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Get an overview of this hotel3.5-star hotel, walk to Sunsphere Tower
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Most business owners can’t wait to outsource marketing as fast as possible. Unfortunately, sometimes in our rush to outsource marketing, we put ourselves at risk of making a few big mistakes. On this episode of Profit Without Worry, we’re diving into the dos and don’ts of outsourcing your marketing.
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\begin{document}
\begin{center}
{\LARGE \textbf{$d^3=0$, $d^2=0$ Differential calculi on certain
non-commutative (super) spaces.}}\\[0pt]\vspace{1cm}
\vspace{1cm}
{\large M. EL BAZ\footnote{{\large moreagl@yahoo.co.uk}}, A. EL HASSOUNI
\footnote{{\large lhassoun@fsr.ac.ma}}, Y. HASSOUNI\footnote{{\large
y-hassou@fsr.ac.ma}}\\\ and E.H. ZAKKARI\footnote{{\large
hzakkari@hotmail.com}}.}\\[0pt]\vspace{0.5cm}\vspace{0.5cm}
Laboratory of Theoretical Physics\\[0pt]PO BOX 1014, University Mohammed V
\\[0pt]Rabat, Morocco.
\end{center}
\bigskip
\vspace {3cm}
\textbf{Abstract:} In this paper, we construct a covariant
differential calculus on a quantum plane with two-parametric
quantum group as a symmetry group. The two cases $d^2=0$ and
$d^3=0$ are completely established.
We also construct differential calculi $n=2$ and $n=3$ nilpotent
on super quantum spaces with one and two-parametric symmetry
quantum supergroup.
\vspace{2cm}
\textbf{Keywords:} non-commutative (super) plane, non-commutative
differential calculi $d^3=0$, $d^2=0$.
\newpage\ \
\section{Introduction:}
A non-commutative quantum (super) space \cite{manin1, manin2} is an unital,
associative algebra with a quantum (super) group as a symmetry group. These
objects \cite{drinfeld, jimbo} have enriched the arena of mathematics and
mathematical physics: they appear in the context of theory of knots and
braids \cite{zachos}, as well as in the study of Yang-Baxter equations \cite
{wadati}. Quantum (super) groups are deformations of the enveloping (super)
algebra of classical Lie groups in the sense that one recovers the classical
(super) commutator when the deformation parameters go to some particular
values. Usually the generators of a quantum (super) group are assumed to
commute with the non-commuting coordinates of the corresponding (super)
plane. As a consequence, the quantum (super) plane admits a quantum group as
a symmetry group with only one parameter: ($GL_q(1/1)$) $GL_q(2)$ \cite
{corrigan}. More generally, one can obtain a multiparametric quantum (super)
group, if one relaxes this property (commutation between space coordinates
and group generators), namely, $GL_{p,q}(1/1)$ \cite{tahri} and $GL_{p,q}(2)$
\cite{cho} respectively in the two dimensional quantum superplane and
quantum plane cases.
Many authors \cite{tahri, kobayachi, wess1, berezinski,
schirrmacher, coquereaux1, coquereaux2, salih1} have also studied
differential calculus with nilpotency $n=2$ on (super) spaces with
one or two-parameter (super) group as symmetry groups. An adequate
way leading to generalization of this ordinary differential
calculus arises from the graded differential algebra
\cite{kerner1, kerner2, kerner3, notre, salih2, salih3}. The
latter involves a complex parameter that satisfies some conditions
allowing to obtain a consistent generalized differential calculus.
The most important property of
this calculus is that the operator $"d"$ satisfies $\{d^n=0$ $/$ $d^l\neq 0$
, $1\leq l\leq n-1\}$ and it contains as a consequence, not only
first differentials $dx^i$ $,$ $i=1...m,$ but involves also higher
order differentials $d^jx^i,$ $j=1...n-1.$
In this paper, we construct covariant differential calculus
$d^3=0$ on certain quantum (super) spaces with one or
two-parametric quantum group as a symmetry groups. We will show
that our differential calculus is covariant under the algebra with
a quantum group structure. The complex $j$, which appears in the
Leibniz rule, is a third-root of unity and will be an interesting
and non trivial aspect of the differential calculus that we will
introduce.
This paper is organized as follows:
In section $2$ we start by recalling the two-parameter quantum group acting
on a two-dimensional quantum plane. We also establish $n=2$ and $n=3$
covariant differential calculi on this space following $R.$ $Coquereaux$
approach \cite{coquereaux1, coquereaux2}. It will be noticed that some
modifications have been brought up to this approach in order to adapt it to
the two-parameter quantum group symmetry and the $n=3$ differential
calculus. In section $3,$ the same method will be applied to construct the $
n=2$, $n=3$ covariant differential calculus on $1+1$-dimensional superspace
with one parameter quantum supergroup as a supersymmetry group. In section $
4 $ we generalize the results of section $3$ by taking the two-parameter
quantum group acting covariantly on the superspace.
\section{Differential calculus on a two-parametric quantum plane.}
\subsection{ Preliminaries}
The two dimensional quantum plane is an associative algebra generated by two
non-commuting coordinates $x$ and $y$ \cite{manin1, manin2, coquereaux1}
satisfying the relation:
\begin{equation}
xy=q \, yx \; , \;\;\; q\neq 0,1 \;\;\;(q\in C).
\end{equation}
In order to have a two-parameter quantum group $GL_{p,q^{^{\prime }}}(2)$ as
a symmetry group of such a space\cite{cho}, one must assume that the
coordinates do not commute in general with elements defining this group. \\%
Indeed, for a generic element $T=\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right)$ of $GL_{p,q^{^{\prime }}}(2)$, the relations between the
matrix entries and the coordinates are assumed to be \cite{cho}:
\begin{equation}
\begin{array}{ccccl}
x \, a & = & q_{11}\;a\,x ~~~~~~~~~~~~ y\,a & = & q_{21}\;a\,y \cr x \, b & =
& q_{12}\;b\,x ~~~~~~~~~~~~ y\,b & = & q_{22}\;b\,y \cr x \, c & = &
q_{13}\;c\,x ~~~~~~~~~~~~ y\,c & = & q_{23}\;c\,y \cr x \, d & = &
q_{14}\;d\,x ~~~~~~~~~~~~ y\,d & = & q_{24}\;d\,y. \cr
\end{array}
\end{equation}
The coordinates $x$ and $y$ transform under $T$ and $^tT$ (transposed
matrix) as:
\[
\left(
\begin{tabular}{l}
$x$ \\
$y$
\end{tabular}
\right) \stackrel{T}{\longrightarrow }\left(
\begin{tabular}{l}
$x^{\prime }$ \\
$y^{\prime }$
\end{tabular}
\right) =\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) \left(
\begin{tabular}{l}
$x$ \\
$y$
\end{tabular}
\right)
\]
\[
\left(
\begin{tabular}{l}
$x$ \\
$y$
\end{tabular}
\right) \stackrel{^tT}{\longrightarrow }\left(
\begin{tabular}{l}
$x^{\prime \prime }$ \\
$y^{\prime \prime }$
\end{tabular}
\right) =\left(
\begin{array}{cc}
a & c \\
b & d
\end{array}
\right) \left(
\begin{tabular}{l}
$x$ \\
$y$
\end{tabular}
\right) .
\]
The requirement that the transformed coordinates obey a similar relation as $
eq(1)$ (not necessarily with the same deformation parameter $q$) i.e.
\begin{eqnarray}
x^{\prime }y^{\prime } &=& \bar{q} \, y^{\prime }x^{\prime },\hspace{1.0in}\bar{
q}\in C \\ x^{\prime \prime }y^{\prime \prime } &=& \;
\stackrel{=}{q}y^{\prime \prime }x^{\prime \prime
},\hspace{0.9in}\stackrel{=}{q}\in C,
\end{eqnarray}
and taking account of the defining relations of
$GL_{p,q^{\prime}}(2)$ \cite {schirrmacher},
\begin{eqnarray}
ab=p \, ba\hspace{1.0in}cd= &&p \, dc \nonumber \\
ac=q^{^{\prime }}\, ca\hspace{1.0in}bd = &&q^{^{\prime }} \, db \\
p \, bc=q^{^{\prime }} \, cb\hspace{0.6in}ad-da= &&(p-\frac 1{q^{^{\prime
}}}) \, bc, \nonumber
\end{eqnarray}
for some non zero $p$, $q^{^{\prime }}$ with $pq^{^{\prime }}\neq -1,$
implicates further constraints on the involved parameters:
\[
\bar{q} = \stackrel{=}{q}
\]
and
\begin{eqnarray}
q_{11}=1\hspace{1.2in}q_{21}&= &qq^{^{\prime }-1}k \nonumber \\
q_{12}=\bar{q}p^{-1}\hspace{1.0in}q_{22}&=&q\bar{q}p^{-1}[\bar{q}
-(p-q^{^{\prime }-1})k] \nonumber \\
q_{13}=\bar{q}q^{^{\prime }-1}\hspace{1.0in}q_{23}&= &q\bar{q}
q^{^{\prime }-1}[\bar{q}-(p-q^{^{\prime }-1})k] \\
q_{14}=\bar{q}q^{^{\prime }-1}k\hspace{1.0in}q_{24}&= &q\bar{q}
^2q^{^{\prime }-1}p^{-1}[\bar{q}-(p-q^{^{\prime }-1})k]. \nonumber
\end{eqnarray}
One can check that the matrix $T=\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) $ is indeed an element of the quantum group $GL_{p,q^{^{\prime
}}}(2) $ and is consistent with Hopf algebra structures \cite{cho}. For
supplementary properties and results concerning the quantum group $
GL_{p,q^{^{\prime }}}(2)$ see for example \cite{schirrmacher}.
It is clear that many quantum planes could be associated to this
two-parameter quantum group, depending on choices of the $q_{ij}$'s. In the
following, we shall confine our selves to the case $\bar{q}=q$,
which corresponds to the standard definition of the quantum plane.
\subsection{Differential calculus with nilpotency $n=2$ $(d^2=0)$.}
Our aim in this section, is to construct a differential calculus on the
previously defined quantum plane. We proceed using the same approach as the
one adopted in \cite{coquereaux1, coquereaux2, notre}.
We start by defining the exterior differential $"d"$ which satisfies the
usual properties, namely:
$i/$ Linearity
$ii/$Nilpotency
\[
d^2=0.
\]
$iii/$ Leibniz rule
\[
d(uv) = d(u) v+(-1)^n ud(v),
\]
where $u\in \Omega ^n.$ $\Omega ^n$ is the space of forms with degree $n,$
\[
d:\Omega ^n\rightarrow \Omega ^{n+1}.
\]
$\Omega ^0$ is the algebra of functions defined on the quantum plane. We
have also:
\begin{equation}
d(x)= dx,\;\; d(y)= dy \;\; \hbox{and} \;\; d(1)=0.
\end{equation}
From (2), we deduce:
\begin{equation}
\begin{array}{ccccl}
(dx)a & = & q_{11} \, a(dx)~~~~~~~~~~~~(dy)a & = & q_{21} \, a(dy) \cr (dx)b &
= & q_{12}\, b(dx)~~~~~~~~~~~~(dy)b & = & q_{22} \, b(dy) \cr (dx)c & = &
q_{13} \,c(dx)~~~~~~~~~~~~(dy)c & = & q_{23} \, c(dy) \cr (dx)d & = & q_{14}
\, d(dx)~~~~~~~~~~~~(dy)d & = & q_{24} \, d(dy). \cr
\end{array}
\end{equation}
One can write \textit{a priori} $xdx,xdy,ydx$, and $ydy$ in terms of $
(dx)x,(dy)x,(dx)y$ and $(dy)y$, by mean of $16$ unknown coefficients \cite
{coquereaux1}. Imposing the covariance of the obtained relations under $
GL_{p,q^{^{\prime }}}(2)$, and differentiating $eq(1)$, permit to fix 15 of
the 16 unknown coefficients. The associativity of the expression $
(xdx)dy=x(dxdy)$ enables us to fix the last unknown parameter.
We notice that in the usual case $Gl_q(2)$ this approach yields directly the
desired differential calculus. However, when $GL_{p,q^{^{\prime }}}(2)$ is a
symmetry group, we obtain additional conditions on the parameters $k$ and $
q^{^{\prime }}$:
\begin{equation}
q^{^{\prime }}=q\;,\;k=\frac{q^{^{\prime }}}p.
\end{equation}
Then $eq(6)$ becomes:
\begin{equation}
q_{11}=q_{13}=1\;,\;q_{12}=q_{14}=q_{21}=q_{23}=qp^{-1}\;,
\;q_{22}=q_{24}=q^2p^{-2}.
\end{equation}
So, the covariant differential calculus is given by:
\begin{eqnarray}
x \, dx &=& \frac 1{pq} \; dx \, x ~~~~~~~~~~~~~~~~~~ x \, dy
=\frac {1}{p} \; dy\, x \nonumber \\ y\, dy &=& \frac {1}{pq}\;
dy\, y ~~~~~~~~~~~~~~~~~~ y\, dx =(\frac {1}{pq} - 1) \; dy\, x+
\frac {1}{q}\; dx\, y \\ dx\, dy &=& -\frac {1}{p} \; dy\, dx
~~~~~~~~~~~~~~ (dx)^2 = (dy)^2=0, \nonumber
\end{eqnarray}
and the differential algebra is $\Omega _{x,y}^{q,p}=\{x,$ $y,$ $dx,$ $dy\}.$
It is remarkable that the differential calculus on the quantum plane with
$GL_q(2)$, as a symmetry group \cite{wess1,berezinski,coquereaux1}, can be
obtained from the two-parameter one $eq(11)$ in the $p\rightarrow q$ limit.
As in the ordinary case, the differential operator $d$ can be realized by
\[
d := dx \, \partial x + dy \, \partial y \, .
\]
Based on this realization one can construct a gauge field theory on the
two-parameter quantum plane. This should be achieved formally as in \cite
{notre}.
The nilpotent differential calculus can be extended to higher orders, as
there is no reason to constrain this one to $n=2$ nilpotency \cite
{kerner1,kerner2,kerner3,notre,coquereaux3,dubois, dubois1, dubois2}.
In the following section, we generalize the differential calculus on the
quantum plane with one-parameter symmetry group \cite{notre} to the
two-parameter one, this is done by extending the $n=2$ differential calculus
obtained here to $n=3$ case.
\subsection{Differential calculus with nilpotency $n=3$ $(d^3=0).$}
Let us introduce the differential operator ''$d$'' that satisfies the
following conditions:
$i/$ Linearity
$ii/$ Nilpotency
\[
d^3=0\;,\;d^2\neq 0.
\]
$iii/$ Leibniz rule
\[
d(uv)=(du)v+(j)^nud(v).
\]
where $j$ is the cubic root of unity: $j=e^{\frac{2i\pi
}3},1+j+j^2=0$. $u$ is an element of $\Omega ^n$, the space of forms with degree $n$. It is a
subspace of the differential algebra $\tilde{\Omega} _{x,y}^{q,p}=
\{x,y,dx,dy,d^2x,d^2y\}.$ The new objects $d^2x$ and $d^2y$ which appear are
defined by:
\[
d(dx)=d^2(x)=d^2x\;\;,\;\;d(dy)=d^2(y)=d^2y,
\]
these are ''forms'' with degree two.
In order to ensure the covariance of the differential calculus under the
two-parameter symmetry group $GL_{p,q^{^{\prime }}}(2$), we proceed as in
the previous section. However, instead of the last step where we have used
the associativity property, we shall use the independence between the two
different $2$-forms $z$ $d^2z^{^{\prime }}$ and $dz$ $dz^{^{\prime }}$,
where $z,$ $z^{^{\prime }}=x,$ $y.$ Below, we will discuss how to recover
this property.
The same constraints on $q^{^{\prime }}$ and $k$ $eq(9)$ are recovered, thus
the $q_{ij}$ 's are the same as in $eq(10).$ The covariant differential
calculus is then given by:
\begin{eqnarray}
x\, dx &=& j^2 \; dx\, x ~~~~~~~~~~~~~~~~~~~ x\, dy = -\frac{jq}{1+qp} \;
dy\, x+\frac{j^2qp-1}{1+qp} \; dx\, y \nonumber \\
y\, dy &=&j^2 \; dy\, y ~~~~~~~~~~~~~~~~~~~ y\, dx = \frac{j^2-qp}{1+qp} \;
dy\, x - \frac{jp}{1+qp} \; dx\, y \nonumber \\
x\, d^2x &=& j^2 \; d^2x\, x ~~~~~~~~~~~~~~~~~ x\, d^2y = -\frac{jq}{1+qp}
\; d^2y \, x + \frac{j^2qp-1}{1+qp} \; d^2x \, y \nonumber \\
y\, d^2y &=& j^2 \; d^2y\, y ~~~~~~~~~~~~~~~~~ y\, d^2x = \frac{j^2-qp}{1+qp}
\; d^2y\, x - \frac{jp}{1+qp}\; d^2x\, y \\
dx\, d^2x &=& j \; d^2x\, dx ~~~~~~~~~~~~~~~ dx\, d^2y = -\frac q{1+qp}\;
d^2y\, dx + \frac{jqp-j^2}{1+qp}\; d^2x\, dy \nonumber \\
dy\, d^2y &=& j \; d^2y\, dy ~~~~~~~~~~~~~~~ dy\, d^2x = \frac{j-j^2qp}{1+qp}
\; d^2y\, dx - \frac p{1+qp} \; d^2x\, dy \nonumber \\
dx\, dy &=& q \; dy\, dx ~~~~~~~~~~~~~~~ d^2x\, d^2y = q \; d^2y\, d^2x.
\nonumber
\end{eqnarray}
Moreover, a realization of $"d"$ in terms of partial derivatives:
\begin{equation}
d=dx\, \partial _x+dy\, \partial _y
\end{equation}
permits us to have $(dx)^3=(dy)^3=0$ \cite{notre}.
\vspace{0.5cm}
We note that the differential algebra $\tilde{\Omega} _{x,y}^{q,p}$, defined
above, is not associative. One can check this statement by first assuming
that this property (associativity) is preserved, then deriving some
inconsistent relations. Especially, one expects, due to this assumption, the
two expressions $(x\, dx)dy$ and $x(dx\, dy)$ to be equal. However, using
(12) and successively moving the parenthesis, one obtains two expressions
which are manifestly not equal, unless $pq=j^2$.
Thus, the differential algebra $\tilde{\Omega} _{x,y}^{q,p}$ is associative
only when $pq=j^2$, otherwise it is not.
Another associative 3-nilpotent differential algebra, for $pq=j$, can be constructed basing on the method already
mentioned in section (2.2), with a proper substitution of the differential
operator $d^2=0$ with the one $d^3=0$, $(d^2 \neq 0)$. It follows from this
method that the commutation relations between the coordinates and their
first order differentials are given (by the first ones) in (11). The first,
second and third differentiations of these relations give rise to the
remaining commutation relations between $x,\, y,\, dx,\, dy,\, d^2x$ and $
d^2y$.
\vspace{0.5cm}
The results of \cite{notre} (i.e., differential calculus on a reduced
quantum plane respectively with $q^3=1$ and $q^N=1$) can be recovered as
limiting cases of the one obtained here (12); this is done by taking the
adequate limit $p\rightarrow q$ (respectively with $q^3=1$ and $q^N=1$).
It is also remarkable that the case $n=3$ differential calculus was applied
to introduce interesting \textit{''Higher order gauge theories''} \cite
{kerner2, kerner3, notre}. Indeed, an interesting manner to do this (in the
present case) is to pursue the same steps of \cite{notre}.
\vspace{0.5cm}
Another important question arises at this step is how to adapt the techniques
applied in subsections $(2.2)$ and $(2.3)$ to the quantum superplane. This will
be developed in the next section.
\section{Differential calculus on a one-parameter quantum superplane.}
\subsection{$n=2$ Differential calculus.}
The $1+1$ dimensional quantum superspace, in $Manin^{\prime }s$ approach
\cite{manin2, corrigan}, is an algebra generated by a bosonic and a
fermionic coordinate satisfying the relations:
\begin{eqnarray}
x \theta &=& q \, \theta x\;\;,\;\;q\neq 0,1 \\
\theta ^2&=&0.
\end{eqnarray}
In analogy with the quantum plane, a symmetry supergroup of this space is $
GL_q(1/1)$, and a generic element of this supergroup is a supermatrix: $
T=\left(
\begin{array}{cc}
a & \beta \\
\gamma & d
\end{array}
\right) ,$ where $a$, $d$ are bosonic elements commuting with $x$ and $
\theta $ while $\beta ,\gamma $ are fermionic elements commuting with $x$,
anticommuting with $\theta $ and obeying the following relations:
\begin{equation}
\begin{array}{rclcc}
a\beta & = & q \, \beta a ~~~~~~~~~~~~~ d\beta & = & q \, \beta d \cr a\gamma
& = & q \, \gamma a ~~~~~~~~~~~~~ d\gamma & = & q \, \gamma d \cr \beta
\gamma +\gamma \beta & = & 0 ~~~~~~~~~~~~~~~~~ \beta ^2 & = & \gamma ^2=0
\cr ad-da & = & (q^{-1}-q)\, \beta \gamma . & & \cr
\end{array}
\end{equation}
These relations can also be obtained by imposing the invariance of $
eqs(14,15)$ under $T$ and $^{st}T$ $=\left(
\begin{array}{cc}
a & -\gamma \\
\beta & d
\end{array}
\right) $(supertranspose).
Many authors studied the differential calculus on this superspace \cite
{tahri, kobayachi,soni}. Here we construct the differential calculus based
on the same technique adopted by \textit{R. Couquereaux} \cite{coquereaux1} which is
used in the previous section, with however, some modifications to adapt it
to this superspace. We introduce an exterior differential operator $"d"$
satisfying the properties:
$i/$ Linearity
\begin{equation}
d(\lambda u)=(-1)^{\stackrel{\wedge }{\lambda }} \; \lambda \, d(u),
\end{equation}
where the parity $\stackrel{\wedge }{\lambda }=0,1$ respectively, if $
\lambda $ is a bosonic or a fermionic element.
$ii/$ Nilpotency
\[
d^2=0.
\]
$iii/$Leibniz rule
\begin{equation}
d(uv)=(du)v+(-1)^{\stackrel{\wedge }{u}}(-1)^{\deg u}u(dv),
\end{equation}
where $\stackrel{\wedge }{u}$ is the parity of $u$ and $degu$ is the degree
of the differential form $u.$
Note that consistency requires that $d\theta $ commutes with $a$, $d$, $\beta
$, $\gamma $ and $dx$ commutes with $a$, $d$ and anticommutes with $\beta ,$ $
\gamma .$
The same method applied in section $2.2$ yields:
\begin{eqnarray}
x\, dx &=& q^{-2} \; dx\, x ~~~~~~~~~~~~ x\, d\theta \; = \; q^{-1} \;
d\theta \, x \nonumber \\
\theta \, d\theta &=& d\theta \, \theta ~~~~~~~~~~~~~~~~~~ \theta \, dx \; =
\; (1-q^{-2}) \; d\theta \, x- q^{-1} \; dx\, \theta \\
dx\, d\theta &=& q^{-1} \; d\theta \, dx ~~~~~~~~~ (dx)^2 \; = \; 0,
\nonumber
\end{eqnarray}
and the associative differential algebra is denoted $\Omega
_{x,\theta }^q=\{x,$ $\theta ,$ $dx,$ $d\theta \}.$
As in section $2.3$, one can apply the same method to generalize the
differential calculus on the superspace to higher orders ($d^3=0$). This is
the aim of the next section.
\subsection{ Differential calculus on superspace with nilpotency \protect\\ $
n=3$ $(d^3=0).$}
We proceed as in section $2.3$, in order to construct the $n=3$ covariant
differential calculus on superspace. We introduce a differential operator $
"d"$ satisfying the usual requirements, namely: linearity is the same as in $
eq(17)$, the nilpotency will be changed to $n=3$ $(d^3=0)$ and the Leibniz
rule, $eq(18)$ becomes:
\begin{equation}
d(uv)=(du)v+(-1)^{\stackrel{\wedge }{u}}(j)^{\deg u}u(dv).
\end{equation}
The resulting differential algebra $\tilde{\Omega }_{x,\theta }^q$ is
generated by, $x,$ $\theta ,$ $dx,$ $d\theta ,$ $d^2x$ and $d^2\theta $
satisfying:
\begin{eqnarray}
x\, dx &=& j^2 \; dx\, x ~~~~~~~~~~~~~~~~~~~ x\, d\theta = -\frac{jq}{1+q^2}
\; d\theta \, x+\frac{j^2q^2-1}{1+q^2} \; dx \, \theta \nonumber \\
\theta \, d\theta &=& d\theta \, \theta ~~~~~~~~~~~~~~~~~~~~~~~ \theta \, dx
= \frac{q^2-j^2}{1+q^2} \;d\theta \, x+\frac{jq}{1+q^2} \; dx \, \theta
\nonumber \\
dx\, d\theta &=& -q \; d\theta \, dx ~~~~~~~~~~~~~~~(d\theta )^2 = 0
\nonumber \\
x\, d^2x &=& j^2 \; d^2x\, x ~~~~~~~~~~~~~~~~\, x\, d^2\theta = -\frac{jq}{
1+q^2}\; d^2\theta \, x+\frac{j^2q^2-1}{ 1+q^2} \; d^2x\, \theta \nonumber
\\
\theta \, d^2\theta &=& -d^2\theta \, \theta ~~~~~~~~~~~~~~~~~~ \theta \,
d^2x = \frac{j^2-q^2}{1+q^2} \; d^2\theta \, x-\frac{jq}{1+q^2}\; d^2x\,
\theta \\
dx\, d^2x &=& j \; d^2x\, dx ~~~~~~~~~~~~~~~ dx\, d^2\theta = \frac q{1+q^2}
\; d^2\theta \, dx+\frac{jq^2-j^2}{1+q^2} \; d^2x\, d\theta \nonumber \\
d\theta \, d^2\theta &=& j^2 \; d^2\theta \, d\theta ~~~~~~~~~~~~~~ d\theta
\, d^2x = \frac{j^2q^2-j}{1+q^2} \; d^2\theta \, dx - \frac q{1+q^2}\;
d^2x\, d\theta \nonumber \\
d^2x\, d^2\theta &=& q \; d^2\theta \, d^2x ~~~~~~~~~~~~~~ (d^2\theta )^2 =
0. \nonumber
\end{eqnarray}
Let us point out that the differential algebra $\tilde{\Omega
}_{x,\theta }^q $ is not associative, unless $q=j$. In the case
$q\neq j$, one can recover
this property by following the same steps mentioned at the end of section $
2.3$ with the adequate modifications.
\section{Differential calculus on a two-parameter quantum superplane.}
\subsection{Differential calculus with nilpotency $n=2$, $(d^2=0)$.}
In this section, we generalize the results of section $3,$ in the sense that
we choose a two-parametric quantum supergroup $GL_{p,q^{\prime }}(1/1)$ as a
symmetry group for the superplane $eqs(14,15)$. This group will be
introduced using the same method as in section $2$ \cite{tahri,cho}.
The entries of a matrix element $T=\left(
\begin{array}{cc}
a & \beta \\
\gamma & d
\end{array}
\right) $ of $\, GL_{p,q^{\prime }}(1/1)$ satisfy the following non trivial
relations:
\begin{equation}
\begin{array}{rclcr}
a\beta & = & p\; \beta a ~~~~~~~~~~~~~ d\beta & = & p \; \beta d \cr a\gamma
& = & q^{^{\prime }} \; \gamma a ~~~~~~~~~~~~~ d\gamma & = & q^{^{\prime }}
\; \gamma d \cr p \; \beta \gamma + q^{^{\prime }} \; \gamma \beta & = & 0
~~~~~~~~~~~~~~~~~~~ \beta ^2 & = & \gamma ^2=0 \cr ad-da & = & (q^{^{\prime
}-1}-p) \; \beta \gamma . & & \cr
\end{array}
\end{equation}
As it is done in section $2$ this superspace is covariant under $T$ and $
^{st}T$ (supertranspose), and the analogous of $eq(2)$ are:
\begin{equation}
\begin{array}{cclcrcc}
x \, a & = & k \;a\,x & ~~~~~~~~~~~~ & \theta\,a & = & q\bar{q}q^{^{\prime
}-1}p^{-1}k \;a\,\theta \cr x \, b & = & \bar{q}p^{-1}k \;b\,x & ~~~~~~~~~~~~
& \theta\,b & = & - q\bar{q}^2q^{\prime -1}p^{-2}k \;b\,\theta \cr x \, c & =
& \bar{q}q^{^{\prime }-1}k \;c\,x & ~~~~~~~~~~~~ & \theta\,c & = & - q\bar{q}
^2q^{^{\prime }-2}p^{-1}k \;c\,\theta \cr x \, d & = & \bar{q}^2q^{^{\prime
}-1}p^{-1}k \;d\,x & ~~~~~~~~~~~~ & \theta\,d & = & q \bar{q}^3q^{^{\prime
}-2}p^{-2}k \;d\,\theta. \cr
\end{array}
\end{equation}
We are interested in establishing a covariant differential calculus on this
superspace in the case $\bar{q}=$ $\stackrel{=}{q}$ $=$ $q.$
To achieve this construction, for $n=2$, we introduce a differential
operator ''$d"$ satisfying the same properties as in section $3.2$ (Linearity
$eq(17)$, nilpotency and Leibniz rule $eq(18)$). The associative differential algebra $
\Omega _{x,\theta }^{p,q}=\{x,$ $\theta ,$ $dx,$ $d\theta \}$ is generated
by the following relations:
\begin{eqnarray}
x\, dx &=& (qp)^{-1}\; dx\, x ~~~~~~~~~~~~ x\, d\theta = p^{-1} \;
d\theta \, x \nonumber \\ \theta \, d\theta &=& d\theta \, \theta
~~~~~~~~~~~~~~~~~~~~~\, \theta \, dx = (1-(qp)^{-1}) \; d\theta \,
x- q^{-1} \; dx\, \theta \\ dx\, d\theta &=& p^{-1} \; d\theta \,
dx ~~~~~~~~~~~~~ (dx)^2 = 0. \nonumber
\end{eqnarray}
We have used $q^{\prime }=q$ and $k=\frac qp$, which, as in $eq(9)$, are
consequences of the requirement of the covariance of $\Omega _{x,\theta
}^{p,q}$ under $GL_{p,q^{\prime }}(1/1)$.
As expected, in the limit $p\rightarrow q$, we recover $\Omega _{x,\theta
}^q $ and relations (19).
\subsection{Differential calculus with nilpotency $n=3$ $(d^3=0).$}
The technique used in sections $(2.3)$ and $(3.3)$, allows us to construct
the $n=3$ differential algebra $\tilde{\Omega }_{x,\theta }^{p,q}=\{x,$ $
\theta ,$ $dx,$ $d\theta ,$ $d^2x,$ $d^2\theta \}:$
\begin{eqnarray}
x\, dx &=& j^2 \; dx\, x ~~~~~~~~~~~~~~~~~~~ x\, d\theta =-\frac{jq}{1+qp}
\; d\theta \, x+\frac{j^2qp-1}{1+qp}\; dx \, \theta \nonumber \\
\theta \, d\theta &=& d\theta \, \theta ~~~~~~~~~~~~~~~~~~~~~~~ \theta \, dx
=\frac{qp-j^2}{1+qp} \; d\theta \, x+\frac{jp}{1+qp} \; dx \, \theta
\nonumber \\
dx\, d\theta &=& -q \; d\theta \, dx ~~~~~~~~~~~~~~~ (d\theta )^2 = 0
\nonumber \\
x\, d^2x &=& j^2 \; d^2x\, x ~~~~~~~~~~~~~~~~ x\, d^2\theta =-\frac{jq}{1+qp}
\; d^2\theta \, x+\frac{j^2qp-1}{1+qp}\; d^2x\, \theta
\\ \theta \, d^2\theta &=&-d^2\theta \, \theta ~~~~~~~~~~~~~~~~
\theta \, d^2x = \frac{j^2-qp}{1+qp} \; d^2\theta \,
x-\frac{jp}{1+qp} \; d^2x\, \theta \nonumber \\ dx\, d^2x &=& j \;
d^2x\, dx ~~~~~~~~~~~~~ dx\, d^2\theta = \frac q{1+qp}\; d^2\theta
\, dx+\frac{jqp-j^2}{1+qp}\; d^2x\, d\theta \nonumber \\ d\theta
\, d^2\theta &=& j^2 \; d^2\theta \, d\theta ~~~~~~~~~~~~~ d\theta
\, d^2x = \frac{j^2qp-j}{1+qp}\; d^2\theta \, dx-\frac p{1+qp}\;
d^2x\, d\theta \nonumber \\ d^2x\, d^2\theta &=& q \; d^2\theta
\, d^2x ~~~~~~~~~~~~~ (d^2\theta )^2=0. \nonumber
\end{eqnarray}
The same limit as in section $2.3,$ namely $p\rightarrow q,$ yields $\tilde{
\Omega }_{x,\theta }^q$. The differential algebra $\tilde{\Omega
}_{x,\theta }^{p,q}$ is not associative. In order to restore this
property we proceed as mentioned at the end of sections $2.3$ and
$3.2$.
One physical application of the differential calculi (sections 3 and 4) is
to construct a supersymmetric gauge field theory on the quantum superplane
(with one or two parameter quantum supergroup as symmetry groups; the latter
will be a generalization of the former). However, this is not
straightforward, since one should firstly start by defining a supersymmetric
covariant derivative.
\section{Conclusion:}
In this paper, we have constructed differential calculi on certain quantum
(super) spaces. Namely, the $n=2$ and $n=3$ nilpotent differential calculi on
the quantum plane with two parametric quantum group ($GL_{p,q}(2)$) as a
symmetry group was obtained. We have also considered two cases of quantum
superplanes related to the one and two-parametric quantum supergroups $
GL_q(1|1)$ and $GL_{p,q}(1|1)$, as symmetry groups, respectively. The
related $n=2$ and $n=3$ differential calculi were also established.
In general, the differential calculus can be applied to formulate gauge
field theories \cite{wess2, wess3, wess4}. As a consequence, the results
obtained here permit us to construct gauge theories on the corresponding
non-commutative spaces \cite{coming}. Indeed, for the quantum space (section
2), this can be done using the same techniques of \cite{notre}, where the
symmetry group is a one-parameter.
The non-commutative supersymmetric case (sections 3 and 4) will be treated
in the same fashion, with however, more care since it is essential first, to
define a covariant supersymmetric derivative \cite{nordine, west}.
We note that the differential calculus was also applied to derive a
corresponding quantum oscillator, where the latter is seen as a
representation of the former \cite{mishra}. It will be interesting to
achieve this with the differential calculus in section 2, as the resulting
quantum oscillator will be two-parameter dependent.
| 21,736
|
TITLE: Transcendental Lonely Runner
QUESTION [2 upvotes]: Four runners start at the same position on a circular track which is 1 km long, and all run in the same direction around and around the track. The runners’ speeds are √5, e, 3, and pi, π all in m/s. What is the first time (measured in seconds after the start) that the speed-e runner is at a distance of at least 250 m from every other runner?
REPLY [3 votes]: We may look at the track from the point of view of the speed-e runner. Thus we consider the speed-e runner to remain at position $1$ on the unit circle in the complex plane, while at time $t$ (measured in units of $1000$ seconds) the other three are at $\exp(2\pi i v_j t)$ where
$v_j = \sqrt{5}-e$, $3-e$ and $\pi - e$. We want $\cos(2 \pi v_j t) \le 0$ for $j = 1, 2, 3$. Plotting these cosines,
we see that the first time is when $2\pi (3-e) t = \pi/2$, at $t = 1/(12-4e)$.
| 26,261
|
by Carsten HAUPTMEIER
Frankfurt am Main (AFP) |
“He is in police custody,” a German police spokesman told AFP, allegedly had “so tightly choked her that her vision went black”, said prosecutor Nadja Niesen, adding that an investigation is continuing on possible “grievous bodily harm”.
“The accused cannot yet be heard. He is still under the influence of significant levels of alcohol and drugs,” she added..
“I have a good feeling in my gut, I feel well. This will be my new start,” Ullrich told Bild.
But he had only arrived back in Germany on Thursday evening to start his therapy before getting caught up in the latest drama, Bild reported.
– From cycling to drink-driving –
The former cyclist’s dramatic fall from grace came two decades after he became the only German to have won the Tour de France, in 1997.
Born in the former communist East Germany, Ullrich’s triumph, which came after reunification, turned caught up in drinking problems.
In 2014, he injured two people in a car crash in Switzerland, and was charged with drink driving.
He was convicted three years later over the case by a Swiss court for drink driving. Sentenced to 21 months in prison, Ullrich was however able to convert that into a suspended sentence of four years plus a fine of 10,000 euros.
Amid his latest woes, his former rival turned friend Armstrong offered his support, the most important is that Jan first allows himself to be helped,” said Hoppe, recounting a telephone conversation with the disgraced US cyclist.
© 2018 AFP
| 21,640
|
A Little Bit of Jack
This is a poem I wrote last week, when I was having a ramble around Ashridge on the most perfect frosty morning. I hope you enjoy my ode to Jack! 😉
Jack was out last night
creating his white magic
waving his wand
over the landscape,
icicles cling
to every surface
highlighting every vein
on every leaf
of every tree
he ran his fingers through
every blade of grass
each one as hard and stiff
as hair which has been gelled
into a thousand spikes.
The world is quiet
the only thing I can hear
is the crunch and crackle
of my footsteps on the ground,
it feels like he froze
time itself.
Even the most barren of sights
now appear beautiful,
it’s incredible what a little bit of Jack
can do. . .
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Good work Becky! 🙂
Thank you! 🙂 x
Oh Becka, that was brilliant! At first I wondered, who is Jack? Will I be able to figure it out? Well as soon as I started reading, there was no doubt as your wonderful poem was such a vibrant portrait of the mystical character himself. We are having a very unusual February with sunshine and it is in the 60’s! No Jack here at the moment! 🙂 x
Ah I’m glad you could work it out Lea, he certainly puts the magic into winter 🙂 Although I wouldn’t say no to some of your sunshine at the moment! Thank you for reading and commenting x
This is just wonderful Becky! I love these poems of yours that come out of your walks out and about, just beautiful. Jack is very talented! 😊 Xxx
I’m glad you liked it Christine, poems like this just come out of nowhere. Nature is a constant inspiration 🙂 thank you! 😀 xxx
Hi Becka–Christine has eagerly introduced me to your blog, which is gorgeous and lovely, and also sent me your photo of birch trees to work up a poem for you. It’s now on my to-do list, so we’ll see what happens–no guarantees, but I can usually do “somethin'”. I’m glad to “meet” you! Mirada
It’s nice to meet you too Mirada! Thank you for the lovely comments you’ve made on my posts, I’ve started following your blog and am looking forward to reading more of your poetry 🙂 hopefully the birch trees will inspire you! x
I will credit and link back to you as soon as I “get the poem”–we have snow as of last evening, so I might be able to work on poetry this afternoon. I thoroughly enjoy doing collaborations, so I’m sure the birch trees will produce something! See you later.
I’d be happy to do another collaboration if you see any other photos on my blog that you like – the birch trees poem was a delight to read. 🙂 x
Wonderful poem Becky! And I love your last photo, simple and beautiful! 🙂
Thank you 😀 I couldn’t decide which photo to use so I ended up including both! x
Beautiful pictures and words. Love how you brought them together.
Thank you very much! 😀 x
A reciprocal visit. It has been a while since i was here last and your blog has come on in leaps and bounds. Your passion for nature shines through and your photographs are spot-on. Great stuff! 🙂
Thanks so much for your lovely comment! I don’t always have time to update my blog as much as I would like to but it’s comments like yours that make it worth while. I appreciate you visiting again, thank you 😀 x
| 114,271
|
Welcome to Glastonburytor.com
Welcome to our website. Here you can find all your relevant information on several topics which include Ararat the Mountain, Glastonbury Abbey, Glastonbury Attractions, and Glastonbury High Street.
Use the search box in order to search for information that is not included on this page. You can also find more relevant information about the following topics: Glastonbury Tor, and Glastonbury Tor in UK.
| 352,315
|
Rossi, Vanishing Point.jpg
41,653pages onAdd New Page
this wiki
this wiki
Rossi,_Vanishing_Point.jpg (369 × 364 pixels, file size: 20 KB, MIME type: image/jpeg)
Crewman Rossi in the mess hall. (ENT: "Vanishing Point")
-...
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Click on a date/time to view the file as it appeared at that time.
| 36,381
|
TITLE: How prove this interesting identity $(y_{1})^2\cdot y_{2}\cdot y_{3}=x^2_{1}\cdot x_{2}\cdot x_{3}$
QUESTION [2 upvotes]: let $0<x_{1}<x_{2}<x_{3}$, and there exsit $a$ such
$$\begin{cases}
y_{1}=x_{1}-\ln{x_{1}}=\dfrac{x^2_{1}}{ax_{1}+\ln{x_{1}}}\\
y_{2}=x_{2}-\ln{x_{2}}=\dfrac{x^2_{2}}{ax_{2}+\ln{x_{2}}}\\
y_{3}=x_{3}-\ln{x_{3}}=\dfrac{x^2_{3}}{ax_{3}+\ln{x_{3}}}
\end{cases}$$
show that:
$$(y_{1})^2\cdot y_{2}\cdot y_{3}=x^2_{1}\cdot x_{2}\cdot x_{3}$$
My try: since
$$(y_{1})^2\cdot y_{2}\cdot y_{3}=x^2_{1}\cdot x_{2}\cdot x_{3}$$
$$\Longleftrightarrow \dfrac{x^4_{1}x^2_{2}x^2_{3}}{(ax_{1}+\ln{x_{1}})(ax_{2}+\ln{x_{2}})(ax_{3}+\ln{x_{3}})}=x^2_{1}\cdot x_{2}\cdot x_{3}$$
$$\Longleftrightarrow (ax_{1}+\ln{x_{1}})(ax_{2}+\ln{x_{2}})(ax_{3}+\ln{x_{3}})=x^2_{1}\cdot x_{2}\cdot x_{3}$$
since
$$\ln{x_{1}}=x_{1}-y_{1},\ln{x_{2}}=x_{2}-y_{2},\ln{x_{3}}=x_{3}-y_{3}$$
then
$$\Longleftrightarrow [(a+1)x_{1}-y_{1}][(a+1)x_{2}-y_{2}][(a+1)x_{3}-y_{3}]=x^2_{1}\cdot x_{2}\cdot x_{3}$$
then I can't Continue ,This problem is from china's college entrance examination simulation.
and this problem not have solution
Thank you
REPLY [2 votes]: Note $x_i>0$, and since $e^x \geq 1+x$ we have $y_i=x_i-\ln x_i \geq 1$. We have
$$y_i=x_i-\ln x_i=\frac{x_i^2}{ax_i+\ln x_i}=\frac{x_i^2}{ax_i+x_i-y_i}=\frac{x_i^2}{(a+1)x_i-y_i}$$
$$(a+1)x_iy_i-y_i^2=x_i^2$$
Let $a+1=2b$, so $2bx_iy_i-y_i^2=x_i^2$.
$$(x_i-by_i)^2=(b^2-1)y_i^2$$
$$x_i=\left(b \pm \sqrt{b^2-1}\right)y_i$$
Lemma: The equation $\frac{x}{x-\ln x}=c$ has exactly one positive real root if $0<c \leq 1$, and at most two positive real roots if $1<c$.
Proof: Let $f(x)=\frac{x}{x-\ln x}$. We see $f'(x)=\frac{1-\ln x}{(x-\ln x)^2}$, so $f$ is strictly increasing on $(0, e]$, has a maximum at $x=e$, and is strictly decreasing on $[e, \infty)$. Furthermore $f(e)=\frac{e}{e-1}>1$, $\lim_{x \to 0}{f(x)}=0$ and $\lim_{x \to \infty}{f(x)}=1$.
It follows that $f$ restricted to $(0, e]$ is a strictly increasing bijection to $(0, \frac{e}{e-1}]$ and $f$ restricted to $[e, \infty)$ is a strictly decreasing bijection to $(1, \frac{e}{e-1}]$. Now the desired conclusion follows easily.
Case 1: $x_i=(b-\sqrt{b^2-1})y_i$.
Since $b-\sqrt{b^2-1} \leq 1$, we have $x_i \leq y_i=x_i-\ln x_i$ so $x_i \leq 1$.
Note $\frac{x_i}{x_i-\ln x_i}=b-\sqrt{b^2-1}$. By the lemma applied to $0<c=b-\sqrt{b^2-1} \leq 1$, we have at most one possible solution for $x_i$.
Case 2: $x_i=(b+\sqrt{b^2-1})y_i$.
Since $b+\sqrt{b^2-1} \geq 1$, we have $x_i \geq y_i=x_i-\ln x_i$ so $x_i \geq 1$.
Note $\frac{x_i}{x_i-\ln x_i}=b+\sqrt{b^2-1}$. By the lemma applied to $1<c=b+\sqrt{b^2-1}$, we have at most two possible solutions for $x_i$.
Since $0<x_1<x_2<x_3$, we must have $x_1$ satisfying Case 1 and $x_2, x_3$ satisfying case 2.
Thus
$$x_1=(b-\sqrt{b^2-1})y_1$$
$$x_2=(b+\sqrt{b^2-1})y_2$$
$$x_3=(b+\sqrt{b^2-1})y_3$$
Thus
$$x_1^2x_2x_3=(b-\sqrt{b^2-1})^2(b+\sqrt{b^2-1})(b+\sqrt{b^2-1})y_1^2y_2y_3=y_1^2y_2y_3$$
| 115,665
|
Fiesta Gray cup and saucer
> #
- 16_854
- Quantity
- 1
- Maker
- Homer Laughlin
- Year
- 1950s - 1973
- Dimensions
- Width: 0 inch
- Height: 0 inch
- Depth: 0 inch
- Weight: 0 pound
-
- Condition
- Good
- Material
- null
| 36,468
|
Not quite the right one?
- Not quite the right one?
Do you like this item?YES to save to your wishlist.
Smocked and ruffled top - desert pink
$118
Select available size
This item runs true to size
Details & fit
This piece is a complete stand-alone top due to its all-over ruffled details, smocked shoulder detail and deep colourway. Say hello to your new charming top.
Article number: 168017_0418
Article number: 168017_0418
Women’s top
Smocked
Ruffled shoulder
Button-down closure
Care: 30 degrees delicate wash
100% Polyester
Smocked
Ruffled shoulder
Button-down closure
Care: 30 degrees delicate wash
100%.
| 49,229
|
A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proved to exist, but many, including 196, are suspected on heuristic and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of "Cheryl", his girlfriend's first name.
The reverse-and-add process produces the sum of a number and the number formed by reversing the order of its digits. For example, 56 + 65 = 121. As another example, 125 + 521 = 646.
Some numbers become palindromes quickly after repeated reversal and addition, and are therefore not Lychrel numbers. All one-digit and two-digit numbers eventually become palindromes after repeated reversal and addition.
About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers:
*56 becomes palindromic after one iteration: 56+65 = 121.
*57 becomes palindromic after two iterations: 57+75 = 132, 132+231 = 363.
*59 becomes a palindrome after three iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111
*89 takes an unusually large (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8,813,200,023,188.
*10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after .
*1,186,060,307,891,929,990 takes ' to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which was a former world record for the . It was solved by Jason Doucette's algorithm and program (using Benjamin Despres' reversal-addition code) on November 30, 2005.
*On January 23, 2017 a Russian schoolboy, Andrey S. Shchebetov, announced on his web site that he had found a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. This sequence was published in OEIS as A281506. This sequence started with 1,186,060,307,891,929,990 - by then the only publicly known number found by Jason Doucette back in 2005. On May 12, 2017 this sequence was extended to 108864 terms in total and included the first 108864 delayed palindromes with 261-step delay. The extended sequence ended with 1,999,291,987,030,606,810 - its largest and its final term.
*On 26 April 2019, Rob van Nobelen computed a new World Record for the Most Delayed Palindromic Number: 12,000,700,000,025,339,936,491 takes ' to reach a 142 digit palindrome.
*On 5 January 2021, Anton Stefanov computed two new Most Delayed Palindromic Numbers: and takes 289 iterations to reach the same 142 digit palindrome as the Rob van Nobelen number.
*On December 14, 2021, Dmitry Maslov computed a new World Record for the Most Delayed Palindromic Number: takes 293 iterations to reach 132 digit palindrome.
*The OEIS sequence A326414 contains 19353600 terms with 288-step delay known at present.
*Any number from A281506 could be used as a primary base to construct higher order 261-step palindromes. For example, based on 1,999,291,987,030,606,810 the following number 199929198703060681000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001999291987030606810 also becomes a 238-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 after 261 steps.
The smallest number that is not known to form a palindrome is 196. It is the smallest Lychrel number candidate.
The number resulting from the reversal of the digits of a Lychrel number not ending in zero is also a Lychrel number.
Let $n$ be a natural number. We define the Lychrel function for a number base $b > 1$ $F_{b} : \mathbb{N} \rightarrow \mathbb{N}$ to be the following:
$$
F_{b}(n) = n + \sum_{i = 0}^{k - 1} d_i b^{k - i - 1}
$$
where $k = \lfloor \log_{b}{n} \rfloor + 1$ is the number of digits in the number in base $b$, and
$$
d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i}
$$
is the value of each digit of the number. A number is a Lychrel number if there does not exist a natural number $i$ such that $F_{b}^{i + 1}(n) = 2 F_{b}^{i}(n)$, where $F^i$ is the $i$-th iteration of $F$
In other bases (these bases are power of 2, like binary and hexadecimal), certain numbers can be proven to never form a palindrome after repeated reversal and addition, but no such proof has been found for 196 and other base 10 numbers.
It is conjectured that 196 and other numbers that have not yet yielded a palindrome are Lychrel numbers, but no number in base ten has yet been proven to be Lychrel. Numbers which have not been demonstrated to be non-Lychrel are informally called "candidate Lychrel" numbers. The first few candidate Lychrel numbers are:
196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997.
The numbers in bold are suspected Lychrel seed numbers (see below). Computer programs by Jason Doucette, Ian Peters and Benjamin Despres have found other Lychrel candidates. Indeed, Benjamin Despres' program has identified all suspected Lychrel seed numbers of less than 17 digits. Wade Van Landingham's site lists the total number of found suspected Lychrel seed numbers for each digit length.
The brute-force method originally deployed by John Walker has been refined to take advantage of iteration behaviours. For example, Vaughn Suite devised a program that only saves the first and last few digits of each iteration, enabling testing of the digit patterns in millions of iterations to be performed without having to save each entire iteration to a file. However, so far no algorithm has been developed to circumvent the reversal and addition iterative process.
The term thread, coined by Jason Doucette, refers to the sequence of numbers that may or may not lead to a palindrome through the reverse and add process. Any given seed and its associated kin numbers will converge on the same thread. The thread does not include the original seed or kin number, but only the numbers that are common to both, after they converge.
Seed numbers are a subset of Lychrel numbers, that is the smallest number of each non palindrome producing thread. A seed number may be a palindrome itself. The first three examples are shown in bold in the list above.
Kin numbers are a subset of Lychrel numbers, that include all numbers of a thread, except the seed, or any number that will converge on a given thread after a single iteration. This term was introduced by Koji Yamashita in 1997.
Because 196 (base-10) is the lowest candidate Lychrel number, it has received the most attention.
In the 1980s, the 196 palindrome problem attracted the attention of microcomputer hobbyists, with search programs by Jim Butterfield and others appearing in several mass-market computing magazines. In 1985 a program by James Killman ran unsuccessfully for over 28 days, cycling through 12,954 passes and reaching a 5366-digit number.
John Walker began his 196 Palindrome Quest on 12 August 1987 on a Sun 3/260 workstation. He wrote a C program to perform the reversal and addition iterations and to check for a palindrome after each step. The program ran in the background with a low priority and produced a checkpoint to a file every two hours and when the system was shut down, recording the number reached so far and the number of iterations. It restarted itself automatically from the last checkpoint after every shutdown. It ran for almost three years, then terminated (as instructed) on 24 May 1990 with the message:
Stop point reached on pass 2,415,836.
Number contains 1,000,000 digits.
196 had grown to a number of one million digits after 2,415,836 iterations without reaching a palindrome. Walker published his findings on the Internet along with the last checkpoint, inviting others to resume the quest using the number reached so far.
In 1995, Tim Irvin and Larry Simkins used a multiprocessor computer and reached the two million digit mark in only three months without finding a palindrome. Jason Doucette then followed suit and reached 12.5 million digits in May 2000. Wade VanLandingham used Jason Doucette's program to reach 13 million digits, a record published in Yes Mag: Canada's Science Magazine for Kids. Since June 2000, Wade VanLandingham has been carrying the flag using programs written by various enthusiasts. By 1 May 2006, VanLandingham had reached the 300 million digit mark (at a rate of one million digits every 5 to 7 days). Using distributed processing, in 2011 Romain Dolbeau completed a billion iterations to produce a number with 413,930,770 digits, and in February 2015 his calculations reached a number with billion digits. A palindrome has yet to be found.
Other potential Lychrel numbers which have also been subjected to the same brute force method of repeated reversal addition include 879, 1997 and 7059: they have been taken to several million iterations with no palindrome being found.
In base 2, 10110 (22 in decimal) has been proven to be a Lychrel number, since after 4 steps it reaches 10110100, after 8 steps it reaches 1011101000, after 12 steps it reaches 101111010000, and in general after 4n steps it reaches a number consisting of 10, followed by n+1 ones, followed by 01, followed by n+1 zeros. This number obviously cannot be a palindrome, and none of the other numbers in the sequence are palindromes.
Lychrel numbers have been proven to exist in the following bases: 11, 17, 20, 26 and all powers of 2.
No base contains any Lychrel numbers smaller than the base. In fact, in any given base b, no single-digit number takes more than two iterations to form a palindrome. For b > 4, if k < b/2, then k becomes palindromic after one iteration: k + k = 2k, which is single-digit in base b (and thus a palindrome). If k > b/2, k becomes palindromic after two iterations.
The smallest number in each base which could possibly be a Lychrel number are :
Lychrel numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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- Musician's Friend Best Selection, Price & Service. Guaranteed.
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>> breakfast!
With a stay at HI Boston, you'll be centrally located in Boston, steps from Cutler Majestic Theatre and within a 5-minute walk of Boston Opera House. Featured amenities include a computer station, express check-out, and complimentary newspapers in the lobby.
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SHUGBOROUGH MILFORD NEAR STAFFORD
Set in 900 acres of English countryside in the heart of Staffordshire, Shugborough has rolling swathes of lush green parkland, historic woodlands and Grade I listed formal gardens which frame a perfect riverside setting.
Shugborough offers a wide range of tours for adult groups, taking in all aspects of The Complete Working Historic Estate, accompanied by scrummy lunches and cream teas in the Garden Room using recipes which include flour, salads & vegetables and even beer produced on the estate itself.
Shugborough is located just ten minutes pleasant drive from Junction 13 of the M6 and is ideal as either a full day out for groups or as a welcome stop on the way up to Liverpool or the Lake District, just 30 minutes from Birmingham and 60 minutes from Liverpool.
Discover all that Shugborough has to offer, enjoy the magnificent Estate at your own pace, enjoy a drink or meal in our Ladywalk Tearoom. There are picnic tables and benches around the estate making it ideal for a leisurely picnic in beautiful surroundings.
SHUGBOROUGH, MILFORD NEAR STAFFORD, ST17 0XB.
For all the latest information relating to Events go to Chester & Cheshire EventsNorth Wales Events Guide
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Tuesday September 29, 2009 9:42 pm
Monday Ratings: Trauma Needs Assistance
Posted by Veronica Santiago Categories: Comedy, Drama, Prime Time, Reality, Sci-Fi/Horror, Talk Shows, ABC, CBS, FOX, NBC, The CW, Dancing With The Stars, Heroes, House, Ratings
NBC needs some help, stat! Yesterday’s series premiere of Trauma did nothing for the Peacock network’s health on Mondays. The medical drama pulled in numbers smaller than last fall’s stinker, My Own Worst Enemy. By night’s end, ABC was on top with the viewers while FOX was 1st in the demos.
8pm
- Dancing with the Stars (16.7 million, 3.7/9 in 18-49) slipped by 800,000.
- House (14.4 million, 5.7/15) had the evening’s best demos.
- How I Met Your Mother (8.7 million, 3.6/10) was down 500,000; Accidentally on Purpose (8.1 million, 3.1/8) lost 900,000.
- Heroes (5.8 million, 2.5/6) gave up 200,000.
- One Tree Hill (2.6 million, 1.2/3) added on 100,000.
| TV by the Numbers
9pm
- Two and a Half Men (13.6 million, 4.7/11 in 18-49) remained steady.
- The Big Bang Theory (13 million, 5.1/12) beat its lead-in in the demos.
- Approximately 7.7 million (2.9/7) watched the debut of Lie to Me.
- Trauma (2.3/6) came to the aid of 6.9 million.
- Gossip Girl (2.3 million, 1.2/3) was up 200,000.
10pm
- CSI: Miami (13.5 million, 4.1/11 in 18-49) dipped by 200,000.
- Castle (9.6 million, 2.3/6) improved by 200,000.
- The Jay Leno Show (5.6 million, 1.8/5) did about the same.
*denotes repeat
(You can review last Monday’s ratings here.)
- Related Tags:
- 18-49, accidentally on purpose, audience, big bang theory, castle, csi miami, csi: miami, dancing with the stars, dancing with the stars 9, demos, dwts, dwts 9, gossip girl, heroes, himym, house, how i met your mother, jay leno show, lie to me, monday ratings, one tree hill, ratings, the big bang theory, the jay leno show, trauma,.
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The Centre conducts a number of workshops each year. Topics of the workshops are decided from time to time through faculty meeting.
Number of participants of each of the workshops vary from 30 at the minimum to 40 at the maximum. Experience shows that recommendations of these workshops have made valuable contribution towards the identification and solution of many important national issues.
PPR Department of the Centre will appoint Workshop Administration and nominate the participants of the workshops according to the theme of the paper of the workshops. Duration of the workshops may range from one day to six days. Nomination to the workshops is sought through separate letters to the organizations concerned.
Male-female distribution of participants of different workshops
Workshop on NIS:
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TITLE: Pigeon Hole Application on a set
QUESTION [0 upvotes]: Find the minimum number of elements that one needs to take from a set $$ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$ to be sure that there exists at least 2 numbers with the sum of 10.
Here are my work so far:
These are the pairs with the sum of 10: (1, 9), (2, 8), (3, 7), (4, 6)
If we select 4 numbers, then at least 2 of them must be in the same pairs above. But then the set also contains 5, so I suppose that we need at least 5 numbers to safely have a pair with a sum of 10 - is this correct?
REPLY [1 votes]: If we take $6$ of them then there will be at least one pair giving a sum of $10$.
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The father-daughter duo – Jayant Saha and Titas Saha – believe that embracing energy conservation has become an important aspect in any manufacturing process.
According to you, what are the noticeable technological advancements in cement manufacturing that have taken place in the last 10 years?
In the last decade, technological advancement in cement industry has taken place with a steady uniform pace. In grinding area, close circuit pre-grinder in combination with close circuit tube mill has gained considerable popularity especially for capacity upgradation. For new grinding installation, vertical roller mill is still the most accepted. Improvement is taking place in this area too. For very high capacity cement grinding, [recently] LOESCHE and RENK together developed COPE (Compact Planetary Electric) drives, especially for powerful vertical roller mills with over 6 MW power. To address exact material friction factor and to stabilise grinding, variable speed (75 per cent to 100 per cent) drive for grinding table is being recommended by a few OEMs with success reference. Development is also being taking place for roller press.
Recently, a lot of research is going on in the field of green cement. In contrast to conventional cement, its production does not involve any burning process. “Slagsrar” is one such cement, produced from granulated slag, sulphate agent and special additives, patented in over 50 countries worldwide.
In fact, improvements are taking place in every area/equipment in a cement plant. In electrical side, use of VFD drive has become a regular feature contributing in saving electrical power consumption. Like every other industries automation, now a days is being applied in any aspect you name like optimisation of process, fuel mix, product mix, power consumption, in laboratory, in condition monitoring and predictive maintenance even from remote, ensuring safety, in cement production accounting and many others.
What is the progress done to reduce overall energy consumption in manufacturing.
In today’s date, embracing energy conservation has become an important aspect in any manufacturing process. Since, cement manufacturing sector ranks third in the consumption of energy worldwide, the reduction of energy consumption becomes integral.
Advances in pyro technology have brought down heat consumption of ~1,400 kCal/kG clinker for wet process kilns to <700 kCal/kG clinker for state-of-the-art dry process kilns. Around 400 kCal/kG clinker requires as heat of reaction for clinkerisation. The rest is needed to cover losses from radiation (~60 kCal/kG clinker), cooler (~100 kCal/kG clinker) and preheater exhaust (~160 kCal/kG clinker). About 30 kCal/kG clinker comes from material and fuel. As a standard practice part of heat from preheater/cooler exhaust is utilised for raw material/fuel drying. To conserve thermal energy, alternate avenue of generation of electrical energy from preheater exit gas as well as cooler exhaust air, WHRS, has got enhanced promotion. Cogeneration potential ranging 20–30 kWh/t clinker exists in different plants, saving ~15–25 kCal/kG clinker.
Other than saving of energy directly by improvement of electrical system like VFD system, considerable energy saving could be achieved by introducing pre grinding concept for raw material and cement grinding, reducing idle run of equipment over improved run factor by higher level of plant maintenance and also plant optimisation through respectable application of plant automation and fuzzy logic.
What developments have occurred in pyro processing?
In pyro section, improvements have been observed for using alternate fuel. To tackle difficult situations, separate combustion chamber has been introduced for pre-calciner. Improvements have been noticed in fuel burners too to tackle alternate fuel and reduce NOx generation. Staged combustion in calciner has been proven effective in reducing NOx generation. Probably maximum work has been done on clinker cooler to improve cooler efficiency consistently as well as to reduce cooler maintenance. Almost all frontline OEMs have come out with new coolers claiming the both. Around 2.2 to 2.3 kG cooling air per kG clinker is being used to cool clinker to ambient plus 65 degree Celsius. At the same time, the grate load has typically been increased from ~40 to ~50 TPD clinker per m2, considering the same clinker temperature. Use of mechanical flow regulator for cooling air is another improvement observed in near past.
What are your comments on the use of Fuzzy logic and expert systems in kiln and mill operations and its propagation?
The concept of Fuzzy logic and expert system came into operation in cement industry a long time ago, in the middle of 1980s. However, it did not gain much popularity in India because it requires high degree of plant maintenance especially in instrumentation area. At moment, it is gaining momentum in mainly cement grinding. In pyro section, it is not much accepted.
What are the changes you can point out in grinding process?
In cement plant, a major portion of total consumed electrical energy goes for raw material and cement grinding. Developments have taken place in the last decade in introducing pre-grinder to close circuit tube mill. Various combinations of vertical roller pre-grinder and roller press with close circuit ball mill for grinding in semi finish and finish modes could save ~5–7 kWh/t in cement grinding. Use of VRM in cement grinding gives more saving. Developments are going in ball mill also. Using thinner liner plate with improved metallurgy creates higher chamber volume to accommodate higher grinding media and hence more power.
What is your take on continuous emission monitoring system (CEMS)?
In recent years, online emission monitoring technology has received attention and interest in context of providing accurate and continuous information on particulate matter/gaseous emission from stacks. There are already available systems for monitoring parameters such as PM, HC L, HF, NH3, SO2, CO, O2, CO2, NOx, VOC, etc.
The Central Pollution Control Board, in 2014, has issued directions under section 18 (1) of the Water and Air Acts to the State Pollution Control Committees for directing the 17 categories of highly polluting industries for installation of online effluent quality and emission monitoring systems to help tracking the discharges of pollutants from these units.
The direction envisage:
–Installation of online emission quality monitoring system
–Installation of surveillance system
–Ensure regular maintenance and operation of the online system with tamperproof mechanism having facilities for online calibration (onsite/offsite; remote)
At the moment, in cement plants, the parameters required to be monitored in the stack emissions using continuous emission monitoring system are:
–Particulate Matter
–NH3 (as Ammonia)
–SO2 (Sulphur Dioxide)
–NOx (Oxides of Nitrogen)
With rapid industrialisation, it is becoming a necessity to regulate compliance by industries with minimal inspection of industries. Therefore, efforts need to be made to bring discipline in the industries to exercise self-monitoring and compliance and transmit (effluent and) emission quality data to SPCBs/PCCs and CPCB on a continuous basis. CEMS plays a vital role in this aspect.
What are the measures taken to reduce gaseous and dust pollution?
Recently, the Central Pollution Control Board has taken serious steps in reducing gaseous pollution, NOx, in cement industry. Latest norm for NOx emission through chimney is 800 mg/Nm3 and 600 mg/Nm3 for old and new installation respectively while the measurement is corrected for 10 per cent O2 and dry basis. Overnight demands for primary abatement for generation of NOx and SNCR (selective non–catalytic reduction) system have increased considerably. Installation of SNCR system for plants where NOx emission is on higher side has been taken up seriously by plant owners. If not properly installed or operated this system will contribute to ammonia emission to atmosphere. This is also to be taken care of though today there is no limit imposed by the Pollution Control Board.
It is relevant to mention here that at least for new plant installations, the owners should target NOx elimination to match today’s norm in European Union and Germany, 200–450 mg/Nm3, keeping in mind the fact that in India, for particulate emission norm started at ~250 mg/Nm3 in 1990s and ultimately came to 30 mg/Nm3 to match the European standard. This is to avoid reinvestment in same area in future.
Dust collection and recycle equipment such as bag house/filter, electrostatic precipitator are commonly used to reduce dust emissions in cement industry. Use of bag house, which ensures uninterrupted and very efficient dust collecting system, is extensively being used for cleaning kiln/raw mill gases. However, for cleaning cooler exhaust air, still the electrostatic precipitator is being preferred, which should be replaced by high reliability bag house/filter in combination with heat exchanger or water cooling system in the system.
How far has been the penetration of robotic labs for quality control? What are your comments specifically on sample collection and real-time analysis.
Advances in automation over the last 10 years are permitting typical cement laboratory to go hi-tech. The use of microprocessor, computer control system, robotics and optics have permitted increased precision and accuracy in testing as well as greater laboratory efficiency. Uniform kiln feed quality is a must for smooth kiln operation and consistent quality of clinker. It requires homogenisation – right from limestone stacking to reclaiming. The next step comes in controlling raw mill feed proportions based on average raw meal sample analysis. The average sample collection from auto sampler on a real-time basis and prompt analysis play a very important role in this aspect. Collection of samples is totally dependent on availability, sincerity, training and also whims of sample boy. In a few cases, it also happens that the sample boy collects sample in one go and then furnishes hourly samples to laboratory from the same lot. The uniformity in analysis misleads kiln operator and makes his life miserable in controlling kiln operations. To avoid such problem and where management understood the importance, robotic lab is gradually started taking its place in Indian cement industry.
A few latest plants set up by industry leaders like Dalmia Bharat, Wonder, JK, Bharati Cements have robotic laboratories.
What about the automation done in the physical testing of cement?
With the increase in plant capacity, a number of units in same plant and a number of testing personnel, it becomes difficult to keep track on sample analysis, analysis procedures followed, maintaining regularity in sampling and storing of data, which has been normally done with the help of plant-generated spreadsheets. To handle this problem, new software are coming into concept, which supports from scheduling and planning, through testing, data acquisition and long-term data storage, to the final conformity report. It ensures one common and uniform interface to all data analysis and test procedures. It supports planning and scheduling of physical tests, collect analytical data, generates works list conformity tests, reports. The data treatment and reporting, operates in accordance with relevant EN/ASTM norms or any other standards, if opted. Ideally a given operator should only see information relevant for him.
What technological gaps you see in plants in India and that in Europe?
Although the Indian cement industry is keeping itself updated with the latest, avant-grade technologies in cement, there is further scope of improvement in certain areas like use of alternate fuel, pre-blending facility of coal, computer operated plant operation and in pollution control seriousness.
Another aspect is in philosophy. India is yet to gather confidence level to apply and improve upon well understood technology without case reference in Europe as a matter any western country. Application of SNCR system in cement plants is one example. Once confidence level is established many developments will definitely get start up in India.
What are the steps taken to reduce dust and mitigation of CO2 emission per unit of cement in the present system or by way of development of a new product?
Generation of fine particulates and dusts are inherent in the process of cement manufacturing. The priority in the cement industry is to minimise the increase in ambient particulate levels by reducing the mass load emitted from the stack, from fugitive emissions and other sources.
Serious measures have already been taken by most of the major cement manufacturers to satisfy norms set by the Pollution Control Board.
For control of fugitive dust:
–Ventilation systems are used in conjunction with hoods and enclosures covering transfer points and conveyors
–Drop distances are minimised by the use of adjustable conveyors
–Dusty areas like roads are wetted down on a regular basis to reduce dust generation
For production of OPC, CO2 generation is around 0.82 kg per kg of cement in best operated cement plants. Contribution from process generation is maximum, approximately 65 per cent followed by approximately 27 per cent and approximately 8 per cent from thermal and electrical energy consumed in production, respectively.
CO2 generation can be reduced by following process:
c -Reduction of clinker/cement ratio in cement
-Utilisation of biomass
.
Is use of simulation-based learning for skill upgradation happening in cement? Please give details.
Only blessed professionals passed through stage-wise proper training programme in the beginning of their career not only to become successful in future life but also to enjoy their jobs.
Effective training programme comprises of following three stages:
a)Classroom training;
b)In-plant training; and
c)On-job training
Probably, simulation-based training can be put in category b) and then onwards. In India, except for may be a very few plants owned by global cement companies, this facility has not become popular yet. Generally, in most plants, the new comers are put directly in category ‘c’ training. A big disadvantage in this process is that its success depends largely on trainer. In most of the cases, training does not become effective because of biased concept of trainers based on their past experience.
Well developed training simulators provide a dynamic simulation model of each process units, which is made up of sub-models, allowing for each customisation. A good training programme comprising of simulator-based training will definitely not only increase skill and produce good operators/supervisors but also in the long run the investment will be paid off in improved productivity. Earlier, the plant management understands the fact better it is for the industry.
What about the advancements in bagging and loading to reduce labour intensity?
Bagging and loading processes have always been labour intensive. In the past couple of years, compelling research has been dedicated to tackle this issue. One good example is the cement dispatch system software (FLS Automation and Ventomatic), which provides solution to most of the prevailing issues faced by dispatch operations. Such automated systems along with new age machines are contributing largely in reducing labour requirements.
What is your call on zero water consumption in cement manufacturing?
In recent past, governments around the world and companies have greatly increased their attention to the world’s supply of fresh water and have recognised access to safe drinking water and sanitation as a human right. In this context, water has now taken up importance as a sustainability issue. Water conservation, water footprint and water management are having a prominent place on the sustainability agendas of many businesses, ranking next to carbon as a finite global resource that requires meticulous management. Many global cement companies like Holcim, Lafarge, CEMEX are developing methodologies to standardise water measurement and management across all the company’s operations.
A modern, dry process cement factory consumes water in three ways:
i)For cooling bearings of large machines;
ii)For injection in process and dust suppression; and
iii)As potable, drinking water.
In total, a 3,000 tpd cement plant might require a bearing cooling flow of 3,600 m3/h. This cooling water is usually recirculated and around 20 per cent is required as make up
Water for injection into the process will vary from one dry process cement factory to another. The major consumers are:
-Dust suppression in crushers;
-Coal storage;
-Gas cooling tower;
-Cooler exhaust gas temperature control; and
-In grinding systems.
A benchmark value for water consumption in process use for a modern, dry process cement plant would be ~0.2/t of cement produced.
The first step towards water less plant or zero water consumption would be to monitor the current consumption. Once the base line value is determined, targets for reduction can be set and measures can be taken to reduce the water reduce the water consumption in gradual steps. Water management is indeed a need for cement plants today to be sustainable in future.
Jayant Saha holds a Masters Degree in Chemical Engineering from IIT Kharagpur. He worked with L&T for a long time, and was Director and CEO at Penta India Cement and Minerals Pvt Ltd. He is now a freelance consultant.
Titas Saha is a Chemical Engineer from Mumbai, and has obtained her Masters Degree from New York, USA. She did her internship with FLSmidth Inc.
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In what would be the biggest acquisition in its long and storied history, Apple is poised to buy a headphone-maker – but not for the headphones.
Apple is in talks to acquire Beats Electronics, the maker of the iconic Beats headphones, according to multiple reports. At a proposed price tag of $3.2-billion (U.S.), the deal would be the largest ever undertaken by Apple.
Since its founding in 2008, Beats has become a cultural phenomenon. The company’s flagship “Beats By Dre” headphones, endorsed by Beats co-founder and hip-hop mogul Dr. Dre, climbed to prominence on the back of myriad celebrity endorsements, mostly by pop stars and professional athletes. Today, the bass-heavy headphones compete with the likes of Bose and Sennheiser in the premium headphone market. According to research group NPD, Beats now controls more than half of that market.
Apple’s apparent willingness to spend $3.2-billion on Beats left some industry observers baffled. The headphone-maker’s best-known asset is its brand – signified by its lower-case-B logo. However Apple normally acquires companies for their technology or talent, rarely for their branding.
But there is one business where Beats has a head start on its suitor – streaming music.
In iTunes, Apple owns what is likely the biggest and most profitable record store on Earth. However the company has relatively little presence in the growing market of streaming music and music subscription services. In that arena, a host of startups such as Spotify and Pandora have built a loyal fan base, as many of the tech industry’s biggest companies struggle to build their own streaming services.
Using Beats Music as a launching pad for its own streaming service would allow Apple to round out its music offerings and open up one or more new revenue models. For example, Apple may give users an ad-supported free music streaming service, similar to Pandora in the U.S., or offer a pay-per-month subscription service that opens up access to a vast music library – the latter option potentially giving Apple a means of turning its consumers from one-time music buyers to recurring monthly customers.
“While the price tag would clearly be a shocker … we think given the large cash position coupled with the reality that [Apple] needs to build a more robust ‘recurring’ revenue stream there may be more logic to the deal than meets the eye,” said RBC Capital Markets analyst Amit Daryanani.
Many streaming music services have yet to turn a profit, but are nonetheless experiencing rapid user growth. Mr. Daryanani pointed to a report from the International Federation of the Phonographic Industry, a music industry group, that estimated streaming music industry revenue jumped 50 per cent last year, while the music downloads offered through services such as iTunes dropped 2 per cent – the first decline in the industry’s history.
Apple’s purchase – should it come to fruition – is not without risk. In 2011, phone-maker HTC Corp. paid $300-million for a 50.1-per cent stake in Beats. However the partnership did little to boost HTC’s fortunes or increase the popularity of its smartphones. By the end of 2013, the company had sold its entire stake in Beats.
But given Apple’s massive store of free cash, the acquisition may have little downside. As Canaccord Genuity analyst T. Michael Walkley noted this week, sales of Apple’s iPhone 5 were weaker in April than the month before, but the company is likely to have a breakout second half of 2014, as it is widely expected to launch a new line of larger smartphones. With Beats Electronics under its corporate umbrella, Apple may be able to supplement those devices with new and improved headphones and expanded music offerings – further inducing its customers to buy all their digital entertainment hardware and software from the same place.
An Apple spokeswoman declined to comment for this story. Dr. Dre himself appeared to confirm the Apple acquisition in a shaky home video posted online and then quickly taken down this week. In the video, the rapper bragged about becoming the hip-hop industry’s first billionaire – he is estimated to hold a 15-per-cent stake in Beats Electronics.Report Typo/Error
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\begin{document}
\begin{abstract}
Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincar\'{e}-Einstein manifold which is $C^{3,\alpha}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=\rho^2g_+)$ via some boundary defining function $\rho$, there are two types of Yamabe constants: $Y(\overline{X},\partial X,[\bar g])$ and $Q(\overline{X},\partial X,[\bar g])$. (See definitions (\ref{def.type1}) and (\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between $Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. In this paper, we first show that the equality holds in Gursky-Han's theorem if and only if $(X^{n},g_+)$ is isometric to the standard hyperbolic space $(\mathbb{H}^{n}, g_{\mathbb{H}})$. Secondly, we derive an inequality between $Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X, [\hat g])$, and show that the equality holds if and only if $(X^{n},g_+)$ is isometric to $(\mathbb{H}^{n}, g_{\mathbb{H}})$. Based on this, we give a simple proof of the rigidity theorem for Poincar\'{e}-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.
\end{abstract}
\maketitle
\section{Introduction}
Let $\overline{X}$ be an $n$-dimensional smooth connected compact manifold with boundary.
Denote by $X$ the interior of $\overline{X}$ and by $\partial X$ the boundary of $\overline{X}$.
Let $g_+$ be a Remannian metric in $X$.
We call
$(X, g_+)$ a $C^{3,\alpha}$ conformally compact Poincar\'{e}-Einstein manifold if $g_{+}$ is a complete metric in $X$ satisfying
$$
\mathrm{Ric}_{g_+}=-(n-1)g_+,
$$
and $\bar g=\rho^2 g_+$ can be $C^{3,\alpha}$ extended to $\overline{X}$ by some boundary defining function $\rho$.
Here $\rho$ is said to be a smooth boundary defining function if
$$
\textrm{$\rho\in C^{\infty}(\overline{X})$, $\rho>0$ in X, $\rho=0$ on $\partial X$ and $\ud \rho\neq 0$ on $\partial X$. }
$$
Thus $(X,g_{+})$ is conformally
compactified to $(\overline{X}, \bar{g})$, which is a compact Riemannian manifold with boundary. It is clear that $\bar g$ and the induced metric $\hat{g}=\bar g|_{\partial X}$ depend on the choice of $\rho$, but their conformal classes do not. In particular, we call
$(\partial X, [\hat{g}])$ the conformal infinity of $(X, g_+)$.
Our work in this paper is motivated by the recent work of Gursky-Han~\cite{GH}. To describe their and our
results explicitly, we first recall the Yamabe constant on the conformal infinity. It is defined as follows:
$$
Y(\partial X,[\hat{g}]):=\inf_{\hat{h} \in[\hat{g}]}
\frac{ \int_{\partial X} R_{\hat{h}} \ud S_{\hat{h}} }
{(\int_{\partial X} \ud S_{\hat{h}} )^{\frac{n-3}{n-1}} },
$$
where $R_{\hat{h}}$ is the scalar curvature of metric $\hat{h}$ on $\partial X$.
This is related to the classical Yamabe problem on closed manifolds, which was completely solved.
See \cite{Au1, Tru, Sch, LP} and many other works.
For a compact manifold with boundary $(\overline{X}, \partial X, [\bar g])$, there are two types of Yamabe constants defined as follows:
\begin{equation}\label{def.type1}
Y(\overline{X},\partial X,[\bar g]):=\inf_{\bar{h} \in[\bar g]}\frac{\int_X R_{\bar{h} }\ud V_{\bar{h} }
+2\int_{\partial X}H_{\bar{h} }\ud S_{\bar{h} }}
{(\int_{ X} \ud V_{ \bar{h} })^{\frac{n-2}{n}}};
\end{equation}
\begin{equation}\label{def.type2}
Q(\overline{X},\partial X,[\bar g]):=\inf_{\bar{h} \in[\bar g]}\frac{\int_X R_{\bar{h} }\ud V_{\bar{h} }+2\int_{\partial X}H_{\bar{h} }\ud S_{\bar{h} }}{(\int_{\pa X} \ud S_{\bar{h} })^{\frac{n-2}{n-1}}}.
\end{equation}
where $R_{\bar{h} }$ is the scalar curvature of metric $\bar{h} $ and $H_{\bar{h} }$ is the mean curvature of the boundary.
The related Yamabe problem for compact manifolds with boundary was intensively studied in the past half century.
For $Y(\overline{X},\partial X,[\bar g])$, Escobar showed in \cite{Es1} that for $n\geq 3$
\begin{equation}\label{ineq.Y}
Y(\overline{X},\partial X,[\bar g])\leq Y(S^n_+, \partial S^n_+, [g_{S^n}]) =n(n-1)\left(\frac{1}{2}\omega_n\right)^{\frac{2}{n}},
\end{equation}
where $S^n_+$ is the upper hemisphere, $g_{S^n}$ is the round metric and $\omega_n$ is the volume of unit $n$-sphere. When the inequality is strict, then $Y(\overline{X},\partial X,[\bar g])$ is attained by a metric $\bar{g}$ of constant scalar curvature with minimal boundary, i.e.,
$$
\begin{aligned}
& R_{\bar{g}}= Y(\overline{X},\partial X,[\bar g]) \mathrm{Vol}(\overline{X}, \bar{g})^{-\frac{2}{n}}, \quad &\mathrm{in}\ X,
\\
& H_{\bar{g}}=0, \quad &\mathrm{on}\ \partial X.
\end{aligned}
$$
Escobar also showed that the inequality in (\ref{ineq.Y}) is strict if $(\overline{X},\partial X, \bar{g})$ is not conformally equivalent to $(S^n_+, \partial S^n_+, g_{S^n})$ and $3\leq n\leq 5$, or $n\geq 6$ and $\partial X$ is not umbilic. In \cite{BCh}, Brendle-Chen considered the remaining case: $n\geq 6$ and $\partial X$ is umbilic. They verified the remaining case subject to the validity of the Positive Mass Theorem (PMT). By \cite{Wi}, the PMT is valid for spin manifold if $n\geq 6$. Sumarizing their results and applying them to $C^{3,\alpha}$ conformal compactification of Poincar\'{e}-Einstein manifolds, which always have umbilic boundary, we get the following:
\begin{theorem}\label{thm.Y}
Let $(X^n, g_+)$ be a $C^{3,\alpha}$ conformally compact Poincar\'{e}-Einstein manifold satisfying one of the followings:
\begin{itemize}
\item[(a)] the dimension $3\leq n\leq 5$;
\item[(b)] the dimension $n\geq 6$ and $X$ is spin.
\end{itemize}
Then there is a conformal compactification $\bar{g}=\rho^2g_+$ satisfying
\begin{itemize}
\item[(1)] the scalar curvature $R_{\bar{g}}$ is constant;
\item[(2)] the boundary is totally geodesic, which implies that $H_{\bar{g}}=0$;
\end{itemize}
moreover the Yamabe constant $Y(\overline{X},\partial X, [\bar{g}])$ is achieved.
We call such $(\overline{X}, \partial X, \bar{g})$ the \textbf{first type Escobar-Yamabe compactification} of $(X^n, g_+)$.
\end{theorem}
For $Q(\overline{X},\partial X,[\bar g])$, in \cite{Es2} Escobar proved that for $n\geq 3$, there holds
\begin{equation}\label{ineq.Q}
Q(\overline{X},\pa X, [\bar g]) \leq Q(\overline{B^n},S^{n-1},[g_{\mathbb{R}^n}]),
\end{equation}
where $B^n$ is the unit Euclidean ball and $g_{\mathbb{R}^n}$ is the Euclidean metric.
Moreover, when the inequality is strict, $Q(\overline{X},\pa X, [\bar g])$ is achieved by a scalar flat metric $\bar{g}$ with boundary having constant mean curvature, i.e.,
$$
\begin{aligned}
&R_{\bar{g}}=0, \quad& \mathrm{in}\ \overline{X},
\\
&H_{\bar{g}}=\frac{1}{2}Q(\overline{X},\partial X,\bar g)\mathrm{Vol}(\pa X,\hat g)^{-\frac{1}{n-1}}, \quad& \mathrm{on}\ \partial X,
\end{aligned}
$$
where $\hat{g}=\bar{g}|_{\partial X}$.
Assuming that $(\overline{X},\pa X,[\bar g])$ is not conformally equivalent to $(\overline{B^n},S^{n-1},[g_{\mathbb{R}^n}])$ and $Q(\overline{X},\partial X,\bar g)> -\infty$, Escobar was able to verify that the inequality \eqref{ineq.Q} is strict if $n=3$; or $n=4,5$ and $\pa X$ is umbilic; or $n\geq 6$ and $X$ is locally conformally flat with umbilic boundary; or $n\geq 6$ and $\pa X$ has a non-umbilic point (in this case, $n=6$ was proved in \cite{Es3}). When $n=4,5$ and $\pa X$ has a non-umbilic point, the strict inequality was verified by Marques \cite{Mar2}. In \cite{Ch}, S. Chen considered the remaining case of $n\geq 6$ and $\pa X$ is umbilic. Moreover, Marques \cite{Mar1} developed an important tool of conformal Fermi coordinates, which plays the same role as conformal normal coordinates in the Yamabe problem. He then constructed appropriate test functions without using PMT in the umbilic boundary case: (i) $n=8$ and the Weyl tensor of $\pa X$ is nonzero at some boundary point; (ii) $n\geq 9$ and the Weyl tensor of $\overline X$ is nonzero at some boundary point.
See also \cite{Al} for a flow approach. The work of S. Chen and that of Almaraz are particularly relevant to our setting. Applying their results, we have
\begin{theorem}\label{thm.Q}
Let $(X^n,g_+)$ be $C^{3,\alpha}$ conformally compact Poincar\'e-Einstein metric satisfying one of the following:
\begin{enumerate}
\item[(a)] the dimension $3 \leq n\leq 7$;
\item[(b)] the dimension $n\geq 8$ and $X$ is spin;
\item[(c)] the dimension $n\geq 8$ and $X$ is locally conformally flat.
\end{enumerate}
Then there exists a conformal compactification $\bar g=\rho^2 g_+$ satisfying
\begin{itemize}
\item[(1)] the scalar curvature $R_{\bar{g}}=0$;
\item[(2)] the boundary mean curvature $H_{\bar g}$ is a constant,
\end{itemize}
and the Yamabe constant $Q(\overline{X},\partial X, [\bar{g}])$ is achieved.
We call such $(\overline{X}, \partial X, \bar{g})$ the \textbf{second type Escobar-Yamabe compactification} of $(X^n, g_+)$.
\end{theorem}
Readers are referred to \cite{ChS} for more details and generalizations on the Yamabe problem for compact manifolds with boundary.
In \cite{GH}, the authors derived an inequality between $Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$ using the first type Escobar-Yamabe compactification of $(X, g_+)$. The inequality involves an isoperimetric ratio $I(\overline{X}, \partial X, \bar{g})={\mathrm{Vol}(\partial {X}, \hat{g})^n} / {\mathrm{Vol}(\overline{X}, \bar{g})^{n-1}}$. An interesting application of this inequality is the nonexistence of Poincar\'{e}-Einstein metric in the unit Euclidean ball $B^8$ with conformal infinity being $(S^7,[g])$ for infinitely many conformal classes $[g]$. Their inequality is the following:
\begin{theorem}[Gursky-Han]\label{thm.GH}
Let $(X^n, g_+)$ be a Poincar\'{e}-Einstein manifold satisfying the hypotheses of Theorem \ref{thm.Y}. Let $(\overline{X}, \partial X, \bar{g})$ be the first type Escobar-Yamabe compactification and $\hat{g}=\bar{g}|_{\partial X}$.
\begin{itemize}
\item If the dimension $n\geq 4$, then
$$
Y(\overline{X},\partial X,[\bar g]) \cdot I(\overline{X}, \partial X, \bar{g})^{\frac{2}{n(n-1)}}
\geq \frac{n}{n-2} Y(\partial X, [\hat{g}]);
$$
\item If the dimension $n=3$, then
$$
Y(\overline{X},\partial X,[\bar g])\cdot I(\overline{X}, \partial X, \bar{g})^{\frac{1}{3}}\geq 12\pi \chi(\partial X).
$$
\end{itemize}
The equality is sharp and achieved when $(\overline{X}, \partial X, \bar{g})$ is the standard hemisphere. Moreover,
if the equality holds then $\bar{g}$ is Einstein and $\hat{g}$ has constant scalar curvature.
\end{theorem}
In this paper, we first show that the equality case in Theorem \ref{thm.GH} actually yields a rigidity theorem. We prove the following
\begin{theorem}\label{thm.2}
Let $(X^n, g_+)$ be a Poincar\'{e}-Einstein manifold satisfying the hypotheses of Theorem \ref{thm.Y}, with first type Escobar-Yamabe compactification $(\overline{X}, \partial X, \bar{g})$. Let $\hat{g}=\bar{g}|_{\partial X}$.
If the equality in Theorem \ref{thm.GH} holds, then $(X^n, g_+)$ is isometric to the standard hyperbolic space $(\mathbb{H}^{n}, g_{\mathbb{H}})$.
\end{theorem}
Secondly, we study the relation between $Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. By working on the second type Escobar-Yamabe compactification, we derive an inequality only involving $Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat g])$. The difference from Gursky-Han's inequality is that no volume ratio is involved and hence the inequality is conformally invariant.
Moreover, we show that the equality holds if and only if $(X, g_+)$ is isometric to the hyperbolic space.
\begin{theorem}\label{thm.1}
Let $(X,g_+)$ be Poincar\'{e}-Einstein metric which satisfies hypotheses of Theorem \ref{thm.Q}.
Let $\bar{g}=\rho^2g_+$ be any conformal compactification and $\hat{g}=\bar{g}|_{\pa X}$ be the induced metric on $\partial X$.
\begin{itemize}
\item If the dimension $n\geq 4$, then
\begin{equation}\label{ineq:Q_Y_n>3}
Y(\pa X, [\hat g])\leq \frac{n-2}{4(n-1)} Q(\overline{X},\pa X,[\bar g])^2;
\end{equation}
\item If the dimension $n=3$, then
\begin{equation}\label{ineq:Q_Y_n=3}
32\pi \chi (\pa X)\leq Q(\overline{X},\pa X,[\bar g])^2.
\end{equation}
\end{itemize}
In both cases, the equality holds if and only if $(X, g_+)$ is isometric to the hyperbolic space $(\mathbb{H}^n,g_{\mathbb{H}})$.
\end{theorem}
Similar to Gursky-Han's result, the assumptions in Theorem \ref{thm.Q} and Theorem \ref{thm.1} are mainly because of the limited validity of positive mass theorem (PMT).
In the proof of Theorem \ref{thm.1} we need to choose the second type Yamabe compactification, i.e. the Escobar-Yamabe metric of zero scalar curvature and constant boundary mean curvature such that $Q(\overline{X},\partial X,[\bar g])$ is achieved. The existence of such metric relies on PMT at the present stage. If the higher dimensional PMT is valid, then this assumption is expected to be removed.
An application of Theorem \ref{thm.1} is to provide a simple proof of the rigidity theorem for Poincar\'{e}-Einstein manifolds.
\begin{corollary}\label{cor1}
Let $(X,g_+)$ satisfy the same hypotheses as in Theoerem \ref{thm.Q}.
If the conformal infinity of $(X,g_+)$ is conformally equivalent to the standard sphere $(S^{n-1},g_{S^{n-1}})$, then $(X,g_+)$ is isometric to the hyperbolic space $(\mathbb{H}^n,g_{\mathbb{H}})$.
\end{corollary}
This rigidity theorem was proved in \cite[Theorem 1.1]{Qi1} for $4 \leq n\leq 7$, eventually it was extended to any dimension in \cite[Corollary 1.5]{LQS}, see also \cite{ST,DJ}. Here as a corollary of Theorem \ref{thm.1}, the proof is much shorter and more straight-forward. The disadvantage here is that we need a pre-assumption as in Theorem \ref{thm.1} such that the PMT in the boundary case works.
Here we would like to mention that a very closely related project to the second Escobar-Yamabe compactification is an isoperimetric inequality over scalar flat conformal class, which was studied by Hang-Wang-Yan \cite{HWY1,HWY2}. They proposed a conjecture therein: Let $(\overline {X^n},\pa X, \bar g)$ be a smooth compact Riemannian manifold of dimension $n \geq 3$ with smooth boundary, if $(\overline {X^n}, \pa X, [\bar g])$ is not conformally diffeomorphic to $(\overline{B^n},S^{n-1},[g_{\mathbb{R}^n}])$, then $\Theta(\overline X,\pa X, [\bar g])>\Theta (\overline{B^n},S^{n-1},[g_{\mathbb{R}^n}]),$ where
$$
\Theta(\overline X,\pa X, [\bar g])=
\sup_{\tilde g \in [\bar g], R_{\tilde g}=0} I(\overline{X}, \partial X, \tilde{g})^{\frac{-1}{n(n-1)}}.
$$
Furthermore, they proved that assuming this conjecture is true, if the first Dirichlet eigenvalue of conformal Laplacian of metric $\bar g$ is positive, then $\Theta(\overline X,\pa X, [\bar g])$ is achieved by some conformal metric with zero scalar curvature. Readers are referred to \cite{JX} for very recent progress in the above conjecture.
This paper is organized as follows. In Section \ref{sec2}, we recall Graham's work for singular Yamabe metrics, as well as an asymptotic calculus for boundary defining function $\rho$. In Section \ref{sec3}, we prove Theorem \ref{thm.2} by a flow analysis on the first type Escobar-Yamabe compactification and a generalization of Obata's rigidity theorem. In Section \ref{sec4}, we study the geometric formulae for the second type Escobar-Yamabe compactification and prove Theorem \ref{thm.1}.
In Section \ref{sec5}, we prove Corollary \ref{cor1} as a simple application of Theorem \ref{thm.1}.
\textbf{Acknowledgement}: The authors want to thank professor Paul Yang for valuable comments on this work.
\vspace{0.2in}
\section{Singular Yamabe and Asymptotics}\label{sec2}
We recall in this section some asymptotic computations by Graham \cite{Gr1} as well as the equations for scalar curvatures and Ricci curvatures under conformal change.
Let $(X^n, g_+)$ be a $C^{3,\alpha}$ conformally compact Poincar\'{e}-Einstein and $\bar{g}=\rho^2 g_+$ is a compactification. Let $\hat{g}=\bar{g}|_{\partial X}$. The normal exponential map
$\mathrm{exp}:[0,\delta)_r\times \partial X\longrightarrow \overline{X}$ relative to $\bar{g}$ is a diffeomorphism onto a neighbourhood of $\partial X$, with respect to which $\bar{g}$ takes the form
$$
\bar{g} = \ud r^2+h_r,
$$
where $r(x)=d_{\bar g}(x,\pa X)$ is the distance to the boundary and $h_{r}$ is a one-parameter family of metrics on $\partial X$. We use $\{\alpha, \beta, \cdots\}$ as indices for objects on $\overline{X}$, $\{i,j, \cdots\}$ for objectors on $\partial X$ and $0$ for the $r$ variable. This means $\alpha$ corresponds to $(0, i)$ relative to the product identification induced by $\mathrm{exp}$. At $r=0$, the derivative $\partial^k_rh_r$ can be expressed in terms of curvature of $\bar{g}$. More explicitly,
\begin{equation*}
\begin{aligned}
&(h_r)|_{r=0} = \hat{g}, \\
&(\partial_rh_r)|_{r=0} =h', \quad h'_{ij}=-2L_{ij}, \\
&(\partial^2_rh_r)|_{r=0}=h'', \quad h''_{ij}=-2\bar{R}_{0i0j} +2L_{ik}L_j^{\ k},
\end{aligned}
\end{equation*}
where $L_{ij}$ is the second fundamental form on the boundary and $\bar{R}_{\alpha\beta\delta\gamma}$ is the Riemannian curvature of $\bar{g}$.
Let $\bar{R}$ and $\hat{R}$ be the scalar curvature of $\bar{g}$ and $\hat{g}$, respectively. Since $g_+$ has constant scalar curvature $-n(n-1)$, the conformal change of scalar curvature induces an equation for $\rho$:
\begin{equation}\label{eq.rho1}
-n(n-1) =-n(n-1)|\bar{\nabla}\rho|^2_{\bar{g}} +2(n-1)\rho \Delta_{\bar{g}}\rho +\rho^2 \bar{R},
\end{equation}
where $\Delta_{\bar{g}}=\bar{g}^{\alpha\beta}\bar{\nabla}_{\alpha}\bar{\nabla}_{\beta}$.
This implies that $\rho$ has an asymptotic expansion:
\begin{equation}\label{eq.asymrho}
\begin{aligned}
&\rho =r+c_2r^2 +c_3r^3+o(r^3),
\\
&c_2 = -\frac{1}{2(n-1)} H,
\\
&c_3 =\frac{1}{6(n-2)}(\hat{R}-|\mathring{L}|^2)-\frac{1}{6(n-1)} (\bar{R}+H^2),
\end{aligned}
\end{equation}
where $\mathring{L}_{ij}$ is the trace free part of $L_{ij}$ and $H$ is the mean curvature of the boundary.
Let $\bar{R}_{\alpha\beta}$ be the Ricci curvature tensor of $\bar{g}$ and $\bar{E}_{\alpha\beta}$ the trace free part of $\bar{R}_{\alpha\beta}$. Since $\mathrm{Ric}_{g_+}=-(n-1)g_+$, the conformal change of Ricci curvature implies that
\begin{equation}\label{eq.tracefreeRic}
\bar{E}=-(n-2)\rho^{-1} \left[\bar{\nabla}^2\rho - \frac{1}{n}(\Delta_{\bar{g}}\rho)\bar{g}\right].
\end{equation}
\vspace{0.2in}
\section{Proof of Theorem \ref{thm.2}}\label{sec3}
In this section, we prove Theorem \ref{thm.2}.
Let $(X^n, g_+)$ be a $C^{3,\alpha}$ conformally compact Poincar\'{e}-Einstein manifold satisfying the hypotheses of Theorem \ref{thm.Y} with first type Escobar-Yamabe compactification $(\overline{X}, \partial X, \bar{g})$. This means that $(\overline{X}, \bar{g})$ has constant scalar curvature $\bar{R}$ with totally geodesic boundary. Let $\hat{g}=\bar{g}|_{\partial X}$.
We further assume the equality in Theorem \ref{thm.GH} holds. Then $\bar{g}$ is Einstein, i.e. the trace free part of Ricci curvature vanishes, and $\hat{g}$ has constant scalar curvature $\hat{R}$ on the boundary.
We will show that $(\overline{X}, \partial X, \bar{g})$ must be isometric to the standard hemisphere up to scalings. The idea here follows closely to Obata \cite{Ob1}, Cheeger-Colding \cite{ChC} and Wu-Ye \cite{WuYe}. Here we apply it in the case of a compact manifold with boundary with some extra conditions.
First, we apply the asymptotics in Section \ref{sec2} to the first type Escobar-Yamabe compactification $\bar{g}$. Since $H=0$, (\ref{eq.asymrho}) implies that in this case the boundary defining function has asymptotics
\begin{equation*}
\begin{gathered}
\rho =r+c_2r^2 +c_3r^3+o(r^3),
\end{gathered}
\end{equation*}
where
$$
c_2 = 0,
\quad
c_3 =\frac{1}{6(n-2)}\hat{R}-\frac{1}{6(n-1)} \bar{R}.
$$
Hence
\begin{equation}\label{eq.asym1}
\lim_{r \to 0} \frac{|\bar{\nabla} \rho|_{\bar g}^2 -1}{\rho}=4c_2=0.
\end{equation}
Rewrite equation (\ref{eq.rho1}) as
\begin{equation}\label{eq.deltarho}
\Delta_{\bar{g}} \rho =\frac{n}{2} \rho^{-1} \left(|\bar{\nabla} \rho|^2_{\bar{g}}-1\right)-\frac{1}{2(n-1)}\bar{R}\rho.
\end{equation}
Then (\ref{eq.asym1}) implies that on the boundary,
$$
|\bar{\nabla}\rho|_{\bar{g}}^2|_{\partial X}=1, \quad
\Delta_{\bar{g}} \rho|_{\partial X}=0.
$$
Since $\bar{E}=0$, by (\ref{eq.tracefreeRic}) we have
\begin{equation}\label{eq.rho2}
\bar{\nabla}^2\rho - \frac{1}{n}(\Delta_{\bar{g}}\rho)\bar{g}=0.
\end{equation}
For simplicity, we also denote
\begin{equation}\label{eq.z}
z=- \frac{1}{n}(\Delta_{\bar{g}}\rho)=-\frac{|\bar{\nabla}\rho|^2-1}{2\rho} +\frac{\bar{R}}{2n(n-1)} \rho .
\end{equation}
Hence $(\rho, z)$ satisfies the equation
\begin{equation}\label{eq.rho3}
\bar{\nabla}^2\rho +z\bar{g}=0.
\end{equation}
Since the boundary is totally geodesic, the Gauss-Codazzi equation
$
\bar{R}=\hat{R}+2\bar{R}_{00} + |L|^2-H^2
$
gives that on the boundary
\begin{equation}\label{eq.scalar}
\hat{R}=\frac{n-2}{n}\bar{R}.
\end{equation}
Notice that up to now the sign of $\bar{R}$ (and hence $\hat{R}$) has not been determined.
Second, we consider a flow generated by the vector field $\bar{\nabla}\rho/|\bar{\nabla}\rho|^2$. Denote by $F(t,p)$ the flow lines of $\bar{\nabla}\rho/|\bar{\nabla}\rho|^2$ starting on $\partial X$, where $p\in \partial X$ is an initial point and $t$ is the time, i.e. $F(0,p)=p.$
Since $|\bar{\nabla}\rho|=1$ on $\partial X$, there exists some $T>0$ such that
$$
F:[0,T)\times \partial X\longrightarrow \overline{X}
$$
is a $C^{3,\alpha}$ diffeomorphism to its image and
$$|\bar{\nabla}\rho|(F(t,p))\neq 0, \quad \forall\ t\in [0,T).$$
This means $F(\{t\}\times\partial X)$ is a regular component of the level set of $\rho$.
We divide the rest part of the proof into the following claims.
\begin{itemize}
\item[(a)] $\rho(F(t,p))=t$ for all $t\in [0,T)$.
\\
\textit{Proof.}
This is because $\rho(p)=0$ for $p \in \pa X$ and
$$
\frac{d}{dt}\rho(F(t,p))=\bar{\nabla}\rho \cdot \frac{\bar{\nabla}\rho}{|\bar{\nabla}\rho|^2}=1.
$$
\item[(b)] $|\bar{\nabla}\rho|(F(t,p))$ is independent of $p\in \partial X$ for all $t\in [0,T)$.
\\
\textit{Proof.}
This is because the equation (\ref{eq.rho3}) is equivalent to
\begin{equation*}\label{eq.vector}
\bar{\nabla}_{V}\bar{\nabla} \rho +z V=0, \quad \forall\ V\in T\overline{X}.
\end{equation*}
If choosing $V$ to be tangent to $F(\{t\}\times\partial X)$, a regular level set of $\rho$, we have
$$
\bar{\nabla}_{V}|\bar{\nabla} \rho|^2 =-2z V\cdot \bar{\nabla}\rho=0.
$$
\item[(c)] $[\triangle_{\bar{g}}\rho](F(t,p))$ and hence $z(F(t,p))$ are independent of $p\in \partial X$ for all $t\in [0,T)$.
\\
\textit{Proof.}
In view of equation (\ref{eq.deltarho}) and by (a) and (b), the function $[\triangle_{\bar{g}}\rho](F(t,p))$ is independent of $p$ for all $t\in [0,T)$.
\item[(d)] $z(F(t,p)) =\bar{R}t/[n(n-1)], \ \forall\ t\in[0, T).$
\\
\textit{Proof.}
By \eqref{eq.z} and \eqref{eq.rho3}, we compute the $t$ derivative as follows
$$
\begin{aligned}
\frac{d}{dt}z(F(t,p)) & = \bar{\nabla}\left( -\frac{|\bar{\nabla}\rho|^2-1}{2\rho}+\frac{\bar{R}}{2n(n-1)}\rho
\right) \cdot \frac{\bar{\nabla}\rho}{|\bar{\nabla}\rho|^2}
\\
& = \left(-\frac{\Delta_{\bar{g}}\rho}{n\rho} +\frac{|\bar{\nabla}\rho|^2-1}{2\rho^2}+\frac{\bar{R}}{2n(n-1)}
\right) \bar{\nabla}\rho\cdot \frac{\bar{\nabla}\rho}{|\bar{\nabla}\rho|^2}
\\
&= \frac{\bar{R}}{n(n-1)},
\end{aligned}
$$
which is a constant. Since $z(F(0,p))= 0$ by \eqref{eq.asym1} and \eqref{eq.deltarho}, the claim then follows.
\item[(e)] The scalar curvature $\bar{R}>0$.
\\
\textit{Proof.}
Let $T_0$ be the maximum choice of $T$ such that $|\bar{\nabla}\rho|(F(t,p))\neq 0$ for all $t\in [0,T)$. $T_0$ is finite since $\rho$ is bounded on $\overline{X}$. Let $\Lambda$ be the limit set of the flow as $t\rightarrow T_0$, i.e.
\begin{equation}\label{limitset}
\Omega=F([0, T_0)\times \partial X), \quad
\Lambda = \overline{\Omega}-\Omega.
\end{equation}
It is obvious that $\Lambda\neq \emptyset$.
$\forall~q\in\Lambda$, we show that $q$ must be a critical point of $\rho$, i.e. $|\bar{\nabla}\rho(q)|=0$. By definition, there exists $(t_k, p_k)\in [0,T_0)\times \partial X$ such that $F(t_k, p_k)\longrightarrow q$. By passing to a subsequence if necessary, we may further assume that $t_k$ increases to $T_0$ and $p_k$ converges to $p_0$. If $|\bar{\nabla}\rho(q)|\neq 0$, then (b) implies that all the flow lines can be extended over $T_0$. This contradicts the maximum choice of $T_0$.
Next suppose $\bar{R}\leq 0$.
If $\bar{R}=0$, then
by (d), $z\equiv 0$ on the flow lines. By taking limit, we have $[\triangle_{\bar{g}}\rho](q)=0$. However, by (\ref{eq.deltarho}),
$[\triangle_{\bar{g}}\rho](q)=-n/[2\rho(q)] <0$, where we have used that $q$ is a critical point. Hence a contradiction.
If $\bar{R}<0$, we consider a global maximum point of $\rho$, denoted by $q_0$. Let
$\max_{\overline{X}}\rho =\rho_0=\rho(q_0)>0.$
Then by (\ref{eq.deltarho})
\begin{align} \notag
-\frac{n}{2} -\frac{1}{2(n-1)}\bar{R}\rho_0^2=&[\rho\triangle_{\bar{g}}\rho](q_0)\leq 0.
\end{align}
This implies that
\begin{align}\notag
[\rho\triangle_{\bar{g}}\rho](q) &= -\frac{n}{2} -\frac{\bar{R}}{2(n-1)}\rho_0^2 +\frac{\bar{R}}{2(n-1)}(\rho_0^2-\rho^2(q))\leq 0.
\end{align}
However, in this case, (d) implies that $z<-C<0$ for all $t>T_0/2$. So $\rho\triangle_{\bar{g}} \rho$ is strictly positive at $q$ by taking limit. This yields a contradiction again.
\item[(f)] All critical points of $\rho$ are non-degenerate local maxima and $\overline{X}=\overline{\Omega}$.
\\
\textit{Proof.}
By (d), (e) and (\ref{eq.rho3}), the Hessian of $\rho$ is negative definite at critical point $q\in \Lambda$, which is defined in (\ref{limitset}). So any point $q\in \Lambda$ is non-degenerate and local maximum. In particular, $\Lambda$ consists of isolated critical points. For any $q\in \Lambda$, we can take a geodesic ball $B_r(q)$ with radius $r$ small enough such that $B_r(q)\cap \Lambda=\{q\}$. Since $(B_r(q)-\{q\}) \cap \Omega\neq \emptyset$, we have $(B_r(q)-\{q\}) \subset \Omega$ , i.e. $q$ is an interior point of $\overline{\Omega}$. This implies $\overline{\Omega}$ is a smooth manifold with the only boundary $\partial X$. Since $\overline{X}$ is connected, $\overline{X}=\overline{\Omega}$.
This also shows that there are no other critical points except those in $\Lambda$.
\item[(g)] $\partial X$ is connected.
\\
\textit{Proof.}
Since $(\overline{X}, \partial X, \bar{g})$ is the first type Escobar-Yamabe compactification, $\bar{R}>0$ implies that $Y(\overline{X}, \partial X, [\bar{g}])>0$. Since the equality in Theorem \ref{thm.GH} holds in our case, we have $Y(\partial X, [\hat{g}])>0$ as well, which implies that $\partial X$ is connected by a theorem of Witten-Yau~\cite{WiYa}, see also ~\cite{CG}.
\item[(h)] $\overline{X}$ is diffeomorphic to a hemisphere and $\rho$ has only one critical point.
\\
\textit{Proof.}
Since $\rho=0$ on $\partial X$, in view of claim (f) and standard Morse theory, we conclude that
\[
\overline{X}\cong (\partial X\times [0,1] )\bigsqcup_{\partial X\times \{1\}} (\sqcup_{i} D^{n}_i),
\]
i.e., $\overline{X}$ is resulted from attaching several copies of handles to $\partial X \times [0,1]$ along $\partial X\times\{1\}$. In this case, all handles are
$n$-disks. Therefore $\partial X$ is diffeomorphic to disjoint union of spheres. In view of (g), there exists only one disk, and the claim follows. Consequently, the only critical point of $\rho$ is where it attains its maximum on $\overline{X}$. Let us denote it by $q_0$.
\item[(i)] Level sets of $\rho$ are geodesic spheres of the unique critical point $q_0$.
\\
\textit{Proof.}
Combining claims (a) and (d), we can now rewrite (\ref{eq.rho3}) as
\begin{align}\label{eq.19}
\bar{\nabla}^2 \rho+ \frac{\bar{R}}{n(n-1)}\rho \bar{g}=0.
\end{align}
Up to a scaling, we may assume that $\bar{R}=n(n-1)$, thus (\ref{eq.19}) becomes to
\begin{align}\label{eq.20}
\bar{\nabla}^2 \rho+ \rho \bar{g}=0.
\end{align}
Taking any unit-speed geodesic $\gamma(s)$ starting from $q_0$, by (\ref{eq.20}) we have
\[
\frac{d^2}{ds^2} \rho(\gamma(s)) + \rho(\gamma(s))=0,
\quad \rho(\gamma(0))=T_0, \quad \frac{d}{ds}\rho(\gamma(s))\vline_{s=0}=0.
\]
It follows that
\[
\rho(\gamma(s))=T_0\cos (s),
\]
which means the geodesic $s$-sphere centered at $q_0$ is exactly the $T_0\cos(s)$-level set of $\rho$.
\end{itemize}
Finally, we piece all information together. According to the construction of the flow, and claims (h) and (i), it follows that
$F: \partial X\times [0,T_0) \to \overline{X}\setminus \{q_0\}$ is a diffeomorphism. By (i), the flow lines coincide with geodesics starting at $q_0$ as they are both perpendicular to the level set of $\rho$. Thus equivalently, the exponential map at $q_0$, $\exp_{q_0}: B_{\frac{\pi}{2}}(0)\subset T_{q_0}\overline{X}\to X$ is a diffeomorphism. Using the coordinates associated with the exponential map at $q_0$, $\bar{g}$ can be expressed as
\[
\bar{g}= \ud s^2+ g_{s},
\] where $g_{s}$ is a smooth family of metrics on $S^{n-1}$, with $\displaystyle\lim_{s\to 0} s^{-2} g_s= g_{S^{n-1}}$.
Here $g_{S^{n-1}}$ is the standard round metric on $S^{n-1}$. Also by abuse of notation, $\rho(s)= T_0 \cos (s)$. As the variation of $g_s$ is the second fundamental form of geodesic $s$-sphere, we infer from (\ref{eq.20})
and $\rho(s)=T_0\cos(s)$ that
\[
\frac{d}{ds} g_s=2\cot(s) g_s.
\]
Hence $g_s=\sin^2 (s) g_0$. Then $(\overline{X}, \bar{g})$ is isometric to the standard hemisphere and $g_{+}= \rho^{-2}\bar{g}$ is the standard hyperbolic metric on $X$. This finishes the proof of Theorem \ref{thm.2}.
\vspace{0.2in}
\section{Proof of Theorem \ref{thm.1}} \label{sec4}
In this section we prove Theorem \ref{thm.1}.
Let $(X^n, g_+)$ be a $C^{3,\alpha}$ conformally compact Poincar\'{e}-Einstein manifold satisfying the hypotheses of Theorem \ref{thm.1} and hence those of Theorem \ref{thm.Q}.
We consider the second type Escobar-Yamabe compactification $(\overline{X}, \partial X, \bar{g})$. This means that $(\overline{X}, \bar{g})$ is scalar flat with constant mean curvature $H$ on the boundary. Such $\bar{g}$ may not be unique if exists. Here we just fix a choice of $\bar{g}=\rho^2g_+$ and let $\hat{g}=\bar{g}|_{\partial X}$.
Notice that in our setting, $(X, g_+)$ is $C^{3,\alpha}$ conformally compact. Hence by the boundary regularity theorem in~\cite{CDLS}, $(\overline{X}, \bar{g})$ has an umbilical boundary, i.e. the trace free part of $L_{ij}$, denoted by $\mathring{L}_{ij}$, vanishes on $\pa X$, and hence
$$
L_{ij}=\frac{H}{n-1}\hat{g}_{ij}.
$$
We apply the asymptotics in Section \ref{sec2} to the second type Escobar-Yamabe
compactification $\bar{g}$.
Since $\bar{R}=0$ and $\mathring{L}_{ij}=0$ for $\bar{g}$ now, (\ref{eq.asymrho}) implies that in this case the boundary defining function has an asymptotical expansion:
$$
\rho=r+c_2r^2 +c_3r^3+O(r^{3+\alpha}),
$$
where $r$ is the distance to the boundary and
\begin{align*}
c_2 =& -\frac{1}{2(n-1)}H,
\\
c_3 =&\frac{1}{6(n-2)}\hat{R}-\frac{1}{6(n-1)} H^2.
\end{align*}
Hence
\begin{equation}\label{eq.asym2}
\lim_{r \to 0} \frac{|\bar{\nabla} \rho|_{\bar g}^2 -1}{\rho}=4c_2=-\frac{2}{n-1}H.
\end{equation}
Because $\bar{R}=0$, we can rewrite equation (\ref{eq.rho1}) for $\rho$, which comes from the conformal change of scalar curvatures, as
\begin{equation}\label{eq.deltarho2}
\Delta_{\bar{g}} \rho =\frac{n}{2} \rho^{-1} \left(|\nabla \rho|^2_{\bar{g}}-1\right).
\end{equation}
Recall $\bar{E}$ is the trace-free part of Ricci tensor of $\bar{g}$ and the conformal change of Ricci curvature gives
\begin{equation}\label{eq:trace-free_Ricci}
\bar{E} = -(n-2)\rho^{-1} \left[ \bar{\nabla}^2\rho - \frac{1}{n}(\Delta_{\bar{g}}\rho)\bar{g}\right].
\end{equation}
Set $X_{\delta}:=\{x \in X| \ud_{\bar{g}}(x, \partial X)\geq \delta \}$ for small $\delta>0$. By \eqref{eq.deltarho2} and \eqref{eq:trace-free_Ricci}, a similar integration by parts argument as in \cite{GH} shows that
\begin{align}\label{eq:Obata1}
\frac{2}{(n-2)^2} \int_{X_{\delta}} |\bar{E}|_{\bar g}^2 \rho \ud V_{\bar{g}} &=
\int_{\partial X_{\delta}} \rho^{-1}\left[ N(|\bar{\nabla}\rho|^2_{\bar{g}}) +\rho^{-1} (1-|\bar{\nabla}\rho|^2_{\bar{g}}) N(\rho)\right] \ud S_{\hat{g}},
\end{align}
where $N$ is the unit outward normal w.r.t. $\bar g$ on $\partial X_{\delta}$.
Clearly, $N=-\partial_r$.
Then a direct computation on the asymptotics of $\rho$ shows that
\begin{align*}
&\rho^{-1}=r^{-1}-c_2+(c_2^2-c_3)r+O(r^{1+\alpha}),
\\
&|\nabla \rho|_{\bar g}^2=1+4c_2 r+(4c_2^2+6c_3) r^2+O(r^{2+\alpha}).
\\
&\frac{|\nabla \rho|_{\bar g}^2 -1}{\rho}=4c_2+6c_3r+O(r^{1+\alpha}),
\\
&N(\rho)=-1-2c_2r-3c_3r^2+O(r^{2+\alpha}),
\\
&N(|\nabla \rho|_{\bar g}^2)=-4c_2-(8c_2^2+12c_3)r+O(r^{1+\alpha}).
\end{align*}
And hence
\begin{align*}
&\rho^{-1}\left[ N(|\nabla\rho|^2_{\bar{g}}) +\rho^{-1} (1-|\nabla\rho|^2_{\bar{g}}) N(\rho)\right]
=-6c_3+O(r^{\alpha})=\frac{1}{n-1} H^2 - \frac{1}{n-2}\hat{R}+O(r^{\alpha}).
\end{align*}
Letting $\delta \to 0$ in \eqref{eq:Obata1}, we obtain
\begin{equation}\label{eq:Obata2}
\frac{2}{(n-2)^2} \int_X |\bar{E}|_{\bar g}^2 \rho \ud V_{\bar{g}}=\int_{\partial X} \left(\frac{1}{n-1} H^2 - \frac{1}{n-2}\hat{R} \right) \ud S_{\hat{g}}.
\end{equation}
We are ready to prove the inequalities in Theorem \ref{thm.1} now:
\begin{itemize}
\item
When $n\geq 4$, it follows from \eqref{eq:Obata2} and Theorem \ref{thm.Q} that
\begin{equation}\label{est:Q_Y_n>3}
\begin{aligned}
Y(\pa X,[\hat g]) &\leq
\left( \int_{\pa X}\hat R \ud S_{\hat g} \right) \mathrm{Vol}(\pa X,\hat g)^{-\frac{n-3}{n-1}}\\
&=\left(\frac{n-2}{n-1}\int_{\partial X} H^2 \ud S_{\hat g}-\frac{2}{n-2} \int_X |\bar{E}|_{\bar g}^2 \rho \ud V_{\bar{g}}\right) \mathrm{Vol}(\pa X,\hat g)^{-\frac{n-3}{n-1}}\\\
&\leq \frac{n-2}{4(n-1)} Q(X,\pa X,[\bar g])^2.
\end{aligned}
\end{equation}
\item
When $n=3$, by Gauss-Bonnet theorem and \eqref{eq:Obata2} we have
\begin{equation}\label{est:Q_Y_n=3}
\begin{aligned}
4\pi \chi(\pa X)=\int_{\pa X}\hat R \ud S_{\hat g}=&\frac{1}{2}\int_{\partial X} H^2 \ud S_{\hat g}-2\int_X |\bar{E}|_{\bar g}^2 \rho \ud V_{\bar{g}}
\leq\frac{1}{8} Q(X,\pa X,[\bar g])^2.
\end{aligned}
\end{equation}
\end{itemize}
Next we consider the situation when the equality occurs. One direction is obvious. That is if $(X,g_+)$ is the standard hyperbolic manifold, then we can take a conformal compactification such that $(\overline{X}, \bar{g})$ is the Euclidean ball. A direct computation shows that the equality occurs in \eqref{ineq:Q_Y_n>3} if $n>3$, or in \eqref{ineq:Q_Y_n=3} if $n=3$. For the converse direction, we need to show that
when the equality occurs, the Escobar-Yamabe metric $\bar{g}$ must be the Euclidean metric and $(\overline{X}, \bar{g})$ must be the Euclidean ball.
We observe from \eqref{est:Q_Y_n>3} and \eqref{est:Q_Y_n=3} that in both cases if the equality holds, then $\bar{E}=0$.
So the Ricci curvature of $\bar{g}$ vanishes, i.e. $\bar{R}_{\alpha\beta}=0$. Recall that the the boundary is umbilic. These with the Gauss-Codazzi equation
\begin{equation*}
\bar{R}=\hat{R}+2\bar{R}_{00}+ |L|_{\hat g}^2-H^2,
\end{equation*}
give that on the boundary
$$
\hat{R}=\frac{n-2}{n-1}H^2.
$$
Hence $\hat{R}\geq 0$ is also a constant. Moreover, in this case, \eqref{eq:trace-free_Ricci} shows that
$$
\bar{\nabla}^2\rho = \frac{1}{n}(\Delta_{\bar{g}}\rho)\bar{g}.
$$
Thus we get
$$
\bar{\nabla}^{\beta}\bar{\nabla}_{\beta}\bar{\nabla}_{\alpha}\rho = \frac{1}{n} \bar{\nabla}_{\alpha}(\Delta_{\bar{g}}\rho).
$$
On the other hand,
$$
\bar{\nabla}^{\beta}\bar{\nabla}_{\beta}\bar{\nabla}_{\alpha}\rho
= \bar{\nabla}_{\alpha}\bar{\nabla}^{\beta}\bar{\nabla}_{\beta}\rho +\bar{R}_{\alpha\beta}\bar{\nabla}^{\beta}\rho
=\bar{\nabla}_{\alpha}(\Delta_{\bar{g}}\rho).
$$
Since $n\geq 3$,
$$
\bar{\nabla} (\Delta_{\bar{g}}\rho) =\frac{1}{n} \bar{\nabla}(\Delta_{\bar{g}}\rho)
\quad \Longrightarrow \quad \bar{\nabla}(\Delta_{\bar{g}}\rho)=0.
$$
Therefore $ \Delta_{\bar{g}}\rho$ is a constant all over $\overline{X}$. By \eqref{eq.asym2} and \eqref{eq.deltarho2} we have
\begin{equation}\label{eq.rho}
\Delta_{\bar{g}}\rho = (\Delta_{\bar{g}}\rho)|_{\partial X}= -\frac{n}{n-1} H.
\end{equation}
This implies that all over $\overline{X}$,
$$
\bar{\nabla}^2\rho = -\frac{1}{n-1}H\bar{g}.
$$
We claim that $H\neq 0$. Otherwise, $\Delta_{\bar{g}}\rho=0$ and $\rho|_{\partial X}=0$ implies that $\rho\equiv 0$ all over $\overline{X}$, which obviously can not happen. Now we can set $u=-(n-1)\rho/(nH)$. Then $u$ satisfies
\begin{equation}\label{eq:u_PDE}
\begin{cases}
\Delta_{\bar g}u=1 &\hbox{~~in~~} X,\\
u=0 &\hbox{~~on~~} \pa X,
\end{cases}
\end{equation}
and $N(u)$ is constant on $\pa X$,
$$
N(u) = \frac{n-1}{nH}.
$$
Now the Reilly's formula in \cite{Reilly} together with \eqref{eq:u_PDE} gives
$$
\begin{aligned}
\frac{n-1}{n}\mathrm{Vol}(\overline{X}, \bar{g})
& =\frac{n-1}{n} \int_X (\Delta_{\bar{g}}u)^2 \ud V_{\bar{g}}
\\
&=\int_X\left[ (\Delta_{\bar{g}}u)^2 -|\bar{\nabla}^2u|^2_{\bar{g}} \right]\ud V_{\bar{g}}
\\
&=\int_{\partial X} H N(u)^2 \ud S_{\hat{g}}
\\
&=\left(\frac{n-1}{n}\right)^2\int_{\partial X} \frac{1}{H} \ud S_{\hat{g}}.
\end{aligned}
$$
Therefore we conclude that
$$
\int_{\partial X} \frac{n-1}{H} \ud S_{\hat{g}}=n\mathrm{Vol}(X, \bar{g}) .
$$
Hence $H>0$.
Then it follows from \cite[Theorem 1]{Ros} that $(\overline{X}, \bar{g})$ is isometric to an Euclidean ball.
Up to a constant scaling, we can assume $H=n-1$. Then solving equation (\ref{eq.rho}) we can get $\rho=(1-r^2)/2$ and $(\overline{X}, \bar{g})$ is the unit ball in Euclidean space. And hence $(X, g_+=\rho^{-2}\bar{g})$ is the standard hyperbolic space $(\mathbb{H}^n,g_{\mathbb{H}})$. We finish the proof of Theorem \ref{thm.1}.
\vspace{0.2in}
\section{Proof of Corollary \ref{cor1}} \label{sec5}
In this section we prove Corollary \ref{cor1}.
Let $(X,g_+)$ satisfy the same hypotheses as in Theoerem \ref{thm.Q}. Assume the conformal infinity of $(X, g_+)$ is conformally equivalent to the standard sphere $(S^{n-1}, g_{S^{n-1}})$.
Then
$$
\begin{aligned}
&Y(\pa X,[\hat g])=Y(S^{n-1},[g_{S^{n-1}}])\quad &\textrm{if $n>3$};
\\
&\chi(\partial X)=\chi(S^2)& \quad \textrm{if $n=3$}.
\end{aligned}
$$
Combing this with the inequalities \eqref{ineq:Q_Y_n>3} and \eqref{ineq:Q_Y_n=3} in Theorem \ref{thm.1}, we obtain the following:
\begin{itemize}
\item
If $n>3$ then
\begin{equation}\label{eq.cor1}
\begin{aligned}
Y(S^{n-1},[g_{S^{n-1}}])=Y(\pa X,[\hat g])\leq& \frac{n-2}{4(n-1)} Q(X,\pa X,[\bar g])^2\\
\leq& \frac{n-2}{4(n-1)} Q(B^n,S^{n-1},[g_{\mathbb{R}^n}])^2=Y(S^{n-1},[g_{S^{n-1}}]).
\end{aligned}
\end{equation}
\item
If $n=3$, then
\begin{equation}\label{eq.cor2}
\begin{aligned}
32\pi\chi (S^2)=32\pi \chi (\pa X)\leq Q(X,\pa X,[\bar g])^2 \leq Q(B^3,S^{2},[g_{\mathbb{R}^3}])^2 = 32\pi\chi (S^2).
\end{aligned}
\end{equation}
\end{itemize}
These force all the inequalities in \eqref{eq.cor1} or \eqref{eq.cor2} to be equalities.
By Theorem \ref{thm.1}, this implies that $(X,g_+)$ is isometric to the hyperbolic space $(\mathbb{H}^n,g_{\mathbb{H}})$.
We finish the proof of Corollary \ref{cor1}.
\vspace{0.2in}
| 11,067
|
Recent Posts
November 16, 2008
Do two half requests...
...a whole one make?
I'm a day behind, so I'll keep this short...
I asked for a few small things, but nothing bold or blog worthy.
Picked up Mr. A from the airport. As we pulled out of the parking lot, I asked him to take some time for himself, to sleep, maybe get a day off work, anything to make up for the 18 hour days he was pulling for the month before and during the conference. But I felt like a (no offense!!) wife, a mom, a nag. I think that's something I need to figure out, as well: How to ask someone to do something for him/herself. Find a job. See a doctor. And yet, I don't think it's really my really place to do that. When can/should I intervene?
Then, that night, we had dinner with his cousins, and I asked their three-year-old daughter for a kiss on the cheek. She's bashful, but turns into a bundle of cuddles once she's comfortable with you.
(I did think about asking for a discount while I was at a mug shop for tourists, but I only remembered after she rang up my purchase. D'oh!)
Gained: a promise to at least consider prioritizing sleep/health/well being, and a cutie's kiss
More like this: Asking for knowledge ·
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| 28,243
|
Mildred W. Allen, 86, passed away Saturday, December 14, 2013 at Medicalodges in Independence. Funeral services will be 10 a.m. Thursday December 19, 2013 at Potts Chapel of Independence, with Chaplin Jeff of Harden Hospice officiating. Burial will follow in Jefferson Cemetery, Jefferson Kansas. Friends may register from 9 a.m. to 8 p.m. with family receiving visitors from 6 p.m. to 8 p.m. at Potts Chapel. Memorial contributions may be made to Harden Hospice and may be left at the chapel.
Mildred was born in Independence Kansas on August, 24th 1927 to John Wesley Cole and Mable Boone Cole . She attended Independence schools. She married Dennis Walter Simmons the couple later divorced. In 1955 she married Jess Allen in Barville, Arkansas, the couple made their home on a farm south of Independence. When Jess passed away in 1988, Mildred moved into town. Mildred was preceded in death by her husband Jess Allen, one sister Elise Jean Houck. Survivors include one son Pete Simmons of Independence, a grandson Jeremy Simmons of Independence, and one greatgranddaughter Jete Littell of Cherryvale three nephews Mike, Ken and John Houck one niece Nancy Winebrenner . Arrangements by Potts Chapel of Independence
To send flowers to Mildred's family, please visit our floral section.
| 294,774
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\begin{document}
\title{Brauer spaces for commutative rings\\ and structured ring spectra}
\author{Markus Szymik}
\newdateformat{mydate}{\monthname\ \THEYEAR}
\mydate
\date{\today}
\maketitle
\renewcommand{\abstractname}{}
\begin{abstract}
\noindent
Using an analogy between the Brauer groups in algebra and the Whitehead groups in topology, we first use methods of algebraic K-theory to give a natural definition of Brauer spectra for commutative rings, such that their homotopy groups are given by the Brauer group, the Picard group and the group of units. Then, in the context of structured ring spectra, the same idea leads to two-fold non-connected deloopings of the spectra of units.
\vspace{\baselineskip}
\noindent Keywords: Brauer groups and Morita theory, Whitehead groups and simple homotopy theory, algebraic K-theory spaces, structured ring spectra
\vspace{\baselineskip}
\noindent MSC: 19C30, 55P43, 57Q10
\end{abstract}
\section*{Introduction}
Let~$K$ be a field, and let us consider all finite-dimensional associative~$K$-algebras~$A$ up to isomorphism. For many purposes, a much coarser equivalence relation than isomorphism is appropriate: Morita equivalence. Recall that two such algebras~$A$ and~$B$ are called {\it Morita equi\-valent} if their categories of modules (or representations) are~$K$-equivalent. While this description makes it clear that we have defined an equivalence relation, Morita theory actually shows that there is a convenient explicit description of the relation that does not involve categories: two~$K$-algebras~$A$ and~$B$ are Morita equivalent if and only if there are non-trivial, finite-dimensional~$K$-vector spaces~$V$ and~$W$ such that~\hbox{$A\otimes_K\End_K(V)$} and~\hbox{$B\otimes_K\End_K(W)$} are isomorphic as~$K$-algebras. Therefore, we may say that Morita equivalence is generated by simple extensions: those from~$A$ to~$A\otimes_K\End_K(V)$. There is an abelian monoid structure on the set of Morita equivalence classes of algebras: The sum of the classes of two~$K$-algebras~$A$ and~$B$ is the class of the tensor product~$A\otimes_KB$, and the class of the ground field~$K$ is the neutral element. Not all elements of this abelian monoid have an inverse, but those algebras~$A$ for which the natural map~\hbox{$A\otimes_KA^\circ\to\End_K(A)$} from the tensor product with the opposite algebra~$A^\circ$ is an isomorphism clearly do. These are precisely the central simple~$K$-algebras. The abelian group of invertible classes of algebras is the Brauer group~$\Br(K)$ of~$K$. These notions have been generalized from the context of fields~$K$ to local rings by Azumaya~\cite{Azumaya}, and further to arbitrary commutative rings~$R$ by Auslander and Goldman~\cite{Auslander+Goldman}.
Our first aim is to use some of the classical methods of algebraic K-theory, recalled in the following Section~\ref{sec:background}, in order to define spaces~$\Brspace(R)$ such that their groups of components are naturally isomorphic to the Brauer groups~$\Br(R)$ of the commutative rings~$R$:
\[
\pi_0\Brspace(R)\cong\Br(R).
\]
Of course, this property does not characterize these spaces, so that we will have to provide motivation why the choice given here is appropriate. Therefore, in Section~\ref{sec:Whitehead}, we review Waldhausen's work~\cite{Waldhausen:Top1} and~\cite{Waldhausen:LNM} on the Whitehead spaces in geometric topology in sufficient detail so that it will become clear how this inspired our definition of the Brauer spaces~$\Brspace(R)$ to be given in Section~\ref{sec:Brauer_for_commutative_rings} in the case of commutative rings~$R$. Thereby we achieve the first aim.
We can then relate the Brauer spaces to the classifying spaces of the Picard groupoids, and prove that we have produced a natural delooping of these, see Theorem~\ref{thm:Pic_identification_rings}. In particular, the higher homotopy groups are described by natural isomorphisms
\[
\pi_1\Brspace(R)\cong\Pic(R)
\]
and
\[
\pi_2\Brspace(R)\cong\GL_1(R).
\]
As will be discussed in Section~\ref{sec:Duskin}, we obtain an arguably more conceptual result than earlier efforts of Duskin~\cite{Duskin} and Street~\cite{Street}. This becomes particularly evident when we discuss comparison maps later. Another bonus of the present approach is the fact that it naturally produces infinite loop space structures on the Brauer spaces, so that we even have Brauer spectra~$\br(R)$ such that~\hbox{$\Omega^\infty\br(R)\simeq\Brspace(R)$} for all commutative rings~$R$.
The following Section~\ref{sec:Brauer_for_commutative_S-algebras} introduces Brauer spaces and spectra in the context of structured ring spectra. The approach presented here is based on and inspired by the same classical algebraic K-theory machinery that we will already have used in the case of commutative rings. These spaces and spectra refine the Brauer groups for commutative~$\Sbb$-algebras that have been defined in collaboration with Baker and Richter, see~\cite{Baker+Richter+Szymik}. Several groups of people are now working on Brauer groups of this type. For example, Gepner and Lawson are studying the situation of Galois extensions using methods from Lurie's higher topos theory~\cite{Lurie:HTT} and higher algebra. For connective rings, these methods are used in~\cite{Antieau+Gepner} for constructions and computations similar to~(but independent of) ours in this case. For example, the Brauer group~$\Br(\Sbb)$ of there sphere spectrum itself is known to be trivial. In contrast, this is not the case for the~(non-connective) chromatic localizations of the sphere spectrum by~\cite[Theorem~10.1]{Baker+Richter+Szymik}, and Angeltveit, Hopkins, and Lurie are making further progress towards the computation of these chromatic Brauer groups.
In the final Section~\ref{sec:relative}, we indicate ways of how to apply the present theory. We first discuss the functoriality of our construction and define relative invariants which domi\-nate the relative invariants introduced in~\cite{Baker+Richter+Szymik}, see Proposition~\ref{prop:relative_relation}. Then we turn to the relation between the Brauer spectra of commutative rings and those of structured ring spectra. The Eilenberg-Mac Lane functor~$\upH$ produces structured ring spectra from ordinary rings, and it induces a homomorphism~\hbox{$\Br(R)\to\Br(\upH R)$} between the corresponding Brauer groups, see~\cite[Proposition 5.2]{Baker+Richter+Szymik}. We provide for a map of spectra which induces the displayed homomorphism after passing to homotopy groups, see~Proposition~\ref{prop:EMmap_spectra}.
Another potential application for Brauer spectra is already hinted at in~\cite{Clausen}: they are the appropriate target spectra for an elliptic~J-homomorphism. Recall that the usual~J-homomorphism can be described as a map $\ko\to\pic(\Sbb)$ from the real connective K-theory spectrum to the Picard spectrum of the sphere by means of the algebraic K-theory of symmetric monoidal categories. An elliptic~J-homomorphism should be a map from the connective spectrum of topological modular forms (or a discrete model thereof) to the Brauer spectrum of the sphere, where~$\br(\Sbb)\simeq\Sigma\pic(\Sbb)$ since~$\Br(\Sbb)=0$. It now seems highly plausible that such a map can be constructed from the algebraic K-theory of $\ko$, which has at least the correct chromatic complexity by results of Ausoni and Rognes~\cite{Ausoni+Rognes}. This will be pursued elsewhere.
\subsection*{Acknowledgment}
This work has been partially supported by the Deutsche Forschungsgemeinschaft~(DFG) through the Sonderforschungsbereich~(SFB)~701 ``Spektrale Strukturen und Topologische Methoden in der Mathe\-matik'' at Bielefeld University, and by the Danish National Research Foundation through the Centre for Symmetry and Deformation at the University of Copenhagen~(DNRF92). I would like to thank the referee of an earlier version for the detailed report.
\section{Background on algebraic K-theory}\label{sec:background}
After Quillen, algebraic K-theory is ultimately built on the passage from categorical input to topological output. Various different but equivalent methods to achieve this can be found in the literature. We will need to recall one such construction here, one that produces spectra from symmetric monoidal categories. This is originally due to Thomason~\cite{Thomason} and Shimada-Shimakawa~\cite{Shimada+Shimakawa}, building on earlier work of Segal~\cite{Segal} in the case where the monoidal structure is the categorical sum, and May~(\cite{May:perm1} and~\cite{May:perm2}) in the case when the category is permutative. See also~\cite{Thomason:Aarhus} and the appendix to~\cite{Thomason:CommAlg}. Here we will follow some of the more contemporary expositions such as the ones given in~\cite{Carlsson},~\cite{Elmendorf+Mandell},~\cite{Mandell} and~\cite{Dundas+Goodwillie+McCarthy}, for example. The reader familiar with Segal's~$\Gamma$-machine can safely skip this section and refer back to it for notation only.
\subsection*{$\Gamma$-spaces}
One may say that the key idea behind Segal's machine is the insight that the combinatorics of abelian multiplications is governed by the category of finite pointed sets and pointed maps between them. A~{\it~$\Gamma$-space} is simply a pointed functor~$G$ from this category to the category of pointed spaces.
For set-theoretic purposes, it is preferable to work with a skeleton of the source category: For any integer~$n\geqslant0$, let~$n_+$ denote the pointed set~$(\{0,1,\dots,n\},0)$. We note that there is a canonical pointed bijection~\hbox{$1_+=\upS^0$} with the~$0$-sphere. The full subcategory category~$\Gamma^\circ$ has objects~$n_+$, for~$n\geqslant0$. These span a skeleton of the category of finite pointed sets that is isomorphic to the opposite of Segal's category~$\Gamma$. This explains the odd notation.
A~$\Gamma$-space~$G$ is called {\it special} if the Segal maps
\[
G(N,n)\longrightarrow\prod_{N\setminus\{n\}} G(\upS^0)
\]
which are induced by the maps with singleton support, are weak equivalences. If this is the case, then there is an induced abelian monoid structure on the set~$\pi_0(G(\upS^0))$ of components of~$G(\upS^0)$, and the~$\Gamma$-space~$G$ is called {\it very special} if, in addition, this abelian monoid is a group.
For the purpose of exposition, let us note that basic examples of very special~$\Gamma$-spaces are given by abelian groups~$A$: We can associate to the finite pointed set~$n_+$ the space~$A^n$, and it should be thought of as the space of pointed maps~$f\colon n_+\to A$, where~$A$ is pointed by the zero element. For each pointed map~\hbox{$\phi\colon m_+\to n_+$} the transfer formula
\[
(\phi_*f)(j)=\sum_{\phi(i)=j}f(i)
\]
induces a pointed map~\hbox{$\phi_*\colon A^m\to A^n$}.
\subsection*{Spectra from~$\Gamma$-spaces}
Every~$\Gamma$-space~$G$ extends to a functor from spectra to spectra. In particular, every~$\Gamma$-space~$G$ has an associated spectrum~$G(\Sbb)$ by evaluation of this extension on the sphere spectrum~$\Sbb$, and hence it also has an associated (infinite loop) space~$\Omega^\infty G(\Sbb)$. For later reference, we need to recall some of the details from~\cite[Section~4]{Bousfield+Friedlander}. First of all, the functor~$G$ is extended to a functor from (all) pointed sets to pointed spaces by left Kan extension. Then it induces a functor from pointed simplicial sets to pointed simplicial spaces, and these can then be realized as spaces again. Finally, the~$n$-th space of~$G(\Sbb)$ is~$G(\upS^n)$, where~$\upS^n$ is the simplicial~$n$-sphere: the simplicial circle~$\upS^1=\Delta^1/\partial\Delta^1$ is a simplical set that has precisely~$n+1$ simplices of dimension~$n$, and~$\upS^n=\upS^1\wedge\dots\wedge\upS^1$ is the smash product of~$n$ simplicial circles.
We state the following fundamental theorem for later reference.
\begin{theorem}\label{thm:group_completion}
{\bf\upshape(\cite[1.4]{Segal}, \cite[4.2,~4.4]{Bousfield+Friedlander})}
If~$G$ is a special~$\Gamma$-space, then the adjoint structure maps
$G(\upS^n)\to\Omega G(\upS^{n+1})$ are equivalences for~$n\geqslant1$, and if~$G$ is very special, then this also holds for~$n=0$, so that under this extra hypothesis~$G(\Sbb)$ is an~$\Omega$-spectrum.
\end{theorem}
Thus, if~$G$ is a special~$\Gamma$-space, then the associated infinite loop space
is canonically identified with~$\Omega G(\upS^1)$, and if~$G$ is a very special~$\Gamma$-space, then the associated infinite loop space is canonically identified with~$G(\upS^0)$ as well.
\subsection*{$\Gamma$-spaces from~$\Gamma$-categories}
In all the preceding definitions, spaces may be replaced by categories: A {\it~$\Gamma$-category} is a pointed functor~$\calG$ from the category of finite pointed sets to the category of pointed~(small) categories. A~$\Gamma$-category~$\calG$ is called {\it special} if the Segal functors are equivalences. If this is the case, then this induces an abelian monoid structure on the set of isomorphism classes of objects of~$\calG(\upS^0)$, and the~$\Gamma$-category~$\calG$ is called {\it very special} if, in addition, this abelian monoid is a group.
A~$\Gamma$-category~$\calG(-)$ defines a~$\Gamma$-space~$|\calG(-)|$ by composition with the geometric realization functor from~(small) categories to spaces.
\subsection*{$\Gamma$-categories from symmetric monoidal categories}
Recall that the data required for a {\it monoidal category} are a category~$\calV$ together with a functor~\hbox{$\Box=\Box_\calV\colon\calV\times\calV\to\calV$}, the {\it product}, and an object~\hbox{$e=e_\calV$}, the {\it unit}, such that the product is associative and unital up to natural isomorphisms which are also part of the data. If, in addition, the product is commutative up to natural isomorphisms, then a choice of these turns~$\calV$ into a {\it symmetric monoidal category}.
Every symmetric monoidal category~$\calV$ gives rise to a~$\Gamma$-category~$\calV(-)$ as follows. Since the functor~$\calV(-)$ has to be pointed, we can assume that we have a pointed set~$(N,n)$ such that~$N\setminus\{n\}$ is not empty. Then, the objects of~$\calV(N,n)$ are the pairs~$(V,p)$, where~$V$ associates an object~$V(I)$ of~$\calV$ to every subset~$I$ of~$N\setminus\{n\}$, and~$p$ associates a map
\[
p(I,J)\colon V(I\cup J)\longrightarrow V(I)\Box_\calV V(J)
\]
in~$\calV$ to every pair~$(I,J)$ of disjoint subsets~$I$ and~$J$ of~$N\setminus\{n\}$, in such way that four conditions are satisfied: the~$(V,p)$ have to be pointed, unital, associative, and symmetric. We refer to the cited literature for details.
\begin{examples}\label{ex:gamma012}
If~$N=0_+$, then the category~$\calV(0_+)$ is necessarily trivial.
If~$N=1_+$, then the category~$\calV(1_+)=\calV(\upS^0)$ is equivalent to original category~$\calV$ via the functor that evaluates at the non-base-point. If~$N=2_+$, then the category~$\calV(2_+)$ is equivalent to the category of triples $(V_{12},V_1,V_2)$ of objects together with morphisms $V_{12}\to V_1\Box_\calV V_2$.
\end{examples}
Let us now describe the functoriality of the categories~$\calV(N,n)$ in~$(N,n)$. If we are given a pointed map~$\alpha\colon(M,m)\to (N,n)$ of finite pointed sets, then a functor
\[
\alpha_*=\calV(\alpha)\colon\calV(M,m)\longrightarrow\calV(N,n)
\]
is defined on the objects of~$\calV(M,m)$ by~$\alpha_*(V,p)=(\alpha_*V,\alpha_*p)$, where the components are defined by~\hbox{$(\alpha_*V)(I)=V(\alpha^{-1}I)$} and~\hbox{$(\alpha_*p)(I,J)=p(\alpha^{-1}I,\alpha^{-1}J)$}, and similarly on the morphisms of~$\calV(M,m)$. It is then readily checked that~$\alpha_*$ is a functor, and that the equations~\hbox{$\id_*=\id$} and~$(\alpha\beta)_*=\alpha_*\beta_*$ hold, so that we indeed have a~$\Gamma$-category~$\calV(-)\colon(N,n)\mapsto\calV(N,n)$.
\subsection*{Algebraic K-theory}
If~$\calV$ is a symmetric monoidal category, then its {\it algebraic K-theory spectrum}~$\bfk(\calV)$ will be the spectrum associated with the~$\Gamma$-category that it determines. Its~$n$-th space is
\[
\bfk(\calV)_n=|\calV(\upS^n)|.
\]
The {\it algebraic K-theory space}~$\bfK(\calV)=\Omega^\infty\bfk(\calV)$ of~$\calV$ is the underlying (infinite loop) space. By Theorem~\ref{thm:group_completion}, there is always a canonical equivalence
\[
\bfK(\calV)\simeq\Omega|\calV(\upS^1)|.
\]
In addition, there is also a canonical equivalence
\[
\bfK(\calV)\simeq|\calV(\upS^0)|\simeq|\calV|
\]
in the cases when the abelian monoid of isomorphism classes of~$\calV$ is a group under~$\Box_\calV$. This condition will be met in all the examples in this paper.
\subsection*{Functoriality}
We finally need to comment on the functoriality of this algebraic K-theory construction. We would like maps~$\bfk(\calV)\to\bfk(\calW)$ of spectra to be induced by certain functors~\hbox{$\calV\to\calW$}.
It is clear that this works straightforwardly for strict functors between symmetric monoidal categories. But, it is useful to observe that it suffices, for example, to have an {\it op-lax symmetric monoidal functor} that is {\it strictly unital}: this is a functor~$F\colon\calV\to\calW$ such that~$F(e_\calV)=e_\calW$ holds, together with a natural transformation
\[
\Phi\colon F(V\Box_\calV V')\longrightarrow F(V)\Box_\calW F(V')
\]
that commutes with the chosen associativity, unitality, and commutativity isomorphisms. Given such a strictly unital op-lax symmetric monoidal functor~$\calV\to\calW$, there is still an induced~$\Gamma$-functor~$\calV(-)\to\calW(-)$ between the associated~$\Gamma$-categories: It is defined on objects by the formula~$F_*(V,p)=(F(V),\Phi\circ F(p))$ that makes it clear how~$\Phi$ is used.
\section{Whitehead groups and Whithead spaces}\label{sec:Whitehead}
In this section, we will review just enough of Waldhausen's work on Whitehead spaces so that it will become clear how it inspired the definition of Brauer spaces to be given in the following Section~\ref{sec:Brauer_for_commutative_rings}.
A geometric definition of the Whitehead group of a space has been suggested by many people, see \cite{Stocker}, \cite{Eckmann+Maumary}, \cite{Siebenmann}, \cite{Farrell+Wagoner}, and \cite{Cohen}. We will review the basic ideas now. Ideally, the space~$X$ will be a nice topological space which has a universal covering, but it could also be a simplicial set if the reader prefers so. One considers finite cell extensions~(cofibrations)~$X\to Y$ up to homeomorphism under~$X$. An equivalence relation coarser than homeomorphism is generated by the so-called elementary extensions~\hbox{$Y\to Y'$}, or their inverses, the elementary collapses. By~\cite{Cohen:article}, this is the same as the equivalence relation generated by the simple maps. (Recall that a map of simplicial sets is {\it simple} if its geometric realization has contractible point inverses, see~\cite{Waldhausen+Jahren+Rognes}.) The sum of two extensions~$Y$ and~$Y'$ is obtained by glueing~$Y\cup_X Y'$ along~$X$, and~$X$ itself is the neutral element, up to homeomorphism. Not all elements have an inverse here, but those~$Y$ for which the structure map~$X\to Y$ is invertible (a homotopy equivalence) do. The abelian group of invertible extensions is called the {\it Whitehead group}~$\Wh(X)$ of~$X$.
The preceding description of the Whitehead group, which exactly parallels the description of the Brauer group given in the introduction, makes it clear that these are very similar constructions.
The Whitehead group~$\Wh(X)$ of a space~$X$, as described above, is actually a homotopy group of the Whitehead space. Let us recall from~\cite[Section 3.1]{Waldhausen:LNM} how this space can be constructed. We denote by~$\calC_X$ the category of the cofibrations under~$X$; the objects are the cofibrations~$X\to Y$ as above, and the morphisms from~$Y$ to~$Y'$ are the maps under~$X$. The superscript~$f$ will denote the subcategory of finite objects, where~$Y$ is generated by the image of~$X$ and finitely many cells. The superscript~$h$ will denote the subcategory of the invertible objects, where the structure map is an equivalence. The prefix~$s$ will denote the subcategory of simple maps. Then there is a natural bijection
\begin{equation}\label{eq:Wh_is_a_group}
\Wh(X)\cong\pi_0|s\calC^{fh}_X|,
\end{equation}
see~\cite[3.2]{Waldhausen+Jahren+Rognes}.
The bijection~\eqref{eq:Wh_is_a_group} is an isomorphism of groups if one takes into account the fact that the category~$\calC_X$ is symmetric monoidal: it has (finite) sums. This leads to a delooping of the space~$|s\calC^{fh}_X|$. Because the abelian monoid~$\pi_0|s\calC^{fh}_X|$ is already a group by~\eqref{eq:Wh_is_a_group}, Waldhausen deduces that there is a natural homotopy equivalence
\begin{equation*}
|s\calC^{fh}_X|\simeq|s\calC^{fh}_X(\upS^0)|\simeq\Omega|s\calC^{fh}_X(\upS^1)|,
\end{equation*}
see~\cite[Proposition 3.1.1]{Waldhausen:LNM}, and he calls~$|s\calC^{fh}_X(\upS^1)|$ the {\it Whitehead space} of~$X$. Thus, the Whitehead space of~$X$ is a path connected space, whose fundamental group is isomorphic to the Whitehead group~$\Wh(X)$ of~$X$.
\begin{remark}
One may find the convention of naming a space after its fundamental group rather odd, but the terminology is standard in geometric topology.
\end{remark}
Because the category~$s\calC^{fh}_X$ is symmetric monoidal, the algebraic K-theory machine that we have reviewed in Section~\ref{sec:background} can be used to produce a spectrum, the {\it Whitehead spectrum} of~$X$, such that the Whitehead space is its underlying infinite loop space.
Since it has proven to be very useful in geometric topology to have Whitehead spaces and spectra rather than just Whitehead groups, will we now use the analogy presented in this Section in order to define Brauer spaces and spectra as homotopy refinements of Brauer groups.
\section{Brauer spectra for commutative rings}\label{sec:Brauer_for_commutative_rings}
In this section, we will complete the analogy between Brauer groups and Whitehead groups by defining Brauer spaces and spectra in nearly the same way as we have described the Whitehead spaces and spectra in the previous section. Throughout this section, the letter~$R$ will denote an ordinary commutative ring, and~\cite[Chapter II]{Bass} will be our standard reference for the facts used from Morita theory. See also~\cite{Bass+Roy}.
\subsection*{The categories~$\calA_R$}
Let~$R$ be a commutative ring. Given such an~$R$, we will now define a category~$\calA_R$. The objects are the associative~$R$-algebras~$A$. It might be useful to think of the associative~$R$-algebra~$A$ as a placeholder for the~$R$-linear category~$\calM_A$ of right~$A$-modules. The morphisms~$A\to B$ in~$\calA_R$ will be the~$R$-linear functors~\hbox{$\calM_A\to\calM_B$}. Composition in~$\calA_R$ is composition of functors and identities are the identity functors, so that it is evident that~$\calA_R$ is a category. In fact, the category~$\calA_R$ is naturally enriched in spaces: The~$n$-simplices in the space of morphisms~\hbox{$A\to B$} are the functors~$\calM_A\times[n]\to\calM_B$. Here~$[n]$ denotes the usual poset category with object set~$\{0,\dots,n\}$ and standard order.
\begin{remark}\label{rem:higher_categories}
It seems tempting to work in a setting where the morphism are given by bimodules instead of functors, as in~\cite[XII.7]{MacLane}. But, composition and identities are then given only up to choices, and this approach does not define a category in the usual sense. Compare Remark~\ref{rem:Street}.
\end{remark}
\subsection*{Decorated variants}
There are full subcategories~$\calA^f_R$ and~$\calA^h_R$ of~$\calA_R$, defined as follows. Recall the following characterization of faithful modules from~\cite[IX.4.6]{Bass}.
\begin{proposition}\label{prop:bass}
For finitely generated projective~$R$-modules~$P$, the following are equivalent.\\
{\upshape(1)} The~$R$-module~$P$ is a generator of the category of~$R$-modules.\\
{\upshape(2)} The rank function~$\mathrm{Spec}(R)\to\ZZ$ of~$P$ is everywhere positive.\\
{\upshape(3)} There is a finitely generated projective~$R$-module~$Q$ such that~\hbox{$P\otimes_RQ\cong R^{\oplus n}$} for some positive integer~$n$.
\end{proposition}
The full subcategory~$\calA^f_R$ consists of those~$R$-algebras~$A$ which, when considered as an~$R$-module, are finitely generated projective, and faithful in the sense of Proposition~\ref{prop:bass}.
An~$R$-algebra~$A$ is in the full subcategory~$\calA^h_R$ if and only if the natural map
\begin{equation*}
A\otimes_RA^\circ\longrightarrow\End_R(A)
\end{equation*}
is an isomorphism. We are mostly interested in the intersection
\begin{equation*}
\calA^{fh}_R=\calA^f_R\cap\calA^h_R.
\end{equation*}
\begin{remark}\label{rem:Azumaya}
An~$R$-algebra~$A$ lies in the full subcategory~$\calA^{fh}_R$ if and only if it is an Azumaya~$R$-algebra in the sense of~\cite{Auslander+Goldman}.
\end{remark}
While~$f$ and~$h$ refer to restrictions on the objects, and therefore define full subcategories, the prefix~$s$ will indicate that we are considering less morphisms: Morphisms~\hbox{$A\to B$} in~$s\calA^{fh}_R$ are those functors~$\calM_A\to\calM_B$ which are~$R$-linear equivalences of categories: Morita equivalences. For the higher simplices in~$s\calA^{fh}_R$, we require that they codify natural isomorphisms rather than all natural transformations. This implies that the mapping spaces are nerves of groupoids. In particular, they satisfy the Kan condition.
\begin{remark}
By Morita theory, the~$R$-linear equivalences~$\calM_A\to\calM_B$ of categories are, up to natural isomorphism, all of the form~\hbox{$X\longmapsto X\otimes_AM$} for some invertible~$R$-symmetric~$(A,B)$-bimodule~$M$. And conversely, all such bimodules define equivalences.
\end{remark}
\subsection*{A symmetric monoidal structure}
A symmetric monoidal structure on~$\calA_R$ and its decorated subcategories is induced by the tensor product
\begin{equation*}
(A,B)\mapsto A\otimes_RB
\end{equation*}
of~$R$-algebras. The neutral object is~$R$. We note that the tensor product is not the categorical sum in~$\calA_R$, because the morphisms in that category are not just the algebra maps.
\begin{proposition}\label{prop:pi0_is_a_group_1}
With the induced multiplication, the abelian monoid~$\pi_0|s\calA^{fh}_R|$ of isomorphism classes of objects is an abelian group.
\end{proposition}
\begin{proof}
The elements of the monoid~$\pi_0|s\calA^{fh}_R|$ are represented by the objects of the category~$s\calA^{fh}_R$, and we have already noted that these are just the Azumaya algebras in the sense of~\cite{Auslander+Goldman}, see~Remark~\ref{rem:Azumaya}. Because each Azumaya algebra~$A$ satisfies~$A\otimes_RA^\circ\cong\End_R(A)$, we have~$[A]+[A^\circ]=[\End_R(A)]$ in~$\pi_0|s\calA^{fh}_R|$, so that~$[A^\circ]$ is an inverse to~$[A]$ in~$\pi_0|s\calA^{fh}_R|$ if there is a path from~$\End_R(A)$ to~$R$ in the category~$|s\calA^{fh}_R|$. But, by Proposition~\ref{prop:bass}, we know that~$A$ is a finitely generated projective generator in the category of~$R$-modules, so that the~$R$-algebras~$\End_R(A)$ and~$R$ are Morita equivalent. This means that there exists an~$R$-linear equivalence, and this gives rise to a~$1$-simplex which connects the two vertices. This shows that the monoid~$\pi_0|s\calA^{fh}_R|$ is in fact a group.
\end{proof}
\begin{proposition}\label{prop:pi0_is_Brauer_1}
The group~$\pi_0|s\calA^{fh}_R|$ is naturally isomorphic to the Brauer group~$\Br(R)$ of the commutative ring~$R$ in the sense of~{\upshape\cite{Auslander+Goldman}}.
\end{proposition}
\begin{proof}
The elements in both groups have the same representatives, namely the Azumaya algebras, and the multiplications and units are also agree on those. Thus, it suffices to show that the equivalence relations agree for both of them. The equivalence relation in the Brauer group is generated by the simple extensions, and the equivalence relation in~$\pi_0|s\calA^{fh}_R|$ is generated by Morita equivalence.
We have already seen in the preceding proof that simple extensions are Morita equivalent. Conversely, if an algebra~$A$ is Morita equivalent to~$R$, then~$A$ is isomorphic to~$\End_R(P)$ for some finitely generated projective generator~$P$ of the category of~$R$-modules, so that~$A$ is a simple extension of~$R$, up to isomorphism.
\end{proof}
\subsection*{Brauer spaces and spectra}
The following definition is suggested by Proposition~\ref{prop:pi0_is_Brauer_1}.
\begin{definition}
Let~$R$ be a commutative ring. The space
\begin{equation*}
\Brspace(R)=|s\calA^{fh}_R|
\end{equation*}
is called the {\it Brauer space} of~$R$.
\end{definition}
By Proposition~\ref{prop:pi0_is_Brauer_1}, there is an isomorphism
\begin{equation}\label{eq:pi_0_Br}
\pi_0\Brspace(R)\cong\Br(R)
\end{equation}
that is natural in~$R$.
As described in Section~\ref{sec:background}, the symmetric monoidal structure on~$s\calA^{fh}_R$ also gives rise to an algebraic K-theory spectrum.
\begin{definition}
Let~$R$ be a commutative ring. The spectrum
\begin{equation*}
\br(R)=\bfk(s\calA^{fh}_R)
\end{equation*}
is called the {\it Brauer spectrum} of~$R$.
\end{definition}
We will now spell out the relation between the Brauer space~$\Brspace(R)$ and the Brauer spectrum~$\br(R)$ in detail.
\begin{proposition}\label{prop:delooping_rings}
There are natural homotopy equivalences
\begin{equation*}
\Omega^\infty\br(R)
\simeq\Omega|s\calA^{fh}_R(\upS^1)|
\simeq|s\calA^{fh}_R(\upS^0)|
\simeq|s\calA^{fh}_R|=\Brspace(R).
\end{equation*}
\end{proposition}
\begin{proof}
As explained in Section~\ref{sec:background}, the first and third of these generally hold for symmetric monoidal categories, and the second uses the additional information provided by Proposition~\ref{prop:pi0_is_a_group_1}, namely that the spaces on the right hand side are already group complete.
\end{proof}
In particular, we also have natural isomorphisms
\[
\pi_0\br(R)\cong\Br(R),
\]
of abelian groups, so that we can recover the Brauer group as the~$0$-th homotopy group of a spectrum. We will now determine the higher homotopy groups thereof.
\subsection*{Higher homotopy groups}
Let us now turn our attention to the higher homotopy groups of the Brauer space (or spectrum) of a commutative ring~$R$.
Recall that an~$R$-module~$M$ is called {\it invertible} if there is another~$R$-module~$L$ and an isomorphism~$L\otimes_RM\cong R$ of~$R$-modules. (We remark that, later on, the ring~$R$ might be graded, and then we will also have occasion to consider graded~$R$-modules, but we will also explicitly say so when this will be the case.) The {\it Picard groupoid} of a commutative ring~$R$ is the groupoid of invertible~$R$-modules and their isomorphisms. The realization~$\Picspace(R)$ of the Picard groupoid can have only two non-trivial homotopy groups:
the group of components
\begin{equation}\label{eq:pi_0_Pic}
\pi_0\Picspace(R)\cong\Pic(R)
\end{equation}
is the Picard group~$\Pic(R)$ of~$R$. The fundamental groups of the Picard space~$\Picspace(R)$ are all isomorphic to the group of automorphisms of the~$R$-module~$R$, which is the group~$\GL_1(R)$ of units in~$R$.
\begin{equation}\label{eq:pi_1_Pic}
\pi_1(\Picspace(R),R)\cong\GL_1(R)
\end{equation}
See~\cite[Cor on~p.~3]{Weibel:Azumaya}, for example.
The multiplication on the set of components comes from the fact that the Picard groupoid is symmetric monoidal with respect to the tensor product~$\otimes_R$. Since the isomorphism classes of objects form a group by~\ref{eq:pi_0_Pic}, this also implies that there is Picard spectrum~$\pic(R)$ such that~$\Picspace(R)\simeq\Omega^\infty\pic(R)$, and there is a connected delooping
\[
\upB\Picspace(R)\simeq\Omega^\infty\Sigma\pic(R)
\]
such that its homotopy groups are isomorphic to those of~$\Picspace(R)$, but shifted up by one.
\begin{theorem}\label{thm:Pic_identification_rings}
The components of~$\Brspace(R)$ are all equivalent to~$\upB\Picspace(R)$.
\end{theorem}
\begin{proof}
All components of an infinite loop space such as~$\Brspace(R)\simeq\Omega^\infty\br(R)$ have the same homotopy type.
Therefore, it suffices to deal with the component of the unit~$R$. But that component is the realization of the groupoid of~$R$-linear self-equivalences of the category~$\calM_R$ and their natural isomorphisms. It remains to be verified that the space of~$R$-linear self-equivalences of the category~$\calM_R$ and their natural isomorphisms is naturally equivalent to the Picard space~$\Picspace(R)$.
On the level of components, this follows from Morita theory, see~\cite{Bass}. On the level of spaces, the equivalence is given by evaluation at the symmetric monoidal unit~$R$. In more detail, if~$F$ is an~$R$-linear equivalence from~$\calM_R$ to itself, then~$F(R)$ is an invertible~$R$-symmetric~$(R,R)$-bimodule, and these are just the invertible~$R$-modules. If~$F\to G$ is a natural isomorphism between two~$R$-linear self-equivalences, this gives in particular an isomorphism~$F(R)\to G(R)$ betweem the corresponding two invertible~$R$-modules. This map induces the classical isomorphism on components, and the natural automorphisms of the identity are given by the units of~(the center of)~$R$, which are precisely the automorphisms of~$R$ as an~$R$-module.
\end{proof}
The preceding result implies the calculation of all higher homotopy groups of the Brauer space as a corollary. We note that a similar description and computation of the higher Whitehead groups of spaces is out of reach at the moment.
\begin{corollary}\label{cor:pis_of_Br(R)}
If~$R$ is a commutative ring, then the Brauer space~$\Brspace(R)$ has at most three non-trivial homotopy groups:
\begin{gather*}
\pi_0\Brspace(R)\cong\Br(R),\\
\pi_1\Brspace(R)\cong\Pic(R),\\
\pi_2\Brspace(R)\cong\GL_1(R),
\end{gather*}
and the results for the Brauer spectrum~$\br(R)$ are the same.
\end{corollary}
\begin{proof}
The first of these is the isomorphism~\eqref{eq:pi_0_Br}, and the second follows from the preceding theorem together with the isomorphisms~\eqref{eq:pi_0_Pic}. For the third, we only need to recall~\ref{eq:pi_1_Pic}: the fundamental groups of the Picard space~$\Picspace(R)$ are all isomorphic to~$\GL_1(R)$. The final statement follows from the equivalence~$\Brspace(R)\simeq\Omega^\infty\br(R)$.
\end{proof}
We remark that Brauer spectra (and spaces) are rather special in the sense that not every~$2$-truncated connective spectrum is equivalent to the Brauer spectrum of a ring. This follows, for example, from the well-known fact that there is no commutative ring which has exactly five units, whereas there are clearly spaces and spectra such that their second homotopy group has order five.
\section{A scholion on the Azumaya complex}\label{sec:Duskin}
In this section, we will review some related work of Duskin and Street. Let again~$R$ be an ordinary commutative ring. In this case, Duskin, in~\cite{Duskin}, has built~a reduced Kan complex~$\Az(R)$, the {\it Azumaya complex}, with~$\pi_1\Az(R)$ isomorphic to the Brauer group of~$R$, with the group~$\pi_2\Az(R)$ isomorphic to the Picard group of~$R$, and with the group~$\pi_3\Az(R)$ isomorphic to the multiplicative group of units in~$R$.~(As~$\Az(R)$ is reduced, we may omit the base-point from the notation.) In fact, he hand-crafts the~$4$-truncation so that the homotopy groups work out as stated, and then he takes its~$4$-co-skeleton.
Here is a sketch of his construction. There is only one~$0$-simplex in~$\Az(R)$. It does not need a name, but it can be thought of as the commutative ring~$R$. We note that also our $s\calA^{fh}_R(0_+)$ is the trivial category. The~$1$-simplices in the Azumaya complex~$\Az(R)$ are the Azumaya~$R$-algebras~$A$. (In particular, the degenerate~$1$-simplex is given by the~$R$-algebra~$R$ itself.) We note that these are precisely the objects in our category~\hbox{$s\calA^{fh}_R(1_+)\simeq s\calA^{fh}_R$}; but, the latter comes with higher homotopy information form the mapping spaces. Now a map $\partial\Delta^2\to\Az(R)$ corresponds to three Azumaya algebras $A_1$, $A_2$, and $A_{12}$, and the $2$-simplices in $\Az(R)$ with this given restriction are defined to be the~$R$-symmetric~$(A_{12},A_1\otimes_RA_2)$-bimodules $F$ which are invertible.~A suggestive notation is~\hbox{$F\colon A_{12}\Rightarrow A_1\otimes_RA_2$}. By Example~\ref{ex:gamma012}, these are essentially the objects of the category $s\calA^{fh}_R(2_+)$, except for the fact that we are working with the actual equivalences defined by the bimodules. Now a map $\partial\Delta^3\to\Az(R)$ corresponds to four bimodules
\begin{align*}
A_{123}&\Longrightarrow A_{12}\otimes_RA_3\\
A_{123}&\Longrightarrow A_1\otimes_RA_{23}\\
A_{12}&\Longrightarrow A_1\otimes_RA_2\\
A_{23}&\Longrightarrow A_2\otimes_RA_3,
\end{align*}
and the $4$-simplices in $\Az(R)$ with this boundary are the isomorphisms $u$ between the two corresponding bimodules
\[
A_{123}\Longrightarrow A_1\otimes_RA_2\otimes_RA_3
\]
that can be obtained by tensoring them in the two meaningful ways. Finally, the $4$-simplices of $\Az(R)$ are uniquely determined by their boundary, and their existence depends on a compatibility condition that we will not recall here.
\begin{remark}\label{rem:Street}
As already mentioned in~\cite{Duskin}, Street has described some catego\-ri\-cal structures underlying Duskin's construction. However, these were published only much later, in~\cite{Street}. Street considers the bicategory whose objects are~$R$-algebras, whose morphism~$M\colon A\to B$ are~$R$-symmetric~$(A,B)$-bimodules, and whose~$2$-cells~\hbox{$f\colon M\Rightarrow N$} are bimodule morphisms; vertical composition is composition of functions and horizontal composition of modules~$M\colon A\to B$ and~\hbox{$N\colon B\to C$} is given by tensor product~$M\otimes_B N\colon A\to C$ over~$B$. The tensor product~$A\otimes_R B$ of algebras is again an algebra, and this makes this category a monoidal bicategory. He then passes to its suspension, the one-object tricategory whose morphism bicategory is the category described before and whose composition is the tensor product of algebras. While this cannot, in general, be rigidified to a 3-category, there is a~$3$-equivalent~Gray category. The Gray subcategory of invertibles consists of the arrows~$A$ which are biequivalences, the~$2$-cells~$M$ which are equivalences, and the~$3$-cells~$f$ which are isomorphisms, so that the morphisms~$A$ are the Azumaya algebras, and the~$2$-cells are the Morita equivalences. The nerve of this Gray subcategory is Duskin's complex~$\Az(R)$.
\end{remark}
In this paper, we have chosen to present an approach that does not involve higher categories, at least none that do not have a well-defined composition. While one may argue that the loop space $\Omega\Az(R)$ would be equivalent to the Brauer space $\Brspace(R)$, the present direct construction seems to be more natural. It certainly seems rather artificial to realize the Brauer group as $\pi_1$ instead of $\pi_0$. In any case, our delooping~$|s\calA^{fh}_R(\upS^1)|$ provides for a space with such a $\pi_1$ as well, if so desired, see Proposition~\ref{prop:delooping_rings}. In fact, the general algebraic K-theory machinery used here yields arbitrary deloopings $|s\calA^{fh}_R(\upS^n)|$ without extra effort. This feature seems to be unavailable in the approach of Duskin and Street.
\section{Brauer spectra for structured ring spectra}\label{sec:Brauer_for_commutative_S-algebras}
We will now transfer the preceding theory from the context of commutative rings to the context of structured ring spectra. There are many equivalent models for this, such as symmetric spectra~\cite{HSS} or~$\Sbb$-modules~\cite{EKMM}, and we will choose the latter for the sake of concordance with~\cite{Baker+Richter+Szymik}. In Section~\ref{sec:Brauer_for_commutative_rings}, we have defined a Brauer space~$\Brspace(R)$ and a Brauer spectrum~$\br(R)$ for each commutative ring~$R$, starting from a category~$\calA_R$ and its subcategory~$s\calA^{fh}_R$. If now~$R$ denotes a commutative~$\Sbb$-algebra, we may proceed similarly. Let us see how to define the corresponding categories.
\subsection*{The categories~$\calA_R$}
Let~$\calA_R$ denote the category of cofibrant~$R$-algebras and~$R$-functors between their categories of modules. This is slightly more subtle than the situation for ordinary rings, as the categories of modules are not just categories, but come with homotopy theories. In order to take this into
account, the model categories of modules will first be replaced by the simplicial categories obtained from them by Dwyer-Kan localization. We note that the categories of modules are enriched in the symmetric monoidal model category of~$R$-modules in this situation. This allows us to use the model structure from~\cite[Appendix A.3]{Lurie:HTT} on these. Then~$\calA_R$ is again a simplicial category: The class of objects~$A$ is still discrete, and the space of morphisms~$A\to B$ is the derived mapping space of~$R$-functors~$\calM_A\to\calM_B$.
\subsection*{Decorated variants}
There is the full subcategory~$\calA^f_R$, where~$A$ is assumed to satisfy the finiteness condition used in~\cite{Baker+Richter+Szymik}: it has to be faithful and dualizable as an~$R$-module.
Also, there is the full subcategory~$\calA^h_R$, where we assume that the natural map
\begin{equation*}
A\wedge_RA^\circ\longrightarrow\End_R(A)
\end{equation*}
is an equivalence. We are mostly interested in the intersection
\begin{equation*}
\calA^{fh}_R=\calA^f_R\cap\calA^h_R,
\end{equation*}
which consists precisely the Azumaya~$R$-algebras in the sense of~\cite{Baker+Richter+Szymik}. While~$f$ and~$h$ again refer to restrictions on the objects, the prefix~$s$ will indicate that we are considering only those~$R$-functors which are equivalences of simplicial categories, and their natural equivalences. The standard references for Morita theory in this context are~\cite{Schwede+Shipley} as well as the expositions~\cite{Schwede} and~\cite{Shipley}. Up to natural equivalence, the~$R$-equivalences are of the form~\hbox{$X\longmapsto X\wedge_AM$} for some invertible~$R$-symmetric~$(A,B)$-bimodule~$M$. Similarly, the higher simplices codify natural equivalences rather than all natural transformations. This ends the description of the simplicial category~$s\calA^{fh}_R$. The following result describes one of its mapping spaces.
\begin{proposition}\label{prop:auto_R}
The space of auto-equivalences of the category of~$R$-modules is naturally equivalent to the space of invertible~$R$-modules.
\end{proposition}
\begin{proof}
This is formally the same as the corresponding result in the proof of Theorem~\ref{thm:Pic_identification_rings}. A map in one direction is given by the evaluation that sends an equivalence to its value on~$R$. On the other hand, given an invertible~$R$-module, the smash product with it defines an equivalence. Compare with~\cite[4.1.2]{Schwede+Shipley}.
\end{proof}
\subsection*{A symmetric monoidal structure}
A symmetric monoidal structure on~$\calA_R$ and its subcategories is induced by the smash product
\begin{equation*}
(A,B)\mapsto A\wedge_RB
\end{equation*}
of~$R$-algebras with neutral element~$R$. We note that this is not the categorical sum, as the morphisms in these categories are not just the algebra maps.
\begin{proposition}\label{prop:Br_is_a_group_2}
With the induced multiplication, the abelian monoid~$\pi_0|s\calA^{fh}_R|$ of isomorphism classes of objects is an abelian group, and this abelian group is isomorphic to the Brauer group~$\Br(R)$ of the commutative~$\Sbb$-algebra~$R$ in the sense of~{\upshape\cite{Baker+Richter+Szymik}}.
\end{proposition}
\begin{proof}
This is formally the same as the proofs of Proposition~\ref{prop:pi0_is_a_group_1} and Proposition~\ref{prop:pi0_is_Brauer_1}.
\end{proof}
\subsection*{Brauer spaces and spectra}
The following definition is suggested by Proposition~\ref{prop:Br_is_a_group_2}.
\begin{definition}
Let~$R$ be a commutative~$\Sbb$-algebra. The space
\begin{equation*}
\Brspace(R)=|s\calA^{fh}_R|
\end{equation*}
is called the {\it Brauer space} of~$R$.
\end{definition}
By Proposition~\ref{prop:Br_is_a_group_2}, there is an isomorphism
\begin{equation}
\pi_0\Brspace(R)\cong\Br(R)
\end{equation}
that is natural in~$R$.
As described in Section~\ref{sec:background}, the symmetric monoidal structure on~$s\calA^{fh}_R$ also gives rise to an algebraic K-theory spectrum.
\begin{definition}
Let~$R$ be a commutative~$\Sbb$-algebra. The spectrum
\begin{equation*}
\br(R)=\bfk(s\calA^{fh}_R)
\end{equation*}
is called the {\it Brauer spectrum} of~$R$.
\end{definition}
We will now spell out the relation between the Brauer space~$\Brspace(R)$ and the Brauer spectrum~$\br(R)$ in detail.
\begin{proposition}
There are natural homotopy equivalences
\begin{equation*}
\Omega^\infty\br(R)
\simeq\Omega|s\calA^{fh}_R(\upS^1)|
\simeq|s\calA^{fh}_R(\upS^0)|
\simeq|s\calA^{fh}_R|=\Brspace(R).
\end{equation*}
\end{proposition}
\begin{proof}
As explained in Section~\ref{sec:background}, the first and third of these generally hold for symmetric monoidal categories, and the second uses the additional information provided by Proposition~\ref{prop:Br_is_a_group_2}, namely that the spaces on the right hand side are already group complete.
\end{proof}
In particular, we also have natural isomorphisms
\[
\pi_0\br(R)\cong\Br(R),
\]
of abelian groups, so that we can recover the Brauer group as the~$0$-th homotopy group of a spectrum. We will now determine the higher homotopy groups thereof.
\subsection*{A review of Picard spaces and spectra}
As for the Brauer space of a commutative ring, also in the context of structured ring spectra, there is a relation to the Picard groupoid of~$R$, and this will be discussed now.
Let~$R$ be a cofibrant commutative~$\Sbb$-algebra. In analogy with the situation for discrete rings, it is only natural to make the following definition, compare~\cite[Remark~1.3]{Ando+Blumberg+Gepner}.
\begin{definition}
The {\it Picard space}~$\Picspace(R)$ of~$R$ is the classifying space of the Picard groupoid: the simplicial groupoid of invertible~$R$-modules and their equivalences.
\end{definition}
We note that the components of the Picard space are the equivalence classes of invertible~$R$-modules, and with respect to the smash product~$\wedge_R$, these components form a group, by the very definition of an invertible module. This is Hopkins' Picard group~$\Pic(R)$ of~$R$:
\begin{displaymath}
\pi_0(\Picspace(R))\cong\Pic(R).
\end{displaymath}
See for example~\cite{Strickland},~\cite{Hopkins+Mahowald+Sadofsky}, and~\cite{Baker+Richter} for more information about this group. In contrast to the case of discrete commutative rings, the Picard space of a commutative~$\Sbb$-algebra is no longer $1$-truncated. There is an equivalence
\[
\Picspace(R)\simeq\Pic(R)\times\upB\GL_1(R),
\]
where $\upB\GL_1(R)$ is the classifying space for the units of $R$.
The Picard category is symmetric monoidal with respect to the smash product~$\wedge_R$. Therefore, as in the case of discrete rings, there is also a Picard spectrum~$\pic(R)$ such that~$\Picspace(R)\simeq\Omega^\infty\pic(R)$, and there is a connected delooping
\[
\upB\Picspace(R)\simeq\Omega^\infty\Sigma\pic(R)
\]
such that its homotopy groups are isomorphic to those of~$\Picspace(R)$, but shifted up by one. We will see now that the Brauer spaces provide for another delooping that is typically non-connected.
\subsection*{Higher homotopy groups}
After this recollection, let us now see how the Picard spaces and spectra relate to the Brauer spaces and spectra defined above.
\begin{theorem}\label{thm:Pic_identification_S-algebras}
If~$R$ is a commutative~$\Sbb$-algebra, then the component of the neutral vertex~$R$ in the Brauer space~$\Brspace(R)$ is naturally equivalent (as an infinite loop space) to the classifying space of the Picard groupoid~$\Picspace(R)$.
\end{theorem}
\begin{proof}
The component of the neutral element~$R$ in~$\Brspace(R)=|s\calA^{fh}_R|$ is equivalent to the classifying space of the automorphism group of $R$ in $s\calA^{fh}_R$. By definition of that category, this is the group-like simplicial monoid of Morita self-equivalences of the category $\calM_R$ of $R$-modules. The result now follows from Proposition~\ref{prop:auto_R}: this is equivalent to the simplicial groupoid of invertible $R$-modules, the Picard groupoid of $R$.
\end{proof}
\begin{corollary}\label{cor:Pic_identification}
There are natural isomorphisms
\begin{equation*}
\pi_n\Brspace(R)\cong\pi_{n-2}\Picspace(R)
\end{equation*}
for~$n\geqslant2$,
\begin{equation*}
\pi_n\Brspace(R)\cong\pi_{n-3}\GL_1(R)
\end{equation*}
for~$n\geqslant3$, and
\begin{equation*}
\pi_n\Brspace(R)\cong\pi_{n-3}(R).
\end{equation*}
for~$n\geqslant4$.
\end{corollary}
\begin{proof}
The first statement is an immediate consequence of the preceding theorem. The second follows from the first and the fact that the Picard space is a delooping of the space of units, and the last statement follows from the second and~$\pi_n\GL_1(R)\cong\pi_n(R)$ for~\hbox{$n\geqslant1$}.
\end{proof}
We note in particular that the Brauer space is~$2$-truncated in the case of an Eilenberg-Mac Lane spectrum. This will be used in~Section~\ref{sec:EM}.
\section{Functoriality}
In this section, we will see how functorality of the K-theory construction immediately leads to spacial and spectral versions of relative Brauer invariants as well as to a characteristic map that codifies the obstructions to pass from topological information about Eilenberg-Mac Lane spectra to algebra.
\subsection*{Relative invariants}\label{sec:relative}
In this section, we define relative Brauer spectra, as these are likely to be easier to compute than their absolute counterparts. We will focus on the case of extensions of commutative~$\Sbb$-algebras, but the case of ordinary commutative rings is formally identical.
To start with, let us first convince ourselves that the construction of the Brauer space (or spectrum) is sufficiently natural.
\begin{proposition}\label{prop:naturality}
If~$R\to S$ is a map of commutative~$\Sbb$-algebras, then there is a map
\begin{equation*}
\br(R)\longrightarrow\br(S)
\end{equation*}
of Brauer spectra, and similarly for Brauer spaces.
\end{proposition}
\begin{proof}
The map is induced by~$A\mapsto S\wedge_RA$. By~\cite[Proposition 1.5]{Baker+Richter+Szymik}, it maps Azumaya algebras to Azumaya algebras. It therefore induces functors between the symmetric monoidal categories used to define the Brauer spectra.
\end{proof}
If~$R\to S$ is a map of commutative~$\Sbb$-algebras, then~$\br(S/R)$ and~$\Brspace(S/R)$ will denote the homotopy fibers of the natural maps in Proposition~\ref{prop:naturality}. We note that there is an equi\-valence~\hbox{$\Brspace(S/R)\simeq\Omega^\infty\br(S/R)$} of infinite loop spaces.
\begin{remark}\label{rem:Thomason}
Thomason's general theory of homotopy colimits of symmetric monoidal categories (\cite{Thomason:Aarhus} and \cite{Thomason:CommAlg}) might provide a first step to obtain a more manageable description of these relative terms.
\end{remark}
The defining homotopy fibre sequences lead to exact sequences of homotopy groups. Together with the identifications from Proposition~\ref{thm:Pic_identification_S-algebras} and Corollary~\ref{cor:Pic_identification}, these read
\begin{gather*}
\dots\to\pi_2\br(S/R)\to\pi_0\GL_1(R)\to\pi_0\GL_1(S)\to
\pi_1\br(S/R)\to \Pic(R)\to\\
\to\Pic(S)\to
\pi_0\br(S/R)\to\Br(R)\to\Br(S)\to\pi_{-1}\br(S/R)\to0.
\end{gather*}
In~\cite[Definition 2.6]{Baker+Richter+Szymik}, the relative Brauer group~$\Br(S/R)$ is defined as the kernel of the natural homomorphism
$\Br(R)\to\Br(S)$.
\begin{proposition}\label{prop:relative_relation}
The relative Brauer group~$\Br(S/R)$ is naturally isomorphic to the cokernel of the natural boundary map~$\Pic(S)\to\pi_0\br(S/R)$.
\end{proposition}
\begin{proof}
This is an immediate consequence of the definition of~$\Br(S/R)$ as the kernel of the natural homomorphism~$\Br(R)\to\Br(S)$ and the long exact sequence above.
\end{proof}
\begin{remark}
We note that the theory of Brauer spaces presented here also has Bousfield local variants, building on~\cite[Definition 1.6]{Baker+Richter+Szymik}, and that it might be similarly interesting to study the behavior of the Brauer spaces under variation of the localizing homology theory.
\end{remark}
\subsection*{Eilenberg-Mac Lane spectra}\label{sec:EM}
Let us finally see how the definitions of the present paper work out in the case of Eilenberg-Mac Lane spectra. Let~$R$ be an ordinary commutative ring, and let~$\upH R$ denote its Eilenberg-Mac Lane spectrum. This means that we have two Brauer groups to compare:~$\Br(R)$ as defined in~\cite{Auslander+Goldman}, and~$\Br(\upH R)$ as defined in~\cite{Baker+Richter+Szymik}, where there is also produced a natural homomorphism
\begin{equation}\label{eq:EMmap}
\Br(R)\longrightarrow\Br(\upH R)
\end{equation}
of groups, see~\cite[Proposition 5.2]{Baker+Richter+Szymik}. Using results from~\cite{Toen}, one can deduce that this homomorphism is an isomorphism if~$R$ is a separably closed field, because both sides are trivial, see~\cite[Proposition 5.5 and Remark 5.6]{Baker+Richter+Szymik}. In general, one may show that it is injective with a cokernel which is isomorphic to the product of~$\upH^1_\mathrm{et}(R;\ZZ)$ and the torsion-free quotient of~$\upH^2_\mathrm{et}(R;\GL_1)$. See~\cite{Johnson} and~\cite[Remark 5.3]{Baker+Richter+Szymik}. In particular, the map is an isomorphism for all fields~$K$.
Using the spaces and spectra defined in~Section~\ref{sec:Brauer_for_commutative_rings} for~$R$ and in Section~\ref{sec:Brauer_for_commutative_S-algebras} for~$\upH R$, the homomorphism~\eqref{eq:EMmap} of abelian groups can now be refined to a map of spectra.
\begin{proposition}\label{prop:EMmap_spectra}
There is a natural map
\begin{equation}\label{eq:EMmap_spectra}
\br(R)\longrightarrow\br(\upH R)
\end{equation}
of spectra that induces the homomorphism~\eqref{eq:EMmap} on components.
\end{proposition}
\begin{proof}
This map is induced by the Eilenberg-Mac Lane functor~$\upH\colon R\mapsto\upH R$. It induces functors between the symmetric monoidal categories used to define the Brauer spectra.
\end{proof}
\begin{theorem}\label{prop:1-truncated}
The homotopy fibre of the map~\eqref{eq:EMmap_spectra} is a~$0$-truncated spectrum. Its only non-trivial homotopy groups are~$\pi_0$ which is infinite cyclic, and~$\pi_{-1}$ which is isomorphic to the cokernel of the map~\eqref{eq:EMmap}, the product of~$\upH^1_\mathrm{et}(R;\ZZ)$ and the torsion-free quotient of~$\upH^2_\mathrm{et}(R;\GL_1)$.
\end{theorem}
\begin{proof}
If~$R$ is a commutative ring, then the natural equivalence
\begin{displaymath}
\gl_1(\upH R)\simeq\upH\GL_1(R)
\end{displaymath}
describes the spectrum of units of the Eilenberg-Mac Lane spectrum. It follows that~$\br(\upH R)$ is~$2$-truncated. As~$\br(R)$ is always~$2$-truncated by Corollary~\ref{cor:pis_of_Br(R)}, so is the homotopy fibre. On~$\pi_2$, the map~\eqref{eq:EMmap_spectra} induces an isomorphism between two groups both isomorphic to the group of units of~$R$. On~$\pi_1$, the map~\eqref{eq:EMmap_spectra} is the map
\begin{equation}\label{eq:Pic_map}
\Pic(R)\longrightarrow\Pic(\upH R)
\end{equation}
induced by the Eilenberg-Mac Lane functor~$\upH$. A more general map has been studied in~\cite{Baker+Richter}, where the left hand side is replaced by the Picard group of graded~$R$-modules, and then the map is shown to be an isomorphism. For the present situation, this means that~\eqref{eq:Pic_map} is a monomorphism with cokernel isomorphic to the group~$\ZZ$ of integral grades. On~$\pi_0$, the map~\eqref{eq:EMmap_spectra} induces the map~\eqref{eq:EMmap} by Proposition~\ref{prop:EMmap_spectra}. As has been remarked at the beginning of this section, this map is injective with the indicated cokernel. The result follows.
\end{proof}
| 2,620
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Q. While accessing a dataset, I selected Surname property in filter control but it is showing only 10000 records in the filter and others are omitted by showing a message "1532 more"
A. In case you are not already aware, once inside a dataset, if you click CTRL and the OrgVue logo shown on the top left of the screen then this will display the Settings Panel:
This enables you to set a number of configurable items, including Filter Limit, which can be adjusted on a per tenant basis.
The default value(s) set will affect performance, and typically any change(s) that increase numbers will be at the expense of performance degradation.
| 26,521
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\begin{document}
\title[The reduction number and degree bound of projective subschemes]{The reduction number and degree bound\\ of projective subschemes}
\author{\fontencoding{T5}\selectfont \DJ o\`an Trung C\uhorn{}\`\ohorn ng}
\address{\fontencoding{T5}\selectfont \DJ o\`an Trung C\uhorn{}\`\ohorn ng. Institute of Mathematics and the Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam.}
\email{dtcuong@math.ac.vn}
\author{Sijong Kwak}
\address{Sijong Kwak. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Republic of Korea}
\email{sjkwak@kaist.ac.kr}
\thanks{\fontencoding{T5}\selectfont \DJ o\`an Trung C\uhorn{}\`\ohorn ng is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2015.26.}
\thanks{Sijong Kwak was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2015R1A2A2A01004545).}
\subjclass[2010]{Primary: 14N05; secondary: 13D02}
\keywords{reduction number, degree, Castelnuovo-Mumford regularity, arithmetically Cohen-Macaulay subscheme, Betti table}
\begin{abstract}
In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay with linear resolutions. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give the possible explicit Betti tables for almost maximal cases. In addition, interesting examples are provided to understand our main results.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and codimension $e$ defined over an algebraically closed field $k$ of arbitrary characteristic with the ideal sheaf $\mathcal I_X$. Let $S_0=k[x_0, \ldots, x_{n+e}]$ and $R=S_0/I_X$ be the homogeneous coordinate ring of $X$ where $I_X=\oplus_{m=0}^\infty H^0(\bP^{n+e}, \mathcal I_X(m))$ is the saturated homogeneous ideal. Among the numerical invariants of $X$ there are the degree $\deg(X)$, the Castelnuovo-Mumford regularity $\reg(R)$ and the reduction number $r(X)$ (which is defined as the reduction number of the homogeneous coordinate ring $R$). The complexity of $R$ is reflected in those invariants. The Castelnuovo-Mumford regularity is the height of the Betti table of $R$ and the reduction number $r(X)$ is the least number among the maximal degrees of minimal generators of $R$ as a module over its Noether normalizations.
There have been several results on the relations between these invariants (see \cite{NVT87}, \cite{Vas96}). For example, we always have
\begin{equation}\label{eq1}
1\leq r(X)\leq \reg(R).
\end{equation}
We can generalize the inequality (\ref{eq1}) as follows: If $R$ is $d$-regular until the $e$-th step in the minimal free resolution, then
\begin{equation}\label{eq2}
r(X)\leq d\leq \reg(R).
\end{equation}
It is interesting to mention that in many cases, the reduction number $r(X)$ is smaller than $d$ as shown in Examples \ref{34} (Ulrich's example), \ref{57} and \ref{58}.
On the other hand, the degree upper bound can be read off from the shape of Betti tables. As examples, projective varieties satisfying property $\textbf{N}_{2,p}$ for $p\ge 1$ have a degree bound (\cite{AS10}, \cite{HK12}). The multiplicity conjecture (\cite{HS98}), which was proved by Eisenbud and Schreyer, Boij and Söderberg (\cite{BS12}, \cite{ES09}, \cite{ES11}, \cite{Pe11}), gives us an upper bound with the maximal cases which are arithmetically Cohen-Macaulay with pure resolutions. More precisely, let $\beta_{ij}^{S_0}(R):=\dim_k\Tor_i^{S_0}(R, k)_{i+j}$ be the $(i,j)$-th graded Betti number and let $d_i:=\max\{i+j: \beta_{ij}^{S_0}(R)\not=0\}$ for $i=1, 2, \ldots, e$. We have a degree upper bound for projective subschemes, not necessarily arithmetically Cohen-Macaulay, namely,
$$\deg(X)\leq \frac{1}{e!}\prod_{i=1}^ed_i,$$
where equality holds if and only if $R$ is Cohen-Macaulay with a pure resolution.
Along this line, the reduction number also provides an upper bound for the degree of a projective subscheme $X\subset \bP^{n+e}$. Actually, in this paper we prove the following theorem.
\begin{theorem}\label{31}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and reduction number $r$. Then
\begin{equation}\label{eq3}
\deg(X)\leq \binom{e+r}{r}.
\end{equation}
Furthermore, $\deg(X)=\binom{e+r}{r}$ if and only if $X$ is arithmetically Cohen-Macaulay with an $(r+1)$-linear minimal free resolution.
\end{theorem}
The second part of Theorem \ref{31} has been proven by Ahn-Kwak \cite{AK15} under the assumption that $X$ is an algebraic set satisfying the $N_{3,e}$-property and $k$ is of characteristic zero. In the present paper, in place of $N_{d,e}$-property we use the reduction number to get a better degree upper bound and give a short proof for the characterization of closed subschemes of maximal degree.
If a projective closed subscheme satisfies the equivalent conditions in Theorem \ref{31}, i.e., $\deg(X)=\binom{e+r}{r}$, we say that it has a {\it maximal degree}. Combining Theorem \ref{31} and its proof with the description of the Betti table of Cohen-Macaulay graded $k$-algebra with linear resolution due to Eisenbud-Goto \cite[Proposition 1.7]{EG84}, we obtain the following consequence.
\begin{corollary}\label{32}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$, codimension $e$ and reduction number $r$. Let $I_X\subset S_0$ be the defining ideal of $X$ and $R=S_0/I_X$. The following statements are equivalent:
\begin{enumerate}[(a)]
\item $X$ has a maximal degree;
\item Suppose the natural homomorphism $S=S_e=k[x_e, \ldots, x_{n+e}] \rightarrow R$ is a Noether normalization with the reduction number $r_S(R)=r$. Then the initial ideal $\ini(I_X)$ with respect to the degree reverse lexicographic order is generated by the set of all monomials in $x_0, x_1, \ldots, x_{e-1}$ of degree $r+1$;
\item The graded Betti numbers of $X$ are
$$\beta_{ij}(X):=\beta_{ij}(R)=\begin{cases}
1 &\mbox{ if } i=j=0,\\
\binom{e+r}{i+r}\binom{i-1+r}{r}&\mbox{ if } j=r, 1\le i\le e,\\
0 &\mbox{ otherwise.}
\end{cases}
$$
\end{enumerate}
\end{corollary}
We then investigate the `almost maximal' cases with $\deg(X)=\binom{e+r(X)}{r(X)}-1$. To begin with, we have the following refined inequality
\begin{equation}\label{eq4}
\deg(X)\leq \mu_S(R)\leq \binom{e+r(X)}{r(X)},
\end{equation}
where $S\hookrightarrow R$ is a Noether normalization which determines the reduction number $r(X)$, and $\mu_S(R)$ is the number of minimal generators of $R$ as an $S$-module. The almost maximal cases consist of arithmetically Cohen-Macaulay subschemes with $\mu_S(R)=\binom{e+r(X)}{r(X)}-1$ and non-arithmetically Cohen-Macaulay subschemes with $\mu_S(R)=\binom{e+r(X)}{r(X)}$. In this paper, we characterize both two cases by describing the structures of the initial ideals with respect to the degree reverse lexicographic order and the syzygies of the Noether normalization $S\hookrightarrow R$ (see Theorem \ref{41} and Corollary \ref{52}).
As an application, we obtain explicit Betti tables of those subschemes in each case by analyzing the Betti tables of the initial ideal of $I_X$, then applying the graded mapping cone sequence (Theorem \ref{26}) and the Cancellation Principle due to M. Green \cite[Corollary 1.21]{Gre98} (see also \cite[Section 3.3]{HH11}). In the first case, we note that a projective subscheme $X$ is arithmetically Cohen-Macaulay if and only if $R$ is a free $S$-module, i.e., no syzygy among generators of $R$ over $S$ (see Corollary \ref{27}). The Betti table of an arithmetically Cohen-Macaulay subscheme of almost maximal degree is obtained in the following theorem.
\begin{theorem}\label{45}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and reduction number $r$. Assume that $X$ is arithmetically Cohen-Macaulay with $\deg(X)=\binom{e+r}{r}-1$. The Betti table of the homogeneous coordinate ring of $X$ is (in the following table, we write only rows with some possibly non-zero entries)
\newpage
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& $\cdots$& $e-1$& $e$\\
\hline
0 & 1& --& --& $\cdots$& --& --\\
r-1 & --& $1$& --&$\cdots$& -- &--\\
r & --& $\beta_{1r}$& $\beta_{2r}$&$\cdots$& $\beta_{e-1,r}$ &$\beta_{e,r}$\\
\end{tabular}
\end{figure}
\noindent where for $i=1, \ldots, e$,
$$\beta_{i,r}=\binom{e+r}{i+r}\binom{r+i-1}{r}-\binom{e}{i}.$$
\end{theorem}
The case of non-arithmetically Cohen-Macaulay subschemes is more complicated as these subschemes might have big projective dimension. We restrict to the smaller category of projective varieties, i.e., reduced, irreducible projective subschemes and show that if $X$ is a projective variety of almost maximal degree then $\depth(R)\geq \dim(X)$ (Theorem \ref{51}). The assumption of being projective variety is actually crucial. Using this result on arithmetic depth, we are able to describe an initial ideal of $I_X$ and as in the previous cases, we obtain explicit Betti tables for varieties of almost maximal degree as in the following theorem.
\begin{theorem}\label{54}
Let $X\subset \bP^{n+e}$ be a non-degenerate projective variety of dimension $n$, codimension $e$ and reduction number $r$. Let $R$ be the homogeneous coordinate ring of $X$. Suppose $\deg(X)= \binom{e+r}{r}-1$ and $X$ is not arithmetically Cohen-Macaulay. Then the Betti table (over $S_0$) of $R$ has one of the following shapes (in the following tables, we write only rows with some possibly non-zero entries):
\begin{enumerate}
\item[(a)] $\reg(R)=r$:
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{1.5cm}|>{\centering}m{1cm} >{\centering}m{1cm}>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& $0$ & $1$& $\ldots$ & $i$& $\ldots$ &$e$& $e+1$\\
\hline
$0$ & $1$ & --& $\ldots$ & --& $\ldots$ &--& --\\
$r$ & --& $\beta_{1,r}$ & $\ldots$ & $\beta_{i,r}$ & $\ldots$ &$\beta_{e,r}$& $1$\\
\end{tabular}
\end{figure}
\noindent where for $1\leq i\leq e+1$,
$$\beta_{i,r}=\binom{e+r}{i+r}\binom{r+i-1}{r}+\binom{e}{i-1}.$$
\item[(b)] $\reg(R)=r+1$:
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{1.5cm}|>{\centering}m{1cm} >{\centering}m{1cm}>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& $0$ & $1$& $\ldots$ & $i$& $\ldots$ &$e$& $e+1$\\
\hline
$0$ & $1$ & --& $\ldots$ & --& $\ldots$ &--& --\\
$r$ & --& $\beta_{1,r}$ & $\ldots$ & $\beta_{i,r}$ & $\ldots$ &$\beta_{e,r}$& --\\
$r+1$ & --& $\beta_{1,r+1}$ & $\ldots$ & $\beta_{i,r+1}$ & $\ldots$ &$\beta_{e,r+1}$& $1$\\
\end{tabular}
\end{figure}
\noindent where for $1\leq i\leq e+1$,
$$\beta_{i, r}-\beta_{i-1,r+1}=\binom{e+r}{i+r}\binom{r+i-1}{r}-\binom{e}{i-2}.$$
\item[(c)] $\reg(R)>r+1$:
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{1.5cm}|>{\centering}m{1cm} >{\centering}m{1cm}>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& $0$ & $1$& $\ldots$ & $i$& $\ldots$ &$e$& $e+1$\\
\hline
$0$ & $1$ & --& $\ldots$ &--& $\ldots$ & --& --\\
$r$ & --& $\beta_{1r}$ & $\ldots$ &$\beta_{ir}$ & $\ldots$ & $\beta_{e,r}$& --\\
$\reg(R)$ & --& $\binom{e}{0}$ & $\ldots$ &$\binom{e}{i-1}$ & $\ldots$ & $\binom{e}{e-1}$& $\binom{e}{e}$\\
\end{tabular}
\end{figure}
\noindent where for $1\leq i\leq e+1$,
$$\beta_{i,r}=\binom{e+r}{i+r}\binom{i+r-1}{r},$$
$$\beta_{i,\reg(R)}=\binom{e}{i-1}.$$
\end{enumerate}
\end{theorem}
We also provide interesting examples with all possible Betti tables to illustrate our main results.
About the structure of this paper, we prove Theorem \ref{31} on degree upper bound with a characterization of the maximal cases and give further consequences of this theorem in Section 3. Projective subschemes of almost maximal degree are studied in the next two sections. Theorem \ref{45} on Betti tables of arithmetically Cohen-Macaulay subschemes of almost maximal degree is proved in Section 4. In Section 5 we focus on projective varieties of almost maximal degree which are not arithmetically Cohen-Macaulay. Theorem \ref{54} is proved in this section.
In this paper, the Betti tables are computed by using Macaulay 2 \cite{GS}.
\medskip
\noindent{\bf Acknowledgments.} The authors thank the anonymous referee for careful reading and many useful comments which help to improve the presentation of the paper. The first author thanks
\fontencoding{T5}\selectfont Nguy\~\ecircumflex n \DJ\abreve ng H\d\ohorn p
for useful discussion on Macaulay 2 and graded Betti numbers of monomial ideals, in particular, for suggesting the short proof of Lemma \ref{44}. This work has been done during the visit of \fontencoding{T5}\selectfont \DJ o\`an Trung C\uhorn{}\`\ohorn ng to the Korea Advanced Institute of Science and Technology (KAIST). He thanks Professor Sijong Kwak and KAIST for support and hospitality during his visit. The second author would like to thank KAIST Grand Challenge 30 Project for financial support which has been the basis of his research activities.
\section{Preliminaries}
\subsection{Reduction number}
Throughout this paper, $k$ is an infinite field. Let $S_i=k[x_i, \ldots, x_{n+e}]$ be the polynomial ring over $k$, for $i=0, 1, \ldots, n+e$. Let $I\subset S_0$ be a homogeneous ideal of codimension $e$ and put $R=S_0/I$. The irrelevant homogeneous ideal of $R$ is denoted by $R_+$. Let $J$ be a minimal homogeneous reduction of $R_+$. The reduction number of $R_+$ with respect to $J$ is
$$r_J(R_+)=\min\{t>0: R_+^{t+1}=JR_+^t\}=\min\{t>0: R_{t+1}=J_{t+1}\}.$$
This number is a measure for the complexity of the algebra $R$. It has deep relations with other invariants of the same type such as the Castelnuovo-Mumford regularity and the a-invariant. The latter is defined by
$$a_{n+1}(R)=\max\{t: H^{n+1}_{R_+}(R)_t\not=0\}+1.$$
N.V. Trung gave the following comparison.
\begin{proposition}{\cite[Proposition 3.2]{NVT87}} \label{21} We have $a_{n+1}(R)+n\leq r_J(R_+)\leq \reg(R)$. In addition, if $R$ is Cohen-Macaulay, then $a_{n+1}(R)+n= r_J(R_+)=\reg(R)$.
\end{proposition}
On the other hand, the ideal $J$ is minimally generated by $n+1$ linear forms and after a change of variables, we may assume that $J=(x_e, \ldots, x_{n+e})$. Let $S=S_e=k[x_e, \ldots, x_{n+e}]$. The natural homomorphism $S\rightarrow R$ is in fact an inclusion and is a Noether normalization of $R$, i.e., $S$ is a polynomial $k$-algebra and $R$ is a finitely generated $S$-module. The reduction number can be computed by the maximal degree of minimal generators of $R$ over $S$. Namely,
\begin{lemma}{\cite[Proposition 5.1.3]{Vas94}}\label{22}
Let $b_1, \ldots, b_s$ be a minimal set of homogeneous generators of $R$ as an $S$-module, then
$$r_J(R_+)=\max\{\deg(b_1), \ldots, \deg(b_s)\}.$$
\end{lemma}
The reduction number of $R$ is the least number among $r_J(R_+)$ for all minimal reduction $J$ of $R_+$ and will be denoted by $r(R)$.
Sometimes we also call $r_J(R_+)$ the reduction number of the Noether normalization $S\hookrightarrow R$ and denote it by $r_S(R)$. The finite $S$-algebra structure on $R$ is particularly interesting and should provide an effective way to understand the structure of the algebra. We have some very first properties of minimal sets of generators of $R$ over $S$.
\begin{proposition}\label{23}
The minimal number of generators of $R$ over $S$ is bounded by
$$\mu_S(R)\leq \binom{e+r_S(R)}{r_S(R)}.$$
\end{proposition}
\begin{proof}
As an $S$-module, $R$ is generated by all monomials in $x_0, \ldots, x_{e-1}$ of degree at most $r_S(R)$. There are $\binom{e+r_S(R)}{r_S(R)}$ such monomials and the inequality follows.
\end{proof}
We can describe precisely a minimal set of monomials generating $R$ considered as an $S$-module. To do this, we fix the degree reverse lexicographic order on the monomial of $S_0$. Then $R$ is minimally generated over $S$ by monomials which formulate a basis of the $k$-vector space $S_0/I+(x_e, \ldots, x_{n+e})$. These are exactly the standard monomials with respect to the ideal $I_X+(x_e, \ldots, x_{n+e})$. Recall that a monomial in $S_0$ is a standard monomial with respect to a homogeneous ideal $J$ if it is not contained in the initial ideal of $J$. We sum up in the following lemma.
\begin{lemma}\label{24}
Let $S=S_e=k[x_e, \ldots, x_{n+e}]$ and suppose that the natural map $S\rightarrow R$ is a Noether normalization of $R$. Then $R$ is minimally generated as a module over $S$ by the set
$$\{\overline{m}\in R: m\in S_0 \text{ is a monomial in $x_0, \ldots, x_{e-1}$ which is standard with respect to } I_X+(x_e, \ldots, x_{n+e})\}.$$
\end{lemma}
Related to this lemma, the following criterion for Cohen-Macaulayness is very useful in the next sections.
\begin{proposition}\label{25}
Let $I\subset S_0$ be a homogeneous ideal and $R=S_0/I$. Suppose $S=S_e=k[x_e, \ldots, x_{n+e}]\rightarrow R$ is a Noether normalization. We fix the degree reverse lexicographic order on the monomial of $S_0$. Then
$$\deg(R)\leq \mu_S(R).$$
Furthermore, the following statements are equivalent:
\begin{enumerate}[(a)]
\item $R$ is Cohen-Macaulay;
\item $\deg(R)=\mu_S(R)$;
\item The initial ideal $\ini(I)$ is minimally generated by a set of monomials in $x_0, \ldots, x_{e-1}$.
\end{enumerate}
\end{proposition}
It should be remarked that the equivalence of $(a)$ and $(c)$ is due to Bermejo-Gimenez \cite[Proposition 2.1]{BG01}. The inequality in Propoposition \ref{25} can be seen as a graded Lech's inequality between length and degree. The equivalence between $(a)$ and $(b)$ should be well-known to experts. It could be proved by localizing things with respect to the maximal homogeneous ideal, then use \cite[Theorem 17.11]{Mat86}. Belows we present another proof using initial ideal and Bermejo-Gimenez's result.
\begin{proof}
We denote the minimal monomial generators of $\ini(I)$ by
$$w_1, \ldots, w_t, u_1v_1, \ldots, u_sv_s,$$
where $w_1, \ldots, w_t, u_1, \ldots, u_s$ are monomials in $x_0, \ldots, x_{e-1}$ of positive degrees and $v_1, \ldots, v_s$ are monomials in $x_e, \ldots, x_{n+e}$, also of positive degrees. Following Lemma \ref{24}, $\mu_S(R)$ is the number of monomials in $x_0, \ldots, x_{e-1}$ not lying in the ideal $(w_1, \ldots, w_t)$. On the other hand, the only minimal prime ideal of $\ini(I)$ is $(x_0, \ldots, x_{e-1})$. Hence
$$\deg(R)=\deg(S_0/\ini(I))=\deg(S_0/(w_1, \ldots, w_t, u_1, \ldots, u_s)).$$
The latter number, as $S_0/(w_1, \ldots, w_t, u_1, \ldots, u_s)$ is Cohen-Macaulay, is the number of monomials in $x_0, \ldots, x_{e-1}$ not lying in $(w_1, \ldots, w_t, u_1, \ldots, u_s)$. Hence
$$\deg(R)\leq \mu_S(R),$$
and equality occurs if and only if $s=0$, or in other words, $\ini(I)$ is minimally generated by some monomials in $x_0, \ldots, x_{e-1}$. The latter holds true if and only if $R$ is Cohen-Macaulay due to $(a)\Leftrightarrow (c)$.
\end{proof}
\subsection{Syzygies and Betti table}
Let $M$ be a finitely generated graded $S_0$-module. The minimal free resolution of $M$ is
$$\cdots \rightarrow F_i=\bigoplus_j S_0(-i-j)^{\beta^{S_0}_{ij}}\rightarrow \cdots \rightarrow F_0=\bigoplus_j {S_0}(-j)^{\beta^{S_0}_{0,j}}\rightarrow 0,$$
where $\beta^{S_0}_{ij}=\dim_k\Tor_i^{S_0}(M, k)_{i+j}$ is the $(i, j)$-th graded Betti number. The number $\beta^{S_0}_i(M)=\mathrm{rank}_{S_0}(F_i)=\sum_{j\in \bZ}\beta_{ij}^{S_0}(M)$ is called the $i$-th Betti number of $M$. The module $M$ is $d$-regular if $\beta_{ij}^{S_0}(M)=0$ for all $i\geq 0$ and all $j> d$. The Castelnuovo-Mumford regularity of $M$ is
$$\reg(M)=\min\{d: M \text{ is } d-\text{regular}\}.$$
The Castelnuovo-Mumford regularity together with the projective dimension are the height and the width of the Betti table.
We say that $R$ has a $d$-linear resolution if $\beta_{ij}^{S_0}(R)=0$ unless $i=j=0$ or $j=d-1$ and $i\geq 1$. Obviously, if $R$ has a $d$-linear resolution then $I$ is generated by a set of forms of degree $d$.
Closely related to the regularity is the $N_{d,p}$-property defined by Eisenbud-Goto \cite{EG84} and Eisenbud-Green-Hulek-Popescu \cite{EGHP05}. We say that $M$ satisfies the $N_{d,p}$-property ($d\geq 2$) if $\beta_{i,j}^{S_0}(M)=0$ for all $i\leq p$ and $j\geq d$. In other words, $M$ satisfies the $N_{d,p}$-property if $M$ is $(d-1)$-regular up to degree $p$. Clearly $M$ satisfies the $N_{d,p}$-property for all $d>\reg(M)$ and all $p\geq 0$.
In the study of the structure of modules with $N_{d,p}$-property, a mapping cone sequence of homology groups has been used effectively. Recall the notation $S_i=k[x_i, \ldots, x_{n+e}]$, $i=1, 2, \ldots, n+e$. Through the inclusion $S_1\subset S_0$, any graded $S_0$-module $M$ is also a graded $S_1$-module. The Koszul complex $K(x_1, \ldots, x_{n+e}; M)$ fits in a short exact sequence of complexes
$$0\rightarrow K(x_1, \ldots, x_{n+e}; M) \rightarrow K(x_1, \ldots, x_{n+e}; K(x_0; M))\rightarrow K(x_1, \ldots, x_{n+e}; M[-1])[-1]\rightarrow 0,$$
where $K(x_0, M)$ is the Koszul complex of $M$ with respect to the single element $x_0$. Then there is a long exact sequence of Koszul homology
\begin{multline*}
\cdots \rightarrow H_i(x_1, \ldots, x_{n+e}; M)_{i+j}\rightarrow H_i(x_0, \ldots, x_{n+e}; M)_{i+j}\rightarrow H_{i-1}(x_1, \ldots, x_{n+e}; M)_{i+j-1}\\
\stackrel{*x_0}\longrightarrow H_{i-1}(x_1, \ldots, x_{n+e}; M)_{i+j}\rightarrow H_{i-1}(x_0, \ldots, x_{n+e}; M)_{i+j}\rightarrow \cdots
\end{multline*}
We have $H_i(x_1, \ldots, x_{n+e}; M)_{i+j}\simeq \Tor^{S_1}_i(M,k)_{i+j}$ and $H_i(x_0, \ldots, x_{n+e}; M)_{i+j}\simeq \Tor^{S_0}_i(M,k)_{i+j}$. This leads to a so-called graded mapping cone sequence obtained by Ahn-Kwak.
\begin{theorem}{\cite[Theorem 3.2]{AK11}}\label{26}
Let $M$ be a graded $S_0$-module. There is a long exact sequence
\begin{multline}\label{eq5}
\cdots \rightarrow \Tor^{S_1}_i(M,k)_{i+j}\rightarrow \Tor^{S_0}_i(M,k)_{i+j}\rightarrow \Tor^{S_1}_{i-1}(M,k)_{i+j-1}\\
\stackrel{\delta}\longrightarrow \Tor^{S_1}_{i-1}(M,k)_{i+j}\rightarrow \Tor^{S_0}_{i-1}(M,k)_{i+j}\rightarrow \cdots
\end{multline}
whose connecting homomorphisms $\delta$'s are induced from the multiplication by $x_0$.
\end{theorem}
The following consequence follows immediately from the exact sequence in the theorem.
\begin{corollary}\label{27}
\begin{enumerate}
\item[(a)] Keep the notations as in Theorem \ref{26}. Denote
$$\chi_m^{S_0}(M)=\sum_{j=0}^m(-1)^j\beta_{m-j, j}^{S_0}(M),$$
for $m\in \bZ$. Then we have an additive formula
$$\chi_m^{S_0}(M)=\chi_m^{S_1}(M)+\chi_{m-1}^{S_1}(M).$$
Furthermore,
$$\chi_m^{S_0}(M)=\sum_{j=0}^e\binom{e}{j}\chi_{m-j}^{S_e}(M).$$
\item[(b)] Fix two indexes $p>0$ and $d\geq 0$. If $\beta_{ij}^{S_1}(M)=0$ for all $i\geq p$ and $j=0,1, \ldots, d$, then $\beta_{i+1, j}^{S_0}(M)=0$ for all $i\geq p$ and $j=0,1, \ldots, d$;
\item[(c)] \cite[Corollary 2.2]{AK15} $\mathrm{proj.dim}_{S_0}(M)=\mathrm{proj.dim}_{S_t}(M)+t$;
\item[(d)] \cite[Proposition 2.3]{AK15} Fix an index $i$. If $\beta_{ij}^{S_0}(M)=0$ for all $j\geq d$, then $\beta_{i-1, j}^{S_1}(M)=0$ for all $j\geq d$. Consequently, if $M$ satisfies the $N_{d, p}$-property over $S_0$ then $M$ satisfies the $N_{d, p-t}$-property over $S_t$ for $t>0$. Furthermore, we have $\reg_{S_0}(M)=\reg_{S_t}(M)$.
\end{enumerate}
\end{corollary}
\begin{proof}
(a) and (b) are induced directly from Theorem \ref{26} and (c) is a consequence of $(b)$.
In order to prove (d), we note that if $\beta_{ij}^{S_0}(M)=0$ then from the exact sequence (\ref{eq5}) in Theorem \ref{26}, $\beta_{i-1, j}^{S_1}(M)\leq \beta_{i-1, j+1}^{S_1}(M)$. If $\beta_{ij}^{S_0}(M)=0$ for all $j\geq d$ then
$$\beta_{i-1, d}^{S_1}(M)\leq \beta_{i-1, d+1}^{S_1}(M)\leq \ldots \leq \beta_{i-1, j}^{S_1}(M)\leq \ldots $$
But $\beta_{i-1, j}^{S_1}(M)=0$ for large enough $j$, thus
$$\beta_{i-1, d}^{S_1}(M)=\beta_{i-1, d+1}^{S_1}(M)=\ldots=0.$$
\end{proof}
As an application of Corollary \ref{27}, we obtain an upper bound for the reduction number which is somehow stronger than Trung's upper bound in Proposition \ref{21}.
\begin{corollary}\label{28}
Let $I\subset S_0$ be a homogeneous ideal of codimension $e$. Suppose $R=S_0/I$ satisfies the $N_{d,e}$-property. Then
$$ r(R)<d $$
where $r(R)$ is the reduction number.
\end{corollary}
\begin{proof}
It should be mentioned that $R$ always satisfies the $N_{d,e}$-property for some $d\leq \reg(R)$. So this corollary gives a stronger upper bound than Trung's.
We may assume that the natural homomorphism $S_e=k[x_e, \ldots, x_{n+e}]\rightarrow R$ is a Noether normalization of $R$. By Corollary \ref{27}(d), $R$ satisfies the $N_{d,0}$-property as an $S_e$-module. In other words, $\beta_{0,j}^{S_e}(R)=0$ for all $j\geq d$ and hence $r(R)<d$ (see Lemma \ref{22}).
\end{proof}
$\blacksquare$ Notations and Conventions \\
\medskip
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ with the ideal sheaf $\mathcal I_X$.
Let $I_X=\oplus_{m=0}^\infty H^0(\bP^{n+e}, \mathcal I_X(m))$ be the saturated homogeneous ideal and $R=k[x_0, \ldots, x_{n+e}]/I_X$ be the homogeneous coordinate ring.
\begin{itemize}
\item The reduction number of $X$ is the same as the reduction number of $R$, i.e., $r(X)=r(R)$.
\item The $(i,j)$-th graded Betti number of $X$ is
$$\beta_{i,j}(X):=\beta_{i,j}^{S_0}(R)=\beta^{S_0}_{i-1,j+1}(I_X)$$
\item The Castelnuovo-Mumford regularity of $X$ is $\reg(X)=\reg(I_X)=\reg(R)+1$.
\item We say that $X$ satisfies the $N_{d,p}$-property if so does $R$. In other words, $X$ satisfies the $N_{d,p}$-property if $R$ is $(d-1)$-regular up to the degree $p$.
\item A projective variety is always assumed to be an irreducible and reduced projective subscheme.
\end{itemize}
Let $X\subset\bP^{n+e}$ be a projective variety of codimension $e$. We say that $X$ has the minimal degree if $\deg(X)=e+1$. The variety $X$ is a del Pezzo variety if it is arithmetically Cohen-Macaulay with $\deg(X)=e+2$ (almost minimal degree).
In the whole paper, we only consider the degree reverse lexicographic order on the monomials. The initial ideal of a homogeneous ideal $I$ with respect to this order is denoted by $\ini(I)$.
\section{Degree upper bound in terms of reduction number and the maximal cases}
The nature of the reduction number is a bound for the complexity of an algebra or the associated scheme as we can see in Proposition \ref{21} and especially Lemma \ref{22}. The upper bound for the degree in terms of the reduction number in Theorem \ref{31} provides more evidence for this observation. In this section we first prove this theorem and then give several consequences and applications.
\begin{proof}[\bf Proof of Theorem \ref{31}]
Let $I_X\subset S_0$ be the saturated homogeneous defining ideal of $X$ and $R=S_0/I_X$. Changing the variables if necessary, we may assume that $Q=(x_e, \ldots, x_{n+e})$ is a minimal reduction of the irrelevant ideal $R_+$ with the reduction number $r_Q(R)=r$. In particular, $S=S_e=k[x_e, \ldots, x_{n+e}]\hookrightarrow R$ is a Noether normalization of $R$. Let $\mu_S(R)=\dim_k(R/S_+R)$ be the minimal number of generators of $R$ over $S$. Then we have
\begin{equation}\label{eq6}
\deg(X)\leq \mu_S(R)\leq \binom{e+r}{r},
\end{equation}
due to Propositions \ref{23} and \ref{25}. This proves the inequality (\ref{eq3}) in Theorem \ref{31}.
Now, if $X$ is arithmetically Cohen-Macaulay with an $(r+1)$-linear resolution then Eisenbud-Goto \cite[Proposition 1.7]{EG84} showed that
$$\deg(X)= \binom{e+r}{r}.$$
Conversely, assume that $\deg(X)=\binom{e+r}{r}$. Then all inequalities in (\ref{eq6}) are actually equalities, i.e.,
$$\mu_S(R)=\deg(X)=\binom{e+r}{r}.$$
Then $R$ is Cohen-Macaulay by using Proposition \ref{25} again.
It remains to show that $X$ has an $(r+1)$-linear resolution. Since $R$ is Cohen-Macaulay, we have by Proposition \ref{21} that
$$\reg(R)=a_{n+1}(R)+n=r_S(R)=r.$$
Hence $\reg(I_X)=\reg(R)+1=r+1$. On the other hand, combining the fact $\mu_S(R)=\binom{e+r}{r}$ with Lemma \ref{24}, we can see that $R$ is minimally generated over $S$ by all the monomials $x_0^{\alpha_0}\ldots x_{e-1}^{\alpha_{e-1}}$ with $\alpha_0+\ldots+\alpha_{e-1}\leq r$. They are all standard monomials with respect to $I_X+(x_e, \ldots, x_{n+e})$ and the degree reverse lexicographic order on the monomials. Note also that, due to Bermejo-Gimenez (see Proposition \ref{25}), $\ini(I_X)$ has a set of generators consisting of monomials in $x_0, \ldots, x_{e-1}$. Therefore
$$\ini(I_X)=(x_0^{\alpha_0}\ldots x_{e-1}^{\alpha_{e-1}}: \alpha_0+\ldots+\alpha_{e-1}= r+1).$$
In particular, the minimal degree of generators of $I_X$ is $r+1$. Combining this with the fact that $\reg(I_X)\leq r+1$, we get that $I_X$ is generated by a set of forms of degree $r+1$, hence $R$ has an $(r+1)$-linear minimal free resolution.
\end{proof}
\begin{example}\label{32-1}({\bf{Zero dimensional schemes with maximal degree}})
Let $\Gamma \subset \bP^e$ be a non-degenerate finite scheme with the reduction number $r$. Since $\Gamma$ is arithmetically Cohen-Maculay, the reduction number $r$ is equal to $\reg(R_{\Gamma})$. Therefore, it is $r$-normal and $\deg(\Gamma)\le \binom{e+r}{r}$ (or using Theorem \ref{31}). In particular, $\deg(\Gamma)=\binom{e+r}{r}$ if and only if there is no hypersurface of degree $r$ containing $\Gamma $ if and only if $R_{\Gamma}$ has an $(r+1)$-linear resolution.
For example, consider two distinct sets of $6$ points in $\bP^2$. One is $\Gamma_1$ not being contained in a conic curve. In this case, $r(\Gamma_1)=2$ and for a Noether normalization $S\rightarrow R_{\Gamma_1}$ which defines the reduction number $r(\Gamma_1)$, we have
$$S(-2)^{\oplus 3}\oplus S(-1)^{\oplus 2} \oplus S\simeq R_{\Gamma_1}.$$
The other is $\Gamma_2$ being contained in a unique conic curve. In this case, $r(\Gamma_2)=3$ and for a Noether normalization $S\rightarrow R_{\Gamma_2}$ which defines the reduction number $r(\Gamma_2)$, we also have
$$S(-3)\oplus S(-2)^{\oplus 2}\oplus S(-1)^{\oplus 2} \oplus S\simeq R_{\Gamma_2}.$$
\end{example}
\begin{example}\label{35}
There are many examples of projective subschemes with maximal degree (see Corollary \ref{32}). For examples, the algebraic set defined by the ideal of maximal minors of a $1$-generic $d \times(e +d -1)$-matrix of linear forms is such a subscheme which is arithmetically Cohen-Macaulay with $d$-linear resolution.
\end{example}
On the other hand, the proof of Theorem \ref{31} and Corollary \ref{32} induce the following combinatorial identity which is presented here for later usage.
\begin{lemma}\label{33}
Let $e, m, r$ be non-negative integers such $e>0$ and $m\leq r+e$. We have
$$\sum_{j=m-r}^e(-1)^{j}\binom{e}{j}\binom{e+m-j-1}{e-1}=
\begin{cases}
(-1)^{m+r}\binom{e+r}{m}\binom{m-1}{r} &\mbox{ if } m>r,\\
0 &\mbox{ if } 0<m\leq r,
\end{cases}
$$
where, by convention, $\binom{a}{b}=0$ if $a<b$.
\end{lemma}
\begin{proof}
Let $I\subset S_0=k[x_0, \ldots, x_{n+e}]$ be a monomial ideal generated by all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$. Then $R=S_0/I$ is a Cohen-Macaulay $k$-algebra with an $(r+1)$-linear minimal free resolution and $\deg(R)=\binom{e+r}r$ (see Corollary \ref{32}). Moreover, $(x_e, \ldots, x_{n+e})$ is a minimal reduction of $R_+$ with the reduction number $r$. In particular, $S=S_e=k[x_e, \ldots, x_{n+e}]\subset R$ is a Noether normalization of $R$.
As an $S$-module, the Betti numbers of $R$ are
$$\beta_{ij}^S(R)=\begin{cases}
\binom{e+j-1}{j} &\mbox{ if } i=0, 0\leq j\leq r,\\
0 &\mbox{ otherwise}.
\end{cases}$$
On the other hand, the Betti numbers of the $S_0$-module $R$ are
$$\beta_{ij}^{S_0}(R)=\begin{cases}
1&\mbox{ if } i=j=0,\\
\binom{e+r}{i+r}\binom{i+r-1}{r} &\mbox{ if } 0\leq i\leq e, j=r,\\
0 &\mbox{ otherwise}.
\end{cases}$$
due to \cite[Proposition 1.7]{EG84}.
Following Corollary \ref{27}, the Betti numbers of $R$ over $S$ and over $S_0$ are related by the equality
$$\chi_m^{S_0}(R)=\sum_{j=0}^e\binom{e}{j}\chi_{m-j}^{S}(R),$$
where
$$\chi_m^{S_i}(R)=\sum_{j=0}^m(-1)^j\beta_{m-j, j}^{S_i}(R),$$
for $m\in \bZ$. From the Betti table of $R$ as an $S$-module above, we have
$$\chi_{m-j}^{S}(R)=(-1)^{m-j}\beta_{0,m-j}^S(R)=
\begin{cases}
(-1)^{m-j}\binom{e+m-j-1}{m-j}&\mbox{ if } j\geq m-r,\\
0&\mbox{ if } j<m-r.
\end{cases}
$$
Therefore, if $0<m\leq r$ then
$$\sum_{j=m-r}^e(-1)^j\binom{e}{j}\binom{e+m-j-1}{e-1}=0,$$
and if $r< m\leq r+e$,
$$\sum_{j=m-r}^e(-1)^j\binom{e}{j}\binom{e+m-j-1}{e-1}=(-1)^{m+r}\binom{e+r}{m}\binom{m-1}{r}.$$
\end{proof}
In general, the reduction number is smaller than the regularity index $d$ in the $N_{d,e}$-property. The following example shows that the reduction number is $1$ and the regularity index can be arbitrarily large.
\begin{example}(due to B. Ulrich)\label{34}
Let $S_0=k[x_0, x_1, x_2, x_3]$. Consider the projective non-reduced line defined by the ideal $I=((x_0, x_1)^2, x_0x_2^t+x_1x_3^t)$ for some $t\geq 2$. Let $R=S_0/I$. Then $(x_2, x_3)$ is a minimal reduction of $R_+$ and $S=k[x_2, x_3]\hookrightarrow R$ is a Noether normalization. Since $R=S+Sx_0+Sx_1$, the reduction number $r(R)=1$.
On the other hand, the Betti table of $R$ as an $S_0$-module is given by
\newpage
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{1.5cm}|>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& $3$\\
\hline
0 & 1& --& --& --\\
1 & --& $3$& $2$& --\\
2 & --& --& --& --\\
$\vdots$ & $\vdots$& $\vdots$& $\vdots$& $\vdots$\\
$t-1$ & --& --& --& --\\
$t$ & --& $1$& $2$& $1$\\
\end{tabular}
\newline
where the symbol $-$ means that the corresponding Betti number is zero.
\end{figure}
\noindent So $\reg(R)=t$, $\mathrm{proj.dim}_{S_0}(R)=3$, $\depth(R)=1$. The Hilbert polynomial of $R$ is $P_R(n)=2n+t-1$ and thus $\deg(R)=2$.
\end{example}
In the second part of this section, we will apply Theorem \ref{31} and its consequences to study projective varieties with small reduction number.
Recall that a projective variety $X\subset\bP^{n+e}$ of codimension $e$ has a minimal degree if $\deg(X)=e+1$.
\begin{corollary}\label{36}
Let $X\subset \bP^{n+e}$ be a non-degenerate projective variety of dimension $n$ and codimension $e$. The following statements are equivalent:
\begin{enumerate}
\item[(a)] $X$ has a minimal degree;
\item[(b)] The reduction number $r(X)=1$;
\item[(c)] The Castelnuovo-Mumford regularity $\reg(X)=2$.
\end{enumerate}
\end{corollary}
\begin{proof}
Let $X$ be a variety with reduction number $r(X)=1$. The degree of $X$ is bounded above by $e+1$ due to Proposition \ref{23}. Hence $\deg(X)=e+1$ and $X$ has a minimal degree. Furthermore, $\reg(X)=2$ by \cite{EG84}. This shows $(b)\Rightarrow (a)$ and $(a)\Rightarrow (c)$.
On the other hand, if $\reg(X)=2$ then $\reg(R)=1$, where $R$ is the homogeneous coordinate ring of $X$. So the implication $(c)\Rightarrow (b)$ follows immediately from Proposition \ref{21}.
\end{proof}
The case of projective closed subschemes of reduction number $2$ is much more complicated. The del Pezzo varieties are interesting examples of such subschemes. Recall that a projective variety $X\subset\bP^{n+e}$ is a del Pezzo variety if it is arithmetically Cohen-Macaulay with $\deg(X)=\mathrm{codim}(X)+2$ (i.e. almost minimal degree).
\begin{example}\label{37}
Let $X\subset \bP^{n+e}$ be a del Pezzo variety of dimension $n$. The Betti table of the homogeneous coordinate ring of $X$ is
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& $\cdots$& $e-1$& $e$\\
\hline
0 & 1& --& --& $\cdots$& --& --\\
1 & --& $\beta_{11}$& $\beta_{21}$&$\cdots$& $\beta_{e-1,1}$ &--\\
2 & --& --& --&$\cdots$& -- &$\beta_{e,2}=1$\\
\end{tabular}
\end{figure}
\noindent where $\beta_{i1}=i\binom{e+1}{i+1}-\binom{e}{i-1}$ (see \cite{Hoa93} and also the next proposition).
So $X$ satisfies the $N_{3,e}$-property but not the $N_{2, e}$-property. The reduction number of $X$ is at most $2$ by Corollary \ref{28}. On the other hand, as a del Pezzo variety does not have a minimal degree, its reduction number is at least $2$ by Corollary \ref{36}. This concludes that $r(X)=2$.
\end{example}
The next proposition provides information on Betti table of any arithmetically Cohen-Macaulay subscheme of reduction number $2$.
\begin{proposition}\label{38}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$. Assume that $X$ is arithmetically Cohen-Macaulay with reduction number $r(X)=2$. Then the Betti table of the homogeneous coordinate ring of $X$ (over $S_0$) is
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& $\cdots$& $e-1$& $e$\\
\hline
0 & 1& --& --& $\cdots$& --& --\\
1 & --& $\beta_{11}$& $\beta_{21}$&$\cdots$& $\beta_{e-1,1}$ &$\beta_{e,1}$\\
2 & --& $\beta_{12}$& $\beta_{22}$&$\cdots$& $\beta_{e-1,2}$ &$\beta_{e,2}$\\
\end{tabular}
\end{figure}
\noindent where $$\beta_{i,2}=\beta_{i+1,1}+\binom{e}{i}\deg(X)-(i+1)\binom{e+2}{i+2}.$$
In particular,
$$\deg(X)=e+1+\beta_{e,2}.$$
\end{proposition}
\begin{proof}
Let $R$ be the homogeneous coordinate ring of $X$. By changing the variables, we might assume that $S=k[x_e, \ldots, x_{n+e}]$ is a Noether normalization of $R$ and reduction number $r_S(R)=r(R)=2$.
By the assumption, the Castelnuovo-Mumford regularity of $R$ is $2$ and almost all Betti numbers of $R$ over $S_0$ vanish except $\beta_{00}^{S_0}(R)=1$ and $\beta_{ij}^{S_0}(R)$ for $i=1, 2, \ldots, e$ and $j=1, 2$. Corollary \ref{27}(d) then implies that $R$ as an $S$-module has the Betti numbers $\beta_{ij}^{S}(R)=0$ except $\beta_{00}^S(R)=1$, $\beta_{01}^S(R)=e$ and
$$\beta_{02}^S(R)=\deg(X)-\beta_{00}^S(R)-\beta_{01}^S(R)=\deg(X)-e-1.$$
Now using the notations and results in Corollary \ref{27}(a), we have
$$\chi_m^{S_0}(R)=\sum_{j=0}^m(-1)^j\beta_{m-j, j}^{S_0}(R)=\beta_{m-2, 2}^{S_0}(R)-\beta_{m-1,1}^{S_0}(R),$$
$$\chi_{m}^S(R)=\sum_{j=0}^{m}(-1)^j\beta_{m-j, j}^S(R)=(-1)^m\beta_{0,m}(R),$$
and
$$\beta_{i, 2}^{S_0}(R)-\beta_{i+1,1}^{S_0}(R)=\binom{e}{i+2}-\binom{e}{i+1}\beta_{01}^S(R)+\binom{e}{i}\beta_{02}^{S}(R).$$
Let $\beta_{ij}:=\beta_{ij}^{S_0}(R)$, then we obtain
\[\begin{aligned}
\beta_{i2}&=\beta_{i+1,1}+\binom{e}{i+2}-\binom{e}{i+1}e+\binom{e}{i}(\deg(X)-e-1)\\
&=\beta_{i+1,1}+\binom{e}{i}\deg(X)-(i+1)\binom{e+2}{i+2}.
\end{aligned}\]
\end{proof}
\section{The arithmetically Cohen-Macaulay subschemes with almost maximal degree}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and the reduction number $r$. The degree of $X$ is always bounded above by $\binom{e+r}{r}$ and we have seen in Theorem \ref{31} a characterization for those subschemes whose degree attains the maximal value. In the next two sections we will investigate the almost maximal cases, namely, when
$$\deg(X)= \binom{e+r}{r}-1.$$
We keep the notations $S_i, I_X, R=S_0/I_X$ as in the previous section. Changing the variables if necessary, we assume that $J=(x_e, \ldots, x_{n+e})$ is a minimal reduction of $(R)_+$ with reduction number $r_J((R)_+)=r(R)=r$. Then $S=S_e=k[x_e, \ldots, x_{n+e}]$ is a Noether normalization of $R$. By Propositions \ref{23} and \ref{25}, there are bounds for the minimal number of generators of $R$ as an $S$-module
\begin{equation}\label{eq41}
\deg(X)\leq \mu_S(R)\leq \binom{e+r}{r}.
\end{equation}
If $\deg(X)=\binom{e+r}{r}-1$ then there are only two possibilities for $\mu_S(R)$, namely,
\begin{enumerate}
\item[1)] $\mu_S(R)= \binom{e+r}{r}-1$: the arithmetically Cohen-Macaulay case (see Proposition \ref{25});
\item[2)] $\mu_S(R)= \binom{e+r}{r}$: the non-arithmetically Cohen-Macaulay case.
\end{enumerate}
This section is devoted to study the arithmetically Cohen-Macaulay case with a proof of Theorem \ref{45}, while the latter will be considered in the next section.
Arithmetically Cohen-Macaulay projective subschemes of almost maximal degree are characterized in the following theorem.
\begin{theorem}\label{41}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and reduction number $r$. Let $I_X\subset S_0$ be the defining ideal of $X$ and $R=S_0/I_X$. Assume that the natural homomorphism $S=S_e=k[x_0, \ldots, x_{e-1}]\rightarrow R$ is a Noether normalization with the reduction number $r=r_S(R)$. The following statements are equivalent:
\begin{enumerate}
\item[(a)] $\deg(X)=\mu_S(R)= \binom{e+r}{r}-1$;
\item[(b)] $X$ is arithmetically Cohen-Macaulay, $\dim_k(I_X)_r=1$ and the truncated ideal $(I_X)_{\ge r+1}$ has a linear minimal resolution;
\item[(c)] The initial ideal $\ini(I_X)$ with respect to the degree reverse lexicographic order is generated by a set of monomials in $x_0, x_1, \ldots, x_{e-1}$ consisting of all monomials of degree $r+1$ and a monomial of degree $r$.
\end{enumerate}
\end{theorem}
\begin{proof}
\noindent $(a) \Rightarrow (b)$: We have $\deg(X)=\mu_S(R)$ and hence $R$ is Cohen-Macaulay by Proposition \ref{25}. Then the Castelnuovo-Mumford regularity of $R$ is the same as the reduction number $r$ (see Proposition \ref{21}). This shows that $(I_X)_{>r}$ has a linear resolution.
Again, we fix the degree reverse lexicographic order on the monomials of $S_0$. There are totally $\mu_S(R)=\binom{e+r}{r}-1$ standard monomials with respect to $\ini(I_X)+(x_e, \ldots, x_{n+e})$ and they all have degrees from $0$ to $r$. This shows that there is a monomial $m$ in $x_0, \ldots, x_{e-1}$ with $\deg(m)\leq r$ which are not contained in the ideal $\ini(I_X)$. Then clearly $\deg(m)=r$ and $(I_X)_r=kg$ for some polynomial $g\in I(X)$ with $\ini(g)=m$.
\medskip
\noindent $(b)\Rightarrow (a)$: Since $X$ is arithmetically Cohen-Macaulay, we have by Propositions \ref{23} and \ref{25},
$$\deg(X)=\mu_S(R)\leq \binom{e+r}{r}.$$
The fact $\dim_k(I_X)_r=1$ implies that the $k$-vector space of degree $r$ component of the initial ideal of $I_X$ has dimension one too. Hence $\mu_S(R)\geq \binom{e+r}{r}-1$ due to Lemma \ref{24}. If $\deg(X)=\mu_S(R)=\binom{e+r}{r}$ then $I_X$ has an $(r+1)$-linear resolution by Theorem \ref{31}. This is impossible because $(I_X)_r\not=0$. Therefore
$$\deg(X)=\mu_S(R)=\binom{e+r}{r}-1.$$
\medskip
\noindent $(b)\Rightarrow (c)$: Suppose that $R$ is Cohen-Macaulay. Due to Bermejo-Gimenez (see Proposition \ref{25}(c)), the initial ideal $\ini(I_X)$ is minimally generated by a set $\mathcal B$ of monomials in $x_0, \ldots, x_{e-1}$. Moreover, the generating set $\mathcal B$ contains a monomial $m$ of degree $r$ since $\dim_k(I_X)_r=1$.
On the other hand, as we have seen in the proof of $(b)\Rightarrow (a)$, the $S$-module $R$ is generated by all monomials in $x_0, \ldots, x_{e-1}$ of degrees at most $r$ except the monomial $m$. As these are all the standard monomials with respect to $\ini(I_X)+(x_e, \ldots, x_d)$, the set $\mathcal B$ contains all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ which are not multiples of $m$. This proves $(c)$.
\medskip
\noindent $(c)\Rightarrow (b)$: Again by Proposition \ref{25}, the $k$-algebra $R$ is Cohen-Macaulay. Hence $\reg(R)=r$ and $(I_X)_{>r}$ has a linear resolution. The fact $\dim_k(I_X)_r=1$ is clear from the assumption.
\end{proof}
In the next we give some examples of arithmetically Cohen-Macaulay projective subschemes of almost maximal degree.
\begin{example}\label{41-1}({\bf Finite points with almost maximal degree})
Let $\Gamma \subset \bP^e$ be a non-degenerate set of finite points with reduction number $r$. In this case, the reduction number $r(\Gamma)=\reg(R_{\Gamma})$ (see Proposition \ref{21}). Therefore, it is $r$-normal and in particular, $\deg(\Gamma)=\binom{e+r}{r}-1$ if and only if there is only one hypersurface of degree $r$ containing $\Gamma $ and $\reg(R_{\Gamma})=r$ (cf. Example \ref{32-1}).
In this case, $\Gamma$ has the following Betti numbers
$$\beta_{i,r}=\binom{e+r}{i+r}\binom{r+i-1}{r}-\binom{e}{i},$$
for $i=1, \ldots, e$ (cf. Theorem \ref{45}).
\end{example}
\begin{example}\label{42}
Interesting examples of arithmetically Cohen-Macaulay closed subschemes of $\bP^{n+e}$ with almost maximal degree consist of Castelnuovo surface, Castelnuovo $3$-fold and their higher dimensional analogues. Let $X$ be the Castelnuovo surface in $\bP^4$. It is smooth and arithmetically Cohen-Macaulay of degree $5$. The defining ideal of $X$ is generated by a quadric and two cubics. The $S_0$-minimal free resolution of the homogeneous coordinate ring $R$ is
$$0\rightarrow S_0(-4)^2\rightarrow S_0(-3)^2\oplus S_0(-2) \rightarrow S_0\rightarrow 0$$ with the Betti table
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2\\
\hline
0 & 1& --& --\\
1 & --& 1& --\\
2 & --& 2& 2
\end{tabular}
\end{figure}
Assume that $S_2=k[x_2,x_3, x_4]$ is a Noether normalization of $R$. Then as an $S_2$-module,
$$R\simeq S_2\oplus S_2(-1)^2\oplus S_2(-2)^2.$$
In particular, $X$ has an almost maximal degree $\binom{2+2}{2}-1$ and $r(X)=2$.
Let $C$ be a generic hyperplane section of $X\subset \bP^4$ which is a smooth curve of degree $5$ and genus $2$
in $\bP^3$. Then, $C$ is also arithmetically Cohen-Macaulay with $r(C)=2$ (see \cite[Example 6.4.2]{Hart77}).
The Betti table of its homogeneous coordinate ring $R_C$ is the same as that of $R$.
\end{example}
If we do not require the subschemes to be reduced and irreducible, examples are found easily. The following example comes from Theorem \ref{41}.
\begin{example}\label{43}
Let $S_0=k[x_0, \ldots, x_{n+e}]$. Let $J=(x_0, \ldots, x_{e-1})$ and $u\in J$ be a monomial of degree $r$. Put $I=(u)+J^{r+1}$. Then $V(I)\subset \bP^{n+e}$ is a subscheme of almost maximal degree.
\end{example}
In the second half of this section, we are going to prove Theorem \ref{45} which give the explicite Betti tables of all arithmetically Cohen-Macaulay subschemes of $\bP^{n+e}$ of almost maximal degree. We first compute the Betti table of the subscheme in Example \ref{43}.
\begin{lemma}\label{44}
Let $S_0$, $u$, $J$, $I$ be as in Example \ref{43}. The Betti numbers of the ideal $I$ are
$$\beta_{ij}^{S_0}(I)=\begin{cases}
1 &\mbox{ if } i=0, j=r,\\
\binom{e+r}{i+r+1}\binom{r+i}{r}-\binom{e}{i+1} &\mbox{ if } 0\leq i\leq e-1, j=r+1,\\
0 &\mbox{ otherwise.}
\end{cases}
$$
\end{lemma}
\begin{proof}
We denote $K=J^{r+1}$. Then $(u)\cap K=uJ$. From the short exact sequence
$$0\rightarrow (u)\rightarrow I\rightarrow I/(u)\rightarrow 0,$$
and the fact that $I/(u)\simeq (u)+K/(u)\simeq K/(u)\cap K=K/uJ$ and $(u)\simeq S_0[-r]$, we obtain a short exact sequence
$$0\rightarrow S_0[-r]\rightarrow I\rightarrow K/uJ\rightarrow 0.$$
This gives rise to a long exact sequence
$$\ldots \rightarrow \Tor_{i+1}^{S_0}(k, K/uJ)_{i+j} \rightarrow \Tor_i^{S_0}(k, S_0)_{i+j-r} \rightarrow \Tor_i^{S_0}(k, I)_{i+j} \rightarrow \Tor_i^{S_0}(k, K/uJ)_{i+j} \rightarrow \ldots$$
Since $S_0[-r]$ has the regularity $r$ and $I$ has the regularity $r+1$, the regularity of $K/uJ$ is $r+1$. This together with the fact $K/uJ$ being generated by degree $(r+1)$-elements imply that $K/uJ$ has an $(r+1)$-linear minimal free resolution.
Now we show that the homomorphism $\Tor_{i+1}^{S_0}(k, K/uJ)_{i+j} \rightarrow \Tor_i^{S_0}(k, S_0)_{i+j-r}$ in the long exact sequence above is actually zero. Indeed, if $j\not= r+2$ then $\Tor_{i+1}^{S_0}(k, K/uJ)_{i+j}=0$. If $j=r+2$ then $\Tor_i^{S_0}(k, S_0)_{i+j-r}=0$. This proves the claim. Consequently, we have a short exact sequence
$$0 \rightarrow \Tor_i^{S_0}(k, S_0)_{i+j-r} \rightarrow \Tor_i^{S_0}(k, I)_{i+j} \rightarrow \Tor_i^{S_0}(k, K/uJ)_{i+j} \rightarrow 0,$$
for all $i, j$. This particularly implies that
$$\beta_{ij}^{S_0}(I)=\beta_{i, j-r}^{S_0}(S_0)+\beta_{ij}^{S_0}(K/uJ).$$
In order to compute the Betti numbers of $K/uJ$, we use the short exact sequence
$$0\rightarrow J[-r]\stackrel{*u}{\longrightarrow} K\rightarrow K/uJ\rightarrow 0.$$
Again, there is a long exact sequence
$$\ldots \rightarrow \Tor_{i+1}^{S_0}(k, K/uJ)_{i+j} \rightarrow \Tor_i^{S_0}(k, J)_{i+j-r} \rightarrow \Tor_i^{S_0}(k, K)_{i+j} \rightarrow \Tor_i^{S_0}(k, K/uJ)_{i+j} \rightarrow \ldots$$
Note that all $J[-r]$, $K$ and $K/uJ$ have $(r+1)$-linear resolutions. By analogous argument as in the first part of the proof, we obtain a short exact sequence
$$0 \rightarrow \Tor_i^{S_0}(k, J)_{i+j-r} \rightarrow \Tor_i^{S_0}(k, K)_{i+j} \rightarrow \Tor_i^{S_0}(k, K/uJ)_{i+j} \rightarrow 0,$$
for all $i, j$. Therefore
$$\beta_{ij}^{S_0}(I)=\beta_{i, j-r}^{S_0}(S_0)+\beta_{ij}^{S_0}(K)-\beta_{i,j-r}^{S_0}(J).$$
Now, using Corollary \ref{32}, finally we get an explicit formula for the Betti numbers of $I$, namely,
$$\beta_{ij}^{S_0}(I)=\begin{cases}
1 &\mbox{ if } i=0, j=r,\\
\binom{e+r}{i+r+1}\binom{r+i}{r}-\binom{e}{i+1} &\mbox{ if } 0\leq i\leq e-1, j=r+1,\\
0 &\mbox{ otherwise.}
\end{cases}
$$
\end{proof}
Lemma \ref{44} does not only give the explicit Betti table for a particular subscheme of almost maximal degree but is also very useful when we compute the Betti table for the general case. Now we use this lemma to prove Theorem \ref{45}.
\begin{proof}[\bf Proof of Theorem \ref{45}]
Let $R=S_0/I_X$ be the homogeneous coordinate ring of $X$. We assume that $S=k[x_e, \ldots, x_{n+e}]\subset R$ is a Noether normalization of $R$ with the reduction number $r_S(R)=r$. We consider $R$ as an $S_0$-module and as an $S$-module.
As an $S$-module, the Betti number of $R$ is
$$\beta_{ij}^S(R)=\begin{cases}
\binom{e+j-1}{j} &\mbox{ if } i=0, 0\leq j\leq r-1,\\
\binom{e+r-1}{r}-1 &\mbox{ if } i=0, j=r,\\
0 &\mbox{ otherwise.}
\end{cases}$$
Thus
$$\chi_m^S(R)=\sum_{j=0}^m(-1)^j\beta_{m-j, j}^S(R)=(-1)^m\beta_{0,m}^S(R)=
\begin{cases}
(-1)^m\binom{e+m-1}{m} &\mbox{ if } 0\leq m\leq r-1,\\
(-1)^m\binom{e+r-1}{r}-1 &\mbox{ if } m=r,\\
0 &\mbox{ otherwise.}
\end{cases}$$
Using Corollary \ref{27}(a) which relates the Betti numbers of $R$ over $S$ with those over $S_0$, we are able to find a nice relation between Betti numbers of $R$ over $S_0$, namely,
\[\begin{aligned}
\chi_{m}^{S_0}(R)
&=\sum_{j=0}^e\binom{e}{j}\chi_{m-j}^{S_e}(R)\\
&=\sum_{j=m-r}^e(-1)^{m-j}\binom{e}{j}\binom{e+m-j-1}{m-j}-(-1)^r\binom{e}{m-r}\\
\end{aligned}\]
On the other hand, by Theorem \ref{41}, the initial ideal $\ini(I_X)$ with respect to the degree reverse lexicographic order is generated by all monomials in $x_0, x_1, \ldots, x_{e-1}$ of degree $r+1$ and a monomial in $x_0, x_1, \ldots, x_{e-1}$ of degree $r$. Lemma \ref{44} applies to $\ini(I_X)$ and we have
$$\beta_{ij}^{S_0}(\ini(I_X))=0,$$
for either $j\not=r, r+1$, or $j=r, i>0$, or $j=r+1, i\geq e$. Now, by comparing the Betti numbers of $I_X$ and its initial ideal (see, for example, \cite[Corollary 1.21]{Gre98}), we get
$$\beta^{S_0}_{ij}(I_X)\leq \beta^{S_0}_{ij}(\ini(I_X)),$$
for any $i, j$. So
$$\beta_{ij}^{S_0}(I_X)=0,$$
for either $j\not=r, r+1$, or $j=r, i>0$, or $j=r+1, i\geq e$. This implies that
$$\chi_{i+r}^{S_0}(R)
=\sum_{j=0}^{i+r}(-1)^j\beta_{r+i-j, j}^{S_0}(R)
=(-1)^r\beta_{i, r}^{S_0}(R).$$
Combining these with Lemma \ref{33}, we therefore obtain
\[\begin{aligned}
\beta_{i, r}^{S_0}(R)
&=\sum_{j=i}^e(-1)^{i-j}\binom{e}{j}\binom{e+i+r-j-1}{i+r-j}-\binom{e}{i}\\
&=\binom{e+r}{i+r}\binom{r+i-1}{i-1}-\binom{e}{i}.
\end{aligned}\]
\end{proof}
\section{The non-arithmetically Cohen-Macaulay varieties with almost maximal degree}
In this section we consider projective subschemes with almost maximal degree which are not arithmetically Cohen-Macaulay. It is much more complicated to explore this class of subschemes than the arithmetically Cohen-Macaulay subschemes in the previous section. Fortunately, in the case of reduced and irreducible projective subschemes, we have the following key theorem.
\begin{theorem}\label{51}
Let $X\subset \bP^{n+e}$ be a non-degenerate projective variety of dimension $n$, codimension $e$ and reduction number $r$. Assume that $\deg(X)=\binom{e+r}{r}-1$. Then the homogeneous coordinate ring of $X$ has depth $\geq n$.
\end{theorem}
\begin{proof}
Let $I\subset S_0=k[x_0, \ldots, x_{n+e}]$ be the defining prime ideal of $X$ and $R=S_0/I$. Assume that $S=S_e=k[x_e, \ldots, x_{n+e}]$ is a Noether normalization of $R$ such that the reduction number $r_S(R)=r$. It is enough to prove the assertion for the case $X$ being not arithmetically Cohen-Macaulay.
The proof consists of several steps. Note that we fix the degree reverse lexicographic order on the monomials of $S_0$.
\medskip
\noindent Step 1. We show that the initial ideal $\ini(I)$ has a minimal set of generators consisting of all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ and some monomials $uv_1, \ldots, uv_s$, where $u$ is a monomial in $x_0, \ldots, x_{e-1}$ of degree $r$ and $v_1, \ldots, v_s$ are monomials in $x_e, \ldots, x_{n+e}$ of positive degree.
Denote the set of all monomials in $x_0, \ldots, x_{e-1}$ of degree $d$ by $T_d$, for $d\geq 0$.
Using Propositions \ref{23} and \ref{25}, we have $\deg(X)\leq \mu_S(R)\leq \binom{r+e}{r}$. Since $X$ is not arithmetically Cohen-Macaulay, the inequality $\deg(X)<\mu_S(R)$ is strict and thus $\mu_S(R)=\binom{e+r}{r}$. It together with Proposition \ref{23} and Lemma \ref{24} shows that the monomials in $T_0\cup T_1\cup\ldots \cup T_r$ are all monomials not being contained in the ideal $\ini(I)+(x_e, \ldots, x_{n+e})$. This also shows that $\ini(I)$ has a minimal set of generators consisting of $T_{r+1}$ and some monomials $u_1v_1, \ldots, u_sv_s$ where $u_i\in T_j$ for some $j\leq r$ and $v_i\in S_+$. Here it is worth noting that $\deg(u_i)>0$ since $S_0/I$ and $S_0/\ini(I)$ have the same Krull dimension.
We proceed to show that $u_1=\ldots=u_s$ by computing the degree of $S_0/\ini(I)$. It suffices to look at the minimal prime ideals of $\ini(I)$. Since
$$\ini(I)=(u_1, u_2v_2, \ldots, u_sv_s, T_{r+1})\cap (v_1, u_2v_2, \ldots, u_sv_s, T_{r+1}),$$
the degree of $S_0/\ini(I)$ is the same as the degree of $S_0/(u_1, u_2v_2, \ldots, u_sv_s, T_{r+1})$. Taking a similar decomposition with respect to $u_2v_2, \ldots, u_sv_s$, we get
$$\deg(S_0/\ini(I))=\deg S_0/(u_1, \ldots, u_s, T_{r+1}).$$
Note that $(u_1, \ldots, u_s, T_{r+1})$ is a monomial ideal and $S_0/(u_1, \ldots, u_s, T_{r+1})$ is a Cohen-Macaulay algebra with a maximal regular sequence $x_e, \ldots, x_{n+e}$. Then, in order to compute the degree of $S_0/(u_1, \ldots, u_s, T_{r+1})$, it suffices to count the monomials in $T_0\cup T_1\cup\ldots\cup T_r$ which are not in $(u_1, \ldots, u_s, T_{r+1})$. On the other hand,
we have totally $\binom{e+r}{r}$ monomials in the set $T_0\cup T_1\cup\ldots\cup T_r$ and
$$\deg(S_0/I)=\deg (S_0/\ini(I))=\deg(S_0/(u_1, u_2, \ldots, u_s, T_{r+1}))=\binom{e+r}{r}-1.$$
So there is only one monomial of degree at most $r$ in $(u_1, u_2, \ldots, u_s, T_{r+1})$
which is not in the ideal $\ini(I)$. This implies that $u_1=u_2=\ldots=u_s$ which is of degree $r$. Denote this monomial by $u$.
\medskip
\noindent Step 2. The aim of this step is to prove that the equivalence classes in $R$ of the monomials in $(T_0\cup T_1\cup \ldots \cup T_r)\setminus \{u\}$ are linearly independent over $S$.
\smallskip
Let's denote the monomials in the set $(T_0\cup T_1\cup \ldots \cup T_r)\setminus \{u\}$ by $u_1, \ldots, u_N$, where $N=\binom{e+r}{r}-1$. Assume that there are $f_1, \ldots, f_N\in S$ which are not all identically zero and satisfy the relation
$$f_1\bar u_1+\ldots+f_N\bar u_N=0.$$
Let $f=f_1u_1+\ldots+f_Nu_N\in I$. We can assume in addition that $f_1, \ldots, f_N$ are homogeneous polynomials such that $f$ is also homogeneous. Obviously the monomials $u_1, \ldots, u_N\in S_0$ are linearly independent over $S$, so $f\not=0$. Let $\ini(f)=\lambda m_1m_2$, where $\lambda\in k$, $\lambda\not=0$, and $m_1\in(T_0\cup T_1\cup \ldots \cup T_r)\setminus \{u\}$, $m_2\in S$. This contradicts to the fact that $\ini(f)$ lies in the ideal $\ini(I)$, the latter is minimally generated over $S$ by $T_{r+1}\cup \{uv_1, \ldots, uv_s\}$. This completes the proof for Step 2.
\medskip
\noindent Step 3. We prove that $\ini(I)$ is minimally generated by $T_{r+1}\cup \{uv_1\}$ using the fact
that $I$ is a prime ideal.
\smallskip
As we have shown in Step 1, the initial ideal $\ini(I)$ is minimally generated by the monomials in $T_{r+1}$ and some monomials $uv_1, \ldots, uv_s$, where $u\in T_r$ and $v_1, \ldots, v_s\in S_+$. For the convenience, we denote the monomials in $T_{r+1}$ by $m_1, \ldots, m_t$, where $t=\binom{e+r}{r+1}$. Let $h_1, \ldots, h_s$ be part of the reduced Gr{\"o}bner basis with $\ini(h_j)=uv_j$. Observe that the polynomials $h_1, \ldots, h_s$ are irreducible as $I$ is a prime ideal.
Since no trailing terms of any polynomial in the Gr{\"o}bner basis lie in the initial ideal $\ini(I)$, we write
$$h_i=uq_i+\sum_{j=1}^N u_jq_{ij},$$
where $q_i, q_{i1}, \ldots, q_{iN}$ are homogeneous polynomials in $S$ and $\ini(q_i)=v_i$. The irreducibility of $h_i$ implies particularly that $q_i, q_{i1}, \ldots, q_{iN}$ have no common factors of positive degree.
Assume that $s\geq 2$. Then
$$q_2h_1-q_1h_2=\sum_{i=1}^Nu_j(q_2q_{1j}-q_1q_{2j})$$
which is in the ideal $I$. By Step 2, the equivalence classes of the monomials $u_1, \ldots, u_N$ in $R$ are linearly independent over $S$. Hence $q_2q_{1j}-q_1q_{2j}=0$ for all $j=1, \ldots, N$. Since $q_i, q_{ij}\in S$ have unique irreducible factorizations and $q_i, q_{i1}, \ldots, q_{iN}$ have no common factors of positive degree for each $i=1, 2$, it implies that $q_1=\lambda q_2$ for some $\lambda\in k$. This is impossible because it implies $v_1=\ini(q_1)=\lambda\ini(q_2)=\lambda v_2$. Therefore $s=1$ and $\ini(I)$ is minimally generated by $T_{r+1}\cup \{uv_1\}$.
\medskip
\noindent Step 4. Finally, we show that $\depth(R)=n$.
\smallskip
By Step 3, the initial ideal $\ini(I)$ is minimally generated by $T_{r+1}\cup\{uv_1\}$. We have a short exact sequence
$$0\rightarrow S_0/(T_{r+1})\stackrel{*uv_1}{\longrightarrow} S_0/(T_{r+1})\rightarrow S_0/\ini(I) \rightarrow 0.$$
Since $S_0/(T_{r+1})$ is Cohen-Macaulay of dimension $n+1$, we obtain
$$\depth(S_0/\ini(I))\geq n.$$
Since $\beta^{S_0}_{ij}(S_0/I)\leq \beta^{S_0}_{ij}(S_0/\ini(I))$ for any $i, j$ (see, for example, \cite[Corollary 1.21]{Gre98}), we obtain in particular $$\mathrm{proj.dim}_{S_0}(S_0/I)\leq \mathrm{proj.dim}_{S_0}(S_0/\ini(I)).$$
The Auslander-Buchsbaum formula then implies that $\depth(S_0/I)\geq \depth(S_0/\ini(I))\geq n$. As $S_0/I$ is not Cohen-Macaulay, $\depth(S_0/I)=n$.
\end{proof}
In Theorem \ref{51}, we really use the fact that $X$ is a projective variety, i.e., a reduced and irreducible projective subscheme. We will see later in Example \ref{59} that this assumption is necessary and can not be omitted.
The following consequence follows from the proof of Theorem \ref{51}.
\begin{corollary}\label{52}
Let $X\subset \bP^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and reduction number $r$. Let $I\subset S_0$ be the defining ideal of $X$ and $R=S_0/I$. Let $S=S_e=k[x_e, \ldots, x_{n+e}]$ and suppose the natural map $S\hookrightarrow R$ is a Noether normalization of $R$ with the reduction number $r_S(R)=r$. Again, we fix the degree reverse lexicographic order on the monomials of $S_0$. The following are equivalent:
\begin{enumerate}
\item[(a)] $\deg(X)=\binom{e+r}{r}-1$ and $X$ is not arithmetically Cohen-Macaulay;
\item[(b)] The initial ideal $\ini(I)$ is generated by all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ and some monomials $uv_1, \ldots, uv_s$, where $u$ is a monomial in $x_0, \ldots, x_{e-1}$ of degree $r$ and $v_1, \ldots, v_s\in S_+$;
\end{enumerate}
If $X$ is a projective variety then the above equivalent statements are equivalent to one of the following statements:
\begin{enumerate}
\item[(c)] $\ini(I)$ is generated by all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ and a monomial $uv$, where $u$ is a monomial in $x_0, \ldots, x_{e-1}$ of degree $r$ and $v\in S_+$;
\item[(d)] $R$, as an $S$-module, has the Betti numbers
$$\beta_i^S(R)=\begin{cases}
\binom{e+r}{r} &\mbox{ if } i=0,\\
1 &\mbox{ if } i=1,\\
0 &\mbox{ if } i>1;
\end{cases}$$
\item[(e)] $R$, as an $S$-module, has the graded Betti numbers
$$\beta_{i,j}^S(R)=\begin{cases}
\binom{e+j-1}{j} &\mbox{ if } i=0, 0\leq j\leq r,\\
1 &\mbox{ if } i=1, j=\reg(R),\\
0 &\mbox{ if } i=1, j\not=\reg(R) \text{ or } i>1.
\end{cases}$$
\end{enumerate}
\end{corollary}
\begin{proof}
We have shown in Steps 1 and 2 of the proof of Theorem \ref{51} that $(a)\Rightarrow (b)$. For $(b)\Rightarrow (a)$, suppose $\ini(I)=(x_0, \ldots, x_{e-1})^{r+1}+(uv_1, \ldots, uv_s)$ as in $(b)$, then $R$ is not Cohen-Macaulay due to Proposition \ref{25}. Furthermore, we have
$$\deg(R)=\deg(S_0/\ini(I))=\deg(S_0/(x_0, \ldots, x_{e-1})^{r+1}+(u))=\binom{e+r}{r}-1.$$
So $(a)$ is equivalent to $(b)$.
Now suppose that $X$ is a projective variety. By Step 3 in the proof of Theorem \ref{51}, we have $(b)\Rightarrow (c)$. The implications $(e)\Rightarrow (d)\Rightarrow (a)$ are obvious. We will show that $(c)\Rightarrow (a)$ and $(a)\Rightarrow (e)$.
\smallskip
\noindent $(c)\Rightarrow (a)$: We denote by $T_d$ the set of the monomials in $x_0, \ldots, x_{e-1}$ of degree $d$.
Assume that $\ini(I)$ has a minimal set of generators consisting of $T_{r+1}$ and some monomials $m_i=uv_i$, $i=1, \ldots, s$, with $u\in T_r$ and $v_i\in S_+$. We have
$$\deg(R)=\deg(S_0/\ini(I))=\deg(S_0/(T_{r+1}, u))=\binom{e+r}{r}-1.$$
On the other hand, since $T_{r+1}$ is a part of a minimal set of generators of $\ini(I)$, the set of standard monomials with respect to $\ini(I)+(x_e, \ldots, x_{n+e})$ is exactly $T_0\cup T_1\cup \ldots \cup T_r$. There are $\binom{e+r}{r}$ such monomials. Using again Proposition \ref{23} and Lemma \ref{24}, we conclude that
$$\mu_S(R)=\binom{e+r}{r}>\deg(R),$$
and $R$ is not Cohen-Macaulay.
\smallskip
\noindent $(a)\Rightarrow (e)$: Suppose $X$ is a projective variety of almost maximal degree, i.e., $\deg(X)=\binom{e+r}{r}-1$, and $X$ is not arithmetically Cohen-Macaulay. We have shown in the proof of Theorem \ref{51} that $R$ is minimally generated over $S$ by all monomials in $x_0, \ldots, x_{e-1}$ of degree from $0$ to $r$. Consequently, we have
$$\sum_{j=0}^r\beta_{0,j}^S(R)=\binom{e+r}{r}.$$
On the other hand, $\beta_{0,j}^S(R)$ is bounded above by the number of monomials in $x_0, \ldots, x_{e-1}$ of degree $j$, i.e.,
$$\beta_{0,j}^S(R)\leq \binom{e+j-1}{j}.$$
Then
$$\sum_{j=0}^r\beta_{0,j}^S(R)\leq \sum_{j=0}^r\binom{e+j-1}{j}=\binom{e+r}{r}.$$
This implies that
$$\beta_{0,j}^S(R)=\binom{e+j-1}{j},$$
for all $j=0, 1, \ldots, r$.
Moreover, we know by Theorem \ref{51} that $R$ has depth $n$. The Auslander-Buchsbaum formula then gives us $\mathrm{proj.dim}_{S_0}(R)=\depth(S_0)-\depth(R)=e+1$. Hence $\mathrm{proj.dim}_S(R)=1$ by Corollary \ref{27}(c). Equivalently, $\beta_{i,j}^S(R)=0$ for all $i>1, j\geq 0$, and there is a positive integer $d>0$ such that
$$\beta_{1,j}^S(R)=\begin{cases}
\mu_S(R)-\deg(R)=1&\mbox{ if } j=d,\\
0&\mbox{ otherwise.}
\end{cases}$$
The minimal free $S$-resolution of $R$ thus is
$$0\rightarrow S(-d-1)\rightarrow \bigoplus_{i=0}^rS(-i)^{\binom{e+i-1}{i}}\rightarrow R\rightarrow 0.$$
It remains to show that $d=\reg(R)$, or equivalently, $d\geq r$. In the proof of Theorem \ref{51}, we have seen that the initial ideal $\ini(I)$ with respect to the degree reverse lexicographic order is minimally generated by all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ and a monomial $uv$ where $v\in S_+$ and $u$ is a monomial in $x_0, \ldots, x_{e-1}$ of degree $r$ (see Step 3 in the proof of Theorem \ref{51}). Moreover, the equivalence classes in $R$ of the monomials in $x_0, \ldots, x_{e-1}$ of degree at most $r$ except $u$ are linearly independent over the ring $S$. Therefore, $d+1\geq \deg(uv)\geq r+1$, or $d\geq r$.
\end{proof}
So over the Noether normalization $S=S_e$ of the homogeneous coordinate ring $R$, the Betti table of a projective variety $X$ with almost maximal degree is described precisely. It depends on whether the variety is arithmetically Cohen-Macaulay (ACM) or non-arithmetically Cohen-Macaulay (non-ACM) and is either
\begin{figure}[!htb]
\captionsetup[subfigure]{width=4cm}
\subfloat[ACM]{\begin{tabular}{>{\centering}m{2cm}| c}
& $0$\\
\hline
$0$ & $1$\\
$1$ & $\binom{e}{1}$\\
$2$ & $\binom{e+1}{2}$\\
$\vdots$ & $\vdots$\\
$r$ & $\binom{e+r-1}{r}$-1\\
\end{tabular}}
\hspace{1cm}
\subfloat[non-ACM, $\reg(R)=r$]{\begin{tabular}{>{\centering}m{1.5cm}| >{\centering}m{1.5cm} c}
& $0$ & $1$\\
\hline
$0$ & $1$ & --\\
$1$ & $\binom{e}{1}$& -- \\
$2$ & $\binom{e+1}{2}$& -- \\
$\vdots$ & $\vdots$& $\vdots$\\
$r$ & $\binom{e+r-1}{r}$& $1$\\
\end{tabular}}
\hspace{1cm}
\subfloat[non-ACM, $\reg(R)>r$]{\begin{tabular}{>{\centering}m{1.5cm}| >{\centering}m{1.5cm} c}
& $0$ & $1$\\
\hline
$0$ & $1$ & --\\
$1$ & $\binom{e}{1}$& -- \\
$2$ & $\binom{e+1}{2}$& -- \\
$\vdots$ & $\vdots$& $\vdots$\\
$r$ & $\binom{e+r-1}{r}$& --\\
$r+1$ & --& --\\
$\vdots$ & $\vdots$& $\vdots$\\
$\reg(R)$ & --& $1$\\
\end{tabular}}
\end{figure}
\noindent From these tables we can recover the Betti table over the ring $S_0$ of the variety $X$. This is stated precisely in Theorem \ref{54}. In the next we will present a proof of this theorem. At first, we need the following lemma which is an analogue of Lemma \ref{44} in the arithmetically Cohen-Macaulay case.
\begin{lemma}\label{53}
Let $S_0=k[x_0, \ldots, x_{n+e}]$ and $J=(x_0, \ldots, x_{e-1})$. Let $u\in J$ be a monomial of degree $r$ and $v$ be a non-constant monomial in $x_e, \ldots, x_{n+e}$. Put $I=(uv)+J^{r+1}$. Then $V(I)\subset \bP^{n+e}$ is a subscheme of almost maximal degree. The Betti numbers of the ideal $I$ are as follows:
\begin{enumerate}[(a)]
\item If $\deg(uv)=r+1$ then
$$\beta_{ij}^{S_0}(I)=\begin{cases}
\binom{e+r}{i+r+1}\binom{r+i}{r}+\binom{e}{i} &\mbox{ if } 0\leq i<e, j=r+1,\\
0 &\mbox{ otherwise.}
\end{cases}
$$
\item If $\deg(uv)>r+1$ then
$$\beta_{ij}^{S_0}(I)=\begin{cases}
\binom{e+r}{i+r+1}\binom{r+i}{r}&\mbox{ if } 0\leq i<e, j=r+1,\\
\binom{e}{i} &\mbox{ if } 0\leq i\leq e, j=\deg(uv),\\
0 &\mbox{ otherwise.}
\end{cases}
$$
\end{enumerate}
\end{lemma}
\begin{proof}
Denote by $G(I)$ the minimal set of monomials generating $I$. Then
$$G(I)=\{uv\}\cup G(J^{r+1}).$$
Since $(uv)\simeq S_0[-\deg(uv)]$, the ideals $(uv)$ and $J^{r+1}$ have linear resolutions. By \cite[Corollary 2.4]{FHT09}, $I=(uv)+J^{r+1}$ is a Betti splitting, that means,
$$\beta_{ij}^{S_0}(I)=\beta_{ij}^{S_0}(uv)+\beta_{ij}^{S_0}(J^{r+1})+\beta_{i-1, j+1}^{S_0}((uv)\cap J^{r+1}),$$
for all $i, j$.
We have $(uv)\cap J^{r+1}=v.uJ\simeq J[-\deg(uv)]$ and $(uv)\simeq S_0[-\deg(uv)]$. Hence
$$\beta_{ij}^{S_0}(I)=\beta_{i,j-\deg(uv)}^{S_0}(S_0)+\beta_{ij}^{S_0}(J^{r+1})+\beta_{i-1, j-\deg(uv)+1}^{S_0}(J).$$
It worth noting that for any $t>0$, the ideal $J^t$ defines a subscheme of maximal degree. Now using Corollary \ref{32}, we easily complete the proof.
\end{proof}
Now we are ready to prove Theorem \ref{54}.
\begin{proof}[\bf Proof of Theorem \ref{54}]
Let $I_X\subset S_0$ be the saturated homogeneous defining ideal of $X$ and $R=S_0/I_X$. Changing the variables if necessary, we may assume that $Q=(x_e, \ldots, x_{n+e})$ is a minimal reduction of the irrelevant ideal $R_+$ with the reduction number $r_Q(R)=r$. In particular, $S=S_e=k[x_e, \ldots, x_{n+e}]\hookrightarrow R$ is a Noether normalization of $R$. The algebra $R$ is particularly a finitely generated module over $S_{e-i}$ for $i=0, 1, \ldots, e$.
By Corollary \ref{52}, the initial ideal $\ini(I)$ with respect to the degree reverse lexicographic order is generated by all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ and a monomial $uv$, where $u$ is a monomial in $x_0, \ldots, x_{e-1}$ of degree $r$ and $v$ is a non-constant monomial in $x_e, \ldots, x_{n+e}$. From the proof of $(a)\Rightarrow (e)$ of Corollary \ref{52} and the usage of Proposition \ref{27}(d), we obtain
$$\reg(R)=\reg_{S_0}(R)=\reg_{S_t}(R)=\deg(uv)-1.$$
By the Cancellation Principle (see \cite[Corollary 1.21]{Gre98} or \cite[Section 3.3]{HH11}), there is for each $m\geq 0$ a complex of $k$-vector spaces
$$V_{\bullet, m}: 0\rightarrow V_{m,m}\rightarrow V_{m-1, m}\rightarrow \ldots\rightarrow V_{1, m}\rightarrow V_{0, m}\rightarrow 0$$
such that
$$V_{i,m}\simeq \Tor_i^{S_0}(\ini(I_X), k)_m,$$
$$H_i(V_{\bullet, m})\simeq \Tor_i^{S_0}(I_X, k)_m,$$
for all $i\geq 0$. Consequently,
$$\beta^{S_0}_{ij}(I_X)\leq \beta^{S_0}_{ij}(\ini(I_X)),$$
for any $i, j$.
Now using Lemma \ref{53}, we obtain the vanishing of Betti numbers, namely,
$$\beta_{ij}^{S_0}(I_X)=0,$$
for either $j\not=r+1, \reg(R)+1$, or $j=r+1, i\geq e$, or $j=\reg(R)+1, i>e$. Moreover, it also shows that in the complex $V_{\bullet, m}$ above, almost all vector spaces are zero except $V_{i, r+1+i}$ and $V_{i, \reg(R)+1+i}$.
To ease the presentation, we consider two cases according to the difference between the regularity and the reduction number.
\smallskip
\noindent Case 1: $\reg(R)$ is either $r$ or $r+1$.
Recall the notation
$$\chi_m^{S_i}(R)=\sum_{j=0}^m(-1)^j\beta_{m-j, j}^{S_i}(R)=\sum_{j=0}^{\reg(R)}(-1)^j\beta_{m-j, j}^{S_i}(R),$$
for $m\in \bZ$. From Corollary \ref{52}(e), we have
$$\chi_m^{S_e}(R)=\begin{cases}
(-1)^m\binom{e+m-1}{m} &\mbox{ if } 0\leq m\leq r,\\
(-1)^{\reg(R)} &\mbox{ if } m=\reg(R)+1,\\
0 &\mbox{ otherwise.}
\end{cases}$$
Combining this with Corollary \ref{27}(a), we have
$$\chi_m^{S_0}(R)=\sum_{j=0}^e\binom{e}{j}\chi_{m-j}^{S_e}(R)
=\begin{cases}
\sum_{j=0}^e\binom{e}{j}(-1)^{m-j}\binom{e+m-j-1}{m-j}& \mbox{ if } m\leq \reg(R),\\
\sum_{j=0}^e\binom{e}{j}(-1)^{m-j}\binom{e+m-j-1}{m-j}+(-1)^{\reg(R)}\binom{e}{m-\reg(R)-1}& \mbox{ if } m> \reg(R),\\
\end{cases}$$
Finally, using Lemma \ref{33} we get
$$\chi_m^{S_0}(R)=\begin{cases}
(-1)^r\binom{e+r}{m}\binom{m-1}{r} & \mbox{ if } m\leq \reg(R),\\
(-1)^r\binom{e+r}{m}\binom{m-1}{r}+(-1)^{\reg(R)}\binom{e}{m-\reg(R)-1} & \mbox{ if } \reg(R)< m\leq e+\reg(R)+1,\\
0& \mbox{ if } m>e+\reg(R)+1.
\end{cases}$$
If $\reg(R)=r$ then $\chi_m^{S_0}(R)=(-1)^r\beta_{m-r,r}^{S_0}(R)$. This proves (a).
On the other hand, if $\reg(R)=r+1$ then $\chi_m^{S_0}(R)=(-1)^r\beta_{m-r,r}^{S_0}(R)+(-1)^{r+1}\beta_{m-r-1, r+1}^{S_0}(R)$. This proves (b).
\medskip
\noindent Case 2: $\reg(R)>r+1$.
This is an application of the Cancellation Principle (see \cite[Corollary 1.21]{Gre98} or \cite[Section 3.3]{HH11}). Using the notations in the first part of the proof, we have $V_{i,m}=0$ for all $m, i\geq 0$ such that $m-i\not=r+1, \reg(R)+1$. In particular, $V_{i, r+2+i}=0$ and $V_{i, \reg(R)+i}=0$. Consequently, we obtain $H_i(V_{\bullet, m})\simeq V_{i, m}$ for all $i, m$, which induces by the Cancellation Principle the equality $\beta_{ij}^{S_0}(I_X)=\beta_{ij}^{S_0}(\ini(I_X))$. Therefore the Betti number of the homogeneous coordinate ring of $X$ can be computed using Lemma \ref{53}, namely,
$$\beta_{ij}^{S_0}(R)=
\begin{cases}
1&\mbox{ if } i=j=0,\\
\binom{e+r}{i+r}\binom{i+r-1}{r} &\mbox{ if } j=r, 1\leq i\leq e,\\
\binom{e}{i-1} &\mbox{ if } j=\reg(R), 1\leq i\leq e+1,\\
0&\mbox{ otherwise.}
\end{cases}$$
This proves (c).
\end{proof}
There are various examples of non-arithmetically Cohen-Macaulay projective varieties of almost maximal degree. Belows we give some of them and show that all the cases (a), (b), (c) in Theorem \ref{54} actually occur.
\begin{example}\label{55}
Let $C_1$ be a smooth elliptic curve in $\bP^3$ of degree $5$. Then, $C_1$ is an isomorphic projection from a complete embedding of an elliptic curve $C_0 \hookrightarrow \bP^4$. So, $C_1$ is $m$-normal for all $m\ge 2$, but not linearly normal. Thus, we have $\beta_{1,1}(C_1)=h^0(\mathcal {I}_{C_1}(2))=0, \reg(C_1)=3$, the reduction number $r(C_1)=2$ and $\deg(C_1)= \binom{e+r}{r}-1=5$.
The Betti table of its homogeneous coordinate ring is
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& 3\\
\hline
0 & 1& --& --& --\\
1 & --& --& --& --\\
2 & --& 5& 5& 1
\end{tabular}
\end{figure}
This Betti table corresponds to (a) in Theorem \ref{54}.
\end{example}
\begin{example}\label{56}
Let $C$ be a smooth curve of degree $9$ and genus $4$ in $\bP^5$. Then, $C$ is a projectively normal embedding and consider its isomorphic projection $C_1\subset\bP^4$ from a center $p\notin Z_2(X)$ where $Z_2(X)$ is the Jacobian scheme defined by $6$ quadrics containing $C$. Note that $Z_2(X)$ is a hypersurface of degree $6$. Then, $C_1$ is $m$-normal for all $m\ge 2$, but not linearly normal
(see \cite[Theorem 2.7]{AR02}). Thus, we have $\beta_{1,1}(C_1)=h^0(\mathcal {I}_{C_1}(2))=0$, the regularity $\reg(C_1)=3$, the reduction number $r(C_1)=2$ and the degree $\deg(C_1)= \binom{e+r}{r}-1=9$. The following Betti table of its homogeneous coordinate ring is type (a) in Theorem \ref{54}:
\newpage
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm}>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& 3& 4\\
\hline
0 & 1& --& --& --&--\\
1 & --& --& --& --&--\\
2 & --& 11& 18& 9& 1
\end{tabular}
\end{figure}
\end{example}
\begin{example}\label{57}
Let $C_2$ be the smooth rational curve in $\bP^3$ defined by $(s, t)
\mapsto (s^5, s^4t+s^3t^2, st^4, t^5)$. Then, the curve $C_2$ is of type (5) in Naito's list in \cite[Theorem 1]{Naito02}, and its Betti table is
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& 3\\
\hline
0 & 1& --& --& --\\
1 & --& --& --& --\\
2 & --& 4& 3& --\\
3 & --& 1& 2& 1
\end{tabular}
\end{figure}
In particular, $C_2$ is not arithmetically Cohen-Macaulay with the reduction number $r(C_2)=2$, $\reg(C_2)=4$ and $\deg(C_2)=5=\binom{2+2}{2}-1$. The Betti table of $C_2$ corresponds to (b) in Theorem \ref{54}.
\end{example}
\begin{example}\label{58}
Let $C_3$ be the rational curve in $\bP^4$ defined as the image of the map $\bP^1\rightarrow \bP^4$,
\[\begin{aligned}
(s,t)\mapsto (&s^9,s^8 t-s t^8,s^7 t^2-s^2 t^7,\\
&s^6 t^3+s^5 t^4+s^4 t^5+s^3 t^6+s^8 t+s^7 t^2,t^9)
\end{aligned}\]
Using Macaulay 2, we find that the Betti table of $C_3$ is
\begin{figure}[!htb]
\begin{tabular}{>{\centering}m{2cm}|>{\centering}m{1cm}>{\centering}m{1cm} >{\centering}m{1cm} >{\centering}m{1cm} c}
& 0& 1& 2& 3 & 4\\
\hline
0 & 1& --& --& --& --\\
1 & --& --& --& -- &--\\
2 & --& 10& 15& 6 & --\\
3 & --& --& --& -- &--\\
4 & --& --& --& -- &--\\
5 & --& 1& 3 & 3 & 1
\end{tabular}
\end{figure}
Let $S_0=k[x_0,x_1,x_2,x_3,x_4]$ and $I$ be the defining ideal of $C_3$. The homogeneous coordinate ring $R=S_0/I$ has a Noether normalization $k[x_0,x_4]\rightarrow R$ and by computation, $R/(x_0,x_4)\simeq k[x_1,x_2,x_3]/(x_1,x_2,x_3)^3$. Hence the reduction number of $C_3$ is $r(C_3)=2$. Since $\deg(C_3)=9$, the curve $C_3$ has an almost maximal degree. Moreover, $C_3$ is not arithmetically Cohen-Macaulay, and $\reg(C_3)=6$ and the Betti table of $C_3$ corresponds to (c) in Theorem \ref{54}.
\end{example}
We have proved in Theorem \ref{51} that if a non-degenerate projective variety has almost maximal degree then its coordinate ring has depth at least the dimension of the variety. In the next example, we will see that projective varieties can not be replaced by projective subschemes in this theorem. The idea of the example comes from the proof of Theorem \ref{51}.
\begin{example}\label{59}
Let $S_0=k[x_0, \ldots, x_{n+e}]$ as from the beginning. Let $r>0$ and let $u$ be a monomial in $x_0, \ldots, x_{e-1}$ of degree $r$. As in the proof of Theorem \ref{51}, we denote the set of all monomials in $x_0, \ldots, x_{e-1}$ of degree $r+1$ by $T_{r+1}$. Let $I$ be the monomial ideal generated by $T_{r+1}$ and $ux_e, \ldots, ux_{n+e}$. Clearly
$$I=(T_{r+1}, u)\cap (T_{r+1}, x_e, \ldots, x_{n+e}).$$
Then
$$\deg(S_0/I)=\deg(S_0/(T_{r+1}, u))=\binom{e+r}{r}-1,$$
and $\depth(S_0/I)=0$.
\end{example}
| 14,939
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Find Loft Conversion Specialists near you
Loft Conversion Specialists in Forest Row
Find loft conversion specialists in Forest Row, East Sussex and get free quotes fast with Bark. Whether it's for a full conversion, installing a new staircase or just boarding Bark will find Forest Row, East Sussex builders for your Loft Conversion fast.
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Floren
Carpentry and building
STAAC
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About The Bloggers
« Use of Songs, Videos in 2012 Presidential Campaign Prompting Legal Challenges |
Main
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Wednesday's Three Burning Legal Questions
Here are today's three burning legal questions, along with the answers provided by the blogosphere.
1) Question: I'm at the public library in Seattle with my grade school-age daughter. There's a guy in the middle of the library looking at hard-core pornography on one of the library's computers! Can he do that?! This is very uncomfortable!
Answer: The Seattle public library "facilitates access to constitutionally protected information," and will not compromise freedom of speech or censor information. The porn stays. (Consumerist, Should Libraries Let People Look At Porn In The Open?)
2) Question: I applied early admission to college. I went online yesterday and was thrilled to learn that I had been admitted! The admission letter instructed me to withdraw my applications to other schools, which I did right away. But when I went online later to check out the letter again there was a notice that the acceptance letter had been a mistake and I was actually rejected. Now what??
Answer: Oooofffff. That hurts, sorry. The same thing happened at Vassar recently, so keep an eye on how things shake out there. (NBC New York, Dozens of Vassar College Applicants Mistakenly Get Admissions Letters)
3) Question: I'm about to walk my two dogs in a national park. Yes, I know I am supposed to have the dogs on a leash but these are tiny lapdogs that won't bother anyone. What will happen if a ranger catches me with the dogs off-leash?
Answer: Ask yourself if walking the dogs off the leash is worth getting shot with a Taser. (Jonathan Turley, Teaching Citizens to Heel: Park Ranger Reportedly Tasers Man Walking Small Dogs Off Leash)
Posted by Bruce Carton on February 1, 2012 at 02:11 PM | Permalink
| 227,161
|
THE contrast between both coaches could not have been greater following Oldham’s 31-26 play-off defeat of Swinton Lions on Sunday.
A game which ebbed and flowed throughout."
Benson, whose patched-up side made their fans suffer before emerging victorious, can now look forward to a final home game of 2009 on Sunday.
Victory over Hunslet would then see them travel to the losers of the Keighley-York encounter, knowing another win would ensure a third consecutive Grand Final.
"Right throughout the game, I just knew we were going to come out on top," said Benson. "Swinton put up a huge fight and they’re loaded with good players across the park. One try made the difference. Credit should go to our boys for putting them under so much pressure because Swinton had a lot of chances in the last 15 minutes and couldn’t capitalise because of the pressure we put on them. I thought the attitude was fantastic for our guys to pull one out of the bag. They showed their never-say-die attitude."
| 60,359
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What do you like about working at Peek?
"The environment of this company is greet, people are very friendly."
Do you have any tips for others interviewing with this company?
"Good at C/C++ is very important. And the knowledge of OS is a plus."
What don't you like about working at Peek?
"Because the company is a start-up, there is no more attention is payed on the specification of software development."
What suggestions do you have for management?
"It would be better if make the software development process meet the specifications."
"Worst place to work!"
What do you like about working at Peek?
"Nothing. It was one of the worst career moves of my life!"
What don't you like about working at Peek?
"The CEO is a super autocrat - and everyone else is a worker bee (with a good chance of getting layed off).Work here means doing PPT pages, not doing any real work."
What suggestions do you have for management?
"There is a reason people are experienced in a certain vertical they live in - a lot of product, organization, etc is about subjectivity and balance.The CEO makes the decisions - there is no flat hierarchy. Lastly, the CEO has no product experience and the company is purely struggling because of his lack of insight."
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| 205,207
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Springtime at the Castle Valley Center
Here we are with another school year coming to a close. As the students anticipate the summer break we at the Castle Valley Center have been busy with many activities this spring. Students and staff look forward to some time off during the summer. We have had a very busy school year with students working hard in the classroom and being involved in their community. It seems that we have something every day.
One of the more enjoyable activities we have been working on is a school choir. Tiffany Parker is our music specialist; she does an excellent job in working with our students and tuning them to perfection as they learn new songs. Our final performance for the school year will be singing as part of our graduation program for three students who will complete their studies at the Castle Valley Center.
We have been out and about these past few months. We have been planting trees for Arbor Day; we have visited local farms to talk to the animals. We have visited the recent dinosaur bone discovery in Emery County. We have held both a Karaoke day and a talent show for students and staff at the Castle Valley Center to share their talents with their peers, co-workers and anyone else who came to see the show.
The CEU auto shop has invited us for a complete tour of their impressive facility. The Carbon County Sheriff's Posse had us come to the Posse grounds to see the horses and riders go through the paces as they showed us routines that show great skill and understanding of horsemanship.
We once again want to express our appreciation to this wonderful community of Price and surrounding areas, we do take care of our own and there is no better place to live and work in than we have right in our area.
| 364,571
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Stay Connected with CECH
Programs and Degrees
School Psychology
Why study School Psychology?
The School Psychology Program at the University of Cincinnati is dedicated to preparing highly competent professional school psychologists from a scientist-practitioner model at the doctoral (PhD) level. Graduates are prepared to make significant contributions to this challenging field through up-to-date and research-based professional practice. Doctoral-level graduates are prepared research-based professional practice plus leadership, systems change, and advanced behavioral theory and research..
For more information, please contact Amanda Carlisle at amanda.carlisle@uc.edu.
Major Maps
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The Program admits students once a year for a fall semester start. Applications are due January 15.
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UC's Doctoral Program in School Psychology is approved by the National Association of School Psychologists (NASP) and accredited by both the American Psychological Association (APA)
Doctor of Philosophy Degree in School Psychology
Full-Time Program Duration
5 Years
Location
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School of Psychology
Cincinnati, OH 45221
Phone: (513) 556-3342
Renee Hawkins
Renee.Hawkins@uc.edu
| 202,938
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See us for allergies treatment in Philadelphia, Pennsylvania, Maple Shade, New Jersey, Bala Cynwyd , Pennsylvania, and the surrounding areas. The Falcone Center for Functional, Cosmetic and Regenerative Medicine uses functional medicine to treat Allergies and other chronic medical conditions.
What are Allergies?
Allergies are the body’s response to something usually dust, mold, trees, weeds, pet dander and even food to which most have no reaction. When someone is allergic to something, the body overproduces antibodies which then produce histamines. Histamines are the culprits that cause the symptoms of allergies.
What Causes Allergies?
The cause of allergies comes from the body’s reaction to allergens. When antigens, which are the protein particles in all sorts of substances from foods, dust, pollen, and other factors, enter the body and cause a reaction they become allergens.
What Are the Symptoms of Allergies?
The symptoms of allergies vary from person to person as well as from the element to which they are allergic. For the most part, people who come into contact with that element experience the following symptoms:
- Watery, itchy eyes
- Runny nose
- Rash
- Stuffy nose or congestion
- Fatigue
- Nausea
- Swelling inside the mouth or bronchial tubes (food allergies)
How Are Allergies Treated with Functional Medicine?
Like any chronic condition, functional medicine seeks to find the cause of your allergies rather than just treat the symptoms as done with traditional medicine. Once we determine the cause, we work to lower and even eliminate the symptoms of allergies by designing an effective treatment plan that decreases the effects that allergens have on your body. Treatment plans vary with each patient but usually involve making modifications to your lifestyle and avoiding the suspected allergens as well as adding nutritional supplements to your diet to support your immune system and calm inflammation.
What Is the Cost of Treating Allergies Can the Falcone Center Help Me with My Allergies?
At the Falcone Center, our.
How Do I Begin Allergy Treatment at the Falcone Center?
To begin allergy treatment, please contact us to schedule an initial consultation and evaluation to determine the cause of the allergy. Additionally, we require patients to fill out our new patient forms and return them before the initial consultation. This allows us to review the information and have a better understanding of the condition. Dr. Falcone will lay out an effective treatment plan to lessen and even eradicate allergic reactions from your system.!
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I don’t like wearing shoes, at least, not all the time. They make my feet feel stuffy and sometimes kind of sweaty. Inevitably, they get dirty and worn out, and eventually, they need to be replaced. It all seems wasteful and pointless. I’ve grown up walking around without shoes on, mostly in the house — I usually take my shoes off as soon as I get inside — but I also enjoy wandering throughout my backyard barefoot, too. Think it’s kind of gross? Or dangerous? Sometimes, it can be, but generally, it’s pretty convenient and comfortable. Read on and find yourself convinced.
It’s cozier indoors
People who walk around in their own homes with boots or shoes on scare me a little bit. Personally, it just doesn’t feel as if I’m really home until I’m barefoot. And how are you supposed to comfortably snuggle up on the couch if you’re wearing shoes? For those of you who are worried about your feet being cold — a very valid concern — I have a groundbreaking suggestion for you: socks. They are pretty revolutionary. Though it can occasionally be dangerous, such as if someone has recently broken glass, often I feel more comfortable and less likely to trip or slip when I’m barefoot. Can you ever really be comfortable when you’re wearing shoes?
It’s a freeing feeling
Sometimes, there is nothing better than just taking your shoes off and running around. Whether it’s been a long day of class or walking from place to place, taking my shoes off is a vital step in relaxing and letting loose. Also, walking outside without shoes on is the easiest way to remind yourself that social conventions really don’t matter. And there’s that tiny thrill in knowing that you might get a splinter or a cut but you’re doing it anyways. There’s something really natural about going outside without shoes on, and it can be pretty rejuvenating to try it after a long day. I’m not saying you should walk all over Berkeley without any form of foot cover – that I would strongly discourage – but walking around in a backyard or through grass is a must.
It’s pretty grounding
It’s easy to get caught up in the world of computers and professionalism, but it’s also important to connect with nature and the outside world. Though going on hikes and camping trips are awesome ways to do this, it’s not always possible. However, just by going outside and taking your shoes off, you can feel grounded and connected to the world without planning an entire vacation around it.
Feet are easier to clean than shoes
I said it. Even if you go outside without shoes, or if your housemates wear shoes indoors so your feet are perpetually covered in dirt, you can always just use soap and water to get clean again. It’s honestly a pretty great option. And if you like wearing socks, your feet are clean again as soon as you take them off. What’s not to love? It can be fun to feel like a kid again and not worry about getting a little grass or mud on your feet if you’re walking around barefoot. And the feeling of clean feet after walking around outside is pretty much unmatched.
There are lots of good reasons to go barefoot. Sometimes, I’m just too lazy to put on shoes. Other times, I’m intentionally trying to feel a little more grounded. If you’re someone who is pretty much always in shoes unless you’re sleeping, think about kicking them off the next time you get home. You just might realize how overrated shoes are.
Contact Elysa Dombro at [email protected].
| 159,784
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.
"Did you see Tony the Tiger's vlog last night?"
A sample vlog on youtube:
A sample vlog on youtube:
by BeckBooBoo January 14, 2009
Talking to the camera like it's a person.
Vlogs are internet videos, typically made by amateurs with the hope of going viral or developing an online following. The connotation of "vlogs" is that it is not mainstream content.
Vlogs are generally in the variety of freak shows, conspiracy theories, or monologue rants.
by Rosebud2939 August 07, 2017
VLOGGER: My life is so interesting. I record every second of it.
Human Dictionary: Define {narcissist.
Human Dictionary: Define {narciss
| 86,815
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\begin{document}
\LARGE
\noindent
\textbf{Tau functions
as Widom constants}
\normalsize
\vspace{1cm}\\
\noindent\textit{M. Cafasso$\,^{a,}$\footnote{cafasso@math.univ-angers.fr},
P. Gavrylenko$\,^{b,c,d,}$\footnote{pasha145@gmail.com},
O. Lisovyy$\,^{e,}$\footnote{lisovyi@lmpt.univ-tours.fr}}
\vspace{0.2cm}\\
$^a$ LAREMA, Université d'Angers, 2 bd Lavoisier, 49045 Angers, France
\vspace{0.1cm}\\
$^b$ Center for Advanced Studies, Skolkovo Institute of Science and Technology, 143026 Moscow, Russia
\vspace{0.1cm}\\
$^c$ National Research University Higher School of Economics, Department of
Mathematics and International Laboratory of Representation Theory and Mathematical
Physics, 119048 Moscow, Russia\vspace{0.1cm}\\
$^d$ Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine
\vspace{0.1cm}\\
$^e$ Laboratoire de Math\'ematiques et Physique Th\'eorique CNRS/UMR 7350, Universit\'e de Tours, Parc de Grandmont,
37200 Tours, France
\begin{abstract}
We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom's determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo-Miwa-Ueno tau function for Riemann-Hilbert problems of isomonodromic origin (Painlev\'e VI, V, III, Garnier system, etc) and the Sato-Segal-Wilson tau function for integrable hierarchies such as Gelfand-Dickey and Drinfeld-Sokolov.
\end{abstract}
\section{Introduction}
Tau functions play a central role in the theory of integrable equations, both in fields of isospectral and isomonodromic deformations. They had been introduced in the 80s by the Kyoto school, with the explicitly stated aim \cite{JMU} to construct a generalization of the theta functions appearing, since Riemann \cite{Riemann}, as particular solutions of some non--linear equations\footnote{See also the work of S. Kovalevskaya \cite{Kov} and the more recent ones on finite-gap integration; e.g. \cite{DMN,Matv} and references therein.}.
In the theory of isomonodromic deformations, tau functions are constructed starting from a certain differential 1-form $\omega_{\text{JMU}}$ defined on the space of the deformation parameters \cite{JMU}. Under the hypothesis that the parameters are of isomonodromic type, the form $\omega_{\text{JMU}}$ is closed and the tau function $\tau_{\text{JMU}}$ is defined (locally and up to a multiplicative constant) by the formula
\beq
\mathrm{d}\ln\tau_{\text{JMU}} := \omega_{\text{JMU}},
\eeq
where $\mathrm{d}$ denotes the total differential.
Quite differently, on the side of isospectral deformations, Sato \cite{Sato} defined the tau function starting from his interpretation of the KP hierarchy in terms of the geometry of Grassmannian manifolds. Namely, to each solution of the KP hierarchy, one can associate a point $W$ in an infinite dimensional Grassmannian, and the related tau function is nothing but the formal series
\beq
\tau_W := \sum_{\mathsf{Y} \in \mathbb{Y}} s_{\mathsf{Y}} W_{\mathsf{Y}},
\eeq
where $ \mathbb{Y}$ is the set of partitions, $\big\{s_{\mathsf{Y}\in \mathbb{Y}}\big\}$ are the Schur polynomials and $\big\{ W_{\mathsf{Y}\in \mathbb{Y}}\big\}$ is the set of the Pl\"ucker coordinates of $W$. In \cite{SW}, Segal and Wilson provided an analytic version of Sato's theory, where formal series are replaced by $L^2$ functions, and rewrote the tau function as the (analytically well-defined) Fredholm determinant of a certain projection operator.
Since the 80s, many generalizations of both definitions had been constructed; giving a complete account of the literature on the subject is out of the scope of this introduction. The generalizations touch different branches of mathematics as diverse as the representation theory of infinite--dimensional Lie algebras \cite{Kac}, Frobenius manifolds \cite{DZ}, instanton counting \cite{Nekrasov,NO}, Riemann--Hilbert boundary value problems \cite{Bertola,ILP} and topological recursion \cite{Eyn}, to name few of them. The reasons of such a flourishing literature, from our point of view, are to be found on the side of applications: while the several different definitions of tau functions could seem very abstract, the explicit computation of some of them are important for a growing mathematical community working on e.g. random matrix theory, statistical models, algebraic and symplectic enumerative geometry.
The aim of this paper is to show that, at least for a very wide array of examples touching both the worlds of isomonodromic and isospectral deformations, tau functions coincide with a pretty simple object whose introduction by Widom goes back to 1976 \cite{Widom2}, before the very first seminal papers of the Kyoto school. Namely, they are the (Szeg\H o-)Widom constants associated to matrix--valued symbols $J:\mathcal C\to\mathrm{GL}\lb N,\Cb\rb$, where $\mathcal C\subset\Cb\Pb^1$ is a circle centered at the origin. Recall that, given a symbol $J\lb z\rb=\sum_{k\in\Zb}J_k z^k$, the associated $n$-th block Toeplitz matrix is defined by
$T_n\left[J\right]:=\lb J_{k-l}\rb_{k,l=1}^n$.
\iffalse
\lb \begin{array}{ccccc}
J_0 & J_{-1} & J_{-2} & \ldots & J_{-K+1} \\
J_1 & J_0 & J_{-1} & \ldots & J_{-K+2} \\
J_2 & J_1 & J_0 & \ldots & J_{-K+3} \\
\vdots &\vdots & \vdots & \ddots & \vdots \\
J_{K-1} & J_{K-2} & J_{K-3} & \ldots & J_0
\end{array}\rb.
\fi
The asymptotics of $T_n[J]$ had been extensively studied in the operator theory literature, very often with motivations coming from applications in statistical mechanics such as, for instance, the Ising model \cite{DIK}.
In particular, a celebrated theorem of Widom \cite{Widom2} states that, under certain analytical conditions on the symbol $J$,
\ben
\lim_{n\to\infty}{G\left[J\right]}^{-n}\operatorname{det}T_n\left[J\right]=\tau\left[J\right].
\ebn
Here
$G\left[J\right]=\exp \ds\frac{1}{2\pi i}\oint_{\mathcal C}
z^{-1}\ln\operatorname{det}J\lb z\rb dz$
and $\tau\left[J\right]$, which is known nowadays as the Widom constant, is the Fredholm determinant (the notations are explained in the next section)
\beq\label{introtauRHP}
\tau\left[J\right]= \operatorname{det}_{H_+}\lb \Pi_+ J^{-1}\Pi_+
J\rb.
\eeq
Indeed, this is a highly non-trivial extension of the celebrated strong Szeg\H o theorem \cite{Sz}, treating the case of scalar Toeplitz determinants.
On the isospectral side, the coincidence between the Widom constant and the Sato-Segal-Wilson tau function had been established, for the so-called Gelfand-Dickey hierarchies, by one of the authors in \cite{Cafasso}, and successively extended to the Drinfeld-Sokolov hierarchies associated to an arbitrary Kac-Moody algebra in \cite{CafassoWu1}. Based on this identification, effective computations had been carried out in \cite{CafassoWu2,CDD} for topological and polynomial tau functions.
In this work we show that, quite surprisingly, the recent results of \cite{GL16,GL17}, inspired by the isomonodromy/CFT/gauge theory correspondence, lead to a Fredholm determinant representation of the isomonodromic tau functions which is ultimately the same as given in \eqref{introtauRHP}. This implies, in particular, that the combinatorial expansion of the Sato's tau function and the much more recent series representations for the isomonodromic tau functions of Painlevé VI, V and III equations \cite{GIL12,GIL13} (at $0$), which originated from the AGT correspondence, are both of the same nature. Namely, they are nothing but the expansions of the Fredholm determinant \eqref{introtauRHP}, their terms being products of Pl\"ucker coordinates of subspaces in the Sato-Segal-Wilson Grassmannian.
It turns out to be very fruitful to consider the symbol $J\lb z\rb$ as a jump matrix for a pair of Riemann-Hilbert problems (RHPs) on the circle $\mathcal C$. To construct Fredholm determinant and series representations for the tau functions of more general isomonodromic problems (e.g. the Garnier system), one needs to consider RHPs set on a union of non-intersecting ovals. In the present paper, we show how the definition \eqref{introtauRHP} of $\tau\left[J\right]$ can be generalized in this case and prove a formula for the log-differential of the appropriate extension of the Widom's determinant with respect to parameters of the jumps, which leads to its identification with the Jimbo-Miwa-Ueno tau function for RHPs of isomonodromic origin.
We expect that the identification between the Widom constants and isomonodromic tau functions will lead to an effective way to compute the latter in the so far unsolved problems. These include, in particular, the construction of explicit asymptotic expansions of irregular type (at $\infty$) for Painlevé~I--V transcendents, cf \cite{BLMST,Nagoya}. Furthermore, the results of \cite{BSh,JNS} on $q$-Painlevé III and $q$-Painlevé VI equation give a hope that our approach may also be adapted to the $q$-difference setting.
Starting from the foundational work \cite{Jimbo}, the standard scheme of asymptotic analysis of Painlevé transcendents \cite{FIKN} is to construct an \textit{approximate} solution of the appropriate RHP from solutions of ``elementary'' RHPs (parametrices), and then to extract the asymptotics of Painlevé functions from this approximation. The main ideological shift of our approach is that it gives an \textit{exact} Fredholm determinant expression for the tau functions in terms of parametrices, which define the relevant integrable kernels. The Fredholm determinant yields, with a relatively little effort, a complete asymptotic expansion of the tau function. The solution of the RHP (exact or approximate) is not needed at all, even though it can also be expressed via the resolvent of the appropriate integral operator.
Let us now briefly describe the organization of the paper. After introducing basic notations and recalling relevant results in Subsection \ref{subsecwidom}, we show that the Widom's determinant $\tau\left[J\right]$ admits a combinatorial series expansion whose individual terms are indexed by tuples of Young diagrams and are given by products of minors of Cauchy-Plemelj operators. In Subsection~\ref{subsec_appl}, we explain how $\tau\left[J\right]$ appears in the isomonodromy theory considering the example of Fuchsian systems with 4 regular singular points and systems with 2 irregular singularities; relations to previously known results on integrable hierarchies are also discussed. Section \ref{secmulti} is devoted to Riemann-Hilbert problems posed on a union of non-intersecting smooth closed curves. Specifically, we propose an extension of $\tau\left[J\right]$ to this case (Definition~\ref{defmultitau}) and establish the differentiation formulae for the so defined tau function with respect to parameters (Theorem~\ref{theomultitau}).
\section{One-circle case\label{seconecircle}}
\subsection{Widom formulas\label{subsecwidom}}
Let $\mathcal C\subset\Cb\Pb^1$ be an anticlockwise oriented cicle centered at the origin, and let $f^{[+]}$ and $f^{[-]}$ denote its interior and exterior. Pick a loop $J:\mathcal C\to\mathrm{GL}\lb N,\Cb\rb$ that can be analytically continued into a fixed annulus $\mathcal A\supset \mathcal C$ and such that $\operatorname{det}J\lb z\rb$ has no winding along $\mathcal C$.
We are going to associate to the pair $\lb \mathcal C,J\rb$ two Riemann-Hilbert problems (RHPs). They ask to find $ \mathrm{GL}\lb N,\Cb\rb$ matrix functions $\Psi_{{\pm}}\lb z\rb$, $\bar\Psi_{{\pm}}\lb z\rb$ analytic in $f^{[\pm]}$ whose boundary values on $\mathcal C$ satisfy
\begin{subequations}\label{RHPs}
\begin{align}
\label{RHPdirect}
\text{direct RHP}:&\quad J\lb z\rb={\Psi_-\lb z\rb}^{-1} \Psi_+\lb z\rb,\\
\label{RHPdual}
\text{dual RHP}:&\quad J\lb z\rb= \bar\Psi_+\lb z\rb{\bar\Psi_-\lb z\rb}^{-1}.
\end{align}
\end{subequations}
It is a classical fact that $J\lb z\rb$ admits Birkhoff factorizations
\beq\label{birkhoff}
J\lb z\rb={Y_-\lb z\rb}^{-1} z^D Y_+\lb z\rb={\bar Y_+\lb z\rb} z^{\bar D} {\bar Y_-\lb z\rb}^{-1},
\eeq
where $Y_{\pm}\lb z\rb $, $\bar Y_{\pm}\lb z\rb $ can be continued to analytic functions in
$f^{[\pm]}\cup \mathcal A$, and $D=\operatorname{diag}\lb d_1,\ldots,d_N\rb$, $\bar D=\operatorname{diag}\lb\bar d_1,\ldots,\bar d_N\rb$ with all $d_k,\bar d_{k'}\in\Zb$ such that $\sum_{k=1}^Nd_k=\sum_{k=1}^N\bar d_k=0$. The sets $\{d_k\}$ and $\{\bar d_k\}$ of partial indices are uniquely determined by $J$. The direct (dual) RHP is solvable iff $D=0$ (resp. $\bar D=0$).
Introduce the Hilbert space $H=L^2\lb \mathcal C,\Cb^N\rb$. Its elements will be regarded as column vector functions. This space can be decomposed as $H=H_+\oplus H_-$, where the functions from $H_+$ ($H_-$) continue analytically inside $\mathcal C$ (resp. outside $\mathcal C$ and vanish at $\infty$). We denote by $\Pi_{\pm}$ the projections on $H_{\pm}$ along $H_{\mp}$.
\begin{defin}\label{tauRHPdef} The tau function of the RHPs defined by $\lb \mathcal C,J\rb$ is defined as Fredholm determinant
\beq\label{tauRHP}
\tau\left[J\right]=\operatorname{det}_{H_+}\lb \Pi_+ J^{-1}\Pi_+
J\rb.
\eeq
\end{defin}
The operator $\Pi_+ J^{-1}\Pi_+
J$ is known to be a trace class perturbation of the identity on $H_+$, which makes the determinant (\ref{tauRHP}) well-defined. The dual RHP is solvable iff the operator $P:=\Pi_+ J^{-1}$ is invertible on $H_+$, in which case its inverse is given by
$P^{-1}=\bar\Psi_+\Pi_+ \bar\Psi_-^{-1}$. Likewise, the direct RHP is solvable iff the operator $Q:=\Pi_+ J$ is invertible, the inverse being equal to $Q^{-1}=\Psi_+^{-1}{\Pi_+ \Psi}_-$. If the direct or dual RHP is not solvable, then either $P$ or $Q$ has a nontrivial kernel and $\tau\left[J\right]$ clearly vanishes.
Suppose that $J\lb z\rb$ admits a direct factorization (\ref{RHPdirect}). Define two Cauchy-Plemelj operators on $H$,
\ben
\mathsf a_H=\Psi_+\Pi_+\Psi_+^{-1}-\Pi_+,\qquad \mathsf d_H=\Psi_-\Pi_-\Psi_-^{-1}-\Pi_-.
\ebn
They can be explicitly written as integral operators
\ben
\lb \mathsf a_H g\rb\lb z\rb=\frac{1}{2\pi i}\oint_{\mathcal C} \mathsf a\lb z,z'\rb g\lb z'\rb dz',\qquad
\lb \mathsf d_H g\rb\lb z\rb=\frac{1}{2\pi i}\oint_{\mathcal C} \mathsf d\lb z,z'\rb g\lb z'\rb dz',
\ebn
where
\beq\label{adops}
\mathsf a\lb z,z'\rb=\frac{\mathbb 1-\Psi_+\lb z\rb
{\Psi_+\lb z'\rb}^{-1}}{z-z'},\qquad
\mathsf d\lb z,z'\rb=\frac{\Psi_-\lb z\rb
{\Psi_-\lb z'\rb}^{-1}-\mathbb 1}{z-z'}.
\eeq
The integral kernels $\mathsf a\lb z,z'\rb$ and $\mathsf d\lb z,z'\rb$ have integrable form, are not singular on the diagonal $z=z'$ and extend to analytic functions on $\mathcal A\times \mathcal A$. Since $\operatorname{im} \mathsf a_H\subseteq H_+\subseteq \operatorname{ker}\mathsf a_H$, $\operatorname{im} \mathsf d_H\subseteq H_-\subseteq \operatorname{ker}\mathsf d_H$, it is convenient to consider the restrictions
\ben
\mathsf a =\mathsf a_H\bigl|_{H_-}: H_-\to H_+,\qquad
\mathsf d =\mathsf d_H\bigl|_{H_+}: H_+\to H_-.
\ebn
\begin{lemma} If the direct RHP (\ref{RHPdirect}) is solvable, then $\tau[J]$ admits Fredholm determinant representation
\beq\label{FR1c}
\tau[J]=\operatorname{det}_H\lb\mathbb 1+L\rb,\qquad
L=\lb \begin{array}{cc}
0 & \mathsf a \\ \mathsf d & 0
\end{array}\rb\in\mathrm{End}\lb H_+\oplus H_-\rb,
\eeq
where integral operators $\mathsf a$ and $\mathsf d$ have block integrable kernels defined by (\ref{adops}).
\end{lemma}
\pf We have
\ben
\operatorname{det}_H\lb\mathbb 1+L\rb=\operatorname{det}_{H_+}
\lb\mathbb 1-\mathsf a\mathsf d\rb
=\operatorname{det}_{H_+}\lb\mathbb 1-
\Psi_+\Pi_+\Psi_+^{-1}\Psi_-\Pi_-
\Psi_-^{-1}\rb=
\operatorname{det}_{H_+}\lb\mathbb 1-
\Pi_+ J^{-1}\Pi_- J\rb,
\ebn
where the last equality is obtained by conjugating the action on $H_+$ by multiplication by $\Psi_+$. Replacing $\Pi_-=\mathbb 1-\Pi_+$ in the last expression, we obtain the determinant (\ref{tauRHP}). \epf
Of course, an analog of the representation (\ref{FR1c}) can also be written for the dual factorization (\ref{RHPdual}). However, from the point of view of applications it is convenient to consider the direct factorization as given; the relevant matrix functions $\Psi_{\pm}$ \textit{define} the jump $J$. The Fredholm determinant (\ref{FR1c}) then yields an explicit representation for $\tau\left[J\right]$, whereas the dual factorization remains to be found.
Let us now recall a formula for derivatives of $\tau\left[J\right]$ with respect to parameters of the jump matrix. It appeared as an intermediate result in the work of Widom \cite{Widom1} in 1974 and was rediscovered in \cite{IJK} more than 30 years later. As we expain below, this result is a precursor of the Jimbo-Miwa-Ueno definition of the isomonodromic tau function \cite{JMU}.
\begin{theo}\label{WidomII} Consider a smooth family of $\,\mathrm{GL}\lb N,\Cb\rb$-loops $(z,t)\mapsto J\lb z,t\rb$ which depend on an additional parameter $t$ and admit both factorizations (\ref{RHPs}). Then
\beq\label{dertauRH}
\partial_t \ln\tau\left[J\right]=\frac1{2\pi i}\oint_{\mathcal C}\operatorname{Tr}\left\{J^{-1}\partial_tJ\left[\partial_z\bar\Psi_-\, {\bar\Psi_-}^{-1}+\Psi_+^{-1}\,\partial_z\Psi_+
\right]\right\}dz.
\eeq
\end{theo}
\pf Using previously defined operators $P=\Pi_+J^{-1}$, $Q=\Pi_+ J$ as well as their inverses $P^{-1}=\bar{\Psi}_+\Pi_+\bar\Psi_-^{-1}$, $Q^{-1}=\Psi_+^{-1}\Pi_+\Psi_-$, we may write
\begin{gather}
\nonumber\partial_t\ln\tau\left[J\right]=
\partial_t\ln\operatorname{det}_{H_+}PQ=
\operatorname{Tr}_{H_+}\lb \partial_t P\,P^{-1}+Q^{-1}\partial_t Q\rb=\\
\label{widom20}
=\operatorname{Tr}_{H_+}\lb-\Pi_+ J^{-1}\partial_t J\,J^{-1}\Pi_+
\bar{\Psi}_+\Pi_+\bar\Psi_-^{-1} +
\Psi_+^{-1}\Pi_+\Psi_-\Pi_+\partial_t J
\rb.
\end{gather}
Let us simplify this expression. First observe that since
$\Pi_+\bar\Psi_+\Pi_+=\bar\Psi_+\Pi_+$ and $\Pi_+\Psi_-\Pi_-=0$,
the expression under trace (recall that this is an operator on $H_+$!) can be rewritten as
\ben
-\Pi_+ J^{-1}\partial_t J\,
\bar{\Psi}_-\Pi_+\bar\Psi_-^{-1} +
\Psi_+^{-1}\Pi_+\Psi_+J^{-1}\partial_t J=
\Pi_+ J^{-1}\partial_t J\,\lb
\bar{\Psi}_-\Pi_-\bar\Psi_-^{-1}-\Pi_-\rb+
\lb\Psi_+^{-1}\Pi_+\Psi_+-\Pi_+\rb J^{-1}\partial_t J .
\ebn
The operators ${\mathsf a}'_{H}=\Psi_+^{-1}\Pi_+\Psi_+-\Pi_+$, $\bar{\mathsf d}_H=\bar\Psi_-\Pi_-\bar\Psi_-^{-1}-\Pi_-$ on $H$ have the properties $\operatorname{im} {\mathsf a}_H'\subseteq H_+$, $H_-\subseteq \operatorname{ker} \bar{\mathsf d}_H$. This allows to extend the trace in \eqref{widom20} to the whole space $H$ and to rewrite it as
\beq\label{widom21}
\partial_t\ln\tau\left[J\right]=
\operatorname{Tr}_{H}\left\{
J^{-1}\partial_t J\lb{\mathsf a}_H'+\bar{\mathsf{d}}_H\rb\right\}.
\eeq
Since $ J^{-1}\partial_t J$ is a multiplication operator, to compute the last trace it suffices to know expressions for the kernels $\mathsf a'\lb z,z'\rb$, $\bar{\mathsf d}\lb z,z'\rb$ of the integral operators $\mathsf a_H'$, $\bar{\mathsf d}_H$ along the diagonal $z=z'$. From the respective counterparts of the formulas (\ref{adops}) it follows that
\ben
\mathsf a'\lb z,z\rb={\Psi_+\lb z\rb}^{-1}
\partial_z\Psi_+\lb z\rb,\qquad
\bar{\mathsf d}\lb z,z\rb=\partial_z\bar\Psi_-\lb z\rb
{\bar\Psi_-\lb z\rb}^{-1}.
\ebn
In combination with (\ref{widom21}), this yields the statement of the theorem.
\epf
\begin{rmk}
Theorem~\ref{WidomII} and its proof above clearly remain valid if we replace the circle $\mathcal C$ of \cite{Widom1,IJK} by any simple closed curve and consider the loops $J$ that continue to its tubular neighborhood. Notice that the right side of (\ref{dertauRH}) is closely related to the Malgrange-Bertola 1-form \cite{Malgrange,Bertola}
\ben
\omega_{\mathrm{MB}}\lb\delta\rb:=\frac1{2\pi i}\oint_{\mathcal C}\operatorname{Tr}\left\{J^{-1}\delta J\, \Psi_+^{-1}\,\partial_z\Psi_+\right\}dz,
\ebn
where $\delta$ is an arbitrary vector field on the space of parameters of $J$.
The curvature of the latter form does not necessarily vanish. Its analog defined by (\ref{dertauRH}) is on the other hand closed by construction, i.e. the contributions to curvature from the direct and dual factorization counterbalance each other.
An attempt to relate the tau function defined by $\omega_{\mathrm{MB}}$ (in those cases where $\omega_{\mathrm{MB}}$ is closed) to Fredholm determinants with integrable kernels was made in \cite{Bertola17}. It corresponds to decomposition of the initial RHP into a sequence of auxiliary RHPs with simpler matrix structure of the jumps, but does not seem to us to be directly related to our construction.
\end{rmk}
\subsection{Combinatorial expansion\label{subseccombi}}
In this subsection we briefly outline some of the results on expansions of the Fredholm determinant $\tau\left[J\right]$. While the previous works \cite{GL16,GL17} focus on RHPs of isomonodromic origin, the combinatorial structure remains the same for generic jump $J$.
\subsubsection{Maya and Young diagrams}
Let $\Zb'=\Zb+\frac12$ be the half-integer lattice, $\Zb'_{\pm}=\Zb'_{\gtrless0}$, and let $\operatorname{Conf}\lb \Zb'\rb=\left\{0,1\right\}^{\Zb'}$ be the set of all finite subsets of $\Zb'$. The elements $X\subset\operatorname{Conf}\lb\Zb'\rb$ determine the positions of particles $\mathsf p_X:=X\cap\Zb'_{+}$ and holes $\mathsf h_X:=X\cap\Zb'_{-}$,
thereby defining point configurations on $\Zb'$. A configuration $X$ may be alternatively represented by
\begin{itemize}
\item A Maya diagram $\mathsf m_X$ obtained by drawing filled circles at
sites ${\lb\left.\Zb'_{+}\right\backslash \mathsf p_X\rb \cup \mathsf h_X}$ and empty circles
at $\mathsf p_X\cup\lb\left.\Zb'_{-}\right\backslash \mathsf h_X\rb$, see Fig.~\ref{Toeplitz_Fig1}. The charge of $\mathsf m_X$ is defined as $\mathsf Q_X=\left| \mathsf p_X\right|-\left| \mathsf h_X\right|$. The set of all Maya diagrams will be denoted by $\mathbb M$.
\item A charged partition $\lb \mathsf Y_X,\mathsf Q_X\rb\in\mathbb Y\times \Zb$ where $\mathbb Y$ denotes the set of partitions $\mathsf Y=\lb\mathsf Y_1\ge\mathsf Y_2\ge\ldots \ge 0\rb$, with all ${\mathsf Y_k\in\Zb_{\ge0}}$. The partitions are identified with Young diagrams in the usual way. The Maya diagram corresponding to a charged partition $\lb \mathsf Y_X,\mathsf Q_X\rb$ can be described by the positions of empty circles, given by $\left\{\mathsf Y_k-k+\frac12+\mathsf Q_X\right\}_{k=1}^{\infty}$, cf Fig.~\ref{Toeplitz_Fig1}.
\end{itemize}
\begin{figure}[h!]
\centering
\includegraphics[height=7cm]{ToeplitzF_Fig1.eps}
\begin{minipage}{0.77\textwidth}
\caption{Correspondence between Maya and Young diagrams. The positions of particles and holes are $\mathsf p_X=\left\{\frac12,\frac52,\frac{13}2\right\}$ and
$\mathsf h_X=\left\{-\frac{21}{2},-\frac{15}2,-\frac{11}2,
-\frac92,-\frac32,-\frac12\right\}$. The charge $\mathsf Q_X=-3$ corresponds to signed distance between the vertical axis and left boundary of the profile of $\mathsf Y_X$.
\label{Toeplitz_Fig1}}
\end{minipage}
\end{figure}
Let $L\in\Cb^{\mathfrak X\times\mathfrak X}$ be a matrix indexed by a discrete set $\mathfrak X$. The latter can be infinite, in which case $L$ is required to be a trace class operator on $\ell^2\lb\mathfrak X\rb$. The determinant $\operatorname{det}\lb \mathbb 1+L\rb$ can be expressed as the sum of principal minors enumerated by all possible subsets of $\mathfrak X$:
\beq\label{vKf}
\operatorname{det}\lb \mathbb 1+L\rb=\sum_{\mathfrak Y\in\left\{0,1\right\}^{\mathfrak X}}\operatorname{det} L_{\mathfrak Y},
\eeq
i.e. $L_{\mathfrak Y}$ is the restriction of $L$ to rows and columns $\mathfrak Y$.
In order to apply this formula to the determinant (\ref{FR1c}), rewrite the integral operators $\mathsf a$ and $\mathsf d$ in the Fourier basis. Their kernels (\ref{adops}) may be expressed as
\beq\label{fourierad}
\mathsf{a}\lb z,z'\rb=\sum_{p,q\in\Zb'_{+}}\mathsf{a}^{\;\;\;\,p}_{-q}
z^{-\frac12+p}z'^{-\frac12+q},\qquad
\mathsf{d}\lb z,z'\rb=\sum_{p,q\in\Zb'_{+}}\mathsf{d}_{\;\;\;\,p}^{-q}
z^{-\frac12-q}z'^{-\frac12-p},
\eeq
where the coefficients $\mathsf{a}^{\;\;\;\,p}_{-q}, \mathsf{d}_{\;\;\;\,p}^{-q}$ are themselves $N\times N$ matrices whose elements we write as
$\mathsf{a}^{\;\;\;\,p;\alpha}_{-q;\beta},
\mathsf{d}_{\;\;\;\,p;\beta}^{-q;\alpha}$. The ``color'' indices $\alpha,\beta=1,\ldots,N$ correspond to $\mathrm{GL}\lb N,\mathbb C\rb$-matrix structure of the RHP defined by the loop $J$. The principal minors of $L$ in (\ref{FR1c}) are therefore labeled by $N$-tuples of Maya diagrams
\ben
\begin{gathered}
\boldsymbol{\mathsf m}=\lb \mathsf m_1,\ldots,\mathsf m_N\rb=
\bigl(\boldsymbol{\mathsf p},\boldsymbol{\mathsf h}\bigr)\in \mathbb M^N,\\
\boldsymbol{\mathsf p}=\mathsf p_1\sqcup\ldots\sqcup \mathsf p_N,\qquad
\boldsymbol{\mathsf h}=\mathsf h_1\sqcup\ldots\sqcup \mathsf h_N.
\end{gathered}
\ebn
Here $\mathsf p_{\alpha}\in \left\{0,1\right\}^{\Zb'_{+}}$, $\mathsf h_{\alpha}\in \left\{0,1\right\}^{\Zb'_{-}}$ denote the positions of particles and holes of color $\alpha\in\left\{1,\ldots,N\right\}$. The minors with $| \boldsymbol{\mathsf p}|\ne| \boldsymbol{\mathsf h}|$ clearly vanish, cf Fig.~\ref{Toeplitz_Fig2}. We may thus restrict the summation to $N$-tuples of Maya diagrams of zero total charge,
\beq\label{combiexp}
\begin{gathered}
\tau\left[J\right]=\sum_{\boldsymbol{\mathsf m}\in\mathbb M^N:\,|\boldsymbol{\mathsf p}|=| \boldsymbol{\mathsf h}|}
Z_{\;\boldsymbol{\mathsf m}}^{[+]}
Z_{\;\boldsymbol{\mathsf m}}^{[-]},\\
Z_{\;\boldsymbol{\mathsf m}}^{[+]}=\operatorname{det}
\mathsf a^{\,\boldsymbol{\mathsf p}}_{\,\boldsymbol{\mathsf h}},\qquad
Z_{\;\boldsymbol{\mathsf m}}^{[-]}=\lb -1\rb^{|\boldsymbol{\mathsf p}|} \operatorname{det}
\mathsf d^{\,\boldsymbol{\mathsf h}}_{\,\boldsymbol{\mathsf p}}.
\end{gathered}
\eeq
The matrices $\mathsf a^{\,\boldsymbol{\mathsf p}}_{\,\boldsymbol{\mathsf h}},\mathsf d^{\,\boldsymbol{\mathsf h}}_{\,\boldsymbol{\mathsf p}}\in\operatorname{Mat}_{|\mathsf p|\times |\mathsf p| }\lb\Cb\rb$ correspond to the upper-right and lower-left block in the principal minor in Fig.~\ref{Toeplitz_Fig2}.
Using the identification of Maya diagrams and charged partitions described above, the individual contributions to (\ref{combiexp}) may also be labeled by an $N$-tuple of partitions
$\boldsymbol{\mathsf Y}\in\mathbb Y^N$ and an integer charge vector
$\boldsymbol{\mathsf Q}\in \mathfrak Q_{N-1}$ from the $A_{N-1}$ root lattice
\ben
\mathfrak Q_{N-1}=\left\{\lb Q_1,\ldots,Q_N\rb\in\Zb^N\Bigl|\Bigr.\,\sum\nolimits_{\alpha=1}^NQ_{\alpha}=0\right\}.
\ebn
Adapting the notation, the combinatorial expansion (\ref{combiexp}) may then be written as
\beq\label{combiexp2}
\tau\left[J\right]=\sum_{\boldsymbol{\mathsf Q}\in\mathfrak Q_{N-1}}
\sum_{\boldsymbol{\mathsf Y}\in\mathbb Y^N}
Z_{\;\boldsymbol{\mathsf Y},\boldsymbol{\mathsf Q}}^{[+]}
Z_{\;\boldsymbol{\mathsf Y},\boldsymbol{\mathsf Q}}^{[-]}.
\eeq
The structure of this series coincides with that of the dual Nekrasov-Okounkov partition functions introduced in \cite{NO}; in fact, in some cases these partition functions can be obtained as specializations of \eqref{combiexp2}.
\begin{figure}[h!]
\centering
\includegraphics[height=5.5cm]{ToeplitzF_Fig2.eps}
\begin{minipage}{0.77\textwidth}
\caption{Example of labeling of principal minors with $N=3$ colors and $|\boldsymbol{\mathsf p}|=
|\boldsymbol{\mathsf h}|=5$.
\label{Toeplitz_Fig2}}
\end{minipage}
\end{figure}
\begin{rmk}
If $L$ is such that all principal minors $\operatorname{det}L_{\mathfrak Y}$ in (\ref{vKf}) are non-negative, then $\operatorname{Prob}\lb\mathfrak Y\rb:=\ds\frac{\operatorname{det}L_{\mathfrak Y}}{\operatorname{det}\lb \mathbb 1+L\rb}$ may be interpreted as a probability measure for a random point process on $\mathfrak X$ called the $L$-ensemble. This process is well-known to be determinantal and to have the correlation kernel $K=\ds\frac{L}{\mathbb 1+L}$. In our case, $L$ comes from a rewrite of the definition (\ref{tauRHP}) of $\tau\left[J\right]$ and
\ben
\mathfrak X\cong \left\{\boldsymbol{\mathsf m}\in\mathbb M^N:
\,|\boldsymbol{\mathsf p}|=| \boldsymbol{\mathsf h}| \right\}\cong \mathfrak Q_{N-1}\times\mathbb Y^N.
\ebn
Explicit formulae for the inverses of $P$ and $Q$, already used in the proof of Theorem~\ref{WidomII}, then allow to express $K \in\mathrm{End}\lb H_+\oplus H_-\rb$ in terms of solutions of the direct and dual RHPs,
\ben
K=\lb\begin{array}{rr}
\Pi_+\Psi_+\bar{\Psi}_-\Pi_- \bar{\Psi}_-^{-1}\Psi_+^{-1}\Pi_+ &
- \Pi_+\Psi_+\bar{\Psi}_-\Pi_- \bar{\Psi}_-^{-1}\Psi_+^{-1}\Pi_- \\
-\Pi_-\Psi_+\bar{\Psi}_-\Pi_+ \bar{\Psi}_-^{-1}\Psi_+^{-1}\Pi_+ &
\Pi_-\Psi_+\bar{\Psi}_-\Pi_+ \bar{\Psi}_-^{-1}\Psi_+^{-1}\Pi_-
\end{array}\rb.
\ebn
\end{rmk}
Denote by $\chi_{\mathfrak J} $ the indicator
function of any subset $\mathfrak J \subseteq \mathbb{Z}'$.
It is worth observing that the component $K_{++} := \chi_{\Zb_+'^N} K\chi_{\Zb_+'^N}$, i.e. the term in the upper-left corner of the matrix representation above, is nothing but the Fredholm operator appearing in the celebrated Borodin-Okounkov formula \cite{BO}, in its matrix version\footnote{For comparison, one should use that $K_{++} = \Pi_+\Psi_-\bar{\Psi}_+\Pi_- \bar{\Psi}_-^{-1}\Psi_+^{-1}\Pi_+$ and compare the factorizations of the jump $\phi$ in \cite{BW} with those of $J^{-1}$ in the present paper. } \cite{BW}. Hence, the (gap) probability of finding no particles of any colour in the sites $\mathfrak J_n=\left\{k+\tfrac12\in\Zb',k\ge n\right\} $ is given by
\beq
\operatorname{Prob}\lb\mathbf{p} \mathop{\cap} \mathfrak J_n^N = \emptyset \rb = \det \lb\mathbb{1} - \chi_{\mathfrak J_n^N}K_{++}\chi_{\mathfrak J_n^N} \rb = \frac{\det T_n\left[ J^{-1}\right]}{\tau\left[J\right]},
\eeq
where the last equality (valid under assumption that $\operatorname{det}J\lb z\rb$ has geometric mean $1$) is precisely the content of the Borodin--Okounkov formula.
\subsubsection{Grassmannian interpretation}\label{Grassmannians}
It is possible to give an interpretation of the formulas above in the setting of the Sato-Segal-Wilson theory of infinite-dimensional Grassmannians. We will start with the analytic theory \cite{SW} and then comment on the relation with Sato's formal definition of tau function \cite{Sato}. Consider the point $W := \Psi_- \cdot H_+$ in the Segal-Wilson Grassmannian $\mathrm{Gr}\lb H\rb$.
The subspace $W$ is spanned by the columns of the (rectangular) matrix
\ben
G^{[-]} := \begin{pmatrix} \Pi_+ \Psi_- \Pi_+ \\ \Pi_- \Psi_- \Pi_+\end{pmatrix}.
\ebn
This is a \emph{frame} for the point $W$. More generally, a frame for $W$ will be a rectangular matrix $(w_+, w_-)^{T}$ whose columns span $W$, and the frame will be called \emph{admissible} if $w_+ - \mathbb{1}$ is of trace class on $H_+$. Of course, in general $G^{[-]}$ will not be admissible. Nevertheless, since $\Pi_+ \Psi_- \Pi_+$ is invertible, there is a canonical way to transform $G^{[-]}$ into an admissible frame by right multiplication:
\ben
G^{[-]} \mapsto G^{[-]}\lb\Pi_+ \Psi_- \Pi_+\rb^{-1} = \begin{pmatrix} \mathbb{1}\\ \Pi_- \Psi_- \Pi_+\Psi^{-1}_- \Pi_+ \end{pmatrix} = \begin{pmatrix} \mathbb{1}\\ -{\mathsf d}\end{pmatrix}
\ebn
In other words, $-\mathsf{d}$ is the map whose graph is equal to $W$. Now, we can act on $W$ with multiplication by $\Psi_+^{-1}$, and the Segal-Wilson tau function $\tau_W\lb \Psi_+\rb$ is defined by the formula
\beq
\tau_W\lb\Psi_+\rb.\Psi_+^{-1}\sigma\lb W\rb = \sigma\lb\Psi_+^{-1}W\rb,
\eeq
where $\sigma$ is the canonical global section of the determinant line bundle $\mathrm{Det}^*$ over $\mathrm{Gr}\lb H\rb$, and the action of $\Psi_+^{-1}$ on $\mathrm{Gr}\lb H\rb$ is extended to $\mathrm{Det}^*$. The reader is referred to \cite{SW} for the details. What is important here is that, since the operator of multiplication by $\Psi_+^{-1}$ has block form of type
\ben
\Psi_+^{-1} = \begin{pmatrix} \Pi_+ \Psi_+^{-1} \Pi_+ & \Pi_+ \Psi_+^{-1} \Pi_-\\ 0 & \Pi_- \Psi_+^{-1} \Pi_- \end{pmatrix},
\ebn
the tau function is given by the Fredholm determinant, see \cite[formula (3.5)]{SW},
\beq\label{SWtau}
\tau_{W}\lb\Psi_+\rb = \mathrm{det}_{H_+}\lb\mathbb{1} - \Pi_+ \Psi_+ \Pi_+ \Psi_+^{-1} \Pi_-\mathsf{d}\rb = \mathrm{det}_{H_+}\lb\mathbb{1} - \mathsf{a}\mathsf{d}\rb,
\eeq
so that finally we have $\tau_W\lb \Psi_+\rb = \tau\left[J\right]$.
If we are willing to work with formal series instead of analytic functions, there is no reason to restrict to admissible frames and, in Sato's style \cite{Sato}, one can simply define the tau function as
\beq\label{Satotau1}
\tau_W\lb\Psi_+\rb = \mathrm{det}_{H_+}\Big(G^{[+]}G^{[-]}\Big),
\eeq
where $G^{[+]}$ and $G^{[-]}$ are given, respectively, by the matrices associated to $\Pi_+ \Psi_+^{-1}$ and $\Psi_- \Pi_+$:
\ben
\Pi_+ \Psi_+^{-1} = G^{[+]} := \begin{pmatrix} \Pi_+ \Psi_+^{-1} \Pi_+ & \Pi_+ \Psi_+^{-1} \Pi_-\end{pmatrix}, \quad \quad \Psi_-\Pi_+ = G^{[-]} := \begin{pmatrix} \Pi_+ \Psi_- \Pi_+ \\ \Pi_- \Psi_- \Pi_+ \end{pmatrix}.
\ebn
Since $ \Pi_+ \Psi_+^{-1} \Pi_+$ and $ \Pi_+ \Psi_- \Pi_+$ are respectively upper and lower triangular with the identity on the main diagonal, the two definitions \eqref{SWtau} and \eqref{Satotau1} are (formally!) the same. Nevertheless, $G^{[+]}G^{[-]} - \mathbb{1}$ is not a trace class operator, therefore the determinant in \eqref{Satotau1} is to be understood as the limit of the determinant whose size goes to infinity. Indeed, this is nothing but $\operatorname{det}T_{n\to\infty}\left[J^{-1}\right]$, and the equality between \eqref{SWtau} and \eqref{Satotau1} is simply a rephrasing of the Widom's theorem stated in the introduction. The way to compute the latter is through the Cauchy-Binet formula. Namely, for any $N$-tuple $\boldsymbol{\mathsf m}$ of Maya diagrams of zero total charge, define the associated Pl\"ucker coordinates $G^{[\pm]}_{\boldsymbol{\mathsf m}}$ as the determinants of the square matrices obtained by choosing the columns/lines of the matrix $G^{[\pm]}$ in correspondence with the filled circles in $\boldsymbol{\mathsf m}$. Then $\tau_W\lb\Psi_+\rb$ is given by
\beq\label{SatoTau}
\tau_W\lb\Psi_+\rb = \sum_{\boldsymbol{\mathsf m}\in\mathbb M^N:\,|\boldsymbol{\mathsf p}|=| \boldsymbol{\mathsf h}|} G^{[+]}_{\boldsymbol{\mathsf m}}G^{[-]}_{\boldsymbol{\mathsf m}}.
\eeq
It is natural to wonder what is the relation between the two expansions \eqref{SatoTau} and \eqref{combiexp}. The answer is that they are actually the same, since $G^{[\pm]}_{\boldsymbol{\mathsf m}} =
Z^{[\pm]}_{\boldsymbol{\mathsf m}}$.
This identity (cf Proposition 2.1 in \cite{EH}) is indeed the main step in the proof of the so-called Giambelli identity, relating Pl\"ucker coordinates associated to an arbitrary Young diagram to the hooked ones. We conclude this subsection by noticing that the integral formula \eqref{adops} for the so-called affine coordinates $\mathsf{a}$, $-\mathsf{d}$ already appeared, in the context of Gelfand-Dickey equations, in \cite{BD}.
\subsubsection{Matrix elements}
The reader might wonder when matrix elements $\mathsf{a}^{\;\;\;\,p}_{-q}, \mathsf{d}_{\;\;\;\,p}^{-q}$ become effectively computable. In applications to integrable hierarchies, the entries of $\mathsf{a}$ are universal (i.e. they do not depend on the solution but just on the hierarchy) and are given explicitly in terms of the elementary Schur polynomials, while $\mathsf{d}$ determines the point in the Grassmannian corresponding to the given solution, see Subsection \ref{subsecIH} below for more details. Another typical situation where such calculation is possible occurs in the context of monodromy preserving deformations. Suppose that $\Psi_{\pm}\lb z\rb$ satisfy
\beq\label{DEcond}
\partial_z\Psi_{\pm}\lb z\rb= \Psi_{\pm}\lb z\rb A_{\pm}\lb z\rb+z^{-1}\Lambda_{\pm}\lb z\rb\Psi_{\pm}\lb z\rb ,
\eeq
with $A_{\pm}\lb z\rb$ rational in $z$ and $\Lambda_{\pm}\lb z\rb$ polynomial in $z^{\pm1}$. The latter condition holds in a number of examples where $\Psi_{\pm}\lb z\rb$ are related to fundamental matrix solutions of linear systems. It should be seen as an analog of the conditions used in \cite{TW93} to derive nonlinear PDEs satisfied by Fredholm determinants of certain scalar integrable kernels.
Introduce the operator $\mathcal L_0=z\partial_z+z'\partial_{z'}+1$. Since $\mathcal L_0\frac{1}{z-z'}=0$, we have
\beq\label{rhsME}
\mathcal L_0 \mathsf a_{\pm}\lb z,z'\rb=\pm\Psi_{\pm}\lb z\rb \mathbf A_{\pm}\lb z,z'\rb{\Psi_{\pm}\lb z'\rb}^{-1}+
\Lambda_{\pm}\lb z\rb \mathsf a_{\pm}\lb z,z'\rb-
\mathsf a_{\pm}\lb z,z'\rb \Lambda_{\pm}\lb z'\rb\pm
\boldsymbol{\Lambda}_{\pm}\lb z,z'\rb,
\eeq
where $\mathsf a_{+}\lb z,z'\rb=\mathsf a\lb z,z'\rb$, $\mathsf a_{-}\lb z,z'\rb=\mathsf d\lb z,z'\rb$ and
\ben
\mathbf A_{\pm}\lb z,z'\rb=\frac{
z'A_{\pm}\lb z'\rb-zA_{\pm}\lb z\rb}{z-z'},\qquad
\boldsymbol{\Lambda}_{\pm}\lb z,z'\rb=\frac{
\Lambda_{\pm}\lb z'\rb-\Lambda_{\pm}\lb z\rb}{z-z'}.
\ebn
There exist $M_{\pm}\in\Zb_{\ge0}$ such that $\mathbf A_{\pm}\lb z,z'\rb=\sum_{m=1}^{M_{\pm}}\varphi_{m,\pm}\lb z\rb\otimes
\bar\varphi_{m,\pm}\lb z'\rb$, where $\varphi_{m,\pm}$ and
$\bar\varphi_{m,\pm}$ are column and raw $N$-vectors. On the other hand, applying $\mathcal L_0$ directly to Fourier expansions (\ref{fourierad}), one has
\beq\label{lhsME}
\begin{aligned}
\mathcal L_0\mathsf{a}\lb z,z'\rb=&\;\;\;\;\,\sum\nolimits_{p,q\in\Zb'_{+}}\lb p+q\rb \mathsf{a}^{\;\;\;\,p}_{-q}
z^{-\frac12+p}z'^{-\frac12+q},\\
\mathcal L_0\mathsf{d}\lb z,z'\rb=&-\sum\nolimits_{p,q\in\Zb'_{+}}\lb p+q\rb \mathsf{d}_{\;\;\;\,p}^{-q}
z^{-\frac12-q}z'^{-\frac12-p}.
\end{aligned}
\eeq
Comparing this expression with (\ref{rhsME}), we obtain a system of linear equations that determine $\mathsf{a}^{\;\;\;\,p}_{-q}$, $\mathsf{d}_{\;\;\;\,p}^{-q}$ in terms of Fourier modes of $\Psi_{\pm}\lb z\rb\varphi_{m,\pm}\lb z\rb$,
$\bar\varphi_{m,\pm}\lb z\rb {\Psi_{\pm}\lb z\rb}^{-1}$ and the coefficients of $\Lambda_{\pm}\lb z\rb\in \operatorname{Mat}_{N\times N}\lb\Cb\left[z^{\pm1}\right]\rb$, $\boldsymbol{\Lambda}_{\pm}\lb z,z'\rb\in\operatorname{Mat}_{N\times N}\lb\Cb\left[z^{\pm1},z'{}^{\pm1}\right]\rb$.
The simplest nontrivial situation corresponds to $M_+=M_-=1$ and $\Lambda_{\pm}$ given by constant diagonal matrices. It occurs, in particular, for generic (non-logarithmic) solutions of Painlevé VI, V and III. In these cases, (\ref{rhsME}) and (\ref{lhsME}) imply that
\ben
\begin{aligned}
\sum\nolimits_{p,q\in\Zb'_{+}}\lb p+q-\mathrm{ad}_{\Lambda_+}\rb\mathsf{a}^{\;\;\;\,p}_{-q}
z^{-\frac12+p}z'^{-\frac12+q}=&\Psi_+\lb z\rb\varphi_{+}\lb z\rb\otimes
\bar\varphi_{+}\lb z'\rb \Psi_+\lb z'\rb^{-1},\\
\sum\nolimits_{p,q\in\Zb'_{+}}\lb p+q+\mathrm{ad}_{\Lambda_-}\rb\mathsf{d}_{\;\;\;\,p}^{-q}
z^{-\frac12-q}z'^{-\frac12-p}=&\Psi_-\lb z\rb\varphi_{-}\lb z\rb\otimes
\bar\varphi_{-}\lb z'\rb \Psi_-\lb z'\rb^{-1}.
\end{aligned}
\ebn The modes $\mathsf{a}^{\;\;\;\,p}_{-q}$, $\mathsf{d}_{\;\;\;\,p}^{-q}$ are therefore given by Cauchy matrices which in turn implies that the determinants
$Z_{\;\boldsymbol{\mathsf Y},\boldsymbol{\mathsf Q}}^{[\pm]}$ have nice factorized expressions.
\subsection{Applications}\label{subsec_appl}
In this subsection, we demonstrate the relation between our Definition~\ref{tauRHPdef} and tau functions of certain classes of isomonodromic systems and integrable hierarchies. The general strategy is to reduce the associated linear problem to a RHP on a circle and make use of Widom's differentiation formula (Theorem~\ref{WidomII}).
\subsubsection{Four regular singularities}\label{subsec4reg}
Our basic example deals with a linear system with four Fuchsian singularities placed at $0,t,1,\infty$. It is given by
\beq \label{fuchs4}
\partial_z\Phi=\Phi A\lb z\rb,\qquad A\lb z\rb=\frac{A_0}{z}+\frac{A_t}{z-t}+
\frac{A_1}{z-1},
\eeq
with $ A_{0,t,1}\in\operatorname{Mat}_{N\times N}\lb \Cb\rb$.
Consider generic situation where $A_{0,t,1}$ and $A_{\infty}:=-A_0-A_t-A_1$ are diagonalizable. For $a=0,t,1,\infty$, fix the diagonalizations $A_{a}=G_{a}^{-1}\Theta_{a} G_{a}$ with diagonal $\Theta_{a}$.
Assume that the eigenvalues of $A_{a}$ are distinct mod $\mathbb Z$. Then there exist unique fundamental matrix solutions $\Phi^{(a)}\lb z\rb$ of (\ref{fuchs4}), holomorphic on the universal covering of $\Cb\backslash\left\{0,t,1\right\}$ and such that
\ben
\Phi^{(a)}\lb z\rb=\begin{cases} \lb a-z\rb^{\Theta_{a}}G^{(a)}\lb z\rb,\qquad &\text{for }a=0,t,1, \\ \lb -z\rb^{-\Theta_{\infty}}G^{(\infty)}\lb z\rb,\qquad &\text{for } a=\infty,
\end{cases}
\ebn
where $G^{(a)}\lb z\rb$ is holomorphic and invertible in a finite open disk around $z=a$ and satisfies the normalization condition ${G^{(a)}\lb a\rb=G_{a}}$.
Further assume for notational simplicity that $t\in\lb 0,1\rb$. The canonical solutions $\Phi^{(0,\infty)}\lb z\rb$ analytically continue to single-valued matrix functions on the cut Riemann sphere $\Cb\Pb^1\backslash\mathbb R_{\ge0}$. Similarly, $\Phi^{(t)}\lb z\rb$
and $\Phi^{(1)}\lb z\rb$ are naturally defined on $\Cb\Pb^1\backslash\left((-\infty,0]\cup[t,\infty)\right)$ and
$\Cb\Pb^1\backslash\left((-\infty,t]\cup[1,\infty)\right)$, respectively.
Take an arbitrary fundamental solution $\Phi\lb z\rb$, defined on $\Cb\Pb^1\backslash\mathbb R_{\ge0}$. The connection matrices
$C_{a,\epsilon}=\Phi\lb z\rb {\Phi^{(a)}\lb z\rb}^{-1}$, with $\epsilon = \operatorname{sgn}\Im z$, are independent of $z$. They satisfy the compatibility conditions
\beq\label{compa}
\begin{gathered}
\begin{aligned}C_{0,+}=C_{0,-},&\qquad C_{\infty,+}=C_{\infty,-},\\
M_0=C_{0,-}e^{2\pi i\Theta_0}C_{0,+}^{-1}=C_{t,-}C_{t,+}^{-1},&\qquad M_{\infty}^{-1}=C_{1,-}e^{2\pi i\Theta_1}C_{1,+}^{-1}=
C_{\infty,-}e^{-2\pi i\Theta_{\infty}}C_{\infty,+}^{-1},
\end{aligned}\\
M_0M_t=\lb M_1 M_{\infty}\rb^{-1}=C_{t,-}e^{2\pi i\Theta_t}C_{t,+}^{-1}=C_{1,-}C_{1,+}^{-1}.\qquad
\end{gathered}
\eeq
where $M_{a}$ denotes anticlockwise monodromy matrix of $\Phi\lb z\rb$ around the Fuchsian singular point $a\in\left\{0,t,1,\infty\right\}$. The connection matrices $\{C_{a,\pm}\}$ and exponents $\left\{\Theta_{a}\right\}$ of local monodromy constitute the monodromy data for the 4-point Fuchsian system (\ref{fuchs4}).
Let us now explain how to transform (\ref{fuchs4}) into a Riemann-Hilbert problem on a circle. This will be achieved in several steps:
\begin{enumerate}
\item Start with the contour $\tilde{\Gamma}$ shown in Fig.~\ref{Toeplitz_Fig3}a by solid black curves. Denote by $D_{a}$ the disk around $z=a$ bounded by $\gamma_{a}$ and define
\ben
\tilde\Psi\lb z\rb=\begin{cases}
G^{(a)}\lb z\rb,\qquad& z\in D_{a},\\
\Phi\lb z\rb, \qquad &z\notin\mathbb R_{\ge0}\cup \bar D_0\cup \bar D_t\cup \bar D_1 \cup \bar D_{\infty}.
\end{cases}
\ebn
Comparing with (\ref{RHPdual}), we see that the matrix function $\tilde{\Psi}\lb z\rb$ solves a dual RHP set on $\tilde \Gamma$ with the jumps indicated in Fig.~\ref{Toeplitz_Fig3}a.
\item Next cancel the constant jump $\lb M_0M_t\rb^{-1}$ on the real segment cut out by the dashed red circles $\mathcal C_{\mathrm{out,in}}$. To this end, let us write $M_0M_t=e^{2\pi i \mathfrak S}$. There is a certain freedom in the choice of $\mathfrak S$; for example, in the generic situation where $\mathfrak S$ may be assumed diagonal, we may add to it any integer diagonal matrix. Denote by $\hat{\mathcal A}$ the open annulus bounded by $\mathcal C_{\mathrm{out,in}}$ and set
\ben
\hat\Psi\lb z\rb=\begin{cases}
\lb -z\rb^{-\mathfrak S}\tilde\Psi\lb z\rb,\qquad& z\in\hat{\mathcal A},\\
\tilde\Psi\lb z\rb, \qquad &z\notin\bar{\hat{\mathcal A}}.
\end{cases}
\ebn
The dual RHP for $\hat{\Psi}\lb z\rb$ is set on the contour $\hat \Gamma$ indicated in Fig.~\ref{Toeplitz_Fig3}b by solid black lines. The jump matrices associated to $\mathcal C_{\mathrm{out}}$ and $\mathcal C_{\mathrm{in}}$ are $\lb -z\rb^{-\mathfrak S}$; on the rest of the contour the jumps are the same as for~$\tilde\Psi\lb z\rb$.
\item The contour $\hat\Gamma$ has two connected components, $\hat\Gamma_-$ and $\hat\Gamma_+$, containing respectively $\mathcal C_{\mathrm{out}}$ and $\mathcal C_{\mathrm{in}}$. Choose $\mathfrak S$ so that $\operatorname{Tr} \mathfrak S=\operatorname{Tr}\lb \Theta_0+\Theta_{t}\rb=-\operatorname{Tr}\lb \Theta_1+\Theta_{\infty}\rb$ (this choice still allows for $\mathfrak Q_{N-1}$-shifts). The RHPs obtained by restricting the initial contour to $\hat \Gamma_-$ or $\hat\Gamma_+$ while keeping the same jumps are then generically solvable. Their solutions are related to fundamental matrices $\Phi_-\lb z\rb$ and $\Phi_+\lb z\rb$ of 3-point Fuchsian systems whose singular poins are $0,t,\infty$ and $0,1,\infty$. Let us denote these solutions by $\Psi_-\lb z\rb$ and $\Psi_+\lb z\rb$. The subscript reminds that these functions are analytic outside $\mathcal C_{\mathrm{out}}$ and inside $\mathcal C_{\mathrm{in}}$, respectively.
Consider an auxiliary circle $\mathcal C$ inside $\hat{\mathcal A}$, indicated by dashed red line in Fig.~\ref{Toeplitz_Fig3}b, and define
\beq\label{psi1c}
\bar\Psi\lb z\rb=
\begin{cases}
{\Psi_+\lb z\rb}^{-1}\hat\Psi\lb z\rb,\qquad & \text{outside }\mathcal C,\\
{\Psi_-\lb z\rb}^{-1}\hat\Psi\lb z\rb,\qquad &
\text{inside }\mathcal C.
\end{cases}
\eeq
The matrix function $\bar\Psi\lb z\rb$ has no jumps except on $\mathcal C$. The jump of the relevant \textit{dual} RHP is written in the form of \textit{direct} factorization,
\beq\label{jump4fuchs}
J\lb z\rb={\Psi_-\lb z\rb}^{-1}\Psi_+\lb z\rb,
\eeq
cf (\ref{RHPdirect}). The problem of solving the 4-point Fuchsian system with a prescribed monodromy is therefore converted into a RHP for $\bar\Psi_{\pm}\lb z\rb$ on a single circle (Fig.~\ref{Toeplitz_Fig3}c), with the jump matrix expressed in terms of 3-point solutions $\Psi_{\pm}\lb z\rb$. The latter will be considered as known, even though their explicit expressions in higher rank $N\ge 3$ are available only in a few special cases (rigid systems, etc).
\end{enumerate}
\begin{figure}[h!]
\centering
\includegraphics[height=6.5cm]{ToeplitzF_Fig3.eps}
\begin{minipage}{0.77\textwidth}
\caption{RH contours for (a) $\tilde \Psi$
(b) $\hat \Psi$ (c) $\bar\Psi$.
\label{Toeplitz_Fig3}}
\end{minipage}
\end{figure}
Let us now apply the results of Subsection~\ref{subsecwidom}. To the 4-point Fuchsian system (\ref{fuchs4}) we associate a tau function $\tau\lb t\rb\equiv \tau\left[J\right]$ defined by (\ref{adops})--(\ref{FR1c}) as a Fredholm determinant with a block integrable kernel given in terms of 3-point solutions. Theorem~\ref{WidomII} then yields
\begin{cor}\label{cor4pt} Let $\tau_{\mathrm{JMU}}\lb t\rb$ denote the Jimbo-Miwa-Ueno tau function of (\ref{fuchs4}) defined by
\beq\label{taujmu1}
\partial_t\ln\tau_{\mathrm{JMU}}\lb t\rb=\frac{\operatorname{Tr}
A_0 A_t}{t}+\frac{\operatorname{Tr}
A_t A_1}{t-1}.
\eeq
It coincides with $\tau\lb t\rb$ defined by (\ref{adops})--(\ref{FR1c}) up to a trivial prefactor,
\beq\label{tau4tauJMU}
\tau_{\mathrm{JMU}}\lb t\rb=\operatorname{const}\cdot t^{\frac12 \operatorname{Tr}\lb \mathfrak S^2-\Theta_0^2-\Theta_t^2\rb}
\tau\lb t\rb.
\eeq
\end{cor}
\pf On the circle $\mathcal C$, we have
\beq\label{special01}
\Psi_{\pm}\lb z\rb=\lb -z\rb^{-\mathfrak S}\Phi_{\pm}\lb z\rb,
\qquad
\bar\Psi_{\pm}\lb z\rb= {\Phi_{\mp}\lb z\rb}^{-1}\Phi\lb z\rb,
\qquad
J\lb z\rb={\Phi_{-}\lb z\rb}^{-1}\Phi_{+}\lb z\rb.
\eeq
It is convenient to choose the normalization of the auxiliary fundamental solutions so that $\Phi_+\lb z\rb\simeq \lb -z\rb^{\mathfrak S}$ as $z\to0$ and $\Phi_-\lb z\rb\simeq \lb -z\rb^{\mathfrak S}$ as $z\to\infty$. In particular, in this normalization $\Phi_+\lb z\rb$ is independent of $t$. Substituting (\ref{special01}) into the
Widom's formula (\ref{dertauRH}), we then obtain
\beq\label{special02}
\partial_t\ln\tau\lb t\rb=\frac1{2\pi i}\int_{\mathcal C}\operatorname{Tr}\lb \partial_t\Phi_-\,\Phi_-^{-1}\frac{\mathfrak S}{z} -\partial_t\Phi_-\,\Phi_-^{-1} \partial_z\Phi\;\Phi^{-1}\rb dz.
\eeq
Observe that the first and second term under trace are meromorphic, respectively, outside and inside $\mathcal C$ with the only possible pole at $z=\infty$ and $z=0,t$. This effectively reduces the above integral to residue calculation. Indeed, the analog of (\ref{fuchs4})
for $\Phi_-$ is given by
\beq \label{fuchs3}
\partial_z\Phi_-=\Phi_- A^{-}\lb z\rb,\qquad A^-\lb z\rb=\frac{A^-_{0}}{z}+\frac{A^-_{t}}{z-t},
\eeq
where $A^-_0$, $A^-_t$ and $A^-_0+A^-_t$ belong to conjugacy classes of $\Theta_0$, $\Theta_t$ and $\mathfrak S$. Conservation of monodromy upon variation of $t$ gives one more equation,
$\partial_t \Phi_-=-\Phi_- A^-_t \lb z-t\rb^{-1}$, which implies that the first term in (\ref{special02}) (given by the residue at $z=\infty$) vanishes. The second term may be rewritten as
\beq\label{special03}
\frac1{2\pi i}\int_{\mathcal C}\operatorname{Tr}\lb \frac{A^-_t}{z-t}\Phi_-^{-1}\Phi A\lb z\rb \Phi^{-1}\Phi_-\rb dz
=\partial_t\ln\tau_{\mathrm{JMU}}\lb t\rb-\frac{\operatorname{Tr}A^-_0A^-_t}{t}+\operatorname{res}_{z=t}
\frac{ \operatorname{Tr}A^-_t\Phi_-^{-1}\Phi A_t \Phi^{-1}\Phi_-}{\lb z-t\rb^2}.
\eeq
Here, the last expression corresponds to the contribution with a 2nd order pole at $z=t$ and the first two are the residues of the rest at simple poles $z=t$ and $z=0$.
Since $2\operatorname{Tr}A^-_0A^-_t=\operatorname{Tr}\lb \mathfrak S^2-\Theta_0^2-\Theta_t^2\rb$, it now suffices to show that the last expression vanishes to finish the proof. Indeed, since
\ben\Phi\lb z\rb^{-1}\Phi_-\lb z\rb=G_{t}^{-1}\left[\mathbb 1+g_{t}\lb z-t\rb+O\lb \lb z-t\rb^2\rb\right] G_{t}^- \qquad
\text{as } z\to t,
\ebn
with some $g_t\in\operatorname{Mat}_{N\times N}\lb \Cb\rb$, the last contribution to (\ref{special03}) is equal to $\operatorname{Tr}g_t\bigl[G_{t}^-A^-_t \lb {G_{t}^-}\rb^{-1},G_{t}A_t G_{t}^{-1}\bigr]$. But we have $G_{t}^-A^-_t \lb{G_{t}^-}\rb^{-1}=G_{t}A_t G_{t}^{-1}=\Theta_t$, and the statement follows. \epf
\subsubsection{Two irregular singularities}
Let us now consider a linear system with two irregular singularities at $0$ and $\infty$ of respective Poincar\'e ranks ${R_0,R_{\infty}\in\Zb_{>0}}$. The general form of such a system reads
\beq\label{IrrS}
\partial_z\Phi=\Phi A\lb z\rb,\qquad A\lb z\rb=\sum_{k=-R_0}^{R_{\infty}}z^{k-1}A_{k},
\eeq
with $A_k\in\operatorname{Mat}_{N\times N}\lb \Cb\rb$ and $\operatorname{Tr} A_k=0$. The coefficients $A_{-R_{0}},A_{R_{\infty}}$ corresponding to the most singular terms at $z=0,\infty$, are assumed to be diagonalizable; we write $A_{-R_0}=G_0^{-1}\Theta^{\lb 0\rb}_{-R_{0}} G_0$,
$A_{R_{\infty}}=G_{\infty}^{-1}\Theta^{\lb \infty\rb}_{R_{\infty}} G_{\infty}$ with diagonal and traceless $\Theta^{\lb 0\rb}_{-R_{0}}$, $\Theta^{\lb \infty\rb}_{R_{\infty}}$.
There exist unique formal fundamental solutions
\begin{align*}
\Phi^{\lb a\rb}_{\mathrm{form}}\lb z\rb=e^{\Theta^{\lb a\rb}\lb z\rb}\hat{\Phi}^{(a)}\lb z\rb G_{a},\qquad a=0,\infty,
\end{align*}
with
\ben
\begin{gathered}
\hat{\Phi}^{\lb 0\rb}\lb z\rb=\mathbb 1+\sum_{k=1}^{\infty}g^{\lb 0\rb}_k z^k,\qquad
\hat{\Phi}^{\lb \infty\rb}\lb z\rb=\mathbb 1+\sum_{k=1}^{\infty}g^{
\lb \infty\rb}_k z^{-k}, \\
\Theta^{\lb 0\rb}\lb z\rb=\sum_{k=-R_0}^{-1}\frac{\Theta^{\lb 0\rb}_k}{k}z^k+\Theta^{\lb 0\rb}_0\ln z,\qquad
\Theta^{\lb \infty\rb}\lb z\rb=\sum_{k=1}^{R_{\infty}}\frac{\Theta^{\lb \infty\rb}_k}{k}z^k-\Theta^{\lb \infty\rb}_0\ln z,
\end{gathered}
\ebn
where all $\Theta^{\lb a\rb}_k$ are given by diagonal matrices. These matrices, together with the coefficients $g^{\lb a\rb}_k$, are uniquely fixed by the linear system \eqref{IrrS}. Genuine canonical solutions $\Phi^{\lb a\rb}_k\lb z\rb$ with $k=1,\ldots,2R_{a}+1$ are asymptotic to $\Phi^{\lb a\rb}_{\mathrm{form}}\lb z\rb$ in $2R_{a}+1$ Stokes sectors $\mathcal S^{\lb a\rb}_k$ around $z=a$, and are related by Stokes matrices $S_k=\Phi^{\lb a\rb}_{k+1}\lb z\rb{\Phi^{\lb a\rb}_k\lb z\rb}^{-1}$ on their overlap.
\begin{figure}[h!]
\centering
\includegraphics[height=7cm]{ToeplitzF_Fig4b.eps}
\begin{minipage}{0.65\textwidth}
\caption{Transformation of RH contour for systems with two irregular singularities of Poincar\'e ranks $R_0=2$, $R_{\infty}=3$.
\label{Toeplitz_Fig4}}
\end{minipage}
\end{figure}
The transformation of (\ref{IrrS}) into a Riemann-Hilbert problem on a circle is carried out similarly to the $4$-point Fuchsian case:
\begin{enumerate}
\item Introduce a function $\tilde{\Psi}\lb z\rb$ which coincides with the canonical solutions $\Phi^{\lb a\rb}_k\lb z\rb$ inside the sectors ${\Omega^{\lb a\rb}_k\subset\mathcal S^{\lb a\rb}_k}$ schematically represented in Fig.~\ref{Toeplitz_Fig4}a. The rays therein belong to overlaps of adjacent Stokes sectors. The function $\tilde{\Psi}\lb z\rb$ solves a dual RHP on the contour $\tilde\Gamma$, indicated in Fig.~\ref{Toeplitz_Fig4}a by solid black curves. Besides the jumps on the rays (given by the Stokes matrices), one has constant jumps on different arcs of the connection circle. All of the latter can be expressed in terms of one connection matrix, e.g. $E=\Phi^{\lb 0\rb}_{1}\lb z\rb{\Phi^{\lb \infty\rb}_1\lb z\rb}^{-1}$. There is also an asymptotic condition
$e^{-\Theta^{\lb a\rb}\lb z\rb}\tilde{\Psi}\lb z\rb=O\lb1\rb$ as $z\to a$ on~$\Cb\Pb^1\backslash\tilde{\Gamma}$.
\item We would now like to cancel the jumps inside the open annulus $\hat{\mathcal A}$ bounded by the circles $\mathcal C_{\mathrm{out},\mathrm{in}}$ indicated in Fig.~\ref{Toeplitz_Fig4}a by dashed red lines. Pick any fundamental matrix solution $\tilde\Phi\lb z\rb$ of (\ref{IrrS}), e.g. $\Phi^{\lb 0\rb}_{1}\lb z\rb$. Let $M_0\in\mathrm{SL}\lb N,\Cb\rb$ be its anticlockwise monodromy around $z=0$. This matrix is determined up to conjugation by the Stokes matrices. Write $M_0=e^{2\pi i \mathfrak S}$ choosing $\mathfrak S$ so that $\operatorname{Tr}\mathfrak S=0$. Define
\ben
\hat\Psi\lb z\rb=\begin{cases}
\lb -z\rb^{-\mathfrak S}\tilde\Phi\lb z\rb,\qquad& z\in\hat{\mathcal A},\\
\tilde\Psi\lb z\rb, \qquad &z\notin\bar{\hat{\mathcal A}}.
\end{cases}
\ebn
The dual RHP for $\hat\Psi\lb z\rb$ is posed on the contour $\hat{\Gamma}$ indicated in Fig.~\ref{Toeplitz_Fig4}b by solid black lines.
\item As before in (\ref{psi1c}), it now suffices to divide $\hat\Psi\lb z\rb$ inside and outside of an auxiliary circle $\mathcal C$ by the solutions $\Psi_-\lb z\rb$ and $\Psi_+\lb z\rb$ of the auxiliary dual RHPs set on the two connected components $\hat \Gamma_-$ and $\hat\Gamma_+$ of $\hat{\Gamma}$, containing $\mathcal C_{\mathrm{out}}$
and $\mathcal C_{\mathrm{in}}$. They are related to the solutions $\Phi_-\lb z\rb$ and $\Phi_+\lb z\rb$ of two auxiliary linear systems. The first one has an irregular singular point of Poincar\'e rank $R_0$ at $z=0$ and a regular singularity at $z=\infty$. In the second, there is a regular singular point at $z=0$ and an irregular singularity of Poincar\'e rank $R_{\infty}$ at $z=\infty$. We thereby obtain a dual RHP for a function $\bar{\Psi}\lb z\rb$, with the jump
\ben
J\lb z\rb= {\Psi_-\lb z\rb}^{-1}\Psi_+\lb z\rb={\Phi_-\lb z\rb}^{-1}\Phi_+\lb z\rb,
\ebn
on $\mathcal C$ written in the form of a direct factorization.
Similarly to the above, we make the identifications
$ \Psi_{\pm}\lb z\rb=\lb -z\rb^{-\mathfrak S}\Phi_{\pm}\lb z\rb$,
$\bar\Psi_{\pm}\lb z\rb= {\Phi_{\mp}\lb z\rb}^{-1}\tilde\Phi\lb z\rb$.
\end{enumerate}
The set $\mathcal T$ of isomonodromic times consists of the diagonal elements of
$\Theta^{\lb a\rb}_k$ with $k\ne 0$. We accordingly decompose it as $\mathcal T=\mathcal T^{\lb 0\rb}\cup \mathcal T^{\lb \infty\rb}$.
The Jimbo-Miwa-Ueno tau function of the system (\ref{IrrS}) is defined by the closed 1-form \cite[eq. (1.23)]{JMU}
\beq\label{tauIrr}
d_{\mathcal T}\ln\tau_{\mathrm{JMU}}\lb \mathcal T\rb=-\sum_{a=0,\infty}
\operatorname{res}_{z=a}
\operatorname{Tr}\lb \partial_z\hat{\Phi}^{\lb a\rb}\lb z\rb
{\hat{\Phi}^{\lb a\rb}\lb z\rb}^{-1}
d_{\mathcal T^{\lb a\rb}}\Theta^{\lb a\rb}\lb z\rb\rb.
\eeq
Let us now make contact between this formula and the construction given in Definition~\ref{tauRHPdef}.
\begin{cor}
Let $\tau^{\lb 0\rb}_{\mathrm{JMU}}\lb \mathcal T^{\lb 0\rb}\rb$ and $\tau^{\lb \infty\rb}_{\mathrm{JMU}}\lb \mathcal T^{\lb \infty\rb}\rb$ be the Jimbo-Miwa-Ueno tau functions of auxiliary linear systems for $\Phi_{-}\lb z\rb$ and $\Phi_+\lb z\rb$, and $\tau\left[J\right]$ be the Fredholm determinant defined by (\ref{adops})--(\ref{FR1c}). Then
\beq\label{tau2irr}
\tau \left[J\right] =\left[\tau^{\lb 0\rb}_{\mathrm{JMU}}\lb \mathcal T^{\lb 0\rb}\rb\tau^{\lb \infty\rb}_{\mathrm{JMU}}\lb \mathcal T^{\lb \infty\rb}\rb\right]^{-1} \tau_{\mathrm{JMU}}\lb \mathcal T\rb.
\eeq
\end{cor}
\pf We choose again the normalization in which $\Phi_+\lb z\rb\simeq \lb -z\rb^{\mathfrak S}$ as $z\to0$ and $\Phi_-\lb z\rb\simeq \lb -z\rb^{\mathfrak S}$ as $z\to\infty$. This implies that $\Phi_-\lb z\rb$ is independent of $\mathcal T^{\lb \infty\rb}$ and $\Phi_+\lb z\rb$ independent of $\mathcal T^{\lb 0\rb}$. From the Widom's differentiation formula then follows an analog of the equation (\ref{special01}),
\beq\label{special11}
d_{\mathcal T^{\lb 0\rb}}\ln\tau\left[J\right]=\frac1{2\pi i}\int_{\mathcal C}\operatorname{Tr}\lb d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1}\frac{\mathfrak S}{z} -d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1} \partial_z\tilde\Phi\;{\tilde\Phi}^{-1}\rb dz,
\eeq
and a similar formula for $d_{\mathcal T^{\lb \infty\rb}}\ln\tau\left[J\right]$. The first term in the integrand of (\ref{special11}) is analytic outside $\mathcal C$ and the corresponding integral reduces to the residue at $z=\infty$. The isomonodromy equation for $\Phi_-$ has the form $d_{\mathcal T^{\lb 0\rb}}\Phi_-=\Phi_-U^-\lb z\rb$ with $U^-\lb z\rb=\sum_{k=-R_0}^{-1}z^{k} U_k^{-}$, which implies that this residue vanishes. The second term in the integrand extends to a meromorphic function inside $\mathcal C$ with the only possible pole at $z=0$, so that
\begin{align*}
d_{\mathcal T^{\lb 0\rb}}\ln\tau\left[J\right]=&
-\operatorname{res}_{z=0}\operatorname{Tr}\lb
d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1} \partial_z\tilde\Phi\;{\tilde\Phi}^{-1}\rb=\\
=&
-\operatorname{res}_{z=0}\operatorname{Tr}\lb
d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1} \partial_z\Theta^{
\lb 0\rb}+ e^{-\Theta^{\lb 0\rb}}
d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1}e^{\Theta^{\lb 0\rb}}\partial_z
\hat{\Phi}^{\lb 0\rb}{\hat{\Phi}^{\lb 0\rb}\,}^{-1}\rb=\\
=& -\operatorname{res}_{z=0}\operatorname{Tr}\lb
d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1} \partial_z\Phi_- \Phi_-^{-1}-
e^{-\Theta^{\lb 0\rb}}d_{\mathcal T^{\lb 0\rb}}\Phi_-\,\Phi_-^{-1} e^{\Theta^{\lb 0\rb}}\partial_z
\hat{\Phi}_-^{\lb 0\rb}{\hat{\Phi}_-^{\lb 0\rb}\,}^{-1}+
d_{\mathcal T^{\lb 0\rb}}\Theta^{\lb 0\rb}\partial_z
\hat{\Phi}^{\lb 0\rb}{\hat{\Phi}^{\lb 0\rb}\,}^{-1}\rb=\\
=& -\operatorname{res}_{z=0}\operatorname{Tr}\lb
U^-\lb z\rb A^-\lb z\rb
- d_{\mathcal T^{\lb 0\rb}}\Theta^{\lb 0\rb}\partial_z
\hat{\Phi}^{\lb 0\rb}_-{\hat{\Phi}_-^{\lb 0\rb}\,}^{-1} +
d_{\mathcal T^{\lb 0\rb}}\Theta^{\lb 0\rb}\partial_z
\hat{\Phi}^{\lb 0\rb}{\hat{\Phi}^{\lb 0\rb}\,}^{-1}
\rb,
\end{align*}
where $A^-\lb z\rb=\Phi_-^{-1}\partial_z\Phi_-$. Since $A^-\lb z\rb=
\sum_{k=-R_0}^{0}z^{k-1} A_k^{-}$, the residue of the first term vanishes, while the second and third yield $-d_{\mathcal T^{\lb 0\rb}}\ln\tau^{\lb 0\rb}_{\mathrm{JMU}}+d_{\mathcal T^{\lb 0\rb}}\ln\tau_{\mathrm{JMU}}$. Similarly computing the differential $d_{\mathcal T^{\lb \infty\rb}}\ln\tau\left[J\right]$, we arrive at the expression (\ref{tau2irr}).
\epf
We have thus shown that $\tau\left[J\right]$ coincides with $\tau_{\mathrm{JMU}}\lb\mathcal T\rb$ up to more elementary factors depending separately on $\mathcal T^{\lb 0\rb}$ and $\mathcal T^{\lb \infty\rb}$. These normalization factors are the tau functions of the auxiliary linear systems arising upon ``decorated pants decomposition'' of the Riemann sphere with 2 irregular punctures into two spheres with 1 irregular and 1 regular puncture. Schematically,
\begin{subequations}
\beq
\begin{gathered}
\tau_{\mathrm{JMU}}\lb\;
\vcenter{\hbox{\includegraphics[height=5.5ex]{CGLirrpants.eps}}}\;\rb=
\tau_{\mathrm{JMU}}\lb\;
\vcenter{\hbox{\includegraphics[height=5.5ex]{CGLirrpantsL.eps}}}\;\rb
\tau_{\mathrm{JMU}}\lb\;
\vcenter{\hbox{\includegraphics[height=5.5ex]{CGLirrpantsR.eps}}}\;\rb\;
\det\lb\begin{array}{cc}
\mathbf 1 & \mathsf a\lb\;
\vcenter{\hbox{\includegraphics[height=5ex]{CGLirrpantsR.eps}}}\;\rb
\\
\mathsf d\lb\;
\vcenter{\hbox{\includegraphics[height=5ex]{CGLirrpantsL.eps}}}\;\rb & \mathbf 1
\end{array}\rb .
\end{gathered}
\eeq
The regular holes here correspond to Fuchsian singularities and cusps represent anti-Stokes directions.
The prefactor $t^{\frac12 \operatorname{Tr}\lb \mathfrak S^2-\Theta_0^2-\Theta_t^2\rb}$ in (\ref{tau4tauJMU}) has a similar interpretation: it represents the isomonodromic tau function of the auxiliary Fuchsian system for $\Phi_-$ having singular points at $0$, $t$ and $\infty$, while the tau function of the auxiliary $3$-point system for $\Phi_+$ is just a constant, so that
\beq
\begin{gathered}
\tau_{\mathrm{JMU}}\lb\;
\vcenter{\hbox{\includegraphics[height=7ex]{CGL4pants.eps}}}\;\rb=
\tau_{\mathrm{JMU}}\lb\;
\vcenter{\hbox{\includegraphics[height=7ex]{CGLpantsL.eps}}}\;\rb
\tau_{\mathrm{JMU}}\lb\;
\vcenter{\hbox{\includegraphics[height=7ex]{CGLpantsR.eps}}}\;\rb\;
\det\lb\begin{array}{cc}
\mathbf 1 & \mathsf a\lb\;
\vcenter{\hbox{\includegraphics[height=6ex]{CGLpantsR.eps}}}\;\rb
\\
\mathsf d\lb\;
\vcenter{\hbox{\includegraphics[height=6ex]{CGLpantsL.eps}}}\;\rb & \mathbf 1
\end{array}\rb .
\end{gathered}
\eeq
\end{subequations}
The idea to associate Riemann surfaces with cusped boundaries to monodromy manifolds of isomonodromic systems in rank $N=2$ first appeared in \cite{CM,CMR}. Our results illustrate the use of the corresponding pictures in the analytic setting.
\subsubsection{Integrable hierarchies\label{subsecIH}}
As a second example, we will show how to apply our definition of tau function to the study of integrable hierarchies. The results outlined in this section are not new (see \cite{Cafasso,CafassoWu1,CafassoWu2,CDD}); the aim is to describe them in a way that makes the comparison with the case of isomonodromic deformations more transparent.
To start with, consider a differential operator of fixed degree $N$
\begin{equation}\label{L}
L := D^{N} + u_{N-2}D^{N-2} + \ldots + u_0,
\end{equation}
where we denoted $D := \partial_x$, and the coefficients $u_0, \ldots, u_{N - 2}$ depend on $x$ and some additional parameters we are now going to describe. The isospectral deformations of $L$ are described by the Lax system
\begin{equation}\label{LaxSystem}
\left\{
\begin{array}{rcl}
L\, \phi &=& z \phi,\\
\partial_{ t_j}\phi &=& \lb L^{j/N}\rb_+\phi, \qquad j \ne 0\;\mathrm{mod}\; N .
\end{array}
\right.
\end{equation}
giving rise to the Gelfand-Dickey equations for the variables $\left\{u_0\lb x,\bt\rb, \ldots, u_{N - 2}\lb x,\bt\rb\right\}$, written in the Lax form as the compatibility conditions of the system \eqref{LaxSystem}:
\begin{equation}
\partial_{t_j} L = \left[\lb L^{j/N}\rb_+,L\right], \qquad j \ne 0\;\mathrm{mod}\; N .
\end{equation}
Here and below $\bt$ denotes the collection of all the deformation parameters $\bt := \left\{t_1,\ldots,t_{N-1},t_{N+1},\ldots \right\}$.
Converting the equations \eqref{LaxSystem} into a 1st order system of size $N$, we get
\begin{equation}\label{LaxSystem2}
\left\{
\begin{array}{rcl}
\partial_x \Phi &=& \Phi \left(\begin{array}{ccccc}
0 & 0 & \ldots & 0 & z - u_0\\
1 & 0 & \ldots & 0 & -u_1 \\
0 & 1 & \ldots & 0 & -u_2\\
\vdots & 0 & \ddots & 0 & \vdots\\
0 & \ldots & \ldots & 1 & 0
\end{array}\right),\\
\partial_{t_j}\Phi &=& \Phi M_j, \qquad j \ne 0\;\mathrm{mod}\; N.
\end{array}
\right.
\end{equation}
where the matrices $M_j$ are completely determined by the coefficients of $\lb L^{j/N}\rb_+$. We fix uniquely a fundamental solution $\Phi\lb x,\bt;z\rb$ by
imposing the asymptotics at infinity
\beq\label{asympGD}
\Phi\lb x,\bt;z\rb = e^{x\Lambda + \sum_{j \ne 0\;\mathrm{mod}\; N}t_j \Lambda^j}\left[\mathbb 1 + O\lb z^{-1}\rb \right] = e^{x\Lambda + \sum_{j \ne 0\;\mathrm{mod}\; N}t_j \Lambda^j}\Phi_-\lb x,\bt;z\rb, \qquad z\to \infty ,
\eeq
where
\ben
\Lambda := \left(\begin{array}{ccccc}
0 & 0 & \ldots & 0 & z \\
1 & 0 & \ldots & 0 & 0 \\
0 & 1 & \ldots & 0 & 0\\
\vdots & 0 & \ddots & 0 & \vdots\\
0 & \ldots & \ldots & 1 & 0
\end{array}\right).
\ebn
Indeed, in some cases $\Phi$ will be a genuine analytic function in a neighborhood of $\mathcal C$, and the corresponding solution belongs to the Segal--Wilson Grassmannian. Otherwise, one can still consider $\Phi$ just as a formal series, and in this case the solution belongs to the Sato Grassmannian.
We now define our direct RHP by imposing
\beq\label{JumpGD}
J\lb x,\bt;z\rb := \Phi^{-1}_-\lb 0,0;z\rb e^{-x\Lambda - \sum_{j \ne 0\;\mathrm{mod}\; N}t_j \Lambda^j} =: \Psi_-^{-1}\lb z\rb\Psi_+\lb x,\bt;z\rb.
\eeq
In \cite{Cafasso,CafassoWu1} it has been proven that the related tau function defined in Definition \ref{tauRHPdef} coincides with the one defined by Segal and Wilson.
Indeed, one can act with the matrix-valued series $\Psi_-\lb z\rb$ on the subspace $H_+$, thus obtaining a subspace $W := H_+\cdot\Psi_- $ in $\mathrm{Gr}\lb H\rb$. The operator $-\mathsf{d} : H_{_+} \to H_-$ is nothing but the operator whose graph gives the subspace $W$ (i.e. the operator $A$ in the notation of \cite{SW}) and the formulas to be compared are \cite[eq. (3.5)]{SW} and
$$\tau[J] = \mathrm{det}_H\lb \mathbb{1} + L\rb = \mathrm{det}_{H_+}\lb \mathbb 1 - \mathsf{a}\mathsf{d}\rb.$$
Concerning the combinatorial expansion in the Subsection \ref{subseccombi}, we start by observing that the matrix $G^{[+]}$ does not depend on the particular solution to be studied, and can be computed explicitly. It reads
\beq\label{SchurMatrix}
G^{[+]} = \left(\begin{array}{cccc|cccc}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\ldots & 1 & s_1 & s_2 & s_3 & s_4 & s_5 & \ldots \\
\ldots & 0 & 1 & s_1 & s_2 & s_3 & s_4 & \ldots \\
\ldots & 0 & 0 & 1 & s_1 & s_2 & s_3 & \ldots
\end{array}\right),
\eeq
where the elementary Schur polynomials $s_j = s_j\lb \bt\rb$ are defined by the relation
\beq
\sum_{j \geq 0} s_j(\bt)z^j = \exp\left( \sum_{j \neq 0 \, \mathrm{mod}\, N} t_j z^j \right).
\eeq
In particular, its minors give, by definition (see for instance \cite{Macdonald}), the Schur polynomials associated to an arbitrary partition. Equivalently, one can compute the same Schur polynomial through the principal minor of $\mathsf{a}$, and the equivalence between the two approaches is given by the Giambelli identity \cite{EH}. As we mentioned before, the graph of the function $-\mathsf{d}$ determines the point $W$ associated to the particular solution we want to study. This means that its minors give the Pl\"ucker coordinates of $W$, which are equivalently computed as the minors of $G^{[-]}$, again via the Giambelli identity. Hence, the combinatorial formula \eqref{SatoTau}, in this case, is the standard tau function expansion in terms of Schur polynomials and Pl\"ucker coordinates.
Note that, in this case, \emph{any} loop $\Psi_-\lb z\rb$ with the prescribed asymptotics $\mathbb{1} + O\lb z^{-1}\rb$ determines a solution of the hierarchy, so that it does not make much sense to ask about the general form of $-\mathsf{d}$. On the other hand, one can wonder how the already known solutions of the Gelfand-Dickey equations are obtained through our procedure, and this had been answered already for quite a few families:
\begin{itemize}
\item Suppose that $\Psi_-(z) = e^{X\lb z\rb}$, with $X(z)$ a nilpotent element of $\mathfrak{sl}_N[z^{-1}]$. Then there is just a finite number of non-zero minors of $\mathsf{d}$. In this case, the combinatorial expansion \eqref{SatoTau} becomes a finite sum and the tau function will be a polynomial obtained by particular linear combinations of Schur polynomials, see \cite{CDD}. These solutions are much studied in the literature, as their zeros evolve according to the (rational) Calogero--Moser hierarchy (see \cite{Wilson} and references therein).
\item More generally, if $\Psi_-(z)$ has a truncated expansion (i.e. just a finite number of non-zero Fourier coefficients, say up to $n$) then, using a result obtained by Widom \cite{Widom1}, one can compute the Szeg\H o-Widom constant of $J^{-1}$ as the (finite-size) determinant $\det T_n\left[J\right]$. Such solutions are the ones associated to rational curves with singularities (see \cite{SW}), i.e. they correspond to multi--solitons, see \cite{Cafasso}.
\item Consider a Riemann surface of type (symmetric $N$-covering)
\beq\label{RiemannSurface}
\lambda^N = \prod_{j = 1}^{Nk +1} \lb z - a_j\rb = P\lb z\rb,
\eeq
and define $N$ functions $\left\{w_1\lb z\rb, \ldots , w_N\lb z\rb \right\}$ by
\beq
w_i\lb z\rb := \left(\frac{P\lb z\rb}{z}\right)^{\frac{i-1}N} \frac{1}{\prod_{j = 1}^{(i-1)k}\lb z-a_j\rb}, \quad i = 1,\ldots,N.
\eeq
The point in the Grassmannian generated by
$\Psi_-\lb z\rb := \mathrm{diag}\lb w_1\lb z\rb,\ldots,w_N\lb z\rb\rb$
belongs to the so-called Krichever locus \cite{SW}. It corresponds to an algebro-geometric solution of the Gelfand-Dickey hiearchy, whose spectral curve is exactly \eqref{RiemannSurface}. The Schur expansion for these tau functions had been studied in great detail in \cite{EH}. A way to compute the tau function in this case is using the Widom's differentiation formula \eqref{dertauRH}. Indeed, here one can reduce the dual RHP \eqref{RHPdual} to a RHP with \emph{constant} jumps on the intervals $[a_1,a_2] \cup \ldots \cup [a_{Nk-1},a_{Nk}] \cup [a_{Nk+1},\infty)$, and solve it explicitly in terms of theta functions associated to the curve \eqref{RiemannSurface}, as described in \cite{Cafasso} for the case $N = 2$ (i.e. for hyper-elliptic curves). The procedure is a classical one, used already in the 80s by the Saint Petersburg school in the context of the so-called finite-gap integration (see for instance \cite{Matv} and references therein). The idea to use it to compute the Szeg\H o-Widom constant associated to matrix-valued algebraic symbols had been developed for the first time in \cite{IJK,IMM}.
\item As an example of solutions not belonging to the Segal-Wilson Grassmannian, we consider \emph{topological} solutions, uniquely fixed by the equations of the hierarchy and the additional string equation
\beq
\left(\sum_{i \neq 0\,\mathrm{mod}\, N }\left( \frac{i+ N}{N} t_{i+
N}-\delta_{i,1}\right)\frac{\partial }{\partial t_{i}}+\frac{1}{2 N}\sum_{ i+j= N}i j t_i t_j\right)\tau({\mathbf t}) =0.
\eeq
For $N = 2$, this is the celebrated Witten-Kontsevich tau function \cite{Kon}, and for $N > 2$ this is the Frobenius potential, to all genera, of the singularity of type $A_{N-1}$. Finding the element $\Psi_-\lb z\rb$ defining the relevant jump \eqref{JumpGD} corresponds to finding the point in the Sato-Segal-Wilson Grassmannian associated to the given solution of the hierarchy. This had been achieved, in the 90s, by Kac and Schwarz \cite{KS}, who proved that $W =H \cdot \Psi_-$ is uniquely fixed by the two conditions\footnote{Note that here we are working on the vector-valued Grassmannian, while in the cited paper the two conditions are (equivalently) stated in the scalar Grassmannian and the Kac-Schwarz operator, in particular, is a scalar one.}
\beq\label{KacSchwarz}
zW \subseteq W, \quad \mathcal R_N W \subseteq W,
\eeq
where $\mathcal R_N$ is the differential operator
\beq
\mathcal R_N = \partial_z - \Lambda + \lb N z\rb^{-1}\rho,
\eeq
and $\rho$ is a diagonal traceless matrix whose coefficients depend on the Cartan matrix of the Lie algebra $\mathfrak{sl}_N$, see for instance \cite[eq. (3.24)]{CafassoWu2}. Another way of stating the second equation in \eqref{KacSchwarz} is to say that $\Psi_-\lb z\rb$ satisfies the so-called \emph{reduced string equation}
\beq\label{reducedSE}
\mathcal R_N\Psi_- = \Psi_-\lb\Psi_-^{-1}\Lambda\Psi_-\rb_+,
\eeq
and it had been proven in \cite{CafassoWu2} that there exists a unique solution $\Psi_-(z) = {\rm e}^{X(z)}$, with $X\lb z\rb$ an element in the loop (sub)algebra $\mathfrak{sl}_N[[z^{-1}]]$ . This solution is in general just a formal series, exactly as the corresponding tau function, whose coefficients give intersection numbers on the Deligne-Mumford moduli space of stable curves. Note that \eqref{reducedSE} is of the same form as \eqref{DEcond}. This is not surprising, as the connection between isomonodromic deformations and string equations goes back to the work \cite{Moore} (see also \cite{ACvM}, where this connection is established using the Kac-Schwarz operators described above).
\end{itemize}
In order to simplify the notations, in this subsection we only considered the case of Gelfand-Dickey hierarchies. Nevertheless, most of the results described above apply as well to the more general case of the Drinfeld-Sokolov hierarchies associated to arbitrary (affine) Kac-Moody algebras \cite{DS}. The idea is to consider direct RHPs as the one in \eqref{JumpGD} above, but with $\Psi_-$ an element of the form $\Psi_-\lb z\rb = {\rm e}^{X\lb z\rb}$, with $X\lb z\rb \in \mathfrak{g}[[z^{-1}]]$, and $\mathfrak{g}$ an arbitrary (finite-dimensional) Lie algebra. The element $\Psi_+$, on the other hand, will be replaced by $$\Psi_+\lb x,\mathbf{t};z\rb = {\rm e}^{-x\Lambda_1 - \sum_{j \in E_+}t_j\Lambda_j},$$ where $E_+$ is the set of the (positive) exponents of the Kac-Moody algebra $\mathfrak g[z,z^{-1}] \oplus \mathbb C c$, and $\left\{\Lambda_j,\, j \in E_+ \right\}$ is (half of) the Heisenberg sub-algebra associated to an arbitrary gradation of the algebra. Polynomial and topological solutions of these hierarchies had been treated, using this formalism, in \cite{CDD, CafassoWu2}. It would be interesting (but technically involved, because of the size of matrix representations) to study algebro-geometric solutions associated to arbitrary Drinfeld-Sokolov hierarchies.
\section{General contour}\label{secmulti}
The Definition~\ref{tauRHPdef} of $\tau\left[J\right]$ makes sense if we replace the circle $\mathcal C$ by a finite collection $\Gamma=\bigcup_{a=1}^M\mathcal C_a$ of non-intersecting smooth closed curves which we sometimes call ovals. However, defining the jump of the relevant dual RHP in the form of direct factorization is no longer natural from the point of view of applications, which makes the Fredholm determinant representation (\ref{FR1c}) and the differentiation formula (\ref{dertauRH}) less useful in this setting. What we would like to have instead are the formulae of the same type but expressed in terms of the direct factorization of the \textit{individual} jumps on each curve~$\mathcal C_a$.
The existence of such expressions is suggested by the recent work \cite{GL16} by two of the authors, where Fredholm determinant and combinatorial series representations were obtained for the tau function of the $n$-point Fuchsian system --- including, in particular, the tau function of the Garnier system $\mathcal G_{n-3}$. The paper \cite{GL16} deals with the special case where the contour is given by a collection of concentric circles coming from a ``linear'' pants decomposition of $\Cb\Pb^1\backslash\left\{n\text{ points}\right\}$. Our aim here is to extend these results to RHPs with more general jumps on arbitrary configurations of ovals such as the one represented in Fig.~\ref{fig_sic}c.
\begin{figure}[h!]
\centering
\includegraphics[height=6cm]{ToeplitzF_Fig6.eps}
\begin{minipage}{0.65\textwidth}
\caption{``Sicilian'' RH contour for $6$-point Fuchsian systems.
\label{fig_sic}}
\end{minipage}
\end{figure}
\subsection{Notations and setting of the RHP\label{subsecnotationmany}}
The complement of $\Gamma$ in $\Cb\Pb^1$ has $M+1$ connected components, which will be called faces (or pants). It admits a unique, up to permutation, 2-coloring by colors $\left\{+,-\right\}$ which will be fixed once and for all. Denote by $\mathsf F_{\pm}$ the set of faces of color $\pm$, by $\mathsf C:=\left\{\mathcal C_1,\ldots, \mathcal C_M\right\}$ the set of all ovals and by $\mathsf C_f$ the set of boundary curves of the face $f$. Let $\phi_{\pm}:\mathsf C\to\mathsf F_{\pm}$ be the map assigning to each curve $\mathcal C\in\mathsf C$ the unique face in $\mathsf F_{\pm}$ having $\mathcal C$ among the boundary components.
The coloring allows to choose a convenient orientation of $\Gamma$ for which all faces of color $+$ (or $-$) are located on the positive (resp. negative) side of their boundary curves. We denote by $\varphi_+\lb \mathcal C\rb$ and $\varphi_-\lb \mathcal C\rb$ the closure of interior and exterior of the curve $\mathcal C$ with respect to the above orientation. Clearly $\phi_{\pm}\lb \mathcal C\rb\subseteq\varphi_+\lb \mathcal C\rb$, but in general $\varphi_\pm\lb \mathcal C\rb$ can contain faces of different colors.
Assign to every boundary $\mathcal C\in\mathsf C$ a pair of functions $\Psi_{\mathcal C,\pm}:\mathcal C\to\mathrm{GL}\lb N,\Cb\rb$ that continue analytically to $\varphi_{\pm}\lb \mathcal C\rb$. Also, to every $\mathcal C\in \mathsf C$ we assign the space of functions $H_{\mathcal C}=L^2\lb \mathcal C,\Cb^N\rb$. It may be decomposed as $H_{\mathcal C}=H_{\mathcal C,+}\oplus H_{\mathcal C,-}$, where $H_{\mathcal C,\pm}$ consist of functions that continue to $\varphi_{\pm}\lb \mathcal C\rb$ (and vanish at $\infty$ on the appropriate side of $\mathcal C$). This means in particular that the columns of $\Psi_{\mathcal C,-}$ do not necessarily belong to $H_{\mathcal C,-}$ but e.g. $\Psi_{\mathcal C,-}H_{\mathcal C,-}\subseteq H_{\mathcal C,-}$. \vspace{0.2cm}\\
\textit{Riemann-Hilbert problem}. We wish to consider the dual RHP posed on $\Gamma=\bigcup_{a=1}^M \mathcal C_a$ in which the jumps are given in the form of direct factorization,
\ben
J_{\mathcal C}\equiv J\bigl|_{\mathcal C}:=\Psi_{\mathcal C,-}^{-1}
\Psi_{\mathcal C,+},\qquad \mathcal C\in\mathsf C.
\ebn
That is, we want to find an analytic invertible matrix function $\bar\Psi\lb z\rb$ on $\Cb\Pb^1\backslash\Gamma$ whose boundary values on $\Gamma$ satisfy $J=\bar\Psi_+{\bar\Psi_-}^{-1}$.\vspace{0.2cm}
Denote by $\Pi_{\mathcal C,\pm}$ the projections on $H_{\mathcal C,\pm}$ along $H_{\mathcal C,\mp}$ and consider
\ben
H=H_+\oplus H_-,\qquad H_{\pm}=\bigoplus_{\mathcal C\in\mathsf C}H_{\mathcal C,\pm}.
\ebn
For $\mathcal C,\mathcal C'\in\mathsf C$, $\mathcal C\in\varphi_{\pm}\lb\mathcal C'\rb$, we define the operators $\Pi_{ \mathcal C\leftarrow\mathcal C',\pm}: H_{\mathcal C'}\to H_{\mathcal C}$ such that $\Pi_{ \mathcal C\leftarrow\mathcal C',\pm}g$ is the restriction to $\mathcal C$ of the analytic continuation of $\Pi_{\mathcal C',\pm} g$ to $\varphi_{\pm}\lb \mathcal C'\rb$. In particular, for $\mathcal C\in \mathsf C_{\phi_{\pm}\lb\mathcal C'\rb}$, $\mathcal C\ne\mathcal C'$ we have $\operatorname{im}\lb \Pi_{ \mathcal C\leftarrow\mathcal C',\pm }\rb\subseteq H_{\mathcal C,\mp}$, whereas for $\mathcal C=\mathcal C'$ one has $\Pi_{ \mathcal C'\leftarrow\mathcal C',\pm}=\Pi_{\mathcal C',\pm}$.
\begin{defin}\label{defmultitau}
The tau function $\tau\left[J\right]$ associated to the above RHP is defined as the Fredholm determinant
\beq\label{taugen}
\tau\left[J\right]=\operatorname{det}_H\lb \mathbb 1+L \rb, \qquad
L=\lb\begin{array}{cc} 0 & \mathsf A_{+-} \\ \mathsf A_{-+} & 0\end{array}\rb
\in\operatorname{End}\lb H_+\oplus H_-\rb,
\eeq
where the operators $\mathsf A_{\pm\mp}:H_\mp\to H_\pm$ are defined by
\begin{align*}
\mathsf A_{\pm\mp}=&\,\sum_{f\in\mathsf F_{\mp}}\sum_{\mathcal C,\mathcal C'\in\mathsf C_f}\mathsf A_{\mathcal C,\pm;\mathcal C',\mp},\\
\mathsf A_{\mathcal C,\pm;\mathcal C',\mp}=&\,
\Psi_{\mathcal C,\pm}
\Pi_{ \mathcal C\leftarrow\mathcal C',\mp}\Psi_{\mathcal C',\pm}^{\;-1}-\delta_{\mathcal C,\mathcal C'}\;\Pi_{\mathcal C',\mp}.
\end{align*}
\end{defin}
One can consider elementary summands $\mathsf A_{\mathcal C,\pm;\mathcal C',\mp}$ as integral operators on $H$,
\ben
\lb\mathsf A_{\mathcal C,\pm;\mathcal C',\mp} g\rb\lb z\rb=
\frac1{2\pi i}\oint_{\mathcal C'}\mathsf A_{\mathcal C,\pm;\mathcal C',\mp}\lb z,z'\rb g\lb z'\rb dz',
\ebn
whose integral kernels have integrable form and are given by
\ben
\mathsf A_{\mathcal C,\pm;\mathcal C'\mp}\lb z,z'\rb
=\pm\chi_{\mathcal C}\lb z\rb \frac{\Psi_{\mathcal C,\pm}\lb z\rb {\Psi_{\mathcal C',\pm}\lb z'\rb}^{-1}-\mathbb 1\;\delta_{\mathcal C,\mathcal C'}}{z-z'}\chi_{\mathcal C'}\lb z'\rb,
\ebn
with $\chi_{\mathcal C}\lb z\rb$ denoting the indicator function of $\mathcal C$. We leave it as an exercise for the reader to verify that $\operatorname{im}\mathsf A_{\mathcal C,\pm;\mathcal C',\mp}\subseteq H_{\pm}\subseteq \operatorname{ker}\mathsf A_{\mathcal C,\pm;\mathcal C',\mp}$. The above expressions provide a many-oval counterpart of the one-circle formula \eqref{FR1c}. The operators $\mathsf A_{+-}$ and $\mathsf A_{-+}$ are analogs of the operators $\mathsf a$ and~$\mathsf d$. They can be regarded as matrices whose operator entries are labeled by pairs of curves in $\mathsf C$; these entries are non-zero only if the relevant curves bound the same face.
\begin{rmk}
One may wonder whether it is also possible to obtain an analog of the Widom's determinant (\ref{tauRHP}), i.e. not to use direct factorization and express $\tau\left[J\right]$ solely in terms of the jumps.
The answer is positive and can be obtained by conjugation of $L$ by the multiplication operator $\lb\bigoplus_{\mathcal C\in\mathsf C}\Psi_{\mathcal C,+}\rb\oplus \lb\bigoplus_{\mathcal C\in\mathsf C}\Psi_{\mathcal C,-}\rb$: one can equivalently write
\beq\label{taugen2}
\begin{gathered}
\tau\left[J\right]=\operatorname{det}_H\lb \mathbb 1+\tilde L \rb, \qquad
\tilde L=\lb\begin{array}{cc} 0 & \tilde{\mathsf A}_{+-} \\ \tilde{\mathsf A}_{-+} & 0\end{array}\rb,\\
\tilde{\mathsf A}_{\pm\mp}=\sum_{f\in\mathsf F_{\mp}}\sum_{\mathcal C,\mathcal C'\in\mathsf C_f}\lb
\Pi_{ \mathcal C\leftarrow\mathcal C',\mp}J_{\mathcal C'}^{\mp1}- \delta_{\mathcal C,\mathcal C'}\;J_{\mathcal C'}^{\mp 1}\rb\Pi_{\mathcal C',\mp}.
\end{gathered}
\eeq
\end{rmk}
\subsection{Differentiation formula\label{subsec43}}
Let us now establish the many-oval counterpart of the differentiation formula given in Theorem~\ref{WidomII}. The first step is the calculation of the inverse $\lb\mathbb 1+L\rb^{-1}$. To this end we first rewrite $\mathbb 1+L$ in the form
\beq\label{rhs1}
\mathbb 1+L=\lb\begin{array}{cc}\sum\limits_{f\in\mathsf F_{+}}\sum\limits_{\mathcal C,\mathcal C'\in\mathsf C_f}
\Psi_{\mathcal C,-}
\Pi_{ \mathcal C\leftarrow\mathcal C',+}\Psi_{\mathcal C',-}^{\;-1} &
\sum\limits_{f\in\mathsf F_{-}}\sum\limits_{\mathcal C,\mathcal C'\in\mathsf C_f}
\Psi_{\mathcal C,+}
\Pi_{ \mathcal C\leftarrow\mathcal C',-}\Psi_{\mathcal C',+}^{\;-1}
\end{array}\rb,
\eeq
where the first and second column correspond to the action of
$\mathbb 1+L$ on $H_+$ and $H_-$. Let us note that
\ben
\Gamma=\bigcup\nolimits_{f\in\mathsf F_\pm}\bigcup\nolimits_{\mathcal C\in\mathsf C_f}\mathcal C.
\ebn
It is useful to interpret the contributions $\mathcal P_{\oplus,\pm}^{[f]}:= \sum\limits_{\mathcal C,\mathcal C'\in\mathsf C_f}
\Psi_{\mathcal C,\mp}
\Pi_{ \mathcal C\leftarrow\mathcal C',\pm}\Psi_{\mathcal C',\mp}^{\;-1}$ of individual faces $f\in\mathsf F_{\pm}$ to the above sums as integral operators acting from $\bigoplus\limits_{\mathcal C\in\mathsf C_f}H_\pm\lb\mathcal C\rb$ to $\bigoplus\limits_{\mathcal C\in\mathsf C_f}H\lb\mathcal C\rb$ by
\beq\label{rhs2}
\lb\mathcal P_{\oplus,\pm}^{[f]} g^{[f]}\rb\lb z\rb=\pm
\sum_{\mathcal C,\mathcal C'\in\mathsf C_f} \frac{1}{2\pi i}\oint_{\mathcal C'} \frac{
\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,\mp}\lb z\rb
\Psi_{\mathcal C',\mp}\lb z'\rb^{\;-1}
g^{[f]}_{\mathcal C'}\lb z'\rb dz'}{z'-z}.
\eeq
The contour of integration is deformed to the face $\phi_{\mp}\lb \mathcal C\rb$ (i.e. outside the face $f$) whenever it becomes necessary to interpret the singular factor $1/\lb z'-z\rb$.
Next we construct in a similar fashion the operators
$\mathcal P_{\Sigma,\pm}:H\to H_{\pm}$ defined by
\beq\label{psigpm}
\lb\mathcal P_{\Sigma,\pm} g\rb\lb z\rb=
\pm
\sum_{\mathcal C,\mathcal C'\in\mathsf C}\Pi_{\mathcal C,\pm} \frac{1}{2\pi i}\oint_{\mathcal C'} \frac{
\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,+}\lb z\rb\bar\Psi_{\mathcal C,-}\lb z\rb
\bar\Psi_{\mathcal C',-}\lb z'\rb^{\;-1}\Psi_{\mathcal C',+}\lb z'\rb^{-1}
g_{\mathcal C'}\lb z'\rb dz'}{z'-z}.
\eeq
The convention for the contour is the same, i.e. it is pushed away slightly to the negative (positive) faces for $\mathcal P_{\Sigma,+}$
(resp. $\mathcal P_{\Sigma,-}$). In contrast to the face operators $\mathcal P_{\oplus,\pm}^{[f]}$, the operators $\mathcal P_{\Sigma,\pm}$ involve the solution $\bar\Psi_{\mp}$ of the dual RHP. Constructing this solution is essentially equivalent to the calculation of the resolvent $\lb \mathbb 1+L\rb^{-1}$ thanks to the following lemma.
\begin{lemma} If the dual RHP is solvable, then
$\lb\mathbb 1+L\rb^{-1}=
\ds\lb\begin{array}{c} \mathcal P_{\Sigma,+} \\ \mathcal P_{\Sigma,-}\end{array}\rb$.
\end{lemma}
\pf Let us compute, for instance, the action of the ``$+-$'' component of the product $\ds\lb\begin{array}{c} \mathcal P_{\Sigma,+} \\ \mathcal P_{\Sigma,-}\end{array}\rb \lb \mathbb 1+L\rb$. Given $g_-\in H_-$, it reads
\begin{align*}
&\Bigl (\mathcal P_{\Sigma,+}\sum\limits_{f\in\mathsf F_{-}}\sum\limits_{\mathcal C,\mathcal C'\in\mathsf C_f}
\Psi_{\mathcal C,+}
\Pi_{ \mathcal C\leftarrow\mathcal C',-}\Psi_{\mathcal C',+}^{\;-1} g_-\Bigr)\lb z\rb=\\&=-\!\!\!
\sum_{\mathcal C,\mathcal C''\in\mathsf C}\sum_{\;\;\mathcal C'\in\mathsf C_{\phi_-\lb \mathcal C''\rb}}\!\!\!\Pi_{\mathcal C,+}\frac{1}{\lb 2\pi i\rb^2}
\oint_{\mathcal C''} \oint_{\mathcal C'}
\frac{
\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,+}\lb z\rb\bar\Psi_{\mathcal C,-}\lb z\rb
\bar\Psi_{\mathcal C',-}\lb z'\rb^{\;-1}\Psi_{\mathcal C'',+}\lb z''\rb^{-1}
g_{\mathcal C'',-}\lb z''\rb dz'dz''}{\lb z'-z\rb \lb z'' -z'\rb}.
\end{align*}
Recall that in this expression, the contour of integration w.r.t. $z''$ is slightly deformed to the face $\phi_+\lb\mathcal C''\rb$, and the contour of integration w.r.t. $z'$ is slightly deformed to the face $\phi_-\lb\mathcal C'\rb=\phi_-\lb\mathcal C''\rb$. Therefore the integration contour in
\ben
\sum_{\quad\mathcal C'\in\mathsf C_{\phi_-\lb \mathcal C''\rb}}\oint_{\mathcal C'} \frac{\bar{\Psi}_{\mathcal C',-}\lb z'\rb^{-1}dz'}{
\lb z'-z\rb \lb z'' -z'\rb}
\ebn
can be collapsed through the face $\phi_-\lb \mathcal C''\rb$ and the corresponding integral vanishes. The vanishing of the ``$-+$'' component may be shown in a similar way using in addition that, by definition of $\bar\Psi$, we have $\Psi_{\mathcal C,-}\bar\Psi_{\mathcal C,+}=
\Psi_{\mathcal C,+}\bar\Psi_{\mathcal C,-}$ and shrinking the contours through positive faces.
For the ``$++$'' component, the analog of the above is
\begin{align*}
&\Bigl( \mathcal P_{\Sigma,+}\sum\limits_{f\in\mathsf F_{+}}\sum\limits_{\mathcal C,\mathcal C'\in\mathsf C_f}
\Psi_{\mathcal C,-}
\Pi_{ \mathcal C\leftarrow\mathcal C',+}\Psi_{\mathcal C',-}^{\;-1} g_+\Bigr)\lb z\rb=\\&=
\sum_{\mathcal C,\mathcal C''\in\mathsf C}\sum_{\;\;\mathcal C'\in\mathsf C_{\phi_+\lb \mathcal C''\rb}}\!\!\!\Pi_{\mathcal C,+}\frac{1}{\lb 2\pi i\rb^2}
\oint_{\mathcal C''} \oint_{\mathcal C'}
\frac{
\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,+}\lb z\rb\bar\Psi_{\mathcal C,-}\lb z\rb
\bar\Psi_{\mathcal C',+}\lb z'\rb^{\;-1}\Psi_{\mathcal C'',-}\lb z''\rb^{-1}
g_{\mathcal C'',+}\lb z''\rb dz'dz''}{\lb z'-z\rb \lb z'' -z'\rb}.
\end{align*}
The contours of integration w.r.t. $z'$ and $z''$ are deformed to $\phi_-\lb \mathcal C'\rb$ and $\phi_-\lb \mathcal C''\rb$, and the former is located to the positive side of the latter on the coinciding faces. Collapsing the contour in the integral
\ben
\sum_{\quad\mathcal C'\in\mathsf C_{\phi_+\lb \mathcal C''\rb}}\oint_{\mathcal C'}\frac{ \bar\Psi_{\mathcal C',+}\lb z'\rb^{\;-1} dz'}{\lb z'-z\rb\lb z''-z' \rb}
\ebn
through the face $\phi_+\lb \mathcal C''\rb$, we eventually pick up a residue at $z'=z$, equal to $\ds\frac{2\pi i \bar\Psi_{\mathcal C',+}\lb z\rb^{\;-1}}{z''-z}$, if $\mathcal C\in \mathsf C_{\phi_+\lb \mathcal C''\rb}$. Using this, reorganize the previous expression as
\ben
\sum_{\mathcal C''\in\mathsf C}\sum_{\;\;\mathcal C\in\mathsf C_{\phi_+\lb \mathcal C''\rb}}\Pi_{\mathcal C,+}\frac{1}{ 2\pi i}
\oint_{\mathcal C''}
\frac{
\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,-}\lb z\rb\Psi_{\mathcal C'',-}\lb z''\rb^{-1}
g_{\mathcal C'',+}\lb z''\rb dz''}{ z'' -z}.
\ebn
For $\mathcal C\ne \mathcal C''$, the integral defines a function of $z$ that analytically continues to $\varphi_+\lb \mathcal C''\rb\supset \phi_-\lb \mathcal C\rb$ and therefore vanishes under the action of $\Pi_{\mathcal C,+}$. There remains a sum
\ben
\sum_{\mathcal C\in\mathsf C}\Pi_{\mathcal C,+}\frac{1}{ 2\pi i}
\oint_{\mathcal C}
\frac{
\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,-}\lb z\rb\Psi_{\mathcal C,-}\lb z''\rb^{-1}
g_{\mathcal C,+}\lb z''\rb dz''}{ z'' -z},
\ebn
where $\mathcal C$ is slightly deformed to $\phi_-\lb\mathcal C\rb$ as to avoid the singularity at $z''=z$. Deforming it instead to $\phi_+\lb\mathcal C\rb$ we obtain a function of $z$ annihilated by $\Pi_{\mathcal C,+}$ at the expense of picking up the residue at $z''=z$, equal to $g_{\mathcal C,+}$. This ultimately yields the expected result $\sum_{\mathcal C\in\mathsf C}\Pi_{\mathcal C,+}\chi_{\mathcal C}g_{\mathcal C,+}=g_+$. The calculation for the ``$--$'' component is completely analogous.
\epf
\begin{theo}\label{theomultitau}
Suppose that the functions $\Psi_{\mathcal C,\pm}\lb z\rb $ appearing in the direct factorization of individual jumps $J_{\mathcal C}\lb z\rb$ smoothly depend on an additional parameter $t$. If the solution $\bar\Psi_{\pm}\lb z\rb$ of the dual RHP exists and smoothly depends on $t$, then
\beq\label{widom_gen}
\partial_t\ln\tau\left[J\right]=\sum_{\mathcal C\in\mathsf C}
\frac1{2\pi i}\oint_{\mathcal C}\operatorname{Tr}
\left\{J_{\mathcal C}^{-1}\partial_tJ_{\mathcal C}\left[\partial_z\bar\Psi_{\mathcal C,-}\, {\bar\Psi_{\mathcal C,-}}^{-1}+\Psi_{\mathcal C,+}^{-1}\,\partial_z\Psi_{\mathcal C,+}
\right]\right\}dz.
\eeq
\end{theo}
\pf We are going to mimick the proof of Theorem~\ref{WidomII}.
Note e.g. that the operators $\mathsf A_{\pm\mp}$ in \eqref{taugen}, or more precisely their conjugates $\tilde{\mathsf A}_{\pm\mp}$ in \eqref{taugen2}, are analogs of the operators
$\Pi_+J^{-1}\Pi_-$ and $\Pi_- J\Pi_+$.
The main difference here is that it becomes convenient to write various projection operators as explicit contour integrals.
Differentiating the determinant yields
\begin{align*}
\partial_t\ln\tau\left[J\right]=\operatorname{Tr}_H\Bigl(\lb\mathbb 1+L\rb^{-1}\partial_t L\Bigr)=
\operatorname{Tr}_H\lb \mathcal P_{\Sigma,-}\bigl|_{H_+}\partial_t\mathsf A_{+-} + \mathcal P_{\Sigma,+}\bigl|_{H_-} \partial_t\mathsf A_{-+} \rb=\operatorname{Tr}_H\lb
\mathcal P_{\Sigma,-}\partial_t\mathsf A_{+-} + \mathcal P_{\Sigma,+} \partial_t\mathsf A_{-+} \rb,
\end{align*}
where the last equality is obtained using that $\operatorname{im}\mathsf A_{\pm \mp}\subseteq H_{\pm}$. Moreover, thanks to the property that $ H_{\pm}\subseteq\operatorname{ker}\mathsf A_{\pm \mp}$, the first projector in the definition (\ref{psigpm}) of $\mathcal P_{\Sigma,\pm}$ may be omitted. The computation of traces then reduces to residue calculation. Indeed, we have
\begin{align*}
\begin{aligned}
\operatorname{Tr}_H \mathcal P_{\Sigma,-}\partial_t\mathsf A_{+-}
=\!\!\!\sum_{\mathcal C,\mathcal C''\in\mathsf C}\sum_{\;\;\mathcal C'\in\mathsf C_{\phi_{-}\lb \mathcal C''\rb}}
\frac{1}{\lb 2\pi i\rb^2}\oint_{\mathcal C'}\oint_{\mathcal C''}
\operatorname{Tr}\Biggl\{&\,
\frac{\chi_{\mathcal C}\lb z\rb\Psi_{\mathcal C,+}\lb z\rb\bar\Psi_{\mathcal C,-}\lb z\rb
\bar\Psi_{\mathcal C',-}\lb z'\rb^{\;-1}\Psi_{\mathcal C',+}\lb z'\rb^{-1}}{z'-z}\times\\
\times&\,\frac{\partial_t\lb
\Psi_{\mathcal C',+}\lb z'\rb \Psi_{\mathcal C'',+}\lb z''\rb^{-1}\rb}{z''-z'}\Bigl|_{z''=z}\Biggr\}\; dz'dz=
\end{aligned}\\
=\sum_{\mathcal C\in\mathsf C}\sum_{\;\;\mathcal C'\in\mathsf C_{\phi_{-}\lb \mathcal C\rb}}
\frac{1}{\lb 2\pi i\rb^2}\oint_{\mathcal C}\oint_{\mathcal C'}
\frac{\operatorname{Tr}\bar\Psi_{\mathcal C,-}\lb z\rb
\bar\Psi_{\mathcal C',-}\lb z'\rb^{\;-1}\left[
\Psi_{\mathcal C,+}\lb z\rb^{-1}\partial_t\Psi_{\mathcal C,+}\lb z\rb- \Psi_{\mathcal C',+}\lb z'\rb^{-1}\partial_t \Psi_{\mathcal C',+}\lb z'\rb\right] dz dz'}{\lb z'-z\rb^2}.
\end{align*}
Recall that the contours $\mathcal C'$ of integration with respect to $z'$ are slightly pushed to positive faces $\phi_+\lb \mathcal C'\rb$ according to the definition of $\mathcal P_{\Sigma,-}$. The contribution of the 1st term under trace is readily computed by collapsing the contours $\mathcal C'$ through negative faces and is given by (minus) the residue at $z'=z$,
\begin{subequations}
\beq\label{sum01}
\frac{1}{2\pi i}\sum_{\mathcal C\in\mathsf C}\oint_{\mathcal C}
\operatorname{Tr}\left\{
\partial_z\bar\Psi_{\mathcal C,-} \bar\Psi_{\mathcal C,-}^{-1}\Psi_{\mathcal C,+}^{-1}\partial_t \Psi_{\mathcal C,+}\right\}\,dz.
\eeq
To compute in a similar way the 2nd term under trace, rearrange the sum $\sum\limits_{\mathcal C\in\mathsf C}\sum\limits_{\;\;\mathcal C'\in\mathsf C_{\phi_{-}\lb \mathcal C\rb}}$ as $\sum\limits_{\mathcal C'\in\mathsf C}\sum\limits_{\;\;\mathcal C\in\mathsf C_{\phi_{-}\lb \mathcal C'\rb}}$. Shrinking afterwards the contours $\mathcal C$ through negative faces we meet no poles and therefore the corresponding integrals sum up to zero.
One can analogously prove that
\beq\label{sum02}
\operatorname{Tr}_H \mathcal P_{\Sigma,+}\partial_t\mathsf A_{-+}
=-\frac{1}{2\pi i}\sum_{\mathcal C\in\mathsf C}\oint_{\mathcal C}
\operatorname{Tr}\left\{
\partial_z\bar\Psi_{\mathcal C,+} \bar\Psi_{\mathcal C,+}^{-1}\Psi_{\mathcal C,-}^{-1}\partial_t \Psi_{\mathcal C,-}\right\}\,dz.
\eeq
\end{subequations}
It remains to show that the sum of (\ref{sum01}) and (\ref{sum02}) coincides with the right side of \eqref{widom_gen}. To this end, note that
\begin{align*}
&\operatorname{Tr}J_{\mathcal C}^{-1}\partial_tJ_{\mathcal C}\left[\partial_z\bar\Psi_{\mathcal C,-}\, {\bar\Psi_{\mathcal C,-}}^{-1}+\Psi_{\mathcal C,+}^{-1}\,\partial_z\Psi_{\mathcal C,+}
\right]=\\
=&\,\operatorname{Tr}\left\{\lb\partial_t\Psi_{\mathcal C,+}-\partial_t\Psi_{\mathcal C,-}\, \Psi_{\mathcal C,-}^{-1}\Psi_{\mathcal C,+}\rb\partial_z\bar\Psi_{\mathcal C,-} \bar\Psi_{\mathcal C,-}^{-1}\Psi_{\mathcal C,+}^{-1}+\lb \partial_t\Psi_{\mathcal C,+}\,\Psi_{\mathcal C,+}^{-1}-\partial_t\Psi_{\mathcal C,-}\,\Psi_{\mathcal C,-}^{-1}\rb\partial_z\Psi_{\mathcal C,+}\,\Psi_{\mathcal C,+}^{-1}\right\}=\\
=&\,\operatorname{Tr}\left\{\Psi_{\mathcal C,+}^{-1}\partial_t\Psi_{\mathcal C,+}\lb \partial_z\bar\Psi_{\mathcal C,-} \bar\Psi_{\mathcal C,-}^{-1}+\Psi_{\mathcal C,+}^{-1}\partial_z\Psi_{\mathcal C,+}\rb-\Psi_{\mathcal C,-}^{-1}\partial_t\Psi_{\mathcal C,-}\lb\Psi_{\mathcal C,-}^{-1}\partial_z\Psi_{\mathcal C,-}+\partial_z\bar{\Psi}_{\mathcal C,+}\bar{\Psi}_{\mathcal C,+}^{-1}\rb\right\},
\end{align*}
where to obtain the first equality, we use that $J_{\mathcal C}=\Psi_{\mathcal C,-}^{-1}\Psi_{\mathcal C,+}$; the second equality is obtained by replacing $\Psi_{\mathcal C,+}=\Psi_{\mathcal C,-}\bar{\Psi}_{\mathcal C,+}\bar{\Psi}_{\mathcal C,-}^{-1}$ in the 4th term under trace. In the last expression, the 1st and 4th term reproduce (\ref{sum01}) and (\ref{sum02}). The 2nd and 3rd term are given by the boundary values of functions analytic in $\varphi_+\lb\mathcal C\rb$ and $\varphi_-\lb\mathcal C\rb$, therefore the corresponding integrals vanish.
\epf
\begin{rmk}
We conclude this subsection by mentioning the work of Palmer \cite{Palmer} on the tau function of the massive Euclidean Dirac operator in the presence of Aharonov-Bohm fluxes. While it may seem unrelated to the present paper, it is the adaptation of the localization ideas of \cite{Palmer} to the chiral case which led to \cite{GL16}. Here we made a substantial further improvement by getting rid of artificial doubling of the RHP contours, generalizing the results to arbitrary oval configurations and providing a concise definition of $\tau\left[J\right]$. It might be interesting to understand whether it is possible to go backwards and apply our results, in particular, series expansions of Subsection~\ref{subseccombi}, to the study of correlation functions of twist fields in free massive QFTs.
\end{rmk}
\subsection{Jimbo-Miwa-Ueno differential}
In this subsection, we explain how to recover from Theorem~\ref{theomultitau} the Jimbo-Miwa-Ueno definition \cite{JMU} of the isomonodromic tau function for systems of linear differential equations with rational coefficients.
Let us start with a Fuchsian system with $n$ regular singular points $a_0=0,a_1,\ldots,a_{n-2},a_{n-1}=\infty$ on $\Cb\Pb^1$,
\beq\label{FS}
\partial_z\Phi=\Phi A\lb z\rb,\qquad A\lb z\rb = \sum_{k=0}^{n-2}\frac{A_{k}}{z-a_{k}},\qquad
A_{k}\in\operatorname{Mat}_{N\times N}\lb \Cb\rb.
\eeq
For simplicity it is assumed that $a_1,\ldots,a_{n-2}\in \mathbb R_{>0}$ and that the singularities are ordered as $a_1<\ldots<a_{n-2}$. The fundamental solution $\Phi\lb z\rb$ can then be considered as a single-valued analytic function on $\Cb\backslash\mathbb{R}_{\ge0}$. Similarly to Subsection~\ref{subsec4reg}, we also assume that $A_0,\ldots,A_{n-2},A_{n-1}:=-\sum_{k=0}^{n-2}A_{k}$ are diagonalizable as $A_{k}=G_{k}^{-1}\Theta_{k} G_{k}$ and have non-resonant eigenvalues, so that in the neigborhood of each singular point we have
\ben
\Phi\lb z\rb= C_{k,\epsilon} \lb a_{k}-z\rb^{\Theta_{k}}G^{\lb k\rb}\lb z\rb, \qquad \epsilon =\operatorname{sgn} \Im z,
\ebn
where $G^{\lb k\rb}\lb z\rb$ are holomorphic invertible and normalized so that $G^{\lb k\rb}\lb a_{k}\rb=G_{k}$ (for $a_{n-1}=\infty$ the formula above should be modified in the obvious manner). The connection matrices $\left\{C_{k,\pm}\right\}$ satisfy the compatibility conditions analogous to \eqref{compa} and, together with local monodromy exponents $\left\{\Theta_{k}\right\}$, encode the monodromy representation of $\pi_1\lb\Cb\Pb^1\backslash\left\{n\text{ points}\right\}\rb$ associated to $\Phi\lb z\rb$.
Different pants decompositions of the $n$-punctured sphere give rise to different RHPs associated with the linear system \eqref{FS} and distinct Fredholm determinant representations of the tau functions, adapted to analysis of different asymptotic regimes. Since at this point we only want to give an example of an $n>4$ analog of the relation \eqref{tau4tauJMU}, let us pick the simplest ``linear''\footnote{One may assign to an arbitrary collection of non-intersecting ovals a tree graph with vertices given by faces in $\mathsf F_{+}\cup \mathsf F_{+}$ and the edges given by their common boundaries. The contour shown in Fig.~\ref{fig_gar}b leads to a linear graph whereas e.g. the contour in Fig.~\ref{fig_sic}c yields a star-shaped graph with 4 vertices.}
pants decomposition leading to a RHP set on a collection of $n-3$ circles $\mathcal C_1,\ldots,\mathcal C_{n-3}$ decomposing $\Cb\Pb^1$ into $n-2$ faces $f^{[1]},\ldots,f^{[n-2]}$ as shown in Fig.~\ref{fig_gar}. By convention, the faces $f^{[2k+1]}$ and $f^{[2k]}$ will be of color $+$ and $-$, respectively.
\begin{figure}[h!]
\centering
\includegraphics[height=6cm]{ToepGarnier.eps}
\begin{minipage}{0.6\textwidth}
\caption{RH contour and coloring by $\left\{+,-\right\}$ associated to linear pants decomposition for Fuchsian systems.
\label{fig_gar}}
\end{minipage}
\end{figure}
Let $M_{k}\in\mathrm{GL}\lb N,\Cb\rb$ ($k=0,\ldots,n-1$) denote the monodromy of $\Phi\lb z\rb$ along the contour starting on the negative real axis and going around $a_{k}$ counterclockwise. These monodromies satisfy the cyclic relation $M_0M_1\ldots M_{n-1}=\mathbb 1$. It will be convenient for us to consider the products $M_{0\to k}:=M_0\ldots M_k$ and suppose that they can be diagonalized as
\ben
M_{0\to k}=S_k^{-1} e^{2\pi i\mathfrak S_k}S_k, \qquad k=0,\ldots, n-2,
\ebn
where the eigenvalues of diagonal matrices $\mathfrak S_k$ are assumed to be pairwise distinct $\operatorname{mod}\;\Zb$. It may also be assumed that $\operatorname{Tr}\mathfrak S_k=\sum_{j=0}^k \operatorname{Tr}\Theta_{j}$ and that $\mathfrak S_0=\Theta_0$, $\mathfrak S_{n-2}=-\Theta_{n-1}$.
Denote by $\Phi^{[k]}\lb z\rb$ ($k=1,\ldots,n-2$) the solution of 3-point Fuchsian system associated to the face $f^{[k]}$ (cf Subsection \ref{subsec4reg}) which has regular singularities at $0$, $a_k$ and $\infty$ characterized by monodromies $M_{0\to k-1}$, $M_{k}$ and
$M_{0\to k}^{-1}$. The local behavior of this solution near the singular points is given by
\ben
\Phi^{[k]}\lb z\rb =
\begin{cases}
S_{k-1} \lb -z\rb^{\mathfrak S_{k-1}} G_0^{[k]}\lb z\rb,\qquad & z\to0, \\
C_{k,\epsilon}\lb a_k-z\rb^{\Theta_k}G_{k}^{[k]}\lb z\rb,\qquad
& z\to a_k,\quad\epsilon=\mathrm{sgn}\Im z, \\
S_{k} \lb -z\rb^{\mathfrak S_{k}} G_{\infty}^{[k]}\lb z\rb,\qquad & z\to \infty,
\end{cases}
\ebn
where $G^{[k]}_{0,k,\infty}\lb z\rb$ are holomorphic invertible in the respective neighborhoods of $0,a_k,\infty$. The 3-point solutions define the jumps on $\mathcal C_1\cup\ldots\cup\mathcal C_{n-3}$: in the notation of the previous subsection, we have
\beq\label{local}
\begin{aligned}
&\Psi_{\mathcal C_{2k-1},+}\lb z\rb=G_0^{[2k]}\lb z\rb = \lb-z\rb^{-\mathfrak S_{2k-1}}S_{2k-1}^{-1}\Phi^{[2k]}\lb z\rb,\\
&\Psi_{\mathcal C_{2k-1},-}\lb z\rb = G_{\infty}^{[2k-1]}\lb z\rb = \lb-z\rb^{-\mathfrak S_{2k-1}}S_{2k-1}^{-1}\Phi^{[2k-1]}\lb z\rb,\\
&\Psi_{\mathcal C_{2k},+}\lb z\rb = G_{\infty}^{[2k]}\lb z\rb = \lb-z\rb^{-\mathfrak S_{2k}}S_{2k}^{-1}\Phi^{[2k]}\lb z\rb,\\
&\Psi_{\mathcal C_{2k},-}\lb z\rb = G_{0}^{[2k+1]}\lb z\rb = \lb-z\rb^{-\mathfrak S_{2k}}S_{2k}^{-1}\Phi^{[2k+1]}\lb z\rb,
\end{aligned}
\eeq
and $\bar{\Psi}\lb z\rb = \Phi^{[k]}\lb z\rb^{-1}\Phi\lb z\rb$ for $z\in f^{[k]}$.
Substitute these expressions into differentiation formula (\ref{widom_gen}) choosing therein $t=a_k$. The only circles that contribute to the sum are $\mathcal C_{k-1}$ and $\mathcal C_k$ (for the others $\partial_t J=0$). Moreover, for instance, for odd $k$ the only $\Psi_{\mathcal C,\epsilon}$ depending on $a_k$ are $\Psi_{\mathcal C_k,-}$ and $\Psi_{\mathcal C_{k-1},-}$, so that in this case
\begin{align*}
&\partial_{a_{k}}\ln\tau\left[J\right]=\\
=&\,-\frac1{2\pi i}\left[\oint_{\mathcal C_{k-1}}\!\!\!\!\!\! \operatorname{Tr}\left\{
\partial_z\bar\Psi_{\mathcal C_{k-1},+} \bar\Psi_{\mathcal C_{k-1},+}^{-1}\Psi_{\mathcal C_{k-1},-}^{-1}\partial_{a_k} \Psi_{\mathcal C_{k-1},-}\right\}dz+\oint_{\mathcal C_{k}} \!\!\!\!\operatorname{Tr}\left\{
\partial_z\bar\Psi_{\mathcal C_{k},+} \bar\Psi_{\mathcal C_{k},+}^{-1}\Psi_{\mathcal C_{k},-}^{-1}\partial_{a_k} \Psi_{\mathcal C_{k},-}\right\}dz\right]=\\
=&\,-\operatorname{res}_{z=a_k}\operatorname{Tr}\left\{\partial_z\lb{\Phi^{[k]}}^{-1}\Phi\rb
\Phi^{-1}\partial_{a_k}\Phi^{[k]}\right\}=\\
=&\,\operatorname{res}_{z=a_k}\operatorname{Tr}\left\{\partial_z\lb{G^{[k]}_k}^{-1}G^{(k)}\rb
{G^{(k)}}^{-1}\lb\frac{\Theta_k}{z-a_k}G^{[k]}_k-\partial_{a_k}G^{[k]}_k\rb\right\}=\\
=&\, \operatorname{res}_{z=a_k}\operatorname{Tr}\left\{\lb\partial_z G^{(k)}\,{G^{(k)}}^{-1}-\partial_z G_k^{[k]}\,{G_k^{[k]}}^{-1}\rb\frac{\Theta_k}{z-a_k}\right\} =\\
=&\, \partial_{a_k}\tau_{\mathrm{JMU}}-\partial_{a_k}\tau_{\mathrm{JMU}}^{[k]},
\end{align*}
where $\tau_{\mathrm{JMU}}$ is the tau function of the $n$-point system \eqref{FS} and $\tau_{\mathrm{JMU}}^{[k]}=\mathrm{const}\cdot a_k^{\frac12\operatorname{Tr}\lb\mathfrak S_k^2-\mathfrak S_{k-1}^2-\Theta_k^2\rb}$ is the tau function of the $3$-point Fuchsian system for $\Phi^{[k]}$. The transition from the 2nd to the 3rd line is obtained using that the integrands continue to the same meromorphic function on $f^{[k]}$ with the only pole at $z=a_k$. The 4th line follows from \eqref{local} and the 5th from the fact that ${G^{[k]}_k}^{-1}G^{(k)}$, ${G^{(k)}}^{-1}$ and $\partial_{a_k}G^{[k]}_k$ are holomorphic in $f^{[k]}$. The final equality follows from the Jimbo-Miwa-Ueno definition of the tau function, cf \cite[eq. (1.23)]{JMU}.
We have thus shown that for Fuchsian systems and linear pants decomposition the tau function defined by the Fredholm determinant \eqref{taugen} coincides with
\beq
\tau\left[J\right]=\frac{\tau_{\mathrm{JMU}}\lb a_1,\ldots,a_{n-2}\rb}{\prod_{k=1}^{n-2}
\tau_{\mathrm{JMU}}^{[k]}\lb a_k\rb}.
\eeq
When the singular points $a_0,\ldots a_{n-1}$ are irregular, one can obtain a similar identification by decomposing the original RHP into e.g. an $n$-point Fuchsian one (with regular singularities at $a_0,\ldots,a_{n-1}$) and $n$ two-point RHPs with one regular and one irregular singularity.
\vspace{0.3cm}
\noindent
{ \small \textbf{Acknowledgements}. The authors would like to thank M. Bertola, T. Grava, Y. Haraoka, N. Iorgov, A. Its, K. Iwaki, H. Nagoya, A. Prokhorov and V. Roubtsov for useful discussions. The present work was supported by the PHC Sakura grant No. 36175WA and CNRS/PICS project ``Isomonodromic deformations and conformal field theory''. The work of P.G. was partially supported the Russian Academic Excellence Project `5-100' and by the RSF grant No. 16-11-10160. In particular, results of Subsection 3.1 have been obtained using support of Russian Science Foundation. P.G. is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. }
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“I wanted to find a place where the voice was so loud that you couldn’t hear the grape,” says Mike Weersing, winegrower at Pyramid Valley, in New Zealand.
After 15 years of searching, Mike and his wife Claudia, who were originally from the US, finally settled on an old sheep farm in Wairkari, North Canterbury. They had spent over a decade studying soil maps and travelling to and from Europe, California, New Mexico, Portugal, and all throughout Australia. Climbing slopes and digging holes, sending countless soils samples back to scientist friends who would analyse them, looking for the perfect combination of limestone and clay on which to plant a brand new vineyard and liquify its geography into wine.
“There are affinity’s between grape varieties and soil types,” says Mike. “The reason for that is that each variety has it’s own characteristics and, when combined with the right soils, it will give you a unique voice… Clay limestone soil combinations are perfect for Pinot because it’s a thin skinned variety, it’s very aromatic, it doesn’t need schist or granite to boost the aromatics, but because it’s thin skinned, high in acid and lean, the clay provides volume and richness and texture, while the limestone gives it structure.
“I was looking for a specific combination of soil and climate and topography that could make a wine that would trump varietal voice,” explains Mike. “I wanted to add a new voice to the chorus of world wine voices that matter most to me… I didn’t want to make a wine that just tasted like the grape.”
The search eventually led Mike and Claudia to New Zealand.
“We’d never thought that the site was going to be in New Zealand.” says Mike. “I got a job making wine at Neudorf, in Nelson, and spent years going over the country with a fine tooth comb. It got to a point where I began looking at soil maps of Uruguay and Claudia was looking at flights back home to the United States.”
“We were looking at all these soils maps that were created in 1942,” says Claudia, “and we’d identify these limestone outcrops, but when we got there, there’d be all this wind blown loess, about 4-5 metres deep, and we started to think that maybe this just isn’t possible.
“That’s that makes Pyramid Valley so special,” continues Claudia. “It doesn’t have any loess covering the limestone. When we actually dug a few holes at the top we found that there was too much limestone, but a bit further down the slope was exactly what we were looking for.”
“I knew I was being pretty anal about it,” admits Mike, “but I also knew that I was old enough and poor enough that I really only had one shot at it.”
Pyramid Valley has been meticulously close planted with only Chardonnay and Pinot Noir grape vines. These vines are planted across four distinct sites, covering a total area of 2.2 ha, whose borders are defined by the existence of a balanced combination of clay and limestone soils.
Rather than plant right up and down the hillside and maximise the amount of wine that could potentially be made from the property, Mike and Claudia want to maximise the quality of each wine that’s grown from the four separate sites. As a result, each vineyard looks, from a distance, like some modern geometrical piece of art that’s been left hidden on a hill side while it waits for its great unveiling.
“We dug loads of holes by hand and marked them with bamboo,” explains Mike, “so we knew how high we could go before you would get only limestone, and how low you could go before you reached only clay. If you go too high on the slope you get only limestone, which isn’t good for soil health or plant health, or wine quality. If you go too low on the slope you get to a place where clay dominates and that’s not good either, for soil health, plant health, or wine quality.
“It would be so much more convenient if Mother Nature decided to work in a rectilinear grid,” says Mike.
From the very beginning, Pyramid Valley has been managed using biodynamics.
“I’ve worked with a range of different producers in Europe and in the New World,” explains Mike. “Some were conventional, and some were organic, and some were biodynamic, and it soon became very clear to me that the biodynamic producers always had the best soil and vine health, and therefore the best vineyards. Maybe not the best sites, but always the most healthy vineyards.”
“I was a sceptical of biodynamics when we first planted the vineyard,” explains Claudia, “and it took me at least four years to be convinced that it had value for growing wine. But, after making a few different compost heaps that had the biodynamic preps in them, I realised that the quality of the compost was so much better in the BD heaps than in the other ones. I kept forcing my ego and expectations on our land until I looked deeper and understood that it’s both our energies and the energy of the vines and the land all working together.”
Pyramid Valley’s Home vineyards and wines are all certified biodynamic by Demeter, and all the biodynamic preparations are made on site from materials collected off the Pyramid Valley farm.
“We make our own preps with offerings from our farm,” says Claudia, “because it’s just like yeast in wine and why you don’t buy packet yeast from Germany and use it to ferment your wine, and then talk about terroir.”
After all this searching high and low, all around the world, for the most perfect site to grow some wine on, planning and planting out the vineyards in such a meticulous way so that you might maximise the site’s potential to sing in it’s own unique voice, then spend the next seven years or so patiently tending to your young vines, encouraging them to grow stronger and healthier by not using any harmful chemicals around them, positioning, pruning, bud rubbing, shoot thinning, leaf plucking and finally fruit thinning, until at last the day arrives when the vine is finally ready to grant you a little wine, you might think you’d be a little nervous.
“The hardest part for me was waiting those 7 to 8 years before we could actually taste a wine from the property,” says Mike. “You know you’ve done as much as you can do but if you taste the wine and it’s banal, then you have to make a decision whether to commit and continue making banal wines from this site, or think about going to work for the council, or something.”
–
Thankfully, for Mike and Claudia (and the entire world of wine) the grapes that took so long to be composed on the slopes of Pyramid Valley posses one of the most beautiful voices in New Zealand. And, this voice – as far as I’ve tasted – has been known to sing some of the most unique and empyreal songs ever written into the song book of world wide wine.
“The grape is never the voice, it’s only the messenger…” says Mike. “Wine has this magical capacity to become a kind of liquid geography that can show the little nuances of a particular time and place, and I feel like it’s a moral failure not to aim for that. You might not get there every time, but you must do the best you can with what you have. Beer can’t do this. So, honour wine by trying to show what’s so special about it.”
D// – The Wine Idealist
Links and Further Reading
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When running a business, it is imperative to consider the various aspects which will determine the success thereof. One of these essential aspects includes the provision made to ensure workplace health and safety.
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TITLE: Find a generator for ideal $(a,b)$ in an euclidean domain [Euclidean $\Rightarrow$ PID]
QUESTION [1 upvotes]: Let $D$ be a euclidean domain and $a, b \in D$. Show that $M = \{xa + yb \ \mid \ x, y \in D\} $ is an ideal of $D$. Find $d \in D$ such that $M = \langle d \rangle$ and prove your claim.
My effort: we can extend Bezout's identity to euclidean domains (I see nothing in the proof that can't be extended here) to find $\alpha, \beta \in D$ such that $d \doteq
\text{gcd } (a,b) = \alpha \cdot a + \beta \cdot b$. Then, given any $x \in D$, we have $x d = x\cdot (\alpha \cdot a + \beta \cdot b) = (x\alpha)a + (x \beta)b \in M$, therefore $\langle d \rangle \subset M$. Conversely, since $a = d m $ and $b = d n$ for some $m, n \in D$, given $xa + yb \in M$, we have $xa + yb = d(xm + yn) \in \langle d \rangle$, whence $M \subset \langle d \rangle$, and then $M = \langle d \rangle$.
Is this alright? I'm always wary of questions that are solved too easily (and I'm usually right for that, since my solutions are usually wrong in those cases), so I'd appreciate a second opinion/corrections.
REPLY [2 votes]: Yes, the proof of the Bezout identity is the same as in $\Bbb Z,\,$ e.g. see here.. We can finish more simply as below, similer to the Euclidean algorithm
$$\begin{align} (a,b) &= (d,a,b)\ \ \ {\rm by} \ \ d\in (a,b)\\
&= (d,\color{#c00}{0,0})\ \ \ {\rm by} \ \ a\equiv \color{#c00}0\equiv b\!\!\!\pmod{\!d}
\end{align}\qquad\qquad$$
Generally it is true that $\ (d,I) = (d,\,I\bmod d)\,$ for any ideal $I\,$ (with easy proof - à la Euclid in $\Bbb Z)$
Remark $ $ The key idea is that ideals are closed under remainder (mod), so the "least" $\,d\in I\,$ must divide every $\,i\in I,\,$ else $\,0\neq i\ {\rm mod}\ d\,$ is in $\,I\,$ and smaller than $\,d,\,$ contra minimality of $\,d.\,$ The descent in this proof can be interpreted constructively as computing a generator of $\,I\,$ by computing the gcd of its elements (by taking repeated remainders).
The idea extends to PIDs: (Dedekind-Hasse criterion)
a domain $\rm\,D\,$ is a PID iff given $\rm\:0\neq a, b \in D,\:$
either $\rm\:a\:|\:b\:$ or some D-linear combination $\rm\:a\,d+b\,c\:$ is "smaller" than $\rm\,a.\,$ In a PID we can choose the number of prime factors as a measure of (Euclidean) size.
REPLY [1 votes]: That looks great to me. This is exactly the power of Bezout's identity to show that finitely generated ideals are principal.
It is perhaps worth noting that you should also be able to do it in even more generality for any ideal, not just finitely generated ideals. You simply choose an element of minimal degree under the Euclidean division algorithm and it will generate the ideal. See for example: https://proofwiki.org/wiki/Euclidean_Domain_is_Principal_Ideal_Domain
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Mim premier trade exhibition for construction materials and equipment in the region, in which Stars Innovation participated with great results. The show took place from 26 – 29 June at the Seaside Arena, Beirut.
“Our sincere congratulations go to Stars Innovation on their 20th anniversary,” said Ms Danna Drion, Senior Marketing Manager EMEA of Mimaki Europe. “Stars Innovation has always been a valued partner of Mimaki in the region. Having benefited from their impeccable knowledge and unbeatable expertise, we believe Stars Innovation is the right partner to help us maintain our position as the market leader in Lebanon. We look forward to continuing our relationship for years to come.”
Mimaki solutions highlighted at Project Lebanon 2018
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- The UCJV300 Series. This latest UV release from Mimaki was the main attraction at the stand of Stars Innovation. UCJV300 printers deliver a remarkable range of applications and versatility with four-layer printing in addition.
- The Tx300P-1800. Engineered exclusively for direct-to-fabric printing, the Mimaki Tx300P-1800 includes many features found in high-end direct-print models, but with a lower cost-of-ownership model that is suitable for users creating samples or short-run pieces. It also features Mimaki’s unique dual ink capability, with which users can simultaneously load both textile pigment and sublimation dye inks on one printer.
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TITLE: Quotient of a smooth complex variety by a finite group with fixed locus of codimention $1$ is smooth
QUESTION [0 upvotes]: Let $X$ be a smooth complex variety and $G$ be a finite group acting on $X$ with the fixed locus of codimension $1$. Then why is $X/G$ a smooth variety? I am confused with this claim because given a disk $D$, and a finite group $G$ acts on $D$ by rotation around a point $x$, then the quotient space is a cone with a cone point from $x$, which makes the quotient space not smooth at the cone point. However, I agree the quotient space is homeomorphic to the disk. On the other hand, given the coordinate $z$ on the holomorphic disk and $G$ acts by $z\mapsto z^n$, so the quotient space is a holomorphic disk with holomorphic coordinate $w=z^n$. (The ring of functions of $X/G$ is $\mathbb C[z^n]\cong \mathbb C[z]$, thus $X/G$ is still a holomorphic disk.)
REPLY [1 votes]: Suppose that the variety is quasi projective. In this case the quotient exists and it is locally given by the spectrum of the algebra of invariants. You can study the problem in the neighborhood of a fixed point $x$ and study the action on the tangent space.
In this case you can apply the Chevalley-Shepperd-Todd Theorem: The quotient is smooth if and only if the group is generated by pseudoreflections.
A pseudoreflection is an diagonalizable element of $GL_n$ with all the eigenvalues but one equal to one (it fixes an hyperplane).
Consequently you see, looking to the tangent plane of the fixed part, that:
-- If $x$ is a smooth point of the fixed divisor (or more in general the cone of it is an hyperplane), then the quotient around it is smooth (direct application of Chevalley-Shepperd-Todd: the tangent space of the fixed part is fixed).
-- By the same argument, the cone of the fixed part cannot be an hyperplane: otherwise the action would fix the tangent space at $x$ of the fixed part which coincides with the tangent space of $X$. So it would fix all of $X$.
I think that the confusion you have is due to the fact that you do not specify in which category you work: What you mean by quotient? the cone you propose is topologically isomorphic to the second interpretation (this is why I started my answer with the local construction of quotient).
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Timmythewop89Member
Content Count965
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About Timmythewop89
- RankGold Member
Contact Methods
- AIMstkahtdoginurass
- Website URLhttp://
Vehicle Information
- VehiclesChevy Camaro
- Modifications383 stroker, aftermarket cam, edelbrock 750 carb, 64cc combustion heads with flap-top pistons, soon to get exhaust upgrade
Question for you cranny
Timmythewop89 replied to willthiswork89's topic in Preludeif it were valve seals, then the blue smoke would most likely show at start up or when you let off the gas after hard acceleration. not trying to sound like hte bringer of bad news here, but blue smoke under hard acceleration usually means piston rings are hurting. A cheaper thing to try is a head gasket, process of elimination, but im pointing towards rings.
been gone for a while
Timmythewop89 replied to Timmythewop89's topic in General Discussionlol nothin beats a big block, but small blocks have a sound of their own. everything with my friend has settled down. like hte parents are doing a lot better and stuff like that. there is a huge memorial at the crash site, me and a couple friends made a huge cross, and left markers so people could sign it, and its filled with signatures. turned out really nice. as for the car, i bougth 4.11 ring and pinion, and as soon as i sell my little dodge ill have aluminum heads too. finally coming along, last summer was rough when it came to making money because i had to pay off the lawn equipment, but this year will be golden. alright peace. tim
been gone for a while
Timmythewop89 posted a topic in General DiscussionWell havent been on in a while. december 1st my best friend flipped his car into woods and died december 3rd, so its been pretty crapty. but anyways i figured i stop by and show some videos of the maro i did with a cell phone. quality sucks but whatever.
You're my first....
Timmythewop89 replied to Well Accorded Blonde's topic in Introductionshi
- lastnight i raced him twice, we raced when we got onto the highway then raced after the u turn back. the first time i started way after him and the second time was dead on. it was a first gear role at 25 mph. beeped 3 times, floored it, went right through my third gear and we were going 110 before i hit the brakes. i won. whoot whoot. the war is over for now and there wil be no more racing for me. he is the only kid i will race really because all the other cars suck. also, when we were leaving the off ramp and heading back to town there was a cop pulling onto the highway. that was close.
- not only is his carb way to high, but he is way overcammed. he has a bigger cam then me butnot only does it csound like crap, no other part in his motor is matched to it. this is a 290 cam and he has a performer intake manifold. it doesnt even idle nice, but because its loud all the gay redneck kdis think its badass but none of htem have a ear fir stuff like that because they dont kno anything. my cams a comp cams 280h and it makes more overlap then his. he doesnt even have aftermarket rockers or springs. as for the chevy/ford comment, when ford builds there crate motors and such, they do incorprate better internal designs that help with the travel of oil and the moving of air in and out, but that is why their parts are more costly. when it comes to durability chevy and ford are right next to each other, but when it comes to working on, ide much rather work on a chevy. it jsut seems to me that every time ive worked on a ford compared to a chevy, even with simple things, has been more of a hassle. the one thing with chevy v8's, is i think oyu get more bankg for your buck. i am getting a rear end out of a chevy truck, with comes stock with 3.73's. really all i need is the rear end, heads, and a all manual valve body with ratchet shift. as for heads, im taking some HO heads off a 305, because of their small combustion chambers. im going to try to achieve about 10:1 compression after porting and milling them and porting the intake manifold ot match. that alone should set me ahead of him, but my tranny is a real slug with its valve body. its great and strong in the manual 1 and 2 gears, but when you drop it in 3rd it doesnt grip fully because it thinks whether it wants to downshift or stay the same. eventually im getting a th-400 tranny but that doesnt have to happen soon, 700r4's are strong tranny's. like i said about the rear end, if i find myself a truck with 3.73's and hopefully disk brakes in teh back, then ill be good to go. and he has stock heads by the way.
- hung dude shut up you act like its really pissing you off. your sitting at your damn computer if it pisses you off turn away. everybody is so dramatic. the guy is asking a technical question who gives a fk if its repetative if u think it is dont answer theres no reason to be a dooshbag, christ people act like if they know something over someone else that they are superior. ignorant people these days, and thats huge coming form me, a very ignorant person.
backfire out carb?
Timmythewop89 replied to garrettacura's topic in Accordwe flew right by that bandwagon after getting a 505 horsepower engine getting 26mpg highway. even tbi in the early 90's was good performing, gave big trucks about 15 mpg which was very good back then. oh and domestics has fuel infection earlier then that. i think 85 or 87 was the first years.
- i dont know i personally dont even want to race it until i get a new rear end and heads. another thing is i dont know if you are familiar with the 700r4 tranny, but its 1st gear is very short, helping a lot for off the line. the only problem with that is my second gear is so big that my cars well out of its power band when i drop it into second. it kicks in at about 3 grand and is at its peak at 4500. well ill keep you posted. i forgot who said it but i would never rip out anything to make the car lighter. if anything ide replace heavy parts for more high quality lighter parts. i dont know hwat else to say so im gonna stop here.
What do you think the problem is?
Timmythewop89 replied to Sneekpeaks's topic in Accordlooks to me like oil from the valve cover gaskets. when oil is introduced to fire, it makes blue smoke. but when its introduced to jsut high heat, it lets off a white steam. looks like chunky oil to me. replace valve cover gaskets.
- oh my bad it is more pricey for v8's. lucky hondas if i wanted to pay the money ide get 4-2-1 headers all the way, unless it was a top end motor.
Running ever so rich... help emissions!
Timmythewop89 replied to HondAtuner0487's topic in Civicput the stock injectors back in if u replaced them, but if you didnt, i really dont kno what to do. its not carbureted so you cant adjust it. messing with your timing can help, but messing with it too much is bound to make it worse.
- i havent lost any friends or relatives. does it may me not as much of a hardas? dont think so, but then again there are good safer places to race around here, not saying its mistake free. and dont get the idea i go around and race. the cars been out all spring and summer and i havent raced a soul. theres many people that would disagree with you along with me about domestics being gay.
- depends on the use really. for more street friendly bottom end, ide prefer a 4-2-1, but they may be more expensive. 4-1's are cheaper, but may hurt bottom end unless the collectors are significantly long. i jsut say go with 4-1, itl save u money and really wont hurt much bottom end.
- dount sound mature u kno if a mazda miata pulled up next to you and reved youde race him. its not "people" planning to race. its me and 1 other person, no one else. ur saying its better to randomely race then plan it? oh yea and i dont have a race track in my back yard. well lets look at the alternatives. i could take a 2 hour drive to seekonk speedway and pay a couple hundred, oh wait for laps on the track is a couple thousand. or no, i could take a 3-4 hour drive to new england dragway. ide rather race on the four lane perfectly straight 1 mile long escape road with flashing traffic lights at each end.
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TITLE: Proof that a sequence is bounded.
QUESTION [0 upvotes]: I am attempting to prove that the infinite sequence;
$1,-2,3,-4,....$
is unbounded to get a better understanding of sequences.
Suppose that the sequence $a_{1},a_{2},a_{3},...$ defined by $a_{n}=(-1)^{n-1} . n$ for $ n \in N $, is bounded by $M$, such that $|a_{n}| \leq M$ for any $n$. where $M$ is a rational number such that $M \ge 0$.
since $M$ is a rational number, there exists an integer p such that $ p \le M < p+1 $, taking absolute values $ |p| \le |M| < |p+1| \le |p|+|1|$.
I need to show that $|p+1|=|a_n|$ for some n... i think. This will be my contradiction.
REPLY [1 votes]: $$|a_{p+1}| = |(-1)^{p} \cdot (p+1)| = |p+1|$$
So, $|M| < |a_{p+1}|$ and the sequence is thus unbounded because your premise that $|a_n| < |M|$ for any $n$ is shown to be false.
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From Kazuma Kiryu’s last outing, Ys VIII’s oft-delayed PC port, and Don’t Starve’s trek on to consoles, this week’s itinerary of new game releases offers several highlights. And while the Nintendo’s Switch receives a healthy number of titles this week, oddly the platform won’t be receiving the Star Wars: The Last Jedi DLC for Pinball FX 3. Potentially, the omission is related to Zen Studio’s commitment for getting the game running at sixty frames-per second.
Header image: Supipara – Chapter 2 Spring Has Come!, PC
PlayStation 4
Don’t Starve Mega Pack (physical and digital, $29.99)
Dusty Raging Fist (digital, $14.99)
Regalia: Of Men and Monarchs – Royal Edition (digital, $24.99)
Rogue Aces (digital, $TBA)
Pinball FX3 – Star Wars: The Last Jedi (DLC, digital, $TBA)
Yakuza 6: The Song of Life (physical and digital, $59.99)
Switch
ACA NeoGeo Gururin (digital, $7.99)
Asdivine Hearts (digital, $12.99)
#Breakforcist Battle (digital, $9.99)
Burly Men at Sea (digital, $9.99)
Don’t Starve Nintendo Switch Edition (digital, $19.99)
Dragon Blaze (digital, $7.99)
Drone Fight (digital, $4.99)
Eternal Edge (digital, $20.00)
It’s Spring Again (digital, $1.79)
Pirates: All Aboard! (digital, $5.99)
Regalia: Of Men and Monarchs – Royal Edition (digital, $24.99)
Rogue Aces (digital, $10.40)
Skies of Fury DX (digital, $19.99)
Streets of Red: Devil’s Dare Deluxe (digital, $7.19)
Word Search by POWGI (digital, $7.99)
Zotrix: Solar Division (digital, $14.99)
Wii U
Shadow Archery (digital, free)
Xbox One
Don’t Starve Mega Pack (physical and digital, $29.99)
Pinball FX3 – Star Wars: The Last Jedi (DLC, digital, $TBA)
Regalia: Of Men and Monarchs – Royal Edition (digital, $24.99)
Ys Origin (digital, $19.99)
PS Vita
Rogue Aces (digital, $TBA)
PC
Dead in Vinland ($17.99)
Double Turn ($9.99)
Pinball FX3 – Star Wars: The Last Jedi (DLC, digital, $TBA)
Super Saurio Fly ($TBA)
Supipara – Chapter 2 Spring Has Come! ($TBA)
The Road to Hades ($13.49)
The Wastes ($11.24)
Ys VIII: Lacrimosa of Dana
Robert’s Pick: With the industry’s weakness for reboots that offer only marginal improvements on previous iterations, sequels motivated by profits rather than expression, the ubiquity of season passes, and those damn loot boxes, it’s easy to become jaded by the current state of gaming. But then there’s the Yakuza series, which in it’s sixth iteration, wraps up Kazuma Kiryu’s story. Twelve years and three generations of hardware later, Kamurocho is still a mesmerizing destination, offering a myriad of recreations that emulate the density and diversity of Tokyo’s legendary entertainment district. Complementing the urban sprawl is a trip to Hiroshima, which offers a pastoral complement to the bustling city. The appearance of Takeshi Kitano is the icing on the cake for me, making Yakuza 6: The Song of Life not just my pick, but an early game of the year contender, that offsets any sense of cynicism for its forty+ hour duration.
Ryan’s Pick: Yakuza 6: The Song of Life all the way. The Yakuza series is very aware of itself, which is what makes the games so memorable and fun. From serious aniki moments and hilarious fist fights, to singing karaoke by yourself, the Yakuza series understands how to entertain. I’m really excited to see that Tatsuya Fujiwara and a bunch of other famous Japanese actors will debut like in the previous games. Being able to go into Club Sega and play old games is also a welcome feature – I’m particularly stoked to see that one of my favorites Space Harrier is included! I remember playing it in the arcade and being scared of it when I was really young…yare yare. Again, very excited that Kiryu is back!
Matt’s pick (Editor, DigitallyDownloaded): Yakuza’s the easy choice, but I reviewed that so long ago it feels like old news now. Ys VIII is a lovely game, but I’d recommend people wait for the Switch version. So what’s really left?
You know what? I’m going to go with Pinball. I’ve been playing Zen Pinball 2, and now Pinball FX3, for at least half a decade now. Probably longer. It’s not an everyday thing, but when I’ve got ten minutes here or there, I’ll boot it up. I’ve loved pinball as a kid, and these games do a great job of capturing the joy and excitement I’ve always got from pinball. Heck, I’ll probably end up spending more time playing this table than anything else on the list that I might end up playing. Yakuza is a play once and then, maybe, come back to it a few years later kind of thing. Pinball FX 3 tables are a commitment for life.
Zack’s Pick (Senior Editor, RPG Site): I’d love to say Yakuza 6 like some of the others, but I actually haven’t touched the series since I covered Yakuza 5 for my site a few years ago. With that in mind, I will have to give it to Ys Origin on the Xbox One, which marks the first time the Ys series has been seen on a Microsoft console.
While I wait to play Ys VIII on PC, I can wholeheartedly recommend my favorite entry in the series. Everything from its lightning-paced combat to its healthy dose of gameplay variety makes this one stand head and shoulders above many of its series’ brethren. That along with its stellar soundtrack makes it a mighty satisfying experience. Oh, and Epona. Epona is the best.
Where the hell is God of War????
Microsoft: “We don’t have Ys VIII but we got Ys Origin! You’re excited, right???”
Xbox One owners: ….
The game title that best describes my sex life this week? Don’t Starve Mega Pack
I guess that’s better than Burly Men at Sea…
The wait for Yakuza 6 is finally over. I can’t wait!
There’s something ugly and weird about that Pinball FX 3 screenshot. The walls look too high for one thing.
Can someone tell me the true meaning of “yare yare” ?
Pirates: All Aboard’s logo reminds me of Sid Meier’s Pirates
Its like one of the rip off apps or Red Box movies.
Yakuza for me. Robert are you and Ryan brothers?
Regalia: Of Men and Monarchs looks like it might be worth checking out.
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Nominents for annual Estonian Theatre Awards have been announced
March 01
Estonian Theatre Association has announced the nominents for annual Estonian Theatre Awards. Awards are given for creative works presented in 2018.
The nominents for the Ballet Award are:
Jevgeni Grib – choreographer-director of the shortballet „Keep a light in the window“ (Estonian National Ballet).
Darja Günter – performance in the short ballet "Keep a light in the window" (Estonian National Ballet).
Marta Navasardjan – performance in the short ballet "Keep a light in the window" (Estonian National Ballet).Vanemuine Ballet – in performances „Man in the Movie“ and „Romeo and Juliet“ (Vanemuine Theatre).
Jury: Enn Suve, Tiit Härm, Jelena Karpova, Saima Kranig and Agnes Oaks.
The winners are to be announced on International Theatre's Day on 27th March. This year the awards gala takes place in Rakvere Theatre.
The award is monetary and is financed by Estonian Cultural Foundation. It includes also a glass-figure - the Theodor's Eye - made by Estonian glass-artist Ivo Lill and financed by the ministry of culture.
Estonian Theatre Union has been giving annual awards since 1961.
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Freaks and Geeks is a 1999 show telling the story of the titular groups in a 1980’s American high school.
Fans and creators of the show often discuss the misfortune of the characters as a stand out feature. Arrested Development or Peep Show heap unrelenting failure upon their protagonists, yet the more variable fortunes of the characters in this show are not only more realistic, but also creates characters that you care more about as people, rather than fictional punching bags. It isn’t joke a minute for sure, but consequently the laughs come in a way that resembles how you would laugh at something in real life.
Multitudes of future famous people show up and it is fun how subtle their parts often are. For example, Seth Rogan’s character is very subdued and sarcastic where you get the sense that today he might put more intensity into the performance. Not all the performances are amazing. Jason Segel mumbles his way through, though this is probably the point due to his character being high all the time and Sam Weir occasionally says thing with an unnatural cadence.
The direction, while never mind blowing, is consistently good and never distracting. The music, on the other had, often steals the show. Perhaps songs that are current are harder to put in TV, or perhaps crappy unknown stuff has producers who are willing to pay more, because shows including Smallville and Pretty Little Liars always have garbage unknown music. For whatever reason, the 80’s setting allows the editors to pull from diverse music from throughout the decade and I don’t recognise half of it, but it all sounds good.
Freaks and Geeks has been called ahead of its time and sure enough I keep struggling to remember that I was 5 or 6 when this show came out and the child actors are all adults now. Perhaps part of it, as well as the edgy writing, is the 80’s setting. The 4:3 aspect ratio and slight low definition fuzz look a bit dated, but the whole aesthetic was always meant to be dated and being a 90’s show it also had the budget and freedom to not actually feel like an 80’s show (perhaps more like an 80’s movie). It literally has a sort of timelessness that still serves it.
It doesn’t feel like too great a tragedy that there are no more seasons of Freaks and Geeks. I have the sense that future seasons could have been consistently enjoyable, but, as with Firefly, being cut short likely contributed to its fame and renown. During the first half of the season, I thought the idea of a follow up film would be terrible, but after reading an interview with Paul Feig where he says: “it was going to become much more of a story of a small town and who gets out and who doesn’t,” a high school reunion movie could be a very good way of doing that and hopefully it would be able to stand alone too.
Connor Cochrane
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TITLE: Derivation of composite bayesian posterior
QUESTION [0 upvotes]: In Trifonov et al. (2013), which uses a bayesian framework to identify prioritized gene lesions, a composite posterior is defined as:
$$P(D|S) \quad \underline{deff} \quad \sum_{i}P(D|M_i) \cdot P(M_i|S) \tag{1}$$
I am trying to derive this posterior from the simple posteriors defined previously in the article, i.e.:
$$P(D|M)= \frac{\delta(D \in M) \cdot P^d(D)}{\sum_{G∈M}P^d(G)}\tag{2}$$
$$P(M|S)= \frac{\delta(M \in S)\cdot P^d(M)}{\sum_{j}P^d(M_j)}\tag{3}$$
My derivation is the following:
$$P(D|S) \quad \underline{deff} \quad \sum_{i}P(D|M_i)·P(M_i|S)\tag{4}$$
Inserting the actual posteriors:
$$P(D|S)= \sum_{i} \frac{\delta(D \in M_i) \cdot P^d(D)}{\sum_{G \in M_i} P^d(G)} \cdot \frac{\delta(M \in S) \cdot P^d(M_i)}{\sum_{j} P^d(M_j)}\tag{5}$$
If $D \in M_i$ then $D \in \bigcup S$ because $M_i \in S$, then:
$$P(D|S) = (\delta(D \in \bigcup S) \cdot P^d(D)) \cdot \sum_{i} \frac{ P^d(M_i)}{\sum_{G \in M_i} P^d(G) \cdot \sum_{j} P^d(M_j)}\tag{6}$$
Same as the previous step, since $M_i \in S$ the first sum at the denominator can be defined with respect to $\bigcup S$:
$$P(D|S) = \frac{\delta(D \in \bigcup S) \cdot P^d(D)}{\sum_{G \in \bigcup S} P^d(G)} \sum_{i} \frac{P^d(M_i)}{\sum_{j} P^d(M_j)}\tag{7}$$
At this point I'm stuck. The original article continues:
To obtain the global posterior $P^g(D|S)$ we assume that $P^d(M) \propto \sum_{G \in M} P^d(G)$ in which case:
$$P^g(D|S) = \frac{\delta(D \in \bigcup S) \cdot P^d(D)}{\sum_{G \in \bigcup S} P^d(G)}\tag{8}$$
What is the passage that lead to the elimination of $\sum_{i} \frac{P^d(M_i)}{\sum_{j} P^d(M_j)}$?
REPLY [0 votes]: It was actually simple. The elimination of $\sum_{i}\frac{P^d(M_i)}{\sum_{j}P^d(M_j)}$ is straightforward once you see that both the numerator and denominator are the sum of the same elements.
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.
This hardly seems fair. The County should follow their own rules. When was the property developed? When were the highway district regulations enacted? Do the regulations apply to all properties or only properties when they are redeveloped or repurchased?
Good comment. The rules require the improvements to be made if changes/improvements are made to the property. I watched changes being made to this property at the same time as businesses in Hayes. Those businesses had to meet the rules and the county basically said do as I say and not as I do.
The rules were in place when the county made the modifications, business owners told me about what they had to do while watching the county ignore the rules.
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Photography
"So now that my hobby has become a business what do I do to unwind and relax, I pick up my camera."
I hope you enjoy my photos as much as I enjoy taking them.
“What I like about photographs is that they capture a moment that’s gone forever, impossible to reproduce.”
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Feature
Vaporizing the Gas Tax Myth
The United States must move away from the gas tax to solutions that charge people for the roads they use, including a VMT fee, congestion pricing for peak hours and toll roads, says Jack Finn of HNTB. Such efforts will encourage Americans to be less dependent on oil, reduce congestion, take public transit and properly invest in infrastructure.
Americans hate the gasoline tax about as much as they love their cars.
At the federal level, money from a gas tax was first placed into the Highway Trust Fund in 1956 as the country embarked on President Dwight Eisenhower's grand vision of establishing a network of interstate highways to spur commerce and aid in the country's defense.
Now, more than 50 years later, Eisenhower's long-ago realized vision is reaching the end of its useful lifespan, and the gas tax itself is running on empty.
This is unwelcome news to the average American driver, already suffering through the current economic downturn and the painful $4 per gallon gas that preceded it.
Photo by flickr user NovaMan396
For years we've comforted ourselves with the notion that filling up at the pump pays for our roads in full. This is nothing more than a myth, a misperception that must end. In fact:
- There is no such thing as a free road. Anyone who has paid off a mortgage knows there are always costs of home ownership. Renovations, expansions and simple upkeep - while necessary - can be expensive. Simply paying off the original financing on a transportation project doesn't mean it's paid for either.
- Roads don't pay for themselves. Research from the Texas Department of Transportation has compared how much gasoline is consumed on a roadway with how much gas tax that generates, revealing that no road completely pays for itself over a 40-year lifespan.
- Weve run up a huge transportation tab. Crumbling roads, rusting bridges and congestion are all signs we've deferred too much maintenance. According to a national commission that studied our surface transportation needs, we need to invest at least $225 billion annually for the next 50 years to repair and upgrade the system. The longer we wait the more expensive it becomes.
- The gas tax isn't what it used to be. The federal gas tax, now set at 18.4 cents per gallon, was last increased in 1993. A combination of inflation, changing driving habits - due in part to higher gas prices - and better fuel economy of our cars has robbed it of much of its purchasing power. In fact, the trust fund is broke, needing infusions from the general treasury totaling more than $15 billion in the last year alone.
The way we fund our roads is at odds with almost every other public policy America has adopted. While proposed climate change legislation, green energy initiatives and even our foreign policy demand that we move away from a dependence on oil, we pay for our transportation system almost entirely by using more of it.
In the short-term, we need to consider an increase in the gas tax. It's a bitter pill to swallow, but it's the only way we can ease the congestion we face. At last count, that congestion costs every traveler in the U.S. $750 a year. A gas tax increase between 5 cents and 8 cents each year during the next five years will cost average Americans only $10 to $20 each month per car.
In Britain and much of Europe the gas tax is nearly $4 per gallon, 20 times the federal tax in the U.S.
In the long-term, we must move away from the gas tax to solutions that actually charge people for the roads they use, including a vehicle miles traveled user fee, congestion pricing for peak hours and more toll roads. We're willing to pay for actual use of other utilities - like electricity, water and natural gas - why not our roads?
Such efforts will encourage Americans to be less dependent on oil, reduce congestion, encourage use of public transit and properly invest in infrastructure.
Jack Finn is National Director of Toll Services for HNTB Corporation.
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\begin{document}
\keywords{$L$-functions, trigonometric sums, Jordan totient function, Euler totient function, mean square averages, Gauss sum, Ramanujan sum, Bernoulli numbers}
\subjclass[2010]{11M06, 11L05, 11L03}
\author[N E Thomas]{Neha Elizabeth Thomas}
\address{Department of Mathematics, University College, Thiruvananthapuram, Kerala - 695034, India}
\email{nehathomas2009@gmail.com}
\author[A Chandran]{Arya Chandran}
\address{Department of Mathematics, University College, Thiruvananthapuram, Kerala - 695034, India}
\email{aryavinayachandran@gmail.com}
\author[K V Namboothiri]{K Vishnu Namboothiri}
\address{Department of Mathematics, Government College, Ambalapuzha, (Affiliated to the University of Kerala, Thiruvananthapuram) Kerala - 688561, INDIA\\Department of Collegiate Education, Government of Kerala, India}
\email{kvnamboothiri@gmail.com}
\begin{abstract}
Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,\chi)$ and $L(4,\chi)$ using certain trigonometric sums and Bernoulli polynomials and express them in terms of the Euler totient function $\phi$ and the Jordan totient function $J_s$.
\end{abstract}
\maketitle
\section{Introduction}
Let $k$ be a natural number $\geq 3$. A Dirichlet character $\chi$ is defined to be odd if $\chi(-1)=-1$ and even if $\chi(-1)=1$. The Dirichlet $L$-function $L(s,\chi)$ is defined by the infinite series $\sum\limits_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ where $s\in\C$ with $Re\,(s)>1$. It is an important function in number theory especially due to its connection with the Rieman zeta function $\zeta(s)$. For rational integer $r$, the problem of computing exact values of
\begin{align}\label{eqn:gen_sum}
\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2
\end{align}
and thus finding the mean square averages of this sum has been attempted in various cases by many.
In 1982, Walum \cite{walum1982exact} gave an exact formula for the sum (\ref{eqn:gen_sum}) with $r=1$. Louboutin (\cite{louboutin1999mean}) computed the sum of $\lchi^2$ over all odd primitive Dirichlet characters modulo $k$. See \cite[Chapter 6]{tom1976introduction} for the definition of primitivity of Dirichlet characters. In \cite{louboutin1993quelques}, Louboutin gave an exact formula for the sum of $\lchi^2$ over all odd Dirichlet characters in terms of the prime divisors of $k$ and the Euler totient function $\phi$. He mainly used the orthogonality properties of characters and some trigonometric identites in his computations. Using the same techniques, in \cite{louboutin2001mean} he derived exact formulae for the general versions of these sums in two cases : $\chi$ even and $\chi$ odd.
E. Alkan in \cite{alkan2011mean} derived exact formulae for the sums $\sum\limits_{\chi \, \text{odd}}|L(1,\chi)|^2$ and $\sum\limits_{\chi \, \text{even}}|L(2,\chi)|^2$ using weighted averages of Gauss and Ramanujan sums. His formulae involved Jordan totient function and Euler totient function (See the next section for definitions). Alkan employed certain exact evaluations of trigonometric sums appearing in \cite{alkan2011values} to enable these computations. The computations he performed in this paper were so extensive and beautiful so that some of the identities he derived during these computations gave rise a sequence of papers starting with those by L. Toth \cite{toth2014averages} and K V Nambooothiri \cite{namboothiri2017certain}.
After Alkan, many other authors also attempted the problem of finding mean square values of $L(r,\chi)$ for various values of $r$. A general formula was provided by T. Okamoto and T. Onozuka in \cite{okamoto2015mean} and \cite{okamoto2017mean} using a technique suggested by S. Louboutin in \cite{louboutin2001mean}. Note that this technique was different from the one used by Alkan in \cite{alkan2011mean}. The mean square values and certain related problems were dealt in some other papers as well, see example, \cite{wu2012mean}, \cite{zhang2015hybrid}, and the recent papers \cite{lin2019mean}, \cite{zhang2020certain}. In \cite{lin2019mean}, X. Lin provided a general inductive formula for computing the sum (\ref{eqn:gen_sum}).
Some other closely related problems were attempted by a few authors. In \cite{zhang2003problem}, W. Zhang derived an asymptotic (not exact) formula for $\lchi^4$ over odd Dirichlet characters. He used Abel's identity and Cauchy's inequality in addition to the orthogonality properties of characters to arrive at his estimate. Alkan in \cite{alkan2013averages} derived the sum $\sum\limits_{\chi \, \text{odd}}(L(1,\chi))^r$ where $r\geq 1$. In \cite{louboutin2015twisted}, S. Louboutin computed $\sum\limits_{\chi \, \text{odd}}\chi(c)|L(1,\chi)|^2$ where c, a positive integer with some extra conditions. Computation of such sums, known as twisted sums is another problem of active interest.
Note that Alkan's techniques in \cite{alkan2011mean} were completely different from the other derivations used in computing the square sums. It was also observed by Alkan that his techniques could be used to determine the sum (\ref{eqn:gen_sum}) for any larger $r(\geq 3)$, but at the cost of increasingly complex computations. We here undertake these complex comuptations and use the same techniques used by Alkan in \cite{alkan2011mean} to derive exact formulae for $\sum\limits_{\chi(-1)=-1}|L(3,\chi)|^2$ and $\sum\limits_{\chi(-1)=1}|L(4,\chi)|^2$.
We also derive some trigonometric identities during these computations that could be of independent interest.
\section{Notations and some elementary results}
In this section we introduce the basic definitions and some identities that we use throughout this paper. The definitions of terms we do not define, but appear in this paper can be found in \cite{tom1976introduction} or \cite{montgomery2006multiplicative}. For a positive integer $k$, if $\chi$ is a Dirichlet character modulo $k$, then $\chi(-1)=(-1)^n$ for some natural number $n$. The parity of $\chi$ is the parity of this $n$. Hence $\chi$ is odd if $n$ is odd and even otherwise.
The Gauss sum $G(z,\chi)$ for any complex number $z$ is defined as
\begin{align}
G(z,\chi):= \sum\limits_{m=1}^{k}\chi(m)e^{\frac{2\pi imz}{k}}.
\end{align} When $\chi=\chi_0$ is the principal character, Gauss sum becomes the Ramanujan sum $R_k(z)$:
\begin{align}
R_k(z):= \sum\limits_{\substack{m=1\\(m, k)=1}}^{k}e^{\frac{2\pi imz}{k}}.
\end{align}
The Jordan totient function $J_k(n)$ is defined by \begin{align}
J_k(n) := n^k\prod_{\substack{p|n\\p\text{ prime}}}\left(1-\frac{1}{p^k}\right).
\end{align}
A similar type of function which we use in this paper is $\phi_k(n)$, defined as the sum of the $k$th powers of numbers $\leq n$ and relatively prime to $n$.
By $B_q$ we mean the $q$\textsuperscript{th} Bernoulli number. See \cite[Chapter 12]{tom1976introduction} for definition and other properties of Bernoulli numbers and Bernoulli polynomials. The values of Bernoulli numbers we use in this paper are $B_0=1$, $B_1=-\frac{1}{2}$, $B_2=\frac{1}{6}$, $B_3=0$, $B_4=-\frac{1}{30}$.
For a positive integer $m$, by $S(m, \chi)$ we mean the sum
\begin{align}
S(m, \chi):=\sum\limits_{j=1}^k(\frac{j}{k})^mG(j, \chi).
\end{align}
Some other important identities that we use in our computation are listed below:
\begin{align}
& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \frac{1}{\sin^2(\frac{\pi m}{k})}=\frac{J_2(k)}{3} \label{eqn:1bysin2}\text{ \cite[Identity 5.16]{alkan2011values}}.\\
&\sum\limits_{j=1}^{k-1}j^re^{\frac{2\pi imj}{k}}=\sum\limits_{j=1}^{k-1}\binom{r}{j} k^j\lim\limits_{w\rightarrow \frac{2\pi im}{k}}\frac{d^{r-j}}{dw^{r-j}} \left(\frac{1}{e^w-1}\right) \label{eqn:jrediff}\text{ \cite[Section 2]{alkan2011values}}.
\end{align}
The following identities can be easily computed using (\ref{eqn:jrediff}):
\begin{align}
&\sum\limits_{s=1}^{k-1}se^{\frac{2\pi ims}{k}}=\frac{k}{e^{\frac{2\pi im}{k}}-1}.\\
&\sum\limits_{s=1}^{k-1}se^{\frac{-2\pi ims}{k}}=-k-\frac{k}{e^{\frac{2\pi im}{k}}-1}.\\
&\sum\limits_{s=1}^{k-1}s^2e^{\frac{2\pi ims}{k}}=\frac{k^2}{e^{\frac{2\pi im}{k}}-1}-\frac{2ke^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}.\\
& \sum\limits_{s=1}^{k-1}s^2e^{\frac{-2\pi ims}{k}}=-k^2-\frac{k^2}{e^{\frac{2\pi im}{k}}-1}-\frac{2ke^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}.
\end{align}
\begin{align}\label{1}
\frac{(-1)^{v+1}kr!}{i^r2^{r-1}\pi^r}L(r,\chi)=\sum\limits_{q=0}^{2[\frac{r}{2}]}\binom{r}{q}
B_q S(r-q,\chi)\text{ \cite[Theorem 1]{alkan2011values} }
\end{align}
where $ \chi$ and $r\geq1$ have same parity.
Now we may derive a useful identity quickly. Using identity (5.12) in \cite{alkan2011values} we have
\begin{align*}
\frac{k^4}{\pi^4}L(4, \chi_0)
&=\frac{2}{3}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}+\frac{1}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{2\pi m}{k})}{\sin^4(\frac{\pi m}{k})}\\
&=\frac{2}{3}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}+\frac{1}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{(1-2\sin^2(\frac{\pi m}{k}))}{\sin^4(\frac{\pi m}{k})}\\
&=\frac{1}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}-\frac{1}{3}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}.
\end{align*}
Using $ \frac{k^4}{\pi^4}L(4, \chi_0)=\frac{k^4}{\pi^4}\zeta(4)\prod\limits_{\substack{p|n\\p\text{ prime}}}\left(1-\frac{1}{p^4}\right)=\frac{J_4(k)}{90}$ \cite[Chapter 11]{tom1976introduction} and identity (\ref{eqn:1bysin2}) above we get
\begin{align}\label{eqn:1bysin4}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\sin^4(\frac{\pi m}{k})}=\frac{J_4(k)}{45}+\frac{2}{9}J_2(k).
\end{align}
We now state four identities.
\begin{lemm}
\begin{align}
&\sum\limits_{j=1}^{k-1}j^3e^{\frac{2\pi imj}{k}}=\frac{k^3}{e^{\frac{2\pi im}{k}}-1}-\frac{3k^2e^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}+\frac{3ke^{\frac{2\pi im}{k}}+3ke^{\frac{4\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^3}.\label{j^3}\\
&\sum\limits_{j=1}^{k-1}j^3e^{\frac{-2\pi imj}{k}}=-\frac{k^3e^{\frac{2\pi im}{k}}}{e^{\frac{2\pi im}{k}}-1}-\frac{3k^2e^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}-\frac{(3ke^{\frac{2\pi im}{k}}+3ke^{\frac{4\pi im}{k}})}{(e^{\frac{2\pi im}{k}}-1)^3}.\label{j^-3}\\
&\sum\limits_{j=1}^{k-1}j^4e^{\frac{2\pi imj}{k}}\label{j^4}=\frac{k^4}{e^{\frac{2\pi im}{k}}-1}-\frac{4k^3e^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}+\frac{6k^2e^{\frac{2\pi im}{k}}+6k^2e^{\frac{4\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^3}\\&-\frac{(4ke^{\frac{2\pi im}{k}}+16ke^{\frac{4\pi im}{k}}+4ke^{\frac{6\pi im}{k}})}{(e^{\frac{2\pi im}{k}}-1)^4}\nonumber. \\
&\sum\limits_{j=1}^{k-1}j^4e^{\frac{-2\pi imj}{k}}\label{j^-4}=-\frac{k^4e^{\frac{2\pi im}{k}}}{e^{\frac{2\pi im}{k}}-1}-\frac{4k^3e^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}-\frac{(6k^2e^{\frac{2\pi im}{k}}+6k^2e^{\frac{4\pi im}{k}})}{(e^{\frac{2\pi im}{k}}-1)^3}\\&-\frac{(4ke^{\frac{2\pi im}{k}}+16ke^{\frac{4\pi im}{k}}+4ke^{\frac{6\pi im}{k}})}{(e^{\frac{2\pi im}{k}}-1)^4} \nonumber.
\end{align}
\end{lemm}
\begin{proof}
The identity (\ref{j^3}) can be derived by putting $r=3$ in (\ref{eqn:jrediff}) and by applying the limit.
Now we derive the second identity as follows:
\begin{align*}
\sum\limits_{j=1}^{k-1}j^3e^{\frac{-2\pi imj}{k}}
&=\sum\limits_{j=1}^{k-1}j^3e^{\frac{2\pi i(k-m)j}{k}}\\
&=\frac{k^3}{e^{\frac{2\pi i(k-m)}{k}}-1}-\frac{3 k^2 e^{\frac{2\pi i(k-m)}{k}}}{(e^{\frac{2\pi i(k-m)}{k}}-1)^2}+\frac{3ke^{\frac{2 \pi i(k-m)}{k}}+3ke^{\frac{4\pi i(k-m)}{k}}}{(e^{\frac{2\pi i(k-m)}{k}}-1)^3}\\
&=\frac{k^3}{e^{\frac{-2\pi im}{k}}-1}-\frac{3 k^2 e^{\frac{-2\pi im}{k}}}{(e^{\frac{-2\pi im}{k}}-1)^2}+\frac{3ke^{\frac{-2 \pi im}{k}}+3ke^{\frac{-4\pi im}{k}}}{(e^{\frac{-2\pi im}{k}}-1)^3}\\
&=-\frac{k^3}{e^{\frac{2\pi im}{k}}-1}-\frac{3 k^2 e^{\frac{2\pi im}{k}}}{(e^{\frac{2\pi im}{k}}-1)^2}-\frac{(3ke^{\frac{2 \pi im}{k}}+3ke^{\frac{4\pi im}{k}})}{(e^{\frac{2\pi im}{k}}-1)^3}.
\end{align*}
The last line above is obtained from the fact that $$e^{\frac{-2\pi im}{k}}-1=\frac{1-e^{\frac{2\pi im}{k}}}{e^{\frac{2\pi im}{k}}}=-\frac{(e^{\frac{2\pi im}{k}}-1)}{e^{\frac{2\pi im}{k}}}.$$
\end{proof}
\section{Main Results and proofs}
We state below the main results we prove in this paper.
\begin{theo}\label{L(3,x)}
The formula
\begin{align*}
\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}|L(3, \chi)|^2 = \frac{\pi^6}{90k^6}\phi(k)\left(\frac{J_6(k)}{21}-J_2(k)
\right)
\end{align*}
holds for all $k\geq3$.
\end{theo}
\begin{theo}\label{L(4,x)}
The formula
\begin{align*}
\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}|L(4, \chi)|^2 = \frac{\pi^8}{27k^8}\phi(k)\left(\frac{J_8(k)}{700}+\frac{J_4(k)}{150}+\frac{2}{21}J_2(k)
\right)
\end{align*}
holds for all $k\geq3$.
\end{theo}
From the above two results, it can be seen that the average values of $|L(3, \chi)|^2$ over all odd characters modulo $k$ is
$\frac{2\pi^6}{90k^6}\left(\frac{J_6(k)}{21}-J_2(k)\right)$ and $|L(4, \chi)|^2$ over all even characters modulo $k$ is
$\frac{2\pi^8}{27k^8}\left(\frac{J_8(k)}{700}+\frac{J_4(k)}{150}+\frac{2}{21}J_2(k)\right)$.
Now we proceed to prove the above two results. The outline of the proof is the following:
\begin{enumerate}
\item Use identity (\ref{1}) connecting $L(r,\chi)$, Bernoulli numbers, and Gauss sums with $r=3, 4$ and expand it.
\item Take the sum of $|L(r,\chi)|^2$ over $\chi$ having same parity as that of $r$, further expand it using the complex number property $|z|^2=z\overline{z}$.
\item The expansion consists of terms of the form $\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^p s^q\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j\pm s)}{k}}$ for certain values of $p,q$. Simplify these terms further.
\item In the simplification of terms obtained in the last step, the exponential power expands to trigonometric identities, necessiating the computation of sums of the trigonometric functions of the form $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\sin^p(\frac{\pi m}{k})}$ where $p$ takes values 2, 4,\ldots. These sums results in certain combinations of the Euler totient function and Jordan totient function which is what we intend to prove.
\end{enumerate}
\begin{proof}[Proof of theorem \ref{L(3,x)}]
Put $r=3$ in equation (\ref{1}) and use the values of the Bernoulli numbers to get
\begin{align}\label{2}
L(3,\chi)=\frac{-2i\pi^3}{3k}
\left[
S(3, \chi)-\frac{3}{2}S(2,\chi)+\frac{1}{2}S(1, \chi)
\right].
\end{align}
If $\chi\neq\chi_0$ and $r\geq1$ are of opposite parity, then
\begin{align}\label{3}
\sum\limits_{q=0}^{2[\frac{r}{2}]}\binom{r}{q}
B_q S(r-q.\chi)=0 \text{ \cite[Theorem 1]{alkan2011values} }.
\end{align}
When $r=2$, we get
\begin{align}
S(2, \chi)-S(1, \chi)+\frac{1}{6}S(0, \chi) =0.\label{4}
\end{align}
Since $S(0, \chi)=\sum\limits_{j=1}^kG(j,\chi)=0$,
\begin{align}\label{5}
S(2, \chi)=S(1, \chi).
\end{align}
From equations (\ref{2}) and (\ref{5}) we have
\begin{align*}
L(3,\chi)=\frac{-2i\pi^3}{3k}
\left[
S(3, \chi)-S(1, \chi)
\right].
\end{align*}
Since $G(k, \chi)= \sum\limits_{m=1}^{k}\chi(m)=0$ for any non principal character \cite[Theroem 6.10]{tom1976introduction}, we have
\begin{align}\label{6}
L(3, \chi)= \frac{-2i\pi^3}{3k^2}
\left[
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^3G(j, \chi)-\sum\limits_{j=1}^{k-1} jG(j, \chi)
\right].
\end{align}
Now we are ready to compute the mean square sum.
Using the above identity we get
\begin{align*}
&\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}|L(3, \chi)|^2 \\
=& \frac{4\pi^6}{9k^4}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}
\left|
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^3G(j, \chi)-\sum\limits_{j=1}^{k-1} jG(j, \chi)
\right|^2\\
=&\frac{4\pi^6}{9k^4}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}
\left(
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^3G(j, \chi)-\sum\limits_{j=1}^{k-1} jG(j, \chi)
\right) \times\\
&\left(
\frac{1}{k^2}\sum\limits_{s=1}^{k-1} s^3\overline{G(s, \chi)}-\sum\limits_{s=1}^{k-1} s\overline{G(s, \chi)}
\right)
\\
=&\frac{4\pi^6}{9k^8}\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{m=1}^{k-1}\sum\limits_{n=1}^{k-1}e^{\frac{2\pi i(mj-ns)}{k}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(m)\overline{\chi(n)}
\\
-& \frac{4\pi^6}{9k^6}\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{m=1}^{k-1}\sum\limits_{n=1}^{k-1}e^{\frac{2\pi i(mj-ns)}{k}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(m)\overline{\chi(n)}\\
-&\frac{4\pi^6}{9k^6}\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js^3\sum\limits_{m=1}^{k-1}\sum\limits_{n=1}^{k-1}e^{\frac{2\pi i(mj-ns)}{k}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(m)\overline{\chi(n)}\\
+&\frac{4\pi^6}{9k^4}\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{m=1}^{k-1}\sum\limits_{n=1}^{k-1}e^{\frac{2\pi i(mj-ns)}{k}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(m)\overline{\chi(n)}.
\end{align*}
There are exactly $\frac{\phi(k)}{2}$ odd characters modulo $k\geq3$. Therefore $\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(u)=\frac{\phi(k)}{2}$ or $-\frac{\phi(k)}{2}$ depending on if $u=1$ or $u=-1$ respectively. If $u$ is neither of these modulo $k$, then $\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(u)=0$. Thus
$\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(m)\overline{\chi(n)}=\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(mn^{-1})=\frac{\phi(k)}{2} \text{ or } -\frac{\phi(k)}{2}$
when $m=n$ or $m=k-n$ repectively and
$\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}\chi(m)\overline{\chi(n)}=0 $
otherwise.
Hence we get
\begin{align}\label{|l(3,x)|}
&\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}|L(3, \chi)|^2 \\ =&\frac{4\pi^6}{9k^8}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \nonumber \\
-&\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
-&\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
+&\frac{4\pi^6}{9k^4}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right).\nonumber
\end{align}
Take the first term in the above expression. If we write $\alpha=e^{\frac{2\pi im}{k}}$ for notational convenience, we have
\begin{align*}
&\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}\\
=& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(\sum\limits_{j=1}^{k-1} j^3\alpha^j
\right)
\left(
\sum\limits_{s=1}^{k-1}s^3\alpha^{-s}
\right)\\
=& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k^3}{\alpha-1}-\frac{3k^2\alpha}{(\alpha-1)^2}+\frac{3k\alpha^2+3k\alpha}{(\alpha-1)^3}
\right) \times \\
&\left(
-\frac{k^3\alpha}{\alpha-1}-\frac{3k^2\alpha}{(\alpha-1)^2}-\frac{3k\alpha^2+3k\alpha}{(\alpha-1)^3}
\right)\\ &(\text{ using identities }(\ref{j^3}) \text{ and }(\ref{j^-3}))\\
=& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl(
-\frac{k^6\alpha}{(\alpha-1)^2}+\frac{3k^5\alpha^2}{(\alpha-1)^3}-\frac{3k^5\alpha}{(\alpha-1)^3}-\frac{3k^4\alpha^3}{(\alpha-1)^4}\\
&+\frac{3k^4\alpha^2}{(\alpha-1)^4}-\frac{3k^4\alpha}{(\alpha-1)^4} -\frac{9k^2\alpha^4}{(\alpha-1)^6}-\frac{9k^2\alpha^2}{(\alpha-1)^6}-\frac{18k^2\alpha^3}{(\alpha-1)^6}
\biggr).
\end{align*}
By similar computations we get
\begin{align*}
&\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}\\ =& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl (
\frac{k^6}{(\alpha-1)^2}-\frac{6k^5\alpha}{(\alpha-1)^3}+\frac{15k^4\alpha^2}{(\alpha-1)^4}+\frac{6k^4\alpha}{(\alpha-1)^4}\\
&-\frac{18k^3\alpha^3}{(\alpha-1)^5}-\frac{18k^3\alpha^2}{(\alpha-1)^5}+\frac{9k^2\alpha^4}{(\alpha-1)^6}+\frac{9k^2\alpha^2}{(\alpha-1)^6}+\frac{18k^2\alpha^3}{(\alpha-1)^6}
\biggr).
\end{align*}
Hence
\begin{align}\label{l(3,x)j^3s^3}
&\frac{4\pi^6}{9k^8}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \\
=& \frac{4\pi^6}{9k^8}\frac{\phi(k)}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl(
-\frac{k^6}{(\alpha-1)^2}-\frac{k^6\alpha}{(\alpha-1)^2}+\frac{3k^5\alpha^2}{(\alpha-1)^3}+\frac{3k^5\alpha}{(\alpha-1)^3}\nonumber\\&-\frac{3k^4\alpha^3}{(\alpha-1)^4} -\frac{12k^4\alpha^2}{(\alpha-1)^4}-\frac{9k^4\alpha}{(\alpha-1)^4}
+ \frac{18k^3\alpha^3}{(\alpha-1)^5}+\frac{18k^3\alpha^2}{(\alpha-1)^5}-\frac{18k^2\alpha^4}{(\alpha-1)^6}\nonumber\\&-\frac{36k^2\alpha^3}{(\alpha-1)^6}-\frac{18k^2\alpha^2}{(\alpha-1)^6}
\biggr).\nonumber
\end{align}
Now we evaluate each term by term.
\begin{align*}
& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}-\frac{k^6}{(\alpha-1)^2}\\
&=-k^6\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{(e^{\frac{2\pi im}{k}}-1)^2}\times\frac{e^{\frac{-2\pi im}{k}}}{e^{\frac{-2\pi im}{k}}}\\
&=-k^6\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{e^{\frac{-2\pi im}{k}}}{(e^{\frac{\pi im}{k}}-e^{\frac{-\pi im}{k}})^2}\\
&=-k^6\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\cos(\frac{2\pi m}{k})-i\sin(\frac{2\pi m}{k})}{(2i)^2 \sin^2(\frac{\pi m}{k})}\\
&=-k^6
\left[
\frac{-1}{4} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{1-2sin^2(\frac{\pi m}{k})}{\sin^2(\frac{\pi m}{k})}
\right)
+\frac{i}{4}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{2\sin(\frac{\pi m}{k})cos(\frac{\pi m}{k})}{\sin^2(\frac{\pi m}{k})}
\right)
\right]\\
&=-k^6
\left[
\frac{-1}{4} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\sin^2(\frac{\pi m}{k})}+\frac{2}{4}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} 1+\frac{2i}{4}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot(\frac{\pi m}{k})
\right].
\end{align*}
Use identity (\ref{eqn:1bysin2}) and the fact that
$ \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot(\frac{\pi m}{k})=0$ as $ \cot(\frac{\pi m}{k})$ and $ \cot(\frac{\pi (k-m)}{k})$ cancel each other to get
\begin{align}\label{1/(a-1)^2}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}-\frac{k^6}{(\alpha-1)^2}
= \frac{k^6 }{12}J_2(k)+\frac{-k^6}{2}\phi(k).
\end{align}
A similar evaluation gives
\begin{align}\label{a/(a-1)^2}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}-\frac{k^6\alpha}{(\alpha-1)^2}=
\frac{k^6 }{12}J_2(k).
\end{align}
Proceeding similarly, we get
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{3k^5\alpha^2}{(\alpha-1)^3}
=3k^5
\left[
\frac{-1}{8i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{\cos(\frac{\pi m}{k})}{\sin^3(\frac{\pi m}{k})}
\right)
+\frac{-i}{8i}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{1}{\sin^2(\frac{\pi m}{k})}
\right)
\right].
\end{align*}
Once again using identity (\ref{eqn:1bysin2}) and
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \frac{\cos(\frac{\pi m}{k})}{\sin^3(\frac{\pi m}{k})}&= \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot(\frac{\pi m}{k})\csc^2(\frac{\pi m}{k})\\&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot(\frac{\pi m}{k}) +\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot^3(\frac{\pi m}{k})
=0,
\end{align*} we get
\begin{align}\label{a^2/(a-1)^3}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{3k^5\alpha^2}{(\alpha-1)^3} =
- \frac{k^5 }{8}J_2(k).
\end{align}
Similar computations can be performed to get
\begin{align}\label{a/(a-1)^3}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{3k^5\alpha}{(\alpha-1)^3}
=
\frac{k^5 }{8}J_2(k).
\end{align}
Proceeding similarly, we get
\begin{align}\label{a^3/(a-1)^4}
& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-3k^4\alpha^3}{(\alpha-1)^4}\\
&=-3k^4
\left[
\frac{1}{16} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}-\frac{2}{16} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}+\frac{2i}{16}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\cos(\frac{\pi m}{k})}{\sin^3(\frac{\pi m}{k})}
\right] \nonumber\\
&=
- \frac{k^4 }{240}J_4(k)+\frac{k^4}{12}J_2(k) \text{ using identity (\ref{eqn:1bysin4})}. \nonumber
\end{align}
It is not very diffcult to evaluate that
\begin{align}\label{a^2/(a-1)^4}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-12k^4\alpha^2}{(\alpha-1)^4}
=
- \frac{k^4 }{60}J_4(k)-\frac{k^4}{6}J_2(k).
\end{align}
Also,
\begin{align}\label{a/(a-1)^4}
& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-9k^4\alpha}{(\alpha-1)^4}\\
&=-9k^4
\left[
\frac{1}{16} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}-\frac{2}{16} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}-\frac{2i}{16}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\cos(\frac{\pi m}{k})}{\sin^3(\frac{\pi m}{k})}
\right]\nonumber\\
&=
- \frac{k^4 }{80}J_4(k)+\frac{k^4}{4}J_2(k)\nonumber.
\end{align}
Now
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{18k^3\alpha^3}{(\alpha-1)^5}
&=18k^3
\left[
\frac{1}{32i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}
+\frac{1}{32}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}
\right].
\end{align*}
\begin{align*}
\text{ Since }&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}
= \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot(\frac{\pi m}{k})\csc^4(\frac{\pi m}{k})\\&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot(\frac{\pi m}{k}) +2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot^3(\frac{\pi m}{k})+\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}} \cot^5(\frac{\pi m}{k})
=0,
\end{align*}
we get
\begin{align}\label{a^3/(a-1)^5}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{18k^3\alpha^3}{(\alpha-1)^5}
=
\frac{k^3 }{80}J_4(k)+\frac{k^3}{8}J_2(k)
\end{align}
and
\begin{align}\label{a^2/(a-1)^5}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{18k^3\alpha^2}{(\alpha-1)^5}
=
-\frac{k^3 }{80}J_4(k)-\frac{k^3}{8}J_2(k).
\end{align}
Similar set of computations using the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}$ yields
\begin{align}\label{a^4/(a-1)^6}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-18k^2\alpha^4}{(\alpha-1)^6}
=
\frac{9k^2}{32} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}
- \frac{k^2}{80}
J_4(k)-\frac{k^2}{8}J_2(k)
\end{align}
and
\begin{align}\label{a^3/(a-1)^6}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-36k^2\alpha^3}{(\alpha-1)^6}
=
\frac{9k^2}{16} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}.
\end{align}
Similarly
\begin{align}\label{a^2/(a-1)^6}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-18k^2\alpha^2}{(\alpha-1)^6}
= \frac{9k^2}{32} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}
- \frac{k^2}{80}
J_4(k)-\frac{k^2}{8}J_2(k).
\end{align}
Putting back all the expansions into (\ref{l(3,x)j^3s^3}) we get
\begin{align*}
&\frac{4\pi^6}{9k^8}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \\
=&\frac{4\pi^6}{9k^8}\frac{\phi(k)}{2}
\biggl[
-\frac{k^6}{2}\phi(k)+\frac{k^6 }{6}J_2(k)- \frac{k^4 }{30}J_4(k)+\frac{k^4}{6}J_2(k)\\&- \frac{k^2}{40}J_4(k)-\frac{k^2}{4}J_2(k)+ \frac{9k^2}{8} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}
\biggr].
\end{align*}
To simplify the above further, we have to evaluate the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}$.
We use equation (\ref{1}) with principal character $\chi=\chi_0$ modulo $k\geq3$ and $r=6$ to arrive at the following formula
\begin{align*}
&\frac{k^6}{\pi^6}L(6, \chi_0)\\
=&-\frac{14}{15}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}-\frac{5}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{2\pi m}{k})}{\sin^4(\frac{\pi m}{k})}+2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\sin(\frac{3\pi m}{k})}{\sin^5(\frac{\pi m}{k})}\\&+\frac{1}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{4\pi m}{k})}{\sin^6(\frac{\pi m}{k})}.
\end{align*}
Now using elementary trigonometric identities on $\sin3\theta$ and $\cos2\theta$, we get
\begin{align*}
\frac{k^6}{\pi^6}L(6, \chi_0)
=\frac{1}{15}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}-\frac{1}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}+\frac{1}{2}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}.
\end{align*}
Using $ \frac{k^6}{\pi^6}L(6, \chi_0)=\frac{k^6}{\pi^6}\zeta(6)\prod\limits_{\substack{p|n\\p\text{ prime}}}\left(1-\frac{1}{p^6}\right)=\frac{J_6(k)}{945}$ \cite[Theorem 11.7]{tom1976introduction},
we conclude that
\begin{align}\label{1bysin6}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}=\frac{2}{945}J_6(k)+\frac{1}{45}J_4(k)+\frac{8}{45}J_2(k).
\end{align}
Therefore
\begin{align}\label{j^3s^3L(3,x)}
&\frac{4\pi^6}{9k^8}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
&=\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
-\frac{k^4}{2}\phi(k)+\frac{k^4 }{6}J_2(k)- \frac{k^2 }{30}J_4(k)+\frac{k^2}{6}J_2(k)-\frac{1}{20}J_2(k)+
\frac{1}{420}J_6(k)
\right).
\end{align}
Now consider the second term on the RHS of (\ref{|l(3,x)|}) and simplify the first sum in bracket to get
\begin{align*}
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}= \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\sum\limits_{j=1}^{k-1} j^3e^{\frac{2\pi imj}{k}}
\right)
\left(
\sum\limits_{s=1}^{k-1}se^{\frac{-2\pi ims}{k}}
\right)
\end{align*} which on further simplification using identity (\ref{j^3}) gives
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}
\\=& \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl(
-\frac{k^4}{\alpha-1}-\frac{k^4}{(\alpha-1)^2}+\frac{3k^3\alpha}{(\alpha-1)^2}+\frac{3k^3\alpha}{(\alpha-1)^3}-\frac{3k^2\alpha^2}{(\alpha-1)^3}\\&-\frac{3k^2\alpha}{(\alpha-1)^3}-\frac{3k^2\alpha^2}{(\alpha-1)^4}-\frac{3k^2\alpha}{(\alpha-1)^4}
\biggr).
\end{align*}
Similar computations give
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}\\&= \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k^4}{(\alpha-1)^2}-\frac{3k^3\alpha}{(\alpha-1)^3}+\frac{(3k^2\alpha^2}{(\alpha-1)^4}+\frac{3k^2\alpha}{(\alpha-1)^4}
\right).
\end{align*}
Thus the second term on the RHS of (\ref{|l(3,x)|}) becomes
\begin{align*}
&\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \\
=&\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl(
-\frac{k^4}{\alpha-1}-\frac{2k^4}{(\alpha-1)^2}+\frac{3k^3\alpha}{(\alpha-1)^2}+\frac{(6k^3-3k^2)\alpha}{(\alpha-1)^3}\\&-\frac{3k^2\alpha^2}{(\alpha-1)^3}-\frac{6k^2\alpha}{(\alpha-1)^4}
-\frac{6k^2\alpha^2}{(\alpha-1)^4}
\biggr).
\end{align*}
From identities (\ref{1/(a-1)^2}), (\ref{a/(a-1)^2}), (\ref{a/(a-1)^3}), (\ref{a^2/(a-1)^3}), (\ref{a/(a-1)^4}), (\ref{a^2/(a-1)^4}) and
\begin{align}\label{1/a-1}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-k^4}{\alpha-1}&=-k^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\alpha-1}\\&=-k^4\left[
\frac{1}{2i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\cos(\frac{\pi m}{k})}{\sin(\frac{\pi m}{k})}-\frac{1}{2} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}1
\right]=\frac{k^4}{2}\phi(k)\nonumber
\end{align}
we get
\begin{align}\label{j^3sL(3,x)}
&\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
&=\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
-\frac{k^4}{2}\phi(k)+\frac{k^4}{6}J_2(k)-\frac{k^2}{60}J_4(k)+\frac{k^2}{12}J_2(k)
\right).
\end{align}
Now we consider the third term on the RHS of (\ref{|l(3,x)|}) and
interchange the variables $j$ and $s$ to get
\begin{align}\label{js^3L(3,x)}
&\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi i(k-m)(s-j)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \nonumber\\
&= \frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^3s\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \nonumber \\
&=\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
-\frac{k^4}{2}\phi(k)+\frac{k^4}{6}J_2(k)-\frac{k^2}{60}J_4(k)+\frac{k^2}{12}J_2(k)
\right).
\end{align}
Now consider the last term on the RHS of (\ref{|l(3,x)|}).
First get
\begin{align*}
&\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}\\
&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\sum\limits_{j=1}^{k-1} je^{\frac{2\pi imj}{k}}
\right)
\left(
\sum\limits_{s=1}^{k-1}se^{\frac{-2\pi ims}{k}}
\right)\\
&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k}{\alpha-1}
\right)
\left(
-k-\frac{k}{\alpha-1}
\right)\\
&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
- \frac{k^2}{\alpha-1}-\frac{k^2}{(\alpha-1)^2}
\right)
\end{align*}
and then
\begin{align*}
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k^2}{(\alpha-1)^2}
\right).
\end{align*}
So
\begin{align}\label{jsL(3,x)}
&\frac{4\pi^6}{9k^4}\frac{\phi(k)}{2}
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}js\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
&= \frac{4\pi^6}{9k^4}\frac{\phi(k)}{2}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
- \frac{k^2}{\alpha-1}-\frac{2k^2}{(\alpha-1)^2}
\right)\nonumber \\
&=\frac{4\pi^6}{9k^6}\frac{\phi(k)}{2}
\left(
-\frac{k^4}{2}\phi(k)+ \frac{k^4}{6}J_2(k)
\right) \text{ using identities } (\ref{1/(a-1)^2}), (\ref{1/a-1}).
\end{align}
From equations (\ref{j^3s^3L(3,x)}), (\ref{j^3sL(3,x)}), (\ref{js^3L(3,x)}), (\ref{jsL(3,x)}) we get
\begin{align*}
&\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}|L(3, \chi)|^2 \\
&=\frac{\pi^6}{90k^6}\phi(k)
\left(
\frac{1}{21}J_6(k) -J_2(k)
\right)
\end{align*} which is what we required.
So the average value of $ \sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ odd}}}|L(3, \chi)|^2$ over all odd characters modulo $k$ is $
\frac{\pi^6}{45k^6}\left(
\frac{1}{21}J_6(k) -J_2(k)\right).
$
\end{proof}
To compute the sum in theorem \ref{L(4,x)}, we need the following identity.
\begin{lemm}\label{phi_4}
\begin{align*}
\phi_4(n)=\frac{n^4}{5}\phi(n)+\frac{n^3}{3}\prod\limits_{p|n}(1-p)-\frac{n}{30}\prod\limits_{p|n}(1-p^3).
\end{align*}
\end{lemm}
\begin{proof}
It is not very difficult to show that
\begin{align*}
\sum\limits_{d|n}\frac{\phi_k(d)}{d^k}=\frac{1^k+\cdots+n^k}{n^k} \text{ \cite[Chapter 2,exercise 15]{tom1976introduction} }.
\end{align*}
If we write $f(n)=\sum\limits_{j=1}^n \frac{j^4}{n^4}$ and $g(n)=\frac{\phi_4(n)}{n^4}$
then $\sum\limits_{d|n}g(d)=f(n)$.
By Mobius inversion we get
\begin{align*}
\frac{\phi_4(n)}{n^4}=&\sum\limits_{d|n}\left(\sum\limits_{j=1}^d \frac{j^4}{d^4}\right)\mu\left(\frac{n}{d}\right)\\
=&\sum\limits_{d|n}\frac{d(d+1)(2d+1)(3d^2+3d-1)}{30}\frac{\mu(\frac{n}{d})}{d^4}\\
=&\frac{1}{5}\sum\limits_{d|n}d\mu\left(\frac{n}{d}\right)+\frac{1}{2}\sum\limits_{d|n}\mu\left(\frac{n}{d}\right)+\frac{1}{3}\sum\limits_{d|n}\frac{\mu(\frac{n}{d})}{d}-\frac{1}{30}\sum\limits_{d|n}\frac{\mu(\frac{n}{d})}{d^3}\\
=&\frac{1}{5}\phi(n)+\frac{1}{3}\sum\limits_{d|n}\frac{\mu(d)}{\frac{n}{d}}-\frac{1}{30}\sum\limits_{d|n}\frac{\mu(d)}{(\frac{n}{d})^3}\\&\text{ since }
\sum\limits_{d|n}d\mu\left(\frac{n}{d}\right)=\phi(n) \text{ and }\sum\limits_{d|n}\mu\left(\frac{n}{d}\right)=0
\\
=&\frac{1}{5}\phi(n)+\frac{1}{3n}\sum\limits_{d|n}d\mu(d)-\frac{1}{30n^3}\sum\limits_{d|n}d^3\mu(d).
\end{align*}
Therefore
\begin{align*} \phi_4(n)=\frac{n^4}{5}\phi(n)+\frac{n^3}{3}\sum\limits_{d|n}d\mu(d)-\frac{n}{30}\sum\limits_{d|n}d^3\mu(d).
\end{align*}
Let $n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}$ be the prime decomposition of $n$.
\begin{align*}
\sum\limits_{d|n}d\mu(d)&=\mu(1)+p_1\mu(p_1)+\cdots +p_r\mu(p_r)+p_1p_2\mu(p_1p_2)\\&+\cdots +p_1p_2\cdots p_r\mu(p_1p_2\cdots p_r)\\
&=1-\sum p_i+\sum p_ip_j-\cdots +(-1)^rp_1\cdots p_r\\
&=\prod\limits_{p|n}(1-p).
\end{align*}
Similarly
\begin{align*}
\sum\limits_{d|n}d^3\mu(d)
&=1-\sum p_i^3+\sum p_i^3p_j^3-\cdots +(-1)^r(p_1\cdots p_r)^3\\
&=\prod\limits_{p|n}(1-p^3).
\end{align*}
Hence, $\phi_4(n)=\frac{n^4}{5}\phi(n)+\frac{n^3}{3}\prod\limits_{p|n}(1-p)-\frac{n}{30}\prod\limits_{p|n}(1-p^3)$.
\end{proof}
Now we proceed to prove theorem \ref{L(4,x)}.
\begin{proof}[Proof of theorem \ref{L(4,x)}]
First note that
\begin{align*}
\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}|L(4, \chi)|^2=|L(4, \chi_0)|^2+\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}\\ \chi \neq \chi_0}}|L(4, \chi)|^2.
\end{align*}
We have $L(4, \chi_0)=\zeta(4)\prod\limits_{\substack{p|n\\p\text{ prime}}}\left(1-\frac{1}{p^4}\right)=\frac{\pi^4J_4(k)}{90k^4}$ \cite[Theorem 11.7]{tom1976introduction}.
Therefore
\begin{align}\label{1*}
\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}|L(4, \chi)|^2=\frac{\pi^8J_4^2(k)}{8100k^8}+\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}\\ \chi \neq \chi_0}}|L(4, \chi)|^2.
\end{align}
Put $r=4$ in equation (\ref{1}), we have
\begin{align}\label{2*}
L(4, \chi)= \frac{-\pi^4}{3k} \biggl[&
B_0S(4, \chi)+4B_1S(3, \chi)+6B_2S(2, \chi)+4B_3S(1, \chi)\\&+B_4S(0, \chi)
\biggr]\nonumber.
\end{align}
Put $r=1$ in equation (\ref{3}), we get
\begin{align}\label{4*}
S(1, \chi)= \sum\limits_{j=1}^k \left(\frac{j}{k}\right)G(j, \chi)=\sum\limits_{j=1}^{k-1} \left(\frac{j}{k}\right)G(j, \chi)=0
\end{align}
since $r=1$ and $\chi$ are of opposite parity and $G(k, \chi)=0$ when $\chi$ is non-principal modulo $k\geq3$.
Also put $r=3$ in equation (\ref{3}), we get
\begin{align*}
0=\sum\limits_{q=0}^{2}\binom{3}{q}
B_q S(3-q,\chi)=
B_0S(3, \chi)+3B_1S(2, \chi)+3B_2S(1, \chi).
\end{align*}
Therefore
\begin{align}\label{5*}
S(3, \chi)=\frac{3}{2}S(2, \chi).
\end{align}
Equations (\ref{2*}), (\ref{4*}), (\ref{5*}) and the fact that $S(0, \chi)=0$ , when $\chi \neq \chi_0$ we get
\begin{align*}
L(4, \chi)&= \frac{-\pi^4}{3k} \left[ S(4, \chi)-2S(2, \chi)
\right]\\
&=\frac{-\pi^4}{3k^3} \left[\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right].
\end{align*}
\begin{align*}
\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}\\ \chi \neq \chi_0}}|L(4, \chi)|^2=\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}\\ \chi \neq \chi_0}}\left|
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right|^2.
\end{align*}
\begin{align}\label{6*}
&\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}\\ \chi \neq \chi_0}}|L(4, \chi)|^2\\ \nonumber =&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\left|
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right|^2\\ \nonumber &-\frac{\pi^8}{9k^6}\left|
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi_0)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi_0)
\right|^2 \nonumber.
\end{align}
Consider the first term in the above diference.
\begin{align*}
&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\left|\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right|^2\\
=&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\left[\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right] \times \\&\left[\frac{1}{k^2}\sum\limits_{s=1}^{k-1} s^4\overline{G(s, \chi)}-2\sum\limits_{s=1}^{k-1} s^2\overline{G(s, \chi)}
\right]\\
=&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\biggl[
\frac{1}{k^4}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^4 s^4G(j, \chi)\overline{G(s, \chi)}-\frac{2}{k^2}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^4 s^2G(j, \chi)\overline{G(s, \chi)}\\&-\frac{2}{k^2}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^2 s^4G(j, \chi)\overline{G(s, \chi)}+4\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^2 s^2G(j, \chi)\overline{G(s, \chi)}
\biggr].
\end{align*}
For convenience write $S=\left(
\sum\limits_{m=1}^{k-1}\chi(m)e^{\frac{2\pi imj}{k}}
\right)
\left(
\sum\limits_{n=1}^{k-1}\overline{\chi(n)}e^{\frac{-2\pi ins}{k}}
\right)$.
Thus we can rewrite the above equation as
\begin{align}\label{|L(4,x)|}
&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\left|\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right|^2\\
=&\frac{\pi^8}{9k^{10}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^4 s^4S-\frac{2\pi^8}{9k^{8}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^4 s^2S\nonumber \\
&-\frac{2\pi^8}{9k^8}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^2 s^4S-\frac{4\pi^8}{9k^{6}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^2 s^2S \nonumber.
\end{align}
Firstly we simplify the term
\begin{align*}
& \frac{\pi^8}{9k^{10}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\sum\limits_{j=1}^{k-1}\sum\limits_{s=1}^{k-1}j^4 s^4S\\
&=\frac{\pi^8}{9k^{10}}\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{m=1}^{k-1}\sum\limits_{n=1}^{k-1}e^{\frac{2\pi i(mj-ns)}{k}}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\chi(mn^{-1})\\
&=\frac{\pi^8}{18k^{10}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)
\end{align*}
where we used the fact that if $u\equiv \pm1$(mod $k$) then $\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\chi(u)=\frac{\phi(k)}{2}$ for $k \geq 3$ and otherwise $\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\chi(u)=0$.
By similar computations we arrive at
\begin{align}\label{L(4,x)1}
&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\left|\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right|^2\\ \nonumber
=&\frac{\pi^8}{18k^{10}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\\ \nonumber
&-\frac{\pi^8}{9k^{8}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\\ \nonumber
&-\frac{\pi^8}{9k^8}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\\ \nonumber
&+\frac{2\pi^8}{9k^{6}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \nonumber.
\end{align}
We simplify each term in this sum. First we consider the term
\begin{align*} \frac{\pi^8}{18k^{10}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right).
\end{align*}
For simplicity, write $\alpha=e^{\frac{2\pi im}{k}}$.
Then
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\left(\sum\limits_{j=1}^{k-1}j^4e^{\frac{2\pi imj}{k}} \right)
\left(
\sum\limits_{s=1}^{k-1}s^4e^{\frac{-2\pi ims}{k}}
\right)\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k^4}{\alpha-1}-\frac{4k^3\alpha}{(\alpha-1)^2}+\frac{6k^2\alpha+6k^2\alpha^2}{(\alpha-1)^3}-\frac{(4k\alpha+16k\alpha^2+4k\alpha^3)}{(\alpha-1)^4}
\right)\\
&\left(
\frac{-k^4\alpha}{\alpha-1}-\frac{4k^3\alpha}{(\alpha-1)^2}-\frac{(6k^2\alpha+6k^2\alpha^2)}{(\alpha-1)^3}-\frac{(4k\alpha+16k\alpha^2+4k\alpha^3)}{(\alpha-1)^4}
\right)
\\ & (\text{ By identities }(\ref{j^4}) \text{ and}(\ref{j^-4}))
\text{ which expands to} \\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl[
-\frac{k^8\alpha}{(\alpha-1)^2}-\frac{4k^7\alpha}{(\alpha-1)^3}+\frac{4k^7\alpha^2}{(\alpha-1)^3}-\frac{6k^6\alpha}{(\alpha-1)^4}+\frac{4k^6\alpha^2}{(\alpha-1)^4}-\frac{6k^6\alpha^3}{(\alpha-1)^4}\\
& -\frac{4k^5\alpha}{(\alpha-1)^5}-\frac{12k^5\alpha^2}{(\alpha-1)^5}+\frac{12k^5\alpha^3}{(\alpha-1)^5}+\frac{4k^5\alpha^4}{(\alpha-1)^5}-\frac{4k^4\alpha^2}{(\alpha-1)^6}+\frac{56k^4\alpha^3}{(\alpha-1)^6}\\&-\frac{4k^4\alpha^4}{(\alpha-1)^6}+\frac{16k^2\alpha^2}{(\alpha-1)^8}+\frac{128k^2\alpha^3}{(\alpha-1)^8}+\frac{288k^2\alpha^4}{(\alpha-1)^8}+\frac{128k^2\alpha^5}{(\alpha-1)^8}+\frac{16k^2\alpha^6}{(\alpha-1)^8}
\biggr].
\end{align*}
By similar computations using equation (\ref{j^4}) we get
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl[
\frac{k^8}{(\alpha-1)^2}-\frac{8k^7\alpha}{(\alpha-1)^3}+\frac{12k^6\alpha}{(\alpha-1)^4}+\frac{28k^6\alpha^2}{(\alpha-1)^4}-\frac{8k^5\alpha}{(\alpha-1)^5}\\&-\frac{80k^5\alpha^2}{(\alpha-1)^5}-\frac{56k^5\alpha^3}{(\alpha-1)^5}
+\frac{68k^4\alpha^2}{(\alpha-1)^6}+\frac{200k^4\alpha^3}{(\alpha-1)^6}+\frac{68k^4\alpha^4}{(\alpha-1)^6} -\frac{48k^3\alpha^2}{(\alpha-1)^7}\\&-\frac{240k^3\alpha^3}{(\alpha-1)^7}-\frac{240k^3\alpha^4}{(\alpha-1)^7}-\frac{48k^3\alpha^5}{(\alpha-1)^7}
+\frac{16k^2\alpha^2}{(\alpha-1)^8}+\frac{256k^2\alpha^4}{(\alpha-1)^8}+\frac{16k^2\alpha^6}{(\alpha-1)^8}\\&+\frac{32k^2\alpha^4}{(\alpha-1)^8}+\frac{128k^2\alpha^3}{(\alpha-1)^8}+\frac{128k^2\alpha^5}{(\alpha-1)^8}
\biggr].
\end{align*}
Thus we get
\begin{align*}
& \frac{\pi^8}{18k^{10}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\\
=&\frac{\pi^8}{18k^{10}}\phi(k)\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\biggl[
\frac{k^8}{(\alpha-1)^2}-\frac{k^8\alpha}{(\alpha-1)^2}-\frac{12k^7\alpha}{(\alpha-1)^3}+\frac{4k^7\alpha^2}{(\alpha-1)^3}+\frac{6k^6\alpha}{(\alpha-1)^4}\\&+\frac{32k^6\alpha^2}{(\alpha-1)^4}-\frac{6k^6\alpha^3}{(\alpha-1)^4}- \frac{12k^5\alpha}{(\alpha-1)^5}-\frac{92k^5\alpha^2}{(\alpha-1)^5}-\frac{44k^5\alpha^3}{(\alpha-1)^5} +\frac{4k^5\alpha^4}{(\alpha-1)^5}\\&+\frac{64k^4\alpha^2}{(\alpha-1)^6}+\frac{256k^4\alpha^3}{(\alpha-1)^6}+\frac{64k^4\alpha^4}{(\alpha-1)^6} -\frac{48k^3\alpha^2}{(\alpha-1)^7}-\frac{240k^3\alpha^3}{(\alpha-1)^7}-\frac{240k^3\alpha^4}{(\alpha-1)^7}\\&-\frac{48k^3\alpha^5}{(\alpha-1)^7}
+\frac{32k^2\alpha^2}{(\alpha-1)^8}+\frac{256k^2\alpha^3}{(\alpha-1)^8}+\frac{576k^2\alpha^4}{(\alpha-1)^8}+\frac{256k^2\alpha^5}{(\alpha-1)^8}+\frac{32k^2\alpha^6}{(\alpha-1)^8}
\biggr].
\end{align*}
Proceeding using the method used for simplifying (\ref{a^3/(a-1)^5}), we have
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-12k^5\alpha}{(\alpha-1)^5}
=&-12k^5\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\cos(\frac{3\pi m}{k})-i\sin(\frac{3\pi m}{k})}{(2i)^5 \sin^5(\frac{\pi m}{k})}\\
=&-12k^5
\biggl[
\frac{4}{32i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos^3(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}-\frac{3}{32i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})} \\
&-\frac{3}{32}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}+\frac{1}{8}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}
\biggr] \\
=&-12k^5
\left[
-\frac{3}{32}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}+\frac{1}{8}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^2(\frac{\pi m}{k})}
\right]\\
&
\text{ since } \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos^3(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}=0 \text{ and } \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}=0\\
=&\frac{k^5}{40}J_4(k)-\frac{k^5}{4}J_2(k) .
\end{align*}
Similarly, using the fact that $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos^3(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^3(\frac{\pi m}{k})\\+\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^5(\frac{\pi m}{k})=0$ and $ \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}=0$ we get
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{4k^5\alpha^4}{(\alpha-1)^5}
=\frac{k^5}{120}J_4(k)-\frac{k^5}{12}J_2(k).
\end{align*}
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-48k^3\alpha^2}{(\alpha-1)^7}=&-48k^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\alpha^2}{(\alpha-1)^7}\\
=&-48k^3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{\cos(\frac{3\pi m}{k})-i\sin(\frac{3\pi m}{k})}{(2i)^7 \sin^7(\frac{\pi m}{k})} \\
=&-48k^3
\biggl[
\frac{-4}{128i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos^3(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}+\frac{3}{128i} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})} \\
&+\frac{3}{128}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}-\frac{1}{32}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}
\biggr] \\
=&-48k^3
\left[
\frac{3}{128}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}-\frac{1}{32}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}
\right]
\end{align*}
Since \begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos^3(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^3(\frac{\pi m}{k})+2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^5(\frac{\pi m}{k}) +\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^7(\frac{\pi m}{k})=0
\end{align*} and
\begin{align*}
&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}\\&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot(\frac{\pi m}{k})+3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^3(\frac{\pi m}{k})+3\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^5(\frac{\pi m}{k})+\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\cot^7(\frac{\pi m}{k})=0
\end{align*}
Hence
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-48k^3\alpha^2}{(\alpha-1)^7}=-\frac{k^3}{420}J_6(k)+\frac{k^3}{720}J_4(k)+\frac{2k^3}{15}J_2(k).
\end{align*}
Using the fact that $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}
=0$ we get
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-240k^3\alpha^3}{(\alpha-1)^7}
=-\frac{k^3}{252}J_6(k)-\frac{k^3}{24}J_4(k)-\frac{k^3}{3}J_2(k)
\end{align*}and
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-240k^3\alpha^4}{(\alpha-1)^7}
=\frac{k^3}{252}J_6(k)+\frac{k^3}{24}J_4(k)+\frac{k^3}{3}J_2(k).
\end{align*}
Now using the fact that $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos^3(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}=0$ and $ \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}=0$ we get
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{-48k^3\alpha^5}{(\alpha-1)^7}
=\frac{k^3}{420}J_6(k)-\frac{k^3}{720}J_4(k)-\frac{2k^3}{15}J_2(k).
\end{align*}
Using $ \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^7(\frac{\pi m}{k})}=0, \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{\cos(\frac{\pi m}{k})}{\sin^5(\frac{\pi m}{k})}=0$ and the identities (\ref{eqn:1bysin4}), (\ref{1bysin6}) we get
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{32k^2\alpha^2}{(\alpha-1)^8}
=&32k^2
\biggl[
\frac{1}{256}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^8(\frac{\pi m}{k})}-\frac{1}{32} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}\\&+\frac{1}{32} \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^4(\frac{\pi m}{k})}
\biggr]\\
=&\frac{k^2}{8}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^8(\frac{\pi m}{k})}-\frac{2k^2}{945}J_6(k)+\frac{2}{45}J_2(k),
\end{align*}
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{256k^2\alpha^3}{(\alpha-1)^8}
=k^2 \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\sin^8(\frac{\pi m}{k})}-\frac{4k^2}{945}J_6(k)-\frac{2k^2}{45}J_4(k)-\frac{16k^2}{45}J_2(k),
\end{align*}
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{576k^2\alpha^4}{(\alpha-1)^8}=\frac{9k^2}{8}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\sin^8(\frac{\pi m}{k})},
\end{align*}
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{256k^2\alpha^5}{(\alpha-1)^8}=k^2 \sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\sin^8(\frac{\pi m}{k})}-\frac{4k^2}{945}J_6(k)-\frac{2k^2}{45}J_4(k)-\frac{16k^2}{45}J_2(k)
\end{align*}and
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{32k^2\alpha^6}{(\alpha-1)^8}=\frac{k^2}{8}\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^8(\frac{\pi m}{k})}-\frac{2k^2}{945}J_6(k)+\frac{2}{45}J_2(k).
\end{align*}
Now as in the case of $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^6(\frac{\pi m}{k})}$, we get
\begin{align*}
\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^8(\frac{\pi m}{k})}=\frac{1}{4725}J_8(k)+\frac{8}{2835}J_6(k)+\frac{14}{675}J_4(k) +\frac{16}{105}J_2(k).
\end{align*}
On substituting the above identities and (\ref{1/(a-1)^2}), (\ref{a/(a-1)^2}), (\ref{a/(a-1)^3}), (\ref{a^2/(a-1)^3}), (\ref{a/(a-1)^4}), (\ref{a^2/(a-1)^4}), (\ref{a^3/(a-1)^4}), (\ref{a^2/(a-1)^5}), (\ref{a^3/(a-1)^5}), (\ref{a^2/(a-1)^6}), (\ref{a^3/(a-1)^6}), (\ref{a^4/(a-1)^6}) and simplifying, we get
\begin{align}\label{j^4s^4L(4,x)}
&\frac{\pi^8}{18k^{10}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
=&\frac{\pi^8}{k^8}\phi(k)
\biggl[
\frac{k^6}{36}\phi(k)-\frac{k^5}{27}J_2(k)+\frac{k^4}{405}J_4(k)+\frac{2k^4}{81}J_2(k)+\frac{k^3}{270}J_4(k)
-\frac{2k^2}{2835}J_6(k)\\&-\frac{k^2}{405}J_4(k)-\frac{4k^2}{405}J_2(k)+\frac{J_8(k)}{18900}+\frac{J_4(k)}{4050}+\frac{2}{567}J_2(k)
\biggr]\nonumber.
\end{align}
Now we consider the term
\begin{align*}
\frac{\pi^8}{9k^{8}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right).
\end{align*}
The first term in this expands to
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\left(\sum\limits_{j=1}^{k-1}j^4e^{\frac{2\pi imj}{k}} \right)
\left(
\sum\limits_{s=1}^{k-1}s^2e^{\frac{-2\pi ims}{k}}
\right)\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k^4}{\alpha-1}-\frac{4k^3\alpha}{(\alpha-1)^2}+\frac{6k^2\alpha+6k^2\alpha^2}{(\alpha-1)^3}-\frac{(4k\alpha+16k\alpha^2+4k\alpha^3)}{(\alpha-1)^4}
\right)\\
& \left(
-k^2- \frac{k^2}{\alpha-1}-\frac{2k\alpha}{(\alpha-1)^2}
\right)(\text{ By identity }(\ref{j^4}))\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl[
- \frac{k^6}{\alpha-1}-\frac{k^6}{(\alpha-1)^2}+\frac{4k^5\alpha}{(\alpha-1)^2}+\frac{2k^5\alpha}{(\alpha-1)^3}-\frac{6k^4\alpha}{(\alpha-1)^3}\\&-\frac{6k^4\alpha^2}{(\alpha-1)^3}-\frac{6k^4\alpha}{(\alpha-1)^4}+\frac{2k^4\alpha^2}{(\alpha-1)^4}+\frac{4k^3\alpha}{(\alpha-1)^4}+\frac{16k^3\alpha^2}{(\alpha-1)^4}+\frac{4k^3\alpha^3}{(\alpha-1)^4}\\&+\frac{4k^3\alpha}{(\alpha-1)^5}+\frac{4k^3\alpha^2}{(\alpha-1)^5}-\frac{8k^3\alpha^3}{(\alpha-1)^5}+\frac{8k^2\alpha^2}{(\alpha-1)^6}+\frac{32k^2\alpha^3}{(\alpha-1)^6}+\frac{8k^2\alpha^4}{(\alpha-1)^6}
\biggr].
\end{align*}
Similarly the second term expands to
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}\\
=&\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl[
\frac{k^6}{(\alpha-1)^2}-\frac{6k^5\alpha}{(\alpha-1)^3}-\frac{6k^4\alpha}{(\alpha-1)^4}+\frac{14k^4\alpha^2}{(\alpha-1)^4}-\frac{4k^3\alpha}{(\alpha-1)^5}\\&-\frac{28k^3\alpha^2}{(\alpha-1)^5}-\frac{16k^3\alpha^3}{(\alpha-1)^5}+\frac{8k^2\alpha^2}{(\alpha-1)^6}+\frac{32k^2\alpha^3}{(\alpha-1)^6}+\frac{8k^2\alpha^4}{(\alpha-1)^6}
\biggr].
\end{align*}
Thus we get
\begin{align*}
&\frac{\pi^8}{9k^{8}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\\
=&\frac{\pi^8}{9k^{8}}\phi(k)\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\biggl[
- \frac{k^6}{\alpha-1}+\frac{4k^5\alpha}{(\alpha-1)^2}-\frac{4k^5\alpha}{(\alpha-1)^3}-\frac{6k^4\alpha}{(\alpha-1)^3}-\frac{6k^4\alpha^2}{(\alpha-1)^3}\\&+\frac{16k^4\alpha^2}{(\alpha-1)^4}+\frac{4k^3\alpha}{(\alpha-1)^4}+\frac{16k^3\alpha^2}{(\alpha-1)^4}+\frac{4k^3\alpha^3}{(\alpha-1)^4}-\frac{24k^3\alpha^2}{(\alpha-1)^5}-\frac{24k^3\alpha^3}{(\alpha-1)^5}\\&+\frac{16k^2\alpha^2}{(\alpha-1)^6}+\frac{64k^2\alpha^3}{(\alpha-1)^6}+\frac{16k^2\alpha^4}{(\alpha-1)^6}
\biggr].
\end{align*}
Using identities (\ref{a/(a-1)^2}), (\ref{a^2/(a-1)^3}), (\ref{a/(a-1)^3}), (\ref{a^3/(a-1)^4}), (\ref{a^2/(a-1)^4}), (\ref{a/(a-1)^4}), (\ref{a^3/(a-1)^5}), (\ref{a^2/(a-1)^5}), (\ref{a^4/(a-1)^6}), (\ref{a^3/(a-1)^6}), (\ref{a^2/(a-1)^6}), (\ref{1/a-1}), we further modify the above to
\begin{align}\label{j^4s^2L(4,x)}
&\frac{\pi^8}{9k^{8}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
=&\frac{\pi^8}{k^{8}}\phi(k)
\biggl[
\frac{k^6}{18}\phi(k)-\frac{k^5}{18}J_2(k)+\frac{k^4}{405}J_4(k)+\frac{2k^4}{81}J_2(k)\\&+\frac{k^3}{270}J_4(k)-\frac{k^2}{2835}J_6(k)-\frac{k^2}{810}J_4(k)-\frac{2k^2}{405}J_2(k)
\biggr]\nonumber.
\end{align}
Next we consider
\begin{align*}
\frac{\pi^8}{9k^8}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^4\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right).
\end{align*}
Interchange the variables $j$ and $s$, we get
\begin{align}\label{j^2s^4L(4,x)}
&\frac{\pi^8}{9k^8}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi i(k-m)(s-j)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \nonumber \\
=& \frac{\pi^8}{9k^8}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}-\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^4s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right)\nonumber \\
=&\frac{\pi^8}{k^{8}}\phi(k)
\biggl[
\frac{k^6}{18}\phi(k)-\frac{k^5}{18}J_2(k)+\frac{k^4}{405}J_4(k)+\frac{2k^4}{81}J_2(k)+\frac{k^3}{270}J_4(k)\\&-\frac{k^2}{2835}J_6(k)-\frac{k^2}{810}J_4(k)-\frac{2k^2}{405}J_2(k)
\biggr]\nonumber.
\end{align}
By a similar sequence of computations, we may see that
\begin{align*}
& \sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}\\&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
-\frac{k^4}{\alpha-1}- \frac{k^4}{(\alpha-1)^2}+\frac{2k^3\alpha}{(\alpha-1)^2}+\frac{4k^2\alpha^2}{(\alpha-1)^4}
\right)
\end{align*}
and
\begin{align*}
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
\frac{k^4}{(\alpha-1)^2}-\frac{4k^3\alpha}{(\alpha-1)^3}+\frac{4k^2\alpha^2}{(\alpha-1)^4}
\right).
\end{align*}
Hence we get
\begin{align}\label{j^2s^2L(4,x)}
&\frac{2\pi^8}{9k^{6}}\phi(k)
\left(
\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j-s)}{k}}+\sum\limits_{j=1}^{k-1} \sum\limits_{s=1}^{k-1}j^2s^2\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}e^{\frac{2\pi im(j+s)}{k}}
\right) \nonumber \\
&=\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\left(
-\frac{k^4}{\alpha-1}+\frac{2k^3\alpha}{(\alpha-1)^2}-\frac{4k^3\alpha}{(\alpha-1)^3}+\frac{8k^2\alpha^2}{(\alpha-1)^4}
\right)\nonumber\\
&=\frac{\pi^8}{k^{8}}\phi(k)
\left(
\frac{k^6}{9}\phi(k)-\frac{2k^5}{27}J_2(k)+\frac{k^4}{405}J_4(k)+\frac{2k^4}{81}J_2(k)
\right) \\& \text{ using identities } (\ref{a/(a-1)^2}), (\ref{a/(a-1)^3}), (\ref{a^2/(a-1)^4}), (\ref{1/a-1})\nonumber.
\end{align}
Hence on substituting (\ref{j^4s^4L(4,x)}), (\ref{j^4s^2L(4,x)}), (\ref{j^2s^4L(4,x)}), (\ref{j^2s^2L(4,x)}) into (\ref{L(4,x)1}), we get
\begin{align*}
&\frac{\pi^8}{9k^6}\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}\left|\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi)
\right|^2\\
&=\frac{\pi^8}{k^{8}}\phi(k)
\left(
\frac{k^6}{36}\phi(k)+\frac{J_8(k)}{18900}+\frac{J_4(k)}{4050}+\frac{2}{567}J_2(k)
-\frac{k^3}{270}J_4(k)
\right).
\end{align*}
Now we have to find
$
\frac{\pi^8}{9k^6}\left|
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi_0)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi_0)
\right|^2
$ for evaluating identity (\ref{6*}).
We have
\begin{align*}
\sum\limits_{j=1}^{k-1} j^2G(j, \chi_0)=\sum\limits_{j=1}^{k-1} j^2R_k(j)&=-\frac{k^2\phi(k)}{2}+\frac{k^3}{\pi^2}L(2, \chi_0)\text{ \cite[identity 3.22]{alkan2011mean}}\\&=-\frac{k^2\phi(k)}{2}+\frac{k}{6}J_2(k).
\end{align*}
Now to find $\sum\limits_{j=1}^{k-1} j^4G(j, \chi_0)=\sum\limits_{j=1}^{k-1} j^4R_k(j)$.
We use the identity $
R_k(j)=\frac{\phi(k)\mu(\frac{k}{(k, j)})}{\phi(\frac{k}{(k,j)})}
$.
\begin{align*}
\sum\limits_{j=1}^{k} j^4R_k(j)&=\sum\limits_{j=1}^{k-1} j^4\frac{\phi(k)\mu(\frac{k}{(k, j)})}{\phi(\frac{k}{(k,j)})}\\
&=\phi(k)\sum\limits_{d|k}\sum\limits_{\substack{1\leq j \leq k\\(k, j)=d}}\frac{j^4\mu(\frac{k}{d})}{\phi(\frac{k}{d})}\\
&=k^4\phi(k)+\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}\sum\limits_{\substack{1\leq j \leq k\\(k, j)=d}}\frac{j^4\mu(\frac{k}{d})}{\phi(\frac{k}{d})}\\
&=k^4\phi(k)+\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}\frac{d^4\mu(\frac{k}{d})}{\phi(\frac{k}{d})}\sum\limits_{\substack{1\leq j \leq k\\(k, j)=d}}\left(\frac{j}{d}\right)^4.
\end{align*}
Therefore
\begin{align}\label{R_k}
\sum\limits_{j=1}^{k-1} j^4R_k(j)=\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}\frac{d^4\mu(\frac{k}{d})}{\phi(\frac{k}{d})}\sum\limits_{\substack{1\leq j \leq k\\(k, j)=d}}\left(\frac{j}{d}\right)^4.
\end{align}
With $n=\frac{k}{d}$ in lemma (\ref{phi_4}) we get
\begin{align} \label{m^4}
\phi_4\left(\frac{k}{d}\right)=\sum\limits_{\substack{1\leq m \leq \frac{k}{d}\\(m, \frac{k}{d})=1}}m^4=\frac{k^4}{5d^4}\phi\left(\frac{k}{d}\right)+\frac{k^3}{3d^3}\prod\limits_{p|(\frac{k}{d})}(1-p)-\frac{k}{30d}\prod\limits_{p|(\frac{k}{d})}(1-p^3).
\end{align}
Denote by $w(n)$, the number of distinct prime divisors of $n$ and $P_s(n)$, product of $s$\textsuperscript{th} powers of all distinct prime divisors of $n$. If $P(n)=P_1(n)$ we have
\begin{align*}
\phi\left(\frac{k}{d}\right)=\left(\frac{k}{d}\right)\prod\limits_{p|(\frac{k}{d})}\left(1-\frac{1}{p}\right)&=\left( \frac{k}{d}\right)\left(\frac{\prod\limits_{p|(\frac{k}{d})}(p-1)}{\prod\limits_{p|(\frac{k}{d})}p}\right)\\&=\left( \frac{k}{d}\right)(-1)^{w(\frac{k}{d})}\left(\frac{\prod\limits_{p|(\frac{k}{d})}(1-p)}{P(\frac{k}{d})}\right).
\end{align*}
Therefore
\begin{align*}
\prod\limits_{p|(\frac{k}{d})}(1-p)=\left(\frac{d}{k}\right)(-1)^{w(\frac{k}{d})}P\left(\frac{k}{d}\right)\phi\left(\frac{k}{d}\right).
\end{align*}
We have
\begin{align*}
J_3\left(\frac{k}{d}\right)=\left(\frac{k}{d}\right)^3\prod\limits_{p|(\frac{k}{d})}\left(1-\frac{1}{p^3}\right)&=\left(\frac{k}{d}\right)^3\left(
\frac{\prod\limits_{p|(\frac{k}{d})}(p^3-1)}{\prod\limits_{p|(\frac{k}{d})}p^3}
\right)\\&=\left( \frac{k^3}{d^3}\right)(-1)^{w(\frac{k}{d})}\left(\frac{\prod\limits_{p|(\frac{k}{d})}(1-p^3)}{P_3(\frac{k}{d})}\right).
\end{align*}
and so
\begin{align*}
\prod\limits_{p|(\frac{k}{d})}(1-p^3)=\left( \frac{d^3}{k^3}\right)(-1)^{w(\frac{k}{d})}P_3\left(\frac{k}{d}\right)J_3\left(\frac{k}{d}\right).
\end{align*}
Hence from equation (\ref{m^4}) we get
\begin{align*}
\sum\limits_{\substack{1\leq m \leq \frac{k}{d}\\(m, \frac{k}{d})=1}}m^4=&\frac{k^4}{5d^4}\phi\left(\frac{k}{d}\right)+\frac{k^2}{3d^2}(-1)^{w(\frac{k}{d})}P\left(\frac{k}{d}\right)\phi\left(\frac{k}{d}\right)\\&-\frac{d^2}{30k^2}(-1)^{w(\frac{k}{d})}P_3\left(\frac{k}{d}\right)J_3\left(\frac{k}{d}\right).
\end{align*}
So
\begin{align*}
\sum\limits_{\substack{1\leq j \leq k\\(k, j)=d}}\left(\frac{j}{d}\right)^4=\sum\limits_{\substack{1\leq m \leq \frac{k}{d}\\(m, \frac{k}{d})=1}}m^4=&\frac{k^4}{5d^4}\phi\left(\frac{k}{d}\right)+\frac{k^2}{3d^2}(-1)^{w(\frac{k}{d})}P\left(\frac{k}{d}\right)\phi\left(\frac{k}{d}\right)\\&-\frac{d^2}{30k^2}(-1)^{w(\frac{k}{d})}P_3\left(\frac{k}{d}\right)J_3\left(\frac{k}{d}\right)
\end{align*}
and thus equation (\ref{R_k}) becomes,
\begin{align*}
&\sum\limits_{j=1}^{k-1} j^4R_k(j)\\
=&\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}\frac{d^4\mu(\frac{k}{d})}{\phi(\frac{k}{d})}\biggl[
\frac{k^4}{5d^4}\phi\left(\frac{k}{d}\right)+\frac{k^2}{3d^2}(-1)^{w(\frac{k}{d})}P\left(\frac{k}{d}\right)\phi\left(\frac{k}{d}\right)\\&-\frac{d^2}{30k^2}(-1)^{w(\frac{k}{d})}P_3\left(\frac{k}{d}\right)J_3\left(\frac{k}{d}\right)
\biggr]\\
=&\left(\frac{k^4}{5}\right)\phi(k)\sum\limits_{\substack{d|k\\d\neq 1}}\mu(d)+\left(\frac{k^2}{3}\right)\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}d^2(-1)^{w(\frac{k}{d})}P\left(\frac{k}{d}\right)\mu\left(\frac{k}{d}\right)\\
&-\left(\frac{1}{30k^2}\right)\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}d^6(-1)^{w(\frac{k}{d})}\frac{P_3(\frac{k}{d})J_3(\frac{k}{d})\mu(\frac{k}{d})}{\phi(\frac{k}{d})}.
\end{align*}
Since
$
\sum\limits_{\substack{d|k\\d\neq 1}}\mu(d)=-1
$
and $
\sum\limits_{d|k}d^2(-1)^{w(\frac{k}{d})}P(\frac{k}{d})\mu(\frac{k}{d})=k^2\prod\limits_{p|k}\left(1+\frac{1}{p}\right)
$
we get
\begin{align*}
\sum\limits_{j=1}^{k-1} j^4R_k(j)=&-\left(\frac{k^4}{5}\right)\phi(k)+\left(\frac{k^2}{3}\right)\phi(k)\left(-k^2+k^2\prod\limits_{p|k}\left(1+\frac{1}{p}\right)\right)\\&-\left(\frac{1}{30k^2}\right)\phi(k)\sum\limits_{\substack{d|k\\d\neq k}}d^6(-1)^{w(\frac{k}{d})}\frac{P_3(\frac{k}{d})J_3(\frac{k}{d})\mu(\frac{k}{d})}{\phi(\frac{k}{d})}.
\end{align*}
Now we have to evaluate $
\sum\limits_{\substack{d|k\\d\neq k}}d^6(-1)^{w(\frac{k}{d})}\frac{P_3(\frac{k}{d})J_3(\frac{k}{d})\mu(\frac{k}{d})}{\phi(\frac{k}{d})}$.
Consider
$\sum\limits_{d|k}d^6(-1)^{w(\frac{k}{d})}\frac{P_3(\frac{k}{d})J_3(\frac{k}{d})\mu(\frac{k}{d})}{\phi(\frac{k}{d})}
$.
Let $f(m)=m^6$ and $g(m)=(-1)^{w(m)}\frac{P_3(m)J_3(m)\mu(m)}{\phi(m)}$.
Thus \begin{align*}\sum\limits_{d|k}d^6(-1)^{w(\frac{k}{d})}\frac{P_3(\frac{k}{d})J_3(\frac{k}{d})\mu(\frac{k}{d})}{\phi(\frac{k}{d})}=(f*g)(k)
\end{align*}where $*$ denotes the usual Dirichlet's product.
Since $f$ and $g$ are multiplicative, \begin{align*}(f*g)(k)=(f*g)\left(\prod\limits_{p|k}p^m\right)=\prod\limits_{p|k}(f*g)(p^m)
\end{align*} where $p^m$ is the exact power of a prime $p$ dividing $k$.
\begin{align*}
&\sum\limits_{d|k}d^6(-1)^{w(\frac{k}{d})}\frac{P_3\left(\frac{k}{d}\right)J_3\left(\frac{k}{d}\right)\mu\left(\frac{k}{d}\right)}{\phi(\frac{k}{d})}\\
=& \prod\limits_{p|k}\left(\sum\limits_{d|k}d^6(-1)^{w(\frac{p^m}{d})}\frac{P_3(\frac{p^m}{d})J_3(\frac{p^m}{d})\mu(\frac{p^m}{d})}{\phi(\frac{p^m}{d})}\right)\\
=& \prod\limits_{p|k}\left((p^m-1)^6p^3\left(\frac{1-\frac{1}{p^3}} {p-1}\right)+(p^m)^6
\right)\\& (\text{ since sum is non-zero if } d=p^{m-1} \text{ or } d=p^m )\\
=& \prod\limits_{p|k}\left(
p^{6m-6}\cdot p^3(p^2+p+1)+p^{6m}
\right)\\
=& \left(\prod\limits_{p|k}p^{6m}\right)\left( \prod\limits_{p|k}\left(1+\frac{1}{p}+\frac{1}{p^2}+\frac{1}{p^3}\right)\right)\\
=&k^6\prod\limits_{p|k}\left(1+\frac{1}{p}+\frac{1}{p^2}+\frac{1}{p^3}\right).
\end{align*}
Therefore
\begin{align*}
\sum\limits_{\substack{d|k\\d\neq k}}d^6(-1)^{w(\frac{k}{d})}\frac{P_3(\frac{k}{d})J_3(\frac{k}{d})\mu(\frac{k}{d})}{\phi(\frac{k}{d})}= -k^6 +k^6\prod\limits_{p|k}\left(1+\frac{1}{p}+\frac{1}{p^2}+\frac{1}{p^3}\right).
\end{align*}
Hence
\begin{align*}
&\sum\limits_{j=1}^{k-1} j^4R_k(j)\\=&-\left(\frac{k^4}{5}\right)\phi(k)+\left(\frac{k^2}{3}\right)\phi(k)\left(-k^2+k^2\prod\limits_{p|k}\left(1+\frac{1}{p}\right)\right)\\&-\left(\frac{1}{30k^2}\right)\phi(k)\left(-k^6 +k^6\prod\limits_{p|k}\left(1+\frac{1}{p}+\frac{1}{p^2}+\frac{1}{p^3}\right)\right)\\
=&-\frac{k^4}{2}\phi(k)+\frac{k^5}{3}\prod\limits_{p|k}\left(1-\frac{1}{p^2}\right)-\frac{k^5}{30}\prod\limits_{p|k}\left(1-\frac{1}{p^4}\right)\\
=&-\frac{k^4}{2}\phi(k)+\frac{k^5}{3}\left(\frac{L(2,\chi_0)}{\zeta(2)}\right)-\frac{k^5}{30}\left(\frac{L(4,\chi_0)}{\zeta(4)}\right) \text{ \cite[Theorem 11.7]{tom1976introduction}}\\
=&-\frac{k^4}{2}\phi(k)+\frac{k^5}{3}\left(\frac{6}{\pi^2}\right)\left(\frac{\pi^2J_2(k)}{6k^2}\right)-\frac{k^5}{30}\left(\frac{90}{\pi^4}\right)\left(\frac{\pi^4J_4(k)}{90k^4}\right)\\
=&-\frac{k^4}{2}\phi(k)+\frac{k^3}{3}J_2(k)-\frac{k}{30}J_4(k).
\end{align*}
Thus,
\begin{align*}
&\frac{\pi^8}{9k^6}\left|
\frac{1}{k^2}\sum\limits_{j=1}^{k-1} j^4G(j, \chi_0)-2\sum\limits_{j=1}^{k-1} j^2G(j, \chi_0)
\right|^2\\&=\frac{\pi^8}{9k^6}\left|
-\frac{k^2}{2}\phi(k)+\frac{k}{3}J_2(k)-\frac{1}{30k}J_4(k)+k^2\phi(k)-\frac{k}{3}J_2(k)
\right|^2\\
&=\frac{\pi^8}{9k^6}\left|
\frac{k^2}{2}\phi(k)-\frac{1}{30k}J_4(k)
\right|^2\\
&=\frac{\pi^8}{k^8}\phi(k)\left(
\frac{k^6}{36}\phi(k)-\frac{k^3}{270}\phi(k)J_4(k)+\frac{1}{8100}\frac{(J_4(k))^2}{\phi(k)}
\right).
\end{align*}
Coming back to equation (\ref{6*})
\begin{align*}
&\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}\\ \chi \neq \chi_0}}|L(4, \chi)|^2\\=&\frac{\pi^8}{k^8}\phi(k)\biggl[
\frac{k^6}{36}\phi(k)+\frac{J_8(k)}{18900}+\frac{J_4(k)}{4050}+\frac{2}{567}J_2(k)
-\frac{k^3}{270}J_4(k)\\&-\frac{k^6}{36}\phi(k)+\frac{k^3}{270}\phi(k)J_4(k)-\frac{1}{8100}\frac{(J_4(k))^2}{\phi(k)}
\biggr]\\
=&\frac{\pi^8}{k^8}\phi(k)\left(\frac{J_8(k)}{18900}+\frac{J_4(k)}{4050}+\frac{2}{567}J_2(k)-\frac{1}{8100}\frac{(J_4(k))^2}{\phi(k)}
\right).
\end{align*}
Hence, equation (\ref{1*}) becomes
\begin{align*}
\sum\limits_{\substack{\chi(\text{mod } k) \\ \chi \text{ even}}}|L(4, \chi)|^2
=\frac{\pi^8}{27k^8}\phi(k)\left(\frac{J_8(k)}{700}+\frac{J_4(k)}{150}+\frac{2}{21}J_2(k)
\right)
\end{align*}which is what we claimed.
Hence the average value of $|L(4, \chi)|^2$ over all odd characters modulo $k$ is
$
\frac{2\pi^8}{27k^8}\left(\frac{J_8(k)}{700}+\frac{J_4(k)}{150}+\frac{2}{21}J_2(k)\right)$.
\end{proof}
\section{Further directions}
Though it is possible to extend our method to find higher mean square averages of $L(r,\chi)$, the major difficulty arises in finding the value of $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}
\frac{1}{\sin^{2r}(\frac{\pi m}{k})}$. It is also possible that our method can be applied to the twisted sum case. But one may have to be ready to undertake longer computations with complexity increasing with respect to the size of $r$.
\section{Acknowledgements}
The first author thanks the Kerala State Council for Science,Technology and Environment, Thiruvananthapuram, Kerala, India for providing financial support for carrying out research work. The second author thanks the University Grants Commission of India for providing financial support for carrying out research work through their Junior Research Fellowship (JRF) scheme.
| 146,241
|
Sir Chris Bonington: 'My climbs were funded by tobacco and Durex'
Mountaineer Sir Chris Bonington, 83, made the first British ascent of the North Wall of the Eiger in 1962 and led the expedition that made the first successful ascent of the south-west face of Everest in 1975.
He was knighted for services to mountaineering in 1996 and in 2015 received the Piolet d’Or, one of the most prestigious awards in mountaineering.
He divides his time between London and Cumbria with his second wife, Loreto. He has two surviving sons from his first marriage to his late wife Wendy.
How did your childhood influence your work ethic and attitude to money?
My parents split up when I was nine months old. To support us my mother initially worked as a secretary, then as a copywriter...
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| 336,057
|
Heresy against the Goracle!
Apparently, climate tracking stations are reporting the fastest global temperature change ever recorded in 2007–the average temperature of the earth DROPPED about 3/4 of a degree Celcius. This explains some of the “must be global warming” posts I’ve made, like the snowfall in Baghdad earlier this year. I’ve had trouble find hard information, but I’ve posted some news articles below:
| 271,677
|
\title{Physics-informed machine learning techniques for edge plasma turbulence modelling in computational theory and experiment}
\author{Abhilash Mathews}
\prevdegrees{B.Sc. Physics, Western University (2017)}
\department{Department of Nuclear Science and Engineering}
\degree{DOCTOR OF PHILOSOPHY IN APPLIED PLASMA PHYSICS}
\degreemonth{May}
\degreeyear{2022}
\thesisdate{May 11, 2022}
\supervisor{Jerry W. Hughes, Ph.D.}{Principal Research Scientist, MIT Plasma Science and Fusion Center}
\reader{Anne E. White, Ph.D.}{Professor and Head of Nuclear Science and Engineering}
\chairman{Ju Li, Ph.D.}{Battelle Energy Alliance Professor of Nuclear Science and Engineering \\Chairman, Department Committee on Graduate Theses}
\maketitle
\cleardoublepage
\setcounter{savepage}{\thepage}
\begin{abstractpage}
\input{abstract}
\end{abstractpage}
\cleardoublepage
\section*{Acknowledgments}
\setlength{\epigraphwidth}{0.8\textwidth}
\epigraph{I suppose in the end, the whole of life becomes an act of letting go, but what always hurts the most is not taking a moment to say goodbye.}{Yann Martel, Life of Pi}
It is only as I stitch together the final pieces of my PhD that I am realizing it is nearly over. A trek that once almost felt endless just a short while ago is now just about finished. And at this end, there are countless people to thank.
To start, this thesis and myself owe immense gratitude to J.W. Hughes. He provided me an example every day of what it means to be a good scientist and an even better human. Freedom is not worth having if it does not include the freedom to make mistakes\footnote{Mahatma Gandhi}, and Jerry aptly provided ample freedom to find my own meaningful problems and make my own mistakes. There may be no greater gift as a student, and Jerry's wit---in life and experiment---made this journey a joy. I thank A.E. White for making this work all possible beginning on the first day I walked into the PSFC as a Cantabrigian by bringing Jerry into her office. While the project evolved over the years, Anne knew how to start it all. And ever since our first serendipitous encounter in Austin, D.R. Hatch was a source of constant encouragement when exploring unknown territory and truly beginning this research. I am also grateful to my thesis defence members D.G. Whyte and J.D. Hare for their time in crafting this dissertation. At every stage, from my NSE admission letter to navigating a pandemic abroad, B. Baker was incredibly supportive throughout the entirety of graduate school.
Upon first meeting Mana on campus, I didn't realize how instrumental he would be to this work as a mentor and a friend, but this document would not exist without him. I can also say essentially all the same about Jim, who truly made these last chapters of the thesis possible. And while he may not be a formal co-author listed on publications, I wholeheartedly thank Ted for always being there at the beginning of my PhD. I could not have asked for a better neighbour. I am also indebted to the contributions of all my collaborators: M. Francisquez taught me to conduct the two-fluid simulations presented in Chapter 2 using the \texttt{GDB} code run on MIT's Engaging cluster and co-developed with B. Zhu and B.N. Rogers; N. Mandell developed and ran the electromagnetic gyrokinetic simulations described in Chapter 3, which were performed on the Perseus cluster at Princeton University and the Cori cluster at NERSC, and based upon the \texttt{Gkeyll} framework led by A. Hakim and G.W. Hammett; B. LaBombard and D. Brunner operated the mirror Langmuir probe utilized in Chapter 4; A.Q. Kuang and M.A. Miller assisted with probe data analysis in Chapter 4; S.G. Baek ran DEGAS2 simulations to inform neutral modelling in Chapters 4 and 5; J.L. Terry and S.J. Zweben operated the GPI diagnostic in Chapters 4 and 5; M. Goto, D. Stotler, D. Reiter, and W. Zholobenko developed the HeI collisional radiative codes employed in Chapters 4 and 5. The content of these pages are enabled by their technical support and camaraderie. Any errors in this thesis are my own.
While this may be the end of my PhD at MIT, the sights of Park Street and Killian Court will forever feel warm. I thank all the wonderful people I ran into on Albany Street and across the world over the past years including Nick, Francesco, Pablo, Rachel, Beatrice, Thanh, Eli, Bodhi, Fernanda, Nima (Harvard's P283B taught me to truly view physics questions as constrained-optimization problems), Lt. Reynolds, Erica, Muni, Christian, Patricio, Yu-Jou, Libby, Cassidy, Alex, Lucio, Aaron, Sam, Evan, Anna (from Switzerland), Anna (from Austria), Manon, Eva, Josh, and the never-faraway Peter. Prior to ever setting foot in Cambridge, I am truly lucky to have met a lifelong teacher and advisor in Martin, who pushed me to this point starting from London. I also thank my best friends in Canada for always making me feel at home wherever I am---from Fenway Park with Katie and Sammy, to sailing into the Charles with Donna, to the great lake with George and Maher (or even adventuring on it in Niagara), to escaping the French secret police with Dylan, to Adam's backyards in Toronto and Montr\'eal and Killarney, to living in a van across Valhalla and sleeping in train stations by Berchtesgaden with Nicholas---no place is ever too distant.
There are also many people without whom this thesis would literally not be possible such as the great folks at Jasper Health Services for helping me to still write, Mississauga Fire Department for allowing me to still breathe, and Procyon Wildlife for inspiring me years ago and still to this day. But beyond all, this PhD is the product of the unconditional love and support I constantly get from my big family---all the way from my dear grandparents to my boisterous cousins who are always full of life. At the core, my parents instilled the values of good work and education in me at an early age. Together with my older brother, they endlessly push me in everything that I do with their time and heart and extraordinary cooking. There can never be enough appreciation for all they've done and for what they mean to me.
And to my littlest brother, Kobe, thank you for teaching me to see the world.
\\
\noindent\rule{15.25cm}{0.4pt}
\\ \\
{\it
\noindent As you set out for Ithaka\\
hope your road is a long one,\\
full of adventure, full of discovery.\\
Laistrygonians, Cyclops,\\
angry Poseidon---don’t be afraid of them:\\
you’ll never find things like that on your way\\
as long as you keep your thoughts raised high,\\
as long as a rare excitement\\
stirs your spirit and your body.\\
Laistrygonians, Cyclops,\\
wild Poseidon---you won’t encounter them\\
unless you bring them along inside your soul,\\
unless your soul sets them up in front of you.\\
\vspace{-0.7cm}
\\ \\
Hope your road is a long one.\\
May there be many summer mornings when,\\
with what pleasure, what joy,\\
you enter harbors you’re seeing for the first time;\\
may you stop at Phoenician trading stations\\
to buy fine things,\\
mother of pearl and coral, amber and ebony,\\
sensual perfume of every kind---\\
as many sensual perfumes as you can;\\
and may you visit many Egyptian cities\\
to learn and go on learning from their scholars.\\
\vspace{-0.7cm}
\\ \\
Keep Ithaka always in your mind.\\
Arriving there is what you’re destined for.\\
But don’t hurry the journey at all.\\
Better if it lasts for years,\\
so you’re old by the time you reach the island,\\
wealthy with all you’ve gained on the way,\\
not expecting Ithaka to make you rich.\\
\vspace{-0.7cm}
\\ \\
Ithaka gave you the marvelous journey.\\
Without her you wouldn't have set out.\\
She has nothing left to give you now.\\
\vspace{-0.7cm}
\\ \\
And if you find her poor, Ithaka won’t have fooled you.\\
Wise as you will have become, so full of experience,\\
you’ll have understood by then what these Ithakas mean.}\vspace{0.4cm}\\{--- C. P. Cavafy (translated by Edmund Keeley)}
\newpage
\section*{List of Publications}
Part of the content included in this thesis has already been published in peer-reviewed journals or is presently under referee review. Permission to reuse text and figures from these articles has been granted and are listed below:
\begin{itemize}
\item {\bf A. Mathews}, J.W. Hughes, J.L. Terry, S.G. Baek, “Deep electric field predictions by drift-reduced Braginskii theory with plasma-neutral interactions based upon experimental images of boundary turbulence” arXiv:2204.11689
(2022)
\item {\bf A. Mathews}, J.L. Terry, S.G. Baek, J.W. Hughes, A.Q. Kuang, B. LaBombard, M.A. Miller, D. Stotler, D. Reiter, W. Zholobenko, and M. Goto, “Deep modelling of plasma and neutral fluctuations from gas puff turbulence imaging” arXiv:2201.09988 (2022)
\item {\bf A. Mathews}, N. Mandell, M. Francisquez, J.W. Hughes, and A. Hakim, “Turbulent field fluctuations in gyrokinetic and fluid plasmas” Physics of Plasmas {\bf 28}, 112301 (2021)
\item {\bf A. Mathews}, M. Francisquez, J.W. Hughes, D.R. Hatch, B. Zhu, and B.N. Rogers, “Uncovering turbulent plasma dynamics via deep learning from partial observations” Physical Review E {\bf 104}, 025205 (2021)
\item {\bf A. Mathews} and J.W. Hughes, “Quantifying experimental edge plasma evolution via multidimensional adaptive Gaussian process regression” IEEE Transactions on Plasma Science {\bf 49}, 12 (2021)
\end{itemize}
\noindent Funding support came from the Natural Sciences and Engineering Research Council of Canada (NSERC) through the doctoral postgraduate scholarship (PGS D), U.S. Department of Energy (DOE) Office of Science under the Fusion Energy Sciences program by contract DE-SC0014264, Joseph P. Kearney Fellowship, and Manson Benedict Fellowship from the MIT Department of Nuclear Science and Engineering.
| 155,643
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In having a successful marriage, nothing is more important than each partner walking in the will of the Lord; next comes forgiveness. We all make mistakes and fail one another at some time or other. Will we forgive and move forward or implode because our rearview mirror is larger than our windshield?
| 115,900
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TITLE: $X=\{x_1,\dots,x_n\}$, $x_ix_j\in X$ whenever $1\leq i \leq j \leq n$. Then $X\leq G$.
QUESTION [2 upvotes]: Let $X$ be a non-empty finite subset of a group $G$; say $X=\{x_1,\dots,x_n\}$ where $n$ is a positive integer. Suppose that $x_ix_j\in X$ whenever $1\leq i \leq j \leq n$.
Prove that $X\leq G$.
I have done the following:
(i) $x_i^m\in X$ for every positive integer $m$ and for every $i=1,\dots,n$.
(ii) For every $i=1,\dots,n$, $x_i$ has finite order and $x_i^{-1}\in X$.
(iii) If $x_iX=X$, then $x_i^{-1}X=X$.
I want to prove by induction on $j$ that for every $j=1,\dots,n$, $x_jX=X$.
For $j=1$, it is clear since $x_1x_k\in X$ for every $k\geq 1$ and hence $x_1X\subseteq X$.
Define $f:X\rightarrow x_1X$ by $f(x)=x_1x$. Then $f$ is one-to-one. Hence $|x_1X|\leq |X|\leq |x_1X|$ hence $x_1X=X$.
Suppose that $x_lX=X$.
I have no idea how to prove $x_{l+1}X=X$ since I still can't verify that $x_{l+1}x_k\in X$ for $l+1>k$.
If this is proven, then we have $X^2=X$. Since $G$ is finite, $X\leq G$.
REPLY [1 votes]: Your inductive assumption is that $x_kX = X$ for $1 \le k \le l$.
So by (iii), which you have proven, for any such $k$ we have $x_k^{-1}X = X$, and hence in particular $x_k^{-1}x_{l+1}^{-1} = (x_{l+1}x_k)^{-1} \in X$.
So by your property (ii), $x_{l+1}x_k \in X$, and you are done.
| 7,748
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Dolphins To Check Out Other WRs Today
The Miami Dolphins have been sit by injuries to their second-tier receivers already in camp, so theyre taking a look at a few more.
In addition to the previously reported scheduled visit with Laurent Robinson, the Dolphins are expected to have other wideouts work out today, according to Barry Jackson of the Miami Herald.
Read more on Pro Football Talk
| 141,787
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Pollination needs’ Category
50+ Flowering Pollen/Nectar Plants For Bumblebees […] […]
| 156,941
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Grandma Primate, What good eyes you have! “The better to see snakes,” evolutionists suggest.
The Snake Detection Theory has been one of more controversial ideas about how the brains of primates and eventually people evolved. The theory recently got a neurobiological boost from the discovery that macaque monkeys that have never even seen a snake have an intense rapid neurological response to snake pictures, suggesting they are hard-wired to detect and avoid snakes.
Snake-sensitive neurons in a visual-relay center of macaque brains outnumbered neurons that responded to other pictures. These neurons also elicited stronger and more rapid responses. Image: iStockphoto/Thinkstock through “Did Snakes Help Build the Primate Brain?” in ScienceNow.
These are the pictures shown to macaques while electrodes in their brains monitored the responses of neurons thought to relay threatening visual images. Forty percent of the 91 neurons were highly responsive to snake pictures, with face-sensitive neurons coming in second. Image: Van Le et al., “Pulvinar neurons reveal neurobiological evidence of past selection for rapid detection of snakes,” PNAS (28 October 2013).
A joint study by behavioral ecologist Lynne Isbell—who came up with the Snake Detection Theory of mammalian and primate brain development—with a team of Japanese and Brazilian neuroscientists asserts that snake-sensitive neurons in the monkeys’ brains are evidence for “the growing evolutionary perspective that snakes have long shaped our primate lineage.”1 The authors of the study published in the Proceedings of the National Academy of Sciences write, “Our findings are unique in providing neuroscientific evidence in support of the Snake Detection Theory, which posits that the threat of snakes strongly influenced the evolution of the primate brain. This finding may have great impact on our understanding of the evolution of primates.”1
Monkeys raised behind high walls, and therefore never exposed to snakes, were the subjects for the study. This species, in the wild, would live in environments inhabited by snakes. Neurobiologists implanted electrodes in the monkeys’ brains to see what sort of images evoked neuronal responses. The monkeys were shown images of angry and calm monkey faces, monkey hands, simple geometric shapes, and snakes. The neurobiologists monitored 91 neurons in the pulvinar—a subcortical portion of primate brains thought to be involved in relaying threatening visual images.1
Snakes, whether coiled or elongated, elicited intense, rapid responses from the largest proportion of the monkeys’ neurons—over 40%. Reaction time of the “snake-best” neurons also outstripped all others, firing 15 to 25 milliseconds faster than neurons responsive to other stimuli. The simple shapes were the least provocative. The neuronal runner-up to “snake-best” were “face-best” neurons, with about 28% responding to angry monkey faces.1
“The results show that the brain has special neural circuits to detect snakes,” says neurobiologist Hisao Nishijo, “and this suggests that the neural circuits to detect snakes have been genetically encoded.”2
Snake Detection Theory, proposed in 2006 by University of California Davis anthropologist Lynne Isbell, was later publicized in her book The Fruit, the Tree and the Serpent: Why We See So Well. Isbell says that constricting snakes were “the first and most persistent predators”3 faced by early rodent-like mammals as they evolved together 100 million years ago. A second “pulse of selective pressure” from the evolution of venomous snakes exerted itself 40 million years later and favored the survival of early primates with superior vision. While snakes are a threat to burrowing ground-based animals as well as primates, those animals often rely on other senses to detect danger. Isbell maintains that the forward-facing eyes of primates and the devotion of a large portion of the primate brain to vision evolved to deal with this danger, making snakes the primordial shaper of the human brain.
Speaking of the chance to team up with neurobiologists to look for evidence to support her theory, Isbell says, “I don't do neuroscience and they don't do evolution, but we can put our brains together and I think it brings a wider perspective to neuroscience and new insights for evolution.”2 The part of the monkey brain selected for study has no corresponding homologous structure for processing visual images in non-primates and is therefore considered a structure that evolved only in primates. “We're finding results consistent with the idea that snakes have exerted strong selective pressure on primates,” she adds. “I don't see another way to explain the sensitivity of these neurons to snakes except through an evolutionary path.”2
Vision is the primary way primates evaluate their environments, relying more heavily on sight than on any other sense. The position and structure of primate eyes as well as “complex and energetically costly neural components”1 devoted to vision “demands an adaptive explanation,”1 Isbell, Nishijo, and colleagues write. The team hypothesized that “primate-specific regions of the pulvinar evolved in part to assist primates in detecting and thus avoiding snakes.”1 They declare the high proportion of snake-sensitive neurons in the pulvinar as “evidence in support of the snake detection theory.”1
Not all primates have vision as sharp as that of the monkeys in the study. Isbell believes this variation is also the result of snake selection. She says that primates in Madagascar, with no endemic venomous snakes, only faced selective pressure from constrictors and therefore “don't have a fovea, a pit in the retina that allows our central vision to be very sharp so that we can see fine detail. So their visual acuity isn't the best. It's still better than other mammals’; it’s just not as good as ours.” She believes that venomous snakes in Madagascar would have prompted them to evolve better vision. Seeing snakes is an important survival skill for primates, Isbell explains. “It's important to be able to see them before they see us so we can stop in time to do that acrobatic jumping away.”
Isbell cautions that their study addressed neuronal responses but not the emotions associated with those responses. Fear may be a subject for another day. “People want to connect what we’ve done with fear and phobias, but in fact we haven't addressed emotion in this study,” Isbell says. “We've only looked at vision. So we've looked at the very first step. How does the image of the snake get into our brains to begin with?”
Vision-dependent animals—like these monkeys—that must contend in the wild with threats like snakes would naturally be more likely to survive if they are able to see snakes in time to react. Thus the development of a population of monkeys wired to see snakes as a threat and respond quickly makes sense in terms of ordinary selection pressures on the species. But natural selection acting on a species of monkeys has nothing to do with millions of years of mammalian and primate evolution.
The only subjects of this study were one living species of monkey. All conclusions about the monkeys’ evolutionary past were mere speculation based on the unverifiable and unobservable presumptions of molecules-to-man evolution. Rodent-like mammals and snakes appear at about the same place in the fossil record. The authors interpret the fossil record in accord with worldview-based biases, accepting without question the idea that mammals evolved from other kinds of animals millions of years ago at the same time as snakes. The fossil record, however, is not the record of when various forms of life evolved. Much of the fossil record, rather, is the record of the order in which organisms were catastrophically buried as the global Flood of Noah’s day, less than 4,500 years ago, overwhelmed various habitats.
Thus the authors have learned something interesting about the brains of macaques. Perhaps God equipped monkeys with the neurological equipment to recognize dangers in the world cursed by mankind’s sin long ago. In any case, however, the discovery is consistent with a possible history of a fairly specific selection pressure on macaques. It can tell us nothing about the evolutionary history of mammals, primates, or people. No biologists—not even neurobiologists or behavioral ecologists like those who did this research—have ever observed animals evolving into new, more complex kinds of animals.
What we observe in biology is consistent with the Scriptural truth that God created all kinds of animals as well as the first two human beings to reproduce and vary only within their created kinds. He did so during the space of two days about 6,000 years ago, according to the Bible. Primates generally have good visual acuity and brains equipped to deal with the visual images their eyes collect, not because they had to evolve that anatomy, but because that’s how God designed!
| 342,999
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Parallax, September 2016
Dear rangefinder enthusiasts and classic photographers,
Thank you for allowing us to share some news and random thoughts with you.
The Medium (Format) is the Message at Photokina 2016. (Our Apologies to Marshall McLuhan)
Gear and gadgets, weirdly wireless and Wi-Fi, megapixels and Wiener schnitzel (really), even cameras that fly, but what stole the show was the reemergence of medium-format photography. Both the Hasselblad X1D and the Fujifilm GFX attracted big crowds and bona fide interest. Taking a page from Apple’s playbook, Leica decided not to announce any significant products. “Ve doo our own events,” Herr Anonymous Hun lectured. “Ztay tuned!” By the winks and nods I saw, at very least it looks like we can expect that a new T series camera and lenses will be announced in 2016.
The most interesting part of Photokina was spending time at the image gallery which, for the third time in a row, was produced and sponsored by the good folks of Leica Camera AG. No less than twenty shows covering an area upwards of 20,000 ft2 were expertly produced, meticulously curated and beautifully lit. Featured were the likes of Kurt Hutton, an oft-overlooked early Leica master documenting life in Old Europe, and Jens Umbach, whose Afghanistan project of landscapes and portraits (reminiscent of the work of Richard Alvedon), provoked deep emotions and considerable debate about Afghanistan and how it has changed and continues to change life both within its borders and around the world.
I was happy to see that a good friend, Ara Güler, was elected to the Leica Hall of Fame. His exhibition, alongside that of Alex Webb, brought back the halcyon days of photojournalism. Others that impressed me were Jacob De Boer, a young Canadian from Toronto whose black-and-white photos on the origins of coffee were gracious and dignified, and Roger Ballen, whose portraits of society’s outcasts are certain to move even the most hardened soul.
Then there was Bruce Gilden, whose fascination with people on the margin and their appearance, seems never-ending. Highly technical, flash lit and shot with a Leica S, I felt as if I had stumbled into a clinical dermatologist’s presentation of unusual teenage and geriatric acne. He calls his show American Made, but for this person who has travelled throughout the U.S.A. for the past 40 years, I can honestly claim to have never encountered the likes of his subjects. I would like to ask this capable Magnum photographer, “Why?”, and “To what purpose?”
The Homecoming! Return to Wetzlar
As I am prone to do, after Photokina I made my way to Wetzlar. An early morning stroll through the old town and along the Lahn River provided insight into Oskar Barnack and his diminutive creation. It is not only to his genius that we owe the Leica camera, but also to the topography of Wetzlar, whose steep hills on both sides of the river must have inspired Barnack on his brisk walks to work.
This trip I took full advantage of an invitation to visit Leitz Park. Intended as a symbol of Leica’s enduring ingenuity and excellence, it is much more than that. There is a tangible sense of renewed pride and purpose in the venerable camera maker. Roland Elbert, a production process engineer with Leica both in Germany and Midland, Ontario, served as my gracious host. He possesses encyclopedic knowledge and, most impressive after 30 years with the company, passion and dedication which are contagious. He could not be happier coming back to Wetzlar from Solms, if only because it puts him within bicycling distance from home.
It is hard to grasp that this German camera manufacturer which changed the world of photography, has no more than 350 workers manufacturing and assembling gear at the facility. There are a few hundred more in administration and marketing, but it is the engineers and craftsmen who are actually responsible for the M, Q, S, SL, X and soon-to-be new T cameras, as well as over 50 different lenses with each element painstakingly ground, polished, coated and hand assembled by people who share the same commitment as Roland Elbert.
In recent times, the M8 and M9 are credited with resuscitating Leica and securing its future for a few years to come. I prefer to look to the arrival of Andreas Kaufmann in 2004, as the decisive moment that made this company what it is today. His vision and dedication to producing fine cameras and lenses while untiringly promoting photography, led directly to the company being in a much better place than it has ever been. Great photographers capture life the way they see it. Today, Leica once again plays an indispensable role in achieving this.
Dr. Andreas Kaufmann
At Leitz Park, Kaufmann’s imagination and sharp focus were instrumental in the creation of a facility that is a marvel of design, beauty and function, where the flow is as seamless and natural as New York’s Guggenheim Museum. It’s a Zen-like space with pleasing shapes and colours and, most importantly, it has soul!
To see Leica’s history and all the cameras on display is something to behold, but the jewel in the crown is the prominent, permanent space exhibiting work of old and new Leica masters as well as a beautifully lit gallery dedicated to Leica’s Hall of Fame photographers.
Oh, let’s not forget the Leica boutique, with a place to expose photographs, a library filled with photo books and a spot to enjoy a coffee. Hmmm…Where have I seen this before?
Rhine Rambler.
***
Always looking forward to your comments.
Photographically yours,
Jean Bardaji and Daniel Wiener
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Dominican Republic at the 1980 Summer Games: Previous Summer Games ▪ Next Summer Games
Host City: Moskva, Soviet Union
Date Started: July 29, 1980
Date Finished: July 29, 1980
Events for Dominican Republic: 1
Participants for Dominican Republic: 1 (1 man and 0 women)
Most Medals for Dominican Republic (Athlete): No medalists
Go to the Weightlifting at the 1980 Moskva Summer Games page
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2032 3388 A Santella /wp-content/uploads/2017/05/wormguides2_banner_5_2017-300x115.png A Santella2018-05-23 21:08:332018-05-23 21:08:33WormGUIDES Board Meeting May 2018
WormGUIDES Board Meeting May 2018
The WormGUIDES advisory board met on May 18th 2018 at Sloan Kettering Cancer Center along with PIs graduate students and postdocs to discuss the new WormGUIDES atlas release and other updates. Thanks to all the board members for their valuable feedback.
Post meeting discussion over lunch.
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I WRITE to make you aware of a young persons’ initiative that is currently taking place at George Street in Hucknall, where young people work with local tenants and residents to carry out litter picks for the area.
As part of this, Ashfield Homes Ltd have recently worked with the local tenants and residents association in the George Street area with regards to a poster campaign for the area. Whereas young people have been designing ‘No Litter’ and ‘Keep George Street Clean’ posters.
This has now been judged by myself and Cath Garrett, who is a local tenant and an active voice within the community.
The winners will be provided with a voucher and an Ashfield Homes Ltd football. These will be presented at the group’s annual general meeting at George Street Club, Hucknall next Monday (October 17).
This is a great initiative.
KEVIN HORNSBY,
Tenancy services
manager,
Ashfield Homes Ltd.
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TITLE: Cauchy's Theorem, Stokes' Theorem, de Rham Cohomology
QUESTION [7 upvotes]: I've been struggling these last couple of days to see the connection, if at all there is one, between the following facts:
For holomorphic functions $f$, $\mathrm{d}(f(z)\mathrm{d}z) = 0$.
In a simply connected domain, a holomorphic function has a primitive, i.e. there exists a function $g$ defined over this domain such that $g'=f$.
The de Rham Cohomology "measures the failure of closed forms to be exact".
Trying to convince myself that Cauchy's theorem is somehow geometrically intuitive led me to the one-line proof where point 1 above is combined with Stoke's theorem, which in turn has led me to wonder if there was something going on at the level of differential forms over $\mathbb{C}$. I apologise if the question is unclear; as I said, I have the feeling like there's a revelation about holomorphic functions dancing just out of my reach.
REPLY [11 votes]: There are easy and hard things to see here. Let me see if I can help.
Complex valued differential forms First of all, throughout this answer, I'll want to work with differential forms that take complex values. So a differential $k$-form, at each point of $\mathbb{C}$, will take as input $k$ tangent vectors and output a complex number. The formalism is exactly the same as the real-valued case. So, if $z=x+iy$, we literally have the equation $dz = dx + i dy$, meaning that if $z$ is the coordinate function on $\mathbb{C}$, with $x$ the real part of $z$ and $y$ the imaginary part of $z$, then $dz$, $dx$ and $dy$ are one forms which obey the relation $dz=dx+i dy$.
For any smooth function $f : \mathbb{C} \to \mathbb{C}$, we have $df = \tfrac{\partial f}{\partial z} dz + \tfrac{\partial f}{\partial \overline{z}} d\overline{z}$, where
$$\frac{\partial f}{\partial z} = \frac{1}{2} \left( \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y} \right) \ \mbox{and} \ \frac{\partial f}{\partial \overline{z}} = \frac{1}{2} \left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} \right).$$
The Cauchy-Riemann equations tell us that $f$ is holomorphic if and only if $\tfrac{\partial f}{\partial x} + i \tfrac{\partial f}{\partial y}=0$, so $f$ is holomorphic if and only if $\overline{\partial}(f)=0$. In that case, $df = \partial f$.
If $g$ is holomorphic, then $d \left( g\ dz \right) = dg \wedge dz = g'(z) dz \wedge dz = 0$.
So, what does this mean for de Rham cohomology of open subsets of $\mathbb{C}$? Let $g(z)$ be holomorphic. From the last paragraph above, $g dz$ is closed. By de Rham's theorem, $g dz$ will be exact if and only if $\oint_{\gamma} g=0$ for every closed path $\gamma$.
Now, all that de Rham's theorem gives us is that $g dz = df$ for some smooth function $f$. But then we have $g dz = \tfrac{\partial f}{\partial z} dz + \tfrac{\partial f}{\partial \overline{z}} d \overline{z}$ so, comparing coefficients of $d \overline{z}$ on both sides, $\tfrac{\partial f}{\partial \overline{z}} =0$ and $f$ will actually be holomorphic.
So just using de Rham's theorem for (complex valued) $1$-forms, and good notation, immediately tells us the statement for holomorphic functions.
A much harder issue A much harder and more interesting question is whether the holomorphic analogue of the de Rham complex will compute cohomology. For some open region $U$ of $\mathbb{C}^n$, let $\Omega^p$ be the $p$-forms of the form $\sum g_{i_1 i_2 \cdots i_p}(z_1, \ldots, z_n) dz_{i_1} \wedge \cdots \wedge dz_{i_p}$, with $g_{i_1 \cdots i_p}$ holomorphic functions. Take the cohomology of the complex $0 \to \Omega^0 \to \Omega^1 \to \cdots \to \Omega^n \to 0$. There is always a map from this cohomology to de Rham, because every class is represented by a closed form, and we can consider that form as a de Rham class. But will that map be an isomorphism? (In the first draft, I called this Dolbeault cohomology, but that's something else. No deep reason for the error, I just wasn't thinking.)
In general, the answer is no! For example, $\mathbb{C}^2 \setminus \{ (0,0) \}$ retracts onto the $3$-sphere, so it has nontrivial $H^3$, but the holomorphic complex only goes up to degree $2$.
There is also a more subtle possible issue -- there are complex $3$-folds with holomorphic $3$-forms which are exact but are not the differential of a holomorphic $2$-form; see Cordero, Fernandez and Gray "The Frölicher spectral sequence and complex compact nilmanifolds".
But for open subsets of $\mathbb{C}$, the answer is yes! In general, the answer is yes for Stein spaces. If you are looking for motivation to get into the sort of harder multivariable complex analysis where the Stein condition comes up, this is a great starting point.
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State of the Union: Trump vows to stop 'endless wars'
Episode 6: Flavours of Iraq
‘Flavours of Iraq’ chronicles a country at war
France will remain 'militarily engaged' in Middle East through 2019
Iraq tracks down last IS group fighters despite tensions on ground
"Flavours of Iraq", episode 1: Apricot Ice Cream
From the top stories of 2018 to bittersweet memories of Iraq, FRANCE 24's week in review
Flavours of Iraq: Feurat Alani's bittersweet memories
Trump visits Iraq ‘not a moment too soon’ leaving behind US’s controversial shutdown
US not global 'policeman', Trump says on first visit to troops in Iraq
Yazidi rape victims face isolation after unwanted pregnancies
Kurdish-led forces retake IS group hub in east Syria
World must 'protect' Yazidis, Murad says in Nobel acceptance speech
Iraq's Anbar rebuilds, one year after defeat of IS group
Cellist Karim Wasfi hopes to breathe new life into Mosul with his music
Music in Mosul: Meet Karim Wasfi, a musician breathing new life into the city
Flavours of Iraq: A colourful look at the country's troubled past
'Age old cities' exhibition: 'It's part of our history that's been destroyed'
Rapping with reason: Mona Haydar's music tackles racism and Islamophobia
US will allow eight importers to keep buying Iranian oil
The risk of standing out: Sudden deaths of prominent Iraqi women raise questions
France to take in 100 Yazidi women stranded in Iraqi Kurdistan
| 56,388
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\section{Numerical Methods for GHNA}
\label{sec:methods}
\subsection{Approximate GHNA}
The GHNA method can quickly become numerically unstable.
This problem arises from the transformation formulas \eqref{eq:transform}
for the construction of a scaled all-pass error transfer function.
It is easy to see that the diagonal matrix
$\Gamma = \check{\Sigma}^{2} - \varsigma_{r+1}^{2}I_{n_{\min}-k}$ can
lead to large numerical errors for small proper Hankel singular values in
further computations.
This happens if either the chosen value $\varsigma_{r+1}$ or the
remaining proper Hankel singular values in $\check{\Sigma}$ are very small.
One preventive measure was the usage of the descriptor system structure
\eqref{eq:hankelsys} to avoid unnecessary scaling by $\Gamma$.
In further considerations, only the case of too small remaining Hankel singular
values is treated.
Small proper Hankel singular values can arise from numerical errors during the
computation of the minimal realization.
Therefor, one approach to solve this problem is to compute a smaller balanced
truncation of the slow subsystem than the minimal realization such that
too small proper Hankel singular values are cut off.
In this case, an additional error is made since the balanced realization is only
an approximation of the original system.
To get a measure for the additional error, let $G_{b}$ be the computed balanced
truncation of order $n_{b}$ of the slow subsystem $G_{sp}$.
Then it has been shown in \cite{morGlo84} that in the Hankel semi-norm it holds
\begin{align} \label{eq:hankelbterror}
\lVert G_{sp} - G_{b} \rVert_{H} & \leq 2\sum\limits_{k = n_{b} + 1}^{n_{f}}
{\varsigma_{k}(G_{sp})},
\end{align}
with $n_{f}$ the order of the slow subsystem $G_{sp}$.
For the overall error, let $G = G_{sp} + P$ be the original descriptor system
and $\tilde{G} = G_{b} + P_{b}$ the balanced realization with $G_{b}$ of order
$n_{b}$.
The generalized Hankel-norm approximation is denoted by $\hat{G} = G_{h} +
P_{b}$, where the $r$-th order standard Hankel-norm approximation $G_{h}$ was
computed from the balanced realization $G_{b}$.
Using \eqref{eq:hankelbterror} one obtains
\begin{align} \label{eq:apprerror}
\begin{aligned}
\lVert G - \hat{G} \rVert_{H} & = \lVert G_{sp} + P - G_{h} - P_{b}
\rVert_{H}\\
& = \lVert G_{sp} - G_{h} \rVert_{H}\\
& = \lVert G_{sp} - G_{b} + G_{b} - G_{h} \rVert_{H} \\
& \leq \lVert G_{b} - G_{h} \rVert_{H} + \lVert G_{sp} - G_{b}
\rVert_{H} \\
& \leq \varsigma_{r + 1}(G_{b}) + 2\sum\limits_{k = n_{b} + 1}^{n_{f}}
{\varsigma_{k}(G_{sp})}.
\end{aligned}
\end{align}
Since balancing the system does not change the Hankel singular values, the
Hankel singular values of $G_{b}$ and $G_{sp}$ are also the proper
Hankel singular values of $G$.
The resulting error can be bounded by
\begin{align*}
\lVert G - \hat{G} \rVert_{H} & \leq \varsigma_{r + 1}(G) +
2\sum\limits_{k = n_{b} + 1}^{n_{f}}{\varsigma_{k}(G)}.
\end{align*}
Concerning the $\mathcal{H}_{\infty}$-norm, the approach in \eqref{eq:apprerror}
can be used to get
\begin{align*}
\lVert G - \hat{G} \rVert_{\mathcal{H}_{\infty}} & \leq
2\sum\limits_{k = r + 1}^{n_{f}}{\varsigma_{k}(G)},
\end{align*}
which is the same error bound as for the exact method.
This approximate version of the GHNA takes advantage of the use of the GBT(SR)
method in form of the adaptive choice of the order $n_{b}$.
It is possible to choose the order $n_{b}$ with respect to the proper Hankel
singular value $\varsigma_{r + 1}$ such that
\begin{align*}
2\sum\limits_{k = n_{b} + 1}^{n_{f}}{\varsigma_{k}(G)} & \ll
\varsigma_{r + 1}(G).
\end{align*}
In this case, the resulting additional error becomes negligible small concerning
the original Hankel semi-norm error.
But the corresponding matrix $\Gamma$ leads to a better conditioned problem.
The algorithmic adjustments in the implementation of the GHNA method are small,
since only the truncation of non-zero proper Hankel singular values has to be
allowed in the generalized balanced truncation method.
In this case, the $\Sigma_{2}$ term in \eqref{eq:svd1} with the undesired proper
Hankel singular values is not zero and only the matrices $U_{1}$, $\Sigma_{1}$,
and $V_{1}$ are used for further computations.
Another advantage of the approximate algorithm can be found in the
computation of the balanced truncation.
The GBT(SR) method needs to scale the transformation matrices \eqref{eq:bttrans}
using the inverse remaining Hankel singular values which is more accurate if the
small proper Hankel singular values are truncated.
Also in the sense of computational costs, this approximate method has
advantages.
The further steps of the algorithm, i.e., the all-pass transformation and
additive decomposition, are extremely costly for large matrices in terms of
computational time and memory usage.
Therefor, it is advantageous to already have a small balanced realization
for the further computations.
\subsection{Application to Sparse Systems}
\label{sec:sparse}
A frequently appearing case in practice is the model reduction of large-scale
sparse descriptor systems.
In this case, the system matrices $E$ and $A$ from the descriptor system
\eqref{eq:desc} are in a large-scale sparse form, i.e., the dimension $n$ is
large, the matrices can be stored using $\mathcal{O}(n)$ memory and the
matrix-vector multiplication can be computed in $\mathcal{O}(n)$ effort.
Often such matrices result from the discretization of partial differential
equations.
The transformation into a balanced realization does not preserve the
sparsity of the system matrices.
Therefor, the GHNA method can only be adapted to sparse systems in the first two
steps.
This concerns the computation of the solutions of the generalized projected
Lyapunov equations \eqref{eq:gcalec}--\eqref{eq:gdaleo}.
It has been observed that the eigenvalues of the symmetric positive semidefinite
solutions of Lyapunov equations with low-rank right-hand sides generally decay
rapidly.
The same result holds for the generalized projected Lyapunov equations
\cite{Sty08}.
Therefor, the system Gramians can be approximated by low-rank Cholesky
factorizations, e.g., $\mathcal{G}_{pc} \approx Z_{pc}Z_{pc}^{T}$ with $Z_{pc}
\in \mathbb{R}^{n \times k}$ and $k \ll n$.
For the proper system Gramians, the computation is done by adapting existing
low-rank methods, e.g., Krylov subspace methods or low-rank ADI methods.
In this case, the right-hand side has to be replaced by the projected form
from the Lyapunov equations \eqref{eq:gcalec}, \eqref{eq:gcaleo}.
Additionally, it is recommended to project the solution back into the
corresponding subspace after some steps of the methods due to a drift-off effect.
In contrast to this, for the improper system Gramians full-rank factorizations
can be constructed explicitly such that $G_{ic} = Z_{ic}Z_{ic}^{T}$ and
$G_{io} = Z_{io}Z_{io}^{T}$, with
\begin{align*}
Z_{ic} & =\begin{bmatrix} Q_{r}A^{-1}B, & A^{-1}EQ_{r}A^{-1}B, &
\ldots, & (A^{-1}E)^{\nu - 1}Q_{r}A^{-1}B \end{bmatrix},\\
Z_{io} & = \begin{bmatrix} Q_{\ell}^{T}A^{-T}C^{T}, & A^{-T}E^{T}
Q_{\ell}^{T}A^{-T}C^{T}, & \ldots, & (A^{-T}E^{T})^{\nu - 1}
Q_{\ell}^{T}A^{-T}C^{T} \end{bmatrix};
\end{align*}
see \cite{Sty08} for more details.
Thereby, the size of the full-rank factorizations is bounded by the number of
inputs $m$ or outputs $p$ times the system's index $\nu$.
This corresponds to the overall bound of the non-zero improper Hankel singular
values \eqref{eq:impbound}.
Still for using these methods, the spectral projections $P_{\ell}$, $P_{r}$,
$Q_{\ell}$ and $Q_{r}$ have to be computed. But for many problems, these spectral
projections can be applied by exploiting the special structure of the
problem; see \cite{Sty08} for some examples.
\subsection{The Projection-Free Approach}
\label{sec:projfree}
In case of unstructured problems, there are no explicit construction formulas for
the spectral projectors $P_{\ell}$, $P_{r}$, $Q_{\ell}$ and $Q_{r}$, so they have
to be explicitly computed for the use in the generalized projected Lyapunov
equations \eqref{eq:gcalec}--\eqref{eq:gdaleo}.
But as for the GBT(SR) method, an alternative approach to the
use of spectral projectors can be given; see \cite{morSty04}.
As already used in the GHNA algorithm, the GBT method can be interpreted as a
decoupling of the original system into the slow and fast subsystems and the
individual reduction of both.
Therefor, consider the following generalized block triangular form.
There are orthogonal matrices $U, V \in \mathbb{R}^{n \times n}$ such that
\begin{align*}
\begin{aligned}
E & = V \begin{bmatrix} E_{f} & E_{u} \\ 0 & E_{\infty} \end{bmatrix}
U^{T} & \text{and} && A & = V \begin{bmatrix} A_{f} & A_{u} \\ 0 &
A_{\infty}\end{bmatrix} U^{T},
\end{aligned}
\end{align*}
where the matrix pencil $\lambda E_{f} - A_{f}$ contains all the finite
eigenvalues of $\lambda E - A$ and the matrix pencil $\lambda E_{\infty} -
A_{\infty}$ has only infinite eigenvalues.
For the computation of a block diagonalization of the system, the coupled
Sylvester equations
\begin{align} \label{eq:coupsylv}
E_{f}Y - ZE_{\infty} & = -{E_{u}},\\
A_{f}Y - ZA_{\infty} & = -{A_{u}},
\end{align}
have to be solved for $Y$ and $Z$; see \cite{morBenQQ05}.
Using all of these matrices for the restricted system equivalence transformation
\begin{align*}
\begin{aligned}
W_{dec} & = V \begin{bmatrix} I_{n_{f}} & 0 \\ -Z^{T} & I_{n_{\infty}}
\end{bmatrix}, &
T_{dec} & = U \begin{bmatrix} I_{n_{f}} & Y \\ 0 & I_{n_{\infty}}
\end{bmatrix}
\end{aligned}
\end{align*}
of the original descriptor system \eqref{eq:desc}, one obtains
\begin{align} \label{eq:diagdesc}
\begin{aligned}
\begin{bmatrix} E_{f} & 0 \\ 0 & E_{\infty} \end{bmatrix}
\dot{\tilde{x}}(t) & = \begin{bmatrix} A_{f} & 0 \\ 0 & A_{\infty}
\end{bmatrix}\tilde{x}(t) + \begin{bmatrix} B_{f} \\ B_{\infty}
\end{bmatrix}u(t), \\
y(t) & = \begin{bmatrix} C_{f} & C_{\infty} \end{bmatrix}\tilde{x}(t)
+ Du(t),
\end{aligned}
\end{align}
where the remaining matrices are constructed as
\begin{align} \label{eq:blkright}
\begin{aligned}
V^{T}B & = \begin{bmatrix} B_{u} \\ B_{\infty} \end{bmatrix}, &
B_{f} & = B_{u} - ZB_{\infty},\\
CU & = \begin{bmatrix} C_{f} \\ C_{u} \end{bmatrix}, &
C_{\infty} & = C_{f}Y + C_{u}.
\end{aligned}
\end{align}
Obviously, the realization in \eqref{eq:diagdesc} decouples into the fast and
slow subsystems of \eqref{eq:desc}.
Since the spectral projectors of the subsystems are identity matrices, the
corresponding Lyapunov equations \eqref{eq:gcalec}--\eqref{eq:gdaleo} simplify to
\begin{align*}
E_{f}X_{pc}A_{f}^{T} + A_{f}X_{pc}E_{f}^{T} + B_{f}B_{f}^{T} & = 0,\\
E_{f}^{T}X_{po}A_{f} + A_{f}^{T}X_{po}E_{f} + C_{f}^{T}C_{f} & = 0,
\end{align*}
for the slow subsystem and
\begin{align*}
A_{\infty}X_{ic}A_{\infty}^{T} - E_{\infty}X_{ic}E_{\infty}^{T} -
B_{\infty}B_{\infty}^{T} & = 0,\\
A_{\infty}^{T}X_{io}A_{\infty} - E_{\infty}^{T}X_{io}E_{\infty} -
C_{\infty}^{T}C_{\infty} & = 0,
\end{align*}
for the fast subsystem.
These Lyapunov equations can be computed without the spectral projections.
The matrices $X_{pc}$ and $X_{po}$ correspond to the parts of the proper
controllability and observability Gramians, which contain the potentially
non-zero proper Hankel singular values.
The same holds for $X_{ic}$, $X_{io}$ and the improper system Gramians.
For the rest of the algorithm, only the transformations have to be restricted to
the subsystems.
The projection-free approach is implemented in the version 3.0 of the MORLAB
toolbox; see \cite{morBenW17a}.
In this special implementation, the block diagonalization of the system is done
by using a block transformation approach based on the following generalization
of Theorem 4.1 from \cite{KagV92}.
\begin{theorem} \label{thm:adtf}
Let $\Gamma \subset \mathbb{C}$ be a region in the complex plane which
contains $n_{1}$ eigenvalues of the matrix pencil $\lambda E - A$.
Let $Q, Z \in \mathbb{R}^{n \times n}$ be orthogonal matrices that
transform the matrix pencil $\lambda E - A$ into the upper block triangular
form:
\begin{align*}
Q^{T}(\lambda E - A)Z = \begin{bmatrix}Q_{1}^{T} \\ Q_{2}^{T}\end{bmatrix}
(\lambda E - A)\begin{bmatrix}Z_{1}, Z_{2} \end{bmatrix} =
\begin{bmatrix}\lambda E_{11}^{(1)} - A_{11}^{(1)} &
\lambda E_{12}^{(1)} - A_{12}^{(1)} \\ 0 & \lambda E_{22}^{(1)} -
A_{22}^{(1)}\end{bmatrix},
\end{align*}
with $\Lambda(A_{11}^{(1)}, E_{11}^{(1)}) \subseteq \Gamma$ and
$\Lambda(A_{11}^{(1)}, E_{11}^{(1)}) \cap
\Lambda(A_{22}^{(1)}, E_{22}^{(1)}) = \emptyset$.
Similarly, let $U, V \in \mathbb{R}^{n \times n}$ be orthogonal matrices that
transform the matrix pencil $\lambda E - A$ into the upper block triangular
form:
\begin{align*}
U^{T}(\lambda E - A)V = \begin{bmatrix}U_{1}^{T} \\
U_{2}^{T}\end{bmatrix}(\lambda E - A)\begin{bmatrix}V_{1}, V_{2}
\end{bmatrix} = \begin{bmatrix}\lambda E_{11}^{(2)} - A_{11}^{(2)} &
\lambda E_{12}^{(2)} - A_{12}^{(2)} \\ 0 & \lambda E_{22}^{(2)} -
A_{22}^{(2)}\end{bmatrix},
\end{align*}
with $\Lambda(A_{22}^{(2)}, E_{22}^{(2)}) \subseteq \Gamma$ and
$\Lambda(A_{11}^{(2)}, E_{11}^{(2)}) \cap \Lambda(A_{22}^{(2)},
E_{22}^{(2)}) = \emptyset$.
Then
\begin{align*}
\begin{aligned}
X & = \begin{bmatrix}U_{2}, & Q_{2} \end{bmatrix} && \mathrm{and} &
Y & = \begin{bmatrix}Z_{1}, & V_{1} \end{bmatrix}
\end{aligned}
\end{align*}
are transformation matrices, such that $X^{T}(\lambda E - A)Y$ has a block
diagonal structure where the upper block contains the $n_{1}$ eigenvalues
lying inside $\Gamma$ and the lower block has the remaining $n - n_{1}$
eigenvalues of $\lambda E - A$ outside of $\Gamma$.
\end{theorem}
\begin{proof}
The proof can be found in \cite[Section~5.2]{morWer16}.
\end{proof}
In contrast to the approach above, it is not necessary to compute the solution
of the coupled Sylvester equations and, due to the block orthogonal structure
of the transformation matrices, the right-hand sides are usually better
conditioned than \eqref{eq:blkright}.
In MORLAB, the right matrix pencil disk function method is used to generate the
block transformation matrices, see \cite{morWer16} for more details on the
implementation.
Additionally, Theorem \ref{thm:adtf} can be used to compute the additive
decomposition in step 9 of Algorithm \ref{alg:ghna} by separating the eigenvalues
with negative and positive real-parts.
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Table of Contents[Hide][Show]
When you think about how to shrink pores naturally, images of face masks and scrubs may come to mind. With so many remedies out there, it’s hard to know what’s actually effective, and what’s hype.
Firstly, we need to look at what causes enlarged pores so we can get to the root cause and have symptom relief.
What Are Pores?
Sweat, pimples, blackheads…our pores make a lot of things. When people talk about reducing the size of pores though they mean sebaceous glands, not sweat glands.
Our sebaceous glands (aka skin pores) make sebum. This oily sebum helps moisturize skin, keeping it soft and supple. Different factors can make our pores seem larger and more obvious.
What Causes Enlarged Pores?
There’s a whole lot of speculation, but here’s what we do know. Factors like age, ethnicity, and health play a role in pore size.
Race
A 2018 article “Facial Skin Pores: A Multiethnic Study,” looked at the differences between women of different races to compare pore size. Chinese and Japanese women have significantly lower pore density. These women had 5-8 times less pore density than any other ethnic group.
By far Indian and Brazilian women had the most noticeable pores. Caucasian women came in slightly behind the Indian women. However, Asian women by far won the least noticeable pores award.
Age
It’s often said pore size increases with age, but that’s not exactly true. While pore density does vary some with age, the differences are usually insignificant. Our ethnicity and genetics play a much bigger role in pore size than our age.
However, skin damage does make pores more visible, and skin damage increases with age.
A 2017 study in Skin Research and Technology notes that while women don’t get larger pores or more pores with age, the shape does change from round to oval.
Hormone Imbalance
Hormones control many functions in the body, from fertility, to growth, to skin changes. As we age, decreasing estrogen results in skin damage and thinning. The epidermal layer gets thinner and dead skin cells stick around longer.
This doesn’t just affect older women though. Hormone imbalance contributes to early signs of aging skin, acne, and other skin issues that make pores more noticeable.
A Good and a Bad Thing
Another hormone, insulin like growth factor (IGF-1), is key to a healthy body. However, too much of a good thing can be harmful. Too much IGF-1 is linked to a shorter lifespan, cancer risk, and other serious issues.
According to a 2010 article in Archives of Dermatological Research, IGF-1 may also affect pores. An overabundance of IGF-1 can make facial pores more noticeable by affecting the skin’s epidermis.
Overall Health
We are what we eat and that includes our skin. Inflammatory foods contribute to hormone imbalance and oxidative damage. Both can wreck skin health. Free radical damage to skin breaks down the collagen and elastin we need for smooth, healthy skin.
Choosing healthy, whole foods not only makes us feel better, but our skin looks better too.
Can You Really Shrink Your Pores?
Yes, and no. Pore size is largely determined by genetics, but what we eat, drink and put on our skin affects our skin’s health. While we can’t technically shrink pores, we can tighten, tone, and reduce the appearance of your pores.
Let’s take a look at some of the commonly recommended treatments for shrinking pores.
Glycolic Acid and Chemical Peels
Conventional solutions include chemical creams to reduce the appearance of large pores. Chemical peels are another option. And yes, it’s exactly like iit sounds. Chemical acids are applied to the face to burn off the top layer(s) of skin.
Alpha-hydroxy peels, like lactic acid and glycolic acid, are common choices. Salicylic acid, often recommended for acne, also helps to clear out pores to diminish their appearance.
How Do Chemical Peels Work?
Glycolic acid chemically strips the skin. It works by deep cleaning pores, removing blackheads, fighting acne, evening out skin tone, brightening skin tone and smoothing out rough skin. All of these benefits help reduce the look of enlarged pores.
Peels can range from milder at home masks, to intense and risky procedures at the dermatologist’s office. While they might reduce the appearance of pores, synthetic chemical peels can also cause redness, burning, and irritation. More severe side effects include scarring, skin infections, and organ damage.
Retinoids
Retinoids are another commonly recommended treatment for large pores. While there’s evidence it is effective, it might not be the safest option. Retinoids are synthetic compounds derived from vitamin A. Retinoids (or Retinol) increase cell turnover for smoother skin and less noticeable pores.
However, there’s evidence retinol can cause cell damage, cell death, and may play a role in heart disease. Too much synthetic vitamin A can cause birth defects. Both Germany and Canada have restricted this chemicals use in skincare products.
Sunscreen
It’s well known that overexposure to UV rays can cause skin damage. It’s no surprise many skincare experts recommend sunscreen to reduce the pore’s appearance. There’s a little more to the story though.
Sunscreens with synthetic chemicals can do more harm than good. That’s one reason why I opt for mineral based sunscreens. My best tip for avoiding too much sun is to use spf clothing and seek out some shade.
It’s important to not see sunshine as the enemy though! We need some sun to make vitamin D for a healthy body, skin, and pores.
How to Shrink Pores Naturally
Aging, environmental pollution, and poor health choices cause a breakdown of collagen and elastin in the skin. Without these, skin becomes damaged and pores can appear enlarged. Pimples, blackheads and other impurities clog pores and also make them more visible. Shrinking pores naturally requires a few steps.
- Clear out pores
- Improve collagen, elastin and cell turnover.
- Change skin from the inside out with healthy foods and hormone balance.
These remedies for how to shrink face pores naturally do just that.
Detox Pores With a Mask
Clogged pores are full of gunk that make pores appear larger. Our skin is one way the body eliminates toxins so it’s important to keep pores clear and open. Bentonite clay, rhassoul clay, and charcoal are all fabulous at pulling impurities out of skin. Here are some of my favorite, simple detoxing face masks.
Too busy to make it?
Alitura clay mask contains a blend of clays, natural powders, and nutrients that go deep down into pores. Along with bentonite, green, and kaolin clay, it also has rhassoul clay. Recent research points to rhassoul clays ability to improve skin’s elasticity and unclog pores.
Egg White Face Mask
Eggs are great for breakfast, baking with, and… rubbing on your face? Yes! The proteins in egg whites are said to help tighten and plump skin to reduce the appearance of pores.
Here’s how to make a simple egg white face mask:
- Whisk an egg white from a free range, pastured egg until smooth. You can also use the whisk attachment on a mixer for faster results.
- Apply the foamy egg white to the face and allow it to soak in.
- After about 15 minutes gently wash the face.
Papaya Face Mask
Papaya has a naturally occurring enzyme, papain, that’s great for skin. Papain helps exfoliate dead skin, repairs skin, and reduces wrinkles by modulating collagen and elastic fibers in skin.
Papaya extract as a supplement has significant skin benefits too.
A 2016 study looked at papaya’s skin benefits. The group given papaya extract had healthier skin than the group given other antioxidants.
To make a papaya face mask:
- Mash the fruit and apply to the face. It can be combined with honey, yogurt, or other skin loving ingredients if you want.
- After 10-15 minutes, gently wash the papaya face mask off.
More Pore-Cleansing Face Mask Recipes
Wash Your Face
A daily cleanser can help prevent breakouts and blemishes that lead to more obvious pores. Here are a few recipes to get you started. (If time is short, the Alitura Pearl Cleanser is one of my favorites!)
Facial Steam
A facial steam helps clear congestion for easier breathing, but it also helps clear out dirt and impurities from pores. Adding herbs amps up the skin cleansing benefits. The steam gently opens pores and carries the herbal properties with it.
Astringent herbs help tighten and tone tissues. Here are some astringent herbs to add to your next pore cleansing facial steam:
- Green or black tea
- Blackberry leaf
- Red raspberry leaf
- Witch hazel
- Yarrow
- Rose
See this post for a step by step tutorial.
Can Apple Cider Vinegar Shrink Pores?
Apple cider vinegar is used for many things in the natural health world, including to benefit skin. Skin’s pH is naturally acidic and ACV can help protect skin’s acid mantle for a healthy pH balance. ACV is also thought to tone skin and reduce the appearance of pores.
To use apple cider vinegar as a toner, dilute vinegar with water, apply to a cotton ball and swab your face. I use 50% water and 50% vinegar, but some do better with a 25% vinegar and 75% water solution.
Pore-Shrinking Toners
Cold water is often used to “close pores” but the effects are temporary. Here are some natural toners to help minimize the appearance of pores.
Make it:
Buy it:
Alpha-hydroxy Acids (AHA)
Alpha-hydroxy acids are commonly used in conventional pore minimizing treatments. While synthetic chemical peels come with risks, AHAs are naturally found in certain foods and have definite skin benefits.
DIY recipes aren’t as strong as a chemical peel (thank goodness!), but help gently and naturally exfoliate skin. This probiotic face mask has yogurt with naturally occuring AHA.
Gently Exfoliate
Exfoliation is important to improve cell turnover and remove dead cells to reveal healthier, glowing skin.
Baking soda is gently exfoliating to skin, but ideally shouldn’t be used more than once a week. Baking soda has a high pH of about 9, while our skin is happiest at a pH of about 5 (or a little lower).
To make a baking soda scrub:
- Combine a few teaspoons of baking soda with enough water to make a paste and gently scrub skin.
- Wash the baking soda off with warm water.
More DIY Exfoliants
Different skin types, like sensitive skin, oily skin, dry skin, and acne-prone skin respond better to different facial care products. You can customize facial scrubs with various skin-loving essential oils (safely diluted of course). Here are plenty of options that you can customize to fit your needs.
Saunas for Healthy Skin
Sweating it out can improve skin tone and function. Saunas increase blood flow to the skin to deliver necessary nutrients. Those who use saunas on a regular basis retain moisture better and have less pore-clogging sebum. Saunas also help strengthen the skin barrier, which is necessary for minimizing pore appearance. Learn more about saunas here.
LLLT or Red Light Therapy
I use Low Level Laser Therapy (LLLT) to improve my health on many levels. Also known as red light therapy, LLLT benefits skin health too. Red light therapy helps improve collagen in the skin to reduce wrinkles and improve skin’s appearance. A 2018 study in Dermatologic Surgery also found it improves the appearance of pores (by up to 54.5 percent)!
Skin Care Routine for Healthy Pores
There are plenty of options when it comes to taking care of our skin and naturally shrinking pores. Having clean skin, using a moisturizer, and eating a healthy, whole foods diet will help our skin and pores be their healthiest!
Your turn! What are your favorite natural cleansers as well as tips and tricks for smaller pores?
Sources:
- Bertuccelli, G., Zerbinati, N., Marcellino, M., Nanda Kumar, N. S., He, F., Tsepakolenko, V., Cervi, J., Lorenzetti, A., & Marotta, F. (2016). Effect of a quality-controlled fermented nutraceutical on skin aging markers: An antioxidant-control, double-blind study. Experimental and therapeutic medicine, 11(3), 909–916.
- Environmental Working Group(N.D.). Retinol (Vitamin A). EWG’s Skin Deep.
- Farage, M. A., Miller, K. W., Elsner, P., & Maibach, H. I. (2013). Characteristics of the Aging Skin. Advances in wound care, 2(1), 5–10.
- Flament, F., Francois, G., Qiu, H., Ye, C., Hanaya, T., Batisse, D., Cointereau-Chardon, S., Seixas, M. D., Dal Belo, S. E., & Bazin, R. (2015). Facial skin pores: a multiethnic study. Clinical, cosmetic and investigational dermatology, 8, 85–93.
- Junnila, R. K., List, E. O., Berryman, D. E., Murrey, J. W., & Kopchick, J. J. (2013). The GH/IGF-1 axis in ageing and longevity. Nature reviews. Endocrinology, 9(6), 366–376.
- Kolen, R. (2013, Aug 14). Guide to Basic Herbal Actions. Mountain Rose Herbs.
- Kwon H., Choi S., Lee W., Jung J., Park G. (2018). Clinical and Histological Evaluations of Enlarged Facial Skin Pores After Low Energy Level Treatments With Fractional Carbon Dioxide Laser in Korean Patients. Dermatol Surg. (3):405-412. doi: 10.1097/DSS.0000000000001313.
- Lambers H., Piessens S., Bloem A., Pronk H., Finkel P. (2006). Natural skin surface pH is on average below 5, which is beneficial for its resident flora. Int J Cosmet Sci. (5):359-70. doi: 10.1111/j.1467-2494.2006.00344.x.
- Mayo Clinic (N.D.). Chemical Peel.
- Palmer, A. (2020, Jan 8). Is it Possible to Shrink Large Pores? Very Well Health.
- Shaiek, A., Flament, F., François, G., Lefebvre-Descamps, V., Barla, C., Vicic, M., Giron, F., Bazin, R. (2017). A new tool to quantify the geometrical characteristics of facial skin pores. Changes with age and a making-up procedure in Caucasian women. Advances in wound care, 2(1), 5–10.
- Sharad J. (2013). Glycolic acid peel therapy – a current review. Clinical, cosmetic and investigational dermatology, 6, 281–288.
- Sugiyama-Nakagiri, Y., Ohuchi, A., Hachiya, A., Kitahara, T. (2010). Involvement of IGF-1/IGFBP-3 signaling on the conspicuousness of facial pores. Arch Dermatol Res. 302(9):661-7. doi: 10.1007/s00403-010-1062-3.
- Surbhi, (2018, Sept 19). 7 benefits of Papaya for skin. Dermatocare.
- University of Leeds (N.D.). Three types of glands: Eccrine/merocrine Sweat Glands. The Histology Guide.
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Who’s the biggest fan of gift cards for special occasions like birthdays, anniversaries and Christmas?
If you guessed your spouse, children, grandchildren, co-workers, friends or business associates … you’d be wrong.
They may like them but the number one fan of gift cards is the retailers who sell them to you.
# 1 Thing You Should Know—Last year, gift card sales hit nearly $80 billion (that’s right, BILLION) and roughly ten percent of that amount, $8 BILLION was NEVER USED.
You would never knowingly pay for an item and leave the business without the product you purchased. But basically that’s what’s happens with a lot of gift cards.
It’s reported that last year Home Depot and Best Buy both added an additional $43 million to their bottom line because of unused gift cards.
# 2 Thing You Should Know—Your gift card becomes immediately worthless if the retailer files for bankruptcy. Gift cards are treated as loans to the company, not cash, so a bankrupt company can refuse to honor them.
Just ask The Sharper Image customers who lost a total of $62 million in gift cards from the retailer. Of course, you can now add Bennigans to that list.
# 3 Thing You Should Know—If you lose or misplace the card, you’re out of luck because there is no way to get a replacement. Kudos to Starbucks; they register their cards and a lost card can be replaced with only a slight fee.
Bottom line – give some thought to the recipient before you buy them a gift card. It may seem like the perfect gift but you want to be sure it’s their “fave” restaurant or they “love” to shop that department store before you bless them with a gift card.
A word or warning…when purchasing a card from a rack of cards like they offer in the grocery and pharmacies these days, choose one off the bottom of the stack. There are thieves who copy down the activation number from cards hanging on the rack and then use them to purchase goods online where it can’t be traced.BLOG COMMENTS POWERED BY DISQUS
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PNF Warm-ups With Young Athletes
By Wil Fleming
Ask coaches what their program should include and invariably the answer sounds like this “Strength, speed, agility, power and oh yeah warm-up“. The warm-up is always tossed in there, but not with much enthusiasm.
All too often our warm-ups occur in singular planes of motion, typically sagittal or frontal, and for certain joints this will not do. The hip and shoulder, in particular require motion that does not only go through these single planes, and in truth requires more than just the addition of motion through the transverse plane.
A great solution to this is to use PNF patterns of movement to truly warm-up the athlete. In using PNF patterns we are able to use patterns that efficiently recruit the most relevant muscle.
PNF or proprioceptive neuromuscular facilitation, is commonly thought of as only a type of stretching pattern done by athletic trainers but is actually an entire system of movement.
In the great book Supertraining, Mel Siff described PNF movement patterns in this way “The importance of these patterns cannot be overestimated, since they can enhance the effectiveness of any training session.”
While the unloaded movement of a “warm-up” cannot satisfy all the necessary pieces to be considered PNF the important foundations of PNF which must be considered are as follows.
-The motion must use spiral and diagonal movement patterns
-The motion must cross the sagittal midline of the body.
-The motion must recruit all movement patterns including, flexion/extension, abduction/adduction, and internal/external rotation.
To use the techniques of PNF in our warm-up we use a lunge matrix and corresponding “reaches”.
Lateral Hip Rotator Lunge w/ Contralateral Reach
Have the athlete stand perpendicular to a start line, flex at the hip and knee with the lead leg. First internally rotate at the hip, move towards external rotation with the lead hip as they step outward as far as possible. Once the lead foot reaches the ground they will raise their opposite arm overhead and come across the midline of the body to reach the instep of their lead leg, the young athletes should follow this movement with their eyes until completion.
Reverse Lunge w/ X Reach
Have the athlete make a reverse lunge movement (that part is simple). While in this split stance they should reach with one hand to their opposite front pocket, move this arm across the midline of the body to an overhead position and rotate the torso. Again the athlete should follow the movement of their arms with their eyes. Do the same movement with the opposite arm and then reverse lunge with the other leg.
These modifications on traditional lunges will add multi direction skill and a more complete neuromuscular warm-up to your young athletes programs.
| 285,909
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Explore Henson Kermit, Kermit Jim, and more!
Explore related topics
Jim Henson & Kermit
Awwww Kermit! Even though green is my favorite color, Kermit professes that it's not easy being green...
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You can email or call us with any questions you might have We are sorry but the shirts come in WHITE only All Sizes are available! The shirts are Pre shrunk Cotton Short Sleeve, Crew Cut Measurem
Wil Wheaton explains why being a nerd is awesome
@youngshizzle ♔
One of the best movies...ever! 'The Princess Bride' (1987) - The 2012 25th reunion photoshoot, L to R: Vizzini/Wallace Shawn; Miracle Max/Billy Crystal & Valerie/Carol Kane; Prince Humperdinck/Chris Sarandon; Westley, the Dread Pirate Roberts & the Man In Black/Cary Elwes & Buttercup, Princess of Florin/Robin Wright; Inigo Montoya/Mandy Pantinkin, Count Tyrone Rugen/Christopher Guest & a picture of Fezzik/André the Giant (who died in 1993)
These early 70s behind the scenes photos of Sesame Street are funny, beautiful, and honestly slightly morbid.
Thank you for accepting me into your neighborhood, Mr. Rogers…
Steve Martin. Everyone had his album "Wild and Crazy Guy." First of the "rock star" comedians.
| 145,945
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TITLE: Where does $\pi^2$ appear spontaneously within Physical Phenomenon and Mathematics Equations?
QUESTION [24 upvotes]: The term $\pi$ is found to appear in many equations and natural phenomenon; however my question is related to $\pi^2$.
While trying to figure out the reason for some $\pi^2$ terms appearing in certain equalities that I came across, I have a question. And the question is this:
In which all mathematics/physics equation or contexts does $\pi^2$ appear inherently?
-- and (now this second part is merely a follow up question that did not form part of the original query but added later) where that $\pi^2$ term can lend some interpretation of the underlying phenomenon, just like does $\pi$ whereby we can interpret (in most cases i.e.) that some type of circular ambulation in 1 dimension is involved??
As you can understand, the $\pi^2$ term is more complex and does not directly lend itself to an interpretation -- as opposed to $\pi$ which is very intuitive.
Thanks
REPLY [6 votes]: The $\text{Riemann Zeta function} \ \ \zeta(s)=\large \sum_{n=1}^{\infty} \frac{1}{n^s}$ is used in many branch of Science and mathematics. Replacing $s=2$, we have $ \zeta(2)=\sum \frac{1}{n^2}$. In $1735$ Leonard Euler showed that $$ \zeta(2)=\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{{\color{blue}{\pi^2}}}{6}.$$
This result leads to Number Theory and probability result as follows:
The $ \ {\color{blue}{ probability}} \ $ of two random number being $ \ {\color{blue}{ relatively \ \ prime}} \ $ is given by the following product over all primes
$$ \prod_{p}^{\infty} \left(1-\frac{1}{p^2} \right)=\left(\prod_{p}^{\infty}\frac{1}{1-p^{-2}} \right)^{-1}=\frac{1}{1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots}=\frac{1}{\zeta(2)}=\frac{6}{{\color{blue}{\pi^2}}} \approx 61 \%.$$
This is an interesting result where $\pi^2$ is involved.
| 105,169
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\begin{document}
\begin{center} {\Large\bf Boundedness of commutators generated by m-th Calder\'{o}n-Zygmund type
singular integrals and local Campanato functions on generalized local Morrey spaces}
\vspace{0.3cm}
\noindent{\large\bf HuiXia MO\footnote[1]{
Correspondence:huixiamo@bupt.edu.cn\\
2010 AMS Mathematics Subject
Classification: 42B20, 42B25}, Hongyang XUE}\\
School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China
\end{center}
\parskip 12pt
\noindent{\bf Abstract}\;\;{\footnotesize
Let $T_m$ be the $m$-th Calder\'{o}n-Zygmund type singular integral. In the paper, we consider the boundedness of
$T_m$ on the generalized product local Morrey spaces
$LM_{p_1,\varphi_1}^{\{x_0\}}\times LM_{p_2,\varphi_2}^{\{x_0\}}\times\dots\times LM_{p_m,\varphi_m}^{\{x_0\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also.}
\vspace{0.3cm}
{\bf Key words}\;\; m-th Calder\'{o}n-Zygmud type singular integral,
commutator, local Campanato function, generalized local Morrey space
\section{\bf Introduction}
In recent years, the multilinear singular integrals have been
attracting attention and great developments have been achieved (see
[1-11]). The study for the multilinear singular integrals is motivated
not only by a mere quest to generalize the theory of linear
operators but also by their natural appearance in analysis.
Meanwhile, the commutators generated by the multilinear singular integral
and BMO functions or Lipschitz functions also attract much
attention, since the commutator is more singular than the singular
integral operator itself.
Moreover, the classical Morrey space $M_{p,\lambda}$ were first introduced by Morrey in \cite{M} to study the local behavior of solutions to second order elliptic partial differential equations. In \cite{LL}, the authors studied the boundedess of the multilinear Calder\'{o}n-Zygmund singular integral on the classical Morrey space $M_{p,\lambda}.$ And, in \cite{BG}, the authors introduced the local generalized Morrey space $LM_{p,\varphi}^{\{x_{0}\}}$, and they also studied the boundedness of the homogeneous singular integrals with rough kernel on these spaces.
Motivated by the works of \cite{LL,BG}, we are going to
consider the boundedness of the multilinear Calder\'{o}n-Zygmund singular integral and its commutator on the local generalized Morrey space $LM_{p,\varphi}^{\{x_{0}\}}$
Now, let us give some related notations.
We are going to be working in $\mathbb{R}^{n}$. Let $m\in\mathbb{N}$
and $K(y_{0},y_{1},\dots,y_{m})$ be a function defined away from the
diagonal $y_{0}=y_{1}=,\dots,=y_{m}$ in $(\mathbb{R}^{n})^{m+1}.$
Let $T_m$ be a multilinear operator which was initially defined on
the m-fold product of Schwartz space $\mathcal{S}(\mathbb{R}^n)$ and
take its values in the space of tempered distributions
$\mathcal{S'}(\mathbb{R}^n)$ and such that for $K$, the integral
representation below is valid:
$$\begin{array}{cl}T_m(\vec{f})(x)=T_m(f_{1},\dots,f_{m})(x) =\dint_{(\mathbb{R}^{n})^{m}}K(x,y_{1},\dots ,y_{m})
f_{1}(y_1) \dots f_{m}(y_m)dy_1\dots dy_m,\end{array}\eqno{(1.1)}$$ whenever $f_{i},$ $i=1,
\dots ,m,$ are smooth functions with compact support and
$x\notin\cap_{i=1}^{m}\mbox{supp}f_{i}.$
Moreover, if the kernel $K$ satisfies the following size and smoothness estimates:
$$\begin{array}{cl}
|K(y_0,y_{1},\dots,y_{m})|\leq
\dfrac{C}{(\sum^{m}_{k,l=0}|y_{k}-y_{l}|)^{mn}},\end{array}\eqno{(1.2)}$$
for all $(y_0,y_{1},\dots,y_{m})\in(\mathbb{R}^{n})^{m+1}$ away from
the diagonal;
$$\begin{array}{cl}|K(y_0,\dots,y_i,\dots,y_{m})-K(y_0,\dots,y'_i,\dots,y_{m})|\leq\dfrac{C|y_i-y'_i|^{\epsilon}}
{(\sum^{m}_{k,l=0}|y_{k}-y_{l}|)^{mn+\epsilon}},\end{array}\eqno{(1.3)}$$
for some $C>0$ and $\epsilon>0,$ whenever $0\leq j\leq m$ and
$|y_i-y'_i|\leq 1/2\max_{0\leq k\leq m}|y_i-y_{k}|,$ then the kernel
is called a m-th Calder\'{o}n-Zygmund kernel and the collection of
such functions is denoted by $m-CZK(C,\epsilon)$. Let $T_m$ be as in
(1.1) with a $m-CZK(C,\epsilon)$ kernel, then $T_m$ is called a m-th
Calder\'{o}n-Zygmund type singular integral and the collection of
these operators is denoted by $m-CZO$.
Now, we define the commutators generated by the m-th multilinear Calder\'{o}n-Zygmund type singular integral as follows.
Let $\vec{b}=(b_{1},\dots,b_{m})$ be a finite family of
locally integrable functions, then the commutators generated by the m-th
Calder\'{o}n-Zygmund type singular integral and $\vec{b}$ is
defined by:
$$T_{m}^{\vec{b}}(\vec{f})(x) =\dint_{(\mathbb{R}^{n})^{m}}K(x,y_{1},\dots ,y_{m})
\prod\limits_{i=1}^{m}(b_{i}(x)-b_{i}(y_{i}))f_{i}(y_i)dy_1\dots dy_m.$$
In the following, we will establish the boundedness of $T_m$ on generalized product local Morrey spaces. And, we also consider the boundedness of the commutators generated by the m-th Calder\'{o}n-Zygmund type singular integral $T_m$
and the local Campanato function on generalized product local Morrey spaces.
\section{\bf Some notations and lemmas}
\textbf{Definition 2.1}\cite{BG}~Let $\varphi(x,r)$ be a positive measurable function on $\mathbb{R}^n\times(0,\infty)$ and $1\leq p\leq\infty.$ For any fixed $\emph{x}_{0}\in\mathbb{R}^n,$ a function $f\in L_{loc}^{q}$ is said to belong to the local Morrey space, if
$$\|f\|_{LM^{\{x_0\}}_{p,\varphi}}=\sup\limits_{r>0}\varphi^{-1}(x_0,r)|B(x_0,r)|^{-\frac{1}{p}}\|f\|_{\emph{L}_p(B(x_0,r))}<\infty.$$
And, we denote
$$\emph{LM}^{\{x_0\}}_{p,\varphi}\equiv\emph{LM}^{\{x_0\}}_{p,\varphi}(\mathbb{R}^n)=\{f\in L_{loc}^{q}(\mathbb{R}^n):\|f\|_{{LM}^{\{x_0\}}_{p,\varphi}}<\infty\}.$$
According to this definition, we recover the local Morrey space$ \emph{LM}^{\{x_0\}}_{p,\lambda}$ under the choice $\varphi(x_0,r)=r^{\frac{\lambda-n}{p}}.$
\textbf{Definition 2.2}\cite{BG}~Let $1\leq q<\infty$ and $0\leq\lambda<1/n.$ A function $f\in L_{loc}^{q}(\mathbb{R}^n)$ is said to belong to the space $LC_{q,\lambda}^{\{x_0\}}$ (local Campanato space), if
$$\|f\|_{LC_{q,\lambda}^{\{x_0\}}}=\sup\limits_{r>0}\biggl(\dfrac{1}{|B(x_0,r)|^{1+\lambda q}}\dint_{B(x_,r)}|f(y)-f_{B(x_0,r)}|^{q}dy\biggr)^{1/q}<\infty,$$
where $$f_{B(x_0,r)}=\dfrac{1}{|B(x_0,r)|}\dint_{B(x_0, r)}f(y)dy.$$
Define $$LC_{q,\lambda}^{\{x_0\}}(\mathbb{R}^n)=\{f\in L_{loc}^{q}(\mathbb{R}^n):\|f\|_{LC_{q,\lambda}^{\{x_0\}}}<\infty\}.$$
{\bf Remark.}\cite{BG} Note that, the central $BMO$ space $CBMO_{q}(\mathbb{R}^n)=LC_{q,0}^{\{0\}}(\mathbb{R}^n),$
$CBMO_{q}^{\{x_0\}}(\mathbb{R}^n)=LC_{q,0}^{\{x_0\}}(\mathbb{R}^n),$
and $BMO_{q}(\mathbb{R}^n)\subset \bigcap_{q>1}CBMO_{q}^{\{x_0\}}(\mathbb{R}^n).$ Moreover, one can imagine that the behavior of $CBMO_{q}^{\{x_0\}}(\mathbb{R}^n)$ may be quite different from that of $BMO(\mathbb{R}^n),$ since there is no analogy of the John-Nirenberg inequality of $BMO$ for the space $CBMO_{q}^{\{x_0\}}(\mathbb{R}^n).$
{\bf Lemma 2.1}~Let $1<q<\infty,$ $0<r_2<r_1$ and $b\in LC_{q,\lambda}^{\{x_0\}},$ then
$$\begin{array}{cl}\bigg(\dfrac{1}{|B(x_0,r_1)|^{1+\lambda q}}\dint_{B(x_0,r_1)}|b(x)-b_{B(x_0,r_2)}|^{q}dx\biggr)^{1/q}\leq C\biggl(1+\ln\dfrac{r_1}{r_2}\biggr)\|b\|_{LC_{q,\lambda}^{\{x_0\}}}.\end{array}\eqno{(2.1)}$$
And, from this inequality, we have
$$\begin{array}{cl}|b_{B(x_0,r_1)}-b_{B(x_0,r_2)}|\leq C\biggl(1+\ln\dfrac{r_1}{r_2}\biggr)|B(x_0,r_1)|^{\lambda}\|b\|_{LC_{q,\lambda}^{\{x_0\}}}.\end{array}\eqno{(2.2)}$$
In this section, we are going to use the following statement on the boundedness of the weighted Hardy operator:
$$\emph{H}_{w}g(t):=\int^\infty_{t}g(s)w(s)ds,~0<t<\infty,$$
where $\emph{w}~$is a fixed function non-negative and measurable on $(0,\infty).$
{\bf Lemma 2.2}\cite{G1,G2}~Let $v_1,v_2$ and $w$ be positive almost everywhere and measurable functions on $(0,\infty).$ The inequality\\
$$\begin{array}{cl}ess ~\sup\limits_{t>0}v(2t)\emph{H}_wg(t)\leq Cess ~\sup\limits_{t>0}v_1(t)g(t)\end{array}\eqno{(2.3)}$$
holds for some $C>0$ and all non-negative and non-decreasing g on$~(0,\infty)~$if and only if
$$B:ess \sup\limits_{t>0}v_2(t)\int^\infty_{t}\frac{w(s)ds}{ess ~\sup_{s<\tau<\infty} v_1(\tau)}ds<\infty.$$ Moreover, if $\tilde{C}$ is the minimum value of $C$ in (2.3), then $\tilde{C}=B$.
{\bf Lemma 2.3}\cite{KS}~\;\;Let $T_m$ be a $m-CZO$.
Suppose that $1\leq p_{1},\cdots,p_{m}<\infty$ and $1/p=1/p_{1}+\cdots +1/p_{m}.$
If $p_{i}>1,i=1,\cdots,m,$ then there exists a constant $C>0,$ such that
$$\|T_{m}\vec{f}\|_{L^{p}}\leq C\prod\limits_{i=1}^{m}\|f_{i}\|_{L^{p_{i}}}.$$
\section{ M-th Calder\'{o}n-Zygmund type singular integral operator on generalized product local Morrey space}
{\bf Theorem 3.1}\;\ Let $x_0\in{\mathbb{R}^n},$ $1<p, p_1,p_2,\dots,p_m<\infty,$ such that $1/p=1/p_1+1/p_2+\dots+p_m.$ Then
the inequality
$$\|T_m(\vec{f})\|_{L^p(B(x_0,r))}\lesssim r^{n/p}\dint^{\infty}_{2r}\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(B(x_0,r))}t^{-n/p-1}dt$$
holds for any ball $B(x_0,r)$ and all $f_i\in L^{p_i}_{loc}(\mathbb{R}^n),$ $i=1,2,\dots,m.$
{\bf Proof.} Without loss of generality, it is suffice to show that the conclusion holds for
$T_{2}(f_{1},f_{2}).$
Let $B=B(x_0,r).$ And, we write $f_1=f^0_1+f^\infty_1$ and $f_2=f^0_2+f^\infty_2,$ where $f^0_i=f_i\chi_{2B},$ $f^\infty_i=f_i\chi_{{(2B)}^c},$ for $i=1.2.$ Thus, we have
$$\begin{array}{cl}
&\|T_2(f_1,f_2)\|_{L^p(B(x_0,r))}\\
\leq&\|T_2(f^0_1,f^0_2)\|_{L^p(B)}+\|T_2(f^0_1,f^\infty_2)\|_{L^p(B)}+\|T_2(f^\infty_1,f^0_2)\|_{L^p(B)}+\|T_2(f^\infty_1,f^\infty_2)\|_{L^p(B)}\\
=:&I+II+III+IV.
\end{array}$$
Using the $L^p$ boundedness of $T_2$(Lemma 2.3), we have
$$\begin{array}{cl}
I \lesssim&\|f_1\|_{L^{p_1}(2B)}\|f_2\|_{L^{p_2}(2B)} \\
\lesssim&r^{\frac{n}{p}}\|f_1\|_{L^{p_2}(2B)}\|f_2\|_{L^{p_2}(2B)}\dint^\infty_{2r}\frac{dt}{t^{\frac{n}{p}+1}}\\
\leq&r^\frac{n}{p}\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{\frac{n}{p}+1}}.\\
\end{array}\eqno{(3.1)}$$
Moreover, when $x\in B(x_0, r)$ and $y\in{(2B)}^c,$ we have $$\begin{array}{cl}\dfrac{1}{2}|x_0-y|\leq|x-y|\leq\dfrac{3}{2}|x_0-y|.\end{array}\eqno{(3.2)}$$
Then, it follows from (1.2) that
$$\begin{array}{cl}
|T_2(f^0_1,f^\infty_2)(x)|
&\lesssim\dint_{\mathbb{R}^n}\dint_{\mathbb{R}^n}\dfrac{|f^0_1(y_1)||f^\infty_2(y_2)|}{{(|x-y_1|+|x-y_2|)}^{2n}}dy_1dy_2\\
&\lesssim\dint_{2B}|f_1(y_1)|dy_1\dint_{{(2B)}^c}\frac{|f_2(y_2)|}{{|x_0-y_2|}^{2n}}dy_2 \\
&\lesssim\dint_{2B}|f_1(y_1)|dy_1\dint_{{(2B)}^c}|f_2(y_2)|\biggl[\dint^\infty_{|x_0-y_2|}\frac{dt}{t^{2n+1}}\biggr]dy_2 \\
&\lesssim\|f_1\|_{L^{p_1}(2B)}{|2B|}^{1-1/p_1}\dint^{\infty}_{2r}\|f_2\|_{L^{p_2}(B(x_0,t))}{|B(x_0,t)|}^{1-1/p_2}\frac{dt}{t^{2n+1}} \\
&\lesssim\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{n/p+1}},
\end{array}\eqno{(3.3)}$$
where $1/p=1/{p_1}+1/{p_2}.$
Thus,
$$\begin{array}{cl}II=\|T_2(f^0_1,f^\infty_2)\|_{L^p(B)} \lesssim r^{n/p}\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{n/p+1}}.\end{array}\eqno{(3.4)}$$
Similarly, we have
$$III=\|T_2(f^\infty_1,f^0_2)\|_{L^p(B)} \lesssim r^{n/p}\int^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{n/p+1}}.$$
Moreover, similar to the estimate of (3.3), we have
$$\begin{array}{cl}
|T_2(f^\infty_1,f^\infty_2)(x)|
&\lesssim\dint_{(2B)^c}\int_{(2B)^c}\frac{|f_1(y_1)||f_2(y_2)|}{{(|x_0-y_1|+|x_0-y_2|)}^{2n}}dy_1dy_2 \\
&\lesssim\dint_{(2B)^c}\int_{(2B)^c}|f_1(y_1)||f_2(y_2)|dy_1dy_2\int^\infty_{|x_0-y_1|+|x_0-y_2|}\frac{dt}{t^{2n+1}} \\
&\lesssim\dint^\infty_{2r}\biggl[\int_{B(x_0,t)}|f_1(y_1)|dy_1\int_{B(x_0,t)}|f_2(y_2)|dy_2\biggr]\frac{dt}{t^{2n+1}} \\
&\lesssim\dint^\infty_{2r}\|f\|_{L^{p_1}(B(x_0,t))}\|f\|_{L^{p_1}(B(x_0,t))}{|B(x_0,t)|}^{2-(1/p_1+1/p_2)}\frac{dt}{t^{2n+1}} \\
&\lesssim\dint^\infty_{2r}\|f\|_{L^{p_1}(B(x_0,t))}\|f\|_{L^{p_1}(B(x_0,t))}\frac{dt}{t^{n/p+1}}.
\end{array}$$
Thus,
$$\begin{array}{cl}IV=\|T_2(f^\infty_1,f^\infty_2)\|_{L^p(B)} \lesssim r^{n/p}\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{n/p+1}}. \end{array}\eqno{(3.5)}$$
Combining the above estimates, we obtain
$$\begin{array}{cl}\|T_2(f_1,f_2)\|_{L^p(B)} \lesssim r^{n/p}\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{n/p+1}}.\end{array}$$
{\bf Theorem 3.2}\;\;Let $x_0\in{\mathbb{R}^n},$ $1<p, p_1, p_2, \dots, p_m<\infty$ such that $1/p=1/p_1+1/p_2+\dots+p_m.$ If functions $\varphi,$ $\varphi_i:$ $\mathbb{R}^n\times(0,\infty)\rightarrow(0,+\infty),(i=1,2,\cdots,m)$
satisfy the condition
$$\begin{array}{cl}\dint_{r}^{\infty}\dfrac{\mbox{ess}\inf\limits_{t<s<\infty}\prod\limits_{i=1}^m\varphi_{i}(x_0,s)s^{n/p}}{t^{n/p+1}}dt\leq C\psi(x_0,r),\end{array}\eqno{(3.6)}$$
where constant $C>0$ doesn't depend on $r.$ Then the operator $T_m$ is bounded from the product space
$LM_{p_1,\varphi_1}^{\{x_0\}}\times LM_{p_2,\varphi_2}^{\{x_0\}}\times\dots\times LM_{p_m,\varphi_m}^{\{x_0\}}$ to $LM_{p,\psi}^{\{x_0\}}.$ Moreover,
the following inequality
$$\|T_m(\vec{f})\|_{LM_{p,\psi}^{\{x_0\}}} \lesssim \prod\limits_{i=1}^{m}\|f_i\|_{LM_{p_i,\varphi_i}^{\{x_0\}}}.$$
holds.
{\bf Proof.} Taking $v_1(r)=\prod\limits_{i=1}^{m}\varphi_i^{-1}(x_0,r)r^{-n/p},$ $v_2(r)=\psi^{-1}(x_0,r),$
$g(r)=\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(B(x_0,r))}$ and $w(r)=r^{-n/p-1},$ then we have
$$ess\sup\limits_{t>0}v_2(t)\dint_{t}^{\infty}\dfrac{w(s)ds}{ess\sup\limits_{s<\tau<\infty}v_1(\tau)}<\infty.$$
Thus, by Lemma 2.2, we have
$$\begin{array}{cl}ess\sup\limits_{t>0}v_2(t)H_wg(t)\leq C ess\sup\limits_{t>0}v_1(t)g(t).\end{array}\eqno{(3.7)}$$
Therefore, from Theorem 3.1 and (3.7), it follows that
$$\begin{array}{cl}
&\|T_m(\vec{f})\|_{LM_{p,\psi}^{\{x_0\}}}\\
=&\sup\limits_{r>0}\psi^{-1}(x_0,r)|B(x_0,r)|^{-1/p}\|T_m(\vec{f})\|_{L^{p}(B(x_0,r))}\\
\lesssim&\sup\limits_{r>0}\psi^{-1}(x_0,r)|B(x_0,r)|^{-1/p}r^{n/p}\dint^{\infty}_{2r}\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(B(x_0,t))}
t^{-n/p-1}dt\\
\lesssim&\sup\limits_{r>0}\prod\limits_i^m\varphi_{i}^{-1}(x_0,r)r^{-n/p}\prod\limits_{i=1}^m\|f_i\|_{L^{p_i}(B(x_0,r))}\\
\lesssim&\sup\limits_{r>0}\prod\limits_{i=1}^m\varphi_{i}^{-1}(x_0,r)r^{-n/p_i}\|f_i\|_{L^{p_i}(B(x_0,r))}\\
=&\prod\limits_{i=1}^{m}\|f_i\|_{LM_{p_i,\varphi_i}^{\{x_0\}}}.
\end{array}$$
\section{Commutators generated by m-th Calder\'{o}n Zygmund type
singular integral operators and local Campanato functions}
{\bf Theorem 4.1}\;\; Let $x_0\in{\mathbb{R}^n},$ $1<p,$ $p_i,$ $q_i<\infty (i=1,2,\dots,m)$ such that $1/p=1/p_1+1/p_2+\dots+1/p_m+1/p_1+1/q_2+\dots+1/q_m$ and $b_i\in LC_{q_i,\lambda_i}^{\{x_0\}}$ for $0<\lambda_i<1/n,$ $i=1,2,\cdots,m.$
Then the inequality
$$\|T_m^{\vec{b}}(\vec{f})\|_{L^p(B(x_0,r))}\lesssim \prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i,\lambda_i}^{\{x_0\}}}\;r^{n/p}
\dint^{\infty}_{2r}\biggl(1+\ln\frac{t}{r}\biggr)^{m}t^{n\sum\limits_{i=1}^{m}\lambda_i-n\sum\limits_{i=1}^{m}1/p_i-1
}\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(B(x_0,t))}dt$$
holds for any ball $B(x_0,r)$ and all $f_i\in L^{p_i}_{loc}(\mathbb{R}^n),$ $i=1,2,\dots,m.$
{\bf Proof.} Without loss of generality, it is suffice for us to show that the conclusion holds for
$m=2.$
Let $B=B(x_0,r),$ $f_1=f^0_1+f^\infty_1$ and $f_2=f^0_2+f^\infty_2,$ where $f^0_i$ and $f^\infty_i$ are as in the proof of Theorem 3.1, for $i=1.2.$ Thus, we have
$$\begin{array}{cl}&T_{2}^{(b_1,b_2)}(f_{1},f_{2})(x)\\
=&T_{2}^{(b_1,b_2)}(f_{1}^0,f_{2}^0)(x)+T_{2}^{(b_2,b_2)}(f_{1}^0,f_{2}^{\infty})(x)+T_{2}^{(b_1,b_2)}(f_{1}^{\infty},f_{2}^{0})(x)
+T_{2}^{(b_1,b_2)}(f_{1}^{\infty},f_{2}^{\infty})(x).
\end{array}$$
So,$$\begin{array}{cl}&\|T_{2}^{(b_1,b_2)}(f_{1},f_{2})\|_{L^{p}(B)}\\
&\leq\|T_{2}^{(b_1,b_2)}(f_{1}^0,f_{2}^0)\|_{L^{p}(B)}+\|T_{2}^{(b_1,b_2)}(f_{1}^0,f_{2}^{\infty})\|_{L^{p}(B)}\\
&\quad+\|T_{2}^{(b_1,b_2)}(f_{1}^{\infty},f_{2}^{0})\|_{L^{p}(B)}+\|T_{2}^{(b_1,b_2)}(f_{1}^{\infty},f_{2}^{\infty})\|_{L^{p}(B)}\\
&=:I+II+III+IV.\end{array}$$
Let us estimate $I, II, III$ and $IV$, respectively.
Since, $$\begin{array}{cl}&(b_1(x)-b_1(y))(b_2(x)-b_2(y))\\
&=(b_1(x)-(b_1)_{B})(b_2(x)-(b_2)_{B})-(b_1(x)-(b_1)_{B})(b_2(y)-(b_2)_{B})\\
&\quad-(b_1(y)-(b_1)_{B})(b_2(x)-(b_2)_{B})+(b_1(y)-(b_1)_{B})(b_2(y)-(b_2)_{B}).\end{array}\eqno{(4.1)}$$
Then,
$$\begin{array}{cl}
&\|T_{2}^{(b_1,b_2)}(f_{1}^0,f_{2}^0)\|_{L^{p}(B)}\\
=&\|(b_1-(b_1)_{B})(b_2-(b_2)_{B})T_{2}(f_{1}^0,f_{2}^0)\|_{L^{p}(B)}+\|(b_1-(b_1)_{B})T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^0)\|_{L^{p}(B)}\\
&+\|(b_2-(b_2)_{B})T_{2}((b_1-(b_1)_{B})f_{1}^0,f_{2}^0)\|_{L^{p}(B)}+\|T_{2}((b_1-(b_1)_{B})f_{1}^0,(b_2-(b_2)_{B})f_{2}^0)\|_{L^{p}(B)}\\
=:&I_1+I_2+I_3+I_4.
\end{array}\eqno{(4.2)}$$
Let $1<\bar{p},\bar{q}<\infty,$ such that $1/\bar{p}=1/p_1+1/p_2$ and $1/\bar{q}=1/q_1+1/q_2.$ Then, using the H\"{o}lder's inequality and Lemma 2.3, we have
$$\begin{array}{cl}
I_1
&\lesssim\|(b_1-(b_1)_{B})(b_2-(b_2)_{B}\|_{L^{\bar{q}}(B)}\|T_{2}(f_{1}^0,f_{2}^0)\|_{L^{\bar{p}}(B)}\\
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_2}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}\|f_{1}\|_{L^{p_1}(2B)}\|f_{2}\|_{L^{p_1}(2B)}\\
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)} \|b_2-(b_2)_{B}\|_{L^{q_2}(B)} r^{(1/p_1+1/p_2)n}\\
&\quad\times\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}\frac{dt}{t^{(1/p_1+1/p_2)n+1}}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}\eqno{(4.3)}$$
Let $1<\tau<\infty,$ such that $1/p=1/q_1+1/\tau.$ Then similarly to the estimate of (4.3), we have
$$\begin{array}{cl}
I_2
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^0)\|_{L^{\tau}(B)}\\
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|f_{1}^0\|_{L^{p_1}(\mathbb{R}^n)}\|(b_2-(b_2)_{2B})f_{2}^0)\|_{L^{s}(\mathbb{R}^n)}\\
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(2B)}\|f_{1}\|_{L^{p_1}(2B)}\|f_{2}
\|_{L^{p_2}(2B)},\\
\end{array}\eqno{(4.4)}$$
where $1<s<\infty,$ such that $1/s=1/p_2+1/q_2=1/\tau-1/{p_1}.$
From Lemma 2.1, it is easy to see that
$$\|b_i-(b_i)_{B}\|_{L^{q_i}(B)}\leq Cr^{n/q_{i}+n\lambda_i}\|b_i\|_{LC_{q_i,\lambda_i}^{\{x_0\}}},$$
and
$$\begin{array}{cl}\|b_i-(b_i)_{B}\|_{L^{q_i}(2B)}\leq \|b_i-(b_i)_{2B}\|_{L^{q_i}(2B)}+\|(b_i)_{B}-(b_i)_{2B}\|_{L^{q_i}(2B)}\leq Cr^{n/q_{i}+n\lambda_i}\|b_i\|_{LC_{q_i,\lambda_i}^{\{x_0\}}},\end{array}\eqno{(4.5)}$$
for $i=1,2.$
Then,
$$\begin{array}{cl}
I_2
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Similarly,
$$\begin{array}{cl}
I_3
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Moreover, let $1<\tau_1,\tau_2<\infty,$ such that $1/\tau_{1}=1/p_1+1/q_1$ and $1/\tau_{2}=1/p_2+1/q_2.$ It is easy to see that
$1/p=1/\tau_{1}+1/\tau_{2}.$
Then by Lemma 2.3, H\"{o}lder's inequality and (4.5), we obtain
$$\begin{array}{cl}
I_4
&\lesssim\|(b_1-(b_1)_{B})f_{1}^0\|_{L^{\tau_1}(\mathbb{R}^n)}\|(b_2-(b_2)_{B})f_{2}^0\|_{L^{\tau_2}(\mathbb{R}^n)}\\
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(2B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(2B)}\|f_{1}\|_{L^{p_1}(2B)}\|f_{2}
\|_{L^{p_2}(2B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}\eqno{(4.6)}$$
Therefore, combining the estimates of $I_1, I_2, I_3$ and $I_4,$ we have
$$\begin{array}{cl}I
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Let us estimate $II.$
It's analogues to (4.2), we have
$$\begin{array}{cl}
&\|T_{2}^{(b_1,b_2)}(f_{1}^0,f_{2}^\infty)\|_{L^{p}(B)}\\
=&\|(b_1-(b_1)_{B})(b_2-(b_2)_{B})T_{2}(f_{1}^0,f_{2}^\infty)\|_{L^{p}(B)}+\|(b_1-(b_1)_{B})T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)\|_{L^{p}(B)}\\
&+\|(b_2-(b_2)_{B})T_{2}((b_1-(b_1)_{B})f_{1}^0,f_{2}^\infty)\|_{L^{p}(B)}+\|T_{2}((b_1-(b_1)_{B})f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)\|_{L^{p}(B)}\\
=:&II_1+II_2+I_3+II_4.
\end{array}\eqno{(4.7)}$$
Let $1<\bar{p},\bar{q}<\infty,$ such that $1/\bar{p}=1/p_1+1/p_2$ and $1/\bar{q}=1/q_1+1/q_2.$ Then, using the H\"{o}lder's inequality and (3.4), we have
$$\begin{array}{cl}
II_1
&\lesssim\|(b_1-(b_1)_{B})(b_2-(b_2)_{2B}\|_{L^{\bar{q}}(B)}\|T_{2}(f_{1}^0,f_{2}^\infty)\|_{L^{\bar{p}}(B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{(1/q_1+1/q_2)n+(\lambda_1+\lambda_2)n}r^{(1/p_1+1/p_2)n}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}\eqno{(4.8)}$$
Moreover, using (1.2) and (3.2), we have
$$\begin{array}{cl}
&|T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\dint_{2B}|f_1(y_1)|dy_1\dint_{{(2B)}^c}\frac{|b_2(y_2)-(b_2)_{B}||f_2(y_2)|}{{|x_0-y_2|}^{2n}}dy_2. \\
\end{array}$$
It's obvious that
$$\begin{array}{cl}\dint_{2B}|f_1(y_1)|dy_1\lesssim \|f_1\|_{L^{p_1}(2B)}{|2B|}^{1-1/p_1},\end{array}\eqno{(4.9)}$$
and
$$\begin{array}{cl}
&\dint_{{(2B)}^c}\frac{|b_2(y_2)-(b_2)_{B}||f_2(y_2)|}{{|x_0-y_2|}^{2n}}dy_2 \\
&\lesssim\dint_{{(2B)}^c}|b_2(y_2)-(b_2)_{B}||f_2(y_2)|\biggl[\dint^\infty_{|x_0-y_2|}\frac{dt}{t^{2n+1}}\biggr]dy_2 \\
&\lesssim \dint^{\infty}_{2r}
\|b_2(y_2)-(b_2)_{B(x_0,t)}\|_{L^{q_2}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}|B(x_0,t)|^{1-(1/p_2+1/q_2)}\frac{dt}{t^{2n+1}}\\
&\quad+\dint^{\infty}_{2r}|(b_2)_{B(x_0,t)}-(b_2)_{B(x_0,r)}
|\|f_2\|_{L^{p_2}(B(x_0,t))}|B(x_0,t)|^{1-1/p_2}\frac{dt}{t^{2n+1}}\\
&\lesssim \|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}
\dint^{\infty}_{2r}|B(x_0,t)|^{1/q_2+\lambda_2}\|f_2\|_{L^{p_2}(B(x_0,t))}|B(x_0,t)|^{1-(1/p_2+1/q_2)}\frac{dt}{t^{2n+1}}\\
&\quad+\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}
\dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)|B(x_0,t)|^{\lambda_2}
\|f_2\|_{L^{p_2}(B(x_0,t))}|B(x_0,t)|^{1-1/p_2}\frac{dt}{t^{2n+1}}\\
&\lesssim\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}
\dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{-n+n\lambda_2-n/p_2-1}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.\\
\end{array}\eqno{(4.10)}$$
Therefore, from (4.9) and (4.10), it follows that
$$\begin{array}{cl}
&|T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}\|f_1\|_{L^{p_1}(2B)}{|2B|}^{1-1/p_1}
\dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{-n+n\lambda_2-n/p_2-1}\|f_2\|_{L^{p_2}(B(x_0,t))}dt\\
&\lesssim\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}} \dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{n\lambda_2-(1/p_1+1/p_2)n-1}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.\\
\end{array}$$
Thus, let $1<\tau<\infty,$ such that $1/p=1/q_1+1/\tau$, then similarly to the estimate of (4.3), we have
$$\begin{array}{cl}
II_2&=\|(b_1-(b_1)_{B})T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)\|_{L^p(B)}\\
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|T_{2}(f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)\|_{L^{\tau}(B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}|B|^{\lambda_1+1/{q_1}+1/\tau}\\
&\quad\times\dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{n\lambda_2-(1/p_1+1/p_2)n-1}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}\eqno{(4.11)}$$
Similarly, we have
$$\begin{array}{cl}
II_3&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Let us estimate $II_4.$
Since,
$$\begin{array}{cl}
&|T_{2}((b_1-(b_1)_{B})f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\dint_{2B}|b_1(y_1)-(b_1)_{B}||f_1(y_1)|dy_1\dint_{{(2B)}^c}\frac{|b_2(y_2)-(b_2)_{B}||f_2(y_2)|}{{|x_0-y_2|}^{2n}}dy_2, \\
\end{array}$$
and
$$\begin{array}{cl}&\dint_{2B}|b_1(y_1)-(b_1)_{B}||f_1(y_1)|dy_1
\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}|B|^{\lambda_1+1-1/p_1}\|f_1\|_{L^{p_1}(2B)}.
\end{array}\eqno{(4.12)}$$
Then, by (4.10) and (4.12), we have
$$\begin{array}{cl}
&|T_{2}((b_1-(b_1)_{B})f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}
\dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{n(\lambda_1+\lambda_2)-n(1/p_1+/p_2)-1}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Therefore,
$$\begin{array}{cl}
II_4&=\|T_{2}((b_1-(b_1)_{B})f_{1}^0,(b_2-(b_2)_{B})f_{2}^\infty)\|_{L^p(B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Combining the estimates of $II_1-II_4,$ we have
$$\begin{array}{cl}
II
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Similarly,
$$\begin{array}{cl}
III&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
For $IV,$ we have
$$\begin{array}{cl}
&\|T_{2}^{(b_1,b_2)}(f_{1}^\infty,f_{2}^{\infty})\|_{L^{p}(B)}\\
\leq&\|(b_1-(b_1)_{B})(b_2-(b_2)_{B})T_{2}(f_{1}^\infty,f_{2}^{\infty})\|_{L^{p}(B)}
+\|(b_1-(b_1)_{B})T_{2}(f_{1}^\infty,(b_2-(b_2)_{B})f_{2}^{\infty})\|_{L^{p}(B)}\\
&\quad+\|(b_2-(b_2)_{B})T_{2}((b_1-(b_1)_{B})f_{1}^\infty,f_{2}^{\infty})\|_{L^{p}(B)}+\|T_{2}((b_1-(b_1)_{B})f_{1}^\infty,(b_2-(b_2)_{B})f_{2}^{\infty})\|_{L^{p}(B)}\\
=:&IV_1+IV_2+IV_3+IV_4.\end{array}$$
Let us estimate $IV_1,$ $IV_2,$ $IV_3$ and $IV_4,$ respectively.
Let $1<\tau<\infty,$ such that $1/p=1/q_1+1/q_2+1/\tau.$ Then, from H\"{o}lder's inequality and (3.5), we get
$$\begin{array}{cl}
IV_1&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|b_2-(b_2)_{B}\|_{L^{q_2}(B)}\|T_{2}(f_{1}^\infty,f_{2}^{\infty})\|_{L^{\tau}(B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}|B|^{(\lambda_1+\lambda_2)+(1/q_1+1/q_2)+1/\tau}\\
&\quad\times\dint^\infty_{2r}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}t^{-n(/p_1+1/p_2)-1}dt\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Moreover, by (1.2) and (3.2), we have
$$\begin{array}{cl}
&|T_{2}(f_{1}^\infty,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\dint_{{(2B)}^c}\dint_{{(2B)}^c}\dfrac{|b_2(y_2)-(b_2)_{B}||f_1(y_1)||f_2(y_2)|}{{(|x_0-y_1|+|x_0-y_2|)}^{2n}}dy_1dy_2\\
&\lesssim\dint_{{(2B)}^c}\dint_{{(2B)}^c}|f_1(y_1)||b_2(y_2)-(b_2)_{B}||f_2(y_2)|\biggl[\dint^\infty_{|x_0-y_1|+|x_0-y_2|}\frac{dt}{t^{2n+1}}\biggr]dy_1dy_2 \\
&\lesssim \dint^{\infty}_{2r}\biggr[\dint_{B(x_0,t)}|f_1(y_1)|dy_1\biggr]\biggr[\dint_{B(x_0,t)}|b_2(y_2)-(b_2)_{B}|
|f_2(y_2)|dy_2\biggr]\frac{dt}{t^{2n+1}}.\\
\end{array}$$
Since,
$$\begin{array}{cl}\dint_{B(x_0,t)}|f_1(y_1)|dy_1\lesssim\|f_1\|_{L^{p_1}(B(x_0,t))}t^{n(1-1/p_1)}, \end{array}$$
and
$$\begin{array}{cl}
&\dint_{B(x_0,t)}|b_2(y_2)-(b_2)_{B}||f_2(y_2)| \\
&\lesssim\|b_2-(b_2)_{B(x_0,t)}\|_{L^{q_2}(B(x_0,t))}\|f_2\|_{L^{p_2}}|B(x_0,t)|^{1-(1/p_2+1/q_2)}\\
&\quad+|(b_2)_{B(x_0,t)}-(b_2)_{B(x_0,r)}
|\|f_2\|_{L^{p_2}}|B(x_0,t)|^{1-1/p_2}\\
&\lesssim \|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}|B(x_0,t)|^{1/q_2+\lambda_2}\|f_2\|_{L^{p_2}}|B(x_0,t)|^{1-(1/p_2+1/q_2)}\\
&\quad+\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}\biggl(1+\ln\dfrac{t}{r}\biggr)|B(x_0,t)|^{\lambda_2} \|f_2\|_{L^{p_2}}|B(x_0,t)|^{1-1/p_2}\\
&\lesssim\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}
\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{n\lambda_2-n/p_2+n}\|f_2\|_{L^{p_2}(B(x_0,t))}.\\
\end{array}$$
Then,
$$\begin{array}{cl}
&|T_{2}(f_{1}^\infty,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}\dint^{\infty}_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{n\lambda_2-(1/p_1+1/p_2)n-1}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.\\
\end{array}\eqno{(4.13)}$$
Let $1<\tau<\infty,$ such that $1/p=1/q_1+1/\tau.$ Then, from H\"{o}lder's inequality and (4.13), we have
$$\begin{array}{cl}
IV_2
&\lesssim\|b_1-(b_1)_{B}\|_{L^{q_1}(B)}\|T_{2}(f_{1}^\infty,(b_2-(b_2)_{B})f_{2}^\infty)\|_{L^{\tau}(B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Similarly,
$$\begin{array}{cl}
IV_3
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Similar to the estimate of (4.13), we have
$$\begin{array}{cl}
&|T_{2}((b_1-(b_1)_{B})f_{1}^\infty,(b_2-(b_2)_{B})f_{2}^\infty)(x)|\\
&\lesssim\dint_{{(2B)}^c}\dint_{{(2B)}^c}|b_1(y_1)-(b_1)_{B}||b_2(y_2)-(b_2)_{B}||f_1(y_1)||f_2(y_2)|\biggl[\dint^\infty_{|x_0-y_1|+|x_0-y_2|}\frac{dt}{t^{2n+1}}\biggr]dy_1dy_2 \\ &\lesssim \dint^{\infty}_{2r}\biggr[\dint_{B(x_0,t)}|b_1(y_1)-(b_1)_{B}||f_1(y_1)|dy_1\biggr]\biggr[\dint_{B(x_0,t)}|b_2(y_2)-(b_2)_{B}||f_2(y_2)|dy_2\biggr]\frac{dt}{t^{2n+1}}\\
&\lesssim \|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}} \|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}\dint^{\infty}_{2r}
\biggl(1+\ln\dfrac{t}{r}\biggr)^2t^{n(\lambda_1+\lambda_2)-n(1/p_1+1/p_2)-1}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}
dt.\\
\end{array}$$
Thus,
$$\begin{array}{cl}
IV_4
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Then, from the estimates of $IV_1-IV_4,$ we deduce that
$$\begin{array}{cl}
IV&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
So, combining the estimates for $I, II, III$ and $IV$, we have
$$\begin{array}{cl}&\|T_{2}^{(b_1,b_2)}(f_{1},f_{2})\|_{L^{p}(B)}\\
&\lesssim\|b_1\|_{LC_{q_1,\lambda_1}^{\{x_0\}}}\|b_2\|_{LC_{q_2,\lambda_2}^{\{x_0\}}}r^{n/p}\\
&\quad\times\dint^\infty_{2r}\biggl(1+\ln\dfrac{t}{r}\biggr)^2{t^{(\lambda_1+\lambda_2)n-(1/p_1+1/p_2)n-1}}\|f_1\|_{L^{p_1}(B(x_0,t))}\|f_2\|_{L^{p_2}(B(x_0,t))}dt.
\end{array}$$
Therefore, we complete the proof of Theorem 4.1.
{\bf Theorem 4.2}\;\; Let $x_0\in{\mathbb{R}^n},$ $1<p, p_i, q_i<\infty,$ for $i=1, 2,\dots,m$ such that $1/p=1/p_1+1/p_2+\dots+1/p_n+1/p_1+1/q_2+\dots+1/q_n.$ Suppose that $0<\lambda_i<1/n$ such that $b_i\in LC_{q_i,\lambda_i}^{\{x_0\}},$ for $0<\lambda_i<1/n,$ $i=1,2,\cdots,m.$
If functions $\varphi,$ $\varphi_i:$ $\mathbb{R}^n\times(0,\infty)\rightarrow(0,+\infty),(i=1,2,\cdots,m)$
satisfy the condition
$$\dint_{r}^{\infty}\biggl(1+\ln\frac{t}{r}\biggr)^{m}t^{n\sum\limits_{i=1}^{m}\lambda_i-n\sum\limits_{i=1}^{m}1/p_i-1}\mbox{ess}\inf\limits_{t<s<\infty}\prod\limits_{i=1}^m\varphi_{i}(x_0,s)s^{n/p_i}dt\leq C\psi(x_0,r),$$
where constant $C>0$ doesn't depend on $r.$ Then the operator $T_m^{\vec{b}}$ is bounded from product space
$LM_{p_1,\varphi_1}^{\{x_0\}}\times LM_{p_2,\varphi_2}^{\{x_0\}}\times\dots\times LM_{p_m,\varphi_m}^{\{x_0\}}$ to $LM_{p,\psi}^{\{x_0\}}.$ Moreover,
the inequality
$$\|T_m^{\vec{b}}(\vec{f})\|_{LM_{p,\psi}^{\{x_0\}}}\lesssim\prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i,\lambda_i}^{\{x_0\}}}\prod\limits_{i=1}^{m}\|f_i\|_{LM_{p_i,\varphi_i}^{\{x_0\}}}.$$
holds.
{\bf Proof.} Taking $v_1(t)=\prod\limits_{i=1}^{m}\varphi_i^{-1}(x_0,t)t^{-n/p_i},$ $v_2(t)=\psi^{-1}(x_0,t),$
$g(t)=\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(B(x_0,t))}$ and $w(t)=(1+\ln\frac{t}{r})^{m}t^{n\sum\limits_{i=1}^{m}\lambda_i-n\sum\limits_{i=1}^{m}1/p_i-1},$
then we have
$$ess\sup\limits_{t>0}v_2(t)\dint_{t}^{\infty}\dfrac{w(s)ds}{ess\sup\limits_{s<\tau<\infty}v_1(\tau)}<\infty.$$
Thus, by Lemma 2.2, we have
$$ess\sup\limits_{t>0}v_2(t)H_wg(t)\leq C ess\sup\limits_{t>0}v_1(t)g(t).$$
So,
$$\begin{array}{cl}
&\|T_m^{\vec{b}}(\vec{f})\|_{LM_{p,\psi}^{\{x_0\}}}\\
=&\sup\limits_{r>0}\psi^{-1}(x_0,r)|B(x_0,r)|^{-1/p}\|T_m(\vec{f})\|_{L^{p}(B(x_0,r))}\\
\lesssim&\prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i,\lambda_i}^{\{x_0\}}}\sup\limits_{r>0}\psi^{-1}(x_0,r)
\dint^{\infty}_{2r}\biggl(1+\ln\frac{t}{r}\biggr)^mt^{n\sum\limits_{i=1}^{m}\lambda_i-n\sum\limits_{i=1}^{m}1/p_i-1}\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(B(x_0,t))}dt\\
\lesssim&\prod\limits_{i=1}^{m}\|b_i\|_{LC_{q_i,\lambda_i}^{\{x_0\}}}\sup\limits_{r>0}\prod\limits_{i=1}^m\varphi_{i}^{-1}(x_0,r)r^{-n/p_i}\|f_i\|_{L^{p_i}(B(x_0,r))}\\
=&\prod\limits_{i=1}^{m}\|f_i\|_{LM_{p_i,\varphi_i}^{\{x_0\}}}. \end{array}$$
Thus we complete the proof of Theorem 4.2.
\noindent{\bf Acknowledgments}\;\;{This work is supported by the National Natural Science Foundation of China (11161042, 11471050)}
\newpage
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\begin{document}
\maketitle
\begin{abstract}
In this article, we are interested in a non-monotone system of logistic reaction-diffusion equations. This system of equations models an epidemics where two types of pathogens are competing, and a mutation can change one type into the other with a certain rate. We show the existence of minimal speed travelling waves, that are usually non monotonic. We then provide a description of the shape of those constructed travelling waves, and relate them to some Fisher-KPP fronts with non-minimal speed.
\end{abstract}
\section{Introduction}
\gr{Epidemics of newly emerged pathogen can have catastrophic consequences}. Among those who have infected humans, we can name the black plague, the Spanish flu, or more recently SARS, AIDS, bird flu or Ebola. Predicting the propagation of such epidemics is a great concern in public health. Evolutionary phenomena play an important role in the emergence of new epidemics\gr{: such epidemics typically start when the pathogen acquires the ability to reproduce in a new host, and to be transmitted within this new hosts population}. Another \gr{phenotype} that \gr{can} often vary rapidly is the virulence of the pathogen, that is how much the parasite is affecting its host; \gr{Field data show that the virulence of newly emerged pathogens changes rapidly, which moreover seems related to} unusual \gr{spatial dynamics} observed \gr{in such
populations} (\cite{Hawley, Phillips}, see also \cite{Lion, Heilmann}). It is unfortunately difficult to set up experiments with a controlled environment to study evolutionary epidemiology phenomena with a spatial structure, we refer to \cite{Bell,Keymer} for current developments in this direction. Developing the theoretical approach for this type of problems is thus especially interesting. Notice finally that many current problems in evolutionary biology and ecology combine evolutionary phenomena and spatial dynamics: the effect of global changes on populations \cite{PY,DSE}, biological invasions \cite{SBP,KT}, cancers or infections \cite{Gerlingeretal,Frostetal}.
In the framework of evolutionary ecology, the virulence of a pathogen can be seen as a life-history trait of the pathogen \cite{Roff, Frank}. To explain and predict the evolution of virulence in a population of pathogens, many of the recent theories introduce a \textit{trade-off} hypothesis, namely a link between the parasite's virulence and its ability to transmit from one host to another, see e.g. \cite{Aletal}. The basic idea behind this hypothesis is that the more a pathogen reproduces (in order to transmit some descendants to other hosts), the more it ''exhausts'' its host. A high virulence can indeed even lead to the premature death of the host, which the parasite \gr{within this host} rarely survives. In other words, by increasing its transmission rate, a pathogen reduces its own life expectancy. There exists then an optimal virulence trade-off, that may depend on the ecological environment. An environment that changes in time \gr{(e.g. if the number of susceptible hosts is heterogeneous in
time and/or space)} can then lead to a Darwinian evolution of the pathogen
population. \gr{For
instance,} in \cite{
Bernetal}, an experiment shows how the composition of a viral population (composed of the phage $\lambda$ and its virulent mutant $\lambda $cl857, which differs from $\lambda$ by a single locus mutation only) evolves in the early stages of the infection of an \textit{E. Coli} culture.
\medskip
The Fisher-KPP equation is a classical model for epidemics, and more generally for biological invasions, when no evolutionary phenomenon is considered. It describes the time evolution of the density $n=n(t,x)$ of a population, where $t\geqslant 0$ is the time variable, and $x\in\mathbb R$ is a space variable. The model writes as follows:
\begin{equation}\label{KPP}
\partial_tn(t,x)-\sigma\Delta n(t,x)=rn(t,x)\left(1-\frac{n(t,x)}K\right).
\end{equation}
It this model, the term $\sigma\Delta n(t,x)=\sigma\Delta_x n(t,x)$ models the random motion of the individuals in space, while the right part of the equation models the logistic growth of the population (see \cite{Verhulst}): when the density of the population is low, there is little competition between individuals and the number of offsprings is then roughly proportional to the number of individuals, with a growth rate $r$\,; when the density of the population increases, the individuals compete for e.g. food, or in our case for susceptible hosts, and the growth rate of the population decreases, and becomes negative once the population's density exceeds the so-called carrying capacity $K$. The model \eqref{KPP} was introduced in \cite{fisher,KPP1937}, and the existence of travelling waves for this model, that is special solutions that describe the spatial propagation of the population, was proven in \cite{KPP1937}. Since then, travelling waves have had important implications in
biology
and physics, and raise many challenging problems. We refer to \cite{Xin} for an overview of this field of research.
\medskip
In this study, we want to model an epidemics, but also take into account the possible diversity of the pathogen population. It has been recently noticed that models based on \eqref{KPP} can be used to study this type of problems (see \cite{Betal,ACR,BM}). Following the experiment \cite{Bernetal} described above, we will consider two populations: a wild type population $w$, and a mutant population $m$. For each time $t\geqslant 0$, $w(t,\cdot)$ and $m(t,\cdot)$ are the densities of the respective populations over a one dimensional habitat $x\in\mathbb R$. The two populations differ by their growth rate in the absence of competition (denoted by $r$ in \eqref{KPP}) and their carrying capacity (denoted by $K$ in \eqref{KPP}). We will assume that the mutant type is more virulent than the wild type, \gr{in the sense that} it will have an increased growth rate in the absence of competition (larger $r$), at the expense of a reduced carrying capacity (smaller $K$). We assume that the dispersal rate of the
pathogen (denoted by $\sigma$ in \eqref{KPP}) is not affected by the mutations, and is then the same for the two types. Finally, when a parent gives birth to an offspring, a mutation occurs with a rate $\mu$, and the offspring will then be of a different type. Up to a rescaling, the model is then:
\begin{equation} \label{rescalled}
\left\{\begin{array}{l}
\partial_tw(t,x)-\Delta_x w(t,x)=w(t,x)\left(1-\left(w(t,x)+m(t,x)\right)\right)
+\mu(m(t, x)-w(t, x)), \\
\partial_tm(t,x)-\Delta_x m(t,x)=r\qg m(t,x)\left(1-\left(\frac{w(t,x)+m(t,x)}{K}\right)\right)
+\mu(w(t,x)-m(t,x)),
\end{array}\right.
\end{equation}
where $t\geq 0$ is the time variable, $x\in\mathbb R$ is a spatial variable, $r>1$, $K<1$ and $\mu>0$ are constant coefficients. In \eqref{rescalled}, $r>1$ represents the fact that the mutant population reproduces faster than the wild type population if many susceptible hosts are available, while $K<1$ represents the fact that the wild type tends to out-compete the mutant if many hosts are infected.
Our goal is to study the travelling wave solutions of \eqref{rescalled}, that is solutions with the
following form :
\[ w(t, x)=w(x-ct), \quad m(t, x)=m(x-ct), \]
with $ c\in \mathbb R$. \eqref{rescalled} can then be re-written as follows, with $x\in\mathbb R$:
\begin{equation}\label{systemefront}
\left\{\begin{array}{l}
-cw'(x)-w''(x)=w(x)\left(1-\left(w(x)+m(x)\right)\right)
+\mu(m(x)-w(x)), \\
-cm'(x)-m''(x)=rm(x)\left(1-\left(\frac{w(x)+m(x)}{K}\right)\right)
+\mu(w(x)-m(x)).
\end{array}\right.
\end{equation}
The existence of planar fronts in higher dimension ($x\in\mathbb R^N$) is actually equivalent to the $1D$ case ($x\in\mathbb R$), our analysis would then also be the first step towards the understanding of propagation phenomena for \eqref{rescalled} in higher dimension.
\medskip
\gr{There exists a large literature on }travelling waves for systems of several interacting species. In some cases, the systems are monotonic (or can be transformed into a monotonic system). Then, sliding methods and comparison principles can be used, leading to methods close to the scalar case \cite{Volpert2,Volpert1,Roquejoffre}. The combination of the inter-specific competition and the mutations prevents the use of this type of methods here.
\gr{Other methods that have been used to study systems of interacting populations include phase plane methods (see e.g. \cite{Tang,Fei}) and singular perturbations (see \cite{Gardner2,Gardner}). }
More recently, a different approach, based on a topological degree argument, has been developed for reaction-diffusion equations with non-local terms \cite{Nadin,ACR}. The method we use here to prove the existence of travelling wave for \eqref{systemefront} will indeed be derived from these methods. Notice finally that we consider here that dispersion, mutations and reproduction occur on the same time scale. This is an assumption that is important from a biological point of view (and which is satisfied in the particular $\lambda$ phage epidemics that guides our study, see \cite{Bernetal}). In particular, we will not use the Hamilton-Jacobi methods that have proven useful to study this kind of phenomena when different time scales are considered (see \cite{Mirrahimi, Betal, BM}).
\medskip
This mathematical study has been done jointly with a biology work, see \cite{GRG}. We refer to this other article for a deeper analysis on the biological aspects of this work, as well as a discussion of the impact of stochasticity for a related individual-based model (based on simulations and formal arguments).
\medskip
We will make the following assumption,
\begin{ass}\label{ass}
$r\in (1,\infty)$, $\mu\in\left(0,\min\left(\frac{r}{2},1-\frac{1}{r},1-K,K\right)\right)$ and $K\in \left(0,\min\left(1,\frac{r}{r-1}\left(1-\frac{\mu}{1-\mu}\right)\right)\right)$.
\end{ass}
This assumption ensures the existence of a unique stationary solution of \eqref{rescalled} of the form $(w,m)(t,x)\equiv (w^*,m^*)\in(0,1)\times (0,K)$ (see Appendix~\ref{appendix_reaction_terms}). It does not seem very restrictive for biological applications, and we believe the first result of this study (Existence of travelling waves, Theorem~\ref{thm:main}) could be obtained under a weaker assumption, namely:
\[r\in (1,\infty),\quad K\in(0,1),\quad \mu\in (0,K).\]
\medskip
Throughout this document we will denote by $f_w$ and $f_m$ the terms on the left hand side of \eqref{systemefront}:
\begin{equation}\label{def_f}
\begin{array}{l}
f_w(w,m):=w(1-(w+m))+\mu(m-w), \\
f_m(w,m):=rm\left(1-\left(\frac{w+m}{K}\right)\right)+
\mu(w-m).
\end{array}
\end{equation}
We structure our paper as follows : in Section~\ref{section:main_results}, we will present the main results of this article, which are three fold: Theorem~\ref{thm:main} shows the existence of travelling waves for \eqref{systemefront}, Theorem~\ref{thm:monotonicity} describes the profile of the fronts previously constructed, and Theorem~\ref{thm:KPP} relates the travelling waves for \eqref{systemefront} to travelling waves of \eqref{KPP}, when $\mu$ and $K$ are small. sections~\ref{section:proof_box}, \ref{sec:monotonicity} and \ref{sec:K_small} are devoted to the proof of the three theorems stated in Section~\ref{section:main_results}.
\section{Main results}\label{section:main_results}
The first result is the existence of travelling waves of minimal speed for the model \eqref{rescalled}, and an explicit formula for this minimal speed. We recall that the minimal speed travelling waves are often the biologically relevant propagation fronts, for a population initially present in a bounded region only (\cite{Bramson}), and it seems to be the one that is relevant when small stochastic perturbations are added to the model (\cite{MMQ}). Although we expect the existence of travelling waves for any speed higher than the minimal speed, we will not investigate this problem here - we refer to \cite{Nadin,ACR} for the construction of such higher speed travelling waves for related models. Notice also that the convergence of the solutions to the parabolic model \eqref{rescalled} towards travelling waves, and even the uniqueness of the travelling waves, remain open problems.
\begin{thm}\label{thm:main}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists a solution $(c,w,m)\in\mathbb R\times C^\infty(\mathbb R)^2$ of
\eqref{systemefront}, such that
\[ \forall x\in\mathbb R,\quad w(x)\in(0,1),\,m(x)\in (0,K), \]
\[ \underset{x\to-\infty}{\liminf}(w(x)+m(x))>0,\quad \underset{x\to\infty}{\lim}(w(x)+m(x))=0, \]
\[ c=c_*, \]
where
\begin{equation}
c_*:=\sqrt{2\left(1+r-2\mu+\sqrt{(r-1)^2+4\mu^2}\right)}
\label{eq:defminc}
\end{equation}
is the minimal speed $c>0$ for which such a travelling wave exists.
\end{thm}
The difficulty of the proof of Theorem~\ref{thm:main} has several origins:
\begin{itemize}
\item The system cannot be modified into a monotone system (see \cite{Tang,Calvez}), which prevents the use of sliding methods to show the existence of traveling waves.
\item The competition term has a negative sign, which means that comparison principles often cannot be used directly.
\end{itemize}
As mentioned in the introduction, new methods have been developed recently to show the existence of travelling wave in models with negative nonlocal terms (see \cite{Nadin, ACR}). To prove Theorem \ref{thm:main}, we take advantage of those recent progress by considering the competition term as a nonlocal term (over a set composed of only two elements : the wild and the virulent type viruses). The method of \cite{Nadin, ACR} are however based on the Harnack inequality \gr{(or related arguments)}, that are not as simple for systems of equations (see \cite{Busca}). We have thus introduced a different localized problem (see \eqref{pbnormbox}), which allowed us to prove our result without any Harnack-type argument.
\medskip
Our second result describes the shape of the travelling waves that we have constructed above. We show that three different shapes at most are possible, depending on the parameters. In the most biologically relevant case, where the mutation rate is small, we show that the travelling wave we have constructed in Theorem~\ref{thm:main} is as follows: the wild type density $w$ is decreasing, while the mutant type density $m$ has a unique global maximum, and is monotone away from this maximum. In numerical simulations of \eqref{rescalled}, we have always observed this situation (represented in Figure~\ref{fig-shape}), even for large $\mu$. This result also allows us to show that behind the epidemic front, the densities $w(x)$ and $m(x)$ of the two pathogens stabilize to $w^*$, $m^*$, which is the long-term equilibrium of the system if no spatial structure is considered. For some results on the monotony of solutions of the non-local Fisher-KPP equation, we refer to \cite{FZ,AC}. For models closer to \eqref{rescalled} (see e.g. \cite{ACR,Betal}), we do not believe any qualitative result describing the shape of the travelling waves exists.
\begin{figure}[h]
\centering
\includegraphics{shape.pdf}
\caption{Numerical simulation of \eqref{rescalled} with \qg{$r=2$, $K=0.5$, $\mu=0.01$}, with a heaviside initial condition for \qg{$w$ and null initial condition for $m$.} The numerical code is based on an \qg{implicit Euler scheme}. For large times, the solution seems to converge to a travelling wave, that we represent here, propagating towards large $x$. In the initial phase of the epidemics, the mutant ($m$, red line) population is dominant, but this mutant population is then quickly replaced by a population almost exclusively composed of wild types ($w$, green line).}
\label{fig-shape}
\end{figure}
\begin{thm}\label{thm:monotonicity}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists a solution $(c,w,m)\in \mathbb R_+\times C^\infty(\mathbb R)^2$ of \eqref{systemefront} such that
\[\lim_{x\to-\infty}(w(x),m(x))=(w^*,m^*),\quad\lim_{x\to\infty}(w(x),m(x))=(0,0),\]
where $(w^*,m^*)$ is the only solution $(w^*,m^*)\in (0,1]\times (0,K]$ of $f_w(w,m)=f_m(w,m)=0$.
The solution $(c,w,m)\in \mathbb R_+\times C^\infty(\mathbb R)^2$ satisfies one of the three following properties:
\begin{description}
\item[(a)] $w$ is decreasing on $\mathbb R$, while $m$ is increasing on $(-\infty,\bar x]$ and decreasing on $[\bar x,\infty)$ for some $\bar x<0$,
\item[(b)] $m$ is decreasing on $\mathbb R$, while $w$ is increasing on $(-\infty,\bar x]$ and decreasing on $[\bar x,\infty)$ for some $\bar x<0$,
\item[(c)] $w$ and $m$ are decreasing on $\mathbb R$.
\end{description}
Moreover, there exists $\mu_0=\mu_0(r,K)>0$ such that if $\mu<\mu_0$, then there exists a solution as above which satisfies $(\mathrm a)$.
\end{thm}
Finally, we consider the special case where the mutant population is small (due to a small carrying capacity $K>0$ of the mutant, and a mutation rate \gr{satisfying} $0<\mu<K$). If we neglect the mutants completely, the dynamics of the wild type would be described by the Fisher-KPP equation \eqref{KPP} (with $\sigma=r=K=1$), and they would then propagate at the minimal propagation speed of the Fisher-KPP equation, that is $c=2$. Thanks to Theorem~\ref{thm:main}, we know already that the mutant population will indeed have a major impact on the minimal speed of the population which becomes $c_*=2\sqrt{r}+\mathcal O(\mu)>2$, and thus shouldn't be neglected. In the next theorem, we show that the profile of $w$ is indeed close to the travelling wave of the Fisher-KPP equation with the non-minimal speed $2\sqrt r$, provided the conditions mentioned above are satisfied (see Figure~\ref{fig-KPP}). The effect of the mutant is then essentially to speed up the epidemics.
\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{shape_smallK_KPP.pdf}\hfill
\includegraphics[scale=0.75]{shape_bigK_KPP.pdf}\\\medskip
\includegraphics[scale=0.75]{shape_bigbigK_KPP.pdf}
\caption{Comparison of the travelling wave solutions of \eqref{rescalled} and the travelling wave solution of the Fisher-KPP equation of (non-minimal) speed $2\sqrt r$. These figures are obtained for \qg{$r=2$, $\mu=0.001$,} and three values of $K$: \qg{$K=0.05,\,0.25,\,0.75$.} We see that the agreement between the density of the wild type ($w$, green line) and the corresponding solution of the Fisher-KPP equation ($u$, dashed blue line) is good as soon as $K\leq 0.25$. The travelling waves solutions of \eqref{rescalled} are obtained numerically as long-time solutions of \eqref{rescalled} (based on an explicit Euler scheme), while the travelling waves solutions of the Fisher-KPP equations (for a the given speed $2\sqrt r$ that is not the minimal travelling speed for the Fisher-KPP model) is obtained thanks to a phase-plane approach, with a classical ODE numerical solver.}
\label{fig-KPP}
\end{figure}
\begin{thm}\label{thm:KPP}
Let $r\in(1,\infty)$, $K\in (0,1)$, $\mu\in(0,K)$ and $(c_*,w,m)\in\mathbb R\times C^0(\mathbb R)^2$ (see Theorem~\ref{thm:main} for the definition of $c_*$), $w,\,m>0$, a solution of \eqref{systemefront} such that
\[ \underset{x\to-\infty}{\liminf}(w(x)+m(x))>0,\quad \underset{x\to\infty}{\lim}(w(x)+m(x))=0.\]
There exists $ C=C(r)>0$, $ \beta\in\left(0, \frac{1}{2}\right) $ and $ \varepsilon>0 $ such that if $ 0<\mu<K<\varepsilon $, then
\[ \Vert w-u\Vert_{L^\infty}\leqslant C K^\beta, \]
where $ u \in C^0(\mathbb R)$ is a traveling wave of the Fisher-KPP equation, that is a solution (unique up to a translation) of
\begin{equation}\label{travelling_wave_KPP}
\left\{\begin{array}{l}
-cu'- u''= u(1-u),\\
\underset{x\to-\infty}{\lim}u(x)=1,\; \underset{x\to\infty}{\lim} u(x)=0,
\end{array}\right.
\end{equation}
with speed $ c=\qg{c_0=2\sqrt r.} $
\end{thm}
The Theorem~\ref{thm:KPP} is interesting from an epidemiological point of view: it describes a situation where the spatial dynamics of a population would be driven by the characteristics of the mutants, even though the population of these mutants pathogens is very small, and thus difficult to sample in the field.
\section{Proof of Theorem \ref{thm:main}}\label{section:proof_box}
We will prove Theorem~\ref{thm:main} in several steps. We refer to Remark~\ref{Rk:endproof} for the conclusion of the proof.
\subsection{A priori estimates on a localized problem}\label{subsection:apriori}
We consider first a restriction of the problem \eqref{systemefront} to a compact interval $[-a,a]$, for $a>0$. More precisely, we consider, for $c\in\mathbb R$,
\begin{equation}
\left\{\begin{array}{l}
w,\, m\in C^0([-a, a]),\\% \;w,\,m\geq 0,\\
-cw'-w''=f_w(w, m)\chi_{w\geq 0}\chi_{m\geq 0}, \\
-cm'-m''=f_m(w, m)\chi_{w\geq 0}\chi_{m\geq 0}, \\
w(-a)=w^*,\, m(-a)=m^*,\, w(a)=m(a)=0,
\end{array}\right. \label{eq:pbbox}
\end{equation}
where we have used the notation \eqref{def_f}, and $(w^*,m^*)$ are defined in the Appendix, see Subsection~\ref{appendix_reaction_terms}.
\subsubsection{Regularity estimates on solutions of \eqref{eq:pbbox}}\label{subsubsection:regularity}
The following result shows the regularity of the solutions of \eqref{eq:pbbox}.
\begin{prop}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass} and $a > 0$. If $ (w, m) \in L^\infty([-a, a]) $ satisfies
\begin{equation}
\left\{\begin{array}{l} -cw'-w''=f_w(w, m), \\ -cm'-m''=f_m(w, m), \end{array}\right.
\label{eq:thmregularity}
\end{equation}
on $ [-a, a]$, where $f_w,\,f_m$ are defined by \eqref{def_f}, and $c\in\mathbb R$, then $ w, m\in C^\infty([-a, a]). $
\label{thm:regularity}
\end{prop}
\begin{proof}[Proof of Proposition \ref{thm:regularity}]
Since $f_w(w, m),\,f_m(w, m)\in L^\infty([-a,a])\subset L^p([-a, a])$ for any $p>1$, the classical theory (\cite{GT98},
theorem 9.15) predicts that the solutions of the Dirichlet problem associated with \eqref{eq:thmregularity} lies in \qg{$ W^{2, p}. $} This shows that $ w, m\in \qg{W^{2, p}} ((-a, a))$ for any $ p>1. $ But
then $ w, m\in C^{1, \alpha} ((-a, a))$ for any $ 0\leqslant \alpha < 1 $ (thanks to Sobolev embeddings). It follows
that $ f(w, m) $ is a $ C^{1, \alpha}((-a, a)) $ function of the variable $ x\in(-a, a)$ (see \eqref{def_f} for the definition of $f$). Let us choose one such $ \alpha\in (0,1)$. Now we can apply classical theory (\cite{GT98},
theorem 6.14) to deduce that $ w, m\in C^{2, \alpha}((-a, a))$. But then $ w''$ and $m'' $ verify some uniformly
elliptic equation of the type
\[ -c(w'')'-(w'')''=g, \]
\[ -c(m'')'-(m'')''=h, \]
with $ g, h\in C^{0, \alpha}((-a, a)) $, and we can apply again (\cite{GT98}, theorem 6.14). This argument can be used
recursively to show that $ w, m\in C^{2n, \alpha} ((-a, a))$ for any $ n\in\mathbb N $, so that finally, $ w, m\in C^\infty((-a, a))$.
\end{proof}
\subsubsection{Positivity and $ L^\infty $ bounds for solutions of \eqref{eq:pbbox}}\label{subsubsection:Linfty}
In this subsection, we prove the positivity of the solutions of \eqref{eq:pbbox}, as well as some $ L^\infty $ bounds.
\begin{prop}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, $a>0$, and $c\in\mathbb R$. If $(w,m)\in C^0([-a,a])^2$ is a solution of \eqref{eq:pbbox}, then $w$ and $m$ satisfy are positive, that is $w(x)>0$ and $m(x)>0$ for all $x\in [-a,a)$.
\label{lem:samesolbox}
\end{prop}
\begin{proof}[Proof of Proposition \ref{lem:samesolbox}]
We observe that
\[ f_w(w, m)=w(1-(w+m))+\mu(m-w)=w(1-\mu-w)+m(\mu-w), \]
so that if $ w\leqslant\min(\mu, 1-\mu), $ then $ f_w(w, m)\chi_{w\geqslant 0}\chi_{m\geqslant 0}
\geqslant 0. $ Let $ x_0\in [-a, a] $ such that $ w(x_0)\leqslant 0, $ and $ [\alpha, \beta] $ the
connex
compound of the set $ \{w\leqslant \min(\mu, 1-\mu)\} $ that contains $ x_0. $ Since $ -cw'-w''\geqslant 0 $
over $(\alpha, \beta) $ and $ w(\alpha), w(\beta)\geqslant 0, $ the weak minimum principle imposes $\underset{(\alpha, \beta)}{\inf}w\geqslant 0$, and thus $ w(x_0)=0. $ But then $ w $ reaches its global minimum at $ x_0, $ so the strong maximum
principle imposes that $ x_0\in\{\alpha, \beta\}, $ or else $ w $ would be constant. We deduce then from
our hypothesis $ (w(-a)>0, w(a)=0) $ that $ x_0=\beta=a. $ That shows that $ w>0 $ in $ [-a, a). $
\medskip
To show that $m>0$, we notice that
\begin{eqnarray*}
f_m(w, m)&=&rm\left(1-\frac{w+m}{K}\right)+\mu(w-m) \\
&=&m\left(r-\mu-\frac{r}{K}m\right)+ w\left(\mu-\frac{r}{K}m\right),
\end{eqnarray*}
so that if $ m\leqslant\min\left(\frac{K}{r}\mu, K\left(1-\frac{\mu}{r}\right)\right), $
then $ f_m(w, m)\chi_{w\geqslant 0}\chi_{m\geqslant 0}
\geqslant 0$. The end of the argument to show the positivity of $w$ can the n be reproduced to show that $ m>0 $.
\end{proof}
\begin{prop}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, $a>0$, and $c\in\mathbb R$. If $(w,m)\in C^0([-a,a])^2$ is a positive solution of \eqref{eq:pbbox}, then $w$ and $m$ satisfy
\[ \forall x\in (-a, a), \quad w(x)< 1, \]
\[ \forall x\in (-a, a), \quad m(x)< K. \]
\label{thm:precisebound}
\end{prop}
\begin{proof}[Proof of Proposition \ref{thm:precisebound}]
Let $ (w,m) $ a positive solution of \eqref{eq:pbbox}.
\begin{itemize}
\item We assume that there exists $ x_0\in(-a, a) $ such that $ w(x_0)>1. $ Let then $ [a_1, a_2] $ the
connex compound of the set $ \{w\geqslant 1\} $ that contains $ x_0. $ Then in $ (a_1, a_2) $ we have
\begin{eqnarray*}
-cw'-w''&=&w(1-\mu-w-m)+\mu m\leqslant w(-\mu-m)+\mu m\\
&=&m(\mu-w)-\mu w\leqslant 0,
\end{eqnarray*}
along with $ w(a_1)=w(a_2)=1, $ so that the weak {maximum} principle states $ w\leqslant 1 $ in
$ (a_1, a_2), $ which is absurd because $ w(x_0)>1. $ Therefore, $ w(x)\leqslant 1$ for all $x\in(-a, a)$
\item We assume that there exists $ x_0\in(-a, a) $ such that $ m(x_0)>K. $ Let then $ [a_1, a_2] $ the
connex compound of the set $ \{m\geqslant K\} $ that contains $ x_0. $ Then in $ (a_1, a_2) $ we have
\begin{eqnarray*}
-cm'-m''&=&m\left(r-\mu-\frac{r}{K}(w+m)\right)+\mu w\leqslant m\left(-\mu-\frac{rw}{K}\right)+
\mu w \\
&=&w\left(\mu-\frac{r}{K}m\right)-\mu m \leqslant 0,
\end{eqnarray*}
Thanks to Assumption~\ref{ass}. Since $ m(a_1)=m(a_2)=K$, the weak {maximum} principle states $ m\leqslant K $ in
$ (a_1, a_2), $ which is absurd because $ m(x_0)>K. $ Therefore, $ m(x)\leqslant K$ for all $x\in(-a, a)$.
\item Now if $ w(x)\in(\max(\mu,1-\mu), 1]$, we still have the estimate
\[ -cw'(x)-w''(x)\leqslant m(x)(\mu-w(x))+w(x)(1-\mu-w(x))\leqslant 0,\]
so that if there exists $x_0\in(-a,a)$ such that $ w(x_0)=1$, then $ w $ is locally equal to $ 1 $ thanks to the strong
maximum principle. But in that case
\[ 0=(-cw'-w'')(x_0)=-m(x_0)+\mu(m(x_0)-1)<0, \]
which is absurd. Hence, $w<1$.
Similarly, if $ m(x_0)=K$, we get
\[ 0=(-cm'-m'')(x_0)=-K\mu+w(x_0)(\mu-r)<0, \]
which is absurd, and thus $m<K$.
\end{itemize}
\end{proof}
\subsubsection{Estimates on solutions of \eqref{eq:pbbox} when $c\geq c^*$ or $c=0$}\label{subsubsection:c}
The next result shows that the solutions of \eqref{eq:pbbox} degenerate when
$ a\rightarrow +\infty $
if the speed $ c$ is larger than a minimal speed $c^*$ (see Theorem \ref{thm:main} for the definition of $c^*$).
\begin{prop}[Upper bound on $c$]
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists $C>0$ such that for $a>0$ and $ c\geqslant c_* $, any solution $(w,m)\in C^0([-a,a])^2$ of \eqref{eq:pbbox} satisfies
$$\forall x\in [-a,a],\quad \max\left(w(x), m(x)\right)\leq Ce^{\frac{-c-\sqrt{c^2-c_*^2}}{2}(x+a)}.$$
\label{lem:refupperbound}
\end{prop}
\begin{proof}[Proof of Proposition \ref{lem:refupperbound}]
Let $ c\geqslant c_*$, and
\[ M:=\left(\begin{matrix} 1-\mu & \mu \\ \mu & r-\mu \end{matrix}\right) .\]
Since $M+\mu\, Id$ is a positive matrix, the Perron-Frobenius theorem implies that $M$ has a principal eigenvalue $h^+$ and a positive principal eigenvector $X$ (that is $X_i>0$ for $i=1,2$), given by
\begin{eqnarray}
&h_+=\frac{1+r-2\mu+\sqrt{(1-r)^2+4\mu^2}}{2},\quad X=\left(\begin{matrix} 1-r+\sqrt{(1-r)^2+4\mu^2} \\ 2\mu \end{matrix}\right).&\label{def:hX}
\end{eqnarray}
The function $\psi_\eta(x):=\eta Xe^{\lambda_-x} $ with $\lambda_-:=\frac{-c-\sqrt{c^2-c_*^2}}{2}$ and $ \eta>0 $ is then a solution of the equation
\[ -c\psi_\eta'-\psi_\eta''=M\psi_\eta=h_+\psi_\eta. \]
We can define $ \mathcal A=\{\eta, (\psi_\eta)_1\geqslant w\textrm{ on }[-a,a]\}\cap\{\eta, (\psi_\eta)_2\geqslant m\textrm{ on }[-a,a]\} $, which is a closed subset of $ \mathbb R^+$. $\mathcal A$ is non-empty since $w$ and $m$ are bounded while $\left(Xe^{\lambda_-x}\right)_i \geq X_ie^{\lambda_-a}>0$ for $i=1,\,2$.
Consider now $ \eta_0:=\inf\mathcal A$. Then $ (\psi_\eta)_1\geq w$, $ (\psi_\eta)_2\geq m$, and there exists $ x_0\in [-a,a] $ such that either $ (\psi_\eta)_1(x_0)=w(x_0) $ or $ (\psi_\eta)_2(x_0)=m(x_0)$. We first consider the case where
$ (\psi_\eta)_1(x_0)=w(x_0)$. Then
\[ -c(w-(\psi_\eta)_1)'(x_0)-(w-(\psi_\eta)_1)''(x_0)\leqslant-w(x_0)\left(w(x_0)+m(x_0)\right)\leqslant 0 \]
over $ [-a, a]$. The weak maximum principle (\cite{GT98}, theorem 8.1) implies that
\[ \underset{[-a, a]}{\sup}(w-(\psi_\eta)_1)=\max((w-(\psi_\eta)_1)(-a), (w-(\psi_\eta)_1)(a)), \]
and then, thanks to the definition of $ \eta_0$, $\underset{[-a, a]}{\sup}(w-(\psi_\eta)_1)=0$. Since $ w(a)=0<(\psi_\eta)_1(a), $ this means that $ (\psi_\eta)_1(-a)=w(-a) $, and thus
\[ \eta_0=\frac{b_w^-}{1-r+\sqrt{(1-r)^2+4\mu^2}}e^{\lambda_-a}. \]
The argument is similar if $ (\psi_\eta)_2(x_0)=m(x_0)$, which concludes the proof.
\end{proof}
The following Proposition will be used to show that $ c\neq 0$.
\begin{prop}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, and $a >a_0:= \frac{\pi}{\sqrt{2(1-\mu)}}$. Every positive solution $ (w,m)\in C^0([-a,a])^2$ of \eqref{eq:pbbox} with $c=0$ satisfies the estimate
\begin{equation}\label{granden0}
\underset{[-a_0, a_0]}{\max}(w+m)\geqslant \frac{K}{2}(1-\mu).
\end{equation}
\label{lem:nosmallc}
\end{prop}
\begin{proof}[Proof of Proposition \ref{lem:nosmallc}]
We assume that $c=0$, $a>a_0$, and that \eqref{granden0} does not hold. We want to show that those assumptions lead to a contradiction.
For $ A\geqslant 0, $ the function defined by
\[ \psi_A(x)=A\cos\left(\sqrt{\frac{1-\mu}{2}}x\right), \]
is a solution of the equation $-\psi_A''=\frac{1-\mu}{2}\psi_A$ over $[-a_0,a_0]$. Since $ w,m>0 $ over $ [-a_0, a_0]$ and are bounded, the set $\mathcal A:=\{A, \forall x\in[-a_0, a_0], \psi_A(x)\leqslant \min(w(x),m(x))\}$ is a closed bounded nonempty set in $(0, +\infty)$.
Let now $ A_0 :=\max\mathcal A. $
We still have
$ \psi_{A_0}\leqslant\min(w,m) $ over $ [-a_0, a_0]$, and then, since \eqref{granden0} does not hold and $K<1$,
\begin{eqnarray}
-(w-\psi_{A_0})''&\geqslant& (1-\underset{[-a_0, a_0]}{\max}(w+m)-\mu)w-\frac{1-\mu}{2}\psi_{A_0}\label{eq1}\\
&\geqslant&
\frac{1-\mu}{2}(w-\psi_{A_0})\geqslant 0.\nonumber
\end{eqnarray}
Similarly, using additionally that $r>1$,
\[ -(m-\psi_{A_0})''\geqslant \frac{1-\mu}{2}(m-\psi_{A_0})\geqslant 0. \]
The weak {minimum} principle (\cite{GT98}, theorem 8.1) then imposes
\begin{align*}
& \min\left(\underset{[-a_0, a_0]}{\inf}(w-\psi_{A_0}),\underset{[-a_0, a_0]}{\inf}(m-\psi_{A_0})\right)\\
&\quad =\min((w-\psi_{A_0})(-a_0), (w-\psi_{A_0})(a_0), (m-\psi_{A_0})(-a_0), (m-\psi_{A_0})(a_0)).
\end{align*}
But the left side of the equation is $ 0 $ by definition of $ A_0, $ while the right side is strictly positive
since $ \psi_{A_0}(-a_0)=\psi_{A_0}(a_0)=0$. This contradiction shows the result.
\end{proof}
\begin{Rk}\label{Rk:gene_subsection_c}
Notice that Propositions~\ref{thm:regularity}, \ref{lem:samesolbox},\ref{thm:precisebound}, \ref{lem:refupperbound} and \ref{lem:nosmallc} also holds if $(c,w,m)\in \mathbb R\times C^0([-a,a])$ is a solution of
\begin{equation}
\left\{\begin{array}{l}
w,\, m\in C^0([-a, a]),\\
-cw'-w''=\left(w(1-(w+\sigma m))+\mu(\sigma m-w)\right)\chi_{w\geq 0}\chi_{m\geq 0}, \\
-cm'-m''=\left(rm\left(1-\left(\frac{\sigma w+m}{K}\right)\right)+\mu(\sigma w-m) \right)\chi_{w\geq 0}\chi_{m\geq 0}, \\
w(-a)=w^*,\, m(-a)=m^*,\, w(a)=m(a)=0,
\end{array}\right. \label{eq:pbbox_sigma}
\end{equation}
where $\sigma\in[0,1]$.
\end{Rk}
\subsection{Existence of solutions to a localized problem}\label{subsection:existence_localized}
To show the existence of travelling waves solutions of \eqref{systemefront}, we will follow the approach of \cite{ACR}. The first step is to show the existence of solutions of \eqref{eq:pbbox} satisfying the additional normalization property $\underset{[-a_0, a_0]}{\max}(w+m)={\nu_0}$, that is the existence of a solution $(c,w, m)$ to
\begin{equation}
\left\{\begin{array}{l}
(c,w, m)\in\mathbb R\times C^0([-a, a])^2, \\
-cw'-w''=f_w(w, m)\chi_{w\geqslant 0}\chi_{m\geqslant 0}, \\
-cm'-m''=f_m(w, m)\chi_{w\geqslant 0}\chi_{m\geqslant 0}, \\
w(-a)=w^*,\, m(-a)=m^*,\, w(a)=m(a)=0,\\
\underset{[-a_0, a_0]}{\max}(w+m)={\nu_0},
\end{array}\right. \label{pbnormbox}
\end{equation}
where $f_w,\,f_m$ are defined by \eqref{def_f}, { $\nu_0=\min\left(\frac{K}{4}(1-\mu), \frac{w^*+m^*}{2}\right) $} and $w^*,\,m^*$ are defined in Appendix \ref{appendix_reaction_terms}.
We introduce next the Banach space $(X,\Vert\cdot\Vert_X)$, with $ X:=\mathbb R\times C^0([-a, a])^2$ and $\Vert(c, w, m)\Vert_X := \max(|c|, \underset{[-a, a]}{\sup} |w|, \underset{[-a, a]}{\sup} |m|)$. We also define the operator
\begin{equation}\label{Ksigma}
\begin{array}{rccl}
K^\sigma: & X & \longrightarrow & \qquad X, \\
& (c,w,m) & \longmapsto & (c+\underset{[-a_0, a_0]}{\max}(\tilde w+\tilde m)-{\nu_0}, \tilde w,
\tilde m)
\end{array}
\end{equation}
where $ (\tilde w, \tilde m)\in C^0([-a, a])^2 $ is the unique
solution of
\begin{equation*}
\left\{\begin{array}{l} -c\tilde w'-\tilde w''= \left[w(1-(w+\sigma m))+\mu(\sigma m-w)\right]\chi_{w\geqslant 0}\chi_{m\geqslant 0}\textrm{ on } (-a, a), \\
-c\tilde m'-\tilde m''=\left[rm\left(1-\left(\frac{\sigma w+m}{K}\right)\right)+\mu(\sigma w-m)\right]\chi_{w\geqslant 0}\chi_{m\geqslant 0} \textrm{ on } (-a, a), \\
\tilde w(-a)=w^*,\,\tilde m(-a)=m^*,\, \tilde w(a)=\tilde m(a)=0.
\end{array}\right.
\end{equation*}
The solutions of \eqref{pbnormbox} with $ c\geqslant 0 $ are then the fixed points of $ K^1 $ in the domain $\{(c, w, m), 0\leqslant w\leqslant 1, 0\leqslant m\leqslant K, c\geqslant 0\}$.
We define
\[ \Omega:=\left\{(c, w, m)\in \mathbb R_+\times C^0([-a,a])^2;\; c\in (0,c_*),\,\forall x\in[-a,a],\,-1<w(x)<1,\, -K<m(x)<K \right\}, \]
where $ c_* $ is defined by \eqref{eq:defminc}.
\begin{lem}\label{Lemma:continuous_operator_family}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, and $a >0$. Then, $(K^\sigma)_{\sigma\in[0, 1]}$,
defined by \eqref{Ksigma}, is a family of compact operators on $(X,\Vert\cdot\Vert_X)$, that is continuous with respect to
$\sigma\in[0, 1]$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{Lemma:continuous_operator_family}]
We can write $ K^\sigma=(\mathcal L_D)^{-1}\circ \mathcal F^\sigma $ where $ (\mathcal L_D)^{-1} $ is defined by
\[ (\mathcal L_D)^{-1}(c, g, h)=(\tilde c, \tilde w, \tilde m), \]
where $ (\tilde c, \tilde w, \tilde m) $ is the unique solution of
\[ \left\{\begin{array}{l} -c\tilde w'-\tilde w''=g\textrm{ on } (-a, a), \\
-c\tilde m'-\tilde m''=h\textrm{ on }(-a, a), \\
\tilde w(-a)=w^*,\, \tilde m(-a)=m^*,\,\tilde w(a)=\tilde m(a)=0, \\
\tilde c=c+\underset{[-a_0, a_0]}{\max}(\tilde w+ \tilde m) - {\nu_0},
\end{array}\right. \]
and $ \mathcal F^\sigma $ is the mapping
\[ \mathcal F^\sigma(c, w, m)=\left(c, w(1-(w+\sigma m))+\mu(\sigma m-w),rm\left(1-\frac{\sigma w+m}{K}\right)+\mu(\sigma w-m)\right). \]
$ \sigma\mapsto\mathcal F^\sigma$ is a continuous mapping from $ [0, 1] $ to $ C^0(\Omega, X)$, and $ (\mathcal L_D)^{-1} $ is a continuous application from $(X,\Vert\cdot\Vert_X)$ into itself (see Lemma \ref{lem:degreecompacity}), it then follows that $ \sigma\mapsto K^\sigma=(\mathcal L_D)^{-1}\circ \mathcal F^\sigma$ is a continuous mapping from $ [0, 1] $ to
$ C^0(\Omega, X)$. Finally, the operator $(\mathcal L_D)^{-1}$ is compact (see Lemma \ref{lem:degreecompacity}), which implies that $ K^\sigma$ is compact for any fixed $ \sigma\in [0,1] $.
\end{proof}
We now introduce the following operator, for $\sigma\in[0,1]$:
\begin{equation}\label{lemma:Ftau}
F^\sigma:=Id-K^\sigma.
\end{equation}
Similarly, we introduce the operator
\begin{equation}\label{Ktau}
\begin{array}{rccl}
K_\tau: & X & \longrightarrow & \qquad X, \\
& (c,w,m) & \longmapsto & (c+\underset{[-a_0, a_0]}{\max}(\tilde w+\tilde m)-{\nu_0}, \tilde w,
\tilde m)
\end{array}
\end{equation}
where $ (\tilde w, \tilde m)\in C^0([-a, a])^2 $ is the unique
solution of
\begin{equation}\label{def_tilde_w_m} \left\{\begin{array}{l} -c\tilde w'-\tilde w''=\tau w(1-\mu -w)\chi_{w\geqslant 0}\chi_{m\geqslant 0} \textrm{ on } (-a, a), \\
-c\tilde m'-\tilde m''=\tau rm\left(1-\qg{\frac{\mu}{r}} -\frac mK\right) \chi_{w\geqslant 0}\chi_{m\geqslant 0} \textrm{ on } (-a, a), \\
\tilde w(-a)=w^*,\,\tilde m(-a)=m^*,\, \tilde w(a)=\tilde m(a)=0.
\end{array}\right.
\end{equation}
The argument of Lemma~\ref{Lemma:continuous_operator_family} can be be reproduced to prove that $(K_\tau)_{\tau\in[0,1]}$ is also a continuous family of compact operators on $(X,\|\cdot\|_X)$, and we can define, for $\tau\in[0,1]$, the operator
\begin{equation}\label{lemma:Ftau2}
F_\tau:=Id-K_\tau.
\end{equation}
Finally, we introduce, for some $\bar c<0$ that we will define later on,
\[ \tilde\Omega:=\left\{(c, w, m)\in \mathbb R_+\times C^0([-a,a])^2;\; c\in (\bar c,c_*),\,\forall x\in[-a,a],\,-1<w(x)<1,\, -K<m(x)<K \right\}.\]
In the next Lemma, we will show that the Leray-Schauder degree of $F_0$ in the domain $\tilde \Omega$ is non-zero as soon as $a>0$ is large enough. We refer to chapter 12 of \cite{Smo} or to chapter 10-11 of \cite{Brown} for more on the Leray-Schauder degree.
\begin{lem}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists $\bar a>0$ such that the Leray-Schauder degree of $F_0$ in the domain $\tilde \Omega$ is non-zero as soon as $a\geqslant \bar a$.
\label{nonzerodegreeF0}
\end{lem}
\begin{proof}[Proof of Lemma \ref{nonzerodegreeF0}]
We first notice that for $\tau=0$, the solution $(\tilde w,\tilde m)$ of \eqref{def_tilde_w_m} is independent of $(w,m)$, and then, \[F_0(c,w,m)=\left({\nu_0}-\underset{[-a_0, a_0]}{\max}(w_{c}+m_{c}),w-w_c,m-m_c\right),\]
where $(w_{c},m_{c})$ is the solution of \eqref{def_tilde_w_m} with $\tau=0$, that is
\[\left(w_c,m_c\right)(x):=\left(w^*\left(\frac{e^{-cx}-e^{-ca}}{e^{ca}-e^{-ca}}\right),m^*\left(\frac{e^{-cx}-e^{-ca}}{e^{ca}-e^{-ca}}\right)\right),\]
for $c\neq 0$, and $(w_c,m_c)(x)=(\frac{a-x}{2a}w^*,\frac{a-x}{2a}m^*)$ for $c=0$.
The solutions of $F_0(c, w, m)=0$ then satisfy $w=w_c$ and $m=m_c$. In particular, the solutions of $F_0(c, w, m)=0$ satisfy $0<w<1$ and $0<m<K$ on $[-a,a)$, and then,
\begin{eqnarray*}
(c, w, m)&\notin&\left\{(\tilde c, \tilde w, \tilde m)\in \mathbb R\times C^0([-a,a])^2;\; \exists x\in[-a, a],\, \tilde w(x)\in\{-1, 1\}\right\} \\
&&\cup \left\{(\tilde c, \tilde w, \tilde m)\in \mathbb R\times C^0([-a,a])^2;\; \exists x\in[-a, a],\, \tilde m(x)\in\{-K, K\}\right\}\Big).
\end{eqnarray*}
The solutions of $F_0(c_*, w, m)=0$ also satisfy
\[\underset{[-a_0, a_0]}{\max}(w_{c_*}+m_{c_*})\leq 2\frac {e^{c_*a_0}}{e^{c_*a}-1},\]
so that $ \underset{[-a_0, a_0]}{\max}(w_{c_*}+m_{c_*})<
{\nu_0}$ if $a>\bar a$ for some $\bar a>0$. It follows that $F_0=0$ has no solution in $\overline{\tilde \Omega}\cap\left(\{c^*\}\times C^0([-a,a])^2\right)$, provided $a>\bar a$.
Finally, for $c\leq 0$, the solutions of $F_0(c, w, m)=0$ satisfy $(w_c,m_c)(x)\geq (w_0,m_0)(x)=\left(-\frac{w^*}{2a}x+\frac{w^*}{2},-\frac{m^*}{2a}x+\frac{m^*}{2}\right)$, so that
\[ \underset{[-a_0, a_0]}{\max}(w_c+m_c)>\underset{[-a_0, a_0]}{\max}(w_0+m_0)= \frac{w^*+m^*}{2}\left(1+\frac{a_0}{a}\right) > {\frac{w^*+m^*}{2}\geq \nu_0}, \]
and $F_0=0$ has no solution in $\overline{\tilde \Omega}\cap\left(\mathbb R_-\times C^0([-a,a])^2\right)$.
\medskip
We notice next that since $c\mapsto \underset{[-a_0, a_0]}{\max}(w_c+m_c)$ is decreasing, there exists a unique $c_0\in (0,c_*)$ such that $\underset{[-a_0, a_0]}{\max}(w_{c_0}+m_{c_0})={\nu_0}.$ We can then define
\[ \Phi_\tau(c, w, m)= \left({\nu_0}-\underset{[-a_0, a_0]}{\max}(w_c+m_c),
w-\left((1-\tau)w_c+\tau w_{c_0}\right) , m-\left((1-\tau)m_c+\tau m_{c_0}\right)\right),\]
which connects continuously $F_0=\Phi_0$ to
\[ \Phi_1(c, w, m)=\left({\nu_0}-\underset{[-a_0, a_0]}{\max}(w_c+m_c),
w-w_{c_0},m-m_{c_0}\right). \]
Notice that $\Phi_\tau(c,w,m)=0$ implies $\underset{[-a_0, a_0]}{\max}(w_c+m_c)={\nu_0}$, which in turn implies that $c=c_0$. For any $\tau\in[0,1]$, the only solution of $ \Phi_\tau(c,w,m)=0 $ is then $(c_0,w_{c_0},m_{c_0})\not\in\partial\tilde \Omega$, which implies that the Leray-Schauder degree $\deg (F_0,\tilde \Omega)$ of $F_0$ is equal to $\deg(\Phi_1,\tilde\Omega)$, which can easily be computed since its variables are separated :
\begin{eqnarray*}
\deg(\Phi_1,\Omega)&=&\deg\left({\nu_0}-\underset{[-a_0, a_0]}{\max}(w_c+m_c),(\bar c,c_*)\right)\\
&&\deg\left(w-w_{c_0},\left\{\tilde w\in C^0([-a,a]);\;-1<\tilde w(x)<1\right\}\right)\\
&&\deg\left(m-m_{c_0},\left\{\tilde m\in C^0([-a,a]);\;-K<\tilde m(x)<K\right\}\right)=1.
\end{eqnarray*}
\end{proof}
Next, we show that the Leray-Schauder degree of $F^0$ in the domain $\Omega$ is also non-zero, as soon as $a>0$ is large enough.
\begin{lem}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists $\bar a>0$ such that the Leray-Schauder degree of $F^0$ in the domain $\Omega$ is non-zero as soon as $a\geqslant \bar a$.
\label{nonzerodegreeF0sigma}
\end{lem}
\begin{proof}[Proof of Lemma \ref{nonzerodegreeF0sigma}]
Thanks to Proposition~\ref{lem:nosmallc} and Remark~\ref{Rk:gene_subsection_c}, any solution $(c,w,m)\in \tilde\Omega$ of \eqref{eq:pbbox_sigma}, and thus any solution $(c,w,m)\in \tilde\Omega$ of $F^0(c,w,m)=0$ satisfies $c>0$, that is $(c,w,m)\in\Omega$. Then,
\begin{equation}\label{eq2}
\deg(F^0,\Omega)=\deg(F^0,\tilde\Omega)=\deg(F_1,\tilde\Omega).
\end{equation}
For $\tau\in[0,1]$, any solution $(c,w,m)\in\tilde \Omega$ of $F_\tau(c,w,m)=0$ satisfies
\[-cw'-w''\geq -\mu w,\quad -cm'-m''\geq -\mu m,\]
and then $w,m\geq \phi_c$, where $\phi_c$ is the solution of $-c\phi_c'-\phi_c''= -\mu\phi_c$ with $\phi_c(-a)=K$, $\phi_c(a)=0$. This solution can easily be computed explicitly, and satisfies (for any fixed $a>0$)
\[\lim_{c\to-\infty}\phi_c(0)=K.\]
we can then choose $-\bar c>0$ large enough for $\phi_{\bar c}(0)\geq \nu_0$ to hold (note that the constant $\bar c\in\mathbb R$ is not independent of $a$). Then, $F_\tau(\bar c,w,m)=0$ implies $\underset{[-a_0, a_0]}{\max}(w+m)\geq 2\phi_{\bar c}(0)>\nu_0$, which implies in turn that $F_\tau(c,w,m)=0$ has no solution on $\left(\{\bar c\}\times C^0([-a,a])^2\right)\cap \overline{\tilde \Omega}$, for any $\tau\in[0,1]$. If $F_\tau(c,w,m)=0$ with $(c,w,m)\in\overline{\tilde \Omega}$, a classical application of the strong maximum principle shows that $0<w<1$ and $0<m<K$ on $(-a,a)$ (notice that $w$ and $m$ are indeed solutions of two uncoupled Fisher-KPP equations on $[-a,a]$). Moreover, the proof of Proposition~\ref{lem:refupperbound} applies to solutions of $F_\tau(c_*,w,m)=0$, which implies that (for any $\tau\in[0,1]$),
\[ \max_{[-a_0, a_0]}(w+m)\leq Ce^{-c_*\frac{a-a_0}2},\]
and thus, $F_\tau(c,w,m)=0$ has no solution on $\left(\{c_*\}\times C^0([-a,a])^2\right)\cap \overline{\tilde \Omega}$ as soon as $a>0$ is large enough (uniformly in $\tau\in[0,1]$).
We have shown that $F_\tau(c,w,m)=0$ has no solution on $\partial \tilde\Omega$ for $\tau\in[0,1]$. Since $\tau\mapsto F_\tau$ is a continuous familly of compact operators on $\tilde \Omega$, this implies that
\[\deg(F_1,\tilde\Omega)=\deg(F_0,\tilde\Omega),\]
which, combined to \eqref{eq2} and Proposition~\ref{nonzerodegreeF0}, concludes the proof.
\end{proof}
\begin{prop}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists $\bar a>0$ such that for $a\geqslant \bar a$, there exists
a solution $(c,w,m)\in \mathbb R\times C^0([-a,a])^2$ of \eqref{pbnormbox} with $c\in(0, c_*)$.
\label{thm:localexistencedegree}
\end{prop}
\begin{proof}[Proof of Proposition \ref{thm:localexistencedegree}]
The first step of the proof is to show that there exists no solution $(c,w,m)\in\partial\Omega$ of $F^\sigma(c,w,m)=0$ with $\sigma\in[0,1]$.
If such a solution exists, then Proposition \ref{lem:nosmallc} (see also Remark~\ref{Rk:gene_subsection_c}) implies that $c\neq 0$, and if $c=c_*$, then Proposition \ref{lem:refupperbound} (see also Remark~\ref{Rk:gene_subsection_c}) implies that
\begin{equation}
\underset{[-a_0, a_0]}{\max}(w+ m)\leqslant Ce^{-c_*\frac{a-a_0}2},
\label{eq:prooflemdegreeFtau}
\end{equation}
where $C>0$ is a positive constant independent from $\sigma\in[0,1]$. If $a$ is large enough (more precisely if $a\geq a_0+\frac 2{c_*}\ln\left({\frac{2C}{\nu_0}}\right)$), then $\underset{[-a_0, a_0]}{\max}(w+m)\leq {\frac{\nu_0}{2}}$, which is a contradiction. Any solution $(c,w,m)\in \overline\Omega$ of $F^\sigma(c,w,m)=0$ then satisfies $c\in(0,c_*)$, as soon as $a>0$ is large enough.
Any solution $(c,w,m)\in \overline\Omega$ of $F^\sigma(c,w,m)=0$ is a solution of \eqref{eq:pbbox_sigma}, Proposition~\ref{lem:samesolbox} and Proposition~\ref{thm:precisebound} (see also Remark~\ref{Rk:gene_subsection_c}) then imply that for any $x\in(-a,a)$, $0< w(x)<1$ and $0< m(x)<K$.
We have shown that $F^\sigma(c,w,m)=0$ had no solution $(c,w,m)\in\partial\Omega$, for $\sigma\in [0,1]$. Since moreover $(F_\sigma)_{\sigma\in[0,1]}$ is a continuous family of compact operators (see Lemma~\ref{Lemma:continuous_operator_family}) this is enough to show that $\deg(F^\sigma,\Omega)$ is independent of $\sigma\in[0,1]$, and then, thanks to Lemma~\ref{nonzerodegreeF0sigma}, as soon as $a>0$ is large enough,
\[\deg(F^1,\Omega)=\deg(F^0,\Omega)\neq 0.\]
which implies in particular that there exists at least one solution $(c,w,m)\in\Omega$ of $F^1(c,w,m)=0$, that is a solution $(c,w,m)$ of \eqref{pbnormbox} in $\Omega$.
\end{proof}
\subsection{Construction of a travelling wave}\label{subsection:whole_line}
\begin{prop}\label{prop:existence_front}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. There exists a solution $(c, w, m)\in(0, c_*]\times C^0(\mathbb R)^2$ of problem \eqref{systemefront} that satisfies $0<w(x)<1$ and $0<m(x)<K$ for $x\in\mathbb R$, as well as $(w+m)(0)={\nu_0}. $
\end{prop}
\begin{proof}[Proof of Proposition \ref{prop:existence_front}]
For $n\geq 0$, let $ a_n:=\bar a+n $ (where $ \bar a>0 $ is defined in Proposition \ref{thm:localexistencedegree}), and $(c_n, w_n, m_n)$ a solution of \eqref{pbnormbox} provided by Proposition \ref{thm:localexistencedegree}.
We denote by $(w_n^k, m_n^k)$ the restriction of $(w_n, m_n)$ to $ [-a_k, a_k] $ $ (k<n). $ From interior elliptic estimates (see e.g. Theorem 8.32 in \cite{GT98}), we know that there exists a constant $C>0$ independent of $k>0$, such that for any $n\geq k+1$,
\[ \max\left(\left\Vert w_n|_{[-a_k, a_k]}\right\Vert_{C^1([-a_k, a_k])},\left\Vert m_n|_{[-a_k, a_k]}\right\Vert_{C^1([-a_k, a_k])}\right)\leqslant C, \]
Since $c_n\in [0,c^*]$ for all $n\in\mathbb N$, we can extract from $(c_n, w_n, m_n)$ a subsequence (that we also denote by $(c_n, w_n, m_n)$), such that $c_n\rightarrow c_0$ for some $c_0\in[0,c^*]$. Since $c_n\in (0,c_*)$ for all $n\geq 3$, the limit speed satisfies $c_0\in[0,c_*]$. Thanks to Ascoli's Theorem, $C^1([-a_k, a_k])$ is compactly embedded in $C^0([-a_k,a_k])$. We can then use a diagonal extraction, to get a subsequence such that $w_n$ and $m_n$ both converge uniformly on every compact interval of $\mathbb R$.
Let $w_0,\, m_0\in C^0(\mathbb R)$ the limits of $(w_n)_n$ and $(m_n)_n$ respectively. Then, thanks to the uniform convergence, we get that
\[\forall x\in\mathbb R,\quad 0\leqslant w_0(x)\leqslant 1,\quad 0\leqslant m_0(x)\leqslant K, \]
\[ -c_0w_0'-w_0''=f_w(w_0, m_0)\textrm{ on }\mathbb R, \]
\[ -c_0m_0'-m_0''=f_m(w_0, m_0)\textrm{ on }\mathbb R, \]
in the sense of distributions. Thanks to Proposition \ref{thm:regularity}, these two functions are smooth and are thus classical solutions of \eqref{systemefront}. Moreover, $ \underset{[-a_0, a_0]}{\max}(w_0+m_0)={\nu_0}$, and Lemma \ref{lem:nosmallc} implies that $c_0\neq 0$. Finally, up to a shift, $w_0(0)+m_0(0)={\nu_0}$.
\end{proof}
In the next proposition, we show that the solution of \eqref{systemefront} obtained in Proposition~\ref{prop:existence_front} are indeed propagation fronts.
\begin{prop}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass} and $(c, w, m)\in\mathbb R\times C^0(\mathbb R)^2$ a solution of \eqref{systemefront} such that $(w+m)(0)={\nu_0}$.
Then $w+m$ is decreasing on $(0,+\infty)$,
\[\lim_{x\to \infty}w(x)=\lim_{x\to \infty}m(x)=0,\]
and $w(x)+m(x)\geq {\nu_0}$ on $(-\infty,0]$.
\label{prop:propright}
\end{prop}
\begin{proof}[Proof of Proposition \ref{prop:propright}]
Assume that $w(x)+m(x)< K$, and $w'(x)+m'(x)\geq 0$. Then, \qg{
\begin{equation}\label{eqconcavity}
-c(w+m)'(x)-(w+m)''(x)=w(x)(1-(w(x)+m(x)))+rm(x)\left(1-\frac{w(x)+m(x)}{K}\right),
\end{equation} }
\qg{with the right side positive, }and then $(w+m)''(x)< 0$.
If there exists $x_0\in\mathbb R$ satisfying $w(x_0)+m(x_0)< K$, and $w'(x_0)+m'(x_0)\geq0$, then we can define $ \mathcal C=\{x\leqslant x_0, \forall y\in[x, x_0], (w+m)''(y)\leqslant 0\}$. Then $\mathcal C\neq\varnothing$ and $\mathcal C$ is closed since $(w+m)''$ is continuous. Let $x_1\in \mathcal C$. Then $(w+m)'$ is decreasing on $[x_1, x_0]$, so that $(w+m)'(x_1)\geqslant (w+m)'(x_0)>0$ and $(w+m)(x_1)\leqslant (w+m)(x_0)$. \eqref{eqconcavity} then implies that $(w+m)''(x_1)<0$, which proves that $ \mathcal C $ is open, and thus $\mathcal C=(-\infty, 0)$. This implies in particular that $w(x)+m(x)<0$ for some $x<x_0$, which is a contradiction. We have then proven that $x\mapsto w(x)+m(x)$ is decreasing on $[x_0,\infty)$ as soon as $w(x_0)+m(x_0)\leq K$. It implies that $w(x)+m(x)\geq {\nu_0}$ for $x\leq 0$, and that $x\mapsto w(x)+m(x)$ is decreasing on $[0,\infty)$.
Then, $\lim_{x\to\infty}w(x)+m(x)=l\in [0,K)$ exists, which implies that $\lim_{x\to\infty}w'(x)+m'(x)=\lim_{x\to\infty}w''(x)+m''(x)=0$, since $w$ and $m$ are regular. Then,
\[\lim_{x\to\infty}\left( -c(w+m)'(x)-(w+m)''(x)\right)=0,\]
which, combined to \eqref{eqconcavity}, proves that $\lim_{x\to\infty}w(x)+m(x)=0$.
\end{proof}
\subsection{Characterization of the speed of the constructed travelling wave}
\begin{lem}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass} and $(c,w,m)\in\mathbb R\times C^0(\mathbb R)^2$ a solution of \eqref{systemefront} such that $(w+m)(0)={\nu_0}$. Then there exists $x_0\in\mathbb R$ and $C>0$ such that
\[\forall x\geqslant x_0,\quad w(x)+m(x)\leqslant C \min(w(x),m(x)).\]
\label{lem:compuvsum}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:compuvsum}]
Let $S(x):=w(x)+m(x)$, and $\alpha>0$. Then
\[-c(S-\alpha w)'-(S-\alpha w)''=((1-S)-(1-S-\mu)\alpha)w+(S-w)\left(r\left(1-\frac{S}{K}\right)-\alpha\mu\right). \]
Let $x_1\in\mathbb R$ such that $S(x)\leqslant \frac{1-\mu}{2}$ for all $x\geq x_1$ ($x_1$ exists thanks to Proposition~\ref{prop:propright}). Then, for $\alpha\geqslant\alpha_0:=\max\left(\frac{2}{1-\mu}, \frac{r}{\mu}\right)$ and $x\geqslant x_1$,
\[-c(S-\alpha w)'(x)-(S-\alpha w)''(x)\leqslant 0. \]
Let $\alpha_1:=\max\left(\frac{S(x_1)}{w(x_1)}, \alpha_0\right)+1$. Then $-c(S-\alpha_1 w)'-(S-\alpha_1 w)''\leqslant 0 $ over $(x_1, +\infty)$. We can then apply the weak maximum principle to show that for any $x_2>x_1$,
\[\max_{[x_1,x_2]} (S-\alpha_1 w)(x)=\max\left((S-\alpha_1 w)(x_1),(S-\alpha_1 w)(x_2)\right).\]
Since $(S-\alpha_1 w)(x_1)\leqslant 0$ and $\underset{x_2\rightarrow+\infty}{\lim}(S-\alpha_1 w)(x_2)=0$, we have indeed shown that $\sup_{[x_1,\infty)} (S-\alpha_1 w)(x)=0$, and then,
\[\forall x\geq x_1,\quad w(x)+m(x)\leq \alpha_1w(x).\]
A similar argument can be used to show that there exists $x_2\in\mathbb R$ and $\alpha_2>0$ such that $w(x)+m(x)\leqslant \alpha_2 m(x)$ for $x\geqslant x_2$, which concludes the proof of the Lemma.
\end{proof}
\begin{prop}\label{prop:vitesse_front}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass} and $(c,w,m)\in\mathbb R_+\times C^0(\mathbb R)^2$ a solution of \eqref{systemefront} such that $(w+m)(0)={\nu_0}$ and $c\leq c_*$. Then, $c=c_*$.
\end{prop}
\begin{Rk}\label{Rk:endproof}
Combined to Proposition~\ref{prop:existence_front} and Proposition~\ref{prop:propright}, this proposition completes the proof of Theorem~\ref{thm:main}.
\end{Rk}
\begin{proof}[Proof of Proposition \ref{prop:vitesse_front}]
Let $(c,w,m)\in[0,c_*]\times C^0(\mathbb R)^2$ a solution of \eqref{systemefront} such that $(w+m)(0)={\nu_0}$. Thanks to Lemma~\ref{lem:compuvsum}, there exists $x_0>0$ and $C>0$ such that
\begin{equation*}
\left\{\begin{array}{l}
-cw'-w''\geq w(1-\mu-C w)+\mu m,\\
-cm'-m''\geq m(r-\mu-C m)+\mu w.
\end{array}
\right.
\end{equation*}
Let now $\varphi_\eta(x+x_1):=\eta e^{-\frac c2x}\sin\left(\frac{\sqrt{4h-c^2}}2x\right)$, where $\eta>0$, $h\geq c^2/4$ and $x_1>x_0$. $\varphi_\eta$ then satisfies $\varphi_\eta(x_1)=\varphi_\eta\left(x_1+2\pi/\sqrt{4h-c^2}\right)=0$, and $-c\varphi_\eta'-\varphi_\eta''=h\varphi_\eta$ on $\left[x_1,x_1+2\pi/\sqrt{4h-c^2}\right]$. $\psi_\eta:=\varphi_\eta X$ ($X$ is defined by \eqref{def:hX}) is then a solution of
\[-c\psi_\eta'-\psi_\eta''=(M+(h-h_+)Id)\psi_\eta,\]
where $h_+$ is defined by \eqref{def:hX}, and we can also write this equality as follows
\begin{equation*}
\left\{\begin{array}{l}
-c(\psi_\eta)_1'-(\psi_\eta)_1''= (\psi_\eta)_1\left(1-\mu+(h-h_+)\right)+\mu (\psi_\eta)_2,\\
-c(\psi_\eta)_2'-(\psi_\eta)_2''= (\psi_\eta)_2\left(r-\mu+(h-h_+)\right)+\mu (\psi_\eta)_1.
\end{array}
\right.
\end{equation*}
Assume now that $c<c_*$. Then, we can choose $c^2/4<h<c_*^2/4=h_+$, and define
\[\bar\eta=\max\left\{\eta>0;\,\forall x\in\left[x_1,x_1+2\pi/\sqrt{4h-c^2}\right],\, (\psi_\eta)_1(x)\leq w(x),\,(\psi_\eta)_2(x)\leq m(x)\right\}.\]
Since $w$ and $m$ are positive bounded function, $\bar\eta>0$ exists, and since
$$(\psi_\eta)_i(x_1)=(\psi_\eta)_i\left(x_1+2\pi/\sqrt{4h-c^2}\right)=0,$$
there exists $\bar x\in \left(x_1,x_1+2\pi/\sqrt{4h-c^2}\right)$ such that either $(\psi_\eta)_1(\bar x)= w(\bar x)$ or $(\psi_\eta)_2(\bar x)= m(\bar x)$. Assume w.l.o.g. that $(\psi_\eta)_1(\bar x)= w(\bar x)$. Then $w-(\psi_\eta)_1$ has a local minimum in $\bar x$, which implies that
\begin{eqnarray*}
0&\geq& -c(w-(\psi_{\bar \eta})_1)'(\bar x)-(w-(\psi_{\bar \eta})_1)''(\bar x)\\
&\geq& \left[w(\bar x)(1-\mu-C w(\bar x))+\mu m(\bar x)\right]-\left[(\psi_{\bar \eta})_1(\bar x)\left(1-\mu+(h-h_+)\right)+\mu (\psi_{\bar \eta})_2(\bar x)\right]\\
&\geq&(\psi_{\bar \eta})_1(\bar x)\left[(h_+-h)-C(\psi_{\bar \eta})_1(\bar x)\right],
\end{eqnarray*}
and then $\bar \eta\geq (h_+-h)/(CX_1)$. A similar argument holds if $(\psi_\eta)_2(\bar x)= m(\bar x)$, so that in any case, $\bar \eta\geq (h_+-h)/(C\max(X_1,X_2))$, and $\psi_{\bar\eta}(x_1+\cdot)\leq m$, $\psi_{\bar\eta}(x_1+\cdot)\leq m$ on $\left[x_1,x_1+2\pi/\sqrt{4h-c^2}\right]$, as soon as $x_1\geq x_0$. In particular, for any $x_1\geq x_0$,
\[w\left(x_1+\pi/\sqrt{4h-c^2}\right)\geq \frac{h_+-h}{C\max(X_1,X_2)}e^{-\frac {c\pi}{2\sqrt{4h-c^2}}}X_1>0,\]
which is a contradiction, since $w(x)\to_{x\to\infty}0$ thanks to Proposition~\ref{prop:propright}.
\end{proof}
\section{Proof of Theorem \ref{thm:monotonicity}}
\label{sec:monotonicity}
\subsection{General case}
\label{sub:general_case}
The proof of the next lemma is based on a phase-plane-type analysis, see Figure~\ref{fig-phase}
\begin{figure}[h]
\centering
\includegraphics{phase_plan.pdf}
\caption{Phase-plane-type representation of a solution of \eqref{eq:pbbox}: we represent (dark line) $x\mapsto (w(x),m(x))\in [0,1]\times[0,K]$. Note that the usual phase-plane for \eqref{eq:pbbox} is of dimension $4$. The blue line represents the set of $(w,m)$ such that $f_w(w,m)=0$ (see \eqref{def_f}), $f_w(w,m)>0$ for $(m,w)$ on the left of the blue curve. The green line represent the set of $(w,m)$ such that $f_m(w,m)=0$ (see \eqref{def_f}), $f_m(w,m)>0$ for $(m,w)$ under the green curve. the gray lines represent several other solutions of \eqref{systemefront} such $(w(-a),m(-a))=(w^*,m^*)$. The dashed dark lines separate this phase plane into four compartiments that will be used in the third step of the proof of Lemma~\ref{lem:plan_phase}. \qg{Finally, the solid black line corresponds to the travelling wave.}}
\label{fig-phase}
\end{figure}
\begin{lem}\label{lem:plan_phase}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. Let $(c,w,m)\in \mathbb R_+\times C^0([-a,a])^2 $ be a solution of \eqref{eq:pbbox}.
Then there exists $\bar x\in[-a,0)$ such that one of the following properties is satisfied:
\begin{itemize}
\item $w$ is decreasing on $[-a,a]$, while $m$ is increasing on $[-a,\bar x]$ and decreasing on $[\bar x,a]$,
\item $m$ is decreasing on $[-a,a]$, while $w$ is increasing on $[-a,\bar x]$ and decreasing on $[\bar x,a]$.
\end{itemize}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem:plan_phase}]
\noindent\emph{Step 1: sign of $f_w$ and $f_m$.}
We recall the definition \eqref{def_f} of $f_w,\,f_m$.
The inequality $f_w(w,m)\geq 0$ is equivalent, for $w\in[0,1]$ and $m\in[0,K]$, to
\begin{equation}\label{def:phiw}
w\leq \phi_w(m):= \frac 12\left[1-\mu-m+\sqrt{(1-\mu-m)^2+4\mu m}\right].
\end{equation}
Notice that $m\in[0,K]\mapsto \phi_w(m)$ is a decreasing function (see Lemma~\ref{lem:assumption2}), that divides the square $\{(w,m)\in[0,1]\times[0,K]\}$ into two parts.
Similarly, $f_m(w,m)\geq 0$ is equivalent, for $w\in[0,1]$ and $m\in[0,K]$, to
\begin{equation}\label{def:phim}
m\leq \phi_m(w):= \frac 12\left[K-\frac{\mu K}{r}-w+\sqrt{\left(K-\frac{\mu K}{r}-w\right)^2+4\frac{\mu K}{r}w}\right].
\end{equation}
Here also, $w\in[0,1]\mapsto \phi_m(w)$ is a decreasing function (see Lemma~\ref{lem:assumption2}), since $\mu\leq 1/2$ (see Assumption \ref{ass}), that divides the square $\{(w,m)\in[0,1]\times[0,K]\}$ into two parts.
\medskip
\noindent\emph{Step 2: possible monotony changes of $w(x),\,m(x)$.}
Let $(c,w,m)\in \mathbb R_+\times C^0([-a,a])^2$ be a solution of \eqref{eq:pbbox}. If $w'(x)\geq 0$ for some $x> -a$, we can define $\bar x:=\inf\{y\geq x;\,w'(y)< 0\}$. Then $w'(\bar x)=0$, and $w''(\bar x)\leq 0$, which implies
$$f_w(w(\bar x),m(\bar x))=-cw'(\bar x)-w''(\bar x) \geq 0,$$
that is $w(\bar x)\leq \phi_w(m(\bar x))$. The symmetric property also holds: if $w'(x)\leq 0$ for some $x> -a$, we can define $\bar x:=\inf\{y\geq x;\,w'(y)> 0\}$, and then, $w(\bar x)\geq \phi_w(m(\bar x))$.
We repeat the argument for the function $m$: let $(c,w,m)\in \mathbb R_+\times C^0([-a,a])^2$ be a solution of \eqref{eq:pbbox}. If $m'(x)\geq0$ for some $x> -a$, we can define $\bar x:=\inf\{y\geq x;\,m'(y)< 0\}$, and then, $m(\bar x)\leq \phi_m(w(\bar x))$. Finally, if $m'(x)\leq 0$ for some $x> -a$, we can define $\bar x:=\inf\{y\geq x;\,m'(y)> 0\}$, and then, $m(\bar x)\geq \phi_m(w(\bar x))$.
\medskip
\noindent\emph{Step 3: phase plane analysis}
Notice that $(w(-a),m(-a))=(w^*,m^*)$, and then,
\begin{equation}\label{phi_bar_mw}
m(-a)=\phi_m(w(-a)),\quad w(-a)=\phi_w(m(-a)).
\end{equation}
We will consider now consider individually the four possible signs of $w'(-a), $ $ m'(-a)$ (the cases where $w'(-a)=0$ or $m'(-a)=0$ will be considered further in the proof):
(i) Assume that $w'(-a)>0$ and $m'(-a)>0$. We define $\bar x:=\inf\{y\geq {\color{red}-a};\,w'(y)< 0\textrm{ or }m'(y)<0\}$. Since $w$ and $m$ are increasing on $[-a,\bar x]$, \eqref{phi_bar_mw} holds and $w\mapsto \phi_m(w)$, $m\mapsto \phi_w(m)$ are decreasing functions, we have
$$w(\bar x)>w(-a)=\phi_w(m(-a))\geq\phi_w(m(\bar x)),$$
$$m(\bar x)>m(-a)=\phi_m(w(-a))\geq\phi_m(w(\bar x)).$$
Then, $f_w(w(\bar x),m(\bar x))<0$ and $f_m(w(\bar x),m(\bar x))<0$. It then follows from Step~2 that $\bar x=a$, which means that $w$ and $m$ are increasing on $[-a,a]$. It is a contradiction, since $0=w(a)<w(-a)=\bar w$.
Notice that the same argument would also work \qg{on $[x,a]$, for any $ (w(x), m(x)) $ that satisfies $ w(x)>\Phi_w(m(x))$, $m(x)> \Phi_m(w(x))$, $w'(x)>0$ and $m'(x)>0$.}
(ii) Assume that $w'(-a)<0$ and $m'(-a)<0$. Let $\bar x:=\inf\{y\geq -a;\,w'(y)> 0\textrm{ or }m'(y)>0\}$. Since $w$ and $m$ are decreasing on $[-a,\bar x]$, \eqref{phi_bar_mw} holds and $w\mapsto \phi_m(w)$, $m\mapsto \phi_w(m)$ are decreasing functions, we have
$$w(\bar x)<w(-a)=\phi_w(m(-a))\leq\phi_w(m(\bar x)),$$
$$m(\bar x)<m(-a)=\phi_m(w(-a))\leq \phi_m(w(\bar x)).$$
It then follows from Step~2 that $\bar x=a$, which means that $w$ and $m$ are non-increasing on $[-a,a]$. Notice that this is not a contradiction, since $w(a)=0<w^*=w(-a)$, $m(a)=0<m^*=m(-a)$.
Notice that the same argument would work \qg{on $[x,a]$, for any $ (w(x), m(x)) $ that satisfies $ \Phi_w(m(x))> w(x) $, $ m(x)<\Phi_m(w(x))$, $w'(x)<0$ and $m'(x)<0$. }
(iii) Assume that $w'(-a)<0$ and $m'(-a)>0$. We define $\bar x:=\inf\{y\geq -a;\,w'(y)> 0\textrm{ or }m'(y)<0\}$. The argument used in the two previous cases cannot be employed here.
We know however that $w(\bar x)<w^*$, $m(\bar x)>m^*$. Since $m(a)=0<m^*$, it implies in particular that $\bar x<a$, and, with the notations of Lemma~\ref{lem:zerosinclusions}, $(w(\bar x),m(\bar x))\in \mathcal D_l$.
If $w'$ changes sign in $\bar x$, then Step 2 implies that $w(\bar x)\geq \phi_w(m(\bar x))$, that is, with the notations of Lemma~\ref{lem:zerosinclusions}, $(w(\bar x),m(\bar x))\in Z_w^-$. Thanks to Lemma~\ref{lem:zerosinclusions}, it follows that $(w(\bar x),m(\bar x))\in Z_w^-\cap \mathcal D_l\subset Z_m^-$, and then $m(\bar x)>\phi_m(w(\bar x))$, which implies $-cm'(\bar x)-m''(\bar x)=f_m(w(\bar x),m(\bar x))<0$. If $m'(\bar x)=0$, then $m''(\bar x)>0$, which is incompatible with the fact that $m'\geq 0$ on $[-a,\bar x)$ and $m'(\bar x)=0$. We have thus shown that $m'(\bar x)>0$. Thanks to the definition of $ \bar x$,
either $w$ is locally increasing near $ \bar x^+, $ or there exists a sequence $ (x_n)\to \bar x^+ $ such that $ w'(x_{2n})>0 $ and $ w'(x_{2n+1})<0. $ In the first case, for $ \varepsilon>0 $ small enough, $ w(\bar x+\varepsilon)>w(\bar x)\geq \Phi_w(m(\bar x))\geq\Phi_w(m(\bar x+\varepsilon))$ along with $ w'(\bar x+\varepsilon)>0. $ In the second case, $ w''(\bar x)=0 $, then $f_w(w, m)(\bar x)=0$, and a simple computation shows that for $ \varepsilon>0 $ small enough,
\[ f_w\left(w(\bar x+\varepsilon), m(\bar x+\varepsilon)\right)=(\mu-w(\bar x))\varepsilon m'(\bar x)+o(\varepsilon)<0, \]
where we have used the fact that $\mu-w(\bar x)<0$ (since $w(\bar x)\geq \phi_w(m(\bar x))\subset \phi_w([0,K])\subset(\mu,\infty)$, see Lemma~\ref{lem:assumption2}). In any case, for some $\varepsilon>0$ arbitrarily small, $w(\bar x+\varepsilon)> \phi_w(m(\bar x+\varepsilon))$, $m(\bar x+\varepsilon)>\phi_m(w(\bar x+\varepsilon))$, along with $w'(\bar x+\varepsilon)>0$ and $m'(\bar x+\varepsilon)>0$. argument (i) can now be applied to $(w,m)|_{[\bar x+\varepsilon,a]}$, leading to a contradiction.
If $m'$ changes sign in $\bar x$, then Step 2 implies that $m(\bar x)\leq \phi_m(w(\bar x))$, that is, with the notations of Remark~\ref{Rk:zerosinclusions}, $(w(\bar x),m(\bar x))\in Z_m^+$. Thanks to Remark~\ref{Rk:zerosinclusions}, it follows that $(w(\bar x),m(\bar x))\in Z_m^+\cap \mathcal D_l\subset Z_w^+$, and then $w(\bar x)<\phi_w(m(\bar x))$, which implies $-cw'(\bar x)-w''(\bar x)=f_w(w(\bar x),m(\bar x))>0$. If $w'(\bar x)=0$, then $w''(\bar x)<0$, which is incompatible with the fact that $w'\leq 0$ on $[-a,\bar x)$ and $w'(\bar x)=0$. We have thus shown that $w'(\bar x)<0$. Thanks to the definition of $ \bar x$,
either $m$ is locally decreasing near $ \bar x^+, $ or there exists a sequence $ (x_n)\to \bar x^+ $ such that $ m'(x_{2n})>0 $ and $ m'(x_{2n+1})<0. $ In the first case, for $ \varepsilon>0 $ small enough, $ m(\bar x+\varepsilon)<m(\bar x)\leq \Phi_m(w(\bar x))\leq\Phi_m(w(\bar x+\varepsilon))$ along with $ m'(\bar x+\varepsilon)>0. $ In the second case, $ m''(\bar x)=0 $, then $f_m(w, m)(\bar x)=0$, and a simple computation shows that for $ \varepsilon>0 $ small enough,
\[ f_m\left(w(\bar x+\varepsilon), m(\bar x+\varepsilon)\right)=\left(\mu-\frac{r}{K}m(\bar x)\right)\varepsilon w'(\bar x))+o(\varepsilon)>0, \]
where we have used the fact that $\mu-\frac{r}{K}m(\bar x)<0$ (since $m(\bar x)>m^*=\phi_m(w^*)\subset \phi_m([0,1])\subset (\mu K/r,\infty)$, see Lemma~\ref{lem:assumption2}). In both cases, argument~(ii) can now be applied to $(w,m)|_{[\bar x+\varepsilon,a]}$, which is not a contradiction, since $w(a)=0<w(\bar x)$, $m(a)=0<m(\bar x)$.
(iv) Assume that $w'(-a)>0$ and $m'(-a)<0$. We define $\bar x:=\inf\{y\geq -a;\,w'(y)< 0\textrm{ or }m'(y)>0\}$. Then $w(\bar x)>w^*$, $m(\bar x)<m^*$. Since $m(a)=0<m^*$, it implies in particular that $\bar x<a$, and, with the notations of Lemma~\ref{lem:zerosinclusions}, $(w(\bar x),m(\bar x))\in \mathcal D_r$.
If $w'$ changes sign in $\bar x$, then Step 2 implies that $w(\bar x)\leq \phi_w(m(\bar x))$, that is, with the notations of Remark~\ref{Rk:zerosinclusions}, $(w(\bar x),m(\bar x))\in Z_w^+$. Thanks to Remark~\ref{Rk:zerosinclusions}, it follows that $(w(\bar x),m(\bar x))\in Z_w^+\cap \mathcal D_r\subset Z_m^+$, and then $m(\bar x)<\phi_m(w(\bar x))$, which implies $-cm'(\bar x)-m''(\bar x)=f_m(w(\bar x),m(\bar x))>0$. If $m'(\bar x)=0$, then $m''(\bar x)<0$, which is incompatible with the fact that $m'\leq 0$ on $[-a,\bar x)$ and $m'(\bar x)=0$. We have thus shown that $m'(\bar x)<0$. Thanks to the definition of $ \bar x$,
either $w$ is locally decreasing near $ \bar x^+, $ or there exists a sequence $ (x_n)\to \bar x^+ $ such that $ w'(x_{2n})>0 $ and $ w'(x_{2n+1})<0. $ In the first case, for $ \varepsilon>0 $ small enough, $ w(\bar x+\varepsilon)<w(\bar x)\leq \Phi_w(m(\bar x))\leq\Phi_w(m(\bar x+\varepsilon))$ along with $ w'(\bar x+\varepsilon)<0. $ In the second case, $ w''(\bar x)=0 $, then $f_w(w, m)(\bar x)=0$, and a simple computation shows that for $ \varepsilon>0 $ small enough,
\[ f_w(w, m)(\bar x+\varepsilon)=(\mu-w(\bar x))\varepsilon m'(\bar x)+o(\varepsilon)>0, \]
where we have used the fact that $\mu-w(\bar x)<0$ (since $w(\bar x)>w^*=\phi_w(m^*)\subset \phi_w([0,1])\subset (\mu,\infty)$, see Lemma~\ref{lem:assumption2}). In both cases, argument~(ii) can now be applied to $(w,m)|_{[\bar x+\varepsilon,a]}$, which is not a contradiction, since $w(a)=0<w(\bar x)$, $m(a)=0<m(\bar x)$.
If $m'$ changes sign in $\bar x$, then Step 2 implies that $m(\bar x)\geq \phi_m(w(\bar x))$, that is, with the notations of Lemma~\ref{lem:zerosinclusions}, $(w(\bar x),m(\bar x))\in Z_m^-$. Thanks to Lemma~\ref{lem:zerosinclusions}, it follows that $(w(\bar x),m(\bar x))\in Z_m^-\cap \mathcal D_r\subset Z_w^-$, and then $w(\bar x)>\phi_w(m(\bar x))$, which implies $-cw'(\bar x)-w''(\bar x)=f_w(w(\bar x),m(\bar x))<0$. If $w'(\bar x)=0$, then $w''(\bar x)>0$, which is incompatible with the fact that $w'\geq 0$ on $[-a,\bar x)$ and $w'(\bar x)=0$. We have thus shown that $w'(\bar x)>0$. Thanks to the definition of $\bar x$,
either $m$ is locally increasing near $ \bar x^+, $ or there exists a sequence $ (x_n)\to \bar x^+ $ such that $ m'(x_{2n})>0 $ and $ m'(x_{2n+1})<0. $ In the first case, for $ \varepsilon>0 $ small enough, $ m(\bar x+\varepsilon)>m(\bar x)\geq \Phi_m(w(\bar x))\geq\Phi_m(w(\bar x+\varepsilon))$ along with $ m'(\bar x+\varepsilon)>0. $ In the second case, $ m''(\bar x)=0 $, then $f_m(w, m)(\bar x)=0$, and a simple computation shows that for $ \varepsilon>0 $ small enough,
\[ f_m(w, m)(\bar x+\varepsilon)=\left(\mu-\frac{r}{K}m(\bar x)\right)\varepsilon w'(\bar x)+o(\varepsilon)<0,\]
where we have used the fact that $\mu-\frac{r}{K}m(\bar x)<0$ (since $m(\bar x)\geq \phi_m(w(\bar x))\subset \phi_m([0,1])\subset(\mu K/r,\infty)$, see Lemma~\ref{lem:assumption2}). In both cases, argument~(i) can now be applied to $(w,m)|_{[\bar x+\varepsilon,a]}$, leading to a contradiction.
\medskip
Let consider now the case where $w'(-a)=0$ or $m'(-a)=0$. If $w'(-a)=m'(-a)=0$, then $w\equiv w^*$, $m\equiv m^*$, which is a contradiction. Assume w.l.o.g. that $w'(-a)\neq 0$. If there exists $\varepsilon>0$ such that for any $x\in [-a,-a+\varepsilon]$, $m'(x)=0$, then $m$ is constant on the interval $[-a,-a+\varepsilon]$, and then $f_m(w(x),m(x))=0$ for $x\in [-a,-a+\varepsilon]$. This implies in turn that $m(x)=\phi_m(w(x))$, and then $w$ is constant on $[-a,-a+\varepsilon]$, since $\phi_m$ is a decreasing function, which is a contradiction. There exists thus a sequence $x_n\to -a$, $x_n>-a$, such that $w(x_n)\neq w^*$, and $\textrm{sgn}(m(x_n)-m^*)=\textrm{sgn}(m'(x_n))\neq 0$, while $\textrm{sgn}(w(x_n)-w^*)=\textrm{sgn}(w'(0))\neq 0$. The above argument (i-iv) can therefore be reproduced for $(w,m)|_{[x_n,a]}$.
Finally, the fact that $\bar x\leq 0$ is a consequence of $w(0)+m(0)<\min (w^*,m^*)$.
\end{proof}
\begin{prop}\label{prop:monotonicity}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. Let $(c,w,m)\in \mathbb R_+\times C^0(\mathbb R)^2$ be a solution of \eqref{systemefront} constructed in Theorem \ref{thm:main}. Then, there exists $\bar x\in[-\infty,0)$ such that
\begin{itemize}
\item either $w$ is decreasing on $\mathbb R$, while $m$ is increasing on $(-\infty,\bar x]$ and decreasing on $[\bar x,\infty)$,
\item or $m$ is decreasing on $\mathbb R$, while $w$ is increasing on $(-\infty,\bar x]$ and decreasing on $[\bar x,\infty)$,
\end{itemize}
Moreover,
$$w(x)\to w^*,\quad m(x)\to m^*\textrm{ as }x\to-\infty.$$
\end{prop}
\begin{proof}[Proof of Proposition \ref{prop:monotonicity}]
The travelling wave $(c,w,m)$ constructed in Theorem \ref{thm:main} is obtained as a limit (in $L^\infty_{loc}(\mathbb R)$) of solutions $(w_n,m_n,c_n)\in \mathbb R_+\times C^0([-a_n,a_n])^2$ of \eqref{eq:pbbox} on $[-a_n,a_n]$, with $a_n\underset{n\to\infty}{\longrightarrow}\infty$. Each of those solutions then satisfy one of the two the monotonicity properties of Lemma~\ref{lem:plan_phase}. In particular, there is at least one of those properties that is satisfied by an infinite sequence of solutions $(w_n,m_n,c_n)$. We may then assume w.l.o.g. that all the solutions $(w_n,m_n,c_n)$ satisfy the first monotonicity property in Lemma~\ref{lem:plan_phase}. We assume therefore that for all $n\in\mathbb N$, there exists $\bar x_n\in[-a_n,0)$ such that $w_n$ is decreasing on $[-a_n,a_n]$, while $m_n$ is increasing on $[-a_n,\bar x_n]$ and decreasing on $[\bar x_n,a_n]$. Up to an extraction, we can define $\bar x:=\lim_{n\to\infty}a_n\in [-\infty,0]$. Then, $w$ is a uniform limit of decreasing function on any bounded interval, and is thus decreasing. Let now $\tilde x>\bar x$. $m_n$ is then a decreasing function on $[\tilde x,\infty)\cap[-a_n,a_n]$ for $n$ large enough, and $m|_{[\tilde x,\infty)}$ is thus a uniform limit of decreasing functions on any bouded interval of $[\tilde x,\infty)$. This implies that $m$ is decreasing on $[\bar x,\infty)$. A similar argument shows that $m$ is increasing on $(-\infty,\bar x]$, if $\bar x>-\infty$. the case where all the solutions $(w_n,m_n,c_n)$ satisfy the second monotonicity property in Lemma~\ref{lem:plan_phase} can be treated similarly.
\medskip
We have shown in particular that $w$, $m$ are monotonic on $(-\infty,\tilde x)$, for some $\tilde x<0$ ($\tilde x=\bar x$ if $\bar x>-\infty$, $\tilde x=0$ otherwise). Since $w$ and $m$ are regular bounded functions, it implies that
\[ f_w(w(x),m(x))=-cw'(x)-w''(x)\to 0, \]
\[ f_m(w(x),m(x))=-cm'(x)-m''(x)\to 0, \]
as $ x\to -\infty.$
This combined to $\liminf_{x\to-\infty}w(x)+m(x)>0$ and $(w,m)\in[0,1]\times [0,K]$ implies that $w(x)\to w^*$ and $m(x)\to m^*$ as $x\to-\infty.$
\end{proof}
\subsection{Case of a small mutation rate}
\label{sub:small_mu}
The result of this subsection shows that if $\mu>0$ is small, then only the first situation described in Lemma \ref{lem:plan_phase}, with $\bar x>-\infty$, is possible.
\begin{prop}\label{prop:monotonicity_mu_petit}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. Let $(c,w,m)\in \mathbb R_+\times C^0(\mathbb R)$ be a solution of \eqref{systemefront} constructed in Theorem \ref{thm:main}. There exists $\bar \mu=\bar \mu(r,K)>0$ such that $\mu<\bar \mu$ implies that $w$ is decreasing on $\mathbb R$, while $m$ is increasing on $(-\infty,\bar x]$ and decreasing on $[\bar x,\infty)$, for some $\bar x\in \mathbb R_-$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{prop:monotonicity_mu_petit}]
Notice that the solution $(c,w,m)$ satisfies the assumptions of Proposition \ref{prop:monotonicity}.
Let us assume that $\|m\|_\infty\leq m^*$. We will show that this assumption leads to a contradiction if $\mu>0$ is small. Let $\bar x=\max\left\{x>-\infty;\, w(x)\geq m^*\right\}$. Then $w$ satisfies $-cw'-w''\leq (1-\mu)w+\mu\,m^*$ on $(-\infty,\bar x]$. Since $(c,w,m)$ satisfies the assumptions of Proposition \ref{prop:monotonicity} and $w(\bar x)=m^*<w^*$, we have that $w(x)\geq m^*$ for all $x\leq \bar x$. $w$ thus satisfies $-cw'-w''\leq w$ on $(-\infty,\bar x]$. We define now
$$\bar w(x)=m^*\, e^{\frac{c-\sqrt{c^2-4}}2(\bar x-x)},$$
which satisfies $-c\bar w'-\bar w''=\bar w$ on $(-\infty,\bar x]$, $\bar w(\bar x)=w(\bar x)$, and $\bar w(y)\geq 1\geq w(y)$ for $y<<0$. Since $w$ is bounded, $\alpha \bar w>w$ for $\alpha>0$ large enough. We can then define $\alpha^*:=\min\{\alpha>0;\, \alpha \bar w>w \textrm{ on }(-\infty,\bar x)\}$. If $\alpha^*>1$, there exists $x^*\in (-\infty,\bar x)$ such that $\alpha^* \bar w(x^*)= w(x^*)$, and then, $-c(\alpha^*\bar w- w)'(x^*)-(\alpha^*\bar w- w)''(x^*)>\alpha^*\bar w(x^*)- w(x^*)=0$, which is a contradiction, since $\alpha^* \bar w> w$ implies that $-c(\alpha^*\bar w- w)'(x^*)-(\alpha^*\bar w- w)''(x^*)\leq 0$. Thus,
\begin{equation}\label{borne_sup_w}
w(x)\leq\bar w(x)
,\textrm{ for }x\in(-\infty,\bar x].
\end{equation}
In particular, if we define
\begin{equation}\label{def:tildex}
\tilde x:=\bar x-\frac 2{c-\sqrt{c^2-4}}\ln\qg{\left(\frac K{m^*}\left(\frac 12-\frac\mu r-\frac 1{2r}\right)-1\right)},
\end{equation}
then \qg{ $ w(x)\leq K\left(\frac 12-\frac \mu r-\frac 1{2r}\right)-m^*$ } on $[\tilde x,\bar x]$. \qg{Notice that $m^*\to 0$ as $\mu\to 0$ (see Lemma~\ref{lem:estmu0}), and then $\frac K{m^*}\left(\frac 12-\frac \mu r-\frac 1{2r}\right)\to \infty$ as $\mu\to 0$; $\tilde x$ is then well defined as soon as $ \mu>0$ is small enough, and $\tilde x-\bar x\to -\infty$ as $\mu\to 0$.} This estimate applied to the equation on $m$ (see \eqref{systemefront}), implies, for $x\in[\tilde x,\bar x]$, that
$$-cm'(x)-m''(x)\geq r\left(1-\frac \mu r-\frac{m^*+w(x)}K\right)m(x)+\mu\,w\geq \frac {1+r}2m+\mu\,m^* ,\textrm{ for }x\in(\tilde x,\bar x],$$
where we have also used the fact that $w\geq m^*$ on $(-\infty,\bar x]$.
\medskip
We define next
$$\bar m_1:=-\frac{\mu\,m^*}{c+2}(x-\bar x)\left(x-(\bar x-1)\right),$$
which satisfies $-c\bar m_1'-\bar m_1''\leq \mu m^*$ as well as $\bar m_1\left(\bar x-1\right)=0\leq m\left(\bar x-1\right)$ and $\bar m_1(\bar x)=0\leq m(\bar x)$. The weak maximum principle (\cite{GT98}, Theorem 8.1) then implies that $ m(x)\geq \bar m_1(x)$ for all \qg{$x\in\left[\bar x-1, \bar x\right]$,} and in particular,
$$m(\bar x-1/2)\geq \qg{ \frac{\mu\,m^*}{4(c+2)}.}$$
We define (we recall the definition \eqref{def:tildex} of $\tilde x$)
$$\bar m_2:= \frac{\mu\, m^*}{4(c+2)}e^{\frac{c-\sqrt{c^2-2(1+r)}}2\left(\bar x-1/2-x\right)}-Ae^{\frac{c}2\left(\bar x-1/2-x\right)},$$
with $A= \frac{\mu\, m^*}{4(c+2)}e^{-\frac{\sqrt{c^2-2(r+1)}}2(\bar x-1/2-\tilde x)}$, so that $\bar m_2(\tilde x)=0$. $\bar m_2$ then satisfies $-c\bar m_2'-\bar m_2''< \frac {(1+r)}2\bar m_2$, since $c(c/2)-(c/2)^2> \frac{1+r}2$ (see \eqref{eq:defminc}). Let now
$$\alpha^*:=\max\{\alpha;m(x)\geq \alpha \bar m_2(x),\,\forall x\in [\tilde x,\bar x-1/2]\}.$$
$\alpha^*>0$, since $\min_{[\tilde x,\bar x-1/2]}m>0$. If $\alpha^*<1$, then $\alpha^*\bar m_2(\bar x-1/2)< \frac{\mu\,m^*}{4(c+2)}\leq m(\bar x-1/2)$, while $\alpha^*\bar m_2(\tilde x)=0<m(\tilde x)$. Then $\alpha^*\bar m_2\leq m$ on $[\tilde x,\bar x-1/2]$, and there exists $x^*\in[\tilde x,\bar x-1/2]$ such that $\alpha^*\bar m_2(x^*)=m(x^*)$, and
$$0\leq -c(\bar m_2-m)'(x^*)-(\bar m_2-m)''(x^*)<\frac {(1+r)}2(\bar m_2-m)(x^*)=0,$$
which is a contradiction. We have thus proven that $m\geq \bar m_2$ on $[\tilde x,\bar x-1/2]$, and in particular, for $\mu>0$ small enough,
$$\|m\|_\infty\geq \bar m_2\left(\tilde x+\frac{2\ln(2)}{\sqrt{c^2-2(1+r)}}\right)=\frac{\mu\,m^*}{4(c+2)}e^{\frac{c\ln 2}{\sqrt{c^2-2(1+r)}}}e^{\frac{c-\sqrt{c^2-2(1+r)}}2\left(\bar x-1/2-\tilde x\right)}.$$
We recall indeed that $\tilde x-\bar x\to -\infty$ as $\mu\to 0$, and then $\tilde x+\frac{2\ln(2)}{\sqrt{c^2-2(1+r)}}\in [\tilde x,\bar x-1/2]$ if $\mu>0$ is small enough. Thanks to the definition of $\tilde x$, this inequality can be written
\begin{align*}
&\ln\left(\frac{4(c+2) \|m\|_\infty}{\mu\,m^*}\right)-\frac{c\ln 2}{\sqrt{c^2-2(1+r)}}\\
&\quad \geq \frac{c-\sqrt{c^2-2(1+r)}}2\left(-1/2+\frac 2{c-\sqrt{c^2-4}}\qg{\ln\left(\frac K{ m^*}\left(\frac 12-\frac \mu r-\frac 1{2r}\right)-1\right)}\right).
\end{align*}
We have assumed that $\|m\|_\infty=m^*$, thus, if we denote by $\mathcal O_{\mu\sim 0^+}(1)$ a function of $\mu>0$ that is bounded for $\mu$ small enough, we get
$$\ln\left(\frac 1\mu\right)+\mathcal O_{\mu\sim 0^+}(1)\geq \frac{c-\sqrt{c^2-2(1+r)}}{c-\sqrt{c^2-4}}\ln\left(\frac 1{ m^*}\right).$$
Moreover, we know that $m^*\leq C \mu$ for some $C> 0$, see Lemma \ref{lem:estmu0}. Then,
$$\ln\left(\frac 1\mu\right)+\mathcal O_{\mu\sim 0^+}(1)\geq \frac{c-\sqrt{c^2-2(1+r)}}{c-\sqrt{c^2-4}}\ln\left(\frac 1{\mu}\right),$$
which is a contradiction as soon as $\mu>0$ is small, since $r>1$.
\medskip
We have thus proved that for $\mu>0$ small enough, we have $\|m\|_\infty> m^*$. This estimate combined to Proposition~\ref{prop:monotonicity} proves Proposition~\ref{prop:monotonicity_mu_petit}.
\end{proof}
\section{Proof of Theorem \ref{thm:KPP}}
\label{sec:K_small}
Notice first that if we chose $\varepsilon>0$ small enough, then $0<\mu<K<\varepsilon$ implies that Assumption~\ref{ass} is satisfied.
We will need the following estimate on the behavior of travelling waves of \eqref{systemefront}:
\begin{prop}\label{prop:w_left}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}. Let $(c,w,m)\in \qg{\mathbb R_+\times C^\infty(\mathbb R)\times C^\infty(\mathbb R)}$ be a solution of \eqref{systemefront}, such that $\liminf_{x\to -\infty}(w(x)+m(x))>0$. Then, $\liminf_{x\to -\infty} w(x)\geq 1-\mu-K$.
Moreover, if $ w(\bar x)<1-K-\mu$ for some $\bar x\in\mathbb R$, then $w$ is decreasing on $[\bar x,\infty)$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{prop:w_left}]
Since $m(x)< K$ for all $x\in\mathbb R$, any local minimum $\bar x$ of $w$ satisfies
\begin{eqnarray}
0&\geq& -cw'(\bar x)-w''(\bar x)=(1-\mu-w(\bar x)-m(\bar x))w(\bar x)+\mu m(\bar x)\nonumber\\
&>&(1-\mu-K-w(\bar x))w(\bar x),\label{control_w}
\end{eqnarray}
and then $w(\bar x)\geq 1-\mu-K$.
Assume that $\liminf_{x\to -\infty}w(x)<1-\mu-K$. Then, $x\mapsto w(x)$ can not have a minimum for $x<<0$, and is thus monotonic for $x<<0$. Then $l:=\lim_{x\to-\infty}w(x)\in [0,1-\mu-K]$ exists and $w'(x)\to_{x\to -\infty}0$, $w''(x)\to_{x\to -\infty}0$. This implies $-cw'(x)-w''(x)\to_{x\to\infty} 0$, which, coupled to \eqref{control_w} implies that $l=0$ or $l=1-\mu-K$. $l=0$ leads to a contradiction, since $\liminf_{x\to -\infty}(w(x)+m(x))>0$, which proves the first assertion.
\medskip
To prove the second assertion, we notice that since $w$ cannot have a minimum $\tilde x\in\mathbb R$ such that $w(\tilde x)<1-K-\mu$, $w$ is monotonic on $\{x\in\mathbb R;\;w(x)<1-K-\mu\}$. This monotony combined to $\liminf_{x\to -\infty}w(x)\geq1-\mu-K>w(\bar x)$ implies that $w$ is decreasing on $[\bar x,\infty)$.
\end{proof}
The main idea of the proof of theorem~\ref{thm:KPP} is to compare $w$ to solutions of modified Fisher-KPP equations, which we introduce in the following lemma:
\begin{lem}
Let $r,\,K,\,\mu$ satisfy Assumptions~\ref{ass}. Let $(c,w,m)\in\qg{\mathbb R_+\times C^\infty(\mathbb R)\times C^\infty(\mathbb R)}$ be a solution of \eqref{systemefront}, with $c\geq 2+K$. Let also $\overline w\in C^\infty(\mathbb R)$, $\underline w\in C^\infty(\mathbb R)$ solutions of
\begin{equation}\label{overline_w}\left\{\begin{array}{l}
-c\overline w'-\overline w''=\overline w(1-\overline w)+K,\\
\overline w(x)\to_{x\to-\infty} \frac{1+\sqrt{1+4K}}{2},\; \overline w(x)\to_{x\to+\infty}-\frac{\sqrt{1+4K}-1}{2},
\end{array}\right.
\end{equation}
\begin{equation}\label{underline_w}\left\{\begin{array}{l}
-c\underline w'-\underline w''=\underline w(1-2K-\underline w),\\
\underline w(x)\to_{x\to-\infty}1-2K,\; \underline w(x)\to_{x\to+\infty}0.\phantom{sfqsqdgdfg}
\end{array}\right.
\end{equation}
Assume $ \underline w(0)\leqslant w(0)\leqslant \overline w(0)$. Then
\[ \forall x\leqslant 0, \qquad \underline w(x)\leqslant w(x)\leqslant \overline w(x). \]
\label{thm:slidingsubsupKPP}
\end{lem}
\begin{Rk}\label{rem:w_KPP}
Notice that $\overline w+\frac{\sqrt{1+4K}-1}2$ and $\underline w$ are solution of a classical Fisher-KPP equation $-cu'-u''=u(a-bu)$ with $a\in(0,1+2K)$, $b>0$, and a speed $c\geq 2\sqrt a$. The existence, uniqueness (up to a translation) and monotony of $\overline w$ and $\underline w$ are thus classical results (see e.g. \cite{KPP1937}). Thanks to those relations, the argument developed in this section can indeed be seen as a precise analysis on the profile of $x\mapsto u(x)$ for $x>0$ large.
\end{Rk}
\begin{proof}[Proof of Lemma \ref{thm:slidingsubsupKPP}] To prove this lemma, we use a sliding method.
\begin{itemize}
\item Let $\underline w_\eta(x):= \underline w(x+\eta)$. Thanks to Proposition~\ref{prop:w_left}, there exists $x_0\in \mathbb R$ such that $w(x)>1-2K=\sup_{\mathbb R}\underline w_\eta$ for all $x\leq x_0$ (we recall that $\mu<K$). Since $\lim_{x\to\infty}\underline w(x)=0$, there exists $x^0>0$ such that $\underline w(x)<\min_{[x_0,0]}w$ for all $x>x^0$. Then, for $\eta\geq x_0+x^0$,
\[\underline w_\eta(x)<w(x),\quad \forall x\leqslant 0. \]
We can then define $ \overline \eta:=\inf\{\eta, \forall x\leqslant 0, \underline w_\eta(x)\leqslant w(x)\}$. We have then $\underline w_{\overline \eta}(x)\leqslant \qg{w(x)}$ for all $x\leq 0$. If $\overline\eta>0$, since $\inf_{(-\infty,x_0]}w>1-2K=\sup_{\mathbb R}\underline w_\eta$ and $\underline w_{\overline \eta}(0)=\overline w(\eta)<\overline w(0)\leq w(0)$ (we recall that $\underline w$ is decreasing, see Remark~\ref{rem:w_KPP}), there exists $ \underline x\in (x_0,0)$ such that $ \underline w_{\overline\eta}(\underline x)=w(\underline x)$. $ \underline x$ is then a minimum of $w-\underline w_{\overline \eta}$, and thus
\begin{eqnarray}
0&\geq& -c(w-\underline w_{\overline \eta})'(\underline x)-(w-\underline w_{\overline \eta})''(\underline x)\nonumber\\
&=&w(\underline x)(1-\mu-m(\underline x)-w(\underline x))+\mu m(\underline x)-\underline w_{\overline \eta}(\underline x)(1-2K-\underline w_{\overline \eta}(\underline x))\nonumber\\
&>&w(\underline x)(1-2K-w(\underline x))-\underline w_{\overline \eta}(\underline x)(1-2K-\underline w_{\overline \eta}(\underline x))=0,\label{est1}
\end{eqnarray}
where we have used the estimate $\|m\|_\infty\leq K$ obtained in Proposition~\ref{thm:precisebound}. \eqref{est1} is a contradiction, we have then shown that $\bar \eta\leq 0$, and thus, for all $x\leq 0$, $\underline w(x)\leq w(x)$.
\item Similarly, let $\overline w_\eta(x):=\overline w(x-\eta)$. Since $\lim_{x\to-\infty}\overline w(x)>1$ and $w$ satisfies the estimate of Proposition~\ref{thm:precisebound}, we have, for $\eta\in \mathbb R$ large enough,
\[ \forall x\leqslant 0,\quad w(x)<1<\overline w_\eta(x). \]
We can then define $ \overline \eta:=\qg{\inf\{\eta, \forall x\leqslant 0, w(x)\leqslant \overline w_\eta(x)\}}. $ We have then $w(x)\leqslant \overline w_{\overline \eta}$ for all $x\leq 0$. If $\overline\eta>0$, since $\sup_{\mathbb R}w<1< \lim_{x\to -\infty} \overline w(x)$ and $w(0)\leq \overline w(0)< \overline w(-\bar \eta)=\overline w_{\bar \eta}(0)$ (we recall that $\overline w$ is decreasing, see Remark~\ref{rem:w_KPP}), there exists $\bar x<0$ such that $w(\underline x)=\overline w_{\overline \eta}(\underline x)$. $\bar x$ is then a minimum of $\overline w_{\overline \eta}-w$, and thus
\begin{eqnarray*}
0&\geq&-c(\overline w_{\overline \eta}-w)'(\bar x)-(\overline w_{\overline \eta}-w)''(\bar x)\\
&=&\overline w_{\overline \eta}(\bar x)(1-\overline w_{\overline \eta}(\bar x))+K-w(\bar x)(1-\mu-w(\bar x)-m(\bar x))-\mu m(\bar x)\\
&>&\overline w_{\overline \eta}(\bar x)(1-\overline w_{\overline \eta}(\bar x))-w(\bar x)(1-w(\bar x))=0,
\end{eqnarray*}
which is a contradiction. We have then shown that $\bar \eta\leq 0$, and thus, for all $x\leq 0$, $w(x)\leq \overline w(x)$.
\end{itemize}
\end{proof}
We also need to compare the solution of the Fisher-KPP equation with speed $c$ to the solutions of the modified Fisher-KPP equations introduced in Lemma~\ref{thm:slidingsubsupKPP}.
\begin{lem}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, and $c\geq 2+K$. Let $(c,u)$, with $u\in C^\infty(\mathbb R)$, be a travelling wave of the Fisher-KPP equation, see \eqref{travelling_wave_KPP}. Let also $\overline w$, $\underline w$ solutions of \eqref{overline_w} and \eqref{underline_w} respectively. Assume $ \underline w(0)\leqslant u(0)\leqslant \overline w(0)$. Then
\[ \forall x\leqslant 0, \qquad \underline w(x)\leqslant u(x)\leqslant \overline w(x). \]
\label{thm:slidingKPPKPP}classical
\end{lem}
The arguments of the proof of Lemma~\ref{thm:slidingsubsupKPP} can be used to prove Lemma \ref{thm:slidingKPPKPP}. We omit the details.
\medskip
We can now prove theorem \ref{thm:KPP}.
\begin{proof}[Proof of Theorem \ref{thm:KPP}]
Notice first that $c_*>2+K$, provided $K,\,\mu>0$ are small enough. Let $ \overline w\in C^\infty(\mathbb R)$ and $ \underline w\in C^\infty(\mathbb R)$ satisfying \eqref{overline_w} and \eqref{underline_w} respectively. $\overline w$ and $\underline w$ are then decreasing (see Remark~\ref{rem:w_KPP}), and we may assume (up to a translation) that they satisfy $\underline w(0)=w(0)=u(0)=\overline w(0)$. Then Lemma \ref{thm:slidingsubsupKPP} and \ref{thm:slidingKPPKPP} imply that $\underline w(x)\leqslant w(x), u(x)\leqslant \overline w(x)$ for $x\leq 0$, and then, $\Vert w-u\Vert_{L^\infty(-\infty, 0]}\leqslant \Vert\overline w-\underline w\Vert_{L^\infty(-\infty, 0]}$.
Let $ \tilde w=\overline w-\underline w\geqslant 0 $, which satisfies
\[ -c\tilde w'-\tilde w''=\tilde w(1-(\overline w+\underline w))+K+2K\underline w. \]
\medskip
We estimate first the maximum of $ \tilde w $ over $\{x\in\mathbb R;\,\underline w(x)\leqslant 3/4-K\}$ to prove the estimate on
$ \Vert \tilde w \Vert_{L^\infty(-\infty, 0]}$ stated in Theorem~\ref{thm:KPP}. If $ \underline w\leqslant 3/4-K$, then
\begin{equation}
-c\underline w'-\underline w''\geqslant \underline w(1/4-K). \label{eq_esti}
\end{equation}
Let
\[ \lambda_+:=\frac{c+\sqrt{c^2-4(1/4-K)}}{2}, \qquad \lambda_-:=\frac{c-\sqrt{c^2-4(1/4-K)}}{2}, \]
and $ \varphi(x):=e^{-\lambda_-x}-e^{-\lambda_+x}$. Then $\varphi$ satisfies $-c\varphi'-\varphi''=(1/4-K)\varphi$, $ \varphi(-\infty)=-\infty$ and $ \varphi(+\infty)=0$. Moreover, $ \varphi$ is
positive when $ x>0 $ and negative when $ x<0$. Finally, the maximum of $ \varphi$ is attained at $\overline x:=\frac{\ln \lambda_+-\ln\lambda_-}{\lambda_+-\lambda_-}>0$. One can show that $\varphi(\bar x)$ is a continuous and positive function of $c$ and $K$, which is uniformly bounded away from $0$ for $K\in (0,1/8)$ and $c\in[2,\infty)$. There exists thus a universal constant $C>0$ such that $\varphi(\bar x)>C>0$, for any $K\in (0,1/8)$, $c\in[2,\infty)$. Let $\gamma\in(0,3/4-K)$ and $\varphi^\gamma$ defined by
\[ \varphi^\gamma(x):=\gamma\frac {\varphi(x)}{\varphi(\bar x)},\]
and $\varphi^\gamma_\eta(x):=\varphi^\gamma(x+\eta)$ for $\eta\in\mathbb R$. Since (for $K>0$ small) $\max_{\mathbb R}\varphi^\gamma\leq 3/4-K<1-2K=\lim_{x\to-\infty}\underline w(x)$ and $\lim_{\eta\to\infty}\varphi^\gamma_\eta(0)=\lim_{x\to\infty}\varphi^\gamma(x)=0<\underline w(0)$, we have that for $\eta>0$ large enough,
\[ \forall x\leqslant 0,\quad \varphi^\gamma_\eta(x)\leqslant \underline w(x).\]
Let $ \tilde \eta:=\inf\{\eta\in\mathbb R;\; \forall x\leqslant 0, \,\varphi^\gamma_\eta(x)\leqslant\underline w(x)\}$. Then
$ \varphi^\gamma_{\tilde\eta}\leq \underline w$ on $ (-\infty, 0]$, and since $\varphi^\gamma_{\tilde \eta}(x)<0$ for $x<<0$, either $\underline w(0)=\varphi^\gamma_{\tilde \eta}(0)$, or there exists $\tilde x\in (-\infty,0)$ such that $\underline w(\tilde x)=\varphi^\gamma_{\tilde \eta}(\tilde x)$. In the latter case, $\tilde x$ is the minimum of $\underline w-\varphi^\gamma_{\tilde\eta}$, and then
\begin{eqnarray*}
0&\geq& -c(\underline w-\varphi^\gamma_{\tilde\eta})'(\tilde x)-(\underline w-\varphi^\gamma_{\tilde\eta})''(\tilde x)\\
&=&(1-2K-\underline w(\tilde x))\underline w(\tilde x)-(1/4-K)\varphi^\gamma_{\tilde\eta}(\tilde x)\\
&\geq&\left(3/4-K-\underline w(\tilde x)\right)\underline w(\tilde x)>0,
\end{eqnarray*}
since $\underline w(\tilde x)=\varphi^\gamma_{\tilde\eta}(\tilde x)\leq \gamma<3/4-K$. The above estimate is a contradiction, which implies $\underline w(0)=\varphi^\gamma_{\tilde \eta}(0)$. Then
\[ (e^{-\lambda_-\tilde \eta}-e^{-\lambda_+\tilde\eta})=\frac{w(0)}{ \gamma }\left(e^{-\lambda_-\overline x}-e^{-\lambda_+\overline x}\right),\]
and then
\[-\lambda_-\tilde \eta\geq\ln\left(\frac{w(0)}{ \gamma }\left(e^{-\lambda_-\overline x}-e^{-\lambda_+\overline x}\right)\right),\]
which implies $\varphi^\gamma_{\eta}(x)\leq \underline w(x)$ for all $x\in(-\infty,0]$, with $\gamma\in(0,3/4-K)$ and $\eta=-\frac{1}{\lambda_-}\ln\left(\frac{w(0)}{\gamma}(e^{-\lambda_-\overline x}-e^{-\lambda_+\overline x})\right)$. Passing to the limit $\gamma\to 3/4-K$, we then get that $\varphi^{3/4-K}_{\bar \eta}(x)\leq \underline w(x)$ for all $x\in(-\infty,0]$, with
\[ \bar \eta:= -\frac{1}{\lambda_-}\ln\left(\frac{w(0)}{3/4-K}\left(e^{-\lambda_-\overline x}-e^{-\lambda_+\overline x}\right)\right).\]
In particular, $\varphi^{3/4-K}_{\bar \eta}\leq \underline w$ implies that $\{x\in (-\infty,0];\;\underline w(x)\leqslant 3/4-K\}\subset [\min(0,\overline x-\bar \eta), 0]\subset[\min(0,-\bar \eta), 0]$ (indeed, $-\bar \eta<0$ if $w(0)$ is small enough).
Since $\sup_{\mathbb R}\underline w=1-2K$, we have
\[ -c\tilde w'-\tilde w''=\tilde w(1-(\underline w+\overline w))+K+2K\underline w<\tilde w+K(3-4K), \]
and $ \tilde w(0)=0$. We can then introduce $\psi(x)=K(3-4K)\left(e^{-\alpha x}-1\right)$, with $\alpha=\frac{c-\sqrt{c^2-4}}{2}$ which satisfies $-c\psi'-\psi''=\psi+K(3-4K)$. A sliding argument (that we skip here) shows that
\[\forall x\leq 0,\quad \tilde w(x)\leqslant \psi(x)= K(3-4K)\left(e^{-\alpha x}-1\right).\]
This estimate implies that
\[ \underset{[-\bar \eta, 0]}{\max}\tilde w\leqslant K(3-4K)\exp\left(-\frac{\alpha}{\lambda_-}\ln\left(\frac{w(0)}{3/4-K}\left(e^{-\lambda_-\overline x}-e^{-\lambda_+\overline x}\right)\right)\right)\leqslant C\,Kw(0)^{-\frac{\alpha}{\lambda_-}}, \]
where $ C>0$ is a universal constant.
\medskip
We consider now the case where the maximum of $\tilde w$ is reached on $[-\infty,0)\setminus\{x\in\mathbb R;\,\underline w(x)\leqslant 3/4-K\}$. If this supremum is a maximum attained in $\bar x$, then $\overline w(\bar x)+\underline w(\bar x)\geq \frac 32-2K>1$ (this last inequality holds if $K$ is small enough), and $-c\tilde w'(\bar x)-\tilde w''(\bar x)\geqslant 0$, which implies
\[ (\overline w+\underline w-1)\tilde w(\overline x) \leqslant K+2K\underline w\leqslant K(3-4K), \]
that is $\tilde w(\overline x)\leqslant \frac{K(3-4K)}{1/2-2K}\leq CK$ for some constant $C>0$, provided $K>0$ is small enough. If the supremum is not a maximum, it is possible to obtain a similar estimate, we skip here the additional technical details.
\medskip
We have shown that
\[\sup_{[-\infty,0]}\tilde w\leq \max\left(CK,CKw(0)^{-\frac{\alpha}{\lambda_-}}\right),\]
We choose now $\beta=(1+\alpha/\lambda_-)^{-1}\in (0,1/2)$
and $w(0)=K^\beta$ (we recall that the solution $(c,w,m)$ is still a solution when $w$ and $m$ are translated). Then, $\sup_{[-\infty,0]}\tilde w\leqslant CK^{\beta}$, and thus
\[ \Vert w-u\Vert_{L^\infty((-\infty, 0])}\leqslant \Vert\tilde w\Vert_{L^\infty((-\infty, 0])}\leqslant CK^\beta. \]
Furthermore, $ w $ and $ u $ are decreasing for $ x\geqslant 0$ thanks to Proposition \ref{prop:w_left}, which implies that
\[ \forall x\geqslant 0, |w-u|(x)\leqslant w(x)+u(x)\leqslant w(0)+u(0)\leqslant 2K^\beta
.\]
\qg{
From \cite{GT98}, theorem 8.33, there exists a universal constant that we denote $ C>0 $ such that
\begin{equation}
\Vert v\Vert_{C^{1, \alpha}}\leqslant C,
\end{equation}
where $ v $ is a solution of \eqref{travelling_wave_KPP}, and this constant $ C $ is uniform in the speed $ c$ in the neighbourhood of $ c_0=2\sqrt r. $ In particular, $ u $ satisfies
\begin{equation}
-c_0u'-u''=(c_*-c_0)u'+u(1-u) = u(1-u)+\mathcal O(K).
\end{equation}
Let $ v $ the solution of \eqref{travelling_wave_KPP} with speed $ c_0 $ and $ v(0)=u(0), $ the above argument can then be reproduced to show that
\begin{equation}
\Vert u-v\Vert_{L^\infty}\leq CK^\beta,
\end{equation}
where $ C $ is a universal constant and $ \beta $ depend only on $ r, $ }
which finishes the proof.
\end{proof}
\bibliography{MaBibli}
\appendix
\section{Appendix}
\subsection{Compactness results}
We provide here two results that are used in the proof of Theorem~\ref{thm:main}.
\begin{lem}[Elliptic estimates]
Let $a,\,b^-,\,b^+\in \mathbb R_+^\ast$, $ g\in L^\infty(-a, a)$, and $ \gamma >0. $ For any
$ b^+, b^-\in\mathbb R $ and
$ c\in [-\gamma, \gamma], $ the Dirichlet problem
\[ \left\{\begin{array}{l} -cw'-w''=g, \quad (-a, a), \\ w(-a)=b^-,\, w(a)=b^+, \end{array}\right. \]
has a unique weak solution $ w. $ In addition we have $ w\in C^{1, \alpha}([-a, a]) $ for all
$ \alpha\in [0, 1)$, and there is a constant $ C>0 $
depending only on $ a $ and $ \gamma $
such that
\[ \Vert w\Vert_{C^{1, \alpha}([-a, a])}\leqslant C(\max(|b^+|,|b^-|)+\Vert g\Vert_{L^\infty}), \]
\label{lem:ellipticestimates}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:ellipticestimates}]
As the domain lies in $ \mathbb R, $ we are not concerned with the regularity
problem near the boundary. Since
\[ L^\infty(-a, a)\subset \underset{p>1}{\bigcap}L^p(-a, a), \]
theorem 9.16 \cite{GT98} gives us existence and uniqueness of a solution $ w\in W^{2, p}, $ for all
$ p > 1. $ We deduce from Sobolev imbedding that $ w\in C^{1, \alpha}([-a, a]) $ for all $ \alpha < 1. $
The classical theory (\cite{GT98}, theorem 3.7) gives us a constant $ C'>0 $ depending only on $ a $ and $ \gamma $
such that
\[ \Vert w\Vert_{L^\infty}\leqslant \max(b^+, b^-)+C'\Vert g\Vert_{L^\infty}. \]
The estimate on the H\"older norm of the first derivative comes now from \cite{GT98}, theorem 8.33,
which states that whenever $ w $ is a $ C^{1, \alpha} $ solution of $ -cw'-w''=g $ with $ g\in L^\infty, $
then
\[ \Vert w\Vert_{C^{1, \alpha}([-a, a])}\leqslant C''(\Vert w\Vert_{C^0([-a, a])}+\Vert g\Vert_{L^\infty}), \]
with a constant $ C''=C''(a, \gamma) $ depending only on $ a $ and $ \gamma. $ That proves the theorem.
\end{proof}
\begin{lem}
Let $a,\,b^-,\,b^+\in \mathbb R_+^\ast$. The operator
$ (L)^{-1}_D:\mathbb R\times C^0([-a,a]) \longrightarrow C^0([-a,a]) $
defined by
\[ L_D^{-1}(c, g)=w, \]
where $w$ is the unique solution of
\[ \left\{\begin{array}{l} -cw'-w''=g, \quad (-a, a), \\
w(-a)=b^+, w(a)=b^-,
\end{array}\right. \]
is continuous and compact.
\label{lem:degreecompacity}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:degreecompacity}]
Let $ (c, g), (\tilde c, \tilde g) \in\mathbb R\times C^0([-a,a])$,
$ \gamma > 0 $ and $ w, \tilde w\in C^0([-a,a]) $
such that $ c, \tilde c \leqslant \gamma $ and
\[ \left\{\begin{array}{l} -cw'-w''=g\textrm{ on }(-a, a), \\
w(-a)=b^+, w(a)=b^-,
\end{array}\right. \]
\[ \left\{\begin{array}{l} -\tilde c\tilde w'-\tilde w''=\tilde g\textrm{ on }(-a, a), \\
\tilde w(-a)=b^+, \tilde w(a)=b^-.
\end{array}\right. \]
Then $ w-\tilde w $ satisfies
\[ \left\{\begin{array}{l} -c(w-\tilde w)'-(w-\tilde w)''=g-\tilde g+(c-\tilde c)\tilde w'
\textrm{ on }(-a, a), \\
(w-\tilde w)(-a)=0, (w-\tilde w)(a)=0.
\end{array}\right. \]
We deduce from Lemma \ref{lem:ellipticestimates} that there exists a constant $ C $ depending only on
$ a>0 $
such that
\[ \Vert w-\tilde w\Vert_{C^0}\leqslant C(\Vert g-\tilde g\Vert_{C^0}+|c-\tilde c|
(\Vert\tilde g\Vert_{C^0}+\max(b^+, b^-))), \]
which shows the pointwise continuity of $ L_D^{-1}. $
Now let $ (c_n, g_n) $ a bounded sequence in $ \mathbb R\times C^0. $ Let $ \gamma=\limsup |c_n|. $ From
Lemma \ref{lem:ellipticestimates}
we deduce the existence of a constant $ C>0 $ depending only on $ a $ and $ \gamma $ such that
\[ \Vert u_n\Vert_{C^1}\leqslant C(\max(b^+, b^-)+\Vert g_n\Vert_{C^0}), \]
where $ u_n=L_D^{-1}(c_n, g_n), $ which shows that $ (g_n) $ is bounded in $ C^1. $ Since $ C^1 $ is
compactly embedded in $ C^0, $
there exists a $ w\in C^0 $ such that $ \Vert u_n-w\Vert_{C^0} \rightarrow 0. $ This shows the compactness of
$ L_D^{-1}. $
\end{proof}
\subsection{Properties of the reaction terms}
\label{appendix_reaction_terms}
The proofs of Theorem~\ref{thm:monotonicity} requires precise estimates on the reaction terms $f_w$ and $f_m$. Here we prove a number of technical lemmas that are necessary for our study.
\begin{lem}\label{lem:assumption2}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, and $\phi_w$, $\phi_m$ defined by \eqref{def:phiw} and \eqref{def:phim} respectively. Then, $\phi_w,\,\phi_m:\mathbb R_+\to \mathbb R$ are decreasing functions such that
\[\phi_w([0,K])\subset (\mu,\infty),\quad \phi_m([0,1])\subset (\mu K/r,\infty).\]
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:assumption2}]
We prove the lemma for $\phi_m$. The results on $\phi_w$ follow since both functions coincide when $r=K=1$.
\delbyqg{Notice that the result on $\phi_w$ can be obtained from the one on $\phi_m$ by setting $ r=K=1$. We will thus only consider the case of $\phi_m$.} The fact that $\phi_m$ is decreasing simply comes from the computation of its derivative:
\[\phi_m'(w)=\frac 12\left[-1+\frac{-2\left(K-\frac{\mu K}{r}-w\right)+4\frac{\mu K}{r}}{2\sqrt{\left(K-\frac{\mu K}{r}-w\right)^2+4\frac{\mu K}{r}w}}\right],
\]
one can check that $\phi_m'(w)<0$ for all $w\geq 0$ as soon as $\mu<\frac r2$. Next, we can estimate $\phi_m(w)$ for $w>0$ large:
\begin{eqnarray*}
\phi_m(w)&=&\frac{w+\frac{\mu K}r-K}2\left(-1+\sqrt{1+\frac{4\mu Kw}{r\left(w+\frac{\mu K}r-K\right)^2}}\right)\\
&=&\frac{w+\frac{\mu K}r-K}2\left(\frac{2\mu Kw}{r\left(w+\frac{\mu K}r-K\right)^2}+o(1/w)\right)\\
&=&\frac{\mu K}r+o(1),
\end{eqnarray*}
that is $\lim_{w\to\infty}\phi_m(w)=\frac{\mu K}r$, which, combined to the variation of $\phi_w$, shows that $\phi_m([0,1])\subset (\mu K/r,\infty)$.
\end{proof}
\begin{lem}
\label{lem:zerosfmfw}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, $\phi_w$, $\phi_m$ defined by \eqref{def:phiw} and \eqref{def:phim} respectively.
\[ Z_w=\{(w, m)\in (0, 1)\times(0, K)/f_w(w,m)=0 \}, \]
\[ Z_m=\{(w,m)\in (0, 1)\times(0, K)/f_m(w, m)=0 \}, \]
and denote
\begin{equation}
\mathcal D=(0, 1)\times(0, K).
\end{equation}
Then:
\begin{enumerate}
\item $ Z_w $ can be described in two ways:
\begin{equation}\label{eq:fwzerosx}
Z_w=\left\{\left(\phi_w(m)
, m\right), m\in(0, K) \right\},
\end{equation}
and
\begin{equation}\label{eq:fwzerosy}
Z_w=\left\{\left(w,\varphi_w(w)
\right),\qg w \in(\mu, 1) \right\} \cap \mathcal D,
\end{equation}
where $\varphi_w(w)=\frac{w(1-\mu-w)}{w-\mu}$.
\item Similarly, $ Z_m $ can be described as:
\begin{equation}
\label{eq:fmzerosx}
Z_m=\left\{\left(w, \phi_m(w)
\right), w\in(0, 1)\right\},
\end{equation}
and
\begin{equation}
\label{eq:fmzerosy}
Z_m=\left\{\left(\varphi_m(m)
, m\right), m\in\left(\frac{\mu K}{r}, K\right) \right\}\cap\mathcal D,
\end{equation}
where $\varphi_m(m):=\frac{m(K-\frac{\mu K}{r}-m)}{m-\frac{\mu K}{r}}$.
\end{enumerate}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:zerosfmfw}]
Notice that point 1 can be obtained from point 2 by setting $ r=K=1. $ Thus, we are only going to prove point 2. We write
\[ f_m(w, m)=rm\left(1-\frac{w+m}{K}\right)+\mu(w-m)=-\frac{r}{K}m^2+\left(r-\mu-\frac{r}{K}w\right)m+\mu w. \]
Since $ \Delta=\left(r-\mu-\frac{r}{K}w\right)^2+4\qg{\frac{\mu r}{K}} w >0 $ for any $ w\geq 0, $ $ f_m(w, m)=0 $ admits only two solutions for $ w\geq 0 $ fixed. Those write:
\[ \frac{1}{2} \left((K-\frac{\mu K}{r}-w)\pm\sqrt{(K-\frac{\mu K}{r}-w)^2+4\frac{\mu K}{r} w}\right),\]
one of those two solutions is negative for all $w\neq 0$, so that $f_m(w,m)=0$ with $(w,m)\in\mathcal D$ implies that $m=\phi_m(w)$,
which leads to \eqref{eq:fmzerosx}.
Thanks to Lemma~\ref{lem:assumption2}, $m>\mu K/r$ on $Z_m$, $f(w,m)=0$ with $(w,m)\in\mathcal D$ then implies $w=\varphi_m(m)$.
For $ m\in\left(0, \frac{\mu K}{r}\right), $ $ \varphi_m(m) $ is decreasing and
\[ \left\{\begin{array}{l} \varphi_m(0)=0, \\ \underset{m\rightarrow\left(\frac{\mu K}{r}\right)^-}{\lim\phantom{fsd}}\varphi_m(m)=-\infty, \end{array}\right. \]
so that $ \varphi_m(m)<0 $ for $ m \in \left(0, \frac{\mu K}{r}\right). $ That proves \eqref{eq:fmzerosy}.
\end{proof}
The next lemma proves that $ f $ admits only one zero in $ \mathcal D, $ and proves some inclusions between $ f_m>0 $ and $ f_w>0. $
\begin{lem}
\label{lem:deczeros}
Let $r,\,K,\,\mu$ satisfy Assumption~\ref{ass}, $\phi_w$, $\phi_m$, $\varphi_w$, $\varphi_m$ defined by \eqref{def:phiw}, \eqref{def:phim}, \eqref{eq:fwzerosy} and \eqref{eq:fmzerosy} respectively. Then $ \phi_w $ and $ \varphi_m $ are convex, strictly decreasing functions over $ (0, K) $ and $ \left(\frac{\mu K}{r}, K\right) $ respectively.
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:deczeros}]
We have already shown that $\phi_w$ is decreasing on $(0,K)$. We compute:
\[ \phi_w'(m)=-\frac{1}{2}\left(1+\frac{1-3\mu-m}{\sqrt{(1-\mu-m)^2+4\mu m}}\right).\]
Computing the second derivative, we find:
\[ \phi_w''(m)=\frac{(1-\mu-m)^2+4\mu m-(1-3\mu-m)^2}{2\left((1-\mu-m)^2+4\mu m\right)^{\frac{3}{2}}} \]
\[ =\frac{2\mu(1-2\mu)}{\left((1-\mu-m)^2+4\mu m\right)^{\frac{3}{2}}} >0, \]
so that $ \phi_w $ is convex over $ \mathbb R_+. $
Thanks to polynomial arithmetics, we compute:
\[ \varphi_m(m)=\frac{m\left(K\left(1-\frac{\mu}{r}\right)-m\right)}{m-\frac{\mu K}{r}}=K\left(1-\frac{2\mu}{r}\right)-m+\frac{\frac{\mu K^2}{r}\left(1-\frac{2\mu}{r}\right)}{m-\frac{\mu K}{r}}, \]
which makes $ \varphi_m $ obviously convex and strictly decreasing on $ \left(\frac{\mu K}{r}, K\right) $.
\end{proof}
\begin{lem}
\label{lem:zerof}
There exists a unique solution to the problem:
\begin{equation}
\label{eq:pbzerof}
f_w(w, m)=f_m(w, m)=0,
\end{equation}
with $ (w, m)\in(0,1)\times(0, K). $
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:zerof}]
We write:
\[ f_w=w(1-\mu-w)+m(\mu-w). \]
Since $ \mu < \frac{1}{2}, $ we have
\[ f_w(\mu, m)=\mu(1-2\mu)>0, \]
so that there cannot be a solution of $ f_w(w, m)=0 $ with $ w=\mu. $ Thus, $ f_w(w, m)=0 $ if and only if
\begin{equation}
\label{eq:mfw}
m=\frac{w(1-\mu-w)}{w-\mu}.
\end{equation}
Substituting \eqref{eq:mfw} in $ f_m(w, m)=0, $ we get:
\[ \eqref{eq:pbzerof}\Rightarrow\underset{A}{\underbrace{r\frac{w(1-\mu-w)}{w-\mu}\left(1-\frac{\frac{w(1-\mu-w)}{w-\mu}+w}{K}\right)}}+\underset{B}{\underbrace{\mu\left(w-\frac{w(1-\mu-w)}{w-\mu}\right)}}=0. \]
We compute:
\[ A=\frac{rw(1-\mu-w)}{K(w-\mu)^2}\left(w(K+2\mu-1)-\mu K\right), \]
\[ B=\mu \frac{w(2w-1)}{w-\mu}. \]
From now on we assume $ w\neq 0. $ Then:
\[ \eqref{eq:pbzerof}\Rightarrow C(w):=\frac{r}{K}(1-\mu-w)(w(K+2\mu-1)-\mu K)+\mu(2w-1)(w-\mu)=0. \]
Now $ C $ is a polynomial function of degree at most 2. We compute:
\[ C(0)=\mu(1-r(1-\mu))<0, \]
\[ C(1)=\mu\left(1-\mu+\frac{r}{K}((1-\mu)(1-K)-\mu)\right)>0, \]
under the following assumptions:
\[ \mu<1-\frac{1}{r}, \]
\[ K<\frac{r}{r-1}\left(1-\frac{\mu}{1-\mu}\right). \]
That proves the uniqueness of a solution of \eqref{eq:pbzerof} with $ w\in(0, 1). $
Now recall the notations of lemmas \ref{lem:zerosfmfw} and \ref{lem:deczeros}. The existence of a solution to problem \eqref{eq:pbzerof} is equivalent to showing $ Z_m\cap Z_w\neq\varnothing. $ Since:
\[ \Phi_w\left(\frac{\mu K}{r}\right)\in\mathbb R, \]
\[ \underset{m\rightarrow \left(\frac{\mu K}{r}\right)^+}{\lim}\varphi_m(m)=+\infty, \]
\[ \Phi_w(K)=\frac{1}{2}\left(1-\mu-K+\sqrt{(1-\mu-K)^2+4\mu K}\right)>0, \]
\[ \varphi_m(K)=-\frac{\mu}{r-\mu}<0, \]
and since $ \Phi_w $ and $ \varphi_m $ are continuous over $ \left(\frac{\mu K}{r}, K\right), $ there exists a solution to $ \varphi_m(m)=\Phi_w(m) $ with $ m\in \left(\frac{\mu K}{r}, K\right). $
Since $ \forall m\in(0, K), 0<\Phi_w(m)<1, $ that gives us a solution to \eqref{eq:pbzerof}, and proves Lemma \ref{lem:zerof}.
\end{proof}
\begin{lem}\label{lem:zerosinclusions}
Let $ \mathcal D=(0, 1)\times(0, K), $
\[ \gr{Z_w=\{f_w=0 \}\cap \mathcal D=\{w=\phi_w(m)\}\cap \mathcal D}, \quad Z_w^-=\{f_w<0\}\cap \mathcal D=\{w>\phi_w(m)\}\cap\mathcal D, \]
\[ \gr{Z_m=\{f_m=0 \}\cap \mathcal D=\{m=\phi_m(w)=0 \}\cap \mathcal D},\quad Z_m^-=\{f_m<0\}\cap\mathcal D=\{m>\phi_m(w)<0\}\cap\mathcal D, \]
and
\[ \mathcal D_l=\{(w, m)\in\mathcal D, w\leq w^*, m\geq m^* \}, \]
\[ \mathcal D_r=\{(w, m)\in\mathcal D, w\geq w^*, m\leq m^* \}, \]
where $ (w^*, m^*) $ is the only solution of $ f_m=f_w=0 $ in $ \mathcal D. $
Then
\begin{equation}
\label{eq:redzeros}
Z_m\cup Z_w\subset \mathcal D_l\cup \mathcal D_r.
\end{equation}
Moreover,
\begin{equation}
\label{eq:zeroinclusionwsm}
Z_w^-\cap \mathcal D_l\subset Z_m^-,
\end{equation}
\begin{equation}
\label{eq:zeroinclusionmsw}
Z_m^-\cap \mathcal D_r\subset Z_w^- .
\end{equation}
\end{lem}
\begin{Rk}\label{Rk:zerosinclusions}
Let
\[Z_w^+=\{f_w>0\}\cap\mathcal D= \{w<\phi_w(m)\}\cap\mathcal D ,\quad Z_m^+=\{f_m>0\}\cap\mathcal D=\{m<\phi_m(w)\}\cap\mathcal D.\]
Lemma~\ref{lem:zerosinclusions} implies that
\[Z_m^+\cap \mathcal D_l\subset Z_w^+,\quad Z_w^+\cap \mathcal D_r\subset Z_m^+.\]
\end{Rk}
\begin{proof}[Proof of Lemma~\ref{lem:zerosinclusions}]
Assertion \eqref{eq:redzeros} comes from the fact that $ \Phi_w $ and $ \varphi_m $ are decreasing.
Assertion \eqref{eq:zeroinclusionwsm} comes from the fact that $ \Phi_w-\varphi_m $ is negative for $ m $ close to $ \left(\frac{\mu K}{r}\right)^+ $ and does not change sign in $ \left(\frac{\mu K}{r}, m^*\right) $ since $ (w^*, m^*) $ is the only solution of \eqref{eq:pbzerof}. A similar argument proves assertion \eqref{eq:zeroinclusionmsw}
\end{proof}
The last thing we need here is an estimate of the behaviour of $ m^*(\mu, r, K) $ when $ \mu\rightarrow 0 $:
\begin{lem}
\label{lem:estmu0}
For $ \mu<1-K, $ we have
\begin{equation}
m^*(\mu, r, K)\leq\frac{\frac{\mu K}{r}(1-\mu)}{1-\mu-K\left(1-\frac{2\mu}{r}\right)}.
\end{equation}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:estmu0}]
Recall the notations of Lemma \ref{lem:deczeros}. From Lemma \ref{lem:zerof} we know that $ m^* $ is the only solution of $ \Phi_w=\varphi_m $ that lies in $ (0, K). $ Since $ m\mapsto \sqrt m $ is increasing and $ 1-\mu-K>0, $ we have:
\begin{equation}
\Phi_w(m)\geq 1-\mu-m.
\end{equation}
We deduce then:
\[ \varphi_m(m)-\Phi_w(m)\leq\varphi_m(m)-(1-\mu-m). \]
Now $ \varphi_m-\Phi_w $ is positive near $ \left(\frac{\mu K}{r}\right)^+, $ and for $ w\in \left(\frac{\mu K}{r}, m^*\right), $
\[ 0<\varphi_m(w)-\Phi_w(w)\leq\varphi_m(w)-(1-\mu-m), \]
which means that if $ \bar m $ satisfies
\begin{equation}
\label{eq:lemestmu0eq1}
\varphi_m(\bar m)=1-\mu-\bar m.
\end{equation}
then $ \bar m\geq m^*. $
A simple computation shows that the only solution of \eqref{eq:lemestmu0eq1} is:
\[ \bar m=\frac{\frac{\mu K}{r}(1-\mu)}{1-\mu-K\left(1-\frac{2\mu}{r}\right)}, \]
which finishes to prove Lemma \ref{lem:estmu0}
\end{proof}
\end{document}
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TITLE: If $f$ is an even function defined on the interval $(-5,5)$ then four real values of $x$ satisfying the equation $f(x)=f(\frac{x+1}{x+2})$ are?
QUESTION [2 upvotes]: If $f$ is an even function defined on the interval $(-5,5)$ then four real values of $x$ satisfying the equation $f(x)=f(\frac{x+1}{x+2})$ are?
I thought that $(x+1)/(x+2)=-x$.But I'm getting only two values by solving this.How do I get the other two values?
And if i put $(x+1)/(x+2)=x$ then the answer is not matching.I dont know why.
REPLY [2 votes]: Don't forget the possibility $(x+1)/(x+2)=+x$.
Setting $\frac{x+1}{x+2}=-x$ leads to $x+1=-x^2-2x$, i.e., $x^2+3x+1=0$, $x=\frac{-3\pm\sqrt{5}}2$. Setting $\frac{x+1}{x+2}=+x$ leads to $x+1=x^2+2x$, i.e., $x^2+x-1=0$, $x=\frac{-1\pm\sqrt{5}}2$.
All four values are well within $(-5,5)$.
| 2,792
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\begin{document}
\title[Some results associated with Bernoulli and Euler numbers with applications]{Some results associated with Bernoulli and Euler numbers with applications}
\author[{ C.-P. Chen }]{ Chao-Ping Chen$^{*}$}
\address{C.-P. Chen: School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City 454000, Henan Province, China}
\email{chenchaoping@sohu.com}
\author[{ R.B. Paris }]{Richard B. Paris}
\address{R.B. Paris: Division of Computing and Mathematics\\
University of Abertay, Dundee, DD1 1HG, UK}
\email{R.Paris@abertay.ac.uk}
\thanks{*Corresponding Author}
\thanks{2010 Mathematics Subject Classification. Primary 11B68; Secondary 26A48, 26D15}
\thanks{Key words and phrases. Bernoulli polynomials and numbers; Euler polynomials and numbers; Completely monotonic functions; Inequality}
\begin{abstract}
In this paper, we present series
representations of the remainders in the expansions for $2/(e^t+1)$, $\sech t$ and $\coth t$.
For example,
we prove that for $t > 0$ and $N\in\mathbb{N}:=\{1, 2, \ldots\}$,
\begin{align*}\label{Thm1-remainder-cosh-OpenProblemsolution1}
\sech t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t)
\end{align*}
with
\begin{align*}
R_N(t)=\frac{(-1)^{N}2t^{2N}}{\pi^{2N-1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+\frac{1}{2})^{2N-1}\Big(t^2+\pi^2(k+\frac{1}{2})^2\Big)},
\end{align*}
and
\begin{align*}
\sech t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+\Theta(t,
N)\frac{E_{2N}}{(2N)!}t^{2N}
\end{align*}
with a suitable $0 < \Theta(t, N) < 1$. Here $E_n$ are the Euler numbers. By using the obtained
results, we deduce some inequalities and completely monotonic
functions associated with the ratio of gamma functions. Furthermore, we give a (presumably new) quadratic recurrence relation for the Bernoulli numbers.
\end{abstract}
\maketitle
\section{Introduction}
The Bernoulli polynomials $B_n(x)$ and Euler polynomials
$E_n(x)$ are defined, respectively, by the generating functions:
\begin{equation*}\label{generalizedBernoullipolynomials}
\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^n}{n!}\quad
(|t|<2\pi)\quad \text{and}\quad
\frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}(x)\frac{t^n}{n!}\quad
(|t|<\pi).
\end{equation*}
The numbers $B_n = B_n(0)$ and $E_n = 2^{n}E_n(\frac{1}{2})$, which are
known to be rational numbers and integers, respectively, are
called Bernoulli and Euler numbers.
It follows from \cite[Chapter 4, Part I, Problem 154]{PS}
that
\begin{equation}\label{Problem}
\sum_{j=1}^{2m}\frac{B_{2j}}{(2j)!}t^{2j}<\frac{t}{e^{t}-1}-1+\frac{t}{2}
<\sum_{j=1}^{2m+1}\frac{B_{2j}}{(2j)!}t^{2j}
\end{equation}
for $t>0$ and $m\in \mathbb{N}_0:=\mathbb{N}\cup \{0\}, \,
\mathbb{N}:=\{1,2,3,\ldots\}$. The inequality \eqref{Problem} can
be also found in \cite{Koumandos1458,Sasvari}.
It is also known \cite[p. 64]{Temme1996} that
\begin{equation}\label{remainder3}
\frac{t}{e^{t}-1}-1
+\frac{t}{2}=\sum_{j=1}^{n}\frac{B_{2j}}{(2j)!}t^{2j}+(-1)^{n}t^{2n+2}\nu_{n}(t)\quad
(n\in \mathbb{N}_0),
\end{equation}
where
\begin{equation}\label{remainder3-series}
\nu_{n}(t)=\frac{2}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{1}{k^{2n}(t^{2}+4\pi^{2}k^{2})}.
\end{equation}
It is easily seen that \eqref{remainder3} implies \eqref{Problem}. Koumandos \cite{Koumandos1458} gave the following integral
representation of $\nu_{n}(t)$:
\begin{equation}\label{remainder-integral}
\nu_{n}(t)=\frac{(-1)^{n}}{(2n+ 1)!}\frac{
1}{e^t-1}\int_{0}^{1}e^{xt}B_{2n+1}(x)\textup{\,d}x.
\end{equation}
\begin{remark}
From \eqref{remainder-integral}, it is possible to deduce
\eqref{remainder3-series} by making use of the expansion \textup{\cite[p. 592, Eq. (24.8.2)]{Olver-Lozier-Boisvert-Clarks2010}}
\begin{equation*}\label{E2m-1-known}\begin{split}
B_{2n+1}(x)=
\frac{(-1)^{n+1}2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac{\sin(2k\pi
x)}{k^{2n+1}}\quad (n\in\mathbb{N}, \quad 0\leq x\leq1).
\end{split}\end{equation*}
We then obtain from \eqref{remainder-integral} that
\begin{align*}
&\nu_{n}(t)=-\frac{
1}{e^t-1}\frac{2}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\int_{0}^{1}\frac{e^{xt}\sin(2k\pi x)}{k^{2n+1}}\textup{\,d}x=\frac{2}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{1}{k^{2n}(t^{2}+4\pi^{2}k^{2})}.
\end{align*}
An alternative derivation of \eqref{remainder3} and another integral representation of the remainder
function $\nu_{n}(t)$ are given in the appendix.
\end{remark}
Binet's first formula \cite[p. 16]{Srivastava2001} for the logarithm of $\Gamma(x)$ states that
\begin{equation}\label{binet}
\ln \Gamma(x)= \left(x-\frac1{2}\right)\ln x-x+\ln\sqrt{2\pi}+\int_{0}^{\infty}\left(\frac{t}{e^{t}-1}-1+\frac{t}{2}\right)\frac{e^{-xt}}{t^2}\textup{\,d} t \quad (x>0).
\end{equation}
Combining \eqref{remainder3} with \eqref{binet}, Xu and Han
\cite{Xu-Han47--51} deduced in 2009 that for every
$m\in\mathbb{N}_0$, the function
\begin{equation}\label{Rm-CLF}
R_{m}(x)=(-1)^{m}\left[\ln \Gamma(x)-\left(x-\frac{1}{2}\right)\ln
x+x-\ln
\sqrt{2\pi}-\sum_{j=1}^{m}\frac{B_{2j}}{2j(2j-1)x^{2j-1}}\right]
\end{equation}
is completely monotonic on $(0, \infty)$. Recall that a function $f(x)$
is said to be completely monotonic on an interval $I$ if it has
derivatives of all orders on $I$ and satisfies the following
inequality:
\begin{equation}\label{cmf-dfn-ineq}
(-1)^{n}f^{(n)}(x)\geq0\quad (x\in I, \quad n\in \mathbb{N}_0).
\end{equation}
For $m=0$, the complete
monotonicity property of $R_{m}(x)$ was proved by Muldoon
\cite{Muldoon54}. Alzer \cite{Alzer373} first proved in 1997 that
$R_{m}(x)$ is completely monotonic on $(0, \infty)$. In 2006,
Koumandos \cite{Koumandos1458} proved the double inequality \eqref{Problem},
and then used \eqref{Problem} and \eqref{binet} to give a simpler proof of the complete monotonicity property of $R_{m}(x)$. In 2009,
Koumandos and Pedersen \cite[Theorem 2.1]{Koumandos-Pedersen33--40}
strengthened this result.
Chen and Paris \cite[Lemma 1]{Chen-Paris514--529}
presented an analogous result to \eqref{Problem} given by
\begin{equation}\label{ThmEn-inequality}\begin{split}
\sum_{j=2}^{2m+1}\frac{(1-2^{2j})B_{2j}}{j}\frac{t^{2j-1}}{(2j-1)!}<\frac{2}{e^t+1}-1+\frac{t}{2}&<\sum_{j=2}^{2m}\frac{(1-2^{2j})B_{2j}}{j}\frac{t^{2j-1}}{(2j-1)!}
\end{split}\end{equation}
for $t>0$ and $m\in\mathbb{N}$. The inequality \eqref{ThmEn-inequality} can also be written for $t>0$ and $m\in\mathbb{N}_0$ as
\begin{equation}\label{ThmEn-inequalityre}\begin{split}
(-1)^{m+1}\left(\frac{2}{e^t+1}-1-\sum_{j=1}^{m}\frac{(1-2^{2j})B_{2j}}{j}\frac{t^{2j-1}}{(2j-1)!}\right)>0.
\end{split}\end{equation}
Based on the inequality \eqref{ThmEn-inequalityre}, Chen and Paris \cite[Theorem 1]{Chen-Paris514--529}
proved
that for every $m\in\mathbb{N}_0$, the function
\begin{equation}\label{En-Fmx}\begin{split}
F_{m}(x)=(-1)^{m}\left[\ln\left(\frac{\Gamma(x+1)}{\Gamma(x+\frac{1}{2})}\right)-\frac{1}{2}\ln
x-\sum_{j=1}^{m}\left(1-\frac{1}{2^{2j}}\right)\frac{B_{2j}}{j(2j-1)x^{2j-1}}\right]
\end{split}\end{equation}
is completely monotonic on $(0,\infty)$. This result is similar to
the complete monotonicity property of $R_{m}(x)$ in \eqref{Rm-CLF}. In analogy with \eqref{remainder3}, these authors also considered \cite[Eq. (2.4)]{Chen-Paris514--529} the remainder $r_{m}(t)$ in the expansion
\begin{equation}\label{remainder-formula-r_m(x)}
\frac{2}{e^t+1}=1+\sum_{j=1}^{m}\frac{(1-2^{2j})B_{2j}}{j\cdot(2j-1)!}t^{2j-1}+r_{m}(t)
\end{equation}
and gave an integral representation for $r_{m}(t)$ when $t>0$.
Chen \cite{Chen790--799} proposed the following conjecture.
\begin{Conjecture}\label{Conjecture1-Gamma-prod}
For $t>0$ and $m\in\mathbb{N}_0$, let
\begin{align}\label{conjecture-Gamma-prod}
\mu_m(t)&=\frac{e^{t/3}-e^{2t/3}}{e^{t}-1}-\sum_{j=0}^{m}\frac{2B_{2j+1}(\frac{1}{3})}{(2j+1)!}t^{2j}
\end{align}
and
\begin{align}\label{conjecture-Gamma-prod-nu}
\nu_m(t)&=\frac{e^{t/4}-e^{3t/4}}{e^{t}-1}-\sum_{j=0}^{m}\frac{2B_{2j+1}(\frac{1}{4})}{(2j+1)!}t^{2j},
\end{align}
where $B_n(x)$ denotes the Bernoulli polynomials.
Then, for $t>0$ and $m\in\mathbb{N}_0$,
\begin{equation}\label{Conjecture1-Gamma-mu}
(-1)^{m}\mu_m(t)>0
\end{equation}
and
\begin{align}\label{Conjecture1-Gamma-nu}
(-1)^{m}\nu_m(t)>0.
\end{align}
\end{Conjecture}
Chen \cite[Lemma 1]{Chen790--799} has proved the statements in Conjecture
\ref{Conjecture1-Gamma-prod} for $m=0, 1, 2$, and $3$.
He has also pointed out in \cite{Chen790--799} that, if Conjecture \ref{Conjecture1-Gamma-prod} is true,
then it follows that the functions
\begin{align}\label{Thm1-asymptotic-ratio-gammas-rewrittenfind}
U_m(x)=(-1)^{m}\left[\ln\frac{\Gamma(x+\frac{2}{3})}{x^{1/3}\Gamma(x+\frac{1}{3})}-
\sum_{j=1}^{m}\frac{B_{2j+1}(\frac{1}{3})}{j(2j+1)}\frac{1}{x^{2j}}\right]
\end{align}
and
\begin{align}\label{Thm1-asymptotic-ratio-gammas-rewrittenfind-Vx}
V_m(x)=(-1)^{m}\left[\ln\frac{\Gamma(x+\frac{3}{4})}{x^{1/2}\Gamma(x+\frac{1}{4})}-
\sum_{j=1}^{m}\frac{B_{2j+1}(\frac{1}{4})}{j(2j+1)}\frac{1}{x^{2j}}\right]
\end{align}
for $m\in\mathbb{N}_0$ are completely monotonic on $(0,\infty)$.
The complete monotonicity properties of $U_m(x)$ and $V_m(x)$ are
similar to the complete monotonicity property of $F_{m}(x)$ in
\eqref{En-Fmx}.
In this paper, we obtain the following results: (i) a series representation of the remainder $r_m(t)$ in \eqref{remainder-formula-r_m(x)} (Theorem \ref{Thm1-remainder}); (ii) a series representation of the remainder in the
expansion of $\sech t$ involving the Euler numbers (Theorem \ref{Thm2-remainder-cosh-OpenProblem}), together with the double
inequality for $t>0$ and $m\in\mathbb{N}_0$,
\begin{equation}\label{Euler-constantSm-inequality1}
\sum_{j=0}^{2m+1}\frac{E_{2j}}{(2j)!}t^{2j}<\sech t<\sum_{j=0}^{2m}\frac{E_{2j}}{(2j)!}t^{2j};
\end{equation}
(iii) the proof of the inequality \eqref{Conjecture1-Gamma-nu} for all $m\in\mathbb{N}_0$, and a demonstration that the
function $V_m(x)$ in \eqref{Thm1-asymptotic-ratio-gammas-rewrittenfind-Vx} is completely monotonic on $(0, \infty)$ (Remark \ref{Remark-completely-function-Vm}); (iv) a series
representation of the remainder in the expansion for $\coth t$ (Theorem \ref{Thm3-remainder-coth}); and finally, (v) a quadratic recurrence relation for the Bernoulli numbers (Theorem \ref{Thm4-remainder}).
\vskip 8mm
\section{Main results}
\begin{theorem}\label{Thm1-remainder}
For $t>0$ and $m\in\mathbb{N}$,
\begin{align}\label{Chen-remainder-rm}
\frac{2}{e^t+1}=1+\sum_{j=1}^{m}\frac{(1-2^{2j})B_{2j}}{j\cdot(2j-1)!}t^{2j-1}+(-1)^{m+1}t^{2m+1}s_m(t),
\end{align}
where $s_m(t)$ is given by
\begin{align}\label{Chen-rm(x)}
s_m(t)=\frac{4}{\pi^{2m}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{2m}\big(t^2+\pi^2(2k+1)^2\big)}.
\end{align}
\end{theorem}
\begin{proof}
Boole's summation formula (see \cite[p. 17, Theorem 1.4]{Temme1996}) for a function $f(t)$ defined on $[0, 1]$
with $k$ continuous derivatives states that, for $k\in\mathbb{N}$,
\begin{equation}\label{Boole-summation-formula}\begin{split}
f(1)=\frac{1}{2}\sum_{j=0}^{k-1}\frac{E_{j}(1)}{j!}\Big(f^{(j)}(1)+f^{(j)}(0)\Big)+\frac{1}{2(k-1)!}\int_{0}^{1}f^{(k)}(x)E_{k-1}(x){\rm
d}x.
\end{split}\end{equation}
Noting \cite[p. 590]{Olver-Lozier-Boisvert-Clarks2010} that
\begin{equation}\label{noting-En}
E_{n}(1)=\frac{2(2^{n+1}-1)}{n+1}B_{n+1}\qquad (n\in\mathbb{N}),
\end{equation}
we see that
\begin{equation*}\label{noting-En}\begin{split}
E_{2j-1}(1)=\frac{(2^{2j}-1)B_{2j}}{j}\quad\text{and}\quad
E_{2j}(1)=0\quad (j\in\mathbb{N}).
\end{split}\end{equation*}
The choice\footnote{It is also possible to choose $k=2m$ in (\ref{Boole-summation-formula}) and to use the Fourier expansion for $E_{2m+1}(x)$ in \cite[p.~16]{Temme1996} to obtain the same result.}
$k=2m+1$ in \eqref{Boole-summation-formula} yields
\begin{align}\label{Boole-summation-formula-k=2m+1}
f(1)-f(0)&=\sum_{j=1}^{m}\frac{(2^{2j}-1)B_{2j}}{j\cdot(2j-1)!}\Big(f^{(2j-1)}(1)+f^{(2j-1)}(0)\Big)\nonumber\\
&\qquad\qquad\quad+\frac{1}{(2m)!}\int_{0}^{1}f^{(2m+1)}(x)E_{2m}(x){\rm
d}x.
\end{align}
Application of the above formula to $f(x)=e^{xt}$
then produces
\begin{equation}\label{Boole-summation-formula-re-obtainnew}\begin{split}
\frac{2}{e^t+1}=1+\sum_{j=1}^{m}\frac{(1-2^{2j})B_{2j}}{j\cdot(2j-1)!}t^{2j-1}+r_m(t),
\end{split}\end{equation}
where
\begin{equation}\label{Boole-summation-formula-re-obtainNew}\begin{split}
r_m(t)=-\frac{1}{e^t+1}\frac{t^{2m+1}}{(2m)!}\int_{0}^{1}e^{xt}E_{2m}(x){\rm
d}x.
\end{split}\end{equation}
Using the following formula \textup{(}see \textup{\cite[p.
16]{Temme1996}):}
\begin{equation}\label{Euler-constant-Fourier-Expansion}
E_{2m}(x)= (-1)^{m}\frac{4(2m)!}{\pi^{2m+1}}\sum_{k=0}^{\infty}\frac{\sin[(2k+1)\pi
x]}{(2k+1)^{2m+1}}\qquad (m\in\mathbb{N}, \quad 0\leq x\leq1),
\end{equation}
we obtain
\begin{align*}
r_m(t)&=\frac{(-1)^{m+1}}{e^t+1} \frac{4t^{2m+1}}{\pi^{2m+1}}\sum_{k=0}^{\infty}\int_{0}^{1}e^{xt}\frac{\sin[(2k+1)\pi x]}{(2k+1)^{2m+1}}{\rm d}x\nonumber\\
&=(-1)^{m+1}\frac{4t^{2m+1}}{\pi^{2m+1}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{2m}\big(t^2+\pi^2(2k+1)^2\big)}.
\end{align*}
This completes the proof of Theorem \ref{Thm1-remainder}.
\end{proof}
\begin{remark}
From \eqref{Chen-remainder-rm} we retrieve
\eqref{ThmEn-inequalityre}.
\end{remark}
\begin{remark}
From \textup{\cite[p. 592, Eq. (24.7.9)]{Olver-Lozier-Boisvert-Clarks2010}} and \textup{\cite[p. 43, Ex. 12(i)]{Wang-Guo1999}} we have
\begin{align*}
E_{2n}(x)=(-1)^n \sin(\pi x) \int_0^\infty \frac{4t^{2n} \cosh(\pi t)}{\cosh(2\pi t)-\cos(2\pi x)}\, {\rm d}t
\qquad (0<x<1,\quad n\in \mathbb{N}_0),
\end{align*}
from which it follows that
\begin{align*}
E_{4m}(x)>0\quad \mbox{and}\quad E_{4m+2}(x)<0\qquad (
0<x<1,\quad m\in \mathbb{N}_0).
\end{align*}
By combining these inequalities with \eqref{Boole-summation-formula-re-obtainnew}
and \eqref{Boole-summation-formula-re-obtainNew} we immediately
obtain \eqref{ThmEn-inequality}.
\end{remark}
\begin{corollary}\label{Thm2-remainder}
For $t>0$ and $m\in\mathbb{N}$,
\begin{align}\label{diff-rm(x)obtain}
(-1)^{m}\left(\frac{2e^t}{(e^t+1)^2}-\sum_{j=1}^{m}\frac{(2^{2j}-1)B_{2j}}{j\cdot(2j-2)!}t^{2j-2}\right)>0.
\end{align}
\end{corollary}
\begin{proof}
Differentiating the expression in \eqref{Chen-remainder-rm}, we find
\begin{align}\label{diff-rm(x)}
-\frac{2}{(e^t+1)^2}e^t=-\sum_{j=1}^{m}\frac{(2^{2j}-1)B_{2j}}{j\cdot(2j-2)!}t^{2j-2}+(-1)^{m+1}\big(t^{2m+1}s_m(t)\big)'.
\end{align}
It is easy to see that
\begin{align*}
t^2s_m(t)+s_{m-1}(t)=\frac{4}{\pi^{2m}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{2m}}=\frac{4}{\pi^{2m}}(1-2^{-2m})\zeta(2m),
\end{align*}
where $\zeta(z)$ is
the Riemann zeta function. This last expression can be written as
\begin{align}\label{Chen-rm(x)-rm-1(x)}
t^2s_m(t)=\frac{4}{\pi^{2m}}(1-2^{-2m})\zeta(2m)-s_{m-1}(t).
\end{align}
Then, since $s_m(t)$ is strictly decreasing for $t>0$, we deduce from
\eqref{Chen-rm(x)-rm-1(x)} that $t^2s_m(t)$ is strictly increasing
for $t>0$. Hence, $t^{2m+1}s_m(t)$ is strictly increasing for
$t>0$, and we then obtain from \eqref{diff-rm(x)} that
\begin{align*}
(-1)^{m}\left(\frac{2e^t}{(e^t+1)^2}-\sum_{j=1}^{m}\frac{(2^{2j}-1)B_{2j}}{j\cdot(2j-2)!}t^{2j-2}\right)=\big(t^{2m+1}s_m(t)\big)'>0
\end{align*}
for $t>0$ and $m\in\mathbb{N}$. The proof is complete.
\end{proof}
\begin{theorem}\label{Thm2-remainder-cosh-OpenProblem}
For $t > 0$ and $N\in\mathbb{N}$, we have
\begin{align}\label{Thm1-remainder-cosh-OpenProblemsolution1}
\sech t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t)
\end{align}
with
\begin{align}
R_N(t)=\frac{(-1)^{N}2t^{2N}}{\pi^{2N-1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+\frac{1}{2})^{2N-1}\Big(t^2+\pi^2(k+\frac{1}{2})^2\Big)},
\end{align}
and
\begin{align}\label{Solution-open}
\sech t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+\Theta(t,
N)\frac{E_{2N}}{(2N)!}t^{2N}
\end{align}
with a suitable $0 < \Theta(t, N) < 1$.
\end{theorem}
\begin{proof}
It follows from \cite[p. 136]{Whittaker-Watson1966} (see also
\cite[p. 458, Eq. (27.3)]{Berndt1998}) that
\begin{equation*}\label{Euler-constantSm-inequality1}
\frac{\pi}{4\cosh\left(\frac{\pi
x}{2}\right)}=\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)}{(2k+1)^{2}+x^2},
\end{equation*}
which can be written as
\begin{equation}\label{cosh-expansion}
\sech t=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)\left(1+\left(\frac{2t}{\pi(2k+1)}\right)^2\right)}.
\end{equation}
Substitution of $x=\frac{1}{2}$ in \eqref{Euler-constant-Fourier-Expansion} leads to
\begin{equation}\label{Euler-constant-Fourier-Expansion-obtain}
\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2j+1}}=\frac{(-1)^{j}\pi^{2j+1}}{2^{2j+2}(2j)!}\,E_{2j}.
\end{equation}
Using the identity
\begin{equation}\label{identity1/(1+z)}
\frac{1}{1+q}=\sum_{j=0}^{N-1}(-1)^{j}q^{j}+(-1)^{N}\frac{q^N}{1+q}\qquad
(q\not=-1)
\end{equation}
and \eqref{Euler-constant-Fourier-Expansion-obtain}, we obtain from
\eqref{cosh-expansion} that
\begin{align*}
\sech t&=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)}\left(\sum_{j=0}^{N-1}(-1)^{j}\left(\frac{2t}{\pi(2k+1)}\right)^{2j}+(-1)^{N}\frac{\left(\frac{2t}{\pi(2k+1)}\right)^{2N}}{1+\left(\frac{2t}{\pi(2k+1)}\right)^2}\right)\nonumber\\
&=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t),
\end{align*}
with
\begin{align*}
R_N(t)=\frac{2}{\pi^{2N-1}}\sum_{k=0}^{\infty}\frac{(-1)^{N+k}}{(k+\frac{1}{2})^{2N-1}}\,\frac{t^{2N}}{\big(t^2+\pi^2(k+\frac{1}{2})^2\big)}.
\end{align*}
Noting that \eqref{Euler-constant-Fourier-Expansion-obtain} holds,
we find that $R_N(t)$ can be written as
\begin{align*}
R_N(t)=\Theta(t,N)\,\frac{E_{2N} t^{2N}}{(2N)!},\qquad \Theta(t,N):=\frac{F(t)}{F(0)},
\end{align*}
where
\begin{align*}
F(t):=\sum_{k=0}^\infty (-1)^k \alpha_k,\qquad \alpha_k:=\frac{1}{(k+\frac{1}{2})^{2N-1}}\,\frac{1}{t^2+\pi^2(k+\frac{1}{2})^2}.
\end{align*}
Then it is easily seen that $\alpha_{2k}>\alpha_{2k+1}$ for $k\in\mathbb{N}_0$, $t>0$ and $N\in\mathbb{N}$; thus $F(t)>0$ for $t>0$.
Differentiation yields
\begin{align*}
F'(t)=-2t\sum_{k=0}^{\infty}\frac{(-1)^{k}\alpha_k}{t^2+\pi^2(k+\frac{1}{2})^2}
\end{align*}
and a similar reasoning shows that $F'(t)<0$ for $t>0$.
Hence, for all $t > 0$ and $N\in\mathbb{N}$, we have $0 < F (t) < F(0)$ and thus $0 < \Theta(t,
N) < 1$. The proof of Theorem \ref{Thm2-remainder-cosh-OpenProblem} is complete.
\end{proof}
\begin{remark}
Recalling that
\begin{align*}
E_{4m} >0\quad \mbox{and}\quad E_{4m+2}<0\qquad ( m\in
\mathbb{N}_0),
\end{align*}
we can deduce \eqref{Euler-constantSm-inequality1}
from \eqref{Solution-open}. Note that the inequality
\eqref{Euler-constantSm-inequality1} can also be written as
\begin{equation}\label{Euler-constantSm-inequality-ie}
(-1)^{m+1}\left(\sech
t-\sum_{j=0}^{m}\frac{E_{2j}}{(2j)!}t^{2j}\right)>0\qquad (t>0,
\,\, m\in\mathbb{N}_0).
\end{equation}
\end{remark}
\begin{remark}\label{Remark-completely-function-Vm}
It was shown in \cite{Chen790--799} that
\eqref{conjecture-Gamma-prod-nu} can be written as
\begin{equation}
\nu_m(t)=-\frac{1}{2\cosh(\frac{t}{4})}+\sum_{j=0}^{m}\frac{E_{2j}}{2(2j)!}\left(\frac{t}{4}\right)^{2j}
\end{equation}
and \eqref{Conjecture1-Gamma-nu} is equivalent to
\eqref{Euler-constantSm-inequality-ie}. Hence, for $t>0$ and
$m\in\mathbb{N}_0$, \eqref{Conjecture1-Gamma-nu} holds true.
It was also shown in \cite{Chen790--799} that
\begin{align}\label{Vm-gammaRatio1/4}
V_{m}(x)&=(-1)^m\Bigg[\int_{0}^{\infty}\left(\frac{e^{t/4}-e^{3t/4}}{e^{t}-1}+\frac{1}{2}\right)\frac{e^{-xt}}{t}{\rm
d}t-\sum_{j=1}^{m}\frac{2B_{2j+1}(\frac{1}{4})}{(2j+1)!}\int_0^\infty
t^{2j-1}e^{-xt}{\rm d}
t\Bigg]\notag\\
&=\int_{0}^{\infty}(-1)^{m}\nu_m(t)\frac{e^{-xt}}{t}\textup{\,d}t.
\end{align}
We obtain from \eqref{Vm-gammaRatio1/4}
that for all $m\in\mathbb{N}_0$,
\begin{align*}
(-1)^{n}V_m^{(n)}(x)=\int_{0}^{\infty}(-1)^{m}\nu_m(t)t^{n-1}e^{-xt}\textup{\,d}t>0
\end{align*}
for $x>0$ and $n\in\mathbb{N}_0$. Hence, the function
$V_m(x)$, defined by
\eqref{Thm1-asymptotic-ratio-gammas-rewrittenfind-Vx}, is
completely monotonic on $(0,\infty)$.
\end{remark}
Sondow and Hadjicostas \cite{Sondowa292--314} introduced and studied
the generalized-Euler-constant function $\gamma(z)$, defined by
\begin{align}\label{gammaz}
\gamma(z)=\sum_{n=1}^{\infty}z^{n-1}\left(\frac{1}{n}-\ln
\frac{n+1}{n}\right),
\end{align}
where the series converges when $|z|\leq 1$. Pilehrood and Pilehrood
\cite{Pilehrood117--131} considered the function $z\gamma(z)$ ($|z|\leq1$). The function
$\gamma(z)$ generalizes both Euler's constant $\gamma(1)$ and the
alternating Euler constant $\ln \frac{4}{\pi} = \gamma(-1)$
\cite{Sondow61--65, Sondow2005}. An interesting comparison by Sondow \cite{Sondow61--65} is the double integral and alternating series
\begin{equation}\label{Sondow-double-integral}\begin{split}
\ln\frac{4}{\pi}=\int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1+xy)\ln(xy)}\textup{d}x\textup{d}y=\sum_{n=1}^{\infty}(-1)^{n-1}\left(\frac{1}{n}-\ln
\frac{n+1}{n}\right).
\end{split}\end{equation}
The formula \eqref{Vm-gammaRatio1/4} can provide integral
representations for the constant $\pi$. For example,
the choice $(x,m)=(1/4, 0)$ in \eqref{Vm-gammaRatio1/4} yields
\begin{equation}\label{new-representationsPi2new}\begin{split}
\int_{0}^{\infty}\left(\frac{e^{t/4}-e^{3t/4}}{e^{t}-1}+\frac{1}{2}\right)\frac{2e^{-t/4}}{t}{\rm
d}t=\ln\frac{4}{\pi},
\end{split}\end{equation}
which provides a new integral representation for the
alternating Euler constant $\ln \frac{4}{\pi}$.
The choice $(x,m)=(3/4, 0)$ in \eqref{Vm-gammaRatio1/4} yields
\begin{equation}\label{new-representationsPi2newnew}\begin{split}
\int_{0}^{\infty}\left(\frac{e^{t/4}-e^{3t/4}}{e^{t}-1}+\frac{1}{2}\right)\frac{2e^{-3t/4}}{t}{\rm
d}t=\ln\frac{\pi}{3}.
\end{split}\end{equation}
Many formulas exist for the representation of $\pi$, and a
collection of these formulas is listed in
\cite{Sofo184--189,SofoJIPAM2005}. For more history of $\pi$ see
\cite{Beckmann1971, Berggren-Borwein-Borwein, Dunham1990}.
Noting \cite[Eq. (3.26)]{Chen790--799} that $B_{2n+1}(\tfrac{1}{4})$
can be expressed in terms of the Euler numbers
\begin{align}\label{B1/4-E2n}
B_{2n+1}(\tfrac{1}{4})=-\frac{(2n+1)E_{2n}}{4^{2n+1}}\qquad (n\in\mathbb{N}_0),
\end{align}
we find that \eqref{Thm1-asymptotic-ratio-gammas-rewrittenfind-Vx}
can be written as
\begin{align}\label{Thm1-asymptotic-ratio-gammas-rewrittenfindVxEuler-numbers}
V_m(x)=(-1)^{m}\left[\ln\frac{\Gamma(x+\frac{3}{4})}{x^{1/2}\Gamma(x+\frac{1}{4})}+
\sum_{j=1}^{m}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{x^{2j}}\right].
\end{align}
From the inequalities $V_m(x)>0$ for $x>0$, we obtain the
following
\begin{corollary}\label{corollary1-Gamma-1/3-2/3-ratio-inequalityVx}
For $x>0$,
\begin{align}\label{Thm1-asymptotic-ratio-gammas-rewrittenfindVx}
x^{1/2}\exp\left(-\sum_{j=1}^{2m}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{x^{2j}}\right)<\frac{\Gamma(x+\frac{3}{4})}{\Gamma(x+\frac{1}{4})}<
x^{1/2}\exp\left(-\sum_{j=1}^{2m+1}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{x^{2j}}\right).
\end{align}
\end{corollary}
The problem of finding new and sharp inequalities for the gamma
function $\Gamma$ and, in particular, for the Wallis ratio
\begin{equation}\label{WallisRatio}\begin{split}
\frac{(2n-1)!!}{(2n)!!}=\frac{1}{\sqrt{\pi}}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}
\end{split}\end{equation}
has attracted the attention of many researchers
(see
\cite{Chen-Paris514--529,ChenQi397-401,Koumandos1365--1367,Lampret328--339,Lampret775--787,
Mortici425--433} and references therein).
Here, we employ the special double
factorial notation as follows:
\begin{align*}
&(2n)!!=2\cdot 4\cdot 6\cdots (2n)=2^n n!,\quad 0!!=1,\qquad (-1)!!=1,\\
&(2n-1)!!=1\cdot 3\cdot 5\cdots
(2n-1)=\pi^{-1/2}2^n\Gamma\left(n+\frac{1}{2}\right);
\end{align*}
see \cite[p. 258]{abram}. For example, Chen and Qi \cite{ChenQi397-401} proved that for $n\in\mathbb{N}$,
\begin{equation}
\label{walthmin}
\frac1{\sqrt{\pi\bigl(n+\frac{4}{\pi}-1\bigr)}}\le\frac{(2n-1)!!}{(2n)!!}<\frac1{\sqrt{\pi\bigl(n+\frac14\bigr)}},
\end{equation}
where the constants $\frac{4}{\pi}-1$ and $\frac14$ are the best
possible. This inequality is a consequence
of the complete monotonicity on $(0, \infty)$ of the function (see \cite{Chen-Qi303--307})
\begin{equation}\label{WallisRatioVx}
V(x)=\frac{\Gamma(x+1)}{\sqrt{x+\frac{1}{4}}\,\Gamma(x+\frac{1}{2})}.
\end{equation}
If we write \eqref{Thm1-asymptotic-ratio-gammas-rewrittenfindVx} as
\begin{align*}
\frac{1}{\sqrt{x}}\exp\left(\sum_{j=1}^{2m+1}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{x^{2j}}\right)<\frac{\Gamma(x+\frac{1}{4})}{\Gamma(x+\frac{3}{4})}<
\frac{1}{\sqrt{x}}\exp\left(\sum_{j=1}^{2m}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{x^{2j}}\right)
\end{align*}
and replace $x$ by $x+\frac{1}{4}$, we find
\begin{align}\label{Thm1-asymptotic-ratio-gammas-rewrittenfindVxwrite-yields}
&\frac{1}{\sqrt{x+\frac{1}{4}}}\exp\left(\sum_{j=1}^{2m+1}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{(x+\frac{1}{4})^{2j}}\right)<\frac{\Gamma(x+\frac{1}{2})}{\Gamma(x+1)}\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad<\frac{1}{\sqrt{x+\frac{1}{4}}}\exp\left(\sum_{j=1}^{2m}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{(x+\frac{1}{4})^{2j}}\right).
\end{align}
Noting that \eqref{WallisRatio} holds, we then deduce from
\eqref{Thm1-asymptotic-ratio-gammas-rewrittenfindVxwrite-yields}
that
\begin{align}\label{Thm1-asymptotic-ratio-gammas-rewrittenfindVxwrite-yieldsWallis-ineq}
&\frac{1}{\sqrt{\pi(x+\frac{1}{4})}}\exp\left(\sum_{j=1}^{2m+1}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{(x+\frac{1}{4})^{2j}}\right)<\frac{(2n-1)!!}{(2n)!!}\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad<\frac{1}{\sqrt{\pi(x+\frac{1}{4})}}\exp\left(\sum_{j=1}^{2m}\frac{E_{2j}}{j\cdot4^{2j+1}}\frac{1}{(x+\frac{1}{4})^{2j}}\right),
\end{align}
which generalizes a recently published result of Chen \cite[Eq.
(3.40)]{Chen790--799}, who proved the inequality
\eqref{Thm1-asymptotic-ratio-gammas-rewrittenfindVxwrite-yieldsWallis-ineq}
for $m=1$.
\begin{theorem}\label{Thm3-remainder-coth}
For $t > 0$ and $N\in\mathbb{N}_0$, we have
\begin{align}\label{Thm2-remainder-coth-expansion}
\coth t=\sum_{j=0}^{N}\frac{2^{2j}B_{2j}}{(2j)!}t^{2j-1}+\sigma_N(t),
\end{align}
where
\begin{align}\label{Thm2-remainder-coth-rN}
\sigma_N(t)=\frac{(-1)^{N}t^{2N+1}}{\pi^{2N}}\sum_{k=1}^{\infty}\frac{2}{k^{2N}(t^2+\pi^2k^2)},
\end{align}
and
\begin{align}\label{Thm2-remainder-coth-theta}
\coth t=\sum_{j=0}^{N}\frac{2^{2j}B_{2j}}{(2j)!}t^{2j-1}+\theta(t,
N)\frac{2^{2N+2}B_{2N+2}}{(2N+2)!}t^{2N+1}
\end{align}
with a suitable $0<\theta(t, N)<1$.
\end{theorem}
\begin{proof}
It follows from \cite[p. 126, Eq.
(4.36.3)]{Olver-Lozier-Boisvert-Clarks2010} that
\begin{equation}\label{coth-expansion}
\coth
t=\frac{1}{t}+2t\sum_{k=1}^{\infty}\frac{1}{\pi^2k^2+t^2}=\frac{1}{t}+\frac{2t}{\pi^2}\sum_{k=1}^{\infty}\frac{1}{k^2\big(1+(\frac{t}{\pi
k})^2\big)}.
\end{equation}
It is well known that
\begin{equation}\label{Zeta-2n}
\sum_{k=1}^{\infty}\frac{1}{k^{2j}}=\frac{(-1)^{j-1}(2\pi)^{2j}B_{2j}}{2(2j)!}.
\end{equation}
Using \eqref{identity1/(1+z)} and \eqref{Zeta-2n}, we obtain from
\eqref{coth-expansion} that
\begin{align*}
\coth
t&=\frac{1}{t}+2t\sum_{k=1}^{\infty}\frac{1}{k^2\pi^2}\left(\sum_{j=0}^{N-1}(-1)^{j}\left(\frac{t}{k\pi}\right)^{2j}+(-1)^{N}\frac{\left(\frac{t}{k\pi}\right)^{2N}}{1+\left(\frac{t}{k\pi}\right)^2}\right)\nonumber\\
&=\frac{1}{t}+\sum_{j=0}^{N-1}\frac{2^{2j+2}B_{2j+2}}{(2j+2)!}t^{2j+1}+\sigma_N(t)\\
&=\sum_{j=0}^{N}\frac{2^{2j}B_{2j}}{(2j)!}t^{2j-1}+\sigma_N(t)
\end{align*}
with
\begin{align*}
\sigma_N(t)=\frac{2(-1)^{N}}{\pi^{2N}}\sum_{k=1}^{\infty}\frac{t^{2N+1}}{k^{2N}(t^2+\pi^2k^2)}.
\end{align*}
Noting that \eqref{Zeta-2n} holds, we can rewrite $\sigma_N(t)$ as
\begin{align*}
\sigma_N(t)=\theta(t,N)\,\frac{2^{2N+2} B_{2N+2}}{(2N+2)!}\,t^{2N+1},
\end{align*}
where
\begin{align*}
\theta(t,N):=\frac{f(t)}{f(0)},\qquad f(t):=\sum_{k=1}^\infty \frac{1}{k^{2N}(t^2+\pi^2k^2)}.
\end{align*}
Obviously, $f(t)>0$ and is strictly decreasing for $t>0$. Hence, for all
$t > 0$, $0 < f (t) < f (0)$ and thus $0 < \theta(t, N) < 1$. The
proof of Theorem \ref{Thm3-remainder-coth} is complete.
\end{proof}
The following expansion for Barnes $G$-function
was established by Ferreira and L\'opez \cite[Theorem
1]{Ferreira298-314}. For $|\textup{arg}(z)|<\pi$,
\begin{equation*}\label{Ferreira-Lopez-Formula}\begin{split}
\ln G(z+1)&=\frac{1}{4}z^{2}+z\ln
\Gamma(z+1)-\left(\frac{1}{2}z^{2}+\frac{1}{2}z+\frac{1}{12}\right)\ln z-\ln A\\
&\quad+\sum_{k=1}^{N-1}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}+\mathcal{R}_{N}(z)\qquad
(N\in\mathbb{N}),
\end{split}\end{equation*}
where $B_{2k+2}$ are the Bernoulli numbers and $A$ is the
Glaisher--Kinkelin constant defined by
\begin{equation}\label{An}
\ln A=\lim_{n \to \infty}\left\{\ln
\left(\prod_{k=1}^{n}k^{k}\right)-\left(\frac{n^{2}}{2}+\frac{n}{2}+\frac{1}{12}\right)\ln
n+\frac{n^{2}}{4}\right\},
\end{equation}
the numerical value of $A$ being $1.282427\ldots$. For $\Re(z)>0$, the remainder
$\mathcal{R}_{N}(z)$ is given by
\begin{equation}\label{G-RN}
\mathcal{R}_{N}(z)=\int_{0}^{\infty}\left(\frac{t}{e^{t}-1}-\sum_{k=0}^{2N}\frac{B_{k}}{k!}t^{k}\right)\frac{e^{-zt}}{t^{3}}d
t.
\end{equation}
Estimates for $|\mathcal{R}_{N}(z)|$ were also obtained by Ferreira and L\'opez
\cite{Ferreira298-314}, showing that the expansion is indeed an
asymptotic expansion of $\ln G(z+1)$ in sectors of the complex plane
cut along the negative axis. Pedersen \cite[Theorem
1.1]{Pedersen171-178} proved that for any $N\geq 1$, the function $
x\mapsto (-1)^{N}\mathcal{R}_{N}(x)$ is completely monotonic on $(0, \infty)$.
Here, we present another proof of this complete monotonicity result.
From \eqref{Thm2-remainder-coth-expansion}, we obtain the following
inequality:
\begin{align*}
\sum_{j=0}^{2m}\frac{2^{2j}B_{2j}}{(2j)!}t^{2j-1}<\coth
t<\sum_{j=0}^{2m+1}\frac{2^{2j}B_{2j}}{(2j)!}t^{2j-1}\qquad (t>0,
\,\, m\in\mathbb{N}_0),
\end{align*}
which is equivalent to
\begin{align}\label{Thm2-remainder-coth-expansion-inequality-ie}
(-1)^{N}\left(\coth
t-\sum_{j=0}^{N}\frac{2^{2j}B_{2j}}{(2j)!}t^{2j-1}\right)>0\qquad
(t>0, \,\, N\in\mathbb{N}_0).
\end{align}
Replacement of $t$ by $t/2$ in
\eqref{Thm2-remainder-coth-expansion-inequality-ie} yields
\begin{align}\label{Thm2-remainder-coth-expansion-inequality-or}
(-1)^{N}\left(\frac{t}{2}\coth
\left(\frac{t}{2}\right)-\sum_{j=0}^{N}\frac{B_{2j}}{(2j)!}t^{2j}\right)>0\qquad
(t>0, \,\, N\in\mathbb{N}_0).
\end{align}
Accordingly, we obtain from \eqref{G-RN} that the function
\begin{equation}\label{G-RN-x}
(-1)^{N}\mathcal{R}_{N}(x)=\int_{0}^{\infty}(-1)^{N}\left(\frac{t}{2}\coth\left(\frac{t}{2}\right)-\sum_{k=0}^{N}\frac{B_{2k}}{(2k)!}t^{2k}\right)\frac{e^{-xt}}{t^{3}}d
t
\end{equation}
is completely monotonic on $(0, \infty)$.
\begin{remark}\label{Remark-deducte-remainder3again}
From \eqref{Thm2-remainder-coth-expansion}, we can deduce
\eqref{remainder3}. In fact, noting that
\begin{align*}
\coth t=\frac{e^t+e^{-t}}{e^t-e^{-t}}=1+\frac{2}{e^{2t}-1},
\end{align*}
we see that \eqref{Thm2-remainder-coth-expansion} can be written as
\begin{align}\label{Thm2-remainder-coth-expansion-re}
x+\frac{2x}{e^{2x}-1}=\sum_{j=0}^{N}\frac{B_{2j}}{(2j)!}(2x)^{2j}+\frac{(-1)^{N}x^{2N+2}}{\pi^{2N}}\sum_{k=1}^{\infty}\frac{2}{k^{2N}(x^2+\pi^2k^2)}.
\end{align}
Replacement of $x$ by $t/2$ in
\eqref{Thm2-remainder-coth-expansion-re} yields \eqref{remainder3}.
\end{remark}
\vskip 8mm
\section{A quadratic recurrence relation for $B_n$}
Euler (see \cite[p. 595, Eq. (24.14.2)]{Olver-Lozier-Boisvert-Clarks2010} and \cite{Wikipedia-contributors-Bernoulli-number}) presented a quadratic recurrence relation for the Bernoulli numbers:
\begin{align}\label{Euler-quadratic-recurrences-Bn}
\sum_{k=0}^{n}\binom{n}{k} B_{k}B_{n-k}=(1-n)B_n-nB_{n-1}\qquad (n\geq1),
\end{align}
which is equivalent to
\begin{align}\label{Euler-quadratic-recurrences-Bn-equivalent}
\sum_{j=1}^{n-1}\binom{2n}{2j} B_{2j}B_{2n-2j}=-(2n+1)B_{2n} \qquad (n\geq2).
\end{align}
The relation \eqref{Euler-quadratic-recurrences-Bn-equivalent} can be used to show by induction that
\begin{align*}
(-1)^{n-1}B_{2n}>0 \quad \text{for all}\quad n\geq1,
\end{align*}
i.e., the even-index Bernoulli numbers have alternating signs. Other quadratic recurrences for the Bernoulli numbers have been given
by Gosper (see \cite[Eq. (38)]{Weisstein-Bn}) as
\begin{align}\label{Gosper-quadratic-recurrences-Bn}
B_n=\frac{1}{1-n}\sum_{k=0}^n (1-2^{1-k})(1-2^{k-n+1})\binom{n}{k} B_{k}B_{n-k}
\end{align}
and by Matiyasevitch \cite{Matiyasevich1997} (see also \cite{Wikipedia-contributors-Bernoulli-number}) as
\begin{align}\label{Matiyasevitch-quadratic-recurrences-Bn}
B_n=\frac{1}{n(n+1)}\sum_{k=2}^{n-2} \left\{n+2-2\binom{n+2}{k} \right\} B_{k}B_{n-k}\qquad (n\geq 4).
\end{align}
Here, we present a (presumably new) quadratic recurrence relation for the Bernoulli numbers.
\begin{theorem}\label{Thm4-remainder}
The Bernoulli numbers satisfy the following quadratic recurrence relation:
\begin{align}\label{Thm3-Result}
B_{n}=\frac{1}{2^{n}-1}\sum_{k=2}^{n-2}(1-2^k)\binom{n}{k}B_kB_{n-k} \quad \quad (n\geq4).
\end{align}
\end{theorem}
\begin{proof}
If we replace $t$ by $t/2$ in \eqref{Chen-remainder-rm}, we find
\begin{equation}\label{Write-Ej-as-bj}
\frac{2}{e^{t/2}+1}=1+\sum_{j=2}^{\infty}b_{j}t^{j-1}, \qquad b_j=\frac{2(1-2^{j})B_{j}}{2^{j-1}\cdot j!}.
\end{equation}
The Bernoulli numbers $B_n$ are defined by the generating function
\begin{equation}\label{generalizedBernoullinumbers}
\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^n}{n!},
\end{equation}
which yields
\begin{equation}\label{Write-Bj-as}
\frac{t/2}{e^{t/2}-1}=\sum_{k=0}^{\infty}\frac{B_k t^k}{2^k k!}.
\end{equation}
It then follows from (\ref{Write-Ej-as-bj}) and (\ref{Write-Bj-as}) that
\begin{align*}
\frac{t}{e^{t}-1}&=\left(1+\sum_{j=2}^{\infty}b_{j}t^{j-1}\right)\sum_{k=0}^{\infty}\frac{B_k t^k}{2^{k} k!}\\
&=\sum_{k=0}^{\infty}\frac{B_k t^k}{2^{k} k!}+\sum_{j=2}^{\infty}b_jt^{j-1}\sum_{k=0}^{\infty}\frac{B_k t^k}{2^{k} k!}\\
&=\sum_{j=0}^{\infty}\frac{B_j t^j}{2^{j}
j!}+\sum_{j=1}^{\infty}\sum_{k=0}^{j-1}b_{k+2}\frac{B_{j-k-1} t^k}{2^{j-k-1}(j-k-1)!},
\end{align*}
that is
\begin{equation}\label{Chen-remainder-rmwriteas-t-2tthatisbj}
\frac{t}{e^{t}-1}=\sum_{j=0}^{\infty}\left(\frac{B_j}{2^{j}
j!}+\sum_{k=0}^{j-1}b_{k+2}\frac{B_{j-k-1}}{2^{j-k-1}(j-k-1)!}\right)t^j.
\end{equation}
Equating coefficients of equal powers of $t$ in
\eqref{generalizedBernoullinumbers} and
\eqref{Chen-remainder-rmwriteas-t-2tthatisbj}, we see that
\begin{align}\label{Chen-remainder-rmwriteas-t-2tthatisbjseeThm3}
\frac{B_j}{j!}=\frac{B_j}{2^{j}\cdot
j!}+\sum_{k=0}^{j-1}b_{k+2}\frac{B_{j-k-1}}{2^{j-k-1}\cdot(j-k-1)!}\qquad
(j\in\mathbb{N}_0).
\end{align}
Substitution of the coefficients $b_j$ in \eqref{Write-Ej-as-bj} into
\eqref{Chen-remainder-rmwriteas-t-2tthatisbjseeThm3} then yields
\begin{align}\label{Chen-remainder-rmwriteas-t-2tthatisbjresultThm3}
B_j=\frac{j!}{2^{j}-1}\sum_{k=0}^{j-1}\frac{2(1-2^{k+2})B_{k+2}B_{j-k-1}}{(k+2)!\cdot
(j-k-1)!}\qquad (j\in \mathbb{N}).
\end{align}
It is easy to see that
\begin{align*}
B_j&=\frac{j!}{2^{j}-1}\left(\sum_{k=0}^{j-3}\frac{2(1-2^{k+2})B_{k+2}B_{j-k-1}}{(k+2)!\cdot
(j-k-1)!}-\frac{(1-2^j)B_j}{j!}+\frac{2(1-2^{j+1})B_{j+1}}{(j+1)!}\right)\\
&=\frac{j!}{2^{j}-1}\sum_{k=0}^{j-3}\frac{2(1-2^{k+2})B_{k+2}B_{j-k-1}}{(k+2)!\cdot
(j-k-1)!}+B_j+\frac{2(1-2^{j+1})B_{j+1}}{(2^{j}-1)(j+1)}.
\end{align*}
We therefore obtain
\begin{align*}
B_{j+1}=\frac{1}{2^{j+1}-1}\sum_{k=0}^{j-3}(1-2^{k+2})\binom{j+1}{k+2}B_{k+2}B_{j-k-1} \quad \quad (n\in \mathbb{N}\setminus \{1, 2\}),
\end{align*}
which, upon replacing $j$ by $n-1$ and $k$ by $k-2$, yields \eqref{Thm3-Result}. This completes the proof of
Theorem \ref{Thm4-remainder}.
\end{proof}
\vskip 4mm
\begin{center}
{ Appendix: An alternative proof of \eqref{remainder3}}
\end{center}
\setcounter{section}{1}
\setcounter{equation}{0}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
Euler's summation formula states that, for $k\in\mathbb{N}$, (see \cite[p.~9]{Temme1996})
\begin{align}\label{Euler-summation-formula}
f(1)&=\int_{0}^{1}f(x)\,{\rm d}
x+\sum_{j=1}^{k}\frac{(-1)^{j}B_{j}}{j!}\Big(f^{(j-1)}(1)-f^{(j-1)}(0)\Big)\nonumber\\
&\quad+\frac{(-1)^{k+1}}{k!}\int_{0}^{1}f^{(k)}(x)B_{k}(x)\,{\rm d}x.
\end{align}
The choice $k=2n$ in \eqref{Euler-summation-formula} yields
\[\frac{f(1)+f(0)}{2}=\int_{0}^{1}f(x)\,{\rm d}
x+\sum_{j=1}^{n}\frac{B_{2j}}{(2j)!}\Big(f^{(2j-1)}(1)-f^{(2j-1)}(0)\Big)\hspace{3cm}\]
\[\hspace{5cm}-\frac{1}{(2n)!}\int_{0}^{1}f^{(2n)}(x)B_{2n}(x)\,{\rm d}x.\]
Application of this formula to $f(x)=e^{xt}$ then yields
\[\frac{t}{e^t-1}=1-\frac{t}{2}+\sum_{j=1}^{n}\frac{B_{2j}}{(2j)!}t^{2j}-\frac{t}{e^t-1}\frac{t^{2n}}{(2n)!}\int_{0}^{1}e^{xt}B_{2n}(x)\,{\rm d}x.\]
Using the following formula (see \cite[p.
5]{Temme1996})
\begin{equation*}
B_{2n}(x)= 2(-1)^{n+1}(2n)!\sum_{k=1}^{\infty}\frac{\cos(2k\pi
x)}{(2k\pi)^{2n}}\qquad (n\in\mathbb{N},\quad 0\leq x\leq1),
\end{equation*}
we have
\begin{align}\label{E2m-known-wehave}
\frac{t}{e^t-1}-\left(1-\frac{t}{2}+\sum_{j=1}^{n}\frac{B_{2j}}{(2j)!}t^{2j}\right)&=-\frac{t}{e^t-1}\frac{t^{2n}}{(2n)!}\int_{0}^{1}e^{xt}B_{2n}(x)\,{\rm d}x\nonumber\\
&=\frac{(-1)^{n}2t^{2n+1}}{e^t-1}\sum_{k=1}^{\infty}\int_{0}^{1}e^{xt}\frac{\cos (2k\pi
x)}{(2k\pi)^{2n}}\,{\rm d}x\nonumber\\
&=\frac{(-1)^{n}2t^{2n+2}}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{1}{k^{2n}(t^2+4\pi^2k^2)}\nonumber\\
&=(-1)^{n}t^{2n+2}\nu_{n}(t).
\end{align}
This gives another derivation of \eqref{remainder3}.
We obtain from \eqref{E2m-known-wehave} that
\begin{equation}\label{remainder-integral-Chen}
\nu_{n}(t)=\frac{(-1)^{n-1}}{t(e^t-1)\cdot(2n)!}\int_{0}^{1}e^{xt}B_{2n}(x)\,{\rm d}x,
\end{equation}
which provides an alternative integral representation of $\nu_{n}(t)$.
\vskip 8mm
| 54,627
|
TITLE: Can I state that a spacetime is homogeneous and isotropic iff $\nabla_\mu R = 0$?
QUESTION [2 upvotes]: If a spacetime is homogenous and isotropic can I say that $\nabla_\mu R =0$?
I was reading this paper https://arxiv.org/abs/astro-ph/0610483 and, I think that is the justification for the authors setting $\nabla_\mu R =0$ (just below Eq. (3)). (Am I correct here?)
I found some references (such as section 5.1 of Wald) that state a spacelike surface that is homogenous and isotropic will have constant curvature, but what bothers me is that the space like surface of curvature scalar K is not the same curvature scalar for a 3+1 universe with curvature scalar R.
But if this is not the case, I don't understand the author's justification for setting $\nabla_\mu R =0$ in the paper above.
REPLY [3 votes]: The condition $\nabla_\mu R=0$ simply means that scalar curvature is constant. It neither implies homogeneity and isotropy nor follows from it.
For example Ricci-flat spacetimes or solutions with cosmologiacal constant (but without matter) would have this condition but they are not necessarily isotropic or homogenous (one example is Kerr-de Sitter solution).
On the other side, spacetimes with homogenous and isotropic $t=const$ slices can have $\nabla_\mu R\ne 0$. Simplest example is FLRW metric with dust-like matter. It has time dependent Ricci scalar $R=-8\pi G \rho(t)$.
As for the paper referenced in question, the condition $\nabla_\mu R=0$ together with $T=0$ was imposed to check that one particular spacetime, the de Sitter vacuum (which has maximal number of spacetime symmetries) is a solution of this model and find the expression for de Sitter Hubble parameter through the model parameter $\mu$.
| 110,460
|
Sunday, 9 January 2011
A new dawn in the East
Maybe it is because I am Swedish and overly rational. Maybe it is because I run a small company where I still feel the effects of the crisis. To follow the Spanish political debate makes me nervous and impatient. But I am not the only one who is disappointed.
The least pleasant aspect of going back to Vilanova i la Geltrú, after having celebrated Christmas in Sweden, was the feeling of leaving behind an atmosphere of optimism and returning to depression. Fredrik Reinfeldt (M), Prime Minister of Sweden for the last four and a half years, today enjoys approval ratings of 72%; a percentage so high that commentators with a sense of humour draw parallels to North Korea. This can be contrasted with the situation in Spain where PSOE has ruled for seven years, but where almost 80% of the population expresses a lack of confidence in Prime Minister José Luis Rodriguez Zapatero.
Recently, Zapatero himself announced that it will take until 2013 before we reach acceptable unemployment levels. I hope he is just as wrong this time as he when, in 2008, he saw the green shoots (brotes verdes in Spanish) before the crisis in Spain – and different to North and Central Europe – had not even really begun.
Some people probably feel sorry for Zapatero when Spain so often is being described as the domino tile which can cause the collapse of the Euro. Public finances here actually showed a surplus as late as 2007, so the state cannot be accused of having had chronic budget deficits. Even today, the public debt to GDP ratio is significantly lower than e.g. in France or Italy. Personally, however, I have a growing understanding of the discontent of the market. Several times, the government has announced reforms to improve the business environment, but put them on ice when the trade unions have protested.
But, lately, I have found reasons for a renewed hope. Until now we have not seen much of the trade unions’ counterpart - the employers’ federation CEOE. Its former President, Gerardo Díaz Ferrán, was deeply involved in legal conflicts with the staff of companies that he used to own, but allowed to go bankrupt, and that poisoned the media image of the entire interest organisation. But since the end of December there is a new leadership under Joan Rosell, who has already reached out a hand to the Spanish finance minister Helena Salgado.
In politics there has also been a big change. After CiU won the elections to the Catalan Parliament the party needs for the whole Spanish economy to rapidly get back on its feet. With the number of seats this group holds in the Spanish parliament, it is by itself big enough to give PSOE a majority for difficult decisions.
I hope that Zapatero will finally accept the inevitable: in order to bring down unemployment without worsening the fiscal outlook, the new jobs need to come from the private sector. That will only occur if someone believes that he or she can earn money by creating them - not by politicians continuing to focus their support on those who already have jobs.
After years of unsuccessful attempts to reach consensus with the trade unions, the government needs new ideas. Joan Rosell has his roots in the SMEs of Catalonia where, at the same time, CiU finds a significant part of their voters. Within Spain, Catalonia is known for its entrepreneurial spirit and manufacturing traditions. Imagine what a psychologic effect it could have for all business people if PSOE dared to approach the interlocutors who represent our perspective. Maybe I am an optimist in extremis, because I believe it will. I believe that we will see a new dawn. And the sun always rises in the east.
- - -
The same post is available in a Swedish version and you can also find translations into Spanish and Catalan.
- - -
Sources and inspiration
Aftonbladet.se: Världens vinnare
LaVozdeAsturias.es: Zapatero apenas mejora en confianza
Expansión.com: Zapatero no espera datos "satisfactorios" de creación de empleo hasta "dentro de 2 años"
CincoDías.com: Rosell ofrece a Salgado colaborar para conseguir "lo mejor para la economía"
LaVanguardia.es: Termina sin acuerdo la reunión entre Gobierno y sindicatos que sigue mañana
| 412,820
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\begin{document}
\title{Smooth densities for Stochastic Differential Equations with Jumps}
\author{THOMAS CASS }
\date{October 2, 2007}
\maketitle
\begin{abstract}
We consider a solution $x_{t}$ to a generic Markovian jump diffusion and
show that for any $t_{0}>0$ the law of $x_{t_{0}}$ has a $C^{\infty }$
density with respect to Lebesgue measure under a uniform version of H\"{o}
rmander's conditions. Unlike previous results in the area the result covers
a class of infinite activity jump processes. The result is accompolished by
using carefully crafted refinements to the classical arguments used in
proving the smoothness of density via Malliavin calculus. In particular, we
provide a proof that the semimartinagale inequality of Norris persists for
discontinuous semimartingales when the jumps are small.
\end{abstract}
\section{Introduction}
This paper focuses on the study of the stochastic differential equation
\begin{equation}
x_{t}=x+\int_{0}^{t}Z(x_{s-})ds+\int_{0}^{t}V(x_{s-})dW_{s}+\int_{0}^{t}
\int_{E}Y(x_{s-},y)(\mu -\nu )(dy,ds), \label{SDE}
\end{equation}
\noindent and addresses the fundamental problem of finding a sufficient
condition for the existence of a smooth ($C^{\infty }$) density for the
solution at positive times. For diffusion processes the pioneering work of
Bismut \cite{bismut} \ and Stroock \cite{stroock1} and \cite{stroock2}
provides a probabilistic framework for establishing such a result under H
\"{o}rmander's conditions on the vector fields. As is pointed out in \cite
{stroock2} it is, given the existence of alternative methods based on
partial differential equations, difficult to justify the effort involved in
the probabilistic proof of this result purely for the sake of diffusion
processes. From the outset it was always understood that this approach
should be used as a template for investigating the smoothness properties for
different probabilistic objects, not amenable to analysis by PDE theory. We
now switch our focus to the question : \textit{when does a solution to the
SDE (\ref{SDE}) admit a smooth density?}
We point out that we are by no means the first to consider this problem and\
several prominent landmarks are worthy of comment. \ The first comprehensive
account of these ideas was presented in \cite{bichteler2}, where a
smoothness result is proved under a uniform ellipticity on the diffusion
vector fields (in fact \cite{bichteler2} also explores how a smooth density
can be acquired through the jump component)$.$ Further progress was made in
\cite{norris1} where existence of the density was shown under a version of H
\"{o}rmander's conditions which are local in the starting point. Both these
works were successful in establishing a criterion for a smooth density
namely that the inverse of \ the (reduced) Malliavin covariance matrix has
finite $L^{p}$ norms for $p\geq 2$.
Verification of this criterion usually occurs by way of subtle estimates on
the reduced covariance matrix which are in general difficult to establish.
In the diffusion case a streamlined approach to obtaining these estimates
has been achieved by a semimartingale inequality known as Norris's lemma
(see \cite{norris} or \cite{Nu06}). This result, interesting in its own
right, provides an estimate for the probability that a continuous
semimartingale is small on a set where its quadratic variation is
comparatively large. Traditionally, this result has been presented as a
quantitative form of the uniqueness of the Doob-Meyer decomposition for
continuous semimartingales, however the appearance of similar estimates in
the context of fractional Brownian motion with $H>1/2$ (not a
semimartingale) (see \cite{BH}) have made it seem as though Norris's lemma
expresses something fundamental rather than anything tied to the particular
structure of continuous semimartingales.
Some recent work in the case of jump diffusions has been undertaken in \cite
{teichmann},\cite{komatsu} and \cite{tak}. The article \cite{teichmann}
proves a smoothness result under uniform H\"{o}rmander conditions and under
the assumption that the underlying jump process is of finite activity. This
is achieved by fixing some $T>0$, conditioning on $N_{T}=n$, the number of
jumps until time $T,$ and noticing that this gives rise to some (random)
interval $[S_{1}(\omega ),S_{2}(\omega ))$ with $0\leq S_{1}<S_{2}<T$ such
that $S_{2}(\omega )-S_{1}(\omega )\geq T(n+1)^{-1}$ and
\begin{equation*}
\left\{ x_{t}^{x}:S_{1}\leq t<S_{2}\right\} \overset{\mathcal{D}}{=}\left\{
\tilde{x}_{t}^{x_{S_{1}}^{x}}:0\leq t<S_{2}-S_{1}\right\}
\end{equation*}
\noindent where\bigskip\ $\tilde{x}_{t}^{x}$ is the diffusion process
\begin{equation*}
\tilde{x}_{t}^{x}=x+\int_{0}^{t}Z(\tilde{x}_{s}^{x})ds+\int_{0}^{t}V(\tilde{x
}_{s}^{x})dW_{s}.
\end{equation*}
\noindent The usual diffusion Norris lemma may be applied to give estimates
for the Malliavin covariance matrix arising from $\tilde{x}_{t}$ on this
interval which can then easily be related to covariance matrix for $x_{t}$.
In this paper we pursue this idea further by proving that the quality of the
estimate which features in Norris's lemma is preserved when jumps are
introduced provided that these jumps are small enough that they do not
interfere too much. \ We then develop the conditioning argument outlined
above by splitting up the sample path into disjoint intervals on which the
jumps are small, and then estimating the Malliavin covariance matrix on the
largest of these intervals. The outcome of this reasoning will be the
conclusion that a solution to (\ref{SDE}) has a smooth density under uniform
H\"{o}rmander conditions (indeed, the same conditions as in \cite{teichmann}
) and subject to some restrictions on the rate at which the jump measure
accumulates small jumps. These conditions are sufficiently flexible to admit
some jump diffusions based on infinite activity jump processes.
This paper is arranged as follows : we first present some preliminary
results and notation on Malliavin calculus. Subsequently, we state and prove
our new version of Norris's lemma and then illustrate how it may be utilized
in concert with classical arguments to verify the $C^{\infty }$ density
criterion for the solution to (\ref{SDE}).
\begin{acknowledgement}
The author would like to thank James Norris and Peter Friz for related
discussions.
\end{acknowledgement}
\section{\protect\bigskip Preliminaries}
\bigskip Let $x_{t}$ denote the solution to the SDE
\begin{equation}
x_{t}=x+\int_{0}^{t}Z(x_{s-})ds+\int_{0}^{t}V(x_{s-})dW_{s}+\int_{0}^{t}
\int_{E}Y(x_{s-},y)(\mu -\nu )(dy,ds), \label{sol}
\end{equation}
\noindent Where $W_{t}=\left( W_{t}^{1},...,W_{t}^{d}\right) $ is an $
\mathbb{R}
^{d}-$valued Brownian motion on some probability space $\left( \Omega ,
\mathcal{F}_{t},P\right) $ and $\mu $ is a $\left( \Omega ,\mathcal{F}
_{t},P\right) -$Poisson random measure on $E\times \lbrack 0,\infty )$ for
some topological \footnote{
we will later need some vector space structure on $E$ and will principally
be concerned with the case $E=\mathbb{R}^{n}$} space $E$ such that $\nu $,
the compensator of $\mu $, is of the form $G(dy)dt$ for some $\sigma $
-finite measure $G.$ The vector fields $Z:
\mathbb{R}
^{e}\rightarrow
\mathbb{R}
^{e},Y\left( \cdot ,y\right) :
\mathbb{R}
^{e}\rightarrow
\mathbb{R}
^{e}$ and $V=(V_{1},...,V_{d}),$ where $V_{i}:
\mathbb{R}
^{e}\rightarrow
\mathbb{R}
^{e}$ for $i\in \left\{ 1,...,d\right\} .$ At times we will write $x_{t}^{x}$
when we wish to emphasize the dependence of the process on its initial
condition.
We introduce some notation, firstly for $p\in
\mathbb{R}
$ let
\begin{equation*}
L_{+}^{p}(G)=\left\{ f:E\rightarrow
\mathbb{R}
^{+}:\int_{E}f(y)^{p}G(dy)<\infty \right\} ,
\end{equation*}
\noindent and define
\begin{equation*}
L_{+}^{p,\infty }(G)=\underset{q\geq p}{\cap }L_{+}^{q}(G).
\end{equation*}
\noindent \noindent We will always assume that at least the following
conditions are in force
\begin{condition}
\label{c1}$Z,V_{1},...,V_{d}\in C_{b}^{\infty }($ $
\mathbb{R}
^{e})$
\end{condition}
\begin{condition}
\label{c2}For some $\rho _{2}\in L_{+}^{2,\infty }(G)$ and every $n\in
\mathbb{N}
$
\begin{equation*}
\underset{y\in E,x\in
\mathbb{R}
^{e}}{\sup }\frac{1}{\rho _{2}(y)}|D _{1}^{n}Y(x,y)|<\infty .
\end{equation*}
\end{condition}
\begin{condition}
\label{c3}$\sup_{y\in E,x\in
\mathbb{R}
^{e}}|\left( I+D _{1}Y(x,y)\right) ^{-1}|<\infty .$
\end{condition}
\noindent We now define the processes $J_{t\leftarrow 0}$ and $
J_{0\leftarrow t}$ considered as linear maps from $
\mathbb{R}
^{e}$ to $
\mathbb{R}
^{e}$ as the solutions to the following SDEs
\begin{eqnarray}
J_{t\leftarrow 0} &=&I+\int_{0}^{t}DZ(x_{s-})J_{s-\leftarrow
0}ds+\int_{0}^{t}DV(x_{s-})J_{s-\leftarrow 0}dW_{s} \label{jacobian} \\
&&+\int_{0}^{t}\int_{E}D_{1}Y(x_{s-},y)J_{s-\leftarrow 0}(\mu -\nu )(dy,ds)
\notag
\end{eqnarray}
\noindent and
\begin{gather}
J_{0\leftarrow t}=I-\int_{0}^{t}J_{0\leftarrow s-}\Bigg(DZ(x_{s-})-\underset{
i=1}{\overset{d}{\sum }}DV_{i}(x_{s-})^{2} \label{ijacobian1} \\
-\int_{E}(I+D_{1}Y(x_{s-},y))^{-1}D_{1}Y(x_{s-},y)^{2}G(dy)\Bigg)
ds-\int_{0}^{t}J_{0\leftarrow s-}DV(x_{s-})dW_{s} \notag \\
-\int_{0}^{t}\int_{E}J_{0\leftarrow
s-}(I+D_{1}Y(x_{s-},y))^{-1}D_{1}Y(x_{s-},y)(\mu -\nu )(dy,ds). \notag
\end{gather}
\noindent The following result may then be verified (see for instance \cite
{norris})
\begin{theorem}
\label{jacob}Under conditions \ref{c1},\ref{c2} and \ref{c3} the system of
SDEs (\ref{sol},\ \ref{jacobian}) and \ (\ref{sol},\ref{ijacobian1}) have
unique solutions with
\begin{equation*}
\underset{0\leq s\leq t}{\sup }|J_{s\leftarrow 0}|\text{ and}\underset{0\leq
s\leq t}{\sup }|J_{0\leftarrow s}|\text{ }\in L^{p}
\end{equation*}
\noindent for all $t\geq 0$ and $p<\infty .$ Moreover,
\begin{equation*}
J_{0\leftarrow t}=J_{t\leftarrow 0}^{-1}\text{ for all }t\geq 0\text{ almost
surely.}
\end{equation*}
\end{theorem}
\noindent We define the reduced Malliavin covariance matrix
\begin{equation*}
C_{0,t}^{x,I}=C_{t}^{x,I}=\int_{0}^{t}\underset{i=1}{\overset{d}{\sum }}
J_{0\leftarrow s-}^{x,I}V_{i}(x_{s-}^{x})\otimes J_{0\leftarrow
s-}^{x,I}V_{i}(x_{s-}^{x})ds
\end{equation*}
\noindent which we will sometimes refer to simply as $C_{t}$ suppressing the
dependence on the initial conditions. The following well known result
provides a sufficient condition for the process $x_{t}$ to have a $C^{\infty
}$ density in terms of the moments of the inverse of $C_{t}.$
\begin{theorem}
\label{criterion}Fix $t_{0}>0$ and $x\in
\mathbb{R}
^{e}$ and suppose that for every $p\geq 2$ $\left\vert C_{t}^{-1}\right\vert
\in L^{p},$ then $x_{t_{0}}^{x}$ has a $C^{\infty }$density with respect to
Lebesgue measure.
\end{theorem}
\section{Norris's lemma}
From now on we set $E=\mathbb{R}^{n}$. The following result provides an
exponential martingale type inequality for a class of local martingales
based on stochastic integrals with respect to a Poisson random measure when
the jumps of the local martingale are bounded. Interesting discussions on
results of this type can be found in \cite{barlow} and \cite{dzh}
\begin{lemma}
\label{EMI}\noindent Let $\mu $ be a Poisson random measure on $E\times
\lbrack 0,\infty )$ with compensator $\nu $ of the form $\nu
(dy,dt)=G(dy)dt. $ Let $f(t,y)$ be a real-valued previsible process having
the property that
\begin{equation*}
\underset{y\in E}{\sup }\text{ }\underset{0\leq s\leq t}{\sup }|f(s,y)|<A
\end{equation*}
\noindent for every $0<t<\infty $ and some $A<\infty $. Then, if $
M_{t}=\int_{0}^{t}\int_{E}f(s,y)(\mu -\nu )(dy,ds)$ the following inequality
holds
\begin{equation*}
P\left( \underset{0\leq s\leq t}{\sup }|M_{s}|\geq \delta ,\left\langle
M\right\rangle _{t}<\rho \right) \leq 2\exp \left( -\frac{\delta ^{2}}{
2(A\delta +\rho )}\right)
\end{equation*}
\end{lemma}
\begin{proof}
Consider $Z_{t}=\exp (\theta M_{t}-\alpha \left\langle M\right\rangle _{t})$
with $0<\theta <A^{-1}$ and $\alpha =2^{-1}\theta ^{2}(1-\theta A)^{-1}.$
Since for any $x\in
\mathbb{R}
$ we have
\begin{equation}
g_{\theta }(x):=e^{\theta x}-1-\theta x=\underset{k=2}{\overset{\infty }{
\sum }}\frac{\theta ^{k}x^{k}}{k!}\leq \frac{\theta ^{2}x^{2}}{2}\underset{
k=0}{\overset{\infty }{\sum }}(\theta A)^{k}=\frac{\theta ^{2}x^{2}}{
2(1-\theta A)}=\alpha x^{2}. \label{expmart}
\end{equation}
\noindent We may deduce that $Z$ is a supermartingale by writing
\begin{eqnarray*}
Z_{t} &=&\exp \left( \theta M_{t}-\int_{0}^{t}\int_{E}g_{\theta
}(f(s,y))G(dy)ds\right) \\
&&\exp \left( \int_{0}^{t}\int_{E}\left( g_{\theta }(f(s,y))-\alpha
f(s,y)^{2}\right) G(dt)ds\right)
\end{eqnarray*}
\noindent and, using It\^{o}'s formula the first term of the product is a
non-negative local martingale (and hence a supermartingale) and the second
term decreases in $t$ by (\ref{expmart}). Define the stopping time $T=\inf
\left\{ s\geq 0:\left\langle M\right\rangle _{s}>\rho \right\} $ then, since
$E[Z_{T}]\leq 1,$ taking $\theta =\delta (\rho +A\delta )^{-1}$ and applying
Chebyshev's inequality gives
\begin{equation*}
P\left( \underset{0\leq s\leq t}{\sup }|M_{s}|\geq \delta ,\left\langle
M\right\rangle _{t}<\rho \right) \leq P\left( \underset{0\leq s\leq T}{\sup }
Z_{s}\geq e^{\delta \theta -\alpha \rho }\right) \leq \exp \left( -\frac{
\delta ^{2}}{2(A\delta +\rho )}\right) .
\end{equation*}
\noindent Finally, we complete the proof by applying the same argument to $
-M.$
\end{proof}
\bigskip
\noindent From now on we will assume that the following technical conditions
on the jump measure $G$ and the jump vector field $Y$ are in force :
\begin{condition}
\label{condition1}$\underset{x\in
\mathbb{R}
^{e}}{\sup }\int_{E}|Y(x,y)|G(dy)<\infty .$
\end{condition}
\begin{condition}
\label{condition2}For some $\kappa \geq n$ we have
\begin{equation}
\underset{\epsilon \downarrow 0}{\lim \sup }\frac{1}{f(\epsilon )}
\int_{|y|>\epsilon }G(dy)\text{ }<\infty \text{ ,} \label{cond2a}
\end{equation}
\noindent where $f:(0,\infty )\rightarrow (0,\infty )$ is defined by
\begin{equation}
f(x)=\left\{
\begin{array}{c}
\log x^{-1}\text{\ if }\kappa =n \\
x^{-\kappa +n}\text{\ \ if }\kappa >n
\end{array}
\right. . \label{cond2b}
\end{equation}
\noindent Moreover, for any $\beta >0$ we have
\begin{equation*}
\int_{E}|y|^{\kappa -n+\beta }G(dy)<\infty ,
\end{equation*}
\noindent and
\begin{equation}
\underset{\epsilon \downarrow 0}{\lim \sup }\frac{1}{\epsilon ^{\beta }}
\int_{|y|<\epsilon }|y|^{\kappa -n+\beta }G(dy)\text{ }<\infty \text{ .}
\label{cond2c}
\end{equation}
\end{condition}
\begin{condition}
\label{condition3}There exists a function $\phi \in L_{+}^{1}(G)$ which has
the properties that for some $\alpha >0$
\begin{equation*}
\underset{y\rightarrow 0}{\lim \sup }\frac{\phi (y)}{|y|^{\kappa -n+\alpha }}
\text{ }<\infty ,
\end{equation*}
\noindent and, for some positive constant $C<\infty $ and every $k\in
\mathbb{N}
\cup \left\{ 0\right\} $
\begin{equation*}
\underset{x\in
\mathbb{R}
^{e}}{\sup }|D _{1}^{k}Y(x,y)|\leq C\phi (y).
\end{equation*}
\end{condition}
\bigskip
Conditions \ref{condition1}, \ref{condition2} and \ref{condition3} may at
first sight appear somewhat opaque, however they will be crucial ingredient
in our subsequent arguments, in particular they enable us quantify the rate
at which the total mass of the jump measure increases near zero. \ To
develop intuition for their implications consider the following
straight-forward example : take $n=1$ and $Y(x,y)=\widetilde{Y}(x)y$ for
some $C^{\infty }-$bounded $\widetilde{Y}:
\mathbb{R}
^{e}\rightarrow
\mathbb{R}
^{e}$ (this puts us in the set up of \cite{teichmann})$.$ Also, define the
measure $G$ on $
\mathbb{R}
$ by taking $G(dy)=|y|^{-\kappa }dy$. We then see what is needed to verify
each of the conditions in turn, firstly, condition \ref{condition1} will be
satisfied provided
\begin{equation*}
\underset{x\in
\mathbb{R}
^{e}}{\sup }\int_{E}|\widetilde{Y}(x)y|G(dy)=\underset{x\in \mathbb{R}^{e}}{
\sup}|\widetilde{Y}(x)| \int_{E}|y|G(dy)=2\underset{x\in \mathbb{R}^{e}}{\sup
}|\widetilde{Y}(x)| \int_{0}^{\infty}y^{1-\kappa }dy<\infty ,
\end{equation*}
\bigskip \noindent which will hold so long as $\kappa <2$. The constraint
that $\kappa \geq 1$ in condition \ref{condition2} ensures that the jump
measure is of infinite activity and (\ref{cond2a}) and (\ref{cond2b}) are
trivially verified by integration. Since we are in the setting $1\leq \kappa
<2$, we may find $\alpha \in (0,1)$ such that $\kappa +\alpha <2$ to ensure
that $\phi (y):=|y|$ is $O(|y|^{\kappa -n+\alpha })$ as $y\rightarrow 0$ and
hence condition \ref{condition3} is also satisfied$.$
Suppose now that $\Upsilon :[0,t_{0}]\times E\rightarrow
\mathbb{R}
.$ is some given, real-valued, previsible process. It will at times be
important for us to impose the following condition on $\Upsilon$ .
\begin{condition}
\label{condition4}Let $G$ satisfy condition \ref{condition2}. Then there
exists some previsible process $D_{t}$ taking values in $[0,\infty )$ with $
\sup_{0\leq t\leq t_{0}}D_{t}\in L^{p}$ for all $p\geq 1,$ and a function $
\phi $ $\in L_{+}^{1}(G)$ such that
\begin{equation}
|\Upsilon (t,y)|\leq D_{t}\phi (y)\text{ \ \ for all }t\in \lbrack 0,t_{0}]
\text{ and }y\in E,
\end{equation}
\noindent and for some $\alpha =\alpha _{\Upsilon }>0$
\begin{equation}
K_{\phi }:=\underset{y\rightarrow 0}{\lim \sup }\frac{\phi (y)}{|y|^{\kappa
-n+\alpha }}\text{ }<\infty . \label{vfconstant}
\end{equation}
\end{condition}
Equipped with these remarks we are now in a position to state and prove the
following lemma which will be fundamental to providing the estimates on the
reduced covariance matrix we need later.
\begin{lemma}
\label{Norris}(Norris-type lemma) Fix $t_{0}>0$ and for every $\epsilon >0$
suppose $\beta^{\epsilon} (t),\gamma^{\epsilon} (t)=(\gamma^{\epsilon}
_{1}(t),...,\gamma^{\epsilon} _{d}(t))$ ,$u^{\epsilon}(t)=(u^{
\epsilon}_{1}(t),...,u^{\epsilon}_{d}(t))$ are previsible processes taking
values in $
\mathbb{R}
,
\mathbb{R}
^{d}$ and $
\mathbb{R}
^{d}$ respectively. Suppose further that $\zeta^{\epsilon} (t,y)$and $
f^{\epsilon}(t,y)$ are real-valued previsible processes satisfying condition
\ref{condition4} such that the functions $\phi^{\zeta}$ and $\phi^{f}$ do
not depend on $\epsilon$ and moreover for every $q\geq 1$
\begin{equation}
\underset{\epsilon >0}{\sup }E\left[ \underset{0\leq t\leq t_{0}}{\sup }
\left( D_{t}^{\zeta,\epsilon}\right) ^{q}+ \underset{0\leq t\leq t_{0}}{\sup
}\left( D_{t}^{f,\epsilon}\right) ^{q}\right] <\infty. \label{epsilonunif}
\end{equation}
Let $\alpha =\min (\alpha _{\zeta }$,$\alpha _{f})$, $\delta >0$, $z=3\delta
(\kappa -n+\alpha )^{-1}$ and define the processes $a^{\epsilon }$ and $
Y^{\epsilon }$ as the solutions to the SDEs
\begin{eqnarray*}
a^{\epsilon }(t) &=&\alpha +\int_{0}^{t}\beta^{\epsilon} (s)ds+\underset{i=1}
{\overset{d}{\sum }}\int_{0}^{t}\gamma^{\epsilon}
_{i}(s)dW_{s}^{i}+\int_{0}^{t}\int_{|y|<\epsilon ^{z}}\zeta^{\epsilon}
(s,y)(\mu -\nu )(ds,dy) \\
Y^{\epsilon }(t) &=&y+\int_{0}^{t}a^{\epsilon}(s)ds+\underset{i=1}{\overset{d
}{\sum }}\int_{0}^{t}u^{\epsilon}_{i}(s)dW_{s}^{i}+\int_{0}^{t}\int_{|y|<
\epsilon ^{z}}f^{\epsilon}(s,y)(\mu -\nu )(ds,dy).
\end{eqnarray*}
\noindent Assume that for some $p\geq 2$ the quantity
\begin{equation}
\underset{\epsilon >0}{\sup }E\left[ \underset{0\leq t\leq t_{0}}{\sup }
\left( |\beta^{\epsilon} (t)|+|\gamma^{\epsilon} (t)|+|a^{\epsilon
}(t)|+|u^{\epsilon}(t)|+\int_{E}(|\zeta^{\epsilon}
(t,y)|^{2}+|f^{\epsilon}(t,y)|^{2})G(dy)\right) ^{p}\right] \label{unif}
\end{equation}
\noindent is finite, and for some $\rho _{1},\rho _{2}\in L_{+}^{2,\infty
}(G)$ we have
\begin{equation*}
\underset{\epsilon >0}{\sup } \left( E\left[ \left( \underset{0\leq t\leq
t_{0}}{\sup }\underset{y\in E}{\sup }\frac{|\zeta^{\epsilon}(t,y)|}{\rho
_{1}(y)}\right) ^{p}\right] +E\left[ \left( \underset{0\leq t\leq t_{0}}{
\sup }\underset{y\in E}{\sup }\frac{|f^{\epsilon}(t,y)|}{\rho _{2}(y)}
\right) ^{p}\right] \right) <\infty .
\end{equation*}
\noindent Then we can find finite constants $c_{1},c_{2}$ and $c_{3}$ which
do not depend on $\epsilon $, such that for any $q>8$ and any $l,r,v,w>0$
with $18r+9v<q-8,$ there exists $\epsilon _{0}=\epsilon _{0}(t_{0},q,r,v,l)$
such that if $\epsilon \leq \epsilon _{0}<1$ and $\delta w^{-1}>\max
(q/2-r+v/2,(\kappa -n+\alpha )/4\alpha )$ we have
\begin{eqnarray*}
P\Bigg(\int_{0}^{t_{0}}\left( Y^{\epsilon }(t)\right) ^{2}dt<\epsilon
^{qw},\int_{0}^{t_{0}}\Bigg(\left\vert a^{\epsilon }(t)-\int_{|y|<\epsilon
^{z}}f^{\epsilon }(t,y)G(dy)\right\vert ^{2}+|u^{\epsilon }(t)|^{2}\Bigg)dt
&\geq &l\epsilon ^{w}\Bigg) \\
\leq c_{1}\epsilon ^{rwp}+c_{2}\epsilon ^{wp/4}+c_{3}\exp \left( -\epsilon
^{-vw/2}\right) . &&
\end{eqnarray*}
\noindent Moreover, we have $\epsilon _{0}(t_{0},q,r,v,l)=t_{0}^{-k}\epsilon
_{0}(q,r,v,l)$ for some $k>0.$
\end{lemma}
\begin{proof}
Let $0<C<\infty $ denote a generic constant which varies from line to line
and which does not depend on $\epsilon .$ We begin with some preliminary
remarks. Firstly, the hypotheses of the theorem are sufficient to imply (by
Theorem A6 of \cite{bichteler}) that
\begin{equation*}
\noindent \sup_{\epsilon >0}\left( \max \left( E\left[ \sup_{0\leq t\leq
t_{0}}|Y^{\epsilon }(t)|^{p}\right] ,E\left[ \sup_{0\leq t\leq
t_{0}}|a^{\epsilon }(t)|^{p}\right] \right) \right) <\infty .
\end{equation*}
\noindent Secondly, by hypothesis we can find previsible processes $
D_{t}^{\zeta ,\epsilon }$ and $D_{t}^{f,\epsilon }$ and functions $\phi
^{\zeta }$ and $\phi ^{f}$ not depending on $\epsilon $ such that
\begin{equation}
|\zeta ^{\epsilon }(t,y)|\leq D_{t}^{\zeta ,\epsilon }\phi ^{\zeta }(y)\text{
\ and }|f^{\epsilon }(t,y)|\leq D_{t}^{f,\epsilon }\phi ^{f}(y).\text{\ }
\label{vfc}
\end{equation}
\noindent \noindent Let $D_{t}^{\epsilon }=\max (D_{t}^{\zeta ,\epsilon },$ $
D_{t}^{f,\epsilon }$ ) and $\phi (y)=\max (\phi ^{\zeta }(y)$ ,$\phi
^{f}(y)) $ and (using the notation of ( \ref{vfconstant}) ) $K=\max
(K_{\zeta },K_{f},1)$, then for some $\epsilon ^{\ast }>0$ we have
\begin{equation}
\phi (y)\leq K|y|^{\kappa -n+\alpha } \label{vfc3}
\end{equation}
\noindent for $|y|\leq \epsilon ^{\ast }.$ Consequently taking $\epsilon
\leq \min (\epsilon ^{\ast },1)$ and using the definition of $z$ we see that
for $|y|<\epsilon ^{z}$
\begin{equation}
\phi (y)\leq K\epsilon ^{z(\kappa -n+\alpha )}=K\epsilon ^{3\delta }.
\label{vfc2}
\end{equation}
\noindent Now, we define
\begin{equation*}
A=\left\{ \int_{0}^{t_{0}}\left( Y^{\epsilon }(t)\right) ^{2}dt<\epsilon
^{qw},\int_{0}^{t_{0}}\Bigg(\left\vert a^{\epsilon }(t)-\int_{|y|<\epsilon
^{z}}f^{\epsilon }(t,y)G(dy)\right\vert ^{2}+|u^{\epsilon }(t)|^{2}\Bigg)
dt\geq l\epsilon ^{w}\right\}
\end{equation*}
\noindent and let
\begin{equation*}
\theta _{t}=|\beta ^{\epsilon }(t)|+|\gamma ^{\epsilon }(t)|+|a^{\epsilon
}(t)|+|u^{\epsilon }(t)|+\int_{|y|<\epsilon ^{z}}(|\zeta ^{\epsilon
}(t,y)|^{2}+|f^{\epsilon }(t,y)|^{2})G(dy).
\end{equation*}
\noindent Taking $\psi =\alpha (\kappa -n+\alpha )^{-1}\leq 1$ we see using (
\ref{vfc}) and (\ref{vfc2}) that on the set \noindent $\left\{ \sup_{0\leq
t\leq t_{0}}|D_{t}^{\epsilon }|\leq K^{-1}\epsilon ^{-\psi \delta }\right\} $
we have
\begin{equation}
\underset{0\leq t\leq t_{0}}{\sup }\max (|\zeta ^{\epsilon
}(t,y)|,|f^{\epsilon }(t,y)|)\leq \epsilon ^{-\psi \delta }\epsilon
^{3\delta }\leq \epsilon ^{2\delta }. \label{bound}
\end{equation}
\noindent Define the stopping time $T=\min (\inf \left\{ s\geq 0:\sup_{0\leq
u\leq s}\theta _{s}>\epsilon ^{-rw}\right\} ,t_{0})$, let $A_{1}=\left\{
T<t_{0}\right\} ,$ $A_{2}=\left\{ \sup_{0\leq t\leq t_{0}}|D_{t}^{\epsilon
}|>K^{-1}\epsilon ^{-\psi \delta }\right\} ,$ $A_{3}=A\cap A_{1}^{c}\cap
A_{2}^{c}$ and observe that
\begin{equation*}
P(A)\leq P(A_{1})+P(A_{2})+P(A_{3}).
\end{equation*}
\noindent Using (\ref{epsilonunif}) ,the finiteness of (\ref{unif}) and
Chebyshev's inequality gives
\begin{equation*}
P(A_{1})\leq \epsilon ^{rwp}E\left[ \underset{0\leq t\leq t_{0}}{\sup }
\theta _{s}^{p}\right] \leq C\epsilon ^{rwp}\text{ and }P(A_{2})\leq
\epsilon ^{\delta \psi p}E\left[ \underset{0\leq t\leq t_{0}}{\sup }D_{t}^{p}
\right] \leq C\epsilon ^{\delta \psi p},
\end{equation*}
\noindent while on the set $A_{3}$ the processes $a^{\epsilon }$ and $
Y^{\epsilon }$ satisfy, by virtue of (\ref{bound}), the SDEs
\begin{eqnarray*}
da^{\epsilon }(t) &=&\beta ^{\epsilon }(t)dt+\underset{i=1}{\overset{d}{\sum
}}\gamma _{i}^{\epsilon }(t)dW_{t}^{i}+\int_{|y|<\epsilon ^{z}}\zeta
^{\epsilon }(t,y)1_{\left\{ |\zeta ^{\epsilon }(t,y)|<\epsilon ^{2\delta
}\right\} }(\mu -\nu )(dt,dy),\text{ } \\
dY^{\epsilon }(t) &=&a^{\epsilon }(t)dt+\underset{i=1}{\overset{d}{\sum }}
u_{i}^{\epsilon }(t)dW_{t}^{i}+\int_{|y|<\epsilon ^{z}}f^{\epsilon
}(t,y)1_{\left\{ |f^{\epsilon }(t,y)|<\epsilon ^{2\delta }\right\} }(\mu
-\nu )(dt,dy),\text{ }
\end{eqnarray*}
\noindent with $a^{\epsilon }(0)=\alpha ,$ $Y^{\epsilon }(0)=y.$ We now
define the following processes
\begin{eqnarray*}
A_{t} &=&\int_{0}^{t}a^{\epsilon }(s)ds\text{, \ }M_{t}=\underset{i=1}{
\overset{d}{\sum }}\int_{0}^{t}u_{i}^{\epsilon }(s)dW_{s}^{i},\text{ \ }
Q_{t}=\underset{i=1}{\overset{d}{\sum }}\int_{0}^{t}A(s)\gamma
_{i}^{\epsilon }(s)dW_{s}^{i},\text{ \ } \\
N_{t} &=&\underset{i=1}{\overset{d}{\sum }}\int_{0}^{t}Y^{\epsilon
}(s-)u_{i}^{\epsilon }(s)dW_{s}^{i},\text{ }P_{t}=\int_{0}^{t}\int_{|y|<
\epsilon ^{z}}f^{\epsilon }(s,y)1_{\left\{ |f^{\epsilon }(s,y)|<\epsilon
^{2\delta }\right\} }(\mu -\nu )(ds,dy),\text{ \ } \\
L_{t} &=&\int_{0}^{t}\int_{|y|<\epsilon ^{z}}Y^{\epsilon }(s-)f^{\epsilon
}(s,y)1_{\left\{ |f^{\epsilon }(s,y)|<\epsilon ^{2\delta }\right\} }(\mu
-\nu )(ds,dy),\text{ } \\
H_{t} &=&\int_{0}^{t}\int_{|y|<\epsilon ^{z}}A(s)\zeta ^{\epsilon
}(s,y)1_{\left\{ |\zeta ^{\epsilon }(s,y)|<\epsilon ^{2\delta }\right\}
}(\mu -\nu )(ds,dy), \\
\text{ \ }J_{t} &=&\int_{0}^{t}\int_{|y|<\epsilon ^{z}}f^{\epsilon
}(s,y)^{2}1_{\left\{ |f^{\epsilon }(s,y)|<\epsilon ^{2\delta }\right\} }(\mu
-\nu )(ds,dy),
\end{eqnarray*}
\noindent and for $\delta _{j}>0,\rho _{j}>0,$ $j\in \left\{ 1,...,7\right\}
$ define the sets
\begin{eqnarray*}
B_{1} &=&\left\{ \left\langle N\right\rangle _{T}<\rho _{1},\underset{0\leq
t\leq T}{\sup }|N_{t}|\geq \delta _{1}\right\} ,\text{ }B_{2}=\left\{
\left\langle M\right\rangle _{T}<\rho _{2},\underset{0\leq t\leq T}{\sup }
|M_{t}|\geq \delta _{2}\right\} , \\
B_{3} &=&\left\{ \left\langle Q\right\rangle _{T}<\rho _{3},\underset{0\leq
t\leq T}{\sup }|Q_{t}|\geq \delta _{3}\right\} ,\text{ }C_{1}=\left\{
\left\langle P\right\rangle _{T}<\rho _{4},\underset{0\leq t\leq T}{\sup }
|P_{t}|\geq \delta _{4}\right\} , \\
C_{2} &=&\left\{ \left\langle L\right\rangle _{T}<\rho _{5},\sup_{0\leq
t\leq T}|L_{t}|\geq \delta _{5}\right\} ,\text{ \ }C_{3}=\left\{
\left\langle N\right\rangle _{T}<\rho _{6},\underset{0\leq t\leq T}{\sup }
|N_{t}|\geq \delta _{6}\right\} , \\
C_{4} &=&\left\{ \left\langle J\right\rangle _{T}<\rho _{7},\underset{0\leq
t\leq T}{\sup }|J_{t}|\geq \delta _{7}\right\} .
\end{eqnarray*}
\noindent The exponential martingale inequality for continuous
semimartingales gives $P(B_{j})\leq 2e^{-\delta _{j}^{2}/2\rho _{j}}$ for $
j=1,2,3.$ Since the jumps in $P$ and $J$ are bounded by $\epsilon ^{2\delta
} $ and $\epsilon ^{4\delta }$ respectively, an application of lemma \ref
{EMI} gives $\ $
\begin{equation*}
P(C_{1})\leq 2\exp \left( \frac{-\delta _{4}^{2}}{2(\epsilon ^{2\delta
}\delta _{4}+\rho _{4})}\right) \text{ \ and }P(C_{4})\leq 2\exp \left(
\frac{-\delta _{7}^{2}}{2(\epsilon ^{4\delta }\delta _{7}+\rho _{7})}\right)
.
\end{equation*}
\noindent \noindent For $C_{2}$ and $C_{3}$ we use the fact that $
\sup_{0\leq t\leq T}|a^{\epsilon }(t)|\in L^{p}$ and $\sup_{0\leq t\leq
T}|Y^{\epsilon }(t)|\in L^{p}$ uniformly in $\epsilon $ to see
\begin{eqnarray*}
P(C_{2}) &\leq &P\left( \left\langle L\right\rangle _{T}<\rho
_{5},\sup_{0\leq t\leq T}|L_{t}|\geq \delta _{5},\sup_{0\leq t\leq
T}|Y^{\epsilon }(t)|\leq \epsilon ^{-\delta }\right) \\
&&+P\left( \sup_{0\leq t\leq T}|Y^{\epsilon }(t)|>\epsilon ^{-\delta }\right)
\\
&\leq &2\left( \frac{-\delta _{5}^{2}}{2(\epsilon ^{\delta }\delta _{5}+\rho
_{5})}\right) +C\epsilon ^{\delta p},
\end{eqnarray*}
\noindent where the second term comes from Chebyshev's inequality and the
first follows from lemma \ref{EMI} in concert with the observation that, on
the set $\left\{ \sup_{0\leq t\leq T}|Y^{\epsilon }(t)|\leq \epsilon
^{-\delta }\right\} $, we have
\begin{equation*}
L_{t}=\int_{0}^{t}\int_{|y|<\epsilon ^{z}}Y^{\epsilon }(s-)f^{\epsilon
}(s,y)1_{\left\{ |f^{\epsilon }(s,y)|<\epsilon ^{2\delta },|Y^{\epsilon
}(s-)|\leq \epsilon ^{-\delta }\right\} }(\mu -\nu )(ds,dy)
\end{equation*}
\noindent for $0\leq t\leq T$. Hence, the jumps in $L$ are bounded by $
\epsilon ^{\delta }$ on this set (the same argument may also be applied to $
C_{3}$). \ We now show that $A_{3}\subset \left( \cup _{j=1}^{3}B_{j}\right)
\cup \left( \cup _{j=1}^{4}C_{j}\right) $ whence on choosing appropriate
values for $\delta _{j}$ and $\rho _{j}$ the proof shall be complete. To do
this suppose that $\omega \notin \left( \cup _{j=1}^{3}B_{j}\right) \cup
\left( \cup _{j=1}^{4}C_{j}\right) ,$ $T(\omega )=t_{0},$ $
\int_{0}^{T}Y_{t}^{\epsilon }(\omega )^{2}dt<\epsilon ^{qw}$ and $
\sup_{0\leq t\leq T}|D_{t}^{\epsilon }(\omega )|<K^{-1}\epsilon ^{-\psi
\delta }.$ \ Then
\begin{equation*}
\left\langle N\right\rangle _{T}=\int_{0}^{T}(Y^{\epsilon
}(t-))^{2}|u^{\epsilon }(t)|^{2}dt<\epsilon ^{(-2r+q)w}=:\rho _{1},
\end{equation*}
\noindent and since $\omega \notin B_{1},$ $\sup_{0\leq t\leq T}\left\vert
\sum_{i=1}^{d}\int_{0}^{t}Y^{\epsilon }(s-)u_{i}^{\epsilon
}(s)dW_{s}^{i}\right\vert <\delta _{1}:=\epsilon ^{q_{1}}$, where $
q_{1}=(q/2-r-v/2)w$. By the same reasoning we have
\begin{equation*}
\left\langle L\right\rangle _{T}=\int_{0}^{T}\int_{|y|<\epsilon
^{z}}Y^{\epsilon }(t-)^{2}f^{\epsilon }(s,y)^{2}1_{\left\{ |f^{\epsilon
}(t,y)|<\epsilon ^{2\delta }\right\} }G(dy)dt<\epsilon ^{(-2r+q)w}=:\rho
_{5},
\end{equation*}
\noindent since $\omega \notin C_{2}$ we may let $\delta _{5}=\epsilon
^{q_{1}}$ to give $\sup_{0\leq t\leq T}|L_{t}|<\delta _{5}.$ Since we also
have
\begin{equation*}
\sup_{0\leq t\leq T}\left\vert \int_{0}^{t}Y^{\epsilon }(s-)a^{\epsilon
}(s)ds\right\vert \leq \left( t_{0}\int_{0}^{T}Y^{\epsilon
}(s-)^{2}a^{\epsilon }(s)^{2}ds\right) ^{1/2}<t_{0}^{1/2}\epsilon
^{(-r+q/2)w},
\end{equation*}
\noindent it follows that
\begin{equation*}
\sup_{0\leq t\leq T}\left\vert \int_{0}^{t}Y^{\epsilon }(s-)dY^{\epsilon
}(s)\right\vert <t_{0}^{1/2}\epsilon ^{(-r+q/2)w}+2\epsilon ^{q_{1}}.
\end{equation*}
\noindent It\^{o}'s formula now gives $Y^{\epsilon
}(t)^{2}=y^{2}+2\int_{0}^{t}Y^{\epsilon }(s-)dY^{\epsilon }(s)+\left\langle
M\right\rangle _{t}+\left[ P\right] _{t},$ and we notice that because
\begin{eqnarray*}
\left\langle J\right\rangle _{T} &=&\int_{0}^{T}\int_{|y|<\epsilon
^{z}}f^{\epsilon }(s,y)^{4}1_{\left\{ |f^{\epsilon }(s,y)|<\epsilon
^{2\delta }\right\} }G(dy)dt \\
&\leq &\epsilon ^{4\delta }\int_{0}^{T}\int_{|y|<\epsilon ^{z}}f^{\epsilon
}(s,y)^{2}1_{\left\{ |f^{\epsilon }(s,y)|<\epsilon ^{2\delta }\right\}
}G(dy)dt\leq \epsilon ^{4\delta -rw}=:\rho _{7},
\end{eqnarray*}
\noindent and since $\omega \notin C_{4}$ we must have $\sup_{0\leq t\leq
T}|J_{t}|=\sup_{0\leq t\leq T}|\left[ P\right] _{t}-\left\langle
P\right\rangle _{t}|\leq \delta _{7}:=\epsilon ^{2\delta -(r+v)w}.$
Consequently,
\begin{equation*}
\left\langle M\right\rangle _{t}+\left\langle P\right\rangle _{t}\leq
Y^{\epsilon }(t)^{2}-y^{2}-2\int_{0}^{t}Y^{\epsilon }(s-)dY^{\epsilon
}(s)+\sup_{0\leq t\leq T}|\left[ P\right] _{t}-\left\langle P\right\rangle
_{t}|
\end{equation*}
\noindent and hence,
\begin{equation*}
\int_{0}^{T}\left\langle M\right\rangle _{t}dt+\int_{0}^{T}\left\langle
P\right\rangle _{t}dt<\epsilon ^{qw}+t_{0}^{3/2}\epsilon
^{(-r+q/2)w}+2t_{0}\epsilon ^{q_{1}}+t_{0}\epsilon ^{2\delta -(r+v)w}.
\end{equation*}
\noindent We notice that $2\delta -(r+v)w>(q-3r)w>q_{1},$ $qw>q_{1}$ and $
(q/2-r)w>q_{1}$ and so provided
\begin{equation*}
\epsilon <\min \left( 1,t_{0}^{-1/\left( 2\delta -(r+v)w-q_{1}\right)
},t_{0}^{-3/2\left( \left( -r+q/2\right) w-q_{1}\right) }\right)
\end{equation*}
we get
\begin{equation*}
\int_{0}^{T}\left\langle M\right\rangle _{t}dt+\int_{0}^{T}\left\langle
P\right\rangle _{t}dt<(2t_{0}+3)\epsilon ^{q_{1}}.
\end{equation*}
\noindent $\left\langle M\right\rangle _{t}$ and $\left\langle
P\right\rangle _{t}$ are increasing processes, so for any $0<\gamma <T$
\begin{equation*}
\gamma \left\langle M\right\rangle _{T-\gamma }<(2t_{0}+3)\epsilon ^{q_{1}}
\text{ \ and \ }\gamma \left\langle P\right\rangle _{T-\gamma
}<(2t_{0}+3)\epsilon ^{q_{1}}.
\end{equation*}
\noindent Since these processes are also continuous we get $\left\langle
M\right\rangle _{T}\leq \gamma ^{-1}(2t_{0}+3)\epsilon ^{q_{1}}+\gamma
\epsilon ^{-2rw}$ and $\left\langle P\right\rangle _{T}\leq \gamma
^{-1}(2t_{0}+3)\epsilon ^{q_{1}}+\gamma \epsilon ^{-2rw}$ . By defining $
\rho _{2}=\rho _{4}:=2(2t_{0}+3)^{1/2}\epsilon ^{-2rw+q_{1}/2}$ and $\gamma
=(2t_{0}+3)^{1/2}\epsilon ^{q_{1}/2}$, we get $\left\langle M\right\rangle
_{T}<\rho _{2}$ and $\left\langle P\right\rangle _{T}<\rho _{4}$, and since $
\omega \notin B_{2}\cup C_{1}$ we have
\begin{equation*}
\sup_{0\leq t\leq T}|M_{t}|<\delta _{2}:=\epsilon
^{(q/8-5r/4-5v/8)w}=:\epsilon ^{q_{2}},\text{ \ }\sup_{0\leq t\leq
T}|P_{t}|<\delta _{4}=\epsilon ^{q_{2}}.
\end{equation*}
\noindent Since $\int_{0}^{T}Y^{\epsilon }(t)^{2}dt<\epsilon ^{qw}$
Chebyshev's inequality gives
\begin{equation*}
Leb\left\{ t\in \lbrack 0,T]:\left\vert Y_{t}^{\epsilon }(\omega
)\right\vert \geq \epsilon ^{qw/3}\right\} \leq \epsilon ^{qw/3}
\end{equation*}
\noindent so that
\begin{equation*}
Leb\left\{ t\in \lbrack 0,T]:\left\vert y+A_{t}(\omega )\right\vert \geq
\epsilon ^{qw/3}+2\epsilon ^{q_{2}}\right\} \leq \epsilon ^{qw/3}.
\end{equation*}
\noindent Then, for each $t\in \lbrack 0,T],$ there exists some $s\in
\lbrack 0,T]$ such that $|s-t|\leq \epsilon ^{qw/3}$ and $|y+A_{s}(\omega
)|<\epsilon ^{qw/3}+2\epsilon ^{q_{2}}$, which yields
\begin{equation*}
|y+A_{t}|\leq |y+A_{s}|+\left\vert \int_{s}^{t}a^{\epsilon }(\tau )d\tau
\right\vert <(1+\epsilon ^{-rw})\epsilon ^{qw/3}+2\epsilon ^{q_{2}}.
\end{equation*}
\noindent In particular we have $|y|<(1+\epsilon ^{-rw})\epsilon
^{qw/3}+2\epsilon ^{q_{2}}$ and, for all $t\in \lbrack 0,T]$, since $
q_{2}<(q/3-r)w$, we have
\begin{equation*}
|A_{t}|<2\left( (1+\epsilon ^{-rw})\epsilon ^{qw/3}+2\epsilon
^{q_{2}}\right) \leq 8\epsilon ^{q_{2}}.
\end{equation*}
\noindent This implies that
\begin{eqnarray*}
\left\langle Q\right\rangle _{T} &=&\int_{0}^{T}A(t)^{2}|\gamma ^{\epsilon
}(t)|^{2}dt<64t_{0}\epsilon ^{2q_{2}-2rw}=:\rho _{3} \\
\left\langle H\right\rangle _{T} &=&\int_{0}^{T}\int_{|y|<\epsilon
^{z}}A(t)^{2}\zeta ^{\epsilon }(t,y)^{2}1_{\left\{ |\zeta ^{\epsilon
}(t,y)|<\epsilon ^{2\delta }\right\} }G(dy)dt\leq \rho _{3}=:\rho _{6},
\end{eqnarray*}
\noindent and since $\omega \notin B_{3}\cup C_{3}$ we must have
\begin{eqnarray*}
\underset{0\leq t\leq T}{\sup }|Q_{t}| &=&\underset{0\leq t\leq T}{\sup }
\left\vert \overset{d}{\underset{i=1}{\sum }}\int_{0}^{t}A(s)\gamma
_{i}^{\epsilon }(s)dW_{i}(s)\right\vert <\delta _{3}:=\epsilon
^{(q/8-9r/4-9v/8)w}=:\epsilon ^{q_{3}} \\
\underset{0\leq t\leq T}{\sup }|H_{t}| &=&\underset{0\leq t\leq T}{\sup }
\left\vert \int_{0}^{t}\int_{|y|<\epsilon ^{z}}A(s)\zeta ^{\epsilon
}(s,y)1_{\left\{ |\zeta ^{\epsilon }(s,y)|<\epsilon ^{2\delta }\right\}
}(\mu -\nu )(ds,dy)\right\vert <\delta _{6}:=\epsilon ^{q_{3}}.
\end{eqnarray*}
\noindent Now we observe using (\ref{vfc}),(\ref{vfc3}), condition \ref
{condition2} , $\sup_{0\leq t\leq T}|D_{t}^{\epsilon }(\omega
)|<K^{-1}\epsilon ^{-\psi \delta }$, the definition of $\psi $, and the fact
that $\phi ^{f}$ does not depend on $\epsilon $
\begin{eqnarray*}
\int_{0}^{t_{0}}\left\vert \int_{|y|<\epsilon ^{z}}f^{\epsilon
}(t,y)G(dy)\right\vert ^{2}dt &\leq &t_{0}\left( \epsilon ^{-\delta \psi
}\int_{|y|<\epsilon ^{z}}|y|^{\kappa -n+\alpha }G(dy)\right) ^{2} \\
&\leq &Ct_{0}\epsilon ^{-2\delta \psi +2z\alpha }=Ct_{0}\epsilon ^{4\delta
\alpha /(\kappa -n+\alpha )}.
\end{eqnarray*}
\noindent An application of \noindent It\^{o}'s formula then gives
\begin{eqnarray*}
&&\int_{0}^{T}\left( \left\vert a^{\epsilon }(t)-\int_{|y|<\epsilon
^{z}}f^{\epsilon }(t,y)G(dy)\right\vert ^{2}+|u^{\epsilon }(t)|^{2}\right) dt
\\
&\leq &2\int_{0}^{T}a^{\epsilon }(t)^{2}dt+\int_{0}^{T}|u^{\epsilon
}(t)|^{2}dt+2\int_{0}^{T}\left\vert \int_{|y|<\epsilon ^{z}}f^{\epsilon
}(t,y)G(dy)\right\vert ^{2}dt \\
&\leq &2\int_{0}^{T}a^{\epsilon }(t)dA(t)+\left\langle M\right\rangle
_{T}+2Ct_{0}\epsilon ^{4\delta \alpha /(\kappa -n+\alpha )} \\
&=&2\Bigg(a^{\epsilon }(T)A(T)-\int_{0}^{T}A(t)\beta ^{\epsilon }(t)dt-
\underset{i=1}{\overset{d}{\sum }}\int_{0}^{T}A(t)\gamma _{i}^{\epsilon
}(t)dW_{t}^{i} \\
&&-\int_{0}^{T}\int_{|y|<\epsilon ^{z}}A(s)\zeta ^{\epsilon }(s,y)1_{\left\{
|\zeta ^{\epsilon }(s,y)|<\epsilon ^{2\delta }\right\} }(\mu -\nu )(ds,dy)
\Bigg)+\left\langle M\right\rangle _{T} \\
&&+2Ct_{0}\epsilon ^{4\delta \alpha /(\kappa -n+\alpha )} \\
&\leq &16(1+t_{0})\epsilon ^{q_{2}-rw}+4\epsilon
^{q_{3}}+4(2t_{0}+3)^{1/2}\epsilon ^{-2rw+q_{1}/2}+2Ct_{0}\epsilon ^{4\delta
\alpha /(\kappa -n+\alpha )} \\
&\leq &l\epsilon ^{w}
\end{eqnarray*}
\noindent provided
\begin{multline}
\epsilon <\min \Bigg(\left( \frac{l}{16(1+t_{0})}\right)
^{(q_{2}-rw-w)^{-1}},\left( \frac{l}{4}\right) ^{(q_{3}-w)^{-1}}, \label{a}
\\
\left( \frac{l}{4\left( 2t_{0}+3\right) ^{1/2}}\right)
^{(-2rw+q_{1}/2-w)^{-1}},\left( \frac{l}{2Ct_{0}}\right) ^{\left( \frac{
4\delta \alpha }{\kappa -n+\alpha }-w\right) ^{-1}}\Bigg). \notag
\end{multline}
\noindent Where the last inequality follows from $q_{2}-rw>w,$ $q_{3}>w,$ $
q_{1}/2-2rw>w$ and $\delta >w(\kappa -n+\alpha )/4\alpha $. \ Finally, by
the choice of $\delta _{j}$ and $\rho _{j}$ and the assumption that $\delta
>(-r+q/2+v/2)w$ (which also implies that $\delta >-rw+q_{1}/2-q_{2}/4$ and $
\delta >2q_{2}-2rw-q_{3}$) we see that $\epsilon ^{2\delta }\delta _{4}<\rho
_{4},$ $\epsilon ^{\delta }\delta _{5}<\rho _{5},$ $\epsilon ^{\delta
}\delta _{6}<\rho _{6}$ and $\epsilon ^{4\delta }\delta _{7}<\rho _{7}.$
Therefore this choice for $\delta _{j}$ and $\rho _{j}$ enable us to deduce
that
\begin{eqnarray*}
P\left( \underset{j=1}{\overset{3}{\cup }}B_{j}\right) &\leq &2\Bigg(\exp
\left( -\frac{1}{2}\epsilon ^{-vw}\right) +\exp \left( -\frac{1}{
4(2t_{0}+3)^{1/2}}\epsilon ^{-vw}\right) \\
&&+\exp \left( -\frac{1}{128t_{0}}\epsilon ^{-vw}\right) \Bigg),
\end{eqnarray*}
\noindent and
\begin{eqnarray*}
P\left( \underset{j=1}{\overset{4}{\cup }}C_{j}\right) &\leq &2\Bigg(2\exp
\left( -\frac{1}{4}\epsilon ^{-vw}\right) +\exp \left( -\frac{1}{
8(2t_{0}+3)^{1/2}}\epsilon ^{-vw}\right) \\
&&+\exp \left( -\frac{1}{256t_{0}}\epsilon ^{-vw}\right) +C\epsilon ^{\psi
\delta p}\Bigg).
\end{eqnarray*}
\noindent The proof is finished on noting that $\delta \psi >w/4,$ and the
dependence of $\epsilon _{0}$ on $t_{0}$ follows immediately from the proof.
\end{proof}
\section{\protect\bigskip Uniform H\"{o}rmander condition}
\bigskip We now present our uniform H\"{o}rmander condition.
\begin{condition}[UH]
\label{UH}Let $V_{0}=Z-\frac{1}{2}\sum_{i=1}^{d}D V_{i}V_{i}$ and assume
that condition \ref{condition1} holds. Recursively define the following
families of vector fields
\begin{gather*}
\mathcal{L}_{0}=\left\{ V_{1},...,V_{d}\right\} \\
\mathcal{L}_{n+1}=\mathcal{L}_{n}\cup \left\{ \lbrack V_{i},K],\text{ }
i=1,...,d:\text{ }K\in \mathcal{L}_{n}\right\} \\
\cup \left\{ \left[ V_{0},K\right] -\int_{E}[Y,K](\cdot ,y)G(dy):K\in
\mathcal{L}_{n}\right\} .
\end{gather*}
\noindent Then there exists some smallest integer $j_{0}\geq 1$ and a
constant $c>0$ such that for any $u\in
\mathbb{R}
^{e}$ with $|u|=1$ we have
\begin{equation*}
\underset{x\in
\mathbb{R}
^{e}}{\inf }\overset{j_{0}}{\underset{j=0}{\sum }}\underset{K\in \mathcal{L}
_{j}}{\sum }\left( u^{T}K(x)\right) ^{2}\geq c
\end{equation*}
\end{condition}
\noindent The next important result is a development of an idea presented in
\cite{teichmann}, it enables us to estimate the Malliavin covariance matrix
on a time interval where the Poisson random measure records no jumps of size
greater than some truncation parameter. As in \cite{teichmann} the key idea
is to make explicit the dependence of the estimate on the length of the time
interval under consideration.
\begin{theorem}
\label{cthm}Let $t>0$ and let $x_{t}$ satisfy the SDE
\begin{equation*}
x_{t}=x+\int_{0}^{t}Z(x_{s-})ds+\int_{0}^{t}V(x_{s-})dW_{s}+\int_{0}^{t}
\int_{E}Y(y,x_{s-})(\mu -\nu )(dy,ds)
\end{equation*}
\noindent and assume that the following conditions are satisfied :
\begin{equation}
Z,V_{1},..,V_{d}\in C_{b}^{\infty }(
\mathbb{R}
^{e}),
\end{equation}
\noindent for every $y\in E$ $Y(\cdot ,y)\in C_{b}^{\infty }(
\mathbb{R}
^{e})$ and, for some $\rho _{2}\in L^{2,\infty }(G)$ and every $n\in
\mathbb{N}
\cup \left\{ 0\right\} $
\begin{equation}
\underset{y\in E,x\in
\mathbb{R}
^{e}}{\sup }\frac{1}{\rho _{2}(y)}|D_{1}^{n}Y(x,y)|<\infty ,
\label{Lpinequalities}
\end{equation}
\begin{equation*}
\underset{y\in E,x\in
\mathbb{R}
^{e}}{\sup }|\left( I+D_{1}Y(x,y)\right) ^{-1}|<\infty \text{ \ and }
\underset{x\in
\mathbb{R}
^{e}}{\sup }|\left( I+D_{1}Y(x,\cdot )\right) ^{-1}|\in L^{\infty }(G).\text{
}
\end{equation*}
\noindent Further assume conditions \ref{condition1}, \ref{condition2} , \ref
{condition3} \ and condition (UH) hold. For some $0<t<t_{0},$ $\delta
,\alpha >0$ and $z=3\delta (\kappa -n+\alpha )^{-1}$ define the set $
A_{t}=A_{t}(\epsilon )$ by
\begin{equation*}
A_{t}=\left\{ \omega :\text{ }\left( \text{supp }\mu (\cdot ,\cdot )\right)
\cap \lbrack 0,t)\times E\subseteq \lbrack 0,t)\times \left\{ |y|\leq
\epsilon ^{z}\right\} \right\} .
\end{equation*}
\noindent Then, $P\left( \left\{ \sup_{0\leq s\leq t}|x_{s}-x_{s}(\epsilon
)|>0\right\} \cap A_{t}\right) =0$, where $x_{t}(\epsilon )$ is the solution
to the SDE
\begin{eqnarray}
dx_{t}(\epsilon ) &=&\left( Z(x_{t-}(\epsilon ))-\int_{|y|\geq \epsilon
^{z}}Y(x_{t-}(\epsilon ),y)G(dy)\right) dt+V(x_{t-}(\epsilon ))dW_{t} \notag
\\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ }+\int_{|y|<\epsilon ^{z}}Y(x_{t-}(\epsilon ),y)(\mu -\nu )(dy,dt),
\label{conditionedprocess}
\end{eqnarray}
Moreover if we let the reduced Malliavin covariance matrix associated with $
x_{t}(\epsilon )$ be denoted by $C_{t}(\epsilon )$ then we have for any $
p\geq 1$ and some $\epsilon _{0}(p)>0,$ $K(p)\geq 1$, that
\begin{equation*}
\underset{|u|=1}{\sup }P(\left\{ u^{T}C_{t}u\leq \epsilon \right\} \cap
A_{t})=\underset{|u|=1}{\sup }P(u^{T}C_{t}(\epsilon )u\leq \epsilon )\leq
\epsilon ^{p}
\end{equation*}
\noindent for $0\leq \epsilon \leq t^{K(p)}\epsilon _{0}(p),$ provided that
\begin{equation*}
16 \delta > \max \left(8-r+\frac{v}{2},\frac{\kappa -n+\alpha }{4\alpha}
\right),
\end{equation*}
where $r,v>0$ are such that $18r + 9v<8$.
\end{theorem}
\begin{proof}
The indistinguishability of the processes $x$ and $x(\epsilon )$ on $A_{t}$
is a trivial. For the remainder of the proof we first note that condition
(UH) uniform enables us to identify a smallest integer $j_{0}$ and a
constant $c>0$ such that, for any $u\in
\mathbb{R}
^{e}$ with $|u|=1$
\begin{equation*}
\underset{x\in
\mathbb{R}
^{e}}{\inf }\overset{j_{0}}{\underset{j=0}{\sum }}\underset{K\in \mathcal{L}
_{j}}{\sum }\left( u^{T}K(x)\right) ^{2}\geq c.
\end{equation*}
\noindent For $j=0,1,...,j_{0}$ set $m(j)=2^{-4j}$ and define
\begin{equation*}
E_{j}=\left\{ \underset{K\in \mathcal{L}_{j}}{\sum }\int_{0}^{t}\left(
u^{T}(\epsilon )J_{0\leftarrow s}(\epsilon )K(x_{s}(\epsilon ))\right)
^{2}ds\leq \epsilon ^{m(j)}\right\} ,
\end{equation*}
\noindent where $J_{t\leftarrow 0}(\epsilon )$ denotes the Jacobian of the
flow associated with $x_{t}(\epsilon )$ and $J_{0\leftarrow t}(\epsilon )$
denotes its inverse (which exists by the assumptions on the vector fields as
in theorem \ref{jacob}). It is straight forward to note, using (\ref
{Lpinequalities}), $L^{p}$ inequalities for stochastic integrals based on
Poisson random measures (see \cite{bichteler}, lemma A.14) and Gronwall's
inequality that for any $p<\infty $
\begin{equation}
\underset{\epsilon \geq 0}{\sup }\mathbb{E}\left[ \underset{0\leq s\leq t}{
\sup }|J_{t\leftarrow 0}(\epsilon )|^{p}\right] <\infty . \label{epsLp}
\end{equation}
Let $C$ denote a constant which varies from line to line and does not depend
on $\epsilon $. Then, as usual we have
\begin{equation*}
\left\{ u^{T}C_{t}(\epsilon )u\leq \epsilon \right\} =E_{0}\subset \left(
E_{0}\cap E_{1}^{c}\right) \cup (E_{1}\cap E_{2}^{c})\cup ...\cup
(E_{j_{0}-1}\cap E_{j_{0}}^{c})\cup F
\end{equation*}
\noindent where $F=E_{0}\cap E_{1}\cap ...\cap E_{j_{0}}.$ Define the
stopping time
\begin{equation*}
S=\min \left( \inf \left\{ s\geq 0:\underset{0\leq z\leq s}{\sup }
|J_{0\leftarrow z}(\epsilon )-I|\geq \frac{1}{2}\right\} ,t\right) ,
\end{equation*}
\noindent and notice that by choosing $0<\beta <m(j_{0})$ we discover that $
P(F)\leq P(S<\epsilon ^{\beta })\leq C\epsilon ^{q\beta /2}$ for $\epsilon
\leq \epsilon _{1}$and any $q\geq 2$ (see \cite{Nu06} and \cite{teichmann}
for details), where as in \cite{teichmann}, $\epsilon _{1}$ satisfies
\begin{equation*}
\epsilon _{1}<\min \left( t^{1/\beta },\left( \frac{c}{4(j_{0}+1)}\right)
^{1/(m(j_{0})-\beta )}\right) .
\end{equation*}
\noindent We notice that for any $K\in C_{b}^{\infty }(
\mathbb{R}
^{e})$ we have
\begin{gather*}
d(u^{T}J_{0\leftarrow t}(\epsilon )K(x_{t}(\epsilon ))=u^{T}J_{0\leftarrow
t-}(\epsilon )\Bigg(\left[ V_{0},K\right] (x_{t-}(\epsilon ))-\int_{E}\left[
Y,K\right] (x_{t-}(\epsilon ),y)G(dy) \\
+\frac{1}{2}\overset{d}{\underset{i=1}{\sum }}\left[ V_{i},\left[ V_{i},K
\right] \right] (x_{t-}(\epsilon )) \\
+\int_{|y|<\epsilon ^{z}}((I+D_{1}Y(x_{t-}(\epsilon
),y)^{-1})K(x_{t-}(\epsilon )+Y(x_{t-}(\epsilon ),y))-K(x_{t-}(\epsilon
))G(dy))\Bigg)dt \\
+u^{T}J_{0\leftarrow t-}(\epsilon )\overset{d}{\underset{i=1}{\sum }}\left[
V_{i},K\right] (x_{t-}(\epsilon ))dW_{t}^{i} \\
+u^{T}J_{0\leftarrow t-}(\epsilon )\int_{|y|<\epsilon^{z}}
((I+D_{1}Y(x_{t-}(\epsilon ),y)^{-1})K(x_{t-}(\epsilon )+Y(x_{t-}(\epsilon
),y))-K(x_{t-}(\epsilon ))(\mu -\nu )(dy,dt).
\end{gather*}
\noindent We now verify the conditions of lemma \ref{Norris} in the case
where
\begin{eqnarray*}
Y^{\epsilon }(t) &=&u^{T}J_{0\leftarrow t}(\epsilon )K(x_{t}(\epsilon )) \\
a^{\epsilon }(t) &=&u^{T}J_{0\leftarrow t}(\epsilon )\Bigg(\left[ V_{0},K
\right] (x_{t}(\epsilon ))-\int_{E}\left[ Y,K\right] (x_{t}(\epsilon
),y)G(dy) \\
&&+\frac{1}{2}\overset{d}{\underset{i=1}{\sum }}\left[ V_{i},\left[ V_{i},K
\right] \right] (x_{t}(\epsilon )) \\
&&+\int_{|y|<\epsilon ^{z}}((I+D_{1}Y(x_{t}(\epsilon
),y)^{-1})K(x_{t}(\epsilon )+Y(x_{t}(\epsilon ),y))-K(x_{t}(\epsilon ))G(dy))
\Bigg). \\
&=:&u^{T}J_{0\leftarrow t}\tilde{K}(x_{t}(\epsilon)),
\end{eqnarray*}
where $\tilde{K}\in C_{b}^{\infty}(\mathbb{R}^{e})$. \noindent To do this we
observe, using the notation of lemma \ref{Norris} that
\begin{equation*}
f^{\epsilon}(t,y)=u^{T}J_{0\leftarrow t-}(\epsilon
)((I+D_{1}Y(x_{t-}(\epsilon ),y)^{-1})K(x_{t-}(\epsilon )+Y(x_{t-}(\epsilon
),y))-K(x_{t-}(\epsilon ))
\end{equation*}
\noindent and hence for some $0<C<\infty $
\begin{eqnarray*}
|f^{\epsilon}(t,y)| &\leq &C\left\vert u^{T}J_{0\leftarrow t-}(\epsilon )
\right\vert \max\left(\underset{x\in \mathbb{R}^{e}}{\sup}\left\vert K(x
)\right\vert ,\underset{x\in \mathbb{R}^{e}}{\sup}\left\vert DK(x
)\right\vert\right) \\
&& \bigg(\underset{x \in \mathbb{R}^{e}, y\in E}{\sup }\left\vert \left(
I+D_{1}Y(x ,y)\right)^{-1}\right\vert |D_{1}Y(x_{t-}(\epsilon ),y)|
+|Y(x_{t-}(\epsilon ),y)|\bigg).
\end{eqnarray*}
\noindent Condition \ref{condition3} then gives that $|f^{\epsilon}(t,y)|
\leq C\left\vert u^{T}J_{0\leftarrow t-}(\epsilon )\right\vert \phi (y)$
where $\phi \in L_{+}^{1}(G)$ does not depend on $\epsilon $, $C=C(K)<\infty
$ and where and for some $\alpha >0$ (which does not depend on $\epsilon $
or $K$!) we have
\begin{equation*}
\underset{y\rightarrow 0}{\lim \sup }\frac{\phi (y)}{|y|^{\kappa -n+\alpha }}
\text{ }<\infty .
\end{equation*}
Finally, using the notation of (\ref{epsilonunif}), we notice that
Cauchy-Schwarz gives
\begin{equation}
|u^{T}J_{0\leftarrow t-}(\epsilon )|\leq \underset{i=1}{\overset{e}{\sum}}
|e_{i}^{T}J_{0\leftarrow t-}(\epsilon)|^{2}=:D_{t}^{f,\epsilon},
\label{uniformity}
\end{equation}
where $e_{i}$ is the standard basis in $\mathbb{R}^{e}$. Hence by (\ref
{epsLp}) we have for any $p<\infty$
\begin{align*}
\underset{\epsilon\geq 0}{\sup}E\left[\underset{0\leq s\leq t}{\sup}
\left(D_{s}^{f,\epsilon}\right)^{p}\right]<\infty.
\end{align*}
\noindent We have therefore verified the conditions of lemma \ref{Norris}
for the process $f^{\epsilon}(t,y)$. They may be also checked for the
process $\zeta^{\epsilon}(t,y)$ in the same manner. The other hypotheses of
lemma \ref{Norris} are trivial to verify so we apply this lemma with $
z=3\delta (\kappa -n+\alpha )^{-1},$ and with $q=16$, $r,v>0$ such that $
18r+9v<8$ and $16\delta >\max \left( 8-r+v/2,(\kappa -n+\alpha )/4\alpha
\right) $ to deduce that for $j\in \left\{ 0,1,...,j_{0}-1\right\} $
\begin{align*}
P(E_{j}\cap E_{j_{+1}}^{c})=P\Bigg(\underset{K\in \mathcal{L}_{j}}{\sum }&
\int_{0}^{t}\left( u^{T}J_{0\leftarrow s}(\epsilon )K(x_{s}(\epsilon
))\right) ^{2}ds\leq \epsilon ^{m(j)}, \\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }&
\text{\ }\underset{K\in \mathcal{L}_{j+1}}{\sum }\int_{0}^{t}\left(
u^{T}J_{0\leftarrow s}(\epsilon )K(x_{s}(\epsilon ))\right) ^{2}ds>\epsilon
^{m(j+1)}\Bigg) \\
\leq \underset{K\in \mathcal{L}_{j}}{\sum }P\Bigg(\int_{0}^{t}(v^{T}&
J_{0\leftarrow s}(\epsilon )K(x_{s}(\epsilon ))^{2}ds\leq \epsilon ^{m(j)},
\\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ }\underset{k=1}{\overset{d}{
\sum }}\int_{0}^{t}(u^{T}J_{0\leftarrow s}(\epsilon )& V_{k}(x_{s}(\epsilon
))^{2}ds+\int_{0}^{t}u^{T}J_{0\leftarrow s-}(\epsilon )\Bigg(\left[ V_{0},K
\right] (x_{s-}(\epsilon )) \\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }-\int_{E}\left[ Y,K\right]
(x_{s-}(\epsilon ),y)& G(dy)+\frac{1}{2}\overset{d}{\underset{i=1}{\sum }}
\left[ V_{i},\left[ V_{i},V_{k}\right] \right] (x_{s-}(\epsilon ))\Bigg)
ds\left. >\right. \frac{\epsilon ^{m(j+1)}}{n(j)}\Bigg).
\end{align*}
\noindent Which \noindent is $o(\epsilon ^{p})$ for $\epsilon \leq \epsilon
_{2}(p)$ where $\epsilon _{2}$ can be chosen as $\epsilon _{3}t^{-k^{\ast }}$
for some $k^{\ast }>0$ and where $\epsilon _{3}$ is independent of $t$.
Setting $\epsilon _{0}=\min (\epsilon _{1},\epsilon _{2})$ and noticing by (
\ref{uniformity}) that all the estimates are uniform over $|u|=1$ gives the
result.
\end{proof}
\section{C$^{\infty }$ density under the H\"{o}rmander condition}
We now state and prove our main result
\begin{theorem}
\bigskip Suppose that $x_{t}$ is the solution to the SDE
\begin{equation*}
x_{t}=x+\int_{0}^{t}Z(x_{s-})ds+\int_{0}^{t}V(x_{s-})dW_{s}+\int_{0}^{t}
\int_{E}Y(y,x_{s-})(\mu -\nu )(dy,ds)
\end{equation*}
\noindent and that the conditions of theorem \ref{cthm} are in force. Then,
for any $t_{0}>0$ the law of $x_{t_{0}}$ has a $C^{\infty }$ density with
respect to Lebesgue measure under the uniform H\"{o}rmander condition \ref
{UH} provided, in the notation of theorem \ref{cthm}, we have
\begin{equation}
16m(j_{0})>3(\kappa -n)\max \left( \frac{8-r+v/2}{\kappa -n+\alpha },\frac{1
}{4\alpha }\right) . \label{bracketcond}
\end{equation}
\noindent
\end{theorem}
\begin{remark}
Note that (\ref{bracketcond}) is always true when $\kappa =n.$
\end{remark}
\begin{proof}
By Theorem \ref{criterion} it suffices to check that $\left\vert
C_{t_{0}}^{-1}\right\vert \in L^{p}$ for all $p\geq 2.$ Let $\Lambda=
\underset{|u|=1}{\inf}u^{T}C_{t_{0}}u$ be the smallest eigenvalue of $
C_{t_{0}}$. Then it is sufficient to show that $\Lambda^{-1} \in L^{p}$ for
all $p\geq 2$. However, we may write
\begin{eqnarray}
E[\Lambda ^{-p}]=C_{1} \int_{0}^{\infty }\epsilon ^{-k}P(\Lambda \leq
\epsilon^{2} )d\epsilon \leq C_{2}+C_{3}\int_{0}^{1}\epsilon ^{-k}P(\Lambda
\leq\epsilon^{2} )d\epsilon , \notag \label{inverse}
\end{eqnarray}
for some $k>1$. By a routine compactness argument we may show (see \cite
{norris}) that
\begin{align*}
P(\Lambda \leq \epsilon) \leq C_{2}\epsilon^{-e}\underset{|u|=1}{\sup}
P(u^{T}C_{t_{0}}u\leq \epsilon),
\end{align*}
so that for some $k^{\prime }>1$
\begin{equation}
E[\Lambda ^{-p}]\leq C_{3}+C_{4}\int_{0}^{1} \epsilon ^{-k^{\prime }}
\underset{|u|=1}{\sup}P(u^{T}C_{t_{0}}u\leq \epsilon^{2})d\epsilon .
\label{lambda}
\end{equation}
Now we define a Poisson process $N_{\epsilon }$ on $
\mathbb{R}
^{+}$ for $\epsilon >0$ by
\begin{equation*}
N_{\epsilon }(t)=\int_{0}^{t}\int_{|y|>\epsilon ^{z}}\mu (dy,ds),
\end{equation*}
\noindent whose rate is given as
\begin{equation*}
\lambda (\epsilon )=\int_{|y|>\epsilon ^{z}}G(dy).
\end{equation*}
By (\ref{cond2a}) we know that
\begin{equation}
\underset{\epsilon \rightarrow 0}{\lim \sup }\frac{\lambda (\epsilon )}{
f\left( \epsilon \right) }<\infty \label{rategrowth}
\end{equation}
We may find a (random) subinterval $[t_{1},t_{2})\subseteq \lbrack 0,t_{0})$
such that $t_{2}-t_{1}\geq t_{0}(N_{\epsilon }(t_{0})+1)^{-1}$ on which the
Poisson random measure $\mu $ records no jumps of absolute value greater
than $\epsilon ^{z}$ and, as such, the underlying process $x_{t}$ solves the
SDE (\ref{conditionedprocess}) started at $x_{t_{1}}$ on this interval. We
emphasize the dependence of $C_{t_{0}}$on the starting point $(x,I)$ of the
process $(x_{t},J_{0\leftarrow t})$. Then, using the fact that $
J_{0\leftarrow t}^{x,V}=VJ_{0\leftarrow t}^{x,I},$ $J_{0\leftarrow
t}=J_{t\leftarrow 0}^{-1}\ $ , the (strong) Markov property, and the two
observations that $t_{2}-t_{1}\geq t_{0}(N_{\epsilon }(t_{0})+1)^{-1}$ and
\begin{equation*}
span\{u^{T}J_{0\leftarrow t}^{x,I}:u\in \mathbb{R}^{e},|u|=1\}=\mathbb{R}
^{e}\,\,\,\,\text{a.s. for every $t>0$ and $x\in \mathbb{R}^{e}$}
\end{equation*}
we see that for any $q<\infty $
\begin{eqnarray}
\underset{|u|=1}{\sup }P(u^{T}C_{t_{0}}^{x,I}u\leq \epsilon ^{2}) &\leq &
\underset{|u|=1}{\sup }P\left(
u^{T}C_{t_{1},t_{2}}^{x_{t_{1}}^{x},J_{0\leftarrow t_{1}}^{x,I}}u\leq
\epsilon ^{2}\right) \notag \\
&=&\underset{|u|=1}{\sup }P\left( u^{T}J_{0\leftarrow
t_{1}}^{x,I}C_{t_{1},t_{2}}^{x_{t_{1}}^{x},I}\left( J_{0\leftarrow
t_{1}}^{x,I}\right) ^{T}u\leq \epsilon ^{2}\right) \notag \\
&=&\underset{|u|=1}{\sup }P\left( \frac{u^{T}J_{0\leftarrow
t_{1}}^{x,I}C_{t_{1},t_{2}}^{x_{t_{1}}^{x},I}\left( J_{0\leftarrow
t_{1}}^{x,I}\right) ^{T}u}{|u^{T}J_{0\leftarrow t_{1}}^{x,I}|^{2}}\leq \frac{
\epsilon ^{2}}{{|u^{T}J_{0\leftarrow t_{1}}^{x,I}|^{2}}}\right) \notag \\
&\leq &\underset{|u|=1}{\sup }P\left(
u^{T}C_{t_{1},t_{2}}^{x_{t_{1}}^{x},I}u\leq \epsilon \right) +\underset{|u|=1
}{\sup }P\left( |u^{T}J_{0\leftarrow t_{1}}^{x,I}|^{-1}\geq \epsilon
^{-1/2}\right) \notag \\
&=&\underset{|u|=1}{\sup }P\left(
u^{T}C_{t_{2}-t_{1}}^{x_{t_{1}}^{x},I}(\epsilon )u\leq \epsilon \right)
+O(\epsilon ^{q}) \notag \\
&\leq &\underset{|u|=1}{\sup }P\left( u^{T}C_{t_{0}(N_{\epsilon
}(t_{0})+1)^{-1}}^{x_{t_{1}},I}(\epsilon )u\leq \epsilon \right) +O(\epsilon
^{q}). \label{chain}
\end{eqnarray}
An application of theorem \ref{cthm} yields
\begin{equation*}
\underset{|u|=1}{\sup }P\left( u^{T}C_{t_{0}(N_{\epsilon
}(t_{0})+1)^{-1}}^{x_{t_{1}},I}(\epsilon )u\leq \epsilon \right) \,\,\,\text{
is}\,\,\,O(\epsilon ^{q})
\end{equation*}
for any $q\geq 2$ \noindent if $\epsilon \leq \epsilon
_{0}t_{0}^{1/K(q)}(N_{\epsilon }(t_{0})+1)^{-1/K(q)}$ provided that \newline
$\delta >\max \left( 8-r+v/2,(\kappa -n+\alpha )/4\alpha \right) .$ From
this, (\ref{lambda}) and (\ref{chain}) we get that
\begin{equation*}
E[\Lambda ^{-p}]\leq C_{5}+C_{6}\int_{0}^{1}\epsilon ^{-k^{\prime }}P\left(
N_{\epsilon }(t_{0})>\left\lfloor t_{0}\left( \frac{\epsilon _{0}}{\epsilon }
\right) ^{1/K(q)}\right\rfloor \right) d\epsilon .
\end{equation*}
From the proof of theorem \ref{cthm} we see that $K(q)=K(q,\epsilon )=\beta
^{-1}$ for $\epsilon $ small enough, where $\beta <m(j_{0})$, and hence to
see that $E[\Lambda ^{-p}]<\infty $ it will suffice to show
\begin{equation*}
P\left( N_{\epsilon }(t_{0})>\left\lfloor t_{0}\left( \frac{\epsilon _{0}}{
\epsilon }\right) ^{\beta }\right\rfloor \right) \text{ \ is \ }o(\epsilon
^{q})\text{ as }\epsilon \rightarrow 0\text{ for any }q>0.
\end{equation*}
\noindent Chebyshev's inequality and (\ref{rategrowth}) yield
\begin{eqnarray*}
P\left( N_{\epsilon }(t_{0})>\left\lfloor t_{0}\left( \frac{\epsilon _{0}}{
\epsilon }\right) ^{\beta }\right\rfloor \right) &\leq &\exp \left(
-t_{0}\left( \frac{\epsilon _{0}}{\epsilon }\right) ^{\beta
}+(e-1)t_{0}\lambda (\epsilon )\right) \\
&\leq &\exp \left( -t_{0}\left( \frac{\epsilon _{0}}{\epsilon }\right)
^{\beta }+C(e-1)t_{0}f(\epsilon )\right) \text{ as }\epsilon \rightarrow 0.
\end{eqnarray*}
\noindent Which, by the definition of $f$ is seen to be $o(\epsilon ^{q})$
for any $q>0$ if
\begin{equation*}
\beta >\frac{3\delta (\kappa -n)}{(\kappa -n+\alpha )}.
\end{equation*}
Since $\beta $ and $\delta $ may take any values subject to the constraints $
\beta <m(j_{0})$ and
\noindent $16\delta >\max \left( 8-r+v/2),(\kappa -n+\alpha )/4\alpha
\right) ,$ this condition becomes
\begin{equation*}
16m(j_{0})>3(\kappa -n)\max \left( \frac{8-r+v/2}{\kappa -n+\alpha },\frac{1
}{4\alpha }\right) .
\end{equation*}
\end{proof}
\bigskip
The condition (\ref{bracketcond}) exposes the qualitative structure of the
problem structure of the problem quite well in that it becomes easier to
satisfy with smaller values of $j_{0}$ (so that $
\mathbb{R}
^{e}$ is spanned with brackets of smaller length), or with smaller values of
$\kappa $ (less intense jumps) or larger values of $\alpha $ (corresponding
to better behaved vector fields). One might think that the use of the lower
bound $t_{0}(m+1)^{-1}$ on the size of the longest interval is somewhat
crude. Indeed, conditional on $N_{\epsilon }(t_{0})=m,$ the distribution
function of the longest interval is known (see Feller \cite{feller}) :
\begin{equation*}
F(x)=\overset{m}{\underset{i=1}{\sum }}(-1)^{-i}\dbinom{m}{i}\left( 1-\frac{
ix}{t_{0}}\right) _{+}^{i-1}
\end{equation*}
\noindent and more explicit calculation may be performed using this, however
they seem to lead to no improvement in the eventual criterion obtained.
Clearly, the use of only part of the covariance matrix in forming the
estimate is an area in which improvement would allow further insight to be
gained.
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{The Advantages and Ways To Select a High-end Rehab Center in Franklin
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TITLE: 2 questions about zero point of function: $f(x) = x^2 \ln(x) - ax + 1$, with unknown parameter $a$.
QUESTION [1 upvotes]: We know a function, $f(x) = x^2 \ln x - ax + 1$, with an unknown parameter $a \in \Bbb R$. There are two questions about this function:
(1) Find the range of $a$, if $f(x) \geq 0$ is always true.
(2) Define a function: $g(x) = f(x) - ax^3 + ax - 1 =x^2 \ln x - ax^3$. If we know that $g(x)$ has two different zero points $x_1, x_2$, prove that $x_1 x_2 > e^2$.
The original question is shown above. And this one is questioned for students 16-17yrs old. They have learned basic calculus with integration and derivative. But without $\epsilon - \delta$ system.
For the first subquestion, I know when $a=1$, the function $f(x)$ has only one zero point because its global lowest value is 0 when $x = 1$, then I found that the answer is $a \in (-\infty, 1]$. But I'm can get this answer only with function graph. Is there a logical way to prove it?
i.e. How to solve $f'(x) = 2x\ln x + x - a = 0$ analytically?
For the second subquestion, the only result I found is that $a \in (0,\frac{1}{e})$ for two different zero points. I tried:
$\ln(x_1) - ax_1 = 0, \ln(x_2) - ax_2 = 0\quad \Rightarrow \ln(x_1 x_2) - a(x_1 + x_2) = 0$
But I failed because the value of $x_1 + x_2$ is hard to handle.
REPLY [1 votes]: (1) $f(x)\geqslant 0$ is equivalent to $a\leqslant \dfrac{x^2\ln(x)+1}{x}$. Let $h(x):=\dfrac{x^2\ln(x)+1}{x}$. Then, one has $h'(x)=\dfrac{x^2-1+x^2\ln(x)}{x^2}$ and $h''(x)=\dfrac{x^2+2}{x^3}\gt 0$. So, $h'(x)$ is increasing with $h'(1)=0$. Since the minimum value of $h(x)$ is $h(1)=1$, one can see that $a\leqslant h(x)$ always holds if and only if $a\leqslant 1$.
(2) $g(x)=0$ is equivalent to $a=\dfrac{\ln x}{x}$. Let $j(x):=\dfrac{\ln x}{x}$. Then, $j'(x)=\dfrac{1-\ln x}{x^2}$. So, $j(x)$ is increasing for $0\lt x\lt e$ and is decreasing for $e\lt x$ with $\displaystyle\lim_{x\to 0^+}j(x)=-\infty, \displaystyle\lim_{x\to\infty}j(x)=0,j(1)=0$ and $j(e)=\dfrac 1e$. Therefore, one has $0\lt a\lt\dfrac 1e,1\lt x_1$ and $1\lt x_2$. Since $a=\dfrac{\ln x_1}{x_1}=\dfrac{\ln x_2}{x_2}$, one finally gets $$x_1x_2=\dfrac{\ln(x_1)\ln(x_2)}{a^2}\gt e^2\ln(x_1)\ln(x_2)\gt e^2$$
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June 18, 2018
**Please note that power has been restored at the below noted locations**
We are currently experiencing a power outage that is affecting Members north west of Westlock within the following LSD's: 60-27, 61-01 & 61-27.
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TITLE: Non-commutative quotient group?
QUESTION [2 upvotes]: If you have a non-abelian group $G$ with some normal subgroup $K$, is it possible to have a non-abelian quotient group $G/K$?
Besides actually sitting down and trying to generate quotient groups through exhaustion, I have been thinking about using the fundamental theory of homomorphisms to pick a small non-abelian group like $D_6$ and find the quotient group it is isomorphic to. Does this seem like a good tactic?
I'm not looking for answers, just confirmation that this is a useful way to be thinking about it.
REPLY [8 votes]: In general, there is the following fact:
If $N$ is a normal subgroup of $G$, then $G/N$ is abelian if and only if $[G,G]$ (the commutator subgroup) is a subgroup of $N$.
So the quotient will not be commutative precisely when your normal subgroup does not contain the commutators.
| 34,516
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TITLE: Visual representation of the domain and range of a function?
QUESTION [6 upvotes]: An excerpt in the book "College Algebra by Michael Sullivan is that:
When the graph of a function is given, its domain may be viewed as the shadow created by the graph on the x- axis by vertical beams of light. Its range can be viewed as the shadow created by the graph on the y-axis by horizontal beams of light.
I did't get it and i also sought in google to get some idea but i can't find one.
Can anyone help me to understand it maybe with some graphics.
Thanks.
REPLY [8 votes]: The naive definition of domain and range are represented in the following figure
the domain is the set of all $x$ values such that $y=f(x)$ is defined (and unique)
the range is the set of all $y$ values such that $\exists x$ (at least one) and $y=f(x)$
REPLY [5 votes]: Consider a particular point on the function. If you cast a vertical light beam through that point it would cast a shadow on the x-axis according to the x value of that point. Since the domain is the set of all possible x values of a function, imagine repeating the vertical light beams on every point on the function. Then the resulting shadow on the x-axis represents the domain. It is a set of values on the x-axis. By a similar process you can represent the range on the y-axis.
REPLY [3 votes]: This is not meant to be difficult. It says that going perpendicularly down/up from each $(x,f(x))$ (parallel to $y $-axis) we reach the $x$ point (point of which $f(x) $ is the image).
So 'projecting' all $(x,f(x))$ onto the $x $-axis gives the domain.
And similarly for the 'shadow' on the $y$-axis.
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Many people ask the question “How can eating something cause me to have an asthma attack in my lungs”.
The answer is simple, but to understand this question, you first must gain an understanding of what asthma is. It is also essential to understand how asthma is developed and the difference between normal respiratory defence mechanisms for fighting foreign particle attacks in the lungs in comparison to what happens in an asthma attack.
Unlike other respiratory conditions like cystic fibrosis, COPD and Bronchiectasis, asthma is actually a hypersensitivity condition. An asthma attack is actually an immune response to enzymes, proteins and anti-enzymes released by a parasite or an antigen.
As a part of the body’s immune response in an asthma attack, apart from the standard respiratory responses of inflammation and excess secretion of mucus, the body also releases a number of immune chemicals including immunoglobulin E or IgE (among others).
The release of IgE induces constriction of the smooth muscles around the outside of the airways, also known as Bronchoconstriction. This constricts the airways more and works towards helping to amplify the cough mechanism and expel the parasite from the lungs.
We have similar defence systems (called mucociliary escalators) in other parts of our bodies, including the large and small intestines of our digestive system. All of these systems are connected through our circulatory system where the immune system works its magic. Let me explain this further and how it all works.
Because it is all connected by the circulatory system, when enzymes from allergens or secretion from a parasite, or chemicals in preservatives, additives, etc… similar to these enzymes reach the walls of the large intestines, small intestines, etc… this stimulates an immune response similar to an asthma attack and can also induce an asthma attack through the circulatory system. defence system, which reacts to new infections or attacks from foreign particles, from then onwards.
What is the Difference Between Normal Defense and Asthmatic Defense?
When a parasite or allergen attacks the host, it secretes a fluid (proteins, enzymes and antienzymes) which break down the bond between the skin cells (called the epithelium) so that the parasite can gain access to the tissue) performs the following 3 steps:
1. Hypersecretion of mucus – The 1st line of defence reaches expel the parasites from the lungs through a process called wind sheering (where the top layer of the mucus containing the parasites is expelled from the body) and a cough also assist in mobilising the contaminated mucus up to the throat to be coughed or swallowed naturally.
Most of the time, dust mites are removed from the body simply by being trapped in the mucus and then by being expelled via the mucociliary escalator. defence uses the same 3 forms of defence mechanisms as mentioned in the “Normal Defense to Parasitic Attack” above, plus 1 more form of defence.
Because of the small number of parasites, the lungs decide to employ a 3rd form of defence to help remove the parasites quicker and help reduce the amount damage to the airway walls, mucus buildup and inflammation of the tissue. This, in turn, has the potential to reducing the damage from the parasitic attack in a shorter period of time.
This form of defence is known as Bronchoconstriction – This is where the muscles of the airways spasm or contract, to assist in the removal of the mucus from the airways, while also assisting in amplifying the effect of the cough, in removing the parasites.
The asthma attack and Bronchoconstriction are only employed if, the way the body reacts to the invasion determines how the body’s adaptive immune system is set up for all future attacks.
In the case of asthma, the adaptive immune system develops memory cells for employing this defence mechanism for this type of parasitic or allergen attack (which caused Bronchoconstriction) for all future attacks. If the body has no further attacks in a set period of time (i.e. a couple of years), then the memory cells may be discarded as they aren’t considered to be needed, but if you receive another in the next couple of years, then the memory cells are retained.
This is similar to a vaccine needing to be delivered multiple times for effectiveness.
Because the defence defence mechanism (i.e. the lungs, pancreas, intestines, even the skin, etc…), then any part of the body can experience an effect, similar to an asthma attack for this part of the body as well as others.
Some examples include:
- An asthma attack in the lungs,
- A bout of vomiting or diarrhea from the large or small intestines of the digestive system,
- A bout of rashes and inflammation of the skin,
It isn’t only limited by the secretions of a parasite, if chemicals in food (i.e. preservatives, additives, etc.… ) are similar to the secretions of the parasite or allergen, then it can induce an asthma attack, make you feel sick or both at the same time.
This is why it is essential to consult with an immunologist to get this important information and understand which allergens, parasites, preservatives and chemicals may affect you.
Another important point is to have healthy lungs with a healthy mucociliary escalator. The reason being that the mucociliary escalator traps and removes up to 90% of foreign particles and is the 1st line of defence.
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Another important point is that it may not actually be pollen which causes asthma attacks, but the chemicals, secretions or allergens whichD,bD, a,d,
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\begin{document}
\begin{abstract}
For any odd $n$, we construct a smooth minimal (i.e. obtained by
adding an irreducible hypersurface) compactification $\mathcal
M_n$ of the quasi-projective homogeneous variety
$S_{n}=\PP\GL(n+1)/\SL(2)$ that parameterizes the rational
normal curves in $\PP^n$. $\mathcal M_{n}$ is isomorphic to a
component of the Maruyama scheme of the semi-stable sheaves on
$\PP^n$ of rank $n$ and Chern polynomial $(1+t)^{n+2}$. This will
allow us to explicitly compute the Betti numbers of $\mathcal
M_n$.
In particular $\mathcal M_{3}$ is isomorphic to the variety of
nets of quadrics defining twisted cubics, studied by G.
Ellinsgrud, R. Piene and S. Str{\o}mme \cite{EPS}.
\end{abstract}
\maketitle
\tableofcontents
\vskip 1 cm
\section{Introduction}
A rational normal curve $C_n$, or equivalently a Veronese curve, is a
smooth, rational, projective
curve of degree $n$, in the complex projective space $\PP^n$: in particular
the Hilbert polynomial of $C_n$ is $P_{C_n}(d)=nd+1$. For a description of some
interesting properties of this curve, see \cite{Harris}.
The set $S_n$ of the rational normal curves is an homogeneous
quasi-projective variety isomorphic to $\PP\GL(n+1)/\SL(2)$. The
purpose of the paper is to describe a nice compactification of such
variety, by considering some vector bundles on $\PP^n$, called
Schwarzenberger bundles \cite{Sch}.
In particular we compute the Euler characteristic of such
compactification and its Betti numbers.
\vskip 5 mm
There are several ways to define a compactification of the variety
$S_n$: probably the most natural way is to consider the
closure $\mathcal H_n$ of the open sub-scheme of the Hilbert scheme
$\Hilb^{P_{C_n}}(\PP^n)$, parameterizing the rational normal curves in
$\PP^n$.
In \cite{PS}, the authors describe such compacti\-fication in the case
$n=3$. In particular, they show that ${\mathcal H_3}\subseteq
\Hilb^{3d+1} (\PP^3)$ is a smooth irreducible variety
of dimension $12$. Only recently, it was proven by M. Martin-Deschamps
and R. Piene \cite{MP} that
$\mathcal H_4$ is singular. Moreover it is not difficult to verify,
with the help of the algorithm described in \cite{NS}, that $\mathcal
H_5$ and $\mathcal H_6$ are singular in the points represented by
the $5-fold$ and $6-fold$ lines respectively. Therefore we can suspect
that $\mathcal H_n$ is singular for any $n\ge 4$ (see also \cite{Kap},
remark 2.6).
Another natural compactification is given by the closure $\mathcal
C_n$ of the quasi projective variety $S_n$ considered as an
open subset of the Chow variety $\mathcal C_{1,n}(\PP^n)$ that parameterizes
the effective cycles of dimension $1$ and degree $n$ in $\PP^n$.
In \cite{EPS}, a third natural compactification $\mathcal M_n$ of
$S_n$ is described:
this is made by considering the space of all the $2\times n$ matrices
with linear forms as entries. In fact all the rational normal curves
in $\PP^n$ is the zero locus of the $2-$minors of such a matrix.
In particular, when $n=3$, $\mathcal M_3$ can be seen as the variety
parametrizing the
nets of quadrics in $\PP^3$ and $\mathcal H_3$ is the blow-up of $\mathcal
M_3$.
In \cite{C}, it is shown that for any odd $n$, the projective
variety $\mathcal M_n$ is isomorphic to a smooth irreducible
component of the Maruyama scheme $\mathcal M_{\PP^n}
(n;c_1,\dots,c_n)$ parameterizing the semi-stable sheaves on
$\PP^n$ of rank $n$ and with Chern polynomial $c_t=\sum c_it^i =
(1+t)^{n+2}$. $\mathcal M_n$ can be seen as the quotient of a
projective space $\PP^N$, by the action of a reductive algebraic
group $G$. This description will allow us to apply a technique of
Bialynicki-Birula \cite{B}, to compute the Betti numbers of
$\mathcal M_n$ (see also \cite{ES}).
\vskip 1 cm
Throughout the paper we will use the following notations:
\begin{itemize}
\item $V$, $W$, $I$ are complex vector spaces of dimension $n+1$,
$m+k$ and $k$ respectively, where $m\ge n$.
\item For any $A\in\PP(\Hom(W,V\otimes I))$, the cokernel $\mathcal
F_A$ of the associated map $$A^*: I\otimes\mathcal
O_{\PP(V)}\longrightarrow W\otimes\mathcal O_{\PP(V)}(1)$$ is a
coherent sheaf of rank $m$. If $A^*$ is injective and $\mathcal
F_A$ is a vector bundle, then it is said Steiner bundle of rank
$m$, and it is contained in the exact sequence:
\begin{equation}\label{suc.esatta}
0\longrightarrow I\otimes\mathcal O_{\PP(V)}\stackrel
{A^*}\longrightarrow W\otimes\mathcal O_{\PP(V)}(1)\longrightarrow
\mathcal F_A\longrightarrow 0.
\end{equation}
Moreover if $k=2$ and $n=m$, then all the Steiner bundles are
Schwarzenberger bundles (see also \cite{C}).
\item $\GG(k,n+1)$ ($\simeq\GG(k-1,\PP^n)$) is the Grassmannian of the
$k$-subspaces of $V$ or equivalently of the $k-1$ subspaces of the
projective space $\PP^n$.
\item Let $G=\SL(I)\times\SL(W)$ and $X=\PP(\Hom(W,V\otimes I))$:
we will study the natural action of $G$ on $X$ and we will denote
by $X^s$ (resp. $X^{ss}$) the open subset of the stable (resp.
semi-stable) points of $X$.
\item For any $A\in X$, $\Stab_G(A)=\{(P,Q)\in G|PAQ^{-1}=kA \text{
for some } k\in \CC^*\}$ is the stabilizer of $A$ by the group
$G$.
\item $\mathcal M_{n,m,k}=X^{ss}//G$ (resp. $X^{s}/G$) is the
categorical (resp. geometric) quotient of $X$ by $G$. In
particular, if $n=m$, we will denote $\mathcal M_{n,k}=\mathcal
M_{n,n,k}$.
\item $V^*=\CC[x_0,\dots,x_n]_1$ is the dual space of $V$.
\item For any $A\in X$, $D(A)$ is the degeneracy locus of
$A$ and $D_0(A)$ is the variety of all the points $x\in \PP^n$
such that $\rank A_x=0$.
\item ${\mathcal S}=\{ A\in X| D(A)=\emptyset\}=\{A\in X|S(\mathcal
F_A)=\emptyset \}\subseteq X^s$.
\item $S_{n,m,k} = {\mathcal S}/G$ is the moduli space
of the rank $m$ Steiner bundles on $\PP^n=\PP(V)$: in particular
$S_{n,k}=S_{n,n,k}$ is the moduli space of the ``classical''
Steiner bundles or rank $n$ on $\PP(V)$.
\item For any matrix $A\in {\mathcal M}(k\times
(m+k),V^*)$, if $A=(a_{i,j})$
we define $i_s(A)=\min\{j=0,\dots,n+k-1|a_{s,j}\neq 0\}$ (we will
often write $i_s$ instead of $i_s(A)$).
\item $j(n)=[\frac {n+3} 2]$ where $[m]$ denotes the integer part of $m$.
\item For any coherent sheaf $\mathcal E$ of rank $r$ on $\PP^n$ and for any $t\in
\ZZ$, we write $\mathcal E(t)$ instead of $\mathcal E\otimes
\OPN(t)$. $\mathcal E_N$ will denote the normalized of $\mathcal
E$, i.e. $\mathcal E_N=\mathcal E(t_0)$ where $t_0\in\ZZ$ is such that
$-r< c_1(\mathcal E(t_0))\le 0$.
\noindent Moreover, we define the slope of $\mathcal E$ as the
number $\mu(\mathcal E)=\frac {c_1(\mathcal E)} r$
and $\mathcal E$ is said to be $\mu$-stable if it is Mumford-Takemoto stable.
\end{itemize}
\vskip 1 cm
In the first part of the paper we describe the (semi-)stable
points of the projective space $\PP(\Hom(W,V\otimes I))$ under
the action of $\SL(I)\times\SL(W)$ (see \cite{MFK} for an
introduction to the geometric invariant theory) and in particular we
will prove that, if $m<\frac{n k} {k-1}$, then all the Steiner
bundles are defined by stable matrices,
i.e. $S_{n,m,k}\subseteq\PP(\Hom(W,V\otimes I))^s/(\SL(I)\times\SL(W))$.
\vskip 5 mm
In the second part of the paper, we investigate some properties of
$S_{n,m,2}$ and in particular of $S_{n,2}$, the moduli space of the
Schwarzenberger bundles.
By the previous correspondence of bundles and curves, $\mathcal
M_{n,n,2}$ gives us a compactification of the set of the
rational normal curves in $\PP^n$.
We define a filtration of $\mathcal M_{n,m,2}$ and we show
that the compactification is obtained by adding an irreducible
hypersurface.
Moreover in \cite{C} it is shown that, if $k=2$ and $m$ is odd,
then $A\in\PP(\Hom(W,V\otimes I)$
is stable if and only if the correspondent coherent sheaf $\mathcal F_A$ is
$\mu-$stable. This yields the theorem:
\begin{thm}\label{main.thm}
$M_{n,m,2}$ is isomorphic to the connected component of the Maru\-yama
moduli space $\mathcal
M_{\PP^n}(m,c_1,\dots,c_n)$ containing the Steiner bundles. Such
component is smooth and irreducible.
\end{thm}
\vskip 4 mm
In the last two sections we compute the Betti
and Hodge numbers of the smooth projective variety $\mathcal
M_{n,m,2}$. This formula will be obtained by studying a natural action of
$\CC^*$ on $\mathcal M_{n,m,2}$: in particular we will describe its
fixed points and we will compute the weights of the action of $\CC^*$
induced in the tangent spaces of the variety at the fixed points.
\vskip 2 cm
\section{The categorical quotient of $\PP(\Hom(W,V\otimes I))$ by
$\SL(I)\times\SL(W)$}
We are interested in the study of the action of $G=\SL(I)\times\SL(W)$
on the projective space $X=\PP(\Hom(W,I\otimes V))$. In fact, as shown
in the introduction, each $A\in X^{ss}$, such that $A^*: I\otimes
\OPN\rightarrow W\otimes \OPN(1)$ is injective, corresponds to a
coherent sheaf $\mathcal F_A$ contained in the exact sequence
(\ref{suc.esatta}).
Furthermore $\mathcal F_A\simeq \mathcal F_B$ if and only if $P A=B Q$
for some $P\in \SL(I)$ and $Q\in \SL(W)$ (see for instance \cite{AO} or
\cite{MT}).
\
\begin{lemma}\label{iniettivo}
Let $A\in X^{ss}$.
Then both $A:W\to I\times V$ and $A^*:I\to W\times V$ are injective.
\end{lemma}
\begin{proo}
Let $A:W\to I\times V$ be non-injective. Then we can suppose that the
first column of $A$ is zero. Let us consider the 1-dimensional
parameter subgroup $\lambda:\CC^*\rightarrow G$
defined by $t\mapsto (\Id,\diag(t^{-(m+k-1)},t,\dots,t))\in
\SL(I)\times\SL(W)$:
then $\lim_{t\to 0}\lambda(t)A = 0$ and, by the
Hilbert-Mumford criterion, the matrix $A$ cannot be semi-stable.
Let us suppose now that $A^*:I\to W\times V$ is not injective:
i.e. the first row of $A$ is zero. In this case it
suffices to consider the 1-dimensional parameter subgroup
$\mu: t\mapsto (\diag(t^{-(k-1)},t,\dots,t),\Id)\in\SL(I)\times\SL(V)$
in order to have $\lim_{t\to 0}\mu(t)A = 0$.
\end{proo}
\
As a direct consequence of the lemma, it follows that for any $A\in
X^{ss}$, the sheaf $\mathcal F_A$ is well-defined as the cokernel of
$A^*$ and is contained in the sequence (\ref{suc.esatta}). Moreover it
results $T_A:= A(W)\in \GG(m+k,I\otimes V)$.
Thus, in order to study the (semi-)stable point of $X$
by the action of $G$, it suffices to study the action of $\SL(I)$ on
the variety $\GG(m+k,I\otimes V)$: in particular we have that the
categorical quotient $\mathcal M_{n,m,k}:= X^{ss}//G$ is isomorphic to
the quotient $\GG(m+k, I\otimes V)^{ss}//\SL(I)$.
Let us recall first the following known result:
\begin{prop}
Let $T\in \GG(m+k,I\otimes V)$. The following are equivalent:
\begin{enumerate}
\item $T$ is semi-stable (resp. stable) under the action of $\SL(I)$;
\item for any non-empty subspace $I'\subsetneq I$
$$\frac {\dim T'} {\dim I'} \le \frac {\dim T} {\dim I}
\quad (\text{resp. }<)$$
where $T'=(I'\otimes V)\cap T$.
\end{enumerate}
\end{prop}
\begin{proof}
See for instance \cite{NT} (prop. 5.1.1)
\end{proof}
As a corollary we get a description of the (semi-)stable
points of $X$ by the action of $G$:
\begin{thm}\label{stabile}
$A\in X$ is not stable under the action of $G$ if and only if
with respect to suitable bases of $W$ and $I$, it results
$i_0(A)\ge i_1(A)\ge\dots\ge i_{k-1}(A)$ and there exists $s\in\{0,\dots,k-1\}$
such that:
\begin{equation}\label{condizione.stabile}
\text { either } \quad i_s(A)\ge \frac {m+k} k (k-1-s) \text{ if }
s\neq k-1 \qquad \text { or } \quad i_{k-1}(A)>0
\end{equation}
\end{thm}
\begin{thm}\label{semistabile}
$A\in X$ is not semi-stable under the action of $G$ if and only if
with respect to suitable bases of $W$ and $I$,
it results $i_0(A)\ge i_1(A)\ge\dots
i_{k-1}(A)$ and there exists $s\in\{0,\dots,k-1\}$
such that:
\begin{equation}\label{condizione.semistabile}
i_s(A)> \frac {m+k} k (k-1-s)
\end{equation}
\end{thm}
\vskip 1 cm
\begin{corol}\label{relat.primi}
$X^s=X^{ss}$ if and only if $(m,k)=1$
\end{corol}
\begin{proo}
If there exists $A\in X$ properly semi-stable, then there exists $s\in
\{0,\dots,k-2\}$ such that
$$i_s(A)= \frac {m+k} k (k-1-s).$$
Since $1\le k-1-s\le k-1$, such $s$ exists if and only if $(m,k)\neq 1$.
\end{proo}
\vskip 2 cm
Now we are interested to study the stability of the matrices defining
the Steiner bundles and thus we will consider all the matrices $A$
such that $\rank A_x = k$ for any
$x\in\PP^n$: in
\cite{AO} it is shown that if $n=m$ (boundary format) then all such
matrices are stable.
We generalize such result with the following:
\begin{thm}\label{vect.bundle} If $m<\frac{nk} {k-1}$ then every
indecomposable vector bundle $\mathcal F_A$ is defined by a
G.I.T. stable matrix $A$.
\end{thm}
Before proving the theorem, we remind the following known lemma:
\begin{lemma}\label{thm2.8}
Let $F$ be a vector bundle of rank $f$ on a smooth projective variety
$X$ such that $c_{f-k+1}(F)\neq 0$ and let $\phi:\mathcal
O_X^k\longrightarrow F$ be a morphism with $k\le f$. Then the
degeneracy locus $D(\phi)=\{x\in X|\rank (\phi_x)\le k-1\}$ is
nonempty and $\codim D(\phi)\le f-k+1$.
\end{lemma}
\vskip 5 mm
\begin{proof}[proof of theorem \ref{vect.bundle}]
Let $\mathcal F_A$ be an indecomposable vector bundle. Then for any
base of $W$ and $I$, $i_{k-1}(A)=0$, otherwise $\mathcal F_A=
\mathcal F'\oplus \OPN(1)$ for some vector bundle $\mathcal F'$.
Let $I'\subseteq I$ of dimension $r$: if $s = \dim(I'\otimes V)\cap
T_A$ and $I''\subseteq I$ is such that
$I'\oplus I''=I$, then the restriction of $A^*$ in $I''$ defines a
morphism of vector bundles $A':\OPN^{k-r}\longrightarrow \OPN(1)^{m+k-s}$.
Let us suppose $s>m-n+r$, then
$c_{(m+k-s)-(k-r)+1}(\OPN(1)^{m+k-s})\neq 0$: lemma \ref{thm2.8} implies that
the degeneracy locus of $A'$ is not empty, which leads to a
contradiction.
Thus:
$$\dim(I'\otimes V)\cap T_A\le m+k-n-k+r = m-n+r;$$
and in particular, if $m<\frac{nk} {k-1}$, it results $\dim(I'\otimes V)\cap
T_A<\frac{r(m+k)} k$, i.e. $A$ is G.I.T. stable.
\end{proof}
\begin{remark}
By lemma \ref{iniettivo} we have that if $A\in X^{ss}$, then
$A:W\hookrightarrow I\otimes V$ is injective, thus it results
$X^{ss}=\emptyset$ if $m>kn$. Furthermore it is easy to see that if
$m=nk$ the only point of $M_{n,kn,k}$ is represented by the vector
bundle $I\otimes T_{\PP^n}$.
\end{remark}
\vskip 2 cm
\section{Compactification of $S_{n,m,2}$}
So far we have studied the G.I.T. compactification of $S_{n,m,k}$ for
any value of $n,m$ and $k$.
From now, we restrict our study to the case $k=2$: in particular we
know that the moduli space $S_{n,2}$ is uniquely composed by Schwarzenberger
bundles and thus it is isomorphic to $$\PP\GL(n+1)/\SL(2).$$ Hence
$\mathcal M_{n,2}$ is a compactification of the set of rational normal
curves in $\PP^n$.
After a short review of the previous section, we define a
$G-$invariant filtration of the space $\mathcal M_{n,m,2}$ and we study
some properties of it.
\
Theorems
\ref{stabile} and \ref{semistabile} become:
\begin{thm}\label{stabile.k2}
Let $j(m)=[\frac {m+3} 2]$. $A\in X$ is not stable if and only if
$$\text{either}\quad A\sim\begin{pmatrix}0 &\dots& 0 & f_{j(m)+1} &\dots&
f_{m+2}\cr g_1 &\dots& g_{j(m)} & g_{j(m)+1} &\dots& g_{m+2}
\end{pmatrix} \quad \text{or} \quad A\sim \begin{pmatrix} 0 *
\dots * \cr 0 * \dots * \end{pmatrix}$$
\end{thm}
\vskip 4 mm
\begin{thm}\label{semistabile.k2}
If $n$ is odd then $X^{ss}=X^{s}$, i.e. there are not properly
semi-stable points in $X$.
If $n$ is even then $A\in X$ is not semi-stable if and only if
$$\text{either}\quad A\sim\begin{pmatrix}0 &\dots& 0 & f_{j(m)+2} &\dots&
f_{m+2}\cr g_1 &\dots& g_{j(m)+1} & g_{j(m)+2} &\dots& g_{m+2}
\end{pmatrix} \quad \text{or} \quad A\sim \begin{pmatrix} 0 *
\dots * \cr 0 * \dots * \end{pmatrix}$$
\end{thm}
\vskip 1 cm
\begin{lemma}\label{lemma1}
Let $m$ be even and for any $i=1,2$ let us define the subspaces
$I^i_f=<f_0^i\dots f^i_{\frac m 2}>$
and $I_g=<g^i_0\dots g^i_{\frac m 2}>$
of $\CC[x_0,\dots,x_n]$ of dimension
${\frac m 2}+1$. Moreover let
$$A^i=\begin{pmatrix} 0&\dots &0& f^i_0&\dots& f^i_{\frac m 2}\cr g^i_0&
\dots& g^i_{\frac m 2}& 0& \dots& 0 \end{pmatrix}\qquad i=1,2$$
Then
\begin{equation}\label{s1}
A^1\sim A^2
\end{equation}
if and only if
\begin{equation}\label{s2}\text{either}\quad
I^1_f=I^2_f \text{ and } I^1_g=I^2_g \quad \text{or}\quad
I^1_f=I^2_g \text{ and } I_g^1=I_f^2
\end{equation}
\end{lemma}
\begin{proof}
Let us suppose that (\ref{s1}) holds, then $A^1$ and $A^2$ have the same
degeneracy locus and this implies:
$$V(I^1_f)\cup V(I^1_g) = V(I^2_f)\cup V(I^2_g).$$
Since $V(I^1_f)$, $V(I^1_g)$, $V(I^2_f)$ and $V(I^2_g)$ are
irreducible, it results $V(I^1_f)=V(I^2_f)$ and $V(I^1_g)=V(I^2_g)$
or $V(I^1_f)=V(I^2_g)$ and $V(I^1_g)=V(I^2_f)$, thus (\ref{s2}) holds.
Vice-versa let us suppose $I^1_f=I^2_f$ and $I^1_g=I^2_g$ and let
$B_1,B_2\in\SL(\frac m 2 +1)$ be the respective base change matrices.
Then
$$A_1\begin{pmatrix} B_2 &
0\cr 0 & B_1\end{pmatrix}= A_2.$$
Otherwise if $I_f=I_g'$ and $I_g=I_f'$ then if $C_1,C_2\in\SL(\frac m 2
+1)$ are the respective base change matrices, then
$$\begin{pmatrix} 0 & 1\cr 1 & 0\end{pmatrix}A_1
\begin{pmatrix} 0 & C_2
\cr C_1 & 0\end{pmatrix}= A_2.$$
Thus (\ref{s1}) holds.
\end{proof}
\vskip 1 cm
\begin{thm}\label{sottoinsieme}
Let $m$ be even. Then
\begin{equation}\label{sottoinsieme.formula}
(X^{ss}\setminus X^s)//G\simeq
S^2\GG\left({\frac m 2},\PP(V)\right).
\end{equation}
\end{thm}
\begin{proof}
Let $A\in X^{ss}\setminus X^s$. Then
$$A\sim
\begin{pmatrix}0 &\dots& 0 & f_{j(m)+1} &\dots&
f_{m+2}\cr g_1 &\dots& g_{j(m)} & g_{j(m)+1} &\dots& g_{m+2}
\end{pmatrix}$$ and thus if we consider the 1-dimensional parameter
subgroup defined by the weights $\beta=(-1,1)$ and
$\gamma=(-1,\dots,-1,1,\dots,1)$, it results:
$$\lim_{t\rightarrow 0}tA =
\begin{pmatrix}0 &\dots& 0 & f_{j(m)+1} &\dots&
f_{m+2}\cr g_1 &\dots& g_{j(m)} & 0 &\dots& 0\end{pmatrix}.$$
Thus the points of $(X^{ss}\setminus X^s)//G$ are in one-one
correspondence with the orbits
of the matrices $\begin{pmatrix} 0&\dots&0&*&\dots&*\\
*&\dots&*&0&\dots&0\end{pmatrix}\in X^{ss}$ by the action of $G$.
The previous lemma implies the isomorphism in
(\ref{sottoinsieme.formula}).
\end{proof}
\vskip 1 cm
\begin{remark} Since $\GG(m+2,2(n+1))\simeq \GG(2n-m,2(n+1))$, it
follows that $\mathcal M_{n,m,2}\simeq \mathcal M_{n,2n-m-2,2}$. In
particular $\mathcal M_{n,2}$ parameterizes the $n\times 2$ matrices
with entries in $V^*$: in fact a rational normal curve is the zero
locus of the minors of such a matrix.
In the case $n=3$, we have that $\mathcal M_{3,2}$ is
isomorphic to the variety of the nets of quadrics that define the
twisted cubics in
$\PP^3$. In \cite{EPS}, the authors describe this variety and they
show that there exists a natural morphism from the Hilbert scheme
compactification $\mathcal H_3$ to $\mathcal M_{3,3,2}$. It would be
interesting to know if there exist a canonical morphism, $\mathcal H_n\to
\mathcal M_{n,n,2}$, for any odd $n$.
\end{remark}
\vskip 1 cm
For any $\omega \in I$ we define $R_\omega=\omega\otimes V\subseteq
I\otimes V$: by theorems \ref{stabile.k2} and \ref{semistabile.k2} we
have that an injective matrix $A: W\hookrightarrow I\otimes V$ is
semi-stable (resp. stable) if and only if
$$\dim R_\omega\cap T_A\le \frac{m+2} 2 \quad (\text{resp.} <)$$
for any $\omega\in I$.
For any $j=0,1,\dots$ we construct the subsets:
\begin{eqnarray*}
S^j &=&\left\{A\in X^{ss}|\exists ~\omega\in I \text{ such that } \dim
R_\omega\cap T_A\ge j+m-n \right\}\subseteq X^{ss}\quad\text{and}\\
\tilde S^j&=&\{A\in X^{ss}|\dim D(A) \ge j - 2\}\subseteq X^{ss}.
\end{eqnarray*}
Such subsets of $X$ define two filtrations:
\begin{eqnarray*}
\emptyset= S^{j_0+1}\subseteq & S^{j_0}\subseteq \dots \subseteq
&S^2\subseteq S^1=X^{ss}\\
\emptyset\subseteq\dots\subseteq \tilde S^{j_0+1}\subseteq &\tilde
S^{j_0}\subseteq \dots\subseteq &\tilde S^2\subseteq \tilde S^1=X
\end{eqnarray*}
where $j_0=j(m)+n-m$. It results
$S^{j_0}=X^{ss}\setminus X^s$ and in particular it is empty if $m$ is odd.
Furthermore we have:
\begin{thm}\label{filtrazione}
\
\begin{enumerate}
\item $S^j\subseteq\tilde S^j\subseteq S^{j-1}$ for any $j\ge 2$;
\item $S^{2}=\tilde S^{2}$;
\item $S^{1}=\tilde S^{1}=X^{ss}$.
\end{enumerate}
In particular such subsets define a unique filtration $G-$invariant:
\begin{eqnarray*}
\emptyset=S^{j_0+1}\subseteq \tilde S^{j_0+1}\subseteq
S^{j_0}\subseteq \tilde S^{j_0}\subseteq\dots\\
\dots\subseteq S^3\subseteq
\tilde S^3\subseteq S^2 =\tilde S^2\subseteq S^1=\tilde S^1 =
X^{ss}
\end{eqnarray*}
\end{thm}
\begin{proo}
See \cite{C} (thm 2.1).
\end{proo}
\vskip .7 cm
\begin{remark} In general $S^i\neq\tilde S^i$: let us consider, for
instance, $n=m=3$ and
$$A=\begin{pmatrix} 0 &0& x_0& x_1& x_2 \cr
x_0& x_1& 0& 0& x_3\end{pmatrix}.$$
Since $D(A)=\{(0:0:t_1:t_2)\}\simeq \PP^1$, $A\in\tilde S^3$; but
$S^3=\emptyset$ (see also prop. \ref{codimensione}).
\end{remark}
\vskip 1 cm
\begin{corol}
If $m$ is odd and $A\in X^s=X^{ss}$ then $\codim D(A)\ge \frac {m+1}
2$.
If $m$ is even and $A\in X^{ss}$ (resp. $X^{s}$) then $\codim D(A)\ge
\frac m 2$ (resp. $>$).
\end{corol}
\begin{proo}
It suffices to notice that the previous theorem implies that $\tilde
S^{j_0+1}=\emptyset$ and that $S^{j_0}$ is the set of the properly
semi-stable points of $X$.
\end{proo}
\vskip .5 cm
\begin{prop}\label{codimensione}
If $m$ is odd, $A\in X$ is stable and $\codim D(A) = {\frac {m+1} 2}$,
then, up to the action of
$\SL(I)\times\SL(W)\times\SL(V)$,
we have
$$A \simeq \begin{pmatrix}x_0 &\dots & x_{t-1} & 0 &\dots &0 &x_t\\ 0
&\dots &0 &x_0 &\dots &x_{t-1} &x_{t+1}
\end{pmatrix},$$
where $t={\frac {m+1} 2}$.
\end{prop}
\begin{proof}
By the proof of theorem \ref{filtrazione} we have that for any
$\omega\in I$, $\dim(\omega\otimes V)\cap T = t$,
where $T$ is the image of $A$ as a subspace of $I\otimes V$.
Thus we have, up to a base change,
$$A \simeq \begin{pmatrix}x_0 &\dots & x_{t-1} & 0 &\dots &0 &x_t\\ 0
&\dots &0 &y_0 &\dots &y_{t-1} &y_{t+1}
\end{pmatrix},$$
where $x_0,\dots, x_t$ and $y_0,\dots, y_t$ are linearly independent.
It is easily checked that $D(A) = V(x_0,\dots, x_t)\cup V(y_0, \dots,
y_t)\cup V(x_0,\dots,x_{t-1},
y_0,\dots,y_t)$ and since $\codim D(A) = t$, it must be $\codim V(x_0,\dots,x_{t-1},
y_0,\dots,y_t)= t$: this implies that $<x_0,\dots, x_{t-1}>=<y_0,\dots,y_{t-1}>$.
Moreover $x_t \neq a y_t$ for any $a\in \CC$ otherwise $A$ cannot be
stable.
\end{proof}
\begin{remark} The matrix above can exist if $n+1\ge t+1 = \frac
{m+3} 2$, i.e. if $m\le 2 n -1$.
Since $A:W\hookrightarrow I\otimes V$ is injective, it must be $m+2\le
2 (n+1)$, i.e. $m\le 2 n$: thus in the odd case, the two requirements
are equivalent.
\end{remark}
\begin{corol}
Let $V_i=X^{ss}\setminus S^i$ e $\tilde V_i=X^{ss}\setminus \tilde
S^i$.
Then such subsets define a $G-$invariant increasing filtration:
$$\emptyset=V_1=\tilde V_1\subseteq V_2= \tilde V_2\subseteq
\tilde V_3\subseteq V_3\subseteq\dots$$
$$\dots\subseteq\tilde V_{j_0}\subseteq
V_{j_0}\subseteq \tilde V_{j_0+1}\subseteq V_{j_0+1}=X^{ss}.$$
In particular $V_2$ is the set of matrices that define vector bundles
and $V_{j_0}$ is the open set of the stable points in $X$.
\end{corol}
\vskip .6 cm
\begin{remark}
If $n$ is odd then $V_{j(m)}=X^s=\tilde V_{j(m)+1} = V_{j(m)+1}=X^{ss}$.
Otherwise if $m$ is even then $S^{j(m)}//G \simeq S^2\GG(\frac m 2,\PP^n)$
(theorem \ref{sottoinsieme}).
\end{remark}
\vskip .6 cm
All these results are needed to prove the following theorem:
\begin{thm}\label{stabilita}
Let $k=2$ and $m\in\NN$ odd.
$A\in \GG(m+2,I\otimes V)$ is G.I.T. stable if and only if $\mathcal F_A$ is $\mu$-stable.
\end{thm}
\begin{proo} See \cite{C} (thm. 3.1).
\end{proo}
Theorem \ref{main.thm} is a direct consequence of this equivalence
within the stability of the maps and the stability of the
cokernels.
\vskip 2 cm
\section{Dimension of $S^j/G$}
For any $j<j(m)$ we calculate the dimension of $S^j/G\subseteq
\mathcal M_{n,m,2}$ and we show that it is irreducible. In particular we
show that $S^2/G$
is the irreducible hypersurface that parameterizes all the sheaves in $
\mathcal M_{n,2}$ that are not bundles or, on the other hand, all the
points added to compactificate the moduli space of the rational normal
curves in $\PP^n$.
\
We remind that:
$$S^j=\{A\in X^{ss}|\exists ~0\neq \omega\in I \text{ such that }
\dim(T_A\cap R_\omega)\ge j+m-n\}.$$
Thus, if $j<j(m)$,
$$\frac{S^j}{\SL(W)} \simeq \{T\in \GG(m+1,\PP(I\otimes V))^{ss}|
\exists ~\omega\in I^* :\dim(T\cap
\PP(R_\omega))\ge j+m-n-1\}.$$
Let us define the incidence correspondence ${\mathcal I}_j\subseteq
\GG(m+1,\PP(I\otimes V))\times \PP(I)$ as:
$${\mathcal I}_j=\{(T,[\omega])|T\in \GG(m+1,\PP(I\otimes V))^{ss},
[\omega]\in \PP(I), \dim(T\cap \PP(R_\omega))\ge j+m-n-1\} $$
and let $p_1$ and $p_2$ be the respective projections.
Since $S^1=X^{ss}$, we can suppose $2\le j<j(m) $.
Let us fix $[\omega]\in \PP(I)$: then
$$p_2^{-1}([\omega])\simeq\{T\in\GG(m+1,\PP(I\otimes V))^{ss}|\dim(T
\cap \PP(\omega\otimes V))\ge j+m-n-1\}$$ and:
\begin{equation*}
\begin{split}
\dim p_2^{-1}([\omega])&=
(n+1-(j+m-n))(j+m-n)+\\
&\qquad+(2(n+1)-(m+2))~(m+2-(j+m-n))=\\
&=2mn-m^2 + 3n-m +(n-m)j + j - j^2.
\end{split}
\end{equation*}
Hence ${\mathcal I}_j$ is irreducible (see \cite{Harris}, theorem 11.14)
of dimension $2mn-m^2 + 3n-m +1+(n-m)j + j - j^2$.
Now, if $T\in p_1(\mathcal I_j)$ is a generic point, $p_1^{-1}(T)$ is
discrete, i.e. $\dim p_1^{-1}(T)=0$ that implies:
$$\dim S_{j}/\SL(W)=\dim p_1({\mathcal I}_j)=2mn-m^2 + 3n-m +1+(n-m)j
+ j - j^2.$$
Furthermore $S^j/\SL(W)$ is irreducible.
Since all the points of $S^j$ are stable under the action of $G$
(we are supposing $j<j(n)$), theorem \ref{main.thm} implies $$\dim
(S^j/G) = \dim S^j - \dim G = \dim (S^j/\SL(W)) - \dim \SL(I).$$
Hence we have:
\begin{thm}
$S^j/G$ is irreducible of codimension $(j+m-n)(j-1)-1$
for any $2\le j< j(m)$.
\end{thm}
In particular:
\begin{corol}If $n=m$ (boundary format)
$S^2/G$ is an irreducible hypersurface of $\mathcal M_{n,2}$
such that $$\mathcal M_{n,2} \setminus (S^2/G)\simeq S_{n,2}.$$
\end{corol}
By theorem \ref{sottoinsieme}, we know that, if $m$ is even,
the variety $\mathcal M_{n,m,2}\setminus (S^{j(m)}//G)$ is isomorphic to
$S^2\GG\left(\frac m 2, \PP(V)\right)$
and thus it is irreducible of dimension $(n-\frac m 2)(\frac m 2 +1)$,
i.e. the G.I.T. quotient $S^{j(m)}//G$ is of codimension
$(n-\frac m 2)(\frac m 2 +1)$.
If $m$ is odd, then $S^{j(m)}=\emptyset.$
\vskip 2 cm
\section{A torus action on $\mathcal M_{n,m,2}$}
In the following two sections we compute the Euler characteristic of
$\mathcal M_{n,m,2}$ and an implicit formula for its Hodge numbers.
For this purpose, we will use the technique of Bialynichi-Birula
\cite{B}, that is based on the study of the action of a torus on a
smooth projective variety: such method was extensively used in the
last decade to compute the Betti numbers of smooth moduli spaces (see
for istance \cite{Kl}).
In fact let an algebraic torus $T$ act on a smooth projective variety
$Z$ and let $Z^T$ be its fixed points set. Then the Euler characteristics
of $Z$ and $Z^T$ are equal. Furthermore if $T=\CC^*$ is
1-dimensional, then all the cohomology
groups of $Z$ and their Hodge decomposition may be reconstructed from
the Hodge structure of the connected components $Z_i^T$ of
$Z^T$. In order to do that, we fix a point $z_i\in Z_i^T$ for any
component and
we consider the action of $T$ on the tangent space $T_{z_i}Z$: let
$n_i$ be the number of positive weights of $T$ acting on $T_{z_i}Z$,
then we have:
\begin{thm}[Bialynichi-Birula]\label{BB}
There is a natural isomorphism:
$$\HH^{p,q}(Z) = \bigoplus_i \HH^{p-n_i,q-n_i}(Z^T_i).$$
\end{thm}
\begin{proof}
See \cite{B} and \cite{G}.
\end{proof}
\vskip .8 cm
Thus let us consider now the action of $T=\CC^*$ on $\mathcal
M_{n,m,2}$ defined by the morphism
$\rho:\CC^*\rightarrow \GL(V)$ with weights
$c=(1,2,2^2,\dots,2^n)$: this choice is motiveted by the fact that
\begin{equation}\label{ci}
c_i-c_j=c_{i'}-c_{j'}\qquad \text{ if and only if } \qquad i = i'\text{ and } j= j'
\end{equation}
that will be useful later on.
For any $t\in \CC^*$, we will write $t(\cdot)$ to denote the image of
$\cdot$ by the map $\rho(t)$.
Let $A=\begin{pmatrix} f_0 \dots f_{m+1} \\ g_0 \dots
g_{m+1}\end{pmatrix}\in \mathcal M_{n,m,2}$ be a fixed point then
$$t(A)=\begin{pmatrix} t(f_0) \dots t(f_{m+1}) \\ t(g_0) \dots
t(g_{m+1})\end{pmatrix}\sim A$$
for any $t\in \CC^*$. Thus it is defined a morphism $\tilde
\rho:\CC^*\rightarrow \Aut(I)\times \Aut(W)$, such that
$\rho(t)(A)=\tilde\rho(t)(A)$ for any $t\in \CC^*$.
Thus for any fixed point $A$, $\rho$ induces an action of $\CC^*$
on $I$ and $W$: let $P(t)$ and $Q(t)$ be the components of $\tilde
\rho$ in $\Aut(I)$ and $\Aut(W)$ respectively, then
$t(A)=P(t) ~ A ~ Q(t)^{-1}$ for any $t$ in $\CC^*$.
We can suppose that such action is diagonal and that
it is defined by the weights $(a_0,a_1)$ and $(b_0,\dots, b_{m+1})$
respectively (at the moment we do not fix any order for such weights,
we will do it later on).
If $f_k = \sum r_i x_i$ then $\sum r_i t^{c_i} x_i=t(f_k)=\sum r_i
t^{a_0 - b_k}x_i$, and
since $c_i\neq c_j$ if $i\neq
j$, it must be $f_k =
r_{i_k}x_{i_k}$ for a suitable $i_k\in\{0,\dots,n\}$ and with
$r_{i_k}\in \CC$; moreover it results $a_0 - b_k = c_{i_k}$
for any $k$ such that $r_{i_k}\neq 0$.
Similarly we have $g_k=s_{j_k}x_{j_k}$ with $s_{j_k}\in \CC$,
$j_k\in\{0,\dots,n\}$ and $a_1 - b_k = c_{j_k}$ for any $k$ such that
$r_{j_k}\neq 0$.
Thus the matrix $A$ is monomial with respect to the bases of $I$
and $W$ chosen. Moreover the weights $(a_0,a_1)$ and
$(b_0,\dots,b_{m+1})$ are the solution of a system:
\begin{equation}\label{spf}
\begin{cases}
a_0 - b_k = c_{i_k} \quad \forall ~k \text{ s.t. } f_{k}\neq 0\\
a_1 - b_k = c_{j_k} \quad \forall ~k \text{ s.t. } g_{k}\neq 0\\
\end{cases}
\end{equation}
Since $A$ is stable, there exists $\tilde k$ such that $f_{\tilde
k},g_{\tilde k}\neq 0$, thus, by (\ref{spf}), it follows that $a_0-a_1 =
c_{i_{\tilde k}} -c_{j_{\tilde k}}$: it is easy to check that if
(\ref{spf}) admits a solution, then such solution is unique up to an
additive constant; for this reason we can suppose $a_0=0$.
Now we can fix an order on the base of $W$ chosen (we did
not do it before): in fact we can suppose $f_k=0$ if and only if $k>
k_0$ where $k_0\in\{1,\dots,m+1\}$: moreover we can take $b_0\ge b_1
\ge\dots\ge b_{k_0}$ and, if $k_0\le m$, we can also take
$b_{k_0+1}\ge b_{k_0+2}\ge\dots \ge b_{m+1}$.
In particular we have $c_{i_0}\le c_{i_1}\le \dots \le c_{i_{k_0}}$ and
$c_{j_{k_0+1}}\le\dots\le c_{j_{m+1}}$, that implies $i_0\le i_1\le
\dots\le i_{k_0}$ and $j_{k_0+1}\le\dots\le j_{m+1}$.
Let $k_1,\dots,k_z\le k_0$ be such that $f_{k_j},g_{k_j}\neq 0$ for
any $j=1,\dots, z$: it must be $z\ge 1$ and
$a_1=c_{j_{k_1}}-c_{i_{k_1}}$.
Thus (\ref{spf}) becomes:
\begin{equation}\label{spf2}
\begin{cases}
b_k= -c_{i_k} \quad &\forall ~k\le k_0\\
b_k=a_1 - c_{j_k} \quad &\forall ~k> k_0\\
a_1=c_{j_{k_s}}-c_{i_{k_s}}\quad &\forall ~s=1,\dots,z
\end{cases}
\end{equation}
By (\ref{ci}) and since
$(f_{k_j},g_{k_j})\neq (f_{k_1},g_{k_1})$ if $s=2,\dots, z$, we can
either suppose $z=1$ or $a_1=0$ that implies $i_{k_s}=j_{k_s}$ for any
$s=1,\dots, z$.
Thus we have to distinguish two cases:
\begin{enumerate}
\item $a_1\neq 0$, $z=1$
\item $a_1=0$, $z\ge 1$
\end{enumerate}
Under each of these hypothesis, it is easy to show that the system
(\ref{spf2}) admits a unique solution that defines a fixed point
$A\in\mathcal M_{n,m,2}$ by the action of
$\rho$.
In order to have a total description of the fixed points, we will
consider each case separately:
\begin{enumerate}
\item Let us define
\begin{equation}\label{primo.caso}
A_{I,J}=\begin{pmatrix}
x_{i_0} & x_{i_1} & \dots & x_{i_t} & 0 & \dots & 0 \\
x_{j_0} & 0 & \dots & 0 & x_{j_1} & \dots & x_{j_t}
\end{pmatrix}
\end{equation}
where $I=(i_0,\dots,i_t)$ and $J=(j_0,\dots,j_t)$, with
$i_1<\dots<i_t$, $j_1<\dots<j_t$, $i_0<j_0$ and $i_0\neq i_s$,
$j_0\neq j_s$ for any $s=1,\dots,t$.
\noindent It is easy to see that under this assumption the matrices
$A_{I,J}$'s are stable and determine uniquely all the fixed point of
$\rho$ with $a_1\neq 0$.
\vskip 5 mm
\item The matrices fixed by $\rho$ with $a_1=0$ are given by
\begin{equation}\label{secondo.caso}
A_{\omega}^i = (\omega_1~x_{i_1},\dots,\omega_{m+2} ~x_{i_{m+2}})
\end{equation}
with $\omega=(\omega_1,\dots,\omega_{m+2})\in I^{m+2}$ and
$i=(i_1,\dots,i_{m+2})$ where $0\le i_1\le\dots\le i_{m+2}\le n$.
\noindent Since $\dim I=2$, if $i_j=i_{j+1}$ then we can suppose
that $\{\omega_{i_j}, \omega_{i_{j+1}}\}$ is the base of $I$ fixed above.
Moreover there cannot exist a $j$ such that $i_{j-1}=i_j=i_{j+1}$
otherwise $A_{\omega}^i$ cannot be stable.
Thus, in particular,
\begin{equation}\label{li}
l(i)=\#\{i_j|i_j\neq i_k \text{ for any } k\neq j\}
\end{equation}
is odd.
\noindent It is easy to check that $A_\omega^i$ and $A_{\omega'}^{i'}$
are contained in the same connected component of $\mathcal M_{n,m,2}^\rho$
if and only if $i=i'$ and that such component is isomorphic to
$\PP(I)^{l(i)}/\SL(2)$. In particular $A_\omega^i$ is stable if and
only if the corresponding poing in $\PP(I)^{l(i)}$ is stable under
the action of $\SL(2)$. The stable points of $\PP(I)^{l(i)}$
are described by:
\end{enumerate}
\vskip .6 cm
\begin{prop}Let $l\in \NN$ be odd and let the group
$\SL(I)$ act on $Y=\PP(I)^l=\PP(I)\times\dots\times\PP(I)$;
for any $\omega\in Y$, let $I_k(\omega) =
\{j\in(1,\dots,n)|\omega_i=\omega_k\}$;
then
$$Y^s=Y^{ss}=\{\omega \in Y|~ \# I_k(\omega) < \frac l 2 \quad \text{
for any } k=1,\dots,l\}.$$
\end{prop}
\begin{proo}
It is a direct consequence of the Hilbert-Mumford criterion for stability.
(see also \cite{MFK}).
\end{proo}
\vskip .8 cm
The Hodge numbers of $M_l=\PP(I)^l/SL(I)$ are given by the following:
\begin{thm}\label{hodge.ml}
Let $l$ be odd. Then:
$$\hh^{p,q}(M_l)=\begin{cases} 0\qquad &\text{if } p\neq q\\
1 + (l-1) + \dots + \begin{pmatrix} l-1\\
\min(p,l-3-p)\end{pmatrix}\qquad &\text{if } p=q \end{cases}.$$
In particular, the Poincar\'e polynomial is:
$$P_t(M_l)=1 + \hh^{1,1}t^2 + \dots + \hh^{j,j}t^{2j}+\dots+t^{2l-6}.$$
and the Euler characteristic is given by:
$$\e(M_l)= \sum_{p=0,\dots, l-3} \hh^{p,p}$$
\end{thm}
\begin{proo}
See \cite{K} pag. 193.
\end{proo}
\vskip 5 mm
By the classification of the fixed points of $\mathcal M_{n,m,2}$, we
thus have:
\begin{corol}
$h^{p,q}(\mathcal M_{n,m,2})=0$ for any $p\neq q$.
\end{corol}
\vskip 5 mm
We are now ready to compute the Euler characteristic of $\mathcal M_{n,m,2}$:
\begin{thm}\label{eul.char}
Let $m$ be odd and let $t=\frac{m+1} 2$. Then the Euler characteristic
of $\mathcal M_{n,m,2}$ is given by:
\begin{equation}\label{eul}
\e(\mathcal M_{n,m,2}) = \begin{pmatrix} n+1\\ 2 \end{pmatrix}
\begin{pmatrix} n \\t\end{pmatrix}^2
+\sum_{d=1}^{n-t} \begin{pmatrix}n+1\\ t-d\end{pmatrix}
\begin{pmatrix} n+1 -t +d\\ 2d +1 \end{pmatrix}
\e(\PP(I)^{2d+1}/\SL(I))\enspace .
\end{equation}
\end{thm}
\begin{proo}
By theorem \ref{BB}, it results $\e(\mathcal M_{n,m,2})=\sum_i \e
(\mathcal M_{n.m,2}^T)_i$, where $(\mathcal M_{n.m,2}^T)_i$ are the
connected components of the fixed point of $\mathcal M_{n,m,2}$ under
the action of the torus $T$ considered.
The points $A_{I,J}$, defined in (\ref{primo.caso}), represent
discrete components of such space and since they are uniquely determined by
$I=(i_0,\dots,i_t)$ and $J=(j_0,\dots,j_t)$, with
$i_1<\dots<i_t$, $j_1<\dots<j_t$ and $i_0<j_0$,
it is easy to compute that they are exactly
\begin{equation}\label{eul1}
\begin{pmatrix} n+1\\ 2 \end{pmatrix}
\begin{pmatrix} n \\t\end{pmatrix}^2.
\end{equation}
On the other hand, the matrices $A^i_\omega$, defined in
(\ref{secondo.caso}), form connected components
determined by $i=(i_1,\dots,i_{m+2})$ and isomorphic to $M_{l(i)}$, where
$l(i)$ is defined in (\ref{li}).
Let $d(i)=m+2-l(i)$ the number of the couples of equal terms in $i$:
for any $d\ge 1$ the number of the admissible vectors
$i=(i_1,\dots,i_{m+2})$ with $d(i)=d$ is given by
\begin{equation*}
\begin{pmatrix}n+1\\d\end{pmatrix} ~ \begin{pmatrix}n+1-d\\m+2-2d\end{pmatrix}
\end{equation*}
thus the Euler characteristic of the set of such matrices is given by
\begin{equation}\label{eul2}
\begin{split}
\sum_{d=m-n+1}^{t-1} \begin{pmatrix}n+1\\d\end{pmatrix} ~
\begin{pmatrix}n+1-d\\m+2-2d\end{pmatrix} ~
\e(M_{m+2-2d}) \\
=\sum_{d=1}^{n-t}\begin{pmatrix}n+1\\ t-d\end{pmatrix}
\begin{pmatrix} n+1 -t +d\\ 2d +1 \end{pmatrix}
~ \e(M_{2d+1}).
\end{split}
\end{equation}
(\ref{eul}) is obtained by summing (\ref{eul1}) with (\ref{eul2}).
\end{proo}
\vskip 2 cm
\section{Betti numbers}
We compute the numbers $n_i$ for any fixed point in a connected
component $(\mathcal M_{n,m,2}^T)_i$. We remind that $n_i$ represents
the number of positive weights of $T=\CC^*$ acting on the tangent
space of $\mathcal M_{n,m,2}$ at the fixed points.
These numbers will yield to the computation of the Betti numbers
of $\mathcal M_{n,m,2}$ for any odd $m$.
In particular we get a topological description of the moduli space of
the rational normal curves on $\PP^n$ for any odd $n$.
\vskip 6 mm
Let $A\in \mathcal M_{n,m,2}$ be a fixed point for $\rho$.
Then $\rho$ induces an action on the tangent space $T_A \mathcal
M_{n,m,2}$. By theorem \ref{main.thm}, such vector space is isomorphic
to the tangent space of the Maruyama scheme
$\mathcal M_{\PP^n}(m;c_1,\dots,c_n)$ at the point corresponding to
the sheaf $\mathcal F_A$ and thus it is isomorphic to $\Ext^1(\mathcal
F_A,\mathcal F_A)$ (see \cite{Mar1} and \cite{Mar2}).
By the sequence (\ref{suc.esatta}) that defines the sheaf $\mathcal
F_A$, it is easily checked that $\Ext^1(\mathcal F_A,
\mathcal F_A)$ is contained in the exact sequence:
$$0\to \Hom(\mathcal F_A,\mathcal F_A)\to \Hom(W\otimes
\OPN(1),\mathcal F_A) \to \Hom(I\otimes \OPN(1),\mathcal F_A) \to
\Ext^1(\mathcal F_A,\mathcal F_A)\to 0.$$
Moreover $\HH^0(\mathcal F_A)=(W\otimes V)/a(I)$ where $a:I\hookrightarrow
W\otimes V$ is the map induced by $A^*:I\otimes \OPN\hookrightarrow
W\otimes \OPN(1)$ and $\HH^0(\mathcal F(-1))=W$.
Thus, it results $\Hom(W\otimes\OPN(1),\mathcal F_A)=W^*\otimes
\HH^0(\mathcal F_A(-1)) = W^*\otimes W$ and $\Hom(I\otimes \OPN, \mathcal
F_A)=I^*\otimes (W\otimes V)/a(I)$.
In particular the weights of the action $\rho$ on $\Ext^1(\mathcal
F_A,\mathcal F_A)$ are easily computed using the sequence:
\begin{equation}\label{pesi.ext}
0\to \CC \to W^*\otimes W \to I^*\otimes \frac {W\otimes V} {a(I)} \to
\Ext^1(\mathcal F_A,\mathcal F_A)\to 0.
\end{equation}
In the previous section we have seen that for any fixed matrix $A\in X$, $\rho$
induces an action on $I$ and $W$ defined by the weights $(a_0,a_1)$ and
$(b_0,\dots, b_{m+1})$ described by (\ref{spf2}), where, we remind,
$c=(1,2,\dots,2^n)$.
For any $A\in (\mathcal M_{n,m,2}^T)_i$ we write $n(A)$ in place of
$n_i$ and moreover we define $n_1(A)$ as the number of the positive
weights of $\rho$ on $W^*\otimes W$ and similarly $n_2(A)$ as the number
of the positive weights on $I^*\otimes (W\otimes V)/a(I)$.
Thus by the sequence (\ref{pesi.ext}), it results $n(A)=n_2(A)-n_1(A)$.
\vskip 5 mm
In order to calculate $n_1(A)$ and $n_2(A)$ for all the fixed matrices
by the action of $\rho$, we need to distinguish the cases
described above:
\begin{prop}\label{ni}
\
\begin{enumerate}
\item Let $A_{I,J}$ be defined as in (\ref{primo.caso}); then:
\begin{align*}
n_1(A_{I,J}) = ~ & 4tn +2t+2n -1 - \sum_{s=0}^t i_s-\sum_{s=0}^t j_s -
\sum_{i_s>i_0} i_s - \sum_{j_s>i_0,s\ge 1} j_s \\
& - \#\{s=1,\dots,t|j_s>j_0\} -
i_0\cdot\#\{s=1,\dots,t|j_s\le i_0\} \\
\intertext{and}\\
n_2(A_{I,J}) = ~ & \begin{pmatrix} m+2 \\ 2 \end{pmatrix}.\\
\\
\intertext{\item Let $A_\omega^i$ be defined as in
(\ref{secondo.caso}); then:}\\
n_1(A_\omega^i) = ~ &2(m+2)n -2\sum_{s=0}^{m+1}i_s~\\
\intertext{and}\\
n_2(A_\omega^i) = ~& \begin{pmatrix}m+2 \\ 2\end{pmatrix} +
\frac{m+2-l(i)} 2.
\end{align*}
\end{enumerate}
\end{prop}
\begin{proof}
It is just a direct computation.
\end{proof}
\vskip 1 cm
Proposition \ref{ni} and theorem \ref{hodge.ml} give us the right
ingredients to apply theorem
\ref{BB} of Bialynichi-Birula. Thus we have an algorithm to compute
the Betti numbers of $\mathcal M_{n,m,2}$ for any $m\ge n$, and in
particular of $\mathcal M_{n,n,2}$ the compactification of the variety
$S_n$ of the rational normal curves.
In fact, let $b_i(n)=\dim\HH^{i}(\mathcal M_{n,n,2},\QQ)$: the
following table provides the values of $b_i(n)$, for $n=2,3,5,7$ and
for all the even $i=0,\dots, 36$.
\vskip 1 cm
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} \hline
$n$ &
$ b_0 $ & $ b_2 $ & $ b_4 $ & $ b_6 $ & $ b_8 $ &
$ b_{10} $ & $ b_{12} $ & $ b_{14} $ & $ b_{16} $ & $ b_{18} $ &
$ b_{20} $ & $ b_{22} $ & $ b_{24} $ & $ b_{26} $ & $ b_{28} $ &
$ b_{30} $ & $ b_{32} $ & $ b_{34} $ & $ b_{36} $
\\\hline\hline
$ 2$ & 1 & 1 & 1 & 1 & 1 & 1 & & & & & & & & & & & & &
\\\hline
$ 3$ & 1 & 1 & 3 & 4 & 7 & 8 & 10 & 8 & 7 & 4 &3 & 1 & 1 & & & & & &
\\\hline
$ 5$ & 1 & 1 & 3 & 4 & 8 &11 & 18 & 24 & 35 & 45 & 61 & 74 & 93
&106 & 122 & 128 & 134 & 128 & 122
\\ \hline
$ 7$ & 1 & 1 & 3 & 4 & 8 &11 & 19 & 26 & 40 & 54 &77 & 100 & 134 &165 &205 &
242 & 289 & 334 & 400
\\ \hline
$ 9$ & 1 & 1 & 3 & 4 & 8 &11 & 19 & 26 & 41 & 56 &82 &110 & 154 &202 &273 &
352 & 461 & 595 & 750
\\ \hline
\end{tabular}
\vskip 1 cm
See also \cite{EPS}, for the computation of the Betti numbers of
$\mathcal M_{3,3,2}$.
\vskip 5 mm
By this table, it seems that, for any $i\ge 0$ and $n>>0$,
the value of $b_i(n)$ is costant. In particular we have:
\begin{prop}
$b_2(n)=h^{1,1}(\mathcal M_{n,n,2})=1$ for any odd $n$.
\end{prop}
\begin{proof} By prop. \ref{ni}, it follows that $n_i(A_\omega^i)\ge
3$ for any $A_\omega^i$ defined as in (\ref{secondo.caso}).
Thus, by theorem \ref{BB}, the points represented by the matrices
$A_\omega^i$, do not give any contribute to $b_2(n)$.
Moreover it is not difficult to see that, if $n>3$, the only matrix
$A_{I,J}$, as in (\ref{primo.caso}), such that $n_i(A_{I,J})=1$ is given by
$I=(n-t,n-t+1,\dots,n)$ and $J=(n-t-2,n-t+1,n-t+2,\dots,n)$, where $t=
\frac{n+1} 2$.
\end{proof}
\vskip 5 mm
\begin{remark}
It would be interesting to have a description of the Chow rings of
$\mathcal M_{n,m,2}$: in \cite{Sc}, the author studies the Chow ring of
the Hilbert compactification $\mathcal H_3$ of the moduli space of the
twisted cubics in $\PP^3$.
\end{remark}
\vskip 2 cm
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}}
| 55,741
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The Problem of Legend And History
“This site also demonstrates one of the great dangers of archeology, not to life and limb, although that does sometimes take place, I’m talking about folklore…” – Indiana Jones, Raiders of the Lost Ark
The most problematic aspect of research into tales such as Oak Island involves the level of historicity, or historical authenticity, that can be deduced from available sources. If the existence of an event, person, or place is unsupported from a historian’s point of view – i.e. little or no reliable evidence can be found in the historical record to confirm whether an event occurred, it may be considered ahistorical.
It should be noted that, while laypersons use the terms myth, legend, and folktale almost interchangeably, folklorists consider each a specific subtype with identifiable characteristics. The Oak Island tale falls most closely into the legend category, which is identified as follows:
Legends are prose narratives that, like myths, are regarded as having happened in some historic or remembered time by their narrators and the audience.
· legends are set in a less remote period than myth, when the world was something like we know it today
· they tend to be more secular than sacred (though there are many legends about religious figures like saints)-their principal characters are human
· legendary topics include migrations, wars and victories, tales about past heroes, succession in dynasty or family, and so forth
· they are the verbal counterpart of written history, but also contain unverifiable elements like buried treasure, fairies, ghosts, saints, and other topics[1]
It must also be remembered that none of these terms are in any way pejorative or insulting. While someone may use the phrase “oh, that’s just a myth” conversationally when making a disparaging remark about a story, the folklorist makes no such judgement. It is not necessary for a legend or myth to be false, and indeed many such tales contain kernels of fact and verifiable detail. This is what makes legends believable – the listener identifies known locations, people, or events that lend credence to the story. But as mentioned above, a legend nearly always contains unverifiable and frequently fantastic elements. The Oak Island story also contains elements of folklore, such as the claims that “strange lights and fires” gave the island a reputation of being haunted, and that men who rowed to investigate such sightings failed to return.
The story also contains kernels of fact, such as the “artificial beach” (a real and interesting feature that deserves proper study by qualified industrial archaeologists) and many of the documented events occurring after roughly 1865. It is the earlier material (from 1795 to 1860) that is problematic and currently ahistorical, as will be demonstrated below. As this is the core of the legend upon which all later elements depend, the whole premise of the treasure hunt is placed on shaky ground.
The first problem is that no primary sources – contemporary, first-hand evidence such as letters, plans, sketches, journals, or even news articles – have been discovered that describe the initial events said to have occurred prior to 1860. As will be discussed in subsequent chapters, the first evidence of a treasure hunt on the island does not emerge until 1849 – a single document involving the grant of a treasure hunting license. Detailed accounts of events prior to 1860 were not published until 1861-63. This is disturbing, since an event as unusual as the discovery of a deeply excavated shaft with a mysteriously “inscribed” stone at the bottom and wooden platforms every ten feet should have found its way into news articles or other media soon after the first major excavation allegedly occurred circa 1804-05. Its failure to make such an appearance is not damning on its own, but it is unusual.
The delay in publication also presents an additional problem. A significant time lag between an event and the creation of a written account describing it introduces the need for caution, since observers’ memories are certain to change over time. Details are lost or jumbled, others are added as the tale is passed from one person to another. First person, eyewitness accounts are just as likely to become confused over time; only if multiple accounts containing similar details are available (that hopefully match the physical evidence) can these be trusted. Even if the event did occur in some form, an account written decades later is certain to contain fabrications, errors, omissions, and other flaws.
The probability of invented evidence being introduced into the Oak Island tale is high for other reasons. Many authors, especially those who produced early accounts of the treasure hunt, had a vested interest in preserving and perpetuating the tale. It is also likely one or more men invented the story out of whole cloth in an effort to hoax or swindle unsuspecting investors, since treasure related hoaxes were very popular in the early 19th century.
The subject matter itself – a supposedly vast buried treasure – is one that invariably involves wild claims and invented details unsupported by objective evidence. It also involves romanticized notions of hidden wealth, secrecy, and discovery that lend the topic to even more invention of detail. Fantastic tales, whether related to treasure, the supernatural, or other events, must always be treated with suspicion by the historian unless a great deal of confirmatory evidence is available. Large portions of the Oak Island tale are also peculiar or unique – in particular, the claim that an extensive set of excavations and flood tunnels was constructed there, and that no other site exhibiting similar features is known to exist elsewhere.
Additionally, many features and artifacts said to have been found on the island – the inscribed stone, pieces of chain recovered from the depths, bits of wood, or the so-called “Spanish” shoe, can be more readily explained by invention, misinterpretation, or mistaken identification. The men who are said to have found these and other objects had fixed expectations regarding the island and its history: they were psychologically predisposed to associate items found in the vicinity of the excavation (and, indeed, across the whole island) with the legendary treasure. Often there was no reason for such associations to be made, and in other cases much more prosaic explanations existed for the presence of certain objects.
For instance, the chain and other artifacts said to have been found buried in the shaft were almost certainly debris that fell into the depths during one of many recorded collapses or floods caused by earlier excavation attempts. Expedition after expedition sank dozens of shafts and horizontal connecting tunnels, largely unrecorded, during the mid 19th century; many of these collapsed or were filled with debris after they were abandoned. Later excavations have since recovered bits of debris from earlier attempts and mistakenly claimed these represented evidence of the “original” excavation. Finds made by these men provided confirmation of preconceived beliefs, justifying their emotional and monetary investment in the validity of the treasure tale.
Taylor cites another historian, Whitney R. Cross, to describe those personality characteristics necessary to early Yankees’ emotional investment in treasure related tales.
[T]hey were credulous in a particular way: they believed only upon evidence. Their observation, to be sure, was often inaccurate and usually incomplete, but when they arrived at a conclusion by presumably foolproof processes their adherence to it was positively fanatic.
Cross’ description, it should be said, is also extremely accurate when describing subsequent generations of treasure hunters even to the present day. Taylor also supplies a description of early treasure hunting that exactly parallels the evolution of the Oak Island tale: “[p]ersistent failure and insistent belief progressively promoted evermore complex techniques and tools in the search for treasure. Unwilling to surrender their treasure beliefs, seekers concluded that they needed more sophisticated methods. They remained confident that, by trial and error, they would ultimately obtain the right combination of conductor, equipment, time, magic circle, and ritual.” [2] It is only necessary to remove references to magic circles when describing modern day treasure hunters, though they continue to rely on supernatural means in order to determine the most likely place to dig.
The Oak Island story is analogous to many other legends involving buried treasure, and such tales are prone to cross-pollination of story elements. Elements of the Money Pit story that may be found in other tales of buried treasure include those described below.
The Marker
The pulley in the tree, the “stone triangle” said to point to the location of the Pit, and the “inscribed stone” allegedly found at the ninety foot level fulfill the role of the ” ’X’ marks the spot” motif found in other tales of treasure.
We also find mention of “marks” denoting treasure sites in period literature. For instance, according to Taylor “a 1729 Philadelphia newspaper essay by Benjamin Franklin and Joseph Breitnal described local treasure seeking’s extent: […] They wander thro’ the Woods and Bushes by Day to discover the Marks and Signs; at Midnight they repair to the hopeful spot with Spades and Pickaxes.” [3]
Mysterious Location
Later versions of the Money Pit story talk of lights and fires on the island, and the disappearance of local men who attempted to investigate these events. These elements do not appear in early versions of the story, thus providing yet another example of the amount of alteration and fabrication that occurred over time. This represents an attempt to justify the initial belief in the presence of treasure on the island, since many pirate tales (both historically accurate and fictional) involve men landing in remote locations to bury their gold under cover of darkness.
Repeated, failed recovery attempts
This is especially critical, since the tradition of treasure hunting makes especially strong use of imagery involving recovery efforts that fail at the last minute due to some error on the part of the excavators. This imagery is used repeatedly in the case of the Money Pit legend, where numerous groups were said to have been “just that close” to recovering the treasure before it was snatched away.
Famous associations
In the early days of the Money Pit legend, the famous pirate Kidd was considered the primary architect of the treasure shaft. Later, once it became known his travels did not include this area of Nova Scotia and his treasure hoard was small, his name disappeared from the tale and was replaced with the generic term “pirate.” Later writers introduced the Knights Templar, Francis Bacon, and other famous names into the story, frequently with no supporting evidence whatsoever.
Traps and impediments
The flooding system said to lie beneath the island represents the primary impediment, aside from the sheer depth of the shaft itself. Water often represents a barrier to treasure recovery, as may be seen in tales of half-flooded caves that can only be reached at certain times of the day or month or stories in which an almost-recovered chest sinks back into quicksand or mud at the last moment.
Involvement of Children
While modern versions of the Oak Island tale use the imagery of children (“boys on an adventure” in many cases) wandering the island and stumbling across the tree and chain, the original story used no such motif; the men said to have found the Money Pit site were adults who owned land on the island. Other treasure-related tales tell of children finding some tell-tale mark that was missed by adults, and the motif of the clever child who locates or holds the key to recovery of an item unobtainable by adults is very common in pirate treasure tales. For example:. [italics mine] [2]
In the above case, the children’s discovery of a “knotted rope” was apparently ignored by adults, who only later realized its significance after the pirate treasure was recovered and spirited away under cover of darkness.
Numerous cases have been discovered in which authors introduced a similar story element into fictional works based on the Money Pit tale, which may explain how the original tale of adults finding the site was transformed into an event involving children. James DeMille, a Canadian author and historian whose involvement in the Oak Island tale is not yet fully understood, made use of this motif in his book The Treasure of the Seas (1872), as well as Old Garth: A Story of Sicily and possibly in other contexts as yet undiscovered.
Fig. 1: frontispiece from DeMille’s The Treasure of the Seas, showing a young boy chatting with a sailor (courtesy canadiana.org)
Breaking a spell
Many treasure recovery folktales include admonitions involving specific activities during the excavation effort. Often a prohibition against speaking or crossing a magic circle drawn around the site is broken, causing the nearly-recovered treasure to “sink out of sight” or a guardian spirit to awaken. The Money Pit tale includes an element involving excavators driving an iron bar into the bottom of the Pit every evening; the last time this rite is performed, they strike what they believe is a treasure chest several feet below and tap on it repeatedly using the bar. The next day, the pit is filled with water. Various authors have asserted that “those taps on the chest loosened some stopper” and caused the water trap to be sprung – this action caused the “spell” to be broken and the treasure to be lost by invoking a “guardian” (in the guise of the flood system) to protect the treasure. Indeed, the term “tapping” or “rapping” is often associated with the presence of spirits or poltergeists in Spiritualist and other supernatural lore.
[1] Fair, Susan, Lecture notes for English 248A, spring semester, 2002
[2] Taylor, Alan, The Early Republic’s Supernatural Economy: Treasure Seeking in the American Northeast, 1780-1830. American Quarterly, Vol 38, No 1. Spring 1986, p. 15.
[3] Taylor, Alan, Treasure Seeking in the American Northeast, 1780-1830 p. 17.
[4] Skinner, Charles, Myths and Legends of Our Own land vol 9
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Benjamin Jim 1 Posted December 24, 2014 Share Posted December 24, 2014 Hi , Just downloaded Evernote. I live in Germany and want the english version. Tried the download from the english website but it still operates in German.Is it at all possible? Hope someone can help out, Kr, Benjamin Jim I use Mac. Link to comment
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A new frontier of collaboration - humans and AI
A new wave of AI is being explored currently, which adds humans to the equation as positive influencers of events rather than passive recipients, thereby creating new possibilities for the future.
In recent popular imagination, there have been only two common (mis)understandings of AI. It is either the great saviour, which will automate all the jobs that humans shouldn’t be doing in the first place. Or, it is a sign of destruction, full of foreboding, entering into the ecosystem of humans with the aim to displace them from it. These understandings are rooted in and portrayed through various fictitious mediums; those who are within the field of AI and ML understand the reality is far from either of two situations - with the recent progress in GTP-3 being the only remarkable thing to straddle both imaginations, and rather ineffectively at that.
With that said, however, there is a new wave of AI being explored currently, which falls in neither of the two visions. Instead of being centered completely around AI, this vision imagines AI as a collaborative effort with humans, and therefore forces us to rethink the inherent possibilities of such a partnership. By adding humans to the equation as positive influencers of events rather than passive recipients, this partnership makes us reimagine the potential of a future that is neither dystopian nor utopian.
This leads us to the question: what are some ways in which this partnership has already been explored? And how is it likely to be explored in the future?
Exploring the Possibility of a Human-AI Partnership
As far back as 2017, Daniela Ras, in a Keynote speech at MIT, spoke of a project in which AI helps vision-impaired commuters navigate through self-driving cars. Her brave quote was this: “I believe people and machines should not be competitors, they should be collaborators.” And it’s not just at MIT that this noble idea was explored - it has also been explored in another part of the world, at Amsterdam, where AI has been used in the process of manufacturing steel, helping identify faults in the various stages of the manufacturing process. These faults then get verified by human experts, with the overall setup constituting another scenario of successful collaboration between humans and AI.
One of the most remarkable instances of AI outsmarting humans had been in the field of chess. When the IBM Computer Deep Blue beat Garry Kasparov in 1997, it ruffled the feathers of all in both the emerging computing industry as well as those in chess. Two decades on, there is a new form of chess - Centaur Chess - which teams up a human and AI against humans or AI or a combination of the two. This new revolutionary form of chess teams up the emotive capacities of humans with the data-crunching capacity of computers, and the results are explosive.
Understanding the Human-AI Relationship in Depth
As the human-ai relationship is explored in different areas through different collaborations, more and more data will be required on how to fit this into the structure of work. For instance, AI is assisting and collaborating with a human workforce in hundreds of different fields right now - in some places as an extension of the human body, while in other places as expert fraud detection mechanisms.
Much work is still to be done in this domain - and the future is anybody’s bet at this point. To further this understanding, Prof. Phanish Puranam of INSEAD is conducting an online survey here, which seeks to illuminate various aspects of a developing consciousness towards AI. This includes the preferred collaboration style when it comes to working with AI, as well as level of trust in multiple collaboration configurations with AI, and aversion and/or acceptance of AI.
On completion, a one page downloadable “cheat sheet” of curated content (links to articles/videos) that will bring readers up to the cutting edge of thinking about how AI is affecting organizations.
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