in I(f)$ and
$\theta^-\cdot f^{-1}(p) > 0$ for every $p\in I(f^{-1})$.
\end{thm}
| 26
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Tim Sherwood’s reign as Tottenham manager is surely at an end yet the head coach might find himself back at White Hart Lane sooner than his detractors might expect – in the opposing dug-out.
Sherwood was expected to be dismissed by Spurs chairman Daniel Levy after leading Spurs to a comfortable victory in their final league game of the campaign. The home side won thanks to a strike from Paulinho, an own-goal from Nathan Baker and a penalty from Emmanuel Adebayor, but any remaining doubts about Sherwood’s future appeared to be dispelled by Levy’s end-of-season message in the club’s match-day programme.
There was no mention whatever of the 45-year-old, even though Spurs have taken 42 points from his 22 league matches – an average of 1.91 per game. Instead, there was plenty of implied criticism from the man in charge.
“Our sense of falling short, felt by all, including the players, is based on some poor performances during the season and knowing we have not performed at the level we know we could have done,” Levy wrote.
Then, the killer line: “Even in games where we gained maximum points, our football was not always what we have come to expect and associate with our club.”
Sherwood has made mistakes during his five months in charge. He has changed players and systems too often, struggled to find a coherent tactical plan and made controversial public and private comments when he might have been wiser to keep quiet – especially when working for a board who like their managers to be seen and not heard.
Yet what did Levy and Co expect? Sherwood had never managed a first team before so, naturally, he was going to make errors. In any are of employment, who gets everything right in their first few months in a given job? Sir Alex Ferguson, Jose Mourinho and Carlo Ancelotti probably got things wrong early in their careers, too.
Best quotes from the 2013/14 Premier League season
Best quotes from the 2013/14 Premier League season
1/23 Steven Gerrard
.
2/23 Manuel Pellegrini
"I think we have a style of play, I think we are an attractive team, I think we score many goals and we are always thinking to score more goals. That to me has the same importance as winning the title. In the way we play, I am very happy to win the title this way" - Manchester City manager Manuel Pellegrini, with his team on the brink of winning the title, is pleased to have performed with such a swagger.
GETTY
3/23 Jose Mourinho
.
GETTY
4/23 Brendan Rodgers
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Sherwood came into the job in mid-season, replacing Andre Villas-Boas, to work with a group of players he had not chosen. He is believed to have been convinced by only one – Christian Eriksen – of Spurs’ seven signings last summer.
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Both Nabil Bentaleb and Harry Kane were given proper chances in the first team and showed they could prosper at the highest level. Their elevation will give hope to other young players at Tottenham, something they have lacked for many seasons. Next season, Sherwood’s successor will have the chance to improve those two, as well as Zeki Fryers, Milos Veljkovic and Alex Pritchard.
The performances against Manchester City and Liverpool were supine yet these were the strongest teams in the country over the season. Spurs were hardly the only sides to be despatched by them. Quietly, however, Sherwood fixed a traditional Spurs problem: poor performances against the Premier League’s lesser sides. Against sides outside the top six, Spurs picked up 39 points from a possible 51 under Sherwood.
This work will have been appreciated elsewhere. Both West Bromwich Albion and Aston Villa could soon be looking for new managers, while Norwich could turn to their former player to try to secure immediate promotion back to the top flight.
While Sherwood has much to learn, he is far from the managerial ingénue his critics – many of them on social media – would have us believe. He has done well enough to earn a chance elsewhere. Wherever Sherwood’s next role might be, don’t bet against him making a success of it.
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Navigation, State Parameter, and Microphysics HRT Data in GENPROI format
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\begin{document}
\title{On Laplacian energy of non-commuting graphs of finite groups }
\author{Parama Dutta and Rajat Kanti Nath\footnote{Corresponding author}}
\date{}
\maketitle
\begin{center}\small{ Department of Mathematical Sciences,\\ Tezpur
University, Napaam-784028, Sonitpur, Assam, India.\\
Emails: parama@gonitsora.com and rajatkantinath@yahoo.com}
\end{center}
\smallskip
\noindent {\small{\textbf{Abstract:} In this paper, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups.
}}
\bigskip
\noindent \small{\textbf{\textit{Key words:}} non-commuting graph, spectrum, L-integral graph, finite group.}
\noindent \small{\textbf{\textit{2010 Mathematics Subject Classification:}} 05C50, 15A18, 05C25, 20D60.}
\section{Introduction} \label{S:intro}
Let ${\mathcal{G}}$ be a graph. Let $A({\mathcal{G}})$ and $D({\mathcal{G}})$ denote the adjacency matrix and degree matrix of the graph respectively. Then the Laplacian matrix of ${\mathcal{G}}$ is given by $L({\mathcal{G}}) = D({\mathcal{G}}) - A({\mathcal{G}})$. Let $\beta_1, \beta_2, \dots, \beta_m$ be the eigenvalues of $L({\mathcal{G}})$ with multiplicities $b_1, b_2, \dots, b_m$. Then the Laplacian spectrum of ${\mathcal{G}}$, denoted by $\L-spec({\mathcal{G}})$, is the set $\{\beta_1^{b_1}, \beta_2^{b_2}, \dots, \beta_m^{b_m}\}$. The Laplacian energy of ${\mathcal{G}}$, denoted by $LE({\mathcal{G}})$, is given by
\begin{equation}\label{Lenergy}
LE({\mathcal{G}}) = \sum_{\mu \in \L-spec({\mathcal{G}})}\left|\mu - \frac{2|e({\mathcal{G}})|}{|v({\mathcal{G}})|}\right|
\end{equation}
where $v({\mathcal{G}})$ and $e({\mathcal{G}})$ are the sets of vertices and edges of the graph $\mathcal{G}$ respectively. A graph $\mathcal{G}$ is called L-integral if $\L-spec({\mathcal{G}})$ contains only integers. Various properties of L-integral graphs and $LE({\mathcal{G}})$ are studied in \cite{Abreu08,Kirkland07,Merries94}.
Let $G$ be a finite non-abelian group with center $Z(G)$. The non-commuting graph of $G$, denoted by ${\mathcal{A}}_G$ is a simple undirected graph such that $v({\mathcal{A}}_G) = G\setminus Z(G)$ and two vertices $x$ and $y$ are adjacent if and only if $xy \ne yx$. Various aspects of non-commuting graphs of different families of finite non-abelian groups are studied in \cite{Ab06,AF14,dbb10,Abd13,tal08}. Note that the complement of ${\mathcal{A}}_G$ is the commuting graph of $G$ denoted by ${\overline{\mathcal{A}}}_G$. Commuting graphs are studied extensively in \cite{amr06,iJ07,mP13,par13}. In \cite{lspac}, the authors have computed the Laplacian spectrum of the non-commuting graphs of several well-known families finite non-abelian groups. In this paper we study the Laplacian energy of those classes of finite groups.
\section{Some Computations}
In this section, we compute Laplacian energy of some families of groups whose central factors are some well-known groups.
\begin{theorem} \label{order-20}
Let $G$ be a finite group and $\frac{G}{Z(G)} \cong Sz(2)$, where $Sz(2)$ is the Suzuki group presented by $\langle a, b : a^5 = b^4 = 1, b^{-1}ab = a^2 \rangle$. Then
\[
LE({\mathcal{A}}_G) = \left(\frac{120}{19}|Z(G)| + 30\right)|Z(G)|.
\]
\end{theorem}
\begin{proof}
It is clear that $|v({\mathcal{A}}_G)| = 19|Z(G)|$. Since $\frac{G}{Z(G)} \cong Sz(2)$, we have
\[
\frac{G}{Z(G)} = \langle aZ(G), bZ(G) : a^5Z(G) = b^4Z(G) = Z(G), b^{-1}abZ(G) = a^2Z(G) \rangle.
\]
Then
\[
\begin{array}{ll}
C_G(a) &= Z(G)\sqcup aZ(G) \sqcup a^2Z(G)\sqcup a^3Z(G)\sqcup a^4Z(G),\\
C_G(ab) &= Z(G)\sqcup abZ(G) \sqcup a^4b^2Z(G)\sqcup a^3b^3Z(G),\\
C_G(a^2b) &= Z(G)\sqcup a^2bZ(G) \sqcup a^3b^2Z(G)\sqcup ab^3Z(G),\\
C_G(a^2b^3) &= Z(G)\sqcup a^2b^3Z(G) \sqcup ab^2Z(G)\sqcup a^4bZ(G),\\
C_G(b) &= Z(G)\sqcup bZ(G) \sqcup b^2Z(G)\sqcup b^3Z(G) \quad \text{ and }\\
C_G(a^3b) &= Z(G)\sqcup a^3bZ(G) \sqcup a^2b^2Z(G)\sqcup a^4b^3Z(G)
\end{array}
\]
are the only centralizers of non-central elements of $G$. Since all these distinct centralizers are abelian, we have
\[
{\overline{{\mathcal{A}}}_G} = K_{4|Z(G)|}\sqcup 5K_{3|Z(G)|}
\]
and hence $|e({\mathcal{A}}_G)| = 150{|Z(G)|}^2$. By Theorem 3.1 of \cite{lspac}, we have
\[
\L-spec({\mathcal{A}}_G) =\{ 0, {(15|Z(G)|)}^{4|Z(G)|-1}, {(16|Z(G)|)}^{15|Z(G)|-5}, {(19|Z(G)|)}^5 \}.
\]
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{300}{19}|Z(G)|$, $\left|15|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{15}{19}|Z(G)|$,\\ $\left|16|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{4}{19}|Z(G)|$, $\left|19|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{61}{19}|Z(G)|$. By \eqref{Lenergy}, we have
\begin{align*}
LE({\mathcal{A}}_G) = & \frac{300}{19}|Z(G)| + (4|Z(G)|-1)\left(\frac{15}{19}|Z(G)|\right) + (15|Z(G)|-5)\left(\frac{4}{19}|Z(G)|\right)\\
& + 5\left(\frac{61}{19}|Z(G)|\right).
\end{align*}
Hence the result follows.
\end{proof}
\begin{theorem}\label{main2}
Let $G$ be a finite group such that $\frac{G}{Z(G)} \cong {\mathbb{Z}}_p \times {\mathbb{Z}}_p$, where $p$ is a prime integer. Then
\[
LE({\mathcal{A}}_G) = 2p(p - 1)|Z(G)|.
\]
\end{theorem}
\begin{proof}
It is clear that $|v({\mathcal{A}}_G)| = (p^2 - 1)|Z(G)|$. Since $\frac{G}{Z(G)} \cong \mathbb{Z}_p\times \mathbb{Z}_p$, we have
$
\frac{G}{Z(G)} = \langle aZ(G), bZ(G) : a^p, b^p, aba^{-1}b^{-1} \in Z(G)\rangle,
$
where $a$, $b\in$ $G$ with $ab\neq ba$. Then for any $z\in Z(G)$
\begin{align*}
C_G(a) & = Z(G)\sqcup aZ(G)\sqcup \dots \sqcup a^{p - 1}Z(G) \text{ for } 1\leq i\leq p - 1 \quad \text{and} \\
C_G(a^jb) &= Z(G)\sqcup a^jbZ(G)\sqcup \dots \sqcup {a^jb}^{p - 1}Z(G) \text{ for } 1\leq j\leq p
\end{align*}
are the only centralizers of non-central elements of $G$. Also note that these
centralizers are abelian subgroups of $G$. Therefore
\[
{\overline{{\mathcal{A}}}_G} = K_{|C_G(a)\setminus Z(G)|}\sqcup (\underset{j = 1}{\overset{p}{\sqcup}} K_{|C_G(a) \setminus Z(G)|}).
\]
Since, $|C_G(a)|$ and $|C_G(a^jb)|$ for $1\leq j\leq p$, we have
\[
{\overline{{\mathcal{A}}}_G} = (p + 1)K_{(p - 1)|Z(G)|}.
\]
and hence $|e({\mathcal{A}}_G)| = \frac{p(p + 1){(p - 1)}^2}{2}{|Z(G)|}^2$. By Theorem 3.2 of \cite{lspac}, we have
\[
\L-spec({\mathcal{A}}_G) = \{ 0, {((p^2-p)|Z(G)|)}^{(p^2-1)|Z(G)|-p-1}, {((p^2-1)|Z(G)|)}^p \}.
\]
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = p(p - 1)|Z(G)|$, $\left|(p^2-p)|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = 0$ and \\ $\left|(p^2-1)|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = (p - 1)|Z(G)|$. By \eqref{Lenergy}, we have
\[
LE({\mathcal{A}}_G) = p(p - 1)|Z(G)| + ((p^2-1)|Z(G)|-p-1)0 + p((p - 1)|Z(G)|).
\]
Hence the result follows.
\end{proof}
\begin{corollary}
Let $G$ be a non-abelian group of order $p^3$, for any prime $p$, then
\[
LE({\mathcal{A}}_G) = 2p^2(p - 1).
\]
\end{corollary}
\begin{proof}
Note that $|Z(G)| = p$ and $\frac{G}{Z(G)} \cong {\mathbb{Z}}_p \times {\mathbb{Z}}_p$. Hence the result follows from Theorem \ref{main2}.
\end{proof}
\begin{theorem}\label{main4}
Let $G$ be a finite group such that $\frac{G}{Z(G)} \cong D_{2m}$, for $m \geq 2$. Then
\[
LE({\mathcal{A}}_G) = \frac{(2m^2 - 3m)(m - 1){|Z(G)|}^2 + m(4m - 3)|Z(G)|}{2m - 1}.
\]
\end{theorem}
\begin{proof}
Clearly, $|v({\mathcal{A}}_G)| = (2m - 1)|Z(G)|$. Since $\frac{G}{Z(G)} \cong D_{2m}$ we have $\frac{G}{Z(G)} = \langle xZ(G), yZ(G) : x^2, y^m, xyx^{-1}y\in Z(G)\rangle$, where $x, y \in G$ with $xy \ne yx$.
It is easy to see that for any $z \in Z(G)$
\begin{align*}
C_G(xy^j) & = C_G(xy^jz) = Z(G) \sqcup xy^jZ(G), 1 \leq j \leq m \quad \text{and} \\
C_G(y) & = C_G(y^iz) = Z(G) \sqcup yZ(G) \sqcup\cdots \sqcup y^{m - 1}Z(G), 1 \leq i \leq m - 1
\end{align*}
are the only centralizers of non-central elements of $G$. Also note that these centralizers are abelian subgroups of $G$ and $|C_G(x^jy)| = 2|Z(G)|$ for $1 \leq j \leq m$ and $|C_G(y)| = m|Z(G)|$. Hence
\[
{\overline{{\mathcal{A}}}_G} = K_{(m - 1)|Z(G)|}\sqcup mK_{|Z(G)|}.
\]
and $|e({\mathcal{A}}_G)| = \frac{3m(m - 1){|Z(G)|}^2}{2}$. By Theorem 3.4 of \cite{lspac}, we have
\begin{align*}
\L-spec({\mathcal{A}_G}) = &\{ 0 , {(m|Z(G)|)}^{(m-1)|Z(G)|-1} , {(2(m-1)|Z(G)|)}^{m|Z(G)|-m},\\
&{((2m-1)|Z(G)|)}^m \}.
\end{align*}
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{3m(m - 1)|Z(G)|}{2m - 1}$, $\left|m|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{m(m - 1)|Z(G)|}{2m - 1}$,\\ $\left|2(m-1)|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{(m - 1)(m - 2)|Z(G)|}{2m - 1}$ and $\left|(2m-1)|Z(G)| - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{(m^2 - m +1)|Z(G)|}{2m - 1}$. By \eqref{Lenergy}, we have
\begin{align*}
LE({\mathcal{A}}_G) = & \frac{3m(m - 1)|Z(G)|}{2m - 1} + ((m-1)|Z(G)|-1)\left(\frac{m(m - 1)|Z(G)|}{2m - 1}\right)\\
& + (m|Z(G)|-m)\left(\frac{(m - 1)(m - 2)|Z(G)|}{2m - 1}\right) + m\left(\frac{(m^2 - m +1)|Z(G)|}{2m - 1}\right)
\end{align*}
and hence the result follows.
\end{proof}
Using Theorem \ref{main4}, we now compute the Laplacian energy of the non-commuting graphs of the groups $M_{2mn}, D_{2m}$ and $Q_{4n}$ respectively.
\begin{corollary}\label{main05}
Let $M_{2mn} = \langle a, b : a^m = b^{2n} = 1, bab^{-1} = a^{-1} \rangle$ be a metacyclic group, where $m > 2$.
\[
LE({\mathcal{A}}_{M_{2mn}}) =
\begin{cases}
\frac{m(2m - 3)(m - 1)n^2 + m(4m - 3)n}{2m - 1}, & \mbox{if m is odd } \\
\frac{m(m - 2)(m - 3)n^2 + m(2m - 3)n}{m - 1}, & \mbox{if m is even}.
\end{cases}
\]
\end{corollary}`
\begin{proof}
Observe that $Z(M_{2mn}) = \langle b^2 \rangle$ or $\langle b^2 \rangle \cup a^{\frac{m}{2}}\langle b^2 \rangle$ according as $m$ is odd or even. Also, it is easy to see that $\frac{M_{2mn}}{Z(M_{2mn})} \cong D_{2m}$ or $D_m$ according as $m$ is odd or even. Hence, the result follows from Theorem \ref{main4}
\end{proof}
\noindent As a corollary to the above result we have the following results.
\begin{corollary}\label{main005}
Let $D_{2m} = \langle a, b : a^m = b^{2} = 1, bab^{-1} = a^{-1} \rangle$ be the dihedral group of order $2m$, where $m > 2$.
\[
LE({\mathcal{A}}_{D_{2m}}) =
\begin{cases}
m^2, & \mbox{ if m is odd } \\
\frac{m(m^2 - 3m + 3)}{m - 1}, & \mbox{if m is even }.
\end{cases}
\]
\end{corollary}
\begin{corollary}\label{q4m}
Let $Q_{4m} = \langle x, y : y^{2m} = 1, x^2 = y^m,yxy^{-1} = y^{-1}\rangle$, where $m \geq 2$, be the generalized quaternion group of order $4m$. Then
\[
LE({\mathcal{A}}_{Q_{4m}}) = \frac{2m(4m^2 - 6m + 3)}{2m - 1}.
\]
\end{corollary}
\begin{proof}
The result follows from Theorem \ref{main4} noting that $Z(Q_{4m}) = \{1, a^m\}$ and $\frac{Q_{4m}}{Z(Q_{4m})} \cong D_{2m}$.
\end{proof}
\section{Some well-known groups}
Now we compute Laplacian energy of the non-commuting graphs of some well-known families of finite non-abelian groups.
\begin{proposition}\label{order-pq}
Let $G$ be a non-abelian group of order $pq$, where $p$ and $q$ are primes with $p\mid (q - 1)$. Then
\[
LE({\mathcal{A}}_G) = \frac{2q(p^2 - 1)(q - 1)}{pq - 1}.
\]
\end{proposition}
\begin{proof}
It is clear that $|v({\mathcal{A}}_G)| = pq - 1$. Note that $|Z(G)| = 1$ and the centralizers of non-central elements of $G$ are precisely the Sylow subgroups of $G$. The number of Sylow $q$-subgroups and Sylow $p$-subgroups of $G$ are one and $q$ respectively. Therefore, we have
\[
{\overline{{\mathcal{A}}}_G} = K_{q-1} \sqcup qK_{p - 1}
\]
and hence $|e({\mathcal{A}}_G)| = \frac{q(p^2 - 1)(q - 1)}{2}$. By Proposition 4.1 of \cite{lspac}, we have
\[
\L-spec({\mathcal{A}}_G) = \{0, {(pq - q)}^{q - 2}, {(pq - p)}^{pq - 2q}, {(pq - 1)^q}\}.
\]
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{p^2q^2 - p^2q - q^2 + q}{pq - 1}$, $\left|pq - q - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{q(q - p)(p - 1)}{pq - 1}$,\\ $\left|pq - p - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{(q - p)(q - 1)}{pq - 1}$ and $\left|pq - 1 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_G)|}\right| = \frac{p^2q + q^2 - 2pq -q + 1}{pq - 1}$. By \eqref{Lenergy}, we have
\begin{align*}
LE({\mathcal{A}}_G) = & \frac{p^2q^2 - p^2q - q^2 + q}{pq - 1} + (q - 2)\left(\frac{q(q - p)(p - 1)}{pq - 1}\right) \\
& + (pq - 2q)\left(\frac{(q - p)(q - 1)}{pq - 1}\right) + q\left(\frac{p^2q + q^2 - 2pq -q + 1}{pq - 1}\right)
\end{align*}
and hence the result follows.
\end{proof}
\begin{proposition}\label{semid}
Let $QD_{2^n}$ denotes the quasidihedral group $\langle a, b : a^{2^{n-1}} = b^2 = 1, bab^{-1} = a^{2^{n - 2} - 1}\rangle$, where $n \geq 4$. Then
\[
LE({\mathcal{A}}_{QD_{2^n}}) = \frac{2^{3n - 3} - 2^{2n} + 3.2^n}{2^{n -1} - 1}.
\]
\end{proposition}
\begin{proof}
It is clear that $Z(QD_{2^n}) = \{1, a^{2^{n - 2}}\}$, so $|v({\mathcal{A}}_{QD_{2^n}})| = 2(2^{n - 1} - 1)$. Note that
\begin{align*}
C_{QD_{2^n}}(a) &= C_{QD_{2^n}}(a^i) = \langle a \rangle \text{ for } 1 \leq i \leq 2^{n - 1} - 1, i \ne 2^{n - 2} \quad \text{and}\\
C_{QD_{2^n}}(a^jb) &= \{1, a^{2^{n - 2}}, a^jb, a^{j + 2^{n - 2}}b \} \text{ for } 1 \leq j \leq 2^{n - 2}
\end{align*}
are the only centralizers of non-central elements of $QD_{2^n}$. Note that these centralizers are abelian subgroups of $QD_{2^n}$. Therefore, we have
\[
\overline{{\mathcal{A}}}_{QD_{2^n}} = K_{|C_{QD_{2^n}}(a)\setminus Z(QD_{2^n})|} \sqcup (\underset{j = 1}{\overset{2^{n - 2}}{\sqcup}} K_{|C_{QD_{2^n}}(a^jb)\setminus Z(QD_{2^n})|}).
\]
Since $|C_{QD_{2^n}}(a)| = 2^{n - 1}, |C_{QD_{2^n}}(a^jb)| = 4$ for $1 \leq j \leq 2^{n - 2}$, we have $\overline{{\mathcal{A}}}_{QD_{2^n}} = K_{2^{n - 1} - 2} \sqcup 2^{n - 2} K_2$. Hence
\[
|e({\mathcal{A}}_{QD_{2^n}})| = \frac{3.2^{2n - 2} - 6.2^{n - 1}}{2}.
\]
By Proposition 4.2 of \cite{lspac}, we have
\[
\L-spec({\mathcal{A}}_{QD_{2^n}}) = \{0, {(2^{n-1})}^{2^{n-1}-3}, {(2^n-4)}^{2^{n-2}}, {(2^n-2)}^{2^{n-2}}\}.
\]
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_{QD_{2^n}})|}{|v({\mathcal{A}}_{QD_{2^n}})|}\right| = \frac{3.2^{n - 1}(2^{n - 1} - 2)}{2.2^{n - 1} - 2}$, $\left|2^{n-1} - \frac{2|e({\mathcal{A}}_{QD_{2^n}})|}{|v({\mathcal{A}}_{QD_{2^n}})|}\right| = \frac{2^{2n - 2} - 4.2^{n - 1}}{2.2^{n - 1} - 2}$,\\ $\left|2^n-4 - \frac{2|e({\mathcal{A}}_{QD_{2^n}})|}{|v({\mathcal{A}}_{QD_{2^n}})|}\right| = \frac{2^{2n - 2} - 6.2^{n - 1} + 8}{2.2^{n - 1} - 2}$ and $\left|2^n-2 - \frac{2|e({\mathcal{A}}_{QD_{2^n}})|}{|v({\mathcal{A}}_{QD_{2^n}})|}\right| = \frac{2^{2n - 2} - 2.2^{n - 1} + 4}{2.2^{n - 1} - 2}$.
By \eqref{Lenergy}, we have
\begin{align*}
LE({\mathcal{A}}_{QD_{2^n}}) = & \frac{3.2^{n - 1}(2^{n - 1} - 2)}{2.2^{n - 1} - 2} + (2^{n-1}-3)\left(\frac{2^{2n - 2} - 4.2^{n - 1}}{2.2^{n - 1} - 2}\right) \\
& + 2^{n-2}\left(\frac{2^{2n - 2} - 6.2^{n - 1} + 8}{2.2^{n - 1} - 2}\right) + 2^{n-2}\left(\frac{2^{2n - 2} - 2.2^{n - 1} + 4}{2.2^{n - 1} - 2}\right)
\end{align*}
and hence the result follows.
\end{proof}
\begin{proposition}\label{psl}
Let $G$ denotes the projective special linear group $PSL(2, 2^k)$, where $k \geq 2$. Then
\[
LE({\mathcal{A}}_G) = \frac{3.2^{6k} - 2.2^{5k} - 7.2^{4k} + 2^{3k} + 4.2^{2k} +2^k}{2^{3k} - 2^k - 1}.
\]
\end{proposition}
\begin{proof}
Clearly, $|v({\mathcal{A}}_G)| = 2^{3k} - 2^k - 1$. Since $G$ is a non-abelian group of order $2^k(2^{2k} - 1)$ and its center is trivial. By Proposition 3.21 of \cite{Ab06}, the set of centralizers of non-trivial elements of $G$ is given by
\[
\{xPx^{-1}, xAx^{-1}, xBx^{-1} : x \in G\}
\]
where $P$ is an elementary abelian \quad $2$-subgroup and $A, \quad B$ are cyclic subgroups of $G$ having order $2^k, 2^k - 1$ and $2^k + 1$ respectively. Also the number of conjugates of $P, A$ and $B$ in $G$ are $2^k + 1, 2^{k - 1}(2^k + 1)$ and $2^{k - 1}(2^k - 1)$ respectively. Hence $\overline{{\mathcal{A}}_{G}}$ of $G$ is given by
\[
(2^k + 1)K_{|xPx^{-1}| - 1} \sqcup 2^{k - 1}(2^k + 1)K_{|xAx^{-1}| - 1} \sqcup 2^{k - 1}(2^k - 1)K_{|xBx^{-1}| - 1}.
\]
That is, $\overline{{\mathcal{A}}}_{G} = (2^k + 1)K_{2^k - 1} \sqcup 2^{k - 1}(2^k + 1)K_{2^k - 2} \sqcup 2^{k - 1}(2^k - 1)K_{2^k}$. Therefore,
\[
e({\mathcal{A}}_G) = \frac{2^{6k} - 3.2^{4k} - 2^{3k} + 2.2^{2k} + 2^k}{2}.
\]
By Proposition 4.3 of \cite{lspac}, we have
\begin{align*}
\L-spec({\mathcal{A}}_{G}) = & \{0, {(2^{3k}-2^{k+1}-1)}^{2^{3k-1}-2^{2k}+2^{k-1}}, {(2^{3k}-2^{k+1})}^{2^{2k}-2^k-2},\\
& {(2^{3k}-2^{k+1}+1)}^{2^{3k-1}-2^{2k}-3.2^{k-1}}, {(2^{3k}-2^k-1)}^{2^{2k}+2^k}\}.
\end{align*}
Now, $\left|0 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_G)|}\right| = \frac{2^{6k} - 3.2^{4k} - 2^{3k} + 2.2^{2k} + 2^k}{2^{3k} - 2^k - 1}$, $\left|2^{3k}-2^{k+1}-1 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_{G})|}\right| = \frac{2^{3k} -2.2^k - 1}{2^{3k} - 2^k - 1}$,\\ $\left|2^{3k}-2^{k+1} - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right| = \frac{2^k}{2^{3k} - 2^k - 1}$, $\left|2^{3k}-2^{k+1}+1 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right| = \frac{2^{3k} - 1}{2^{3k} - 2^k - 1}$\\ and $\left|2^{3k}-2^k-1 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right| = \frac{2 ^{4k} - 2^{3k} - 2^{2k} + 2^k + 1}{2^{3k} - 2^k - 1}$. By \eqref{Lenergy}, we have
\begin{align*}
LE({\mathcal{A}}_G) = & \frac{2^{6k} - 3.2^{4k} - 2^{3k} + 2.2^{2k} + 2^k}{2^{3k} - 2^k - 1} + (2^{3k-1}-2^{2k}+2^{k-1})\left(\frac{2^{3k} -2.2^k - 1}{2^{3k} - 2^k - 1}\right) \\
& + (2^{2k}-2^k-2)\left(\frac{2^k}{2^{3k} - 2^k - 1}\right) + (2^{3k-1}-2^{2k}-3.2^{k-1})\left(\frac{2^{3k} - 1}{2^{3k} - 2^k - 1}\right) \\
& + (2^{2k}+2^k)\left(\frac{2 ^{4k} - 2^{3k} - 2^{2k} + 2^k + 1}{2^{3k} - 2^k - 1}\right)
\end{align*}
and hence the result follows.
\end{proof}
\begin{proposition}
Let $G$ denotes the general linear group $GL(2, q)$, where $q = p^n > 2$ and $p$ is a prime. Then
\[
LE({\mathcal{A}}_G) = \frac{q^9 - 2q^8 - q^7 + 2q^6 + 2q^5 + q^4 - 4q^3 +2q^2 + q}{q^4 - q^3 - q^2 +1}.
\]
\end{proposition}
\begin{proof}
Clearly, $|v({\mathcal{A}}_G)| = q^4 - q^3 - q^2 +1$. We have $|G| = (q^2 -1)(q^2 - q)$ and $|Z(G)| = q - 1$. By Proposition 3.26 of \cite{Ab06}, the set of centralizers of non-central elements of $GL(2, q)$ is given by
\[
\{xDx^{-1}, xIx^{-1}, xPZ(GL(2, q))x^{-1} : x \in GL(2, q)\}
\]
where $D$ is the subgroup of $GL(2, q)$ consisting of all diagonal matrices, $I$ is a cyclic subgroup of $GL(2, q)$ having order $q^2 - 1$ and $P$ is the Sylow $p$-subgroup of $GL(2, q)$ consisting of all upper triangular matrices with $1$ in the diagonal. The orders of $D$ and $PZ(GL(2, q))$ are $(q - 1)^2$ and $q(q - 1)$ respectively. Also the number of conjugates of $D, I$ and $PZ(GL(2, q))$ in $GL(2, q)$ are $\frac{q(q + 1)}{2}, \frac{q(q - 1)}{2}$ and $q + 1$ respectively. Hence the commuting graph of $GL(2, q)$ is given by
\[
\frac{q(q + 1)}{2}K_{|xDx^{-1}| - q + 1} \sqcup \frac{q(q - 1)}{2}K_{|xIx^{-1}| - q + 1} \sqcup (q + 1)K_{|xPZ(GL(2, q))x^{-1}| - q + 1}.
\]
That is, ${\overline{{\mathcal{A}}}_G} = \frac{q(q + 1)}{2}K_{q^2 - 3q + 2} \sqcup \frac{q(q - 1)}{2}K_{q^2 - q} \sqcup (q + 1)K_{q^2 - 2q + 1}$. Hence $e({\mathcal{A}}_G) = \frac{q^8 - 2q^7 - 2q^6 + 5q^5 + q^4 - 4q^3 + q}{2}$. By Proposition 4.4 of \cite{lspac}, we have
\begin{align*}
\L-spec({\mathcal{A}}_G) = & \{0, {(q^4-q^3-2q^2+2q)}^{q^3-q^2-2q}, {(q^4-q^3-2q^2+q+1)}^{\frac {q^4-2q^3+q}{2}},\\ & {(q^4-q^3-2q^2+3q-1)}^{\frac {q^4-2q^3-2q^2+q}{2}}, {(q^4-q^3-q^2+1)}^{q^2+q}\}.
\end{align*}
Now, $\left|0 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_G)|}\right| = \frac{q^8 - 2q^7 - 2q^6 + 5q^5 + q^4 - 4q^3 + q}{q^4 - q^3 - q^2 +1}$, $\left|q^4-q^3-2q^2+2q - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right|$\\ $= \frac{q^3 - 2q^2 + q}{q^4 - q^3 - q^2 +1}$, $\left|q^4-q^3-2q^2+q+1 - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_{G})|}\right| = \frac{q^5 - 2q^4 - q^3 + 3q^2 - 1}{q^4 - q^3 - q^2 +1}$,\\ $\left|q^4-q^3-2q^2+3q-1 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right| = \frac{q^5 - 2q^4 + q^3 - q^2 + 2q - 1}{q^4 - q^3 - q^2 +1}$ and \\$\left|q^4-q^3-q^2+1 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right|= \frac{q^6 - 3q^5 + 2q^4 + 2q^3 - q^2 - q + 1}{q^4 - q^3 - q^2 +1}$. By \eqref{Lenergy}, we have
\begin{align*}
LE({\mathcal{A}}_G) = & \frac{q^8 - 2q^7 - 2q^6 + 5q^5 + q^4 - 4q^3 + q}{q^4 - q^3 - q^2 +1} + (q^3-q^2-2q)\left(\frac{q^3 - 2q^2 + q}{q^4 - q^3 - q^2 +1}\right)\\
& + \left(\frac {q^4-2q^3+q}{2}\right)\left(\frac{q^5 - 2q^4 - q^3 + 3q^2 - 1}{q^4 - q^3 - q^2 +1}\right) \\
& + \left(\frac{q^4-2q^3-2q^2+q}{2}\right)\left(\frac{q^5 - 2q^4 + q^3 - q^2 + 2q - 1}{q^4 - q^3 - q^2 +1}\right)
\end{align*}
and hence the result follows.
\end{proof}
\begin{proposition}\label{Hanaki1}
Let $F = GF(2^n), n \geq 2$ and $\vartheta$ be the Frobenius automorphism of $F$, i. e., $\vartheta(x) = x^2$ for all $x \in F$. If $G$ denotes the group
\[
\left\lbrace U(a, b) = \begin{bmatrix}
1 & 0 & 0\\
a & 1 & 0\\
b & \vartheta(a) & 1
\end{bmatrix} : a, b \in F \right\rbrace
\]
under matrix multiplication given by $U(a, b)U(a', b') = U(a + a', b + b' + a'\vartheta(a))$, then
\[
LE({\mathcal{A}}_G) = 2^{2n + 1} - 2^{n + 2}.
\]
\end{proposition}
\begin{proof}
It is clear that $v({\mathcal{A}}_G) = 2^n(2^n - 1)$. Note that $Z(G) = \{U(0, b) : b\in F\}$ and so $|Z(G)| = 2^n$. Let $U(a, b)$ be a non-central element of $G$. The centralizer of $U(a, b)$ in $G$ is $Z(G)\sqcup U(a, 0)Z(G)$. Hence $\overline{{\mathcal{A}}}_{G} = (2^n - 1)K_{2^n}$ and $|e({\mathcal{A}}_G)| = \frac{2^{4n} - 3.2^{3n} + 2.2^{2n}}{2}$. By Proposition 4.5 of \cite{lspac}, we have
\[
\L-spec({\mathcal{A}}_G) = \{ 0, {(2^{2n}-2^{n+1})}^{{(2^n-1)}^2}, {(2^{2n}-2^n)}^{2^n-2} \}.
\]
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_G)|}\right| = 2^{2n} - 2.2^n$, $\left|2^{2n}-2^{n+1} - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_{G})|}\right| = 0$ and \\ $\left|2^{2n}-2^n - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right| = 2^n$. By \eqref{Lenergy}, we have
\[
LE({\mathcal{A}}_G) = 2^{2n} - 2.2^n + ({(2^n-1)}^2)0 + (2^n-2)2^n
\]
and hence the result follows.
\end{proof}
\begin{proposition}\label{Hanaki2}
Let $F = GF(p^n)$ where $p$ is a prime. If $G$ denotes the group
\[
\left\lbrace V(a, b, c) = \begin{bmatrix}
1 & 0 & 0\\
a & 1 & 0\\
b & c & 1
\end{bmatrix} : a, b, c \in F \right\rbrace
\]
under matrix multiplication $V(a, b, c)V(a', b', c') = V(a + a', b + b' + ca', c + c')$,
then
\[
LE({\mathcal{A}}_G) = 2(p^{3n} - p^{2n}).
\]
\end{proposition}
\begin{proof}
Clearly, $v({\mathcal{A}}_G) = p^n(p^{2n} - 1)$. We have $Z(G) = \{V(0, b, 0) : b \in F\}$ and so $|Z(G)| = p^n$. The centralizers of non-central elements of $A(n, p)$ are given by
\begin{enumerate}
\item If $b, c \in F$ and $c \ne 0$ then the centralizer of $V(0, b, c)$ in $G$ is\\ $\{V(0, b', c') : b', c' \in F\}$ having order $p^{2n}$.
\item If $a, b \in F$ and $a \ne 0$ then the centralizer of $V(a, b, 0)$ in $G$ is\\ $\{V(a', b', 0) : a', b' \in F\}$ having order $p^{2n}$.
\item If $a, b, c \in F$ and $a \ne 0, c \ne 0$ then the centralizer of $V(a, b, c)$ in $G$ is \\$\{V(a', b', ca'a^{-1}) : a', b' \in F\}$ having order $p^{2n}$.
\end{enumerate}
It can be seen that all the centralizers of non-central elements of $A(n, p)$ are abelian. Hence,
\[
\overline{{\mathcal{A}}}_G = K_{p^{2n} - p^n}\sqcup K_{p^{2n} - p^n}\sqcup (p^n - 1)K_{p^{2n} - p^n} = (p^n + 1)K_{p^{2n} - p^n}.
\]
and $e({\mathcal{A}}_G) = \frac{p^{6n} - p^{5n} - p^{4n} + p^{3n}}{2}$. By Proposition 4.6 of \cite{lspac}, we have
\[
\L-spec({\mathcal{A}}_{A(n, p)}) = \{ 0, {(p^{3n}-p^{2n})}^{p^{3n}-2p^{n}-1}, {(p^{3n}-p^{n})}^{p^n} \}.
\]
Therefore, $\left|0 - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_G)|}\right| = p^{3n} - p^{2n}$, $\left|p^{3n}-p^{2n} - \frac{2|e({\mathcal{A}}_G)|}{|v({\mathcal{A}}_{G})|}\right| = 0$ and \\ $\left|p^{3n}-p^{n} - \frac{2|e({\mathcal{A}}_{G})|}{|v({\mathcal{A}}_{G})|}\right| = p^{2n} - p^n$. By \eqref{Lenergy}, we have
\[
LE({\mathcal{A}}_G) = p^{3n} - p^{2n} + (p^{3n}-2p^{n}-1)0 + p^n(p^{2n} - p^n)
\]
and hence the result follows.
\end{proof}
\section{Some consequences}
For a finite group $G$, the set $C_G(x) = \{y \in G : xy = yx\}$ is called the centralizer of an element $x \in G$. Let $|\cent(G)| = |\{C_G(x) : x \in G\}|$, that is the number of distinct centralizers in $G$. A group $G$ is called an $n$-centralizer group if $|\cent(G)| = n$. The study of these groups was initiated by Belcastro and Sherman \cite{bG94} in the year 1994. The readers may conf. \cite{Dutta10} for various results on these groups. In this section, we compute Laplacian energy of the non-commuting graphs of non-abelian $n$-centralizer finite groups for some positive integer $n$. We begin with the following result.
\begin{proposition}\label{4-cent}
If $G$ is a finite $4$-centralizer group, then
\[
LE({\mathcal{A}}_G) = 4|Z(G)|.
\]
\end{proposition}
\begin{proof}
Let $G$ be a finite $4$-centralizer group. Then, by \cite[Theorem 2]{bG94}, we have $\frac{G}{Z(G)} \cong {\mathbb{Z}}_2 \times {\mathbb{Z}}_2$. Therefore, by Theorem \ref{main2}, the result follows.
\end{proof}
\noindent Further, we have the following result.
\begin{corollary}
If $G$ is a finite $(p+2)$-centralizer $p$-group for any prime $p$, then
\[
LE({\mathcal{A}}_G) = 2p(p - 1)|Z(G)|.
\]
\end{corollary}
\begin{proof}
Let $G$ be a finite $(p + 2)$-centralizer $p$-group. Then, by \cite[Lemma 2.7]{ali00}, we have $\frac{G}{Z(G)} \cong {\mathbb{Z}}_p \times {\mathbb{Z}}_p$. Therefore, by Theorem \ref{main2}, the result follows.
\end{proof}
\begin{proposition}\label{5-cent}
If $G$ is a finite $5$-centralizer group, then
\[
LE({\mathcal{A}}_G) = 12|Z(G)| \text{ or } \frac{18{|Z(G)|}^2 + 27|Z(G)|}{5}.
\]
\end{proposition}
\begin{proof}
Let $G$ be a finite $5$-centralizer group. Then by \cite[Theorem 4]{bG94}, we have $\frac{G}{Z(G)} \cong {\mathbb{Z}}_3 \times {\mathbb{Z}}_3$ or $D_6$. Now, if $\frac{G}{Z(G)} \cong {\mathbb{Z}}_3 \times {\mathbb{Z}}_3$, then by Theorem \ref{main2}, we have
\[
LE({\mathcal{A}}_G) = 12|Z(G)|.
\]
If $\frac{G}{Z(G)} \cong D_6$, then by Theorem \ref{main4} we have
\[
LE({\mathcal{A}}_G) = \frac{18{|Z(G)|}^2 + 27|Z(G)|}{5}.
\]
\end{proof}
Let $G$ be a finite group. The commutativity degree of $G$ is given by the ratio
\[
\Pr(G) = \frac{|\{(x, y) \in G \times G : xy = yx\}|}{|G|^2}.
\]
The origin of the commutativity degree of a finite group lies in a paper of Erd$\ddot{\rm o}$s and Tur$\acute{\rm a}$n (see \cite{Et68}). Readers may conf. \cite{Caste10,Dnp13,Nath08} for various results on $\Pr(G)$. In the following few results we shall compute various energies of the commuting graphs of finite non-abelian groups $G$ such that $\Pr(G) = r$ for some rational number $r$.
\begin{proposition}
Let $G$ be a finite group and $p$ the smallest prime divisor of $|G|$. If $\Pr(G) = \frac{p^2 + p - 1}{p^3}$, then
\[
LE({\mathcal{A}}_G) = 2p(p - 1)|Z(G)|.
\]
\end{proposition}
\begin{proof}
If $\Pr(G) = \frac{p^2 + p - 1}{p^3}$, then by \cite[Theorem 3]{dM74}, we have $\frac{G}{Z(G)}$ is isomorphic to ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$. Hence the result follows from Theorem \ref{main2}.
\end{proof}
As a corollary we have
\begin{corollary}
Let $G$ be a finite group such that $\Pr(G) = \frac{5}{8}$. Then
\[
LE({\mathcal{A}}_G) = 4|Z(G)|.
\]
\end{corollary}
\begin{proposition}
If $\Pr(G) \in \{\frac{5}{14}, \frac{2}{5}, \frac{11}{27}, \frac{1}{2}\}$, then
\[
LE({\mathcal{A}}_G) = 9,\frac{28}{3},25 \text{ or } \frac{126}{5}.
\]
\end{proposition}
\begin{proof}
If $\Pr(G) \in \{\frac{5}{14}, \frac{2}{5}, \frac{11}{27}, \frac{1}{2}\}$, then as shown in \cite[pp. 246]{Rusin79} and \cite[pp. 451]{Nath13}, we have $\frac{G}{Z(G)}$ is isomorphic to one of the groups in $\{D_6, D_8, D_{10} D_{14}\}$. Hence the result follows from Corollary \ref{main005}.
\end{proof}
\begin{proposition}\label{order16}
Let $G$ be a group isomorphic to any of the following groups
\begin{enumerate}
\item ${\mathbb{Z}}_2 \times D_8$
\item ${\mathbb{Z}}_2 \times Q_8$
\item $M_{16} = \langle a, b : a^8 = b^2 = 1, bab = a^5 \rangle$
\item ${\mathbb{Z}}_4 \rtimes {\mathbb{Z}}_4 = \langle a, b : a^4 = b^4 = 1, bab^{-1} = a^{-1} \rangle$
\item $D_8 * {\mathbb{Z}}_4 = \langle a, b, c : a^4 = b^2 = c^2 = 1, ab = ba, ac = ca, bc = a^2cb \rangle$
\item $SG(16, 3) = \langle a, b : a^4 = b^4 = 1, ab = b^{-1}a^{-1}, ab^{-1} = ba^{-1}\rangle$.
\end{enumerate}
Then
\[
LE({\mathcal{A}}_G) = 16.
\]
\end{proposition}
\begin{proof}
If $G$ is isomorphic to any of the above listed groups, then $|G| = 16$ and $|Z(G)| = 4$. Therefore, $\frac{G}{Z(G)} \cong {\mathbb{Z}}_2 \times {\mathbb{Z}}_2$. Thus the result follows from Theorem~\ref{main2}.
\end{proof}
Recall that genus of a graph is the smallest non-negative integer $n$ such that the graph can be embedded on the surface obtained by attaching $n$ handles to a sphere. A graph is said to be planar if the genus of the graph is zero. We conclude this paper with the following result.
\begin{theorem}
Let $\Gamma_G$ be the commuting graph of a finite non-abelian group $G$. If $\Gamma_G$ is planar then
\[
LE({\mathcal{A}}_G) = \frac{28}{3} \text{ or } 9.
\]
\end{theorem}
\begin{proof}
From \cite[Proposition 2.3]{AF14}, if ${\mathcal{A}}_G$ is planner then $G \cong S_3, D_8 \text{ or } Q_8$. From Corollary \ref{main005} and Corollary \ref{q4m}, if $G\cong D_8 \text{ or } Q_8$ then $LE({\mathcal{A}}_G) = \frac{28}{3}$ and if $G \cong S_3$ then $LE({\mathcal{A}}_G) = 9$.
\end{proof}
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779 – Sunday 16th July 2017: Preparations
The big day has come – the start of Summer Euro 2017 (not footie) mini tour! Finished laundrifying at Royal Rothbury, filled with LPG in Guildford and started to a wild camping spot on the coast near Folkestone. Well, the lanes got narrower and narrower (are we in Sicilia?) – several farm vehicles and no suitable sites – so, back to good old proper aire at Canterbury New Dover Road… and only £4.00.
Zzzzzs for the night./
780 – Monday 17th July: And We’re Off
The train terminal was busy even without the school hols – this week to commence. La Belle France – joy – straight on to free motorways. Robin and Kensey awaited (Oscar’s latest diary preempted this – he couldn’t wait…). K and R went to town and wined themselves (K: Only 1 glass – a rather pleasant little Bourgone Aligote). Supper – and evening saw 30+ motorhomes in situ!!! Obviously the place to overnight close to Calais.
781 – Tuesday 18th July: Bearing Up in Bergues
Nice early run – trying to up the mileage (or kilometerage) – successfully… Coffee in town and then R and Kensey left for Calais. So nice here, we decided to stay until tomorrow – knitting, reading and resting under the awning. Parfait! Wouldn’t normal roll out the awning on an aire, but it was blinkin hot and quite a few had done so. K even got the twin tub out … we will need to replenish our water soon. So warm that the laundry all dried within hours. This sun is what we came for after the cold and blowy Scottish islands.
782 – Wednesday 19th July: Popping into Poperinge
Mini shop, diesel fill, an aire on the border with 100ltr water for ERU3, and off to Poperinge we popped to a nice compact and central car park that takes motorhomes for a max of 48 hours. This town is a the centre of a hop growing region and WW1 troop transit and field hospital areas. We visited a local bar for Belgian beer (a new experience – the beer not the bar – we passed the ‘Bar’ exams years ago!). Are we “Baristas” or just old socks, sorry – soaks. The Talbot House provided front line troops with a haven – a piece of home for a few days – chapel on the top level. The UK couple who volunteer there for short periods said that people have detected a ‘presence’ in the house occasionally – not surprising as many of the young men who visited never came back.
The local brewery museum was fascinating – hundreds of beer varieties – up to 12% alcohol by volume!!!
We took turns at the Hop Museum to hop (!) knob with the rather merry locals.
The original building was the hop drying building. The museum thoroughly explained the whole process using an audio guide.
Talbot House: Originally a private house, it was rented to provide a sanctuary for all soldiers, regardless of class or rank. The strongest beverage served was tea. The towns streets apparently were awash with drunken brawls and prostitutes.
Will the cowboy in the reflection, take heed of these rules?
The Chapel up some steep stairs.
The rear garden, which at times had soldiers sprawled everywhere. A real sense of peace. Tea and coffee served by volunteers, who also provide breakfast for overnight guests.
Check out our different cafe choices! Not a pang of regret on either side.
783 – Thursday 20th July: Bopping in Poperinge
Walking in a town park, K spotted a notice for a party tonight in the same park – instant decision to stay an extra night – because we can! Obligatory coffee in town square. The Chinese Labour Museum told the highly unusual tale of almost 95,000 Chinese labourers recruited by the British army to work in Belgium on the war effort – most evocative stories through original photographs. Now , we don’t really mind rain (San Sebastian, Gibraltar, et al) but it was a great excuse to shelter whilst having a good lunch and sampling some more Belgian socks – damn, that old soaks thing again – beer! Wunderbar….
The Park Party was – incredible! Loads of families – beer ,food and good chatting to locals – inspired by our gentle “hund” in Flemish – dog, Oscar. It was ever thus. Professional dancers led and taught many local couples in Swing, Lindy Hop, Waltz and Jive. Again we commented how mad UK Health and Safety has gone. Candles lit on table in jam jars, logs burning in open braziers and beer served in real glass! We left at 1030-ish still in some daylight.
We are really coming around to the Belgian people – lots have been extremely friendly and helpful and they are super dog friendly … Oscar is allowed in museums! One local lady gave a recommendation to visit the Trappist Monastery – makers of world class beer – and holy men too! That’s our destination sorted for tomorrow then!
Mister Ghylbe satirising the long standing cloth wars with Ypres, Ghent and Brugges – riding a donkey backwards with spoons for spurs.
Quite sleepy now and hard to envisage all the troops marching through.
We saw the jail where rowdy soldiers were incarcerated with their graffiti carved onto the walls. It also was the place of the last night for 4 soldiers who were executed in the square of the town hall. Overall 3,080 British soldiers were sentenced to death during WW!, but ‘only’ 346 executions took place. 90% of them have since been pardoned with posthumous apologies to families. 600 men were executed by the French (not sure of this includes the 2 random men shot per regiment when they refused to advance after the awful losses at Verdun), only 48 German and 750 Italians. No recorded executions for the Australian army as they so not use it as a form of military punishment. Many of the deserters were put back with their regiments, who were unsympathetic. 77% of the executed soldiers were for desertion. Other crimes were cowardice (5%), mutiny, insubordination, falling asleep on your watch, throwing away your weapon and striking a superior officer.
This poem translates as:
Light, bleak dawn.
The worn out night bursting open in my chest and fading.
My hands holding the glass –my last one.
The priest bringing his God, the doctor his opiates.
Mother of God.
Out there she’s warming her feet against the coal.
Out there she’s turning in her sleep.
Do not aim at me, lads.
Aim at the white cloth on my chest.
Light, bleak light etching words, bare words in the wall. Erwin Mortier
This was the actual wall. We only knew this as we ear wigged a knowledgable English chap doing a tour for some friends. He’d served and lost friends in Afghanistan, been a policeman in Guildford and now worked for United National Peace Keeping.
Back to beer as an aperitif … comparing Blonde and Brun Leffe.
Our own Strictly Come Dancing – they got quite few folk up learning steps.
Some locals were in shorts and a light jacket … J and I, from more Northern climes, had full winter layers. Oscar coming up for a tummy rub!
No, it’s a full sit on.
784 – Friday 21st July: Must Taste – Best Beer in the World
We visited the Lijssenhoek Military Cemetery and we were immediately saddened to see the grave memorials – from age 19 to 42 years young. So many lives lost in this “War to end all Wars’ – did it? No, tragically. A modern visitor centre with audio witness accounts … very moving. A famous war poem comes to mind…
“They shall grow not old, as we that are left grow old
Age shall not weary them, nor the years condemn
At the going down of the sun and in the morning
we will remember them.” Robert Laurence Binyon
We arrived at a brand new aire in Westvleteren near the above recommended Saint Sixtus Abbey Trappist Brewery/Monastery….. Euro12 with electric is reasonable value – highly recommended, as only 1.9km from the beer. We supped prime Trappist beer at the Cafe across the road from the Monastery – the Brewery and Monastery are closed visitors. They only brew enough beer to support themselves, and yet the 10.2% proof version is in high demand having been voted the best beer for several years. It is only sold commercially the the cafe or via the small shop there. Today the shop had signs up “No beer for sale today” and reading Trip Advisor, you have to be quick when it is available as queues form quickly. If you really want to buy some, you can ring the hot line, but the monks only answer it occasionally and book for months ahead to come to the gate to collect your two cases limit.
11,800 + graves.
Some countries repatriate their fallen .. French gaps on the left.
Only 6 breweries in Belgium brew Trappist beer and it has to be on the premises. Bottles are unlabelled, but the caps are colour coded and contain legal information.
J went for the blond and I threw cation to the wind and went for the Westvleteren 12 … 33 ml bottles so equivalent to a third of a bottle of wine.
Spotted a Belgian Welshie … we saw another one in Ypres, but he went for Oscar … must be a breeding nest of them nearby.
On the camper stop, steps made out of the beer crates.
We found the beer to strong so think you very brave, sticking to sherry her in London. Will be following Euro 2017 with enthusiasm and wish you both a wonderful journey.
Lovely thank you.xxx
Sent from my iPad
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Bought this with the top for vacation to HI. Very nice fit, comfortable, and did not ride-up at all. I'm 5'1" and 135, and my belly isn't very flat, but this suit doesn't make me feel self-conscious about it! I'm not crazy about the decorative twisty belt part because it's too big and I wish it was stitched to the suit, so it hugged my body; the knot looks lumpy under a shirt/skirt/dress. But, that's minor considering the suit is a great fit otherwise..
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Pingback: Time to Harvest Vegetables! « Gardora.net
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TITLE: Find the higher order expansion for det(I+ϵA) where ϵ is small
QUESTION [0 upvotes]: For small $\epsilon$ I want to prove
\begin{align}
\det(I+\epsilon A) = 1 + \epsilon tr(A) + \frac{\epsilon^2}{2} \left((tr(A))^2 - tr(A^2) \right) + O(\epsilon^3)
\end{align}
Furthermore I want to know how to calculate higher order terms in general.
For the linear order I found one in Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.
How about in general high order terms?
REPLY [1 votes]: Since $\frac12\left(\operatorname{tr}^2(A)-\operatorname{tr}(A^2)\right)
=\frac12\left[(\sum_i\lambda_i)^2-\sum_i\lambda_i^2\right]=\sum_{i_1<i_2}\lambda_{i_1}\lambda_{i_2}$ and
\begin{aligned}
&\det(I+\epsilon A)=\prod_{i=1}^n(1+\epsilon\lambda_i)\\
&=1
+\epsilon\operatorname{tr}(A)
+\epsilon^2\left(\sum_{i_1<i_2}\lambda_{i_1}\lambda_{i_2}\right)
+\cdots
+\epsilon^{n-1}\left(\sum_{i_1<i_2<\cdots<i_{n-1}}\lambda_{i_1}\lambda_{i_2}\cdots\lambda_{i_{n-1}}\right)
+\epsilon^n\det(A),
\end{aligned}
the result follows. In theory, one may write the coefficients $\sum_{i_1<i_2<\cdots<i_k}\lambda_{i_1}\lambda_{i_2}\cdots\lambda_{i_k}$ in terms of the traces of the powers of $A$ by using Newton's identities, but for higher-order terms, such trace expressions can be quite complicated.
| 199,277
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TITLE: What actually is the idea behind the condensed mathematics?
QUESTION [26 upvotes]: Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead different expressions of the same concepts (similarly to the way in which different human languages can express the same thing). -wiki
Before I learned that category theory is said to be the unifying idea of mathematics which is based on noting that generally the act of studying a mathematical object is equivalent to studying it's relation with all other objects in the category of that mathematical object. So, what exactly is the idea in condensed mathematics which helps us unify mathematics beyond how we do it in Category theory?
REPLY [36 votes]: I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written that you can get some sense of the main idea just from studying the first few pages.
The starting point is the observation that the traditional way of endowing something with both a topology and an algebraic structure has some shortcomings. The simplest example is that topological abelian groups do not form an abelian category. We know from long experience with category theory that an excellent indication of whether you have a "good" definition of an object is that the category of all your objects (and the maps between them) has nice properties, and topological abelian groups fail to have some of those nice properties. In condensed mathematics, the category of topological abelian groups is replaced by the category of condensed abelian groups, which is an abelian category.
Although the goal is easy enough to state and motivate, the method of achieving it was initially not obvious even to Scholze, one of the architects (along with Clausen) of condensed mathematics. A key role is played by a category that might seem unpromising at first glance: the category $\mathcal{S}$ of profinite sets, a.k.a. totally disconnected compact Hausdorff spaces (or Stone spaces), with finite jointly surjective families of maps as covers. Most people's first impression of totally disconnected spaces is that they're weird, and they have some trouble even thinking of examples other than the discrete topology. However, categorically speaking, $\mathcal{S}$ has some nice properties. A condensed abelian group, roughly speaking, a special kind of (contravariant) functor from $\mathcal{S}$ to the category of abelian groups (somewhat more precisely, it is a sheaf of abelian groups on $\mathcal{S}$). That is, the way a topological structure is imposed on abelian groups is not in the classical way (i.e., by taking a set and defining a group operation and a topology on it in isolation), but by taking certain functors from this funny-looking topological category $\mathcal{S}$ to your algebraic category.
There's nothing about this story that is peculiar to abelian groups; by replacing "abelian group" with "set" or "ring" you get condensed sets and condensed rings and so forth.
The benefit of this shift from classical structures to condensed structures is not just aesthetic. One of the nicest applications is that it leads to new proofs of certain classical results in algebraic geometry. A longstanding puzzle (if you want to call it that) is that certain theorems about complex varieties that "feel algebraic" seem to be provable only via "transcendental methods"; i.e., by invoking analysis in a seemingly essential way. Condensed mathematics provides new proofs of some of these classical theorems that are more algebraic. See Condensed Mathematics and Complex Geometry by Clausen and Scholze for more details.
| 160,121
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TITLE: A (possible) error in Ahlfors' text (analytic continuation along arcs)
QUESTION [4 upvotes]: In Ahlfors' complex analysis text, page 289 he discusses analytic continuation along arcs. I will start with some background
Let $\mathbf{f}$ be a global analytic function, with corresponding Riemann surface $\mathfrak{S_0} (\mathbf{f})$ (which is a connected a component of the sheaf $\mathfrak{S}$ of germs of analytic functions in the complex plane). Ahlfors defines an analytic continuation, along an arc $\gamma:[a,b] \to \mathbb C$ to be a continuous function $\overline{\gamma}:[a,b] \to \mathfrak{S_0}(\mathbf{f})$, such that $\pi \circ \bar{\gamma}=\gamma$, where $\pi$ is the standard projection map.
Later in the same page, he investigates the case where analytic continuation is impossible along an arc $\gamma:[a,b] \to \mathbb{C}$, starting at the initial germ $\mathbf{f}_{\zeta(a)}$ (I think that's a typo, and should read $\mathbf{f}_{\gamma(a)}$ instead). He notes that analytic continuation is possible if we restrict ourselves to subarcs $\gamma \big|_{[a,t_0]}$ for small enough $t_0$. Next, he considers the supremum of the set (which I'll call $E$) of all such numbers $t_0$, and denotes it $\tau$.
The following claims are made about $\tau$, which I'm having trouble agreeing with:
$a<\tau<b$
continuation will be possible for $t_0 < \tau$, impossible for $t_0 \geq \tau$.
Regarding the first claim, I agree that $a<\tau$, since $E$ contains numbers greater than $a$ - however, I can't see why $\tau$ is strictly lesser than $b$.
Regarding the second claim, I agree that analytic continuation is possible for $t_0<\tau$ since these are members of $E$. I agree also that analytic continuation is impossible for $t_0>\tau$, by the properties of the supremum - however, why is analytic continuation never possible for the point $t_0=\tau$ itself?
Please help me settle this. Thanks!
REPLY [1 votes]: Regarding $a < \tau < b$, yes, it is absolutely possible that $\tau = b$. That's a mistake, it should be $a < \tau \leqslant b$.
As for
however, why is analytic continuation never possible for the point $t_0 = \tau$ itself
if you have $\overline{\gamma}(t) \in \mathfrak{S}_0$, then that is a germ of an analytic function, and that means there is an anaytic function in a neighbourhood $U$ of $\gamma(t)$ whose germ in $\gamma(t)$ is $\overline{\gamma}(t)$, and that function allows extending $\overline{\gamma}$ a little bit beyond $t$, since $\gamma([t,t+\varepsilon]) \subset U$ for small enough $\varepsilon > 0$.
Concerning the $\mathbf{f}_{\zeta(a)}$ issue, I think Ahlfors uses $\zeta(a)$ to denote a generic point above $\gamma(a)$. Note that the surface may have several points above any point of the plane, so one needs to specify which point of $\mathfrak{S}_0$ one is talking about. It would be desirable if that was made entirely clear.
| 211,777
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July 10, 2016
I took the suggestion from another review. I had my deep fryer on 375 and cooked the hot dogs 3 minutes 20 second. They came out fantastic. My autistic son even ate them doing his usual thing of cutting off anything charred. I ate mine with fresh diced onions and BBQ sauce. Gave the bacon a smokey flavor. I also used soft French rolls to handle a larger dog and to hold any condiments applied. served with baked beans and fries. Definitely will make again.
Fried Bacon Wrapped Hot Dog
8
10 Ratings
| 31,939
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\begin{document}
\title{Polynomial Invariants, Knot Homologies, and Higher Twist Numbers of Weaving Knots $W(3,n)$}
\author{
Rama Mishra
\thanks{We thank the office of the Dean of the College of Arts and Sciences,
New Mexico State University, for arranging a visiting appointment.}
\\
Department of Mathematics
\\
IISER Pune
\\
Pune, India
\and
Ross Staffeldt
\thanks{We thank the office of the Vice-President for Research, New Mexico
State University, for arranging a visit to IISER-Pune to discuss
implementation of an MOU between IISER-Pune and NMSU.}
\\Department of Mathematical Sciences\\
New Mexico State University\\
Las Cruces, NM 88003 USA}
\maketitle
\begin{abstract}
We investigate several conjectures in
geometric topology by assembling computer data obtained by studying
weaving knots, a doubly infinite family $W(p,n)$ of examples of
hyperbolic knots.
In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass $W(3,n)$ of this
family.
We use these knot invariants to conclude that all knots $W(3,n)$ are fibered
knots and provide estimates for some geometric invariants of these
knots. Finally, we study the asymptotics of the ranks of their Khovanov homology groups.
Our investigations provide evidence for our conjecture that, asymptotically as $n$ grows large,
the ranks of Khovanov homology groups of $W(3,n)$ are normally distributed.
\end{abstract}
\section{Introduction} \label{Introduction}
Distinguishing knots and links up to ambient isotopy is the central problem in knot theory.
The recipe that a knot theorist uses is to compute some knot invariants and see if one of them can be of help.
Classically one used the Alexander polynomial, which has a topological
definition
\cite[p.160]{Rolfsen}, to distinguish knots.
Over the last 35 years tremendous progress has been made in the development of several new knot
invariants, starting with the Jones polynomial and the HOMFLY-PT polynomial
\cite{Jones_poly86}.
Recently even more sophisticated invariants such as Heegard-Floer homology groups \cite{Manolescu}
and Khovanov homology groups \cite{BarNatan_Categorification} have been added to the toolkit.
These homology theories are called {\it categorifications} of polynomial invariants, because
passage to an appropriate Euler characteristic retrieves a polynomial invariant.
Though these invariants are well understood to a great extent,
it is very difficult to compute them for a given knot whose crossing number is only moderately large.
Moreover, some of these invariants yield a very large amount of numerical data, and the problem arises of
parsing, or summarizing, the data in a reasonable way.
Since the Alexander polynomial can clearly be related to topology, it is natural to want to relate the
new invariants to topological or geometric features of knots or links.
In the 1980s William Thurston's seminal result
\cite[Corollary 2.5]{Thurston82}
that most knot complements have
the structure of a hyperbolic manifold,
combined with Mostow's rigidity theorem
\cite[Theorem 3.1]{Thurston82}
giving uniqueness of such structures,
establishes a strong connection between hyperbolic geometry and knot theory, since knots are determined by their complements.
Indeed, any geometric invariant of a knot complement, such as the hyperbolic volume, becomes a topological invariant of the knot.
Thus, investigating if data derived from the new knot invariants is related to natural differential
geometric invariants becomes another natural problem.
In this paper, we take up the family of weaving
knots $W(p,n)$, where $p$ and $n$ are positive integers. We compute the
signature for the general knot $W(p,n)$, and compute the polynomial knot invariants for the subfamily $W(3,n)$.
We explore the two problems just mentioned using our computations.
In particular, Dasbach and Lin \cite{DasbachLin} have provided some bounds on the hyperbolic volume of alternating knots
in terms of coefficients of the Jones polynomial. They also define {\it higher twist numbers} for knots in terms of coefficients
of the Jones polynomials and suggested that these may have some correlation with the hyperbolic volume of the knots.
We investigate this idea for the $W(3,n)$ knots. As for parsing the enormous amount of numerical information
yielded by our methods, we explore the approximation of normalized Khovanov homology by normal distributions
from mathematical statistics. A preprint \cite{Distributions} developing this idea further is in preparation.
Let us pause for more explanation of our decision to focus on weaving knots.
According to \cite{Weaving_vol}, these knots have recently attracted interest,
because it was conjectured that their complements would have the largest hyperbolic volume for a fixed crossing number.
Concerning the conjecture about the volume, we cite the following theorem.
\begin{theorem*}
[Theorem 1.1, \cite{Weaving_vol}]
If $p \geq 3$ and $q \geq 7$, then
\begin{equation} \label{CKPbounds}
v_{{\rm oct}}(p-2)\,q\,\biggl(1 - \frac{(2\pi)^2}{q^2}\biggr)^{3/2} \leq {\rm vol}(W(p,q))
< \bigl(v_{{\rm oct}}(p-3) + 4\,v_{{\rm tet}}) q.
\end{equation}
\end{theorem*}
Here $v_{{\rm oct}}$ and $v_{{\rm tet}}$ denote the volumes of the ideal octahedron and ideal tetrahedron respectively.
Champanerkar, Kofman, and Purcell call these bounds asymptotically sharp because their ratio approaches 1,
as $p$ and $q$ tend to infinity. Since the crossing number of $W(p,q)$ is known to be $(p{-}1)q$, the volume
bounds in the theorem imply
\begin{equation*}
\lim_{p,q \to \infty}\frac{{\rm vol}(W(p,q))}{c(W(p,q))} = v_{{\rm oct}} \approx 3.66
\end{equation*}
Their study raises the general question of examining the asymptotic behaviour of other
invariants of weaving knots. We also investigate the efficiency of the upper and lower bounds given in the theorem
for weaving knots $W(3,n)$.
Turning to practical matters, here is the weaving knot $W(3,4)$.
\begin{figure}[h!]
\begin{equation*}
\xygraph{ !{0;/r1.5pc/:}
!{\hcap}[u]
!{\hcap[3]}[u]
!{\hcap[5]}[llllllll]
!{\xcaph[-8]@(0)}[dl]
!{\xcaph[-8]@(0)}[dl]
!{\xcaph[-8]@(0)}[uul]
!{\hcap[-5]}[d]
!{\hcap[-3]}[d]
!{\hcap[-1]}[d]
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
}
\end{equation*} \vspace{-8em}
\end{figure}
\newline
Enumerating strands $1, \ldots, p$ from the outside inward, our example is
the closure of the braid $(\sigma_1 \sigma_2^{-1})^4$ on three strands.
The weaving knot $W(p,n)$ is obtained from the torus knot $T(p,n)$
by making a standard diagram of the torus knot alternating. Symbolically, $T(p,n)$ is the
closure of the braid $(\sigma_1 \sigma_2 \cdots \sigma_{p-1})^n$, and
$W(p,n)$ is the closure of the braid $(\sigma_1 \sigma_2^{-1} \cdots \sigma_{p-1}^{\pm 1})^n$.
Obviously, the parity of $p$ is important. If the greatest common divisor $\gcd(p,q) > 1$,
then $T(p,n)$ and $W(p,n)$ are both links with $\gcd(p,n)$ components. In general we
are interested only in the cases when $W(p,n)$ is an actual knot.
The first invariant that we compute for $W(p,n)$ is the signature $\sigma(W(p,n))$
using a combinatorial method useful for alternating knot diagrams discussed
in \cite{Lee_Endo04}. We then focus on the knots $W(3,n)$ which are closures of $3$-strand braids.
From early in the development of Khovanov homology, computer experimentation has played a role in advancing the theory.
For example, \cite{BarNatan_Categorification} provided {\em Mathematica} routines to
compute Khovanov homology of knots of up to 10 crossings and provided tables of Betti numbers. Based on the computations,
he makes a number of observations and conjectures about the structure of Khovanov homology.
Subsequently, \cite{Khovanov_Patterns} recorded some observations about patterns in Khovanov homology and a remarkable
relationship between the volume of a knot complement and the total rank of the knot's Khovanov homology. Some of the
conjectures on patterns are proved in \cite{Lee_Endo04}, on which our results depend.
In this paper we start a study of the asymptotic behavior of Khovanov homology of weaving knots. With the assistance
of the computer algebra system {\em Maple}, we provide data on
the Khovanov homology of weaving knots $W(3,n)$ with up to 652 crossings. The Khovanov homology groups are truly enormous,
so we approximate the distribution of dimensions using probability density functions. We find that
normal distributions fit the data surprisingly well.
In more detail, this paper is organized as follows.
In section \ref{Weaving}, we discuss the generalities of weave knots and compute the signature $\sigma(W(p,n))$.
We also make observation on Rasmussen's invariant \cite{Rasmussen} for these knots.
In section \ref{Hecke} we prepare to follow the development of polynomial invariants in \cite{Jones_poly86}, starting from
representations of braid groups into Hecke algebras. For weaving knots $W(3,n)$, which are naturally represented as the closures
of braids on three strands, we develop recursive formulas for their representations in the Hecke algebras.
We note that the survey \cite{BirmanMenasco} collects a number of facts and tools to facilitate computations
of invariants of knots and links that are the closures of 3-strand braids, as well as classifying the prime knots that are
closure of 3-strand braids but not of 2-strand braids. We would also like to point out that
\cite{Takahashi} studies the Jones polynomials of knots that are the closures of general 3-strand braids,
but the method is based on the skein relation satisfied by the Jones polynomial.
Our formulas are used not only
in computer calculations of the polynomial invariants we need, but also in the development of information about particular
coefficients of these polynomials.
Section \ref{Polys} builds on the recursion formulas
to develop information about the Alexander polynomial $\Delta_{W(3,n)}(t)$ and Jones polynomial $V_{W(3,n)}(t)$.
As an application we exploit the relation between the Heegard-Floer homology associated to an alternating knot
and its Alexander polynomial given in \cite[Theorem 1.3]{OSFloer},
as well as properties of Heegard-Floer homology,
to prove that the complements of the knots $W(3,n)$ are fibered over $S^1$.
In section \ref{TwistNumbersVolume} we investigate for knots $W(3,n)$ the higher twist numbers defined by Dasbach and Lin
in \cite{DasbachLin} in terms of the Jones polynomial. They ask if there is a correlation between the values of the higher
twist numbers and the hyperbolic volume of the knot complements. As new results we have formulas for the second and
third twist numbers of the knots $W(3,n)$, as well as conjectures for even higher twist numbers. We believe that
improved lower bounds on the volume of knots $W(3,n)$ can be derived from the higher twist numbers, displaying
results of computer experiments to support this idea. We also exhibit plots of higher twist numbers against volume
that support the idea that values of higher twist numbers are correlated with volume.
In section \ref{Jones-to-Khovanov} we explain how to obtain the two-variable Poincar\'{e} polynomial for Khovanov homology
with rational coefficients, and
we present the results of calculations in a few relatively small examples.
Using recent results of Shumakovitch \cite{Khovanovtorsion} that explain how, for alternating knots, the rational Khovanov homology
determines the integral Khovanov homology, we provide a display of the integral Khovanov homology of the knot $W(3,4)$.
Concerning rational Khovanov homology, we observe that the distributions of
dimensions in Khovanov homology resemble normal distributions. We explore this further in section \ref{Data}, where we present
tables displaying summaries of calculations for weaving knots $W(3,n)$ for selected values of $n$ satisfying $\gcd(3, n) = 1$
and ranging up to $n=326$. The standard deviation $\sigma$ of the normal distribution we attach to the Khovanov homology
of a weaving knot is a significant parameter. The geometric significance of this number is an open question.
In section \ref{PolynomialsExtra} we display expressions for a few polynomials arising from the Hecke algebra representations
of braid representations of $W(3,n)$, as they are used in section \ref{TwistNumbersVolume}, and values of the Jones polynomial,
Alexander polynomial and HOMFLY-PT polynomial for knots $W(3,4)$, $W(3,5)$, $W(3,10)$ and $W(3, 11)$. Finally, in section \ref{ComputerNotes}
we provide some information about the computer experiments we have performed with the knots $W(3,n)$ and how the experimental
results influenced the formulations of propositions and theorems.
Finally, we thank several colleagues, especially Dr.~Joan Birman, for advice and suggestions concerning the exposition.
\section{Generalities on Weaving knots} \label{Weaving}
We have already mentioned that weaving knots are alternating by definition.
Various facts about alternating knots facilitate our calculations of the Khovanov homology of weaving knots $W(3,n)$.
For example, we appeal first to the following theorem of Lee.
\begin{theorem}[Theorem 1.2, \cite{Lee_Endo04}] \label{locateKHLee}
For any alternating knot $L$ the Khovanov invariants
${\mathcal H}^{i,j}(L)$
are supported in two lines
\begin{equation*}
j = 2i -\sigma(L) \pm 1. \qed
\end{equation*}
\end{theorem}
We will see that this result also has several practical implications.
For example, to obtain a vanishing result for a particular alternating
knot, it suffices to compute the signature.
Likewise, in connection with Heegard-Floer homology for $W(p,n)$,
\cite[Theorem 1.3]{OSFloer}
essentially says that Heegard-Floer homology for an alternating knot is completely determined
by the coefficients of its Alexander polynomial and the signature.
Thus, it is important to compute the signature.
Indeed, it turns out that there is a combinatorial formula for the
signature of oriented non-split alternating links. To state the formula requires the following terminology.
\begin{definition} \label{crossings}
For a link diagram $D$ let $c(D)$ be the number of crossings of $D$, let $x(D)$ be number of
negative crossings, and let $y(D)$ be the number of positive crossings. For an
oriented link diagram, let $o(D)$ be the number
of components of $D(\emptyset)$, the diagram obtained by $A$-smoothing every crossing.
\end{definition}
\vspace{-3em}
\begin{figure}[h!]
\begin{minipage}[h!]{0.5\linewidth}
\centering
\begin{equation*}
\UseComputerModernTips \xygraph{ !{0;/r3pc/:}
!{\htwist=>}[rr]
!{\htwistneg=>}[rr]
}
\end{equation*}
\caption{Positive and negative crossings}
\end{minipage}
\begin{minipage}[h!]{0.5\linewidth}
\centering
\begin{equation*}
\UseComputerModernTips \xygraph{ !{0;/r3pc/:}
!{\huntwist}[rr]
!{\vuntwist}
}
\end{equation*}
\caption{$A$-smoothing a positive, resp., negative, crossing}
\end{minipage}
\label{fig:crossings_smoothings}
\end{figure}
In words, $A$-regions in a neighborhood of a crossing are the regions swept out as the upper strand sweeps
counter-clockwise toward the lower strand. An $A$-smoothing removes the crossing to connect these regions.
With these definitions, we may cite the following proposition.
\begin{proposition}[Proposition 3.11, \cite{Lee_Endo04}]
\label{basic_signature}
For an oriented non-split alternating link $L$ and a reduced alternating diagram $D$ of $L$,
$\sigma(L) = o(D) - y(D) -1$. \qed
\end{proposition}
We now use this result to compute the signatures of weaving knots.
For a knot or link $W(m, n)$ drawn in the usual way, the number of crossings $c(D) = (m{-}1)n$.
In particular, for $W(2k{+}1,n)$, $c\bigl(W(2k{+}1,n)\bigr)= 2kn$; for $W(2k, n)$, $c\bigl( W(2k, n) \bigr) = (2k{-}1)n$.
Evaluating the other quantities in definition \ref{crossings}, we calculate the signatures of weaving knots.
\begin{proposition} \label{weavingsignature}
For a weaving knot $W(2k{+}1,n)$, $\sigma\bigl( W(2k{+}1,n) \bigr) = 0$,
and for $W(2k, n)$, $\sigma\bigl( W(2k, n) \bigr) = -n{+}1$.
\end{proposition}
\begin{corollary}
For a weaving knot $W(2k{+}1, n)$, the Rasmussen $s$-invariant is zero. For
a weaving knot $W(2k, n)$, the Rasmussen $s$-invariant is $-n{+}1$.
\end{corollary}
\begin{proof}
For alternating knots, it is known that the $s$-invariant coincides with the
signature \cite[Theorems 1--4]{Rasmussen}.
\end{proof}
\begin{figure}[h!]
\centering
\begin{equation*}
\xygraph{ !{0;/r1.5pc/:}
!{\hcap}[u]
!{\hcap[3]}[u]
!{\hcap[5]}[llllllll]
!{\xcaph[-8]@(0)}[dl]
!{\xcaph[-8]@(0)}[dl]
!{\xcaph[-8]@(0)}[uul]
!{\hcap[-5]}[d]
!{\hcap[-3]}[d]
!{\hcap[-1]}[d]
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}
}
\end{equation*} \vspace{-9em}
\caption{The weaving knot $W(3,4)$}
\label{fig:w34}
\end{figure}
\begin{figure}[h!]
\centering
\begin{equation*}
\xygraph{ !{0;/r1.5pc/:}
!{\hcap}[u]
!{\hcap[3]}[u]
!{\hcap[5]}[llllllll]
!{\xcaph[-8]@(0)}[dl]
!{\xcaph[-8]@(0)}[dl]
!{\xcaph[-8]@(0)}[uul]
!{\hcap[-5]}[d]
!{\hcap[-3]}[d]
!{\hcap[-1]}[d]
!{\xcaph[1]@(0)}[dl]
!{\huntwist}[u]
!{\huncross}[ddl]
!{\xcaph[1]@(0)}[uu]
!{\xcaph[1]@(0)}[dl]
!{\huntwist}[u]
!{\huncross}[ddl]
!{\xcaph[1]@(0)}[uu]
!{\xcaph[1]@(0)}[dl]
!{\huntwist}[u]
!{\huncross}[ddl]
!{\xcaph[1]@(0)}[uu]
!{\xcaph[1]@(0)}[dl]
!{\huntwist}[u]
!{\huncross}[ddl]
!{\xcaph[1]@(0)}[uu]
}
\end{equation*} \vspace{-9em}
\caption{The $A$-smoothing of $W(3,4)$}
\label{fig:w34a}
\end{figure}
\begin{proof}[Proof of Proposition \ref{weavingsignature}]
Consider first the example $W(3,n)$, illustrated by figures \ref{fig:w34} and \ref{fig:w34a} for $W(3,4)$.
After $A$-smoothing the diagram, the outer ring of crossings produces a circle bounding the
rest of the smoothed diagram.
On the inner ring of crossings the $A$-smoothings produce $n$ circles in a cyclic arrangement. Therefore
$o\bigl( W(3,n) \bigr) = 1 + n$. The outer ring of crossings consists of positive crossings
and the inner ring of crossings consists of negative crossings, so $x(D) = y(D) = n$.
Applying the formula of proposition \ref{basic_signature}, we obtain the result $\sigma\bigl( W(3,n) \bigr)=0$.
For the general case $W(2k{+}1, n)$, we have the following considerations.
The crossings are organized into $2k$ rings. Reading from the outside toward the center, we have
first a ring of positive crossings, then a ring of negative crossings, and so on, alternating
positive and negative. Thus, $y(D) = kn$. Considering the $A$-smoothing of the diagram of $W(2k{+}1,n)$,
as in the special case, a bounding circle appears from the smoothing of the outer ring.
A chain of $n$ disjoint smaller circles appears inside the second ring. No circles appear in the third ring, nor in any odd-numbered ring
thereafter. On the other hand, chains of $n$ disjoint smaller circles appear in each even-numbered ring. Since there are $k$ even-numbered
rings, we have $o(D) = 1 + kn$.
Applying the formula of proposition \ref{basic_signature}
\begin{equation*}
\sigma\bigl( W(2k{+}1, n)\bigr) = o(D) - y(D) -1 = (1+kn) - kn -1 = 0.
\end{equation*}
\begin{figure}[h!]
\centering
\begin{equation*}
\xygraph{ !{0;/r1.2pc/:}
!{\hcap}[u]
!{\hcap[3]}[u]
!{\hcap[5]}[u]
!{\hcap[7]}[lllllllllll]
!{\xcaph[-11]@(0)}[dl]
!{\xcaph[-11]@(0)}[dl]
!{\xcaph[-11]@(0)}[dl]
!{\xcaph[-11]@(0)}[uuul]
!{\hcap[-7]}[d]
!{\hcap[-5]}[d]
!{\hcap[-3]}[d]
!{\hcap[-1]}[d]
!{\xcaph[2]@(0)}[dl]
!{\xcaph[1]@(0)}[dl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}[u]
!{\htwist}[ddl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}[ul]
!{\xcaph[1]@(0)}
!{\htwist}[ddl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}[ul]
!{\xcaph[1]@(0)}
!{\htwist}[ddl]
!{\htwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\htwistneg}[ul]
!{\xcaph[1]@(0)}
!{\htwist}[ddl]
!{\htwist}[d]
!{\xcaph[2]@(0)}[uul]
!{\htwistneg}[ul]
!{\xcaph[1]@(0)}
!{\htwist}[ddl]
!{\xcaph[1]@(0)}
}
\end{equation*} \vspace{-9em}
\caption{The weaving knot $W(4,5)$}
\label{fig:w45}
\end{figure}
\begin{figure}[h!]
\centering
\begin{equation*}
\xygraph{ !{0;/r1.2pc/:}
!{\hcap}[u]
!{\hcap[3]}[u]
!{\hcap[5]}[u]
!{\hcap[7]}[lllllllllll]
!{\xcaph[-11]@(0)}[dl]
!{\xcaph[-11]@(0)}[dl]
!{\xcaph[-11]@(0)}[dl]
!{\xcaph[-11]@(0)}[uuul]
!{\hcap[-7]}[d]
!{\hcap[-5]}[d]
!{\hcap[-3]}[d]
!{\hcap[-1]}[d]
!{\xcaph[2]@(0)}[dl]
!{\xcaph[1]@(0)}[dl]
!{\huntwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\huncross}[u]
!{\huntwist}[ddl]
!{\huntwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\huncross}[ul]
!{\xcaph[1]@(0)}
!{\huntwist}[ddl]
!{\huntwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\huncross}[ul]
!{\xcaph[1]@(0)}
!{\huntwist}[ddl]
!{\huntwist}[d]
!{\xcaph[1]@(0)}[uul]
!{\huncross}[ul]
!{\xcaph[1]@(0)}
!{\huntwist}[ddl]
!{\huntwist}[d]
!{\xcaph[2]@(0)}[uul]
!{\huncross}[ul]
!{\xcaph[1]@(0)}
!{\huntwist}[ddl]
!{\xcaph[1]@(0)}
}
\end{equation*} \vspace{-9em}
\caption{The $A$-smoothing of $W(4,5)$}
\label{fig:w45a}
\end{figure}
For the case $W(2k, n)$, we show $W(4,5)$ in figures \ref{fig:w45} and \ref{fig:w45a} as an example.
Our standard diagram may be organized into $2k{-}1$ rings of crossings.
In each ring there are $n$ crossings, so the total number of crossings is $c(D) = (2k{-}1)n$.
In our standard representation, there is an outer ring of $n$ positive crossings, next a ring of $n$ negative crossings, alternating
until we end with an innermost ring of $n$ positive crossings. There are thus $k$ rings of $n$ positive crossings and $k{-}1$ rings
of $n$ negative crossings. Therefore, $y(D) = kn$ and $x(D) = (k{-}1)n$.
Considering the $A$-smoothing of the diagram, a bounding circle appears from the smoothing of the outer ring. As before, a chain of $n$ disjoint
smaller circles appears inside the second ring and in each successive even-numbered ring. As previously noted, there are $k{-}1$ of these
rings. No circles appear in odd-numbered rings, until we reach the last ring, where an inner bounding circle appears.
Thus, $o(D) = 1 + (k{-}1)n + 1 = (k{-}1)n + 2$.
Consequently,
\begin{equation*}
\sigma\bigl( W(2k, n)\bigr) = o(D) - y(D) -1 = \bigl((k{-}1)n + 2\bigr) - kn - 1
= -n{+}1. \qedhere
\end{equation*}
\end{proof}
\begin{theorem}
\label{locateKH}
For a weaving knot $W(2k{+}1,n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k{+}1, n) \bigr)$ lies on the
lines
\begin{equation*}
j = 2i \pm 1.
\end{equation*}
For a weaving knot $W(2k, n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k, n) \bigr)$ lies on the lines
\begin{equation*}
j = 2i + n -1 \pm 1
\end{equation*}
\end{theorem}
\begin{proof}
Substitute the calculations made in proposition \ref{weavingsignature} into
the formula of theorem \ref{locateKHLee}.
\end{proof}
\section{Recursion in the Hecke algebra} \label{Hecke}
We review briefly the definition of the Hecke algebra $H_{N+1}$ on generators $1$ and $T_1$ through $T_N$,
and we define the representation of the braid group $B_3$ on three strands in $H_3$.
Theorem \ref{heckerecursion} sets up recursion relations for the coefficients in the expansion
of the image in $H_3$ of the braid $(\sigma_1 \sigma_2^{-1})^{n}$,
whose closure is the weaving knot $W(3,n)$.
These coefficients are polynomials in a parameter $q$, which is built into the definition \ref{Heckealgebras} of
the Hecke algebra. The recursion relations are essential for automating
the calculation of the Jones polynomial for the knots $W(3,n)$.
Proposition \ref{C121} uses the relations developed in theorem \ref{heckerecursion}
to prove a vanishing result for one of the coefficients. Being able to ignore one of
the coefficients speeds up the computations slightly.
Proposition \ref{trailing} evaluates the constant terms of the families of polynomials,
proposition \ref{degreeone} evaluates the degree one coefficients, and
theorem \ref{palindromes} proves certain identities satisfied by the polynomials.
These identities imply symmetry properties of the coefficients and enable calculation
of the degrees of the polynomials in corollary \ref{degrees1}.
\begin{definition}
\label{Heckealgebras}
Working over the ground field $K$ containing an element $q \neq 0$, the Hecke
algebra $H_{N+1}$ is the associative algebra with $1$ on generators $T_1$, \ldots, $T_N$ satisfying these relations.
\begin{align}
T_iT_j &= T_jT_i, \quad \text{whenever $\abs{i-j} \geq 2$,} \label{commutativity}
\\
T_iT_{i+1}T_i &= T_{i+1}T_iT_{i+1}, \quad \text{for $1 \leq i \leq N{-}1$,} \label{interchange}
\intertext{and, finally,}
T_i^2 &= (q{-}1)T_i + q,\quad \text{for all $i$.} \label{inverse}
\end{align}
It is well-known
\cite{Jones_poly86}
that $(N{+}1)!$ is the dimension of $H_{N+1}$ over $K$.
\end{definition}
Recasting the relation
$T_i^2 = (q{-}1)T_i + q$ in the form $q^{-1} \bigl(T_i - (q-1)\bigr)\cdot T_i = 1$ shows that
$T_i$ is invertible in $H_{N+1}$ with
$T_i^{-1} = q^{-1}\bigl( T_i - (q-1) \bigr)$.
Consequently, the specification
$\rho(\sigma_i) = T_i$, combined with relations \eqref{commutativity} and \eqref{interchange}, defines a homomorphism
$\rho \colon B_{N+1} \ra H_{N+1} $ from $B_{N+1}$, the group of braids on $N{+}1$ strands,
into the multiplicative monoid of $H_{N+1}$.
For work in $H_3$, choose the ordered basis
$\{1, T_1, T_2, T_1T_2, T_2T_1, T_1T_2T_1\}$.
The word in the Hecke algebra corresponding to the knot $W(3,n)$ is formally
\begin{equation}
\label{eq:basicw3n}
\rho\bigl( (T_1T_2^{-1})^n\bigr)
= q^{-n}\bigl(C_{n,0}+ C_{n,1}\cdot T_1 + C_{n,2} \cdot T_2 + C_{n,12} \cdot T_1T_2
+ C_{n,21} \cdot T_2T_1 + C_{n,121} \cdot T_1T_2T_1 \bigr),
\end{equation}
where the coefficients $C_{n,*} = C_{n,*}(q)$ of the monomials in $T_1$ and $T_2$ are polynomials in $q$.
For $n = 1$,
\begin{equation*}
\rho ( \sigma_1 \sigma_2^{-1}) = T_1T_2^{-1} = q^{-1}\cdot\bigl( T_1 ( -(q{-}1) + T_2) \bigr)
= q^{-1}\bigl( -(q{-}1)\cdot T_1 + T_1T_2 \bigr),
\end{equation*}
so we have initial values
\begin{equation} \label{initialCs}
C_{1,0}(q) = 0, \; C_{1,1}(q) = -(q{-}1), \; C_{1,2}(q) = 0, \; C_{1,12}(q) = 1,
\; C_{1,21}(q) = 0, \; \text{and} \; C_{1,121}(q) = 0.
\end{equation}
\begin{theorem}
\label{heckerecursion}
These polynomials satisfy the following recursion relations.
\begin{align}
C_{n,0}(q) &= q^2\cdot C_{n-1,21}(q) - q(q{-}1)\cdot C_{n-1,1}(q) \label{cn0}
\\
C_{n,1}(q) &= - (q{-}1)^2\cdot C_{n-1,1}(q) - (q{-}1)\cdot C_{n-1,0}(q) + q^2\cdot C_{n-1,121}(q) \label{cn1}
\\
C_{n,2}(q) &= q\cdot C_{n-1,1}(q) \label{cn2}
\\
C_{n,12}(q) &= (q{-}1)\cdot C_{n-1,1}(q) + C_{n-1,0}(q) \label{cn12}
\\
\begin{split}
C_{n, 21}(q) &= -(q{-}1)\cdot C_{(n-1),2}(q) + q\cdot C_{n-1,12}(q)\\
& \hspace{3em} - (q{-}1)^2\cdot C_{n-1,21}(q) + q(q{-}1)\cdot C_{n-1, 121}(q) \label{cn21}
\end{split}
\\
C_{n,121}(q) &= C_{n-1,2}(q) + (q{-}1)\cdot C_{n-1,21}(q) \label{cn121}
\end{align}
\end{theorem}
\begin{proof}
We have
\begin{multline} \label{exp0}
\rho( T_1T_2^{-1} )^n = \rho( T_1T_2^{-1} )^{n-1} \cdot \rho (T_1T_2^{-1})
\\
= q^{-n+1}\bigl(C_{n-1,0}+ C_{n-1,1}\cdot T_1 + C_{n-1,2} \cdot T_2 + C_{n-1,12} \cdot T_1T_2
+ C_{n-1,21} \cdot T_2T_1 + C_{n-1,121}\cdot T_1T_2T_1 \bigr)
\\ \cdot q^{-1}\bigl( -(q{-}1)\cdot T_1 + T_1T_2 \bigr)
\\
= q^{-n}\biggl( -(q{-}1)C_{n-1,0}\cdot T_1 -(q{-}1)C_{n-1,1}\cdot T_1^2 -(q{-}1)C_{n-1,2} \cdot T_2T_1
\\
-(q{-}1)C_{n-1,12} \cdot T_1T_2T_1 -(q{-}1)C_{n-1,21} \cdot T_2T_1^2 -(q{-}1)C_{n-1,121}\cdot T_1T_2T_1^2
\\
+ C_{n-1,0} \cdot T_1T_2+ C_{n-1,1}\cdot T_1^2T_2 + C_{n-1,2} \cdot T_2T_1T_2
\\
+ C_{n-1,12} \cdot T_1T_2T_1T_2 + C_{n-1,21} \cdot T_2T_1^2T_2 + C_{n-1,121}\cdot T_1T_2T_1^2T_2 \biggr)
\\
= q^{-n}\biggl( \Bigl( -(q{-}1)C_{n-1,0}\cdot T_1 -(q{-}1)C_{n-1,2} \cdot T_2T_1 -(q{-}1)C_{n-1,12} \cdot T_1T_2T_1 + C_{n-1,0} \cdot T_1T_2\Bigr)
\\
+ \Bigl\{-(q{-}1)C_{n-1,1}\cdot T_1^2 -(q{-}1)C_{n-1,21} \cdot T_2T_1^2 -(q{-}1)C_{n-1,121}\cdot T_1T_2T_1^2 + C_{n-1,1}\cdot T_1^2T_2
\\
+C_{n-1,2} \cdot T_2T_1T_2 + C_{n-1,12} \cdot T_1T_2T_1T_2 + C_{n-1,21} \cdot T_2T_1^2T_2 + C_{n-1,121}\cdot T_1T_2T_1^2T_2 \Bigr\} \biggr)
\end{multline}
after collecting powers of $q$ and expanding. In the last grouping, the first four terms inside the parentheses $( \ )$
involve only elements of the preferred basis;
the second eight terms in the pair of braces $\{\ \}$ all require further expansion, as follows.
\begin{align}
\begin{split} -(q{-}1)C_{n-1,1}\cdot T_1^2 &= -(q{-}1)C_{n-1,1}\cdot ((q-1)T_1 + q)
\\ &= -(q-1)^2C_{n-1,1}\cdot T_1 - q(q-1)C_{n-1,1} \end{split} \label{expa}
\\
\begin{split} -(q{-}1)C_{n-1,21} \cdot T_2T_1^2 &= -(q{-}1)C_{n-1,21} \cdot T_2((q-1)T_1 + q)
\\ &= -(q-1)^2C_{n-1,21}\cdot T_2T_1 - q(q-1)C_{n-1,21} \cdot T_2 \end{split} \label{expb}
\\
\begin{split} -(q{-}1)C_{n-1,121}\cdot T_1T_2T_1^2 &= -(q{-}1)C_{n-1,121} \cdot T_1T_2((q-1)T_1 + q)
\\ & = -(q-1)^2C_{n-1,121}\cdot T_1T_2T_1 - q(q-1)C_{n-1,121} \cdot T_1T_2 \end{split} \label{expc}
\\
\begin{split} C_{n-1,1}\cdot T_1^2T_2 &= C_{n-1,1}\cdot( (q{-}1) T_1 + q) T_2
\\ &= (q{-}1)C_{n-1,1}\cdot T_1T_2 + qC_{n-1,1} \cdot T_2
\end{split} \label{expd}
\\
C_{n-1,2} \cdot T_2T_1T_2 &= C_{n-1,2} \cdot T_1T_2T_1 \label{expe}
\\
\begin{split}
C_{n-1,12} \cdot T_1T_2T_1T_2 &= C_{n-1,12}\cdot T_1^2T_2T_1 = C_{n-1,12}((q{-}1)T_1 + q)T_2T_1
\\ &= (q{-}1)C_{n-1,12}\cdot T_1T_2T_1 + qC_{n-1,12}\cdot T_2T_1
\end{split} \label{expf}
\\
\begin{split}
C_{n-1,21} \cdot T_2T_1^2T_2 &= C_{n-1,21}\cdot T_2 ((q{-}1)T_1 + q) T_2
\\ &= (q{-}1)C_{n-1,21} \cdot T_2T_1T_2 + qC_{n-1,21} \cdot T_2^2
\\ &= (q{-}1)C_{n-1,21} \cdot T_1T_2T_1 + qC_{n-1,21}\cdot ((q{-}1)T_2 + q)
\\ &= (q{-}1)C_{n-1,21} \cdot T_1T_2T_1 + q(q{-}1)C_{n-1,21}\cdot T_2 + q^2C_{n-1,21})
\end{split} \label{expg}
\\
\begin{split}
C_{n-1,121}\cdot T_1T_2T_1^2T_2 &= C_{n-1,121} \cdot T_1T_2((q{-}1)T_1 + q)T_2
\\ &= (q{-}1)C_{n-1,121} \cdot T_1T_2T_1T_2 + qC_{n-1,121} \cdot T_1T_2^2
\\ &= (q{-}1)C_{n-1,121} \cdot T_1^2T_2T_1 + qC_{n-1,121}\cdot T_1((q{-}1)T_2+ q)
\\ &= (q{-}1)C_{n-1,121} \cdot ((q{-}1)T_1+ q)T_2T_1
\\ & \hspace{3em} + qC_{n-1,121}\cdot T_1((q{-}1)T_2+ q)
\\ &= (q{-}1)^2C_{n-1,121}\cdot T_1T_2T_1 + q (q{-}1)C_{n-1,121}\cdot T_2T_1
\\ & \hspace{3em} + q(q{-}1)C_{n-1,121}\cdot T_1T_2+ q^2C_{n-1,121} \cdot T_1
\end{split} \label{exph}
\end{align}
Collecting the constant terms from \eqref{expa} and \eqref{expg}, we get
\begin{align*}
C_{n,0} &= - q(q-1)C_{n-1,1} + q^2C_{n-1,21}.
\intertext{Collecting coefficients of $T_1$ from \eqref{exp0}, \eqref{expa}, \eqref{exph}, we get}
C_{n,1} &= -(q{-}1)C_{n-1,0} -(q-1)^2C_{n-1,1} + q^2C_{n-1,121}.
\intertext{Collecting coefficients of $T_2$ from \eqref{expb}, \eqref{expd}, and \eqref{expg}, we get}
C_{n,2} &= - q(q-1)C_{n-1,21} + qC_{n-1,1} + q(q{-}1)C_{n-1,21} = qC_{n-1,1}.
\intertext{Collecting coefficients of $T_1T_2$ from \eqref{exp0}, \eqref{expc}, \eqref{expd}, and \eqref{exph}, we get}
C_{n,12} &= C_{n-1,0} - q(q-1)C_{n-1,121} + (q{-}1)C_{n-1,1} + q(q{-}1)C_{n-1,121} = C_{n-1,0} + (q{-}1)C_{n-1,1}
\intertext{Collecting coefficients of $T_2T_1$ from \eqref{exp0}, \eqref{expb}, \eqref{expf}, and \eqref{exph}, we get}
C_{n,21} &= -(q{-}1)C_{n-1,2} -(q-1)^2C_{n-1,21} + qC_{n-1,12} + q (q{-}1)C_{n-1,121}.
\intertext{Collecting coefficients of $T_1T_2T_1$ from \eqref{exp0}, \eqref{expc}, \eqref{expe}, \eqref{expf}, \eqref{expg}, and \eqref{exph},
we get}
C_{n,121} &= -(q{-}1)C_{n-1,12} -(q-1)^2C_{n-1,121} + C_{n-1,2}
\\
&\hspace{3em}+ (q{-}1)C_{n-1,12} + (q{-}1)C_{n-1,21} + (q{-}1)^2C_{n-1,121}
\\
&= C_{n-1,2} + (q{-}1)C_{n-1,21}
\end{align*}
Up to simple rearrangements and expansion of notation, these are formulas \eqref{cn0} through ~\eqref{cn121}.
\end{proof}
\begin{example} \label{secondCs}
Applying the recursion formulas just proved to the table of initial polynomials,
or by computing $\rho\bigl( (\sigma_1 \sigma_2^{-1})^2 \bigr)$ directly from the definitions,
we find
\begin{align}
C_{2,0}(q) &= q^2 \cdot C_{1,21}(q) - q(q{-}1)\cdot C_{1,1}(q) = q(q{-}1)^2,
\\
C_{2,1}(q) &= -(q{-}1)^2\cdot C_{1,1}(q)-(q{-}1)\cdot C_{1,0}(q) = (q{-}1)^3,
\\
C_{2,2}(q) &= q \cdot C_{1,1}(q) = -q(q{-}1),
\\
C_{2,12}(q) &= (q{-}1)\cdot C_{1,1}(q) + C_{1,0}(q) = -(q{-}1)^2,
\\
C_{2,21}(q) &= -(q{-}1)\cdot C_{1,2}(q) + q \cdot C_{1,12}(q) - (q{-}1)^2\cdot C_{1,21}(q) = q,
\\
C_{2,121}(q) &=0.
\end{align}
\end{example}
As a first application, we have the following vanishing result.
\begin{proposition}
\label{C121}
For all $n$, $C_{n,121}(q) = 0$.
\end{proposition}
\begin{proof}
For $n \geq 1$, we claim $C_{n+1,121}(q) = 0$.
Make the inductive assumption that $C_{k,121}(q) = 0$ for $1 \leq k \leq n$. Apply \eqref{cn121}, \eqref{cn21}, and the
inductive hypothesis to write
\begin{multline*}
C_{n+1,121}(q) = C_{n,2}(q) + (q{-}1)\cdot C_{n,21}(q)
\\
\shoveleft = C_{n,2}(q)
\\
+ (q{-}1)\Bigl( -(q{-}1)\cdot C_{n-1,2}(q) + q\cdot C_{n-1,12}(q) - (q{-}1)^2\cdot C_{n-1,21}(q)
+ q(q{-}1)\cdot C_{n-1, 121} \Bigr)
\\
= C_{n,2}(q) +
(q{-}1)\bigl( -(q{-}1)\cdot C_{n-1,2}(q) + q\cdot C_{n-1,12}(q) -
(q{-}1)^2\cdot C_{n-1,21}(q) \bigr).
\end{multline*}
Using \eqref{cn2} to replace the first term $C_{n,2}(q)$ and \eqref{cn12} to replace the third term factor $C_{n-1,12}(q)$ on the right,
\begin{align*}
C_{n+1, 121}(q) &= q\cdot C_{n-1,1}(q) - (q{-}1)^2C_{n-1,2}(q) + q(q{-}1)\bigl( (q{-}1)C_{n-2,1}(q)+C_{n-2,0}(q) \bigr)
\\
&\hspace{3em} - (q{-}1)^3C_{n,21}(q)
\\
&= q\cdot C_{n-1,1}(q) - (q{-}1)^2C_{n-1,2}(q) + (q{-}1)^2\Bigl( q C_{n-2,1}(q) \Bigr) + q(q{-}1)C_{n-2,0}(q)
\\
&\hspace{3em} - (q{-}1)^3C_{n,21}(q)
\\
&= q\cdot C_{n-1,1}(q) - (q{-}1)^2C_{n-1,2}(q) + (q{-}1)^2C_{n-1,2}(q)+ q(q{-}1)C_{n-2,0}(q)
\\
&\hspace{3em} - (q{-}1)^3C_{n,21}(q),
\end{align*}
where we use \eqref{cn2} in reverse to rewrite the term $q C_{n-2,1}(q) $. Making the obvious cancellation,
\begin{align*}
C_{n+1, 121} &= q\cdot C_{n-1,1}(q)+ q(q{-}1) \cdot C_{n-2,0}(q) - (q{-}1)^3 \cdot C_{n,21}(q)
\\
&=q\bigl( C_{n-1,1}+ (q{-}1)C_{n-2,0}\bigr) - (q{-}1)^3 \cdot C_{n-1, 21}
\\
&= q\Bigl(\bigl(-(q{-}1)^2 \cdot C_{n-2,1} - (q{-}1) \cdot C_{n-2,0}\bigr) + (q{-}1) \cdot C_{n-2,0}\Bigr) - (q{-}1)^3 \cdot C_{n-1, 21},
\end{align*}
since
\begin{align*}
C_{n-1,1}(q) &= - (q{-}1)^2\cdot C_{n-2,1}(q) - (q{-}1)\cdot C_{n-2,0}(q) + q^2\cdot C_{n-2,121}(q)
\\
& = - (q{-}1)^2\cdot C_{n-2,1}(q) - (q{-}1)\cdot C_{n-2,0}(q)
\end{align*}
by \eqref{cn1} and the inductive hypothesis. Therefore,
\begin{align*}
C_{n+1,121}(q) &= -q(q{-}1)^2\cdot C_{n-2,1}(q) - (q{-}1)^3\cdot C_{n-1, 21}(q)
\\
&= -(q{-}1)^2\cdot C_{n-1,2}(q) - (q{-}1)^3\cdot C_{n-1,21}(q),
\intertext{using \eqref{cn2} in the form $ C_{n-1,2}(q) = q\cdot C_{n-2,1}(q) $,}
&= -(q{-}1)^2\bigl( C_{n-1, 2}(q) - (q{-}1)\cdot C_{n-1,21} (q)\bigr)
\\
&= -(q{-}1)^2\cdot C_{n, 121}(q) = 0,
\end{align*}
using \eqref{cn121} and the inductive hypothesis.
\end{proof}
\begin{proposition}
\label{trailing}
For $n \geq 1$, the degree $0$ terms in the non-vanishing polynomials $C_{n,-}$ are as follows:
\begin{gather}
C_{n,0}(0) = c_{n,0,0} = 0,\;C_{n,1}(0) = c_{n,1,0} = (-1)^{n-1},\; C_{n,2}(0) = c_{n,2,0}=0,\notag
\\
C_{n,12}(0) =c_{n,12,0}= (-1)^{n-1},\; C_{n,21}(0) = c_{n,21,0}=0. \label{trailingcoefftable}
\end{gather}
\end{proposition}
Since proposition \ref{C121} proves that $C_{n, 121}(q)$ is identically zero, it does not appear in the list just given or anywhere
in the later parts of this paper.
\begin{proof}
Examinination of the formulas for the polynomials $C_{1,-}(q)$ given in equations \eqref{initialCs}
and for $C_{2, -}(q)$ given in example \ref{secondCs} starts the inductive proof.
Substituting $q{=}0$ in the recursive formulas \eqref{cn0} and \eqref{cn2} immediately yields
$C_{n,0}(0) = 0$ and $C_{n,2}(0) = 0$. Substituting $q{=}0$ into formula \eqref{cn1} yields
\begin{equation*}
c_{n,1,0} = C_{n,1}(0) = - (0{-}1)^2\cdot C_{n-1,1}(0) - (0{-}1)\cdot C_{n-1,0}(0) = -(-1)^{n-2} + 0 = (-1)^{n-1},
\end{equation*}
by the inductive hypothesis. Similarly, substituting $q{=}0$ into formula \eqref{cn12} yields
\begin{equation*}
c_{n,12,0} = C_{n,12}(0) = (0{-}1)\cdot C_{n-1,1}(0) + C_{n-1,0}(0) = (-1)\cdot(-1)^{n-2} + 0 = (-1)^{n-1}.
\end{equation*}
Finally, substituting $q{=}0$ into formula \eqref{cn21} yields
\begin{multline*}
c_{n,21,0}= C_{n,21}(0) = -(0{-}1)\cdot C_{n-1,2}(0) + 0\cdot C_{n-1,12}(0) - (0{-}1)^2\cdot C_{n-1,21}(0)
\\
= 1\cdot 0 + 0\cdot(-1)^{n-2} - 1\cdot 0 = 0. \qedhere
\end{multline*}
\end{proof}
\begin{proposition} \label{degreeone}
For $n \geq 2$ the degree one coefficients in the non-vanishing polynomials $C_{n,-}(q)$ are as follows:
\begin{gather}
c_{n,0,1} = (-1)^n, \quad c_{n,1,1} = (-1)^n(n{+}1), \quad c_{n,2,1} = (-1)^n, \notag
\\
c_{n,12,1} = (-1)^nn, \quad c_{n,21,1} = (-1)^n. \label{degonetable}
\end{gather}
\end{proposition}
\begin{proof}
These are all handled in the same manner. Namely, differentiate the recursive relations
for each successive polynomial, substitute $q{=}0$, and use the values from proposition \ref{trailing} as appropriate.
Concerning $C_{n,0}(q)$, differentiate \eqref{cn0} with respect to $q$, obtaining
\begin{equation*}
C_{n,0}'(q) = 2q \cdot C_{n-1,21}(q) + q^2 C_{n-1,21}'(q) - (2q{-}1)\cdot C_{n-1,1}(q) - q(q{-}1)\cdot C_{n-1,1}'(q),
\end{equation*}
whence
\begin{equation*}
c_{n,0,1} = C_{n,0}'(0) = -(-1)\cdot C_{n-1,1}(0) = (-1)^{n-2}= (-1)^n.
\end{equation*}
Concerning $C_{n,1}(q)$, differentiate \eqref{cn1} with respect to $q$, obtaining
\begin{equation*}
C_{n,1}'(q) = -2(q{-}1) \cdot C_{n-1,1}(q) - (q{-}1)^2\cdot C_{n-1,1}'(q) - C_{n-1,0}(q) - (q{-}1)\cdot C_{n-1,0}'(q),
\end{equation*}
whence
\begin{multline*}
c_{n,1,1} = C_{n,1}'(0) = 2\cdot C_{n-1,1}(0) - C_{n-1,1}'(0) -
C_{n-1,0}(0) + C_{n-1,0}'(0)
\\
= 2\cdot(-1)^{n-2}-c_{n-1,1,1} - 0 + (-1)^{n-1}.
\end{multline*}
Thus, we have the recursive formula $c_{n,1,1} = (-1)^n - c_{n-1,1,1}$. Starting from $C_{2,1}(q) = (q-1)^3$ and $c_{2,1,1} = 3$, we obtain
the closed form expression $c_{n,1,1} = (-1)^n(n{+}1)$.
Concerning $C_{n,2}(q)$, differentiate \eqref{cn2} with respect to $q$, obtaining
\begin{equation*}
C_{n,2}'(q) = C_{n-1,1}(q) + q \cdot C_{n-1,1}'(q),
\end{equation*}
whence
\begin{equation*}
c_{n,2,1} = C_{n,2}'(0) = C_{n-1,1}(0),\quad \text{and} \quad c_{n,2,1} = c_{n-1,1,0} = (-1)^{n-2}.
\end{equation*}
Concerning $C_{n,12}(q)$, differentiate \eqref{cn12} with respect to $q$, obtaining
\begin{equation*}
C_{n,12}'(q) = C_{n-1,1}(q) + (q{-}1)\cdot C_{n-1,1}'(q) + C_{n-1,0}'(q),
\end{equation*}
whence
\begin{multline*}
c_{n,12,1} = C_{n,12}'(0) = C_{n-1,1}(0) - C_{n-1,1}'(0) + C_{n-1,0}'(0)
\\
= c_{n-1,1,0} - c_{n-1,1,1} + c_{n-1,0,1} = (-1)^{n-2} -(-1)^{n-1}n + (-1)^{n-1} = (-1)^nn.
\end{multline*}
At last, concerning $C_{n,21}(q)$, we have from \eqref{cn21}
\begin{multline*}
C_{n,21}'(q) = - C_{n-1,2}(q) - (q{-}1)\cdot C_{n-1,2}'(q)
\\
+ C_{n-1,12}(q) + q\cdot C_{n-1,12}'(q)
\\
-2(q{-}1)\cdot C_{n-1,21}(q) - (q-1)^2\cdot C_{n-1,21}'(q),
\end{multline*}
so, since $C_{n-1,2}(0) = 0$ and $C_{n-1,21}(0) = 0$,
\begin{align*}
c_{n, 21,1} &= C_{n,21}'(0) = -(-1)\cdot C_{n-1,2}'(0) + C_{n-1,12}(0) - (-1)^2\cdot C_{n,21}'(0)
\\
&= c_{n-1,2,1} + c_{n-1,12,0} - c_{n-1,21,1} = (-1)^{n-3} + (-1)^{n-2} - c_{n-1,21,1}
\\
&= -c_{n-1,21,1}.
\end{align*}
Starting from $C_{2,21}(q) = q$ and $c_{2,21,1} = 1$, we deduce $c_{n,21,1}= (-1)^n$.
\end{proof}
\begin{theorem}
\label{palindromes}
The following identities are satisfied by the polynomials $C_{n,-}(q)$.
\begin{gather}
C_{n,0}(q) = q^{2n}C_{n,0}(q^{-1}), \quad
C_{n,1}(q) = -q^{2n-1}C_{n,1}(q^{-1}), \quad
C_{n,2}(q) = -q^{2n-1}C_{n,2}(q^{-1}), \notag
\\
C_{n,12}(q) = q^{2n-2} C_{n,12}(q^{-1}),\quad
C_{n,21}(q) = q^{2n-2}C_{n,21}(q^{-1}). \label{palindromicidentities}
\end{gather}
In terms of the coefficients of the various polynomials, we have
\begin{gather}
c_{n,0,i} = c_{n,0,2n-i}, \quad c_{n,1,i} = -c_{n,1,2n-1-i}, \quad c_{n,2,i} = -c_{n,2,2n-1-i}, \notag
\\
c_{n,12,i} = c_{n,12, 2n-2-i}, \quad c_{n,21,i} = c_{n,21,2n-2-i}. \label{coeffidentities}
\end{gather}
\end{theorem}
These identities reflect certain palindromic properties of the polynomials and permit us to compute their degrees in
corollary \ref{degrees1}. We will say a polynomial $p(x) = a_0 + a_1\, x + \cdots + a_n\,x^n$ of degree $n$ is {\em palindromic} if
\begin{equation*}
p(x) = x^np(x^{-1}) = x^n(a_0 + a_1\,x^{-1} + \cdots + a_n\,x^{-n}) = a_n + a_{n-1}\,x + \cdots + a_0\, x^n.
\end{equation*}
We say a polynomial of degree $n$ is {\em skew-palindromic} if $p(x) = - x^np(x^{-1})$.
Obviously, for a palindromic or a skew-palindromic polynomial the leading and trailing coefficients are both non-vanishing.
\begin{proof}
These are all proved by induction, using the recursive formulas from theorem \ref{heckerecursion}.
For the first identity,
\begin{align*}
C_{n,0}(q) &= q^2\cdot C_{n-1,21}(q) - q(q{-}1)\cdot C_{n-1,1}(q)
\\
& = q^2 \cdot q^{2n-4}C_{n-1,21}(q^{-1}) - q(q{-}1)\cdot \bigl(-q^{2n-3}C_{n-1,1}(q^{-1})\bigr)
\\
&= q^{2n}\cdot\bigl(q^{-2}C_{n-1,21}(q^{-1})\bigr) - q^{2n}\cdot\bigl(q^{-1}(q^{-1}-1)\cdot C_{n-1,1}(q^{-1})\bigr) = q^{2n}\cdot C_{n,0}(q^{-1}).
\end{align*}
For the second identity,
\begin{align*}
C_{n,1}(q) &= - (q{-}1)^2\cdot C_{n-1,1}(q) - (q{-}1)\cdot C_{n-1,0}(q)
\\
&= -(q{-}1)^2\cdot (-1) \cdot q^{2n-3}C_{n-1,1}(q^{-1}) -(q{-}1)\cdot q^{2n-2}C_{n-1,0}(q^{-1})
\\
&= q^{2n-1}\cdot\bigl((1{-}q^{-1})^2\cdot C_{n-1,1}(q^{-1}) + (q^{-1}{-}1)\cdot C_{n-1,0}(q^{-1})\bigr)
\\
&= -q^{2n-1}\cdot C_{n,1}(q^{-1}).
\end{align*}
For the third identity,
\begin{multline*}
C_{n,2}(q) =q\cdot C_{n-1,1}(q)
=q\cdot(-1)\cdot q^{2n-3}\cdot C_{n-1,1}(q^{-1})
\\
= -q^{2n-1}\cdot\bigl( q^{-1}\cdot C_{n-1,1}(q^{-1})
= -q^{2n-1}\cdot C_{n,2}(q^{-1}).
\end{multline*}
For the fourth identity,
\begin{align*}
C_{n,12}(q) &= (q{-}1)\cdot C_{n-1,1}(q) + C_{n-1,0}(q)
\\
&=(q{-}1)\cdot (-1)\cdot q^{2n-3} \cdot C_{n-1,1}(q^{-1}) + q^{2n-2}\cdot C_{n-1,0}(q^{-1})
\\
&=q^{2n-2}\cdot \bigl(q^{-1}{-}1)\cdot C_{n-1,1}(q^{-1}) + C_{n-1,0}(q^{-1}) \bigr)
\\
&= q^{2n-2}\cdot C_{n,12}(q^{-1}).
\end{align*}
Finally, for the fifth identity,
\begin{align*}\begin{split}
C_{n,21}(q) &= -(q{-}1)\cdot C_{n-1,2}(q) + q\cdot C_{n-1,12}(q) - (q{-}1)^2\cdot C_{n-1,21}(q)
\\
&= -(q{-}1)\cdot (-1)\cdot q^{2n-3}\cdot C_{n-1,2}(q^{-1}) + q \cdot q^{2n-4} \cdot C_{n-1,12}(q^{-1})
\\
&\hspace{18em} - (q{-}1)^2\cdot q^{2n-4}\cdot C_{n-1,21}(q^{-1})
\\
&= q^{2n-2}\cdot \bigl( q^{-1}{-}1)^2 \cdot C_{n-1,2}(q^{-1}) + q^{-1}\cdot C_{n-1,12}(q^{-1}) - (q^{-1}{-}1)^2\cdot C_{n-1,21}(q^{-1})\bigr)
\\
&= q^{2n-2}\cdot C_{n,21}(q^{-1}). \qedhere
\end{split}
\end{align*}
\end{proof}
Now we state the implications for the polyomials $C_{n,*}(q)$.
\begin{corollary}
\label{degrees1}
The degrees of the polynomials $C_{n,*}(q)$ are as follows.
\begin{gather}
\deg(C_{n,0}) = 2n{-}1, \quad \deg(C_{n,1}) = 2n{-}1 , \quad \deg(C_{n,2}) = 2n{-}2, \notag
\\
\deg(C_{n,12}) = 2n{-}2, \quad \text{and} \quad \deg(C_{n,21}) = 2n{-}3. \label{degreevalues}
\end{gather}
\end{corollary}
\begin{proof}
Consider first $C_{n,0}(q)$. We know from proposition \ref{trailing} that $C_{n,0}(0){=}0$, so $q$ is a
factor. That is, $P_{n,0}(q) = q^{-1}C_{n,0}(q)$ is also a polynomial, and its trailing coefficient is $c_{n,0,1} \neq 0$ by proposition
\ref{degreeone}. We observe
\begin{equation*}
P_{n,0}(q) = q^{-1}C_{n,0}(q) = q^{-1}\cdot q^{2n}C_{n,0}(q^{-1}) = q^{2n-2}\cdot q\,C_{n,0}(q^{-1}) = q^{2n-2}\cdot P_{n,0}(q^{-1}).
\end{equation*}
Thus, $C_{n,0}(q)$ is $q$ times a palindromic polynomial of degree $2n{-}2$, which means $C_{n,0}(q)$ has degree $2n{-}1$.
Similarly, we conclude that $C_{n,2}(q)$ and $C_{n,21}(q)$ are, respectively, $q$ times a skew-palindromic polynomial of
degree $2n{-}3$ and $q$ times a palindromic polynomial of degree $2n{-}4$. Thus $C_{n,2}(q)$ has degree $2n{-}2$ and
$C_{n,21}(q)$ has degree $2n{-}3$. Since $C_{n,1}(0){\neq}0$ and $C_{n,12}(0){\neq}0$, the identities stated in theorem
yield that $C_{n,1}(q)$ is a skew-palindromic polynomial of degree $2n{-}1$ and that $C_{n,12}(q)$ is a palindromic polynomial
of degree $2n{-}2$.
\end{proof}
Accordingly, set
\begin{align}
C_{n,0}(q) &= \sum_{i=1}^{2n-1} c_{n,0,i}q^i, &
C_{n,1}(q) &= \sum_{i=0}^{2n-1} c_{n,1,i}q^i, &
C_{n,2}(q) &= \sum_{i=1}^{2n-2} c_{n,2,i}q^i, \notag
\\
C_{n,12}(q) &= \sum_{i=0}^{2n-2} c_{n,12,i}q^i,
& &\text{and} &
C_{n,21}(q) &= \sum_{i=1}^{2n-3} c_{n,21,i}q^i. \label{Cexpansions}
\end{align}
\section{Obtaining the Polynomial Invariants} \label{Polys}
Following the construction given in
\cite[p.288]{Jones_poly86}
we work over the function field $K = \bC(q,z)$
and follow their recipes to obtain expressions for the two-variable HOMFLY-PT polynomials,
the one-variable Jones polynomials $V_{W(3,n)}(t)$,
and the Alexander polynomials $\Delta_{W(3,n)}(t)$.
The expressions are subsequently refined to incorporate information obtained in section \ref{Hecke}.
From this point we evaluate the span of the Jones polynomial $V_{W(3,n)}(t)$ in proposition \ref{span}, a
result already known to Kauffman \cite[Theorem~2.10]{States}, where we demonstrate how to use
equations \eqref{trailingcoefftable} and \eqref{coeffidentities}.
Let $H_{N+1}$ be the Hecke algebra over $K$ corresponding
to $q$ with $N$ generators as in definition \ref{Heckealgebras}. The
starting point is the following theorem.
\begin{theorem}
\label{traces}
For $N \geq 1$ there is a family of trace functions $\Tr \colon H_{N+1} \ra K$ compatible with the inclusions $H_N \ra H_{N+1}$ satisfying
\begin{enumerate}
\item $\Tr(1) = 1$,
\item $\Tr$ is $K$-linear and $ \Tr(ab) = \Tr(ba)$,
\item If $a, b \in H_N$, then $\Tr (aT_Nb) = z\Tr(ab)$.
\end{enumerate}
\end{theorem}
Property 3 enables the calculation of $\Tr$ on basis elements of $H_{N+1}$ through use of the defining relations and induction.
For $H_3$, note that
\begin{equation*}
\Tr(T_1) = \Tr(T_2) = z, \quad \Tr(T_1T_2) = \Tr(T_2T_1) = z^2, \quad \Tr(T_1T_2T_1) = z \Tr(T_1^2) = z \bigl((q{-}1)z + q\bigr),
\end{equation*}
and we put $w=1{-}q{+}z$.
The next step toward the polynomial invariants of the knot that is the closure of
the braid $\alpha \in B_{N+1}$ is given by the formula
\begin{equation*}
V_{\alpha}(q,z) = \Bigl(\frac{1}{z}\Bigr)^{(N + e(\alpha))/2}
\cdot \Bigl( \frac{q}{w} \Bigr)^{(N-e(\alpha))/2}\cdot \Tr\bigl(\rho(\alpha)\bigr),
\end{equation*}
where $e(\alpha)$ is the exponent sum of the word $\alpha$. The expression defines an element in the quadratic extension $K(\sqrt{q/zw})$.
For the weaving knot $W(3,n)$, viewed as the closure of $(\sigma_1\sigma_2^{-1})^n$, we have the exponent sum $e=0$, and $N=2$, and
\begin{equation*}
\rho\bigl( (\sigma_1\sigma_2^{-1})^n \bigr)= (T_1T_2^{-1})^n
= q^{-n}\bigl( C_{n,0}(q) + C_{n,1}(q)\cdot T_1 + C_{n,2}(q) \cdot T_2 + C_{n,12}(q) \cdot T_1T_2 + C_{n,21}(q) \cdot T_2T_1\bigr) ,
\end{equation*}
thanks to proposition \ref{C121}, which says the expression for $(T_1T_2^{-1})^n$ requires only the use of the basis elements
$1$, $T_1$, $T_2$, $T_1T_2$ and $T_2T_1$. Then we have
\begin{multline}
V_{(\sigma_1\sigma_2^{-1})^n}(q,z) \\ = \Bigl(\frac{1}{z}\Bigr)\cdot \Bigl( \frac{q}{w} \Bigr)\cdot q^{-n}
\Tr \bigl(C_{n,0}(q) + C_{n,1}(q)\cdot T_1 + C_{n,2}(q) \cdot T_2 + C_{n,12}(q) \cdot T_1T_2
+ C_{n,21}(q) \cdot T_2T_1 \bigr)
\\
= \Bigl(\frac{q}{zw}\Bigr)\cdot q^{-n} \cdot \bigl( C_{n,0}(q) + C_{n,1}(q) \cdot z + C_{n,2}(q) \cdot z + C_{n,12}(q) \cdot z^2
+ C_{n,21} (q) \cdot z^2 \bigr), \label{Heckeoutput}
\end{multline}
using the facts that $\Tr T_1 = \Tr T_2 = z$ and $\Tr T_1T_2 = \Tr T_2T_1 = z^2$.
This expression is the starting point for our manipulations.
Following \cite{Jones_poly86}, we point out that the universal skein invariant
$P_{W(3,n)}(\ell, m)$, an element of the Laurent polynomial ring $\bZ[\ell, \ell^{-1}, m , m^{-1}]$, is obtained by rewriting
$V_{(\sigma_1\sigma_2^{-1})^n}(q,z)$ in terms of $\ell = i(z/w)^{1/2}$ and $m = i(q^{-1/2} - q)$, a task
easily managed in a computer algebra system by simplifying $V_{(\sigma_1\sigma_2^{-1})^n}(q,z)$ with
respect to side relations. Starting from
$P_{W(3,n)}(\ell, m)$, the Jones polynomial $V_{W(3,n)}(t)$ is obtained by
setting $\ell = it $ and $m = i (t^{1/2}{-}t^{-1/2}) $, the Alexander polynomial $\Delta_{W(3,n)}(t)$ is obtained
by setting $\ell = i$ and $m = i (t^{1/2}{-}t^{-1/2})$, and the HOMFLY-PT polynomial is obtained by
setting $\ell = ia$ and $m = iz$. We have no specific use for the HOMFLY-PT polynomial
in this paper, so we content ourselves with a few values in section \ref{PolynomialsExtra}.
To obtain the Alexander polynomial from $V_{(\sigma_1\sigma_2^{-1})^n}(q,z)$, it is useful to first rewrite
\begin{equation*}
V_{(\sigma_1\sigma_2^{-1})^n}(q,z)
= q^{-n+1} \cdot \bigl( C_{n,0}(q)\cdot(zw)^{-1} + \bigl(C_{n,1}(q){+}C_{n,2}(q)\bigr) \cdot w^{-1}
+ \bigl(C_{n,12}(q) {+} C_{n,21} (q)\bigr) \cdot zw^{-1} \bigr).
\end{equation*}
First set $q=t$ and make the substitutions
\begin{equation*}
z = \frac{\ell^2(t-1)}{1+\ell^2}, \quad w = 1- q + z = 1- t + z = \frac{-1(t{-}1)}{1+\ell^2}
\end{equation*}
to obtain an expression
\begin{multline*}
t^{-n+1}\cdot \Bigl( C_{n,0}(t) \cdot \biggl(\frac{(1+\ell^2)}{\ell^2}\biggr)^2\cdot \frac{1}{(t-1)t}
+ \bigl(C_{n,1}(t) + C_{n,2}(t)\bigr)\cdot \frac{(-1)\dot( 1 + \ell^2)}{t-1}
\\
+ \bigl(C_{n,12}(t) + C_{n,21}(t)\bigr)\cdot (-1) \cdot \ell^2\Bigr)
\end{multline*}
Now make the substitution $\ell = i$ and we arrive at
\begin{equation}
\label{eq:alexander}
\Delta_{W(3,n)}(t) = t^{-n+1} \bigl(C_{n, 12}(t) + C_{n, 21}(t)\bigr).
\end{equation}
Evidently a lot of information from the braid representation of $W(3,n)$ has been lost.
To see what remains, corollary \ref{degrees1} says that the degree of $C_{n,12}(t)$ is $2n{-}2$ and
the degree of $C_{n,21}(t)$ is $2n{-}3$. It follows that the degree of $\Delta_{W(3,n)}(t)$ is
$(2n{-}2) - n + 1= n{-}1$. Moreover, the lowest order non-vanishing coefficients are
$c_{n, 12, 0} = (-1)^n $ and $c_{n,21,1} = (-1)^n$. By theorem \ref{palindromes} we also have
$c_{n, 12, 2n-2} = c_{n, 12, 0} = (-1)^n$ and $c_{n, 21, 2n-3} = c_{n, 21, 1}= (-1)^n$. Thus,
\begin{equation}
\label{eq:alexanderw3n}
\Delta_{W(3,n)}(t) = a_0 + \sum_{s>0} a_s(t^s+t^{-s}) = (-1)^n \cdot t^{-n+1} + \cdots + (-1)^n\cdot t^{n-1}.
\end{equation}
\begin{theorem}
\label{HFhomology}
The Seifert genus of $W(3,n)$ is $n{-}1$, and the complement of $W(3,n)$ is fibered over $S^1$.
\end{theorem}
\begin{proof}
We know the signature of $W(3,n)$ is zero, by Proposition \ref{weavingsignature}, so
we apply \cite[Theorem 1.3]{OSFloer} relating the coefficients of the Alexander polynomial and the signature
of $W(3,n)$ to the ranks of the Heegard-Floer homology groups of $S^3$ associated to $W(3,n)$. The result is
\begin{equation*}
\widehat{HFK}_s(S^3, W(3,n), s) =
\begin{cases}
\bZ^{\abs{a_s}}, \quad \text{$0 \leq s \leq n{-}1$},
\\
0, \quad \text{else}.
\end{cases}
\end{equation*}
By \cite[Theorem 1.2]{OSFloer2}, Seifert genus of $W(3,n)$ is $n{-}1$.
Since we have explicitly
\begin{equation*}
\widehat{HFK}_s(S^3, W(3,n), n{-}1) \iso \bZ,
\end{equation*}
\cite[Theorem 2.5]{Manolescu} says that the complement of $W(3,n)$ is fibered over $S^1$.
\end{proof}
Turning to the Jones polynomial, we follow a similar scheme, but the details are necessarily more complicated. To start, the substitutions
\begin{equation*}
q = t, \quad z = \frac{t^2}{1+t}, \quad w = \frac{1}{1+t}
\end{equation*}
in \eqref{Heckeoutput} lead to the one-variable Jones polynomial
\begin{multline*}
V_{W(3,n)}(t) = \frac{t(1{+}t)^2}{t^2} \cdot t^{-n} \cdot
\Bigl( C_{n,0}(t) + (C_{n,1}(t)+C_{n,2}(t))\cdot \frac{t^2}{1{+}t} + (C_{n,12}(t) + C_{n,21}(t)) \cdot \frac{t^4}{(1{+}t)^2}\Bigr)
\\
= t^{-n-1}\cdot\bigl( (1{+}t)^2\cdot C_{n,0}(t) + (1{+}t)\cdot( C_{n,1}(t) + C_{n,2}(t) )\cdot t^2 + (C_{n,12}(t) + C_{n,21}(t))\cdot t^4 \bigr).
\end{multline*}
\begin{example}
For $W(3,1)$, which is the unknot, we have
\begin{align*}
V_{W(3,1)}(t)
&= t^{-2}\cdot\bigl( (1{+}t)^2\cdot C_{1,0}(t) + (1{+}t)\cdot( C_{1,1}(t) + C_{1,2}(t) )\cdot t^2 + (C_{1,12}(t) + C_{1,21}(t))\cdot t^4 \bigr)
\\
&= t^{-2}\cdot\bigl( (1{+}t)^2\cdot 0 + (1{+}t)\cdot(-(t-1) + 0 )\cdot t^2 + (1 + 0 )\cdot t^4 \bigr)
\\
&= t^{-2}\cdot ( (1{-}t^2) t^2 + t^4 ) = 1.
\end{align*}
\end{example}
\begin{example} \label{jonesfig8knot}
For $W(3,2)$, which is the figure-8 knot, we have
\begin{align*}
V_{W(3,2)}(t)
&=t^{-3}\cdot \bigl( (1{+}t)^2\cdot C_{2,0}(t) + (1{+}t)\cdot( C_{2,1}(t) + C_{2,2}(t) )\cdot t^2
+ (C_{2,12}(t) + C_{2,21}(t))\cdot t^4 \bigr)
\\
&=t^{-3}\cdot \bigl( (1{+}t)^2\cdot t(t{-}1)^2 +(1{+}t)\cdot( (t{-}1)^3 - t(t{-}1) ) \cdot t^2
+( -(t{-}1)^2+ t ) \cdot t^4 \bigr)
\\
&= t^{-3}\cdot \bigl( t^5 - t^4 + t^3 -t^2 + t \bigr) = t^2 - t + 1 -t^{-1} + t^{-2}
\end{align*}
\end{example}
Now we take a closer look at the formal expression
\begin{multline*}
V_{W(3,n)}(t) =
\\
= t^{-n-1}\cdot\bigl( (1{+}t)^2\cdot C_{n,0}(t) + (1{+}t)\cdot( C_{n,1}(t) + C_{n,2}(t) )\cdot t^2 + (C_{n,12}(t) + C_{n,21}(t))\cdot t^4 \bigr)
\end{multline*}
for the Jones polynomial of the weaving knot $W(3,n)$.
Incorporating the formal expansions given in equations \eqref{Cexpansions}, we have
\begin{align}
V_{W(3,n)}(t)
&= t^{-n-1}\cdot \bigl( (1{+}t)^2\cdot C_{n,0}(t) + (t^2{+}t^3)\cdot( C_{n,1}(t) + C_{n,2}(t) ) + t^4 \cdot (C_{n,12}(t) + C_{n,21}(t)) \bigr) \notag
\\
\begin{split}
&= t^{-n-1}\cdot \Biggl( (1{+}t)^2\cdot \biggl(\sum_{i=1}^{2n-1} c_{n,0,i}t^i\biggr)
\\
& \hspace{0.15\linewidth} + (t^2{+}t^3)\cdot \Bigl( \sum_{i=0}^{2n-1} c_{n,1,i}t^i + \sum_{i=1}^{2n-2} c_{n,2,i}t^i \Bigr)
\\
& \hspace{0.30\linewidth} + t^4 \cdot \biggl(\sum_{i=0}^{2n-2} c_{n,12,i}t^i + \sum_{i=1}^{2n-3} c_{n,21,i}t^i\biggr) \Biggr). \label{Vformal1}
\end{split}
\\
&= t^{-n-1}\cdot P(t) = t^{-n-1}\cdot( p_0 + p_1\,t + p_2 \, t^2 + p_3\, t^3 + p_4\, t^4 + \cdots). \label{Vformal2}
\end{align}
The first piece of information about the Jones polynomial $V_{W(3,n)}(t)$ we obtain by using results about the $C_{n,-}$ is the following
fact, due to Kauffmann \cite[Theorem~2.10]{States}.
\begin{proposition} \label{span}
The span of the Jones polynomial $V_{W(3,n)}(t)$ is $2n$, and the trailing and leading coefficients are $(-1)^n$.
\end{proposition}
\begin{proof}
We observe that $ p_0 = P(0) = 1\cdot c_{n,0,0} = 0 $ by proposition \ref{trailing}, so the lowest non-zero term in $V_{W(3,n)}(t)$ is
$t^{-n-1}\cdot p_1\,t$. Clearly, $p_1 = c_{n,0,1} = (-1)^{n}$ by proposition \ref{degreeone}.
One identifies the top degree term in the polynomial factor of \eqref{Vformal1} as the term of degree $2n{+}2$ with coefficient
\begin{equation*}
c_{n,1,2n-1} + c_{n,12,2n-2} = -c_{n,1,0} + c_{n,12,0} = -(-1)^{n-1} + (-1)^{n-1} = 0,
\end{equation*}
where we use the palindromic equations \eqref{coeffidentities} and the table of trailing coefficients
\eqref{trailingcoefftable} to do the computation.
Turning to the term of degree $2n{+}1$, we find the coefficient is
\begin{align*}
c_{n,0,2n-1} &+ c_{n,1,2n-1} + c_{n,1, 2n-2} + c_{n,2,2n-2} + c_{n,12, 2n-3} + c_{n,21,2n-3}
\\
&= c_{n,0,1} - c_{n,1,0} - c_{n,1,1} -c_{n,2,1} + c_{n,12,1} + c_{n,21,1}
\\
&= (-1)^n - (-1)^{n-1} - (-1)^n(n{+}1) - (-1)^n + (-1)^nn + (-1)^n
\\
&= (-1)^n,
\end{align*}
using first \eqref{coeffidentities} and then the tables \eqref{trailingcoefftable} and \eqref{degonetable} to complete the evaluation.
\end{proof}
Thus, we may write the Jones polynomial in the form
\begin{equation*}
V_{W(3,n)}(t) = (-1)^n\, t^{-n} + v_{-n+1}t^{-n+1} + v_{-n+2}t^{-n+2} + \cdots + v_{n-2}t^{n-2} + v_{n-1}t^{n-1} + (-1)^n\,t^n,
\end{equation*}
where it is known that $v_{-n+i} = v_{n-i}$, since $W(p,q)$ for $p$ odd is amphicheiral.
The twist number of $W(3,n)$ is $\abs{v_{-n+1}}{+}\abs{v_{n-1}}$ according to \cite{DasbachLin}.
We will now recompute the twist number for $n \geq 3$ from the information we have gathered
about the coefficients of the Jones polynomial.
First, observe that
\begin{equation}\label{vfirst}
v_{-n+1}t^{-n+1} = t^{-n-1}\cdot p_2 \, t^2,
\end{equation}
and $p_2\,t^2$ is computed from
\begin{equation*}
(1+2t)\cdot(c_{n,0,1}\,t + c_{n,0,2}\,t^2) + t^2\cdot c_{n,1,0} = c_{n,0,1}\, t + (c_{n,0,2}{+}2\,c_{n,0,1}{+}c_{n,1,0})\cdot t^2,
\end{equation*}
and no other terms from the expansion \eqref{Vformal1}, because $c_{n,2,0} = 0$ by proposition \ref{trailing}.
We have
\begin{align*}
p_2 &= c_{n,0,2}+2\,c_{n,0,1}+c_{n,1,0} = c_{n,0,2} + 2 \cdot (-1)^n + (-1)^{n-1}
\intertext{by propositions \ref{trailing} and \ref{degreeone},}
&= c_{n,0,2} + (-1)^n.
\end{align*}
We identify $c_{n,0,2}$ by reducing the recursive description \eqref{cn0} mod $q^3$, obtaining
\begin{align*}
c_{n,0,0} + c_{n,0,1} \, q + c_{n,0,2}\, q^2 &\equiv q^2\cdot (c_{n-1, 21,0}) + (- q^2 + q)\cdot(c_{n-1,1,0} + c_{n-1,1,1}\, q)
\\
&\equiv c_{n-1,1,0}\,q + (c_{n-1,21,0} +c_{n-1,1,1}{-}c_{n-1,1,0})\cdot q^2 \mod q^3
\end{align*}
After extracting the coefficient of $q^2$,
\begin{align}
c_{n,0,2} &= c_{n-1,21,0} +c_{n-1,1,1}{-}c_{n-1,1,0} \notag
\\
&= 0 + (-1)^{n-1}((n{-}1)+1) - (-1)^{n-2} = (-1)^{n-1}(n+1), \quad \text{for $n \geq 3$,} \label{cn0deg2}
\end{align}
by propositions \ref{trailing} and \ref{degreeone}. Since we have used the formula for $c_{n-1,1,1}$ in \eqref{degonetable},
we must assume $n{-}1 \geq 2$.
Referring to \eqref{vfirst},
\begin{equation*}
v_{-n+1} = p_2= c_{n,0,2}+(-1)^n = (-1)^{n-1}n + (-1)^{n-1} + (-1)^n = (-1)^{n-1}n.
\end{equation*}
Thus, we have reproved the following formula given in theorem 5.1 of \cite{DasbachLin}.
\begin{proposition} \label{twistnumber}
For $n \geq 3$, the twist number of $W(3,n)$ is $\abs{v_{-n+1}}{+}\abs{v_{n-1}} {=} n {+} n {=} 2n$. \qed
\end{proposition}
\section{Higher Twist Numbers and Volume} \label{TwistNumbersVolume}
In \cite{DasbachLin}, Dasbach and Lin define higher twist numbers of a knot in terms of the Jones polynomial,
with the idea that these invariants also correlate with the hyperbolic volume of the knot complement.
If
\begin{equation*}
V_K(t) = \lambda_{-m}t^{-m} + \lambda_{-m+1}t^{-m+1} + \cdots + \lambda_{n-1}t^{n-1} + \lambda_nt^n,
\end{equation*}
then the $j$th twist number of $K$ is $T_j(K) = \abs{\lambda_{-m+j}} + \abs{\lambda_{n-j}}$.
Note that twist numbers $T_j(K)$ are only defined for $j$ within the span of the Jones polynomial. In the
case of weaving knots $W(3,n)$, the relevant twist numbers are defined for $1 \leq j \leq n{-}1$.
In proposition \ref{twistnumber} we have recomputed the first twist number of $W(3,n)$ using our results from section \ref{Hecke}.
In theorems \ref{twistnumber2} and \ref{twistnumber3} we extend the technique to compute the second and third twist numbers.
In the appendix to \cite{DasbachLin} one finds a scatter plot generated from a table of alternating knots of 14
crossings by plotting along a horizontal axis the higher twist numbers of the knots and along the vertical axis
the volume of the complement. The authors also construct similar plots starting from a table of non-alternating
knots of 14 crossings. In both cases, there appears to be some correlation between these combinatorial invariants and the
geometric invariant. We are going to explore how well higher twist numbers and volume correlate as the number of crossings
increases.
We supplement our rigorous calculations of $T_2\bigl(W(3,n)\bigr)$ and $T_3\bigl(W(3,n)\bigr)$
with some conjectural calculations in the following table of higher twist numbers.
To obtain these results, we use {\em Mathematica} or {\em Maple}
to extract the coefficients $\lambda_{-n+k}$ of $t^{-n+k}$ in $V_{W(3,n)}(t)$ for $k = 4$, $5$, $6$, and $7$
associated to weaving knots $W(3,n)$ starting near $n = 2k$.
We conjecture that the $k$th twist number $T_k\bigl(W(3,n)\bigr)$ is a polynomial in $n$ of degree $k$.
Taking iterated differences of the coefficient sequences, we find they are
consistent with the conjecture as long as $n$ is sufficiently large. The following formulas for the twist numbers
$T_k\bigl(W(3,n)\bigr) = 2 \abs{\lambda_{-n+k}}$ for $k = 4$, $5$, $6$, and $7$ were produced by fitting polynomials
to sufficiently large selections of coefficients $\lambda_{-n+k}$ and comparing polynomial
values with computed coefficients for different values of $n$.
\begin{table}[h!]
\caption{Higher twist numbers for $W(3,n)$}
\label{highertwists}
\centering \renewcommand{\arraystretch}{1.25}
\begin{tabular}[h]{|l|l|}\hline
$k$ & $T_k\bigl(W(3,n)\bigr)$
\\ \hline
2 & $-n+n^2$
\\ \hline
$3$ & $ n ( n{-}1 ) ( n{-}2 )/3 +2\,n$
\\ \hline
$4$ & $-(9/2)\,n+(35/12)\,{n}^{2} -(1/2)\,{n^{3}}+(1/12)\,{n^{4}}$
\\ \hline
$5$ & $ (42/5) \,n - (35/6)\,{n}^{2} + (19/12)\,{n}^{3} - (1/6)\,n^{4} + (1/60)\,{n}^{5}$
\\ \hline
$6$ & $- (52/3) \,n + (2237/180) \,n^2 - (29/8) \,n^3 + (41/72) \,n^4 - (1/24)\,n^{5} + (1/360) \, n^{6} $
\\ \hline
$7$ & $(254/7) \,n - (413/15) \,n^2 + (1541/180)\,n^3 $
\\
& $\qquad \qquad \qquad - (35/24)\,n^4 + (11/72) \,n^5 - (1/120)\, n^6 + (1/2520)\, n^7$ \\ \hline
\end{tabular}
\end{table}
Figures \ref{fig:secondtwistvsvolume}, \ref{fig:thirdtwistvsvolume}, and \ref{fig:fourthtwistvsvolume}
plot horizontally values of the twist numbers $T_2$, $T_3$, and the conjectured $T_4$ and vertically
values of the volume of the link complement. We used the program SnapPy \cite{SnapPy} to compute estimates of
the volume.
In addition,
one can ask how efficient are the bounds given in \eqref{CKPbounds} for the volume of weaving knots $W(p,q)$.
For weaving knots $W(3,n)$ the bounds simplify to
\begin{equation*}
v_{{\rm oct}}\,n\,\biggl(1 - \frac{(2\pi)^2}{n^2}\biggr)^{3/2} \leq {\rm vol}(W(3,n)) < 4\,v_{{\rm tet}}\cdot n.
\end{equation*}
If we consider the volume relative to the crossing number ${\rm vol}(W(3,n))/2n$, then we have the chain
\begin{equation} \label{CKPboundspecialrelative}
\frac{v_{{\rm oct}}}{2}\biggl(1 - \frac{(2\pi)^2}{n^2}\biggr)^{3/2} \leq \frac{{\rm vol}(W(3,n))}{2n} < 2\,v_{{\rm tet}}
\end{equation}
For a fixed value of $n$ there is a gap between the upper and lower bounds.
We can ask whether or not better bounds on the relative volume of weaving knots $W(3,n)$ can be teased out of the
higher twist numbers of these knots.
To obtain some information on the question we appeal to algorithms in the program SnapPy to generate estimates of the
volume of these knots.
We perform the following manipulations on the formula for $T_k\bigl(W(3,n)\bigr)$. First take the $k$th root
of the expression and then divide by the crossing number $2n$ to obtain an expression whose limit as
$n$ tends to infinity is finite. Then multiply by a normalization constant $C_k$ so that
\begin{equation*}
\lim_{n \ra \infty}C_k \frac{\sqrt[k]{T_k\bigl(W(3,n)\bigr)}}{2n} = 2 \, v_{{\rm tet}}.
\end{equation*}
In figure \ref{fig:comparingbounds}
we show the upper bound from equation \eqref{CKPboundspecialrelative} as a horizontal line at the top of the
plot and the lower bound as the lowest curve. Values ${\rm vol}\bigl(W(3,n)\bigr)/2n$ according to SnapPy
are plotted as points.
We also plot $C_k \cdot \sqrt[k]{T_k\bigl(W(3,n)\bigr)}/2n$ for $k=2$, $3$, and $4$. We see that all three of these
curves provide better lower bounds on the relative volume than the lower bound given in \eqref{CKPboundspecialrelative}.
Indeed, for $n$ sufficiently large the lower bound from $T_2$ is better than the bound from $T_3$, which is, in turn,
better than the bound from $T_4$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{comparebounds_31jan.eps}
\caption{Comparing bounds}
\label{fig:comparingbounds}
\end{figure}
\begin{theorem}
\label{twistnumber2}
For $n \geq 5$, the second twist number of $W(3,n)$ is $n(n-1)$.
\end{theorem}
\begin{proof}
Comparing \eqref{Vformal1} with \eqref{Vformal2}, and noting that $C_{n,12}(t)$ and $C_{n,21}(t)$ start in degrees $0$ and $1$, respectively,
we want to compute the term $p_3\,t^3$ from the truncated expansion
\begin{multline*}
p_1\, t + p_2\,t^2 + p_3\,t^3
\\
= (1+t)^2(c_{n,0,0} + c_{n,0,1}t + c_{n,0,2}\,t^2 + c_{n,0,3}\,t^3)
+ (t^2+t^3)\bigl((c_{n,1,0} + c_{n,1,1}\,t) + (c_{n,2,0} + c_{n,2,1}\,t)\bigr).
\end{multline*}
Extracting the coefficient of $t^3$ and substituting from \eqref{trailingcoefftable} and \eqref{degonetable} as well as
equation \eqref{cn0deg2} yields
\begin{multline} \label{p3}
p_3 = (c_{n,0,3} + 2 c_{n,0,2} + c_{n,0,1}) + (c_{n,1,0} + c_{n,1,1}) + (c_{n,2,0} + c_{n,2,1})
\\
=( c_{n,0,3} + 2(-1)^{n-1}(n+1)+ (-1)^n) + \bigl( (-1)^{n-1} + (-1)^n(n+1) \bigr) + \bigl(0+ (-1)^{n} \bigr)
\\
= c_{n,0,3} + (-1)^{n-1}n. \hspace{12em}
\end{multline}
We reduce the recursive formula \eqref{cn0} modulo $t^4$ to compute $c_{n,0,3}$.
\begin{align*}
C_{n,0}(t) &= t^2\cdot C_{n-1,21}(t) - t(t-1)\cdot C_{n-1,1}(t)
\\
& \equiv t^2(c_{n-1,21,0} + c_{n-1,21,1}\, t) - t^2(c_{n-1,1,0} + c_{n-1,1,1}\,t) + t(c_{n-1,1,0} + c_{n-1,1,1}\,t + c_{n-1,1,2}\,t^2)
\\
&\quad \mod t^4,
\end{align*}
so after extracting the coefficient of $t^3$, we have
\begin{multline} \label{cn0deg3}
c_{n,0,3} = c_{n-1,21,1} - c_{n-1,1,1} + c_{n-1,1,2}
\\
= (-1)^{n-1}-(-1)^{n-1}n + c_{n-1,1,2} = (-1)^{n}(n{-}1) + c_{n-1,1,2}.
\end{multline}
Thus, we need a formula for $c_{n-1,1,2}$. For this return to the recursive formula \eqref{cn1} and differentiate twice, obtaining
\begin{equation*}
C_{n,1}^{(2)}(q) = -2\cdot C_{n-1,1}(q) - 4(q-1)\cdot C_{n-1,1}'(q) - (q-1)^2\cdot C_{n-1,1}^{(2)}(q) - 2 C_{n-1,0}'(q) -(q-1)\cdot C_{n-1,0}^{(2)}(q).
\end{equation*}
Substituting $q=0$, we get
\begin{align*}
2 c_{n,1,2} &= -2 c_{n-1,1,0} + 4 c_{n-1,1,1} - 2 c_{n-1,1,2} - 2 c_{n-1,0,1} +2 c_{n-1,0,2},
\\
&=-2\cdot (-1)^{n-2} + 4 \cdot (-1)^{n-1}n - 2 c_{n-1,1,2} - 2(-1)^{n-1} + 2 (-1)^{n-2}\,n,
\end{align*}
applying propositions \ref{trailing} and \ref{degreeone} and formula \eqref{cn0deg2}. Note that the use of
\eqref{cn0deg2} requires $n{-}1 \geq 3$, so we have to have $n\geq 4$.
Rewriting this expression, we get
\begin{equation}
c_{n,1,2} + c_{n-1,1,2} = (-1)^{n-1}n, \quad \text{for $n \geq 4$.} \label{cn1deg2recursion}
\end{equation}
Now we create a closed form expression for $c_{n,1,2}$ by forming a telescoping sum.
\begin{align*}
c_{n,1,2} + c_{n-1,1,2} &= (-1)^{n-1}n
\\
- c_{n-1,1,2} - c_{n-2,1,2} &= (-1)^{n-1}(n{-}1)
\\ &\cdots
\\
(-1)^k c_{n-k,1,2} + (-1)^k c_{n-k-1,1,2} &= (-1)^k(-1)^{n-k-1}(n{-}k)
\\
&\cdots
\\
(-1)^{n-5} c_{5,1,2} + (-1)^{n-5}c_{4,1,2} &= (-1)^{n-1}5
\\
(-1)^{n-4} c_{4,1,2} + (-1)^{n-4}c_{3,1,2} &= (-1)^{n-1}4.
\end{align*}
Adding these equations yields
\begin{multline*}
c_{n,1,2} + (-1)^{n-4}c_{3,1,2} = (-1)^{n-1}\sum_{k=4}^n k
\\
= (-1)^{n-1}\biggl( \frac{n(n+1)}{2} - \frac{3(3+1)}{2} \biggr)
= (-1)^{n-1}\bigl( n(n+1)/2 - 6\bigr) .
\end{multline*}
From section \ref{PolynomialsExtra},
$C_{3,1}(q) = 1 - 4\,q + 7\,q^2 - 7\,q^3 + 4 \, q^4 - q^5$, so $c_{3,1,2} = 7$ and we obtain
\begin{equation}
\label{cn1deg2}
c_{n,1,2} = (-1)^{n-1}\Bigl( \frac{n(n+1)}{2}+1\Bigr), \quad \text{for $n \geq 4$.}
\end{equation}
Substituting the results of \eqref{cn0deg3} and \eqref{cn1deg2} into \eqref{p3}, we obtain
\begin{align}
\label{p3final}
v_{-n+2} = p_3 &= (-1)^{n-1}n + c_{n,0,3} \notag
\\
&= (-1)^{n-1}n + \bigl( (-1)^{n}(n{-}1) + c_{n-1,1,2}\bigr) \notag
\\
&= \bigl( (-1)^{n-1}n + (-1)^{n}(n{-}1)\bigr) + (-1)^{n-2}(n-1)n/2 +(-1)^{n-2} \notag
\\
&= (-1)^n(n-1)n/2.
\end{align}
for $n{-}1 \geq 4$, or $n\geq 5$.
We conclude the second twist number for $W(3,n)$ is $\abs{v_{-n+2}} + \abs{v_{n-2}}= n(n-1)$.
\end{proof}
\begin{theorem} \label{twistnumber3}
For $n \geq 5$ the coefficient of $t^{-n+3}$ in the Jones polynomial $V_{W(3,n)}(t)$ is
\begin{equation*}
v_{-n+3} = (-1)^{n-1}\bigl(n(n-1)(n-2)/6 + n\bigr),
\end{equation*}
so third twist number for $W(3,n)$ is $\abs{v_{-n+3}} + \abs{v_{n-3}}= n(n-1)(n-2)/3 + 2n$.
\end{theorem}
\begin{proof}
The essential point is to compute the coefficient $p_4$ in the expansion \eqref{Vformal2}. Starting from the
truncated polynomial expression
\begin{multline*}
(1+2t+t^2)(c_{n,0,1}t+c_{n,0,2}t^2+ c_{n,0,3}t^3 + c_{n,0,4}t^4)
\\
+ (t^2 + t^3)\bigl[(c_{n,1,0} + c_{n,1,1}t + c_{n,1,2}t^2) + (c_{n,2,0} + c_{n,2,1}t + c_{n,2,2}t^2)\bigr]
\\
+ t^4(c_{n,12,0}+c_{n,21,0}),
\end{multline*}
we extract the coefficient of $t^4$, obtaining
\begin{align}
p_4 &= (c_{n,0,4}+2\,c_{n,0,3} + c_{n,0,2}) + \bigl[(c_{n,1,1}+c_{n,1,2}) + (c_{n,2,1}+c_{n,2,2})\bigr] + (c_{n,12,0}+c_{n,21,0})\notag
\\
&= (c_{n,0,4}+2\,c_{n,0,3} + c_{n,0,2}) + \bigl[((-1)^n(n+1)+c_{n,1,2}) + ((-1)^n+c_{n,2,2})\bigr] + (-1)^{n-1} \notag
\intertext{by \eqref{trailingcoefftable} and \eqref{degonetable},}
&= (c_{n,0,4}+ 2\,c_{n,0,3} +(-1)^{n-1}(n{+}1)) + \bigl[((-1)^n(n+1)+c_{n,1,2}) + c_{n,2,2})\bigr] \notag
\intertext{evaluating $c_{n,0,2} = (-1)^{n-1}(n{+}1)$ for $n \geq 3$ by \eqref{cn0deg2}, }
&= c_{n,0,4} + 2\, c_{n,0,3} + c_{n,1,2} + c_{n,2,2} \notag
\\
&= c_{n,0,4} +2 \bigl( (-1)^{n}(n-1) + c_{n-1,1,2} \bigr) + c_{n,1,2} + c_{n-1,1,1} \notag
\intertext{substituting for $c_{n,0,3}$ from \eqref{cn0deg3} and using \eqref{cn2} which implies $c_{n,2,2}=c_{n-1,1,1}$, }
&= c_{n,0,4} + 2(-1)^{n}(n-1) + c_{n-1,1,2} + (-1)^{n-1}n + (-1)^{n-1}n \notag
\intertext{since $ c_{n,1,2} + c_{n-1,1,2} = (-1)^{n-1}n$ by \eqref{cn1deg2recursion} and $c_{n-1,1,1} = (-1)^{n-1}n$ by \eqref{degonetable}, }
&= c_{n,0,4} +c_{n-1,1,2} + 2(-1)^{n-1} \label{p4v1}
\end{align}
Compute $c_{n,0,4}$ from the recursion formula \eqref{cn0} reduced modulo $q^5$, which yields
\begin{multline*}
c_{n,0,0} + c_{n,0,1}\,q + c_{n,0,2}\,q^2 + c_{n,0,3}\,q^3 + c_{n,0,4}\,q^4
\\
= q^2\bigl( c_{n-1,21,0} + c_{n-1,21,1}\, q + c_{n-1,21,2}\, q^2 \bigr)
- q^2\bigl( c_{n-1,1,0} + c_{n-1,1,1}\, q + c_{n-1,1,2}\,q^2 \bigr)
\\
+ q\bigl( c_{n-1,1,0} + c_{n-1,1,1}\, q + c_{n-1,1,2}\,q^2 + c_{n-1,1,3}\,q^3\bigr)
\end{multline*}
Extracting the coefficients of $q^4$ gives
\begin{equation}
\label{cn04step1}
c_{n,0,4} = c_{n-1,21,2} - c_{n-1,1,2} + c_{n-1,1,3},
\end{equation}
so we have
\begin{align}
p_4 &= c_{n,0,4} +c_{n-1,1,2} + 2(-1)^{n-1} = (c_{n-1,21,2} - c_{n-1,1,2} + c_{n-1,1,3})+c_{n-1,1,2} + 2(-1)^{n-1} \notag
\\
&= c_{n-1,21,2} +c_{n-1,1,3} + 2(-1)^{n-1} \label{p4v2}
\end{align}
We now deal with $c_{n-1,21,2}$
by reducing the recurrence relation \eqref{cn21} mod $q^3$.
We get
\begin{multline*}
c_{n,21,0} + c_{n,21,1}\,q + c_{n,21,2}\,q^2
\\
\equiv (-q{+}1)\bigl(c_{n-1,2,0} + c_{n-1,2,1}\,q + c_{n-1,2,2}\,q^2\bigr) + q \bigl(c_{n-1,12,0}+ c_{n-1,12,1}\, q \bigr)
\\
-(q{-}1)^2\bigl( c_{n-1,21,0} + c_{n-1,21,1} \, q + c_{n-1,21,2}\, q^2 \bigr) \mod q^3.
\end{multline*}
Extracting the coefficient of $q^2$ gives
\begin{equation*}
c_{n,21,2} = c_{n-1,2,2} - c_{n-1,2,1} + c_{n-1,12,1}
+ (-c_{n-1,21,2} + 2\,c_{n-1,21,1} - c_{n-1,21,0})
\end{equation*}
or, since $C_{n-1,2}(q) = q\cdot C_{n-2,1}(q)$ by \eqref{cn2}, we have $c_{n-1,2,2}= c_{n-1,1,1}$, so
\begin{align}
c_{n,21,2} &= c_{n-2,1,1}- c_{n-1,2,1} + c_{n-1,12,1} + (-c_{n-1,21,2} + 2\,c_{n-1,21,1} - c_{n-1,21,0}) \notag
\\
&= (-1)^{n-2}(n-1) - (-1)^{n-1} + (-1)^{n-1}(n-1) - c_{n-1,21,2} + 2(-1)^{n-1} - 0, \notag
\intertext{substituting from \eqref{degonetable} and \eqref{trailingcoefftable},}
&= - c_{n-1,21,2} + (-1)^{n-1}.
\end{align}
We compute an alternating sum of another sequence of equalities
\begin{align*}
c_{n,21,2} + c_{n-1,21,2} &= (-1)^{n-1}
\\
(-1)\bigl( c_{n,21,2} + c_{n-1,21,2}\bigr) &= (-1)(-1)^{n-2}
\\
&\cdots
\\
(-1)^{k-1}\bigl(c_{n-k+1,21,2} + c_{n-k,21,2}\bigr) &= (-1)^{k-1}(-1)^{n-k}
\\
(-1)^k\bigl(c_{n-k,21,2} + c_{n-k-1,21,2}\bigr) &= (-1)^k(-1)^{n-k-1}
\end{align*}
Adding the equations we get
\begin{equation*}
c_{n,21,2} + (-1)^kc_{n-k-1,21,2} = (-1)^{n-1}(k{+}1),
\; \text{or} \;
c_{n,21,2} + (-1)^{n-j-1}c_{j,21,2} = (-1)^{n-1}(n{-}j),
\end{equation*}
if we write $k=n-j-1$, so that $j=n-k-1$.
Referring to section \ref{PolynomialsExtra}, the first $j$ for which $c_{j,21,2} \neq 0$ is $j=3$,
and $C_{3,21}(q) = -q+ 2\,q^2 - q^3$, so $c_{3,21,2} = 2$. Substituting and rearranging,
\begin{equation}
\label{cn21deg2}
c_{n,21,2} = (-1)^{n-1}(n{-}3) - (-1)^nc_{3,21,2} = (-1)^{n-1}\bigl((n-3)+2 \bigr) = (-1)^{n-1}(n-1),
\end{equation}
and this holds for $n \geq 3$. Substituting into \eqref{p4v2}, we get
\begin{equation}
\label{p4v3}
p_4 = c_{n-1,1,3} + (-1)^{n-2}(n-2) + 2(-1)^{n-1} = c_{n-1,1,3} + (-1)^nn + 4(-1)^{n-1}.
\end{equation}
The most straightforward approach to computing $c_{n-1,1,3}$ is through the recursion relation \eqref{cn1}.
Reducing the relation mod $q^4$ gives
\begin{multline*}
c_{n,1,0} + c_{n,1,1}\, q + c_{n,1,2}\,q^2 + c_{n,1,3}\, q^3
\\
(-1+ 2q - q^2)(c_{n-1,1,0} + c_{n-1,1,1}\,q + c_{n-1,1,2}q^2 + c_{n-1,1,3}\,q^3)
\\
+ (1-q) (c_{n-1,0,0} + c_{n-1,0,1}\,q + c_{n-1,0,2}\,q^2 + c_{n-1,0,3}\,q^3) \mod q^4,
\end{multline*}
and extracting the coefficient of $q^3$ gives
\begin{align*}
c_{n,1,3} &= -c_{n-1,1,3} + 2\, c_{n-1,1,2} - c_{n-1,1,1} + c_{n-1,0,3} - c_{n-1,0,2}
\\&=-c_{n-1,1,3} + 2\, c_{n-1,1,2} + c_{n-1,0,3},
\end{align*}
since $c_{n-1,1,1} + c_{n-1,0,2} = (-1)^{n-1}n + (-1)^{n-2}n = 0$ for $n \geq 4$ by
\eqref{degonetable} and \eqref{cn0deg2}.
Making the substitution for $c_{n-1,0,3}$ from \eqref{cn0deg3},
\begin{align} \begin{split}
c_{n,1,3} + c_{n-1,1,3} &= 2\,(-1)^{n-2}\Bigl(\frac{n(n-1)}{2}+1\Bigr) + c_{n-1,0,3}
\\
&= (-1)^{n-2}\bigl( n(n-1)+2\bigr) + (-1)^{n-1}(n-2) + c_{n-2,1,2}
\\
&= (-1)^{n-2}\bigl( n(n-1)+2\bigr) + (-1)^{n-1}(n{-}2) + (-1)^{n-3}\bigl((n{-}1)(n{-}2)/2 + 1\bigr)
\\
&= (-1)^n\frac{n^2}{2} + (-1)^{n-1}\frac{n}{2} + (-1)^n2
\end{split}
\end{align}
Now generate a telescoping sum from the chain of equalities
\begin{align*}
c_{n,1,3} + c_{n-1,1,3} &= (-1)^n\biggl[\frac{n^2}{2} - \frac{n}{2} +2\biggr]
\\
(-1)\bigl[ c_{n-1,1,3} + c_{n-2,1,3}\bigr] &= (-1)(-1)^{n-1}\biggl[\frac{(n{-}1)^2}{2} -\frac{n{-}1}{2} + 2\biggr]
\\
&\cdots
\\
(-1)^k\bigl[c_{n-k,1,3} + c_{n-k-1,1,3}\bigr] &=(-1)^k (-1)^{n-k}\biggl[\frac{(n{-}k)^2}{2} - \frac{(n{-}k)}{2} + 2 \biggr]
\end{align*}
Adding these equalities gives
\begin{equation*}
c_{n,1,3} + (-1)^kc_{n-k-1,1,3} = (-1)^n\biggl[\frac{1}{2}\sum_{j=n-k}^n j^2 - \frac{1}{2} \sum_{j=n-k}^n j + (k{+}1)2\biggr]
\end{equation*}
and, if we write $n-k-1 = \ell$, so that $n-k = \ell + 1$ and $k+1 = n - \ell$,
\begin{align}
\begin{split}
c_{n,1,3} + (-1)^{n-\ell -1} c_{\ell,1,3}
&= (-1)^n\biggl[\frac{1}{2}\sum_{j=\ell+1}^n j^2 - \frac{1}{2} \sum_{j=\ell+1}^n j + (n{-}\ell)2\biggr]
\\
&= (-1)^n\biggl[\frac{1}{2}\frac{n(n{+}1)(2n{+}1)}{6} - \frac{1}{2}\frac{\ell(\ell{+}1)(2\ell{+}1)}{6}
\\ &\hspace{6em}
- \frac{1}{2} \frac{n(n{+}1)}{2} + \frac{1}{2}\frac{\ell(\ell{+}1)}{2} + (n{-}\ell)2 \biggr]
\end{split}
\end{align}
Taking $\ell=4$, so that, from section \ref{PolynomialsExtra},
$ C_{4,1}(q) = -1 + 5\,q - 11 \, q^2 + 16\, q^3 - \cdots$ and $c_{4,1,3} = 16$,
we evaluate
\begin{align*}
\begin{split}
c_{n,1,3} - (-1)^n(16) &=
(-1)^n\biggl[\frac{1}{2} \frac{2n^3}{6} + \frac{1}{2}\frac{3n^2}{6} + \frac{1}{2} \frac{n}{6} - \frac{1}{2}\frac{4\cdot 5 \cdot 9}{6}
\\
&\hspace{8em} - \frac{1}{2}\frac{n^2}{2} - \frac{1}{2}\frac{n}{2} + \frac{1}{2}\frac{4\cdot 5}{2} + 2(n-4)\biggr]
\\
&= (-1)^n \biggl[\frac{n^3}{6} - \frac{n}{6} - 10 + 2n -8 \biggr]
\\
&= (-1)^n \biggl[ \frac{n(n-1)(n+1)}{6} + 2n - 18 \biggr]
\end{split}
\end{align*}
Therefore,
\begin{equation}
\label{cn1deg3}
c_{n,1,3} = (-1)^n\biggl[ \frac{n(n-1)(n+1)}{6} + 2n - 18 + 16 \biggr] = (-1)^n\biggl[ \frac{n(n-1)(n+1)}{6} + 2n - 2\biggr],
\end{equation}
a formula valid for $n \geq 4$, but failing for $n = 3$, so
\begin{align}
\lambda_{-n+ 3} = p_4 &= c_{n-1,1,3} + (-1)^nn + 4(-1)^{n-1} \notag
\\
&= (-1)^{n-1}\biggl[ \frac{n(n-1)(n-2)}{6} + 2(n-1) - 2\biggr] - (-1)^{n-1}n + 4(-1)^{n-1} \notag
\\
&= (-1)^{n-1}\bigl( n(n-1)(n-2)/6 + n \bigr), \label{p4v4}
\end{align}
which is consequently valid for $n \geq 5$. This ends the proof.
\end{proof}
\begin{figure}[h!]
\centering
\includegraphics[width = 0.3\linewidth]{secondtwistvsvolume_22jan.eps}
\caption{Second twist number versus volume}
\label{fig:secondtwistvsvolume}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width = 0.3\linewidth]{thirdtwistvsvolume_22jan.eps}
\caption{Third twist number versus volume}
\label{fig:thirdtwistvsvolume}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width = 0.3\linewidth]{fourthtwistvsvolume_22jan.eps}
\caption{Fourth twist number versus volume}
\label{fig:fourthtwistvsvolume}
\end{figure}
\section{From the Jones Polynomial to Khovanov homology} \label{Jones-to-Khovanov}
In this section we amplify Theorem \ref{locateKH}, at least the first part of it. \addtocounter{section}{-3} \addtocounter{theorem}{4}
\begin{theorem}
For a weaving knot $W(2k{+}1,n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k{+}1, n) \bigr)$ lies on the
lines
\begin{equation*}
j = 2i \pm 1.
\end{equation*}
For a weaving knot $W(2k, n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k, n) \bigr)$ lies on the lines
\begin{equation*}
j = 2i + n -1 \pm 1
\end{equation*}
\end{theorem}
We have the following definition of the bi-graded Euler characteristic
associated to Khovanov homology.
\begin{equation*}
Kh(L)(t,Q) \stackrel{{\rm def}}{=} \sum t^iQ^j \dim {\mathcal H}^{i,j}(L)
\end{equation*}
\addtocounter{section}{3} \addtocounter{theorem}{-5}
\begin{theorem}[Theorem 1.1, \cite{Lee_Endo04}] \label{Lee1}
For an oriented link $L$, the graded Euler characteristic
\begin{equation*}
\sum_{i,j \in \bZ} (-1)^iQ^j \dim {\mathcal H}^{i,j}(L)
\end{equation*}
of the Khovanov invariant ${\mathcal H}(L)$ is equal to $(Q^{-1}{+}Q)$
times the Jones polynomial $V_L(Q^2)$ of $L$.
In terms of the associated polynomial $Kh(L)$,
\begin{equation} \label{JonesfromKhovanov}
Kh(L)(-1, Q) = (Q^{-1}+Q)V_L(Q^2). \qed
\end{equation}
\end{theorem}
\begin{theorem}[Compare Theorem 1.4 and subsequent remarks from \cite{Lee_Endo04}] \label{Lee4}
For an alternating knot $L$, its Khovanov invariants ${\mathcal H}^{i,j}(L)$ of
degree difference $(1,4)$ are paired except in the $0$th cohomology group.
\qed \end{theorem}
This fact may be expressed in terms of the polynomial $Kh(L)$, as follows.
There is another polynomial $Kh'(L)$ in one variable and an equality
\begin{equation} \label{polynomialperiodicity}
Kh(L)(t, Q) = Q^{-\sigma(L)}\bigl\{ (Q^{-1}{+}Q) + (Q^{-1}+tQ^2\cdot Q)\cdot Kh'(L)(tQ^2) \bigr\}
\end{equation}
When we combine theorems \ref{Lee1} and \ref{Lee4}, we find that the bi-graded Euler characteristic and
the Jones polynomial of an alternating link determine one another. Obviously, the equality \eqref{JonesfromKhovanov}
shows that one knows $V_L$ if one knows $Kh(t,Q)$.
To obtain $Kh(t,Q)$ from $V_L(Q^2)$ requires a certain amount of manipulation. Implementing these
manipulations in {\em Maple} and {\em Mathematica} is an important step in our experiments.
Setting $t{=}-1$ in \eqref{polynomialperiodicity} and combining with equation \eqref{JonesfromKhovanov}, one has
\begin{align*}
(Q^{-1} + Q) \cdot V_L(Q^2) &= Q^{-\sigma(L)}\bigl\{ (Q^{-1}{+}Q) + (Q^{-1}- Q^3)\cdot Kh'(L)(-Q^2) \bigr\}.
\intertext{Consequently,}
V_L(Q^2) &= Q^{-\sigma(L)}\bigl\{ 1 + \frac{(Q^{-1}- Q^3)}{(Q^{-1}{+}Q)}\cdot Kh'(L)(-Q^2) \bigr\}
\\
&= Q^{-\sigma(L)}\bigl\{ 1 + (1 - Q^2)\cdot Kh'(L)(-Q^2)\bigr\}.
\intertext{Furthermore,}
Q^{\sigma(L)} \cdot V_L(Q^2) -1 &= (1 - Q^2)\cdot Kh'(L)(-Q^2),
\intertext{or}
Kh'(L)(-Q^2) &= (1 - Q^2)^{-1}\cdot \bigl(Q^{\sigma(L)} \cdot V_L(Q^2) -1\bigr) .
\end{align*}
Replacing $Q^2$ in the last equation by $-tQ^2$ is the last step to obtain $Kh'(L)$ from the Jones polynomial.
Within a computer algebra system, one must first replace $Q^2$ by $-X$ and then replace $X$ by $tQ^2$. Once one has
$Kh'(L)(tQ^2)$, one obtains $Kh(t,Q)$ directly from equation~\eqref{polynomialperiodicity}.
\begin{example}
We have computed $V_{W(3,2)}(t) = t^{-2} - t^{-1} + 1 - t + t^2$ in example \ref{jonesfig8knot}, so
\begin{align*}
Kh'\bigl(W(3,2)\bigr)(-Q^2) &= (1- Q^2)^{-1} \cdot \bigl( Q^0 \cdot ( Q^{-4} - Q^{-2} - Q^2 + Q^4) \bigr)
\\
&= (1-Q^2)^{-1} \cdot \bigl( (1 - Q^2) \cdot ( Q^{-4} - Q^2) \bigr)
\\
&= Q^{-4} - Q^2.
\end{align*}
It follows that $Kh'\bigl( W(3,2) \bigr) (tQ^2) = t^{-2}Q^{-4} + tQ^2$, and
\begin{multline*}
Kh\bigl(W(3,2)\bigr) (t, Q) = (Q+Q^{-1})+ (Q^{-1} + tQ^3)(t^{-2}Q^{-4} + tQ^2)
\\
= t^{-2}Q^{-5} + t^{-1}Q^{-1} + Q^{-1} + Q + tQ + t^2Q^5.
\end{multline*}
\end{example}
\section{Khovanov homology examples} \label{Khovanov}
Once one has the Khovanov polynomial one can make a plot of the Khovanov
homology in an $(i,j)$-plane as in this example. The Betti number $\dim {\mathcal H}^{i,j}\bigl( W(3,11) \bigr)$ is plotted
at the point with coordinates $(i,j)$.
\begin{figure}[h]
\centering
\includegraphics[width=6in]{KHW3_10.eps}
\caption{Khovanov homology of $W(3,10)$}
\label{fig:w310}
\end{figure}
Clearly, as $n$ gets larger, it is going to be harder to make sense of such
plots. Notice that the ``knight move'' $(1,4)$-periodicity
of the Khovanov homology for these knots essentially makes the information on one of the lines $j-2i = \pm 1$ redundant.
Before we continue to explore Khovanov homology with rational coefficients,
we observe we can also compute the integral Khovanov homology.
By Corollary 5 of \cite{Khovanovtorsion} there is only torsion of order 2 in the integral Khovanov homology.
Even better, there are rules for calculating the number of $\bZ/2\bZ$-summands
present. We have already noted how rational Khovanov homology spaces are related by so-called ``knight moves.''
Except when one lands or starts in a space ${\mathcal H}^{0,*}\bigl(W(3,n)\bigr)$, a move from
${\mathcal H}^{i,j}\bigl(W(3,n)\bigr)$ to ${\mathcal H}^{i+1,j+4}\bigl(W(3,n)\bigr)$ is a move from one space
into another space of the same dimension.
If one reduces the dimensions of the spaces ${\mathcal H}^{0,*}\bigl(W(3,n)\bigr)$ by one, then this phenomenon persists without
qualification on the bidegrees. Having made this adjustment,
Shumakovitch \cite[1.G~Definitions]{Khovanovtorsion} provides the following rules for
computing the integral Khovanov homology:
If $\dim {\mathcal H}^{i,2i+1}\bigl(W(3,n)\bigr) = a \neq 0$, then
${\mathcal H}^{i,2i+1}\bigl(W(3,n); \bZ \bigr)$ is torsion-free of rank $a$.
Moreover, for every non-zero pair linked by a knight move,
${\mathcal H}^{i-1,2i-3}\bigl(W(3,n); \bZ \bigr)$ and ${\mathcal H}^{i,2i+1}\bigl(W(3,n); \bZ \bigr)$ have the same
rank, but groups along the line $j{-}2i = -1$ may have torsion.
In fact, the two-torsion part of ${\mathcal H}^{i,2i-1}\bigl(W(3,n)\;\bZ \bigr)$
is an abelian $2$-group $(\bZ/2\bZ)^a$. The table \ref{IKHW34} shows the rules in operation
for the knot $W(3,4)$, also identified as 8\_18 in standard knot tables. Integer entries along
the line $j{-}2i = 1$ indicate
the ranks of free abelian groups and an entry $r, a_2$ along the line
$j {-} 2i= -1$ indicates a free abelian group of rank $r$ summed with a 2-group $(\bZ/2\bZ)^a$.
\begin{table}
\caption{Integral Khovanov homology of $W(3,4)$}\label{IKHW34}
\begin{center}
\begin{tabular}[h]{||c|c|c|c|c|c|c|c|c|c||} \hline \hline
& -4 & -3 & -2 & -1 & 0 & 1 &2 & 3 & 4
\\ \hline
9 & & & & & & & & & 1
\\ \hline
7 & & & & & & & & 3 & $1_2$
\\ \hline
5 & & & & & & & 3 & 1, $3_2$ &
\\ \hline
3 & & & & & & 4 & 3, $3_2$ & &
\\ \hline
1 & & & & & 5 & 3 ,$4_2$ & & &
\\ \hline
-1 & & & & 3 & 5, $4_2$ & & & &
\\ \hline
-3 & & & 3 & 4, $3_2$ & & & & &
\\ \hline
-5 & & 1 & 3, $3_2$ & & & & & &
\\ \hline
-7 & & 3, $1_2$ & & & & & & &
\\ \hline
-9 & 1 & & & & & & & &
\\ \hline \hline
\end{tabular}
\end{center}
\end{table}
Returning to rational Khovanov homology,
we take advantage of the ``knight move'' periodicity and simplify by recording the Betti numbers from only along the line $j- 2i = 1$.
In order to study the asymptotic behavior of Khovanov homology we have to normalize the data. This is done
by computing the total rank of the Khovanov homology along the line and dividing each Betti number by the
total rank. We obtain normalized Betti numbers that sum to one.
This raises the possibility of approximating the distribution of normalized Betti numbers by a probability distribution.
For our baseline experiments we choose to use the normal $N(\mu, \sigma^2)$ probability density function
\begin{equation*}
f_{\mu, \sigma^2}(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \Bigl( - \frac{(x-\mu)^2}{2\sigma^2} \Bigr)
\end{equation*}
Fit a quadratic function
$q_n(x)= -(\alpha \, x^2 - \beta\, x + \delta)$
to the logarithms of the normalized Khovanov dimensions along the line $j=2i+1$
and exponentiate the quadratic function.
Since the total of the normalized dimensions is 1, we normalize the
exponential, obtaining
\begin{equation*}
\rho_n(x) = A_n e^{q_n(x)} \quad \text{satisfying} \quad \int_{-\infty}^{\infty} \rho_n(x) \; dx = 1.
\end{equation*}
To obtain a formula for $A_n$, complete the square
\begin{equation*}
q_n(x) = -\alpha \cdot \bigl( x - (\beta/2\alpha) \bigr)^2 +\bigl( (\beta^2/4\alpha) - \delta\bigr).
\end{equation*}
Then consider
\begin{align*}
1 &= A_n \int_{-\infty}^{\infty} \exp q_n(x) \; dx
\\
&= A_n \cdot
\int_{-\infty}^{\infty} \exp \bigl((\beta^2/4\alpha) - \delta \bigr)
\cdot \exp \bigl( -\alpha \cdot \bigl( x -(\beta/2\alpha) \bigr)^2\bigr) \; dx
\\
&= A_n \cdot \exp \bigl((\beta^2/(4\alpha) - \delta \bigr) \cdot
\int_{-\infty}^{\infty} \exp \bigl( -\alpha \cdot \bigl( x -(\beta/2\alpha) \bigr)^2\bigr) \; dx
\\
&= A_n \cdot \exp \bigl((\beta^2/4\alpha) - \delta\bigr) \cdot \sqrt{\pi/\alpha}
\end{align*}
Thus, the expression for $A_n$ is
\begin{equation*}
A_n = \exp\bigl( -\bigl((\beta^2/4\alpha) - \delta\bigr)\bigr) \cdot \sqrt{\alpha/\pi}.
\end{equation*}
Equate the expressions
\begin{equation*}
\rho_n(x) = \frac{1}{\sigma_n \sqrt{2\pi}} \exp \Bigl( - \frac{(x-\mu_n)^2}{2\sigma_n^2} \Bigr)
\quad \text{and} \quad
\rho_n(x) = A_n \exp ( q_n(x)),
\end{equation*}
and observe $\mu_n = \beta/2\alpha$ by equating the two expressions for the location of the local maximum
of $\rho_n(x)$. Then the efficient way to the parameter $\sigma_n$ is to solve the equation
\begin{equation*}
\frac{1}{\sigma_n \sqrt{2\pi}} = \rho_n(\beta/2\alpha) = A_n \exp( q_n(\beta/2\alpha))
= \exp\bigl( -\bigl((\beta^2/4\alpha) - \delta\bigr)\bigr) \cdot \sqrt{\alpha/\pi} \cdot
\exp\bigl( (\beta^2/4\alpha) - \delta \bigr),
\end{equation*}
obtaining $\sigma_n = 1/ \sqrt{2\alpha} $.
Working this out for $W(3,10)$, and carrying only 3 decimal places,
the raw dimensions are
\begin{center}
\begin{small}
\begin{tabular}[h!]{c|c|c|c|c|c|c|c|c|c|c}
$i$ & -9 & -8 & -7 & -6 & -5 & -4 & -3& -2 &-1 & 0
\\
$\dim$ & 1& 9& 36& 94& 196& 346& 529& 721& 879& 970
\\
$i$ & 1& 2& 3& 4& 5& 6& 7& 8& 9& 10
\\
$\dim$ & 971& 879& 721& 529& 346& 196& 94& 36& 9& 1
\end{tabular}
\end{small}
\end{center}
and, to three significant digits, the logarithms of the normalized dimensions are
\begin{center}
\begin{small}
\begin{tabular}[h!]{c|c|c|c|c|c|c|c|c|c|c}
$i$ & -9 & -8 & -7 & -6 & -5 & -4 & -3& -2 &-1 & 0
\\
& -17.9& -15.7& -14.3& -13.3& -12.6& -12.0& -11.6& -11.3& -11.1& -11.0
\\
$i$ & 1& 2& 3& 4& 5& 6& 7& 8& 9& 10
\\
& -11.0& -11.1& -11.3& -11.6& -12.0& -12.6& -13.3& -14.3& -15.7& -17.9
\end{tabular}
\end{small}
\end{center}
Fitting a quadratic to this information, we get
\begin{equation*}
q_{10}(x) = -10.7 + 0.0720\, x - 0.0720\, x^2,\quad \alpha = \beta = 0.0720, \quad \delta = 10.7.
\end{equation*}
To three significant digits $\mu_{10} = 0.500$ and $\sigma_{10} = 2.64$.
By the symmetry of Khovanov homology, the mean $\mu_n$ approaches $1/2$ rapidly, so this parameter is of little interest.
On the other hand, relating the parameter $\sigma_n$ to some geometric quantity, say, some hyperbolic invariant
of the complement of the link, is a very interesting problem.
For $W(3,10)$, the density function is
\begin{multline*}
\rho_{10}(x) = 11686.8431618280538\,\sqrt {{\pi }^{-1}} \\
\cdot \exp ({- 10.7018780565714309+ 0.0716848579220777243\,x- 0.0716848579220778631\,{x}^{2}})
\end{multline*}
When placed into standard form, $\mu_{10} =0.5000054030 $ and $\sigma_{10} = 2.640882970 $.
Figure \ref{fig:w310comp} compares the plot of the density function $\rho_{10}$ with a point plot of normalized dimensions.
\begin{figure}[h]
\centering
\includegraphics[height=2.0in]{khw3_10_11apr17.eps}
\caption{normalized homology of $W(3,10)$ compared with density function}
\label{fig:w310comp}
\end{figure}
\newline
For the knot $W(3,11)$
the expression for the density function is
\begin{multline*}
\rho_{11}(x) =29676.8676257830375\,\sqrt {{\pi }^{-1}} \\
\cdot \exp ({- 11.6724860231789886+ 0.0661625395821569817\,x- 0.0661623073574252735\,{x}^{2}})
\end{multline*}
When placed into standard form, $\mu_{11} =0.5000017550 $ and $\sigma_{11} = 2.749031276$.
Figure \ref{fig:w311comp} compares the plot of the density function $\rho_{11}$ with a point plot of normalized dimensions.
\begin{figure}[h!]
\centering
\includegraphics[height=2.0in]{khw3_11_11apr17.eps}
\caption{normalized homology of $W(3,11)$ compared with density function}
\label{fig:w311comp}
\end{figure}
\newline
For $W(3,22)$, the density function is
\begin{multline*}
\rho_{22}(x) =833596689.149608016\,\sqrt {{\pi }^{-1}} \\
\cdot \exp ({- 22.2219365040983057+ 0.0353061029354434300\,x- 0.0353061029347388616\,{x}^{2}})
\end{multline*}
When placed into standard form, $\mu_{22} =0.500000000 $ and $\sigma_{22} = 3.763224354$.
Figure \ref{fig:w322comp} compares the plot of the density function $\rho_{22}$ with a point plot of normalized dimensions.
\begin{figure}[h!]
\centering
\includegraphics[height=2.0in]{khw3_22_11apr17.eps}
\caption{normalized homology of $W(3,22)$ compared with density function}
\label{fig:w322comp}
\end{figure}
\newline
For $W(3,23)$, the density function is
\begin{multline*}
\rho_{23}(x) = 2113964949.23002362\,\sqrt {{\pi }^{-1}} \\
\cdot exp( {- 23.1731352596503442+ 0.0338545815354610105\,x- 0.0338545815348441914\,{x}^{2}})
\end{multline*}
When placed into standard form, $\mu_{23} =0.5000000000 $ and $\sigma_{23} =3.843052143 $.
Figure \ref{fig:w323comp} compares the plot of the density function $\rho_{23}$ with a point plot of normalized dimensions.
\begin{figure}[h!]
\centering
\includegraphics[height=2.0in]{khw3_23_11apr17.eps}
\caption{normalized homology of $W(3,23)$ compared with density function}
\label{fig:w323comp}
\end{figure}
\newline
{\em Maple} worksheets and, later, {\em Mathematica} notebooks will be available at URL \cite{software} prepared by the second-named author.
\section{Data Tables}\label{Data}
This section contains tables of data generated using {\em Maple} to implement some of the results of earlier sections.
The first table collects data for weaving knots $W(3,n)$ with $n \equiv 1 \mod 3$; the second table does the same for
weaving knots $W(3,n)$ with $n \equiv 2 \mod 3$.
In each table the first column lists the value of $n$;
the second column lists the total dimension of the Khovanov homology lying along the line $j = 2i{+}1$; and the third
column lists the dimension of the vector space ${\mathcal H}^{0,1}\bigl( W(3,n)\bigr)$.
Columns four and five display measures of the deviation of the proposed normal distributions from
the actual distribution of normalized dimensions.
In section \ref{Khovanov} we have approximated a distribution of normalized
Khovanov dimenstions by a standard normal distribution, and we have displayed
graphics comparing an actual distribution with its approximation.
To quantify those visual impressions, we compute and tabulate an $L^1$- and an $L^2$-deviation. Let
\begin{equation*}
\text{Total dimension} = \sum_{i=-2n}^{2n+1} \dim {\mathcal H}^{i, 2i+1}\bigl( W(3,n) \bigr).
\end{equation*}
For the $L^2$-comparison, we compute
\begin{equation*}
\Biggl( \sum_{i = -2n}^{2n+1} \biggl( \rho_n(i)- \frac{\dim {\mathcal H}^{i, 2i+1}\bigl( W(3,n) \bigr)}{\text{Total dimension}} \biggr)^2 \Biggr)^{1/2}
\end{equation*}
For the $L^1$-comparison, we compute
\begin{equation*}
\sum_{i = -2n}^{2n+1} \Babs{ \rho_n(i) - \frac{\dim {\mathcal H}^{i, 2i+1}\bigl( W(3,n) \bigr)}{\text{Total dimension} } }
\end{equation*}
The $L^2$ comparisons appear to tend to 0, whereas the $L^1$ comparisons
appear to be growing slowly.
\begin{table} \caption{Data for $W(3,n)$ with $n \equiv 1 \mod 3$}
\begin{tabular}[h!]{|c|c|c|c|c|c|}
$n$ & Total dimension & $\dim {\mathcal H}^{0,1}$ & $\sigma$ & $L^2$-comparison & $L^1$ comparison
\\
10 & 7563 & 970 & 2.64088 & 0.040510 & 0.134828
\\
13 & 135721 & 15418 & 2.95616 & 0.041133 & 0.150599
\\
16 & 2435423 & 250828 & 3.24564 & 0.040792 & 0.155995
\\
19 & 43701901 & 4146351 & 3.51339 & 0.040145 & 0.161336
\\
22 & 784198803 & 69337015 & 3.76322 & 0.039413 & 0.165763
\\
25 & 14071876561 & 1169613435 & 3.99810 & 0.038678 & 0.167576
\\
28 & 252509579303 & 19864129051 & 4.22032 & 0.037971 & 0.167790
\\
31 & 4531100550901 & 339205938364 & 4.43167 & 0.037303 & 0.170736
\\
34 & 81307300336923 & 5818326037345 & 4.63358 & 0.036676 & 0.172392
\\
37 & 1459000305513721 & 100173472277125 & 4.82719 & 0.036089 & 0.173119
\\
40 & 26180698198910063 & 1730135731194046 & 5.01342 & 0.035541 & 0.173178
\\
43 & 469793567274867421 & 29963026081609060 & 5.19306 & 0.035028 & 0.173812
\\
46 & 8430103512748703523 & 520131503664409798 & 5.36674 & 0.034546 & 0.175052
\\
49 & $1.51272\cdot 10^{20}$ & $ 9.04765\cdot10^{18}$ & 5.53502 & 0.034093 & 0.175779
\\
52 & $ 2.71447\cdot 10^{21}$ & $ 1.57670\cdot 10^{20} $ & 5.69838 & 0.033667 & 0.176100
\\
55 & $ 4.87091\cdot 10^{22} $ & $ 2.75210\cdot 10^{21} $ & 5.85721 & 0.033265 & 0.176098
\\
58 & $ 8.74050\cdot 10^{23} $ & $ 4.81071\cdot 10^{22} $ & 6.01187 & 0.032885 & 0.175898
\\
61 & $ 1.56842\cdot 10^{25} $& $ 8.42017\cdot 10^{23}$ & 6.16267 & 0.032524 & 0.176778
\\
64 & $ 2.81441\cdot 10^{26} $ & $ 1.47552\cdot 10^{25} $ & 6.30989 & 0.032182 & 0.177369
\\
67 & $5.05026\cdot 10^{27}$ & $ 2.58843\cdot 10^{26} $ & 6.45376 & 0.031857 & 0.177716
\\
70 &$ 9.06233\cdot 10^{28} $& $4.54520\cdot 10^{27}$ & 6.59451 & 0.031547 & 0.177859
\\
73 & $1.62617\cdot 10^{30}$ & $7.98842\cdot 10^{28}$ & 6.73233 & 0.031251 & 0.177831
\\
76 & $ 2.91804\cdot 10^{31}$ & $1.40517 \cdot 10^{30}$ & 6.86740 & 0.030968 & 0.177657
\\
79 & $5.23621\cdot 10^{32}$ & $ 2.47359 \cdot 10^{31}$ & 6.99986 & 0.030697 & 0.177995
\\
82 & $9.39600\cdot 10^{33} $& $4.35747 \cdot 10^{32}$& 7.12988 & 0.030437 & 0.178445
\\
85 & $1.68604\cdot 10^{35} $& $7.68116 \cdot 10^{33}$ & 7.25757 & 0.030188 & 0.178746
\\
88 & $ 3.02548\cdot 10^{36} $ & $1.35483 \cdot 10^{35}$ & 7.38305 & 0.029948 & 0.178918
\\
91 & $ 5.42901\cdot 10^{37} $ & $ 2.39106 \cdot 10^{36} $ & 7.50645 & 0.029718 & 0.178976
\\
94 & $ 9.74196\cdot 10^{38}$ & $ 4.22211 \cdot 10^{37} $ & 7.62786 & 0.029496 & 0.178935
\\
97 & $ 1.74812\cdot 10^{40} $ & $7.45910 \cdot 10^{38}$ & 7.74736 & 0.029282 & 0.178807
\\
100 & $3.13688\cdot 10^{41} $ & $1.31840 \cdot 10^{40} $& 7.86506 & 0.029075 & 0.178890
\\
121 & $1.87923\cdot 10^{50} $& $ 7.18477\cdot 10^{48} $ & 8.64424 & 0.027805 & 0.179577
\\
142 & $1.12580\cdot 10^{59} $ & $ 3.97500 \cdot 10^{57} $ & 9.35886& 0.026769& 0.180247
\\
163 & $6.74436\cdot 10^{67} $ & $2.22337 \cdot 10^{66}$ & 10.0227& 0.025900& 0.180596
\\
184 & $4.04037\cdot 10^{76}$ & $ 1.25398 \cdot 10^{75} $ & 10.6453& 0.025156& 0.180629
\\
205 & $ 2.42049\cdot 10^{85}$ & $7.11854 \cdot 10^{83} $ & 11.2334& 0.024508& 0.180907
\\
247 & $8.68689\cdot 10^{102}$ & $2.32816 \cdot 10^{101} $ & 12.3258& 0.023423& 0.181027
\\
289 & $3.11764\cdot 10^{120} $ & $7.72623 \cdot 10^{118} $ & 13.3289& 0.022542& 0.181268
\end{tabular}
\end{table}
\begin{table} \caption{Data for $W(3,n)$ with $n \equiv 2 \mod 3$}
\begin{tabular}{c|c|c|c|c|c}
$n$ & Total dimension & $\dim {\mathcal H}^{0,1}$ & $\sigma$ & $L^2$-comparison & $L^1$ comparison
\\
11 & 19801&2431& 2.74903& 0.040906& 0.141925
\\
14&355323&38983& 3.05533& 0.041079& 0.153170
\\
17&6376021&637993& 3.33710& 0.040595& 0.156595
\\
20&114413063&10591254& 3.59850& 0.039905& 0.163190
\\
23&2053059121&177671734& 3.84305& 0.039166& 0.166596
\\
26&36840651123&3004390818& 4.07348& 0.038438& 0.167789
\\
29&661078661101&51124396786& 4.29190& 0.037744& 0.168941
\\
32&11862575248703&874400336044& 4.49997 & 0.037089& 0.171411
\\
35&212865275815561&15018149469823& 4.69899& 0.036476& 0.172723
\\
38&3819712389431403&258853011125599& 4.89004& 0.035903& 0.173203
\\
41&68541957733949701&4474997964407374& 5.07400& 0.035366& 0.173083
\\
44&1229935526821663223& 77563025486587315& 5.25158& 0.034864& 0.174290
\\
47&22070297525055988321& 1347390412214087833 & 5.42341 & 0.034392& 0.175346
\\
50& $ 3.96035\cdot 10^{20} $& $ 2.34525 \cdot 10^{19} $ & 5.59000& 0.033949& 0.175926
\\
53& $ 7.10657\cdot 10^{21} $& $ 4.08927 \cdot 10^{20} $& 5.75181 & 0.033531 & 0.176131
\\
56& $ 1.27522\cdot 10^{23} $& $ 7.14133\cdot 10^{21} $& 5.90921 & 0.033136 & 0.176037
\\
59& $ 2.28829\cdot 10^{24} $& $ 1.24888 \cdot 10^{23} $& 6.06255& 0.032763 & 0.176227
\\
62& $ 4.10617\cdot 10^{25} $& $ 2.18679\cdot 10^{24} $& 6.21213& 0.032408& 0.177005
\\
65& $ 7.36823\cdot 10^{26} $& $ 3.83347\cdot 10^{25} $& 6.35821& 0.032072& 0.177510
\\
68& $ 1.32218\cdot 10^{28} $& $ 6.72713\cdot 10^{26} $& 6.50102& 0.031752& 0.177785
\\
71& $ 2.37255\cdot 10^{29} $& $ 1.18163\cdot 10^{28} $& 6.64077 & 0.031446 & 0.177867
\\
74& $ 4.25736\cdot 10^{30} $& $ 2.07736\cdot 10^{29} $& 6.77765 & 0.031155 & 0.177787
\\
77& $ 7.63953\cdot 10^{31} $& $ 3.65504 \cdot 10^{30} $& 6.91183 & 0.030876 & 0.177602
\\
80& $ 1.37086\cdot 10^{33} $& $ 6.43571 \cdot 10^{31} $& 7.04347 & 0.030609 & 0.178163
\\
83& $ 2.45990\cdot 10^{34} $& $ 1.13397 \cdot 10^{33} $& 7.17269 & 0.030353 & 0.178561
\\
86& $ 4.41412\cdot 10^{35} $& $ 1.99933 \cdot 10^{34} $& 7.29963 & 0.030107 & 0.178817
\\
89& $ 7.92082\cdot 10^{36} $& $ 3.52717\cdot 10^{35} $& 7.42441 & 0.029871 & 0.178949
\\
92& $ 1.42133\cdot 10^{38} $& $ 6.22605\cdot 10^{36} $& 7.54714 & 0.029643& 0.178972
\\
95& $ 2.55048\cdot 10^{39} $& $ 1.09958\cdot 10^{38} $& 7.66790 & 0.029424 & 0.178901
\\
98& $ 4.57665\cdot 10^{40} $& $ 1.94290\cdot 10^{39} $& 7.78679 & 0.029212 & 0.178747
\\
119& $ 2.74175\cdot 10^{49} $& $ 1.05696\cdot 10^{48} $& 8.57308& 0.027914& 0.179650
\\
140& $ 1.64251\cdot 10^{58} $& $ 5.84051 \cdot 10^{56} $& 9.29316& 0.026859& 0.180257
\\
161& $ 9.83989\cdot 10^{66} $& $ 3.26385\cdot 10^{65} $& 9.96138& 0.025977& 0.180552
\\
182& $ 5.89483\cdot 10^{75} $& $ 1.83951\cdot 10^{74} $& 10.5875& 0.025223& 0.180539
\\
203& $ 3.53144\cdot 10^{84} $& $ 1.04367\cdot 10^{83} $& 11.1787& 0.024566& 0.180926
\\
245& $ 1.26740\cdot 10^{102} $& $ 3.41053\cdot 10^{100} $& 12.2759& 0.023469& 0.181064
\\
287& $ 4.54858\cdot 10^{119} $& $ 1.13115\cdot 10^{118} $& 13.2829& 0.022580& 0.181221
\\
329& $ 1.63244\cdot 10^{137} $& $ 3.79224\cdot 10^{135} $& 14.2187& 0.021838& 0.181399
\end{tabular}
\end{table}
\newpage
\section{More on Polynomials} \label{PolynomialsExtra}
First we collect basic values of polynomials $C_{n,*}(q)$, some of which are
referred to in sections \ref{Hecke}, \ref{Polys}, and \ref{TwistNumbersVolume}.
Recall the initilization values:
\begin{equation*}
C_{1,0}(q) = 0, \; C_{1,1}(q) = -(q{-}1), \; C_{1,2}(q) = 0, \; C_{1,12}(q) = 1,
\; C_{1,21}(q) = 0, \; \text{and} \; C_{1,121}(q) = 0.
\end{equation*}
\begin{align*}
C_{2,0}(q) &= q(q{-}1)^2 = q - 2\,q^2 + q^3 & C_{3,0}(q) &= -q + 4\, q^2 - 5\,q^3 + 4\,q^4 - q^5
\\
C_{2,1}(q) &= (q{-}1)^3 = - 1 + 3\, q - 3\,q^2 + q^3 & C_{3,1}(q) &= 1 - 4\, q + 7\,q^2 - 7q^3 + 4\, q^4 - q^5
\\
C_{2,2}(q) &= -q(q{-}1) = q - q^2 & C_{3,2}(q) &= - q + 3\, q^2 - 3\, q^3 + q^4
\\
C_{2,12}(q) &= -(q{-}1)^2 = -1 + 2\,q -q^2 & C_{3,12}(q) &= 1 - 3\, q + 4\,q^2 - 3\,q^3 + q^4
\\
C_{2,21}(q) &= q & C_{3,21}(q) &= -q + 2\,q^2 - q^3
\end{align*}
\begin{align*}
C_{4,0}(q) &= q- 5\,q^2+ 10 \,q^3 -12\,q^4 +10\,q^5 - 5\,q^6 + q^7
\\
C_{4,1}(q) &= -1 + 5\,q - 11 \, q^2 + 16\, q^3 - 16\, q^4 + 11\, q^5 - 5 \, q^6 + q^7
\\
C_{4,2}(q) &= q - 4\,q^2 + 7\,q^3 - 7\,q^4 + 4\,q^5 - q^6
\\
C_{4,12}(q) &= -1 + 4\,q - 7\,q^2 + 9\,q^3 - 7\,q^4 + 4 \,q^5 - q^6
\\
C_{4,21}(q) &= q - 3\,q^2 + 4\,q^3 - 3 \, q^4 + q^5
\end{align*}
\begin{table}[h!]
\caption{Alexander polynomials for $W(3,n)$}
\label{alexpolys}
\centering \renewcommand{\arraystretch}{1.25}
\begin{tabular}{|c|c|} \hline
$n$ & $\Delta_{W(3, n)}(t)$
\\ \hline
4 & $-t^{3}+5\,t^{2}-10\,t+13-10\,t^{-1}+5\,t^{-2}-t^{-3}$
\\ \hline
5 & $t^{4}-6\,t^{3}+15\,t^{2}-24\,t+29-24\,t^{-1}+15\,t^{-2}-6\,t^{-3}+t^{-4}$
\\ \hline
10 & $ -t^{9}+11\,t^{8}-55\,t^{7}+174\,t^{6}-409\,t^{5}+777\,t^{4} -1243\,t^{3}$
\\
& $+1716\,t^{2}-2073\,t+2207-2073\,t^{-1}+1716\,t^{-2}$
\\
& $-1243\,t^{-3}+777\,t^{-4}-409\,t^{-5}+174\,t^{-6}-55\,t^{-7}+11\,t^{-8}-t^{-9}$
\\ \hline
11 & $t^{10}-12\,t^{9}+66\,t^{8}-230\,t^{7}+593\,t^{6}-1232\,t^{5}+2157\,t^{4}-3268\,t^{3}$
\\
& $+4356\,t^{2}-5158\,t+5455 -5158\,t^{-1}+4356\,t^{-2}$
\\
& $ -3268\,t^{-3}+2157\,t^{-4}-1232\,t^{-5} +593\,t^{-6}-230\,t^{-7}+66\,t^{-8}-12\,t^{-9}+t^{-10}$
\\ \hline
\end{tabular}
\end{table}
\begin{table}[h!]
\caption{Jones polynomials for $W(3,n)$}
\label{Jonesw3n}
\centering \renewcommand{\arraystretch}{1.25}
\begin{tabular}{|c|c|} \hline
$n$ & $V_{W(3,n)}(t)$
\\ \hline
4 & $t^{4}-4\,t^{3}+6\,t^{2}-7\,t+9-7\,t^{-1}+6\,t^{-2}-4\,t^{-3}+t^{-4}$
\\ \hline
5 & $-t^{5}+5\,t^{4}-10\,t^{3}+15\,t^{2}-19\,t+21-19\,t^{-1}+15\,t^{-2}-10\,t^{-3}+5\,t^{-4}-t^{-5}$
\\ \hline
10 & $t^{10}-10\,t^{9}+45\,t^{8}-130\,t^{7}+290\,t^{6}-542\,t^{5}+875\,t^{4}$
\\
& $-1250\,t^{3}+1600\,t^{2}-1849\,t+1941-1849\,t^{-1} +1600\,t^{-2}-1250\,t^{-3}$
\\
& $+875\,t^{-4}-542\,t^{-5}+290\,t^{-6}-130\,t^{-7}+45\,t^{-8}-10\,t^{-9}+t^{-10}$
\\ \hline
11 & $-t^{11}+11\,t^{10}-55\,t^{9}+176\,t^{8}-429\,t^{7}+869\,t^{6}-1518\,t^{5}+2343\,t^{4}$
\\
& $-3245\,t^{3}+4070\,t^{2}-4652\,t+ 4863-4652\,t^{-1}+4070\,t^{-2}-3245\,t^{-3}$
\\
& $+2343\,t^{-4}-1518\,t^{-5}+869\,t^{-6}-429\,t^{-7}+176\,t^{-8}-55\,t^{-9}+11\,t^{-10}-t^{-11}$
\\ \hline
\end{tabular}
\end{table}
\newpage
\begin{table}[h!]
\caption{HOMFLY-PT polynomials for $W(3,n)$}
\label{homflyptw3n}
\centering \renewcommand{\arraystretch}{1.25}
\begin{tabular}{|c|c|} \hline
$n$ & $H_{W(3,n)}(a, z)$
\\ \hline
4 & $a^2\bigl( {z}^{4}+{z}^{2}-1 \bigr)
+ \bigl(-{z}^{6}-3\,{z}^{4}-{z}^{2}+3 \bigr)
+ a^{-2}\bigl( {z}^{4}+{z}^{2}-1 \bigr)$
\\ \hline
5 & $ a^2\bigl( -{z}^{6}-2\,{z}^{4}+{z}^{2}+2 \bigr)
+ \bigl( {z}^{8}+4\,{z}^{6}+3\,{z}^{4}-4\,{z}^{2}-3\bigr)
+ a^{-2}\bigl( -{z}^{6}-2\,{z}^{4}+{z}^{2}+2 \bigr)$
\\ \hline
10 & $a^2\bigl( {z}^{16}+7\,{z}^{14}+14\,{z}^{12}-2\,{z}^{10}-29\,{z}^{8}-11\,{z}^{6}+ 18\,{z}^{4}+6\,{z}^{2}-3 \bigr)$
\\
& $+ \bigl( -{z}^{18}-9\,{z}^{16}-28\,{z}^{14}-26\,{z}^{12}+33\,{z}^{10}+69\,{z}^{8}+4\,{z}^{6}-42\,{z}^{4}-9\,{z}^{2}+7\bigr)$
\\
&$ + a^{-2}\bigl( {z}^{16}+7\,{z}^{14}+14\,{z}^{12}-2\,{z}^{10}-29\,{z}^{8}-11\,{z}^{6}+ 18\,{z}^{4}+6\,{z}^{2}-3 \bigr)$
\\ \hline
11 & $a^2\bigl( -{z}^{18}-8\,{z}^{16}-20\,{z}^{14}-6\,{z}^{12}+40\,{z}^{10}+34\,{z}^{8}-25\,{z}^{6}-24\,{z}^{4}+6\,{z}^{2}+4 \bigr) $
\\
& $+ \bigl( {z}^{20}+10\,{z}^{18}+36\,{z}^{16}+46\,{z}^{14}-28\,{z}^{12}-114\,{z}^{10}-43\,{z}^{8}+74\,{z}^{6}+42\,{z}^{4}-16\,{z}^{2}-7 \bigr) $
\\
& $ + a^{-2}\bigl( -{z}^{18}-8\,{z}^{16}-20\,{z}^{14}-6\,{z}^{12}+40\,{z}^{10}+34\,{z}^{8}-25\,{z}^{6}-24\,{z}^{4}+6\,{z}^{2}+4 \bigr)$
\\ \hline
\end{tabular}
\end{table}
\section{Notes on Computing}
\label{ComputerNotes}
This file contains some remarks on the roles played by {\em Mathematica} and
{\em Maple} experiments in generating data, conjectures, and results.
Our initial interest was in the Khovanov homology of weaving knots, which we knew was determined in a straightforward
manner by the Jones polynomials. It also turns out that the Khovanov homology of our knots can be determined by knowing
half of the Khovanov homology, essentially. Instead of having to keep track of a bigraded object, the study of Khovanov
homology of weaving knots is reduced to the study of a graded object. We normalized our examples by dividing each dimension
in the graded object by the total dimension and plotted the results for a large number of the knots. In the plots
bell-shaped curves appear as envelopes of the plots of the normalized dimensions. First, this led us to conjecture
that the standard deviations of the bell curves may be an interesting invariant for the family of knots $W(3, n)$.
As mentioned, the Jones polynomial of a weaving knot $W(3,n)$ determines the two-variable Khovanov polynomial of
the bi-graded Khovanov homology. To simplify matters, we studied the Jones polynomial on its own terms.
We knew that the Jones polynomials have the form
\begin{equation*}
V_{W(3,n)}(t) = \pm t^{-n} + \lambda_{-n+1}t^{-n+1} + \cdots \lambda_{n-1}t^{n-1} + \pm t^n,
\end{equation*}
so we conjectured that $\lambda_{-n+k} = \lambda_{n-k}$ is a polynomial
function of degree $k$ in $n$. The basis for this conjecture is the well-known binomial distribution approximating
the standard normal distribution. To investigate
this conjecture further, we detoured through another round of experiments.
During a visit to the University of Osnabr\"{u}ck in Germany, the second-named author was tutored in {\em Mathematica}
by Prof.~Dr.~Karl-Heinz Spindler. During the demonstrations of techniques for manipulating polynomials, Dr.~Spindler
asked if we knew explanations of the patterns we were observing. These questions led to the formulation and
proof of the palindromic properties of the building block polynomials $C_{n,-}(q)$ stated in theorem \ref{palindromes}.
For a large sample of computed Jones polynomials, we extracted the coefficients $\lambda_{-n+k}$ obtaining sequences of
integers upon which {\em Mathematica} routines computed iterated differences. In accordance with the conjectured
behavior, we observed the differences vanishing after the expected number of iterations. From these experiments
it was possible to generate formulas for the numbers $\lambda_{-n+2}$ and $\lambda_{-n+3}$
eventually proved in theorems \ref{twistnumber2} and \ref{twistnumber3}.
One may also obtain expressions for
two-variable HOMFLY-PT polynomial $H_{(W(3,n)}(a, z)$ normalized to $H({\rm Unknot})(a,z) = 1$.
This amounts to applying a different sequence of substitutions to the Hecke algebra output
$ V_{(\sigma_1\sigma_2^{-1})^n}(q,z) $ given in \eqref{Heckeoutput}.
Since we have no immediate use for these gadgets, we offer the brief table \ref{homflyptw3n}.
Turning to the volume computations, we thank Ilya Kofman for a significant
improvement of our first script for computing volumes using SnapPy \cite{SnapPy}. According to our
data, the volume of the complement of $W(3,n)$ is growing roughly linearly with $n$, so it is not so
surprising that the volume is strongly correlated with the higher twist numbers. Another feature
of the SnapPy data is that, although the volume is growing linearly, the number of simplices used
by SnapPy to compute the volume is growing quite irregularly.
\bibliographystyle{plain}
\bibliography{knotinvariantslist}
\end{document}
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Pond Spring, the General Joe Wheeler Home
- Location: : 12280 Hwy. 20, Hillsboro, AL 35643
- Phone: (256) 637-8513
AboutLocated near Courtland, Alabama, Pond Spring was home to Joseph Wheeler, former Major General of Cavalry of the Confederate western army, The Army of Tennessee. Wheeler was also a long-time U.S. Congressman following the Civil War, and yet again became a Major General during the Spanish-American War (U.S. Army Volunteers).. Pond Spring, the Gen. Joe Wheeler Home is currently undergoing restoration and preservation work, this managed by the Alabama Historical Commission. The grounds, cemeteries, and three of the outbuildings at Pond Spring, The General Joe Wheeler Home, are open by appointment for group tours. Admission rates are $8 for adults, $5 for Seniors, college students, military, $3 for children 6-18, and Free for children under 6 years old. Group Rate (10+ people) $1 less than admission prices. Please call the Site Director at 256-637-8513 to make an appointment for your group to visit Pond Spring. Hours of operation: Wed. - Sat. 9:00 a.m. - 4:00 p.m. with tours at 9, 10, 11 a.m. and 1, 2, 3 p.m. and Sunday 1:00 - 5:00 p.m. . No reservations are required UNLESS it is a group.
Features
Amenities
| 389,013
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Jehan le Taintenier or Jean Teinturier, Latinised in Johannes Tinctoris (aka Jean de Vaerwere) (c. 1435 - 1511) was a Renaissance composer and music theorist from the Low Countries.;[1][2] certainly Tinctoris must at least have known the elder Burgundian there. Tinctoris went to Naples about survived shows a love for complex, smoothly flowing polyphony, as well as a liking for unusually low tessituras, occasionally descending in the bass voice to the C two octaves below middle C (showing an interesting similarity to Ockeghem in this regard). Tinctoris wrote masses, motets and a few chansons.
Tinctoris was also known as a cleric, a poet, a mathematician, and a lawyer; there is even one reference to him as an accomplished painter.[]
From his third book on counterpoint.
Rule #1 Begin and finish with perfect consonance. It is, however, not wrong if the singer is improvising a counterpoint and ends with imperfect consonance, but in that case, the movement should be many-voiced. Sixth or octave doubling of the bass is not allowed.
Rule #2 Follow together with ténor up and down in imperfect and perfect consonances of the same kind. (Parallels at the third and sixth are recommended, fifth and octave parallels are forbidden.)
Rule #3 If ténor remains on the same note, you can add both perfect and imperfect consonances.
Rule #4 The counterpointed part should have a melodic closed form even if ténor makes big leaps.
Rule #5 Do not put cadence on a note if it ruins the development of the melody.
Rule #6 It is forbidden to repeat the same melodic turn above a cantus firmus, especially if the cantus firmus contains that same repetition.
Rule #7 Avoid two or more consecutive cadences of the same pitch even if cantus firmus allows it.
Rule #8 In all counterpoint, try to achieve manifoldness and variety by altering measure, tempo, and cadences. Use syncopes, imitations, canons, and pauses. But remember that an ordinary chanson uses fewer different styles than a motet and a motet uses fewer different styles than a mass.
| 257,176
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One Stabbed To Death In Suspected Cult War In Diobu
One person has been stabbed to death in suspected cult war at Azikiwe street smile 2,Diobu,PortHarcourt,Rivers state,Nigeria this evening
We learnt that a disagreement broke out between two suspected cult group drinking at a bar in 40 Azikiwe street at about 8:00pm this night
Our source said one of the suspected cultists stabbed his rival known as Chimezie to death
We learnt the victim died on his way to hospital
Meanwhile the Police has taken over the place as at press time
The spokesperson of Rivers state police command Omoni Nnamdi is yet to respond to text message sent to him by Revelation Agents
As at press time Some close associate of the deceased confirmed the incident to Revelation Agents
| 374,828
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TITLE: Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
QUESTION [5 upvotes]: Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori of smaller dimensions;
The group $T(\mathbb{F}_{\!q})$ contains (or even isomorphic to) a large subgroup of the form $(\mathbb{Z}/n)^m$ for some naturals $n, m > 1$.
REPLY [1 votes]: $\def\FF{\mathbb{F}}\def\ZZ{\mathbb{Z}}\def\Id{\text{Id}}$The question is a little vague as to what large means, but I will show that, for any prime $\ell$ and any exponent $n$, the group $T(\FF_q)$ can contain $(\ZZ/\ell \ZZ)^n$. As Will Sawin writes in comments, this corresponds to constructing a matrix $M$ of finite order in $GL_d(\ZZ)$ such that $M$ is not decomposable as $M_1 \oplus M_2$ and such that $M-q \Id$ has an $n$-dimensional kernel modulo $\ell$.
Before I do the general case, an example to show that $T(\FF_q)$ can contain $(\ZZ/2 \ZZ)^2$. Let $\omega$ be a primitive cube root of unity. Note that we have an injection
$$\ZZ[X]/(X^6-1) \ZZ[X] \hookrightarrow \ZZ \oplus \ZZ[\omega] \oplus \ZZ[\omega] \oplus \ZZ$$
by $X \mapsto (1,\omega,-\omega,-1)$. Let $A \subset \ZZ \oplus \ZZ[\omega] \oplus \ZZ[\omega] \oplus \ZZ$ be the subring
$$\{ (\alpha, \beta, \gamma, \delta) : \alpha \bmod 3 \equiv \beta \bmod 1-\omega,\ \beta \bmod 2 \equiv \gamma \bmod 2,\ \gamma \bmod 3 \equiv \delta \bmod 1-\omega \}$$
of $\ZZ \oplus \ZZ[\omega] \oplus \ZZ[\omega] \oplus \ZZ$.
Then $A$ is an indecomposable $\ZZ[X]/(X^6-1)\ZZ[X]$-module and, as a $\ZZ$-module, $A \cong \ZZ^6$. So the action of $X$ on $A$ gives a $6 \times 6$ integer matrix $M$ whose $6$-th power is $1$, and which can't be put in block diagonal form.
Now, let $q$ be any odd prime; we count solutions of $Mx = qx$ in $A/2A$. Of course, this is just the same as $Mx=x$.
We have
$$A/2A = \FF_2 \times (\text{nilpotent extension of $\FF_4$}) \times \FF_2$$
and $M$ acts on $A/2A$ by $(1,\omega, 1)$. The point is that the condition modulo $1-\omega$ disappears modulo $2$.
Then the $1$-eigenspace of $M$ is two dimensional.
Now, we need to do the general case. The notation is awful, I'm afraid.
For any positive integer $d$, let $\zeta_d$ be a primitive $d$-th root of unity. Then $\ZZ[X]/(X^r-1) \ZZ[X]$ injects into the ring $\prod_{d|r} \ZZ[\zeta_d]$. I'll write $(\alpha_d)_{d|r}$ for coordinates on the product $\ZZ[\zeta_d]$. Let $p^k s$ be a divisor of $r$ with $k>1$ and $p$ not dividing $s$. Then, on the image of $\ZZ[X]/(X^r-1) \ZZ[X]$, we have $\alpha_{p^{k-1} s} \bmod \langle 1-\zeta_{p^{k-1}}, p \rangle \equiv \alpha_{p^k s} \bmod \langle 1-\zeta_{p^{k}}, p \rangle$.
Let $E$ be any set of pairs of the form $(p^{k-1} s, p^k s)$ for $p^k s| r$ and $p$ not dividing $s$. Let $A$ be the subring of $\prod_{d|r} \ZZ[\zeta_d]$ where we impose that $\alpha_{p^{k-1} s} \bmod \langle p,1-\zeta_{p^{k-1}} \rangle \equiv \alpha_{p^k s} \bmod \langle p,1-\zeta_{p^k}\rangle$ for $(p^{k-1} s, p^k s) \in E$. Then $A$ is a $\ZZ[X]/(X^r-1) \ZZ[X]$-module. If the graph with edge set $E$ is connected, then $A$ is not a direct sum. So, for any such connected subgraph $E$, we get an $r \times r$ matrix with $M^r = 1$ which is not a direct sum.
In particular, take $r = p \ell^{n-1}$ for some auxilliary prime $p \neq \ell$. Let $E$ be the set of pairs of the form $(\ell^i, p \ell^i)$, $0 \leq i \leq n-1$ and $(p \ell^{j-1}, p \ell^j)$ for $1 \leq j \leq n-1$.
We want to understand $A/\ell A$. We can localize $p$ first, which make the conditions modulo $\langle p,1-\zeta_{p^k} \rangle$ vanish. So $A[p^{-1}] \cong \bigoplus_{j=0}^{n-1} \ZZ[\zeta_{\ell^j}] \oplus (\text{something else})$ and then $A/\ell A \cong A[p^{-1}]/\ell A[p^{-1}] \cong \bigoplus_{j=0}^{n-1} \ZZ[\zeta_{\ell^j}]/\ell \ZZ[\zeta_{\ell^j}] \oplus (\text{something else})/\ell(\text{something else})$.
Then each $\ZZ[\zeta_{\ell^j}]/\ell \ZZ[\zeta_{\ell^j}]$ gives a one dimension $1$-eigenspace for $X$, so the space of solutions to $Mx = x \bmod \ell$ is at least $n$-dimensional. Taking $q$ to be a prime $\equiv 1 \bmod \ell$, we see that $T(\FF_q)$ can contain $(\ZZ/\ell \ZZ)^n$.
| 76,444
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We’re always on the lookout for cool, summer sandals and the moment we saw this rope version from Toteme we knew we had to snatch them up immediately for our suitcase (and we did!). I mean, these are the kind of thing we would usually search high and low for on our travels…and here they were, right in our own backyard! They’re perfect to wear in nautical spots like Martha’s Vineyard or Nantucket (paired with a fabulous denim skirt), or in more far away destinations like Greece or Turkey (styled with a bikini and sundress). Either way your friends will totally think you found them in some exotic outdoor market or tucked away in a quaint, little boutique in some remote, seaside town. Don’t worry, we’ll keep your secret (wink, wink).
| 242,841
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Growing Future has joined the ranks of those who want to raise awareness about well intended, poorly structured “help.” This cyber-censorship project is getting a lot of attention, because people want the freedom to use the websites they want to use. The Internet was created as a “wild frontier,” where people could say what they wanted to say – it was up to the readers to choose to visit, or not, to share, or not. This “trust” was intrinsic in the development of the “World Wide Web.” That only works if people use their good sense, and if people are held accountable for saying the truth.
There are many websites that don’t report the truth, but that should be something that can be handled without “censorship.” A rating system similar to “Politico.com” is one idea. People need to learn to think before they believe what they read. But the government making that decision isn’t helpful. It just lets one person control another person.
It’s hard for some people to see correlations between farms and tech, but actually technology started with farming. The first person to push into the soil to plant something, instead of broadcasting it, was using technology. The hoe, the plow, irrigation are just a few ideas of how “tech” started on the farm. But there is a similarity much more pertinent. The American people have let the government decide with a simple set of regulations, what food is safe, and what isn’t. We’ve gotten so “trusting” (read as ignorant) that many people eat bad food in the name of “healthy.” Rather than teach people what is “enough,” we make processed food so people can eat more useless calories all in the name of “being full.” We, as a people, have abdicated our choices to the USDA and FDA, and as a result we are accepting highly processed, genetically modified foods instead of helping people access better food close to home.
The SOPA/PIPA protest is one way to get people to understand what censorship could do, and it’s easier than asking people not to eat for the day. But in the case of SOPA/PIPA it is a good intent with bad outcomes, rather than giving people the tools to make better choices. Something has to be done about piracy, but I think somehow we have to stress “intent” rather than action. If a person has a financial goal or a political one, and misleading people is part of that agenda, that’s a different issue than a person who just wants to share good song or a relevant website. One is malevolent, the other is well-intentioned. The government may have good intent here, but as usual they are looking for the easy way out, as lead by the lobbyists, instead of listing to people who actually use, or produce, the content.
Encouraging knowledge seems more efficient than swatting websites, or small farmers.
| 149,542
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Talking to Mike Schur About ‘Parks and Rec’, ‘Brooklyn Nine-Nine’, and Writing with Bill Murray, [...]
, premieres tonight on Fox. Schur co-created the ensemble show with longtime Parks writer Dan Goor, and it’s one of the most promising comedies of the new TV season. I had the chance to talk to Mike Schur last week about what to expect from Parks and Rec and Brooklyn Nine-Nine this season, why he hated playing Mose on The Office, adapting Infinite Jest into a movie, and getting to write a sketch with Bill Murray while at SNL.
Brooklyn Nine-Nine has an incredibly diverse cast for a network show. How did that group come together and was the diversity an intentional choice?
We sort of did race blind casting. We got Andy onboard, and then, we got Andre Braugher onboard as the captain, but we’d already met with Terry Crews. We essentially just designed that part for Terry. Like, the character’s name in the script was Terry ‘cause he was kind of the only person we wanted to play the role. And then after that, the other four cast members, we had very generic names for them. Their names were like Amy and Megan and Bill or something. Allison Jones is our casting director, who’s like the greatest casting director of all time, and we told her we don’t care. “Age, size, weight, height, anything, ethnicity. Just find funny people.” So she brought in Joe Lo Truglio, who she had loved for a long time and cast in a bunch of stuff and who we knew. Then, she found Melissa Fumero and Stephanie Beatriz. She knew Stephanie really well, and Stephanie had been on Modern Family a couple times. And then, Chelsea Peretti was a writer on Parks and Rec for us. We just really wanted to design a part for her too. At the end of the day, it sort of just shook out the way it shook out. We ended up with two Latina actresses, one Italian guy, one half-Italian half-Jewish lady, two African American guys, and Andy Samberg.
I feel it’s just rare to see a cast like that. Most sitcoms just have one token non-white person and that’s it.
Yeah, well, different shows do it differently. Some shows are like family shows, right? And the family is generally speaking the same ethnicity. And there are workplace shows that are set in parts of the world where you try to represent the general population of that part of the world or whatever. But if you’re gonna set a show in New York City, it would seem silly if everybody was a white dude. That doesn’t make any sense to me.
Did you and Dan write the main part for Andy Samberg, or did the character exist before he came onboard?
It was actually a little bit of both. We came up with the character, and we had the whole pitch worked out and stuff. We had this character description who’s like 32 and kind of a wise-ass and copes with the stress of his job by making jokes and didn’t take stuff that seriously when he should probably take it a little more seriously. We heard that Andy was leaving SNL and went to Fox and were like, “Hey, if Andy Samberg were interested…” and they were like, “Yeah yeah yeah. Go go go.” The character existed, and once Andy signed on, we tailored it a little bit to him, but it was already pretty much right in his wheelhouse. In terms of his character traits and his age and his outlook on life, it was pretty similar to what Andy already is. The model in our heads at the beginning was Hawkeye Pierce on M*A*S*H. Hawkeye made a lot of jokes and did a lot of funny stuff, but he was salvaged as a character because he was a great surgeon and a good person. We sort of felt like, if we make this guy a good cop and fundamentally a nice person, then he can goof around and wise off and you won’t get angry with him for not doing his job well …
The trick is — 10 years later, it’s impossible to imagine anyone playing Michael Scott except for Steve Carell and it’s impossible to imagine anyone playing Leslie Knope except for Amy Poehler. I think it’s already getting very hard to imagine anyone playing Jake Peralta other than Andy Samberg, and it’s just because after a while, you just learn the rhythms of the actors’s voice and you learn how to write better jokes for them. The character and the actor merge at some point. It’s hard to say exactly when that happens, but the characters and actors, as time goes on, merge into one big conglomerate that makes it so it’s impossible to imagine anybody else doing it.
That’s great that’s already starting to happen with Andy Samberg on the show.
Yeah, I think it is. We’re shooting episode six right now, and it’s really fun — it’s the whole cast, really. The whole cast is really great, and they’re really strong. They just kind of hit the ground running, which makes our lives so much easier as writers. They’re just so talented that it takes some of the pressure off.
Chelsea Peretti has written for Parks before. Did you guys encourage her to get in front of the camera or was that something she wanted to do?
Well, she wrote for us for one year and then she left because she is a standup and a performer. She really wanted to focus on her performing career. Dan Goor and I knew her to be an insanely talented and funny person, so when we were deciding this show, it sort of felt like we’d be crazy not to try to grab her. Her character wasn’t in the very first concept of the script that we came up with, and then when we realized we had a chance to grab her, we sort of just designed a part for her. She’s a civilian administrator. She’s basically an office secretary. We realized, after coming up with the first story and laying out the characters, that it would be a good idea to have one character who wasn’t a police officer, who wasn’t a detective, just to give another dimension to the ensemble. When we came up with that idea, it was like, ‘Oh, she’s the perfect person.’ She’s just a great shit-stirrer. The idea behind her character is that she’s a civilian. She’s kind of punching a clock. She has no interest in the police work. She doesn’t think of them as heroes or New York’s finest. She’s not particularly impressed by anything they do, which is a very fun dynamic to play as they’re going about their very high pressure jobs.
It seems like on your shows, you tend to have a lot of writers who cross over into acting or who happen to be talented performers themselves, whether it be Harris Wittels or Joe Mande or Katie Dippold. Is that just a coincidence?
No, it’s not. It’s very intentional. I think it really ultimately comes from SNL because on SNL, the line between writer and performer is very blurry and the performers write a lot of their own sketches. Sometimes, the writers appear on the show. In general, the collaboration is very intense between writers and performers. A lot of the writers are also performing at UCB every week and doing improv and stuff like that. That spirit was taken to The Office. That was an explicit idea that Greg Daniels had for The Office was that there would be crossover. That’s why he hired Mindy as a writer-performer; he hired BJ Novak that way; he basically held Paul Lieberstein down and forced him to be in the show against Paul’s will; and I played Mose.
Pretty much every writer at some point appeared on the show in some way. That was so fun. It was really fun to feel like there was no division between the writers and the actors, which sometimes happens on TV shows. Sometimes on TV shows, the writing is done in LA and the shooting is in New York and the cast and the writers never even really meet. Greg explicitly wanted to remove that barrier, and we sort of took that spirit over to Parks and Rec. A number of writers have appeared on the show, sometimes in speaking roles like Harris and Joe Mande. Ali Rushfield was on the show last year, and Alan Yang is one of the members of Andy Dwyer’s band, Mouse Rat. It’s a fun aspect of the show that we cram writers into the cast whenever we can.
Were you reluctant to play Mose on The Office, and what was your performing background before that?
I am a terrible actor. I swore off acting for good when I was in college. My feeling was that I always knew how words should sound coming out of my mouth; I could just never make them sound that way. And I finally after being in another play where I felt like I was a solid B minus, it was like, ‘Oh, I don’t have to do this anymore. I can write the words and have more talented people say them.’ I just completely switched over to becoming a writer and a director, which was a very good call on my part. Mose was just basically a sight gag. I was probably on the show a dozen times; I think I’d said a total of eleven words. Thankfully, I did a part that did not require me to have any actual acting skill.
After you started running Parks and Rec, was it a hassle have to go back and play Mose just for a two-second scene?
Oh yeah, it was miserable. I despised it because it was always like I’d have to wake up at 4:30 in the morning and drive to someplace deep in the Simi Valley, where it was 138 degrees, and put on wool clothes and run around and do something embarrassing with a neck beard on. And it would take seven hours and then I would end up appearing in the show for four seconds. [Laughs] It was a huge pain in the butt, but I was sort of weirdly flattered and honored that in the finale, Mose actually had an important role in the plot. There’s an actual storyline for him, which was exciting.
Is there a piece of advice or something that stands out that you’ve learned from an older writer on a show you’ve written for?
Oh, there’s a million. Yeah, I mean, the only real way you can learn how to write and produce TV is by watching people who are better than you do it. My career has been this insanely fortunate sequence of events, starting at SNL where Lorne Michaels, who’s arguably the greatest television producer, was my boss. Then, the other bosses I had there were people like Steve Higgins, who’s now Jimmy Fallon’s sidekick/announcer guy of his show, Tim Herlihy who writes Adam Sandler movies, and Mike Shoemaker, who is on Jimmy Fallon’s show and is gonna produce Seth Meyers’s show. Those are the guys who were producing the show and my bosses. Most of what I learned early on was just by watching them and learning from them. Then, I got plucked out of SNL and put on The Office with Greg Daniels, who is a master of half-an-hour television comedy, so I got to apprentice to him for four years. Then, he and I built this show together. A certain amount of TV production and writing is instinct and practice, and a lot of it is just osmosis..
When you were moving from SNL to The Office, did you have to send a writing sample over to The Office?
Yeah, the way it works is you write a spec script. You pick a show that you like and you write a sample episode of that show, or ultimately, you can write an original pilot or something or a movie. Different people want to read different things. I wrote a Curb Your Enthusiasm spec script in 2003, and Greg Daniels read it and hired me off of that.
As far as what you guys are looking for from writer submissions on your shows, do you prefer to read pilots or specs of existing shows?
It’s personal preference, and different people look for different things. I prefer reading the original material because I find it hard to judge — if someone’s written an episode of Big Bang Theory, I sometimes find it hard to know whether they are really funny and a good writer or whether they’re really good at mimicking the style of an existing show. I prefer to read original material, but I’ll read anything. I’ve hired people from their Twitter feeds, I’ve hired people off of movies, off of short stories. Rachel Axler is a writer that I really liked who I hired just based off of a short story she had written. Megan Amram was hired here because of her Twitter feed.
Did she send a writing sample over or was it purely off her Twitter feed?
No, I was such a fan of hers off Twitter and I met with her for like an hour and just could tell that her comedy brain was operating at an extremely high level so I just really hired her off that.
It’s interesting how technology is changing things.
Yeah, definitely. I mean, that’s the perfect example. There are many, many more ways for young writers and performers to be seen. You can make a short with your friends, you can have a blog, you can have a Twitter feed, you can do Funny or Die shorts. There’s just a million ways now that didn’t exist even ten years ago for people to get their name out there. One of Amy Poehler’s proudest accomplishments, I would say, is co-founding the UCB Theatre. Now, there’s this incredible biomass of comedians that are gathered in one place every day, and people can do shows and they can use it to get their names out there. They can keep their skills sharp after they’ve already gotten on TV or into a movie or whatever. The level of comedy that you can see at the UCB Theatre in LA or New York on any given day is remarkable. Back when I started coming up, there was The Groundlings and there was Second City. Then, there were little smaller shops like Improv Olympic or places like that. Now, there’s just a million venues for people. It’s really great. There’s a comedy boom happening right now, and I think a lot of it is because of the internet, because of the way that technology is changing things.
How has balancing Parks and Brooklyn Nine-Nine been going so far?
It’s been a little hard, but it’s fun. They’re produced on the same lot, so I kinda go back and forth. I split my time. Some days, I’ll spend whole days at one place or the other. Some days, I’ll split my time. There’s a lot of extremely talented people who are also working on both shows. Dan Goor is running Brooklyn full time. He’s there every minute of every day. We have a lot of great writers at Parks and Rec — some of whom have been here from the beginning, some were just brought in recently. They’re doing a really great job in moving the ball forward and stuff. So far, it’s been pretty manageable. It’s a little more stressful ‘cause there’s two different sets of stuff to keep in my brain, but so far so good, I would say.
You have a lot of great guest stars lined up on Parks and Rec this season. Do you have a dream guest star you haven’t been able to land yet?
Well, we’ve said it before, but our dream guest star is still Bill Murray. The mayor of the town has been mentioned a number of times, but we’ve never seen him. I don’t know if we ever will, but we really want Bill Murray to play the mayor, so if somehow Bill Murray’s reading this, call me.
[Laughs] Have you made attempts to get him on the show? I know he has that toll-free voicemail line that producers and directors offer him parts on..
Have you ever interacted with him? Did you cross paths if maybe he came by SNL when you were working there?
Yeah, he hosted when I was there, and it was awesome. I wrote a sketch that he was in, and it was like a career highlight for me.
What sketch was it?
I don’t think it actually aired. I co-wrote it with a guy named Scott Wainio, and I believe it was Scott’s idea. I should fully attribute the idea. It was an E! entertainment news celebrity gossip show that took place at an insane asylum, so it was just three crazy people in straightjackets just giving celebrity gossip that was complete nonsense. It was a crazy sketch, but for whatever reason, he really liked it, which meant the world to us. I think it got cut after dress rehearsal, but just being able to rehearse a sketch with Bill Murray in it was amazing. Like I said, it was a career highlight to just be in the same room as him. And at one point, I think he wanted to do a rewrite of the sketch, and Scott and I sat in his dressing room for like an hour and just went line by line.
That’s got to be a crazy experience, especially for being new on the show at the time.
Yeah, exactly, This was like my dream. It was literally a dream. He had always been my favorite of those early years guys. He was always my favorite. There was something about him that I found so compelling and silly and great. If we had sat in his dressing room in silence for five minutes, it would have been amazing, but to actually work on a sketch with him was incredibly cool.
So with Rob Lowe and Rashida Jones leaving Parks and Rec, do you have any plans to introduce new cast members, or are you just going to focus more on everyone that you already have?
I don’t think we’re going to be adding anyone permanently. We have a lot of fun guest stars in the beginning of the year, in part because we’re absent Chris Pratt for a number of episodes because he’s off shooting that Guardians of the Galaxy movie. You may have seen it, but we brought in Sam Elliott and June Raphael and Billy Eichner and Kristen Bell is doing a guest spot and we have Lucy Lawless back for two episodes. But I think that when the dust settles — Rob and Rashida head out after episode 13 — I don’t think we have any specific plans at this point to really replace them. It’s not about replacing them, as much as it is reshuffling a little bit and then focusing on the core characters that we have left. And who knows? That could change down the line. The one thing I know is, for example, Ann Perkins is Leslie Knope’s best friend in the world, and their relationship will have been 103 episodes long. You can’t just magically switch her out and bring in a new person and now say, “Oh, this is now Leslie’s best friend.” That would [feel] very false to me, so it won’t be anything close to that if we do get new cast members.
Have you thought about how much longer the show’s lifespan might be?
Not recently. I mean, we’ve had an attitude on the show almost from the very beginning, which was you never know how much time you have left really, with very few exceptions. With monstrous hit shows, they might have some idea of how many episodes they have left or they can see five years into the future. Not having that luxury, we sort of treat every season like we’re gonna tell every story we want to tell. We’re not gonna hold anything back.
The characters are very aggressive in the way that they want to change their lives. People get married and they have kids and they change jobs. They’re very sort of forward-thinking. Largely, that’s just part of who they are. Leslie Knope isn’t the kind of person who’s ever really satisfied. She’s a very goal-oriented person, so it would also ring false if she just were completely satisfied with just being the Deputy Director of the Parks and Recreation Department forever, so it sort of dovetailed nicely that our feeling is that we should be very aggressive in our storytelling and not leave anything in reserve or on the table. We’ll just sort of continue to do that as long as they let us make episodes.
Do you remember the first time you ever saw Nick Offerman perform?
Yeah, I do very well. The reason he’s on this show is I wrote an episode of The Office in season two, where Michael Scott went to New York and met a bunch of the managers from the other Dunder Mifflin branches — from the Albany branch and the Buffalo branch and stuff. There was a role in the episode for one of the branch managers, and the idea was that this guy was a bigger buffoon than Michael Scott. The idea was that it would go a long way towards explaining how it is that Michael Scott still has his job. There are other people who are way worse than he is out there. Nick auditioned for it and was just great. He was so funny and so great and so interesting as a performer. We tried to book him, but he had booked a Will & Grace episode that same week, and he couldn’t do it. I wrote the words “Nick Offerman” down on a Post-it note and stuck it to me computer just as a reminder to myself that there was this guy out there who I thought was really great and talented. And then, when Greg and I were writing the pilot for Parks and Rec, he was like, “Oh yeah, there’s that guy.” I still have that Post-it note, and I gave it Nick as a present at the end of last season.
Was the episode of Parks and Rec last season with all the Infinite Jest references something that you had wanted to do for a little while, or did that just come up spontaneously?
It wasn’t like I set out to do it or anything. As a little Infinite Jest reference when we introduced Ben Wyatt as a character … there’s a character in Infinite Jest named Ortho Stice who’s from Partridge, Kansas, so we just made up Partridge, Minnesota, as his hometown. We knew we wanted to do this episode where he went back to his hometown, and Dave King, who wrote the episode, is also a fan of Infinite Jest, so we sort of just said, “Well, as long as we’re visiting Partridge, which is itself a reference, we might as well make everything in the episode a reference to Infinite Jest.” So we just loaded it up. It also just so happened that the B-story of that episode was that Ron was being sued by Councilman Jamm for punching him in the face at Leslie’s wedding. It was like, we have to see a law firm and a law firm is just a collection of names, so every name in the law firm was a character from the book. Every time you saw a building, the name of the building was a character’s name from Infinite Jest. It’s not like I always wanted to do that. It just sort of occurred naturally because we set the episode in a place that was a reference to the book.
And you own the film rights to Infinite Jest, right?
I do. Yeah.
Do you ever intend to do something with that, or is that more just to prevent someone from making the movie and ruining it?
I wouldn’t have optioned them if I didn’t have any plans to try to do something with it. It’s obviously an incredibly challenging piece of work, and I’m not sure exactly what the end result will be, but it’s certainly on my mind as something I’d like to try to do someday.
Have you worked on it at all?
No, it’s gonna be a long slow development process if I can pull it off. It’s gonna take some time to figure out what the right place is to do it and how to do it and all that sort of stuff. You’re not gonna see Infinite Jest next month, the movie.
You’ve been very successful at television. Do you have any desire to start writing and directing films at some point?
I like new challenges. That was part of what made taking on a new show like Brooklyn Nine-Nine exciting is that it’s a new thing; it’s a new challenge. But you can also stretch yourself too thin. I met Mike White a long time ago, and he said something kind of smart and beautiful, which was, “The goal of a writer in Hollywood isn’t to make things. It’s to make good things.” I think there’s a little bit of a balance you have to strike between trying to do everything in the world that you have the barest inkling of doing and making sure the actual things that you’re doing are good. I think that was really good advice that he had for me a long time ago. I think it’s always good to try to stretch yourself and do different kinds of things and take new challenges and stuff like that, but I don’t think you want to do that at the expense of having any of them actually be interesting and entertaining.
That makes sense. You were writing a vehicle for Will Arnett a few years ago, weren’t you?
Yeah, many years ago. We sold a movie together. A long time ago now. That’s probably eight years ago now.
What was the movie about?
It was a movie called The Ambassador, and basically, he was the son of the Vice President or the former Vice President who’s just a spoiled rich kid disaster human being. There’s some high-up people in the government who wanted to piss off all of Europe and get them out of an international treaty, so as an ambassador to the European Union, they sent Will’s character. He was a stooge; they wanted him intentionally to screw everything up, which he very successfully did for a good part of the movie until he finally figured out their evil plot and tried to undo it. Like many things in Hollywood, it sort of stalled out, but it was really funny. I love Will. I think he’s so funny. I’m rooting for his new show. I hope it really is great and it works well.
Who are some of your favorite writers or directors right now?
Oh God, there’s a million of ‘em. In comedy, honestly, one of my favorite writers in the world is Seth Meyers. I think he’s a truly great writer. I love Joss Whedon, I love Aaron Sorkin, I love Vince Gilligan. I mean, most of the shows that I watch on TV are dramas, weirdly. I think it’s ‘cause when you work in comedy, it’s hard to watch comedy at home. It’s just like, “Enough already.” There’s so much good writing on TV that it’s almost scary. There’s so many great shows, so many incredibly well-written shows. Probably because we have more outlets then ever that’re actually making them. I’m currently in a state of existential despair over the fact that there’s only three episodes of Breaking Bad left. I don’t know what I’m gonna do with myself. It’s incredibly sad.
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A quiet revolution is underway. It’s made up of rumbling conversation, growing louder, and small shifts in societal perceptions, rules, norms.
I am talking about menopause.
Menopause is a word that’s still sometimes used in a derogatory fashion (we have all no doubt on occasion heard a reference to someone being “menopausal”, in connection to their irritability and unpredictability). And it’s one of the final frontiers for women in many ways. In most countries, we have won the right to vote, to participate in the workforce, to not be discriminated against based on our gender, and in many places we now have certain pregnancy and maternity rights as well.
But women continue to suffer through menopause in silence for the most part.
A third of women who have suffered from symptoms of menopause say they hid them at work, and many think there remains a stigma around talking about the subject. I recently asked my own mother about what menopause was like for her, and whether she suffered from all the classic symptoms like hot flashes. The answer was yes, but when I asked how she dealt with it and why we didn’t know anything about it at the time, she simply said she just got on with it, “because there was a lot of other stuff going on in life.”
But there is change underfoot.
For starters, there is a more and more open debate about this. The notion of perimenopause has only recently become more widely known and understood, notably the fact that it may start sooner than experts originally thought: it has been known to kick off just after the age of 40 for some women. This is leading many women to re-evaluate symptoms they previously couldn’t explain. I am not just hearing this from my friends. There are more and more public figures coming forward to talk about perimenopause and menopause, notably Michelle Obama recently talking about the pressure of having to keep going with busy jobs in professional clothes while having severe hot flashes or acute period cramps that “feel like a knife being stabbed and turned”. There are campaigns cropping up, pushing for more open debate and awareness about this issue, like the current #ownyourmenopause campaign by The Menopause Charity. All of this dialogue is slowly helping to shift the narrative about menopause, from something that used to be trivialized and seen as embarrassing, to something understood as a natural process affecting half the population.
Women are not just talking about menopause, they are also increasingly taking the issue to court: According to the latest UK data, there were five employment tribunals referencing the claimant’s menopause in 2018, six in 2019 and sixteen in 2020. There have been 10 in the first six months of 2021 alone. Perhaps most importantly, there is now a possibility of enshrining this into laws. Caroline Nokes, an MP who is leading an inquiry into menopause discrimination, recently stated that strengthening legislation on this front shouldn’t be ruled out. Specifically, the question is whether menopause should be a protected characteristic under the Equality Act, which would put it on the list alongside age, disability, gender, race, religion, and a whole host of other things based on which discrimination is illegal.
Why does this matter?
Menopause can affect women’s ability to perform in the workplace, and it deserves adequate medical attention and care. If women are not properly supported through menopause, they are not able to participate in public life on equal terms. Not to mention that it is an economic issue: women of menopausal age are typically at the peak of their careers. Women aged 50 to 64 are the fastest-growing, economically active group in the UK, for example. If you are an employer, you want to retain this segment of the population in the workforce, not push them out through poor policies. And yet, 10% of women in the UK have left their jobs because of menopausal symptoms.
What does this look like in practice?
We are seeing more signals of this emerging. It looks like Carolyn Harris, deputy leader of Welsh Labour, launching a private member’s bill to try and make hormone replacement therapy (HRT) exempt from NHS prescription charges in England, as is currently the case in both Wales and Scotland. It looks like Channel 4 putting in place a menopause policy to support its female employees, which includes giving women access to flexible working arrangements and paid leave if they feel unwell because of the side-effects. And it looks like The Menopause Café, which provides an open space for discussion about menopause – the necessary mental health support which is often overshadowed by more biomedical approaches.
Because surely calling menopause out as a key condition women face during their lifetime, and creating societal norms and rules that adequately recognize that, is a step towards creating a more just world.
Explore more
- A third of women hide menopause symptoms at work
- What Your Mother Never Told You About Health with Dr. Sharon Malone -The Michelle Obama Podcast
- The Menopause Charity
- Menopause at centre of increasing number of UK employment tribunals
- Equality laws could be changed to protect women in menopause, says MP
- Channel 4 launches menopause policy for employees
- Menopause Cafe
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My mile eater has developed a droning noise in the front end like a worn wheel bearing but I haven't determined that is the problem yet. I am pretty good and fixing things but not so great at diagnosing some problems.
So far I jacked up the right wheel and it seemed hard to rotate, then I jacked up the left and it rotated well so I went back to the right hand side to investigate it some more and it had loosened up and seemed to rotate well so I was at a bit of a loss. After that I tore out the front wheel well liners which have been rubbing on the tires for the past few years but that did not help with the sound. The droning seems to develop after I hit 40 km/hr (25 mph) and it is kind of worrisome so I'd like to figure it out. The sound can be heard when I'm on the gas and coasting.
Can this sound come from the CV axles or the transmission? It doesn't seem to be a typical worn wheel bearing where the wheel is wobbly when you lift up the car and start tilting the wheel. Front end is all original @ 250,000 miles and I do not swerve for pot holes.
Cheers, Steve
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