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\begin{document}
\begin{abstract}
We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and the first author. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at $\infty$.
\end{abstract}
\maketitle
\section{Introduction}
A modular form $f(z)$ of level $N$ and weight $k$ is a function which is holomorphic on the upper half plane, satisfies the equation
\[f\paren{\frac{az+b}{cz+d}} = (cz+d)^k f(z) \text{ for all } \sm{a}{b}{c}{d} \in \Gamma_0(N),\]
and is holomorphic at the cusps of $\Gamma_0(N).$ Letting $q=e^{2 \pi i z},$ these functions have Fourier series representations of the form $f(z) = \sum_{n = 0}^\infty a(n)q^n.$ A {\em weakly holomorphic} modular form is a modular form that is allowed to be meromorphic at the cusps. We define $M_k^\sharp (N)$ to be the space of weakly holomorphic modular forms of weight $k$ and level $N$ that are holomorphic away from the cusp at $\infty,$ and define $M_k^\flat(N)$ similarly, but for forms holomorphic away from the cusp at 0.
The coefficients of many modular forms have interesting arithmetic properties; for instance, the coefficients $c(n)$ of the $j$-invariant $j(z) = q \inv + 744 + \sum_{n=1}^\infty c(n)q^n$ appear as linear combinations of dimensions of irreducible representations of the Monster group. Also, modular form coefficients often satisfy certain congruences. Lehner \cite{lehner1,lehner2} proved that the $c(n)$ satisfy the congruence
\[c(2^a3^b5^c7^dn) \equiv 0 \pmod{2^{3a+8}3^{2b+3}5^{c+1}7^d}.\]
Many others have obtained similar congruences, in particular for forms in the space $M_0^\sharp(N).$ Kolberg \cite{kolberg1,kolberg2}, Aas \cite{aas}, and Allatt and Slater \cite{allatt} strengthened Lehner's congruences for $j(z)$, and Griffin \cite{griffin} extended Kolberg's and Aas's results to all elements of a canonical basis for $M_0^\sharp(1).$ The first author, Andersen, and Thornton \cite{andersen,thornton1} proved congruences for Fourier coefficients of canonical bases for $M_0^\sharp(p)$ with $p = 2,$ $3,$ $5,$ and $7.$ A natural question is whether similar congruences hold for coefficients of canonical bases for spaces where we allow poles at some other cusp.
Recall that the cusps of $\Gamma_0(2)$ are $0$ and $\infty.$ Taking $\eta(z) = q^{\frac{1}{24}} \prod_{n=1}^\infty \left(1 - q^n \right)$ to be the Dedekind eta function, a Hauptmodul for $\Gamma_0(2)$ is
\[\phi(z) = \paren{\frac{\eta(2z)}{\eta(z)}}^{24} = q+24q^2+\cdots,\]
which vanishes at $\infty$ and has a pole only at 0. Note that the functions $\phi(z)^m$ for $m \geq 0$ are a basis for $M_0^\flat(2).$ Andersen and the first author used powers of $\phi(z)$ to prove congruences involving $ \psi = \frac{1}{\phi} = q\inv - 24 + \cdots \in M_0^\sharp(2)$ in \cite{andersen}, and made the following remark: ``Additionally, it appears that powers of the function $[\phi(z)]$ have Fourier coefficients with slightly weaker divisibility properties... It would be interesting to more fully understand these congruences.'' In this paper, we prove congruences for these Fourier coefficients.
Write $\phi(z)^m$ as $\sum_{n=m}^\infty a(m,n)q^n$. The main result of this paper is the following theorem.
\begin{restatable}{theorem}{congruence}
\label{congruence}
Let $n = 2^\alpha n'$ where $2 \nmid n'.$ Express the binary expansion of $m$ as $a_r\dots a_2a_1,$ and consider the rightmost $\alpha$ digits $a_\alpha\dots a_2 a_1,$ letting $a_i = 0$ for $i > r$ if $\alpha > r.$ Let $i'$ be the index of the rightmost 1, if it exists. Let
\[
\gamma(m,\alpha) =
\begin{cases}
\#\set{i\ | \ a_i = 0, i > i'}+1 &\text{if } i' \text{ exists},\\
0 &\text{otherwise.}
\end{cases}
\]
Then
\[a(m,2^\alpha n') \equiv 0 \pmod{2^{3\gamma(m,\alpha)}}.\]
\end{restatable}
\noindent
That the structure of the binary expansion of $m$ appears in the modulus of this congruence is a surprising result. We note that this congruence is not sharp. For $m = 1,$ Allatt and Slater in \cite{allatt} proved a stronger result that provides an exact congruence for many $n$.
As an example, the binary expansion for $m = 40$ is $m = \cdots 000101000.$ As we increment $\alpha,$ the $\gamma$ function gives the values in Table \ref{gammaexample}.
\begin{table}[H]
\label{gammaexample}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\alpha$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & $\cdots$ & $\alpha$ & $\cdots$\\
\hline
$\gamma(40,\alpha)$ & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 3 & 4 & 5 & $\cdots$ & $\alpha - 4$ & $\cdots$\\ \hline
\end{tabular}
\caption{Values of $\gamma(m,\alpha)$ for $m = 40$}
\end{table}
\noindent
Notice that once $\alpha$ surpasses 6---and the leftmost 1 in the binary expansion of $m$ occurs in the 6th place---$\gamma$ always increases by 1 as $\alpha$ increases by 1. This illustrates that $\gamma(m,\alpha)$ is unbounded for a fixed $m$.
We also prove the following result on the parity of $a(1,n).$
\begin{restatable}{theorem}{oddsquares}
\label{oddsquares}
The $n$th coefficient $a(1,n)$ of $\phi(z)$ is odd if and only if $n$ is an odd square.
\end{restatable}
Section 2 contains the machinery and definitions we use in the proof of Theorem \ref{congruence}. The proof of Theorem \ref{congruence} is in Section 3, and the proof of Theorem \ref{oddsquares} is in Section 4.
\section{Preliminary Lemmas}
The operator $U_p$ on a function $f(z)$ is given by
\[U_p f(z) = \dfrac{1}{p} \sum_{j = 0}^{p-1} f\paren{\frac{z+j}{p}}.\]
We have $U_p : M_k^!(N) \to M_k^!(N)$ if $p$ divides $N$. If $f(z)$ has the Fourier expansion $\sum^\infty_{n = n_0}a(n)q^n$, then the effect of $U_p$ is given by $U_p f(z) = \sum^\infty_{n = n_0}a(pn)q^n$.
The following result describes how $U_p$ applied to a modular function behaves under the Fricke involution. This will help us in Lemma \ref{polynomial} to write $U_2\phi^m$ as a polynomial in $\phi$.
\begin{lemma}
\label{fricke}
\cite[Theorem~4.6]{apostol} Let $p$ be prime and let $f(z)$ be a level $p$ modular function. Then
\[p(U_pf)\paren{\frac{-1}{pz}} = p(U_pf)(pz)+f\paren{\frac{-1}{p^2z}}-f(z).\]
\end{lemma}
The Fricke involution $\sm{0}{-1}{2}{0}$ swaps the cusps of $\Gamma_0(2)$, which are 0 and $\infty.$ We will use this fact in the proof of Lemma \ref{polynomial}, and the following relations between $\phi(z)$ and $\psi(z)$ will help us compute this involution.
\begin{lemma}
\label{phipsi}
\cite[Lemma~3]{andersen}
The functions $\phi(z)$ and $\psi(z)$ satisfy the relations
\begin{gather*}
\phi\paren{\frac{-1}{2z}} = 2^{-12}\psi(z),\\
\psi\paren{\frac{-1}{2z}} = 2^{12}\phi(z).
\end{gather*}
\end{lemma}
The following lemma is a special case of a result from one of Lehner's papers \cite{lehner2}. It provides a polynomial whose roots are modular forms used in the proof of Theorem \ref{betterbound}.
\begin{lemma}
\label{polyrelation}
\cite[Theorem~2]{lehner2}
There exist integers $b_j$ such that
\[U_2\phi(z) = 2 (b_1 \phi(z)+ b_2 \phi(z)^2).\]
Furthermore, let $h(z) = 2^{12}\phi(z/2)$, $g_1(z) = 2^{14}\paren{ b_1 \phi(z) + b_2 \phi(z)^2 }$, and $g_2(z) = -2^{14}b_2 \phi(z)$. Then
\[h(z)^2 - g_1(z)h(z)+g_2(z) = 0.\]
\end{lemma}
In the following lemma, we extend the result from the first part of Lemma \ref{polyrelation}, writing $U_2\phi^m$ as an integer polynomial in $\phi$. In particular, we give the least and greatest powers of the polynomial's nonzero terms.
\begin{lemma}
\label{polynomial}
For all $m \geq 1$, $U_2 \phi^m \in \Z[\phi].$ In particular,
\[U_2\phi^m = \sum_{j = \ceil{m/2}}^{2m} d(m,j)\phi^j\] where $d(m,j) \in \Z$, and $d(m, \ceil{m/2})$ and $d(m,2m)$ are not 0.
\end{lemma}
\begin{proof}
Using Lemmas \ref{fricke} and \ref{phipsi}, we have that
\begin{align*}
U_2\phi(-1/2z)^m
&= U_2\phi(2z)^m + 2\inv\phi(-1/4z)^m-2\inv \phi(z)^m \\
&= U_2\phi(2z)^m + 2^{-1-12m} \psi(2z)^m - 2\inv \phi(z)^m\\
&= 2^{-1-12m}q^{-2m} + O(q^{-2m+2})\\
2^{1+12m}U_2\phi(-1/2z)^m &= q^{-2m} + O(q^{-2m+2}).
\end{align*}
Because $\phi(z)^m$ is holomorphic at $\infty,$ $U_2 \phi(z)^m$ is holomorphic at $\infty$. So $U_2 \phi(-1/2z)^m$ is holomorphic at 0 and, since it starts with $q^{-2m}$, must be a polynomial of degree $2m$ in $\psi(z).$ Let $b(m,j) \in \Z$ such that
\[
2^{1+12m}U_2\phi(-1/2z)^m = \sum_{j = 0}^{2m} b(m,j)\psi(z)^j,
\]
and we note that $b(m, 2m)$ is not 0. Now replace $z$ with $-1/2z$ and use Lemma \ref{phipsi} to get
\[
2^{1+12m}U_2\phi(z)^m = \sum_{j = 0}^{2m} b(m,j)2^{12j}\phi(z)^j,
\]
which gives
\[
U_2\phi(z)^m = \sum_{j = 0}^{2m} b(m,j)2^{12(j-m)-1}\phi(z)^j.
\]
If $m$ is even, the leading term of the above sum is $q^{m/2}$, and if $m$ is odd, the leading term is $q^{(m+1)/2}$, so the sum starts with $j = \ceil{m/2}$ as desired. Notice that $b(m,j)2^{12(j-m)-1}$ is an integer because the coefficients of $\phi(z)^m$ are integers.
\end{proof}
We may repeatedly use Lemma \ref{polynomial} to write $U_2^\alpha \phi^m$ as a polynomial in $\phi$. Let
\begin{equation}
\label{fdefn}
f(\ell) = \ceil{\ell/2},\ f^0(\ell) = \ell, \text{ and } f^k(\ell) = f(f^{k-1}(\ell)).
\end{equation}
Using Lemma \ref{polynomial}, the smallest power of $q$ appearing in $U_2^\alpha \phi^m$ is $f^\alpha(m).$ Lemma \ref{binarygammma} provides a connection between $\gamma(m,\alpha)$ and the integers $f^\alpha(m).$
\begin{lemma}
\label{binarygammma}
The function $\gamma(m,\alpha)$ as defined in Theorem \ref{congruence} is equal to the number of odd integers in the list
\[m, f(m), f^2(m), \dots, f^{\alpha -1}(m). \]
\end{lemma}
\begin{proof}
Write the binary expansion of $m$ as $a_r\dots a_2 a_1$, and consider its first $\alpha$ digits, $a_\alpha \dots a_2 a_1$, where $a_ i = 0$ for $i > r$ if $\alpha > r.$ If all $a_i = 0$, then all of the integers in the list are even. Otherwise, suppose that $a_i = 0$ for $1 \leq i < i'$ and $a_{i'} =1.$ Apply $f$ repeatedly to $m$, which deletes the beginning 0s from the expansion, until $a_{i'}$ is the rightmost remaining digit; that is, $f^{i'-1}(m) = a_\alpha \dots a_{i'-1}a_{i'}.$ In particular, this integer is odd. Having reduced to the odd case, we now treat only the case where $m$ is odd.
If $m$ in the list is odd, then $a_1 = 1,$ which corresponds to the $+1$ in the definition of $\gamma(m,\alpha).$ Also, $f(m) = \ceil{m/2} = (m+1)/2.$ Applied to the binary expansion of $m$, this deletes $a_1$ and propagates a 1 leftward through the binary expansion, flipping 1s to 0s, and then terminating upon encountering the first 0 (if it exists), which changes to a 1. As in the even case, we apply $f$ repeatedly to delete the new leading 0s, producing one more odd output in the list once all the 0s have been deleted. Thus, each 0 to the left of $a_{i'}$ corresponds to one odd number in the list.
\end{proof}
\section{Proof of the Main Theorem}
Theorem \ref{congruence} will follow from the following theorem.
\begin{restatable}{theorem}{betterbound}
\label{betterbound}
Let $f(\ell)$ be as in (\ref{fdefn}). Let $\gamma(m,\alpha)$ be as in Theorem \ref{congruence}, and let $\alpha \geq 1$. Define
\[
c(m,j,\alpha) =
\begin{cases}
-1 & \text{if } f^{\alpha - 1}(m) \text{ is even and is not }2j, \\
0 & \text{otherwise}.
\end{cases}
\]
Write $U_2^\alpha \phi^m =\sum\limits_{j = f^\alpha(m)}^{2^\alpha m}d(m,j,\alpha)\phi^j.$ Then
\begin{equation}
\label{goodresult}
\nu_2(d(m,j,\alpha)) \geq 8(j - f^\alpha(m)) + 3 \gamma(m,\alpha) + c(m,j,\alpha).
\end{equation}
\end{restatable}
\noindent
The $\alpha$ of Theorem \ref{betterbound} corresponds to the $\alpha$ in $n=2^\alpha n'$ of Theorem \ref{congruence}, and because our methods use the $U_2$ operator, they do not give meaningful congruences for the case when $\alpha = 0.$ Theorem \ref{betterbound} is an improvement on the following result by Lehner \cite{lehner2}.
\begin{theorem}
\label{lehnerbound}
\cite[Equation~3.4]{lehner2}
Write $U_2 ^\alpha \phi^m$ as $\sum d(m,j,\alpha)\phi^j \in \Z[\phi].$ Then $\nu_2(d(m,j,\alpha)) \geq 8(j-1) + 3(\alpha-m+1)+(1-m).$
\end{theorem}
\noindent
In particular, Lehner's bound sometimes only gives the trivial result that the 2-adic valuation of $d(m,j,\alpha)$ is greater than some negative integer.
We prove Theorem \ref{betterbound} by induction on $\alpha$. The base case is similar to Lemma 6 from \cite{andersen}, which gives a subring of $\Z[\phi]$ which is closed under the $U_2$ operator. The polynomials are useful because their coefficients are highly divisible by 2. Here, we employ a similar technique to prove divisibility properties of the polynomial coefficients in Lemma \ref{polynomial}. We then induct to extend the divisibility results to the polynomials that arise from repeated application of $U_2.$
\\
\noindent
{\em Proof of Theorem \ref{betterbound}.} For the base case, we let $\alpha = 1$, and seek to prove the statement
\[U_2\phi^m = \sum_{j = \ceil{m/2}}^{2m} d(m,j,1)\phi^j\]
with
\begin{equation}
\label{alpha1bound}
\nu_2(d(m,j,1)) \geq 8(j-\ceil{m/2}) + c(m,j)
\end{equation}
where
\begin{equation*}
c(m,j) =
\begin{cases}
3 & m \text{ is odd},\\
0 & m =2j,\\
-1&\text{otherwise.}
\end{cases}
\end{equation*}
The term $c(m,j)$ combines $c(m,j,\alpha)$ and $3\gamma(m,\alpha)$ for notational convenience. We prove (\ref{alpha1bound}) by induction on $m$.
We follow the proof techniques used in Lemmas 5 and 6 of \cite{andersen}. From the definition of $U_2$, we have
\[U_2\phi^m = 2\inv\paren{\phi\paren{\frac{z}{2}}^m + \phi\paren{\frac{z+1}{2}}^m} = 2^{-1-12m}\paren{h_0(z)^m+h_1(z)^m}\]
where $h_\ell(z) = 2^{12}\phi\paren{\frac{z+\ell}{2}}.$ To understand this form, we construct a polynomial whose roots are $h_0(z)$ and $h_1(z)$. Let $g_1(z) = 2^{16}\cdot 3\phi(z)+2^{24}\phi(z)^2$ and $g_2(z) = -2^{24}\phi(z).$ Then by Lemma \ref{polyrelation}, the polynomial $F(x) = x^2 -g_1(z)x+g_2(z)$ has $h_0(z)$ as a root. It also has $h_1(z)$ as a root because under $z \mapsto z+1,$ $h_0(z) \mapsto h_1(z)$ and the $g_\ell$ are fixed.
Recall Newton's identities for the sum of powers of roots of a polynomial. For a polynomial $\prod_{i = 1}^n (x - x_i),$ let $S_\ell = x_1^\ell + \cdots + x_n^\ell$ and let $g_\ell$ be the $\ell$th symmetric polynomial in the $x_1, \dots, x_n.$ Then
\[
S_\ell = g_1S_{\ell-1}-g_2S_{\ell-2}+\cdots+(-1)^{\ell+1}\ell g_\ell.
\]
We apply this to the polynomial $F(x),$ which has only two roots, to find that
\[h_0(z)^m + h_1(z)^m =S_m = g_1S_{m-1}-g_2S_{m-2}.\]
Furthermore,
\begin{equation}
\label{u2phipoly}
U_2\phi^m = 2^{-1-12m}S_m.
\end{equation}
Lastly, let $R$ be the set of polynomials of the form $d(1) \phi(z) + \sum_{n=2}^N d(n) \phi(z)^n$ where for $n \geq 2$, $\nu_2(d(n)) \geq 8(n-1).$
Now we rephrase the theorem statement in terms of $S_m$ and elements of $R$. When $m$ is odd, we wish to show that for some $r \in R$, $U_2\phi^m = 2^{-8(\ceil{m/2}-1)+3}r.$ Performing straightforward manipulations using (\ref{u2phipoly}), this is equivalent to $S_m = 2^{8(m+1)} r$ for some $r \in R.$ Similarly, when $m$ is even and is not $2j$, we wish to show that $U_2\phi^m = 2^{-8(\ceil{m/2}-1)-1}r$ for some $r \in R.$ This again reduces to showing that $S_m = 2^{8(m+1)}r$ for some $r \in R.$ If $m = 2j$, then (\ref{alpha1bound}) gives $8(j-\ceil{2j/2})+0 = 0,$ which means the polynomial has integer coefficients, which is true by Lemma \ref{polynomial}.
When $m=1$ or $2,$ we have that $S_m=2^{8(m+1)}r$ for some $r \in R$, as
\begin{gather*}
S_1 = g_1 = 2^{8(2)}(3 \phi + 2^{8}\phi^2),\\
S_2 = g_1S_1 - 2g_2 = 2^{8(3)}(2 \phi + 2^{8}3^2\phi^2+2^{17}\phi^3+2^{24}\phi^4).
\end{gather*}
Now assume the equality is true for positive integers less than $m$ with $m$ at least 3. Then for some $r_1, r_2 \in R$,
\begin{align*}
S_m &= g_1S_{m-1}-g_2S_{m-2}\\
&= (2^{16}(3\phi + 2^8 \phi^2))(2^{8m}r_1)+(2^{24}\phi)(2^{8(m-1)}r_2)\\
&= 2^{8(m+1)}[(3\cdot2^8\phi+2^{16}\phi^2)r_1+2^8\phi r_2],
\end{align*}
completing the proof where $\alpha = 1.$
Assume the theorem is true for $U_2^\alpha \phi^m =\sum\limits_{j = s}^{2^\alpha m}d(j)\phi^j,$ meaning
\begin{equation}
\label{inductivehypothesis}
\nu_2(d(j)) \geq 8(j - f^\alpha(m)) + 3 \gamma(m,\alpha) + c(m,j,\alpha).
\end{equation}
Note that $s = f^\alpha(m).$ Letting $s' = f(s)$ and $U_2 \phi^j = \sum_{i = \ceil{j/2}}^{2j} b(j,i) \phi^i,$ we define $d'(j)$ as the integers satisfying the following equation:
\begin{align}
U_2^{\alpha + 1} \phi^m &= U_2 \left( \sum^{2^{\alpha} m}_{j = s} d(j) \phi^j \right) \nonumber\\
&= \sum^{2^{\alpha} m}_{j = s} d(j) U_2 \phi^j \nonumber\\
&= \sum^{2^{\alpha} m}_{j = s} \sum^{2 j}_{i = \ceil{j/2}} d(j)b(j,i) \phi^i \nonumber\\
&= \sum^{2^{\alpha + 1} m}_{j = s'} d'(j) \phi^j \label{djdjprime}.
\end{align}
We wish to prove that
\begin{equation}
\label{inductiveconclusion}
\nu_2(d'(j)) \geq 8(j - f^{\alpha+1}(m))+3\gamma(m,\alpha+1) + c(m,j,\alpha+1).
\end{equation}
We will prove inequalities that imply (\ref{inductiveconclusion}). Observe that
\begin{flalign*}
&&
c(m,j,\alpha + 1) &=
\begin{cases}
-1 &\text{if } s \text{ is even and not } 2j,\\
0 &\text{if } s\text{ is odd or }s = 2j,
\end{cases} &&\\
\text{and} && & &&
\\
&& \gamma(m,\alpha+1)&=
\begin{cases}
\gamma(m,\alpha) &\text{if } s \text{ is even,}\\
\gamma(m,\alpha)+1 & \text{if } s \text{ is odd.}
\end{cases} &&
\end{flalign*}
\noindent
Also, $c(m,s,\alpha) = 0$ because if $f^{\alpha-1}(m)$ is even, then $s = f^{\alpha-1}(m)/2$ so $f^{\alpha - 1}(m) = 2s.$ Therefore, $\nu_2(d(s)) \geq 3\gamma(m,\alpha)$ by (\ref{inductivehypothesis}).
If $s$ is even, we will show that
\begin{equation}
\label{theineq}
\nu_2 (d'(j)) \geq \max \left\{8\left(j - s'\right) -1 + \nu_2 (d(s)), \nu_2 (d(s)) \right\},
\end{equation}
because then if $j = s'$, we have
\begin{align*}
\nu_2(d'(s')) &\geq \nu_2(d(s))\\
&\geq 8(s' - s') + 3\gamma(m,\alpha) + c(m, s', \alpha + 1),
\end{align*}
and for all $j$,
\begin{align*}
\nu_2(d'(j))&\geq 8(j - s')+3\gamma(m,\alpha)+c(m,j,\alpha+1)\\
&= 8(j - f^{\alpha+1}(m))+3\gamma(m,\alpha+1)+c(m,j,\alpha+1),
\end{align*}
so that (\ref{theineq}) implies (\ref{inductiveconclusion}). If $s$ is odd we will show that
\begin{equation}
\label{theineq2}
\nu_2 (d'(j)) \geq 8\left(j - s'\right) +3 + \nu_2 (d(s)),
\end{equation}
because then
\begin{align*}
\nu_2(d'(j)) &\geq 8(j-s')+3\gamma(m,\alpha)+3\\
&= 8(j-s')+3(\gamma(m,\alpha)+1)\\
&=8(j-f^{\alpha+1}(m))+3\gamma(m,\alpha+1)+c(m,j,\alpha+1),
\end{align*}
which is (\ref{inductiveconclusion}).
For the sake of brevity, we treat here only the case where $s$ is odd. The case where $s$ is even has a similar proof. This case breaks into subcases. We will only show the proof where $j \leq 2s$, but the other cases are $2s < j \leq 2^{\alpha-1}m$ and $2^{\alpha - 1}m < j \leq 2^{\alpha+1}m$, using the same subcases for when $s$ is even. These subcases are natural to consider because in the first range of $j$-values, the $d(s)$ term is included for computing $d'(j)$, in the second range, there are no $d(s)$ or $d({2^{\alpha}m})$ terms, and in the third range, there is a $d({2^{\alpha}m})$ term.
Let $j \leq 2s$. Using (\ref{djdjprime}), we know that $d'(j) = \sum^{2j}_{i = s}d(i) b(i,j)$ by collecting the coefficients of $\phi^j$. Let $\delta(i)$ be given by
\[\delta(i) = \nu_2(d(i)) + \nu_2(b(i, j)).\]
Let $D = \set{\delta(i) \ | \ s \leq i \leq 2j}.$ Therefore we have
\begin{align*}
\nu_2 (d'(j)) &\geq \min \set{ \nu_2(d(i)) + \nu_2(b(i,j)) \mid s \leq i \leq 2j} \\
&= \min D.
\end{align*}
We claim that $\delta(i)$ achieves its minimum with $\delta(s),$ which proves (\ref{theineq2}). For that element of $D$, we know by inequality (\ref{alpha1bound}) that
\[\delta(s) \geq \nu_2(d(s)) + 8(j - s') + 3.\]
Now suppose $i > s$. Then every element of $D$ satisfies the following inequality:
\begin{align*}
\delta(i) &= \nu_2(d(i)) + 8\left(j - \ceil{i/2}\right) + c(i,j)\\
&\geq 8\left( i - s \right) - 1 + \nu_2(d(s)) + 8\left(j - \ceil{i/2}\right) + c(i,j) \\
&\geq 8\left(s + 1 - s + j - \ceil{(s+1)/2} \right) - 2 + \nu_2(d(s)) \\
&= 8\left(j - s' \right) + 6 + \nu_2(d(s)),
\end{align*}
but this is clearly greater than $\delta(s)$. Therefore, if $j \leq 2s$ and $s$ is odd, then $\nu_2 (d'(j)) \geq 8\left(j - s'\right) +3 + \nu_2 (d(s))$. The other cases are similar.
\qed
\\
Now Theorem \ref{congruence} follows easily from Theorem \ref{betterbound}.
\congruence*
\begin{proof}
Letting $j = f^\alpha(m)$ in (\ref{goodresult}), the right hand side reduces to
\[3\gamma(m,\alpha) + c(m,f^\alpha(m),\alpha).\]
Notice that $c(m,f^\alpha(m),\alpha)=0,$ because if $f^{\alpha-1}(m)$ is even, then $f^\alpha(m)=f^{\alpha-1}(m)/2$ so $f^{\alpha - 1}(m) = 2f^\alpha(m).$ The right hand side of (\ref{goodresult}) is minimized when $j = f^\alpha(m),$ so we conclude that $\nu_2(a(m,2^\alpha n')) \geq 3 \gamma(m,\alpha).$
\end{proof}
\section{The Parity of $a(1,n)$}
Table \ref{oddcoeffs} contains all odd coefficients of $\phi(z) = \sum_{n = 1}^\infty a(1,n)q^n$ up to $n = 225.$ The table shows that, up to $n = 225,$ the coefficient $a(1,n)$ is odd if and only if $n$ is an odd square. This holds in general.
\oddsquares*
\begin{proof}
Substitute $\eta(z)$ into the definition of $\phi(z):$
\[\phi(z) = \left(\frac{\eta(2z)}{\eta(z)}\right)^{24} = \left(\frac{ q^{2/24} \prod\limits_{n=1}^{\infty} (1-q^{2n}) }{ q^{1/24} \prod\limits_{n=1}^{\infty} (1-q^{n})}\right)^{24}.\]
By recognizing that $(1-q^{2n}) = (1- q^n)(1+ q^n)$ and simplifying, it is easy to see that
\[ \phi(z) = q \prod_{n=1}^{\infty} (1+q^{n})^{24}.\]
Reducing this mod 2, the odd coefficients will be the only nonzero terms. But $\binom{24}{i}$ is odd if and only if $i = 0,8,16,24$. It follows that
\[ \phi(z) \equiv q \prod_{n=1}^{\infty} (1+q^{8n} + q^{16n} + q^{24n}) \pmod{2}.\]
Immediately, it is clear that the coefficient of $q^n$ in the Fourier expansion of $\phi(z)$ is even if $n \not \equiv 1 \pmod{8}$.
Note that the coefficient of $q^n$ in the product $\prod_{n=1}^{\infty} (1+q^{n} + q^{2n} + q^{3n})$ is odd if and only if the coefficient of $q^{8n + 1}$ is odd in the Fourier expansion of $\phi(z)$. Furthermore, this product can be interpreted as the generating function for the number of partitions of $n$ where each part is repeated at most 3 times. The $n$th coefficient of the generating function is equivalent mod 2 to $T_n$ of \cite{partitions}. Theorem 2.1 of \cite{partitions} shows that $n$ is a triangular number if and only if $T_n$ is odd. Therefore, the coefficient of $q^n$ is odd in the Fourier expansion of $\phi(z)$ if and only if
\[n = 8\frac{k(k+1)}{2} + 1 = 4k^2 + 4k + 1 = (2k+1)^2,\]
meaning that $n$ is an odd square.
\end{proof}
\begin{table}[hpbt]
\begin{tabular}{|l|l|}
\hline
$n$ & $a(1,n)$ \\ \hline
1 & 1 \\ \hline
9 & 10400997 \\ \hline
25 & 254038914924791 \\ \hline
49 & 8032568516459357451913 \\ \hline
81 & 288274504516836871723618295721 \\ \hline
121 & 11156646861439805613118172199024038253 \\ \hline
169 & 453988290543887189391963063089337222684846687 \\ \hline
225 & 19146547947132951990683661128349583597266368489785587 \\ \hline
\end{tabular}
\caption{All odd coefficients of $\phi(z)$ up to $n = 225.$}
\label{oddcoeffs}
\end{table}
\bibliographystyle{amsplain} | 166,466 |
TITLE: Unspecific limits in a summation sign
QUESTION [0 upvotes]: I'm working on a probability problem, displayed in image 1, in which I encounter a problem. The solution is on the left and the original assignment on the right. The problem is that I don't get the final step in the solutions of part 1 and 2 of 11.1a and 11.b. The summation sign has lower limit l=1 and upper limit l=k-1. Because, as all probabilities for every number k=1,...,N are equal to 1/N, p(k)=1/N, there is no longer any l in the summation sign in which you can fill in the limits.
Anyone who can help me?
Problem
REPLY [0 votes]: what ever $a$ is, it holds $\sum_{i=1}^{k-1}a=\underbrace{a+a+...+a}_{k-1 \text{ times}}=a\cdot(k-1)$
REPLY [0 votes]: This is just the definition of the summation symbol. You probably know
$$\sum_{i=1}^n x_i:=x_1+x_2\ldots + x_n.$$ Now if $x_i$ is independent of $i$, i.e. $x_i=c$ for all $i=1,\ldots, n$ for some constant $c$, well then $\sum_{i=1}^nx_i=\sum_{i=1}^n c=c+c+\ldots+c$ ($n$-times in total), so $\sum_{i=1}^n=nc$.
In words, for some fixed $k\leq n$ in $\mathbb{Z}$, "$\sum_{i=k}^n [\text{something}]$" means:
Start by setting $i=k$ and evaluate $\text[\text{something}]$ for this $i$
Increase $i$ by one
Evaluate $\text[\text{something}]$ for $i$ and add it to the previous result
Repeat 2. & 3. until $i=n$, then stop.
Remark 1: So in total you do $n-k+1$ steps (check by counting).
Remark 2: If $[\text{something}]$ is independent of $i$, then there's not much to evaluate, you just take the $[\text{something}]$ as it is. Otherwise nothing in the above strategy changes.
This means that $$\sum_{l=k}^n c=c+c+\ldots+c=(n-k+1)c.$$
This means for example in 11.1a where $k$ is fixed:
$$\sum_{l=k-6}^{6}\left(\frac{1}{6}\cdot\frac{1}{6}\right)=\sum_{l=k-6}^{6}\frac{1}{36}=(6-(k-6)+1)\frac{1}{36}=\frac{13-k}{36}$$ | 166,662 |
Below is an extract from my forthcoming novel Samantha. The story concerns Samantha, a young woman forced into prostitution by her brutal pimp Barry. The below contains strong language. If you are offended by such language please read no further
---
Sam pressed the buzzer for flat 22. A fuzzy image of a man’s face appeared, “Hello?” “Its Angel” (as always Sam used her working name. She wanted clients
to know as little as possible about the real Samantha Parker-Jones). “Come up. I’m on the second floor”. The door clicked open and Sam entered a well lit
entrance hall. A shiny metal lift faced her. Sam pressed the call button.
The doors opened immediately. She entered pressing the button for the second floor. The lift rose soundlessly gliding open on the second floor. Sam stepped
out into a carpeted corridor. Flat 22 was to Sam’s left, at the far end of the corridor as she exited the lift. “Dear Christ please let him not be a nutter”
Sam thought as she pressed the doorbell.
The door was opened by a man in his mid to late thirties with receeding brown hair and a pair of hazel eyes which looked like pin pricks. “God a fucking
coke head” Sam thought. She recognised the classic signs of a heavy cocaine user, the eyes like pin pricks, the constant twitching of Nick’s hands and
his flushed face. “Come in” Nick said opening the door just enough to permit Sam to wriggle through. Once Sam had entered Nick double locked the door and
placed a heavy brass security chain across it for good measure. Sam moistened her dry lips, “Can I have the cash now please then we can have some fun”.
Nick reached into the pocket of his blue silk dressing gown and handed Sam an envelope. Sam opened it and counted the money. £150 for 2 hours, the money
was correct. “Thanks sweetie I’ll just call my driver and let him no that everything is OK then we can relax together”. She called Barry’s mobile. He picked
up on the first ring, he’d obviously been waiting with his mobile at the ready for her call. “Everythings fine”. “OK I’ll get one of the drivers to pick
you up in 2 hours. He’ll be waiting where I dropped you”. Barry ended the call. Sam returned the mobile to her handbag and giving Nick what she hoped approximated
to a genuine smile said “What would you like to do?” “Lets go to the bedroom”.
Sam followed Nick into a big bedroom. Her feet sank into an expensive blue carpet. “I like the bed. We can have some fun rolling about on that” she said.
To Sam her voice sounded false, however Nick appeared not to notice. “Want some coke?” he said pointing to several lines of cocaine neatly laid out on
a small mirror lying on the dressing table. “No thanks honey”. “Come on Barry told me that you are a good time girl, have some fucking Charlie” he said.
The muscles in Nick’s neck tightened and flecks of spittle flew from his mouth. “I would normally darling but I’m pregnant and I don’t want to harm the
baby”. It was a lie but it had the desired effect. “I understand. My friend’s girlfriend is having a baby and she’s given up smoking. Sorry I didn’t realise
that you where pregnant. Congratulations”. “Thanks. Would you like a nice massage?” Nick nodded and flinging his dressing gown over a chair which stood
next to the bed lay down on his stomach. Sam produced a bottle of jasmine oil from her bag and began massaging Nick’s back. As her hands neaded and rubbed
at Nick’s body Sam’s thoughts returned to Peter. He was a lovely man and she so wanted a decent guy in her life. But how could she think about relationships?
Her life was a fucking mess. She couldn’t look after herself let alone anyone else. Even if she did see Peter again how could she tell him that she was
a whore? “Oh my darling I’ve something important to tell you. I’m not really a nurse. I lied about that. I’m a tart. I sleep with guys for cash. Well,
actually I don’t keep much of what I earn. My pimp, Barry takes most of it. Do you still love me?” Sam laughed bitterly. “What is it?” “Nothing sweetie.
Just a frog in my throat. You relax and enjoy yourself. Turn over for me”. Nick turned over. “I want to fuck”. “OK babe but you need to be hard first.
Angel will make you big”. Mechanically Sam rubbed and stroked Nick’s penis. His cock was soon erect. Sam took a condom from the bedside table where she
had left it when she’d entered the room. With an expertise grown of long practice Sam rolled it down over Nick’s erection. She undid her skirt and removed
her knickers. Sam sat on Nick’s penis and began to girate her hips. Nick moaned with pleasure. “Is that good babe?” Sam said her mind elsewhere. She so
wanted to be back in Peter’s gentle embrace. She bit her lip fighting back tears.
With a grunt Nick came. Sam removed the durex. “Can I use your bathroom please?” “It’s just opposite the bedroom” Nick said. “Thanks I’ll take a shower
if that’s OK?” “Fine” Nick replied.
After her shower Sam returned to the bedroom. “Nice meeting you Nick. My driver will be here in five minutes. I’ll go down and wait for him outside if that’s
OK?” Without a word Nick got up, threw on his dressing gown, lead the way to the front door and unfastened it. “Thanks sweetie” Sam said giving Nick a
kiss on the cheek. Once the door had closed behind her Sam gave vent to her feelings. Her body shook with huge wretching sobs. “Driver will be waiting.
Pull yourself together girl” she said to herself. Blowing her nose on a tissue Sam headed purposefully for the lift.
(To be continued) | 198,159 |
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Nick is a Growth Marketer working for email marketing software Moosend. When he is not writing articles, you’ll find him running his SEO magic site-wide, or making friends from blogs and companies discussing the expert power of Email Marketing.
Let’s assume that you own a business. And let’s assume that you have already picked out what your business is about and who your target audience is. You’re all set and ready to begin and you’ve already decided that an email campaign is essential. That’s a great call. In this day and age, email marketing […] | 274,381 |
Whoever sings this song probably likes snakes. They probably have, like, three as pets. They probably feed them stuff that's not dead yet, because they think the snakes would like that. And they sing like a snake, the drums dance like a twisting and weaving head, the beep beep is sung with flashing eyes. Its weakness lies in that I never remember the song after it's finished; it's new every time. This is not actually a weaknessssssssssssssss. [Buy]
--
Bodies of Water - "I Guess I'll Forget the Sound, I Guess, I Guess"
The first half of this song is being led into a wooded area (where sunlight drips, Pony da Look lives). The second half of the song is a promenade in a circle, ceremonial (where hands are held, Tilly and the Wall reads Updike). Bodies of Water are new children, nobody is just like their brothers and sisters. [site]
--
let me ask you some questions about myself.Posted by Dan at December 29, 2005 4:07 AM
re. your interview:
dan, if you like fassbinder, you should check out "katzellmacher". (that might be spelled incorrectly; my german is shabby.) it's a really beautiful film. you might also like "satan's brew" (the cover of which will really impress that video store girl).Posted by: sarah at December 30, 2005 12:56 AM
hey sarah, happy new year. yeah, i saw parts of katzelmacher and that's what made me want to check out more. i'll get satan's brew as soon as i can.Posted by: dan at December 30, 2005 1:08 AM
cool! let me know what you think.Posted by: sarah at December 31, 2005 12:32 AM | 324,670 |
Annex 10: Characterisation of New Fuel Qualities
Final Report
Characterization of new fuel qualities
Background
Many standardised tests to evaluate fuel properties have originally been designed for screening hydrocarbon products. In the case of fuels blended with new components or treated with additives traditional test methods may give misleading results. The objective of the task was to evaluate the correlation between the results obtained by standardised testing and the real-life serviceability of new diesel fuel qualities. The following properties were studied: combustion properties, properties affecting exhaust emissions, low-temperature operability and stability and diesel fuel lubricity.
The diesel fuel matrix comprised conventional diesel fuels, low-emission diesel fuel, conventional diesel fuel blended with rapeseed methyl ester, tall oil methyl ester and ethanol. Some of the fuels were tested with and without ignition improver and cold flow improver additives to find out the response of additive in different base fuels.
The results showed that the traditional cetane number measurement describes well ignition delay of heavy-duty engine at low and medium loads, but is more suitable for hydrocarbon fuels than for alternative fuels. Cetane number does not describe combustion process with advanced light-duty vehicles. Cetane number overestimates the effect of cetane improvers, especially for biodiesels. Esters were found to act as effective lubricity additives according to HFRR tests.
Participants
- Belgium
- Canada
- Finland
- Japan
- Netherlands
- Sweden
- USA
Operating Agent
Päivi Aakko
VTT Energy, Engine Technology, Espoo, Finland | 312,058 |
Airtel Mobile Recharge is a one stop solution for providing easy airtel mobile rechargefor airtel prepaid mobile phones through the online, you can buy top up voucher anywhere from the world through online. There is no additional charge or hidden costs for recharging your prepaid mobile at airtelrecharge.in. No need to physically go to shops to buy recharge voucher you can recharge your airtel mobile phone without having to go to a dealer. Online mobile recharge airtel is faster and easier than ever before. All you need to do is select recharge type and enter your number and to proceed. Airtelrecharge offering you easy way to recharge your airtel mobile online by using Credit Cards/Debit Cards or online netbanking etc. | 315,706 |
Job Information
Kelly Services Material Handler - Columbia, MO in Columbia, Missouri
Kelly is hiring for 30 material handler and shipping receiving positions in Columbia, MO
Please apply to this posting or call 417-883-0830, option 3 to apply.
Job Details:
Shift: Monday- Friday occasional Saturdays
1st 6:00am- 2:00pm $15.00
2nd 2:00pm- 10:30pm $16.00
3rd 10:30pm- 6:30am $16.00
Work Environment: Cold storage facility typically around 45 degrees,
-Material Handler
The Material Handler is an entry level manufacturing position for the production team. The Material Handler plays a key-supporting role in the production line/process, being a safe and productive employee and packing and aiding in the making, preservation and shipping of Beyond Meat products.
-Ability to work 8-12 hour shifts
-Experience in Food manufacturing preferred
-Flexibility to work any/all shifts to support a high growth work environment
Please. | 213,892 |
\begin{document}
\maketitle
\section{Introduction}
In \cite{reimann.slaman:2007,reimann.slaman:2008}, Reimann and Slaman
raise the question ``For which infinite binary sequences $X$ do there
exist continuous probability measures $\mu$ such that $X$ is
effectively random relative to $\mu$?''
\subsection{Randomness relative to continuous measures}
We begin by reviewing the basic definitions needed to precisely
formulate this question.
\begin{notation}
\begin{itemize}
\item For $\sigma\in2^{<\omega}$, $[\sigma]$ is the basic open subset of
$2^\omega$ consisting of those $X$'s which extend $\sigma$.
Similarly, for $W$ a subset of $2^{<\omega}$, let $[W]$ be the open
set given by the union of the basic open sets $[\sigma]$ such that
$\sigma\in W$.
\item For $U\subseteq 2^\omega$, $\lambda(U)$ denotes the
measure of $U$ under the uniform distribution. Thus,
$\lambda([\sigma])$ is $1/2^\ell$, where $\ell$ is the length of
$\sigma$.
\end{itemize}
\end{notation}
\begin{definition}
A \textit{representation} $m$ of a probability measure $\mu$ on
$2^\omega$ provides, for each $\sigma\in 2^{<\omega}$, a sequence of
intervals with rational endpoints, each interval containing
$\mu([\sigma])$, and with lengths converging monotonically to 0.
\end{definition}
\begin{definition} Suppose $Z \in 2^\omega$. A \emph{test relative to
$Z$}, or \emph{$Z$-test}, is a set $W \subseteq \omega\times
2^{<\omega}$ which is recursively enumerable in $Z$. For
$X\in2^\omega$, $X$ \emph{passes} a test $W$ if and only if there is
an $n$ such that $X\not\in[W_n]$.
\end{definition}
\begin{definition} Suppose that $m$ represents the measure $\mu$ on
$2^\omega$ and that $W$ is an $m$-test.
\begin{itemize}
\item $W$ is \emph{correct for $\mu$} if and only if for all
$n$, $\sum_{\sigma \in W_n} \mu([\sigma]) \leq 2^{-n}.$
\item $W$ is \emph{Solovay-correct for $\mu$} if and only if
$\sum_{n\in\omega}\mu([W_n])<\infty$.
\end{itemize}
\end{definition}
\begin{definition}
$X\in2^\omega$ is \emph{$1$-random relative to a representation $m$ of
$\mu$} if and only if $X$ passes every $m$-test which is correct for
$\mu$. When $m$ is understood, we say that $X$ is 1-random relative
to $\mu$.
\end{definition}
By an argument of Solovay, see \cite{nies:2009}, $X$ is $1$-random
relative to a representation $m$ of $\mu$ if an only if for every
$m$-test which is Solovay-correct for $\mu$, there are infinitely
many $n$ such that $X\not\in[W_n]$.
\begin{definition}
$X\in\NCR_1$ if and only if there is no representation $m$ of a
continuous measure $\mu$ such that $X$ is 1-random relative to the
representation $m$ of $\mu$.
\end{definition}
In \cite{reimann.slaman:2008}, Reimann and Slaman show that if $X$ is
not hyperarithmetic, then there is a continuous measure $\mu$ such
that $X$ is 1-random relative to $\mu$. Conversely,
Kj{\o}s-Hanssen and Montalb\'an, see \cite{montalban:2005},
have shown that if $X$ is an element of a countable $\Pi^0_1$-class,
then there is no continuous measure for which $X$ is 1-random. As the
Turing degrees of the elements of countable $\Pi^0_1$-classes are
cofinal in the Turing degrees of the hyperarithmetic sets, the
smallest ideal in the Turing degrees that contains the degrees
represented in $\NCR_1$ is exactly the Turing degrees of the
hyperarithmetic sets.
In \citet{reimann.slaman:ta}, Reimann and Slaman pose the problem to
find a natural $\Pi^1_1$-norm for $\NCR_1$ and to understand its
connection with the natural norm mapping a hyperarithmetic set $X$ to
the ordinal at which $X$ is first constructed. As of the writing of
this paper, this problem is open in general, but completed in
\cite{reimann.slaman:ta} for $X\in\Delta^0_2$.
Suppose that $X\in\Delta^0_2$ and that for all $n$,
$X(n)=\lim_{t\to\infty}X_t(n)$, where $X_t(n)$ is a computable
function of $n$ and $t$. Let $g_X$ be the convergence function for
this approximation, that is for all $n$, $g_X(n)$ is the least $s$
such that for all $t\geq s$ and all $m\leq n$, $X_t(m)=X(m)$. Let
$f_X$ be function obtained by iterated application of $g_X$:
$f_X(0)=g_X(0)$ and $f_X(n+1)=g_X(f_X(n))$.
For a representation $m$ of a continuous measure $\mu$, the
granularity function $s_m$ maps $n\in\omega$ to the least $\ell$ found
in the representation of $\mu$ by $m$ such that for all $\sigma$ of
length $\ell$, $\mu([\sigma])<1/2^n$. Note that, $s_m$ is
well-defined by the compactness of $2^\omega$.
\begin{theorem}[Reimann and Slaman~\cite{reimann.slaman:ta}]\label{1.7}
If $X$ is 1-random relative the representation $m$ of $\mu$, then
the granularity function $s_m$ for $\mu$ is eventually bounded by
$f_X$.
\end{theorem}
Thus, there is a continuous measure relative to which $X$ is 1-random
if and only if there is a continuous measure whose granularity is
eventually bounded by $f_X$. The latter condition is arithmetic,
again by a compactness argument.
\subsection{$K$-triviality}
$K$-triviality is a property of sequences which characterizes another
aspect of their being far from random. We briefly review this notion and
the results surrounding it. A full treatment is given in
Nies~\cite{nies:2009}.
For $\sigma\in2^{<\omega}$, let $K(\sigma)$ denote the prefix-free
Kolmogorov complexity of $\sigma$. Intuitively, given a universal
computable $U$ with domain an antichain in $2^{<\omega}$, $K(\sigma)$
is length of the shortest $\tau$ such that $U(\tau)=\sigma$.
Similarly, for $X\in2^\omega$, let $K^X(\sigma)$ denote the prefix-free
Kolmogorov complexity of $\sigma$ relative to $X$. That is, $K^X$ is
determined by a function universal among those computable relative to $X$.
\begin{definition}\label{1.8}
A sequence $X\in2^\omega$ is \emph{$K$-trivial} if and only if there is a
constant $k$ such that for every $\ell$, $K(X\restriction\ell) \leq
K(0^\ell)+k$, where $0^\ell$ is the sequence of $0$'s of length
$\ell$.
\end{definition}
By early results of Chaitin and Solovay and later results of Nies and
others, there are a variety of equivalents to $K$-triviality and a
variety of properties of the $K$-trivial sets. For example, $X$ is
$K$-trivial if and and only if for every sequence $R$, $R$ is 1-random
for $\lambda$ if and only if $R$ is 1-random for $\lambda$ relative to
$X$.
In the next section, we will apply the following.
\begin{theorem}[Nies~\cite{nies:2009}, strengthening
Chaitin~\cite{chaitin:1976}]\label{1.9}
If $X$ is $K$-trivial, then there is a computably enumerable and
$K$-trivial set which computes $X$.
\end{theorem}
The following theorem follows from the work of Nies and others \cite{nies:2009}.
Some versions of this property have been used by Ku\v{c}era extensively, e.g.\ in \cite{MR820784}.
\begin{theorem}\label{1.10} Suppose $X$ is
$K$-trivial and $\{U_e^X:e\in\om\}$ a uniformly $\Sigma^{0,X}_1$
family of sets. Then, there is a computable function $g$ and a
$\Sigma^0_1$ set $V$ of measure less than 1 such for every $e$, if
$\lambda(U_e^Z)<2^{-g(e)}$ for every oracle $Z$, then $U_e^X\subseteq V$.
\end{theorem}
\begin{proof} (George Barmpalias)
Let $\big((E_i^e)\big)_{e\in\Nat}$ be a uniform sequence of all oracle \ml
tests. A standard construction of a universal oracle \ml test $(T_i)$
(e.g.\ see \cite{nies:2009}) gives a recursive function $f$
such that $\forall Z\subseteq \om\ (E_{f(i,e)}^{e,Z}\subseteq T_i^Z)$ for all $e,i\in\Nat$.
Let $T:=T_2$ and $f(e):=f(2,e)$ for all $e\in\Nat$, so that $\mu( T^Y) \leq 2^{-2}$ for all $Y \in 2^\omega$ and $E_{f(e)}^e\subseteq T$ for all $e\in\Nat$.
In \cite{MR2336587} it was shown that $X$ is $K$-trivial iff for some member $T$ of a universal oracle Martin-L\"of test, there is a $\Sigma^0_1$ class $V$ with $T^X \subseteq V$ and $\mu(V) < 1$.
Now given a uniform enumeration $(U_e)$ of oracle \sz classes we have the following property of $T$:
\begin{quote}\label{Tproperty}
{There is a recursive function $g$ such that for each $e$, \\ either
$\exists Z\subseteq \om\ (\mu(U_e^{Z})\geq 2^{-g(e)-1})$, or $\forall Z\subseteq \om\ (U_e^Z\subseteq T^Z)$.}
\end{quote}
To see why this is true, note that every $U_e$ can be effectively
mapped to the oracle \ml test $(M_i)$ where $M_i^Z=U_e^Z[s_i]$ and $s_i$
is the largest stage such that $\mu(U_e^Z[s_i])<2^{-i-1}$ (which could be infinity).
Effectively in $e$ we can get an index $n$ of $(M_i)$.
It follows that if $\mu(U^Z_e)<2^{-f(n)-1}$ for all $Z$, then $U_e^X = M_{f(n)}^X = E^{n,X}_{f(n)} \subseteq T^X \subseteq V$.
So $g(e)=f(n)+1$ is as wanted.
\end{proof}
\subsection{\texorpdfstring{$X$ is $K$-trivial implies $X\in\NCR_1$}{K-trivial implies NCR-1}}
Intuitively, $X\in\NCR_1$ asserts that $X$ is not effectively random
relative to any continuous measure and $X$ is $K$-trivial asserts that
relativizing to $X$ does change the evaluation of randomness relative
to the uniform distribution. In the next section, we connect the two
notions by showing that if $X$ is $K$-trivial then $X\in\NCR_1$.
\section{The Main Theorem}
\begin{theorem}\label{2.1}
Every $K$-trivial set belongs to $\NCR_1$.
\end{theorem}
\begin{proof}
Let $Y$ be $K$-trivial and let $\mu$ be a continuous measure with
representation $m$; we want to show $Y$ is not $\mu$-random. By
Theorem~\ref{1.9}, let $X$ be a computably enumerable $K$-trivial
sequence that computes $Y$. Let $f$ be the iterated convergence
function as defined above for the computable approximation to $Y$
given by approximating $X$'s computation of $Y$. Since $X$ is
computably enumerable, $X$ can compute the convergence function for
its own enumeration and hence $f$ is computable from $X$.
Let $s_m$ be the granularity function for $\mu$ as represented by
$m$. By Theorem~\ref{1.7}, $f$ eventually dominates $s_m$. By
changing finitely many values of $f$, we may assume that $f$
dominates $s_m$ everywhere. So, we have that for every $n$
\[
\mu([Y\upto f(n)])\leq 2^{-n}.
\]
Further, we may assume that $f$ can be obtained as the limit of a
computable function $f(n,s)$ such that for all $s$, $f(n-1,s)\leq
f(n,s)\leq f(n,s+1)$.
We will build an $m$-test $\{S_i:i\in\om\}$ which is Solovay-correct
for $\mu$ and which $Y$ does not pass, thereby concluding that $Y$
is not $\mu$-random. That is, we plan to build $\{S_i:i\in\om\}$ to
be a uniformly $\Sigma^{0,m}_1$ sequence of sets such that
$\sum_{i\in\omega}\mu(S_i)$ is bounded and such that there are
co-finitely $i$ for which $Y\in[S_i]$. Our construction will not be
uniform.
$X$'s $K$-triviality is exploited in the form of Theorem~\ref{1.10}.
Let $V$ and $g$ be given by Theorem~\ref{1.10} where $\{U_e^X:e\in\om\}$ is a listing of all $\Sigma^{0,X}_1$ sets.
We will build an oracle $\Sigma^0_1$ class $U$ along the construction.
We use the recursion theorem to assume that in advance we know an index $e$ such that $U=U_e$.
During the construction we will make sure that for every oracle $Z$, $\lambda(U^Z)<2^{-g(e)}$.
Theorem~\ref{1.10} then implies that $U^X\subseteq V$ where $V$ is a $\Sigma^0_1$ class of measure less than 1.
To simplify our notation, let $a$ denote $g(e)$.
Furthermore, assume $a$ is large enough so that $\lambda(V)+2^{-a}<1$.
We use the approximation to $X$ as a computably enumerable set to
enumerate approximations to initial segments of $Y$ into the sets
$S_i$; we rely on the $K$-triviality of $X$ to keep the total
$\mu$-measure of the $S_i$'s bounded.
For each $n>a$ we have a requirement $R_n$ whose task is to
enumerate $Y\upto f(n)$ into $S_n$. Let $y_{n,s}=Y_s\upto f(n,s)$
the stage $s$ approximation to $Y\upto f(n)$. Let $x_{n,s}$ be the
initial segment of $X_s$ necessary to compute $y_{n,s}$ and
$f(n,s)$. So, if $y_{n,s+1}\neq y_{n,s}$, it is because
$x_{n,s+1}\neq x_{n,s}$. In this case, $x_{n,s+1}$ is not only
different than $x_{n,s}$, but also incomparable. At stage $s$,
$R_n$ would like to enumerate $y_{n,s}$ into $S_n$, but before doing
that it will {\em ask for confirmation} using the fact that
$U^X\subseteq V$. Since we are constrained to keep $\lambda(U^X)$
less than or equal to $2^{-a}$, we will restrict $R_n$ to enumerate
at most $2^{-n}$ measure into $U^X$. The reason why we need a
bit of security before enumerating a string in $S_n$ is that we have
to ensure that $\sum_i\mu(S_i)$ is bounded. For this purpose, we
will only enumerate mass into $S_n$ when we see an equivalent mass
going into $V$.
{\bf Action of requirement $R_n$:}
\begin{enumerate}
\item The first time after $R_n$ is initialized, $R_n$ chooses a
clopen subset of $2^{\om}$, $\si_n$, of $m$-measure $2^{-n}$, that
is disjoint form $V_s$ and $U_s^{X_s}$. Note that since $V$ and
$U^{X_s}$ have measure less than $\lambda(V)+2^{-a}<1$, we can always
find such a clopen set. Furthermore we can chose $\si_n$ to be
different from the $\si_i$ chosen by other requirements $R_i$,
$i>a$. We note the value of $\si_n$ might change if $R_n$ is
initialized.
\item To {\em confirm} $x_{n,s}$, requirement $R_n$ enumerates $\si_n$
into $U^{x_{n,s}}$. Requirement $R_n$ will not be allowed to
enumerate anything else into $U^{X_s}$ unless $X_s$ changes below
$x_{n,s}$. This way $R_n$ is always responsible for at most
$2^{-n}$ measure enumerated in $U^{X_s}$.
\item Then, we wait until a stage $t>s$ such that
\begin{enumerate}
\item either $x_{n,s}\not\subseteq x_{n,t}$ (as strings),
\item or $\si_n\subseteq V_t$.
\end{enumerate}
Observe that if $x_{n,s}$ is actually an initial segment of $X$,
then we will have $\si_n\subseteq U^X\subseteq V$. So, we will
eventually find such a stage $t$.
\begin{itemize}
\item In Case 3(a), we start over with $R_n$. Note that in
this case $\si_n$ has come out of $U^{X_t}$, and hence $R_n$ is
responsible for no measure inside $U^{X_t}$ at stage $t$.
\item In Case 3(b), if $\mu([y_{n,t}])\leq 2^{-n}$, enumerate
$y_{n,t}$ into $S_n$. (Recall that we are allowed to use the
representation of $\mu$ as an oracle when enumerating $S_n$.)
\end{itemize}
\end{enumerate}
Since we only enumerate $y_{n,t}$ of $\mu$-measure less than $2^{-n}$
when $\si_n$ is enumerated in $V$, we have that
\[
\sum_i\mu(S_i) \leq \lambda(V)<1.
\]
It is not hard to check that $\lambda(U^X) \leq \sum_{n=a+1}^\infty 2^{-n}
=2^{-a}$, so we actually have that $U^X\subseteq V$. Also notice that
once $x_{n,s}$ is a initial segment of $X$, we will eventually
enumerate $\si_n$ into $V$ and an initial segment of $Y$ into $S_n$.
\end{proof}
\bibliographystyle{alpha} | 216,712 |
If there were one chocolate chip cookie recipe I could bake up and eat for the rest of my life, this would be it! These chocolate chip cookies are giant, soft on the inside, slightly crunchy on the outside, perfectly buttery, and everything a chocolate chip cookie should be. Thanks to my wonderful Mother-and-Grandma-in-love for sharing this recipe with me, and making them numerous times over the years! I don’t think any family get together is complete without these cookies. For a wedding gift my sweet Grandma-in-Law gave me a binder full of family recipes, with this one in it, and I think it’s safe to say I’ve made this recipe the most.
I’m so excited to be able to share this recipe with you all. Not only is this recipe easy to follow it’s also fun to customize to your liking. We like the cookies best with a half and half mixture of white chocolate chips and semi-sweet or milk chocolate chips, but they can be made with just one kind if you prefer. They’re also great with nuts like walnuts, or macadamia nuts. One tip I learned from my MIL is to add extra vanilla for kicks, it makes them even more delicious. In this batch I halved the entire recipe, because there’s only two of us and we definitely don’t need to eat 20 cookies each, but I still used the full amount of vanilla, and they were so delicious and full of flavor! 😉 You can’t go wrong with this recipe.
First off, cream together the wet ingredients, then combine with the dry ingredients.
Mix in those chips and try your best to not eat all of the dough 😉
Scoop them out and bake them up…
Enjoy your perfect cookies, as many as you want in one sitting, or one a day, or have one for breakfast and 5 for dinner. You do you, I won’t judge. 🙂
Eric loves his cookies warm out of the oven and thinks I’m so weird because I like mine best at room temperature. I told him it’s because when the cookies are warm it’s all about the chocolate, but when they’re room temp you get all the flavors from the butter, salt, vanilla, brown sugar and chocolate chips.. needless to say he thinks I’m a weirdo but I like what I like, which includes him 😉
Hope you enjoy this Chocolate Chip Cookie recipe as much as our family does and share them with all your loved ones. 🙂
Chocolate Chip Cookies
Makes about 40 cookies
Preheat oven to 325º F
Ingredients
1 pound salted butter, at room temp.
2 cups brown sugar
1 and ½ cups sugar
2 tablespoons vanilla extract
3 eggs
6 cups all-purpose flour
1 and ½ teaspoons baking soda
1 and ½ teaspoons salt
4 cups white or semi-sweet or milk chocolate chips or combination
2 cups walnuts (optional)
Preparation
In a large mixing bowl using a hand-mixer or in a stand-mixer cream together butter, sugar, eggs, and vanilla. In a separate mixing bowl mix together flour, salt and baking soda. Mix the dry ingredient mix into the wet mix in a few batches, making sure it’s all thoroughly mixed together. Lastly, using a rubber spatula fold in the chocolate chips. At this point the mixture should be sticky and dry.
Use an ice cream scoop to scoop out leveled-out balls of dough onto a nonstick cookie sheets. Place the cookie dough balls about 2 inches apart, they’ll spread out quite a bit.
Place the cookie sheets into the oven and bake for 6 minutes. At the 6 minute mark switch the cookie sheets, so the bottom sheet will then go on the top rack and the top sheet will go on the bottom rack. Bake for an additional 6 minutes. Total bake time: 12 minutes
Remove cookies from oven, they’ll be really soft, give them a few minutes to firm up before moving them to cooling racks. Enjoy warm or place them in an airtight container for up to a week.
Thanks for stopping by CB EATS!
xo CB
4 thoughts on “Chocolate Chip Cookies” | 138,919 |
\begin{document}
\maketitle
\begin{abstract}
Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\RR^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for $n$ and $d$ with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of some known results in fixed dimension to the case where dimension tends to infinity are described.
\end{abstract}
\tableofcontents
\section{Introduction}
The convex hull of $n$ i.i.d.\ random points in $\RR^d$ is a well understood random geometric object in fixed dimension $d$ for a large variety of distributions such as Gaussian or uniform distribution in a smooth convex body or its boundary. There exists an extensive literature on the properties of these polytopes, as surveyed in \cite{Barany_book, Hug_book, weil_book}. The most well-studied characteristics are the expected number of faces and intrinsic volumes. In fixed dimension, there are also many known results on the asymptotic behavior of functionals of these random polytopes as the number of points $n$ tends to infinity \cite{barany_vu_2007, reitzner_2005, vu_2006}.
The results focus mainly on concentration around the mean and central limit theorems for the volume and the number of faces.
There is also increasing interest in the asymptotic behavior of random polytopes as the dimension $d$ tends to infinity. This high dimensional regime is relevant to applications in statistics (e.g. \cite{Cai_Fan_Jiang_2011, Candes_Tao_2007}), compressed sensing \cite{Candes_Tao_2006, Donoho_2006}, and information theory \cite{infotheoryBook, Shannon_1948}.
For the case of the convex hull of i.i.d.\ points, recent developments in high dimensions include an asymptotic formula as $n$ and $d$ tend to infinity for the the expected number of facets of Gaussian random polytopes in \cite{Boroczky_Lugosi_Reitzner_2018} and threshold phenomena for the volume of beta random polytopes as dimension grows in \cite{Bonnet_2018}.
Central limit theorems for the volume of random simplices in high dimensions were proved in \cite{Thale_asymptoticnormal_2019, Thale_simplices_2019}.
The geometry of these random polytopes in high dimensions have also been studied using techniques from the field of asymptotic geometric analysis.
For example, the isotropic constant of random polytopes
was studied in \cite{alonso2008,alonso2016,dafnis2010,klartag2009,Proncho_Thale_2017, Proncho_Thale_2019}.
Other random polytopes studied in high dimensions include particular cells in Poisson hyperplane tessellations \cite{Voronoi_2008, Hug_Thale_2015,oreilly2018bit} and Poisson Voronoi tessellations \cite{oreilly2018thin}.
In this paper, we are interested in the $(d-1)$-dimensional faces, or facets, of the random polytope generated as the convex hull of $n$ i.i.d.\ points chosen uniformly from unit sphere $\mathbb{S}^{d-1}$.
Formulas for the expected number of faces as well as the surface area and mean-width of this random polytope were first obtained in \cite{buchta_stochastical_1985}. These are recovered in \cite{kabluchko2019, kabluchko_beta_2018} which provide formulas for the expected values of all intrinsic volumes and number of $k$-dimensional faces for the classes of beta and beta-prime polytope.
Additionally, concentration and a central limit theorem for the volume was proved in \cite{thale2018} for the convex hull of i.i.d.\ points chosen uniformly on the boundary of a smooth convex body, which includes the case of a sphere.
This work has been extended to all intrinsic volumes in \cite{turchi2018}.
In both cases the results hold only in fixed dimension $d$. Here, we consider both the expected number of facets as well as the height of the facets, as both the number of points $n$ tends to infinity and the dimension is either fixed or allowed to grow.
To formally present the problems under consideration, we first define some notation.
Let $ X_1 , \ldots , X_n $ be i.i.d.\ unit vectors uniformly distributed on the sphere $\mathbb{S}^{d-1}$, $ n > d \geq 2$, and denote by $ P_{n,d} = [ X_1 , \ldots , X_n ] $ the convex hull of these points.
We say that a facet of $P_{n,d}$ has height $h\in[-1,1]$ if its supporting hyperplane has the form $\{ x \in \RR^d : \langle x , u \rangle = h \} $ for some unit vector $u\in \mathbb{S}^{d-1}$ and the polytope $P$ is contained in the half space $\{ x \in \RR^d : \langle x , u \rangle \leq h \} $.
Note that a facet can have a negative height.
In fact, a polytope contains the origin in its interior precisely when all facets have positive height.
In this paper we investigate the heights of the facets of $ P_{n,d} $, as $n \to \infty $ and $ d $ is either constant or tends to $ \infty $.
In particular, we are interested by the following three problems.
\medskip
First, consider the typical facet of $ P_{n,d} $. This is a random $ (d-1) $ dimensional simplex with vertices on the unit sphere which has the same distribution as $ [ X_1 , \ldots , X_d ] $ conditioned on the event that it is a facet of $ P_{n,d} $. The \textit{typical height} $ \typheight $ is the random variable defined as the height of the typical facet of $ P_{n,d} $.
We are interested by the \textbf{distribution of the typical height $ \typheight $}, given in
\[ \PP ( \typheight \in \cdot )
= \PP ( [X_1,\ldots,X_d] \text{ has height } \in \cdot \mid [X_1,\ldots,X_d] \text{ is a facet of } P_{n,d} ) . \]
Second, we will find a tight \textbf{range containing the heights of all the facets of $P_{n,d}$}. For heights $ -1 < h_1 < h_2 < 1$, denote the \textit{expected number of facets with height in the range $[h_1,h_2]$} by
\[ \facets{h_1}{h_2}
= \EE \# \{ \text{facets of $P_{n,d}$ with height in } [h_1,h_2] \} . \]
We will find heights $ -1 < h_1 < h_2 < 1$, with $ h_i $ depending on $ n $ and $ d $, such that both $ \facets{-1}{h_1} \to 0 $ and $ \facets{h_2}{1} \to 0 $. In particular, this implies that the heights of all the facets belongs to the range $ [ h_1 , h_2 ] $, with probability tending to $1$.
Finally, we consider the \textbf{expected number of facets} $\facets{-1}{1}$, for which we are interested in an asymptotic expression.
The computation of this asymptotic will be facilitated by the results of the second question since $ \facets{-1}{1} = \facets{h_1}{h_2} + o(1) $.
\medskip
There are a various asymptotic regimes for the dimension $d$ and number of points $n$ we consider that will produce different results on the behavior of the facets of the polytope as $n$ grows to infinity. In order to briefly describe these regimes let us introduce some notation. Here and in the rest of the paper we consider that $\NN \ni n\to\infty$ and $d=d(n)\in\{2,\ldots,n-1\}$ is a function of $n$ which is either constant or tends to infinity. We use the classical Landau notation. For any sequence $f(n)$, a term $O(f(n))$ (resp. $o(f(n))$) represents a sequence $g(n)$ such that $g(n)/f(n)$ is bounded (resp. tends to $0$). When $g(n) =o(f(n))$ we also write $g\ll f$ or $f\gg g$. When $f(n)/g(n)$ is both lower and upper bounded by positive constants, we write $f=\Theta(g)$. Finally, $f\sim g$ means $f(n)/g(n)\to 1$.
The regimes can first be divided into two main categories. We will call all regimes where $n \gg d$ the fast regimes, and the regimes where $n = O(d)$ are called the slow regimes.
Within the slow regimes, we first have the \textit{sublinear} regime where $n-d \ll d $. In this case, the heights of the facets will approach zero faster than $d^{-1/2}$. Second is the
\textit{linear} regime where $ (n - d)/d \to \rho $ for some $\rho \in (0, \infty)$. In this case, the heights of the facets approach zero on the order of $d^{-1/2}$. In addition, we are able to identify optimal $r_{u}$, $r_{\ell}$ such that $F_{[-1,1]} = F_{[r_{\ell}/\sqrt{d}, r_u/\sqrt{d}]} + o(1)$.
For the fast regimes, the first is the \textit{subexponential} regime, where $ \ln n \ll d \ll n$. This regime includes $n = \Theta(d^{\alpha})$ for any $\alpha > 1$.
In this regime, the heights approach zero on the order of $\sqrt{(\ln n)/d}$.
Then, we have the exponential regimes where $(\ln n)/d\to \rho$. In this case, the heights of all the facets approach a positive constant less than one as $n$ increases.
Finally, we have the \textit{super exponential} regime, where $\ln n \gg d$. This includes the case when $d$ is fixed. In this regime, we show that the heights of all the facets approach one, the diameter of the ball.
In \cite{Cai_Fan_Jiang_2013}, the authors consider a very related question in this setting, proving results on the minimum and maximum angles between any two of $n$ points uniformly distributed on the unit sphere as both $n$ and dimension $d$ grows. Their work was motivated by studying the coherence of random matrices with particular applications to hypothesis testing for spherical distributions and constructing matrices for compressed sensing \cite{Cai_Fan_Jiang_2011}. It is interesting to note that their results are divided into the same asymptotic regimes for $n$ and $d$ as in our work, since a small minimum angle between vectors corresponds to facets with heights close to one and large minimum angles corresponds to facets with heights close to zero.
The organization of the paper is as follows.
In sections \ref{s:typheight}, \ref{s:range}, and \ref{s:expnum}, we present our results for each problem we consider. In section \ref{s:fixed}, we describe related results from the literature in fixed dimension, and describe how our results extend these formulas to the case when $d$ tends to infinity. Finally, in section \ref{s:proofs} we present the proofs in increasing order of the asymptotic regimes for $n$.
\section{Typical height} \label{s:typheight}
Recall that the number of points $n$ goes to infinity and the dimension $d$ is either fixed or goes to infinity.
In this paper we use the notation $\xrightarrow{D}$ and $\xrightarrow{P}$ for convergence in distribution and probability, respectively.
First we consider the regime where $(n-d)/d \to \rho \in [0, \infty)$. The lower bound $\rho \geq 0$ comes from the assumption that $n \geq d+1$ to ensure we have a full-dimensional polytope, with probability $1$. Also note that in this regime, when $n \to \infty$, $d \to \infty$ also. The first two results cover the case when $\rho = 0$, i.e., when $n - d = o(d)$.
\begin{theorem} \label{thm:typheight_smalln2}
Assume $(n-d)/\sqrt{d} \to \rho \in [0, \infty)$.
Then,
\[ d \typheight - \rho\sqrt{2/\pi} \xrightarrow{D} Z,\]
where $Z$ is a standard normal random variable.
\end{theorem}
Now, in the case where $n-d \gg \sqrt{d}$ and $n-d$ still grows slower than $d$, the typical height will scale like $(n-d)d^{-3/2}$, which is $o(d^{-1/2})$ but grows faster than $d^{-1}$, which is the scaling of the typical height in Theorem \ref{thm:typheight_smalln2}. The precise result is as follows.
\begin{theorem} \label{thm:typheight_smalln}
Assume $\sqrt{d} \ll n-d \ll d$. Then,
\[\frac{d^{3/2}}{n-d}\typheight \xrightarrow{P} \sqrt{2/\pi}.\]
\end{theorem}
Next we consider the case when $n-d = \rho d + o(d)$ for a finite constant $\rho$ for $\rho > 0$.
\begin{theorem} \label{thm:typheight_nlin}
Fix $\rho > 0$ and assume $ (n-d)/d \to \rho $. Define the function
\[f_{\rho}(r) = \rho \ln \Phi(r)-\frac{r^2}{2}, \qquad r \in \RR,\]
where $\Phi(r)$ is the CDF of a standard normal random variable, and let $r_{\rho} := \argmax f_{\rho} \in (0, \infty)$. Then,
\[ \sqrt{d}\typheight \xrightarrow{P} r_{\rho}.\]
\end{theorem}
Next we consider all asymptotic regimes for $n$ and $d$ such that $n\gg d$. There are sub-regimes with different asymptotic behaviors for $\typheight$, but the unifying property of this regime is that $\typheight$ either approaches a positive constant in $(0,1]$ or tends to zero slowly enough so that the quantity $(1 - \typheight^2)^{(d-1)/2}$ will approach zero.
The following result shows that the height of the typical facet, scaled appropriately for each regime, will converge in probability to a constant.
\begin{theorem} \label{thm:typheight_largen}
Assume that $d\ll n$.
\begin{enumerate}
\item[(i)] If $\ln n \ll d$, then $\typheight$ is approaching zero, and more precisely,
\[ \sqrt{\frac{d}{\ln (n/d)}}\typheight
\xrightarrow{P} \sqrt{2}.\]
\item[(ii)] If $(\ln n)/d \to \rho > 0$, then
\[ \typheight \xrightarrow{P} \sqrt{1 - e^{-2\rho}}.\]
\item[(iii)] If $\ln n \gg d$, then $\typheight$ is approaching one, and more precisely,
\[ -\frac{d-1}{\ln n}\ln(1 - \typheight^2) \xrightarrow{P} 2.\]
\end{enumerate}
\end{theorem}
The last result of this section is on the asymptotic law of the typical height in the sub-regime of the super exponential regime where $n$ grows fast enough so that $\ln n \gg d \ln d$ holds. This regime includes the case when $d$ is fixed.
We show that an appropriate renormalization of the typical height is close, in total variation distance (denoted by $d_{TV}$), to a $\Gamma_{d-1}$-distributed random variable, i.e. a positive random variable with density proportional to $e^{-t} t^{d-2}$. When $d$ tends to infinity this implies a Central Limit Theorem.
\begin{theorem} \label{thm:CLT}
Assume that $\ln n \gg d \ln d$.
For $k\in\NN$, set $ X_{d-1} $ to be a $\Gamma_{d-1} $ distributed random variable.
Then
\[ d_{TV} \left( X_{d-1} \,,\, n \frac{ \Gamma (\frac{d}{2}) }{ 2 \sqrt{\pi} \Gamma (\frac{d+1}{2}) } (1-\typheight^2)^{\frac{d-1}{2}} \right)
\to 0 . \]
It implies that
\begin{itemize}
\item[(i)] if $d$ is fixed, then
\[ n \frac{ \Gamma (\frac{d}{2}) }{ 2 \sqrt{\pi} \Gamma (\frac{d+1}{2}) } (1-\typheight^2)^{\frac{d-1}{2}}
\xrightarrow{d_{TV}} X_{d-1} ,\]
\item[(ii)] if $d\to\infty$, then
\[ \frac{n}{2\sqrt{\pi} d} (1-\typheight^2)^{\frac{d}{2}} - \sqrt{d}
\xrightarrow{d_{TV}} Z ,\]
where $Z$ is a random variable with standard normal distribution.
\end{itemize}
\end{theorem}
\section{Range containing the heights of all facets of \texorpdfstring{$P_{n,d}$}{P(n,d)}.} \label{s:range}
For the regime where $ n-d \ll d$, the facets will have heights approaching zero faster than $1/\sqrt{d}$, as stated in the following result.
\begin{theorem} \label{thm:range_sublin}
If $n-d \ll d$, then for all fixed $r > 0$,
\[\facets{-1}{-r/\sqrt{d}} \to 0
\text{ and } \facets{r/\sqrt{d}}{1} \to 0.\]
\end{theorem}
In the case that $ (n-d)/d\to \rho $ for $\rho \in (0, \infty)$, all of the facets are $O(d^{-1/2})$, and the following result gives a precise range of facet heights such that the expected number of facets with a height outside this range goes to zero.
\begin{theorem} \label{thm:range_nlin}
Fix $\rho$ such that $\rho > 0$ and assume $(n-d)/d \to \rho$.
Define the function
\[ g_{\rho}(r)
:= (\rho + 1)\ln (\rho + 1) - \rho \ln \rho - \frac{r^2}{2} + \rho \ln\Phi(r), \qquad r \in \RR,\]
where $\Phi(r)$ is the CDF of a standard normal random variable. Then there exist $r_{\ell}, r_u \in \RR $, defined as
\[ r_{\ell}
:= \inf\{r \in \RR : g_{\rho}(r) > 0\}
\qquad \mathrm{and} \qquad
r_{u} := \sup\{r \in \RR : g_{\rho}(r) > 0\} , \]
such that
\[ \lim_{n \rightarrow \infty} \facets{-1}{r/\sqrt{d}} =
\begin{cases}
\infty, & r > r_{\ell} \\
0, & r < r_{\ell},
\end{cases}
\qquad \mathrm{and} \qquad
\lim_{n \rightarrow \infty} \facets{r/\sqrt{d}}{1} =
\begin{cases}
\infty, & r < r_{u} \\
0, & r > r_{u}.
\end{cases} \]
\end{theorem}
\Remark{By Wendel's theorem \cite{Wendel_1962}, it is in this regime that we see a threshold for the probability that the origin is contained in the convex hull of $n$ i.i.d. radially symmetric random points. Indeed, for $n - d = \rho d + o(d)$, it can be shown that
\[ \PP(0 \notin [X_1, \ldots, X_n]) \to
\begin{cases}
1, & \rho < 1 \\
0, & \rho > 1 .
\end{cases} \]
However, from the proof of Theorem \ref{thm:range_nlin} (see Figure \ref{fig:g_rho}), $F_{[-1,0]} \to \infty$ for all $\rho < \rho_0 \simeq 3.4$. This means there is a range for $\rho$ for which the probability that there are facets of negative height goes to zero, but the expected number of facets with negative height goes to infinity as dimension grows.}
For the regime where $n \gg d$, we define a precise range $ [h_1,h_2] \subset [-1,1] $, such that, all of the facets lie at height within this range with probability tending to one.
The heights $h_1$ and $h_2$ depend on the number of vectors $n$ and the space dimension $d$.
There are different regimes with different asymptotic behaviors for $h_1$ and $h_2$.
\begin{theorem} \label{thm:height_range}
Assume that $n\gg d$.
Define
\begin{equation} \label{eq:h0h2}
h_1 = \sqrt{1 - \left(\frac{r_1 d (\ln (n/d))^{3/2}}{n}\right)^{\frac{2}{d-1}}}
\qquad\text{ and }\qquad
h_2 = \sqrt{1 - \left(\frac{r_2 d}{n}\right)^{\frac{2(d+1)}{(d-1)^2}}}.
\end{equation}
Then, for fixed positive constants $r_1$ sufficiently large and $r_2$ sufficiently small,
\[ \facets{-1}{1}
= \facets{h_1}{h_2} + o(1) . \]
\end{theorem}
Whenever we mention the heights $h_1$ and $h_2$, as defined above, we will implicitly assume that $n/d$ is large enough so that these quantities are well defined.
\section{Expected number of facets} \label{s:expnum}
We now present the asymptotic expression for the expected number of facets in each of these regimes.
\begin{theorem} \label{thm:nbfacets_smalln}
Assume $n-d\ll d$. Then,
\[ \facets{-1}{1}
= \binom{n}{d}\frac{2}{2^{n-d}}e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)} . \]
\end{theorem}
Note that when $n - d = o(\sqrt{d})$, the expression simplifies to $\binom{n}{d} 2^{-n+d+1} e^{o(1)}$. Next, we consider the case where $n - d = \rho d + o(d)$ for $\rho \in (0, \infty)$, and in this regime the expected number of facets grows exponentially with speed $d$ and rate function that depends on $\rho$.
\begin{theorem} \label{thm:nbfacets_nlin}
Fix $\rho > 0$ and assume $ (n-d)/d \to \rho$. Then,
\[\facets{-1}{1}
= e^{d g_{\rho}(r_\rho) + o(d)} ,\]
where \( g_{\rho}(r_\rho) := \max_{r\in\RR} \{ (\rho + 1)\ln (\rho + 1) - \rho \ln \rho - \frac{r^2}{2} + \rho \ln\Phi(r) \} > 0 \).
\end{theorem}
The next results show that when $n \gg d$, the expected number of facets grows super exponentially.
\begin{theorem} \label{thm:nbfacets_sub}
Assume $\ln n \ll d \ll n$, i.e. $n=n(d)$ grows with a regime strictly more than linear and strictly less than exponential. Then,
\[ \facets{-1}{1}
= \left[ (4\pi + o(1))\ln (n/d)\right]^{\frac{d-1}{2}} . \]
\end{theorem}
\Remark{Notice the similarity between the previous three results and Theorems 1.1 and 1.3 in \cite{Boroczky_Lugosi_Reitzner_2018}. This is to be expected since in high dimension Gaussian random vectors are close to a sphere of radius $\sqrt{n}R$, with high probability, and so if the number of vectors grows slowly enough with dimension, these polytopes have a similar facet structure to that of a polytope with points chosen uniformly on a sphere.}
\begin{theorem} \label{thm:nbfacets_exp}
Assume that $n=n(d)$ grows exponentially with $d$, i.e.\ $(\ln n)/d \to \rho$ for some $\rho \in (0,\infty)$. Then,
\[ \facets{-1}{1}
= \left[2 \pi \left(e^{2\rho} - 1\right)d\left(1 + o(1)\right)\right]^{\frac{d-1}{2}} . \]
\end{theorem}
Lastly, in the regime where $\ln n \gg d$, we obtain a more precise asymptotic approximation.
\begin{theorem} \label{thm:nbfacets}
If $(\ln n)/ d \to \infty$, then
\[ \facets{-1}{1}
\sim n K_d h_*^{d-1} , \]
where
\[ K_d
= \frac{ 2^d \pi^{\frac{d}{2}-1} }{ d (d-1)^2 } \frac{ \Gamma( \frac{d^2-2d+2}{2} ) }{ \Gamma ( \frac{d^2-2d+1}{2} ) } \left( \frac{ \Gamma( \frac{d+1}{2} ) }{ \Gamma (\frac{d}{2}) } \right)^{d-1} , \]
and $h_* = \sqrt{1 - d^{3/(d-1)}n^{-2/(d-1)}}$.
If, in addition, $\ln n \gg d \ln d$, i.e.\ where $n^{1/d}/d \to \infty$ (including the case where $d$ is fixed), then $\facets{-1}{1} \sim n K_d $.
\end{theorem}
\section{Related results from the literature in fixed dimension} \label{s:fixed}
In this section, we review some relevant results from the literature on the asymptotic behavior of some quantity related to the facets of spherical random polytopes in fixed dimension as the number of points $n$ tends to infinity. For each of these results we show an extension or a related result, in the setting where the dimension $d$ is also allowed to grow, using the asymptotic formulas presented in this paper.
\subsection{Expected number of facets}
The quantity $\facets{-1}{1}$ is the expected number of all the facets, regardless of their positions.
In fixed dimension, Buchta, M{\"u}ller, Tichy \cite{buchta_stochastical_1985} obtained a first asymptotic approximation of this quantity, as $n\to\infty$.
Kabluchko, Th{\"a}le and Zaporozhets \cite[Thm. 1.7]{kabluchko_beta_2018}) showed in a recent work the following more precise estimate
\[ K_d
:= \lim_{n\to\infty} n^{-1} \facets{-1}{1} = \frac{2^d \pi^{\frac{d}{2}-1}}{d (d-1)^2} \frac{\Gamma(\frac{d^2-2d+2}{2})}{ \Gamma(\frac{d^2-2d+1}{2})} \left(\frac{\Gamma(\frac{d+1}{2})}{ \Gamma(\frac{d}{2})} \right)^{d-1} .\]
Theorems \ref{thm:nbfacets_smalln}-\ref{thm:nbfacets} generalize this asymptotic formula for the expected number of facets to the case when $d$ is allowed to grow to infinity.
\subsection{Hausdorff distance}
The Hausdorff distance between the convex hull $P_{n,d}$ and the unit ball, denoted $ d_H ( P_{n,d} , B^d ), $ equals $ 1 - H_{\min} $, where $ H_{\min} $ is the smallest height of the facets of $P_{n,d}$. In fixed dimension, the asymptotic of the Hausdorff distance as the number of points becomes large is quite well understood. We cite here two results.
Glasauer and Schneider \cite[Theorem 4]{glasauer_asymptotic_1996} gave the precise asymptotic of the Hausdorff distance between a smooth convex body and the convex points of i.i.d. points on its boundary.
Applying this result to the sphere, we get
\begin{equation} \label{eq:GlasauerSchneider96}
d_H ( P_{n,d} , B^d ) \Big/ c_d \left( \frac{\ln n}{n} \right)^{\frac{2}{d-1}}
\overset{d}{\rightarrow} 1 ,
\end{equation}
where
$ 2 c_d
= \left( 2 \sqrt{\pi} \Gamma(\frac{d+1}{2}) / \Gamma(\frac{d}{2}) \right)^{2/(d-1)} $ and $\overset{d}{\rightarrow}$ denotes the convergence in distribution.
Richardson and Vu \cite[Lemma 4.2]{richardson_inscribing_2008} obtained a large deviation result stating that, for a given convex body $K$ with smooth boundary, there exist constants $c$ and $c'$ such that for $n$ large enough and $ \ee \geq c' \ln n / n $, the floating body $K_\ee$ is not contained in the convex hull of $n$ i.i.d.\ uniform points on the boundary of $K$ with a probability at most $\exp(-c \ee n) $.
In fixed dimension, it is easy to see that the $\ee$ floating body of the unit ball is a ball of radius $r$ satisfying $\ee \sim (\kappa_{d-1} / d) (1-r)^{(d+1)/2} $, as $\ee\to0$.
Therefore, for $n$ large enough and $\delta \geq 1$,
\begin{equation} \label{eq:RichardsonVu}
\PP\left( d_H(P_{n,d},B^d) > c \left( \delta \frac{\ln n}{n}\right)^\frac2{d+1} \right)
\leq \PP\left( P_{n,d} \not\supset \tilde{c} \delta \frac{\ln n}{n} B^d \right)
\leq \exp(-c' \delta \ln n) ,
\end{equation}
where $ c $, $\tilde{c}$ and $c'$ are non explicit constants depending only on the dimension.
Now, note that if $h_1$ and $h_2$ are such that $ \facets{-1}{h_1} \to 0 $ and $\facets{h_2}{1} \to 0 $, then $ 1 - h_1 \leq d_H ( P_{n,d} , B^d ) \leq 1 - h_2 $ with probability tending to $1$.
In the fast regimes, we have found this range and the asymptotic behavior for $1 - h_i$ is the same for $i = 1,2$, and hence we can understand the asymptotic behavior of the Hausdorff distance in this regime.
In particular, for this distance to tend to zero, we will need to be in the super exponential regimes, i.e.\ where $\ln n \gg d$.
Theorems \ref{thm:range_sublin}, \ref{thm:range_nlin}, \ref{thm:height_range}, and Lemma \ref{lem:h0} give the following corollary.
\begin{corollary}
Choose $n$ points uniformly from the unit sphere $S^{d-1}$ and denote their convex hull by $P_{n,d}$.
\begin{enumerate}
\item[(i)] Suppose $\ln n \gg d$. This condition allows for fixed $d$ or $d \to \infty$. Then,
\[ d_H(P_{n,d},B^d) = 1 - H_{\min} \xrightarrow{P} 0,\]
and if additionally $\ln \ln n \ll d$, then
\[ 2n^{\frac{2}{d-1}} d_H(P_{n,d},B^d) \xrightarrow{P} 1. \]
\item[(ii)] Suppose $(\ln n)/d\to \rho$ for $\rho \in (0,\infty)$. Then,
\[d_H(P_{n,d},B^d) = 1 - H_{\min} \xrightarrow{P} 1 - \sqrt{1 - e^{-2\rho}}.\]
\item[(iii)] Suppose $\ln n \ll d$ for $\rho \in (0,\infty)$. Then,
\[d_H(P_{n,d},B^d) = 1 - H_{\min} \xrightarrow{P} 1.\]
\end{enumerate}
\end{corollary}
\subsection{Delaunay triangulation of the sphere}
Almost surely all the faces of the random polytope $P_{n,d} = [X_1,\ldots,X_n]$ are simplices and their collection forms a simplicial complex.
By taking the projection $x \mapsto x / \| x \|$ onto the unit sphere of each of the simplices one obtains the so-called \textit{spherical Delaunay simplicial complex}.
Considering this complex is motivated by Edelsbrunner and Nikitenko in \cite{edelsbrunner_random_2017} where they explain an interesting connection with the Fisher information metric.
Let us describe further this setting in order to present one of their results and then translate it back in terms of facet heights.
For a given facet $[X_{i_1},\ldots,X_{i_d}]$ with supporting hyperplane $H$, one of the two half spaces bounded by $H$ contains the polytope and the other is empty of points. We call the empty half space $H^+$. The spherical cap $H^+ \cap S^{d-1}$ is called the \textit{circumscribed cap} to the spherical Delaunay simplex with vertices $X_{i_1},\ldots,X_{i_d}$.
Note that a circumscribed cap corresponding to facet of height $h$ has geodesic \textit{radius}
\begin{equation} \label{e:relrh}
r = \arcsin\left(\sqrt{1 - h^2}\right) .
\end{equation}
In the aforementioned paper the authors work in fixed dimension and study asymptotics, as $n\to\infty$, for the number of simplices of dimension $j\in\{1,\ldots,d-1\}$ in a random Delaunay triangulation of the sphere, with or without restriction on their radii.
Their primary focus is when the number of points is Poisson distributed with intensity $\rho>0$, but they also show in the appendix that their results still hold when the number of points is not random.
The only adaption to do is to replace the expected number of vertices $\rho \, \omega_d$, by the non random number of vertices $n$.
In particular their Corollary $2$, applied with $j=d-1$, says that the geodesic radius $R_{\mathrm{typ}}$ of the typical facet satisfies, for any fixed $\overline{\eta_0}>0$,
\begin{equation*}
\PP \left[ R_{\mathrm{typ}} \Bigl( \frac{n}{\omega_d} \Bigr)^{\frac{1}{d-1}} \leq \overline{\eta_0} \right] \to \PP \left[ X_{d-1} \leq \overline{\eta_0}^{d-1} \kappa_{d-1} \right] ,
\quad \text{as $n\to\infty$} ,
\end{equation*}
where $X_{d-1}$ is a Gamma distributed random variable with parameter $d-1$, i.e.\ has density $\1(t\geq 0) e^{-t} t^{d-2} / \Gamma(d-1) $.
Using the relation \eqref{e:relrh} between height and radius, rearranging the terms and using the fact that $\omega_d = 2 \pi^{d/2} / \Gamma(d/2) $ and $ \kappa_{d-1} = \pi^{(d-1)/2} / \Gamma ( (d+1)/2 ) $, this can be reformulated as
\[ n \frac{ \Gamma (\frac{d}{2}) }{ 2 \sqrt{\pi} \Gamma (\frac{d+1}{2}) } (1-\typheight^2)^{\frac{d-1}{2}} \xrightarrow{D} X_{d-1} , \quad \text{as $n\to\infty$.} \]
With our Theorem \ref{thm:CLT} we recover this result with a stronger kind of convergence (total variation) and provide an extension in the setting where the dimension goes to infinity and the number of vertices $n=n(d)$ grows super exponentially fast.
\section{Proofs}\label{s:proofs}
It is well known (see for example Theorem 1.2 in \cite{kabluchko_beta_2018}) that the expected number of facets of $P_{n,d}$ is equal to
\begin{equation} \label{e:nbfacets}
\binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} \int_{-1}^{1} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h ,
\end{equation}
where both normalizing constants $ c_\alpha $ are such that $c_\alpha \int_{-1}^1 (1-t^2)^\alpha \dint t = 1 $, or more explicitly
\begin{equation} \label{eq:c_asymp}
c_{\frac{d^2-2d-1}{2}}
= \frac{ \Gamma( \frac{d^2-2d+2}{2} ) }{ \sqrt{\pi} \Gamma ( \frac{d^2-2d+1}{2} ) }
\sim \frac{d}{\sqrt{2\pi}} ,
\text{ and }
c_{\frac{d-3}{2}}
= \frac{ \Gamma( \frac{d}{2} ) }{ \sqrt{\pi} \Gamma ( \frac{d-1}{2} ) }
\sim \sqrt{\frac{d}{2\pi}} ,
\end{equation}
where the asymptotics hold if $d$ goes to infinity.
Detailed proofs can be found in \cite{bonnet2017monotonicity,kabluchko_beta_2018}. They rely on very classical integral geometric arguments. The idea is to compute the probability that $ [ X_1 , \ldots , X_d ] $ is a facet of the polytope, or equivalently, that all the $n-d$ remaining points belong to the same half-space cut by the affine hull of the points $X_1, \ldots , X_d$. This probability turns out to be the quantity \eqref{e:nbfacets} without the binomial coefficient. The variable $h$ represents the height of the (potential) facet $ [ X_1 , \ldots , X_d ] $.
Therefore we see that the probability that $ [ X_1 , \ldots , X_d ] $ is a facet with height in $ [ h_1 , h_2 ] $ equals $ 2 c_{\frac{d^2-2d-1}{2}} I_{[h_1,h_2]} $, where
\begin{equation} \label{e:defIh1h2}
I_{[h_1,h_2]}
:= \int_{h_1}^{h_2} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h ,
\end{equation}
and the expected number of facets with height between $h_1$ and $h_2$ is given by
\begin{equation} \label{eq:inth1h2}
\facets{h_1}{h_2}
= \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} I_{[h_1,h_2]}
= \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} \int_{h_1}^{h_2} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h .
\end{equation}
Recall that the typical height $\typheight$ is the height of $[X_1,\ldots,X_d]$ conditioned on $[X_1,\ldots,X_d]$ to be a facet, and thus its distribution is described by
\begin{equation*}
\PP ( \typheight \in [h_1,h_2] )
= \frac{ I_{[h_1,h_2]} }{ I_{[-1,1]} } .
\end{equation*}
Thus, the proofs of all of the results in this paper rely on estimates of the integral $I_{[h_1,h_2]}$ for appropriately chosen $h_1$ and $h_2$, depending on $n$ and $d$ for each regime. While the results were presented in order of the problems, the proofs will be ordered by regime for ease of presentation, since the various results within each regime rely on the same approximations.
We present first a small lemma that will be used for estimation in different regimes.
\begin{lemma} \label{lem:exbnd}
If $ 0 \leq x/n \leq 1/2 $, then $ e^{-x-x^2/n} \leq (1-x/n)^n \leq e^{-x} $.
\end{lemma}
\begin{proof}
To see why the upper bound holds, one only need to write $(1-x/n)^n$ as $\exp(n \ln(1-x/n))$ and use the upper bound $\ln(1+t)\leq t$.
It remains to show the lower bound.
For this we write $ (1-x/n)^n / e^{-x -x^2/n} $ as $ \exp [ n \ln ( 1 - x/n ) + x + x^2/n ) = \exp [ n ( \ln ( 1 - y ) + y + y^2 ) ] $ with $y = x/n$.
But $ \ln ( 1 - y ) + y + y^2 \geq 0 $ for $ 0 \leq y \leq y_0 \simeq 0.68... $, which is the case for $ y = x/n \leq 1/2 $. The lower bound follows directly.
\end{proof}
\subsection{Slow regimes} \label{s:small_n}
In this section, we provide proofs for the regime where $(n-d)/d \to \rho \in [0, \infty)$.
In the case when $(n - d)/\sqrt{d} \to \rho \in [0, \infty)$, an application of the Dominated Convergence Theorem gives the asymptotic formulas for the integrals. For the remaining cases, the proof strategy is to approximate the integrand of $I_{[h_1, h_2]}$ with a function of the form $e^{-g(d) f(h)}$, where $g(d) \to \infty$ as $d \to \infty$. We then use Laplace's method to find an asymptotic approximation of the integral of this function around its peak. This approximation is obtained after scaling the heights through a change of variable.
First, recall that Laplace's method says the following.
Assume that a function $f$ achieves a unique maximum on $[a,b]$ and let $r^*$ be such that $f(r^*) = \max_{h \in [a,b]} f(h)$. First, assume $r^* \in (a,b)$ and that $f$ is twice differentiable in a neighborhood of $r^*$ with $f''(r^*) < 0$. Then, as $x \to \infty$,
\begin{equation} \label{eq:Laplace_int}
\int_a^b g(h)e^{xf(h)} \dint h
\sim g(r^*)e^{xf(r^*)}\sqrt{\frac{2\pi}{x \lvert f''(r^*) \rvert }}.
\end{equation}
Also, if $r^* = a$ and $f$ is differentiable with $f'(h) < 0$ for $h \in [a,b]$ or $r^* = b$ and $f'(h) > 0$ for $h \in [a,b]$, then as $x \to \infty$,
\begin{equation} \label{eq:Laplace_end}
\int_a^b g(h)e^{xf(h)} \dint h
\sim g(r^*)e^{xf(r^*)}\frac{1}{x \, \lvert f'(r^*) \rvert}.
\end{equation}
For a general reference on Laplace's method, we refer the reader to \cite{wong2001asymptotic}.
Another approximation we will use is that for the constant $ c_{\frac{d-3}{2}} = \Gamma(d/2)/ [\sqrt{\pi}\Gamma((d-1)/2)]$.
By Gautschi's inequality \cite{Gautschi_1959},
\begin{align} \label{e:c_approx}
c_{\frac{d-3}{2}} = \sqrt{\frac{d}{2\pi}}(1 + O(d^{-1})), \qquad \text{as } d \to \infty.
\end{align}
\subsubsection{Sub-linear regimes: Proofs of Theorems \ref{thm:typheight_smalln2}, \ref{thm:typheight_smalln}, \ref{thm:range_sublin} and \ref{thm:nbfacets_smalln} \label{s:n-d=o(d)}}
The first lemma we present gives a good approximation for the inner integral in $I_{[h_1,h_2]}$ in the case where $n-d = o(d)$.
\begin{lemma} \label{l:sublin_bnd}
Assume that $n$ and $d$ tend to infinity. Let $h\in\RR$ depending on $n$ and $d$ with $h=o(d^{-1/2})$.
Then, as $d\to\infty$,
\begin{align*}
\left( 2c_{\frac{d-3}{2}} \int_{-1}^{h} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d}
= e^{(n-d) d^{1/2} h \sqrt{2/\pi} \left(1 + O(d^{-1}) + O(h d^{1/2})\right)} .
\end{align*}
\end{lemma}
\begin{proof}
First, observe that
\begin{align*}
2c_{\frac{d-3}{2}} \int_{-1}^{h} (1-s^2)^{\frac{d-3}{2}} \dint s
&= 1 + F_d(h),
\end{align*}
where $F_d(h) = 2c_{\frac{d-3}{2}} \int_{0}^{h} (1-s^2)^{\frac{d-3}{2}} \dint s$.
Now, by the Taylor expansion of the integral $\int_{0}^{h} (1-s^2)^{\frac{d-3}{2}} \dint s$ at $h = 0$,
\[ \int_{0}^{h} (1-s^2)^{\frac{d-3}{2}} \dint s
= h \left( 1 + O(d h^2) \right) .\]
Multiplying by the normalizing constant $2c_{\frac{d-3}{2}}$, which is approximated by \eqref{e:c_approx}, gives
\[ F_d (h) = \sqrt{\frac{2}{\pi}} d^{\frac{1}{2}} h \left(1 + O(d^{-1}) + O(d h^2)\right) . \]
Note that the error factor $\left(1 + O(d^{-1}) + O(d h^2)\right)$ tends to one because of the assumption $h = o(d^{-1/2})$.
In particular $F_d(h) = O(d^{1/2} h)$.
Now, by the fact that
$ \ln(1+t) = t \left( 1 + O(t) \right)$,
\begin{align*}
\ln (1 + F_d(h))
&= \sqrt{\frac{2}{\pi}} d^{\frac{1}{2}} h \left(1 + O(d^{-1}) + O(d h^2)\right) \left( 1 + O \left( d^{\frac{1}{2}} h \right) \right)
\end{align*}
which simplifies to
\[ \ln (1 + F_d(h))
= \sqrt{\frac{2}{\pi}} d^{\frac{1}{2}} h \left(1 + O(d^{-1}) + O \left( d^{\frac{1}{2}} h \right) \right) . \]
Multiplying by $(n-d)$ and taking the exponential ends the proof.
\end{proof}
The next lemma gives us the asymptotic approximation of both $I_{[-1, r/d]}$ and $I_{[-1, 1]}$ in the regimes where $n-d$ is of order $\sqrt{d}$ or lower.
This is the key to prove Theorem \ref{thm:typheight_smalln2} and an essential part of the proof of Theorem \ref{thm:nbfacets_smalln}.
\begin{lemma}\label{lem:I_smalln1}
Assume $(n-d)/\sqrt{d} \to \rho $ for some fixed $ \rho \in [0, \infty)$. Then,
\begin{equation} \label{e:I_smalln_1}
I_{[-1,1]}
\sim \frac{\sqrt{2\pi}}{2^{n-d}d}e^{\rho^2/\pi},
\end{equation}
and for any fixed $r \in \RR$,
\begin{equation} \label{e:I_smalln_2}
I_{[-1, r/d]}
\sim \frac{\sqrt{2\pi}}{2^{n-d}d}e^{\rho^2/\pi}\PP(Z_{\rho} \leq r) ,
\end{equation}
where $Z_{\rho} \sim \mathcal{N}(\rho\sqrt{2/\pi},1)$.
\end{lemma}
\begin{proof}
By the linear substitution $h \to h/d$,
\begin{align*}
I_{[-1,1]}
&= \int_{-1}^{1} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h \\
&= \frac{1}{2^{n-d}d} \int_{-\infty}^\infty \1 (h\in[-d,d]) \left(1-\frac{h^2}{d^2}\right)^{\frac{d^2-2d-1}{2}} \left( 2c_{\frac{d-3}{2}} \int_{-1}^{h/d} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h.
\end{align*}
With this renormalization, we will now see that
the integrand converges pointwise to the function $e^{-h^2/2 + h \rho \sqrt{2/\pi}}$ and is uniformly bounded by the integrable function $e^{-h^2+C h}$, where $C$ is a sufficiently large constant.
For the first part of the integrand, we have for any fixed $h$
\[ \lim_{n \to \infty} \1 \left( h \in [-d,d] \right) \left(1-\frac{h^2}{d^2}\right)^{\frac{d^2-2d-1}{2}}
= \lim_{n \to \infty} \left(1-\frac{h^2}{d^2}\right)^{\frac{d^2}{2}}
= e^{-h^2/2} , \]
and for any $d\geq 3$ and any $h\in \RR $
\begin{equation*}
\left(1-\frac{h^2}{d^2}\right)^{\frac{d^2-2d-1}{2}}
\leq \left(1-\frac{h^2}{d^2}\right)^{\frac{d^2}{9}}
\leq e^{-h^2/9} ,
\end{equation*}
The second part is approximated by Lemma \ref{l:sublin_bnd} which tells us
\[ \left( 2c_{\frac{d-3}{2}} \int_{-1}^{h/d} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d}
= e^{(n-d) d^{-1/2} h \sqrt{2/\pi} \left(1 + O(d^{-1}) + O( h d^{-1/2})\right)}
,\]
which converges to $e^{h\rho\sqrt{2/\pi}}$ because of the assumption $(n-d)/\sqrt{d} \to \rho$.
From the same approximation we conclude also that this second part of the integrand is bounded by $e^{C h}$.
Therefore we can apply the Dominated Convergence Theorem if the integrand converges pointwise,
\[I_{[-1,1]} \sim \frac{1}{2^{n-d}d}\int_{-\infty}^{\infty} e^{-h^2/2 + h\rho \sqrt{2/\pi}} \dint h
= \frac{e^{\rho^2/\pi}}{2^{n-d}d}\int_{-\infty}^{\infty} e^{-(h - \rho\sqrt{2/\pi})^2/2} \dint h
= \frac{\sqrt{2\pi}}{2^{n-d}d}e^{\rho^2/\pi} . \]
This matches the claim since in this case $e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^2}{d^3}\right)} \to e^{\rho^2/\pi}$.
We show \eqref{e:I_smalln_2} in a similar way.
Following the same steps as above we obtain
\[I_{[-1,r/d]}
\sim \frac{e^{\rho^2/\pi}}{2^{n-d}d}\int_{-\infty}^{r} e^{-(h - \rho\sqrt{2/\pi})^2/2} \dint h
= \frac{\sqrt{2\pi}}{2^{n-d}d}e^{\rho^2/\pi}\PP(Z_{\rho} \leq r) . \]
\end{proof}
In the next lemma we move up to the regime where $n-d$ is growing much faster than $\sqrt{d}$ but still slower than $d$. Similarly as in Lemma \ref{lem:I_smalln1} we provide an asymptotic approximation of both $I_{[-1,1]}$ and $I_{[-1, \frac{n-d}{d^{3/2}} r]}$, which we will use in the proofs of Theorems \ref{thm:typheight_smalln} and \ref{thm:nbfacets_smalln}.
\begin{lemma} \label{lem:I_smalln2}
Assume $\sqrt{d} \ll n-d \ll d$. Then,
\begin{equation} \label{e:I_smalln_4}
I_{[-1,1]}
= \frac{\sqrt{2\pi}}{2^{n-d}d}e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)},
\end{equation}
and for any fixed $r>0$,
\begin{equation} \label{e:I_smalln_3}
I_{\left[0,r\frac{n-d}{d^{3/2}}\right]} =
\begin{cases}
\frac{\sqrt{2\pi }}{2^{n-d}d}e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)}, & r > \sqrt{2/\pi} , \\
\frac{(n-d)}{2^{n-d}d^{1/2}\left(\sqrt{2/\pi} - r\right)}e^{\frac{(n-d)^2}{d} f(r) + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)}, & r < \sqrt{2/\pi} ,
\end{cases}
\end{equation}
where $f(r):= r \sqrt{2/\pi} - r^2/2$.
\end{lemma}
\begin{proof}
We start by showing \eqref{e:I_smalln_3}.
For this we split the integral $I_{[-1, r(n-d)d^{-3/2}]}$ as the sum $I_{[-1,0]} + I_{[0,r(n-d)d^{-3/2}]}$.
First we compute the asymptotic of second term of this sum and later we will show that the first term is negligible.
By the change of variable $h \to h (n-d) d^{-3/2}$,
\begin{align*}
I_{\left[0,r\frac{n-d}{d^{3/2}}\right]} &= \int_{0}^{r\frac{n-d}{d^{3/2}}} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h \\
&= \frac{(n-d)}{d^{3/2}2^{n-d}} \int_{0}^{r} \left(1-\frac{h^2(n-d)^2}{d^3}\right)^{\frac{d^2-2d-1}{2}} \left( 2c_{\frac{d-3}{2}} \int_{-1}^{h\frac{n-d}{d^{3/2}}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h .
\end{align*}
By the assumption on $n-d$, it follows that $(n-d) d^{-3/2} = o(d^{1/2})$. Then, Lemma \ref{l:sublin_bnd} implies that for all $h \in [0,r]$,
\[ \left( 2c_{\frac{d-3}{2}} \int_{-1}^{h\frac{n-d}{d^{3/2}}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d}
= e^{\frac{(n-d)^2}{d}h\sqrt{\frac{2}{\pi}}\left(1 - O\left(h\frac{n-d}{d}\right)\right)}
= e^{\frac{(n-d)^2}{d}\sqrt{\frac{2}{\pi}}h - O\left(r^2 \frac{(n-d)^3}{d^{2}}\right)} . \]
Also, by Lemma \ref{lem:exbnd}, for all $h \in [0,r]$,
\[ e^{-\frac{h^2(n-d)^2}{2d} - \frac{r^4(n-d)^4}{2d^4}} \leq \left(1-\frac{h^2(n-d)^2}{d^3}\right)^{\frac{d^2-2d-1}{2}}
\leq e^{-\frac{h^2(n-d)^2}{2d} + r^2 \frac{(n-d)^2}{d^2}\left(1 + \frac{1}{2d}\right)}.\]
Since $n-d = o(d)$ in this regime, these bounds give
\[ \left(1-\frac{h^2(n-d)^2}{d^3}\right)^{\frac{d^2- 2d - 1}{2}}
= e^{-\frac{h^2(n-d)^2}{2d} + o(r^2) + o(r^4)}. \]
Thus,
\[ I_{\left[0,r\frac{n-d}{d^{3/2}}\right]}
= \frac{(n-d)}{2^{n-d}d^{3/2}}e^{O\left(\frac{(n-d)^3}{d^2}\right)} \int_{0}^{r} e^{\frac{(n-d)^2}{d}\left(h\sqrt{\frac{2}{\pi}} - \frac{h^2}{2}\right)} \dint h \]
The integral is now of a form for which we can apply Laplace's method to obtain an asymptotic approximation. The maximum of the function $f(h) := h \sqrt{\frac{2}{\pi}} - \frac{h^2}{2}$ occurs at $r^* := \sqrt{\frac{2}{\pi}}$, and thus by \eqref{eq:Laplace_int} and \eqref{eq:Laplace_end},
\[ \int_{0}^{r} e^{\frac{(n-d)^2}{d}\left(h\sqrt{\frac{2}{\pi}} - \frac{h^2}{2}\right)} \dint h \sim
\begin{cases}
\frac{\sqrt{2\pi d}}{n-d}e^{\frac{(n-d)^2}{\pi d} }, & r > r^* \\
\frac{d}{(n-d)^2\left|\sqrt{2/\pi} - r \right|}e^{\frac{(n-d)^2}{d}f(r) }, & r < r^*.
\end{cases} \]
Then,
\begin{equation} \label{eq:DomLaplace}
I_{\left[0,r\frac{n-d}{d^{3/2}}\right]} =
\begin{cases}
\frac{\sqrt{2\pi }}{2^{n-d}d}e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^3}{d^2}\right) +o(1)}, & r > r^* \\
\frac{(n-d)}{2^{n-d}d^{1/2}\left|\sqrt{2/\pi} - r\right|}e^{\frac{(n-d)^2}{d}f(r) + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)}, & r < r^*.
\end{cases}
\end{equation}
This approximation fits with the one of $I_{[-1, \frac{n-d}{d^{3/2}} r]}$ in the lemma and therefore we only have to show that $I_{[-1,0]}$ is negligible in order to prove \eqref{e:I_smalln_3}.
For this we use the rough bound
\[ c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s
\leq c_{\frac{d-3}{2}} \int_{-1}^0 (1-s^2)^{\frac{d-3}{2}} \dint s
= \frac{1}{2}, \quad h\in[-1,0], \]
which comes from the fact that $\1(s\in[-1,1]) c_{\alpha} (1-s^2)^{\alpha}$ is the density of a symmetric random variable, for any $\alpha>-1$.
Because of the same fact we also have that $\int_{-1}^{0} (1-h^2)^{\frac{d^2-2d-1}{2}} \dint h = (c_{\frac{d^2 - 2d - 1}{2}})^{-1} \sim d/ \sqrt{2\pi}$, as $d\to\infty$.
Therefore we have for any fixed $r > 0$,
\begin{equation} \label{eq:neg_o}
I_{\left[-1,0\right]}
= \int_{-1}^{0} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h
= O \left( \frac{\sqrt{2\pi}}{2^{n-d}d} \right)
= o\left(I_{\left[0,r\frac{n-d}{d^{3/2}}\right]}\right).
\end{equation}
It remains to compute the asymptotic of $I_{[-1,1]}$.
Let $r$ be large enough such that $\tilde{f}(r) := r - r^2/9 < 1/\pi = f(r^*) $.
We are going to show now that the term $I_{[r\frac{n-d}{d^{3/2}}, 1]}$ is negligible.
Recall that
\[ I_{[r\frac{n-d}{d^{3/2}}, 1 ]}
= \frac{1}{2^{n-d}} \int_{r}^{\frac{d^{3/2}}{n-d}} \left(1-\frac{h^2(n-d)^2}{d^3}\right)^{\frac{d^2-2d-1}{2}} \left( 2c_{\frac{d-3}{2}} \int_{-1}^{h\frac{n-d}{d^{3/2}}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h . \]
For $d\geq 3$ the exponent $(d^2-2d-1)/2$ is more than $d^2/9$, and therefore
\[ \left(1-\frac{h^2(n-d)^2}{d^3}\right)^{\frac{d^2-2d-1}{2}}
\leq e^{ - \frac{(n-d)^2}{d} \frac{h^2}{9} },
\quad h \in \left[ r , \frac{d^{3/2}}{n-d} \right] . \]
Also, Lemma \ref{l:sublin_bnd} gives that for $d$ large enough
\[ \left( 2c_{\frac{d-3}{2}} \int_{-1}^{h\frac{n-d}{d^{3/2}}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d}
\leq e^{\frac{(n-d)^2}{d} h},
\quad h\in \left[ r , \frac{d^{3/2}}{n-d} \right] .\]
Therefore we have
\[ I_{[r\frac{n-d}{d^{3/2}}, 1 ]}
\leq \frac{1}{2^{n-d}} \int_{r}^{\frac{d^{3/2}}{n-d}} e^{\frac{(n-d)^2}{d} \tilde{f}(h) } \dint h , \]
where $\tilde{f}(h)= h - h^2/9$.
Note that the function $ \tilde{f} $ is strictly decreasing on $[r,\infty)$.
Therefore with Laplace method as in \eqref{eq:Laplace_end}, the approximation \eqref{eq:DomLaplace} of $I_{\left[0,r\frac{n-d}{d^{3/2}}\right]}$ and the assumption $\tilde{f}(r) < f(r^*)$, we get
\[ I_{[r\frac{n-d}{d^{3/2}}, 1 ]}
\sim \frac{ e^{\frac{(n-d)^2}{d} \tilde{f}(r) }}{ r\, \lvert \tilde{f}'(r)\rvert }
= o\left( e^{\frac{(n-d)^2}{d} f(r^*)} \right)
= o\left( I_{\left[0,r\frac{n-d}{d^{3/2}}\right]} \right) .\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:typheight_smalln2}]
By Lemma \ref{lem:I_smalln1},
\begin{align*}
\PP(d \typheight \leq r) = \PP(\typheight \in [-1, r/d]) = \frac{I{[-1, r/d]}}{I{[-1, 1]}} \to \PP(Z_{\rho} \leq r), \text{ as } n \to \infty,
\end{align*}
where $Z_{\rho} \sim \mathcal{N}(\rho\sqrt{2/\pi}, 1)$.
Hence, $d \typheight - \rho\sqrt{2/\pi}$ converges in distribution to $Z \sim \mathcal{N}(0,1)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:typheight_smalln}]
Let $r>0$.
Recall that by definition
\[ \PP\left(\frac{d^{3/2}}{n-d}\typheight \leq r \right)
= \PP\left(\typheight \in \left[-1, \frac{n-d}{d^{3/2}}r\right]\right)
= \frac{I_{[-1, \frac{n-d}{d^{3/2}}r]}}{I_{[-1,1]}} . \]
It is now a direct consequence of Lemma \ref{lem:I_smalln2} that
\[ \PP\left(\frac{d^{3/2}}{n-d}\typheight \leq r \right)
= e^{\frac{(n-d)^2}{d}\left(f(\min\{r, \sqrt{2/\pi}\} - f(\sqrt{2/\pi})\right) + o\left(\frac{(n-d)^2}{d}\right)} \to
\begin{cases}
0, & 0 < r < \sqrt{2/\pi} \\
1, & r > \sqrt{2/\pi},
\end{cases} \]
where $f(r) = r \sqrt{2/\pi} - r^2/2$.
This implies the conclusion of the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:nbfacets_smalln}]
By Lemmas \ref{lem:I_smalln1} and \ref{lem:I_smalln2} and \eqref{eq:c_asymp},
\[ \facets{-1}{1}
= \binom{n}{d} 2c_{\frac{d^2 - 2d-1}{2}}I_{[-1,1]}
= \binom{n}{d} \frac{2d}{\sqrt{2\pi}} \frac{\sqrt{2\pi}}{2^{n-d}d} e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)}
= \binom{n}{d}2^{d-n + 1}e^{\frac{(n-d)^2}{\pi d} + O\left(\frac{(n-d)^3}{d^2}\right) + o(1)} . \]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:range_sublin}]
Let $r > 0$. We have to bound the quantities
\[ \facets{-1}{-r/\sqrt{d}}
= \binom{n}{d}2 c_{\frac{d^2-2d-1}{2}}\int_{-1}^{-r/\sqrt{d}} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( \int_{-1}^h c_{\frac{d-3}{2}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h . \]
and
\[ \facets{r/\sqrt{d}}{1}
= \binom{n}{d}2 c_{\frac{d^2-2d-1}{2}}\int_{-r/\sqrt{d}}^{1} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( \int_{-1}^h c_{\frac{d-3}{2}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h . \]
Using the substitution $h\to -h$, it is easy to see that $\facets{-1}{-r/\sqrt{d}} \leq \facets{r/\sqrt{d}}{1} $ and therefore we only need to consider the latter.
We bound the inner integral by one, do a linear substitution and recall that the coefficient $c_{\frac{d^2-2d-1}{2}}$ is of order $d$.
This gives
\begin{equation} \label{e:concentration}
\facets{r/\sqrt{d}}{1}
= \binom{n}{d} O(\sqrt{d}) \int_{r}^{\sqrt{d}} \left(1-\frac{h^2}{d}\right)^{\frac{d^2-2d-1}{2}} \dint h .
\end{equation}
Using the trivial inequalities $ 1-x\leq e^{-x} $ and $(d^2-2d-1)/2 > (d^2/2)-2d$ we upper bound the last integrand by $ g(h) e^{d f(h)} $ where $g(h) = e^{2h^2}$ and $f(h)=-h^2/2$.
Thus, with Laplace's method \eqref{eq:Laplace_end} we get
\[ \int_{r}^{\sqrt{d}} \left(1-\frac{h^2}{d}\right)^{\frac{d^2-2d-1}{2}} \dint h
\leq \int_{r}^{\infty} g(h) e^{d f(h)} \dint h
= e^{2r^2} e^{-d \frac{r^2}{2}} \frac{1}{d r} e^{o(1)}
= e^{-d\frac{r^2}{2} + O(1)}. \]
Therefore we only need to show that the binomial coefficient in \eqref{e:concentration} grows less than exponentially fast.
For this we use the assumption $n-d = o(d)$ which implies
\[ \binom{n}{d}
\leq \left( \frac{n}{n-d} \right)^{n-d}
= e^{(n-d) \ln \left( 1 + \frac{d}{n-d} \right)}
= e^{o(d)} . \]
Thus we have found that $ \facets{-1}{-r/\sqrt{d}} \leq \facets{r/\sqrt{d}}{1} = e^{-d r^2 / 2 + o(d)} \to 0 $.
\end{proof}
\subsubsection{Linear regimes: Proofs of Theorems \ref{thm:typheight_nlin}, \ref{thm:nbfacets_nlin} and \ref{thm:range_nlin}} \label{s:linear_proofs}
In the section we present the proofs for the regime when $n-d = \rho d + o(d)$ as $d \to \infty$. The proofs in this section rely on approximating the integrand of $I_{[h_1, h_2]}$ with the density and CDF of a normal random variable.
The first lemma will provide bounds showing this approximation and illuminates the similarity to approximations in the case of Gaussian polytopes in \cite{Boroczky_Lugosi_Reitzner_2018}.
Let us first set up some useful notation. Define the Gaussian CDF and density
\[ \Phi(h)
:= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^h e^{-s^2/2} \dint s ,
\text{ and }
\phi(h)
:= \Phi'(h) = \frac{1}{\sqrt{2\pi}} e^{-h^2/2} .\]
For any $\alpha>0$ and $h\in[-\sqrt{\alpha},\sqrt{\alpha}]$, define
\[ \Phi_\alpha(h)
:= \frac{a_\alpha}{\sqrt{2\pi}} \int_{-\sqrt{\alpha}}^h \left(1-\frac{s^2}{\alpha}\right)^{\alpha/2} \dint s ,
\text{ and }
\phi_\alpha(h)
:= \Phi'_\alpha(h) = \frac{a_\alpha}{\sqrt{2\pi}} \left(1-\frac{h^2}{\alpha}\right)^{\alpha/2} \1 \left( h \in [-\sqrt{\alpha},\sqrt{\alpha}] \right) ,\]
where $a_\alpha$ is the normalizing constant
\[ a_\alpha
:= \left( \frac{1}{\sqrt{2\pi}} \int_{-\sqrt{\alpha}}^{\sqrt{\alpha}} \left(1-\frac{s^2}{\alpha}\right)^{\alpha/2} \dint s \right)^{-1}
= \frac{\Gamma(\frac{\alpha+3}{2})}{\Gamma(\frac{\alpha}{2}+1)\sqrt{\frac{\alpha}{2}}}.\]
The constant $a_\alpha$ is similar to the constants $c_{\frac{d^2-2d-1}{2}}$ and $c_{\frac{d-3}{2}}$ except that the normalization is different. To illustrate this, note that $a_\alpha \to 1 $ as $\alpha\to\infty$. In fact, Gautschi's inequality implies
\begin{equation} \label{e:a_approx}
a_{\alpha}
= 1 + O\left(\frac{1}{\alpha}\right), \text{ as } \alpha \to \infty.
\end{equation}
The following lemma gives an approximation of $\Phi$ by $\Phi_{\alpha}$.
\begin{lemma}\label{l:phi_bnd}
For any $h\in[0,\sqrt{\alpha}]$,
\[ \Phi(h) \leq \Phi_{\alpha}(h) \leq a_{\alpha}\Phi(h),\]
and for any $h\in[-\sqrt{\alpha},0]$,
\[ \Phi(h) \geq \Phi_{\alpha}(h) \geq \frac{1}{2}(1 - a_{\alpha}) + a_{\alpha} \Phi(h) .\]
\end{lemma}
\begin{proof}
Using the inequality $1 - x \leq e^x$, we see that $ \phi_\alpha \leq a_\alpha \phi $.
Moreover we have that $\Phi(0)=\Phi_{\alpha}(0)=1/2$.
Thus for positive $h$ we get
\[ \Phi_\alpha(h) - \frac{1}{2}
= \int_0^h \phi_\alpha(s) \dint s
\leq a_\alpha \int_0^h \phi(s) \dint s
= a_\alpha \left( \Phi(h) - \frac{1}{2} \right) .\]
Since $a_\alpha>1$, this implies the inequality $\Phi_{\alpha}(h) \leq a_{\alpha}\Phi(h)$ for positive $h$.
Similarly, if $h$ is negative, we get $\Phi_\alpha(h) - \frac{1}{2} \geq a_\alpha \left( \Phi(h) - \frac{1}{2} \right) $ which is equivalent to $\Phi_{\alpha}(h) \geq \frac{1}{2}(1 - a_{\alpha}) + a_{\alpha} \Phi(h)$.
To show that $\Phi(h) \leq \Phi_{\alpha}(h)$ when $h\geq0$, we start by comparing the corresponding densities.
We have
\[ \phi_{\alpha}(0) - \phi(0) = \frac{a_{\alpha} - 1}{\sqrt{2\pi}} > 0
\text{ , and }
\phi_{\alpha}(\sqrt{\alpha}) - \phi(\sqrt{\alpha}) = 0 - (2\pi)^{-1/2} e^{-\alpha/2} < 0 .\]
Moreover the equation $\phi_{\alpha}(h) - \phi(h) = 0$ has a unique solution $h_0$ in the interval $[0,\sqrt{\alpha}]$.
Indeed, by definition of $\phi_{\alpha}$ and $\phi$ and taking the logarithm this equation can be rewritten as $\ln a_{\alpha} + (\alpha/2) \ln(1-h^2/\alpha) = -h^2/2 $ which leads to $\ln(1-x)+x+b_{\alpha}=0$ where $x$ and $b_{\alpha}$ stand for $h^2/\alpha$ and $(2\ln a_{\alpha})/\alpha$, respectively. It is easy to see the unicity of the solution with this last formulation.
Because of the continuity of $\phi_{\alpha}$ and $\phi$ it follows that $\phi_{\alpha}(h)-\phi(h)\geq 0$ for $h\in[0,h_0]$ and $\phi_{\alpha}(h)-\phi(h)\leq 0$ in $[h_0,\sqrt{\alpha}]$. Therefore $ h \in [0,\sqrt{\alpha}] \mapsto \Phi_{\alpha}(h) - \Phi(h) = \int_0^h \phi_{\alpha}(s) - \phi(s) \dint s $ is unimodular with its maximum at $h_0$. In particular it is always bigger than $ \min \{ \Phi_{\alpha}(0)-\Phi(0) , \Phi_{\alpha}(\sqrt{\alpha})-\Phi(\sqrt{\alpha}) \} = \min \{ (1/2) - (1/2) , 1 - \Phi(\sqrt{\alpha}) \} = 0 $.
This proves $\Phi(h) \leq \Phi_{\alpha}(h)$ for any $h\in [0,\sqrt{\alpha}]$.
By symmetry this same argument gives the bound $\Phi(h) \geq \Phi_{\alpha}(h)$ for $h \leq 0$.
This completes the proof of the lemma.
\end{proof}
We will also need the following technical lemma.
\begin{lemma}\label{l:f_alpha}
Define the function $f_{\rho}(r) := \rho\ln \Phi(r) - r^2/2$ as in Theorem \ref{thm:typheight_nlin} for fixed $\rho > 0$.
Then, $f_{\rho}$ is strictly concave on $[0, \infty)$ and has a unique maximum at some $r_{\rho} \in (0,\infty)$. In addition, $f_{\rho}$ is strictly increasing on $(-\infty, 0]$.
\end{lemma}
\begin{proof}
Fix $\rho > 0$. The first derivative of $f_{\rho}$ is
\[f'_{\rho}(r) = \frac{\rho\phi(r)}{\Phi(r)} - r.\]
Note that $f'_{\rho}$ is continuous, $f'_{\rho}(r) > 0$ for $r \in (-\infty, 0]$.
Also, the second derivative is
\begin{align*}
f''_{\rho}(r) = - \rho \left[\frac{r \phi(r) \Phi(r) + \phi(r)^2}{\Phi(r)^2} \right] - 1.
\end{align*}
Then, the claim follows from the fact that $f''_{\rho} (r) < 0$ for all $r \in [0,\infty)$.
\end{proof}
The following lemma gives asymptotic approximations of the integrals $I_{[-1, r/\sqrt{d}]}$ and $I_{[-1,1]}$, which will be used in the proofs of Theorems \ref{thm:typheight_nlin}, \ref{thm:range_nlin}, and \ref{thm:nbfacets_nlin}.
\begin{lemma} \label{lem:I_nlin}
Define the function $f_{\rho}$ as in Lemma \ref{l:f_alpha}, and define $r_{\rho} := \argmax f_{\rho}$. Let $n = n(d)$ be such that $n -d = \rho d + o(d)$ for a finite constant $\rho > 0$ as $d \to \infty$. Then, for any fixed $r \in \RR$,
\[ I_{[-1, r/\sqrt{d}]}
= e^{df_{\rho}(\min\{r, r_{\rho}\}) + o(d)}
\qquad \text{ and } \qquad
I_{[-1,1]}
= e^{df_{\rho}(r_{\rho}) + o(d)}. \]
\end{lemma}
\begin{proof}
First, fix $r \in \RR$ and choose an $\ee > 0$ depending on $\rho$ and $r$ such that $r - \ee < r_{\rho}$. Then, divide the integral $I_{[-1, r/\sqrt{d}]}$ in the following way:
\[ I_{[-1,r/\sqrt{d}]}
= I_{[-1,(r-\ee)/\sqrt{d}]} + I_{[(r-\ee)/\sqrt{d},r/\sqrt{d}]}.\]
We show the asymptotic formula is determined by the second term of this sum.
By the linear substitution $h \to h/\sqrt{d}$,
\begin{align*}
I_{[(r-\ee)/\sqrt{d},r/\sqrt{d}]}
&=\int_{\frac{r-\ee}{\sqrt{d}}}^{\frac{r}{\sqrt{d}}} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h \\
&= \frac{1}{\sqrt{d}} \int^{r}_{r - \ee} \left(1-\frac{h^2}{d}\right)^{\frac{d^2-2d-1}{2}}\left( c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}} (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n -d} \dint h.
\end{align*}
Since $\sqrt{d} > \sqrt{d-3}$, we have the following upper bound on the inner integral:
\[ \int_{-1}^{h/\sqrt{d}} (1-s^2)^{\frac{d-3}{2}} \dint s
\leq \int_{-1}^{h/\sqrt{d-3}} (1-s^2)^{\frac{d-3}{2}} \dint s . \]
By a change of variable,
\[ \int_{-1}^{h/\sqrt{d-3}} (1-s^2)^{\frac{d-3}{2}} \dint s
= \frac{1}{\sqrt{d-3}} \int_{-\sqrt{d-3}}^{h} \left(1-\frac{s^2}{d-3}\right)^{\frac{d-3}{2}} \dint s
= \frac{1}{a_{d-3}}\sqrt{\frac{2\pi}{d-3}}\Phi_{d-3}(h). \]
Then \eqref{e:c_approx} and Lemma \ref{l:phi_bnd} imply, for $h \in \RR$,
\begin{equation} \label{e:inner_upper_bnd}
c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}} (1-s^2)^{\frac{d-3}{2}} \dint s
\leq
(1 + O(d^{-1}) ) \Phi(h).
\end{equation}
After a change of variable, we also have the following lower bound for the integral:
\[ \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s
\geq \frac{1}{\sqrt{d}} \int_{-\sqrt{d}}^{h} \left(1-\frac{s^2}{d}\right)^{\frac{d}{2}} \dint s
= \frac{1}{a_d}\sqrt{\frac{2\pi}{d}}\Phi_d(h) .
\]
Then, \eqref{e:c_approx}, \eqref{e:a_approx}, and Lemma \ref{l:phi_bnd} imply that for $h \geq 0$,
\[ c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s
\geq (1 + O(d^{-1})) \Phi(h), \]
and for $h < 0$,
\[ c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s
\geq O(d^{-1}) + (1 + O(d^{-1}) )\Phi(h) . \]
Since we consider this integral only for $h$ in the fixed interval $[r-\ee,r]$, we can combine the term $\Phi(h)^{-1}O(d^{-1})$ with the $O(d^{-1})$ term that does not depend on $h$. That is,
\[ c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s
\geq (1 + O(d^{-1}))\Phi(h), \qquad h \in [r-\ee, r] . \]
Combining the above upper and lower bound we get
\begin{equation} \label{e:inner_upper_bnd2}
\left(c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s\right)^{n-d}
= (1 + O(d^{-1}))^{n-d} \Phi(h)^{n-d}
= \Theta (1) \Phi(h)^{n-d} ,
\quad h\in[r-\ee,r]
\end{equation}
where $\Theta (1)$ is a term bounded by positive constants which might depend on $\rho$ but independent of $n$, $d$ and $h$. The last equality is a consquence of our assumption $n-d=\rho d + o(d)$.
For $h$ in a fixed bounded interval we have that $\Phi(h)=\Theta(1)$, thus $\Phi(h)^{n-d} = \Theta(1)^{o(d)} \Phi(h)^{\rho d} = e^{o(d)} \Phi(h)^{\rho d} $ because of our assumption on the growth of $n$. The last equation can be rewritten as
\begin{equation}\label{e:inner_upper_bnd3}
\left(c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s\right)^{n-d}
= e^{o(d)} \Phi(h)^{\rho d} ,
\quad h\in[r-\ee,r]
\end{equation}
Now we approximate the other term in the integrand. With the help of Lemma \ref{lem:exbnd} it is easy to see that
\[ \left(1-\frac{h^2}{d}\right)^{\frac{d^2-2d-1}{2}}
= \Theta(1) e^{-\frac{d h^2}{2}} ,
\quad h \in [r - \ee,r] \]
where $\Theta(1)$ is a term bounded by positive constants which depend on $r$ but are independent from $n$ and $d$.
Therefore we have shown
\begin{equation} \label{e:outer_approx}
I_{[(r-\ee)/\sqrt{d},r/\sqrt{d}]}
= e^{o(d)} \int_{r-\ee}^{r} e^{d\left(\rho \ln \Phi(h) -h^2/2\right)} \dint h.
\end{equation}
Recall the definition of the function $f_{\rho}(r) := \rho \ln \Phi(r) - r^2/2$. By Lemma \ref{l:f_alpha} and Laplace's method \eqref{eq:Laplace_int} and \eqref{eq:Laplace_end},
\[ \int_{r-\ee}^{r}e^{df_{\rho}(h)} \dint h \sim
\begin{cases}
\sqrt{\frac{2\pi}{d \, \lvert f_{\rho}''(r_{\rho}) \rvert }} e^{df_{\rho}(r_{\rho})}, & r > r_{\rho} \\
\frac{1}{d f_{\rho}'(r)} e^{df_{\rho}(r)}, & r < r_{\rho} .
\end{cases} \]
This implies that
\[ I_{[(r-\ee)/\sqrt{d},r/\sqrt{d}]}
= e^{df_{\rho}(\min\{r,r_{\rho}\}) + o(d)} . \]
It remains to show $I_{[-1, (r-\ee)/\sqrt{d}]} = o(I_{[(r-\ee)/\sqrt{d},r/\sqrt{d}]})$.
First, we note that we can extend equation \eqref{e:inner_upper_bnd3} to the full interval $[-\sqrt{d},\sqrt{d}]$ at the cost of replacing the equality by an inequality, that is
\[ \left(c_{\frac{d-3}{2}} \int_{-1}^{h/\sqrt{d}}(1 - s^2)^{\frac{d-3}{2}} \dint s\right)^{n-d}
\leq e^{o(d)} \Phi(h)^{\rho d} ,
\quad h\in[-\sqrt{d},\sqrt{d}]. \]
We would like to also extend equation \eqref{e:outer_approx} to the full $[-\sqrt{d},\sqrt{d}]$, but the expression on the right hand side of \eqref{e:outer_approx} turns out to be too small. Instead we will use the bound
\[ \left( 1 - \frac{h^2}{d} \right)^{\frac{d^2-2d-1}{2}}
\leq e^{-\frac{dh^2}{2}+2h^2} ,\]
which follows from the simple inequalities $(d^2-2d-1)/2 \geq (d^2/2)- 2 d$ and $(1-x)\leq e^{-x}$.
Therefore using again the substitution $h \to h/\sqrt{d}$ we obtain
\[ I_{[-1, (r-\ee)/\sqrt{d}]}
\leq e^{o(d)} \int^{r-\ee}_{-\infty} e^{d (\rho \ln \Phi(h) -\frac{h^2}{2})}e^{2h^2}\dint h
= e^{df_{\rho}(r - \ee) + o(d)} , \]
where the last equality follows from the assumption $r - \ee < r_{\rho}$ and the Laplace method \eqref{eq:Laplace_end}.
The equality $I_{[-1, (r-\ee)/\sqrt{d}]} = o\left(I_{[(r-\ee)/\sqrt{d},r/\sqrt{d}]}\right)$ follows since $f_{\rho}(r-\ee) < f_{\rho}(\min\{r,r_{\rho}\})$.
This concludes the proof of the first part of the lemma.
Next we turn to the asymptotic formula for $I_{[-1,1]}$.
Fix an $r>r_\rho$ and split the integral as $I_{[-1,1]} = I_{[-1,r/\sqrt{d}]} + I_{[r/\sqrt{d},1]}$.
Because of the first part of the lemma we already know that $I_{[-1,r/\sqrt{d}]} = e^{df_\rho(r_\rho)+o(d)}$ and it is sufficient to show that $I_{[r/\sqrt{d},1]}=o(I_{[-1,r/\sqrt{d}]})$.
This is done following the same lines as above when we bounded the term $I_{[-1,(r-\ee)/\sqrt{d}]}$, and thus the proof is now complete.
\end{proof}
Now we can prove the main results.
\begin{proof}[Proof of Theorem \ref{thm:typheight_nlin}]
Letting $f_{\rho}$ be defined as in Lemma \ref{l:f_alpha}, Lemma \ref{lem:I_nlin} implies
\[ \PP(\typheight \leq r/\sqrt{d})
= \frac{I_{[-1, r/\sqrt{d}]}}{I_{[-1, 1]}}
= e^{d \left(f_{\rho}(\min\{r, r_{\rho}\}) - f_{\rho}(r_{\rho})\right) + o(d)}
\to \begin{cases} 0, & r < r_{\rho} \\ 1, & r > r_{\rho},\end{cases} \qquad \text{ as } n \to \infty, \]
where $r_{\rho} := \argmax f_{\rho}$. This gives the conclusion of the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:nbfacets_nlin}]
First, recall that $\facets{-1}{1} = \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}}I_{[-1,1]}$.
We start by approximating the binomial coefficient.
By Stirling formula,
\[ \binom{n}{d}
\sim \frac{1}{\sqrt{2\pi}}\left(\frac{n}{(n-d)d}\right)^{1/2}\left(\frac{n}{n-d}\right)^{n-d}\left(\frac{n}{d}\right)^d .\]
Using the assumption $n- d = \rho d + o(d)$ we see from the two last factors that we can only approximate the binomial coefficient up to an error factor $e^{o(d)}$. The two first factors above are of smaller order and thus we have
\begin{equation} \label{e:binom_approx_lin}
\binom{n}{d}
= e^{o(d)} \left(\frac{\rho+1}{\rho}\right)^{\rho d} \left(\rho+1\right)^d .
\end{equation}
Recall that $c_{\frac{d^2-2d-1}{2}}$ is of order $d$ which implies that it is negligible in front of the error factor $e^{o(d)}$. Then, letting $f_{\rho}$ be defined as in Lemmata \ref{l:f_alpha} and \ref{lem:I_nlin}, the latter lemma implies
\[ \facets{-1}{1}
= \left(\frac{(\rho+1)^{\rho+1}}{\rho^{\rho}}\right)^d e^{d f_{\rho}(r_{\rho}) + o(d)}
= e^{dg_{\rho}(r_{\rho}) + o(d)}, \]
where $g_{\rho}(r) = f_{\rho}(r) + (\rho + 1)\ln (\rho + 1) - \rho \ln \rho$ and $r_{\rho} := \argmax f_{\rho} = \argmax g_{\rho}$.
It only remains to check that $g_{\rho}(r_{\rho}) > 0$.
It suffices to show there exists an $r > 0$ such that $g_{\rho}(r) > 0$, since $g_{\rho}(r_{\rho}) \geq g_{\rho}(r)$ for all $r \geq 0$.
First we set $r':= \Phi^{-1}(\rho/(\rho+1))$.
In particular $r'$ satisfies
\begin{equation} \label{e:h_bar}
\rho + 1 = \frac{1}{1 - \Phi(r')}.
\end{equation}
Then,
\[ g_{\rho}(r')
= (\rho +1 )\ln(\rho + 1) - \rho \ln \rho - \frac{(r')^2}{2} + \rho \ln \left(\frac{\rho}{\rho +1}\right)
= \ln (\rho + 1) - \frac{(r')^2}{2} = - \ln(1 - \Phi(r')) - \frac{(r')^2}{2} . \]
Now, we have the following upper bound: since $t/r' > 1$ for all $t > r'$,
\[ 1 - \Phi(r')
= \frac{1}{\sqrt{2\pi}}\int_{r'}^{\infty}e^{-\frac{t^2}{2}} dt \leq \frac{1}{r'\sqrt{2 \pi} } \int_{r'}^{\infty} t e^{-\frac{t^2}{2}}
= \frac{1}{r'\sqrt{2 \pi} } e^{-\frac{(r')^2}{2}}.\]
Thus,
\[ g_{\rho}(r')
\geq - \ln\left(\frac{1}{r'\sqrt{2 \pi} } e^{-\frac{(r')^2}{2}}\right) - \frac{(r')^2}{2}
= \ln(r' \sqrt{2\pi}). \]
So, $g_{\rho}(r') > 0$ if $r' > 1/\sqrt{2 \pi}$.
Since $(1 - \Phi(r))^{-1}$ is increasing in $r$, \eqref{e:h_bar} implies that $r' > 1/\sqrt{2 \pi}$ if and only if
\[ \rho + 1
\geq \frac{1}{1 - \Phi(1/\sqrt{2\pi})} \approx 2.9 . \]
Thus, $g_{\rho}(r_{\rho}) > 0$ for $\rho \geq 2$.
To show $g_{\rho}(r_{\rho}) > 0$ for $\rho \in (0,2)$, we see that letting $r = 0$ gives
\[ g_{\rho}(0)
= (\rho+1) \ln (\rho + 1) - \rho \ln \rho + \rho \ln (1/2)
= (\rho + 1)\ln (\rho + 1) - \rho \ln \rho - \rho \ln 2 > 0, \]
for all $\rho \leq 2$, see Figure \ref{fig:g_rho}.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
axis lines = middle,
xlabel = $\rho$,
]
\addplot [
color=red,
domain=0:4,
samples=100,
color=red,
]
{(x+1)*ln(x+1) - x*ln(x) - x*ln(2)};
\addlegendentry{$g_\rho(0)$}
\end{axis}
\end{tikzpicture}
\caption{Plot of the function $(0,\infty)\ni\rho\mapsto g_\rho(0)$.}
\label{fig:g_rho}
\end{figure}
Thus, $g_{\rho}(r_{\rho}) > 0$ for all $\rho > 0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:range_nlin}]
First, recall that $\facets{-1}{r/\sqrt{d}} = \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} I_{[-1, r/\sqrt{d}]}$.
Then, by Lemma \ref{lem:I_nlin} and \eqref{e:binom_approx_lin},
\[ \facets{-1}{h/\sqrt{d}}
= e^{d g_{\rho}(\min\{r,r_{\rho}\}) + o(d)},\]
where $g_{\rho}(r) := \rho \ln \Phi(r) - (r^2/2) + (\rho + 1) \ln (\rho + 1) - \rho \ln \rho$, and $r_{\rho} := \argmax g_{\rho}$.
Thus, $\facets{-1}{r/\sqrt{d}}$ will approach zero for all fixed $r$ such that $g_{\rho}(\min\{r, r_{\rho}\}) < 0$.
Similarly, by Lemma \ref{lem:I_nlin},
\[ \facets{r/\sqrt{d}}{1}
= e^{dg_{\rho}(\max\{r, r_{\rho}\}) + o(d)}, \]
and so $\facets{r/\sqrt{d}}{1}$ will approach zero for all fixed $r$ such that $g_{\rho}(\max\{r, r_{\rho}\}) < 0$.
If $g_{\rho}(r_{\rho}) > 0$, then since $g_{\rho}(r) \rightarrow -\infty$ as $r \rightarrow +\infty$ and as $r \to - \infty$, the continuity of $g_{\rho}$ implies the existence of a $r_{\ell} \in (-\infty, r_{\rho})$ and $r_u \in (r_{\rho}, \infty)$ such that $g_{\rho}(r_{\ell}) = g_{\rho}(r_u) = 0$, and for all $r \notin [r_{\ell}, r_u]$, $g_{\rho}(r) < 0$.
This implies the conclusion of the theorem.
\end{proof}
\subsection{Fast regimes} \label{s:large_n}
We now turn to the proofs of results in the regime where $n \gg d$.
The following lemma gives the asymptotic behavior of a height depending on $n$ and $d$ in a particular way that will be used in the approximations in this regime.
\begin{lemma} \label{lem:h0}
Let $f(n,d)$ be a function of $n$ and $d$ such that $\ln f(n,d) = o(\ln(n/d))$.
Assume that $n \gg d$ and let
\[h := h(n,d) = \sqrt{1 - \left(\frac{d}{n} f(n,d)\right)^{\frac{2}{d-1}}}.\]
Then,
\begin{enumerate}
\item[(i)] If $\ln n \ll d $ then $h \sim \sqrt{2\ln (n/d)/d}$.
\item[(ii)] If $(\ln n) / d \to \rho$ for $\rho \in (0,\infty)$, then $h \to \sqrt{1 - e^{-2\rho}}$.
\item[(iii)] If $\ln n \gg d$ then $h \to 1$, and $-\ln(1-h^2) \sim \frac{2 \ln n}{d-1}$.
\end{enumerate}
In particular, note that in general in these regimes, $h \gg d^{-1/2}$ and $\left(1 - h^2\right)^{(d-1)/2} \to 0$.
\end{lemma}
\begin{proof}
We set $A$ to be the quantity such that $h=\sqrt{1-\exp(A)}$, that is
\[ A = A(n,d)
:= \frac{2}{d-1} \ln \left( \frac{d}{n} f(n,d) \right)
= - \frac{2}{d-1} \ln \left( \frac{n}{d} \right) (1+o(1)), \]
where the asymptotic given by the right hand side is equivalent with the assumption on $f$.
If $\ln n \ll d \ll n $, then $A$ tends to $0$ and thus
\[ h^2
= 1-\exp A
= -A (1+o(1)) ,\]
from which $(i)$ follows.
If $\ln n = \rho d + o(d)$, then $A$ tends to $-2\rho$ which gives us directly $(ii)$.
Finally we consider the case $\ln n \gg d$.
Here we have that $A$ tends to $-\infty$ and $-\ln (1 - h^2) = -A$, from which (iii) follows.
\end{proof}
Next we have a technical lemma which provides approximation for the integral $\int_h^1 (1-s^2)^{\frac{d-3}{2}} \dint s$.
Note that the bounds of this lemma are good when $h = o(D^{-1/2})$, which will make it a good approximation in the fast regimes.
\begin{lemma} \label{lem:approxint2}
For any $D\in\RR_{> -1}$ and $ h \in (0,1)$, we have
\[ 1 - \frac{1-h^2}{2h^2 (D+2)}
\leq \left( \int_h^1 (1-s^2)^D \dint s \right) \left( \frac{(1-h^2)^{D+1}}{2h(D+1)} \right)^{-1}
\leq 1 . \]
\end{lemma}
\begin{proof}
With the substitution $u=(s^2-h^2)/(1-h^2)$ one gets
\begin{equation} \label{eq:substitution1}
\int_h^1 (1-s^2)^D \dint s
= \frac{(1-h^2)^{D+1}}{2 \, h} \int_0^1 (1-u)^D \left( 1 + \frac{1-h^2}{h^2} u \right)^{-\frac{1}{2}} \dint u .
\end{equation}
It is easy to see that $ (1+x)^{-1/2} \geq 1 - x/2 $ for $ x \geq 0 $.
In particular
\begin{equation} \label{eq:bounds1}
1 - \frac{1-h^2}{2 h^2} u
\leq \left( 1 + \frac{1-h^2}{h^2} u \right)^{-\frac{1}{2}}
\leq 1 ,
\end{equation}
for $h$ and $u$ between $0$ and $1$.
The upper bound of Lemma \ref{lem:approxint2} follows from plugging the upper bound of \eqref{eq:bounds1} in \eqref{eq:substitution1} and using the fact that ${ \int_0^1 (1-u)^D \dint u = 1/(D+1) }$.
Now, we will compute the lower bound.
From the equations above, we have
\begin{equation*}
\int_h^1 (1-s^2)^D \dint s
\geq
\frac{(1-h^2)^{D+1}}{2 \, h} \left( \int_0^1 (1-u)^D \dint u - \frac{1-h^2}{2 h^2} \int_0^1 (1-u)^D u \, \dint u \right) .
\end{equation*}
In the last expression the first integral is equal to $1/(D+1)$ and the second integral is the beta function $B(D+1,2)$ which evaluates as $\Gamma\left(D+1\right) \Gamma(2) /\Gamma(D+3) = 1 / [(D+1)(D+2)]$.
Therefore
\begin{equation*}
\int_h^1 (1-s^2)^{D} \dint s
\geq \frac{(1-h^2)^{D+1}}{2(D+1)h} \left( 1 - \frac{1-h^2}{2 (D+2) h^2} \right) ,
\end{equation*}
which is precisely the lower bound of Lemma \ref{lem:approxint2}.
\end{proof}
\subsubsection{Proof of Theorem \ref{thm:height_range}}
Let $r_1$ and $r_2$ be positive numbers and set
\begin{equation} \label{eq:h0h2bis}
h_1 = \sqrt{1 - \left(\frac{r_1 d (\ln (n/d))^{3/2}}{n}\right)^{\frac{2}{d-1}}}
\text{ and }
h_2 = \sqrt{1 - \left(\frac{r_2 d}{n}\right)^{\frac{2(d+1)}{(d-1)^2}}}.
\end{equation}
Assume that $n\gg d$. Theorem \ref{thm:height_range} states that $\facets{-1}{h_1} \to 0$ if $r_1$ is sufficiently large, and $\facets{h_2}{1}\to 0$ if $r_2$ is sufficiently small.
These are precisely the statements of the next two lemmas.
Note that for all fixed $r_1$ and $r_2$, $h_1$ and $h_2$ will be strictly positive for all $n$ large enough.
\begin{lemma} \label{lem:boundh0}
Assume that $n\gg d$ and consider $h_1$ as in \eqref{eq:h0h2bis}.
If $r_1$ is a sufficiently large constant, then \( \facets{-1}{h_1} \to 0\).
\end{lemma}
\begin{proof}
First, by Lemma \ref{lem:approxint2},
\[ \int_h^1 (1-s^2)^{\frac{d-3}{2}} \dint s
\geq \left(1 - \frac{1- h^2}{h^2(d+1)}\right) \frac{(1-h^2)^{\frac{d-1}{2}}}{h(d-1)}. \]
Then since $1-t \leq \ln(1/t)$ for all $t > 0$,
\begin{equation} \label{e:h0sqrtd}
h_1
\leq \sqrt{\ln \left(\frac{n}{r_1 d (\ln (n/d))^{3/2}} \right)^{\frac{2}{d-1}}}
\leq \sqrt{ \frac{2}{d-1} \ln \left(\frac{n}{d}\right) } ,
\end{equation}
where the second inequality holds when $n/d$ is sufficiently big so that $r_1 (\ln(n/d))^{3/2} \geq 1$, which eventually happens thanks to the assumption $n\gg d$.
Now, recall that by \eqref{eq:inth1h2},
\[ \facets{-1}{h_1}
= \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} \int_{-1}^{h_1} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( c_{\frac{d-3}{2}} \int_{-1}^h (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h . \]
Bounding the inner integral by its evaluation for $h=h_1$, using the fact that $ c_{\frac{d^2-2d-1}{2}} \int_{-1}^{h_1} (1-h^2)^{\frac{d^2-2d-1}{2}} \dint h < 1 $ and bounding the binomial coefficient by $n^d$, we have
\begin{equation} \label{e:UBsimple}
\facets{-1}{h_1}
\leq n^d \left( 1 - A \right)^{n-d} ,
\end{equation}
where $ A $ is defined as
\[ A = A (n,d,r_1)
:= 1 - c_{\frac{d-3}{2}} \int_{-1}^{h_1} (1-s^2)^{\frac{d-3}{2}} \dint s
= c_{\frac{d-3}{2}} \int_{h_1}^1 (1-s^2)^{\frac{d-3}{2}} \dint s , \]
and the second equality follows from the definition of the normalizing constant $c_{\frac{d-3}{2}}$.
Now, Lemma \ref{lem:approxint2} provides the lower bound
\[ A \geq c_{\frac{d-3}{2}} \left(1 - \frac{1- h_1^2}{h_1^2(d+1)}\right) \frac{(1-h_1^2)^{\frac{d-1}{2}}}{h_1(d-1)} . \]
Lemma \ref{lem:h0} tells us that $h_1\gg d^{-1/2}$, and thus the expression in the first pair of brackets goes to $1$.
Using also that $c_{\frac{d-3}{2}}$ is of order $\sqrt{d}$, there exists a positive constant $C$ such that
\[ A \geq C \frac{(1-h_1^2)^{\frac{d-1}{2}}}{h_1 \sqrt{d}}
\geq C \frac{1}{\sqrt{d}} \frac{r_1 d (\ln (n/d))^{3/2}}{n} \Big/ \sqrt{\frac{2}{d-1}\ln\left(\frac{n}{d}\right)}
\geq C \frac{r_1 d \ln(n/d)}{n}
\geq C \frac{r_1 d \ln(n)}{n-d}.\]
For the second inequality, we used \eqref{eq:h0h2bis} to rewrite the term $(1-h_1^2)^{(d-1)/2}$ and \eqref{e:h0sqrtd} to bound $h_1$.
Note that the constant $C$ varies from line to line and can be chosen so that it depends only on $r_1$.
Therefore \eqref{e:UBsimple} gives
\[ \facets{-1}{h_1}
\leq n^d \exp \left( - C r_1 d \ln(n) \right)
= n^{(1-C r_1)d} . \]
For $r_1>C^{-1}$ this upper bound goes to $0$ and thus the lemma is proved.
\end{proof}
While the previous lemma proves the first part of Theorem \ref{thm:height_range}, the next one shows the second part of the theorem. Note that Lemma \ref{lem:UBH21} applies to a larger setting than the one of the aforementioned theorem since the condition $n\gg d$ is not required.
\begin{lemma} \label{lem:UBH21}
Consider $h_2$ as in \eqref{eq:h0h2bis}. If $r_2$ is a sufficiently small constant, then $\facets{h_2}{1} \to 0$.
\end{lemma}
\begin{proof}
By upper bounding the inner integral of \eqref{eq:inth1h2} by $1$ we obtain
\[ \facets{h_2}{1}
\leq \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} \int_{h_2}^{1} (1-h^2)^{\frac{d^2-2d-1}{2}} \dint h .\]
Using Stirling's formula we can bound the binomial coefficient by $(n e / d)^d$.
Recall that $c_{\frac{d^2-2d-1}{2}}$ is of order $d$ and that Lemma \ref{lem:approxint2} provides a bound of the last integral.
Thus there exists a positive constant $C$ such that
\begin{equation} \label{e:UBFh2}
\facets{h_2}{1}
\leq \left( \frac{n e}{d} \right)^d C d \frac{ (1-h_2^2)^{\frac{(d-1)^2}{2}}}{h_2 (d-1)^2}
\leq \left( \frac{n e}{d} \right)^d \frac{C}{\sqrt{d}} (1-h_2^2)^{\frac{(d-1)^2}{2}}
= \frac{\sqrt{d}}{ne}\left( r_2 e \right)^{d+1} C,
\end{equation}
where the second inequality follows from $h_2^{-1} = O(\sqrt{d})$,
which can be checked directly from the definition \eqref{eq:h0h2bis} of $h_2$, and the equality is another consequence of the same definition.
If $d$ is upper bounded, $\sqrt{d}/(ne) \to 0$, otherwise the term $(r_2 e)^{d+1}$ goes to $0$ exponentially fast.
In both cases the right hand side of \eqref{e:UBFh2} tends to $0$.
This concludes the proof.
\end{proof}
\subsubsection{Proof of Theorem \ref{thm:typheight_largen}}
\begin{proof}
When $\ln n \gg d \ln d $, Theorem \ref{thm:typheight_largen} is actually a corollary of the more precise Theorem \ref{thm:CLT}.
We will now see that in the specific case where $d$ is fixed and only $n$ goes to infinity.
In that setting Theorem \ref{thm:CLT} says that the random variable $Y_n :=
(1 - \typheight^2)^{(d-1)/2} n \Gamma(d/2) / [ 2\sqrt{\pi}\Gamma((d+1)/2)]
$ converges to $\Gamma_{d-1}$ distributed random variable $X_{d-1}$. Therefore, for any $\varepsilon > 0$,
\begin{align*}\PP \left( -\frac{(d-1)}{\ln n}\ln(1 - \typheight^2) \in [2-\ee,2+\ee] \right)
&= \PP \left( Y_n \in \left[ \frac{\Gamma(\frac{d}{2})}{2\sqrt{\pi}\Gamma(\frac{d+1}{2})} n^{-\ee/2}, \frac{\Gamma(\frac{d}{2})}{2\sqrt{\pi}\Gamma(\frac{d+1}{2})} n^{\ee/2}\right] \right) \\
&\to \PP (X_{d-1} \in (0, \infty) )
= 1 .
\end{align*}
Therefore $(iii)$ of Theorem \ref{thm:typheight_largen} is proven in the constant $d$ setting.
For the rest of the proof we assume that $d\to\infty$ and $n\gg d$.
Consider $h_1$ and $h_2$ as in \eqref{eq:h0h2} and write them in the form
\[ h_1 = \sqrt{1 - \left(\frac{d}{n}f_1(n,d)\right)^{\frac{2}{d-1}}}
\text{ and }
h_2 = \sqrt{1 - \left(\frac{d}{n}f_2(n,d)\right)^{\frac{2}{d-1}}} , \]
where $f_1(n,d) := r_1 \ln(n/d)^{3/2}$ and $f_2(n,d) := r_2^{(d+1)/(d-1)} (d/n)^{2/(d-1)}$ for some positive constants $r_1$ and $r_2$.
Assume that $r_1$ is sufficiently large and $r_2$ sufficiently small so that by Theorem \ref{thm:height_range},
\[ \PP ( \typheight \in [h_1,h_2] )
= \facets{h_1}{h_2}/ \facets{-1}{1}
\to 1 , \]
and therefore we only have to show that $h_1$ and $h_2$ have the correct asymptotic.
More precisely we only need to check that for $i=1,2$,
\begin{enumerate}
\item if $\ln n \ll d \ll n $ then $\sqrt{d / \ln (n/d)} \, h_i \to \sqrt{2}$,
\item if $ (\ln n)/d \to \rho > 0 $ then $\sqrt{1-h_i^2} \to e^{-\rho}$,
\item if $ \ln n \gg d $ then $-((d-1)/\ln n) \ln(1-h_i^2) \to 2$.
\end{enumerate}
These three statements are the conclusion of Lemma \ref{lem:h0} which applies here because $\ln f_i(n,d) = o(\ln (n/d))$, for $i = 1,2$.
This ends the proof.
\end{proof}
\subsubsection{Proofs of Theorems \ref{thm:nbfacets_sub} and \ref{thm:nbfacets_exp}}
In this section we obtain asymptotic formulas for the expected number of facets $\facets{-1}{1}$ in the large $n$ regime. The main idea of the approximation is to renormalize the integrand $I_{[h_1,h_2]}$ so that it approaches the density of $ \Gamma_{d-1} $ random variable.
The next lemma, which holds in all regimes, is the first step in that direction, and gives a general estimate of the integral $I_{[h_1, h_2]}$ in terms of the probability that a Gamma distributed random variable is within an interval depending on $h_1$ and $h_2$.
\begin{lemma} \label{lem:approxI1}
Assume that $2 c_{\frac{d-3}{2}} / (d-1) < h_1 \leq h_2 \leq 1 $ and set $X_{d-1}$ to be a Gamma($d-1$) distributed random variable.
Then
\[ I_{[h_1,h_2]}
= \beta\alpha^{d-1}C\PP(X_{d-1} \in [V_2,V_1]), \]
where $\alpha = \alpha(h_1, h_2, d)$ and $\beta = \beta(h_1, h_2, n, d)$ satisfies the inequalities
\[ h_1
\leq \alpha
\leq h_2\left(1 - \frac{1-h_1^2}{h_1^2 (d+1)}\right)^{-1} ,
\quad \text{ and } \quad
\frac{e^{-V_1^2/n}}{h_2}
\leq \beta
\leq \frac{e^{V_1d/n}}{h_1} , \]
and where $C = C(n,d) $ and $V_i = V_i(n,d,h_1,h_2)$, $i=1,2$, are defined as
\[ C
= \frac{(d-1)^{d-2} \Gamma(d-1)}{ (nc_{\frac{d-3}{2}})^{d-1} },
\quad \text{ and } \quad
V_i
= \frac{nc_{\frac{d-3}{2}} (1-h_i^2)^{\frac{d-1}{2}} }{ \alpha (d-1) } . \]
\end{lemma}
\begin{proof}
Recall that $ I_{[h_1,h_2]} $ is defined by
\[ I_{[h_1,h_2]}
= \int_{h_1}^{h_2} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( 1 - c_{\frac{d-3}{2}} \int_h^1 (1-s^2)^{\frac{d-3}{2}} \dint s \right)^{n-d} \dint h .\]
An approximation of the inner integral is given by Lemma \ref{lem:approxint2} applied with $D = (d-3)/2$,
\[ I_{[h_1,h_2]}
= \int_{h_1}^{h_2} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( 1 - \frac{ c_{\frac{d-3}{2}} (1-\theta(h) ) }{ h (d-1) } (1-h^2)^{\frac{d-1}{2}} \right)^{n-d} \dint h , \]
where $\theta(h)$ is an error term satisfying $0 \leq \theta (h) \leq (1-h^2) / [h^2 (d+1)]$.
Since $(1-h^2)/h^2$ is a decreasing function of $h$, we can upper bound $\theta(h)$ by $ (1-h_1^2)/[h_1^2(d+1)] $ for any $h \in [h_1,h_2]$.
By the intermediate value theorem, there exist $ \hat{\theta} $ and $ \hat{h} $ depending on $ h_1 $, $ h_2 $, $ n $ and $ d $ with the properties
$0 \leq \hat{\theta} \leq (1-h_1^2)/[h_1^2 (d+1)] $ and $ h_1 \leq \hat{h} \leq h_2 $, such that
\begin{equation*}
I_{[h_1,h_2]}
= \int_{h_1}^{h_2} (1-h^2)^{\frac{d^2-2d-1}{2}} \left( 1 - \frac{ c_{\frac{d-3}{2}} (1-\hat{\theta}) }{ \hat{h} (d-1) } (1-h^2)^{\frac{d-1}{2}} \right)^{n-d} \dint h .
\end{equation*}
Applying the substitution $ u = 1 - h^2 $ and letting $\alpha = \hat{h}/(1 - \hat{\theta})$, we get
\[ I_{[h_1,h_2]}
= \int_{1-h_2^2}^{1-h_1^2} u^{\frac{d^2-2d-1}{2}} \left( 1 - \frac{ c_{\frac{d-3}{2}}}{\alpha (d-1) } u^{\frac{d-1}{2}} \right)^{n-d} \frac{1}{2 \sqrt{1-u}} \dint u , \]
and observe that $\alpha$ satisfies the bound of the Lemma.
Using the intermediate value theorem once more see that there exists a $ \hat{h'} $ between $ h_1 $ and $ h_2 $ such that
\[ I_{[h_1,h_2]}
= \frac{1}{2 \hat{h'}} \int_{1-h_2^2}^{1-h_1^2} u^{\frac{d^2-2d-1}{2}} \left( 1 - \frac{ c_{\frac{d-3}{2}} }{ \alpha(d-1) } u^{\frac{d-1}{2}} \right)^{n-d} \dint u
= \frac{\alpha^{d-1} C}{\hat{h'} \Gamma(d-1)}\int_{V_2}^{V_1} v^{ d - 2 } \left( 1 - \frac{ v }{ n} \right)^{n-d} \dint v . \]
where the last equality follows from the substitution $ v = n c_{\frac{d-3}{2}} u^{\frac{d-1}{2}} / [\alpha (d-1)]$ and where $C$, $V_1$ and $V_2$ are defined as in the Lemma.
Observe that $V_1$ is less than $n/2$.
Thus we can approximate the term $(1-v/n)^{n-d}$ in the last integrand with the help of Lemma \ref{lem:exbnd} which gives, for any $v\in[V_2,V_1]$,
\[ e^{-V_1^2} e^{-v}
\leq e^{-v - \frac{v^2}{n}}
\leq \left(1 - \frac{v}{n}\right)^{n}
\leq \left(1 - \frac{v}{n}\right)^{n-d}
\leq e^{-v + v\frac{d}{n}}
\leq e^{V_1 \frac{d}{n}} e^{-v} .\]
Using this to bound the integrand in the last integral concludes the proof.
\end{proof}
In order to prove the theorems, we will add restrictions on $ h_1 $, $h_2$, $n$ and $d$ such that we can handle the error terms $\alpha$ and $\beta$ of the previous lemma.
\begin{proof}[Proof of Theorem \ref{thm:nbfacets_sub}]
Let $h_1$ and $h_2$ be defined as in the assumptions of Theorem \ref{thm:height_range}, i.e.
\begin{equation} \label{eq:h1h2}
h_1 = \sqrt{1 - \left(\frac{r_1 d (\ln (n/d))^{3/2}}{n}\right)^{\frac{2}{d-1}}}
\text{ and }
h_2 = \sqrt{1 - \left(\frac{r_2 d}{n}\right)^{\frac{2(d+1)}{(d-1)^2}}} ,
\end{equation}
where $r_1$ and $r_2$ are positive numbers.
We assume that $r_1$ sufficiently large and $r_2$ sufficiently small so that from Theorem \ref{thm:height_range}, $F_{[-1,1]} - F_{[h_1,h_2]} \to 0$.
Thus with \eqref{eq:inth1h2} we have
\[ F_{[-1,1]}
\sim \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} I_{[h_1,h_2]}. \]
We know from Lemma \ref{lem:h0} that $h_1$ and $h_2$ have the same asymptotic $\sqrt{2\ln(n/d)/d}\,(1 + o(1))$, and that in particular $d^{-1/2}\ll h_1 < h_2$.
Thus we can apply Lemma \ref{lem:approxI1} which says that
\begin{equation} \label{e:Ih1h2}
I_{[h_1,h_2]}
= \beta \alpha^{d-1}C\PP(X_{d-1} \in [V_2, V_1])
\end{equation}
where $\alpha = \alpha(h_1, h_2, d)$ and $\beta = \beta(h_1, h_2, n, d)$ satisfies the inequalities
\begin{equation} \label{e:alphabeta}
h_1
\leq \alpha
\leq h_2\left(1 - \frac{1-h_1^2}{h_1^2 (d+1)}\right)^{-1} ,
\quad \text{ and } \quad
\frac{e^{-V_1^2/n}}{h_2}
\leq \beta
\leq \frac{e^{V_1d/n}}{h_1} ,
\end{equation}
and where $C = C(n,d) $ and $V_i = V_i(n,d,h_1,h_2)$, $i=1,2$, are defined as
\begin{equation} \label{e:CVi}
C
= \frac{(d-1)^{d-2} \Gamma(d-1)}{ (nc_{\frac{d-3}{2}})^{d-1} },
\quad \text{ and } \quad
V_i
= \frac{nc_{\frac{d-3}{2}} (1-h_i^2)^{\frac{d-1}{2}} }{ \alpha (d-1) } .
\end{equation}
The next step is to get simpler approximations of the terms above.
Using the asymptotic of $h_1$ and $h_2$ obtained from Lemma \ref{lem:h0} and recalled above we get
\[ \alpha \sim \sqrt{\frac{2\ln (n/d)}{d}} ,
\quad \text{and} \quad
\beta = \sqrt{\frac{d}{2\ln(n/d)}} e^{O(V_1^2/n) + O(V_1 d/n) + o(1)} . \]
With this approximation of $\alpha$, the definitions of $h_1$ and $h_2$ and the approximation $c_{\frac{d-3}{2}} \sim \sqrt{d/(2\pi)}$, we compute
\[ V_1
\sim \frac{r_1 d \ln(n/d)}{2\sqrt{\pi}},
\text{ and }
V_2
\sim \frac{r_2 d^{1+2/(d+1)}}{2\sqrt{\pi} n^{2/(d-1)} \ln(n/d)}
\sim \frac{r_2 d}{2\sqrt{\pi} \ln(n/d)} , \]
where the last approximation follows from the assumption $ \ln n \ll d $.
From this and the assumption $n\gg d$ we can bound the error terms appearing in our approximation of $\beta$,
\[ \frac{V_1^2}{n}
= O \left(\frac{\ln (n/d)^2}{(n/d)} d \right)
= o(d)
\qquad \text{and} \qquad
\frac{V_1 d}{n}
= O \left( \frac{\ln (n/d)}{(n/d)} d \right)
= o(d) . \]
Therefore our approximation of $\beta$ takes now the simpler form $\beta = \sqrt{d/[2\ln(n/d)]} e^{o(d)}$, or equivalently $\beta = \alpha^{-1} e^{o(d)}$.
As an another consequence of the above approximation of $V_1$, we easily see that $V_1/(d-1) \to \infty$ and $V_2/(d-1)\to 0$ thanks to the assumption $d\ll n$.
But, on the other hand, a basic property of Gamma distributions tells us that $X_{d-1}/(d-1)$ converges in distribution to the constant random variable $1$.
Therefore the probability in \eqref{e:Ih1h2} tends to $1$ and this equation simplifies to
\[ I_{[h_1,h_2]}
= \alpha^{d-2} C e^{o(d)}.\]
Finally, by the approximations $\binom{n}{d} \sim n^d/d!$, $c_{\frac{d^2-2d-1}{2}} \sim d/\sqrt{2\pi}$, and $C = (d-3)!\left(2\pi d\right)^{\frac{d-1}{2}}e^{o(d)}/n^{d-1}$, the expected number of facets is given by
\[ F_{[-1,1]}
\sim \binom{n}{d} 2c_{\frac{d^2 - 2d - 1}{2}}I_{[h_1,h_2]}
= n\left(2\pi d\right)^{\frac{d-1}{2}}\alpha^{d-2}e^{o(d)}
= \left(2\pi d \alpha^2 \right)^{\frac{d-1}{2}}e^{o(d)}, \]
where the last equality follows from the assumption $\ln n\ll d$ which says precisely that $n=e^{o(d)}$ and implies that $\alpha=e^{o(d)}$.
Using the above asymptotic for $\alpha$ gives the conclusion of the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:nbfacets_exp}]
The proof follows the same lines as in the one of Theorem \ref{thm:nbfacets_sub} with small variations appearing because of the different assumption on the regime.
As in the previous proof we can write
\[ F_{[-1,1]}
\sim \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} I_{[h_1,h_2]}
= \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} \beta \alpha^{d-1}C\PP(X_{d-1} \in [V_2, V_1]), \]
where $h_1$ and $h_2$ are defined as in \eqref{eq:h1h2} and $\alpha$, $\beta$, $C$, $V_1$ and $V_2$ satisfy \eqref{e:alphabeta} and \eqref{e:CVi}.
Recall that now $(\ln n)/d \to \rho$.
Thus Lemma \ref{lem:h0} (ii) tells us that both $h_1$ and $h_2$ tend to $\sqrt{1 - e^{-2\rho}}$.
From this and elementary computation, the equations \eqref{e:alphabeta} and \eqref{e:CVi} provide the asymptotics
\[ \alpha \to \sqrt{1-e^{-2\rho}} ,
\quad \beta \to \frac{1}{\sqrt{1-e^{-2\rho}}} ,
\quad V_1 \sim \frac{r_1 \rho^{3/2}}{\sqrt{ 2 \pi (1-e^{-2\rho})}} d^2 ,
\text{ and } V_2 \sim \frac{ r_2 e^{-2\rho}}{\sqrt{ 2 \pi (1-e^{-2\rho})}} \sqrt{d} .\]
The asymptotic of $V_1$ and $V_2$ imply $\PP(X_{d-1} \in [V_1, V_2]) \to 1$ for the same reasons as in the previous proof.
Using similar ideas as in the proof of Theorem \ref{thm:nbfacets_sub},
\[ F_{[-1,1]}
= n \left(2\pi d \left(1 - e^{-2\rho}\right)\left(1 + o(1)\right)\right)^{\frac{d-1}{2}} . \]
With the assumption $(\ln n)/d \to \rho$ we can rewrite $n$ as $(e^{2\rho}(1 + o(1)) )^{ (d-1)/2 }$ and therefore the asymptotic of Theorem \ref{thm:nbfacets_exp} follows.
\end{proof}
\subsubsection{Proofs of Theorems \ref{thm:nbfacets} and \ref{thm:CLT}}
In the super-exponential regime, we first have the following lemma that will allow for a quick proof of the main result.
\begin{lemma} \label{lem:approxIalphabeta}
Assume $\ln n \gg d$. Let
\[ h_1
= \sqrt{ 1 - \frac{ d^{\frac{3}{d-1}}\gamma_{n,d} }{ n^{\frac{2}{d-1}} } }
\qquad \text{ and } \qquad
h_2
= \sqrt{ 1 - \frac{d^{\frac{3}{d-1}} \gamma_{n,d}^{-1}}{ n^{\frac{2}{d-1}} } } , \]
where $ \gamma_{n,d} \geq 1$ is some function of $n$ and $d$ satisfying
\begin{equation} \label{eq:propgamma}
1
\ll ( \gamma_{n,d} - 1 ) d
\ll (\ln n) / d ,
\end{equation}
for example $\gamma_{n,d} = 1 + \sqrt{(\ln n)/d^3}$.
Then,
\begin{equation*}
\begin{aligned}
I_{[h_1,h_2]}
\sim h_*^{d-1} \frac{ (d-1)^{d-2} }{ (n c_{\frac{d-3}{2}})^{d-1} } \Gamma(d-1),
\end{aligned}
\end{equation*}
where $h_* = \sqrt{1 - d^{\frac{3}{d-1}}n^{-\frac{2}{d-1}}}$.
\end{lemma}
\begin{proof}
Recall that Lemma \ref{lem:approxI1} says that
\begin{equation*}
I_{[h_1,h_2]}
= \beta \alpha^{d-1}C\PP(X_{d-1} \in [V_2, V_1])
\end{equation*}
where $\alpha = \alpha(h_1, h_2, d)$ and $\beta = \beta(h_1, h_2, n, d)$ satisfies the inequalities
\begin{equation*}
h_1
\leq \alpha
\leq h_2\left(1 - \frac{1-h_1^2}{h_1^2 (d+1)}\right)^{-1} ,
\quad \text{ and } \quad
\frac{e^{-V_1^2/n}}{h_2}
\leq \beta
\leq \frac{e^{V_1d/n}}{h_1} ,
\end{equation*}
and where $C = C(n,d) $ and $V_i = V_i(n,d,h_1,h_2)$, $i=1,2$, are defined as
\begin{equation*}
C
= \frac{(d-1)^{d-2} \Gamma(d-1)}{ (nc_{\frac{d-3}{2}})^{d-1} },
\quad \text{ and } \quad
V_i
= \frac{nc_{\frac{d-3}{2}} (1-h_i^2)^{\frac{d-1}{2}} }{ \alpha (d-1) } .
\end{equation*}
Thus we only have to show that both $\beta$ and the probability above tend to $1$ and that $\alpha^{d-1} \sim h_*^{d-1}$.
First, note that the assumption $(\gamma_{n,d}-1)d \gg 1$ implies $d \ln \gamma_{n,d} \to \infty$ and thus $\gamma_{n,d}^{(d-1)/2} \to \infty$. Also, by the inequality $1+x \leq e^{x}$, we have \[\gamma_{n,d}^{\frac{d-1}{2}} \leq e^{\Theta(d(\gamma_{n,d}-1))} = e^{o((\ln n)/d)},\]
where the equality follows from the assumption $(\gamma_{n,d}-1)d \ll (\ln n)/d$.
This assumption also means $(\gamma_{n,d} - 1) \ll n^{2/(d-1)}$, and therefore
\[\frac{d^{\frac{3}{d-1}}\gamma_{n,d}}{n^{\frac{2}{d-1}}}
= \frac{d^{\frac{3}{d-1}}}{n^{\frac{2}{d-1}}} + \frac{d^{\frac{3}{d-1}}(\gamma_{n,d} - 1)}{n^{\frac{2}{d-1}}}
\to 0,
\qquad \text{ and } \qquad
\frac{d^{\frac{3}{d-1}}}{n^{\frac{2}{d-1}}\gamma_{n,d}}
\leq \frac{d^{\frac{3}{d-1}}}{n^{\frac{2}{d-1}}}
\to 0. \]
This implies $h_i \to 1$, $i= 1,2$.
It follows that $\alpha \to 1$, and therefore
\begin{equation*}
V_1
= \frac{c_{\frac{d-3}{2}}d^{3/2} \gamma_{n,d}^{\frac{d-1}{2}}}{ \alpha (d-1) }
= \Theta(d \gamma_{n,d}^{\frac{d-1}{2}}),
\qquad \text{ and } \qquad
V_2
= \frac{ c_{\frac{d-3}{2}}d^{3/2} \gamma_{n,d}^{-\frac{d-1}{2}}}{\alpha (d-1)}
= \Theta (d\gamma_{n,d}^{-\frac{d-1}{2}}).
\end{equation*}
These asymptotics imply $\PP(X_{d-1} \in [V_2, V_1]) \to 1$ as in the previous proofs, because $V_1/(d-1) = \Theta( \gamma_{n,d}^{(d-1)/2}) \to \infty$ and $V_2/(d-1) = \Theta( \gamma_{n,d}^{-(d-1)/2}) \to 0$.
The estimates for $V_1$ and $V_2$ and the above upper bound on $\gamma_{n,d}^{\frac{d-1}{2}}$ also imply that $V_1^2/n \to 0$ and $V_1d/n \to 0$, giving the limit $\beta \to 1$.
It remains to show that $\alpha^{d-1} \sim h_*^{d-1}$.
Using the definitions of $h_1$ and $h_*$ and the fact that $d^{3/(d-1)}=O(1)$, we observe that
\[ \left(\frac{h_1}{h_*}\right)^2 - 1
= \frac{h_i^2-h_*^2}{h_*^2}
= O\left( \frac{\gamma_{n,d} - 1}{n^{\frac{2}{d-1}}} \right)
= o\left( \frac{\ln n^{\frac{1}{d}}}{n^{\frac{2}{d-1}} d} \right)
= o\left( \frac{1}{d} \right) , \]
where the third equality follows from the upper bound assumption on $\gamma_{n,d}$, and the fourth equality is a consequence of $\ln n\gg d$.
From this we deduce that $h_1^{d-1} \sim h_*^{d-1}$.
Similarly we find the same asymptotic for $h_2^{d-1}$.
Thus we have
\[ h_*^{d-1}
\sim h_1^{d-1}
\leq \alpha^{d-1}
\leq h_2^{d-1} \left(1 - \frac{1-h_1^2}{h_1^2 (d+1)}\right)^{-(d-1)}
\sim h_*^{d-1} \left( 1 + o\left(\frac{1}{d}\right) \right)^{-d}
\sim h_*^{d-1} , \]
and therefore $\alpha^{d-1} \sim h_*^{d-1}$ which was the only remaining point to show.
\end{proof}
Now it is easy to prove the theorem.
\begin{proof}[Proof of Theorem \ref{thm:nbfacets}]
Let $h_1$ and $h_2$ be as in Lemma \ref{lem:approxIalphabeta}. By the same lemma, we have that that $ I_{[h_1,h_2]} \sim [h_* (d-1)]^{d-2} \Gamma ( d-1 ) / ( n c_{\frac{d-3}{2}} )^{d-1} $.
The remaining steps of the proof are the following:
\begin{enumerate}
\item Consider $ h_0 = \sqrt{1 - \left(\frac{r_0 d (\ln (n/d))^{3/2}}{n}\right)^{\frac{2}{d-1}}}$ defined as the $h_1$ appearing in Theorem \ref{thm:height_range} and show that $ I_{[h_0, h_1]} \ll I_{[h_1,h_2]} $.
\item Show that $ I_{[h_2,1]} \ll I_{[h_1,h_2]} $.
\item Conclude that $ \facets{-1}{1} \sim \facets{h_1}{h_2} \sim n K_d h_*^{d-1}$.
\end{enumerate}
\textbf{Step 1}:
We use Lemma \ref{lem:approxI1} again to obtain
\[ I_{[h_0,h_1]}
= \beta\alpha^{d-1}C\PP(X_{d-1} \in [V_1,V_0]), \]
where $\alpha = \alpha(h_0, h_1, d)$ and $\beta = \beta(h_0, h_1, n, d)$ satisfies the inequalities
\[ h_0
\leq \alpha
\leq h_1\left(1 - \frac{1-h_0^2}{h_0^2 (d+1)}\right)^{-1} ,
\quad \text{ and } \quad
\frac{e^{-V_0^2/n}}{h_1}
\leq \beta
\leq \frac{e^{V_0d/n}}{h_0} , \]
and where $C = C(n,d) $ and $V_i = V_i(n,d,h_0,h_1)$, $i=0,1$, are defined as
\[ C
= \frac{(d-1)^{d-2} \Gamma(d-1)}{ (nc_{\frac{d-3}{2}})^{d-1} },
\quad \text{ and } \quad
V_i
= \frac{nc_{\frac{d-3}{2}} (1-h_i^2)^{\frac{d-1}{2}} }{ \alpha (d-1) } . \]
Using similar estimates as in the proof of Lemma \ref{lem:approxIalphabeta} we find that $\beta\to 1$, $\alpha^{d-1} = O(h_1^{d-1})$ and $V_1\gg d$.
Therefore, with the approximation given by Lemma \ref{lem:approxIalphabeta} we get
\[ \frac{ I_{[h_0,h_1]} }{ I_{[h_1,h_2]} }
= O \left( \Bigl( \frac{h_1}{h_*} \Bigr)^{d-1} \right) \PP(X_{d-1} \geq V_1)
= O(1) \PP(X_{d-1} \geq V_1)
\to 0 , \]
where the last equality follows from the fact that $h_1 \leq h_*$, and the limit is a consequence of the concentration of the Gamma distribution concentrated around $(d-1)$ while $V_1\gg d$.
\textbf{Step 2}: Similarly as in the first step we find
\[ \frac{ I_{[h_2,1]} }{ I_{[h_1,h_2]} }
= O \left( \Bigl( \frac{h_2}{h_*} \Bigr)^{d-1} \right) \PP(X_{d-1} \leq V_2) , \]
with $V_2\ll d$.
We need this time to be a bit more careful to conclude because we cannot ignore the fraction $h_2/h_*$ which is bigger than $1$.
Nevertheless we know that it tends to $1$ so we can bound the big O term by $e^{d-1}$.
This gives
\[ \frac{ I_{[h_2,1]} }{ I_{[h_1,h_2]} }
\leq \frac{e^{d-1}}{\Gamma(d-1)} \int_0^{V_2} x^{d-2} e^{-x} \dint x
\leq \frac{(e V_2)^{d-1}}{\Gamma(d)}
\leq \left( \frac{e^2 V_2}{d-1} \right)^{d-1}
= o(1)^{d-1}
\to 0 , \]
where we use the lower bound $\Gamma(k+1) = k! \geq (k/e)^k$ with $k=d-1$, and the above observation $V_2\ll d$.
\textbf{Step 3}: Now we combine the above results. But first we recall from Lemma \ref{lem:boundh0} that $\facets{-1}{h_0} \to 0 $, thus
\[ F_{[-1,1]}
\sim F_{[h_0,1]}
\overset{\eqref{eq:inth1h2}}{=} \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} I_{[h_0,1]} . \]
But with Steps 1 and 2 we have that $ I_{[h_0,1]} \sim I_{[h_1,h_2]} $ which is approximated in the previous lemma.
This gives
\[ F_{[-1,1]}
\sim \binom{n}{d} 2 c_{\frac{d^2-2d-1}{2}} \frac{[h_* (d-1)]^{d-2} }{( n c_{\frac{d-3}{2}} )^{d-1} } \Gamma ( d-1 ) . \]
Doing elementary computation and approximation, we get the desired result.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:CLT}]
First we observe that the main statement implies the particular cases. If $d$ is fixed there is nothing to do.
If $d\to\infty$, it suffices to observe that $ \Gamma (d/2) / \Gamma( (d+1)/2 ) \sim \sqrt{2/d} $ and that $ d^{-1/2} X_{d-1} - \sqrt{d} \xrightarrow{d_{TV}} Z $.
Now we start with the proof of the main statement.
We begin by setting some notation and reducing the problem to a setting which will allow us later ignore the event $\{ \typheight \leq 0 \}$. Let
\[ \widehat{Y_{n,d}}
= n \frac{ \Gamma (\frac{d}{2}) }{ 2 \sqrt{\pi} \Gamma (\frac{d+1}{2}) } (1-\typheight^2)^{\frac{d-1}{2}}
\text{ and }
Y_{n,d}
= \widehat{Y_{n,d}} \1 ( \typheight \geq 0 ) . \]
Since $ \PP ( \typheight \leq 0 ) \to 0 $, we have that $ d_{TV} ( Y_{n,d} , \widehat{Y_{n,d}} ) \to 0 $ and thus we only have to show that $ d_{TV} (X_{d-1},Y_{n,d}) \to 0 $.
Considering $Y_{n,d}$ rather than $\widehat{Y_{n,d}} $ has the advantage that we can rewrite
\begin{align*}
\PP ( h_b \leq H_{typ} \leq h_a )
= \PP (a \leq Y_{n,d} \leq b )
\end{align*}
where
\[ h_t
= \sqrt{1 - \left(\frac{t 2 \sqrt{\pi} \Gamma(\frac{d+1}{2})}{n \Gamma(\frac{d}{2})} \right)^{\frac{2}{d-1}}} . \]
With arguments similar as in the proof of Theorem \ref{thm:nbfacets} we see that for a sequence $ b_{n,d} >0 $ such that $ b_{n,d} / d \to \infty $, we have $ \PP ( Y_{n,d} \geq b_{n,d} ) \to 0 $.
Also, we have been able to get a very precise approximation of $ I_{[h_1,h_2]} $ in certain settings.
Here this means that we get good approximation of $ \PP ( a \leq Y_{n,d} \leq b ) $ if $[a,b]\subset [0 ,b_{n,d}]$ and
$b_{n,d}$ deviates sufficiently slowly from $ d $.
In the next steps we will exploit these facts with well chosen $ b_{n,d}$.
Let $b_{n,d}$ be a sequence such that
\[ b_{n,d} n^{-\frac{2}{d-1}}
\to 0
\text{ and }
b_{n,d}d^{-1}
\to \infty.\]
Note that this sequence exists under the condition that $\ln n \gg d \ln d \to \infty$ since this implies that $n^{-2/(d-1)}d \to 0$.
Now set
\[ A_1 = \left[ 0 , b_{n,d}\right]
\,,\,
A_2 = \left[ b_{n,d} ,\infty \right],\]
and for $ i \in \{ 1 , 2 \} $ and for random variables $ X $ and $ Y $, we define
\[ d_{TV,A_i} (X,Y)
= \sup_{ A \in \mathcal{B} (A_i) } \lvert \PP ( X \in A ) - \PP ( Y \in A ) \rvert . \]
It is easy to see that $ d_{TV} \leq d_{TV,A_1} + d_{TV,A_2}$ and thus we only have to show $ d_{TV,A_i} (X_{d-1},Y_{n,d}) \to 0 $ for $ i \in \{ 1 , 2\} $.
For $ i = 2$ we use the trivial bound
\[ d_{TV,A_2} (X_{d-1},Y_{n,d})
\leq \PP ( X_{d-1} \in A_2 ) + \PP (Y_{n,d} \in A_2 ) .\]
Now, by Markov's inequality and by the assumption on $b_{n,d}$,
\[ \PP ( X_{d-1} \in A_2 )
= \PP(X_{d-1} \geq b_{n,d})
\leq \frac{\EE(X_{d-1})}{b_{n,d}}
= \frac{d-1}{b_{n,d}}
\to 0,\]
Then, using the fact that $I_{[-1,1]} \sim (d-1)^{d-2} (n c_{\frac{d-3}{2}})^{-(d-1)} \Gamma(d-1)$ in this regime and the approximation given by Lemma \ref{lem:approxIalphabeta} combined with similar estimations as in the proof of Theorem \ref{thm:nbfacets}, we get
\[ \PP (Y_{n,d} \in A_2 )
\leq \PP\left( 0 \leq H_{typ} \leq h_{b_{n,d}} \right)
= \frac{I_{[0,h_{b_{n,d}}]}}{I_{[-1,1]}}
\leq \frac{1+o(1)}{\Gamma(d-1)}\int_{V_b}^{\infty} v^{d-2}e^{-v} \dint v
\to 0 ,\]
where $V_b$ is a term depending on $b$, $n$ and $d$ and has property that $V_b\gg d$, which implies the last limit.
Thus, $d_{TV,A_2} (X_{d-1},Y_{n,d}) \to 0$.
It remains to show $d_{TV,A_1} (X_{d-1},Y_{n,d}) \to 0$.
We will actually prove the stronger statement
\begin{equation} \label{eq:strongerstatement}
\lvert \PP ( X_{d-1} \in A ) - \PP ( Y_{n,d} \in A ) \rvert
\leq \ee_{n,d} \PP (X_{d-1} \in A) \text{ for any } A \in \mathcal{B}(A_1) ,
\end{equation}
where $ \ee_{n,d} \to 0 $ is independent from $A$. We see that this is indeed a stronger statement by upper bounding the probability on the right hand side by $1$ and taking the supremum over all $ A \in \mathcal{B}(A_1) $.
Note that the inequality \eqref{eq:strongerstatement} is stable under disjoint union in the sense that if it holds for any $ A $ in a collection $ \{ B_i \}_{i\in\NN} $ of pairwise Borel sets then it is also true for $ A = \cup B_i $.
This is a simple consequence of the triangular inequality and the sigma additivity of $\PP$.
In particular we only need to show \eqref{eq:strongerstatement} for intervals $ A = [a,b] \subset A_1 $.
For both random variables $X_{d-1}$ and $Y_{n,d}$, we need to evaluate the probability that it is contained in $[a,b]$.
For $X_{d-1}$ this is simply $\Gamma(d-1)^{-1} \int_a^b e^{-t} t^{d-2} \dint t$.
For $Y_{n,d}$, we have
\[ \PP ( Y_{n,d} \in [a,b] )
= \PP ( \typheight \in [h_b,h_a] )
= \frac{ I_{[h_b,h_a]} }{I_{[-1,1]}}.\]
Then, by Lemma \ref{lem:approxI1}, there is an $\alpha$, $\beta$ such that
\[ \PP ( \typheight \in [h_b,h_a] )
= \frac{ I_{[h_b,h_a]}}{ I_{[-1,1]} }
\sim \beta \alpha^{d-1} \PP \left(X_{d-1} \in \left[ \frac{a}{\alpha}, \frac{b}{\alpha} \right] \right) \]
where $\alpha$ and $\beta$ satisfy
\[ h_b
\leq \alpha
\leq h_a\left(1 - \frac{1-h_b^2}{h_b^2(d+1)}\right)^{-1}
\quad\text{ and }\quad
\frac{e^{-b^2/(n\alpha^2)}}{h_a}
\leq \beta
\leq \frac{e^{bd/(\alpha n)}}{h_b} . \]
Applying a linear substitution $v = \alpha t$ we get
\begin{align*}
\PP ( Y_{n,d} \in [a,b] )
&\sim \frac{\beta \alpha^{d-1} }{\Gamma(d-1)} \int_{a/\alpha}^{b/\alpha} t^{d-2} e^{-t} dt
= \frac{\beta}{\Gamma(d-1)} \int_a^b v^{d-2}e^{-v}e^{v(1 - \alpha^{-1})}\dint v .
\end{align*}
Therefore,
for any $[a,b] \subseteq A_2$,
\begin{align*}
\lvert \PP ( X_{d-1} \in [a,b] ) - \PP ( Y_{n,d} \in [a,b] ) \rvert
&= \left\lvert \frac{1}{\Gamma(d-1)} \int_{a}^{b} v^{d-2} e^{-v} [1 - (1 + o(1))\beta e^{v(1- \alpha^{-1})} ] \dint v \right\rvert
\\&\leq \PP ( X_{d-1} \in [a,b] ) \max_{v\in[a,b]} \left\lvert 1 - (1 + o(1))\beta e^{v(1-\alpha^{-1})} \right\rvert \\
&
\leq \PP ( X_{d-1} \in [a,b] ) \left\lvert 1 - (1 + o(1)) \beta e^{b_{n,d}(1-\alpha^{-1})} \right\rvert .
\end{align*}
Now, by Lemma \ref{lem:h0}, $\alpha^{-1} -1 \sim 1 - \alpha \leq 1-h_b =
O(n^{-2/(d-1)})$.
Then, by the assumption on $b_{n,d}$, $b_{n,d}(\alpha^{-1} - 1) \to 0$. Also, since $1 - \alpha = O(n^{-2/(d-1)}) = o(1/d) $ by assumption,
we get that $ \alpha^{d-1} \to 1 $.
Thus \eqref{eq:strongerstatement} holds with $\ee_{n,d} = b_{n,d}\left[1- \alpha^{-1} \right] $.
\end{proof}
\bibliographystyle{plain}
\bibliography{biblio}
\end{document} | 58,956 |
Madeira Beach attorney Ray Blacklidge raised another $13,800 between his campaign and committee over the past two weeks, building on his fundraising edge over Republican primary opponent Jeremy Bailie.
Blacklidge and Bailie are running to replace exiting Republican Rep. Kathleen Peters in Pinellas County’s House District 69. Peters announced last year that she would not seek a fourth term in the district and would instead run for a seat on the Pinellas County Commission.
The new reports, covering Aug. 11 through Aug. 23, show Blacklidge added $6,700 in hard money, including max checks from Florida Beer Wholesalers, Great Bay Distributors and the Associated Industries of Florida, among others.
The other $7,100 was collected by his political committee, Friends of Ray Blacklidge, which deposited a $5,000 check from the Property Casualty Insurers Association of America Political Account and a $2,000 check from Windhaven Insurance Company. The other $100 came in from Jacksonville contractor Ruben Lavarias.
Spending for the reporting period came in at more than $76,000 and included nearly $64,000 in payments to Front Line Strategies. Those payments were marked down as “campaign consulting” fees, though the Tallahassee firm provides media buying and direct mail services, which could explain large expenditures.
All told, Blacklidge has raised more than $229,000 for his campaign since he filed for the race in mid-2017 and finished the reporting period with just $7,578 left to spend — not much considering that even if he proves successful Tuesday, the seat will be hotly contested in November.
Bailie raised $2,510 in new money, including $1,000 checks from the Florida Medical Association and the Realtors Political Advocacy Committee, as well as a $500 check from JP Morgan Chase. Bailie’s bid for the seat has been endorsed by the Florida Realtors.
The meager fundraising was coupled with $21,373 in spending, including a $19,410 payment to Strategic Image Management for printing and mailing work.
As of Aug. 23, Bailie had raised a total of $80,748 for his campaign and had $13,295 left to spend during the home stretch.
Bailie made headlines not long ago for being caught on video snagging pro-Blacklidge flyers off of doors while he was out canvassing. He has since publicly apologized for the stunt.
A recent poll of the race, conducted after that incident, showed Blacklidge with a 25-point lead among likely Republican primary voters. Among the two-fifths of respondents who said they had already voted, he led by 35 percentage points.
The winner of Tuesday’s election will move on to face Democratic nominee Jennifer Webb in November.
Webb was also the Democratic nominee in the 2016 cycle but lost to Peters by 13 points on Election Day. The district has a small Republican advantage, and Webb has built a sizable war chest and earned some major endorsements for her second bid to flip the seat.
HD 69 covers part of southern Pinellas County including the coastal communities from Redington Shores southward as well as a piece of mainland Pinellas. The district has a slim Republican advantage. | 230,694 |
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Download Link | 139,134 |
TITLE: If $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then $R^{\times}=\mathbb Z\big/6\mathbb Z$
QUESTION [2 upvotes]: How to show that if $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then the group of units $R^{\times}=\mathbb Z\big/6\mathbb Z$
Now since $-3\equiv1\mod 4$ the ring of integers are $\mathbb Z\left[\frac{1+\sqrt{-3}}{2}\right]$
So any element in the ring is of the form $a+b\left(\frac{1+\sqrt{-3}}{2}\right)$ with $a,b\in\mathbb Z$
but I can always find another $2$ elements $\tilde{a},\tilde{b}\in\mathbb Z$ with the same parity such that
$\displaystyle a+b\left(\frac{1+\sqrt{-3}}{2}\right)=\frac{\tilde{a}+\tilde{b}\sqrt{-3}}{2}$
Now the norm is easier to examine, if I set it equal to $1$;
$N(\frac{\tilde{a}+\tilde{b}\sqrt{-3}}{2})=\frac{\tilde{a}^2+3\tilde{b}^2}{4}=1$
$\implies\tilde{a}=\pm2,\tilde{b}=0\quad$ or $\quad\tilde{a}=\pm1,\tilde{b}=\pm1$
So there are $6$ possibilities, but how is it isomorphic to $\mathbb Z\big/6\mathbb Z$ ?
REPLY [1 votes]: I provide an alternative solution with a bit less appeal to abstract results and a focus on a concrete and strictly number-theoretic approach. Note the norm is most easily written as
$$N(a+b\zeta_3)=(a+b\zeta_3)(a+b\zeta_3^2)=a^2+b^2-ab$$
since $\zeta_3^2+\zeta_3+1=0$. The Cauchy-Schwarz inequality shows that this is always positive, so we need only examine when $a^2+b^2-ab=1$, and if $a$ and $b$ have opposite signs, the norm is $\ge 3$ as then $a^2+b^2-ab\ge 1+1+1$. So we may as well assume $a,b\ge 0$ and know that we get twice as many units as satisfy this by multiplying $a+b\zeta_3$ by $-1$. Since
$$a\ge b\ge 0\implies a^2+b^2-ab=b^2+a(a-b)\ge b^2.$$
(the last inequality follows from $a-b\ge 0$)
As such $b\in\{0,1\}$ are all we need check, and so also $a\in\{0,1\}$ also, by size considerations. The total pairs being $(a,b)\in \{(0,0),(1,0),(1,1),(0,1)\}$. Clearly $(0,0)$ is out, but the others are all seen to work, producing $3$ units apiece. Since we noted earlier we could double this number by negating all units arising in this way, there are $6$ in all.
We can also find a generator quite explicitly. Since $\zeta_3$ satisfies $\zeta_3^3=1$ and no smaller power works, then we know that $(-\zeta_3)^6=1$ and further that
$$(-\zeta_3)^3=(-1)^3(\zeta_3)^3=-1$$
so that it's order is exactly $6$. Indeed, if we think about it, we can see that $-\zeta_3=\zeta_6$, a primitive $6^{th}$ root of unity, but this is not necessary. Since there are $6$ units, and $-\zeta_3$ has order $6$ and is clearly a unit, the group is generated by this element. | 152,930 |
MUSKEGON, MI – A Muskegon judge denied bond Friday for Terry Wesley Raap, who’s charged with trying to murder two Muskegon County Jail deputies in a failed escape attempt Aug. 29.
It’s not like Raap is going anywhere anytime soon, anyway. He’s lodged as a maximum-security state prisoner in Ionia Correctional Facility on an alleged parole violation.
Raap, 26, of Twin Lake was arraigned Sept. 20 in Muskegon County’s 60th District Court by video from the Ionia prison. Appearing on camera, Raap said little, simply answering “yes” to the judge’s questions.
In his first court appearance in connection with the deputy assault, Raap was arraigned on two counts of assault with intent to murder, one count of attempted escape from a jail through violence and one count of possessing a weapon in jail..
Judge Andrew Wierengo III granted a request from Timothy M. Maat, chief assistant Muskegon County prosecutor, that bond be denied.
Maat argued that Raap’s attack on the two deputies showed him to be a danger if allowed out on bond. “Mr. Raap tried to kill two deputies ... and was nearly successful had they not been able to prevent that from happening,” Maat said. “And he certainly took a lot of time planning and took a lot of action in terms of making that occur.”
Maat also noted that Raap was lodged in the jail at the time of the attack for a felony – methamphetamine manufacture – that was alleged to have occurred while Raap was out on parole after serving an earlier prison term.
And, Maat said, Raap has been convicted of two prior “violent felonies,” second-degree criminal sexual conduct, and resisting and obstructing a police officer.
Wierengo scheduled a preliminary hearing for Oct. 1.
Injured were veteran deputies John Jenkins and Thomas Geoghan, according to the prosecutor’s office.
Prosecutors allege that Raap fashioned a clubbing weapon out of a brick-like chunk of concrete that he the meth charge, and he also faces federal gun charges arising from that case.
The jail incident happened around 3:45 p.m. Aug. 29 near a housing unit on the second floor.
John S. Hausman covers courts, the environment and local government for MLive/Muskegon Chronicle. Email him at [email protected] and follow him on Twitter. | 300,141 |
Tales of Britain
By Jem Roberts
The finest, funniest stories of England, Scotland & Wales, refreshed for the 21st century. By Brother Bernard, as told to Jem Roberts.
Wednesday, 24 April 2019
The Quest For Tales of Britain
Greetings! As today's Folklore Thursday theme is 'quests, missions and adventures', this is the simplest blog we have had to post hitherto. We may have told the story of our quest to put an all-new anthology of British folktales on bookshelves in text form in the past, but now we can see what a long quest it has been – and our adventure is far from over...
Six years ago, a jumble of retold folktales labouring under the misleading but terribly alliterative title of Brother Bernard's Big Book of British Bedtime Ballads was in search of a publisher, any publisher...
Two years ago, an official trailer was launched with Unbound...
Tales of Britain Unbound Trailer (2017)
And now, in 2019, the book is available at last. Brother Bernard has been travelling up and down the island telling his tales – once you've popped into your local indie bookshop and asked about getting copies in, why not look up our Live page to see if there's a Tales show near you, or even invite us to your town for a show of your own?
Tales of Britain Trailer (2019)
Long may our quest continue! | 219,538 |
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Attention California residents: This product contains lead, a chemical known to the State of California to cause cancer and birth defects or other reproductive harm. | 78,734 |
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We really appreciated the extra effort you put in to get our loan settled so quickly. Your professionalism was well noted too – we always received a phone call from you when you said you’d call, regardless of whether things had changed. We will be using you again if we ever need finance.
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“I was taken prisoner by the rebels the morning after the assault on Fort Wagner, South Carolina, July 19, 1863,” U.S. Navy Surgeon John Tarleton Luck wrote in a letter to the editor of the Army and Navy Journal in October of 1865. As Luck was being escorted to the Charleston prisoner’s hospital by confederate soldiers, he noticed Colonel Robert Gould Shaw’s dead body lying next to his deceased black sergeant.
Luck’s letter continues, “Brigadier-General Hagood, commanding the rebel forces, said to me: ‘Had [Shaw] been in command of white troops I should have given him an honorable burial. As it is, I shall bury him in the common trench, with the negroes that fell with him.’”
Luck, who spent his post-Navy days practicing medicine in Union Hill (now Union City) and his final hours in his home in Weehawken, thus became a Civil War whistleblower of sorts.
“[The 54th Massachusetts Regiment infantry] lost every field officer, every captain, and a lieutenant only remained to order it back.” – The New York Tribune, 1863
____________
In further controversy, the nature of Hagood’s alleged statement (which has seen both more harsh and more proper permutations in various publications and historical accounts) lends a decidedly racist tone to what was considered “dishonorable.”
Defining ‘dishonor’
Shaw was the commander of the famous 54th Regiment Massachusetts Infantry, made up of the Civil War’s first assembled black troops. The epic Second Battle of Fort Wagner was memorialized in director Edward Zwick’s 1989 movie “Glory,” and the very scene which Luck alludes to in his letter serves as the film’s dramatic, slow-motion climax.
Racism is clearly implied in Hagood’s alleged statement, as it could be interpreted to mean that a burial alongside black soldiers compounds the dishonor of being buried in a mass grave. However, one must take into account both Shaw’s and his family’s sentiments to see the other side.
In “War and Remembrance from the Iliad to Vietnam,” author James Tatus writes, “Shaw’s family thought it the most glorious internment possible.” In addition, Shaw’s willingness to command the first black regiment in and of itself, in addition to the movie’s screenplay based in part on Shaw’s letters, suggests he would have preferred to be buried with his soldiers.
The 54th
The 54th Regiment Massachusetts was authorized in March of 1863 by Massachussetts Gov. John A. Andrew, just after President Abraham Lincoln issued the Emancipation Proclamation.
Fort Wagner was a Confederate battery that stood south of Charleston Harbor and was widely considered one of the toughest of the Confederate States’ beachhead defenses. Hagood and Brigadier General William B. Taliaferro were in command.
On July 18, 1863, the 54th led a bloody assault on the fort, during which Shaw was shot through the chest and killed during battle with 29 of his men. Twenty-four died later from wound complications, 149 were wounded, 15 were captured, and 52 were never accounted for, bringing the casualty total to 272.
“The 54th Mass (colored reg’t), went into that awful carnage under its noble commander, Col. Shaw, when a Rebel bullet struck as he gained the parapet” the July 27, 1863 edition of the New York Tribune stated. “It lost every field officer, every captain, and a lieutenant only remained to order it back.”
They lost one fourth of their men, which was the highest loss for the infantry during the war. Though the Union was unable to take the fort, the 54th received recognition from Lincoln for its valor and for contributing to their final victory.
From Iowa to war to home
Luck, who is assumed to have been born in Iowa in 1838 as one of 12 children, graduated from Harvard Medical School and Columbia University and practiced in his home state until the civil War began.
Luck entered the Navy on Jan. 29, 1862, and according to Luck’s obituary in the1909 Daily Dispatch, he was “one of the first to answer the call and he entered the navy and was proud of the fact that his commission was signed by Lincoln.”
After he was captured in Charleston in 1863, he was moved to Columbia and confined in a prison there for three months. He was then sent to Libby Prison, notorious for its mistreatment of prisoners. He was released two weeks later.
Luck relocated to Bull’s Ferry Road in Weehawken and set up a medical practice on Bergenline Avenue in Union Hill. He served as post commander of the Grand Army of the Republic Union Hill, was a consulting physician to the North Hudson Hospital, and served as the town’s school trustee and library commissioner for many years.
He died on Jan. 4, 1909 at North Hudson after a long and painful illness, according to the Hudson Dispatch obituary.
Stumbling upon Luck
Union City park enthusiast and local historical preservation advocate Tony Squire came upon Luck’s letter to the editor while researching another Civil War veteran at the New York Public Library. A former Navy corpsman himself who has worked in the medical field, Squire’s attention was caught. At first, he found no biography, diaries, or relatives associated with Luck, then was fortunate enough to discover bits of information through online forums that led to secondary sources and newspaper accounts.
In honor of Memorial Day, Squire contacted the Reporter and passed on his research, and with it he hopes to inspire two things: first, to encourage Civil War researchers to further explore Luck’s military life; and second, that Union City’s Historical Preservation Committee will consider erecting a plaque that lists Union Hill and West Hoboken Civil War veterans.
Gennarose Pope may be reached at [email protected] | 199,749 |
\begin{document}
\title[The Picard group of the universal moduli space of vector bundles over $\CMg$]{The Picard group of the universal moduli space\\ of vector bundles on stable curves.}
\author{Roberto Fringuelli}
\begin{abstract}We construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of Poincar\'e bundles over the universal curve of an open substack of the rigidification, generalizing a result of Mestrano-Ramanan.
\end{abstract}
\maketitle
\tableofcontents
\section*{Introduction.}
Let $\Vc^{(s)s}$ be the moduli stack of (semi)stable vector bundles of rank $r$ and degree $d$ on smooth curves of genus $g$. It turns out that the forgetful map $\Vc^{ss}\to\Mg$ is universally closed, i.e. it satisfies the existence part of the valuative criterion of properness. Unfortunately, if we enlarge the moduli problem, adding slope-semistable (with respect to the canonical polarization) vector bundles on stable curves, the morphism to the moduli stack $\CMg$ of stable curves is not universally closed anymore. There exists two natural ways to make it universally closed. The first one is adding slope-semistable torsion free sheaves and this was done by Pandharipande in \cite{P96}. The disadvantage is that such stack, as Faltings has shown in \cite{Fa96}, is not regular if the rank is greater than one. The second approach, which is better for our purposes, is to consider vector bundles on semistable curves: see \cite{Gie84}, \cite{K}, \cite{NS} in the case of a fixed irreducible curve with one node, \cite{Cap94}, \cite{Mel09} in the rank one case over the entire moduli stack $\CMg$ or \cite{Sc}, \cite{T98} in the higher rank case over $\CMg$. The advantages are that such stacks are regular and the boundary has normal-crossing singularities. Unfortunately, for rank greater than one, we do not have an easy description of the objects at the boundary. We will overcome the problem by constructing a non quasi-compact smooth stack $\CVc$, parametrizing properly balanced vector bundles on semistable curves (see \S\ref{pbvb} for a precise definition). In some sense, this is the right stacky-generalization in higher rank of the Caporaso's compactification $\overline{J}_{d,g}$ of the universal Jacobian scheme. Moreover it contains some interesting open substacks, like:
\begin{itemize}
\item[-] The moduli stack $\Vc$ of (not necessarily semistable) vector bundles over smooth curves.
\item[-] The moduli stack $\CVc^{P(s)s}$ of vector bundles such that their push-forwards in the stable model of the curve is a slope-(semi)stable torsion free sheaf.
\item[-] The moduli stack $\CVc^{H(s)s}$ of H-(semi)stable vector bundles constructed by Schmitt in \cite{Sc}.
\item[-] The moduli stack of Hilbert-semistable vector bundles (see \cite{T98}).
\end{itemize}
The main result of this paper is computing and giving explicit generators for the Picard groups of the moduli stacks $\Vc$ and $\CVc$ for rank greater than one, generalizing the results in rank one obtained by Melo-Viviani in \cite{MV}, based upon a result of Kouvidakis (see \cite{Kou91}). As a consequence, we will see that there exist natural isomorphisms of Picard groups between $\Vc^{ss}$ and $\Vc$, among $\CVc^{Hss}$, $\CVc^{Pss}$ and $\CVc$ and between $\CVc^{Ps}$ and $\CVc^{Hs}$.\\
The motivation for this work comes from the study of modular compactifications of the moduli stack $\Vc^{ss}$ and the coarse moduli space $U_{r,d,g}$ of semistable vector bundles on smooth curves from the point of view of the log-minimal model program (LMMP). One would like to mimic the so called Hassett-Keel program for the moduli space $\overline{M}_g$ of stable curves, which aims at giving a modular interpretation to the every step of the LMMP fot $\overline{M}_g$. In other words, the goal is to construct compactifications of the universal moduli space of semistable vector bundles over each step of the minimal model program for $\overline{M}_g$. In the rank one case, the conjectural first two steps of the LMMP for the Caporaso's compactification $\overline{J}_{d,g}$ have been described by Bini-Felici-Melo-Viviani in \cite{BFMV}. From the stacky point of view, the first step (resp. the second step) is constructed as the compactified Jacobian over the Schubert's moduli stack $\CMg^{ps}$ of pseudo-stable curves (resp. over the Hyeon-Morrison's moduli stack $\CMg^{wp}$ of weakly-pseudo-stable curves). In higher rank, the conjectural first step of the LMMP for the Pandharipande's compactification $\widetilde{U}_{r,d,g}$ has been described by Grimes in \cite{Gr14}: using the torsion free approach, he constructs a compactification $\widetilde{U}^{ps}_{r,d,g}$ of the moduli space of slope-semistable vector bundles over $\overline{M}^{ps}_g$.
In order to construct birational compact models for the Pandharipande compactification of $U_{r,d,g}$, it is useful to have an explicit description of its rational Picard group which naturally embeds into the rational Picard group of the moduli stack $\TF$ of slope-semistable torsion free sheaves over stable curves. Indeed our first idea was to study directly the Picard group of $\TF$ . For technical difficulties due to the fact that such stack is not smooth, we have preferred to study first $\CVc$, whose Picard group contains $\Pic(\TF)$, and we plan to give a description of $\TF$ in a subsequent paper.\vspace{0.2cm}
In Section \ref{cvc}, we introduce and study our main object: \emph{the universal moduli stack $\CVc$ of properly balanced vector bundles of rank $r$ and degree $d$ on semistable curves of arithmetic genus $g$}. We will show that it is an irreducible smooth Artin stack of dimension $(r^2+3)(g-1)$. The stacks of the above list are contained in $\CVc$ in the following way
\begin{equation}\label{chainopen}
\begin{array}{lllll}
\CVc^{Ps}\subset&\CVc^{Hs}\subset&\CVc^{Hss}\subset&\CVc^{Pss}\subset&\CVc\\
\cup & &\cup & &\cup\\
\Vc^s&\subset&\Vc^{ss}&\subset&\Vc.
\end{array}
\end{equation}
The stack $\CVc$ is endowed with a morphism $\overline{\phi}_{r,d}$ to the stack $\CMg$ which forgets the vector bundle and sends a curve to its stable model. Moreover, it has a structure of $\mathbb G_m$-stack, since the group $\mathbb G_m$ naturally injects into the automorphism group of every object as multiplication by scalars on the vector bundle. Therefore, $\CVc$ becomes a $\mathbb G_m$-gerbe over the $\mathbb G_m$-rigidification $\CVr:=\CVc\fatslash \mathbb G_m$. Let $\nu_{r,d}:\CVc\to\CVr$ be the rigidification morphism. Analogously, the open substacks in (\ref{chainopen}) are $\mathbb G_m$-gerbes over their rigidifications
\begin{equation}\label{chainopenrig}
\begin{array}{lllll}
\CVr^{Ps}\subset&\CVr^{Hs}\subset&\CVr^{Hss}\subset&\CVr^{Pss}\subset&\CVr\\
\cup & &\cup & &\cup\\
\Vr^s&\subset&\Vr^{ss}&\subset&\Vr.
\end{array}
\end{equation}
The inclusions (\ref{chainopen}) and (\ref{chainopenrig} give us the following commutative diagram of Picard groups:
\begin{equation}\label{compic}
\xymatrix @C=0.01em @R=0.7em{
& \Pic\left(\CVc\right)\ar@{-}[d]\ar@{->>}[rr] & & \Pic\left(\CVc^{Pss}\right)\ar@{-}[d]\ar@{->>}[rr] & & \Pic\left(\CVc^{Ps}\right)\ar@{->>}[dd]\\
\Pic\left(\CVr\right)\ar@{^(->}[ur]\ar@{->>}[rr]\ar@{->>}[dddd] &\ar@{->>}[ddd] & \Pic\left(\CVr^{Pss}\right)\ar@{^(->}[ur]\ar@{->>}[rr]\ar@{->>}[dd] &\ar@{->>}[d] & \Pic\left(\CVr^{Ps}\right)\ar@{^(->}[ur]\ar@{->>}[dd]\\
& & & \Pic\left(\CVc^{Hss}\right) \ar@{-}[d]\ar@{-}[r]&\ar@{->>}[r] & \Pic\left(\CVc^{Hs}\right)\ar@{->>}[dd]\\
& & \Pic\left(\CVr^{Hss}\right)\ar@{^(->}[ur]\ar@{->>}[rr]\ar@{->>}[dd] & \ar@{->>}[d]& \Pic\left(\CVr^{Hs}\right)\ar@{^(->}[ur]\ar@{->>}[dd]\\
& \Pic\left(\Vc\right)\ar@{-}[r] &\ar@{->>}[r] & \Pic\left(\Vc^{ss}\right)\ar@{-}[r] &\ar@{->>}[r] & \Pic\left(\Vc^{s}\right)\\
\Pic\left(\Vr\right)\ar@{^(->}[ur]\ar@{->>}[rr] & & \Pic\left(\Vr^{ss}\right)\ar@{^(->}[ur]\ar@{->>}[rr] & & \Pic\left(\Vr^{s}\right)\ar@{^(->}[ur]
}
\end{equation}
where the diagonal maps are the inclusions induced by the rigidification morphisms, while the vertical and horizontal ones are the restriction morphisms, which are surjective because we are working with smooth stacks. We will prove that the Picard groups of diagram (\ref{compic}) are generated by the boundary line bundles and the tautological line bundles, which are defined in Section \ref{linechow}.\\
In the same section we also describe the irreducible components of the boundary divisor $\CVc\backslash\Vc$. Obviously the boundary is the pull-back via the morphism $\overline{\phi}_{r,d}:\CVc\to\CMg$ of the boundary of $\CMg$. It is known that $\CMg\backslash \Mg=\bigcup_{i=0}^{\lfloor g/2\rfloor}\delta_i$, where $\delta_0$ is the irreducible divisor whose generic point is an irreducible curve with just one node and, for $i\neq 0$, $\delta_i$ is the irreducible divisor whose generic point is the stable curve with two irreducible smooth components of genus $i$ and $g-i$ meeting in one point. In Proposition \ref{boundary}, we will prove that $\widetilde\delta_i:=\overline{\phi}^*_{r,d}\left(\delta_i\right)$ is irreducible if $i=0$ and, otherwise, decomposes as $\bigcup_{j\in J_i} \widetilde\delta_i^j$, where $J_i$ is a set of integers depending on $i$ and $\widetilde\delta_i^j$ are irreducible divisors. Such $\widetilde\delta_i^j$ will be called \emph{boundary divisors}. For special values of $i$ and $j$, the corresponding boundary divisor will be called \emph{extremal boundary divisor}. The boundary divisors which are not extremal will be called \emph{non-extremal boundary divisors} (for a precise description see \S\ref{boundiv}). By smoothness of $\CVc$, the divisors $\{\widetilde\delta_i^j\}$ give us line bundles. We will call them \emph{boundary line bundles} and we will denote them with $\{\oo(\widetilde\delta_i^j)\}$. We will say that $\oo(\widetilde\delta_i^j)$ is a \emph{(non)-extremal boundary line bundle} if $\widetilde\delta_i^j$ is a (non)-extremal boundary divisor.
The irreducible components of the boundary of $\CVr$ are the divisors $\nu_{r,d}(\widetilde\delta_i^j)$. The associated line bundles are called boundary line bundles of $\CVr$. We will denote with the the same symbols used for $\CVc$ the boundary divisors and the associated boundary line bundles on $\CVr$. \\
In \S\ref{tautbun} we define the \emph{tautological line bundles}. They are defined as determinant of cohomology and as Deligne pairing (see \S\ref{dcdp} for the definition and basic properties) of particular line bundles along the universal curve $\overline{\pi}:\overline{Vec}_{r,d,g,1}\to\CVc$. More precisely they are
$$
\begin{array}{rcl}
K_{1,0,0} &:=&\langle \omega_{\overline{\pi}},\omega_{\overline{\pi}}\rangle,\\
K_{0,1,0}&:=&\langle \omega_{\overline{\pi}},\det\mt E\rangle,\\
K_{-1,2,0}&:=&\langle \det\mt E,\det\mt E\rangle,\\
\Lambda(m,n,l) & :=& d_{\overline{\pi}}(\omega_{\overline{\pi}}^m\otimes (\det \mt E)^n\otimes \mathcal E^l).
\end{array}
$$
where $\omega_{\overline{\pi}}$ is the relative dualizing sheaf for $\pi$ and $\mt E$ is the universal vector bundle on $\overline{Vec}_{r,d,g,1}$. Following the same strategy of Melo-Viviani in \cite{MV}, based upon the work of Mumford in \cite{Mum83}, we apply Grothendieck-Riemann-Roch theorem to the morphism $\pi:\overline{Vec}_{r,d,g,1}\to\CVc$ in order to compute the relations among the tautological line bundles in the rational Picard group. In particular, in Theorem \ref{relations} we prove that all tautological line bundles can be expressed in the (rational) Picard group of $\CVc$ in terms of $\Lambda(1,0,0)$, $\Lambda(0,1,0)$, $\Lambda(1,1,0)$, $\Lambda(0,0,1)$ and the boundary line bundles.\vspace{0.2cm}
Finally we can now state the main results of this paper. In Section \ref{robba}, we prove that all Picard groups on diagram (\ref{compic}) are free and generated by the tautological line bundles and the boundary line bundles. More precisely, we have the following.
\begin{theoremalpha}\label{pic}Assume $g\geq 3$ and $r\geq 2$.
\begin{enumerate}[(i)]
\item The Picard groups of $\Vc$, $\Vc^{ss}$, $\Vc^s$ are freely generated by $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$ and $\Lambda(0,0,1)$.
\item The Picard groups of $\CVc$, $\CVc^{Pss}$, $\CVc^{Hss}$ are freely generated by $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$, $\Lambda(0,0,1)$ and the boundary line bundles.
\item The Picard groups of $\CVc^{Ps}$, $\CVc^{Hs}$ are freely generated by $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$, $\Lambda(0,0,1)$ and the non-extremal boundary line bundles.
\end{enumerate}
\end{theoremalpha}
Let $v_{r,d,g}$ and $n_{r,d}$ be the numbers defined in the Notations \ref{notations} below. Let $\alpha$ and $\beta$ be (not necessarily unique) integers such that $\alpha(d+1-g)+\beta(d+g-1)=-\frac{1}{n_{r,d}}\cdot\frac{v_{1,d,g}}{v_{r,d,g}}(d+r(1-g))$. We set
$$\Xi:=\Lambda(0,1,0)^{\frac{d+g-1}{v_{1,d,g}}}\otimes\Lambda(1,1,0)^{-\frac{d-g+1}{v_{1,d,g}}},\quad
\Theta:=\Lambda(0,0,1)^{\frac{r}{n_{r,d}}\cdot \frac{v_{1,d,g}}{v_{r,d,g}}}\otimes \Lambda(0,1,0)^{\alpha}\otimes \Lambda (1,1,0)^{\beta}.$$
\begin{theoremalpha}\label{picred}Assume $g\geq 3$ and $r\geq2$.
\begin{enumerate}[(i)]
\item The Picard groups of $\Vr$, $\Vr^{ss}$, $\Vr^{s}$ are freely generated by $\Lambda(1,0,0)$, $\Xi$ and $\Theta$.
\item The Picard groups of $\CVr$, $\CVr^{Pss}$, $\CVr^{Hss}$ are freely generated by $\Lambda(1,0,0)$, $\Xi$, $\Theta$ and the boundary line bundles.
\item The Picard groups of $\CVr^{Ps}$ and $\CVr^{Hs}$ are freely generated by $\Lambda(1,0,0)$, $\Xi$, $\Theta$ and the non-extremal boundary line bundles.
\end{enumerate}
\end{theoremalpha}
If we remove the word ''freely'' from the assertions, the above theorems hold also in the genus two case. This will be shown in appendix \ref{genus2}, together with an explicit description of the relations among the generators.
We sketch the strategy of the proofs of the Theorems \ref{pic} and \ref{picred}. First, in \S\ref{indipendece}, we will prove that the boundary line bundles are linearly independent. Since the stack $\CVc$ is smooth and it contains quasi-compact open substacks which are "large enough" and admit a presentation as quotient stacks, we have a natural exact sequence of groups
\begin{equation}\label{quasiex}
\bigoplus_{i=0,\ldots,\lfloor g/2\rfloor}\oplus_{j\in J_i}\langle\oo(\widetilde\delta_i^j)\rangle\longrightarrow \Pic(\CVc)\rightarrow \Pic(\Vc)\longrightarrow 0
\end{equation}
In Theorem \ref{indbou}, we show that such sequence is also left exact. The strategy that we will use is the same as the one of Arbarello-Cornalba for $\CMg$ in \cite{AC87} and the generalization for $\CJc$ done by Melo-Viviani in \cite{MV}. More precisely, we will construct morphisms $B\to\CVc$ from irreducible smooth projective curves $B$ and we show that the intersection matrix between these test curves and the boundary line bundles on $\CVc$ is non-degenerate.\\
Furthermore, since the homomorphism of Picard groups induced by the rigidification morphism $\nu_{r,d}:\CVc\to\CVr$ is injective and it sends the boundary line bundles of $\CVr$ to the boundary line bundles of $\CVc$, we see that also the boundary line bundles in the rigidification $\CVr$ are linearly independent (see Corollary \ref{boundrig}). In other words we have an exact sequence:
\begin{equation}\label{exrig}
0\longrightarrow\bigoplus_{i=0,\ldots,\lfloor g/2\rfloor}\oplus_{j\in J_i}\langle\oo(\widetilde\delta_i^j)\rangle\longrightarrow \Pic(\CVr)\longrightarrow \Pic(\Vr)\longrightarrow 0.
\end{equation}
We will show that the sequence (\ref{quasiex}, (resp. (\ref{exrig}), remains exact if we replace the middle term with the Picard group of $\CVc^{Pss}$ (resp. $\CVr^{Pss}$) or $\CVc^{Hss}$ (resp. $\CVr^{Hss}$). This reduces the proof of Theorem \ref{pic}(ii) (resp. of Theorem \ref{picred}(ii)) to proving the Theorem \ref{pic}(i) (resp. to Theorem \ref{picred}(i)). While for the stacks $\CVc^{Ps}$ and $\CVc^{Hs}$ (resp. $\CVr^{Ps}$ and $\CVr^{Hs}$) the sequence (\ref{quasiex}) (resp. (\ref{exrig})) is exact if we remove the extremal boundary line bundles. This reduces the proof of Theorem \ref{pic}(iii) (resp. of Theorem \ref{picred}(iii)) to proving the Theorem \ref{pic}(i) (resp. the Theorem \ref{picred}(i)).\vspace{0.2cm}
The stack $\Vc$ admits a natural map $det$ to the universal Jacobian stack $\Jc$, which sends a vector bundle to its determinant line bundle. The morphism is smooth and the fiber over a polarized curve $(C,\mt L)$ is the irreducible moduli stack $\mathcal Vec_{=\mathcal L,C}$ of pairs $(\mt E,\varphi)$, where $\mt E$ is a vector bundle on $C$ and $\varphi$ is an isomorphism between $\det\mt E$ and $\mt L$ (for more details see \S\ref{fibre}). Hoffmann in \cite{H} showed that the pull-back to $\mathcal Vec_{=\mathcal L,C}$ of the tautological line bundle $\Lambda(0,0,1)$ on $\CVc$ freely generates $\Pic\left(\mathcal Vec_{=\mathcal L,C}\right)$ (see Theorem \ref{fibers}).
Moreover, as Melo-Viviani have shown in \cite{MV}, the tautological line bundles $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$ freely generate the Picard group of $\Jc$ (see Theorem \ref{picjac}). Since the the Picard groups of $\Vc$, $\Vc^{ss}$, $\Vc^{s}$ are isomorphic (see Lemma \ref{redsemistable}), Theorem \ref{pic}(i) (and so Theorem \ref{pic}) is equivalent to prove that we have an exact sequence of groups
\begin{equation}\label{theoAi}
0\lra \Pic(\Jc)\lra \Pic(\Vc^{ss})\lra \Pic(\mathcal Vec^{ss}_{=\mathcal L,C})\lra 0
\end{equation}
where the first map is the pull-back via the determinant morphism and the second one is the restriction along a fixed geometric fiber. We will prove this in \S\ref{liscio}. If we were working with schemes, this would follow from the so-called seesaw principle: if we have a proper flat morphism of varieties with integral geometric fibers then a line bundle on the source is the pull-back of a line bundle on the target if and only if it is trivial along any geometric fiber. We generalize this principle to stacks admitting a proper good moduli space (see Appendix \ref{App}) and we will use this fact to prove the exactness of (\ref{theoAi}).\\
In \S\ref{comparing}, we use the Leray spectral sequence for the lisse-\'etale sheaf $\mathbb{G}_m$ with respect to the rigidification morphism $\nu_{r,d}:\Vc\lra\Vr$, in order to conclude the proof of Theorem \ref{picred}. Moreover we will obtain, as a consequence, some interesting results about the properties of $\CVr$ (see Proposition \ref{poincare}). In particular we will show that the rigidified universal curve $\mt V_{r,d,g,1}\to\Vr$ admits a universal vector bundle over an open substack of $\Vr$ if and only if the integers $d+r(1-g)$, $r(d+1-g)$ and $r(2g-2)$ are coprime, generalizing the result of Mestrano-Ramanan (\cite[Corollary 2.9]{MR85}) in the rank one case.\vspace{0.2cm}
The paper is organized in the following way. In Section \ref{cvc}, we define and study the moduli stack $\CVc$ of properly balanced vector bundles on semistable curves. In \S\ref{pbvb}, we give the definition of a properly balanced vector bundle on a semistable curve and we study the properties. In \S\ref{univ} we prove that the moduli stack $\CVc$ is algebraic. In \S\ref{Sc} we focus on the existence of good moduli spaces for an open substack of $\CVc$, following the Schmitt's construction. In \S\ref{prop} we list some properties of our stacks and we introduce the rigidified moduli stack $\CVr$. We will use the deformation theory of vector bundles on nodal curves for study the local structure of $\CVc$ (see \S\ref{locstr}). In Section \ref{linechow}, we resume some basic facts about the Picard group of a stack. In \S\ref{boh} we explain the relations between the Picard group and the Chow group of divisors of stacks. We illustrate how to construct line bundles on moduli stacks using the determinant of cohomology and the Deligne pairing (see \S\ref{dcdp}). Then we recall the computation of the Picard group of the stack $\CMg$, resp. $\Jc$, resp. $\mathcal Vec_{=\mathcal L,C}$ (see \S\ref{cmg}, resp. \S\ref{jc}, resp. \S\ref{fibre}). In \S\ref{boundiv} we describe the boundary divisors of $\CVc$, while in \S\ref{tautbun} we define the tautological line bundles and we study the relations among them. Finally, in Section \ref{robba}, as explained before, we prove Theorems \ref{pic} and \ref{picred}. The genus two case will be treated separately in the Appendix \ref{genus2}. In Appendix \ref{App}, we recall the definition of a good moduli space for a stack and we develop, following the strategy adopted by Brochard in \cite[Appendix]{Br2}, a base change cohomology theory for stacks admitting a proper good moduli space.\\
\textbf{Acknowledgements:} The author would like to thank his advisor Filippo Viviani, for introducing the author to the problem, for his several suggests and comments without which this work would not have been possible.
\subsection*{Notations.}
\begin{notations}\label{notations}
Let $g\geq 2$, $r\geq 1$, $d$ be integers. We will denote with $g$ the arithmetic genus of the curves, $d$ the degree of the vector bundles and $r$ their rank. Given two integers $s$, $t$ we will denote with $(s,t)$ the greatest common divisor of $s$ and $t$. We will set
$$
n_{r,d}:=(r,d),\; v_{r,d,g}:=\left(\frac{d}{n_{r,d}}+\frac{r}{n_{r,d}}(1-g),\,d+1-g,\,2g-2\right),\; k_{r,d,g}:=\frac{2g-2}{\left(2g-2,d+r(1-g)\right)}.
$$
Given a rational number $q$, we denote with $\lfloor q\rfloor$ the greatest integer such that $\lfloor q\rfloor\leq q$ and with $\lceil q\rceil$ the lowest integer such that $q\leq\lceil q\rceil$.
\end{notations}
\begin{notations}
We will work with the category $Sch/k$ of (not necessarily noetherian) schemes over an algebraically closed field $k$ of characteristic $0$. When we say commutative, resp. cartesian, diagram of stacks we will intend in the $2$-categorical sense. We will implicitly assume that all the sheaves are sheaves for the site lisse-\'etale, or equivalently for the site lisse-lisse champ\^etre (see \cite[Appendix A.1]{Br1}).\\
The choice of characteristic is due to the fact the explicit computation of the Picard group of $\CMg$ is known to be true only in characteristic $0$ (if $g\geq 3$). Also the computation of $\Jc$ in \cite{MV} is unknown in positive characteristic, because its computation is based upon a result of Kouvidakis in \cite{Kou91} which is proved over the complex numbers. If these two results could be extended to arbitrary characteristics then also our results would automatically extend.
\end{notations}
\section{The universal moduli space $\CVc$.}\label{cvc}
Here we introduce the moduli stack of properly balanced vector bundles on semistable curves. Before giving the definition, we need to define and study the objects which are going to be parametrized.
\begin{defin}
A \emph{stable} (resp. \emph{semistable}) curve $C$ over $k$ is a projective connected nodal curve over $k$ such that any rational smooth component intersects the rest of the curve in at least $3$ (resp. $2$) points. A \emph{family of (semi)stable curves over a scheme $S$} is a proper and flat morphism $C\rightarrow S$ whose geometric fibers are (semi)stable curves. A \emph{vector bundle on a family of curves $C\rightarrow S$} is a coherent $S$-flat sheaf on $C$ which is a vector bundle on any geometric fiber.
\end{defin}
To any family $C\rightarrow S$ of semistable curves, we can associate a new family $C^{st}\rightarrow S$ of stable curves and an $S$-morphism $\pi:C\rightarrow C^{st}$, which, for any geometric fiber over $S$, is the stabilization morphism, i.e. it contracts the rational smooth subcurves intersecting the rest of the curve in exactly $2$ points. We can construct this taking the $S$-morphism $\pi: C\rightarrow \mathbb{P}(\omega_{C/S}^{\otimes 3})$ associated to the relative dualizing sheaf of $C\rightarrow S$ and calling $C^{st}$ the image of $C$ through $\pi$.
\begin{defin}
Let $C$ be a semistable curve over $k$ and $Z$ be a non-trivial subcurve. We set $Z^c:=\overline{C\backslash Z}$ and $k_Z:=|Z\cap Z^c|$. Let $\mathcal E$ be a vector bundle over $C$. If $C_1,\ldots C_n$ are the irreducible components of $C$, we call \emph{multidegree} of $\mathcal E$ the $n$-tuple $(\deg\mt E_{C_1},\ldots,\deg\mt E_{C_n})$ and \emph{total degree} of $\mt E$ the integer $d:=\sum \deg\mt E_{C_i}$.
\end{defin}
With abuse of notation we will write $\omega_Z:=\deg(\omega_C|_{Z})=2g_Z-2+k_Z$, where $\omega_C$ is the dualizing sheaf and $g_Z:=1-\chi(\oo_Z)$. If $\mt E$ is a vector bundle over a family of semistable curves $C\to S$, we will set $\mathcal E(n):=\mathcal E\otimes\omega^n_{C/S}$. By the projection formula we have
$$
R^i\pi_*\mathcal E(n):=R^i\pi_*(\mathcal E\otimes\omega^n_{C/S})\cong R^i\pi_*(\mathcal E)\otimes\omega^n_{C^{st}/S}
$$
where $\pi$ is the stabilization morphism.
\subsection{Properly balanced vector bundles.}\label{pbvb}
We recall some definitions and results from \cite{K}, \cite{Sc} and \cite{NS}.
\begin{defin} A \emph{chain of rational curves} (or \emph{rational chain}) $R$ is a connected projective nodal curve over $k$ whose associated graph is a path and whose irreducible components are rational. The $\emph{lenght}$ of $R$ is the number of irreducible components.
\end{defin}
Let $R_1,\ldots,R_k$ be the irreducible components of a chain of rational curves $R$, labeled in the following way: $R_i\cap R_j\neq \emptyset$ if and only if $|i-j|\leq 1$. For $1\leq i\leq k-1$ let $x_i:=R_i\cap R_{i+1}$ be the nodal points and $x_0\in R_1$, $x_k\in R_k$ closed points different from $x_1$ and $x_{k-1}$. Let $\mathcal E$ be a vector bundle on $R$ of rank $r$. By \cite[Proposition 3.1]{T91b}, any vector bundle $\mt E$ over a chain of rational curves $R$ decomposes in the following way
$$
\mt E\cong \bigoplus_{j=1}^r\mt L_j,\text{ where } \mt L_j \text{ is a line bundle for any }j=1,\ldots,r.
$$
Using these notations we can give the following definitions.
\begin{defin} Let $\mt E$ be a vector bundle of rank $r$ on a rational chain $R$ of lenght $k$.
\begin{itemize}
\item $\mt E$ is \emph{positive} if $\deg \mt L_{j|R_i}\geq 0$ for any $j\in\{1,\ldots,r\}$ and $i\in\{1,\ldots,k\}$,
\item $\mt E$ is \emph{strictly positive} if $\mt E$ is positive and for any $i\in\{1,\ldots,k\}$ there exists $j\in\{1,\ldots,r\}$ such that $\deg \mt L_{j|R_i}> 0$,
\item $\mt E$ is \emph{stricly standard} if $\mt E$ is strictly positive and $\deg \mt L_{j|R_i}\leq 1$ for any $j\in\{1,\ldots,r\}$ and $i\in\{1,\ldots,k\}$.
\end{itemize}
\end{defin}
\begin{defin}Let $R$ be a chain of rational curves over $k$ and $R_1,\ldots,R_k$ its irreducible components. A strictly standard vector bundle $\mt E$ of rank $r$ over $R$ is called \emph{admissible}, if one of the following equivalent conditions (see \cite[Lemma 2]{NS} or \cite[Lemma 3.3]{K}) holds:
\begin{itemize}
\item $h^0(R,\mathcal E(-x_0))=\sum \deg\mathcal E_{R_i}=\deg \mt E$,
\item $H^0(R,\mathcal E(-x_0-x_k))=0$,
\item $\mt E=\bigoplus_{i=1}^r\mt L_i$, where $\mt L_i$ is a line bundle of total degree $0$ or $1$ for $i=1,\ldots,r$.
\end{itemize}
\end{defin}
\begin{defin} Let $C$ be a semistable curve $C$ over $k$.
The subcurve of all the chains of rational curves will be called \emph{exceptional curve} and will be denoted with $C_{exc}$ and we set $\widetilde C:=C_{exc}^c$. A connected subcurve $R$ of $C_{exc}$ will be called \emph{maximal rational chain} if there is no rational chain $R'\subset C$ such that $R\subsetneq R'$.
\end{defin}
\begin{defin}Let $C$ be a semistable curve and $\mathcal E$ be a vector bundle of rank $r$ over $C$. $\mathcal E$ is \emph{(strictly) positive}, resp. \emph{strictly standard}, resp. \emph{admissible vector bundle} if the restriction to any rational chain is (strictly) positive, resp. strictly standard, resp. admissible.
Let $C\rightarrow S$ be a family of semistable curves with a vector bundle $\mathcal E$ of relative rank $r$. $\mathcal E$ is called \emph{(strictly) positive}, resp. \emph{strictly standard}, resp. \emph{admissible vector bundle} if it is (strictly) positive, resp. strictly standard, resp. admissible for any geometric fiber.
\end{defin}
\begin{rmk}\label{lenght} Let $(C,\mt E)$ be a semistable curve with a vector bundle. We have the following sequence of implications: $\mt E$ is admissible $\Rightarrow$ $\mt E$ is strictly standard $\Rightarrow$ $\mt E$ is strictly positive $\Rightarrow$ $\mt E$ is positive. Moreover if $\mt E$ is admissible of rank $r$ then any rational chain must be of lenght $\leq r$.
\end{rmk}
The role of positivity is summarized in the next two propositions.
\begin{prop}\label{stabil}\cite[Prop 1.3.1(ii)]{Sc} Let $\pi:C'\rightarrow C$ be a morphism between semistable curves which contracts only some chains of rational curves. Let $\mathcal E$ be a vector bundle on $C'$ positive on the contracted chains. Then $R^i\pi_*(\mathcal E)=0$ for $i>0$. In particular $H^j(C',\mathcal E)=H^j(C,\pi_*\mathcal E)$ for all $j$.
\end{prop}
\begin{prop}\label{NSlemma4} Let $C\rightarrow S$ be a family of semistable curves, $S$ locally noetherian scheme and consider the stabilization morphism
$$
\xymatrix{
C\ar[rd]\ar[rr]^{\pi} & & C^{st}\ar[ld]\\
& S &}
$$
Suppose that $\mathcal E$ is a positive vector bundle on $C\rightarrow S$ and for any point $s\in S$ consider the induced morphism $\pi_{s*}:C_s\rightarrow C^{st}_s$. Then
$$
\pi_*(\mathcal E)_{C^{st}_s}=\pi_{s*}(\mathcal E_{C_s}).
$$
Moreover $\pi_*\mathcal E$ is $S$-flat.
\end{prop}
\begin{proof}It follows from \cite[Lemma 4]{NS} and \cite[Remark 1.3.6]{Sc}.
\end{proof}
The next results gives us a useful criterion to check if a vector bundle is strictly positive or not.
\begin{prop}\label{amplenesso}\cite[Proposition 1.3.3]{Sc}. Let $C$ be a semistable curve containing the maximal chains $R_1,\ldots,R_k$. We set $\widetilde C_j:=R^c_{j}$, and let $p_1^j$, $p_2^j$ be the points where $R_j$ is attached to $\widetilde C_j$, for $j=1,\ldots,k$. Suppose that $\mt E$ is a strictly positive vector bundle on $C$ which satisfies the following conditions:
\begin{enumerate}[(i)]
\item $H^1\left(\widetilde C_j,\mt I_{p_1^j,p_2^j}\mt E_{\widetilde C_j}\right)=0$ for $j=1,\ldots,k$.
\item The homomorphism
$$
H^0\left(\widetilde C_j,\mt I_{p_1^j,p_2^j}\mt E_{\widetilde C_j}\right)\lra \left(\mt I_{p_1^j,p_2^j}\mt E_{\widetilde C_j}\right)\Big/\left( \mt I^2_{p_1^j,p_2^j}\mt E_{\widetilde C_j}\right)
$$
is surjective for $j=1,\ldots,k$.
\item For any $x\in\widetilde C\backslash\{p_1^j,p_2^j, j=1,\ldots,k\}$, the homomorphism
$$
H^0\left(C,\mt I_{C_{exc}}\mt E\right)\lra \mt E_{\widetilde C}\Big/\left(\mt I^2_x\mt E_{\widetilde C}\right)
$$
is surjective.
\item For any $x_1\neq x_2\in\widetilde C\backslash\{p_1^j,p_2^j, j=1,\ldots,k\}$, the evaluation homomorphism
$$
H^0\left(C,\mt I_{C_{exc}}\mt E\right)\lra \mt E_{\{x_1\}}\oplus \mt E_{\{x_2\}}
$$
is surjective.
\end{enumerate}
Then $\mt E$ is generated by global sections and the induced morphism in the Grassmannian
$$
C\hookrightarrow Gr(H^0(C,\mathcal E),r)
$$
is a closed embedding.
\end{prop}
Using \cite[Remark 1.3.4]{Sc}, we deduce the following useful criterion
\begin{cor}\label{ampleness} Let $\mathcal E$ be a vector bundle over a semistable curve $C$. $\mathcal E$ is strictly positive if and only if there exists $n$ big enough such that the vector bundle $\mt E(n)$ is generated by global sections and the induced morphism in the Grassmannian $C\to Gr(H^0(C,\mathcal E(n)),r)$ is a closed embedding.
\end{cor}
\begin{rmk} Let $\mathcal F$ be a torsion free sheaf over a nodal curve $C$. By \cite[Huitieme Partie, Proposition 3]{Se82}, the stalk of $\mt F$ over a nodal point $x$ is of the form
\begin{itemize}
\item $\oo_{C,x}^{r_0}\oplus\oo_{C_1,x}^{r_1}\oplus\oo_{C_2,x}^{r_2}$, if $x$ is a meeting point of two irreducible curves $C_1$ and $C_2$.
\item $\oo_{C,x}^{r-a}\oplus m_{C,x}^{a}$, if $x$ is a nodal point belonging to a unique irreducible component.
\end{itemize}
If $\mathcal F$ has uniform rank $r$ (i.e. it has rank $r$ on any irreducible component of $C$), we can always write the stalk at $x$ in the form $\oo_{C,x}^{r-a}\oplus m_{C,x}^{a}$ for some $\alpha$. In this case we will say that $\mathcal F$ is \emph{of type $a$ at $x$}.
\end{rmk}
Now we are going to describe the properties of an admissible vector bundle. The following proposition (and its proof) is a generalization of \cite[Proposition 5]{NS}.
\begin{prop}\label{prop5NS}Let $\mathcal E$ be a vector bundle of rank $r$ over a semistable curve $C$, and $\pi:C\rightarrow C^{st}$ the stabilization morphism, then:
\begin{enumerate}[(i)]
\item $\mathcal E$ is admissible if and only if $\mt E$ is strictly positive and $\pi_*\mt E$ is torsion free.
\item Let $R$ be a maximal chain of rational curves and $x:=\pi(R)$. If $\mathcal E$ is admissible then $\pi_*\mathcal E$ is of type $\deg\mathcal E_R$ at $x$.
\end{enumerate}
\end{prop}
\begin{proof}
Part (i). By hypothesis $\mathcal E$ is strictly positive. Let $\widetilde{C}$ be the subcurve of $C$ complementary to the exceptional one. Consider the exact sequence: $$0\lra\mt I_{\widetilde C}\mathcal E\lra \mathcal E\lra \mathcal E_{\widetilde C}\lra 0.$$
We can identify $\mt I_{\widetilde C}\mathcal E$ with $\mt I_{D}\mathcal E_{C_{exc}}$, where $D:=C_{exc}\cap \widetilde C$ with its reduced scheme structure. Then we have:
$$0\lra \pi_*(\mt I_D\mathcal E_{C_{exc}}) \lra \pi_*\mathcal E\lra \pi_*(\mathcal E_{\widetilde C}).$$
Now $\pi_*(\mathcal E_{\widetilde C})$ is a torsion-free sheaf and $\pi_*(\mt I_D\mathcal E_{C_{exc}})$ is a torsion sheaf, because its support is $D$. So $\pi_*\mathcal E$ is torsion free if and only if $\pi_*(\mt I_D\mathcal E_{C_{exc}})=0$. Let $R$ be a maximal rational chain which intersects the rest of the curve in $p$ and $q$ and $x:=\pi(R)$. By definition the stalk of the sheaf $\pi_*(\mt I_D\mathcal E_{C_{exc}})$ at $x$ is the $k$-vector space $H^0(R,\mt I_{p,q}\mathcal E_R)=H^0(R,\mathcal E_R(-p-q))$. Applying this method for any rational chain we have that $\pi_*\mathcal E$ is torsion free if and only if for any chain $R$ if a global section $s$ of $\mathcal E_R$ vanishes on $R\cap R^c$ then $s\equiv 0$. In particular, if $\mathcal E$ is admissible then $\pi_*\mathcal E$ is torsion free and $\mathcal E$ is strictly positive.\\
Conversely, suppose that $\pi_*\mathcal E$ is torsion free and $\mathcal E$ is strictly positive. The definition of admissibility requires that the vector bundle must be strictly standard, so a priori it seems that the viceversa should not be true. However we can easily see that if $\mt E$ is strictly positive but not strictly standard then there exists a chain $R$ such that $H^0(R,\mt I_{p,q}\mathcal E_R)\neq 0$. So $\pi_*\mt E$ cannot be torsion free, giving a contradiction. In other words, if $\pi_*\mathcal E$ is torsion free and $\mathcal E$ is strictly positive then $\mt E$ is strictly standard. By the above considerations the assertion follows.\\
Part (ii). Let $R$ be a maximal chain of rational curves. By hypothesis and part (i), $\pi_*\mt E$ is torsion free and we have an exact sequence:
$$
0\lra\pi_*\mt E\lra \pi_*(\mt E_{\widetilde C})\lra R^1\pi_*(\mt I_D\mt E_{C_{exc}})\lra 0.
$$
The sequence is right exact by Proposition \ref{stabil}. Using the notation of part (i), we have that the stalk of the sheaf $R^1\pi_*(\mt I_D\mt E_{C_{exc}})$ at $x$ is the $k$-vector space $H^1(R,\mt E_{R}(-p-q))$.
If $\deg(\mt E_R)=r$ is easy to see that $H^1(R,\mt E_R(-p-q))=0$, thus $\pi_*\mt E$ is isomorphic to $\pi_*(\mt E_{\widetilde C})$ locally at $x$. The assertion follows by the fact that $ \pi_*(\mt E_{\widetilde C})$ is a torsion free sheaf of type $\deg(\mt E_R)=r$ at $x$. Suppose that $\deg(\mt E_R)=r-s<r$. Then we must have that $\mt E_R=\oo_R^s\oplus\mt F$. Using the sequence
$$0\lra \mathcal E\lra \mathcal E_{R^c}\oplus \mathcal E_R\lra \mt E_{\{p\}}\oplus \mt E_{\{q\}}\lra 0$$
we can found a neighbourhood $U$ of $x$ in $C^{st}$ such that $\mt E_{\pi^{-1}(U)}=\oo_{\pi^{-1}(U)}^s\oplus\mt E'$ reducing to the case $\deg(\mt E_R)=r$.
\end{proof}
The proposition above has some consequence, which will be useful later. For example in \S\ref{Sc}, where we will prove that a particular subset of the set of admissible vector bundles over a semistable curve is bounded. The following results are generalizations of \cite[Remark 4]{NS}.
\begin{cor}\label{NSrmk}\noindent
\begin{enumerate}[(i)]
\item Let $C$ be a stable curve, $\pi:N\rightarrow C$ a partial normalization and $\mt F_1$, $\mt F_2$ two vector bundles on $N$. Then
$$
\mbox{Hom}_{\oo_N}(\mt F_1,\mt F_2)\cong \text{Hom}_{\oo_C}(\pi_*(\mt F_1) ,\pi_*(\mt F_2)).
$$
In particular $\mt F_1\cong\mt F_2\iff\pi_*(\mt F_1) \cong\pi_*(\mt F_2)$.
\item Let $C$ be a semistable curve with an admissible vector bundle $\mt E$, let $R$ be a subcurve composed only by maximal chains. We set $\widetilde C:=R^c$ and $D$ the reduced subscheme $R\cap \widetilde C$. Let $\pi:C\rightarrow C^{st}$ be the stabilization morphsim and $D^{st}$ be the reduced scheme $\pi(D)$. Then
$$
\pi_*\left(\mt I_R \mt E_R\right)=\pi_*\left(\mt I_D\mt E_{\widetilde C}\right)=\mt I_{D^{st}}\left(\pi_*\mt E\right).
$$
\item We set $\widetilde C:=C^c_{exc}$. We have that $\pi_*\mt E$ determines $\mt E_{\widetilde C}$, i.e. consider two pairs $(C,\mt E)$, $(C',\mt E')$ of semistable curves with admissible vector bundles such that $(C^{st},\pi_*\mt E)\cong (C'^{st},\pi'_*\mt E')$ , then $(\widetilde C,\mt E_{\widetilde C})\cong (\widetilde C',\mt E'_{\widetilde C'})$.
Observe that $C_{exc}$ and $C'_{exc}$ can be different.
\end{enumerate}
\end{cor}
\begin{proof}
Part (i). Adapting the proof of \cite[Remark 4(ii)]{NS} to our more general case, we obtain the assertion.\\
Part (ii). Consider the following exact sequence
$$
0\lra \mt I_{R}\mt E\lra \mt E\lra \mt E_R\lra 0.
$$
We can identify $\mt I_R\mt E$ with $\mt I_{D}\mt E_{\widetilde C}$. Applying the left exact functor $\pi_*$, we have
$$
0\lra \pi_*\left(\mt I_{D}\mt E_{\widetilde C}\right)\lra \pi_*(\mt E)\lra \pi_*(\mt E_R)\lra 0.
$$
The sequence is right exact because $\mt E$ is positive. Moreover $\pi_*(\mt E_R)$ is supported at $D^{st}$ and annihilated by $\mt I_{D^{st}}$. By Proposition \ref{prop5NS}(ii), the morphism $\pi_*(\mt E)\lra \pi_*(\mt E_R)$ induces an isomorphism of vector spaces at the restriction to $D^{st}$. This means that $\pi_*\left(\mt I_D\mt E_{\widetilde C}\right)=\mt I_{D^{st}}\left(\pi_*\mt E\right)$.\\
Part (iii). Suppose that $(C^{st},\pi_*\mt E)\cong (C'^{st},\pi'_*\mt E')$, i.e. there exist an isomorphism of curves $\psi:C^{st}\rightarrow C'^{st}$ and an isomorphism of sheaves $\phi:\pi_*\mt E\cong \psi^*\pi_*\mt E'$. By (ii), we have
$$
\pi_*(\mt I_D\mt E_{\widetilde C})\cong \psi^*\pi_*(\mt I_{D'}\mt E'_{\widetilde C'}).
$$
First we observe that $\widetilde C$ and $\widetilde C'$ are isomorphic and $\psi$ induces an isomorphism $\widetilde\psi$ between them, such that
$$
\psi^*\pi_*(\mt I_{D'}\mt E'_{\widetilde C'}).\cong \pi_*(\mt I_D\widetilde\psi^*\mt E'_{\widetilde C'}).
$$
Now by (i), we obtain an isomorphism of vector bundles $\mt I_D\mt E_{\widetilde C}\cong \mt I_D\widetilde\psi^*\mt E'_{\widetilde C'}$. Twisting by $\mt I_D^{-1}$, we have the assertion.
\end{proof}
\begin{defin}Let $\mathcal E$ be a vector bundle of rank $r$ and degree $d$ on a semistable curve $C$. $\mathcal E$ is \emph{balanced} if for any subcurve $Z\subset C$ it satisfies the \emph{basic inequality}:
$$
\left|\deg \mt E_Z-d\frac{\omega_Z}{\omega_C}\right|\leq r\frac{k_Z}{2}.
$$
$\mathcal E$ is \emph{properly balanced} if is balanced and admissible. If $C\rightarrow S$ is a family of semistable curves and $\mt E$ is a vector bundle of relative rank $r$ for this family, we will call it \emph{(properly) balanced} if is (properly) balanced for any geometric fiber.
\end{defin}
\begin{rmk}\label{balcon} We have several equivalent definitions of balanced vector bundle. We list some which will be useful later:
\begin{enumerate}[(i)]
\item $\mt E$ is balanced;
\item the basic inequality is satisfied for any subcurve $Z\subset C$ such that $Z$ and $Z^c$ are connected;
\item for any subcurve $Z\subset C$ such that $Z$ and $Z^c$ are connected, we have the following inequality
$$
\deg\mt E_Z-d\frac{\omega_Z}{\omega_C}\leq r\frac{k_Z}{2};
$$
\item for any subcurve $Z\subset C$ such that $Z$ and $Z^c$ are connected, we have the following inequality
$$\frac{\chi(\mt F)}{\omega_Z}\leq \frac{\chi(\mt E)}{\omega_C},$$
where $\mt F$ is the subsheaf of $\mt E_Z$ of sections vanishing on $Z\cap Z^c$;
\item for any subcurve $Z\subset C$ such that $Z$ and $Z^c$ are connected, we have $\chi(\mathcal G_Z)\geq 0$, where $\mt G$ is the vector bundle
$$
\left(\det\mathcal E\right)^{\otimes 2g-2}\otimes\omega_{C/S}^{\otimes -d+r(g-1)}\oplus\oo_C^{\oplus r(2g-2)-1}.
$$
\end{enumerate}
\end{rmk}
\begin{lem}Let $(p:C\rightarrow S, \mathcal E)$ be a vector bundle of rank $r$ and degree $d$ over a family of reduced and connected curves. Suppose that $S$ is locally noetherian. The locus where $C$ is a semistable curve and $\mt E$ strictly positive, resp. admissible, resp. properly balanced, is open in $S$.
\end{lem}
\begin{proof}We can suppose that $S$ is noetherian and connected. Suppose that there exists a point $s\in S$ such that the geometric fiber is a properly balanced vector bundle over a semistable curve. It is known that the locus of semistable curves is open on $S$ (see \cite[Chap. X, Corollary 6.6]{ACG11}). So we can suppose that $C\rightarrow S$ is a family of semistables curves of genus $g$. Up to twisting by a suitable power of $\omega_{C/S}$ we can assume, by Corollary \ref{ampleness}, that the rational $S$-morphism
$$
i:C\dashrightarrow Gr(p_*\mathcal E,r)
$$
is a closed embedding over $s$. By \cite[Lemma 3.13]{K}, there exists an open neighborhood $S'$ of $s$ such that $i$ is a closed embedding. Equivalently $\mathcal E_{S'}$ is strictly positive by Corollary \ref{ampleness}. We denote as usual with $\pi:C\rightarrow C^{st}$ the stabilization morphism. By Proposition \ref{NSlemma4}, the sheaf $\pi_*(\mt E_{S'})$ is flat over $S'$ and the push-forward commutes with the restriction on the fibers. In particular, it is torsion free at the fiber $s$, and so there exists an open subset $S''$ of $S'$ where $\pi_*(\mt E_{S''})$ is torsion-free over any fiber (see \cite[Proposition 2.3.1]{HL}). By Proposition \ref{prop5NS}, $\mt E_{S''}$ is admissible. Putting everything together, we obtain an open neighbourhood $S''$ of $s$ such that over any fiber we have an admissible vector bundle over a semistable curve. Let $0\leq k\leq d$, $0\leq i\leq g$ be integers. Consider the relative Hilbert scheme
$$
Hilb_{C/S''}^{\oo_{C}(1),P(m)=km+1-i}
$$
where $\oo_{C}(1)$ is the line bundle induced by the embedding $i$. We call $H_{k,i}$ the closure of the locus of semistable curves in $Hilb_{C/S''}^{\oo_{C}(1),P(m)=km+1-i}$ and we let $Z_{k,i}\hookrightarrow C\times_{S''}H_{k,i}$ be the universal curve. Consider the vector bundle $\mathcal G$ over $C\rightarrow S''$ as in Remark \ref{balcon}(v). Let $\mathcal G^{k,i}$ its pull-back on $Z_{k,i}$. The function
$$\chi: h\mapsto \chi(\mathcal G^{k,i}_h)$$
is locally constant on $H_{k,i}$. Now $\pi: H_{k,i}\rightarrow S''$ is projective. So the projection on $S''$ of the connected components of
$$
\bigsqcup_{0\leq k\leq d\atop 0\leq i\leq g}\\ H_{k,i}
$$
such that $\chi$ is negative is a closed subscheme. Its complement in $S$ is open and, by Remark \ref{balcon}(v), it contains $s$ and defines a family of properly balanced vector bundles over semistable curves.
\end{proof}
\subsection{The moduli stack of properly balanced vector bundles $\CVc$.}\label{univ}
Now we will introduce our main object of study: \emph{the universal moduli stack $\CVc$ of properly balanced vector bundles of rank $r$ and degree $d$ on semistable curves of arithmetic genus $g$}. Roughly speaking, we want a space such that its points are in bijection with the pairs $(C,\mt E)$ where $C$ is a semistable curve on $k$ and $\mt E$ is a properly balanced vector bundle on $C$. This subsection is devoted to the construction of such space as Artin stack.
\begin{defin}Let $r\geq 1$, $d$ and $g\geq 2$ be integers. Let $\CVc$ be the category fibered in groupoids over $Sch/k$ whose objects over a scheme $S$ are the families of semistable curves of genus $g$ with a properly balanced vector bundle of relative total degree $d$ and relative rank $r$. The arrows between the objects are the obvious cartesian diagrams.
\end{defin}
The aim of this subsection is proving the following
\begin{teo}\label{teo1}$\CVc$ is an irreducible smooth Artin stack of dimension $(r^2+3)(g-1)$. Furthermore, it admits an open cover $\{\overline{\mathcal U}_n\}_{n\in\mathbb Z}$ such that $\overline{\mathcal U}_n$ is a quotient stack of a smooth noetherian scheme by a suitable general linear group.
\end{teo}
\begin{rmk}\label{capo} In the case $r=1$, $\overline{\mt Vec}_{1,d,g}$ is quasi compact and it corresponds to the compactification of the universal Jacobian over $\CMg$ constructed by Caporaso \cite{Cap94} and later generalized by Melo \cite{Mel09}. Following the notation of \cite{MV}, we will set $\CJc:=\overline{\mt Vec}_{1,d,g}$.
\end{rmk}
The proof consists in several steps, following the strategies adopted by Kausz \cite{K} and Wang \cite{W}. First, we observe that $\CVc$ is clearly a stack for the Zariski topology. We now prove that it is a stack also for the fpqc topology (defined in \cite[Section 2.3.2]{FAG}). With that in mind, we will first prove the following lemma which allows us to restrict to families of semistable curves with properly balanced vector bundles over locally noetherian schemes.
\begin{lem}\label{rednoet}Let $\mathcal E$ be a properly balanced vector bundle over a family of semistable curves $p:C\rightarrow S$. Suppose that $S$ is affine. Then there exists
\begin{itemize}
\item a surjective morphism $\phi:S\rightarrow T$ where $T$ is a noetherian affine scheme,
\item a family of semistable curves $C_T\rightarrow T$,
\item a properly balanced vector bundle $\mt E_T$ over $C_T\rightarrow T$,
\end{itemize}
such that the pair $(C\rightarrow S,\mathcal E)$ is the pull-back by $\phi$ of the pair $(C_T\rightarrow T,\mathcal E_T)$.
\end{lem}
\begin{proof}We can write $S$ as a projective limit of affine noetherian $k$-schemes $(S_{\alpha})$. By \cite[8.8.2 (ii)]{EGAIV} there exists an $\alpha$, a scheme $C_{\alpha}$ and a morphism $C_{\alpha}\rightarrow S_{\alpha}$ such that $C$ is the pull-back of this scheme by $S\rightarrow S_{\alpha}$. By \cite[8.10.5 (xii)]{EGAIV} and \cite[11.2.6 (ii)]{EGAIV} we can assume that $C_{\alpha}\rightarrow S_{\alpha}$ is flat and proper. By \cite[8.5.2 (ii)]{EGAIV} there exists a coherent sheaf $\mathcal E_{\alpha}$ on $C_{\alpha}$ such that its pull-back on $S$ is $\mathcal E$. Moreover, by \cite[11.2.6 (ii)]{EGAIV} we may assume that $\mathcal E_{\alpha}$ is $S_{\alpha}$-flat. Set $S_{\alpha}=:T$, $C_{\alpha}=:C_T$ and $\mt E_{\alpha}=:\mt E_T$. Now the family $C_T\rightarrow T$ will be a family of semistable curves. The vector bundle $\mt E$ is properly balanced because this condition can be checked on the geometric fibers.
\end{proof}
\begin{prop}\label{descenteffective}Let $S'\rightarrow S$ be an fpqc morphism of schemes, set $S'':=S'\times_S S'$ and $\pi_i$ the natural projections. Let $(C\rightarrow S',\mathcal E')\in\CVc(S')$. Then every descent data
$$
\varphi:\pi_1^*(C\rightarrow S',\mt E)\cong\pi_2^*(C\rightarrow S',\mt E)
$$
is effective.
\end{prop}
\begin{proof}First we reduce to the case where $S'$ and $S$ are noetherian schemes. By \cite[(8.8.2)(ii), (8.10.5)(vi), (8.10.5)(viii) (11.2.6)(ii)]{EGAIV} there exists an fpqc morphism of noetherian affine schemes $S'_0\rightarrow S_0$ and a morphism $S\rightarrow S_0$, such that the diagram
$$
\xymatrix{
S'\ar[r]\ar[d] & S\ar[d]\\
S'_0\ar[r] & S_0
}$$
is cartesian. By Lemma \ref{rednoet}, there exists a pair $(C_0\rightarrow S'_0,\mt E'_0)\in\CVc(S'_0)$ such that its pull-back via $S'\rightarrow S'_0$ is isomorphic to $(C\rightarrow S',\mathcal E')\in\CVc(S')$. By \cite[(8.8.2)(i), (8.5.2)(i), (8.8.2.4), (8.5.2.4)]{EGAIV} we can assume that $\varphi$ comes from a descent data
$$
\varphi_0:\pi_1^*(C\rightarrow S'_0,\mt E'_0)\cong\pi_2^*(C\rightarrow S'_0,\mt E'_0).
$$
So we can assume that $S$ and $S'$ are noetherian. By the properly balanced condition, up to twisting by some power of the dualizing sheaf, we can suppose that $\det\mt E'$ is relatively ample on $S'$, in particular $\varphi$ induces a descent data for $(C\rightarrow S',\det\mt E')$ and this is effective by \cite[Theorem. 4.38]{FAG}. So there exists a family of curves $C\rightarrow S$ such that its pull-back via $S'\rightarrow S$ is $C'\rightarrow S'$. In particular, $C'\rightarrow C$ is an fpqc cover and $\varphi$ induces a descent data for $\mt E'$ on $C'\rightarrow C$, which is effective by \cite[Theorem. 4.23]{FAG}.
\end{proof}
\begin{prop}\label{diagonalrep}Let $S$ be an affine scheme. Let $(C\rightarrow S,\mt E),\;(C'\rightarrow S,\mt E')\in\CVc(S)$. The contravariant functor
$$
(T\rightarrow S)\mapsto \text{Isom}_T((C_T,\mt E_{T}),(C'_T,\mt E'_{T}))
$$
is representable by a quasi-compact separated $S$-scheme. In other words, the diagonal of $\CVc$ is representable, quasi-compact and separated.
\end{prop}
\begin{proof}Using the same arguments above, we can restrict to the category of locally noetherian schemes. Suppose that $S$ is an affine connected noetherian scheme. Consider the contravariant functor
$$
\left(T\rightarrow S\right)\mapsto \text{Isom}\left(C_T,C'_T\right).
$$
This functor is represented by a scheme $B$ (see \cite[pp. 47-48]{ACG11}). More precisely: let $Hilb_{C\times_S C'/S}$ be the Hilbert scheme which parametrizes closed subschemes of $C\times_S C'$ flat over $S$. $B$ is the open subscheme of $Hilb_{C\times_S C'/S}$ with the property that a morphism $f:T\rightarrow Hilb_{C\times_S C'/S}$ factorizes through $B$ if and only if the projections $\pi: Z_T\rightarrow C_T$ and $\pi': Z_T\rightarrow C'_T$ are isomorphisms, where $Z_T$ is the closed subscheme of $C\times_S C'$ represented by $f$. Consider the universal pair
$$
\left(Z_B,\varphi:=\pi'\circ \pi^{-1}: C_B\cong Z_B\cong C'_B\right).
$$
Now we prove that $B$ is quasi-projective. By construction it is enough to show that $B$ is contained in $Hilb_{C\times_S C'/S}^{P,\mt L}$, which parametrizes closed subschemes of $C\times_S C'/S$ with Hilbert polynomial $P$ respect to the relatively ample line bundle $\mt L$ on $C\times_S C'/S$. Let $\mt L$ (resp. $\mt L'$) be a relatively very ample line bundle on $C/S$ (resp. $C'/S$). We can take $\mt L=(\det\mt E)^m$ and $\mt L'=(\det\mt E')^m$ for $m$ big enough. Then the sheaf $\mt L\boxtimes_S\mt L'$ is relatively very ample on $C\times_S C'/S$. Using the projection $\pi$ we can identify $Z_B$ and $C_B$. The Hilbert polynomial of $Z_B$ with respect to the polarization $\mt L\boxtimes_S\mt L'$ is
$$
P(n)=\chi((\mt L\boxtimes_S\mt L')^n)=\chi(\mt L^n\otimes \varphi^*\mt L'^n)=\deg(\mt L^n)+\deg(\mt L'^n)+1-g.
$$
It is clearly independent from the choice of the point in $B$ and from $Z_B$, proving the quasi-projectivity. In particular, $B$ is quasi-compact and separated over $S$. The proposition follows from the fact that the contravariant functor
$$
\left(T\rightarrow B\right)\mapsto\text{Isom}_{C_T}\left(\mt E_T,\varphi^*\mt E'_T\right)
$$
is representable by a quasi-compact separated scheme over $B$ (see the proof of \cite[Theorem 4.6.2.1]{LMB}).
\end{proof}
Putting together Proposition \ref{descenteffective} and Proposition \ref{diagonalrep}, we get
\begin{cor}$\CVc$ is a stack for the fpqc topology.
\end{cor}
We now introduce a useful open cover of the stack $\CVc$. We will prove that any open subset of this cover has a presentation as quotient stack of a scheme by a suitable general linear group. In particular, $\CVc$ admits a smooth surjective representable morphism from a locally noetherian scheme. Putting together this fact with Proposition \ref{diagonalrep}, we get that $\CVc$ is an Artin stack locally of finite type.
\begin{prop}For any scheme $S$ and any $n\in \mathbb Z$, consider the subgrupoid $\overline{\mathcal U}_n(S)$ of $\CVc (S)$ of pairs $(p:C\rightarrow S,\mathcal E)$ such that
\begin{enumerate}
\item $R^ip_*\mathcal E(n)=0$ for any $i>0$,
\item $\mathcal E(n)$ is relatively generated by global sections, i.e. the canonical morphism $p^*p_*\mathcal E(n)\rightarrow\mathcal E(n)$ is surjective, and the induced morphism $C\rightarrow Gr(p_*\mathcal E(n),r)$ is a closed embedding.
\end{enumerate}
Then the sheaf $p_*\mathcal E(n)$ is flat on $S$ and $\mathcal E(n)$ is cohomologically flat over $S$. In particular, the inclusion $\overline{\mathcal U}_n\hookrightarrow\CVc$ makes $\overline{\mathcal U}_n$ into a fibered full subcategory.
\end{prop}
\begin{proof}We set $\mt F:=\mt E(n)$. By \cite[Proposition 4.1.3]{W}, we know that $p_*\mathcal F$ is flat on $S$ and $\mathcal F$ is cohomologically flat over $S$. Consider the following cartesian diagram
$$\xymatrix{
C_T \ar[d]^{p_T}\ar[r] & C\ar[d]^p\\
T\ar[r]^{\phi} & S
}$$
By \emph{loc. cit.}, we have that $R^ip_{T*}(\mathcal F_T)=0$ for any $i>0$ and that $\mathcal F_T$ is relatively generated by global sections. It remains to prove that the induced $T$-morphism $C_T\rightarrow Gr(p_{T*}\mathcal F_T,r)$ is a closed embedding. This follows easily by cohomological flatness and the base change property of the Grassmannian.
\end{proof}
\begin{lem} The subcategories $\left\{ \overline{\mt U}_n\right\}_{n\in\mathbb Z}$ form an open cover of $\CVc$.
\end{lem}
\begin{proof}Let $S$ be a scheme, $(p:C\rightarrow S,\mathcal E)$ an object of $\CVc(S)$ and $n$ an integer. We must prove that exists an open $U_n\subset S$ with the universal property that $T\rightarrow S$ factorizes through $U_n$ if and only if $\mathcal E_T$ is an object of $\overline{\mathcal U}_n(T)$.\\
We can assume $S$ affine. Lemma \ref{rednoet} implies that the morphism $S\rightarrow \overline{\mathcal U}_n$ factors through a noetherian affine scheme. So we can suppose that $S$ is affine and noetherian. Let $\mathcal F:=\mathcal E(n)$ and $U_n$ the subset of points of $S$ such that:
\begin{enumerate}
\item $H^i(C_s,\mathcal F_s)=0$ for $i>0$,
\item $H^0(C_s,\mathcal F_s)\otimes\oo_{C_s}\rightarrow \mathcal F_s$ is surjective,
\item the induced morphism in the Grassmannian $C_s\rightarrow Gr(H^0(C_s,\mathcal F_s),r)$ is a closed embedding.
\end{enumerate}
We must prove that $U_n$ is open and it satifies the universal property. As in the proof of \cite[Lemma 4.1.5]{W}, consider the open subscheme $V_n\subset S$ satisfying the first two conditions above. By definition it contains $U_n$ and it satisfies the universal property that any morphism $T\rightarrow S$ factorizes through $V_n$ if and only if $R^ip_{T*}\mathcal F_T=0$ for any $i>0$ and $\mathcal F_T$ is relatively generated by global sections. By \cite[Proposition 4.1.3]{W}, $\mt F_{V_n}$ is cohomologically flat over $V_n$. This implies that the fiber over a point $s$ of the morphism
$$
C_{V_n}\rightarrow Gr(p_{V_n*}\mathcal F_{V_n},r)
$$
is exactly $C_s\rightarrow Gr(H^0(C_s,\mathcal F_s),r)$. Since the property of being a closed embedding for a morphism of proper $V_n$-schemes is an open condition (see \cite[Lemma 3.13]{K}), it follows that $U_n$ is an open subscheme and $\mathcal F_{U_n}\in\overline{\mathcal U}_n(U_n)$.\\
Viceversa, suppose now that $\phi:T\rightarrow S$ is such that $\mathcal F_T\in\overline{\mathcal U}_n(T)$. The morphism factors through $V_n$ and for any $t\in T$
$$
C_t\rightarrow Gr(H^0(C_t,\mathcal F_t),r)
$$
is a closed embedding. Since the morphism $\phi$ restricted to a point $t\in T$ onto is image $\phi(t)$ is fppf, by descent the morphism $C_\phi(t)\rightarrow Gr(H^0(C_{\phi(t)},\mathcal F_{\phi(t)}),r)$ is a closed embedding, or in other words $\phi(t)\in U_n$.\\
It remains to prove that $\{\mathcal U_n\}$ is a covering. It is sufficient to prove that for any point $s$ exists $n$ such that $\mathcal E_s(n)$ satisfies the conditions (1), (2) and (3). By Proposition \ref{NSlemma4}, the push-forward of $\mathcal E$ in the stabilized family is $S$-flat and the cohomology groups on the fibers are the same, so for any point $s$ in $S$ there exists $n$ big enough such that (1) is satisfied, and by Corollary \ref{ampleness} the same holds for (2) and (3).
\end{proof}
\begin{rmk}\label{wang}
As in \cite[Remark 4.1.7]{W} for a scheme $S$ and a pair $(p:C\rightarrow S,\mathcal E)\in\overline{\mathcal U_n}(S)$, the direct image $p_*(\mathcal E(n))$ is locally free of rank $d+r(2n-1)(g-1)$. By cohomological flatness, locally on $S$ the morphism in the Grassmannian becomes $
C\hookrightarrow Gr(V_n,r)\times S
$, where $V_n$ is a $k$-vector space of dimension $P(n):=d+r(2n-1)(g-1)$.
\end{rmk}
We are now going to obtain a presentation of $\overline{\mt U}_n$ as a quotient stack. Consider the Hilbert scheme of closed subschemes on the Grassmannian $Gr(V_n,r)$
$$
Hilb_n:=Hilb_{Gr(V_n,r)}^{\oo_{Gr(V_n,r)}(1), Q(m)}
$$
with Hilbert polinomial $Q(m)=m(d+nr(2g-2))+1-g$ relative to the Plucker line bundle $\oo_{Gr(V_n,r)}(1)$. Let $\mathscr{C}_{(n)}\hookrightarrow Gr(V_n,r)\times Hilb_n$ be the universal curve. The Grassmannian is equipped with a universal quotient $V_n\times \oo_{Gr(V_n,r)}\to \mt E$, where $\mt E$ is the universal vector bundle. If we pull-back this morphism on the product $Gr(V_n,r)\times Hilb_n$ and we restrict to the universal curve, we obtain a surjective morphism of vector bundles $q:V_n\otimes\oo_{\mathscr {C}_{(n)}}\rightarrow \mathscr E_{(n)}$. We will call $\mathscr E_{(n)}$ (resp. $q:V_n\otimes\oo_{\mathscr {C}_{(n)}}\rightarrow \mathscr E_{(n)}$) the \emph{universal vector bundle (resp. universal quotient) on $\mathscr {C}_{(n)}$}. \label{Hn}Let $H_n$ be the open subset of $Hilb_n$ consisting of points $h$ such that:
\begin{enumerate}
\item $\mathscr {C}_{(n)h}$ is semistable,
\item $\mathscr {E}_{(n)h}$ is properly balanced,
\item $H^i(\mathscr {C}_{(n)h},\mathscr {E}_{(n)h})=0$ for $i>0$,
\item $H^0(q_h)$ is an isomorphism.
\end{enumerate}
The restriction of the universal curve and of the universal vector bundle on $H_n$ defines a morphism of stacks $\Theta: H_n\rightarrow \overline{\mathcal U}_n$. Moreover, the Hilbert scheme $Hilb_n$ is equipped with a natural action of $GL(V_n)$ and $H_n$ is stable for this action.
\begin{prop}The morphism of stacks
$$
\Theta: H_n\rightarrow \overline{\mathcal U}_n
$$
is a $GL(V_n)$-bundle (in the sense of \cite[2.1.4]{W}).
\end{prop}
\begin{proof}We set $GL:=GL(V_n)$. First we prove that $\Theta$ is $GL$-invariant, i.e.
\begin{enumerate}
\item the diagram
$$
\xymatrix{
H_n\times GL\ar[r]^m\ar[d]^{pr_{1}} & H_n\ar[d]^{\Theta}\\
H_n\ar[r]^{\Theta} & \overline{\mathcal U}_n
}
$$
where $pr_1$ is the projection on $H_n$ and $m$ is the multiplication map is commutative. Equivalently, there exists a natural transformation $\rho:pr_1^*\Theta\rightarrow m^*\Theta$.
\item $\rho$ satifies an associativity condition (see \cite[2.1.4]{W}).
\end{enumerate}
In our case $\rho$ is the identity and it is easy to see that the second condition holds. We will fix a pair $(p:C\rightarrow S,\mathcal E)\in\overline{\mathcal U}_n(S)$ and let $f:S\rightarrow\overline{\mathcal U}_n$ be the associated morphism. It remains to prove that morphism $f^*\Theta$ is a principal $GL$-bundle. More precisely, we will prove that there exists a $GL$-equivariant isomorphism over $S$
$$H_n\times_{\overline{\mathcal U}_n}S\cong Isom( V_n\otimes\oo_S,p_{S*}\mathcal E(n)).$$
For any $S$-scheme $T$, a $T$-valued point of $H_n\times_{\overline{\mathcal U}_n}S$ corresponds to the following data:
\begin{enumerate}[(1)]
\item a morphism $T\rightarrow H_n$,
\item a $T$-isomorphism of schemes $\psi:C_T\cong \mathscr {C}_{(n)T}$,
\item an isomorphism of vector bundles $\psi^*\mathscr {E}_{(n)T}\cong\mathcal E_T (n)$.
\end{enumerate}
Consider the pull-back of the universal quotient of $H_n$ through $T\rightarrow H_n$
$$
q_T:V_n\otimes\oo_{\mathscr {C}_{(n)T}}\rightarrow \mathscr {E}_{(n)T}.
$$
If we pull-back by $\psi$ and compose with the isomorphism of (3), we obtain a surjective morphism
$$
V_n\otimes\oo_{C_T}\rightarrow \mathcal E_T(n).
$$
We claim that the push-forward $V_n\otimes \oo_T\rightarrow p_{T*}(\mathcal E_T(n))\cong p_*(\mathcal E(n))_T$ is an isomorphism, or in other words it defines a $T$-valued point of $Isom( V_n\otimes \oo_S,p_{S*}\mathcal E(n))$. As explained in Remark \ref{wang}, the sheaf $p_{T*}(\mathcal E (n)_T)$ is a vector bundle of rank $P(n)$, so it is enough to prove the surjectivity. We can suppose that $T$ is noetherian and by Nakayama lemma it suffices to prove the surjectivity on the fibers. On a fiber the morphism is
$$
V_n\otimes\oo_{C_t}\rightarrow H^0(\mathscr {E}_{(n)t})\cong H^0(\mathcal E_t(n))
$$
which is an isomorphism by the definition of $H_n$.\\
Conversely, let $T$ be a scheme and $V_n\otimes\oo_T\rightarrow p_*(\mathcal E(n))_T$ a $T$-isomorphism of vector bundles.
By hypothesis, $\mathcal E_T (n)$ is relatively generated by global sections and the induced morphism in the Grassmannian is a closed embedding. Putting everything together, we obtain a surjective map
$$
V_n\otimes\oo_{C_T}\cong p_T^*p_{T*}\mathcal E_T(n)\rightarrow \mathcal E_T(n)
$$
and a closed embedding $C_T\hookrightarrow Gr(V_n,r)\times T$ which defines a morphism $T\rightarrow H_n$. If we set $\psi$ equal to the identity $C_T=\mathscr {C}_{(n)T}$, we have a unique isomorphism of vector bundles $\psi^*\mathscr {E}_{(n)T}\cong\mathcal E_T(n)$. Then we have obtained a $T$-valued point of $H_n\times_{\overline{\mathcal U}_n} S$. The two constructions above are inverses of each other, concluding the proof.
\end{proof}
\begin{prop}\label{qts}The map $\Theta: H_n\rightarrow \overline{\mathcal U}_n$ gives an isomorphism of stacks
$$
\overline{\mathcal U}_n\cong [H_n/GL(V_n)]
$$
\end{prop}
\begin{proof}This follows from \cite[Lemma 2.1.1.]{W}.
\end{proof}
From the above presentation of $\overline{\mt U}_n$ as a quotient stack, we can now prove the smoothness of $\CVc$ and compute its dimension. This will conclude the proof of Theorem \ref{teo1} except for the irreducibility of $\CVc$ which will be proved in Lemma \ref{finteo1}.
\begin{cor}The scheme $H_n$ and the stack $\CVc$ are smooth of dimension respectively $P(n)^2+(r^2+3)(g-1)$ and $(r^2+3)(g-1)$.
\end{cor}
\begin{proof}
We set $Gr:=Gr(V_n,r)$. Arguing as in \cite[Proposition 3.1.3.]{Sc}, we see that for any $k$-point $h:=[C\hookrightarrow Gr]\in H_n$ the co-normal sheaf $\mathcal I_C/\mathcal I^2_C$ is locally free and we have an exact sequence:
$$
0\lra\mathcal I_C/\mathcal I^2_C\lra\Omega^1_{Gr}|_C\lra\Omega^1_{C}\lra 0.
$$
Applying the functor $\text{Hom}_{\oo_C}(-,\oo_C)$, we obtain the following exact sequence of vector spaces
$$
0\longrightarrow \text{Hom}_{\oo_C}(\Omega^1_C,\oo_C)\longrightarrow H^0(C,T_{Gr}|_C)\longrightarrow \text{Hom}_{\oo_C}(\mathcal I_C/\mathcal I^2_C,\oo_C)\longrightarrow \text{Ext}^1_{\oo_C}(\Omega^1_C,\oo_C)\longrightarrow 0
$$
Now $\text{Hom}_{\oo_C}(\mathcal I_C/\mathcal I^2_C,\oo_C)$ is the tangent space of $H_n$ at $h$. We can prove that its dimension is $P(n)^2+(r^2+3)(g-1)$ by using the sequence above as in the proof of \emph{loc. cit.} This implies that $H_n$ is smooth of dimension $P(n)^2+(r^2+3)(g-1)$. The assertion for the stack $\CVc$ follows immediately from Proposition \ref{qts}.
\end{proof}
\subsection{The Schmitt compactification $\CU$.}\label{Sc}
In this section we will resume how Schmitt in \cite{Sc}, generalizing a result of Nagaraj-Seshadri in \cite{NS}, constructs via GIT an irreducible projective variety, which is a good moduli space (for the definition see Appendix \ref{App}) for an open substack of $\CVc$.\vspace{0.2cm}
First we recall the Seshadri's definition of slope-(semi)stable sheaf for a stable curve in the case of the canonical polarization.
\begin{defin}
Let $C$ be a stable curve and let $C_1,\ldots,C_s$ be its irreducible components. We will say that a sheaf $\mt E$ is \emph{P-(semi)stable} if it is torsion free of uniform rank $r$ and for any subsheaf $\mt F$ we have
$$
\frac{\chi(\mt F)}{\sum s_i\omega_{C_i}}\underset{(\leq)}<\frac{\chi(\mt E)}{r\omega_C}
$$
where $s_i$ is the rank of $\mt F$ at $C_i$.
A P-semistable sheaf has a Jordan-Holder filtration with P-stable factors. Two P-semistable sheaves are \emph{equivalent} if they have the same Jordan-Holder factors. Two equivalence classes are said to be \emph{aut-equivalent} if they differ by an automorphism of the curve.
\end{defin}
Consider the stack $\mt TF_{r,d,g}$ of torsion free sheaves of uniform rank $r$ and Euler characteristic $d+r(1-g)$ on stable curves of genus $g$. Pandharipande has proved in \cite{P96} that exists an open substack $\mt TF^{ss}_{r,d,g}$ which admits a projective irreducible variety as good moduli space. More precisely, this variety is a coarse moduli space for the aut-equivalence classes of P-semistable sheaves over stables curves (see \cite[Theorem 9.1.1]{P96}). This is the reason why we prefer the "P" instead of "slope" in the definition above.
Consider the open substack $\CVc^{P(s)s}\subset\CVc$ of pairs $(C,\mt E)$ such that the sheaf $\pi_*\mt E$ over the stabilized curve $C^{st}$ is P-(semi)stable. Sometimes we will simply say that the pair $(C^{st},\pi_*\mt E)$ is P-(semi)stable. As we will see in the next proposition, the set of such pairs is bounded.
\begin{prop}\label{qc}The stack $\CVc^{Pss}$ is quasi-compact.
\end{prop}
\begin{proof}By construction, it is sufficient to prove that there exists $n$ big enough such that $\CVc^{Pss}\subset \overline{\mt U}_n$. It is enough showing that there exists $n$ big enough such that $\mt E(n)$ satisfies the conditions of Proposition \ref{amplenesso}, for any $k$-point $(C,\mt E)$ in $\CVc^{Pss}$.\\
Consider the set $\{(C,\mt E)\}$ of $k$-points in $\CVc^{Pss}$. Denote with $\pi:C\rightarrow C^{st}$ the stabilization morphism. Let $C$ be a semistable curve, let $R\subset C$ be a subcurve obtained as union of some maximal chains. We set, as usual, $\widetilde C=R^c$ and $D:=|\widetilde C\cap R|$. Consider the exact sequence
$$
0\lra\mt I_{D^{st}}(\pi_*\mt E)\lra \pi_*\mt E\lra (\pi_*\mt E)_{D^{st}}\lra 0,
$$
Observe that the cokernel is a torsion sheaf. By construction $\chi((\pi_*\mt E)_{D^{st}})=h^0((\pi_*\mt E)_{D^{st}})\leq 2rN$, where $N$ is the number of nodes on $C^{st}$. A stable curve of genus $g$ can have at most $3g-3$ nodes. By \cite{P96}, the set of P-semistable torsion free sheaves with $\chi=d+r(1-g)$ on stable curves of genus $g$ is bounded. This allows us, using the theory of relative Quot schemes, to construct a quasi-compact scheme which is the fine moduli space for the pairs $(X,q:\mt P\to \mt F)$ where $X$ is a stable curve of genus $g$, $q$ is a surjective morphism of sheaves on $X$, $P$ is a P-semistable torsion free and $\mt F$ is a sheaf with constant Hilbert polynomial less or equal than $6r(g-1)$. In particular, up to twisting by a suitable power of the canonical bundle, we can assume that the sheaf $\mt I_{D^{st}}(\pi_*\mt E)$ is generated by global sections and that $H^1(C^{st},\mt I_{D^{st}}(\pi_*\mt E))=0$ for any $k$-point $(C,\mt E)$ in $\CVc^{Pss}$ and any collection $R$ of maximal chains in $C_{exc}$. By Corollary \ref{NSrmk}(ii), we have
$$\mt I_{D^{st}}(\pi_*\mt E)\cong \pi_*(\mt I_{D}\mt E_{\widetilde C})=\pi_*(\mt I_R\mt E).$$
Observe that $H^i(C^{st}, \pi_*(\mt I_{D}\mt E_{\widetilde C}))=H^i(\widetilde C,\mt I_{D}\mt E_{\widetilde C})$ for $i=0,1$. In particular $\mt E$ satisfies the condition (i) of Proposition \ref{amplenesso}. Suppose that $R$ is a maximal chain, $D=\{p,q\}$ and $D^{st}=x$. So, the fact that $H^0(C^{st},\pi_*\left(\mt I_{D}\mt E_{\widetilde C}\right))$ generates $\pi_*\left(\mt I_{D}\mt E_{\widetilde C}\right)$ implies that
$$H^0(\widetilde C,\mt I_{p,q}\mt E_{\widetilde C})\to \pi_*\left(\mt I_{p,q}\mt E_{\widetilde C}\right)_{\{x\}}=\left(\mt I_{p,q}\mt E_{\widetilde C}\right)_{\{p\}}\oplus\left(\mt I_{p,q}\mt E_{\widetilde C}\right)_{\{q\}}$$
is surjective. In other words $\mt E$ satisfies the condition (ii) of \emph{loc. cit.}\\
For the rest of the proof $R$ will be the exceptional curve $C_{exc}$. Set $\mt G:=\mt I_{D}\mt E_{\widetilde C}$. Let $p$ and $q$ (not necessarily distinct) points on $C\backslash R=\widetilde C\backslash D$. Consider the exact sequence of sheaves on $\widetilde C$.
$$
0\lra \mt I_{p,q}\mt G\lra\mt G\lra \mt G/\mt I_{p,q}\mt G\lra 0,
$$
where, when $p=q$, we denote with $\mt I_{p,p}\mt G$ the sheaf $\mt I^2_{p}\mt G$. If we show that $H^1(\widetilde C,\mt I_{p,q}\mt G)=H^1(C^{st},\pi_*(\mt I_{p,q}\mt G))$ is zero for any $k$-point $(C,\mt E)$ in $\CVc^{Pss}$ then the conditions (iii) and (iv) are satisfied for any pair in $\CVc^{Pss}(k)$. We have already shown that the pairs $(C^{st},\pi_*\mt G)$ are bounded. As before the sheaf $G/\mt I_{p,q}$ is torsion and its Euler characteristic is $2r$ (not depend from the choice of $p$ and $q$). Arguing as above, we can conclude that $H^1(C^{st},\pi_*(\mt I_{p,q}\mt G))=0$.
\end{proof}
Schmitt proves in \cite{Sc} that there exists an open substack $\CVc^{Hss}\subset\CVc^{Pss}$ which admits a good moduli space $\overline{U}_{r,d,g}$. We recall briefly the conctruction of such space following \cite[pp. 174-175]{Sc}.\\
Gieseker has shown in \cite{Gie} that the coarse moduli space of stable curves $\overline{M}_g$ can be constructed via GIT. More precisely $\overline M_g\cong H_g\sslash_{\mathcal L_{H_g}}SL(W)$, where $H_g$ is the Hilbert scheme of stable curves embedded with $\omega^{10}$ in $\mathbb P(W)=\mathbb P^{10(2g-2)-g}$, while $\mt L_{H_g}$ is a suitable $SL(W)$-linearized ample line bundle on $H_g$.
Let $C_g\rightarrow H_g$ be the universal curve. Consider the relative Quot scheme
$$
\rho:Q:=Quot(C_g/H_g,V_n\otimes\oo_{C_g},\omega^{10}_{C_g/H_g})\rightarrow H_g.
$$
We have a natural action of $SL(V_n)\times SL(W)$, linearized with respect to a suitable $\rho$-ample line bundle $\mathcal L_{Q}$. With an abuse of notation, we will denote again with $Q$ the open (and closed) subscheme of $Q$ consisting of sheaves with Euler characteristic equal to $P(n)=\dim V_n$ and uniform rank $r$. We set $\mathcal L_a:=\mathcal L_{Q}\otimes\rho^*\mathcal L_{H_g}^a$. For $a\gg 0$ the GIT-quotient $\overline{ Q}:=Q\sslash_{\mathcal L_a}SL(W)$ exists and it is the coarse moduli space for the functor which sends a scheme $S$ to the set of isomorphism classes of pairs $(C_S\rightarrow S,q_S: V_n\otimes\oo_{C_S}\rightarrow \mathcal E)$ where $C_S\rightarrow S$ is a family of stable curve and $q_S$ is a surjective morphism of $S$-flat sheaves with $\chi(E_s)=P(n)$ and uniform rank $r$. Moreover $\overline{Q}$ is equipped with a $SL(V_n)$-linearized line bundle $\mathcal L_{\overline Q}$.\\
Consider now the scheme $H_n$ defined at page \pageref{Hn}. It has a natural $SL(V_n)$-linearized line bundle $\mathcal L_{Hilb}$, the semistable points for this linearized action are called \emph{Hilbert semistable} points and their description is an open problem (see \cite{T98} for some partial results in this direction). Let $$(\mathscr C_{(n)},q:V_n\otimes\oo_{\mathscr C_{(n)}}\rightarrow\mathscr E_{(n)})$$
be the universal pair on $H_n$. Consider the stabilized curve $\pi:\mathscr C_{(n)}\rightarrow\mathscr C_{(n)}^{st}$.
The push-forward $\pi_*(q)$ (as in \cite[p. 180]{Sc}) defines a morphism $H_n\to\overline Q$. The closure of the graph
$
\overline\Gamma\hookrightarrow H_n\times \overline Q
$
gives us a $SL(V_n)$-linearized ample line bundle $\mathcal L_{Hilb}^m\boxtimes\mathcal L_{\overline Q}^a$. For $a\gg 0$, Schmitt has proved that the semistable points are contained in the graph (see \cite[Theorem 2.1.2]{Sc}). Therefore, we can view such semistable points inside $H_n$ and call them \emph{H-semistable}.
\begin{rmk}\label{defhsem}
An H-semistable point has the following properties (see \cite[Def. 2.2.10]{Sc}): let, as usual, $\pi:C\rightarrow C^{st}$ be the stabilization morphism and $(C,\mt E)$ is a pair in $\overline{\mathcal U}_n$.
\begin{enumerate}[(i)]
\item Suppose that $C$ is smooth. Then $(C,\mt E)$ is $H$-(semi)stable if and only if $(C,\mt E)$ is $P$-(semi)stable. In this case we will say just $(C,\mt E)$ is \emph{(semi)-stable}.
\item We have the following chain of implications:\\
$(C^{st},\pi_*\mt E)$ P-stable $\Rightarrow$ $(C,\mt E)$ H-stable $\Rightarrow$ $(C,\mt E)$ H-semistable $\Rightarrow$ $(C^{st},\pi_*\mt E)$ P-semistable.
\item Suppose that $(C^{st},\pi_*E)$ is strictly P-semistable. Then $(C,\mt E)$ is H-semistable if and only if for every one-parameter subgroup $\lambda$ of $SL(V_n)$ such that $(C^{st},\pi_*\mathcal E)$ is strictly P-semistable with respect to $\lambda$ then $(C,\mt E)$ is Hilbert-semistable with respect to $\lambda$.
\end{enumerate}
\end{rmk}
A priori the H-semistability is a property of points in $H_n$, i.e. $\left[C\hookrightarrow Gr(V_n,r)\right]$. However it is easy to see that it depends only on the curve and the restriction of universal bundle to the curve.\\
In his construction Schmitt just requires that a vector bundle must be admissible, but not necessarily balanced. The next lemma proves that the vector bundles appearing in his construction are indeed also properly balanced.
\begin{lem}If $(C^{st},\pi_*\mt E)$ is P-semistable then $\mt E$ is properly balanced.
\end{lem}
\begin{proof}By considerations above, we must prove that $\mt E$ is balanced. By Remark \ref{balcon}(iv), we have to prove that for any connected subcurve $Z\subset C$ such that $Z^c$ is connected, we have
$$
\frac{\chi(\mt F)}{\omega_Z}\leq \frac{\chi(\mt E)}{\omega_C},
$$
where $\mt F$ is the subsheaf of $\mt E_Z$ of sections that vanishes on $Z\cap Z^c$. Observe that $\mt F$ is also a subsheaf of $\mt E$. The hypothesis and the fact that the push-forward is left exact imply
$$
\frac{\chi(\pi_*\mt F)}{\omega_{Z^{st}}}\leq \frac{\chi(\pi_*\mt E)}{\omega_{C^{st}}}=\frac{\chi(\mt E)}{\omega_C},
$$
where $Z^{st}$ is the reduced subcurve $\pi(Z)$. It is clear that $\omega_{Z^{st}}=\omega_Z$. We have an exact sequence of vector spaces
$$
0\lra H^1\left(Z^{st},\pi_*\mt F\right)\lra H^1\left(Z,\mt F\right)\lra H^0(Z^{st},R^1\pi_*\mt F)\lra 0.
$$
This implies $\chi(\mt F)\leq\chi(\pi_*\mt F)$, concluding the proof.
\end{proof}
\subsection{Properties and the rigidified moduli stack $\CVr$.}\label{prop}
The stack $\CVc$ admits a \emph{universal curve} $\overline{\pi}: \overline{\mt Vec}_{r,d,g,1}\rightarrow \CVc$, i.e. a stack $\overline{\mt Vec}_{r,d,g,1}$ and a representable morphism $\overline{\pi}$ with the property that for any morphism from a scheme $S$ to $\CVc$ associated to a pair $(C\rightarrow S,\mt E)$ there exists a morphism $C\rightarrow \overline{\mt Vec}_{r,d,g,1}$ such that the diagram
$$
\xymatrix{
C\ar[r]\ar[d] &\overline{\mt Vec}_{r,d,g,1}\ar[d]^{\overline{\pi}}\\
S\ar[r] & \CVc}
$$
is cartesian. Furthermore, the universal curve admits a \emph{universal vector bundle}, i.e. for any morphism from a scheme $S$ to $\CVc$ associated to a pair $(C\rightarrow S,\mt E)$, we associate the vector bundle $\mt E$ on $C$. This allows us to define a coherent sheaf for the site lisse-\'etale on $\overline{\mt Vec}_{r,d,g,1}$ flat over $\CVc$.
The stabilization morphism induces a morphism of stacks
$$\overline{\phi}_{r,d}:\CVc\lra \CMg$$
which forgets the vector bundle and sends the curve in its stabilization. We will denote with $\Vc$ (resp. $\mt U_n$) the open substack of $\CVc$ (resp. $\overline{\mt U}_n)$ of pairs $(C,\mt E)$ where $C$ is a smooth curve. In the next sections we will often need the restriction of $\overline{\phi}_{r,d}$ to the open locus of smooth curves $$\phi_{r,d}:\Vc\lra \Mg.$$
The group $\mathbb G_m$ is contained in a natural way in the automorphism group of any object of $\CVc$, as multiplication by scalars on the vector bundle. There exists a procedure for removing these automorphisms, called \emph{$\mathbb{G}_m$-rigidification} (see \cite[Section 5]{ACV}). We obtain an irreducible smooth Artin stack $\CVr:=\CVc\fatslash \mathbb{G}_m$ of dimension $(r^2+3)(g-1)+1$, with a surjective smooth morphism $\nu_{r,,d}:\CVc\rightarrow\CVr$. The forgetful morphism $\overline{\phi}_{r,d}:\CVc\lra \CMg$ factorizes through the forgetful morphism $\overline{\phi}_{r,d}:\CVr\rightarrow\CMg$. Over the locus of smooth curves we have the following diagram
$$
\xymatrix{
\Vc\ar[d]^{det}\ar[rr]^{\nu_{r,d}} & & \Vr\ar[d]^{\widetilde{det}}\\
\Jc\ar[rd]\ar[rr]^{^{\nu_{1,d}}} & & \Jr\ar[ld]\\
& \Mg
}
$$
where $det$ (resp. $\widetilde{det}$) is the determinant morphism, which send an object $(C\rightarrow S,\mt E)\in\Vc(S)$ (resp. $\in\Vr(S)$) to $(C\rightarrow S,\det\mt E)\in\Jc(S)$ (resp. $\in\Jr(S)$). Observe that the obvious extension on $\CVc$ of the determinant morphism does not map to the compactified universal Jacobian $\CJc$, because the basic inequalities for $\CJc$ are more restrictive.
\subsection{Local structure.}\label{locstr}
The local structure of the stack $\CVc$ is governed by the deformation theory of pairs $(C,\mt E)$, where $C$ is a semistable curve and $\mt E$ is a properly balanced vector bundle. Therefore we are going to review the necessary facts. First of all, the deformation functor $\text{Def}_C$ of a semistable curve $C$ is smooth (see \cite[Proposition 2.2.10(i), Proposition 2.4.8]{Ser06}) and it admits a miniversal deformation ring (see \cite[Theorem 2.4.1]{Ser06}), i.e. there exists a formally smooth morphism of functors of local Artin $k$-algebras
$$
\text{Spf }k\llbracket x_1,\ldots,x_{N}\rrbracket\rightarrow \text{Def}_C, \text{ where } N:=\text{ext}^1(\Omega_C,\oo_C)
$$
inducing an isomorphism between the tangent spaces. Moreover, if $C$ is stable its deformation functor admits a universal deformation ring (see \cite[Corollary 2.6.4]{Ser06}), i.e. the morphism of functors above is an isomorphism. Let $x$ be a singular point of $C$ and $\hat{\oo}_{C,x}$ the completed local ring of $C$ at $x$. The deformation functor $\text{Def}_{\text{Spec}\hat{\oo}_{C,x}}$ admits a miniversal deformation ring $k\llbracket t\rrbracket$ (see \cite[pag. 81]{DM69}). Let $\Sigma$ be the set of singular points of $C$. The morphism of Artin functors
$$
loc:\text{Def}_C\rightarrow \prod_{x\in\Sigma}\text{Def}_{\text{Spec}\hat{\oo}_{C,x}}
$$
is formally smooth (see \cite[Proposition 1.5]{DM69}). For a vector bundle $\mt E$ over $C$, we will denote with $\text{Def}_{(C,\mt E)}$ the deformation functor of the pair (for a more precise definition see \cite[Def. 3.1]{CMKV}). As in \cite[Def. 3.4]{CMKV}, the automorphism group $\text{Aut}(C,\mt E)$ (resp. $\text{Aut}(C)$) acts on $\text{Def}_{(C,\mt E)}$ (resp. $\text{Def}_C$). Using the same argument of \cite[Lemma 5.2]{CMKV}, we can see that the multiplication by scalars on $\mt E$ acts trivially on $\text{Def}_{(C,\mt E)}$. By \cite[Theorem 8.5.3]{FAG}, the forgetful morphism
$$
\text{Def}_{(C,\mt E)}\rightarrow \text{Def}_C
$$
is formally smooth and the tangent space of $\text{Def}_{(C,\mt E)}$ has dimension $\text{ext}^1(\Omega_C,\oo_C)+\text{ext}^1(\mt E,\mt E)$.
Let $h:=[C\hookrightarrow Gr(V_n, r)]$ be a $k$-point of $H_n$. Let $\hat{\oo}_{H_n,h}$ be the completed local ring of $H_n$ at $h$. Clearly, the ring $\hat{\oo}_{H_n,h}$ is a universal deformation ring for the deformation functor $\text{Def}_{h}$ of the closed embedding $h$. Moreover
\begin{lem}\label{111}The natural morphism $$\text{Def}_h\rightarrow \text{Def}_{(C,\mt E)}$$ is formally smooth.
\end{lem}
\begin{proof}For any $k$-algebra $R$, we will set $Gr(V_n,r)_R:=Gr(V_n,r)\times_k \text{Spec}R$. We have to prove that given
\begin{enumerate}
\item a surjection $B\rightarrow A$ of Artin local $k$-algebras,
\item a deformation $h_A:=[C_A\hookrightarrow Gr(V_n,r)_A]$ of $h$ over $A$
\item a deformation $(C_B,\mt E_B)$ of $(C,\mt E)$ over $B$, which is a lifting of $(C_A,\mt E_A)$,
\end{enumerate}
then there exists an extension $h_B$ over $B$ of $h_A$ which maps on $(C_B,\mt E_B)$. Since by hypothesis $H^1(C,\mt E(n))=0$, we can show that the restriction map $res:H^0(C_B,\mt E_B(n))\rightarrow H^0(C_A,\mt E_A(n))$ is surjective. Now $h_A$ only depends on the vector bundle $\mt E_A$ and on the choiche of a basis for $H^0(C_A,\mt E_A(n))$. We can lift the basis, using the map $res$, to a basis $\mt B$ of $H^0(C_B,\mt E_B(n))$. The basis $\mt B$ induces a morphism
$
C_B\rightarrow Gr(V_n,r)_B
$
which is a lifting for $h_A$.
\end{proof}
The next lemma concludes the proof of Theorem \ref{teo1}.
\begin{prop}\label{finteo1}The stack $\CVc$ is irreducible.
\end{prop}
\begin{proof}Since the morphism $loc:\text{Def}_C\rightarrow \prod_{x\in\Sigma}\text{Def}_{\text{Spec}\hat{\oo}_{C,x}}$ is formally smooth, Lemma \ref{111} implies that the morphism $Def_h\rightarrow \prod_{x\in\Sigma}\text{Def}_{\text{Spec}\hat{\oo}_{C,x}}$ is formally smooth. In particular, any semistable curve with a properly balanced vector bundle can be deformed to a smooth curve with a vector bundle. In other words, the open substack $\Vc$ is dense in $\CVc$; hence $\CVc$ is irreducible if and only if $\Vc$ is irreducible. And this follows from the fact that $\Mg$ is irreducible and that the morphism $\Vc\to\Mg$ is open (because is flat and locally of finite presentation) with irreducible geometric fibers (by \cite[Corollary A.5]{Ho10}).
\end{proof}
We are now going to construct a miniversal deformation ring for $\text{Def}_{(C,\mt E)}$ by taking a slice of $H_n$.
\begin{lem}\label{locstrc} Let $h:=[C\hookrightarrow Gr(V_n,r)]$ a $k$-point of $H_n$ and let $\mt E$ be the restriction to $C$ of the universal vector bundle. Assume that $\text{Aut}(C,\mt E)$ is smooth and linearly reductive. Then the following hold.
\begin{enumerate}[(i)]
\item There exists a slice for $H_n$. More precisely, there exists a locally closed Aut$(C,\mt E)$-invariant subset $U$ of $H_n$, with $h\in U$, such that the natural morphism
$$
U\times_{Aut(C,\mt E)}GL(V_n)\rightarrow H_n
$$
is \'etale and affine and moreover the induced morphism of stacks
$$
[ U /Aut(C,\mt E)]\rightarrow \overline{\mt U}_n
$$
is affine and \'etale.
\item The completed local ring $\hat{\oo}_{U,h}$ of $U$ at $h$ is a miniversal deformation ring for $\text{Def}_{(C,\mt E)}$.
\end{enumerate}
\end{lem}
\begin{proof}The part (i) follows from \cite[Theorem 3]{Al10}.
We will prove the second one following the strategy of \cite[Lemma 6.4]{CMKV}. We will set $F\subset Def_h$ as the functor pro-represented by $\hat{\oo}_{U,h}$, $G:=GL(V_n)$ and $N:=\text{Aut}(C,\mt E)$. Since $\text{Def}_h\rightarrow \text{Def}_{(C,\mt E)}$ is formally smooth, it is enough to prove that the restriction to $F(A)$ of $\text{Def}_h(A)\rightarrow \text{Def}_{(C,\mt E)}(A)$ is surjective for any local Artin $k$-algebra $A$ and bijective when $A=k[\epsilon]$.
Let $\mathfrak{g}$ (resp. $\mathfrak{n}$) be the deformation functor pro-represented by the completed local ring of $G$ (resp. $N$) at the identity. There is a natural map $\mathfrak{g}/\mathfrak{n}\rightarrow\text{Def}_h$ given by the derivative of the orbit map. More precisely, for a local Artin $k$-algebra $A$:
$$
\begin{array}{ccc}
\mathfrak{g}/\mathfrak{n}(A) & \rightarrow &\text{Def}_h(A)\\
\left[g\right]&\mapsto & g.v^{triv}
\end{array}
$$
where $v^{triv}$ is the trivial deformation over Spec$A$. First of all we will construct a morphism $\text{Def}_h\rightarrow \mathfrak{g}/\mathfrak{n}$ such that the derivative of the orbit map defines a section. The construction is the following: up to \'etale base change, the morphism $U\times_N G\rightarrow H_n$ of part $(i)$, admits a section locally on $h$. The morphism, obtained composing this section with the morphism $U\times_NG\rightarrow G/N$, which sends a class $[(u,g]]$ to $[g]$, induces a morphism of Artin functors
$$
\text{Def}_h\rightarrow \mathfrak{g}/\mathfrak{h}
$$
with the desired property. By construction, if $A$ is a local Artin $k$-algebra then the inverse image of $0\in \mathfrak{g}/\mathfrak{n}(A)$ is $F(A)$. If $v\in \text{Def}_h(A)$ maps to some element $[g]\in\mathfrak{g}/\mathfrak{n}(A)$ then $g^{-1}v\in F(A)$. Because both $v$ and $g^{-1}v$ map to the same element of $\text{Def}_{(C,\mt E)}$, we can conclude that $F(A)\rightarrow \text{Def}_{(C,\mt E)}(A)$ is surjective.\\
It remains to prove the injectivity of $F(k[\epsilon])\rightarrow \text{Def}_{(C,\mt E)}(k[\epsilon])$. We consider the following complex of $k$-vector spaces
$$
0\rightarrow \mathfrak{g}/\mathfrak{n}\rightarrow Def_h(k[\epsilon])\rightarrow Def_{(C,\mt E)}(k[\epsilon])\rightarrow 0
$$
where the first map is the derivative of the orbit map. We claim that this is an exact sequence, which would prove the injectivity of $F(k[\epsilon])\rightarrow \text{Def}_{(C,\mt E)}(k[\epsilon])$ by the definition of $F$. The only non obvious thing to check is the exactness in the middle. Suppose that $h_{k[\epsilon]}\in \text{Def}_h(k[\epsilon])$ is trivial in $\text{Def}_{(C,\mt E)}(k[\epsilon])$, i.e. if $q_{\epsilon}:V_n\otimes\oo_{C_{\epsilon}}\rightarrow\mt E_{\epsilon}$ represents the embedding $h_{k[\epsilon]}$, then there exists an isomorphism with the trivial deformation on $k[\epsilon]$: $\varphi:C_{\epsilon}\cong C[\epsilon]$ and $\psi:\varphi_*\mt E_{\epsilon}\cong \mt E[\epsilon]$. Consider the morphism
$$
g_{\epsilon}:=\psi\circ\varphi_*q_{\epsilon}:V_n\otimes\oo_{C[\epsilon]}\rightarrow\mt E[\epsilon]
$$
which represents the same class $h_{k[\epsilon]}$. By definition of $H_n$, the push-forward of $g_{\epsilon}$ on $k[\epsilon]$ is an isomorphism
$$
V_n\otimes k[\epsilon]\rightarrow H^0(C,\mt E(n))\otimes k[\epsilon]
$$
and it defines uniquely the class $h_{k[\epsilon]}$. We can choose basis for $V_n$ and $H^0(C,\mt E(n))$ such that $g_{\epsilon}$ differs from the trivial deformation of $\text{Def}_{h}(k[\epsilon])$ by an invertible matrix $g\equiv Id$ mod $\epsilon$, which concludes the proof.
\end{proof}
\section{Preliminaries about line bundles on stacks.}\label{linechow}
\subsection{Picard group and Chow groups of a stack.}\label{boh}
We will recall the definitions and some properties of the Picard group and the Chow group of an Artin stack. Some parts contains overlaps with \cite[Section 2.9]{MV}. Let $\mathcal X$ be an Artin stack locally of finite type over $k$.
\begin{defin}\cite[p.64]{Mum65} A \emph{line bundle} $\mathcal L$ on $\mathcal X$ is the data consisting of a line bundle $\mathcal L(F_S)\in \text{Pic}(S)$ for every scheme $S$ and morphism $F_S:S\rightarrow\mathcal X$ such that:
\begin{itemize}
\item For any commutative diagram
$$
\xymatrix{
S \ar[rd]_{F_S}\ar[rr]^{f} & & T\ar[ld]^{F_T}\\
& \mathcal X &
}
$$
there is an isomorphism $\phi(f):\mathcal L(F_S)\cong f^*\mathcal L(F_T)$.
\item For any commutative diagram
$$
\xymatrix{
S \ar[rd]_{F_S}\ar[r]^{f} & T\ar[d]^{F_T}\ar[r]^g & Z\ar[ld]^{F_Z}\\
& \mathcal X &
}
$$
we have the following commutative diagram of isomorphisms
$$
\xymatrix{
\mathcal L(F_S)\ar[r]^{\phi(f)}\ar[d]^{\phi(g\circ f)} & f^*\mathcal L(F_T)\ar[d]^{f^*\phi(g)}\\
(g\circ f)^*\mathcal L(F_Z)\ar[r]^{\cong} & f^* g^*\mathcal L(F_Z)
}
$$
\end{itemize}
The abelian group of isomorphism classes of line bundles on $\mathcal X$ is called the \emph{Picard group} of $\mathcal X$ and is denoted by $\text{Pic}(\mathcal X)$.
\end{defin}
\begin{rmk}The definition above is equivalent to have a locally free sheaf of rank $1$ for the site lisse-\'etale (\cite[Proposition 1.1.1.4.]{Br1}).
\end{rmk}
If $\mathcal X$ is a quotient stack $[X/G]$, where $X$ is a scheme of finite type over $k$ and $G$ a group scheme of finite type over $k$, then $\text{Pic}(\mathcal X)\cong \text{Pic}(X)^G$ (see \cite[Chap. XIII, Corollary 2.20]{ACG11}), where $\text{Pic}(X)^G$ is the group of isomorphism classes of $G$-linearized line bundles on $X$.\vspace{0.2cm}
In \cite[Section 5.3]{EG98} (see also \cite[Definition 3.5]{Edi12}) Edidin and Graham introduce the operational Chow groups of an Artin stack $\mathcal X$, as generalization of the operational Chow groups of a scheme.
\begin{defin}A \emph{Chow cohomology class} $c$ on $\mathcal X$ is the data consisting of an element $c(F_S)$ in the operational Chow group $A^*(S)=\oplus A^i(S)$ for every scheme $S$ and morphism $F_S:S\rightarrow\mathcal X$ such that for any commutative diagram
$$
\xymatrix{
S \ar[rd]_{F_S}\ar[rr]^{f} & & T\ar[ld]^{F_T}\\
& \mathcal X &
}
$$
we have $c(F_S)\cong f^*c(F_T)$, with the obvious compatibility requirements. The abelian group consisting of all the $i$-th Chow cohomology classes on $\mathcal X$ together with the operation of sum is called the \emph{$i$-th Chow group} of $\mathcal X$ and is denoted by $A^i(\mathcal X)$.
\end{defin}
If $\mathcal X$ is a quotient stack $[X/G]$, where $X$ is a scheme of finite type over $k$ and $G$ a group scheme of finite type over $k$, then $A^i(\mathcal X)\cong A^i_G(X)$ (see \cite[Proposition 19]{EG98}), where $A^i_G(X)$ is the operational equivariant Chow group defined in \cite[Section 2.6]{EG98}. We have a homomorphism of groups $c_1:\text{Pic}(\mathcal X)\rightarrow A^1(\mathcal X)$ defined by the first Chern class.\vspace{0.2cm}
The next theorem resumes some results on the Picard group of a smooth stack, which will be useful for our purposes.
\begin{teo}\label{picchow}Let $\mt X$ be a (not necessarily quasi-compact) smooth Artin stack over $k$. Let $\mt U\subset\mt X$ be an open substack.
\begin{itemize}
\item[\textit{(i)}] The restriction map $\Pic(\mt X)\rightarrow \Pic(\mt U)$ is surjective.
\item[\textit{(ii)}] If $\mt X\backslash\mt U$ has codimension $\geq 2$ in $\mt X$, then $\Pic(\mt X)=\Pic(\mt U)$.
\end{itemize}
Suppose that $\mt X=[X/G]$ where $G$ is an algebraic group and $X$ is a smooth quasi-projective variety with a $G$-linearized action
\begin{itemize}
\item[\textit{(iii)}] The first Chern class map $c_1:\Pic(\mathcal X)\rightarrow A^1(\mathcal X)$ is an isomorphism.
\item[\textit{(iv)}] If $\mt X\backslash\mt U$ has codimension $1$ with irreducible components $\mt D_i$, then we have an exact sequence
$$
\bigoplus_i\mathbb{Z}\langle \oo_{\mathcal X}(\mt D_i)\rangle\lra \Pic(\mt X)\lra \Pic(\mt U)\lra 0
$$
\end{itemize}
\end{teo}
\begin{proof}The first two points are proved in \cite[Lemma 7.3]{BH12}.The third point follows from \cite[Corollary 1]{EG98}. The last one follows from \cite[Proposition 5]{EG98}
\end{proof}
\subsection{Determinant of cohomology and Deligne pairing.}\label{dcdp}
There exists two methods to produce line bundles on a stack parametrizing nodal curves with some extra-structure (as our stacks): the determinant of cohomology and the Deligne pairing. We will recall the main properties of these construction, following the presentation given in \cite[Chap. XIII, Sections 4 and 5]{ACG11} and the resume in \cite[Section 2.13]{MV}.\\
Let $p:C\rightarrow S$ be a family of nodal curves. Given a coherent sheaf $\mt F$ on $C$ flat over $S$, the \emph{determinant of cohomology} of $\mathcal F$ is a line bundle $d_{p}(\mt F)\in \Pic(S)$ defined as it follows. Locally on $S$ (by Proposition \ref{lasvolta}), there exists a complex of vector bundles $f:V_0\rightarrow V_1$ such that $\text{ker} f=p_{*}(\mt F)$ and $\text{coker} f=R^1p_{*}(\mt F)$ and then we set
$$
d_{p}(\mt F):=\det V_0\otimes (\det V_1)^{-1}.
$$
This definition does not depend on the choice of the complex $V_0\rightarrow V_1$; in particular this defines a line bundle globally on $S$. The proof of the next theorem can be found in \cite[Chap. XIII, Section 4]{ACG11}.
\begin{teo}\label{detcoh}Let $p:C\rightarrow S$ be a family of nodal curves and let $\mathcal F$ be a coherent sheaf on $C$ flat on $S$.
\begin{enumerate}[(i)]
\item The first Chern class of $d_p(\mt F)$ is equal to
$$
c_1(d_{p}(\mt F))=c_1(p_!(\mt F)):=c_1(p_*(\mt F))-c_1(R^1p_*(\mt F)).
$$
\item Given a cartesian diagram
$$
\xymatrix{
C\times_S T\ar[r]^g\ar[d]_q & C\ar[d]^p\\
T\ar[r]^f & S
}
$$
we have a canonical isomorphism
$$
f^*d_p(\mt F)\cong d_q(g^*\mt F).
$$
\end{enumerate}
\end{teo}
Given two line bundles $\mt M$ and $\mt L$ over a family of nodal curves $p:C\rightarrow S$, the \emph{Deligne pairing} of $\mt M$ and $\mt L$ is a line bundle $\langle\mt M,\mt L\rangle_p\in Pic(S)$ which can be defined as
$$
\langle\mt M,\mt L\rangle_p:=d_p(\mt M\otimes\mt L)\otimes d_p(\mt M)^{-1}\otimes d_p(\mt L)^{-1}\otimes d_p(\oo_{C}).
$$
The proof of the next theorem can be found in \cite[Chap. XIII, Section 5]{ACG11}.
\begin{teo}\label{delpair}Let $p:C\rightarrow S$ be a family of nodal curves.
\begin{enumerate}[(i)]
\item The first Chern class of $\langle\mt M,\mt L\rangle_p$ is equal to
$$
c_1(\langle\mt M,\mt L\rangle_p)=p_*(c_1(\mt M)\cdot c_1(\mt L)).
$$
\item Given a Cartesian diagram
$$
\xymatrix{
C\times_S T\ar[r]^g\ar[d]_q & C\ar[d]^p\\
T\ar[r]^f & S
}
$$
we have a canonical isomorphism
$$
f^*\langle\mt M,\mt L\rangle_p\cong \langle g^*\mt M,g^*\mt L\rangle_q
$$
\end{enumerate}
\end{teo}
\begin{rmk}\label{rmk2}By the functoriality of the determinant of cohomology and of the Deligne pairing, we can extend their definitions to the case when we have a representable, proper and flat morphism of Artin stacks such that the geometric fibers are nodal curves.
\end{rmk}
\subsection{Picard group of $\CMg$.}\label{cmg}
The universal family $\overline{\pi}:\overline{\mt M}_{g,1}\rightarrow \CMg$ is a representable, proper, flat morphism with stable curves as geometric fibers. In particular we can define the relative dualizing sheaf $\omega_{\overline{\pi}}$ on $\overline{\mathcal M}_{g,1}$ and taking the determinant of cohomology $d_{\overline{\pi}}(\omega^n_{\overline{\pi}})$ we obtain line bundles on $\CMg$. The line bundle $\Lambda:=d_{\overline{\pi}}(\omega_{\overline{\pi}})$ is called the \emph{Hodge line bundle}.
Let $C$ be a stable curve and for every node $x$ of $C$, consider the partial normalization $C'$ at $x$. If $C'$ is connected then we say $x$ node of type $0$, if $C'$ is the union of two connected curves of genus $i$ and $g-i$, with $i\leq g-i$ (for some $i$), then we say that $x$ is a node of type $i$.
The boundary $\CMg\slash \Mg$ decomposes as union of irreducible divisors $\delta_i$ for $i=0,\ldots,\lfloor g/2\rfloor$, where $\delta_i$ parametrizes (as stack) the stable curves with a node of type $i$. The generic point of $\delta_0$ is an irreducible curve of genus $g$ with exactly one node, the generic point of $\delta_i$ for $i=1,\ldots,\lfloor g/2\rfloor$ is a stable curve formed by two irreducible smooth curves of genus $i$ and $g-i$ meeting in exactly one point. We set $\delta:=\sum\delta_i$. By Theorem \ref{picchow} we can associate to any $\delta_i$ a unique (up to isomorphism) line bundle $\oo(\delta_i)$. We set $\oo(\delta)=\bigotimes_i\oo(\delta_i)$.
The proof of the next results for $g\geq 3$ can be found in \cite[Theorem. 1]{AC87} based upon a result of \cite{Har83}. If $g=2$ see \cite{Vis98} for $\Pic(\mt M_2)$ and \cite[Proposition 1]{Cor07} for $\Pic(\overline{\mt M}_2)$.
\begin{teo}\label{picmg}Assume $g\geq 2$. Then
\begin{enumerate}[(i)]
\item $\Pic(\Mg)$ is freely generated by the Hodge line bundle, except for $g=2$ in which case we add the relation $\Lambda^{10}=\oo_{\mt M_2}$.
\item $\Pic(\CMg)$ is freely generated by the Hodge line bundle and the boundary divisors, except for $g=2$ in which case we add the relation $\Lambda^{10}=\oo\left(\delta_0+2\delta_1\right)$.
\end{enumerate}
\end{teo}
\subsection{Picard Group of $\Jc$.}\label{jc}
The universal family $\pi:\mt Jac_{d,g,1}\rightarrow \Jc$ is a representable, proper, flat morphism with smooth curves as geometric fibers. In particular, we can define the relative dualizing sheaf $\omega_{\pi}$ and the universal line bundle $\mt L$ on $\mt Jac_{d,g,1}$. Taking the determinant of cohomology $\Lambda(n,m):=d_{\pi}(\omega^n_{\pi}\otimes\mt L^m)$, we obtain several line bundles on $\Jc$.
The proof of next theorem can be found in \cite[Theorem A(i) and Notation 1.5]{MV}, based upon a result of \cite{Kou91}.
\begin{teo}\label{picjac}Assume $g\geq 2$. Then $\Pic(\Jc)$ is freely generated by $\Lambda(1,0)$, $\Lambda(1,1)$ and $\Lambda(0,1)$, except in the case $g=2$ in which case we add the relation $\Lambda(1,0)^{10}=\oo_{\Jc}$.
\end{teo}
\subsection{Picard Groups of the fibers.}\label{fibre}
Fix now a smooth curve $C$ with a line bundle $\mathcal L$. Let $\mathcal Vec_{=\mathcal L,C}$ be the stack whose objects over a scheme $S$ are the pairs $(\mt E,\varphi)$ where $\mt E$ is a vector bundle of rank $r$ on $C\times S$ and $\varphi$ is an isomorphism between the line bundles $\det\mt E$ and $\mt L\boxtimes \oo_S$. A morphism between two objects over $S$ is an isomorphism of vector bundles compatible with the isomorphism of determinants. $\mathcal Vec_{=\mathcal L,C}$ is a smooth Artin stack of dimension $(r^2-1)(g-1)$. We denote with $\mt Vec_{=\mt L,C}^{(s)s}$ the open substack of (semi)stable vector bundles. Since the set of isomorphism classes of semistable vector bundles on $C$ is bounded, the stack $\mt Vec_{=\mt L,C}^{ss}$ is quasi-compact. Consider the set of equivalence classes (defined as in Section \ref{Sc}) of semistable vector bundles over the curve $C$ with determinant isomorphic to $\mt L$. There exists a normal projective variety $U_{\mt L,C}$ which is a coarse moduli space for this set. Observe the stack $\mathcal Vec_{=\mathcal L,C}$ is the fiber of the determinant morphism $det:\Vc\to\Jc$ with respect to the $k$-point $(C,\mt L)$.
\begin{teo}\label{fibers}Let $C$ be a smooth curve with a line bundle $\mathcal L$. Let $\mathcal E$ be the universal vector bundle over $\pi:\mathcal Vec_{=\mt L,C}\times C\to\mathcal Vec_{=\mt L,C}$ of rank $r$ and degree $d$. Then:
\begin{enumerate}[(i)]
\item We have natural isomorphisms induced by the restriction
$$\langle(d_{\pi}(\mathcal E)\rangle\cong Pic(\mathcal Vec_{=\mathcal L,C})\cong Pic(\mathcal Vec^{ss}_{=\mathcal L,C}).$$
\item $U_{\mathcal L,C}$ is a good moduli space for $\mathcal Vec^{ss}_{=\mathcal L,C}$.
\item The good moduli morphism $\mt Vec^{ss}_{=\mt L,C}\rightarrow U_{\mt L, C}$ induces an exact sequence of groups
$$
0\rightarrow Pic(U_{\mathcal L,C}) \rightarrow Pic(\mathcal Vec^{ss}_{=\mathcal L,C})\rightarrow\mathbb Z/\tfrac{r}{n_{r,d}}\mathbb Z\rightarrow 0
$$
where the second map sends $d_{\pi}(\mt E)^k$ to $k$.
\end{enumerate}
\end{teo}
\begin{proof} Part (i) is proved in \cite[Theorem 3.1 and Corollary 3.2]{H}. Part (ii) follows from \cite[Section 2]{H}. Part (iii) is proved in \cite[Theorem 3.7]{H}.
\end{proof}
\begin{rmk}\label{caso2schifo}By \cite[Corollary 3.8]{H}, the variety $U_{\mt L,C}$ is locally factorial. Moreover, except the cases when $g=r=2$ and $\deg\mt L$ is even, the closed locus of strictly semistable vector bundles is not a divisor. So, by Theorem \ref{picchow}, when $(r,g,d)\neq (2,2,0)\in \mathbb Z\times\mathbb Z\times(\mathbb Z/2\mathbb Z)$ we have that $\Pic(\mathcal Vec_{=\mathcal L,C})\cong \Pic(\mathcal Vec^s_{=\mathcal L,C})$ and, since $U_{\mt L,C}$ is locally factorial, $\Pic(U_{\mathcal L,C})\cong\Pic(U^s_{\mathcal L,C})$.
\end{rmk}
\subsection{Boundary divisors.}\label{boundiv}
The aim of this section is to study the boundary divisors of $\CVc$. We first introduce some divisors contained in the boundary of $\CVc$.
\begin{defin}\label{boundef}The \emph{boundary divisors} of $\CVc$ are:
\begin{itemize}
\item $\widetilde\delta_0:=\widetilde\delta_0^0$ is the divisor whose generic point is an irreducible curve $C$ with just one node and $\mt E$ is a vector bundle of degree $d$,
\item if $k_{r,d,g}|2i-1$ and $0< i < g/2$:\\
$\widetilde\delta_i^{j}$ for $0\leq j\leq r$ is the divisor whose generic point is a curve $C$ composed by two irreducible smooth curves $C_1$ and $C_2$ of genus $i$ and $g-i$ meeting in one point and $\mt E$ a vector bundle over $C$ with multidegree
$$
(\deg\mt E_{C_1},\deg\mt E_{C_2})=\left(d\frac{2i-1}{2g-2}-\frac{r}{2}+j,d\frac{2(g-i)-1}{2g-2}+\frac{r}{2}-j\right),
$$
\item if $k_{r,d,g}\nmid 2i-1$ and $0< i < g/2$:\\
$\widetilde\delta_i^{j}$ for $0\leq j\leq r-1$ is the divisor whose generic point is a curve $C$ composed by two irreducible smooth curves $C_1$ and $C_2$ of genus $i$ and $g-i$ meeting in one point and $\mt E$ a vector bundle over $C$ with multidegree
$$
(\deg\mt E_{C_1},\deg\mt E_{C_2})=\left(\left\lceil d\frac{2i-1}{2g-2}-\frac{r}{2}\right\rceil +j,\left\lfloor d\frac{2(g-i)-1}{2g-2}+\frac{r}{2}\right\rfloor-j\right),
$$
\item if $g$ is even:\\
$\widetilde\delta_{\frac{g}{2}}^{j}$ for $0\leq j\leq \lfloor \frac{r}{2}\rfloor$ is the divisor whose generic point is a curve $C$ composed by two irreducible smooth curves $C_1$ and $C_2$ of genus $g/2$ meeting in one point and $\mt E$ a vector bundle over $C$ with multidegree
$$
(\deg\mt E_{C_1},\deg\mt E_{C_2})=\left(\left\lceil \frac{d-r}{2}\right\rceil +j,\left\lfloor \frac{d+r}{2}\right\rfloor-j\right).
$$
\end{itemize}
If $i<g/2$ and $k_{r,d,g}|2i-1$ (resp. $g$ and $d+r$ even) we will call $\widetilde\delta_i^0$ and $\widetilde\delta_i^r$ (resp. $\widetilde\delta_{\frac{g}{2}}^0)$ the \emph{extremal boundary divisors}. We will call \emph{non-extremal boundary divisors} the boundary divisors which are not extremal.\\
By Theorem \ref{picchow}, we can associate to $\widetilde\delta_i^j$ a line bundle on $\overline{\mt U}_n$ for any $n$, which glue to a line bundle $\oo(\widetilde\delta_i^j)$ on $\CVc$, we will call them \emph{boundary line bundles}. Moreover, if $\widetilde\delta_i^j$ is a (non)-extremal divisor, we will call $\oo(\widetilde\delta_i^j)$ \emph{(non)-extremal boundary line bundle}.
\end{defin}
Indeed, it turns out that the boundary of $\CVc$ is the union of the above boundary divisors.
\begin{prop}\label{boundary}\noindent
\begin{enumerate}[(i)]
\item The boundary $\widetilde\delta:=\CVc\slash\Vc$ of $\CVc$ is a normal crossing divisor and its irreducible components are $\widetilde\delta_i^j$ for $0\leq i\leq g/2$ and $j\in J_i$ where
$$J_i=
\begin{cases}
0 &\mbox{if } i=0,\\
\{0,\ldots,r \} & \mbox{if } k_{r,d,g}|2i-1 \mbox{ and } 0<i<g/2,\\
\{0,\ldots,r-1\} & \mbox{if } k_{r,d,g}\nmid 2i-1 \mbox{ and } 0<i<g/2,\\
\{0,\ldots,\lfloor r/2\rfloor \}& \mbox{if g even and } i=g/2.
\end{cases}
$$
\item Let $\overline{\phi}_{r,d}:\CVc\rightarrow\CMg$ be the forgetful map. For $0\leq i\leq g/2$, we have
$$
\overline{\phi}_{r,d}^*\oo(\delta_i)=\oo\left(\sum_{j\in J_i}\widetilde\delta_i^j\right).
$$
\end{enumerate}
\end{prop}
\begin{proof}Part (i). Observe that $\widetilde\delta:=\CVc\backslash\Vc=\overline{\phi}_{r,d}^{-1}(\CMg\backslash \Mg)$. Clearly, we have a set-theoretically equality
$$
\overline{\phi}_{r,d}^{-1}(\delta_i)=\bigcup_{j\in J_i}\widetilde\delta_i^j.
$$
We can easily see that $\delta_i^j=\delta_t^k$ if and only if $j=k$ and $i=t$. Now we are going to prove that they are irreducible. Let $\widetilde\delta^*$ be the locus of $\widetilde\delta$ of curves with exactly one node. As in \cite[Corollary 1.9]{DM69} we can prove that $\widetilde\delta$ is a normal crossing divisor and $\widetilde\delta^*$ is a dense smooth open substack in $\widetilde\delta$. Moreover, setting $\widetilde\delta_i^{*j}:=\widetilde\delta^*\cap\widetilde\delta_i^j$, we see that $\widetilde\delta_i^j$ is irreducible if and only if $\widetilde\delta_i^{*j}$ is irreducible. It can be shown also that they are disjoint, i.e. $\delta_i^{*j}\cap\delta_t^{*k}\neq\emptyset$ if and only if $j=k$ and $i=t$.\\
Consider the forgetful map $\phi:\widetilde\delta_i^{*j}\rightarrow \delta_i^*$, where $\delta_i^*$ is the open substack of $\delta_i$ of curves with exactly one node. In \S\ref{locstr}, we have seen that the morphism of Artin functors $\text{Def}_{(C,\mt E)}\to\text{Def}_C$ is formally smooth for any nodal curve. This implies that the map $\phi$ is smooth, in particular is open. Since $\delta_i^*$ is irreducible (see \cite[pag. 94]{DM69}), it is enough to show that the geometric fibers of $\phi$ are irreducible.\\
Let $C$ be a nodal curve with two irreducible components $C_1$ and $C_2$, of genus $i$ and $g-i$, meeting at a point $x$, this defines a geometric point $[C]\in\delta^*_i$. Consider the moduli stack $\widetilde\delta_C^{j}$ of vector bundles on $C$ of multidegree
$$
(d_1,d_2):=(\deg_{C_1}\mt E,\deg_{C_2}\mt E)=\left(\left\lceil d\frac{2i-1}{2g-2}-\frac{r}{2}\right\rceil+j,\left\lfloor d\frac{2(g-i)-1}{2g-2}+\frac{r}{2}\right\rfloor-j\right).
$$
It can be shown that there exists an isomorphism of stacks $\widetilde\delta_C^j\to\phi^*([C])$. Observe that defining a properly balanced vector bundle on $\widetilde\delta_C^j$ is equivalent to giving a vector bundle on $C_1$ of degree $d_1$, a vector bundle on $C_2$ of degree $d_2$ and an isomorphism of vector spaces between the fibers at the node. Consider the moduli stack $\mt Vec_{r,d_1,C_1}$ parametrizing vector bundles on $C_1$ of degree $d_1$ and rank $r$. Let $\mt E$ be the universal vector bundle on $\mt Vec_{r,d_1,C_1}\times C_1$. We fix an open (and dense) substack $\mt V$ such that $\mt E_{\mt V\times \{x\}}$ is trivial. Analogously, let $\mt W$ be an open subset of the moduli stack $\mt Vec_{r,d_2,C_2}$, parametrizing vector bundles on $C_2$ of degree $d_2$ and rank $r$, such that the universal vector bundle on $\mt Vec_{r,d_2,C_2}\times C_2$ is trivial along $\mt W\times \{x\}$.
Via glueing procedure, we obtain a dominant morphism
$
\mt V\times \mt W\times GL_r\lra\widetilde\delta^j_C.
$
The source is irreducible (because $\mt V$ and $\mt W$ are irreducible by \cite[Corollary A.5]{Ho10}), so the same holds for the target $\widetilde\delta^j_C$.\\
Part (ii). By part (i), for $0\leq i\leq g/2$ we have
$$
\overline{\phi}_{r,d}^*\oo(\delta_i)=\oo\left(\sum_{j\in J_i}a_i^j\widetilde\delta_i^j\right)
$$
where $a_i^j$ are integers. We have to prove that the coefficients are $1$. We can reduce to prove it locally on $\widetilde\delta$. The generic element of $\widetilde\delta$ is a pair $(C,\mt E)$ such that $C$ is stable with exactly one node and $\Aut(C,\mt E)=\mathbb{G}_m$. By Lemma \ref{locstrc}, locally at such $(C,\mt E)$, $\overline{\phi}_{r,d}$ looks like
$$
\left[\Spf\, k\llbracket x_1,\ldots,x_{3g-3},y_1,\ldots,y_{r^2(g-1)+1}\rrbracket/\mathbb{G}_m\right]\rightarrow \left[\Spf\, k\llbracket x_1,\ldots,x_{3g-3}\rrbracket/\Aut(C)\right].
$$
We can choose local coordinates such that $x_1$ corresponds to smoothing the unique node of $C$. For such a choice of the coordinates, we have that the equation of $\delta_i$ locally on $C$ is given by $(x_1=0)$ and the equation of $\widetilde\delta_i^j$ locally on $(C,\mt E)$ is given by $(x_1=0)$. Since $\overline{\phi}_{r,d}^*(x_1)=x_1$, the theorem follows.
\end{proof}
With an abuse of notation we set $\widetilde\delta_i^j:=\nu_{r,d}(\widetilde\delta_i^j)$ for $0\leq i\leq g/2$ and $j\in J_i$, where $\nu_{r,d}:\CVc\rightarrow \CVr$ is the rigidification map. From the above proposition, we deduce the following
\begin{cor}\label{boundarycor} The following hold:
\begin{enumerate}
\item The boundary $\widetilde\delta:=\CVr\slash\Vc$ of $\CVr$ is a normal crossing divisor, and its irreducible components are $\widetilde\delta_i^j$ for $0\leq i\leq g/2$ and $j\in J_i$.
\item For $0\leq i\leq g/2$, $j\in J_i$ we have
$
\nu_{r,d}^*\oo(\widetilde\delta_i^j)=\oo(\widetilde\delta_i^j)
$.
\end{enumerate}
\end{cor}
\subsection{Tautological line bundles.}\label{tautbun}
In this subsection, we will produce several line bundles on the stack $\CVc$ and we will study their relations in the rational Picard group of $\CVc$. Consider the universal curve $\overline{\pi}:\overline{\mt Vec}_{r,d,g,1}\rightarrow \CVc$. The stack $\overline{\mt Vec}_{r,d,g,1}$ has two natural sheaves, the dualizing sheaf $\omega_{\overline{\pi}}$ and the universal vector bundle $\mathcal E$. As explained in \S\ref{dcdp}, we can produce the following line bundles which will be called \emph{tautological line bundles}:
$$
\begin{array}{rcl}
K_{1,0,0} &:=&\langle \omega_{\overline{\pi}},\omega_{\overline{\pi}}\rangle,\\
K_{0,1,0}&:=&\langle \omega_{\overline{\pi}},\det\mt E\rangle,\\
K_{-1,2,0}&:=&\langle \det\mt E,\det\mt E\rangle,\\
\Lambda(m,n,l) & :=& d_{\overline{\pi}}(\omega_{\overline{\pi}}^m\otimes (\det \mt E)^n\otimes \mathcal E^l).
\end{array}
$$
With an abuse of notation, we will denote with the same symbols their restriction to any open substack of $\CVc$. By Theorems \ref{detcoh} and \ref{delpair}, we can compute the first Chern classes of the tautological line bundles:
$$
\begin{array}{rcccl}
k_{1,0,0}&:=&c_1(K_{1,0,0})&=&\overline{\pi}_*\left(c_1(\omega_{\overline{\pi}})^2\right),\\
k_{0,1,0}&:=&c_1(K_{0,1,0})&=&\overline{\pi}_*\left(c_1(\omega_{\overline{\pi}})\cdot c_1(\mt E)\right)\\
k_{-1,2,0}&:=&c_1(K_{-1,2,0})&=&\overline{\pi}_*\left(c_1(\mt E)^2\right)\\
\lambda(m,n,l)&:=&c_1(\Lambda(m,n,l))&=&c_1\left(\overline{\pi}_!\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes \mt E^l\right)\right)
\end{array}
$$
\begin{teo}\label{relations}The tautological line bundles on $\CVc$ satisfy the following relations in the rational Picard group $Pic(\CVc)\otimes \mathbb Q$.
\begin{enumerate}[(i)]
\item $K_{1,0,0}=\Lambda(1,0,0)^{12}\otimes \oo(-\widetilde\delta).$
\item $K_{0,1,0}=\Lambda(1,0,1)\otimes \Lambda(0,0,1)^{-1}=\Lambda(1,1,0)\otimes \Lambda(0,1,0)^{-1}.$
\item $K_{-1,2,0}=\Lambda(0,1,0)\otimes \Lambda(1,1,0)\otimes\Lambda(1,0,0)^{-2}.$
\item For $(m,n,l)$ integers we have:\begin{eqnarray*}
\Lambda(m,n,l) &=&\Lambda(1,0,0)^{r^l(6m^2-6m+1-n^2-l)-2r^{l-1}nl-r^{l-2}l(l-1)}\otimes\\
& & \otimes\Lambda(0,1,0)^{r^l\left(-mn+{n+1\choose 2}\right)+r^{l-1}l\left(n -m\right)+r^{l-2}{l\choose 2}}\otimes\\
& &\otimes\Lambda(1,1,0)^{r^l\left(mn+{n\choose 2}\right)+r^{l-1}l\left(m+n\right)+r^{l-2}{l\choose 2}}\otimes\\
&&\otimes\Lambda(0,0,1)^{r^{l-1}l}\otimes\oo\left(-r^l{m\choose 2}\widetilde\delta\right).
\end{eqnarray*}
\end{enumerate}
\end{teo}
\begin{proof}As we will see in the Lemma \ref{redsemistable}, we can reduce to proving the equalities on the quasi-compact open substack $\CVc^{Pss}$. We follow the same strategy in the proof of \cite[Theorem 5.2]{MV}. The first Chern class map is an isomorphism by Theorem \ref{picchow}. Thus it is enough to prove the above relations in the rational Chow group $A^1\left( \CVc^{Pss}\right)\otimes\mathbb Q$. Applying the Grothendieck-Riemann-Roch Theorem to the universal curve $\overline{\pi}:\overline{\mt Vec}_{r,d,g,1}\rightarrow \CVc$, we get:
\begin{equation}\label{1}
\ch\left(\overline{\pi}_!\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes\mt E^l\right)\right)=\overline{\pi}_*\left(\ch\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes\mt E^l\right)\cdot \Td\left(\Omega_{\overline{\pi}}\right)^{-1}\right)
\end{equation}
where $\ch$ is the Chern character, $\Td$ the Todd class and $\Omega_{\overline{\pi}}$ is the sheaf of relative Kahler differentials. Using Theorem \ref{detcoh}, the degree one part of the left hand side becomes
\begin{equation}\label{2}
\ch\left(\overline{\pi}_!\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes\mt E^l\right)\right)_1=c_1\left(\overline{\pi}_!\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes\mt E^l\right)\right)=c_1\left(\Lambda(m,n,l)\right)=\lambda(m,n,l).
\end{equation}
In order to compute the right hand side, we will use the fact that $c_1\left(\Omega_{\overline{\pi}}\right)=c_1\left(\omega_{\overline{\pi}}\right)$ and $\overline{\pi}_*\left(c_2\left(\Omega_{\overline{\pi}}\right)\right)=\widetilde\delta$ (see \cite[p. 383]{ACG11}. Using this, the first three terms of the inverse of the Todd class of $\Omega_{\overline{\pi}}$ are equal to
\begin{equation}\label{3}
\text{Td}\left(\Omega_{\overline{\pi}}\right)^{-1}=1-\frac{c_1\left(\Omega_{\overline{\pi}}\right)}{2}+\frac{c_1\left(\Omega_{\overline{\pi}}\right)^2+c_2\left(\Omega_{\overline{\pi}}\right)}{12}+\ldots=1-\frac{c_1\left(\omega_{\overline{\pi}}\right)}{2}+\frac{c_1\left(\omega_{\overline{\pi}}\right)^2+c_2\left(\Omega_{\overline{\pi}}\right)}{12}+\ldots
\end{equation}
By the multiplicativity of the Chern character, we get
\begin{multline}\label{4}
ch\left(\omega_{\overline{\pi}}^m\otimes( \det\mt E)^n\otimes\mt E^l\right)=\ch\left(\omega_{\overline{\pi}}\right)^m\ch(\det\mt E)^n\ch\left(\mt E\right)^l=\\
\shoveleft=\left(1+c_1\left(\omega_{\overline{\pi}}\right)+\frac{c_1\left(\omega_{\overline{\pi}}\right)^2}{2}+\ldots\right)^m\cdot\left(1+c_1\left(\mt E\right)+\frac{c_1\left(\mt E\right)^2}{2}+\ldots\right)^n\cdot\\
\shoveright{\cdot\left(r+c_1\left(\mt E\right)+\frac{c_1\left(\mt E\right)^2-2c_2\left(\mt E\right)}{2}+\ldots\right)^l=}\\
\shoveleft=\left(1+mc_1\left(\omega_{\overline{\pi}}\right)+\frac{m^2}{2}c_1\left(\omega_{\overline{\pi}}\right)^2+\ldots\right)\cdot\left(1+nc_1\left(\mt E\right)+\frac{n^2}{2}c_1\left(\mt E\right)^2+\ldots\right)\cdot\\
\shoveright{\cdot\left(r^l+lr^{l-1}c_1\left(\mt E\right)+\frac{lr^{l-2}}{2}\left((r+l-1)c_1\left(\mt E\right)^2-2rc_2\left(\mt E\right))\right)+\ldots\right)=}\\
\shoveleft{=r^l+\left[rmc_1\left(\omega_{\overline{\pi}}\right)+(rn+l)c_1\left(\mt E\right)\right]r^{l-1}+\left[r^l\frac{m^2}{2}c_1\left(\omega_{\overline{\pi}}\right)^2+r^{l-1}m\left(rn+l\right)c_1\left(\omega_{\overline{\pi}}\right)c_1\left(\mt E\right)\right.}\\
\left.+ \frac{r^{l-2}}{2}\left(r^2n^2+lr(2n+1)+l(l-1)\right)c_1\left(\mt E\right)^2-lr^{l-1}c_2\left(\mt E\right)\right].
\end{multline}
Combining (\ref{3}) and (\ref{4}), we can compute the degree one part of the right hand side of (\ref{1}):
\begin{multline}
\left[\overline{\pi}_*\left(ch\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes\mt E^l\right)\cdot \text{Td}\left(\Omega_{\overline{\pi}}\right)^{-1}\right)\right]_1=\overline{\pi}_*\left(\left[ch\left(\omega_{\overline{\pi}}^m\otimes(\det\mt E)^n\otimes\mt E^l\right)\cdot \text{Td}\left(\Omega_{\overline{\pi}}\right)^{-1}\right]_2\right)=\\
\shoveleft{=\overline{\pi}_*\left(\frac{r^l}{12}(6m^2-6m+1)c_1(\omega_{\overline{\pi}})^2+\frac{r^{l-1}}{2}(rn+l)(2m-1)c_1\left(\omega_{\overline{\pi}}\right)c_1\left(\mt E\right)+\right.}\\
\shoveright{\left.+\frac{r^{l-2}}{2}\left(r^2n^2+lr(2n+1)+l(l-1)\right)c_1\left(\mt E\right)^2-lr^{l-1}c_2\left(\mt E\right)+\frac{r^l}{12}c_2\left(\Omega_{\overline{\pi}}\right)\right)=}\\
\shoveleft{=\frac{r^l}{12}(6m^2-6m+1)k_{1,0,0}+\frac{r^{l-1}}{2}(rn+l)(2m-1)k_{0,1,0}+}\\
+\frac{r^{l-2}}{2}\left(r^2n^2+lr(2n+1)+l(l-1)\right)k_{-1,2,0}-lr^{l-1}\overline{\pi}_*c_2\left(\mt E\right)+\frac{r^l}{12}\widetilde\delta.
\end{multline}
Combining with (\ref{2}), we have:
\begin{multline}\label{5}
\lambda(m,n,l)=\frac{r^l}{12}(6m^2-6m+1)k_{1,0,0}+\frac{r^{l-1}}{2}(rn+l)(2m-1)k_{0,1,0}+\\
+\frac{r^{l-2}}{2}\left(r^2n^2+lr(2n+1)+l(l-1)\right)k_{-1,2,0}-lr^{l-1}\overline{\pi}_*c_2\left(\mt E\right)+\frac{r^l}{12}\widetilde\delta.
\end{multline}
As special case of the above relation, we get
\begin{equation}\label{6}
\lambda(1,0,0)=\frac{k_{1,0,0}}{12}+\frac{\widetilde\delta}{12}.
\end{equation}
If we replace (\ref{6}) in (\ref{5}), then we have
\begin{multline}\label{7}
\lambda(m,n,l)=r^l(6m^2-6m+1)\lambda(1,0,0)+\frac{r^{l-1}}{2}(rn+l)(2m-1)k_{0,1,0}+\\
+\frac{r^{l-2}}{2}\left(r^2n^2+lr(2n+1)+l(l-1)\right)k_{-1,2,0}-lr^{l-1}\overline{\pi}_*c_2\left(\mt E\right)-r^l{m\choose 2}\widetilde\delta.
\end{multline}
Moreover from (\ref{7}) we obtain:
\begin{equation}\label{8}
\begin{cases}
\lambda(0,1,0)=\lambda(1,0,0)-\frac{k_{0,1,0}}{2}+\frac{k_{-1,2,0}}{2}\\
\lambda(1,1,0)=\lambda(1,0,0)+\frac{k_{0,1,0}}{2}+\frac{k_{-1,2,0}}{2}\\
\lambda(0,0,1)=r\lambda(1,0,0)-\frac{k_{0,1,0}}{2}+\frac{k_{-1,2,0}}{2}-\overline{\pi}_*c_2\left(\mt E\right)\\
\lambda(1,0,1)=r\lambda(1,0,0)+\frac{k_{0,1,0}}{2}+\frac{k_{-1,2,0}}{2}-\overline{\pi}_*c_2\left(\mt E\right)
\end{cases}
\end{equation}
which gives
\begin{equation}\label{9}
\begin{cases}
k_{0,1,0}=\lambda(1,0,1)-\lambda(0,0,1)=\lambda(1,1,0)-\lambda(0,1,0)\\
k_{-1,2,0}=-2\lambda(1,0,0)+\lambda(0,1,0)+\lambda(1,1,0)\\
\overline{\pi}_*c_2\left(\mt E\right)=(r-1)\lambda(1,0,0)+\lambda(0,1,0)-\lambda(0,0,1).
\end{cases}
\end{equation}
Substituing in (\ref{7}), we finally obtain
\begin{eqnarray}
r^{2-l}\lambda(m,n,l)&= &\Big(r^2(6m^2-6m+1-n^2-l)-2rnl-l(l-1)\Big)\lambda(1,0,0)+\\
\nonumber & &+\left(r^2\left(-mn+{n+1\choose 2}\right)+rl\left(n -m\right)+{l\choose 2}\right)\lambda(0,1,0)+\\
\nonumber& &+\left(r^2\left(mn+{n\choose 2}\right)+rl\left(m+n\right)+{l\choose 2}\right)\lambda(1,1,0)+\\
\nonumber & &+\,rl\lambda(0,0,1)-r^2{m\choose 2}\widetilde\delta.
\end{eqnarray}
\end{proof}
\begin{rmk}As we will see in the next section the integral Picard group of $\Pic(\CVc)$ is torsion free for $g\geq 3$. In particular the relations of Theorem \ref{relations} hold also for $\Pic(\CVc)$.
\end{rmk}
\section{The Picard groups of $\CVc$ and $\CVr$.}\label{robba}
The aim of this section is to prove the Theorems \ref{pic} and \ref{picred}. We will prove them in several steps. For the rest of the paper we will assume $r\geq 2$.
\subsection{Independence of the boundary divisors.}\label{indipendece}
The aim of this subsection is to prove the following
\begin{teo}\label{indbou}Assume that $g\geq 3$. We have an exact sequence of groups
$$
0\longrightarrow\bigoplus_{i=0,\ldots,\lfloor g/2\rfloor}\oplus_{j\in J_i}\langle\oo(\widetilde\delta_i^j)\rangle\longrightarrow Pic(\CVc)\longrightarrow Pic(\Vc)\longrightarrow 0
$$
where the right map is the natural restriction and the left map is the natural inclusion.
\end{teo}
For the rest of this subsection, with the only exceptions of Proposition \ref{T} and Lemma \ref{sHs}, we will always assume that $g\geq 3$. We recall now a result from \cite{T95}.
\begin{prop}\label{T}\cite[Proposition 1.2]{T95}. Let $C$ a nodal curve of genus greater than one without rational components and let $\mt E$ be a balanced vector bundle over $C$ with rank $r$ and degree $d$. Let $C_1,\ldots,C_s$ be its irreducible components. If $\mt E_{C_i}$ is semistable for any $i$ then $\mt E$ is P-semistable. Moreover if the basic inequalities are all strict and all the $\mt E_{C_i}$ are semistable and at least one is stable then $\mt E$ is P-stable.
\end{prop}
\begin{rmk}\label{gvb}Recall that for a smooth curve of genus greater than $1$ the generic vector bundle is stable. On the other hand for an elliptic curve the stable locus is not empty if and only if the degree and the rank are coprime. In this case any semistable vector bundle is stable. In general for an elliptic curve the generic vector bundle of degree $d$ and rank $r$ is direct sum of $n_{r,d}$ stable vector bundles of degree $d/n_{r,d}$ and rank $r/n_{r,d}$; in particular it will be semistable.
\end{rmk}
We deduce from this
\begin{lem}\label{T2}The generic point of $\widetilde\delta_i^j$ is a curve $C$ with exactly one node and a properly balanced vector bundle $\mt E$ such that
\begin{enumerate}[(i)]
\item if $i=0$ the pull-back of $\mt E$ at the normalization is a stable vector bundle,
\item if $i=1$ the restriction $\mt E_{C_1}$ is direct sum of stable vector bundles with same rank and degree and $\mt E_{C_2}$ is a stable vector bundle.
\item if $2\leq i\leq \lfloor g/2\rfloor$ the restrictions $\mt E_{C_1}$ and $\mt E_{C_2}$ are stable vector bundles.
\end{enumerate}
Furthermore the generic point of $\widetilde\delta_i^j$ is a curve with exactly one node with a P-stable vector bundle if $\widetilde\delta_i^j$ is a non-extremal divisor and a curve with exactly one node with a strictly P-semistable vector bundle if $\widetilde\delta_i^j$ is an extremal divisor.
\end{lem}
\begin{proof}The case $i=0$ is obvious. We fix $i\in \{1,\ldots,\lfloor g/2\rfloor\}$ and $j\in J_i$. By definition the generic point of $\widetilde\delta_i^j$ is a curve with two irreducible components $C_1$ and $C_2$ of genus $i$ and $g-i$ meeting at one point and a vector bundle $\mt E$ with multidegree
$$
(\deg_{C_1}\mt E,\deg_{C_2}\mt E)=\left(\left\lceil d\frac{2i-1}{2g-2}-\frac{r}{2}\right\rceil+j,\left\lfloor d\frac{2(g-i)-1}{2g-2}+\frac{r}{2}\right\rfloor-j\right).
$$
As observed in Remark \ref{gvb} the generic vector bundle over a smooth curve of genus $>1$ (resp. $1$) is stable (resp. direct sum of stable vector bundles). Giving a vector bundle over $C$ is equivalent to give a vector bundle on any irreducible component and an isomorphism of vector spaces between the fibers at the nodes. With this in mind, it is easy to see that we can deform any vector bundle $\mt E$ in a vector bundle $\mt E'$ which is stable (resp. is a direct sum of stable vector bundles with same rank and degree) over any component of genus $>1$ (resp. $1$). By Proposition \ref{T}, the generic point of $\widetilde\delta_i^j$ is P-semistable. Moreover if $\widetilde\delta_i^j$ is a non-extremal divisor the basic inequalities are strict. By the second assertion of \emph{loc. cit.}, if $\widetilde\delta_i^j$ is a non-extremal divisor the generic point of $\widetilde\delta_i^j$ is P-stable. It remains to prove the assertion for the extremal divisors. Suppose that $\widetilde\delta_i^0$ is an extremal divisor, the proof for the $\widetilde\delta_i^r$ is similar. It is easy to prove that
$$
\deg_{C_1}\mt E=d\frac{2i-1}{2g-2}-\frac{r}{2}\iff
\frac{\chi\left(\mt E_{C_1}\right)}{\omega_{C_1}}=\frac{\chi\left(\mt E\right)}{\omega_{C}}.
$$
In other words, $\mt E_{C_1}$ is a destabilizing quotient for $\mt E$, concluding the proof.
\end{proof}
\begin{lem}\label{redsemistable}
The Picard group of $\Vc$ $($resp. $\Vr)$, is naturally isomorphic to the Picard group of the open substacks $\Vc^{ss}$ $($resp. $\Vr^{ss})$ and $\mt U_n$ $($resp. $\mt U_n\fatslash\mathbb{G}_m)$ for $n$ big enough.\\
The Picard group of $\CVc$ $($resp. $\CVr)$, is naturally isomorphic to the Picard group of the open substacks $\CVc^{Pss}$ $($resp. $\CVr^{Pss})$ and $\overline{\mt U}_n$ $($resp. $\overline{\mt U}_n\fatslash\mathbb{G}_m)$ for $n$ big enough.
\end{lem}
\begin{proof}We have the following equalities
\begin{eqnarray*}
\dim\Vc &=&\dim\Jc+\dim\mt Vec_{=\mt L,C},\\
\dim\left(\Vc\backslash\Vc^{ss} \right)& \leq&\dim\Jc+\dim\left(\mt Vec_{=\mt L,C}\backslash\mt Vec^{ss}_{=\mt L,C}\right).
\end{eqnarray*}
Thus $\cod(\Vc\backslash\Vc^{ss},\Vc)\geq\cod(\mt Vec_{=\mt L,C}\backslash\mt Vec^{ss}_{=\mt L,C},\mt Vec_{=\mt L,C})\geq 2$ (see proof of \cite[Corollary 3.2]{H}).
By Proposition \ref{qc}, there exists $n_*\gg 0$ such that $\CVc^{Pss}\subset \overline{\mt U}_n$ for $n\geq n_*$. In particular $\cod(\mt U_n\backslash\Vc^{ss},\mt U_n)\geq 2$. Suppose that $\cod(\CVc\backslash\CVc^{Pss},\CVc)=1$, so
$\CVc\backslash\CVc^{Pss}$ contains a substack of codimension $1$. By the observations above this stack must be contained in some irreducible components of $\widetilde\delta$. The generic point of any divisor $\widetilde\delta_i^j$ is P-semistable by Lemma \ref{T2}, then we have a contradiction. So $\cod(\CVc\backslash\CVc^{Pss},\CVc)\geq\cod(\overline{\mt U}_n\backslash\CVc^{Pss},\overline{\mt U}_n)\geq 2$. The same holds for the rigidifications. By Theorem \ref{picchow}, the lemma follows.
\end{proof}
By Lemma \ref{redsemistable}, Theorem \ref{indbou} is equivalent to proving that there exists $n_*\gg 0$ such that for $n\geq n_*$ we have an exact sequence of groups
$$
0\lra\bigoplus_{i=0,\ldots,\lfloor g/2\rfloor}\oplus_{j\in J_i}\langle\oo(\widetilde\delta_i^j)\rangle\lra \Pic(\overline{\mt U}_n)\lra \Pic(\mt U_n)\lra 0.
$$
By Theorem \ref{picchow}, the sequence exists and it is exact in the middle and at right. It remains to prove the left exactness.
The strategy that we will use is the same as the one of Arbarello-Cornalba for $\CMg$ in \cite{AC87} and the generalization for $\overline{\mt Jac}_{r,g}$ done by Melo-Viviani in \cite{MV}. More precisely, we will construct morphisms $B\to\overline{\mt U}_n$ from irreducible smooth projective curves $B$ and we compute the degree of the pull-backs of the boundary divisors of $\Pic(\overline{\mt U}_n)$ to $B$. We will construct liftings of the families $F_h$ (for $1 \leq h \leq (g - 2 ) /2$), $F$ and $F'$ used by Arbarello-Cornalba in \cite[pp. 156-159]{AC87}. Since $\CVc\cong\overline{\mathcal Vec}_{r,d',g}$ if $d\equiv d'$ mod $(r(2g-2))$, in this section we can assume that $0\leq d<r(2g-2)$.\\\\
\textbf{The Family $\widetilde F$}.\\
Consider a general pencil in the linear system $H^0(\mathbb{P}^2,\oo(2))$. It defines a rational map $\mathbb{P}^2\dashrightarrow\mathbb{P}^1$, which is regular outside of the four base points of the pencil. Blowing the base locus we get a conic bundle $\phi:X\rightarrow\mathbb{P}^1$. The four exceptional divisors $E_1$, $E_2$, $E_3$, $E_4\subset X$ are sections of $\phi$. It can be shown that the conic bundle has 3 singular fibers consisting of rational chains of length two. Fix a smooth curve $C$ of genus $g-3$ and $p_1,p_2,p_3,p_4$ points of $C$. Consider the following surface
$$
Y=\left(X\amalg(C\times\mathbb P^1)\right)/(E_i\sim \{p_i\}\times \mathbb P^1).
$$
We get a family $f:Y\rightarrow \mathbb P^1$ of stable curves of genus $g$. The general fiber of $f$ is as in Figure \ref{Figure1} where $Q$ is a smooth conic.
\begin{figure}[h!]
\begin{center}
\unitlength .65mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(85.55,50.324)(0,110)
\qbezier(12.78,153.574)(19.905,97.574)(34.53,116.574)
\qbezier(34.53,116.574)(46.155,136.574)(56.28,116.574)
\qbezier(56.28,116.574)(71.53,96.699)(78.78,154.324)
\put(10.828,145.765){\makebox(0,0)[cc]{$C$}}
\put(4.625,120.248){\line(1,0){.21}}
\put(4.835,120.248){\line(1,0){88.715}}
\put(88.504,122.561){\makebox(0,0)[cc]{$Q$}}
\end{picture}
\end{center}
\caption{The general fiber of $f:Y\to \mathbb P^1$}\label{Figure1}
\end{figure}\\
While the 3 special components are as in Figure 2 where $R_1$ and $R_2$ are rational curves.
\begin{figure}[ht]
\begin{center}
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\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(80.75,50.75)(0,150)
\put(10.25,182.25){\line(-1,0){.25}}
\put(10,182.25){\line(1,0){.25}}
\multiput(10.25,182.25)(.04485138004,-.03370488323){942}{\line(1,0){.04485138004}}
\multiput(35,150.75)(.04842799189,.03372210953){986}{\line(1,0){.04842799189}}
\qbezier(12.25,196)(19.375,140)(34,159)
\qbezier(34,159)(45.625,179)(55.75,159)
\qbezier(55.75,159)(71,139.125)(78.25,196.75)
\put(24.5,176.25){\makebox(0,0)[cc]{$R_1$}}
\put(62.75,175){\makebox(0,0)[cc]{$R_2$}}
\put(8,196.5){\makebox(0,0)[cc]{$C$}}
\end{picture}
\end{center}
\caption{The three special fibers of $f:Y\to \mathbb P^1$}\label{Figure2}
\end{figure}\\
Choose a vector bundle of degree $d$ on $C$, pull it back to $C\times \mathbb P^1$ and call it $E$. Since $E$ is trivial on $\{p_i\}\times \mathbb P^1$, we can glue it with the trivial vector bundle of rank $r$ on $X$ obtaining a vector bundle $\mt E$ on $f:Y\rightarrow\mathbb P^1$ of relative rank $r$ and degree $d$.
\begin{lem}$\mt E$ is properly balanced.
\end{lem}
\begin{proof}$\mt E$ is obviously admissible because is defined over a family of stable curves. Since being properly balanced is an open condition, we can reduce to check that $\mt E$ is properly balanced on the three special fibers. By Remark \ref{balcon}, it is enough to check the basic inequality for the subcurves $R_1\cup R_2$, $R_1$ and $R_2$. And by the assumption $0\leq d<r(2g-2)$ is easy to see that the inequalities holds.
\end{proof}
We call $\widetilde F$ the family $f:X\rightarrow\mathbb P^1$ with the vector bundle $\mt E$. It is a lifting of the family $F$ defined in \cite[p. 158]{AC87}. So we can compute the degree of the pull-backs of the boundary bundles in $\Pic(\CVc)$ to the curve $\widetilde F$. Consider the commutative diagram
$$
\xymatrix{
\mathbb P^1\ar[rd]^{F}\ar[r]^{\widetilde F} & \CVc\ar[d]^{\overline{\phi}_{r,d}}\\
&\CMg
}$$
By Proposition \ref{boundary}, we have $\deg_{\widetilde F}\oo(\widetilde\delta_0)=\deg_F\oo(\delta_0)$ and $\deg_F\oo(\delta_0)=-1$ by \cite[p. 158]{AC87}. Since $\widetilde F$ does not intersect the other boundary divisors, we have:
$$
\begin{cases}
\deg_{\widetilde F}\oo(\widetilde\delta_0)=-1, &\\
\deg_{\widetilde F}\oo(\widetilde\delta_i^j)=0 & \mbox{if } i\neq 0\mbox{ and } j\in J_i.
\end{cases}
$$\\
\textbf{The Families $\widetilde F_1'^j$ and $\widetilde F_2'^j$}(for $j\in J_1$).\\
We start with the same family of conics $\phi:X\rightarrow \mathbb P^1$ and the same smooth curve $C$ used for the family $\widetilde F$. Let $\Gamma$ be a smooth elliptic curve and take points $p_1\in \Gamma$ and $p_2,p_3,p_4\in C$. We construct a new surface
$$
Z=\left(X\amalg(C\times\mathbb P^1)\amalg(\Gamma\times\mathbb P^1)\right)/(E_i\sim \{p_i\}\times \mathbb P^1).
$$
We obtain a family $g:Z\rightarrow \mathbb P^1$ of stable curves of genus $g$. The general fiber is as in Figure \ref{Figure3} where $Q$ is a smooth conic. The three special fibers are as in Figure \ref{Figure4} where $R_1$ and $R_2$ are rational smooth curves.
\begin{figure}[ht]
\begin{center}
\unitlength .6mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(131,70.125)(0,115)
\qbezier(18.5,176)(37.5,96.75)(53.5,158.5)
\qbezier(53.5,158.5)(72.75,214.125)(83,130.25)
\put(13,146.75){\line(1,0){1.75}}
\put(14.75,146.75){\line(1,0){112.5}}
\qbezier(123.5,176)(131,165.5)(103.5,155)
\qbezier(103.5,155)(93,147)(103.5,138)
\qbezier(103.5,138)(130.625,123.625)(121.25,114.75)
\put(127.25,173.75){\makebox(0,0)[cc]{$\Gamma$}}
\put(125,150){\makebox(0,0)[cc]{$Q$}}
\put(20.75,177){\makebox(0,0)[cc]{$C$}}
\end{picture}
\end{center}
\caption{The general fibers of $g:Z\to \mathbb P^1$.}\label{Figure3}
\end{figure}
\begin{figure}[ht]
\begin{center}
\unitlength .6mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(131,60.125)(0,120)
\qbezier(18.5,176)(37.5,96.75)(53.5,158.5)
\qbezier(53.5,158.5)(72.75,214.125)(83,130.25)
\qbezier(123.5,176)(131,165.5)(103.5,155)
\qbezier(103.5,155)(93,147)(103.5,138)
\qbezier(103.5,138)(130.625,123.625)(121.25,114.75)
\put(128.25,173.75){\makebox(0,0)[cc]{$\Gamma$}}
\put(20.75,177){\makebox(0,0)[cc]{$C$}}
\put(11,169.5){\line(1,0){.25}}
\multiput(11.25,169.5)(.05160550459,-.03371559633){1090}{\line(1,0){.05160550459}}
\put(51.5,133.5){\line(0,1){.25}}
\multiput(51.5,133.75)(.125,.0337370242){578}{\line(1,0){.125}}
\put(10.75,165){\makebox(0,0)[cc]{$R_1$}}
\put(122.5,147.5){\makebox(0,0)[cc]{$R_2$}}
\end{picture}
\end{center}
\caption{The three special fibers of $g:Z\to \mathbb P^1$.}\label{Figure4}
\end{figure}\\
Let $j$ be an integer. We choose two vector bundles of degree $d-j$ and $d-3j$ on $C$, pull them back to $C\times \mathbb P^1$ and call them $G_1^j$ and $G_2^j$. We choose a vector bundle of degree $j$ on $\Gamma$, pull it back to $\Gamma\times\mathbb P^1$ and call it $M^j$. We glue the vector bundle $G_1^j$ (resp. $G_2^j$) on $C\times\mathbb P^1$, the vector bundle $M^j$ on $\Gamma\times\mathbb P^1$ and the vector bundle $\oo_X^r$ (resp. $\phi^*\oo_{\mathbb P^1}(j)\otimes\omega_{X/\mathbb P^1}^{-j}\oplus\oo_X^{r-1}$), obtaining a vector bundle $\mt G_1^j$ (resp. $\mt G_2^j$) on $Z$ of relative rank $r$ and degree $d$.
\begin{lem} Let $j$ be an integer such that $$\left|j-\frac{d}{2g-2}\right|\leq\frac{r}{2}$$
Then $\mt G_1^j$ is properly balanced if $0\leq d\leq r(g-1)$ and $\mt G_2^j$ is properly balanced if $r(g-1)\leq d<r(2g-2)$.
\end{lem}
\begin{proof}As before we can check the condition on the special fibers. By Remark \ref{balcon} we can reduce to check the inequalities for the subcurves $\Gamma,C,R_1$ and $R_2\cup\Gamma$. Suppose that $0\leq d\leq r(g-1)$ and consider $\mt G_1^j$. The inequality on $\Gamma$ follows by hypothesis. The inequality on $C$ is
$$
\left|d-j-d\frac{2g-5}{2g-2}\right|\leq\frac{3}{2}r \iff \left|j-d\frac{3}{2g-2}\right|\leq\frac{3}{2}r,
$$
and this follows by these inequalities (true by hypothesis on $j$ and $d$)
$$
\left|j-d\frac{3}{2g-2}\right|\leq\left|j-\frac{d}{2g-2}\right|+\left|\frac{d}{g-1}\right|\leq \frac{r}{2}+r.
$$
The inequality on $R_1$ is
$$
\left|\frac{d}{2g-2}\right|\leq\frac{3}{2}r,
$$
and this follows by the hypothesis on $d$. Finally the inequality on $R_2\cup \Gamma$ is
$$
\left|j-\frac{d}{g-1}\right|\leq r,
$$
and this follows by the following inequalities (true by hypothesis on $j$ and $d$)
$$
\left|j-\frac{d}{g-1}\right|\leq\left|j-\frac{d}{2g-2}\right|+\left|\frac{d}{2g-2}\right|\leq \frac{r}{2}+\frac{r}{2}.
$$
Suppose next that $r(g-1)\leq d<r(2g-2)$ and consider $\mt G_2^j$. The inequality on $\Gamma$ follows by hypothesis. On $C$, the inequality gives
$$
\left|d-3j-d\frac{2g-5}{2g-2}\right|\leq\frac{3}{2}r \iff \left|j-\frac{d}{2g-2}\right|\leq\frac{r}{2},
$$
which follows by hypothesis on $j$. The inequality on $R_1$ is
$$
\left|j-\frac{d}{2g-2}\right|\leq\frac{3}{2}r,
$$
and this follows by hypothesis on $j$. The inequality on $R_2\cup \Gamma$ is
$$
\left|2j-\frac{d}{g-1}\right|\leq r,
$$
and this follows by the inequalities (true by hypothesis on $j$)
$$
\left|2j-\frac{d}{g-1}\right|\leq 2\left|j-\frac{d}{2g-2}\right|\leq r.
$$
\end{proof}
Let $k\in J_1$. If $0\leq d\leq r(g-1)$, we call $\widetilde {F'}_1^k$ the family $g:Z\rightarrow\mathbb P^1$ with the properly balanced vector bundle $\mt G_1^{\lceil\frac{d}{2g-2}-\frac{r}{2}\rceil+k}$. If $r(g-1)\leq d<r(2g-2)$ we call $\widetilde {F'}_2^k$ the family $g:Z\rightarrow\mathbb P^1$ with the properly balanced vector bundle $\mt G_2^{\lceil\frac{d}{2g-2}-\frac{r}{2}\rceil+k}$. As before we compute the degree of boundary line bundles to the curves $\widetilde {F'}_1^k$ and $\widetilde {F'}_2^k$ (in the range of degrees where they are defined) using the fact that they are liftings of the family $F'$ in \cite[p. 158]{AC87}.
If $0\leq d\leq r(g-1)$ then we have
$$
\begin{cases}
\deg_{\widetilde{F'}_1^k}\oo(\widetilde\delta_1^k)=-1, &\\
\deg_{\widetilde{F'}_1^k}\oo(\widetilde\delta_1^j)=0 & \mbox{if }j\neq k,\\
\deg_{\widetilde{F'}_1^k}\oo(\widetilde\delta_i^j)=0 & \mbox{if }i> 1,\mbox{ for any }j\in J_i.
\end{cases}
$$
Indeed the first two relations follow from
$$\deg_{\widetilde{F'}_1^k}\oo\left(\sum_{j\in J_i}\widetilde\delta_1^j\right)=\deg_{F'}\oo(\delta_1)=-1$$ (see \cite[p. 158]{AC87}) and the fact that $\widetilde{F'}_1^k$ does not meet $\widetilde\delta_1^j$ for $k\neq j$. The last follows by the fact that $\widetilde{F'}_1^k$ does not meet $\widetilde\delta_i^j$ for $i>1$. Similarly for $\widetilde{F'}_2^k$ we can show that for $r(g-1)\leq d< r(2g-2)$, we have
$$
\begin{cases}
\deg_{\widetilde{F'}_2^k}\oo(\delta_1^k)=-1, &\\
\deg_{\widetilde{F'}_2^k}\oo(\delta_1^j)=0 & \mbox{if }j\neq k,\\
\deg_{\widetilde{F'}_2^k}\oo(\delta_i^j)=0 & \mbox{if }i> 1.
\end{cases}
$$\\
\textbf{The Families $\widetilde F_{h}^j$} (for $1\leq h\leq \frac{g-2}{2}$ and $j\in J_h$).\\
Consider smooth curves $C_1$, $C_2$ and $\Gamma$ of genus $h$, $g-h-1$ and
$1$, respectively, and points $x_1\in C_1$, $x_2\in C_2$ and $\gamma\in \Gamma$. Consider the surface
$Y_2$ given by the blow-up of $\Gamma\times \Gamma$
at $(\gamma, \gamma)$. Let $p_2:Y_2\to \Gamma$ be the map given by composing the blow-down
$Y_2\to \Gamma\times \Gamma$ with the second projection, and $\pi_1: C_1\times\Gamma\to \Gamma$ and $\pi_3:C_2\times\Gamma\to \Gamma$ be the projections along the second factor.
As in \cite[p. 156]{AC87} (and \cite{MV}), we set (see also Figure \ref{Figure 5}):
$$\begin{aligned}
&A=\{x_1\}\times \Gamma, \\
&B=\{x_2\}\times \Gamma, \\
&E= \text{ exceptional divisor of the blow-up of } \Gamma\times \Gamma \text{ at } (\gamma,\gamma),\\
&\Delta= \text{ proper transform of the diagonal in } Y_2, \\
&S= \text{ proper transform of } \{\gamma\}\times \Gamma \text{ in } Y_2,\\
& T= \text{ proper transform of } \Gamma \times \{\gamma\}\text{ in } Y_2.
\end{aligned}
$$
\begin{figure}[ht]
\begin{center}
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\begin{picture}(150,60.75)(30,65)
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\put(22.75,66.5){\framebox(43.5,48.25)[cc]{}}
\put(156.25,66.25){\framebox(43.5,48.25)[cc]{}}
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\put(23,102){\line(1,0){43.25}}
\put(44,104){\makebox(0,0)[cc]{$A$}}
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\put(109.25,63.5){\makebox(0,0)[cc]{}}
\put(43.5,63.5){\makebox(0,0)[cc]{}}
\put(177,63.25){\makebox(0,0)[cc]{}}
\put(109.5,63){\makebox(0,0)[cc]{$\Gamma$}}
\put(43.75,63){\makebox(0,0)[cc]{$\Gamma$}}
\put(177.25,62.75){\makebox(0,0)[cc]{$\Gamma$}}
\put(18.25,86.5){\makebox(0,0)[cc]{$\pi_1$}}
\put(151.75,86.25){\makebox(0,0)[cc]{$\pi_3$}}
\put(61.75,86.5){\makebox(0,0)[cc]{}}
\put(128.75,87.25){\makebox(0,0)[cc]{}}
\put(63,87.25){\makebox(0,0)[cc]{}}
\put(196.5,87){\makebox(0,0)[cc]{}}
\put(63,86.5){\makebox(0,0)[cc]{$C_1$}}
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\put(123.75,98){\line(0,-1){31.5}}
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\put(177,81.75){\makebox(0,0)[cc]{$B$}}
\put(94.25,104.5){\makebox(0,0)[cc]{$S$}}
\put(83.5,86.75){\makebox(0,0)[cc]{$\Gamma$}}
\put(101.25,86.75){\makebox(0,0)[cc]{$\Delta$}}
\put(119.5,78){\makebox(0,0)[cc]{$T$}}
\put(113.5,108.75){\makebox(0,0)[cc]{$E$}}
\put(196.75,87.25){\makebox(0,0)[cc]{$C_2$}}
\end{picture}
\end{center}
\caption{Constructing $f:X\to \Gamma$.}\label{Figure 5}
\end{figure}\\
Consider the line bundles $\oo_{Y_2}$, $\oo_{Y_2}(\Delta)$, $\oo_{Y_2}(E)$ over the surface $Y_2$. From \cite[p. 16-17]{MV}, we obtain the Table \ref{table1}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
& $deg_E$ & $deg_T$ & restriction to $\Delta$ & restriction to $S$ \\
\hline
$\oo_{Y_2}$ & $0$ & $0$ & $\oo_{\Gamma}$ & $\oo_{\Gamma}$ \\
$\oo_{Y_2}(\Delta)$ & $1$ & $0$ & $\oo_{\Gamma}(-\gamma)$ & $\oo_{\Gamma}$ \\
$\oo_{Y_2}(E)$ & $-1$ & $1$ & $\oo_{\Gamma}(\gamma)$ & $\oo_{\Gamma}(\gamma)$ \\
\hline
\end{tabular}\caption{}\label{table1}
\end{center}
\end{table}\\
We construct a surface $X$ by identifying $S$ with $A$ and $\Delta$ with $B$. The surface $X$ comes
equipped with a projection $f:X\to \Gamma$. The fibers over all the points $\gamma'\neq \gamma$ are
shown in Figure \ref{Figure6}, while the fiber over the point $\gamma$ is shown in Figure \ref{Figure7}.
\begin{figure}[h!]
\begin{center}
\unitlength .65mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(122.75,50.5)(0,145)
\qbezier(49.25,173.5)(76.875,158.125)(102,164.25)
\qbezier(87,153.5)(95.125,175.625)(122.75,183.25)
\qbezier(17.5,153.25)(52.125,160.125)(62.25,184.5)
\put(17.75,157){\makebox(0,0)[cc]{$C_1$}}
\put(30.75,153.75){\makebox(0,0)[cc]{$h$}}
\put(72.5,167){\makebox(0,0)[cc]{$\Gamma$}}
\put(64.25,162.5){\makebox(0,0)[cc]{$1$}}
\put(113.25,183.75){\makebox(0,0)[cc]{$C_2$}}
\put(115,173.75){\makebox(0,0)[cc]{$g-h-1$}}
\end{picture}
\end{center}
\caption{The general fiber of $f:X\to \Gamma$.}\label{Figure6}
\end{figure}
\begin{figure}[h!]
\begin{center}
\unitlength .65mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(93.5,55)(0,135)
\put(21.5,170.5){\line(1,0){72}}
\qbezier(-1.75,140.5)(43.375,159.375)(28,182.75)
\qbezier(51.25,184.25)(39.5,162.25)(52.75,134.25)
\qbezier(81.25,185)(59.125,172.125)(92.5,134.75)
\put(88.25,173){\makebox(0,0)[cc]{$E$}}
\put(1.25,147.25){\makebox(0,0)[cc]{$C_1$}}
\put(52.75,145.75){\makebox(0,0)[cc]{$C_2$}}
\put(87.75,145.5){\makebox(0,0)[cc]{$\Gamma$}}
\end{picture}
\end{center}
\caption{The special fiber of $f:X\to \Gamma$.}\label{Figure7}
\end{figure}\\
Let $j$, $k$, $t$ be integers. Consider a vector bundle on $C_1$ of rank $r-1$ and degree $j$, we pull-back it on $C_1\times \Gamma$ and call it $H^j$. Similarly consider a vector bundle on $C_2$ of rank $r-1$ and degree $k$, we pull-back it on $C_2\times \Gamma$ and call it $P^k$.
Consider the following vector bundles
$$
\begin{array}{lcl}
M_{C_1\times \Gamma}^{j,k,t}:=H^j\oplus\pi_1^*\oo_{\Gamma}(t\gamma) & \text{on} & C_1\times \Gamma ,\\
M_{C_2\times \Gamma}^{j,k,t}:=P^k\oplus\pi_3^*\oo_{\Gamma}(j+k+t-d)\gamma) &\text{on} & C_2\times \Gamma,\\
M_{Y_2}^{j,k,t}:=\oo_{Y_2}^{r-1}\oplus\oo_{Y_2}\left((d-j-k)\Delta+ tE)\right) &\text{on}& Y_2.
\end{array}
$$
By Table \ref{table1} we have $M_{C_1\times \Gamma}^{j,k,t}|_A\cong M^{j,k,t}_{Y_2}|_S \text{ and } M_{C_2\times \Gamma}^{j,k,t}|_B\cong M_{Y_2}^{j,k,t}|_{\Delta}$. So we can glue the vector bundles in a vector bundle $\mt M_{h}^{j,k,t}$ on the family $f:X\rightarrow \Gamma$. Moreover, by Table \ref{table1}, on the special fiber we have
$$\begin{cases}
& \deg_{C_1}(\mt M_{h}^{j,k,t}|_{f^{-1}(\gamma)})=\deg_{\pi_1^{-1}(\gamma)}(M^{j,k,t}_{C_1\times\Gamma})=j,\\
& \deg_{C_2}(\mt M^{j,k,t}_{h}|_{f^{-1}(\gamma)})=\deg_{\pi_3^{-1}(\gamma)}(M^{j,k,t}_{C_2\times\Gamma})=k,\\
& \deg_{\Gamma}(\mt M^{j,k,t}_{h}|_{f^{-1}(\gamma)})=\deg_{T}(M^{j,k,t}_{Y_2})=t,\\
& \deg_E(\mt M^{j,k,t}_{h}|_{f^{-1}(\gamma)})=\deg_E(M^{j,k,t}_{Y_2})=d-j-k-t.
\end{cases}$$
In particular $\mt M_{h}^{j,k,t}$ has relative degree $d$.
\begin{lem} If $j$, $k$, $t$ satisfies:
$$\left| j-d\frac{2h-1}{2g-2}\right|\leq\frac{r}{2};\:\left|k-d\frac{2g-2h-3}{2g-2}\right|\leq\frac{r}{2};\; \left|t-\frac{d}{2g-2}\right|\leq\frac{r}{2}$$
then $\mt M_h^{j,k,t}$ is properly balanced.
\end{lem}
\begin{proof}We can reduce to check the condition just on the special fiber. By Remark \ref{balcon}, it is enough to check the inequalities on $C_1$, $C_2$ and $\Gamma$; this follows easily from the numerical assumptions.
\end{proof}
For any $1\leq h\leq \frac{g-2}{2}$ choose $j(h)$, resp. $t(h)$, satisfying the first, resp. third, inequality of lemma (observe that such numbers are not unique in general). For every $k\in J_{h+1}$ we call $\widetilde {F}_h^k$ the family $f:X\rightarrow\Gamma$ with the properly balanced vector bundle $$\mt M_{h}^{j(h),\lfloor d\frac{2g-2h-3}{2g-2} +\frac{r}{2}\rfloor-k, t(h)}.$$ As before we compute the degree of the boundary line bundles to the curves $\widetilde {F}_{h}^k$ using the fact that they are liftings of families $F_h$ of \cite[p. 156]{AC87}. We get
$$
\begin{cases}
\deg_{\widetilde{F}_{h}^k}\oo(\widetilde\delta_{h+1}^k)=-1,\\
\deg_{\widetilde{F}_{h}^k}\oo(\widetilde\delta_{h+1}^j)=0 & \mbox{if }j\neq k,\\
\deg_{\widetilde{F}_{h}^k}\oo(\delta_i^j)=0 & \mbox{if } h+1<i,\mbox{ for any }j\in J_i.
\end{cases}
$$
Indeed, the first two relations follow by
$$
\deg_{\widetilde{F}_{h}^k}\oo\left(\sum_{j\in J_{h+1}}\widetilde\delta_{h+1}^j\right)=\deg_{F_h}\oo(\delta_{h+1})=-1
$$
(see \cite[p. 157]{AC87}) and the fact that $\widetilde{F}_h^k$ does not meet $\widetilde\delta_{h+1}^j$ for $j\neq k$. The last follows from the fact $\widetilde{F}_h^k$ does not meet $\widetilde\delta_i^j$ for $i>h+1$.\\\\
\begin{dimo} \emph{\ref{indbou}.} We know that there exists $n_*$ such that $\CVc$ and $\overline{\mt U}_{n}$ have the same Picard groups for $n\geq n_*$. We can suppose $n_*$ big enough such that families constructed before define curves in $\overline{\mathcal U}_n$ for $n\geq n_*$. Suppose that there exists a linear relation
$$
\oo\left(\sum_i\sum_{j\in J_i}a_i^j\widetilde\delta_i^j\right)\cong\oo\in \Pic(\overline{\mt U}_n)
$$
where $a_i^j$ are integers. Pulling back to the curve $\widetilde F\rightarrow \overline{\mt U}_n$ we deduce $a_0=0$. Pulling back to the curves $\widetilde{F'}_1^j\rightarrow \overline{\mt U}_n$ and $\widetilde{F'}_2^j\rightarrow \overline{\mt U}_n$ (in the range of degrees where they are defined) we deduce $a_1^j=0$ for any $j\in J_1$. Pulling back to the curve $\widetilde{F}_{h}^j\rightarrow \overline{\mt U}_n$ we deduce $a_{h+1}^j=0$ for any $j\in J_{h+1}$ and $1\leq h\leq \frac{g-2}{2}$. This concludes the proof.
\end{dimo}\\
We have a similar result for the rigidified stack $\CVr$.
\begin{cor}\label{boundrig}
We have an exact sequences of groups
$$
0\longrightarrow\bigoplus_{i=0,\ldots,\lfloor g/2\rfloor}\oplus_{j\in J_i}\langle\oo(\widetilde\delta_i^j)\rangle\longrightarrow \Pic(\CVr)\longrightarrow \Pic(\Vr)\longrightarrow 0
$$
where the right map is the natural restriction and the left map is the natural inclusion.
\end{cor}
\begin{proof}As before the only thing to prove is the independence of the boundary line bundles in $\Pic(\CVr)$. By Theorem \ref{indbou} and Corollary \ref{boundarycor}, we can reduce to prove the injectivity of $\nu_{r,d}^*:\Pic(\CVr)\rightarrow \Pic(\CVc)$. A quick way to prove this it is using the Leray spectral sequence associated to the rigidification morphism $\nu_{r,d}:\CVc\rightarrow\CVr$ as in the \S\ref{comparing}.
\end{proof}
\begin{rmk}\label{hstable}
As observed before we have that the boundary line bundles are independent on the Picard groups of $\CVc$, $\CVc^{Pss}$, $\CVr$, $\CVr^{Pss}$. A priori we do not know if $\widetilde\delta_i^j$ is a divisor of $\CVc^{Hss}$ for any $i$ and $j\in J_i$, because it can be difficult to check when a point $(C,\mt E)$ is H-semistable if $C$ is singular. But as explained in Remark \ref{defhsem}, if $(C,\mt E)$ is P-stable then it is also H-stable. By Proposition \ref{T}, we know that if $\widetilde\delta_i^j$ is a non-extremal divisor the generic point of $\widetilde\delta_i^j$ is P-stable, in particular it is H-stable.
\end{rmk}
The end of the section is devoted to prove that also the extremal divisors are in $\CVc^{Hss}$, more precisely the generic points of the extremal divisors in $\CVc$ are strictly $H$-semistable. To this aim, we will use the following criterion to prove strictly H-semistability.
\begin{lem}\label{sHs}Assume that $g\geq 2$. Let $(C,\mt E)\in \CVc$ such that $C$ has two irreducible smooth components $C_1$ and $C_2$ of genus $1\leq g_{C_1}\leq g_{C_2}$ meeting at
$N$ points $p_1,\ldots,p_N$. Suppose that $\mt E_{C_1}$ is direct sum of stable vector bundles with the same rank $q$ and same degree $e$ such that $e/q$ is equal to the slope of $\mt E_{C_1}$ and $\mt E_{C_2}$ is a stable vector bundle. If $\mt E$ has multidegree
$$
(deg_{C_1}\mt E,deg_{C_2}\mt E)=\left(d\frac{\omega_{C_1}}{\omega_C}-N\frac{r}{2},d\frac{\omega_{C_2}}{\omega_C}+N\frac{r}{2}\right)\in \mathbb Z^2
$$
then $(C,\mt E)$ is strictly P-semistable and strictly H-semistable.
\end{lem}
\begin{proof}By Proposition \ref{T}, $\mt E$ is P-semistable. We observe that multidegree condition is equivalent to $\omega_C\chi(\mt E_{C_1})=\omega_{C_1}\chi(\mt E)$, so $\mt E$ is strictly P-semistable. Suppose that $\mt M$ is a destabilizing subsheaf of $\mt E$ of multirank $(m_1,m_2)$. Consider the exact sequence
$$
\xymatrix{
0\ar[r] & \mt E_{C_2}(-\sum_1^N p_i)\ar[r] &\mt E\ar[r] &\mt E_{C_1}\ar[r] & 0\\
0\ar[r] & \mt M_2\ar[r]\ar@{^{(}->}[u] &\mt M\ar[r]\ar@{^{(}->}[u] &\mt M_1\ar[r]\ar@{^{(}->}[u] & 0
}
$$
From this we have
$$
\chi(\mt M)=\chi(\mt M_1)+\chi(\mt M_2)\leq \frac{m_1}{r}\chi (\mt E_{C_1})+\frac{m_2}{r}\chi (\mt E_{C_2}(-\sum p_i))=\frac{m_1\omega_{C_1}+m_2\omega_{C_2}}{r\omega_C}\chi (\mt E).
$$
By hypothesis, $\mt E_{C_2}$ stable. So we have two possibilities: $\mt M_2$ is $0$ or $\mt E_{C_2}(-\sum_1^N p_i)$, because otherwise the inequality above is strict. Suppose that $\mt M_2=0$. Then $\mt M=\mt M_1$ which implies that $\mt M\subset \mt E_{C_1}(-\sum p_i)$ so the inequality above is strict. Thus we have just one possibility: if $\mt M$ is destabilizing sheaf then $\mt E_{C_2}(-\sum_1^N p_i)\subset \mt M$.\\\\
In \cite[\S 2.2]{Sc} there is the following criterion to check if a point is H-semistable. A point $(C,\mt E)$ is H-semistable if and only if $(C^{st},\pi_*\mt E)$ is P-semistable and for any one-parameter subgroup $\lambda$ such that $\mt E$ is strictly P-semistable for $\lambda$ then $(C,\mt E)$ is Hilbert-semistable for $\lambda$. Observe that, in our case, $(C,\mt E)=(C^{st},\pi_*\mt E)$.\\
Let $n$ be a natural number big enough such that $\CVc^{Pss}\subset\overline{\mt U}_n$, set $V_n:=H^0(C,\mt E(n))$ and let $B_n:=\{v_1,\ldots,v_{\dim V_n}\}$ be a basis for $V_n$ such that $\lambda$ is given with respect to this basis by the weight vector
$$
\sum_{i=1}^{\dim V_n-1}\alpha_i(\underbrace{i-\dim V_n,\ldots,i-\dim V_n}_i,\underbrace{i, \ldots,i}_{i-\dim V_n})
$$
where $\alpha_i$ are non-negative rational numbers. $\mt E$ is strictly P-semistable with respect to $\lambda$ if and only if there exists a chain of subsheaves $\mt F_1\subset\ldots\subset\mt F_k$ such that
\begin{itemize}
\item $\sum_i^k\alpha_i\left(\chi (\mt E(n)) \left(\sum_j rk\left(\mt F_{i|_{C_j}}\right)\omega_{C_j}\right)-\chi (\mt F_i(n))r\omega_C\right)=0$, in other words $\mt F_j$ are destabilizing sheaves.
\item $H^0(q_C)(Z_j)=H^0(\mt F_j(n))$ where $q_C:V_n\otimes \oo_C\rightarrow \mt E(n)$ is any surjective morphism of vector bundles and $Z_{\bullet}$ is the filtration induced by the one-parameter $\lambda$.
\end{itemize}
Now, we fix the morphism $q_C$ and we set $\det\mt E(n):=\mt L_n$. Consider the morphism
$
S^m\bigwedge^rV_n\rightarrow H^0(C,\mt L_n^m)
$
induced by $q_c$. The one-parameter subgroup $\lambda$ acts on this morphism. Let $w(m)$ be the minimum among the sums of the weights of the elements of the basis $S^m\bigwedge^r B_n$ of $S^m\bigwedge^rV_n$ which induce, by using $q_C$, a basis of $H^0(C,\mt L_n^m)$. So $(C,\mt E)$ is Hilbert-semistable for $\lambda$ if and only if $w(m)\leq 0$ for $m\gg 0$ (see \cite[Recall 1.5]{T98}).
It is enough to check the Hilbert-semistability for the one-parameter subgroups $\lambda$ such that the associated chain of destabilizing sheaves is maximal. By hypothesis $\mt E_{C_1}=\bigoplus_{i=0}^k\mt G_i$, where $\mt G_0=0$ and $\mt G_i$ stable bundle of rank $q$ and same slope of $\mt E_{C_1}$. Observe that $\mt F_j/\mt E_{C_2}(-\sum_i^Np_i)\cong\bigoplus_{i=0}^j\mt G_i$. Moreover if we set $\widetilde Z_j:=\langle v_{\dim Z_{j-1}+1},\ldots,v_{\dim Z_j}\rangle$ for $j=1,\ldots,k$ and $\widetilde Z_0:=Z_0$ we have
$$
\bigwedge^rV_n=\bigoplus_{\rho_0,\ldots,\rho_{k}|\sum\rho_j=r}W_{\rho_0,\ldots,\rho_{k}}, \quad\text{ where }\quad
W_{\rho_0,\ldots,\rho_{k}}:=\bigwedge^{\rho_0}\widetilde Z_0\otimes\ldots\otimes\bigwedge^{\rho_{k}}\widetilde Z_{k}.
$$
An element of the basis $\bigwedge^rB_n$ contained in $W_{\rho_1,\ldots,\rho_{k}}$ has weight $w_{\rho_0,\ldots,\rho_{k}}(n)=\rho_0\gamma_0(n)+\ldots+\rho_{k}\gamma_{k}(n)$. Where $\gamma_j(n)$ is the weight of an element of $B_n$ inside $\widetilde Z_j$, i.e.
$$
\gamma_j(n)=\sum_{i=0}^{k-1}\alpha_i\chi(\mt F_i(n))-\sum_{i=j}^{k}\alpha_i\chi (\mt E(n))\quad \text{where}\quad\alpha_k=0
$$
As in \cite[p. 186-187]{Sc} the space of minimal weights which produces sections which do not vanish on $C_1$ is $W^1_{min}:=W_{0,q,\ldots,q}$. The associated weight is
$$
w_{1,min}(n):=\sum_{i=0}^{k-1}\alpha_i\left(\chi(\mt F_i(n))r-\chi(\mt E(n))iq\right).
$$
Moreover a general section of $W^1_{min}$ does not vanish at $p_i$. By \cite[Corollary 2.2.5]{Sc} the space $S^mW^1_{min}$ generates $
H^0\left(C_1,\left(\mt L_{n|C_1}\right)^m\right)
$, so that the elements of $S^m\bigwedge^rB_n$ inside $S^mW^1_{min}$ will contribute with weight
$$
K_1(n,m):=m(m(\deg\mt E_{C_1}+nr\omega_{C_1})+1-g_{C_1})w_{1,min}(n)
$$
to a basis of $H^0\left(C_1,\left(\mt L_{n|C_1}\right)^m\right)$.\\
On the other hand the space of minimal weights which produces sections which do not vanish on $C_2$ is $W^2_{min}:=W_{r,0,\ldots,0}$. The associated weight is
$$w_{2,min}(n):=\sum_{i=0}^{k-1}\alpha_i(\chi(\mt F_i(n))-\chi(\mt E(n)))r$$
A general section of $W^2_{min}$ vanishes at $p_i$ with order $r$. By \cite[Corollary 2.2.5]{Sc}, the space $S^mW^2_{min}$ generates $H^0\left(C_1,\left(\mt L_{n|C_2}\left(-r\sum p_i\right)\right)^m\right)$, in particular the elements of $S^m\bigwedge^rB_n$ inside $S^mW^2_{min}$ will contribute with weight
$$
K_2(n,m):=m(m(\deg\mt E_{C_2}-rN+nr\omega_{C_2})+1-g_{C_2})w_{2,min}(n)
$$
to a basis of $H^0\left(C_2,\left(\mt L_{n|C_2}\right)^m\right)$. It remains to find the elements in $S^m\bigwedge^rB_n$ which produce sections of minimal weight in $H^0\left(C_2,\left(\mt L_{n|C_2}\right)^m\right)$ vanishing with order less than $mr$ on $p_i$ for $i=1,\ldots,N$. By a direct computation, we can see that the space of minimal weights which gives us sections with vanishing order $r-s$ at $p_i$ such that $tq\leq s\leq (t+1)q$ (where $0\leq t\leq k-1$) is
$$
\mathbb O_{r-s}:=W_{r-s,\underbrace{q\ldots,q}_t,s-tq,0,\ldots,0}.
$$
The associated weight is
$$
w_{2,p_i}^{r-s}(n):=\sum_{i=0}^{k-1}\alpha_i\left(\chi(\mt F_i(n))-\chi(\mt E(n))\right)r+\sum_{i=0}^{t}(s-iq)\alpha_i\chi(\mt E(n)).
$$
For any $0\leq \nu\leq mr-1$ and $1\leq i\leq N$, we must find an element of minimal weight in $S^m\bigwedge^rB_n$ which produces a section in $H^0\left(C_2,\left(\mt L_{n|C_2}\right)^m\right)$ vanishing with order $\nu$ at $p_i$. Observe first that we can reduce to check it on the subspace
$$
S^m\mathbb O=\bigoplus_{m_0,\ldots,m_r|\sum m_i=m}S^{m_0}\mathbb O_0\otimes\ldots\otimes S^{m_r}\mathbb O_r.
$$
A section in $S^{m_0}\mathbb O_0\otimes\ldots\otimes S^{m_r}\mathbb O_r$ vanishes with order at least $\nu=m_1+2m_2+\ldots+rm_r$ at $p_i$ and we can find some with exactly that order. As explained in \cite[p. 191-192]{Sc}, an element of $S^m\bigwedge^rB_n$ of minimal weight, such that it produces a section of order $\nu$ at $p_i$, lies in
$$
S^j\mathbb O_{t-1}\otimes S^{m-j}\mathbb O_t
$$
where $\nu=mt-j$ and $1\leq j\leq m$. So the mininum among the sums of the weights of the elements in $S^m\bigwedge^r B_n$ which give us a basis of
$$
H^0\left(C_2,\left(\mt L_{n|C_2}\right)^m\right)/H^0\left(C_2,\left(\mt L_{n|C_2}(-r\sum p_i)\right)^m\right)
$$
is
\begin{eqnarray*}
D_2(n,m)&=&N\left(m^2(w_{2,p_i}^1(n)+\ldots+w_{2,p_i}^r(n))+\frac{m(m+1)}{2}(w_{2,p_i}^0(n)-w_{2,p_i}^r(n))\right)=\\
&=&N\sum_{i=0}^{k-1}\alpha_i\left(m^2\left(\chi(\mt F_i(n))r^2-\chi(\mt E(n))\left(r^2-\frac{(r-iq-1)(r-iq)}{2}\right) \right)\right. +\\
& & \left.+ \frac{m(m+1)}{2}(r-iq)\chi(\mt E(n))\right).
\end{eqnarray*}
Then a basis for $H^0\left(C_2,\left(\mt L_{n|C_2}\right)^m\right)$ will have minimal weight $K_2(n,m)+D_2(n,m)$.\\\\
As in \cite[p.192-194]{Sc} we obtain that $(C,\mt E)$ will be Hilbert-semistable for $\lambda$ if and only if exists $n^*$ such that for $n\geq n^*$
$$
P(n,m)=K_1(n,m)+K_2(n,m)+D_2(n,m)-mNw_{1,min}(n)\leq 0
$$
as polynomial in $m$. A direct computation shows that $P(n,m)\leq 0$ as polynomial in $m$. So $(C,\mt E)$ is H-semistable.\\
It remains to check that $(C,\mt E)$ is not H-stable. It is enough to construct a one-parameter subgroup $\lambda$ such that $(C,\mt E)$ is strictly P-semistable respect to $\lambda$ and $P(n,m)\equiv0$ as polynomial in $m$. Fix a basis of $W_n:=H^0(C,\mt E(n)_{C_2}(-\sum p_i))$ and complete to a basis $B_n:=\{v_1,\ldots,v_{\dim V_n}\}$ of $V_n:=H^0(C,\mt E(n))$. We define the one-paramenter subgroup $\lambda$ of $SL(V_n)$ diagonalized by the basis $B_n$ with weight vector
$$
(\underbrace{\dim W_n-\dim V_n,\ldots,\dim W_n-\dim V_n}_{\dim W_n},\underbrace{\dim W_n, \ldots,\dim W_n}_{\dim W_n-\dim V_n}).
$$
A direct computation shows $P(n,m)\equiv 0$ (observe that it is the case when $\alpha_1=1$ and $\alpha_i=0$ for $2\leq i\leq k-1$ in the previous computation), which implies that $(C,\mt E)$ is stricly H-semistable.
\end{proof}
\begin{prop}\label{hextr}The generic point of an extremal boundary divisor is strictly P-semistable and strictly H-semistable.
\end{prop}
\begin{proof}Fix $i\in\{0,\ldots,\lfloor g/2\rfloor\}$ such that $\widetilde\delta_i^0$ is an extremal divisor. By Lemma \ref{T2} the generic point of the extremal boundary $\widetilde\delta_i^0$ is a curve $C$ with two irreducible smooth components $C_1$ and $C_2$ of genus $i$ and $g-i$ and a vector bundle $\mt E$ such that $\mt E_{C_1}$ is a stable vector bundle (or direct sum of stable vector bundles with same slope of $\mt E_{C_1}$ if $i=1$) and $\mt E_{C_2}$ is stable vector bundle. By Lemma \ref{sHs} the generic point of $\widetilde\delta_i^0$ is strictly P-semistable and stricly H-semistable.\\
Suppose now that $i\neq g/2$ and consider the extremal boundary divisor $\widetilde\delta_i^r$. Take a point $(C,\mt E)\in \widetilde\delta_i^0$ as above. Consider the destabilizing subsheaf $\mt E_{C_2}(-p)\subset \mt E$, where $p$ is the unique node of $C$. Fix a basis of $W_n:=H^0(C,\mt E(n)_{C_2}(-p))$ and complete to a basis $\mt V:=\{v_1,\ldots,v_{dim V_n}\}$ of $V_n=H^0(C,\mt E(n))$. We define the one-parameter subgroup $\lambda$ of $SL(V_n)$ given with respect to the basis $\mt V$ by the weight vector
$$
(\underbrace{\dim W_n-\dim V_n,\ldots,\dim W_n-\dim V_n}_{\dim W_n},\underbrace{\dim W_n, \ldots,\dim W_n}_{\dim W_n-\dim V_n}).
$$
We have seen in the proof of Lemma \ref{sHs} that the pair $(C,\mt E)$ is strictly H-semistable respect to $\lambda$. In particular the limit respect to $\lambda$ is strictly H-semistable. The limit will be a pair $(C',\mt E')$ such that $C'$ is a semistable model for $C$ and $\mt E'$ a properly balanced vector bundles such that the push-forward in the stabilization is the P-semistable sheaf $$\mt E_{C_2}(-p)\oplus\mt E_{C_1}.$$
By Corollary \ref{NSrmk}(iii), $\mt E'_{C_1}\cong \mt E_{C_1}$ and $\mt E'_{C_2}\cong \mt E_{C_2}(-p)$. In particular, $\mt E$ has multidegree
$$
\left(\deg\mt E'_{C_1},\deg\mt E'_R,\deg\mt E'_{C_2}\right)=\left(\deg_{C_1}\mt E,r,\deg_{C_2}\mt E-r\right)=\left(d\frac{2i-1}{2g-2}-\frac{r}{2}, r,d\frac{2(g-i)-1}{2g-2}-\frac{r}{2}\right)
$$
Smoothing all nodal points on the rational chain $R$ except the meeting point $q$ between $R$ and $C_2$, we obtain a generic point $(C'',\mt E'')$ in $\widetilde\delta_i^r$. It is H-semistable by the openess of the semistable locus. Let $W''_n$ be a basis for $H^0\left(C'',\mt E''(n)_{C_1}(-q)\right)$ and complete to a basis $\mt V''_n$ of $H^0(C'',\mt E''(n))$. Let $\lambda''$ be the one parameter subgroup defined by the weight vector (with respect to the basis $B$)
$$
(\underbrace{\dim W''_n-\dim V''_n,\ldots,\dim W''_n-\dim V''_n}_{\dim W''_n},\underbrace{\dim W''_n, \ldots,\dim W''_n}_{\dim W''_n-\dim V''_n}).
$$
As in the proof of Lemma \ref{sHs}, a direct computation shows that $(C'',\mt E'')$ is strictly H-semistable respect to $\lambda''$, then also strictly P-semistable concluding the proof.
\end{proof}
Using this, we obtain
\begin{cor}\label{boundrigH}
We have an exact sequences of groups
$$
0\longrightarrow\bigoplus_{i=0,\ldots,\lfloor g/2\rfloor}\oplus_{j\in J_i}\langle\oo(\widetilde\delta_i^j)\rangle\longrightarrow Pic(\CVc^{Hss})\longrightarrow Pic(\Vc^{ss})\longrightarrow 0
$$
where the right map is the natural restriction and the left map is the natural inclusion. The same holds for the rigidification $\CVr^{Hss}$.
\end{cor}
\subsection{Picard group of $\Vc$.}\label{liscio}
In this section we will prove Theorem \ref{pic}. Note that the first three line bundles on the theorem are free generators for the Picard group of $\Jc$ (see Theorem \ref{picjac}) and the fourth line bundle restricted to $\mt Vec_{=\mathcal L,C}$ freely generates its Picard group (see Theorem \ref{fibers}). By Lemma \ref{redsemistable} together with Theorem \ref{fibers}(i) and Remark \ref{caso2schifo}, we see that Theorem \ref{pic}(i) is equivalent to:
\begin{teo}\label{picsmooth}Assume that $g\geq 2$. For any smooth curve $C$ and $\mathcal L$ line bundle of degree $d$ over $C$ we have an exact sequence.
$$
0\lra Pic(\Jc)\lra Pic(\Vc^{ss})\lra Pic(\mathcal Vec^{ss}_{=\mathcal L,C})\lra 0
$$
\end{teo}
For the rest of the subsection we will assume $g\geq 2$. Observe that the above theorem together with Lemma \ref{redsemistable}, Theorem \ref{indbou} and Corollary \ref{boundrigH} imply Theorem \ref{pic}(ii). Using Remark \ref{hstable} together with Proposition \ref{hextr}, we deduce Theorem \ref{pic}(iii).\vspace{0.2cm}
Let $\jc$ (resp. $\jr$) the open substack of $\Jc$ (resp. $\Jr$) which parametrizes the pairs $(C,\mt L)$ such that $\Aut(C,\mt L)=\mathbb G_m$.
Note that $\jr$ is a smooth irreducible variety, more precisely it is a moduli space of isomorphism classes of line bundle of degree $d$ over a curve $C$ satisfying the condition above.
\begin{lem}There are isomorphisms
$$\Pic(\Jc)\cong\Pic(\jc),\quad \Pic(\Jr)\cong\Pic(\jr)$$
induced by the restriction maps.
\end{lem}
\begin{proof}We will prove the lemma for $\jr$, the assertion for $\jc$ will follow directly. We set $\Jr^*:=\Jr\backslash\jr$. By Theorem \ref{picchow}, it is enough to prove that the closed substack $\Jr^*$ has codimension $\geq 2$. First we recall some facts about curves with non-trivial automorphisms: the closed locus $\Jr^{Aut}$ in $\Jr$ of curves with non-trivial automorphisms has codimension $g-2$ and it has a unique irreducible component $\mt J\mt H_g$ of maximal dimension corresponding to the hyperelliptic curves (see \cite[Remark 2.4]{GV}). Moreover in $\mt J\mt H_{d,g}$ the closed locus $\mt{JH}_{d,g}^{extra}$ of hyperelliptic curves with extra-automorphisms has codimension $2g-3$ and it has a unique irreducible component of maximal dimension corresponding to the curves with an extra-involution (for details see \cite[Proposition 2.1]{GV}).\\
By definition, $\Jr^*\subset \Jr^{Aut}$. By the facts above, it is enough to check the dimension of $\Jr^*\cap\mt J\mt H_g\subset\mt J\mt H_g$. With an abuse of notation, the stack $\Jr^*\cap\mt J\mt H_{d,g}$ , i.e. the locus of pairs $(C,\mt L)$ such that $C$ is hyperelliptic and $\Aut(C,\mt L)\neq\mathbb G_m$, will be called $\Jr^*$.\\
If $g\geq 4$, $\mt{JH}_{d,g}$ has codimension $\geq2$, then the lemma follows. If $g=3$ then $\mt J\mt H_{d,3}$ is an irreducible divisor. It is enough to show that $\mt J_{d,3}^*\neq\mt{JH}_{d,3}$ and it is easy to check. If $g=2$, then all curves are hyperelliptic, $\dim\mt J_{d,2}=5$ and $\mt{JH}_{d,g}^{extra}$ has codimension $1$. Consider the forgetful morphism $\mt J_{d,2}^*\rightarrow\mt M_2$.
The fiber at $C$, when is non empty, is the closed subscheme of the Jacobian $J^d(C)$ where the action of $\Aut(C)$ is not free. If $C$ is a curve without extra-automorphisms then the fiber has dimension $0$. In particular if the open locus of such curves is dense in $\mt J_{d,2}^*$ then $\dim\mt J_{d,2}^*\leq\dim \mt M_2=3$ and the lemma follows. Otherwise, $\mt J_{d,2}^*$ can have an irreducible component of maximal dimension which maps in the divisor $\mt {H}^{extra}_{2}\subset\mt M_2$ of curves with an extra-involution. In this case $\dim\mt J_{d,2}^*<\dim \mt {H}^{extra}_{2}+\dim J^d(C)=4$, which concludes the proof.
\end{proof}
We denote with $\vc$ (resp. $\vr$) the open substack of $\Vc^{ss}$ (resp. $\Vr^{ss}$) of pairs $(C,\mt E)$ such that $\Aut(C,\det\mt E)=\mathbb G_m$. By lemma above, Theorem \ref{picsmooth} is equivalent to prove the exactness of
$$
0\lra \Pic(\jc)\lra \Pic(\vc)\lra \Pic(\mathscr V ec^{ss}_{=\mt L,C})\lra 0.
$$
The morphism $det:\vc\longrightarrow \jc$ is a smooth morphism of Artin stacks. Let $\Lambda$ be a line bundle over $\vc$, which is obviously flat over $\jc$ by flatness of the map $det$. The first step is to prove the following
\begin{lem}Suppose that $\Lambda$ is trivial over any geometric fiber. Then $det_*\Lambda$ is a line bundle on $\jc$ and the natural map $det^*det_*\Lambda\longrightarrow \Lambda$ is an isomorphism.
\end{lem}
\begin{proof}
Consider the cartesian diagram
$$
\xymatrix{
\vc\ar[d] &V_H\ar[l] \ar[d]\\
\jc & H\ar[l] }
$$
where the bottom row is an atlas for $\jc$. We can reduce to control the isomorphism locally on $V_H\to H$. Suppose that the following conditions hold
\begin{enumerate}[(i)]
\item $H$ is an integral scheme,
\item the stack $V_H$ has a good moduli scheme $U_H$,
\item $U_H$ is proper over $H$ with geometrically irreducible fibers.
\end{enumerate}
Then, by Seesaw Principle (see Corollary \ref{seesaw}), we have the assertion. So it is enough to find an atlas $H$ such that the conditions (i), (ii) and (iii) are satisfied.\\
We fix some notations: since the stack $\vc$ is quasi-compact, there exists $n$ big enough such that $\vc\subset \overline{\mt U}_n=[H_n/GL(V_n)]$. So we can suppose $d$ big enough such that $\vc\subset\overline{\mt U}_0$. Let $Q$ be the open subset of $H_0$ such that $\vc=[Q/G]$, where $G:=GL(V_0)$. Analogously, we set $\jc=[H/\Gamma]$. Denote by $Z(\Gamma)$ (resp. $Z(G)$) the center of $\Gamma$ (resp. of $G$) and set $\widetilde{G}=G/Z(G)$, $\widetilde{\Gamma}=\Gamma /Z(\Gamma)$. Note that $Z(G)\cong Z(\Gamma)\cong\mathbb G_m$. As usual we set $\mt BZ(\Gamma):=[\Spec\, k/Z(\Gamma)]$. Since $\jc$ is integral then $H$ is integral, satisfying the condition (i).
We have the following cartesian diagrams
$$
\xymatrix@!0@R=3pc@C=5pc{
& Q\ar[d] & & Q\times_{\jr} H\ar[ll]\ar[d] & & Q\times_{\jc} H\ar[ll]_{\pi}\ar[d]\\
& \vc\ar[dl]\ar@{-}[d] & & [Q\times_{\jr}H/G]\ar[ll]\ar[dl]^q\ar[ddd] & & [Q\times_{\jc}H/G]\cong V_H\ar[ddd]\ar[ll]_>>>>>>>{p}\\
\vr\ar[d]&\ar@{-}[d] & [Q\times_{\jr}H/\widetilde G]\ar[ll]\ar[d]\\
U^o_{r,d,g}\ar[dd] &\ar[d] & U_H\ar[dd]\ar[ll]\\
& \jc\ar[dl]&\ar[l] & H\times \mt BZ(\Gamma)\ar[dl]\ar@{-}[l] & &H\ar[ll]\\
\jr & & H\ar[ll]\ar@{=}[urrr]
}
$$
where $U^o_{r,d,g}$ is the open subscheme in $\overline{U}_{r,d,g}$ of pairs $(C,\mt E)$ such that $C$ is smooth and $\Aut(C,\det\mt E)=\mathbb G_m$. Note that $U_H$ is proper over $H$, because $U_{r,d,g}^o\rightarrow \jr$ is proper. In particular, the geometric fiber over a $k$-point of $H$ which maps to $(C,\mt L)$ in $ \jr$ is the irreducible projective variety $U_{\mt L,C}$.\\
So it remains to prove that $U_H$ is a good moduli space for $V_H$. Since $V_H$ is a quotient stack, it is enough to show that $U_H$ is a good $G$-quotient of $Q\times_{\jc}H$. The good moduli morphisms are preserved by pull-backs \cite[Proposition 3.9]{Al}, in particular $U_H$ is a good $G$-quotient of $Q\times_{\jr}H$.
Consider the commutative diagram
$$
\xymatrix{
Q\times_{\jc}H\ar[r]^\pi\ar[rd]^\beta & Q\times_{\jr} H\ar[r]\ar[d]^\alpha & U_H\ar[ld]\\
& H &}
$$
\underline{Claim:} the horizontal maps makes $U_H$ a categorical $G$-quotient of $Q\times_{Jac^0}H$.\\ Suppose that the claim holds. Then $U_H$ is a good $G$-quotient also for $Q\times_{\jc}H$, because the horizontal maps are affine (see \cite[1.12]{Mum94}), and we have done.\\
It remains to prove the claim. The idea for this part comes from \cite[Section 2]{H}. Since the map $Q\rightarrow \jr$ is $G$-invariant then $Q\times_{\jr}H\rightarrow H$ is $G$-invariant. In particular we can study the action of $Z(G)$ over the fibers of $\alpha$. Fix a geometric point $h$ on $H$ and suppose that its image in $\jc$ is the pair $(C,\mathcal L)$. Then the fiber of $\beta$ (resp. of $\alpha$) over $h$ is the fine moduli space of the triples $(\mathcal E,B,\phi)$ (resp. of the pairs $(\mathcal E,B)$), where $\mathcal E$ is a semistable vector bundle on $C$, $B$ a basis of $H^0(C,\mathcal E)$ and $\phi$ is an isomorphism between the line bundles $\det\mt E$ and $\mathcal{L}$. If $g\in Z(G)$ we have $g.(\mathcal E,B,\phi)=(\mathcal E,gB,\phi)$ and $g.(\mathcal E,B)=(\mathcal E,gB)$. Observe that the isomorphism $g.Id_{\mathcal E}$ gives us an isomorphism between the pairs $(\mathcal E, B)$ and $(\mathcal E, gB)$ and between the triples $(\mathcal E, B,\phi)$ and $(\mathcal E, gB, g^r\phi)$. So $g.(\mathcal E,B,\phi)=(\mathcal E,B,g^{-r}\phi)$ and $g.(\mathcal E,B)=(\mathcal E,B)$. On the other hand, $\pi:Q\times_{\jc}H\rightarrow Q\times_{\jr}H$ is a principal $Z(\Gamma)$-bundle and the group $Z(\Gamma)$ acts in the following way: if $\gamma\in Z(\Gamma)$ we have $\gamma.(\mathcal E,B,\phi)=(\mathcal E,B,\gamma\phi)$ and $\gamma.(\mathcal E,B)=(\mathcal E,B)$.\\
This implies that the groups $Z(G)/\mu_r$ (where $\mu_r$ is the finite algebraic group consisting of $r$-roots of unity) and $Z(\Gamma)$ induce the same action on $Q\times_{\jc}H$. Since $\pi:Q\times_{\jc}H\rightarrow Q\times_{\jr}H$ is a principal $Z(\Gamma)$-bundle, any $G$-invariant morphism from $Q\times_{\jc}H$ to a scheme factorizes uniquely through $Q\times_{\jr}H$ and so uniquely through $U_H$ concluding the proof of the claim.
\end{proof}
The next lemma conclude the proof of Theorem \ref{picsmooth}.
\begin{lem}Let $\Lambda$ be a line bundle on $\vc$. Then
$\Lambda$ is trivial on a geometric fiber of $det$ if and only if $\Lambda$ is trivial on any geometric fiber.
\end{lem}
\begin{proof} Consider the determinant map $det:\vc\rightarrow \jc$. Let $T$ be the set of points $h$ (in the sense of \cite[Chap. 5]{LMB}) in $\jc$ such that the restriction $\Lambda_h:=\Lambda_{det^*h}$ is the trivial line bundle.
By Theorem \ref{fibers}(iii), the inclusion
$$
\Pic(U_{\mt L,C})\hookrightarrow\Pic(\mathcal Vec^{ss}_{=\mathcal L,C})\cong\mathbb Z
$$
is of finite index. The variety $U_{\mt L,C}$ is projective, in particular any non-trivial line bundle on it is ample or anti-ample. This implies that $\chi(\Lambda_h^n)$, as polynomial in the variable $n$, is constant if and only if $\Lambda_h$ is trivial. So $T$ is equal to the set of points $h$ such that the polynomial $\chi(\Lambda^n_h)$ is constant. Consider the atlas defined in the proof of precedent lemma $H\rightarrow \jc$. The line bundle $\Lambda$ is flat over $\jc$ so the function
$$
\chi_n:H\rightarrow \mathbb{Z}:\,h=(C,\mathcal L,B)\mapsto\chi(\Lambda^n_h)
$$
is locally constant for any $n$, then constant because $H$ is connected. Therefore, the condition $\chi_n=\chi_m$ for any $n,\, m\in\mathbb Z$ is either always satisfied or never satisfied, which concludes the proof.
\end{proof}
\subsection{Comparing the Picard groups of $\CVc$ and $\CVr$.}\label{comparing}
Assume that $g\geq 2$. Consider the rigidification map $\nu_{r,d}:\Vc\rightarrow\Vr$ and the sheaf of abelian groups $\mathbb G_m$. The Leray spectral sequence
\begin{equation}\label{leray}
H^p(\Vr,R^q\nu_{r,d*}\mathbb{G}_m) \Rightarrow H^{p+q}(\Vc,\mathbb G_m)
\end{equation}
induces an exact sequence in low degrees
$$
0\rightarrow H^1(\Vr,\nu_{r,d*}\mathbb G_m)\rightarrow H^1(\Vc,\mathbb G_m)\rightarrow H^0(\Vr,R^1\nu_{r,d*}\mathbb G_m)\rightarrow H^2(\Vr,\nu_{r,d*}\mathbb G_m).
$$
We observe that $\nu_{r,d*}\mathbb{G}_m=\mathbb G _m$ and that the sheaf $R^1\nu_{r,d*}\mathbb G_m$ is the constant sheaf $H^1(\mt B\mathbb G_m,\mathbb G_m)\cong \Pic(\mt B\mathbb G_m)\cong \mathbb Z$. Via standard coycle computation we see that exact sequence becomes
\begin{equation}\label{lss}
0\longrightarrow \Pic(\Vr)\longrightarrow \Pic(\Vc)\xrightarrow{res} \mathbb{Z}\xrightarrow{obs} H^2(\Vr,\mathbb G_m)
\end{equation}
where $res$ is the restriction on the fibers (it coincides with the weight map defined in \cite[Def. 4.1]{H07}), $obs$ is the map which sends the identity to the $\mathbb G_m$-gerbe class $[\nu_{r,d}]\in H^2(\Vr,\mathbb G_m)$ associated to $\nu_{r,d}:\Vc\rightarrow\Vr$ (see \cite[IV, $\S$3.4-5]{Gi}).
\begin{lem}We have that:
$$
\begin{cases}
res(\Lambda(1,0,0)) =0,\\
res(\Lambda(0,0,1))=d+r(1-g),\\
res(\Lambda(0,1,0))=r(d+1-g),\\
res(\Lambda(1,1,0))=r(d-1+g).
\end{cases}
$$
\end{lem}
\begin{proof}Using the functoriality of the determinant of cohomology, we get that the fiber of $\Lambda(1,0,0)=d_{\pi}(\omega_{\pi})$ over a point $(C,\mt E)$ is canonically isomorphic to
$\det H^0(C,\omega_C)\otimes \det^{-1} H^1(C,\omega_C)$. Since $\mathbb G_m$ acts trivially on $H^0(C,\omega_C)$ and on
$H^1(C, \omega_C)$, we get that $res(\Lambda(1,0,0))=0$.\\
Similarly, the fiber of $\Lambda(0,0,1)$ over a point $(C,\mt E)$ is canonically isomorphic to
$\det H^0(C,\mt E)\otimes \det^{-1} H^1(C,\mt E)$. Since $\mathbb G_m$ acts with weight one on the vector spaces
$H^0(C, \mt E)$ and $H^1(C,\mt E)$, Riemann-Roch gives that
$$res(\Lambda(0,0,1))=h^0(C,\mt E)-h^1(C,\mt E)=\chi(\mt E)=d+r(1-g).$$
The fiber of $\Lambda(0,1,0)$ over a point $(C,\mt E)$ is canonically isomorphic to
$\det H^0(C,\det\mt E)\otimes \det^{-1} H^1(C,\det\mt E)$. Now $\mathbb G_m$ acts with weight $r$ on the vector spaces
$H^0(C, \det\mt E)$ and $H^1(C,\det\mt E)$, so that Riemann-Roch gives
$$res(\Lambda(0,1,0))=r\cdot h^0(C,\det\mt E)-r\cdot h^1(C,\det\mt E)=r\cdot\chi(\det\mt E)=r(d+1-g).$$
Finally, the fiber of $\Lambda(1,1,0)$ over a point $(C,\mt E)$ is canonically isomorphic
to $\det H^0(C,\omega_C\otimes\det\mt E)\otimes \det^{-1} H^1(C,\omega_C\otimes\det\mt E)$. Since $\mathbb G_m$ acts with weight $r$ on the vector spaces $H^0(C, \omega_C\otimes\det\mt E)$ and $H^1(C,\omega_C\otimes\det\mt E)$, Riemann-Roch gives that
$$res(\Lambda(1,1,0))=r\cdot h^0(C,\omega_C\otimes\det\mt E)-r\cdot h^1(C,\omega_C\otimes\det\mt E)=
r\cdot\chi(\omega_C\otimes\det\mt E)=r(d-1+g). $$
\end{proof}
Combining the Lemma above with Theorem \ref{pic}(i) and the exact sequence (\ref{lss}), we obtain
\begin{cor}\label{eccocequasi}
\noindent
\begin{enumerate}[(i)]
\item The image of $\Pic(\Vc)$ via the morphism $res$ of (\ref{lss}) is the subgroup of $\mathbb Z$ generated by $$n_{r,d}\cdot v_{r,d,g}=\left(d+r(1-g), r(d+1-g), r(d-1+g)\right).$$
\item The Picard group of $\Vr$ is (freely) generated by the line bundles $\Lambda(1,0,0)$, $\Xi$ and $\Theta$ (when $g\geq 3$).
\end{enumerate}
\end{cor}
Now we are ready for\\\\\begin{dimo} \emph{\ref{picred}}. Corollary \ref{eccocequasi}(ii) says that the Theorem \ref{picred}(i) is true for the stack $\Vr$. Using the Leray spectral sequence for the (semi)stable locus, we see that the Corollary \ref{eccocequasi} holds also for the stack $\Vr^{(s)s}$, concluding the proof of Theorem \ref{picred}(i).\\
By Corollary \ref{boundrig} and Theorem \ref{picred}(i), the Theorem \ref{picred}(ii) holds for $\CVr$. Using the Lemma \ref{redsemistable}, the same is true for $\CVr^{Pss}$. Finally by Corollary \ref{boundrigH}, Theorem \ref{picred}(ii) holds also for $\CVr^{Hss}$.\\
The Theorem \ref{picred}(iii) follows using the previous parts together with Remark \ref{hstable} and Proposition \ref{hextr}.
\end{dimo}
\begin{rmk}\label{kou}Let $U_{r,d,g}$ be the coarse moduli space of aut-equivalence classes of semistable vector bundles on smooth curves. Suppose that $g\geq 3$. Kouvidakis in \cite{Kou93} gives a description of the Picard group of the open subset $U^*_{r,d,g}$ of curves without non-trivial automorphisms. As observed in Section $3$ of \emph{loc. cit.}, such locus is locally factorial. Since the locus of strictly semistable vector bundles has codimension at least $2$, we can restrict to study the open subset $U^{\star}_{r,d,g}\subset U_{r,d,g}$ of stable vector bundles on curves without non-trivial automorphisms. The good moduli morphism $\Psi_{r,d}:\Vr^{ss}\lra U_{r,d,g}$ is an isomorphism over $U^\star_{r,d,g}$. In other words, we have an isomorphism $\Vr^{\star}:=\Psi_{r,d}^{-1}(U^\star_{r,d,g})\cong U^{\star}_{r,d,g}$. Therefore, we get a natural surjective homomorphism
$$
\psi:\Pic(\Vr^{ss})\cong\Pic(\Vr^{s})\twoheadrightarrow \Pic(\Vr^\star)\cong\Pic(U^\star_{r,d,g})
$$
where the first two homomorphisms are the restriction maps.\\
When $g\geq 4$ the codimension of $\Vr^s\backslash\Vr^\star$ is at least two (see \cite[Remark 2.4]{GV}). Then the map $\psi$ is an isomorphism by Theorem \ref{picchow}.
If $g=3$, the locus $\mt V^{(s)s}_{r,d,3}\backslash\mt V^*_{r,d,3}$ is a divisor in $\mt V^{(s)s}_{r,d,3}$ (see \cite[Remark 2.4]{GV}). More precisely is the pull-back of the hyperelliptic (irreducible) divisor in $\mt M_3$. As line bundle, it is isomorphic to $\Lambda^9$ in the Picard group of $\mt M_3$ (see \cite[Chap. 3, Sec. E]{HM98}). Therefore, by Theorem \ref{picred}(i), we get that $\Pic(U^*_{r,d,3})$ is the quotient of $\Pic(\mt V^{(s)s}_{r,d,3})$ by the relation $\Lambda(1,0,0)^9$.\\
In particular, (when $g\geq 3$) the line bundle $\Theta^s\otimes\Xi^{t}\otimes \Lambda (1,0,0)^u$, where $(s,t,u)\in\mathbb Z^3$, on $U^*_{r,d,g}$ has the same properties of the canonical line bundle $\mathscr L_{m,a}$ in \cite[Theorem 1]{Kou93}, where $m=s\cdot \frac{v_{1,d,g}}{v_{r,d,g}}$ and $a=-s(\alpha+\beta)-t\cdot k_{1,d,g}$.
\end{rmk}
As explained in Section \ref{Sc}, $\CVc^{Hss}$ admits a projective variety as good moduli space. This means, in particular, that the stacks $\CVc^{Hss}$ and $\CVr^{Hss}$ are of finite type and universally closed. Since any vector bundle contains the multiplication by scalars as automorphisms, $\CVc^{Hss}$ is not separated. The next Proposition tell us exactly when the rigidification $\CVr^{Hss}$ is separated.
\begin{prop}\label{poincare}The following conditions are equivalent:
\begin{enumerate}[(i)]
\item $n_{r,d}\cdot v_{r,d,g}=1$, i.e. $n_{r,d}=1$ and $v_{r,d,g}=1$.
\item There exists a universal vector bundle on the universal curve of an open substack of $\CVr$.
\item There exists a universal vector bundle on the universal curve of $\CVr$.
\item The stack $\CVr^{Hss}$ is proper.
\item All H-semistable points are H-stable.
\item $\CVr^{Hss}$ is a Deligne-Mumford stack.
\item The stack $\CVr^{Pss}$ is proper.
\item All P-semistable points are P-stable.
\item $\CVr^{Pss}$ is a Deligne-Mumford stack.
\end{enumerate}
\end{prop}
\begin{proof}The strategy of the proof is the following$$$$
$$
\xymatrix{
(iii)\ar@{=>}[d] & (i)\ar@{<=>}[d]\ar@{=>}[l] &(ix)\ar@{=>}[r] &(v)\ar@{=>}@/_2pc/[ll]\ar@{<=>}[r] \ar@{<=>}[d]&(iv)\\
(ii)\ar@{=>}[ru]& (viii)\ar@{=>}[r]\ar@{=>}[ur] &(vii)\ar@{=>}[ur]& (vi)
}
$$
$(i) \Rightarrow (iii)$. By Corollary \ref{eccocequasi}(i) and the exact sequence (\ref{lss}), any line bundle on $\CVc$ must have weight equal to $c\cdot n_{r,d}\cdot v_{r,d,g}$, where $c\in\mathbb Z$. In particular the condition $(i)$ is equivalent to have a line bundle $\mt L$ of weight $1$ on $\CVc$. Let $\left(\overline{\pi}:\overline{\mt Vec}_{r,d,g,1}\rightarrow \CVc,\mt E\right)$ be the universal pair, we see easily that $\mt E\otimes \overline{\pi}^*\mt L^{-1}$ descends to a vector bundle on $\CVr$ with the universal property.\\
$(iii) \Rightarrow (ii)$ Obvious.\\
$(ii)\Rightarrow (i)$ Suppose that there exists a universal pair $(\mt S_1\rightarrow \mt S,\mt F)$ on some open substack $\mt S$ of $\CVr$. We can suppose that all the points $(C,\mt E)$ in $\mt S$ are such that $\Aut(C,\mt E)=\mathbb G_m$. Let $\nu_{r,d}:\mt T:=\nu_{r,d}^{-1}\mt S\rightarrow \mt S$ be the restriction of the rigidification map and $(\overline{\pi}:\mt T_1\rightarrow \mt T,\mt E)$ the universal pair on $\mt T\subset \CVc$. Then $$\overline{\pi}_*\left(Hom\left(\nu_{r,d}^*\mt F,\mt E\right)\right)$$ is a line bundle of weight $1$ on $\mt T$ and, by smoothness of $\CVc$, we can extend it to a line bundle of weight $1$ on $\CVc$.\\
$(iv) \iff (v)$. If all H-semistable points are H-stable, then by \cite[Corollary 2.5]{Mum94} the action of $GL(V_n)$ on the H-semistable locus of $H_n$ is proper, i.e. the morphism $PGL\times H_n^{Hss}\rightarrow H_n^{Hss}\times H_{n}^{Hss}:(A,h)\mapsto (h,A.h)$ is proper (for $n$ big enough). Consider the cartesian diagram
$$
\xymatrix{
PGL\times H_n^{Hss}\ar[r]\ar[d] & H_n^{Hss}\times H_n^{Hss}\ar[d]\\
\CVr^{Hss}\ar[r] &\CVr^{Hss}\times \CVr^{Hss}
}
$$
this implies that the diagonal is proper, i.e. the stack is separated. We have already seen that it is always universally closed and of finite type, so it is proper. Conversely, if the diagonal is proper the automorphism group of any point must be finite, in particular there are no strictly H-semistable points.\\
$(v)\iff (vi)$. By \cite[Theorem 8.1]{LMB}, $\CVr^{Hss}$ is Deligne-Mumford if and only if the diagonal is unramified, which is also equivalent to the fact that the automorphism group of any point is a finite group (because we are working in characteristic $0$). As before, this happens if and only if all semistable points are stable.\\
$(v),(viii) \Rightarrow (i)$. It is known that, on smooth curves, $n_{r,d}=1$ if and only if all semistable vector bundles are stable. So we can suppose that $n_{r,d}=1$, so that $v_{r,d,g}=(2g-2, d+1-g, d+r(1-g))=(2g-2,d+r(1-g))$. If $v_{r,d,g}\neq 1$ we have $k_{r,d,g}<2g-2$, we can construct a nodal curve $C$ of genus $g$, composed by two irreducible smooth curves $C_1$ and $C_2$ meeting at $N$ points, such that $\omega_{C_1}=k_{r,d,g}$. In particular $(d_1,d_2):=(d\frac{\omega_{C_1}}{\omega_C}-N\frac{r}{2},d\frac{\omega_{C_2}}{\omega_C}+N\frac{r}{2})$ are integers. So we can construct a vector bundle $\mt E$ on $C$ with multidegree $(d_1,d_2)$ and rank $r$ satisfying the hypothesis of Lemma \ref{sHs}. This implies that the pair $(C,\mt E)$ must be strictly P-semistable and strictly H-semistable.\\
$(i)\Rightarrow (viii)$. Suppose that there exists a point $(C,\mt E)$ in $\CVr$ such that $(C^{st},\pi_*\mt E)$ is strictly P-semistable. If $C$ is smooth then $n_{r,d}\neq 1$ and we have done. Suppose that $n_{r,d}=1$ and $C$ singular. By hypothesis there exists a destabilizing subsheaf $\mt F\subset \pi_*\mt E$, such that
$$
\frac{\chi(\mt F)}{\sum s_i\omega_{C_i}}=\frac{\chi(\mt E)}{r\omega_C}.
$$
The equality can exist if and only if $(\chi(\mt E),r\omega_C)=(d+r(1-g),r(2g-2))\neq 1$. We have supposed that $d$ and $r$ are coprime, so $(d+r(1-g),r(2g-2))=(d+r(1-g), 2g-2)=(2g-2,d+1-g,d+r(1-g))=v_{r,d,g}$, which concludes the proof.\\
$(viii)\Rightarrow (vii),(ix)$. By hypothesis $\CVr^{Pss}=\CVr^{Ps}=\CVr^{Hss}=\CVr^{Hs}$, so $(vi)$ and $(viii)$ hold by what proved above.\\
$(vii), (ix)\Rightarrow (v)$. Suppose that $(v)$ does not hold, then there exists a strictly H-semistable point with automorphism group of positive dimension. Thus $\CVr^{Hss}$, and in particular $\CVr^{Pss}$, cannot be neither proper nor Deligne-Mumford.
\end{proof}
\appendix
\renewcommand*{\thesection}{\Alph{section}}
\section{Genus Two case.}\label{genus2}
\setcounter{equation}{0}
In this appendix we will extend the Theorems \ref{pic} and \ref{picred} to the genus two case. The main results are the following
\begin{teoa}\label{pic2}Suppose that $r\geq 2$.
\begin{enumerate}[(i)]
\item The Picard groups of $\Vtc$ and $\Vtc^{ss}$ are generated by $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$ and $\Lambda(0,0,1)$ with the unique relation
\begin{equation}\label{relsm}
\Lambda(1,0,0)^{10}=\oo.
\end{equation}
\item The Picard groups of $\CVtc$ and $\CVtc^{Pss}$ are generated by $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$, $\Lambda(0,0,1)$ and the boundary line bundles with the unique relation
\begin{equation}\label{relcp}
\Lambda(1,0,0)^{10}=\oo\left(\widetilde\delta_0+2\sum_{j\in J_1}\widetilde\delta^j_1\right)
\end{equation}
\end{enumerate}
\end{teoa}
Let $v_{r,d,2}$ and $n_{r,d}$ be the numbers defined in the Notations \ref{notations}. Let $\alpha$ and $\beta$ be (not necessarily unique) integers such that $\alpha(d-1)+\beta(d+1)=-\frac{1}{n_{r,d}}\cdot\frac{v_{1,d,2}}{v_{r,d,2}}(d-r)$. We set
$$\Xi:=\Lambda(0,1,0)^{\frac{d+1}{v_{1,d,2}}}\otimes\Lambda(1,1,0)^{-\frac{d-1}{v_{1,d,2}}},\quad
\Theta:=\Lambda(0,0,1)^{\frac{r}{n_{r,d}}\cdot \frac{v_{1,d,2}}{v_{r,d,2}}}\otimes \Lambda(0,1,0)^{\alpha}\otimes \Lambda (1,1,0)^{\beta}.$$
\begin{teoa}\label{pic2red}Suppose that $r\geq2$.
\begin{enumerate}[(i)]
\item The Picard groups of $\Vr$ and $\Vr^{ss}$ are generated by $\Lambda(1,0,0)$, $\Xi$ and $\Theta$, with the unique relation (\ref{relsm}).
\item The Picard groups of $\CVtr$ and $\CVtr^{Pss}$ are generated by $\Lambda(1,0,0)$, $\Xi$, $\Theta$ and the boundary line bundles with the unique relation (\ref{relcp}).
\end{enumerate}
\end{teoa}
Unfortunately, at the moment we can not say if the Theorems \ref{pic} and \ref{picred} hold also for the other open substacks in the assertions.
\begin{rmka}\label{g22}
Observe that, using Proposition \ref{T}, we can prove that Lemma \ref{redsemistable} holds also in genus two case. In particular, by Theorem \ref{picchow}, we have that $\Pic(\CVtc)\cong\Pic(\CVtc^{Pss})\cong\Pic(\overline{\mt U}_n)$ and $\Pic(\Vtc)\cong\Pic(\Vtc^{ss})\cong\Pic(\mt U_n)$ for $n$ big enough.
\end{rmka}
We have analogous isomorphisms for the rigidified moduli stacks.\vspace{0.2cm}\\
\begin{dimo}\ref{pic2}(i) and \ref{pic2red}(i).
By the precedent observation, it is enough to prove the theorems for the semistable locus. Let $(C,\mt L)$ be a $k$-point of $\mt Jac_{d,2}$. We recall that Theorem \ref{picsmooth} says that the complex of groups
$$
0\lra \Pic(\mt Jac_{d,2})\lra \Pic(\Vtc^{ss})\lra \Pic(\mathcal Vec^{ss}_{=\mathcal L,C})\lra 0
$$
is exact. By Theorem \ref{fibers}, the cokernel is freely generated by the restriction of the line bundle $\Lambda(0,0,1)$ on the fiber $\mathcal Vec^{ss}_{=\mathcal L,C}$. In particular the Picard groups of $\Vtc$ and $\Vtc^{ss}$ decomposes in the following way
$$
\Pic(\mt Jac_{d,2})\oplus\langle\Lambda(0,0,1)\rangle.
$$
By Theorem \ref{jc}, Theorem \ref{pic2}(i) follows. By Corollary \ref{eccocequasi}, Theorem \ref{pic2red}(i) also holds.
\end{dimo}\\
Now we are going to prove the Theorems \ref{pic2}(ii) and \ref{pic2red}(ii). First of all, by Theorems \ref{pic2}(i) and \ref{picchow}, we know that the Picard group of $\CVtc$ is generated by $\Lambda(1,0,0)$, $\Lambda(1,1,0)$, $\Lambda(0,1,0)$, $\Lambda(0,0,1)$ and the boundary line bundles. Consider the forgetful map $\overline{\phi}_{r,d}:\CVtc\to\overline{\mt M}_2$. By Theorem \ref{picmg}, the Picard group of $\overline{\mt M}_2$ is generated by the line bundles $\delta_0$, $\delta_1$ and the Hodge line bundle $\Lambda$, with the unique relation $\Lambda^{10}=\oo(\delta_0+2\delta_1)$. By pull-back along $\overline{\phi}_{r,d}$ we obtain the relation (\ref{relcp}). So for proving Theorem \ref{pic2}(ii), it remains to show that we do not have other relations on $\Pic(\CVtc)$.\vspace{0.2cm}
Suppose there exists another relation, i.e.
\begin{equation}\label{relmore}
\Lambda(1,0,0)^a\otimes\Lambda(1,1,0)^b\otimes\Lambda(0,1,0)^c\otimes\Lambda(0,0,1)^d\otimes\oo\left(e_0\widetilde\delta_0+\sum_{j\in J_1} e^j_1\widetilde\delta^j_1\right)=\oo
\end{equation}
where $a,b,c,d,e_o,e_1^j\in\mathbb Z$. By Theorem \ref{pic2}(i), the integers $b,c,d$ must be $0$ and $a$ must be a multiple of $10$. We set $a=10t$. Combining the equalities (\ref{relcp}) and (\ref{relmore}) we obtain:
\begin{equation}\label{rel}
\oo\left((e_0-t)\widetilde\delta_0+\sum_{j\in J_1}(e_1^j-2t)\widetilde\delta_1^j\right)=\oo
\end{equation}
where the integers $(e_0-t),(e_1^j-2t)$ cannot be all equal to $0$, because we have assumed that the two relations are independent. In other words the existence of two independent relations is equivalent to show that does not exist any relation among the boundary line bundles. We will show this arguing as in \S\ref{indipendece}. Observe that, arguing in the same way, we can arrive at same conclusions for the rigidified moduli stack $\CVtr$. \\\\
\textbf{The Family $\widetilde G$.}\\
Consider a double covering $Y'$ of $\mathbb P^2$ ramified along a smooth sextic $D$. Consider on it a general pencil of hyperplane sections. By blowing up $Y'$ at the base locus of the pencil we obtain a family $\varphi:Y\rightarrow \mathbb P^1$ of irreducible stable curves of genus two with at most one node. Moreover the two exceptional divisors $E_1$, $E_2\subset Y$ are sections of $\varphi$ trough the smooth locus of $\varphi$. The vector bundle $\mt E:=\oo_Y\left(d E_1\right)\oplus\oo_Y^{r-1}$ is properly balanced of relative degree $d$. We call $G$ (resp. $\widetilde G$) the family of curves $\varphi:Y\rightarrow\mathbb P^1$ (resp. the family $\varphi$ with the vector bundle $\mt E$). We claim that
$$
\begin{cases}
\deg_{\widetilde G}\oo(\widetilde \delta_0)=30,\\
\deg_{\widetilde G}\oo(\widetilde \delta_1^j)=0 &\mbox{for any }j\in J_1.
\end{cases}
$$
The second result comes from the fact that all fibers of $\varphi$ are irreducible. We recall that, as \S\ref{indipendece}: $\deg_{\widetilde G}\oo(\widetilde \delta_0)=\deg_G\oo(\delta_0)$. So our problem is reduced to check the degree on $\overline{\mt M}_2$. Observe also that $Y$ is smooth and the generic fiber of $\varphi$ is a smooth curve. Since any fiber of $\varphi:Y\to\mathbb P^{-1}$ can have at most one node and the total space $Y$ is smooth, by \cite[Lemma 1]{AC87}, $\deg_{\widetilde G}\oo(\widetilde \delta_0)$ is equal to the number of singular fibers of $\varphi$. We can count them using the morphism $\varphi_D:D\rightarrow \mathbb P^1$, induced by the pencil restricted to the sextic $D$. By the generality of the pencil, we can assume that over any point of $\mathbb P^1$ there is at most one ramification point and that its ramification index at this point is $2$. So $\deg_{\widetilde G}\oo(\widetilde \delta_0)$ is equal to the degree of the ramification divisor in $D$. Using the Riemann-Hurwitz formula for the degree six morphism $\varphi_{D}$ we obtain the first equality.\\\\
\textbf{The Families $\widetilde G_1^j$.}\\
Consider a general pencil of cubics in $\mathbb P^2$. Blowing up the nine base points of the pencil, we obtain a family of irreducible stable elliptic curves $\phi:X\rightarrow\mathbb P^1$. The nine exceptional divisors $E_1,\ldots, E_9\subset X$ are sections of $\phi$ trough the smooth locus of $\phi$. The family will have twelve singular fibers consisting of irreducible nodal elliptic curves. Fix a smooth elliptic curve $\Gamma$ and a point $\gamma\in\Gamma$. We construct a surface $Y$ by setting
$$
Y=\left(X\coprod\left(\Gamma\times\mathbb P^1\right)\right)/\left(E_1\sim \{\gamma\}\times\Gamma\right)
$$
We get a family $f:X\rightarrow\mathbb P^1$ of stable curves of genus two. The general fiber is as in Figure \ref{Figure8} where $C$ is a smooth elliptic curve. While the twelve special fibers are as in Figure \ref{Figure9} where $C$ is a nodal irreducible elliptic curve.
\begin{figure}[h!]
\begin{center}
\unitlength .65mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(122.75,50.5)(0,145)
\qbezier(47,153.5)(85.125,175.625)(122.75,183.25)
\qbezier(0,183)(25,185)(90,153)
\put(25,183.75){\makebox(0,0)[cc]{$C$}}
\put(113.25,183.75){\makebox(0,0)[cc]{$\Gamma$}}
\end{picture}
\end{center}
\caption{The general fiber of $f:X\to \mathbb P^1$.}\label{Figure8}
\end{figure}
\begin{figure}[h!]
\begin{center}
\unitlength .65mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(122.75,50.5)(0,145)
\qbezier(77,153.5)(95.125,175.625)(122.75,183.25)
\qbezier(50,183)(25,180)(100,163)
\qbezier(0,153)(75,180)(50,183)
\put(5,160.75){\makebox(0,0)[cc]{$C$}}
\put(113.25,183.75){\makebox(0,0)[cc]{$\Gamma$}}
\end{picture}
\end{center}
\caption{The special fibers of $f:X\to \mathbb P^1$.}\label{Figure9}
\end{figure}
Choose a vector bundle $M^j$ of degree $\left\lceil \frac{d-r}{2}\right\rceil+j$ on $\Gamma$, pull it back to $\Gamma\times\mathbb P^1$ and call it again $M^j$. Since $M^j$ is trivial on $\{\gamma\}\times\mathbb P^1$, we can glue it with the vector bundle
$$
\oo_X\left(\left(\left\lfloor\frac{d+r}{2}\right\rfloor-j\right)E_2\right)\oplus\oo_X^{r-1}
$$
on $X$ obtaining a vector bundle $\mt E^j$ on $f:X\to\mathbb P^1$ of relative rank $r$ and degree $d$. The next lemma follows easily
\begin{lema}The vector bundle $\mt E^j$ is a properly balanced for $j\in J_1=\{0,\ldots,\lfloor r/2\rfloor\}$.
\end{lema}
We call $G_1$ (resp. $\widetilde G_1^j$) the family of curves $f:X\to\mathbb P^1$ (resp. the family $f$ with the vector bundle $\mt E^j$). Moreover $\widetilde G_1^j$ does not intersect $\widetilde\delta_1^k$ for $j\neq k$. In particular $\deg_{\widetilde G_1^j}\oo(\widetilde \delta_1^j)=\deg_{G_1}\oo(\delta_1)$. By \cite[Lemma 1]{AC87}, the divisor $\oo(\delta_1)$ restricted to the family $G_1$ is isomorphic to the tensor product between the normal bundle of $E_1$ in $X$ and the normal bundle of $\gamma\times\mathbb P^1$ in $\Gamma\times\mathbb P^1$, i.e. $N_{E_1/X}\otimes N_{\{\gamma\}\times\mathbb P^1/\Gamma\times\mathbb P^1}$. The first factor has degree $-1$, while the second is trivial. Putting all together, we get
$$
\begin{cases}
\deg_{\widetilde G_1^k}\oo(\widetilde\delta_1^k)=-1,\\
\deg_{\widetilde G_1^k}\oo(\widetilde\delta_1^j)=0 &\mbox{if } j\neq k.
\end{cases}
$$
Now we can finally conclude the proof of Theorems \ref{pic2} and \ref{pic2red}.\vspace{0.2cm}\\
\begin{dimo} \ref{pic2}(ii) and \ref{pic2red}(ii). Suppose there exists a non-trivial relation $\oo(a_0\widetilde\delta_0+\sum a_1^j\widetilde\delta_1^j)=\oo$. If we restrict this equality on $\widetilde G$ we have $a_0=0$. Pulling back to $G_1^j$ we deduce $a_1^j=0$ for any $j\in J_1$. This concludes the proof of \ref{pic2}(ii). Repeating the same arguments for the rigidified moduli stack $\CVtr$ we prove Theorem \ref{pic2red}(ii).
\end{dimo}
\section{Base change cohomology for stacks admitting a good moduli space.}\label{App}
We will prove that the classical results of base change cohomology for proper schemes continue to hold again (not necessarily proper) stacks, which admit a proper scheme as good moduli space (in the sense of Alper). The propositions and proofs are essentially equal to ones in \cite[Appendix A]{Br2}, but we rewrite them, because our hypothesis are weaker.\\
In this section, $\mathcal X$ will be an Artin stack of finite type over a scheme $S$, and a sheaf $\mathcal F$ will be a sheaf for the site lisse-\'etale defined in \cite[Sec. 12]{LMB} (see also \cite[Appendix A]{Br1}).
Recall first the definition of good moduli space.
\begin{defina}\cite[def 4.1]{Al} Let $S$ be a scheme, $\mathcal X$ be an Artin Stack over $S$ and $X$ an algebraic space over $S$ . We call an $S$-morphism $\pi:\mathcal X\rightarrow X$ a good moduli space if
\begin{itemize}
\item $\pi$ is quasi-compact,
\item $\pi_*$ is exact,
\item The structural morphism $\oo_{ X}\rightarrow\pi_*\oo_{\mathcal X}$ is an isomorphism.
\end{itemize}
\end{defina}
\begin{rmka}\label{goodquotient} Let $\mt X$ be a quotient stack of a quasi-compact $k$-scheme $X$ by a smooth affine linearly reductive group scheme $G$. Suppose that $\mt L$ is a $G$-linearization on $X$. By\cite[Theorem 13.6 and Remark 13.7]{Al}, the GIT good quotient $X^{ss}_{\mt L}\sslash_{\mt L} G$ is a good moduli space for the open substack $\left[X^{ss}_{\mt L}/ G\right]$.\\
Conversely, suppose that there exists an open $U\subset X$ such that the open substack $\left[U/G\right]$ admits a good moduli space $Y$. By \cite[Theorem 11.14]{Al}, there exists a $G$-linearized line bundle $\mt L$ over $X$ such that $U$ is contained in $X^{ss}_{\mt L}$, $[U/G]$ is saturated respect to the morphism $[X^{ss}_{\mt L}/G]\to X^{ss}_{\mt L}\sslash_{\mt L}G$ and $Y$ is the GIT good quotient $U\sslash_{\mt L}G$.
\end{rmka}
Before stating the main result of this Appendix, we need to recall the following
\begin{lema}$($\cite[Lemma 1, II]{Mum70}, see also \cite[Lemma 4.1.3]{Br2}$)$.
\begin{enumerate}[(i)]
\item Let $A$ be a ring and let $C^{\bullet}$ be a complex of $A$-modules such that $C^p\neq 0$ only if $0\leq p\leq n$. Then there exists a complex $K^{\bullet}$ of $A$-modules such that $K^p\neq 0$ only if $0\leq p\leq n$ and $K^p$ is free if $1\leq p\leq n$, and a quasi-isomorphism of complexes $K^{\bullet}\rightarrow C^{\bullet}$. Moreover, if the $C^p$ are flat, then $K^0$ will be $A$-flat too.
\item If $A$ is noetherian and if the $H^i(C^\bullet)$ are finitely generated $A$-modules, then the $K^p$'s can be chosen to be finitely generated.
\end{enumerate}
\end{lema}
\begin{propa}\label{lasvolta}Let $\mathcal X$ be a quasi-compact Artin stack over an affine scheme (resp. noetherian affine scheme) $S=Spec(A)$. Let $\pi:\mathcal X\rightarrow X$ be a good moduli space with $X$ separated (resp. proper) scheme over $S$. Let $\mathcal F$ be a quasi-coherent (resp. coherent) sheaf on $\mathcal{X}$ that is flat over $S$. Then there is a complex of flat $A$-modules (resp. of finite type)
$$
0\longrightarrow M^0\longrightarrow M^1\longrightarrow\dots\longrightarrow M^n\longrightarrow 0
$$
with $M^i$ free over $A$ for $1\leq i\leq n$, and isomorphisms
$$
H^i(M^{\bullet}\otimes_A A')\longrightarrow H^i(\mathcal{X}\otimes_A A',\mathcal{F}\otimes_A A')
$$
functorial in the $A$-algebra $A'$.
\end{propa}
\begin{proof} We consider the Cech complex $C^{\bullet}(\mathcal{U},\pi_*\mathcal F)$ associated to an affine covering $\mathcal U=(U_i)_{i\in I}$ of $X$. It is a finite complex of flat (by \cite[Theorem 4.16(ix)]{Al}) $A$-modules. Moreover, since $X$ is separated, then we have $H^i(C^{\bullet}(\mathcal{U},\pi_*\mathcal F))\cong H^i(X,\pi_* \mathcal F)$. If $A'$ is an $A$-algebra, then the covering $\mathcal U\otimes_A A'$ is still affine by $S$-separateness of $X$. This implies that
$$
H^i(C^{\bullet}(\mathcal{U},\pi_*\mathcal F))\otimes_A A'\cong H^i(X\otimes_A A',(\pi_* \mathcal F)\otimes_A A').
$$
By \cite[Proposition 4.5]{Al}, we have
$$
H^i(X\otimes_A A',(\pi_* \mathcal F)\otimes_A A')\cong H^i( X\otimes_A A', \pi_*(\mathcal F\otimes_A A')).
$$
Since $\pi_*$ is exact, the Leray-spectral sequence $H^i(X\otimes_A A',R^j\pi_*(\mt F\otimes_A A'))\Rightarrow H^{i+j}(\mt X\otimes_A A',\mt (F\otimes_A A'))$ (see \cite[Theorem. A.1.6.4]{Br1}) degenerates in the isomorphisms $H^i(X\otimes_A A',\pi_*(\mathcal F\otimes_A A'))\cong H^i(\mt X\otimes_A A',\mathcal F\otimes_A A')$. Putting all together:
$$
H^i(C^{\bullet}(\mathcal{U},\pi_*\mathcal F))\otimes_A A'\cong H^i(\mt X\otimes_A A',\mathcal F\otimes_A A').
$$
It can be check that such isomorphisms are functorial in the $A$-algebra $A'$. Observe that if $\mathcal F$ is coherent then also $\pi_*\mathcal F$ is coherent (see \cite[Theorem 4.16(x)]{Al}). So if $X$ is proper, then the modules $H^i(\mathcal{X},\mathcal{F})$ are finitely generated. In particular, the cohomology modules of the complex $C^{\bullet}(\mathcal{U},\pi_*\mathcal F)$ are finitely generated. We can use the precedent lemma for conclude the proof.
\end{proof}
From the above results, we deduce several useful Corollaries.
\begin{cora}\label{cohoflat}Let $S$ be a scheme and let $q:\mathcal X\rightarrow S$ be a quasi-compact Artin Stack with an $S$-separated scheme $X$ as good moduli space. Let $\mathcal F$ be a quasi-coherent sheaf on $\mathcal X$ flat over $S$. If all sheaves $R^iq_*\mathcal F$ are flat over $S$ then $\mathcal F$ is cohomologically flat.
\end{cora}
\begin{proof} See \cite[Corollary 2.6]{Br2}
\end{proof}
The proofs of next results are the same of \cite[II.5]{Mum70}.
\begin{cora}Let $\mathcal X\rightarrow X$ be a good moduli space over a scheme $S$, $X$ proper scheme over $S$ and $\mathcal F$ coherent sheaf over $\mathcal X$ flat over $S$. Then we have:
\begin{enumerate}[(i)]
\item for any $ p\geq 0$ the function $S\rightarrow \mathbb Z$ defined by $s\mapsto dim_{k(s)}H^i(\mathcal X_s,\mathcal F_s)$ is upper semicontinuous on $S$.
\item The function $S\rightarrow \mathbb Z$ defined by $s\mapsto \chi (\mathcal{F}_s)$ is locally constant.
\end{enumerate}
\end{cora}
\begin{cora}\label{TFAE}
Let $\mathcal X\rightarrow X$ be a good moduli space over an integral scheme $S$, $X$ proper scheme over $S$ and $\mathcal F$ coherent sheaf over $\mt X$ flat over $S$. The following conditions are equivalent
\begin{enumerate}[(i)]
\item $s\mapsto dim_{k(s)}H^i(\mathcal X_s,\mathcal F_s)$ is a constant function,
\item $R^iq_*(\mathcal F)$ is locally free sheaf on $S$ and for any $s\in S$ the map
$$
R^iq_*(\mathcal F)\otimes k(s)\rightarrow H^i(\mathcal X_s,\mathcal F_s)
$$
is an isomorphism.
\end{enumerate}
If these conditions are satisfied, then we have an isomorphism
$$
R^{i-1}q_*(\mathcal F)\otimes k(s)\rightarrow H^{i-1}(\mathcal X_s,\mathcal F_s)
$$
\end{cora}
\begin{cora}Let $\mathcal X\rightarrow X$ be a good moduli space over a scheme $S$, $X$ proper scheme over $S$ and $\mathcal F$ coherent sheaf over $\mathcal X$ flat over $S$. Assume for some $i$ that $H^i(\mathcal X_s,\mathcal F_y)=(0)$ for any $s\in S$. Then the natural map
$$
R^{i-1}q_*(\mathcal F)\otimes_{\oo_S}k(s)\rightarrow H^{i-1}(\mathcal X_s,\mathcal F_y)
$$
is an isomorphism for any $s\in S$.
\end{cora}
\begin{cora}Let $\mathcal X\rightarrow X$ be a good moduli space over a scheme $S$, $X$ proper scheme and $\mathcal F$ coherent sheaf over $\mt X$ flat over $S$. If $R^i q_*(\mathcal F)=(0)$ for $i\geq i_0$ then $H^i(\mathcal X_s,\mathcal F_s)=(0)$ for any $s\in S$ and $i\geq i_0$.
\end{cora}
\begin{cora}\label{seesaw}$[$The SeeSaw Principle$]$.\\
Let $\mathcal X\rightarrow X$ be a good moduli space over an integral scheme $S$ and $\mathcal L$ be a line bundle on $\mt X$. Suppose that $q:\mt X\to S$ is flat and that $X\rightarrow S$ is proper with integral geometric fibers. Then the locus
$$
S_1=\{s\in S|\mathcal L_s \cong \oo_{\mathscr X_s}\}
$$
is closed in $S$. Moreover if we call $q_1:\mathcal X\times_{S} S_1 \rightarrow S_1$ the restriction of $q$ on this locus, then $q_{1*}\mathcal L$ is a line bundle on $S$ and the natural morphism $q_1^*q_{1*}\mathcal{L}\cong \mathcal L$ is an isomorphism.
\end{cora}
\begin{proof}A line bundle $\mathcal M$ on a stack $\mathcal X$ with a proper integral good moduli space $X$ is trival if and only if $h^0(\mathcal M)>0$ and $h^0(\mathcal M^{-1})>0$. The necessity is obvious. Conversely suppose that these conditions hold. Then we have two non-zero homomorphisms $s:\oo_{\mathcal X}\rightarrow \mathcal M$, $t:\oo_{\mathcal X}\rightarrow \mathcal M^{-1}$. If we dualize the second one and compose with the first one, we have a non-zero morphism $h:\oo_{\mathcal X}\rightarrow \oo_{\mathcal X}$. Now $X$ is an integral proper scheme then $H^0(X,\oo_X)=k$ so $H^0(\mathcal X,\oo_{\mathcal X})=k$. Hence $h$ is an isomorphism. This implies that also $s$ and $t$ are isomorphisms. As a consequence, we have
$$
S_1=\{s\in S|h^0(\mathcal X_s,\mathcal L_s)>0,\,h^0(\mathcal X_s,\mathcal L_s^{-1})>0\}.
$$
In particular, $S_1$ is closed by upper semicontinuity. Up to restriction we can assume $S=S_1$, so the function $s\mapsto h^0(\mathcal X_s,\mathcal L_s)=1$ is constant. By Corollary \ref{TFAE}, $q_*\mathcal L$ is a line bundle on $S$ and the natural map $q_*\mathcal L\otimes_{\oo_S} k(s)\rightarrow H^0(\mathcal X_s,\mathcal L_s)$ is an isomorphism. Consider the natural map $\pi:q^*q_*\mathcal L\rightarrow \mathcal L$. Its restriction on any fiber $\mathcal X_s$
$$
\oo_{\mathcal X_s}\otimes H^0(\mathcal X_s,\mathcal L_s)\rightarrow \mathcal L_s
$$
is an isomorphism. In particular $\pi$ is an isomorphism for any geometric point $x\in \mt X$. Since it is a map between line bundles, by Nakayama lemma, it is an isomorphism.
\end{proof} | 155,514 |
TITLE: Double expected value
QUESTION [1 upvotes]: Let $m$ be a probability measure on $\mathbb{R}^n$, so that $m(\mathbb{R}^n) = 1$.
Consider two measurable functions $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, and $g : \mathbb{R}^n \rightarrow \mathbb{R}$. Assume that $f$ and $g$ are uniformly bounded by an integrable function.
I am wondering about the following inequality.
$$ \mathbb{E}_w[\mathbb{E}_v[ f(v,w) - g(w) ]] \ \leq \ \mathbb{E}_w[ f(w,w) - g(w) ]$$
Comments. The expected value $\mathbb{E}$ is defined as follows. $\mathbb{E}_w[g(w)]:= \int_{\mathbb{R}^n} g(w) m(dw)$. Therefore $\displaystyle \mathbb{E}[\mathbb{E}[ f(v,w) - g(w) ]] = \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} \left( f(v,w) - g(w) \right) m(dv) \right) m(dw). $
REPLY [2 votes]: Although you can easily derive the answer from the perfect hint of Tim, let me provide some intuition why the inequality doesn't have to be true. Namely, let is imagine the idea of such inequality came to our mind for the first time, and let's examine this idea, so that the next time it would be easier for you to come with a count example by yourself.
First of all, note that for the LHS
$$
\iint(f(v,w) - g(v))m(\mathrm dw)m(\mathrm dv) = \iint f(v,w)m(\mathrm dw)m(\mathrm dv) - \int g(v)m(\mathrm dv)
$$
and for the RHS
$$
\int (f(w,w) - g(w))m(\mathrm dw) = \int f(w,w)m(\mathrm dw) - \int g(w)m(\mathrm dw)
$$
and so LHS$-$RHS does not depend on $g$ at all. Thus, your inequality is equivalent to
$$
\iint f(v,w)m(\mathrm dw)m(\mathrm dv) - \int f(w,w)m(\mathrm dw)\leq0 \tag{1}.
$$
Note that if $f$ satisfies $(1)$, then $-f$ shall also satisfy $(1)$ which by the claim holds at least for any bounded $f$. Thus, $(1)$ has to be even an equality for all bounded $f$. And the latter fact clearly does not have to be true: see below.
Even regardless of the trick with inequality turning into an equality, critical glance at $(1)$ shall tell us that it does not have to be true. The point is that you can always decompose $f$ into a diagonal part and the off-diagonal one, that is let
$$
\Delta\subset \Bbb R^n\times \Bbb R^n = \{(x,x):x\in \Bbb R^n\}
$$
be the diagonal, then $f = 1_\Delta f_1 + 1_{\Delta^c}f_2$ and the decomposition is clearly unique: we just use $f_1$ for values of $f$ on the diagonal, and $f_2$ for everywhere else. As a result, $(1)$ turns into
$$
\iint (1_\Delta f_1(w,v)+1_{\Delta^c}f_2(w,v))m(\mathrm dw)m(\mathrm dv)\leq \int f_1(w,w)m(\mathrm dw)
$$
and if you choose $m$ with, say continuous density, then $m\otimes m(\Delta) = 0$, so
$$
\iint f_2(w,v)m(\mathrm dw)m(\mathrm dv)\leq \int f_1(w,w)m(\mathrm dw).
$$
Now, since $f_1$ and $f_2$ can be chosen in a totally free way, nothing actually tells us that the latter inequality is true, and you can easily come up with a counterexample. E.g. let $f_1$ be negative everywhere, and let $f_2$ be positive everywhere. This method of decomposing $f$ into two parts would also work for a lot of modifications of your inequality. Namely, if you would ask about
$$
\Bbb E_w[\Bbb E_v[(f(v,w) - g(w))^2]]\leq \Bbb E_w[(f(w,w) - g(w))^2]
$$
then points $1.$ and $2.$ are not applicable anymore. However, if you ask the latter inequality to holds for all bounded measurable functions, it has to hold at least for the case when $g = 0$, and then you can apply again the decomposition. | 211,418 |
fleur du mal
Mini Tux China
- Dry clean only
- Hidden button front closures
- Belted waist with bow accent
- Side slit pockets
- Back flap pocket
- Our Style No. FLEF-WR7
- Manufacturer Style No. JS0041
- Model is wearing size 2. View detailed measurements of this item.. | 55,484 |
Nutritional Info
- Servings Per Recipe: 3
- Amount Per Serving
- Calories: 230.0
- Total Fat: 2.9 g
- Cholesterol: 8.7 mg
- Sodium: 489.8 mg
- Total Carbs: 27.8 g
- Dietary Fiber: 4.2 g
- Protein: 25.3 g
View full nutritional breakdown of Veggie and Parmesan Frittata calories by ingredient
Veggie and Parmesan FrittataSubmitted by: MICRONERDCHICK
Number of Servings: 3
Ingredients
2.5 cup sliced mushrooms
1 red bell pepper, chopped
1 green bell pepper, chopped
1 medium onion, chopped
1 zucchini, chopped
3 cloves garlic, minced
2 cups egg substitute
1/4 cup milk
1/4 cup grated parmesan cheese
cooking spray
salt and pepper to taste
Directions
Spray a pan with cooking spray. Sautee all vegetables together (with garlic) until soft. Mix egg substitute, cheese, and milk. Add to vegetables. Bake @ 350F for 20 minutes or until cooked thoroughly. Serves 3.
Number of Servings: 3
Recipe submitted by SparkPeople user MICRONERDCHICK.
Number of Servings: 3
Recipe submitted by SparkPeople user MICRONERDCHICK. | 354,602 |
Links
Information below is listed for reference only. Behind the Orange Curtain is not affiliated with and does not endorse any program, therapist, literature or rehabilitation program. It is strongly suggested that you conduct a thorough due-diligence before entering into any rehabilitation or detox program. Detoxification should medically supervised by a physician in a hospital setting. If someone you know is overdosing call 911 immediately.
California Youth Services – a Non-Profit organization 949-663-6755
The Newport Academy, Jamison Monroe, Jr.
The Meadows – Arizona | 203,572 |
TITLE: What is the probability that both of the scores is a $3$?
QUESTION [0 upvotes]: Question
Someone rolls two identical fair 6-sided dice but does not know the result. Someone was told that one of the numbers on the die was a $3$. Given this information, what is the probability that both of the numbers on Someone's die was a $3$?
Answer
$$ \frac{1}{11} $$
Given that the first number was a $3$, isn't the probability of the $2$nd die being $3$ equal to $\frac{1}{6}$? How is $\frac{1}{11}$ obtained?
REPLY [2 votes]: Given that the first number was a $3$, isn't the probability of the $2$nd die being $3$ equal to $\frac{1}{6}$?
No.
Using a Probability Diagram, draw all of the possible results:
$1$
$2$
$3$
$4$
$5$
$6$
$1$
1,1
2,1
3,1
4,1
5,1
6,1
$2$
1,2
2,2
3,2
4,2
5,2
6,2
$3$
1,3
2,3
3,3
4,3
5,3
6,3
$4$
1,4
2,4
3,4
4,4
5,4
6,4
$5$
1,5
2,5
3,5
4,5
5,5
6,5
$6$
1,6
2,6
3,6
4,6
5,6
6,6
Now highlight all of the results with at least one $3$:
$1$
$2$
$3$
$4$
$5$
$6$
$1$
1,1
2,1
3,1
4,1
5,1
6,1
$2$
1,2
2,2
3,2
4,2
5,2
6,2
$3$
1,3
2,3
3,3
4,3
5,3
6,3
$4$
1,4
2,4
3,4
4,4
5,4
6,4
$5$
1,5
2,5
3,5
4,5
5,5
6,5
$6$
1,6
2,6
3,6
4,6
5,6
6,6
If you count them, there are $11$ possible results with a $3$, and there is only one result which has two $3$s (3,3).
\begin{align}
\therefore P(\text{both of the numbers on the die was a 3}) &= \frac{n(\text{number of results with two 3s})}{n(\text{number of results with a 3})} \\
&= \frac{1}{11}
\end{align} | 201,512 |
Fatwa ID: 02105
Answered by Ustadha Mahdiyah Siddique
Question:
I married on April 2016 and my wife asking me for divorce (khula). I am very simple and kind person, she blamed me that I tell everything lies to her, she blamed me that I am not fulfilling her wishes and not performing husband duty, I and my family member trying to convince her regular, but she want divorce. As a husband I fulfilled all her wishes perform as a husband, what I do, please suggest me.
Answer:
In response to your question;
If your wife has came to the decision that she would like to appeal for a Khula’ then that is her decision and one that she is entitled to. As you have mentioned you and your family members have tried to convince her to no avail, thus the only other advice would be to set up a final attempt of mediation between the two of you by choosing a trustworthy member from both your and her family to try and help overcome the differences.
This is something we can take from the Quran, as Allah the Almighty states,)
To conclude, if the aforementioned endeavour does not bring about any resolution then as per her right she would be entitled to Khula’ and thus be set free from the marriage, on the basis of the statement of Allah the Almighty, ‘The divorce is twice, after that, either you retain her on good terms or release her with kindness. And it is not lawful for you (men) to take back (from your wives) any of that which you have given them (Mahr), except when both parties fear that they would be unable to keep the limits ordained by Allah. Then if you fear that they would not be able to keep the limits ordained by Allah, then there is no sin on either of them regarding that which she sets herself free with (if she gives back the Mahr or part of it for her Khula’). These are the limits ordained by Allah, so do not transgress them. And whoever transgresses the limits of Allah; then those are the wrong-doers.’ (Surah al-Baqarah: Verse 229)
Only Allah Knows Best
Written by Ustadha Mahdiyah Siddique
Checked and approved by Mufti Mohammed Tosir Miah
Darul Ifta Birmingham | 267,776 |
The water levels of Lake Mead are rising. For the first time in over ten years, the water is headed up not down and although no one wants to celebrate too soon, it’s worth popping a little bubbly for.
The water problem in Las Vegas has been a concern for years and the main solution has been water conservation. In 2007 we blogged about a Water Smart Congress and Las Vegas has been focusing their efforts to ease the water problem. Clark County continues to be applauded and awarded for it’s various conservation practices including the tax credits for converting to desert landscaping and the enforcement of water schedules and practices. Way to go!
Analysts believe that the rising levels of Lake Mead will only continue until the end of the year. Heavy snowfall has accounted for the level increase and only time and Mother Nature will tell what will happen this next season. As it stands with the current rate of water rising, Lake Mead will be around 50% capacity. Not yet enough to start splurging on our water usage.
Conservation programs are continually being tweaked and new ideas are being birthed as the promise to climb out of this decade of drought is uncertain. My contribution? Take fewer baths and shorter showers, keep the inside instead of the outside of my car squeaky clean and order beer instead of tap water at the local watering hole.
Tags: Environment, Featured, Geography, Life, Local, Utilities | 144,179 |
TITLE: How to find all values for $\alpha$ and $\beta$ such that $\int _0^{\infty }f\left(x\right)$ converge
QUESTION [1 upvotes]: $f(x) = \begin{cases} x^{\alpha }\left(1-cos\left(1-x\right)\right)^{\beta } & \text{if $\;\;\;0<x<1$} \\ \frac{1}{x^{\alpha }+x^{\beta }} & \text{if $\;\;\;1\ge x$} \end{cases}$
I split into two integrals:
$\int _0^1x^{\alpha }\left(1-cos\left(1-x\right)\right)^{\beta }dx\:+\:\int _1^{\infty }\left(\frac{1}{x^{\alpha }+x^{\beta }}\right)dx$
I want to use the comparison test. but I find it hard to choose $g(x)$ that will give me an answer for $\alpha$ and $\beta$
REPLY [0 votes]: The convergence is obtained for
$$
\alpha>-1 \quad \text{and} \quad \beta>-1/2 \quad \text{and} \quad \max(\alpha,\beta)>1.
$$
Hint. Potential issues are as $x \to 0^+$, as $x \to 1^-$ and as $x \to \infty.$
As $x \to 0^+$, one has $$
f(x)=x^\alpha(1-\cos1)^\beta+O(x^{\alpha+1}) $$ and $ \displaystyle
\int_0^\epsilon\!\!f(x)dx$ converges ($0<\epsilon<1$) iff $ \displaystyle
\int_0^\epsilon\!\!x^\alpha dx$ converges, that is it converges iff $\alpha>-1.$
As $x \to 1^-$, one has $$
f(x)=(1-x)^{2\beta}2^{-\beta}+O((x-1)^{2\beta+1}) $$ and $ \displaystyle
\int_{1-\epsilon}^1\!\!f(x)dx$ converges ($0<\epsilon<1$) iff $ \displaystyle
\int_{1-\epsilon}^1\!\!(1-x)^{2\beta} dx$ converges, that is it converges iff
$\beta>-1/2.$
As $x \to \infty$, one has $$
f(x)=\frac{1}{x^{\max(\alpha,\beta)}}+O\left(\frac{1}{x^{\max(\alpha,\beta)+(\max(\alpha,\beta)-\min(\alpha,\beta))}}\right)
$$ and $ \displaystyle \int_1^\infty\!\!f(x)dx$ converges iff $
\displaystyle \int_0^1\!\!\frac{1}{x^{\max(\alpha,\beta)}} dx$
converges, that is it converges iff $\max(\alpha,\beta)>1.$ | 199,631 |
TITLE: Let $X$ be a non-empty set of ordinals. What are the necessary and sufficient conditions for $X$ to be an ordinal?
QUESTION [0 upvotes]: $X$ is ordinal if and only if X is transitive and well-ordered under $\in$.
Let $X$ be a non-empty set of ordinals. What are the necessary and sufficient conditions for $X$ to be an ordinal?
I found that the answer is quite simple in case $X$ is finite.
Assume $X$ has $n$ elements $a_1,a_2,\cdots,a_n$. First of all, $X$ is well-ordered under $\in$, so WLOG we can safely assume that $a_1\in a_2\in a_3\in\cdots\in a_n$, hence $a_i\in a_n$ for all $0<i<n$. It is easily to prove that $a_1=\emptyset$. As a result, $X$ is ordinal $\implies$ $\emptyset\in X$ and $\exists c\in X,X\setminus\{c\}\subsetneq c$. One can easily verify that $\emptyset\in X$ and $\exists c\in X,X\setminus\{c\}\subsetneq c\implies X$ is ordinal.
Please help me in case $X$ is infinite!
REPLY [2 votes]: A set $X$ of ordinals is itself an ordinal if and only if $X$ is transitive, i.e. every element of $X$ is a subset of $X$.
Since, for every ordinal $\alpha$, $\alpha$ is transitive this is clearly necessary.
But it is also sufficient: A set $Y$ is an ordinal iff it is transitive an strictly totally ordered by $\in$. Since $X$ is a set of ordinals, it is strictly totally ordered by $\in$ and by our requirement it is transitive as well. | 112,402 |
Why Do Excel Grid Lines Disappear in Some Places?on January 13th, 2012 at 11:13 AM
You may notice that grid lines in Excel disappear from time to time. This generally occurs as you are altering the formatting. If all of your excel grid lines have disappeared, you probably have them turned off. This solution is for the situation you see here where only some of the grid lines
are missing in a workbook. This is usually caused by cell formatting. You will notice here that the culprit is actually a cell fill. The cell fill has been set to white. The normal setting is no fill. Select all the cells with no border and set the the fill to no fill and your grid lines should return. Now you can get back to those formulas and finish your project on time. | 261,034 |
\begin{document}
\maketitle
\begin{abstract}
The concepts of evaluation and interpolation are extended from univariate skew polynomials to multivariate skew polynomials, with coefficients over division rings. Iterated skew polynomial rings are in general not suitable for this purpose. Instead, multivariate skew polynomial rings are constructed in this work as follows: First, free multivariate skew polynomial rings are defined, where multiplication is additive on degrees and restricts to concatenation for monomials. This allows to define the evaluation of any skew polynomial at any point by unique remainder division. Multivariate skew polynomial rings are then defined as the quotient of the free ring by (two-sided) ideals that vanish at every point. The main objectives and results of this work are descriptions of the sets of zeros of these multivariate skew polynomials, the families of functions that such skew polynomials define, and how to perform Lagrange interpolation with them. To obtain these descriptions, the existing concepts of P-closed sets, P-independence, P-bases (which are shown to form a matroid) and skew Vandermonde matrices are extended from the univariate case to the multivariate one.
\textbf{Keywords:} Derivations, free polynomial rings, Lagrange interpolation, Newton interpolation, skew polynomials, Vandermonde matrices.
\textbf{MSC:} 08B20, 11C08, 12E10.
\end{abstract}
\section{Introduction} \label{sec intro}
\textit{Univariate skew polynomial rings}, introduced in \cite{ore}, are those ``non-commutative polynomial rings'', over some coefficient ring, whose addition is the usual one, but whose multiplication is arbitrary with the following restrictions: The $i$th power (being $ i $ a natural number) of the variable $ x $ corresponds to the monomial ``$ x^i $'', and the degree of a product of two arbitrary polynomials is the sum of their degrees. Adding the commutativity property yields the \textit{conventional} polynomial ring, that is, the monoid ring of the natural numbers over the coefficient ring.
An extension of the concept of \textit{evaluation} to skew polynomials over division rings was first given in \cite{lam} and further developed in \cite{algebraic-conjugacy, lam-leroy}. Since a skew polynomial ring (over a division ring) is a right-Euclidean domain \cite{ore}, the evaluation of $ F(x) $ on a point $ a $ is defined in \cite{lam,lam-leroy} as the remainder of the Euclidean division of $ F(x) $ by $ x - a $ on the right. Such extension is natural in the sense that it is based on the ``Remainder Theorem'' for conventional polynomials and it is analogous to projecting on a quotient ring defined by a maximal ideal, as in algebraic geometry. This concept of evaluation helps unify the study of Vandermonde, Moore and Wronskian matrices \cite{lam, lam-leroy} and further matrix types (see \cite[p. 604]{linearizedRS} for instance), and gives a natural framework for Hilbert 90 Theorems \cite{hilbert90} and pseudolinear transformations \cite{leroy-pol}, which unify semilinear and differential transformations (see also \cite[Sec. 8.4]{cohn}). It has also provided error-correcting codes with good minimum Hamming distance \cite{skewcyclic1}, maximum rank distance codes \cite{gabidulin}, and maximum sum-rank distance codes \cite{linearizedRS} with finite-field sizes that are not exponential in the code length, in constrast with \cite{gabidulin} (see \cite[Sec. 4.2]{linearizedRS}).
Extending this concept of evaluation to multivariate skew polynomials is not straightforward. In general, unique remainder algorithms \cite[Sec. 4]{lenstra} do not hold for \textit{iterated} skew polynomials since they do not satisfy Jategaonkar's condition \cite{jategaonkar} for $ n > 1 $ variables (see \cite[Prop. 4.7]{lenstra} and also \cite[Sec. 8.8]{cohn}). Recently in \cite{skewRM}, it is proposed to evaluate certain iterated skew polynomials (those forming Poincar{\'e}-Birkhoff-Witt extensions following \cite[Def 2.1]{zhangPBW}) at points $ (a_1, a_2, \ldots, a_n) $ where $ x_1 - a_1 $, $ x_2 - a_2 $, $ \ldots $, $ x_n - a_n $ form a G{\"o}bner-Shirshov basis, since then a unique remainder algorithm exists. However, this does not include all iterated skew polynomial rings or affine points (see \cite[Ex. 3.5]{skewRM}) and the important concepts and results from \cite{lam, algebraic-conjugacy, lam-leroy} do not seem to hold.
In this work, we overcome these issues by considering an alternative construction. We start by defining \textit{free multivariate} skew polynomial rings (using the free monoid with basis $ x_1, x_2, \ldots, x_n $) following Ore's idea: The product of two monomials consists in appending them, and the degree of a product of two skew polynomials is the sum of their degrees. Over fields, adding commutativity between constants and variables (that is, turning the ring into an algebra) yields the conventional free algebra \cite[Sec. 0.11]{cohn} as a particular case. Thanks to this definition, we show that we may define the evaluation of any (free) skew polynomial $ F(x_1, x_2, \ldots, x_n) $ at any affine point $ (a_1, a_2, \ldots, a_n) $ as the unique remainder of the Euclidean division of $ F(x_1, x_2, \ldots, x_n) $ by $ x_1 - a_1 $, $ x_2 - a_2 $, $ \ldots $, $ x_n - a_n $ on the right. Once this is done, we may define general (\textit{nonfree}) skew polynomial rings, where evaluation is still natural at every point, as quotients of the free ring by two-sided ideals of skew polynomials that vanish at every point (Definition \ref{def skew polynomial rings}). Reasonably behaved iterated skew polynomial rings are also quotients of the introduced free multivariate skew polynomial rings, and evaluations by unique remainder division as in \cite{skewRM} are recovered by the proposed evaluations (Remark \ref{remark iterated are also quotients}), although the converse does not seem to hold.
Our main objective is to describe the \textit{functions} obtained by evaluating multivariate skew polynomials over division rings, under some finiteness conditions (see the paragraph below). This problem is closely related to that of interpolation in the sense of Lagrange, which has been studied previously in the univariate case in \cite{eric, lam, lam-leroy, skew-interpolation, zhang}.
Our main results are as follows: We obtain a description of the family of such functions, when defined on a finitely generated set of zeros (\textit{P-closed set}), as a left vector space over the division ring of coefficients, and we find its dimension and a left basis (Theorem \ref{th describing evaluation as vector space}). For this, we first obtain a Lagrange-type interpolation theorem (Theorem \ref{th lagrange interpolation}) on P-closed sets. To this end, we need to extend first the concept of \textit{P-independence} and \textit{P-basis} from \cite[Sec. 4]{algebraic-conjugacy}, which naturally form a matroid (Proposition \ref{prop matroid}). See \cite{oxley} for more details on matroid theory. For that purpose, we need to introduce \textit{ideals of zeros}, whose properties are based on extensions to the multivariate case of tools from \cite{lam, algebraic-conjugacy, lam-leroy}: A multiplication that is additive on degrees (Theorem \ref{th multiplication is additive}), an iterative evaluation on monomials (Theorem \ref{th fundamental functions}) and a product rule (Theorem \ref{th product rule}).
Apart from its own interest, our main motivations to develop this theory come from the theory of error-correcting codes over finite fields, in view of \cite{skewcyclic1, gabidulin, linearizedRS}, as explained above. A definition of skew Reed-Muller codes has been recently proposed \cite{skewRM}, based on evaluating certain iterated skew polynomials at certain points as noted previously. However, the core properties of skew polynomial evaluation codes rely on the matroid given by P-independence and evaluation on P-bases (see \cite{linearizedRS}), which we introduce in the multivariate case in this work. Apart from applications in coding theory, it has been recently shown in \cite{lin-multivariateskew} that Hilbert's Theorem 90 can be naturally stated and proven using the framework of this paper for general Galois extensions of fields (as considered by Noether) using arbitrary generators and relations of the Galois group (note that univariate skew polynomials restrict Hilbert 90 Theorems to a single generator \cite{hilbert90}, as originally stated by Kummer and Hilbert). A differential or more general version of such a Hilbert's Theorem 90 can be similarly put in this framework. Further applications in Galois theory or partial differential equations (such as a study of multivariate Moore or Wronskian matrices) may be possible and of interest.
The organization is as follows. In Section \ref{sec matrix morphisms}, we show which multiplications are additive on degrees over ``free multivariate polynomial rings'' (Theorem \ref{th multiplication is additive}), extending \cite[Eq. (3), (4) \& (5)]{ore}. In Section \ref{sec evaluations}, we show how to define evaluations as remainders of Euclidean divisions and give a recursive formula for monomials (Theorem \ref{th fundamental functions}), extending \cite[Lemma 2.4]{lam-leroy} and \cite[Eq. (2.3)]{lam-leroy}. In Section \ref{sec product rule}, we show how the product of two skew polynomials is preserved after evaluation (Theorem \ref{th product rule}), extending \cite[Th. 2.7]{lam-leroy}. In Section \ref{sec zeros}, we define P-closed sets and ideals of zeros, and give their basic properties. Using them, we define in Section \ref{sec general skew} \textit{nonfree} multivariate skew polynomial rings (Definition \ref{def skew polynomial rings}). In Section \ref{sec P-bases}, we extend the crucial concepts of \textit{P-independence} and \textit{P-bases} from \cite[Sec. 4]{algebraic-conjugacy} to our context. In Section \ref{sec lagrange interpolation}, we show the existence of Lagrange interpolating skew polynomials (Theorem \ref{th lagrange interpolation}). In Section \ref{sec image and ker}, we obtain the dimension and left bases of the left vector space of skew polynomial functions over a finitely generated P-closed set (Theorem \ref{th describing evaluation as vector space}). In Section \ref{sec skew vandermonde}, we give explicit computational methods to find such dimensions and bases and to perform Lagrange interpolation, via an extension of the Vandermonde matrices considered in \cite{lam, lam-leroy}. The complexity for finding ranks and P-bases is exponential in general, but given a P-basis, the complexity of finding Lagrange interpolating polynomials is polynomial.
\section*{Notation}
Unless otherwise stated, $ \mathbb{F} $ will denote a division ring.
Assuming $ \mathbb{F} $ to be finite (thus a field \cite{weddeburn}) avoids all other finiteness assumptions.
For positive integers $ m $ and $ n $, $ \mathbb{F}^{m \times n} $ will denote the set of $ m \times n $ matrices over $ \mathbb{F} $, and $ \mathbb{F}^n $ will denote the set of column vectors of length $ n $ over $ \mathbb{F} $. That is, $ \mathbb{F}^n = \mathbb{F}^{n \times 1} $.
On a non-commutative ring $ \mathcal{R} $, we will denote by $ (A) \subseteq \mathcal{R} $ the left ideal generated by a set $ A \subseteq \mathcal{R} $, and on a left vector space $ \mathcal{V} $ over $ \mathbb{F} $, we will denote by $ \langle B \rangle \subseteq \mathcal{V} $ the $ \mathbb{F} $-linear left vector space generated by a set $ B \subseteq \mathcal{V} $. We use the simplified notation $ (F_1, F_2, \ldots, F_n) = (\{ F_1, F_2, \ldots, F_n \}) $ and $ \langle F_1, F_2, \ldots, F_n \rangle = \langle \{ F_1, F_2, \ldots, F_n \} \rangle $.
All rings in this work will be assumed to have multiplicative identity.
\section{Free skew polynomial rings, matrix morphisms and vector derivations} \label{sec matrix morphisms}
In this section, we show which multiplications over a free non-commutative polynomial ring consist in appending monomials and are additive on degrees. See Remark \ref{remark why variables dont commute} to see why we cannot assume that variables commute with each other, unless we are dealing with conventional multivariate polynomials over fields. See Remarks \ref{remark why not iterated skew polynomials} and \ref{remark iterated are also quotients} to see why we do not consider iterated skew polynomial rings.
Fix a positive integer $ n $ from now on, let $ x_1, x_2, \ldots, x_n $ be $ n $ distinct characters, and denote by $ \mathcal{M} $ the set of all finite strings using these characters, that is, the free monoid with basis $ x_1, x_2, \ldots, x_n $ (see \cite[Sec. 6.5]{cohn}). The empy string will be denoted by $ 1 $. A character $ x_i $ will be called a \textit{variable}, an element $ \mathfrak{m} \in \mathcal{M} $ will be called a \textit{monomial}, and we will define its \textit{degree}, denoted by $ \deg(\mathfrak{m}) $, as its length as a string.
Let $ \mathcal{R} $ be the left vector space over $ \mathbb{F} $ with basis $ \mathcal{M} $. That is, every element $ F \in \mathcal{R} $ can be expressed uniquely as a linear combination (with coefficients on the left)
$$ F = \sum_{\mathfrak{m} \in \mathcal{M}} F_\mathfrak{m} \mathfrak{m}, $$
where $ F_\mathfrak{m} \in \mathbb{F} $, for $ \mathfrak{m} \in \mathcal{M} $, and $ F_\mathfrak{m} = 0 $ except for a finite number of monomials.
An element $ F \in \mathcal{R} $ will be called a \textit{(multivariate) skew polynomial}, and we will define its \textit{degree}, denoted by $ \deg(F) $, as the maximum degree of a monomial $ \mathfrak{m} \in \mathcal{M} $ such that $ F_\mathfrak{m} \neq 0 $, if $ F \neq 0 $. We will define $ \deg(F) = \infty $ if $ F = 0 $.
Formally, our objective is to provide $ \mathcal{R} $ with an inner product $ \mathcal{R} \times \mathcal{R} \longrightarrow \mathcal{R} $ that turns it into a non-commutative ring with $ 1 $ as multiplicative identity, restricts to the operation $ \mathcal{M} \times \mathcal{M} \longrightarrow \mathcal{M} $ that consists in appending strings, and where the degree of a product of two skew polynomials is the sum of their degrees.
First observe that, by identifying $ a \in \mathbb{F} $ with $ a 1 \in \mathcal{R} $, we may assume that $ \mathbb{F} \subseteq \mathcal{R} $, with the elements in $ \mathbb{F} $ called \textit{constants}. Furthermore, $ \mathbb{F} $ is a subring of $ \mathcal{R} $ as long as $ 1 $ is the multiplicative identity. Next, by inspecting constants and variables, we see that we need functions
$$ \sigma_{i,j} : \mathbb{F} \longrightarrow \mathbb{F}, \quad \textrm{and} \quad \delta_i : \mathbb{F} \longrightarrow \mathbb{F}, $$
for $ i,j = 1,2, \ldots, n $, such that
\begin{equation}
x_i a = \sum_{j=1}^n \sigma_{i,j}(a) x_j + \delta_i(a),
\label{eq def inner product}
\end{equation}
for $ i = 1,2, \ldots, n $, and for all $ a \in \mathbb{F} $. This defines two maps
\begin{equation}
\sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} : a \mapsto \left( \begin{array}{cccc}
\sigma_{1,1}(a) & \sigma_{1,2}(a) & \ldots & \sigma_{1,n}(a) \\
\sigma_{2,1}(a) & \sigma_{2,2}(a) & \ldots & \sigma_{2,n}(a) \\
\vdots & \vdots & \ddots & \vdots \\
\sigma_{n,1}(a) & \sigma_{n,2}(a) & \ldots & \sigma_{n,n}(a) \\
\end{array} \right),
\label{eq def matrix morphism}
\end{equation}
and
\begin{equation}
\delta : \mathbb{F} \longrightarrow \mathbb{F}^n : a \mapsto \left( \begin{array}{c}
\delta_1(a) \\
\delta_2(a) \\
\vdots \\
\delta_n(a)
\end{array} \right).
\label{eq def vector derivation}
\end{equation}
With this more compact notation, we may write Equation (\ref{eq def inner product}) as
\begin{equation}
\mathbf{x} a = \sigma(a) \mathbf{x} + \delta(a),
\label{eq def inner product compact}
\end{equation}
where $ \mathbf{x} $ is a column vector containing $ x_i $ in the $ i $th row, for $ i = 1,2, \ldots, n $. We have the following result, which extends the discussion in the case $ n = 1 $ given at the beginning of \cite{ore}. See also \cite[Th. 10.1]{cohn}.
\begin{theorem} \label{th multiplication is additive}
If an inner product in $ \mathcal{R} $ turns it into a non-commutative ring with multiplicative identity $ 1 $, consists in appending monomials when restricted to $ \mathcal{M} $ and is additive on degrees, then it is given on constants and variables as in (\ref{eq def inner product}), the map $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ in (\ref{eq def matrix morphism}) is a ring morphism, and the map $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $ in (\ref{eq def vector derivation}) is additive and satisfies that
\begin{equation}
\delta(ab) = \sigma(a) \delta(b) + \delta(a) b,
\label{eq multiplicative property vector derivations}
\end{equation}
for all $ a,b \in \mathbb{F} $.
Conversely, for any two such maps $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ and $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $, there exists a unique inner product in $ \mathcal{R} $ satisfying the properties in the previous paragraph. Furthermore, two such inner products are equal if, and only if, the corresponding maps are equal.
\end{theorem}
\begin{proof}
First assume that a given inner product in $ \mathcal{R} $ satisfies the properties given in the first paragraph. The additive properties of $ \sigma $ and $ \delta $ then follow from
$$ x_i(a+b) = (x_ia) + (x_ib), $$
for all $ a,b \in \mathbb{F} $ and all $ i = 1,2, \ldots, n $, their multiplicative properties follow from
$$ x_i(ab) = (x_ia)b, $$
for all $ a,b \in \mathbb{F} $ and all $ i = 1,2, \ldots, n $, and $ \sigma(1) = I $ follows from $ x_i 1 = 1 x_i $ (since $ 1 $ is a multiplicative identity) for all $ i = 1,2, \ldots, n $.
Next, the uniqueness and equality properties in the second paragraph are straightforward using Equations (\ref{eq def inner product}) or (\ref{eq def inner product compact}).
Finally, given a ring morphism $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ and an additive map $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $ satisfying (\ref{eq multiplicative property vector derivations}), we may define the desired inner product in $ \mathcal{R} $ as follows.
First, constants in $ \mathbb{F} $ act on the left as scalars ($ (a 1) F = a F $, for all $ F \in \mathcal{R} $). Now given $ \mathfrak{m}, \mathfrak{n} \in \mathcal{M} $, we define recursively on $ \mathfrak{m} $ the products
$$ (\mathfrak{m} x_i)(a \mathfrak{n}) = \sum_{j=1}^n \mathfrak{m} (\sigma_{i,j}(a) (x_j \mathfrak{n})) + \mathfrak{m}(\delta_i(a) \mathfrak{n}), $$
for all $ i = 1,2, \ldots, n $ and all $ a \in \mathbb{F} $, where $ \mathfrak{m} x_i $ and $ x_j \mathfrak{n} $ denote appending of monomials. Observe that this already defines, recursively on $ \mathfrak{m} $, the products of monomials as appending them, by choosing $ a = 1 $.
Finally, given general skew polynomials $ F = \sum_{\mathfrak{m} \in \mathcal{M}} F_\mathfrak{m} \mathfrak{m} $ and $ G = \sum_{\mathfrak{n} \in \mathcal{M}} G_\mathfrak{n} \mathfrak{n} $, where $ F_\mathfrak{m}, G_\mathfrak{m} \in \mathcal{R} $, for all $ \mathfrak{m} \in \mathcal{M} $, we define
$$ FG = \sum_{\mathfrak{m} \in \mathcal{M}} \sum_{\mathfrak{n} \in \mathcal{M}} F_\mathfrak{m} \left( \mathfrak{m} \left( G_\mathfrak{n} \mathfrak{n} \right) \right). $$
Note that this product is well-defined, since $ \deg(F) = d $ and $ \deg(G) = e $ imply that the coefficient of $ \mathfrak{m} $ in $ FG $ is zero whenever $ \deg(\mathfrak{m}) > d+e $, for all $ F, G \in \mathcal{R} $ and all $ \mathfrak{m} \in \mathcal{M} $. The properties of such an inner product stated in the theorem are all trivial, except for associativity, whose verification is left to the reader.
\end{proof}
This motivates the following definitions:
\begin{definition}[\textbf{Matrix morphisms and vector derivations}] \label{def matrix mor and vector der}
We call every ring morphism $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ a matrix morphism (over $ \mathbb{F} $), and we say that a map $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $ is a $ \sigma $-vector derivation (over $ \mathbb{F} $) if it is additive and satisfies
$$ \delta(ab) = \sigma(a) \delta(b) + \delta(a)b, $$
for all $ a,b \in \mathbb{F} $.
\end{definition}
\begin{definition}[\textbf{Free multivariate skew polynomial rings}]
Given a matrix morphism $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ and a $ \sigma $-vector derivation $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $, we define the free (multivariate) skew polynomial ring corresponding to $ \sigma $ and $ \delta $ as the unique ring $ \mathcal{R} = \mathbb{F}[\mathbf{x}; \sigma, \delta] $ with the inner product given by (\ref{eq def inner product}).
\end{definition}
Observe that the conventional free multivariate polynomial ring (called free algebra over $ \mathbb{F} $ when $ \mathbb{F} $ is commutative, see \cite[Sec. 0.11]{cohn} and \cite[Sec. 6.5]{cohn}) on the variables $ x_1, x_2, \ldots, $ $ x_n $ is obtained by choosing $ \sigma = {\rm Id} $ and $ \delta = 0 $, where we define $ {\rm Id}(a) = aI $, for all $ a \in \mathbb{F} $. Moreover, observe that this is the only case where constants and variables commute, which coincides with the only case where $ \mathbb{F}[\mathbf{x}; \sigma, \delta] $ is an algebra over $ \mathbb{F} $ when $ \mathbb{F} $ is commutative (here by \textit{algebra} we mean a ring $ \mathcal{R} $ that is a vector space over $ \mathbb{F} $ and whose inner product is $ \mathbb{F} $-bilinear, as in \cite{cohn}). Finally, observe also that $ \mathbb{F}[\mathbf{x}; \sigma, \delta] $ can still be characterized by a universal property similar to that of the free algebra. We only need to replace in the universal property the commutativity of constants and variables on free algebras by the rule (\ref{eq def inner product}). We leave the details to the reader.
We conclude the section with some particular instances of matrix morphisms and vector derivations of interest:
\begin{example} \label{example diagonal case}
A matrix morphism $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ satisfies $ \sigma_{i,j}(a) = 0 $, for all $ a \in \mathbb{F} $ and all $ i \neq j $ if, and only if, there exist ring endomorphisms $ \sigma_i : \mathbb{F} \longrightarrow \mathbb{F} $, for $ i = 1,2, \ldots, n $, such that
$$ \sigma(a) = \left( \begin{array}{cccc}
\sigma_1(a) & 0 & \ldots & 0 \\
0 & \sigma_2(a) & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & \sigma_n(a) \\
\end{array} \right), $$
for all $ a \in \mathbb{F} $. It is trivial to check that the family of $ \sigma $-vector derivations in this case are precisely those such that $ \delta_i $ is a $ \sigma_i $-derivation, for $ i = 1,2, \ldots, n $. An example is $ \mathbb{F} = k(t_1, t_2, \ldots, t_n) $, where $ k $ is a field, $ t_1, t_2, \ldots, t_n $ are algebraically independent variables, $ \sigma_i = {\rm Id} $ and $ \delta_i = \frac{\partial}{\partial t_i} $ is the conventional $ i $th partial derivative, for $ i = 1,2, \ldots, n $.
\end{example}
\begin{example} \label{example vector derivations}
Let $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ be a matrix morphism, and let $ \boldsymbol\beta \in \mathbb{F}^n $. The map $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $ defined by
$$ \delta(a) = \sigma(a) \boldsymbol\beta - \boldsymbol\beta a, $$
for all $ a \in \mathbb{F} $, is a $ \sigma $-vector derivation. When $ n = 1 $, these vector derivations are called \textit{inner derivations} in the literature.
\end{example}
\section{Evaluations of multivariate skew polynomials} \label{sec evaluations}
In this section, we show how to define \textit{evaluation maps} $ E_{\mathbf{a}} : \mathbb{F}[\mathbf{x}; \sigma, \delta] \longrightarrow \mathbb{F} $, for all $ \mathbf{a} \in \mathbb{F}^n $, that can be considered natural or standard. We will first require that these maps are left linear forms over $ \mathbb{F} $. We may then define the \textit{total evaluation map} as
\begin{equation}
E : \mathbb{F}[\mathbf{x}; \sigma, \delta] \longrightarrow \mathbb{F}^{\mathbb{F}^n}: F \mapsto \left( E_{\mathbf{a}}(F) \right) _{\mathbf{a} \in \mathbb{F}^n},
\label{eq def total evaluation map}
\end{equation}
which is again left linear. By linearity, we have that
$$ E_{\mathbf{a}} \left( \sum_{\mathfrak{m} \in \mathcal{M}} F_\mathfrak{m} \mathfrak{m} \right) = \sum_{\mathfrak{m} \in \mathcal{M}} F_\mathfrak{m} N_\mathfrak{m}(\mathbf{a}), $$
for all $ \mathbf{a} \in \mathbb{F}^n $, all $ F_\mathfrak{m} \in \mathbb{F} $, and for functions
$$ N_\mathfrak{m} : \mathbb{F}^n \longrightarrow \mathbb{F} : \mathbf{a} \longrightarrow E_\mathbf{a}(\mathfrak{m}) , $$
where $ \mathfrak{m} \in \mathcal{M} $. Therefore, giving a total evaluation map $ E $ is equivalent to giving the family of functions $ (N_\mathfrak{m})_{\mathfrak{m} \in \mathcal{M}} $, thus these will be called \textit{fundamental functions} of the evaluation $ E $. When $ n=1 $, the fundamental functions $ N_i = N_{x^i} $, for $ i = 0,1,2, \ldots $, coincide with those in \cite{lam, lam-leroy}.
As stated in Section \ref{sec intro}, a standard way of understanding evaluations of multivariate conventional polynomials is by giving a canonical ring isomorphism
$$ \mathbb{F} [x_1, x_2, \ldots, x_n] / \left( x_1-a_1, x_2-a_2, \ldots, x_n - a_n \right) \longrightarrow \mathbb{F}, $$
for all $ a_1, a_2, \ldots, a_n \in \mathbb{F} $, due to the ``Remainder Theorem''. The same idea is used in classical algebraic geometry to define evaluations as projections to a quotient ring given by a maximal ideal, which would be isomorphic to the so-called residue field.
To obtain such an isomorphism, we give a Euclidean-type division for skew polynomials of the type $ x_1-a_1, x_2-a_2, \ldots, x_n-a_n $:
\begin{lemma} \label{th euclidean division}
For any $ a_1, a_2, \ldots , a_n \in \mathbb{F} $ and any $ F \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $, there exist unique $ G_1, G_2, \ldots, $ $ G_n \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $ and $ b \in \mathbb{F} $ such that
\begin{equation}
F = \sum_{i = 1}^n G_i (x_i - a_i) + b.
\label{eq euclidean division}
\end{equation}
\end{lemma}
\begin{proof}
Existence is proven by a Euclidean division algorithm as usual. We next prove the uniqueness property. We only need to prove that if
\begin{equation}
\sum_{i = 1}^n G_i (x_i - a_i) + b = 0,
\label{eq euclidean division proof}
\end{equation}
then $ G_1 = G_2 = \ldots = G_n = b = 0 $. Assume the opposite. Without loss of generality, we may assume that $ G_n \neq 0 $ and $ \deg(G_n) \geq \deg(G_i) $, for all $ i $ with $ G_i \neq 0 $.
Let $ \prec $ denote the graded lexicographic (from right to left) ordering in $ \mathcal{M} $ with $ x_1 \prec x_2 \prec \ldots \prec x_n $, and denote by $ {\rm LM}(G) \in \mathcal{M} $ the leading monomial of a skew polynomial $ G \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $ with respect to $ \prec $. Then we see that the monomial $ {\rm LM}(G_n (x_n - a_n)) = {\rm LM}(G_n) x_n $ cannot be cancelled by any other monomial on the left-hand side of (\ref{eq euclidean division proof}). This is absurd and thus $ G_i = 0 $, for all $ i = 1,2, \ldots, n $. Hence $ b = 0 $ and we are done.
\end{proof}
\begin{remark}
Observe that the facts that the product in $ \mathbb{F}[\mathbf{x}; \sigma, \delta] $ consists in appending monomials and is additive on degrees are crucial in the proof of the previous lemma, since they allow us to state that $ {\rm LM}(G_n (x_n - a_n)) = {\rm LM}(G_n) x_n $ for the graded lexicographic ordering. These properties also ensure that the division algorithm does not run indefinitely. Note moreover that $ \mathbb{F} $ can be an arbitrary ring, since the leading coefficients of $ x_1-a_1, x_2-a_2, \ldots, x_n-a_n $ are all $ 1 $.
\end{remark}
\begin{remark} \label{remark why variables dont commute}
Observe that (being $ \mathbb{F} $ a division ring) we cannot guarantee that Lemma \ref{th euclidean division} (uniqueness of remainders) holds if we allow the variables to commute, unless we are dealing with multivariate conventional polynomials over fields.
Assume that $ n > 1 $ and add to the ring $ \mathcal{R} $ in Section \ref{sec matrix morphisms} the commutativity property on the variables: $ x_i x_j = x_j x_i $, for all $ i,j = 1,2, \ldots, n $. Observe that the rest of the properties of $ \mathcal{R} $ still imply the existence of the matrix morphism $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ and the $ \sigma $-vector derivation $ \delta : \mathbb{F} \longrightarrow \mathbb{F}^n $ by inspecting constants and variables.
Next take $ a_1, a_2, \ldots, a_n \in \mathbb{F} $. For fixed $ 1 \leq i < j \leq n $, we have that
$$ x_j (x_i - a_i) - x_i (x_j - a_j) = x_i a_j - x_j a_i = \sum_{k=1}^n \left( \sigma_{i,k}(a_j) - \sigma_{j,k}(a_i) \right) (x_k - a_k) $$
\begin{equation}
+ \left( \sum_{k=1}^n \left( \sigma_{i,k}(a_j) - \sigma_{j,k}(a_i) \right) a_k \right) + \delta_i(a_j) - \delta_j(a_i).
\label{eq term for noncommutative}
\end{equation}
Then the term (\ref{eq term for noncommutative}) equals $ 0 $ for all $ a_1, a_2, \ldots, a_n \in \mathbb{F} $ and all $ 0 \leq i < j \leq n $ if, and only if, $ \mathbb{F} $ is commutative, $ \sigma = {\rm Id} $ and $ \delta = 0 $. We leave the proof to the reader.
In particular, unique remainder as in Lemma \ref{th euclidean division} can only be guaranteed in this case (variables commute and $ \mathbb{F} $ is a division ring) if $ \mathbb{F} $ is commutative, $ \sigma = {\rm Id} $ and $ \delta = 0 $. Therefore, evaluation by unique remainder division (thus ``plug-in'' evaluation, see Remark \ref{remark pluging non-commutative}) does not exist even for multivariate conventional polynomials with commutative variables over non-commutative division rings (as considered in \cite{amitsur}, for instance).
However, one may usually define non-trivial relations between variables while preserving evaluation properties. See Section \ref{sec general skew}.
\end{remark}
\begin{remark} \label{remark why not iterated skew polynomials}
Since variables commute in many reasonable iterated skew polynomial rings, they do not satisfy the uniqueness of remainders as in Lemma \ref{th euclidean division}. Thus we do not consider iterated skew polynomials in this paper, in contrast with \cite{skewRM}. Take for instance any iterated skew polynomial ring $ \mathcal{S} = (\mathbb{F}[x_1;\sigma_1,\delta_1])[x_2;\sigma_2,\delta_2] $, where $ \delta_1 = \delta_2 = 0 $, $ \sigma_2(x_1) = x_1 $ and $ \sigma_1 \sigma_2 = \sigma_2 \sigma_1 $. Then $ x_2 x_1 = x_1 x_2 $. This can be easily extended to any number of variables. See also Remark \ref{remark iterated are also quotients}.
\end{remark}
We may now define a standard evaluation as follows, which extends the case $ n = 1 $ from \cite{lam, lam-leroy}:
\begin{definition}[\textbf{Standard evaluation}] \label{def standard evaluation}
For $ \mathbf{a} = (a_1, a_2, \ldots, a_n) \in \mathbb{F}^n $ and a skew polynomial $ F \in \mathbb{F} [\mathbf{x} ; \sigma,\delta] $, we define its $ (\sigma,\delta) $-evaluation, denoted by
\begin{equation}
F(\mathbf{a}) = E_{\mathbf{a}}^{\sigma, \delta}(F),
\label{eq def standard evaluation}
\end{equation}
as the unique element $ F(\mathbf{a}) \in \mathbb{F} $ such that
$$ F - F(\mathbf{a}) \in \left( x_1-a_1, x_2 - a_2, \ldots, x_n - a_n \right). $$
We denote the corresponding total evaluation map by $ E^{\sigma, \delta} $, and we use the notations $ E_{\mathbf{a}} $ and $ E $ when there is no confusion about $ \sigma $ and $ \delta $.
\end{definition}
These evaluation maps are well-defined and left linear by Lemma \ref{th euclidean division}. To conclude, we give a recursive formula on the fundamental functions of the total evaluation map $ E^{\sigma, \delta} $, which is of computational interest. This result is an extension of the case $ n = 1 $ given in \cite[Lemma 2.4]{lam-leroy} and \cite[Eq. (2.3)]{lam-leroy}.
\begin{theorem} \label{th fundamental functions}
The fundamental functions $ N_\mathfrak{m}^{\sigma, \delta} = N_\mathfrak{m} : \mathbb{F}^n \longrightarrow \mathbb{F} $, for $ \mathfrak{m} \in \mathcal{M} $, of the $ (\sigma,\delta) $-evaluation $ E^{\sigma, \delta} $ in Definition \ref{def standard evaluation} are given recursively as follows: $ N_1(\mathbf{a}) = 1 $, and
\begin{equation}
\left( \begin{array}{c}
N_{x_1 \mathfrak{m}}(\mathbf{a}) \\
N_{x_2 \mathfrak{m}}(\mathbf{a}) \\
\vdots \\
N_{x_n \mathfrak{m}}(\mathbf{a}) \\
\end{array} \right) = \sigma(N_\mathfrak{m}(\mathbf{a})) \mathbf{a} + \delta (N_\mathfrak{m}(\mathbf{a})),
\label{eq fundamental functions standard ev}
\end{equation}
for all $ \mathfrak{m} \in \mathcal{M} $ and all $ \mathbf{a} \in \mathbb{F}^n $.
\end{theorem}
\begin{proof}
We will use the compact matrix/vector notation in (\ref{eq def inner product compact}), and we proceed recursively on $ \mathfrak{m} \in \mathcal{M} $, for fixed $ \mathbf{a} \in \mathbb{F}^n $.
Obviously, $ N_1(\mathbf{a}) = 1 $. Assume now that it is true for a monomial $ \mathfrak{m} \in \mathcal{M} $. Therefore, there exist skew polynomials $ P_1, P_2, \ldots, P_n \in \left( x_1-a_1, x_2 - a_2, \ldots, x_n - a_n \right) $ such that, if we denote by $ \mathbf{P} $ the column vector whose $ i $th row is $ P_i $, for $ i = 1,2, \ldots, n $, then
$$ \mathbf{x} \mathfrak{m} = \mathbf{P} + \mathbf{x} N_\mathfrak{m}(\mathbf{a}) = \mathbf{P} + \sigma (N_\mathfrak{m}(\mathbf{a})) \mathbf{x} + \delta(N_\mathfrak{m}(\mathbf{a})) $$
$$ = \mathbf{P} + \sigma(N_\mathfrak{m}(\mathbf{a})) (\mathbf{x} - \mathbf{a}) + (\sigma(N_\mathfrak{m}(\mathbf{a})) \mathbf{a} + \delta(N_\mathfrak{m}(\mathbf{a}))), $$
and the result follows by Lemma \ref{th euclidean division}.
\end{proof}
\begin{remark} \label{remark pluging non-commutative}
Note that, when $ \sigma = {\rm Id} $ and $ \delta = 0 $, Theorem \ref{th fundamental functions} states that evaluation by unique remainder coincides with evaluation performed by ``plugging values'' in the variables but with reversed orders (see also Remark \ref{remark why variables dont commute}). For instance, the evaluation of $ x_1x_2 $ at $ (a_1,a_2) $ would be $ a_2a_1 $.
\end{remark}
We recall that in the case $ n=1 $ and $ \delta = 0 $, we have that $ N_i(a) = N_{x^i}(a) = \sigma^{i-1}(a) \cdots \sigma(a) a $, for $ i = 1,2, \ldots $, hence the notation $ N_\mathfrak{m} $ is a reminder of its similarity with the \textit{norm} function.
It has been recently shown \cite{lin-multivariateskew} that norms as in (\ref{eq fundamental functions standard ev}) allow to naturally state Hilbert's Theorem 90 for general Galois extensions of fields (as considered by Noether) using arbitrary generators and relations of the Galois group.
\section{Conjugacy and the product rule} \label{sec product rule}
From the previous section, we know that the $ (\sigma, \delta) $-evaluation $ E^{\sigma, \delta} $ is left linear. In this section, we will use the multiplicative properties of $ \sigma $, $ \delta $ and the fundamental functions of $ E^{\sigma, \delta} $ to show that it preserves products of skew polynomials in a certain way. This property will be used in the next section to define ideals of zeros and to characterize which of them are two-sided (Proposition \ref{prop two-sided ideals}). It will be especially important in Section \ref{sec lagrange interpolation} for constructing skew polynomials of restricted degree with a given set of zeros.
We need the concept of conjugacy, where the case $ n = 1 $ was given in \cite[Eq. (2.5)]{lam-leroy}.
\begin{definition}[\textbf{Conjugacy}]
Given $ \mathbf{a} \in \mathbb{F}^n $ and $ c \in \mathbb{F}^* $, we define the $ (\sigma,\delta) $-conjugate, or just conjugate if there is no confusion, of $ \mathbf{a} $ with respect to $ c $ as
\begin{equation}
\mathbf{a}^c = \sigma(c) \mathbf{a} c^{-1} + \delta(c) c^{-1} \in \mathbb{F}^n .
\label{eq def conjugate}
\end{equation}
\end{definition}
We have the following properties, which extend the case $ n = 1 $ given in \cite[Eq. (2.6)]{lam-leroy}.
\begin{lemma}
Given $ \mathbf{a}, \mathbf{b} \in \mathbb{F}^n $ and $ c,d \in \mathbb{F}^* $, the following properties hold:
\begin{enumerate}
\item
$ \mathbf{a}^1 = \mathbf{a} $ and $ (\mathbf{a}^c)^d = \mathbf{a}^{dc} $.
\item
The relation $ \mathbf{a} \thicksim \mathbf{b} $ if, and only if, there exist $ e \in \mathbb{F}^* $ with $ \mathbf{b} = \mathbf{a}^e $, is an equivalence relation on $ \mathbb{F}^n $.
\end{enumerate}
\end{lemma}
If $ n = 1 $, $ \sigma = {\rm Id} $ and $ \delta = 0 $, then the previous notion of conjugacy coincides with the usual one on the multiplicative monoid of $ \mathbb{F} $, which explains the terminology.
As noted in \cite{hilbert90}, Hilbert's Theorem 90 can be understood as any effective criterion for conjugacy, which in its classical form (cyclic Galois extensions) is given by the classical norm function. The same idea can be used to reinterpret Hilbert's Theorem 90 over any Galois extension of fields \cite{lin-multivariateskew}, where the norm function is replaced by the fundamental functions from the last section.
We may now establish and prove the product rule. The case $ n = 1 $ was first given in \cite[Th. 2.7]{lam-leroy}. We follow their proof using our matrix/vector notation.
\begin{theorem}[\textbf{Product rule}] \label{th product rule}
Given skew polynomials $ F, G \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $ and $ \mathbf{a} \in \mathbb{F}^n $, if $ G(\mathbf{a}) = 0 $, then $ (FG)(\mathbf{a}) = 0 $, and if $ c = G(\mathbf{a}) \neq 0 $, then
\begin{equation}
(FG)(\mathbf{a}) = F(\mathbf{a}^c) G(\mathbf{a}).
\label{eq product rule}
\end{equation}
\end{theorem}
\begin{proof}
It is obvious from Lemma \ref{th euclidean division} and Definition \ref{def standard evaluation} that, if $ G(\mathbf{a}) = 0 $, then $ (FG)(\mathbf{a}) = 0 $. Now assume that $ c = G(\mathbf{a}) \neq 0 $. First observe that
$$ \left( \mathbf{x} - \mathbf{a}^c \right) c = \sigma(c) (\mathbf{x} - \mathbf{a}). $$
Second, by Definition \ref{def standard evaluation} there exist skew polynomials $ P_i, Q_i \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $, for $ i = 1,2, \ldots, n $, such that
$$ F = \mathbf{P}^T \left( \mathbf{x} - \mathbf{a}^c \right) + F \left( \mathbf{a}^c \right), \textrm{ and } G = \mathbf{Q}^T (\mathbf{x} - \mathbf{a}) + G(\mathbf{a}), $$
where $ \mathbf{P} $ and $ \mathbf{Q} $ denote the column vectors whose $ i $th rows are $ P_i $ and $ Q_i $, respectively, for $ i = 1,2, \ldots, n $. Combining these facts, we obtain that
$$ FG = F \mathbf{Q}^T (\mathbf{x} - \mathbf{a}) + F G(\mathbf{a}) $$
$$ = \left( F \mathbf{Q}^T + \mathbf{P}^T \sigma(c) \right) (\mathbf{x} - \mathbf{a}) + F(\mathbf{a}^c)G(\mathbf{a}), $$
and we are done.
\end{proof}
This theorem can be stated when $ \mathbb{F} $ is an arbitrary ring by considering only the cases where $ c=0 $ or $ c $ is a unit. The fact that only one of these two cases happen when $ \mathbb{F} $ is a division ring will be crucial in Proposition \ref{prop two-sided ideals} and from Section \ref{sec lagrange interpolation} onwards.
Note that $ \mathbf{a}^c \neq \mathbf{a} $ in general when $ \mathbb{F} $ is non-commutative even if $ \sigma = {\rm Id} $ and $ \delta = 0 $. Thus the product rule is still of value for conventional polynomials over division rings. In particular, it may be that $ F(\mathbf{a}) = 0 $ and $ (FG)(\mathbf{a}) \neq 0 $ even for conventional polynomials, when $ \mathbb{F} $ is non-commutative.
\section{Zeros of multivariate skew polynomials} \label{sec zeros}
In this section, we will define and give the basic properties of sets of zeros of multivariate skew polynomials and, conversely, sets of skew polynomials that vanish at a certain set of affine points, which will be crucial in Section \ref{sec lagrange interpolation} for Lagrange interpolation. Conceptually, they will also be important in Section \ref{sec general skew} to define general skew polynomial rings with relations on the variables (\textit{nonfree}) and where evaluation still works in a natural way.
Observe that at this point our theory loses most of its analogies with the univariate case \cite{lam, algebraic-conjugacy, lam-leroy}, since $ \mathbb{F}[\mathbf{x}; \sigma, \delta] $ is not a principal ideal domain if $ n > 1 $, hence the use of \textit{minimal skew polynomials} as in \cite{lam, algebraic-conjugacy} is not possible. On the other hand, we gain analogy with respect to classical algebraic geometry:
\begin{definition} [\textbf{Zeros of skew polynomials}]
Given a set $ A \subseteq \mathbb{F}[\mathbf{x}; \sigma, \delta] $, we define its zero set as
$$ Z(A) = \{ \mathbf{a} \in \mathbb{F}^n \mid F(\mathbf{a}) = 0, \forall F \in A \}. $$
And given a set $ \Omega \subseteq \mathbb{F}^n $, we define its associated ideal as
$$ I(\Omega) = \{ F \in \mathbb{F}[\mathbf{x}; \sigma, \delta] \mid F(\mathbf{a}) = 0, \forall \mathbf{a} \in \Omega \}. $$
\end{definition}
Observe that the ideal associated to a subset of $ \mathbb{F}^n $ is indeed a left ideal:
\begin{proposition}
For any $ \Omega \subseteq \mathbb{F}^n $, it holds that $ I(\Omega) \subseteq \mathbb{F}[\mathbf{x}; \sigma, \delta] $ is a left ideal.
\end{proposition}
\begin{proof}
It follows directly from the product rule (Theorem \ref{th product rule}). Alternatively, it can be proven by noting that $ I(\Omega) = \bigcap_{\mathbf{a} \in \Omega} (x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n) $.
\end{proof}
We next list some basic properties of zero sets and ideals of zeros that follow from the definitions, in the same way as in classical algebraic geometry.
\begin{proposition} \label{prop properties of zeros}
Let $ \Omega, \Omega_1, \Omega_2 \subseteq \mathbb{F}^n $ and $ A, A_1, A_2 \subseteq \mathbb{F}[\mathbf{x}; \sigma, \delta] $ be arbitrary sets. The following properties hold:
\begin{enumerate}
\item
$ I(\{ \mathbf{a} \}) = \left( x_1-a_1, x_2 - a_2, \ldots, x_n - a_n \right) $ and $ Z(x_1-a_1, x_2 - a_2, \ldots, x_n - a_n) = \{ \mathbf{a} \} $, for all $ \mathbf{a} = (a_1, a_2, \ldots, a_n) \in \mathbb{F}^n $.
\item
$ I(\varnothing) = \left( 1 \right) $ and $ Z( 1 ) = \varnothing $.
\item
$ I(\mathbb{F}^n) \subseteq I(\Omega) $ and $ Z(\{ 0 \}) = \mathbb{F}^n $. That is, $ I(\mathbb{F}^n) $ is the minimal ideal of zeros.
\item
If $ \Omega_1 \subseteq \Omega_2 $, then $ I(\Omega_2) \subseteq I(\Omega_1) $.
\item
If $ A_1 \subseteq A_2 $, then $ Z(A_2) \subseteq Z(A_1) $.
\item
$ I(\Omega_1 \cup \Omega_2) = I(\Omega_1) \cap I(\Omega_2) $.
\item
$ Z(A) = Z(\left( A \right)) $ and $ Z(A_1 \cup A_2) = Z(\left( A_1 \right) + \left( A_2 \right)) = Z(A_1) \cap Z(A_2) $.
\item
$ \Omega \subseteq Z(I(\Omega)) $, and equality holds if, and only if, $ \Omega = Z(B) $ for some $ B \subseteq \mathbb{F}[\mathbf{x}; \sigma, \delta] $.
\item
$ A \subseteq \left( A \right) \subseteq I(Z(A)) $, and equality holds if, and only if, $ A = I(\Psi) $ for some $ \Psi \subseteq \mathbb{F}^n $.
\end{enumerate}
\end{proposition}
Item 8 in the previous proposition motivates the definition of \textit{P-closed sets}, where the case $ n = 1 $ was given in \cite{lam, lam-leroy}:
\begin{definition}[\textbf{P-closures}]
Given a subset $ \Omega \subseteq \mathbb{F}^n $, we define its P-closure as
$$ \overline{\Omega} = Z(I(\Omega)), $$
and we say that $ \Omega $ is P-closed if $ \overline{\Omega} = \Omega $.
\end{definition}
By Proposition \ref{prop properties of zeros}, Item 8, P-closed sets correspond to sets of zeros of sets of skew polynomials, and we have the following:
\begin{lemma}
Given a subset $ \Omega \subseteq \mathbb{F}^n $, it holds that $ \overline{\Omega} $ is the smallest P-closed subset of $ \mathbb{F}^n $ containing $ \Omega $.
\end{lemma}
\section{General and minimal skew polynomial rings} \label{sec general skew}
In this section, we define \textit{general skew polynomial rings} as those with a set of relations on the variables and where evaluation is still as in Definition \ref{def standard evaluation}. In particular, by considering a maximum set of such relations, we may define \textit{minimal skew polynomial rings}.
Note that the whole space $ \mathbb{F}^n $ is P-closed, and Item 3 in Proposition \ref{prop properties of zeros} says that, for evaluation purposes, we may just consider the quotient left module
$$ \mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\mathbb{F}^n), $$
which is a ring if $ I(\mathbb{F}^n) $ is a two-sided ideal, and in such a case we obtain the above mentioned minimal skew polynomial ring where the $ (\sigma, \delta) $-standard evaluation is still defined.
In the following proposition, we characterize when an ideal of zeros is two-sided, which includes in particular the ideal $ I(\mathbb{F}^n) $:
\begin{proposition} \label{prop two-sided ideals}
Given a subset $ \Omega \subseteq \mathbb{F}^n $, the following are equivalent:
\begin{enumerate}
\item
$ I(\Omega) $ is a two-sided ideal.
\item
If $ F \in I(\Omega) $ and $ c \in \mathbb{F} $, then $ Fc \in I(\Omega) $.
\item
If $ \mathbf{a} \in \overline{\Omega} $, then $ \mathbf{a}^c \in \overline{\Omega} $, for all $ c \in \mathbb{F}^* $.
\item
If $ \mathbf{a} \in \Omega $, then $ \mathbf{a}^c \in \overline{\Omega} $, for all $ c \in \mathbb{F}^* $.
\end{enumerate}
In particular, $ I(\mathbb{F}^n) $ is a two-sided ideal.
\end{proposition}
\begin{proof}
We prove the following implications:
$ 1) \Longrightarrow 2) $: Trivial.
$ 2) \Longrightarrow 3) $: Let $ \mathbf{a} \in \overline{\Omega} $, $ F \in I(\Omega) $ and $ c \in \mathbb{F}^* $. First, it holds that $ I(\Omega) = I(\overline{\Omega}) $ by Items 8 and 9 in Proposition \ref{prop properties of zeros}, and $ Fc \in I(\Omega) $ by hypothesis. Thus
$$ 0 = (Fc)(\mathbf{a}) = F(\mathbf{a}^c) c $$
by the product rule (Theorem \ref{th product rule}). Hence $ \mathbf{a}^c \in Z(I(\Omega)) = \overline{\Omega} $.
$ 3) \Longrightarrow 4) $: Trivial from $ \Omega \subseteq \overline{\Omega} $.
$ 4) \Longrightarrow 1) $: Let $ F \in I(\Omega) $ and $ G \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $, fix $ \mathbf{a} \in \Omega $ and define $ c = G(\mathbf{a}) $. If $ c = 0 $, then $ (FG)(\mathbf{a}) = 0 $ by the product rule. If $ c \neq 0 $, by hypothesis and the product rule, we have that
$$ (FG)(\mathbf{a}) = F(\mathbf{a}^c) G(\mathbf{a}) = 0, $$
since $ \mathbf{a}^c \in \overline{\Omega} $ and $ F \in I(\Omega) = I(\overline{\Omega}) $. Hence $ (FG)(\mathbf{a}) = 0 $ for any $ \mathbf{a} \in \Omega $ and thus $ FG \in I(\Omega) $, and we are done.
\end{proof}
Observe that, to prove $ 4) \Longrightarrow 1) $, we use that $ \mathbb{F} $ is a division ring, since we use that every $ c \in \mathbb{F} \setminus \{ 0 \} $ is invertible.
We may now define (nonfree) general skew polynomial rings and, in particular, a minimal one.
\begin{definition} [\textbf{Skew polynomial rings}] \label{def skew polynomial rings}
For any two-sided ideal $ I \subseteq I(\mathbb{F}^n) $, we say that the quotient ring
$$ \mathbb{F}[\mathbf{x}; \sigma, \delta] / I $$
is a skew polynomial ring with matrix morphism $ \sigma $ and vector derivation $ \delta $. The minimal skew polynomial ring with matrix morphism $ \sigma $ and vector derivation $ \delta $ is defined as that obtained when $ I = I(\mathbb{F}^n) $.
\end{definition}
This is exactly what happens with multivariate conventional polynomial rings (the case $ \sigma = {\rm Id} $ and $ \delta = 0 $) over fields. One may consider the free multivariate polynomial ring and define the conventional evaluation on it, either by plugging values in the variables or equivalently by unique remainder division. When $ \mathbb{F} $ is a field, $ I(\mathbb{F}^n) $ contains the two-sided ideal $ J $ generated by $ x_ix_j - x_jx_i $, for $ 1 \leq i < j \leq n $. If $ \mathbb{F} $ is infinite, then $ J = I(\mathbb{F}^n) $, whereas
\begin{equation}
I(\mathbb{F}^n) = J + ( x_1^q - x_1, x_2^q - x_2, \ldots, x_n^q - x_n )
\label{eq min conventional skew pol ring for finite}
\end{equation}
if $ \mathbb{F} $ is finite and has $ q $ elements. The results in the following sections will be proven for the free multivariate skew polynomial ring. By projecting onto the quotient, they can also be stated for any multivariate skew polynomial ring.
We conclude this section by showing that skew polynomial rings can form iterated sequences of rings by adding variables, even though we do not consider iterated skew polynomial rings in the standard way, as shown in Remark \ref{remark why not iterated skew polynomials}. The proof of the following result is straightforward.
\begin{proposition}
Let $ 0 < r < n $ be a positive integer, let $ \tau : \mathbb{F} \longrightarrow \mathbb{F}^{r \times r} $ and $ \nu : \mathbb{F} \longrightarrow \mathbb{F}^{(n-r) \times (n-r)} $ be matrix morphisms and let $ \delta_\tau : \mathbb{F} \longrightarrow \mathbb{F}^r $ and $ \delta_\nu : \mathbb{F} \longrightarrow \mathbb{F}^{(n-r)} $ be a $ \tau $-vector derivation and a $ \nu $-vector derivation, respectively. Define $ \sigma : \mathbb{F} \longrightarrow \mathbb{F}^{n \times n} $ and $ \delta_\sigma : \mathbb{F} \longrightarrow \mathbb{F}^n $ by
$$ \sigma(a) = \left( \begin{array}{cc}
\tau(a) & 0 \\
0 & \nu(a)
\end{array} \right) \quad \textrm{and} \quad \delta_\sigma(a) = \left( \begin{array}{c}
\delta_\tau(a) \\
\delta_\nu(a)
\end{array} \right), $$
for all $ a \in \mathbb{F} $. Then $ \sigma $ is a matrix morphism and $ \delta_\sigma $ is a $ \sigma $-vector derivation. Consider now the natural inclusion map
$$ \rho : \mathbb{F}[x_1, x_2, \ldots, x_r ; \tau, \delta_\tau] \longrightarrow \mathbb{F}[x_1, x_2, \ldots, x_n ; \sigma, \delta_\sigma]. $$
The following properties hold:
\begin{enumerate}
\item
$ \rho $ is a one to one ring morphism.
\item
For all $ F \in \mathbb{F}[x_1, x_2, \ldots, x_r ; \tau, \delta_\tau] $, all $ \mathbf{a}_\tau \in \mathbb{F}^r $ and all $ \mathbf{a}_\nu \in \mathbb{F}^{n-r} $, it holds that
$$ E_{\mathbf{a}_\tau}^{\tau, \delta_\tau}(F) = E_{\mathbf{a}_\sigma}^{\sigma, \delta_\sigma}(\rho(F)), $$
where $ \mathbf{a}_\sigma = (\mathbf{a}_\tau, \mathbf{a}_\nu) \in \mathbb{F}^n $.
\item
For any two-sided ideal $ J \subseteq I(\mathbb{F}^n) $, it holds that $ \rho^{-1}(J) \subseteq I(\mathbb{F}^r) $ is a two-sided ideal and $ \rho $ can be restricted to a one to one ring morphism
$$ \rho : \mathbb{F}[x_1, x_2, \ldots, x_r ; \tau, \delta_\tau] / \rho^{-1}(J) \longrightarrow \mathbb{F}[x_1, x_2, \ldots, x_n ; \sigma, \delta_\sigma] / J. $$
This holds in particular choosing $ J = I(\mathbb{F}^n) $, which implies that $ \rho^{-1}(J) = I(\mathbb{F}^r) $.
\end{enumerate}
\end{proposition}
In particular, if $ \sigma $ and $ \delta $ are given as in Example \ref{example diagonal case}, then $ \mathbb{F}[\mathbf{x};\sigma, \delta] $ contains a sequence of $ n $ nested skew polynomial rings, where the first one is the univariate skew polynomial ring $ \mathbb{F}[x_1; \sigma_1, \delta_1] $.
\begin{remark} \label{remark iterated are also quotients}
Iterated skew polynomial rings such that $ \delta_i(\mathbb{F}) \subseteq \mathbb{F} + \mathbb{F}x_1 + \cdots + \mathbb{F}x_{i-1} $, for $ i = 1,2, \ldots, n $, are also quotients of free multivariate skew polynomial rings since they satisfy the rules (\ref{eq def inner product}), setting $ \sigma_{i,i}(a) = \sigma_i(a) $ and $ \sigma_{i,j}(a) $ as the coefficient of $ x_j $ in $ \delta_i(a) $ for $ j<i $ (note that necessarily $ \sigma_i(\mathbb{F}) \subseteq \mathbb{F} $, for $ i = 1,2, \ldots, n $). Use for instance the universal property, as explained in Section \ref{sec matrix morphisms}. Examples include those in Remark \ref{remark why not iterated skew polynomials} or \cite[Ex. 2.3]{skewRM}, and important rings such as Weyl algebras \cite{galligo-diff} or solvable iterated skew polynomial rings \cite{kandri, zhangPBW}. In particular, when evaluation can be given for such iterated skew polynomials by unique remainder as in \cite{skewRM}, it must coincide with our notion of evaluation (Definition \ref{def standard evaluation}). What happens is that these iterated skew polynomial rings are in general quotients by a two-sided ideal $ J $ that satisfies that $ J \setminus I(\mathbb{F}^n) \neq \varnothing $ and $ I(\mathbb{F}^n) \setminus J \neq \varnothing $.
\end{remark}
\section{P-generators, P-independence and P-bases} \label{sec P-bases}
The main feature of P-closed sets is that they can be ``generated'' by certain subsets, called \textit{P-bases}, that control the possible values given by a function defined by a skew polynomial on such sets, as we will show in the next section. P-bases are given by a \textit{P-independence} notion and naturally form a matroid (Proposition \ref{prop matroid}). P-independence was defined for the case $ n=1 $ in \cite{lam, algebraic-conjugacy}. We start with the main definitions:
\begin{definition}[\textbf{P-generators}] \label{def P-generators}
Given a P-closed set $ \Omega \subseteq \mathbb{F}^n $, we say that $ \mathcal{G} \subseteq \Omega $ generates $ \Omega $ if $ \overline{\mathcal{G}} = \Omega $, and it is then called a set of P-generators for $ \Omega $. We say that $ \Omega $ is finitely generated if it has a finite set of P-generators.
\end{definition}
\begin{definition}[\textbf{P-independence}]
We say that $ \mathbf{a} \in \mathbb{F}^n $ is P-independent from $ \Omega \subseteq \mathbb{F}^n $ if it does not belong to $ \overline{\Omega} $.
A set $ \Omega \subseteq \mathbb{F}^n $ is called P-independent if every $ \mathbf{a} \in \Omega $ is P-independent from $ \Omega \setminus \{ \mathbf{a} \} $. P-dependent means not P-independent.
\end{definition}
\begin{definition}[\textbf{P-bases}]
Given a P-closed set $ \Omega \subseteq \mathbb{F}^n $, we say that a subset $ \mathcal{B} \subseteq \Omega $ is a P-basis of $ \Omega $ if it is P-independent and $ \overline{\mathcal{B}} = \Omega $.
\end{definition}
The following is the main result of this section, where Item 3 will be crucial in order to perform Lagrange interpolation recursively.
\begin{proposition} \label{prop characterizations P-bases}
Given sets $ \mathcal{B} \subseteq \Omega \subseteq \mathbb{F}^n $, where $ \Omega = \overline{\mathcal{B}} $, the following are equivalent:
\begin{enumerate}
\item
$ \mathcal{B} $ is a P-basis of $ \Omega $.
\item
If $ \mathcal{G} \subseteq \mathcal{B} $ and $ \overline{\mathcal{G}} = \Omega $, then $ \mathcal{G} = \mathcal{B} $. That is, $ \mathcal{B} $ is a minimal set of P-generators of $ \Omega $.
\item
(If $ \mathcal{B} $ is finite) For any ordering $ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M $ of the elements in $ \mathcal{B} $ and for $ i = 0,1,2, \ldots, M-1 $, it holds that $ \mathbf{b}_{i+1} $ is P-independent from $ \mathcal{B}_i = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, $ $ \mathbf{b}_i \} $, where $ \mathcal{B}_0 = \varnothing $.
\end{enumerate}
\end{proposition}
\begin{proof}
We prove each implication separately:
$ 1) \Longrightarrow 2) $: Assume that there exists $ \mathcal{G} \subsetneqq \mathcal{B} $ with $ \overline{\mathcal{G}} = \Omega $ and let $ \mathbf{a} \in \mathcal{B} \setminus \mathcal{G} $. Then
$$ \mathbf{a} \in \Omega = \overline{\mathcal{G}} = \overline{\mathcal{B} \setminus \{ \mathbf{a} \}}, $$
hence Item 1 does not hold.
$ 2) \Longrightarrow 1) $: Assume that $ \mathcal{B} $ is not P-independent and take $ \mathbf{a} \in \mathcal{B} $ with $ \mathbf{a} \in \overline{\mathcal{B} \setminus \{ \mathbf{a} \}} $. Define $ \mathcal{G} = \mathcal{B} \setminus \{ \mathbf{a} \} \subsetneqq \mathcal{B} $. It holds that
$$ \mathcal{B} = \{ \mathbf{a} \} \cup (\mathcal{B} \setminus \{ \mathbf{a} \}) \subseteq \overline{\mathcal{G}}, $$
hence $ \overline{\mathcal{G}} = \Omega $ and Item 2 does not hold.
$ 1) \Longrightarrow 3) $: Assume that $ \mathbf{b}_{i+1} $ is P-dependent from $ \mathcal{B}_i $ for a given $ i $ and a given ordering of $ \mathcal{B} $. Then
$$ \mathbf{b}_{i+1} \in \overline{\mathcal{B}_i} \subseteq \overline{\mathcal{B} \setminus \{ \mathbf{b}_{i+1} \}}, $$
hence Item 1 does not hold.
$ 3) \Longrightarrow 1) $: Assume that $ \mathbf{a} $ is P-dependent from $ \mathcal{B} \setminus \{ \mathbf{a} \} $ and order the $ M $ elements in $ \mathcal{B} $ in such a way that $ \mathbf{b}_M = \mathbf{a} $. Then $ \mathbf{b}_M $ is P-dependent from $ \mathcal{B}_{M-1} $ and Item 3 does not hold.
\end{proof}
We have the following important immediate consequence of Item 2 in the previous proposition:
\begin{corollary} \label{corollary finite basis}
If a P-closed set is finitely generated, then it admits a finite P-basis.
\end{corollary}
Finally, we observe that the family of P-independent sets forms a matroid \cite[Sec. 1.1]{oxley} whose bases \cite[Sec. 1.2]{oxley} are precisely the family of P-bases. The proof requires results from the following sections, but we will state the observation in this section for clarity.
\begin{proposition} \label{prop matroid}
For every finitely generated P-closed set $ \Omega \subseteq \mathbb{F}^n $, the pair $ (\mathcal{P}(\Omega), \mathcal{I}_\Omega) $ forms a matroid, where $ \mathcal{P}(\Omega) $ is the collection of all subsets of $ \Omega $, and $ \mathcal{I}_\Omega $ is the collection of P-independent subsets of $ \Omega $. Furthermore, the bases of the matroid $ (\mathcal{P}(\Omega), \mathcal{I}_\Omega) $ are precisely the P-bases of $ \Omega $.
\end{proposition}
\begin{proof}
First, it is trivial to see that $ \varnothing \in \mathcal{I}_\Omega $ and, if $ \mathcal{A}^\prime \subseteq \mathcal{A} $ and $ \mathcal{A} \in \mathcal{I}_\Omega $, then $ \mathcal{A}^\prime \in \mathcal{I}_\Omega $. The augmentation property of matroids is the first statement in Lemma \ref{lemma adding one to P-independent}, proven in Section \ref{sec image and ker}. Finally, the fact that bases (that is, maximal independent sets) and P-bases coincide is the second statement in Lemma \ref{lemma adding one to P-independent}.
\end{proof}
\section{Skew polynomial functions and Lagrange interpolation} \label{sec lagrange interpolation}
In this section, we give the main result of this paper. We show, in a Lagrange-type interpolation theorem, what values a function given by a skew polynomial can take when evaluated on a finitely generated P-closed set (Theorem \ref{th lagrange interpolation}). This result will be crucial in the following sections to describe the image and kernel of the evaluation map defined below (Theorem \ref{th describing evaluation as vector space}), to prove later that P-independent sets form a matroid (Lemma \ref{lemma adding one to P-independent}) and that the rank of a P-closed set is the rank of the corresponding skew Vandermonde matrix (Proposition \ref{prop rank vandermonde}). On the way, we derive other important results on P-closed sets.
Observe first that the total $ (\sigma,\delta) $-evaluation gives a left linear map
\begin{equation}
E^{\sigma,\delta}_{\Omega} : \mathbb{F}[\mathbf{x}; \sigma, \delta] \longrightarrow \mathbb{F}^{\Omega},
\label{eq restricted total evaluation}
\end{equation}
when restricted to evaluating over a subset $ \Omega \subseteq \mathbb{F}^n $ or, in other words, by composing $ E^{\sigma,\delta}_{\Omega} = \pi_\Omega \circ E^{\sigma,\delta} $, where $ \pi_\Omega : \mathbb{F}^{\mathbb{F}^n} \longrightarrow \mathbb{F}^\Omega $ is the canonical projection map.
Hence $ E^{\sigma, \delta}_{\Omega} $ gives a correspondence between multivariate skew polynomials $ F \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $ and some particular functions $ f = E^{\sigma, \delta}_{\Omega}(F) : \Omega \longrightarrow \mathbb{F} $. Such functions will be called \textit{multivariate skew polynomial functions} over $ \Omega $.
Formally, the objective of this section and the next one is to describe the kernel and image of the map $ E^{\sigma, \delta}_{\Omega} $ when $ \Omega $ is P-closed and finitely generated. We start with the following lemma, which is a key tool in Lagrange interpolation:
\begin{lemma} \label{lemma previous to lagrange}
Let $ \mathcal{B} \subseteq \mathbb{F}^n $ be a finite P-independent set and let $ \mathbf{b} \notin \overline{\mathcal{B}} $. There exists $ F \in I(\mathcal{B}) \setminus I(\mathcal{B} \cup \{ \mathbf{b} \}) $ such that $ \deg(F) \leq \# \mathcal{B} $.
\end{lemma}
\begin{proof}
First we prove that $ I(\mathcal{B}) \setminus I(\mathcal{B} \cup \{ \mathbf{b} \}) \neq \varnothing $. Assume the opposite. Then
$$ \overline{\mathcal{B}} = Z(I(\mathcal{B})) = Z(I(\mathcal{B} \cup \{ \mathbf{b} \})), $$
and $ \mathcal{B} \cup \{ \mathbf{b} \} \subseteq Z(I(\mathcal{B} \cup \{ \mathbf{b} \})) $ by Item 8 in Proposition \ref{prop properties of zeros}. Thus $ \mathbf{b} \in \overline{\mathcal{B}} $, which is a contradiction.
Now let $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} $ with $ M = \# \mathcal{B} $, let $ \prec $ be any ordering of $ \mathcal{M} $ preserving degrees, and take $ F \in I(\mathcal{B}) \setminus I(\mathcal{B} \cup \{ \mathbf{b} \}) $ such that $ {\rm LM}(F) $ is minimum possible with respect to $ \prec $. Assume that $ \deg(F) \geq M+1 $, which implies that $ \deg({\rm LM}(F)) \geq M+1 $ by the choice of the ordering $ \prec $.
Let $ {\rm LM}(F) = \mathfrak{m} x_{i_1} x_{i_2} \cdots x_{i_{M+1}} $, for some $ \mathfrak{m} \in \mathcal{M} $. By the product rule (Theorem \ref{th product rule}), we may choose elements $ a_1, a_2, \ldots, a_{M+1} \in \mathbb{F} $ such that
$$ G = \mathfrak{m} (x_{i_1} - a_1) (x_{i_2} - a_2) \cdots (x_{i_{M+1}} - a_{M+1}) $$
satisfies that $ G(\mathbf{b}_i) = 0 $, for $ i = 1,2, \ldots, M+1 $, denoting $ \mathbf{b}_{M+1} = \mathbf{b} $. In particular, there exists $ a \in \mathbb{F} $ such that $ H = F - aG $ satisfies $ {\rm LM}(H) \prec {\rm LM}(F) $, since $ {\rm LM}(F) = {\rm LM}(G) $. Now, by the definition of $ G $, it holds that
$$ H = F - aG \in I(\mathcal{B}) \setminus I(\mathcal{B} \cup \{ \mathbf{b} \}), $$
which is absurd by the minimality of $ {\rm LM}(F) $. Therefore $ \deg(F) \leq M $ and we are done.
\end{proof}
\begin{remark}
Note that, to construct $ G $ in the previous proof, we are implicitly using that $ \mathbb{F} $ is a division ring and we are implicitly applying Theorem \ref{th product rule} in its full form. Note however that constructing $ G $ is only needed to ensure the bound $ \deg(F) \leq \# \mathcal{B} $, but the existence of $ F $ still holds if $ \mathbb{F} $ is an arbitrary ring. The bound on $ \deg(F) $ will only be used to define skew Vandermonde matrices in Section \ref{sec skew vandermonde}. We do not investigate the full validity of Lemma \ref{lemma previous to lagrange} when $ \mathbb{F} $ is an arbitrary ring.
\end{remark}
The main result of this section is a Lagrange-type interpolation theorem in $ \mathbb{F}[\mathbf{x}; \sigma, \delta] $, whose proof is given by an iterative Newton-type algorithm thanks to Item 3 in Proposition \ref{prop characterizations P-bases}. This result extends the case $ n = 1 $ given in \cite[Th. 8]{lam} (see also the beginning of \cite[Sec. 5]{lam-leroy}). Newton-type iterative algorithms have been given in \cite{zhang} for univariate skew polynomials, and in \cite{skew-interpolation} for their free left modules.
\begin{theorem}[\textbf{Lagrange interpolation}] \label{th lagrange interpolation}
Let $ \Omega \subseteq \mathbb{F}^n $ be a finitely generated P-closed set with finite P-basis $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} $. The following hold:
\begin{enumerate}
\item
If $ E^{\sigma, \delta}_{\mathcal{B}}(F) = E^{\sigma, \delta}_{\mathcal{B}}(G) $, then $ E^{\sigma, \delta}_{\Omega}(F) = E^{\sigma, \delta}_{\Omega}(G) $, for all $ F,G \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $. That is, the values of a skew polynomial function $ f : \Omega \longrightarrow \mathbb{F} $ are uniquely given by $ f(\mathbf{b}_1), f(\mathbf{b}_2), \ldots, f(\mathbf{b}_M) $.
\item
For every $ a_1, a_2, \ldots, a_M \in \mathbb{F} $, there exists $ F \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $ such that $ \deg(F) < M $ and $ F(\mathbf{b}_i) = a_i $, for $ i = 1,2, \ldots, M $.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove each item separately.
\begin{enumerate}
\item
We just need to prove that $ E^{\sigma, \delta}_{\mathcal{B}}(F) = 0 $ implies $ E^{\sigma, \delta}_{\Omega}(F) = 0 $. By definition, $ \mathcal{B} \subseteq Z(F) $, and by Proposition \ref{prop properties of zeros}, it holds that $ I(Z(F)) \subseteq I(\mathcal{B}) $ and
$$ \Omega = \overline{\mathcal{B}} = Z(I(\mathcal{B})) \subseteq Z(I(Z(F))) = Z(F), $$
and the result follows.
\item
Let $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} $ as in Proposition \ref{prop characterizations P-bases}, Item 3. We prove the result iteratively for each of the P-independent sets $ \mathcal{B}_i $, $ i = 1,2, \ldots, M $, as in Newton's algorithm.
We start by defining the skew polynomial $ F_1 = a_1 $, which obviously satisfies $ F_1(\mathbf{b}_1) = a_1 $ and $ \deg(F_1) < 1 $. Now assume that $ M > 1 $, $ 1 \leq i \leq M-1 $ and there exists a skew polynomial $ F_i $ such that $ F_i(\mathbf{b}_j) = a_j $, for $ j = 1,2, \ldots, i $, and $ \deg(F_i) < i $. By Lemma \ref{lemma previous to lagrange}, there exists
$$ G \in I(\{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_i \}) \setminus I(\{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_{i+1} \}) $$
such that $ \deg(G) < i+1 $. The skew polynomial
$$ F_{i+1} = F_i + \left( a_{i+1} - F_i(\mathbf{b}_{i+1}) \right) G(\mathbf{b}_{i+1})^{-1} G $$
satisfies that $ F_{i+1}(\mathbf{b}_j) = a_j $, for $ j = 1,2, \ldots, i+1 $, and $ \deg(F_{i+1}) < i+1 $.
\end{enumerate}
\end{proof}
In the rest of the section, we derive some important consequences of this theorem. We start with the concept of dual P-bases.
\begin{definition} [\textbf{Dual P-bases}]
Given a finite P-basis $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} $ of a P-closed set $ \Omega \subseteq \mathbb{F}^n $, we say that a set of skew polynomials
$$ \mathcal{B}^* = \{ F_1, F_2, \ldots, F_M \} \subseteq \mathbb{F}[\mathbf{x}; \sigma, \delta] $$
is a dual P-basis of $ \mathcal{B} $ if $ F_i(\mathbf{b}_j) = \delta_{i,j} $ for all $ i,j = 1,2, \ldots, M $.
\end{definition}
We have the following immediate consequence of Theorem \ref{th lagrange interpolation} on the existence and uniqueness of dual P-bases:
\begin{corollary} \label{corollary existence dual P-bases}
Any finite P-basis, with $ M $ elements, of a P-closed set $ \Omega $ admits a dual P-basis consisting of $ M $ skew polynomials of degree less than $ M $. Moreover, any two dual P-bases of the same P-basis define the same skew polynomial functions over $ \Omega $.
\end{corollary}
An important consequence of Theorem \ref{th lagrange interpolation} is the following result on the sizes of P-bases:
\begin{corollary} \label{corollary all P-bases same size}
Any two P-bases of a finitely generated P-closed set are finite and have the same number of elements.
\end{corollary}
\begin{proof}
Given a finite P-basis $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} $ of size $ M $ of a P-closed set $ \Omega $, we will show first that $ {\rm Im}(E^{\sigma, \delta}_{\Omega}) $ is a left vector subspace of $ \mathbb{F}^{\Omega} $ with basis
\begin{equation}
\left\lbrace E^{\sigma, \delta}_{\Omega}(F_1), E^{\sigma, \delta}_{\Omega}(F_2), \ldots, E^{\sigma, \delta}_{\Omega}(F_M) \right\rbrace ,
\label{eq dual P-basis is basis}
\end{equation}
for any dual P-basis $ \{ F_1, F_2, \ldots, F_M \} $ of $ \mathcal{B} $. Assume that there exist $ \lambda_1, \lambda_2, \ldots, \lambda_M \in \mathbb{F} $ such that $ \sum_{i=1}^M \lambda_i E_\Omega^{\sigma, \delta}(F_i) = 0 $. Defining $ F = \sum_{i=1}^M \lambda_i F_i $, it follows that
$$ E_\Omega^{\sigma, \delta}(F) = E_\Omega^{\sigma, \delta} \left( \sum_{i=1}^M \lambda_i F_i \right) = \sum_{i=1}^M \lambda_i E_\Omega^{\sigma, \delta}(F_i) = 0, $$
thus $ \lambda_i = F(\mathbf{b}_i) = 0 $, for $ i = 1,2, \ldots, M $, and the set in (\ref{eq dual P-basis is basis}) is left linearly independent. Now, given $ F \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $, define $ G = F - \sum_{i=1}^M F(\mathbf{b}_i) F_i $. By definition, we have that $ E_\mathcal{B}^{\sigma, \delta}(G) = 0 $. Therefore $ E_\Omega^{\sigma, \delta}(G) = 0 $ by Theorem \ref{th lagrange interpolation}, thus
$$ E_\Omega^{\sigma, \delta}(F) = \sum_{i=1}^M F(\mathbf{b}_i) E_\Omega^{\sigma, \delta}(F_i), $$
and we conclude that the set in (\ref{eq dual P-basis is basis}) is a left basis of $ {\rm Im}(E^{\sigma, \delta}_{\Omega}) $.
In particular, $ \dim( {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) ) = M $ and the result follows for finite P-bases, since $ \dim ( {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) ) $ is independent of the choice of finite P-basis (since $ \mathbb{F} $ is a division ring).
Finally, if there exists an infinite P-basis $ \mathcal{B}^{\prime} $ of $ \Omega $, we may take a P-independent subset $ \mathcal{C} \subseteq \mathcal{B}^{\prime} $ of size $ M+1 $, define $ \Psi = \overline{\mathcal{C}} \subseteq \Omega $ and we would have that $ \dim( {\rm Im} ( E^{\sigma, \delta}_{\Psi} ) ) = M+1 $, and the canonical projection map
$$ \pi_\Psi : {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) \longrightarrow {\rm Im} ( E^{\sigma, \delta}_{\Psi} ) $$
is onto. This is absurd (since $ \mathbb{F} $ is a division ring) and the result follows.
\end{proof}
We conclude with the following natural definition, which is motivated by the previous corollary. It is an extension of the case $ n = 1 $ given in \cite{lam, lam-leroy}.
\begin{definition} [\textbf{Rank of P-closed sets}]
Given a finitely generated P-closed set $ \Omega \subseteq \mathbb{F}^n $, we define its rank, denoted by $ {\rm Rk}(\Omega) $, as the size of any of its P-bases.
\end{definition}
\begin{remark}
If $ \Omega $ is a finitely generated P-closed set, then $ {\rm Rk}(\Omega) $ coincides with the rank of the matroid $ (\mathcal{P}(\Omega), \mathcal{I}_\Omega) $ from Proposition \ref{prop matroid} (see \cite[Sec. 1.3]{oxley}). Note that we make use of Corollary \ref{corollary all P-bases same size} to prove later in Lemma \ref{lemma adding one to P-independent} that $ (\mathcal{P}(\Omega), \mathcal{I}_\Omega) $ is indeed a matroid.
\end{remark}
\section{The image and kernel of the evaluation map} \label{sec image and ker}
In this section, we describe the left vector space of skew polynomial functions and, to that end, we obtain the dimensions and some left bases of the image and kernel of the evaluation map (Theorem \ref{th describing evaluation as vector space}). As a conclusion to the section, we also deduce a finite-dimensional left vector space description of quotients of a skew polynomial ring, which includes the minimal skew polynomial ring when $ \mathbb{F} $ is a finite field.
We need the following auxiliary lemmas. The first can be seen as a refinement of Item 1 in Theorem \ref{th lagrange interpolation}:
\begin{lemma} \label{lemma for skew vandermonde}
Let $ \Omega \subseteq \mathbb{F}^n $ be a finitely generated P-closed set and let $ \mathcal{G} \subseteq \Omega $. It holds that $ \Omega = \overline{\mathcal{G}} $ if, and only if,
$$ \dim( {\rm Im} ( E^{\sigma, \delta}_{\Omega} )) = \dim( {\rm Im} ( E^{\sigma, \delta}_{\mathcal{G}} )). $$
\end{lemma}
\begin{proof}
First, recall that the given dimensions are finite due to the proof of Corollary \ref{corollary all P-bases same size}. The direct implication is in essence Item 1 in Theorem \ref{th lagrange interpolation}. For the reversed implication, the equality on dimensions implies that the projection map $ {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) \longrightarrow {\rm Im} ( E^{\sigma, \delta}_{\mathcal{G}} ) $ is a left vector space isomorphism. Thus $ I(\mathcal{G}) = I(\Omega) $, which implies that
$$ \overline{\mathcal{G}} = Z(I(\mathcal{G})) = Z(I(\Omega)) = \Omega. $$
\end{proof}
The next lemma is a further refinement of Proposition \ref{prop characterizations P-bases}:
\begin{lemma} \label{lemma adding one to P-independent}
If $ \mathcal{B} \subseteq \mathbb{F}^n $ is finite and P-independent, and $ \mathbf{a} \in \mathbb{F}^n \setminus \overline{\mathcal{B}} $, then $ \mathcal{B}^\prime = \mathcal{B} \cup \{ \mathbf{a} \} $ is P-independent.
As a consequence, a finite subset $ \mathcal{B} \subseteq \mathbb{F}^n $ is a P-basis of a finitely generated P-closed set $ \mathcal{B} \subseteq \Omega \subseteq \mathbb{F}^n $ if, and only if, the following property holds: $ \mathcal{B} $ is P-independent, and if $ \mathcal{B} \subseteq \mathcal{G} \subseteq \Omega $ and $ \mathcal{G} $ is P-independent, then $ \mathcal{G} = \mathcal{B} $. That is, $ \mathcal{B} $ is a maximal P-independent set in $ \Omega $.
\end{lemma}
\begin{proof}
Since $ \mathbf{a} \notin \overline{\mathcal{B}} $, it holds that $ I(\mathcal{B}) \setminus I(\mathcal{B}^\prime) \neq \varnothing $, as in the proof of Lemma \ref{lemma previous to lagrange}. Thus
$$ \dim({\rm Im}(E^{\sigma, \delta}_{\mathcal{B}^\prime})) \geq \dim({\rm Im}(E^{\sigma, \delta}_{\mathcal{B}})) + 1. $$
By the previous lemma and the proof of Corollary \ref{corollary all P-bases same size}, we conclude that $ {\rm Rk}(\overline{\mathcal{B}^\prime}) = {\rm Rk}(\overline{\mathcal{B}}) + 1 $. Again by Corollary \ref{corollary all P-bases same size} and its proof, we conclude that $ \mathcal{B}^\prime $ is a P-basis of $ \overline{\mathcal{B}^\prime} $ and, in particular, it is P-independent.
\end{proof}
Before giving the main result of this section, we need another consequence of Theorem \ref{th lagrange interpolation}, which will allow us to define the concepts of complementary P-closed sets and complementary P-bases:
\begin{corollary}
Let $ \Psi \subseteq \Omega \subseteq \mathbb{F}^n $ be P-closed sets. If $ \Omega $ is finitely generated, then so is $ \Psi $. Moreover, for any finite P-basis $ \mathcal{B} $ of $ \Psi $, there exists a finite P-independent set $ \mathcal{C} \subseteq \Omega $ such that $ \mathcal{B} \cap \mathcal{C} = \varnothing $ and $ \mathcal{B} \cup \mathcal{C} $ is a P-basis of $ \Omega $. In particular, if $ \Phi = \overline{\mathcal{C}} $, then
$$ {\rm Rk}(\Omega) = {\rm Rk}(\Psi) + {\rm Rk}(\Phi). $$
\end{corollary}
\begin{proof}
Assume that $ \Psi $ is not finitely generated. Using Lemma \ref{lemma adding one to P-independent}, we may construct iteratively a P-independent set $ \mathcal{D} \subseteq \Psi $ of size $ {\rm Rk}(\Omega) + 1 $. This is absurd by the same argument as in the proof of Corollary \ref{corollary all P-bases same size}.
Now, we may extend $ \mathcal{B} $ to a maximal P-independent subset of $ \Omega $ by adding iteratively to it elements $ \mathbf{c}_1, \mathbf{c}_2, \ldots, \mathbf{c}_N \in \Omega $, again by Lemma \ref{lemma adding one to P-independent}, which would be a P-basis of $ \Omega $ by maximality (again by Lemma \ref{lemma adding one to P-independent}). By defining $ \mathcal{C} = \{ \mathbf{c}_1, \mathbf{c}_2, \ldots, \mathbf{c}_N \} $, the rest of the claims in the corollary follow.
\end{proof}
\begin{definition} [\textbf{Complementary P-closed sets and P-bases}]
If $ \Psi \subseteq \Omega \subseteq \mathbb{F}^n $ are finitely generated P-closed sets and $ \mathcal{B} $ and $ \mathcal{C} $ are as in the previous Corollary, then we say that $ \Phi = \overline{\mathcal{C}} \subseteq \Omega $ is a complementary P-closed set of $ \Psi $ in $ \Omega $, and $ \mathcal{C} $ is a complementary P-basis of $ \mathcal{B} $ in $ \Omega $.
\end{definition}
We may now state and prove the second main result of the paper, which describes the image and kernel of $ E^{\sigma, \delta}_{\Omega} $ as left vector spaces over $ \mathbb{F} $ with some particular left bases.
\begin{theorem} \label{th describing evaluation as vector space}
Given a finitely generated P-closed set $ \Omega \subseteq \mathbb{F}^n $ with finite P-basis $ \mathcal{B} $, we have that
\begin{enumerate}
\item
$ {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) $ is a left vector space over $ \mathbb{F} $ of dimension $ M = {\rm Rk}(\Omega) $ with left basis
$$ E^{\sigma,\delta}_{\Omega}(\mathcal{B}^*) = \left\lbrace E^{\sigma, \delta}_{\Omega}(F_1), E^{\sigma, \delta}_{\Omega}(F_2), \ldots, E^{\sigma, \delta}_{\Omega}(F_M) \right\rbrace , $$
where $ \mathcal{B}^* = \{ F_1, F_2, \ldots, F_M \} $ is a dual P-basis of $ \mathcal{B} $. Observe that, by Corollary \ref{corollary existence dual P-bases}, $ E^{\sigma,\delta}_{\Omega}(\mathcal{B}^*) $ depends only on $ \mathcal{B} $ and not on the choice of the dual P-basis.
\item
If $ \mathbb{F}^n $ is finitely generated as P-closed set, $ \mathcal{C} $ is a complementary P-basis of $ \mathcal{B} $ in $ \mathbb{F}^n $, and $ \mathcal{C}^* = \{ G_1, G_2, \ldots, G_N \} $ is a dual P-basis of $ \mathcal{C} $ that is part of a dual P-basis $ (\mathcal{B} \cup \mathcal{C})^* $ of $ \mathcal{B} \cup \mathcal{C} $, then
$$ {\rm Ker} \left( E^{\sigma, \delta}_{\Omega} \right) = I(\mathbb{F}^n) \oplus \langle G_1, G_2, \ldots, G_N \rangle, $$
as left vector spaces over $ \mathbb{F} $, and $ G_1, G_2, \ldots, G_N $ are left linearly independent over $ \mathbb{F} $.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof of Item 1 was given in the proof of Corollary \ref{corollary all P-bases same size}. Now we prove Item 2:
First, $ G_1, G_2, \ldots, G_N $ are left linearly independent over $ \mathbb{F} $ by Item 1, since so are their evaluations over $ \Phi = \overline{\mathcal{C}} $.
Now we show that, if $ F \in I(\mathbb{F}^n) \cap \langle G_1, G_2, \ldots, G_N \rangle $, then $ F = 0 $. To that end, write $ F = \sum_{i=1}^N \lambda_i G_i $, for some $ \lambda_i \in \mathbb{F} $ and all $ i = 1,2, \ldots, N $. Since $ F \in I(\mathbb{F}^n) $, it holds that $ E_\Phi^{\sigma, \delta}(F) = 0 $, and since $ E^{\sigma, \delta}_{\Phi}(G_1), E^{\sigma, \delta}_{\Phi}(G_2), \ldots, E^{\sigma, \delta}_{\Phi}(G_M) $ are left linearly independent by Item 1, we conclude that $ F = 0 $.
Next let $ F \in {\rm Ker} ( E^{\sigma, \delta}_{\Omega} ) $. Then by Theorem \ref{th lagrange interpolation}, it holds that
$$ F - \sum_{i=1}^N F(\mathbf{c}_i) G_i \in I(\mathbb{F}^n), $$
since this skew polynomial vanishes at $ \mathcal{B} \cup \mathcal{C} $, and this set is a P-basis of $ \mathbb{F}^n $. Hence $ F \in I(\mathbb{F}^n) \oplus \langle G_1, G_2, \ldots, G_N \rangle $.
Conversely, let $ F \in I(\mathbb{F}^n) \oplus \langle G_1, G_2, \ldots, G_N \rangle $. By the assumptions, we have that $ F(\mathbf{b}) = 0 $, for all $ \mathbf{b} \in \mathcal{B} $. Hence $ F \in {\rm Ker}( E^{\sigma, \delta}_{\Omega} ) $ again by Theorem \ref{th lagrange interpolation}.
\end{proof}
We conclude with the following consequence, which describes the quotient left modules over the ideal associated to a finitely generated P-closed set. Such quotient left modules include the minimal skew polynomial ring if $ \mathbb{F}^n $ is finitely generated, which is the case if $ \mathbb{F} $ is finite.
\begin{corollary}
If $ \{ F_1, F_2, \ldots, F_M \} $ is a dual P-basis of a finitely generated P-closed set $ \Omega \subseteq \mathbb{F}^n $, then
$$ \mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\Omega) \cong \langle F_1, F_2, \ldots, F_M \rangle $$
as left vector spaces, where the isomorphism is given by inverting the projection to the quotient ring. Moreover, $ F_1, F_2, \ldots, F_M $ are left linearly independent, and hence
$$ \dim \left( \mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\Omega) \right) = {\rm Rk}(\Omega). $$
In particular, the minimal skew polynomial ring $ \mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\mathbb{F}^n) $ is a finite-dimensional left vector space over $ \mathbb{F} $ of dimension $ {\rm Rk}(\mathbb{F}^n) $ if $ \mathbb{F}^n $ is finitely generated.
\end{corollary}
\begin{proof}
Again, it follows directly from Item 1 in Theorem \ref{th describing evaluation as vector space} that $ F_1, F_2, \ldots, F_M $ are left linearly independent over $ \mathbb{F} $, since so are their evaluations over $ \Omega $.
Now consider the left linear projection map $ \rho : \langle F_1, F_2, \ldots, F_M \rangle \longrightarrow \mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\Omega) $. To show that it is onto, it suffices to observe that, given $ F \in \mathbb{F}[\mathbf{x}; \sigma, \delta] $, it holds that
$$ \rho(F) = \rho \left( \sum_{i=1}^M F(\mathbf{b}_i) F_i \right), $$
by Item 1 in Theorem \ref{th lagrange interpolation}, where $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} $ is the P-basis of $ \Omega $ associated to $ \{ F_1, F_2, \ldots, $ $ F_M \} $.
Finally, the evaluation map $ \mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\Omega) \longrightarrow {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) $ is a left vector space isomorphism by definition, thus by Item 1 in Theorem \ref{th describing evaluation as vector space}, it holds that
$$ \dim ( \langle F_1, F_2, \ldots, F_M \rangle ) = M = \dim ( {\rm Im} ( E^{\sigma, \delta}_{\Omega} ) ) = \dim (\mathbb{F}[\mathbf{x}; \sigma, \delta] / I(\Omega)). $$
Hence $ \rho $ is a left vector space isomorphism, and we are done.
\end{proof}
As shown in Equation (\ref{eq min conventional skew pol ring for finite}), if $ \mathbb{F} $ is finite and has $ q $ elements, then
$$ \dim(\mathbb{F}[\mathbf{x}] / I(\mathbb{F}^n)) = q^n = {\rm Rk}(\mathbb{F}^n), $$
since $ {\rm Rk}(\mathbb{F}^n) = \# \mathbb{F}^n = q^n $ in the conventional case. Hence the previous corollary extends this well-known result for finite fields.
\section{Skew Vandermonde matrices and how to find P-bases} \label{sec skew vandermonde}
In the univariate case ($ n = 1 $), Vandermonde matrices are a crucial tool to explicitly compute Lagrange interpolating polynomials. The multivariate case works similarly, although only existence of interpolating skew polynomials may be derived, and not their uniqueness. This is due to the non-square form of multivariate Vandermonde matrices.
In this section, we extend the concept of \textit{skew Vandermonde matrix} from the univariate case in \cite{lam, lam-leroy} to the multivariate case. As applications and thanks to the recursive formula in Theorem \ref{th fundamental functions}, we show how to explicitly compute P-bases, dual P-bases and Lagrange interpolating skew polynomials over finitely generated P-closed sets.
The case $ n = 1 $ in the following definition was given in \cite[Eq. (4.1)]{lam-leroy}, and previously in \cite{lam}:
\begin{definition} [\textbf{Skew Vandermonde matrices}]
Let $ \mathcal{N} \subseteq \mathcal{M} $ be a finite set of monomials and let $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_M \} \subseteq \mathbb{F}^n $. We define the corresponding $ (\sigma, \delta) $-skew Vandermonde matrix, denoted by $ V_{\mathcal{N}}^{\sigma, \delta}(\mathcal{B}) $, as the $ ( \# \mathcal{N} ) \times M $ matrix over $ \mathbb{F} $ whose rows are given by
$$ (N_\mathfrak{m}(\mathbf{b}_1), N_\mathfrak{m}(\mathbf{b}_2), \ldots, N_\mathfrak{m}(\mathbf{b}_M)) \in \mathbb{F}^M, $$
for all $ \mathfrak{m} \in \mathcal{N} $ (given certain ordering in $ \mathcal{M} $). If $ d $ is a positive integer, we define $ \mathcal{M}_d $ as the set of monomials of degree less than $ d $, and we denote
$$ V_{d}^{\sigma, \delta}(\mathcal{B}) = V_{\mathcal{M}_d}^{\sigma, \delta}(\mathcal{B}). $$
\end{definition}
An important consequence of Theorem \ref{th lagrange interpolation} is finding the rank and a P-basis of a given finitely generated P-closed set:
\begin{proposition} \label{prop rank vandermonde}
Given a finite set $ \mathcal{G} \subseteq \mathbb{F}^n $ with $ M $ elements, and $ \Omega = \overline{\mathcal{G}} $, it holds that
$$ {\rm Rk} \left( V_{M}^{\sigma, \delta}(\mathcal{G}) \right) = {\rm Rk}(\Omega). $$
Moreover, a subset $ \mathcal{B} \subseteq \mathcal{G} $ is a P-basis of $ \Omega $ if, and only if, $ \# \mathcal{B} = {\rm Rk}(\Omega) = {\rm Rk} ( V_{\# \mathcal{B}}^{\sigma, \delta}(\mathcal{B}) ) $.
Hence applying Gaussian elimination to the matrix $ V_{M}^{\sigma, \delta}(\mathcal{G}) $, we may find the rank of $ \Omega $ and at least one of its P-bases.
\end{proposition}
\begin{proof}
First, by Corollary \ref{corollary existence dual P-bases} and Theorem \ref{th describing evaluation as vector space}, it holds that $ {\rm Im} ( E_{\Omega}^{\sigma, \delta} ) $ is the left vector space generated by the evaluations $ (N_\mathfrak{m}(\mathbf{a}))_{\mathbf{a} \in \Omega} \in \mathbb{F}^\Omega $, for $ \mathfrak{m} \in \mathcal{M}_M $. By Lemma \ref{lemma for skew vandermonde}, to calculate $ \dim({\rm Im} ( E_{\Omega}^{\sigma, \delta} )) $, we may restrict such evaluations to points in $ \mathcal{G} $, and the first claim follows.
Now we prove the second claim. If $ \mathcal{B} $ is a P-basis of $ \Omega $, then $ \# \mathcal{B} = {\rm Rk}(\Omega) $ by definition, and $ {\rm Rk}(\Omega) = {\rm Rk} ( V_{\# \mathcal{B}}^{\sigma, \delta}(\mathcal{B}) ) $ by the first claim.
Conversely, if $ \# \mathcal{B} = {\rm Rk}(\Omega) = {\rm Rk} ( V_{\# \mathcal{B}}^{\sigma, \delta}(\mathcal{B}) ) $, then by Theorem \ref{th describing evaluation as vector space}, it holds that
$$ \dim( {\rm Im} ( E^{\sigma, \delta}_{\Omega} )) = {\rm Rk}(\Omega) = {\rm Rk} ( V_{\# \mathcal{B}}^{\sigma, \delta}(\mathcal{B}) ) \leq \dim( {\rm Im} ( E^{\sigma, \delta}_{\mathcal{B}} )). $$
Since the opposite inequality always holds, it follows from Lemma \ref{lemma for skew vandermonde} that $ \overline{\mathcal{B}} = \Omega $. Now, $ \mathcal{B} $ is a minimal set of P-generators of $ \Omega $, since $ \# \mathcal{B} = {\rm Rk}(\Omega) $, all minimal sets of P-generators are P-bases by Proposition \ref{prop characterizations P-bases} and all have the same size by Corollary \ref{corollary all P-bases same size}. Hence we conclude that $ \mathcal{B} $ is a P-basis of $ \Omega $.
\end{proof}
A classical way of stating the Lagrange interpolation theorem is as the invertibility of Vandermonde matrices. This result is an immediate consequence of Theorem \ref{th lagrange interpolation}:
\begin{corollary}
Let $ \Omega \subseteq \mathbb{F}^n $ be a finitely generated P-closed set with P-basis $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, $ $ \mathbf{b}_M \} $. There exists a solution to the linear system
\begin{equation}
(F_\mathfrak{m})_{\mathfrak{m} \in \mathcal{M}_M} V_{M}^{\sigma, \delta}(\mathcal{B}) = (a_1, a_2, \ldots, a_M),
\label{eq system for interpolation}
\end{equation}
for any $ a_1, a_2, \ldots, a_M \in \mathbb{F} $ (that is, $ V_{M}^{\sigma, \delta}(\mathcal{B}) $ is left invertible). For any solution, the corresponding skew polynomial $ F = \sum_{\mathfrak{m} \in \mathcal{M}_M} F_\mathfrak{m} \mathfrak{m} $ satisfies that $ F(\mathbf{b}_i) = a_i $, for $ i = 1,2, \ldots, M $, and $ \deg(F) < M $.
\end{corollary}
Another important immediate consequence is the following:
\begin{corollary}
Given a P-basis $ \mathcal{B} $, with $ M $ elements, of a P-closed set, one can obtain a dual P-basis of $ \mathcal{B} $, consisting of skew polynomials of degree less than $ M $, by solving $ M $ systems of $ M $ linear equations whose coefficients are taken from left linearly independent rows in $ V_{M}^{\sigma, \delta}(\mathcal{B}) $.
\end{corollary}
In conclusion, to find a P-basis of a P-closed set $ \Omega \subseteq \mathbb{F}^n $ with $ M = {\rm Rk}(\Omega) $ and generated by a finite set $ \mathcal{G} $, we need to find $ M $ linearly independent columns in $ V_{M}^{\sigma, \delta}(\mathcal{G}) $. Using Gaussian elimination, such method has exponential complexity in $ M $ if $ n > 1 $, since the number of rows in $ V_M^{\sigma, \delta}(\mathcal{G}) $ is $ \# \mathcal{M}_M $, which is exponential in $ M $.
Fortunately, if we are given or have precomputed a P-basis of $ \Omega $, we may find Lagrange interpolating skew polynomials over $ \Omega $ with complexity $ \mathcal{O}(M^3) $, and find a dual P-basis with complexity $ \mathcal{O}(M^4) $.
\section{Conclusion and open problems}
In this paper, we have introduced free multivariate skew polynomials with coefficients over rings, although we have focused on division rings. We have given a natural definition of evaluation (Definition \ref{def standard evaluation}), which extends the univariate case studied in \cite{lam, algebraic-conjugacy, lam-leroy}, and we have obtained a product rule (Theorem \ref{th product rule}). With these notions and assumptions, we were able to define general nonfree multivariate skew polynomial rings (Definition \ref{def skew polynomial rings}), where evaluation is still natural. We have described (by giving dimensions and left bases) in Theorem \ref{th describing evaluation as vector space} the left vector spaces of functions defined by multivariate skew polynomials, when defined over a finitely generated P-closed set (set of zeros). This has been done thanks to a Lagrange-type interpolation theorem (Theorem \ref{th lagrange interpolation}). The following problems are left open:
\begin{enumerate}
\item
Find explicit descriptions of general multivariate skew polynomial rings. In other words, find explicit descriptions of matrix morphisms, vector derivations and two-sided ideals contained in $ I(\mathbb{F}^n) $.
\item
The previous item is particularly interesting in the case of finite fields, where the minimal skew polynomial ring is generated by a finite collection of skew polynomials.
\item
Although we have given computational methods to find ranks, P-bases, dual P-bases and Lagrange interpolating skew polynomials, it would be interesting to obtain explicit formulas for such objects. Algorithms for finding P-bases with polynomial complexity are also interesting, as well as reducing the complexity of finding Lagrange interpolating skew polynomials.
\item
Investigate how to perform Euclidean-type divisions over multivariate skew polynomial rings, which would extend Lemma \ref{th euclidean division}. A notion of Gr{\"o}bner basis may be possible and useful in this context.
\end{enumerate}
\section*{Acknowledgement}
The authors wish to thank the anonymous reviewer for their very helpful comments. The first author gratefully acknowledges the support from The Independent Research Fund Denmark (Grant No. DFF-4002-00367, Grant No. DFF-5137-00076B ``EliteForsk-Rejsestipendium'', and Grant No. DFF-7027-00053B).
\small | 170,610 |
TITLE: Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
QUESTION [35 upvotes]: Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not surjective. As we learned in this question, there are models of $ZF$ where $V \to V^{\ast \ast}$ is an isomorphism when $V$ has a countable basis. I think the same argument shows that it is consistent with ZF that this is an isomorphism whenever $V$ has a basis.
Is it consistent with $ZF$ that $V \to V^{\ast \ast}$ is an isomorphism for all vector spaces $V$?
I ask because I'm teaching a rigorous undergrad analysis class. My students keep asking me whether they have to believe that $V \to V^{\ast \ast}$ can fail to be an isomorphism. Of course, I'm trying to change their intuition to point out why most mathematicians find the failure of isomorphism plausible and point out that there are more subtle ways to salvage the claim, such as Hilbert spaces, but I'd also love to be able to give them a choice free proof that there is some vector space where this issue comes up.
REPLY [46 votes]: No, it’s not consistent.
Let $V=k^{(\omega)}$ be the vector space of finite sequences of elements of $k$. Then $V^*$ can be identified with the vector space $k^\omega$ of all sequences, and elements of the image of the natural map $V\to V^{**}$, considered as maps $k^\omega\to k$, are determined by their restriction to $k^{(\omega)}$.
So if $V\to V^{**}$ is an isomorphism, then, taking $W=k^\omega/k^{(\omega)}$, there are no nonzero linear maps $W\to k$, and hence $W^{**}=0$. But $W$ is nonzero.
So the map to the double dual must fail to be an isomorphism either for $V$ or for $W$.
REPLY [6 votes]: Not an answer to your question but a variant which is perhaps more à propos given that you said you are teaching a rigorous undergrad analysis class.
Let $V=\oplus_{\mathbb{N}}\mathbb{R}$ be the space with countable basis which motivated your question. Another way to salvage the isomorphism $V\simeq V^{\ast\ast}$ (without giving up any cherished axiom) is to think of $V$ as equipped with the finest locally convex topology and consider duals as topological duals always given the strong topology. It is easy to see that $V^{\ast}\simeq \prod_{\mathbb{N}}\mathbb{R}$ with the product topology. Although perhaps counterintuitive, when one takes the proper (i.e., strong topological) dual of $\prod_{\mathbb{N}}\mathbb{R}$, then one gets back to $\oplus_{\mathbb{N}}\mathbb{R}$. | 215,354 |
Michael Kay writes: > > If the XSLT writes to standard output, yes. but if I use eg the Saxon > > extension to create a new file, I expect that to be created > > in the same directory as the input. > > Saxon's <xsl:document> doesn't yet fully conform to the XSLT 1.1 draft: the > filename is interpreted relative to the current directory rather than the > base URI. This is a documented restriction. I don't use xsl:document (I reverted back to 6.0.2 and saxon:output), because I cannot see any way to write a stylesheet using it that any other processor today will accept. Is there a way? so my problem will go away with final xsl:document, I gather. thats good. Sebastian XSL-List info and archive: | 171,308 |
Date My Family bachelor Okafor lights fuse on SA-Nigeria ties
Date My Family bachelor, Nigerian Arinze Okafor, ruffled feathers in the latest episode on Sunday.
Okafor angered many social media commentators who perceived him to have been "rude" and "disrespectful" on the show.
His demeanor was heavily criticised and perpetuated a perception that some Nigerian men did not take South African women seriously.
Okafor did not seem to think before he spoke as he shocked tweeps when he asked the first family, with the mother of the first potential date present, if they thought their daughter would satisfy him sexually.
He went further and referred to the mother as a "slay queen". Slay queen is slang for a young woman who uses glamour to impress rich men for financial gain.
He was labelled a player after visiting the third family where he was more interested in the potential date's cousin.
The highlight of the show was when he chose a potential date. He almost kicked himself when he realised he chose a woman who did not have all the qualities he wanted.
Okafor had to drag himself to hug the poor woman who was waiting for him.
On their way to their date, the two were not talking much.
Abdulmalik Tumelo tweeted: "No offence but Nigerian men disrespect South African women. Look at... #DateMyFamilly."
Ozzie Mthembu wrote: "Today's episode was a proper demonstration of how African brothers view SA women."
KhudaniNekhwev1 said: "They should have called this week episode sexual edition???? #DateMyFamilly."
Kaytee_scorpio summed it up by saying: "Apart from the fact that the guy was disrespectful throughout the show, his date was just as disappointed as he was in her. A mess of a show yesterday." | 163,526 |
TITLE: Class of sets is a semiring, but not a ring
QUESTION [0 upvotes]: Let $E$ be a nonempty finite set and let $\Omega:=E^\mathbb{N}$ be the set of all $E$-valued sequences $(w_n)_{n\in{\mathbb{N}}}$. For any $w_1,...,w_n\in E$, let
$[w_1,...,w_n]_:=\{w'\in\Omega\colon w_i'=w_i,\ \forall i=1,..,n\}$
be the set of all sequences whose first $n$ values are $w_1,...,w_n$. Let $\mathcal{A}_0=\{\emptyset\}$ and define
$\mathcal{A}_n:=\{[w_1,...,w_n]\colon w_1,...,w_n\in E\}$
Finally, let $\mathcal{A}:=\bigcup_{n=0}^\infty \mathcal{A}_n$
I tried to show that this is a semiring, but not a ring for $|E|>1$.
Definition: A class of sets $\mathcal{A}\subset 2^{\Omega}$ is called a semiring if
$\emptyset \in \mathcal{A}$
for any two sets $A,B\in\mathcal{A}$, the difference set $B\setminus A$ is a finite union of mutually disjoint sets in $\mathcal{A}$
$\mathcal{A}$ is closed under finite intersections.
First, the closed intersection: I pick two elements of $\mathcal{A}$, being $\mathcal{A}_n$ and $\mathcal{A}_m$ for which i assume that $n<m$. But no elements $\omega_n = [w_1,...,w_n], \omega_m=[w_1,...,w_m]$ are identical. To see this, consider elements of $[w_1,...,w_m]$ which are identical up to index $n$ (for elements not identical up to $n$ the intersection is empty). In $[w_1,...,w_m]$ we still have indices $w_{n+1},...,w_{m}$ which are fixed for all elements in $[w_1,...,w_m]$, but not for the ones in $[w_1,...,w_n]$. Thus, no two $\omega_n, \omega_m$ are identical and $\mathcal{A}_n\cap\mathcal{A}_m=\emptyset\in\mathcal{A}$.
A similar argument holds for the second condition, where we always have that for $\mathcal{A}_n,\mathcal{A}_m\in\mathcal{A}, \ n\le m$ that $\mathcal{A}_n\setminus\mathcal{A}_m=\emptyset$
The first condition is trivial, since it follows from the definition of $\mathcal{A}_0$.
Is my reasoning correct?
Now I want to show that $\mathcal{A}$ is a ring for $|E|>1$. For this it is enough (from what I have shown so far), to disprove the closedness under unions. I will not make a rigorous proof for this, but only say that the union of two $\mathcal{A}_n,\mathcal{A}_m$ will have elements which can not be produced by a another single $\mathcal{A}_l$.
Is this in principle correct?
REPLY [1 votes]: You are confusing $\bigcup \mathcal A_n$ with $\bigcup \{\mathcal A_n\}$. The elements of $\mathcal A=\bigcup_\Bbb N \mathcal A_n$ (as you defined it) are the sets $x$ such that $x\in\mathcal A_n$ for some $n\in\Bbb N$. In other words, $\mathcal A$ consists of the sets $[w_1,\dots,w_n]$ such that $n\in\Bbb N$ and $w_i\in E$ for each $i$. So we have that $\mathcal A_n\subseteq\mathcal A$, and not that $\mathcal A_n\in\mathcal A$.
We can check the three conditions for being a semiring:
We have $\varnothing\in \mathcal A$, since $\varnothing\in\mathcal A_0\subseteq\mathcal A$.
Let $n\leq m$, and let $W=[w_1,\dots,w_n]\in\mathcal A_n$ and $V=[v_1,\dots,v_m]\in\mathcal A_m$, then these are elements of $\mathcal A$. We want to show that $W\setminus V$ and $V\setminus W$ are both finite unions of disjoint sets in $\mathcal A$.
Let's start with the case where $w_i\neq v_i$ for some $i\leq n$. Then $[w_1,\dots,w_n]\cap[v_1,\dots,v_m]=\varnothing$, and thus $W\setminus V=W=\bigcup\{W\}$ and $V\setminus W=V=\bigcup\{V\}$
In the other case $w_i=v_i$ for each $i\leq n$, and thus $V\subseteq W$. This gives us that $V\setminus W=\varnothing=\bigcup\{\varnothing\}$. On the other hand, the set $W\setminus V$ is now the union of the sets $[w_1,\dots,w_n,u_{n+1},\dots,u_m]$ such that $u_{i}\neq v_i$ for some $n<i\leq m$. How many of these sets are there? Well, there are $|E|^{m-n}$ sequences $u_{n+1},\dots,u_m$, and only one of them is equal to $v_{n+1},\dots,v_m$, thus there are $|E|^{m-n}-1$ of such sets. Since $|E|$ is finite, and all $[w_1,\dots,w_n,u_{n+1},\dots,u_m]$ are disjoint from each other, we see that $W\setminus V$ is the union of a finite family of disjoint sets.
Again, let $n\leq m$, and let $[w_1,\dots,w_n]\in\mathcal A_n$ and $[v_1,\dots,v_m]\in\mathcal A_m$, then there are two possibilities:
$w_i\neq v_i$ for some $i\leq n$. We saw that $[w_1,\dots,w_n]\cap[v_1,\dots,v_m]=\varnothing$, and this is an element of $\mathcal A$.
$w_i=v_i$ for every $i\leq n$. In this case you will see that $[v_1,\dots,v_m]\subseteq[w_1,\dots,w_n]$, giving that $[w_1,\dots,w_n]\cap[v_1,\dots,v_m]=[w_1,\dots,w_n]$, which is in $\mathcal A$. | 5,785 |
PDHPE newsletter - Term 3 2016
K-6 PDHPE news – update on Draft directions for syllabus development PDHPE K-10, decommissioning of curriculum support website and national health and physical education (HPE) day.
Newsletter contains:
- Things to know about the K-10 Syllabus development process for PDHPE
- How to access files no longer available through curriculum support
- Purpose, learning intentions and success criteria for physical education lessons
- NSW physical literacy continuum
- 150 minutes of physical activity a week
- Health resources
- National HPE day.
Share this | 6,902 |
TITLE: Shortest supersequence of all permutations of $n$ elements
QUESTION [21 upvotes]: Given an alphabet with $n$ characters, what is the shortest sequence that contains all $n!$ permutations as subsequences?
A subsequence can be obtained from a sequence by deleting any characters, thus it's different from a substring, whose elements have to be contiguous in the original sequence. I say this because the similar problem of finding the shortest sequence having all permutations as substrings seems to be more studied, but it's different from what I'm asking here.
Some examples of shortest supersequences:
$n=2\quad-\quad121$ (length 3)
$n=3\quad-\quad1213121$ (length 7)
$n=4\quad-\quad1234123142134$ (length 13 - not proven to be shortest).
It's easy to see that $n^2$ is an upper bound, since a sequence
$$12\ldots n\, 12\ldots n\, \ldots\, 12\ldots n $$
($n$ times) contains all permutations. A simple lower bound is $n(n+1)/2$, basically because
$$12\ldots n\; 12\ldots (n-1)\; 12\ldots (n-2)\;\ldots $$
is too short (this can be proven rigorously). Is anything more known about this problem?
The question was asked on stack exchange, but the answer there is far from satisfactory since it gives only a broken link and no reference.
REPLY [0 votes]: This is a suggestion for further development, as opposed to an answer. It seems to hold
much promise.
Note that the n^2 upper bound can easily be shortened by 2, since any permutation not
beginning with 1 and not ending in n does not need those letters in the example, and otherwise
encodes a shorter permutation which does not need the first n or last n letters of the example.
More generally, the only permutation that "needs" all n^2 letters is the permutation with all
descending letters: if a_i less than a_{i+1} then put those in the ith run, and now you can make
do with one less run of letters. I believe you can extend this to chop off n-1 letters from either
end when n is sufficiently large.
Even more generally, replace one of the middle runs 1...n with a shorter decreasing run
n-1...2: if now there is a decrease of three or more consecutive elements, the longest or
perhaps the "most central"
such falling run can be placed
in the middle, and the rest of the permutation on either side.
I believe a careful analysis of up down sequences in permutations will show a
nice upper bound of n(n - sqrt(n)) using an example of sqrt(n) falling runs
interspersed with n- 2sqrt(n) rising runs. | 107,998 |
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\begin{document}
\title{Namioka spaces and strongly Baire spaces}
\author{V.V.Mykhaylyuk}
\address{Department of Mathematics\\
Chernivtsi National University\\ str. Kotsjubyn'skogo 2,
Chernivtsi, 58012 Ukraine}
\email{[email protected]}
\subjclass[2000]{Primary 54C08, 54C30, 54C05}
\commby{Ronald A. Fintushel}
\keywords{separately continuous functions, Namioka property, Baire space, caliber of topological space}
\begin{abstract}
A notion of strongly Baire space is introduced. Its definition is a transfinite development of some equivalent reformulation of the Baire space definition. It is shown that every strongly Baire space is a Namioka space and every $\beta-\sigma$-unfavorable space is a strongly Baire space.
\end{abstract}
\maketitle
\section{Introduction}
Investigation of the discontinuity points set of separately continuous functions, that is functions which are continuous with respect every variable, was beginning by R.Baire in [1] and was continued by many mathematicians. A Namioka's result [2] on the continuity points set of separately continuous functions, defined on the product of two spaces one of which is compact, has become a new impulse to development of this investigation.
A topological space $X$ is called {\it strongly countably complete}, if there exists a sequence $({\mathcal U}_n)_{n=1}^{\infty}$ of open covers of $X$ such that for every centered sequence $(F_n)^{\infty}_{n=1}$ of closed in $X$ sets $F_n$ such that for every $n\in \mathbb N$ there exists $U\in {\mathcal U}_n$
such that $F_n\subseteq U$, the intersection $\bigcap\limits_{n=1}^{\infty}F_n$ is nonempty.
\begin{theorem}[Namioka] Let $X$ be a strongly countably complete space, $Y$ be a compact space and $f:X\times Y \to \mathbb R$ be a separately continuous function. Then there exists an everywhere dense in $X$ $G_{\delta}$-set $A\subseteq X$ such that the function $f$ is continuous at every point of set $A\times Y$.
\end{theorem}
The following notions were introduced in [3].
A mapping $f:X\times Y \to \mathbb R$ {\it has the Namioka property} if there exists a dense in $X$ $G_{\delta}$-set $A\subseteq X$ such that $A\times Y\subseteq C(f)$, where by $C(f)$ we denote the set of the joint continuity points set of mapping $f$.
A Baire space $X$ is called {\it Namioka space}, if for every compact space $Y$ every separately continuous function $f:X\times Y\to \mathbb R$ has the Namioka property.
It was obtained in [4] that Namioka spaces are closely connected with topological games.
Let ${\mathcal P}$ be a system of subsets of topological space $X$. We define a $G_{\mathcal P}$-game on $X$, in which the players $\alpha$ and $\beta$ participate.
A nonempty open in $X$ set $U_0$ is the first move of $\beta$ and a nonempty open in $X$ set $V_1\subseteq U_0$ and set $P_1\in {\mathcal P}$ are the first move of $\alpha$. Further $\beta$ chooses a nonempty open in $X$ set $U_1\subseteq V_1$ and $\alpha$ chooses a nonempty open in $X$ set $V_2\subseteq U_1$ and a set $P_2\in {\mathcal P}$ and so on. The player $\alpha$ wins if $(\bigcap\limits_{n=1}^{\infty}V_n)\bigcap (\overline{\bigcup\limits_{n=1}^{\infty}P_n})\ne\O$. Otherwise
$\beta$ wins.
A topological space $X$ is called {\it $\alpha$-favorable in the $G_{{\mathcal P}}$-game} if $\alpha$ has a winning strategy in this game. A topological space $X$ is called {\it $\beta$-unfavorable in the $G_{{\mathcal P}}$-game} if $\beta$ has no winning strategy in this game. Clearly, any $\alpha$-favorable topological space $X$ is a $\beta$-unfavorable space.
In the case of ${\mathcal P}=\{X\}$ the game $G_{{\mathcal P}}$ is the classical Choquet game and $X$ is $\beta$-unfavorable in this game if and only if $X$ is a Baire space (see [3]). If ${\mathcal P}$ is the system of all finite (or one-point) subsets of $X$ then $G_{{\mathcal P}}$-game is called a {\it $\sigma$-game}.
J.~Saint-Raymond shows in [3] that for usage the topological games method in these investigations it is enough to require a weaker condition of $\beta$-unfavorability instead of the $\alpha$-favorability. He proved that every $\beta-\sigma$-unfavorable space is a Namioka space and generalized the Christensen result.
A further development of this technique leads to a consideration of another topological games which based on wider systems ${\mathcal P}$ of subsets of a topological space $X$.
Let $T$ be a topological space and ${\mathcal K}(T)$ be a collection of all compact subsets of $T$. Then $T$ is said to be {\it ${\mathcal K}$-countably-determined} if there exist a subset $S$ of the topological space ${\mathbb N}^{\mathbb N}$ and a mapping $\varphi:S\to {\mathcal K}(T)$ such that for every open in $T$ set $U\subseteq T$ the set $\{s\in S: \varphi(s)\subseteq U\}$ is open in $S$ and $T=\bigcup\limits_{s\in S} \varphi (s)$; and it is called {\it ${\mathcal K}$-analytical} if there exists such a mapping $\varphi$ for the set $S={\mathbb N}^{\mathbb N}$.
A set $A$ in a topological space $X$ is called {\it bounded} if for any continuous function $f:X\to \mathbb R$ the set $f(A)=\{f(a): a\in A\}$ is bounded.
The following theorem gives further generalizations of Saint-Raymond result.
\begin{theorem}\label{th:1.2} Any $\beta$-unfavorable in $G_{\mathcal P}$-game topological space $X$ is a Namioka space if :
$(i)$ ${\mathcal P}$ is the system of all compact subsets of $X$ ({\bf M.~Talagrand} \cite{T});
$(ii)$ ${\mathcal P}$ is the system of all ${\mathcal K}$-analytical subsets of $X$ ({\bf G.~Debs} \cite{D});
$(iii)$ ${\mathcal P}$ is the system of all bounded subsets of $X$ ({\bf O.~Maslyuchenko} \cite{Ma});
$(iv)$ ${\mathcal P}$ is a system of all ${\mathcal K}$-countable-determined subsets of $X$ ({\bf V.~Rybakov} \cite{R}).
\end{theorem}
It is easy to see that $(iv)\Rightarrow (ii) \Rightarrow (i)$ and $(iii) \Rightarrow (i)$.
It was interesting in this connection to obtain the characterization of Namioka spaces using their internal structure without topoplogical games (analogously as in the classical definition od Baire space). In this paper we study the notion of strongly Baire space. It introduced in the form which is a transfinite generalization of one reformulation of the definition of Baire space. We show that every $\beta-\sigma$-unfavorable space is strongly Baire space and every strongly Baire space is a Namioka space. Moreover, we study sufficient conditions on a topological space $X$ which provide the metrizability of every compact $Y\subseteq C_p(X)$.
\section{Strongly Baire space and Namioka property}
It this section we introduce the notion of strongly Baire space and prove some results on relations between strongly Baire spaces and Namioka spaces and
$\beta-\sigma$-unfavorable spaces.
Let $\alpha$ be an ordinal. We say that {\it a net $(B_{\xi}:\xi<\alpha)$ of sets $B_{\xi}$ occupies a net $(A_{\xi}:\xi<\alpha)$ of sets $A_\xi$} if
$(i)$ $A_{\xi}\subseteq B_{\xi}$ for every $\xi<\alpha$;
$(ii)$ $A_{\beta}\subseteq \bigcup\limits_{\xi<\beta} B_{\xi}$ for every limited ordinal $\beta<\alpha$.
A topological space $X$ is called {\it strongly Baire}, if for every ordinal $\alpha$, an increasing net $(A_{\xi}:\xi<\alpha)$ of closed in $X$ sets $A_{\xi}$ and an increasing net $(B_{\xi}:\xi<\alpha)$ of closed in $X$ sets $B_{\xi}$ which occupies the net $(A_{\xi}:\xi<\alpha)$ the following condition holds
$$
{\rm int}(\bigcup\limits_{\xi<\alpha} A_{\xi}) \subseteq \overline{\bigcup\limits_{\xi<\alpha} {\rm int}(B_{\xi})},
$$
where by ${\rm int}(C)$ we denote the interior of $C$. For $\alpha=\omega_0$ we obtain that
$${\rm int}(\bigcup\limits_{n\in \mathbb N} A_{n}) \subseteq \overline{\bigcup\limits_{n\in\mathbb N} {\rm int}(A_{n})}$$
for every sequence $(A_n)^{\infty}_{n=1}$ of closed in $X$ sets $A_n$. This is equivalent to the fact that $X$ is Baire. Hence the definition of strongly Baire space is a transfinite strengthening of the Baire condition.
We will use the following two proposition from [9]. The first of they illustrates a property of the dependence on some coordinates of continuous functions defined on compacts. The second proposition illustrates a relation between the dependence and the Namioka property of separately continuous functions. Together these propositions lead to consider the nets from the definition of strongly Baire space.
\begin{proposition}\label{p:2.1} Let $Y\subseteq {\mathbb R}^T$ be a compact space, $Z$ be a metric space, $f:Y\to Z$ be a separately continuous function, $\varepsilon \geq 0$ and set $S\subseteq T$ such that $|f(y')-f(y'')|_Z\leq\varepsilon$ for every $y',y''\in Y$ with $y'|_S=y''|_S$. Then for every $\varepsilon'>\varepsilon$ there exist a finite set $S_0\subseteq S$ and $\delta>0$ such that $|f(y')-f(y'')|_Z\leq \varepsilon'$ for every $y',y''\in Y$ with $|y'(s)-y''(s)|<\delta$ for every $s\in S_0$.
\end{proposition}
\begin{proposition}\label{p:2.2} Let $X$ be a Baire space, $Y\subseteq {\mathbb R}^T$ be a compact space, $f:X\times Y\to \mathbb R$ be a separately continuous function. Then the following conditions are equivalent:
$(i)$ $f$ has the Namioka property;
$(ii)$ for every open in $X$ nonempty set $U$ and real $\varepsilon >0$ there exist an open in $X$ nonempty set $U_0\subseteq U$ and at most countable set $S_0\subseteq T$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon$ for every $x\in U_0$ and $y',y''\in Y$ with $y'|_{S_0}=y''|_{S_0}$.
\end{proposition}
\begin{theorem}\label{th:2.3} Every strongly Baire space is a Namioka space.
\end{theorem}
\begin{proof} Let $X$ be a strongly Baire space, $Y\subseteq {\mathbb R}^T$ be a compact space, $f:X\times Y\to \mathbb R$ be a separately continuous function, $U$ be an open in $X$ nonempty set and $\varepsilon>0$. According to Theorem 2.2, it is enough to prove that there exist an at most countable set $S_0\subseteq T$ and an open in $X$ nonempty set $U_0\subseteq U$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon$ for every $x\in U_0$ and $y',y''\in Y$ with $y'|_{S_0}=y''|_{S_0}$.
Suppose the contrary, that is for every at most countable set $S\subseteq T$ and open in $X$ nonempty set $U'\subseteq U$ there exist $x\in U'$ and $y',y''\in Y$ with $y'|_{S}=y''|_{S}$ such that $|f(x,y')-f(x,y'')|> \varepsilon$. Let $\omega$ be the first ordinal with the cardinality $|T|$, $T=\{t_{\alpha}:\alpha<\omega\}$ and $(\varepsilon_n)^{\infty}_{n=0}$ be a strictly increasing sequence of reals $\varepsilon_n>0$ which tends to $\varepsilon$.
For every $\alpha<\omega$ we denote by $A^{(1)}_{\alpha}$ the set of all $x\in \overline{U}$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_0$ for every $y',y''\in Y$ with $y'(t_{\xi})=y''(t_{\xi})$ for $\xi<\alpha$ and by $B^{(1)}_{\alpha}$ we denote the set of all $x\in\overline{U}$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_1$ for every $y',y''\in Y$ with $y'(t_{\xi}=y''(t_{\xi})$ for $\xi\leq\alpha$. It follows from the continuity of $f$ with respect to $x$ that all sets $A_{\alpha}^{(1)}$ and $B^{(1)}_{\alpha}$ are closed, besides $A^{(1)}_{\alpha}\subseteq B^{(1)}_{\alpha}$ for every $\alpha<\omega$ and the sequences $(A^{(1)}_{\alpha}:\alpha<\omega)$ and $(B^{(1)}_{\alpha}:\alpha<\omega)$ are increasing. Moreover, it follows from Proposition 2.1 that for every limited ordinal $\alpha<\omega$ and every $x\in A^{(1)}_{\alpha}$ there exists an ordinal $\xi<\alpha$ such that $x\in B^{(1)}_{\xi}$, that is $A^{(1)}_{\alpha}\subseteq \bigcup\limits_{\xi<\alpha}B^{(1)}_{\xi}$. Thus, the sequence $(B^{(1)}_{\alpha}:\alpha<\omega)$ occupies the sequence $(A^{(1)}_{\alpha}:\alpha<\omega)$. It follows from Proposition 2.1 that $U\subseteq\bigcup\limits_{\alpha<\omega}A^{(1)}_{\alpha}$.
Since $X$ is strongly Baire, $\bigcup\limits_{\alpha<\omega}{\rm int}(B^{(1)}_{\alpha})\ne\O$. Therefore there exist an ordinal $\beta_1<\omega$ and an open in $X$ nonempty set $U_1\subseteq U$ such that $\overline{U_1}\subseteq B^{(1)}_{\beta}$. Note that $\beta_1$ is a uncountable ordinal. Really, otherwise the at most countable set $S=\{t_{\alpha}:\alpha\leq\beta_1\}$ and the open set $U_1$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_1\leq\varepsilon$ for every $x\in U_1$ and $y',y''\in Y$ with $y'|_S=y''|_S$, a contradiction.
Put $S_1=\{t_{\alpha}:\alpha\leq\beta_1\}$. Let $\gamma_1$ be the first ordinal with the cardinality $|S_1|$ and $S_1=\{s^{(1)}_{\alpha}:\alpha<\gamma_1\}$. Clearly that $\gamma_1\leq\beta_1$. Note that $|f(x,y')-f(x,y'')|\leq \varepsilon_1$ for every $x\in\overline{U}_1$ and $y',y''\in Y$ with $y'|_{S_1}=y''|_{S_1}$. For every $\alpha<\gamma_1$ we denote by $A^{(2)}_{\alpha}$ the set of all $x\in \overline{U}_1$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_2$ for every $y',y''\in Y$ with $y'(s^{(1)}_{\xi})=y''(s^{(1)}_{\xi})$ for $\xi<\alpha$ and by $B^{(2)}_{\alpha}$ we denote the set of all $x\in\overline{U}_1$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_3$ for every $y',y''\in Y$ with $y'(s^{(1)}_{\xi})=y''(s^{(1)}_{\xi})$ for $\xi\leq\alpha$.
All sets $A^{(2)}_{\alpha}$ and $B^{(2)}_{\alpha}$ for $\alpha<\gamma_1$ are closed. It follows from 2.1 that the sequence $(B^{(2)}_{\alpha}:\alpha<\gamma_1)$ occupies the sequence $(A^{(2)}_{\alpha}:\alpha<\gamma_1)$ and $U_1\subseteq\bigcup\limits_{\alpha<\gamma_1}A^{(2)}_{\alpha}$. Since $X$ is strongly Baire, there exist an ordinal $\beta_2<\gamma_1$ and an open in $X$ nonempty set $U_2\subseteq U_1$ such that $\overline{U}_2\subseteq B^{(2)}_{\beta_2}$. We can to show analogously as for the ordinal $\beta_1$ that $\beta_2$ is a uncountable ordinal.
Put $S_2=\{s^{(1)}_{\alpha}:\alpha<\beta_2\}$. Let $\gamma_2$ be the first ordinal of the cardinality $|S_2|$ and $S_2=\{s^{(2)}_{\alpha}:\alpha<\gamma_2\}$. Clearly that $\gamma_2\leq\beta_2$. For every $\alpha<\gamma_2$ we denote by $A^{(3)}_{\alpha}$ the set of all $x\in\overline{U}_2$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_4$ for every $y',y''\in Y$ with $y'(s^{(2)}_{\xi})=y''(s^{(2)}_{\xi})$ for $\xi<\alpha$ and by $B^{(3)}_{\alpha}$ we denote the set of all $x\in\overline{U}_2$ such that $|f(x,y')-f(x,y'')|\leq \varepsilon_5$ for every $y',y''\in Y$ with $y'(s^{(2)}_{\xi})=y''(s^{(2)}_{\xi})$ for $\xi\leq\alpha$. It follows from Proposition 2.1 that the sequence $(B^{(3)}_{\alpha}:\alpha<\gamma_2)$ occupies the sequence $(A^{(3)}_{\alpha}:\alpha<\gamma_2)$ and $U_2\subseteq\bigcup\limits_{\alpha<\gamma_2}A^{(3)}_{\alpha}$. Therefore there exist an at most countable ordinal $\beta_3<\gamma_2$ and an open in $X$ nonempty set $U_3\subseteq U_2$ such that $\overline{U}_3\subseteq B^{(3)}_{\beta_3}$.
Continuing this process to infinity we obtain a sequence of ordinals
$$\beta_1\geq\gamma_1>\beta_2
\geq\gamma_2>\beta_3\geq\gamma_3>\dots,
$$
a contradiction.
\end{proof}
\begin{theorem}\label{th:2.4} Every $\beta-\sigma$-unfavorable space is a strongly Baire space.\end{theorem}
\begin{proof} Let a topological space $X$ is not strongly Baire. That is there exist an ordinal $\omega$ and increasing nets $(A_{\alpha}:\alpha<\omega)$ and
$(B_{\alpha}:\alpha<\omega)$ of closed in $X$ sets $A_{\alpha}$ and $B_{\alpha}$ such that the net $(B_{\alpha}:\alpha<\omega)$ occupies the net $(A_{\alpha}:\alpha<\omega)$ and $$U_0={\rm int}(\bigcup\limits_{\alpha<\omega}A_{\alpha})\setminus \overline{\bigcup\limits_{\alpha<\omega}{\rm int}(B_{\alpha})}\ne \O.$$
Clearly that $\omega$ is a limited ordinal. According to [10, Theorem 10, p.282] the ordinal $\omega$ is confinal to the first ordinal $\omega'$ with the same confinality. If $\omega'=\omega_0$, then $X$ is not Baire. That is the space $X$ is a $\beta$-unfavorable space, in particular, $X$ is $\beta-\sigma$-unfavorable.
We consider the case of $\omega'>\omega_0$. We describe an winner strategy $\tau$ for the player $\beta$ in $\sigma$-game. Let $\alpha_0<\omega$, $U_0$ is the first move of $\beta$, $V_1\subseteq U_0$ be an open in $X$ nonempty set and $x_1\in X$. If $x_1\in U_0$, then $x_1\in \bigcup\limits_{\alpha<\omega}A_{\alpha}$. Therefore there exists an ordinal $\alpha_1<\omega$ such that $x_1\in A_{\alpha_1}$. If $x_1\not \in U_0$, then we put $\alpha_1=1$. The closed set $B_{\alpha_1}$ is nowhere dense in $U_0$. Therefore the open set $U_1=\tau(U_0,V_1,x_1)=V_1\setminus B_{\alpha_1}$ is nonempty.
Let $V_2\subseteq U_1$ be an open in $X$ nonempty set and $x_2\in X$. If $x_2\in U_0$, then we choose an ordinal $\alpha_2$ such that $\alpha_1<\alpha_2<\omega$ and $x_2\in A_{\alpha_2}$. If $x_2\not \in U_0$, then we put $\alpha_2=\alpha_1+1$. Now we put $U_2=\tau(U_0,V_1,x_1,U_1,V_2,x_2)= V_2\setminus B_{\alpha_2}$.
Continuing this process to infinity we obtain an strictly increasing sequence $(\alpha_n)^{\infty}_{n=1}$ of ordinals $\alpha_n$, an sequence $(x_n)^{\infty}_{n=1}$ of $x_n\in X$ and decreasing sequences $(U_n)^{\infty}_{n=0}$ and $(V_n)^{\infty}_{n=1}$ of open in $X$ nonempty sets $U_n$ and $V_n$ respectively such that $V_n\subseteq U_{n-1}$, $U_n=V_{n-1}\setminus B_{\alpha_n}$, where $\alpha_n=\alpha_{n-1}+1$, if $x_n\not \in U_0$, and $\alpha_n>\alpha_{n-1}$ such that $x_n\in A_{\alpha_n}$, if $x_n\in U_0$.
We show that $(\bigcap\limits^{\infty}_{n=0}U_n)\bigcap\overline{\{x_n:n\in\mathbb N\}}=\O$. Put $\gamma={\rm sup}\,\alpha_n$. Since $\omega'>\omega_0$, $\gamma<\omega$. We consider the sets $A=\{x_n:n\in\mathbb N, x_n\not\in U_0\}$ and $B=\{x_n:n\in\mathbb N, x_n\in U_0\}$. Note that $A\cap U_0=\O$. Therefore
$\overline{A}\cap(\bigcap\limits^{\infty}_{n=0}U_n)=\O$. Since the sequence $(A_{\alpha}:\alpha<\omega)$ is increasing, according to the choice of ordinals $\alpha_n$ we have $B\subseteq A_{\gamma}$. Recall that the sequence $(B_{\alpha}:\alpha<\omega)$ occupies the sequence $(A_{\alpha}:\alpha<\omega)$. Therefore for limited ordinal $\gamma$ we have
$$
\overline{B}\subseteq A_{\gamma}\subseteq\bigcup\limits_{\alpha<\gamma}B_{\alpha}=
\bigcup\limits_{n=1}^{\infty}B_{\alpha_n}.
$$
On other hand, according to the construction, we have $U_n\cap B_{\alpha_n}=\O$. Therefore $(\bigcap\limits_{n=0}^{\infty}U_n)\cap(\bigcup\limits^{\infty}_{n=1}B_{\alpha_n})=\O$.
Hence, $\overline{B}\cap(\bigcap\limits_{n=0}^{\infty}U_n)=\O$.
Thus, $(\bigcap\limits^{\infty}_{n=0}U_n)\cap\overline{\{x_n:n\in\mathbb
N\}}=\O$. Hence, the strategy $\tau$ is a winner strategy for $\beta$ in $\sigma$-game and $X$ is a $\beta-\sigma$-favorable space.
\end{proof}
\section{Properties of strongly Baire spaces}
In this section we investigate properties of strongly Baire spaces which related with the calibers of these spaces. The proof of the following statement is obvious.
\begin{proposition}\label{p:3.1} Íåõàé ${\rm int}(\bigcup\limits_{\xi<\alpha} A_{\xi}) \subseteq \overline{\bigcup\limits_{\xi<\alpha} {\rm int}(A_{\xi})}$ for every increasing sequence $(A_{\xi}:\xi<\alpha)$ of closed in topological space $X$ sets $A_{\xi}$. Then $X$ is a strongly Baire space.
\end{proposition}
Recall (see [11 p.16]) that a cardinal $\aleph$ is called {\it a caliber of the topological space $X$}, if for every family $(U_{\alpha}:\alpha\in A)$ of nonempty open in $X$ sets $U_{\alpha}$ ç $|A|=\aleph$ there exists a set $B\subseteq A$ such that $|B|=\aleph$ and $\bigcap\limits_{\alpha\in B}U_{\alpha}\ne\O$.
\begin{proposition}\label{p:3.2} Let a Baire space $X$ such that every regular uncountable cardinal is a caliber of $X$. Then ${\rm int}(\bigcup\limits_{\xi<\alpha} A_{\xi}) \subseteq \overline{\bigcup\limits_{\xi<\alpha} {\rm int}(A_{\xi})}$ for every increasing sequence $(A_{\xi}:\xi<\alpha)$ of closed in $X$ sets $A_{\xi}$, in particular, $X$ is strongly Baire.
\end{proposition}
\begin{proof} Let $(A_{\xi}:\xi<\alpha)$ be a increasing sequence of closed in $X$ sets $A_{\xi}$. If $\alpha$ is not limited ordinal, that is $\alpha=\beta+1$, then
$$
{\rm int}(\bigcup\limits_{\xi<\alpha} A_{\xi}) = {\rm int}(A_{\beta}) \subseteq \overline{\bigcup\limits_{\xi<\alpha} {\rm int}(A_{\xi})}.
$$
Let $\alpha$ is a limited ordinal. According to [10, Theorem 10, p.282], the cardinal $\alpha$ is confinal to the first ordinal $\omega$ with the same confinality. This implies that the cardinal $\aleph=|\omega|$ is regular.
We choose an strictly increasing sequence $(\xi_{\gamma}:\gamma<\omega$ of ordinals $\xi_{\gamma}$ such that $\sup\limits_{\gamma<\omega}\,\xi_{\gamma}=\alpha$. For every $\gamma<\omega$ we put $F_{\gamma}=A_{\xi_{\gamma}}$. Since the sequence $(A_{\xi}:\xi<\alpha)$ is increasing, the sequence $(F_{\gamma}:\gamma<\omega)$ is increasing too, $\bigcup\limits_{\xi<\alpha}A_{\xi}=\bigcup\limits_{\gamma<\omega}F_{\gamma}$ and $\bigcup\limits_{\xi<\alpha}{\rm int}(A_{\xi}) = \bigcup\limits_{\gamma<\omega}{\rm int}(F_{\gamma})$.
If $\omega=\omega_0$, then it follows from the fact that $X$ is Baire that
$${\rm int}(\bigcup\limits_{\gamma<\omega_0}F_{\gamma})\subseteq
\overline{\bigcup\limits_{\gamma<\omega_0}{\rm int}(F_{\gamma})},$$
that is ${\rm int}(\bigcup\limits_{\xi<\alpha}A_{\xi})\subseteq \overline{\bigcup\limits_{\xi<\alpha}{\rm int}(A_{\xi})}$.
Let $\omega$ is an uncountable ordinal. Suppose that $U=\bigcup\limits_{\gamma<\omega}{\rm int}(F_{\gamma})\not\subseteq \overline{\bigcup\limits_{\gamma<\omega}{\rm int}(F_{\gamma})}$. Then $U\not\subseteq F_{\gamma}$, that is $U_{\gamma}=U\setminus F_{\gamma}\ne \O$ for every $\gamma<\omega$. Since $\aleph$ is a caliber of $X$, for the family $(U_{\gamma}:\gamma<\omega)$ there exists $x_0\in X$ such that the set $\Gamma=\{\gamma<\omega: x_0\in U_{\gamma}\}$ has the cardinality $\aleph$. Since $\omega$ is the first ordinal of the cardinality $\aleph$, $\sup\Gamma=\omega$. Note that $(U_{\gamma}:\gamma<\omega)$ is decreasing sequence, therefore $\bigcap\limits_{\gamma<\omega}U_{\gamma}=\bigcap\limits_{\gamma\in\Gamma}U_{\gamma}\ni x_0$. Thus, $\bigcap\limits_{\gamma<\omega}U_{\gamma}\ne\O$, that is $U\not\subseteq\bigcup\limits_{\gamma<\omega}F_{\gamma}$, in particular, $x_0\in U$ and $x_0\not\in\bigcup\limits_{\gamma<\omega}F_{\gamma}$. But this contradicts to $U={\rm int}(\bigcup\limits_{\gamma<\omega}F_{\gamma})$.
\end{proof}
It easy to see that for every separable space $X$ every infinite regular cardinal is a caliber of $X$. Moreover, according to [11, Theorem 0.3.13, p. 16], the product of multipliers with caliber $\aleph$ has the caliber $\aleph$ too. Hence, the following result is true.
\begin{proposition} \label{p:3.3} Let a topological product $X=\prod\limits_{t\in T}X_t$ is a Baire space, besides all spaces $X_t$ are separable. Then $X$ is strongly Baire.
\end{proposition}
Further, we shall use the following auxiliary statement (see [12, p.185]), which often be used for the investigation of product properties.
\begin{lemma}[Shanin]\label{l:3.4} Let $\aleph$ be an uncountable regular cardinal, $(T_{\gamma}:\gamma\in \Gamma)$ be a family of finite sets $T_{\gamma}$, moreover $|\Gamma|=\aleph$. Then there exist a set $\Delta\subseteq\Gamma$ and a finite set $S\subseteq \bigcup\limits_{\gamma\in\Gamma}T_{\gamma}$ such that $|\Delta|=\aleph$ and $T_{\beta}\cap T_{\gamma}=S$ for every distinct $\beta, \gamma \in \Delta$.
\end{lemma}
\begin{proposition}\label{p:3.5} Let $(X_t:t\in T)$ be a family of separable metric spaces $(X_t,|\cdot-\cdot|_t)$, $X\subseteq \prod\limits_{t\in T}X_{t}$ be a Baire space, which is dense in the space $Y= \prod\limits_{t\in T}X_{t}$ with the topology of uniform convergence on $T$. Then $X$ is a strongly Baire space.
\end{proposition}
\begin{proof} Let $\aleph$ be an uncountable regular cardinal, $(V_{\gamma}:\gamma\in \Gamma)$ be a family of nonempty open in $X$ basic sets $V_{\gamma}$, besides $|\Gamma|=\aleph$ and $(U_{\gamma}:\gamma\in \Gamma)$ be a family of nonempty open in $\prod\limits_{t\in T}X_{t}$ basic sets $U_{\gamma}=\prod\limits_{t\in T}U_{\gamma}^{(t)}$ such that $V_{\gamma}=U_{\gamma}\bigcap\prod\limits_{t\in T}X_{t}$ for every $\gamma\in \Gamma$. Put $T_{\gamma}=\{t\in T:U_{\gamma}^{(t)}\ne X_t\}$ for every $\gamma\in \Gamma$. Taking into account the regularity of $\aleph$ and lemma 3.4 we can propose that all finite sets $T_{\gamma}$ have the same cardinality and $T_{\gamma}\cap T_{\beta}= S$ for every distinct $\gamma, \beta \in \Gamma$ and some finite set $S\subseteq T$. For every $\gamma\in \Gamma$ we choose points $x_{\gamma}^{(t)}\in X_t$ for $t\in T_{\gamma}$ and real $\delta_{\gamma}> 0$ such that $\{x_t\in X_t: |x_t-x_{\gamma}^{(t)}|_t<\delta_{\gamma}\}\subseteq U_{\gamma}^{(t)}$. Since $\aleph$ is an uncountable regular cardinal, there exist $n\in\mathbb N$ and $\Gamma'\subseteq \Gamma$ such that $|\Gamma'|=\aleph$ and $\delta_{\gamma}\geq \frac{2}{n}$ for every $\gamma\in\Gamma'$. In separable metric space $\tilde{X}= \prod\limits_{s\in S}X_{s}$ we find a point $\tilde x\in \tilde X$ such that $|\tilde x(s)-x_{\gamma}^{(s)}|_s<\frac{1}{n}$ for every $s\in S$ and $\gamma\in \Gamma''$, where $\Gamma'' \subseteq \Gamma'$ is a set with $|\Gamma''|=\aleph$.
We put $T'=\bigcup\limits_{\gamma \in \Gamma''}S_{\gamma}$, where $S_{\gamma}=T_{\gamma}\setminus S$. Note that $T'=\O$ or $|T'|=\aleph$. If $T'=\O$, that is $T_{\gamma}=S$ for every $\gamma\in \Gamma''$, then we choose a point $y_0\in Y$ such that $y_0(s)=\tilde{x}(s)$ for every $s\in S$. Since the set $X$ is dense in $Y$, there exists a point $x_0\in X$ such that $|x_0(t)-y_0(t)|_t<\frac{1}{n}$ for every $t\in T$. Then for every $\gamma\in \Gamma''$ and $t\in T_{\gamma}$ we have
$$
|x_0(t)-x_{\gamma}^{(t)}|_t\leq |x_0(t)-y_0(t)|_t +
|y_0(t)-x_{\gamma}^{(t)}|_t =
$$
$$
=|x_0(t)-y_0(t)|_t + |\tilde x(t)-x_{\gamma}^{(t)}|_t< \frac{1}{n}
+ \frac{1}{n} \leq \delta_{\gamma}.
$$
Thus, $x_0(t)\in U_{\gamma}^{(t)}$ for every $\gamma\in \Gamma''$ and $t\in T_{\gamma}$, that is $x_0\in V_{\gamma}$ for every $\gamma \in \Gamma''$.
Let $|T'|=\aleph$. We choose a point $y_0\in Y$ such that $y_0(s)=\tilde{x}(s)$ for every $s\in S$ and $y_0(s)=x_{\gamma}^{(s)}$ for every $\gamma \in \Gamma''$ and $s\in S_{\gamma}$. Then, analogously as in the previous case, for some point $x_0\in X$ such that $|x_0(t)-y_0(t)|_t<\frac{1}{n}$ for every $t\in T$, we have $x_0\in V_{\gamma}$ for every $\gamma \in \Gamma''$.
Thus, every uncountable cardinal $\aleph$ is the caliber of the Baire space $X$. According to Proposition 3.2, $X$ is a strongly Baire space.
\end{proof}
\begin{corollary}\label{c:3.6} Let $X\subseteq [0,1]^T$ be a Baire space which is dense in the space $Y=[0,1]^T$ with the topology of the uniform convergence on $T$. Then $X$ is a strongly Baire space.
\end{corollary}
\begin{proposition} \label{p:3.7} Let $X$ be a strongly Baire space, $Y$ be a topological space, $f:X\to Y$ be a continuous surjective mapping such that for every nowhere dense in $Y$ set $B$ the set $f^{-1}(B)$ is nowhere dense in $X$. Then $Y$ is a strongly Baire space.
\end{proposition}
\begin{proof} Let $\alpha$ be a limited ordinal, $(A_{\xi}:\xi<\alpha)$ and $(B_{\xi}:\xi<\alpha)$ be increasing sequences of closed in $Y$ sets $A_{\xi}$ and $B_{\xi}$ such that $(B_{\xi}:\xi<\alpha)$ occupiers $(A_{\xi}:\xi<\alpha)$.
Show that
$$
{\rm int}(\bigcup\limits_{\xi<\alpha} A_{\xi}) \subseteq \overline{\bigcup\limits_{\xi<\alpha} {\rm int}(B_{\xi})}.
$$
We take a point $y_0\in {\rm int}(\bigcup\limits_{\xi<\alpha} A_{\xi})$ and a closed in $Y$ neighborhood $V$ of $y_0$. For every $\xi<\alpha$ we put
$\tilde{A}_{\xi}=f^{-1}(A_{\xi}\cap V)$, $\tilde{B}_{\xi}=f^{-1}(B_{\xi}\cap V)$. Clearly $\tilde{A}_{\xi}$ and $\tilde{B}_{\xi}$ are closed in $X$, moreover $\tilde{A}_{\xi}\subseteq \tilde{B}_{\xi}$ for every $\xi<\alpha$. Besides, for every limited ordinal $\beta<\alpha$ we have
$$
\tilde{A}_{\beta}=f^{-1}(A_{\beta}\cap V)\subseteq f^{-1}(\bigcup\limits_{\xi<\beta}(B_{\xi}\cap V))=\bigcup\limits_{\xi<\beta}f^{-1}(B_{\xi}\cap
V)=\bigcup\limits_{\xi<\beta}\tilde{B}_{\xi}.
$$
Thus, the sequence $(\tilde{B}_{\xi}:\xi<\alpha)$ occupies the sequence $(\tilde{A}_{\xi}:\xi<\alpha)$. Since a mapping $f$ is continuous and the set
$\bigcup\limits_{\xi<\alpha}(A_{\xi}\cap V)$ is a neighborhood of $y_0$, the set $f^{-1}(\bigcup\limits_{\xi<\alpha}(A_{\xi}\cap V))=\bigcup\limits_{\xi<\alpha}\tilde{A}_{\xi}$ is a neighborhood of every point $x\in f^{-1}(y_0)$, in particular, ${\rm int}(\bigcup\limits_{\xi<\alpha}\tilde{A}_{\xi})\ne\O$. Taking into account that $X$ is a strongly Baire space, we obtain that $\bigcup\limits_{\xi<\alpha}{\rm int}(\tilde{B}_{\xi})\ne\O$, that is there exists $\gamma<\alpha$ such that ${\rm int}(\tilde{B}_{\gamma})\ne\O$. The set $\tilde{B}_{\gamma}$ is not nowhere dense in $X$, therefore the Proposition conditions imply that the set $B_{\gamma}\cap V$ is nowhere dense in $Y$. Since $B_{\gamma}\cap V$ is closed, ${\rm int}(B_{\gamma}\cap V)\ne\O$. Thus, ${\rm int}(B_{\gamma})\cap V\ne \O$ and $y_0\in \overline{\bigcup\limits_{\xi<\alpha}{\rm int}(B_{\xi})}$.
\end{proof}
\section{Metrizable compacts in space of continuous functions}
In this section we investigate sufficient conditions on a topological space $X$ for the metrizability of every compacts $Y\subseteq C_p(X)$.
Recall that {\it a topological space $X$ has countable chain condition} if every disjoint system of open sets is at most countable.
\begin{theorem}\label{th:4.1} Let $X$ be a strongly Baire space with the countable chain condition, in particular, a Baire space for which every regular uncountable cardinal is the caliber. Then every compact $Y\subseteq C_p(X)$ is metrizable.\end{theorem}
\begin{proof} Firstly, we note that for a completely regular space $X$ this theorem follows from Theorem 2.3 and [13, Theorem 2 and Proposition 5].
In general case for a Namioka space $X$ with the countable chain condition and a compact $Y\subseteq C_p(X)$, using Theorem 2.2 it easy to construct an at most countable set $A\subseteq X$ such that $y'|_A\ne y''|_A$ for every distinct $y',y''\in Y$. This implies the metrizability of $Y$.
\end{proof}
On other hand, the following result is true. It is an analog of Theorem 3.1 from [14] where a similar relation between the weight of a compact space $X$ and the caliber of $C_p(X)$ is obtained.
\begin{theorem}\label{th:4.2} Let $X$ be a topological space, $Y\subseteq C_p(X)$ be a nonmetrizable compact and $\aleph={\rm cof}(w(Y))>\aleph_0$. Then $\aleph$ is not caliber of $X$.
\end{theorem}
\begin{proof} Let $T$ be a set with $|T|=w(Y)$, $Z\subseteq\mathbb R^T$ be a compact such that there exists a homeomorphism $\varphi: Z\to Y$.
We consider a separately continuous function $f:X\times Z\to\mathbb R$, $f(x,z)=\varphi(z)(x)$. Let $\omega$ is the first ordinal with the cardinality $\aleph$ and $(T_{\alpha}:\alpha<\omega)$ is an increasing sequence of sets $T_{\alpha}\subseteq T$ such that $\bigcup\limits_{\alpha<\omega}T_{\alpha}=T$ and $|T_{\alpha}|<|T|$ for every $\alpha<\omega$. For each $\alpha<\omega$ we put
$$A_{\alpha}=\{x\in X: f(x,z')=f(x,z'')\mbox{\,\,for\,\,every\,\,}z', z''\in Z
\mbox{\,\,with\,\,}z'|_{T_{\alpha}}=z''|_{T_{\alpha}}\}.$$
It follows from the continuity of $f$ with respect to $x$ that all sets $A_{\alpha}$ are closed in $X$. Since $f$ is continuous with respect to the second variable, for every $x\in X$ there exists an at most countable set $S\subseteq T$ such that $f(x,z')=f(x,z'')$ for every $z',z''\in Z$ with $z'|_S=z''|_S$. Therefore $X=\bigcup\limits_{\alpha<\omega}A_{\alpha}$, that is $\bigcap\limits_{\alpha<\omega}U_{\alpha}=\O$, where $U_{\alpha}=X\setminus A_{\alpha}$. Taking into account that the sequence $(U_{\alpha}:\alpha< \omega)$ is decreasing and $\omega$ is the first ordinal with cardinality $\aleph$, we obtain that
$\bigcap\limits_{\alpha\in I}U_{\alpha}=\O$ for every set of ordinals $I\subseteq[0,\omega)$ with $|I|=\aleph$. Thus, $\aleph$ is not caliber of $X$.
\end{proof}
Thus, the following question naturally arises. It is an analog of Question 3.6 from [14].
\begin{question}\label{q:4.3} Let $X$ be a topological space for which every uncountable cardinal is caliber. Is every compact $Y\subseteq C_p(X)$ metrizable?\end{question}
Note that analogously as in [14, Theorem 3.7] it can proved using Theorem 4.2 that this question has a positive answer in the case $\aleph_2=2^{\aleph_1}$.
\begin{proposition}\label{p:4.4} Let $Y$ be a compact space such that for every set $B\subseteq Y$ with $|B|\leq \aleph_1$ the compact set $\overline{B}$ is metrizable. Then $Y$ is metrizable.\end{proposition}
\begin{proof} It is enough to prove that $Y$ is separable. Suppose that $Y$ is not separable. Then it easy to construct a sequence $(y_{\xi}: \xi<\aleph_1)$ of points $y_{\xi}\in Y$ such that $y_{\alpha}\not \in \overline{\{y_{\xi}:\xi<\alpha\}}$ for every $\alpha<\aleph_1$. Since the space $Z=\overline{\{y_{\xi}:\xi<\aleph_1\}}$ is metrizable, there exists an at most countable subset of the set $\{y_{\xi}:\xi<\aleph_1\}$ which is dense in $Z$. But this contradicts to the choice of $y_{\xi}$.\end{proof}
\begin{theorem}\label{th:4.5} Let $\aleph_i$ be the first uncountable cardinal with ${\rm cof}(\aleph_i)=\aleph_0$. Suppose that $\aleph_i>2^{\aleph_1}$. Then for every topological space $X$ for which every regular cardinal $\aleph\in [\aleph_1, 2^{\aleph_1}]$ is its caliber, every compact $Y\subseteq C_p(X)$ is metrizable.\end{theorem}
\begin{proof} For every compact space $Y\subseteq C_p(X)$ with $d(Y)\leq\aleph_1$ we have $w(Y)\leq 2^{\aleph_1}$. It follows from Theorem 4.2 and the condition $\aleph_i>2^{\aleph_1}$ that $w(Y)=\aleph_0$, that is $Y$ is metrizable. It remains yo use Proposition 4.4.
\end{proof}
The following result can be proved analogously.
\begin{theorem} \label{th:4.6} Let $\aleph_i$ be the first uncountable cardinal with ${\rm cof}(\aleph_i)=\aleph_0$ and $X$ be a topological space such that $d(X)<\aleph_i$ and every regular cardinal $\aleph$ is a caliber of $X$. Then every compact $Y\subseteq C_p(X)$ is metrizable.
\end{theorem}
\begin{proof} Note that for every compact space $Y\subseteq C_p(X)$ and every dense in $X$ set $A$ the mapping $\varphi: Y\to C_p(A)$, $\varphi(y)=y|_A$, is a homeomorphic embedding. Therefore $w(Y)\leq d(X)<\aleph_i$ and according to Theorem 4.2, $Y$ is metrizable.
\end{proof}
\bibliographystyle{amsplain} | 35,286 |
\begin{document}
\title[Symmetries of differential equations]
{\Large Modular forms, Schwarzian conditions, and symmetries of differential equations in physics}
\vskip .3cm
\author{Y. Abdelaziz, J.-M. Maillard$^\pounds$}
\address{$^\pounds$ LPTMC, UMR 7600 CNRS,
Universit\'e de Paris 6, Tour 24,
4\`eme \'etage, case 121,
4 Place Jussieu, 75252 Paris Cedex 05, France}
\ead{[email protected]}
\begin{abstract}
We give examples of infinite order rational
transformations that leave linear differential
equations covariant. These examples are
non-trivial yet simple enough illustrations
of exact representations of the renormalization group.
We first illustrate covariance properties on order-two
linear differential operators associated with identities
relating the same $\, _2F_1$ hypergeometric function with different
rational pullbacks. These rational transformations
are solutions of a differentially algebraic
equation that already emerged in a paper by Casale on the
Galoisian envelopes. We provide two new and more general results of
the previous covariance by rational functions: a new Heun
function example and a higher genus $\, _2F_1$ hypergeometric
function example. We then focus on identities relating
the same $\, _2F_1$ hypergeometric function with two different
algebraic pullback transformations: such remarkable identities
correspond to modular forms, the algebraic transformations
being solution of another differentially algebraic
Schwarzian equation that also emerged in Casale's paper. Further,
we show that the first differentially algebraic
equation can be seen as a subcase of the last Schwarzian differential
condition, the restriction corresponding to a factorization condition
of some associated order-two linear differential operator.
Finally, we also explore generalizations of these results, for instance,
to $\, _3F_2$, hypergeometric functions, and show that
one just reduces to the previous
$\, _2F_1$ cases through a Clausen identity. The question of the reduction
of these Schwarzian conditions to
modular correspondences remains an open question.
In a $ \, _2F_1$ hypergeometric framework the Schwarzian condition
encapsulates all the modular forms
and modular equations of the
theory of elliptic curves, but these two conditions are actually richer than elliptic
curves or $\, _2F_1$ hypergeometric functions, as can be seen
on the Heun and higher genus example.
This work is a strong incentive to develop more differentially algebraic
symmetry analysis in physics.
\end{abstract}
\vskip .3cm
\noindent {\bf PACS}: 05.50.+q, 05.90.+m, 05.10.-a, 02.30.Hq, 02.30.lk,02.30.Gp, 02.40.Xx
\noindent {\bf AMS Classification scheme numbers}: 34M55, 47Exx, 32Hxx, 32Nxx, 34Lxx, 34Mxx, 14Kxx, 14H52
\vskip .3cm
{\bf Key-words}: Square Ising model, Schwarzian derivative,
infinite order rational symmetries of ODEs,
Fuchsian linear differential equations, Gauss and generalized hypergeometric functions,
Heun function,
globally nilpotent linear differential operators, isogenies of elliptic curves, Hauptmoduls,
elliptic functions, modular forms, modular equations,
modular correspondences,
mirror maps, renormalization group, Malgrange pseudo-group,
Galoisian envelope, Latt\`es transformations.
\vskip .1cm
\section{Introduction: infinite order symmetries.}
\label{int}
In its simplest form, the concept of symmetries in physics
corresponds to a (univariate) transformation
$\, x \, \rightarrow \, \, R(x)$ preserving some structures. Whether
these structures are linear differential equations, or more complicated
mathematical objects (systems of differential equations, functional
equations, etc ...), they must be {\em invariant} or {\em covariant} under
the previous transformations $\, x \, \rightarrow \, \, R(x)$. Of course,
these transformation symmetries can be studied, per se,
in a discrete dynamical perspective\footnote[1]{In their
pioneering work Julia, Fatou and Ritt the theory of iteration
of rational functions was seen as a method for investigating
functional equations~\cite{Fatou,Ritt,Fatou2}. More generally,
one can try to find all pairs of {\em commuting rational functions},
see~\cite{Eremenko}.}. Along this iteration line, or more generally,
{\em commuting transformations} line, there is no need
to underline the success of the renormalization group revisited
by Wilson~\cite{Migdal,Fisher} seen as a fundamental
symmetry in lattice statistical mechanics or field theory.
\vskip .2cm
The renormalization of the one-dimensional Ising model without
a magnetic field (even if it can also be performed with
a magnetic field~\cite{Hindawi}), which corresponds to the simple
(commuting) transformations $\, x \, \rightarrow \, x^n$
(where $\, x\, = \,\, \tanh(K)$), is usually seen as the heuristic
``student'' example of {\em exact} renormalization in physics, but it is
trivial being one-dimensional. For less academical models one
could think that no exact\footnote[2]{For instance, a Migdal-Kadanoff
decimation can introduce, in a finite-dimensional parameter space
of the model, rational transformations that can be seen as efficient
approximations of the generators of the renormalization group, hoping
that the basin of attraction of the fixed points of the
transformation is ``large enough''.} closed form representation of the
renormalization group exists, but can one hope to find anything better ?
For Yang-Baxter integrable models~\cite{broglie,bo-ha-ma-ze-07b}
with a canonical genus-one
parametrization~\cite{Automorphisms,Baxterization,BeMaVi92} (elliptic
functions of modulus $\, k$) {\em exact} representations
of the generators of the renormalization group
happen to exist. Such exact symmetry transformations must have
$\, k \, = \, 0$ and $\, k \, = \, 1$ as a fixed point, be
compatible with the Kramers-Wannier duality
$\, k \, \leftrightarrow \, 1/k$, and, most importantly,
be compatible with the {\em lattice of periods} of the elliptic
functions parametrizing the model. Thus, these exact generators
must be the {\em isogenies}~\cite{Heegner,buium} of the elliptic functions
(of modulus $\, k$).
The simplest example of a transformation carrying these properties
is the {\em Landen transformation}~\cite{bo-ha-ma-ze-07b,Heegner}
\begin{eqnarray}
\label{Landen}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
k \, \quad \longrightarrow \, \quad k_L \, = \, \,
{{2 \sqrt{k}} \over {1+k}},
\end{eqnarray}
with the {\em critical point} of the
square Ising model (resp. Baxter model) given by the fixed
point of the transformation: $\, k\,= \, 1$.
This algebraic transformation corresponds
to multiplying
({\em or dividing} because of the
modular group symmetry $\tau \,\leftrightarrow \, 1/\tau$)
the ratio $\tau$ of the two periods of the elliptic curves
$ \, \,\tau \, \, \longleftrightarrow \, \, 2\, \tau$.
The other (isogeny) transformations\footnote[9]{See
for instance (2.18) in~\cite{Canada}.}
correspond to
$\tau \,\leftrightarrow \, N \cdot \tau$,
for various integers $\, N$.
Setting out to find the precise covariance of some of the physical quantities
related to the 2-D Ising model, like the partition function
per site, the correlation functions, the $\, n$-fold correlations
$\, \chi^{(n)}$ associated with the full
susceptibility~\cite{High,ze-bo-ha-ma-05b,bo-gu-ha-je-ma-ni-ze-08,higher3},
with respect to transformations of the Landen type (\ref{Landen}), is a
difficult task. An easier goal would be to find a covariance, not
on the selected\footnote[5]{They are not only Fuchsian, the
corresponding linear differential operators are globally nilpotent
or $\, G$-operators~\cite{bo-bo-ha-ma-we-ze-09,Andre,Andre2}.} linear
differential operators that these quantities satisfy,
but on the different {\em factors} of these operators. Luckily the factors
of the operators associated with these physical quantities are linear
differential operators whose solutions can be expressed in terms of
{\em elliptic functions, modular forms}~\cite{bo-bo-ha-ma-we-ze-09}
(and beyond $\, _4F_3$ hypergeometric functions associated with
{\em Calabi-Yau ODEs}~\cite{IsingCalabi,IsingCalabi2}, etc ...).
\vskip .2cm
Let us give an illustration of the precise action of non-trivial symmetries
like (\ref{Landen})
on some elliptic functions that actually occur in the 2-D Ising
model~\cite{IsingCalabi,IsingCalabi2,Christol}: weight-one {\em modular forms}.
\vskip .2cm
Let us introduce the $\, j$-invariant\footnote[8]{The $j$-invariant~\cite{Heegner,Canada}
(see also Klein's modular invariant) regarded as a function of a complex
variable $\, \tau$, is a modular function of weight zero for $\, SL(2,\, \mathbb{Z})$.}
of the elliptic curve and its
transform by the Landen transformation
\begin{eqnarray}
\label{jjprime}
\hspace{-0.95in}&& \quad \quad \quad
j(k) \, = \, \, \, \, 256
\cdot {{(1-k^2+k^4)^3} \over {k^4 \cdot (1-k^2)^2}},
\quad \quad j(k_L) \, = \, \, \, \,
16 \cdot {\frac { (1+14\,{k}^{2}+{k}^{4})^3}{
(1-{k}^{2})^{4} \cdot {k}^{2} }}.
\end{eqnarray}
and let us also introduce the two corresponding
{\em Hauptmoduls}~\cite{Heegner}
\begin{eqnarray}
\label{Haupt}
\hspace{-0.95in}&& \quad \quad \quad\quad \quad
x \, \, = \, \, \, {{1728} \over {j(k)}},
\quad \quad \quad \,
y \, \, = \,\, \, {{1728} \over {j(k_L)}},
\end{eqnarray}
with the two Hauptmoduls being related by the
{\em modular equation}~\cite{Andrews,Atkin,Hermite,Hanna,Morain,Weisstein}:
\begin{eqnarray}
\label{modularcurve}
\hspace{-0.95in}&& \quad
1953125\,{x}^{3}{y}^{3} \, \, -187500\,{x}^{2}{y}^{2} \cdot \, (x+y) \, \,
+375\, xy \cdot \, (16\,{x}^{2}-4027\,xy+16\,{y}^{2})
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad
\quad
\, -64\, \, (x+y) \cdot \, ({x}^{2}+1487\,xy+{y}^{2})
\,\, +110592\,xy
\, \, \,= \, \,\, \, \, 0.
\end{eqnarray}
The transformation
$\, x \, \rightarrow \, \, y(x) \, = \, y$, where $\, y$
is given by the modular equation (\ref{modularcurve}),
is an {\em algebraic} transformation
{\em which corresponds to the Landen transformation}
(as well as the inverse Landen transformation: it is {\em reversible}
because of the $\,x \, \leftrightarrow \, y$ symmetry
of (\ref{modularcurve})).
The emergence of a {\em modular form}~\cite{IsingCalabi,IsingCalabi2,Christol}
corresponds to the remarkable identity on the {\em same}
hypergeometric function but where the pullback $\, x$ is changed
$\, x \, \rightarrow \, \, y(x) \, = \, y$ according to the
modular equation (\ref{modularcurve}) corresponding to the
Landen transformation, or inverse Landen transformation
\begin{eqnarray}
\label{modularform2explicit}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
_2F_1\Bigl([{{1} \over {12}}, \, {{5} \over {12}}], \, [1], \, y \Bigr)
\, = \, \, \, \,\,
{\cal A}(x) \cdot \,
_2F_1\Bigl([{{1} \over {12}}, \, {{5} \over {12}}], \, [1], \, x \Bigr),
\end{eqnarray}
where $\, {\cal A}(x)$ is an algebraic function given by:
\begin{eqnarray}
\label{wherecalA}
\hspace{-0.95in}&& \quad \quad \quad
1024\,\,{\cal A}(x)^{12} \, \,
-1152\,\,{\cal A}(x)^{8} \, \, +132\,\,{\cal A}(x)^{4}
\, \, +125\,x \, \, -4 \, \, \, \,= \, \, \, \, \, 0.
\end{eqnarray}
The emergence of a modular form is thus associated with
a selected hypergeometric function having an
{\em exact covariance property}~\cite{Stiller,Zudilin}
{\em with respect to an infinite order algebraic transformation},
corresponding here to the Landen transformation, which is
precisely what we expect for an exact representation
of the renormalization group of the square Ising
model~\cite{Hindawi,Heegner}.
With the example of the Ising model one sees that
the exact representation of the
renormalization group immediately requires considering the
isogenies of {\em elliptic curves}~\cite{Heegner}, and
thus transformations, corresponding to the {\em modular equations},
$\, x \, \rightarrow \, y(x)$ which are (multivalued)
{\em algebraic} functions.
In a previous paper~\cite{Hindawi}, we studied simpler
examples of identities on $\, _2F_1$ hypergeometric functions
where the transformations $\, x \, \rightarrow \, y(x)$ were
{\em rational functions}. In that paper we found
that the rational functions $\, y(x)$
are {\em differentially algebraic}\footnote[2]{All the non-linear
differential equations we consider in this paper (see (\ref{mad}),
see the Schwarzian equations (\ref{condition1}), (\ref{condition3F2}), ...
below) are differentially algebraic~\cite{Selected,IsTheFull}, i.e.
$\, y(x)$ is a solution of a polynomial equation $P(x,y,y',y'',...)$,
as a consequence of the fact that the functions $\, A_R(x)$
or $\, W(x)$ in these equations are {\em rational functions}
instead of general meromorphic functions (see (\ref{cas2}),
(\ref{Casale}) below).
}~\cite{Selected,IsTheFull}: they
verify a (non-linear) differential equation
\begin{eqnarray}
\label{mad}
\hspace{-0.95in}&& \quad \quad \quad \quad
A(y(x)) \cdot \, y'(x)^2 \,\, = \,\, \, \,\,
A(x) \cdot \, y'(x) \, \, \, + y''(x).
\end{eqnarray}
where $\, A(x)$ is a rational function (which is in
fact a log-derivative~\cite{Hindawi}).
This non-trivial condition coincides exactly with
one of the conditions G. Casale
obtained~\cite{Casale,Casale2,Casale3,Casale4,Casale5,Casale6,Casale7}
in a classification of
Malgrange's $\, {\cal D}$-envelope and $\, {\cal D}$-groupoids on
$\mathbb{P}_1$. Denoting $\, y'(x)$, $\, y''(x)$ and $\, y'''(x)$ the
first, second and third derivative of $\, y(x)$ with respect to $\, x$,
these conditions read respectively\footnote[5]{More generally see
the concept of differential algebraic invariant of
isogenies in~\cite{buium}.}
\begin{eqnarray}
\label{cas2}
\hspace{-0.95in}&& \quad \quad \quad \quad
\mu(y) \cdot \, y'(x) \,\, -\mu(x) \,\, + \, {{y''(x)} \over{ y'(x)}}
\, \,\, = \, \, \, \, 0,
\\
\label{Casale}
\hspace{-0.95in}&& \quad \quad \quad \quad
\nu(y) \cdot \, y''(x)^2 \, \, -\nu(x) \, \, \,
+ \, {{y'''(x)} \over{ y'(x)}} \,\,
-{{3} \over {2}} \cdot \, \Bigl({{y''(x)} \over{ y'(x)}}\Bigr)^2
\, \, = \, \,\, \, 0,
\end{eqnarray}
together with $\, \gamma(y) \cdot \, y'(x)^n \, - \, \gamma(x)
\, \, = \, \, \, 0$
and $\, h(y) \, \, = \, \, \, h(x)$,
corresponding respectively to rank two, rank three,
together with rank one and rank nul groupo\"ids,
where $\, \nu(x)$, $\, \mu(x)$, $\, \gamma(x)$
are {\em meromorphic} functions ($h(x)$ is holomorph).
Clearly Casale's condition (\ref{cas2}) is
{\em exactly the same condition as} the one we already
found in~\cite{Hindawi}, and this is not a coincidence !
In this paper we will refer to Casale's first condition (\ref{cas2})
as the ``rank-two condition'', and to the
Casale's second condition (\ref{Casale}) as
the ``rank-three condition'', or the
``Schwarzian condition''. When our paper~\cite{Hindawi} was
published we had no example corresponding to a
{\em Schwarzian condition} like (\ref{Casale}).
Without going into the details of
Malgrange's pseudo-groups~\cite{Casale2,Casale7}, Galoisian envelopes,
$\, {\cal D}$-envelopes of a germ of foliation~\cite{Casale6},
and $\, {\cal D}$-groupo\"ids, let us
just say that these concepts are built in order to generalize
the idea of differential Galois groups to {\em non-linear}~\cite{Malgrange}
ODEs\footnote[2]{In the case of linear ODEs the $\, {\cal D}$-envelope
gives back the differential Galois group of the linear ODEs.} or
{\em non-linear} functional equations\footnote[1]{The typical example is
the (non-linear) functional equation $\, f(x+1) \, = \, \, y(f(x))$, which is
such that its Malgrange pseudo-group (generalization of the
Galois group) will be ``small enough'' if and only if, there
exists a rational function $\, \nu(x)$, such that the
Schwarzian condition (\ref{Casale}) is satisfied.}
(see~\cite{Paul}). In an experimental
mathematics pedagogical approach, we will provide more
examples of {\em rational} transformations verifying rank-two
condition (\ref{cas2}), and new pedagogical examples of {\em algebraic}
transformations verifying {\em Schwarzian conditions} like
in (\ref{Casale}). We hope that these (slightly obfuscated for
physicists) Galoisian envelope conditions will become clearer
in a framework of {\em identities on hypergeometric functions}. In
a {\em modular form} perspective, we will show that the infinite
number of algebraic transformations corresponding to the
{\em infinite number of the modular equations}, are
solutions of a {\em unique} Schwarzian condition
(\ref{Casale}) with $\, \nu(x)$ a {\em rational function}.
\vskip .1cm
The paper is organized as follows. We first recall the $\, _2F_1$
results in~\cite{Hindawi} which correspond to rational transformations
and rank-two condition (\ref{cas2}) on these rational
transformations. We then display a set of new results also
corresponding to rational transformations with condition (\ref{cas2}).
Then focusing on a modular form hypergeometric identity, we
show that it actually
provides a first heuristic example of a Schwarzian condition
(\ref{Casale}) where $\, \nu(x)$ is a rational function
and analyze them in detail.
We then show that the rank-two condition (\ref{cas2})
is a subcase of the rank-three Schwarzian condition
(\ref{Casale}), the restriction corresponding to a
{\em factorization condition} of some associated order-two
linear differential operator.
We then explore generalizations of the hypergeometric identity
to $\, _3F_2$, $\, _2F_2$ and $\, _4F_3$ hypergeometric functions,
and show that the $\, _3F_2$ attempt, in fact, just reduces to
the previous $\, _2F_1$ cases through a Clausen identity.
\vskip .1cm
\vskip .2cm
\section{Recalls: rational transformation and $\, _2F_1$ hypergeometric functions }
\label{recalls}
We recall a few examples and results from~\cite{Hindawi} on the
hypergeometric examples displayed in~\cite{Vidunas}.
The hypergeometric function
\begin{eqnarray}
\label{vid}
\hspace{-0.95in}&& \quad \quad \quad
Y(x) \,\, = \, \,\,\, x^{1/4} \cdot \,
_2F_1\Bigl( [{{1} \over {2}}, {{1} \over {4}}],
[{{5} \over {4}}];\, x\Bigr)
\,\, = \, \, \, \,
{{1} \over {4}} \cdot \,
\int_0^{x} \, t^{-3/4} \cdot \, (1-t)^{-1/2} \cdot \, dt
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\, = \, \, \,
x^{1/4} \cdot \, (1-x)^{-1/2} \cdot \, \, \,
_2F_1\Bigl( [{{1} \over {2}}, {{1} \over {4}}], [{{5} \over {4}}];
\, {{-4\, x} \over {(1-x)^2}} \Bigr),
\end{eqnarray}
is the integral of an algebraic function.
It has a simple covariance property
with respect to the {\em infinite order
rational} transformation
$\, x \, \rightarrow \, -4\, x/(1-x)^2$:
\begin{eqnarray}
\label{F}
Y\Bigl({{-4\, x} \over {(1-x)^2}} \Bigr)
\,\, \, = \, \, \, \,\,\, (-4)^{1/4} \cdot Y(x).
\end{eqnarray}
This hypergeometric function can be seen
as an 'ideal' example of physical functions,
covariant by an exact (rational)
transformation. Three other hypergeometric functions
with similar covariant properties were
analyzed in~\cite{Hindawi}:
\begin{eqnarray}
\label{YM3first}
\hspace{-0.95in}&&
Y(x) \, \, = \, \, \,
x^{1/3} \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \, x\Bigr)
\, \, = \, \, \,
{{1} \over {3}} \cdot \,
\int_0^{x} \, t^{-2/3} \cdot \, (1-t)^{-2/3} \cdot \, dt
\\
\hspace{-0.95in}&&
\, = \, \, \, (-8)^{-1/3} \cdot \,
R(x)^{1/3} \cdot \,
_2F_1\Bigl( [{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}],
\, R(x)\Bigr),
\quad \, \hbox{with:} \quad \quad
R(x) \, = \, \, {{x \cdot \, (x \, -2)^3} \over {(1 \, -2 \, x)^3}},
\nonumber
\end{eqnarray}
as well as
\begin{eqnarray}
\label{YM6first}
\hspace{-0.95in}&&
Y(x) \, \, = \, \, \, \,
x^{1/6} \cdot \,
_2F_1\Bigl([{{1} \over {2}}, \, {{1} \over {6}}], \, [{{7} \over {6}}], \, x\Bigr)
\, \, = \, \, \, \,
{{1} \over {6}} \cdot \,
\int_0^{x} \, t^{-5/6} \cdot \, (1-t)^{-1/2} \cdot \, dt
\\
\hspace{-0.95in}&& \quad
\, = \, \, \, (-27)^{-1/6} \cdot \,
R(x)^{1/6} \cdot \,
_2F_1\Bigl( [{{1} \over {2}}, \, {{1} \over {6}}], \, [{{7} \over {6}}],
\, R(x)\Bigr),
\quad \hbox{with:} \quad \quad
R(x) \, = \, \, {{ -27 \, x} \over { (1\, -4 \, x)^3}},
\nonumber
\end{eqnarray}
which can be seen as a particular subcase ($\alpha \, = \, \, 1/2$)
of the identity on hypergeometric functions:
\begin{eqnarray}
\hspace{-0.95in}&& \quad \quad \quad \quad
_2F_1\Bigl([\alpha, \, {{1-\alpha} \over {3}}], \, [{{4\, \alpha \, +5} \over {6}}], \, x\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
\, \, = \, \, \, (1\, -4\, x)^{-\alpha} \cdot \,
_2F_1\Bigl([{{\alpha} \over {3}}, \, {{\alpha+1} \over {3}}], \, [{{4\, \alpha \, +5} \over {6}}], \,
{{-\, 27 \, x} \over {(1\, -4\, x)^3}}\Bigr),
\end{eqnarray}
and, finally, the simple function $\, Y(x) \, = \, \, tanh^{-1}(x^{1/2})$
that one represents as a hypergeometric function:
\begin{eqnarray}
\label{zero}
\hspace{-0.95in}&& \quad
Y(x)\,\, = \, \,\,\, \, x^{1/2} \, \cdot \,
_2F_1\Bigl([1,\, {{1} \over {2}}],[{{3} \over {2}}], \, x\Bigr)
\,\, = \, \,\,\,
{{1} \over {2}} \cdot \,
\int_0^{x} \, t^{-1/2} \cdot \, (1-t)^{-1} \cdot \, dt
\\
\hspace{-0.95in}&& \quad \quad
\, = \, \, \, (4)^{-1/2} \cdot \,
R(x)^{1/2} \cdot \,
_2F_1\Bigl( [1,\, {{1} \over {2}}],[{{3} \over {2}}], \,
\, R(x)\Bigr),
\quad \hbox{with:} \quad \quad
R(x) \,\, = \, \,\,\, \,
{{4 \, x } \over {(1\, +x)^2}}.
\nonumber
\end{eqnarray}
Though not mentioned in~\cite{Hindawi}, two other hypergeometric
functions, also covariant under a rational transformation, could
have been deduced from the previous hypergeometric examples
using Goursat and Darboux
identities (see \ref{more2F1GoursatDarboux}, and
especially (\ref{newidR1R2R3})):
\begin{eqnarray}
\label{more1}
\hspace{-0.95in}&& \quad
Y(x) \, = \, \, \, x^{1/4} \cdot \, (1\, -x)^{1/4} \cdot \,
_2F_1\Bigl([{{1} \over {2}}, \, 1], \, [ {{5} \over {4}}], \, x\Bigr)
\,\, = \, \,\,\,
{{1} \over {4}} \cdot \,
\int_0^{x} \, t^{-3/4} \cdot \, (1-t)^{-3/4} \cdot \, dt
\nonumber \\
\hspace{-0.95in}&& \quad \quad
\, \, = \, \, \, (-4)^{-1/4} \cdot \,
R(x)^{1/4} \cdot \, (1\, -R(x))^{1/4} \cdot \,
_2F_1\Bigl([{{1} \over {4}}, \, {{1} \over {2}}], \, [ {{5} \over {4}}], \,
R(x) \Bigr)
\\
\label{more1pull}
\hspace{-0.95in}&& \quad \quad
\quad \hbox{where:} \quad \quad \quad \quad \quad
R(x) \, \, = \, \, \,
{{ -4 \cdot \, x \cdot \, (1\, -x)} \over { (1 \, -2 \, x)^2}},
\end{eqnarray}
and
\begin{eqnarray}
\hspace{-0.96in}&&
\label{QQQ8first}
Y(x) \, = \, \, x^{1/6} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \, x\Bigr)
\,\, = \, \,\,\,
{{1} \over {6}} \cdot \,
\int_0^{x} \, t^{-5/6} \cdot \, (1-t)^{-2/3} \cdot \, dt
\\
\hspace{-0.96in}&&
\, = \, \, (64)^{-1/6} \cdot R(x)^{1/6} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \, R(x)\Bigr)
\quad \hbox{with:} \, \quad R(x) \, = \, \,
{{64 \, x } \over{ (1\, +18 \, x \, -27 \, x^2)^2}}.
\nonumber
\end{eqnarray}
\vskip .1cm
These six hypergeometric functions are incomplete integrals that
are canonically associated with an algebraic curve
$\,\, u^N \, - \, P(t) \, = \, \, 0\,\,$ of {\em genus one}
for (\ref{vid}), (\ref{YM3first}), (\ref{YM6first}), (\ref{more1})
(\ref{QQQ8first}), and genus zero for (\ref{zero})
\begin{eqnarray}
\label{form}
\hspace{-0.95in}&&
Y(x)\,\, = \, \,\, {{1} \over {N}} \cdot \, \int_0^{x} \, {{dt} \over { u(t) }}
\,\, = \, \,\,
{{1} \over {N}} \cdot \, \int_0^{x} \, {{dt} \over {\sqrt [N]{P(t)} }},
\quad \, \, \hbox{or:} \quad \quad
N \cdot \, Y'(x) \, = \, \, {{1} \over { u(x)}},
\end{eqnarray}
and are solutions of a
second order linear differential operator:
\begin{eqnarray}
\label{Omega}
\hspace{-0.95in}&& \quad \quad \quad \,\,
\Omega
\,\,\, = \,\,\, \,\, \omega_1 \cdot D_x,
\qquad \,\,\quad \hbox{with:} \quad \quad \quad \quad \,
\omega_1 \, = \, \,\, \,\,
D_{x} \,\, + A_R(x),
\end{eqnarray}
where $\, D_x$ denotes $\, d/dx$,
where a rational function $\, A_R(x)$
is\footnote[1]{The fact that $\, A_R(x)$ is
the log-derivative of the $\, N$-th root of a rational function, here
a polynomial, is a consequence of the fact that $\, \Omega$
is a globally nilpotent linear differential operator~\cite{bo-bo-ha-ma-we-ze-09}.}
the logarithmic derivative of a simple algebraic\footnote[2]{Note that
$\, u(x)$ being an algebraic function,
these examples are such that $\, N \cdot Y'(x) \, = \, 1/u(x)$,
$\, Y'(x)$ is holonomic but also its reciprocal $\, 1/Y'(x)$.} function
$\, u(x) \, = \, \, \sqrt [N]{P(x)}$.
The expressions of the rational functions $\, A_R(x)$ read
respectively for the four
hypergeometric examples (\ref{vid}), (\ref{YM3first}), (\ref{YM6first})
and (\ref{zero})
\begin{eqnarray}
\label{respecAR}
\hspace{-0.95in}&& \quad \,\,
{{1} \over {4}} \, {{3 \, -5\,x} \over {x \cdot \, (1\,-x)}}, \quad \quad
{{2} \over {3}} \cdot {\frac {1-2\,x}{ x \cdot \,(1\, -x) }}, \quad
\quad {{1} \over {6}} \cdot {\frac {5-8\,x}{ x \cdot \,(1\,-x) }}, \quad \quad
\, {{1} \over {2}} \cdot {\frac {1 \, -3\,x}{ x \cdot \, (1\,-x) }}.
\end{eqnarray}
and for the two new examples (\ref{more1}) and (\ref{QQQ8first}):
\begin{eqnarray}
\label{respecARmore}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
{{3} \over { 4}} \cdot \, {{1 \, -2 \, x} \over {x \cdot \, (1 \, -x) }},
\,\, \quad \quad \quad \quad
{{1} \over { 6}} \cdot \, {{5 \, -9 \, x} \over {x \cdot \, (1 \, -x) }}.
\end{eqnarray}
\vskip .1cm
In the interesting cases emerging
in physics~\cite{bo-ha-ma-ze-07b,IsingCalabi,IsingCalabi2,Christol},
the operator $\, \Omega$ happens to be globally
nilpotent~\cite{bo-bo-ha-ma-we-ze-09}, in which case
$\, A_R(x)$ is the log-derivative
of the $\, N$-th root of a rational function. At first
we do not require $\, \Omega$ to be globally
nilpotent\footnote[9]{Imposing the
global nilpotence generates additional relations
(see section (\ref{assuminglob}) below).}, then we will see
what this assumption entails.
\vskip .2cm
Let us consider a rational transformation
$\, x \, \rightarrow \, R(x)$ and the order-one operator
$\, \, \omega_1 \, = \, \, D_x \, \, + \, A_R(x)$.
The change of variable $\, x \, \rightarrow \, R(x) \, $
on the order-one operator $\, \omega_1$ reads:
\begin{eqnarray}
\label{RHSL1}
\hspace{-0.95in}&& \quad \, \, \quad \quad
D_x \, + \, \, A_R(x) \quad \longrightarrow \quad \, \, \, \,
1/R'(x) \cdot D_x \, + \, \, A_R(R(x))
\nonumber \\
\hspace{-0.95in}&& \quad \, \quad \, \quad \quad \quad \quad
\, \, = \, \, \, \,
\gamma(x) \cdot {\cal L}_1
\, \, = \, \, \, \,
\gamma(x) \cdot \Bigl( D_x \, + \, \, A_R(R(x)) \cdot \, R'(x) \Bigr),
\end{eqnarray}
with $\, \gamma(x) \, = \, \, 1/R'(x)$.
Now imposing the order-one operator $\, {\cal L}_1$ of the RHS expression (\ref{RHSL1})
to be equal to the conjugation by $\, \gamma(x)$ of
$\, \, \omega_1 \, = \, \, D_x \, \, + \, A_R(x)$, namely
\begin{eqnarray}
\hspace{-0.95in}&& \quad \quad \quad
\gamma(x) \cdot
\Bigl(D_x \, + \, \, A_R(x)\Bigr) \cdot {{1} \over {\gamma(x)}}
\, \, \, = \, \, \,\, D_x \, \, \, - \, {{ d\ln(\gamma(x))} \over {dx}} \,\, + \, A_R(x),
\nonumber
\end{eqnarray}
one deduces a rank-two functional equation~\cite{Hindawi}
on $\, A_R(x)$ and $\, R(x)$:
\begin{eqnarray}
\label{mad}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
A_R(R(x)) \cdot \, R'(x)^2 \,\, = \,\, \, \,\,
A_R(x) \cdot \, R'(x) \, \, \, + \, R''(x) .
\end{eqnarray}
This condition is {\em exactly} the first rank-two condition
of Casale given by (\ref{cas2}). Using the chain rule formula of
derivatives for the composition of functions, one can show,
for a given rational function $\, A_R(x)$, that the composition
$\, R_1(R_2(x))$ {\em verifies condition} (\ref{mad})
if two rational functions $\, R_1(x)$
and $\, R_2(x)$ verify condition (\ref{mad}). In
particular if $\, R(x)$ verifies condition (\ref{mad}),
all the iterates of $\, R(x)$ also verify that
condition\footnote[5]{This is in agreement with the fact
that (\ref{mad}) is the condition for $\, \Omega\, =\,\,$
$ (D_x\, + \, A_R(x)) \cdot D_x$ to be
covariant by $x \, \rightarrow \, R(x)$: this condition is
obviously preserved by the composition of $\, R(x)$'s
(for $\, A(x)$ fixed).}: $\, R(x) \, $
$\longrightarrow \,\, \, R(R(x))$,
$\, R(R(R(x))), \,\, \cdots$
\vskip .2cm
Keeping in mind the well-known example of the
parametrization of the standard map
$\, x \, \rightarrow \,4\, x \cdot (1-x)$ with
$\, x \, = \, \sin^2(\theta)$, yielding
$\, \theta \, \rightarrow \, 2 \, \theta$,
let us seek a ({\em transcendental})
parametrization $\, \,x \, = \, P(u)\,$
such that\footnote[2]{This is the idea of Siegel's
linearization~\cite{Siegel,Siegel2,Almost}
(or Koenig's linearization theorem see~\cite{Milnor}).}
\begin{eqnarray}
\hspace{-0.95in}&& \quad \quad \quad \quad
R_{a_1}\Bigl(P(u) \Bigr) \, = \, \, P(a_1\, u)
\qquad \hbox{or:} \quad \quad \quad \, \,\, \,
R_{a_1}\, = \, \, P \circ H_{a_1} \circ \, Q,
\end{eqnarray}
where $\, H_{a_1}$ denotes the scaling
transformation $\, \,x \, \rightarrow \, \, a_1 \cdot x\,$
and $\, Q \, = \, \, P^{-1}$ denotes the composition inverse
of $\, P$. One can also
verify an essential property that we expect to be true
for a representation of the renormalization group,
namely that two $\, R_{a_1}(x)$ for different
values of $\, a_1$ commute,
the result corresponding to the product of these two $\, a_1$:
\begin{eqnarray}
\label{commute}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
R_{a_1}\Bigl(R_{b_1} (x) \Bigr) \,\, \, = \, \, \, \,
R_{b_1}\Bigl(R_{a_1} (x) \Bigr)
\,\, \, = \, \, \, \, R_{a_1 \cdot b_1} (x).
\end{eqnarray}
The neutral element of this abelian group
corresponds to $\, a_1 \, = \, 1$,
giving the identity transformation
$\, R_{1}(x) \, = \, \, x$.
Performing the composition inverse of $\, R_{a_1}(x)$
amounts to changing
$\, a_1$ into its inverse $\, 1/a_1$.
The structure of the (one-parameter) group and
the extension of the composition of $\, n$ times a rational
function $\, R(x)$ (namely $\, R(R( \cdots R(x) \cdots ))$)
to $\, n$ {\em any complex number},
is a straight consequence of this relation. For example,
in the case of the $\, _2F_1$ hypergeometric
function (\ref{YM3first}), the one-parameter series
expansion of $\, R_{a_1}(x)$ reads:
\begin{eqnarray}
\label{mad2oneseriesfirst}
\hspace{-0.95in}&& \quad \quad
R(a, \, x) \,\, = \,\, \, \,\,
a \cdot \, x \, \, \, + \, a \cdot \, (a-1) \cdot \, S_a(x)
\quad \quad \quad \quad \quad \hbox{where:}
\\
\hspace{-0.95in}&& \quad \quad
S_a(x) \,\, = \,\, \, \,\,
- {{1} \over {2}} \cdot \, {x}^{2} \, \, \,
+ {{1} \over {28}} \cdot \, (5\,a-9) \cdot \, {x}^{3}
\,\, -{\frac {
\, (3\,{a}^{2}-12\,a+13) }{56}} \, \cdot {x}^{4}
\, \, \, \, + \, \,\, \cdots
\nonumber
\end{eqnarray}
This one-parameter series (\ref{mad2oneseriesfirst})
is a family of commuting one-parameter series solution of
the rank-two condition (\ref{mad}), and these solution series
have {\em movable singularities} (more details
in \ref{more2F1examples}). Defining some
``infinitesimal composition''
($Q \, = \, P^{-1}$, $\epsilon \, \simeq \, 0$)
\begin{eqnarray}
\label{R1pluseps}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \,
R_{1\, + \, \epsilon}(x)\, \, = \, \, \, \, \,
P \circ \,\, H_{1\, + \, \epsilon} \circ \, Q(x)
\,\, = \, \, \,\, \, x \,\, \, + \, \epsilon \cdot F(x)
\,\,\, \,\, + \,\, \cdots
\end{eqnarray}
we see, from (\ref{commute}), that $\, R_{a_1}(R_{1\, + \, \epsilon}(x)) \,= \, \,
R_{1\, + \, \epsilon}(R_{a_1} (x))$. Using (\ref{R1pluseps}) and Taylor expansion
one gets the following relations
between $\, R_{a_1}(x)$ and the function\footnote[1]{Generically,
$ \, F(x)$ is a transcendental function,
not a rational nor an algebraic function. } $\, F(x)$:
\begin{eqnarray}
\label{commuteinf}
\hspace{-0.95in}&&
R_{a_1}\Bigl(R_{1\, + \, \epsilon}(x) \Bigr) \, = \, \,
R_{a_1}\Bigl (x \, + \, \epsilon \cdot F(x) + \, \cdots \Bigr)
\, \, = \, \, \, R_{a_1}(x)
\, \, + \, {{dR_{a_1}(x) } \over {dx}} \cdot \epsilon \cdot F(x)
\, \, + \,\, \cdots
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad
= \,\, R_{1\, + \, \epsilon}\Bigl(R_{a_1} (x) \Bigr) \,\, \, = \, \, \,
R_{a_1} (x) \, \, + \, \, \epsilon \cdot F(R_{a_1} (x)) \, \, + \, \, \, \cdots
\end{eqnarray}
which gives at the first order in $\, \epsilon$:
\begin{eqnarray}
\label{Ra1}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
{{dR_{a_1}(x) } \over {dx}}
\cdot \, F(x) \,\, = \, \,\,\, F(R_{a_1}(x)).
\end{eqnarray}
For $\, R(x)$ and for the $\, n$-th iterates of the rational function
$\, R(x)$ (which are in the one-parameter family $\, R_{a_1}(x)$)
relation (\ref{Ra1}) reduces to:
\begin{eqnarray}
\hspace{-0.95in}&& \quad \, \,\,
R'(x)
\cdot \, F(x) \,\, = \, \,\, F(R(x)),
\quad \quad \quad {{dR^{(n)}(x) } \over {dx}}
\cdot \, F(x) \,\, = \, \,\, F(R^{(n)}(x)),
\\
\hspace{-0.95in}&& \quad \quad \quad \quad \, \,\,
\hbox{where:} \qquad \quad \quad \quad \quad \, \,\,
R^{(n)}(x) \, = \, \, \, R(R(\cdots R(x)) \cdots ).
\nonumber
\end{eqnarray}
From (\ref{R1pluseps}) one gets
$\, P(Q(x) \, + \, \epsilon \cdot \, Q(x)) \, =
\, x \, + \, P'(Q(x)) \cdot \, \epsilon \cdot \, Q(x) \, + \, \cdots \,
= \, x \, + \, \epsilon \cdot \, F(x) \, + \, \cdots \, $
and also $\, P \circ H_{1\, + \, \epsilon}(x) \, = \, P(x \, + \epsilon \cdot \, x)$
$ \, = \, P(x) \, + \, P'(x) \cdot \, \epsilon \cdot \, x \, + \, \cdots \, $
$\, = \, P(x) \, + \, \epsilon \cdot \, F(P(x)) \, + \,\cdots \, \, $
yielding respectively
\begin{eqnarray}
\label{covP}
\hspace{-0.95in}&& \quad \quad
Q(x) \cdot \, P'(Q(x)) \, = \, \, F(x)
\quad \quad \hbox{and thus:} \quad \quad \quad
x \cdot P'(x) \,\, = \, \, \, \,\, F(P(x)).
\end{eqnarray}
This last relation yields $\, Q(x) \cdot P'(Q(x)) \,\, = \, \, \, \,\, F(x)$,
which can also be written using $\, P \circ \, Q(x) \, = \, x$
(and thus $\, P'(Q(x)) \cdot \, Q'(x) \, = \, 1$):
\begin{eqnarray}
\label{QF}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad \quad
Q'(x) \cdot F(x) \,\, = \, \, \, \,\, Q(x).
\end{eqnarray}
\vskip .1cm
Inserting (\ref{R1pluseps}) in the rank-two condition (\ref{mad})
one immediately finds (at the first order in $\, \epsilon$) that
$ \, F(x)$ is a {\em holonomic function},
solution of a second order linear differential operator
$\,\Omega^{*}$ which can be seen to be the
{\em adjoint} of the second order operator
$\, \Omega$ defined by (\ref{Omega}):
\begin{eqnarray}
\label{Fholo}
\hspace{-0.95in}&& \quad \, \quad \quad \quad
\Omega^{*} \, \, \, = \, \, \, \, \,
D_x^2 \,\, - \, A_R(x) \cdot \, D_x \,\, - \, A'_R(x)
\, \, \, = \, \, \, \, \,
D_x \cdot \Bigl(D_x \, - \, A_R(x)\Bigr).
\end{eqnarray}
\subsection{New results: $\, Q(x)$ and $\, P(x)$ as differentially algebraic functions \\}
\label{resnewPQ}
\vskip .1cm
The two functions $\, Q(x)$ and its composition inverse
$\, P(x) = \, \, Q^{-1}(x)$ are
{\em differentially algebraic functions}~\cite{Selected,IsTheFull}
as can be seen in~\cite{Hindawi}. The function $\, Q(x)$
is solution of the differentially algebraic equation:
\begin{eqnarray}
\label{logderiv1first}
\hspace{-0.97in}&& \quad
A_R(x) \cdot \, G(x) \cdot \, G'(x)
\,\, - \, \, A'_R(x) \cdot \, G(x)^2
\, \,+2 \, G'(x)^2
\,\, - \, G(x) \cdot \, G"(x)
\, \, = \,\, \, 0,
\end{eqnarray}
where $\, G(x)$ is the log-derivative\footnote[2]{With an extra log-derivative
step equation (\ref{logderiv1first}) can be written in an even simpler
form. Introducing $\, H(x) \, = \, \, G'(x)/G(x)$, equation (\ref{logderiv1first}) becomes
$\, A_R'(x) \, -A_R(x)\cdot \, H(x) \, + \, H'(x) \, - \, H(x)^2 \, = \, \, 0$.}
of $\, Q(x)$, i.e. $\, G(x) \, = \, \, Q'(x)/Q(x)$. While equation (\ref{QF})
means that $\, F(x) \, = \, \, 1/G(x)$,
equation (\ref{logderiv1first}) is immediately obtained by
imposing $\, F(x) \, = \, \, 1/G(x)$ to be a
solution of $\, \Omega^{*}$.
\vskip .2cm
One remarks that this non-linear differential equation
corresponds to a {\em homogeneous} quadratic
equation in $\, G(x)$ and its derivatives. In terms
of $\, Q(x)$ this equation corresponds to
a homogeneous cubic equation in $\, Q(x)$
and its derivatives:
\begin{eqnarray}
\label{QDAfirst}
\hspace{-0.95in}&& \quad \quad
A_R(x) \cdot \, \Bigl( Q'(x)^2 \,
-Q(x) \cdot \, Q''(x)\Bigr)
\cdot \, Q'(x) \, \,
+ \, A'_R(x) \cdot \, Q(x) \cdot \, Q'(x)^2
\\
\hspace{-0.95in}&& \quad \quad \quad \quad
+Q''(x) \cdot \, Q'(x)^2 \,
+Q(x) \cdot \, Q'''(x) \cdot \, Q'(x)
\,\, -2 \, Q(x) \cdot \, Q''(x)^2
\, \,= \, \, \, 0.
\nonumber
\end{eqnarray}
\vskip .2cm
The function $\, P(x)$, being the composition inverse
of a differentially algebraic function, is solution of the
{\em differentially algebraic}~\cite{Selected,IsTheFull}
equation\footnote[5]{Equation (\ref{simplerelation61first})
can be obtained using the Fa\`a di Bruno formulas for the
higher derivatives of inverse functions.}:
\begin{eqnarray}
\label{simplerelation61first}
\hspace{-0.95in}&& \, \quad \quad \,
A_R(P(x)) \cdot \, P'(x)^2 \cdot \,
\Bigl( x \cdot \, P''(x) \, + P'(x)\Bigr)
\,\,\, + \, \, x \cdot \, A_R'(P(x))
\cdot \, P'(x)^4 \, \,
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
+ x \cdot \, P"(x)^2 \, \, \,
-x \cdot \, P'(x) \cdot \, P'''(x)
\, \,\, - P'(x) \cdot \, P"(x)
\,\, \, = \, \, \, \, 0.
\end{eqnarray}
For instance, for the hypergeometric function (\ref{vid}), one verifies
straightforwardly that $\, P(x) \, = \, \, sn^4(x, \, (-1)^{1/2})$,
given in~\cite{Hindawi}, verifies (\ref{simplerelation61first})
with $\, A_R(x)$ given by the first
rational function in (\ref{respecAR}).
\vskip .1cm
\subsection{Assuming that $\, \Omega$ is globally nilpotent}
\label{assuminglob}
The rank-two condition (\ref{mad}) turns out to identify exactly with
the first Casale condition (\ref{cas2}), the only difference
being that $\, A_R(x)$
is not meromorphic as in Casale's condition (\ref{cas2}), but a
{\em rational function}: in lattice statistical mechanics and
enumerative combinatorics, the differential operators are
linear differential operators with polynomial coefficients. In fact,
the operators emerging in lattice models are not only Fuchsian,
but {\em globally nilpotent} operators~\cite{bo-bo-ha-ma-we-ze-09},
or $\, G$-operators~\cite{Andre}, thus their wronskians are the
$\, N$-th root of a rational function~\cite{bo-bo-ha-ma-we-ze-09}. This
naturally leads us to examine the case where $\, \Omega$ is taken to
be globally nilpotent. Given $\, \Omega$ globally nilpotent, there
exists an algebraic function $\, u(x)$ ($N$-th root
of a rational function) such that $\, A_R(x)$
is the log-derivative of $\, u(x)$. Consequently
$\, \Omega$ and $\, \Omega^{*}$,
which read respectively
$\,\Omega \, = \, \, $
$ u(x)^{-1} \cdot \, D_x \cdot \, u(x) \cdot \, D_x$
and $\, \Omega^{*} \, = \, \, $
$ D_x \cdot \, u(x) \cdot \, D_x \cdot \, u(x)^{-1}$,
are related by the simple conjugation:
\begin{eqnarray}
\label{simplerelati}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
\,\, \Omega^{*} \cdot \, u(x)
\,\, \, = \, \, \,\, u(x) \cdot \, \Omega.
\end{eqnarray}
Thus, $ \, F(x)$ and $\, Y(x)$ are related through the simple
equation:
\begin{eqnarray}
\label{simplerelation}
\hspace{-0.95in}&& \quad \quad \quad \, \,
\quad \quad \quad \quad \quad
u(x) \cdot \, Y(x) \, \, = \, \, \, \, F(x).
\end{eqnarray}
\vskip .1cm
The fact that the holonomic function $\, Y(x)$ is solution of $\, \Omega$,
amounts to writing that the log-derivative of $\, Y'(x)$ is
equal to $\, -A_R(x)$. If $\, \Omega$ is globally nilpotent then
$\, -A_R(x)$ is the log-derivative of the reciprocal $\, 1/u(x)$, and
the logarithm of $\, Y'(x)$ is equal to the logarithm of $\, 1/u(x)$,
up to a constant of integration $\, \ln(\alpha)$, and thus:
\begin{eqnarray}
\label{deduces}
\hspace{-0.95in}&& \, \, \, \quad \, \,
\alpha \cdot \, {{d Y(x)} \over { d x}}
\, \, = \, \, \, \, {{1} \over {u(x)}}
\quad \quad \quad \, \, \hbox{or:} \quad \quad \quad \quad \,
\alpha \cdot \, {{ Y'(x)} \over {Y(x)}} \, \, = \, \, \, \,
{{1} \over {u(x) \cdot \, Y(x)}}.
\end{eqnarray}
\vskip .1cm
Recalling the fact that the rank-two condition (\ref{mad}) gives (\ref{QF}),
namely that the log-derivative of $\, Q(x)$ is equal to $\, 1/F(x)$, one
deduces by combining (\ref{simplerelation}) with (\ref{deduces}):
\begin{eqnarray}
\label{combining}
\hspace{-0.95in}&& \, \, \, \, \, \quad
{{Q'(x) } \over {Q(x)}} \, \, = \, \, \, \, {{1} \over {F(x)}}
\, \, = \, \, \, \, \alpha \cdot \, {{ Y'(x)} \over {Y(x)}}
\quad \quad \, \, \, \, \hbox{i.e.} \quad \quad \quad
Q(x) \, \, = \, \, \, \lambda \cdot \, Y(x)^{\alpha}.
\end{eqnarray}
Note that, without any loss of generality, one can restrict $\, \lambda$
to $\, \lambda \, = \, \, 1$.
\vskip .1cm
$\, F(x)$ is solution of $\, \Omega^{*}$ as a consequence of the
rank-two condition (\ref{mad}). This second order linear differential
equation can be integrated into
$\, F'(x) \, -A_R(x) \cdot \, F(x) \, = \, \, u(x) \cdot \, Y'(x)$,
and taking into account (\ref{deduces}) this gives:
\begin{eqnarray}
\label{integrated}
\hspace{-0.95in}&& \, \, \quad \quad
\quad \quad \quad \quad \quad \quad
F'(x) \, \,\, -A_R(x) \cdot \, F(x)
\, \,\, = \, \, \, \, \, {{1} \over {\alpha}}.
\end{eqnarray}
\vskip .1cm
For the new results (see sections (\ref{moreHeun}) and (\ref{more2F1higher}) below),
corresponding to a rank-two condition (\ref{mad}) like the hypergeometric examples
seen in the beginning of this section, the holonomic function $\, Y(x)$ is
of the form (\ref{form}). Thus the constant $\, \alpha$ is actually
equal to a {\em positive integer} $\, N$ (see the case where
$\, N \, = \, 3$ in \ref{more2F1examples} for a worked example). Further
one deduces from (\ref{combining})
that $\, Q(x)$ is {\em always a holonomic function}:
$\, Q(x) \, = \, \, \lambda \cdot \, Y(x)^N$, for instance,
for the hypergeometric functions
(\ref{vid}), (\ref{YM3first}), (\ref{YM6first}), (\ref{zero}),
(\ref{more1}) and (\ref{QQQ8first}), we have $\, Q(x)\, = \, \, Y(x)^N$
with $\, N= \, 4, \, 3, \, 6, \, 2, \, 4, \, 6$ respectively.
\vskip .1cm
Without assuming (\ref{form}), the constant $\, \alpha$ is not necessarily
a positive integer, thus $\, Q(x)$ has no reason to be
holonomic: it is just {\em differentially algebraic} (see (\ref{QDAfirst})).
The log-derivatives of $ \, Q(x)$ and $\, Y(x)$ being
equal up to a multiplicative factor $\, \alpha$ (see (\ref{combining})),
one deduces from the fact that (\ref{logderiv1first}) is a
{\em homogeneous} (quadratic) condition in $\, G(x)$ and its derivatives,
that $ \, Q(x)$ and $\, Y(x)$ verify necessarily the {\em same}
differentially algebraic condition (\ref{QDAfirst}).
\vskip .1cm
With this global nilpotence assumption, the differentially algebraic
function $\, P(x)$ is, in fact, solution of much simpler non-linear
ODEs. From $\,\, u(x) \cdot \, Y(x) \, = \, F(x)\,$ one gets
using (\ref{covP}):
\begin{eqnarray}
\label{muchsimpler}
\hspace{-0.95in}&& \, \, \, \, \quad \quad \quad \quad
u(P(x))^{\alpha} \cdot \, Y(P(x))^{\alpha} \,\,\, = \, \,\, F(P(x))^{\alpha}
\, \,\, = \, \, \,\Bigl(x \cdot \, P'(x)\Bigr)^{\alpha}.
\end{eqnarray}
Using $\,\, Q(x) = \, \lambda \cdot \, Y(x)^{\alpha}$,
and $\, \,Q(P(x)) \, = \, x$, one deduces:
\begin{eqnarray}
\label{muchsimpler}
\hspace{-0.95in}&& \, \, \quad \, \, \, \quad \quad \quad \quad \quad
x \cdot \, u(P(x))^{\alpha}
\, \,\, = \, \, \, \, \lambda \cdot \,
\Bigl(x \cdot \, P'(x)\Bigr)^{\alpha}.
\end{eqnarray}
\vskip .1cm
\section{More rational transformations: an identity on a Heun function }
\label{moreHeun}
In this section we write an identity similar to the $\, _2F_1$
hypergeometric identities (\ref{vid}), (\ref{YM3first}),
(\ref{YM6first}), but, this time, on a {\em Heun function}, that is
a holonomic function with {\em four} singularities instead of the
well-known three singularities $\, 0$, $\, 1$, $\, \infty$
of the hypergeometric functions.
\vskip .1cm
Let us consider the rational transformation\footnote[5]{Emerging
as a symmetry of the complete elliptic integrals of the third kind
in the anisotropic Ising model (see~\cite{Barry}).}
\begin{eqnarray}
\label{next}
\hspace{-0.95in}&& \, \quad \quad \quad
\quad \quad \quad \quad
x \quad \longrightarrow \quad \quad
4 \cdot \,\frac{x \cdot \, (1-x)
\cdot \, (1 \, -k^2 \, x)}{(1 \, -k^2 \, x^2)^2},
\end{eqnarray}
where one recognizes the transformation\footnote[2]{The general
case $\,\,\theta \, \rightarrow \, \, p \, \theta \,\,$
is laid out in \ref{MiscellHeun}.}
$\, \theta \, \rightarrow \, \, 2 \, \theta$ on the
square of the elliptic sine $\, x \, = \, \, sn(\theta, \, k)^2$:
\begin{eqnarray}
\label{doubling}
\hspace{-0.95in}&& \quad \quad \quad
sn(\theta, \, k)^2 \, \,
\quad \longrightarrow \, \, \, \quad \quad
sn(2 \, \theta, \, k)^2 \, \, \, =
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \qquad
\, \, \, \, = \, \, \, \,
4 \cdot \, {{ sn(\theta, \, k)^2 \cdot \,
(1\, - sn(\theta, \, k)^2) \cdot \,
(1\, - \, k^2 \cdot \, sn(\theta, \, k)^2)
} \over {(1 \, -k^2 \cdot \, sn(\theta, \, k)^4 )^2 }}.
\end{eqnarray}
Denoting $\, M \, = \, \, 1/k^2$, the transformation
(\ref{next}) yields:
\begin{eqnarray}
\label{Aadoubling}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
R(x) \, \, = \, \, \, 4 \cdot \,\frac{x \cdot \, (1-x)
\cdot \, (1 \, -\, x/M)}{(1 \, - \, x^2/M)^2}.
\end{eqnarray}
For a given $\, M$, the transformations
$\, \theta \, \rightarrow \, \, p \, \theta \, $
give rational transformations $\,x \, \rightarrow \, R_p(x)$
on the square of the elliptic sine,
$\, x \, = \, \, sn(\theta, \, k)^2$,
which are sketched for the first primes $\, p \, \, $
in \ref{MiscellHeun}. The series expansions of these rational
transformations read
$\, R_p(x) \, = \, \, p^2 \cdot \, x \, + \, \cdots \, $
With these rational functions $\, R_p(x)$
we have the following identity on a Heun function\footnote[9]{The Heun
function is the Heun {\em general} function, HeunG function in Maple,
not a confluent Heun function.}:
\begin{eqnarray}
\label{Fdoublingidefirst}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
R_p(x) \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, R_p(x) \Bigr)^2
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
\, \, = \, \, \,
p^2 \cdot \, x \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)^2.
\end{eqnarray}
Using the formalism introduced
in section (\ref{recalls}), we write
\begin{eqnarray}
\label{AadoublingAa}
\hspace{-0.95in}&&
A_R(x) \, = \, \, {{ u'(x)} \over {u(x)}} \, = \, \,
-{{1} \over {2 \, (M\, -x)}}
\, +{{2 \, x \, -1} \over { 2 \, x \, (x-1)}}
\,\, \, = \,\, \, \,
{{1} \over {2}} \cdot \,
{{3\, x^2 \, -2\, (M+1)\, x\, +M } \over {
x \cdot \, (1-x) \cdot \, (M \, -x) }},
\nonumber \\
\hspace{-0.95in}&& \, \quad \quad \quad \quad
\hbox{where:} \quad \quad \quad \quad \, \, \,
u(x) \, \, = \, \, \, \Bigl( x \cdot \, (1-x) \cdot \, (1 \, -x/M) \Bigr)^{1/2}.
\end{eqnarray}
The Liouvillian solution of the
operator $\,\, \Omega \, = \, \, (D_x \, +A_R(x)) \cdot \, D_x\,$
corresponds to the
{\em incomplete elliptic integral of the first kind}
(introducing $\, u \, =\, \, sin^2(\theta)$ and
$\, x \, =\, \, sin^2(\phi)$):
\begin{eqnarray}
\label{incomplete}
\hspace{-0.95in}&&
F(\phi, \, m) \,\, = \, \, \,
\int_{0}^{\phi} \, {{ d \theta } \over { (1\, - \, m \cdot \, sin^2(\theta))^{1/2}}}
\,\, = \, \, \, {{1} \over {2}} \cdot \,
\int_{0}^{x} \,
{{ d u} \over {
u^{1/2} \cdot \, (1\, -u)^{1/2} \cdot \, (1\, - \, m \cdot \, u)^{1/2}}}.
\nonumber
\end{eqnarray}
This corresponds to a Heun function,
or equivalently to the {\em inverse Jacobi sine \footnote[5]{The Jacobi sine function
arises from the inversion of the incomplete elliptic integral of the first kind.}}:
\begin{eqnarray}
\label{otherwords}
\hspace{-0.95in}&&
x^{1/2} \cdot \, Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)
\, = \, \,
InverseJacobiSN\Bigl(x^{1/2}, \, {{1} \over {M^{1/2}}}\Bigr).
\end{eqnarray}
The Heun solution of $\, \Omega$ reads with
$\, x \, =\, \, sin^2(\phi)$:
\begin{eqnarray}
\label{Fdoubling}
\hspace{-0.95in}&& \quad \quad
Y(x) \, = \, \,\,
x^{1/2} \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)
\, = \, \,\, \, {{1} \over {M^{1/2}}}
\cdot \, F\Bigl(\phi, \, {{1} \over {M}}\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
\,\, = \, \, \, {{1} \over {2}} \cdot \,
\int_0^{x} \, {{ d u} \over { u^{1/2} \cdot \, (1\, -u)^{1/2} \cdot \, (M\, -u)^{1/2}}}.
\end{eqnarray}
The Heun identity (\ref{Fdoublingidefirst}) amounts to writing
a covariance on this Heun function given by:
\begin{eqnarray}
\label{Ydoublingidentity}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
Y\Bigl(R_p(x)\Bigr) \, \, \, = \, \, \, \,
p \, \cdot \, Y(x).
\end{eqnarray}
\vskip .1cm
The adjoint operator
$\,\, \Omega^{*} \, = \, \, D_x \cdot (D_x \, -A_R(x))\,$
has the following Heun function solution:
\begin{eqnarray}
\label{Fdoubling}
\hspace{-0.95in}&&
F(x) \, \, = \, \, \, \,
x \cdot \, (1 \, -x)^{1/2} \cdot \,
\Bigl(1 \, -{{x} \over {M}} \Bigr)^{1/2} \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \,
{{1} \over {2}}, \, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr).
\end{eqnarray}
All the rational transformations $\, R_p(x)$ verify
a rank-two condition (\ref{mad}) with
$\, A_R(x)$ given by (\ref{AadoublingAa}).
More generally, the one-parameter series solution of
the rank-two condition (\ref{mad}) are, again,
{\em commuting} series:
\begin{eqnarray}
\label{Heunonepara}
\hspace{-0.95in}&&
R(a, \, x) \, \, = \, \, \, \,
a \cdot \, x \, \,\,\, + a \cdot (a-1) \cdot \, S_a(x)
\quad \quad \quad \quad \quad \hbox{where:}
\\
\hspace{-0.95in}&&
S_a(x) \, \, = \, \, \, \,-\,{\frac { (M\, +1) }{ 3 \, M}} \cdot \, x^2 \, \, \,
\, + \, \, {\frac { (2\, \cdot \, (M^{2}+1)
\cdot \, (a \, -4) \, + \, (13\,a \, -7) \cdot \, M) }{
45 \cdot \, M^{2}}} \cdot \, x^3 \, \,
\nonumber \\
\hspace{-0.95in}&& \, \, \,
- \, {{(M\, +1)} \over { 315 \cdot \, M^3 }} \cdot
\Bigl((M^{2}+1) \cdot \, (a \, -4) \cdot \, (a \, -9)
\, + \, \, (29\,a^2 \,-62\,a \,-6) \cdot \, M\Bigr)
\cdot \, x^4
\, \, + \, \,\cdots
\nonumber
\end{eqnarray}
with $\,\,\, R(a_1, \, R(a_2, \, x)) \, = \, \, $
$R(a_2, \, R(a_1, \, x)) \, = \, \, R( a_1\,a_2, \, \, x)$.
The one-parameter series (\ref{Heunonepara}) reduces to the
series expansion of the rational functions $\, R_p(x)$ for $\, a \, = \, p^2$
{\em for every integer} $\, p$. One thus sees that the rank-two
condition (\ref{mad}) with $\, A_R(x)$ given by (\ref{AadoublingAa}),
{\em encapsulates an infinite number of commuting rational transformations}
$\, R_p(x)$.
Finally, as far as the Koenig-Siegel
linearization~\cite{Siegel,Siegel2,Almost,Milnor}
of the one-parameter series is concerned, one has
$ \,\, Q(x) \, \, = \, \, Y(x)^2\,$ and:
\begin{eqnarray}
\label{Pdoubling}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
P(x) \, = \, \, \, sn\Bigl(x^{1/2}, \, {{1} \over {M^{1/2}}}\Bigr)^2.
\end{eqnarray}
One easily verifies that this exact expression (\ref{Pdoubling})
in terms of the elliptic sine is solution of the
differentially algebraic equation (\ref{simplerelation61first})
with $\, A_R(x)$ given by (\ref{AadoublingAa}).
\vskip .1cm
One can verify (though it is not totally straightforward) that the rational
function (\ref{Aadoubling}), and more generally the $\, R_p(x)$,
have the decomposition
\begin{eqnarray}
\label{decompo}
\hspace{-0.95in}&&
\, \, \,\, \, 4 \cdot \,\frac{x \cdot \, (1 \, -x)
\cdot \, (1 \, -\, x/M)}{(1 \, - \, x^2/M)^2}
\, \, = \,\, \, P(4 \cdot Q(x)),
\quad \, \, \, \, \,
R_p(x) \, = \,\, \, P(p^2 \cdot Q(x)),
\end{eqnarray}
with $\, P(x)$ and $\, Q(x)$ given respectively by (\ref{Pdoubling})
and $ \,\, Q(x) \, \, = \, \, Y(x)^2$.
\vskip .2cm
\subsection{$\, _2F_1$ hypergeometric functions deduced from the Heun example}
\label{more2F1Heun}
\vskip .2cm
We know from~\cite{maier-05,Belyi3} for example, that selected
Heun functions can reduce to pullbacked $\, _2F_1$
hypergeometric functions. This is also the case
for the Heun function (\ref{Fdoubling})
in section (\ref{moreHeun}) for selected
values of $\, M$. For $\, M\, = \, \, 2$ we have:
\begin{eqnarray}
\label{M2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad
\quad \quad \quad \quad \quad \quad \quad
\, \, = \, \, \, (1\, -x)^{-1/4} \cdot \,
_2F_1\Bigl([{{1} \over {4}}, \, {{3} \over {4}}], \, [{{5} \over {4}}],
\, {{-x^2} \over { 4 \, \cdot \, (1 \, -x)}}\Bigr),
\end{eqnarray}
for $\, M\, = \, -1$:
\begin{eqnarray}
\label{Mm1}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad
\quad \quad \quad \quad \quad
\, \, = \, \, \, (1\, -x^2)^{-1/4} \cdot \,
_2F_1\Bigl([{{1} \over {4}}, \, {{3} \over {4}}], \, [{{5} \over {4}}],
\, {{-x^2} \over { 1 \, -x^2}}\Bigr),
\end{eqnarray}
and for $\, M\, = \, 1/2$:
\begin{eqnarray}
\label{M1over2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad
\quad \quad \quad \quad \quad \quad
\, \, = \, \, \, (1\, -2\, x)^{-1/4} \cdot \,
_2F_1\Bigl([{{1} \over {4}}, \, {{3} \over {4}}], \, [{{5} \over {4}}],
\, {{-x^2} \over { 1 \, -2\, x}}\Bigr).
\end{eqnarray}
Besides, the three previous values of $\, M \, = \, 1/k^2$
such that the Heun function
(or the inverse Jacobi sine, InverseJacobiSN in Maple) reduces
to pullbacked hypergeometric functions, correspond to a
{\em complex multiplication value} of
the $\, j$-function~\cite{j},
namely~\cite{Heegner} $\, j \, = \, \, (12)^3 \, = \, 1728$:
\begin{eqnarray}
\label{jfunc}
\hspace{-0.95in}&& \quad \quad
j \, \, = \, \, \,
256 \cdot \,{\frac { ({M}^{2}-M+1)^{3}}{ M^{2} \cdot \, (M \, -1)^{2}}},
\quad \quad \, \, \,
j \, =\, 1728 \,
\quad \longleftrightarrow \quad
M \, = \, \, \, 2, \,\, {{1} \over {2}}, \,\, -1.
\end{eqnarray}
The other complex multiplication values (Heegner numbers
see~\cite{Heegner}) do not seem to correspond to a reduction of
the Heun function to pullbacked hypergeometric functions.
\vskip .2cm
Recalling (\ref{M2}), (\ref{Mm1}), (\ref{M1over2}),
and specifying the Heun identity (\ref{Fdoublingidefirst}),
or (\ref{Ydoublingidentity}),
for $\, M\, = \, 2$, $\, M\, = \, -1$, and $\, M\, = \, 1/2 \, $
respectively, one gets three identities on the
hypergeometric function $\, _2F_1([1/4,3/4],[5/4],x)$. These three
identities are in fact consequences of the simple identity:
\begin{eqnarray}
\label{NEWident}
\hspace{-0.95in}&& \, Y(x) \, = \, \, \, \, x^{1/4} \cdot \,
_2F_1\Bigl([{{1} \over {4}}, \, {{3} \over {4}}], \, [{{5} \over {4}}], \, x\Bigr)
\, = \, \, \,
{{1} \over {2}} \cdot \, {\cal P}(x)^{1/4} \cdot \,
_2F_1\Bigl([{{1} \over {4}}, \, {{3} \over {4}}], \, [{{5} \over {4}}], \, {\cal P}(x) \Bigr),
\end{eqnarray}
where
\begin{eqnarray}
\label{pullNEWident}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
{\cal P}(x) \, \, = \, \, \,
16 \cdot \, \,{\frac {x \cdot \, (1 \, -x) }{ (1 \,+4\,x\,-4\,{x}^{2})^{2}}},
\end{eqnarray}
together with the ``transmutation'' relations
\begin{eqnarray}
\label{transmut}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \, \,
{\cal P}(p_k(x)) \, \, = \, \, \, p_k({\cal P}_k(x)),
\quad \quad \quad \quad
k \, \, = \, \, 1, \, 2, \, 3,
\end{eqnarray}
where the pullbacks $\, {\cal P}(p_k(z))$ are transformation
(\ref{Aadoubling}) for respectively $\, M\, = \, 2$,
$\, M\, = \, -1$, and $\, M\, = \, 1/2$
\begin{eqnarray}
\label{transmutwhere}
\hspace{-0.95in}&& \quad \quad \quad
{\cal P}_1(x) \, \, = \, \, \,
8 \cdot \,
{\frac { x \cdot \, (1\, - x) \cdot \, (2\, -x) }{ ({x}^{2}-2)^{2}}},
\quad \quad \quad
{\cal P}_2(x) \, \, = \, \, \,
4 \cdot \,{\frac {x \cdot \, (1\, -x^2) }{ (1 + \,{x}^{2})^{2}}},
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\quad
{\cal P}_3(x) \, \, = \, \, \,
4 \cdot \,{\frac {x \cdot \, (1\, -x)
\cdot \, (1 \, -2\,x) }{ (1 \, - 2\,{x}^{2})^{2}}},
\end{eqnarray}
and where $\, p_k(x)$ are the pullbacks emerging in the $\, _2F_1$
representations (\ref{M2}), (\ref{Mm1}), (\ref{M1over2}) of the
Heun function:
\begin{eqnarray}
\label{transmutwhere2}
\hspace{-0.95in}&& \, \quad
p_1(x) \, \, = \, \, \,
- {{1} \over {4}} \cdot \,{\frac {{x}^{2}}{1 \, -x}},
\quad \quad
p_2(x) \, \, = \, \, \,
-{\frac {{x}^{2}}{1 \, -{x}^{2}}},
\quad \quad
p_3(x) \, \, = \, \, \,
-{\frac {{x}^{2}}{1 \, -2\,x}}.
\end{eqnarray}
The hypergeometric function $\, Y(x)$ given by (\ref{NEWident}),
is solution of the order-two linear differential operator
$\,\, \Omega \, = \, \, (D_x \, + \, A_R(x)) \cdot \, D_x \,\,$
where
\begin{eqnarray}
\label{Y2F11over4Az}
\hspace{-0.95in}&& \quad \, \quad \quad \quad \quad \quad \quad
A_R(x) \, \, = \, \, \,
{{3} \over {4}} \cdot \, {\frac {1 \, -2\,x}{x \cdot \, (1 \, -x) }},
\end{eqnarray}
verifies the rank-two condition (\ref{mad}):
\begin{eqnarray}
\label{Y2F11over4Rota}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
A_R({\cal P}(x)) \cdot \, {\cal P}'(x)^2
\,\, = \,\, \, \,\, A_R(x) \cdot \, {\cal P}'(x) \, \, \,
+ \, {\cal P}''(x).
\end{eqnarray}
One notes that the hypergeometric functions (\ref{more1}) and
(\ref{NEWident}) are associated with the same $\, A_R(x)$
given by (\ref{Y2F11over4Az}): one can easily show
that these two hypergeometric
functions are equal. Therefore (\ref{NEWident}) shares the
same rank-two condition (\ref{mad}) with (\ref{Y2F11over4Az}),
a condition that is also verified for the rational
transformation (\ref{more1pull}) together
with the pullback (\ref{pullNEWident}), with (\ref{more1pull})
and (\ref{pullNEWident}) {\em commuting}. The hypergeometric
function (\ref{NEWident}) verifies an identity with the pullback
(\ref{more1pull}), namely $\, Q(R(x)) \, = \, \, -4 \cdot \, Q(x)$
with $\, R(x)$ given by (\ref{more1pull}), where
$\, Q(x) \, = \, \, Y(x)^4$.
\vskip .2cm
{\bf Remark:} For these selected values of $\, M$,
one could be surprised that the function $\, Q(x)$ in the case of the
Heun function is such that $\, Q(x) \, = \, Y(x)^2$, when the $\, Q(x)$
for the hypergeometric function (\ref{NEWident}) closely related to this
Heun function (see identities (\ref{M2}), (\ref{Mm1}), (\ref{M1over2}))
is such that $\, Q(x) \, = \, Y(x)^4$. This difference comes from the
pullbacks (\ref{transmutwhere2}): the pullbacked hypergeometric
functions (\ref{M2}), (\ref{Mm1}), (\ref{M1over2}) also
correspond to $\, Q(x) \, = \, Y(x)^2$.
\vskip .2cm
\subsection{A comment on the non globally bounded character of the Heun function }
\label{moreHeuncomment}
Heun functions with generic parameters
are generally not reducible to $\, _2F_1$ hypergeometric functions
with one or several pullbacks\footnote[5]{This corresponds
to the emergence of a modular form represented
as a $\, _2F_1$ hypergeometric functions with two
possible pullbacks~\cite{Christol}:
the series expansion can be recast into a series with
{\em integer coefficients}~\cite{Kratten}.}.
Unlike $\, _2F_1$ functions the corresponding linear
differential Heun operators are generally not globally nilpotent,
and the series of Heun functions are
not globally bounded. While, for Heun functions, the reducibility
to pullbacked $\, _2F_1$ hypergeometric functions,
the global nilpotence, and the global boundedness implicate
each other in general, this is not true
when the corresponding linear differential operator {\em factors}.
Note that the series (\ref{Fdoubling}) as well as the
series $\, Q(x)\, = \, \, Y(x)^N$ for the various hypergeometric
functions ((\ref{vid}), (\ref{YM3first}), (\ref{YM6first}),
... with $\, N= \, 4, \, 3, \, 6$, ...)) are
{\em not globally bounded}\footnote[2]{A globally bounded
series is a series that can be recast into a series
with {\em integer coefficients}~\cite{Christol}.}.
\vskip .1cm
In this light, the fact that the series (\ref{Fdoubling})
as well as the series $\, Q(x)\, = \, \, Y(x)^N$
are not globally bounded, does not seem to be in agreement with
the previous modular form emergence and the previous remarkable
identities (\ref{Ydoublingidentity}), or
$\,\, Q(R(x)) \, = \, \, 4 \cdot \, Q(x)$. The series
\begin{eqnarray}
\label{Gdoubling}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
G(x) \,\, = \, \, \, \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, \, 4 \, M \, x\Bigr),
\end{eqnarray}
might not be globally bounded, yet it is ``almost globally bounded'':
the denominator of the coefficients of $\, x^n$ are of the form
$\, 2 \, n \, +1$. Therefore one finds that the
closely related series
\begin{eqnarray}
\label{Udoubling}
\hspace{-0.96in}&& \, \, \,
{\tilde G}(x) \, \, = \, \, \, \,
2 \, \, x \cdot \, G'(x) \,\, + \, \, G(x)
\nonumber \\
\hspace{-0.96in}&& \,\,\,\, = \,
1\,
+ 2 \, (M+1) \cdot \, x
\,+ 2 \, (3\,{M}^{2} +2\,M +3) \cdot \, {x}^{2}
\,+4\, (M+1) \, (5\,{M}^{2} -2\,M +5) \cdot \, {x}^{3}
\nonumber \\
\hspace{-0.96in}&& \quad \quad \quad \quad
\,+ \, 2 \cdot \, (35\,{M}^{4} +20\,{M}^{3} +18\,{M}^{2} +20\,M +35)
\cdot \, {x}^{4}
\,\, \, \, + \,\, \,\cdots
\end{eqnarray}
is actually globally bounded for any rational number
value of $\, M$: the coefficient of $\, x^n$ is a
polynomial in $\, M$ {\em with integer coefficients}
of degree $\, n$. ${\tilde G}(x)$ is solution of
an order-one linear differential operator
and is an algebraic
function:
$\, {\tilde G}(x) \,\, = \, \,\, \,$
$ (1 \, -4 \, x)^{-1/2} \, \cdot \, (1 \, -4 M \, x)^{-1/2}$.
Thus the series (\ref{Udoubling}) is globally bounded
for any rational number $\, M$.
\vskip .1cm
{\bf Remark:} To be globally bounded~\cite{Christol} is a property
that is preserved by operator homomorphisms: the transformation
by a linear differential operator of
a globally bounded series is also globally bounded, however,
{\em it is not preserved by integration}.
\vskip .1cm
\vskip .1cm
\section{$\, _2F_1$ hypergeometric function: a higher genus case }
\label{more2F1higher}
The $\, _2F_1$ hypergeometric examples (\ref{vid}),
(\ref{YM3first}), (\ref{YM6first}),
and (\ref{zero}) are associated with
{\em elliptic or rational} (see (\ref{zero}))
{\em curves}. It is tempting to imagine the rank-two conditions (\ref{mad})
to be {\em only} associated with hypergeometric functions connected to
{\em elliptic curves}, and with pullbacks given by
{\em rational functions}\footnote[1]{Casale showed in~\cite{Casale} that the only
{\em rational} functions from $\mathbb{P}_1$ to $\mathbb{P}_1$ with a non-trivial
$\, {\cal D}$-envelope are Chebyshev polynomials
and {\em Latt\`es transformations}. Latt\`es transformations
are rational transformations associated with elliptic
curves (see for instance~\cite{Eremenko2}).}.
This is not the case though, as we shall see in the next {\em genus-two}
hypergeometric example with {\em algebraic} function pullbacks.
\vskip .1cm
Let us consider the hypergeometric function
\begin{eqnarray}
\label{Ygenus}
\hspace{-0.95in}&& \, \, \, \, \,\, \,
Y(x) \,\, = \, \, \, \,
x^{1/6} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [{{7} \over {6}}], \, x \Bigr)
\,\, = \, \, \,\,
{{1} \over {6}} \cdot \, \int_0^{x} \, (1\, -t)^{-1/3} \cdot \, t^{-5/6} \cdot \, dt,
\end{eqnarray}
solution of the (factorized) order-two operator
$\, \Omega \, = $
$\, \, (D_x \, + \, A_R(x)) \cdot \, D_x$
where:
\begin{eqnarray}
\label{aAgenus}
\hspace{-0.95in}&& \,
A_R(x) \, = \, \, \,
{{1} \over { 6}} \,{\frac {7\,x \, -5}{x \cdot \, (x \, -1) }}
\, = \, \, \, {{u'(x)} \over {u(x)}}
\quad \, \, \, \, \, \hbox{where:} \quad \quad
u(x) \, \, = \, \, \, (1\, -x)^{1/3} \cdot \, x^{5/6},
\end{eqnarray}
and one gets $\, 6 \cdot \, Y'(x) \, = \, \, 1/u(x)$. Introducing
$\, u \, = \, \, 6 \cdot \, Y'(x)$, one can
canonically associate
to (\ref{aAgenus}) the algebraic curve
\begin{eqnarray}
\label{curvegenus}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
u^6 \, \,\, - \, (1\, -x)^2 \cdot \, x^5
\, \,\, = \, \,\, \, 0,
\end{eqnarray}
which is a {\em genus-two algebraic curve}.
We are seeking an identity on this
hypergeometric function (\ref{Ygenus}) of the form:
\begin{eqnarray}
\label{Ygenusrewrit}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
{\cal A}(x) \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [{{7} \over {6}}], \, x \Bigr)
\, \, = \, \, \, \,
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [{{7} \over {6}}], \, y(x) \Bigr).
\end{eqnarray}
Introducing the order-two linear differential operators
annihilating respectively the LHS and RHS of (\ref{Ygenusrewrit}),
the identification of the wronskians of these two operators
gives the algebraic function $\, {\cal A}(x)$ in terms
of the pullback $\, y(x)$:
\begin{eqnarray}
\label{YgenusrewritcalA}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
{\cal A}(x)\, \,\, = \, \, \,
\Bigl({{ -27 \cdot \, x} \over { y(x)}}\Bigr)^{1/6}.
\end{eqnarray}
The pullback $\, y(x)$ must be some symmetry (isogeny) of the
genus-two algebraic curve (\ref{curvegenus}). At first sight, this
{\em seems to exclude rational function} pullbacks similar to
the ones previously introduced. In fact, remarkably, there
exists a simple identity on this
(higher genus) hypergeometric function:
\begin{eqnarray}
\label{identityparam2}
\hspace{-0.95in}&& \quad \quad \, \, \,
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [{{7} \over {6}}], \,\, \,
-27 \cdot \, \,
{\frac {v \cdot \, (1-v) \cdot \, (1+v)^{4}}{
(1 \, +3\,v) \cdot \, ( 1 \, -3\,v)^{4}}} \Bigr)^6
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad
\, \, \, \, = \, \, \, \, \,
{\frac { (1 \, +3\,v)^2 \cdot \, (1 \, -3\,v)^4 }{
(1 \, -v)^2 \cdot \,(1+v)^{4} }} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [{{7} \over {6}}], \, \, \,
{\frac {v \cdot \, (1 \, +3\,v) }{1 \, -v}} \Bigr)^6.
\end{eqnarray}
The two pullbacks in this remarkable identity (\ref{identityparam2})
yield the simple rational parametrization
\begin{eqnarray}
\label{param}
\hspace{-0.95in}&& \quad \quad \quad \quad
x \, \, = \, \, \, {\frac {v \cdot \, ( 1 \, +3\,v) }{1 \, -v}},
\quad \quad \quad
y \, \, = \, \, \,
-27 \cdot \,{\frac {v \cdot \, (1 \, -v)
\cdot \, (1 \, +v)^{4}}{ (1 \, +3\,v) \cdot \, ( 1 \, -3\,v)^{4}}},
\end{eqnarray}
which parametrizes the following
{\em genus-zero (i.e. rational) curve}\footnote[9]{One
should not confuse these two algebraic curves: the genus-two
curve (\ref{curvegenus}) is associated with integrant of the
hypergeometric integral (\ref{Ygenus}), when
the rational curve (\ref{curverationalgenus}) is associated with
the pullback in the hypergeometric identity (\ref{Ygenusrewrit}).
}:
\begin{eqnarray}
\label{curverationalgenus}
\hspace{-0.95in}&& \quad \quad
-27\, \cdot \, x \cdot \, (x \, -1)^{4} \cdot \, ({y}^{2} \, +1) \, \,
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \,
- \, ({x}^{6} -12\,{x}^{5}
+807\,{x}^{4} +2504\,{x}^{3} +807\,{x}^{2} -12\,x +1)
\cdot \, y
\, \, \, = \, \, \, \, 0.
\end{eqnarray}
The algebraic function $\, y \, = \, y(x)$, defined by the
genus-zero curve (\ref{curverationalgenus}),
is an example of a
pullback\footnote[5]{This is a consequence of identity (\ref{identityparam2}).}
$\, y(x)$ occurring in
the $\, _2F_1$ hypergeometric identity (\ref{Ygenusrewrit}).
This (multivalued) algebraic function $\, y \, = \, y(x)$
has the following series expansions:
\begin{eqnarray}
\label{seriescurverationalgenus}
\hspace{-0.95in}&& \quad \quad
y_1 \, \, \, = \, \, \,
-27\,x \,\, \, -216\,{x}^{2}\, \,
-648\,{x}^{3}\, \,\, -1944\,{x}^{4}\,\,
-648\,{x}^{5}\,\, -27864\,{x}^{6}\, +203256\,{x}^{7}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad\,
-2123928\,{x}^{8}\,\, +21844728\,{x}^{9}\,\, -233611992\,{x}^{10}
\, \, \, + \, \, \, \cdots
\\
\hspace{-0.95in}&& \quad \quad
\label{y2genus}
y_2 \, \, \, = \, \, \, \, -{{1} \over { 27 \, x}}
\,\, +{\frac {8}{27}}\,\,\, -{\frac {40\,x}{27}}\,\,
+{\frac {200\,{x}^{2}}{27}}\,\,
-{\frac {1192\,{x}^{3}}{27}}\,\,
+{\frac {8456\,{x}^{4}}{27}}\, \, -{\frac {68264\,{x}^{5}}{27}}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \,\,
+{\frac {604360\,{x}^{6}}{27}} \,\, \, -{\frac {5722664\,{x}^{7}}{27}}
\, \,\, +{\frac {6332872\,{x}^{8}}{3}} \,
\, \,\, + \, \, \cdots
\end{eqnarray}
Note that the rational curve (\ref{curverationalgenus}) has
the obvious symmetry $\, y \, \leftrightarrow \, \, 1/y$ (as well
as the $\, x \, \leftrightarrow \, \, 1/x$ symmetry, consequence
of the palindromic form of (\ref{curverationalgenus})), therefore
the series (\ref{y2genus}) is
the reciprocal of
(\ref{seriescurverationalgenus}): $\, y_2 \, = \, 1/y_1$.
Clearly $\, x$ and $\, y$ are not on the same footing.
The composition inverse of the previous series
gives the series
\begin{eqnarray}
\label{seriesreversiongenus}
\hspace{-0.95in}&& \quad
{{ - \, y} \over { 27}} \, \, \, -{\frac {8\,{y}^{2}}{729}} \, \, \,
-{\frac {104\,{y}^{3}}{19683}} \, \, \,
-{\frac {1672\,{y}^{4}}{531441}} \, \, \,
-{\frac {30248\,{y}^{5}}{14348907}} \, \, \,
-{\frac {196568\,{y}^{6}}{129140163}} \,\, \,
\, + \, \, \, \cdots
\\
\hspace{-0.95in}&& \quad
\label{reverse2}
{{-27} \over {y}} \, \, +8 \, \,+{\frac {40\,y}{27}} \, \,
+{\frac {520\,{y}^{2}}{729}} \, \,
+{\frac {8552\,{y}^{3}}{19683}} \,
\,+{\frac {158344\,{y}^{4}}{531441}} \, \,
+{\frac {3151144\,{y}^{5}}{14348907}} \,\,
\, + \, \, \cdots,
\end{eqnarray}
the second series being the reciprocal of the first
one\footnote[9]{Note that the rational curve (\ref{curverationalgenus})
provides additional Puiseux series.}.
\vskip .1cm
Furthermore the two series
(\ref{seriescurverationalgenus}) and (\ref{y2genus})
verify\footnote[1]{These two series are related by
$\, y \, \leftrightarrow \, \, 1/y$. Note that
$\, y \, \leftrightarrow \, \, 1/y$ is not a symmetry
of (\ref{madgenus}) in general.
} the rank-two condition (\ref{mad}) with $\, A_R(x)$ given by
(\ref{aAgenus}):
\begin{eqnarray}
\label{madgenus}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
A_R(y(x)) \cdot \, y'(x)^2
\,\, = \,\, \, \,\,
A_R(x) \cdot \, y'(x) \, \, \, + \, y''(x).
\end{eqnarray}
Do note that the series, corresponding to the
composition inverse of
these two series (\ref{seriescurverationalgenus})
and (\ref{y2genus}) (namely (\ref{seriesreversiongenus})
and (\ref{reverse2}) where one changes $\, y$ into $ \, x$),
also verify the rank-two condition (\ref{mad}) with $\, A_R(x)$ given by
(\ref{aAgenus}). For example, introducing $\, Q(x) \, =\, \, Y(x)^6$,
one finds that $\, Q(y(x)) \, \, =\, \, \, -27 \cdot \, Q(x)$
for $\, y(x)$ the algebraic function corresponding to series
(\ref{seriescurverationalgenus}). The
composition inverse of series\footnote[2]{Namely series
(\ref{seriesreversiongenus}) where one changes
$\, y$ into $ \, x$.} (\ref{seriescurverationalgenus})
gives the (reversed) result:
$\, Q(x) \, =\, \, - 27 \cdot \, Q(y(x))$.
\vskip .1cm
\vskip .1cm
{\bf Remark:} The rank-two condition (\ref{madgenus})
with $\, A_R(x)$ given by (\ref{aAgenus}) has a one-parameter
family of {\em commuting} solution series:
\begin{eqnarray}
\label{madseriesgenus}
\hspace{-0.95in}&& \quad \quad
R(a, \, x) \, \, = \, \, \, \,
a \cdot \, x \, \,\, + a \cdot (a-1) \cdot \, S_a(x)
\quad \quad \ \quad \quad \hbox{where:}
\\
\hspace{-0.95in}&& \quad \quad
S_a(x) \, \, = \, \, \, \, -{{2} \over {7}} \cdot \, x^2 \, \,
+{{17\,a \, -87} \over {637}} \cdot \, x^3 \,\,
+{{2 \cdot \, (113\,a^2 \,-856\,a \,+3438)} \over {84721}} \cdot \, x^4
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
-{{3674\,a^3 +121194\,a^2 -552261\,a +2095059 } \over {38548055}} \cdot \, x^5
\, \,\,\, + \, \, \, \cdots
\end{eqnarray}
with $\,\,\, R(a_1, \, R(a_2, \, x)) \, = \, \, $
$R(a_2, \, R(a_1, \, x)) \, = \, \, R( a_1\,a_2, \, \, x)$,
where (\ref{madseriesgenus}) reduces to the algebraic series
(\ref{seriescurverationalgenus}) and (\ref{seriesreversiongenus})
for $\, a \, = \, -27$ and $\, a \, = \, -1/27$
respectively. Consequently the occurrence of a {\em higher genus}
curve like (\ref{curvegenus})
is not an obstruction to the existence of a family of
one-parameter {\em abelian} series.
\vskip .2cm
\section{Schwarzian condition on an algebraic
transformation: $\, _2F_1$ representation of a modular form}
\label{Schwarz}
The typical situation emerging in physics with
{\em modular forms}~\cite{IsingCalabi,IsingCalabi2,Diagselect}
is that some ``selected'' hypergeometric
function $\,\, _2F_1([\alpha, \, \beta], \, [\gamma], \, x)$
verifies an identity with
{\em two different pullbacks}\footnote[5]{The modular
forms occuring in physics often correspond to cases
where the two different pullbacks $\, p_1(x)$ and $\, p_2(x)$
are rational functions, but they can also be
algebraic functions~\cite{Christol,Morain}.
} related
by an {\em algebraic} curve, the {\em modular equation} curve
$\,\, M(p_1(x), \,p_2(x)) \, = \, \, \,0$:
\begin{eqnarray}
\label{modularform}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, p_1(x) \Bigr)
\, = \, \, \, \,\, {\tilde A}(x) \cdot \,
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, p_2(x) \Bigr),
\end{eqnarray}
where $\, {\tilde A}(x)$ is an algebraic function.
This representation of {\em modular forms} in terms of
hypergeometric functions with {\em many pullbacks}, is well
described in Maier's papers~\cite{SuperMaier,BelyiMaier}. It
is different from the ``mainstream'' mathematical definition of
modular forms as (complex) analytic functions on the upper half-plane
satisfying functional equations with respect to the group action
of the modular group. {\em However, this hypergeometric
representation is the one we do need in
physics}~\cite{IsingCalabi,IsingCalabi2}. The reason why
this hypergeometric function representation of modular forms
exists is a consequence of a not very well-known equality
between the Eisenstein~\cite{Sebbar} series $\, E_4$
(of weight four under the modular group), and a
hypergeometric function of the (weight zero) modular
$\,j$-invariant~\cite{Heegner,Canada}
(see Theorem 3 page 226 in~\cite{Stiller}, see also page 216 of~\cite{Shen}):
\begin{eqnarray}
\label{paradox}
\hspace{-0.95in}&& \quad \quad
E_4(\tau) \, \, \, = \, \, \, \,
1 \, + \, 240 \, \sum_{n=1}^{\infty} \, n^3\cdot \, {{q(\tau)^n} \over {1-q(\tau)^n}}
\, \, \, = \, \, \, \,
_2F_1\Bigl([{{1} \over {12}}, \, {{5} \over {12}}],
\, [1], {{1728} \over {j(\tau)}}\Bigr)^4.
\end{eqnarray}
In terms of $\, k$ the modulus of the elliptic functions, the $\, E_4$
Eisenstein series (\ref{paradox}) can also be written as:
\begin{eqnarray}
\label{paradoxE4}
\hspace{-0.95in}&&
_2F_1\Bigl([{{1} \over {12}}, {{5} \over {12}}], [1], \,
{{27} \over {4}} \,
{\frac { {k}^{4} \cdot \, (1 \, -{k}^{2})^{2}}{ ({k}^{4}-{k}^{2}+1)^{3}}}\Bigr)^4
\, = \,\, (1-k^2+k^4) \cdot \, _2F_1\Bigl([{{1} \over {2}}, {{1} \over {2}}],
[1], \, k^2\Bigr)^4.
\end{eqnarray}
Another relation between hypergeometric functions and modular forms
corresponds to the representation of the Eisenstein series $\, E_6$ in
terms of the hypergeometric functions\footnote[9]{One easily verifies
that the expressions (\ref{paradoxE4}) and (\ref{paradoxE6})
for respectively $\, E_4$ and $\, E_6$, are
such that $\, (E_4^3 \, -E_6^2)/E_4^3 \, $ is actually the well-known
expression of the Hauptmodul $\, 1728/j$ given as a rational function
of the modulus $\, k$ (see (\ref{jjprime}) and
(\ref{Haupt})).
} (\ref{paradoxE4}) (see page 216 of~\cite{Shen}):
\begin{eqnarray}
\label{paradoxE6}
\hspace{-0.95in}&& \quad \quad
E_6 \, \, = \,\, \, \,
(1+k^2)\cdot \, (1 \, -2\,k^2) \cdot \, \Bigl(1 \, -{{k^2} \over {2}}\Bigr)
\cdot \, _2F_1\Bigl([{{1} \over {2}}, \, {{1} \over {2}}],\, [1], \, k^2\Bigr)^6
\\
\label{paradoxE6bis}
\hspace{-0.95in}&& \quad \quad \quad \quad
\, \, \,\, = \,\, \, \,
(1+k^2) \cdot \, (1 \, -2\,k^2) \cdot \, \Bigl(1 \, -{{k^2} \over {2}}\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
\, \, \,\, \,\, \, \,
\times (1-k^2+k^4)^{-3/2} \cdot \,
_2F_1\Bigl([{{1} \over {12}}, {{5} \over {12}}], [1], \,
{{27} \over {4}} \,
{\frac { {k}^{4} \cdot \, (1 \, -{k}^{2})^{2}}{ ({k}^{4}-{k}^{2}+1)^{3}}}\Bigr)^6.
\end{eqnarray}
One can rewrite a remarkable hypergeometric identity
like (\ref{modularform}) in the form
\begin{eqnarray}
\label{modularform2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
{\cal A}(x) \cdot \,
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, x \Bigr)
\, = \, \, \, \,\,
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, y(x) \Bigr),
\end{eqnarray}
where $\, {\cal A}(x)$ is an algebraic function
and where $\, y(x)$ is an algebraic function
corresponding to the previous modular curve
$\,\, M(x, \, y(x)) \, = \,\, \, 0$.
The Gauss hypergeometric function
$\,\, _2F_1([\alpha, \, \beta], \, [\gamma], \, x) \, $
is solution of the second order linear
differential operator\footnote[1]{Note that $\, A(x)$ is
the log-derivative of
$\, \, u(x) \, \,= \,\, \, x^{\gamma} \cdot \, (1 \, -x)^{\alpha+\beta+1-\gamma}$.}:
\begin{eqnarray}
\label{Gaussdiff}
\hspace{-0.95in}&& \quad \quad \quad \quad
\Omega \,\, = \, \, \, \,
D_x^2 \, \, + \, A(x) \cdot \, D_x \, \, + \, B(x),
\quad \quad \quad \quad \quad \hbox{where:}
\nonumber \\
\hspace{-0.95in}&& \quad
A(x) \,\, = \, \, \,
{{ (\alpha +\beta+1) \cdot \, x \, \, -\gamma} \over { x \cdot \, (x\, -1)}}
\, \, = \, \, \, {{u'(x)} \over { u(x)}},
\quad \quad \quad
B(x) \,\, = \, \, \, {{\alpha \, \beta } \over {x \cdot \, (x\, -1) }}.
\end{eqnarray}
We would like now, to identify the two order-two linear differential
operators of the LHS and RHS of identity (\ref{modularform}).
A straightforward calculation
enables us to find the algebraic function $\, {\cal A}(x)$
in terms of the algebraic function pullback $\, y(x)$
in (\ref{modularform2}):
\begin{eqnarray}
\label{modularform3}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
{\cal A}(x) \, \, = \, \, \,
\Bigl( {{u(x)} \over { u(y(x)) }}
\cdot \, y'(x) \Bigr)^{-1/2}.
\end{eqnarray}
Expression (\ref{modularform3}) for $\, {\cal A}(x)$
is such that the two order-two linear differential operators
(of a similar form as (\ref{Gaussdiff}))
have the same $\, D_x$ coefficient. The identification
of these two operators thus corresponds
(beyond (\ref{modularform3})) to just one condition that can
be rewritten (after some algebra ...) in the following
Schwarzian form:
\begin{eqnarray}
\label{condition1}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
or:
\begin{eqnarray}
\label{condition}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
{{W(x)} \over {y'(x)}}
\, \, \, \,-W(y(x)) \cdot \, y'(x)
\, \, \, \,+ \, {{ \{ y(x), \, x\} } \over {y'(x)}}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
where
\begin{eqnarray}
\label{wherecond}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
W(x) \, \, = \, \, \, \, \, A'(x) \, \, \, + \, \,
{{A(x)^2} \over {2 }} \, \, \, \, -2 \cdot \, B(x),
\end{eqnarray}
and where $\,\{ y(x), \, x\}$ denotes the
{\em Schwarzian derivative}~\cite{What}:
\begin{eqnarray}
\label{Schwa}
\hspace{-0.95in}&& \, \, \, \,
\{ y(x), \, x\} \, \, = \, \, \, \, \,
{{y'''(x) } \over{ y'(x)}}
\, \, - \, \, {{3} \over {2}}
\cdot \, \Bigl({{y''(x)} \over{y'(x)}}\Bigr)^2
\, \, = \, \, \, \, \,
{{ d } \over { dx }} \Bigl( {{y''(x) } \over{ y'(x)}} \Bigr)
\, \, - {{1} \over {2}} \cdot \, \Bigl( {{y''(x) } \over{ y'(x)}} \Bigr)^2.
\end{eqnarray}
In the identity (\ref{modularform}), characteristic of modular forms,
the two pullbacks $\, p_1(x)$ and $\, p_2(x)$ are clearly on the
{\em same footing}, while identity (\ref{modularform2}) breaks
this fundamental symmetry, seeing $\, y$ as a function of
$\, x$. We can perform the same calculations
seeing the variable $\, x$ as a function of $\, y$
in (\ref{modularform2}).
Despite the simplicity of condition (\ref{condition})
it is not clear whether $\, x$ and $\, y$ are on the same footing
in condition (\ref{condition}). This is actually the case,
since if one considers $\, x$ as a function of $\, y$, we
have the well-known classical
result that the Schwarzian derivative of $\, x$ with
respect to $\, y$ is simply related to (\ref{Schwa}),
the Schwarzian derivative of $\, y$ with respect to $\, x$:
\begin{eqnarray}
\label{Schwarew}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\{ y(x), \, x\} \, \, \, = \, \, \, \,
-y'(x)^2 \cdot \, \{ x(y), \, y\}.
\end{eqnarray}
In other words, if one introduces the following Schwarzian bracket
\begin{eqnarray}
\label{Schwabis}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
[ y, \, x ] \, \, = \, \, \, \, \,
{{ \{ y(x), \, x\} } \over {y'(x)}} \, \, = \, \, \, \, \,
{{y'''(x) } \over{ y'(x)^2 }}
\, \, \, - \, \, {{3} \over {2}} \cdot \, {{y''(x)^2} \over{y'(x)^3}},
\end{eqnarray}
it is antisymmetric: $\, [ y, \, x ] \, = \, \, -\, [ x, \, y ]$.
With this appropriate notation,
$\, x$ and $\, y$ {\em can be seen on the same footing}.
With this in mind we can now rewrite condition (\ref{condition})
in a balanced way:
\begin{eqnarray}
\label{conditionNEW}
\hspace{-0.95in}&& \quad \quad \quad \quad
2 \cdot \, W(x) \cdot \, {{ d x} \over { dy}}
\, \,
\, \, \,+ \, [ y, \, x]
\, \,\, \, = \,\, \, \, \,
2 \cdot \, W(y) \cdot
\, {{ d y} \over { dx}} \, \, \, \, + \, [ x, \, y].
\end{eqnarray}
If one denotes by $\, L(x, \, y)$ the LHS of (\ref{conditionNEW})
\begin{eqnarray}
\label{L}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
L(x, \, y)\,\, \, = \,\, \, \, \,
{{ 2 \cdot \, W(x) \, + \, \, \{ y, \, x\} } \over {y'}},
\end{eqnarray}
the Schwarzian condition (\ref{condition}), or (\ref{conditionNEW}),
reads $\, \, L(x, \, y) \, = \, \, L(y, \, x)$.
Being the result of the covariance (\ref{modularform2}), a
Schwarzian identity like (\ref{conditionNEW}) {\em has to be compatible}
with the composition of functions. For instance,
from (\ref{modularform2}) one immediately deduces:
\begin{eqnarray}
\label{modularform22}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, y(y(x)) \Bigr)
\, = \, \, \, \,\,
{\cal A}(y(x)) \cdot \,
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, y(x) \Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad
\quad \quad \quad \quad \quad \quad \quad \quad
\, = \, \, \, \,\,
{\cal A}(y(x)) \cdot \, {\cal A}(x) \cdot \,
_2F_1\Bigl([\alpha, \, \beta], \, [\gamma], \, x \Bigr).
\end{eqnarray}
One thus expects condition (\ref{condition}) to be compatible with
the composition of function (similarly to the previous compatibility
of the rank-two condition (\ref{mad}) with the iteration
of $\, x \, \rightarrow \, R(x)$): this is actually the case.
Recalling the (well-known) chain rule for the Schwarzian
derivative of the composition of functions
\begin{eqnarray}
\label{chainrule}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\{ z(y(x)), \, x\} \, \, \,= \, \, \, \, \,
\{ z(y), \, y \} \cdot y'(x)^2
\, \,\, + \, \, \{ y(x), \, x\},
\end{eqnarray}
it is straightforward to show directly (without referring to
the covariance (\ref{modularform2})) that condition (\ref{condition})
is actually compatible with the composition of functions
(see \ref{Schwarzcomp}
for a demonstration).
\vskip .1cm
The Schwarzian derivative is the perfect tool~\cite{What}
to describe the {\em composition of functions} and the
{\em reversibility} of an iteration (the previously mentioned
$\, x \, \longleftrightarrow \, y$
symmetry): it is not a surprise to see
the emergence of a Schwarzian derivative in the description
of the modular forms~\cite{Schwarzian2,Schwarzian,Chakra}
corresponding to identities like (\ref{modularform2}). We
are going to see, for a given (selected ...) hypergeometric
function $\,\, _2F_1([\alpha, \, \beta], \, [\gamma], \, x)$,
that the condition (\ref{condition}) ``encapsulates''
all the isogenies corresponding to all the modular equations
associated to transformations on the ratio of periods
$\, \tau \, \rightarrow \, N \cdot \, \tau$
(resp. $\, \tau \, \rightarrow \, \, \tau/N$),
for various values of the integer $\, N$ corresponding
to the different modular equations.
\vskip .2cm
\subsection{Schwarzian condition and the simplest example of modular forms: a series viewpoint}
\label{Schwarzsimplest}
Let us focus on an example of a modular form that emerged
many times in the analysis of $\, n$-fold integrals of the
square Ising model~\cite{broglie,bo-ha-ma-ze-07b,Heegner,Christol}. Let
us recall the simplest example of a modular form and of
a modular equation curve
\begin{eqnarray}
\label{modularform2explicit}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
{\cal A}(x) \cdot \,
_2F_1\Bigl([{{1} \over {12}}, \, {{5} \over {12}}], \, [1], \, x \Bigr)
\, = \, \, \, \,\,
_2F_1\Bigl([{{1} \over {12}}, \, {{5} \over {12}}], \, [1], \, y \Bigr),
\end{eqnarray}
where $\, {\cal A}(x)$ is an algebraic function
and where $\, y \, = \, \, y(x)$ is an algebraic function
corresponding to the modular equation (\ref{modularcurve}).
The algebraic function $\, y \, = \, \, y(x)$ is a
multivalued function, but we can single out the series
expansion\footnote[1]{This series (\ref{seriesmodularcurve})
has a radius of convergence $\, 1$, even if the discriminant
of the modular equation (\ref{modularcurve}) which vanishes
at $\, x \, = \, 1$, vanishes for values inside the
unit radius of convergence, for instance at
$\, x \, = \, -64/125$.}:
\begin{eqnarray}
\label{seriesmodularcurve}
\hspace{-0.95in}&& \quad \quad \quad
y \, \, \, = \, \, \, \, \,
{\frac {1}{1728}} \cdot \, {x}^{2} \, \, \,
+{\frac {31}{62208}} \cdot \, {x}^{3}\,
\, +{\frac {1337}{3359232}} \cdot \,{x}^{4} \, \,\,
+{\frac {349115}{1088391168}} \cdot \,{x}^{5}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
+{\frac {20662501}{78364164096}} \cdot \, {x}^{6} \,\, \,
+{\frac {1870139801}{8463329722368}} \cdot \,{x}^{7} \, \,
\, + \, \, \cdots
\end{eqnarray}
One verifies easily that the Schwarzian
condition (\ref{condition}) is verified with:
\begin{eqnarray}
\label{whereconda}
\hspace{-0.95in}&&
W(x) \, = \,
-{\frac {32\,{x}^{2}-41\,x+36}{72 \cdot \, {x}^{2} \cdot \, (x \, -1)^{2}}},
\, \,
A(x) \, = \,
{{ 3 \cdot \, x \, \, -2} \over { 2 \, x \cdot \, (x\, -1)}},
\, \,
B(x) \, = \,
{{ 5} \over { 144 \, x \cdot \, (x\, -1) }}.
\end{eqnarray}
\subsubsection{Other algebraic transformations: other modular equations \\}
\label{other}
\vskip .1cm
\vskip .1cm
\vskip .1cm
The {\em modular equations} $\, {\cal M}_N(x, \, y) \, = \, \, 0$,
corresponding to the transformation
$\, \tau \, \rightarrow \, N \cdot \, \tau$,
or $\, \tau \, \rightarrow \, \, \tau/N$, define algebraic
transformations (isogenies) $\, x \, \, \rightarrow \, \, y$
for the identity (\ref{modularform2explicit})
with $\, {\cal A}(x)$ given by an algebraic function.
Let us consider another important modular equation.
The modular equation of order three corresponding to
$\, \, \tau \, \rightarrow \, 3 \cdot \, \tau$,
or $\, \tau \, \rightarrow \, \, \tau/3$,
reads\footnote[5]{Legendre already knew (1824) this
order three modular equation in the form
$\, (k \lambda)^{1/2} + (k' \lambda')^{1/2}
= 1$, where $\, k$ and $\, k'$, and $\,\lambda$, $\,\lambda'$
are pairs of complementary moduli $\, k^2+k'^2=1$,
$\, \lambda^2+\lambda'^2=1$, and Jacobi derived that
modular equation~\cite{Jacobi,Nova}.}:
\begin{eqnarray}
\label{orderthree}
\hspace{-0.95in}&& \quad \quad \quad \quad
{k}^{4} \,\, +12\,{k}^{3}\lambda \, \, \, +6\,{k}^{2}{\lambda}^{2} \,\,
+12\,k{\lambda}^{3}\,\, +{\lambda}^{4} \, \, \, \,
-16\,{k}^{3}{\lambda}^{3} \, -16\,k\lambda
\,\, \, \, = \, \,\, \, 0.
\end{eqnarray}
Recalling that
\begin{eqnarray}
\label{orderthreexy}
\hspace{-0.95in}&& \,
x \,\, = \, \,\, {{27} \over {4}} \cdot \,
{\frac { {k}^{4} \cdot \, (1 \, -{k}^{2})^{2}}{ ({k}^{4}-{k}^{2}+1)^{3}}}
\,\, = \, \,\, {{1728} \over {j(k)}}, \quad \,
y \, \, = \, \,\,{{27} \over {4}} \cdot \, {\frac {{\lambda}^{4} \cdot \,
\left( 1-{\lambda}^{2} \right)^{2}}{({\lambda}^{4}-{\lambda}^{2}+1)^{3}}}
\,\, = \, \,\, {{1728} \over {j(\lambda)}},
\end{eqnarray}
the modular equation
(\ref{orderthree}) becomes the modular curve:
\begin{eqnarray}
\label{orderthreemod}
\hspace{-0.95in}&& \quad
262144000000000 \cdot \, {x}^{3}{y}^{3} \cdot \, (x+y) \, \,
+4096000000 \cdot \, {x}^{2}{y}^{2}
\cdot \, (27\,{x}^{2}-45946\,xy+27\,{y}^{2})\,
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad
+15552000 \cdot \, xy \cdot \,
(x+y) \cdot \, ({x}^{2}+241433\,xy+{y}^{2})
\nonumber \\
\hspace{-0.95in}&& \quad \, \quad \quad
+729\,{x}^{4} \, -779997924\,{x}^{3}y \, \,
+1886592284694\,{x}^{2}{y}^{2} \, \, -779997924\,x{y}^{3} \,
+729\,{y}^{4}
\nonumber \\
\hspace{-0.95in}&& \quad \, \quad \quad
+2811677184 \cdot \,xy \cdot \, (x+y) \,
\, -2176782336 \cdot \,x\, y \, \, \, = \, \, \, \, 0.
\end{eqnarray}
which gives the series expansion:
\begin{eqnarray}
\label{orderthreey}
\hspace{-0.95in}&& \quad
y \,\, = \, \,\,\, {\frac {{x}^{3}}{2985984}}
\, \,\, +{\frac {31 \,x^4 }{71663616}} \, \,
+{\frac {36221\, x^5 }{82556485632}} \,
\,\, +{\frac {29537101 \, {x}^{6}}{71328803586048}}
\,\, \, \, + \, \,\, \cdots
\end{eqnarray}
\vskip .2cm
One can easily get the the polynomial
with integer coefficients $\, {\cal M}_4(x, \, y)$, in the modular equation
$\, {\cal M}_4(x, \, y) \, = \, 0$ corresponding to the transformation
$\, \tau \, \rightarrow \, 4 \cdot \, \tau$,
or $\, \tau \, \rightarrow \, \, \tau/4$, as follows:
if one denotes by $\, {\cal M}_2(x, \, y)$ the LHS of the
modular equation (\ref{modularcurve}),
the polynomial $\, {\cal M}_4(x, \, y)$
is straightforwardly obtained by calculating the resultant
of $\, {\cal M}_2(x, \, z)$ and $\, {\cal M}_2(z, \, y)$
in $\, z$, which factorizes in the form\footnote[8]{The
exact expression of $\, {\cal M}_4(x, \, y)$ is a bit too large
to be given here.} $\, (x\, -y)^2 \cdot \, {\cal M}_4(x, \, y)$.
The modular equation $\,\, {\cal M}_4(x, \, y) \, = \, \, 0\, $
defines several algebraic series corresponding to the different
branches\footnote[1]{These series can be obtained using the
command ``algeqtoseries'' in the ``gfun'' package of Maple.
} of the (multivalued) algebraic function transformation
$\, x \, \rightarrow \, \, y$. We find Puiseux series
and two analytic series at $\, x=\,0$ given by
\begin{eqnarray}
\label{back}
\hspace{-0.95in}&& \quad \quad
y \, \, = \, \, \, \, {\frac {{x}^{4}}{5159780352}} \, \,
+{\frac {31\,{x}^{5}}{92876046336}} \, \,
+{\frac {43909\,{x}^{6}}{106993205379072}} \,
\,\,\, + \, \, \cdots
\end{eqnarray}
which is clearly similar to the previous series
(\ref{seriesmodularcurve}) and (\ref{orderthreey}),
but also an {\em involutive}\footnote[2]{The series (\ref{othersolution}) is the
{\em only involutive series} of the form $\,\, - x \, + \, \, \cdots \, $
{\em which verifies the Schwarzian condition} (\ref{condition}).
} series of radius of convergence $\, 1$, of the (quite unexpected)
simple form $\, \, -x \, + \, \cdots \, $ namely:
\begin{eqnarray}
\label{othersolution}
\hspace{-0.95in}&&
y \, \, = \, \, \, \,
-x \, \, \, \, -{\frac {31\,{x}^{2}}{36}}\, \,
-{\frac {961 }{1296}} \cdot \, x^3 \, \,
-{\frac {203713 }{314928}}\cdot \, x^4 \, \,
-{\frac {4318517 }{7558272}} \cdot \, x^5\, \,
-{\frac {832777775}{1632586752}}\cdot \, x^6\,
\nonumber \\
\hspace{-0.95in}&& \, \, \, \,
-{\frac {729205556393 }{1586874322944}} \cdot \,{x}^{7}
-{\frac {2978790628903 }{7140934453248}} \cdot \,{x}^{8}
-{\frac {43549893886943 }{114254951251968}} \cdot \,{x}^{9}
\, \, + \, \dots
\end{eqnarray}
One easily verifies that all these series (\ref{seriesmodularcurve}),
(\ref{orderthreey}), (\ref{back}), (\ref{othersolution})
(as well as the other Puiseux series) are solutions
of the Schwarzian condition
(\ref{condition}), and that the series (\ref{seriesmodularcurve}),
(\ref{orderthreey}), (\ref{back}) {\em commute} when composed, while (\ref{back})
and (\ref{othersolution}) {\em do not} ! This
is a consequence of the fact that they correspond to the various commuting
isogenies $\, \tau \, \rightarrow \, N \cdot \, \tau \, $
(resp. $ \tau \, \rightarrow \, \, \tau/N$).
\subsubsection{A one-parameter solution series of the Schwarzian condition\\}
\label{oneparam}
\vskip .1cm
\vskip .1cm
\vskip .1cm
\vskip .1cm
Let us first seek solution-series of the
Schwarzian condition (\ref{condition}) of the form
$\, \, e \cdot \, x \,\, + \, \cdots\, \, $
with $\, W(x)$ given by (\ref{whereconda}).
One finds that the Schwarzian condition
(\ref{condition}) has a {\em one-parameter family} of
solution-series as well of the form
$\, \, e \cdot \, x \, \, + \, \cdots\, \,\, $
namely\footnote[5]{The one-parameter series (\ref{seriesmodcurve1a})
is {\em completely defined} by the fact that it is a series of the form
$\, e \cdot \, x \, + \, \, \cdots \, $ commuting with
the algebraic series (\ref{othersolution})
and the hypergeometric series (\ref{seriesmodcurve1aeps}),
{\em without referring to the Schwarzian condition}
(\ref{condition}).}:
\begin{eqnarray}
\label{seriesmodcurve1a}
\hspace{-0.95in}&& \, \, \quad \, \quad \quad
y(e, \, x) \, \, \, = \, \, \, \,
e \cdot \, x \, \,\, \, + \, e \cdot \, (e-1) \cdot \, S_e(x),
\quad \quad \quad \quad \quad \hbox{where:}
\\
\hspace{-0.95in}&& \, \, \quad \quad \quad \quad \quad \, \,
S_e(x)\, \, = \, \, \, \,
-{\frac {31}{72}} \cdot \, {x}^{2} \,
\, \, \, +{\frac { (9907\,e -20845) }{82944}} \cdot \, {x}^{3}
\nonumber \\
\hspace{-0.95in}&& \, \, \, \, \, \,
\quad \quad \quad \quad \quad \quad \quad
\,\, -{\frac {
(4386286\,{e}^{2}-20490191\,e +27274051)
}{161243136}}\cdot \, {x}^{4}
\,\, \, \, \,+ \, \,\, \cdots
\end{eqnarray}
The series (\ref{seriesmodcurve1a}) is a one-parameter
family of {\em commuting series}:
\begin{eqnarray}
\label{seriesmodcurve1axx}
y(e, \, y(\tilde{e}, \, x))
\, \, \, = \, \, \, \,
y(\tilde{e}, \, y(e, \, x))
\, \, \, = \, \, \, \,y(e \, \tilde{e}, \, x),
\end{eqnarray}
and in the $\, e \, \rightarrow \, \, 1$ limit
of the one-parameter family (\ref{seriesmodcurve1a}),
one has:
\begin{eqnarray}
\label{seriesmodcurve1aeps}
\hspace{-0.95in}&& \quad \quad
y(e, \, x) \, \, \, = \, \, \, \,
x \, \, \, + \, \, \, \epsilon \cdot \, F(x)\, \, \,
+ \, \,\epsilon^2 \cdot \, G(x) \, \,\, \,+ \, \, \cdots
\qquad \quad \, \quad \, \, \hbox{where:}
\\
\label{holo1eps}
\hspace{-0.95in}&&
F(x)\, \, = \, \, x \cdot \, (1\, -x)^{1/2} \cdot \,
_2F_1\Bigl([{{1} \over{12}}, \, {{5} \over{12}}], \, [1], \, x\Bigr)^2,
\quad \,\,
G(x) \, = \, \,\,
{{1} \over {2}} \cdot \, F(x) \cdot \, (F'(x) \, -1).
\nonumber
\end{eqnarray}
\subsubsection{Other one-parameter solution series of the Schwarzian condition\\}
\label{oneparam}
\vskip .1cm
\vskip .1cm
\vskip .1cm
Clearly the analytic series (\ref{seriesmodularcurve}),
(\ref{orderthreey}), (\ref{back})
corresponding to the various isogenies
$\, \tau \, \rightarrow \, N \cdot \, \tau$,
are not series of the form
$\, \, e \cdot \, x \, + \, \cdots\, \, $, instead they are
solution-series of the
Schwarzian condition (\ref{condition}) of the form
$\, \, a \cdot \, x^N \, + \, \cdots\, \, $
In order to generalize the solution-series (\ref{seriesmodularcurve}),
we will first seek solution-series of the
Schwarzian condition (\ref{condition}) of the form
$\, \, a \cdot \, x^2 \, + \, \cdots\, \, $
A straightforward calculation gives a one-parameter
family of solution-series of (\ref{condition}) of the form
$\, \, a \cdot \, x^2 \, + \, \cdots\, \, $:
\begin{eqnarray}
\label{seriesmodcurvea}
\hspace{-0.96in}&&
y_2 \, \, = \, \, \,
a \cdot \,{x}^{2} \,
+{\frac {31 \cdot \, a{x}^{3}}{36}}
\,-{\frac {a \cdot \,
\left( 5952\,a-9511 \right) }{13824}}\cdot \, {x}^{4}
\,-{\frac {a \cdot \,
\left( 14945472\,a-11180329 \right) }{20155392}} \cdot \, {x}^{5}
\nonumber \\
\hspace{-0.96in}&& \quad \quad \,
\,+{\frac {a \cdot \,
\left( 88746430464\,{a}^{2}-677409785856\,a+338926406215 \right)
}{743008370688}} \cdot \, {x}^{6}
\, \, \,\, + \, \, \cdots
\end{eqnarray}
which actually reduces to (\ref{seriesmodularcurve}) for
$\, a \, = \, \, 1/1728$.
Similarly, one also finds a one-parameter
family of solution-series of (\ref{condition}) of the form
$\, \, b \cdot \, x^3 \, + \, \cdots\, \, $:
\begin{eqnarray}
\label{seriesmodcurve3a}
\hspace{-0.95in}&& \quad \quad \,
y_3 \, \, \, = \, \, \, \, \,
b \cdot \, {x}^{3} \, \,
+{\frac {31\,b }{24}} \cdot \, {x}^{4}\, \,
+{\frac {36221\,b }{27648}} \cdot \, {x}^{5} \, \, \,
-{\frac {b \cdot \, \left( 23141376\,b-66458485 \right)
}{53747712}} \cdot \, {x}^{6}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \,
\, -{\frac {b \cdot \, (183649959936\,b-187769367601)
}{165112971264}} \cdot \, {x}^{7}
\, \, \, \, + \, \, \cdots
\end{eqnarray}
which reduces to (\ref{orderthreemod}) for
$\, b \, = \, \, 1/2985984 \, = \, 1/1728^2$,
and another one-parameter
family of solution-series of (\ref{condition}) of the form
$\, \, c \cdot \, x^4 \, + \, \cdots\, \, $:
\begin{eqnarray}
\label{corresponding}
\hspace{-0.95in}&& \quad \quad \quad \quad
y_4 \, \, = \, \, \, \,\, c \cdot \, {x}^{4} \, \, \,
+{\frac {31\,c }{18}}\cdot \,{x}^{5}
\,\, \, +{\frac {43909\,c }{20736}} \cdot \,{x}^{6} \, \, \,
+{\frac {46242779\,c }{20155392}} \cdot \, {x}^{7}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \, \, \, \, \,
+{\frac {c \cdot \, (869687301215 -159953190912\,c) }{371504185344}}
\cdot \, {x}^{8}\,
\, \,\, \, + \, \, \, \cdots
\end{eqnarray}
which reduces to (\ref{back}) for
$\, c \, = \, 1/5159780352 \, = \, \,1/1728^3$.
The series (\ref{seriesmodcurvea}), (\ref{seriesmodcurve3a}),
(\ref{corresponding}), {\em do not commute}. The
composition of the one-parameter series (\ref{seriesmodcurve3a})
with the one-parameter series (\ref{seriesmodcurvea})
gives the series\footnote[2]{If one seeks for the solution series
of the Schwarzian condition (\ref{condition}) of the form
$\, d \cdot \, x^6 \, + \, \, \cdots \, \,$
one recovers the one-parameter family (\ref{curious2}).}:
\begin{eqnarray}
\label{curious2}
\hspace{-0.95in}&& \quad \quad \,\,
y_2(y_3(x)) \, \, = \, \, \, \, \, d \cdot \, {x}^{6}
\, \,\,
+{\frac {31\,d \cdot \, {x}^{7}}{12}} \, \,
\,+{\frac {59285\,d }{13824}} \cdot \, {x}^{8} \, \,
\,+{\frac {19676177\,d }{3359232}} \cdot \, {x}^{9}
\nonumber \\
\hspace{-0.95in}&& \quad \,\, \quad \quad \quad \quad
\,+{\frac {197722802303\,d }{27518828544}} \cdot \, {x}^{10} \, \,
\,\, +{\frac {8173747929317\,d }{990677827584}}\cdot \, {x}^{11}
\,\,\, \,\, + \,\,\, \cdots
\end{eqnarray}
where $\, d \, \, = \, \, a \cdot \, b^2$. The composition of
the one-parameter series (\ref{seriesmodcurvea})
with the one-parameter series (\ref{seriesmodcurve3a})
gives a similar result where, now,
$\, d \, \, = \, \, b \cdot \, a^3$.
These two one-parameter series commute when
$\, a \cdot \, b^2 \, \, = \, \, b \cdot \, a^3$, i.e.
$\, b \, = \, a^2$, and the modular equation series
corresponds to $\,\, b \, = \, a^2\,$ with $\, a \, = \, \, 1/1728$.
The composition of the one-parameter series (\ref{seriesmodcurvea})
with the one-parameter series (\ref{seriesmodcurve1a})
gives the series (\ref{seriesmodcurvea}) for $\, a \, e$ and
$\, a \, e^2 \, $ respectively:
\begin{eqnarray}
\label{curious5}
\hspace{-0.95in}&& \quad \quad \quad \quad
y_1(e, \, y_2(a, \, x)) \, \, = \, \, y_2(a \, e, \, x),
\qquad y_2(a, \,y_1(e, \, x)) \, \, = \, \, y_2(a \, e^2, \, x).
\end{eqnarray}
In other words if one introduces the modular equation series $ \, Y_2(x)$
given by (\ref{seriesmodularcurve}), corresponding to
$ \, y_2(a, \, x)$ for $\, a \, = \, \, 1/1728$,
the one-parameter series $\, y_2(a, \, x)$ given by (\ref{seriesmodcurvea}),
can be obtained as $\,\, y_1(1728 \,a, \, Y_2(x))\,$ or as
$\, \, Y_2(y_1((1728 \,a)^{1/2}, \, x))$.
\vskip .2cm
Therefore, all the one-parameter families (\ref{seriesmodcurvea}),
(\ref{seriesmodcurve3a}), (\ref{corresponding}), are nothing
but the {\em isogeny-series} (\ref{seriesmodularcurve}),
(\ref{orderthreey}), (\ref{back}) {\em transformed by the one-parameter series}
(\ref{seriesmodcurve1a}).
\vskip .2cm
\subsection{The equivalent of $\, P(z)$ and $ \, Q(z)$ for the Schwarzian condition: the mirror maps \\}
\label{Schwarzmirror}
\vskip .2cm
Let us recall the concept of
{\em mirror map}~\cite{IsingCalabi,IsingCalabi2,Candelas,Doran,Doran2,LianYau}
relating the reciprocal of the $\,j$-function and the nome,
with the well-known series with integer coefficients:
\begin{eqnarray}
\label{mirror}
\hspace{-0.95in}&& \quad
\tilde{X}(q) \, \, = \, \, \, \, q \,\, \, -744\,{q}^{2}
\,\, +356652\,{q}^{3} \,\, -140361152\,{q}^{4} \,\, +49336682190\,{q}^{5}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \,
-16114625669088\,{q}^{6} \,
\, +4999042477430456\,{q}^{7} \, \,-1492669384085015040\,{q}^{8}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \, \,
+432762759484818142437\,{q}^{9} \, \, \, + \, \, \cdots
\end{eqnarray}
and\footnote[1]{In Maple the series (\ref{mirror})
can be obtained substituting $\, L=EllipticModulus(q^{1/2})^2$,
in $ 1/j \, = \, \,$
$ \,L^2 \cdot \,(L-1)^2/(L^2-L+1)^3/256$. See https://oeis.org/A066395
for the series (\ref{mirror})
and https://oeis.org/A091406 for the series (\ref{mirror2}).}
its composition inverse:
\begin{eqnarray}
\label{mirror2}
\hspace{-0.95in}&&
\tilde{Q}(x) \, \, = \, \, \, \,
x \, \, \, +744\,{x}^{2} \, \, +750420\,{x}^{3}
\, \, +872769632\,{x}^{4} \, \,
+1102652742882\,{x}^{5}
\nonumber \\
\hspace{-0.95in}&& \quad \, \, +1470561136292880\,{x}^{6}
\, +2037518752496883080\,{x}^{7} \,
+2904264865530359889600\,{x}^{8}
\nonumber \\
\hspace{-0.95in}&& \quad \, \,
+4231393254051181981976079\,{x}^{9}\, \, \, + \, \, \cdots
\end{eqnarray}
These series correspond to $\, x$ being the reciprocal of the
$\, j$-function: $\, 1/j$ . In this paper, as a consequence of the (modular
form) hypergeometric identities (\ref{modularform2explicit})
(see (\ref{Haupt}), (\ref{modularcurve}) and also (\ref{paradox})),
we need $\, x$ to be identified
with the {\em Hauptmodul} $\, 1728/j$. Consequently we introduce
$\, X(q) \, = \, 1728 \cdot \, \tilde{X}(q)$
and $\, Q(x) \, = \, \tilde{Q}(x/1728)$. With these appropriate
changes of variables one finds that the series (\ref{seriesmodcurve1a})
is nothing but $\, \, \, \, X( e \cdot \, Q(x))$.
Thus an interpretation of the one-parameter series
(\ref{seriesmodcurve1a}) through the prism of the mirror map, is
that the one-parameter series
amounts to the multiplication of the nome of
elliptic functions~\cite{Heegner} by an arbitrary complex
number $\, e$: $\, q \, \, \longrightarrow \, \, e \cdot \, q$.
The isogenies correspond to $\, \, q \, \longrightarrow \, \, q^N$
(resp. $\,\, q \, \longrightarrow \, \, q^{1/N}$) for an integer $\, N$
and the one parameter families we have encountered
(namely (\ref{seriesmodcurvea}),
(\ref{seriesmodcurve3a}))
correspond to the composition of
$\,\, q \, \, \longrightarrow \, \, e \cdot \, q\, $
and $\,\, q \, \longrightarrow \, \, q^N$
(resp. $\,\, q \, \longrightarrow \, \, q^{1/N}$),
namely $\,\, q \, \, \longrightarrow \, \, e \cdot \, q^N$
(resp. $\,\, q \, \longrightarrow \, \, e \cdot \, q^{1/N}$).
The series $\, X(q) \, = \, \, 1728 \cdot \, \tilde{X}(q)$
(with $\tilde{X}(q)$ given by (\ref{mirror})) is solution of
the Schwarzian equation
\begin{eqnarray}
\label{Harnad11}
\hspace{-0.95in}&& \quad \,
\{X(q), \, q \} \, \,\, -{{1} \over {2 \, q ^2}} \,\, \, \,
+ \, {{1} \over {72}} \, \cdot \,
{{ 32 \, X(q)^2 \, -41 \, X(q) \, +36} \over {
X(q)^2 \cdot \, (1\, - X(q))^2 }} \cdot \,
\Bigl( {{ d X(q)} \over {d q}} \Bigr)^2
\, \, = \,\, \, 0.
\end{eqnarray}
which is nothing but:
\begin{eqnarray}
\label{Harnad111}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
\{X(q), \, q \} \, \, \,\, -{{1} \over { 2 \, q ^2}} \, \, \, \,
- \, W(X(q))
\cdot \, \Bigl( {{ d X(q)} \over {d q}} \Bigr)^2
\, \, \, = \,\,\, \, 0.
\end{eqnarray}
The series $\, Q(x) \, = \, \tilde{Q}(x/1728)$
(with $\,\tilde{Q}(x)$ given by (\ref{mirror2})) is solution
of the Schwarzian equation
\begin{eqnarray}
\label{Harnad213}
\hspace{-0.95in}&&
-\, \{Q(x), \, x \} \, \, \,
- {{1} \over {2 \cdot \, Q(x)^2 }}
\cdot \Bigl({{ d Q(x)} \over {d x}} \Bigr)^2
\, + \, {{1} \over {72}} \cdot \,
\Bigl( {\frac {32\, x^{2} \, -41 \, x +36}{
x^{2} \cdot \, (1-\, x)^{2}}} \Bigr)
\, \, = \, \, \, \, \, 0,
\end{eqnarray}
equivalently written as:
\begin{eqnarray}
\label{Harnad214}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
\, \{Q(x), \, x \} \, \, \, \,
+ {{1} \over {2 \cdot \, Q(x)^2 }}
\cdot \Bigl({{ d Q(x)} \over {d x}} \Bigr)^2
\, \, + \, W(x) \, \, \, = \, \, \, \, \, 0.
\end{eqnarray}
The two mirror map series (\ref{mirror}), (\ref{mirror2})
thus correspond to differentially
algebraic~\cite{Selected,IsTheFull} functions,
and are solutions of simple Schwarzian equations like in
(\ref{condition1}).
These differentially algebraic mirror maps transformations
$\, Q(x)$ and $\, X(q)$ are the well-suited changes of variables
such that the transformation $\, x \, \longrightarrow \, \, y(x)$
verifying the Schwarzian equation (\ref{condition1}) become
simple transformations, ``simple'' meaning
transformations like
$\, q \, \longrightarrow \, \, S(q) \, = \, \, e \cdot \, q^N$
(or $\, S(q) \, = \, \, e \cdot \, q^{1/N}$)
in the nome $\, q$ of elliptic functions~\cite{Heegner}. Generalizing
the Koenig-Siegel
linearization~\cite{Siegel,Siegel2,Almost,Milnor},
we thus decompose $\, y(x)$ as
$\,y(x) \, = \, \, X(S(Q(x)))$.
The Schwarzian conditions (\ref{Harnad111}), (\ref{Harnad214}) are
essentially the well-known Schwarzian equation discovered
by Jacobi~\cite{Jacobi,Nova} on the $\, j$-function (see for instance
equation (1.26) in~\cite{HarnadHalphen}). The compatibility of the
Schwarzian equations (\ref{Harnad111}), (\ref{Harnad214})
on the mirror maps with the Schwarzian condition (\ref{condition1})
on $\, y(x)$ emerging from a more general Malgrange's pseudo-group
perspective~\cite{Casale,Casale2,Casale3,Casale4}, is
shown in \ref{Schwarzmirror}. The fact that the {\em same function} $\, W(x)$
occurs in the Schwarzian conditions (\ref{Harnad111}), (\ref{Harnad214})
on the mirror maps, and on the Schwarzian condition (\ref{condition1}),
is crucial for this demonstration and is not a mere coincidence.
\subsection{The general case: $\, _2F_1([\alpha, \, \beta], \, [\gamma],x)$ hypergeometric function.}
\label{general}
\subsubsection{The $\, _2F_1([1/6,1/3],[1],x)$ hypergeometric function. \\}
\label{16131}
\vskip .1cm
\vskip .1cm
\vskip .1cm
We have analyzed in some detail in section (\ref{Schwarzsimplest})
the modular form example (\ref{modularform2explicit}). For other values
of the $\, [[\alpha, \, \beta], \, [\gamma]]\, $ parameters of
the $\, _2F_1$ (see (\ref{modularform2}), (\ref{Gaussdiff}))
one can easily find series expansions of the solution
$\, y(x)$ of the Schwarzian condition. A set of values like
$\, [[1/2, \, 1/2], \, [1]]$,
$\, [[1/4, \, 1/4], \, [1]]$,
$\, [[1/3, \, 1/3], \, [1]]$, $\, [[1/3, \, 2/3], \, [1]]$
or $\, [[1/3, \, 1/6], \, [1]]$
(see for instance~\cite{Christol,SuperMaier} and Ramanujan's cubic theory
of alternative bases~\cite{Canada}) which are known to yield
modular form hypergeometric identities like (\ref{modularform2explicit})
with algebraic pullbacks $\, y(x)$ associated with modular equations.
For these values of the $\, [[\alpha, \, \beta], \, [\gamma]]\, $
parameters one finds a set of one-parameter series totally similar to
what is described in section (\ref{Schwarzsimplest}). The example of the
$\, _2F_1([1/6,1/3],[1],x)$ hypergeometric function is sketched in \ref{16131app}.
\subsubsection{The general case: $\, _2F_1([\alpha, \, \beta], \, [\gamma],x) \, $ hypergeometric function. \\}
\label{general2}
\vskip .1cm
\vskip .1cm
\vskip .1cm
Let us now consider arbitrary parameters of the Gauss hypergeometric
function $ \, [[\alpha, \, \beta], \, [\gamma]] \,$ that are not
in the previous selected set, and are different from the
cases given in sections (\ref{recalls}), (\ref{more2F1Heun}),
and (\ref{more2F1higher}) corresponding to the rank-two condition.
\vskip .1cm
A simple calculation shows that one always finds a series of the form
$\, e \cdot \, x \, + \, \cdots \, $ (like (\ref{seriesmodcurve1a})
or (\ref{361})), solution of the Schwarzian condition,
{\em but it is only for} $\, \gamma \, = \, 1$ that series
of the form $\, a \cdot \, x^2 \, + \, \cdots$,
$\, b \cdot \, x^3 \, + \, \cdots$, etc ... (like
(\ref{seriesmodcurvea}) or (\ref{seriesmodcurve3a})) can be
solutions of the Schwarzian condition.
When $\, \gamma \, = \, 1 \,$ one gets the following series
of the form $\, a \cdot \, x^2 \, + \, \cdots \, \,$
solution of the Schwarzian condition
\begin{eqnarray}
\label{cequalone}
\hspace{-0.96in}&& \quad
y_2(u, \, x) \,\, = \, \,\, \,\, a \cdot \, {x}^{2}
\, \,\,\,
-2 \, a \cdot \, (2\,\alpha \beta \, -\alpha -\beta) \cdot \, {x}^{3}
\, \,\,+ \, \, {{a} \over {2}} \cdot \, C_4 \cdot {x}^{4} \,\, \,
+ \,\, \cdots \quad \quad \hbox{with:} \
\nonumber \\
\hspace{-0.96in}&& \, \, \,
C_4 \, = \, \,\, \,
2\,\,(2\,\alpha \beta \, -\alpha -\beta) \cdot a \, \,
+ (\alpha \beta \, -1) (\alpha \beta \, -\alpha -\beta) \,
+\, \, 5\,\, (2\,\alpha \beta\, -\alpha -\beta)^{2},
\end{eqnarray}
and one also gets the following series
of the form $\,\, b \cdot \, x^3 \, + \, \cdots \, \,\,$
solution of the Schwarzian condition
\begin{eqnarray}
\label{cequalone}
\hspace{-0.95in}&& \, \, \quad \quad
y_3(v, \, x) \, \, = \, \,\, \,\, b \cdot \, {x}^{3}
\, \,\,\, -3 \, b \cdot \, (2\,\alpha \beta\, -\alpha -\beta) \cdot \, {x}^{4} \,\,
\\
\hspace{-0.95in}&& \quad \quad \quad \quad \, \, \, \, \, \,
\, \, +\,{{3 \, b} \over {4}} \cdot \,
\Bigl( (\alpha \beta \, -1) \cdot \, (\alpha \beta\, -\alpha -\beta) \,
+7 \, \, (2\,\alpha \beta\, -\alpha -\beta)^{2} \Bigr) \cdot {x}^{5}
\, \, \, \, + \,\, \cdots \nonumber
\end{eqnarray}
together with the one-parameter family of commuting series
of the form $\, e \cdot \, x \, + \cdots \, $
\begin{eqnarray}
\label{cequalone}
\hspace{-0.95in}&&
y_1(e, \, x) \, = \, \, e \cdot \, x
\,\,
+ e \cdot \, \, (e\, -1) \cdot \,
(2\,\alpha \beta\, -\alpha -\beta) \cdot \, {x}^{2}
\, \,\, +\,{{ e \cdot \, (e-1)} \over {4}}
\cdot \, C_3 \cdot {x}^{3} \, \,
+ \,\, \cdots
\nonumber \\
\hspace{-0.95in}&& \hbox{with:} \quad \quad
C_3 \, = \, \,
(\alpha \beta -1) (\alpha \beta -\alpha -\beta) \cdot \, (e\, +1)
\, + (2\,\alpha \beta -\alpha -\beta)^{2} \cdot \, (5\, e -3).
\end{eqnarray}
Again one has the equalities
\begin{eqnarray}
\label{suchsuch}
\hspace{-0.95in}&& \quad \quad \quad
y_1(e, \,y_2(a, \, x)) \, = \, \, y_2(a\, e, \, x),
\quad \quad \quad
y_2(a, \,y_1(e, \, x)) \, = \, \, y_2(a \, e^2, \, x),
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad
y_1(e, \,y_3(b, \, x)) \, = \, \, y_3(b \, e, \, x),
\quad \quad \quad
y_3(b, \,y_1(e, \, x)) \, = \, \, y_3(b \, e^3, \, x),
\end{eqnarray}
and, again, the two series $\, y_2(a, \, x)$ and $\, y_3(b, \, x)$
commute for $\, b \, = \, a^2$. As far as series analysis is concerned we
have {\em exactly the same structure} (\ref{suchsuch}) as the one
previously described (see (\ref{Schwarzsimplest}) and (\ref{16131}))
where {\em modular correspondences}~\cite{Goro} take place.
However, it is not clear if such one-parameter series
can reduce to algebraic functions for some selected
values of the parameter $\, a, \, b, \cdots \, $ In other words,
are these series modular correspondences, or are they just
``similar'' to modular correspondences ?
{\em The question of the reduction of these Schwarzian conditions to
modular correspondences remains an open question}.
\vskip .1cm
When $ \, \gamma \, \ne \, 1$ the situation is drastically
different\footnote[2]{Recall that {\em globally bounded} $\, _nF_{n-1}$
series of ``weight zero''~\cite{Heckman} (no ``down'' parameter
is equal to $1$ or to an integrer,
i.e. in the case of globally bounded $\, _2F_1$
series, $\, \gamma \, \, $ is
different from an integer), are {\em algebraic functions}.}: one does
not have solution of the Schwarzian equation of the form
$\, a \cdot \, x^2 \, + \, \cdots \,\, $
or $\, b \cdot \, x^3 \, + \, \cdots \,\, $ etc ...
One only has a one-parameter family of commuting series:
\begin{eqnarray}
\label{357}
\hspace{-0.95in}&& \, \, \, \,
y(e, \, x) \, \, \, = \, \, \, \,
e \cdot \, x \, \,\, \,
- \, e \cdot \, (e\, -1) \cdot \,
{{ \gamma^2 \, -(\alpha+\beta+1)\cdot \, \gamma \, +2\, \alpha \, \beta } \over {
\gamma \cdot \, (\gamma\,-2) }} \cdot \, x^2
\, \, \, \, +\, \, \cdots
\end{eqnarray}
Again, it is not clear to see if such a one-parameter series
can reduce to algebraic functions for some selected
values of the parameter $\, e$.
\vskip .1cm
\vskip .1cm
\section{Rank-two condition on the rational transformations as a subcase of the Schwarzian condition}
\label{subcase}
\vskip .1cm
\subsection{Preliminary result: factorization
of the order-two linear differential operator}
\label{prelim}
When
\begin{eqnarray}
\label{Bx}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
B(x) \, \, = \, \, \, \, {{C(x)} \over {4}} \cdot \, (2\,A(x) \, -C(x))
\, \,\, + \, \, {{1} \over {2}} \cdot \, C'(x),
\end{eqnarray}
the second order linear differential operator
\begin{eqnarray}
\label{Omeg}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\Omega \,\, = \, \, \, \, \,
D_x^2 \,\, \, + \, A(x) \cdot \, D_x \, \, + \, B(x),
\end{eqnarray}
factorizes as follows:
\begin{eqnarray}
\label{OmegFact}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\Omega \,\, = \, \, \, \, \,
\Bigl(D_x \, + \, A(x) \, -{{C(x)} \over {2}}\Bigr)
\cdot \, \Bigl(D_x \, +\, {{C(x)} \over {2}}\Bigr).
\end{eqnarray}
Let us assume that $\, C(x)$ is a log-derivative:
\begin{eqnarray}
\label{Assume}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
C(x) \,\, = \, \, \, \, \,
2 \cdot \, {{ d \ln(\rho(x))} \over {dx}},
\end{eqnarray}
one immediately finds that a conjugation of
(\ref{OmegFact}) factors as follows:
\begin{eqnarray}
\label{Immediat}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\rho(x) \cdot \, \Omega \cdot \, {{1} \over {\rho(x)}}
\,\,\, = \, \, \, \, \,
\Bigl(D_x \, + \, A(x) \, - C(x)\Bigr) \cdot \, D_x.
\end{eqnarray}
Therefore the $\, A_R(x)$ in the rank-two condition
(\ref{mad}) is not the $\, A(x)$ in (\ref{Omeg}) but
$\, A_R(x) \,\, = \,\, A(x) \, - C(x) \,\, $
where $\, B(x)$ is of the form (\ref{Bx}).
The rank-two condition reads:
\begin{eqnarray}
\label{rotaA}
\hspace{-0.95in}&& \quad \quad \quad
y''(x) \,\, \, = \,\, \,
(A(y(x)) \, - \, C(y(x))) \cdot \, y'(x)^2 \, \,
- \, (A(x) \, - \, C(x)) \cdot \, y'(x),
\end{eqnarray}
to be compared with the Schwarzian condition
\begin{eqnarray}
\label{condition1n}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
where:
\begin{eqnarray}
\label{wherecondn}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
W(x) \, \, = \, \, \, \, \, A'(x) \, \, \, + \, \,
{{A(x)^2} \over {2 }} \, \, \, \, -2 \cdot \, B(x).
\end{eqnarray}
{\bf Remark:} For a general Gauss hypergeometric function
$\, _2F_1([\alpha, \, \beta], \, [\gamma], \, x)$,
$\, A(x)$ and $\, B(x)$ are given by (\ref{Gaussdiff}). The
factorization condition (\ref{Bx}) can be satisfied
only for selected values of the $[[\alpha, \, \beta], \, [\gamma]]$
parameters\footnote[1]{For these conditions on the parameters the function $\, C(x)$
read respectively $\, C(x)=\, 2\,\alpha/x$, $\, C(x)=\, 2\,\beta/x$,
$\, C(x)=\, 2\,(\beta\,x -\gamma+1)/x/(x-1)$,
$\, C(x)=\, 2\,(\alpha\,x-\gamma+1)/x/(x-1)$,
$\, C(x)=\, 2\,\alpha/(x-1)$, $\, C(x)=\,2\, \gamma\, \beta/x/(x-1)$,
$\, C(x)=0$, $\, C(x)=0$.}:
$\, \gamma= \, \alpha\, +1$, $\, \gamma= \, \beta\, +1$, $\, \gamma= \,1$,
$\, \beta= \,1$, $\gamma \, = \, \beta$,
$ \, \gamma \, = \, \alpha$, $ \,\alpha=0 \,$ and $\, \beta=0$.
\vskip .1cm
\subsection{Condition on the rational transformation as a subcase of the Schwarzian condition}
\label{subcase}
Let us assume that the rank-two condition (\ref{rotaA})
is satisfied, then we can use
it to express the second derivative $\, y''(x)$
in terms of $\, y(x)$ and the first derivative the $\, y'(x)$.
One finds that the Schwarzian condition (\ref{condition1n})
is automatically
verified provided $\, A(x)$, $\, B(x)$, $\, C(x)$ are related
though the condition (\ref{Bx}) which amounts to
a factorization condition for the second order linear differential
operator (\ref{Omeg}).
The $\, A_R(x)$ in the rank-two condition (see (\ref{mad})):
\begin{eqnarray}
\label{rota}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
y''(x) \,\, = \,\, \,
A_R(y(x)) \cdot \, y'(x)^2 \, \, - \, A_R(x) \cdot \, y'(x),
\end{eqnarray}
is nothing but $\, A_R(x) \, = \, \, A(x) \, - \, C(x)$,
or after rearranging $\, A(x) \, = \, \, A_R(x) \, +C(x)$. Now
substituting (\ref{Bx}) in (\ref{wherecondn}) one gets:
\begin{eqnarray}
\label{wherecondn2before}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
W(x) \,\, = \,\, \,\, A'_{R}(x)\, \, \, + \frac{A(x)^2}{2}
\,\, - C(x) \, A(x) \,\, \, - \frac{C^2(x)}{2},
\end{eqnarray}
with the last three terms being equivalent to $ \, A_{R}(x)^2$. Thus one finds
that $\, W(x)$ is {\em only a function of} $\, A_R(x)$:
\begin{eqnarray}
\label{wherecondn2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
W(x) \, \, = \, \,\, \, \, A'_R(x)
\, \, \, + \, \, {{A_R(x)^2} \over {2 }}.
\end{eqnarray}
With this expression (\ref{wherecondn2}) of $\, W(x)$
the Schwarzian condition reads:
\begin{eqnarray}
\label{condition1nbis}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
In order to see the compatibility of the rank-two condition (\ref{rota})
with the Schwarzian condition (\ref{condition1nbis}) when the
function $\, W(x)$ is given by (\ref{wherecondn2}),
let us rewrite the rank-two condition (\ref{rota}) as
\begin{eqnarray}
\label{Rotarewr}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
{{y''(x)} \over {y'(x)}}
\, \, = \, \, \, \, A_R(y(x)) \cdot \, y'(x) \,\,\, -A_R(x).
\end{eqnarray}
Using (\ref{Rotarewr}), one can rewrite the Schwarzian
derivative as
\begin{eqnarray}
\label{Schwarwrbefore}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\{y(x), \, x\} \, \, = \, \, \, \,
{{d } \over { dx}} \Bigl({{y''(x)} \over {y'(x)}}\Bigr)
\,\, -{{1} \over {2}} \cdot \, \Bigl({{y''(x)} \over {y'(x)}}\Bigr)^2
\end{eqnarray}
as $\,\, W(y(x)) \cdot \, y'(x)^2 \,\,\, -W(x)\, + \, \Delta$, where $\, W(x)$
is given by (\ref{wherecondn2}),
and where $\, \Delta$ is given by:
\begin{eqnarray}
\label{Schwarwr}
\hspace{-0.95in}&& \, \,
\Delta \, \, \, = \, \, \,
A_R(y(x)) \cdot \, y''(x) \, \, - \, A_R(y(x))^2 \cdot \, y'(x)^2 \, \,
\, + \, A_R(x) \cdot \, A_R(y(x)) \cdot \, y'(x).
\end{eqnarray}
Note that $\, \Delta$ is clearly zero when the rank-two condition
is fulfilled. This shows that the Schwarzian condition (\ref{condition1nbis})
when the function $\, W(x)$ is given by (\ref{wherecondn2}), actually reduces to
the rank-two condition (\ref{rota}), as expected.
\vskip .1cm
\vskip .1cm
{\bf Remark:} The Heun function case of section (\ref{moreHeun})
was a case where the rank-two condition was verified with
$\, A_R(x)$ given by (\ref{AadoublingAa}). One also verifies that
the rational transformation (\ref{Aadoubling}), and more generally
the rational transformations $\, R_p(x)$
(pullbacks on the Heun function, see (\ref{Fdoublingidefirst})),
are solutions of a Schwarzian equation (\ref{condition1nbis})
with $\, W(x)$ deduced from (\ref{wherecondn2}) with $\, A_R(x)$
given by (\ref{AadoublingAa}),
namely:
\begin{eqnarray}
\label{AadoublingbisW}
\hspace{-0.95in}&&
W(x) \, \, = \, \, \, -{{3} \over {8 \cdot \, (x-M)^2}}
\, \,
-{{1} \over {4}} \cdot \, {\frac {2\,x \, -1}{ (M \, -x) \cdot \, x \cdot \, (x-1) }}
\, \,
-{{1} \over {8}} \cdot \, {\frac {4\,{x}^{2}-4\,x+3}{ x^{2} \cdot \, (x-1)^{2}}}.
\end{eqnarray}
\vskip .1cm
In the previous case where the rank-two condition can be seen as a subcase of the
Schwarzian condition (\ref{condition1nbis}) on $\, y(x)$, it is tempting to imagine,
in a Koenig-Siegel linearization perspective, that the differentially algebraic
function $\, Q(x)$ (see (\ref{QDAfirst})) also verifies a Schwarzian condition
similar to the Schwarzian condition (\ref{Harnad214})
on $\, Q(x)$ now seen as a mirror map and we show in \ref{subcase}
that this is actually the case.
\vskip .1cm
\section{Schwarzian condition for generalized hypergeometric functions}
\label{Schwarzgeneralized}
\vskip .1cm
\subsection{Schwarzian condition and $\, _3F_2$ hypergeometric identities}
\label{Schwarz3F2}
Generalizing the modular form identity considered in section (\ref{int}),
let us seek a $\, _3F_2$ hypergeometric identity of the form
\begin{eqnarray}
\label{modularform3F2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
{\cal A}(x) \cdot \, _3F_2\Bigl([a, \, b, \, c], \, [d, \, e], \, x \Bigr)
\, = \, \, \, \,\,
_3F_2\Bigl([a, \, b, \, c], \, [d, \, e], \, y(x) \Bigr),
\end{eqnarray}
where $\, {\cal A}(x)$ is an algebraic function.
Similarly to what has been performed in section (\ref{int}),
we consider the two order-three linear differential operators
associated respectively to the LHS and RHS of
(\ref{modularform3F2}).
A straightforward calculation
enables us to find (from the equality of the wronskians of these
two operators) the algebraic function $\, {\cal A}(x)$
in terms of the algebraic function pullback $\, y(x)$
in (\ref{modularform3F2}):
\begin{eqnarray}
\label{modularform3F2calA}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
{\cal A}(x) \, \, = \, \, \,
\Bigl( {{ y(x)^{\eta} \cdot \,
(1 \, -y(x))^{\nu} } \over {
x^{\eta} \cdot \, (1 \, -x)^{\nu} }} \Bigr)
\cdot \, \Bigl( {{d y(x)} \over {dx }} \Bigr)^{-1},
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
\quad
\eta \, \, = \, \, \, {{d+e+1} \over {3}},
\quad \quad \quad \quad
\nu \, \, = \, \, \, {{a+b+c+2-d-e} \over { 3}},
\end{eqnarray}
The identification
of the $\, D_x$ coefficients of these two linear differential
operators, gives (beyond (\ref{modularform3F2calA}))
a first condition that can be rewritten
in the following Schwarzian form:
\begin{eqnarray}
\label{condition3F2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
where $\, W(x)$ reads:
\begin{eqnarray}
\label{condition3F2W}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
W(x) \,\, \, = \,\, \, \,
{{1} \over {6}} \cdot \, {{P_W(x)}\over { x^2 \cdot \, (1\, -x)^2}},
\quad \quad \quad \quad \quad \quad \hbox{where:}
\\
\hspace{-0.95in}&&
P_W(x)\,\, \, = \,\, \,
({a}^{2} \, +{b}^{2} \, +{c}^{2} \, - a b \, - \, a c \, \, -bc \, -3) \cdot \, {x}^{2}
\nonumber \\
\hspace{-0.95in}&&
+ \, ( 3\, (a b \, +\, a c \, +\,bc \, +\,de \, +1)
-2\, (a d \, +\, a e \, \, +\,bd \, +\,be \,+\,cd \,+\,ce) \, \, +a+b+c) \cdot \, x
\nonumber \\
\hspace{-0.95in}&& \quad \quad
\, \, +{d}^{2}\, +{e}^{2} \, -de \, -d \, -e \, -2.
\end{eqnarray}
The identification
of the coefficients with no $\, D_x$ of these two linear differential
operators gives a second condition where the fourth derivative of
$\, y(x)$ takes place. The analysis of this set of conditions
corresponds to tedious but straightforward differential algebra
calculations which are performed in \ref{Reduc3F2}.
One finds that all the
conditions on the parameters $\, a, \, b, \, c, \, d, \, e$
of the $\, _3F_2$ hypergeometric function associated with
$\, Q(x) \, = \, \, \, 0$,
correspond to cases where the order-three operator
is the symmetric square of a second order operator having
$\, _2F_1$ solutions. In other words this situation correspond to
the {\em Clausen identity}, the $\, _3F_2$ hypergeometric function
reducing to the square of a $\, _2F_1$ hypergeometric function:
\begin{eqnarray}
\label{Clausen}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
_3F_2\Bigl([2\,a, \, a\, +b, \, 2\, b], \,
[a\, +b\, +{{1} \over {2}}, \, 2\,a \, +2 \, b], \, x \Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad
\quad \quad \quad \quad \quad \quad \, \, \,
\, = \, \, \,
_2F_1\Bigl([a, \, b], \, [a\, +b\, +{{1} \over {2}}], \, y(x) \Bigr)^2.
\end{eqnarray}
In that Clausen identity case, the Schwarzian condition (\ref{condition3F2})
we found for the $\, _3F_2$ is nothing but the Schwarzian condition
on the underlying $\, _2F_1$.
\vskip .1cm
\subsubsection{The intriguing $\, _3F_2([1/9,4/9,5/9],[1/3,1],x)$ case \\}
\label{intring}
\vskip .1cm
\vskip .1cm
\vskip .1cm
Beyond the trivial transformation $\, y(x) \, = \, \, x$
one hopes to find a condition (\ref{modularform3F2})
where the pullback $\, y \, = \, \, y(x)$ is an algebraic function.
For the intriguing hypergeometric function
$\, _3F_2([1/9,4/9,5/9],[1/3,1],x)$, known to be a globally
bounded\footnote[5]{The series
$ \, _3F_2([1/9,4/9,5/9],[1/3,1], \, 3^5 \, x) \, $
is a series with {\em integer coefficients}~\cite{Christol}.}
series~\cite{Christol}, one does not know if it is the
{\em diagonal of a rational function},
or not. It is natural to apply the previous conditions
to see if we could have an identity like (\ref{modularform3F2}) generalizing
the identities one gets for modular forms. The occurrence of a series with
{\em integer coefficients} is a strong argument for a
``modular form interpretation'' of this intriguing $\, _3F_2$
hypergeometric function. Therefore, it is tempting
to imagine that a remarkable identity like (\ref{modularform3F2}) exists
for this $\, _3F_2$ hypergeometric function.
The corresponding order-three
operator has a differential Galois group that is an
extension\footnote[1]{See the Boucher-Weil criterion~\cite{Boucher}. The
symmetric square and exterior square of a normalized order-three
operator has no rational solutions. One sees also clearly that
this order-three operator is not homomorphic to its adjoint.} of
$\, SL(3, \, \mathbb{C})$. Therefore,
this operator cannot be homomorphic to the symmetric square
of an order-two operator: {\em an identity of the Clausen type
is thus excluded for this $\, _3F_2$ hypergeometric function}. The previous
calculations showing that an identity like (\ref{modularform3F2})
exists only when the $\, _3F_2$ hypergeometric function reduces to
square of $\, _2F_1$ hypergeometric functions discards an identity
like (\ref{modularform3F2}) for $\, _3F_2([1/9,4/9,5/9],[1/3,1],x)$.
This is easily seen: for
this hypergeometric function the ``invariant''
$\,{\cal I}(x) \, = \, \, {\cal I}y(x)$ (see \ref{Reduc3F2}), and
the rational function $\, W(x)$ in the Schwarzian condition
read respectively
\begin{eqnarray}
\label{invariant3F2}
\hspace{-0.95in}&&
{\cal I}(x) = \, \,
{{p_8^3} \over {(140\,{x}^{3} +81\,{x}^{2}+2403\,x-864)^8 }},
\quad
W(x) = \, \,
-{\frac {230\,{x}^{2}-261\,x+207}{486\,{x}^{2} \left( x-1 \right) ^{2}}},
\end{eqnarray}
where:
\begin{eqnarray}
\label{invariant3F2more}
\hspace{-0.95in}&& \quad \,
p_8 \, \, = \, \, \, 254800\,{x}^{8}\, +7247520\,{x}^{7}\,
+223006266\,{x}^{6} \,\, -533339127\,{x}^{5}\,\, -62800191\,{x}^{4}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
+1082145339\,{x}^{3}\,\, -244855791\,{x}^{2}\, \, -290993472\,x \,\, +26873856.
\end{eqnarray}
Reinjecting the invariance condition $\,{\cal I}(x) \, = \, \, {\cal I}y(x)$
with (\ref{invariant3F2})
in the Schwarzian condition (\ref{condition3F2}), one finds that there
is no (algebraic) solution $\, y(x)$ except the trivial solution
$\,\, y(x) \, = \, \, x$.
\vskip .1cm
\subsection{Schwarzian condition and other generalized hypergeometric functions}
\label{Schwarz4F3}
\vskip .1cm
In \ref{Schwarz4F32F2} we seek an identity of the form
(\ref{modularform3F2}) but where the $\, _3F_2$ hypergeometric function
is replaced by a $\, _4F_3$ hypergeometric function known to correspond
to a {\em Calabi-Yau ODE}~\cite{IsingCalabi,IsingCalabi2},
or a hypergeometric function with {\em irregular} singularities
namely a simple $\, _2F_2$ hypergeometric function. One finds, unfortunately,
that the only solution, for these two examples sketched respectively
in \ref{Schwarz4F3} and \ref{Schwarz2F2}, is the trivial solution
$\, y(x) \, = \, \, x$. Keeping in mind the non trivial results previously
obtained on a Heun function, or on a $\, _2F_1$ hypergeometric function
associated with a higher genus curve, these two negative results should
rather be seen as an incentive to find more non trivial examples of these
extremely rich and deep Schwarzian equations.
\vskip .1cm
\vskip .1cm
\vskip .1cm
\section{Conclusion}
\label{Conclusion}
In this paper we focus essentially on identities relating the same
hypergeometric function with two different algebraic pullback
transformations related by modular equations. This corresponds
to the modular forms that emerged so many
times in physics~\cite{IsingCalabi,IsingCalabi2,Christol}: these
algebraic transformations can be seen as simple illustrations
of exact representations of the renormalization group~\cite{Hindawi}.
Malgrange's pseudo-group approach aims at
generalizing differential Galois theory to non-linear differential
equations. In his analysis of Malgrange's pseudo-group Casale found two
non-linear differential equations (\ref{cas2}) and (\ref{Casale})
yet these two conditions were presented separately with no explicit link.
In a previous paper~\cite{Hindawi}, where we gave simple
examples of exact representations of the renormalization group,
associated with selected linear differential operators covariant by
rational pullbacks, we found simple exact examples
of Casale's condition (\ref{cas2}). Building on this work we revisited
these previous examples and provided non-trivial new examples
associated with a Heun function and a $ \, _2F_1$ hypergeometric function
associated with higher genus curves. Then we instantiated, for the first
time, Casale's second condition (\ref{Casale}) with the examples
given in section (\ref{Schwarz}). Furthermore we found that
Casale's condition (\ref{cas2}) can be seen as a subcase of the
Schwarzian condition (\ref{Casale}), corresponding to a factorization
of a linear differential operator $\, \Omega$.
Seemingly, this Schwarzian condition (\ref{Casale}) is seen
to ``encapsulate'' in one differentially algebraic (Schwarzian)
equation, all the {\em modular forms}
and {\em modular equations} of the theory of elliptic curves.
The Schwarzian condition (\ref{Casale}) can thus be seen as some
quite fascinating ``pandora box'', which encapsulates an infinite number of
highly remarkable modular equations, and a whole ``universe'' of
{\em Belyi-maps}\footnote[2]{Belyi-maps~\cite{Belyi3,Belyi,Belyi2,Belyi4,Belyi5}
are central to Grothendieck's program
of ``dessins d'enfants''.}. Furthermore we found, only when
$\, \gamma\, = \, 1$, that one-parameter series starting with quadratic,
cubic, or higher order terms satisfy the rank-three condition. The
question of a modular correspondence interpretation of these series
is an open question.
Recalling the two previous higher-genus and Heun
examples, it is important to underline that these conditions (\ref{cas2})
and (\ref{Casale}) are actually richer than just elliptic curves, and
go beyond ``simple'' restriction to $\, _2F_1$ hypergeometric functions.
This paper provides a simple and pedagogical illustration of
such exact non-linear symmmetries in physics (exact representations of the
renormalization group transformations like the Landen transformation for
the square Ising model, ...) and is a strong incentive to discover more
differentially algebraic equations involving fundamental symmetries,
developping more differentially algebraic analysis
in physics~\cite{Selected,IsTheFull}, beyond obvious candidates
like the full susceptibility of the
square-lattice Ising model~\cite{IsTheFull,Automat}.
\vskip .1cm
\vskip .4cm
\vskip .4cm
{\bf Acknowledgments:}
We would like to thank
S. Boukraa, G. Casale, S. Hassani, E. Paul, C. Penson
and J-A. Weil for very fruitful
discussions. We would like to thank the anonymous referee
for his very careful reading of our manuscript and his
valuable suggestions and corrections.
This work has been performed
without any ERC, ANR, PES or MAE financial support.
\vskip .5cm
\vskip .5cm
\appendix
\section{$\, _2F_1$ hypergeometric example: $\, N\, = \, 3$ }
\label{more2F1examples}
\vskip .2cm
Recalling Vidunas paper~\cite{Vidunas} one introduces the
following hypergeometric function:
\begin{eqnarray}
\label{YM3}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
Y(x) \, \, = \, \, \,
x^{1/3} \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \, x\Bigr),
\end{eqnarray}
for which one has the following exact expressions
for $\, A_R(x)$, $\, u(x)$ and $\, R(x)$:
\begin{eqnarray}
\label{Aa3}
\hspace{-0.96in}&& \quad \,
A_R(x) \, \, = \, \, \,
{{2} \over {3 }} \cdot \,
{{2 \, x \, -1 } \over { x \cdot \, (x \, -1)}}\, \, = \, \, \, {{u'(x)} \over {u(x)}},
\, \, \, \quad \, \, \hbox{where:} \quad \quad \, \, \,
u(x) \, \, = \, \, \, x^{2/3} \cdot \, (1-x)^{2/3},
\nonumber \\
\hspace{-0.96in}&& \quad \quad \quad \quad \quad \quad \quad
R(x) \,\, = \, \, \,
{{x \cdot \, (x \, -2)^3} \over { (1\, -2 \, x)^3}}.
\end{eqnarray}
One verifies that
$\, Q(x)\, = \, \, Y(x)^3$:
\begin{eqnarray}
\hspace{-0.95in}&&
{{d Q(x)} \over { dx}}/Q(x)
\,\, = \, \, \, 3 \cdot \, {{d Y(x)} \over { dx}}/Y(x)
\,\, = \, \, \, {{1} \over {F(x)}},
\quad \hbox{where:} \quad F(x) \,\, = \, \, \,
u(x) \cdot Y(x).
\end{eqnarray}
One has the identity:
\begin{eqnarray}
\label{Q8}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
Q(R(x)) \,\, = \, \,\,\, -8 \cdot \, Q(x) \, = \, \, \, \,
-8 \cdot \, x \, \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \, x\Bigr)^3
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad \, \,
\,\, = \, \, \, \,
{{x \cdot \, (x \, -2)^3} \over { (1\, -2 \, z)^3}} \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \,
{{x \cdot \, (x \, -2)^3} \over { (1\, -2 \, x)^3}}\Bigr)^3.
\nonumber
\end{eqnarray}
The rational function\footnote[1]{Note a typo in~\cite{Vidunas}: the
$\, R(x)$ in equation (64) of~\cite{Vidunas} is $\, -R(x)$. }:
\begin{eqnarray}
\label{other}
\hspace{-0.95in}&& \quad \quad
\quad \quad \quad \quad \quad \quad
\tilde{R}(x) \, = \, \,
{{ 27 \, x \cdot \, (1\, -x) \, (1 \, -x \, +x^2)^3} \over {
(1\, +3\, x -6\, x^2\, +\, x^3)^3 }},
\end{eqnarray}
commutes with $\, R(x)$ given by (\ref{Aa3}). Also note that
$\, R(x)$ given by (\ref{Aa3}) commutes with the two known symmetries
of the hypergeometric function, namely
$\,\, R(x) \, = \, \, 1 \, -x \,$ and $\, R(x) \, = \, \, 1/x$.
These last two transformations yield the involution
$\, R(x) \, = \, -x/(1\, -x)\, $ which commutes with the two previous
rational transformations (\ref{Aa3}), (\ref{other}),
and corresponds to $\, Q(-x/(1\, -x)) \, = \, Q(x)$.
The composition of $\, R(x) \, = \, -x/(1\, -x)\,\, $
with (\ref{Aa3}) and (\ref{other}) gives respectively:
\begin{eqnarray}
\label{other2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
{{x \cdot \, (2\, -x)} \over {(1\, -x) \cdot \, (1\, +x)^3 }},
\quad \quad \quad
{{ -27 \, x \cdot \, (1\, -x) \, (1 \, -x \, +x^2)^3} \over {
(1\, -6\, x +3 \, x^2\, +\, x^3)^3 }}.
\end{eqnarray}
Note that $\,\, R(x) \,= \, 1/x\,$ and $\,\, R(x) \,= \, 1\, -x\,$
also verify the rank-two condition.
As we can see, the one-parameter family of solution of
\begin{eqnarray}
\label{mad2one}
\hspace{-0.95in}&& \quad \quad \quad \quad
\Bigl({{ d R(a, \, x)} \over {dx}}\Bigr)^2 \cdot A(R(a, \,x))
\,\, = \,\, \, \,\,
{{ d R(a, \,x)} \over {dx}} \cdot A(x) \, \, \,
+ {{ d^2 R(a, \,x)} \over {dx^2}},
\end{eqnarray}
namely the differentially algebraic series
\begin{eqnarray}
\label{mad2oneseries}
\hspace{-0.95in}&& \,
R(a, \, x) \,\, = \,\, \, \,\,\,
a \cdot \, x \, \, \,\,\,
- {{1} \over {2}} \,a \cdot \, (a-1) \cdot \, {x}^{2} \, \, \, \,
+ {{1} \over {28}}\,a \cdot \, (a-1) \cdot \, (5\,a-9) \cdot \, {x}^{3}
\\
\hspace{-0.95in}&& \quad
\,\, \, \, -{\frac {a \cdot \, (a-1)
\, (3\,{a}^{2}-12\,a+13) }{56}} \, \cdot {x}^{4}
\,\, \, + \, \, \cdots \, \, \,
+ \, a \cdot \, (a-1) \cdot {{P_{18}(a)} \over {D_{20}}} \cdot \, x^{20}
\, \,\, + \,\, \cdots
\nonumber
\end{eqnarray}
corresponds to {\em movable singularities}. For (an infinite number of)
selected values of the
parameter $\, a$, this series becomes a rational function, for instance
(\ref{Aa3}) for $\, a \, = \, -8$, (\ref{other}) for $\, a \, = \, 27$,
(\ref{other2}) for $\, a \, = \, 8$ and $\, a \, = \, -27$. For a generic
parameter $\, a$ the series is much more complex, it is not globally bounded.
For instance, $\, P_{18}(a)$ in (\ref{mad2oneseries})
is a polynomial with integer coefficients of degree $\, 18$ in $\, a$,
and the denominator $\, D_{20} \, = \, \, 1277610230161807653119590400$
is an integer that factors in many primes: $\, \, D_{20} \, = \, \,$
$2^{17} \cdot \, 5^2 \cdot \, 7^9 \cdot \, 13^4 \cdot \, 19^3 \,
\cdot \, 31 \cdot \, 37 \cdot \, 43 $.
One verifies easily on this series that the two differentially
algebraic series $\, R(a, x)$ and $\, R(b, x)$ commute and that
\begin{eqnarray}
\label{mad2oneseries}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
R(a, \, R(b, \, x)) \,\, = \,\, \, \,\,
R(b, \, R(a, \, x)) \,\, = \,\, \, \,\,
R(a\, b, \, x).
\end{eqnarray}
Note that the $\, a \, \rightarrow \, \, 1$
limit of the one-parameter series (\ref{mad2oneseries})
gives as expected
\begin{eqnarray}
\label{mad2oneseries}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
R(1\, + \, \epsilon \cdot \, x)
\,\, = \,\, \, \,\, \,
x \,\, \,+ \, \, \epsilon \cdot \, F(x) \, \,\, + \, \, \cdots
\end{eqnarray}
where:
\begin{eqnarray}
\label{FM3}
\hspace{-0.95in}&&
F(x) \, \, = \, \, \,
x \cdot \, (1 \, -x)^{2/3} \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \, x\Bigr)
\, \, \, = \, \, \,\,
x \,\, \,\, -{\frac {x^2}{2}} \, \, \,
-{\frac {x^3}{7}} \,
\, \, + \,\, \cdots
\end{eqnarray}
\vskip .1cm
\vskip .1cm
\section{$\, _2F_1$ hypergeometric functions deduced from Goursat and Darboux identity}
\label{more2F1GoursatDarboux}
\vskip .2cm
\subsection{$\, _2F_1$ hypergeometric functions deduced from the quadratic identity \\}
\label{more2F1quadra}
Using the quadratic identity
\begin{eqnarray}
\hspace{-0.95in}&& \quad \, \, \,
_2F_1\Bigl([\alpha, \, \beta], \, [ {{\alpha+\beta+1} \over {2}}], \, x\Bigr)
\, \, = \, \, \, \,
_2F_1\Bigl([{{\alpha} \over {2}}, \, {{\beta} \over {2}}], \, [ {{\alpha+\beta+1} \over {2}}], \,
4 \, x \, (1\, -x) \Bigr),
\end{eqnarray}
one deduces:
\begin{eqnarray}
\hspace{-0.95in}&& \quad \quad \quad \quad \, \,
_2F_1\Bigl([{{1} \over {2}}, \, 1], \, [ {{5} \over {4}}], \, x\Bigr)
\, \, = \, \, \, \,
_2F_1\Bigl([{{1} \over {4}}, \, {{1} \over {2}}], \, [ {{5} \over {4}}], \,
4 \, x \, (1\, -x) \Bigr).
\end{eqnarray}
The previously described relations on $\, _2F_1([1/4,\, 1/2], [5/4], x)$,
together with the rational function $\, R(x) $
$\, = \,\, -4 \,x/(1\, -x)^2$, yields the new identity
\begin{eqnarray}
\label{newidR1R2R3}
\hspace{-0.95in}&& \quad \quad \,
(1 \, -2 \, x) \cdot \,
_2F_1\Bigl([{{1} \over {2}}, \, 1], \, [ {{5} \over {4}}], \, x\Bigr)
\, \, \,= \, \, \,\,
_2F_1\Bigl([{{1} \over {2}}, \, 1], \, [ {{5} \over {4}}], \,
- \, 4\,{\frac {x \cdot \, (1 \, -x) }{ (1 \, -2\,x)^2}}\Bigr),
\end{eqnarray}
where we have used the relation
$ \, \, R_3(R_1(x)) \, = \, \, R_2(R_3(x))\, \, $
with:
\begin{eqnarray}
\label{transmutation}
\hspace{-0.95in}&&
R_1(x) \, \, = \, \, \,
- \, 4\,{\frac {x \cdot \, (1 \, -x) }{ (1 \, -2\,x)^2 }},
\quad
R_2(x) \,\, = \, \,\, {{ - \, 4 \, x} \over { (1\, -\, x)^2}},
\quad
R_3(x) \,\, = \, \,\, 4 \, x \cdot \, (1\, -x).
\end{eqnarray}
Introducing
\begin{eqnarray}
\label{newYY}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
Y(x) \, \,\, = \, \, \, \,
x^{1/4} \cdot \, (1 \, -x)^{1/4} \cdot \,
_2F_1\Bigl([{{1} \over {2}}, \, 1], \, [ {{5} \over {4}}], \, x\Bigr),
\end{eqnarray}
one sees that it is solution of
$\,\,\, \Omega \, = \, (D_x \, + \, A_R(x)) \cdot\, D_x\,\, $ with:
\begin{eqnarray}
\label{newOmega}
\hspace{-0.95in}&& \quad \quad \, \,
A_R(x) \, \, = \, \, \,
{{3} \over { 4}} \cdot \,{\frac {2\,x-1}{x \left( x-1 \right) }}
\, \, = \, \, \, {{u'(x)} \over { u(x)}},
\quad \quad \, \,
u(x) \, \, = \, \, \, x^{3/4} \cdot \, (1 \, -x)^{3/4}.
\end{eqnarray}
The rank-two condition is verified with $\, A_R(x)$
given by (\ref{newOmega})
and $\, R(x)$ given by $\, R_1(x)$ in (\ref{transmutation}).
\vskip .2cm
\subsection{$\, _2F_1$ hypergeometric functions deduced from the Goursat identity \\}
\label{more2F1Goursat}
Using the Goursat identity
\begin{eqnarray}
\hspace{-0.96in}&&
_2F_1\Bigl([\alpha, \, \beta], \, [ 2\, \beta], \, x\Bigr)
\, \, = \, \, \, (1\, -\, x/2)^{-a} \cdot \,
_2F_1\Bigl([{{\alpha} \over {2}}, \, {{\alpha+1} \over {2}}], \, [ \beta \, + \, {{1} \over {2}}], \,
{{ \, x^2} \over {(2 \, -\, x)^2}}\Bigr).
\end{eqnarray}
for $\, \alpha \, = \, 1/3$, $\, \beta \, = \, 2/3$,
one gets:
\begin{eqnarray}
\hspace{-0.95in}&& \quad \quad
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \, x\Bigr)
\, \, = \, \, \,\, (1\, -\, x/2)^{-1/3} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \,
{{ \, x^2} \over {(2 \, -\, x)^2}}\Bigr).
\end{eqnarray}
Combining this last identity with (\ref{Q8}) one gets:
\begin{eqnarray}
\hspace{-0.95in}&&
\label{QQ8}
{{x \cdot \, (x \, -2)^3} \over { (1\, -2 \, x)^3}} \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \,
{{x \cdot \, (x \, -2)^3} \over { (1\, -2 \, x)^3}}\Bigr)^3
\, = \, \, \, -8 \cdot \, x \, \cdot \,
_2F_1\Bigl([{{1} \over {3}}, \, {{2} \over {3}}], \, [{{4} \over {3}}], \, x\Bigr)^3
\nonumber\\
\hspace{-0.95in}&& \quad
\, \, = \, \, \, \,
{{16 \, x} \over {x \, -2}} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \,
{{ \, x^2} \over {(2 \, -\, x)^2}}\Bigr)^3
\\
\hspace{-0.95in}&& \quad \, = \, \, \, \,
{{-2 \, x \cdot \, (x-2)^3} \over {x^4+10 \, x^3-12\, x^2+4\, x-2}} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \,
{{ \, x^2 \cdot \, (x-2)^6} \over {(x^4+10 \, x^3-12\, x^2+4\, x-2)^2}} \Bigr)^3.
\nonumber
\end{eqnarray}
It yields the identity on this new hypergeometric function:
\begin{eqnarray}
\hspace{-0.95in}&&
\label{QQQ8}
\quad \quad \quad \quad \, \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \,
{{64 \, x } \over{ (1\, +18 \, x \, -27 \, x^2)^2}}\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad \, \,
\, = \, \, \, \, (1+18\,x-27\,x^2)^{1/3} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{2} \over {3}}], \, [{{7} \over {6}}], \, x\Bigr).
\end{eqnarray}
We have used the relation
$ \, \, R_3(R_1(x)) \, = \, \, R_2(R_3(x))\, \, $
with:
\begin{eqnarray}
\label{transmutation}
\hspace{-0.96in}&&
R_1(x) \, = \, \,
\,{\frac {x \cdot \, (x \, -2)^3 }{ (1 \, -2\,x)^3 }},
\, \, \,\,
R_2(x) \, = \, \,
{{ 64 \, x} \over { (1\, +18 \, x \, -27 \, x^2)^2}},
\, \, \, \,
R_3(z) \, = \, \,
{{ x^2} \over {(2\, -x)^2 }}.
\end{eqnarray}
\vskip .1cm
\section{Miscellaneous rational functions for the covariance of a Heun function}
\label{MiscellHeun}
Let us consider the well-known formula for the addition on elliptic sine:
\begin{eqnarray}
\label{additionellipticsinus}
\hspace{-0.95in}&& \quad \quad \quad \quad
sn(u \,+ \, v) \,\,\, = \, \, \, \, {{
sn(u) \, cn(v) \, dn(v) \,\, + \, \, sn(v) \, cn(u) \, dn(u)
} \over { 1 \, \, - k^2 \, sn(u)^2 \, sn(v)^2
}}.
\end{eqnarray}
Introducing the variables $\, x \, = \, sn(u)^2$, $\, y \, = \, sn(v)^2$
and $\, z \, = \, sn(u \, +v)^2$, and $\, M \, = \, \, 1/k^2$,
the previous addition formula
(\ref{additionellipticsinus}) for the elliptic sine reads:
\begin{eqnarray}
\label{additionellipticsinusbis}
\hspace{-0.95in}&& \quad \quad \quad \quad
(M-xy)^{2}\cdot \, z^{2} \, \, \, +2\,M \cdot \, \Bigl(2\,x y \cdot \, (M \, +1) \,
- \, (x+y) \cdot \, (xy \, +M) \Bigr) \cdot \, z
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad \, \,
+ \,(x-y)^{2} \cdot \, M^{2}
\, \, = \, \, \, 0.
\end{eqnarray}
Note that, since $\, y$ is the square of the elliptic sine,
$\, y \, = \, sn(v)^2\, = \, sn(-v)^2$,
the ``master equation'' (\ref{additionellipticsinusbis})
{\em is also a representation of the difference on elliptic sine}:
$\, x \, = \, sn(u)^2$, $\, y \, = \, sn(v)^2$, $\, z \, = \, sn(u \, -v)^2$.
Actually $\, x$, $\, y$ and $\, z$ are on the same footing in this
``master equation'' (\ref{additionellipticsinusbis}) that can be rewritten
in a symmetric way as an algebraic surface:
\begin{eqnarray}
\label{mastersym}
\hspace{-0.95in}&& \quad \quad \quad \, \quad
x^2 \, y^2 \, z^2 \, \, \,
-2\cdot \, M \cdot \, (x \,+y \, +z) \cdot \, x \, y \, z \, \, \,
+4 \cdot \, M \cdot \, (M+1) \cdot \, x \, y \, z
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \, \, \quad \quad \quad
\, \, +M^2 \cdot \, ((x \,+y \, +z)^2 \, \, -4\cdot \, (x \, y \, +x \, z \, +y \, z))
\, \, \,\, = \, \, \,\, \, 0.
\end{eqnarray}
For every fixed $\, z$ and $\, M$ (except $z \, = \, 0, \, 1, \, M, \, \infty$
and $\, M= \, 0, \, 1, \, \infty$),
condition (\ref{mastersym}) reduces to an
algebraic curve of {\em genus one}. The algebraic surface (\ref{mastersym})
is thus foliated in elliptic curves\footnote[2]{In mathematics, an
{\em elliptic surface} is a surface that has an elliptic fibration: almost
all fibers are smooth curves of genus $\,1$.}. This algebraic surface is
left invariant by an {\em infinite set of birational transformations}
generated by the three involutions:
\begin{eqnarray}
\label{involution}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \,\,
(x, \,\, y ,\, \, z) \quad \, \,\, \,\longrightarrow \, \quad \quad \,
\Bigl( {{M^2\cdot \,(y-z)^2} \over {(M\,-y\,z)^2 \cdot \, x}}, \,\, y, \,\, z \Bigr),
\end{eqnarray}
and the two other ones corresponding to the permutation of $\, x$, $\, y$ and $\, z$.
\vskip .1cm
{\bf Remark:} For fixed $\, z$ condition (\ref{mastersym}) is an elliptic
curve (except $\, M=0, \, 1, \, \infty$). If one calculates its
$\, j$-invariant\footnote[1]{In
Maple use with(algcurves) and the command $\, j\_invariant$.}
one gets the same result as (\ref{jfunc}) namely
\begin{eqnarray}
\label{jfuncbis}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
j \, \, = \, \, \,
256 \cdot \,{\frac { ({M}^{2}-M+1)^{3}}{ M^{2} \cdot \, (M \, -1)^{2}}}.
\end{eqnarray}
{\em which does not depend on} $\, z$. Of course one gets the same result for the
elliptic curves corresponding to condition (\ref{mastersym}) for fixed $\, x$
or fixed $\, y$.
The rational transformation (\ref{Aadoubling}) corresponding to
$\, \theta \, \rightarrow \, \, 2 \, \theta \,$ is obtained by imposing
$\,\, y \, = \, x \,$ in (\ref{additionellipticsinusbis}). For
$\,\, y \, = \, x \,$ the relation (\ref{additionellipticsinusbis})
factorizes\footnote[9]{Other cases of factorizations are,
up to permutations in $\, x$, $\, y$ and $\, z$: $\, y \, =$
$ \, 0, \, 1, \, M, \, \infty$,
$\, y \, = \, M/x$, $\, y \, = \, (M-x)/(1-x)$,
$\, y \, = \, M \cdot \,(1-x)/(M-x)$.} into:
\begin{eqnarray}
\label{additionellipticsinusbisfacto}
\hspace{-0.95in}&& \quad \quad \quad \quad
z \cdot \, \Bigl( (M-x^2)^2 \cdot \, z \,\,
-4\cdot \, M \cdot \, x \cdot \, (1-x) \cdot \, (M-x) \Bigr)
\,\, = \,\,\, \, 0.
\end{eqnarray}
Discarding the trivial solution $\, z \, = \, \, 0$, one gets:
\begin{eqnarray}
\label{additionellipticsinusbisfacto}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
z \, \, = \, \, \,
4 \cdot \, {{ x \cdot \, (1\, -x) \cdot \, (1-x/M)} \over { (1\, -x^2/M)^2}},
\end{eqnarray}
which is exactly (\ref{Aadoubling}).
Imposing in (\ref{additionellipticsinusbis}) $\, y$ to be equal
to (\ref{Aadoubling}) one deduces the rational transformation
corresponding to $\, \theta \, \rightarrow \, \, 3 \, \theta $, and
one can deduce from the ``master'' equation (\ref{additionellipticsinusbis})
all the rational transformations corresponding to
$\, \theta \, \rightarrow \, \, p \, \theta $. When $\, p$ is a
prime number different from $\, p\, = \, 2$, the
corresponding rational transformations have a simple form.
Introducing the square of the elliptic sine
$\, x \, = \, \, sn(\theta, \, k)^2$, the rational transformations
corresponding to
$\, \,\theta \, \rightarrow \, \, p \, \theta \,$
give for a given $\, M$:
\begin{eqnarray}
\label{Aaprime}
\hspace{-0.95in}&& \quad \quad \quad \quad \,\,
R_p(x, \, M) \, \, = \, \, \,
x \cdot \, \Bigl({{ P_p(x, \, M)} \over { Q_p(x, \, M)}} \Bigr)^2,
\quad \quad \quad \quad \quad \hbox{where:}
\\
\hspace{-0.95in}&& \quad \quad \quad \quad \,\,
Q_p(x, \, M) \, \, = \, \, \, \, \,
x^{(p^2-1)/2} \cdot \, M^{(p^2-1)/4} \cdot \,
P_p\Bigl({{1} \over {x}}, \, \, {{1} \over {M}}\Bigr),
\nonumber
\end{eqnarray}
where $ \,P_p(x, \, M)$ are polynomials in $\, x$ and $\, M$
of degree $\, (p^2-1)/2 \, $ in $\, x$ and of degree $\, (p^2-1)/4 \,$
in $\, M$. For instance, $ \,P_3(x, \, M)$ reads:
\begin{eqnarray}
\label{P3zM}
\hspace{-0.95in}&& \quad \quad \quad \quad
P_3(x, \, M) \, \, = \, \, \, \, \,
{x}^{4} \, \,\, -6\,M \cdot \, {x}^{2} \,\,
+4 \cdot \,M \cdot \, (M \,+1) \cdot \, x \,\, \, -3\,{M}^{2}.
\end{eqnarray}
The polynomial $\, P_p(x, \, M)$ reads for $\, p \, = \, 5$:
\begin{eqnarray}
\label{ppfirstprime}
\hspace{-0.95in}&&
P_5(z, \, M) \, \, = \, \, \, {x}^{12} \,
-50\,M\, {x}^{10}\, +140\,M \, (M+1) \cdot \, {x}^{9}\,
-5\,M \, (32\,{M}^{2}+89\,M+32) \cdot \, {x}^{8}\,
\nonumber \\
\hspace{-0.95in}&& \quad \quad
+16\,M \, (M+1) \, (4\,{M}^{2}+31\,M+4) \cdot \, {x}^{7}
\, \, \,
-60\,{M}^{2} \, (4\,{M}^{2}+13\,M+4) \cdot \, {x}^{6}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \,
+360\,{M}^{3} \, (M+1) \cdot \, {x}^{5} \, \,\,
-105\,{M}^{4}\cdot \, {x}^{4}\, \, \,
-80\,{M}^{4} \, (M+1) \cdot \, {x}^{3}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \,
+2\,{M}^{4} \, (8\,{M}^{2}+47\,M+8) \cdot \, {x}^{2}
\,\, \,
-20\,{M}^{5} \, (M+1) \cdot \, x
\, \, \, +5\,{M}^{6},
\end{eqnarray}
It is straightforward to calculate the next $\, P_p(z, \, M)$
for $\, p\, = \, 7, \, 11, \, 13, \, \cdots$, but the
expressions become quickly too large to be given here.
As expected, the two rational functions
(\ref{Aaprime}) {\em commute} for different primes $\, p$.
The series expansion of these rational transformations
read:
\begin{eqnarray}
\label{Aaprimeseries}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
R_p(x) \, \, = \, \, \, \, \,
p^2 \cdot \, x \, \, \, \,
- \, {{ p^2 \cdot \, (p^2-1)} \over {3}}
\cdot \, {{M\, +1} \over {M}} \cdot \, x^2
\, \, \, \, + \, \cdots
\end{eqnarray}
When $\, p$ is not a prime the rational functions $\, R_p(x)$
corresponding to $\, \theta \, \rightarrow \, \, p \, \theta$,
are no longer of the form (\ref{Aaprime}) but they still have
the series expansion (\ref{Aaprimeseries}).
We have the following identity on a Heun function
where $\, R_p(x)$ are the previous rational functions
(\ref{Aaprime}):
\begin{eqnarray}
\label{Fdoublingideapp}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
R_p(x) \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, R_p(x) \Bigr)^2
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
\, \, = \, \, \,
p^2 \cdot \, x \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)^2.
\end{eqnarray}
Note that the Heun identity (\ref{Fdoublingideapp}) is valid
even when the integer $\, p$ is no longer a prime, $\, R_p(x)$ being
a rational function representation of
$\,\,\, \theta \, \rightarrow \, \, p \cdot \, \theta$, and that
all these (commuting) rational transformations are solutions
of the rank-two condition.
\vskip .2cm
\section{The Schwarzian conditions are compatible with the composition of functions}
\label{Schwarzcomp}
We want to have
\begin{eqnarray}
\label{condition1zy}
\hspace{-0.95in}&& \quad \quad \quad \quad \,
W(x)
\, \, \, \,-W(z(y(x))) \cdot \, \Bigl({{d z(y(x))} \over { dx}}\Bigr)^2
\, \, \, \,+ \, \{ z(y(x)), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
which reads using the derivative of composition of function and
the previous chain rule (\ref{chainrule}):
\begin{eqnarray}
\label{condition1zy}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \,
W(x)
\, \, \, \, -W(z(y(x))) \cdot \,
\Bigl({{d z(y)} \over { dy}} \Bigr)^2 \cdot \, y'(x)^2
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad \quad
\, \, \, \,+ \, \{ z(y), \, y \} \cdot y'(x)^2
\, \,\,\, + \, \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
from
\begin{eqnarray}
\label{condition1y}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \, \,
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
and
\begin{eqnarray}
\label{condition1z}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \, \,
W(y)
\, \, \, \,-W(z(y)) \cdot \, z'(y)^2
\, \, \, \,+ \, \{ z(y), \, y\}
\, \,\, \, = \,\, \, \, \, 0.
\end{eqnarray}
Let us multiply the previous relation (\ref{condition1z})
by $\, y'(x)^2$ one gets:
\begin{eqnarray}
\label{condition1y2z}
\hspace{-0.95in}&& \quad
W(y) \cdot \, y'(x)^2
\, \, \, \,-W(z(y)) \cdot \, z'(y)^2 \cdot \, y'(x)^2
\, \, \, \,+ \, \{ z(y), \, y\} \cdot \, y'(x)^2
\, \,\, \, = \,\, \, \, \, 0.
\end{eqnarray}
Adding (\ref{condition1y}) to (\ref{condition1y2z})
one gets:
\begin{eqnarray}
\label{condition1y2z}
\hspace{-0.95in}&& \quad \quad \, \, \,
W(x) \, \,\, + W(y) \cdot \, y'(x)^2
\, \, \, \,-W(z(y)) \cdot \, z'(y)^2 \cdot \, y'(x)^2
\, \, \, \,+ \, \{ z(y), \, y\} \cdot \, y'(x)^2
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0.
\end{eqnarray}
which gives after simplification nothing
but (\ref{condition1zy}). Q. E. D.
\vskip .1cm
\section{Compatibility of the three Schwarzian conditions
(\ref{Harnad111}), (\ref{Harnad214}) and (\ref{condition1})\\}
\label{Schwarzmirror}
\vskip .1cm
The Schwarzian equation on the $\, j$-invariant are known to be invariant
by the group of modular transformations (see for instance equation (1.26)
in~\cite{HarnadHalphen} or (1.13) in~\cite{Harnad}). More remarkably
(and less known) the Schwarzian equation (\ref{Harnad214}) on the
nome\footnote[1]{In the Schwarzian equation (\ref{Harnad214}) the nome
is seen as a function of the Hauptmodul.} is {\em invariant}
under the transformations\footnote[2]{See the concept of
{\em replicable functions}~\cite{Replicable}.}
$\,\, q \, \longrightarrow \, \, S(q) \, = \, \, e \cdot \, q^N$. Equation
(\ref{Harnad214}) is clearly invariant under the rescaling
$\, Q(x) \, \rightarrow \, e \cdot \, Q(x)$, and one can verify easily,
using the chain rule for the Schwarzian derivative of a composition, that
the sum of the first two terms in the LHS of (\ref{Harnad214}), namely
$\,\{Q(x), \, x \} \, + \, Q'(x)^2/Q(x)^2/2\,\, $ is actually invariant by
$\, Q(x)\, \rightarrow \, \, Q(x)^N$. Therefore we also have the equation:
\begin{eqnarray}
\label{Harnad214bis}
\hspace{-0.95in}&& \quad \quad
\, \{S(Q(x)), \, x \} \, \, \, \,
+ {{1} \over {2 \cdot \, S(Q(x))^2 }}
\cdot \Bigl({{ d S(Q(x))} \over {d x}} \Bigr)^2
\, \, + \, W(x) \, \, \, = \, \, \, \, \, 0.
\end{eqnarray}
Equation (\ref{Harnad111}) yields
\begin{eqnarray}
\label{Harnad111bis}
\hspace{-0.95in}&&
\{X(S(Q(q))), \, S(Q(x)) \} \, \, -{{1} \over { 2 \, S(Q(x))^2}} \, \,
- \, W(X(S(Q(q))))
\cdot \, \Bigl( {{ d X(S(Q(x)))} \over {d S(Q(x))}} \Bigr)^2
\, = \,\, \, 0,
\nonumber
\end{eqnarray}
and thus:
\begin{eqnarray}
\label{Harnad111ter}
\hspace{-0.95in}&& \quad \quad
\{X(S(Q(x))), \, S(Q(x)) \} \cdot \, \Bigl({{ d S(Q(x))} \over {d x}} \Bigr)^2
\, \, \,\, -{{1} \over { 2 \, S(Q(x))^2}}
\cdot \Bigl({{ d S(Q(x))} \over {d x}} \Bigr)^2
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad
\, \, \, \,
- \, W(X(S(Q(x))))
\cdot \, \Bigl( {{ d X(S(Q(x)))} \over {d S(Q(x))}} \Bigr)^2
\cdot \Bigl({{ d S(Q(x))} \over {d x}} \Bigr)^2
\, \, \, = \,\,\, \, 0.
\end{eqnarray}
Using the chain rule for Schwarzian derivative of
the composition of functions
\begin{eqnarray}
\label{Harnad111chain}
\hspace{-0.95in}&& \,
\{X(S(Q(x))), \, x\} \, \, \, = \, \, \, \,
\{X(S(Q(x))), \, S(Q(x)) \} \cdot
\Bigl({{ d S(Q(x))} \over {d x}} \Bigr)^2 \, + \, \,
\, \, \{S(Q(x)), \, x \},
\nonumber
\end{eqnarray}
we see immediately that the sum of (\ref{Harnad214bis})
and (\ref{Harnad111ter}) gives:
\begin{eqnarray}
\label{condition1ter}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
\vskip .1cm
\section{The $\, _2F_1([1/6,1/3],[1],x)$ hypergeometric function. }
\label{16131app}
Let us consider the Schwarzian condition in the case of the
$\, _2F_1([1/6,1/3],[1],x)$ hypergeometric function.
The one-parameter
family of commuting series solution of the Schwarzian condition
reads:
\begin{eqnarray}
\label{361}
\hspace{-0.95in}&& \, \, \quad \quad \quad
y_1(e, \, x) \, \, \, = \, \, \, \,
e \cdot \, x \, \,\, \, + \, e \cdot \, (e-1) \cdot \, S_e(x),
\quad \quad \quad \quad \quad \hbox{where:}
\nonumber \\
\hspace{-0.95in}&& \, \, \quad \quad \quad \quad \quad \quad
S_e(x)\, \, = \, \, \, \,
-{\frac {7}{18}} \cdot \, {x}^{2} \,
\, \, \, +{\frac { (109 \,e -283) }{1296}} \cdot \, {x}^{3}
\, \, \, \,+ \, \,\, \cdots
\end{eqnarray}
The series of the form $\, a \cdot \, x^2 \, + \, \cdots \,\, $ reads
\begin{eqnarray}
\label{361y2}
\hspace{-0.95in}&&
y_2(a, \, x) \, = \, \, a \cdot x^{2}
+ {{7 \, a} \over { 9}} \cdot x^{3}
- a \cdot {{84 a -127 } \over {216 }} \cdot x^{4}
- a \cdot {{47628 a - 36049} \over {78732}} \cdot x^{5} + \cdots
\end{eqnarray}
and is such that:
\begin{eqnarray}
\label{such}
\hspace{-0.95in}&& \quad \quad \quad
y_1(e, \,y_2(a, \, x)) \, = \, \, y_2(a\, e, \, x),
\quad \quad \quad
y_2(a, \,y_1(e, \, x)) \, = \, \, y_2(a\, e^2, \, x).
\end{eqnarray}
The series of the form $\, b \cdot \, x^3 \, + \, \cdots \,\, $
reads
\begin{eqnarray}
\label{361y2}
\hspace{-0.95in}&& \,
y_3(b, \, x) \, \, = \, \,\, \, b \cdot x^{3} \, \,
+{{7 \, b} \over {6}} \cdot x^{4} \, \,
+{{479 \, b} \over {432}} \cdot x^{5} \, \,
+ b \cdot \, {{81648 b - 210031} \over {209952 }} \cdot x^{6}
\, \, \, + \, \, \cdots
\end{eqnarray}
and is such that
\begin{eqnarray}
\label{such2}
\hspace{-0.95in}&& \quad \quad \quad
y_1(e, \,y_3(b, \, x)) \, = \, \, y_3(b\, e, \, x),
\quad \quad \quad
y_3(b, \,y_1(e, \, x)) \, = \, \, y_3(b\, e^3, \, x).
\end{eqnarray}
The two series $\, y_2(a, \, x)$ and $\, y_3(b, \, x)$
commute for $\, b \, = \, a^2$. For $\, a \, = \, 1/108$
the series (\ref{361y2}) becomes the series expansion
\begin{eqnarray}
\label{such2}
\hspace{-0.95in}&&
y \, \, = \, \, \, {\frac {{x}^{2}}{108}} \,\, \, +{\frac {7\,{x}^{3}}{972}} \, \, \,
+{\frac {71\,{x}^{4}}{13122}} \,\, \,+{\frac {4451\,{x}^{5}}{1062882}} \,\,
\, +{\frac {63997\,{x}^{6}}{19131876}} \,\, \,
+{\frac {1417505\,{x}^{7}}{516560652}}
\,\, \, + \, \, \cdots
\end{eqnarray}
which corresponds to the modular equation (A.3) in~\cite{IsingCalabi2}:
\begin{eqnarray}
\label{A3}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
4\,x^3\,y^3 \, \,\, -12\,x^2\,y^2 \cdot \,(x+y) \,\, \, \,
+3\, x \,y \cdot \,(4\,x^2-127\,x\,y+4\,y^2) \,
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad
\quad \quad \quad \quad \quad
-4 \cdot \,(x+y)\cdot\,(x^2+83\,x\,y+y^2)\, \,\, +432\,x\,y
\, \,\, \, = \, \, \, \, 0,
\end{eqnarray}
This modular equation has a rational parametrization: it corresponds to the
relation between two rational pullbacks in the hypergeometric
identity (A.11) in~\cite{Christol}:
\begin{eqnarray}
\label{A11}
\hspace{-0.95in}&& \quad \quad \quad
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [1], \, \, 108 \, v^2 \cdot \, (1+4v)\Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\, \,= \, \, \, (1 \, -12\, v)^{-1/2} \cdot \,
_2F_1\Bigl([{{1} \over {6}}, \, {{1} \over {3}}], \, [1], \, \,
-{{ 108 \, v \cdot \, (1+4v)^2} \over { (1 \, -12\, v)^3 }} \Bigr).
\end{eqnarray}
\vskip .1cm
\section{The solutions $\, Q(x)$ of the non-linear conditions (\ref{QDAfirst})
seen as solutions of the Schwarzian conditions on the mirror maps}
\label{subcase}
In all the cases recalled in section (\ref{recalls}), the differentially algebraic
function $\, Q(x)$ was of the form\footnote[1]{The constant $\,N$ being
a positive integer $\, Q(x)$ was, in fact, holonomic.} $\, Y(x)^N$. From
$\, Q(x) \, = \, \, Y(x)^N$ or even
$\, Q(x) \, = \, \, \alpha \cdot \, Y(x)^N$, one can rewrite the Schwarzian
derivative on $\, Q(x)$ with respect to $\, x$:
\begin{eqnarray}
\label{rewritewithresp}
\hspace{-0.95in}&& \quad \quad
\{ Q(x), \, x\} \, \, = \, \,\, \{ Y(x)^N, \, x\} \, \, = \, \,\,
-{{N^2\, -1} \over {2}} \cdot \, \Bigl({{Y'(x)} \over {Y(x)}}\Bigr)^2
\, + \,\,\{ Y(x), \, x\}.
\end{eqnarray}
Since $\, Y(x)$ is a solution of the operator $\, \Omega$,
the ratio $\, Z(x) \, = \, \, Y"(x)/Y'(x)$ (log-derivative of $\, Y'(x)$)
is in fact a rational function,
namely $\, -A_R(x)$. The Schwarzian derivative $\, \{ Y(x), \, x\}$
can also be written as:
\begin{eqnarray}
\label{rewritewithresp}
\hspace{-0.95in}&& \quad \quad \quad
\{ Y(x), \, x\} \, \, = \, \,
Z'(x) \,\, -\, {{Z(x)^2} \over {2}} \, \, = \, \,
-\, A'_R(x) \, -\, {{A_R(x)^2} \over {2}}
\, \, = \, \, -W(x).
\end{eqnarray}
From $\, Q(x) \, = \, \, \alpha \cdot \, Y(x)^N$ one deduces immediately
the relation between the log-derivative of $\, Q(x)$ and $\, Y(x)$,
namely $\, Q'(x)/Q(x) \, = \, N \cdot Y'(x)/Y(x)$.
Equation (\ref{rewritewithresp}) can be rewritten
using $\, Q'(x)/Q(x) \, = \, N \cdot Y'(x)/Y(x)$
and (\ref{rewritewithresp}),
as\footnote[2]{One recovers the Schwarzian condition
(\ref{Harnad214}) in the $\, N \, \rightarrow \, \infty$ limit.}:
\begin{eqnarray}
\label{rewritewithrespN}
\hspace{-0.95in}&& \quad \quad \quad \quad
\{ Q(x), \, x\} \, \, \,
+{{N^2\, -1} \over {2 \, N^2}} \cdot \, \Bigl({{Q'(x)} \over {Q(x)}}\Bigr)^2
\,\, \, + \,\, W(x) \, \, \, = \,\, \, \, 0.
\end{eqnarray}
For instance, one verifies immediately that $\, Q(x)$ given
by $\, Q(x) \, = \, Y(x)^N$ and $\, Y(x)$ given by
(\ref{vid}), (\ref{YM3first}), (\ref{YM6first}), (\ref{zero}),
(\ref{more1}) (which identifies with (\ref{NEWident})) and (\ref{QQQ8first})
are actually solutions of the Schwarzian condition
(\ref{rewritewithrespN})
for the corresponding $\, A_R(x)$ given in (\ref{respecAR})
for respectively $\, N= \, 4, \, 3, \, 6, \, 2, \, 4, \, 6$.
Note that the higher-genus case hypergeometric function (\ref{Ygenus})
is {\em also such that} $\, Q(x) \, = \, Y(x)^6$
{\em is solution of the Schwarzian condition}
(\ref{rewritewithrespN}) with $\, N\, = \, 6$ and $ \, W(x)$ deduced from
$\, A_R(x)$ given by (\ref{aAgenus}).
\vskip .1cm
\vskip .1cm
One gets immediately the Schwarzian condition for the composition
inverse $\, P(x) \, = \, Q^{-1}(x)$, namely:
\begin{eqnarray}
\label{rewritewithrespNrev}
\hspace{-0.95in}&& \quad \quad \quad \quad
\{ X(q), \, q\} \, \, \,
-{{N^2\, -1} \over {2 \, N^2}} \cdot \, {{1} \over { q^2}}
\,\, \, - \,\, W(X(q)) \cdot \, \Bigl( {{ d X(q)} \over {d q}} \Bigr)^2
\, \, \, = \,\, \, \, 0.
\end{eqnarray}
\vskip .1cm
{\bf Remark:} One verifies straightforwardly for the Heun function
example of section (\ref{moreHeun}) that
\begin{eqnarray}
\label{onefindsfi}
\hspace{-0.95in}&& \quad \quad \quad \quad
Q(x) \, = \, \, Y(x)^2 \, = \, \, x \cdot \,
Heun\Bigl(M, \, {{M\, +1} \over {4}}, \, {{1} \over {2}},
\, 1, \, {{3} \over {2}}, \,{{1} \over {2}}, \, x\Bigr)^2,
\end{eqnarray}
is {\em actually solution of the Schwarzian condition}
(\ref{rewritewithrespN}) with $\, N \, = \, \, 2$,
with $\, W(x)$ given by (\ref{AadoublingbisW}). The composition
inverse of the holonomic function $\, Q(x)$
given by (\ref{onefindsfi}) is
\begin{eqnarray}
\label{onefindsinv}
\hspace{-0.95in}&& \quad \quad \quad \, \,
P(x) \,\, = \, \, \, sn\Bigl(x^{1/2}, \, {{1} \over {M^{1/2}}}\Bigr)^2
\\
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\,\, \,\, = \, \, \, \, x \, \, \, \, \,
- {{1} \over {3}} \,{\frac { M \, +1}{M}} \cdot \, x^2 \, \, \, \,
+\, {{1} \over {45}} \,{\frac { 2\,{M}^{2}+13\,M+2}{{M}^{2}}} \cdot \, x^3
\,\, \, \, +\, \, \, \cdots
\nonumber
\end{eqnarray}
It is solution of (\ref{rewritewithrespNrev}) with $\, N\, = \, 2$:
\begin{eqnarray}
\label{rewritewithrespNrevP}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
\{ P(x), \, x\} \, \, \,\,
-{{3} \over {8}} \cdot \, {{1} \over { x^2}}
\,\, \, \,- \,\, W(P(x)) \cdot \, P'(x)^2
\, \, \, = \,\, \, \, 0.
\end{eqnarray}
\vskip .2cm
\subsection{From the Schwarzian condition (\ref{rewritewithrespN})
back to the differential algebraic condition (\ref{simplerelation61first}) on $\, Q(x)$\\}
\label{backto}
If one compares the Schwarzian condition (\ref{rewritewithrespN}) with
the differentially algebraic condition (\ref{simplerelation61first})
on $\, Q(x)$, one finds that they both have a third derivative $\, Q'''(x)$
but one condition depends on a constant $\, N$, while the other one
is ``universal'': let us try to understand the compatibility between
these two conditions. If one eliminates the third derivative $\, Q'''(x)$
between these two equations one finds a remarkably factorized condition
$\, \, E_{+} \cdot \, E_{-} \, = \, \, 0\, \, $ where:
\begin{eqnarray}
\label{factorised}
\hspace{-0.95in}&& \, \, \, \, \,
E_{\pm} \, = \, \, {\cal F}'(x) \, \,
- \, A_R(x) \cdot \, \, {\cal F}(x) \, \, \pm \, {{1} \over { N}}
\quad \quad \quad \hbox{with:} \, \, \quad \quad
{\cal F}(x) \, = \, \, {{Q(x)} \over {Q'(x)}}.
\end{eqnarray}
Recalling (\ref{QF}) we see that $\, {\cal F}(x)$ is nothing but $\, F(x)$.
The holonomic function $\, F(x)$ is known to be solution of
$\, \Omega^{*}$, which can be rewritten, after one integration step,
as $\, F'(x) \, - \, A_R(x) \cdot \, F(x) \, = \, \, Cst$,
which is actually (\ref{factorised}). The compatibility of
the Schwarzian condition (\ref{rewritewithrespN})
with the differentially algebraic condition (\ref{simplerelation61first})
thus corresponds to $\, F(x)$ being annihilated by $\, \Omega^{*}$.
\vskip .1cm
\section{Reduction of $\, _3F_2$ identities to $\, _2F_1$ Schwarzian conditions}
\label{Reduc3F2}
Performing the derivative of the Schwarzian condition
(\ref{condition3F2}) one can eliminate this fourth derivative of $\, y(x)$,
and then, in a second step, eliminate the third derivative of $\, y(x)$
between the previous result and the Schwarzian condition
(\ref{condition3F2}), and so on. One finally gets the following
relation that can be seen as the compatibility condition between
the two previous conditions:
\begin{eqnarray}
\label{compat}
\hspace{-0.95in}&& \quad \quad \quad
x^3 \cdot \, (1\, -x)^3 \cdot \, Q(y(x))
\cdot \, y'(x)^3
\, \, = \, \, \, \,
y(x)^3 \cdot \, (1\, -y(x))^3 \cdot \, Q(x),
\end{eqnarray}
where the polynomial $\, Q(x)$ reads:
\begin{eqnarray}
\label{compatQ}
\hspace{-0.95in}&& \, \, \,
Q(x) \, \, = \, \, \, \,
-2\, \cdot \, (b+c -2\,a) \, (a+ c -2\,b)
\, (a+b -2\,c) \cdot \, {x}^{3}
\nonumber \\
\hspace{-0.95in}&& \quad \quad \, \,
+3 \cdot \, q_2 \cdot \, {x}^{2} \, \, \, +3 \cdot \, q_1 \cdot \, x
\, \, \, -2\, \cdot \, (1+d-2\,e) \, (d+e-2) \, (2\,d -e-1),
\end{eqnarray}
where
\begin{eqnarray}
\label{compatQ2}
\hspace{-0.95in}&& \quad
q_2 \, = \, \, \, 6\,{a}^{2}b\, +6\,{a}^{2}c\, -4\,{a}^{2}d\,\, -4\,{a}^{2}e \,\,
+6\, a {b}^{2} \, \, -18\, a bc \,\, -2\, a bd \,
-2\, a be \,\, +6\, a {c}^{2} \,
\nonumber \\
\hspace{-0.95in}&& \quad \quad
-2\, a cd \,\, -2\, a ce \, \,
+6\, a de \,\, +6\,{b}^{2}c \,\, -4\,{b}^{2}d \, \,
-4\,{b}^{2}e \,\, +6\,b{c}^{2} \,\, -2\,bcd \,\, -2\,bce
\nonumber \\
\hspace{-0.95in}&& \quad \, \quad
+6\,bde \,\, -4\,{c}^{2}d\, -4\,{c}^{2}e \, \,
+6\,cde \,\, +2\,{a}^{2} \,\, + a b \,\,
+ \, a c\, +2\,{b}^{2}\, +bc\, +2\,{c}^{2}
\nonumber \\
\hspace{-0.95in}&& \quad \quad
\, -9\,de \, \, -3\,a\, -3\,b\, -3\,c\, +6\,d\, +6\,e \,\, -3,
\end{eqnarray}
\begin{eqnarray}
\label{compatQ1}
\hspace{-0.97in}&&
q_1 \, = \, \,18\, a bc \, -6\, a bd \, -6\, a be \,
-6\, a cd \, -6\, a ce \, +4\, a {d}^{2}
\, +2\, a de \, +4\, a {e}^{2} \,-6\,bcd \, -6\,bce \,
\nonumber \\
\hspace{-0.97in}&& \quad
+4\,b{d}^{2} \, +2\,bde+4\,b{e}^{2}+4\,c{d}^{2}+2\,cde
+4\,c{e}^{2} \,-6\,{d}^{2}e \,
-6\,d{e}^{2} +3\, a b+3\, a c\, -4\, a d
\nonumber \\
\hspace{-0.97in}&& \quad
\, -4\, a e \, +3\,bc \, -4\,bd \, -4\,be-4\,cd-4\,ce+
21\,de \, +a+b+c \, -6\,d-6\,e+3.
\end{eqnarray}
The condition (\ref{compat}) can be seen as a an equality on
a one-form and the same one-form where $\, x$
has been changed into:
\begin{eqnarray}
\label{compatQ1}
\hspace{-0.95in}&&
\quad \quad \quad \quad
Q(x)^{1/3} \cdot \, {{ dx} \over { x \cdot \, (1\, -x) }}
\, \, = \, \, \, {{ dx} \over { u}}
\, \, = \, \, \, \,
Q(y)^{1/3} \cdot \, {{ dy} \over { y \cdot \, (1\, -y) }}.
\end{eqnarray}
This one-form is clearly associated with the algebraic curve:
\begin{eqnarray}
\label{curveAA}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad \quad
Q(x)\cdot u^3 \, \, = \, \,\, \, x \cdot \, (1\, -x).
\end{eqnarray}
One actually finds that this algebraic curve (\ref{curveAA})
is a {\em genus-one curve}.
One can go a step further by eliminating all
the derivatives $\, y'(x)$,$\, y''(x)$,$\, y'''(x)$,
from the confrontation of the Schwarzian condition
(\ref{condition3F2}) with the compatibility condition
(\ref{compat}). One gets that way (after some calculation)
a condition reading
\begin{eqnarray}
\label{curveone}
\hspace{-0.95in}&& \quad \quad \quad \quad
{\cal I}(x) \, = \, \, \, {\cal I}(y(x))
\quad \quad \quad \hbox{where:}
\quad \quad \quad \quad {\cal I}(x)
\, = \, \, \, {{Q(x)^8 } \over { P_8(x)^3 }},
\end{eqnarray}
where $\, P_8(x)$ is a (quite large) polynomial
of degree $\, 8$ in $\, x$, sum of 4724 terms.
We are seeking for non-trivial pullbacks $\, y(x)$ being
different from the obvious solution $\, y(x) \, = \, \, x$.
The interesting cases for physics are the one where
$\, x \, \, \rightarrow \, \, y(x)$ is an infinite order
transformation. In such cases one has
\begin{eqnarray}
\label{curvetwo}
\hspace{-0.95in}&& \quad \quad \quad \quad
{\cal I}(x) \, = \, \, \, {\cal I}(y(x))\, = \, \, \,{\cal I}(y(y(x)))
\, = \, \, \,{\cal I}(y(y(y(x)))) \, \, = \, \, \, \cdots
\end{eqnarray}
which amounts to saying that $\, {\cal I}(x)$ must be a constant.
The cases where $\, Q(x)^8 \, = \, \, \, \lambda \cdot \, P_8(x)^3$
correspond to a set of extremely large conditions on the
parameters $\, a, \, b, \, c, \, d, \, e$ of the $\, _3F_2$
hypergeometric function, that is difficult to study
because of the size of polynomial $\, P_8(x)$.
However a simple case can fortunately be analyzed, namely
$ \, {\cal I}(x) \, = \, \, \, 0$, which corresponds to
$\, Q(x) \, = \, \, \, 0$.
In such a case the two conditions are compatible, and one just has one
condition: the Schwarzian condition (\ref{condition3F2}) with
the additional condition being automatically verified (see (\ref{compat})).
One finds that all the
conditions on the parameters $\, a, \, b, \, c, \, d, \, e$
of the $\, _3F_2$ hypergeometric function associated with
$\, Q(x) \, = \, 0\, $
in fact correspond to cases where the order-three operator
is exactly the symmetric
power of a second order operator have $\, _2F_1$ solutions.
In other words this situation corresponds to
the {\em Clausen identity}, the $\, _3F_2$ hypergeometric function
reducing to the square of a $\, _2F_1$ hypergeometric function:
\begin{eqnarray}
\label{Clausen}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad
_3F_2\Bigl([2\,a, \, a\, +b, \, 2\, b], \,
[a\, +b\, +{{1} \over {2}}, \, 2\,a \, +2 \, b], \, x \Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad \quad \quad
\quad \quad \quad \quad \quad \, \, \,
\, = \, \, \,
_2F_1\Bigl([a, \, b], \, [a\, +b\, +{{1} \over {2}}], \, x \Bigr)^2.
\end{eqnarray}
In this Clausen identity case, we found that the Schwarzian
condition (\ref{condition3F2}) for $\, _3F_2$ is nothing but the
Schwarzian condition for the underlying $\, _2F_1$.
If one considers the other case $\, P_8(x)\, = \,\, 0$ the
vanishing condition of the $\, x^8$ coefficient
and the
vanishing condition of the constant coefficient in $\, x$ read
respectively:
\begin{eqnarray}
\label{C8}
\hspace{-0.95in}&& \quad
({a}^{2}-ab-ac+{b}^{2}-bc+{c}^{2})
\cdot \, (a \, +c -2\,b)^{2} \cdot \, (a+b\, -2\,c)^{2}
\cdot \, (2\,a \, -c-b)^{2} \, \, = \, \, \, 0,
\nonumber \\
\label{C0}
\hspace{-0.95in}&& \quad
({d}^{2}-de+{e}^{2}-d-e+1) \cdot \,
(1+d \,-2\,e)^{2} \cdot \, \, (d +e \, -2)^{2} \cdot \, (2\,d -e-1)^{2}
\, \, = \, \, \, 0.
\nonumber \
\end{eqnarray}
These two conditions are, respectively, very similar to the
vanishing condition of the $\, x^3$ and constant coefficient
of $\, Q(x)$, the other coefficients of $\, P_8(x)$ being
more involved. The vanishing condition of all the
$\, x^n$ coefficients of $\, P_8(x)$ yields more relations on the
$\, a, \, b, \, c, \, d, \, d, \, e$ parameters. All these
miscellaneous cases correspond to cases where the order-three
linear differential operator reduces to the symmetric square
of an order-two operator, and to the Clausen identities
of the form (\ref{Clausen}). More simply on can verify that
for parameters such that $\, Q(x) \, = \, \, 0$
(for which a Clausen reduction take place (\ref{Clausen}))
are also such that $\, P_8(x)\, = \,\, 0$
(the invariant $\, {\cal I}(x)$
in (\ref{curveone}) is thus of the form $\, 0/0$).
\vskip .1cm
\section{Schwarzian condition and other generalised hypergeometric functions}
\label{Schwarz4F32F2}
\vskip .1cm
\subsection{Schwarzian condition and $\, _4F_3$ hypergeometric functions}
\label{Schwarz4F3}
\vskip .1cm
Let us consider a $\, _4F_3$ hypergeometric known to correspond
to a Calabi-Yau ODE~\cite{IsingCalabi,IsingCalabi2} and seek
an identity of the form:
\begin{eqnarray}
\label{modularform4F3}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
{\cal A}(x) \cdot \, _4F_3\Bigl(
[{{1} \over {2}}, \, {{1} \over {2}}, \, {{1} \over {2}}, \, {{1} \over {2}}],
\, [1, \, 1, \, 1], \, x \Bigr)
\nonumber \\
\hspace{-0.95in}&& \quad \quad
\quad \quad \quad \quad \quad \quad \quad \quad
\, = \, \, \, \,\,
_4F_3\Bigl(
[{{1} \over {2}}, \, {{1} \over {2}}, \, {{1} \over {2}}, \, {{1} \over {2}}],
\, [1, \, 1, \, 1], \, y(x) \Bigr)
\end{eqnarray}
where $\, {\cal A}(x)$ is an algebraic function.
Again we introduce the order-four linear differential operator
annihilating the LHS and RHS of identity (\ref{modularform4F3}).
The equality of the wronskians of these two linear differential
operators enables us to get the expression of $\, {\cal A}(x)$
in terms of $\, y(x)$, namely:
\begin{eqnarray}
\label{A4F3}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
{\cal A}(x) \, \, = \, \, \,
\Bigl( {{ (1-y(x))\cdot \, y(x)^3 } \over {
(1\, -x) \cdot \, x^3 \cdot \, y'(x)^3}} \Bigr)^{1/2}.
\end{eqnarray}
After eliminating $\, {\cal A}(x)$ from (\ref{A4F3}), the identification
of the $\, D_x^2$ coefficients for these two linear differential operators
of order four gives the Schwarzian condition
\begin{eqnarray}
\label{condition4F3}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
W(x)
\, \, \, \,-W(y(x)) \cdot \, y'(x)^2
\, \, \, \,+ \, \{ y(x), \, x\}
\, \,\, \, = \,\, \, \, \, 0,
\end{eqnarray}
where $\, W(x)$ reads:
\begin{eqnarray}
\label{condition4F3W}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad
W(x)\, \, = \, \, \,
-\, {{1} \over { 10}} \cdot \,
{\frac {5\,{x}^{2} \, -7\,x\, +5}{{x}^{2} \cdot \, (1\, - x)^{2}}}.
\end{eqnarray}
The condition corresponding to the identification
of the $\, D_x$ coefficient can be seen to be compatible with the previous
Schwarzian condition: it can be seen to be a consequence of
condition (\ref{condition4F3}), corresponding to
a combination of (\ref{condition4F3})
with the derivative of (\ref{condition4F3}). One
finds, unfortunately, that the only solution is the trivial solution
$\, y(x) \, = \, \, x$, the other solutions being spurious solutions
$\, y \, +5 \, = \, 0$, $\, 5\,y \, +1 \, = \, 0$,
$\, \,y \, -1 \, = \, 0$, etc ...
\vskip .1cm
\vskip .1cm
{\bf Remark:} Similarly to what has been performed
in section (\ref{Schwarz4F3}) one can imagine
to seek for an identity (\ref{modularform4F3}) but, now,
for the general $\, _4F_3$ hypergeometric function
$\, _4F_3([a,\, b, \, c, \, d], \, [e, \, f, \, g], \, x)$. These
calculations are really too large.
\vskip .1cm
\subsection{Schwarzian condition and hypergeometric functions with irregular singularities}
\label{Schwarz2F2}
\vskip .1cm
The $\, n$-fold integrals emerging in
lattice statistical mechanics or enumerative combinatorics
are naturally diagonal of rational functions~\cite{Christol},
the corresponding linear differential operators being
globally nilpotent, and in particular Fuchsian. In such a lattice
framework only $\, _nF_{n-1}$ hypergeometric functions~\cite{Heckman}
with {\em regular} singularities occur. Of course {\em irregular}
singularities can also occur in physics~\cite{penson1,penson2,penson3},
in particular in the {\em scaling limit} of lattice models~\cite{scaling,Holo}
(modified Bessel functions, etc ...).
Let us consider a hypergeometric function with an {\em irregular}
singularity, namely
a simple $\, _2F_2$ hypergeometric function
solution of an order-three linear differential operator.
We seek an identity of the form:
\begin{eqnarray}
\label{modularform4F3bis}
\hspace{-0.95in}&& \quad \quad \quad \quad
{\cal A}(x) \cdot \, _2F_2\Bigl(
[{{1} \over {2}}, \, {{1} \over {2}}],
\, [1, \, 1], \, x \Bigr)
\,\,\, = \, \, \, \,\,
_2F_2\Bigl(
[{{1} \over {2}}, \, {{1} \over {2}}],
\, [1, \, 1], \, y(x) \Bigr).
\end{eqnarray}
The calculations are the same as in section (\ref{Schwarz3F2}). The identification
of the wronskians of the two operators (the pullbacked order-three linear
differential operator and the conjugated one) gives
\begin{eqnarray}
\label{A2F2}
\hspace{-0.95in}&& \quad \quad \quad \quad \quad \quad \quad
{\cal A}(x) \, \, = \, \, \, \exp\Bigl( {{x\, -y(x)} \over {3}} \Bigr) \cdot \,
{{ y(x) } \over {
\, x \cdot \, y'(x)}},
\end{eqnarray}
and the Schwarzian equation:
\begin{eqnarray}
\label{condition2F2bis}
\hspace{-0.95in}&&
W(x)
\, \,-W(y(x)) \cdot \, y'(x)^2
\, \,+ \, \{ y(x), \, x\}
\, \, = \,\, \, 0, \quad \hbox{with:}\quad
W(x)\, = \,\, {{1} \over { 6}} \cdot \, {{x^2 \, -3 } \over {x^2}}.
\end{eqnarray}
However, combining equation (\ref{condition2F2bis}) with the last
condition emerging from the identification of the terms with no $\, D_x$
in the two operators, one finds that there is no pullback $\, y(x)$
for (\ref{modularform4F3bis}) except the trivial solution
$\, y(x) \, = \, x$.
\vskip .2cm
\vskip .3cm
\vskip .5cm
\vskip .5cm
\vskip .5cm | 219,878 |
This site is about the subversive, the strange, the peculiar around us.
It is about things that we do not see, until they are captured with a camera. For many years I have been hunting this world of absurd to share it with you. So, welcome and enjoy the show!
Irakly Shanidze | 238,184 |
An average intern at the European Commission is 26 years old, speaks four languages and has two diplomas.
Interns working in Brussels recently (July 2013) held a protest for better work conditions and pay. Some people make more than EUR 1,000 a month, but most of them work without pay for months. A report on ‘life as an intern’ from the capital of the EU.
Each year, the European Commission employs 600 “Blue Book trainees,” considered to be the cream of the crop of interns, for a period of five months and a salary of a bit more than EUR 1,100 a month. The European Parliament, the European Central Bank, the Economic and Social Committee and the European Ombudsman offer internships with similar conditions, but to much fewer people. Having a university degree is a must everywhere, and experience indicates that for an application to be successful, other foreign language skills are necessary beyond English or French. At least one-third of the salary is spent on housing and another one-third on food. Those who manage their finances well and forgo luxuries such as evenings out or a visit home with the family, are left with around EUR 200-300 at the end of the month. Most people spend the money they saved on traveling after the program ends, mostly since it is rare to land a job right after the internship ends.
The political jobs: European Parliament
The European Parliament groups, as well as the EP delegations of national groups also have their own internship programs. Conditions are no worse here than at the official programs of the EU institutions, but political-ideological alignment is required and applications are unlikely to be successful without a contact on the inside. Steven started working with an MEP when he was 23: “As with many other things, the work hours also depend on the given representative. Officially, work hours are from 8:30 a.m. to 5:30 p.m. on weekdays and to 2:30 p.m. on Friday. If I remember correctly, not once was I able to keep to these hours, as I assisted a workaholic MEP, so there were even 12-hour days. But no matter, someone has to compensate for some other interns who showed up at 10 with a hangover and went to siesta after lunch (and never came back),” he added.
Money matters
However, most of the several thousand interns in Brussels work for less money than this, or even for free, working as much as 10–12 hours a day at lobby firms, professional or political institutes and companies established around EU institutions. The interns finance their expense of living in Brussels, which comes to at least EUR 600–700 a month, either partly or fully themselves. At the end of the internship, there is usually no opportunity to move up within the organization’s ranks.
According to the organizers of the Brussels protest, most interns do the same level of work as employees that work for a full salary. Moreover, many of the interns not only have the knowledge and language skills they learned at university, but have also gained work experience during their university years and are able to work independently or as part of an organization.
The protest’s organizers admit that even so, interns do not count as fully-fledged employees and that the goal of the internship is primarily for juniors to learn to apply the skills learned at university. Nonetheless, they also point out that this often takes place amid humiliating circumstances.
Sandwich for lunch?
This is also what the protest’s slogan symbolizes: when did you last have something other than a sandwich for lunch? According to Steven, quoted above, “the knowledge you gain at university is useful but not always necessary, especially if representatives use interns to serve them coffee or to fetch the lipstick they left behind at the office, because that’s what is really needed to make them look good on the umpteenth photograph.”
Even so, Brussels remains attractive to fresh graduates dealing with EU or international affairs since the international work experience looks good on their CV, the city itself is busting with life, and it is easier to build the contacts needed to land a good job later than anywhere else.
When work ends…
Many believe that what really matters starts only after the work ends. According to a former intern, “there are some people who only see Brussels at night. But this is also a good time to get to know the city, which by the way is not that lively. When I began my stay in Brussels, I opted to follow the path of party-sightseeing-party-sightseeing-party-exhaustion.”
However, unpaid internships or ones that pay less than is needed to finance living expenses lead to adverse selection. Those people whose families are unable to finance the internship fall out of the system and start off with a disadvantage on the labor market. It is also not unusual for highly trained and experienced young people to be unable to break out of life as an intern for years, as full employment is expensive and the effects of the economic crisis are felt in Brussels as well.
Image credit: | 177,932 |
You can’t make this stuff up.
“I know I had told you I’d give you a heads up when I launched so you could see your amazing work in action. Things got a little crazy on my end so I just went live with it last week. The feedback has been amazing so far – everyone is wowed by how beautiful and professional it looks, and that’s all thanks to you. I’m so thrilled with how it came out! Thanks again for all your wonderful hard work.”
“Shari was wonderful! She over delivered with my blog project! Shari was professional, polite, and super responsive. I’ve had a few digital projects designed in the past and nothing compares to this experience! Shari has exceptional customer service and she delivered my project earlier than I expected. I will definitely use Little Blue Deer again in the future!”
“I can’t tell you how happy I am to have found you. THANK YOU doesn’t begin to cut it! You have been such a joy to work with and truly set the tone for my new blog. It’s very exciting to have something so beautiful and professional to put out into the world so that in return people will look at me seriously and get a sense for what I’m trying to convey. I couldn’t be more appreciative of your patience and help and I am thrilled to continue utilizing your expertise in the future!”
| 399,047 |
I graduated from the University of Maryland in May and I recently moved to Washington, D.C.to start my life. Over the summer, I took time to enjoy my last free summer for the next 43 years (assuming I retire at age 65), and now I’ve started to work full time, making a respectable amount of money as a Systems Analyst, whatever that means.
Being Frugal
I’ve always believed in not paying full price. That used to mean just looking for coupons before making big purchases, but it slowly turned into watching the prices of external hard drives drop while the amount of space offered increased drastically. Of course, by the time I decided that I had found a good price, it was time for a new computer, which came with more space than I knew what to do with. I’ve also learned how to reduce monthly bills and have had literally thousands of dollars taken off various AT&T bills, some because my brother decided to put his SIM card in an iPhone without paying for a data plan, and others because I simply thought that our family shouldn’t be paying for things like text messages. Or minutes.
Although I may not have too much life experience to offer, and I certainly won’t be writing about real estate or whether to purchase an annuity, I know a lot about wasting money and making poor decisions with money. We all make mistakes like paying for that magazine from the kid who comes door to door trying to “pay for college”, or the Razor Scooter that all your friends have but you’ll never use. I see people wasting money every day on things they don’t need, and it drives me crazy. I want to help, and rather than calling up the phone and cable companies for each of my friends, I think this is a better way of reaching people.
About a year ago, as I was preparing my first serious job search, I started reading personal finance articles. This started as reading the articles in Yahoo! and eventually led me to blogs such as I Will Teach You to be Rich and Get Rich Slowly, among others. To the right is a list of blogs I read daily and have been the start of me falling in love with the idea of personal finance. It just seems to fit in with how I already live my life.
Thanks for subtly referencing me… | 318,139 |
\begin{document}
\maketitle
\begin{center}
\textbf{\author{Brijesh Kumar Tripathi and $ ^{*} $V. K. Chaubey}}
\end{center}
\begin{center}
Department of Mathematics, L.D. College of Engineering, Ahmedabad \\* (Gujarat)-380015, India, E-mail:[email protected]
\end{center}
\begin{center}
$ ^{*} $ Department of Applied Sciences, Buddha Institute of Technology, GIDA \\* Gorakhpur, U.P., 273209, India, E-Mail: [email protected]
\end{center}
\begin{abstract}
Igarashi introduce the concept of $(\alpha, \beta)$-metric in Cartan space $\ell^{n}$ analogously to one in Finsler space and obtained the basic important geometric properties and also investigate the special class of the space with $(\alpha, \beta)$-metric in $\ell^{n}$ in terms of $ 'invariants' $. In the present paper we determine the $ 'invariants' $ in two different cases of deformed infinite series metric which characterize the special classes of Cartan spaces $\ell^{n}$.
\end{abstract}
\textbf{Mathematics Subject Classifications:} 53C60,53B40\\
\textbf{Keywords:} Finsler space, Cartan Space, $(\alpha,\beta)$-metrics, Riemannian metric, One form metric, Infinite series metric.
\section{Introduction}
\hspace{10pt} In 2004 Lee and Park \cite{LP} introduced the concept of r-th series $(\alpha,\beta)$-metric where r is varies from $ 0, 1, 2, ..., \infty $ and give very interesting example of special $(\alpha, \beta)$-metric for the different values of r such as one-form metric, Randers metric, combination of Kropina and Randers metric, infinite series metric etc.\\
In 1994, Igarashi \cite{Iga1, Iga2} introduce the concept of $(\alpha, \beta)$-metric in Cartan space $\ell^{n}$ analogously to one in Finsler space and obtained the basic important geometric properties and also investigate the special class of the space with $(\alpha, \beta)$-metric in $\ell^{n}$ in terms of $ 'invariants' $. The classes which he obtained includes the spaces corresponding to Randers and Kropina space. Further he characterizes these spacial classes by means of $ 'invariants' $ in case of Finsler theory.\\
In the present paper we determine the $ 'invariants' $ in two different cases of deformed infinite series metric in which firest metric is defined as the product of infinite series and Riemannian metric another one is the product of infinite series and one-form metric. Further we characterize the special classes of Cartan spaces $\ell^{n}$ in case of these two metrics and also investigate the relation under which "invariants" are characterized as the special classes of $\ell^{n}$
\section{Preliminaries}
E. Cartan \cite{Car} introduced the concept of a Cartan Space, where the measure of its hypersurface element $(x,y)$ is given a priori by homogeneous function $F(x,y)$ of degree one in y,i.e.,the "area" of a domain on hypersurface $S_{n-1} : x^{i} = x^{i}(v^{1},v^{2},v^{3},.....,v^{n}), \;\;\; i=1,2,3,....,n$ is given by
\begin{equation}
S=\underbrace{\idotsint}F(x,y)dv^{1}dv^{2}dv^{3}.....dv^{n-1}
\end{equation}
where $ y = (y_{i})$ is the determinant of $ (n-1, n-1)$ minor matrix omitted ith row of $(n, n-1)$ matrix $(\frac{\partial x^{i}}{\partial v^{\alpha}})$, \;\;\; $\alpha=1,2,3.....n-1$. In this space, we obtain the fundamental tensor by
\begin{equation}
g^{ij}=G^{-\frac{1}{n-1}},\;\;\; G=det\lVert G^{ij}\rVert, \;\;\; G^{ij}=\frac{\partial^{2}(\frac{1}{2}F^{2})}{\partial y_{i}\partial y_{j}}, \;\;\; (y_{i})\neq 0.
\end{equation}
As the special case for the fundamental tensor $a_{ij}$ of Riemannian space, we can find the $(n-1)$ dimensional area of a domain on hypersurface such that
\begin{center}
$S=\underbrace{\idotsint}\sqrt{det\lVert a_{ij}(x)\frac{\partial x^{i}}{\partial v^{\alpha}}\frac{\partial x^{j}}{\partial v^{\beta}}\rVert}dv^{1}dv^{2}dv^{3}.....dv^{n-1}$
\end{center}
hence it is clear that Riemannian space is a special case of Cartan space. On the other hand, Cartan space is considered the dual notion of Finsler Space. Further the relation between both spaces is studied by L. Berwald \cite{Be} in early days, afterwards, by H. Rund \cite{Ra} and F.Brickel \cite{Br}. Recently R. Miron \cite{Mi1, Mi2} established new Carton geometry which shows totally different feature in the form of particularization the Hamilton space which defined as:
\begin{definition}
A Cartan space is a Hamilton space $\mathcal{H}^{n}=\{M,\underbrace{H(x,y)}\}$ in which the fundamental function $ H(x,y)$ is positively 2-homogeneous in $ y_{i}$ on $T^{*}M$. We denote it by $\ell^{n}$.
\end{definition}
The fundamental tensor field of $\ell^{n}$ and its reciprocal $g_{ij}(x,y)$ is given by
\begin{equation}
g^{ij}(x,y)=\frac{1}{2}\dot{\partial^{i}}\dot{\partial^{j}}H,
\end{equation}
\begin{equation}
g_{ij}(x,y)g^{jk}(x,y)=\delta_{i}^{k}
\end{equation}
The homogeneity of $H(x,y)$ is expressed by
\begin{equation}
y_{j}\dot{\partial^{j}}H=2H,\,\, which\,\, also\,\, implies\,\, H=g^{ij}y_{i}y_{j}
\end{equation}
where $g^{ij}(x,y)$ and its reciprocal $g_{ij}(x,y)$ are both symmetric and homogeneous of degree 0 in $y_{i}$.\\\\
On the other hand, the Finsler spaces with $(\alpha,\beta)$-metric were considered by G. Randers \cite{Ra}, V. K. Kropina \cite{Kro} and M. Matsumoto \cite{M1, M2, M3}, especially the last paper shows the great success for investigation of these spaces.\\
In \cite{Mi1}, R. Miron expected the existence of Randers type metric:
\begin{equation}
H(x,y)=\lbrace\alpha(x,y)+\beta(x,y)\rbrace^{2}
\end{equation}
and of Kropina's one :
\begin{equation}
H(x,y)=\{\frac{[\alpha(x,y)]^{2}}{\beta(x,y)}\}^{2},\,\,\,\,\,\,\,\,\beta(x,y)\neq 0.
\end{equation}
in Cartan spaces. Here we have put as
\begin{equation}
\alpha^{2}(x,y)=a^{ij}(x) y_{i} y_{j},\,\,\,\,\beta(x,y)=b^{i}(x)y_{i},
\end{equation}
$a^{ij}(x)$ being a Riemannian metric on the base manifold M and $b^{i}(x)$ a vector field on M such that $\beta>0$ on a region of $T^{*}M\stackrel{def}{=} T^{*}M-\lbrace 0 \rbrace $.
\section{Cartan spaces with $(\alpha,\beta)$-metric.}
Cartan spaces with $(\alpha,\beta)$-metric \cite{Iga1, Iga2, Mi1} can be defiend as
\begin{definition}
A Cartan space $\ell^{n}=\{M, H(x,y)\}$ is known as Cartan space with $(\alpha, \beta)$-metric if its fundamental metric $H(x,y)$ is a function of $\alpha(x,y)$ and $\beta(x,y)$ only i.e.
\begin{equation}
H(x,y)=\breve{H}\{\alpha(x,y), \beta(x,y)\}
\end{equation}
\end{definition}
It is clear that $\breve{H}$ satisfy the conditions imposed to the function $H(x,y)$ as a fundamental function for $\ell^{n}$. Then
\begin{equation}
t\alpha(x,y)=\alpha(x,ty),\;\; t\beta(x,y)=\beta(x,ty),\;\; H(x,ty)=t^{2}H(x,y)\;\; t>0.
\end{equation}
It follows:
\begin{proposition}
The function $\breve{H}(\alpha(x,y),\beta(x,y))$ is positively homogeneous of degree 2 in both $\alpha$ and $\beta$.
\end{proposition}
By this reason, there maybe no confusion if we adopt the notation $H(\alpha,\beta)$ itself instead of $\breve{H}(\alpha,\beta)$. Also we write such that
\begin{equation}
H_{\alpha}=\frac{\partial H}{\partial \alpha},\,\,H_{\beta}=\frac{\partial H}{\partial \beta},\,\,H_{\alpha\beta}=\frac{\partial^{2} H}{\partial \alpha\partial \beta},\,\,etc
\end{equation}
\begin{proposition}
The following identities hold :
\begin{equation}
\alpha H_{\alpha\alpha}+\beta H_{\beta\beta}=H,\,\,\alpha H_{\alpha\beta}+\beta H_{\beta\beta}=H_{\beta},
\end{equation}
\begin{equation}
\alpha^{2}H_{\alpha \alpha}+2\alpha\beta H_{\alpha\beta}+\beta^{2} H_{\beta\beta}=\alpha H_{\alpha}+\beta H_{\beta}=2H \nonumber .
\end{equation}
\end{proposition}
Differentiating $ \alpha $ and $ \beta $ with respect to $y_{i}$ we have
\begin{equation}
\dot{\partial^{i}} \alpha=\alpha^{-1}a^{ij}y_{j}=\alpha^{-1}Y^{i},\;\;\;\;\;\dot{\partial^{i}}\beta=b^{i}(x),
\end{equation}
where
\begin{equation}
Y^i(x,y)=a^{ij}(x)y_{i},\;\;\; \{Y=(Y^{i})\}\ne 0
\end{equation}
and the vector field $ Y^{i}$ satisfies the relation
\begin{equation}
Y^{i}y_{i}=\alpha^{2},
\end{equation}
Further let
\begin{equation}
B_{i}(x)=a_{ij}(x)b^{j}(x),\;\;\;B^{2}(x)=a^{ij}B_{i}B_{j}=a_{ij}b^{i}b^{j},
\end{equation}
\begin{equation}
y^{i}(x,y)=g^{ij}(x,y)y_{j}=\frac{1}{2}\frac{\partial H}{\partial y_{i}}
\end{equation}
Also, we have the relation similar to (3.7):
\begin{equation}
y^{i}y_{i}=H(x,y).
\end{equation}
Differentiating (3.6) and (3.9) by $y_{i}$ succeedingly, we have
\begin{equation}
\dot{\partial ^{j}}Y^{i}=a^{ij}(x),\;\;\; \dot{\partial ^{k}}\dot{\partial^{j}}Y^{i}=0,
\end{equation}
\begin{equation}
\dot{\partial^{j}}y^{i}=g^{ij},\;\;\; \dot{\partial^{k}}\dot{\partial^{j}}y^{i}=\dot{\partial^{k}}g^{ij}=-2C^{ijk},
\end{equation}
And using the same manner for (3.5), we get
\begin{equation}
\dot{\partial^{j}}\dot{\partial^{i}}\alpha=\alpha^{-1} a^{ij}(x)-\alpha^{-3}Y^{i}Y^{j},\;\;\dot{\partial^{j}}\dot{\partial^{i}}\beta=0,\;\;\;\dot{\partial^{j}}\dot{\partial^{i}}(\frac{1}{2}\alpha^{2})=a^{ij}.
\end{equation}
On account of (3.5) and $y^{i}=\frac{1}{2}(H_{\alpha}\dot{\partial^{i}}\alpha+H_{\beta}\dot{\partial^{i}}\beta)$, we have
\begin{lemma}
The Liouville vector field $y^{i}$ is expressed in the form
\begin{equation}
y^{i}=\rho_{1}b^{i}+\rho Y^{i},
\end{equation}
where
\begin{equation}
\rho_{1}=\frac{1}{2}H_{\beta},\;\;\;\; \rho=\frac{1}{2\alpha}H_{\alpha}
\end{equation}
\end{lemma}
Taking into account that the relation $\dot{\partial^{i}}\rho =\frac{\partial\rho}{\partial \alpha}\dot{\partial^{i}}\alpha +\frac{\partial\rho}{\partial \beta}\dot{\partial^{i}}\beta $ holds, we have
\begin{lemma}
The quantities $\rho_{1}$ and $\rho$ satify the relations
\begin{equation}
\dot{\partial^{i}}\rho_{1} =\rho_{0}b^{i}+ \rho_{-1}Y^{i},\;\;\;\; \dot{\partial^{i}}\rho =\rho_{-1}b^{i}+ \rho_{-2}Y^{i}
\end{equation}
respectively, where
\begin{equation}
\rho_{0}=\frac{1}{2}H_{\beta\beta},\;\;\; \rho_{-1}=\frac{1}{2\alpha}H_{\alpha\beta},\;\;\;\rho_{-2}=\frac{1}{2\alpha^{2}}(H_{\alpha\alpha}-\alpha^{-1}H_{\alpha})
\end{equation}
\end{lemma}
Contracting $\dot{\partial^{i}}\rho_{1}$,\;\;$ \dot{\partial^{i}}\rho$ in (3.16) by $y_{i}$,we have\\
\begin{equation}
y_{i}\dot{\partial^{i}}\rho_1=\alpha^{2}\rho_{-1}+\beta\rho_{0}=\rho_{1},\;\;\;y_{i}\dot{\partial^{i}}\rho=\alpha^{2}\rho_{-2}+\beta\rho_{-1}=0.
\end{equation}
Analogously to deduction of Lemma (3.2), we have also
\begin{lemma}
The quantities $\rho_{0}$ and $\rho_{-1}$ satisfy the relations
\begin{equation}
\dot{\partial^{i}}\rho_{0} =r_{-1}b^{i}+ r_{-2}Y^{i},\;\;\;\; \dot{\partial^{i}}\rho_{-1} =r_{-2}b^{i}+ r_{-3}Y^{i},
\end{equation}
respectively, where
\begin{equation}
r_{-1}=\frac{1}{2}H_{\beta\beta\beta},\;\;\; r_{-2}=\frac{1}{2\alpha}H_{\alpha\beta\beta},\;\;\;r_{-3}=\frac{1}{2\alpha^{2}}(H_{\alpha\alpha\beta}-\alpha^{-1}H_{\alpha\beta}).
\end{equation}
\end{lemma}
\;\;\; Corresponding to (3.18), it follows
\begin{equation}
y_{i}\dot{\partial^{i}}\rho_0=\alpha^{2}r_{-2}+\beta r_{-1}=0,\;\;\;y_{i}\dot{\partial^{i}}\rho_{-1}=\alpha^{2} r_{-3}+\beta r_{-2}=-\rho_{-1}.
\end{equation}
Furthermore for $\rho_{-2}$, we have
\begin{lemma}
The quantities $\rho_{-2}$ satisfies the relations
\begin{equation}
\dot{\partial^{i}}\rho_{-2} =r_{-3}b^{i}+ r_{-4}Y^{i}
\end{equation}
\begin{equation}
r_{-4}=\frac{1}{2\alpha^{3}}(H_{\alpha\alpha\alpha}-3\alpha^{-1}H_{\alpha\alpha}+3\alpha^{-2}H_{\alpha}).
\end{equation}
\end{lemma}
Also following homogeneity holds good:
\begin{equation}
y_{i}\dot{\partial^{i}}\rho_{-2}=\alpha^{2}r_{-4}+\beta r_{-3}=-2\rho_{-2}.
\end{equation}
It is easy to conclude for the scalars (or invariants) $ \rho_{1},\rho,\rho_{0},\rho_{-1},.......,r_{-1}, r{-2},......$ in the above lemmas that The subscript of $\rho$,s and $r$,s represent degree of their own homogeneity in $(\alpha,\beta)$ or $y_{i}$,where the $\rho$ without subscript means of degree 0.\\
We have these properties from expressions in (3.18),(3.19) and (3.21) and the following relations
\begin{equation}
y_{i}\dot{\partial^{i}}r_{-1}=-r_{-1},\;\;\;y_{i}\dot{\partial^{i}}r_{-2}=-2r_{-2},
\end{equation}
\begin{equation}
y_{i}\dot{\partial^{i}}r_{-3}=-3r_{-3},\;\;\;y_{i}\dot{\partial^{i}}r_{-4}=-4r_{-4},\nonumber
\end{equation}
because of the homogeneity of
\begin{equation}
\alpha H_{\alpha\alpha\alpha\alpha}+\beta H_{\alpha\alpha\alpha\beta}=-H{\alpha\alpha\alpha},\;\;\; \alpha H_{\alpha\alpha\alpha\beta}+\beta H_{\alpha\alpha\beta\beta}=-H{\alpha\alpha\beta},
\end{equation}
\begin{equation}
\;\;\alpha H_{\alpha\alpha\beta\beta}+\beta H_{\alpha\beta\beta\beta}=-H{\alpha\beta\beta},\;\;\; \alpha H_{\alpha\beta\beta\beta}+\beta H_{\beta\beta\beta\beta}=-H{\beta\beta\beta}.\nonumber
\end{equation}
We shall use the previous results for study the fundamental geometric objects of the space $\ell^{n}$ with $(\alpha,\beta)$-metric.All these scalar functions $ \rho_{1},\rho,\rho_{0},\rho_{-1},......$ as well as $r_{-1},r_{-2},.....$ will be called the invariants of the Cartan space $\ell^{n}$ with the fundamental function $H(\alpha,\beta)$.\\
\section{The fundamental tensor of the space $\ell^{n}$ with $(\alpha,\beta)$-metric.}
We need to derive the fundamental tensor from the fundamental function $H(x,y)$ of the Cartan space $\ell^{n}$.
\begin{theorem}
The fundamental tensor $g^{ij}$ of Cartan space $\ell^{n}$ with $(\alpha,\beta)$-metric is given by
\begin{equation}
g^{ij}=\rho a^{ij}+\rho_{0}b^{i}b^{j}+\rho_{-1}(b^{i}Y^{j}+b^{j}Y^{i})+\rho_{-2}Y^{i}Y^{j},
\end{equation}
where $\rho,\rho_{0},\rho_{-1}\rho_{-2}$ are the invariants given by (2.15) and (2.17).
\end{theorem}
\textbf{{Proof.}}\;\;\;
Making use of (3.12) and (3.14),we have
\begin{center}
$ g^{ij}=\dot{\partial^{j}}y^{i}=\dot{\partial^{j}}(\rho_{1}b^{i}+\rho Y^{i})=(\dot{\partial^{j}}\rho_{1})b^{i}+(\dot{\partial^{j}}\rho)Y^{i}+\rho a^{ij}$.
\end{center}
Taking into account Lemma (3.2), we have (3.1).\;\;\;\;\;\;\;\;\;\;Q.E.D.\\
In order to check the fitness of this tensor $g^{ij}$ for the fundamental tensor of $(\alpha,\beta)$-metric,we verify the homogeneity of $g^{ij}$.Contracting $g^{ij}$ by $y_{i}$ and $y_{j}$,we have
\begin{center}
$g^{ij}y_{i}y_{j}=\frac{1}{2}\lbrace \alpha^{-1}H_{\alpha}a^{ij}y_{i}y_{j}+(\alpha^{-2}H_{\alpha\alpha}-\alpha^{-3}H_{\alpha})Y^{i}Y^{j}y_{i}y_{j}$
$+\alpha^{-1}H_{\alpha\beta}(Y^{i}y_{i}b^{j}y_{j}+b^{i}y_{i}Y^{j}y_{j})+H_{\beta\beta}b^{i}y_{i}b^{j}y_{j}\rbrace$
$=\frac{1}{2}\lbrace \alpha H_{\alpha}+(\alpha^{-2}H_{\alpha\alpha}-\alpha^{-3}H_{\alpha})\alpha^{4}+\alpha^{-1}H_{\alpha\beta}.2\alpha^{2}\beta+H_{\beta\beta} \}$\\
$=\frac{1}{2}.2H=H.$
\end{center}
which shows that our conclusion is right.\\
Let us rewrite this expression in the form
\begin{equation}
g^{ij}=A^{ij}+C^{i}C^{j},
\end{equation}
where
\begin{equation}
A^{ij}=\rho a^{ij},\;\;\;C^{i}=q_{0}b^{i}+q^{-1}Y^{i},
\end{equation}
\begin{equation}
\rho_{0}=q^{2}_{0},\;\;\;\; \rho_{-1}=q_{0}q_{-1},\;\;\; \rho_{-2}=q_{-1}^{2}
\end{equation}
and
\begin{equation}
\rho_{0}\rho_{-2}=\rho_{-1}^{2}
\end{equation}
The reciprocal tensor $g_{ij}$ of $g^{ij}$ are given by
\begin{equation}
g_{ij}=A_{ij}-\frac{1}{1+C^{2}}C_{i}C_{j},
\end{equation}
where
\begin{equation}
det\lVert A_{ij}\rVert=(1+C^{2})det\lVert A^{ij}\rVert\;\;\;\;(if 1+C^{2}\ne 0),
\end{equation}
and $A_{ij}$,$C_{i}$ are given by
\begin{equation}
A_{ij}A^{jk}=\delta^{i}_{k},\;\;\; C^{i}C_{j}=\delta^{i}_{j}C^{2},\;\;\; C^{i}=A^{ij}C_{j},\;\;\;C_{i}=A_{ij}C^{j}.
\end{equation}
Consequently, we have $g^{ij}(x,y)g_{jk}(x,y)=\delta^{i}_{k}$,and\\
{(3.2')}\space\;\;\;\;\;\;\;\;\; $rank\lVert g^{ij}(x,y)\rVert=n$\\
because of\\
{(3.2'')\;\;\;\;\;\;$det\lVert g^{ij}(x,y)\rVert=(1+C^{2})det\lVert A^{ij}\rVert=(1+C^{2})det\lVert a^{ij}(x)\rVert\ne 0$\\
The following relations are useful afterwards :
\begin{equation}
A_{ij}=\frac{1}{\rho}a_{ij},\;\;\;det\lVert A^{ij}\rVert=\rho^{n}det\lVert a^{ij}\rVert ,\;\;(A=det\lVert a^{ij}\rVert\ne 0),
\end{equation}
\begin{equation}
C^{2}=\frac{1}{\rho}(\rho_{0}B^{2}+\rho_{-1}\beta),\;\;\;det\lVert g^{ij}\rVert=\rho^{n-1}\tau,
\end{equation}
where we use the notations
\begin{equation}
B^{2}=a^{ij}b_{i}b_{j},\;\;\; \tau=\rho+\rho_{0} B^{2}+\rho_{-1}\beta
\end{equation}
Therefore we can prove without difficulty:
\begin{proposition}
The covariant form of the fundamental tensor is given by
\begin{equation}
g_{ij}=\sigma a_{ij}-\sigma_{0} B_{i}B_{j}+\sigma_{-1}(B_{i}y_{j}+B_{j}y_{i})+\sigma_{-2}y_{i}y_{j},
\end{equation}
where we use the notations
\begin{equation}
\sigma=\frac{1}{\rho},\;\; \sigma_{0}=\frac{\rho_{0}}{\rho\tau},\;\;\; \sigma_{-2}=\frac{\rho_{-2}}{\rho\tau}.
\end{equation}
\end{proposition}
We can get another result from the Lemma (3.4) such that
\begin{theorem}
The Cartan tensor $C^{ijk}$ of a Cartan space $\ell^n$ with $(\alpha,\beta)$-metric is given by
\begin{eqnarray}
C^{ijk}=\frac{-1}{2}[r_{-1}b^{i}b^{j}b^{k}+\varPi _{ijk}\lbrace\rho_{-1}a^{ij}b^{k}+\rho_{-2}a^{ij}Y^{k}+r_{-2}b^{i}b^{j}Y^{k} \\\nonumber + r_{-3}b^{i}Y^{j}Y^{k}\rbrace + r_{-4}Y^{i}Y^{j}Y^{k}]
\end{eqnarray}
\end{theorem}
where the notation $\varPi_{ijk}$ means the cyclic symmetrization of the quantity in the brackets with respect to indices i, j, k.\\
We can deduce the other important geometric object fields for $\ell^{n}$ with $(\alpha,\beta)$-metric, for instance, $N_{ij}$,$H_{jk}^{i}$,$C_{i}^{jk}$ etc. without difficulty.
\section{Cartan spaces with infinite series of $(\alpha,\beta)$-metric}
In 2004 Lee and Park \cite{LP} introduced a r-th series $(\alpha,\beta)$-metric
\begin{equation}
L(\alpha,\beta)=\beta\sum_{k=0}^{r}(\frac{\alpha}{\beta})^{k},
\end{equation}
where they assume $\alpha <\beta$.
If $ r = 1$ then $ L =\alpha +\beta$ is a Randers metric. If $ r = 2$ then $ L = \alpha + \beta +\frac{\alpha^{2}}{\beta}$ is a combination of Randers metric and Kropina metric. If $ r = \infty$
then above metric is expressed as
\begin{equation}
L(\alpha,\beta)=\frac{\beta^{2}}{\beta-\alpha}
\end{equation}
and the metric (5.2) named as infinite series $(\alpha, \beta)$-metric. This metric is very remarkable because it is the difference of Randers and Matsumoto metric.\\
In this section we consider two cases of Cartan Finsler spaces with special $(\alpha,\beta)$-metrics of deformed infinite series metric which are defined as
\begin{flushleft}
\textbf{I} $ \;\;\; H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha} $ i.e. the product of infinite series and Riemannian metric. \\
\textbf{II} $ \;\; H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha} $ i.e. the product of infinite series and one-form metric.
\end{flushleft}
\subsection{Cartan space $\ell^{n}$ for $H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha}$}
In the first case, partial derivatives of the fundamental function $H(\alpha,\beta)$ lead us the followings:\\
\begin{center}
$H_{\alpha}=\frac{\beta^{3}}{(\beta-\alpha)^{2}}$,\;\;\; $H_{\beta}=\frac{\alpha\beta^{2}-2\alpha^{2}\beta}{(\beta-\alpha)^{2}}$\\
$H_{\alpha\alpha}=\frac{2\beta^{3}}{(\beta-\alpha)^{3}}$\;\;\;$H_{\alpha\beta}=\frac{\beta^{3}-3\alpha\beta^{2}}{(\beta-\alpha)^{3}}$\;\;\;$H_{\beta\beta}=\frac{2\alpha^{3}}{(\beta-\alpha)^{3}}$\\
$H_{\alpha\alpha\alpha}=\frac{6\beta^{3}}{(\beta-\alpha)^{4}}$\;\;\;$H_{\alpha\alpha\beta}=\frac{-6\alpha\beta^{2}}{(\beta-\alpha)^{4}}$\;\;\;$H_{\beta\beta\beta}=\frac{-6\alpha^{3}}{(\beta-\alpha)^{4}}$\;\;\;$H_{\alpha\beta\beta}=\frac{6\alpha^{2}\beta}{(\beta-\alpha)^{4}}$
\end{center}
Using equation (3.15) and (3.17) we have following invariants
\begin{equation}
\rho_{1}=\frac{\alpha\beta^{2}-2\alpha^{2}\beta}{2(\beta-\alpha)^{2}},\;\;\rho=\frac{\beta^{3}}{2(\beta-\alpha)^{2}},\;\;\rho_{0}=\frac{\alpha^{3}}{(\beta-\alpha)^{3}},\;\;\;
\end{equation}
\begin{equation}
\rho_{-1}=\frac{\beta^{3}-3\alpha\beta^{2}}{2\alpha(\beta-\alpha)^{3}}\;\;\; \rho_{-2}=\frac{\beta^{3}(3\alpha-\beta)}{2\alpha^{3}(\beta-\alpha)^{3}}\nonumber
\end{equation}
\begin{proposition}
The invariants $\rho$ never vanishes in a Cartan space $\ell^{n}$ equipped with deformed infinite series metric function $H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha}$ -metric on $\widetilde{T^{*}M}$. Converesely, we have $H_{\alpha}\ne 0$ on $\widetilde{T^{*}M}$.
\end{proposition}
Again using equation (3.20) and (3.23) we have following invariants
\begin{equation}
r_{-1}=\frac{-3\alpha^{3}}{(\beta-\alpha)^{4}},\;\;\;\;\;\;\;\; r_{-2}=\frac{3\alpha\beta}{(\beta-\alpha)^{4}},
\end{equation}
\begin{equation}
r_{-3}=\frac{4\alpha\beta^{3}-\beta^{4}-9\alpha^{2}\beta^{2}}{2\alpha^{3}(\beta-\alpha)^{4}}\;\;\;r_{-4}=\frac{15\alpha^{2}\beta^{3}+3\beta^{5}-12\alpha\beta^{4}}{2\alpha^{5}(\beta-\alpha)^{4}}\nonumber
\end{equation}
\begin{proposition}
The invariants of Cartan tensor $ C^{ijk} $ in Cartan space $\ell^{n}$ which equipped with deformed infinite series metric function $H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha}$ is given by (5.2).
\end{proposition}
The invariants of equations (5.1) and (5.2) satisfies the following relations
\begin{equation}
\alpha^{2}\rho_{-1}+\beta\rho_{0}=\rho_{1},\;\;\;\alpha^{2}\rho_{-2}+\beta\rho_{-1}=0,
\end{equation}
\begin{equation}
\alpha^{2} r_{-2}+\beta r_{-1}=0,\;\;\;\alpha^{2} r_{-3}+\beta r_{-2}=-\rho_{-1},\nonumber
\end{equation}
\begin{equation}
\alpha^{2} r_{-4}+\beta r_{-3}=-2\rho_{-2},\nonumber
\end{equation}
\begin{theorem}
The Cartan space $\ell^{n}$ equipped with deformed infinite series metric function $H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha}$ has the invariants in equations (5.1) and (5.2) are satifies the relations in (5.3).
\end{theorem}
From equation (5.1) and (5.2), The fundamental tensor $g^{ij}(x,y)$ is of the form
\begin{equation}
g^{ij}(x,y)=\frac{\beta^{3}}{2(\beta-\alpha)^{2}} a^{ij}+\frac{\alpha^{3}}{(\beta-\alpha)^{3}} b^{i}b^{j}+\frac{\beta^{3}-3\alpha\beta^{2}}{2(\beta-\alpha)^{3}}(b^{i}Y^{j}+b^{j}Y^{i})
\end{equation}
\begin{equation}
+\frac{\beta^{3}(3\alpha-\beta)}{2\alpha^{3}(\beta-\alpha)^{3}}Y^{i}Y^{j}\nonumber
\end{equation}
\begin{corollary}
The fundamental tensor $g^{ij}(x, y)$ of the space $\ell^{n}$ endoewd with the metric function $H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha}$ is given by the equation (5.4).\\
\end{corollary}
Converesely we obtain
\begin{theorem}
The Cartan space with $(\alpha,\beta)$-metric which have the invariants such that (5.1)and (5.2) is the spaces $\ell^{n}$ with the fundamental function $H(\alpha,\beta)=\frac{\alpha\beta^{2}}{\beta-\alpha}$, i.e. deformed infinite series metric.
\end{theorem}
\subsection{Cartan space $\ell^{n}$ for $H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha}$}
In the second case, partial derivatives of the fundamental function $H(\alpha,\beta)$ lead us the followings:\\
\begin{center}
$H_{\alpha}=\frac{\beta^{3}}{(\beta-\alpha)^{2}}$,\;\;\; $H_{\beta}=\frac{2\beta^{3}-3\alpha\beta^{2}}{(\beta-\alpha)^{2}}$\\
$H_{\alpha\alpha}=\frac{2\beta^{3}}{(\beta-\alpha)^{3}}$\;\;\;$H_{\alpha\beta}=\frac{\beta^{3}-3\alpha\beta^{2}}{(\beta-\alpha)^{3}}$\;\;\;$H_{\beta\beta}=\frac{2\beta^{3}-6\alpha\beta^{3}+6\alpha^{2}\beta}{(\beta-\alpha)^{3}}$\\
$H_{\alpha\alpha\alpha}=\frac{6\beta^{3}}{(\beta-\alpha)^{4}}$\;\;\;$H_{\alpha\alpha\beta}=\frac{-6\alpha\beta^{2}}{(\beta-\alpha)^{4}}$\;\;\;$H_{\beta\beta\beta}=\frac{-6\alpha^{3}}{(\beta-\alpha)^{4}}$\;\;\;$H_{\alpha\beta\beta}=\frac{6\alpha^{2}\beta}{(\beta-\alpha)^{4}}$
\end{center}
Using equation (3.15) and (3.17) we have following invariants
\begin{equation}
\rho_{1}=\frac{2\beta^{3}-3\alpha\beta^{3}}{2(\beta -\alpha)^{2}},\;\;\rho=\frac{\beta^{3}}{2(\beta-\alpha)^{2}},\;\;\rho_{0}=\frac{\beta^{3}-3\alpha \beta^{2}+3\alpha^{2}\beta}{(\beta-\alpha)^{3}},\;\;\;
\end{equation}
\begin{equation}
\rho_{-1}=\frac{\beta^{3}-3\alpha\beta^{2}}{2\alpha(\beta-\alpha)^{3}}\;\;\; \rho_{-2}=\frac{\beta^{3}(3\alpha-\beta)}{2\alpha^{3}(\beta-\alpha)^{3}}\nonumber
\end{equation}
\begin{proposition}
The invariants $\rho$ never vanishes in a Cartan space $\ell^{n}$ equipped with deformed infinite series metric function $H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha}$ -metric on $\widetilde{T^{*}M}$. Converesely, we have $H_{\alpha}\ne 0$ on $\widetilde{T^{*}M}$.
\end{proposition}
Again using equations (3.20) and (3.23) we have following invariants
\begin{equation}
r_{-1}=\frac{-3\alpha^{3}}{(\beta-\alpha)^{4}},\;\;\;\;\;\;\;\; r_{-2}=\frac{3\alpha\beta}{(\beta-\alpha)^{4}},
\end{equation}
\begin{equation}
r_{-3}=\frac{4\alpha\beta^{3}-\beta^{4}-9\alpha^{2}\beta^{2}}{2\alpha^{3}(\beta-\alpha)^{4}}\;\;\;r_{-4}=\frac{15\alpha^{2}\beta^{3}+3\beta^{5}-12\alpha\beta^{4}}{2\alpha^{5}(\beta-\alpha)^{4}}\nonumber
\end{equation}
\begin{proposition}
The invariants of Cartan tensor $ C^{ijk} $ in Cartan space $\ell^{n}$ which equipped with deformed infinite series metric function $H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha}$ is given by (5.6).
\end{proposition}
The invariants of equations (5.5) and (5.6) satisfies the following relations
\begin{equation}
\alpha^{2}\rho_{-1}+\beta\rho_{0}=\rho_{1},\;\;\;\alpha^{2}\rho_{-2}+\beta\rho_{-1}=0,
\end{equation}
\begin{equation}
\alpha^{2} r_{-2}+\beta r_{-1}=0,\;\;\;\alpha^{2} r_{-3}+\beta r_{-2}=-\rho_{-1},\nonumber
\end{equation}
\begin{equation}
\alpha^{2} r_{-4}+\beta r_{-3}=-2\rho_{-2},\nonumber
\end{equation}
\begin{theorem}
The Cartan space $\ell^{n}$ equipped with deformed infinite series metric function $H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha}$ has the invariants in equations (5.5) and (5.6) are satifies the relations in (5.7).
\end{theorem}
From equation (5.5) and (5.6), The fundamental tensor $g^{ij}(x,y)$ is of the form
\begin{equation}
g^{ij}(x,y)=\frac{\beta^{3}}{2(\beta-\alpha)^{2}} a^{ij}+\frac{\beta^{3}-3\alpha \beta^{2}+3\alpha^{2}\beta}{(\beta-\alpha)^{3}} b^{i}b^{j}
\end{equation}
\begin{equation}
+\frac{\beta^{3}-3\alpha\beta^{2}}{2(\beta-\alpha)^{3}}(b^{i}Y^{j}+b^{j}Y^{i})+\frac{\beta^{3}(3\alpha-\beta)}{2\alpha^{3}(\beta-\alpha)^{3}}Y^{i}Y^{j}\nonumber
\end{equation}
\begin{corollary}
The fundamental tensor $g^{ij}(x,y)$ of the space $\ell^{n}$ endoewd with the metric function $H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha}$ is given by the equation (5.8).
\end{corollary}
Converesely we obtain
\begin{theorem}
The Cartan space with $(\alpha,\beta)$-metric which have the invariants such that (5.5)and (5.6) is the spaces $\ell^{n}$ with the fundamental function $H(\alpha,\beta)=\frac{\beta^{3}}{\beta-\alpha}$,i.e. deformed infinite series metric.
\end{theorem}
\section{Conclusions}
In this work, we consider the infinite series $(\alpha,\beta)$-metric, Riemannian metric and 1-form metric we determine relations with the "invarints" which characterize the special classes in Cartan Finsler frames . But, in Finsler geometry, there are many($\alpha$,$\beta$)-metrics, in future work we can determine the frames for them also.
\pagebreak | 214,062 |
TITLE: Calculating $ \lim_{n \rightarrow \infty} \int_{[0, \infty)} \frac{\sin(e^x) }{1+nx^2}\,dx$
QUESTION [2 upvotes]: I want to calculate the limit of following Lebesgue-integral:
$$ \lim_{n \rightarrow \infty} \int_{[0, \infty)} \frac{\sin(e^x) }{1+nx^2}\,\mathrm dx$$
Therefore I wanted to apply Lebesgue's dominated convergence theorem.
$f_n(x)$ is measurable and $ f_n \rightarrow 0$ pointwise. Now it holds:
$$ \left|\frac{\sin(e^x) }{1+nx^2}\right| \leq \frac{1}{1+x^2} :=g(x) $$
The improper integral over does converge. That means f is lebesgue integrable.
Therefore
$$ \lim_{n \rightarrow \infty} \int_{[0, \infty)} \frac{\sin(e^x) }{1+nx^2}\,\mathrm dx = \int_{[0, \infty)} \lim_{n \rightarrow \infty} \frac{\sin(e^x) }{1+nx^2}\,\mathrm dx =0$$
Consider $ f_n(0) = sin 1$ does not converge to $ 0$. So I can't apply the theorem, can I ?
REPLY [6 votes]: You are on the right track: apply Lebesgue's dominated convergence with $g(x)=\frac{1}{1+x^2}$ which is Lebesgue integrable in $[0,+\infty)$. Since $f_n(x)=\frac{\sin(e^x) }{1+nx^2}\to 0$ for all $x>0$ the sequence $(f_n)_n$ converges to zero almost everywhere on $[0,+\infty)$, that's enough for dominated convergence, and we may conclude that the limit of $\int_0^{\infty} f_n(x)\,dx$ is zero.
Alternative way (without dominated convergence):
$$\begin{align}\left|\int_{[0, \infty)} \frac{\sin(e^x) }{1+nx^2}\, dx \right|&\leq \int_0^{+\infty} \frac{|\sin(e^x) |}{1+nx^2}\, dx\\
&\leq \int_0^{+\infty} \frac{dx}{1+nx^2}=\left[\frac{\arctan(\sqrt{n}x)}{\sqrt{n}}\right]_0^{+\infty}=\frac{\pi}{2\sqrt{n}}.\end{align}$$
So, again, the limit as $n\to\infty$ is zero. | 213,133 |
TITLE: Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$
QUESTION [4 upvotes]: Let $n,k,r$ be positive integers. The number of all nonnegative solutions of the Diophantine Equation $x_1+x_2+\dots+x_k=n$ is $\binom{n+k-1}{n}$. Is there a general formula for the number of solutions of the equation $x_1+x_2+\dots+x_k=n$ with $x_i\le r$ for every $i\in \{1,2,\dots,k\}$?
If one defines $A_i$ to be the number of solutions with $x_i>r$ then the answer will be $\binom{n+k-1}{n}-|A_1\cup\dots\cup A_k|$. I think it can give a complicated formula. What is the formula?
REPLY [3 votes]: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
Is there a general formula for the number of solutions of the equation
$\ds{x_{1} + \cdots + x_{k} =n}$ with $\ds{0 \leq x_{i} \leq r\quad}$ for every
$\ds{\quad i \in \braces{1,\ldots,k}}$ ?.
The solution is given by
$\ds{\mc{N} \equiv
\sum_{x_{\large 1} = 0}^{r}\ldots\sum_{x_{\large k} = 0}^{r}\bracks{z^{n}}
z^{x_{\large 1} + \cdots + x_{\large k}}}$.
$\ds{\bracks{z^{n}}\mrm{f}\pars{z}}$ denotes the coefficient of $\ds{z^{n}}$ in the $\ds{\mrm{f}\pars{z}}$ power expansion.
With the above definition, $\ds{\bracks{z^{n}}
z^{x_{\large 1} + \cdots + x_{\large k}}}$ is equal to $\ds{\color{#f00}{1}}$ whenever $\ds{x_{1} + \cdots + x_{k} = n}$ and $\ds{\color{#f00}{0}}$ otherwise.
Indeed, it's equivalente to the Kronecker Delta
$\ds{\delta_{n,x_{\large1} + \cdots + x_{\large k}}}$.
Then, each term adds $\ds{\color{#f00}{1}}$ to the sum whenever the condition
$\ds{x_{1} + \cdots + x_{k} = n}$ is satisfied.
Therefore,
\begin{align}
\mc{N} & \equiv \sum_{x_{\large 1} = 0}^{r}\ldots
\sum_{x_{\large k} = 0}^{r}\bracks{z^{n}}
z^{x_{\large 1} + \cdots + x_{\large k}} =
\bracks{z^{n}}\pars{\sum_{x = 0}^{r}z^{x}}^{k} =
\bracks{z^{n}}\pars{z^{r + 1} - 1 \over z - 1}^{k} =
\bracks{z^{n}}{\pars{1 - z^{r + 1}}^{k} \over \pars{1 - z}^{k}}
\\[5mm] & =
\bracks{z^{n}}\bracks{\sum_{i = 0}^{k}{k \choose i}\pars{-z^{r + 1}}^{i}}
\bracks{\sum_{j = 0}^{\infty}{-k \choose j}\pars{-z}^{j}}
\\[5mm] & =
\bracks{z^{n}}\sum_{i = 0}^{k}\pars{-1}^{i}{k \choose i}
\sum_{j = 0}^{\infty}
\bracks{\pars{-1}^{j}{k + j - 1 \choose j}\pars{-1}^{j}}z^{\pars{r + 1}i + j}
\\[5mm] & =
\sum_{i = 0}^{k}\pars{-1}^{i}{k \choose i}\sum_{j = 0}^{\infty}
{k + j - 1 \choose k - 1}\bracks{\pars{r + 1}i + j = n}
\qquad\pars{~\bracks{\cdots}\ \mbox{is the}\ Iverson\ Bracket~}
\\[5mm] & =
\sum_{i = 0}^{k}\pars{-1}^{i}{k \choose i}\sum_{j = 0}^{\infty}
{k + j - 1 \choose k - 1}\bracks{j = n - \pars{r + 1}i}
\\[5mm] & =
\sum_{i = 0}^{k}\pars{-1}^{i}{k \choose i}
{k + n - \pars{r + 1}i - 1 \choose k - 1}\bracks{n - \pars{r + 1}i \geq 0}
\\[5mm] & =
\sum_{i = 0}^{k}\pars{-1}^{i}{k \choose i}
{k + n - \pars{r + 1}i - 1 \choose k - 1}\bracks{i \leq {n \over r + 1}}
\\[5mm] & =
\bbox[15px,#ffe,border:1px dotted navy]{\ds{%
\sum_{i = 0}^{m}\pars{-1}^{i}{k \choose i}
{k + n - \pars{r + 1}i - 1 \choose k - 1}\,,\qquad
m \equiv \min\braces{k,\left\lfloor\,{n \over r + 1}\,\right\rfloor}}}
\end{align} | 155,768 |
Jim Fishwick
Jim Fishwick
Ensemble Cast | Teacher
Joined Impro Melbourne: 2017
Performer Blurb
Jim is an award-winning director, performer and 2017 Maestro National Champion Improviser. He began improvising at the sorry age of seventeen with Impro Australia, and went on to train at the iO Theater, Chicago, and at the Loose Moose Theatre, Calgary. He directed and produced Fresh and the First Thursday Impro Club from 2011-2014, directed Theatresports™ at the University of Sydney from 2015-2017, and directed the National Theatresports™ Championships at the Enmore Theatre.
Teacher Blurb
Jim is passionate about exploring improv that is artful, articulate, and attentive. He champions presence, support, inclusion and freedom in his craft. He taught improvisation at the University of Sydney for three years, is a faculty member at Improv Theatre Sydney, and has taught independent workshops around the Australia and the world since 2012.
Student testimonials:
“Jim is a caring, nurturing and inclusive teacher. I always learn so much in his classes and leave wanting to improve my own practice. His feedback is always bang on and productive. The gold standard in teaching, knowledge and people.”
Reviews:
“Jim Fishwick brings an exciting visceral dimension to the intellectual work, with an avant-garde spirit that injects a sense of adventure and daring to the oft too polite Australian stage.” – Suzy Goes See
Teaching experience:
Performance arts schools
Improvisation Companies – Improv Theatre Sydney
Improvisation Festivals – Improvention, New Zealand Improv Festival
Select Impro credits:
Assistant Director, Grand Exotic Budapest Hotel, (Impro Melbourne, 2018)
Cast Member, How The West Was Improvised (Impro Melbourne, 2018)
Director/Cast Member, Unplotted Potter (Jetpack Theatre, Sydney, 2017)
Director, The Intrepid Bazaar (NZIF, Wellington, 2017)
Director, Theatresports National Championships (Impro Australia, Sydney 2017)
Theatre credit: Director, Art Heist (2017, Jetpack Theatre)
TV credit: Question Writer on Only Connect (BBC2)
Awards:
‘Mr Miyagi’ Award for Inspiring Teacher, New Zealand Improv Festival (Wellington, 2017)
Compelling Concept and Immersive Production Award, Sydney Arts Guide (Sydney, 2015)
Impro Motto or Quote: “We can’t pretend wrong. We can pretend better, but we can’t pretend wrong.” – Craig Uhlir
Impro Mentor: Linda Calgaro
Favourite Impro Melbourne show or role: Maestro, in a heartbeat. The freedom to fail spectacularly is a delight
Comic inspirations: Victoria Wood, Stewart Lee, Maddie Parker, Buster Keaton
My little secret: (something about you like ‘preparation for a show’, ‘do you get nervous’ etc) I love ‘waking up’ from a scene. When I was so sucked into playing a character I wasn’t there anymore. When people can say “I like that scene where you were a pirate” and I can only say “thank you! What happened in it?”
Links:
Jim's website
Twitter @fimjishwick
| 214,478 |
TITLE: $5^a - 5^b$ is divisible by $n$ (prove)
QUESTION [0 upvotes]: Prove that for every n natural number exist natural numbers $a,b \leq 4n, a\not= b $, which accomplish, that number $ 5^a - 5^b $ is divisible by n. How many of these pairs exist?
Help please, I'm stuck with this problem. Any help is appreciated.
REPLY [1 votes]: Let $n=5^km$ with $gcd(5,m)=1$.
Now, the pair $(a,b)$ is a solution if and only if $(b,a)$ is a solution. So it suffices to look for the solutions where $a>b$. Don't forget at the end to double the number of solutions.
Then
$$n|5^a-5^b =5^b(5^{a-b}-1)) \Leftrightarrow k \leq b \mbox{ and } m | 5^{a-b}-1$$
The existence is easy then, pick $b =k $ and $a-b =\phi(m) \leq m \leq n$.
For the number of solutions, you need
$$k \leq b \leq 4n$$
Thus you have $4n-k$ choices for $b$.
Morever, for each $b$, if $j$ is the order of $5$ modulo $m$, you must have
$$j |a-b \,.$$
Thus, $a-b$ has to be a multiple of $j$ between $1$ and $4n-b$. From here you figure out how many choices of $a$ you have for each $b$. Add them, double and you are done. | 102,060 |
\begin{document}
\newcommand{\rdg}{\hfill $\Box $}
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{corollary}[definition]{Corollary}
\newtheorem{remark}[definition]{Remark}
\newtheorem{example}[definition]{Example}
\newtheorem{exercise}[definition]{Exercises}
\newcommand{\tp}{\otimes}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\op}{\oplus}
\newcommand{\n}{\underline n}
\newcommand{\es}{{\frak S}}
\newcommand{\ef}{\frak F}
\newcommand{\qu}{\frak Q} \newcommand{\ga}{\frak g}
\newcommand{\la}{\lambda}
\newcommand{\ig}{\frak Y}
\newcommand{\te}{\frak T}
\newcommand{\cok}{{\sf Coker}}
\newcommand{\Hom}{{\sf Hom}}
\newcommand{\im}{{\sf Im}}
\newcommand{\ext}{{\sf Ext}}
\newcommand{\ho}{{\sf H_{AWB}}}
\newcommand{\HH}{{\sf Hoch}}
\newcommand{\adu}{{\rm AWB}^!}
\def\Im{\operatorname{Im}}
\def\Ker{\operatorname{Ker}}
\def\id{\operatorname{id}}
\def\Coker{\operatorname{Coker}}
\def\Der{\operatorname{Der}}
\def\hom{\operatorname{Hom}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\ad}{\operatorname{ ad}}
\newcommand{\ele}{\cal L} \newcommand{\as}{\cal A} \newcommand{\ka}{\cal
K}\newcommand{\eme}{\cal M} \newcommand{\pe}{\cal P}
\newcommand{\pn}{\par \noindent}
\newcommand{\pbn}{\par \bigskip \noindent}
\bigskip\bigskip
\centerline {\Large {\bf A non-abelian tensor product of Hom-Lie algebras}}
\
\centerline {\bf J. M. Casas$^{(1)}$, E. Khmaladze$^{(2)}$ and N. Pacheco Rego$^{(3)}$}
\bigskip \bigskip
\centerline{$^{(1)}$Dpto. Matem\'atica Aplicada I, Univ. de Vigo, 36005 Pontevedra, Spain}
\centerline{e-mail address: \tt [email protected]}
\medskip
\centerline{$^{(2)}$A. Razmadze Math. Inst. of I. Javakhishvili Tbilisi State University,}
\centerline{Tamarashvili Str. 6, 0177 Tbilisi, Georgia and}
\centerline{Dpto. Matem\'atica Aplicada I, Univ. de Vigo, 36005 Pontevedra, Spain}
\centerline{e-mail address: \tt [email protected]}
\medskip
\centerline{$^{(3)}$IPCA, Dpto. de Ciências, Campus do IPCA,
Lugar do Aldão}
\centerline{4750-810 Vila Frescainha, S. Martinho, Barcelos,
Portugal}
\centerline{e-mail address: \tt [email protected]}
\bigskip \bigskip
\par
{\bf Abstract:} Non-abelian tensor product of Hom-Lie algebras is constructed and studied. This tensor product is used to describe universal ($\alpha$-)central extensions of Hom-Lie algebras and to establish a relation between cyclic and Milnor cyclic homologies of Hom-associative algebras satisfying certain additional condition.
\bigskip
{\it Key words:} Hom-Lie algebra, Hom-action, semi-direct product, derivation, non-abelian tensor product, universal ($\alpha$)-central extension.
\bigskip
{\it A. M. S. Subject Class. (2010):} 17A30, 17B55, 17B60, 18G35, 18G60
\section*{Introduction}
The concept of a Hom-Lie algebra was initially introduced in \cite{HLS} motivated by discretization of vector fields via twisted derivations.
A Hom-Lie algebra is a non-associative algebra satisfying the skew symmetry and the Jacobi identity twisted by a map. When this map is the identity map, then
the definition of a Lie algebra is recovered. Thus, it is natural to seek for possible generalizations of known theories from Lie to Hom-Lie algebras.
In this context, recently there have been several works dealing with the study of Hom-Lie structures (see \cite{CaInPa}, \cite{JL}, \cite{MS} - \cite{Yau2}).
In this paper we introduce a non-abelian tensor product of Hom-Lie algebras generalizing the non-abelian tensor product of Lie algebras \cite{El1} and investigate its properties. In particular, we study its relation to the low dimensional homology of Hom-Lie algebras developed in \cite{Yau1, Yau2}. We use this tensor product in the description of the universal ($\alpha$-) central extensions of Hom-Lie algebras considered in \cite{CaInPa}. We give an application of our non-abelian tensor product of Hom-Lie algebras to cyclic homology of Hom-associative algebras \cite{Yau1}. Namely, for Hom-associative algebras satisfying an additional condition, which we chose to call $\alpha$-identity condition, we establish a relation between cyclic and Milnor cyclic homologies in terms of exact sequences.
Note that not all classical results can be generalized from Lie to Hom-Lie algebras, for example, results on universal central extensions of Lie algebras cannot be extended directly to Hom-Lie algebras and they are divided between universal central and universal $\alpha$-central extensions of Hom-Lie algebras (see Section \ref{section}). Further, in order to obtain Hom-algebra version of Guin's result relating cyclic and Milnor cyclic homology of associative algebras \cite{Gu}, we need to consider a subclass of Hom-associative algebras defined by the $\alpha$-identity condition (see Definition \ref{alfa condition}).
\subsection*{Notations} Throughout this paper we fix $\mathbb{K}$ as a ground field. Vector spaces are considered over $\mathbb{K}$ and linear maps are $\mathbb{K}$-linear maps. We write $\otimes$ (resp. $\wedge$) for the tensor product $\otimes_\mathbb{K}$ (resp. exterior product $\wedge_\mathbb{K}$ ) over $\mathbb{K}$. For any vector space (resp. Hom-Lie algebra) $L$, a subspace (resp. an ideal) $L'$ and $x\in L$ we write $\overline{x}$ to denote the coset $x+L'$.
\section{Hom-Lie algebras}
In this section we review some terminology and recall notions used in the paper. We mainly follow \cite{HLS, JL, MS, Yau1}, although with some modifications.
\subsection{Basic definitions}
\begin{definition}\label{def}
A Hom-Lie algebra $(L, \alpha_L)$ is a non-associative algebra $L$ together with a linear
map $\alpha_L : L \to L$ satisfying
\begin{align*}
&[x,y] = - [y,x], & \text{ (skew-symmetry)}\\
& [\alpha_L(x),[y,z]]+[\alpha_L(z),[x,y]]+[\alpha_L(y),[z,x]]=0 & \text{(Hom-Jacobi identity)}
\end{align*}
for all $x, y, z \in L$, where $[-,-]$ denotes the product in $L$.
\end{definition}
In this paper we only consider (the so called multiplicative) Hom-Lie algebras $(L, \alpha_L)$ such that $\alpha_L$ preserves the product, i.e. $\alpha_L[x,y]=[\alpha_L(x), \alpha_L(y)]$ for all $x, y \in L$.
\begin{example}\label{ejemplo 1} \
\begin{enumerate}
\item[a)] Taking $\alpha_L = \id_L$, Definition \ref{def} gives us the definition of a Lie algebra. Hence any Lie algebra $L$ can be considered as a Hom-Lie algebra $(L,\id_L)$.
\item[b)] Let $V$ be a vector space and $\alpha_V:V\to V$ a linear map, then the pair $(V , \alpha_V)$ is called Hom-vector space. A Hom-vector space $(V,\alpha_V)$ together with the trivial product $[-,-]$ (i.e. $[x,y] = 0$ for any $x,y \in V$) is a Hom-Lie algebra $(V, \alpha_V)$ and it is called abelian Hom-Lie algebra.
\item[c)] Let $L$ be a Lie algebra, $[-,-]$ be the product in $L$ and $\alpha:L \to L$ be a Lie algebra endomorphism. Define $[-,-]_{\alpha} : L \otimes L \to L$ by $[x,y]_{\alpha} = \alpha_L[x,y]$, for all $x, y \in L$. Then $(L, \alpha)$ with the product $[-,-]_{\alpha}$ is a Hom-Lie algebra \cite[Theorem 5.3]{Yau1}.
\item[d)] Any Hom-associative algebra \cite{MS} becomes a Hom-Lie algebra (see Section \ref{section 6} below).
\end{enumerate}
\end{example}
Hom-Lie algebras form a category ${\tt HomLie}$ whose {\it{morphisms}} from $(L,\alpha_L)$ to $(L',\alpha_{L'})$ are algebra homomorphisms $f : L \to L'$ such that $f \circ \alpha_L = \alpha_{L'} \circ f$. Clearly there is a full embedding ${\tt Lie} \hookrightarrow{\tt HomLie}$, $L\mapsto (L,\id_L)$, where ${\tt Lie}$ denotes the category of Lie algebras.
It is a routine task to check that ${\tt HomLie}$ satisfies the axioms of a semi-abelian category \cite{BB}. Consequently, the well-known Snake Lemma is valid for Hom-Lie algebras and we will use it in the sequel. Below we give the ad-hoc definitions of ideal, center, commutator, action and semi-direct product of Hom-Lie algebras and of course these notions agree with the respective general notions in the context of semi-abelian categories (see e.g. \cite{BoJaKe}).
\begin{definition}
A Hom-Lie subalgebra $(H,\alpha_{H})$ of a Hom-Lie algebra $(L, \alpha_L)$ is a vector subspace $H$ of $L$ closed under the product, that is $[x,y] \in H$ for all $x, y \in H$, together with the endomorphism $\alpha_{H}:H\to H$ being the restriction of $\alpha_{L }$ on $H$. In such a case we may write $\alpha_{L\mid}$ for $\alpha_{H}$.
A Hom-Lie subalgebra $(H,\alpha_{L\mid})$ of $(L, \alpha_L)$ is said to be an ideal if $[x, y] \in H$ for any $x \in H$, $y\in L$.
If $(H,\alpha_{L\mid})$ is an ideal of a Hom-Lie algebra $(L,\alpha_L)$, then $(L/H, \overline{\alpha}_L)$, where $\overline{\alpha }_L: L/H \to L/H$ is induced by $\alpha_L$, naturally inherits a structure of Hom-Lie algebra and it is called quotient Hom-Lie algebra.
Let $(H,\alpha_{L\mid})$ and $(K,\alpha_{L\mid})$ be ideals of a Hom-Lie algebra $(L,\alpha_L)$. The commutator (resp. sum) of $(H,\alpha_{L\mid})$ and $(K,\alpha_{L\mid})$, denoted by $([H,K], \alpha_{{L\mid}})$ (resp. $(H+K, \alpha_{{L\mid}})$ ), is the Hom-Lie subalgebra of $(L,\alpha_L)$ spanned by the elements $[h,k]$ (resp. $h+k$), $h \in H$, $k \in K$.
\end{definition}
The following lemma is an easy exercise.
\begin{lemma}\label{ideales} Let $(H,\alpha_{L\mid})$ and $(K,\alpha_{L\mid})$ be ideals of a Hom-Lie algebra
$(L,\alpha_L)$. The following statements hold:
\begin{enumerate} \label{ideales}
\item[a)] $(H \cap K, \alpha_{{L\mid}})$ and $(H+K, \alpha_{{L\mid}})$ are ideals of $(L,\alpha_L)$;
\item[b)] $[H,K] \subseteq H \cap K$;
\item[c)] If $\alpha_L$ is surjective, then $([H,K],\alpha_{{L\mid}})$ is an ideal of $(L,\alpha_L)$;
\item[d)] $([H,K],\alpha_{{L\mid}})$ is an ideal of $(H,\alpha_{L\mid})$ and $(K,\alpha_{L\mid})$. In particular, $([L,L],\alpha_{{L\mid}})$ is an ideal of $(L,\alpha_L)$;
\item[e)] $(\alpha_L(L),\alpha_{L{\mid}})$ is a Hom-Lie subalgebra of $(L,\alpha_L)$;
\item[f)] If $H, K \subseteq \alpha_L(L)$, then $([H,K],\alpha_{{L\mid}})$ is an ideal of $(\alpha_L(L),\alpha_{{L\mid}})$.
\end{enumerate}
\end{lemma}
\begin{definition}
The center of a Hom-Lie algebra $(L,\alpha_L)$ is the vector subspace
\[
Z(L) = \{ x \in L \mid [x, y] =0 \ \text{ for all}\ y \in L\}.
\]
\end{definition}
\begin{remark}
When $\alpha_L : L \to L$ is a surjective endomorphism, then $(Z(L),\alpha_{{L\mid}} )$ is an abelian Hom-Lie algebra and an ideal of $(L, \alpha_L)$.
\end{remark}
\subsection{Hom-action and semi-direct product}
\begin{definition} \label{action}
Let $(L,\alpha_L)$ and $(M, \alpha_M)$ be Hom-Lie algebras. A Hom-action of $(L,\alpha_L)$ on $(M, \alpha_M)$ is a linear map $L \otimes M \to M,$ $x \otimes m\mapsto {}^xm$, satisfying the following properties:
\begin{enumerate}
\item[a)] ${}^{[x,y]} \alpha_M(m) = {}^{\alpha_L(x)}({}^y m) - {}^{\alpha_L(y)} ({}^x m)$,
\item [b)] ${}^{\alpha_L(x)} [m,m'] = [{}^x m, \alpha_M(m')]+[\alpha_M (m), {}^x m']$,
\item [c)] $\alpha_M({}^x m) = {}^{\alpha_L(x)} \alpha_M(m)$
\end{enumerate}
for all $x, y \in L$ and $m, m' \in M$.
The Hom-action is called trivial if ${}^xm=0$ for all $x\in L$ and $m\in M$.
\end{definition}
\begin{remark}
If $(M, \alpha_M)$ is an abelian Hom-Lie algebra enriched with a Hom-action of $(L,\alpha_L)$, then $(M, \alpha_M)$ is nothing else but a Hom-module over $(L,\alpha_L)$ (see \cite{Yau1} for the definition).
\end{remark}
\begin{example}\label{ejemplo 2} \
\begin{enumerate}
\item[a)] Let $(L, \alpha_{L})$ be a Hom-subalgebra of a Hom-Lie algebra $(K,\alpha_{K})$ and $(H,\alpha_{H})$ an ideal of $(K,\alpha_{K})$. Then there exists a Hom-action of $(L,\alpha_{L})$ on $(H,\alpha_{H})$ given by the product in $K$. In particular, there is a Hom-action of $(L,\alpha_{L})$ on itself given by the product in $L$.
\item[b)] Let $L$ and $M$ be Lie algebras. Any Lie action of $L$ on $M$ (see e.g. \cite{El1}) defines a Hom-action of $(L, \id_{L})$ on $(M, \id_{M})$.
\item[c)] Let $L$ be a Lie algebra and $\alpha : L \to L$ be an endomorphism. Suppose $M$ is an $L$-module in the usual sense and the action of $L$ on $M$ satisfies the condition ${}^{\alpha(x)}m = {}^x m$, for all $x \in L$, $m \in M$. Then $(M, \id_M)$ is a Hom-module over the Hom-Lie algebra $(L, \alpha)$ considered in Example \ref{ejemplo 1} c).
\noindent As an example of such $L$, $\alpha$ and $M$ we can consider $L$ to be the $2$-dimensional vector space with basis $\{ e_1,e_2 \}$, together with the product $[e_1,e_2]=-[e_2,e_1]=e_1$ and zero elsewhere, $\alpha$ to be represented by the matrix $\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right)$, and $M$ to be the ideal of $L$ generated by $\{e_1\}$.
\item [d)] Any homomorphism of Hom-Lie algebras $(L,\alpha_L)\to (M,\alpha_M)$ induces a Hom-action of $(L,\alpha_L)$ on $(M,\alpha_M)$ in
the standard way by taking images of elements of $L$ and product in $M$.
\item [e)] Let
$0 \to (M,\alpha_M) \stackrel{i}\to (K,\alpha_K) \stackrel{\pi}\to (L,\alpha_L) \to 0$ be a split short exact sequence of Hom-Lie algebras, that is, there exists a homomorphism of Hom-Lie algebras $s:(L,\alpha_L)\to (K,\alpha_K)$ such that $\pi \circ s=\id_L$. Then there is a Hom-action of $(L,\alpha_L)$ on $(M,\alpha_M)$ defined in the standard way: ${}^{x}m=i^{-1}[s(x),i(m)]$, $x\in L$, $m\in M$.
\end{enumerate}
\end{example}
\begin{definition}
Given a Hom-action of a Hom-Lie algebra $(L,\alpha_L)$ on a Hom-Lie algebra $(M,\alpha_M)$ we define the semi-direct product Hom-Lie algebra, $(M \rtimes L, {\alpha_{\rtimes}})$, with the underlying vector space $M \oplus L$, endowed with the product
\[
[(m_1,x_1),(m_2,x_2)]= ([m_1,m_2]+{^{\alpha_L(x_1)}}{m_2} - {^{\alpha_L(x_2)}}{m_1},[x_1,x_2]),
\]
together with the endomorphism ${\alpha_{\rtimes}} : M \rtimes L \to M \rtimes L$ given by ${\alpha_{\rtimes}} (m, x) = \left(\alpha_M(m), \alpha_L(x)\right)$ for all $x,x_1,x_2 \in L$ and $m, m_1, m_2 \in M$.
\end{definition}
Straightforward calculations show that $(M \rtimes L, {\alpha_{\rtimes}})$ indeed is a Hom-Lie algebra and there is a short exact sequence of Hom-Lie algebras
\begin{equation}\label{semi-direct}
0 \to (M,\alpha_M) \stackrel{i}\to (M \rtimes L,\alpha_{\rtimes}) \stackrel{\pi}\to (L,\alpha_L) \to 0,
\end{equation}
where $i(m) = (m,0)$, $\pi(m,l)= l$. Moreover, $(M,\alpha_M)$ is an ideal of $(M \rtimes L, {\alpha_{\rtimes}})$ and this sequence splits by $s: (L,\alpha_L) \to (M \rtimes L,\alpha_{\rtimes})$, $s(l)=(0,l)$.
Then, as in Example \ref{ejemplo 2} {\it e)}, the above sequence defines a Hom-action of $( L,\alpha_{L})$ on $(M, \alpha_M)$ given by
\[
{^l}m = i^{-1}\left[ \left( 0,l\right) ,\left( m,0\right)
\right] =i^{-1}\left( {^{\alpha_{L}\left(
l\right)}} m,0\right) = {^{\alpha_{L}\left( l\right)}} m.
\]
So, in general, the Hom-action of $\left( L,\alpha_{L}\right)$ on $\left( M,\alpha_{M}\right)$ defined by the split short exact sequence (\ref{semi-direct}) does not coincide with the initial Hom-action of $(L,\alpha_{L})$ on $( M,\alpha_{M})$, but it coincides with the induced Hom-action of $(\alpha_{L}(L),\alpha_{L\mid})$ on $( M,\alpha_{M})$.
\begin{definition}
Let $(M,\alpha_M)$ be a Hom-module over a Hom-Lie algebra $(L,\alpha_L)$. A derivation from $(L,\alpha_L)$ to $(M,\alpha_M)$ is a
linear map $d : L \to M$ satisfying
\begin{enumerate}
\item[a)] $d[x,y] = {^{\alpha_L(x)}}d(y) - {^{\alpha_L(y)}}d(x)$ for all $x, y \in L$,
\item [b)] $\alpha_M \circ d = d \circ \alpha_L$.
\end{enumerate}
We denote by $\Der_{\alpha}\left(L, M\right)$ the vector space of all
derivations from $(L, \alpha_{L})$ to $(M,\alpha_{M})$.
\end{definition}
The following lemma is straightforward.
\begin{lemma}\label{derivation}
Let $(M,\alpha_M)$ be a Hom-module over a Hom-Lie algebra $(L,\alpha_L)$. Then the projection $\theta : M \rtimes L \to M$, $\theta(m,l)=m$, is a derivation, where $(M,\alpha_M)$ is considered as a Hom-module over $(M \rtimes L,\alpha_{\rtimes})$ via the homomorphism $\pi$ in (\ref{semi-direct}).
\end{lemma}
\begin{proposition} \label{correpresentabilidad}
Let $(M,\alpha_M)$ be a Hom-module over a Hom-Lie algebra $(L,\alpha_L)$. For every homomorphism of Hom-Lie algebras $f:(K,\alpha_K) \to (L,\alpha_L)$ and every derivation
$d:(K,\alpha_K) \to (M,\alpha_M)$ there exists a unique homomorphism of Hom-Lie algebras $h : (K,\alpha_K) \to (M \rtimes
L,\alpha_{\rtimes})$ such that $\pi \circ h = f$ and $ \theta \circ h = d$, that is, the following diagram is commutative:
\[
\xymatrix{
& \left( K,\alpha_{K} \right) \ar[dr]^{f} \ar[d]^{h} \ar[ld]_{d} & \\
\left( M,\alpha_{M}\right) \ar[r]^{i\ } & \ar@/^0.3pc/[l]^{\theta \ } \left( M \rtimes L,{\alpha_{\rtimes}}\right) \ar[r]^{\ \pi} & \left( L,\alpha_{L}\right). }\]
Here $(M,\alpha_M)$ is regarded as a Hom-module over $(K,\alpha_K)$ via $f$.
Conversely, every homomorphism of Hom-Lie algebras $h:\left( K,\alpha_{K}\right) \longrightarrow \ $ $\left( M \rtimes L,\alpha_{\rtimes} \right)$, determines a homomorphism of Hom-Lie algebras $f=\pi \circ h:\left(
K,\alpha_{K}\right) \longrightarrow\left( L,\alpha_{L}\right)$ and a derivation
$d=\theta \circ h:\left( K,\alpha_{K}\right) \longrightarrow\left( M,\alpha
_{M}\right)$.
\end{proposition}
{\it Proof.}
Define $h:(K,\alpha_K)\to (M \rtimes L,\alpha_{\rtimes})$ by $h(x) = \left( d\left( x\right) ,f\left( x\right) \right)$, $x \in K$. Then everything can be readily checked. \rdg
\medskip
By taking $(K,\alpha_K)=(L,\alpha_L)$ and $f=\id_L$ we get:
\begin{corollary} Let $(M,\alpha_M)$ be a Hom-module over a Hom-Lie algebra $(L,\alpha_L)$.
The set of derivations from $(L,\alpha_L)$ to $(M,\alpha_M)$ is in a bijective correspondence with the set of
homomorphisms $h : (L,\alpha_L) \to (M \rtimes L,\alpha_{\rtimes})$ such that $\pi \circ h = \id_L$.
\end{corollary}
\begin{theorem} \label{sucesion}
Let $0 \to (N,\alpha_N) \stackrel{i}\to (K,\alpha_K)
\stackrel{\pi} \to (L,\alpha_L) \to 0$ be a short exact sequence of Hom-Lie algebras and $(M,\alpha_M)$ a Hom-module over $(L,\alpha_L)$
(and so a Hom-module over $(K,\alpha_K)$ via $\pi$ )
such that the Hom-action satisfies the condition ${^{\alpha_L(l)}} m = {^l} m$, $l \in L$ and $m \in M$ (e.g. see Example \ref{ejemplo 2} {\it c)}). Denote by $(N^{\rm ab},\overline{\alpha}_{N})$ the quotient of $(N,\alpha_N)$ by the ideal $([N,N],\alpha_{N\mid})$. Then $(N^{\rm ab},\overline{\alpha}_{N})$ has a Hom-module structure over $(L,\alpha_L)$ and there is a natural exact sequence of vector spaces
\[
0 \to \Der_{\alpha}(L, M) \stackrel{\Delta}\longrightarrow \Der_{\alpha}(K, M) \stackrel{\rho}\longrightarrow \hom_L(N^{\rm ab},M),
\]
where $\hom_L(N^{\rm ab},M)=\{ f : (N^{\rm ab},\overline{\alpha}_{N}) \to (M, \alpha_M) \mid f({^l}\overline{n}) = {^l}f(\overline{n}) \}$.
\end{theorem}
{\it Proof.} It is easy to see that the equality ${}^l\overline{n}=i^{-1}[x_l, i(n)]$, where $n\in N$, $l\in L$ and $x_l\in K$ such that $\pi(x_l)=l$, defines a Hom-module structure over $(L,\alpha_L)$ on $(N^{\rm ab},\overline{\alpha}_{N})$.
Let $\Delta(d) = d \circ \pi$, for $d \in \Der_{\alpha}(L,M)$. Obviously $\Delta$ is injective. For any $\delta \in \Der_{\alpha}(K,M)$, $\delta \circ i : (N,\alpha_N) \to (M, \alpha_M)$ is a homomorphism of Hom-Lie algebras that vanishes on $[N,N]$ and so induces a homomorphism of abelian Hom-Lie algebras $\rho : (N^{\rm ab}, \overline{\alpha}_N) \to (M, \alpha_M)$. Now the remaining details are straightforward. \rdg
\medskip
Let us note that, of course the results above recover the well-known classical facts on semi-direct product of Lie algebras (see e.g. \cite{HS}).
\subsection{Homology} \
\noindent The homology of Hom-Lie algebras, generalizing the classical Chevalley-Eilenberg homology of Lie algebras, is constructed in \cite{Yau1, Yau2} (see also \cite{CaInPa}). Let us recall that the homology $ H_{*}^{\alpha}\left( L,M\right)$ of a Hom-Lie algebra $(L, \alpha_{L})$ with coefficients in a Hom-module $(M,\alpha_{M})$ over $(L, \alpha_{L})$ is defined as the homology of the chain complex $(C_{*}^{\alpha}(L,M), d_*)$, where
\[
C_{n}^{\alpha}(L,M)=M\otimes \wedge^n L,\quad n\geq 0
\]
and the boundary map $d_{n}:C_{n}^{\alpha}(L,M) \longrightarrow C_{n-1}^{\alpha}(L,M)$, $n\geq 1$, is given by
\begin{align*}
d_{n}&( m\otimes x_{1}\wedge\cdots\wedge
x_{n}) =\overset{n}{\underset{i=1}
{\displaystyle\sum}}
(-1)^{i} \ \ {}^{x_{i}} m \otimes\alpha_{L}(x_{1})
\wedge\!\cdots\!\wedge\widehat{\alpha_{L}(x_{i})}\wedge\!\cdots\!\wedge \alpha_{L}(x_{n}) +\\
&\!\!\underset{1\leq i<j\leq n}{\sum}\!\!\left( -1\right)
^{i+j}\alpha_{M}\left( m\right) \otimes\left[ x_{i},x_{j}\right]
\wedge\alpha_{L}\left( x_{1}\right) \wedge\!\cdots\!\wedge\widehat{\alpha
_{L}\left( x_{i}\right) }\wedge\!\cdots\!\wedge\widehat{\alpha_{L}\left(
x_{j}\right) }\wedge\!\cdots\!\wedge\alpha_{L}\left( x_{n}\right).
\end{align*}
As usual $\widehat{\alpha_{L}(x_{i})}$ means that the variable $\alpha_{L}(x_{i})$ is omitted.
Let us remark that for a Lie algebra $L$ and an $L$-module $M$, the chain complex
$C_{*}^{\alpha}(L,M)$ is exactly the Chevalley-Eilenberg complex that defines the
Lie algebra homology of $L$ with coefficients in the $L$-module $M$.
Easy computations of low-dimensional cycles and boundaries provide the following results:
\[
H_{0}^{\alpha
}\left( L,M\right) ={\Ker\left( d_{0}\right) }/{\Im\left( d_{1}\right) }={M}/{^{L}M} \ ,
\]
where $^{L}M=\left\{ {}^x m \mid m\in M, \ x\in L\right\}$. Moreover, if $(M,\alpha_M)$ is a trivial Hom-module over $(L,\alpha_L)$, i.e. ${}^x m=0$ for all $x\in L$ and $m\in M$, then
\[
H_{1}^{\alpha}\left( L,M\right)={\Ker\left( d_{1}\right) }/{\Im
\left( d_{2}\right) } ={(M\otimes L)}/{\big(\alpha_M\left( M\right) \otimes\left[
L,L\right]\big) }.
\]
In particular, if $M = \mathbb{K}$, then $H_{1}^{\alpha}\left( L,\mathbb{K} \right) ={L}/{\left[
L,L\right] }$.
Below we use the notation $H_{n}^{\alpha}\left( L \right) $ for $H_{n}^{\alpha}\left( L,\mathbb{K} \right)$.
\section{Non-abelian tensor product of Hom-Lie algebras}
In this section we introduce a non-abelian tensor product of Hom-Lie algebras which generalizes the non-abelian tensor product of Lie algebras \cite{El2}, and study its properties.
\begin{definition}
Let $(M,\alpha_M)$ and $(N,\alpha_N)$ be Hom-Lie algebras with Hom-actions on each other. The Hom-actions are said to be compatible if
\[
^{(^mn)} m'=[m',^nm] \quad \text{and} \quad ^{(^nm)}n'=[n',^mn]
\]
for all $m,m'\in M$ and $n,n'\in N$.
\end{definition}
\begin{example}
If $(H,\alpha_H)$ and $(H',\alpha_{H'})$ both are ideals of a Hom-Lie algebra $(L,\alpha_L)$, then the Hom-actions of $(H,\alpha_H)$ and $(H',\alpha_{H'})$ on each other, considered in Example \ref{ejemplo 2} {a)}, are compatible.
\end{example}
Let $(M,\alpha_M)$ and $(N,\alpha_N)$ be Hom-Lie algebras acting on each other compatibly.
Consider the Hom-vector space $(M\otimes N,\alpha_{M\otimes N})$ given by the tensor product $M\otimes N$ of the underlying vector spaces and the linear map $\alpha_{M \otimes N} : M \otimes N \to M \otimes N$, $\alpha_{M \otimes N}(m \otimes n) = \alpha_M(m) \otimes \alpha_N(n)$. Denote by $D(M,N)$ subspace of $M \otimes N$ generated by all elements of the form
\begin{enumerate}
\item[{\it a)}] $[m,m']\otimes \alpha_N(n) -\alpha_M(m)\otimes {}^{m'}n +\alpha_M(m') \otimes {}^mn$,
\item[{\it b)}] $\alpha_M(m)\otimes [n,n'] - {}^{n'}m\otimes\alpha_N(n)+{}^{n}m\otimes\alpha_N(n')$,
\item[{\it c)}] ${}^{n}m\otimes {}^{m}n$,
\item[{\it d)}]${}^{n}m\otimes {}^{m'}n' +{}^{n'}m'\otimes {}^{m}n$,
\item[{\it e)}]$[{}^{n}m,{}^{n'}m']\otimes \alpha_N({}^{m''}n'')+[{}^{n'}m',{}^{n''}m'']\otimes \alpha_N({}^{m}n)
+[{}^{n''}m'',{}^{n}m]\otimes \alpha_N({}^{m'}n')$,
\end{enumerate}
for $m,m',m''\in M$ and $n,n',n''\in N$.
\begin{proposition}
The quotient vector space $(M\otimes N)/D(M,N)$ with the product
\begin{equation}\label{pr_in_tensor}
[m\otimes n, m'\otimes n']=-{}^{n}m\otimes {}^{m'}n'
\end{equation}
and together with the endomorphism $(M\otimes N)/D(M,N)\to (M\otimes N)/D(M,N)$ induced by $\alpha_{M\otimes N}$, is a Hom-Lie algebra.
\end{proposition}
{\it Proof.}
It is clear that $\alpha_{M\otimes N}$ preserves the elements of $D(M,N)$ as well as the product defined by (\ref{pr_in_tensor}).
Routine calculations show that this product is compatible with the defining relations of $(M\otimes N)/D(M,N)$ and can be extended from generators to any elements. Since the actiones of $(M,\alpha_M)$ and $(N,\alpha_N)$ on each other are compatible, it follows by direct calculations that the product (\ref{pr_in_tensor}) satisfies the skew-symmetry and the Hom-Jacobi identity. \rdg
\begin{definition}
The above Hom-Lie algebra structure on $(M\otimes N)/D(M,N)$ is called the non-abelian tensor product of Hom-Lie algebras $(M,\alpha_M)$ and $(N,\alpha_N)$ (or Hom-Lie tensor product for short). It will be denoted by $(M\star N, \alpha_{M\star N})$ and the equivalence class of $m\otimes n$ will be denoted by $m\star n$.
\end{definition}
\begin{remark}
Note that if $\alpha_M=\id_M$ and $\alpha_N=\id_N$ then $M\star N$ is the non-abelian tensor product of Lie algebras $M$ and $N$ given in \cite{El2} (see also \cite{El1, InKhLa}).
\end{remark}
The Hom-Lie tensor product can also be defined by a universal property in the following way.
\begin{definition}\label{paring} Let $(M,\alpha_M)$ and $(N,\alpha_N)$ be Hom-Lie algebras acting on each other.
For any Hom-Lie algebra $(L,\alpha_L)$, a bilinear map $h : (M\times N,\alpha_M\times\alpha_N) \to (L,\alpha_L)$ is said to be a Hom-Lie pairing if the following properties are satisfied:
\begin{enumerate}
\item [a)] $h([m,m'],\alpha_N(n)) = h(\alpha_M(m), {}^{m'} n) - h(\alpha_M(m'), {}^m n)$,
\item [b)] $h(\alpha_M(m),[n,n']) = h({}^{n'} m, \alpha_N(n)) - h({}^n m, \alpha_N(n'))$,
\item [c)] $h({}^n m, {}^{m'} n') = - [h(m,n), h(m',n')]$,
\item [d)] $h \circ (\alpha_M \times \alpha_N) = \alpha_L \circ h$,
\end{enumerate}
for all $m, m' \in M$, $n, n' \in N$.
\end{definition}
\begin{example}\
\begin{enumerate}
\item[a)] If $\alpha_L=\id_L, \alpha_M= \id_M$ and $\alpha_N= \id_N$, then Definition \ref{paring} recovers the definition of Lie paring given in \cite{El1}.
\item[b)] Let $(M,\alpha_M)$ and $(N,\alpha_N)$ be ideals of a Hom-Lie algebra $(L,\alpha_L)$, then the bilinear map $h : (M\times N,\alpha_M\times\alpha_N) \to (M \cap N,\alpha_{M \cap N})$, given by $h(m,n)=[m,n]$, is a Hom-Lie pairing.
\end{enumerate}
\end{example}
\begin{definition}
A Hom-Lie pairing $h : (M\times N,\alpha_M\times\alpha_N) \to (L,\alpha_L)$ is said to be universal if for any other Hom-Lie pairing $h' : (M\times N,\alpha_M\times\alpha_N) \to (L',\alpha_{L'})$ there is a unique homomorphism of Hom-Lie algebras $\theta : (L,\alpha_L) \to (L',\alpha_{L'})$ such that
$\theta \circ h = h'$.
\end{definition}
Clearly, if $h$ is universal, then $(L,\alpha_{L})$ is determined up to isomorphism by $(M,\alpha_M), (N,\alpha_N)$ and the Hom-actions. Moreover, it is straightforward to show the following
\begin{proposition} Let $(M,\alpha_M)$ and $(N,\alpha_N)$ be Hom-Lie algebras acting on each other compatibly. The map
\[
h : (M\times N,\alpha_M\times\alpha_N) \to (M\star N, \alpha_{M\star N}),\quad (m, n) \mapsto m \star n
\]
is a universal Hom-Lie paring.
\end{proposition}
The Hom-Lie tensor product is symmetric in the sense of the following isomorphism of Hom-Lie algebras
\[
(M\star N, \alpha_{M\star N})\overset{\approx}{\longrightarrow} (N\star M, \alpha_{N\star M}), \quad m\star n\mapsto n\star m.
\]
This follows by the fact that $h : (M\times N,\alpha_M\times\alpha_N) \to (N\star M, \alpha_{N\star M})$, $(m, n)\mapsto n\star m$ is a Hom-Lie pairing and the universal property of $(M\star N, \alpha_{M\star N})$ thus yields a homomorphism $(M\star N, \alpha_{M\star N})\to (N\star M, \alpha_{N\star M})$, the inverse of which is defined similarly.
\
Sometimes the Hom-Lie tensor product can be described as the tensor product of vector spaces. In particular, we have the following
\begin{proposition}\label{proposition 5.10}
If the Hom-Lie algebras $(M,\alpha_M)$ and $(N, \alpha_N)$ act trivially on each other and both $\alpha_M$, $\alpha_N$ are epimorphisms, then there is an isomorphism of abelian Hom-Lie algebras
\[
(M\star N, \alpha_{M\star N})\approx (M^{ab}\otimes N^{ab}, \alpha_{M^{ab}\otimes N^{ab}}),
\]
where $M^{ab}=M/[M,M]$, $N^{ab}=N/[N,N]$ and $\alpha_{M^{ab}\otimes N^{ab}}$ is induced by $\alpha_M$ and $\alpha_N$.
\end{proposition}
{\it Proof.}
Since the Hom-actions are trivial, the relation (\ref{pr_in_tensor}) enables us to see that $(M\star N, \alpha_{M\star N})$ is an abelian Hom-Lie algebra. Further, since $\alpha_M$ and $\alpha_N$ are epimorphisms, the defining relations of the Hom-Lie tensor product say that the vector space $M\star N$ is the quotient of $M\otimes N$ by the relations
$[m,m']\otimes n=0=m\otimes [n,n']$ for all $m,m'\in M$, $n,n'\in N$. The later is isomorphic to $M^{ab}\otimes N^{ab}$ and this isomorphism commutes with the endomorphisms $\alpha_{M^{ab}\otimes N^{ab}}$ and $\alpha_{M\star N}$. \rdg
\medskip
The Hom-Lie tensor product is functorial in the following sense: if $f:(M,\alpha_M)$ $\to (M',\alpha_{M'})$ and $g:(N,\alpha_N)\to (N',\alpha_{N'})$ are homomorphisms of Hom-Lie algebras together with compatible Hom-actions of $(M, \alpha_M)$ (resp. $(M', \alpha_{M'})$) and $(N, \alpha_N)$ (resp. $(N', \alpha_{N'})$) on each other such that $f$, $g$ preserve these Hom-actions, that is
\[
f({}^nm)={}^{g(n)}f(m), \quad g({}^mn)={}^{f(m)}g(n), \qquad m\in M,\ n\in N,
\]
then there is a homomorphism of Hom-Lie algebras
\[
f\star g:(M\star N,\alpha_{M\star N})\to (M'\star N',\alpha_{M'\star N'})
\]
defined by $(f\star g)(m\star n)=f(m)\star g(n)$.
\begin{proposition}\label{exact-tensor-1}
Let $0\to (M_1,\alpha_{M_1})\overset{f}{\to}(M_2,\alpha_{M_2})\overset{g}{\to}(M_3,\alpha_{M_3})\to 0$ be a short exact sequence of Hom-Lie algebras. Let $(N,\alpha_{N})$ be a Hom-Lie algebra together with compatible Hom-actions of $(N,\alpha_{N})$ and $(M_i,\alpha_{M_i})$ $(i=1,2,3)$ on each other and $f$, $g$ preserve these Hom-actions.
Then there is an exact sequence of Hom-Lie algebras
\[
(M_1\star N,\alpha_{M_1\star N})\overset{f\star \id_N}{\longrightarrow}(M_2\star N,\alpha_{M_2\star N})\overset{g\star \id_N}{\longrightarrow}(M_3\star N,\alpha_{M_3\star N})\longrightarrow 0.
\]
\end{proposition}
{\it Proof.}
Clearly $g\star \id_N$ is an epimorphism and $\Im(f\star \id_N) \subseteq \Ker(g\star \id_N)$. Now $\Im(f\star \id_N)$ is generated by all elements of the form $f(m_1)\star n_1$ with $m_1\in M_1$, $n_1\in N$ and it is an ideal in $(M_2\star N,\alpha_{M_2\star N})$ since we have
\[
[f(m_1)\star n_1, m_2\star n_2]=-f({}^{n_1}m_1)\star {}^{m_2}n_2\in \Im(f\star \id_N)
\]
for any generator $m_2\star n_2\in M_2\star N$. Thus, $g\star \id_N$ yields a factorization
\[
\xi:\big((M_2\star N) /\Im(f\star \id_N), \overline{\alpha}_{M_2\star N}\big)\to (M_3\star N,\alpha_{M_3\star N}).
\]
In fact this is an isomorphism of Hom-Lie algebras with the inverse map
\[
\xi':(M_3\star N,\alpha_{M_3\star N}) \to \big((M_2\star N) /\Im(f\star \id_N), \overline{\alpha}_{M_2\star N}\big)
\]
given on generators by $\xi'(m_3\star n)=\overline{m_2\star n}$, where $m_2\in M_2$ such that $g(m_2)=m_3$.
The remaining details are straightforward calculations and we leave to the reader. \rdg
\begin{proposition}\label{exact-tensor-2}
If $(M,\alpha_M)$ is an ideal of a Hom-Lie algebra $(L, \alpha_L)$, then there is an exact sequence of Hom-Lie
algebras
\[
\big((M\star L)\rtimes (L\star M), \alpha_{\rtimes}\big) \overset{\sigma} {\longrightarrow}(L\star L,\alpha_{L\star L})\overset{\tau}{\longrightarrow}({L}/{M}\star {L}/{M},\alpha_{{L}/{M}\star {L}/{M}})\longrightarrow 0.
\]
\end{proposition}
{\it Proof.}
First we note that $\tau$ is the functorial homomorphism induced by the projection $(L,\alpha_L) \twoheadrightarrow (L/M,\alpha_{L/M})$ and clearly it is surjective. Let $\sigma':(M\star L,\alpha_{M\star L})\to (L\star L,\alpha_{L\star L})$ and $\sigma'':(L\star M,\alpha_{L\star M}) \to (L\star L,\alpha_{L\star L})$ be the functorial homomorphisms induced by the inclusion $(M,\alpha_M)\hookrightarrow (L,\alpha_L)$ and by the identity map $(L,\alpha_L)\to (L,\alpha_L)$. Let $\sigma (x,y)=\sigma'(x)+\alpha_{L \star M} \circ \sigma''(y)$ for all $x\in M\star L$ and $y\in L\star M$. It is straightforward to see that $\sigma$ is a homomorphism of Hom-Lie algebras and $\tau \circ \sigma$ is the trivial homomorphism. Clearly $\Im(\sigma)$ is generated by the elements $m\star l$ and $\alpha_L(l)\star \alpha_M(m)$ for $m\in M$, $l\in L$ and, by the formula (\ref{pr_in_tensor}), it is an ideal of $(L\star L,\alpha_{L\star L})$. Let us define a homomorphism of Hom-Lie algebras $\tau': ({L}/{M}\star {L}/{M},\alpha_{{L}/{M}\star {L}/{M}})\to (L\star L,\alpha_{L\star L})/\Im(\sigma)$ by $\tau'(\overline{l}\star \overline{l'})=\overline{l\star l'}$, $l,l'\in L$. It is easy to see that $\tau'$ is well-defined and it has an inverse homomorphism induced by $\tau$. \rdg
\begin{lemma} \label{action-on-tensor}
Let $(M,\alpha_M)$ and $(N,\alpha_N)$ be Hom-Lie algebras with compatible actions on each other.
\begin{enumerate}
\item[a)] There are homomorphisms of Hom-Lie algebras
$\begin{array}{rl}
&\psi_M:(M\star N, \alpha_{M \star N}) \to (M, \alpha_{M }), \quad \psi_M(m\star n)= -{}^nm,\\
&\psi_N:(M\star N, \alpha_{M \star N}) \to (N, \alpha_N),\quad \psi_N(m\star n)= {}^mn.
\end{array}$
\item[b)]
There is a Hom-action of $(M, \alpha_{M })$ (resp. $(N, \alpha_{N })$) on the Hom-Lie tensor product ($M\star N, \alpha_{M \star N}$) given, for all $m,m'\in M$, $n,n'\in N$, by
\[
\begin{array}{rl}
&{}^{m'}(m\star n)=[m',m]\star \alpha_N(n)+\alpha_M(m)\star {}^{m'}n \\
\big( \text{resp.} \ &{}^{n'}(m\star n)={}^{n'}m\star \alpha_N(n)+\alpha_M(m)\star [n',n] \big)
\end{array}
\]
\item[c)] $\Ker(\psi_M)$ (resp. $\Ker(\psi_N)$) is contained in the center of $(M\star N, \alpha_{M \star N})$.
\item[d)] The induced Hom-action of $\Im(\psi_1)$ (resp. $\Im(\psi_2)$) on $\Ker(\psi_1)$ (resp. $\Ker(\psi_2)$) is trivial.
\item[e)] $\psi_M$ and $\psi_N$ satisfy the following properties for all $m, m' \in M$, $n, n' \in N$:
\begin{enumerate}
\item[i)] $\psi_M(^{m'}(m \star n)) = [\alpha_M(m'), \psi_M(m \star n)]$,
\item[ii)] $\psi_N(^{n'}(m \star n)) = [\alpha_N(n'), \psi_N(m \star n)]$,
\item[iii)] ${^{\psi_M(m \star n)}}(m' \star n') = [\alpha_{M \star N}(m \star n), m' \star n'] = {^{\psi_N(m \star n)}} (m' \star n')$.
\end{enumerate}
\end{enumerate}
\end{lemma}
{\it Proof.}
Everything can be readily checked thanks to the compatibility conditions and the relation (\ref{pr_in_tensor}). \rdg
\begin{remark}
If $\alpha_M = id_M$ and $\alpha_N = id_N$, then both $\psi_M$ and $\psi_N$ in Theorem \ref{action-on-tensor} are crossed modules of Lie algebras (see \cite{El1}).
\end{remark}
\begin{definition}
A Hom-Lie algebra $(L, \alpha_L)$ is said to be perfect if $L=[L, L]$.
\end{definition}
\begin{theorem} \label{sucesion exacta}
Let $(M,\alpha_M)$ be an ideal of a perfect Hom-Lie algebra $(L,\alpha_L)$. Then there is an exact sequence of vector spaces
\[
\Ker(M\star L\overset{\psi_{M}\ }\longrightarrow M)\to H_2^{\alpha}(L)\to H_2^{\alpha}({L}/{M})\to {M}/{[L,M]}\to 0
\]
\end{theorem}
{\it Proof.} Thanks to Proposition \ref{exact-tensor-2} there is a commutative diagram of Hom-Lie algebras with exact rows
{\footnotesize\[
\xymatrix{
& \big((M\star L) \rtimes (L\star M), \alpha_{\rtimes}\big) \ar[r] \ar[d]^{\psi}& (L \star L,\alpha_{L \star L}) \ar[r]^{\pi \star \pi \ \ \ \ \ \ \ \ } \ar[d]^{\psi_L} & ({L}/{M} \star {L}/{M}, {\alpha}_{{L}/{M} \star {L}/{M}}) \ar[r] \ar[d]^{{\psi_{L/M}}}& 0\\
0 \ar[r] & (M, \alpha_M) \ar[r] & (L,\alpha_L) \ar[r]^{\pi} & ({L}/{M},{\alpha}_{L/M}) \ar[r] & 0,
}
\]}
where $\psi((m_1,l_1),(l_2,m_2))=[m_1,l_1]+[\alpha_L(l_2),\alpha_M(m_2)]$. Then, by using the Snake Lemma, the assertion follows from Remark \ref{H2} below and the fact that there is a surjective
map $\Ker(\psi) \to \Ker(\psi_{M})$. \rdg
\begin{remark}
If $\alpha_L = \id_L$, then the exact sequence in Theorem \ref{sucesion exacta} is part of the six-term exact sequence in \cite{El2}.
\end{remark}
\section{Application in universal ($\alpha$-)central extensions of Hom-Lie algebras} \label{section}
In this section we complement by new results the investigation of universal central extensions of Hom-Lie algebras done in \cite{CaInPa}. We also describe universal ($\alpha$-)central extensions via Hom-Lie tensor product.
\begin{definition} \label{alfacentral} A central (resp. $\alpha$-central) extension of a Hom-Lie algebra $(L, \alpha_L)$ is an exact sequence of Hom-Lie algebras
\[
(\mathfrak{K}): \ \ 0 \longrightarrow (M, \alpha_M) \longrightarrow (K,\alpha_K) \stackrel{\pi} \longrightarrow (L, \alpha_L) \longrightarrow 0
\]
such that $[M, K] = 0 $, i.e. $M \subseteq Z(K)$ (resp. $[\alpha_M(M), K] = 0$, i.e. $\alpha_M(M) \subseteq Z(K)$).
A central extension $(\mathfrak{K})$ is called universal central (resp. universal $\alpha$-central) extension if, for every central (resp. $\alpha$-central) extension $(\mathfrak{K'})$ of $(L,\alpha_L)$ there exists one and only one homomorphism of Hom-Lie algebras $h : (K,\alpha_K) \to (K',\alpha_{K'})$ such that $\pi' \circ h = \pi$.
\end{definition}
\begin{remark}
Obviously every central extension is an $\alpha$-central extension and these notions coincide when $\alpha_M = \id_M$. On the other hand, every universal $\alpha$-central extension is a universal central extension and these notions coincide when $\alpha_M = \id_M$. Let us also observe that if a universal ($\alpha$-)central extension exists then it is unique up to isomorphism.
\end{remark}
The category ${\tt HomLie}$ is an example of a semi-abelian category which does not satisfy universal central extension condition in the sense of \cite{CaTim}, that is, the composition of central extensions of Hom-Lie algebras is not central in general, but it is an $\alpha$-central extension (see Theorem \ref{teorema} {\it a)} below). This fact does not allow complete generalization of classical results to Hom-Lie algebras and the well-known properties of universal central extensions are divided between universal central and universal $\alpha$-central extensions of Hom-Lie algebras. In particular, the assertions in the following theorem are proved in \cite{CaInPa}.
\begin{theorem}\label{teorema} \
\begin{enumerate}
\item[a)] Let $(K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ and $(F,\alpha_F) \stackrel{\rho} \twoheadrightarrow (K, \alpha_K)$ be central extensions with $(K, \alpha_K)$ a perfect Hom-Lie algebra. Then the composition extension $(F,\alpha_F)$ $\stackrel{\pi \circ \rho} \twoheadrightarrow (L, \alpha_L)$ is an $\alpha$-central extension.
\item[b)] Let $(K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ and $(K',\alpha_{K'}) \stackrel{\pi'} \twoheadrightarrow (L, \alpha_L)$ be two central extensions of $(L,\alpha_L)$. If $(K,\alpha_K)$ is perfect, then there exists at most one homomorphism of Hom-Lie algebras $f : (K,\alpha_K) \to (K', \alpha_{K'})$ such that $\pi' \circ f = \pi$.
\item[c)] If $ (K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ is a universal $\alpha$-central extension, then $(K,\alpha_K)$ is a perfect Hom-Lie algebra and every central extension of $(K,\alpha_K)$ splits.
\item[d)] If $(K,\alpha_K)$ is a perfect Hom-Lie algebra and every central extension of $(K,\alpha_K)$ splits, then any central extension $(K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ is a universal central extension.
\item[e)] A Hom-Lie algebra $(L, \alpha_L)$ admits a universal central extension if and only if $(L, \alpha_L)$ is perfect.
Furthermore, the kernel of the universal central extension is canonically isomorphic to the second homology $H_2^{\alpha}(L)$.
\item[f)] If $(K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ is a universal $\alpha$-central extension, then $H_1^{\alpha}(K) = H_2^{\alpha}(K) = 0$.
\item[g)] If $H_1^{\alpha}(K) = H_2^{\alpha}(K) = 0$, then any central extension $(K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ is a universal central extension.
\end{enumerate}
\end{theorem}
It follows from Lemma \ref{action-on-tensor} that for any Hom-Lie algebra $(L,\alpha_L)$ the homomorphism
\[
\psi:(L\star L, \alpha_{L \star L}) \twoheadrightarrow ([L,L], \alpha_{L \mid}), \quad \psi(l \star l')=[l,l'],
\]
is a central extension of the Hom-Lie algebra $([L,L], \alpha_{L \mid})$.
\begin{theorem} \label{teor}
If $(L,\alpha_L)$ is a perfect Hom-Lie algebra, then the central extension \linebreak $(L\star L, \alpha_{L \star L})\overset{\psi}\twoheadrightarrow (L, \alpha_L)$ is the universal central extension of $(L,\alpha_L)$.
\end{theorem}
{\it Proof.} Let $(C, \alpha_C) \overset{\phi}\twoheadrightarrow (L, \alpha_L)$ be a central extension of $(L, \alpha_L)$. Since $\Ker(\phi)$ is in the center of $(C,\alpha_C)$, we get a well-defined homomorphism of Hom-Lie algebras $f:(L\star L, \alpha_{L\star L})\to (C, \alpha_C)$ given on generators by $f(l\star l')=[c_l,c_{l'}]$, where $c_l$ and $c_{l'}$ are any elements in $\phi^{-1}(l)$ and $\phi^{-1}(l')$, respectively. Obviously $\phi \circ f= \psi$ and $f \circ \alpha_{L \star L} = \alpha_C \circ f$, having in mind that $\alpha_C(c_l) \in \phi^{-1}(\alpha_L(l))$ for all $l \in L$. Since $L$ is perfect, then by equality (\ref{pr_in_tensor}), so is $L\star L$. Hence the homomorphism $f$ is unique by Theorem \ref{teorema} {\it b)}. \rdg
\begin{remark} \label{H2}
If the Hom-Lie algebra $(L,\alpha_L)$ if perfect, by Theorem \ref{teorema} {e)} we have that $H_2^{\alpha}(L)\approx \Ker(L\star L\overset{\psi}\to L )$.
\end{remark}
Now we obtain a condition for the existence of the universal $\alpha$-central extensions. We need the following notion.
\begin{definition}
A Hom-Lie algebra $(L, \alpha_L)$ is said to be $\alpha$-perfect if $L=[\alpha_L(L),$ $\alpha_L(L)]$.
\end{definition}
\begin{example}
Consider the situation when the ground field $\K$ is the field of complex numbers. Let $L$ be the three-dimensional vector space with basis $\{e_1,$ $e_2,e_3\}$. Define product in $L$ by $[e_1,e_2]= - [e_2,e_1] = e_3$, $[e_2,e_3]= - [e_3,e_2] = e_1$, $[e_3,e_1]= - [e_1,e_3] = e_2$ and zero elsewhere. Take the endomorphism $\alpha_L:L\to L$ represented by the matrix $\left( \begin{array}{ccc} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ 0 & -1 & 0 \\ \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} \end{array} \right)$. Then $(L,\alpha_L)$ is an $\alpha$-perfect Hom-Lie algebra.
\end{example}
\begin{remark}\label{alfa perfecta} \
\begin{enumerate}
\item[a)] When $\alpha_L= \id_L$, the notions of perfect and $\alpha$-perfect Hom-Lie algebras are the same.
\item[b)] Obviously, if $\left( L,\alpha_{L}\right)$ is an $\alpha$-perfect Hom-Lie algebra, then it is perfect. Nevertheless the converse is not true in general. For example, the three-dimensional (as a vector space) Hom-Lie algebra $\left( L, \alpha_{L}\right)$ with linear basis $\{e_1, e_2, e_3\}$, product given by $[e_1,e_2]=-[e_2,e_1]=e_3$, $[e_1,e_3]=-[e_3,e_1]=e_2$, $[e_2,e_3]=-[e_3,e_2]=e_1$ and zero elsewhere, and endomorphism $\alpha_L=0$ is perfect, but it is not $\alpha$-perfect.
\item[c)] If $\left( L,\alpha_{L}\right)$ is $\alpha$-perfect, then $L = \alpha_L(L)$, i.e. $\alpha_L$ is surjective. Nevertheless the converse is not true. For instance, consider the two-dimensional (as a vector space) Hom-Lie algebra with linear basis $\{e_1, e_2\}$, bracket given by $[e_1, e_2] = - [e_2, e_1] = e_2$ and zero elsewhere, and endomorphism $\alpha_L$ represented by the matrix $\left( \begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right)$. Obviously the endomorphism $\alpha_L$ is surjective, but $[\alpha_L(L), \alpha_L(L)] = \langle \{e_2 \} \rangle$.
\end{enumerate}
\end{remark}
\begin{lemma} \label{lema 10}
Let $(M, \alpha_M) \rightarrowtail (K,\alpha_K) \stackrel{\pi} \twoheadrightarrow (L, \alpha_L)$ be a central extension and $(K,\alpha_K)$ be an $\alpha$-perfect Hom-Lie algebra. Let $ (M',\alpha_{M'}) \rightarrowtail (K',\alpha_{K'}) \stackrel{\pi'} \twoheadrightarrow (L, \alpha_L)$ be an $\alpha$-central extension. Then there exists at most one homomorphism of Hom-Lie algebras $f : (K,\alpha_K) \to (K', \alpha_{K'})$ such that $\pi' \circ f = \pi$.
\end{lemma}
{\it Proof.} Let us assume that there are homomorphisms $f_1$ and $f_2$ such that $\pi' \circ f_1 = \pi = \pi' \circ f_2$. Then for any $k\in K$ we have $f_1(k) = f_2(k)+m'_k$, for some $m'_k \in M'$. By using the condition $\alpha_{M'}(M') \subseteq Z(K')$ we have
\begin{align*}
f_1[\alpha_K(k_1),\alpha_K(k_2)]&=[\alpha_{K'} f_1(k_1), \alpha_{K'} f_1(k_2)] \\
& = [\alpha_{K'} f_2(k_1) \!+\! \alpha_{K'}(m'_{k_1}), \alpha_{K'} f_2(k_2)\! +\! \alpha_{K'}(m'_{k_2})]\\
& = [\alpha_{K'} f_2(k_1) , \alpha_{K'} f_2(k_2)]\\
& = f_2[\alpha_K(k_1),\alpha_K(k_2)]
\end{align*}
for any $k_1,k_2\in K$. This implies that $f_1=f_2$, since $(K,\alpha_K)$ is $\alpha$-perfect. \rdg
\begin{theorem}\label{alfa uce}
An $\alpha$-perfect Hom-Lie algebra admits a universal $\alpha$-central extension.
\end{theorem}
{\it Proof.}
Given an $\alpha$-perfect Hom-Lie algebra $(L,\alpha_L)$ we construct a universal $\alpha$-central extension
\begin{equation}\label{eq}
0 \to \Ker (u_{\alpha}) \to (\frak{uce}_{\alpha}(L), \widetilde{\alpha}) \stackrel{u_{\alpha}} \to (L, \alpha_L) \to 0
\end{equation}
as follows. We consider the quotient vector space $\frak{uce}_{\alpha}(L) =\big({\alpha_L(L) \wedge \alpha_L(L)}\big)/{I_L}$, where $I_L$ is the vector subspace of $\alpha_L(L) \wedge \alpha_L(L)$ spanned by the elements of the form
$$-[x_1,x_2] \wedge \alpha_L(x_3) + [x_1,x_3] \wedge \alpha_L(x_2) - [x_2,x_3] \wedge \alpha_L(x_1)$$
for all $x_1, x_2, x_3 \in L$. Here we observe that every summand of the form $[x_1,x_2] \wedge \alpha_L(x_3)$ is an element of $\alpha_L(L) \wedge \alpha_L(L)$, since $L$ is $\alpha$-perfect and so $[x_1,x_2] \in L = [\alpha_L(L), \alpha_L(L)] \subseteq \alpha_L(L)$.
We denote by $\{\alpha_L(x_1), \alpha_L(x_2)\}$ the equivalence class of $\alpha_L(x_1) \wedge \alpha_L(x_2)$. The product in $\frak{uce}_{\alpha}(L)$ is defined by
\[[\{\alpha_L(x_1),\alpha_L(x_2)\}, \{\alpha_L(y_1),\alpha_L(y_2)\}] = \{ [\alpha_L(x_1),\alpha_L(x_2)], [\alpha_L(y_1),\alpha_L(y_2)]\}
\]
and the endomorphism $\widetilde{\alpha} : \frak{uce}_{\alpha}(L) \to \frak{uce}_{\alpha}(L)$ is given by
\[
\widetilde{\alpha}(\{\alpha_L(x_1),\alpha_L(x_2)\}) = \{\alpha_L^2(x_1),\alpha_L^2(x_2)\}.
\]
The map $u_{\alpha}$ is defined by $u_{\alpha}(\{\alpha_L(x_1), \alpha_L(x_2)\})= [\alpha_L(x_1), \alpha_L(x_2)]$.
Straightforward calculations show that $(\frak{uce}_{\alpha}(L), \widetilde{\alpha})$ is indeed a Hom-Lie algebra and
$u_{\alpha}$ is a homomorphism of Hom-Lie algebras. Moreover, $u_{\alpha}$ is surjective, because $(L,\alpha_L)$ is $\alpha$-perfect.
Obviously the sequence (\ref{eq}) is a central extension. Moreover, it is a universal $\alpha$-central extension.
Indeed, consider any $\alpha$-central extension $(M,
\alpha_M) \rightarrowtail (K, \alpha_K) \stackrel{\pi} \twoheadrightarrow (L,
\alpha_L)$. We define $\Phi : (\frak{uce}_{\alpha}(L), \widetilde{\alpha}) \to (K, \alpha_K)$ by
$\Phi(\{\alpha_L(x_1), \alpha_L(x_2)\})$ $= [\alpha_K(k_1),
\alpha_K(k_2)]$, where $k_1, k_2 \in K$ such that $\pi(k_1)=x_1$, $\pi(k_2)=x_2$. It is well defined because of the equality $[\alpha_M(M) , K ]= 0$.
Moreover, direct calculations show that $\Phi$ is a homomorphism of Hom-Lie algebras and $\pi \circ \Phi=u_{\alpha}$. To prove the uniqueness of such $\Phi$, by Lemma \ref{lema 10} it is enough to check that $\frak{uce}_{\alpha}(L)$ is
$\alpha$-perfect. For this later we do the following calculations:
\[
[\widetilde{\alpha} \{\alpha_L(x_1), \alpha_L(x_2)\}, \widetilde{\alpha} \{\alpha_L(y_1), \alpha_L(y_2)\}] = \{[\alpha_L^2(x_1), \alpha_L^2(x_2)],[\alpha_L^2(y_1), \alpha_L^2(y_2)]\},
\]
which implies that $[\widetilde{\alpha} (\frak{uce}_{\alpha}(L)),
\widetilde{\alpha}( \frak{uce}_{\alpha}(L))] \subseteq
\frak{uce}_{\alpha}(L)$. Conversely, having in mind that $L=[\alpha_L(L),\alpha_L(L)]$ and hence every element $x\in L$ can be written as $x=\displaystyle \sum_i \lambda_i [\alpha_L(l_{i_1}), \alpha_L(l_{i_2})] $ for some $\lambda_i\in \mathbb{K}$ and $l_{i_1}, l_{i_2}\in L$, we get
\begin{align*}
\{\alpha_L&(x_1), \alpha_L(x_2)\} = \left \{\alpha_L \left( \sum_i \lambda_i [\alpha_L(l_{i_1}), \alpha_L(l_{i_2})] \right), \alpha_L \left( \sum_j \lambda'_j [\alpha_L(l'_{j_1}), \alpha_L(l'_{j_2})] \right)\right \}\\
&=\sum_{i,j} \lambda_i \lambda'_j \left\{ [\alpha^2_L(l_{i_1}),\alpha^2_L(l_{i_2})], [\alpha^2_L(l'_{j_1}),\alpha^2_L(l'_{j_2})] \right \}\\
&= \sum_{i,j} \lambda_i \lambda'_j \left[ \{\alpha^2_L(l_{i_1}),\alpha^2_L(l_{i_2})\} , \{\alpha^2_L(l'_{j_1}),\alpha^2_L(l'_{j_2})\} \right] \\
&=\sum_{i,j} \lambda_i \lambda'_j \left[ \widetilde{\alpha} \{\alpha_L(l_{i_1}),\alpha_L(l_{i_2})\}, \widetilde{\alpha} \{\alpha_L(l'_{j_1}),\alpha_L(l'_{j_2})\} \right] \in [\widetilde{\alpha} (\frak{uce}_{\alpha}(L)),\widetilde{\alpha} (\frak{uce}_{\alpha}(L))]
\end{align*}
for any $\{\alpha_L(x_1), \alpha_L(x_2)\}\in \frak{uce}_{\alpha}(L)$. \rdg
\begin{theorem}
If $(L,\alpha_L)$ is an $\alpha$-perfect Hom-Lie algebra, then the homomorphism of
Hom-Lie algebras $\varphi:(\alpha_L(L)\star \alpha_L(L), \alpha_{\alpha_L(L) \star \alpha_L(L)})\twoheadrightarrow (L, \alpha_L)$ given by $\varphi(\alpha_L(l_1) \star \alpha_L(l_2))=[\alpha_L(l_1), \alpha_L(l_2)]$, is the universal $\alpha$-central extension of $(L,\alpha_L)$.
Moreover, there is an isomorphism of Hom-Lie algebras
\[
(\alpha_L(L) \star \alpha_L(L), \alpha_{\alpha_L(L) \star \alpha_L(L)}) \approx (\frak{uce}_{\alpha}(L), \widetilde{\alpha}),\quad \alpha_L(l) \star \alpha_L(l') \mapsto \{\alpha_L(l),\alpha_L(l')\}.
\]
\end{theorem}
{\it Proof.} This is similar to the proof of Theorem \ref{teor} and we leave to the reader. \rdg
\section{Application in cyclic homology of Hom-associative algebras}\label{section 6}
Throughout this section we assume that $\K$ is a field of characteristic 0.
\begin{definition}
By a Hom-associative algebra (see e.g. \cite{MS}) we mean a pair $(A,\alpha_A)$ consisting of a vector space $A$ and a linear map $\alpha_A:A\to A$, together with a linear map (multiplication) $A\otimes A\to A$, $a\otimes b\mapsto ab$, such that
\begin{align*}
&\alpha_A(a)(bc) = (ab)\alpha_A(c),\\
& \alpha_A(ab)=\alpha_A(a)\alpha_A(b)
\end{align*}
for all $a,b,c\in A$.
\end{definition}
The Hom version of the classical cyclic bicomplex (see e.g. \cite{Lo}) is constructed in \cite{Yau1} and
the cyclic homology of a Hom-associative algebra is defined as the homology of its total complex. A reformulation of this cyclic homology via Connes's complex for Hom-associative algebra is also given in \cite[Proposition 4.7]{Yau1}. It follows that, given a Hom-associative algebra $(A,\alpha_A)$, the first cyclic homology $HC^{\alpha}_1(A)$ is the kernel of the homomorphism of vector spaces
\[
\psi:A\otimes A/J(A,\alpha)\to [A,A], \quad a\otimes b\mapsto ab-ba,
\]
where $[A,A]$ is the subspace of $A$ generated by the elements $ab-ba$, and $J(A,\alpha)$ is the subspace of $A\otimes A$ generated by the elements
\[
a\otimes b+b\otimes a \quad\text{and}\quad ab\otimes \alpha_A(c)- \alpha_A(a)\otimes bc+ca\otimes \alpha_A(b),
\]
for all $a,b,c\in A$
Given a Hom-associative algebra $(A,\alpha_A)$, then it is endowed with a Hom-Lie algebra structure with the induced product $[a,b]=ab-ba$, $a,b\in A$ and the endomorphism $\alpha_A$. Moreover,
there is a Hom-Lie algebra structure on $A\otimes A/J(A,\alpha)$ given by
\[
[a\otimes b, a'\otimes b']=[a,b]\otimes [a',b']
\]
and the endomorphism induced by $\alpha_A$. We denote this Hom-Lie algebra by $(L^{\alpha}(A),\overline{\alpha}_A)$. In fact $(L^{\alpha}(A),\overline{\alpha}_A)$ is the quotient of the Hom-Lie tensor product $(A\star A,\alpha_{A\star A})$ by the ideal generated by the elements
$a\star b+b\star a$ and $ab\star \alpha_A(c)- \alpha_A(a)\star bc+ca\star \alpha_A(b)$, for all $a,b,c\in A$.
\begin{definition} \label{alfa condition}
We say that a Hom-associative algebra $(A,\alpha_A)$ satisfies the $\alpha$-identity condition if
\begin{equation}\label{condition}
[A, \Im(\alpha_A-\id_A)]=0,
\end{equation}
where $[A, \Im(\alpha_A-\id_A)]$ is the subspace of $A$ spanned by all elements $ab-ba$ with $a\in A$ and $b\in \Im(\alpha_A-\id_A)$.
\end{definition}
Note that $\alpha$-identity condition is equivalent to the condition $[a,b]=[\alpha_A(a),b]$ for all $a,b\in A$.
\newpage
\begin{example} \
\begin{enumerate}
\item[a)] Any Hom-associative algebra $(A, \alpha_A)$ with $\alpha_A=\id_A$ (i.e. an associative algebra) satisfies $\alpha$-identity condition.
\item[b)] Any commutative Hom-associative algebra $(A, \alpha_A)$ (i.e. $ab=ba$ for all $a,b\in A$) with $\alpha_A=0$ satisfies $\alpha$-identity condition.
\item[c)] Consider the Hom-associative algebra $(A,\alpha_A)$, where as vector space $A$ is 2-dimensional with basis $\{e_1,e_2\}$, the multiplication is given by $e_1e_1=e_2$ and zero elsewhere, $\alpha_A$ is represented by the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right)$. Then
$(A,\alpha_A)$ satisfies $\alpha$-identity condition.
\item[d)] Consider the Hom-associative algebra $(A,\alpha_A)$, where as vector space $A$ is 3-dimensional with basis $\{e_1,e_2,e_3\}$, the multiplication is given by $e_1e_1=e_2$, $e_1e_2=e_3$, $e_2e_1=e_3$ and zero elsewhere, $\alpha_A$ is represented by the matrix $\left( \begin{array}{ccc} 1 & 0 &0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{array} \right)$. Then $(A,\alpha_A)$ satisfies $\alpha$-identity condition.
\end{enumerate}
\end{example}
\begin{lemma}\label{lema 6.4}
Let $(A,\alpha_{ A})$ be a Hom-associative algebra.
\begin{enumerate}
\item[a)] There are Hom-actions of Hom-Lie algebras $(A,\alpha_A)$ and $(L^{\alpha}(A),\overline{\alpha}_A)$ on each other. Moreover, these Hom-actions are compatible if $(A,\alpha_A)$ satisfies the $\alpha$-identity condition (\ref{condition}).
\item[b)] There is a short exact sequence of Hom-Lie algebras
\[
0\longrightarrow (HC^{\alpha}_1(A),\alpha_{HC})\overset{i}\longrightarrow (L^{\alpha}(A),\overline{\alpha}_A) \overset{\psi}\longrightarrow \big([A,A], \alpha_{A\mid}\big)\longrightarrow 0,
\]
where $(HC^{\alpha}_1(A),\alpha_{HC})$ is an abelian Hom-Lie algebra with $\alpha_{HC}$ induced by $\alpha_A$, $\alpha_{A\mid}$ is the restriction of $\alpha_A$ and $\psi(a\otimes b)=[a,b]$.
\item[c)] The induced Hom-action of $(A,\alpha_A)$ on $(HC^{\alpha}_1(A),\alpha_{HC})$ is trivial. Moreover, if $(A,\alpha_A)$ satisfies the $\alpha$-identity condition (\ref{condition}), then both $i$ and $\psi$ preserve the Hom-actions of the Hom-Lie algebra $(A,\alpha_A)$.
\end{enumerate}
\end{lemma}
{\it Proof.}
{\it a)} The Hom-action of $(A,\alpha_A)$ on $(L^{\alpha}(A),\overline{\alpha}_A)$ is given by
\[
{}^{a'}(a\otimes b) = [a',a]\otimes \alpha_A( b) + \alpha_A(a)\otimes [a',b],
\]
while the Hom-action of $(L^{\alpha}(A),\overline{\alpha}_A)$ on $(A,\alpha_A)$ is defined by
\[
{}^{(a\otimes b)}{a'}=[[a,b],a']
\]
for all $a',a,b\in A$. Straightforward calculations show that these are indeed Hom-actions of Hom-Lie algebras, which are compatible if $(A,\alpha_A)$ satisfies $\alpha$-identity condition (\ref{condition}).
{\it b)} and {\it c)} are immediate consequences of the definitions above. \rdg
By complete analogy to the Dennis-Stein generators \cite{DeSt}, we define the first Milnor cyclic homology for Hom-associative algebras as follows.
\begin{definition}
Let $(A,\alpha_A)$ be a Hom-associative algebra. The first Milnor cyclic homology $HC_1^M(A,\alpha_A)$ is the quotient vector space of $A\otimes A$ by the relations
\begin{align*}
& a\otimes b +b\otimes a=0,\\
& ab\otimes \alpha_A(c)- \alpha_A(a)\otimes bc+ca\otimes \alpha_A(b)=0,\\
& \alpha_A(a)\otimes bc -\alpha_A(a)\otimes cb =0
\end{align*}
for all $a,b,c\in A$.
\end{definition}
Of course for $\alpha_A=\id_A$ this is the definition of the first Milnor cyclic homology of the associative algebra
$A$ in the sense of \cite{Lo} (see also \cite{InKhLa}).
Note also that $HC_1^M(A,\alpha_A)$ coincides with $HC^{\alpha}_1(A)$ when $(A,\alpha_A)$ is commutative.
\begin{theorem}\label{application}
Let $(A,\alpha_A)$ be a Hom-associative (non-commutative) algebra satisfying the $\alpha$-identity condition (\ref{condition}). Then there is an exact sequence of vector spaces
\begin{align*}
A \star \! HC_1^{\alpha}(A) &\to \Ker\big(A\!\star \!L^{\alpha}(A)\to L^{\alpha}(A)\big) \to \Ker\big(A\!\star \![A,A]\to [A,A]\big)\\
&\to HC^{\alpha}_1(A)\to HC_1^M(A,\alpha_A)\to \frac{[A,A]}{[A,[A,A]]}\to 0.
\end{align*}
\end{theorem}
{\it Proof.}
By using Lemma \ref{lema 6.4} and Proposition \ref{exact-tensor-1} we have the commutative diagram of Hom-Lie algebras (written without $\alpha$ endomorphisms)
\[
\xymatrix{
& A\star HC_1^{\alpha}(A) \ar[r] \ar[d]^{\psi_1}& A \star L^{\alpha}(A) \ar[r] \ar[d]^{\psi_2} & A\star [A,A] \ar[r] \ar[d]^{{\psi_3}}& 0\\
0 \ar[r] & HC_1^{\alpha}(A) \ar[r] & L^{\alpha}(A) \ar[r]^{\psi} & [A,A] \ar[r] & 0.
}
\]
Since $\Coker(\psi_3)= [A,A]/[A,[A,A]]$, $\Coker(\psi_2)=HC_1^M(A,\alpha_A)$, $\Coker(\psi_1)=HC_1^{\alpha}(A)$
and $\Ker(\psi_1)=A\star HC_1^{\alpha}(A)$, the assertion is a consequence of the Snake Lemma. \rdg
\medskip
Let us remark that if $\alpha_A$ is an epimorphism, then the term $A \star HC_1^{\alpha}(A)$ in the exact sequence of Theorem \ref{application} can be replaced by $A/[A,A] \otimes \! HC_1^{\alpha}(A)$ since they are isomorphic by Proposition \ref{proposition 5.10}. In particular, if $\alpha_A=\id_A$, the exact sequence in Theorem \ref{application} coincides with that of \cite[Theorem 5.7]{Gu}.
\newpage
\centerline{\bf Acknowledgements}
First and second authors were supported by Ministerio de Economía y Competitividad (Spain) (European FEDER support included), grant MTM2013-43687-P. Second author was supported by Xunta de Galicia, grants EM2013/016 and GRC2013-045 (European FEDER support included) and by Shota Rustaveli National Science Foundation, grant DI/12/5-103/11.
\begin{center} | 100,561 |
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She's a free spirit. She's on the wrong side of the tracks. She has some friends you should meet.
This driving/walking tour is my personal favorite. It was one of my first, and it includes my own neighborhood, the ever-changing, always appealing Bywater.
We start at Crescent Park, on the levee of the Mississippi river just below the French Quarter. The view is unparalleled. Bring your camera!
Then, on to lunch--choose between Elizabeth's or The Joint, (both Zagat rated and very popular with locals), while we go over the rest of our itinerary, which will include sights in the Ninth Ward and Lower Ninth Ward, including the St. Claude Avenue corridor (the new downtown "Main Street"), the St. Roch Cemetery, sites related to Katrina and much more. . . this is the kind of tour that can only be enjoyed as a private tour. . larger tours are not allowed in some of the areas we will visit. . . it is well worth the trip. I will return you to your downtown hotel at the end of the tour.
Itinerary
Crescent Park
Elizabeth's or The Joint
Bywater/Downtown Main Street
St. Roch Cemetery
Musicians' Village
Holy Cross Neighborhood, including the "steamboat houses"
Industrial Canal Levee Marker (Site of Katrina Breech)
Make it Right Nine (an outdoor exhibit about the neighborhood, Katrina, and the recovery)
The House of Dance and Feathers (if available-you have to see this one for yourself--it will restore your heart)
Meeting Location + Tour Duration
Meeting location: Your downtown hotel or other accommodations
End location: Back to the starting point in my private car
Duration: 4 hours
Weather may affect tour duration.
Transportation
Private Car: 2008 Scion XB, seats 5
What’s Included
I will pick you up in my own car (see photos). Most of the tour is driving.
Estimated Local Cash Needed
40 USD - Lunch is on the tour, but not included in the tour price.
What’s Extra
Lunch
Optional Donation to "House of Dance and Feathers", if visited (subject to availability)
A Recent Review of this tour
Read More Reviews For This TourRead More Reviews For This Tour
This was the only private tour exclusively of the Ninth Ward we could find. This is the guide's stomping grounds so he had plenty of first hand knowledge of the area, the history around the area, and the people. Very fun and personable guy.
Nicole Splawn
-
Share This Tour | 302,779 |
theory Cyclic_Group_Ext imports
CryptHOL.CryptHOL
"HOL-Number_Theory.Cong"
begin
context cyclic_group begin
lemma generator_pow_order: "\<^bold>g [^] order G = \<one>"
proof(cases "order G > 0")
case True
hence fin: "finite (carrier G)" by(simp add: order_gt_0_iff_finite)
then have [symmetric]: "(\<lambda>x. x \<otimes> \<^bold>g) ` carrier G = carrier G"
by(rule endo_inj_surj)(auto simp add: inj_on_multc)
then have "carrier G = (\<lambda> n. \<^bold>g [^] Suc n) ` {..<order G}"
using fin by(simp add: carrier_conv_generator image_image)
then obtain n where n: "\<one> = \<^bold>g [^] Suc n" "n < order G" by auto
have "n = order G - 1" using n inj_onD[OF inj_on_generator, of 0 "Suc n"] by fastforce
with True n show ?thesis by auto
qed simp
lemma pow_generator_mod: "\<^bold>g [^] (k mod order G) = \<^bold>g [^] k"
proof(cases "order G > 0")
case True
obtain n where n: "k = n * order G + k mod order G" by (metis div_mult_mod_eq)
have "\<^bold>g [^] k = (\<^bold>g [^] order G) [^] n \<otimes> \<^bold>g [^] (k mod order G)"
by(subst n)(simp add: nat_pow_mult nat_pow_pow mult_ac)
then show ?thesis by(simp add: generator_pow_order)
qed simp
lemma int_nat_pow:
assumes "a \<ge> 0"
shows "(\<^bold>g [^] (int (a ::nat))) [^] (b::int) = \<^bold>g [^] (a*b)"
using assms
proof(cases "a > 0")
case True
show ?thesis
using int_pow_pow by blast
next case False
have "(\<^bold>g [^] (int (a ::nat))) [^] (b::int) = \<one>" using False by simp
also have "\<^bold>g [^] (a*b) = \<one>" using False by simp
ultimately show ?thesis by simp
qed
lemma pow_generator_mod_int: "\<^bold>g [^] ((k :: int) mod order G) = \<^bold>g [^] k"
proof(cases "order G > 0")
case True
obtain n :: int where n: "k = order G * n + k mod order G"
by (metis div_mult_mod_eq mult.commute)
then have "\<^bold>g [^] k = \<^bold>g [^] (order G * n) \<otimes> \<^bold>g [^] (k mod order G)"
using int_pow_mult nat_pow_mult by (metis generator_closed)
then have "\<^bold>g [^] k = (\<^bold>g [^] order G) [^] n \<otimes> \<^bold>g [^] (k mod order G)"
using int_nat_pow by (simp add: int_pow_int)
then show ?thesis by(simp add: generator_pow_order)
qed simp
lemma pow_gen_mod_mult:
shows"(\<^bold>g [^] (a::nat) \<otimes> \<^bold>g [^] (b::nat)) [^] ((c::int)* int (d::nat)) = (\<^bold>g [^] a \<otimes> \<^bold>g [^] b) [^] ((c*int d) mod (order G))"
proof-
have "(\<^bold>g [^] (a::nat) \<otimes> \<^bold>g [^] (b::nat)) \<in> carrier G" by simp
then obtain n :: nat where n: "\<^bold>g [^] n = (\<^bold>g [^] (a::nat) \<otimes> \<^bold>g [^] (b::nat))"
by (simp add: monoid.nat_pow_mult)
also obtain r where r: "r = c*int d" by simp
have "(\<^bold>g [^] (a::nat) \<otimes> \<^bold>g [^] (b::nat)) [^] ((c::int)*int (d::nat)) = (\<^bold>g [^] n) [^] r" using n r by simp
moreover have"... = (\<^bold>g [^] n) [^] (r mod (order G))" using pow_generator_mod_int pow_generator_mod
by (metis int_nat_pow int_pow_int mod_mult_right_eq zero_le)
moreover have "... = (\<^bold>g [^] a \<otimes> \<^bold>g [^] b) [^] ((c*int d) mod (order G))" using r n by simp
ultimately show ?thesis by simp
qed
lemma pow_generator_eq_iff_cong:
"finite (carrier G) \<Longrightarrow> \<^bold>g [^] x = \<^bold>g [^] y \<longleftrightarrow> [x = y] (mod order G)"
by(subst (1 2) pow_generator_mod[symmetric])(auto simp add: cong_def order_gt_0_iff_finite intro: inj_onD[OF inj_on_generator])
lemma cyclic_group_commute:
assumes "a \<in> carrier G" "b \<in> carrier G"
shows "a \<otimes> b = b \<otimes> a"
(is "?lhs = ?rhs")
proof-
obtain n :: nat where n: "a = \<^bold>g [^] n" using generatorE assms by auto
also obtain k :: nat where k: "b = \<^bold>g [^] k" using generatorE assms by auto
ultimately have "?lhs = \<^bold>g [^] n \<otimes> \<^bold>g [^] k" by simp
then have "... = \<^bold>g [^] (n + k)" by(simp add: nat_pow_mult)
then have "... = \<^bold>g [^] (k + n)" by(simp add: add.commute)
then show ?thesis by(simp add: nat_pow_mult n k)
qed
lemma cyclic_group_assoc:
assumes "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G"
shows "(a \<otimes> b) \<otimes> c = a \<otimes> (b \<otimes> c)"
(is "?lhs = ?rhs")
proof-
obtain n :: nat where n: "a = \<^bold>g [^] n" using generatorE assms by auto
obtain k :: nat where k: "b = \<^bold>g [^] k" using generatorE assms by auto
obtain j :: nat where j: "c = \<^bold>g [^] j" using generatorE assms by auto
have "?lhs = (\<^bold>g [^] n \<otimes> \<^bold>g [^] k) \<otimes> \<^bold>g [^] j" using n k j by simp
then have "... = \<^bold>g [^] (n + (k + j))" by(simp add: nat_pow_mult add.assoc)
then show ?thesis by(simp add: nat_pow_mult n k j)
qed
lemma l_cancel_inv:
assumes "h \<in> carrier G"
shows "(\<^bold>g [^] (a :: nat) \<otimes> inv (\<^bold>g [^] a)) \<otimes> h = h"
(is "?lhs = ?rhs")
proof-
have "?lhs = (\<^bold>g [^] int a \<otimes> inv (\<^bold>g [^] int a)) \<otimes> h" by simp
then have "... = (\<^bold>g [^] int a \<otimes> (\<^bold>g [^] (- a))) \<otimes> h" using int_pow_neg[symmetric] by simp
then have "... = \<^bold>g [^] (int a - a) \<otimes> h" by(simp add: int_pow_mult)
then have "... = \<^bold>g [^] ((0:: int)) \<otimes> h" by simp
then show ?thesis by (simp add: assms)
qed
lemma inverse_split:
assumes "a \<in> carrier G" and "b \<in> carrier G"
shows "inv (a \<otimes> b) = inv a \<otimes> inv b"
by (simp add: assms comm_group.inv_mult cyclic_group_commute group_comm_groupI)
lemma inverse_pow_pow:
assumes "a \<in> carrier G"
shows "inv (a [^] (r::nat)) = (inv a) [^] r"
proof -
have "a [^] r \<in> carrier G"
using assms by blast
then show ?thesis
by (simp add: assms nat_pow_inv)
qed
lemma l_neq_1_exp_neq_0:
assumes "l \<in> carrier G"
and "l \<noteq> \<one>"
and "l = \<^bold>g [^] (t::nat)"
shows "t \<noteq> 0"
proof(rule ccontr)
assume "\<not> (t \<noteq> 0)"
hence "t = 0" by simp
hence "\<^bold>g [^] t = \<one>" by simp
then show "False" using assms by simp
qed
lemma order_gt_1_gen_not_1:
assumes "order G > 1"
shows "\<^bold>g \<noteq> \<one>"
proof(rule ccontr)
assume "\<not> \<^bold>g \<noteq> \<one>"
hence "\<^bold>g = \<one>" by simp
hence g_pow_eq_1: "\<^bold>g [^] n = \<one>" for n :: nat by simp
hence "range (\<lambda>n :: nat. \<^bold>g [^] n) = {\<one>}" by auto
hence "carrier G \<subseteq> {\<one>}" using generator by auto
hence "order G < 1"
by (metis One_nat_def assms g_pow_eq_1 inj_onD inj_on_generator lessThan_iff not_gr_zero zero_less_Suc)
with assms show "False" by simp
qed
lemma power_swap: "((\<^bold>g [^] (\<alpha>0::nat)) [^] (r::nat)) = ((\<^bold>g [^] r) [^] \<alpha>0)"
(is "?lhs = ?rhs")
proof-
have "?lhs = \<^bold>g [^] (\<alpha>0 * r)"
using nat_pow_pow mult.commute by auto
hence "... = \<^bold>g [^] (r * \<alpha>0)"
by(metis mult.commute)
thus ?thesis using nat_pow_pow by auto
qed
end
end | 63,112 |
\section{Conclusion}
Density functional theory is a powerful tool for investigating the behavior of fluids and surfaces, taking into account molecular forces at the nanoscale. The Variation Free DFT approach developed in this work for finding the equilibrium density is of interest for the study of intricate systems: systems with additional constrains, complex fluids. Besides, the combination of VF-DFT, which uses stochastic optimization methods, with the classical DFT with Picard iteration made it possible to significantly speed up the calculation for high relative pressure cases while maintaining the solution quality as in the classical DFT. At low relative pressures, the gain in calculation speed is not as significant as at high pressures, because of the value $\gamma$ parameter in Picard iteration method and specific of pattern extraction from dataset for the high relative pressure. The combined algorithm can be applied to speed up calculations of the equilibrium fluid density at high pressures, in particular, to speed up the calculation of the adsorption isotherm or pore stresses. It should be noted that the VF-DFT solution's speed and quality directly depend on the type of basis functions. The size of the basis and the type of basis functions can be changed, improved, and refined to obtain a better solution with a minimum cost of calculation time. In addition to the basis, the speed of VF-DFT operation is influenced by setting the optimization algorithms' parameters. The optimization algorithms GA and PSO considered in this work showed promising results, but PSO has a higher quality and speed of finding a solution. In the future, it is possible to investigate the variation free approach with other optimization methods, investigate systems for which the Helmholtz free energy has a complex form of systems with specific constraints, and use it to calculate the adsorption kernels in the problem of pore size distribution reconstruction. | 152,477 |
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Coniston, United Kingdom
1 Yewdale Road , Coniston, LA21 8DU, United Kingdom – Show map
Excellent location — rated 9.3/10! (score from 63 reviews)
Rated by guests after their stay at The Black Bull Inn and Hotel.
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13 properties in Coniston
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63 verified reviews
We verify all reviews from guests who have stayed at The Black Bull Inn and Hotel. Find out more.
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Website and customer service in English (UK) and 41 other languages
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Outdoors
Garden, Terrace
Food & Drink
Restaurant, Bar
Internet
Free!
WiFi is available in all areas and is free of charge.
Parking
Free!
Free public parking is possible on site (reservation is not needed).
Services
Packed lunches, Laundry, Fax/photocopying
General.
There is no capacity for extra beds in the room.
Pets
Pets are allowed on request. Charges may be applicable.
Cards accepted at this property
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Ideal alternatives for you
Based on 63 reviews
Cleanliness
8.4
Comfort
8.2
Location
9.3
Facilities
8.2
Staff
8.7
Value for money
7.6
WiFi
6.6
Show reviews from:
Group
I do hope to return here again!!
My stay was too short!
Couple
Good room, full of character in an inn at the heart of the village Good choice of drinks and quality food
Wifi issues, slightly unhelpful staff member
Family
The accommodation booked through Booking.com which enjoyed a rooftop view of Coniston village and the fells behind. The food which was delicious and the Coniston Brewery selection of ales.
There was nothing not to like. This was a high quality week-end. I share the views of Paul,Redland,Bristol on the subject of breakfast.
Solo traveller
good food plentyfull of it great choice aswell friendly staff good choice of local beer has it"s own off road parking
bad bed slept on springs room not that clean when I arrived dog hairs everywhere and got an early morning wakeup call from the brewery next door the banging began at 7am sharp for 3 days not amused there was no imformation about the area the shower was not that good either
Solo traveller
Fantastic breakfast and dog friendly. Also a great place to have an evening meal. Central location so close to shops and visitor attractions.
Although the cooked breakfast was excellent, there could have been a larger choice of fresh fruit etc.
Solo traveller
Pleasant clean accommodation with a good breakfast and friendly staff with the added bonus of their ales brewed on site.
Outside of annex accommodation some sort of machinery/compressor intermittently making noise during the night.
Couple
I spend a lot of time working in London and Black Bull Inn is the perfect place to getaway and forget about the city..... Friendly staff and great great beer
Nothing!!!
Solo traveller
The staff were friendly and polite, the meals were usual pub meals and reasonably priced with good service. Excellent choice of beers. Room was as expected for a 17th century building, clean and tidy with good TV, tea and coffee making facilities, nice clean en suite bathroom and it was quiet enough after 10pm. I would stay here again.
WiFi non existent had to go to a different pub to connect which was disappointing. Car parking first come first served. I had to park elsewhere 2 nights out of 3 and Should be reserved for residents only in my opinion.
Group
Only issue was the bar service in the evening This was not the fault of the two staff who were as Helpful as they could be but needed another body to prevent long wait for service
Couple
The breakfast was decent.
The room. Tiny, cramped, stuffy, too hot.
Couple
Location, food and beer.
Couple
We arrived in the bar, nobody smiled, looked up or spoke to us. We were given the wrong room. Our room was very dirty the next day having supposedly bn cleaned. I complained, 4 hours later it was still dirty, then asked to speak to manager who himself ended up cleaning the room.
Couple
Liked everything about it, very helpful staff, nipping clean, great showers and fab cask beers etc. Forgot, cracking location and good breakfast choices. Items rated good only refer to rather cramped bedroom and the fact that at home we use a more superior mattress.
Not a thing to dislike
Couple
food excellent location
bedroom doorway and ceilings very low. had to cancel 1 night 10 days before due to stay but still tried to charge cancellation fee blamed this on booking.com. friendly but look out for the owner
Couple
Very dog friendly. Outstanding beers.
Group
Friendliness and good food with excellent selection of beer.
Rooms could have done with a good clean
Couple
Location...convenient, as we were traveling by train & bus. Also, lovely sound of flowing stream outside the window in bedroom. Bathroom...loved the shower! Convenient electrical plugs for CPAP machine becide the bed and hairdryer beside mirror. Excellent Staff...very helpful and kind. Great service!
Solo traveller
Excellent breakfast and service at breakfast
Room had very low ceiling ; cut my head three times . Pub so crowded that service was poor at the bar and in delivery of food orders. Ate elsewhere after the first night ! Had no English mustard for steak
Group
Staff friendliness, location, accommodation (tidy, comfortable, clean, a WORKING heated towel rail, hot showers, quaint decor). Food, the breakfast included was very tasty and provided a good start to long walks, the Inn's evening menu is also excellent! Free parking. Loved my stay. Would stay again.
Wifi - it is free and its signal showed up on both my devices (phone and tablet) and there's no password! All good right? But upon trying to establish a connection, both devices failed and the connection always 'timed out'. The devices have never had a problem connecting to wifi elsewhere. Getting staff to reset the router didn't solve the problem either. If it's offered, please make sure everyone can connect - especially as phone and roaming signals are both very poor in the area (with my provider anyway). It would have been nice to have the option of using wifi for contacting people at home. However, not imperative and did not detract from my stay (as I spent most of my time outdoors anyway!)
Family
Great room, great food, friendly staff. Will be going back as soon as we can.
Couple
Had a lovely stay over Easter weekend, we were met by the manager Les (I think was his name) he was always very friendly and helpful, the breakfasts were very nice, the waitress in the mornings was lovely . The room was clean and fantastic shower. we would definitely stay again.
some of the bar staff didn't seem interested and weren't aware who was next in the que.
Couple
Having known the pub for a number of years we were looking forward to our stay. The room well appointed, food was very good and staff friendly and helpful.
The mattress was thin and didn't give us a very good nights sleep.
Solo traveller
Very comfortable relaxed accomodation, staff were friendly and helpful
Solo traveller
In some inns there is noise from the bar until quite late, but my room was very quiet. The food was excellent. I enjoyed the alternative options to 'full English' at breakfast. My room was kept very clean. I was walking each day - never got seriously wet but the heated towel rail was handy for slightly damp items. If I had got soaked I would have asked if there's a drying room.
The online receipt for my initial booking made no mention of breakfast. I sent an email to check that breakfast was included and had a swift reply. It may seem obvious to you that the booking includes breakfast but I think you should mention it, particularly given the very good menu.
Couple
The friendliness of the staff. Quality catering. Vibrant atmosphere/
Poor Wifi. | 113,821 |
She Gets It From HIS Side Of The Family
Watching my daughter turn 17 this past week, I realize she has developed a lot of the same characteristics as I possess (all GOOD traits, needless to say).
Ahem.
An affinity for technology.
A love of animals.
The pleasure of reading.
To name a few.
Unfortunately, she has also inherited a certain er, let's call it "idiosyncrasy" from hubby's side of the family.
Yes. Good characteristics from me. Idiosyncrasies from him. My blog. My opinion.
It's something that I had suspected for awhile now, but the events of the other night simply confirmed it:
0100: Being the weekend, the entire family hit the sack quite late.
0215: Because I am a notoriously light sleeper and/or the "Mom Instinct" kicked in, I awoke when daughter got up to use the bathroom. Fearing the flu had hit yet again, I heard her shuffling around in the medicine cabinet.
0230: Seemingly fine, she returned to her bedroom for the rest of the night.
Sometime later I finally drifted back to sleep, reassured she was okay.
In the morning I was already awake (and yes, at the laptop blogging as usual) when daughter emerged from her room looking confused. She told me she remembered taking out her contacts before bed last night, but when she got up, they were back in again.
She had no memory of getting up in the middle of the night.
She had no memory of putting in her contacts.
She had no memory of placing all her supplies back in perfect order in the medicine cabinet.
Yup. She was sleepwalking last night.
So when hubby finally awoke, I blamed him. For you see, his mother has often recited stories of family sleepwalkers.
Thanks dear. Thanks so much for that contribution to the gene pool.
But noting how meticulous she was, I can't help but think it's really a shame she doesn't clean her room in her<<
26 People would rather be commenting:
LOL!, I always blame the oddities in my daughters on their fathers genetics. I know for a fact that all of the good stuff came from my side.
Ha, ha! I used to be a sleep walker as a kid and then did it again for a while as an adult and got food to eat from the kitchen, which I then found half eaten in the bed in the morning. Lots of crumbs in the bed and an unbelieving husband. Luckily, I am over that now.
Sleep walking is one of those things I find fascinating and a wee bit frightening. Hmmm, I wonder if she saw her dreams any clearer ... ?
Oh I do this. I used to get up and get totally dressed for school and be walking out the back door for the bus...at 3 AM. I kept my mom on her toes. Now my husband gets to deal with my nighttime wanderings. Elevated when I am stressed which is all the time nowadays. Lucky him.
That is crazy. I've never known anyone who sleptwalk. Does it make you nervous?
Wow, that is amazing! My friend's 6 year old sleep walks and she's a bit freaked out by it. He usually goes to the washroom in his sleep (ie. thinks he's made it to the toilet, but doesn't really). She's had to lead him to the washroom when she catches him sleepwalking so she doesn't have a big mess in the morning everytime he's sleepwalking!
That's hilarious! I used to sleepwalk when I was a kid, but I don't think I ever did anything like that...just wondered around the house and freaked my mom (also a light sleeper) out when she found me standing in the middle of a room in the middle of the night. LOL
I've always said all the good stuff comes from my side and all the annoying pain-in-the-neck stuff comes from hubby's side! We've experienced night-terrors but never sleep walking.
JT: So it's not just me? Good....
Nora: I do hope she outgrows it too. However, I have to laugh at the image of half-eaten food in your bed!
Kathy: I'll have to ask her!
Vegas Princess: That's just like what she was doing; I think she thought it was morning and she was getting ready for school! Weird.
ShannanB: No, I'm not nervous... but I am going to make sure I pay more attention when I hear her get up in the middle of the night!
Karen Meg: Yikes! Thank goodness THAT doesn't happen!!
Heather: Yeah, that WOULD be freaky; especially in the middle of the night!
KSD: Luckily, this is a rare thing; and thank goodness no screaming!
That was hilarious! I had to share it with Mrs. Phoenix... and I read it verbatim from your post. She laughed as hard as I did! Your writing never ceases to entertain me! I've never understood why people think the good traits come from the mother's side and the bad stuff comes from the dad's... I gave MY kids all kinds of good traits! ;-)
Oh man - I wish ****I**** could learn to clean in my sleep!!!!!!! Putting in contacts doesn't seem nearly as practical, but..y'know...guess she was ready freddy in the morning!!!!
Oh, I'm with you there...I'm sure there must be some scientific research (or non-scientific would work here too) to prove your theory. It's true in my family too! Maybe it's genetic...good traits from mom; not-so-good traits from dad :)
Hm, I would think sleep-cleaning would be kind of cool. Your house is clean but since you don't remember doing it, it sort of feels like you had a maid come in and do it.
Sleep-contact-inserting seems an especially cool "idiosyncracy." Did she get each one in the correct eyeball?
I'd feel gypped if I awoke to discover I'd been sleep-cleaning. To really feel the satisfaction the job, I must remember doing it!
JD at I Do Things
You too?? I would stake money that my daughter's bad habits all come from my hubby's side of the family. How odd to meet someone with the same problem as me! Ha!
Wow...I think I'd be hoping she'd do my whole house! Talk about paying attention to detail!
That is the most well-organized medicine cabinet I've ever seen! Tell her to sleep walk over here and get to cleaning!
That's an opportunity knocking! I say put that sleepwalking to good use...leave the vacuum out!
I thought sleepwalking only happened in the movies or on cartoons.
It's a strange thing
holy hanna...I have always worried about that happening with my gremlins too. Isn't it funny that it was her contacts she had to put in?? LOL
Phoenix5: Ha! Let's hear from your wife first about that ... ;)
Jill: It was so funny, really... she comes out of her room wondering why the heck she could see so well, as they are strong prescriptions so it is easy for her to tell when they are in!
Toners: Agreed. Wholeheartedly. But we would have to make sure the research is conducted by WOMEN not men. ;)
Pand0ra Wilde: Yeah! Especially if my daughter did it for me!
JD: Yes, she put them in perfectly. And I try to forget cleaning whenever humanly possible...
Stephanie: Ha! See? I do believe in my non-scientific poll that the wives win by a landslide; we DO pass on the good qualities in our kids.
Momo Fali: I'll pass it on, but don't hold your breath. I'm still waiting for that miracle in her own room...
VE: Ha! If I left the vacuum out, hubby would be sure to trip over it, not knowing what the heck that contraption was...
Creative-Type Dad: Nope, it happens IRL too... as long as she doesn't grab the car keys in the middle of the night, I can deal with it.
Canadian Flake: I know! We were dying laughing in the morning. It was weird, but oh so funny at the same time. Of course I had to ask her permission to blog about it!
Too funny!
My son Austin is a sleepwalker. We often have to corral him back to bed when we find him wandering around in the middle of the night. When he was very little (around 4) we had to be really quick because he would think various things were the toilet, and we'd often find him with his pants down getting ready to pee in the garbage, or flower pot, or toy box.
I think the courts have ruled that you can't be held liable for anything you do while sleep walking......hope she dosen't become a serial killer or a burglar.
Oh what a funny thing to do, not just sleepwalking but actually putting contacts in... weird..lol
Jeff: Oh Yikes!!!! I'll take putting in contacts over random midnight peeing anyday!!
Lotus07: Um hope not. But now I'm starting to worry about you...
Lady Banana: Yeah, it was unique all right!!!
How funny!! I knew someone who would get up and eat everything in the fridge in the middle of the night and remember nothing. I also have been told I sleepwalk from time to time. Matt said one night I got up and sat on the nightstand and stayed there a while. hehe. I wish I'd clean in my sleep, too!
Ha! Annie! That would have been priceless to see. Now, if he was a blogger, he would have taken your picture.... ;)
Which reminds me, I think I'll need to keep the camera at the ready the next time daughter gets up in the night! | 138,844 |
TITLE: Nonexistence of infinite subdirectly irreducible algebras
QUESTION [3 upvotes]: I am trying to prove a theorem of Quackenbush (Theorem 3.8 in Chapter V of Burris & Sankappanavar):
If $V$ is a locally finite variety with only finitely many finite subdirectly irreducible members (up to isomorphism), then $V$ has no infinite subdirectly irreducible members.
The proof ends with the following situation: Let $A$ be any algebra in $V$, and $A \in ISP(V^*)$, where $V^*$ are the finitely many finite s.i. algebras. From this it should follow that $A \in IP_S(V^*)$. This is stated without further comment, so I suppose I am overlooking something very easy. How do I prove this fact?
REPLY [3 votes]: We have $A \subseteq \prod_{i\in I} B_i$, with $B_i\in V^*$ for all $i\in I$. For all $i$, let $C_i = \pi_i[A] \subseteq B_i$, where $\pi_i$ is the projection to $B_i$ (so $\pi_i$ is surjective onto $C_i$). Let $C_i \subseteq \prod_{j\in J_i} D_{i,j}$ be the subdirect representation of $C_i$ by its subdirectly irreducible quotients, and again let $\pi_j$ be the projection to $D_{i,j}$ (this time we know the $\pi_j$ are surjective onto $D_{i,j}$). Since each $B_i$ is finite, each $C_i$ is finite, so all the $D_j$ are finite, and hence they're all in $V^*$.
Now for all $i\in I$ and $j\in J_i$, the map $\pi_j\circ \pi_i\colon A\to D_{i,j}$ is surjective. And for $a\neq b$ in $A$, there is some $i\in I$ such that $\pi_i(a) \neq \pi_i(b)$ in $C_i$, and there is some $j\in J_i$ such that $\pi_j(\pi_i(a)) \neq \pi_j(\pi_i(b))$ in $D_{i,j}$, so the induced map from $A$ to $\prod_{i\in I}\prod_{j\in J_i} D_{i,j}$ is an embedding. | 57,861 |
The Worlds most famous networking company, Facebook has opened its first headquarters in Africa. They choose Johannesburg in South Africa and appointed Nunu Ntshingila-Njeke as head of Facebook Africa. The company looks forward to adding to its existing 120 million users on the African continent. The new office in Johannesburg will focus on growing markets in Kenya, Nigeria and South Africa.
They estimate about one in five people in Africa as a whole have internet access, but almost double that figure are expected to have mobile internet connections by 2020. About 80% of those who use Facebook in Africa access the site by mobile phone. Nigeria is a huge part of it.
Nunu Ntshingila-Njeke head of Facebook Africa
.
We hope they open more offices in Africa and not just one in South Africa. Another great part of Africa to have an office thus bum business is Nigeria and Ghana. Do you agree?
Written by: Noellin Imoh
Source: BBC | 373,083 |
Although I get asked, “which is better, soy or whey protein?” … my question back is, “better for what?” Each are excellent sources of protein, and each have their own benefits. I suggest to use at least both sources of protein in order to obtain the benefits each provide. We need to consume protein in order to make and replace protein; and athletes and body-builders are very familiar with whey protein as an excellent source of “bio-available” protein.
Protein is essential for producing antibodies, hormones, new muscle tissue, and the oxygen-carrying protein in blood, hemoglobin. All protein lost or destroyed within the body must be replaced by bio-available protein in order for new tissue to be constructed.
Our bodies are able to manufacture many of the amino acids that are used to produce protein; however, there are nine “essential” amino acids that we cannot manufacture, but must obtain from the protein in our food. Not all protein sources provide these essential amino acids. For example, whey is an excellent source of glutathione and the branched chain essential amino acids L-leucine, L-valine and L-isoleucine.
The protein in most beans and vegetables may contain all the essential amino acids, but they are not naturally concentrated in foods, and thus vegans often do not readily obtain adequate amounts of protein, particularly the branch-chained amino acids. However, this can be compensated for by consuming concentrated protein sources, such as found within Nutrimeals.
Almost daily, I drink one or two Nutrimeal meal-replacement drinks. These drinks not only contain a blend of both soy and whey proteins, they also contain protein from two additional sources, that of rice and pea. By obtaining a blend of proteins from these four sources one is obtaining all the essential amino acids and the benefits that each provide.
Soybeans contain high amounts of protein. Soy protein and soy isoflavones have been found to help reduce the symptoms of menopause, help reduce the risk of osteoporosis, and to help prevent a number of hormone-related diseases, such as endometrial cancer, breast cancer, and prostate cancer. (Neither soy nor soy isoflavones increase the risk of breast cancer; in fact quite the opposite, they help maintain breast health.)
In addition, soy has been observed to help maintain heart health. Even the Food and Drug Administration (FDA) has stated 25 grams of soy protein per day can reduce the chances of developing heart disease.
Soy protein has also been shown to help the thyroid, which can help with obtaining a leaner body. In the case of a soy allergy, the opposite would be true …reduced metabolism and weight gain.
If you are not allergic to soy, there are very few side effects to including soy in your diet. The most common side-effect of soy is the production of intestinal gas. Flatulence is a common side-effect of all beans (including soy), due to the bowel bacteria’s fermenting effect on the indigestible sugars contained within beans. Humans do not have the enzyme alpha-galactosidase necessary to break down the sugars that the bowel bacteria feast upon and produce gas.
Beano, purchased over-the-counter, contains alpha galactosidase, and regular use may be able to reduce gas production by breaking down the oligosaccharides (bean sugars) before the bacteria in the large bowel has a chance to ferment the sugar.
For those who are allergic to soy, gas would not be the only problem present, but significant diarrhea and abdominal bloating, hives, skin rash, and worst case, breathing problems. Soy isoflavones are not the same as pure soy protein; and even if one is allergic to soy it wound not mean they were allergic to soy isoflavones, as soy protein (the allergen of the allergy) is not found in soy isoflavones.
Whey protein is used by athletes and body builders because of the higher level of essential amino acids, particularly the branched chain amino acids that are metabolized in the muscle not the liver. Protein is critical in repairing not only muscle, but many other body tissues. Whey is helpful for weight loss and building muscle in those who work out, but will do little to help build muscle in those with sedentary lifestyles.
Whey protein affects the digestive tract in much the same way as yogurt. Therefore, it is considered to be a natural remedy for many intestinal issues. In fact, it is often used in Sweden to help prevent bowel problems, gas, and constipation. However, since whey is obtained from a dairy source (it is the liquid by-product of curdled milk … the solid becomes cheese, and the liquid protein part is dried as a source for whey). Therefore, those who are lactose intolerant should avoid whey protein, and steer towards soy protein only, as gas, constipation, and bloating can be significant. Over-the-counter Lactaid is available to help provide the enzyme necessary to break down dairy-derived lactose sugar found in whey products.
Since both soy and whey protein may lead to constipation, it is important to find a meal-replacement drink, such as Nutrimeals that provide adequate fiber to overcome this side-effect.
Whey protein makes a good alternative to those who are allergic to soy, and vice-versa, but the blend of soy and whey will render the user with the benefits of each, particularly if blended with adequate fiber, as mentioned. While companies selling protein supplements tout the benefits of whatever they’re selling as “the best,” whether it is soy, whey, or a combination of rice and pea protein, which together the last two alone hit numbers between 85 to 90% bioavailable protein, it is good to know that USANA Health Sciences, Inc. has been so wise as to combine all four protein sources in their meal-replacement Nutrimeals, along with fiber, low-glycemic sugar, and vitamins and minerals. The balance is its greatest strength as a perfect meal replacement.
A summary of the benefits of Soy and Whey Proteins, and that you can have BOTH:
Soy Protein
– Soy protein has been found to be higher in non-essential amino
– The consumption of 25-50 grams of soy protein daily may enhance production of thyroid stimulation hormones that regulate the metabolic rate, thereby making it easier for us to lose both body weight and fat and create a leaner body.
– Soy is good for athletes: in a study from Romania, in which endurance athletes experienced lean body mass, increased strength, and decrease fatigue while training. (Revue Roumaine de Physiologie 29, 3-4:63-70, 1992)
– It contains more protein by weight than beef, fish or chicken, and contains less fat (especially saturated fat than meat).
– The FDA has approved the following statement: “Diets low in saturated fat and cholesterol that include 25 grams of soy protein a day may reduce the risk of heart disease.”
– Other studies show that soy protein isolate has the ability to effectively lower LDL cholesterol and triglyceride levels in the blood
– Soy may improve kidney functioning
Whey Protein
– Whey protein assists in losing excess weight and maintaining optimal weight
– Whey protein, combined with resistance training, even those who have immunosuppressive disorders (AIDS) can increase body cell mass, muscle mass and muscle strength, according to a study in AIDS (15, 18:2431-40, 2001).
– Whey protein is superior to other proteins when it comes to anabolic response. It has consistently been shown to stimulate the anabolic hormones after a workout. In other words, whey protein improves athletic performance.
– It mixes well and is low in fat and lactose, and has a superior amino acid profile
– Whey protein lacks no essential amino acids. It needs no fortification or additive to make it complete. It is complete in its natural form
– Whey enhances the immune system because it raises glutathione levels. Glutathione is a powerful antioxidant that helps our immune cells stay charged to help ward off cancer, bacterial infection and viruses. In other words, it helps improve the immune system.
– Whey is also very high in glutamine and the branch chain amino acids L-leucine, L-valine and L-isoleucine, important aminos for repairing muscle
– Whey acts as a natural antibacterial or anti-viral
– Whey reduces the symptoms of Chronic Fatigue Syndrome
– Whey reduces liver damage
– Whey improves blood pressure
– Whey improves the function of the digestive system
– Whey reduces gastric mucosal injury seen in ulcerative colitis
Article Source: | 120,114 |
\begin{document}
\title
[Quantum stochastic convolution cocycles]
{Quantum stochastic convolution cocycles III}
\author[Lindsay]{J.\ Martin Lindsay}
\author[Skalski]{Adam G.\ Skalski}
\footnote{\emph{Permanent address of AGS}.
Department of Mathematics, University of \L\'{o}d\'{z}, ul.
Banacha 22, 90-238 \L\'{o}d\'{z}, Poland.}
\subjclass[2000]{Primary 46L53, 81S25; Secondary 22A30, 47L25, 16W30} \keywords{Noncommutative
probability, quantum stochastic, locally compact quantum group, $C^*$-bialgebra, $W^*$-bialgebra,
stochastic cocycle, quantum L\'{e}vy process}
\begin{abstract}
The theory of quantum L\'{e}vy processes on a compact quantum group,
and more generally
quantum stochastic convolution cocycles on a $C^*$-bialgebra,
is extended to
locally compact quantum groups and multiplier $C^*$-bialgebras.
Strict extension results obtained by Kustermans,
and automatic strictness properties developed here,
are exploited to obtain
existence and uniqueness for
coalgebraic quantum stochastic differential equations
in this setting.
Working in the universal enveloping von Neumann bialgebra,
the stochastic generators of Markov-regular,
completely positive, respectively *-homomorphic,
quantum stochastic convolution cocycles are characterised.
Every Markov-regular quantum
L\'evy process on a multiplier $C^*$-bialgebra is shown to be
equivalent to one governed by a
quantum stochastic differential equation, and
the generating functionals of norm-continuous convolution semigroups
on a multiplier $C^*$-bialgebra are characterised.
Applying a recent result of Belton's,
we give a thorough treatment of the
approximation of quantum stochastic convolution cocycles
by quantum random walks.
\end{abstract}
\maketitle
\section*{Introduction}
\label{Section: Introduction}
The principal aim of this paper is the extension of the theory of
\emph{quantum L\'evy processes}, and more generally
quantum stochastic evolutions with
tensor-independent identically distributed increments on a $C^*$-bialgebra,
to the context of \emph{locally compact quantum semigroups}---in other words
multiplier $C^*$-bialgebras.
The notion of quantum L\'evy process
generalises that of classical L\'evy process on a semigroup.
It was first introduced by Accardi, Sch\"urmann and von Waldenfels,
in the purely algebraic framework of $^*$-bialgebras (\cite{asw}),
and was further developed by Sch\"urmann and others (\cite{Schurmann}, \cite{Uwe})
who, in particular,
extended it to other quantum notions of independence (free, boolean and monotone),
still in the algebraic context.
Inspired by Sch\"urmann's reconstruction theorem,
which states that
every quantum L\'evy process on a $^*$-bialgebra
can be equivalently realised on a symmetric Fock space,
we first showed how the algebraic theory of quantum L\'evy processes can
be extended to the natural setting of
\emph{quantum stochastic convolution cocycles} (\cite{QSCC1}).
These are families of linear maps $(l_t)_{t\geq 0}$
from a $^*$-bialgebra $\Blg$ to operators on the symmetric Fock space $\FFock$,
over a Hilbert space of the form $L^2(\Rplus; \kil)$,
satisfying the following cocycle identity
with respect to the ampliated CCR flow $(\sigma_t)_{t\geq 0}$:
\[
l_{s+t} = l_s \star (\sigma_s \circ l_t), \;\;\; s,t \geq 0,
\]
together with regularity and adaptedness conditions.
Our approach enabled us to then establish a theory of
quantum L\'evy processes on \emph{compact quantum groups}
and, more generally,
quantum stochastic convolution cocycles on
operator space coalgebras (\cite{QSCC2}).
The recent development of a satisfactory
theory of \emph{locally compact} quantum groups (\cite{kuv})
provides the challenge which is addressed in the current work, namely to
extend our analysis to the locally compact realm.
On the algebraic level the theories of
quantum stochastic convolution cocycles on
compact and locally compact quantum semigroups look similar,
however their analytic aspects have a rather different nature.
Whereas the coproduct on a compact quantum semigroup $\blg$
takes values in the spatial tensor product $\blg \ot \blg$,
which led us to an operator-space theoretic development of the theory
and enabled us to establish the main results in the corresponding
natural category of operator-space coalgebras,
a \emph{noncompact} locally compact quantum semigroup $\blg$
is a nonunital $C^*$-algebra
whose coproduct takes values in the multiplier algebra of $\blg \ot \blg$.
Consequently, $C^*$-algebraic methods come more to the fore,
with the strict topology (\cite{Lance}), strict maps (\cite{Kus})
and enveloping von Neumann algebras all playing crucial roles.
The modern approach to quantum stochastics
involves matrix spaces
as a natural tool for combining $C^*$-algebraic quantum state spaces
with von Neumann algebraic quantum noise (\cite{LWexistence}),
and this was successfully exploited in~\cite{QSCC2}.
In the context of the present paper,
the strict topology on the initial $C^*$-algebra has to be
harnessed to the matrix-space technology,
so that both may be exploited in tandem.
For this reason the first part of the paper
(Section~\ref{Section: Extensions})
is devoted to
a careful analysis of relations between
the strict topology and the matrix-space topology,
compatibility between
extensions of maps continuous with respect to the different topologies,
automatic strictness of certain completely bounded maps and
connections with the enveloping von Neumann algebra and ultraweak extension.
The structure of the rest of the paper is as follows.
In Section~\ref{Section: Bialgebras} basic properties of
multiplier $C^*$-bialgebras are described and
a universal enveloping construction is given;
in particular, the $R$-maps
which play a central role are introduced.
Section~\ref{Section: QS} contains a brief summary of
the background quantum stochastics needed here.
Weak and strong
coalgebraic quantum stochastic differential equations
are treated in Section~\ref{Section: QSDE}, where
an automatic strictness result is used to establish uniqueness
of weak solutions.
Quantum stochastic convolution cocycles are analysed in
Section~\ref{Section: Cocycles}, where
Markov-regular completely positive contraction cocycles are shown
to satisfy quantum stochastic differential equations,
and the form of the stochastic generator is given ---
also for *-homomorphic cocycles.
\emph{Markov-regularity} means that
the Markov semigroup of the cocycle is norm-continuous.
In Section~\ref{Section: Levy}
quantum L\'evy processes are defined in our setting and
are shown to be realisable as Fock-space convolution cocycles
when they are Markov-regular.
This leads to two characterisations of the generating functionals of
norm-continuous convolution semigroups of states on a
locally compact quantum semigroup.
The final section contains a thorough analysis of a
natural discrete approximation scheme for
the type of cocycles treated in Section~\ref{Section: Cocycles},
founded on a recent approximation theorem of Belton (\cite{Bel}).
The method of approximation closely mirrors the structure of
the solution of the corresponding quantum stochastic differential equation
via a (sophisticated) form of Picard iteration.
The results of Sections~\ref{Section: QSDE} to~\ref{Section: Levy}
incorporate and extend
corresponding results for unital $C^*$-bialgebras in~\cite{QSCC2},
and those of Section~\ref{Section: Approximation} do likewise
for~\cite{FrS}.
Our most satisfactory results are obtained
in the case of Markov-regular quantum L\'evy processes, where
the Markov (convolution) semigroup of the process is norm-continuous.
A general theory of weakly continuous convolution semigroups
of functionals on multiplier $C^*$-bialgebras is initiated in~\cite{discrete}.
In that paper every such semigroup of \emph{states}
on a multiplier $C^*$-bialgebra of discrete type
is shown to be norm-continuous
so that all the results of this paper apply directly in that case.
Also Theorem~\ref{significant} of this paper
is used in~\cite{discrete} to derive a classical result
on conditionally positive-definite functions on a compact group.
\emph{General notations.}
In this paper the multiplier algebra and universal enveloping von Neumann
algebra of a $C^*$-algebra $\Al$ are denoted by $\Altilde$ and $\Alol$
respectively.
The symbols $\otul$, $\ot$ and $\otol$ are used
respectively for linear/algebraic, spatial/minimal and ultraweak tensor products,
of spaces and respecively,
linear, completely bounded and ultraweakly continuous completely bounded maps.
For any Hilbert space $\hil$,
we have the ampliation and Hilbert space, given respectively by
\begin{equation}
\label{iotahatDelta}
\iota_\hil: B(\Hil; \Kil)\to B(\Hil\ot\hil; \Kil\ot\hil), T\mapsto T\ot I_\hil, \
\text{ and }
\hilhat:= \Comp\oplus\hil.
\end{equation}
where context determines the Hilbert spaces $\Hil$ and $\Kil$.
\section{Strict extensions, tensor products and $\chi$-structure maps}
\label{Section: Extensions}
In this section we recall some definitions and relevant facts about
Hilbert $C^*$-modules (\cite{Lance}), strict topologies (\cite{Kus}),
tensor products and $\hil$-$\kil$-matrix spaces (\cite{LWexistence}).
We establish an automatic strictness result and show how strict
tensor product constructions compare with
$\hil$-matrix space constructions over a multiplier $\Cstar$-algebra.
The section ends by recalling a central concept for quantum L\'evy
processes, namely that of $\chi$-structure maps.
\subsection{Hilbert $\Cstar$-modules and multiplier
algebras}
For Hilbert $C^*$-modules $E$ and $F$ over a $C^*$-algebra $\Cil$,
$\Adjointable(E;F)$ denotes the space of adjointable operators $E\to F$.
Hilbert $\Cstar$-modules are endowed with a natural operator space structure
under which $M_n(\Adjointable(E;F))$ is identified with
$\Adjointable(E^n;F^n)$, where the \emph{column} direct sums $E^n$ and
$F^n$ are also Hilbert $\Cil$-modules, and
$\Adjointable(E;F) \subset CB_\Cil(E;F)$, the space of right $\Cil$-linear
completely bounded maps $E\to F$ (\cite{BlM}).
The \emph{strict topology} on $\Adjointable(E;F)$ is the locally convex
topology generated by the seminorms
$T\mapsto \|Te\| + \|T^*f\|$ ($e\in E, f\in F$);
it is Hausdorff and complete.
The closed subspace of
$\Adjointable(E;F)$ generated by the elementary maps
$| f\ra \la e|: x \mapsto f \la e, x\ra$
($e\in E$, $f\in F$) is denoted $\KAdjointable(E;F)$.
The unit ball of $\KAdjointable(E;F)$ is strictly dense in that
of $\Adjointable(E;F)$, $\KAdjointable(E)$ is a $\Cstar$-algebra
and $\Adjointable(E)$ is a model for its multiplier algebra.
In particular, viewing a $\Cstar$-algebra $\Al$ as a Hilbert
$\Cstar$-module over itself, $\KAdjointable(\Al) = \Al$ so that
$\Adjointable(\Al)$ is a model for the multipler algebra $\Altilde$.
A net of positive contractions $(e_\lambda)$ in $\Al$
is an approximate identity for $\Al$ if and only if
$e_\lambda \to 1_{\Altilde}$
strictly.
When $\Al$ is unital the strict topology coincides with the norm topology.
For Hilbert spaces $\hil$ and $\kil$,
$|\hil\ra := B(\Comp;\hil)$ and $|\kil\ra$ are Hilbert
$\Cstar$-modules over $\Comp$, $\Adjointable(|\hil\ra ; |\kil\ra)$ and
$\KAdjointable(|\hil\ra ; |\kil\ra)$ are naturally identified with
$B(\hil;\kil)$ and $K(\hil;\kil)$ respectively, and the strict topology on
$\Adjointable(|\hil\ra ; |\kil\ra)$ corresponds to the strong*-topology on
$B(\hil;\kil)$.
When a $\Cstar$-algebra $\Al$ acts nondegenerately on a Hilbert space $\hil$,
the multiplier algebra $\Altilde$
is realised as the double centraliser of $\Al$ in
$B(\hil)$: $\{x\in B(\hil): \forall_{a\in\Al} \ xa, ax \in \Al\}$,
the inclusion $\Altilde \subset \Al''$ holds and bounded strictly convergent
nets in $\Altilde$ converge (strong*- and thus) $\sigma$-weakly.
Two elementary classes of strictly continuous maps that feature below
are component maps
$\varepsilon_{kl}:
\Adjointable (E_1\oplus E_2) \to \Adjointable(E_k;E_l)$,
$T = [T_{ij}] \mapsto T_{kl}$, where the column direct sum
$E_1\oplus E_2$ is a Hilbert $\Cil$-module, and multiplication operators
$\Adjointable(E;F) \to \Adjointable(E';F')$, $T\mapsto X^*TY$, where
$X\in \Adjointable(F';F)$ and $Y\in \Adjointable(E';E)$ for Hilbert
$\Cil$-modules $E'$ and $F'$.
\subsection{Strict maps and extensions}
There is a more prevalent notion in the theory than strict continuity. A
bounded operator $\varphi$ from
$\KAdjointable = \KAdjointable(E;F)$ to
$\Adjointable' = \Adjointable(E';F')$
where $E'$ and $F'$ are Hilbert $\Cstar$-modules over a $\Cstar$-algebra
$\Cil'$, is called \emph{strict} if it is strictly continuous on bounded
sets; the collection of such maps, denoted
$\Bstrict(\KAdjointable; \Adjointable')$, is a closed subspace of
$B(\KAdjointable; \Adjointable')$; we describe some of its contents below.
Here we are particularly interested in the classes $\Bstrict(\Al;
B)$ for $\Cstar$-algebras $\Al$ and spaces $B$ of the form $B(\hil;\kil)$
where $\hil$ and $\kil$ are Hilbert spaces.
An important general class of strict maps is the set of *-homomorphisms
$\varphi: \Al \to \Adjointable(E)$, for a $\Cstar$-algebra $\Al$ and
Hilbert $\Cstar$-module $E$,
which are \emph{nondegenerate} in the sense that
$\Linol\, \varphi(\Al)E = E$.
For a $C^*$-algebra $\Al$,
let $\Alol$ denote its universal enveloping von Neumann algebra,
let $\rho$ be the embedding $\Al \to \Alol$ and
let $\iota$ be the inclusion/natural map
$\Alol_* \to \Alol^* = (\Alol_*)^{**}$.
The map
$\iota^*\circ \rho^{**}: \Al^{**} \to (\Alol_*)^* = \Alol$
is a *-isomorphism for the common Arens product on $\Al^{**}$
and a weak*-$\sigma$-weak homeomorphism.
Since $\Al$ acts nondegenerately in the universal representation,
$\Altilde$ may be viewed
as a subalgebra of $\Alol$. All of this is well-known. For ease of
reference we collect together some extension properties which will play an
important role here.
The notation $\Buw$ stands for bounded ultraweakly continuous.
\begin{thm}
\label{Theorem 1.1}
Let $\Al$ be a $C^*$-algebra with multiplier algebra $\Altilde$
and universal enveloping von Neumann algebra $\Alol$.
\begin{alist}
\item
Let $\varphi \in \Bstrict(\Al;\Adjointable)$ where
$\Adjointable = \Adjointable(E;F)$ for $C^*$-modules $E$ and $F$ over a
$\Cstar$-algebra $\Cil$.
Then $\varphi$ has a unique strict
extension $\varphitilde: \Altilde \to \Adjointable$, moreover
$\varphitilde$ is bounded and $\|\varphitilde\| = \|\varphi\|$.
\item
Let $\psi\in B(\Al;B)$ where $B = B(\hil;\kil)$ for Hilbert spaces $\hil$
and $\kil$. Then $\psi$ has a unique normal extension $\psiol \in
\Bsigma(\Alol; B)$, moreover $\|\psiol\|= \|\psi\|$.
\item
Let $\phi \in \Bstrict(\Al;B)$ where
$B=B(\hil;\kil)=\Adjointable(|\hil\ra;|\kil\ra)$ for
Hilbert spaces $\hil$ and $\kil$.
Then $\phitilde = \phiol|_{\Altilde}$.
\end{alist}
In \tu{(}a\tu{)},
$\widetilde{\varphi^\dagger} = \widetilde{\varphi}^\dagger$
where
$\varphi^\dagger: \Al \to \Adjointable(F;E)$ is defined by
$\varphi^\dagger(a^*) = \varphi(a)^*$; similarly, in \tu{(}b\tu{)}
$\ol{\psi^\dagger} = \psiol^\dagger$.
When $F=E$,
$\varphitilde$ is positive/completely positive/multiplicative if $\varphi$
is, and likewise for $\psi$ and $\psiol$ when $\kil = \hil$.
\end{thm}
\begin{proof}
(a) is proved in~\cite{Kus} in the case $E=F=\Cil$. The general case is
obtained by applying this case with $\Cil = \KAdjointable (E\oplus F)$ and
composing with the strict map $\Adjointable(E\oplus F) \to
\Adjointable(E;F)$, $T=[T_{i,j}]\mapsto T_{21}$.
(b) is well-known: set
$\psiol := \iota^* \circ \psi^{**}\circ j$ where $\iota$ is the natural
map/inclusion
$B_* \to (B_*)^{**} = B^*$ and $j$ is the natural isometric isomorphism
$\Alol \to \Al^{**}$.
Since the unit ball of $\Al$ is strictly dense in that of $\Altilde$
(\cite{Lance}, Proposition 1.4), (c)
follows from the fact that strictly convergent bounded nets converge
$\sigma$-weakly.
The last part follows from Kaplansky's density theorem
and its Hilbert $C^*$-module counterpart (just used),
and the separate strict
(respectively $\sigma$-weak) continuity of multiplication
and corresponding continuity of the adjoint operation, in a multiplier
algebra (respectively von Neumann algebra).
\end{proof}
\begin{rems}
(i)
The extensions commute with matrix liftings:
\begin{align*}
&
\widetilde{\varphi^{(n)}} = \varphitilde^{(n)}:
\widetilde{M_n(\Al)} = M_n(\Altilde) \to
M_n(\Adjointable)= \Adjointable(E^n;F^n)
\\ &
\ol{\psi^{(n)}} = \psiol^{(n)}:
\ol{M_n(\Al)} = M_n(\Alol) \to
M_n(B)= B(\hil^n;\kil^n),
\end{align*}
so $\|\varphitilde\|_\cb = \|\varphi\|_\cb$ when
$\varphi\in \CBstrict(\Al;\Adjointable)$
and $\|\phiol\|_\cb = \|\phi\|_\cb$ when
$\psi\in CB(\Al;B)$.
(ii)
Clearly
the range of $\varphitilde$ is contained in the strict closure of
the range of $\varphi$, and
the range of $\psiol$ is contained in the $\sigma$-weak closure of
the range of $\psi$.
(iii)
As a consequence of (a), strict maps may be composed in the
following sense: if
$\varphi_1\in \Bstrict(\Al_1;\Altilde_2)$ and
$\varphi_2\in \Bstrict(\Al_2;\Altilde_3)$,
for $C^*$-algebras $\Al_1, \Al_2$ and $\Al_3$,
then
$a\mapsto \varphitilde_2 ( \varphi_1 (a))$
is strict
with unique strict extension $\varphitilde_2 \circ \varphitilde_1$;
following the widely adopted convention (e.g.\ \cite{Lance}),
it is denoted $\varphi_2 \circ \varphi_1$.
For a nondegenerate *-homomorphism
$\varphi: \Al \to \Ciltilde$, $\varphitilde$ is a unital *-homomorphism,
and conversely every
nondegenerate *-homomorphism $\varphi: \Al \to \Ciltilde$ is the
restriction of a strict unital *-homomorphism
$\Altilde \to \Ciltilde$.
\end{rems}
\noindent
\emph{Warning.}
We now write $\Bstrict(\Altilde;\Adjointable)$ for
the class of strict maps $\Altilde \to \Adjointable$ where, for us,
$\Adjointable$ will always be \emph{either} of the form
$B(\hil;\kil) = \Adjointable(|\hil\ra; |\kil\ra)$ for Hilbert spaces
$\hil$ and $\kil$ \emph{or} of the form $\Ciltilde$ for a $\Cstar$-algebra
$\Cil$ (or \emph{both}:
$B(\hil) = \Adjointable(|\hil\ra) = \widetilde{K(\hil)}$).
Use of this notation then always needs to reflect the algebras of which the
source (and target) multiplier algebras are.
We note that the theorem delivers a commutative diagram of isometric
isomorphisms:
\begin{equation} \label{diag B}
\xymatrix{
\Bstrict(\Al;B) \ar [dr]\ar@{->}[rr] && \Buw(\Alol;B) \\
& \Bstrict(\Altilde;B) \ar[ur]&
}
\end{equation}
for any $\Cstar$-algebra $\Al$ and space $B$ of the form $B(\hil;\kil)$
for Hilbert spaces $\hil$ and $\kil$.
\begin{defn}
Let $\varphi\in\Bstrict(\Al; \Adjointable(E))$ for a $\Cstar$-algebra
$\Al$ and Hilbert $\Cstar$-module $E$. We call $\varphi$
\emph{preunital} if its strict extension is unital:
$
\varphitilde(1) = I.
$
\end{defn}
\begin{rem}
For *-homomorphisms this is equivalent to nondegeneracy;
in general it is equivalent to
$\varphi(e_\lambda)\to 1$ for some/every $\Cstar$-approximate identity
$(e_\lambda)$ for $\Al$, but is stronger than the condition
$\Linol \varphi(\Al)E = E$ (\cite{Lance}, Proposition 2.5; Corollary 5.7).
\end{rem}
\subsection{Automatic strictness and strict tensor products}
In the next theorem we establish an automatic strictness property and
identify a natural class of maps for strict tensoring.
First some notation. When $\Al$ is the spatial tensor product
$\Al_1 \ot \Al_2$, for $C^*$-algebras $\Al_1$ and $\Al_2$,
$\Altilde$ is denoted $\Al_1 \ottilde \Al_2$. Note the relation
\begin{equation} \label{ottilde}
\Altilde_1 \ot \Altilde_2 \subset \Al_1 \ottilde \Al_2
\end{equation}
\begin{thm} \label{P.B}
Let $\Al$, $\Al_1$ and $\Al_2$ be $C^*$-algebras.
\begin{alist}
\item
Let $\varphi \in CB(\Al;B)$ where $B=B(\hil;\kil)$ for a
Hilbert spaces $\hil$ and $\kil$. Then $\varphi$ is strict and
$\varphitilde = \varphiol|_{\Altilde}$.
In particular, all *-homomorphisms $\Al \to B(\hil)$ are strict.
\item
Let
$\varphi_i\in \Lin \CPstrict\big(\Al_i; \Ciltilde_i\big)$ for
$C^*$-algebras
$\Cil_1$ and $\Cil_2$.
Then,
there is a unique map
$\varphi_1\ot \varphi_2 \in
\Lin \CPstrict(\Al_1\ot\Al_2; \Cil_1\ottilde\Cil_2)$
extending the algebraic tensor product map $\varphi_1 \otul \varphi_2$
\end{alist}
\end{thm}
\begin{proof}
(a)
In view of Theorem~\ref{Theorem 1.1} it suffices to prove that $\varphi$
is
strict. It follows from the Wittstock-Paulsen-Haagerup Decomposition
Theorem (\cite{EfR}, Theorem 5.3.3)
that $\varphi = \psi \circ \pi$ where $\pi$ is a *-homomorphism
$\Al \to B(\Hil)$ for some Hilbert space $\Hil$ and
$\psi: B(\Hil) \to B$ is of the form $X\mapsto R^*XS$, for some
operators $R\in B(\kil;\Hil)$ and $S\in B(\hil;\Hil)$.
Moreover, replacing $\Hil$ by $\Hil':= \Linol \pi(\Al)\Hil$,
$\pi$ by its compression to $\Hil'$
$R$ and $S$ by $RP$ and $SP$
where $P$ is the orthogonal projection $\Hil \to \Hil'$, if necessary,
we may suppose that $\pi$ is
nondegenerate and therefore strict, when viewed as a map into
$\Adjointable(|\Hil\ra)$. Since $\psi$ is strict
$\Adjointable(|\Hil\ra) \to \Adjointable(|\hil\ra;|\kil\ra)$,
$\varphi$ is too.
(b)
By linearity we may suppose that $\varphi_1$ and $\varphi_2$ are
completely positive. The result then follows easily from Kasparov's
extension of the Stinespring Decomposition Theorem (\cite{Lance}, Theorem
5.6).
\end{proof}
\begin{rems}
It follows from Part (a) and the remarks after Theorem~\ref{Theorem 1.1}
that~\eqref{diag B} restricts to a commutative diagram of completely
isometric isomorphisms
\begin{equation} \label{diag CB}
\xymatrix{
CB(\Al;B) \ar [dr]\ar@{->}[rr] && \CBuw(\Alol;B) \\
& \CBstrict(\Altilde;B) \ar[ur]&
}
\end{equation}
where $\Al$ and $B$ are as in~\eqref{diag B}. In particular,
we have complete isometries
\begin{equation}
\label{Equation: duals} \Al^* \cong \Altilde_\beta^* \cong \Alol_*.
\end{equation}
In the terminology of~\cite{GL}, the theorem implies that strict
completely positive maps are \emph{completely strict}.
By operator space considerations
$\Ran(\varphi_1\ot\varphi_2) \subset \Ciltilde_1\ot\Ciltilde_2$.
Part (b) leads to the following useful notation. For
$\varphi_i\in \Lin \CPstrict\big(\Al_i; \Ciltilde_i\big)$ ($i=1,2$) we
denote the unique strict extension of
$\varphi_1\ot \varphi_2$ by
$\varphi_1\ottilde \varphi_2$. Thus
\begin{equation} \label{tensor tilde}
\varphi_1\ottilde \varphi_2 \in \Lin \CPstrict
\big(\Al_1\ottilde\Al_2;\Cil_1\ottilde\Cil_2\big).
\end{equation}
\end{rems}
Note the following consequence of
Part (a) and its proof,
which provides a source for tensoring as in Part (b).
\begin{cor}
\label{cor: CB=LinCPbeta}
For a $C^*$-algebra $\Al$ and Hilbert space $\hil$,
\begin{equation}
\label{CB=LinCPbeta}
CB\big(\Al; B(\hil)\big) = \Lin \, \CPstrict\big(\Al; B(\hil)\big).
\end{equation}
\end{cor}
\begin{rem}
This `strict decomposability' property is very useful. For a
general multiplier algebra target space, completely bounded maps
need neither be strict nor be linear combinations of completely
positive maps.
\end{rem}
\subsection{$\hil$-$\kil$-Matrix spaces}
Let $\Vil$ be an operator space in $B(\Hil;\Kil)$, let
$B = B(\hil;\kil)$ for two further Hilbert spaces $\hil$ and $\kil$
with total subsets $S$ and $T$, and let
$Z\in B(\Hil\ot\hil;\Kil\ot\kil) = B(\Hil;\Kil)\otol B$.
Then the following are equivalent:
\begin{align*}
&
E^\xi Z E_\eta \in \Vil \text{ for all } \xi\in T, \eta \in S;
\\ &
(\id_{B(\Hil;\Kil)} \otol \omega)(Z) \in\Vil \text{ for all } \omega \in B_*;
\end{align*}
where, for a Hilbert space vector $\xi$,
\begin{equation}
\label{E notation}
E_{\xi} := I \ot |\xi\ra : u \mapsto u \ot \xi \text{ and }
E^{\xi} := (E_\xi)^* = I \ot \la\xi |,
\end{equation}
and $I$ is the identity operator on the appropriate Hilbert space. The collection of operators $Z$
enjoying this property is an operator space which is denoted $\Vil\otM B$ and called the (right)
$\hil$-$\kil$-\emph{matrix space over} $\Vil$. It is situated between the norm-spatial and
ultraweak-spatial tensor products:
\[
\Vil\ot B \subset
\Vil\otM B \subset
\ol{\Vil}\otol B
\]
and the latter inclusion is an equality if and only if $\Vil$ is
$\sigma$-weakly closed.
If $\varphi \in CB(\Vil;\Vil')$ for another concrete operator space
$\Vil'$
then there is a unique map, its $\hil$-$\kil$-\emph{lifting},
denoted $\varphi \otM \id_{B}$, from
$\Vil\otM B$ to $\Vil'\otM B$ satisfying
$E^\xi(\varphi\otM\id_B)(Z)E_\eta = \varphi(E^\xi ZE_\eta)$
for all $\xi\in\kil$, $\eta \in \hil$ and $Z \in \Vil\otM B$; it
is completely bounded, with $\|\varphi\otM\id_{B}\|_\cb =
\|\varphi\|_\cb$ and completely isometric if $\varphi$ is (unless
$B = \{0\}$), moreover it extends
$\varphi \ot \id_{B}$, and it coincides with
$\varphi \otol \id_{B}$ when $\Vil$ is $\sigma$-weakly
closed and $\varphi$ is $\sigma$-weakly continuous.
The next proposition confirms the compatibility of $\hil$-$\kil$-matrix spaces and
$\hil$-$\kil$-liftings on the one hand, and strict tensor products of
algebras and strict maps on the other.
First note the identity
\begin{equation}
\label{matrix ket id}
\|\eta\|^2 R E_\eta X =
R\big( X \ot |\eta \ra\la\eta|\big) E_\eta,
\end{equation}
for Hilbert space operators
$R\in B(\Hil\ot\hil;\Hil)$ and $X\in B(\Hil)$
and vectors $\eta\in\hil$.
\begin{propn}
Let $\Al$ be a $\Cstar$-algebra and
let $B=B(\hil)$ and $K=K(\hil)$ for a Hilbert space $\hil$.
Then, in any faithful nondegenerate representation of $\Al$
\tu{(}such as its universal representation\tu{)},
\[
\Al\ottilde K \subset \Altilde\otM B,
\]
for the induced concrete realisations of
$\Altilde$ and $\Al\ottilde K$.
Moreover, if $\psi\in\Lin\CPstrict(\Al;\Ciltilde)$
for another $\Cstar$-algebra $\Cil$,
also faithfully and nondegenerately represented,
then
\[
\psi \ottilde \id_{K} \subset
\psitilde \otM \id_{B}.
\]
\end{propn}
\begin{proof}
Let $T\in \Al\ottilde K$ and $\zeta, \eta \in \hil$.
First note that~\eqref{matrix ket id} implies that
\begin{equation}
\label{to get strict}
\|\eta\|^2 E^\zeta T E_\eta a =
E^\zeta T \big( a \ot |\eta \ra\la\eta| \big) E_\eta,
\quad a \in \Al;
\end{equation}
similarly,
\begin{equation}
\label{2 to get strict}
\|\zeta\|^2 a E^\zeta T E_\eta =
E^\zeta \big( a \ot |\zeta \ra\la\zeta| \big) T E_\eta,
\quad a \in \Al;
\end{equation}
and so $E^\zeta T E_\eta \in \Altilde$.
Thus $T\in\Altilde\otM B$.
This proves that $\Al \ottilde K \subset \Altilde\otM B$,
and~\eqref{to get strict} and~\eqref{to get strict}
now imply that the map
\[
T \in \Al \ottilde K \mapsto E^\zeta T E_\eta \in \Altilde
\]
is strictly continuous.
Therefore the maps
\[
E^\zeta \big( \psitilde \otM \id_B \big) (\cdot ) E_\eta =
\psitilde (E^\zeta \cdot E_\eta) \text{ and }
E^\zeta \big( \psi \ottilde \id_K \big) (\cdot ) E_\eta
\]
are strictly continuous $\Al \ottilde K \to \Ciltilde$
and agree on the strictly dense subspace $\Al \ot K$.
They therefore agree on $\Al \ottilde K$ and the result follows.
\begin{comment}OLD PROOF
For $\xi, \eta\in\hil\setminus\{0\}$ and
$a\in\Al$,
$T(a\ot|\eta\ra\la\eta|)\in\Al\ot K(\hil)$ so
\[
E^\xi TE_\eta a =
\|\eta\|^{-2} E^\xi (T(a \ot |\eta\ra\la\eta|) E_\eta \in \Al.
\]
Thus $E^\xi TE_\eta \in \Altilde$ for all $\xi, \eta \in \hil$,
and so $T\in\Altilde\otM B(\hil)$.
The second part follows from the fact that, for $\xi, \eta \in \hil$ and
$a,a'\in\Al_2$, the map
\[
\big( a' \ot |\xi\ra \la\xi | \big)
\big( \psitilde \otM \id_{B(\hil)} \big)( \cdot )
\big( a \ot |\eta\ra \la\eta | \big)
=
a'\psitilde \big( E^\xi \cdot E_\eta \big)
\big( a \ot | \xi \ra \la \eta | \big)
\]
is manifestly strict: $\Al_1 \ottilde K(\hil) \to \Al_2 \ottilde K(\hil)$,
and agrees with
\[
\big( a' \ot |\xi\ra \la\xi | \big)
\big( \psi \ottilde \id_{K(\hil)} \big)( \cdot )
\big( a \ot |\eta\ra \la\eta | \big)
\]
on $\Al_1 \ot K$.
\end{comment}
\end{proof}
\begin{rem}
In the universal representation of $\Al$ we have
the further compatibility relations,
\[
\Altilde\otM B(\hil) \subset \Alol \otol B(\hil)
\text{ and }
\psitilde \otM \id_{B(\hil)} \subset \psiol \otol \id_{B(\hil)}.
\]
\end{rem}
\subsection{$C^*$-algebras with character}
The following notion plays an important role in the theory.
Recall the notation~\eqref{iotahatDelta}.
\begin{defn}
A $\chi$-\emph{structure map on} a $\Cstar$-algebra with
character
$(\Al,\chi)$
is a linear map $\varphi : \Al \to B(\hilhat)$, for some Hilbert
space $\hil$, satisfying
\begin{equation}
\label{Equation: chi structure relation}
\varphi (a^*b) =
\varphi(a)^* \chi (b) + \chi(a)^* \varphi(b) +
\varphi (a)^* \Delta \varphi (b),
\end{equation}
where $\Delta:=
\left[\begin{smallmatrix} 0 & \\ & I_{\hil} \end{smallmatrix}\right]
\in B(\hilhat)$ (no relation to coproducts).
\end{defn}
The following result is established in~\cite{QSCC2}
\begin{thm} \label{established}
Every $\chi$-structure
map $\varphi$ is \emph{implemented} by a pair $(\pi, \xi)$ consisting of a
*-homomorphism $\pi: \Al \to B(\hil)$ and vector $\xi\in\hil$, that is
$\varphi$ has block matrix form
\[
\begin{bmatrix} \la \xi | \\ I_{\hil} \end{bmatrix}
\nu( \cdot )
\begin{bmatrix} | \xi \ra & I_{\hil} \end{bmatrix}
\text{ where }
\nu := \pi - \iota_\hil \circ \chi,
\]
in other words,
\begin{equation}
\label{Equation: block}
\begin{bmatrix} \gamma & \la \xi | \nu( \cdot ) \\
\nu( \cdot ) | \xi \ra & \nu \end{bmatrix},
\text{ where }
\gamma:= \omega_\xi \circ \nu.
\end{equation}
Moreover $\varphi$ is necessarily strict, and
$\pi$ is nondegenerate if and only if $\varphitilde (1) = 0$.
\end{thm}
\begin{proof}
The first part is Theorem A6 of~\cite{QSCC2}.
It implies that $\varphi$ is completely bounded and so,
by Theorem~\ref{P.B}, $\varphi$ is strict.
After strict extension,
the last part now follows by inspection.
\end{proof}
\begin{rems}
It is easily seen that conversely, any map with such a block matrix
representation is a $\chi$-structure map.
By the separate strict/$\sigma$-weak continuity of multiplication,
it follows that if $\varphi$ is a $\chi$-structure map then
$\varphitilde$ is a $\chitilde$-structure map
and
$\varphiol$ is a $\chiol$-structure map.
\end{rems}
We shall need the following result in Section~\ref{Section: Levy}.
\begin{lemma} \label{condpos}
Let $(\Al,\chi)$ be a $C^*$-algebra with character.
Then, for any functional $\gamma \in \Al^*$,
if $\gamma$ is positive on $\Ker \chi$ then
$\wt{\gamma}$ is positive on $\Ker \chitilde$
and $\ol{\gamma}$ is positive on $\Ker \chiol$.
\end{lemma}
\begin{proof}
It suffices to prove that
$\Al_+ \cap \Ker \chi$ is strictly dense in $\Altilde_+ \cap \Ker \chitilde$
and
$\sigma$-weakly dense in $\Alol_+ \cap \Ker \chiol$.
Let $a \in \Altilde_+ \cap \Ker \chitilde$.
The Kaplansky Density Theorem for multiplier algebras (\cite{Lance}, Proposition 1.4)
implies that there is a bounded net $(c_i)_{i \in I}$
of selfadjoint elements in $\Al$ converging strictly to $a^{1/4}$.
Set $a_i = b_i^*b_i$ where
\[
b_i :=
c_i \big(c_i - \chi (c_i)\big) \in \Ker \chi.
\]
Then $a_i \in \Al_+ \cap \Ker \chi$ and
separate strict continuity of multiplication on bounded subsets of $\Al$, and
strictness of $\chi$, imply that $(a_i)_{i \in I}$ converges strictly to $a$.
The ultraweak density of
$\Al_+ \cap \Ker \chi$ in $\Alol_+ \cap \Ker \chiol$
is proved similarly, by appealing to the standard Kaplansky Density Theorem
for von Neumann algebras.
\end{proof}
\section{Multiplier $C^*$-bialgebras}
\label{Section: Bialgebras}
It is convenient to consider bialgebras in both the $\Cstar$- and
$W^*$- categories and a universal enveloping operation linking the two.
\begin{defn}
A (\emph{multiplier}) \emph{$C^*$-bialgebra} is a
$C^*$-algebra $\Bil$ with \emph{coproduct},
that is a nondegenerate *-homomorphism
$\coproduct: \Bil \to \Bil\ottilde\Bil$ satisfying the coassociativity
conditions
\[
(\id_\Bil \ot \coproduct)\circ \coproduct =
(\coproduct \ot \id_\Bil)\circ \coproduct.
\]
A \emph{counit} for $(\Bil,\coproduct)$ is a character $\counit$ on $\Bil$
satisfying the counital property:
\[
(\id_\Bil \ot \counit)\circ \coproduct =
(\counit \ot \id_\Bil)\circ \coproduct = \id_\Bil.
\]
\end{defn}
\begin{rems}
The above definitions extend those for unital $C^*$-bialgebras, for which
$\Biltilde = \Bil$ and $\Bil\ottilde\Bil = \Bil\ot\Bil$.
The strict extension of a coproduct is a unital *-homomorphism and the
strict extension of a counit is a character on $\Biltilde$.
Note however that, in general, $(\Biltilde, \coproducttilde)$ is
\emph{not} itself a $\Cstar$-bialgebra as the inclusion
$\Biltilde \ot \Biltilde \subset \Bil \ottilde \Bil$
is usually proper.
\end{rems}
Examples of counital $\Cstar$-bialgebras include locally compact quantum
groups in the universal setting (\cite{Kus2}), in particular all
coamenable locally compact quantum groups are included.
If the assumptions on the coproduct $\coproduct$ are weakened to it being completely
positive, strict and preunital
then the resulting structure is called a (\emph{multiplier})
$\Cstar$-\emph{hyperbialgebra}
(cf.\ \cite{ChV}).
Let $\Bil$ be a $\Cstar$-bialgebra.
The \emph{convolute} of
$\phi_1\in\Lin \CPstrict(\Bil;\Altilde_1)$ and
$\phi_2\in\Lin \CPstrict(\Bil;\Altilde_2)$
for $\Cstar$-algebras $\Al_1$ and $\Al_2$
is defined by
\[
\phi_1\conv\phi_2 =
(\phi_1\ot\phi_2)\circ \coproduct
\in \Lin \CPstrict(\Bil;\Al_1\ottilde\Al_2).
\]
We denote its strict extension by
$\phi_1\convtilde\phi_2$.
Associativity of both of these convolutions follows from associativity of
$\ottilde$ and coassociativity of $\coproducttilde$.
For each $C^*$-algebra $\Al$ define a map
\[
R_\Al: \Lin \CPstrict(\Bil;\Altilde) \to \CBstrict(\Bil;\Bil\ottilde\Al),
\quad
\phi \to
\id_\Bil \conv\phi = ( \id_\Bil \ot \phi ) \circ \coproduct.
\]
In case $\Al=\Comp$, $\Lin \CPstrict (\Bil;\Altilde)$ is simply $\Bil^*$
and we have
\[
R_{\Comp}(\varphi_1\conv\varphi_2) =
R_{\Comp}\varphi_1
\circ R_{\Comp}\varphi_2,
\quad \varphi_1, \varphi_2 \in \Bil^*.
\]
When $\Bil$ is counital each $R_\Al$ has left-inverse
\begin{equation}
\label{Equation: counit circ Rmap}
E_\Al: \CBstrict(\Bil;\Bil\ottilde\Al) \to \CBstrict(\Bil;\Altilde), \quad
\psi \to (\counit \ot \id_\Bil)\circ \psi.
\end{equation}
\begin{rems}
By the complete positivity and strictness of the coproduct
\[
R_\Al \big( \CPstrict (\Bil; \Altilde) \big) \subset
\CPstrict \big( \Bil; \Bil\ottilde \Al \big)
\]
for any $\Cstar$-algebra $\Al$.
In particular, by~\eqref{CB=LinCPbeta},
\[
R_{K(\hil)} \big( CB ( \Bil; B(\hil) ) \big) \subset
\Lin \CPstrict \big( \Bil; \Bil\ottilde K(\hil) \big).
\]
Note also that, when $\varphi_1 \in CB(\Bil;B(\hil_1))$ and
$\varphi_2 \in CB(\Bil;B(\hil_2))$
for Hilbert spaces $\hil_1$ and $\hil_2$,
\[
\varphi_1\conv\varphi_2 \in CB(\Bil;B(\hil_1\ot\hil_2)).
\]
\end{rems}
For convenience we summarise
useful properties of the $R$-maps next.
\begin{propn}
\label{Proposition: R-map}
Let $\Bil$ be a $\Cstar$-bialgebra and let $\Al$ be a $\Cstar$-algebra.
Then $R_\Al$ is a completely contractive map into
$\CBstrict(\Bil; \Bil\ottilde\Al)$ with image in the subspace
$\Lin CP_\beta(\Bil; \Bil \ottilde \Al)$ and,
after strict extension, $R_{\Comp}$ is furthermore a
homomorphism of Banach algebras:
$(\Bil^*,\conv) \cong
\big((\Biltilde)^*_\beta, \convtilde \big) \to \CBstrict(\Biltilde)$.
When $\Bil$ is counital, $R_\Al$ is completely isometric with completely
contractive left-inverse $E_\Al$ and $R_{\Comp}$ is furthermore a unital algebra morphism.
\end{propn}
We now turn briefly to the $W^*$-category.
\begin{defn}
A \emph{von Neumann bialgebra} is a von Neumann algebra $\Mil$ with
coproduct, that is
a normal unital *-homomorphism $\coproduct: \Mil \to \Mil\otol\Mil$ which
is coassociative:
\[
(\id_\Mil\otol\coproduct)\circ \coproduct =
(\coproduct\otol \id_\Mil)\circ \coproduct.
\]
A counit for $(\Mil,\coproduct)$ is a normal character $\counit$ on $\Mil$
satisfying
\[
(\id_\Mil\otol\counit)\circ \coproduct =
(\counit\otol \id_\Mil)\circ \coproduct = \id_\Mil.
\]
\end{defn}
Convolution in this category is straightforward. Let
$\phi_1\in\CBuw(\Mil; Z_1)$ and $\phi_2\in\CBuw(\Mil; Z_2)$
for $\sigma$-weakly closed concrete operator spaces
$Z$, $Z_1$ and $Z_2$, then
\[
\phi_1\conv\phi_2 =
(\phi_1\otol\phi_2)\circ \coproduct
\in \CBuw(\Mil; Z_1 \otol Z_2),
\]
so that we may define a map
\[
R^\sigma_Z: \CBuw( \Mil; Z ) \to \CBuw( \Mil; \Mil \otol Z ), \quad
\phi \mapsto \id_\Mil\conv\phi =
\big( \id_\Mil \otol \phi \big) \circ \coproduct.
\]
In particular,
\[
R^\sigma_\Comp(\varphi_1\conv\varphi_2) =
R^\sigma_\Comp\varphi_1 \circ R^\sigma_\Comp\varphi_2 \in CB_\sigma(\Mil),
\text{ for } \varphi_1, \varphi_2\in\Mil_*.
\]
When $\Mil$ is counital
$R^\sigma_Z$ has left-inverse
\[
E^\sigma_Z: \CBuw( \Mil; \Mil \otol Z ) \to \CBuw( \Mil; Z ), \quad
\psi \mapsto (\counit \otol \id_Z )\circ \psi.
\]
\begin{propn}
Let $(\Bil, \coproduct)$ be a $\Cstar$-bialgebra.
Then $(\Bilol, \coproductol)$ is a von Neumann bialgebra.
Moreover, if $\counit$ is a counit for $\Bil$ then $\counitol$ is a counit
for $\Bilol$.
\end{propn}
\begin{proof}
The map $\coproductol$ is a normal, unital *-homomorphism and the normal
maps
\[
(\id_{\Bilol}\otol\coproductol)\circ \coproductol \text{ and }
(\coproductol\otol \id_{\Bilol})\circ \coproductol.
\]
agree on $\Bil$, which is $\sigma$-weakly dense in $\Bilol$, and so
coincide. In the counital case, $\counitol$ is a normal character on
$\Bilol$ and the normal maps
\[
\big( \id_{\Bilol} \otol \counitol \big) \circ \coproductol, \
\big( \counitol\otol \id_{\Bilol} \big) \circ \coproductol
\text{ and } \id_{\Bilol}
\]
agree on $\Bil$ and so coincide.
\end{proof}
Naturally, we refer to $(\Bilol, \coproductol)$, respectively
$(\Bilol, \coproductol, \counitol)$
as the \emph{universal enveloping von Neumann bialgebra}
(resp. \emph{counital von Neumann bialgebra}) \emph{of} $\Bil$.
\begin{rem}
The two forms of $\mathcal{R}$-map enjoy an easy compatibility:
if $\phi \in \Lin CP_\beta\big(\Bil;\Altilde\big)$ for a $\Cstar$-algebra $\Al$
then
\begin{equation}
\label{compatibility of R maps}
\ol{R_\Al \phi} = R^\sigma_{\Alol} \,\phiol,
\end{equation}
and similarly for the $E$ maps in the counital case.
\end{rem}
\emph{From now on} we shall denote all maps of the form
$R^\sigma_Z$, respectively $E^\sigma_Z$, by
$\Rbar$, respectively $\Ebar$,
and similarly abbreviate all maps of the form $R_\Al$ and $E_\Al$ to
$\Rmap$ and $\Emap$.
\section{Quantum stochastics}
\label{Section: QS}
\emph{Fix now, and for the rest of the paper},
a complex Hilbert space $\kil$ referred to as the
\emph{noise dimension space}.
For $c\in \kil$ define $\chat:=\binom{1}{c}\in \kilhat$;
and for any function $g$ with values in $\kil$
let $\ghat$ denote the corresponding
function with values in $\kilhat$, defined by $\ghat(s):= \widehat{g(s)}$.
Let $\FFock$ denote the symmetric Fock space over $L^2(\Rplus;\kil)$, let
$\Step$ denote the linear span of $\{d_{[0,t[}: d\in \kil, t\in\Rplus\}$
in $L^2(\Rplus;\kil)$ (for purposes of evaluating, we always take these
right-continuous versions) and
let $\Exps$ denote the linear span of $\{\ve(g): g\in\Step\}$ in $\FFock$,
where $\ve(g)$ denotes the exponential vector
$\big((n!)^{-\frac{1}{2}}g^{\ot n}\big)_{n\geq 0}$.
(There will be no danger of confusion with the inverse of an $R$-map!)
Also define
\begin{equation}
\label{Delta QS}
e_0 :=
\binom{1}{0}\in\kilhat
\text{ and }
\dQS := P_{\{0\}\oplus\kil} =
\begin{bmatrix} 0 & \\ & I_{\kil}
\end{bmatrix} \in B(\kilhat).
\end{equation}
\subsection{Quantum stochastic processes, differential equations and cocycles}
A detailed summary of the relevant results from QS analysis
([$\text{LW}_{1-4}$],~\cite{LSqsde}) is given in~\cite{QSCC2}.
We shall therefore be brief here.
For operator spaces $\Vil$ and $\Wil$, with $\Wil$ concrete,
$\Proc(\Vil \to \Wil)$ denotes the space of adapted proceses
$k = (k_t)_{t\geq 0}$, thus
$k_t \in L\big(\Exps; L(\Vil; \Wil\otM |\FFock\ra)\big)$ for
$t\in\Rplus$, written $\varepsilon\mapsto k_{t,\varepsilon}$.
As in~\cite{QSCC2}, we abbreviate to $\Procstar(\Vil)$ when $\Wil=\Comp$.
Its associated maps
$\kappa^{f,g}_t: \Vil\to\Wil$ ($f,g \in \Step, s,t \in \Rplus$)
are defined by
\[
\kappa^{f',f}_t(x) =
\big(\id_\Wil \otM \la\varepsilon'|\big) k_{t,\varepsilon}(x),
\quad x\in\Vil,
\]
where $\varepsilon = \varepsilon(f_{[0,t[})$ and
$\varepsilon' = \varepsilon(f'_{[0,t[})$.
For us here
the pair $(\Vil,\Wil)$ will be either $(\Mil,\Comp)$ or $(\Mil,\Mil)$
for a von Neumann algebra $\Mil$, or $(\Al,\Comp)$ or
$(\Altilde,\Comp)$ for a $C^*$-algebra $\Al$ with multiplier algebra
$\Altilde$. Thus $\Wil \otM |\Fock\ra$ is either $|\Fock\ra$ or
$\Mil \otol |\Fock\ra$.
The process $k$ is \emph{weakly initial space bounded} if each
$\kappa^{f,g}_t$ is bounded, and
\emph{weakly regular} if further
$\sup \big\{ \| \kappa^{f,g}_s \| : s\in [0,T] \big\} < \infty$,
for all $T \geq 0$. Here `column-boundedness' usually obtains:
$k_{t,\varepsilon} \in B(\Vil; \Wil\otM |\Fock\ra)$ or
$CB(\Vil; \Wil\otM |\Fock\ra)$,
for each $t\in\Rplus$ and $\varepsilon\in\Exps$.
For von Neumann algebras $\Mil$ and $\Nil$,
a process $k \in \Proc(\Mil\to\Nil)$
is called \emph{normal} if each
$\kappa^{f,g}_t$ is normal.
It follows that if $k\in\Proc(\Mil\to\Nil)$ is bounded (meaning that each
$k_t$ is bounded) and normal then each map $k_t$ is normal
$\Mil \to \Nil\otol B(\Fock)$.
Note that, by Theorem~\ref{P.B},
any completely bounded process $l \in \Procstar(\Al)$ on a $C^*$-algebra $\Al$
is necessarily strict in the sense that each map $l_t: \Al\to B(\FFock)$ is strict.
For $\phi\in L( \kilhat; CB(\Vil; \Vil\otM |\kilhat\ra)$
and $\kappa \in CB(\Vil;\Wil)$, where both $\Vil$ and $\Wil$ are concrete
operator spaces, $k^{\phi,\kappa}$ denotes the unique weakly regular
process $k\in\Proc(\Vil\to\Wil)$ satisfying the quantum stochastic
differential equation
\begin{equation}
\label{QSDE}
dk_t = k_t \fullcomp d\Lambda_\phi (t), \quad
k_0 = \iota_{\FFock} \circ \kappa ,
\end{equation}
where $\iota_\FFock$ denotes the ampliation $\Wil\to\Wil\otM B(\FFock)$
$x\mapsto x\ot I_\FFock$.
The solution is given by
\[
k_{t,\ve} =
\sum_{n\geq 0} \Lambda^n_{t,\ve} \circ (\kappa \circ \phi_n)
\]
where $\phi_n$ is an $n$-fold composition of matrix liftings of $\phi$
and the sum is norm-convergent in
$CB(\Vil; \Wil \otM |\FFock\ra)$.
When $\Wil = \Vil$ and $\kappa = \id_\Vil$, $k$ is a weak quantum
stochastic cocycle on $\Vil$ (denoted $k^\phi$), that is it satisfies
$k_0 = \iota_\FFock$ and
for $s,t\in\Rplus$ and $f,g\in\Step$,
\begin{equation}
\label{Equation: weak standard cocycle}
\kappa^{f,g}_0 = \id_\Vil, \quad
\kappa^{f,g}_{s+t} =
\kappa^{f,g}_s \circ
\kappa^{S^*_sf,S^*_sg}_t \quad
(f,g \in \Step, s,t \in \Rplus)
\end{equation}
where $(S_t)_{t\geq 0}$
is the isometric semigroup of right-shifts on $L^2(\Rplus; \kil)$.
Let $(\sigma_t)_{t\geq 0}$ denote the induced endomorphism
semigroup on $B(\FFock)$, ampliated to $\Wil \otM B(\FFock)$.
Then, when $k$ is a completely bounded process, the cocycle relation
simpiflies to
\[
k_{s+t} = k_s \fullcomp \sigma_s \circ k_t,
\]
where the extended composition notation (which we do not need to go into
here) is explained in~\cite{QSCC2}.
\section{Coalgebraic quantum stochastic differential equations}
\label{Section: QSDE}
\emph{For this section we fix a $C^*$-bialgebra} $\Bil$,
which we do not assume to be counital, and
consider the coalgebraic quantum stochastic differential equation
\begin{equation}
\label{Equation: coalg QSDE}
dl_t = l_t \conv d\Lambda_\varphi (t), \quad
l_0 = \iota_{\FFock} \circ \eta ,
\end{equation}
where $\varphi\in SL(\khat, \khat; \Bil^*)$ and $\eta \in \Bil^*$.
\begin{defn}
By a \emph{form solution} of~\eqref{Equation: coalg QSDE}
is meant a family
$\big\{ \lambda^{f,g}_t \big| f, g \in \Step ,t \in \Rplus\big\}$
in $\Bil^*$ satisfying
\begin{rlist}
\item
the map
$s \mapsto \big( \lambda^{f,g}_s \conv \varphi_{\fhat(s),\ghat(s)} \big) (b)$
\ is locally integrable;
\label{i}
\item
$\lambda^{f,g}_t (b) - e^{\la f,g\ra} \eta (b) =
\int^t_0 \, ds \,
\big( \lambda^{f,g}_s \conv \varphi_{\fhat(s),\ghat(s)} \big) (b)$
\label{ii}
\end{rlist}
for all $f,g \in \Step$, $t \in \Rplus$ and $b \in \Bil$.
\end{defn}
\begin{rems}
Let $f,g \in \Step$ and $b \in \Bil$.
By automatic strictness of bounded linear functionals on $\Bil$,
(\ref{i}) makes sense.
By (\ref{ii})
it follows that $\lambda^{f,g}_t(b)$ is continuous in $t$,
and so is locally bounded. Therefore, by the Banach-Steinhaus Theorem,
$\lambda^{f,g}_t$ is locally bounded in $t$ and
(\ref{ii}) therefore implies that (\ref{i}) refines to
\begin{itemize}
\item[(i)$'$]
the map $s \mapsto \lambda^{f,g}_s$ is continuous,
\end{itemize}
which in turn implies that (\ref{ii}) refines to
\begin{itemize}
\item[(ii)$'$]
$\lambda^{f,g}_t - e^{\la f,g\ra} \eta =
\int^t_0 \, ds \,
\lambda^{f,g}_s \conv \varphi_{\fhat(s),\ghat(s)}$,
\end{itemize}
the integrand being piecewise norm-continuous $\Real_+ \to \Bil^*$.
\end{rems}
The following automatic strictness property
is needed to establish uniqueness for form solutions.
\begin{lemma}
\label{lemma 5.1}
Let $\varphi\in SL(\khat, \khat; \Bil^*)$ and $\eta \in \Bil^*$.
Then every form solution
$\big\{ \lambda^{f,g}_t \big| f, g \in \Step ,t \in \Rplus\big\}$
of~\eqref{Equation: coalg QSDE}
is \emph{strict} in the sense that it satisfies
\begin{itemize}
\item[(ii)$\,\wt{}\ $]
$\ \widetilde{\lambda}^{f,g}_t - e^{\la f,g\ra} \widetilde{\eta} =
\int^t_0 \, ds \,
\lambda^{f,g}_s \convtilde \varphi_{\fhat(s),\ghat(s)}$
for all $f,g \in \Step$ and $t \in \Rplus$,
\end{itemize}
where
$\widetilde{\lambda}^{f,g}_t := \big( \lambda^{f,g}_t\big)\,\wt{}$.
\end{lemma}
\noindent
Note that the integrand in (ii)$\,\tilde{}$ is piecewise continuous
$\Rplus \to (\Biltilde)^*_\beta$.
\begin{proof}
Let
$\big\{ \lambda^{f,g}_t \big| f, g \in \Step ,t \in \Rplus\big\}$
be a form solution of~\eqref{Equation: coalg
QSDE}
and let $f,g \in \Step$ and $t \in \Rplus$.
Define bounded linear functionals
\[
\Phi :=
\int^t_0 \, ds \,
\lambda^{f,g}_s \conv \varphi_{\fhat(s),\ghat(s)} \text{ and }
\Psi :=
\int^t_0 \, ds \,
\lambda^{f,g}_s \convtilde \varphi_{\fhat(s),\ghat(s)}
\]
on $\Bil$ and $\Biltilde$ respectively. Note that each Riemann
approximant $\Psi_\mathcal{P}$ of $\Psi$ equals
$(\Phi_\mathcal{P})\,\wt{}$ where
$\Phi_\mathcal{P}$ is the corresponding Riemann approximant of $\Phi$.
The extension map $\Bil^* \to \Biltilde^*$ is (isometric and thus)
continuous therefore
\[
\Psi = \lim \Psi_\mathcal{P} =
(\lim \Psi_\mathcal{P})\,\widetilde{} = \Phi\,\widetilde{}.
\]
Since
$\Phi = \lambda^{f,g}_t - e^{\la f, g\ra} \eta$
it follows that
$\Psi = (\lambda^{f,g}_t)\,\widetilde{} - e^{\la f, g\ra}
\widetilde{\eta}$.
Thus
form solution is strict.
\end{proof}
With this we have uniqueness as well as existence for
form solutions.
\begin{thm}
\label{Theorem: SLQSDE}
Let $\varphi\in SL(\khat, \khat; \Bil^*)$ and $\eta \in \Bil^*$, for a
$C^*$-bialgebra $\Bil$. Then the quantum stochastic differential
equation~\eqref{Equation: coalg QSDE}
has a unique form solution.
\end{thm}
\begin{proof}
For each $c,d\in\kil$ let
$(p^{c,d}_t)_{t\geq 0}$ denote the norm-continuous one-parameter
semigroup
generated by $\varphi_{\chat, \dhat} \in \Bil^*$,
in the unitisation of the Banach algebra $(\Bil^*, \conv)$.
For $f,g\in\Step$ and $t\in\Rplus$,
the prescription
\[
\lambda^{f,g}_t := \eta \conv p^{c_0,d_0}_{t_1-t_0} \conv \cdots \conv
p^{c_{n},d_n}_{t_{n+1}-t_n}
\]
(in which $t_0 = 0$, $t_{n+1} = t$,
$\{t_1 < \cdots < t_n\}$ is the (possibly empty) set of points in $]0,t[$
where $f$ or $g$ is discontinuous and
$(c_i,d_i) = (f(t_i),g(t_i))$ for $i=0, \cdots , n$),
defines an element $\lambda^{f,g}_t$ of $\Bil^*$.
It is easily verified that the resulting family
$\big\{ \lambda^{f,g}_t \big| f, g \in \Step ,t \in \Rplus\big\}$
is a form solution
of~\eqref{Equation: coalg QSDE}.
Suppose now that $\mu$ is the difference of two form
solutions, and let $f,g\in\Step$ and $t\in\Rplus$.
Then Lemma~\ref{lemma 5.1} yields
the identity
\[
\big(\mu^{f,g}_t\big)\,\wt{} =
\int^t_0 \, ds \,
\mu^{f,g}_s \convtilde \varphi_{\fhat(s),\ghat(s)},
\]
which may be iterated. Estimating after repeated iteration
(and using the isometry $\Biltilde^*_\beta \cong \Bil^*$) we have
\[
\|\mu^{f,g}_t\| \leq
\frac{t^n}{n!} \sup_{s\in[0,t]} \|\mu^{f,g}_s\|
\max \big\{ \| \varphi_{\chat,\dhat} \| :c \in \Ran f , d \in \Ran g
\big\}^n
\]
which tends to $0$ as $n \to \infty$.
Thus $\mu = 0$, proving uniqueness.
\end{proof}
We now show how stronger forms of solution are obtained when the
coefficient of the quantum stochastic differential equation is a
bounded mapping rather than just a form.
Below the following natural inclusions are invoked:
\begin{align*}
&B\big(\Bil; B(\hil; \hil')\big) \cong
B\big( \ol{\hil'}, \hil; \Bil^* \big) \subset
SL\big(\hil', \hil; \Bil^*\big),
\\ &
\varphi \mapsto
\big( (\zeta,\eta) \mapsto \varphi_{\zeta,\eta} :=
\la \zeta, \varphi(\cdot)\eta \ra \big),
\end{align*}
for Hilbert spaces $\hil$ and $\hil'$.
Recall the notation for the solution of a
QS differential equation introduced above
equation~\eqref{QSDE}.
\begin{thm}
\label{Theorem: lphieta}
Let $\varphi \in CB \big( \Bil ; B(\khat) \big)$
and $\eta \in \Bil^*$, for a $C^*$-bialgebra $\Bil$. Set
\[
\widetilde{l}^{\varphi, \eta} :=
k^{\widetilde{\phi}, \widetilde{\eta}} \text{ and }
\ol{l}^{\varphi, \eta} := k^{\ol{\phi}, \ol{\eta}}
\text{ where } \phi :=
\Rmap \varphi.
\]
Thus $\widetilde{\phi}
\in \CBstrict\big(\Biltilde; \Bil \ottilde K(\khat)\big)$ and
$\ol{\phi}
=
\Rbar
\ol{\varphi}
\in \CBuw\big(\Bilol; \Bilol \otol B(\khat)\big)$.
\begin{alist}
\item
Abbreviating
$\ol{l}^{\varphi, \eta}$ to $\ol{l}$ and
$\wt{l}^{\varphi, \eta}$ to $\wt{l}$
we have,
for all $\varepsilon \in \Exps$ and $t\in\Rplus$,
\begin{rlist}
\item
$\ol{l}_{t,\varepsilon} \in
\CBuw\big(\Bilol; |\FFock\ra\big)$\tu{;}
\item
$\wt{l}_{t,\varepsilon} =
\ol{l}_{t,\varepsilon}|_{\Biltilde}$\tu{;}
\item
$\wt{l}_{t,\varepsilon} \in
\CBstrict\big(\Biltilde; |\FFock\ra\big)$.
\end{rlist}
\item
For all $f,g\in\Step$ and $t\in\Rplus$,
setting
\begin{equation}
\label{Equation: lambda and kappa}
\ol{\lambda}^{f,g}_t :=
\omega_{\varepsilon(f_{[0,t[}), \varepsilon(g_{[0,t[})}
\circ \ol{l}^{\varphi, \eta}_t
\text{ and }
\kappa^{f,g}_t :=
E^{\varepsilon(f_{[0,t[})} k_t^{\ol{\phi}}( \cdot )
E_{\varepsilon(g_{[0,t[})},
\end{equation}
\begin{rlist}
\item
$\big\{ \ol{\lambda}^{f,g}_t|_{\Bil}\, \big|\ f, g \in \Step ,t \in \Rplus\big\}$
is the unique form solution of~\eqref{Equation: coalg QSDE};
\item
$\ol{\lambda}^{f,g}_t = \ol{\eta} \circ \kappa^{f,g}_t$ and
$
\Rbar
\,\ol{\lambda}^{f,g}_t =
(\Rbar
\,\ol{\eta}) \circ \kappa^{f,g}_t$.
\end{rlist}
\end{alist}
\end{thm}
\begin{proof}
Fix $\ve\in\Exps$ and $t\geq 0$.
By linearity we may assume that $\varepsilon = \varepsilon(g)$.
(a)
(i)
The operator
$\ol{l}_{t,\varepsilon}$ is a norm-convergent sum,
in $CB\big(\Bilol; |\FFock\ra\big)$,
of terms of the form
$\Lambda^n_{t,\ve} \circ \big(\eta\convol \varphi^{\convol n}\big)$
($n\in\ZZ_+$),
and each map $\eta\convol \varphi^{\convol n}$ is $\sigma$-weakly
continuous.
Since $\CBuw\big(\Bilol; |\FFock\ra\big)$ is a norm-closed subspace of
$CB\big(\Bilol; |\FFock\ra\big)$ it remains only to show that the bounded
operator
$\Lambda^n_{t,\ve}:
B\big(\khat^{\ot n}\big) \to |\FFock\ra$
is $\sigma$-weakly continuous. By the Krein-Smulian Theorem
it suffices to prove this on bounded sets.
This follows
from the following identity for multiple QS integrals:
\[
\big\la \varepsilon(f), \Lambda^n_t(A)\varepsilon(g) \big\ra =
\int_{\Delta^n_t} d\mathbf{s}
\, \big\la \pi_{\fhat}(\mathbf{s}), A \pi_{\ghat}(\mathbf{s}) \big\ra
e^{\la f, g \ra}, \quad A\in B(\khat^{\ot n}),
\]
since the integrand is a step function on
$\Delta^n_t :=
\{ s \in \mathbb{R}^n : 0 \leq s_1 \leq \cdots \leq s_n \leq t \}$.
(ii)
Since
$\wt{l}_{t,\varepsilon}$
is a norm-convergent sum, in $CB(\Biltilde; |\FFock\ra)$,
of terms of the form
$\Lambda^n_{t,\ve} \circ (\eta\convtilde \varphi^{\convtilde n})$,
this follows from (i) and the identity
$
\eta\convol \varphi^{\convol n}|_{\Biltilde} =
\eta\convtilde \varphi^{\convtilde n}
$
($n\in\ZZ_+$).
(iii)
This follows from (i) and (ii) since, for any map
$\alpha \in \CBuw( \Bilol; | \FFock \ra )$,
$\wt{\alpha|_{\Bil}}
= \alpha|_{\Biltilde}$
(see~\eqref{diag CB}).
(b)
(i)
This follows from the identity
\[
\omega_{\varepsilon, \varepsilon'}\circ k^{\ol{\phi}, \ol{\eta}}_s
\circ \ol{\phi}_{\chat,\dhat}
=
\omega_{\varepsilon, \varepsilon'}\circ k^{\ol{\phi}, \ol{\eta}}_s
\, \convol \, \ol{\varphi}_{\chat,\dhat},
\quad
\varepsilon, \varepsilon'\in\Exps, c,d\in\kil, s\in\Rplus,
\]
where
$\ol{\phi}_{\chat,\dhat}:=
\big(\id_{\Bilol}\otol \omega_{\chat,\dhat}\big)\circ \ol{\phi}$.
(ii)
The first identity expresses the general relation between
$k^{\ol{\phi}, \ol{\eta}}$ and $k^{\ol{\phi}}$
(\cite{LSqsde}).
By (i), it follows from the proof of Theorem~\ref{Theorem: SLQSDE}
that
$\ol{\lambda}^{f,g}_t$ may be written in the form
\[
\ol{\eta} \, \convol \, \ol{p}^{c_0,d_0}_{t_1-t_0} \convol \cdots
\convol \ol{p}^{c_{n},d_n}_{t_{n+1}-t_n},
\]
where $\ol{p}^{c,d}_t$ denotes the normal extension of $p^{c,d}_t$.
Thus
$
\Rbar
\ol{\lambda}^{f,g}_t$ equals
\[
\Rbar
\ol{\eta} \circ \ol{P}^{c_0,d_0}_{t_1-t_0} \circ \cdots
\circ \ol{P}^{c_{n},d_n}_{t_{n+1}-t_n},
\]
where
$\ol{P}^{c,d}_t := \exp t\big(\phiol_{\chat,\dhat}\big) =
\Rbar
\ol{p}^{c,d}_t$.
(ii)
therefore now follows from the semigroup representation of the standard
QS cocycle $k^{\ol{\phi}}$ (\cite{LWjfa}).
\end{proof}
\begin{notn}
Setting $l^{\varphi ,\eta}_{t, \varepsilon} =
\ol{l}^{\varphi ,\eta}_{t, \varepsilon}|_{\Bil}$
($t\in\Rplus, \varepsilon\in\Exps$)
defines a process
$l^{\varphi ,\eta} \in \Procstar(\Bil)$,
which we denote by $l^{\varphi}$ when $\Bil$ is counital and
$\eta = \counit$.
This extends the notation introduced in~\cite{QSCC2} for the unital case.
\end{notn}
\begin{rems}
(i)
In view of the identity
\[
\big( k_s^{\phiol,\ol{\eta}}\circ \phi \big)_{\ve(f), \fhat(s)} =
\ol{l}_{s,\ve(f)}^{\varphi,\eta} \convol \varphiol_{\fhat(s)},
\]
$\ol{l}^{\varphi,\eta}$ satisfies
\[
\ol{l}_t = \iota_{\FFock} \circ \ol{\eta} +
\int_0^t \ol{l}_s \convol \varphiol \, d\Lambda(s), \quad t\in\Real_+.
\]
In this sense, $l^{\varphi,\eta}$ is a
strong solution of~\eqref{Equation: coalg QSDE}.
(ii)
The proper hypothesis for the theorem above is
$\varphi \in L (\khat ; CB ( \Bil ; | \khat \ra ))$,
in which case,
$\phi \in
L(\khat;
\CBstrict(\Bil; \Adjointable( \Bil; \Bil \ot | \khat \ra )))$
and
$
\phi_{\chat} =
\Rmap \varphi_{\chat}
$ ($c \in \kil$).
However, we have no need of this generality here.
\item
\end{rems}
\begin{rem}
Only the coalgebraic structure of $\Bil$ has been used so far,
not its algebraic structure.
\end{rem}
We end this section by noting some correspondence between convolution
processes and associated standard processes.
Recall the notation for QSDE solutions
introduced above equation~\eqref{Equation: weak standard cocycle}.
\begin{propn}
\label{Proposition: XX}
Let $l = l^\varphi$ and $\ol{l} = \ol{l}^\varphi$,
where $\varphi\in CB(\Bil; B(\khat))$
for a counital $\Cstar$-bialgebra $\Bil$, and set
$k = k^{\ol{\phi}}$ where $\phi := \Rmap \varphi$.
Then
\begin{alist}
\item
$\ol{l}$ is unital if and only if $k$ is.
\item
$l$ is completely bounded \tu{(}respectively, completely positive
or *-homomorphic\tu{)} if and only if $k$ is, in which
case
\[
k_t =
\Rbar
\ol{l}_t, \quad
\ol{l}_t =
\Ebar
k_t \text{ and }
\|l_t\|_\cb = \|k_t\|_\cb, \quad t\in\Rplus.
\]
\end{alist}
\end{propn}
\begin{proof}
In the notations~\eqref{Equation: lambda and kappa},
Theorem~\ref{Theorem: lphieta}(b)(ii) implies that,
\[
\ol{\lambda}^{f,g}_t =
\Ebar
\kappa^{f,g}_t \text{ and }
\kappa^{f,g}_t =
\Rbar
\ol{\lambda}^{f,g}_t,
\quad f,g\in\Step, t\in\Rplus.
\]
Thus (a) follows from the unitality of the maps
$\counitol$ and $\coproductol$.
Moreover, if $k$ is completely bounded then, since
\[
\omega_{\ve,\ve'}
\circ \ol{l}^{\varphi, \eta}_t =
\ol{\lambda}^{f,g}_t =
\Ebar
\kappa^{f,g}_t =
\omega_{\ve,\ve'}
\circ
\Ebar
k_t,
\]
where $\ve = \varepsilon(f_{[0,t[}$ and $\ve' = \varepsilon(f'_{[0,t[}$,
for all $f,f'\in\Step$ and $t\in\Rplus$, it follows that
$l_t =
\Ebar
k_t$
($t\in\Rplus$), in particular $l$ is completely bounded.
Conversely, if $l$ is completely bounded then
$\ol{l_t} = \ol{l}_t$ ($t\in\Rplus$) and
\[
\big( \id_{\Bilol}\otol\omega_{\ve,\ve'} \big) \circ
\Rbar
\ol{l}_t
=
\Rbar
\ol{\lambda}^{f,f'}_t =
\kappa^{f,f'}_t =
\big( \id_{\Bilol}\otol\omega_{\ve,\ve'} \big) \circ
k_t
\]
for all $f,f'\in\Step$ and $t\in\Rplus$, so
$k_t =
\Rbar
\ol{l}_t$ ($t\in\Rplus$), therefore
$k$ is completely bounded.
The rest follows from the fact that $\coproductol$ and
$\counitol \otol \id_{B(\FFock)}$ are *-homomorphisms.
\end{proof}
\section{Quantum stochastic convolution cocycles}
\label{Section: Cocycles}
For this section \emph{we fix a counital $C^*$-bialgebra} $\Bil$.
\begin{defn}
A family
$
\big\{ \lambda^{f,g}_t
\big| f, g \in \Step ,t \in \Rplus\big\}$
in $\Bil^*$ is a \emph{form quantum stochatic cocycle}
on $\Bil$ if it satisfies
\[
\lambda^{f,g}_0 = \counit, \quad
\lambda^{f,g}_{s+t} =
\lambda^{f,g}_s \conv
\lambda^{S^*_sf,S^*_sg}_t, \quad
f,g \in \Step, s,t \in \Rplus,
\]
where
$(S_t)_{t\geq 0}$
is the isometric shift semigroup on $L^2(\Rplus; \kil)$.
\end{defn}
Note that for such a cocycle
\[
p_t^{c,d}:= \lambda_t^{c_{[0,t[}, d_{[0,t[}}
\]
defines one-parameter semigroups $\{p^{c,d}\}_{c,d\in\kil}$
in the unital Banach algebra $(\Bil^*,\conv)$ which we refer to as the
\emph{associated convolution semigroups} of the cocycle. The cocycle is
said to be \emph{Markov-regular} if each of its associated semigroups is
norm-continuous.
\begin{defn}
A process $l\in \Procstar(\Bil)$ is a
(\emph{weak}) \emph{QS convolution cocycle on} $\Bil$
if its associated family
$\{
\omega_{\varepsilon(f_{[0,t[}), \varepsilon(f'_{[0,t[})} \circ l_t |
\ f,f'\in\Step, t\in\Rplus
\}$
is a form QS cocycle on $\Bil$.
\end{defn}
\begin{rems}
(i)
Let $l\in\Procstar(\Bil)$ be a completely bounded QS convolution cocycle on
$\Bil$.
Then $l$ is a QS convolution cocycle in the full sense:
\[
l_{s+t} = l_s \conv \big( \sigma_s \circ l_t \big), \quad
l_0 = \iota_\FFock \circ \counit, \quad
s,t\in\Rplus,
\]
where $(\sigma_s)_{s\geq 0}$ is the injective *-homomorphic
semigroup of right shifts on $B(\FFock)$ and the identification
\[
B(\FFock) =
B(\FFock_{[0,s[})
\otol
\sigma_s\big( B(\FFock)\big)
\]
is invoked.
(ii)
It follows from the proof of Theorem~\ref{Theorem: SLQSDE} that,
for $\varphi \in SL\big(\khat,\khat; \Bil^* \big)$,
the unique form solution of the QS differential equation
\begin{equation}
\label{Equation: bialg QSDE}
dl_t = l_t \conv d\Lambda_\varphi (t), \quad
l_0 = \iota_{\FFock} \circ \counit ,
\end{equation}
is a Markov-regular weak QS convolution cocycle on $\Bil$.
(iii)
Form-cocycles may equally be defined on $\Biltilde$ and
$\Bilol$ with the requirement of strictness/normality,
and $\counit$ replaced by $\counittilde$, respectively $\counitol$.
From the correspondence~\eqref{diag CB} it follows that any one of these
uniquely determines the others.
\end{rems}
Our essential strategy for analysing QS convolution cocycles is
to work in the universal enveloping von Neumann bialgebra $\Bilol$
and, by transferring between convolution and standard QS cocycles
using the maps $\Rbar$ and $\Ebar$,
to apply the theory developed in
[$\text{LW}_{1-4}$], and~\cite{LSqsde}.
We first establish a converse to Remark (ii) above.
\begin{propn}
\label{Proposition: YY}
Let $l$ be a Markov-regular, completely positive, contractive quantum
stochastic convolution cocycle on $\Bil$.
Then there is a unique map $\varphi \in CB(\Bil; B(\khat))$ such
that $l = l^\varphi$.
\end{propn}
\begin{proof}
Set $k := \big(
\Rbar
\ol{l}_t \big)_{t\geq 0}$,
where $\ol{l}_t := \ol{l_t}$ ($t\geq 0$).
Then $k$ is a standard quantum stochastic cocycle on $\Bilol$ which is
Markov-regular, completely positive, contractive and normal.
Therefore, by Theorem 5.10 of~\cite{LWjfa} and Theorem 5.3 of~\cite{LWptrf},
$k$ has a stochastic generator
$\ol{\phi} \in \CBuw\big(\Bilol; \Bilol \otol B(\khat) \big)$,
moreover for $c, d \in \kil$, its associated semigroup $P^{c,d}$ has
generator
$(\id_{\Bilol}\otol \omega_{\chat,\dhat}) \circ \ol{\phi}$.
Set
$\varphiol :=
\Ebar
\ol{\phi} \in
\CBuw\big(\Bilol; B(\khat) \big)$.
Since $\ol{l}_t =
\Ebar
k_t$, the associated
convolution semigroup $p^{c,d}$ of $\ol{l}$ has generating functional
\[
\counitol \circ
\big( \id_{\Bilol} \otol \omega_{\chat,\dhat}\big) \circ \ol{\phi}
=
\omega_{\chat,\dhat} \circ \ol{\varphi}
\]
which equals the generating functional of
the associated convolution semigroup of the QS convolution cocycle
$\ol{l}^\varphi$.
It follows that $\ol{l} = \ol{l}^\varphi$
where $\varphi:= \varphiol|_\Bil$ and so $l = l^\varphi$.
\end{proof}
We refer to $\varphi$ as the \emph{stochastic generator} of the
QS convolution cocycle $l$.
The proof of the next result now proceeds similarly
to those of Theorems 5.1 and 6.1 in~\cite{QSCC2}.
\begin{thm}
\label{Theorem: cocycle}
Let $l$ be Markov-regular quantum stochastic convolution cocycle on $\Bil$.
Then the following equivalences hold:
\begin{alist}
\item
\begin{rlist}
\item
$l$ is completely positive and contractive;
\item
there is $\psi\in CP(\Bil; B(\khat))$
and $\zeta\in\khat$ such that $l = l^\varphi$ where
\begin{equation}
\label{Equation: phi = psi +}
\varphi = \psi - \counit(\cdot)
\big( \dQS + |\zeta\ra\la e_0| + |e_0\ra\la \zeta| \big)
\end{equation}
and $\varphitilde (1) \leq 0$.
\end{rlist}
In this case, $l$ is preunital if and only if $\varphitilde(1) = 0$.
\item
\begin{rlist}
\item
$l$ is completely positive and preunital;
\item
there is a nondegenerate *-representation $(\rho, \Kil)$ of $\Bil$
\tu{(}as $C^*$-algebra\tu{)}, an
isometry $D\in B(\kil;\Kil)$ and vector $\xi\in\Kil$ such that
$l = l^\varphi$ where
\begin{equation}
\label{Equation: instead+}
\varphi =
\begin{bmatrix} \la \xi | \\ D^* \end{bmatrix}
\nu( \cdot )
\begin{bmatrix} | \xi \ra & D \end{bmatrix}
\text{ for }
\nu := \rho - \iota_\Kil \circ \counit,
\end{equation}
\end{rlist}
\item
\begin{rlist}
\item
$l$ is *-homomorphic;
\item
$l = l^\theta$ where $\theta$ is an $\counit$-structure map;
\item
there is a *-homomorphism $\pi: \Bil \to B(\kil)$
and vector $c\in\kil$ such that
\begin{equation}
\label{Equation: instead++}
\theta =
\begin{bmatrix} \la c | \\ I_\kil \end{bmatrix}
\nu( \cdot )
\begin{bmatrix} | c \ra & I_\kil \end{bmatrix}
\text{ where }
\nu := \pi - \iota_\kil \circ \counit,
\end{equation}
\end{rlist}
\noindent
In this case, the cocycle $l$ is nondegenerate if and only if
the *-representation
$\pi$ is.
\end{alist}
\end{thm}
\begin{proof}
In case (i) of (a), (b) and (c) we let $\varphi$ be the stochastic
generator of $l$, let
$\ol{l} = \ol{l}^\varphi = \big( \ol{l_t} \big)_{t\geq 0}$, and set
$k = k^{\ol{\phi}}$ where
$\ol{\phi} =
\Rbar
\varphiol \in
\CBuw\big( \Bilol; \Bilol \otol B(\khat) \big)$.
Thus $k$ is a Markov-regular standard QS cocycle on $\Bilol$ and
$\varphiol =
\Ebar
\ol{\phi}$.
(a)
If (i) holds then $k$ is completely positive and contractive,
by Proposition~\ref{Proposition: XX},
and normal. Therefore, by Theorem 5.10 of~\cite{LWjfa}, there is a map
$\Phi \in \CPuw( \Bilol; \Bilol \otol B(\khat))$ and operator
$Z \in \Bilol \otol \la \khat |$ such that
\begin{equation}
\label{Equation: phibar and Z}
\ol{\phi}(x) = \Phi(x) -
\big( x \ot \dQS + Z^* (x \ot \la e_0 |) + (x \ot | e_0 \ra) Z \big)
\quad
(x \in \Bilol)
\end{equation}
and $\ol{\phi}(1)\leq 0$.
It follows that $\varphiol(1)\leq 0$ and
\[
\varphiol =
\Psi -
\counitol(\cdot)
\big( \dQS + | \zeta \ra \la e_0| + | e_0 \ra \la \zeta | \big)
\]
where
$\Psi =
\Ebar
\Phi$
and
$\la \zeta | =
\big( \counitol \otol \id_{\la \khat |} \big)(Z)$.
Thus (ii) holds with $\psi= \Psi|_{\Bil}$,
moreover if $l$ is preunital then $\ol{l}$ is unital and so $k$ is
too, therefore $\ol{\phi}(1)=0$ so $\varphiol(1)=0$ also.
Conversely, if (ii) holds then, taking normal extensions,
\[
\varphiol =
\ol{\psi} -
\counitol(\cdot)
\big( \dQS + | \zeta \ra \la e_0| + | e_0 \ra \la \zeta | \big)
\]
and so~\eqref{Equation: phibar and Z} holds with
$\Phi =
\Rbar
\ol{\psi}$ and
$Z = 1_{\Bilol} \ot \la \zeta |$. Therefore,
by~\cite{LWptrf} Theorem 5.3,
$k$ is completely positive and contractive and so, by
Proposition~\ref{Proposition: XX}, $l$ is too. Similarly, if
$\varphiol(1)=0$
then
$\ol{\phi}(1)=0$ so $k$ is unital,
thus $\ol{l}$ is too,
and therefore $l$ is preunital. This proves (a).
(b)
If (i) holds then, choosing $\psi$ and $\zeta$ as in (a), let
\[
\begin{bmatrix} \la \xi | \\ D^* \end{bmatrix}
\rho( \cdot )
\begin{bmatrix} | \xi \ra & D \end{bmatrix}
\]
be a minimal Stinespring decomposition of $\psi$.
Thus $(\rho, \Kil)$ is a nondegenerate representation of $\Bil$ and
\[
\big( \dQS + |\zeta\ra\la e_0| + |e_0\ra\la \zeta| \big)
= \wt{\psi}(1) =
\begin{bmatrix}
\|\xi\|^2 & \la \xi | D \\
D^* | \xi \ra & D^*D
\end{bmatrix}
\]
so $D$ is isometric and (ii) holds. Conversely, suppose that (ii) holds
then $\varphi$ has the form~\eqref{Equation: phi = psi +} with
$\zeta = \binom{\frac{1}{2}\|\xi\|^2}{D^*\xi}$ and $\varphiol(1)=0$ so (i)
holds, by (a). This proves (b).
(c)
If (i) holds then $k$ is *-homomorphic so,
by~\cite{LWptrf} Proposition 6.3,
$\ol{\phi}$ is a structure map:
\begin{equation}
\label{Equation: (b)}
\ol{\phi} (x^*y) =
\ol{\phi}(x)^* \iota(y) +
\iota(x)^* \ol{\phi}(y) +
\ol{\phi} (x)^* (1_{\Bilol} \ot \dQS) \ol{\phi} (y)
\quad (x,y\in \Bilol)
\end{equation}
where $\iota$ denotes the ampliation map $x \mapsto x \ot I_\FFock$.
Since
$\counitol \otol \id_{B(\khat)}$ is a unital *-homomorphism this implies
that
\begin{equation}
\label{Equation: varphibar structure}
\varphiol (x^*y) =
\varphiol(x)^* \counitol (y) + \counitol(x)^* \varphiol(y) +
\varphiol (x)^* \dQS \,\varphiol (y)
\quad
(x,y\in\Bilol)
\end{equation}
and so (ii) holds.
Suppose conversely that (ii) holds. By separate $\sigma$-weak continuity
of multiplication in $\Bilol$ it follows
that~\eqref{Equation: varphibar structure} holds and a brief calculation
confirms the identity
\[
\Omega(u^*v) =
\big( \id_{\Bilol} \otol \counitol \big)(u)^*
\Omega(v) +
\Omega(u)^*
\big( \id_{\Bilol} \otol \counitol \big)(v) +
\Omega(u)^*
\big( 1_{\Bilol} \ot \dQS \big)
\Omega(v),
\]
where $\Omega := \big( \id_{\Bilol} \otol \varphiol \big)$,
for simple tensors $u,v\in\Bilol\otul\Bilol$. Since both sides are
separately $\sigma$-weakly continuous the identity is valid for all $u$
and $v$ in $\Bilol \otol \Bilol$.
Substituting in $u = \coproductol x$ and $v = \coproductol y$ we see that
$\ol{\phi}$ satisfies~\eqref{Equation: (b)}.
Therefore, by Corollary 4.2 of
by~\cite{LWhomomorphic},
$k$ is *-homomorphic thus, by
Proposition~\ref{Proposition: XX}, $l$ is too
and therefore (ii) holds. The equivalence of
(ii) and (iii) is the general form of an $\counit$-structure map
(see~\eqref{Equation: block}).
In view of (a), the last part is easily seen from the representation (iii).
This completes the proof.
\end{proof}
\begin{rem}
The proper hypothesis for Parts (a) and (b) above is that $\Bil$ be a
(multiplier) $\Cstar$-\emph{hyperbialgebra}, since the multiplicative
property of $\coproduct$ is not used in their proof.
The above result therefore generalises Theorems 5.1 and 6.2
of~\cite{QSCC2} to the locally compact category.
\end{rem}
\section{Quantum L\'evy processes on multiplier $C^*$-bialgebras}
\label{Section: Levy}
In this section we extend the definition of weak quantum L\'evy process to
multiplier $C^*$-bialgebras and establish a reconstruction theorem which is
analogous to Sch\"urmann's for purely algebraic bialgebras (\cite{Schurmann})
and extends ours, proved for unital $C^*$-bialgebras in \cite{QSCC2}.
Throughout this section $\Bil$ \emph{denotes a fixed counital $C^*$-bialgebra}.
\begin{defn} \label{wLp}
A \emph{weak quantum L\'{e}vy process} on $\Bil$ over a $C^*$-algebra-with-a-state $(\Al , \omega)$
is a family $\big(j_{s,t}\! :\Bil \to \Altilde\big)_{0 \leq s \leq t}$ of nondegenerate
*-homomorphisms for which the functionals $\lambda_{s,t}:= \omega \circ j_{s,t}$
satisfy the following conditions, for $0\leq r \leq s \leq t$:
\begin{rlist}
\item
$\lambda_{r,t} = \lambda_{r,s} \star \lambda_{s,t}$\tu{;}
\item
$\lambda_{t,t} = \Cou $\tu{;}
\item
$\lambda_{s,t} = \lambda_{0,t-s}$\tu{;}
\item
\[
\wt{\omega} \left( \prod^n_{i=1} j_{s_i,t_i} (x_i) \right) = \prod^n_{i=1}
\lambda_{s_i,t_i} (x_i)
\]
whenever $n \in \bn$, $x_1, \ldots, x_n \in\Bil$ and
the intervals $[s_1,t_1[,\ldots ,[s_n, t_n[$ are disjoint\tu{;}
\item
$\lambda_{0,t} \to \Cou$ pointwise as $t \to 0$.
\end{rlist}
A weak quantum L\'{e}vy process is called
\emph{Markov-regular} if
$\lambda_{0,t} \to\Cou$ in norm, as $t \to 0$.
\end{defn}
\begin{rems}
In the case of unital $C^*$-bialgebras we did not insist
that the *-algebra $\Al$ was a $C^*$-algebra.
As in the unital case, we refer to the weakly continuous convolution semigroup
$(\lambda_t := \lambda_{0,t})_{t \geq 0}$ on $\Bil$ as the
\emph{one-dimensional distribution} of the process, and call the process
\emph{Markov-regular} if this is norm-continuous, in which case we refer to the
convolution semigroup generator as the \emph{generating functional} of the process (\cite{discrete}).
Moreover, as in the unital case, we call two weak quantum L\'evy processes \emph{equivalent}
if their one-dimensional distributions coincide.
\end{rems}
\begin{comment}
\begin{rems}
Note that the above definition of a weak quantum L\'evy process, in
contrast to the definition
of a quantum L\'evy process on an algebraic *-bialgebra, does not yield a
recipe for expressing the joint moments of the process increments
corresponding to overlapping time intervals, such as
\[
\omega(j_{r,t}(x) j_{s,t} (y)) \text{ where } r,s < t.
\]
To achieve the latter, one would have to formulate the weak convolution
increment property (wQLP\ref{wQLPi}) in greater generality and assume
certain commutation relations between the increments corresponding to
disjoint time intervals.
For other investigations of the notion of independence in noncommutative
probability, in the absence of commutation relations being imposed, we
refer to the recent paper \cite{hkk}.
\end{rems}
\end{comment}
The generating functional $\gamma$ of a Markov-regular weak quantum L\'evy process,
being the generator of a norm-continuous convolution semigroup of states,
is \emph{real},
that is $\gamma = \gamma^\dagger$ where $\gamma^\dagger(a):= \overline{\gamma(a^*)}$,
\emph{conditionally positive},
that is positive on the ideal $\Ker \counit$,
and its strict extension satisfies
$\gammatilde (1) = 0$.
Note that if $l\in \Procstar(\Bil)$ is a QS convolution cocycle on $\Bil$,
with noise dimension space $\kil$,
which is *-homomorphic and preunital then, setting
$\Al := K(\FFock)$, $\omega:= \omega_{\ve(0)}$,
and $j_{s,t} := \sigma_s \circ l_{t-s}$ for all $0\leq s\leq t$,
we obtain a weak quantum L\'evy process on $\Bil$,
called a \emph{Fock space quantum L\'evy process}, which is Markov-regular if $l$ is.
Our goal now is to establish a converse, in other words to extend the
reconstruction theorem of~\cite{QSCC2} to the nonunital case.
We give an elementary self-contained proof, independent of
automatic implementability/complete boundedness properties of
$\chi$-structure maps.
Recall Lemma~\ref{condpos}.
\begin{thm} \label{Crecon}
Let $\gamma\in\Bil^*$ be real,
conditionally positive and satisfy $\wt{\gamma}(1)=0$.
Then there is a \tu{(}Markov-regular\tu{)}
Fock space quantum L\'{e}vy process with generating functional $\gamma$.
\end{thm}
\begin
{proof}
By Theorem~\ref{Theorem: cocycle} it suffices to show that there is a
Hilbert space $\kil$ and an
$\counit$-structure map $\varphi: \Bil \to B(\kilhat)$ of the form
$\left[\begin{smallmatrix}\gamma & *\\ * & * \end{smallmatrix}\right]$
satisfying $\varphitilde (1) = 0$.
Set $\gammatilde_0 := \gammatilde|_{\Ker \counittilde}$ and let
$\psi$ be the map
$\Bil \to \Biltilde$, $b \mapsto b - \counit (b)1$.
By Theorem~\ref{Theorem 1.1} and Lemma~\ref{condpos},
$\gammatilde$ is real and $\gammatilde_0$ is positive.
Since also $\gammatilde (1) = 0$,
\begin{equation} \label{7.2i}
q:(a,b) \mapsto
\gamma(a^*b) - \gamma(a)^*\counit(b) - \counit(a)^*\gamma(b) =
\gammatilde_0 \big(\psi(a)^*\psi(b)\big)
\end{equation}
defines a nonnegative sesquilinear form on $\Bil$.
Let $\kil$ and $d: \Bil \to \kil$ be respectively
the Hilbert space and induced map obtained by
quotienting $\Bil$ by the null space of $q$ and completing,
so that
\[
\ol{d(\Bil)} = \kil \text{ and } \la d(a), d(b) \ra = q(a,b), \quad a,b \in \Bil,
\]
and let $\delta$ be the linear map
$\Bil \to | \kil \ra$, $b\mapsto |d(b)\ra$.
Then, by the complete boundedness of $\gammatilde$ and $\psi$,
\[
\big\| \delta^{(n)}(A) u \big\|^2 =
\big\la u,
(\gammatilde_0)^{(n)}\big(\psi^{(n)}(A)^*\psi^{(n)}(A) \big) u \big\ra
\leq
\|\gammatilde\|_{\cb}\|\psi\|^2_{\cb} \|A\|^2\|u\|^2,
\]
for all $n\in\mathbb{N}$, $A\in M_n(\Bil)$ and $u \in \Comp^n$,
so $\delta$ is completely bounded and we have
\begin{equation}
\label{7.A}
\delta(a)^*\delta(b) =
\gamma(a^*b) - \gamma(a)^*\counit(b) - \counit(a)^*\gamma(b),
\quad a,b\in\Bil.
\end{equation}
Now
\begin{align*}
\big\| d(ab) - \counit(b)d(a) \big\|^2 &=
\gammatilde_0 \big( \psi(b)^* a^*a \psi(b) \big)
\\ &\leq
\| a \|^2 \,\gammatilde_0\big( \psi(b)^*\psi(b)\big) = \| a \|^2 \| d(b) \|^2,
\quad a,b \in \Bil,
\end{align*}
so there are bounded operators $\pi (a)$ on $\kil$ satisfying
\begin{equation} \label{7.2ii}
\pi(a)d(b) = d(ab) - \counit(b)d(a), \quad a,b\in\Bil.
\end{equation}
Using the density of $d(\Bil)$ it is straightforward to verify that
the map $a\mapsto \pi(a)$ defines a *-representation of $\Bil$ on $\kil$.
From~\eqref{7.2ii},
$\delta$ is a $(\pi,\counit)$-derivation and so,
from~\eqref{7.A},
$\varphi :=
\left[ \begin{smallmatrix}
\gamma & \delta^\dagger \\ \delta & \pi - \iota_{\kil}
\end{smallmatrix} \right]$
defines an $\counit$-structure map $\Bil \to B(\kilhat)$,
and therefore it only remains to prove that $\varphitilde(1)=0$.
Since $\gammatilde(1)=0$, this follows from the identities
\[
\delta(a)^*\delta(b) =
\gamma\big(a^*b - \counit(a)^*b - a^*\counit(b)\big)
\text{ and }
\pi(a)\delta(b) = \delta(ab) - \delta(a)\counit(b),
\quad a,b\in\Bil,
\]
and the density of $\bigcup\{\Ran \delta(b): b\in\Bil\} = d(\Bil)$ in $\kil$.
\end{proof}
\begin{comment}
\begin{proof}
The functional $\wt{\gamma}\in \Biltilde^*_{\beta}$ vanishes at $1_{\Biltilde}$, is real by part
(c) of Theorem \ref{Theorem 1.1} and is positive on the kernel of $\wt{\Cou}$ by Lemma
\ref{condpos}. The GNS-style construction for $\wt{\gamma}|_{\Ker(\wt{\Cou})}$ as in~\cite{QSCC2}
provides a Hilbert space $\kil$, a unital representation $\pi:\Biltilde\to B(\kil)$ and a
$(\pi,\Cou)$-derivation $\delta:\Biltilde\to |\kil\ra$ satisfying the equation ($x,y \in \Bil$):
\[
\delta (x)^* \delta (y) =
\gamma (x^*y) - \gamma (x)^*\epsilon (y) - \epsilon (x)^* \gamma (y).
\]
Theorem A.6 in the Appendix of~\cite{QSCC2} therefore implies that the map from $\Biltilde$ to
$B(\kilhat)$, with block matrix form
\[\begin{bmatrix} \gamma & \delta^{\dagger} \\ \delta & \pi - \iota_{\kil} \circ \Cou \end{bmatrix}
\]
is completely bounded. Denote its restriction to $\Bil$ by $\varphi$ Setting $l = l^\varphi$,
Theorem \ref{Theorem: cocycle} (c) implies that the Markov-regular weak QS convolution cocycle $l$
is preunital and
*-homomorphic. Since $\varphi^0_0 =\gamma$ the result follows.
\end{proof}
\end{comment}
This has two significant consequences.
\begin{cor}
Every Markov-regular weak quantum L\'{e}vy process is equivalent to a Fock
space quantum L\'{e}vy process.
\end{cor}
The second consequence uses the deeper fact that every
$\counit$-structure map is implemented (see Theorem~\ref{established}).
\begin{thm}
\label{significant}
Let $\gamma \in \Bil^*$. Then the following are equivalent:
\begin{rlist}
\item
$\gamma$ is the generating functional
of a \tu{(}necessarily norm-continuous\tu{)}
convolution semigroup of states on $\Bil$\tu{;}
\item
$\gamma$ is real, conditionally positive and
satisfies $\gammatilde(1)=0$\tu{;}
\item
There is a nondegenerate representation $(\pi,\hil)$ of $\Bil$
and vector $\eta\in\hil$ such that
$\gamma = \omega_\eta \circ (\pi - \iota_\hil \circ \counit)$.
\end{rlist}
\end{thm}
In~\cite{QSCC2} we also introduced a stronger notion of
\emph{product system quantum L\'evy
processes} on a unital and counital $C^*$-bialgebra $\Bil$
and established the following two facts:
each Fock space quantum L\'evy process on $\Bil$ is
in particular a product system quantum L\'evy process and
each product system quantum L\'evy process determines in a natural way
a weak quantum
L\'evy process on $\Bil$ with the same finite-dimensional distribution.
The definition of a product
system quantum L\'evy process extends naturally to the nonunital case,
with the assumption of unitality of $^*$-homomorphisms constituting the process
replaced by nondegeneracy, and the proofs of the above two facts remain valid.
\section{Approximation by discrete evolutions}
\label{Section: Approximation}
\emph{For this section $\Bil$ is again a fixed counital $C^*$-bialgebra}.
We show that any Markov-regular, completely positive,
contractive quantum stochastic convolution cocycle on $\Bil$
may be approximated in a strong sense by discrete
completely positive evolutions, and that the discrete evolutions
may be chosen to be preunital and/or *-homomorphic, if the cocycle is.
This extends and strengthens results of~\cite{FrS} for
quantum L\'evy processes on compact quantum semigroups.
We first note that Belton's condition for discrete approximation of
standard Markov-regular QS cocycles (\cite{Bel}) readily translates to
the convolution context using the techniques of this paper. We denote
by $\toy_n^{(h)}$
($h>0, n\in\Nat$)
the injective *-homomorphism
\[
B\big(\khat^{\ot n}\big) = B(\khat)^{\otol n}
\to
B\big(\FFock_{[0,hn[}\big) \ot
I_{\FFock_{[hn,\infty[}} =
\Big(\, \ol{\bigotimes}_{j=1}^{\, n}
\,B\big(\FFock_{[(j-1)h,jh[}\big)\,\Big)
\ot
I_{\FFock_{[hn,\infty[}}
\]
arising from the discretisation of Fock space. Thus
\[
\toy_n^{(h)}:
A \mapsto D_n^{(h)} A D_n^{(h)*} \ot I_{\FFock_{[hn,\infty[}}
\]
where
\begin{align*}
&
D_n^{(h)} :=
\bigotimes_{j=1}^n D_{n,j}^{(h)},
\text{ for the isometries }
\\ &
D_{n,j}^{(h)}:
\khat \mapsto \FFock_{[(j-1)h,jh[}, \
\binom{z}{c} \mapsto
\big(z, h^{-1/2}c_{[(j-1)h, jh[}, 0, 0, \cdots \big).
\end{align*}
Also write
$\toy_{n,\ve}^{(h)}$ for the complete contraction
\[
\toy_n^{(h)}(\cdot ) |\ve\ra: B(\kilhat^{\ot n}) \to |\FFock\ra,
\quad
h > 0 , n\in\mathbb{N}, \ve\in\Exps.
\]
For a map $\Psi\in CB\big( \Vil; \Vil\otM B(\khat)\big)$,
where $\Vil$ is a concrete operator space, its \emph{composition iterates}
are defined by
\[
\Psi_0 := \id_\Vil, \quad
\Psi_n:= \big( \Psi_{n-1} \otM \id_{B(\khat)} \big) \circ \Psi
\in CB\big( \Vil; \Vil\otM B(\khat^{\ot n})\big),
\quad n\in\Nat.
\]
Similarly, for a map $\psi \in CB\big(\Bil; B(\khat)\big)$,
its convolution iterates are defined by
\[
\psi_0 := \counit, \quad \psi_n = \psi_{n-1} \conv \psi \in
CB\big( \Bil; B(\khat^{\ot n})\big)
\quad (n\in\Nat).
\]
As usual we are viewing
$B(\khat^{\ot n})$ as the multiplier algebra of
$K(\khat^{\ot n})$
here, and automatic strictness is being invoked to ensure meaning for
$\psi_{n-1} \conv \psi$.
In short,
$\psi_n = \psi^{\conv n}$ for $n\in\ZZ_+$.
Note the easy compatibility between the two methods of iteration:
for $\psi \in CB\big(\Bil; B(\khat)\big)$,
\begin{equation} \label{Equation: ease}
\Rmap \psi_n =
\Psi_n \text{ where }
\Psi :=
\Rmap \psi.
\end{equation}
We need the following block matrices, on a Hilbert space
of the form $\Hilhat$:
\begin{equation}
\label{Equation: Dh}
\Dh
:= \begin{bmatrix}
h^{-1/2} & \\ & I_\Hil
\end{bmatrix},
\quad h > 0;
\end{equation}
we set $\cDh$ equal to the conjugation by $\Dh$:
$X \mapsto \Dh X \Dh$.
Conjugation provides
the correct scaling for quantum random-walk approximation (\cite{LPrw}).
\begin{thm}
\label{Theorem: 1}
Let $\varphi\in CB(\Bil; B(\khat))$. Suppose that there is a family of maps
$(\psi^{(h)})_{0<h\leq C}$
in $CB(\Bil; B(\khat))$ for some $C>0$, satisfying
\[
\cDh \circ
\big( \psi^{(h)} - \iota_{\khat} \circ \counit \big)
\to \varphi
\text{ in } CB(\Bil; B(\khat)) \text{ as } h\to 0^+.
\]
Then the convolution iterates
$\{
\psi^{(h)}_n: n\in \ZZ_+, 0 < h < C
\}$
satisfy
\begin{equation}
\label{Equation: rw approx}
\sup_{t\in[0,T]}
\Big\|
\toy^{(h)}_{[t/h],\ve} \circ
\psi^{(h)}_{[t/h]} - l^{\varphi}_{t,\ve}
\Big\|_{CB(\Bil; |\FFock\ra)}
\to 0 \text{ as } h \to 0^+,
\end{equation}
for all $T\in\Rplus$ and $\varepsilon \in \Exps$.
\end{thm}
\begin{proof}
Let $\varphiol, \ol{\psi^{(h)}} \in \CBuw\big(\Bilol; B(\khat)\big)$
denote respectively the normal extensions of $\varphi$ and $\psi^{(h)}$ to
the universal enveloping von Neumann bialgebra of $\Bil$ and set
$\ol{\phi} :=
\Rbar
\varphiol$ and
$\Psi^{(h)} :=
\Rbar
\ol{\psi^{(h)}}$
in
$\CBuw\big( \Bilol; \Bilol\otol B(\khat)\big)$.
Then, by the complete isometry of the map
$\Rbar$
and the complete isometry expressed in~\eqref{diag CB},
\[
\big\|
\big( \id_{\Bilol} \otol \cDh \big) \circ
\big( \Psi^{(h)} - \iota_{\khat} \big)
- \ol{\phi} \big\|_\cb
=
\big\|
\cDh \circ
\big( \psi^{(h)} - \iota_{\khat} \circ \counit \big)
- \varphi \big\|_\cb
\]
which tends to $0$ as $h\to 0^+$. Therefore, by Theorem 7.6 of \cite{Bel}, it follows that
\[
\sup_{t\in[0,T]}
\Big\|
\big( \id_{\Bilol} \otol \toy^{(h)}_{[t/h],\ve} \big) \circ
\Psi^{(h)}_{[t/h]}
- k^{\ol{\phi}}_{t,\varepsilon}
\Big\|_{CB( \Bilol; \Bilol \otol | \FFock \ra )}
\to 0 \text{ as } h \to 0^+.
\]
Now recall that
$
l^{\varphi}_{t,\varepsilon} =
\ol{l}^{\varphi}_{t,\varepsilon}|_{\Bil}
$ where
$
\ol{l}^{\varphi}_{t,\varepsilon} =
k^{\ol{\phi}, \counitol}_{t,\varepsilon} =
\Ebar
k^{\ol{\phi}}_{t,\varepsilon}
$
($t\in\Rplus, \ve\in\Exps$).
The result therefore follows from the fact that
$
\Ebar
\Psi_n^{(h)}|_\Bil
=
\psi^{(h)}_n
$
($n\in\ZZ_+$), which is evident from~\eqref{Equation: ease}.
\end{proof}
\begin{rems}
(i)
Since multiplicativity of the coproduct plays no role in the above proof,
the proper hypothesis for the theorem is that $\Bil$ be a (multiplier)
$\Cstar$-hyperbialgebra.
(ii)
Belton's result allows the hypothesis to be weakened to a condition on
columns (like that of the conclusion), namely
$\varphi \in L \big( \khat ; CB\big(\Bil ; | \khat \ra \big) \big)$ and,
for all $c\in\kil$,
\[
\cDh \circ
\big( \psi^{(h)} - \iota_{\khat} \circ \counit \big)
( \cdot )
| \chat\, \ra
\to
\varphi_{\chat}
\text{ in } CB(\Bil; |\khat\ra) \text{ as } h \to 0^+.
\]
The proof requires only minor modification, however the result given is
adequate for our purposes here.
\end{rems}
In the next two propositions the coproduct plays no role.
Recall Theorem~\ref{established} on the automatic implementability of
$\chi$-structure maps.
\begin{propn}
\label{Proposition: Y}
Let $(\Al,\chi)$ be a $\Cstar$-algebra with character and let
$\varphi: \Al \to B(\hilhat)$ be a $\chi$-structure map.
Set
\[
U_\xi^{(h)} :=
\begin{bmatrix} c_{h,\xi} & -s_{h,\xi}^* \\
s_{h,\xi} & c_{h,\xi} Q_\xi + Q_\xi^\perp
\end{bmatrix},
\text{ for } h > 0 \text{ such that } h \|\xi\|^2\leq 1,
\]
where
$(\pi,\xi)$ is an implementing pair for $\varphi$,
and with
$\xi' := \| \xi \|^{-1} \xi$ \tu{(}or $0$ if $\xi = 0$\tu{)},
\[
c_{h,\xi} = c\big( h^{1/2}\|\xi\| \big) := \sqrt{1 - h \|\xi\|^2}, \
s_{h,\xi} = h^{1/2} | \xi \ra \text{ and }
Q_\xi:= P_{\Comp\,\xi} = | \xi' \ra \la \xi' |.
\]
Then each $U_\xi^{(h)}$ is a unitary operator on $\hilhat$
and the family of *-representations $\Bil \to B(\hilhat)$
\begin{equation}
\label{Equation: rho h}
\Big(
\psi^{(h)} = \wh{\pi}^{(h)}_{\xi} :=
U_\xi^{(h)\,*} (\chi \op \pi)(\cdot )U_\xi^{(h)}
\Big)_{h>0, h\|\xi\|^2\leq 1}
\end{equation}
satisfies
\begin{equation}
\label{Equaton: difference}
\varphi -
\cDh \circ
\big( \psi^{(h)} -
\iota_{\hilhat} \circ \chi \big)
=
\frac{h}{1 + c_{h,\xi}}\, \varphi_1 -
\frac{h^2}{( 1 + c_{h,\xi})^2}\, \varphi_2
\end{equation}
for completely bounded maps
$\varphi_1, \varphi_2: \Al \to B(\hilhat)$
independent of $h$.
Moreover each *-representation
$\psi^{(h)}$ is nondegenerate if \tu{(}and only if\tu{)} $\pi$ is.
\end{propn}
\begin{proof}
Unitarity of $U_\xi^{(h)}$ is evident from the identities
\[
c_{h,\xi}^2 + s_{h,\xi}^* s_{h,\xi} = 1, \
s_{h,\xi}^*Q_\xi^\perp = 0 \text{ and }
s_{h,\xi} s_{h,\xi}^* = \big( 1 - c_{h,\xi}^2 \big) Q_\xi.
\]
Set $\nu = \pi - \iota_\hil \circ \chi$ and
$\gamma = \omega_\xi \circ \nu$
so that $\varphi$ has block matrix
form~\eqref{Equation: block},
and $d_{h,\xi} := c_{h,\xi} - 1$.
Then, noting the identities
\[
d_{h,\xi} = - h \big( 1 + c_{h,\xi} \big)^{-1} \|\xi\|^2,\quad
\|\xi\|^2 Q_\xi = |\xi\ra\la\xi|, \quad
c_{h,\xi}Q_\xi + Q_\xi^\perp = d_{h,\xi}Q_\xi + I_\hil,
\]
we have
\begin{align*}
&
\psi^{(h)}(a) - \chi(a) I_{\hilhat}
\\
&=
U_\xi^{(h)\,*} \begin{bmatrix} 0 & \\ & \nu(a) \end{bmatrix} U_\xi^{(h)}
\\&{}
\\
&=
\begin{bmatrix} 0 & h^{1/2} \la\xi| \\
0 & d_{h,\xi}Q_\xi + I_\hil \end{bmatrix}
\begin{bmatrix} 0 & 0 \\
h^{1/2}\nu(a)|\xi\ra & d_{h,\xi}\nu(a)Q_\xi + \nu(a) \end{bmatrix}
\\&{}
\\
&=
\begin{bmatrix}
h\gamma(a) &
h^{1/2} \la\xi| \nu(a) \big[ d_{h,\xi}Q_\xi + I_\hil \big] \\
& \\
h^{1/2}\big[d_{h,\xi}Q_\xi + I_\hil\big]\nu(a) |\xi\ra &
d_{h,\xi}^2Q_\xi\nu(a)Q_\xi +
d_{h,\xi}\big( Q_\xi\nu(a) + \nu(a)Q_\xi \big) + \nu(a)
\end{bmatrix}
\\&{}
\\
&=
\begin{bmatrix} h^{1/2} & \\ & I_\hil \end{bmatrix}
\Big(
\varphi(a) - h\big(1 + c_{h,\xi}\big)^{-1}\, \varphi_1(a)
+ h^2\big(1 + c_{h,\xi}\big)^{-2}\, \varphi_2(a)
\Big)
\begin{bmatrix} h^{1/2} & \\ & I_\hil \end{bmatrix}
\end{align*}
where
\[
\varphi_1 =
\begin{bmatrix} 0 & \gamma(\cdot)\la\xi| \\
\gamma(\cdot)|\xi\ra & X\nu(\cdot) + \nu(\cdot)X \end{bmatrix}
\text{ and }
\varphi_2 = \gamma(\cdot)
\begin{bmatrix} 0 & \\ & X \end{bmatrix}
\text{ for } X = |\xi\ra\la\xi|,
\]
from which the result follows.
\end{proof}
Note that
$U_\xi^{(h)}$ and $\wh{\pi}^{(h)}_{\xi}$
are norm-continuous in $h$ and
converge to $I_{\hilhat}$ and $\chi\op\pi$ respectively as
$h\to 0^+$.
Note also that, for the simplest class of $\chi$-structure map, namely
\[
\varphi =
\begin{bmatrix} 0 & \\ & \nu \circ \chi \end{bmatrix},
\quad
\text{ where } \nu = \pi - \iota_\hil
\text{ for a *-homomorphism } \pi: \Al \to B(\hil),
\]
$\wh{\pi}^{(h)}_0 = ( \chi \op \pi )$ and so
\[
\wh{\pi}^{(h)}_0 - \iota_{\hilhat}
\circ \chi
=
\cDh \circ \big(
\wh{\pi}^{(h)}_0 -
\iota_{\hilhat} \circ \chi \big)
= \varphi
\text{ for all } h.
\]
\begin{rems}
(i)
Unwrapping
$\wh{\pi}^{(h)}_\xi(a)$:
\[
\begin{bmatrix}
\chi(a) + h\gamma(a) & &
s_{h,\xi}^*\big( \nu(a) - \frac{h \gamma(a)}{1+c_{h,\xi}} I_\hil \big) \\
\big( \nu(a) - \frac{h \gamma(a)}{1+c_{h,\xi}} I_\hil \big) s_{h,\xi} & &
\pi(a) - \frac{h}{1+c_{h,\xi}}\big(X\nu(a)+\nu(a)X\big) +
\frac{h^2\gamma(a)}{(1+c_{h,\xi})^2} X
\end{bmatrix}
\]
reveals the vector-state realisation
\[
\omega_{e_0} \circ
\wh{\pi}^{(h)}_\xi
=
\omega_{\Omega^{(h)}_\xi} \circ
(\chi \op \pi)
\text{ where } \Omega^{(h)}_\xi := U_\xi^{(h)} e_0
\]
for the state $\chi + h \gamma =
\chi + h\|\xi\|^2\big(\omega_{\xi'} \circ \pi - \chi\big)$.
Indeed, finding such a representation was the strategy of proof
in~\cite{FrS}.
(ii)
The next remark will be used in the proof of
Theorem~\ref{Theorem: discrete approximation}.
If instead of being a $\chi$-structure map, $\varphi$ is given by
\begin{equation}
\label{Equation: instead}
\begin{bmatrix} \la \xi | \\ D^* \end{bmatrix}
\nu( \cdot )
\begin{bmatrix} | \xi \ra & D \end{bmatrix}
\text{ where }
\nu := \pi - \iota_\Hil \circ \chi,
\end{equation}
for a nondegenerate representation $\pi: \Al \to B(\Hil)$, vector
$\xi\in\Hil$ and isometry $D\in B(\hil;\Hil)$ then, replacing the
unitaries $U_\xi^{(h)}$ by the isometries
$
V_{\xi,D}^{(h)}:= U_\xi^{(h)}
\left[\begin{smallmatrix} 1 & \\ & D \end{smallmatrix}\right]
\in B\big(\hilhat; \Hilhat\big)$
in the above proof yields a
family of completely positive preunital maps
$
\big( \psi^{(h)} = \psi_{\pi,\xi,D}^{(h)} :=
V_{\xi,D}^{(h)\,*} (\chi \op \pi)(\cdot )
V_{\xi,D}^{(h)} \big)_{0< h, h\|\xi\|^2\leq 1}
$
satisfying~\eqref{Equaton: difference}, with
completely bounded maps
\[
\varphi_1 =
\begin{bmatrix} 0 & \gamma(\cdot)\la\eta| \\
\gamma(\cdot)|\eta\ra & Y^*\nu(\cdot)D + D^*\nu(\cdot)Y \end{bmatrix}
\text{ and }
\varphi_2 = \gamma(\cdot)
\begin{bmatrix} 0 & \\ & X \end{bmatrix}
\]
where $Y=|\xi\ra\la\eta|$ and $X = |\xi\ra \la\xi|$ for
$\eta = D^*\xi\in\hil$.
\end{rems}
Recall the notations~\eqref{Delta QS}.
\begin{propn}
\label{Proposition: X}
Let $(\Al,\chi)$ be a $\Cstar$-algebra with character and let
$\varphi \in CB\big(\Al;B(\khat)\big)$
satisfy $\varphiol(1)\leq 0$ and be expressible in the form
\begin{equation}
\label{Equation: decomposition}
\varphi_1-\varphi_2 \text{ where }
\varphi_1 \in CP\big(\Al;B(\khat)\big) \text{ and }
\varphi_2 =
\chi(\cdot) \big( \dQS + |\zeta\ra\la e_0| + |e_0\ra\la\zeta| \big),
\end{equation}
for a vector $\zeta\in\khat$.
Then there is a family of completely positive contractions
$\big(\psi^{(h)}: \Al \to B(\kilhat)\big)_{0<h\leq C}$,
for some $C>0$, such that
\begin{equation}
\label{Equation: dagger}
\cDh \circ
\big( \psi^{(h)} -
\iota_{\hilhat} \circ \chi \big)
\to
\varphi \text{ in } CB\big(\Al; B(\khat)\big) \text{ as } h\to 0^+.
\end{equation}
\end{propn}
\begin{proof}
It follows from Proposition 4.3 and Theorem 4.4 of~\cite{S}, and their
proofs, that there is a Hilbert space $\hil$ containing $\kil$ and a
$\chi$-structure map $\theta: \Al \to B(\hilhat)$ such that
$\varphi$ is the compression of $\theta$ to $B(\khat)$.
By Proposition~\ref{Proposition: Y}, there is a family of *-homomorphisms
$\big(
\wh{\pi}^{(h)}_{\xi}: \Al\to B(\hilhat)\big)_{0<h\leq C}$ for some
$C>0$, satisfying
\[
\cDh \circ
\big( \wh{\pi}^{(h)}_{\xi} -
\iota_{\hilhat} \circ \chi \big)
\to \theta \text{ in } CB\big(\Al; B(\hilhat)\big)
\text{ as } h\to 0^+.
\]
It follows that~\eqref{Equation: dagger} holds for the compressions
$\psi^{(h)}$
of $\wh{\pi}^{(h)}_{\xi}$ to $B(\khat)$,
which are manifestly completely positive and contractive.
\end{proof}
\begin{rem}
When $\Bil$ and $\kil$ are assumed to be separable,
there is an alternative proof
via standard QS cocycles (\cite{GL1}).
\end{rem}
Combining the above results we obtain
the following discrete approximation result for quantum
stochastic convolution cocycles which, in particular,
gives natural quantum random walk approximation
for Markov-regular quantum L\'evy processes
on a locally compact quantum semigroup.
For more details on quantum random walks on quantum groups,
see~\cite{FrS}
and references therein.
\begin{thm}
\label{Theorem: discrete approximation}
Let $l$ be a Markov-regular, completely positive, contractive quantum
stochastic convolution cocycle on a counital $\Cstar$-bialgebra $\Bil$.
Then there is a family of completely positive contractions
$\big( \psi^{(h)}:\Bil\to B(\khat)\big)_{0< h \leq C}$ for some $C>0$,
whose convolution iterates satisfy
\[
\sup_{t\in[0,T]}
\Big\|
\toy^{(h)}_{[t/h],\ve} \circ
\psi^{(h)}_{[t/h]} - l_{t,\ve}
\Big\|_{CB(\Bil; |\FFock\ra)}
\to 0 \text{ as } h \to 0^+,
\]
for all $T\in\Rplus$ and $\varepsilon\in\Exps$.
Moreover if $l$ is preunital, and/or *-homomorphic, then each
$\psi^{(h)}$ may be chosen to be so too.
\end{thm}
\begin{proof}
By Theorem~\ref{Theorem: cocycle}
we know that $l=l^\varphi$ for some $\varphi\in CB\big(\Bil;
B(\khat)\big)$ which has a decomposition of the
form~\eqref{Equation: decomposition}. The first part therefore follows
from Proposition~\ref{Proposition: X} and Theorem~\ref{Theorem: 1}.
If $l$ is preunital then $\varphi$ may be expressed in the
form~\eqref{Equation: instead+} and so, by the remark
containing~\eqref{Equation: instead},
it follows that the completely
positive maps $\psi^{(h)}$ may be chosen to be preunital.
Now suppose that $l$ is *-homomorphic.
Then, by Theorem~\ref{Theorem: cocycle}, $\varphi$ is an
$\counit$-structure map.
By Theorem~\ref{established}, $\varphi$ has an implementing pair
$(\pi,\xi)$, with $\pi$ nondegenerate if $l$ is.
By Proposition~\ref{Proposition: Y} the maps $\psi^{(h)}$ may be chosen to
be *-homomorphic---and also nondegenerate if $l$ is. This completes the
proof.
\end{proof} | 206,613 |
We have raised the Channel Limit from 40 to 60. If you were at the maximum channel limit before, you may now join another 20 channels.
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great!
Finally, thanks usrbingeek! | 115,757 |
Ron Koppelberger
A Wolf embracing the dayChristian Forge had traveled from loves embrace to breaths of dry desolation, desert sands to mushroom strewn forests in bloom, from cinder block abodes to straw and stick foundations. He had loved, laughed and sang praises to heaven as well as cursing the demons that lay just beyond the twilight horizon.
Christian disturbed the ease of calm harbors and gentle asylum, preferring the danger in adventure and exploration. The shack was buried by the palm fronds and briar scrub surrounding it. He had managed the tangle of weeds
And the soft squish of swampy morass for the undressed wont of expectation, a secret will, a mistress in fanged trust, overwhelming, never sated with the human condition.
He had entered the tumble with a cautious desire. The herbs and juju the swamp witch had arranged on the patch of dry dirt floor had enticed his passions. He had touched the wolf-like figurine and flinched, a sharp edge tore his fingertip and the soil drank in his blood, hungry, sanguine and in need, in magic allure. Homeward bound, he thought as he devoured the sacred meal of herbs and wolf-thyme. Just a touch of crimson, coppery, salty and sleek as the tear drizzled into the mystic brew. He made a face at the taste, bitter in test, the blood a flavored liquor, a foothold on what was human.
Soon after, he collapsed and dreamed of wild freedoms and carnal delights. The sleep of wolfs and babes. Near evening-tide he awoke to the rhythm of his breath, his even forceful exhalations in wolf bred, magnified sense. His paws flexed and he growled, the evidence of his rebuke lay in tattered
Torn clothing and vesture. He was refined in the enveloping allure of wolf suspiration and he wanted, in tense posture. He wanted the hunt; a whip-o-will sounded and the keenness of his soul elevated him to heights of unbridled desire. From human to wolf, from the certain sustenance of civil
Union to primal forests and the grace of wily need. Christian would know the will of wolves because he was on the heal of evolution, The balance between man and wolf. | 357,064 |
TITLE: Algebraic closure of $\mathbb{Q}$ in $\mathbb{Q}_p$
QUESTION [2 upvotes]: Algebraic elements in $\mathbb{Q}_p$ are well studied and there was a question on mathoverflow about information regarding this.
I want to know following information:
Let $K_p=\overline{\mathbb{Q}}\cap \mathbb{Q}_p$. So we have extensions
$$\mathbb{Q} \subset K_p \subset \mathbb{Q}_p.$$
The first one is algebraic extension (of $\mathbb{Q}$) and next one is transcendental extension (of $K_p$).
Q. What is known about degrees of these extensions: $[K_p:\mathbb{Q}]$ and $[\mathbb{Q}_p:K_p]$?
REPLY [2 votes]: To summarise some comments into an answer:
A. $[K_p:\Bbb Q]$ is countably infinite.
Namely, since the polynomial ring $\Bbb Q[x]$ is countable, so is $\overline{\Bbb Q}$ and so is $K_p$, so $[K_p:\Bbb Q]$ is at most countably infinite.
To see that it's not finite, notice the following: For $x \in \Bbb Z_p^\times$, $\Bbb Q_p$ contains an $n$-th root of $x$ for all $n$ which are coprime to $p(p-1)$. (E.g. by Hensel's lemma; it is maybe noteworthy that this characterises the set $\Bbb Z_p^\times$ within $\Bbb Q_p$, which can be used to prove e.g. that all automorphisms of $\Bbb Q_p$ are continuous hence trivial.)
Now let's for example look at the element $\alpha:=1+p \in \Bbb Z_p^\times \cap \Bbb Z$. Notice that $\alpha$ can be a perfect $\ell$-th power $k^\ell$ with $\ell$ prime and $k \in \Bbb Q$ (necessarily $k \in \Bbb N$), for at most finitely many primes $\ell_1, ..., \ell_r$, and hence for all $n$ with $gcd(n, 2\ell_1 ... \ell_r) =1$ the polynomial $x^n - \alpha$ is irreducible over $\Bbb Q$. So, for all $n$ with $gcd(n, 2\ell_1 ... \ell_r p(p-1)) =1$, the subextension $\Bbb Q(\sqrt[n]{\alpha}) \vert \Bbb Q$ of $K_p\vert \Bbb Q$ has degree $n$, where $\sqrt[n]{\alpha}$ denotes the $n$-th root of $\alpha$ existing in $\Bbb Q_p$ according to the previous paragraph. Since such $n$ can be arbitrarily high, $[K_p:\Bbb Q]$ must indeed be (countable) infinite.
B. $[\Bbb Q_p: K_p]$ is uncountable, more precisely, has the cardinality of $\Bbb Q_p$ which is $2^{\aleph_0}$.
This is just a cardinality argument using again that $K_p$ is countable, and so would be any countable-dimensional extension of it. | 156,901 |
Seeing how it is getting mighty close to one of our favorite holidays (gobble gobble!), the team thought it would be nice to reflect on what we are grateful for in this beautiful life. All too often life can begin to shrink into nothing but deadlines and to do lists- this Thanksgiving we are striving to be present and aware of the precious blessings that occur in the ordinary. What we may find mundane in the daily routine can easily be small miracles, if only our hearts and minds were attentive to them. So, out of hearts of humble gratitude, here are few of the countless things we are thankful for:
- Our big God
- Freedom
- Our wonderful Country
- Our Families
- Our little Mountain Town
- This Dream Job
- Team Prayers
- Coffee Breaks
- Twinkle Lights
- Giggle Fits
- Siracha Mayo (no judging!)
Have a cozy friday!
XO
Lis and Bobbi | 152,917 |
\begin{document}
\title{Breaking `128-bit Secure' Supersingular Binary Curves\thanks{The second author acknowledges the support
of the Swiss National Science Foundation, via grant numbers 206021-128727 and 200020-132160,
while the third author acknowledges the support of the Irish Research Council, grant number ELEVATEPD/2013/82.}}
\subtitle{(or how to solve discrete logarithms in ${\mathbb F}_{2^{4 \cdot 1223}}$ and ${\mathbb F}_{2^{12 \cdot 367}}$)}
\author{Robert Granger\inst{1} \and Thorsten Kleinjung\inst{1} \and Jens Zumbr\"agel\inst{2}}
\institute{Laboratory for Cryptologic Algorithms, EPFL, Switzerland\\
\and
Institute of Algebra, TU Dresden, Germany\\
\email{[email protected], [email protected], [email protected]}}
\maketitle
\begin{abstract}
In late 2012 and early 2013 the discrete logarithm problem (DLP) in finite fields of small characteristic underwent a dramatic series of breakthroughs,
culminating in a heuristic quasi-polynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thom\'e.
Using these developments, Adj, Menezes, Oliveira and Rodr\'iguez-Henr\'iquez analysed the concrete security of the DLP, as it arises
from pairings on (the Jacobians of) various genus one and two supersingular curves in the literature, which were originally thought to be $128$-bit
secure. In particular, they suggested that the new algorithms have no impact on the security of a genus one curve over ${\mathbb F}_{2^{1223}}$, and
reduce the security of a genus two curve over ${\mathbb F}_{2^{367}}$ to $94.6$ bits.
In this paper we propose a new field representation and efficient general descent principles which together make the new
techniques far more practical. Indeed, at the `128-bit security level' our analysis shows that the aforementioned genus one curve
has approximately $59$ bits of security, and we report a total break of the genus two curve.
\end{abstract}
\keywords{Discrete logarithm problem, finite fields, supersingular binary curves, pairings}
\section{Introduction}\label{sec:intro}
The role of small characteristic supersingular curves in cryptography has been a varied and an interesting one.
Having been eschewed by the cryptographic community for succumbing spectacularly to the subexponential
MOV attack in 1993~\cite{MOV}, which maps the DLP from an elliptic curve (or more generally, the Jacobian of a higher genus curve) to
the DLP in a small degree extension of the base field of the curve, they made a remarkable comeback with the advent of pairing-based
cryptography in 2001~\cite{sakai,tripartite,bonehfranklin}. In particular, for the latter it was reasoned that the existence of a subexponential
attack on the DLP does not {\em ipso facto} warrant their complete exclusion; rather, provided that the
finite field DLP into which the elliptic curve DLP embeds is sufficiently hard, this state of affairs would be acceptable.
Neglecting the possible existence of native attacks arising from the supersingularity of these curves, much research effort
has been expended in making instantiations of the required cryptographic operations on such curves as efficient as
possible~\cite{barreto,stevenTate,duursmalee,granger04,granger05,ssab,1223,multicore,Chatter,ghosh,aranha,hasan}, to name but a few,
with the associated security levels having been estimated using Coppersmith's algorithm from 1984~\cite{coppersmith,Lenstra}.
Alas, a series of dramatic breakthrough results for the DLP in finite fields of small characteristic have potentially rendered all of these
efforts in vain.
The first of these results was due to Joux, in December 2012, and consisted of a more efficient method --- dubbed `pinpointing' --- to obtain relations
between factor base elements~\cite{Joux13a}.
For medium-sized base fields, this technique has heuristic complexity as low as
$L(1/3,2^{1/3}) \approx L(1/3,1.260)$\footnote{The original paper states a complexity of
$L(1/3,(8/9)^{1/3}) \approx L(1/3,0.961)$; however, on foot of recent communications the constant should be as stated.},
where as usual $L(\alpha,c) = L_Q(\alpha,c) = \text{exp}((c + o(1)) (\log{Q})^{\alpha} (\log{\log{Q} })^{1- \alpha})$, with $Q$ the
cardinality of the field. This improved upon the previous best complexity of $L(1/3,3^{1/3}) \approx L(1/3,1.442)$ due to Joux and Lercier~\cite{JL06}.
Using this technique Joux solved example DLPs in fields of bitlength $1175$ and $1425$, both with prime base fields.
Then in February 2013, G\"olo\u{g}lu, Granger, McGuire and Zumbr\"agel used a specialisation of the Joux-Lercier doubly-rational
function field sieve (FFS) variant~\cite{JL06}, in order to exploit a well-known family of `splitting polynomials', i.e., polynomials which
split completely over the base field~\cite{GGMZ13a}. For fields of the form $\F_{q^{kn}}$ with $k \ge 3$ fixed ($k =2$ is even simpler)
and $n \approx dq$ for a fixed integer $ d \ge 1$,
they showed that for binary (and more generally small characteristic) fields,
relation generation for degree one elements runs in heuristic {\em polynomial time},
as does finding the logarithms of degree two elements (if $q^k$ can be written as $q'^{k'}$ for $k' \ge 4$), once degree one
logarithms are known. For medium-sized base fields
of small characteristic a heuristic complexity as low as $L(1/3,(4/9)^{1/3}) \approx L(1/3,0.763)$ was attained; this approach was
demonstrated via the solution of example DLPs in the fields $\F_{2^{1971}}$~\cite{1971Ann} and $\F_{2^{3164}}$.
After the initial publication of~\cite{GGMZ13a}, Joux released a preprint~\cite{Joux13b}
detailing an algorithm for solving the discrete logarithm problem for fields of the form
$\F_{q^{2n}}$, with $n \le q + d$ for some very small $d$, which was used to solve a DLP in $\F_{2^{1778}}$~\cite{1778Ann}
and later in $\F_{2^{4080}}$~\cite{4080Ann}.
For $n \approx q$ this algorithm has heuristic complexity $L(1/4 + o(1),c)$ for some undetermined $c$, and also has a heuristic polynomial
time relation generation method, similar in principle to that in~\cite{GGMZ13a}.
While the degree two element elimination method in~\cite{GGMZ13a} is arguably superior -- since elements can be eliminated on the fly --
for other small degrees Joux's elimination method is faster, resulting in the stated complexity.
In April 2013 G\"olo\u{g}lu {\em et al.} combined their approach with Joux's to solve an example DLP in the field $\F_{2^{6120}}$~\cite{6120Ann}
and later demonstrated that Joux's algorithm can be tweaked to have heuristic complexity $L(1/4,c)$~\cite{GGMZ13b}, where
$c$ can be as low as $(\omega/8)^{1/4}$~\cite{SACtalk}, with $\omega$ the linear algebra constant, i.e., the exponent of matrix multiplication.
Then in May 2013, Joux announced the solution of a DLP in the field $\F_{2^{6168}}$~\cite{6168Ann}.
Most recently, in June 2013, Barbulescu, Gaudry, Joux and Thom\'e announced a {\em quasi-polynomial time} for solving the DLP~\cite{barb},
for fields $\F_{q^{kn}}$ with $k \ge 2$ fixed and $n \le q + d$ with $d$ very small, which for $n \approx q$ has heuristic complexity
\begin{equation}\label{complexity}
(\log q^{kn})^{O(\log{\log{q^{kn}}})}.
\end{equation}
Since~(\ref{complexity}) is smaller than $L(\alpha,c)$ for any $\alpha > 0$, it is asymptotically the most efficient algorithm known for
solving the DLP in finite fields of small characteristic, which can always be embedded into a field of the required form.
Interestingly, the algorithmic ingredients and analysis of this algorithm are much simpler than for Joux's $L(1/4 + o(1),c)$ algorithm.
Taken all together, one would expect the above developments to have a substantial impact on the security of small characteristic parameters appearing
in the pairing-based cryptography literature. However, all of the record DLP computations mentioned above used Kummer or twisted Kummer
extensions (those with $n$ dividing $q^k \mp 1$), which allow for a reduction in the
size of the factor base by a factor of $kn$ and make the descent phase for individual logarithms relatively easy. While such parameters are
preferable for setting records (most recently in $\F_{2^{9234}}$~\cite{9234Ann}), none of the parameters featured in the
literature are of this form, and so it is not {\em a priori} clear whether the new techniques weaken existing pairing-based protocol parameters.
A recent paper by Adj, Menezes, Oliveira and Rodr\'iguez-Henr\'iquez has begun to address this very issue~\cite{AMOR}. Using the time required to
compute a single multiplication modulo the cardinality of the relevant prime order subgroup as their basic unit of time, which we denote by $M_r$,
they showed that the DLP in the field $\F_{3^{6 \cdot 509}}$ costs at most $2^{73.7}$ $M_r$. One can arguably interpret this result to mean that
this field has $73.7$ bits of security\footnote{The notion of bit security is quite fuzzy; for the elliptic curve DLP it is usually intended to mean the
logarithm to the base $2$ of the expected number of group operations, however for the finite field DLP different authors have used different units,
perhaps because the cost of various constituent algorithms must be amortised into a single cost measure. In this work we time everything in seconds,
while to achieve a comparison with~\cite{AMOR} we convert to $M_r$.}.
This significantly reduces the intended security level of $128$ bits (or $111$ bits as
estimated by Shinohara {{\em et al.}}~\cite{japan}, or $102.7$ bits for the Joux-Lercier FFS variant with pinpointing, as estimated in~\cite{AMOR}).
An interesting feature of their
analysis is that during the descent phase, some elimination steps are performed using the method
from the quasi-polynomial time algorithm of Barbulescu {\em et al.}, when one might have expected these steps to only come into play at much higher
bitlengths, due to the high arity of the arising descent nodes.
In the context of binary fields, Adj {\em et al.} considered in detail the DLP in the field $\F_{2^{12 \cdot 367}}$,
which arises via a pairing from the DLP on the Jacobian of a supersingular genus two curve over $\F_{2^{367}}$, first proposed in~\cite{aranha},
with embedding degree $12$.
Using all of the available techniques they provided an upper bound of $2^{94.6}$ $M_r$ for the cost of breaking the DLP in the embedding field,
which is some way below the intended $128$-bit security level. In their conclusion Adj {\em et al.} also suggest that a commonly implemented
genus one supersingular curve over $\F_{2^{1223}}$ with embedding degree~$4$~\cite{1223,multicore,Chatter,ghosh,hasan}, is not weakened by the
new algorithmic advances, i.e., its security remains very close to $128$ bits.
In this work we show that the above security estimates were incredibly optimistic. Our techniques and results are summarised as follows.
\begin{itemize}
\item \textbf{Field representation:} We introduce a new field representation that can have a profound effect on the
resulting complexity of the new algorithms. In particular it permits the use of a smaller $q$ than before, which not only speeds up the computation
of factor base logarithms, but also the descent (both classical and new).
\vspace{2mm}
\item \textbf{Exploit subfield membership:} During the descent phase we apply a {\em principle of parsimony}, by which one
should always try to eliminate an element in the target field, and only when this is not possible should one embed it into an extension field.
So although the very small degree logarithms may be computed over a larger field, the descent cost is {\em greatly reduced} relative to
solving a DLP in the larger field.
\vspace{2mm}
\item \textbf{Further descent tricks:} The above principle also means that elements can automatically be rewritten in terms of elements of smaller degree,
via factorisation over a larger field, and that elements can be eliminated via Joux's Gr\"obner basis computation method~\cite{Joux13b}
with $k=1$, rather than $k > 1$, which increases its degree of applicability.
\vspace{2mm}
\item \textbf{`128-bit secure' genus one DLP:} We show that the DLP in $\F_{2^{4 \cdot 1223}}$ can be solved in approximately $2^{40}\ \text{s}$,
or $2^{59}$ $M_r$, with $r$ a $1221$-bit prime.
\vspace{2mm}
\item \textbf{`128-bit secure' genus two DLP:} We report a total break of the DLP in $\F_{2^{12 \cdot 367}}$ (announced in~\cite{4404Ann}),
which took about $52240$ core-hours.
\vspace{2mm}
\item $\mathbf{L(1/4,c)}$ \textbf{technique only:} Interestingly, using our approach the elimination steps \`a la
Barbulesu {\em et al.}~\cite{barb} were not necessary for the above estimate and break.
\end{itemize}
The rest of the paper is organised as follows. In \S\ref{sec:setup} we describe our field representation and our target fields.
In \S\ref{sec:relgen} we present the corresponding polynomial time relation generation method for degree one elements and
degree two elements (although we do not need the latter for the fields targeted in the present paper),
as well as how to apply Joux's small degree elimination method~\cite{Joux13b} with the new representation.
We then apply these and other techniques to $\F_{2^{4 \cdot 1223}}$ in \S\ref{sec:genus1} and to $\F_{2^{12 \cdot 367}}$ in \S\ref{sec:genus2} .
Finally, we conclude in \S\ref{sec:conclude}.
\section{Field Representation and Target Fields}\label{sec:setup}
In this section we introduce our new field representation and the fields whose DLP security we will address. This representation, as well as some
preliminary security estimates, were initially presented in~\cite{ECCtalk}.
\subsection{Field Representation}\label{sec:fieldrep}
Although we focus on binary fields in this paper, for the purposes of generality, in this section we allow for extension fields of
arbitrary characteristic. Hence let $q = p^l$ for some prime $p$, and let $\K = \F_{q^{kn}}$ be the field under consideration, with $k \ge 1$.
We choose a positive integer $d_h$ such that $n \le q d_h + 1$, and then choose $h_0,h_1 \in \F_{q^k}[X]$ with
$\max \{\text{deg}(h_0),\text{deg}(h_1)\} = d_h$ such that
\begin{equation}\label{fieldrep}
h_1(X^q) X - h_0(X^q) \equiv 0 \pmod{I(X)},
\end{equation}
where $I(X)$ is an irreducible degree $n$ polynomial in $\F_{q^k}[X]$. Then $\K = \F_{q^k}[X]/(I(X))$.
Denoting by $x$ a root of $I(X)$, we introduce the auxiliary variable $y = x^q$, so that one has two isomorphic representations
of $\K$, namely $\F_{q^k}(x)$ and $\F_{q^k}(y)$, with $\sigma: \F_{q^k}(y) \rightarrow \F_{q^k}(x): y \mapsto x^q$.
To establish the inverse isomorphism, note that by~(\ref{fieldrep}) in $\K$ we have $h_1(y)x - h_0(y) = 0$, and hence
$\sigma^{-1}: \F_{q^k}(x) \rightarrow \F_{q^k}(y): x \mapsto h_0(y)/h_1(y)$.
The knowledgeable reader will have observed that our representation is a synthesis of two other useful representations: the one used by
Joux~\cite{Joux13b}, in which one searches for a degree $n$ factor $I(X)$ of $h_1(X)X^q - h_0(X)$;
and the one used by G\"olo\u{g}lu {\em et al.}~\cite{GGMZ13a,GGMZ13b}, in which one searches for a degree $n$ factor $I(X)$ of $X - h_0(X^q)$.
The problem with the former is that it constrains $n$ to
be approximately $q$. The problem with the latter is that the polynomial $X - h_0(X^q)$ is insufficiently general to represent all degrees $n$
up to $q d_h$. By changing the coefficient of $X$ in the latter from $1$ to $h_1(X^q)$, we greatly increase the probability of overcoming the second
problem, thus combining the higher degree coverage of Joux's representation with the higher degree possibilities
of~\cite{GGMZ13a,GGMZ13b}.
The {\em raison d'\^etre} of using this representation rather than Joux's representation is that for a given $n$, by choosing $d_h > 1$, one
may use a smaller $q$. So why is this useful? Well, since the complexity of the new descent methods is typically a function of $q$, then
subject to the satisfaction of certain constraints, one may use a smaller $q$, thus reducing the complexity of solving the DLP.
This observation was our motivation for choosing field representations of the above form.
Another advantage of having an $h_1$ coefficient (which also applies to Joux's representation) is that it increases the chance of there being
a suitable $(h_1,h_0)$ pair with coefficients defined over a proper subfield of $\F_{q^k}$, which then permits one to apply the factor base reduction technique
of~\cite{JL06}, see~\S\ref{sec:genus1} and~\S\ref{sec:genus2}.
\subsection{Target Fields}\label{sec:target}
For $i \in \{0,1\}$ let $E_i/\F_{2^p}: Y^2 + Y = X^3 + X + i$. These elliptic curves are supersingular and can have prime or nearly prime order only
for $p$ prime, and have embedding degree $4$~\cite{stevenSS,barreto,stevenTate}.
We focus on the curve
\begin{equation}\label{curve1}
E_{0} / \F_{2^{1223}}: Y^2 + Y = X^3 + X,
\end{equation}
which has a prime order subgroup of cardinality $r_{1} = (2^{1223} + 2^{612} +1)/5$, of bitlength $1221$.
This curve was initially proposed for $128$-bit secure protocols~\cite{1223} and has enjoyed several optimised
implementations~\cite{multicore,Chatter,hasan,ghosh}.
Many smaller $p$ have also been proposed in the literature (see~\cite{ssab,stevenSS}, for instance), and are clearly weaker.
For $i \in \{0,1\}$ let $H_i/\F_{2^p}: Y^2 + Y = X^5 + X^3 + i$. These genus two hyperelliptic curves are supersingular and can have a nearly
prime order Jacobian only for $p$ prime (note that $13$ is always a factor of $\#\text{Jac}_{H_0}(\F_{2^p})$, since $\#\text{Jac}_{H_0}(\F_2) = 13$),
and have embedding degree $12$~\cite{ssab,stevenSS}.
We focus on the curve
\begin{equation}\label{curve2}
H_0 / \F_{2^{367}}: Y^2 + Y = X^5 + X^3,
\end{equation}
with $\#\text{Jac}_H(\F_{2^{367}}) = 13\cdot 7170258097 \cdot r_{2}$, and $r_2 = (2^{734} + 2^{551} + 2^{367} + 2^{184} + 1)/(13 \cdot 7170258097)$ is
a $698$-bit prime,
since this was proposed for $128$-bit secure protocols~\cite{aranha}, and whose security was analysed in depth by Adj {\em et al.} in~\cite{AMOR}.
\section{Computing the Logarithms of Small Degree Elements}\label{sec:relgen}
In this section we adapt the polynomial time relation generation method from~\cite{GGMZ13a} and
Joux's small degree elimination method~\cite{Joux13b} to the new field representation as detailed in~\S\ref{sec:fieldrep}.
Note that henceforth, we shall refer to elements of $\F_{q^{kn}} = \F_{q^k}[X]/(I(X))$ as field elements or as polynomials, as appropriate,
and thus use $x$ and $X$ (and $y$ and $Y$) interchangeably. We therefore freely apply polynomial ring concepts,
such as degree, factorisation and smoothness, to field elements.
In order to compute discrete logarithms in our target fields we apply
the usual index calculus method. It consists of a precomputation
phase in which by means of (sparse) linear algebra techniques one
obtains the logarithms of the factor base elements, which will consist
of the low degree irreducible polynomials. Afterwards, in the
individual logarithm phase, one applies procedures to recursively rewrite
each element as a product of elements of smaller degree, in this
way building up a {\em descent} tree, which has the target element as its root and factor base elements
as its leaves. This proceeds in several stages,
starting with a continued fraction descent of the target element,
followed by a special-$Q$ lattice descent (referred to as degree-balanced classical
descent, see~\cite{GGMZ13a}), and finally using Joux's Gr\"obner basis descent~\cite{Joux13b} for the
lower degree elements. Details of the continued fraction and classical descent steps are
given in \S\ref{sec:genus1}, while in this section we provide details of how to find the logarithms
of elements of small degree.
We now describe how the logarithms of degree one and two elements (when needed) are to be computed.
We use the relation generation method from~\cite{GGMZ13a}, rather than Joux's method~\cite{Joux13b}, since it automatically
avoids duplicate relations.
For $k \ge 2$ we first precompute the set $\mathcal{S}_k$, where
\[
\mathcal{S}_k = \{ (a,b,c) \in (\F_{q^k})^3 \mid X^{q+1} + a X^q + bX + c \ \ \text{splits completely over} \ \F_{q^k} \}.
\]
For $k=2$, this set of triples is parameterised by $(a,a^q, \F_{q} \ni c \ne a^{q+1})$, of which there are precisely $q^3 - q^2$ elements.
For $k \ge 3$, $\mathcal{S}_k$ can also be computed very efficiently, as follows. Assuming $ c \ne ab$ and
$b \ne a^q$, the polynomial $X^{q+1} + a X^q + bX + c$ may be transformed (up to a scalar factor) into the polynomial
$f_B(\overline{X}) = \overline{X}^{q+1} + B\overline{X} + B$, where $B = \frac{(b - a^{q})^{q+1}}{(c - ab)^{q}}$,
and $X = \frac{c - ab}{b - a^{q}} \overline{X} - a$. The set $\mathcal{L}$ of $B \in \F_{q^k}$ for which $f_B$ splits completely over $\F_{q^k}$
can be computed by simply testing for each such $B$ whether this occurs, and there are precisely $(q^{k-1}-1)/(q^2 - 1)$ such $B$ if $k$ is odd, and
$(q^{k-1}-q)/(q^2 - 1)$ such $B$ if $k$ is even~\cite{Bluher}. Then for any $(a,b)$ such that $b \ne a^q$ and for each $B \in \mathcal{L}$,
we compute via $B = \frac{(b - a^{q})^{q+1}}{(c - ab)^{q}}$ the corresponding (unique) $c \in \F_{q^{k}}$, which thus ensures that
$(a,b,c) \in \mathcal{S}_k$. Note that in all cases we have $|\mathcal{S}_k| \approx q^{3k-3}$.
\subsection{Degree $1$ Logarithms}\label{sec:degree1}
We define the factor base $\mathcal{B}_1$ to be the set of linear elements in $x$, i.e., $\mathcal{B}_1 = \{ x - a \mid a \in \F_{q^k}\}$.
Observe that the elements linear in $y$ are each expressible in $\mathcal{B}_1$, since $(y - a) = (x - a^{1/q})^q$.
As in~\cite{JL06,GGMZ13a,GGMZ13b}, the basic idea is to consider elements of the form $xy + ay + bx + c$ with $(a,b,c) \in \mathcal{S}_k$. The above
two field isomorphisms induce the following equality in $\K$:
\begin{equation}\label{relation1}
x^{q+1} + a x^q + bx + c = \frac{1}{h_1(y)} \big(yh_0(y) + ay h_1(y) + bh_0(y) + ch_1(y) \big).
\end{equation}
When the r.h.s. of~(\ref{relation1}) also splits completely over $\F_{q^k}$, one obtains a relation between elements of $\mathcal{B}_1$
and the logarithm of $h_1(y)$. One can either adjoin $h_1(y)$ to the factor base, or simply use an $h_1(y)$ which splits completely over $\F_{q^k}$.
We assume that for each $(a,b,c) \in \mathcal{S}_k$ that the r.h.s. of~(\ref{relation1}) -- which has degree $d_h+1$ -- splits completely over $\F_{q^k}$ with probability $1/(d_h+1)!$. Hence in order for there to be sufficiently many relations we require that
\begin{equation}\label{cond1}
\frac{q^{3k - 3}}{(d_h+1)!} > q^k, \ \ \text{or equivalently} \ \ q^{2k-3} > (d_h + 1)!.
\end{equation}
When this holds, the expected cost of relation generation is $(d_h+1)! \cdot q^{k} \cdot S_{q^k}(1,d_h+1)$, where $S_{q^k}(m,n)$ denotes
the cost of testing whether a degree $n$ polynomial is $m$-smooth, i.e., has all of its irreducible factors of degree $\le m$,
see Appendix~B. The cost of solving the resulting linear system using sparse linear algebra techniques is
$O(q^{2k+1})$ arithmetic operations modulo the order $r$ subgroup in which one is working.
\subsection{Degree $2$ Logarithms}\label{sec:degree2}
For degree two logarithms, there are several options. The simplest is to apply the degree one method over a quadratic extension of $\F_{q^k}$,
but in general (without any factor base automorphisms) this will cost $O(q^{4k+1})$ modular arithmetic operations.
If $k \ge 4$ then subject to a condition on $q$, $k$ and $d_h$,
it is possible to find the logarithms of irreducible degree two elements on the fly, using the techniques of~\cite{GGMZ13a,GGMZ13b}.
In fact, for the DLP in $\F_{2^{12 \cdot 367}}$ we use both of these approaches, but for different base fields, see~\S\ref{sec:genus2}.
Although not used in the present paper, for completeness we include here the analogue in our field representation of Joux's
approach~\cite{Joux13b}. Since this approach forms the basis of the higher degree elimination steps in the quasi-polynomial time
algorithm of Barbulescu {\em et al.}, its analogue in our field representation should be clear.
We define $\mathcal{B}_{2,u}$ to be the set of irreducible elements of $\F_{q^k}[X]$ of the form $X^2 + uX + v$. For each $u \in \F_{q^k}$
one expects there to be about $q^k/2$ such elements\footnote{For binary fields there are precisely $q^k/2$ irreducibles, since
$X^2 + uX + v$ is irreducible if and only if $\text{Tr}_{\F_{q^k}/\F_2}(v/u^2) = 1$.}.
As in~\cite{Joux13b}, for each $u \in \F_{q^k}$ we find the logarithms of all
the elements of $\mathcal{B}_{2,u}$ simultaneously.
To do so, consider~(\ref{relation1}) but with $x$ on the l.h.s. replaced with $Q = x^2 + ux$. Using the field isomorphisms
we have that $Q^{q+1} + a Q^q + bQ + c$ is equal to
\begin{align*}\label{relation2}
&(y^2 \!+\! u^qy)((h_0(y)/h_1(y))^2 \!\!+\! u(h_0(y)/h_1(y))) \!+\! a(y^2 \!\!+\! u^qy) \!+\! b((h_0(y)/h_1(y))^2 \!\!+\! u(h_0(y)/h_1(y))) \!+\! c\\
&= \frac{1}{h_1(y)^2} \big( (y^2 \!\!+\! u^qy)(h_0(y)^2 \!\!+\! uh_0(y)h_1(y)) \!+\! a(y^2 \!\!+\! u^qy)h_1(y)^2 \!+\! b(h_0(y)^2 \!\!+\! uh_0(y)h_1(y)) \!+\! ch_1(y)^2\big).
\end{align*}
The degree of the r.h.s. is $2(d_h+1)$, and when it splits completely over $\F_{q^k}$ we have a relation between elements of $\mathcal{B}_{2,u}$ and degree one elements,
whose logarithms are presumed known, which we assume occurs with probability $1/(2(d_h+1))!$.
Hence in order for there to be sufficiently many relations we require that
\begin{equation}\label{cond2}
\frac{q^{3k - 3}}{(2(d_h+1))!} > \frac{q^k}{2}, \ \ \text{or equivalently} \ \ q^{2k-3} > (2(d_h + 1))!/2.
\end{equation}
Observe that~(\ref{cond2}) implies~(\ref{cond1}). When this holds, the expected cost of relation generation is
$(2(d_h+1))! \cdot q^{k} \cdot S_{q^k}(1,2(d_h+1))/2$.
The cost of solving the resulting linear system using sparse linear algebra techniques is again $O(q^{2k+1})$
modular arithmetic operations, where now both the number of variables and the average weight is halved relative to the degree one case.
Since there are $q^k$ such $u$, the total expected cost of this stage is $O(q^{3k+1})$ modular arithmetic operations, which may of course
be parallelised.
\subsection{Joux's Small Degree Elimination with the New Representation}\label{sec:smallelim}
As in~\cite{Joux13b}, let $Q$ be a degree $d_Q$ element to be eliminated, let
$F(X) = \sum_{i = 0}^{d_F} f_i X^i, G(X) = \sum_{j = 0}^{d_G} g_j X^j \in \F_{q^k}[X]$
with $d_F+d_G+2 \ge d_Q$, and assume without loss of generality $d_F \ge d_G$.
Consider the following expression:
\begin{equation}\label{corerelation}
G(X) \prod_{\alpha \in \F_q} ( F(X) - \alpha \, G(X) ) = F(X)^q G(X) - F(X) G(X)^q
\end{equation}
The l.h.s. is $\max(d_F,d_G)$-smooth.
The r.h.s. can be expressed modulo $h_1(X^q)X-h_0(X^q)$
in terms of $Y=X^q$ as a quotient of polynomials of relatively low degree
by using
\[
F(X)^q = \sum_{i = 0}^{d_F} f_{i}^q Y^i, \ G(X)^q = \sum_{j = 0}^{d_G} g_{j}^q Y^j
\ \text{and} \ X \equiv \frac{h_0(Y)}{h_1(Y)}.
\]
Then the numerator of the r.h.s. becomes
\begin{eqnarray}\label{fieldrelation}
\bigg( \sum_{i = 0}^{d_F} f_{i}^q Y^i \bigg)
\bigg( \sum_{j = 0}^{d_G} g_{j}^q h_0(Y)^j h_1(Y)^{d_F-j} \bigg)
-
\bigg( \sum_{i = 0}^{d_F} f_{i}^q h_0(Y)^i h_1(Y)^{d_F-i} \bigg)
\bigg( \sum_{j = 0}^{d_G} g_{j}^q Y^j \bigg).
\end{eqnarray}
Setting~(\ref{fieldrelation}) to be $0$ modulo $Q(Y)$ gives
a system of $d_Q$ equations over $\F_{q^k}$ in the $d_F+d_G+2$ variables $f_0,\ldots,f_{d_F},g_0, \ldots,g_{d_G}$.
By choosing a basis for $\F_{q^{k}}$ over $\F_q$ and
expressing each of the $d_F+d_G+2$ variables $f_0,\ldots,f_{d_F},g_0, \ldots,g_{d_G}$ in this
basis, this system becomes a bilinear quadratic system\footnote{The bilinearity makes finding solutions to this system easier~\cite{span},
and is essential for the complexity analysis in~\cite{Joux13b} and its variant in~\cite{GGMZ13b}.}
of $kd_Q$ equations in $(d_F+d_G+2)k$ variables.
To find solutions to this system, one can specialise $(d_F+d_G+2-d_Q)k$ of the variables in order to make the resulting system generically zero-dimensional
while keeping its bilinearity, and then compute the corresponding Gr\"obner basis, which may have no solution, or a small number of solutions.
For each solution, one checks whether~(\ref{fieldrelation}) divided by $Q(Y)$ is $(d_Q-1)$-smooth: if so then
$Q$ has successfully been rewritten as a product of elements of smaller degree; if no solutions give a $(d_Q-1)$-smooth cofactor,
then one begins again with another specialisation.
The degree of the cofactor of $Q(Y)$ is upper bounded by $d_F (1 + d_h) - d_Q$, so assuming that it behaves as a uniformly
chosen polynomial of such a degree one can calculate the probability $\rho$ that it is $(d_Q-1)$-smooth using standard combinatorial techniques.
Generally, in order for $Q$ to be eliminable by this method with good probability, the number of solutions to the initial bilinear system must
be greater than $1/\rho$.
To estimate the number of solutions, consider the action of $\Gl_2(\F_{q^{k}})$ on the set of pairs $(F,G)$.
The subgroups $\Gl_2(\F_q)$ and $\F_{q^{k}}^{\times}$ (via diagonal embedding) both
act trivially on the set of relations, modulo multiplication by elements in $\F_{q^{k}}^{\times}$.
Assuming that the set of $(F,G)$ quotiented out by the action of the compositum of these subgroups
(which has cardinality $\approx q^{k+3}$), generates distinct relations,
one must satisfy the condition
\begin{equation}\label{GBelimprob}
q^{(d_F+d_G +1 - d_Q)k - 3} > 1/ \rho\ .
\end{equation}
Note that while~(\ref{GBelimprob}) is preferable for an easy descent, one may yet violate it and still
successfully eliminate elements by using various tactics, as demonstrated in~\S\ref{sec:genus2}.
\section{Concrete Security Analysis of $\F_{2^{4\cdot 1223}}$}
\label{sec:genus1}
In this section we focus on the DLP in the $1221$-bit prime order $r_1$ subgroup of $\F_{2^{4\cdot 1223}}^{\times}$,
which arises from the MOV attack applied to the genus one supersingular curve~(\ref{curve1}).
By embedding $\F_{2^{4\cdot 1223}}$ into its degree two
extension $\F_{2^{8\cdot 1223}} = \F_{2^{9784}}$ we show that, after
a precomputation taking approximately $2^{40}\ \text{s}$,
individual discrete logarithms can be computed in less than $2^{34}\ \text{s}$.
\subsection{Setup}
We consider the field $\F_{2^{8\cdot 1223}} = \F_{q^n}$ with $q = 2^8$
and $n = 1223$ given by the irreducible factor of degree $n$ of
$h_1(X^q)X - h_0(X^q)$, with
\[ h_0 = X^5 + tX^4 + tX^3 + X^2 + tX + t \:, \quad
h_1 = X^5 + X^4 + X^3 + X^2 + X + t \:, \]
where $t$ is an element of $\F_{2^2} \setminus \F_2$.
Note that the field of definition of this representation is $\F_{2^2}$.
Since the target element is contained in the subfield
$\F_{2^{4\cdot 1223}}$, we begin the classical descent over
$\F_{2^4}$, we switch to $\F_q = \F_{2^8}$, i.e., $k=1$, for the
Gr\"obner basis descent, and, as explained below, we work over
$\F_{q^k}$ with either $k=1$ or a few $k>1$ to obtain the logarithms of
all factor base elements.
\subsection{Linear Algebra Cost Estimate}
In this precomputation we obtain the logarithms of all elements of
degree at most four over $\F_q$. Since the degree $1223$ extension is defined over
$\F_{2^2}$ in our field representation, by the action of
the Galois group $\Gal(\F_q/\F_{2^2})$ on the factor base, the number of irreducible elements of degree $j$
whose logarithms are to be computed can be reduced to about $2^{8j}/(4j)$ for $j \in \{1,2,3,4\}$.
One way to obtain the logarithms of these elements is to carry out the degree 1 relation generation method from~\S\ref{sec:degree1},
together with the elementary observation that an irreducible polynomial of degree $k$ over $\F_q$ splits completely over $\F_{q^k}$.
First, computing degree one logarithms over $\F_{q^3}$ gives the logarithms of
irreducible elements of degrees one and three over $\F_q$. Similarly,
computing degree one logarithms over $\F_{q^4}$ gives the logarithms of
irreducible elements of degrees one, two, and four over $\F_q$.
The main computational cost consists in solving the latter system arising from $\F_{q^4}$, which has size $2^{28}$ and an
average row weight of~$256$.
However, we propose to reduce the cost of finding these logarithms by using $k = 1$ only, in the following easy way.
Consider~\S\ref{sec:smallelim}, and observe that for each polynomial pair $(F,G)$ of degree at most~$d$, one obtains
a relation between elements of degree at most $d$ when the numerator of the r.h.s. is $d$-smooth (ignoring factors of $h_1$).
Note that we are not setting the r.h.s. numerator to be zero modulo $Q$ or computing any Gr\"obner bases.
Up to the action of $\Gl_2(\F_q)$ (which gives equivalent relations) there
are about $q^{2d-2}$ such polynomial pairs. Hence, for $d\ge 3$ there
are more relations than elements if the smoothness probability of the
r.h.s. is sufficiently high. Notice that $k=1$ implies that the r.h.s. is
divisible by $h_1(Y)Y-h_0(Y)$, thus increasing its smoothness
probability and resulting in enough relations for $d=3$ and for $d=4$.
After having solved the much smaller system for $d=3$ we know the
logarithms of all elements up to degree three, so that the average row
weight for the system for $d=4$ can be reduced to about $\frac14 \cdot 256 =
64$ (irreducible degree four polynomials on the l.h.s.). As above the
size of this system is $2^{28}$.
The cost for generating the linear systems is negligible compared to
the linear algebra cost.
For estimating the latter cost we consider
Lanczos' algorithm to solve a sparse $N\times N$,
$N=2^{28}$, linear system with average row weight $W=64$. As noted
in~\cite{Popovyan,GGMZ13b} this algorithm can be
implemented such that
\begin{equation}\label{eq:lanczos}
N^2 \,(2\,W\,\text{ADD} + 2\,\text{SQR} + 3\,\text{MULMOD})
\end{equation} operations are used.
On our benchmark system, an AMD Opteron 6168 processor at $1.9\,$GHz,
using~\cite{gmp} our implementation of these operations took 62 ns, 467 ns and 1853 ns
for an ADD, a SQR and a MULMOD, respectively,
resulting in a linear algebra cost of $2^{40}\ \text{s}$.
As in~\cite{AMOR}, the above estimate ignores communication costs and other possible slowdowns which may arise in practice.
An alternative estimate can be obtained by considering a problem of a similar size over $\F_2$ and
extrapolating from~\cite{RSA768}. This gives an estimated time of $2^{42}\ \text{s}$, or for newer hardware slightly less.
Note that this computation was carried out using the block Wiedemann algorithm~\cite{wiedemann}, which we recommend
in practice because it allows one to distribute the main part of the computation.
For the sake of a fair comparison with~\cite{AMOR} we use the former estimate of $2^{40}\ \text{s}$.
\subsection{Descent Cost Estimate}\label{1223descent}
We assume that the logarithms of elements up to degree four are known, and that computing these logarithms with a lookup table is free.
\subsubsection{Small Degree Descent.}
We have implemented the small degree descent of \S\ref{sec:smallelim} in Magma~\cite{magma}
V2.20-1, using Faugere's F4 algorithm~\cite{F4}. For each degree from $5$ to $15$, on the same AMD Opteron
6168 processor we timed the Gr\"obner basis computation between $10^6$ and $100$ times, depending on the degree.
Then using a bottom-up recursive strategy we estimated the following average running times in seconds for a full logarithm computation,
which we present to two significant figures:
\[ C[5,\ldots,15] =
[ \: 0.038 \,,\, 2. 1 \,,\, 2.1 \,,\, 93 \,,\, 95 \,,\, 180 \,,\,
190 \,,\, 3200 \,,\, 3500 \,,\, 6300 \,,\, 11000 \: ] \:. \]
\subsubsection{Degree-Balanced Classical Descent.}
From now on, we make the conservative assumption that a degree $n$ polynomial which is
$m$-smooth, is a product of $n/m$ degree $m$ polynomials. In practice the descent
cost will be lower than this, however, the linear algebra cost is dominating, so this issue is
inconsequential for our security estimate. The algorithms we used for smoothness testing are detailed in Appendix B.
For a classical descent step with degree balancing we consider
polynomials $P(X^{2^a},Y) \in \F_q[X,Y]$ for a suitably chosen integer
$0 \le a \le 8$. It is advantageous to choose $P$ such that its
degree in one variable is one; let $d$ be the degree in the other
variable. In the case $\deg_{X^{2^a}}(P)=1$, i.e.,
$P=v_1(Y)X^{2^a}+v_0(Y)$, $\deg v_i \le d$, this gives rise to the
relation
\[
L_v^{2^a} = \bigg( \frac{R_v}{h_1(X)^{2^a}} \bigg)^{2^8}
\quad \text{where} \quad
\begin{array}{l}
L_v = \tilde{v}_1(X^{2^{8-a}})X + \tilde{v}_0(X^{2^{8-a}}) \:, \\
R_v = v_1(X)h_0(X)^{2^a}+v_0(X)h_1(X)^{2^a}
\end{array}
\]
in $\F_q[X]/(h_1(X^q)X-h_0(X^q))$
with $\deg L_v \le 2^{8-a}d+1$, $\deg R_v \le d+5 \cdot 2^a$, and
$\tilde{v}_i$ being $v_i$ with its coefficients powered by $2^{8-a}$, for $i=0,1$.
Similarly, in the case $\deg_{Y}(P)=1$, i.e., $P=w_1(X^{2^a})Y+w_0(X^{2^a})$,
$\deg w_i \le d$, we have the relation
\[
L_w^{2^a} = \bigg(\frac{R_w}{h_1(X)^{2^a d}}\bigg)^{2^{8}}
\quad \text{where} \quad
\begin{array}{l}
L_w = \tilde{w}_1(X)X^{2^{8-a}}+ \tilde{w}_0(X) \:, \\
R_w = h_1(X)^{2^a d} \big(w_1 \big( \big(\frac{h_0(X)}{h_1(X)} \big)^{2^a}\big)X
+ w_0 \big( \big( \frac{h_0(X)}{h_1(X)} \big)^{2^a} \big) \big)
\end{array}
\]
with $\deg L_w \le d+2^{8-a}$, $\deg R_w \le 5 \cdot 2^a d+1$ and
again $\tilde{w}_i$ being $w_i$ with its coefficients powered by $2^{8-a}$, for $i=0,1$.
The polynomials $v_i$ (respectively $w_i$) are chosen in such
a way that either the l.h.s.~or the r.h.s.~is
divisible by a polynomial $Q(X)$ of degree $d_Q$. Gaussian reduction
provides a lattice basis $(u_0,u_1),(u_{0}',u_{1}')$ such that the
polynomial pairs satisfying the divisibility condition above are given
by $ru_i+su_{i}'$ for $i=0,1$, where $r,s \in \F_q[X]$. For nearly all
polynomials $Q$ it is possible to choose a lattice basis of
polynomials with degree $\approx d_Q/2$ which we will assume for
all $Q$ appearing in the analysis; extreme cases can be avoided by
look-ahead or backtracking techniques. Notice that a polynomial $Q$
over $\F_{2^4} \subset \F_q$ can be rewritten as a product of
polynomials which are also over $\F_{2^4} $, by choosing the basis as well as
$r$ and $s$ to be over $\F_{2^4}$. This will be done in all steps of the
classical descent. The polynomials $r$ and $s$
are chosen to be of degree four, resulting in $2^{36}$ possible pairs
(multiplying both by a common non-zero constant gives the same relation).
In the final step of the classical eliminations (from degree $26$ to $15$)
we relax the criterion that the l.h.s. and r.h.s. are $15$-smooth,
allowing also irreducibles of even degree up to degree $30$, since these can each be split over
$\F_q$ into two polynomials of half the degree, thereby
increasing the smoothness probabilities. Admittedly, if we follow our worst-case analysis stipulation that all
polynomials at this step have degree $26$, then one could immediately split each of them into two degree $13$ polynomials.
However, in practice one will encounter polynomials of all degrees $\le 26$ and we therefore carry out the analysis
without using the splitting shortcut, which will still provide an overestimate of the cost of this step.
In the following we will state the logarithmic cost (in seconds) of a
classical descent step as $c_l + c_r + c_s$, where $2^{c_l}$ and $2^{c_r}$ denote
the number of trials to get the left hand side and the right hand side
$m$-smooth, and $2^{c_s}\ \text{s}$ is the time required for the corresponding smoothness
test. See Table~\ref{smoothness} for the smoothness timings that we
benchmarked on the AMD Opteron 6168 processor.
\begin{itemize}
\item $\mathbf{d_Q = 26}$ \textbf{to} $\mathbf{m = 15}$\textbf{:}
We choose $\deg_{X^{2^a}}P=1$, $a=5$, $Q$ on the right, and we have $d=17$,
$(\deg(L_v), \deg(R_v)) = (137, 151)$,
and logarithmic cost $13.4 + 15.6 - 9.0$, hence
$2^{20.0}\ \text{s}$; the expected number of factors is $19.2$,
so the subsequent cost will be less than $2^{17.7}\ \text{s}$.
Note that, as explained above, we use the splitting shortcut for irreducibles of even degree up to $30$,
resulting in the higher than expected smoothness probabilities.
\vspace{2mm}
\item $\mathbf{d_Q = 36}$ \textbf{to} $\mathbf{m = 26}$\textbf{:}
We choose $\deg_{X^{2^a}}P=1$, $a=5$, $Q$ on the right, and we have $d=22$,
$(\deg(L_v), \deg(R_v)) = (177, 146)$,
and logarithmic cost $18.7 + 13.6 - 9.0$, hence
$2^{23.3}\ \text{s}$; the expected number of factors is $12.4$,
so the subsequent cost will be less than $2^{23.9}\ \text{s}$.
\vspace{2mm}
\item $\mathbf{d_Q = 94}$ \textbf{to} $\mathbf{m = 36}$\textbf{:}
We choose $\deg_{Y}P = 1$, $a = 0$, $Q$ on the left, and we
have $d=51$, $(\deg(L_w), \deg(R_w)) = (213, 256)$, and
logarithmic cost $15.0 + 20.3 - 7.5$, hence $2^{27.8}\ \text{s}$; the
expected number of factors is $13.0$, so the subsequent cost
will be less than $2^{28.4}\ \text{s}$.\medskip
\end{itemize}
\vspace{-3mm}
\begin{table}[h]
\caption{Timings for testing a degree $n$ polynomial over $\F_{2^4}$
for $m$-smoothness.}
\begin{center}\label{smoothness}
\begin{tabular}{c|c|c}
$n$ & $m$ & time \\
\midrule
$137$ & $30$ & 1.9\,ms \\
$146$ & $26$ & 1.9\,ms \\
$213$ & $36$ & 5.1\,ms \\
~$611$~ & ~$94$~ & ~94\,ms \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\vspace{-5mm}
\subsubsection{Continued Fraction Descent.}
For the continued fraction descent we multiply the target element by random powers of the generator
and express the product as a ratio of two polynomials of degree at most $611$. For each such expression we test
if both the numerator and the denominator are $94$-smooth.
The logarithmic cost here is $17.7 + 17.7 - 3.4$, hence the cost is $2^{32.0}\ \text{s}$.
The expected number of degree $94$ factors on both sides will be
$13$, so the subsequent cost will be less than $2^{32.8}\ \text{s}$.
\subsubsection{Total Descent Cost}
The cost for computing an individual logarithm is therefore upper-bounded by $2^{32.0}\ \text{s} + 2^{32.8}\ \text{s} < 2^{34}\ \text{s}$.
\subsection{Summary}
The main cost in our analysis is the linear algebra computation which takes about $2^{40}\ \text{s}$,
with the individual logarithm stage being considerably faster.
In order to compare with the estimate in \cite{AMOR}, we write the main cost in terms of $M_r$ which gives
$2^{59}$ $M_r$, and thus an improvement by a factor of $2^{69}$.
Nevertheless, solving a system of cardinality $2^{28}$ is still a formidable
challenge, but perhaps not so much for a well-funded adversary.
For completeness we note that if one wants to avoid a linear algebra step of this size, then
one can work over different fields, e.g., with $q = 2^{10}$ and $k=2$, or $q = 2^{12}$ and $k=1$.
However, while this allows a partitioning of the linear algebra into smaller steps as described in~\S\ref{sec:degree2}
but at a slightly higher cost, the resulting descent cost is expected to be significantly higher.
\section{Solving the DLP in $\F_{2^{12\cdot 367}}$}\label{sec:genus2}
In this section we present the details of our solution of a DLP in the $698$-bit prime order $r_2$ subgroup of
$\F_{2^{12\cdot 367}}^{\times} =\F_{2^{4404}}^{\times}$, which arises from the MOV attack applied to the Jacobian of the genus
two supersingular curve~(\ref{curve2}). Magma verification code is provided in Appendix A.
Note that the prime order elliptic curve $E_1 / \F_{2^{367}} : Y^2 + Y = X^3 + X + 1$ with embedding degree 4
also embeds into $\F_{2^{4404}}$, so that logarithms on this curve could have easily been computed as well.
\subsection{Setup}
To compute the target logarithm, as stated in~\S\ref{sec:intro} we applied
a principle of parsimony, namely, we tried to solve all intermediate logarithms in $\F_{2^{12 \cdot 367}}$,
considered as a degree $367$ extension of $\F_{2^{12}}$, and only when this was not possible did we embed
elements into the extension field $\F_{2^{24 \cdot 367}}$ (by extending the base field to $\F_{2^{24}}$) and
solve them there.
All of the classical descent down to degree 8 was carried out over $\F_{2^{12 \cdot 367}}$, which we
formed as the compositum of the following two extension fields.
We defined $\F_{2^{12}}$ using the irreducible polynomial $U^{12} + U^3 + 1$ over $\F_2$, and defined $\F_{2^{367}}$ over $\F_2$
using the degree $367$ irreducible factor of $h_1( X^{64} )X - h_0( X^{64} )$, where $h_1 = X^5 + X^3 + X + 1$, and $h_0 = X^6 + X^4 + X^2 + X + 1$.
Let $u$ and $x$ be roots of the extension defining polynomials in $U$ and $X$ respectively, and let $c = (2^{4404}-1)/r_2$.
Then $g = x + u^7$ is a generator of $\F_{2^{4404}}^{\times}$ and $\bar{g} = g^c$ is a generator of the subgroup of order $r_2$.
As usual, our target element was chosen to be $\bar{x}_{\pi} = x_{\pi}^{c}$ where
\[
x_{\pi} = \sum_{i = 0}^{4403} (\lfloor \pi \cdot 2^{i+1} \rfloor \bmod 2) \cdot u^{11-(i \bmod 12)} \cdot x^{ \lfloor i / 12 \rfloor}.
\]
The remaining logarithms were computed using a combination of tactics, over $\F_{2^{12}}$ when possible, and over $\F_{2^{24}}$ when not.
These fields were constructed as degree 2 and 4 extensions of $\F_{2^{6}}$, respectively.
To define $\F_{2^{6}}$ we used the irreducible polynomial $T^6 + T +1$.
We then defined $\F_{2^{12}}$ using the irreducible polynomial $V^2 + tV + 1$ over $\F_{2^{6}}$, and
$\F_{2^{24}}$ using the irreducible polynomial $W^4 + W^3 + W^2 + t^3$ over $\F_{2^{6}}$.
\subsection{Degree 1 Logarithms}
It was not possible to find enough relations for degree 1 elements over $\F_{2^{12}}$, so in accordance
with our stated principle, we extended the base field to $\F_{2^{24}}$ to compute the logarithms of all $2^{24}$
degree 1 elements. We used the polynomial time relation generation from~\S\ref{sec:degree1}, which took 47 hours.
This relative sluggishness was due to the r.h.s. having degree $d_h + 1 = 7$, which must split over $\F_{2^{24}}$.
However, this was faster by a factor of $24$ than it would have been otherwise, thanks to $h_0$ and $h_1$ being
defined over $\F_{2}$. This allowed us to use the technique from~\cite{JL06} to reduce the size of the factor base
via the automorphism $(x + a) \mapsto (x + a)^{2^{367}}$, which fixes $x$ but has order $24$ on all non-subfield elements
of $\F_{2^{24}}$, since $367 \equiv 7 \bmod 24$ and $\text{gcd}(7,24) = 1$. This reduced the factor base size to $699252$ elements, which
was solved in 4896 core hours on a 24 core cluster using Lanczos' algorithm, approximately $24^2$ times faster than if we had not used
the automorphisms.
\subsection{Individual Logarithm}
We performed the standard continued fraction initial split followed by degree-balanced classical descent as in~\S\ref{1223descent},
using Magma~\cite{magma} and NTL~\cite{ntl}, to reduce the target element to an 8-smooth product in $641$ and
$38224$ core hours respectively. The most interesting part of the descent was the elimination of the elements of degree up to $8$
over $\F_{2^{12}}$ into elements of degree one over $\F_{2^{24}}$, which we detail below.
This phase was completed using Magma and took a further $8432$ core hours. However, we think that the combined time of
the classical and non-classical parts could be reduced significantly via a backwards-induction analysis of the elimination
times of each degree.
\subsubsection{Small Degree Elimination}
As stated above we used several tactics to achieve these eliminations. The
first was the splitting of an element of even degree over $\F_{2^{12}}$ into two elements of half the degree (which had the same logarithm
modulo $r_2$) over the larger field.
This automatically provided the logarithms of all degree 2 elements over $\F_{2^{12}}$.
Similarly elements of degree 4 and 8 over $\F_{2^{12}}$ were rewritten as elements of degree 2 and 4 over $\F_{2^{24}}$,
while we found that degree 6 elements were eliminable more efficiently by initially continuing the descent over $\F_{2^{12}}$, as with degree
$5$ and $7$ elements.
The second tactic was the application of Joux's Gr\"obner basis elimination method from~\S\ref{sec:smallelim} to elements
over $\F_{2^{12}}$, as well as elements over $\F_{2^{24}}$. However, in many cases condition~(\ref{GBelimprob}) was violated, in which
case we had to employ various recursive strategies in order to eliminate elements. In particular, elements of the same degree were allowed
on the r.h.s. of relations, and we then attempted to eliminate these using the same (recursive) strategy. For degree $3$ elements over
$\F_{2^{12}}$, we even allowed degree $4$ elements to feature on the r.h.s. of relations, since these were eliminable via the factorisation into
degree $2$ elements over $\F_{2^{24}}$.
In Figure 1 we provide a flow chart for the elimination of elements of degree up to $8$ over $\F_{2^{12}}$, and
for the supporting elimination of elements of degree up to $4$ over $\F_{2^{24}}$. Nearly all of the arrows in Figure 1 were necessary
for these field parameters (the exceptions being that for degrees 4 and 8 over $\F_{2^{12}}$ we could have initially continued the
descent along the bottom row, but this would have been slower). The reason this `non-linear' descent arises is due to $q$ being
so small, and $d_H$ being relatively large, which increases the degree of the r.h.s. cofactors, thus decreasing the smoothness probability.
Indeed these tactics were only borderline applicable for these parameters; if $h_0$ or $h_1$ had
degree any larger than $6$ then not only would most of the descent have been much harder, but it seems that one would be forced to compute
the logarithms of degree 2 elements over
$\F_{2^{24}}$ using Joux's linear system method from~\S\ref{sec:degree2}, greatly increasing the required number of core hours.
As it was, we were able to eliminate degree 2 elements over $\F_{2^{24}}$ on the fly, as we describe explicitly below.
Finally, we note that our descent strategy is considerably faster than the alternative of embedding the DLP into $\F_{2^{24 \cdot 367}}$
and performing a full descent in this field, even with the elimination on the fly of degree 2 elements over $\F_{2^{24}}$, since much of the resulting
computation would constitute superfluous effort for the task in hand.
\begin{figure}[t]\label{GBpic}
\tikzstyle{line} = [draw, -latex']
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.8cm,
thick,main node/.style={circle,draw,font=\sffamily\Large\bfseries}, scale=1, transform shape]
\node[main node] (1) {1};
\node[main node] (2) [right of=1] {2};
\node[main node] (3) [right of=2] {3};
\node[main node] (4) [right of=3] {4};
\node[main node] (1a) [below of=1] {1};
\node[main node] (2a) [below of=2] {2};
\node[main node] (3a) [below of=3] {3};
\node[main node] (4a) [below of=4] {4};
\node[main node] (5a) [right of=4a] {5};
\node[main node] (6a) [right of=5a] {6};
\node[main node] (7a) [right of=6a] {7};
\node[main node] (8a) [right of=7a] {8};
\node (f24) [left of=1] {\Large{$\F_{2^{24}}$}};
\node (f12) [left of=1a] {\Large{$\F_{2^{12}}$}};
\path [line] (f12) edge [-] (f24);
\path [line] (1a) edge node [left] {$\iota$} (1);
\path [line,dotted] (3a) edge node [left] {$\iota$} (3);
\path [line] (2a) edge node [right] {\hspace{0.7mm}$s$} (1);
\path [line,dashed] (3a) edge [loop below] node {} (3a);
\path [line,dashed] (3a) edge node {} (4a);
\path [line] (4a) edge [bend right=10] node [right] {\hspace{2mm}$s$} (2);
\path [line,dashed] (6a) edge node {} (5a);
\path [line,dashed] (7a) edge node {} (6a);
\path [line] (8a) edge [bend right=10] node [right] {\hspace{6mm}$s$} (4);
\path [line,dotted] (5a) edge [bend right] node {} (4a);
\path [line,dashed] (4) edge [loop above] node {} (4);
\path [line,dashed] (4) edge node {} (3);
\path [line,dashed] (5a) edge node {} (4a);
\path [line,dashed] (2) edge [loop above] node {} (2);
\path[every node/.style={font=\sffamily\small}]
(2) edge node {} (1)
(3) edge node {} (2)
(2) edge node {} (1)
(3a) edge node {} (2a)
(6a) edge [bend left ] node {} (4a)
(5a) edge [bend left ] node {} (3a)
(4) edge [ bend right] node {} (2)
(7a) edge [ bend left] node {} (5a);
\end{tikzpicture}
\caption{This diagram depicts the set of strategies employed to eliminate elements over $\F_{2^{12}}$ of degree up to $8$.
The encircled numbers represent the degrees of elements over $\F_{2^{12}}$ on the bottom row, and over $\F_{2^{24}}$ on the top row.
The arrows indicate how an element of a given degree is rewritten as a product of elements of other degrees, possibly over the larger field.
Unadorned solid arrows indicate the maximum degree of elements obtained on the l.h.s. of the Gr\"obner basis elimination method; likewise dashed
arrows indicate the degrees of elements obtained on the r.h.s. of the Gr\"obner basis elimination method, when these are greater than those obtained
on the l.h.s. Dotted arrows indicate a fall-back strategy when the initial strategy fails.
An $s$ indicates that the element is to be split over the larger field into two elements of half the degree.
An $\iota$ indicates that an element is promoted to the larger field. Finally, a loop indicates that one must use a recursive strategy in which further
instances of the elimination in question must be solved in order to eliminate the element in question.
}
\end{figure}
\subsubsection{Degree 2 Elimination over $\F_{2^{24}}$}
Let $Q(Y)$ be a degree two element which is to be eliminated, i.e., written as a product of degree one elements.
As in~\cite{GGMZ13a,GGMZ13b} we first precompute the set of $64$ elements $B \in \F_{2^{24}}$ such that the polynomial
$f_B(X) = X^{65} + BX + B$ splits completely over $\F_{2^{24}}$
(in fact these $B$'s happen to be in $\F_{2^{12}}$, but this is not relevant to the method).
We then find a Gaussian-reduced basis of the lattice $L_{Q(Y)}$ defined by
\[
L_{Q(Y)} = \{(w_0(Y),w_1(Y)) \in \F_{2^{24}}[Y]^2 : w_0(Y) \,h_0(Y) + w_1(Y)\, h_1(Y) \equiv 0 \pmod{Q(Y)}\} \:.
\]
Such a basis has the form $(u_{0}, Y+u_{1}), (Y+v_{0}, v_{1})$, with $u_i,v_i \in \F_{2^{24}}$,
except in rare cases, see Remark 1.
For $s\in\F_{2^{24}}$ we obtain lattice elements $(w_0(Y), w_1(Y)) = (Y + v_{0} + su_{0}, sY + v_{1} + su_{1})$.
Using the transformation detailed in \S\ref{sec:relgen}, for each $B \in \F_{2^{24}}$ such that $f_B$ splits completely over $\F_{2^{24}}$
we perform a Gr\"obner basis computation to find the set of $s \in \F_{2^{24}}$ that satisfy
\[
B = \frac{( s^{64} + u_{0}s + v_{0})^{65}}
{(u_{0}s^2 + (u_{1} + v_{0})s+ v_{1} )^{64}} \:,
\]
by first expressing $s$ in a $\F_{2^{24}} / \F_{2^{6}}$ basis, which results in a quadratic system in $4$ variables.
This ensures that the l.h.s. splits completely over $\F_{2^{24}}$.
For each such $s$ we check whether the r.h.s. cofactor of $Q(Y)$, which has degree $5$, is $1$-smooth.
If this occurs, we have successfully eliminated $Q(Y)$.
However, one expects on average just one $s$ per $B$, and so the probability of $Q(Y)$ being eliminated in this way is
$1 - (1 - 1/5!)^{64} \approx 0.415$, which was borne out in practice to two decimal places.
Hence, we adopted a recursive strategy in which we stored all of the r.h.s. cofactors
whose factorisation degrees had the form $(1,1,1,2)$ (denoted type 1), or $(1,2,2)$ (denoted type 2). Then for each type 1
cofactor we checked to see if the degree 2 factor was eliminable by the above method. If none were eliminable
we stored every type 1 cofactor of each degree 2 irreducible occurring in the list of type 1 cofactors of $Q(Y)$.
If none of these were eliminable (which occurred with probability just $0.003$), then we reverted to the type 2
cofactors, and adopted the same strategy just specified for each of the degree 2 irreducible factors.
Overall, we expected our strategy to fail about once in every $6 \cdot 10^6$ such $Q(Y)$. This happened just once during our descent,
and so we multiplied this $Q(Y)$ by a random linear polynomial over
$\F_{2^{24}}$ and performed a degree 3 elimination, which
necessitates an estimated 32 degree 2 polynomials being simultaneously eliminable by the above method, which
thanks to the high probability of elimination, will very likely be successful for any linear multiplier.
\subsection{Summary}
Finally, after a total of approximately 52240 core hours (or $2^{48}$ $M_{r_2}$),
we found that $\bar{x}_{\pi} = \bar{g}^{\text{log}}$, with $\text{log} =$
\begin{align*}
&40932089202142351640934477339007025637256140979451423541922853874473604
\\[-.5mm]
&39015351684721408233687689563902511062230980145272871017382542826764695
\\[-.5mm]
&59843114767895545475795766475848754227211594761182312814017076893242 \:.
\\[-.5mm]
\end{align*}
\begin{remark}
During the descent, we encountered several polynomials $Q(Y)$ that were apparently not eliminable via the Gr\"obner basis method.
We discovered that they were all factors of $h_1(Y) \cdot c + h_0(Y)$ for $c \in \F_{2^{12}}$ or $\F_{2^{24}}$, and hence
$h_0(Y)/h_1(Y) \equiv c \pmod{Q(Y)}$.
This implies that~(\ref{fieldrelation}) is equal to $F(c)G^{(q)}(Y)+F^{(q)}(Y)G(c)$ modulo $Q(Y)$,
where $G^{(q)}$ denotes the Frobenius twisted $G$ and
similarly for $F^{(q)}$.
This cannot become $0$ modulo $Q(Y)$ if the degrees
of $F$ and $G$ are smaller than the degree of $Q$, unless $F$ and $G$ are both constants.
However, thanks to the field representation, finding the logarithm of these $Q(Y)$ turns out to be easy. In particular, if
$h_1(Y) \cdot c + h_0(Y) = Q(Y) \cdot R(Y)$ then
$Q(Y) = h_1(Y) \cdot ((h_0/h_1)(Y) + c)/R(Y) = h_1(Y) \cdot (X + c) / R(Y)$, and thus modulo $r_2$ we have
$\log(Q(y)) \equiv \log(x+c) - \log(R(y))$, since $\log(h_1(y)) \equiv 0$. Since $(x + c)$ is in the factor base, if we are able to compute
the logarithm of $R(y)$, then we are done. In all the cases we encountered, the cofactor $R(y)$ was solvable by the above methods.
\end{remark}
\section{Conclusion}\label{sec:conclude}
We have introduced a new field representation and efficient descent principles which together make the recent DLP advances far more practical.
As example demonstrations, we have applied these techniques to two binary fields of central interest to pairing-based cryptography, namely
$\F_{2^{4 \cdot 1223}}$ and $\F_{2^{12 \cdot 367}}$, which arise as the embedding fields of (the Jacobians of) a genus one and a genus two
supersingular curve, respectively. When initially proposed, these fields were believed to be $128$-bit secure, and even in light of the recent
DLP advances, were believed to be $128$-bit and $94.6$-bit secure. On the contrary, our analysis indicates that the former field has approximately
$59$ bits of security and we have implemented a total break of the latter.
\bibliographystyle{plain}
\bibliography{crypto}
\section*{Appendix A}\label{sec:verify}
The following Magma script verifies the solution of the chosen DLP in the order $r_2$ subgroup of $\F_{2^{12 \cdot 367}}^{\times}$:
{
\small{
\begin{verbatim}
// Field setup
F2 := GF(2);
F2U<U> := PolynomialRing(F2);
F2_12<u> := ext< F2 | U^12 + U^3 + 1 >;
F2_12X<X> := PolynomialRing(F2_12);
modulus := (2^734 + 2^551 + 2^367 + 2^184 + 1) div (13 * 7170258097);
cofactor := (2^4404 - 1) div modulus;
h1 := X^5 + X^3 + X + 1;
h0 := X^6 + X^4 + X^2 + X + 1;
temp1 := Evaluate(h1, X^64) * X + Evaluate(h0, X^64);
temp2 := X^17 + X^15 + X^14 + X^13 + X^12 + X^11 + X^10 + X^6 + 1;
polyx := temp1 div temp2;
Fqx<x> := ext< F2_12 | polyx >;
// This is a generator for the entire multiplicative group of GF(2^4404).
g := x + u^7;
// Generate the target element.
pi := Pi(RealField(1500));
xpi := &+[ (Floor(pi * 2^(i+1)) mod 2) * u^(11-(i mod 12)) * x^(i div 12) : i in [0..4403]];
log := 4093208920214235164093447733900702563725614097945142354192285387447\
36043901535168472140823368768956390251106223098014527287101738254282676469\
559843114767895545475795766475848754227211594761182312814017076893242;
// If the following is true, then verification was successful
(g^cofactor)^log eq xpi^cofactor;
\end{verbatim}
}}
\section*{Appendix B}\label{sec:smoothness}
This section provides the algorithmic details of the smoothness testing function used in
\S\ref{sec:genus1} and \S\ref{sec:genus2}.
Given a polynomial $f(X)$ of degree $n$ over $\F_q$, in order to test
its $m$-smoothness we compute
\[ t(X) \::=\: f'(X) \prod_{\lfloor m/2 \rfloor + 1}^m
(X^{q^i} - X) \mod f \:. \]
Let $R$ be the quotient ring $\F_q[X] / \langle f \rangle$ (so that $R
\cong \F_q^n$ as vector spaces), and denote a residue class in $R$ by
$[a(X)]$. A multiplication in $R$ can be computed using $2n^2$
$\F_q$-multiplications.
In order to obtain the above product our main task is to compute
$[X^{q^i}]$ for $i \in \{ \lfloor m/2 \rfloor + 1, \dots, m\}$, after
which we can compute $t(X)$ using $\lceil m/2 \rceil$
$R$-multiplications.
\subsection*{How to Compute a Power $[X^{p^{rs}}]$}
First we explain a method how to obtain a general power $[ X^{p^{rs}}
]$, where $p$ is the characteristic of $\F_q$.
We precompute $[ X^{p^r} ], [ X^{2p^r} ], \dots, [ X^{(n-1)p^r} ]$ by
consecutively multiplying by $[ X ]$ (i.e., shifting). This requires
$(n-1)(p^r-1)$ shifts, each using $n$ $\F_q$-multiplications, so less
than $n^2 p^r$ $\F_q$-multiplications in total.
With this precomputation we then can compute $p^r$-powering in $R$,
i.e., one application of the map $\varphi:R\to R$,
$\alpha\to\alpha^{p^r}$, in the following way:
\[ \big[ \sum_{i=0}^{n-1} a_i X^i \big] ^ {p^r} =
\sum_{i=0}^{n-1} a_i^{p^r} \big[ X^{ip^r} \big] \]
This requires $n$ $p^r$-powering operations in $\F_q$ (which we ignore) and $n$
scalar multiplications in $R$, hence $n^2$ $\F_q$-multiplications.
Finally, we compute the powers $[ X^{p^{ri}} ]$ by repeatedly
applying the map~$\varphi$, i.e., $[ X^{p^{ri}} ] = \varphi^i([X]) =
\varphi^{i-1}([X^{p^r}])$, for $i\in\{2,\dots,s\}$, which requires
$(s-1)$ $p^r$-powerings in $R$. Altogether we can compute $[
X^{p^{rs}} ]$ in less than $n^2(p^r + s)$ $\F_q$-operations.
\medskip
For an alternative method of computing $[ X^{p^r} ], [ X^{2p^r} ],
\dots, [ X^{(n-1)p^r} ]$ we assume that $[ X^{p^r} ]$ is already
known. First, by multiplying by $[X]$ we obtain $[ X^{p^r+1} ], [
X^{p^r+2} ], \dots, [ X^{p^r+(n-1)} ]$ using $(n-1)$ shifts, hence
less than $n^2$ $\F_q$-multiplications. With this we can compute a
multiplication by $X^{p^r}$, i.e.,
\[ \big[ \sum_{i=0}^{n-1} a_i X^i \big] \cdot [ X^{p^r} ] =
\sum_{i=0}^{n-1} a_i \big[ X^{p^r+i} \big] \:, \]
using $n^2$ $\F_q$-multiplications. We apply this multiplication map
repeatedly in order to compute $[ X^{p^r} ], [ X^{2p^r} ], \dots, [
X^{(n-1)p^r} ]$; instead of $n^2 p^r$ $\F_q$-multiplications,
this method requires $n^3$ $\F_q$-multiplications.
\subsection*{Computing the Powers $[X^{q^i}]$}
We outline two strategies to compute the powers $[X^{q^i}]$ for
$i\in\{1,\dots,m\}$.
\paragraph{Strategy 1}
Write $q = (p^r)^s = p^{rs}$. As in the method outlined above
we do a precomputation in order to represent the $p^r$-powering
map in $R$. We then apply this map repeatedly
in order to compute $[ X^{p^{rj}} ]$ for $j\in\{2,\dots,sm\}$,
and obtain this way the powers $[X^{q^i}] = [ X^{p^{rsi}} ]$.
This method requires about $n^2(p^r+sm)$ $\F_q$-multiplications.
\paragraph{Strategy 2}
First we compute $[X^q]$ by writing $q = p^{rs}$ and using the above
method, which requires $n^2(p^r+s)$ $\F_q$-multiplications.
With this we can use the alternative method outlined above
for precomputing the $q$-powering map in $R$; here
we let $s=1$, i.e., $q=p^r$. We then apply this map
repeatedly to obtain the powers $[X^{q^i}]$.
This method requires about $n^2(p^r+s+n+m)$ $\F_q$-multiplications, and corresponds to the smoothness test in the Adj {\em et al.}
paper; but the version here has an improved running time
(the previous one was $n^2(2n+m+4\log q)$ $\F_q$-multiplications).
\subsection*{Examples}
In the case $q = 2^8$ the running time (in $\F_q$-multiplications) using Strategy~1 and $s = 2$ is $n^2(16+2m)$, while using Strategy~2 and $s = 4$ it is $n^2(8+n+m)$. When $q = 2^4$ the running time using Strategy~1 and $s = 1$ is $n^2(16+m)$, and using Strategy~2 and $s = 2$ is $n^2(6+n+m)$. Hence, for typical values of $n$ and $m$ we prefer and implement Strategy~1. For example, if $q = 2^4$, $n = 611$, $m = 94$ (see \S4.3) we need $110 n^2$ $\F_q$-multiplications.
\begin{remark}
Recall that in either case, in order to obtain $t(X)$ and thus to
complete the smoothness test, we have to consider the final $\lceil
m/2 \rceil$ $R$-multiplications. This requires an additional cost of
about $n^2m$ $\F_q$-multiplications.
\end{remark}
\end{document} | 37,297 |
A cubic foot of gold weighs approximately 1,206 pounds. A cubic inch of gold weighs approximately 0.7 pounds. Gold is a dense, heavy metal, and if a 1-gallon milk container was filled with gold, it would weigh about 161 pounds.Continue Reading
A standard bar of gold as held in the United States Mint measures 7 inches long by 3 5/8 inches wide by 1 3/4 inches tall and weighs approximately 400 ounces or 27.5 pounds. In the 21st century, the U.S. Mint held a maximum of 649.6 million ounces of gold in 1941. As of 2014, it had gold holdings of 147.3 million ounces, according to About.com.Learn more about Measurements | 93,467 |
I had so much fun doing Thursday’s tutorial post that I had to show off some more. I just love learning new things and today’s post is full of cool tips & new techniques. Enjoy!
Accessories: Melanie submitted her reversible ribbon headband tutorial. A reversible headband made out of ribbon instead of fabric.
Design: Kara submitted her tutorial on how to do tape transparency; use packing tape, laser print, then voila DIY transparencies.
Home Decor: Here’s Ashley’s article on Don’t Be Afraid of Paint: A painting tutorial for beginners. It’s a tutorial of how to paint furniture for beginners. Where she shares tips of how to do things frugally and easily. With 4 basic tips for buying and painting furniture along with a thorough tutorial on painting her hutch.
Home Decor: How perfect is this! First paint your furniture using Ashley’s tips (above) next head on over to Holly’s blog and learn how to use gold furniture glazing. Here’s Holly’s plain black consignment buffet dressed up for a party.
Photography: Lolli has hooked us up with her Top 10 Tips to Improve your Photography in 2009. It’s 10 easy-to-implement tips for the everyday photographer. No fancy camera needed!
Scrap Craft: Here’s Katie’s tutorial on how to decoupage a vase. They turned out amazing.
Sewing: Rita submitted her sewn note cards tutorial. It shows how to dress up note cards with a little machine sewing stitch.
Sewing: I’ve seen a lot of crayon rolls but take a look at Jenny’s tutorial on how to make a crayon & note book cozy. It’s a great gift for a little one in your life.
Thanks Ladies!! These are amazing and I can’t wait to get started.
Yay! Featured! I am loving that headband! Gotta love reversible ANYTHING!
I loved the photography tips. Thank you for posting it.
According to my own analysis, billions of persons on our planet receive the business loans from different banks. Thence, there’s good chances to find a bank loan in any country. | 149,957 |
"Discussion and Possible Action From Town Council Regarding The Funding and Construction of Additional Sports Fields and Related Improvements for Naranja Park."
You might ask, what is the connection between dipping into the General Fund Contingency Reserves to fund Community Center upgrades (see Part 1. of this article) and funding the construction of sports fields at Naranja Park? The answer is in the text of the agenda item. The financing plan for the sports fields at Naranja calls for a secondary property tax.
The mayor and all the sitting council members ran on a pledge of no property tax but we all know why this property tax discussion is suddenly needed. If the Town had not entered into the fiscally irresponsible decision to purchase the Community Center and Golf Courses from HSL, the .5% increase in the sales tax could have paid for the Naranja Park ball fields instead.
Sowing the seeds
Now the seeds of a property tax are starting to sprout. We finally see the results of the wasted dollars that were squandered by the purchase of the El Con Community Center and Golf Courses. The mayor and council have stated time and time again how Oro Valley can function without a property tax. Mayor Hiremath actually focused one of his State of the Town Addresses on how Oro Valley proudly does not have a property tax and was founded on the promise of no property tax.
Forcing people to subsidize other people’s golf games and sticking us with a Community Center that does not contain any of the Top 10 Amenities that residents requested in a Community Center apparently is not enough for this mayor and council. They’re now considering a property tax to cover the losses from their ill-advised, money-sucking golf course purchase.
The cost of poor leadership: $28 million dollars
The Town is suggesting General Obligation Bond funding (GO) of $17 million to finance these improvements. This will be paid back with property tax revenue of $1.4 million per year for 20 years for a total principal and interest repayment of $28 million. The secondary property tax will sunset at 20 years.
You can view the text of the agenda item here. (The property tax request is under the item labeled “Fiscal Impact.”)
How much has the El Con Real Estate Purchase cost the People so far?
Besides increasing the sales tax .5% to raise an additional $2 million per year, this fiasco has cost us $1 million for the purchase, $1.2 million to start the Community Center Fund, and thousands of dollars in illegal use of HURF funds (Highway User Revenue Funds) for Capital improvements. This is YOUR money. The Community Center fund is over $500,000 in the red with no chance in sight of becoming a positive balance.
Since its inception in 2015, the increased sales tax has raised $3,792,543. This is YOUR money that went to the El Con-HSL debacle.
---
Editor’s Note: Tonight’s Town Council Meeting begins at 6 PM in Town Council Chambers. Although there is no Public Hearing scheduled for the Naranja Park – Secondary Property Tax Agenda Item, residents opposed to this property tax are urged to attend the meeting to make your presence known. A packed room always signifies citizen awareness of a controversial agenda item.
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\title[$\AF ^1$-connectivity v.s. rational equivalence]
{\bf $\AF ^1$-connectivity on Chow monoids v.s. rational equivalence of algebraic cycles}
\author{Vladimir Guletski\u \i }
\date{07 November 2015}
\begin{abstract}
\noindent Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. Choosing an $r$-cycle of minimal degree gives rise to a chain of embeddings of Chow schemes, whose colimit is the connective Chow monoid $C_r^{\infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{\infty }(X)$ be the classifying motivic space of this monoid. In the paper we establish an isomorphism between the Chow group $CH_r(X)_0$ of degree $0$ dimension $r$ algebraic cycles modulo rational equivalence on $X$, and the group of sections of the Nisnevich sheaf of $\AF ^1$-path connected components of the loop space of $BC_r^{\infty }(X)$ at $\Spec (k)$. Equivalently, $CH_r(X)_0$ is isomorphic to the group of sections of the stabilized motivic fundamental group $\Pi _1^{S^1\wedge \AF ^1}(BC_r^{\infty }(X))$ at $\Spec (k)$.
\end{abstract}
\subjclass[2010]{14C25, 14F42, 18G55}
\keywords{}
\maketitle
\section{Introduction}
\label{intro}
Algebraic cycles are linear combinations of closed integral subschemes in algebraic varieties over a field. Two algebraic cycles $A$ and $B$ on a variety $X$ are said to be rationally equivalent if there exists an algebraic cycle $Z$ on $X\times \PR ^1$, such that, for two fundamental points $0$ and $\infty $ on $\PR ^1$, the cycle-theoretic fibres $Z(0)$ and $Z(\infty )$ coincide with $A$ and $B$ respectively. Rational equivalence is a fundamental notion in algebraic geometry, which substantially depends on the intersection multiplicities tacitly involved into the definition above. Intersection multiplicities are well controlled in cycles which are cascade intersections of cycles starting from codimension one. This is not always the case, of course. For example, if $X$ is a $K3$-surface, the Chow group of $0$-cycles modulo rational equivalence on $X$ is large, in the sense that it cannot be parametrized by an abelian variety over the ground field, \cite{Mumford}. On the other hand, its subgroup generated by divisorial intersections on $X$ is just $\ZZ $, see \cite{BeauvilleVoisin}. This example tells us that intersection multiplicities are geometrically manageable only for a small fraction of all algebraic cycles appearing in nature.
Another difficulty with algebraic cycles is that they are originally given in terms of groups, i.e. positive and negative multiplicities can appear in a cycle simultaneously. The use of negative numbers was questionable for mathematicians dealing with algebraic equations in sixteenth century. In modern terms, the concern can be expressed by saying that the completion of a monoid is a too formal construction. The problem might seem to be not that funny when passing to the completions of Chow monoids, i.e. gatherings of Chow varieties parametrizing effective cycles on projective varieties embedded into projective spaces. The Chow monoids themselves are geometrically given in terms of Cayley forms, whereas their completions are less visible.
These two things taken together have an effect that, in contrast to rational connectivity, rational equivalence is difficult to deform in a smooth projective family over a base, cf. \cite{KMM}. As a consequence, the deep conjectures on rational equivalence are hard to approach, and by now they are solved in a small number of cases (see, for example, \cite{Sur les zero-cycles}). The state of things would be possibly better if we could recode rational equivalence into more effective (i.e. positive) data, appropriate for deformation in smooth projective families over a nice base. The purpose of the present paper is to investigate whether the $\AF ^1$-homotopy type can help in finding such data.
More precisely, let $X$ be a projective variety over a field $k$, and fix an embedding of $X$ into the projective space $\PR ^m$. To avoid the troubles with representability of Chow sheaves in positive characteristic, we must assume that $k$ is of characteristic zero. Effective algebraic cycles of dimension $r$ and degree $d$ on $X$, considered with regard to the embedding $X\subset \PR ^m$, are represented by the Chow scheme $C_{r,d}(X)$ over $k$. Let $Z_0$ be an $r$-cycle of minimal degree $d_0$ on $X$. For example, if $r=0$ then $Z_0$ can be a point, and if $r=1$ then $Z_0$ can be a line on $X$. The cycle $Z_0$ gives rise to a chain of embeddings $C_{r,d}(X)\subset C_{r,d+d_0}(X)$, whose colimit $C_r^{\infty }(X)$ is the {\it connective Chow monoid} of effective $r$-cycles on $X$. Let $C_r^{\infty }(X)^+$ be the group completion of $C_r^{\infty }(X)$ in the category of set-valued simplicial sheaves on the smooth Nisnevich site over $k$. Let also $\Pi _0^{\AF ^1}$ be the functor of $\AF ^1$-connected components and $\Pi _1^{\AF ^1}$ be the functor of the $\AF ^1$-fundamental group on simplicial Nisnevich sheaves, see \cite{MorelVoevodsky} or \cite{AsokMorel}.
Now, consider the Chow group $CH_r(X)_0$ of degree zero $r$-cycles modulo rational equivalence on $X$, where the degree of cycle classes is given with regard to the embedding of $X$ into $\PR ^m$ over $k$. Any finitely generated field extension $K$ of the ground field $k$ is the function field $k(Y)$ of an irreducible variety $Y$ over $k$. For a simplicial sheaf $\bcF $, let $\bcF (K)$ be the stalk of $\bcF $ at the generic point $\Spec (K)$ of the variety $Y$. In the paper we establish a canonical (up to a projective embedding) isomorphism
$$
CH_r(X_K)_0\simeq \Pi _0^{\AF ^1}(C_r^{\infty }(X)^+)(K)\; ,
$$
for an arbitrary finitely generated field extension $K$ over $k$ (Theorem \ref{main1}).
Let, furthermore, $BC_r^{\infty }(X)$ be the motivic classifying space of the connective Chow monoid $C_r^{\infty }(X)$. We also prove that
$$
CH_r(X_K)_0\simeq
\Pi _0^{\AF ^1}(\Omega \Ex BC_r^{\infty }(X))(K)
\; ,
$$
where $\Omega $ is right adjoint to the simplicial suspension $\Sigma $ in the pointed category of simplicial Nisnevich sheaves, and $\Ex $ is a fibrant replacement functor for simplicial Nisnevich sheaves (Corollary \ref{main2.5}). Another reformulation of the main result is in terms of $S^1\wedge \AF ^1$-fundamental groups, where $S^1$ is the simplicial circle. Namely,
$$
CH_r(X_K)_0\simeq
\Pi _1^{S^1\wedge \AF ^1}
(BC_r^{\infty }(X))(K)\; ,
$$
i.e. the Chow group of $r$-cycles of degree zero modulo rational equivalence on $X$ is isomorphic to the stalk at $\Spec (K)$ of the $S^1\wedge \AF ^1$-fundamental group of the motivic classifying space of the Chow monoid $C_r^{\infty }(X)$ (Corollary \ref{main3}). The smashing by $S^1$ is a sort of stabilization, and not yet fully understood.
The use of the second isomorphism is that it encodes rational equivalence on $r$-cycles in terms of $\AF ^1$-path connectedness on the motivic space $L_{\AF ^1}\Omega \Ex BC_r^{\infty }(X)$. The localization functor $L_{\AF ^1}$ is a transfinite machine, which can be described in terms of sectionwise fibrant replacement, the Godement resolution, homotopy limit of the corresponding cosimplicial simplicial sheaves and the Suslin-Voevodsky's singularization functor. The quadruple operation $L_{\AF ^1}\Omega \Ex B$ is a bigger machine recoding rational equivalence into $\AF ^1$-path connectivity, at some technical cost, of course.
The proof of the main result (Theorem \ref{main1}) is basically a gathering of known facts in $\AF ^1$-homotopy theory of schemes and Chow sheaves, collected in the right way. The substantial arguments are Lemma \ref{Pi0&completion} and the use of Proposition 6.2.6 from the paper \cite{AsokMorel} by Asok and Morel. In Section \ref{concommon} we introduce the needed tools from simplicial sheaves on a small site and the functor $\Pi _0$. Section \ref{localization} is devoted to the Bousfield localization of simplicial sheaves by an interval and to proving Lemma \ref{Pi0&completion} which says that the group completion commutes with the localized $\Pi _0$. In Section \ref{Chowsheaves} we pass to Nisnevich sheaves on schemes and recall the theory of Chow sheaves following \cite{SV-ChowSheaves}. The main results appear in Section \ref{rateq-ratcon}, where we prove the existence of the canonical (up to a projective embedding) isomorphisms between the Chow groups and the stalks of the corresponding motivic homotopy groups of $C_r^{\infty }(X)^+$ and $BC_r^{\infty }(X)$. In Appendix we collect the needed basics from homotopical algebra, in order to make the text more self-contained.
\medskip
{\sc Acknowledgements.} The paper is written in the framework of the EPSRC grant EP/I034017/1. The author is grateful to Aravind Asok for pointing out a drawback in the proof of Corollary \ref{main3} in the first version of the paper, to Sergey Gorchinskiy for explaining how to remove the degree $1$ cycle assumption from the statement of Theorem \ref{main1}, and to both for their interest and useful comments via email and skype.
\section{$\Pi _0$ and monoids in simplicial sheaves}
\label{concommon}
Let $\Delta $ be the simplex category, i.e. the category whose objects are finite sets $[n]=\{ 0,1,\dots ,n\} $, for all $n\in \NN $, and morphisms $[m]\to [n]$ are order-preserving functions from $[m]$ to $[n]$. Let $\bcS $ be a cartesian monoidal category with a terminal object $*$. The category $\deop \bcS $ of simplicial objects in $\bcS $ is the category of contravariant functors from $\Delta $ to $\bcS $. Since $[0]$ is the terminal object in $\Delta $, the functor $\Gamma :\deop \bcS \to \bcS $, sending $\bcX $ to $\bcX _0$, is the functor of global sections on simplicial objects in $\bcS $ considered as presheaves on $\Delta $. The functor $\Gamma $ admits left adjoint $\Const :\bcS \to \deop \bcS $ sending an object $\bcX $ in $\bcS $ to the constant presheaf on $\Delta $ determined by $\bcX $.
Assume, moreover, that $\bcS $ is cocomplete. For any object $\bcX $ in $\deop \bcS $, let $\Pi _0(\bcX )$ be the coequalizer of the morphisms $\bcX _1 \rightrightarrows \bcX _0$ induced by the two morphisms from $\Delta [0]$ to $\Delta [1]$. This gives a functor $\Pi _0:\deop \bcS \to \bcS $ and the canonical epimorphism $\Psi :\Gamma \to \Pi _0$. If $\bcY $ is an object in $\bcS $, and $f:\bcX \to \Const (\bcY )$ is a morphism in $\deop \bcS $, the precompositions of $f_0:\bcX _0\to \bcY $ with the two morphisms from $\bcX _1$ to $\bcX _0$ coincide. By universality of the coequalizer, we obtain the morphism $f':\Pi _0(\bcX )\to \bcY $. The correspondence $f\mapsto f'$ is one-to-one and natural in $\bcX $ and $\bcY $. In other words, $\Pi _0$ is left adjoint to $\Const $. Since products in $\deop \bcS $ are objectwise, the functor $\Pi _0$ preserves finite products. Certainly, if $\bcC $ is the terminal category, then $\Pi _0$ is the usual functor of connected components on simplicial sets.
Let $\bcC $ be an essentially small category and let $\tau $ be a subcanonical topology on it. Assume also that $\bcC $ contains all finite products and let $*$ be the terminal object in it. Let $\bcP $ be the category of presheaves of sets on $\bcC $ and let $\bcS $ be the full subcategory of set valued sheaves on $\bcC $ in the topology $\tau $. Since $\tau $ is subcanonical, the Yoneda embedding $h:\bcC \to \bcP $, sending an object $X$ to the representable presheaf $h_X=\Hom _{\bcC }(-,X)$, and a morphism $f:X\to Y$ to the morphism of presheaves $h_f=\Hom _{\bcC }(-,f)$, takes its values in the category of sheaves $\bcS $. If $*$ is the terminal object in $\bcC $, then $h_*=\Hom _{\bcC }(-,*)$ is the terminal object in $\bcP $ and $\bcS $. Limits in $\bcS $ are limits in $\bcP $. In particular, we have objectwise finite products in $\bcS $ and the category $\bcS $ is Cartesian monoidal.
For a presheaf $\bcX $, let $\bcX ^{\shf }$ be the sheaf associated to $\bcX $ in $\tau $. Since $\bcP $ is complete, the sheafification of colimits in $\bcP $ shows that $\bcS $ is cocomplete too. In order to make a difference between $\Pi _0$ in $\deop \bcS $ and $\Pi _0$ in $\deop \bcP $, we shall denote the latter functor by $\pi _0$, so that, for a simplicial sheaf $\bcX $, one has $\Pi _0(\bcX )=\pi _0(\bcX )^{\shf }$. As the coequalizer $\pi _0$ is sectionwise, $\Pi _0(\bcX )$ is the sheafififcation of the presheaf sending $U$ to $\pi _0(\bcX (U))$.
Let $\SSets $ be the category of simplicial sets. For a natural number $n$ let $\Delta [n]$ be the representable functor $\Hom _{\Delta }(-,[n])$. For any sheaf $\bcF $ on $\bcC $ let $\Delta _{\bcF }[n]$ be the simplicial sheaf defined by the formula
$$
(\Delta _{\bcF }[n])_m(U)=
\bcF (U)\times \Hom _{\Delta }([m],[n])\; ,
$$
for any $U\in \Ob (\bcC )$ and any natural number $m$. This gives the full and faithful embeddings $\Delta _{?}[n]:\bcS \to \deop \bcS $ and $\Delta _{\bcF }[?]:\Delta \to \deop \bcS $. If $\bcF $ is $h_X$, for some object $X$ in $\bcC $, then we write $\Delta _X[n]$ instead of $\Delta _{\bcF }[n]$, and use $\Delta [n]$ instead of $\Delta _*[n]$. To simplify notation further, we shall identify $\bcC $ with its image in $\deop \bcS $ under the embedding $\Delta _{?}[0]$. For example, for any object $X$ in $\bcC $ it is the same as the corresponding simplicial sheaf $\bcX =\Delta _X[0]=\Const (h_X)$, and the same on morphisms in $\bcC $. The cosimplicial object $\Delta [?]:\Delta \to \deop \bcS $ determines the embedding of simplicial sets into $\deop \bcS $, so that we may also identify $\SSets $ with its image in $\deop \bcS $. This gives the structure of a simplicial category on $\deop \bcS $, such that, for any two simplicial sheaves $\bcX $ and $\bcY $,
$$
\bHom (\bcX ,\bcY )=
\Hom _{\deop \bcS }(\bcX \times \Delta [?],\bcY )\; .
$$
The corresponding (right) action of $\SSets $ on $\deop \bcS $ is given by the formula
$$
(\bcX \times K)_n(U)=\bcX _n(U)\times K_n\; ,
$$
for any simplicial sheaf $\bcX $ and simplicial set $K$. For simplicity of notation, we shall write $\Delta [n]$ instead of $\Delta [n]$. Then $\Delta _X[n]$ is the product of $\Delta _X[0]$ and $\Delta [n]$.
Looking at $\deop \bcS $ as a symmetric monoidal category with regard to the categorical product in it, one sees that it is closed symmetric monoidal. The internal Hom, bringing right adjoint to the Cartesian products, is given by the formula
$$
\cHom (\bcX ,\bcY )_n(U)=
\Hom _{\deop \bcS }(\bcX \times \Delta _U[n],\bcY )\; .
$$
Throughout the paper we will be working with monoids in $\deop \bcS $. All monoids and groups will be commutative by default. If $\bcX $ is a monoid in $\deop \bcS $, let $\bcX ^+$ be the group completion of $\bcX $ in $\deop \bcS $. The terms of $\bcX ^+$ are the sheaves associated with the sectionwise completions of the terms of $\bcX $ in $\bcP $. One has a morphism from $\bcX \times \bcX $ to $\bcX ^+$, which is an epimorphism in $\deop \bcS $. When no confusion is possible, the termwise and section-wise completion of $\bcX $ in $\deop \bcP $ will be denoted by the same symbol $\bcX ^+$.
Monoids form a subcategory in $\bcP $. The corresponding forgetful functor has left adjoint sending presheaves to free monoids with concatenation as monoidal operation. The notion of a cancellation monoid in $\bcP $ is standard and sectionwise. A free monoid in $\bcP $ is a cancellation monoid. As limits and colimits in $\deop \bcS $ are termwise, the functors $\Gamma $ and $\Const $ preserve monoids and groups and $\Gamma (\bcX ^+)$ is the same as $\Gamma (\bcX )^+$. Since $\Pi _0$ commutes with finite products, it follows that $\Pi _0$ also preserves monoids and groups.
The monoid of natural numbers $\NN $ is a simplicial sheaf on $\bcC $. A pointed monoid in $\deop \bcS $ is a pair $(\bcX ,\iota )$, where $\bcX $ is a monoid in $\deop \bcS $ and $\iota $ is a morphism of monoids from $\NN $ to $\bcX $. A graded pointed monoid is a triple $(\bcX ,\iota ,\sigma )$, where $(\bcX ,\iota )$ is a pointed monoid and $\sigma $ is a morphism of monoids from $\bcX $ to $\NN $, such that $\sigma \circ \iota =\id _{\NN }$, see page 126 in \cite{MorelVoevodsky}. Notice that to define a morphism from $\NN $ to $\bcX $ is equivalent to choose an element in $\bcX _0(*)$.
Let $(\bcX ,\iota ,\sigma )$ be a pointed graded monoid in $\deop \bcS $. Since $\sigma \circ \iota =\id _{\NN }$, it follows that, for any natural $n$ and any object $U$ in $\bcC $, we have two maps $\iota _{U,n}:\NN \to \bcX _n(U)$ and $\sigma _{U,n}:\bcX _n(U)\to \NN $. It implies that $\bcX _n(U)$ is the coproduct of the sets $\sigma _{U,\, n}^{-1}(d)$, for all $d\geq 0$. The sets $\sigma _{U,\, n}^{-1}(d)$ give rise to the simplicial sheaf which we denote by $\bcX ^d$. Then $\bcX $ is the coproduct of $\bcX ^d$ for all $d\geq 0$. The addition of $\iota (1)$ in $\bcX $ induces morphisms of simplicial sheaves $\bcX ^d\to \bcX ^{d+1}$ for all $d\geq 0$. Let $\bcX ^{\infty }$ be the colimit
$$
\bcX ^{\infty }=
\colim (\bcX ^0\to \bcX ^1\to \bcX ^2\to \dots )
$$
in $\deop \bcS $. Equivalently, $\bcX ^{\infty }$ is the coequalizer of the addition of $\iota (1)$ in $\bcX $ and the identity automorphism of $\bcX $.
Since now we shall assume that the topos $\bcS $ has enough points, and the category $\bcC $ is Noetherian. Since filtered colimits commute with finite products, $\bcX ^{\infty }$ is the colimit taken in the category of simplicial presheaves, i.e. there is no need to take its sheafification. The commutativity of filtered colimits with finite products also yields the canonical isomorphism between the colimit of the obvious diagram composed by the objects $\bcX ^d\times \bcX ^{d'}$, for all $d,d'\geq 0$, and the product $\bcX ^{\infty }\times \bcX ^{\infty }$. Since the colimit of that diagram is the colimit of its diagonal, this gives the canonical morphism from $\bcX ^{\infty }\times \bcX ^{\infty }$ to $\bcX ^{\infty }$. The latter defines the structure of a monoid on $\bcX ^{\infty }$, such that the canonical morphism
$$
\pi :\bcX =\coprod _{d\geq 0}\bcX ^d\to \bcX ^{\infty }
$$
is a homomorphism of monoids in $\deop \bcS $. We call $\bcX ^{\infty }$ the {\it connective} monoid associated to the pointed graded monoid $\bcX $.
Notice that the category of simplicial sheaves is exhaustive. In particular, if all the morphisms $\bcX ^d\to \bcX ^{d+1}$ are monomorphisms, the transfinite compositions $\bcX ^d\to \bcX ^{\infty }$ are monomorphisms too. This happens if $\bcX $ is a termwise sectionwise cancelation monoid, in which case $\bcX ^{\infty }$ is a termwise sectionwise cancelation monoid too.
The above homomorphisms $\pi $ and $\sigma $ give the homomorphism $(\pi ,\sigma )$ from $\bcX $ to $\bcX ^{\infty }\times \NN $. Passing to completions we obtain the homomorphism $(\pi ^+,\sigma ^+)$ from $\bcX ^+$ to $(\bcX ^{\infty })^+\times \ZZ $.
\begin{lemma}
\label{goraimysh}
Assume $\bcX $ is a sectionwise cancelation monoid. Then
$$
(\pi ^+,\sigma ^+):\bcX ^+\to (\bcX ^{\infty })^+\times \ZZ
$$
is an isomorphism.
\end{lemma}
\begin{pf}
Since the site $\bcS $ has enough points, it suffices to prove the lemma sectionwise and termwise. Then, without loss of generality, we may assume that $\bcX $ is a set-theoretical pointed graded cancelation monoid. Clearly, $\iota ^+$ is an injection, $\pi ^+$ is a surjection, and $\pi ^+\iota ^+=0$. Since $\bcX $ is a cancelation monoid, $\bcX ^+$ is the quotient-set of the set $\bcX \times \bcX $ modulo an equivalence relation
$$
(x_1,x_2)\sim (x_1',x_2')\Leftrightarrow
x_1+x_2'=x_2+x_1'\; .
$$
For any element $(x_1,x_2)$ in $\bcX \times \bcX $ let $[x_1,x_2]$ be the corresponding equivalence class. Since $\bcX $ is a cancelation monoid, so is the monoid $\bcX ^{\infty }$ too. If $\pi ^+[x_1,x_2]$ is zero, that is $[\pi (x_1),\pi (x_2)]=[0,0]$ in $(\bcX ^{\infty })^+$, it is equivalent to say that $\pi (x_1)=\pi (x_2)$. The latter equality means that there exists a positive integer $n$, such that $x_2=x_1+n\iota (1)$, i.e. $[x_1,x_2]=[0,n\iota (1)]$ in $\bcX ^+$. The element $[0,n\iota (1)]$ sits in the image of $\iota ^+$.
\end{pf}
\section{Homotopy completion and localization of $\Pi _0$}
\label{localization}
All the above considerations were categorical. Let us now switch to homotopical algebra and consider the injective model structures on $\deop \bcS $. Recall that a point $P$ of a topos $\bcT $ is an adjoint pair of functors, $P^*:\bcT \to \Sets $ and $P_*:\Sets \to \bcT $, such that $P^*$ is left adjoint to $P_*$ and preserves finite limits in $\bcT $. If $\bcX $ is an object of $\bcT $, then $\bcX _P=P^*(X)$ is the stalk of $\bcX $ at the point $P$. We will assume that the topos $\bcS $ has enough points. Recall that it means that there exists a set of points $P(\bcS )$ of the topos $\bcS $, such that a morphism $f:\bcX \to \bcY $ in $\bcS $ is an isomorphism in $\bcS $ if and only if, for any point $P\in P(\bcS )$, the morphism $f_P:\bcX _P\to \bcY _P$, induced on stalks, is an isomorphism in the category $\Sets $. Respectively, a morphism $f:\bcX \to \bcY $ in $\deop \bcS $ is an isomorphism in $\deop \bcS $ if and only if, for each $P\in P(\bcS )$, the morphism $f_P:\bcX _P\to \bcY _P$ is an isomorphism in $\deop \Sets $.
Now, a morphism $f:\bcX \to \bcY $ in the category of simplicial sheaves $\deop \bcS $ is a weak equivalence in $\deop \bcS $ if and only if for any point $P^*:\bcS \to \Sets $ of the topos $\bcS $ the induced morphism $\deop P^*(f)$ on stalks is a weak equivalence of simplicial sets. Cofibrations are monomorphisms, and fibrations are defined by the right lifting property in the standard way, see Definition 1.2 on page 48 in \cite{MorelVoevodsky}. The pair $(\bcS ,\bcM )$ is then a model category of simplicial sheaves on $\bcC $ in $\tau $. Notice that the model structure $\bcM $ is left proper, see Remark 1.5 on page 49 in loc.cit. One can also show that it is cellular. Let $\Ho $ be the homotopy category $Ho(\deop \bcS )$ of the category $\deop \bcS $ with regard to $\bcM $. For any two simplicial sheaves $\bcX $ and $\bcY $ the set of morphisms from $\bcX $ to $\bcY $ in $\Ho $ will be denoted by $[\bcX ,\bcY ]$.
The simplicial structure on $\deop \bcS $ is compatible with the model one, so that $\bcS $ is a simplicial model category. Since $$
[\bcX ,\bcY ]\simeq \pi _0\bHom (\bcX ,\bcY )
$$
and
$$
\bHom (\Delta _U[0],\bcX )\simeq \bcX (U)\; ,
$$
$\Pi _0(\bcX )$ is the sheafififcation of the presheaf
$$
\pi _0(\bcX ):U\mapsto \pi _0\bHom (\Delta _U[0],\bcX )=
[\Delta _U[0],\bcX ]=[\Const (h_U),\bcX ]
$$
on $\bcC $ in the topology $\tau $. The multiplication of simplicial sheaves and their morphisms by a simplicial set admits right adjoint, so that it commutes with colimits. In particular, $\Pi _0(\Delta _X[n])\simeq \Delta _X[0]$.
A pointed simplicial sheaf $(\bcX ,x)$ is a pair consisting of a simplicial sheaf $\bcX $ and a morphism $x$ from $*$ to $\bcX $. The definition of a morphism of pointed simplicial sheaves is obvious. Let $\deop \bcS _*$ be the category of pointed simplicial sheaves. The corresponding forgetful functor has the standard left adjoint sending $\bcX $ to the coproduct $\bcX _+$ of $\bcX $ and $*$. The model structure $\bcM $ induces the corresponding model structure on $\deop \bcS _*$, such that the above adjunction is a Quillen adjunction. Having two pointed simplicial sheaves $(\bcX ,x)$ and $(\bcY ,y)$, their wedge product $(\bcX ,x)\vee (\bcY ,y)$ is the coproduct in $\deop \bcS _*$, and the smash product $(\bcX ,x)\wedge (\bcY ,y)$ is the contraction of the wedge product in $(\bcX \times \bcY ,x\times y)$.
Let now $S^1$ be the simplical circle $\Delta [1]/\partial \Delta [1]$ pointed by the image of the boundary $\partial \Delta [1]$ in then quotient simplicial set, and let $S^1$ be its image in $\deop \bcS _*$. Define the simplicial suspension endofunctor $\Sigma $ on $\deop \bcS _*$ sending $(\bcX ,x)$ to $S^1\wedge (\bcX ,x)$. Its left adjoint is the simplicial loop functor $\Omega $ sending $(\bcX ,x)$ to $\cHom _*(S^1,(\bcX ,x))$, where $\cHom _*(-,-)$ is the obvious internal Hom in $\deop \bcS _*$.
Let $\bcX $ be a monoid in $\deop \bcS $. For any object $U$ in $\bcC $ and any positive integer $n$ let $N(\bcX _n(U))$ be the nerve of $\bcX _n(U)$, and let $B\bcX $ be the diagonal of the bisimplicial sheaf $\deop \times \deop \to \bcS $ sending $[m]\times [n]$ to the sheaf $U\mapsto N(\bcX _n(U))_m$. Then $(B\bcX )_n$ is $\bcX _n^{\times n}$ for $n>0$ and, by convention, $(B\bcX )_0$ is the terminal object $*$ in $\bcS $, see page 123 in \cite{MorelVoevodsky}. If $\bcC $ is a terminal category, then $B\bcX $ is the usual classifying space of a simplicial monoid $\bcX $ (that is, a monoid in the category of simplicial sets $\deop \Sets $). Just as in topology, there exists a canonical morphism from $\bcX $ to $\Omega B(\bcX )$, which is a weak equivalence if $\bcX $ is a group, loc.cit.
Following Quillen, \cite{QuillenGroupCompletion}, we will say that a simplicial monoid $\bcX $ is good if the morphism $B\bcX \to B\bcX ^+$, induced by the canonical morphism from $\bcX $ to $\bcX ^+$, is a weak equivalence in $\deop \Sets $. If $X$ is a set-theoretical monoid, then $X$ is good if the corresponding constant simplicial monoid $\bcX =\Const (X)$ is good as a simplicial monoid. If $X$ is a free monoid in $\Sets $, then $\bcX =\Const (X)$ is good in $\deop \Sets $, see Proposition Q.1 in loc.cit.
Recall that, for any point $P$ of the topos $\bcS $, the functor $P^*:\bcS \to \Sets $ preserves finite limits. It follows that, if $\bcX $ is a simplicial sheaf monoid, then the stalk $(B\bcX )_P$ of the classifying space $B\bcX $ at $P$ is canonically isomorphic to the classifying space $B(\bcX _P)$ of the stalk $\bcX _P$ of the simplicial sheaf $\bcX $ at $P$. We will say that a simplicial sheaf monoid $\bcX $ is {\it pointwise good}, if the morphism $(B\bcX )_P\to (B\bcX ^+)_P$, is a weak equivalence of simplicial sets for each point $P$ in $P(\bcS )$. This is, of course, equivalent to saying that the morphism $B\bcX \to B\bcX ^+$ is a weak equivalence in $\deop \bcS $, with regard to the model structure $\bcM $.
Now, if $\bcX $ is a monoid in $\bcS $, we will say that $\bcX $ is {\it pointwise free} if $\bcX _P$ is a free monoid in $\Sets $ for each point $P$ in $P(\bcS )$. If $\bcX $ is pointwise free, it does not necessarily mean that $\bcX $ is a free monoid in the category $\bcS $. It is important, however, that if $\bcX _0$ is a pointwise free monoid in $\bcS $, the corresponding constant simplicial sheaf monoid $\bcX =\Const (\bcX _0)$ is pointwise good, which is a straightforward consequence of the first part of Proposition Q.1 in \cite{QuillenGroupCompletion}.
Similarly, we will say that a monoid $\bcX $ in $\bcS $ is a {\it pointwise cancellation} monoid if $\bcX _P$ is a cancellation monoid in $\Sets $ for each point $P$ in $P(\bcS )$. If $\bcX _0$ is a pointwise cancellation monoid, then the simplicial sheaf monoid $\bcX =\Const (\bcX _0)$ is pointwise good by the second part of Quillen's proposition above.
Let $\Ex $ be the fibrant replacement functor $\ExG $, for the model structure $\bcM $, constructed by taking the composition of the sectionwise fibrant replacement of simplicial sets, the Godement resolution and the homotopy limit of the corresponding cosimplicial simplicial sheaf, as in Section 2.1 in \cite{MorelVoevodsky}. Since $\Ex $ preserves finite limits, it preserves monoids and groups. For the same reason, $\Ex $ commutes with taking the classifying spaces of monoids and groups. The right derived functor of $\Omega $ can be computed by precomposing it with $\Ex $. We will need the following variation of Lemma 1.2 on page 123 in \cite{MorelVoevodsky}.
\begin{lemma}
\label{keylemma}
If $\bcX $ is pointwise good, there is a canonical isomorphism
$$
\bcX ^+\simeq \Omega \Ex B(\bcX )
$$
in the homotopy category $\Ho $.
\end{lemma}
\begin{pf}
Since $\bcX $ is pointwise good, the morphism from $B\bcX $ to $B\bcX ^+$, induced by the canonical morphism from $\bcX $ to $\bcX ^+$, is a weak equivalence in $\deop \bcS $. Applying the right derived functor $\RD \Omega $ to the weak equivalence $B\bcX \to B\bcX ^+$ and reverting the corresponding isomorphism in $\Ho $, we obtain the canonical isomorphism from $\Omega \Ex B(\bcX ^+)$ to $\Omega \Ex B(\bcX )$, in the homotopy category $\Ho $. The composition of the canonical morphism $\Sigma \bcX ^+\to B\bcX ^+$ with the weak equivalence $B\bcX ^+\to \Ex B\bcX ^+$ corresponds to the morphism $\bcX ^+\to \Omega \Ex B\bcX ^+$ under the adjunction between $\Sigma $ and $\Omega $. The latter morphism is the composition of the canonical morphism $\bcX ^+\to \Omega B\bcX ^+$ and the morphism $\Omega B\bcX ^+\to \Omega \Ex B\bcX ^+$. The morphism $\bcX ^+\to \Omega B\bcX ^+$ is a weak equivalence because $\bcX ^+$ is a group. Since any simplicial sheaf of groups $\bcG $ can be replaced, up to a weak equivalence, by a fibrant simplicial sheaf of groups, without loss of generality we may assume that $\bcX ^+$ is fibrant (see, for example, Lemma 2.32 on page 83 in \cite{MorelVoevodsky}). Replacing the functor $B$ by the universal cocycle construction $\overline W$, we see that $B$ preserves, up to a weak equivalence, fibrant objects by Theorem 31 in \cite{JardineFieldsLectures}. Then the morphism $\Omega B\bcX ^+\to \Omega \Ex B\bcX ^+$ is a weak equivalence too. Thus, we obtain an isomorphism from $\bcX ^+$ to $\Omega \Ex B\bcX ^+$ in $\Ho $.
\end{pf}
Next, let $A$ be an object of $\bcC $, and let $\bcA $ be the corresponding constant simplicial sheaf $\Delta _A[0]=\Const (h_A)$ in $\deop \bcS $. As in Appendix below, let
$$
S=\{ \bcX \wedge \bcA \to \bcX \; |\; \bcX \in \dom(I)\cup \codom (I)\}
$$
be the set of morphisms induced by the morphism from $\bcA $ to $*$, where $\dom(I)$ and $\codom (I)$ are the sets of domains and codomains of the generating cofibrations in $\bcM $ on $\bcS $. As $\deop \bcS $ is left proper simplicial cellular model category, there exists the left Bousfield localization of $\bcM $ by $S$ in the sense of Hirschhorne, see \cite{Hirsch}. Denote the localized model structure by $\bcM _A$, and let $L_A$ be the corresponding $S$-localization functor, which is a fibrant approximation in $\bcM _A$ on $\deop \bcS $, see Section 4.3 in \cite{Hirsch}, and the earlier work \cite{GoerssJardineLoc}. Let
$$
l:\Id _{\deop \bcS }\to L_A
$$
be the corresponding natural transformation. For any simplicial sheaf $\bcX $ the morphism $l_{\bcX }:\bcX \to L_A(\bcX )$ is a weak cofibration and $L_A(\bcX )$ is $A$-local, i.e. fibrant in $\bcM _A$. The basics on localization functors see Section 4.3 in Hirschhorn's book \cite{Hirsch} and Appendix below.
Let $\Ho _A$ be the homotopy category of simplicial sheaves converting weak equivalences in $\bcM _A$ into isomorphisms. As simplicial sheaves with respect to $\bcM $ form a simplicial closed cartesian monoidal model category, so is the category of simplicial sheaves with respect to $\bcM _A$. All simplicial sheaves are cofibrant, in $\bcM $ and in $\bcM _A$. It follows that the canonical functors from simplicial sheaves to $\Ho $ and $\Ho _A$ are monoidal. See Appendix for more details on all such things. For any two simplicial sheaves $\bcX $ and $\bcY $ let $[\bcX ,\bcY ]_A$ be the set of morphisms from $\bcX $ to $\bcY $ in $\Ho _A$.
Recall that an object $I$ of a category $\bcD $ with a terminal object $*$ is called an interval if there exists a morphism
$$
\mu :I\wedge I\to I
$$
and two morphisms $i_0,i_1:*\rightrightarrows I$, such that
$$
\mu \circ (\id _I\wedge i_0)=i_0\circ p
\quad \hbox{and}\quad
\mu \circ (\id _I\wedge i_1)=\id _I\; ,
$$
where $p$ is the unique morphism from $I$ to $*$, and $i_0\coprod i_1:*\coprod *\to I$ is a monomorphism in $\bcD $, see \cite{MorelVoevodsky}. Certainly, the object $A$ is an interval in $\bcC $ if and only if the object $\bcA $ is an interval in $\deop \bcS $.
Since now we shall assume that $A$ is an interval in $\bcC $. The monoidal multiplication by $\bcA $ is a natural cylinder functor on $\deop \bcS $. If $f,g:\bcX \rightrightarrows \bcY $ are two morphisms from $\bcX $ to $\bcY $ in $\deop \bcS $, a left $A$-homotopy from $f$ to $g$ is a morphism $H:\bcX \times \bcA \to \bcY $, such that $H\circ (\id _{\bcX }\times i_0)=f$ and $H\circ (\id _{\bcX }\times i_1)=g$. Since all simplicial sheaves are cofibrant in both model structures $\bcM $ and $\bcM _A$, $A$-homotopy is an equivalence relation on the set $\Hom _{\deop \bcS }(\bcX ,\bcY )$, see \cite{Hovey}, Proposition 1.2.5 (iii). Let $\Hom _{\deop \bcS }(\bcX ,\bcY )_A$ be the set of equivalence classes modulo this equivalence relation. Whenever $\bcY $ is $A$-local, the set $[\bcX ,\bcY ]_A$ is in the natural bijection with the set $\Hom _{\deop \bcS }(\bcX ,\bcY )_A$.
A point of a simplicial sheaf $\bcX $ is, by definition, a morphism from the terminal simplicial sheaf to $\bcX $. Such morphisms can be identified with the set $\bcX _0(*)$. Two points on $\bcX $ are said to be $A$-path connected if and only if they are left homotopic with respect to $\bcA $.
Since $A$ is an interval, the $A$-localizing functor $L_A$ can be chosen to be more explicit than the construction given in \cite{Hirsch}. Following \cite{MorelVoevodsky}, see page 88, we consider the cosimplicial sheaf
$$
\Delta _{A^{\bullet }}[0]:\Delta \to \bcS
$$
sending $[n]$ to the $n$-product
$$
(\Delta _A[0])^n=\Delta _{A^n}[0]
$$
and acting on morphisms as follows. For any morphism $f:[m]\to [n]$ define a morphism of sets
$$
f':\{ 1,\dots ,n\} \to \{ 0,1,\dots ,m+1\}
$$
setting
$$
f'(i)=\left\{
\begin{array}{ll}
\min \{ l\in \{ 0,\dots ,m\} \; |\; f(l)\geq i\} \; , & \mbox{if this set is nonempty} \\
m+1 & \mbox{otherwise. }
\end{array}
\right.
$$
If now $\pr _k:A^n\to A$ is the $k$-th projection and $p:A^n\to *$ the unique morphism to the terminal object, where $A^n$ is the $n$-fold product of $A$, then
$$
\pr _k\circ \Delta _{A^{\bullet }}[0](f)=\left\{
\begin{array}{ll}
\pr _{f'(k)}\; , & \mbox{if}\; f'(k)\in \{ 1,\dots ,m\} \\
i_0\circ p \; , & \mbox{if}\; f'(k)=m+1 \\
i_1\circ p\; , & \mbox{if}\; f'(k)=0\; .
\end{array}
\right.
$$
For any $\bcX $ let $\Sing _A(\bcX )$ be the Suslin-Voevodsky simplicial sheaf
$$
[n]\mapsto \cHom (\Delta _{A^n}[0],\Delta _{\bcX _n}[0])\; ,
$$
where the internal $\cHom $ is taken in the category of sheaves $\bcS $. It is functorial in $\bcX $ and $p:A^n\to *$ induces the morphism
$$
s:\Id _{\deop \bcS }\to \Sing _A\; .
$$
Notice that, although the virtue of $A$ to be an interval is not explicitly used in the Suslin-Voevodsky's construction above, it is used in proving the numerous nice properties of the functor $\Sing _A$, see \cite{MorelVoevodsky}. In particular, each morphism $s_{\bcX }$ from $\bcX $ to $\Sing _A(\bcX )$ is an $A$-local weak equivalence, i.e. a weak equivalence with regard to the model structure $\bcM _A$, see Corollary 3.8 on page 89 in loc.cit.
As it is shown in \cite{MorelVoevodsky}, there exists a sufficiently large ordinal $\omega $, such that $L_A$ can be taken to be the composition
$$
L_A=(\Ex \circ \Sing _A)^{\omega }\circ \Ex \; ,
$$
where $\Ex $ is the functor $\ExG $, i.e. the composition of the sectionwise fibrant replacement, the Godement resolution and the homotopy limit of the corresponding cosimplicial simplicial sheaf (see above). Such constructed localization functor $L_A$ is quite explicit, which gives a clearer picture of what are the functors $\pi _0^A$ and $\Pi _0^A$.
The canonical functor from $\deop \bcS $ to $\Ho $ preserves products. In other words, if $\bcX \times \bcY $ is the product of two simplicial sheaves, the same object $\bcX \times \bcY $, with the homotopy classes of the same projections, is the product of $\bcX $ and $\bcY $ in $\Ho $ and in $\Ho _A$ (see Appendix). The advantage of the above explicit $L_A$ is that it commutes with finite products, see Theorem 1.66 on pages 69 - 70 and the remark on page 87 in \cite{MorelVoevodsky}. Most likely, the general Hirschhorne's construction (see Section 4.3 in \cite{Hirsch}) also enjoys this property, but we could not find the proof in the literature.
\begin{remark}
\label{filin}
{\rm The left derived to any localization functor $L_A$ from $\deop \bcS $ to $A$-local objects in $\deop \bcS $ is left adjoint to the right derived of the forgetful functor in the opposite direction on the homotopy level, see Theorem 2.5 on page 71 in \cite{MorelVoevodsky}. This implies, in particular, that any two localizations $L_A$ and $L'_A$ are weak equivalent to each other. Therefore, in all considerations up to (pre-$A$-localized) weak equivalence in $\deop \bcS $ we may freely exchange the localization functor $L_A$ considered in \cite{Hirsch} by the concrete Suslin-Voevodsky's one, and vice versa.
}
\end{remark}
\begin{lemma}
\label{triton}
For any simplicial sheaf $\bcX $ the canonical map
$$
\Hom _{\deop \bcS }(*,\bcX )_A
\to
\Hom _{\deop \bcS }(*,L_A(\bcX ))_A
$$
is surjective.
\end{lemma}
\begin{pf}
We know that the natural transformation $l:\Id \to L_A$ induces the epimorphism $\Pi _0\to \Pi _0^A$ by Corollary 3.22 in \cite{MorelVoevodsky}. The morphism $\Psi :\Gamma \to \Pi _0$ is an epimorphism too. This gives that the map
$$
\Hom _{\bcS }(*,\bcX _0)
\to
\Hom _{\bcS }(*,\Pi _0(\bcX ))
\to
\Hom _{\bcS }(*,\Pi _0^A(\bcX ))
$$
is surjective. By adjunction, $\Hom _{\bcS }(*,\bcX _0)\simeq \Hom _{\deop \bcS }(*,\bcX )$, and since $L_A(\bcX )$ is $A$-local, $\Hom _{\bcS }(*,\Pi _0^A(\bcX ))$ is isomorphic to $\Hom _{\deop \bcS }(*,L_A(\bcX ))_A$.
\end{pf}
Now, define the $A$-localized functor $\Pi ^A_0$ from $\deop \bcS $ to $\bcS $ by setting $\Pi ^A_0(\bcX )$ to be the sheaf associated to the presheaf
$$
U\mapsto [\Const (h_U),\bcX ]_A\; .
$$
Then $\Pi ^A_0(\bcX )$ is canonically isomorphic to $\Pi _0(L_A(\bcX ))$, and the morphism $l$ induces the epimorphism $\Pi _0\to \Pi _0^A$, see Corollary 3.22 on page 94 in \cite{MorelVoevodsky}. As $L_A$ is monoidal,
$$
\begin{array}{rcl}
\Pi ^A_0(\bcX \times \bcY )
&=&
\Pi _0(L_A(\bcX \times \bcY )) \\
&=&
\Pi _0(L_A(\bcX )\times L_A(\bcY )) \\
&=&
\Pi _0(L_A(\bcX ))\times \Pi _0(L_A(\bcY )) \\
&=&
\Pi ^A_0(\bcX )\times \Pi ^A_0(\bcY )\; .
\end{array}
$$
This gives that $\Pi _0^A$ preserves monoids and groups.
\begin{lemma}
\label{bloha}
For any monoid $\bcX $ in $\deop \bcS $, one has a canonical isomorphism
$$
\Pi _0(\bcX )^+\simeq \Pi _0(\bcX ^+)
$$
in $\bcS $.
\end{lemma}
\begin{pf}
Since $\Gamma (\bcX ^+)=\Gamma (\bcX )^+$ and $\Pi _0(\bcX )^+$ are completions, one has the universal morphisms $\gamma $ from $\Gamma (\bcX ^+)$ to $\Pi _0(\bcX )^+$ and $\delta $ from $\Pi _0(\bcX )^+$ to $\Pi _0(\bcX ^+)$. Since $\Gamma (\bcX )=\bcX _0$, $\Gamma (\bcX ^+)=\bcX ^+_0$ and $\gamma \circ \Gamma (\nu _{\bcX })=\nu _{\Pi _0(\bcX )}\circ \Psi $, where $\nu $ stays for the corresponding canonical morphisms from the monoids to their completions, the two compositions $\bcX ^+_1\rightrightarrows \bcX ^+_0\stackrel{\gamma }{\to }\Pi _0(\bcX )^+$ coincide, which gives the universal morphism $\varepsilon $ from $\Pi _0(\bcX ^+)$ to $\Pi _0(\bcX )^+$. Since $\Psi $ is an epimorphism, and using the uniqueness of the appropriate universal morphisms, we show that $\delta $ and $\varepsilon $ are mutually inverse isomorphisms of groups in $\bcS $.
\end{pf}
Let $\CMon (\deop \bcS )$ be the category of commutative monoids in $\deop \bcS $. Suppose that all cofibrations in $\deop \bcS $ are symmetrizable, see \cite{GorchinskiyGuletskii}. Then the simplicial model structure on $\deop \bcS $ gives rise to a simplicial model structure on $\CMon (\deop \bcS )$, compatible with Bousfield localizations, see \cite{PavlovScholbach1}, \cite{PavlovScholbach2}, \cite{PavlovScholbach3}, \cite{White1} and \cite{White2}. A morphism in $\CMon (\deop \bcS )$ is a weak equivalence (respectively, fibration) if and only if it is a weak equivalence (respectively, fibration) in $\deop \bcS $, loc.cit. The classifying space functor $B$ is then a functor from the model category $\CMon (\deop \bcS )$ to the model category $\deop \bcS $. Lemma 2.35 on page 85 in \cite{MorelVoevodsky}, and the universality of a left localization of a model structure (see part (b) of the Definition 3.1.1 on pp 47 - 48 of \cite{Hirsch}), being applied to the functor $B$, yield a (simplicial) weak equivalence
$$
(B\circ L_A)(\bcX )\simeq (L_{S^1\wedge A}\circ B)(\bcX )\; ,
$$
for any commutative monoid $\bcX $ in $\deop \bcS $.
\begin{lemma}
\label{Pi0&completion}
For any pointwise good commutative monoid $\bcX $ in $\deop \bcS $, one has a canonical isomorphism
$$
\Pi ^A_0(\bcX )^+\simeq
\Pi ^A_0(\bcX ^+)\; .
$$
in $\bcS $.
\end{lemma}
\begin{pf}
Since $\bcX $ is pointwise good, one has the isomorphism
$$
\bcX ^+\simeq (\Omega \circ \Ex \circ B)(\bcX )
$$
in $\Ho $ by Lemma \ref{keylemma}, where $\Omega $ is the simplicial loop functor and $\Ex $ is the (pre-$A$-localized) fibrant replacement for simplicial sheaves. Applying $L_A$ we get the isomorphism
$$
L_A(\bcX ^+)\simeq
L_A((\Omega \circ \Ex \circ B)(\bcX ))\; .
$$
By Theorem 2.34 on page 84 in \cite{MorelVoevodsky},
$$
L_A((\Omega \circ \Ex \circ B)(\bcX ))\simeq
(\Omega \circ \Ex \circ L_{S^1\wedge A})(B(\bcX ))\; .
$$
Since $B\circ L_A\simeq L_{S^1\wedge A}\circ B$, we obtain the isomorphism
$$
L_A(\bcX ^+)\simeq (\Omega \circ \Ex \circ B)(L_A(\bcX ))
$$
in $\Ho $. Let $\Phi =\Phi _{Mon}$ be the functor constructed in Lemma 1.1 on page 123 in \cite{MorelVoevodsky}, i.e. the cofibrant replacement functor in $\CMon (\deop \bcS )$. Since the morphism
$$
\Phi (L_A(\bcX ))\to L_A(\bcX )
$$
is a weak equivalence in $\deop \bcS $, we get the isomorphism
$$
L_A(\bcX ^+)\simeq (\Omega \circ \Ex \circ B)(\Phi (L_A(\bcX )))
$$
in $\Ho $. The monoid $\Phi (L_A(\bcX ))$ is termwise free. Therefore,
$$
(\Omega \circ \Ex \circ B)(\Phi (L_A(\bcX )))\simeq
(\Phi (L_A(\bcX )))^+
$$
by Lemma 1.2 on page 123 in \cite{MorelVoevodsky}. Applying $\Pi _0$ and using Lemma \ref{bloha}, we obtain the isomorphisms
$$
\begin{array}{rcl}
\Pi ^A_0(\bcX ^+)
&=&
\Pi _0(L_A(\bcX ^+)) \\
&=&
\Pi _0((\Phi (L_A(\bcX )))^+) \\
&=&
\Pi _0(\Phi (L_A(\bcX )))^+ \\
&=&
\Pi _0(L_A(\bcX ))^+ \\
&=&
\Pi ^A_0(\bcX )^+ \\
\end{array}
$$
in the category of sheaves $\bcS $.
\end{pf}
\section{Chow monoids in Nisnevich sheaves}
\label{Chowsheaves}
Now we turn from homotopy algebra to algebraic geometry. Throughout all schemes will be separated by default. Let $k$ be a field, $\Sm $ the category of smooth schemes of finite type over $k$, and let $\goN $ be the category of all noetherian schemes over $k$, not necessarily of finite type. We are going to specialize the abstract material of the previous sections to the case when $\bcC $ is $\Sm $, the topology $\tau $ is the Nisnevich topology on $\bcC $, and $A$ is the affine line $\AF ^1$ over $k$.
The standard Yoneda construction gives the functor $h$ sending any scheme $X$ from $\goN $ to the functor $\Hom _{\goN }(-,X)$, and the same on morphisms. This is a functor to the category of sheaves in \'etale topology, and so in the Nisnevich one, see \cite{SGA4-2}, page 347, i.e. the Nisnevich topology is subcanonical. Composing $h$ with the constant functor $\Const =\Delta _?[0]$ from $\bcS $ to $\deop \bcS $ we obtain the embedding of $\goN $ into $\deop \bcS $. We identify the categories $\goN $ and $\SSets $ with their images under the corresponding embeddings into $\deop \bcS $.
The scheme $\Spec (k)$ is the terminal object in $\bcC $. The affine line $\AF ^1$ over $k$ is an interval in $\deop \bcS $ with two obvious morphisms $i_0$ and $i_1$ from $\Spec (k)$ to $\AF ^1$. As above, the interval $\AF ^1$ gives the natural cylinder and the corresponding notion of left homotopy on morphisms in $\deop \bcS $. The set of points on a simplicial sheaf $\bcX $ is the set $\Hom _{\deop \bcS }(\Spec (k),\bcX )$ of $k$-points on $\bcX $, and it coincides with the set $\bcX _0(k)$. The set of $\AF ^1$-path connected components on $k$-points is the set $\Hom _{\deop \bcS }(\Spec (k),\bcX )_{\AF ^1}$. If $\bcX $ is fibrant in $\bcM _{\AF ^1}$, then $\Hom _{\deop \bcS }(\Spec (k),\bcX )_{\AF ^1}$ can be identified with $[\Spec (k),\bcX ]_{\AF ^1}$.
Let $\bcX $ be a monoid in $\deop \bcS $. Its completion $\bcX ^+$ is a group object, so that $\Hom _{\deop \bcS }(\Spec (k),\bcX ^+)$ is a group in $\deop \bcS $. The morphism $\bcX \to \bcX ^+$ induces a map from $\Hom _{\deop \bcS }(\Spec (k),\bcX )$ to $\Hom _{\deop \bcS }(\Spec (k),\bcX ^+)$. By the universality of group completion, there exists a unique map from $\Hom _{\deop \bcS }(\Spec (k),\bcX )^+$ to $\Hom _{\deop \bcS }(\Spec (k),\bcX ^+)$ with the obvious commutativity.
\begin{lemma}
\label{muravei}
For a simplicial Nisnevich sheaf monoid $\bcX $, the canonical map from $\Hom _{\deop \bcS }(\Spec (k),\bcX )^+$ to $\Hom _{\deop \bcS }(\Spec (k),\bcX ^+)$ is bijective, and, repspectively, the map from $({\Hom _{\deop \bcS }(\Spec (k),\bcX )_{\AF ^1}})^+$ to $\Hom _{\deop \bcS }(\Spec (k),\bcX ^+)_{\AF ^1}$ is a surjection.
\end{lemma}
\begin{pf}
Since $\Spec (k)$ is Henselian, the set $\Hom _{\deop \bcS }(\Spec (k),\bcX ^+)$ is the quotient of the Cartesian square $\Hom _{\deop \bcS }(\Spec (k),\bcX )^2$. The set $\Hom _{\deop \bcS }(\Spec (k),\bcX )^+$ is also the quotient of the same Cartesian square. The maps from $\Hom $-sets to the sets of $\AF ^1$-path connected components are surjective.
\end{pf}
Next, let $K$ be a field extension of the ground field $k$, and let $\bcS _K$ be the category of set valued Nisnevich sheaves on the category $\bcC _K$ of smooth schemes over $K$. Let $\bcM _K$ be the injective model structure on the category $\deop \bcS _K$, obtained in the same way as the model structure $\bcM $ for the category $\deop \bcS $ over $\Spec (k)$. Let $f:\Spec (K)\to \Spec (k)$ be the morphism induced by the extension $k\subset K$, and let $f^*:\deop \bcS \to \deop \bcS _K$ be the scalar extension functor induced by sending schemes over $k$ to their fibred products with $\Spec (K)$ over $\Spec (k)$, and then using the fact that any sheaf is a colimit of representable ones. As the morphism $f$ is smooth, there are two standard adjunctions
$$
f_{\# } \dashv f^*\dashv f_*
$$
for the functor $f^*$, see, for example, \cite{Morel2}.
\begin{lemma}
\label{ext}
For any field extension $K$ of the ground field $k$, one can choose the localization functors $L_{\AF ^1}$ and $L_{\AF ^1_K}$ in $\deop \bcS $ and $\deop \bcS _K$ respectively, to have a canonical isomorphism
$$
f^*L_{\AF ^1}\simeq L_{\AF ^1_K}f^*\; .
$$
\end{lemma}
\begin{pf}
Let $\ExG\!\!_K$ be the fibrant replacement in $\deop \bcS _K$ obtained in the same way as $\ExG $ was constructed for $\deop \bcS $, see page 70 in \cite{MorelVoevodsky}. Let also $\Sing _K$ be the Suslin-Voevodsky endofunctor on $\deop \bcS _K$. Straightforward verifications show that $f^*\ExG \simeq \ExG\!\!_Kf^*$ and $f^*\circ \Sing \simeq \Sing _K\circ f^*$. Choose $L_{\AF ^1}$ (respectively, $L_{\AF ^1_K}$) to be the transfinite compositions of the functors $\ExG $ and $\Sing $ (respectively, $\ExG\!\!_K$ and $\Sing _K$) in $\deop \bcS $ (respectively, in $\deop \bcS _K$).
\end{pf}
We now need to refresh some things from \cite{SV-ChowSheaves}. For a scheme $X$ let $\bcC (X)$ be the free commutative monoid generated by points of $X$, and let $\bcZ (X)$ be the group completion of $\bcC (X)$. An algebraic cycle $\zeta $ is an element in $\bcZ (X)$. As such, $\zeta $ is a finite linear combination $\sum m_i\zeta _i$ of points $\zeta _i$ on $X$ with integral coefficients $m_i$. The cycle $\zeta $ is said to be effective if and only if $m_i\geq 0$ for all $i$. This is equivalent to say that $\zeta $ is an element of $\bcC (X)$.
The support $\supp (\zeta )$ of $\zeta $ is the union of the Zariski closures of the points $\zeta _i$ with the induced reduced structures on them. The correspondence between points on $X$ and the reduced irreducible closed subschemes of $X$ allows to consider algebraic cycles as linear combinations $Z=\sum m_iZ_i$, where $Z_i$ is the Zariski closure of the point $\zeta _i$, for each $i$. Then $\supp (Z)$ is the same thing as $\supp (\zeta )$. The points $\zeta _i$, or the corresponding reduced closed subschemes $Z_i$, are prime cycles on $X$. The dimension of a point in $X$ is the dimension of its Zariski closure in $X$. Let then $\bcC _r(X)$ be the submonoid in $\bcC (X)$ generated by points of dimension $r$, and, respectively, let $\bcZ _r(X)$ be the subgroup in the abelian group $\bcZ (X)$ generated by points of dimension $r$ in $X$.
Let $S$ be a Noetherian scheme, let $k$ be a field, and let
$$
P:\Spec (k)\to S
$$
be a $k$-point of $S$. Recall that a fat point of $S$ over $P$ is an ordered pair $(P_0,P_1)$ of two morphisms of schemes
$$
P_0:\Spec (k)\to \Spec (R)\quad \hbox{and}\quad P_1:\Spec (R)\to S\; ,
$$
where $R$ is a discrete valuation ring with the residue field $k$, such that
$$
P_1\circ P_0=P\; ,
$$
the image of $P_0$ is the closed point of $\Spec (R)$, and $P_1$ sends the generic point $\Spec (R_{(0)})$ into the generic point of $S$.
Let $X\to S$ be a scheme of finite type over $S$, and let
$$
Z\to X
$$
be a closed subscheme in $X$. Let $R$ be a discrete valuation ring, $D=\Spec (R)$, and let
$$
f:D\to S
$$
be an arbitrary morphism of schemes from $D$ to $S$. Let also
$$
\eta =\Spec (R_{(0)})
$$
be the generic point of $D$,
$$
X_D=X\times _SD\; ,\quad Z_D=Z\times _SD\quad \hbox{and}\quad
Z_{\eta }=Z\times _S\eta \; .
$$
Then there exists a unique closed embedding
$$
Z'_D\to Z_D\; ,
$$
such that its pull-back $Z'_{\eta }\to Z_{\eta }$, with respect to the morphism $Z_{\eta }\to Z_D$, is an isomorphism, and the composition
$$
Z'_D\to Z_D\to D
$$
is a flat morphism of schemes, see Proposition 2.8.5 in \cite{EGAIV(2)}.
In particular, we have such a ``platification" if $(P_0,P_1)$ is a fat point over the $k$-point $P$ and $f=P_1$. Let then $X_P$ be the fibre of the morphism $X_D\to D$ over the point $P_0$,
$$
Z_P=Z_D\times _{X_D}X_P
\quad \hbox{and}\quad
Z'_P=Z'_D\times _{Z_D}Z_P\; .
$$
Since the closed subscheme $Z'_D$ of $X_D$ is flat over $D$, one can define the pull-back
$$
(P_0,P_1)^*(Z)
$$
of the closed subscheme $Z$ to the fibre $X_P$ of the morphism $X\to S$, with regard to the fat point $P$, as the cycle associated to the closed embedding $Z'_P\to X_P$ in the standard way (consult \cite[1.5]{Fulton} for what ``the standard way" means).
In particular, if $Z$ is a prime cycle on $X$, then we have the pull-back cycle $(P_0,P_1)^*(\zeta )$ on $X_P$. Extending by linearity we obtain a pull-back homomorphism
$$
(P_0,P_1)^*:\bcZ (X)\to \bcZ (X_P)\; .
$$
Following \cite{SV-ChowSheaves}, we say that an algebraic cycle $\zeta =\sum m_i\zeta _i$ on $X$ is a relative cycle on $X$ over $S$ if the images of the points $\zeta _i$ under the morphism $X\to S$ are the generic points of the scheme $S$, and, for any $k$-point $P$ on $S$, and for any two fat points extending $P$, the pull-backs of the cycle $Z=\bar \zeta $ to $X_P$, with regard to these two fat points, coincide, see Definition 3.1.3 in loc.cit.
Notice that any cycle, which is flat over $S$, is a relative cycle for free. But not any relative cycle on $X/S$ is flat. This is why we need the ``platification" above.
Let $\bcC _r(X/S)$ be the abelian submonoid in $\bcC (X)$ generated by relative cycles of relative dimension $r$ over $S$. It is important that whenever $S$ is a regular Noetherian scheme and $X$ is of finite type over $S$, then $\bcC _r(X/S)$ is a free commutative monoid generated by closed integral subschemes in $X$ which are equidimensional of dimension $r$ over $S$, see Corollary 3.4.6 on page 40 in \cite{SV-ChowSheaves}. Let also
$$
\bcZ _r(X/S)=\bcC _r(X/S)^+
$$
be the group completion of the monoid $\bcC _r(X/S)$.
Now, fix a Noetherian reduced scheme $T$, and let $\goN $ be the category of Noetherian schemes over $T$. Let $X\to T$ be a scheme of finite type over $T$. For any object $S\to T$ in $\goN $ let
$$
\bcZ _r(X/T)(S)=\bcZ _r(X\times _TS/S)\; .
$$
If $f:S'\to S$ is a flat morphism of Noetherian schemes over $T$, the induced morphism $\id _X\times _Tf:X\times _TS'\to X\times _TS$ is also flat, and one has the standard flat pull-back homomorphism
$$
(\id _X\times _Tf)^*:\bcZ _r(X\times _TS/S)\to
\bcZ _r(X\times _TS'/S')\; .
$$
If $f$ is not flat, then the situation is more difficult.
However, if $T$ is a regular scheme, due to the Suslin-Voevodsky's definition of relative cycles given above, the correct pull-back exists for any morphism $f$, see Proposition 3.3.15 in \cite{SV-ChowSheaves}.
This all aggregates, when $T$ is a regular scheme, into the presheaf $\bcZ _r(X/T)$, which is nothing else but the sectionwise completion of the presheaf $\bcC _r(X/T)$ of relative effective cycles on the category $\goN $.
Since now we will assume that $T$ is regular of characteristic $0$. Let $X\to T$ be a projective scheme over $T$, and fix a closed embedding
$$
X\to \PR ^m_T
$$
over $T$. If $Z$ is a relative equidimensional cycle on $X\times _TS/S$, its pullback $Z_P$ to the fibre $X_P$ of the morphism $X\times _TS\to S$ over a point $P$ on $S$ has its degree $\deg (Z_P)$, computed with regard to the induce embedding of $X\times _TS$ into $\PR ^m_S$ over $S$. Since $Z$ is a relative cycle, the degree $\deg (Z_P)$ is locally constant on $S$, see Proposition 4.4.8 in \cite{SV-ChowSheaves}. It follows that, if $S$ is connected, then $\deg (Z_P)$ does not depend on $P$, see Corollary 4.4.9 in loc.cit. Therefore, the degree of $Z$ over $S$ is correctly defined, and we may consider a subpresheaf
$$
\bcC _{r,d}(X/T)\subset \bcC _r(X/T)\; ,
$$
whose sections on $S$ are relative cycles of degree $d$ on $X\times _TS/S$.
The integer $d$ is non-negative, and there is only one cycle in the set $\bcC _{r,0}(X/T)(S)$, namely the cycle $0$ whose coefficients are all zeros. The grading by degrees gives the obvious structure of a graded monoid on the presheaf $\bcC _r(X/T)$ whose neutral element is the cycle $0$ sitting in $\bcC _{r,0}(X/T)(S)$.
It follows from the results in \cite{SV-ChowSheaves} (see also \cite{Kollar}) that the presheaves $\bcC _{r,d}(X/T)$ are representable by a scheme
$$
C_{r,d}(X/T)\; ,
$$
the so-called {\it Chow scheme} of effective relative cycles of relative dimension $r$ and degree $d$ over $T$. This Chow scheme is projective over $T$, i.e. there exist a structural morphism from $C_{r,d}(X/T)$ to $T$, and a closed embedding of $C_{r,d}(X/T)$ into $\PR ^N_T$ over $T$, arising from the representability above.
Notice that the Chow sheaves are representable because $T$ is a regular scheme of characteristic zero. If $T$ would be of positive characteristic, then only $h$-representability takes place, see \cite{SV-ChowSheaves}. If $T$ is the spectrum of an algebraically closed field of characteristic zero, then $C_{r,d}(X/T)$ is the classical Chow scheme of effective $r$-cycles of degree $d$ on $X$.
Since now we will assume that $T=\Spec (k)$, where $k$ is a field of characteristic zero, and systematically drop the symbol $/T$ from the notation. According to our convention to identify schemes and the corresponding representable sheaves, we will write $C_{r,d}(X)$ instead of $h_{C_{r,d}(X)}$. Certainly, the latter sheaf $C_{r,d}(X)$ is isomorphic to, and should be identified with the sheaf $\bcC _{r,d}(X)$.
Let
$$
C_r(X)=\coprod _{d\geq 0}C_{r,d}(X)\; ,
$$
where the coproduct is taken in the category $\bcS $, not in $\goN $. Such defined $C_r(X)$ is also a coproduct in $\bcP $. If we would consider the coproduct of all Chow schemes $C_{r,d}(X)$ in $\goN $ first, and then embed it into $\bcS $ by the Yoneda embedding, that would be a priori a different sheaf, as Yoneda embedding in general does not commute with coproducts. However, the canonical morphism from the above sheafification to this second sheaf is an isomorphism on the Henselizations of the local rings at points of varieties over $k$. Therefore, the two constructions are actually isomorphic in $\bcS $. This also gives that the coproduct of $C_{r,d}(X)$, for all $d\geq 0$, in $\goN $ represents $\bcC _r(X)$.
Identifying $\bcS $ with its image in $\deop \bcS $ under the functor $\Const $, we consider $C_r(X)$ as the graded {\it Chow monoid} in the category of simplicial sheaves on the smooth Nisnevich site over $\Spec (k)$. The completion $C_r(X)^+$ of $C_r(X)$ in $\bcS $ is the sheafification of the completion of $C_r(X)$ as a presheaf. The latter is sectionwise.
Let $\bcO ^h_{P,Y}$ be the Henselization of the local ring $\bcO _{P,Y}$ of a smooth algebraic scheme $Y$ over $k$ at a point $P\in Y$. Since $\bcO _{P,Y}$ is a regular Noetherian ring, the ring $\bcO _{P,Y}^h$ is regular and Noetherian too. As we mentioned above, the set
$$
\bcC _r(X\times _{\Spec (k)}\Spec (\bcO ^h_{P,Y})/\Spec (\bcO ^h_{P,Y}))
$$
is a free commutative monoid generated by closed integral subschemes in the scheme $X\times _{\Spec (k)}\Spec (\bcO _{P,Y}^h)$, which are equidimensional of dimension $r$ over $\Spec (\bcO _{P,Y}^h)$, by Corollary 3.4.6 in \cite{SV-ChowSheaves}. Then we see that the monoid $C_r(X)$ is pointwise free, and hence it is a pointwise cancellation monoid in the category $\bcS $. It follows also that the Chow monoid $C_r(X)$ is pointwise good in the category $\deop \bcS $, and the canonical morphism from $C_r(X)$ to $C_r(X)^+$ is a monomorphism.
Let $K$ be a finitely generated field extension of $k$. Since $\Spec (K)$ is Henselian, $C_r(X)^+(K)$ is the same as the group completion $(C_r(X)(K))^+$. On the other hand, the same group $C_r(X)^+(K)$ can be also identified with the group of morphisms from $\Spec (K)$ to $C_r(X)^+$, in the category of simplicial Nisnevich sheaves $\deop \bcS $.
Let $d_0$ be the minimal degree of positive $r$-cycles on $X$, where the degree is computed with regard to the fixed embedding of $X$ into $\PR ^m$. Choose and fix a positive $r$-cycle $Z_0$ with $\deg (Z_0)=d_0$. For any natural number $d$ the $d$-multiple $dZ_0$ is an effective dimension $r$ degree $dd_0$ cycle on $X$. This gives a morphism $\alpha $ from $\NN $ to $C_r(X)$ sending $1$ to $Z_0$. Since $C_r(X)$ is the coproduct of $C_{r,dd_0}(X)$, for all $d\geq 0$, we also have the obvious morphism $f$ from $C_r(X)$ to $\NN $, such that $f\circ \alpha =\id _{\NN }$. In other words, $Z_0$ gives the structure of a pointed graded monoid on $C_r(X)$. Automatically, we obtain the connective Chow monoid $C_r^{\infty }(X)$ associated to $C_r(X)$. By Lemma \ref{goraimysh}, we also have the canonical isomorphism of group objects
$$
C_r(X)^+\simeq C_r^{\infty }(X)^+\times \ZZ
$$
in $\deop \bcS $. The sheaf $C_r^{\infty }(X)$ can be also understood as the ind-scheme arising from the chain of Chow schemes
$$
C_{r,0}(X)\subset C_{r,d_0}(X)\subset C_{r,2d_0}(X)\subset \ldots \subset C_{r,dd_0}(X)
\subset \ldots
$$
induced by the cycle $Z_0$ of degree $d_0$ on $X$.
As the category $\bcC $ is Noetherian, the category $\bcS $ is exhaustive. Since $C_r(X)$ is a pointwise cancellation monoid in $\bcS $, and the latter category is exhaustive, $C_r^{\infty }(X)$ is a pointwise cancellation monoid in $\bcS $ too. Then both monoids, $C_r(X)$ and $C_r^{\infty }(X)$ are pointwise good monoids in the category $\deop \bcS $. Moreover, the canonical morphism from $C_r^{\infty }(X)$ to $C_r^{\infty }(X)^+$ is a monomorphism in $\bcS $ and in $\deop \bcS $.
\section{Rational equivalence as $\AF ^1$-path connectivity}
\label{rateq-ratcon}
For any algebraic scheme $X$ over $k$ let $CH_r(X)$ be the Chow group of $r$-dimensional algebraic cycles modulo rational equivalence on $X$. In this section we prove our main theorem and deduce three corollaries, which give something close to the desired effective interpretation of Chow groups in terms of $\AF ^1$-path connectivity on loop spaces of classifying spaces of the Chow monoid $C_r^{\infty }(X)$. We leave it for the reader to decide which of the obtained three isomorphisms is more useful for understanding of Chow groups.
\begin{theorem}
\label{main1}
Let $X$ be a projective algebraic variety with a fixed embedding into a projective space over $k$. For any finitely generated field extension $K$ of the ground field $k$, there is a canonical isomorphism
$$
CH_r(X_K)\simeq
\Pi _0^{\AF ^1}(C_r(X)^+)(K)\; .
$$
\end{theorem}
\medskip
\begin{pf}
Without loss of generality, we may assume that $d_0=1$. Consider the obvious commutative diagram
$$
\diagram
\Hom _{\deop \bcS }
(\Spec (k),C_r(X)^+)_{\AF ^1}
\ar[r]^-{(l^+)_*} &
\Hom _{\deop \bcS }
(\Spec (k),L_{\AF ^1}C_r(X)^+)_{\AF ^1} \\ \\
{\Hom _{\deop \bcS }
(\Spec (k),C_r(X))_{\AF ^1}}^+
\ar[uu]_-{\alpha } \ar[r]^-{(l_*)^+} &
{\Hom _{\deop \bcS }
(\Spec (k),L_{\AF ^1}C_r(X))_{\AF ^1}}^+
\ar[uu]_-{\beta }
\enddiagram
$$
where $l=l_{C_r(X)}$. Since $L_{\AF ^1}C_r(X)^+$ is $\AF ^1$-local, the group in the top right corner is canonically isomorphic to the group $\Pi _0^{\AF ^1}(C_r(X)^+)(k)$. By the same reason, the group in the bottom right corner is canonically isomorphic to the group $\Pi _0^{\AF ^1}(C_r(X))(k)^+$. Since $\Spec (k)$ is Henselian, the latter group is nothing but the group $\Pi _0^{\AF ^1}(C_r(X))^+(k)$. Then Lemma \ref{Pi0&completion} gives that $\beta $ is an isomorphism.
Let $q_0:\Spec (k)\to C_r(X)$ and $q_1:\Spec (k)\to C_r(X)$
be two $k$-points on $C_r(X)$, and suppose $q_0$ is connected to $q_1$ by an $\AF ^1$-path $H:\AF ^1\to L_{\AF ^1}C_r(X)$ on $L_{\AF ^1}C_r(X)$. For any $d$ let $(C_r(X))_d$ be the coproduct $\coprod _{i=0}^dC_{r,i}(X)$. Then $(C_r(X))_d$ is canonically embedded into the coproduct $(C_r(X))_{d+1}$. Consider the chain of the embeddings
$$
(C_r(X))_0\subset \ldots \subset (C_r(X))_d\subset (C_r(X))_{d+1}
\subset \ldots
$$
Applying Proposition 4.4.4 on page 77 in \cite{Hirsch} (see also Remark \ref{filin} in Section \ref{localization}) we see that $L_{\AF ^1}C_r(X)$ is canonically isomorphic to the colimit of the chain of the embeddings
$$
L_{\AF ^1}((C_r(X))_0)\subset \ldots \subset
L_{\AF ^1}((C_r(X))_d)\subset L_{\AF ^1}((C_r(X))_{d+1})
\subset \ldots
$$
Since $\AF ^1$ is a compact object in the category $\deop \bcS $, it follows that the homotopy $H$ factorizes through $L_{\AF ^1}((C_r(X))_d)$, for some degree $d$. If $Z_0$ is a degree $1$ algebraic cycle of dimension $r$ on $X$, then $Z_0$ induces the corresponding embeddings
$$
C_{r,0}(X)\subset \ldots \subset C_{r,d}(C)\; .
$$
This gives the epimorphism from the coproduct $(C_r(X))_d$ onto $C_{r,d}(X)$. Composing the homotopy $H:\AF ^1\to L_{\AF ^1}((C_r(X))_d)$ with the induced morphism from $L_{\AF ^1}((C_r(X))_d)$ to $L_{\AF ^1}(C_{r,d}(X))$, we obtain the homotopy
$$
H:\AF ^1\to L_{\AF ^1}(C_{r,d}(X))\; .
$$
Since $X$ is proper and of finite type over $k$, Proposition 6.2.6 in \cite{AsokMorel} gives that the points $q_0$ and $q_1$ are $\AF ^1$-chain connected, and so $\AF ^1$-path connected on $C_{r,d}(X)$. It means that the map
$$
l_*:\Hom _{\deop \bcS }(\Spec (k),C_r(X))_{\AF ^1}
\to
\Hom _{\deop \bcS }
(\Spec (k),L_{\AF ^1}C_r(X))_{\AF ^1}
$$
is injective. Since $l_*$ is surjective by Lemma \ref{triton}, it is bijective. Then $(l^+)_*$ is an isomorphism as well.
Since $\beta $ and $(l^+)_*$ are isomorphisms, and $\alpha $ is an epimorphism by Lemma \ref{muravei}, we see that $\alpha $ is an isomorphism, and then all the maps in the commutative square above are isomorphisms.
Let now $A$ and $A'$ be two $r$-dimensional algebraic cycles on $X$. If $A$ is rationally equivalent to $A'$ on $X$, there exists an effective relative cycle $Z$ on the scheme $X\times _{\Spec (k)}\AF ^1/\AF ^1$ of relative dimension $r$, and an effective dimension $r$ algebraic cycle $B$ on $X$, such that
$$
Z(0)=A+B\quad \hbox{and}\quad Z(1)=A'+B
$$
on $X$. Let $h_Z$ and $h_{B\times \AF ^1}$ be two regular morphisms from $\AF ^1$ to the Chow scheme $C_r(X)$ over $\Spec (k)$ corresponding to the relative cycles $Z$ and $B\times \AF ^1$ on $X\times _{\Spec (k)}\AF ^1/\AF ^1$ respectively. Let
$$
h:\AF ^1\to C_r(X)\times C_r(X)
$$
be the product of $h_Z$ and $h_{B\times \AF ^1}$ in the category $\deop \bcS $. Let
$$
H:\AF ^1\to C_r(X)\times C_r(X)\to C_r(X)^+\; ,
$$
be the composition of $h$ and the morphism from $C_r(X)\times C_r(X)$ to the completion $C_r(X)^+$, in $\deop \bcS $. Then $H_0=A$ and $H_1=A'$, where $H_0$ and $H_1$ are the precompositions of $H$ with $i_0$ and $i_1$ respectively. It means that the cycles $A$ and $A'$ are $\AF ^1$-path connected on $C_r(X)^+$.
Vice versa, suppose we have a morphism
$$
H:\AF ^1\to C_r(X)^+
$$
in $\bcS $, and let $H_0$ and $H_1$ be the compositions of $H$ with $i_0$ and $i_1$ respectively. Since $\Spec (k)$ is Henselian, $H_0$ is represented by two morphisms $H_{0,1}$ and $H_{0,2}$ from $\Spec (k)$ to $C_r(X)$. Similarly, $H_1$ is represented by two morphisms $H_{1,1}$ and $H_{1,2}$ from $\Spec (k)$ to $C_r(X)$. Since $\alpha $ is an isomorphism and $H_0$ is $\AF ^1$-path connected to $H_1$, it follows that there exist two morphisms $f$ and $G$ from $\Spec (k)$ to $C_r(X)$, such that $H_{0,1}+F$ is $\AF ^1$-path connected to $H_{0,2}+G$ and $H_{1,1}+F$ is $\AF ^1$-path connected to $H_{1,2}+G$ on $C_r(X)$. In terms of algebraic cycles on $X$, it means that the effective $r$-cycle $H_{0,1}+F$ is rationally equivalent to the effective $r$-cycle $H_{0,2}+G$, and, similarly, the cycle $H_{1,1}+F$ is rationally equivalent to $H_{1,2}+G$ on $C_r(X)$. Then the cycle $H_0=H_{0,1}-H_{0,2}$ is rationally equivalent to the cycle $H_1=H_{1,1}-H_{1,2}$ on $C_r(X)$.
Thus, the Chow group $CH_r(X)$ is isomorphic to $\Hom _{\deop \bcS }(\Spec (k),C_r(X)^+)_{\AF ^1}$, i.e. the group in the top left corner of the diagram above. Since, moreover, $(l^+)_*$ is an isomorphism, and the group in the top right corner is canonically isomorphic to $\Pi _0^{\AF ^1}(C_r(X)^+)(k)$, we obtain the required isomorphism in case when $L$ is the ground field $k$.
To prove the theorem for an arbitrary finitely generated field extension $K$ of the ground field $k$, we observe that $f^*C_r(X)$ is $C_r(X_K)$, whence
$$
f^*L_{\AF ^1}C_r(X)^+=
L_{\AF ^1_K}f^*C_r(X)^+=
L_{\AF ^1_K}C_r(X_K)^+
$$
by Lemma \ref{ext}. Therefore,
$$
\begin{array}{rcl}
\Pi _0^{\AF ^1}(C_r(X)^+)(K)
&=&
\Pi _0(L_{\AF ^1}C_r(X)^+)(K) \\
&=&
f^*\Pi _0(L_{\AF ^1}C_r(X)^+)(K) \\
&=&
\Pi _0(f^*L_{\AF ^1}C_r(X)^+)(K)\\
&=&
\Pi _0(L_{\AF ^1_K}C_r(X_K)^+)(K)\\
&=&
\Pi _0^{\AF ^1_K}(C_r(X_K)^+)(K) \\
&\simeq &
CH_r(X_K)_0\; .
\end{array}
$$
\end{pf}
\begin{remark}
{\rm Lemma \ref{Pi0&completion} provides that the monoidal completion in Theorem \ref{main1} can be taken before or after computing the $\AF ^1$-connected component functor. Since the monoidal completion is sectionwise on stalks, we obtain the canonical isomorphisms
$$
\begin{array}{rcl}
CH_r(X)
&\simeq &
\Pi _0^{\AF ^1}(C_r(X)^+)(k) \\
&\simeq &
\Pi _0^{\AF ^1}(C_r(X))^+(k) \\
&\simeq &
\Pi _0^{\AF ^1}(C_r(X))(k)^+\; .
\end{array}
$$
}
\end{remark}
The embedding $X\hra \PR ^m$ gives the degree homomorphism from $CH_r(X)$ to $\ZZ $. Let $CH_r(X)_0$ be its kernel, i.e. the Chow group of degree $0$ cycles of dimension $r$ modulo rational equivalence on $X$. Then,
$$
CH_r(X)\simeq CH_r(X)_0\times \ZZ \; .
$$
Let $Z_0$ be a positive $r$-cycle of minimal degree on $X$. As we have seen above, this gives the structure of a pointed graded cancellation monoid on $C_r(X)$, and $C_r^{\infty }(X)$ is a cancelation monoid too.
\begin{corollary}
\label{main2}
In terms above,
$$
CH_r(X_K)_0\simeq
\Pi _0^{\AF ^1}(C_r^{\infty }(X)^+)(K)\; .
$$
\end{corollary}
\begin{pf}
By Lemma \ref{goraimysh},
$$
C_r(X)^+\simeq C_r^{\infty }(X)^+\times \ZZ
$$
Since the functor $\Pi _0^{\AF ^1}$ is monoidal and $\Pi _0^{\AF ^1}(\ZZ )=\ZZ $, we get the formula
$$
\Pi _0^{\AF ^1}(C_r(X)^+)\simeq
\Pi _0^{\AF ^1}(C_r^{\infty }(X)^+)\times \ZZ \; .
$$
Then apply Theorem \ref{main1} and the isomorphism $CH_r(X)\simeq CH_r(X)_0\times \ZZ $.
\end{pf}
\begin{warning}
{\rm If $CH_r(X)_0\simeq \Pi _0^{\AF ^1}(C_r^{\infty }(X))(k)^+=0$, it does not imply that the monoid $\Pi _0^{\AF ^1}(C_r^{\infty }(X))(k)$ vanishes, as this monoid is by no means a pointwise cancellation monoid. One of the reasons for that is that the Chow schemes $C_{r,d}(X)$ can have many components over $k$.
}
\end{warning}
\begin{corollary}
\label{main2.5}
In terms above,
$$
CH_r(X_K)_0\simeq
\Pi _0^{\AF ^1}
(\Omega \Ex BC_r^{\infty }(X))(K)\; .
$$
\end{corollary}
\begin{pf}
The Chow monoid $C_r^{\infty }(X)$ is pointwise good. Lemma \ref{keylemma} gives an isomorphism
$$
C_r^{\infty }(X)^+\simeq \Omega \Ex BC_r^{\infty }(X)
$$
in $\Ho $, whence
$$
\Pi _0^{\AF ^1}(C_r^{\infty }(X)^+)\simeq
\Pi _0^{\AF ^1}(\Omega \Ex BC_r^{\infty }(X))\; .
$$
Corollary \ref{main2} completes the proof.
\end{pf}
Recall that, for a pointed simplicial Nisnevich sheaf $(\bcX ,x)$, its motivic, i.e. $\AF ^1$-fundamental group $\Pi _1^{\AF ^1}(\bcX ,x)$ is, by definition, the Nisnevich sheaf associated to the presheaf sending a smooth scheme $U$ to the set $[S^1\wedge U_+,(\bcX ,x)]_{\AF ^1}$, where the symbol $[-,-]_{\AF ^1}$ stays now for the sets of morphisms in the pointed homotopy category $\Ho _{\AF ^1}$, see \cite{MorelVoevodsky} or \cite{AsokMorel}. Similarly, one can define, for a pointed simplicial Nisnevich sheaf $(\bcX ,x)$, the fundamental group $\Pi _1^{S^1\wedge \AF ^1}(\bcX ,x)$, where $\AF ^1$ is pointed at any $k$-rational point on it. This is the Nisnevich sheaf associated to the presheaf sending a smooth scheme $U$ to the set
$$
[S^1\wedge U_+,(\bcX ,x)]_{S^1\wedge \AF ^1}\; ,
$$
where the symbol $[-,-]_{S^1\wedge \AF ^1}$ stays for the sets of morphisms in the pointed homotopy category $\Ho _{S^1\wedge \AF ^1}$.
\begin{lemma}
\label{onemorelemma}
Let $\bcX $ be a pointwise good simplicial sheaf monoid. Then, for a scheme $U$,
$$
\Pi _0^{\AF ^1}(\bcX ^+)(U)\simeq
\Pi _1^{S^1\wedge \AF ^1}(B\bcX )(U)\; .
$$
\end{lemma}
\begin{pf}
Since $\bcX $ is pointwise good, there is a isomorphism between $\bcX ^+$ and $\Omega \Ex B\bcX $ in the homotopy category $\Ho $, by Lemma \ref{keylemma}. Since the classifying space $B\bcX $ is pointed connected, the canonical morphism
$$
L_{\AF ^1}\Omega \Ex B\bcX \to
\Omega \Ex L_{S^1\wedge \AF ^1}B\bcX
$$
is a simplicial (i.e. pre-$\AF ^1$-localized) weak equivalence and
$$
\Omega \Ex L_{S^1\wedge \AF ^1}B\bcX
$$
is $\AF ^1$-local by Theorem 2.34 on page 84 in loc.cit. This allows us to make the following identifications:
$$
\begin{array}{rcl}
\Pi _0^{\AF ^1}(\bcX ^+)(U)
&\simeq &
[U,\bcX ^+]_{\AF ^1} \\
&\simeq &
[U,\Omega \Ex B\bcX ]_{\AF ^1} \\
&\simeq &
[U,L_{\AF ^1}\Omega \Ex B\bcX ]_{\AF ^1} \\
&\simeq &
[U,\Omega \Ex L_{S^1\wedge \AF ^1}B\bcX ]_{\AF ^1} \\
&\simeq &
[U,\Omega \Ex L_{S^1\wedge \AF ^1}B\bcX ] \\
&\simeq &
[S^1\wedge U_+,L_{S^1\wedge \AF ^1}B\bcX ] \\
&\simeq &
[S^1\wedge U_+,B\bcX ]_{S^1\wedge \AF ^1} \\
&\simeq &
\Pi _1^{S^1\wedge \AF ^1}(B\bcX )(U)\; .
\end{array}
$$
\end{pf}
\begin{corollary}
\label{main3}
In terms above,
$$
CH_r(X_K)_0\simeq
\Pi _1^{S^1\wedge \AF ^1}(BC_r^{\infty }(X))(K)\; .
$$
\end{corollary}
\begin{pf}
This is a straightforward consequence of Corollary \ref{main2} and Lemma \ref{onemorelemma}.
\end{pf}
\begin{example}
{\rm Let $X$ be a nonsingular projective surface over $k$, where $k$ is algebraically closed of characteristic zero. Assume that $X$ is of general type and has no transcendental second cohomology group, i.e. the cycle class map from $CH^1(X)$ to the second Weil cohomology group $H^2(X)$ is surjective. In that case the irregularity of $X$ is zero. Bloch's conjecture predicts that $CH_0(X)=\ZZ $. In other words, any two closed points on $X$ are rationally equivalent to each other. Fixing a point on $X$ gives the Chow monoid $C_0^{\infty }(X)$, which is nothing else but the the infinite symmetric power $\Sym ^{\infty }(X)$ of the smooth projective surface $X$. By Corollary \ref{main2.5}, Bloch's conjecture holds for $X$ if and only if all $k$-points on the motivic space $L_{\AF ^1}\Omega \Ex B\Sym ^{\infty }(X)$ are $\AF ^1$-path connected. Bloch's conjecture holds, for example, for the classical Godeaux surfaces, \cite{Sur les zero-cycles}, and for the Catanese and Barlow surfaces, see \cite{Barlow2} and \cite{VoisinCataneseBarlowSurfaces}.
}
\end{example}
The above vision of Chow groups should be compared with the results of Friedlander, Lawson, Lima-Filho, Mazur and others, who considered topological (i.e. not motivic) homotopy completions of Chow monoids working over $\CC $, see \cite{FriedlanderMazur} and \cite{Lima-Filho}. A nice survey of this topic, containing many useful references, is the article \cite{Lawson}. The topological homotopy completions of Chow monoids are helpful to understand algebraic cycles modulo algebraic equivalence relation, i.e. the groups $A_r(X)$ of algebraically trivial $r$-cycles cannot be catched by the topological methods. In contrast, the motivic, i.e. $\AF ^1$-homotopy completions of Chow monoids, considered above, can give the description of $A_r(X)$, working over an arbitrary ground field of characteristic zero, as the previous examples show. Theorem \ref{main1} also suggests that the {\it motivic Lawson homology} groups can be defined by the formula
$$
L_rH^{\bcM }_n(X)=\Pi _{n-2r}^{\AF ^1}(C_r(X)^+)(k)\; .
$$
\medskip
\section{Appendix: homotopical algebra}
For the convenience of the reader, we collect here the needed extractions from homotopical algebra. Let first $\bcC $ be a symmetric monoidal category with product $\otimes $ and unit $\uno $. The monoidal product $\otimes $ is called to be closed, and the category $\bcC $ is called closed symmetric monoidal, if the product $\otimes :\bcC \times \bcC \to \bcC $ is so-called adjunction of two variables, i.e. there is bifunctor $\cHom $ and two functorial in $X$, $Y$, $Z$ bijections
$$
\Hom _{\bcC }(X,\cHom (Y,Z))\simeq
\Hom _{\bcC }(X\otimes Y,Z)\simeq
\Hom _{\bcC }(Y,\cHom (X,Z))\; .
$$
If $\bcC $ has a model structure $\bcM $ in it, an adjunction of two variables on $\bcC $ is called Quillen adjunction of two variables, or Quillen bifunctor, if, for any two cofibrations $f:X\to Y$ and $f':X'\to Y'$ in $\bcM $ the push-out product
$$
f\square f':(X\otimes Y')\coprod _{X\otimes X'}
(Y\otimes X')\to Y\wedge Y'
$$
is also a cofibration in $\bcM $, which is trivial if either $f$ and $f'$ is. The model category $(\bcC ,\bcM )$ is called closed symmetric monoidal model category if $\otimes $ is a Quillen bifunctor and the following extra axiom holds. If $q:Q\uno \to \uno $ is a cofibrant replacement for the unit object $\uno $, then the morphisms $q\wedge \id :Q\uno \wedge X\to \uno \wedge X$ and $\id \wedge q:X\wedge Q\uno \to X\wedge \uno $ are weak equivalences for all cofibrant objects $X$. If we consider the cartesian product $\bcM \times \bcM $ of the model structure $\bcM $ as a model structure on the cartesian product $\bcC \times \bcC $, then $\otimes $ and $\cHom $ induce left derived functor $\otimes ^L$ from $Ho(\bcC \times \bcC )$ to $Ho(\bcC )$, and right derived functor $R\cHom $ from $Ho(\bcC \times \bcC )$ to $Ho(\bcC )$. It is well known that passing to localization commutes with products of categories,
so that we have the equivalence between $Ho(\bcC \times \bcC )$ and $Ho(\bcC )\times Ho(\bcC )$. This gives the left derived functor
$$
\otimes ^L:Ho(\bcC )\times Ho(\bcC )\to Ho(\bcC )
$$
and the right derived functor
$$
R\cHom :Ho(\bcC )\times Ho(\bcC )\to Ho(\bcC )\; .
$$
As it was shown in \cite{Hovey}, the left derived $\otimes ^L$ and the right derived $R\cHom $ give the structure of a closed symmetric monoidal category on the homotopy category $Ho (\bcC )$. Since we assume that all objects in $\bcC $ are cofibrant in $\bcM $, it is easy to see that the canonical functor from $\bcC $ to $Ho (\bcC )$ is monoidal.
An important particular case is when the symmetric monoidal product $\otimes $ is given by the categorical product in $\bcC $, i.e. when $\bcC $ is the cartesian symmetric monoidal category. Since $Ho(\bcC )$ admits products, and products in $\bcC $ are preserved in $Ho (\bcC )$,
for any three objects $X$, $Y$ and $Z$ in $\bcC $ one has the canonical isomorphism
$$
[X,Y]\times [X,Z]\simeq [X,Y\times Z]\; .
$$
Let now $\bcC $ be a left proper cellular simplicial model category with model structure $\bcM =(W,C,F)$ in it, let $I$ and $J$ be the sets of, respectively, generating cofibrations and generating trivial cofibrations in $\bcC $, and let $S$ be a set of morphisms in $\bcC $. For simplicity we will also be assuming that all objects in $\bcC $ are cofibrant, which will always be the case in applications. An object $Z$ in $\bcC $ is called $S$-local if it is fibrant, in the sense of the model structure $\bcM $, and for any morphism $g:A\to B$ between cofibrant objects in $S$ the induced morphism from $\bHom (B,Z)$ to $\bHom (A,Z)$ is a weak equivalence in $\SSets $. A morphism $f:X\to Y$ in $\bcC $ is an $S$-local equivalence if the induced morphism from $\bHom (Y,Z)$ to $\bHom (X,Z)$ is a weak equivalence in $\SSets $ for any $S$-local object $Z$ in $\bcC $. Then there exists a new left proper cellular model structure $\bcM _S=(W_S,C_S,F_S)$ on the same category $\bcC $, such that $C_S=C$, $W_S$ consists of $S$-local equivalences in $\bcC $, so contains $W$, and $F_S$ is standardly defined by the right lifting property and so is contained in $F$. The model structure $\bcM _S$ is again left proper and cellular with the same set of generating cofibrations $I$ and the new set of generating trivial cofibrations $J_S$. The model category $(\bcC ,\bcM _S)$ is called the (left) Bousfield localization of $(\bcC ,\bcM _S)$ with respect to $S$. This all can be found in \cite{Hirsch}.
Notice that the identity adjunction on $\bcC $ is a Quillen adjunction and induces the derived adjunction $L\Id :Ho(\bcC )\dashv Ho(\bcC _S):R\Id $, where $Ho(\bcC _S)$ is the homotopy category of $\bcC $ with respect to the model structure $\bcM _S$. Since cofibrations remain the same and, according to our assumption, all objects are cofibrant, the functor $L\Id $ is the identity on objects and surjective on Hom-sets. To describe $R\Id $ we observe the following. Since $F_S$ is smaller than $F$, the fibre replacement functor in $(\bcC ,\bcM )$ is different from the fibre replacement functor in $(\bcC ,\bcM _S)$. Taking into account that $\bcC $ is left proper and cellular, one can show that there exists a fibrant replacement $\Id _{\bcC }\to L_S$ in $\bcM _S$, such that, if $X$ is already fibrant in $\bcM $, then $L_S(X)$ can be more or less visibly constructed from $X$ and $S$, see Section 4.3 in \cite{Hirsch} (or less abstract presentation in \cite{DrorFarjoun}). The right derived functor $R\Id $, being the composition of $Ho(L_S)$ and the functor induced by the embedding of $S$-local, i.e. cofibrant in $\bcM _S$, objects into $\bcC $, identifies $Ho(\bcC _S)$ with the full subcategory in $Ho(\bcC )$ generated by $S$-local objects of $\bcC $.
Since $(\bcC ,\bcM )$ is a simplicial model category, then so is $(\bcC ,\bcM _S)$, see Theorem 4.1.1 (4) in \cite{Hirsch}. Suppose that $\bcC $ is, moreover, closed symmetric monoidal with product $\otimes $, and that the monoidal structure is compatible with the model one in the standard sense, i.e. $\bcC $ is a symmetric monoidal model category (see above). The new model category $(\bcC ,\bcM _S)$ is monoidal model, i.e. the model structure $\bcM _S$ is compatible with the existing monoidal product $\otimes $, if and only if for each $f$ in $S$ and any object $X$ in the union of the domains $\dom (I)$ and codomains $\codom (I)$ of generating cofibrations $I$ in $\bcC $ the product $\id _X\otimes f$ is in $W_S$.
This is exactly the case when the set $S$ is generated by a morphism $p:A\to \uno $, where $A$ is an object in $\bcC $ and $\uno $ is the unit object for the monoidal product $\otimes $, i.e.
$$
S=\{ X\wedge A\stackrel{\id _X\wedge p}{\lra }X\; |\; X\in \dom(I)\cup \codom (I)\} \; .
$$
In that case the model structure $\bcM _S$ is compatible with the monoidal one, so that $(\bcC ,\bcM _S)$ is a simplicial closed symmetric monoidal model category, which is left proper cellular.
Let us write $\bcM _A$ and $L_A$ instead of, respectively, $\bcM _A$ and $L_S$ when $S$ is generated by $A$ in the above sense. One of the fundamental properties of the localization functor $LA$ is that, for any two objects $X$ and $Y$ in $\bcC $, the object $L_A(X\times Y)$ is weak equivalent to the object $L_A(X)\times L_A(Y)$, in the sense of the model structure $\bcM _A$. The proof of this fact in topology is given on page 36 of the book \cite{DrorFarjoun}, and it can be verbally transported to abstract setting. All we need is the Quillen adjunction in two variables in $\bcC $, and the fact saying that if $Y$ is $S$-local, then $\cHom (X,Y)$ is $S$-local for any $X$ in $\bcC $, which is also the consequence of adjunction.
\bigskip
\begin{small} | 186,446 |
TITLE: The Best Way to Choose Spaces for Infection
QUESTION [1 upvotes]: Consider a 5x7 patch of 1x1 square units. I can choose any 3 of the units in the patch to be infected. After 1 day, an infected square will spread to all the squares adjacent to it (no diagonal spread). My question is: "What 3 squares should i pick such that the entire 5x7 patch is infected in the fastest time possible?" (you can answer in coordinates or as a matrix of x's and o's) Can I also have the proof as well? Thanks!
REPLY [1 votes]: Suppose the patch is a grid from $(1,1)$ to $(5,7)$. Infecting $(3,1),(3,4),(3,7)$ initially ensures that all cells are infected after $3$ days.
To see why a $2$-day solution cannot exist, note that two cells separated by a giraffe or $(4,1)$ leap
$$\begin{array}{ccccc}
\circ&\circ&\circ&\circ&B\\
A&\circ&\circ&\circ&\circ\end{array}$$
cannot both be infected by one initial infection after $2$ days. Now consider the points $S=\{(1,1),(5,2),(1,6),(5,7)\}$, which are all at least a giraffe leap apart. Any single initial infection can infect at most one cell in $S$ after $2$ days, but there are $4$ cells in $S$ and only $3$ initial infections, so $S$ cannot be completely infected after $2$ days. | 74,032 |
05 Feb North Carolina Society of Engineers 2018 Winter Meeting & 100th Anniversary
The North Carolina Society of Engineers, the oldest engineering society in North Carolina, will celebrate 100 years as an organization on Friday, February 23, 2018. This society was founded in Durham on February 18, 1918, by a group of engineers from across the state of North Carolina. The 2018 Winter Meeting will culminate with a banquet at the Park Alumni Center on North Carolina State University’s campus beginning at 6:00 PM.
The agenda for the banquet will include a keynote address by State House Representative Scott Stone, PE, as well as the presentation of the annual Outstanding Engineer of the Year award, first presented in 1937. This year’s recipient is Mr. Lonnie C. Poole Jr., founder of Waste Industries and an NC State Engineering graduate. The banquet will also feature other recognitions, including the newly created Van Crotts Scholarship and Society Service award. Installation of new officers and directors will complete the banquet. Many state dignitaries will be in attendance to celebrate this monumental occasion.
For more information about the North Carolina Society of Engineers or this event, visit ncsocietyofengineers.org or contact H.E. “Tony” Withers III, PE at [email protected]. | 80,915 |
Hang tight!
Recipe by Barbara
"Family recipe that I have never seen anywhere else. It is our favorite jam."
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3 cups
cranberries
1 1/2 cups
water
2 cups
mashed bananas
7 cups
white sugar
1/2 (6 fluid ounce) container
liquid pectin
1 teaspoon
lemon juice
PREP
COOK
READY IN
When I tasted this jam, I thought I'd died and gone to heaven - it's THAT good! I gave a jar to my brother, and he said it was the best jam he'd ever tasted! Really easy to make, too. (But let it start to set before you put into jars, or the banana floats to the top!)
Well, I thought this was not very good at all. The texture was ultimately weird and inconsistent, sometimes grainy like cranberry sauce and sometimes jammy. I also thought the cranberry/banana flavors clashed instead of mixing well.
66 Ratings
This is the yummiest jam I've ever tasted! This was my first attempt at canning anything and it was so easy. I used a waterbath canner and processed the jars for 10 minutes in it. The amazing thing about this jam is that it really, really, really does taste like strawberry! This is perfect for my stepdaughter who is allergic to strawberry but loves the taste of strawberries! I would let it set for about 3 minutes before pouring into jars just because my first poured jar had the banana and jelly part seperate a bit, but all the rest were perfect, so I'm thinking the pectin just needs a moment to set before you begin pouring. If you mash your bananas up well, it mixes just fine, and comes out a lovely rosy red color perfect for the holidays! This is a great first recipe for someone who has never attempted a jam before.
If you've never made jam before, start with this recipe. The ingredients do the work themselves. The result is beautiful, delicious and unique. I have made bunches of this for Christmas gifts! To can: wait for jam to gel a bit (this took mine about 45 minutes,) put jam in your canning jars, place in large pot, fill with water to one inch above lids, and boil 15 minutes. Remove and place on towel to dry. The lids will suction down while cooling.
Great taste and easy recipe. I found it to be a bit too sweet so I may cut the sugar next time. Also, I didn't taste banana at all. I thought this jam tasted more like strawberry than cranberry - maybe because it's so sweet? Also, after the initial 10 simmer of the cranberries, they were still mostly whole so I mashed them up with a spoon. Maybe that's the difference? Possibly, if I had left them whole, I would have gotten a more tart taste and would have been able to distinguish between the banana and the cranberry. But, since I mashed the cranberries, they just blended with the bananas and created a strawberry taste. Overall a good, quick and easy recipe.
This is such a beautiful jam that if it didn't taste so well, I would never open it! But the taste is so good I can't resist! I am wrapping these up for Christmas gifts ... makes 5 half pint jars. I was not sure if I would like this recipe because I am not crazy about cranberries but I am sure glad I tried this recipe out!
I've made this jam for several years to give as holiday gifts. My friends eagerly look forward to it--and some even place "orders" long before cranberry season arrives!!
Good starter recipe. As is, way too sweet but it's easily bendable to cut the sugar back, if you so choose. Which I did. That's really the only issue I had was just too much sugar. Very simple, so simple that I was able to make this with my youngest son and really have no issues at all.
* Percent Daily Values are based on a 2,000 calorie diet.
Cranana Jam
Serving Size: 1/80 of a recipe
Servings Per Recipe: 80
Amount Per Serving
Calories: 75 delicious homemade jam with cranberries and bananas.
See how to make super-simple strawberry freezer jam from scratch.
This recipe for fresh jam couldn't be easier—just three basic ingredients! | 213,745 |
TITLE: Infinite, totally ordered and well ordered sets
QUESTION [6 upvotes]: It's quite easy to show that a finite set is well ordered iff it is totally ordered.
Is the converse also true? That is: is it true that a set is infinite iff it admits a total order which is not a well order? (For the sake of brevity, I shall write t.o. and w.o. for total order and well order, respectively)
The if part follows from the previous observation (t.o.+finiteness implies w.o.), so the question becomes
Is it true that a set is infinite only if it admits a total order which is not a well order?
I know that every set admits a w.o. (and thus a t.o.); still, answering the question requires to prove (or to disprove) that t.o.≠w.o., and this is out of my reach.
The question supposes ZFC but every other set theory is accepted.
REPLY [4 votes]: It is easy to prove that $A$ is finite if and only if there is an order $<$ on $A$, such that $(A,<)$ and the reverse order $(A,>)$ are both well-orders.
Now, note that the reverse order of a total order is again a total order, and so if $A$ is infinite, there is a well-ordering on $A$, $<$, such that $(A,>)$ is not a well-order itself. (Of course, one can show that this is true for any well-ordering of an infinite set.)
Just a remark on the Axiom of Choice, the above holds for $\sf ZF$ as well, although it is consistent with $\sf ZF$ that there are sets which do not admit any total orders, let alone well-orders.
But a curious thing is that if we change "there exists an order ..." to "every well-order $<$ satisfies ...", then any set which cannot be well-ordered satisfies the definition vacuously. And therefore the Axiom of Choice is equivalent to the statement "a set $A$ is finite if and only if for every well-order on $A$, the reverse order is also a well-order". | 89,154 |
DMSO
LY2090314 is a potent inhibitor of glycogen synthase kinase-3 (GSK-3) with IC50s of 1.5 nM and 0.9 nM for GSK-3α and GSK-3β respectively.
IC50 Value: 1.5 nM (GSK-3α); 0.9 nM (GSK-3β) [1]
Target: GSK-3α; GSK-3β
LY2090314 is a potent inhibitor of glycogen synthase kinase-3 (GSK-3) which plays an important role in many pathways, including initiation of protein synthesis, cell proliferation, cell differentiation, and apoptosis.
in vitro: LY2090314 selectively inhibits the activity of GSK-3 by interrupting ATP binding. LY2090314 is able to stabilize β-catenin. LY2090314 shows limited efficacy as monotherapy. LY3090314 enhances the efficacy of cisplatin and carboplatin in solid tumor cancer cell lines in vitro.
in vivo:, with the notable exception of dog-unique LY2090314 glucuronide [1].
Toxicity:
Clinical trial: A Study of LY-2090314 and Chemotherapy in Participants With Metastatic Pancreatic Cancer. Phase1/2
[1]. Zamek-Gliszczynski MJ, et al. Pharmacokinetics, metabolism, and excretion of the glycogen synthase kinase-3 inhibitor LY2090314 in rats, dogs, and humans: a case study in rapid clearance by extensive metabolism with low circulating metabolite exposure. Drug Metab Dispos. 2013 Apr;41(4):714-26. | 181,179 |
KIDD, Sandra Winifred (2005). Secondary students in two former mining communities: possibilities for the self. Doctoral, Sheffield Hallam University.
Abstract
This thesis was prompted by student working class underachievement in GCSE examinations at two secondary schools in two former mining communities, located in North East Derbyshire and North Nottinghamshire during the period 1995-2002. Reid’s analysis of social class in Britain, (1998), whilst acknowledging a connection between educational achievement and individual ability, states, ‘Educational experience and achievement are clearly related to social class’ (page 187). I propose that a particular working class culture was the all encompassing meaning making framework in locally produced identities, masculinities and femininities. The sense of self/identity of the research participants, notably that of the students, framed by the lived experience of class (the socio-economic, cultural, emotional, psychic reality of a particular class experience) maintained by resilient and powerful local language/discourse, at particular temporal and spatial points, is central to my exploration of the opening up of possibilities for the self in the life chances of the students. The study compares and contrasts the experiences of majority working class students to minority working class students and middle class students via an examination of the inter-relationship between sense of self/identity and local perspectives upon time, space and personal narrativity. The thesis argues that students who combined immersion in local productions of self/identity, such as localised definitions of masculinity and femininity, with a predominantly episodic relationship to schooling/education time and a predominantly territorial relationship to space, the majority working class view,points to both a lack of educational engagement and academic success at GCSE, and thence to the closing down of life chances. The conceptual framework for this thesis is provided by postmodern and feminist thought which foregrounds our existential reality as crucial to understanding our way of being in the world. Bourdieu’s concept of the habitus is utilised to demonstrate how,when a particular habitus interacts with a particular relationship to time and space, possibilities for the self are opened up or closed down. Since being and becoming in time and space are a central focus of this research, much ethnographic data is drawn on; the methodology used is therefore almost wholly qualitative.
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O Jesus, may I lead a good life; may I die a happy death. May I receive Thee before I die. May I say when I am dying, "Jesus, Mary, Joseph, I give you my heart and my soul."
Hail Mary . . .
Please click the play button below to listen to Mary's Patronage Prayer now.
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\begin{document}
\title[]{Free Convolution Powers \\via Roots of Polynomials}
\subjclass[2020]{46L54, 26C10}
\keywords{Free Probability, Roots of Polynomials, Free Additive Convolution.}
\thanks{S.S. is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.}
\author[]{Stefan Steinerberger}
\address{Department of Mathematics, University of Washington, Seattle}
\email{[email protected]}
\begin{abstract} Let $\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $\mu^{\boxplus k}$ for any real $k \geq 1$. The purpose of this short note is to give a formal proof of an elementary description of $\mu^{\boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials. \end{abstract}
\maketitle
\section{Introduction}
\subsection{Free Convolution.} The notion of free convolution $\mu \boxplus \nu$ of two compactly supported probability measures is due to Voiculescu \cite{voic}. A definition is as follows: for any compactly supported probability measure, we can consider its Cauchy transform $G_{\mu}:\mathbb{C} \setminus \mbox{supp}(\mu) \rightarrow \mathbb{C}$ defined via
$$ G_{\mu}(z) = \int_{\mathbb{R}} \frac{1}{z-x} d\mu(x).$$
For a compactly supported measure, $G_{\mu}(z)$ tends to 0 as $|z| \rightarrow \infty$.
Given $G_{\mu}$, we define the $R-$transform $R_{\mu}(s)$ for sufficiently small complex $s$ by demanding that
$$ \frac{1}{G_{\mu}(z)} + R_{\mu}(G_{\mu}(z)) = z$$
for all sufficiently large $z$. The free convolution $\mu \boxplus \nu$ is then the unique compactly supported measure for which
$$ R_{\mu \boxplus \nu}(s) = R_{\mu}(s) + R_{\nu}(s)$$
for all sufficiently small $s$. A fundamental result due to Voiculescu is the \textit{free central limit theorem}: if $\mu$ is a compactly supported probability measure with mean 0 and variance 1, then suitably rescaled copies of $\mu^{\boxplus k}$ converge to the semicircular distribution. This notion can be extended to real powers.
\begin{thm}[Fractional Free Convolution Powers exist, \cite{berc, nica}] Let $\mu$ be a compactly supported probability measure on $\mathbb{R}$ and assume $k \geq 1$ is real. Then there exists a unique compactly supported probability measure $\mu^{\boxplus k}$ such that
$$ R_{\mu^{\boxplus k}}(s) = k \cdot R_{\mu}(s) \qquad \quad \mbox{for all $s$ sufficiently small.}$$
\end{thm}
This was first shown for $k$ sufficiently large by Bercovici \& Voiculescu \cite{berc} and then by Nica \& Speicher \cite{nica} for all $k \geq 1$. We also refer to \cite{ans, bel0, bel1, bel2, berc1, hiai, huang, mingo, nica2, shlya, williams}. The purpose of this note is to (formally) prove an elementary description of $\mu^{\boxplus k}$ in terms of polynomials and the density of the roots of their derivatives.
\subsection{Polynomials.} Roots of polynomials are a classical subject and there are many results we do not describe here, see \cite{byun, farmer, gauss, granero, han, huang, kab, kab2, kornik, lucas, nica, or, or3, or4, or2, pem, pem2, pol, ravi, red, sub, steini, riesz}. Our problem will be as follows: let $\mu$ be a compactly supported probability measure on the real line and suppose $x_1, \dots, x_n$ are $n$ independent random variables sampled from $\mu$ (which we assume to be sufficiently nice).
We then associate to these numbers the random polynomial
$$ p_n(x) = \prod_{k=1}^{n}{(x-x_k)}$$
having roots exactly in these points. What can we say about the behavior of the roots of the derivative $p_n'$? There is an interlacing phenomenon and the roots of $p_n$ are also distributed according to $\mu$ as $n \rightarrow \infty$. The same is true for the second derivative $p_n''$ and any finite derivative. However, once the number of derivatives is proportional to the degree, the distribution will necessarily change.
\begin{quote}
\textbf{Question.} Fix $0 < t < 1$. How are the roots of $p_n^{\left\lfloor t \cdot n\right\rfloor}$ distributed?
\end{quote}
The question was raised by the author in \cite{steini}.
The answer, if it exists, should be another measure $u(t,x)dx$. Note that, since this measure describes the distribution of roots of polynomials of degree $(1-t) \cdot n$, as $n \rightarrow \infty$, we have
$$ \int_{\mathbb{R}} u(t,x) dx = 1-t.$$
Relatively little is known about the evolution of $u(t,x)$: \cite{steini} established, on a formal level, a PDE for $u(t,x)$. This PDE is given by
$$ \frac{\partial u }{\partial t} + \frac{1}{\pi} \frac{\partial}{\partial x} \arctan\left( \frac{Hu}{u} \right) = 0 \qquad \mbox{on}~\mbox{supp}(u),$$
where
$$ Hf(x) = \mbox{p.v.}\frac{1}{\pi} \int_{\mathbb{R}}{\frac{f(y)}{x-y} dy} \qquad \mbox{is the Hilbert transform.}$$
\begin{figure}[h!]
\begin{minipage}[l]{.45\textwidth}
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width = 0.7\textwidth]{pic1.pdf}};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[r]{.45\textwidth}
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width = 0.7\textwidth]{pic2.pdf}};
\end{tikzpicture}
\end{minipage}
\caption{The densities of two evolving measures $u(t,x)dx$. They shrink and vanish at time $t=1$.}
\end{figure}
It is also known that it has to satisfy the conservation laws
$$\int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy.$$
Hoskins and the authors \cite{hoskins} established a universality result for large derivatives of polynomials with random roots: such derivatives behave like random shifts of Hermite polynomials. Hermite polynomials, in turn, have roots whose density is given by a semicircle and this leads one to believe that $u(t,x)$ should, for $t$ close to 1, look roughly like a semicircle (and this has also been observed numerically).
There are two explicit closed-form solutions, derived in \cite{steini}, a shrinking semicircle and a one-parameter solution that lies in the Marchenko-Pastur family (see Fig. 1).
Numerical simulations in \cite{hoskins} also suggested that the solution tends to become smoother. O'Rourke and the author \cite{or2} derived an analogous transport equation
for polynomials with roots following a radial distribution in the complex plane.
\section{The Result}
\subsection{An Equivalence.}
We can now state our main observation: both the free convolution of a measure with itself, $\mu^{\boxplus k}$, and the density of roots of derivatives of polynomials, $u(t,x)$, are described by the same underlying process.
\begin{thm} At least formally, if $\mu = u(0,x)dx$ and $\left\{x : u(0,x) > 0\right\}$ is an interval, then for all real $k\geq 1$
$$ \mu^{\boxplus k} =u\left(1- \frac{1}{k}, \frac{x}{k} \right) dx.$$
\end{thm}
We first clarify the meaning of `formally'. In a recent paper, Shlyakhtenko \& Tao \cite{shlya} derived, formally, a PDE for the evolution of the $\mu^{\boxplus k}$. This PDE happens to be the same PDE (expressed in a different coordinate system) that was formally derived by the author for the evolution of $u(t,x)$ \cite{steini}. The derivation in \cite{steini} is via a mean-field limit approach, the `microscopic' derivation is missing. In particular, the derivation in \cite{steini} assumed the existence of $u(t,x)$ and a crystallization phenomenon for the roots; such a crystallization phenomenon has been conjectured for a while, there is recent progress by Gorin \& Kleptsyn \cite{gorin}.\\
\begin{figure}[h!]
\includegraphics[width=0.6\textwidth]{flat.pdf}
\caption{Evolution of $u(t,x)$ (from \cite{hoskins}) starting with random and uniformly distributed roots: the evolution smoothes and we see a semicircle before it vanishes.}
\end{figure}
Naturally, this has a large number of consequences since it allows us to go back and forth between results from
free probability and results regarding polynomials and their roots.
As an illustration, we recall that for the semicircle law $\mu_{sc}^{\boxplus k}$ is a another semicircle law stretched by a factor $k^{1/2}$, thus
$$ \mu_{sc}^{\boxplus k} = \frac{2}{\pi} \sqrt{ \frac{1}{k} - \frac{x^2}{k^2}}dx$$
Conversely, as was computed in \cite{steini}, the evolution of densities of polynomials when beginning with a semicircle behaves as
$$ u(t,x) = \frac{2}{\pi}\sqrt{1-t- x^2}.$$
One immediate consequence of the equivalence is that it provides us with a fast algorithm to approximate $\mu^{\boxplus k}$ when $\mu = f(x)dx$ and $f$ is smooth. This may be useful in the study of the semigroup $\mu^{\boxplus k}$. Using the logarithmic derivative $p_n'/p_n$, it is possible quickly differentiate real-rooted polynomials $p_n$ a large number of times, $t \cdot n$, even when the degree is as big as $n \sim 100.000$: this was done in \cite{hoskins} using a multipole method (a modification of an algorithm due to Gimbutas, Marshall \& Rokhlin \cite{gimb}). Fig. 2 shows an example computed using 80.000 roots: we observe the initial smoothing and the eventual convergence to a semicircle.
\subsection{Some Connections.} Some connections are as follows.\\
\textit{The Free Central Limit Theorem.} Voiculescu \cite{voic} proved that $\mu^{\boxplus k}$ (suitably rescaled) approaches a semicircle distribution in the limit. Motivated by high-precision numerics, Hoskins and the author \cite{hoskins} conjectured that $u(t,x)$ starts looking like a semicircle for $t$ close to 1 and proved a corresponding universality result for polynomials with random roots: if $p_n$ is a polynomial with random roots (from a probability measure $\mu$ whose moments are all finite), then, for fixed $\ell \in \mathbb{N}$ and $n \rightarrow \infty$, we have for $x$ in a compact interval
$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + \gamma_n),$$
where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial and $\gamma_n$ is a random variable converging to the standard $\mathcal{N}(0,1)$ Gaussian as $n \rightarrow \infty$. Hermite polynomials have roots that are asymptotically distributed like a semicircle. A result in the deterministic setting has recently been provided by Gorin \& Kleptsyn \cite{gorin}. \\
\textit{Conservation Laws.} The author showed that the evolution $u(t,x)$ satisfies the algebraic relations
\begin{align*}
\int_{\mathbb{R}}{ u(t,x) ~ dx} = 1-t, \qquad \qquad \int_{\mathbb{R}}{ u(t,x) x ~ dx} = \left(1-t\right)\int_{\mathbb{R}}{ u(0,x) x~ dx}, \qquad\\
\int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy.
\end{align*}
These are derived from Vieta-type formulas that express elementary symmetric polynomials in terms of power sums. Equivalently, we have
$ \kappa_n ( \mu^{\boxplus k}) = k^n \kappa_n(\mu),$
where $\kappa_n$ is the $n-$th free cumulant providing a large number of conservation laws.\\
\textit{Monotone Quantities.}
Voiculescu \cite{voic2} introduced the free entropy
$$ \chi(\mu) = \int_{\mathbb{R}} \int_{\mathbb{R}} \log{ |s-t|} d\mu(s) d\mu(t) + \frac{3}{4} + \frac{\log{(2\pi)}}{2}$$
and the free Fisher information
$$\Phi(\mu) = \frac{2\pi^2}{3} \int_{\mathbb{R}} \left( \frac{d\mu}{dx} \right)^3 dx.$$
Shlyakhtenko \cite{shlya0} proved that $\chi$ increases along free convolution of $\mu$ with itself whereas $\Phi$ decreases (both suitably rescaled). Shlyakhtenko \& Tao \cite{shlya} showed monotonicity along the entire flow $\mu^{\boxplus k}$ for real $k \geq 1$. Conversely, on the side of polynomials, it is known that
$$ \frac{\left|\left\{x \in \mathbb{R}: u(t,x) > 0 \right\}\right|}{1-t} \qquad \mbox{is non-decreasing in time.}$$
Another basic result for polynomials is commonly attributed to Riesz \cite{farmer, riesz}: denoting the smallest gap of a polynomial $p_n$ having $n$ real roots $\left\{x_1, \dots, x_n\right\}$ by
$$ G(p_n) = \min_{i \neq j}{|x_i - x_j|},$$
we have $G(p_n') \geq G(p_n)$: the minimum gap grows under differentiation. A simple proof is given by Farmer \& Rhoades \cite{farmer}. This would suggest that
the maximal density cannot increase over time.\\
\textit{The Minor Process.} Shlyakhtenko \& Tao \cite{shlya} connect the evolution to the minor process: trying to understand how the eigenvalues of the $n \times n$ minor of a large random Hermitian matrix $N \times N$ behave. This answers a question numerically verified by Hoskins and the author \cite{hoskins}.\\
\textit{Related Results.} There are several other papers in the literature that seem to be connected to this circle of ideas. We mention Gorin \& Marcus \cite{gorin0}, Marcus \cite{marcus0}, Marcus, Spielman \& Srivastava \cite{marcus}.
\section{Proof}
\begin{proof} Shlyakhtenko \& Tao \cite{shlya} derive that if
$$ d\mu^{\boxplus k} = f_k(x) dx$$
and if we substitute $k=1/s$ (thus $0< s < 1$) and $f := f_{1/s}$, then on a purely formal level
$$ \left(-s \frac{\partial}{\partial s} + x \frac{\partial}{ \partial x}\right) f = \frac{1}{\pi} \frac{\partial}{\partial x} \arctan \left( \frac{f}{Hf} \right).$$
On the other hand, the author derived \cite{steini}, also on a formal level, that as long as $\left\{x: u(t,x) > 0\right\}$ is an interval
$$ \frac{\partial u }{\partial t} + \frac{1}{\pi} \frac{\partial}{\partial x} \arctan\left( \frac{Hu}{u} \right) = 0 \qquad \mbox{on}~\mbox{supp}(u).$$
We note that Huang \cite{huang} showed that the number of connected components in the support of $\mu^{\boxplus k}$ is non-decreasing in $k$ which shows that once the support is an interval, this property is preserved.
We want to show that the solutions of these two PDEs are related via a change of variables: since both evolutions obey the same PDE, they must coincide. We observe that one nonlinear term seems to be
the reciprocal of the other, however, this compensates for the different sign.
We compute
\begin{align*}
\frac{\partial}{\partial x} \arctan\left( \frac{f}{Hf} \right) = \frac{1}{1+\frac{f^2}{(Hf)^2}} \partial_x \frac{f}{Hf} =\frac{ f_x (Hf) - f (Hf)_x }{f^2 + (Hf)^2}
\end{align*}
and compare it to
\begin{align*}
\frac{\partial}{\partial x} \arctan\left( \frac{Hf}{f} \right) = \frac{1}{1+\frac{(Hf)^2}{f^2}} \partial_x \frac{Hf}{f} = \frac{ (Hf)_x f - f_x (Hf) }{f^2 + (Hf)^2}
\end{align*}
and see that it is the same term with opposite sign. This allows us to write
$$ \frac{\partial u }{\partial t} = \frac{1}{\pi} \frac{\partial}{\partial x} \arctan\left( \frac{u}{Hu} \right).$$
We now claim that
$$ f(s,x) = u(1-s, s x).$$
Note that the left-hand side transforms
\begin{align*}
(-s \partial_s + x \partial_x) f = -s \left( \frac{\partial u}{\partial t} (-1) + \frac{\partial u}{\partial x} x\right) + x \frac{\partial u}{\partial x} s = s \frac{\partial u}{\partial t}.
\end{align*}
It remains to understand how the right-hand side transforms. The Hilbert transform commutes with dilations and thus
$$\arctan\left( \frac{f}{Hf} \right)= \arctan\left( \frac{u(1-s, sx)}{H \left[ u(1-s, sx)\right]} \right)= \arctan\left( \frac{u(1-s, sx)}{\left[H u (1-s, \cdot) \right] (sx)}\right)$$
whose derivative scales exactly by a factor of $s$.
\end{proof}
\textbf{Remarks.} We see that, both derivations being purely formal, many problems remain. Indeed, this connection suggests
many interesting further avenues to pursue.
Roots of polynomials seem to regularize under differentiation at the micro-scale: if one
were to take a polynomial with random (or just relatively evenly spaced roots), then the roots of the $(\varepsilon \cdot n)-$th derivative are conjectured to behave
locally like arithmetic progressions up to a small error. Results of this flavor date back to Polya \cite{pol} for analytic functions, see also Farmer \& Rhoades \cite{farmer}
and Pemantle \& Subramanian \cite{pem2}. In the converse direction, it could be interesting to study the behavior of $u(t,x)$ when the initial conditions are
close to a semi-circle: despite the equation being both non-linear and non-local, its linearization around the semicircle seems to diagonalize nicely under
Chebychev polynomials -- can PDE techniques be used to get convergence rates for the free central limit theorem? | 206,922 |
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\documentclass[11pt,twoside]{article}
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}
\def\thetitle{A Bernstein - von Mises Theorem for growing parameter dimension}
\def\thanksa
{The author is partially supported by
Laboratory for Structural Methods of Data Analysis in Predictive Modeling, MIPT,
RF government grant, ag. 11.G34.31.0073.
Financial support by the German Research Foundation (DFG) through the Collaborative
Research Center 649 ``Economic Risk'' is gratefully acknowledged.
The author thanks the AE and two anonymous referees for very useful comments and suggestions.}
\def\theruntitle{credible sets for growing parameter dimension}
\def\theabstract{
The prominent Bernstein -- von Mises (BvM) Theorem claims that the posterior distribution
is asymptotically normal and its mean is nearly the maximum likelihood estimator (MLE),
while its variance is nearly the inverse of the total Fisher information matrix,
as for the MLE. This paper revisits the classical result from different viewpoints.
Particular issues to address are: nonasymptotic framework with just one finite sample,
possible model misspecification, and a large parameter dimension.
In particular, in the case of an i.i.d. sample,
the BvM result can be stated for any smooth parametric family provided
that the dimension \( \dimp \) of the parameter space satisfies the condition
``\( \dimp^{3}/\nsize \) is small''.
}
\def\kwdp{62F15}
\def\kwds{62F25}
\def\thekeywords{posterior, concentration, bracketing, Gaussian approximation}
\def\thankstitle{}
\def\authora{Vladimir Spokoiny}
\def\runauthora{spokoiny, v.}
\def\addressa{
Weierstrass-Institute, \\ Humboldt University Berlin, \\ Moscow Institute of
Physics and Technology
\\
Mohrenstr. 39, 10117 Berlin, Germany, \\
}
\def\emaila{[email protected]}
\def\affiliationa{Weierstrass-Institute and Humboldt University Berlin}
\input mydef
\input statdef
\input myfront
\input{pa_BvM_2013_09}
\ifGauss{}
{
\Chapter{Conclusion and Outlook}
The presented study confirms once again that the famous statistical facts
like Fisher, Wilks, Bernstein - von Mises Theorems are mainly based on the local
quadratic structure of the log-likelihood process.
This explains why these results can be extended far beyond the classical parametric
set-up.
If one succeeds in local quadratic approximation of the log-likelihood,
the classical result follow almost automatically and are nearly sharp.
\cite{SP2011} offers a majorization device which can be used for such a local quadratic
approximation (majorization) under possible model misspecification and for finite
samples and large parameter dimension.
The approach does not require to specify a particular model like i.i.d. or regression
with additive errors etc.
This paper extends the methods and results to the penalized MLE and applies to the
Bayes set-up yielding the Bernstein - von Mises Theorem for non-informative and
Gaussian priors.
The results are stated under quite general conditions.
However, we tried everywhere to simplify the presentation by sacrificing
the generality in favor of readability.
This especially concerns the global conditions which can be restrictive in some
situations.
As an example, mention the i.i.d. case where one can apply very strong and
powerful results on concentration of measure; see e.g. \cite{BoMa2011}.
Anyway, even in the presented form, the results cover and extend the classical
regular parametric case with a compact parameter space; see examples in \cite{SP2011},
Section~5.
The study can be continued in several directions.
A forthcoming paper will discuss the semiparametric estimation problem.
A sieve or smooth model selection problem is an important issue which can be
considered as the problem of estimating the model parameter.
Bernstein - von Mises result for hyperpriors can be naturally linked to Bayesian
model selection.
A serious limitation of the proposed approach is that it is limited to the smooth
regular case.
An extension of the approach to the sparse estimation in a high dimensional
parameter space is still a challenge.
}
\bibliography{../bib/exp_ts,../bib/listpubm-with-url}
\end{document} | 213,439 |
Sub-theme 21: Institutionally-embedded practice-based learning in multinationals
Call for Papers
Knowledge transfer within multinational enterprises (MNEs) has been widely studied by institutional theorists (e.g. Guler et al., 2002; Kostova and Roth, 2002). However, the aim of such studies has been typically to highlight the isomorphic pressures to adopting practices to gain legitimacy where organizational learning is projected as the flow of abstract knowledge. By contrast, studies that pay systematic attention to the influence of social institutions on learning where knowledge is grounded in practical consciousness are rare (exceptions include Hong et al., 2005). If institutions are considered at all, the interest remains on their impact on knowledge transfer, rather than learning, leading to similar (e.g. Kostova and Roth, 2002) or diverse (e.g. Ferner et al., 2005) patterns of work organization across foreign subsidiaries. By the same token, if there is an interest in organizational learning, this is, more often than not, conceptualized as knowledge transfer that is divorced from the role of human agency (e.g. Macharzina et al., 2001; Uhlenbruck et al., 2003). The social perspective on organizational learning has been widely canvassed outside the MNE literature (e.g. Blackler, 1993; Lane, 1993; Brown and Duguid, 1991). However, the view of organizational learning in multinational settings is commonly a structuralist one, where learning refers to a process of transferring discrete best practices commonly divorced from the broader institutional contexts (e.g. Barkema and Vermeulen, 1998; Zahra et al., 2000). This calls for a need to introduce contextual and practice-based understanding of organizational learning to the research on the multinational firm. We welcome papers that highlight the role of agency in MNE learning that is embedded in wider institutional contexts. Our aim is to integrate, at the very least, two strands of literature – institutional theory and organizational learning – to examine the broader institutional arrangements and micro-organizational processes that lead to similar or diverse practices or that highlight how new practices are created from activity innovations (see Lounsbury and Crumley, 2007).
Contributions to this sub-theme should adopt a contextual and practice-based understanding of learning (e.g. Cook and Brown, 1999; Nicolini et al., 2003; Gherardi, 2000), and seek to highlight the ways in which institutional influences interact with orientations of actors to enacting acquired knowledge in multinationals. We encourage contributions that draw on different theoretical streams beyond organizational learning and institutional theory, adopt diverse research methodologies and examine multiple levels of analysis.
Some of the questions that can be addressed are:
- How does knowledge transfer differ from organizational learning within multinationals? Are the facilitators of learning different from those of knowledge transfer?
- What are the effects of broader institutions on learning processes within multinationals?
- To what extent are organizational practices shaped by macro-level institutions and micro-level agency? Under what conditions do these play a role? How do the two forces interact?
- How can meaning systems be altered and learning be catalyzed where actors are locked into path-dependent methods of operating in particular institutional contexts?
- How do path dependencies of different sorts (related to home country, host country or enterprise specific practices and institutions) impinge on one another and lead to original creation of practice and knowledge?
- What opportunities do multinational contexts provide for practice-based learning? Do choice of country and modes of operating affect the degree to which an organization learns?
- What implications does the conceptualization of organizational learning as practice-based have for MNE models and institutional theory, in particular the spread of new practices? | 241,611 |
“
I would have to agree, UR. These concepts are retrograde and sadly overused. We might do well to counter Mencken’s sentiment with the maxim of another Cryp luminary, La Rochefoucauld: “A quickness in believing evil without having sufficiently examined it is the effect of pride and laziness. We wish to find the guilty, and we do not wish to trouble ourselves in examining the crime.”
Love it, hal. Thank you! — YUR | 288,278 |
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TITLE: Two different formulas for standard error of difference between two means
QUESTION [3 upvotes]: I mostly see this formula when searching for a formula for the estimate of the standard error in difference between two means, and it is also used in this video.
$$\Delta=\sqrt{s_1^2/N_1+s_2^2/N_2}$$
But I've also seen this one (and this is the one my book uses):
$$\Delta'=\sqrt{\dfrac{\left(N_1-1\right)s_1^2+\left(N_2-1\right)s_2^2}{N_1+N_2-2}\left(\dfrac{1}{N_1}+\dfrac{1}{N_2}\right)}$$
As these are two very different formulas, how come they are used seemingly interchangeably?
REPLY [2 votes]: There are two different versions of the two-sample t test in common usage.
Pooled. The assumption, often unwarranted in practice, is made that the two populations have the same variance $\sigma_1^2 = \sigma_2^2.$ In that case one seeks to estimate the common population variance, using both of the sample variances, to obtain what is called a pooled estimate $s_p^2$.
If the two sample sizes are equal, $n_1 = n_2,$ then this is simply $(s_1^2 + s_2^2)/2.$ But if sample sizes differ, then greater weight is put on the sample variance from the larger sample. The weights use the degrees of freedom $\nu_i = n_i - 1)$ instead of the $n_i.$
The first factor under the radical in your $\Delta^\prime$ is $s_p^2.$ Under the assumption of equal population variances,
the standard deviation of $\bar X_1 - \bar X_2$ (estimated standard error)
is your $\Delta^\prime$.
Consequently, the $T$-statistic is
$T = (\bar X_1 - \bar X_2)/\Delta^\prime$. Under the null hypothesis that population means $\mu_1$ and $\mu_2$ are equal, this $T$-statistic has Student's T distribution with $n_1 + n_2 - 2$ degrees of freedom.
Separate variances (Welch). The assumption of equal population variances is not made. Then the variance of $\bar X_1 - \bar X_2$
is $\sigma_1^2/n_1 + \sigma_2^2/n_2.$ This variance is estimated by $s_1^2/n_1 + s_2^2/n_2.$ So the (estimated) standard error is
$\Delta = \sqrt{s_1^2/n_1 + s_2^2/n_2}.$ So your first formula
is has typos and is incorrect. This may account for "ludicrous" difference you are getting. If $n_1 - n_2$, then you should get $\Delta = \Delta^\prime.$ But the two (estimated) standard errors will not necessarily be equal if sample sizes differ.
An crucial difference between the pooled and Welch t tests is that the Welch test uses a rather complicated formula involving both sample sizes and sample variances for the degrees of freedom (DF). The Welch DF is always between the minimum of $n_1 - 1$ and $n_2 - 1$ on the one hand and $n_1 + n_2 - 2$ on the other. So if both sample sizes are moderately large both $T$-statistics will be nearly normally distributed when $\mu_1 = \mu_2.$
The Welch $T$-statistic is only approximate, but simulation studies have shown that it is a very accurate approximation over a large variety of sample sizes (equal and not) and population variances (equal or not).
The current consensus among applied statisticians is always to use the Welch t test and not worry about whether population variances are equal. Most statistical computer packages use the Welch procedure by default and the pooled procedure only if specifically requested. | 106,291 |
BOULDER, Colo., May 31, 2007 /PRNewswire-FirstCall/ -- Pharmion Corporation reported today that data from 18 abstracts of studies investigating the company's marketed and pipeline products will be presented or published at the American Society of Clinical Oncology's (ASCO) 43rd Annual Meeting in Chicago (June 1-5, 2007). These abstracts include summaries of data from studies of each of the six key products in the company's commercial and development portfolio, for multiple indications, including Myelodysplastic Syndromes (MDS), multiple myeloma, Hodgkin's lymphoma, small cell lung cancer and advanced hormone-refractory prostate cancer.
"ASCO 2007 provides further evidence of Pharmion's progress," said Patrick J. Mahaffy, president and chief executive officer of Pharmion. "Data from six of our products will be presented or published, including Phase III data for Satraplatin and Thalidomide, encouraging Phase II data for Amrubicin, Vidaza and MGCD0103, and human bioavailability data for oral azacitidine. These data demonstrate the strength of the Pharmion business model; we are focused solely on oncology, we have multiple compounds demonstrating clinical benefit, and we have a U.S. and international infrastructure to support those products. Pharmion is well-positioned to take full advantage of the strong portfolio of products in our pipeline."
"The data at ASCO affirm Pharmion's leadership in developing epigenetic therapies as a promising approach to treating cancer," said Andrew R. Allen, Pharmion's chief medical officer. "MGCD0103, our class-selective HDAC inhibitor, is showing meaningful activity as monotherapy in advanced Hodgkin's lymphoma. Furthermore, as we begin to understand the way in which multiple epigen etic mechanisms co-operate to silence key tumor suppressor genes in cancer, we are able to choose epigenetic drugs from within our portfolio to combine and attack these silencing systems simultaneously, with encouraging clinical data emerging in the context of acute myeloid leukemia."
Pharmion has three epigenetic products in its portfolio, including Vidaza(R) (azacitidine for injection), MGCD0103 and oral Azacitidine. Pharmion currently markets Vidaza, its parenteral formulation of Azacitidine, in the U.S. and several additional countries for the treatment of patients with MDS. Abstracts presented at this year's ASCO meeting will address the pharmacokinetics of oral Azacitidine administration, and report data from studies investigating Vidaza's utility in MDS, including alternative dosing schedules and in the treatment of MDS patients with marrow fibrosis.
Additionally, accepted abstracts will investigate the use of Vidaza as part of combination treatment regimens with other epigenetic therapies, including interim results from an ongoing study with MGCD0103, for the treatment of high-risk MDS and acute myeloid leukemia (AML), and in combination with valproic acid in advanced solid tumors.
An oral presentation on a study of MGCD0103 in relapsed/refractory Hodgkin's lymphoma will be presented during ASCO, as will a poster on the use of MGCD0103 as a single agent in patients with MDS or leukemias. Pharmion has a licensing and collaboration agreement with MethylGene for MGCD0103, as well as MethylGene's pipeline of second-generation HDAC inhibitor compounds for oncology indications.
Data from a first-line study evaluating the addition of Thalidomide to the standard of care for elderly multiple myeloma patients will be the subject of an oral presentation at the 2007 ASCO meeting, where a total of three oral presentations and six posters on Thalidomide studies will be presented. Data from the oral presentation reinforce the substantial body of evidence that demonstrate that the addition of Thalidomide to front-line melphalan/prednisone therapy represents a safe and effective treatment for even very elderly patients with multiple myeloma in Europe, where melphalan/prednisone (MP) is currently the standard of care.
Satraplatin, will be the subject of an oral presentation at the Conference. The oral presentation will discuss the progression-free survival results of the randomized Phase III SPARC study (Satraplatin and Prednisone Against Refractory Cancer) in hormone-refractory prostate cancer patients. In addition, Pharmion and GPC Biotech will co-sponsor a satellite symposium on the evening of Friday, June 1, titled "Emerging Strategies for the Management of Advanced Prostate Cancer."
Interim data on a Phase II study of Amrubicin appears in the ASCO 2007 conference publication, and will be discussed at Pharmion's Investor and Analyst Event to be held Monday, June 4, from 6:00 to 7:30pm in Chicago. The meeting will be webcast on the company's website.
The following clinical data will be presented in poster sessions (unless otherwise noted) during the ASCO 2007 annual meeting:
Vidaza(R)
Tolerability and hematologic improvement assessed using three alternative dosing schedules of azacitidine (Vidaza) in patients with MDS -- R. Lyons, US Oncology; Abstract #7083; June 2, 2007; 8:00am-12:00pm; McCormick Place Convention Center, S Hall A2
An oral dosage formulation of azacitidine: A pilot pharmacokinetic study -- R. Ward, Pharmion; Abstract #7084; June 2, 2007; 8:00am-12:00pm; McCormick Place Convention Center, S Hall A2
Response to azacitidine in patients with myelodysplastic syndrome with marrow fibrosis -- R. Juvvadi, Western Penn, Pennsylvania; Abstract #7089; June 2, 2007; 8:00am-12:00pm; McCormick Place Convention Center, S Hall A2
The combination of 5-azacytidine and valproic acid is safe and active in advanced solid tumors -- A.O. Soriano, MD Anders on Cancer Center; Abstract #3547; June 2, 2007; 2:00-6:00pm; McCormick Place Convention Center, S102a
MGCD0103
Oral Presentation: A Phase II Study of a Novel Oral Isotype-Selective Histone Deacetylase (HDAC) Inhibitor in Patients With Relapsed or Refractory Hodgkin's lymphoma -- A. Younes, MD, MD Anderson Cancer Center; Abstract #8000; June 2, 2007; 3:00-3:15pm; McCormick Convention Center, E450bam-12:00pm; McCormick Convention Center, S Hall A2
A Phase I study of MGCD0103 given as a twice weekly oral dose in patients with advanced leukemias or myelodysplastic syndromes (MDS) -- J. Lancet, MD, H. Lee Moffitt Cancer Center; Abstract #2516; June 4, 2007; 2:00-6:00pm; McCormick Convention Center, S102a
Thalidomide
Oral Presentation: Long-term responses to thalidomide and rituximab in Waldenstrom's macroglobulinemia -- J. Soumera; Abstract #8017; June 3, 2007; 1:00-1:15pm;pm; McCormick Place Convention Center, E354b
An analysis of erythro po
A pharmacogenetic study of docetaxel and thalidomide in patients with androgen-independent prostate cancer (AIPC) using targeted human DMET genotyping platform -- J. Deeken, MD; Abstract #3580; June 3, 2007; 8:00am-
Satraplatin
Oral Presentation: Satraplatin demonstrates significant clinical benefits for the treatment of patients with HRPC: Results of a randomized phase III trial -- C. Sternberg; Abstract #5019; June 4, 2007; 11:00-11:15am; Arie Crown Theater
Amrubicin
A randomized phase II trial of Amrubicin vs. Topotecan as second-line treatment in extensive disease small-cell lung cancer (SCLC) sensitive to platinum-based first line chemotherapy. -- R.M. Jotte; Abstract #18064, 2007, its Annual Report on Form 10-K for the y ear144 or+1-720-564-9143; or Tara May of Pharmion Corporation, On-site mediacontact, +1-303-646-7832
Web site:
Ticker Symbol: (NASDAQ-NMS:PHRM)
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Anyone that follows my blog will know that Titanfall is a game that i am really looking forward too, so when i was told that a beta was to be released i was very keen on being a part of it. I was one of the fortunate people that managed to get onto it a couple of days before it became open, and since then it’s pretty much all that i’ve played.
The beta features two maps, Angel City, a small futuristic metropolis, which is the more close quarter of the two, and Fracture, which is quite a bit bigger, it still has a lot of close quarters combat but it is more suited suited to long range weapons than Angel City.
There are three game modes to choose from on the beta, but i have stuck almost exclusively to Attrition, which is a basic team deathmatch, the other two modes that are available are Last Titan Standing, which is pretty self explanatory, and Hardpoint, which is a domination type mode in which teams fight to control three points on the map.
Weapon wise it’s your standard FPS gun selection, you have the choice of an assault rifle, shotgun, sub machine gun, sniper rifle, and then there’s the smart pistol, a pistol which locks onto targets, which at first seemed a bit overpowered to me, but in truth, with the beta at least, the whole game seems really well balanced and the gunplay is solid
My personal favourite setup after a few hours of play is the SMG paired with the extended mags and optic sight, but that’s because i’m that guy you always see running into battle like a headless chicken.
The other side to the battle is the Titans, these can be unlocked fairly easily which i think is good, you basically start off with a couple of minutes until your titan is ready, then for killing enemies and engaging in battles the timer goes down quicker, basically the more you engage the enemy, the quicker you’ll get your Titan. I love this idea because not only does it deter players from ‘camping’, it also means you don’t have to be amazing to get your Titan, i’d say 9 times out of 10 you WILL get a Titan at least once a game.
The movement of the Pilot is another great aspect of this game, you are able to leap from building to building using your equipment and the wall running is as smooth as butter, the whole game just flows so well and fits in incredibly well with the fast paced nature of the battle.
As you’d imagine if you’re a foot soldier and you come across a Titan the best policy tends to be to run, but usually Titans end up going against each other rather than picking of the pilots, this doesn’t mean that a pilot can’t down a Titan though, you do have an anti Titan weapon, but they are best used when a titan is pre occupied with another, as in a one on one you’re always going to struggle to match the Titans firepower, you can also jump on top of opposition Titans and shoot them straight where it hurts, but again getting close enough to do so can be problematic.
When you call your Titan in it feels great, you place your marker down and as it plummets to the Earth, you can hear the sound of it ripping through the atmosphere as it hurtles towards you, when it lands you jump into it in which starts a neat animation of you being placed into the Titan, it’s only a two second thing, but it all feels very satisfying, and when you’re in control you really do feel powerful, big, and when you’re battling against the other Titans it really feels epic.
The whole game feels very fast paced and is undoubtedly the best FPS i’ve played in a very long time, the biggest compliment i can pay it is that you really can tell that it’s the people that were behind Call Of Duty 4, one of the most coveted shooters in history, and having played a few hours of Titanfall i really do think it has the potential to be as good as Call Of Duty 4, if this game is done right, it could set a new benchmark, and become a classic.
The one flaw it does seem to have at the moment is the AI, when you go into battle it’s a 6 vs 6, but you have to add the AI onto that, you have Grunts, the basic low level foot soldiers, and the Spectres who are a level above them, but the point is, they are both useless, now i don’t know if this is the worst thing in the world, mowing down the Grunts is fun, and it can make a bad player feel like he’s doing well, it also means everyone can contribute to the team, so in my opinion although they need improving, i don’t think they should make them too powerful.
Since playing the beta i’ve gone and pre ordered my copy and it’s now only a few weeks away, i really can’t wait to get my hands on the full game, i think this might be one of those games that prevents me from seeing any sunlight, i really hope the full game is as much fun as it looks like it’s going to be, Ladies and Gentlemen, could this be the COD killer we’ve been waiting for? I vote yes
Titanfall beta trailer, from Youtube User: Official EA UK
4 Comments
I haven’t been following this one (more interested in Destiny to be honest), but you make a strong case that I should be! This is only on Xbox One and PC though right?
It’s on Xbox One, PC, and Xbox 360, the beta closed yesterday unfortunately, I’m gutted, got to wait until its out now!
Good to hear that your first impressions of Titanfall are so positive. This is one of the next gen titles I’ve been most looking forward to, so I’m glad it’s proved to be fun! In particular, I’m really glad to hear that it might even turn out to be a COD replacement, because honestly I think it’s about time. Not dissing the franchise, but I think variety’s everything! Unfortunately I’m on PS4, so I guess it’s PC or 360 for me!
Yeah I was quite relieved myself! I’ve been looking forward to it for a while now so i was really hoping the hype wouldn’t be too much, but from the beta I’d say it’s living up to it! Just hope the full game delivers! | 77,858 |
Taking its cue from New York’s bistro-bar culture; encyclopaedia sets, heady black and white prints and a bluesy soundtrack set the scene for intimate dinners or jovial group gatherings.
For more information visit the Boo Radley's Website. | 413,104 |
TITLE: For which fields does the isogeny theorem hold
QUESTION [9 upvotes]: Let $k$ be a field. We say that the isogeny theorem holds over $k$ if, for any abelian variety $A$ over $k$, there are only finitely many $k$-isomorphism classes of abelian varieties $B$ over $k$ which are $k$-isogenous to $A$.
Here are some examples of fields for which the isogeny theorem hold.
$k$ is a finite field.
$k$ is a number field (Faltings)
Does the isogeny theorem hold for $k$ a function field over an algebraically closed field of characteristic zero?
Does the isogeny theorem hold for $k$ a function field over a finite field?
In the number field case, by results of Serre-Tate, the Shafarevich conjecture for abelian varieties implies the isogeny theorem.
Does a similar implication hold over $\mathbf C(t)$? (Of course, the naive analogue of the Shafarevich conjecture is false, but Faltings proves a "correct" version in his paper: Arakelov's theorem for abelian varieties.)
REPLY [6 votes]: Over $\mathbb{C}(T)$ the Isogeny theorem doesn't hold, even if you throw away isotrivial components.
In his paper "Arakelov's theorem for Abelian Varieties" Faltings proves the following analogue of Shafarevich:
Let $B$ be complete smooth complex curve and $S\subset B$ a finite subset of points, and $f:X\rightarrow B\backslash{S}$ a family of Abelian varieties. Let $G$ be the fundamental group of $B\backslash{S}$. Fix a point $p\in B\backslash S$ so that we get an action of $G$ on
$H_1(X_p,\mathbb{Z})$ by monodromy.
Now, say $X$ satisfies (A) if
$$End(X)=End_{G}(H_1(X_p,\mathbb{Z})).$$ Note that the LHS also injects into the RHS.
Faltings proves there are finitely many isomorphisms classes $X$ satisfying (A) of a fixed dimension, and he also gives an example of an 8-dimensional family $X$ which doesn't satisfy (A). Now we'll use this $X$ to give a negative answer to the Isogeny theorem:
By a standard argument (see Milne's Abelian Varieties online notes, Chapter IV, Theorem 2.5) if both $X$ and $X\times X$ have at most finitely many isogenous abelian varieties up to isomorphism, by an isogeny of degree a power of some prime $l$, then
$$End(X)\otimes\mathbb{Q}_l=End_G(H_1(X_p,\mathbb{Z})\otimes\mathbb{Q}_l)$$
where the second endomorphism ring is as $\mathbb{Q}_l$ vector spaces. However, the latter is false for our $X$ by assumption, since taking group invariants commutes with base change over fields, hence either $X$ or $X\times X$ don't satisfy the isogeny theorem.
$\textbf{However},$ if $X$ is a family of abelian varieties over the generic point of $B$ which extends to a family away from $S$ (in other words, it only has bad reduction at points of $S$) then every abelian varietiy isogenous to $X$ shares this property, since good reduction is an isogeny invariant. Hence the isogeny therem is true for all $X$ satisfying (A), and this includes all $X$ of dimension at most 3 by a theorem of Deligne
(see Hodge II, 4.4.13), as long as you insist $X$ has no isotrivial subvarieties. | 167,429 |
TITLE: Show that $\int_0^1|x-\mu|f(x)dx\le \frac{1}{2}, \text { where } \mu=\int_0^1xf(x)dx.$
QUESTION [2 upvotes]: Question: Let $f:[0,1]\to(0,\infty)$ be a function satisfying $$\int_0^1f(x)dx=1.$$ Show that the integral $$\int_0^1(x-a)^2f(x)dx\text{ is minimized when } a=\int_0^1xf(x)dx.$$ Hence or otherwise show that $$\int_0^1|x-\mu|f(x)dx\le \frac{1}{2}, \text { where } \mu=\int_0^1xf(x)dx.$$
My approach: Let $g:\mathbb{R}\to\mathbb{R}$ be such that $$g(a)=\int_0^1(x-a)^2f(x)dx, \forall a\in\mathbb{R}.$$
Thus $$g(a)=\int_0^1 x^2f(x)dx-2a\int_0^1xf(x)dx+a^2\int_0^1f(x)dx\\=\int_0^1 x^2f(x)dx-2a\int_0^1xf(x)dx+a^2, \forall a\in\mathbb{R}.$$
Observe that $g$ is differentiable $\forall a\in\mathbb{R}$.
Now $$g'(a)=-2\int_0^1xf(x)dx+2a, \forall a\in\mathbb{R}.$$
Thus $$g'(a)=0\iff -2\int_0^1xf(x)dx+2a=0\iff a=\int_0^1xf(x)dx.$$
Again observe that $g'$ is differentiable $\forall a\in\mathbb{R}$ and $$g''(a)=2, \forall a\in\mathbb{R}.$$
This implies that $$g''\left(\int_0^1xf(x)dx\right)=2>0.$$
Hence by the double derivative test we can conclude that $g$ has a local minimum at $a=\int_0^1xf(x)dx$.
Now observe that $$\lim_{a\to+\infty}g(a)=+\infty\text{ and }\lim_{a\to-\infty}g(a)=+\infty.$$
Thus we can conclude that $g$ attain it's global minimum at $$a=\int_0^1 xf(x)dx.$$
Hence, we are done with the first part of the question.
For the second part I was trying to solve using the Cauchy-Schwarz inequality for integrals, but haven't found anything meaningful yet. Someone please help me in proceeding.
REPLY [2 votes]: By Cauchy-Schwarz inequality we have $$\left|\int_0^1|x-\mu|f(x)dx\right|=\int_0^1|x-\mu|f(x)dx=\int_0^1\left(|x-\mu|\sqrt{f(x)}\right)\left(\sqrt{f(x)}\right)dx\\\le \sqrt{\int_0^1|x-\mu|^2f(x)dx \int_0^1f(x)dx}\\= \sqrt{\int_0^1|x-\mu|^2f(x)dx}\\=\sqrt{\int_0^1(x-\mu)^2f(x)dx}\\=\sqrt{\int_0^1x^2f(x)dx-\mu^2}.$$
Now we have $$\left(\mu-\frac{1}{2}\right)^2\ge 0 \implies \mu^2-\mu+\frac{1}{4}\ge 0\implies -\mu^2\le -\mu+\frac{1}{4}.$$
Thus we have $$\int_0^1x^2f(x)dx-\mu^2\le \int_0^1x^2f(x)dx-\mu+\frac{1}{4}\\=\int_0^1x^2f(x)dx-\int_0^1xf(x)dx+\frac{1}{4}\int_0^1f(x)dx\\=\int_0^1\left(x^2-x+\frac{1}{4}\right)f(x)dx\\=\int_0^1\left(x-\frac{1}{2}\right)^2f(x)dx.$$
Now since $$0\le x\le 1 \implies -\frac{1}{2}\le x-\frac{1}{2}\le \frac{1}{2}\implies \left|x-\frac{1}{2}\right|\le \frac{1}{2}\\\implies \left|x-\frac{1}{2}\right|^2=\left(x-\frac{1}{2}\right)^2\le \frac{1}{4}.$$
Thus $\forall x\in[0,1]$, we have $$\left(x-\frac{1}{2}\right)^2f(x)\le \frac{1}{4}f(x)\\\implies \int_0^1\left(x-\frac{1}{2}\right)^2f(x)dx\le \frac{1}{4}\int_0^1f(x)dx=\frac{1}{4}.$$
Hence we have $$\int_0^1x^2f(x)dx-\mu^2\le \frac{1}{4}\\\implies \sqrt{\int_0^1x^2f(x)dx-\mu^2}\le \frac{1}{2}.$$
This in turn implies that $$\int_0^1|x-\mu|f(x)dx\le \sqrt{\int_0^1x^2f(x)dx-\mu^2}\le \frac{1}{2}\\\implies \int_0^1|x-\mu|f(x)dx\le \frac{1}{2}.$$ | 175,170 |
TITLE: Derivatives of Schwarz functions at $x=0$.
QUESTION [1 upvotes]: Problem: Given an arbitrary sequence $a_n$ of positive numbers, is it possible to find some Schwarz function $f$ on $\mathbb R$ such that
$$|f^{(n)}(0)| \geq a_n \, \forall n ?$$
If not, how fast can $b_n=f^{(n)}(0)$ grow?
Note: It is clear that the above is not true for analytic functions. But it is not clear to me if it could actually hold for Schwarz functions.
REPLY [2 votes]: As r9m said the best answer is Borel's lemma saying the map sending smooth functions to their formal Taylor series at $0$ is surjective.
Otherwise there is a non-constructive proof
If there exists $(b_n)$ such that for all Schwartz function $f^{(n)}(0)/b_n \to 0$ let $$g_N(x)=\sum_{n=0}^N \frac{(-1)^n}{|b_{2n}| 2^n} \delta^{(2n)}(x)$$ then $$\lim_{n \to \infty} \langle g_n,\phi\rangle$$ converges for all Schwartz function $\phi$, it means so does $$\lim_{n \to \infty} \langle g_n,\hat{\phi}\rangle=\lim_{n \to \infty} \langle \hat{g_n},\phi\rangle=\langle \hat{g_\infty},\phi\rangle$$
Where
$$\hat{g_\infty}(\xi)=\sum_{n=0}^\infty \frac{1}{|b_{2n}|2^n} (2\pi \xi)^{2n}$$
It is obviously wrong because $\hat{g_\infty}$ grows faster than polynomials so that $\frac1{1+\hat{g_\infty}}$ is Schwartz and clearly $$\langle \frac1{1+\hat{g_\infty}},\hat{g_\infty}\rangle= \infty$$ | 161,986 |
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White / III / Article #100063729002
Product Reviews | 369,332 |
\section{$N_\infty$-operads and $N_\infty$-diagrams}
Given a topological space $X$ equipped with a multiplication $m: X \times X \longrightarrow X$, we would like to say that its multiplication is \emph{homotopy commutative} if the diagram
\[
\xymatrix{ X \times X \ar[d]_-{\tau} \ar[rr]^-m & & X \\
X \times X \ar[rru]_-m & &
}
\]
commutes up to homotopy, where $\tau$ is the twist map permuting the two factors. With this in place one would now have to take care of coherence, that is, the chosen homotopy between $m$ and $m \circ \tau$ needs to be compatible with multiplying three or more copies of $X$. Such coherence issues are neatly packaged in the theory of operads~\cite{loop}.
\begin{definition}
A (topological) \emph{symmetric operad} is a collection $\mathcal{O}=\{\mathcal{O}(n)\}_{n \ge 0}$ of topological spaces $\mathcal{O}(n)$ equipped with a (right) $\Sigma_n$-action together with maps
\[
\mathcal{O}(n) \times \mathcal{O}(i_1) \times \cdots \times \mathcal{O}(i_n) \longrightarrow \mathcal{O}(i_1 + \cdots + i_n)
\]
such that the expected coherence diagrams hold with regards to associativity, unitality and the symmetric group actions.
\end{definition}
An \emph{algebra over an operad} $\mathcal{O}$ is a space $X$ together with multiplication maps
\[
\mathcal{O}(n) \times_{\Sigma_n} X^n \longrightarrow X
\]
satisfying the expected coherence diagrams. Here, $\Sigma_n$ acts on $X^n$ by permuting factors. This information is equivalent to a morphism of symmetric operads
\[
\mathcal{O} \longrightarrow \operatorname{End}(X),
\]
where $\operatorname{End}(X)$ denotes the endomorphism operad of $X$, i.e.,
\[
\operatorname{End}(X)(n)=\operatorname{Hom}(X^n, X).
\]
We can think of the space $\mathcal{O}(n)$ as the different possibilities of multiplying $n$ elements in our space. For instance, if $\mathcal{O}(n)=\ast$ for all $n$, then there is a unique way of multiplying $n$ elements
\[
\ast \times_{\Sigma_n} X^n \cong X^n/\Sigma_n \longrightarrow X.
\]
In particular, this means that our space $X$ is a strictly commutative object.
If instead we suppose that $\mathcal{O}(n)\simeq \ast$ for all $n$, then there is one way of multiplying $n$ elements ``up to homotopy'', which leads to the following definition.
\begin{definition}
An \emph{$E_\infty$-operad} is a symmetric operad $\mathcal{O}$ such that the action of $\Sigma_n$ on each space is free, and every $\mathcal{O}(n)$ is $\Sigma_n$-equivariantly contractible.
\end{definition}
There are many different $E_\infty$-operads, each of them having their own technical advantages and disadvantages. Thankfully, all $E_\infty$-operads are weakly equivalent, indeed, there is a Quillen model structure on the category of topological symmetric operads where the weak equivalences are those maps that are levelwise homotopy equivalences of spaces~\cite{BM}. In particular, we can think of this as having one unique (up to homotopy) notion of homotopy commutativity.
Now that we have outlined the theory of homotopy commutativity in the non-equivariant case, we move towards to the more complex setting of $G$-spaces for $G$ some finite group. We now need to consider multiplication maps of the form
\[
\prod\limits_{T} X \longrightarrow X
\]
where $T$ is a $G$-set with $n=|T|$ elements. The $G$-action induces a group homomorphism $G \to \Sigma_n$. This means that the $\mathcal{O}(n)$ spaces should not be thought of merely as $\Sigma_n$-spaces, but as $(G \times \Sigma_n)$-spaces. Note that simply putting a trivial $G$-action on the $\mathcal{O}(n)$ would not allow for multiplications of the above kind for any $T$ with more than one element.
We shall now work towards the theory of $N_\infty$-operads, which allows us to fix this issue.
\begin{definition}
A \emph{graph subgroup} $\Gamma$ of $G \times \Sigma_n$ is a subgroup such that $\Gamma \cap (1 \times \Sigma_n)$ is trivial. (Here $1$ denotes the trivial group.)
\end{definition}
Any graph subgroup is of the form
\[
\Gamma = \{ (h, \sigma(h)) \,\,| \,\, h \in H \},
\]
with $H \leq G$ and $\sigma \colon H \longrightarrow \Sigma_n$ a group homomorphism. Moreover, given a finite $H$-set $T$ with $n$ elements we obtain a graph subgroup
\[
\Gamma(T)=\{ (h, \sigma(h)) \,\,|\,\, h \in H \},
\]
where $\sigma \colon H \longrightarrow \Sigma_n$ represents the $H$-action on $T$. Conversely, we can view any graph subgroup as one of the form $\Gamma(T)$, as for
\[
\Gamma = \{ (h, \sigma(h)) \,\,| \,\, h \in H \},
\]
we can set $T$ to be a set of $n$ elements with the $H$-action given by $\sigma$.
\begin{definition}
An \emph{$N_\infty$-operad} is a symmetric operad $\mathcal{O}$ in the category of $G$-spaces (that is, a collection of $G \times \Sigma_n$-spaces $\mathcal{O}(n), n \ge 0$) satisfying the following conditions.
\begin{itemize}
\item For all $n \ge 0$, $\mathcal{O}(n)$ is $\Sigma_n$-free.
\item For every graph subgroup $\Gamma$ of $G \times \Sigma_n$, the space $\mathcal{O}(n)^\Gamma$ is either empty or contractible.
\item $\mathcal{O}(0)^G$ and $\mathcal{O}(2)^G$ are both nonempty.
\end{itemize}
\end{definition}
The last condition ensures that the operad possesses an equivariant multiplication and an equivariant `point'.
The second point together with the operad structure implies that each $\mathcal{O}(n)$ is a classifying space for a family of subgroups which satisfy some further properties forced by operad structure. This information can be distilled into the theorem below.
\begin{theorem}
Up to weak equivalence, every $N_\infty$-operad determines and is determined by a set $X=\{ N_K^H \}$, where $K<H$ are subgroups of $G$, satisfying the following properties and their conjugacies.
\begin{itemize}
\item (Transitivity) If $N_K^H \in X$ and $N_H^L\in X$, then $N_K^L \in X$.
\item (Restriction) If $N_K^H \in X$ and $L \leq G$, then $N_{K \cap L}^{H \cap L} \in X$.
\end{itemize}
We will call such a set a \emph{transfer system} and the objects $N_K^H$ will sometimes be referred to as \emph{norm maps}.
\end{theorem}
Blumberg and Hill showed that every operad determines an ``indexing system''~\cite{BH}. Rubin~\cite{Rubin}, Gutierrez-White~\cite{GW} and Bonventre-Pereira~\cite{BP} independently showed that for every such indexing system one can construct a corresponding operad. Barnes-Balchin-Roitzheim~\cite{BBR} showed that indexing systems are equivalent to the transfer systems given in the above version of this theorem.
\begin{corollary}
There are as many homotopy types of $N_\infty$-operads for a fixed finite group $G$ as there are transfer systems for $G$. \qed
\end{corollary}
In particular, there can be only finitely many $N_\infty$-operads for a finite group $G$, and as such, it makes sense to count them. We will denote by $N_\infty(G)$ the set of all $N_\infty$-operads on $G$. For $G=C_{p^n}$, the number of $N_\infty$-operads plus some additional structure has been determined in~\cite{BBR}.
Before continuing, let us assess the first non-trivial case. We will choose to display indexing systems as graphs whose vertices are the subgroups of $G$, and there is an edge $H \to K$ if $N_H^K \in X$.
\begin{example}
Let $G=C_p$ for some prime $p$, then there are two $N_\infty$-operads which have the following graph representations.
\begin{figure}[h!]
\begin{tikzpicture}[->, node distance=2cm, auto]
\node (1) at (6.000000,0) {$C_{p^1}$};
\node (2) at (4.000000,0) {$C_{p^0}$};
\node at (3.5,0) {\Huge{(}};
\node at (6.5,0) {\Huge{)}};
\node (11) at (10.000000,0) {$C_{p^1}$};
\node (22) at (8.000000,0) {$C_{p^0}$};
\node at (7.5,0) {\Huge{(}};
\node at (10.5,0) {\Huge{)}};
\draw (22) to (11);
\end{tikzpicture}
\end{figure}
\end{example}
The key ingredient in the result of~\cite{BBR} is an operation
\[
\odot \colon N_\infty(C_{p^i}) \times N_\infty(C_{p^j}) \to N_\infty(C_{p^{i+j+2}}).
\]
In particular it is proved that every $N_\infty$-operad for $C_{p^{i+j+2}}$ is of the form $X \odot Y$ for $X \in N_\infty(C_{p^i})$ and $Y \in N_\infty(C_{p^j})$. This then allows an inductive strategy of proof of the main result.
\begin{theorem}[{\cite[Theorem 1]{BBR}}]
For $n \geq 1$ we have
\[
|N_\infty(C_{p^{n}})| = \mathsf{Cat}(n+1)
\]
where $\mathsf{Cat}(n)$ is the $n^{th}$ Catalan number.
\end{theorem}
The result goes a bit further than just an enumeration result. We can put a partial order on the set of all $N_\infty$-diagrams for a fixed group by saying that one $N_\infty$-operad $X$ is smaller than another $N_\infty$-operad $Y$ if it is a subset of $Y$. On the other side, there is a wealth of objects enumerated by Catalan numbers. One of them is the set of rooted binary trees. We can put a partial order on the set of rooted binary trees with $n$ leaves by saying that one tree is larger than another if it can be obtained from the latter by rotating a branch to the right. Balchin-Barnes-Roitzheim found that, indeed, $N_\infty$-diagrams for $G=C_{p^n}$ and rooted binary trees with $n+2$ leaves are isomorphic as posets~\cite{BBR}. However, we do not wish to elaborate on this result here.
The goal of this paper is to study the set $N_\infty(G)$ for $G$ a group of the form $C_{p_1 \cdots p_n}$ for $p_i$ distinct primes where the situation is somewhat more complicated. Note that, in particular, the subgroup lattice is an $n$-dimensional cube. | 42,650 |
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