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Saif, Mashal
Position
Assistant Professor of Religion
Office: 214 Hardin
Education
Ph.D., Duke University's Graduate Program in Religion.
Dr. Saif's research interests include Islam in contemporary South Asia; the trans-temporal dynamics between medieval and modern Islamic discourses; contemporary Muslim political theology; the intersection of religious studies and postcolonial theory; and the anthropology of the state.
Dr. Saif's first monograph The 'Ulama in Contemporary Pakistan: Contesting and Cultivating an Islamic State is forthcoming with Cambridge University Press. Her co-edited volume State and Subject Formation in South Asia is under contract with Oxford University Press. She is currently working on her second monograph, tentatively titled, Traditional Islam and Modernity in Pakistan: An Intimate Account.
Dr. Saif's scholarly articles have appeared in a variety of journals: Modern Asian Studies, The Journal of Shi'a Islamic Studies, Islamic Studies, Fieldwork in Religion, Annali and Thinking About Religion. She also authored several book chapters, book reviews and encyclopedia articles.
At Clemson University, Saif has received multiple research awards this year (2020), including a CU SEED Grant and a Lightsey Fellowship. And, this year, within the College of Architecture, Arts and Humanities, she was the sole recipient of the Dean's Excellence in Research Award and was the CAAH nominee for the University's Junior Researcher of the Year. | 314,845 |
Attend with respect to great majority " certificate of qualification of English oral interpretation " examinee, the " of obstacle of the biggest " that obtains this letter is certificate exam the oral interpretation part of the 2nd phase, the fierce tiger that its " eats off the examinee number of " to want over scene in relief hillock to go up counts hundredfold! Although oneself are inferior to fierce Song Mengwei, but close watch sympathizes with the students of a gathering of things or people of a gathering of things or people that join " of " tiger's mouth-jaws of death to be being swallowed by " of obstacle of " of oral interpretation exam, wish the English that him part comes more than 20 years and experience of oral interpretation teaching, to arrowy annals Yu Pan is ascended " certificate of oral interpretation qualification " the oral interpretation lover of " of this " high mountain people compare notes chats, discuss a few path that had entered " of obstacle of certificate exam " and method together.
Generally speaking, the English oral interpretation that should adopt high administrative levels takes an exam, the person that consult must have 3 requirements:
1, basic training of sound Chinese and English language (Linguistic Proficiency Both In Chinese And English Languages) ;
2, wide intellectual range (Encyclopedic Knowledge) ;
3, master oral interpretation fundamental skill (Mastery Of Interpretation Techniques/skills) .
Say first the a bitth, namely basic skill of Chinese and English language. In oral interpretation domain, basic skill of Chinese and English language points to namely master adroitly and apply Chinese and English two kinds of languages. Specific and character, capacity of organization of the audition of the person that oral interpretation consults, vocabulary, sentence and oral expression ability should reach quite high level. In education and exam, I discover most examinee to be deficient in badly in the ability of these a few respects. Some people listen to simple stuff to still can be dealt with, but audition material is nominal a bit more difficult, grow when English sentence especially a few, or English former a sentential structure is a few more complex, they two bemused, apprentice sighs but. Inside reason, it is audition level not quite tall.
A paragraph of word listens come down, get the fragmentary fragment information such as word, phrase only, is not the complete meaning of paragraphs of whole word. As to the vocabulary, I think one of the student general situation is, everybody " 4, pass a barrier of 6 " of class " exam after behead general " , it is certain to read a vocabulary to have accumulate, also understand the basic meaning of these vocabularies, but because be used rarely, everybody is not to the head medium vocabulary does not know firstly only secondly, know to apply them adroitly far from namely. Often teach " in " later, students the ground like " of ability " as if wakening from a dream store " of activation of a string of in him head " that string together a word rises, if realize the ground somewhat,nod again and again. The main reason that creates this kind of situation grows period not to use English namely, the vocabulary that as a result carries on the back at past work laboriously not close rise. Accordingly, should raise level of English oral interpretation or annals pricks the person that English oral interpretation takes an exam at rushing, be necessary constant ground, again and again reviews original vocabulary, make oneself read a vocabulary to be changed into audition vocabulary and colloquial vocabulary. If can make oneself commonly used vocabulary maintains,be in 8000 to 10000 between, so, be able to achieve success one way or another can be accomplished when oral interpretation, do a job with skill and ease.
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In an interview with the AP, Stanton said he knew he wanted to juxtapose retro music with this futuristic setting, but discovered "a perfect fit" to his narrative when he stumbled upon the "Hello, Dolly!" repertoire and the lyric "out there."
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And in those first images of planets and stars, "you're meeting Wall-E's dreams before you ever get to meet Wall-E. And I love that. That was just so poetic to me," Stanton said. retreats to his evening hide-out, where he uses an iPod to watch a videotape of.."
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\begin{document}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}
\begin{abstract}
This paper considers the problem of implementing a previously proposed distributed direct coupling quantum observer for a closed linear quantum system. By modifying the form of the previously proposed observer, the paper proposes a possible experimental implementation of the observer plant system using a non-degenerate parametric amplifier and a chain of optical cavities which are coupled together via optical interconnections. It is shown that the distributed observer converges to a consensus in a time averaged sense in which an output of each element of the observer estimates the specified output of the quantum plant.
\end{abstract}
\section{Introduction} \label{sec:intro}
In this paper we build on the results of \cite{PET14Ca} by providing a possible experimental implementation of a direct coupled distributed quantum observer. A number of papers have recently considered the problem of constructing a coherent quantum observer for a quantum system; e.g., see \cite{VP9a,MEPUJ1a,MJP1}. In the coherent quantum observer problem, a quantum plant is coupled to a quantum observer which is also a quantum system. The quantum observer is constructed to be a physically realizable quantum system so that the system variables of the quantum observer converge in some suitable sense to the system variables of the quantum plant. The papers \cite{PET14Aa,PET14Ba,PET14Ca,PET14Da} considered the problem of constructing a direct coupling quantum observer for a given quantum system.
In the papers \cite{MJP1,VP9a,PET14Aa,PET14Ca}, the quantum plant under consideration is a linear quantum system. In recent years, there has been considerable interest in the modeling and feedback control of linear quantum systems; e.g., see \cite{JNP1,NJP1,ShP5}.
Such linear quantum systems commonly arise in the area of quantum optics; e.g., see
\cite{GZ00,BR04}. In addition, the papers \cite{PeHun1a,PeHun2a} have considered the problem of providing a possible experimental implementation of the direct coupled observer described in \cite{PET14Aa} for the case in which the quantum plant is a single quantum harmonic oscillator and the quantum observer is a single quantum harmonic oscillator. For this case, \cite{PeHun1a,PeHun2a} show that a possible experimental implementation of the augmented quantum plant and quantum observer system may be constructed using a non-degenerate parametric amplifier (NDPA) which is coupled to a beamsplitter by suitable choice of the amplifier and beamsplitter parameters; e.g., see \cite{BR04} for a description of an NDPA. In this paper, we consider the issue of whether a similar experimental implementation may be provided for the distributed direct coupled quantum observer proposed in \cite{PET14Ca}.
The paper \cite{PET14Ca} proposes a direct coupled distributed quantum observer which is constructed via the direct connection of many quantum harmonic oscillators in a chain as illustrated in Figure \ref{F0}. It is shown that this quantum network can be constructed so that each output of the direct coupled distributed quantum observer converges to the plant output of interest in a time averaged sense. This is a form of time averaged quantum consensus for the quantum networks under consideration. However, the experimental implementation approach of \cite{PeHun1a,PeHun2a} cannot be extended in a straightforward way to the direct coupled distributed quantum observer \cite{PET14Ca}. This is because it is not feasible to extend the NDPA used in
\cite{PeHun1a,PeHun2a} to allow for the multiple direct couplings to the multiple observer elements required in the theory of \cite{PET14Ca}. Hence, in this paper, we modify the theory of \cite{PET14Ca} to develop a new direct coupled distributed observer in which there is direct coupling only between the plant and the first element of the observer. All of the other couplings between the different elements of the observer are via optical field couplings. This is illustrated in Figure \ref{F1}. Also, all of the elements of the observer except for the first one are implemented as passive optical cavities. The only active element in the augmented plant observer system is a single NDPA used to implement the plant and first observer element. These features mean that the proposed direct coupling observer is much easier to implement experimentally that the observer which was proposed in \cite{PET14Ca}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{F0.eps}
\end{center}
\caption{Distributed Quantum Observer of \cite{PET14Ca}.}
\label{F0}
\end{figure}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=16cm]{F1.eps}
\end{center}
\caption{Distributed Quantum Observer Proposed in This Paper.}
\label{F1}
\end{figure*}
We establish that the distributed quantum observer proposed in this paper has very similar properties to the distributed quantum observer proposed in \cite{PET14Ca} in that each output of the distributed observer converges to the plant output of interest in a time averaged sense. However, an important difference between the observer proposed in \cite{PET14Ca} and the observer proposed in this paper is that in \cite{PET14Ca} the output for each observer element corresponded to the same quadrature whereas in this paper, different quadratures are used to define the outputs with a $90^\circ$ phase rotation as we move from observer element to element along the chain of observers.
\section{Quantum Linear Systems}
In the distributed quantum observer problem under consideration, both the quantum plant and the distributed quantum observer are linear quantum systems; see also \cite{JNP1,GJ09,ZJ11}.
The quantum mechanical behavior of a linear quantum system is described in terms of the system \emph{observables} which are self-adjoint operators on an underlying infinite dimensional complex Hilbert space $\mathfrak{H}$. The commutator of two scalar operators $x$ and $y$ on ${\mathfrak{H}}$ is defined as $[x, y] = xy - yx$.~Also, for a vector of operators $x$ on ${\mathfrak H}$, the commutator of ${x}$ and a scalar operator $y$ on ${\mathfrak{H}}$ is the vector of operators $[{x},y] = {x} y - y {x}$, and the commutator of ${x}$ and its adjoint ${x}^\dagger$ is the matrix of operators
\[ [{x},{x}^\dagger] \triangleq {x} {x}^\dagger - ({x}^\# {x}^T)^T, \]
where ${x}^\# \triangleq (x_1^\ast\; x_2^\ast \;\cdots\; x_n^\ast )^T$ and $^\ast$ denotes the operator adjoint.
The dynamics of the closed linear quantum systems under consideration are described by non-commutative differential equations of the form
\begin{eqnarray}
\dot x(t) &=& Ax(t); \quad x(0)=x_0
\label{quantum_system}
\end{eqnarray}
where $A$ is a real matrix in $\rbb^{n
\times n}$, and $ x(t) = [\begin{array}{ccc} x_1(t) & \ldots &
x_n(t)
\end{array}]\trp$ is a vector of system observables; e.g., see \cite{JNP1}. Here $n$ is assumed to be an even number and $\frac{n}{2}$ is the number of modes in the quantum system.
The initial system variables $x(0)=x_0$
are assumed to satisfy the {\em commutation relations}
\begin{equation}
[x_j(0), x_k(0) ] = 2 \imath \Theta_{jk}, \ \ j,k = 1, \ldots, n,
\label{x-ccr}
\end{equation}
where $\Theta$ is a real skew-symmetric matrix with components
$\Theta_{jk}$. In the case of a
single quantum harmonic oscillator, we will choose $x=(x_1, x_2)^T$ where
$x_1=q$ is the position operator, and $x_2=p$ is the momentum
operator. The
commutation relations are $[q,p]=2 i$.
In general, the matrix $\Theta$ is assumed to be of the form
\begin{equation}
\label{Theta}
\Theta=\diag(J,J,\ldots,J)
\end{equation}
where $J$ denotes the real skew-symmetric $2\times 2$ matrix
$$
J= \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0
\end{array} \right].$$
The system dynamics (\ref{quantum_system}) are determined by the system Hamiltonian
which is a self-adjoint operator on the underlying Hilbert space $\mathfrak{H}$. For the linear quantum systems under consideration, the system Hamiltonian will be a
quadratic form
$\mathcal{H}=\half x(0)\trp R x(0)$, where $R$ is a real symmetric matrix. Then, the corresponding matrix $A$ in
(\ref{quantum_system}) is given by
\begin{equation}
A=2\Theta R \label{eq_coef_cond_A}.
\end{equation}
where $\Theta$ is defined as in (\ref{Theta});
e.g., see \cite{JNP1}.
In this case, the system variables $x(t)$
will satisfy the {\em commutation relations} at all times:
\begin{equation}
\label{CCR}
[x(t),x(t)^T]=2\imath \Theta \ \mbox{for all } t\geq 0.
\end{equation}
That is, the system will be \emph{physically realizable}; e.g., see \cite{JNP1}.
\begin{remark}
\label{R1}
Note that that the Hamiltonian $\mathcal{H}$ is preserved in time for the system (\ref{quantum_system}). Indeed,
$ \mathcal{\dot H} = \frac{1}{2}\dot{x}^TRx+\frac{1}{2}x^TR\dot{x} = -x^TR\Theta R x + x^TR\Theta R x = 0$ since $R$ is symmetric and $\Theta$ is skew-symmetric.
\end{remark}
\section{Direct Coupling Distributed Coherent Quantum Observers}
In our proposed direct coupling coherent quantum observer, the quantum plant is a single quantum harmonic oscillator which is a linear quantum system of the form (\ref{quantum_system}) described by the non-commutative differential equation
\begin{eqnarray}
\dot x_p(t)
&=& A_px_p(t); \quad x_p(0)=x_{0p}; \nonumber \\
z_p(t) &=& C_px_p(t)
\label{plant}
\end{eqnarray}
where $z_p(t)$ denotes the vector of system variables to be estimated by the observer and $ A_p \in \rbb^{2
\times 2}$, $C_p\in \rbb^{1 \times 2}$.
It is assumed that this quantum plant corresponds to a plant Hamiltonian
$\mathcal{H}_p=\half x_p(0)\trp R_p x_p(0)$. Here $x_p = \left[\begin{array}{l}q_p\\p_p\end{array}\right]$ where
$q_p$ is the plant position operator and $p_p$ is the plant momentum operator. As in \cite{PET14Ca}, in the sequel we will assume that $A_p= 0$.
We now describe the linear quantum system of the form (\ref{quantum_system}) which will correspond to the distributed quantum observer; see also \cite{JNP1,GJ09,ZJ11}.
This system is described by a non-commutative differential equation of the form
\begin{eqnarray}
\dot x_o(t) &=& A_ox_o(t);\quad x_o(0)=x_{0o};\nonumber \\
z_o(t) &=& C_ox_o(t)
\label{observer}
\end{eqnarray}
where the observer output $z_o(t)$ is the distributed observer estimate vector and $ A_o \in \rbb^{n_o
\times n_o}$, $C_o\in \rbb^{\frac{n_o}{2} \times n_o}$. Also, $x_o(t)$ is a vector of self-adjoint
non-commutative system variables; e.g., see \cite{JNP1}. We assume the distributed observer order $n_o$ is an even number with $N=\frac{n_o}{2}$ being the number of elements in the distributed quantum observer. We also assume that the plant variables commute with the observer variables.
We will assume that the distributed quantum observer has a chain structure and is coupled to the quantum plant as shown in Figure \ref{F1}.
Furthermore, we write
\[
z_o = \left[\begin{array}{l}z_{o1}\\z_{o2}\\\vdots\\z_{oN}\end{array}\right]
\]
where
\[
z_{oi} = C_{oi}x_{oi} \mbox{ for }i=1,2,\ldots,N.
\]
Note that $C_{oi} \in \rbb^{1 \times 2}$.
The augmented quantum linear system consisting of the quantum plant (\ref{plant}) and the distributed quantum observer (\ref{observer}) is then a quantum system of the form (\ref{quantum_system}) described by equations of the form
where
\begin{eqnarray}
\left[\begin{array}{l}\dot x_p(t)\\\dot x_{o1}(t)\\\dot x_{o2}(t)\\\vdots\\\dot x_{oN}(t)\end{array}\right] &=&
A_a\left[\begin{array}{l} x_p(t)\\ x_{o1}(t)\\x_{o2}(t)\\\vdots\\x_{oN}(t)\end{array}\right];\nonumber \\
z_p(t) &=& C_px_p(t);\nonumber \\
z_o(t) &=& C_ox_o(t)
\label{augmented_system}
\end{eqnarray}
where
\[
C_o = \left[\begin{array}{llll}C_{o1} & & &\\
& C_{o2}& 0 &\\
& 0 & \ddots & \\
&&& C_{oN}
\end{array}\right].
\]
We now formally define the notion of a direct coupled linear quantum observer.
\begin{definition}
\label{D1}
The {\em distributed linear quantum observer} (\ref{observer}) is said to achieve time-averaged consensus convergence for the quantum plant (\ref{plant}) if the corresponding augmented linear quantum system (\ref{augmented_system}) is such that
\begin{equation}
\label{average_convergence}
\lim_{T \rightarrow \infty} \frac{1}{T}\int_{0}^{T}(\left[\begin{array}{l}1\\1\\\vdots\\1\end{array}\right]z_p(t)
- z_o(t))dt = 0.
\end{equation}
\end{definition}
\section{Implementation of a Distributed Quantum Observer} \label{sec:intro}
We will consider a distributed quantum observer which has a chain structure and is coupled to the quantum plant as shown in Figure \ref{F1}.
In this distributed quantum observer, there a direct coupling between the quantum plant and the first quantum observer. This direct coupling is determined by a coupling Hamiltonian which defines the coupling between the quantum plant and the first element of the distributed quantum observer:
\begin{equation}
\label{Rc}
\mathcal{H}_c = x_{p}(0)\trp R_{c} x_{o1}(0).
\end{equation}
However, in contrast to \cite{PET14Ca}, there is field coupling between the first quantum observer and all other quantum observers in the chain of observers. The motivation for this structure is that it would be much easier to implement experimentally than the structure proposed in \cite{PET14Ca}. Indeed, the subsystem consisting of the quantum plant and the first quantum observer can be implemented using an NDPA and a beamsplitter in a similar way to that described in \cite{PeHun1a,PeHun2a}; see also \cite{BR04} for further details on NDPAs and beamsplitters. This is illustrated in Figure \ref{F2}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{F2.eps}
\end{center}
\caption{NDPA coupled to a beamsplitter representing the quantum plant and first quantum observer.}
\label{F2}
\end{figure}
Also, the remaining quantum observers in the distributed quantum observer are implemented as simple cavities as shown in Figure \ref{F3}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{F3.eps}
\end{center}
\caption{Optical cavity implementation of the remaining quantum observers in the distributed quantum observer.}
\label{F3}
\end{figure}
The proposed quantum optical implementation of a distributed quantum observer is simpler than that of \cite{PET14Ca}. However, its dynamics are somewhat different than those of the distributed quantum observer proposed in \cite{PET14Ca}. We now proceed to analyze these dynamics. Indeed, using the results of \cite{PeHun2a}, we can write down quantum stochastic differential equations (QSDEs) describing the plant-first observer system shown in Figure \ref{F2}:
\begin{eqnarray}
\label{augmented1}
d x_p &=& 2 J \alpha\beta^T x_{o1} dt;\nonumber \\
d x_{o1} &=& 2\omega_1 J x_{o1} dt -\frac{1}{2}\kappa_{1b} x_{o1} dt + 2J\beta\alpha^T x_p dt\nonumber \\
&& - \sqrt{\kappa_{1b}}dw_{1b}; \nonumber \\
dy_{1a} &=& \sqrt{\kappa_{1b}} x_{o1} dt + dw_{1b}
\end{eqnarray}
where $x_p = \left[\begin{array}{c} q_p \\ p_p \end{array}\right]$ is the vector of position and momentum operators for the quantum plant and $x_{o1} = \left[\begin{array}{c} q_1 \\ p_1 \end{array}\right]$ is the vector of position and momentum operators for the first quantum observer.
Here, $\alpha \in \rbb^2$, $\beta \in \rbb^2$ and $\kappa_{1b} > 0$ are parameters which depend on the parameters of the beamsplitter and the NDPA. The parameters $\alpha$ and $\beta$ define the coupling Hamiltonian matrix defined in (\ref{Rc}) as follows:
\begin{equation}
\label{Rc_def}
R_c = \alpha\beta^T.
\end{equation}
In addition, the parameters of the beamsplitter and the NDPA need to be chosen as described in
\cite{PeHun1a,PeHun2a} in order to obtain QSDEs of the required form (\ref{augmented1}).
The QSDEs describing the $ith$ quantum observer for $i=2,3,\ldots,N-1$ are as follows:
\begin{eqnarray}
\label{cavity_i}
d x_{oi} &=& 2\omega_i J x_{oi} dt -\frac{\kappa_{ia}+\kappa_{ib}}{2} x_{oi} dt\nonumber \\
&&- \sqrt{\kappa_{ia}}dw_{ia}- \sqrt{\kappa_{ib}}dw_{ib}; \nonumber \\
dy_{ia} &=& \sqrt{\kappa_{ib}} x_{oi} dt + dw_{ib}; \nonumber \\
dy_{ib} &=& \sqrt{\kappa_{ia}} x_{oi} dt + dw_{ia}
\end{eqnarray}
where $x_{oi} = \left[\begin{array}{c} q_i \\ p_i \end{array}\right]$ is the vector of position and momentum operators for the $ith$ quantum observer; e.g., see \cite{BR04}. Here $\kappa_{ia} >0$ and $\kappa_{ib} > 0$ are parameters relating to the reflectivity of each of the partially reflecting mirrors which make up the cavity.
The QSDEs describing the $Nth$ quantum observer are as follows:
\begin{eqnarray}
\label{cavity_N}
d x_{oN} &=& 2\omega_N J x_{oN} dt -\frac{\kappa_{Na}}{2} x_{oN} dt - \sqrt{\kappa_{Na}}dw_{Na}; \nonumber \\
dy_{Nb} &=& \sqrt{\kappa_{Na}} x_{oN} dt + dw_{Na}
\end{eqnarray}
where $x_{oN} = \left[\begin{array}{c} q_N \\ p_N \end{array}\right]$ is the vector of position and momentum operators for the $Nth$ quantum observer. Here $\kappa_{Na} >0$ is a parameter relating to the reflectivity of the partially reflecting mirror in this cavity.
In addition to the above equations, we also have the following equations which describe the interconnections between the observers as in Figure \ref{F1}:
\begin{eqnarray}
\label{interconnection_i}
w_{(i+1)a} &=& - y_{ia}; \nonumber \\
w_{ib} &=& y_{(i+1)b}
\end{eqnarray}
for $i = 1,2,\ldots,N-1$.
In order to describe the augmented system consisting of the quantum plant and the quantum observer, we now combine equations (\ref{augmented1}), (\ref{cavity_i}), (\ref{cavity_N}) and (\ref{interconnection_i}). Indeed, starting with observer $N$, we have from (\ref{cavity_N}), (\ref{interconnection_i})
\[
dy_{Nb} = \sqrt{\kappa_{Na}} x_{oN} dt - dy_{(N-1)a}.
\]
But from (\ref{cavity_i}) with $i= N-1$,
\[
dy_{(N-1)a} = \sqrt{\kappa_{(N-1)b}} x_{o(N-1)} dt + dw_{(N-1)b}.
\]
Therefore,
\begin{eqnarray*}
dy_{Nb} &=& \sqrt{\kappa_{Na}} x_{oN} dt - \sqrt{\kappa_{(N-1)b}} x_{o(N-1)} dt - dw_{(N-1)b}\nonumber \\
&=& \sqrt{\kappa_{Na}} x_{oN} dt - \sqrt{\kappa_{(N-1)b}} x_{o(N-1)} dt - dy_{Nb}
\end{eqnarray*}
using (\ref{interconnection_i}). Hence,
\begin{equation}
\label{yNb}
dy_{Nb} = \frac{\sqrt{\kappa_{Na}}}{2} x_{oN} dt - \frac{\sqrt{\kappa_{(N-1)b}}}{2} x_{o(N-1)} dt.
\end{equation}
From this, it follows using (\ref{cavity_N}) that
\begin{eqnarray*}
dw_{Na} &=& -\sqrt{\kappa_{Na}} x_{oN} dt + dy_{Nb}\nonumber \\
&=& -\frac{\sqrt{\kappa_{Na}}}{2} x_{oN} dt - \frac{\sqrt{\kappa_{(N-1)b}}}{2} x_{o(N-1)} dt.
\end{eqnarray*}
Then, using (\ref{cavity_N}) we obtain the equation
\begin{equation}
\label{xN}
d x_{oN} = 2\omega_N J x_{oN} dt + \frac{\sqrt{\kappa_{(N-1)b}\kappa_{Na}}}{2} x_{o(N-1)} dt.
\end{equation}
We now consider observer $N-1$. Indeed, it follows from (\ref{cavity_i}) and (\ref{interconnection_i}) with $i = N-1$ that
\begin{eqnarray}
\label{xN-1a}
d x_{o(N-1)} &=& 2\omega_{N-1} J x_{o(N-1)} dt \nonumber \\
&&-\frac{\kappa_{(N-1)a}+\kappa_{(N-1)b}}{2} x_{o(N-1)} dt\nonumber \\
&&- \sqrt{\kappa_{(N-1)a}}dw_{(N-1)a}- \sqrt{\kappa_{(N-1)b}}dy_{Nb} \nonumber \\
&=& 2\omega_{N-1} J x_{o(N-1)} dt \nonumber \\
&& -\frac{\kappa_{(N-1)a}+\kappa_{(N-1)b}}{2} x_{o(N-1)} dt\nonumber \\
&&- \sqrt{\kappa_{(N-1)a}}dw_{(N-1)a}\nonumber\\
&&- \frac{\sqrt{\kappa_{Na}\kappa_{(N-1)b}}}{2} x_{oN} dt \nonumber \\
&&+ \frac{\kappa_{(N-1)b}}{2} x_{o(N-1)} dt \nonumber \\
&=& 2\omega_{N-1} J x_{o(N-1)} dt -\frac{\kappa_{(N-1)a}}{2} x_{o(N-1)} dt\nonumber \\
&&- \frac{\sqrt{\kappa_{Na}\kappa_{(N-1)b}}}{2} x_{oN} dt \nonumber \\
&&- \sqrt{\kappa_{(N-1)a}}dw_{(N-1)a}
\end{eqnarray}
using (\ref{yNb}). Now using (\ref{cavity_i}) and (\ref{interconnection_i}) with $i = N-2$, it follows that
\begin{eqnarray*}
dy_{(N-2)a} &=& \sqrt{\kappa_{(N-2)b}} x_{o(N-2)} dt + dw_{(N-2)b} \nonumber \\
&=& \sqrt{\kappa_{(N-2)b}} x_{o(N-2)} dt + dy_{(N-1)b} \nonumber \\
&=& \sqrt{\kappa_{(N-2)b}} x_{o(N-2)} dt + \sqrt{\kappa_{(N-1)a}} x_{o(N-1)} dt \nonumber \\
&&+ dw_{(N-1)a}
\end{eqnarray*}
using (\ref{cavity_i}) with $i = N-1$.
Hence using (\ref{interconnection_i}) with $i = N-2$, it follows that
\begin{eqnarray*}
dy_{(N-2)a} &=& \sqrt{\kappa_{(N-2)b}} x_{o(N-2)} dt + \sqrt{\kappa_{(N-1)a}} x_{o(N-1)} dt \nonumber \\
&&- dy_{(N-2)a}.
\end{eqnarray*}
Therefore
\[
dy_{(N-2)a} = \frac{\sqrt{\kappa_{(N-2)b}}}{2} x_{o(N-2)} dt + \frac{\sqrt{\kappa_{(N-1)a}}}{2} x_{o(N-1)} dt.
\]
Substituting this into (\ref{xN-1a}), we obtain
\begin{eqnarray}
\label{xN-1}
d x_{o(N-1)} &=& 2\omega_{N-1} J x_{o(N-1)} dt \nonumber \\
&&- \frac{\sqrt{\kappa_{(N-1)b}\kappa_{Na}}}{2} x_{oN} dt \nonumber \\
&&+ \frac{\sqrt{\kappa_{(N-2)b}\kappa_{(N-1)a}}}{2} x_{o(N-2)} dt.
\end{eqnarray}
Continuing this process, we obtain the following QSDEs for the variables $x_{oi}$:
\begin{eqnarray}
\label{xi}
d x_{oi} &=& 2\omega_i J x_{oi} dt \nonumber \\
&&- \frac{\sqrt{\kappa_{ib}\kappa_{(i+1)a}}}{2} x_{o(i+1)} dt \nonumber \\
&&+ \frac{\sqrt{\kappa_{(i-1)b}\kappa_{ia}}}{2} x_{o(i-1)} dt
\end{eqnarray}
for $i=2,3, \ldots, N-1$. Finally for $x_{o1}$, we obtain
\begin{eqnarray}
\label{x1}
d x_{o1} &=& 2\omega_1 J x_{o1} dt - \frac{\sqrt{\kappa_{1b}\kappa_{(2a}}}{2} x_{o2} dt \ + 2J\beta\alpha^T x_p dt.\nonumber \\
\end{eqnarray}
We now observe that the plant equation
\begin{eqnarray}
\label{xp}
d x_p &=& 2 J \alpha\beta^T x_{o1} dt\nonumber \\
\end{eqnarray}
implies that the quantity
\[
z_p = \alpha^T x_p
\]
satisfies
\[
d z_p = 2 \alpha^T J \alpha\beta^T x_{o1} dt = 0
\]
since $J$ is a skew-symmetric matrix. Therefore,
\begin{equation}
\label{zp_const}
z_p(t) = z_p(0) = z_p
\end{equation}
for all $t\geq 0$.
We now combine equations (\ref{x1}), (\ref{xi}), (\ref{xN}) and write them in vector-matrix form. Indeed, let
\[
x_o = \left[\begin{array}{c}x_{o1}\\x_{o2}\\\vdots\\x_{oN}\end{array}\right].
\]
Then, we can write
\[
\dot x_o = A_o x_o + B_o z_p
\]
where
{\small
\begin{eqnarray}
\label{Ao}
\lefteqn{A_o =}\nonumber \\
&&2\left[\begin{array}{rrrrrr} \omega_1 J & -\mu_2 I & & & & \\
\mu_2 I & \omega_2 J& -\mu_3 I & & 0 & \\
& \mu_3 I & \omega_3 J& -\mu_4 I && \\
& & \ddots & \ddots & \ddots & \\
0 & & & \mu_{N-1} I & \omega_{N-1} J & -\mu_N I\\
& & & & \mu_N I & \omega_N J
\end{array}\right], \nonumber \\
\lefteqn{B_o =}\nonumber \\
&& 2\left[\begin{array}{r}J \beta \\ 0 \\ \vdots \\ 0 \end{array}\right]
\end{eqnarray}}
and
\[
\mu_i = \frac{1}{4} \sqrt{\kappa_{(i-1)b}\kappa_{ia}} > 0
\]
for $i = 2, 3, \ldots , N$.
To construct a suitable distributed quantum observer, we will further assume that
\begin{eqnarray}
\label{alphabeta}
\beta &=& -\mu_1 \alpha,\nonumber \\
C_p&=& \alpha^T,
\end{eqnarray}
where $\mu_1 > 0$
and
{\small
\begin{eqnarray}
\label{Co}
\lefteqn{C_{o} =}\nonumber \\
&&\frac{1}{\|\alpha\|^2}\left[\begin{array}{llllll}\alpha^T &&&&&\\
& -J\alpha^T &&& 0 &\\
&& -\alpha^T &&&\\
&&& J\alpha^T && \\
&0&&& \ddots & \\
&&&&& (-J)^{N-1}\alpha^T
\end{array}\right].\nonumber \\
\end{eqnarray}}
This choice of the matrix $C_o$ means that different quadratures are used for the outputs of the elements of the distributed quantum observer with a $90^\circ$ phase rotation as we move from observer element to element along the chain of observers.
In order to construct suitable values for the quantities $\mu_i$ and $\omega_i$, we require that
\begin{equation}
\label{xobareqn}
A_o\bar x_o +B_oz_p = 0
\end{equation}
where
\[
\bar x_o = \left[\begin{array}{c}\alpha\\J\alpha\\-\alpha \\ -J\alpha \\ \alpha \\ \vdots\\(J)^{N-1}\alpha\end{array}\right]z_p.
\]
This will ensure that the quantity
\begin{equation}
\label{xe}
x_e = x_o - \bar x_o
\end{equation}
will satisfy the non-commutative differential equation
\begin{equation}
\label{xedot}
\dot{x}_e = A_o x_e.
\end{equation}
This, combined with the fact that
\begin{eqnarray}
\label{Coxbar}
C_o\bar{x}_o&=& \frac{1}{\|\alpha\|^2}\left[\begin{array}{llll}\alpha^T &&&\\
& -J\alpha^T & 0 &\\
&0& \ddots & \\
&&& (-J)^{N-1}\alpha^T
\end{array}\right]\nonumber \\
&& \times \left[\begin{array}{l}\alpha\\J\alpha\\ \vdots\\(J)^{N-1}\alpha\end{array}\right]z_p \nonumber \\
&=& \left[\begin{array}{l}1\\1\\\vdots\\1\end{array}\right]z_p
\end{eqnarray}
will be used in establishing condition (\ref{average_convergence})
for the distributed quantum observer.
Now, we require
\begin{eqnarray*}
\lefteqn{A_o\left[\begin{array}{c}\alpha\\J\alpha\\-\alpha \\ -J\alpha \\ \alpha \\ \vdots\\(J)^{N-1}\end{array}\right]+B_o}\nonumber \\
&=&
2\left[\begin{array}{c}
\omega_1J\alpha-\mu_2J\alpha-\mu_1J\alpha\\
\mu_2 \alpha- \omega_2\alpha+ \mu_3\alpha\\
\mu_3J\alpha-\omega_3J\alpha+\mu_4J\alpha \\
\vdots\\
\mu_N(J)^{N-2} \alpha+\omega_NJ^N\alpha
\end{array}\right]\nonumber \\
&=& 0.
\end{eqnarray*}
This will be satisfied if and only if
\[
\left[\begin{array}{c}
\omega_1-\mu_2-\mu_1\\
\mu_2- \omega_2+ \mu_3\\
\mu_3- \omega_3+\mu_4 \\
\vdots\\
\mu_N-\omega_N
\end{array}\right] = 0.
\]
That is, we will assume that
\begin{equation}
\label{mui}
\omega_i=\mu_i+\mu_{i+1}
\end{equation}
for $i=1,2,\ldots,N-1$ and
\begin{equation}
\label{muN}
\omega_N=\mu_N.
\end{equation}
To show that the above candidate distributed quantum observer leads to the satisfaction of the condition
(\ref{average_convergence}),
we first note that $x_e$ defined in (\ref{xe}) will satisfy (\ref{xedot}). If we can show that
\begin{equation}
\label{xeav}
\lim_{T \rightarrow \infty} \frac{1}{T}\int_{0}^{T} x_e(t)dt = 0,
\end{equation}
then it will follow from (\ref{Coxbar}) and (\ref{xe}) that (\ref{average_convergence}) is satisfied. In order to establish (\ref{xeav}), we first note that we can write
\[
A_o = 2\Theta R_o
\]
where
\begin{small}
\begin{eqnarray*}
\lefteqn{R_o = }\nonumber \\
&\left[\begin{array}{rrrrrr}\omega_1I & \mu_2J & & & &\\
-\mu_2J & \omega_2I & \mu_3J& & 0 & \\
& -\mu_3J & \omega_3I & \mu_4J &&\\
& & \ddots & \ddots & \ddots &\\
& 0 && -\mu_{N-1}J& \omega_{N-1}I & \mu_NJ\\
&&&& -\mu_NJ & \omega_NI
\end{array}\right].
\end{eqnarray*}
\end{small}
We will now show that the symmetric matrix $R_o$ is positive-definite.
\begin{lemma}
\label{L1}
The matrix $R_o$ is positive definite.
\end{lemma}
\begin{proof}
In order to establish this lemma, let
\[
x_o = \left[\begin{array}{l}x_{o1}\\x_{o2}\\\vdots\\x_{oN}\end{array}\right] \in \rbb^{2N}
\]
where $x_{oi} = \left[\begin{array}{c} q_i \\ p_i \end{array}\right] \in \rbb^2$ for $i = 1,2, \ldots, N$. Also, define the complex scalars $a_i = q_i + \imath p_i$ for $i=1,2,\ldots,N$. Then it is straightforward to verify that
\begin{eqnarray*}
x_o^TR_ox_o &=& \omega_1\|x_{o1}\|^2-2\mu_2x_{o1}^T\alpha x_{o2}^T\alpha+\omega_2\|x_{o2}\|^2\nonumber \\
&& -2 \mu_3 x_{o2}^T\alpha x_{o3}^T\alpha+\omega_3\|x_{o3}\|^2\nonumber \\
&& \vdots \nonumber \\
&&-2 \mu_N x_{oN-1}^T\alpha x_{oN}^T\alpha+\omega_N\|x_{oN}\|^2\nonumber \\
&= & \omega_1a_1^*a_1-\imath\mu_2a_1^*a_2+\imath\mu_2a_2^*a_1+\omega_2a_2^*a_2\nonumber \\
&& -\imath \mu_3 a_2^*a_3+\imath \mu_3 a_3^*a_2+\omega_3a_3^*a_3\nonumber \\
&& \vdots \nonumber \\
&&-\imath \mu_N a_{N-1}^*a_N+\imath \mu_N a_N^*a_{N-1}+\omega_Na_N^*a_N\nonumber \\
&=& a_o^\dagger\tilde R_o a_o
\end{eqnarray*}
where
\[
a_o = \left[\begin{array}{l}a_{1}\\a_{2}\\\vdots\\a_{N}\end{array}\right] \in \cbb^N
\]
and
\[
\tilde R_o= \left[\begin{array}{rrrrr}\omega_1 & -\imath \mu_2 & & &\\
\imath \mu_2 & \omega_2 & -\imath \mu_3& & 0 \\
& \imath \mu_3 & \omega_3 & \ddots &\\
0 & & \ddots & \ddots & -\imath \mu_N\\
&&& \imath \mu_N & \omega_N
\end{array}\right].
\]
Here $^\dagger$ denotes the complex conjugate transpose of a vector.
From this, it follows that the real symmetric matrix $R_o$ is positive-definite if and only if the complex Hermitian matrix $\tilde R_o$ is positive-definite.
To prove that $\tilde R_o$ is positive-definite, we first substitute the equations (\ref{mui}) and (\ref{muN}) into the definition of $\tilde R_o$ to obtain
\begin{eqnarray*}
\tilde R_o&=& \left[\begin{array}{rrrrr}\mu_1+\mu_2 & -\imath \mu_2 & & &\\
\imath \mu_2 & \mu_2+\mu_3 & -\imath \mu_3& & 0 \\
& \imath \mu_3 & \mu_3+\mu_4 & \ddots &\\
0 & & \ddots & \ddots & -\imath \mu_N\\
&&& \imath \mu_N & \mu_N
\end{array}\right]\nonumber \\
&=& \tilde R_{o1} + \tilde R_{o2}
\end{eqnarray*}
where
\[
\tilde R_{o1} = \left[\begin{array}{rrrr}
\mu_1 & 0 &\ldots & 0\\
0 & 0 &\ldots & 0\\
\vdots & & & \vdots\\
0 & 0 &\ldots & 0
\end{array}\right] \geq 0
\]
and
\[
\tilde R_{o2} = \left[\begin{array}{rrrrr} \mu_{2} & - \imath \mu_2 & & &\\
+\imath \mu_2 & \mu_2+ \mu_{3} & -\imath \mu_3& & 0 \\
& +\imath\mu_3 & \mu_3+ \mu_{4} & \ddots &\\
0 & & \ddots & \ddots & - \imath \mu_N\\
&&& +\imath \mu_N & \mu_N
\end{array}\right].
\]
Now, we can write
\begin{eqnarray*}
a_o^\dagger\tilde R_{o2} a_o &=&
\mu_2 a_1^*a_1-\imath\mu_2a_1^*a_2+\imath\mu_2a_2^*a_1+\mu_2a_2^*a_2\nonumber \\
&&+\mu_3a_2^*a_2 -\imath \mu_3 a_2^*a_3+\imath \mu_3 a_3^*a_2+\mu_4a_3^*a_3\nonumber \\
&& \vdots \nonumber \\
&&+\mu_{N-1}a_{N-1}^*a_{N-1}-\imath \mu_N a_{N-1}^*a_N\nonumber \\
&&+\imath \mu_N a_N^*a_{N-1}+\mu_Na_N^*a_N\nonumber \\
&=& \mu_2 (-\imath a_1^* + a_2^*)(\imath a_1 + a_2)\nonumber \\
&&+\mu_3 (-\imath a_2^* + a_3^*)(\imath a_2 + a_3)\nonumber \\
&&\vdots \nonumber \\
&&+\mu_N (-\imath a_{N-1}^* + a_N^*)(\imath a_{N-1} + a_N)\nonumber \\
&\geq& 0.
\end{eqnarray*}
Thus, $\tilde R_{o2} \geq 0$. Furthermore, $a_o^\dagger\tilde R_{o2} a_o = 0$ if and only if
\begin{eqnarray*}
a_2 &=& -\imath a_1;\nonumber \\
a_3 &=& -\imath a_2;\nonumber \\
&\vdots& \nonumber \\
a_N &=& -\imath a_{N-1}. \nonumber \\
\end{eqnarray*}
That is, the null space of $\tilde R_{o2}$ is given by
\[
\mathcal{N}(\tilde R_{o2}) = \mbox{span}\{\left[\begin{array}{l}1\\-\imath\\ -1\\ \imath \\ 1 \\\vdots\\(-\imath)^{N-1}\end{array}\right]\}.
\]
The fact that $\tilde R_{o1} \geq 0$ and $\tilde R_{o2} \geq 0$ implies that $\tilde R_{o} \geq 0$. In order to show that $\tilde R_{o} > 0$, suppose that $a_o$ is a non-zero vector in $\mathcal{N}(\tilde R_{o})$. It follows that
\[
a_o^\dagger \tilde R_{o}a_o = a_o^\dagger\tilde R_{o1}a_o+a_o^\dagger\tilde R_{o2}a_o = 0.
\]
Since $\tilde R_{o1} \geq 0$ and $\tilde R_{o2} \geq 0$, $a_o$ must be contained in the null space of $\tilde R_{o1}$ and the null space of $\tilde R_{o2}$. Therefore $a_o$ must be of the form
\[
a_o = \gamma \left[\begin{array}{l}1\\-\imath\\ -1\\ \imath \\ 1 \\\vdots\\(-\imath)^{N-1}\end{array}\right]
\]
where $\gamma \neq 0$. However, then
\[
a_o^\dagger \tilde R_{o1}a_o = \gamma^2 \tilde \mu_1 \neq 0
\]
and hence $a_o$ cannot be in the null space of $\tilde R_{o1}$. Thus, we can conclude that the matrix $\tilde R_{o}$ is positive definite and hence, the matrix $R_{o}$ is positive definite. This completes the proof of the lemma.
\end{proof}
We now verify that the condition (\ref{average_convergence}) is satisfied for the distributed quantum observer under consideration. This proof follows along very similar lines to the corresponding proof given in \cite{PET14Ca}. We recall from Remark \ref{R1} that the quantity $\half x_e(t)\trp R_o x_e(t)$
remains constant in time for the linear system:
\[
\dot{ x}_e = A_o x_e= 2\Theta R_o x_e.
\]
That is
\begin{equation}
\label{Roconst}
\half x_e(t) \trp R_o x_e(t) = \half x_e(0) \trp R_o x_e(0) \quad \forall t \geq 0.
\end{equation}
However, $ x_e(t) = e^{2\Theta R_ot} x_e(0)$ and $R_o > 0$. Therefore, it follows from (\ref{Roconst}) that
\[
\sqrt{\lambda_{min}(R_o)}\|e^{2\Theta R_ot} x_e(0)\| \leq \sqrt{\lambda_{max}(R_o)}\| x_e(0)\|
\]\
for all $ x_e(0)$ and $t \geq 0$. Hence,
\begin{equation}
\label{exp_bound}
\|e^{2\Theta R_ot}\| \leq \sqrt{\frac{\lambda_{max}(R_o)}{\lambda_{min}(R_o)}}
\end{equation}
for all $t \geq 0$.
Now since $\Theta $ and $R_o$ are non-singular,
\[
\int_0^Te^{2\Theta R_ot}dt = \half e^{2\Theta R_oT}R_o^{-1}\Theta ^{-1} - \half R_o^{-1}\Theta ^{-1}
\]
and therefore, it follows from (\ref{exp_bound}) that
\begin{eqnarray*}
\lefteqn{\frac{1}{T} \|\int_0^Te^{2\Theta R_ot}dt\|}\nonumber \\
&=& \frac{1}{T} \|\frac{1}{2}e^{2\Theta R_oT}R_o^{-1}\Theta ^{-1} - \frac{1}{2}R_o^{-1}\Theta ^{-1}\|\nonumber \\
&\leq& \frac{1}{2T}\|e^{2\Theta R_oT}\|\|R_o^{-1}\Theta ^{-1}\| \nonumber \\
&&+ \frac{1}{2T}\|R_o^{-1}\Theta ^{-1}\|\nonumber \\
&\leq&\frac{1}{2T}\sqrt{\frac{\lambda_{max}(R_o)}{\lambda_{min}(R_o)}}\|R_o^{-1}\Theta ^{-1}\|\nonumber \\
&&+\frac{1}{2T}\|R_o^{-1}\Theta ^{-1}\|\nonumber \\
&\rightarrow & 0
\end{eqnarray*}
as $T \rightarrow \infty$. Hence,
\begin{eqnarray*}
\lefteqn{\lim_{T \rightarrow \infty} \frac{1}{T}\|\int_{0}^{T} x_e(t)dt\| }\nonumber \\
&=& \lim_{T \rightarrow \infty}\frac{1}{T}\|\int_{0}^{T} e^{2\Theta R_ot} x_e(0)dt\| \nonumber \\
&\leq& \lim_{T \rightarrow \infty}\frac{1}{T} \|\int_{0}^{T} e^{2\Theta R_ot}dt\|\| x_e(0)\|\nonumber \\
&=& 0.
\end{eqnarray*}
This implies
\[
\lim_{T \rightarrow \infty} \frac{1}{T}\int_{0}^{T} x_e(t)dt = 0
\]
and hence, it follows from (\ref{xe}) and (\ref{Coxbar}) that
\[
\lim_{T \rightarrow \infty} \frac{1}{T}\int_{0}^{T} z_o(t)dt = \left[\begin{array}{l}1\\1\\\vdots\\1\end{array}\right]z_p.
\]
Also, (\ref{zp_const}) implies
\[
\lim_{T \rightarrow \infty} \frac{1}{T}\int_{0}^{T} z_p(t)dt = \left[\begin{array}{l}1\\1\\\vdots\\1\end{array}\right]z_p.
\]
Therefore, condition (\ref{average_convergence}) is satisfied. Thus, we have established the following theorem.
\begin{theorem}
\label{T1}
Consider a quantum plant of the form (\ref{plant}) where $A_p = 0$. Then the distributed direct coupled quantum observer defined by equations (\ref{observer}), (\ref{Rc}), (\ref{Rc_def}), (\ref{Ao}), (\ref{alphabeta}), (\ref{Co}), (\ref{mui}), (\ref{muN}) achieves time-averaged consensus convergence for this quantum plant.
\end{theorem} | 162,562 |
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04/06/13 (Sat)
Trip
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Eli Tollin
11:45 AM
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\section{Moment Generating Function of Exponential Distribution}
Tags: Moment Generating Functions, Exponential Distribution
\begin{theorem}
Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with an [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{> 0}$.
Then the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\displaystyle \map {M_X} t = \frac 1 {1 - \beta t}$
for $t < \dfrac 1 \beta$, and is undefined otherwise.
\end{theorem}
\begin{proof}
From the definition of the [[Definition:Exponential Distribution|Exponential distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\displaystyle \map {f_X} x = \frac 1 \beta e^{-\frac x \beta}$
From the definition of a [[Definition:Moment Generating Function|moment generating function]]:
:$\displaystyle \map {M_X} t = \expect {e^{t X} } = \int_0^\infty e^{t x} \map {f_X} x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = \map {M_X} t
| r = \frac 1 \beta \int_0^\infty e^{x \paren {-\frac 1 \beta + t} } \rd x
| c = [[Exponential of Sum]]
}}
{{eqn | r = \frac 1 {\beta \paren {-\frac 1 \beta + t} } \sqbrk {e^{x \paren {-\frac 1 \beta + t} } }_0^\infty
| c = [[Primitive of Exponential Function]]
}}
{{end-eqn}}
Note that if $t > \dfrac 1 \beta$, then $\displaystyle e^{x \paren {-\frac 1 \beta + t} } \to \infty$ as $x \to \infty$ by [[Exponential Tends to Zero and Infinity]], so the integral diverges in this case.
If $t = \dfrac 1 \beta$ then the integrand is identically $1$, so the integral similarly diverges in this case.
If $t < \dfrac 1 \beta$, then $\displaystyle e^{x \paren {-\frac 1 \beta + t} } \to 0$ as $x \to \infty$ from [[Exponential Tends to Zero and Infinity]], so the integral converges in this case.
Therefore, the function is only well defined for $t < \dfrac 1 \beta$.
Proceeding:
{{begin-eqn}}
{{eqn | l = \frac 1 {\beta \paren {-\frac 1 \beta + t} } \sqbrk {e^{x \paren {-\frac 1 \beta + t} } }_0^\infty
| r = \frac 1 {\beta \paren {-\frac 1 \beta + t} } \paren {0 - 1}
| c = [[Exponential Tends to Zero and Infinity]], [[Exponential of Zero]]
}}
{{eqn | r = \frac 1 {\beta \paren {\frac 1 \beta - t} }
}}
{{eqn | r = \frac 1 {1 - \beta t}
}}
{{end-eqn}}
{{qed}}
[[Category:Moment Generating Functions]]
[[Category:Exponential Distribution]]
9omebrbdg9jyyp53lp3kisnerl3psqr
\end{proof}
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Canticle of the Sun
Canticle of the Sun
$ 115.00 USD
On Sale
Watertown, MA: Kurbel Books, 1984. (19) pp. Approximately 2 1/8 x 2 1/2". LIMITED EDITION of 500 copies, this being #21. Paper band with title label, blue cloth over boards, postage stamps, ribbon bookmark with coin token, marbled endpapers, accordion fold, all housed in a clear plastic box, calligraphy and illumination by E. Helene Sherman, layout by Robert E. Massmann, binding by Bela and Mariana Blau, in fine condition. (Bradbury, Kurbel Books #2)
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- Publisher
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- Binding
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TITLE: For which values of $n$ (natural number) is $(x^n)-1$ divisible by $(x^2)+1$
QUESTION [0 upvotes]: For which values of a natural number $n$ is
$(x^n)-1$ divisible by $(x^2)+1$ ?
REPLY [0 votes]: For any polynomial $\mathrm{p}$, the Factor Theorem says that $x-a$ divides $\mathrm{p}$ if, and only if, $\mathrm{p}(a)=0$.
For $x^2+1$ to divide $\mathrm{p}$, we need both $x-\mathrm{i}$ and $x+\mathrm{i}$ to divide $\mathrm{p}$ since notice that $x^2+1 \equiv (x+\mathrm{i})(x-\mathrm{i})$.
The Factor Theorem tells us that $x^2+1$ divides $\mathrm{p}$ if, and only if, $\mathrm{p}(\mathrm{i}) = \mathrm{p}(-\mathrm{i}) = 0$.
Let $\mathrm{p}(x) = x^n-1$. Consider separately the cases where $n$ is even and odd.
If $n$ is even then $\mathrm{p}(x) = x^{2k}-1$. It follows that
$$\mathrm{p}(\pm\mathrm{i}) = (\pm\mathrm{i})^{2k}-1=\left((\pm\mathrm{i})^2\right)^k-1=(-1)^k-1$$
Hence, $\mathrm{p}(\pm\mathrm{i}) = 0$ if, and only if, $k$ is even, i.e. $n=2k$ is divisible by $4$.
If $n$ is odd then $\mathrm{p}(x) = x^{2k+1}-1$. It follows that
$$\mathrm{p}(\pm\mathrm{i}) = (\pm\mathrm{i})^{2k+1}-1=(\pm\mathrm{i})(\pm\mathrm{i})^{2k}-1 = (\pm\mathrm{i})(-1)^k-1$$
This has non-zero real part for all $k$, and so $\mathrm{p}(\pm\mathrm{i}) \neq 0$ for all odd $n$.
It follows that $x^2+1$ divides $x^n-1$ if, and only if, $n$ is divisible by $4$. | 19,156 |
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\begin{document}
\maketitle
\let\thefootnote\relax
\footnotetext{The authors were partially supported by grants 2017SGR358,
2017SGR1725 (Generalitat de Catalunya) and PGC2018-095998-B-100 (Ministerio de Econom\'{\i}a y Competitividad).}
\begin{abstract}
In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities
with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them. \\*[5pt]
{\bf Keywords:} Convex set, Visual angle, Densities, Invariant measures.\\*[3pt]
{\bf Mathematics Subject Classification (2010):} 52A10, 53A04.
\end{abstract}
\section{Introduction}
Throughout this paper $K$ will be a compact convex set in $\R^{2}$ with boundary of class ${\mathcal C}^{2}$. We will denote by $F$ the area of $K$ and by $L$ the length of its boundary.
In 1868 Crofton showed (\cite{crofton}), using arguments that nowadays belong to Integral Geometry, the well known formula
\begin{equation}\label{21maig-2}
2\int_{P\notin K }(\omega-\sin\omega)\,dP+2\pi F=L^{2},
\end{equation}
where
$\omega=\omega(P)$ is the \emph{visual angle} of $K$ from the point $P$, that is the angle between the two tangents from $P$ to the boundary of $K$. In terms of Integral Geometry
both sides of this formula represent the measure of pairs of lines meeting~$K$. In fact the measure of all pairs of lines meeting $K$ is $L^{2}$,
twice the integral of $\omega-\sin\omega$ with respect to the area element $dP$
is the measure of those pairs of lines intersecting themselves outside $K$ and $2\pi F$ is the measure of those intersecting themselves in~$K$.
Later on, Hurwitz in $1902$, in his celebrated paper \cite{Hurwitz1902} on the application of Fourier series to geometric problems,
considers the integral of some new functions of the visual angle.
Concretely he proves
\begin{equation}\label{21maig-3}
\int_{P\notin K}f_{k}(\omega)\,dP=L^{2}+(-1)^{k}\pi^{2}(k^{2}-1)c_{k}^{2},
\end{equation}
where
\begin{equation}\label{maig9-3}
f_{k}(\omega)=-2\sin\omega+\frac{k+1}{k-1}\sin ((k-1)\omega)-\frac{k-1}{k+1}\sin((k+1)\omega), \quad k\geq 2,
\end{equation}
and $c_{k}^{2}=a_{k}^{2}+b_{k}^{2}$, with $a_{k},b_{k}$ the Fourier coefficients of the support function of~$K$.
In the particular case $k=2$ formula \eqref{21maig-3} gives
\begin{equation}\label{24maig}
\int_{P\notin K}\sin^{3}\omega\,dP=\frac{3}{4}L^{2}+\frac{9}{4}\pi^{2}c_{2}^{2}.
\end{equation}
\medskip
Masotti in $1955$ (\cite{masotti2}) states without proof the following Crofton's type formula
\begin{equation}\label{21maig-4}
\int_{P\notin K}(\omega^{2}-\sin^{2}\omega)\,dP=-\pi^{2}F+\frac{4L^{2}}{\pi}+8\pi\sum_{k\geq 1}\left(\frac{1}{1-4k^{2}}\right)c_{2k}^{2}.
\end{equation}
In \cite{CGR} a unified approach that encompasses the previous results is provided. As well the following formula for the integral of any power of the sine function of the visual angle, that generalises \eqref{24maig}, is given:
\begin{multline}\label{21maig-5}
\int_{P\notin K}\sin^{m}(\omega)\,dP=\frac{m!}{2^{m}(m-2)\Gamma(\frac{m-1}{2})^{2}}\, L^{2} \\*[5pt]
+\frac{m!\pi^{2}}{2^{m-1}(m-2)}\sum_{k\geq 2, \text{ even}}\frac{(-1)^{\frac{k}{2}+1}(k^{2}-1)}{\Gamma(\frac{m+1+k}{2})\Gamma(\frac{m+1-k}{2})}c_{k}^{2}.
\end{multline}
In this paper we deal with a general type of integral formulas of the visual angle including those we have just commented above, from the point of view of Integral Geometry according to Crofton and Santal\'o \cite{santalo}. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities
with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.
\medskip
For each straight line $G$ of the plane that does not pass through the origin let $P$ be the point of $G$ at a minimum distance from the origin. We take as coordinates
for $G$ the polar coordinates $(p, \varphi)$ of the point $P$, with $p>0$ and $0\leq \varphi <2\pi$.
The invariant measure in the set of lines of the plane not containing the origin
is given by a constant multiple of
$dG=dp\,d\varphi$. In the space of ordered pairs of lines we consider the canonical measure $dG_{1}\, dG_{2}$. This measure is, except for a constant factor, the only one invariant under Euclidean motions (see \cite{santalo}).
For every function $\tilde{f}(G_{1},G_{2})$
integrable with respect to $dG_{1}\, dG_{2}$
we can consider the measure with density $\tilde{f}$, that is $\tilde{f}(G_{1},G_{2})\,dG_{1}\, dG_{2}$.
We prove in Proposition \ref{maig9b} that this measure is invariant under Euclidean motions if and only if $\tilde{f}(G_{1},G_{2})=f(\varphi_{2}-\varphi_{1})$ with $f$ a $\pi$-periodic function on $\R$.
For such densities, and under some additional hypothesis, it follows from Theorem \ref{aagg} and Corollary \ref{centredreta} that
\begin{multline}\label{21maig}
A_{0}L^{2}+\pi^{2}\sum_{n\geq 1}c_{2n}^{2}A_{2n}= \int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\, dG_{2} \\
= 2H(\pi) F +2\int_{P\notin K} H(\omega)\,dP,
\end{multline}
where $A_{k}$, $k\geq 0$, are the Fourier coefficients of $f$ corresponding to
$\cos(k\varphi)$, and $H(x)$ is a ${\mathcal C}^{2}$ function on $[0,\pi]$ satisfying $H''(x)=f(x)\sin (x)$, $x\in [0,\pi]$, and $H(0)=H'(0)=0$.
The above
two equalities are the main tools to obtain both new proofs of the formulas discussed above and their interpretation as integrals of densities with respect to the canonical measure in the space of pairs of lines. As concerning to this second point, in section \ref{21maig-7} one obtains the following formulas.
\begin{itemize}
\item[-] {\em Crofton's formula}
$$
\int_{P\notin K}(\omega-\sin\omega)\,dP=-\pi F+\frac{1}{2}\int_{G_{i}\cap K\neq\emptyset}dG_{1}\, dG_{2} .
$$
\end{itemize}
\begin{itemize}
\item[-] {\em Hurwitz's formula}
$$
\int_{P\notin K}f_{k}(\omega)\,dP=\int_{G_{i}\cap K\neq\emptyset}(1+(-1)^{k}(k^{2}-1)\cos(k(\varphi_{2}-\varphi_{1})))\,dG_{1}\, dG_{2} .
$$
\end{itemize}
\begin{itemize}
\item[-] {\em Masotti's formula}
$$
\int_{P\notin K}(\omega^{2}-\sin^{2}\omega)\,dP=-\pi^{2} F+2\int_{G_{i}\cap K\neq\emptyset}|\sin(\varphi_{2}-\varphi_{1})|\,dG_{1}\, dG_{2} .
$$
\end{itemize}
\begin{itemize}
\item[-] {\em Power sine formula}
\begin{multline*}
\!\!\int_{P\notin K}\sin^{m}\omega\,dP\\*[5pt]
\!=\frac{1}{2}\int_{G_{i}\cap K\neq\emptyset}\!\!\left(m(m-1)|\mathrm{sin}^{m-3}(\varphi_{2}-\varphi_{1})|\!-\!m^{2}|\mathrm{sin}^{m-1}(\varphi_{2}-\varphi_{1})|\right)\,dG_{1}\, dG_{2}. \!\!\!
\end{multline*}
\end{itemize}
Moreover using the first equality in \eqref{21maig} one gets the announced new proofs of formulas
\eqref{21maig-2}, \eqref{21maig-4} and \eqref{21maig-5}.
Concerning Hurwitz's integral, when we apply the methods here developed, it appears a different behavior according to $k$
is either even or odd. For $k$ even using~\eqref{21maig} one gets a new proof of \eqref{21maig-3}. Nevertheless when $k$ is odd the density associated
to the Hurwitz integral is not $\pi$-periodic since the function $\cos(kx)$ is not, and so we cannot use \eqref{21maig}. In this case appealing to Proposition \ref{antipi} one obtains a new result
that involves a decomposition of the visual angle $\omega$ into two parts $\omega=\omega_{1}+\omega_{2}$
that also have a geometrical interpretation.
In this setting
it plays a role the function $f_{k}(\omega)+2(\sin\omega-\omega)$, that is the sum
of the functions of Hurwitz and Crofton. In spite of $\int_{P\notin K}(f_{k}(\omega)+2(\sin\omega-\omega))\,dP$ depends on $k$, the surprising fact is that, for $k$ odd, decomposing the visual angle~$\omega$
into the two parts $\omega_{1}$, $\omega_{2}$
and adding the corresponding integrals
leads to
\begin{equation*}
\int_{P\notin K} \left(f_{k}(\omega_{1})+2(\sin \omega_{1}-\omega_{1})+ f_{k}(\omega_{2})+2(\sin \omega_{2}-\omega_{2})\right)\,dP = 2\pi F,
\end{equation*}
for each $k\geq 3$ odd, as a consequence of Proposition \ref{21maig-6}.
Moreover it will appear that the functions of Crofton and Hurwitz are in some sense a basis for the integral of any $\pi$-periodic or anti $\pi$-periodic density
with respect to the measure $dG_{1}\,dG_{2}$ over the set of pairs of lines
meeting a given compact convex set.
\section{Densities in the space of pairs of lines }
For every function $\tilde{f}(G_{1},G_{2})$
defined on the space of pairs of lines integrable with respect to the measure $dG_{1}\, dG_{2}$ we consider the measure with density $\tilde{f}$, that is the measure $\tilde{f}(G_{1},G_{2})\,dG_{1}\, dG_{2}$.
The measure of a set~$A$ of pairs of lines in the plane
is then given by
$$
\int_{A}\tilde{f}(G_{1}, G_{2})\,dG_{1}\, dG_{2}.
$$
We want now to determine when this measure is invariant under Euclidean motions.
\begin{proposition}\label{maig9b}
The measure $\tilde{f}(G_{1},G_{2})\,dG_{1}\, dG_{2}$
is invariant under the group of Euclidean motions if and only if
$\tilde{f}(G_{1},G_{2})=\tilde{f}(p_{1},\varphi_{1},p_{2},\varphi_{2})=f(\varphi_{2}-\varphi_{1})$
with~$f$ a $\pi$-periodic function on $\R$, where $(p_{i},\varphi_{i})$ are the coordinates of $G_{i}$.
\end{proposition}
\begin{proof}
The invariance of the measure is equivalent to the equality $\tilde{f}(p_{1},\!\varphi_{1},p_{2},\!\varphi_{2}) \!=\!\!$ $\tilde{f}(p'_{1},\varphi'_{1},p'_{2},\varphi'_{2})$
for each Euclidean motion sending the lines with coordinates $(p_{1},\varphi_{1}, p_{2},\varphi_{2})$ to the lines with coordinates $(p'_{1},\varphi'_{1}, p'_{2},\varphi'_{2})$.
First of all let us show that $\tilde{f}$
does not depend on $p_{1}$, $p_{2}$.
In fact, for every straight line $G=G(p,\varphi)$ and an arbitrary $a>0$ there is a parallel
line to $G$ with coordinates $(a,\varphi).$ Given two straight lines $G_{1}=G(p_{1},\varphi_{1})$, $G_{2}=G(p_{2},\varphi_{2})$ and two numbers $a_{1},a_{2}>0$ let $G'_{1}$ and $G'_{2}$ be the corresponding parallel lines with coordinates $(a_{1},\varphi_{1})$, $(a_{2},\varphi_{2}).$ Performing the translation that sends the point $G_{1}\cap G_{2}$ to the point $G'_{1}\cap G'_{2}$ we have that $\tilde{f}(p_{1},\varphi_{1},p_{2},\varphi_{2})=\tilde{f}(a_{1},\varphi_{1},a_{2},\varphi_{2})$ and so $\tilde{f}$ does not depend on~$p_{1}$ and~$p_{2}.$
Given now the line $G(p,\varphi)$ if we perform, for instance, the translation given by the vector $-(p+\epsilon)(\cos\varphi,\sin\varphi)$, $\epsilon>0$, the translated line has coordinates $(\epsilon,\varphi+\pi)$. Therefore the function $\tilde{f}$ must be $\pi$-periodic with respect to the arguments $\varphi_{1}$, $\varphi_{2}.$
Finally due to the invariance under rotations
it follows that $\tilde{f}(p_{1},\varphi_{1},p_{2},\varphi_{2})=\tilde{f}(p_{1},0,p_{2},\varphi_{2}-\varphi_{1})$ and so
$\tilde{f}(p_{1},\varphi_{1},p_{2},\varphi_{2})=f(\varphi_{2}-\varphi_{1})$ with $f$ a $\pi$-periodic function.
\end{proof}
Our goal is now to integrate a measure given by a density over the set of pairs of lines meeting $K$. In view of Proposition \ref{maig9b} we shall only consider densities which depend on the angle of the two lines, that is of the form $\tilde{f}(G_{1},G_{2})=f(\varphi_{2}-\varphi_{1})$, with $G_{i}=G_{i}(p_{i},\varphi_{i})$, $i=1,2$. Note that $\varphi_{2}-\varphi_{1}$ gives one of the two angles between the lines $G_{1}$ and $G_{2}$.
\medskip
We give a formula to compute the integral of the measure $\tilde{f}(G_{1},G_{2})\,dG_{1}\, dG_{2}=f(\varphi_{2}-\varphi_{1})\,dG_{1}\, dG_{2}$, with $f$ a $2\pi$-periodic function extended to the pairs of lines meeting $K$ in terms of both the Fourier coefficients of $f$ and of the support function of $K$. Recall that when the origin of coordinates is an interior point of $K$, a hypothesis that we will assume from now on, the support function $p(\varphi)$
is given by the distance to the origin of the tangent to $K$
whose normal makes and angle~$\varphi$ with the positive part of the real axis (see \cite{santalo}).
\begin{theorem}
\label{aagg} Let $K$ be a compact convex set with boundary of length $L$.
Let $f$ be a $2\pi$-periodic continuous function on $\R$ with Fourier expansion
$$
f(\varphi)=\sum_{n\geq 0}A_{n}\cos(n\varphi)+B_{n}\sin(n\varphi).
$$
Then
\begin{equation}\label{eq31}
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=A_{0}L^{2}+\pi^{2}\sum_{n\geq 1}c_{n}^{2}A_{n},
\end{equation}
with $c_{n}^{2}=a_{n}^{2}+b_{n}^{2}$ where $a_{n}$, $b_{n}$ are the Fourier coefficients of the support
function~$p(\varphi)$ of $K$.
\end{theorem}
\begin{proof}
We have
\begin{equation}\label{14g}
\begin{split}
\int_{G_{i}\cap K\neq\emptyset}\! f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}&=\!\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{p(\varphi_{1})}\int_{0}^{p(\varphi_{2})}\!f(\varphi_{2}-\varphi_{1})\,dp_{1}\,dp_{2}\,d\varphi_{1}\,d\varphi_{2}\\*[5pt]
&=\!\int_{0}^{2\pi}\int_{0}^{2\pi}p(\varphi_{1})p(\varphi_{2})f(\varphi_{2}-\varphi_{1})\,d\varphi_{1}\,d\varphi_{2}.
\end{split}
\end{equation}
Performing the change of variables $\varphi_{2}-\varphi_{1}=w$, $\varphi_{1}=u$
the integral \eqref{14g} becomes
\begin{equation}\label{14g2}
\int_{0}^{2\pi}p(u)\int_{-u}^{2\pi-u}p(u+w)f(w)\,dw\,du.
\end{equation}
The Fourier development of $p(u+w)$ in terms of
the Fourier coefficients $a_{n}$, $b_{n}$ of~$p(u)$ is
\begin{multline*}
p(u+w)=a_{0}+\sum_{n\geq 1}\biggl((a_{n}\cos(nu)+b_{n}\sin(nu))\cos(nw)\\ +(-a_{n}\sin(nu)+b_{n}\cos(nu))\sin(nw)\biggr).
\end{multline*}
By the Plancherel identity the integral \eqref{14g2}
is equal to
\begin{multline*}
\int_{0}^{2\pi}p(u)\biggl[2\pi \,A_{0}\,a_{0}+\pi\sum_{n\geq 1}A_{n}(a_{n}\cos(nu)+b_{n}\sin(nu))\\*[5pt]
+B_{n}(-a_{n}\sin(nu)+b_{n}\cos(nu)) \biggr]\,du \\*[5pt]
=
\int_{0}^{2\pi}p(u)\biggl[2\pi \,A_{0}\,a_{0}+\pi\sum_{n\geq 1}(A_{n}a_{n}+B_{n}b_{n})\cos(nu)+(A_{n}b_{n}-B_{n}a_{n})\sin(nu)) \biggr]\,du \\*[5pt]
=
4\pi^{2}A_{0}\,a_{0}^{2}+\pi^{2}\sum_{n\geq 1}(A_{n}a_{n}+B_{n}b_{n})a_{n}+(A_{n}b_{n}-B_{n}a_{n})b_{n}\\*[5pt]
=
4\pi^{2}A_{0}\,a_{0}^{2}+\pi^{2}\sum_{n\geq 1}A_{n}(a_{n}^{2}+b_{n}^{2})=
A_{0}L^{2}+\pi^{2}\sum_{n\geq 1}A_{n}c_{n}^{2},
\end{multline*}
where we have used that $L=2\pi a_{0}$, which is a consequence of the equality $L=\int_{0}^{2\pi}p(\varphi)\,d\varphi$ (see for instance \cite{santalo}), and the Theorem is proved.
\end{proof}
As it is well known (see \cite{Hurwitz1902}) the quantities $c_{k}^{2}=a_{k}^{2}+b_{k}^{2}$, $k\geq 2$, are invariant under Euclidean motions of $K$.
However $c_{1}^{2}$ changes when moving $K$. So
the integral in \eqref{eq31} is invariant under Euclidean motions of $K$ if and only if $A_{1}=0$. In particular this is the case when $f$ is $\pi$-periodic.
\medskip
For a density given by a $\pi$-periodic function $f$ and a compact set of constant width the measure of the pairs of lines that intersect $K$ is proportional to $L^{2}$. More precisely we have
\begin{corollary}
Let $K$ be a compact convex set of constant width and $f$ a continuous $\pi$-periodic function. Then
$$
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=\lambda L^{2},
$$
where $\lambda=(1/\pi)\int_{0}^{\pi}f(\varphi)\,d\varphi$.
\end{corollary}
\begin{proof}
Since $K$ is of constant width the Fourier development of $p(\varphi)$
has only odd terms (see for instance \S 2 of \cite{CGR}). Moreover the Fourier development of $f$ has only even terms because it is $\pi$-periodic. Hence \eqref{eq31} gives
$$
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=A_{0}L^{2},
$$
with $A_{0}=(1/\pi)\int_{0}^{\pi}f(\varphi)\,d\varphi$.
\end{proof}
\section{Integral formulas of the visual angle in terms of densities in the space of pairs of lines}
In \cite{CGR} there is a unified approach to several classical formulas involving integrals of functions of the visual angle of a compact convex set $K$.
Among them one can find the integrals of Crofton, Masotti, powers of sine, and Hurwitz.
The original proof of Crofton's formula, via Integral Geometry, involves a measure
on the space of pairs of lines. The aim of this section is to interpret the formulas in \cite{CGR} in terms of integrals of measures given by densities
in the space of pairs of lines.
To begin with let us consider Hurwitz's formula
\begin{equation}\label{maig9-2}
\int_{P\notin K}f_{k}(\omega)\,dP=L^{2}+(-1)^{k}\pi^{2}(k^{2}-1)c_{k}^{2},
\end{equation}
where $f_{k}(\omega)$ is given in \eqref{maig9-3}. For a proof of \eqref{maig9-2} see \cite{Hurwitz1902} or \cite{CGR}.
Comparing this equality with \eqref{eq31} one gets immediately the following result.
\begin{proposition}\label{maig17}
Let $f_{k}$ be the Hurwitz function defined in \eqref{maig9-3}.
Then
\begin{equation*}\label{maig9}
\int_{P\notin K}f_{k}(\omega)\,dP=\int_{G_{i}\cap K\neq\emptyset}(1+(-1)^{k}(k^{2}-1)\cos(k(\varphi_{2}-\varphi_{1})))\,dG_{1}\, dG_{2}.
\end{equation*}
\end{proposition}
Nevertheless for the other quoted integral formulas it is not clear at all what density must be chosen. We shall provide a general method to find the densities corresponding to integrals of general functions of the visual angle.
\subsection{A change of variables}\label{maig7-2}
The classical proof of Crofton's formula is based on the
change of variables in the space of pairs of lines given by
$$
(p_{1},\varphi_{1},p_{2},\varphi_{2})\longrightarrow (P, \alpha_{1},\alpha_{2}),
$$
where $P$ is the intersection point of the two straight lines and $\alpha_{i}\in [0, \pi]$ are the angles which determine the directions of the lines.
More precisely the angle $\alpha$ associated to a line through a given point $P$ is defined in the following way. Let $\vec{u}$ be a unitary vector orthogonal to $\overrightarrow{OP}$ where $O$ is the origin of coordinates, and such that the basis $(\vec{u},\overrightarrow{OP})$ is positively oriented.
Let $G$ be a line through $P$ with unitary director vector $\vec{v}$ such that the basis $(\vec{u},\vec{v})$ is positively oriented. Then $\alpha=\alpha(G)$ is defined by
$\cos\alpha=\vec{u}\cdot \vec{v}$ and $0<\alpha<\pi$.
From now on we shall say that $\alpha$ is the {\em direction} of the line $G$.
\begin{figure}[h]
\centering
\includegraphics[width=.5\textwidth]{alpha.png}
\caption{{\em Direction} of a line.}
\label{fig:direction}
\end{figure}
\noindent
With these new coordinates, proceeding as in \cite{santalo}, one has
\begin{equation}\label{5abr1}
dG_{1}\, dG_{2}=|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\, d\alpha_{2}\, dP.
\end{equation}
We have used the fact that $\varphi_{2}-\varphi_{1}=\alpha_{2}-\alpha_{1}+\epsilon\pi$ where $\epsilon= \epsilon(P,\alpha_{1},\alpha_{2})=0,\pm 1$, according to the position with respect to the origin of the lines $G_{1}$, $G_{2}$. As a consequence
if $f$ is a $\pi$-periodic function we have
\begin{equation}\label{5abril}
f(\varphi_{2}-\varphi_{1})\,dG_{1}\, dG_{2}=f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\, d\alpha_{2}\, dP.
\end{equation}
\enlargethispage{3.5mm}
\vspace*{-9pt}
\subsection{Integrals of functions of pairs of lines meeting a convex set}\label{maig13} For a point
$P\notin K$ let $\alpha$, $\beta$ be the directions we have introduced corresponding to the support lines of $K$ through $P$, with $0<\alpha<\pi/2$ and $\pi/2<\beta<\pi$.
Then $\omega =\beta-\alpha$ is the visual angle of $K$ from $P$.
This is the reason why we have slightly modified
the definition of the direction angle given by Santal\'o in \cite{santalo}
as the angle between the line through $P$ and the positive $x$ axis, because with this definition one could have $\omega =\beta-\alpha$ or $\omega =\pi-(\beta-\alpha)$; see Figure~\ref{fig:omega}.
We shall provide now a general formula to calculate the integral of the right-hand side of \eqref{5abril}.
\begin{proposition}\label{propdreta}
Let $f$ be a $2\pi$-periodic continuous function on $\R$, and $H$ a ${\mathcal C}^{2}$~function on $[-\pi,\pi]$ satisfying the conditions $H''(x)=f(x)\cdot \sin (x)$, $x\in [-\pi,\pi]$, and $ H(0)=H'(0)=0.$
Denote by $\alpha_{i}$ the direction of the line $G_{i}$. Then
\begin{multline*}\label{25g2}
\int_{G_{i}\cap K\neq\emptyset}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\,d\alpha_{2}\,dP\\*[-7pt]
=(H(\pi)-H(-\pi))F+\int_{P\notin K}(H(\omega)-H(-\omega))\,dP,
\end{multline*}
where $\omega=\omega(P)$ is the visual angle of $K$ from $P$.
\end{proposition}
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{omega1.png}\includegraphics[width=.5\textwidth]{omega2.png}
\caption{Visual angle of a convex set.}
\label{fig:omega}
\end{figure}
\begin{proof}
For a given point $P$ in the plane there are angles $\alpha(P)$, $\beta(P)$ such that the pairs of lines $G_{1}$, $G_{2}$ through $P$ that intersect the convex set $K$ are those satisfying $\alpha(P)\leq \alpha_{i}\leq \beta(P)$, where $\alpha_{i}=\alpha(G_{i})$. When $P\in K$ we have $\alpha(P)=0$ and $\beta(P)=\pi$.
We need to integrate the function $f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|$ over $[\alpha,\beta]^{2}$ with $\alpha=\alpha(P)$ and $\beta=\beta(P)$. In order to perform this integral we divide $[\alpha,\beta]^{2}$ into the union of the regions $\mathcal{R}_{1}=\{(\alpha_{1},\alpha_{2})\in [\alpha,\beta]^{2}: \alpha_{2}\geq \alpha_{1}\}$ and $\mathcal{R}_{2}=\{(\alpha_{1},\alpha_{2})\in [\alpha,\beta]^{2}: \alpha_{2}< \alpha_{1}\}.$ Therefore
\begin{multline*}
\int_{[\alpha,\beta]^{2}}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1} \,d\alpha_{2}\\
=\int_{\mathcal{R}_{1}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1} \,d\alpha_{2}-\int_{\mathcal{R}_{2}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1} \,d\alpha_{2}\\
=\int_{\alpha}^{\beta}\left(\int_{\alpha}^{\alpha_{2}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\right)\,d\alpha_{2}\\
-\int_{\alpha}^{\beta}\left(\int_{\alpha}^{\alpha_{1}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{2}\right)\,d\alpha_{1}\\
=\int_{\alpha}^{\beta}\left[-H'(\alpha_{2}-\alpha_{1})\right]_{\alpha}^{\alpha_{2}}\,d\alpha_{2}-\int_{\alpha}^{\beta}\left[H'(\alpha_{2}-\alpha_{1})\right]_{\alpha}^{\alpha_{1}}\,d\alpha_{1}
\\
=\left[H(\alpha_{2}-\alpha)\right]_{\alpha}^{\beta}-\left[H(\alpha-\alpha_{1})\right]_{\alpha}^{\beta}
=H(\beta-\alpha)-H(\alpha-\beta).
\end{multline*}
Hence
\begin{multline*}
\int_{G_{i}\cap K\neq\emptyset}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\,d\alpha_{2}\,dP \\ =\biggl(\int_{P\in K}+\int_{P\notin K}\biggr) (H(\beta-\alpha)-H(\alpha-\beta))\,dP.
\end{multline*}
Taking into account that the visual angle $\omega(P)$ is given by $\beta(P)-\alpha(P)$ the result follows.
\end{proof}
In the next result we assume the additional hypothesis that $f(x)$ is an even function.
\begin{proposition}\label{prop54}
Let $f$ be a $2\pi$-periodic continuous function on $\R$, with $f(-x)=f(x)$, $x\in\R$, and $H$ a ${\mathcal C}^{2}$ function on $[0,\pi]$ satisfying the conditions $H''(x)=f(x)\cdot \sin (x)$, $x\in [0,\pi]$, and $ H(0)=H'(0)=0.$
Denote by $\alpha_{i}$ the direction of the line $G_{i}$. Then
\begin{equation*}\label{25g}
\int_{G_{i}\cap K\neq\emptyset}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\,d\alpha_{2}\,dP=2H(\pi) F +2\int_{P\notin K} H(\omega)\,dP,
\end{equation*}
where $\omega=\omega(P)$ is the visual angle of $K$ from $P$.
\end{proposition}
\begin{proof}
Just proceed as in the above proof taking into account that
\begin{multline*}
\int_{[\alpha,\beta]^{2}}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1} \,d\alpha_{2}\\ =2\int_{\alpha}^{\beta}\left(\int_{\alpha}^{\alpha_{2}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\right)\,d\alpha_{2}.
\end{multline*}
\end{proof}
For the special case where $f$ is a $\pi$-periodic function one has
\begin{corollary}\label{centredreta}
Let $f$ be a $\pi$-periodic continuous function on $\R$, and $H$ a ${\mathcal C}^{2}$ function on $[-\pi,\pi]$ satisfying the conditions $H''(x)=f(x)\cdot \sin (x)$, $x\in [-\pi,\pi]$, and $ H(0)=H'(0)=0.$ Then
\begin{equation*}
\label{25g4}
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}\!=\! \left((H(\pi)-H(-\pi))F+\!\int_{P\notin K}(H(\omega)-H(-\omega))\,dP\right).\!
\end{equation*}
If moreover $f(-x)=f(x)$ and $H(x)$ is ${\mathcal C}^{2}$ on $[0,\pi]$ with $H''(x)=f(x)\cdot \sin (x)$, $x\in [0,\pi]$, and $ H(0)=H'(0)=0,$ one has
\begin{equation}\label{25gg}
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=2H(\pi) F +2\int_{P\notin K} H(\omega)\,dP.
\end{equation}
\end{corollary}
\begin{proof}
When $f$ is a $\pi$
-periodic function we have equality \eqref{5abril} and the result is then a consequence of Proposition \eqref{propdreta} and Proposition \eqref{prop54}.
\end{proof}
Integral formulas as those given in \eqref{eq31} and \eqref{25gg} open the possibility to prove interesting relations for quantities linked to convex sets. For instance when applied to the function $f(x)=\cos kx$ they give Hurwitz's formula \eqref{maig9-2} for $k$ even (see section \ref{hurwitzsec}).
For odd values of $k$ the Corollary \ref{centredreta} does not apply because $f(x)=\cos kx$ is not a $\pi$-periodic function. In this case we have $f(x+\pi)=-f(x)$ and we say that $f$ is an \emph{anti $\pi$-periodic} function.
For this type of functions we can modify the above proofs to obtain a new result that involve a decomposition of the visual angle $\omega$ into $\omega=\omega_{1}+\omega_{2}$ where $\omega_{1}$ and $\omega_{2}$ are defined in the following way.
Given a point $P\notin K$ we have considered in section \ref{maig13} the directions $0<\alpha<\pi/2<\beta<\pi$ of the support lines of $K$ through $P$ and the visual angle $\omega=\beta-\alpha$. Let us take $\omega_{1}=\pi/2-\alpha$ and $\omega_{2}=\beta-\pi/2$. Then we have
\begin{figure}[h]
\centering
\includegraphics[width=.4\textwidth]{omega12}
\caption{Angles $\omega_{1}$ and $\omega_{2}$.}
\label{fig:omega12}
\end{figure}
\begin{proposition}\label{antipi} Let $f$ be an anti $\pi$-periodic continuous function on $\R$ such that $f(x)=f(-x)$ and $H$ a ${\mathcal C}^{2}$ function on $[0,\pi]$ with $H''(x)=f(x)\cdot \sin (x)$, $x\in [0,\pi]$, and $ H(0)=H'(0)=0.$ Then
\begin{multline}\label{formanti}
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}\\
=2(2H(\pi/2)-H(\pi)) F +2\int_{P\notin K} (2H(\omega_{1})+2H(\omega_{2})-H(\omega))\,dP.
\end{multline}
\end{proposition}
\begin{proof}
In section \ref{maig7-2} we have seen that $\varphi_{2}-\varphi_{1}=\alpha_{2}-\alpha_{1}+\epsilon\pi$ where $\epsilon= \epsilon(P,\alpha_{1},\alpha_{2})=0,\pm 1$. Then
\begin{multline*}
\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}\\
=\int_{P\in\R^{2}}\int_{[\alpha(P),\beta(P)]^{2}}(-1)^{\epsilon}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\, d\alpha_{2}\, dP.
\end{multline*}
If $P\notin K$ we consider the regions
\begin{align*}
\mathcal{R}_{1}&=\{(\alpha_{1},\alpha_{2}): \alpha\leq \alpha_{1}<\alpha_{2}\leq \pi/2\},\\
\mathcal{R}_{2}&=\{(\alpha_{1},\alpha_{2}): \pi/2\leq \alpha_{1}<\alpha_{2}\leq \beta\},\\
\mathcal{R}_{3}&=\{(\alpha_{1},\alpha_{2}): \alpha\leq \alpha_{1}<\pi/2<\alpha_{2}\leq \beta\}.
\end{align*}
In $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ we have $\epsilon=1$ and $\epsilon=-1$ in region $\mathcal{R}_{3}$. Therefore, for $P\notin K$
\begin{multline*}
\int_{\alpha}^{\beta}\int_{\alpha}^{\beta}(-1)^{\epsilon}f(\alpha_{2}-\alpha_{1})|\mathrm{sin}(\alpha_{2}-\alpha_{1})|\,d\alpha_{1}\, d\alpha_{2}\\*[5pt]
=2\left(\int_{\mathcal{R}_{1}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\, d\alpha_{2}+
\int_{\mathcal{R}_{2}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\, d\alpha_{2}\right.\\*[5pt]
=\left. -\int_{\mathcal{R}_{3}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\, d\alpha_{2}\right).
\end{multline*}
The integrals over $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are easily computed and their values are $2H(\omega_{1})$ and $2H(\omega_{2})$ respectively. Let us compute the third integral.
\begin{multline*}
\int_{\mathcal{R}_{3}}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\, d\alpha_{2} \\*[5pt]
=\int_{\pi/2}^{\beta}\int_{\alpha}^{\pi/2}f(\alpha_{2}-\alpha_{1})\sin(\alpha_{2}-\alpha_{1})\,d\alpha_{1}\, d\alpha_{2})\\*[5pt]
=2\int_{\pi/2}^{{\beta}}\left[-H'(\alpha_{2}-\alpha_{1})\right]_{\alpha_{1}=\alpha}^{{\alpha_{1}=\pi/2}}\,d\alpha_{2}=
2\int_{\pi/2}^{\beta}(H'(\alpha_{2}-\alpha)-H'(\alpha_{2}-\pi/2))\,d\alpha_{2}\\*[5pt]
=2\left[H(\alpha_{2}-\alpha)-H(\alpha_{2}-\pi/2)\right]_{\alpha_{2}=\pi/2}^{\alpha_{2}=\beta}=2\left(H(\omega)-H(\beta-\pi/2)-H(\pi/2-\alpha)\right)\\*[5pt]
=2\left(H(\omega)-H(\omega_{2})-H(\omega_{1})\right).
\end{multline*}
Finally, for $G_{1}\cap G_{2}=P\notin K$ we have
\begin{equation}\label{pnotinantip}
\int_{P\notin K}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=2\int_{P\notin K}(2H(\omega_{1})+2H(\omega_{2})-H(\omega))\, dP.
\end{equation}
When $P\in K$ we do the same computations but now $\alpha=0$, $ \beta=\pi$ and $\omega=\beta-\alpha=\pi$ and so $\omega_{1}=\pi/2=\omega_{2}$. Thus
\begin{equation}\label{pinantip}
\int_{G_{1}\cap G_{2}\in K}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=2(4H(\pi/2)-H(\pi))F.
\end{equation}
Joining \eqref{pnotinantip} and \eqref{pinantip} the Proposition follows.
\end{proof}
\subsection{Interpretation in terms of densities of the formulas of Crofton, Masotti and powers of sine}\label{21maig-7}
In this section we will give an interpretation of the integrals of the visual angle appearing in the formulas of Crofton, Masotti and power of sine in terms of integrals of densities in the space of pairs of lines. For Hurwitz's formula this was done in Proposition \ref{maig17}.
\subsubsection*{Crofton's formula}
Taking $H(x)=x-\sin(x)$ it follows that $f=1$ in Corollary \ref{centredreta} and since $H(\pi)=\pi$ using \eqref{25gg} we get
\begin{proposition}\label{prop45} The following equality holds.
$$
\int_{G_{i}\cap K\neq \emptyset}\,dG_{1}\,dG_{2}=2\pi F+2\int_{P\notin K}(\omega-\sin\omega)\,dP.
$$
\end{proposition}
\subsubsection*{Masotti's formula}
Taking $H(x)=x^{2}-\sin^{2}(x)$ one gets $H''(x)/\sin(x)=4\sin(x)$. So the function $f(x)=4|\mathrm{sin}(x)|$, $x\in \R$, satisfies the hypothesis of Corollary \ref{centredreta} and equation \eqref{25gg} gives
\begin{proposition}\label{prop46} The following equality holds
\begin{equation*}
2\int_{G_{i}\cap K\neq\emptyset}|\mathrm{sin}(\varphi_{2}-\varphi_{1})|\,dG_{1}\,dG_{2}=\pi^{2} F +\int_{P\notin K} (\omega^{2}-\sin^{2}\omega)\,dP.
\end{equation*}
\end{proposition}
\subsubsection*{Powers of sine formula}
Finally, in an analogous way we can interpretate the integral of any power of the sine of the visual angle. Effectively for $H(x)= \sin^m(x)$ it follows that
$$
H''(x)/\sin(x)=m(m-1)\sin^{m-3}(x)-m^{2}\sin^{m-1}(x).
$$
So taking $f(x)=m(m-1)|\mathrm{sin}^{n-3}(x)|-m^{2}|\mathrm{sin}^{m-1}(x)|$
the hypothesis of Corollary~\ref{centredreta} are satisfied and by \eqref{25gg} we have
\begin{proposition}\label{prop47}The following equality holds
\begin{multline*}
2\int_{P\notin K}\sin^{m}(\omega)\,dP\\*[5pt]
=\int_{G_{i}\cap K\neq\emptyset}\left(m(m-1)|\mathrm{sin}^{m-3}(\varphi_{2}-\varphi_{1})|-m^{2}|\mathrm{sin}^{m-1}(\varphi_{2}-\varphi_{1})|\right)\,dG_{1}\,dG_{2}.
\end{multline*}
\end{proposition}
\section{New proofs of classical formulas}\label{seccio4}
Combining the results of the previous section with Theorem \ref{aagg} new proofs of the formulas of Masotti and the powers of sine can be obtained, in the spirit of the classical proof of Crofton's formula via Integral Geometry
To begin with we note that Theorem \ref{aagg} implies
the equality $\int_{G_{i}\cap K\neq \emptyset}\,dG_{1}\,dG_{2}=L^{2}$ which is also
an immediate consequence of the well known Cauchy--Crofton's formula (see \cite{santalo}). Now this equality together
with Proposition \ref{prop45} gives Crofton's formula
\begin{equation}\label{crofton18}
L^{2}=2\pi F+2\int_{P\notin K}(\omega-\sin\omega)\,dP.
\end{equation}
\subsubsection*{Masotti's formula}
A simple calculation shows that the Fourier expansion
of the function $|\mathrm{sin}(t)|$ is
\begin{equation}\label{8abril}
|\mathrm{sin}(t)|=\frac{2}{\pi}+\frac{4}{\pi}\sum_{n\geq 1}\frac{\cos(2nt)}{1-4n^{2}}.
\end{equation}
So by Theorem \ref{aagg},
$$
\int_{G_{i}\cap K\neq\emptyset} |\mathrm{sin}(\varphi_{2}-\varphi_{1})|\,dG_{1}\,dG_{2}=\frac{2L^{2}}{\pi}+4\pi\sum_{n\geq 1}\frac{c_{2n}^{2}}{1-4n^{2}},
$$
and using Proposition \ref{prop46}
one gets
\begin{equation*}\label{20jb}
\int_{P\notin K}(\omega^{2}-\sin^{2}\omega)\,dP=-\pi^{2}F+\frac{4L^{2}}{\pi}+8\pi\sum_{n\geq 1}\frac{c_{2n}^{2}}{1-4n^{2}},
\end{equation*}
which is Masotti's formula \eqref{21maig-4}.
\subsubsection*{Another example}
In the preceding sections we have interpreted integral formulas of some functions of the visual angle in terms of densities in the space of pairs of lines.
But one can also proceed in the reverse sense, that is to start from a density and to look for the corresponding function of the visual angle.
For instance the proof of Masotti's formula leads to compute
$\int_{G_{i}\cap K}|\mathrm{sin}(\varphi_{2}-\varphi_{1})|\,dG_{1}dG_{2}$.
If we consider now the density function $|\mathrm{cos}(\varphi_{2}-\varphi_{1})|$, using Theorem~\ref{aagg} and that
\begin{equation*}
|\mathrm{cos}(t)|=\frac{2}{\pi}+\frac{4}{\pi}\sum_{n\geq 1}\frac{(-1)^{n}\cos(2nt)}{1-4n^{2}}
\end{equation*}
we get
$$
\int_{G_{i}\cap K\neq\emptyset} |\mathrm{cos}(\varphi_{2}-\varphi_{1})|\,dG_{1}\,dG_{2}=\frac{2L^{2}}{\pi}+4\pi\sum_{n\geq 1}\frac{(-1)^{n}c_{2n}^{2}}{1-4n^{2}}.
$$
The function $H$ appearing in Corollary \ref{centredreta}
is in this case
$$
H(\omega)=
\begin{cases}
\frac{1}{4}(\omega-\sin\omega\cos\omega)& 0\leq\omega\leq \pi/2,\\[.3cm]
\frac{1}{4}(3\omega-\pi+\sin\omega\cos\omega)&\pi/2\leq \omega\leq \pi.
\end{cases}
$$
Hence, by \eqref{25gg} we have
$$
\int_{G_{i}\cap K\neq\emptyset} |\mathrm{cos}(\varphi_{2}-\varphi_{1})|\,dG_{1}\,dG_{2}=\pi F+ 2\int_{P\notin K}H(\omega)\,dP.
$$
\subsubsection*{Powers of sine formula}
In order to apply Theorem \ref{aagg} to the right-hand side of the equality in Proposition~\ref{prop47} we need to compute
the Fourier coefficients of the function $f(x)=m(m-1)|\mathrm{sin}^{m-3}(x)|-m^{2}|\mathrm{sin}^{m-1}(x)|$.
It is clear that $A_{k}=0$ for $k$ odd. For $k$ even we have
\begin{multline}\label{maig}
A_{k}=\frac{1}{\pi}\int_{0}^{2\pi}f(x)\cos(kx)\,dx\\*[5pt]
=\frac{1}{\pi}\left[2m(m-1)\int_{0}^{\pi}\sin^{m-3}x\cos(kx)\,dx
-2m^{2}\int_{0}^{\pi}\sin^{m-1}x\cos(kx)\,dx\right]\\*[5pt]
=\frac{1}{\pi}[2m(m-1)I_{m-3,k}-2m^{2}I_{m-1,k}],
\end{multline}
where
$$
I_{m,k}=\int_{0}^{\pi}\sin^{m}(x)\cos(kx)\,dx=(-1)^{k/2}\frac{2^{-m}m!\pi}{\Gamma(1+\frac{m-k}{2})\Gamma(1+\frac{m+k}{2})},
$$
(see, for instance, \cite{grads}, p. 372).
Substituting this expression in \eqref{maig} it follows
$$
A_{k}=\frac{m!}{2^{m-2}(m-2)}\frac{(-1)^{\frac{k}{2}+1}(k^{2}-1)}{\Gamma(\frac{m+1+k}{2})\Gamma(\frac{m+1-k}{2})}.
$$
Finally using Theorem \ref{aagg} we get
\begin{multline*}\label{21maig-5}
\int_{P\notin K}\sin^{m}(\omega)\,dP=\frac{m!}{2^{m}(m-2)\Gamma(\frac{m-1}{2})^{2}}\, L^{2} \\
+\frac{m!\pi^{2}}{2^{m-1}(m-2)}\sum_{k\geq 2, even}\frac{(-1)^{\frac{k}{2}+1}(k^{2}-1)}{\Gamma(\frac{m+1+k}{2})\Gamma(\frac{m+1-k}{2})}c_{k}^{2}.
\end{multline*}
Note that for $m$ odd the index $k$ in the sum runs only from $2$ to $m-1$.
This formula, which was first obtained by a different method in \cite{CGR}, provides an interpretation of the coefficients of $c_{k}^{2}$ as the Fourier coefficients of the above function $f$.
\subsubsection*{Crofton-Hurwitz's integral} \label{hurwitzsec}
In the above two previous sections
we have strongly used
equality \eqref{25gg} of Corollary~\ref{centredreta} that depends
on the fact that the function $f(x)$ is $\pi$-periodic, a fact that is crucial in order that equality \eqref{5abril} holds.
Consider now the function $f(x)=\cos kx$ with $k>1$. This function satisfies the hypothesis of Corollary~\ref{centredreta} for $k$ even and the hypothesis of Proposition~\ref{antipi} for $k$~odd. We have that
\begin{equation}\label{Hhurwitzk}
H_k(x)=\frac{1}{2(k^{2}-1)}\left(f_{k}(x)+2(\sin x-x)\right),
\end{equation}
with $f_{k}(x)$ the Hurwitz's function given in \eqref{maig9-3}, satisfies the equation $H_k''(x)=\cos kx\cdot \sin x,$ $x\in [0,\pi],$ and $H_k(0)=H_k'(0)=0.$
Therefore, for $k$ even, equalities~\eqref{eq31} and~\eqref{25gg} give
$$
\pi^{2}c_{k}^{2}= \int_{G_{i}\cap K\neq\emptyset}\cos(k(\varphi_{2}-\varphi_{1}))\,dG_{1}\,dG_{2} =-\frac{\pi F}{k^{2}-1}+2\int_{P\notin K}H_{k}(\omega)\,dP,
$$
and using Crofton's formula \eqref{crofton18} one gets a new proof of Hurwitz's formula \eqref{maig9-2} for $k$ even.
When $k$ is odd equation $\eqref{formanti}$ gives
\begin{multline*}
\int_{G_{i}\cap K\neq\emptyset}\cos k(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}=\\ =-{2\pi F\over k^{2}-1}+
2\int_{P\notin K} (2H_k(\omega_{1})+2H_k(\omega_{2})-H_k(\omega))\ dP.
\end{multline*}
Using the equality \eqref{eq31} one deduces that
$$
\int_{P\notin K}H_k(\omega)\,dP=-{\pi^{2}c_{k}^{2}\over 2}-{2\pi F\over k^{2}-1}+\int_{P\notin K} (H_k(\omega_{1})+H_k(\omega_{2})\,dP.
$$
Now by \eqref{Hhurwitzk} and Crofton's formula we obtain
\begin{equation}\label{hwtzsenar}
\int_{P\notin K}f_{k}(\omega)\,dP=L^{2}-\pi^{2}(k^{2}-1)c_{k}^{2}-2\pi F+2(k^{2}-1)\int_{P\notin K} (H_k(\omega_{1})+H_k(\omega_{2}))\,dP.
\end{equation}
Since we do not know the value of $\int_{P\notin K} (H_k(\omega_{1})+H_k(\omega_{2}))\,dP$ we are not able to prove Hurwitz formula in the case of $k$ odd. But from \eqref{maig9-2} we get the following result.
\begin{proposition}\label{21maig-6} Let $K$ be a compact convex set of area $F$. Then
\begin{equation}\label{maig13-2}
(k^{2}-1)\int_{P\notin K} (H_k(\omega_{1})+H_k(\omega_{2})\,dP=\pi F
\end{equation}
for each $k\geq 3$ odd, where $H_{k}$ is given in \eqref{Hhurwitzk}.
\end{proposition}
Notice that the above equation is equivalent to
\begin{equation}\label{maig15}
\int_{P\notin K} \left(f_{k}(\omega_{1})+2(\sin \omega_{1}-\omega_{1})+ f_{k}(\omega_{2})+2(\sin \omega_{2}-\omega_{2})\right)\,dP = 2\pi F.
\end{equation}
The function $H_{k}$ is the sum, except for a constant, of Hurwitz's function and Crofton's function and so are the terms in the above integrand.
The integral of the sum of Crofton's
and Hurwitz's functions of the visual angle is
$$
\int_{P\notin K}(f_{k}(\omega)+2(\sin \omega-\omega))\,dP= 2\pi F+ (-1)^{k}\pi^{2}(k^{2}-1)c_{k}^{2}, \quad k\geq 2.
$$
The surprising fact is that, for $k$ odd, decomposing the visual angle $\omega$
into the two parts $\omega=\omega_{1}+\omega_{2}$
and adding the corresponding integrals
one gets \eqref{maig15}
in which the right-hand side does not depend on $k$.
\medskip
In concluding we make the following remark. Theorem \ref{aagg} states that the integral $\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}$ depends only on the integrals $\int_{G_{i}\cap K\neq\emptyset}\cos k(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}$. So, by the results of section \ref{maig7-2} we are lead to calculate the functions~$H_{k}(x)$ such that $H_{k}''(x)=\cos(kx)\sin(x)$ with $H_{k}(0)=H_{k}'(0)=0$. These functions appear to be the sum of the functions of Hurtwitz and Crofton given in~\eqref{Hhurwitzk}, that is
$$
H_k(x)=\frac{1}{2(k^{2}-1)}\left(f_{k}(x)+2(\sin x-x)\right),\quad k\geq 2,
$$
and
$H_{1}(x)=(1/8)(2x-\sin(2x)).$
As a consequence when $f$ is a $\pi$-periodic density, according to Corollary \ref{centredreta},
the integral $\int_{G_{i}\cap K\neq\emptyset}f(\varphi_{2}-\varphi_{1})\,dG_{1}\,dG_{2}$ is a linear combination of integrals extended outside $K$
of the functions of the visual angle $H_{k}(\omega)$.
Likewise when the density~$f$ is anti $\pi$-periodic, according to Proposition
\ref{antipi}, the corresponding integral of the density
is
a linear combination of integrals extended outside $K$
of the functions~$H_{k}(\omega)$, $H_{k}(\omega_{1})$ and $H_{k}(\omega_{2})$.
\medskip
Summarizing, it appears that the functions of Crofton and Hurwitz are some kind of basis for the integral of any $\pi$-periodic or anti $\pi$-periodic density
with respect to the measure $dG_{1}\,dG_{2}$ over the set of pairs of lines
meeting a given compact convex set.
\bibliographystyle{plain}
\bibliography{CGRdP} | 43,096 |
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\begin{document}
\baselineskip=17pt
\title[Salem numbers]{ SURVEY ARTICLE: Seventy years of Salem numbers}
\author[C. J. Smyth]{Chris Smyth}
\address{School of Mathematics and Maxwell Institute for Mathematical Sciences\\University of Edinburgh\\Edinburgh, EH9 3JZ, Scotland}
\email{[email protected]}
\date{}
\begin{abstract}
I survey results about, and recent applications of, Salem numbers.
\end{abstract}
\subjclass[2010]{Primary 11R06}
\keywords{Salem number}
\maketitle
\section{Introduction} In this article I state and prove some basic results about Salem numbers, and then survey
some of the literature about them. My intention is to complement
other general treatises on these numbers, rather than to repeat their
coverage. This applies particularly to the work of Bertin and her
coauthors \cite{Bsurvey,Bbook,Blectnotes} and to the application-rich
Salem number survey of Ghate and Hironaka \cite{GH}. I have, however, quoted some fundamental results from Salem's classical monograph \cite{Sa}.
Recall that a complex number is an {\it algebraic integer} if it is the zero of a polynomial with integer coefficients and leading coefficient $1$. Then its (Galois) {\it conjugates} are the zeros of its {\it minimal polynomial}, which is the lowest degree polynomial of that type that it satisfies. This degree is the {\it degree} of the algebraic integer.
A {\it Salem number} is a real algebraic integer $\t>1$ of degree at
least $4$, conjugate to $\t^{-1}$, all of whose conjugates,
excluding $\t$ and $\t^{-1}$, have modulus $1$. Then $\t+\t^{-1}$ is a
real algebraic integer $>2$, all of whose conjugates $\ne
\t+\t^{-1}$lie in the real interval $(-2,2)$. Such numbers are easy
to find: an example is $\t+\t^{-1}=\frac12(3+\sqrt{5})$, giving
$(\t+\t^{-1}-\frac32)^2=\frac54$, so that $\t^4-3\t^3+3\t^2-3\t+1=0$ and
$\t=2.1537\dots$. We note that this polynomial is a so-called
(self)-reciprocal polynomial: it satisfies the equation $z^{\deg
P}P(z^{-1})=P(z)$. This means that its coefficients form a
palindromic sequence: they read the same backwards as forwards. This
holds for the minimal polynomial of every Salem number. It is simply
a consequence of $\t$ and $\t^{-1}$ having the same minimal
polynomial. Salem numbers are named after Rapha\" el Salem, who, in 1945, first defined and studied them \cite[Section 6]{S45}.
Salem numbers are usually defined in an apparently more general way,
as in the following lemma. It shows that this
apparent greater generality is illusory.
\begin{lem}[{{Salem \cite[p.26]{Sa}}}] Suppose that $\t>1$ is a real algebraic integer, all of whose conjugates $\ne\t$ lie in the closed unit disc $|z|\le 1$, with at least one on its boundary $|z|=1$. Then $\t$ is a Salem number (as defined above).
\end{lem}
\bepr Taking $\t'$ to be a conjugate of $\t$ on $|z|=1$, we have that $\overline{\t'}=\t'^{-1}$ is also a conjugate $\t''$ say, so that $\t'^{-1}=\t''$. For any other conjugate $\t_1$ of $\t$ we can apply a Galois automorphism mapping $\t''\mapsto \t_1$ to deduce that $\t_1=\t_2^{-1}$ for some conjugate $\t_2$ of $\t$. Hence the conjugates of $\t$ occur in pairs $\t',\t'^{-1}$. Since $\t$ itself is the only conjugate in $|z|>1$, it follows that $\t^{-1}$ is the only conjugate in $|z|<1$, and so all conjugates of $\t$ apart from $\t$ and $\t^{-1}$ in fact lie on $|z|=1$.
\enpr
It is known that an algebraic integer lying with all its conjugates on the unit circle must be a root of unity (Kronecker \cite{Kr}). So in some sense Salem numbers are the algebraic integers that are `the nearest things to roots of unity'. And, like roots of unity, the set of all Salem numbers is closed under taking powers.
\begin{lem}[{{Salem \cite[p.169]{S45}}}]\label{L-2} If $\t$ is a Salem number of degree $d$, then so is $\t^n$ for all $n\in \N$.
\end{lem}
\bepr If $\t$ is conjugate to $\t'$ then $\t^n$ is conjugate to $\t'^n$. So $\t^n$ will be a Salem number of degree $d$ unless some of its conjugates coincide: say $\t_1^n=\t_2^n$ with $\t_1\ne\t_2$. But then, by applying a Galois automorphism mapping $\t_1\mapsto \t$, we would have $\t^n=\t_3^n$ say, where $\t_3\ne \t$ is a conjugate of $\t$, giving $|\t^n|>1$ while $|\t_3^n|\le 1$, a contradiction.
\enpr
Which number fields contain Salem numbers? Of course one can simply choose a list of Salem numbers $\t,\t',\t'',\dots$ say, and then the number field $\Q(\t,\t',\t'',\dots)$ certainly contains $\t,\t',\t'',\dots$. However, if one is interested only in finding all Salem numbers in a field $\Q(\t)$ for $\t$ a given Salem number, we can be much more specific.
\begin{prop}[{{Salem \cite[p.169]{S45}}}]\label{P-3} \bei\item[(i)] A number field $K$ is of the form $\Q(\t)$ for some Salem number $\t$ if and only if $K$ has a totally real subfield $\Q(\al)$ of index $2$, and $K=\Q(\t)$ with $\t+\t^{-1}=\al$, where $\al>2$ is an irrational algebraic integer, all of whose conjugates $\ne\al$ lie in $(-2,2)$.
\item[(ii)] Suppose that $\t$ and $\t'$ are Salem numbers with $\t'\in\Q(\t)$. Then $\Q(\t')=\Q(\t)$ and, if $\t'>\t$, then $\t'/\t$ is also a Salem number in $\Q(\t)$.
\item[(iii)] If $K=\Q(\t)$ for some Salem number $\t$, then there is a Salem number $\t_1\in K$ such that the set of Salem numbers in $K$ consists of the powers of $\t_1$.
\eni
\end{prop}
\bepr \bei\item[(i)] Suppose that $K=\Q(\t)$ for some Salem number $\t$. Then, defining $\al=\t+\t^{-1}$, every conjugate $\al'$ of $\al$ is of the form $\al_1=\t_1+\t_1^{-1}$, where $\t_1$, a conjugaste of $\t$, either is one of $\t^{\pm 1}$ or has modulus $1$. Hence $\al_1$ is real, and so $\al$ is totally real with $\al>2$ and all other conjugates of $\al$ in $(-2,2)$. Also, because $\Q(\al)$ is a proper subfield of $\Q(\t)$ and $\t^2-\al\t+1=0$ we have $[\Q(\t):\Q(\al)]=2$.
Conversely, suppose that $K$ has a totally real subfield $\Q(\al)$ of index $2$, and $K=\Q(\t)$, where $\al>2$ is an irrational algebraic integer, all of whose conjugates $\ne\al$ lie in $(-2,2)$, and
$\t+\t^{-1}=\al$. Then for every conjugate $\t_1$ of $\t$ we have $\t_1+\t_1^{-1}=\al_1$ for some
conjugate $\al_1$ of $\al$, and for every conjugate $\al_1$ of $\al$ we have $\t_1+\t_1^{-1}=\al_1$ for some
conjugate $\t_1$ of $\t$. If $\al_1=\al$ then $\t_1=\t^{\pm 1}$. Otherwise, $\al_1\in(-2,2)$ and so $|\t_1|=1$. Furthermore, as $\al$ is irrational it does indeed have a conjugate $\al_1\in(-2,2)$ for which the corresponding $\t_1$ has modulus $1$. Hence $\t$ is a Salem number.
\item[(ii)] By (i) we can write $\t'$ as $\t'=p(\al)+\t q(\al)$, where $p(z),q(z)\in\Q[z]$ and $\al=\t+\t^{-1}$. If $\t'$ had lower degree than $\t$, say $[\Q(\t):\Q(\t')]=k>1$ then, for the $d=[\Q(\t):\Q]$ conjugates $\t_i$ of $\t$, $k$ of the values $p(\t_i+\t_i^{-1})+\t_i q(\t_i+\t_i^{-1})$ would equal $\t'$ and another $k$ of them would equal ${\t'}^{-1}$. In particular, $p(\t_i+\t_i^{-1})+\t_i q(\t_i+\t_i^{-1})$ would be real for some nonreal $\t_i$, giving $q(\t_i+\t_i^{-1})=0~$ and hence, on applying a suitable automorphism, that $q(\al)=0$. Thus we would have $\t'=p(\al)$, a contradiction, since $p(\al)$ is totally real. So $k=1$ and $\Q(\t')=\Q(\t)$.
Since $\t'\in \Q(\t)$, it is a polynomial in $\t$. Therefore any Galois automorphism taking $\t\mapsto\t^{-1}$ will map $\t'$ to a real conjugate of $\t'$, namely $\t'^{\pm 1}$. But it cannot map $\t'$ to itself for then, as $\t$ is also a polynomial in $\t'$, $\t$ would be mapped to itself, a contradiction. So $\t'$ is mapped to $\t'^{-1}$ by this automorphism. Hence $\t'\t^{-1}$ is conjugate to its reciprocal. So the conjugates of $\t'\t^{-1}$ occur in pairs $x, x^{-1}$. Again, because $\t'$ is a polynomial in $\t$, any automorphism fixing $\t$ will also fix $\t'$, and so fix $\t'\t^{-1}$. Likewise, any automorphism fixing $\t'$ will also fix $\t$.
Next consider any conjugate of $\t'\t^{-1}$ in $|z|>1$. It will be of the form
$\t_1'\t_1^{-1}$, where $\t_1'$ is a conjugate of $\t'$ and $\t_1$ is a conjugate of $\t$. For this to lie in $|z|>1$, we must either have $|\t_1'|>1$ or $|\t_1|<1$, i.e., $\t_1'=\t'$ or $\t_1=\t^{-1}$. But in the first case, as we have seen, $\t_1=\t$, so that $\t_1'\t_1^{-1}=\t'\t^{-1}$, while in the second case $\t_1'=\t'^{-1}$, giving $\t_1'\t_1^{-1}=\t\t'^{-1}<1$. Hence
$\t'\t^{-1}$ itself is the only conjugate of $\t'\t^{-1}$ in $|z|>1$. It follows that all conjugates of $\t'\t^{-1}$ apart from $(\t'\t^{-1})^{\pm 1}$ must lie on $|z|=1$, making $\t'\t^{-1}$ a Salem number.
\item[(iii)] Consider the set of all Salem numbers in $K=\Q(\t)$. Now the number of Salem numbers $<\t$ in $K$ is clearly finite, as there are only finitely many possibilities for the minimal polynomials of such numbers. Hence there is a smallest such number, $\t_1$ say. For any Salem number, $\t'$ say, in $K$ we can choose a positive integer $r$ such that $\t_1^r\le \t'<\t_1^{r+1}$. But if $\t_1^r< \t'$ then, by (ii) and Lemma \ref{L-2}, $\t'\t_1^{-r}$ would be another Salem number in $K$ which, moreover, would be less than $\t_1$, a contradiction.
Hence $\t'=\t_1^r$.
\eni
\enpr
The statements (ii) and (iii) above in fact represent a slight strengthening of Salem's results, as they do not assume that all Salem numbers in $K$ have degree $[K:\Q]$.
We now show that the powers of Salem numbers have an unusual property.
\begin{prop} [{{Essentially Salem \cite{S45}, reproduced in \cite{Sa}}}]\label{P-Teps} For every Salem number $\t$ and every $\eps>0$ there is a real number $\la>0$ such that the distance $\|\la\t^n\|$ of $\la\t^n$ to the nearest integer is less than $\eps$ for all $n\in\N$.
\end{prop}
The result's first explicit appearance seems to be in Boyd \cite{B-bad}, who credits Salem. A proof comes from an easy modification the proof in \cite[pp 164-166]{S45}.
\bepr We consider the standard embedding of the algebraic integers $\Z(\t)$ as a lattice in $\R^d$ defined for $k=0,1,\dots,d-1$ by the map
\[
\t^k\mapsto(\t^k,\t^{-k},\Re\t_2^k,\Im\t_2^k,\Re\t_3^k,\Im\t_3^k,\dots,\Re\t_{d/2}^k,\Im\t_{d/2}^k),
\]
where $\t^{\pm 1},\t_j^{\pm 1}\,(j=2,\dots,d/2)$ are the conjugates of $\t$. As this is a lattice of full dimension $d$, we know that for every $\eps'>0$ there are lattice points in the `slice' $\{(x_1,\dots,x_n)\in\R^n\, :\, |x_i|<\eps'\,(i=2,\dots,d)\}$. Such a lattice point corresponds to an element $\la(\t)$ of $\Z(\t)$ with conjugates $\la_i$ satisfying $|\la_i|<\sqrt{2}\eps'\,(i=2,\dots,d)$.
Next, consider the sums
\[
\s_n=\la(\t)\t^n+\la(\t^{-1})\t^{-n}+\la(\t_2)\t_2^n+\la(\t_2^{-1})\t_2^{-n}+\dots \la(\t_{d/2})\t_{d/2}^n+\la(\t_{d/2}^{-1})\t_{d/2}^{-n},
\]
where $\la(x)\in\Z[x]$. Since $\s_n$ is a symmetric function of the conjugates of $\t$, it is rational. As it is an algebraic integer, it is in fact a rational integer. Since all terms $\la(\t^{-1})\t^{-n}$, $\la(\t_2)\t_2^n$, $\la(\t_2^{-1})\t_2^{-n}$, \dots, $\la(\t_{d/2})\t_{d/2}^n$, $\la(\t_{d/2}^{-1})\t_{d/2}^{-n}$ are $<\sqrt{2}\eps'$ in modulus, we see that
\[
|\s_n-\la(\t)\t^n|<(d-1)\sqrt{2}\eps'.
\]
Hence, choosing $\eps'=\eps/((d-1)\sqrt{2})$, we have $\|\la\t^n\|\le|\s_n-\la(\t)\t^n|<\eps$.
\enpr
In fact, this property essentially characterises Salem (and Pisot) numbers among all real numbers. Recall that a {\it Pisot number} is an algebraic integer greater than $1$ all of whose conjugates, excluding itself, all lie in the open unit disc $|z|<1$. Pisot \cite{Pi} proved that if $\la$ and $\t$ are real numbers such that
\bee\label{EPi}
\|\la\t^n\|\le \frac{1}{2e\t(\t+1)(1+\log \la)}\qquad (= B \text{ say}),
\ene
for all integers $n\ge 0$ then $\t$ is either a Salem number or a Pisot number and $\la\in\Q(\t)$.
The denominator in this result was later improved by Cantor \cite{Ca} to $2e\t(\t+1)(2+\sqrt{\log \la})$, and then by Decomps-Guilloux and Grandet-Hugot \cite{DG} to
$e(\t+1)^2(2+\sqrt{\log \la})$.
However, Vijayaraghavan \cite{V} proved that for each $\eps>0$ there are uncountably many real numbers $\al>1$ such that $\|\al^n\|<\eps$ for all $n\ge 0$. To be compatible with \eqref{EPi}, it is clear that such $\al$ that are not Pisot or Salem numbers must be large (depending on $\eps$). Specifically, if $\al>(2e\eps)^{-1/2}$ then there is no contradiction to \eqref{EPi}. Furthermore, Boyd \cite{B-bad} proved that if the bound $B$ in \eqref{EPi} is replaced by $10B$ then $\t$ can be transcendental.
For more results concerning the distribution of the fractional parts of $\la\t^n$ for $\t$ a Salem number, see Dubickas \cite{D2}, Za\"\i mi \cite{Z1,Z2,Z3}, and Bugeaud's monograph \cite[Section 2.4]{Bu}.
\section{A smallest Salem number?} Define the polynomial $L(z)$ by
\[
L(z)=z^{10}+z^9-z^7-z^6-z^5-z^4-z^3+z+1.
\]
This is the minimal polynomial of the Salem number $\t_{10}=1.176\dots$, discovered by
D. H. Lehmer \cite{L1} in 1933 (i.e., before Salem numbers had been defined!). Curiously, the polynomial $L(-z)$ had appeared a year
earlier in Reidemeister's book \cite{Re} as the Alexander polynomial of the $(-2,3,7)$
pretzel knot. Lehmer's paper seems to be the first where what is now called the {\it Mahler measure} of a polynomial
appears: the Mahler measure $M(P)$ of a monic one-variable polynomial $P$ is the product
$\prod_i\max(1,|\al_i|)$ over the roots $\al_i$ of the polynomial. For a survey of Mahler measure see \cite{Ssurv}; for its multivariable generalisation, see Boyd \cite{Boyd-spec} and
Bertin and Lal\'\i n \cite{Bertin-Lalin}.
Lehmer also asked whether the Mahler measure of any nonzero noncyclotomic
irreducible polynomial with integer coefficients is bounded below by some constant $c>1$.
This is now commonly referred to as `Lehmer's problem' or (inaccurately) as `Lehmer's conjecture' --- see \cite{Ssurv}. If indeed there were such a bound,
then certainly Salem numbers would be bounded away from $1$, but this would not immediately imply that there is a smallest Salem number (but see the end of Section \ref{SS-3.1}). However, the `strong version' of `Lehmer's conjecture'
states that one can take $c=\t_{10}$, implying that there is indeed a smallest Salem number, namely $\t_{10}$. A consequence of this strong version is the following.
\begin{conj}\label{CS} Suppose that $d\in\N$ and $\al_1,\al_2,\dots,\al_d$ are real numbers with
$\al_1\in(2,\t_{10}+\t_{10}^{-1})$ and $\al_2,\al_3,\dots,\al_d\in(-2,2)$.
Then $\prod_{i=1}^d(x+\al_i)\not\in\Z[x]$.
\end{conj}
(Note that $\t_{10}+\t_{10}^{-1}=2.026\dots$.) For if there were $\al_1,\al_2,\dots,\al_d$ in the intervals stated with $\prod_{i=1}^d(x+\al_i)\in\Z[x]$, then the algebraic integer $\t>1$ defined by $\t+\t^{-1}=\al_1$ would be a Salem number less than $\t_{10}$.
\section{Construction of Salem numbers}
\subsection{Salem's method}\label{SS-3.1}
Salem \cite[Theorem IV, p.30]{Sa} found a simple way to construct infinite sequences of Salem numbers from Pisot numbers. Now if $P(z)$ is the minimal polynomial of a Pisot number, then, except possibly for some small values of $n$, the polynomials $S_{n,P,\pm 1}(z)=z^nP(z)\pm z^{\deg P}P(z^{-1})$ factor as the minimal polynomial of a Salem number, possibly multiplied by some cyclotomic polynomials. In particular, for $P(z)=z^3-z-1$, the minimal polynomial of the smallest Pisot number, $S_{8,P,-1}=(z-1)L(z)$. Salem's construction shows that every Pisot number is the limit on both sides of a sequence of Salem numbers. (The construction has to be modified slightly when $P$ is reciprocal; this occurs only for certain $P$ of degree $2$.)
Boyd \cite{B1} proved that all Salem numbers could be produced by Salem's construction, in fact with $n=1$. It turns out that many different Pisot numbers can be used to produce the same Salem number. These Pisot numbers can be much larger than the Salem number they produce.
In particular, on taking $P(z)=z^3-z-1$ and $\eps=-1$, the minimal polynomial of the smallest Pisot number $\theta_0=1.3247\dots$, Salem's method shows that there are infinitely many Salem numbers less than $\theta_0$. This fact motivates the next definition, due to Boyd.
Salem numbers less than $1.3$ are called {\it small}. A list of $39$ such numbers was compiled by Boyd \cite{B1}, with later additions of four each by Boyd \cite{B2} and Mossinghoff \cite{Mo1}, making $47$ in all -- see \cite{Mo2}. (The starred entries in this list are the four Salem numbers found by Mossinghoff. They include one of degree $46$.) Further, it was determined by Flammang, Grandcolas and Rhin \cite{FGR} that the table was complete up to degree $40$. This was extended up to degree $44$ by Mossinghoff, Rhin and Wu \cite{MRW} as part of a larger project to find small Mahler measures. Their result shows that if Conjecture \ref{CS} is false then any counterexample to that conjecture must have degree $d\ge 23$.
In \cite{B3} Boyd showed how to find, for a given $n\ge 2$, $\eps=\pm 1$ and real interval $[a,b]$, all Salem numbers in that interval that are roots of $S_{n,P,\eps}(z)=0$ for some Pisot number having minimal polynomial $P(z)$. In particular, of the four new small Salem numbers that he found, two were discovered by this method. The other two he found in \cite{B3} are not of this form: they are roots only of some $S_{1,P,\eps}(z)=0$.
Boyd and Bertin \cite{BB0,BB} investigated the properties of the polynomials $S_{1,P,\pm 1}(z)$ in detail. For a related, but interestingly different, way of constructing Salem numbers, see Boyd and Parry \cite{BP}.
Let $T$ denote the set of all Salem numbers (Salem's notation). (It couldn't be called $S$, because that is used for the set of all Pisot numbers. The notation $S$ here is in honour of Salem, however: Salem \cite{Sclosed} had proved the magnificent result that the Pisot numbers form a closed subset of the real line.) Salem's construction shows that the derived set (set of limit points) of $T$ contains $S$. Salem \cite[p.31]{Sa} wrote `We do not know whether numbers of $T$ have limit points other than $S$'. Boyd \cite[p. 327]{B1} conjectured that there were no other such limit points, i.e., that the derived set of $S\cup T$ is $S$.
(Not long before, he had conjectured \cite{B15} that $S\cup T$ is closed -- a conjecture that left open the possibility that some numbers in $T$ could be limit points of $T$.)
Salem's construction shows that every Pisot number is a limit from below of Salem numbers. So if the derived set of $S\cup T$ is indeed $S$, then any limit point of Salem numbers from above is also a limit point of Salem numbers from below. Hence Boyd's conjecture implies that there must be a smallest Salem number.
\subsection{Salem numbers and matrices}
One strategy that has been used to try to solve Lehmer's Problem is to attach some combinatorial object (knot, graph, matrix,\dots) to an algebraic number (for example, to a Salem number). But it is not clear whether
the object could throw light on the (e.g.) Salem number, or, on the contrary, that the Salem number could throw light on the object.
Typically, however, such attachment constructions seem to work only for a restricted class of algebraic numbers, and not in full generality. For example, McKee, Rowlinson, and Smyth considered star-like trees as the objects for attachment. this was extended by McKee and Smyth to more general graphs \cite{McKSm3} and then to integer symmetric matrices \cite{McKSm-ISM,McKSm-IMRN}. (These can be considered as generalisations of graphs: one can identify a graph with its adjacency matrix -- an integer symmetric matrix having all entries $0$ or $1$, with only zeros on the diagonal.) The main tool for their work was the following classical result, which deserves to be better known.
\begin{thm} [{{Cauchy's Interlacing Theorem -- for a proof see for instance \cite{Fisk}}}] Let $M$ be a real $n\times n$ symmetric matrix, and $M'$ be the matrix obtained from $M$ by removing the $i$th row and column. Then the eigenvalues $\la_1,\dots,\la_n$ of $M$ and the eigenvalues $\mu_1,\dots,\mu_{n-1}$ of $M'$ interlace, i.e.,
\[
\la_1\le\mu_1\le\la_2\le\mu_2\le\dots\le\mu_{n-1}\le\la_n.
\]
\end{thm}
We say that an $n\times n$ integer symmetric matrix $M$ is {\it cyclotomic} if all its eigenvalues lie in the interval $[-2,2]$. It is so-called because then its associated reciprocal polynomial
\[
P_M(z)=z^n\det\left((z+z^{-1})I-M\right)
\]
has all its roots on $|z|=1$ and so (Kronecker again) is a product of cyclotomic
polynomials. Here $I$ is the $n\times n$ identity matrix.
The cyclotomic graphs are very familiar.
\bt [{{ J.H. Smith \cite{Smi}}}] The connected cyclotomic graphs consist of the (not necessarily proper) induced subgraphs of the Coxeter graphs $\tilde A_n(n\ge 2)$, $\tilde D_n(n\ge 4)$, $\tilde E_6$, $\tilde E_7$, $\tilde E_8$, as in Figure \ref{F-except}.
\et
\begin{figure}[h]
\begin{center}
\leavevmode
\psfragscanon
\psfrag{A}[l]{$\tilde A_n$}
\psfrag{D}[l]{$\tilde D_n$}
\psfrag{6}[l]{$\tilde E_6$}
\psfrag{7}{$\tilde E_7$}
\psfrag{8}{$\tilde E_8$}
\includegraphics[scale=0.4]{exceptional804.eps}
\end{center}
\caption{The Coxeter graphs $\tilde E_6, \tilde E_7,
\tilde E_8, \tilde A_n (n\geq 2)$ and $ \tilde
D_n (n\geq 4)$. (The number of vertices is $1$ more than the index.)}\label{F-except}
\end{figure}
(These graphs also occur in the theory of Lie algebras, reflection groups, Lie groups, Tits geometries, surface singularities, subgroups of $\operatorname{SU}_2(\C)$ (McKay correspondence),\dots).
McKee and Smyth \cite{McKSm-ISM} describe all the cyclotomic matrices, of which the cyclotomic graphs form a small subset. They prove in \cite{McKSm-IMRN} that the strong version of Lehmer's conjecture is true for the set of polynomials $P_M$: namely, if $M$ is not a cyclotomic matrix, then $P_M$ has Mahler measure at least $\t_{10}=1.176\dots$, the smallest known Salem number. In fact they show that the smallest three known Salem numbers are all Mahler measures of $P_M$ for some integer symmetric matrix $M$, while the fourth-smallest known Salem number is not.
Most constructions of families of Salem numbers produce a monic reciprocal integer polynomials having roots $\t>1$, $1/\t<1$ and all other roots on the unit circle. The final stage of the construction requires that each polynomial be expressed as a product of irreducible factors. Then, by Kronecker's Theorem \cite{Kr}, the irreducible factor having $1/\t$ as a root must also have $\t$ as a root, and any other factors must be cyclotomic polynomials. For $\t$ to be a Salem number, the first-mentioned factor must also have a root of modulus $1$. To determine its degree, it is usually necessary to determine the degrees of the cyclotomic factors. One method, used by Smyth \cite{Sm2} and McKee and Yatsyna \cite{McKY} (see also Beukers and Smyth \cite{BeSm}), is to make use of the fact \cite[Lemma 2.1]{Sm2} that every root of unity $\om$ is conjugate to one of $\om^2$, $-\om^2$ or $-\om$. A second method is to use results of Mann \cite{Mann} on sums of roots of unity. This method was used by Gross, Hironaka and McMullen \cite{GHM} for polynomials coming from modified Coxeter diagrams, and by Brunotte and Thuswaldner \cite{BT} for polynomials coming from star-like trees.
For other construction methods for Salem numbers see Lakatos \cite{L1,L2,L3} and also \cite{McKSm1,McKSm2,McKSm3,McKSm4,MRS,Sm1,Sm2}. In particular, in \cite{L1,L3} Lakatos shows that Salem numbers arise as the spectral radius of Coxeter transformations of certain oriented graphs containing no oriented cycles.
\subsection{Traces of Salem numbers}
McMullen \cite[p.230]{mcm1} asked whether there are any Salem numbers of trace less than $-1$. McKee and Smyth \cite{McKSm1,McKSm2} found examples of Salem numbers of trace $-2$, and indeed showed that there are Salem numbers of every trace.
It is known \cite{Sm2} that the smallest degree of a Salem number of trace $-1$ is $8$, and there are Salem numbers of trace $-1$ for every even degree at least $8$. Recently McKee and Yatsyna \cite{McKY} have shown that there are Salem numbers of trace $-2$ for every even degree at least $38$.
If $\t$ has degree $d\ge 10$ then its trace is known to be at least at least $\lfloor 1-d/9\rfloor$ -- see \cite{McKSm2}. Conversely, Salem numbers having trace $-T\le -2$ must have degree $d\ge 2\lfloor 1+\frac92 T\rfloor$ (corrected from \cite[p. 35]{McKSm2}). For $-T=-2$ this bound is attained, it having been shown in \cite{McKSm1} that there are two Salem numbers of trace $-2$ and degree $20$.
For a summary of the status of even degrees $20<d<38$ for which there are known to be or not to be Salem numbers of degree $d$ and trace $-2$, see \cite[Section 3]{McKY}.
For $-T=-3$ the above bound gives $d\ge 28$, later improved by Flammang \cite{Fla} to $d\ge 30$. In the other direction El Omani, Rhin and Sac-\' Ep\' ee \cite{EORSE} have recently found Salem number examples of trace $-3$ and degree $34$. Their computations strongly suggest that $34$ is in fact the smallest degree for Salem numbers of trace $-3$.
Recently Liang and Wu \cite{LW} have shown that a Salem number of trace $-4$ must have degree at least $40$, and a Salem number of trace $-5$ must have degree at least $50$.
All of these lower bounds for the degree of a Salem number of given negative trace make use of the fact that, for a Salem number $\t$, the number $\t+1/\t+2$ is a totally positive algebraic integer. Thus known lower bounds of the type $\text{trace}(\alpha)>\la d'$ for totally positive algebraic integers
$\al$ of degree $d'\ge 5$ can be used. Specifically, one readily deduces that, from such an inequality, a Salem number of degree $d\ge 10$ has trace at least $\lfloor 1-(1-\la/2)d\rfloor$, and a Salem number of trace $-T\le -2$ has degree at least $2\lfloor\frac{T}{2-\la}\rfloor+2$. (The results quoted above are for $\la=16/9$.)
For an interesting survey of the trace problem for totally positive algebraic integers see Aguirre and Peral \cite{AP}.
\subsection{Distribution modulo 1 of the powers of a Salem number}\label{S-dense}
Let $\t>1$ be a Salem number of degree $d$. Salem \cite[Theorem V, p.33]{Sa} proved that although the powers $\t^n\,\text{mod }1$ of $\t$ are everywhere dense on $(0,1)$, they are not uniformly distributed on this interval. In fact, the asymptotic distribution function of $\t^n\,\text{mod }1$ is that of a sum $2\sum_{j=1}^{d/2-1}\cos{\boldsymbol\theta}_j$, where the ${\boldsymbol\theta}_j$ are independent random variables uniformly distributed on $[0,\pi]$. In the simplest case, $d=4$, a routine calculation shows that for $x\in[0,1]$
\begin{align*}
\text{Prob}(&\t^n\,\text{mod }1\le x)=\\ &\tfrac12+\frac{\arcsin(\frac{x+1}{2})+\arcsin(\frac{x}{2})+\arcsin(\frac{x-1}{2})+\arcsin(\frac{x-2}{2})}{\pi}.
\end{align*}
In particular, $\text{Prob}(1/4\le \t^n\,\text{mod }1\le 3/4)=0.4134038362$, significantly less than $1/2$.
More generally, Akiyama and Tanigawa \cite{AT} gave a quantitative description of how far the sequence $\t^n\,\text{mod }1$ is from being uniformly distributed. They show,
for $\tau$ a Salem number of degree $2d'\ge 8$ and $A_N(\{\tau^n\},I)$ being the number of $n\le N$ for which the fractional part $\{\t^N\}$ lies in a subinterval $I$ of $[0,1]$,
that $\lim_{N\to\infty} \frac{1}{N}A_N(\{\tau^n\},I)$ exists and satisfies
\[
\left|\lim_{N\to\infty} \tfrac{1}{N}A_N(\{\tau^n\},I)-|I|\right|\le 2\zeta\left(\tfrac12(d'-1)\right)(2\pi)^{1-d'}|I|.
\]
Here $|I|$ is the length of $I$.
Note that this difference tends to $0$ as $d'\to\infty$, so that $\{\tau^n\}$ is more nearly uniformly distributed mod $1$ for $\t$ of large degree.
See also \cite{DMR}.
\subsection{Sumsets of Salem numbers} Dubickas \cite{D1} shows that a sum of $m\ge 2$ Salem numbers cannot be a Salem number, but that for every $m\ge 2$ there are $m$ Salem numbers whose sum is a Pisot number and also $m$ Pisot numbers whose sum is a Salem number.
\subsection{Galois group of Salem number fields} Lalande \cite{Lal} and Christopoulos and McKee \cite{CMcK}
studied the Galois group of a number field defined by a Salem
number. Let $\t$ be a Salem number of degree $2n$, $K=\Q(\t)$ and
$L$ be its Galois closure. Then it is known that $G:=\Gal(L/\Q)\le
C_2^n\rtimes S_n$. Conversely, if $K$ is a real number field of
degree $2n>2$ with exactly $2$ real embeddings, and, for its Galois
closure $L$, that $G\le C_2^n\rtimes S_n$, then Lalande proved that
$K$ is generated by a Salem number.
Now, for a Salem number $\t$, let $K'=\Q(\t+\t^{-1})$, $L'$ be its
Galois closure and $N\subset G$ be the fixing group of $L'$. Then
Christopoulos and McKee showed that $G$ is isomorphic to $N\rtimes
\Gal(L'/\Q)$, where $N$ is isomorphic to either $C_2^n$ or
$C_2^{n-1}$. The latter case is possible only when $n$ is odd.
Amoroso \cite{A} found a lower bound, conditional on the Generalised
Riemann Hypothesis, for the exponent of the class group of such
number fields $L$.
\subsection{The range of polynomials $\Z[\t]$}
P. Borwein and Hare \cite{BH} studied the `spectrum' of values $a_0+a_1\t+\dots+a_n\t^n$ when the $a_i\in\{-1,1\}$, $n\in\N$ and $\t$ is a Salem number. They showed that if $\t$ was a Salem number defined by being the zero of a polynomial of the form $z^m-z^{m-1}-z^{m-2}-\cdots-z^2-z+1$ for some $m\ge 4$
, then this spectrum is discrete. They also asked \cite[Section 7]{BH}
\bei
\item Are these the only Salem numbers with this spectrum discrete?
\item Are the only $\t$ where this spectrum is discrete and $M(\t)<2$ necessarily Salem numbers or Pisot numbers?
\eni
Hare and Mossinghoff \cite{HM} show, given a Salem number $\t<\frac12(1+\sqrt{5})$ of degree at most $20$, that some sum of distinct powers of $-\t$ is zero, so that $-\t$ satisfies some Newman polynomial.
Feng \cite{Fe2} remarked that it follows from Garsia \cite[Lemma 1.51]{Ga} that, given a Salem number $\t$ and $m\in\N$ there exists $c>0$ and $k\in\N$ (depending on $\t$ and $m$) such that for each $n\in\N$
there are no nonzero numbers $\xi=\sum_{i=0}^{n-1} a_i\t^i$ with $a_i\in\Z$, $|a_i|\le m$ and $|\xi|<\frac{c}{n^k}$. He asks whether, conversely, if $\t$ is any non-Pisot number in $(1,m+1)$ with this property, then must $\t$ necessarily be a Salem number?
\subsection{Other Salem number studies} Salem \cite{S45}, \cite[p. 35]{Sa} proved that every Salem number is the quotient of two Pisot numbers.
For connections between small Salem numbers and exceptional units, see Silverman \cite{Sil}.
Dubickas and Smyth \cite{DS} showed that any line in $\mathbb C$ containing two nonreal conjugates of a Salem number cannot contain the Salem number itself.
Akiyama and Kwon \cite{AK} constructed Salem numbers satisfying polynomials whose coefficients are nearly constant.
For generalisations of Salem numbers, see Bertin \cite{BK1,BK2}, Cantor \cite{Ca2}, Kerada \cite{Ke}, Meyer \cite{Me}, Samet \cite{Sam}, Schreiber \cite{Schr} and Smyth \cite{Sm1}. Note the correction made to \cite{Sam} in \cite{Sm1}. See also Section \ref{s1} below for $2$-Salem numbers.
\section{Salem numbers outside Number Theory}
The survey of Ghate and Hironaka \cite{GH} contains many applications of Salem numbers, for the period up to 1999. Only a few of the applications they describe are briefly recalled here, in subsections \ref{s1}, \ref{s2} and \ref{s3}. Otherwise, I concentrate on developments since their paper appeared.
For some of these applications, the restriction that Salem numbers should have degree at least $4$ can be dropped: the results also hold for reciprocal Pisot numbers, whose minimal polynomials are $x^2-ax+1$ for $a\in\N$, $n\ge 3$. Some authors include these numbers in the definition of Salem numbers. Accordingly, I will allow these numbers to be (honorary!) Salem numbers in this section.
(Note, however, that for $\t$ such a `quadratic Salem number', the fractional parts of the sequence $\{\t^n\}_{n\in\N}$ tend to $1$ as $n\to\infty$, whereas for true Salem numbers this sequence is dense in $(0,1)$, as stated in Section \ref{S-dense}. Furthermore, neither Proposition \ref{P-3}(ii) nor (iii) would hold, since the proof of Lemma \ref{P-3}(ii) depends on a Salem number having a nonreal conjugate. Indeed, for the example Salem number $\t=2.1537\dots$ in the introduction, the field $\Q(\t)$ contains the Salem numbers $\t^n\,(n=1,2,\dots)$, with $\t$ the smallest Salem number in this field. However, $\Q(\t)$ also contains the `quadratic Salem number' $\t+\t^{-1}=\frac12(3+\sqrt{5})$, which is not a power of $\t$.)
\subsection{Growth of groups}\label{s1} For a group $G$ with finite generating set $S=S^{-1}$, we define its {\it growth series} $F_{G,S}(x)=\sum_{n=0}^\infty a_nx^n$, where $a_n$ is the number of elements of $G$ that can be represented as the product of $n$ elements of $S$, but not by fewer. For certain such groups, $F_{G,S}(x)$ is known to be a rational function. Then expanding $F_{G,S}(x)$ out in partial fractions leads to a closed formula for the $a_n$. See \cite[Section 4]{GH} for a detailed description, including references. See also \cite{BCT}.
In particular, let $G$ be a Coxeter group generated by reflections in $d\ge 3$ geodesics in the upper half plane, forming a polygon with angles $\pi/p_i\,(i=1,2,\dots,d)$, where $\sum_i\pi/p_i<\pi$. Taking $S$ to be the set of these reflections, it is known (Cannon and Wagreich, Floyd and Plotnick, Parry) that then the denominator of $F_{G,S}(x)$ -- call it $\Delta_{p_1,p_2,\dots,p_d}(x)$ -- is the minimal polynomial of a Salem number, $\tau$ say, possibly multiplied by some cyclotomic polynomials. Then the $a_n$ grow exponentially with growth rate $\lim_{n\to\infty}a_{n+1}/a_n=\tau$. Hironaka \cite{Hi} proved that among all such $\Delta_{p_1,p_2,\dots,p_d}(x)$, the lowest growth rate was achieved for
$\Delta_{p_1,p_2,p_3}(x)$, which is Lehmer's polynomial $L(x)$, with growth rate $\t_{10}=1.176\dots$.
To generalise a bit, define a real $2$-Salem number to be an algebraic integer $\al>1$ which has exactly one conjugate $\al'\ne \al$ outside the closed unit disc, and at least one conjugate on the unit circle. Then all conjugates of $\al$ apart from $\al^{\pm 1}$ and ${\al'}^{\pm 1}$ have modulus $1$.
Zerht and Zerht-Liebensd\" orfer \cite{ZZ} give examples of infinitely many cocompact Coxeter groups (``Coxeter Garlands'') in $\mathbb H^4$ with the property that their growth function has denominator
\begin{align*}
D_n(z)&=p_n(z)+nq_n(z)\\
&=z^{16}\! -\! 2z^{15}\! +\! z^{14}\! -\! z^{13}\! +\! z^{12}\! -\! z^{10}\! +\! 2z^9\! -\! 2z^8\! +\! 2z^7\! -\! z^6\! +\! z^4\! -\! z^3\! +\! z^2\! -\! 2z\! +\! 1\\
&\qquad+nz(-2z^{14}+z^{12}+z^{10}+z^9+2z^7+z^5+z^4+z^2-2),
\end{align*}
which, if irreducible, would be the minimal polynomial of a $2$-Salem number.
Umemoto \cite{Um} showed that $D_1(t)$ is irreducible\footnote{In fact, one can show that $D_n(z)$ is irreducible for all $n\ge 1$. A sketch is as follows: comparison with the table \cite{Mo2} shows that neither root of $D_n(z)$ in $|z|>1$ can be a Salem number. Then putting $z=e^{it}$, the fact that $e^{-8it}p_n(e^{it})/q_n(e^{it})$ is real and $>0$ for small $t>0$ shows that $D_n(z)$ has no cyclotomic factors. (Selberg \cite[p. 705]{Sel} remarks that he has always found a sketch of a proof much more informative than a complete proof.)},
and also produced infinitely many cocompact Coxeter groups whose growth rate is a $2$-Salem number of degree $18$. The growth rate in these examples is the larger of the two $2$-Salem conjugates that are outside the unit circle. This is compatible with a conjecture of Kellerhals and Perren \cite{KP} that the growth rate of a Coxeter group acting on hyperbolic $n$-space should be a Perron number (an algebraic integer $\al$ whose conjugates different from $\al$ are all of modulus less than $|\al|$.) This has been verified for $n=3$ for so-called generalised simplex groups by Komori and Umemoto \cite{KU}.
For some other recent papers on non-Salem growth rates see \cite{Kel}, \cite{KK}, \cite{Ko}.
\subsection{Alexander Polynomials}\label{s2} A result of Seifert tells us that a polynomial $P\in\Z[x]$ is the Alexander polynomial of some knot iff it is monic and reciprocal, and $P(1)=\pm 1$. In particular, Hironaka \cite{Hi} showed that
$\Delta_{p_1,p_2,\dots,p_d}(-x)$ is the Alexander polynomial of the $(p_1,p_2,\dots,p_d,-1)$ pretzel knot. Hence, from the result of the previous section, we see that Alexander polynomials are sometimes Salem polynomials (albeit in $-x$).
Indeed, Silver and Williams \cite{SW}, in their study of Mahler measures of Alexander polynomials, found families of links whose Alexander polynomials had Mahler measure equal to a Salem number. The first family $l(q)$ was obtained \cite[Example 5.1]{SW} from the link $7^2_1$ by giving $q$ full right-handed twists to one of the components as it passed through the other component (the trivial knot). The Mahler measure of the Alexander polynomials of these links produced a decreasing sequence of Salem numbers for $q=1,2,\dots,11$. For $q=10$ the Salem number 1.18836\dots (the second-smallest known) was produced, with minimal polynomial$$x^{18}-x^{17}+x^{16}-x^{15}-x^{12}+x^{11}-x^{10}+x^{9}-x^{8}+x^{7}-x^{6}-x^{3}+x^{2}-x+1,$$ while $q=11$ gave the Salem number $M(L(x))=1.17628\dots$. For $q>11$ Salem numbers were not produced.
The second example was obtained in a similar way \cite[Example 5.8]{SW}, using the link formed from the knot $5_1$ by an adding the trivial knot encircling two strands of the knot, and then giving these strands $q$ full right-hand twists. For increasing $q\ge 3$ this gave a monotonically increasing sequence of Salem numbers tending to the smallest Pisot number $\theta_0=1.3247\dots$. These Salem numbers are equal to the Mahler measure $M(x^{2(q+1)}(x^3-x-1)+x^3+x^2-1)$. Furthermore, $M(x^{n}(x^3-x-1)+x^3+x^2-1)$ is also a Salem number for $n\ge 9$ and odd. Silver (private communication) has shown that these Salem numbers are also Mahler measures of Alexander polynomials: ``Putting an odd number of half-twists in the rightmost arm of the pretzel knot produces 2-component links rather than knots.
Their Alexander polynomials have two variables. However, setting the two variables equal to each other produces the so-called 1-variable
Alexander polynomials, and indeed the `odd' sequence of Salem polynomials \dots results.''
\subsection{Lengths of closed geodesics}\label{s3} It is known that there is a bijection between the set of Salem numbers and the set of closed geodesics on certain arithmetic hyperbolic surfaces. Specifically, the length of the geodesic is $2\log\tau$, where $\tau$ is the Salem number corresponding to the geodesic. Thus there is a smallest Salem number iff there is a geodesic of minimal length among all closed geodesics on all arithmetic hyperbolic surfaces. See Ghate and Hironaka \cite[Section 3.4]{GH} and also Maclachlan and Reid \cite[Section 12.3]{MR} for details.
\subsection{Arithmetic Fuchsian groups} Neumann and Reid \cite[Lemmas 4.9, 4.10]{NR} have shown that Salem numbers are precisely the spectral radii of hyperbolic elements of arithmetic Fuchsian groups. See also \cite{GH}, \cite[pp. 378--380]{MR} and \cite[Theorem 9.7]{Lei}.
The following result is related.
\bt[{{ Sury \cite{Su} }}] The set of Salem numbers is bounded away from $1$ iff there is some neighbourhood $U$ of the identity in $\SL_2(\R)$ such that, for each arithmetic cocompact Fuchsian group $\Gamma$, the set $\Gamma\cap U$ consists only of elements of finite order.
\et
A Fuchsian group is a subgroup $\Gamma$ of $\PSL_2(\R)$ that acts discontinuously on the upper half-plane $\mathbb H$ (i.e., for $z\in\mathbb H$ no orbit $\Gamma z$ has an accumulation point).
\subsection{Dynamical systems}
\subsubsection{} For given $\beta>1$, define the map $T_\beta:[0,1]\to[0,1)$ by $T_\beta x=\{\beta x\}$, the fractional part of $\beta x$. Then from $x=\frac{\lfloor \beta x\rfloor}{\beta}+\frac{T_\beta x}{\beta}$ we obtain the identity $x=\sum_{n=1}^\infty \frac{\lfloor \beta T_\beta^{n-1} x\rfloor}{\beta^n}$, the (greedy) {\it $\beta$-expansion of $x$} \cite{Pa}.
Klaus Schmidt \cite{Sch} showed that if the orbit of $1$ is eventually periodic for all $x\in\Q\cap[0,1)$ then $\beta$ is a Salem or Pisot number. He also conjectured that, conversely, for $\beta$ a Salem number, the orbit of $1$ is eventually periodic. This conjecture was proved by Boyd \cite{B4} to hold for Salem numbers of degree $4$. However, using a heuristic model in \cite{B5}, his results indicated that while Schmidt's conjecture was likely to also hold for Salem numbers of degree $6$, it may be false for a positive proportion of Salem numbers of degree $8$.
As Boyd points out, the basic reason seems to be that, for $\beta$ a Salem number of degree $d$, this orbit corresponds to a pseudorandom walk on a $d$-dimensional lattice. Under this model, but assuming true randomness, the probability of the walk intersecting itself is $1$ for $d\le 6$, but is less than $1$ for $d>6$.
Recently, computational degree-8 evidence relating to Boyd's model was compiled by Hichri \cite{Hic}, \cite{Hic2}.
Hare and Tweddle \cite[Theorem 8]{HT} give examples of Pisot numbers for which the sequences of Salem numbers from Salem's construction that tend to the Pisot number from above have eventually periodic orbits.
See also \cite{B4.5}.
For a survey of $\beta$-expansions when $\beta$ is a Pisot or Salem number, see \cite{Ha-conf}. See also Vaz, Martins Rodrigues, and Sousa Ramos \cite{VMP}.
\subsubsection{} Lindenstrauss and Schmidt \cite [Theorem 6.3]{LS} showed that there exists ``a connection between two-sided beta-shifts of Salem numbers and the nonhyperbolic ergodic toral automorphisms defined by
the companion matrices of their minimal polynomials. However, this connection is
much more complicated and tenuous than in the Pisot case'' -- see \cite{Sch2} and \cite[Theorem 6.1]{LS}.
\subsection{Surface automorphisms }
A {\it K3 surface} is a simply-connected compact complex surface $X$ with trivial canonical bundle. The intersection form on $H^2(X,\Z)$ makes it into an even unimodular 22-dimensional lattice of signature $(3,19)$; see \cite[p.17]{mcm5}. Now let $F:X\to X$ be an automorphism of positive entropy of a K3 surface $X$. Then McMullen \cite[Theorem 3.2]{mcm1} has proved that the spectral radius $\la(F)$ (modulus of the largest eigenvalue) of $F$ acting by pullback on this lattice is a Salem number. More specifically,
the characteristic polynomial $\chi(F)$ of the induced map $F^*|H^2(X,\R)\to H^2(X,\R)$ is the minimal polynomial of a Salem number
multiplied by $k\ge 0$ cyclotomic polynomials. Since $\chi(F)$ has degree $22$, the degree of $\la(F)$ is at most 22. (If $X$ is projective, $X$ has Picard group of rank at most 20, and so $\la(F)$ has degree at most 20.)
It is an interesting problem to describe which Salem numbers arise in this way.
McMullen \cite{mcm1} found 10 Salem numbers of degree 22 and trace $-1$, also having some other properties, from which he was able to construct from each of these Salem numbers a K3 surface automorphism having a Siegel disc. (These were the first known examples having Siegel discs.)
Gross and McMullen \cite{GM} have shown that if the minimal polynomial $S(x)$ of a Salem number of degree 22 has $|S(-1)|=|S(1)|=1$ (which they call the {\it unramified} case) then it is the characteristic polynomial of an automorphism of some (non-projective) K3 surface $X$.
(If the entropy of $F$ is $0$ then this characteristic polynomial is simply a product of cyclotomic polynomials.)
It is known (see \cite [p.211]{mcm1} and references given there) that the topological entropy $h(F)$ of $F$ is equal to $\log\la(F)$, so is either $0$ or the logarithm of a Salem number.
For each even $d\ge 2$ let $\t_d$ be the smallest Salem number of degree $d$.
McMullen \cite[Theorem 1.2]{mcm3} has proved that if $F : X \to X$ is an automorphism of {\it any} compact complex surface $X$ with
positive entropy, then $h(F) \ge \log \t_{10}=\log(1.176\dots)=0.162\dots$.
Bedford and Kim \cite{BK} have shown that this lower bound is realised by a particular rational surface automorphism. McMullen
\cite{mcm4} showed that it was realised for a non-projective K3 surface automorphism, and later \cite{mcm5} that it was realised for a projective K3 surface automorphism. He showed that the value $\log\t_d$ was realised for a projective K3 surface automorphism
for $d=2, 4, 6, 8, 10$ or $18$, but not for $d=14$, $16$, or $20$.
(The case $d=12$ is currently undecided.)
Oguiso \cite{O2} remarked that, as for K3 surfaces (see above), the characteristic polynomial of an automorphism of arbitrary compact K\" ahler surface is also the minimal polynomial of a Salem number multiplied by $k\ge 0$ cyclotomic polynomials. This is because McMullen's proof for K3 surfaces in \cite{mcm1} is readily generalised. In another paper \cite{O1} he proved that this result also held for automorphisms of hyper-K\" ahler manifolds. Oguiso \cite{O2} also constructed an automorphism $F$ of a (projective) K3 surface
with $\la(F)=\t_{14}$. Here the K3 surface was projective, contained an $E_8$ configuration of rational curves, and the automorphism also had a Siegel disc.
Reschke \cite{R} studied the automorphisms of two-dimensional complex tori. He showed that the entropy of such an automorphism, if positive, must be a Salem number of degree at most 6, and gave necessary and sufficient conditions for such a Salem number to arise in this way.
\subsection{Salem numbers and Coxeter systems} Consider a Coxeter system $(W,S)$, consisting of a multiplicative group $W$ generated by a finite set $S=\{s_1,\dots,s_n\}$, with relations $(s_is_j)^{m_{ij}}=1$ for each $i,j,$ where $m_{ii}=1$ and $m_{ij}\ge 2$ for $i\ne j$. The $s_i$ act as reflections on $\mathbb R^n$.
For any $w\in W$ let $\la(w)$ denote its spectral radius. This is the modulus of the largest eigenvalue of its action on $\R^n$.
Then McMullen \cite[Theorem 1.1]{mcm2} shows that when $\la(w)>1$ then $\la(w)\ge \t_{10}=1.176\dots$. This could be interpreted as circumstantial evidence for $\t_{10}$ indeed being the smallest Salem number.
The Coxeter diagram of $(W, S)$ is the weighted graph whose vertices are the
set $S$, and whose edges of weight $m_{ij}$ join $s_i$ to $s_j$ when $m_{ij} \ge 3$.
Denoting by $Y_{a,b,c}$ the Coxeter system whose diagram is a tree with 3 branches of
lengths $a$, $b$ and $c$, joined at a single node, McMullen also showed that
the smallest Salem numbers of degrees 6, 8 and 10 coincide with $\la(w)$ for
the Coxeter elements of $Y_{3,3,4}$, $Y_{2,4,5}$ and $Y_{2,3,7}$ respectively.
In particular, $\la(w)=\t_{10}$ for the Coxeter elements of $Y_{2,3,7}$.
\subsection{Dilatation of pseudo-Anosov automorphisms} For a closed connected oriented surface $\mathcal S$ having a pseudo-Anosov automorphism that is a product of two positive multi-twists, Leininger \cite[Theorem 6.2]{Lei} showed that its dilatation is at least $\t_{10}$. This follows from McMullen's work on Coxeter systems quoted above. The case of equality is explicitly described (in particular, $\mathcal S$ has genus 5).
(However, on surfaces of genus $g$ there are examples of pseudo-Anosov automorphisms having dilatations equal to $1+O(1/g)$ as $g\to\infty$. These are not Salem numbers when $g$ is sufficiently large.)
\subsection{Bernoulli convolutions} Following Solomyak \cite{So}, let $\la\in(0,1)$, and $Y_\la=\sum_{n=0}^\infty\pm \la^n$, with the $\pm$ chosen independently `$+$' or `$-$' each with probability $\frac12$. Let $\nu_\la(E)$ be the probability that $Y_\la\in E$, for any Borel set $E$. So it is the infinite convolution product of the means $\frac12(\delta_{-\la^n}+\delta_{\la^n})$ for $n=0,1,2,\dots,\infty$, and so is called a {\it Bernoulli convolution}. Then $\nu_\la(E)$ satisfies the self-similarity property
\[
\nu_\la(E)=\frac12\left(\nu_\la(S_1^{-1}E)+\nu_\la(S_2^{-1}E)\right),
\]
where $S_1x=1+\la x$ and $S_2x=1-\la x$. It is known that the support of $\nu_\la$ is a Cantor set of zero length when $\la\in(0,\frac12)$, and the interval $[-(1-\la)^{-1},(1-\la)^{-1}]$ when $\la\in(\frac12,1)$. When $\la=\frac12$, $\nu_\la$ is the uniform measure on $[-2,2]$. Now the Fourier transform $\hat\nu_\la(\xi)$ of $\nu_\la$ is equal to $\prod_{n=0}^\infty \cos(\la^n\xi)$. Salem \cite[p. 40]{Sam} proved that if $\la\in(0,1)$ and $1/\la$ is not a Pisot number, then $\lim_{\xi\to\infty}\hat\nu_\la(\xi)=0$. This contrasts with an earlier result of Erd\H os that if $\la\ne \frac12$ and $1/\la$ is a Pisot number, then $\hat\nu_\la(\xi)$ does not tend to $0$ as $\xi\to\infty$. Furthermore Kahane \cite{Ka} (as corrected in \cite[p. 10]{PSS}) proved that if $1/\la$ is a Salem number then for each $\eps>0$
\begin{equation}\label{E-xi}
\limsup_{\xi\to\infty} |\hat\nu_\la(\xi)|\,|\xi|^\eps=\infty.
\end{equation}
The only property of Salem numbers used in the proof of \eqref{E-xi} is Proposition \ref{P-Teps}.
Now consider the set $S_{a,\gamma}$ of all $\la\in(a,1)$ such that $\int_{-\infty}^\infty |\hat\nu_\la(\xi)|^2\,|\xi|^{2\gamma} d\xi<\infty$. Then Peres, Schlag and Solomyak \cite[Proposition 5.1]{PSS}
use this set, Lemma \ref{L-2} and \eqref{E-xi}, to give a sufficient condition for the set $T$ of Salem numbers to be bounded away from $1$. Specifically, they prove that if there exist some $\gamma>0$ and $a<1$ such that $S_{a,\gamma}$ is a so-called {\it residual set} (i.e., it is the intersection of countably many sets with dense interiors), then $1$ is not a limit point of $T$.
Concerning the behaviour of $\hat\nu_\la(\xi)$ as $\xi\to\infty$, where $1/\la=\theta>1$ is an algebraic integer: if $\theta$ has another conjugate $\theta'\ne \theta$ outside the unit circle, then Bufetov and Solomyak \cite[Corollary A.3]{BS} recently proved that $\hat\nu_\la(\xi)$ is bounded by a negative power of $\log\xi$ as $\xi\to\infty$.
On the other hand, it is a result of Erd\H os \cite{Er} that when $\theta$ is a Pisot number then $\limsup_{\xi\to\infty}\hat\nu_\la(\xi)$ is positive, showing that such a bound is not possible for Pisot numbers. Bufetov and Solomyak ask, however, whether a bound of that kind might hold for Salem numbers (the only remaining undecided case for an algebraic integer $\theta>1$).
Recently Feng \cite{Fe} has also studied $\nu_\la$ when $1/\la$ is a Salem number, proving that then the corresponding measure $\nu_\la$ is a multifractal measure satisfying the multifractal formalism in all of the increasing part of its multifractal spectrum.
For two very readable surveys of Bernoulli convolutions, including connections with Salem numbers, see Kahane's `Reflections on Paul Erd\H os \dots' article \cite{K-AMS} and the much longer survey by Peres, Schlag and Solomyak \cite{PSS} referred to above.
{\bf Acknowledgements.} This paper is an expanded version of a talk
that I gave at the meeting `Growth and Mahler measures in geometry
and topology' at the Mittag-Leffler Institute, Djursholm, Sweden,
in July 2013. I would like to thank the organisers Eriko Hironaka
and Ruth Kellerhals for the invitation to attend the meeting, and to
thank them, the Institute staff and fellow participants for making it such a stimulating
and pleasant week.
I also thank David Boyd, Yann Bugeaud, Nigel Byott, Eriko Hironaka, Aleksander Kolpakov, James McKee, Curtis McMullen, Andrew Ranicki, Georges Rhin, Dan Silver, Joe Silverman, Boris Solomyak, Timothy Trudgian and especially the referee for their very helpful comments and corrections on earlier drafts of this survey. | 106,474 |
TITLE: $\delta=0$ in Lyapunov's condition of CLT
QUESTION [1 upvotes]: In my textbook, Lyapunov's condition is shown to imply Lindeberg's condition by:
$$\sum_{k=1}^nE1_{(|X_{nk}|>c)}X_{nk}^2\le\sum_{k=1}^nE1_{(|X_{nk}|>c)}\tfrac{1}{c^\delta}|X_{nk}|^{2+\delta}\le\frac{1}{c^\delta}\sum_{k=1}^nE|X_{nk}|^{2+\delta}$$
Hence if $\sum_{k=1}^nE|X_{nk}|^{2+\delta}$ converges, so does $\sum_{k=1}^nE1_{(|X_{nk}|>c)}X_{nk}^2$.
What I don't get is why we need $\delta>0$. It seems like the above argument works just as well for $\delta=0$? Yet any source I can find requires $\delta>0$ which gives them a lot of problems with existence of moments.
Am I missing something totally obvious here?
REPLY [2 votes]: Lindeberg's condition is $1/s_n^2$ times your left side (with $c = \epsilon s_n$) $\to 0$ as $n \to \infty$. The $1/c^\delta$ (which becomes $1/(\epsilon s_n)^\delta$) with $\delta > 0$ is needed to make this go to $0$.
Let's look at the simple case where $X_{nk}$ are iid, and $s_n = \sqrt{n} \sigma$ where $\sigma$ is the standard deviation of $X_{nk}$. With $\delta = 0$, $\sum_{k=1}^n E[ X_{nk}^2] = n$, and after dividing by $s_n^2$ all you get is boundedness rather than convergence to $0$. | 114,063 |
TITLE: A relation involving many integrals
QUESTION [1 upvotes]: I need help to proof the following: Consider the multiple integrals below:
$$ \int_{a}^{b} dx_{1} \int_{a}^{b} dx_{2} \cdots \int_{a}^{b} dx_{n} \ f(x_{1}, x_{2}, ..., x_{n}).$$
Here, $f$ is a function that is unchanged if we exchange two of your variables, that is
$$f(..., x_{i}, ..., x_{j}, ...) = f(..., x_{j}, ..., x_{i}, ...).$$
I have to prove that this integral is equal to
$$n! \int_{a}^{b} dx_{1} \int_{a}^{x_{1}} dx_{2} \cdots \int_{a}^{x_{n-1}} dx_{n} \ f(x_{1}, x_{2}, ..., x_{n}).$$
It may look likes weird, but I am studying many-body perturbation theory using Feynman diagrams and this relation is very useful. I know that this is true, but I can't find anywhere a good proof or justification for it.
REPLY [1 votes]: You are integrating over the region where $ b \geq x_1 \geq x_2 \geq x_3 \geq \ldots \geq x_n \geq a $.
By the condition that the function is symmetric in the variables, it follows that this is $\frac{1}{n!}$ of the original integral. Hence the equality holds. | 105,547 |
What was the wisest decision you made this year, and how did it play out?
(Author: Susannah Conway)
Another easy one! I made a big big big decision this year - I got married! I'm confident that not only was the decision wise (i.e, showing good judgment) but that I also I made the decision prudently.
As of tomorrow we will married six months so it seems a little premature to talk about how this decision played out. But, so far so good :)
Thanks for taking the time to share your thoughts. Every comment brings a smile to my face! | 125,735 |
TITLE: congruence equations and inverse
QUESTION [1 upvotes]: I got two questions that im wondering, one main question and one "bonus" question I guess
1) what is the general method to solve congruence equations like ax ≡ b (mod m) ?
take for example 3x≡2 (mod 5) how would I go about solving this?
I was able to solve it using trial and error and found the answer to be 4, but Im looking for a better and faster way to solve problems like these for larger m 's..
also im wondering sometimes you can solve these with just the inverse of a, but when does that apply? only when b=1 and a and m are coprime? example 9x≡1 (mod 14) for instance?
2) can someone please show me how I can find the modular inverse of 9 in mod14 ?, im familiar with the method used and Im able to find the inverse in other examples, but im not sure why this one is casuing me so much trouble, answer should be 11 I think
REPLY [1 votes]: A few methods to solve $ax\equiv b\pmod{m}$:
$1)\ $ Checking $x\equiv \{0,1,2,\ldots,m-1\}\pmod{m}$, i.e. using brute force.
$2)\ $ Doing something similar to this: $$3x\equiv 2\equiv -3\pmod{5}\stackrel{3}\iff x\equiv -1\equiv 4\pmod{5}$$
$$9x\equiv 1\equiv -27\pmod{14}\stackrel{:9}\iff x\equiv -3\equiv 11\pmod{14}$$
$3)\ $ Using inverses / Extended Euclidean Algorithm.
Inverses can always solve $ax\equiv b\pmod{m}$ when $\gcd(a,m)$. The solution is $x\equiv ba^{-1}\pmod{m}$.
You can find $a^{-1}\bmod m$ by either using Extended Euclidean Algorithm or solving $ax\equiv 1\pmod{m}$.
Using Extended Euclidean Algorithm:
$$\begin{array}\\14=14(1)+9(0)\\ 9=14(0)+9(1)\\ 5=14(1)+9(-1)\\4=14(-1)+9(2)\\1=14(2)+9(-3)\end{array}$$
Therefore $9^{-1}\equiv -3\equiv 11\pmod{14}$.
As for your comment: $ax\equiv b\pmod{m}$ has a solution if and only if $\gcd(a,m)\mid b$. This follows from Bézout's Lemma. | 144,399 |
\section{Preliminaries}\label{sec:bkgrnd}
Let $[n]$ denote the set $\{1,2,\ldots ,n\}$ and let $\p{1}_d$, $\p{0}_d$ denote the all $1$'s vector and all $0$'s vector in $\R^d$ respectively.
For any $\p{x} , \p{y} \in \R^d$, we denote the Euclidean ($\ell_2$) distance between them as $\| \p{x} - \p{y} \|_2$. For any vector $\p{x} \in \R^d$, $x_i$ denotes its $i$-th coordinate.
For any $\p{c} \in \R^d$, and $r >0$, let $B_d(\p{c}, r)$ denote a $d$-dimensional $\ell_2$ ball of radius $r$ centered at $\p{c}$. Also, let $S^{d-1}$ denote the unit sphere about $\p{0}_d$.
Let $\p{e_i} \in \R^d$ denote the $i$-th standard basis vector which has $1$ in the $i$-th position and $0$ everywhere else. Also, for any prime power $q$, let $\mathbb{F}_q$ denote a finite field with $q$ elements.
For a discrete set of points $C \subset \R^d$, let $\conv(C)$ denote the convex hull of points in $C$, \ie
$\conv(C) := \left\{ \sum_{\p{c} \in C} a_c \p{c} \mid a_c \ge 0, \sum_{\p{c} \in C} a_c = 1 \right\}$.
Suppose $\p{w} \in \R^d$ be the parameters of a function to be learned (such as weights of a neural network). In each step of the SGD algorithm, the parameters are updated as $\p{w} \leftarrow \p{w} - \eta \p{\hat{g}}$, where $\eta$ is a possibly time-varying learning rate and $\p{\hat{g}}$ is a stochastic unbiased estimate of $\p{g}$, the true gradient of some loss function with respect to $\p{w}$. The assumption of unbiasedness is crucial here, that implies $\E \p{\hat{g}} = \p{g}$.
The goal of any gradient quantization scheme is to reduce cost of communicating the gradient, i.e., to act as an first-order oracle, while not compromising too much on the \emph{quality} of the gradient estimate.
The quality of the gradient estimate is measured in terms the convergence guarantees it provides.
In this work, we will develop a scheme that is an {\em almost surely bounded oracle} for gradients, i.e., $ \|\p{\hat{g}}\|^2_2 \le B$ with probability 1, for some $B>0$. The convergence rate of the SGD algorithm for any convex function $f$ depends on the upper bound of the norm of the unbiased estimate, i.e., $B$, cf. any standard textbook such as \cite{shalev2014understanding}.
Although we provide an almost surely bounded oracle as our quantization scheme, previous quantization schemes, such as \cite{qsgd}, provides a {\em mean square bounded oracle}, i.e., an unbiased estimate $\p{\hat{g}}$ of $\p{g}$ such that $\E\|\p{\hat{g}}\|_2^2 \le B$ for some $B>0$. It is known that, even with a mean square bounded oracle, SGD algorithm for a convex function converges with dependence on the upper bound $B$ (see \cite{bubeck2017convex}). As discussed in \cite{qsgd}, one can also consider the variance of $\p{\hat{g}}$ without any palpable difference in theory or practice. Therefore, below we consider the variance of the estimate $\p{\hat{g}}$ as the main measure of error.
In distributed setting with $N$ worker nodes, let $\p{g_i}$ and $\p{\hat{g_i}}$ are the local true gradient and its unbiased estimate computed at the $i$th compute node for some $i \in \{1, \dots, N\}$. For $\p{g} =\frac{1}{N}\sum_i \p{g_i}$, the variance of the estimate $\p{\hat{g}} = \frac{1}{N}\sum_i \p{\hat{g_i}}$ is defined as
$$
\text{Var}(\p{\hat{g}}) := \E \left[ \| \frac{1}{N}\sum_{i=1}^N\p{g_i} -\frac{1}{N}\sum_{i=1}^N\p{\hat{g_i}} \|_2^2 \right]
= \frac{1}{N^2} \sum_{i=1}^N\E\left[ \| \p{g_i} - \p{\hat{g_i}} \|_2^2 \right].
$$
In this work, our goal is to design quantization schemes to efficiently compute unbiased estimate $\p{\hat{g}_i}$ of $\p{g}_i$ such that $\text{Var}(\p{\hat{g}})$ is minimized.
For the privacy preserving gradient quantization schemes, we consider the standard notion of $(\eps, \delta)$-differential privacy (DP) as defined in \cite{privacybook}.
Consider data-sets from a domain $\calx$. Two data-sets $U, V \in \calx$, are \emph{neighboring} if they differ in at most one data point.
\begin{definition}\label{def:privacy}
A randomized algorithm $\calm$ with domain $\calx$ is $(\eps, \delta)$-differentially private (DP) if for all $S \subset \text{Range}(\calm)$ and for all neighboring data sets $U, V \in \calx$,
\[
\Pr[\calm(U) \in S ] \le e^{\eps} \Pr[ \calm(V) \in S] + \delta,
\]
where, the probability is over the randomness in $\calm$. If $\delta=0$, we say that $\calm$ is $\eps$-DP.
\end{definition}
We will need the notion of an $\varepsilon$-nets subsequently.
\begin{definition}[$\varepsilon$-net]\label{def:epsnet}
A set of points $N(\varepsilon) \subset {\cal S}^{d-1}$ is an $\varepsilon$-net for the unit sphere ${\cal S}^{d-1}$ if for any point $\p{x} \in {\cal S}^{d-1}$ there exists a net point $\p{u} \in N(\varepsilon)$ such that $\|\p{x}-\p{u}\|_2 \le \varepsilon$.
\end{definition}
There exist various constructions for $\varepsilon$-net over the unit sphere in $\R^d$ of size at most $\left( 1+2/\varepsilon\right)^d$ \cite{cohen1997covering}.
\begin{definition}[Hadamard Matrix]\label{def:hadamard}
A Hadamard matrix $H_n$ of order $n$ is a $n \times n$ square matrix with entries from $\pm 1$ whose rows are mutually orthogonal. Therefore, it satisfies $HH^T = nI_n$, where $I_n$ is the $n \times n$ identity matrix.
\end{definition}
Sylvester's construction \cite{georgiou2003hadamard} provides a recursive technique to construct Hadamard matrices for orders that are powers of $2$ which can be defined as follows. Let $H_1 = [1]$ be the Hadamard matrix of order $2^0$, and let $H_p$ denote the Hadamard matrix of order $2^p$, then a Hadamard matrix of order $2^{p+1}$ can be constructed as
\[
H_{p+1} = \begin{bmatrix}
H_p & H_p \\
H_p & -H_p
\end{bmatrix}
\]
\section{Quantization Scheme}\label{sec:quant}
We first present our quantization scheme in full generality. Individual quantization schemes with different tradeoffs are then obtained as specific instances of this general scheme.
Let $C = \{\p{c_1}, \dots, \p{c}_{m}\} \subset \R^d$ be a discrete set of points such that its convex hull, $\conv(C)$ satisfies
\begin{equation}\label{cond:hull}
B_d(\p{0}_d,1) \subset \conv(C)
\subseteq B_d(\p{0}_d,R), R > 1.
\end{equation}
\iffalse
\setlength{\belowcaptionskip}{-10pt}
\begin{wrapfigure}{l}{0.20\textwidth}
\begin{center}
\includegraphics[width=0.18\textwidth]{exp/chull.jpeg}
\caption{\small{Condition~\ref{cond:hull}}}
\label{fighull}
\end{center}
\end{wrapfigure}
\fi
Let $\p{v} \in B_d(\p{0}_d,1)$. Since $B_d(\p{0}_d,1) \subseteq \conv(C)$, we can write $\p{v}$ as a convex linear combination of points in $C$. Let
$\p{v} = \sum_{i=1}^{m} a_i \p{c_i},
\mbox{ where }
a_i \ge 0, \sum_{i=1}^{m} a_i = 1.$
We can view the coefficients of the convex combination
$(a_1, \ldots, a_{m})$
as a probability distribution
over points in $C$.
Define the quantization of $\p{v}$ with respect to the set of points $C$ as follows:
\[
Q_C(\p{v}) := \p{c_i} \mbox{ with probability } a_i
\]
It follows from the definition of the quantization that $Q_C(\p{v})$ is an unbiased estimator of $\p{v}$.
\begin{lemma}\label{lem:unbiased}
$\E[Q_C(\p{v})] = \p{v}$.
\end{lemma}
\begin{proof}
$\E[Q_C(\p{v})] = \sum_{i=1}^{|C|} a_i \cdot \p{c_i} = \p{\tilde{v}} = \p{v}.$
\end{proof}
We assume that $C$ is fixed in advance and is known to the compute nodes and the parameter server.
\begin{remark}
Communicating the quantization of any vector $\p{v}$, amounts to sending a floating point number $\|\p{v}\|_2$, and the index of point $Q_C(\p{v})$ which requires $\log |C|$ bits.
For many loss functions, such as Lipschitz functions, the bound on the norm of the gradients is known to both the compute nodes and the parameter server. In such settings we can avoid sending $\|v\|_2$ and the cost of communicating the gradients is then exactly $\log |C|$ bits.
\end{remark}
Any point set $C$ that satisfies Condition~\eqref{cond:hull} gives the following bound on the variance of the quantizer.
\begin{lemma}\label{lem:hull}
Let $C \subset \R^d$ be a point set satisfying Condition~\eqref{cond:hull}. For any $\p{v} \in B_d(\p{0}_d,1)$, let $\p{\hat{v}} := Q_C(\p{v})$. Then, $\|\p{\hat{v}}\|_2^2 \le R^2$ almost surely, and
$\E \left[ \| \p{v} - \p{\hat{v}} \|_2^2 \right] \le R^2$.
\end{lemma}
\begin{proof}
From the definition of the quantization function,
\begin{align*}
\E[ \|\p{v} - Q_C(\p{v}) \|_2^2] &= \E[\|Q_C(\p{v})\|^2] - \|\p{v}\|^2 \le R^2.
\end{align*}
This is true as $C$ satisfies Condition~\eqref{cond:hull} and therefore, each point $\p{c_i} \in C$ has a bounded norm, $\|\p{c_i}\| \le R$.
\end{proof}
\begin{remark}
If, for any vector $\p{v}$, we send the floating point number $\|\p{v}\|_2$ separately, instead of there being a known upper bound on gradient, we can just assume without loss of generality that $\p{v} \in S^{d-1}$.
In this case, the subsequent bounds on variance $\E \left[ \| \p{v} - \p{\hat{v}} \|_2^2 \right] = \E \left[ \| \p{\hat{v}} \|_2^2 \right] - \|\p{v} \|_2^2$ can be replaced by $R^2-1$.
\end{remark}
From the above mentioned properties, we get a family of quantization schemes depending on the choice of point set $C$ that satisfy Condition~\eqref{cond:hull}. For any choice of quantization scheme from this family, we get the following bound regarding the convergence of the distributed SGD.
\begin{theorem}\label{thm:hull}
Let $C \subset \R^d$ be a point set satisfying Condition~\eqref{cond:hull}.
Let $\p{g_i} \in \R^d$ be the local gradient computed at the $i$-th node,
Define $\p{\hat{g}} := \frac1N \sum_{i=1}^N \p{\hat{g}_i}$, where $\p{\hat{g}_i} := \|\p{g_i}\| \cdot Q_C(\p{g_i} / \|\p{g_i}\|)$. Then,
\begin{align*}
\E[\p{\hat{g}} ] = \p{g} \quad \mbox{ and} \quad \E \left[ \| \p{g} - \p{\hat{g}} \|_2^2 \right] \le (R/N)^2 \sum_i \| \p{g_i}\|^2.
\end{align*}
\end{theorem}
\begin{proof}
Since $\p{\hat{g}}$ is the average of $N$ unbiased estimators, the fact that $\E[\p{\hat{g}}] = \p{g}$ follows from Lemma~\ref{lem:unbiased}.
For the variance computation, note that
\begin{align*}
\E[\|\p{g} - \p{\hat{g}}\|_2^2] &=\frac{1}{N^2} \left( \sum_{i=1}^N \E[\| \p{g_i} - \p{\hat{g_i}}\|_2^2 ] \right) \qquad (\mbox{ since } \p{\hat{g_i}}
\mbox{ is an unbiased estimator of $\p{g}$ })\\
&\le \frac{R^2}{N^2} \sum_{i=1}^N \| g_i\|^2 \qquad \mbox{(from Lemma~\ref{lem:hull})}.
\end{align*}
\end{proof}
\begin{remark}
Computing the quantization $Q_C(.)$ amounts to solving a system of $d+1$ linear equations in $\R^{|C|}$.
For general point sets $C$, this takes about $O(|C|^3)$ time (since $|C| \ge d+1$). However, we show that for certain structured point sets, the quantization $Q_C(.)$ can be computed in linear time.
\end{remark}
From Theorem~\ref{thm:hull} we observe that the communication cost of the quantization scheme depends on the cardinality of $C$ while the convergence is dictated by the circumradius $R$ of the convex hull of $C$. In the Section~\ref{sec:constructions}, we present several constructions of point sets which provide varying tradeoffs between communication and variance of the quantizer.
\vspace{-5pt}\paragraph{Reducing Variance: }\label{sec:repetition}
In this section, we propose a simple repetition technique to reduce the variance of the quantization scheme.
For any $s > 1$, let $Q_C(s, \p{v}) := \frac 1s \sum_{i=1}^s Q_C^{(i)}(\p{v})$ be the average over $s$ independent applications of the quantization $Q_C(\p{v})$. Note that even though $Q_C(s, \p{v})$ is not a point in $C$, we can communicate $Q_C(s, \p{v})$ using an equivalent representation as a tuple of $s$ independent applications of $Q_C(\p{v})$ that requires $s \log |C|$ bits.
Using this repetition technique
we see that the variance reduces by factor of $s$ while the communication increases by the exact same factor.
\begin{proposition}\label{prop:var}
Let $C \subset \R^d$ be a point set satisfying Condition~\eqref{cond:hull}. For any $\p{v} \in B_d(\p{0}_d,1)$, and any $s \ge 1$, let $\p{\hat{v}} := Q_C(s, \p{v})$. Then,
$\E \left[ \| \p{v} - \p{\hat{v}} \|_2^2 \right] \le R^2/s$.
\end{proposition}
\begin{proof}
The proof follows simply by linearity of expectations.
\begin{align*}
\E \left[ \| \p{v} - \p{\hat{v}} \|_2^2 \right] &= \E \left[ \| \frac1s \sum_{i=1}^s \left(\p{v} - Q_C(\p{v}) \right) \|^2 \right] \le \frac{1}{s} \cdot R^2 \qquad \mbox{(from Lemma~\ref{lem:hull})}.
\end{align*}
\end{proof} | 50,150 |
TITLE: What does discharging an assumption mean in Natural Deduction?
QUESTION [1 upvotes]: I've also noticed as in the question here that it seems that many references I've read say "discharging an assumption" and assume the reader that we know what that means. It's funny because formal logic has very clear definitions of everything.
Regardless, I think my confusion stems from many things. I will try to outline them:
To understand what "discharging as assumption" means, I have to understand what the word assumption means. Does it mean axiom or hypothesis or something else? The closest thing to a definition (even if its informal since thats a start) is that it's a "local axiom". Something that we assume true for the sake of a subproof. But eventually, it has to be shown true or otherwise whats the point!
I need to know what discharging means. Looking at the answer that I referenced from mathoverflow it seems that it has a relation with the deduction theorem from metalogic. Let's recall it:
$$ T, P \vdash Q \text{ iff } T \vdash P \to Q $$
However, it's weird to me because it seems that the role of discharging is nearly the same as "establishing what has a proof already". However, when I write the statement $P \to Q$ I think of it as an implication, so I don't assume that $P$ is already true. It also doesn't tell me how it relates to the axioms.
Idk if I'm confused because I am more used to thinking of starting from the axioms then we can reach statements and that is the only thing that is true. But here things seem to be a little different. Can anyone clarify what is going on? At the very least precise statements of what "discharing an assumption" and "assumptions" mean would be a fantastic start since I precise definition of those are not explicitly found (mostly implied) from what I've read.
In addition, I heard the following comment about discharge:
Discharge function maps each leaf of the tree to an ancestor as allowed
by the inference rules.
which isn't 100% clear to me what it meant.
Cross-posted:
Quora: https://qr.ae/TDmouP
Reddit: https://www.reddit.com/r/logic/comments/evqlgh/what_does_discharging_an_assumption_mean_in/
REPLY [1 votes]: To understand what "discharging as assumption" means, I have to understand what the word assumption means. Does it mean axiom or hypothesis or something else? The closest thing to a definition (even if its informal since thats a start) is that it's a "local axiom". Something that we assume true for the sake of a subproof. But eventually, it has to be shown true or otherwise whats the point!
An assumption is sometimes called an hypothesis.
An assumption does not need be shown to be true. All that is required is that derivations from the assumption are understood to be contingent.
However, eventually the assumption needs to be discharged so that something may be inferred without that assumption. (Unless it is a premise, an hypothesis of the proof, not intended to be discharged.)
Consider the following Fitch style proof. The indentations of the subproofs show where assumptions are made (lines 2,3) and discharged (lines 6,7).
$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline #2\end{array}}\fitch{1.~(P\wedge Q)\to R\hspace{5ex}\text{Premise}}{\fitch{2.~P\hspace{13ex}\text{Assumption}}{\fitch{3.~Q\hspace{9.5ex}\text{Assumption}}{4.~P\wedge Q\hspace{5.5ex}\text{Conjunction Introduction (2,3)}\\5.~R\hspace{10ex}\text{Conditional Elimination (1,4)}}\\6.~Q\to R\hspace{8ex}\text{Conditional Introduction (3-5)}}\\7.~P\to(Q\to R)\hspace{4ex}\text{Conditional Introduction (2-6)}}$$
Note, there are other rules of inference that discharge assumptions: notable negation introduction. | 58,177 |
\begin{document}
\title{\bf On the $\gamma$-reflected Processes with fBm Input}
\bigskip
\author{ Peng Liu\thanks{ School Mathematical Sciences and LMPC, Nankai University, Tianjin 300071, China},
Enkelejd Hashorva\thanks{Department of Actuarial Science,
University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland},
and Lanpeng Ji$^\dagger$}
\maketitle
\vskip -0.61 cm
\centerline{\today{}}
\bigskip
{\bf Abstract:} Define a $\gamma$-reflected process $W_\Ga(t)=Y_H(t)-\Ga\inf_{s\in[0,t]}Y_H(s)$, $t\ge0$ with input process $\{Y_H(t), t\ge 0\}$ which is a fractional Brownian motion with Hurst index $H\in (0,1)$ and a negative linear trend. In risk theory
$R_\gamma(t)=u-W_\Ga(t), t\ge0$ is referred to as the risk process with tax of a loss-carry-forward type, whereas in queueing theory
$W_1$ is referred to as the queue length process. In this paper, we investigate
the ruin probability and the ruin time of the risk process $R_\gamma$ over a surplus dependent time interval $[0, T_u]$.
{\bf Key Words:} $\gamma$-reflected process; risk process with tax; ruin time; maximum losses; fractional Brownian motion; Piterbarg constant;
Piterbarg's theorem; Pickands constant.\\
{\bf AMS Classification:} Primary 60G15; secondary 60G70
\section{Introduction}
Let $\{X_H(t), t\ge0\}$ be a standard fractional Brownian motion (fBm) with Hurst index $H\in(0,1)$ meaning that $X_H$ is
a centered Gaussian process with almost surely continuous sample paths and covariance function
\BQNY
Cov(X_H(t),X_H(s))=\frac{1}{2}(\abs{t}^{2H}+\abs{s}^{2H}-\mid t-s\mid^{2H}),\quad t,s\ge0
\EQNY
We define a $\gamma$-reflected process with input process $Y_H(t)= X_H(t)- ct, c>0$ by
\BQN\label{Wgam}
W_\Ga(t)=Y_H(t)-\Ga\inf_{s\in[0,t]}Y_H(s), \ \ t\ge0, \label{PCW}
\EQN
where $\gamma \in [0,1]$ is the reflection parameter. \\
Motivations for studying $W_\Ga$ come from its wide applications in the fields of queuing, insurance, finance and telecommunication. For instance, in queuing theory $W_1$ is referred to as the queue length process (or the workload process); see, e.g., \cite{Harr85, Asm87, ZeeGly00, Whitt02, AwaGly09} among many others.
In risk theory the process $R_\gamma(t)=u-W_\Ga(t), t\ge0,u\ge 0$ is referred to as
the risk process with tax payments of a loss-carry-forward type; see, e.g., \cite{AsmAlb10}. We refer to \cite{DeMan03, DebickiRol02, HP99, HP08, HZ08, dieker2005extremes} for some recent studies of $W_0$.
For any $u\ge0$, define the {\it ruin time} of the $\gamma$-reflected process $W_\gamma$ by
\BQN\label{eq:tau1}
\tau_{\Ga,u}=\inf\{t\ge0: W_\gamma(t)>u\}\ \ (\text{with}\ \inf\{\emptyset\}=\IF).
\EQN
Further let $T_u, u\ge0$ be a positive function and define the {\it ruin probability} over a surplus dependent time interval $[0,T_u]$ by
\BQNY
\psi_{\Ga, T_u}(u):=\pk{\tau_{\Ga,u}\le T_u}.
\EQNY
Hereafter, $\psi_{\Ga, \IF}(u)$ denotes the ruin probability over an infinite-time horizon.
The ruin time and the ruin probability for the case that $T_u\equiv T\in(0,\IF)$ and the case that $T_u = \IF$ are studied in \cite{HJP13, HJ13}; see also \cite{DebickiRol02, HP99, HP08, HZ08}. In \cite{HJP13} the exact asymptotics of $\psi_{\Ga, T}(u)$ and $\psi_{\Ga, \IF}(u)$ are derived, which combined with the results in \cite{HP99} and \cite{DebickiRol02} lead to the following asymptotic equivalence
\BQN\label{eq:aymequ1}
\psi_{\Ga, T}(u)=\mathcal{C}_{H, \gamma}\psi_{0,T}(u) \ooo, \ \ \ u\to\IF
\EQN
for any $T\in(0,\IF]$, with $\mathcal{C}_{H, \gamma}$ some known positive constant.
The recent contribution \cite{HJ13} investigates the approximation of the conditional ruin time $\tau_{\Ga,u} \lvert (\tau_{\Ga,u}<\IF)$.
As shown therein
the following convergence in distribution (denoted by $\todis$)
\BQN \label{eq:asmNor1}
\frac{\tau_{\Ga,u}-t_0 u}{A(u)} \Big\lvert (\tau_{\Ga,u}<\IF)\ \todis \ \NN
\EQN
holds as $u\to\IF$ for any $\Ga\in[0,1)$, where $\NN$ is an $N(0,1)$ random variable and
\BQN\label{t0Au}
t_0=\frac{H}{c(1-H)},\ \ \ A(u)=\frac{H^{H+1/2}u^H}{(1-H)^{H+\frac{1}{2}}c^{H+1}}.
\EQN
See also \cite{HP08, HZ08, Kobel11, DHJ13a} for related results. We note in passing that the ruin time and the ruin probability are also studied extensively in the framework of other stochastic processes;
see, e.g., \cite{Grif13, GrifMal12, EKM97, AsmAlb10}.
With motivation from \cite{BoPiPiter09} and \cite{DebKo2013}, as a continuation of the investigation of the aforementioned papers we shall
analyze the ruin probability and the conditional ruin time of $W_\Ga$ over the surplus dependent time interval $[0,T_u]$ letting $u\to \IF$.
In the literature, commonly the case $T=\IF$ is considered since for many models explicit calculations of the ruin probability is possible.
From a practical point of view, a more interesting quantity is the finite-time ruin probability. The case of the surplus dependent
time interval lies thus in between and is of both practical and theoretical interests; results for the ruin probability in this case (with $\Ga=0$) are initially derived in \cite{BoPiPiter09}, see Theorem \ref{Coro1} below.
\\
The novel aspect of this paper is that $T_u$ will be a function changing with $u$ according to three different scenarious which we can define with help of (4).
\COM{following three scenarios (to be specified in Section 2) capturing the behaviour of $T_u$ will be discussed in detail:
\begin{itemize}
\item[i)] The short time horizon;
\item[ii)] The intermediate time horizon;
\item[iii)]The long time horizon.
\end{itemize}
}
In \netheo{Thm01} below we show that similar asymptotic equivalence as in \eqref{eq:aymequ1} still holds for all the three scenarios. In \netheo{Thm1} we derive a truncated Gaussian approximation for the (scaled) conditional ruin time over the long time horizon,
whereas for the short and the intermediate time horizons
an exponential approximation is possible.
\COM{
As discussed in \cite{BoPiPiter09} investigation of the {\it maximum losses} given that ruin occurs is also interesting.
In our setup, the maximum losses is defined as
\BQN\label{defL}
L(\Ga, u):=\left(\sup_{t\in[0,T_u]}W_\Ga(t)-u\right)\Bigg\lvert (\tau_u\le T_u).
\EQN
In \cite{BoPiPiter09} the average losses $\E{L(0,u)}$ is discussed
for the 0-reflected process by specifying $T_u$ in the short and intermediate time horizon. In \netheo{Thm3} we shall consider the approximation of the distribution of $L(\Ga, u)$ as $u\to\IF$. It turns out that it is asymptotically exponential for all the three scenarios.
As discussed in \cite{BoPiPiter09} investigation of the {\it moments of losses} given that ruin occurs is also an interesting quantity.
In our setup, for any $p\in(0, \infty)$ the $p$th moment of losses is defined as
\BQN\label{defL}
L_p(T_u, \Ga; u):=\E{\left(\sup_{t\in[0,T_u]}W_\Ga(t)-u\right)^p\Bigg\lvert (\tau_u\le T_u)}.
\EQN
In \cite{BoPiPiter09} $L_1(T_u, \Ga;u)$ is called the average losses.
The aforementioned contribution derived for the 0-reflected process the asymptotics of $L_1(T_u,0;u)$ as $u\to\IF$ by specifying $T_u$ in the short and intermediate time horizon. In \netheo{Thm2} we show that
\BQN \label{lk:asym1}
L_p(T_u, \gamma; u)=\Gamma(p+1) \left(L_1(T_u, 0; u)\right)^p \ooo
\EQN
holds as $u\to\IF$ for any $p\in(0,\IF)$, with $\Gamma(\cdot)$ the Euler Gamma function.
}
Organization of the rest of the paper: The main results are presented in Section 2 followed then by a section dedicated to the proofs.
In Appendix we present a variant of the celebrated Piterbarg's theorem, which is of interest for further theoretical developments.
\section{Main Results}
In this contribution, three scenarios of $T_u$ will be distinguished. In view of
\eqref{eq:asmNor1} the asymptotic mean of the ruin time (equal to $t_0 u$) and the asymptotic standard deviation $A(u)$ (see \eqref{t0Au}) should be used as a scaling parameter for $T_u$ leading to the
definition of the following three scenarios:
\begin{itemize}
\item[i)] The short time horizon: $\lim_{u\to\IF} T_u/u=0$;
\item[ii)] The intermediate time horizon: $\lim_{u\to\IF} T_u/u=s_0\in(0,t_0)$;
\item[iii)]The long time horizon: $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x\in (-\IF, \IF]$.
\end{itemize}
Next we introduce two well-known constants appearing in the asymptotic theory of Gaussian processes. Let therefore $\{B_\alpha(t),t\ge0 \}$ be a standard fBm with Hurst index
$\alpha/2 \in (0,1]$.
The {\it Pickands constant} is defined by
\BQNY\label{pick}
\mathcal{H}_\alpha=\lim_{S\rightarrow\infty}\frac{1}{S} \E{ \exp\biggl(\sup_{t\in[0,S]}\Bigl(\sqrt{2}B_\alpha(t)-t^{\alpha}\Bigr)\biggr)}\in(0,\IF),\ \alpha \in (0,2]
\EQNY
and the {\it Piterbarg constant} is given by
\BQNY
\mathcal{P}_\alpha^b=\lim_{S\rw\IF}\E{ \exp\biggl(\sup_{t\in[0,S]}\Bigl(\sqrt{2}B_\alpha(t)-(1+b) t ^{\alpha}\Bigr)\biggr)} \in (0,\IF),\ \alpha \in (0,2],\ b>0.
\EQNY
We refer to \cite{Pit96, debicki2002ruin, DeMan03, debicki2008note, albin2010new, HJP13, DikerY} for properties and extensions of the Pickands and Piterbarg constants.
In what follows denote by $\Phi(\cdot)$ the distribution function of an $N(0,1)$ random variable, write $\mathcal{N}$ for an $N(0,1)$ random variable
and put $\Psi(\cdot):=1-\Phi(\cdot)$. Before displaying our main results, we include below a key finding of \cite{BoPiPiter09} concerning the ruin probability of the 0-reflected process $W_0$.
\BT\label{Coro1} Let $W_0$ be the 0-reflected process given as in \eqref{Wgam} with $H\in(0,1)$. We have
i) If $\lim_{u \rightarrow \infty} {T_u}/{u} = s_0 \in[0, t_0)$, then
\BQN\label{psi01}
\psi_{0, T_u}(u)=D_{H}\left(\frac{u+cT_u}{T^H_u}\right)^{(\frac{1-2H}{H})_{+}}\Psi\left(\frac{u+cT_u}{T^H_u}\right)\ooo
\EQN
as $u\rw\IF$, where
$$
\mathcal{D}_{H}=\left\{
\begin{array}{ll}
2^{-\frac{1}{2H}}(H-c_0)^{-1}\mathcal{H}_{{2H}}, & \hbox{if } H <1/2 ,\\
\frac{4(1-c_0)^2}{(1-2c_0)(2-2c_0)}, & \hbox{if } H =1/2,\\
1& \hbox{if } H >1/2,
\end{array}
\right.\ \ \ \text{with}\ c_0=\frac{cs_0}{1+cs_0}.
$$
ii) If $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x \in (-\IF, \IF] $, then
\BQN\label{psi2}
\psi_{0, T_u}(u)=\psi_{0, \IF}(u)\Phi(x)\ooo
\EQN
as $u\to\IF$, where the infinite-time ruin probability $\psi_{0,\IF}(u)$ is given by
\BQN
\psi_{0,\IF}(u)=2^{\frac{1}{2}-\frac{1}{2H}}\frac{\sqrt{\pi}}{\sqrt{H(1-H)}} \mathcal{H}_{{2H}}\left(\frac{c^H u^{1-H}}{H^H (1-H)^{1-H}}\right)^{1/H-1}\Psi\left(\frac{c^H u^{1-H}}{H^H (1-H)^{1-H}}\right)(1+o(1)).\label{eq:HP}
\EQN
\ET
The next theorem shows the asymptotic relations between the ruin probability of the $\Ga$-reflected process $W_\Ga$ and that of the 0-reflected process $W_0$.
Therefore in the light of Theorem \ref{Coro1} we obtain the exact asymptotics of the ruin probability
over the surplus dependent interval $[0,T_u]$ of the $\Ga$-reflected process $W_\Ga$.
\BT\label{Thm01} Let $W_\Ga$ be the $\Ga$-reflected process given as in \eqref{Wgam} with $H\in(0,1)$ and $\Ga\in(0,1)$. We have
i) If $\lim_{u \rightarrow \infty} {T_u}/{u} = s_0 \in[0, t_0)$, then
\BQN\label{psi1}
\psi_{\gamma, T_u}(u)=\mathcal{M}_{H,\Ga} \psi_{0, T_u}(u)\ooo
\EQN
as $u\to\IF$, where
$$
\mathcal{M}_{H,\Ga}=\left\{
\begin{array}{ll}
\piter_{2H}^{\frac{1-\Ga}{\Ga}} , & \hbox{if } H <1/2 ,\\
\frac{ 2 -2c_0}{2-2c_0-\Ga}, & \hbox{if } H =1/2,\\
1& \hbox{if } H >1/2.
\end{array}
\right.
$$
ii) If $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x \in (-\IF, \IF] $, then
\BQN\label{psiGa}
\psi_{\Ga, T_u}(u)=\piter_{2H}^{\frac{1-\Ga}{\Ga}} \psi_{0, T_u}(u) \ooo
\EQN
as $u\to\IF$.
\ET
{\bf Remarks.}
{\it a) For the case that $\Ga=1$ we can add: Under the statement $i)$ above
similar arguments as in the proof of \netheo{Thm01} show that \eqref{psi1} holds
as $u\to\IF$, with
$$
\mathcal{M}_{H,1}=\left\{
\begin{array}{ll}
2^{-\frac{1}{2H}}(H-c_0)^{-1}\mathcal{H}_{{2H}}, & \hbox{if } H <1/2 ,\\
\frac{ 2 -2c_0}{1-2c_0}, & \hbox{if } H =1/2,\\
1& \hbox{if } H >1/2.
\end{array}
\right.
$$
For $ii)$, depending on the values of $x$ different asymptotics will appear; those derivations are more involved and will therefore be omitted here.
b) Another scenario of $T_u$ which is between the cases i) and ii) is that $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=-\IF.$ This case can not be dealt with in general;
more conditions should be imposed for the asymptotic behaviour of $T_u$ around $t_0 u$.
c) As discussed in \cite{BoPiPiter09} also of interest is the investigation of the maximum losses given that ruin occurs,
which, in our setup, is defined as
\BQN\label{defL}
L(\Ga, u):=\left(\sup_{t\in[0,T_u]}W_\Ga(t)-u\right)\Bigg\lvert (\tau_u\le T_u).
\EQN
Under the assumptions of \netheo{Thm01}, we have by an application of \netheo{Coro1} and \netheo{Thm01} that
if $i)$ is satisfied, then
\BQNY
\frac{(u+cT_u) L(\Ga,u)}{T_u^{2H}} \stackrel{d}{\to}\ \mathcal{E},\ \ \ u\to\IF,
\EQNY
and if $ii)$ is valid, then
\BQNY
\frac{c^{2H}(1-H)^{2H-1} L(\Ga,u)}{H^{2H}u^{2H-1}} \stackrel{d}{\to}\ \mathcal{E},\ \ \ u\to\IF.
\EQNY
Here (and in the sequel) $\mathcal{E}$ denotes a unit exponential random variable. Note in passing that the last convergence in distribution is clear when $\Ga=0$ and $T_u=\IF$ since it is known that the random variable $\sup_{t\in[0,\IF)}W_0(t)$ is exponentially distributed with parameter $2c$.}
Below we shall establish asymptotic approximations for the ruin times considering all three scenarios for $T_u$.
It turns out that for the long time horizon the (scaled) conditional ruin time can be approximated by a truncated Gaussian random variable.
Surprisingly, this is no longer the case for the short and the intermediate time horizons where the (scaled) conditional ruin time is approximated by an exponential random variable.
\BT\label{Thm1} Let $W_\Ga$ be the $\Ga$-reflected process given as in \eqref{Wgam} with $H\in(0,1)$ and $\Ga\in(0,1)$,
and let $\tau_{\Ga,u}$ be the ruin time defined as in \eqref{eq:tau1}. We have
i) If $\limit u {T_u}/{u}=0$, then
\BQNY\label{eq:asym1}
\frac{H u^2(T_u-\tau_{\Ga,u})}{T^{2H+1}_u}\Bigl\lvert\LT(\tau_{\Ga,u}\leq T_u\RT)\ \stackrel{d}{\to}\ \mathcal{E},\ \ \ u\to\IF;
\EQNY
ii) If $\limit u {T_u}/{u}=s_0\in(0, t_0)$, then
\BQNY\label{eq:asym2}
\frac{(1+cs_0)(H-(1-H)cs_0)(T_u-\tau_{\Ga,u})}{s_0^{2H+1} u^{2H-1}}\Bigl\lvert\LT(\tau_{\Ga,u} \leq T_u\RT)\ \stackrel{d}{\to}\ \mathcal{E},\ \ \ u\to\IF;
\EQNY
iii) If $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x \in (-\IF, \IF]$, then
\BQNY\label{eq:asym3}
\frac{\tau_{\Ga,u}-t_0u}{A(u)}\Bigl\lvert\LT(\tau_{\Ga,u}\leq T_u\RT)\ \stackrel{d}{\to}\ \NN \Bigl\lvert (\NN<x),\ \ \ u\to\IF.
\EQNY
\ET
{\bf Remark.}
{\it As expected, the approximation of the conditional ruin time does not involve the reflection constant $\gamma$, since in view of the proof of \netheo{Thm1} the terms with $\Ga$ are canceled out because of the conditional event.
}
\COM{
Our last investigation concerns the conditional maximum losses of the $\Ga$-reflected process $W_\Ga$.
\BT\label{Thm3}
Let $W_\Ga$ be the $\Ga$-reflected process given as in \eqref{Wgam} with $H\in(0,1)$ and $\Ga\in(0,1)$, and let $L(\Ga, u)$ be the maximum losses given that ruin occurs defined as in \eqref{defL}. We have
i) If $\lim_{u \rightarrow \infty} {T_u}/{u} = s_0 \in[0, t_0)$, then
\BQNY
\frac{(u+cT_u) L(\Ga,u)}{T_u^{2H}} \stackrel{d}{\to}\ E,\ \ \ u\to\IF;
\EQNY
ii) If $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x \in (-\IF, \IF] $, then
\BQNY
\frac{c^{2H}(1-H)^{2H-1} L(\Ga,u)}{H^{2H}u^{2H-1}} \stackrel{d}{\to}\ E,\ \ \ u\to\IF.
\EQNY
\ET
As in \cite{BoPiPiter09}, in order to obtain neat asymptotics for the moments of losses, for the short and intermediate scenarios of $T_u$ we assume that
$T_u=bu + \lambda u^\alpha, b\in[0, t_0], \alpha \in [0, 1), \lambda \in (-\infty, \infty).$ Thus, the following three cases will be considered:
\COM{
(I) The above $T_u$ with $b=0$ and $\lambda \in (0,\infty)$;\\
(II) The above $T_u$ with $b\in (0, t_0)$ and $\lambda \in (-\infty, \infty)$;\\
(III) $T_u$ such that $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x$ holds for some $x\in (-\IF,\IF]$.
}
\begin{itemize}
\item[(I)] The above $T_u$ with $b=0$ and $\lambda \in (0,\infty)$;
\item[(II)] The above $T_u$ with $b\in (0, t_0)$ and $\lambda \in (-\infty, \infty)$;
\item[(III)] $T_u$ such that $\lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x$ holds for some $x\in (-\IF,\IF]$.
\end{itemize}
In the deep contribution \cite{BoPiPiter09}, the following results for the average losses was derived.
\BT\label{Coro2}
Let $W_0$ be the $0$-reflected process given as in \eqref{Wgam} with $H\in(0,1)$, and let $L_1(T_u, 0; u)$ be the average losses defined as in \eqref{defL}. If the positive function $T_u, u\ge0$ satisfies one of the conditions in (I)--(III), then as $u\to\IF$
\BQN\label{eq:LTu0u}
L_1(T_u, 0; u)=\ooo \left\{
\begin{array}{ll}
\frac{\lambda^{2H}}{1-\alpha H} u^{2\alpha H-1}, & \hbox{for case (I)},\\
\frac{b^{H}}{(1+bc)^2(1-H)}u^{2H-1}, & \hbox{for case (II)},\\
\frac{H^{2H}}{c^{2H}(1-H)^{2H-1}}u^{2H-1}, & \hbox{for case (III)}.
\end{array}
\right.
\EQN
\ET
The next theorem gives the relations between the $p$th moment of losses $L_p(T_u, \gamma; u)$ and
the average losses $L_1(T_u, 0; u)$.
\BT\label{Thm2}
Let $W_\Ga$ be the $\Ga$-reflected process given as in \eqref{Wgam} with $H\in(0,1)$ and $\Ga\in(0,1)$,
and let $L_p(T_u, \Ga; u)$, with $p\in(0,\IF)$, be the $p$th moment of losses given as in \eqref{defL}. If the positive function $T_u, u\ge0$ satisfies one of the conditions in (I)--(III), then \eqref{lk:asym1} holds as $u\rightarrow \infty$.
\ET
The above theorem has the following implication: Let $\eta_u,u>0$ be random variables defined on the same probability space such that
the following stochastic representation holds
\BQN
\eta_u\equaldis \frac{1}{L_1(T_u, 0; u)}\left(\sup_{t\in[0,T_u]}W_\Ga(t)-u\right)\Bigg\lvert (\tau_u\le T_u).
\EQN
Under the assumptions of \netheo{Thm2} we have thus
$$ \limit{u}\E{\eta_u^p} = \Gamma(p+1)$$
for any $p>0$. Consequently, since the exponential distribution is determined by its moments and the $p$th moment of a uint exponential random variable equals $\Gamma(p+1)$ for any $p>0$, then we conclude that $\eta_u$ can be approximated as $u\to \IF$ by a unit exponential
random variable $E$, i.e.,
\BQN
\eta_u \todis E, \quad u\to \IF.
\EQN
{\bf ?? If the above holds, we should have that for any $x>0$
\BQNY
\pk{\eta_u>x}\to \exp(-x)
\EQNY
which means that (set $L_u=L_1(T_u, 0; u)$)
\BQN
\psi_{\Ga,T_u}(u+xL_u)/ \psi_{\Ga,T_u}(u)\to \exp(-x)
\EQN
In fact it follows from Thm 2.1 and Thm 2.2 that
\BQN
\psi_{\Ga,T_u}(u+xL_u)/ \psi_{\Ga,T_u}(u)\to \left\{
\begin{array}{ll}
\exp(-\frac{1}{1-\alpha H} x), & \hbox{for case (I)},\\
\exp(-\frac{1}{(1+bc)(1-H)} x), & \hbox{for case (II)},\\
\exp(-x), & \hbox{for case (III)}.
\end{array}
\right.
\EQN
I think the contradiction comes because $\E{\eta_u^p}$ is different from
\BQN
L_p(T_u, \gamma; u)/ \left(L_1(T_u, 0; u)\right)^p
\EQN
(the difference comes from the conditional event)
}
}
\section{Proofs}
In this section, we shall present the proofs of all the theorems. We start with the proof of \netheo{Thm01}. First note that for any $u>0$
\BQNY
\psi_{\gamma, T_u}(u)&=&\mathbb{P}\left(\sup_{t\in[0,T_u]}W_\Ga(t)>u\right)\\
&=&\mathbb{P}\left(\sup_{0\leq s \leq t\leq T_u}\Bigl( Z(s,t)-c(t-\Ga s)\Bigr)>u\right),
\EQNY
where $Z(s,t):=X_H(t)-\Ga X_H(s), s,t\ge0.$
Using the self-similarity of the fBm $X_H$, we further have
\BQN\label{eq:psi1}
\psi_{\gamma, T_u}(u)&=&\mathbb{P}\left(\sup_{0\leq s \leq t\leq1}Y_u(s,t)>\frac{u}{T_u^H}\right),
\EQN
where, for any $u>0$
\BQN \label{eq:Yu}
Y_u(s,t)=\frac{Z(s,t)}{1+\frac{c T_u}{u}(t-\Ga s)},\ \ s,t\ge0.
\EQN
In order to prove statement $i)$ in \netheo{Thm01}, we give the following crucial lemma.
\BEL\label{LE2}
Let $\{Y_u(s,t),s,t\ge0\}, u>0$ be a family of Gaussian random fields defined as in \eqref{eq:Yu} with $H\in(0,1)$ and $\Ga\in(0,1)$.
Assume that the condition of statement $i)$ in \netheo{Thm01} is satisfied.
Then, for any $u$ large enough, the variance function $V_{Y_u}^2(s,t) = \E{Y_u^2(s,t)}$ of the Gaussian random field $Y_u$ attains its maximum over the set $A:=\{(s, t): 0\leq s \leq t \leq 1\}$ at the unique point $(0,1)$. Moreover,
$$ V_{Y_u}(0,1)=\frac{u}{u+cT_u}.$$
\EEL
\prooflem{LE2} We only present the main ideas of the proof omitting thus some tedious and straightforward calculations.
By solving the two equations
$$\frac{\partial V_{Y_u}^2(s,t)}{\partial s}=0, \quad \frac{\partial V_{Y_u}^2(s,t)}{\partial t}=0$$
we have that $s=t$. Therefore, the maximum of $V_{Y_u}^2(s,t)$ over $A$ must be attained on the following three lines $l_1=\{(0,t), 0\leq t \leq 1\}$, $l_2=\{(s,t), 0\leq s=t \leq 1\}$ or $l_3=\{(s, 1), 0\leq s \leq 1\}$. It can be shown that on $l_1$ the maximum is attained uniquely at $(0,1)$ and on $l_2$ the maximum is attained uniquely at $(1,1)$. Obviously, both of the two points are on the line $l_3$. Consequently, the maximum point of $V_{Y_u}^2(s,t)$ over $A$ must be on $l_3$. Moreover, we have that
$$ \left(V_{Y_u}^2(s,1)\right)'= \frac{2c\Ga T_u}{u}\left(1+\frac{cT_u}{u}(1-\Ga s)\right)^{-3}f_{\frac{cT_u}{u}}(s),$$
where, for any $d>0$
\BQN \label{eq:fd}
f_d(s)&=&1-\Ga-(\Ga-\Ga^2)s^{2H}+\Ga(1-s)^{2H}-\frac{H}{d}(1+d-d\Ga s)\nonumber\\
&&\times \left((1-\Ga)s^{2H-1}+(1-s)^{2H-1}\right),\ \ s\ge0.
\EQN
Thus from the following technical lemma and the fact that
$$\lim_{u\to\IF}\frac{cT_u}{u}=cs_0<\frac{H}{1-H}$$
we conclude that
$$
\left(V_{Y_u}^2(s,1)\right)'<0,\ \ \ \ \forall s\in(0, 1)
$$
implying that the maximum of $V_{Y_u}^2(s,t)$ over the set $A$ is attained at the unique point $(0,1)$. This completes the proof. \QED
\BEL\label{LE1} Let $f_d(s), s\ge0$ be given as in \eqref{eq:fd} with
$\Ga\in[0, 1)$ and $d\in [0, \frac{H}{1-H})$. Then
$$
f_d(s)<0,\ \ \ \forall s\in(0,1).
$$
\EEL
\prooflem{LE1} First rewrite $f_d(s)$ as
\BQNY
f_d(s)&=&(1-\Ga)+\Ga(1-H)(1-s)^{2H}-\Ga(1-\Ga)(1-H)s^{2H}\\
&&-H\LT(1+\frac{1}{d}-\Ga\RT)(1-s)^{2H-1}-\frac{H}{d}(1+d)(1-\Ga)s^{2H-1}.
\EQNY
Further, we have
$$1-\Ga<(1-\Ga)(1-s)^{2H-1}+(1-\Ga)s^{2H-1},\ \ s\in(0, 1).$$
Then, replacing $1-\Ga$ by $(1-\Ga)(1-s)^{2H-1}+(1-\Ga)s^{2H-1}$ in the above equation we obtain
\BQNY
f_d(s)&<&(1-\Ga)(1-s)^{2H-1}+(1-\Ga)s^{2H-1}+\Ga(1-H)(1-s)^{2H}-\Ga(1-\Ga)(1-H)s^{2H}\\
&&-H\LT(1+\frac{1}{d}-\Ga\RT)(1-s)^{2H-1}-\frac{H}{d}(1+d)(1-\Ga)s^{2H-1}\\
&<&\LT(1-H-\frac{H}{d}\RT)((1-s)^{2H-1}+(1-\Ga)s^{2H-1})-\Ga(1-\Ga)(1-H)s^{2H},
\EQNY
where in the second inequality we used the fact that
$$\Ga(1-H)(1-s)^{2H}\leq\Ga(1-H)(1-s)^{2H-1}, \quad \forall s\in(0,1).$$
Since for any $d\in [0, \frac{H}{1-H})$
$$1-H < \frac{H}{d}$$
we conclude that $f_d(s)<0$ holds for all $s\in(0,1)$, establishing the proof.\QED
\prooftheo{Thm01} $i)$. First, note that (\ref{eq:psi1}) can be rewritten as
$$\psi_{\gamma, T_u}(u)=\mathbb{P}\left(\sup_{0\leq s \leq t\leq1}\frac{Y_u(s,t)}{V_{Y_u}(0,1)}> \frac{u+cT_u}{T_u^H}\right).$$
Next, for any fixed large $u$, we give expansion of $\frac{V_{Y_u}(s,t)}{V_{Y_u(0,1)}}$ at the point $(0, 1)$. It follows that
\BQN\label{eqcov1}
\frac{V_{Y_u}(s,t)}{V_{Y_u}(0,1)}
&=&\left\{
\begin{array}{ll}
1-(H-c(u))(1-t)-\Ga(H-c(u))s + o(1-t+s), & H>1/2 ,\\
1-(\frac{1}{2}-c(u))(1-t)-\Ga (1-\frac{\Ga}{2}-c(u))s+o(1-t+s), & H=1/2,\\
1-(H-c(u))(1-t)-\frac{\Ga-\Ga^2}{2}s^{2H}+o(1-t+s^{2H}), & H<1/2
\end{array}
\right.
\EQN
holds as $(s,t)\rw (0,1)$, where $c(u)=\frac{cT_u}{u+cT_u}.$ Furthermore, we have that
\BQN\label{eqcov2}
1-Cov\LT(\frac{Y_u(s,t)}{V_{Y_u}(s,t)}, \frac{Y_u(s',t')}{V_{Y_u}(s',t')}\RT)=\frac{1}{2}\left(\mid t-t'\mid^{2H}+\Ga^2\mid s-s'\mid^{2H}\right)(1+o(1))
\EQN
holds as $(s,t), (s',t')\rw (0,1)$.
In addition, there exists a positive constant $\CC$ such that, for all $u$ large enough
\BQNY
\E{\left(\frac{Y_u(s,t)}{V_{Y_u}(0,1)}-\frac{Y_u(s',t')}{V_{Y_u}(0,1)}\right)^2}\leq
\CC (|t-t'|^{2H}+|s-s'|^{2H})
\EQNY
holds for all $(s,t)\in A$. Therefore, by the fact that
$$\lim_{u\to\IF}c(u)=c_0=\frac{cs_0}{1+cs_0}<H$$
and using \netheo {ThmPiter} (see Appendix), we obtain that
\BQN\label{eq:TuGa}
\psi_{\Ga, T_u}(u)=D_{H,\Ga}\left(\frac{u+cT_u}{T^H_u}\right)^{(\frac{1-2H}{H})_{+}}\Psi\left(\frac{u+cT_u}{T^H_u}\right)\ooo
\EQN
as $u\rw\IF$, where
$$
\mathcal{D}_{H,\Ga}=\left\{
\begin{array}{ll}
2^{-\frac{1}{2H}}(H-c_0)^{-1}\mathcal{H}_{{2H}}\piter_{2H}^{\frac{1-\Ga}{\Ga}}, & \hbox{if } H <1/2 ,\\
\frac{4(1-c_0)^2}{(1-2c_0)(2-2c_0-\Ga)}, & \hbox{if } H =1/2,\\
1& \hbox{if } H >1/2.
\end{array}
\right.
$$
Combining the above formula with \eqref{psi01} we obtain (\ref{psi1}). \\
Next, we present the proof of statement $ii)$.\\
Assume first that $ \lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=x \in \mathbb{R}$. We have from \eqref{eq:asmNor1} that
$$\mathbb{P}\LT(\frac{\tau_{\Ga,u}-t_0u}{A(u)}\le x \Bigl\lvert\tau_{\Ga,u}<\IF\RT)\ \rw\ \Phi(x)$$
holds as $u\rw\IF$.
Further note that the above is equivalent to
$$
\limit{u}\frac{\mathbb{P}\LT(\sup_{0\leq t\leq t_0u+xA(u)}W_\Ga(t)>u\RT)}{\mathbb{P}\LT(\tau_{\Ga,u}<\IF\RT)}= \Phi(x).$$
Thus, we obtain that
$$
\psi_{\Ga,T_u}(u)\ =\ \psi_{\Ga,\IF}(u)\Phi(x) \ooo
$$
as $u\to\IF$, which together with \eqref{psi2} and Theorem 1.1 in \cite{HJP13} yields the validity of (\ref{psiGa}).
Finally, assume that $ \lim_{u\to\IF}\frac{T_u-t_0u}{A(u)}=\IF$. For any positive large $M$
$$\psi_{\Ga,t_0u+MA(u) }(u)\leq\psi_{\Ga,T_u}(u)\leq\psi_{\Ga,\IF}(u)$$
holds for all $u$ large enough, hence
$$\Phi(M)\leq \liminf_{u\rw\IF}\frac{\psi_{\Ga,T_u}(u)}{\psi_{\Ga,\IF}(u)}\leq\limsup_{u\rw\IF}\frac{\psi_{\Ga,T_u}(u)}{\psi_{\Ga,\IF}(u)}\leq 1.$$
Letting $M\rw\IF$ in the above we conclude that
$$
\psi_{\Ga,T_u}(u)\ =\ \psi_{\Ga,\IF}(u) \ooo
$$
holds as $u\to\IF$, which further implies that (\ref{psiGa}) is valid, establishing the proof. \QED
\prooftheo{Thm1} We start with the proof of statement $i)$. It follows from \eqref{eq:TuGa} that, for any $x>0$
\BQNY
\mathbb{P}\left(\frac{u^2(T_u-\tau_{\Ga,u})}{T^{2H+1}_u }> x \Bigl\lvert\tau_{\Ga,u}\leq T_u\right)&=&\frac{\mathbb{P}\left(\sup_{0\leq t\leq T_x(u)}W_{\Ga}(t)>u\right)}{\mathbb{P}\left(\sup_{0\leq t\leq T_u}W_{\Ga}(t)>u\right)}\\
&=&\frac{D_{H, \gamma}\left(\frac{u+cT_x(u)}{(T_x(u))^H}\right)^{(\frac{1-2H}{H})_{+}}\Psi\left(\frac{u+cT_x(u)}{(T_x(u))^H}\right)}{D_{H, \gamma}\left(\frac{u+cT_u}{T_u^H}\right)^{(\frac{1-2H}{H})_{+}}\Psi\left(\frac{u+cT_u}{T_u^H}\right)}\oo\\
&=&\exp\LT(-\frac{\LT(\frac{u+cT_x(u)}{(T_x(u))^H}\RT)^2-\LT(\frac{u+cT_u}{T_u^H}\RT)^2}{2}\RT)\oo\\
&\rw&\exp(-H x)
\EQNY
holds as $u\to\IF$, where $T_x(u)=T_u- {xT^{2H+1}_u }/{u^2}$. Therefore the claim follows. \\
Next, we give the proof of
statement $ii)$. Similar arguments as above yield that, for any $x>0$
\BQNY
\mathbb{P}\LT(\frac{T_u-\tau_{\Ga,u}}{u^{2H-1}}>x\Bigl\lvert\tau_{\Ga,u} \leq T_u\RT)&=&\exp\LT(-\frac{\LT(\frac{u+c(T_u-xu^{2H-1})}{(T_u-xu^{2H-1})^H}\RT)^2-\LT(\frac{u+cT_u}{T_u^H}\RT)^2}{2}\RT)\oo\\
&\rw&\exp(-\lambda x)
\EQNY
holds as $u\to\IF$, where $\lambda=\frac{(1+cs_0)(H-(1-H)cs_0)}{s_0^{2H+1}}$.
Finally, since by \eqref{psiGa} for any $y\le x$
\BQNY
\mathbb{P}\LT(\frac{\tau_{\Ga,u}-t_0u}{A(u)}< y\Bigl\lvert\tau_{\Ga,u}\leq T_u\RT)&=&\frac{\mathbb{P}\LT(\sup_{0\leq t\leq t_0u+yA(u)}W_{\Ga}(t)>u\RT)}{\mathbb{P}\LT(\sup_{0\leq t\leq T_u}W_{\Ga}(t)>u\RT)}\\
& \rw& \frac{\Phi(y)}{\Phi(x)}
\EQNY
holds as $u\to\IF$, the claim of statement $iii)$ follows, and thus the proof is complete. \QED
\COM{
In order to prove \netheo {Thm2} we need the following lemma which is a minor extension of Lemma 4 in \cite{BoPiPiter09}.
\BEL\label{LE3}
Let $p\in(0,\IF)$ be a positive constant, and let $X_x, x\ge0$ be a sequence of random variables such that $\limit{x} x^p \mathbb{P}(X_x>x)= 0$ and
\BQNY
\mathbb{P}(X_x>x)= a(x)\exp\LT(-Cx^h\RT)(1+o(1)), \quad x\to \IF,
\EQNY
with some positive constants $h, C$, and some positive measurable function $a(\cdot)$. Suppose that
(1) For some measurable function $B(\cdot)$
\BQN\label{eqB}
\lim_{x\to\IF}\frac{a(x(1+w/(Cx^h))^{1/h})}{a(x)}=B(w)
\EQN
holds for all $w\ge 0$.
(2)
There exists some non-negative measurable
function $B_1(\cdot)$ such that $\int_0^\IF(1+w)^{p(1/h-1)_+}w^{p-1}B_1(w)e^{-w}dw < \IF$, where $a_+:=max(a,0)$, and that
$$\frac{a(x(1+w/(Cx^h))^{1/h})}{a(x)} \leq B_1(w),\ \ \ \forall w>0 $$
holds for all sufficiently large $x$.
Then, as $x\rw\IF$
$$\E{(X_x-x)^p|X_x>x}=p\LT(\frac{1}{Ch}x^{1-h}\RT)^p\int_0^\IF w^{p-1}B(w)e^{-w}dw(1+o(1)).$$
\EEL
{\bf Remarks}: a) If the function $B(\cdot)$ in \eqref{eqB} is a constant, then $\pk{X_x>x}$ is in the Gumbel max-domain of attraction,
and therefore the claim of the lemma follows without imposing $(1)$ and $(2)$. The proof for
$p$ integer can be found in \cite{Berman92}, Theorem 12.2.4, see Lemma 4.2 in \cite{EH06} or Lemma 6 in \cite{Bala13} for the case $p>0$.\\
b) Clearly, if $X_x^*,x\ge0$ is a sequence of random variables such that
$\mathbb{P}(X_x^*>x)= \mu\mathbb{P}(X_x>x)\oo$ holds with $\mu$ some positive constant, then under the assumptions of \nelem{LE3}
\BQN\label{eq:Xstar}
\E{(X_x^*-x)^p|X_x>x}=\E{(X_x-x)^p|X_x>x}\oo,\ \ \ x\to\IF.
\EQN
\prooflem{LE3} The proof is based on the idea in \cite{BoPiPiter09}. It follows that
\BQNY
\E{(X_x-x)^p|X_x>x}&=&
\frac{p\int_x^\IF(y-x)^{p-1}\mathbb{P}(X_x>y)dy}{\mathbb{P}(X_x>x)}\\
&=&\frac{p\int_x^\IF(y-x)^{p-1}a(y)\exp(-Cy^h)dy}{a(x)\exp(-Cx^h)}(1+o(1))
\EQNY
as $u\to\IF.$
Changing variables $w=C(y^h-x^h)$ in the right-hand side of the last equality we obtain
\BQNY
&&\E{(X_x-x)^p|X_x>x}\\&&
=p\LT(\frac{x^{1-h}}{Ch}\RT)\int_0^\IF \frac{a(x(1+w/(Cx^h))^{1/h})}{a(x)}\LT(1+\frac{w}{Cx^h}\RT)^{1/h-1}\LT((w/C+x^h)^{1/h}-x\RT)^{p-1}e^{-w}dw(1+o(1)).
\EQNY
Further note that
\BQNY
\LT((w/C+x^h)^{1/h}-x\RT)^{p-1}&=&\LT(\frac{x^{1-h}}{Ch}\RT)^{p-1}w^{p-1}\LT(\frac{(1+w/(Cx^h))^{1/h}-1}{w/(Chx^h)}\RT)^{p-1}\\
&=&\LT(\frac{x^{1-h}}{Ch}\RT)^{p-1}w^{p-1}(1+\theta w/(Cx^h))^{(1/h-1)(p-1)}
\EQNY
holds for some $\theta\in(0,1)$. Thus we have
\BQNY
\E{(X_x-x)^p|X_x>x}
&=&p\LT(\frac{x^{1-h}}{Ch}\RT)^{p}\int_0^\IF \frac{a(x(1+w/(Cx^h))^{1/h})}{a(x)}w^{p-1}\LT(1+w/(Cx^h)\RT)^{1/h-1}\\
&&\times (1+\theta w/(Cx^h))^{(1/h-1)(p-1)}e^{-w}dw(1+o(1))
\EQNY
as $u\to\IF$. Since further $\int_0^\IF(1+w)^{p(1/h-1)_+}w^{p-1}B_1(w)e^{-w}dw < \IF$, applying the dominated convergence theorem to the last formula we conclude that
\BQNY
\lim_{x\to\IF}\frac{\E{(X_x-x)^p|X_x>x}}{p\LT(\frac{x^{1-h}}{Ch}\RT)^{p}}=\int_0^\IF w^{p-1}B(w)e^{-w}dw.
\EQNY
The claim in \eqref{eq:Xstar} follows easily from the above arguments, and thus the proof is complete. \QED
\COM{
{\bf Remark:}
We assume that $X_x$ is the random variable satisfying the conditions of \nelem{LE3} with $B_1(w)$ and $B(w)$. $X_x^*$ is another random variable (with $B_1^*(w)$ and $B^*(w)$) satisfying $\mathbb{P}(X_x^*>x)\sim D\mathbb{P}(X_x>x)$ with $D$ a positive constant. It is easy to know that as $x\rw\IF,$
$$\mathbb{P}(X_x^*>x)= DA(x)exp\LT(-Cx^h\RT)(1+o(1)).$$
In the light of the definition of $B_1^*(w)$ and $B^*(w)$, we can choose $B_1^*(w)=B_1(w)$ and have $B^*(w)=B(w)$. Therefore, as $x\rw\IF,$
$$\E{(X_x^*-x)^p|X_x>x}\sim\E{(X_x-x)^p|X_x>x}.$$
}
\prooftheo{Thm2}
First, let $X_u=\sup_{t\in[0,T_u]}W_0(t)$, with $T_u$ satisfying the assumptions of \netheo{Thm2}. We have from the arguments in Section 2 in \cite{BoPiPiter09} that
the conditions in \nelem{LE3} are satisfied by $X_u$ for all the three cases (I)--(III), where
$B_1(w)=B(w)=1$ for case (I), $B_1(w)=e^{-\epsilon w},$ with $\epsilon\in(0,1)$ and $B(w)=1$ for case (II), and $B_1(w)=B(w)=1$ for case (III). Thus, it follows from \nelem{LE3} that
\BQN\label{eq:l2}
L_p(T_u, 0; u)=\Gamma(p+1) \LT(\frac{1}{Ch}u^{1-h}\RT)^p \oo
\EQN
holds as $u\rw\IF$, where $C,h$ are known constants which are different for the three cases (I)--(III), see \eqref{eq:LTu0u} for the corresponding values. Clearly, equation (\ref{eq:l2}) gives that
\BQNY
L_{p}(T_u, 0; u)=\Gamma(p+1) \LT(L_1(T_u, 0; u)\RT)^p\oo.
\EQNY
Next, in order to complete the proof we show that
\BQN\label{eq:l1}
L_{p}(T_u, \gamma; u)=L_{p}(T_u, 0; u)\oo
\EQN
holds as $u\rw\IF$. Indeed,
it follows from \netheo{Thm1} that, for all the three cases
$$ \psi_{\Ga,T_u}(u)= \mathcal{Q}_{H,\Ga}\ \psi_{0,T_u}(u)\oo$$
holds as $u\rw\IF$, where $\mathcal{Q}_{H,\Ga}$ is some known positive constant depending only on $H, \Ga$. Therefore, by \eqref{eq:Xstar} together with
the above argument for $X$ we conclude that \eqref{eq:l1} holds establishing thus the proof. \QED
\prooftheo{Thm3} For $i)$, let
$$
S_u=\frac{T_u^{2H}}{u+cT_u},\ \ \ \ u>0.
$$
Since, for any $x>0$
\BQNY
\pk{L(\Ga,u)>S_u x}=\frac{\pk{\sup_{t\in[0,T_u]}W_\Ga(t)>u+S_u x}}{\pk{\sup_{t\in[0,T_u]}W_\Ga(t)>u}}
\EQNY
and further
$$
\lim_{u\to\IF}\frac{T_u}{u+S_u x}=s_0
$$
we conclude by an application of \netheo{Coro1} and \netheo{Thm01} that the claim in $i)$ holds as $u\to\IF$. Similarly, the claim in $ii)$ can be established. The proof is complete. \QED
}
\section{Appendix: Piterbarg's Theorem for Non-homogeneous Gaussian Fields}
We present below a generalization of Theorem D.3 and Theorem 8.2 in \cite{Pit96}, which is tailored for the proof of the main results. We first introduce a generalization of the
Piterbarg constant given by
\BQNY
\tilde{\mathcal{P}}_\alpha^b=\lim_{S\rw\IF}\E{ \exp\biggl(\sup_{t\in[-S,S]}\Bigl(\sqrt{2}B_\alpha(t)-(1+b)\abs{t}^{\alpha}\Bigr)\biggr)} \in (0,\IF),\ \alpha \in (0,2],\ b>0,
\EQNY
where $\{B_\alpha(t), t\in\R\}$ is a standard fBm defined on $\R$. Let $D=\{(s,t), 0\le s\le t\le 1\}$, and let $\{\eta_u(s,t), (s,t)\in D\}, u\ge0$ be a family of Gaussian random fields satisfying the following three assumptions:
{\bf A1:} The variance function $\sigma_{\eta_u}^2(s,t)$ of $\eta_u$ attains its muximum on the set $D$ at some unique point $(s_0,t_0)$ for any $u$ large enough, and further there exist four positive constants $A_i, \beta_i, i=1,2$ and two functions $A_i(u), i=1,2$ satisfying $\lim_{u\rw\IF}A_i(u)=A_i, i=1,2$ such that $\sigma_{\eta_u}(s,t)$ has the following expansion around $(s_0,t_0)$ for all $u$ large enough
\BQNY
\sigma_{\eta_u} (s,t)=1-A_1(u)\abs{s-s_0}^{\beta_1}(1+o(1))-A_2(u)\abs{t-t_0}^{\beta_2}(1+o(1)), \ \ (s,t)\to(s_0,t_0).
\EQNY
{\bf A2:} There exist four constants $B_i>0, \alpha_i\in(0,2], i=1,2$ and two functions $B_i(u), i=1,2$ satisfying $\lim_{u\rw\IF}B_i(u)=B_i, i=1,2$ such that the correlation function $r_{\eta_u}(s,t;s',t')$ of $\eta_u$ has the following expansion around $(s_0,t_0)$ for all $u$ large enough
\BQNY
r_{\eta_u} (s,t;s',t')=1-B_1(u)\abs{s-s'}^{\alpha_1}(1+o(1))-B_2(u)\abs{s-s'}^{\alpha_2}(1+o(1)), \ \ (s,t), (s',t')\to (s_0,t_0).
\EQNY
{\bf A3:} For some positive constants $\mathbb{Q}$ and $\gamma$, and all $u$ large enough
$$
\E{\eta_u(s,t)-\eta_u(s',t')}^2\le \mathbb{Q}(\abs{s-s'}^\gamma+\abs{t-t'}^\gamma)
$$
for any $(s,t), (s',t')\in D$.
\BT\label{ThmPiter}
If $\{\eta_u(s,t), (s,t)\in D\}$, $u\ge0$ is a family of Gaussian random fields satisfying {\bf A1-A3}, then
$$
\pk{\sup_{(s,t)\in D}\eta_u(s,t)>u}= \FF^{(1)}_{\alpha,\beta}(u)\ \FF^{(2)}_{\alpha,\beta}(u)\ \Psi(u),\ \ \text{as}\ u\rw\IF,
$$
where
$$
\FF^{(i)}_{\alpha,\beta}(u)=\left\{
\begin{array}{ll}
\widehat{I_i}\mathcal{H}_{\alpha_i} B_i^{\frac{1}{\alpha_i}} A_i^{-\frac{1}{\beta_i}}\Gamma\LT(\frac{1}{\beta_i}+1\RT)\ u^{\frac{2}{\alpha_i}-\frac{2}{\beta_i}}, & \hbox{if } \alpha_i<\beta_i,\\
\widehat{\mathcal{P}}_{\alpha_1}^{\frac{A_i}{B_i}}, & \hbox{if } \alpha_i=\beta_i,\\
1& \hbox{if } \alpha_i>\beta_i,
\end{array}
\right.\ \ \ i=1,2,
$$
with $\Gamma(\cdot)$ the Euler Gamma function and
\BQNY
&&\widehat{\mathcal{P}}_{\alpha_1}^{\frac{A_1}{B_1}}=\left\{
\begin{array}{ll}
\widetilde{\mathcal{P}}_{\alpha_1}^{\frac{A_1}{B_1}}, & \hbox{if } s_0\in(0,1),\\
\mathcal{P}_{\alpha_1}^{\frac{A_1}{B_1}}& \hbox{if } s_0=0\ \text{or}\ 1,
\end{array}
\right.\ \
\widehat{\mathcal{P}}_{\alpha_2}^{\frac{A_2}{B_2}}=\left\{
\begin{array}{ll}
\widetilde{\mathcal{P}}_{\alpha_2}^{\frac{A_2}{B_2}}, & \hbox{if } t_0\in(0,1),\\
\mathcal{P}_{\alpha_1}^{\frac{A_2}{B_2}}& \hbox{if } t_0=0\ \text{or}\ 1,
\end{array}
\right.\\
&& \widehat{I_1} =\left\{
\begin{array}{ll}
2, & \hbox{if } s_0\in(0,1),\\
1& \hbox{if } s_0=0\ \text{or}\ 1,
\end{array}
\right.\ \
\widehat{I_2} =\left\{
\begin{array}{ll}
2, & \hbox{if } t_0\in(0,1),\\
1& \hbox{if } t_0=0\ \text{or}\ 1.
\end{array}
\right.
\EQNY
\ET
\prooftheo{ThmPiter} It follows from the assumptions {\bf A1-A2} that for any $\vn>0$ and for $u$ large enough we have
\BQNY
(A_1-\vn)\abs{s-s_0}^{\beta_1}+(A_2-\vn)\abs{t-t_0}^{\beta_2}\le 1-\sigma_{\eta_u} (s,t)\le (A_1+\vn)\abs{s-s_0}^{\beta_1}+(A_2+\vn)\abs{t-t_0}^{\beta_2}
\EQNY
as $ (s,t)\to (s_0,t_0)$, and
\BQNY
(B_1-\vn)\abs{s-s'}^{\alpha_1}+(B_2-\vn)\abs{t-t'}^{\alpha_2}\le 1-r_{\eta_u} (s,t;s',t')\le (B_1+\vn)\abs{s-s'}^{\alpha_1}+(B_2+\vn)\abs{t-t'}^{\alpha_2}
\EQNY
as $ (s,t), (s',t')\to (s_0,t_0)$.
Therefore, in the light of Theorem 8.2 in \cite{Pit96} we can get asymptotical upper and lower bounds, and thus the claims follow by letting $\vn\rw0$. The proof is complete. \QED
\bigskip
{\bf Acknowledgement}: The authors kindly acknowledge partial
support from the Swiss National Science Foundation Project 200021-140633/1,
and the project RARE -318984 (an FP7 Marie Curie IRSES Fellowship).
\bibliographystyle{plain}
\bibliography{gausbibR}
\end{document} | 72,364 |
\begin{document}
\begin{center}
{\Large \bf The O(3,2) Symmetry derivable from the \\[2mm]
Poincar\'e Sphere} \\
\vspace{3mm}
Y. S. Kim\\
Department of Physics, University of Maryland,\\
College Park, MD 20742, U.S.A. email: [email protected]
\end{center}
\vspace{3ex}
\begin{abstract}
Henri Poincar\'e formulated the mathematics of the Lorentz transformations,
known as the Poincar\'e group. He also formulated the Poincar\'e sphere
for polarization optics. It is noted that his sphere contains the symmetry
of the Lorentz group applicable to the momentum-energy four-vector of
a particle in the Lorentz-covariant world. Since the particle mass is a
Lorentz-invariant quantity, the Lorentz group does not allow its variations.
However, the Poincar\'e sphere contains the symmetry corresponding to the
mass variation, leading to the $O(3,2)$ symmetry. An illustrative
calculation is given.
\end{abstract}
\vspace{20mm}
\begin{flushleft}
Included in the Nova Editorial Book: Relativity, Gravitation, Cosmology:
Foundations, edited by Valeriy Dvoeglazov (2015).
\end{flushleft}
\newpage
\section{Introduction}\label{intro}
The Poincar\'e sphere is a mathematical instrument for studying polarization
of light waves~\cite{born80,bross98}. This sphere contains the symmetry of
the Lorentz group~\cite{hkn97}. In addition, the sphere allows
us to extend the $O(3,1)$ symmetry of the Lorentz group to the $O(3,2)$
symmetry of the de Sitter group~\cite{bk06jpa,kn13symm,bk13book}.
\par
In the Lorentz-covariant world, the energy and momentum are combined into a
four-vector, and the particle mass remains invariant under Lorentz
transformations. Thus, it is not possible to change the particle mass in
the Lorentzian world. However, in the de Sitter space of $O(3,2)$, there
are two energy variables allowing two mass variables $m_1$ and $m_2$, which
can be written as \cite{bk06jpa}
\begin{equation}\label{mass01}
m_1 = m \cos\chi, \quad\mbox{and}\quad m_2 = m \sin\chi,
\end{equation}
respectively, with
\begin{equation}\label{mass02}
m^2 = m_{1}^2 + m_{2}^2 .
\end{equation}
For a given momentum whose magnitude is $p$, the energy variables are
\begin{equation}\label{mass03}
E_1 = \sqrt{ m^2\cos^{2}\chi + p^2}, \quad\mbox{and}\quad
E_2 = \sqrt{ m^2\sin^{2}\chi + p^2} .
\end{equation}
\par
While the Lorentz group is originally formulated in terms of the
four-by-four matrices applicable to one time and three space coordinates,
it is possible to use two-by-two matrices to perform the same Lorentz
transformation~\cite{hkn97,naimark54}. In this representation,
the four-vector takes the form of a two-by-two Hermitian matrix with four
elements. The determinant of this momentum-energy matrix is the $(mass)^2$
of this determinant. Indeed, the Lorentz-transformation in this
representation consists of determinant-preserving transformations.
\par
When Einstein was formulating his special relativity, he did not consider
internal space-time structures or symmetries of the particles. It was not
until 1939 when Wigner considered the space-time symmetries applicable to
the internal space-time symmetries. For this purpose, Wigner in 1939
considered the subgroups of the Lorentz group whose transformations leave
the four-momentum of a given particle invariant~\cite{wig39,knp86}.
These subgroups are called Wigner's little groups, and they define the
internal space-time symmetries of the particles.
\par
For a massive particle, the internal space-time symmetry is like the
three-dimensional rotation group leading to the particle spin. For a
massless particle, the little group has a cylindrical symmetry with one
rotational and one translational degrees of freedom, corresponding to
the helicity and gauge transformation respectively~\cite{kiwi90jmp}.
These Wigner's symmetry problems can also be framed into the two-by-two
formulation of the Lorentz group~\cite{bk13book}.
\par
It is known that the four Stokes parameters are needed for the complete
description of the Poincar\'e sphere~\cite{hkn97,bk06jpa}. These parameters
can also be placed into a two-by-two matrix. It is noted that phase shifts,
rotations, and amplitude change lead to determinant-preserving
transformations, just like in the case of Lorentz transformations.
\par
However, the determinant of the Stokes parameters becomes smaller as
the two transverse components loses their coherence. Since the determinant
of the two-by-two four-momentum matrix is the $(mass)^2$, this decoherence
could play as an analogy for variations in the mass. The purpose of
this paper is precisely to study this decoherence mechanism in detail.
\par
Sec.~\ref{pew}, it is shown possible to study Wigner's little groups using
the two-by-two representation. Wigner's little groups dictate the internal
space-time symmetries of elementary particles. They are the subgroups of
the Lorentz group whose transformations leave the four-momentum of the
given particle invariant~\cite{wig39,knp86}.
\par
In Sec.~\ref{poincs}, we first note that the same two-by-two matrices are
applicable to two-component Jones vectors and Stokes parameters, which allow
us to construct the Poincar\'e spheres. It is shown that the radius of the
sphere depends on the degree of coherence between two transverse electric
components. The radius is maximum when the system is fully coherent and is
minimum when the system is totally incoherent.
\par
In Sec.~\ref{o32}, it is noted that the variation of the determinant of the
Stokes parameters can be formulated in terms of the symmetry of the $O(3,2)$
group~\cite{bk06jpa}. This allows us to study the extra-Lorentz symmetry which allows
variations of the particle mass. We study in detail the mass variation while
the momentum is kept constant. The energy takes different values when
the mass changes.
\section{Poincar\'e Group, Einstein, and Wigner}\label{pew}
The Lorentz group starts with a group of four-by-four matrices
performing Lorentz transformations on the Minkowskian vector
space of $(t, z, x, y),$ leaving the quantity
\begin{equation}\label{4vec02}
t^2 - z^2 - x^2 - y^2
\end{equation}
invariant. It is possible to perform this transformation using two-by-two
representations~\cite{hkn97,knp86,naimark54}. This mathematical aspect is
known as the $SL(2,c)$ as the universal covering group for the Lorentz group.
\par
In this two-by-two representation, we write the four-vector as a matrix
\begin{equation}
X = \pmatrix{t + z & x - iy \cr x + iy & t - z} .
\end{equation}
Then its determinant is precisely the quantity given in Eq.(\ref{4vec02}).
Thus the Lorentz transformation on this matrix is a determinant-preserving
transformation. Let us consider the transformation matrix as
\begin{equation}\label{g22}
G = \pmatrix{\alpha & \beta \cr \gamma & \delta}, \quad\mbox{and}\quad
G^{\dagger} = \pmatrix{\alpha^* & \gamma^* \cr \beta^* & \delta^*} ,
\end{equation}
with
\begin{equation}
\det{(G)} = 1.
\end{equation}
This matrix has six independent parameters. The group of these $G$
matrices is known to be locally isomorphic to the group of four-by-four
matrices performing Lorentz transformations on the four-vector
$(t, z, x, y)$~\cite{hkn97,knp86,naimark54}. For each matrix of this
two-by-two transformation, there is a four-by-four matrix performing
the corresponding Lorentz transformation on the four-dimensional Minkowskian
vector.
\par
The matrix $G$ is not a unitary matrix, because its Hermitian conjugate is
not always its inverse. The group can have a unitary subgroup called
$SU(2)$ performing rotations on electron spins. This $G$-matrix formalism
explained in detail by Naimark in 1954~\cite{naimark54}.
We shall see first that this representation is convenient for studying
the internal space-time symmetries of particles. We shall then note that
this two-by-two representation is the natural language for the Stokes
parameters in polarization optics.
\par
With this point in mind, we can now consider the transformation
\begin{equation}\label{naim}
X' = G X G^{\dagger} .
\end{equation}
Since $G$ is not a unitary matrix, it is not a unitary transformation.
For this transformation, we have to deal with four complex numbers. However,
for all practical purposes, we may work with two Hermitian matrices
\begin{equation}\label{herm11}
Z(\delta) = \pmatrix{e^{-i\phi/2} & 0 \cr 0 & e^{i\phi/2}},
\quad\mbox{and}\quad
R(\phi) = \pmatrix{\cos(\theta/2) & -\sin(\theta/2) \cr
\sin(\theta/2) & \cos(\theta/2)} ,
\end{equation}
plus one symmetric matrix
\begin{equation}\label{symm11}
B(\mu) = \pmatrix{e^{\mu/2} & 0 \cr 0 & e^{-\mu/2}} .
\end{equation}
The two Hermitian matrices in Eq.(\ref{herm11}) lead to rotations around
the $z$ and $y$ axes respectively. The symmetric matrix of Eq.(\ref{symm11})
performs Lorentz boosts along the $z$ direction.
Repeated applications of these three matrices will lead to the most general
form of the $G$ matrix of Eq.(\ref{g22}) with six independent parameters.
\par
It was Einstein who defined the energy-momentum four vector, and showed
that it also has the same Lorentz-transformation law as the space-time
four-vector. We write the energy-momentum four-vector as
\begin{equation}\label{momen11}
P = \pmatrix{E + p_z & p_x - ip_y \cr p_x + ip_y & E - p_z} ,
\end{equation}
with
\begin{equation}
\det{(P)} = E^2 - p_x^2 - p_y^2 - p_z^2,
\end{equation}
which means
\begin{equation}\label{mass07}
\det{{p}} = m^2,
\end{equation}
where $m$ is the particle mass.
\par
Now Einstein's transformation law can be written as
\begin{equation}
P' = G M G^{\dagger} ,
\end{equation}
or explicitly
\begin{equation}\label{lt03}
\pmatrix{E' + p_z' & p_x' - ip_y' \cr p'_x + ip'_y & E' - p'_z}
= \pmatrix{\alpha & \beta \cr \gamma & \delta}
\pmatrix{E + p_z & p_x - ip_y \cr p_x + ip_y & E - p_z}
\pmatrix{\alpha^* & \gamma^* \cr \beta^* & \delta^*} .
\end{equation}
\par
Later in 1939~\cite{wig39}, Wigner was interested in constructing subgroups
of the Lorentz group whose transformations leave a given four-momentum
invariant, and called these subsets ``little groups.'' Thus, Wigner's
little group consists of two-by-two matrices satisfying
\begin{equation}\label{wigcon}
P = W P W^{\dagger} .
\end{equation}
This two-by-two $W$ matrix is not an identity matrix, but tells about
internal space-time symmetry of the particle with a given energy-momentum
four-vector. This aspect was not known when Einstein formulated his special
relativity in 1905.
\par
If its determinant is a positive number, the $P$ matrix can be
brought to the form
\begin{equation}\label{massive}
P = \pmatrix{1 & 0 \cr 0 & 1},
\end{equation}
corresponding to a massive particle at rest.
\par
If the determinant if zero, we may write $P$ as
\begin{equation}\label{massless}
P = \pmatrix{1 & 0 \cr 0 & 0} ,
\end{equation}
corresponding to a massless particle moving along the $z$ direction.
\par
For all three of the above cases, the rotation matrix $Z(\phi)$ of
Eq.(\ref{herm11}) will satisfy the Wigner condition of Eq.(\ref{wigcon}).
This matrix corresponds to rotations around the $z$ axis.
\begin{table}
\caption{Wigner's Little Groups. The little groups are the subgroups of
the Lorentz group whose transformations leave the four-momentum of a given
particle invariant. They thus define the internal space-time symmetries of
particles. The four-momentum remains invariant under the rotation around it.
In addition, they remain invariant under the following transformations. They
are different for massive and massless particles.}\label{tab11}
\begin{center}
\begin{tabular}{lcl}
\hline
\hline \\[0.5ex]
Particle mass & Four-momentum & Transform matrices \\[1.0ex]
\hline\\
massive & $\pmatrix{1 & 0 \cr 0 & 1}$
&
$\pmatrix{\cos(\theta/2) & -\sin(\theta/2)\cr \sin(\theta/2) & \cos(\theta/2)}$
\\[4ex]
Massless &
$\pmatrix{1 & 0 \cr 0 & 0}$
& $\pmatrix{1 & \gamma \cr 0 & 1}$
\\[4ex]
\hline
\hline\\[-0.8ex]
\end{tabular}
\end{center}
\end{table}
\par
For the massive particle with the four-momentum of Eq.(\ref{massive}),
the two-by-two rotation matrix $R(\theta)$ also leaves the $P$ matrix of
Eq.(\ref{massive}) invariant. Together with the $Z(\phi)$ matrix, this
rotation matrix lead to the subgroup consisting of unitary subset of the
$G$ matrices. The unitary subset of $G$ is $SU(2)$ corresponding to the
three-dimensional rotation group dictating the spin of the
particle~\cite{knp86}.
\par
For the massless case, the transformations with the triangular matrix of
the form
\begin{equation}
\pmatrix{1 & \gamma \cr 0 & 1}
\end{equation}
leaves the momentum matrix of Eq.(\ref{massless}) invariant. The physics of
this matrix has a stormy history, and the variable $\gamma$ leads to gauge
transformation applicable to massless particles~\cite{kiwi90jmp,hks82}.
\par
Table~\ref{tab11} summarizes the transformation matrices for Wigner's
subgroups for massive and massless particles. Of course, it is a
challenging problem to have one expression for both cases, and this
problem has been addressed in the literature~\cite{bk10jmo}.
\section{Geometry of the Poincar\'e Sphere}\label{poincs}
The geometry of the Poincar\'e sphere for polarization optics is determined
the four Stokes parameters. In order to construct those parameters, we
have to start from the two-component Jones vector.
\par
In studying the polarized light propagating along the $z$ direction, the
traditional approach is to consider the $x$ and $y$ components of
the electric fields. Their amplitude ratio and the phase difference
determine the state of polarization. Thus, we can change the polarization
either by adjusting the amplitudes, by changing the relative phase,
or both. For convenience, we call the optical device which changes
amplitudes an ``attenuator'' and the device which changes the relative
phase a ``phase shifter.''
\par
The traditional language for this two-component light is the Jones-matrix
formalism which is discussed in standard optics textbooks.
In this formalism, the above two components are combined into one column
matrix with the exponential form for the sinusoidal function
\begin{equation}\label{jvec11}
\pmatrix{\psi_1(z,t) \cr \psi_2(z,t)} =
\pmatrix{a \exp{\left\{i\left(kz - \omega t + \phi_{1}\right)\right\}} \cr
b \exp{\left\{i\left(kz - \omega t + \phi_{2}\right)\right\}}} .
\end{equation}
This column matrix is called the Jones vector. To this vector we can
apply the following two diagonal matrices.
\par
\begin{equation}\label{shif11}
Z(\phi) = \pmatrix{e^{-i\phi/2} & 0 \cr 0 & e^{i\phi/2}} , \qquad
B(\mu) = \pmatrix{e^{\mu/2} & 0 \cr 0 & e^{-\mu/2}} .
\end{equation}
which leads to a phase shift and a change in the amplitudes respectively. The
polarization axis can rotate around the $z$ axis, and it can be carried out
by the rogation matrix
\begin{equation}\label{rot11}
R(\theta) = \pmatrix{\cos(\theta/2) & -\sin(\theta/2)
\cr \sin(\theta/2) & \cos(\theta/2)} .
\end{equation}
These two-by-two matrices perform transform clearly defined in optics, while
they play the same role in Lorentz transformations as noted in Sec.~\ref{pew}.
Their role in the two different fields of physics are tabulated in
Table~\ref{tab22}.
The physical instruments leading to these matrix operations are mentioned
in the literature~\cite{hkn97,bk13book}. With these operations, we can
obtain the most general form given in Eq.(\ref{jvec11}) by applying the
matrix $B(\mu)$ of Eq.(\ref{shif11})
to
\begin{equation}\label{jvec33}
\pmatrix{\psi_1(z,t) \cr \psi_2(z,t)} =
\pmatrix{a \exp{\left\{i(kz - \omega t - \phi/2)\right\}} \cr
a \exp{\left\{i(kz - \omega t + \phi/2)\right\}}} .
\end{equation}
Here both components have the same amplitude.
\par
However, the Jones vector alone cannot tell whether the two components are
coherent with each other. In order to address this important degree of
freedom, we use the coherency matrix defined as~\cite{born80,bross98}
\begin{equation}\label{cocy11}
C = \pmatrix{S_{11} & S_{12} \cr S_{21} & S_{22}},
\end{equation}
with
\begin{equation}
<\psi_{i}^* \psi_{j}> = \frac{1}{T} \int_{0}^{T}\psi_{i}^* (t + \tau) \psi_{j}(t) dt,
\end{equation}
where $T$ is for a sufficiently long time interval, is much larger
than $\tau$. Then, those four elements become~\cite{hkn97}
\begin{eqnarray}
&{}& S_{11} = <\psi_{1}^{*}\psi_{1}> =a^2 , \qquad
S_{12} = <\psi_{1}^{*}\psi_{2}> = a^2~e^{-(\sigma +i\phi)} , \nonumber \\[1ex]
&{}& S_{21} = <\psi_{2}^{*}\psi_{1}> = a^2~e^{-(\sigma -i\phi)} , \qquad
S_{22} = <\psi_{2}^{*}\psi_{2}> = a^2 .
\end{eqnarray}
The diagonal elements are the absolute values of $\psi_1$ and $\psi_2$
respectively. The off-diagonal elements could be smaller than the product
of $\psi_1$ and $\psi_2$, if the two transverse components are not
completely coherent. The $\sigma$ parameter specifies the degree of
coherence.
\par
\begin{table}[ht]
\caption{Polarization optics and special relativity sharing the same
mathematics. Each matrix has its clear role in both optics and relativity.
The determinant of the Stokes or the four-momentum matrix remains invariant
under Lorentz transformations. It is interesting to note that the
decoherency parameter (least fundamental) in optics corresponds to the mass
(most fundamental) in particle physics.}\label{tab22}
\vspace{2mm}
\begin{center}
\begin{tabular}{lcl}
\hline
\hline \\[0.5ex]
Polarization Optics & Transformation Matrix & Particle Symmetry \\[1.0ex]
\hline \\
Phase shift $\phi$ &
$\pmatrix{e^{-i\phi/2} & 0\cr 0 & e^{i\phi/2}}$
& Rotation around $z$
\\[4ex]
Rotation around $z$ &
$\pmatrix{\cos(\theta/2) & -\sin(\theta/2)\cr \sin(\theta/2) & \cos(\theta/2)}$
& Rotation around $y$
\\[4ex]
Squeeze along $x$ and $y$ &
$\pmatrix{e^{\mu/2} & 0\cr 0 & e^{-\mu/2}}$
& Boost along $z$
\\[4ex]
$(a)^{4} \left(1 - e^{-\sigma}\right)$ & Determinant & (mass)$^2$
\\[4ex]
\hline
\hline\\[-0.8ex]
\end{tabular}
\end{center}
\end{table}
If we start with the Jones vector of the form of Eq.(\ref{jvec11}), the
coherency matrix becomes
\begin{equation}\label{cocy22}
C = a^4\pmatrix{1 & e^{-(\sigma + i\phi)} \cr e^{-(\sigma - i\phi)} & 1} .
\end{equation}
This is a Hermitian matrix and can be diagonalized to
\begin{equation}\label{cocy55}
D = a^4\pmatrix{1 + e^{-\sigma} & 0 \cr 0 & 1 - e^{-\sigma} } .
\end{equation}
\par
For the purpose of studying the Poincar\'e sphere, it is more convenient to
make the following linear combinations.
\begin{eqnarray}\label{stokes11}
&{}& S_{0} = \frac{S_{11} + S_{22}}{\sqrt{2}}, \qquad
S_{3} = \frac{S_{11} - S_{22}}{\sqrt{2}}, \nonumber \\[2ex]
&{}& S_{1} = \frac{S_{12} + S_{21}}{\sqrt{2}}, \qquad
S_{2} = \frac{S_{12} - S_{21}}{\sqrt{2} i}.
\end{eqnarray}
These four parameters are called Stokes parameters, and they are like a
Minkowskian four-vector which are Lorentz-transformed by the four-by-four
matrices constructed from the two-by-two matrices applicable to the
coherency matrix~\cite{hkn97}.
\par
We now have the four-vector $\left(S_0, S_3, S_1, S_2\right)$, and the
sphere defined in the three-dimensional space of
$\left(S_0, S_3, S_1, S_2\right)$ is called the Poincar\'e sphere.
If we start from the Jones vector of Eq.(\ref{jvec33}) with the same
amplitude for both components, $S_{11} = S_{22}$, and thus $S_{3} = 0.$
The Poincar\'e sphere becomes a two-dimensional circle. The radius of
this circle is
\begin{equation}
R = \sqrt{S_{1}^2 + S_{2}^2}.
\end{equation}
This radius takes its maximum value $S_{0}$ when the system is completely
coherent with $\sigma = 0$, and it vanishes when the system is totally
incoherent with $\sigma = \infty.$ Thus, $R$ can be written as
\begin{equation}
R = S_{0} e^{-\sigma} .
\end{equation}
\par
Let us go back to the four-momentum matrix of Eq.(\ref{momen11}). Its
determinant is $m^{2}$ and remains invariant under Lorentz transformations
defined by the Hermitian matrices of Eq.(\ref{herm11}) and the symmetric
matrix of Eq.(\ref{symm11}). Likewise, the determinant of the coherency
matrix of Eq.\ref{cocy22} should also remain invariant.
The determinant in this case is
\begin{equation}
S_0^2 - R^2 = a^{4} \left(1 - e^{-2\sigma}\right) .
\end{equation}
However, this quantity depends on the $\sigma$ variable which measures
decoherency of the two transverse components. This aspects is illustrated
in Table~\ref{tab22}.
\par
While the decoherency parameter is not fundamental and is influenced by
environment, it plays the same mathematical role as in the particle mass
which remains as the most fundamental quantity since Isaac Newton,
and even after Einstein.
\section{O(3,2) symmetry}\label{o32}
The group $O(3,2)$ is the Lorentz group applicable to a five-dimensional
space applicable to three space dimensions and two time dimensions.
Likewise, there are two energy variables, which lead to a five-component
vector
\begin{equation}\label{desi03}
\left(E_1, E_2, p\right) = \left(E_{1}, E_{2}, p_z, p_x, p_y \right) .
\end{equation}
In order to study this group, we have to use five-by-five matrices, but
we are interested in its subgroups. First of all, there is a
three-dimensional Euclidean space consisting of $p_z, p_x,$ and $p_y$,
to which the $O(3)$ rotation group is applicable, as in the case of the
$O(3,1)$ Lorentz group.
\par
If the momentum is in the z direction, this five-vector becomes
\begin{equation}\label{desi05}
\left(E_1, E_2, p\right) = \left(E_{1}, E_{2}, p, 0, 0 \right).
\end{equation}
As for these two energy variables, they take the form
\begin{equation}\label{desi07}
E_{1} = \sqrt{p^2 + m^2 \cos^2\chi}, \quad\mbox{and}\quad
E_{2} = \sqrt{p^2 + m^2 \sin^2\chi} ,
\end{equation}
as given in Eq.(\ref{mass03}), and they maintain
\begin{equation}\label{desi09}
E_{1}^2 + E_{2}^2 = m^{2} + 2p^{2},
\end{equation}
which remains constant for a fixed value of $p^2$. There is thus a
rotational symmetry in the two-dimensional space of $E_{1}$ and $E_{2}$.
In this section, we are interested in this symmetry for a fixed value
of the momentum as described in Fig.~\ref{mvari}.
\par
For the present purpose, the most important subgroups are two Lorentz
subgroups applicable to the Minkowskian spaces of
\begin{equation}
\left(E_1, p, 0, 0\right), \quad\mbox{and}\quad \left(E_2, p, 0, 0\right) .
\end{equation}
Then, in the two-by-two matrix representation, these four-momenta
take the form
\begin{equation}\label{fvec21}
\pmatrix{E_1 + p & 0 \cr 0 & E_1 - p}, \quad\mbox{and}\quad
\pmatrix{E_2 + p & 0 \cr 0 & E_2 - p},
\end{equation}
with their determinant are $m^2\cos^2\chi$ and $m^2\sin^2\chi$
respectively.
With this understanding, we can now concentrate only on the matrix with
$E_{2}$. For $\chi = 0$, we are dealing with the massless particle, while
the particle mass takes its maximum value of $m$.
\par
Indeed, this matrix is in the form of the diagonal matrix for the coherency
matrix given in Eq.(\ref{cocy55}). Thus, we can study the property of
the four-vector matrix of Eq.(\ref{fvec21}) in terms of the coherency
matrix, whose determinant depends on the decoherency parameter $\sigma$.
Let us now take the ratios of the two diagonal elements for these matrices
and write
\begin{equation}
\frac{1 - e^{-\sigma}}{1 + e^{-\sigma}} =
\frac{\sqrt{p^2 + m^2\sin^{2}\chi} - p}{\sqrt{p^2 + m^2\sin^{2}\chi} + p} ,
\end{equation}
which becomes
\begin{equation}\label{fvec25}
\frac{\tanh(\sigma/2)}{1 - \tanh^{2}(\sigma/2)} =
\left(\frac{m}{2p}\right)^2 (\sin\chi)^2 .
\end{equation}
The right and left sides of this equation consist of the variable of
the coherency matrix and that of the four-momentum respectively.
\par
\begin{figure}
\centerline{\includegraphics[scale=0.7]{kimnova11.eps}}
\caption{Energy-momentum hyperbolas for different values of the mass. The
Lorentz group does not allow us to jump from one hyperbola to another, but
it is possible within the framework of the $O(3,2)$ de Sitter symmetry.
This figure illustrate the transition while the magnitude of the momentum
is kept constant.}\label{mvari}
\end{figure}
\par
If $\sigma = 0$, the optical system is completely coherent, and this leads
to the zero particle mass with $\chi = 0.$ The resulting two-by-two
matrices are proportional to the four-momentum matrix for a massless
particle given in Eq.(\ref{massless})
\par
The right side reaches its maximum value of $(m/2p)^{2}$ when
$\chi = 90^{o}$, while the left side monotonically increases as $\sigma$
becomes larger. It becomes infinite as $\sigma$ becomes infinite. For
the left side, this is possible only for vanishing values of momentum.
The resulting two-by-two matrices are proportional to the four-momentum
matrix for a massive particle at rest given in Eq.(\ref{massive}).
\par
The variable $\sigma$ has a concrete physical interpretation in
polarization optics. The variable $\chi$ cannot be explained in the world
where the particle mass remains invariant under Lorentz transformations.
However, this variable has its place in the $O(3,2)$-symmetric world.
\section*{Concluding Remarks}
In this report, it was noted first that the group of Lorentz transformations
can be formulated in terms of two-by-two matrices. In this formalism, the
momentum four-vector can be written in the form of a two-by-two matrix. This
two-by-two formalism can also be used for transformations of the coherency
matrix in polarization optics. Thus, the set of four Stokes parameters is
like a Minkowskian four-vector subject to Lorentz transformations. The
geometry of the Poincar\'e sphere can be extended to accommodate these
transformations.
\par
The radius of the Poincar\'e sphere depends on the degree of coherence
between the two transverse components of electric fields of the optical
beam. If the system is completely coherent, the Stokes matrix is like
that for the four-momentum of a massless particle. When the system is
completely incoherent, the matrix corresponds to that for a massive
particle at rest. The variation of the decoherence parameter corresponds
to the variation of the mass.
\par
This mass variation is not possible in the $O(3,1)$ Lorentzian world, but
is possible if the world is extended to that of the $O(3,2)$ de Sitter
symmetry. In this paper, a concrete calculation is presented for the mass
variation with a fixed momentum in the de Sitter space.
\section*{Acknowledgments}
This report is in part based on a paper presented at the International
Conference ``Spins and Photonic Beams at Interface,'' honoring Academician
F. I. Fedorov held in Minsk, Belarus (2011). I am grateful to professor
Sergei Kilin for inviting me to this conference. I would also like
to thank Sibel Ba\c{s}kal and Marilyn Noz for their many years of
cooperation with me on this subject. Finally, I thank Valeriy Dvoeglazov
for inviting me to submit this paper and for pointing out an error in one
of the rotation angles. | 188,449 |
TITLE: A metric space on Turing machines
QUESTION [0 upvotes]: Let $L$ be a decidable language. Let $X^L$ be the set of deterministic Turing machines which decide $L$. Define two machines $A,B\in X^L$ to be time-equivalent if $t_A(w) = t_B(w)$ for all $w \in \Sigma^*$. Define a metric on the set of equivalence classes as folows:
Let $R>0$ be some big number. $d(A,B) = \sup_{w\in \Sigma^*} | t_A(w) -t_B(w)|$ If this number is bigger than $R$, set $d(A,B) = R$. This defines a complete metric space on the set of equivalence classes of $X^L$. Let $f:X^L \rightarrow X^L$ be the function which to each equivalence class of dTM maps it to its "linear speedup-theorem"-Machine. Suppose that for each machine $M$ we have: $t_{f(M)}(w) = 1/2 t_M(w) + |w| + 2$. Then this map $f$ is a contraction. By the banach fix-point theorem there must exist an equivalence class $M$ such that $f(M) = M$ . But then $t_M(w) = t_{f(M)}(w) = 1/2 t_M(w) + |w| + 2$ and we get $t_M(w) = 2 |w| + 4$. However this seems very absurd. Does it contradict something known? The only assumption which I made is that we have equality instead of $\le$ in the linear speedup theorem.
REPLY [3 votes]: The map $f$ is not a contraction. To see why, let's take for concreteness $R = 100$. Suppose $w$ is a word and $A,B$ are two machines such that $t_A(w) = 200$ and $t_B(w) = 400$. Then $d(A,B) = 100$. Now $t_{f(A)}(w) = 100 + |w| + 2$ and $t_{f(B)}(w) = 200 + |w| + 2$, so $|t_{f(A)}(w) - t_{f(B)}(w)| = 100$ and $d(f(A),f(B)) = 100$ also. But to have a contraction in the sense of the Banach fixed point theorem (also called a strict contraction) you need to have a constant $c < 1$ such that $d(f(A), f(B)) \le c d(A,B)$, so this can't be achieved. | 202,164 |
TITLE: Bernoulli Numbers
QUESTION [5 upvotes]: I've read that Bernoulli Numbers are defined by the series
$$ \frac{z}{e^z-1}\equiv \sum\limits_{n=0}^{\infty}B_n\frac{z^n}{n!},$$
So if I start with $0$ I get
$$ B_0\frac{1}{1}=B_0{1}. $$
My question is, why is there a $B_0$ in the term...is it of any significance? Or just a "marker" or something to indicate that this is the $B_0$ term?
If I find the second term I get
$$ B_1\frac{z}{1}=B_1z $$
What's the $z$? I've read it must be $\left|z\right|<2\pi$, but how does one get $\frac{-1}{2}$?
REPLY [6 votes]: First of all, using the Taylor series for $e^z$ we have
$$
\frac{e^z-1}{z} = 1 + \frac{z}{2} + \frac{z^2}{6} + \frac{z^3}{24} + \cdots.
$$
Multiplying this by the power series for $z/(e^z-1)$ and comparing coefficients (the product should be $1$) we get
$$
\begin{align*}
1 &= B_0 \\
0 &= B_1 + 1/2 \\
0 &= (2B_2) + (1/2) B_1 + (1/6) B_0 \\
0 &= (6B_3) + (1/2) (2B_2) + (1/6) B_1 + (1/24) B_0
\end{align*}
$$
and so on. Therefore $B_0 = 1$, $B_1 = -1/2$, $B_2 = 1/6$, $B_3 = -1/30$, and so on.
If you look at the function
$$ f(z) = \frac{z}{e^z-1} + \frac{z}{2} $$
then you find out that
$$
\begin{align*}
f(-z) = \frac{-z}{e^{-z}-1} - \frac{z}{2} =
\frac{ze^z}{e^z-1} - \frac{z}{2} =
\frac{z}{e^z-1} + z - \frac{z}{2} = f(z).
\end{align*}
$$
Therefore $f(z)$ is even and all the odd coefficients in its power series vanish. This shows that apart from $B_1 = -1/2$, all other odd-indexed Bernoulli numbers vanish. Why do we need them, then? They're just the sequence whose exponential generating series is $z/(e^z-1)$. | 68,247 |
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\begin{document}
\title{Hirzebruch-Riemann-Roch for global matrix factorizations}
\begin{abstract}We prove a Hirzebruch-Riemann-Roch type formula for global matrix factorizations.
This is established by an explicit realization of the abstract Hirzebruch-Riemann-Roch type formula
of Shklarov. We also show a Grothendieck-Riemann-Roch type theorem.
\end{abstract}
\author[B. Kim]{Bumsig Kim}
\address{Korea Institute for Advanced Study\\
85 Hoegiro, Dongdaemun-gu \\
Seoul 02455\\
Republic of Korea}
\email{[email protected] }
\thanks{The work is supported by KIAS individual grant MG016404.}
\subjclass[2010]{Primary 14A22; Secondary 16E40, 18E30}
\keywords{Hirzebruch-Riemann-Roch, Matrix factorizations, Hochschild homology, Hodge cohomology, Pushforward}
\maketitle
\section{Introduction}\label{sec: intro}
Let $k$ be a field of characteristic zero and let
$\GG$ be either the group $\ZZ$ or $\ZZ/ 2$.
We consider a $\GG$-graded dg enhancement $\MFdg (X, w)$ of the derived category of matrix factorizations for $(X, w)$.
Here $X$ is an $n$-dimensional nonsingular variety over $k$ and $w$ is a regular function on $X$.
An object of $\MFdg (X, w)$ is a $\GG$-graded vector bundle $E$ on $X$ equipped with
a degree $1$, $\cO_X$-linear homomorphism $\delta _E : E \to E$ such that $\delta _E ^2 = w \cdot \mathrm{id}_E$.
The structure sheaf $\cO_X$ is by definition $\GG$-graded but concentrated in degree $0$.
The degree of $w$ is 2. If $w$ is nonzero, then $\GG$ is forced to be $\ZZ/2$.
Assume that the critical locus of $w$ is set-theoretically in $w^{-1}(0)$ and proper over $k$.
The Hochschild homology $HH _* (\MFdg (X, w) )$ of $\MFdg (X, w)$ is naturally isomorphic to \[\HH ^{-*} (X, (\Omega ^{\bullet} _{X}, -dw )) ; \]
see \cite{LP, Platt}.
The isomorphism is called the Hochschild-Kostant-Rosenberg (in short HKR) type isomorphism, denoted by $I_{HKR}$.
Here $(\Omega ^{\bullet} _X, -dw )$ is a $\GG$-graded complex $\bigoplus _{p\in \ZZ} \Omega_X ^p [p]$ with the differential $-dw\wedge$.
Let $\ch (E) \in \HH ^{0} (X, (\Omega ^{\bullet} _X, -dw )) $ be
the image of the categorical Chern character $\Ch (E) \in HH _0 (\MFdg (X, w) )$ of $E$ under $I_{HKR}$.
In this paper we prove the following Hirzebruch-Riemann-Roch type formula.
\begin{Thm}\label{thm: main} For matrix factorizations $P$ and $Q$ in $\MFdg (X, w)$ we have
\begin{equation}\label{eqn: main} \sum_{i \in \GG} (-1)^i \dim \RR ^i \Hom (P, Q)
= (-1)^{ \binom{n+1}{2}} \int_X \ch (P) ^{\vee} \wedge \ch (Q) \wedge \td (X) , \end{equation}
where $\td (X) \in \oplus _{p\in \ZZ} H ^0 (X, \Omega ^p _{X} [p])$ is the Todd class of $X$.
\end{Thm}
We explain notation in the above theorem.
Firstly, the operation $\wedge$ is the wedge product inducing
\begin{equation} \label{eqn: wedge} \HH ^{*} (X, (\Omega ^{\bullet} _X, -dw )) \ot \HH ^{*} (X, (\Omega ^{\bullet} _X, dw ))
\ot \HH^{*}(X, (\Omega ^{\bullet} _X, 0)) \xrightarrow{wedge } \oplus _p H^{*} _c(X, \Omega ^p _X[p]) ; \end{equation}
see \ref{def: Wedge} for details.
Secondly, $ \int_X$ is the composition
\begin{equation} \label{eqn: theta}\oplus _{p\in \ZZ} H^{*} _c(X, \Omega ^p _X[p]) \xrightarrow{proj}
H^0_c (X, \Omega ^n _X [n]) \xrightarrow{\tr _X} k \end{equation}
of the projection and the canonical trace map $\tr _X$ for the properly supported cohomology; see \S~\ref{def int}. Thirdly, $\vee$ is induced from
a chain map
\[ (\Omega ^{\bullet}_X, dw ) \to (\Omega ^{\bullet}_X, -dw), \text{ defined by } (-1)^p\mathrm{id}: \Omega ^p _X \to \Omega ^p _X \]
in each component.
For a proper dg category $\cA$ there is an abstract Hirzebruch-Riemann-Roch formula \eqref{eqn: abs HRR} due to Shklyarov \cite{Shk: HRR}.
By an explicit realization of the formula for $\cA = \MFdg (X, w) $ we will obtain Theorem~\ref{thm: main}.
Let \[\lan , \ran _{can} : HH_* (\cA) \ot HH_* (\cA^{op}) \to k \] be
the so-called canonical pairing for $\cA$. Here $\cA ^{op}$ is the opposite category of $\cA$ and there is
an isomorphism $\cA ^{op} \cong \MFdg (X,- w)$; see \S~\ref{sub: geom diag}. This yields
\begin{equation}\label{diag: vv} \xymatrix{ HH_* (\cA ) \ar[r]_(.4){I_{HKR}} \ar[d]_{\vee} & \HH ^{-*} (X, (\Om _X , - dw )) \ar[d]_{\vee} \\
HH_* (\cA ^{op}) \ar[r]_(.4){I_{HKR}} & \HH ^{-*} (X, (\Om _X , dw )) , } \end{equation}
where the left $\vee$ is defined to make the diagram commute.
Shklyarov's formula says that the left-hand side of \eqref{eqn: main}
is equal to $\lan \Ch (Q), \Ch (P)^{\vee} \ran_{can}$.
Therefore Theorem~\ref{thm: main} is reduced to an explicit realization
of the pairing.
\begin{Thm}\label{thm: exp pairing}
The canonical pairing $\lan , \ran_{can}$ under $I_{HKR}$
corresponds to \[ (-1)^{\binom{n+1}{2}} \int _X (\cdot \wedge \cdot \wedge \td (X)) . \]
\end{Thm}
The following formula for $\ch (E)$ is established in
\cite{CKK, KP, Platt}. Let $\fU = \{ U_i \}_{i \in I}$ be an affine open covering of $X$ and let $\nabla _i$ be a connection of $E|_{U_i}$.
In the \v{C}ech hypercohomology $\check{\HH} ^* (\fU, (\Om _X, (-1)^i dw\wedge ))$
\[ \ch (E) = \str \exp (-([\nabla _i, \delta _E])_{i\in I} - (\nabla _i - \nabla _j)_{i, j \in I, i < j }) . \]
Here $\str$ means the supertrace and the products in the exponential are Alexander-\v{C}ech-Whitney cup products in the \v{C}ech complex
$\vC ^* (\fU, (\End (E) \ot \Om _X))$; see \cite{CKK} for details.
Remarks on others' related works are in order.
When $\GG = \ZZ$ and $X$ is a projective variety over $\CC$,
there is a natural isomorphism between Hodge cohomology and the singular (or equivalently $C^{\infty}$-de Rham) cohomology of the associated complex manifold $X^{an}$:
$\oplus _{p, q} H^q (X, \Omega ^p_X[p]) \overset{\phi}{\ra} H^{-\bullet} (X^{an}, \CC)$.
Let $\mathrm{tw}$ be an automorphism of $\oplus _{p, q} H^q (X, \Omega ^p_X[p]) $ sending
a $(p, q)$-form $\gamma ^{p, q}$ to $(\frac{1}{2\pi i })^{p} \gamma ^{p, q}$, then
$\phi (\mathrm{tw} ( \ch (E)))$ coincides with the topological Chern character $\ch _{top} (E)$ of $E$.
The right-hand side of \eqref{eqn: main} becomes \[ \int _{X^{an}} \ch _{top }(P^{\vee}) \ch _{top} (Q) \td _{top}(X) . \]
Here $\int _{X^{an}}$ denotes the usual integration and $\td _{top} (X)$ is the usual Todd class of $X^{an}$.
Hence Theorem \ref{thm: main} is the usual Hirzebruch-Riemann-Roch theorem \cite{Hir}.
When $\GG = \ZZ$ and $k = \CC$, Theorem \ref{thm: main} is the O'Brian -- Toledo -- Tong theorem for algebraic coherent sheaves \cite{OTT}.
When $\GG = \ZZ$, Theorem \ref{thm: main} and its generalization Corollary \ref{cor: gen} coincide with Theorem 4 of Markarian \cite{Mark} and
some works of C\u{a}ld\u{a}raru -- Willerton \cite{CW} and Ramadoss \cite{Ram: HRR}, respectively.
When $\GG = \ZZ /2$, $X$ is an open subscheme of $\AAA ^n _k $ containing the origin, and $w$ has only one singular point at the origin,
the composition of wedge products \eqref{eqn: wedge} and $\int _X$ in \eqref{eqn: theta}
is a residue pairing as shown in \cite[Proposition 4.34]{BW: ShkConj}.
Thus in this case, Theorem \ref{thm: main} is the Polishchuk -- Vaintrob theorem \cite[Theorem 4.1.4]{PV: HRR}
and Theorem \ref{thm: exp pairing} is the Brown -- Walker theorem \cite[Theorem 1.8]{BW: ShkConj}
proving a conjecture of Shklyarov \cite[Conjecture 3]{Shk: Residue}.
It is natural to consider the stacky version of Theorem \ref{thm: main}. It will be treated elsewhere \cite{CKS}.
\medskip
\noindent{\em Conventions}: Unless otherwise stated a dg category is meant to be a $\GG$-graded dg category over $k$.
For a variety $X$ over $k$, we write simply $\Omega ^p_{X}$ for the sheaf $\Omega ^p_{X/k}$ of relative differential $p$-forms of $X$ over $k$.
For a homogenous element $a$ in a $\GG$-graded $k$-space, $|a|$ denotes the degree of $a$.
For a dg category $\cA$ we often write $\cA (x, y)$ instead of the Hom complex $\Hom _{\cA} (x, y) $ between objects $x, y$ of $\cA$.
We write $x \in \cA$ if $x$ is an object of $\cA$.
For a dg algebra $A$, $C(A)$ denotes the Hochschild $\GG$-graded complex $\bigoplus _{n\ge 0} A \ot A[1]^{\ot n}$ with differential $b$.
Similarly for a dg category $\cA$, $C(\cA)$ denotes the Hochschild complex of $\cA$; see for example \cite{BW: ShkConj, Shk: HRR}.
For $a_i \in \cA (x_{i+1}, x_i) $, $i=1, ..., n$, $x_i \in \cA$, we write $a_0[a_1| ... | a_n] $ for $a_0 \ot sa_1 \ot ... \ot sa_n$,
where $s$ is the suspension so that $|sa| = |a| -1$. The symbol $\str$ stands for the supertrace.
By a coherent factorization for $(X, w)$ we mean a $\GG$-graded coherent $\cO_X$-sheaf $E$ with a curved differential $\delta _E$ such that
$\delta _E ^2 = w \cdot \mathrm{id}_E$.
\medskip
\noindent{\em Acknowledgements}: The author thanks David Favero, Taejung Kim, and Kuerak Chung for
useful discussions.
\section{Abstract Hirzebruch-Riemann-Roch}
Following mainly \cite{PV: HRR, Shk: HRR} we review the abstract Hirzebruch-Riemann-Roch theorem
in the framework of Hochschild homology theory.
\subsection{Categorical Chern characters}
For a $\GG$-graded dg category $\cA$ over $k$ let $C (\cA)$ be the Hochschild complex of $\cA$.
For $x \in \cA$, the identity morphism $1_x$ of $x$ is a $0$-cycle element and hence it defines a class
\[ \Ch (x) := [1_x] \in HH _0 ( \cA ) := H ^0 ( C (\cA) ) , \]
which is called the categorical Chern character of $x$.
For an object $y$ of the dg category $ \Perf \cA$ of perfect right $\cA$-modules, we also regard $\Ch (y)$ as
an element of $HH _0 (\cA)$ by the canonical isomorphism $HH_*(\Perf \cA) \cong HH_* (\cA) $.
\subsection{K\" unneth isomorphism}
Let $\cA$, $\cB$ be dg categories.
We define a natural chain map over $k$
\begin{align*} C (\cA) \ot C (\cB) & \to C (\cA \ot \cB) \\
a_0[a_1| ... | a_n] \ot b_0 [b_1| ... | b_m] & \mapsto (-1)^{|b_0| ( \sum_{i=1}^n (|a_i| -1))} a_0 \ot b_0 \sum_{\sigma} {\pm}
[c_{\sigma (1)} | ... | c_{\sigma (n+m)} ] ,
\end{align*}
where $\sigma$ runs for all $(n, m)$-shuffles, $c_1 = a_1 \ot 1, ..., c_n = a_n \ot 1, c_{n+1} = 1\ot b_1, ..., c_{n+m} = 1 \ot b_{m}$, and
the rule of sign $\pm$ is determined by the Koszul sign rule.
Here after each shuffle, each $1$ is uniquely replaced by an appropriate identity morphism so that
the outcomes make sense as elements of the Hochschild complex $C (\cA \ot \cB) $.
The Eilenberg-Zilber theorem says that the chain map is a quasi-isomorphism.
We call the induced isomorphism
\[ HH_* (\cA) \ot HH_* (\cB) \to HH_* (\cA \ot \cB) \]
the K\" unneth isomorphism, denoted by $\Kun$.
\subsection{The diagonal bimodule}
We denote by $\cA ^{op}$ the opposite category of $\cA$. For $x\in \cA$ we write
$x^{\vee}$ for the object of $\cA ^{op}$ corresponding to $x$.
Let $\Com _{dg} k$ be the dg category of complexes over $k$.
The diagonal $\cA$-$\cA$-bimodule $\Delta _{\cA}$ of a dg category $\cA$ is defined to
be the dg functor \[ \Delta _{\cA} : \cA \ot \cA ^{op} \to \Com _{dg} k ; \ \ y \ot x ^{\vee} \mapsto \Hom_{\cA} (x, y) . \]
Since $\cA \ot \cA ^{op} \cong ( \cA ^{op} \ot \cA )^{op}$, $\Delta _A$ is a right $\cA ^{op} \ot \cA $-module.
Assume that $\cA$ is proper, i.e.,
\[\sum _{i \in \GG} \dim H^i (\Hom _{\cA} (x, y)) < \infty \] for all $x, y \in \cA$.
Then we may replace the codomain of $\Delta _{\cA}$ by the dg category $\Perf k $ of perfect dg $k$-modules.
\subsection{The canonical pairing} For a proper dg category $\cA$,
the canonical pairing $\lan\ ,\ \ran _{can} $ is defined as the composition
\begin{equation*} HH_*(\cA) \ti HH_* (\cA ^{op} ) \xrightarrow{\Kun} HH_* (\cA \ot \cA ^{op}) \xrightarrow{\Delta _*} HH_* (\Perf k) \cong k, \end{equation*}
where $\Delta _*$ is the homomorphism in Hochschild homology level induced from the dg functor $\Delta _{\cA}$. Here we use
the canonical isomorphism $ HH_*(\Perf k) \cong k $ making a commuting diagram for $C \in \Perf k$
\begin{align}\label{eqn: perf k tr} \xymatrix{ HH_*(\Perf k) & \ar[l]_{\cong} HH_* (k) \cong k , \\
Z\End (C) \ar[u]^{natural} \ar[ru]_{\str} & } \end{align}
where $Z\End (C)$ is the graded $k$-space of closed endomorphisms of $C$ and $\str$ denotes the supertrace.
Since $HH_*(\Perf k) = HH_0 ( \Perf k)$, the pair $\lan \gamma, \gamma ' \ran$ for $\gamma \in HH_{p} (\cA)$, $\gamma ' \in HH_{p'} (\cA ^{op})$
can be nontrivial only when $p+ p' = 0$.
\subsection{A proposition}
Let $\cA$ be a proper dg category.
Let $M$ be a perfect right $\cA ^{op} \ot \cB$ module, in other words, a perfect $\cA$-$\cB$-bimodule.
Denote by $T_M$ the dg functor \[ \ot _{\cA} M : \Perf \cA \to \Perf \cB \ \text{ sending } N \mapsto N \ot _{\cA} M \]
and denote the induced map in Hochschild homology by \[ (T_M)_*: HH _* (\cA) \to HH_* (\cB) . \]
\begin{Prop} \label{prop: char can} \cite[Proposition 4.2]{Shk: HRR}
If we write $\Ch (M) = \sum _i t_i \ot t^i \in HH_* (\cA ^{op} ) \ot HH_* (\cB ) \cong HH_* (\cA ^{op} \ot \cB) \cong \HH _* (\Perf (\cA ^{op} \ot \cB ))$ via the K\"unneth isomorphism and the canonical isomorphism, then for every $\gamma \in HH _p (\cA )$ we have
\[ (T_M)_* (\gamma ) = \sum _i \lan \gamma , t_i \ran _{can} \, t^i \in HH _p (\cB). \]
\end{Prop}
\begin{proof} The proof given in \cite{Shk: HRR} also works for dg categories. \end{proof}
Furthermore assume that $\cA$ is smooth, i.e, the diagonal bimodule $\Delta_{\cA}$ is perfect. Then the Hochschild homology of $\cA$ is finite
dimensional and hence Proposition \ref{prop: char can} can be rewritten as a commuting diagram
\begin{equation*} \xymatrix{ HH_* (\cA ) \ar[r]^{\lan , \ran _{can}} \ar[rd]_{(T_M)_*}& HH_* (\cA ^{op}) ^* \ar[d]^{\Ch (M)} \\
& HH_* (\cB) . }
\end{equation*}
Since $T_{\Delta _{\cA}} = \mathrm{id}_{\cA}$, the above diagram for $M = \Delta _{\cA}$ shows that $\lan, \ran _{can}$ is non-degenerate
and the canonical pairing is characterized as follows.
Since $\Delta _{\cA} \in \Perf (\cA ^{op} \ot \cA)$, via the K\"unneth isomorphism we can write
\[\Ch (\Delta _{\cA}) = \sum_i T^i \ot T_i, \text{ for some } T^i \in HH_* (\cA ^{op} ) , \ T_i \in HH_* (\cA ) . \]
Then
$\lan , \ran _{can}$ is a unique nondegenerate $k$-bilinear map $\lan , \ran : HH_*(\cA) \ti HH_* (\cA ^{op} ) \to k $ satisfying
\begin{equation}\label{eqn: char pairing}
\sum _i \lan \gamma , T^i \ran \lan T_i, \gamma ' \ran = \lan \gamma, \gamma ' \ran , \text{ for every } \gamma \in HH_* (\cA ), \gamma '\in HH _* (\cA ^{op}) .
\end{equation}
\subsection{The chain map $\vee$}\label{sub: dual} Define an isomorphism of complexes
\begin{align*} \vee : (C (\cA ), b) & \to (C (\cA ^{op}), b) \\
a_0[a_1| ... | a_n] & \mapsto (-1)^{ n + \sum_{1\le i < j \le n} (|a_i|-1)(|a_j|-1) } a_0 [a_n | ... | a_1 ] .\end{align*}
\begin{Rmk} Using the quasi-Yoneda embedding and the HKR-type isomorphism
it is straightforward to check that the chain map $\vee$ in \S~\ref{sub: dual} fits in diagram~\eqref{diag: vv}; see for example \cite{CKK}.
\end{Rmk}
\subsection{Abstract generalized HRR}
For a proper dg category $\cA$ we may consider a sequence of natural maps
\begin{equation}\label{eqn: diag kun}
\xymatrix{ HH_* (\cA) \ot HH_* (\cA) \ar[r]_{\mathrm{id} \ot \vee \ \ }^{\cong \ \ \ } & HH_* (\cA) \ot HH_* (\cA ^{op}) \ar[d]_{\Kun}^{\cong} \ar[rd]^{\lan , \ran _{can}} & \\
& HH_* (\cA \ot \cA ^{op}) \ar[r]_{\Delta _*} & HH_* (\Perf k ) \cong k . } \end{equation}
For two closed endomorphisms $b \in \End_{\cA} (y)$, $a \in \End_{\cA} (x)$, we
define an endomorphism $L_b \circ R_a$ of $\Hom_{\cA} (x, y)$ by sending $c \in \Hom_{\cA} (x, y)$ to $(-1)^{|a||c|} b \circ c \circ a$.
Note that $\Delta _* (b\ot a) = L_b \circ R_a$. Hence from \eqref{eqn: diag kun} and \eqref{eqn: perf k tr} we obtain
\begin{equation}\label{eqn: abs general HRR} \str (L_b \circ R_a ) = \lan [b], [a] ^{\vee} \ran_{can} , \end{equation}
Here $[b], [a]$ are the homology classes in $HH_0 (\cA)$ represented by $b, a$, respectively.
\subsection{Abstract HRR}
When $a= 1_x$, $b=1_y$, \eqref{eqn: abs general HRR} yields the abstract Hirzebruch-Riemann-Roch theorem \cite{PV: HRR, Shk: HRR} for Hochschild homology:
\begin{equation}\label{eqn: abs HRR} \sum_{i\in \GG} (-1)^i \dim H^i (\Hom_{\cA} (x, y)) ) = \lan \Ch (y), \Ch (x)^{\vee} \ran_{can} . \end{equation}
This tautological HRR theorem can be useful when one expresses the right-hand side of \eqref{eqn: abs HRR} in an explicit form.
\section{Proofs of Theorems}
In this section we prove Theorems \ref{thm: main} and \ref{thm: exp pairing}. As in \S~\ref{sec: intro}
let $X$ be an $n$-dimensional nonsingular variety over $k$ and $w$ is a function on $X$ such that
the critical locus of $w$ is in $w^{-1}(0)$ and proper over $k$.
\subsection{A geometric realization of $\Delta _{\cA}$}\label{sub: geom diag}
Let $\cA = \MFdg (X, w) $. It is proper and smooth.
There is the duality functor \[ D: \cA^{op} \to \MFdg (X, -w) ; (E, \delta _E) \mapsto (\Hom _{\cO_X} (E , \cO _X) , \delta _E ^{\vee}) , \]
which is an isomorphism. Hence we have the HKR type isomorphism \[ HH_* (\cA ^{op} ) \cong \HH^{-*} (\Omega ^{\bullet}_X, dw) . \]
Let $X'$ be another nonsingular variety with a global function $w'$. Assume that the critical locus of $w'$ is proper over $k$ and located on the zero locus of $w'$.
Let $\cB$ denote $\MFdg (X', w')$.
Let $\widetilde{w} := w\ot 1 - 1 \ot w' $ a global function on $Z:=X\ti X'$.
We consider a dg functor \[ \Psi : \MFdg (Z, -\widetilde{w}) \to \Perf (\cA ^{op} \ot \cB) \] defined by letting
\begin{align*} \Psi (z) : \cA \ot \cB ^{op} & \to \Com _{dg} (k) ; \\
y\ot x ^{\vee} & \mapsto \Hom _{\MFdg (Z, -\widetilde{w})} (D(y) \boxtimes x, z) \end{align*}
for $y \in \cA$, $x \in \cB$, $z \in \MFdg (Z, -\widetilde{w})$. Since $\cA ^{op} \ot \cB $ are saturated by \cite{LP},
we may apply Proposition 3.4 of \cite{Shk: Serre} to see that $\Psi (z)$ is indeed a perfect right $\cA ^{op} \ot \cB $-module.
Let $f: X \to X'$ be a proper morphism such that $f^* w' = w$. Then there is a dg functor
\[ \RR f_* : \MFdg (X, w) \to \MFdg (X', w') \] by derived pushforward; see \cite[\S~2.2]{CFGKS}.
Define \[ \Delta _{\RR f_*} : \cA \ot \cB ^{op} \to \Com _{dg} k ; \ y \ot x^{\vee} \to \Hom _{\cB} (x, \RR f_* y) . \]
Again by Proposition 3.4 of \cite{Shk: Serre}, we see that $\Delta _{\RR f_*} $ is a perfect right $\cA ^{op} \ot \cB$-module.
Let $\Gamma _f \subset X \ti X'$ denote the graph of $f$.
Since $Z = X \ti X'$ is nonsingular, there is an object $\cO ^{\widetilde{w}}_{\Gamma _f }$ in $\MFdg (Z, -\widetilde{w})$ which is quasi-isomorphic to
the coherent factorization $\cO _{\Gamma _f}$ for $(Z, \widetilde{w})$.
Since \[ \Delta _{\RR f_*} ( y \ot x ^{\vee} ) = \cB (x, \RR f_* y ) \underset{qiso}{\simeq} \Hom_{\MFdg (Z, -\widetilde{w})} (D(y) \boxtimes x, \cO _{\Gamma _f} ^{\tilde{w}} ) , \]
by the projection formula \cite[\S~2.2]{CFGKS},
$\Delta _{\RR f_*} $ and $\Psi (\cO ^{\widetilde{w}}_{\Gamma _f }) $ are isomorphic in the derived category of right $\cA ^{op} \ot \cB$-modules.
Hence $\Ch ( \Delta _{\RR f_*} ) = \Ch (\Psi (\cO ^{\widetilde{w}}_{\Gamma _f })) $.
Consider a dg functor
\[ \boxtimes: \cA ^{op} \ot \cB \to \MFdg (Z, -\widetilde{w}) ; u^{\vee} \ot v \mapsto D(u) \boxtimes v . \]
The following commutative diagram of natural isomorphisms transforms the abstract terms to the concrete terms:
{\tiny \begin{equation}\label{diag: big comm} \xymatrix{
HH_* (\cA ^{op} ) \ot HH_* (\cB) \ar[dd]_{I_{HKR}}^{\cong} \ar[r]_{\Kun}^{\cong} & HH_* ( \cA ^{op} \ot \cB ) \ar[d]_{\boxtimes} \ar[rd]^{Yoneda}_{\cong} \\
& HH_* ( \MFdg (Z, -\widetilde{w})) \ar[d]_{I_{HKR}}^{\cong} \ar[d] \ar[r]_{\Psi} & HH_* (\Perf (\cA ^{op} \ot \cB)) \\
\HH^{-*} (\Omega ^{\bullet}_X, dw) \ot \HH ^{-*} (\Omega ^{\bullet}_{X'}, -dw)
\ar[r]^(.6){\cong}_(.6){\text{\em K\"unneth}} & \HH^{-*} (\Omega _{Z}^{\bullet}, d\widetilde{w}) . & } \end{equation}}
The commutativity of the triangle is straightforward. The commutativity of the rectangle can be seen as follows. Using the Mayer-Vietoris sequence argument,
we reduce it to the case when $X$ and $X'$ are affine. We further reduce it to the curved smooth algebra case. In the curved smooth algebra case, the commutativity
of a corresponding diagram for Hochschild complexes of the second kind is straightforward; see for example \cite{CKK}.
We conclude that
\begin{equation}\label{eqn: ch Rf} \ch ( \Delta _{\RR f_*} ) = \ch (\cO ^{\widetilde{w}}_{\Gamma _f}) \in \HH ^{0} (\Omega ^{\bullet}_{Z} , d\widetilde{w}) \end{equation}
by the compatibility of the K\"unneth isomorphisms and the HKR type isomorphisms in \eqref{diag: big comm}.
In particular for $f=\mathrm{id}_X$ we have
\begin{equation}\label{eqn: ch diag} \ch ( \Delta _{\cA} ) = \ch (\cO ^{\widetilde{w}}_{\Delta _X}) \in \HH ^{0} (\Omega ^{\bullet}_{X^2} , d\widetilde{w}) \end{equation}
if the subscript $\Delta _X$ denote $\Gamma _{\mathrm{id}_X}$.
\subsection{Some definitions}
\begin{Def}\label{def: Td} Considering a vector bundle $F$ as an object in the derived category of coherent sheaves on $X$, we have
the categorical Chern character of $F$ and hence $\ch (F) \in \oplus_p H ^0 (X, \Omega ^p_X [p]) \underset{I_{HKR}}{\cong} HH _0 (D^b (\mathrm{coh} (X)))$. Using this
and the Todd class formula in terms of Chern roots we define
$\td (F) \in \bigoplus _{p} H^0 (X, \Omega ^p_X[p])$, which we call the {\em Todd class} of $F$ valued in Hodge cohomology.
We write $\td (X)$ for $\td (T_X)$, called the Todd class of $X$. Similarly, we define the $i$-th Chern class $c_i (F)$ of $F$ valued in Hodge cohomology.
\end{Def}
\begin{Def}\label{def: Wedge} Let $w_i \in \Gamma (X, \cO _X)$, $i=1, 2$ and let $Z_i$ be the critical locus of $w_i$.
The {\em wedge product} $\wedge$ of twisted Hodge cohomology classes is defined by the composition of
\begin{align*}
& \bigoplus _{q_1+q_2 = q \in \GG} \HH ^{q_1} (X, (\Omega ^{\bullet} _X, dw_1)) \ot \HH ^{q_2} (X, (\Omega ^{\bullet} _X, dw_2)) \\
& \qquad \qquad \xrightarrow{\text{\em K\"unneth}} \quad \HH ^{q} (X^2 , (\Omega ^{\bullet} _X, dw_1 ) \boxtimes (\Omega ^{\bullet} _X, dw_2 )) \\
& \qquad \qquad \qquad \qquad \xrightarrow{\Delta ^*_X} \quad \HH ^{q} (X, (\Omega ^{\bullet} _X, dw_1 ) \otimes _{\cO_X} (\Omega ^{\bullet} _X, dw_2 )) \\
& \qquad \qquad \qquad \qquad \qquad \xrightarrow{ wedge } \quad \HH ^{q}_{Z_1\cap Z_2} (X, (\Omega ^{\bullet} _X, dw_1+ dw_2 )) . \end{align*}
Here $\Delta ^*_X$ is the pullback of the diagonal morphism $X \to X\ti X$. We sometimes omit the symbol $\wedge$ for the sake of simplicity.
\end{Def}
\begin{Def}\label{def: ( )} Let $Z$ denote the critical locus of $w$.
Consider a sequence of maps
\begin{multline} \HH ^{*} (X, (\Omega ^{\bullet}_X, -dw)) \ti \HH ^{*} (X, (\Omega ^{\bullet}_X, dw))
\xrightarrow{\cdot \wedge \cdot } \HH ^{*} _Z(X, (\Omega ^{\bullet}_X, 0)) \\
\xrightarrow{\wedge \td (X) } \HH ^{*} _Z(X, (\Omega ^{\bullet}_X, 0))
\xrightarrow{proj} \HH ^0_Z (X, \Omega ^n _X [n] )
\xrightarrow{} \HH ^0_c (X, \Omega ^n _X [n] )
\xrightarrow{(-1)^{{n+1}\choose{2}} \int _X} k . \end{multline}
Denote the composition $ (-1)^{ {n+1}\choose{2} } \int _X( \cdot \wedge \cdot \wedge \td (X))$ by $\lan , \ran$.
\end{Def}
\subsection{Proof of Theorem \ref{thm: exp pairing} }\label{pf exp pairing}
Since $\td (X)$ is invertible, the nondegeneracy of $\lan , \ran $ follows from Serre's duality; see \cite[\S~4.1]{FK: GLSM}.
Therefore it is enough to show that $\lan , \ran $ satisfies \eqref{eqn: char pairing} under the HKR-type isomorphism in \eqref{diag: big comm}.
Recalling \eqref{eqn: ch diag},
we write \[\ch (\cO ^{\widetilde{w}}_{\Delta _X}) = \sum_i t^i \ot t_i \in \bigoplus _{q \in \GG } \HH^{q} (X, (\Omega ^{\bullet}_X, dw)) \ot \HH ^{-q} (X, (\Omega ^{\bullet}_X, -dw )) .\]
For $\gamma \in \HH ^{*} (\Omega ^{\bullet}_{X}, -dw)$ and $\gamma ' \in \HH ^{*} (\Omega ^{\bullet}_{X}, dw) $, we have
\begin{align}
\sum _i \lan \gamma , t^i \ran \lan t_i, \gamma ' \ran
& = \int _{X\ti X} ( \gamma \ot \gamma ') \wedge \ch (\cO ^{\widetilde{w}}_{\Delta _X}) \wedge (\td (X) \ot \td (X) ) , \label{eqn: ch Delta}
\end{align}
since $\int _X \ot _k \int _X = \int _{X \ti X} \circ \text{\it K\" unneth} $.
Since $\cO ^{\widetilde{w}}_{\Delta _X}$ is supported on the diagonal $\Delta _X \subset X \ti X$, we will apply the deformation of $X\ti X$ to
the normal cone of $\Delta _X$. The normal cone is isomorphic to the tangent bundle $T_X$ of $X$. Let $\pi$ denote the projection $T_X \to X$.
We claim a sequence of equalities
\begin{align*}
\text{ RHS of }\eqref{eqn: ch Delta}
& \overset{(\dagger )}{=} \int _{T_X} \pi^* (\gamma \wedge \gamma ') \wedge \ch (\mathrm{Kos} (s)) \wedge \pi^* \td (X) ^2 \\
& \overset{(\dagger\dagger)}{=} (-1)^{\binom{n+1}{2}} \int _{X} (\gamma \wedge \gamma ' \wedge \td (X))
= \lan \gamma, \gamma ' \ran ,
\end{align*} whose proof will be given below.
Here $s$ is the `diagonal' section of $\pi ^*T_X$ defined by $s(v) = (v, v) \in \pi^* T_X $ for $v\in T_X$
and $\mathrm{Kos} (s)$ is the Koszul complex $(\bigwedge ^{\bullet} \pi^* T_X ^{\vee} , \iota _s )$ associated to $s$.
For $(\dagger)$ consider the deformation space $M^{\circ}$ of $X\ti X$ to the normal cone of the diagonal $\Delta_{X}$; see \cite{Fulton}.
It is a variety with morphisms $h: M^{\circ} \to X\ti X$ and $pr: M^{\circ} \to \PP^1$, satisfying that (i)
the preimages of general points of $\PP^1$ are $X \ti X$, (ii) the preimage of a special point $\infty$ of $\PP ^1$ is
the normal cone $N_{\Delta _X / X^2} = T_X$,
(iii) $pr$ is a flat morphism, and (iv) $h|_{T_{X}} $ coincides with the composition $\Delta \circ \pi$.
The morphism $\Delta \ti \mathrm{id}_{\mathbb{A}^1} : X \ti \mathbb{A}^1 \to X \ti X \ti \mathbb{A}^1$ extends to a closed immersion $f: X \ti \PP ^1 \to M^{\circ}$.
For a closed point $p$ of $\PP^1$ let $M^{\circ}_p$ denote the fiber $pr ^{-1} (p)$ and consider the commuting diagram
\[ \xymatrix{ X \ar[r] \ar[d]_{ \Delta = f_0} & X\ti \PP ^1 \ar[d]^{f} & \ar[l] X \ar[d]^{f_{\infty} = \text{ zero section}} \\
X^2 = M^{\circ}_0 \ar[r]^{g_0} \ar[d] & M^{\circ} \ar[rd]^{h} \ar[d]^{pr} & \ar[l]_{g_{\infty}} M^{\circ}_{\infty} = T_X \ar[d]^{ \Delta \circ \pi } \\
0 \ar[r] & \ \PP ^1 & X^2 } \]
with three fiber squares.
Since $X\ti \PP^1$ and $M^{\circ}_p$ are Tor independent over $M^{\circ}$, we have
\begin{equation}\label{eqn: Tor} \mathbb{L} g_p^* f_* \cO _{X\ti \PP ^1} \underset{qiso}{\sim} (f_{p})_{ *} \cO _{X} , \end{equation}
i.e., they are quasi-isomorphic as coherent factorizations for $(M_p, -h^* \widetilde{w} |_{M_p})$.
Note that $h^*\widetilde{w}|_{\PP (T_{X} \oplus \cO_{X} )} = 0$.
Since $s$ is a regular section with the zero locus $X \subset T_X$,
two factorizations $(f_{\infty})_{ *}\cO _{X} $ and $ \mathrm{Kos} (s) $
are quasi-isomorphic to each other
as coherent factorizations for $( T _{X} , 0)$:
\begin{equation}\label{eqn: Kos bar} (f_{\infty})_{ *}\cO _{X} \underset{qiso}{\sim} \mathrm{Kos} (s) . \end{equation}
For $\rho = ( \gamma \ot \gamma ') \wedge (\td (X) \ot \td (X)) $, we have a sequence of equalities
\begin{align*} & \int _{X \ti X} \rho \wedge \ch ( (f_{0})_* \cO _{X}) & \\
= & \int _{X \ti X} \rho \wedge \ch ( \mathbb{L}g_0^* f_* \cO _{X \ti \PP ^1} )) & \text{ by \eqref{eqn: Tor}} \\
= & \int _{X \ti X} g_0^*(h^* \rho \wedge \ch ( f_* \cO _{X \ti \PP ^1} )) & \text{ by the functoriality of $\ch$} \\
= & \int _{T_{X} } g_{\infty}^*(h^* \rho \wedge \ch ( f_* \cO _{X \ti \PP ^1} )) &
\text{ by Lemma \ref{lem: fiber base}} \\
= & \int _{T_{X} } \pi ^* \Delta^* \rho \wedge \ch ( \mathrm{Kos} (s)) & \text{ by \eqref{eqn: Tor} \& \eqref{eqn: Kos bar}} ,
\end{align*}
which shows $(\dagger)$.
The equality $(\dagger\dagger)$ immediately follows from some basic properties of the proper pushforward \eqref{def: gen push}
in Hodge cohomology: the functoriality \eqref{eqn: fun}, the projection formula \eqref{eqn: proj},
and \eqref{eqn: rel comp}.
\subsection{Proof of Theorem \ref{thm: main}}
For $\ka \in \oplus _{i \in \GG} \RR^i \End (P)$ and $\kb \in \oplus _{i \in \GG} \RR^i\End (Q)$, let us define
\[L_{\kb} \circ R_{\ka} : \oplus _{i \in \GG}\RR ^i \Hom (P, Q) \to \oplus _{i \in \GG} \RR ^i \Hom (P, Q) , \ c \mapsto (-1)^{|\ka||c|} \kb\circ c\circ \ka .\]
Since $\ka$ and $\kb$ are cycle classes of $C (\cA)$, they can be considered as elements of $HH_* (\cA)$.
We denote by $\tau (\ka)$, $\tau (\kb)$ be the image of $\ka$, $\kb$ under the HKR map. The map $\tau$ is sometimes called the boundary-bulk map.
Combining \eqref{eqn: abs general HRR} and Theorem \ref{thm: exp pairing} we obtain this.
\begin{Cor}\label{cor: gen} (The Cardy Condition) We have
\begin{equation}\label{eqn: general HRR} \str (L_{\kb} \circ R_{\ka} ) = (-1)^{ \binom{n+1}{2} } \int _X \tau (\kb ) \wedge \tau (\ka) ^{\vee} \wedge \td (X) .\end{equation}
In particular, Theorem \ref{thm: main} holds.
\end{Cor}
Corollary \eqref{eqn: general HRR} is the matrix factorization version
of Theorem 16 of \cite{CW} and the explicit Cardy condition in \cite{Ram: HRR}.
Let $\fU = \{ U_i \}_{i\in I} $ be an affine open covering of $X$ and let $\nabla _i$ be a connection of $P|_{U_i}$, which always exists.
By \cite{CKK, KP, Platt} the following formula for $\tau (\ka)$ in the \v{C}ech cohomology $\check{\HH}^0 (\fU, (\Omega ^{\bullet}_X, dw )$
is known:
\[ \tau (\ka) = \str \left( \big( \exp (-([\nabla _i, \delta _E])_i - (\nabla _i - \nabla _j)_{i < j} ) \big)\check{\ka} \right) , \]
where $\check{\ka}$ is a \v{C}ech representative of $\ka$.
Here we recall that $\Omega ^{\bullet}_X = \oplus _{p=0}^n \Omega _X ^p [p] $ is $\GG$-graded.
In the local case, i.e., $X$ is an open neighborhood of the origin $0$ in $\mathbb{A} ^n_k$
and $w$ has a critical point only at $0$ with $w(0)=0$, we can relate the canonical pairing with a residue pairing.
Let $x=(x_1, ..., x_n)$ be a local coordinate system
and let $\partial _i w = \frac{\partial w}{\partial x_i}$.
Proposition 4.34 of \cite{BW: ShkConj} shows that
\[ \int _X ( \tau (\kb) \wedge \tau (\ka) ^{\vee} ) = \underset{x=0}{\mathrm{Res}} \left[ \frac{g(x) f(x) }{\partial _1 w, ..., \partial _n w} \right] \]
for $\tau (\ka) = f(x) dx_1 ... dx_n$, $\tau (\kb) = g(x) dx_1 ... x_n$ in $\Omega ^n _{X} / dw \wedge \Omega ^{n-1}_X$.
Hence from Theorem \ref{thm: exp pairing} and $\tau (\ka ^{\vee}) = \tau (\ka )^{\vee}$ we immediately obtain this.
\begin{Cor} \cite{BW: ShkConj, PV: HRR}
In the local case we have \[ \lan \kb , \ka ^{\vee} \ran _{can} = (-1)^{ \binom{n+1}{2}} \underset{x=0}{\mathrm{Res}} \left[ \frac{g(x) f(x)}{\partial _1 w, ..., \partial _n w} \right] . \]
\end{Cor}
The corollary above reproves a conjecture of Shklyarov \cite[Conjecture 3]{Shk: Residue}.
\subsection{GRR type theorem}\label{sub: GRR}
Consider the proper morphism $f: X \to X'$ in \S~\ref{sub: geom diag}, inducing the dg functor $\RR f_*$ and the module $\Delta _{\RR f_*} \in \Perf (\cA ^{op} \ot \cB)$.
They together make a commutative diagram
\begin{equation*} \xymatrix{ \ar[d]_{Yoneda} \cA := \MFdg (X, w) \ar[r]^{ \RR f_* } & \cB := \MFdg (X' , w') \ar[d]^{Yoneda} \\
\Perf \cA \ar[r]_{T_{\Delta _{\RR f_*}}} & \Perf \cB . } \end{equation*}
The paring defined by the composition
\[ \HH ^{*} (X', (\Omega ^{\bullet}_{X'}, -dw')) \ot \HH ^{*} (X', (\Omega ^{\bullet}_{X'}, dw')) \xrightarrow{\wedge} \HH ^{*} _c (X', (\Omega ^{\bullet}_{X'}, 0)) \xrightarrow{\int _{X'}} k \]
is nondegenerate by the Serre duality; see \cite[\S~4.1]{FK: GLSM}.
Using the paring we define the pushforward for $q\in \GG$
\[ \int _f : \HH ^{q} (X, (\Omega ^{\bullet}_{X}, -dw)) \to \HH ^{q} (X', (\Omega ^{\bullet}_{X'}, -dw')) \]
by the projection formula requirement
\[ \int _{X'} ( \int _f \ka ) \wedge \beta = \int _{X} \ka \wedge f^* \beta \]
for every $\beta \in \HH ^{-q} (X', (\Omega ^{\bullet}_{X'}, dw'))$.
Let $n= \dim X$ and $m= \dim X'$. Denote by $HH (\RR f_* )$ the map in Hochschild homology level from $\RR f_*$.
Let $K_0 (\cA)$, $K_0 (\cB)$ be the Grothendieck group of the homotopy category of $\cA$, $\cB$, respectively.
\begin{Thm}
The diagram
\begin{equation*} \xymatrix{ K_0 (\cA ) \ar[d]_{\Ch} \ar[rr]^{\RR f_*} & & K_0 (\cB) \ar[d]^{\Ch} \\
\ar[d]_{I_{HKR}} HH_* (\cA) \ar[rr]^{HH(\RR f_*)} & & HH_* (\cB ) \ar[d]^{I_{HKR}} \\
\HH ^{-*} (X, (\Omega ^{\bullet}_{X}, -dw)) \ar[rr]_{ (-1)^{\sharp } \int _{f} \cdot \wedge \td (T_f) } & & \HH ^{-*} (X', (\Omega ^{\bullet}_{X'}, -dw')) } \end{equation*}
is commutative. Here $\td (T_f) : = \td (X) / f^* \td (X')$ and $\sharp = \binom{n+1}{2} - \binom{m+1}{2} $.
\end{Thm}
\begin{proof} By the definition of categorical Chern characters the upper rectangle is commutative.
Consider $\gamma \in HH_* (\cA) $. Let $\ka := I_{HKR} (\gamma )$ and $\ka ' := I_{HKR} (HH (\RR f_* ) (\gamma )) $.
If we write $\ch (\Delta _{\RR f_*}) = \sum_i T^i \ot T_i \in \HH ^{*} (X, (\Omega ^{\bullet}_{X}, dw)) \ot \HH ^{*} (X', (\Omega ^{\bullet}_{X'}, -dw)) $,
then by Proposition \ref{prop: char can} and Theorem \ref{thm: exp pairing} we have
for $\beta \in \HH ^{-*} (X', (\Omega ^{\bullet}_{X'}, dw'))$
\begin{equation}\label{eqn: Rf 1}
\int _{X'} \ka ' \wedge \beta \wedge \td (X') = (-1)^{{n+1}\choose{2}} \sum _i \int _X \ka \wedge T^i \wedge \td (X) \int _{X'} T_i \wedge \beta \wedge \td (X') . \end{equation}
By \eqref{eqn: ch Rf} and a normal-cone deformation argument as in \S~\ref{pf exp pairing} we have
\begin{align*} & \mathrm{RHS} \text{ of } \eqref{eqn: Rf 1} \\
& = (-1)^{{n+1}\choose{2}} \int _{X\ti X' } (\ka \ot \beta) \wedge \ch (\cO ^{\widetilde{w}}_{\Gamma _f }) \wedge (\td (X) \ot \td (X')) \\
& = (-1)^{{n+1}\choose{2}} \int _{f^*T_{X'}} \pi ^* (\ka \wedge f^* \beta \wedge \td (f^* T_{X'}) \wedge \td (X)) \wedge \ch (\mathrm{Kos} (s)) \\
& = (-1)^{\sharp } \int _{X} \ka \wedge f^* \beta \wedge \td (X) = (-1)^{\sharp } \int _{X'} (\int _f \ka \wedge \td (X) ) \wedge \beta ,\end{align*}
where $\pi$ denotes the projection $f^*T_{X'} \to X$ and $s$ is the diagonal section
of $\pi^*f^*T_{X'}$ on $f^*T_{X'} $.
Hence $\mathrm{LHS}\text{ of } \eqref{eqn: Rf 1}$ equals $ (-1)^{\sharp} \int _{X'} (\int _f \ka \wedge \td (X) ) \wedge \beta $, which shows the commutativity of the lower rectangle.
\end{proof}
\subsection{Pushforward in Hodge cohomology}
We collect some properties of pushforwards in Hodge cohomology that are used in \S~\ref{pf exp pairing}. For lack of a suitable reference we provide their proofs.
Throughout this subsection $f : X \to Y$ will be a morphism between varieties $X$, $Y$ with dimensions $n$, $m$ respectively.
Let $d= n - m$.
\subsubsection{Definition of $f_*$}
Suppose that $f$ is a proper locally complete intersection (l.c.i) morphism.
Let $E$ be a perfect complex on $Y$. Denote by \[ \tau _f : \RR f_* f^! E \to E \]
the duality map in the derived category $D^+_{qc} (\cO_Y)$ of cohomologically bounded below quasi-coherent sheaves; see
for example \cite[\S~4]{Lipman: Found}.
Since $f$ is l.c.i, $f^! \cO_Y$ is taken to be an invertible sheaf up to shift and there is
a canonical isomorphism $\LL f^*E \ot f^!\cO _Y \cong f^! E $.
For $q\in \ZZ$, let $\delta \in \HH ^q (X, f^! E)$, which can be considered as a map $\delta : \cO _X [-q] \to f^! E $ in the derived category.
We have a composition of maps
\[ \cO _Y [-q] \xrightarrow{natural} \RR f_* f^* \cO _Y [-q] \xrightarrow{\RR f_* (\delta ) } \RR f_* f^! E \xrightarrow{ \tau _f } E , \]
denoted by $f_* (\delta )$. This yields a homomorphism
\[ f_* : \HH^q (X, f^! E) \to \HH ^q (Y , E) . \]
Let $g: Y \to Z$ be a proper l.c.i. morphism between varieties.
The uniqueness of adjunction implies the functoriality of the pushforward
\begin{equation}\label{eqn: gen fun} ( g \circ f )_* = g_* \circ f_*: \HH ^q (X, ( g \circ f )^! F ) \to \HH ^{q} (Z, F ) \end{equation}
for $F$ in $D^+_{qc} (\cO_Z)$
\subsubsection{Definitions of $\int _{f}$ and $\int _X$}\label{def int}
Let $f : X \to Y$ be a morphism between nonsingular varieties.
For $p\ge 0$ with $p-d \ge 0$ we have a natural homomorphism
\begin{multline*}
\Omega ^{p}_{X} [q] \cong \bigwedge^{n - p } T_X [ q] \ot f^* \Omega _Y^m [-d] \ot f^! \cO _Y \\
\xrightarrow{} \bigwedge^{n - p } f^* T_Y [ q] \ot f^* \Omega _Y^m [-d] \ot f^! \cO _Y \cong f^* \Omega _Y^{p-d} [q-d] \ot f^! \cO _Y .
\end{multline*} denoted by $\mathscr{T} _f.$
We define {\em Hodge cohomology with proper supports along $f$} as the direct limit:
\[ H^q_{cf} (X, \Omega ^{p} _{X}) := \lim_{\longrightarrow} H^q_Z (X, \Omega ^{p} _{X}) , \]
where $Z$ runs over all closed subvarieties of $X$ that are proper over $Y$.
By Nagata's compactification and the resolution of singularities
there is a nonsingular variety $\bar{X}$ including $X$ as an open subvariety
and a proper morphism $\bar{f}: \bar{X} \to Y$ extending $f$.
Recall the fact that if $Z$ is a closed subvariety of $X$ that is proper over $Y$, then $Z$ is a closed subvariety of $\bar{X}$. Let
\[ nat: H^q_{\natural _1} (X, \Omega ^{p} _{X}) \to H^q_{\natural _2} (\bar{X}, \Omega ^{p}_{\bar{X}}) \] be the natural map where $(\natural _1, \natural _2)$ is either $(c, c)$
or $(cf, \emptyset)$.
We define the pushforward (for $p\ge 0$ with $p-d \ge 0$)
\begin{equation}\label{def: gen push} \int _{f} : H^q _{\natural _1} (X, \Omega ^{p} _{X}) \to H^{q-d}_{\natural _2} (Y, \Omega ^{p-d} _Y ) ; \gamma \mapsto \bar{f}_* ( \mathscr{T}_f (nat (\gamma ))) . \end{equation}
Using the functoriality \eqref{eqn: gen fun}, we note that $\int _{f} $ is independent of the choices of $\bar{X}$, an open immersion $X\hookrightarrow \bar{X}$, and an extension $\bar{f}$.
When $Y=\Spec k$, we also write $\int _{X}$ for $\int _{f}$
If $ v : X' \to X$ be a proper morphism between nonsingular varieties, we have the natural pullback map
\[ v^* : H^q_c (X, \Omega ^p_{X}) \to H^q_c (X' , \Omega ^p_{X'}) . \]
\subsubsection{Base change I}
Consider a fiber square diagram of varieties
\begin{equation}\label{diag: fiber diag} \xymatrix{ X' \ar[r]^{v} \ar[d]_g & X \ar[d]^f \\
Y' \ar[r]_u & Y . } \end{equation}
Assume that $f$ is a flat, proper, l.c.i morphism.
Then from the base change \cite[\S~4.4]{Lipman: Found} we
obtain a base change formula, for $\delta \in H^q (X, f^! \cO_Y ) $
\begin{align}\label{eqn: base} g_* (\LL v^* (\delta )) = \LL u^* ( f_* ( \delta ) ) \end{align}
in $H^q (Y', \cO _{Y'})$.
Here $\LL v^*(\delta ) \in H^ q (X' , g^! \cO _Y) $ is the naturally induced map
\[ \cO_{X'} [-q] \to \LL v^* f^! \cO_Y \cong g^! \LL u^* \cO_Y = g^! \cO _{Y'} \] in the derived category.
Furthermore suppose that all varieties $X, Y, X'$ are nonsingular and $Y'$ is a closed point of $Y$. Then for $\gamma \in H^d _{c} (X, \Omega ^d _{X})$ we easily check that
\[ v^* (\gamma ) = \LL v^* (\mathscr{T}_f (\gamma )) \] in $H^d _{c} (X', \Omega ^d _{X'}) = H^0 _{c} (X', g^! \cO_{Y'}) $.
Hence \eqref{eqn: base} for $\delta = \mathscr{T}_f (\gamma)$ means that
\begin{align}\label{eqn: smooth base} \int _{X'} v^* (\gamma) = u^* (\int _f \gamma ) . \end{align}
\subsubsection{Base change II}
Let $Y$ be a connected nonsingular complete curve and let $Y'$ be a closed point of $Y$.
Consider the fiber square diagram \eqref{diag: fiber diag} of nonsingular varieties.
Assume that $f$ is flat but possibly non-proper.
\begin{lemma} \label{lem: fiber base}
For $ \gamma \in H^d _c (X, \Omega ^d _X)$ we have
\begin{equation}\label{eqn: fiber base} \int _{X'} v ^* (\gamma ) = \int _{f} \gamma \quad \in k . \end{equation}
\end{lemma}
\begin{proof}
By Nagata's compactification $f$ is extendible to a proper flat morphism
$\bar{f}: \bar{X} \to Y$ with an open immersion $X\hookrightarrow \bar{X}$. By the resolution of singularities we can make that $\bar{X}$ is nonsingular
and the closure $\bar{X'}$ of $X'$ in $\bar{X}$ is also nonsingular. Let $\bar{v} : \bar{f}^{-1} (Y') \to \bar{X}$ be the induced morphism and let $\bar{v}_{\circ} := \bar{v} |_{\bar{X'}} $.
Thus we have a commutative diagram
\begin{equation}\label{diag: comp fiber diag} \xymatrix{ \bar{X'}\ \ar@{^{(}->}[r] \ar[rd]_{\bar{g}_{\circ} } \ar@/^1.3pc/[rr]^{\bar{v}_{\circ}}
& \bar{f}^{-1}(Y') \ar[r]_(.6){\bar{v}} \ar[d]_{\bar{g}} & \bar{X} \ar[d]^{\bar{f}} \\
& Y' \ar[r]_u & Y , } \end{equation} with a fiber square.
To show \eqref{eqn: fiber base} we may assume $ \gamma \in H^d _Z (X, \Omega ^d _X)$ for some complete subvariety $Z$ of $X$.
Let $nat$ denote the natural map $H^d _Z (X, \Omega _X^d) \to H^d (\bar{X} , \Omega _{\bar{X}}^d)$. Then by the support condition of $\gamma$
we have
\begin{equation}\label{eqn: LH RH}
(\bar{g}_{\circ})_* \LL \bar{v}_{\circ}^* (\mathscr{T}_{\bar{f}} ( nat (\gamma ) ))
= \bar{g}_* \LL \bar{v}^* (\mathscr{T}_{\bar{f}} ( nat (\gamma ) )) \in k . \end{equation}
Since $ \bar{v}_{\circ}^* ( nat ( \gamma )) = \LL \bar{v}_{\circ}^* (\mathscr{T}_{\bar{f}} (nat(\gamma )) ) $ under $\Omega ^d _{\bar{X'}} \cong g^! \cO _{Y'}$,
LHS of \eqref{eqn: LH RH} becomes $\int _{\bar{X'}} \bar{v}_{\circ}^* ( nat ( \gamma )) $, which equals to LHS of \eqref{eqn: fiber base} by the support condition of $\gamma$.
On the other hand by \eqref{eqn: base}, RHS of \eqref{eqn: LH RH} becomes $u^* \int _{\bar{f}} nat (\gamma) $, which equals to RHS of \eqref{eqn: fiber base} by the support condition and
$H^0 (Y, \cO_Y) = k$.
\end{proof}
\subsubsection{Projection formula}
Let $X$, $Y$, $Z$ be nonsingular varieties and let $f: X \to Y$, $g: Y\to Z$ be morphisms. Let $d' = \dim Y - \dim Z$.
The uniqueness of adjunction implies the functoriality of the pushforward, for $p\ge 0$ with $p-d \ge 0$ and $p-d - d' \ge 0$
\begin{equation}\label{eqn: fun} \int_{g\circ f} = \int _g \circ \int _f : H^q_c (X, \Omega ^{p} _X) \to H^{q-d- d'} _c (Z, \Omega ^{p-d-d'} _Z ) . \end{equation}
Let $f:X \to Y$ be a (possibly non-proper) morphism between nonsingular varieties.
Then for $\gamma \in H^d _{cf} (X, \Omega ^d _{X})$ and $\sigma \in H^{q} (Y, \Omega ^{p} _{Y})$
the projection formula
\begin{align}\label{eqn: proj} \int _f ( f^* \sigma \wedge \gamma ) = \sigma \wedge \int _f \gamma \end{align}
holds in $H^q (Y, \Omega ^p _{Y} )$. This can be verified as follows. We may assume that $f$ is proper. Consider the commuting diagram
\[ \xymatrix{ \Omega _{Y} ^{p} \ar[r]
& \RR f_* \Omega _{X} ^{p} \ar[rr]^{\RR f_* (\cdot \wedge \gamma ) \ \ \ \ \ } &
& \RR f_* ( \Omega _{X} ^{p} \wedge \Omega ^{d}_{X} [d]) \ar[rr]^{\RR f_*(\mathscr{T}_f ) } & & \Omega ^{p}_{Y} \ot \RR f_* f^!\cO_Y \\
\cO_Y [-q] \ar[u]^{\sigma} \ar[r] & \RR f_* \cO _X \ar[u]^{\RR f_* f^* \sigma } \ar@/^.7pc/[urr]_(0.4){\ \ \ \RR f_* (f^* \sigma \wedge \gamma)}
\ar@/_1pc/[rrrru]_{\ \ \RR f_* (\mathscr{T}_f (f^*\sigma \wedge \gamma))}
& & & & } \] We note that the composition of the maps in the top horizontal line is $\mathrm{id}_{\Omega ^p _Y} \ot \RR f_* ( \mathscr{T}_f ( \gamma))$
using the generic smoothness of $f$ and local coordinate systems for compatible bases of $\Omega ^1 _{X}$ and $\Omega ^1_Y$.
The clockwise compositions of maps starting from $\cO _Y[-q]$ followed by $\tau _f$ yields LHS of \eqref{eqn: proj} and
the counterclockwise compositions of maps followed by $\tau _f$ yields RHS of \eqref{eqn: proj}.
\subsubsection{Some computations}
Let $Q$ be the tautological quotient bundle on the projective space $\PP^n$. We want to compute $\int _{\PP^n}$ of the top Chern class $c_{n} (Q) \in H^0 (\PP ^n, \Omega ^n _X [n ] )$.
The class $c_{n} (Q)$ is equal to $(-1)^nc_1(\cO (-1))^n$. Let $U_i = \{ x_i \ne 0 \}$ where $x_0, ..., x_n$ are homogeneous coordinates.
On each $U_i$, we may identify $\cO (-1)$ with the $i$-th component of $\cO _{\PP^n} ^{\oplus n+1}$ by the tautological monomorphism
$\cO (-1) \to \cO _{\PP^n} ^{\oplus n+1}$. This yields connections $\nabla _i$ on $\cO (-1) |_{U_i}$.
Let $z_i = x_i / x_0$.
Note that $\nabla _0 - \nabla _i = - \frac{dz_i}{z_i}$.
Hence $\nabla _i - \nabla _j = \frac{dz_i}{z_i} - \frac{dz_j}{z_j}$ on
$U_0 \cap U_i \cap U_j$.
By the $n$-th fold Alexander-\v{C}ech-Whitney cup product of a \v{C}ech representative $(\nabla _i - \nabla _j ) _{i<j}$ of $c_1(\cO (-1))$ we conclude that
$ c_n (Q) $ is representable by a \v{C}ech cycle
\begin{equation*} (-1)^{ \binom{n+1}{2}} \frac{dz_1 ... dz_n}{z_1 ... z_n} \in \Omega ^n _{\PP^n} (U _0 \cap ... \cap U_n) . \end{equation*}
Here the sign contribution of $ \binom{n}{2}$ among $ \binom{n+1}{2}$
comes from the exchanges of odd \v{C}ech `elements' and differential one forms $\frac{dz_i}{z_i}$; see \cite{BW: ShkConj, CKK}.
Thus \begin{equation}\label{eqn: comp} \int _{\PP ^n_k} c_n (Q) = (-1)^{ \binom{n+1}{2}} \mathrm{res} [ \frac{dz_1 ... dz_n}{z_1 ... z_n} ] = (-1)^{ \binom{n+1}{2}} . \end{equation}
Let $E$ be a rank $n$ vector bundle on a nonsingular variety $X$ and let $\pi: E\to X$ be the projection.
We have the diagonal section $s$ of $\pi ^* E$ by letting $s(e) = (e, e)$.
Let $\bar{\pi} : \PP (E \oplus \cO_X ) \to X$ be the projection, which is a proper extension of $\pi$:
\[ \PP (E \oplus \cO_X ) = \PP (E) \sqcup E \supset E \supset \PP (\cO_X) = X . \]
Let $\cQ$ be the tautological quotient bundle on $\PP (E \oplus \cO_X )$.
It has a section $\bar{s}$ by the composition $\cO \xrightarrow{(0, -\mathrm{id})} \pi ^*E \oplus \cO \xrightarrow{quot} \cQ$.
Note that the zero locus $\bar{s}$ is $\PP (\cO_X)$, since $(0, - \mathrm{id})$ is factored through the kernel of $quot$ exactly on $\PP (\cO _X)$.
Note that the composition $quot \circ (\mathrm{id}, 0) |_{E} : \pi ^* E \to \cQ |_{E}$ is an isomorphism sending $s$ to $\bar{s}|_E$. Therefore we have
\begin{multline}\label{eqn: rel comp}
\int _{\pi} \ch (\mathrm{Kos} (s)) \td (\pi^* E)
= \int _{\bar{\pi}} \ch (\mathrm{Kos} (\bar{s})) \td (\cQ) \\
= \int _{\bar{\pi}} c_{n} (\cQ) \text{ (by letting $\bar{s}=0$)}
= (-1)^{ \binom{n+1}{2}} \text{ (by \eqref{eqn: smooth base} \& \eqref{eqn: comp})}. \end{multline} | 187,030 |
The following information provides an awareness of problems that might be avoided in the future. The information is based on final reports by official investigative authorities on aircraft accidents and incidents.
Jets
Antiskid System Failed
Beech Premier 1A. Destroyed. Five fatalities, two serious injuries.
Affected by an acute sleep loss and inadequate knowledge of the business jet’s systems, the pilot did not react to an indication that the antiskid braking system had failed, according to investigators. He continued the approach but likely realized shortly after touchdown that he would not be able to bring the airplane to a stop on the runway.
The pilot attempted to go around but neglected to retract the landing gear and the wing lift-dump system (spoilers). Struggling to climb, the airplane struck a utility pole and crashed, killing all five passengers and seriously injuring the pilot and copilot.
The accident occurred the night of Feb. 20, 2013, during a business flight from Nashville, Tennessee, U.S., to Thomson, Georgia. During the flight, the copilot cautioned the pilot that he was “blowing through” an assigned altitude and reminded him to adjust his altimeter setting.
“The pilot responded to the altimeter reminder by stating, ‘Say, I’m kind of out of the loop or something. … I appreciate you looking after me,’” said the report by the U.S. National Transportation Safety Board (NTSB).
Investigators determined that the pilot likely was suffering from fatigue. He had slept only five hours early the previous night and had received only about four hours of rest, interrupted by several telephone calls, during the subsequent 18 hours preceding the accident.
Night visual meteorological conditions (VMC) prevailed at Thomson-McDuffie County Airport, and the pilots conducted a visual approach to Runway 10. Shortly after the landing gear was extended, the “ANTI SKID FAIL” annunciator illuminated. The copilot called out the warning to the pilot.
“The pilot continued the approach,” the report said. “He did not respond to the copilot and did not refer to the abnormal procedures … checklist to address the antiskid system failure message.”
The airplane’s flaps had been extended to 30 degrees for the approach, a configuration that is not authorized for landing with the antiskid system inoperative. “According to the checklist, the pilot should move the antiskid switch to ‘OFF’ and plan for a flaps-10 or flaps-up landing,” which increase landing distance by 130 percent and 89 percent, respectively, the report said. However, in either of the approved configurations, the Premier’s required landing distance exceeded the 5,208 ft (1,587 m) available for landing on Runway 10.
About seven seconds after touchdown, the pilot initiated a go-around. Although neither pilot later could recall the reason for the go-around, “it is likely that the pilot recognized that the airplane was not slowing as he expected and might not stop before the end of the runway,” the report said.
The airplane lifted off near the end of the runway and was about 63 ft above the ground when the left wing struck a 72-ft (22-m) concrete utility pole about 1,835 ft (559 m) from the departure threshold. The flaps were in transit, but the landing gear and spoilers were still extended on impact. “The airplane continued another 925 ft [282 m] before crashing in a wooded area” at 2006 local time, the report said.
Although the seat belt sign was illuminated and the chime was sounded during approach, none of the passengers had their seat belts or shoulder harnesses fastened. They succumbed to multiple traumatic injuries, the report said.
The report noted that the utility pole was among several that did not meet U.S. Federal Aviation Administration (FAA) airport-obstruction standards; the utility company had not notified the FAA, as required, before erecting the poles in 1989.
The NTSB concluded that the probable causes of the accident were “the pilot’s failure to follow airplane flight manual procedures for an antiskid failure in flight and his failure to immediately retract the lift-dump after he elected to attempt a go-around on the runway.” The report said that contributing factors were “the pilot’s lack of systems knowledge and his fatigue due to acute sleep loss.”
Struck by Lightning
Embraer 145LR. Minor damage. No injuries.
Significant convective activity prevailed along the regional jet’s route from Dallas–Fort Worth (Texas, U.S.) International Airport to Madison, Wisconsin, the evening of Jan. 24, 2012. The airplane encountered thunderstorms shortly after departing with 50 passengers and three crewmembers.
“The [flight] crew attempted to identify areas where they could divert to avoid the convective activity, but the controller was unable to approve significant deviations because of other traffic flows in the vicinity,” the NTSB report said.
About 15 minutes after takeoff, while discussing the possibility of turning back to Dallas–Fort Worth, the airplane was struck by lightning. The crew declared an emergency and requested clearance to divert to Little Rock, Arkansas. The controller immediately approved the request.
“After landing, the airplane was inspected and found to have sustained damage from the lightning strike requiring repairs to the left wing skin and rivets, and replacement of the left wing tip and aileron,” the report said.
The incident was included as an example in a subsequent recommendation by NTSB that the FAA improve the transmission of real-time lightning data to controllers and pilots.
Hydraulic Failure
Gulfstream GV-SP. Substantial damage. No injuries.
Day VMC prevailed when the Gulfstream departed from Appleton, Wisconsin, U.S., on Feb. 14, 2011, for a post-maintenance test flight. After conducting a landing and a low approach at nearby airports, the flight crew returned to Appleton for a landing on Runway 30, which is 6,501 ft (1,982 m) long.
The GV was on final approach when an amber caution light illuminated, indicating low fluid quantity in the left hydraulic system. Subsequently, the left hydraulic system failure warning light illuminated.
“The pilot not flying (PNF) pulled out the checklist to accomplish the left hydraulic system failure procedures and then suggested a go-around because the landing runway was about 500 feet [152 m] shorter than the recommended minimum runway length indicated in the checklist,” the NTSB report said.
However, the pilot flying (PF) decided to continue the approach. “Both the PF and PNF thought the auxiliary hydraulic system could support normal spoilers, brakes and nosewheel steering,” the report said.
After touchdown, the PF deployed the operative thrust reverser on the right engine and applied manual braking. However, he perceived no braking action and initiated a go-around.
The PNF saw the indicated airspeed stagnate at 100 kt and felt no acceleration, so he pulled the throttles back to reject the go-around. The PF then deployed the right thrust reverser again in attempt to slow the airplane.
The landing gear collapsed and the left wing was substantially damaged after the GV overran the runway at about 95 kt, but the pilots and their passenger were not injured. Examination of the airplane revealed a hydraulic leak emanating from a fractured connecting tube on the nose landing gear swivel assembly. The assembly was found to be misaligned, resulting in wear and eventual seizure.
Turboprops
Stall During a Go-Around
Beech King Air E90. Destroyed. Two fatalities.
The 1,100-hour private pilot had owned a Cessna 414 piston twin before purchasing the King Air and had received about 58 hours of dual instruction in the twin-turboprop. He had not flown for two months before departing with his flight instructor from Marana, Arizona, U.S., the morning of Feb. 6, 2013. He told a line service worker that they were “going out to practice for about an hour,” the NTSB report said.
A witness who later saw the airplane approaching Runway 05 at the airport in Casa Grande, Arizona, recalled that it pulled up into vertical flight, banked left, pitched nose-down and then struck the ground.
“It is likely that the pilot was attempting a go-around, pitched up the airplane excessively and subsequently lost control, which resulted in the airplane [stalling and] impacting flat desert terrain about 100 feet [30 m] north of the active runway at about the midfield point in a steep nose-down, left-wing-low attitude,” the report said. “A post-accident examination of the airframe and both engines revealed no anomalies that would have precluded normal operation.”
Toxicological testing revealed the presence of tetrahydrocannabinol in the flight instructor’s body, at levels indicating that he “most recently used marijuana at least several hours before the accident,” the report said. “However, the effects of marijuana use on the flight instructor’s judgment and performance at the time of the accident could not be determined.”
Wheels-Up Landing
Fairchild Metro III. Substantial damage. No injuries.
The flight crew was conducting a functional check flight the night of Feb. 15, 2012, following the completion of maintenance work on the left engine fuel flow indicating system at Brisbane, Queensland, Australia.
After takeoff, the crew maneuvered the aircraft for an instrument landing system (ILS) approach to Runway 19. The report by the Australian Transport Safety Bureau (ATSB) did not specify the prevailing weather conditions.
“Upon selection of the landing gear handle to the down position, there were no indications or sounds to indicate that the landing gear had extended,” the report said. After receiving clearance from air traffic control (ATC) to enter a holding pattern, the crew consulted the quick reference handbook (QRH).
Activation of the emergency landing gear release lever resulted in sounds of increased airflow but no indications that the gear was down and locked. The pilots then attempted to manually extend the gear but were able to cycle the hand pump only a few times before it resisted further movement.
The crew proceeded to an “additional procedure” recommended by the QRH. “This procedure required the crew to reduce airspeed to just above the flight idle speed, cycle the gear handle and then return the system to the emergency extension mode,” the report said. “The crew reported that these actions were carried out but the landing gear did not extend.”
Consulting by radio with maintenance personnel, the pilots “cycled the gear handle while conducting a series of aircraft manoeuvres in an attempt to force the gear to extend due to in-flight loading,” the report said. These actions also were unsuccessful in extending the gear, as confirmed by maintenance personnel during low passes over the runway.
The crew then confirmed that the landing gear was fully retracted and conducted a wheels-up landing according to the QRH, shutting down both engines and feathering the propellers before touchdown. The Metro skidded to a stop on two propeller blades on each engine and a tail-mounted navigation antenna.
Investigators found that an electrical wire on the landing gear selector had separated from a connector, preventing normal operation of the gear. “The investigation also identified an out-of-rig condition in the landing gear emergency extension system, which prevented correct operation of that system,” the report said.
Prop Strikes Jet Bridge
Embraer Brasilia. Minor damage. No injuries.
The captain was focusing his attention on the marshaller, and the first officer was looking out the right side window to ensure clearance between the wing and a parked fuel truck as the airplane was taxied to the gate at Los Angeles International Airport the morning of Feb. 16, 2010.
“The marshaller reported that he was concentrating on the airplane’s right side … and was concerned about the [fuel] truck’s proximity to the airplane,” the NTSB report said. “Because of where his attention was focused, he misjudged the stop line marked for EMB [Embraer] airplanes and instead signaled the airplane to stop on the line marked for [Bombardier] CRJ airplanes.” The markings were about 18 ft (5 m) apart.
Just as the marshaller was crossing his wands to signal the pilots to stop the airplane, the left propeller struck the jet bridge. Two of the four blades dented the jet bridge and shed fragments that damaged the airplane’s fuselage and left engine nacelle.
“An oil spray residue was observed on the left side of the fuselage, and a pool of oil was present below the left engine and wing,” the report said. The damage was characterized as minor. None of the three people aboard the airplane, nor the marshaller, was injured.
Noting that the captain would have been able to see part of the left engine and propeller as the airplane neared the jet bridge, the NTSB said that the probable cause of the incident was “the flight crew’s failure to maintain clearance from the jet bridge during taxi.” The report said that a contributing factor was “the ramp marshaller’s diverted attention and failure to signal the flight crew to stop at the correct position.”
Piston Airplanes
Fire on Takeoff
Convair 440-38. Destroyed. Two fatalities.
VMC prevailed when the airplane departed from San Juan, Puerto Rico, the morning of March 15, 2012, for a cargo flight to Saint Martin, Netherlands Antilles. Shortly after takeoff from Runway 10, the flight crew declared an emergency and requested a left turn to return to Luis Muñoz International Airport.
ATC approved the request and cleared the crew to land on Runway 28. The crew asked if the controller saw smoke; the controller said that he did not see smoke coming from the airplane.
“Radar data shows that the airplane was heading south at an altitude of about 520 ft when it began a descending turn to the right to line up with Runway 28,” the NTSB report said. “The airplane continued to bank to the right until radar contact was lost.”
Both pilots were killed when the Convair struck trees and crashed in a lagoon about 1 nm (2 km) east of the approach end of Runway 28.
Investigators estimated that airspeed was about 88 kt — 9 kt below stall speed in level flight and 1 kt above the minimum control speed with one engine inoperative — when ATC lost radar contact with the airplane. “However, minimum control speeds increase substantially for a turn into the inoperative engine, as the accident crew did in the final seconds of the flight,” the report said.
Examination of the wreckage revealed signs of an in-flight fire that erupted in the vicinity of the junction between the augmentor assemblies and the exhaust muffler. The source of ignition was not determined.
The examination also revealed that the right engine had been shut down but the propeller had not been feathered, and that the left propeller had been feathered although the engine controls remained set for takeoff.
“The accident airplane was not equipped with a flight data recorder or a cockpit voice recorder (nor was it required to be so equipped),” the report said. “Hence, the investigation was unable to determine at what point in the accident sequence the flight crew shut down the right engine and at what point they feathered the left propeller, or why they would have done so.”
The switch for the autofeather system, which would have feathered the propeller on the engine losing power and prevented the propeller on the operating engine from feathering, was found in the “OFF” position. Interviews with company pilots indicated that the captain generally did not use either the autofeather system or the engine antidetonation (water-injection) system.
The report said that, on takeoff, the airplane was about 6,800 lb (3,084 kg) above the authorized maximum gross takeoff weight with the autofeather and antidetonation systems inoperative.
The NTSB concluded that the probable cause of the accident was “the flight crew’s failure to maintain adequate airspeed after shutting down the right engine due to an in-flight fire in one of the right augmentors, [which] resulted in either an aerodynamic stall or a loss of directional control.”
Wrong Lever
Beech 58 Baron. Substantial damage. No injuries.
Shortly after the Baron touched down on the runway at Rustenburg, South Africa, the morning of Feb. 13, 2014, the landing gear collapsed. The propeller blades, the engines and the lower fuselage skin were damaged as the aircraft slid to a stop, but none of the four occupants was injured.
“The investigation found that the accident occurred because the pilot was attempting to put the flaps control ‘UP’ after landing and moved the landing gear control instead,” said the report by the South African Civil Aviation Authority (CAA).
“The inadvertent movement of the landing gear control was attributed to the pilot’s being more accustomed to flying aircraft in which these two controls were in exactly opposite locations,” the report said.
Helicopters
Retreating-Blade Stall
Eurocopter MBB-BK117. Minor damage. No injuries.
The helicopter encountered light to moderate turbulence and a tail wind shortly after departing from Port Pirie, South Australia, for a medevac flight to Adelaide, South Australia, the afternoon of Feb. 15, 2013.
About 12 minutes after takeoff, the helicopter abruptly pitched nose-up and rolled left. The helicopter descended from 5,000 ft to about 800 ft before the pilot regained control. The crew returned to Port Pirie and landed without further incident.
ATSB investigators found that the helicopter’s high gross weight and high airspeed, along with the turbulence and high density altitude, were conducive to the onset of a retreating-blade stall. “The pilot’s instinctive action of pushing the cyclic control forward delayed recovery from the stall,” the report said.
Wrong Switch
Robinson R44. Substantial damage. No injuries.
The siren-equipped helicopter departed from Setlakgole, South Africa, for a game-culling flight the morning of Feb. 12, 2014. The pilot said that the helicopter was being flown at 40 kt and about 200 ft above the ground when the engine lost power about 35 minutes after takeoff.
The pilot landed the helicopter straight ahead in dense brush. The R44 touched down hard, and the tail rotor blades struck and severed the tail boom. The pilot and his passenger were not hurt.
“Post-accident investigation did not identify any defects of the engine which could have contributed to the accident,” said the report by the South African CAA, which concluded that the pilot had inadvertently disengaged the hydraulic system in flight. The CAA’s report noted that the switches for both the hydraulic system and the siren are mounted on the cyclic. | 318,951 |
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TITLE: Initial conditions in Lagrangian mechanics
QUESTION [0 upvotes]: If we consider physical systems where we have initial conditions, such as a rolling circle having initial velocity $v$, what does our Lagrangian look like?
REPLY [3 votes]: You do not need the initial condition to write down the Lagrangian. The beauty of Lagrangian is that it gives you the differential equation of motion. When you want to solve the differential equation of motion, you need the initial condition. | 116,303 |
\begin{document}
\title{Eight classes of new Hopf algebras of dimension $128$ without the Chevalley property}
\author{Naihong Hu\thanks{Email:\,[email protected]}\\{\small Department of Mathematics, SKLPMMP,} \\ {\small East China Normal University,
Shanghai 200062, China}
\and Rongchuan Xiong\thanks{Email:\,[email protected]}
\\{\small Department of Mathematics, SKLPMMP,} \\ {\small East China Normal University,
Shanghai 200062, China}
}
\maketitle
\newtheorem{question}{Question}
\newtheorem{defi}{Definition}[section]
\newtheorem{conj}{Conjecture}
\newtheorem{thm}[defi]{Theorem}
\newtheorem{lem}[defi]{Lemma}
\newtheorem{pro}[defi]{Proposition}
\newtheorem{cor}[defi]{Corollary}
\newtheorem{rmk}[defi]{Remark}
\newtheorem{Example}{Example}[section]
\theoremstyle{plain}
\newcounter{maint}
\renewcommand{\themaint}{\Alph{maint}}
\newtheorem{mainthm}[maint]{Theorem}
\theoremstyle{plain}
\newtheorem*{proofthma}{Proof of Theorem A}
\newtheorem*{proofthmb}{Proof of Theorem B}
\newcommand{\K}{\mathds{k}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\BN}{\mathcal{B}}
\newcommand{\Lam}{\lambda}
\newcommand{\HYD}{{}^{H}_{H}\mathcal{YD}}
\newcommand{\As}{^\ast}
\newcommand{\N}{\mathds{N}}
\newcommand{\Pp}{\mathcal{P}}
\newcommand{\LA}{\Lambda^5(\mu)}
\newcommand{\LAA}{\Lambda^6(\mu)}
\begin{abstract}
Classifying Hopf algebras of a given dimension is a hard and open question. Using the generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf algebra $H$ of dimension $16$ without the Chevalley property and the corresponding infinitesimal braidings are simple objects in $\HYD$. In particular, we figure out $8$ classes of new Hopf algebras of dimension $128$ without the Chevalley property.
\bigskip
\noindent {\bf Keywords:} Nichols algebra; Hopf algebra; generalized lifting method.
\end{abstract}
\section{Introduction}
Let $\K$ be an algebraically closed field of characteristic zero. The question of classification of all Hopf algebras over $\K$ of a given dimension up to isomorphism was posed by I. Kaplansky in 1975. He conjectured that every Hopf algebra over $\K$ of prime dimension must be isomorphic to a group algebra which was proved by Y. Zhu \cite{Z94} in 1994. Since then, more and more mathematicians have been trying to classify finite-dimensional Hopf algebras of a given dimension and have made some progress. As the aforementioned, the classification of Hopf algebras of prime dimension $p$ has been completed by Y. Zhu \cite{Z94} and all of them are isomorphic to the cyclic group algebra of dimension $p$. Further results have completed the classification of Hopf algebras of dimension $p^2$ for $p$ a prime (see
\cite{Ma96,AS,Ng02}), of dimension $2p$ for $p$ an odd prime (see \cite{Ma95,Ng05}), and of
dimension $2p^2$ for $p$ an odd prime (see \cite{HN09}). For survey of the classification for dimensions up to $100$, we refer to \cite{BG13} and the references therein. Even though the classification of finite-dimensional pointed Hopf algebras as a special class (with abelian groups as the coradicals) has made some astonishing breakthrough under the assumption that the order of the finite abelian groups are prime to $210$
(\cite{AS10}, etc.) in terms of the lifting method (only as one kind approach of the so-called principle realization) introduced by Andruskiewitsch-Schneider in \cite{AS98}, there are few unequivocal results and solving methods in the classification of Hopf algebras of a given dimension. This constitutes actually a great challenge after Heckenberger et al's studies on the level of Nichols algebras (of group type).
Comparison with other newly designed approaches, we would prefer to the lifting method since any new method inevitably encounters the common hard point question: how to derive the explicit relations involving generators. Let us briefly recall it. Let $A$ be a finite dimensional Hopf algebra such that the coradical $A_0$ is a Hopf subalgebra, which implies that the coradical filtration $\{A_n\}_{n=0}^{\infty}$ is a Hopf algebra filtration where $A_n=A_0\bigwedge A_{n-1}$. Let $G=\text{gr}\, A$ be the associated graded Hopf algebra, that is,
$G=\oplus_{n=0}^{\infty}G(n)$ with $G(0)=A_0$ and $G(n)=A_n/A_{n-1}$. Denote by $\pi: G\rightarrow A_0$ the Hopf
algebra projection of $G$ onto $A_0=G(0)$, then $\pi$ splits the inclusion $i: A_0\hookrightarrow G$ and thus by a theorem of Radford \cite{R85}, $G\cong R\sharp A_0$, where $R=G^{co\pi}=\{h\in H\mid (id\otimes \pi)\Delta_G(h)=h\otimes 1\}$ is a braided Hopf algebra in ${}^{A_0}_{A_0}\mathcal{YD}$. Moreover, $R=\oplus_{n=0}^{\infty}R(n)=R\cap G(n)$ with $R(0)=\K$ and $R(1)=\Pp(R)$, the space of primitive elements of $R$, which is a braided vector space called the infinitesimal braiding. In particular, the subalgebra generated as a braided Hopf algebra by $V$ is so-called Nichols algebra over $V$ denoted by $\BN(V)$, which plays a key role in the classification of pointed Hopf algebra under the following
\begin{conj} $($\text{\rm Conjecture\,2.7 \cite{AS02}}$)$
Any finite-dimensional braided Hopf algebra $R$ in ${}^{A_0}_{A_0}\mathcal{YD}$
satisfying $\Pp(R)=R(1)$ is generated by $R(1)$.
\end{conj}
Usually, if we fix a Hopf algebra $H$, then the main steps of the lifting method for classification of all finite-dimensional Hopf algebras $A$ such that $A_0\cong H$ are:
\begin{itemize}
\item Determine all objects $V$ in $\HYD$ such that Nichols algebras $\BN(V)$ are finite-dimensional and
describe $\BN(V)$ explicitly in terms of generators and relations.
\item For such $V$, determine all possible finite-dimensional Hopf algebras $A$ such that the associated graded
Hopf algebras $\text{gr}\,A\cong \BN(V)\sharp H$. We call $A$ a lifting of $V$ $($or $\BN(V)$$)$ over $H$.
\item Prove that any finite-dimensional Hopf algebra over $H$ is generated by the first term of the coradical filtration.
\end{itemize}
So far, the lifting method has produced many striking results of the classification of pointed or copointed Hopf algebras. For more details about the results, we refer to \cite{A14,BG13} and the references therein.
If $A$ is a Hopf algebra without the Chevalley property, then the coradical filtration $\{A_n\}_{n=0}^{\infty}$ is not a Hopf algebra filtration such that the associated graded coalgebra is no longer a Hopf algebra. To overcome this obstacle, Andruskiewitsch and Cuadra \cite{AC13} extended the lifting method by replacing the coradical filtration $\{A_n\}_{n=0}^{\infty}$ by the standard filtration $\{A_{[n]}\}_{n=0}^{\infty}$, which is defined recursively by
\begin{itemize}
\item $A_{[0]}$ to be the subalgebra generated by the coradical $A_0$;
\item $A_{[n]}=A_{[n-1]}\bigwedge A_{[0]}$.
\end{itemize}
Especially, if $A_0$ is a Hopf algebra, then $A_{[0]}=A_0$ and standard filtration is just the coradical filtration. Under the assumption that $S_A(A_{[0]})\subseteq A_{[0]}$, it turns out that the standard filtration is a Hopf algebra filtration, and the associated graded coalgebra $\text{gr}\,A=\oplus_{n=0}^{\infty}A_{[n]}/A_{[n-1]}$ with $A_{[-1]}=0$
is also a Hopf algebra. If we denote as before, $G=\text{gr}\,A$ and $\pi:G\rightarrow A_0$ splits the inclusion $i:A_0\hookrightarrow G$. Thus by a theorem of Radford, $G\cong R\sharp A_0$, where $R=G^{co\pi}=\{h\in G\mid(id\otimes \pi)\Delta_G(h)=h\otimes 1\}$ is a braided Hopf algebra in ${}^{A_0}_{A_0}\mathcal{YD}$. Moreover,
$R=\oplus_{n=0}^{\infty}R(n)=R\cap G(n)$ with $R(0)=\K$ and $R(1)=\Pp(R)$, which is also a braided vector space called the infinitesimal braiding. This is summarized in the following
\begin{thm} $($\text{\rm Theorem\,1.3 \cite{AC13}}$)$
Any Hopf algebra with injective antipode is a deformation of the bosonization of a Hopf algebra generated by a
cosemisimple coalgebra by a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter.
\end{thm}
In order to produce new Hopf algebras by using the generalized lifting method, one needs to consider the following questions:
\begin{itemize}
\item {\text{\rm Question\,\uppercase\expandafter{\romannumeral1} (\cite{AC13})}} Let $C$ be a cosemisimple coalgebra and $\mathcal{S} : C \rightarrow C$
an injective anti-coalgebra morphism. Classify all Hopf algebras $H$ generated by $C$ such that $S|_C = \mathcal{S}$.
\item {\text{\rm Question\,\uppercase\expandafter{\romannumeral2} (\cite{AC13})}} Given $H$ as in the previous item, classify all connected graded
Hopf algebras $R$ in ${}_H^H\mathcal{YD}$.
\item {\text{\rm Question\,\uppercase\expandafter{\romannumeral3} (\cite{AC13})}} Given $H$ and $R$ as in previous items, classify all liftings, that is, classify all Hopf algebras $A$ such that $\text{gr}\,A\cong R\sharp H$.
\end{itemize}
If $A$ is a Hopf algebra satisfying $A_{[0]}$=$H$, where $H$ is an arbitrary finite-dimensional Hopf algebra, then we also call $A$ is a Hopf algebra over $H$.
Following this generalized lifting method, G. A. Garcia and J. M. J. Giraldi \cite{GG16} determined all
finite-dimensional Hopf algebras over a Hopf algebra of dimension $8$ without the Chevalley property, and the corresponding infinitesimal braidings areirreducible objects and obtained some new Hopf algebras of dimension $64$.
Motivated by their work, the authors in \cite{HX16} found all Hopf algebras of dimension $72$ over a Hopf algebra of dimension $12$ without the Chevalley property, and the corresponding infinitesimal braidings are simple objects.
The present paper is in some sense a sequel to \cite{AC13,GG16, HX16} and classifies all finite-dimensional Hopf algebras over a Hopf algebra $H$ of dimension $16$ without the Chevalley property. Here $H$ as an algebra is generated by the elements $a$, $b$, $c$, $d$ satisfying
\begin{gather*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd.
\end{gather*}
and the coalgebra structure is given by
\begin{align*}
\Delta(a)=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b.
\end{align*}
In order to find new Nichols algebras in $\HYD$, we need to compute simple objects in $\HYD$. Following the fact that the category ${}_{\D}\mathcal{M}$ is equivalent to the category $\HYD$, we first determine the structure of the Drinfeld double
$\D=\D(H^{cop})$ and study the irreducible representations of $\D$. In fact, we show in Theorem
$\ref{thmsimplemoduleD}$, there exist $16$ one-dimensional modules
$\K_{\chi_{i,j,k}}$ for $0\leq i,j<2,0\leq k<4$ and $36$ two-dimensional modules
$V_{i,j,k,\iota}$ for $(i,j,k,\iota)\in \Lambda$. Then we translate the simple $\D$-modules to the simple objects in $\HYD$ and describe explicitly their structures as Yetter-Drinfeld modules and their braidings. Using the braidings, we show that there exist $16$ Nichols algebas of dimension $8$ that are isomorphic to quantum linear spaces as algebras but as coalgebras are more complicated. In fact, we have
\begin{mainthm}
The Nichols algebra $\BN(V)$ over a simple object $V$ in $\HYD$ is finite-dimensional if and only if $V$ is isomorphic either to $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$ or $V_{i,j,k,\iota}$ with $(i,j,k,\iota)\in\cup_{3\leq\ell\leq 6}\Lambda^{\ell}$.
\end{mainthm}
Moreover, we present them by generators and relations (see section $\ref{secNicholsalgebaH}$ for more details). To our knowledge, these Nichols algebras are new. Finally, we need to study the bosonizations of these Nichols algebras and their deformations. As a consequence, we obtain some nontrivial liftings in the following
\begin{mainthm}\label{thm:A}\label{liftingsoverH}
Let $A$ be a finite-dimensional Hopf algebra over $H$ such that the infinitesimal braiding is a simple object $V\in\HYD$. Then $A$ is isomorphic to one of the following
\begin{itemize}
\item $\bigwedge\K_{\chi_{i,j,k}}\sharp H$ with $(i,j,k)\in\Lambda^0$, which has dimension $32$;
\item $\BN(V_{i,j,k,\iota})\sharp H$ with $(i,j,k,\iota)\in \Lambda^3\cup\Lambda^4$, which has dimension $128$;
\item $\LA$ with $(i,j,k,\iota)\in \Lambda^5$ and some $\mu\in\K$, which has dimension $128$;
\item $\LAA$ with $(i,j,k,\iota)\in \Lambda^6$ and some $\mu\in\K$, which has dimension $128$.
\end{itemize}
\end{mainthm}
This paper is organized as follows:
In section $\ref{Preliminary}$ we first recall some basic definitions and facts about Yetter-Drinfeld module,
Nichols algebra, Drinfeld double and Hopf $2$-cocycle. In section $\ref{secDrinfelddouble}$,
we give a detailed description of the Hopf algebra structure of $H$ and determine the Drinfeld
double $\D=\D(H^{cop})$ in terms of generators and relations. In section $\ref{secPresentation}$,
we study the irreducible representations of the Drinfeld double $\D(H^{cop})$. In section $\ref{secHYD}$, we determine the simple objects in $\HYD$ by using the equivalence ${}_{\D}\mathcal{M}\simeq \HYD$, and also describe
their braidings. In section $\ref{secNicholsalgebaH}$, we study the Nichols algebras over simple objects in $\HYD$. We determine all finite-dimensional Nichols algebras and describe their structures in terms of generators and relations.
In section $\ref{secHopfalgebraH}$, we determine all finite-dimensional Hopf algebras $A$ over $H$ and the corresponding infinitesimal braidings are simple objects in $\HYD$. In particular, we figure out $8$ classes of new Hopf algebras of
dimension $128$ without the Chevalley property.
\section{Preliminaries}\label{Preliminary}
\paragraph{Conventions.} Throughout the paper, the ground field $\K$ is an algebraically closed field of characteristic zero. Our references for Hopf algebra theory are \cite{M93} and \cite{R11}.
The notation for a Hopf algebra $H$ is standard: $\Delta$, $\epsilon$, and $S$ denote the comultiplication, the counit and the antipode.
We use Sweedler's notation for the comultiplication and coaction, for example, for any $h\in H$, $\Delta(h)=h_{(1)}\otimes h_{(2)}$,
$\Delta^{(n)}=(\Delta\otimes id^{\otimes n})\Delta^{(n-1)}$. Given a Hopf algebra $H$ with bijective antipode, we denote by $H^{op}$
the Hopf algebra with the opposite multiplication, $H^{cop}$ the Hopf algebra with the opposite comultiplication, and $H^{bop}$ the Hopf algebra
$H^{op\,cop}$. If $V$ is a $\K$-vector space, $v\in V$ and $f\in V\As$, we use either $f(v)$, $\langle f$, $v\rangle$, or $\langle v,
f\rangle$ to denote the evaluation.
For any $n>1$, $M_n(\K)$ and $M_n\As(\K)$ denote matrix algebra and matrix coalgebra. If a simple coalgebra $C\cong M_n\As(\K)$,
we call the basis $(c_{ij})_{1\leq i,j\leq n}$ a comatrix basis if $\Delta(c_{i,j})=\sum_{k=1}^{n}c_{ik}\otimes c_{kj}$ and $\epsilon(c_{ij})=\delta_{i,j}$.
\subsection{Yetter-Drinfeld module and Nichols algebra}
\begin{defi}
Let $H$ be a Hopf algebra with bijective antipode. A Yetter-Drinfeld module over $H$ is a left $H$-module and a left $H$-comodule with comodule structure denoted by $\delta: V\mapsto H\otimes V, v\mapsto v_{(-1)}\otimes v_{(0)}$, such that
\begin{align*}
\delta(h\cdot v)=h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}v_{(0)},
\end{align*}
for all $v\in V,h\in H$. Let ${}^{H}_{H}\mathcal{YD}$ be the
category of Yetter-Drinfeld modules over $H$ with $H$-linear and
$H$-colinear maps as morphisms.
\end{defi}
The category ${}^{H}_{H}\mathcal{YD}$ is monoidal, braided. Indeed, if $V,W\in {}^{H}_{H}\mathcal{YD}$, $V\otimes W$ is the
tensor product over $\mathbb{C}$ with the diagonal action and coaction of $H$ and braiding
\begin{align}\label{equbraidingYDcat}
c_{V,W}:V\otimes W\mapsto W\otimes V, v\otimes w\mapsto v_{(-1)}\cdot w\otimes v_{(0)},\forall\,v\in V, w\in W.
\end{align}
Moreover, ${}^{H}_{H}\mathcal{YD}$ is rigid. That is, it has left dual and right dual, if we take $V\As$ and ${}\As V$ to the dual of
$V$ as vector space, then the left dual and the right dual are defined by
\begin{align*}
\langle h\cdot f,v\rangle=\langle f,S(h)v\rangle,\quad f_{(-1)}\langle f_{(0)},v\rangle=S^{-1}(v_{(-1)})\langle f, v_{(0)}\rangle,\\
\langle h\cdot f,v\rangle=\langle f,S^{-1}(h)v\rangle,\quad f_{(-1)}\langle f_{(0)},v\rangle=S(v_{(-1)})\langle f, v_{(0)}\rangle.
\end{align*}
We consider Hopf algebra in ${}^{H}_{H}\mathcal{YD}$. If $R$ is a Hopf algebra in ${}^{H}_{H}\mathcal{YD}$,
the space of primitive elements $P(R)=\{x\in R|\delta(x)=x\otimes 1+1\otimes x\}$ is a Yetter-Drinfeld
submodule of $R$. Moreover, for any finite-dimensional graded Hopf algebra in ${}^{H}_{H}\mathcal{YD}$, it satisfies the Poincar\'{e} duality:
\begin{pro}$\cite[Proposition\,3.2.2]{AG99}$
Let $R=\oplus_{n=0}^N R(i)$ be a finite-dimensional graded Hopf algebra in ${}^{H}_{H}\mathcal{YD}$, and suppose that $R(N)\neq 0$. Then $\dim R(i)=\dim R(N-i)$ for any $0\leq i<N$.
\end{pro}
\begin{defi}
Let $V\in{}^{H}_{H}\mathcal{YD}$ and $I(V)\subset T(V)$ be the largest $\mathds{N}$-graded ideal and coideal
such that $I(V)\cap V=0$. We call $\mathcal{B}(V)=T(V)/I(V)$ the Nichols algebra of $V$. Then $\mathcal{B}(V)=\oplus_{n\geq 0}\mathcal(B)^n(V)$
is an $\mathds{N}$-graded Hopf algebra in ${}^{H}_{H}\mathcal{YD}$.
\end{defi}
\begin{lem}$\cite{AS02}$
The Nichols algebra of an object $V\in{}^{H}_{H}\mathcal{YD}$ is the
(up to isomorphism) unique $\mathds{N}$-graded Hopf algebra $R$ in
${}^{H}_{H}\mathcal{YD}$ satisfying the following properties:
\begin{align*}
&R(0)=\mathds{k}, \quad R(1)=V,\\
&R(1)\ \text{\ generates the algebra R},\\
&P(R)=V.
\end{align*}
\end{lem}
Nichols algebras play a key role in the classification of pointed Hopf algebras, and
we close this subsection by giving the explicit relation between $V$ and $V\As$ in ${}^{H}_{H}\mathcal{YD}$.
\begin{pro}$\cite[Proposition\,3.2.30]{AG99}$\label{proNicholsdual}
Let $V$ be an object in ${}_H^H\mathcal{YD}$. If $\BN(V)$ is finite-dimensional, then $\BN(V\As)\cong \BN(V)\As$.
\end{pro}
\subsection{Radford biproduct construction}
Let $R$ be a bialgebra (resp. Hopf algebra) in ${}^{H}_{H}\mathcal{YD}$ and denote the coproduct
by $\Delta_R(r)=r^{(1)}\otimes r^{(2)}$. We define the Radford biproduct $R\#H$. As a vector space,
$R\#H=R\otimes H$ and the multiplication and comultiplication are given by the smash product and smash-coproduct, respectively:
\begin{align}
(r\#g)(s\#h)&=r(g_{(1)}\cdot s)\#g_{(2)}h,\\
\Delta(r\#g)&=r^{(1)}\#(r^{(2)})_{(-1)}g_{(1)}\otimes (r^{(2)})_{(0)}\#g_{(2)}.
\end{align}
Clearly, the map $\iota:H\rightarrow R\#H, h\mapsto 1\# h,\ \forall h\in H$, and the map
$\pi:R\#H\rightarrow H,r\#h\mapsto \epsilon_R(r)h,\ \forall r\in R, h\in H$ such that $\pi\circ\iota=id_H$. Moreover, $R=(R\#H)^{coH}$.
Let $R, S$ be bialgebras (resp. Hopf algebra) in ${}^{H}_{H}\mathcal{YD}$ and $f:R\rightarrow S$ be a bialgebra
morphisms in ${}^{H}_{H}\mathcal{YD}$. $f\#id:R\#H\rightarrow S\#H$ defined by $(f\#id)(r\#h)=f(r)\#h,
\forall r\in R, h\in H$. In fact, $R\rightarrow R\#H$ and $f\mapsto f\# id$ describes a
functor from the category of bialgebras (resp. Hopf algebras) in ${}^{H}_{H}\mathcal{YD}$ and their morphisms
to the category of usual bialgebras (resp. Hopf algebras).
Conversely, if $A$ is a bialgebra (resp. Hopf algebra) and $\pi:A\rightarrow H$ a bialgebra admitting a bialgebra
section $\iota:H\rightarrow A$ such that $\pi\circ\iota=id_H$ (we call $(A,H)$ a Radford pair for convenience),
$R=A^{coH}=\{a\in A\mid(id\otimes\pi)\Delta(a)=a\otimes 1\}$ is a bialgebra (resp. Hopf algebra) in ${}^{H}_{H}\mathcal{YD}$
and $A\simeq R\#H$, whose Yetter-Drinfeld module and coalgebra structures are given by:
\begin{align*}
h\cdot r&=h_{(1)}rS_A(h_{(2)}),\quad
\delta(r)=(\pi\otimes id)\Delta_A(r),\\
\Delta_R(r)&=r_{(1)}(\iota S_H(\pi(r_{(2)})))\otimes r_{(3)},\quad
\epsilon_R=\epsilon_A|_R,\\
S_R(r)&=(\iota\pi(r_{(1)}))S_A(r_{(2)}),\quad\text{if $A$ is a Hopf algebra}.
\end{align*}
\subsection{Drinfeld double}
\begin{defi}
Let $H$ be a finite-dimensional Hopf algebra with bijective antipode $S$ over $\K$. The Drinfeld double $\D(H)=H^{\ast\,cop}\otimes H$
is a Hopf algebra with the tensor product coalgebra structure and algebra structure defined by
\begin{align}\label{equDrinfelddouble}
(p\otimes a)(q\otimes b)=p\langle q_{(3)}, a_{(1)}\rangle q_{(2)}\otimes a_{(2)}\langle q_{(1)}, S^{-1}(a_{(3)})\rangle.
\end{align}
\end{defi}
By $\cite[Proposition 10.6.16]{M93}$, the category
${}_{\D(H)}{\M}$ of left modules is equivalent to the category ${}_H\mathcal{YD}^H$ of Yetter-Drinfeld
modules. But ${}_H\mathcal{YD}^H$ is equivalent
to the category ${}_{H^{cop}}^{H^{cop}}\mathcal{YD}$ of Yetter-Drinfeld modules.
Thus we have the following
result.
\begin{pro}$\cite{M93}$\label{proDouble}
Let $H$ be a finite-dimensional Hopf algebra with bijective antipode $S$ over $\K$.
Then the category ${}_{\D(H^{cop})}\M$ of left modules is equivalent to the category ${}_H^H\mathcal{YD}$
of Yetter-Drinfeld modules.
\end{pro}
\subsection{Hopf $2$-cocycle deformation}
Let $(H,m,1,\Delta,\epsilon,S)$ be a Hopf algebra. The convolution invertible bilinear form $\sigma: H\otimes H\mapsto
\K$ is called a (left) normalized Hopf $2$-cocycle of $H$ if
\begin{gather*}
\sigma(a,1)=\sigma(1,a)=\epsilon(a),\quad \forall a\in H,\\
\sum\sigma(a_{(1)}, b_{(1)})\sigma(a_{(2)}b_{(2)},c)=\sum\sigma(b_{(1)}, c_{(1)})\sigma(a, b_{(2)}c_{(2)}),\quad\forall a,b,c\in H.
\end{gather*}
Denote by $\sigma^{-1}$ the convolution inverse of $\sigma$. We can construct a new Hopf algebra $(H^{\sigma},m^{\sigma},1,\Delta,\epsilon, S^{\sigma})$, where $H^{\sigma}=H$ as coalgebras,and
\begin{align*}
m^{\sigma}(a\otimes b)=\sum\sigma(a_{(1)},b_{(1)})a_{(2)}b_{(2)}\sigma^{-1}(a_{(3)},b_{(3)}),\quad\forall a,b\in H,\\
S^{\sigma}(a)=\sum\sigma(a_{(1)},S(a_{(2)}))S(a_{(3)})\sigma^{-1}(S(a_{(4)}),a_{(5)}),\quad\forall a\in H.
\end{align*}
We denote by $\mathcal{Z}^2(H,\mathds{k})$ the set of normalized Hopf $2$-cocycles on $H$.
And we will use the following equivalence of categories of Yetter-Drinfeld modules which is due originally to Majid and Oeckl \cite[Theorem\;2.7]{MO99}.
\begin{defi}
Let $M$ be a Yetter-Drinfeld module over $H$ and $\sigma$ is a Hopf $2$-cocycle of $H$. Then there is a corresponding Yetter-Drinfeld module over $H^{\sigma}$ denoted by $M^{\sigma}$ defined as: it is $M$ as a comodule, and the $H^{\sigma}$-action is given by
\begin{align*}
a\cdot^{\sigma}m=\sum \sigma((a_{(2)}\cdot m_{(0)})_{(1)},a_{(1)})(a_{(2)}\cdot m_{(0)})_{(0)}\sigma^{-1}(a_{(3)}\otimes m_{(1)}),
\end{align*}
\end{defi}\noindent
for all $a\in H$ and $m\in M$.
\begin{pro}\cite[Theorem\;2.7]{MO99}\label{proCocycledeformation}
Let $\sigma$ be a Hopf $2$-cocycle on the Hopf algebra $H$. Then the categories $\HYD$ and ${}_{H^{\sigma}}^{H^{\sigma}}\mathcal{YD}$ are monoidally equivalent under the functor
\begin{align*}
F_{\sigma}:\HYD\mapsto {}_{H^{\sigma}}^{H^{\sigma}}\mathcal{YD},
\end{align*}
which is the identity on homomorphisms and on the objects is given by $F_{\sigma}(M)=M^{\sigma}$.
\end{pro}
\section{A Hopf algebra H of dimension $16$ and the Drinfeld double $\D$}\label{secDrinfelddouble}
In this section we describe explicitly the structure of $H$ and present the Drinfeld double $\D=\D(H^{cop})$ by generators and relations.
Throughout the paper, we fix $\xi$ a primitive $4$-th root of unity. For the classification of Hopf algebra of dimension $16$, the semisimple case was classified by Y.~Kashina \cite{K00}, the pointed nonsemisimple case was given by S.~Caenepeel, S.~D\u{a}sc\u{a}lescu, and S.~Raianu \cite{CDR00}, and the full classification was done by G. A. Garc\'{\i}a and C. Vay \cite{GV10}. Now, we choose one pointed Hopf algebra listed in \cite[Section\;2.5]{CDR00}
\begin{defi}
\begin{gather*}
\A:=\langle g,h,x\mid g^4=1, h^2=1, hg=gh, hx=-xh,gx=xg, x^2= 1-g^2 \rangle.\\
\Delta(g)=g\otimes g,\quad \Delta(h)=h\otimes h, \quad\Delta(x)=x\otimes 1+gh\otimes x.
\end{gather*}
\end{defi}
\begin{rmk}\label{rmkGAcocycledefor}
Let $\Gamma\cong Z_4\times Z_2$ be an abelian group with generators $g,h$ and $V=\K\{v\}\in {}_{\Gamma}^{\Gamma}\mathcal{YD}$ given by
\begin{align*}
h\cdot v=\Lam(h)v=-v,\quad g\cdot v=\Lam(g)v=v;\quad\delta(v)=gh\otimes v.
\end{align*}
Then the Nichols algebra $\BN(V)=\K(v)/(v^2)$. Let $G=\BN(V)\sharp \K\Gamma$, then $G$ is a Hopf algebra generated as an algebra by $g$, $h$ and $x$ satisfying
\begin{align*}
g^4=1,\quad h^2=1,\quad hg=gh,\quad hx=-xh,\quad gx=xg,\quad x^2=0,
\end{align*}
and the coalgebra structure given by
\begin{align*}
\Delta(g)=g\otimes g,\quad \Delta(h)=h\otimes h,\quad \Delta(x)=x\otimes 1+gh\otimes x.
\end{align*}
In particular, $A$ is a lifting of the Nichols algebra $\BN(V)$ by deforming the relation $x^2=0$. By \cite[Theorem\;A.1]{Ma08}, $A$ is a Hopf $2$-cocycle deformation of $G$. That is, there exists some $\sigma\in \mathcal{Z}^2(G,\mathds{k})$ such that $A\cong G^{\sigma}$.
\end{rmk}
In order to describe explicitly the structure of $H$ as the dual Hopf algebra of $A$, we first need to compute the irreducible representations of $A$.
\begin{lem}\label{lem1}
There are four one-dimensional $\A-$modules denoted by $\chi_{i,j}$, $i,j\in Z_2$ given by
\begin{align*}
\chi_{i,j}(g)=(-1)^i,\quad \chi_{i,j}(h)=(-1)^j,\quad \chi_{i,j}(x)=0.
\end{align*}
and two two-dimensional simple $\A-$modules denoted by $\rho_i$, $i\in Z_2$ given by
\begin{align*}
\rho_1(g)&=\left(\begin{array}{ccc}
\xi & 0\\
0 & \xi
\end{array}\right),\quad
\rho_1(h)=\left(\begin{array}{ccc}
1 & 0\\
0 & -1
\end{array}\right),\quad
\rho_1(x)=\left(\begin{array}{ccc}
0 & \sqrt 2\\
\sqrt 2& 0
\end{array}\right);\\
\rho_2(g)&=\left(\begin{array}{ccc}
\xi^3 & 0\\
0 & \xi^3
\end{array}\right),\quad
\rho_2(h)=\left(\begin{array}{ccc}
1 & 0\\
0 & -1
\end{array}\right),\quad
\rho_2(x)=\left(\begin{array}{ccc}
0 & \sqrt 2\\
\sqrt 2& 0
\end{array}\right).
\end{align*}
\end{lem}
Let $(\K^2, \rho_i)_{i=1,2}$ be the $2$-dimensional representations given in Lemma $\ref{lem1}$. Let ${(E_{ij})}_{i,j=1,2}$ be the coordinate functions of $\M(2,\K)$. And let $c_{ij}:=E_{ij}\circ \rho_1, d_{ij}:=E_{ij}\circ \rho_2$, we can regard $\E_C:=\{C_{ij}\}_{i,j=1,2}$ and $\E_D:=\{D_{ij}\}_{i,j=1,2}$ as comatrix basis of the simple subcoalgebras of $H$ isomorphic to $C$ and $D$ respectively. The following Lemma shows some useful relations of the elements of $\E_C$ and $\E_D$ and the proof is much similar with that in \cite[Lemma\;3.3]{GV10}.
\begin{lem}
The elements of $\E_C$ and $\E_D$ satisfy:
\begin{align*}
S(C_{12})=\xi D_{21},\quad S(C_{21})=\xi^3 D_{12}, \quad
S(C_{11})=D_{11},\quad S(C_{22})=D_{22},\\
S(D_{12})=\xi^3 C_{21},\quad S(D_{21})=\xi C_{12},\quad
S(D_{11})=C_{11},\quad S(D_{22})=C_{22},\\
C_{11}^2=C_{22}^2=\chi_{1,0},\quad C_{11}C_{22}=C_{22}C_{11}=\chi_{1,1},\,
C_{11}C_{12}=\xi C_{12}C_{11},\\
C_{12}^2=0=C_{21}^2,\quad C_{12}C_{21}=0=C_{21}C_{12}, \quad C_{11}C_{21}=\xi C_{21}C_{11}, \\
C_{22}C_{12}=-\xi C_{12}C_{22},\quad C_{22}C_{21}=-\xi C_{21}C_{22},\quad C_{12}C_{22}=C_{21}C_{11},\\ C_{12}C_{11}=C_{21}C_{22},\quad C_{11}C_{12}=-C_{22}C_{21},\quad C_{22}C_{12}=-C_{11}C_{21}.
\end{align*}
\end{lem}
\begin{rmk}\label{rmkHindependent}
After an easy computation, the elements
\begin{gather*}
C_{11}^3,\, C_{22}^3,\,
C_{11}^2=\chi_{1,0},\, C_{11}C_{22}=\chi_{1,1},\, C_{22}^2C_{11}^2=\epsilon,\,
C_{11}^3C_{22}=\chi_{0,1},\\
C_{11},\, C_{22},\,
C_{12},\, C_{21},\, C_{11}C_{12},\, C_{11}C_{21},\, C_{11}^2C_{12},\, C_{11}^2C_{21},\,
C_{11}^3C_{12},\, C_{11}^3C_{21}.
\end{gather*}
are linearly independent. Thus $H$ is generated by the simple subcoalgebra $C$ since $\dim H=\dim \A=16$.
\end{rmk}
Now for convenience, let $a=C_{11}$, $b=C_{12}$, $c=C_{21}$ and $d=C_{22}$. Then we have the following Proposition.
\begin{pro}\label{proStructureofH}
\begin{enumerate}
\item $H$ as an algebra is generated by the elements $a$, $b$, $c$, $d$ satisfying the relations
\begin{gather*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd.
\end{gather*}
\item A linear basis of $H$ is given by
\begin{align*}
\{1, a,\,a^2,\, a^3,\,d, \, da,\, da^2,\, da^3,\, b,\, c,\, ba,\, ca,\, ba^2,\, ca^2,\, ba^3,\, ca^3\}.
\end{align*}
\item The coalgebra structure of $H$ is given by
\begin{align*}
\Delta(a)=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(a^2)=a^2\otimes a^2,\quad \Delta(a^3)=a^3\otimes a^3+ba^2\otimes ca^2,\\
\Delta(da)=da\otimes da,\quad \Delta(da^2)=da^2\otimes da^2+ca^2\otimes ba^2,\\
\Delta(da^3)=da^3\otimes da^3,\quad \Delta(ba)=ba\otimes da+a^2\otimes ba,\\
\Delta(ca)=ca\otimes a^2+da\otimes ca,\, \Delta(ba^2)=ba^2\otimes da^2+a^3\otimes ba^2,\\
\Delta(ca^2)=ca^2\otimes a^3+da^2\otimes ca^2,\,\Delta(ba^3)=ba^3\otimes da^3+1\otimes ba^3,\\
\Delta(ca^3)=ca^3\otimes 1+da^3\otimes ca^3,\quad \Delta(1)=1\otimes 1, \\
\epsilon(a)=1=\epsilon(d),\quad \epsilon(b)=0=\epsilon(c).
\end{align*}
\item The antipode of $H$ is given by
\begin{align*}
S(a)=a^3,\quad S(d)=d^3,\quad S(b)=\xi ca^2, \quad S(c)=\xi^3ba^2.
\end{align*}
\end{enumerate}
\end{pro}
\begin{rmk}\label{rmkHdualtoA}
Denote by $\{(a^i)\As,(ba^i)\As,(ca^i)\As,(da^i)\As,\;0\leq i\leq 3\}$ the basis of the dual Hopf algebra $H\As$.
Let
\begin{gather*}
\widetilde{x}=\sum_{i=0}^3 (ba^i)^{\ast}+\sum_{i=0}^3 (ca^i)^{\ast},\\
\widetilde{g}=\sum_{i=0}^3 \xi^{i}(a^i)^{\ast}+\sum_{i=0}^3 \xi^{i+1}(da^i)^{\ast},\quad
\widetilde{h}=\sum_{i=0}^3 (a^i)^{\ast}-\sum_{i=0}^3(da^i)^{\ast}.
\end{gather*}
Then using the multiplication table induced by the relations of $H$ given in Proposition $\ref{proStructureofH}$ and after a tedious computation, we have that
\begin{gather*}
\widetilde{g}^4=1,\quad \widetilde{h}^2=1,\quad \widetilde{h}\widetilde{g}=\widetilde{g}\widetilde{h},\quad \widetilde{g}\widetilde{x}=\widetilde{x}\widetilde{g},\quad
\widetilde{h}\widetilde{x}=-\widetilde{x}\widetilde{h},\\
\Delta(\widetilde{x})=\widetilde{x}\otimes \epsilon+\widetilde{g}\widetilde{h}\otimes \widetilde{x},\quad
\Delta(\widetilde{g})=\widetilde{g}\otimes \widetilde{g},\quad
\Delta(\widetilde{h})=\widetilde{h}\otimes \widetilde{h}.
\end{gather*}
In particular, $G(H\As)\cong Z_4\times Z_2$ with generators $\widetilde{g}$ and $\widetilde{h}$.
\end{rmk}
In order to compute the structure of the Drinfeld double $D(H^{cop})$ of $H^{cop}$ in terms of generators and relations, we have the following Lemmawhich builds the isomorphism $\A\cong H^{\ast}$ explicitly.
\begin{lem}\label{lemAtoHdual}
The algebra map $\psi:\A\mapsto H^{\ast}$ given by
\begin{align*}
\psi(g)&=\widetilde{g}=\sum_{i=0}^3 \xi^{i}(a^i)^{\ast}+\sum_{i=0}^3 \xi^{i+1}(da^i)^{\ast},\\
\psi(h)&=\widetilde{h}=\sum_{i=0}^3 (a^i)^{\ast}-\sum_{i=0}^3(da^i)^{\ast},\\
\psi(x)&=\sqrt 2\widetilde{x}=\sqrt 2\sum_{i=0}^3 (ba^i)^{\ast}+\sqrt 2\sum_{i=0}^3 (ca^i)^{\ast}.
\end{align*}
is a Hopf algeba isomorphism.
\end{lem}
\begin{proof}
By Remark $\ref{rmkHdualtoA}$, $\psi$ is a coalgebra map and $\psi(A)$ contains properly $G(H\As)$. Thus by the Nichols-Zoeller theorem, $\psi$ is epimorphic. Since $\dim A=\dim H\As=16$, $\psi$ is isomorphic.
\end{proof}
\begin{rmk}\label{rmkAtoHdual}
Let $\{ g^j, g^jh, g^jx, g^jhx\}_{0\leq j<4}$ be a linear basis of $\A$. We have
\begin{align*}
\psi(g^j)=\sum_{i=0}^3 \xi^{ij}(a^i)^{\ast}+\sum_{i=0}^3 \xi^{ij+j}(da^i)^{\ast},\\
\psi(g^jh)=\sum_{i=0}^3 \xi^{ij}(a^i)^{\ast}-\sum_{i=0}^3 \xi^{ij+j}(da^i)^{\ast},\\
\psi(g^jx)=\sqrt 2\sum_{i=0}^3 \xi^{ij+j}(ba^i)^{\ast}+\sqrt 2\sum_{i=0}^3 \xi^{ij+j}(ca^i)^{\ast},\\
\psi(g^jhx)=\sqrt 2\sum_{i=0}^3 \xi^{ij+j}(ba^i)^{\ast}-\sqrt 2\sum_{i=0}^3\xi^{ij+j} (ca^i)^{\ast}.
\end{align*}
\end{rmk}
Now we try to describe the Drinfeld double $\D:=\D(H^{cop})$ of $H^{cop}$.
\begin{pro}
$\D$ as a $\K$-coalgebra is isomorphic to $\A^{bop}\otimes H^{cop}$, and as a $\K$-algebra is generated by the elements $g$, $h$, $x$, $a$, $b$, $c$, $d$ satisfying the relations in $H^{cop}$, the relations in $\A^{bop}$ and
\begin{align*}
ag=ga,\quad ah=ha,\quad dg=gd,\quad dh=hd,\\
bg=gb,\quad bh=-hb,\quad cg=gc,\quad ch=-hc,\\
ax+\xi xa=\sqrt 2\xi(c-ghb),\, dx-\xi xd=\sqrt 2\xi(ghc-b),\\
bx+\xi xb=\sqrt 2\xi(d-gha),\, cx-\xi xc=\sqrt 2\xi(ghd-a).
\end{align*}
\end{pro}
\begin{proof}
Note that
\begin{align*}
\Delta_{\A^{bop}}^{2}(g)&=g\otimes g\otimes g,\quad
\Delta_{\A^{bop}}^{2}(h)=h\otimes h\otimes h,\\
\Delta_{\A^{bop}}^{2}(x)&=1\otimes 1\otimes x+1\otimes x\otimes gh+x\otimes gh \otimes gh,\\
\Delta_{H^{cop}}^{2}(a)&=a\otimes a\otimes a+a\otimes c\otimes b+c\otimes b\otimes a+c\otimes d\otimes b,\\
\Delta_{H^{cop}}^{2}(b)&=b\otimes a\otimes a+b\otimes c\otimes b+d\otimes b\otimes a+d\otimes d\otimes b,\\
\Delta_{H^{cop}}^{2}(c)&=a\otimes a\otimes c+a\otimes c\otimes d+c\otimes d\otimes d+c\otimes b\otimes c,\\
\Delta_{H^{cop}}^{2}(d)&=d\otimes d\otimes d+d\otimes b\otimes c+b\otimes a\otimes c+b\otimes c\otimes d.
\end{align*}
We have that
\begin{align*}
ag&=\langle g,a\rangle ga\langle g,S(a)\rangle=ga,\quad
ah=\langle h,a\rangle ha\langle h,S(a)\rangle=ha,\\
dg&=\langle g,d\rangle gd\langle g,S(d)\rangle=gd,\quad
dh=\langle h,d\rangle hd\langle h,S(d)\rangle=hd,\\
bg&=\langle g,d\rangle gb\langle g,S(a)\rangle=gb,\quad
bh=\langle h,d\rangle hb\langle h,S(a)\rangle=-hb,\\
cg&=\langle g,a\rangle gc\langle g,S(d)\rangle=gc,\quad
ch=\langle h,a\rangle hc\langle h,S(d)\rangle=-hc,\\
ax&=\langle 1,a\rangle c\langle x,S(b)\rangle+\langle 1,a\rangle xa\langle gh,S(a)\rangle
+\langle x,c\rangle ghb\langle gh,S(a)\rangle \\&
=\sqrt 2\xi c-\xi xa-\sqrt 2\xi ghb,\\
dx&=\langle 1,d\rangle b\langle x,S(c)\rangle+\langle 1,d\rangle xd\langle gh,S(d)\rangle
+\langle x,b\rangle ghc\langle gh,S(d)\rangle \\&
=\sqrt 2\xi^3 b+\xi xd+\sqrt 2\xi ghc,\\
bx&=\langle 1,d\rangle d\langle x,S(b)\rangle+\langle 1,d\rangle xb\langle gh,S(a)\rangle
+\langle x,b\rangle gha\langle gh,S(a)\rangle \\&
=\sqrt 2\xi d-\xi xb-\sqrt 2\xi gha,\\
cx&=\langle 1,a\rangle a\langle x,S(c)\rangle+\langle 1,a\rangle xc\langle gh,S(d)\rangle
+\langle x,c\rangle ghd\langle gh,S(d)\rangle \\&
=\sqrt 2\xi^3 a+\xi xc+\sqrt 2\xi ghd.
\end{align*}
\end{proof}
\section{Presentation of the Drinfeld double $\D$}\label{secPresentation}
In this section, we compute the irreducible representations of $\D$. We begin this section by describing the one-dimensional $\D$-modules.
\begin{lem}\label{lemOnesimpleD}
There are $16$ non-isomorphic one-dimensional simple modules $\K_{\chi_{i,j,k}}$ given by the characters $\chi_{i,j,k},\,0\leq i,j<2,0\leq k<4$, where
\begin{align*}
\chi_{i,j,k}(g)&=(-1)^i,\quad\chi_{i,j,k}(h)=(-1)^j,\quad \chi_{i,j,k}(x)=0,\\
\chi_{i,j,k}(a)&=\xi^k,\, \chi_{i,j,k}(b)=0,\,\chi_{i,j,k}(c)=0,\,\chi_{i,j,k}(d)=(-1)^i(-1)^j\xi^k.
\end{align*}
Moreover, any one-dimensional $\D$-module is isomorphic to $\K_{\chi_{i,j,k}}$ for some $0\leq i,j<2,0\leq k<4$.
\end{lem}
\begin{proof}
Let $\chi\in G(\D^{\ast})=\hom(\D,\K)$. Since $a^4=1=g^4$ and $d^4=1=h^2$, we have that $\chi(a)^4=1=\chi(g)^4$ and $\chi(d)^4=1=\chi(h)^2$. From $b^2=0$, $c^2=0$ and $hx=-xh$, we have that $\chi(b)=\chi(x)=\chi(c)=0$, and whence $\chi(g)^2=1$ since $x^2=1-g^2$. From the relation $bx+\xi xb=\sqrt 2\xi(d-gha)$, we have $\chi(d)=\chi(g)\chi(g)\chi(a)$. Thus $\chi$ is completely determined by $\chi(a)$, $\chi(g)$ and $\chi(h)$. Let $\chi(a)=\xi^k$ for some $k\in Z_4$, $\chi(g)=(-1)^i$, $\chi(h)=(-1)^j$. It is clear that these modules are pairwise non-isomorphic and any one-dimensional $\D$-module is isomorphic to $\K_{\chi_{i,j,k}}$ where $0\leq i,j<2,0\leq k<4$.
\end{proof}
Next, we describe two-dimensional simple $\D$-modules. For this, consider the finite set given by
\begin{align*}
\Lambda=\{(i,j,k,\iota)\mid i\in Z_4, j=1,3, k,\iota\in Z_2\}.
\end{align*}
Clearly, $|\Lambda|=32$.
\begin{lem}\label{lemTwosimpleD}
For any pair $(i,j,k)\in\Lambda$, there exists a simple left $\D$-module $V_{i,j,k}$ of dimension $2$. If we denote $\Lam_1=\xi^i$, $\Lam_2=\xi^j$, $\Lam_3=(-1)^k$ and $\Lam_4=(-1)^\iota$ the action on a fixed basis is given by
\begin{align*}
[a]&=\left(\begin{array}{ccc}
-\Lam_4\Lam_1 & 0\\
0 & \xi \Lam_4\Lam_1
\end{array}\right),\,
[d]=\left(\begin{array}{ccc}
\Lam_1 & 0\\
0 & \xi \Lam_1
\end{array}\right),\,
[b]=\left(\begin{array}{ccc}
0 & \Lam_4\\
0 & 0
\end{array}\right),\\
[c]&=\left(\begin{array}{ccc}
0 & 1\\
0 & 0
\end{array}\right),\,
[g]=\left(\begin{array}{ccc}
\Lam_2 & 0\\
0 & \Lam_2
\end{array}\right),\,
[h]=\left(\begin{array}{ccc}
\Lam_3 & 0\\
0 & -\Lam_3
\end{array}\right),\\
[x]&=\left(\begin{array}{ccc}
0 & \frac{\sqrt 2}{2}\xi \Lam_1^3(\Lam_2 \Lam_3 -\Lam_4)\\
\sqrt 2\xi \Lam_1(\Lam_2\Lam_3+\Lam_4) & 0
\end{array}\right),
\end{align*}
\end{lem}
\begin{proof}
Since the elements $g$, $h$, $a$, $d$ commute each other and $g^4=d^2=a^4=d^4=1$, we can choose a basis of the two dimensional simple $\D$-module $V$ such that the matrices defining the action on $V$ are of the form
\begin{align*}
[g]&=\left(\begin{array}{ccc}
g_1 & 0\\
0 & g_2
\end{array}\right),\,
[h]=\left(\begin{array}{ccc}
h_1 & 0\\
0 & h_2
\end{array}\right),\,
[x]=\left(\begin{array}{ccc}
x_1 & x_2\\
x_3 & x_4
\end{array}\right),\,
[a]=\left(\begin{array}{ccc}
a_1 & 0\\
0 & a_2
\end{array}\right),\\
[d]&=\left(\begin{array}{ccc}
d_1 & 0\\
0 & d_2
\end{array}\right),\quad
[b]=\left(\begin{array}{ccc}
b_1 & b_2\\
b_3 & b_4
\end{array}\right),\quad
[c]=\left(\begin{array}{ccc}
c_1 & c_2\\
c_3 & c_4
\end{array}\right),
\end{align*}
where $a_1^4=1=a_2^4$, $d_1^4=1=d_2^4$, $g_1^4=1=g_2^4$, $h_1^2=1=h_2^2$. From the relations $xh=-hx$, $bh=-hb$ and $ch=-hc$, we have that
\begin{align*}
x_1&=0=x_4, \quad(h_1+h_2)x_2=0=(h_1+h_2)x_3, \\
b_1&=0=b_4, \quad(h_1+h_2)b_2=0=(h_1+h_2)b_3, \\
c_1&=0=c_4, \quad(h_1+h_2)c_2=0=(h_1+h_2)c_3.
\end{align*}
If $h_1+h_2\neq 0$, then we have that $x_2=0=x_3$, $b_2=0=b_3$, $c_2=0=c_3$ and therefore $[b]$, $[c]$, $[x]$ are zero metrices, which implies that $V$ is can be decomposed as a $\D$-module, a contradiction. Thus we have $h_1=-h_2$.
From the relations $gx=xg$, $bg=gb$ and $cg=gc$, we have the relations
\begin{align*}
(g_1-g_2)x_2=0=(g_1-g_2)x_3,\\
(g_1-g_2)b_2=0=(g_1-g_2)b_3, \\
(g_1-g_2)c_2=0=(g_1-g_2)c_3,
\end{align*}
which implies that $g_1=g_2$, otherwise $[b]$, $[c]$, $[x]$ are zero matrices and whence $V$ is not simple.
From the relations $b^2=0=c^2$ and $bc=0=cb$, we have that
\begin{align*}
b_2b_3=0=c_2c_3,\quad b_2c_3=0=b_3c_2,\quad c_2b_3=0=c_3b_2.
\end{align*}
And from the relations $ax+\xi xa=\sqrt 2\xi(c-ghb)$ and $dx-\xi xd=\sqrt 2\xi(ghc-b)$, we have the relations
\begin{align}
a_1x_2+\xi a_2x_2=\sqrt 2\xi(c_2-g_1h_1b_2),\quad a_2x_3+\xi a_1x_3=\sqrt 2\xi(c_3-g_2h_2b_3),\label{eq1}\\
d_1x_2-\xi d_2x_2=\sqrt 2\xi(g_1h_1c_2-b_2),\quad d_2x_3-\xi d_1x_3=\sqrt 2\xi(g_2h_2c_3-b_3).\label{eq2}
\end{align}
By permuting the elements of the basis, we may assume that $b_3=0=c_3$. Now we claim that $b_2\neq 0$ and $b_3\neq 0$. Indeed, if $b_2=0=b_3$, then it is clear that $V$ is simple if and only if $x_2x_3\neq 0$. By equations $\eqref{eq1}$, $\eqref{eq2}$, we have that
\begin{align*}
a_1x_2+\xi a_2x_2=0,\, a_2x_3+\xi a_1x_3=0,\, d_1x_2-\xi d_2x_2=0,\, d_2x_3-\xi d_1x_3=0,
\end{align*}
which imply that $a_1+\xi a_2=0$ and $a_2+\xi a_1=0$ and therefore $a_1=0=a_2$, a contradiction. Thus the claim follows. We may also assume that $c_2=1$.
From the relations $ab=\xi ba$, $ac=\xi ca$, $bd=\xi db$ and $cd=\xi dc$, we have that
\begin{align*}
(a_1-\xi a_2)b_2=0=(a_2-\xi a_1)b_3,\quad (d_2-\xi d_1)b_2=0=(d_1-\xi d_2)b_3,
\end{align*}
which implies that $a_1=\xi a_2$, $d_2=\xi d_1$.
From the relations $bd=ca$ and $ba=cd$, we have that
\begin{align*}
b_2d_2=a_2c_2,\quad b_3d_1=c_3a_1,\quad b_2a_2=c_2d_2,\quad b_3a_1=c_3d_1,
\end{align*}
which implies that $b_2^2=1$ and $a_2=b_2d_2$.
From the relations $bx+\xi xb=\sqrt 2\xi(d-gha)$ and $cx+\xi xc=\sqrt 2\xi(ghd-a)$, we have that
\begin{align*}
b_2x_3+\xi b_3x_2&=\sqrt 2\xi(d_1-g_1h_1a_1),\, b_3x_2+\xi b_2x_3=\sqrt 2\xi(d_2-g_2h_2a_2),\\
c_2x_3-\xi c_3x_2&=\sqrt 2\xi(g_1h_1d_1-a_1),\, c_3x_2-\xi c_2x_3=\sqrt 2\xi(g_2h_2d_2-a_2),
\end{align*}
which implies that $x_3=\sqrt 2\xi d_1(b_2+g_1h_1)$. By equations $\eqref{eq1}$ and $\eqref{eq2}$, we also have that $x_2=\frac{\sqrt 2}{2}\xi d_1^3(g_1h_1-b_2)$. Notice that from the relations $x^2=1-g^2$ and $a^2d^2=1$, we have that $x_2x_3=1-g_1^2$ and $a_1^2d_1^2=1=a_2^2d_2^2$. Indeed, since $a_2=b_2d_2$, $a_1=\xi a_2$ and $d_2=\xi d_1$, $a_1=-b_2d_1$ and therefore the relation $a_1^2d_1^2=1=a_2^2d_2^2$ holds. After a direct computation, we also have that the relation $x_2x_3=1-g_1^2$ holds.
From the discussion above, the matrices defining the action on V are of the form
\begin{align*}
[a]&=\left(\begin{array}{ccc}
-\Lam_4\Lam_1 & 0\\
0 & \xi \Lam_4\Lam_1
\end{array}\right),\,
[d]=\left(\begin{array}{ccc}
\Lam_1 & 0\\
0 & \xi \Lam_1
\end{array}\right),\,
[b]=\left(\begin{array}{ccc}
0 & \Lam_4\\
0 & 0
\end{array}\right),\\
[c]&=\left(\begin{array}{ccc}
0 & 1\\
0 & 0
\end{array}\right),\,
[g]=\left(\begin{array}{ccc}
\Lam_2 & 0\\
0 & \Lam_2
\end{array}\right),\,
[h]=\left(\begin{array}{ccc}
\Lam_3 & 0\\
0 & -\Lam_3
\end{array}\right),\\
[x]&=\left(\begin{array}{ccc}
0 & \frac{\sqrt 2}{2}\xi \Lam_1^3(\Lam_2 \Lam_3 -\Lam_4)\\
\sqrt 2\xi \Lam_1(\Lam_2\Lam_3+\Lam_4) & 0
\end{array}\right),
\end{align*}
where, $\Lam_1^4=1$, $\Lam_2^4=1$, $\Lam_3^2=1$ and $\Lam_4^2=1$. And it is clear that $V$ is simple if and only if $\Lam_2^2\neq 1$. If we set $\Lam_1=\xi^i$, $\Lam_2=\xi^j$, $\Lam_3=(-1)^k$ and $\Lam_4=(-1)^\iota$, then we have that $j=1,3$ and $(i,j,k,\iota)\in\Lambda$.
Now we claim that $V_{i,j,k,\iota}\cong V_{p,q,r,\kappa}$ if and only if $(i,j,k,\iota)=(p,q,r,\kappa)$ in $\Lambda$.
Suppose that $\Phi:V_{i,j,k,\iota}\mapsto V_{p,q,r,\kappa}$ is an $\D$-module isomorphism, and denote by
$[\Phi]=(p_{i,j})_{i,j=1,2}$ the matrix of $\Phi$ in the given basis. As a module morphism, we have $[c][\Psi]=[\Psi][c]$ and $[a][\Psi]=[\Psi][a]$, which imply $p_{21}=0,\,p_{11}=p_{22}$ and $(\xi^p-\xi^i)p_{11}=0,\,(\xi^p-\xi^{i+1})p_{12}=0$. Thus we have $\xi^i=\xi^p$ and then yield that $p_{12}=0$ and $[\Phi]=p_{11}I$ where $I$ is the identity matrix since $\Psi$ is an isomorphism. Similarly, we have $\xi^j=\xi^q$, $k=r$, $\iota=\kappa$ and then the claim follows.
\end{proof}
\begin{rmk}
For a left $\D$-module $V$, there exists a left dual module denoted by $V\As$ with module structure given by $(h\rightharpoonup f)(v)=f(S(h)\cdot v)$ for all $h\in \D, v\in V, f\in V\As$. We claim that $V_{i,j,k,\iota}\As\cong V_{-i-1,-j,k+1,\iota+1}$ for all $(i,j,k,\iota)\in \Lambda$. Indeed, denote by $\{v_1,v_2\}$ and $\{f_1,f_2\}$ the basises of $V_{i,j,k,\iota}$ and $V_{i,j,k,\iota}\As$, and denote by $\{w_1, w_2\}$ the basis of $V_{-i-1,-j,k+1,\iota+1}$. After a direct computation, one can show that the map $\varphi: V_{-i-1,-j,k+1,\iota+1}\mapsto V_{i,j,k,\iota}\As $ given by
\begin{align*}
\varphi(w_1)=\xi\Lam_1^2\Lam_4f_2,\quad \varphi(w_2)=f_1,
\end{align*}
is a left $\D$-module isomorphism.
\end{rmk}
Now we show that if $V$ is a $\D$-simple module, then $\dim V=1,2$.
\begin{lem}
For any simple $\D$-module $V$, $\dim V<3$.
\end{lem}
\begin{proof}
Note that by Proposition $\ref{proDouble}$, ${}_{\D(G)}\mathcal{M}\simeq {}_G^G\mathcal{YD}$ and ${}_{\D}\mathcal{M}\simeq \HYD$ as categories and by \cite[Proposition\;2.2.1]{AG99}, $\HYD \simeq {}_A^A\mathcal{YD}$. By Remark $\ref{rmkGAcocycledefor}$, we know that $A\cong G^{\sigma}$ for some $\sigma\in \mathcal{Z}^2(G,\mathds{k})$. Then we have that ${}_A^A\mathcal{YD}\simeq {}_G^G\mathcal{YD}$ by Proposition $\ref{proCocycledeformation}$. Thus the claim holds in ${}_{\D}\mathcal{M}$ if and only if it holds in ${}_{\D(G)}\mathcal{M}$.
From the proof in \cite[Theorem\,2.5]{B03}, the Drinfeld double $\text{gr}\;\D(G)\cong \BN(U)\,\sharp\, \K[\widehat{\Gamma}\times\Gamma]$, where $U=\K\{x,y\}\in{}_{\widehat{\Gamma}\times\Gamma}^{\widehat{\Gamma}\times\Gamma}\mathcal{YD}$ with appropriate actions and coactions and $\BN(U)=T(U)/J$ is a quantum plane where $J$ is the ideal generated by $x^2=0=y^2, xy+yx=0$. Moreover, $\D(G)$ can be regarded as the lifting of the quantum plane $\BN(U)$ by deforming the linking relation $xy+yx=0$. Thanks to \cite[Theorem\,3.5]{AB04}, we can get that if $V$ is a simple $\D(G)$-module, $\dim V<3$.
\end{proof}
Finally, we describe all the simple modules of $\D$ up to isomorphism.
\begin{thm}\label{thmsimplemoduleD}
There exist $48$ simple left $D$-modules pairwise non-isomorphic, among which $16$ one-dimensional modules are given in Lemma $\ref{lemOnesimpleD}$ and $32$ two-dimensional simple modules are given in Lemma $\ref{lemTwosimpleD}$.
\end{thm}
\section{Yetter-Drinfeld modules category $\HYD$}\label{secHYD}
In this section, we translate the simple $\D$-modules to the objects in $\HYD$ following the fact that the category ${}_{\D}\mathcal{M}$ is equivalent to the category $\HYD$. In order to study the Nichols algebras over the simple objects in $\HYD$, We describe explicitly the simple objects in $\HYD$ and also their braidings.
\begin{pro}
Let $\K_{\chi_{i,j,k}}=\K v$ be a one dimensional $\D$-module with $0\leq i,j<2,0\leq k<4$. Then $\K_{\chi_{i,j,k}}\in\HYD$ with the module structure and comodule structure given by
\begin{align*}
a\cdot v&=\xi^k v,\, b\cdot v=0, \, c\cdot v=0,\, d\cdot v=(-1)^i(-1)^j\xi^k v;\\
\delta(v)&=\begin{cases}
a^{2i}\otimes v &\text{~if~} j=0,\\
da^{2i+3}\otimes v & \text{~if~} j=1.
\end{cases}
\end{align*}
\end{pro}
\begin{proof}
Since $\K_{\chi_{i,j,k}}=\K v$ is a one-dimensional $\D$-module with $0\leq i,j<2,0\leq k<4$, the $H$-action shall be given by the restriction of the character of $\D$ given by Lemma $\ref{lemOnesimpleD}$ and the coaction must be of the form $\delta(v)=t\otimes v$, where $t\in G(H)=\{1,a^2,da,da^3\}$ such that $\langle g, t\rangle v=(-1)^iv$ and $\langle h, t\rangle v=(-1)^jv$. Then the claim follows.
\end{proof}
The following Proposition gives the braiding of $\K_{\chi_{i,j,k}}=\K v$ for all $0\leq i,j<2$, $0\leq k<4$.
\begin{pro}\label{braidingone}
The braiding of the one-dimensional YD-module $\K_{\chi_{i,j,k}}=\K v$ is
\begin{align*}
c(v\otimes v)=\begin{cases}
(-1)^{ik}v\otimes v, &\text{~if~} j=0;\\
-(-1)^{(i+1)k}v\otimes v, & \text{~if~} j=1.
\end{cases}
\end{align*}
\end{pro}
\begin{lem}
Let $V_{i,j,k,\iota}=\K\{v_1,v_2\}$ be a two-dimensional simple $\D$-module with $(i,j,k,\iota)\in\Lambda$. If we denote $\Lam_1=\xi^i$, $\Lam_2=\xi^j$, $\Lam_3=(-1)^k$ and $\Lam_4=(-1)^\iota$, then $V_{i,j,k,\iota}\in\HYD$ with the module structure given by
\begin{align*}
a\cdot v_1&=-\Lam_4\Lam_1v_1,\quad b\cdot v_1=0,\quad c\cdot v_1=0,\quad d\cdot v_1=\Lam_1v_1,\\
a\cdot v_2&=\xi\Lam_4\Lam_1 v_2,\quad b\cdot v_2=\Lam_4v_1,\quad c\cdot v_2=v_1,\quad d\cdot v_2=\xi\Lam_1v_2,
\end{align*}
and the comodule structure given by
\begin{itemize}
\item for $k=0$, i.e., $\Lam_3=1:$
\begin{align*}
\delta(v_1)&=a^j\otimes v_1+\xi(\Lam_1\Lam_4+\Lam_1\Lam_2)ba^{j-1}\otimes v_2,\\
\delta(v_2)&=da^{j-1}\otimes v_2+\frac{1}{2}\xi(\Lam_1^3\Lam_2-\Lam_1^3\Lam_4)ca^{j-1}\otimes v_1,
\end{align*}
\item for $k=1$, i.e., $\Lam_3=-1:$
\begin{align*}
\delta(v_1)&=da^{j-1}\otimes v_1+\xi(\Lam_1\Lam_4-\Lam_1\Lam_2)ca^{j-1}\otimes v_2,\\
\delta(v_2)&=a^{j}\otimes v_2-\frac{1}{2}\xi(\Lam_1^3\Lam_2+\Lam_1^3\Lam_4)ba^{j-1}\otimes v_1.
\end{align*}
\end{itemize}
\end{lem}
\begin{proof}
Note that by Lemma $\ref{lemAtoHdual}$ and Remark $\ref{rmkAtoHdual}$, we have that
\begin{align*}
(g^l)^\ast&=\frac{1}{8}\sum_{i=0}^3 \xi^{-il}a^i+\xi^{-(i+1)l}da^i ,\quad
(g^lh)^\ast=\frac{1}{8} \sum_{i=0}^3 \xi^{-il}a^i -\xi^{-(i+1)l} da^i ,\\
(g^lx)^\ast&=\frac{1}{8\sqrt 2} \sum_{i=0}^3 \xi^{-(i+1)l}ba^i + \xi^{-(i+1)l} ca^i ,\\
(g^lhx)^\ast&=\frac{1}{8\sqrt 2} \sum_{i=0}^3 \xi^{-(i+1)l}ba^i -\xi^{-(i+1)l}ca^i .
\end{align*}
Denote by $\{h_i\}_{1\leq i\leq 16}$ and $\{h^i\}_{1\leq i\leq 12}$ a basis of $H$ and its dual basis respectively. Then the comodule structure is given by $\delta(v)=\sum_{i=1}^{16}c_i\otimes c^i\cdot v$ for any $v\in V_{i,j,k}$. Thus
\begin{align*}
\delta(v_1)&=\sum_{l=0}^{3}\sum_{n=0}^1(g^lh^n)\As\otimes g^lh^n\cdot v_1+(g^lh^nx)\As\otimes g^lh^nx\cdot v_1\\
&=\sum_{l=0}^{3}\sum_{n=0}^1\Lam_3^n\Lam_2^l(g^lh^n)\As\otimes v_1
+\Lam_3^n(\Lam_2)^l(g^lh^nx)\As\otimes x_2v_2\\
&=\frac{1}{2}[(1+\Lam_3)a^j+(1-\Lam_3)da^{j-1}]\otimes v_1+\frac{1}{2\sqrt 2}x_2(1+\Lam_3)ba^{j-1}\otimes v_2\\&\quad
+\frac{1}{2\sqrt 2}x_2(1-\Lam_3)ca^{j-1}\otimes v_2,\\
\delta(v_2)&=\sum_{l=0}^{3}\sum_{n=0}^1((g^l)(g^lh^n)\As\otimes g^lh^n\cdot v_2+(g^lh^nx)\As\otimes g^lh^nx\cdot v_2\\
&=\sum_{l=0}^{3}\sum_{n=0}^1(-\Lam_3)^n\Lam_2^l(g^lh^n)\As\otimes v_2+(-\Lam_3)^n(\Lam_2)^l(g^lh^nx)\As\otimes x_1v_1\\
&=\frac{1}{2}[(1-\Lam_3)a^j+(1+\Lam_3)da^{j-1}]\otimes v_2+\frac{1}{2\sqrt 2}x_1(1-\Lam_3)ba^{j-1}\otimes v_1\\&\quad
+\frac{1}{2\sqrt 2}x_1(1+\Lam_3)ca^{j-1}\otimes v_1,
\end{align*}
where $x_1=\frac{\sqrt 2}{2}\xi \Lam_1^3(\Lam_2 \Lam_3 -\Lam_4)$ and $x_2=\sqrt 2\xi \Lam_1(\Lam_2\Lam_3+\Lam_4)$. Now we describe the coaction explicitly case by case.
If $\Lam_3=1$, then $x_1=\frac{\sqrt 2}{2}\xi(\Lam_1^3\Lam_2-\Lam_1^3\Lam_4)$, $x_2=\sqrt 2\xi(\Lam_1\Lam_2+\Lam_1\Lam_4)$. In such a case, we have $\delta(v_1)=a^j\otimes v_1+\xi(\Lam_1\Lam_2+\Lam_1\Lam_4)ba^{j-1}\otimes v_2$ and $\delta(v_2)=da^{j-1}\otimes v_2+\frac{1}{2}\xi(\Lam_1^3\Lam_2-\Lam_1^3\Lam_4)ca^{j-1}\otimes v_1$.
If $\Lam_3=-1$, then $x_1=-\frac{\sqrt 2}{2}\xi(\Lam_1^3\Lam_2+\Lam_1^3\Lam_4)$, $x_2=\sqrt 2\xi(\Lam_1\Lam_4-\Lam_1\Lam_2)$. In such a case, we have $\delta(v_1)=da^{j-1}\otimes v_1+\xi(\Lam_1\Lam_4-\Lam_1\Lam_2)ca^{j-1}\otimes v_2$ and $\delta(v_2)=a^{j}\otimes v_2-\frac{1}{2}\xi(\Lam_1^3\Lam_2+\Lam_1^3\Lam_4)ba^{j-1}\otimes v_1$.
\end{proof}
After a direct computation using the braiding in $\HYD$, we describe the braidings of the simple modules $V_{i,j,k,\iota}\in\HYD$.
\begin{pro}\label{probraidsimpletwo}
Let $V_{i,j,k,\iota}=\K\{v_1,v_2\}$ be a two-dimensional simple $\D$-module with $(i,j,k,\iota)\in\Lambda$. If we denote $\Lam_1=\xi^i$, $\Lam_2=\xi^j$, $\Lam_3=(-1)^k$ and $\Lam_4=(-1)^\iota$, then $V_{i,j,k,\iota}\in\HYD$. The braiding of $V_{i,j,k,\iota}=\K\{v_1,v_2\}$ is given by
\begin{itemize}
\item If $k=0$, then $c(\left[\begin{array}{ccc} v_1\\v_2\end{array}\right]\otimes\left[\begin{array}{ccc} v_1~v_2\end{array}\right])=$
\begin{align*}
\left[\begin{array}{ccc}
-\Lam_4\Lam_1^jv_1\otimes v_1 & \Lam_4(\xi\Lam_1)^jv_2\otimes v_1+[(\xi\Lam_1)^j-\Lam_4\Lam_1^j]v_1\otimes v_2\\
\Lam_1^jv_1\otimes v_2 &(\xi\Lam_1)^jv_2\otimes v_2+\frac{1}{2}(\xi\Lam_1)^j\Lam_1^2(\Lam_2-\Lam_4)v_1\otimes v_1
\end{array}\right].
\end{align*}
\item If $k=1$, then $ c(\left[\begin{array}{ccc} v_1\\v_2\end{array}\right]\otimes\left[\begin{array}{ccc} v_1~v_2\end{array}\right])=$
\begin{align*}
\left[\begin{array}{ccc}
\Lam_1^jv_1\otimes v_1 & (\xi\Lam_1)^jv_2\otimes v_1+[\Lam_4(\xi\Lam_1)^j+\Lam_1^j]v_1\otimes v_2\\
-\Lam_4\Lam_1^jv_1\otimes v_2
&\Lam_4(\xi\Lam_1)^jv_2\otimes v_2-\frac{1}{2}(\xi\Lam_1)^j\Lam_1^2(\Lam_2\Lam_4+1)v_1\otimes v_1
\end{array}\right].
\end{align*}
\end{itemize}
\end{pro}
\section{Nichols algebra in $\HYD$}\label{secNicholsalgebaH}
In this section, we determine all finite-dimensional Nichols algebras over simple objects in $\HYD$ and present them by generators and relations.
First, we study the Nichols algebras over one-dimensional simple modules. For convenience, let
\begin{align*}
\Lambda^0=\{(1,0,1),(1,0,3),(0,1,0),(0,1,2),(1,1,0),(1,1,1),(1,1,2),(1,1,3)\}.
\end{align*}
By Proposition $\ref{braidingone}$, the following result follows immediately.
\begin{lem}\label{lemNicholsgeneratedbyone}
The Nichols algebra $\BN(\K_{\chi_{i,j,k}})$ over $\K_{\chi_{i,j,k}}=\K v$ with $0\leq i,j<2,0\leq k<4$ is
\begin{align*}
\BN(\K_{\chi_{i,j,k}})=\begin{cases}
\bigwedge \K_{\chi_{i,j,k}} & (i,j,k)\in\Lambda^0,\\
\K[v] & \text{others}.
\end{cases}
\end{align*}
Moreover, let $V=\oplus_{\ell\in I}V_{\ell}$, where $V_{\ell}\cong \K_{\chi_{i,j,k}}$ for some $(i,j,k)\in\Lambda^0$, and $I$ is a finite index set. Then $\BN(V)=\bigwedge V\cong \otimes_{i\in I}\BN(V_i)$.
\end{lem}
Next, we analyze the Nichols algebras associated to two-dimensional simple modules. For convenience, denote by $\Lambda^i$ with $1\leq i\leq 6$ the finite subsets of $\Lambda$ given by
\begin{align*}
\Lambda^1&=\{(2,1,0,0),(0,1,1,0),(2,3,0,0),(0,3,1,0),\\&\quad~~(0,1,0,1),(0,1,1,1),(0,3,0,1) (0,3,1,1)\},\\
\Lambda^2&=\{(1,3,1,1),(3,3,0,1),(1,1,1,1),(3,1,0,1),\\&\quad~~(3,3,1,0),(3,3,0,0),(3,1,1,0),(3,1,0,0)\},\\
\Lambda^3&=\{(0,1,0,0),(0,3,0,0),(2,1,0,1),(2,3,0,1)\},\\
\Lambda^4&=\{(2,1,1,0),(2,1,1,1),(2,3,1,0),(2,3,1,1)\},\\
\Lambda^5&=\{(1,3,0,1),(1,3,0,0),(1,1,0,1),(1,1,0,0)\},\\
\Lambda^6&=\{(3,3,1,1),(1,3,1,0),(3,1,1,1),(1,1,1,0)\}.
\end{align*}
It is clear that $\Lambda=\cup_{i=1}^6\Lambda_i$.
\begin{lem}
$\dim\BN(V_{i,j,k,\iota})=\infty$ for all $(i,j,k,\iota)\in\Lambda^1\cup \Lambda^2$.
\end{lem}
\begin{proof}
By Proposition $\ref{probraidsimpletwo}$, the braiding of $V_{i,j,k,\iota}$ for $(i,j,k,\iota)\in\Lambda^1$
has an eigenvector of eigenvalue $1$. Indeed, $c(v_1\otimes v_1)=v_1\otimes v_1$ in the above cases. And for any element $(i,j,k,\iota)\in\Lambda^2$, there exists one element $(p,q,r,\mu)\in\Lambda^2$ such that $V_{i,j,k,\iota}\As\cong V_{p,q,r,\mu}$ in $\HYD$, then by Proposition $\ref{proNicholsdual}$, the claim follows.
\end{proof}
Then we will show that Nichols algebras $\BN(V_{i,j,k,\iota})$ are finite-dimensional for all $(i,j,k,\iota)\in \Lambda^i$ for $3\leq i\leq 6$, and describe them in terms of generators and relations.
\begin{pro}\label{proV3}
$\BN(V_{i,j,k,\iota}):=\K\langle v_1, v_2\mid v_1^2=0, v_1v_2-\xi^jv_2v_1=0, v_2^4=0\rangle$ for $(i,j,k,\iota)\in\Lambda^3$. In particular, $\dim\BN(V_{i,j,k,\iota})=8$ for $(i,j,k,\iota)\in\Lambda^3$.
\end{pro}
\begin{proof}
In this case, note that $\Lam_1=\Lam_4=\Lam_1^j$, $\delta(v_1)=a^j\otimes v_1+\xi(1+\xi^j\Lam_1)ba^{j-1}\otimes v_2$, $\delta(v_2)=da^{j-1}\otimes v_2+\frac{1}{2}\xi\Lam_1^3(\xi^j-\Lam_4)ca^{j-1}\otimes v_1$, and the braiding is given by
\begin{align*}
c(\left[\begin{array}{ccc} v_1\\v_2\end{array}\right]\otimes\left[\begin{array}{ccc} v_1~v_2\end{array}\right])=
\left[\begin{array}{ccc}
-v_1\otimes v_1 & \xi^jv_2\otimes v_1+(\xi^j\Lam_1^j-1)v_1\otimes v_2\\
\Lam_1^jv_1\otimes v_2 & \xi^j\Lam_1^jv_2\otimes v_2+\frac{1}{2}\xi^j\Lam_1^j(\xi^j-\Lam_4)v_1\otimes v_1
\end{array}\right].
\end{align*}
Using the braiding, we have that
\begin{align*}
\Delta(v_1^2)&=v_1^2\otimes 1+v_1^2\otimes 1,\\
\Delta(v_1v_2)&=v_1v_2\otimes 1+\xi^j\Lam_1^jv_1\otimes v_2+\xi^jv_2\otimes v_1+1\otimes v_1v_2,\\
\Delta(v_2v_1)&=v_2v_1\otimes 1+ \Lam_1^jv_1\otimes v_2+v_2\otimes v_1+1\otimes v_2v_1,\\
\Delta(v_2^2)&=v_2^2\otimes 1+(1+\xi^j\Lam_1^j)v_2\otimes v_2-\frac{1}{2}(\Lam_1^j+\xi^j)v_1\otimes v_1+1\otimes v_2^2,
\end{align*}
which give us the relations $x^2=0$, and $v_1v_2-\xi^jv_2v_1=0$. And since
\begin{align*}
c(v_2\otimes v_2^2)&=da^j\cdot v_2^2\otimes v_2+\frac{1}{2}\xi\Lam_1^3(\xi^j-\Lam_4)ca^{j-1}\cdot v_2^2\otimes v_1\\
&=(da^{j-1}\cdot v_2)(da^{j-1}\cdot v_2)\otimes v_2+(ca^{j-1}\cdot v_2)(ba^{j-1}\cdot v_2)\otimes v_2
\\&\ +\frac{1}{2}\xi\Lam_1^3(\xi^j{-}\Lam_4)[(ca^{j-1}\cdot v_2)(a^j\cdot v_2)+(da^{j-1}\cdot v_2)(ca^{j-1}\cdot v_2)]\otimes v_1\\
&=-v_2^2\otimes v_2+\Lam_4v_1^2\otimes v_2+\frac{1}{2}(1{-}\xi^j\Lam_1^3)(\Lam_1v_2v_1+v_1v_2)\otimes v_1\\
&=-v_2^2\otimes v_2+\Lam_1v_2v_1\otimes v_1,
\end{align*}
we have
\begin{align*}
\Delta(v_2^3)&=(v_2\otimes 1+1\otimes v_2)\Delta(v_2^2)\\
&=v_2^3\otimes 1+(1+\xi^j\Lam_1^j)v_2^2\otimes v_2-\frac{1}{2}(\Lam_1^j+\xi^j)v_2v_1\otimes v_1+v_2\otimes v_2^2-v_2^2\otimes v_2
\\&\quad+\Lam_1v_2v_1\otimes v_1+(\xi^j\Lam_1^j{-}1)v_2\otimes v_2^2-\xi^jv_1\otimes v_1v_2-\frac{1}{2}(1{+}\xi^j\Lam_1^j)v_1\otimes v_2v_1\\&\quad+1\otimes v_2^3\\
&=v_2^3\otimes 1+1\otimes v_2^3+\xi^j\Lam_1^jv_2^2\otimes v_2+\frac{1}{2}(\Lam_1^j-\xi^j)v_2v_1\otimes v_1+\xi^j\Lam_1^jv_2\otimes v_2^2\\&\quad+\frac{1}{2}(1{-}\xi^j\Lam_1^j)v_1\otimes v_2v_1.
\end{align*}
Similarly, after a direct computation, we have that
\begin{align*}
\Delta(v_2^4)=(v_2\otimes 1+v_2\otimes v_1)\Delta(v_2^3)=v_2^4\otimes 1+1\otimes v_2^4,
\end{align*}
since $c(v_2\otimes v_2v_1)=\xi^jv_2v_1\otimes v_2$, and $c(v_2\otimes v_2^3)=\frac{1}{2}(\xi^j-\Lam_4)v_1\otimes v_2^2v_1-\xi^j\Lam_1v_2^3\otimes v_2$. Thus, we get relation $x_2^4=0$.
Thus there exists a graded Hopf algebra epimorphism $\pi:R=T(V_{i,j,k,\iota})/I\twoheadrightarrow \BN(V_{i,j,k,\iota})$ in $\HYD$, where $I$ is the ideal generated by the relations $v_1^2=0, v_1v_2-\xi^jv_2v_1=0, v_2^4=0$.
Note that $R(5)=0, R(1)=V_{3,1}, R(0)=\K$ and $R(4)\neq 0$. Then by the Poincar\'{e} duality, we have that $\dim R(4)=\dim R(0)=1$, and $\dim R(3)=\dim R(1)=2$. And it is clear that $R(2)=\K\{v_2^2,v_1v_2\}$ and whence $\dim R(2)=2$. Since $\dim \BN^5(V_{i,j,k,\iota})=0$ and $\pi$ is injective in degree 0 and 1, we have $\dim R(4)=\dim\BN^4(V_{i,j,k,\iota})$ and $\dim R(3)=\dim\BN^3(V_{i,j,k,\iota})$. Moreover, it is clear that $\dim \BN^2(V_{i,j,k,\iota})=2$ which implies $\dim R=\dim\BN(V_{i,j,k,\iota})$. Then the claim follows.
\end{proof}
\begin{pro}\label{proV4}
$\BN(V_{i,j,k,\iota}):=\K\langle v_1, v_2\mid v_1^2=0, v_1v_2+\xi^jv_2v_1=0, v_2^4=0\rangle$ for $(i,j,k,\iota)\in\Lambda^4$. In particular, $\dim\BN(V_{i,j,k,\iota})=8$ for $(i,j,k,\iota)\in\Lambda^4$.
\end{pro}
\begin{proof}
In such a case, note that $\Lam_1=-1=\Lam_3$,
$\delta(v_1)=da^{j-1}\otimes v_1+\xi(\Lam_2-\Lam_4)ca^{j-1}\otimes v_2$,
$\delta(v_2)=a^{j}\otimes v_2+\frac{1}{2}\xi(\Lam_2+\Lam_4)ba^{j-1}\otimes v_1$. And the braiding is given by
\begin{align*}
c(\left[\begin{array}{ccc} v_1\\v_2\end{array}\right]\otimes\left[\begin{array}{ccc} v_1~v_2\end{array}\right])=
\left[\begin{array}{ccc}
-v_1\otimes v_1 & -\xi^jv_2\otimes v_1-(\Lam_4\xi^j+1)v_1\otimes v_2\\
\Lam_4v_1\otimes v_2
&-\Lam_4\xi^jv_2\otimes v_2+\frac{1}{2}\xi^j(\Lam_2\Lam_4+1)v_1\otimes v_1
\end{array}\right].
\end{align*}
Using the braiding, we have that
\begin{align*}
\Delta(v_1^2)&=v_1^2\otimes 1+1\otimes v_1^2,\\
\Delta(v_1v_2)&=v_1v_2\otimes 1-\xi^jv_2\otimes v_1-\Lam_4\xi^jv_1\otimes v_2+1\otimes v_1v_2,\\
\Delta(v_2v_1)&=v_2v_1\otimes 1+v_2\otimes v_1+\Lam_4v_1\otimes v_2+1\otimes v_2v_1,\\
\Delta(v_2^2)&=v_2^2\otimes 1+(1-\Lam_4\xi^j)v_2\otimes v_2+\frac{1}{2}(\xi^j-\Lam_4)v_1\otimes v_1+1\otimes v_2^2,
\end{align*}
which gives us the relations $v_1^2=0$ and $v_1v_2+\xi^jv_2v_1=0$. Since $c(v_2\otimes v_2^2)=-v_2^2\otimes v_2+\Lam_4v_2v_1\otimes v_1$, we have that
\begin{align*}
\Delta(v_2^3)&=v_2^3\otimes 1-\Lam_4\xi^jv_2^2\otimes v_2-\Lam_4\xi^jv_2\otimes v_2^2 +\frac{1}{2}(\xi^j+\Lam_4)v_2v_1\otimes v_1\\&\quad+\frac{1}{2}(\Lam_2\Lam_4+1)v_1\otimes v_2v_1+1\otimes v_2^3.
\end{align*}
Similarly, after a direct computation, we also have that
\begin{align*}
\Delta(v_2^4)=(v_2\otimes 1+1\otimes v_2)\Delta(v_2^3)=v_2^4\otimes 1+1\otimes v_2^4,
\end{align*}
since
\begin{align*}
c(v_2\otimes v_2v_1)&=-\xi^jv_2v_1\otimes v_2-\frac{1}{2}\xi^j(\Lam_2+\Lam_4)v_1^2\otimes v_1=-\xi^jv_2v_1\otimes v_2,\\
c(v_2\otimes v_2^3)&=\Lam_4\xi^jv_2^3\otimes v_2-\frac{1}{2}(\xi^j+\Lam_4)v_2^2v_1\otimes v_1.
\end{align*}
This gives us relation $v_2^4=0$.
\end{proof}
\begin{pro}\label{proV5}
$\BN(V_{i,j,k,\iota}):=\K\langle v_1, v_2\mid v_1^4=0, v_1v_2+\Lam_4v_2v_1=0, v_1^2+2\Lam_4v_2^2\rangle$ for $(i,j,k,\iota)\in\Lambda^5$. In particular, $\dim\BN(V_{i,j,k,\iota})=8$ for $(i,j,k,\iota)\in\Lambda^5$.
\end{pro}
\begin{proof}
In such a case, note that $\Lam_1=\xi$,
$\delta(v_1)=a^{j}\otimes v_1-(\Lam_2+\Lam_4)ba^{j-1}\otimes v_2$,
$\delta(v_2)=da^{j-1}\otimes v_2+\frac{1}{2}(\Lam_2-\Lam_4)ca^{j-1}\otimes v_1$. And the braiding is given by
\begin{align*}
c(\left[\begin{array}{ccc} v_1\\v_2\end{array}\right]\otimes\left[\begin{array}{ccc} v_1~v_2\end{array}\right])=
\left[\begin{array}{ccc}
-\Lam_4\xi^jv_1\otimes v_1 & -\Lam_4v_2\otimes v_1-(\Lam_4\xi^j+1)v_1\otimes v_2\\
\xi^jv_1\otimes v_2 &-v_2\otimes v_2+\frac{1}{2}(\Lam_2-\Lam_4)v_1\otimes v_1
\end{array}\right].
\end{align*}
Using the braiding, we have that
\begin{align*}
\Delta(v_1^2)&=v_1^2\otimes 1+(1-\Lam_4\xi^j)v_1\otimes v_1+1\otimes v_1^2,\\
\Delta(v_1v_2)&=v_1v_2\otimes 1-\Lam_4v_2\otimes v_1-\Lam_4\xi^jv_1\otimes v_2+1\otimes v_1v_2,\\
\Delta(v_2v_1)&=v_2v_1\otimes 1+v_2\otimes v_1+\xi^jv_1\otimes v_2+1\otimes v_2v_1,\\
\Delta(v_2^2)&=v_2^2\otimes 1+\frac{1}{2}(\xi^j-\Lam_4)v_1\otimes v_1+1\otimes v_2^2,
\end{align*}
which gives us the relations $v_1^2+2\Lam_4v_2^2=0$ and $v_1v_2+\Lam_4v_2v_1=0$. Since $c(v_1\otimes v_1^2)=-v_1^2\otimes v_1$ and $c(v_1\otimes v_1^3)=\xi^j\Lam_4v_1^3\otimes v_1$, we have that
\begin{align*}
\Delta(v_1^3)&=v_1^3\otimes 1+1\otimes v_1^3-\Lam_4\xi^jv_1^2\otimes v_1-\Lam_4\xi^jv_1\otimes v_1^2,\\
\Delta(v_1^4)&=(v_1\otimes 1+1\otimes v_1)\Delta(v_1^3)=v_1^4\otimes 1 +1\otimes v_1^4,
\end{align*}
which implies that $v_1^4=0$.
\end{proof}
\begin{pro}\label{proV6}
$\BN(V_{i,j,k,\iota}):=\K\langle v_1, v_2\mid v_1^4=0, v_1v_2+\Lam_4v_2v_1=0, v_1^2+2\Lam_4v_2^2=0\rangle$ for $(i,j,k,\iota)\in\Lambda^6$. In particular, $\dim\BN(V_{i,j,k,\iota})=8$ for $(i,j,k,\iota)=(i,j,k,\iota)\in\Lambda^6$.
\end{pro}
\begin{proof}
In such a case, note that $\Lam_1\Lam_4=\xi$, $\Lam_1^2=-1$ and $\Lam_1\xi=-\Lam_4$.
$\delta(v_1)=da^{j-1}\otimes v_1+\xi\Lam_1(\Lam_4-\Lam_2)ca^{j-1}\otimes v_2$,
$\delta(v_2)=a^{j}\otimes v_2-\frac{1}{2}\xi\Lam_1^3(\Lam_2+\Lam_4)ba^{j-1}\otimes v_1$. And the braiding is given by
\begin{align*}
c(\left[\begin{array}{ccc} v_1\\v_2\end{array}\right]\otimes\left[\begin{array}{ccc} v_1~v_2\end{array}\right])=
\left[\begin{array}{ccc}
\Lam_1^jv_1\otimes v_1 & (\xi\Lam_1)^jv_2\otimes v_1+(\Lam_1^j-1)v_1\otimes v_2\\
-\xi^jv_1\otimes v_2 &-v_2\otimes v_2+\frac{1}{2}\xi^j(\Lam_1^j-1)v_1\otimes v_1
\end{array}\right].
\end{align*}
Using the braiding, we have that
\begin{align*}
\Delta(v_1^2)&=v_1^2\otimes 1+(1+\Lam_1^j)v_1\otimes v_1+1\otimes v_1^2,\\
\Delta(v_1v_2)&=v_1v_2\otimes 1+(\xi\Lam_1)^jv_2\otimes v_1+\Lam_1^jv_1\otimes v_2+1\otimes v_1v_2,\\
\Delta(v_2v_1)&=v_2v_1\otimes 1+v_2\otimes v_1-\xi^jv_1\otimes v_2+1\otimes v_2v_1,\\
\Delta(v_2^2)&=v_2^2\otimes 1+\frac{1}{2}\xi^j(\Lam_1^j-1)v_1\otimes v_1+1\otimes v_2^2\\
&=v_2^2\otimes 1+\frac{1}{2}\xi^j\Lam_1^j(1+\Lam_1^j)v_1\otimes v_1+1\otimes v_2^2,
\end{align*}
which gives us the relations $v_1^2-2\xi^j\Lam_1^j v_2^2=0$ and $v_1v_2-(\Lam_1\xi)^jv_2v_1=0$. Since $c(v_1\otimes v_1^2)=-v_1^2\otimes v_1$ and $c(v_1\otimes v_1^3)=-\Lam_1^jv_1^3\otimes v_1$, we have that
\begin{align*}
\Delta(v_1^3)&=v_1^3\otimes 1+1\otimes v_1^3+\Lam_1^jv_1^2\otimes v_1+\Lam_1^jv_1\otimes v_1^2,\\
\Delta(v_1^4)&=(v_1\otimes 1+1\otimes v_1)\Delta(v_1^3)=v_1^4\otimes 1 +1\otimes v_1^4,
\end{align*}
which implies that $v_1^4=0$.
\end{proof}
Finally we show that the Nichols algebra $\BN(V)$ over a simple object $V$ in $\HYD$ is finite-dimensional if and only if $V$ is isomorphic either to $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$ or $V_{i,j,k,\iota}$ with $(i,j,k,\iota)\in\cup_{3\leq\ell\leq 6}\Lambda^{\ell}$.
\begin{proofthma}
For any simple object $V\in\HYD$ such that $\dim\BN(V)<\infty$. If $\dim V=1$, then $V\cong \K_{\chi_{i,j,k}}$ where $(i,j,k)\in\Lambda^0$ by Lemma $\ref{lemNicholsgeneratedbyone}$. If $\dim V=2$, then $V\cong V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \cup_{3\leq\ell\leq 6}\Lambda^{\ell}$ by Propositions $\ref{proV3}$, $\ref{proV4}$, $\ref{proV5}$ and $\ref{proV6}$. And it is clear that these Nichols algebras are pairwise non-isomorphic, since their infinitesimal braidings are pairwise non-isomorphic in $\HYD$.
\end{proofthma}
\section{Hopf algebras over $H$}\label{secHopfalgebraH}
In this section, we determine all finite-dimensional Hopf algebras $A$ over $H$ and the corresponding infinitesimal braidings $V$ are isomorphic to $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$, or $V_{i,j,k,\iota}\in\cup_{3\leq \ell\leq 6}\Lambda^{\ell}$.
First, we show that such Hopf algebras mentioned above are generated in degree one with respect to the standard filtration, i.e., $\text{gr}\,A\cong \BN(V)\sharp H$.
\begin{lem}
Let $A$ be a Hopf algebra such that $A_{[0]}\cong H$ and the corresponding infinitesimal braiding $V$ is either the simple modules $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$, and $V_{i,j,k,\iota}\in\cup_{3\leq \ell\leq
6}\Lambda^{\ell}$. Then $\text{gr}\,A\cong \BN(V)\sharp H$. That is, $A$ is generated by the first term of the standard filtration.
\end{lem}\label{lemGenerataionindegreeone}
\begin{proof}
Recall that $T=\text{gr}\,A=\oplus_{i\geq 0}A_{[i]}/A_{[i+1]}=R\sharp H$, where $A_{[0]}\cong H$ and $R=T^{coA_{[0]}}$. In order to show that $gr\,A\cong \BN(V)\sharp H$, i.e., $R\cong \BN(V)$, let $S=R\As$ be the graded dual of $R$ and by the duality principle in $\cite[Lemma\;2.4]{AS02}$, $S$ is generated by $S(1)$ since $\Pp(R)=R(1)$. Thus there exists a surjective morphism $S\twoheadrightarrow \BN(W)$ where $W=S(1)$. Thus $S$ is a Nichols algebra if $\Pp(S)=S(1)$, which implies $R$ is a Nichols algebra, i.e., $R=\BN(V)$. To show that $\Pp(S)=S(1)$, it is enough to prove that the relations of $\BN(V)$ also hold in $S$.
Assume $W=\K_{\chi_{i,j,k}}=\K[v]/(v^2)$ with $(i,j,k)\in\Lambda^0$ and then $\BN(W)=\bigwedge \K_{\chi_{i,j,k}}$ for $(i,j,k)\in\Lambda^0$. In such a case, if $v^2\in S$, then $v^2$ is a primitive element and $c(v^2\otimes v^2)=v^2\otimes v^2$. Since as the graded dual of $R$, $S$ must be finite-dimensional, thus $v^2=0$. Then the claim follows.
Assume that $W=V_{i,j,k,\iota}$ with $(i,j,k,\iota)\in\Lambda^3$, then by Proposition $\ref{proV3}$, we know that as an algebra $\BN(W):=\K\langle v_1, v_2|v_1^2=0, v_1v_2-\xi^jv_2v_1=0, v_2^4=0\rangle$ and the relations of $\BN(W)$ are all primitive elements. Thus we need to show that $c(r\otimes r)=r\otimes r$ for $r=v_1^2$, $v_1v_2-\xi^jv_2v_1$ and $v_2^4$. Since
\begin{align*}
\delta(v_1)&=a^j\otimes v_1+\xi(1+\xi^j\Lam_1)ba^{j-1}\otimes v_2, \\
\delta(v_2)&=da^{j-1}\otimes v_2+\frac{1}{2}\xi\Lam_1^3(\xi^j-\Lam_2)ca^{j-1}\otimes v_1,
\end{align*}
after a direct computation, we have that
\begin{gather*}
\delta(v_1^2)=a^2\otimes v_1^2+\xi(\Lam_2-\Lam_1)ba\otimes(v_1v_2-\xi^jv_2v_1),\\
\delta(v_1v_2-\xi^jv_2v_1)=da\otimes (v_1v_2-\xi^jv_2v_1),\quad
\delta(v_2^4)=1\otimes v_2^3.
\end{gather*}
Thus by the braiding of $\HYD$, we have
\begin{gather*}
c(v_1^2\otimes v_1^2)=v_1^2\otimes v_1^2,\quad c(v_2^4\otimes v_2^4)=v_2^4\otimes v_2^4,\\
c((v_1v_2-\xi^jv_2v_1)\otimes (v_1v_2-\xi^2v_2v_1))=(v_1v_2-\xi^2v_2v_1)\otimes (v_1v_2-\xi^2v_2v_1).
\end{gather*}
Then the claim follows.
Similarly, the claim follows when $W=V_{i,j,k,\iota}$ for $(i,j,k,\iota)\in\Lambda^4\cup\Lambda^5\cup\Lambda^6$.
\end{proof}
Next, we shall show that there do not exist non-trivial liftings for the bosonizations of the Nichols algebras over $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$, and $V_{i,j,k,\iota}$ with $(i,j,k,\iota)\in\Lambda^3\cup\Lambda^4$.
\begin{pro}\label{proLiftingnon0}
Let $A$ be a finite-dimensional Hopf algebra over $H$ such that its infinitesimal braiding $V$ is isomorphic to $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$. Then $A\cong \bigwedge \K_{\chi_{i,j,k}}\sharp H$.
\end{pro}
\begin{proof}
Note that $\text{gr}\, A\cong \BN(V)\sharp H$, where $V$ is isomorphic to $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$. We prove that the relations in $\BN(V)$ also hold in $H$. Indeed, let $\bigwedge \K_{\chi_{i,j,k}}=\K[v]/(v^2)$,
If $j=0$, then $\delta(v)=a^2\otimes v$. In such a case, we have that
\begin{align*}
\Delta_A(v)=v\otimes 1+a^2\otimes v,\quad
\Delta_A(v^2)=v^2\otimes 1+ 1\otimes v^2.
\end{align*}
If $j=1$, then $\delta(v)=da^{2i+3}\otimes v$. In such a case, we have that
\begin{align*}
\Delta(v)=v\otimes 1+da^{2i+3}\otimes v,\quad
\Delta(v^2)=v^2\otimes 1 +1\otimes v^2.
\end{align*}
But since $A$ is a finite-dimensional Hopf algebra so that $A$ cannot contain any primitive element. Therefore the relation $v^2=0$ must hold in $A$.
\end{proof}
\begin{pro}\label{proLiftingnon3}
Let $A$ be a finite-dimensional Hopf algebra over $H$ such that the infinitesimal braiding $V$ is isomorphic either to $V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^3$. Then $A\cong \BN(V_{i,j,k,\iota})\sharp H$ where $(i,j,k,\iota)\in \Lambda^3$.
\end{pro}
\begin{proof}
Note that $\text{gr}\, A\cong \BN(V_{i,j,k,\iota})\sharp H$, where $(i,j,k,\iota)\in \Lambda^3$. For convenience, let $W:=V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^3$.
In such a case, the bosonization $\BN(W)\sharp H$ is generated by $x, y, a, b,c,d$ satisfying the relations
\begin{align*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd,\quad ax=-xa,\\ bx=-xb,\quad cx=\Lam_1xc,\quad dx=\Lam_1xd,\quad xy-\xi^jyx=0,\quad x^2=0,\quad y^4=0,\\
ay-\xi ya=\Lam_4xc,\quad by-\xi yb=\Lam_4xd,\quad
cy-\xi\Lam_1yc=xa,\quad dy-\xi\Lam_1 yd=xb.
\end{align*}
The coalgebra structure is given by
\begin{align*}
\Delta(a)&=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)&=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(x)&=x\otimes 1+a^j\otimes x+\xi(1+\xi^j\Lam_1)ba^{j-1}\otimes y,\\
\Delta(y)&=y\otimes 1+da^{j-1}\otimes y+\frac{1}{2}\xi\Lam_1^3(\xi^j-\Lam_4)ca^{j-1}\otimes x.
\end{align*}
Assume that $A$ is a finite-dimensional Hopf algebra such that $\text{gr}\,A\cong \BN(W)\sharp H$. After a direct computation, we have that
\begin{gather*}
\Delta(x^2)=x^2\otimes 1+a^2\otimes x^2+\xi(\xi^j-\Lam_1)ba\otimes (xy-\xi^jyx),\\
\Delta(xy-\xi^jyx)=(xy-\xi^jyx)\otimes 1+da\otimes(xy-\xi^jyx).
\end{gather*}
From the second equation, we have that
\begin{align*}
xy-\xi^jyx\in\Pp_{1,da}(\BN(W)\sharp H)=\Pp_{1,da}(H)=\K\{1-da\},
\end{align*}
which implies that $xy-\xi^jyx=\mu(1-da)$ for some $\mu\in\K$. Then from the first equation, we get that
\begin{align*}
\Delta(x^2+\xi(\xi^j-\Lam_1)\mu ba)&=x^2\otimes 1+a^2\otimes x^2+\xi(\xi^j-\Lam_1)ba\otimes \mu(1-da)\\&\quad\,
+\xi(\xi^j-\Lam_1)\mu ba\otimes da+\xi(\xi^j-\Lam_1)\mu a^2\otimes ba\\
&=(x^2+\xi(\xi^j{-}\Lam_1)\mu ba)\otimes 1+a^2\otimes(x^2+\xi(\xi^j{-}\Lam_1)\mu ba),
\end{align*}
which implies that there exists $\nu\in\K$ such that
\begin{align*}
x^2+\xi(\xi^j-\Lam_1)\mu ba=\nu(1-a^2).
\end{align*}
Since $ax^2=x^2a$, $bx^2=x^2b$ and $ab=\xi ba$, we get that $\mu=0=\nu$ and whence the relations $x^2=0$ and $xy-\xi^jyx=0$ must hold in $A$. Then after a tedious computation, we have that
\begin{align*}
\Delta(y^4)=\Delta(y)^4=y^4\otimes 1+1\otimes y^4,
\end{align*}
which implies that relation $y^4=0$ holds in $A$. Thus the claim follows.
\end{proof}
\begin{pro}\label{proLiftingnon4}
Let $A$ be a finite-dimensional Hopf algebra over $H$ such that the infinitesimal braiding $V$ is isomorphic either to $V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^4$. Then $A\cong \BN(V_{i,j,k,\iota})\sharp H$ where $(i,j,k,\iota)\in \Lambda^4$.
\end{pro}
\begin{proof}
Note that $\text{gr}\, A\cong \BN(V_{i,j,k,\iota})\sharp H$, where $(i,j,k,\iota)\in \Lambda^4$. For convenience, let $W:=V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^4$.
In such a case, the bosonization $\BN(W)\sharp H$ is generated by $x, y, a, b,c,d$ satisfying the relations
\begin{align*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd,\quad ax=\Lam_4xa,\\
bx=\Lam_4xb,\quad cx=-xc,\quad dx=-xd,\quad xy+\xi^jyx=0,\quad x^2=0,\quad y^4=0,\\
ay+\xi\Lam_4 ya=\Lam_4xc,\quad by+\xi\Lam_4 yb=\Lam_4xd,\quad
cy+\xi yc=xa,\quad dy+\xi yd=xb.
\end{align*}
the coalgebra structure is given by
\begin{align*}
\Delta(a)&=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)&=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(x)&=x\otimes 1+da^{j-1}\otimes x+\xi(\Lam_2-\Lam_4)ca^{j-1}\otimes y,\\
\Delta(y)&=y\otimes 1+a^{j}\otimes y+\frac{1}{2}\xi(\Lam_2+\Lam_4)ba^{j-1}\otimes x.
\end{align*}
Assume that $A$ is a finite-dimensional Hopf algebra such that $\text{gr}\,A\cong \BN(W)\sharp H$. After a direct computation, we have that
\begin{gather*}
\Delta(x^2)=x^2\otimes 1+a^2\otimes x^2+\xi(1+\xi^j\Lam_4)ba\otimes (xy+\xi^jyx),\\
\Delta(xy+\xi^jyx)=(xy+\xi^jyx)\otimes 1+da\otimes(xy+\xi^jyx).
\end{gather*}
From the second equation, we have that
\begin{align*}
xy+\xi^jyx\in\Pp_{1,da}(\BN(W)\sharp H)=\Pp_{1,da}(H)=\K\{1-da\},
\end{align*}
which implies that $xy+\xi^jyx=\mu(1-da)$ for some $\mu\in\K$. Then from the first equation, we get that
\begin{align*}
\Delta(x^2+\xi(1{+}\xi^j\Lam_4)\mu ba)&=x^2\otimes 1+a^2\otimes x^2+\xi(1+\xi^j\Lam_4)ba\otimes \mu(1-da)\\&\quad\,
+\xi(1+\xi^j\Lam_4)\mu ba\otimes da+\xi(1+\xi^j\Lam_4)\mu a^2\otimes ba\\
&=(x^2{+}\xi(1{+}\xi^j\Lam_4)\mu ba)\otimes 1+a^2\otimes(x^2{+}\xi(1{+}\xi^j\Lam_4)\mu ba),
\end{align*}
which implies that there exists $\nu\in\K$ such that
\begin{align*}
x^2+\xi(1+\xi^j\Lam_4)\mu ba=\nu(1-a^2).
\end{align*}
Since $ax^2=x^2a$, $bx^2=x^2b$ and $ab=\xi ba$, we get that $\mu=0=\nu$ and whence the relations $x^2=0$ and $xy+\xi^jyx=0$ must hold in $A$. Then after a tedious computation, we have that
\begin{align*}
\Delta(y^4)=\Delta(y)^4=y^4\otimes 1+1\otimes y^4,
\end{align*}
which implies that relation $y^4=0$ holds in $A$. Thus the claim follows.
\end{proof}
Now we define two families of Hopf algebras $\LA$ and $\LAA$ and show that they are indeed liftings of the Nichols algebras $\BN(V_{i,j,k,\iota})$ for $(i,j,k,\iota)\in \Lambda^5\cup\Lambda^6$.
\begin{defi}
For $\mu\in\K$ and $(i,j,k,\iota)\in \Lambda^5$, Let $\LA$ be the algebra generated by $x$, $y$, $a$, $b$, $c$, $d$ satisfying the relations
\begin{align*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd.\\
ax=-\Lam_4\xi xa,\quad bx=-\Lam_4\xi xb,\quad cx=\xi xc,\quad dx=\xi xd,\\
ay+\Lam_4 ya=\Lam_4xc,\quad by+\Lam_4 yb=\Lam_4xd,\quad
cy+yc=xa,\quad dy+yd=xb,\\
x^2+2\Lam_4y^2=\mu(1-a^2),\quad xy+\Lam_4yx=\frac{1}{2}(\Lam_4-\Lam_2)(\Lam_2+1)\mu ca,\quad x^4=0,
\end{align*}
the coalgebra structure is given by
\begin{align*}
\Delta(a)&=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)&=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(x)&=x\otimes 1+a^{j}\otimes x-(\Lam_2+\Lam_4)ba^{j-1}\otimes y,\\
\Delta(y)&=y\otimes 1+da^{j-1}\otimes y+\frac{1}{2}\xi(\Lam_2-\Lam_4)ca^{j-1}\otimes x.
\end{align*}
\end{defi}
\begin{rmk}
It is clear that $\Lambda^5(0)\cong \BN(V_{i,j,k,\iota})\sharp H$ for $(i,j,k,\iota)\in\Lambda^5$. Moreover, $\LA\cong T(V_{i,j,k,\iota})\sharp H/J^5$ where $J^5$ is the ideal generated by the elements the last row of the equations.
\end{rmk}
\begin{defi}
For $\mu\in\K$ and $(i,j,k,\iota)\in \Lambda^6$. Let $\LAA$ be the algebra generated by $x$, $y$, $a$, $b$, $c$, $d$ satisfying the relations
\begin{align*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd,\\
ax=-\xi xa,\quad bx=-\xi xb,\quad cx=\Lam_1 xc,\quad dx=\Lam_1 xd,\\
ay+ya=\Lam_4xc,\quad by+yb=\Lam_4xd,\quad
cy+\Lam_4yc=xa,\quad dy+\Lam_4yd=xb,\\
x^2+2\Lam_4y^2=\mu(1-a^2),\quad xy+\Lam_4yx=\Lam_4\mu ca,\quad x^4=0,
\end{align*}
the coalgebra structure is given by
\begin{align*}
\Delta(a)&=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)&=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(x)&=x\otimes 1+da^{j-1}\otimes x+(\Lam_2\Lam_4-1)ca^{j-1}\otimes y,\\
\Delta(y)&=y\otimes 1+a^{j}\otimes y-\frac{1}{2}(\Lam_2\Lam_4+1)ba^{j-1}\otimes x.
\end{align*}
\end{defi}
\begin{rmk}
It is clear that $\Lambda^6(0)\cong \BN(V_{i,j,k,\iota})\sharp H$ for $(i,j,k,\iota)\in\Lambda^6$. Moreover, $\LAA\cong T(V_{i,j,k,\iota})\sharp H/J^6$ where $J^6$ is the ideal generated by the elements the last row of the equations.
\end{rmk}
In the following Lemma, we show that $\LA$ and $\LAA$ are finite dimensional Hopf algebras over $H$.
\begin{lem}\label{lemLALAAoverH}
For $\mu\in\K$, $\ell=0,1$ and $(i,j,k,\iota)\in \Lambda^{\ell}$, $\Lambda^{\ell}(\mu)$ is finite-dimensional Hopf algebra over $H$.
\end{lem}
\begin{proof}
We prove the assertion for $\LA$, being the proof for $\LAA$ completely analogous.
Let $\Lambda_0$ be the subalgebra of $\LA$ for some $(i,j,k,\iota)\in\Lambda^5$ generated by the subcoalgebra $C=\K\{a,b,c,d\}$. We claim that $\Lambda_0\cong H$. Indeed, consider the Hopf algebra map $\psi:H\mapsto \LA$ given by the composition $H\hookrightarrow T(V_{i,j,k,\iota})\sharp H\twoheadrightarrow \LA\cong T(V_{i,j,k,\iota})\sharp H/J^5$. It is clear that $\psi(C)\cong C$ as coalgebras and $\psi(H)\cong \Lambda_0$ as Hopf algebras. From Remark $\ref{rmkHindependent}$ the elements $a$, $b$, $c$, $d$ are linearly-independent in $H$, thus they are also linearly-independent in $\LA$ which implies that $\dim\psi(H)> 4$. Then $\dim\psi(H)=8$ or $16$ by the Nichols-Zoeller Theorem. If $\dim\psi(H)=8$, then $\psi(H)$ must be the unique Hopf algebra of dimension $8$ without Chevalley property and it is impossible by the relations in \cite[Proposition\,2.1]{GG16}. Hence $\dim\psi(H)=16$ and $\psi(H)\cong H$ which implies that the claim follows.
Let $\Lambda_1=H\{x,y\}$, $\Lambda_2=\Lambda_1+H\{x^2,xy\}$, $\Lambda_3=\Lambda_2+H\{x^3,x^2y\}$ and $\Lambda_4=\Lambda_3+H\{x^3y\}$. After a direct computation, one can show that $\{\Lambda_{\ell}\}_{\ell=0}^4$ is a coalgebra filtration of $\LA$. Hence $(\LA)_0\subseteq H$ and $(\LA)_{[0]}\cong H$ i.e., $\LA$ is a Hopf algebra over $H$. Moreover, $\LA$ is a finite-dimensional Hopf algebra which is free as $H-$module with a set of generators $\{1,x,y,x^2,xy,x^3,x^2y,x^3y\}$ and $\dim \LA\leq 8\dim H=128$.
\end{proof}
Now we show that the algebras $\LA$ and $\LAA$ are liftings of of the bosonizations $\BN(V_{i,j,k,\iota})\sharp H$ for $(i,j,k,\iota)\in \Lambda^5$ and $\Lambda^6$. The proof is much similar with that in \cite[Lemma\;5.9]{GG16}.
\begin{lem}
For $\ell=5,6$, $\text{gr}\,\Lambda^{\ell}(\mu)\cong \BN(V_{i,j,k,\iota})\sharp H$ where $(i,j,k,\iota)\in\Lambda^{\ell}$.
\end{lem}
\begin{pro}\label{proLiftingnon5}
Let $A$ be a finite-dimensional Hopf algebra over $H$ such that the infinitesimal braiding $V$ is isomorphic either to $V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^5$. Then $A\cong \LA$ where $(i,j,k,\iota)\in \Lambda^5$.
\end{pro}
\begin{proof}
Note that $\text{gr}\; A\cong \BN(V_{i,j,k,\iota})\sharp H$, where $(i,j,k,\iota)\in \Lambda^5$. For convenience, let $W:=V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^5$.
In such a case, the bosonization $\BN(W)\sharp H$ is generated by $x, y, a, b, c, d$ satisfying the relations
\begin{align*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd.\\
ax=-\Lam_4\xi xa,\quad bx=-\Lam_4\xi xb,\quad cx=\xi xc,\quad dx=\xi xd,\\
ay+\Lam_4 ya=\Lam_4xc,\quad by+\Lam_4 yb=\Lam_4xd,\quad
cy+yc=xa,\quad dy+yd=xb,\\
x^2+2\Lam_4y^2=0,\quad x^4=0, \quad xy+\Lam_4yx=0.
\end{align*}
The coalgebra structure is given by
\begin{align*}
\Delta(a)&=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)&=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(x)&=x\otimes 1+a^{j}\otimes x-(\Lam_2+\Lam_4)ba^{j-1}\otimes y,\\
\Delta(y)&=y\otimes 1+da^{j-1}\otimes y+\frac{1}{2}\xi(\Lam_2-\Lam_4)ca^{j-1}\otimes x.
\end{align*}
Assume that $A$ is a finite-dimensional Hopf algebra such that $\text{gr}\,A\cong \BN(W)\sharp H$. After a direct computation, we have that
\begin{align*}
\Delta(x^2+2\Lam_4y^2)&=(x^2+2\Lam_4y^2)\otimes 1+a^2\otimes (x^2+2\Lam_4y^2),\\
\Delta(xy+\Lam_4yx)&=(xy+\Lam_4yx)\otimes 1+\frac{1}{2}(\Lam_2-\Lam_4)(\Lam_2+1)ca\otimes (v_1^2+2\Lam_4v_2^2)\\&\quad
+da\otimes(xy+\Lam_4yx).
\end{align*}
From the first equation, we have that
\begin{align*}
x^2+2\Lam_4y^2\in\Pp_{1,a^2}(\BN(W)\sharp H)=\Pp_{1,a^2}(H)=\K\{1-a^2\},
\end{align*}
which implies that $x^2+2\Lam_4y^2=\mu(1-a^2)$ for some $\mu\in\K$. Then from the second equation, we get that
\begin{align*}
\Delta(xy+\Lam_4yx+\omega ca)
&=(xy+\Lam_4yx)\otimes 1+da\otimes(xy+\Lam_4yx)+\omega ca\otimes (1-a^2)\\&\quad
+\omega ca\otimes a^2+\omega da\otimes ca\\
&=(xy+\Lam_4yx+\omega ca)\otimes 1+da\otimes (xy+\Lam_4yx+\omega ca),
\end{align*}
where $\omega=\frac{1}{2}(\Lam_2-\Lam_4)(\Lam_2+1)\mu $.
Thus there exists $\nu\in\K$ such that
\begin{align*}
xy+\Lam_4yx+\frac{1}{2}(\Lam_2-\Lam_4)(\Lam_2+1)\mu ca=\nu(1-da).
\end{align*}
Since
\begin{align*}
\nu(1-da)c&=\nu c(1-da),\\
c(xy+\Lam_4yx)&=cxy+\Lam_4yx=\xi xcy+\Lam_4cyx\\
&=\xi x(xa-yc)+\Lam_4(xa-yc)x=-\xi(xy+\Lam_4yx)c,
\end{align*}
we get that $\nu=0$ and whence $xy+\Lam_4yx=\frac{1}{2}(\Lam_4-\Lam_2)(\Lam_2+1)\mu ca$. Finally, for $x^4$, we have that
\begin{align*}
\Delta(x^4)=\Delta(x)^4=x^4\otimes 1+1\otimes x^4,
\end{align*}
which implies that the relation $x^4=0$ must hold in $A$.
\end{proof}
\begin{pro}\label{proLiftingnon6}
Let $A$ be a finite-dimensional Hopf algebra over $H$ such that the infinitesimal braiding $V$ is isomorphic either to $V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^6$. Then $A\cong \LAA$ where $(i,j,k,\iota)\in \Lambda^6$.
\end{pro}
\begin{proof}
Note that $\text{gr}\, A\cong \BN(V_{i,j,k,\iota})\sharp H$, where $(i,j,k,\iota)\in \Lambda^6$. For convenience, let $W:=V_{i,j,k,\iota}$ where $(i,j,k,\iota)\in \Lambda^6$.
In such a case, the bosonization $\BN(W)\sharp H$ is generated by $x, y, a, b, c, d$ satisfying the relations
\begin{align*}
a^4=1,\quad b^2=0,\quad c^2=0, \quad d^4=1,\quad a^2d^2=1,\quad ad=da,\quad bc=0=cb,\\
ab=\xi ba,\quad ac=\xi ca,\quad bd=\xi db,\quad cd=\xi dc,\quad bd=ca,\quad
ba=cd.\\
ax=-\xi xa,\quad bx=-\xi xb,\quad cx=\Lam_1 xc,\quad dx=\Lam_1 xd,\\
ay+ya=\Lam_4xc,\quad by+yb=\Lam_4xd,\quad
cy+\Lam_4yc=xa,\quad dy+\Lam_4yd=xb,\\
x^2+2\Lam_4y^2=0,\quad x^4=0, \quad xy+\Lam_4yx=0.
\end{align*}
the coalgebra structure is given by
\begin{align*}
\Delta(a)&=a\otimes a+b\otimes c,\quad \Delta(b)=a\otimes b+b\otimes d,\\
\Delta(c)&=c\otimes a+d\otimes c,\quad \Delta(d)=d\otimes d+c\otimes b,\\
\Delta(x)&=x\otimes 1+da^{j-1}\otimes x+(\Lam_2\Lam_4-1)ca^{j-1}\otimes y,\\
\Delta(y)&=y\otimes 1+a^{j}\otimes y-\frac{1}{2}(\Lam_2\Lam_4+1)ba^{j-1}\otimes x.
\end{align*}
Assume that $A$ is a finite-dimensional Hopf algebra such that $\text{gr}\,A\cong \BN(W)\sharp H$. After a direct computation, we have that
\begin{gather*}
\Delta(x^2+2\Lam_4y^2)=(x^2+2\Lam_4y^2)\otimes 1+a^2\otimes (x^2+2\Lam_4y^2),\\
\Delta(xy+\Lam_4yx)=(xy+\Lam_4yx)\otimes 1+da\otimes(xy+\Lam_4yx)-\Lam_4ca\otimes (v_1^2+2\Lam_4v_2^2).
\end{gather*}
From the first equation, we have that
\begin{align*}
x^2+2\Lam_4y^2\in\Pp_{1,a^2}(\BN(W)\sharp H)=\Pp_{1,a^2}(H)=\K\{1-a^2\},
\end{align*}
which implies that $x^2+2\Lam_4y^2=\mu(1-a^2)$ for some $\mu\in\K$. Then from the second equation, we get that
\begin{align*}
\lefteqn{\Delta(xy+\Lam_4yx-\Lam_4\mu ca)}\hspace*{2mm}\\
&=(xy+\Lam_4yx)\otimes 1+da\otimes(xy+\Lam_4yx)-\Lam_4ca\otimes \mu(1-a^2)\\&\quad
-\mu\Lam_4ca\otimes a^2-\mu\Lam_4da\otimes ca\\&
=(xy+\Lam_4yx-\Lam_4\mu ca)\otimes 1+da\otimes (xy+\Lam_4yx-\Lam_4\mu ca),
\end{align*}
which implies that there exists $\nu\in\K$ such that
\begin{align*}
xy+\Lam_4yx-\Lam_4\mu ca=\nu(1-da).
\end{align*}
After a direct computation, we have that $c(xy+\Lam_4yx)=-\xi(xy+\Lam_4)c$ and $c(1-da)=(1-da)c$ which implies that $\nu=0$ and whence $xy+\Lam_4yx=\Lam_4\mu ca$. Finally, for $x^4$ we have that
\begin{align*}
\Delta(x^4)=\Delta(x)^4=x^4\otimes 1+1\otimes x^4,
\end{align*}
which implies that the relation $x^4=0$ holds in $A$.
\end{proof}
Finally, we have the classification of finite-dimensional Hopf algebras over $H$ such that their infinitesimal braidings are simple objects in $\HYD$.
\begin{proofthmb}
Since $A$ is a finite-dimensional Hopf algebra over $H$, that is, $A_{[0]}\cong H$, by Lemma $\ref{lemNicholsgeneratedbyone}$, we get that $\text{gr}\,A=\BN(V)\sharp H$. If $V$ is isomorphic either to $\K_{\chi_{i,j,k}}$ with $(i,j,k)\in\Lambda^0$, or $V_{i,j,k,\iota}$ with $(i,j,k,\iota)\in\Lambda^3\cup\Lambda^4$, then $A$ is isomorphic either to $\bigwedge\K_{\chi_{i,j,k}}\sharp H$ with $(i,j,k)\in\Lambda^0$, or $\BN(V_{i,j,k,\iota})\sharp H$ with $(i,j,k,\iota)\in \Lambda^3\cup\Lambda^4$ by Proposition $\ref{proLiftingnon0}$, $\ref{proLiftingnon3}$ and $\ref{proLiftingnon4}$.
If $V$ is isomorphic either to $V_{i,j,k,\iota}$ with $(i,j,k,\iota)\in\Lambda^5\cup\Lambda^6$, then $A$ is isomorphic either to $\LA$ or $\LAA$, with $(i,j,k,\iota)\in \Lambda^3\cup\Lambda^4$ by Propositions $\ref{proLiftingnon5}$ and $\ref{proLiftingnon6}$. And these Hopf algebras are pairwise non-isomorphic since their infinitesimal braidings are pairwise non-isomorphic as Yetter-Drinfeld modules over $H$.
\end{proofthmb}
\vskip10pt \centerline{\bf ACKNOWLEDGMENT}
\vskip10pt The paper is supported by the NSFC (Grant No. 11271131).
The authors are grateful to Andruskiewitsch for his kind comments. | 50,088 |
Craig Freshwater, explains how in this new blog entry, PHP can be used to template your site with a few quick and easy steps.
PHP can be very useful in your website.
He tells it more from an "outsider's perspective", for someone not really familiar with PHP. There's a lot of introductory explaination and installation instructions on getting PHP working, but he does eventually get to the (very) simple templating example. | 24,057 |
Video News: United States Defense arrives for retrial Raffaele Sollecito arrived at a Florence court on Thursday (January 9) as his defense team prepares to deliver the final summing up in a retrial of the murder case into the death of British student Meredith Kercher. Sollecito was accompanied by his father as he arrived at the courtroom. He has pleaded for the judge to uphold his acquittal in the case. American student Amanda Knox, 26, and her Italian former boyfriend Sollecito, 29, were convicted of murdering Kercher in 2009, in a verdict overturned in a subsequent trial. Now the case is being tried again in a courtroom in Florence after Italy's Supreme Court quashed the acquittals in March, citing inconsistencies in the case. Both Knox and Sollecito have always maintained their innocence. Kercher's half-naked body was found in 2007 with more than 40 stab wounds and a deep gash in her throat in the apartment she shared with Knox in Perugia, a picturesque town where both were studying as university exchange stude © DailyMotion - Thursday, January 9, 2014 | 311,945 |
TITLE: Solve the PDE by the method of characteristics.
QUESTION [1 upvotes]: I am trying to figure out where my solution went wrong. I am off by a factor of two.
$$ u_x + u_y + u = e^{x+2y}$$
I first found that the characteristic curves are determined by $$\frac{dy}{dx} = 1 \implies y-x = C.$$
I then solved the ODE $$\frac{du}{dx} + u = e^{x+2y}$$
I found $u = \frac12 e^{x+2y} + e^{-x}K(C)$ giving $$u = \frac{e^{x+2y}}{2} + e^{-x}F(y-x)$$ as the general solution where $F$ is an arbitrary function. Where does my work go wrong?
The end solution should be half of what it currently is.
REPLY [1 votes]: $$\begin{align}
u_x + u_y + u &= e^{x + 2y} \\
\implies u_x + u_y &= e^{x + 2y} - u \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*)\\
\end{align} $$
Setting $u = u(x(s),y(s))$ we find
$$\begin{align}
\frac{d}{ds} u &= \frac{\partial u}{\partial x} \cdot \frac{dx}{ds} + \frac{\partial u}{\partial y} \cdot \frac{dy}{ds} \\
&= \frac{\partial u}{\partial x} \cdot 1 + \frac{\partial u}{\partial y} \cdot 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) \\
&= e^{x + 2y} - u
\end{align}$$
Where $(1)$ comes from our original PDE at $(*)$. Equating, we find
$$\begin{align}
\frac{dy}{ds} &= 1 \\
\frac{dx}{ds} &= 1 \implies \frac{dx}{dy} = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) \\
\frac{du}{ds} &= e^{x + 2y} - u \implies \frac{du}{dy} + u = e^{x + 2y} \ \ \ \ \ \ \ \ \ \ \ \ \ (3) \\
\end{align}$$
Solving $(2)$ and $(3)$
$$x(y) = x_0 + y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$$
$$\begin{align}
\frac{du}{dy} + u &= e^{x + 2y} \\
&= e^{x_0 + 3y} \\
\implies (e^{y}u)' &= e^{x_0 + 4y} \\
\implies e^{y}u &= \frac{e^{x_0 + 4y}}{4} + f(x_0) \\
\implies u &= e^{-y} \bigg(\frac{e^{x_0 + 4y}}{4} + f(x_0) \bigg) \\
&= \frac{e^{x_0 + 3y}}{4} + e^{-y}f(x_0) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5) \\
\end{align}$$
and using $(4) \implies x_0 = x - y$ we find
$$u = \frac{e^{x + 2y}}{4} + e^{-y}f(x - y)$$ | 53,370 |
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TITLE: How do I prove convergence from multiple dependent sequences?
QUESTION [0 upvotes]: How do I prove that $a_n$ and $b_n$ converges if and only if
$a_n - b_n$ and $a_n + b_n$ converges?
REPLY [1 votes]: ($\Rightarrow$). Let $a_n$ and $b_n$ be two convergent sequences. Then, $\lim_{n \to \infty}a_n = L$ and $\lim_{n \to \infty}b_n = M$. Now consider the sequence $c_n = a_n + b_n$. Then, $$\lim_{n \to \infty} c_n = \lim_{n \to \infty}(a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty}b_n = L + M,$$ hence $c_n$ converges. Can you prove the other case yourself?
($\Leftarrow$). Given is that the sequences $a_n+b_n$ and $a_n-b_n$ converge. Then, \begin{align}
\lim_{n \to \infty}a_n & = \lim_{n \to \infty} \frac{1}{2}(2a_n) \\
& = \lim_{n \to \infty} \frac{1}{2}(a_n + b_n + a_n - b_n) \\
& = \lim_{n \to \infty} \frac{1}{2}(a_n + b_n) + \lim_{n \to \infty} \frac{1}{2} (a_n - b_n) \\
& = \frac{1}{2}\lim_{n \to \infty}(a_n + b_n) + \frac{1}{2}\lim_{n \to \infty}(a_n - b_n) \\ & = \frac{1}{2}(L_1 + L_2), \end{align} so $a_n$ converges. Can you do the proof for $b_n$ yourself? | 205,014 |
Addressing ‘hidden costs’ in consolidating super
A client relationship manager has urged advisers to be mindful of the hidden costs associated with consolidating clients’ super accounts.
As the new financial year nears, Sean Johnston, client relationship manager at Heffron, said it’s worth considering the benefits of consolidating clients’ super, paying extra attention to any financial headwinds.
“As we are hurtling full speed towards 30 June again, many of us are taking the opportunity to simplify our financial affairs. It stands to reason that many of our clients are also doing the same,” Mr Johnston said in a recent blog post.
“... Whilst an ordinary rollover between funds doesn’t usually have too many complications (SuperStream notwithstanding), there are a number of cases where rolling over certain types of superannuation benefits can create some unforeseen problems.”
A key area to be mindful of surrounds untaxed funds, according to Mr Johnston.
“Certain types of superannuation funds (usually funds related to government bodies or defence-related superannuation) don’t, and in fact can’t, pay tax when members contribute or when they generate earnings along the way. We call these funds ‘untaxed funds’,” he explained.
“To make up for the fact that these funds aren’t taxed along the way, the members in these funds pay tax on their withdrawals when they start taking money out of the fund.
“Unfortunately, a rollover of these benefits to another fund counts as a withdrawal and tax will be levied on the amount rolled over.
“In 2021/22, the first $1,615,000 of any rollover will be taxed at 15 per cent when it enters the new fund – this amount is known as the untaxed plan cap amount. Anything above this amount will be taxed at 47 per cent in the hands of the transferring fund before the net benefit is rolled over to the new fund (as a small concession the residual 53 per cent is rolled over as a tax-free component).”
According to Mr Johnston, another type of superannuation fund that can have some hiccups is a defined benefit fund.
“A defined benefit fund is one where the member does not have an account balance in the fund. Rather their retirement benefit is determined by a formula,” he said.
“Because a member of a defined benefit interest does not have an account balance, it is practically impossible for the government to apply the same rules as it would in a normal fund.
“One example of this is Division 293 tax. Div 293 tax is a modified rate of tax on super contributions for high income earners. In practice it means that people with income over $250,000 pay tax at 30 per cent on their contributions rather than 15 per cent. The first 15 per cent tax is levied to the fund, and the second 15 per cent is levied to the individual. In normal funds, the member can apply to withdraw enough from super to cover this second 15 per cent.
“Because members of defined benefit funds don’t have an account balance from which to deduct money to pay a Div 293 tax assessment, they are often given the choice: Pay the extra tax PERSONALLY as they go, or accrue the tax against their final benefit.”
If a client is accruing the tax against their defined benefit account, then they will need to pay this when they start to make withdrawals, Mr Johnston noted. | 94,563 |
Runes; An Introduction
Manchester: University Press (1971), 1971. Third printing. Hardcover. 8vo. [6], vii-xvi,124, [4], xxiv (pages of plates), pp. Green cloth with gold lettering on the spine. Price of £1.80 net on the front flap of the dust jacket. Illustrated with twenty-four black and white photograph plates of Dark Age artifacts and runic inscriptions from Scandinavia and Great Britain. A study of Anglo-Saxon runes, which were far fewer in number than rune inscriptions remaining in Scandinavia from the sixth century A.D. and onward. Elliott's book also gives context to a Europe which was shifting from Pagan beliefs to the Christian religion. Very Good / Very Good+. Item #00009359
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Price: $25.00 | 155,897 |
At our congregation, we get the great privilege of learning Paul Gerhadt’s hymn Why Should Cross and Trial Grieve Me during the month of September. Some background on Paul Gerhardt can add more meaning to this text. Gerhardt was a German Lutheran pastor in the 17th century. He survived the horrors of the Thirty Year’s War, four out of his five children did not survive childhood, and his wife died when his surviving child was only 6 years old. As a pastor, he remained steadfast against the pressures of his day and this cost him his position at St. Nicholas’ Church in Berlin. He refused to sign a document agreeing not to teach on subjects where Calvinists and Lutherans disagreed, and he was fired for this.
Out of this great suffering, Gerhardt emerges as one of the great writers of Christian hymnody. His hymns are comforting and personal, yet they remain biblically literate hymns that proclaim solid Scriptural teaching.
Stanza 1:
Why should cross and trial grieve me?
Christ is near
With His cheer;
Never will He leave me.
Who can rob me of the heaven
That God’s Son
For me won
When His life was given?
What is a cross? Jesus told us to take up our crosses and follow Him, after all. A cross is when we suffer in this life. More precisely, it is when we suffer as a consequence of living out the Christian faith. God allows these sufferings into our lives in order to refine our faith in much the same way as precious metal is refined in the fire. Peter speaks of these things in 1st Peter chapter 1. This is a longer section, but it is worth reading:.
As St. Peter says, we have a living hope because of Jesus’ resurrection from the dead and the inheritance that is in heaven for us. God allows various difficult periods, challenges, and events into our lives that test our faith, prove it to be genuine, and lead us to rejoice with a joy that is inexpressible. Nothing can rob us of this great treasure.
Stanza 2:
When life’s troubles rise to meet me,
Though their weight
May be great,
They will not defeat me.
God, my loving Savior, sends them;
He who knows
All my woes
Knows how best to end them.
Yes, this stanza does say that God, our Savior, sends our troubles into our life. How can this be? First of all, remember that God is not the ultimate source of evil and trouble; that is the result of sin, death, and the devil. If you look into the book of Job, however, you will see that God does allow Satan to bring various trials into our lives. What are we to do, then? Are we to fear God and ask Him to leave us alone so that these trials will also cease? No! God cares for you! Jesus died for you to reconcile you to Himself! He cares for you as a Father. Rather, bring all your cares and anxieties to Him, because He cares for you, loves you, and will only bring suffering into your life for your own good. Remember, the last line of stanza 2 and know that God will also bring an end to your sufferings.
8And we know that for those who love God all things work together for good, for those who are called according to his purpose. (Romans 8:28))
29Take my yoke upon you, and learn from me, for I am gentle and lowly in heart, and you will find rest for your souls. 30For my yoke is easy, and my burden is light.” (Matthew 11)
Stanza 3:
God gives me my days of gladness,
And I will
Trust Him still
When He sends me sadness.
God is good; His love attends me
Day by day
Come what may,
Guides me and defends me.
With the assurance of the promises of the first two stanzas well in hand, we can boldly join in this proclamation of faith in stanza 3. God is good. When the sadness of life comes, remember that God is good, He loves you, defends you, and guides you. We can rejoice at all times, as James exhorts us to do in James 1:
2 Count it all joy, my brothers, when you meet trials of various kinds, 3for you know that the testing of your faith produces steadfastness. 4And let steadfastness have its full effect, that you may be perfect and complete, lacking in nothing.
Stanza 4:
From God’s joy can nothing sever,
For I am
His dear lamb,
He, my Shepherd ever.
I am His because He gave me
His own blood
For my good,
By His death to save me.
The faith and comfort springing from this stanza are absolutely incredible. It gives us a reason why we can trust that God will never fail us. It’s because our Savior, our Shepherd, has saved us and made us His own by His death! Jesus says as much in John 10:
11 I am the good shepherd. The good shepherd lays down his life for the sheep.
Paul says much the same in Romans 8:
31What then shall we say to these things? If God is for us, who can be against us? 32 He who did not spare his own Son but gave him up for us all, how will he not also with him graciously give us all things?
Now in Christ, death cannot slay me,
Though it might,
Day and night,
Trouble and dismay me.
Christ has made my death a portal
From the strife
Of this life
To His joy immortal!
Having battled the sorrows and trials of this life, we now come face to face with death itself. This great enemy can do nothing to hurt us, for Christ has already won the victory. We are now blessed to know that when we die we enter into eternal joy, where there will be no more trials, no more crosses to bear, nor more sin, and no more sorrow.
42 So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. 43It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power.
56The sting of death is sin, and the power of sin is the law. 57But thanks be to God, who gives us the victory through our Lord Jesus Christ. (1 Corinthians 15)
Yes, thanks be to God! | 414,917 |
Active forest fires in Central Africa are greater than the Amazon fires, and have a potential to be more dangerous, Greenpeace has warned.
Termed “the fire continent” by NASA, Africa is home to at least 70% of fires burning worldwide, during the global fire season between June and September.
The fires, which spread from Angola to Gabon, are close to the Congo Basin, the second-largest tropical rainforest in the globe, which spans the Democratic Republic of Congo, Congo, Equatorial Guinea, Gabon, Cameroon and the Central Africa Republic. Like the Amazon, the Congo Basin absorbs a large percent of the earth’s carbon dioxide, and is responsible for abating the negative effects of climate change.
The Global Forest Watch shows massive fires currently burning in Central and Southern African nations. Fires in both continents are as a result of “slash and burn” farming where land is cleared using fire. The farming method requires natural vegetation to be cut down and burned to clear the land for cultivation. Rainforest Saver describes it as a form of “shifting agriculture” that rapidly leads to poverty, infertile lands, climate change and the destruction of the rainforest, including wildlife. Once farming strips the soil of its nutrients, rendering it infertile, farmers move to a new land area and repeat the process.
The worst of Africa’s fires is contained in Angola, Zambia, Malawi, Mozambique, Madagascar and South Africa. Although fires in Central Africa have not spread to the Congo Basin forest area, fires in South America have spread to the Amazon rainforest, prompting problems for the people and the biodiverse wildlife for whom this is home.
Slash and burn techniques have been criticised by experts. Many, including World Agroforestry and Rainforest Saver, have published environmental alternatives to slash and burn. “The richness of the rainforest is in the trees”, Rainforest Saver has argued. However, with the Amazon rainforest burning, the earth is losing a big part of its richness, including its biodiversity. Despite African forest fires being substantially higher in number, the global focus on the Amazon rainforest stems from a concern that the fires there have not been controlled, unlike African fires.
Working on Fire, under South Africa’s Department of Environmental Affairs, writes that: “Fires are, and always have been, a part of the South African landscape. They occur as a natural phenomenon in grasslands, woodlands, fynbos, and sometimes in indigenous forests.
.”
Despite similarities in agricultural techniques, fire management in both regions is the difference between safe and unsafe forest fires.
“There are fire management questions in these (African) ecosystems, but fire is part of their ecology,” Sally Archibald, a professor at the School of Animal, Plant and Environmental Sciences, at The University of Witwatersrand, Johannesburg, said.
“In South America, the equivalent non-forest woodlands have been largely converted to soybean agriculture already, but in Africa they are largely untransformed.
“The main message is: yes we have a lot of fire, but it’s not bad and can be very good for the ecology. We don’t know how many deforestation fires we have but the best evidence is that our forests are not decreasing, they are in fact increasing.”
While the slash and burn farming practice was sustainable in the past, rising population makes it impossible to abandon burnt land for two decades while it replenishes its nutrients. Other problems associated with rising population, such as rising temperatures, decreased rain from climate change and industrialised practices create the perfect circumstances for slash and burn fires to burn out of control.
Experts, consequently, are now focused on the Amazon’s forest fires, over Africa’s fires. “The question now is to what extent we can compare,” opined Philippe Verbelen, a Greenpeace forest campaigner.
Last month, French president, Emmanuel Macron, voiced concern that the Central African forest fires were not receiving as much media attention as the Amazon forest fires. In the same month, the G7 nations pledged $20 million to the Amazon fires, sending fire-fighting aircraft and firefighter volunteers.
Bolivian authorities have warned that fires in the Chiquitania forest area have left more than a quarter of the country under “extreme risk from the forest fires”. The fires have also destroyed parts of the Amazon and Bolivia’s Pantanal region which borders Bolivia, Brazil and Paraguay.
Bolivian President Evo Morales, who previously rejected international help, has last week, accepted foreign aid efforts.
“Any cooperation is welcome, whether it comes from international organisations, celebrities or from the presidents who offered to help,” he said. He also recently cancel his re-election campaign to join the firefighting efforts.
Data from Brazil’s Space Research Centre (INPE) proves there has been an 83% increase in forest fires in the Amazon since records began in 2013. “The number of forest fires is higher in the Amazon regions most affected by deforestation practices, as fires are one of the main tools used for deforestation, including by farmers,” Greenpeace said.
“Forest fires and climate change operate in a vicious cycle: as the number of fires increase, so do greenhouse gas emissions, increasing the planet’s overall temperature and the occurrence of extreme weather events, such as major droughts,” the NGO continued.
Morales criminalised the slash and burn farming practice last month, despite his very government permitting it in the first place to create revenue from expanding agro-business in Bolivia.
“This month’s devastating fires are the all-too-predictable consequence of the Morales government’s decree authorising new land claims on cleared land”, said Carwil Bjork-James, an anthropology professor at Vanderbilt University.
The Pan-Amazon indigenous organisation COICA has accused Morales and Brazilian president, Jair Bolsorano, of “gutting every environmental and social strategy” designed to protect the Amazon, for the benefit of huge capitalistic profit.
In response to the President Macron’s concerns, President Bolsorano accused the French president of treating the Amazon region “as if it were a colony”.
“We cannot accept that a president, Macron, unleashes unreasonable and gratuitous attacks on the Amazon, nor that he disguises his intentions behind the idea of a G-7 ‘alliance’ to ‘save’ the Amazon, as if we were a colony or a no-man’s-land,” Bolsonaro tweeted.
Western arguments against South America’s governmental policies in the Amazon has led to criticisms of ‘environmental colonialism‘. South American governments are sceptical about environmental aid and advice from Western governments, because of the West’s unethical history with environmentalism.
By prioritising financial gain over global environmental concerns, the United States and Western Europe saw great economic successes in the twentieth and twenty-first century. Under European colonisation, for example, 21.7% of Africa’s tropical rainforest was deforested. Today, US President Trump continues to deny climate change as a real problem, attempting to repeal 84 climate change regulations that stem the profits of ultra-wealthy American businesspersons.
The Amazon forest fires reveal a great global power shift, with the fate of the planet resting on the survival of a continent that the West has historically plundered and destroyed for its benefit. How South America responds to the protection of the Amazon in the future will be a great determining factor in whether we can survive, and even reverse climate change. | 412,155 |
TITLE: Determining center of Ellipse with limited Data Points
QUESTION [0 upvotes]: The dataset I am using only has 200 degrees of the ellipse. The ellipse is not centered at (0,0). The data in this case ranges from 110 degrees to 310 degrees. I need to determine the center of the partial ellipse so that I can calculate the x and y radius to complete the ellipse.
I have tried using the Max and Min values of the dataset to determine the center, but that does not work correctly; (Max + Min)/2
The main focus is to complete the ellipse, so if someone has a better approach please let me know. I know if I obtain the center of the ellipse I can complete the ellipse, so that is the approach I have taken.
REPLY [0 votes]: You could represent your ellipse as a general conic: $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ with unknown $A$, $B$, $C$, $D$, $E$, $F$. Plug in five points $(x,y)$, and you have enough equation to compute the coefficients, up to the obvious homegeneity. If you have more than five points, and there is some measurement uncertainty, you could solve the overdetermined system as a least squares problem instead. | 29,452 |
Key to the Past
Anne Ritchie Ware Waring ’46 donates her mother's 1917 Phi Beta Kappa key. Cornelia Frost Ware ’17 was one of the College’s first five inductees into the nation’s most prestigious academic honor societies.
“The College means so much to us. My mother loved it there and it just seems right to have it return home.”
From the President
Randolph College will never forget its past as Randolph-Macon Woman's College. Richard Dawkins event.
Pay It Forward: Giving Back
Retired Captain’s support is tribute to.
The Randolph College Board of Trustees
Liberal arts education provides a foundation for Carolyn Burgess Featheringill ’69.
View Randolph on your iPhone, Blackberry or other mobile device. Visit > > >
View Randolph in Flash e-Magazine format or download a PDF version. > > > | 316,971 |
\begin{document}
\title{A Resonance Problem in Relaxation of Ground States of Nonlinear Schr\"odinger Equations}\author{Zhou Gang\footnote{Partially supported by NSF grant DMS 1308985, 1443225} \
} \maketitle
\centerline{\small{ Department of Mathematics, California Institute of Technology, Mail Code 253-37,
Pasadena, California, 91125 U.S.A.}}
\setlength{\leftmargin}{.1in}
\normalsize \vskip.1in \setcounter{page}{1}
\setlength{\leftmargin}{-.2in} \setlength{\rightmargin}{-.2in}
\section*{Abstract}
In this paper we consider a resonance problem, in a generic regime, in the consideration of relaxation of ground states of semilinear Schr\"odinger equations. Different from previous results, our consideration includes the presence of resonance, resulted by overlaps of frequencies of different states. All the known key results, proved under non-resonance conditions, have been recovered uniformly. These are achieved by better understandings of normal form transformation and Fermi Golden rule. Especially, we find that if certain denominators are zeros (or small), resulted by the presence of resonances (or close to it), then cancellations between terms make the corresponding numerators proportionally small. \vfil\eject
\tableofcontents
\section{Introduction}
We consider the following 3-dimensional semilinear Schr\"odinger equations
\begin{align}
i\partial_{t}\psi(x,t)=&-\Delta\psi(x,t)+V(x)\psi(x,t)+|\psi(x,t)|^{2}\psi(x,t), \label{eq:NLS}\\
\psi(x,0)=&\psi_{0}(x)\in \mathcal{H}^2(\mathbb{R}^3)\ \text{small}.\nonumber
\end{align}
where $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is
the external potential.
Such equations arise in the theory of Bose-Einstein condensation,
nonlinear optics, theory of water waves and in other areas.
We start with formulating the problem. The potential $V:\ \mathbb{R}^{3}\rightarrow \mathbb{R}$ is a smooth, and rapidly decaying function. And if it is trapping potential, namely
\begin{align}
\inf_{\|f\|_2=1}\langle f,\ (-\Delta+V)f\rangle=-e_0<0,
\end{align} then this linear unbounded self-adjoint operator $-\Delta+V$, mapping $\mathcal{L}^2$ into $\mathcal{L}^2$,
has a ground state $\phi\in \mathcal{L}^2$ with eigenvalue $-e_0$. Moreover the eigenvalue must be simple. Besides the ground states the linear operator might have some other finitely many neutral modes with nonpositive eigenvalues $-e_k ,\ k=1,\cdots, N$.
Its continuous spectrum spans the interval $[0,\ \infty).$
It is well known that for the type of potential $V$ we chose, there is no positive eigenvalues, see e.g. \cite{RSIII}.
In the nonlinear setting, the ground state bifurcates into a family of solitary wave solutions, see e.g. \cite{TsaiYau02},
\begin{align}
\psi(x,t)=e^{i\lambda t}\phi^{\lambda}(x)
\end{align} with $\lambda\in \mathbb{R}$ being close to $e_0$ and $\phi^{\lambda}=C \sqrt{|e_0-\lambda|}\phi+O(|e_0-\lambda|^{\frac{3}{2}}).$
There is a rich literature on studying the orbital stability and
asymptotic stability of the soliton manifold. By results in
\cite{Wein1985, Wei86,GSS87} it is well known that that the ground state manifold is orbital stable in the
$\mathcal{H}^{1}$ space. After these, many attempts were made on
proving the asymptotic stability of the ground state manifold, see e.g. \cite{BP1, RoWe,TsaiYau02, MR1992875, BuSu, GS07, Cuccagna:03, SW-PRL:05, G1, NakanTsai12, ZwHo07}.
In \cite{BP1, RoWe,BuSu,TsaiYau02, MR1992875, SW-PRL:05}, it is assumed that the linear operator $-\Delta+V$ has a ground state (with eigenvalue $-e_0<0$), and only one simple neutral mode with eigenvalue $-e_1<0$ satisfying $2e_1<e_0.$
In \cite{Tsai2003}, multiple neutral modes was considered. Their eigenvalues $-e_k,\ k=1,\cdots,N$ must satisfy two conditions: (1) $2e_k< e_0$, and (2) the so-called non-resonance condition, namely there do not exist $n_k\in \mathbb{Z}, \ k=0,1,\cdots, N$ such that
\begin{align}
\sum |n_k|\not= 0\ \text{and}\ \sum_{k} n_k e_k=0,\label{eqn:nonRe1}
\end{align} see also the non-resonance conditions for multiple neutral modes in \cite{Cucca08, NakanTsai12}.
On the technical level, the condition \eqref{eqn:nonRe1} was needed to prevent small denominator from appearing.
In \cite{GaWe, GaWe2011}, the author, together with M. Weinstein, improved the above results by studying degenerate neutral modes, i.e.
\begin{align}
e_k=e_1, k=1,2,\cdots, N,\label{eqn:nonRe2}
\end{align} or nearly degenerate.
The main purpose of the present paper is to include the presence of the resonance, specifically
by removing the conditions \eqref{eqn:nonRe1} and \eqref{eqn:nonRe2}, and to show all the proved results still hold, uniformly.
A graphic illustration is in Figure \ref{abc}.
\begin{figure}[!htb]
\centering
\includegraphics[width=11.5cm, height=2.5cm]{axisSchro.eps}
\caption{spectrum of $-\Delta+V$}
\label{abc}
\end{figure}
On the technical level, we achieve this by re-defining normal form transformation and Fermi Golden rule. Especially we show that if some denominators are small, caused by the resonances, then their corresponding numerators are proportionally small, resulted by cancellations between terms. For the details, we refer to Sections \ref{sec:FGR} and \ref{NormalForms}, and Proposition \ref{prop:smallDivisor}.
To the best of our knowledge, our result and techniques are new.
{\bf{Related problems.}} The main motivation of the present work is to understand certain types of small divisor problems. In many literatures, for example in \cite{BourWang2008, Bene1988, Bovier86, BourgWang04}, in studying problems arising from statistical mechanics and PDE, the small divisor was avoided, by choosing initial conditions, for example.
The technical advantage in studying the present problem is that we only need to expand the solutions, in certain small parameters, finitely many times (two times), instead of infinitely many times as in \cite{BourWang2008}. We expect the normal form transformation invented here, which makes it easy to see the crucial cancellations between terms, together with algebraic structures observed in \cite{GS07, G1, Cucca08}, for higher order iterations, can be applied to the other problems. These will be addressed in subsequent papers.
{\bf{Structure of the paper.}}
The paper is organized as follows. Some basic properties of the equation are studied in Section \ref{HaGWP}. The linearized operator $L(\lambda)$, obtained by linearizing the ground state solution, is studied in Section \ref{sec:OperL}. The Fermi Golden rule condition is in Section \ref{sec:FGR}. The main Theorem is stated in Section \ref{MainTHM}. The decomposition of the solutions, and the governing equations for various parameters and functions are in Section \ref{SEC:effective}. The main Theorem is reformulated into Theorem \ref{GOLD:maintheorem} in Section \ref{sec:refor}. The proof of different parts of Theorem \ref{GOLD:maintheorem} are in Sections \ref{sec:tildeR}, \ref{NormalForms} and \ref{sec:NFT}. The main theorem is proved in Section \ref{ProveMain}.
Most parts of the paper follow the steps in the previous papers \cite{GaWe, GaWe2011}. Technically the main differences are in normal form transformation in Section \ref{NormalForms} and subsequent sections.
\section*{Acknowledgments} The author wishes to thank Avy Soffer and Wei-Min Wang for a discussion on small divisor problems.
The author is partly supported by NSF Grant DMS 1308985, 1443225.
\newpage
\section{Notation}\label{notation}
\begin{itemize}
\item[(1)]\
$
\alpha_+ = \max\{\alpha,0\},\ \ [\tau]=\max_{\tilde\tau\in Z}\ \{\tilde\tau\le\tau\}
$
\item[(2)]\ $\Re z$ = real part of $z$,\ \ $\Im z$ = imaginary part of $z$
\item[(3)] Multi-indices
\begin{align}
z&=(z_1,\dots, z_N)\ \in C^N,\ \bar{z}=(\overline{z_1},\dots, \overline{z_N})\\
&a\in \mathbb{N}^{N},\ z^a=z_1^{a_1}\cdot\cdot\cdot z_N^{a_N}\nonumber\\
|a|\ &=\ |a_1|\ +\ \dots\ +\ |a_N|
\nonumber
\end{align}
\item[(4)]\ $\mathcal{H}^s$\ =\ Sobolev space of order $s$
\item[(5)] \begin{equation}
J\ =\ \left(\begin{array}{cc} 0 & 1\\ -1 & 0\end{array}\right),
\ \ H\ =\ \left(\begin{array}{cc} L_+ & 0\\ 0 & L_-\end{array}\right),\ \
L=JH=\left(\begin{array}{cc} 0 & L_-\\ -L_+ & 0\end{array}\right)
\nonumber\end{equation}
\item[(6)] $\sigma_{ess}(L)=\sigma_c(L)$ is the essential (continuous) spectrum of $L$,\\ $\sigma_d=\C - \sigma_c(L)$ is the discrete spectrum of $L$.
\item[(7)]\ $P_d(L)$ bi-orthogonal projection onto the discrete spectral part of $L$
\item[(8)]\ $P_c(L)=I-P_d(L)$, bi-projection onto the continuous spectral part of $L$
\item[(9)]\ $\langle f,g\rangle = \int\ f\ \bar{g} $
\item[(10)]\ $\| f\|_p=$ $L^p$ norm,\ \ $1\le p\le\infty$.
\end{itemize}
\vfil\eject
\section{Basic Properties}\label{HaGWP}
Equation \eqref{eq:NLS} is a Hamiltonian system on Sobolev space
$\mathcal{H}^{1}(\mathbb{R}^{3},\mathbb{C})$ viewed as a real space
$\mathcal{H}^{1}(\mathbb{R}^{3},\mathbb{R})\oplus
\mathcal{H}^{1}(\mathbb{R}^{3},\mathbb{R})$. The
Hamiltonian functional is: $$\mathcal{E}(\psi):=\int
[\frac{1}{2}(|\nabla\psi|^{2}+V|\psi|^{2})+\frac{1}{4}|\psi|^{4}].$$
Equation \eqref{eq:NLS} has the time-translational and gauge
symmetries which imply the following conservation laws: for any
$t\geq 0,$ we have
\begin{enumerate}
\item[(CE)] Conservation of energy:\ \ \
$\mathcal{E}(\psi(t))=\mathcal{E}(\psi(0));$
\item[(CP)]
Conservation of particle number:\\
$\mathcal{N}(\psi(t))=\mathcal{N}(\psi(0)),$ where\ \ \ \
$\mathcal{N}(\psi):=\int
|\psi|^{2}.$
\end{enumerate}
In what follows we review the results of the existence of
soliton and their properties.
The following arguments are almost identical
to those in \cite{RoWe,BuSu,TsaiYau02} except that here we have multiple neutral modes (or
excited states), hence we state the results
without proof. We assume that the linear operator $-\Delta+V$ has
the following properties
\begin{enumerate}
\item[(NL)]
The linear operator $-\Delta+V$ has eigenvalues $-e_{0}<-e_{1}\leq \cdots\leq -e_{N}$
satisfying $e_{0}<2e_{1}$. $-e_{0}$ is the lowest eigenvalue with
ground state $\phi>0$, the eigenvalue $-e_{1},\ \cdots,\ -e_{N}$ might be degenerate with eigenvectors
$\xi_{1}^{lin},\xi_{2}^{lin},\cdot\cdot\cdot,\xi_{N}^{lin}.$
\end{enumerate}
In the nonlinear setting the ground state bifurcates into a family of solitary wave solutions of \eqref{eq:NLS}, see e.g. \cite{TsaiYau02},
\begin{align}
\psi(x,t)=e^{i\lambda t}\phi^{\lambda}(x)\label{eq:soli2}
\end{align}
and the function $\phi^{\lambda}>0$ has the following properties, see e.g. \cite{TsaiYau02}.
\begin{lemma}\label{LM:groundNon}
Suppose that the linear operator $-\Delta+V$ satisfies the
conditions in [NL] above. Then there exists a constant
$\delta_{0}>0$ such that for any $\lambda \in [e_{0}-\delta,
e_{0})$ \eqref{eq:NLS} has solutions of the form
$\psi(x,t)=e^{i\lambda t}\phi^{\lambda}(x)\in \mathcal{L}^{2}$ with
\begin{align}\label{eq:phiAsy}
\phi^{\lambda}=\delta\phi+\cO(\delta^3)
\end{align} and
$\delta=(\int
\phi^{4}(x)dx)^{-\frac{1}{2}}(e_{0}-\lambda)^{1/2}+o((e_{0}-\lambda)^{1/2}),$
moreover
$$\partial_{\lambda}\phi^{\lambda}=O((e_{0}-\lambda)^{-1/2})\phi+o((e_{0}-\lambda)^{1/2}).$$
\end{lemma}
\section{The Linearized Operator}\label{sec:OperL}
After linearizing the solution around solitary wave solution \eqref{eq:soli2}, namely considering the solution of \eqref{eq:NLS} $\psi(x,t)=e^{i\lambda t} [\phi^{\lambda}(x)+R(x,t)]$, then the linear part of the equation for $R(x,t)$ is
\begin{align}
\partial_t \vec{R}=L(\lambda)\vec{R}
\end{align} with $\vec{R}:=(ReR,\ ImR)^{T},$
and the linearized operator
$L(\lambda)$ is defined as
\begin{align}\label{eq:defLLambda}
L(\lambda):=\left[
\begin{array}{cc}
0& L_{-}(\lambda)\\
-L_{+}(\lambda) &0
\end{array}
\right]
\end{align} with $L_{\pm}(\lambda)$ being linear Schr\"odinger operators defined as
\begin{align*}
L_{-}(\lambda):=-\Delta+V+\lambda+(\phi^{\lambda})^2,\ \text{and}\ L_{+}(\lambda):=-\Delta+V+\lambda+3(\phi^{\lambda})^2.
\end{align*}
By general result (Weyl's Theorem) on stability of the essential spectrum for localized perturbations of $J(-\Delta)$
\cite{RSIV},
$$\sigma_{ess}(L(\lambda))=(-i\infty,-i\lambda]\cap
[i\lambda,i\infty)$$ if the potential $V$ in Equation
\eqref{eq:NLS} decays at $\infty$ sufficiently rapidly.
Next we study the eigenvalues and eigenvectors of $L(\lambda).$ The proof can be found in \cite{GaWe}, hence we omit it.
\begin{lemma}\label{LM:NearLinear}
Let $L(\lambda)$, or more explicitly, $L(\lambda(\delta),\delta)$
denote the linearized operator about the the bifurcating state
$\phi^\lambda, \lambda=\lambda(\delta)$. Note that $\lambda(0)=
e_0$.
It has an eigenvector $\left(
\begin{array}{cc}
0\\
\phi^{\lambda}
\end{array}
\right)$ and an associated eigenvector $\left(
\begin{array}{cc}
\partial_{\lambda}\phi^{\lambda}\\
0
\end{array}
\right)$ with eigenvalue $0$:
\begin{align}
L(\lambda)\left(
\begin{array}{cc}
0\\
\phi^{\lambda}
\end{array}
\right)=0,\ \ \ \ L(\lambda)\left(
\begin{array}{cc}
\partial_{\lambda}\phi^{\lambda}\\
0
\end{array}
\right)=\left(
\begin{array}{cc}
0\\
\phi^{\lambda}
\end{array}
\right).
\end{align}
Corresponding to the (possibly degenerate) eigenvalue, $-e_1,\ -e_2,\ \cdots, \ -e_{N}$, of
$-\Delta+V$, the matrix operator $$L(\lambda=e_0,\delta=0)$$ has eigenvalues
\begin{align}
\pm iE_{n}(e_0)=\pm i(e_0-e_n),\ n=1,2,\cdots, N.\label{eq:Enen}
\end{align}
For $\delta>0$ and small, these bifurcate to
(possibly degenerate) eigenvalues, of the operator $L(\lambda),$ $\pm iE_1(\lambda),\dots,$ $\pm
iE_N(\lambda)$ with eigenvectors
\begin{align}
\left(
\begin{array}{lll}
\xi_{1}\\
\pm i\eta_{1}
\end{array}
\right),\ \left(
\begin{array}{lll}
\xi_{2}\\
\pm i\eta_{2}
\end{array}
\right),\ \cdot\cdot\cdot, \left(
\begin{array}{lll}
\xi_{N}\\
\pm i\eta_{N}
\end{array}
\right)\label{eq:eigenf}
\end{align} with $\xi_n,\ \eta_n$ being real functions and
\begin{align}
\langle \phi^{\lambda},\ \xi_n\rangle=\langle \partial_{\lambda}\phi^{\lambda},\ \eta_n\rangle=0,\ \langle \xi_{n},\eta_{m}\rangle=\delta_{m,n}.\label{eq:orthogonality}
\end{align}
Moreover, for $\delta$ sufficiently small $2E_n(\lambda)>\lambda,\
n=1,2,\cdot\cdot\cdot,N,$ (resonance at second order with
radiation).
\end{lemma}
Furthermore we need the following condition on the threshold resonances.
\begin{definition}
A function $h$ is called a threshold resonance function of
$-\Delta+V$ at $0$, the endpoint of the essential spectrum,
$|h(x)|\leq c\langle x\rangle^{-1_+}$ and $h$ is $C^{2}$ and
solves the equation
$$(-\Delta+V)h=0.$$
A function $h$ is called a threshold resonance function of
$L(\lambda)$ at $\mu=\pm i\lambda$, the endpoint of the essential spectrum,
$|h(x)|\leq c\langle x\rangle^{-1_+}$ and $h$ is $C^{2}$ and
solves the equation
$$(L(\lambda)-\mu)h=0.$$
\end{definition}
In this paper we make the following assumption:
{\bf
\begin{enumerate}
\item[(SA)] $-\Delta+V$ has no threshold resonance at $0$.
\end{enumerate}
} This assumption is generic since it is known that the threshold resonance is unstable, see e.g. \cite{RSIII}. Based on this assumption it is well known that
\begin{lemma}
If $|e_0-\lambda|$ is sufficiently small and the assumption (SA) holds, then $L(\lambda)$ has no threshold resonances at $\mu=\pm i\lambda$, and $L(\lambda)$ has no other eigenvectors and eigenvalues besides the ones listed in Lemma \ref{LM:NearLinear}.
\end{lemma}
We denote the projection onto the essential spectrum of linear operator $L(\lambda)$ is
$P_c^{\lambda}=1-P_{d}^{\lambda}.$ In the following we give the explicit form of the projection $P_{d}$, whose proof for $N=1$ can be found in
~\cite{MR2187292}, the proof of the general cases are similar, hence
omitted.
\begin{proposition}\label{Riesz-project}
For the non self-adjoint operator $L(\lambda)$ the (Riesz)
projection onto the discrete spectrum subspace of $L(\lambda)$, $P_d=P_d(L(\lambda))=P_d^\lambda$, is given by
\begin{equation}\label{eq:PdProjection}
\begin{array}{lll}
P_{d}&=&\frac{2}{\partial_{\lambda}\| \phi^{\lambda}\|^{2}}\left(\ \left|
\begin{array}{lll}
0\\
\phi^{\lambda}
\end{array}
\right\rangle \left\langle
\begin{array}{lll}
0\\
\partial_{\lambda}\phi^{\lambda}
\end{array}
\right|\ +\ \left|
\begin{array}{lll}
\partial_{\lambda}\phi^{\lambda}\\
0
\end{array}
\right\rangle \left\langle
\begin{array}{lll}
\phi^{\lambda}\\
0
\end{array}
\right|\ \right)\\
& &\\
& &-i\displaystyle\sum_{n=1}^{N}\left(\ \left|
\begin{array}{lll}
\xi_{n}\\
i\eta_{n}
\end{array}
\right\rangle\left\langle
\begin{array}{lll}
-i\eta_{n}\\
\xi_{n}
\end{array}
\right| \ -\ \left|
\begin{array}{lll}
\xi_{n}\\
-i\eta_{n}
\end{array}
\right\rangle\left\langle
\begin{array}{lll}
i\eta_{n}\\
\xi_{n}
\end{array}
\right|\ \right).
\end{array}
\end{equation}
\end{proposition}
We define the projection onto the continuous spectral subspace of $L(\lambda)$ by
\begin{align}
P_c\ =\ P_c(L(\lambda))\ =\ P_c^\lambda\ \equiv\ I\ -\ P_d\label{Pcdef}.
\end{align}
\section{The Negativity of Fermi Golden Rule in Matrix Form}\label{sec:FGR}
The Fermi Golden rule plays an essential role in determining the
decay rate of the neutral modes. For simple neutral modes, as in
\cite{SW99, TsaiYau02, BuSu, SW04},
the form is simple because the number is only one, or if more than one then they
can be separated into independent ones by a near identity
transformation (see ~\cite{Tsai2003, Cucca08}). The problem of multiple neutral modes is more
involved due to the fact that multiple coupled parameters appear and
they can not be separated.
Next we define the new Fermi Golden Rule condition.
Define a function $e:\mathbb{R}^3\times \mathbb{R}^3\rightarrow \mathbb{C}$ as
\begin{align}
e(x,k):=\big[1+ (-\Delta_x-|k|^2-i0)^{-1}V(x) \big]^{-1}e^{ik\cdot x}.
\end{align}
It is known that for any fixed $k\not=0$, this function is well defined by the type of potential $V$ we chose, see e.g. \cite{RSIII}. And it satisfies the equation
\begin{align*}
[-\Delta_x+V(x)-|k|^2]\ e(x,k)=0.
\end{align*}
We define complex functions $\Psi_{m,n}$ on the 2-dimensional unit sphere $\mathbb{S}^2$. For any $\sigma\in \mathbb{S}^2$,
\begin{align}
\Psi_{m,n}(\sigma)=\Psi_{n,m}(\sigma):= \int_{\mathbb{R}^3} e(x,\ |k|_{m,n} \sigma)\phi(x)\ \xi^{lin}_m(x)\xi^{lin}_n(x)\ dx\label{eq:Psimn}
\end{align} with $|k|_{m,n}\in \mathbb{R}^{+}$ defined as
\begin{align*}
|k|_{m,n}:=(e_0-e_m-e_n)^{\frac{1}{2}},
\end{align*} where, recall that we assume that $2e_{l}<e_0,\ l=1,2,\cdots, N,$ and $\phi$, $\xi_{m}^{lin}$ are eigenvectors of $-\Delta+V$ with eigenvalues $-e_0$ and $-e_{m},\ m=1,\cdots,\ N.$
Now we state our Fermi-Golden-rule condition.
\begin{itemize}
\item[(FGR)] For any scalar vector $z=(z_1,z_2,\cdots,z_n)\in \mathbb{C}^{N}$ satisfying $|z|=1$, the functions defined on the unit sphere $\mathbb{S}^2$,
$ \displaystyle\sum_{m,n=1}^{N}\Psi_{m,n}(\sigma) z_{m}\bar{z}_{n} $ is not identically zero.
\end{itemize}
Its important ramification is that there exists some constant $C>0$, such that for any $z=(z_1,\ z_2,\cdots,\ z_n)\in \mathbb{C}^{N},$
\begin{align}
\Gamma(z,\bar{z}):=\| \sum_{m,\ n} \Psi_{m,n}(\sigma)z_m z_n\|_{\mathcal{L}^2(\mathbb{S}^2)}^2\geq C |z|^4. \label{Gammadef}
\end{align} Here we use that $\displaystyle\sum_{m,\ n} \Psi_{m,n}(\sigma)z_m z_n$ is smooth in $\sigma$, hence if it is not identically zero, we have the estimate above.
\begin{remark}
If the set of eigenvalues $\{e_{k}\ | k=1,\cdots,N\}$ can be grouped into well separated clusters, namely
\begin{align}
\{e_{k}|\ k=1,\cdots,N\}=\cup_{l} A_{l}
\end{align} with properties that, for some constant $c_0=\cO(1)$, $$|e_n-e_m|\geq c_0\ \text{if} \ e_n\in A_l, \ e_m\in A_k\ \text{with} \ l\not=k.$$ Then Fermi Golden rule assumption can be relaxed: for each fixed $l$, the each of the functions
$
\displaystyle\sum_{
e_m,\ e_n\in A_l
}\Psi_{m,n}(\sigma)\ z_m z_n
$ is not identically zero, with $\displaystyle\sum_{e_m\in A_l}|z_m|^{2}\not=0.$
\end{remark}
\section{Main Theorem}\label{MainTHM}
In this section we state precisely the main theorem of this paper. The key fact is that despite of the possible presence of resonance, the main results remain the same as in \cite{GaWe,GaWe2011}.
For
technical reasons we impose the following conditions on the external potential $V$ of \eqref{eq:NLS}:
\begin{enumerate}
\item[(VA)] $V$ decays exponentially fast at $\infty.$
\end{enumerate}
Recall the notations $\xi=(\xi_1,\cdots,\xi_N)$ and $\eta=(\eta_1,\dots,\eta_N)$ for components of the neutrally stable modes of frequencies $\pm iE_n(\lambda),\ n=1,\cdots,N,$ of the linearized operator $L(\lambda)$ defined in \eqref{eq:defLLambda}.
\begin{theorem}\label{THM:MainTheorem}
Suppose that Conditions (NL) in Section \ref{HaGWP}, (SA) in Section \ref{sec:OperL}, (FGR) in Section \ref{sec:FGR} and (VA) above are
satisfied. Let $\nu>0$ be fixed and sufficiently large.
\\ Then there exist constants
$c,\epsilon_{0}>0$ such that, if
\begin{equation}\label{InitCond}
\inf_{\gamma\in
\mathbb{R}}
\left\|\psi_0-e^{i\gamma}
\left(\phi^{\lambda_{0}}+\ (Re\ z^{(0)})\cdot \xi+i\ (Im\ z^{(0)})\cdot\eta
\right)\right\|_{\mathcal{H}^{3,\nu}}\ \leq\
c\ | z^{(0)}| \le \epsilon_0,
\end{equation}
then there exist smooth functions
\begin{align}
&\lambda(t):\mathbb{R}^{+}\rightarrow
\mathcal{I},\ \ \ \gamma(t): \mathbb{R}^{+}\rightarrow
\mathbb{R},\ z(t):\mathbb{R}^{+}\rightarrow \mathbb{C}^d,\nn\\
&\ \ R(x,t):\mathbb{R}^{d}\times\mathbb{R}^{+}\rightarrow
\mathbb{C}
\nn\end{align}
such that the solution of NLS evolves in the form:
\begin{align}\label{Decom}
\psi(x,t)&\ =\ e^{i\int_{0}^{t}\lambda(s)ds}e^{i\gamma(t)}\nn\\
&\ \ \ \ \ \ \ \times[\phi^{\lambda}+a_{1}(z,\bar{z})
\D_\lambda\phi^{\lambda}+ia_{2}(z,\bar{z})\phi^{\lambda} + (Re\ \tilde{z})\cdot\xi + i(Im \tilde{z})\cdot\eta + R ],\end{align}
where $\lim_{t\rightarrow \infty}\lambda(t)=\lambda_{\infty},$
for some $\lambda_{\infty}\in \mathcal{I}$.\\
Here, $a_{1}(z,\bar{z}),\ a_{2}(z,\bar{z}): \mathbb{C}^N\times\mathbb{C}^N\rightarrow \mathbb{R}$ and $\tilde{z}-z: \mathbb{C}^N\times\mathbb{C}^N\rightarrow \mathbb{C}^N$
are polynomials of $z$ and $\bar{z}$, beginning with terms of order $|z|^{2}$. Their explicit definitions will be given in \eqref{eq:pkmn}.\\
Moreover:
\begin{enumerate}
\item[(A)] $|z(t)|\leq c(1+t)^{ -\frac{1}{2} }$ and, there exists a polynomial $F(z,\bar{z})=O(|z|^4)\in \mathbb{R}$ such that
$z$ satisfies the initial value problem
\begin{equation}\label{eq:detailedDescription1}
{\partial_t}[ |z|^2+F(z,\bar{z})] =-C\Gamma( z ,\bar{ z }) +\ \cO((1+t)^{-\frac{12}{5}})
\end{equation} where $C>0$ is a constant, $\Gamma( z ,\bar{ z })=\cO(|z|^4)$ is a positive quantity defined in \eqref{Gammadef}.
\item[(B)] $\vec{R}(t)=( Re R(t), Im R(t))^T$ lies in the essential spectral part of $L(\lambda(t))$. Equivalently, $R(\cdot,t)$ satisfies the symplectic orthogonality conditions:
\begin{align}\label{s-Rorthogonal}
&\omega\langle R,i\phi^{\lambda}\rangle\ =\ \omega\langle
R,\partial_{\lambda}\phi^{\lambda}\rangle\ =\ 0\nn\\
&\omega\langle
R,i\eta_{n}\rangle=\omega\langle R,\xi_{n}\rangle=0,\
n=1,2,\cdot\cdot\cdot,N,
\end{align}
where $\omega(X,Y)=Im\ \int X\bar{Y}$
and satisfies the decay estimate:
\begin{equation}
\|(1+x^{2})^{-\nu}\vec{R}(t)\|_{2}\leq c(1+|t|)^{-1}.
\label{Rdecay}
\end{equation}
\end{enumerate}
\end{theorem}
The main theorem will be reformulated into Theorem \ref{GOLD:maintheorem} below.
\section{The Effective
Equations for $\dot{z},\ \dot\lambda,\ \dot\gamma$ and $
R$}\label{SEC:effective}
In this section we derive equations for $\dot{z},\ \dot\lambda,\ \dot\gamma$ and $R$.
We decompose the solution as
\begin{align}
\psi(x,t)&\ =\ e^{i\int_{0}^{t}\lambda(s)ds}e^{i\gamma(t)}\nn\\
&\ \ \ \ \ \times\left[\phi^{\lambda}+a_{1}
\phi_{\lambda}^{\lambda}+ia_{2}\phi^{\lambda}+\sum_{n=1}^{N}(\alpha_{n}+p_{n})\xi_{n}+
i\sum_{n=1}^{N}(\beta_{n}+q_{n})\eta_{n}+R\right]\nn\\
&\ =\ e^{i\int_{0}^{t}\lambda(s)ds}e^{i\gamma(t)}\left[\phi^{\lambda}+a_{1}
\phi_{\lambda}^{\lambda}+ia_{2}\phi^{\lambda}+(\alpha+p)\cdot\xi+
i(\beta+q)\cdot\eta+R\right]
\label{Decom1}
\end{align}
Here and going forward, we'll use the notations:
\begin{align}
\alpha=(\alpha_1,\dots,\alpha_N)^T,\ \ \beta=(\beta_1,\dots,\beta_N)^T,\nn\\
\xi=(\xi_1,\dots,\xi_N)^T,\ \ \eta=(\eta_1,\dots,\eta_N)^T.\nn
\end{align}
Introducing
\begin{equation} z=\alpha+i\beta,\nn\end{equation}
we have
\begin{equation}
\alpha=\frac{1}{2}(z+\overline{z}),\ \ \beta=\frac{1}{2i}(z-\overline{z}),
\nn\end{equation}
and we seek $a_j=a_j(z,\overline{z})=\cO(|z|)^2,\ j=1,2$ and $p_j=p_j(z,\overline{z})=\cO(|z|^2)$, polynomials in $z$ and $\overline{z}$, which are of degree larger than or equal to two, and are real.
Substitution of the Ansatz \eqref{Decom1} into NLS,
equation \eqref{eq:NLS}, we have the following system of equations
for $\vec{R}$, defined as
\begin{align}
\vec{R}:= \left(
\begin{array}{lll}
R_{1}\\
R_{2}
\end{array}
\right),\ R_{1}:= Re R,\ \ R_{2}:= Im R:
\nn
\end{align}
as
\begin{align}
\partial_{t}\vec{R}=&L(\lambda)\vec{R}+\dot\gamma
J\vec{R}-J\vec{N}(\Vec{R},z)
-\left(
\begin{array}{lll}
\partial_{\lambda}\phi^{\lambda}[\dot\lambda+\partial_{t}a_{1}]\\
\phi^{\lambda} [ \dot{\gamma}+\partial_{t} a_{2}-a_{1} ]
\end{array}
\right)\nn\\
& +\ \left(
\begin{array}{lll}
\xi\cdot[E(\lambda)(\beta+q)-\partial_{t}(\alpha+p)]\\
-\eta\cdot[E(\lambda)(\alpha+p)+\partial_{t}(\beta+q)]
\end{array}
\right)\label{Eq:R} \\
&+\dot\gamma\ \left(
\begin{array}{lll}
(\beta+q)\cdot\eta\\
- (\alpha+p)\cdot\xi
\end{array}\right)
-\dot\lambda \left(
\begin{array}{lll}
a_{1}\partial_{\lambda}^{2}\phi^{\lambda}+
(\alpha+p)\cdot\partial_{\lambda}\xi\\
a_{2}\partial_{\lambda}\phi^{\lambda}+
(\beta+q)\cdot\partial_{\lambda}\eta
\end{array}
\right),\nonumber
\end{align}
Here,
\begin{align}
J\vec{N}(\Vec{R},z):=\left(
\begin{array}{lll}
ImN(\vec{R},z)\\
-ReN(\vec{R},z)
\end{array}
\right) \label{JvecN}
\end{align}
with
\begin{align}
ImN(\Vec{R},z)&:=
|\phi^{\lambda}+I_{1}+iI_{2}|^{2} I_{2}-(\phi^{\lambda})^{2} I_{2}\nn\\
Re N(\Vec{R},z)&:=
[|\phi^{\lambda}+I_{1}+iI_{2}|^{2}-(\phi^{\lambda})^{2}](\phi^{\lambda}+I_{1})
-2(\phi^{\lambda})^{2}I_{1}\nn\\
I_{1}\ &=\ A_{1}+A_{2}+R_{1},\ \ \ \
I_{2}\ =\ B_{1}\ +\ B_{2}\ +\ R_{2}\nn\\
A_{1}\ &=\ \alpha\cdot\xi, \ \ \ \ \ A_{2}\ =\ a_{1}\partial_{\lambda}\phi^{\lambda}+
p\cdot\xi,\nn\\
B_{1}\ &=\ \beta\cdot \eta,\ \ \ \ \
B_{2}\ =\ a_{2}\phi^{\lambda}\ +\ q\cdot\eta.\label{eq:A12B12}
\end{align}
From the system of equations \eqref{Eq:R} and the orthogonality conditions
\eqref{eq:orthogonality} and \eqref{s-Rorthogonal} we obtain equations for $
\dot\lambda,\ \dot\gamma$ and $z_{n}=\alpha_n+i\beta_n,\ n=1,\dots,N: $
\begin{align}
\partial_{t}(\alpha_{n}+p_{n})-E_n(\lambda)(\beta_{n}+q_{n})+\langle ImN(\Vec{R},z),\eta_{n}\rangle=&F_{1n};\label{eq:z1} \\
\partial_{t}(\beta_{n}+q_{n})+E_n(\lambda)(\alpha_{n}+p_{n})-\langle ReN(\Vec{R},z),\xi_{n}\rangle=&F_{2n};\label{eq:z2}\\
\dot\gamma+\partial_{t}a_{2}-a_{1}-\frac{1}{\langle
\phi^{\lambda},\phi^{\lambda}_{\lambda}\rangle}\langle
ReN(\Vec{R},z),\phi^{\lambda}_{\lambda}\rangle=& F_{3};\label{eq:gamma}\\
\dot\lambda+\partial_{t}a_{1}+\frac{1}{\langle
\phi^{\lambda},\phi^{\lambda}_{\lambda}\rangle}\langle
ImN(\Vec{R},z),\phi^{\lambda}\rangle=&F_{4}.\label{eq:lambda}
\end{align}
Finally, the scalar functions $F_{j,n},\ j=1,2, F_3,
F_4,$ are defined as
\begin{align}
F_{1n}:= &\dot\gamma\langle
(\beta+q)\cdot\eta,\eta_{n}\rangle-\dot{\lambda}a_{1}\langle
\partial_{\lambda}^{2}\phi^{\lambda},\eta_{n}\rangle
-\dot\lambda\langle
(\alpha+p)\cdot\partial_{\lambda}\xi,\eta_{n}\rangle\nn\\
&-\dot\gamma\langle
R_{2},\eta_{n}\rangle
+\dot\lambda\langle
R_{1},\partial_{\lambda}\eta_{n}\rangle,\nn\\
F_{2n} := &-\dot\gamma\langle
(\alpha+p)\cdot\xi,\xi_{n}\rangle
-\dot{\lambda}a_{2}\langle\phi^{\lambda}_{\lambda},
\xi_{n}\rangle
-\dot\lambda\langle(\beta+q)\cdot\partial_{\lambda}\eta,\xi_{n}\rangle\nn\\
&+\dot\gamma\langle
R_{1},\xi_{n}\rangle+\dot\lambda\langle
R_{2},\partial_{\lambda}\xi_{n}\rangle,\nn\\
F_{3} := &\frac{1}{\langle
\phi^{\lambda},\phi^{\lambda}_{\lambda}\rangle}\times\nn\\
&
\left[\ \dot\lambda \langle
R_{2},\phi_{\lambda\lambda}^{\lambda}\rangle -\dot\gamma\langle
R_{1},\phi_{\lambda}^{\lambda}\rangle
-\langle\dot\gamma(\alpha+p)\cdot\xi+
\dot{\lambda}a_{2}\phi^{\lambda}_{\lambda}
+\dot\lambda (\beta+q)\cdot\partial_{\lambda}\eta,\phi^{\lambda}_{\lambda}\rangle\ \right],\nn\\
F_{4}:=&\frac{1}{\langle
\phi^{\lambda},\phi^{\lambda}_{\lambda}\rangle}\times\nn\\
&
\left[\ \dot\lambda\langle
R_{1},\phi_{\lambda}^{\lambda}\rangle +\dot\gamma\langle
R_{2},\phi^{\lambda}\rangle
+\langle
\dot\gamma (\beta+q)\cdot \eta-\dot{\lambda}a_{1}
\partial_{\lambda}^{2}\phi^{\lambda}-\dot\lambda
(\alpha+p)\cdot\partial_{\lambda}\xi,\phi^{\lambda}\ \rangle\right].
\end{align}
To facilitate later discussions we cast \eqref{eq:z1} and \eqref{eq:z2} into a convenient form. Since $\alpha_n$ and $\beta_n$ are real parameters, it is equivalent to study the complex parameters $z_n:=\alpha_n+i\beta_n$. Compute \eqref{eq:z1}$+i$\eqref{eq:z2} to find
\begin{align}
\partial_t z_n+iE_{n}(\lambda)z_n=&-\partial_{t} (p_n+iq_n)-iE_{n}(\lambda)(p_n+iq_n)-\langle ImN(\Vec{R},z),\eta_{n}\rangle\nonumber\\
&+i\langle ReN(\Vec{R},z),\xi_{n}\rangle
+F_{1n}+iF_{2n}.\label{eq:z}
\end{align}
Note that \eqref{eq:z1} and \eqref{eq:z2} can be recovered from the equation above by considering the real and imaginary parts of \eqref{eq:z}.
\begin{remark}
\begin{itemize}
\item[(a)] Recall the estimate of $Remainder$ in \eqref{remainder}. By \eqref{eq:z1}-\eqref{eq:lambda} we have
\begin{equation}\label{eq:EstRough}
\dot\lambda,\ \dot\gamma,\
\partial_{t}z_{n}+iE_n(\lambda)z_{n}=O(|z|^{2})+Remainder.
\end{equation}
\item[(b)] The functions $a_j(z,\overline{z}), j=1,2,\ p_n(z,\overline{z}),\
q_n(z,\overline{z}), n=1,\dots,N$ will be chosen to eliminate ``non-resonant'' terms: $z^a\overline{z}^b, \ \ 2\le a+b\le3$.
\end{itemize}
\end{remark}
Finally, we derive an equation for
\begin{equation}
\vec{R}=P_c^{\lambda(t)}\vec{R}=P_c\vec{R},
\nn\end{equation}
the continuous spectral part of the solution, relative to the operator, $L(\lambda(t))$. Applying $P_c=P_c^{\lambda(t)}$ to \eqref{Eq:R} to use that (see \eqref{eq:PdProjection})
\begin{align*}
P_c\left(
\begin{array}{lll}
\xi_n\\
\pm i \eta_n
\end{array}
\right)=P_c\left(
\begin{array}{lll}
0\\
\phi^{\lambda}
\end{array}
\right)=P_c\left(
\begin{array}{lll}
\partial_{\lambda}\phi^{\lambda}\\
0
\end{array}
\right)=0
\end{align*} to remove many terms on the right hand side,
and using the commutator identity:
\begin{align}
P_{c}\partial_t\vec{R}=\partial_t\vec{R}-\dot\lambda
\D_\lambda P_{c}\vec{R}
\label{Pccomdt}
\end{align}
we obtain
\begin{align}\label{RAfProj}
\partial_t\vec{R}\ =\
L( \lambda(t) )\vec{R}\ -\ P_{c}^{\lambda(t)}J\vec{N}(\Vec{R},z)\ +\ L_{(\dot\lambda,\dot\gamma)}\vec{R}\ +\ \mathcal{G}.
\end{align}
Here the operator $L_{(\dot\lambda,\dot\gamma)}$ and the
vector function $\mathcal{G}$ are defined
as
\begin{align}
& L_{(\dot\lambda,\dot\gamma)}\ =\ \dot\lambda
\D_\lambda P_{c}^{\lambda(t)}+\dot\gamma P_{c}^{\lambda(t)}J,\label{Lgdld}\\
& \mathcal{G}\ =\ P_{c}^{\lambda(t)}\left(
\begin{array}{lll}
\dot\gamma (\beta+q)\cdot\eta-\dot{\lambda}a_{1}
\partial_{\lambda}^{2}\phi^{\lambda}
-\dot\lambda (\alpha+p)\cdot\partial_{\lambda}\xi\\
-\dot\gamma (\alpha+p)\cdot\xi-\dot{\lambda}a_{2}\phi^{\lambda}_{\lambda}-\dot\lambda
(\beta+q)\cdot\partial_{\lambda}\eta
\end{array}
\right).\label{Gdef}
\end{align}
\section{Reformulation of The Main Theorem}\label{sec:refor}
The proof of Theorem ~\ref{THM:MainTheorem} we use the following result, which gives a more detailed characterization of the terms in the decomposition.
\begin{theorem}\label{GOLD:maintheorem}
The function $R$ in \eqref{Decom} of Theorem ~\ref{THM:MainTheorem}
can be decomposed as
\begin{equation}\label{eq:expanR}
\vec{R}=\displaystyle\sum_{ |m|+|n|=2}R_{m, n}(\lambda)z^{m}\bar{z}^n+\tilde{R}
\end{equation} where $R_{m,n}$ are functions of the form
$$R_{m,n}=\big[L(\lambda)+iE(\lambda)\cdot (m-n)-0\big]^{-1}\phi_{m,n}$$
$\phi_{m,n}$ are smooth, spatially exponentially decaying functions. The function
$\tilde{R}$ satisfies the equation
\begin{equation}\label{eq:tildeR}
\begin{array}{lll}
\partial_t\tilde{R}&=&L(\lambda)\tilde{R}+M_{2}(z,\bar{z})\tilde{R}+P_{c}N_{2}(\vec{R},z)+P_{c}S_{2}(z,\bar{z}),
\end{array}
\end{equation} where
\begin{enumerate}
\item[(1)] $S_{2}(z,\bar{z})=\cO(|z|^{3})$ is a polynomial in $z$ and
$\bar{z}$ with $\lambda$-dependent coefficients, and each coefficient can be written as the sum of
functions of the form
\begin{align}\label{eq:unusual}
[L(\lambda)\pm i(E_m(\lambda)+E_n(\lambda))-0]^{-k}P_{c}\phi_{\pm k}(\lambda),
\end{align}
where $k=0,1,2$ and the functions $\phi_{\pm k}(\lambda)$ are smooth
and decay exponentially fast at spatial $\infty$;
\item[(2)] $M_{2}(z,\bar{z})$ is an operator defined by
\begin{equation}\label{eq:M2}
M_{2}(z,\bar{z}):=\dot{\gamma}P_{c}J+\dot{\lambda}P_{c\lambda}+X,
\end{equation}
where $X$ is a $2\times 2$ matrix, satisfying the bound
$$|X|\leq c|z|e^{-\epsilon_{0}|x|}.$$
\item[(3)] $N_{2}(\vec{R},z)$ can be separated into
localized term and nonlocal term
\begin{equation}\label{eq:N2Decomposition}
N_{2}=Loc+NonLoc
\end{equation} where $Loc$ consists of terms spatially
localized (exponentially) function of $x\in \mathbb{R}^{3}$ as a factor and satisfies the estimate
\begin{equation}
\|\langle x\rangle^{\nu}(-\Delta+1)Loc\|_{2}+\|Loc\|_{1}+
\|Loc\|_{\frac{4}{3}}\leq c(|z|^{3}(t)+|z|(t)\|\langle x\rangle^{-\nu}\vec{R}\|_{2}),
\label{eq:Loc-est}\end{equation}
and $NonLoc$ is given by
\begin{equation}\label{eq:NonlocDef}
NonLoc:=(R_{1}^{2}+R_{2}^{2})J\vec{R}.
\end{equation}
Here $\nu$ is the same as in Theorem
~\ref{THM:MainTheorem}.
\end{enumerate}
In the rest of the paper we denote by $Remainder(t)$ any quantity which satisfies the estimate:
\begin{align}\label{remainder}
|Remainder(t)|\lesssim |z(t)|^{4}+\|\langle
x\rangle^{-\nu}\vec{R}(t)\|_{2}^{2}+\|\vec{R}(t)\|_{\infty}^{2}
+|z(t)|\ \|\langle
x\rangle^{-\nu}\tilde{R}(t)\|_{2}.
\end{align}
The functions $\lambda$, $\gamma,$ $z$ have the following properties
\begin{enumerate}
\item[(A)]
\begin{equation}\label{ExpanLambda}
\dot\lambda=Remainder(t);
\end{equation}
\item[(B)]
\begin{equation}\label{ExpanGamma}
\dot\gamma=\Upsilon+Remainder(t)
\end{equation} with
\begin{equation}\label{eq:Gamma11}
\Upsilon:=\frac{\langle
(\phi^{\lambda})^{2}[\frac{3}{2}|z\cdot\xi|^{2}+\frac{1}{2}|z\cdot\eta|^{2}],\partial_{\lambda}\phi^{\lambda}\rangle}{\langle\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle};
\end{equation}
\item[(C)]
there exists a polynomial $F(z,\bar{z})=\cO(|z|^4)\in \mathbb{R}$ such that the vector $z$ satisfies the equation
\begin{equation}\label{eq:detailedDescription}
{\partial_t}[|z|^2+F(z,\bar{z})]=-\Gamma(z,\bar{z})+|z|Remainder(t)
\end{equation} where $\Gamma(z,\bar{z})$ is a positive quantity defined in \eqref{Gammadef}.\end{enumerate}
\end{theorem}
The definition of $R_{m,n}$ in \eqref{eq:expanR} will be in Section \ref{SEC:effective}, the proof of \eqref{eq:tildeR} will be in Section \ref{sec:tildeR}, \eqref{ExpanLambda} and \eqref{ExpanGamma} will be reformulated into Proposition \ref{Prop:ExplicitPolyno}, \eqref{eq:detailedDescription} will be proved in Section \ref{sec:NFT}.
\section{Proof of \eqref{eq:tildeR}}\label{sec:tildeR}
Observe that in the equation for $\Vec{R}$ in \eqref{RAfProj}, the term on the right hand side, specifically $J\Vec{N}(\Vec{R},z)$, contains terms quadratic in $z$ and $\bar{z}$.
Hence for fixed $z(t)\in\C^N$, the equation for $\vec{R}(t)$ is forced by terms
of order $\cO(|z(t)|^2)$.
In what follows, we extract the quadratic in $z,\overline{z}$ part of $\vec{R}(t)$.
Observe that the quadratic terms generated by the nonlinearity are of the form:
\begin{align}
\sum_{|m|+|n|=2}\ JN_{m,n}z^{m}\bar{z}^n\ =&\left(
\begin{array}{ccc}
2\phi^{\lambda}A_{1}B_{1}\\
-3\phi^{\lambda}A_{1}^{2}-\phi^{\lambda}B_{1}^{2}
\end{array}\right).\label{eq:SecondOrderTerm}
\end{align}
where $A_{1}=\alpha\cdot\xi,\ B_{1}=\beta\cdot\eta$, and recall the definition of $JN$ from \eqref{JvecN}.
Substitute this into the equation for $\Vec{R}$ in
\eqref{RAfProj} and decompose $\Vec{R}$ in the next results:
\begin{theorem}
Define
\begin{equation}\label{eq:Rform}
R_{m,n}:=[L(\lambda)+iE(\lambda)\cdot (m-n)-0]^{-1}P_{c}JN_{m,n},
\end{equation}
and decompose $\vec{R}(t)$ as
\begin{equation}
\Vec{R}\ =\ \sum_{|m|+|n|=2}\ R_{m,n}z^{m}\bar{z}^{n}\ +\ \tilde{R}
\label{RtR}
\end{equation}
Then the vector-function $\tilde{R}(x,t)$
satisfies \eqref{eq:tildeR}.
\end{theorem}
The proof is the same to that in \cite{GaWe}, and skipping it will not affect understanding the main part of this paper. Hence we choose to omit this part.
To facilitate later discussions, we further decompose $$J\vec{N}_{>2}=J\vec{N}(\vec{R},z)-\sum_{|m|+|n|=2}\ JN_{m,n}z^{m}\bar{z}^n.$$ We extract the third order terms of $J\vec{N}_{>2}:$
\begin{equation}\label{eq:JNm+n=3}
\sum_{|m|+|n|=3}JN_{m,n}z^{m}\bar{z}^{n}=\sum_{|m|+|n|=2}X R_{m,n}z^m\bar{z}^{n}+X\left(
\begin{array}{lll}
A_{2}\\
B_{2}
\end{array}
\right)+\left(
\begin{array}{lll}
(A_{1}^{2}+B_{1}^{2})B_{1}\\
-(A_{1}^{2}+B_{1}^{2})A_{1}
\end{array}
\right)
\end{equation} where
$X$ is a $2\times 2$ matrix defined as
\begin{equation}\label{eq:XDef}
X:=\left(
\begin{array}{lll}
2\phi^{\lambda}B_{1}& 2\phi^{\lambda}A_{1}\\
-6\phi^{\lambda}A_{1}&
-2\phi^{\lambda}B_{1}
\end{array}
\right),
\end{equation} and $A_1,\ B_1$ and $A_2,\ B_2$ are defined in \eqref{eq:A12B12}.
\section{Normal Form Transformon, Proofs of \eqref{ExpanLambda} and \eqref{ExpanGamma}}\label{NormalForms}
In this section we present the proofs of equations \eqref{ExpanLambda} and \eqref{ExpanGamma}, governing $\dot\lambda$ and $\dot\gamma$, crucial to controlling the large time behavior. The main result is Proposition \ref{Prop:ExplicitPolyno}.
This part is different from \cite{GaWe, GaWe2011}, in that we have to define a new normal form transformation, some of whose parameters are defined as solutions to systems of linear equations and their existence has to be justified.
Moreover some small denominators will appear and we have to prove the numerators are also small.
Now we present the idea. Central to our claim about the large time dynamics of NLS, is that the solution settles into
an asymptotic solitary wave, $\phi^{\lambda_\infty}$, where $\lambda(t)\to\lambda_\infty$. We achieve this by showing $|\dot\lambda(t)|\lesssim \epsilon_0 (1+t)^{-1-\delta}$ for some $\delta>0$ and small $\epsilon_0>0.$
Since we expect the neutral mode amplitudes, $z(t)$, to decay with a rate $t^{-\frac{1}{2}}$, we require that there be no $\cO(|z(t)|^2)$ on the right hand side of the equation \eqref{eq:lambda}:
$$\dot\lambda(t) = -\D_t a_1(z,\overline{z})-\frac{1}{\langle
\phi^{\lambda},\phi^{\lambda}_{\lambda}\rangle}\langle
ImN(\Vec{R},z),\phi^{\lambda}\rangle+ \dots.$$ The strategy is
to choose the quadratic part of the polynomial $a_1(z,\overline{z})$
so as to eliminate all quadratic terms. There are two types of such terms, (1) the terms $z_k z_l$ and $\bar{z}_k \bar{z}_l$, and they are oscillatory with frequencies $\sim -E_k-E_l$ or $\sim E_k+E_l$, which stay away from zero. And the margins are large enough so that we can easily remove them by a normal form transformation, utilizing that $z_k z_l\approx \partial_{t} \frac{1}{-i (E_k+E_l)} z_k z_l.$ (2)
The terms $z_k\overline{z_m}$ have frequencies $\sim -E_k+E_m$, which might be of small, or zero, frequencies. The key observation is that is that if the frequency of $z_{k}\bar{z}_l$ is small, then the coefficient is proportionally small! This allows us to define normal form transformation. The calculation is carried out below; see
Lemma~\ref{coordinate-lemma}, especially \eqref{defA11}.
Similarly we choose $p_n,\ q_n,\ n=1,2,\cdots,N,$ and $a_2$ to remove most of the lower order terms in the equations for $\dot\gamma,\ \dot\alpha_n$ and $\dot\beta_n$. It turns out some terms can not be removed, for example the term $|z_n|^{2}$ in the equation for $\dot\gamma$, and $|z_n|^2 z_n$ terms in the equation for $\partial_{t}z$. On the other hand, there terms either play a favorable role in our analysis, or will not affect it, namely it does not matter if $\gamma$ is not convergent, since $e^{i\gamma(t)}$ is only a phase factor.
In defining $p_n,\ q_n$ we have to solve system of linear equations, see e.g. \eqref{eq:pq20and02}. The existence and uniqueness of solutions have to be addressed.
In what follows we use the notations $N^{Im}_{m,n},\ N^{Re}_{m,n}$
to stand for functions satisfying
$$\left(
\begin{array}{lll}
N^{Im}_{m,n}\\
-N^{Re}_{m,n}
\end{array}
\right)=JN_{m,n},$$ where, recall the definition of $JN_{m,n},\ |m|+|n|=2,3,$ from \eqref{eq:SecondOrderTerm} and \eqref{eq:JNm+n=3}.
In what follows define the polynomials $a_{1}$, $a_{2}$, $p_{k}$ and $q_{k},\
k=1,2,\cdot\cdot\cdot,N$ in \eqref{Decom} by defining their coefficients:
\begin{align}
a_{1}(z,\bar{z}):=&\displaystyle\sum_{|m|+|n|=2,3}A^{(1)}_{m,n}(\lambda)z^{m}\bar{z}^{n},
\nonumber\\
a_{2}(z,\bar{z}):=&\displaystyle\sum_{|m|+|n|=2,3,\ |m|\not=|n|}A^{(2)}_{m,n}(\lambda)z^{m}\bar{z}^{n},\nonumber\\
p_{k}(z,\bar{z}):=&\displaystyle\sum_{|m|+|n|=
2,3}P^{(k)}_{m,n}(\lambda)z^{m}\bar{z}^{n}, \label{eq:pkmn}\\
q_{k}(z,\bar{z}):=&\displaystyle\sum_{|m|+|n|=
2,3}Q^{(k)}_{m,n}(\lambda)z^{m}\bar{z}^{n},\nonumber
\end{align}
with $m,n\in (\mathbb{Z}^{+}\cup \{0\})^{N}$.
In what follows we use the notation $$m\cdot E(\lambda)=\sum_{k}m_kE_k(\lambda).$$
We start with defining $A_{m,n}^{(1)}$.
For $|m|=2,3$
\begin{align}
A_{m,0}^{(1)}:=&\frac{1}{im\cdot E(\lambda)}\frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}\langle
N^{Im}_{m,0},\phi^{\lambda}\rangle.
\end{align} For $|m|=2,\ |n|=1,$
\begin{align}
A_{m,n}^{(1)}:=&\frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}\frac{1}{i (m-n)\cdot E(\lambda)}\times \nn\\
&\big[\langle
N^{Im}_{m,n},\phi^{\lambda}\rangle-\frac{i}{2}\sum_{{\small{
\begin{array}{lll}
|k|=|r|=|n|=1,\\
k+r=m
\end{array}
}}}\Upsilon_{k,n}\langle
r\cdot\eta,\phi^{\lambda}\rangle\big], \label{eq:A1}
\end{align}
here $\Upsilon_{m,n}$ is from the expansion of $\Upsilon=\Upsilon(z,\bar{z})$, defined in \eqref{eq:Gamma11}:
\begin{align*}
\Upsilon=\sum_{|m|=|n|=1} \Upsilon_{m,n}z^{m}\bar{z}^{n}
\end{align*} and the vector $r $ is in $(\mathbb{Z}^{+}\cup \{0\})^{N}.$
For $|m|= |n|=1$, we define
\begin{align}
A_{m,n}^{(1)}:=&\frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}\frac{1}{i (m-n)\cdot E(\lambda)}\langle
N^{Im}_{m,n},\phi^{\lambda}\rangle\label{defA11}\\
=&\frac{1}{4} \frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}[\langle n\cdot \eta, \ m\cdot\eta\rangle+\langle n\cdot\xi,\ m\cdot \xi\rangle].
\end{align}
Here in the first line it is possible that the denominator $(m-n)\cdot E(\lambda)$ equals to zero or arbitrarily small, for example $m=n$, which might make the term ill-defined. In the second line we indicate that, by using Lemma \ref{coordinate-lemma} below, if the denominator is small, then the numerator is proportionally small. Hence $A_{m,n}^{(1)}$ are always well defined, a similar calculation was in \cite{GaWe2011}.
After defining various terms above, the other terms in $A^{(1)}_{m,n}$ are determined by the relations
$$A_{k,l}^{(1)}:=\overline{A_{l,k}^{(1)}}\ \text{for}\ |k|+|l|=2,3.$$
Now we define various terms in $A^{(2)}_{m,n},\ |m|+|n|=2,3, \ |m|\not=|n|$.
For $|m|=2,3,\ |n|=0$, we define $A_{m,0}^{(2)}$ as
\begin{align}
-i m\cdot E(\lambda)A_{m,0}^{(2)}+A_{m,0}^{(1)}:=&\frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}\langle
N^{Re}_{m,0},\partial_{\lambda}\phi^{\lambda}\rangle.
\end{align}
For $|m|=2,\ |n|=1$,
\begin{align}
&-i(m-n)\cdot E(\lambda)A_{m,n}^{(2)}+A_{m,n}^{(1)}\nn\\
:=&\frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}[\langle
N^{Re}_{m,n},\partial_{\lambda}\phi^{\lambda}\rangle
-\frac{1}{2}\sum_{|k|=|n|=|r|=1, k+r=m}\Upsilon_{k,n}\langle
r\cdot\xi,\partial_{\lambda}\phi^{\lambda}\rangle],
\end{align} here the definition of $\Upsilon_{m,n}$ and the convention on $r$ are as in \eqref{eq:A1}. Especially in solving for $A_{m,n}^{(2)}$ we need $\frac{1}{(m-n)\cdot E(\lambda)}$ to be uniformly bounded, indeed this is true by the facts $E_{k}(\lambda)\approx e_0-e_{k}$ in \eqref{eq:Enen} and $2e_k<e_0$, see Condition (NL) in Section \ref{HaGWP}.
The other terms in $A_{k,l}^{(2)}$ are determined by the relations
$$A_{k,l}^{(2)}:=\overline{A_{l,k}^{(2)}}\ \text{for}\ |k|+|l|=2,3.$$
Next we define coefficients $P^{(k)}_{m,n}$ and $Q^{(k)}_{m,n}$ for the polynomials $p_k$ and $q_k,\ k=1,2,\cdots, N,$ see \eqref{eq:pkmn}.
For $|m|=2,3,\ |n|=0,$ we define $P_{m,0}^{(k)}$ and $Q_{m,0}^{(k)}$ to be solutions to the linear equations
\begin{align}
-im\cdot E(\lambda)P_{m,0}^{(k)}-E_k(\lambda)Q_{m,0}^{(k)}:=&-\langle
N_{m,0}^{Im},\eta_{k}\rangle,\nonumber\\
-i m\cdot E(\lambda)Q_{m,0}^{(k)}+E_k(\lambda)P_{m,0}^{(k)}:=&\langle
N_{m,0}^{Re},\xi_{k}\rangle.\label{eq:pq20and02}
\end{align} Here the solutions exist and are unique because the corresponding $2\times 2$ matrices
\begin{align}
D:=\left[
\begin{array}{cc}
-im\cdot E(\lambda) & -E_k(\lambda)\\
E_k(\lambda) & -im \cdot E(\lambda)
\end{array}
\right]\label{Dmatrix}
\end{align}
are uniformly invertible, or equivalently they determinants stay away from zero by a uniform margin. To see this, compute directly to obtain
\begin{align}
DetD=-[m\cdot E(\lambda)]^2+E_{k}^2(\lambda)=-[m\cdot E(\lambda)+E_k(\lambda)]\ [m\cdot E(\lambda)-E_k(\lambda)].
\end{align} Next we relate the quantities on the right hand side to $e_0$ and $e_l,\ l=1,2,\cdots,N,$ by \eqref{eq:Enen}, namely $$\text{for any}\ l=1,2,\cdots, N,\ E_{l}(\lambda)\approx e_0-e_l.$$ Compute directly and use $2e_{l}< e_0$ to see
\begin{align}
m\cdot E(\lambda)-E_k(\lambda)&\approx \sum_{l=1}^{N} m_l(e_0-e_l)-(e_0-e_k)\nonumber\\
&=(|m|-1)e_0+e_k-\sum_{l=1}^{N} m_l e_l \label{eq:awayZero}
\end{align} is positive and stays way from zero for any $m\in \{\mathbb{Z}^{+}\cup \{0\}\}^{N}$ and $|m|=2,3$.
This together with that $m\cdot E(\lambda)+E_k(\lambda)$ is more positive implies the desired result: $Det D$ stays away from zero.
For $|m|=1$ and $|n|=2,$ we define $P_{m,n}^{(k)}$ and $Q_{m,n}^{(k)}$ to satisfy the equation
\begin{align}
&iP_{m,n}^{(k)}-Q_{m,n}^{(k)}:=\nonumber\\
&\frac{-\langle
N_{m,n}^{Im},\eta_{k}\rangle+i\langle N_{m,n}^{Re},\xi_{k}\rangle+i\displaystyle\sum_{|m|=|k|=|r|=1, k+r=n}\Upsilon_{m,k}[\langle r\cdot \eta,\eta_{k}\rangle-\langle r\cdot\xi,\xi_{k}\rangle]}{E_k-E(\lambda)\cdot (m-n)},\label{eq:PQkmn}
\end{align} here the denominator $E_k-E(\lambda)\cdot (m-n)$ stays away from zero by a uniform margin, by the same justification as in \eqref{eq:awayZero}, and the definition of $\Upsilon_{m,n}$ and the convention on $r$ is as in \eqref{eq:A1}, $r\cdot \eta:=\sum_{k}r_k\eta_k$ and $r\cdot \xi:=\sum_{k}r_k\xi_k$. Note that at this moment \eqref{eq:PQkmn} does not give unique solutions. This will be become clear in a moment.
For $|m|=2$ and $|n|=1,$ we define
\begin{align}
i P_{m,n}^{(k)}-Q_{m,n}^{(k)}=0.
\end{align}
After defining $iP_{m,n}^{(k)}-Q_{m,n}^{(k)}$ for $(|m|,\ |n|)=(1,2), \ (2,1)$ above, it is not hard to see that these together with the relations $P_{m,n}^{(k)}=\overline{P_{n,m}^{(k)}}$ and $Q_{m,n}^{(k)}=\overline{Q_{n,m}^{(k)}}$ determine unique solutions for the linear equations.
We continue to define $P^{(k)}_{m,n},\ Q^{(k)}_{m,n}$ for $|m|=|n|=1,$
\begin{align}
-i(m-n)\cdot E(\lambda)\ P^{(k)}_{m,n}-E_k(\lambda)Q_{m,n}^{(k)}:=&-\langle N_{m,n}^{Im},\eta_{k}\rangle,\nonumber\\
-i(m-n)\cdot E(\lambda)\ Q^{(k)}_{m,n}+E_k(\lambda)P_{m,n}^{(k)}:=&\langle N_{m,n}^{Re},\xi_{k}\rangle.
\end{align} The solutions are well defined and unique since the matrix
\begin{align}\left(
\begin{array}{cc}
-i(m-n)\cdot E(\lambda)& -E_k(\lambda) \\
E_k(\lambda) & -i(m-n) \cdot E(\lambda)
\end{array}\right)
\end{align} is uniformly invertible by the same arguments in showing the invertibility of the matrix in \eqref{Dmatrix}.
We complete defining all the relevant terms by requiring that
$$P_{m,n}^{(k)}:=\overline{P_{n,m}^{(k)}},\ Q_{m,n}^{(k)}:=\overline{Q_{n,m}^{(k)}}.$$
By now we have finished defining the polynomials $a_1,\ a_2,\ p_k,\ q_k,\ k=1,2,\cdots,N.$
Next we study the equation for $\dot{z}_k.$ By the definitions of coefficients $P_{m,n}^{(k)}, \ Q_{m,n}^{(k)}$
we removed the following terms from the $\dot{z}_k$-equation: $z^{m}\bar{z}^{n}$ if $|m|+|n|=2,3$ and $(|m|, \ |n|)\not=(2,1).$ The result is:
\begin{proposition}\label{Prop:ExplicitPolyno}
Define the polynomials $a_{1}(z,\bar{z}),\ a_{2}(z,\bar{z}),\ p_{n}(z,\bar{z}),\ q_{n}(z,\bar{z})$ as
above. Then,
\eqref{ExpanLambda}-\eqref{ExpanGamma} holds and
moreover for $k=1,2,\cdots, N$
\begin{align}\label{eq:ZNequation}
\partial_{t}z_{k}+iE_k(\lambda)z_{k}=&-\left< \sum_{|m|=2,\ |n|=1}JN_{m,n} z^{m}\bar{z}^{n},
\left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}\right)\right>\\
&+
\frac{1}{2}\Upsilon\displaystyle\sum_{m=1}^{N}z_{m}\left<\left(
\begin{array}{lll}
-i\eta_{m}\\ \xi_{m} \end{array}\right),
\left(
\begin{array}{lll}
\eta_{k}\\
i\xi_{k}
\end{array}
\right)\right>+Remainder(t).\nn
\end{align}
\end{proposition}
\begin{proof}
Recall the convention that $Remainder$ represents any quantity satisfying
\begin{align}
\lesssim |z(t)|^{4}+\|\langle
x\rangle^{-\nu}\vec{R}(t)\|_{2}^{2}+\|\vec{R}(t)\|_{\infty}^{2}
+|z(t)|\ \|\langle
x\rangle^{-\nu}\tilde{R}(t)\|_{2}.
\end{align}
We start with casting the $\dot\lambda-$ and $\dot\gamma-$eqns in
\eqref{eq:gamma}, \eqref{eq:lambda} into a matrix form
\begin{align}\label{eq:lambdagamma}
[Id+M(z,\vec{R},p,q)]\left(
\begin{array}{lll}
\dot\lambda\\
\dot\gamma-\Upsilon
\end{array}
\right)=\Omega+Remainder
\end{align} where, the vector $\Omega$ is defined as
\begin{align}
\Omega:=\left(
\begin{array}{lll}
-\frac{1}{\langle
\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}[\ \langle
ImN,\phi^{\lambda}\rangle+\frac{i}{2}\Upsilon\langle\ (z-\bar{z})\cdot\eta,\phi^{\lambda}\ \rangle\ ]-\partial_{t}a_{1}\\
\frac{1}{\langle\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}[\langle
ReN,\partial_{\lambda}\phi^{\lambda}\rangle-\frac{1}{2}\Upsilon\langle\
(z+\bar{z})\cdot\xi,\partial_{\lambda}\phi^{\lambda}\ \rangle]-\Upsilon-\partial_{t}a_{2}+a_{1}
\end{array}
\right)
\end{align}
the term $Remainder$ is produced by the term
$\frac{\Upsilon}{\langle\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}\left(
\begin{array}{lll}
-\langle R_{1},\partial_{\lambda}\phi^{\lambda}\rangle\\
\langle R_{2},\phi^{\lambda}\rangle
\end{array}
\right),$ $Id$ is the $2\times 2$ identity matrix,
$M(z,\vec{R},p,q)$ is a matrix depending on $z,\vec{R},p$ and $q$
and satisfying the estimate
\begin{equation}\label{eq:Mterm}
\|M(z,\vec{R},p,q)\|= \cO(|z|)+Remainder.
\end{equation}
The smallness of the matrix $M$ makes $\big[Id+M\big]^{-1}$ uniformly bounded, hence by \eqref{eq:lambdagamma}
\begin{align}
|\dot\lambda|,\ |\dot\gamma-\Upsilon|\lesssim |\Omega|+Remainder.\label{eq:LamGam}
\end{align}
Next we estimate $\Omega$, and start with casting it into a convenient form.
The purpose of defining $a_{1}$ and $a_{2}$ in \eqref{eq:pkmn} is to
remove the lower order terms, in $z$ and $\bar{z}$, from $\langle
ImN,\phi^{\lambda}\rangle-\frac{i}{2}\Upsilon\langle\
(z-\bar{z})\cdot\eta,\phi^{\lambda}\rangle$ and $\langle
ReN,\partial_{\lambda}\phi^{\lambda}\rangle+\frac{1}{2}\Upsilon\langle\
(z+\bar{z})\cdot\xi,\partial_{\lambda}\phi^{\lambda}\rangle$ to get
\begin{align}
\Omega=D_{1}+D_{2}\label{eq:d1d2}
\end{align} with
$$D_{1}:=\frac{1}{\langle\phi^{\lambda},\partial_{\lambda}\phi^{\lambda}\rangle}\left(
\begin{array}{lll}
-\langle ImN-\displaystyle\sum_{|m|+|n|=2,3}N_{m,n}^{Im} z^{m}\bar{z}^{n},\phi^{\lambda}\rangle\\
\langle
ReN-\displaystyle\sum_{|m|+|n|=2,3}N_{m,n}^{Re} z^{m}\bar{z}^{n},\partial_{\lambda}\phi^{\lambda}\rangle
\end{array}
\right),$$ and
$$D_{2}:=-\displaystyle\sum_{|m|+|n|=2,3}\left(
\begin{array}{lll}
\partial_{t}(A_{m,n}^{(1)}z^m\bar{z}^{n})+iE(\lambda)\cdot (m-n)A_{m,n}^{(1)}z^m\bar{z}^{n}\\
\partial_{t}(A_{m,n}^{(2)}z^m\bar{z}^{n})+iE(\lambda)\cdot (m-n) A_{m,n}^{(2)}z^m\bar{z}^{n}
\end{array}
\right).$$ It is easy to see that
\begin{align}
D_{1}=Remainder.\label{eq:estD1}
\end{align}
To control $D_2$ we use a preliminary estimate from the $z_n-$equation in \eqref{eq:z}
\begin{align}
\partial_{t}z_{n}+iE_n(\lambda)z_{n}=\cO(|z|^{2})+Remainder\label{eq:preli}
\end{align} to obtain
\begin{align}
D_{2}=&-\dot\lambda\displaystyle\sum_{|m|+|n|=2,3}\left(
\begin{array}{lll}
\partial_{\lambda}A_{m,n}^{(1)}\ z^m\bar{z}^{n}\\
\partial_{\lambda}A_{m,n}^{(2)}\ z^m\bar{z}^{n}
\end{array}
\right)+\cO(|z|^{3})+Remainder\nn\\
=&\cO(|z|^{3})+Remainder\label{eq:estD2}
\end{align} here the term $\cO(|z|^3)$ is from the term $\cO(|z|^2)$ in \eqref{eq:preli}.
Collect the estimates above to obtain
\begin{align}
\dot\lambda,\ \dot\gamma-\Upsilon=&\cO(|z|^{3})+Remainder.\label{eq:LambdaGammaRough}
\end{align}
These estimates are still worse than the desired \eqref{ExpanLambda}-\eqref{ExpanGamma}. The reason is that their derivations depend on the non-optimal \eqref{eq:preli}. Next we improve it using \eqref{eq:LambdaGammaRough}.
Choose $p_{n}$ and $q_{n}$ as in
\eqref{eq:pq20and02} to remove the following lower order terms: $z^{m}\bar{z}^{n}$ satisfying $|m|+|n|=2,3$ and $(|m|,\ |n|)\not=(2,1),$ to obtain
\begin{align}\label{eq:ZnRough}
\partial_{t}z_{n}+iE_n(\lambda)z_{n}&=-\left\langle \sum_{|m|=2,\ |n|=1}JN_{m,n}z^{m}\bar{z}^{n}
+\frac{1}{2}\Upsilon\left(
\begin{array}{lll}
iz\cdot\eta\\
z\cdot\xi
\end{array}
\right), \left(
\begin{array}{lll}
\eta_{n}\\
-i\xi_{n}
\end{array}
\right)\right\rangle\nn\\
&+ D_{3}(n)+Remainder
\end{align}
where $D_{3}(n)$ is defined as
\begin{align}
D_{3}(n):=&-\sum_{|k|+|l|=2,3}[\partial_{t}(P_{k,l}^{(n)}z^{k}\bar{z}^{l})+i(k-l)\cdot E(\lambda)P_{k,l}^{(n)}z^{k}\bar{z}^{l}]\nonumber\\
&-i\sum_{|k|+|l|=2,3}[\partial_{t}(Q_{k,l}^{(n)}z^{k}\bar{z}^{l})+i(k-l)\cdot E(\lambda)Q_{k,l}^{(n)}z^{k}\bar{z}^{l}]\nonumber\\
=&-\dot\lambda \ \sum_{|k|+|l|=2,3}[\partial_{\lambda}P_{k,l}^{(n)}-\partial_{\lambda}Q_{k,l}^{(n)}]z^{k}\bar{z}^{l}\nonumber\\
&-\sum_{|k|+|l|=2,3}[P_{k,l}^{(n)}-Q_{k,l}^{(n)}]\ [\partial_{t}(z^{k}\bar{z}^{l})+i(k-l)\cdot E(\lambda)z^{k}\bar{z}^{l}]\nonumber\\
=& \Gamma_1+\Gamma_2
\end{align} and $\Gamma_1$ and $\Gamma_2$ naturally defined.
For $\Gamma_1$ the estimate for $\dot\lambda$ in \eqref{eq:LambdaGammaRough} implies that
\begin{align}
\Gamma_1=Remainder.\label{eq:estGamma1}
\end{align}
For $\Gamma_2$, the preliminary estimate in \eqref{eq:preli} implies
$$\Gamma_2=\cO(|z|^{3})+Remainder.$$ This, in turn, implies an estimate better than \eqref{eq:preli}
\begin{align}
\partial_t z_{n}+iE_n(\lambda)z_{n}=\cO(|z|^{3})+Remainder.\label{eq:betterZn}
\end{align} Sine the estimates derived for $\Gamma_2$ depends on non-optimal \eqref{eq:preli}, this optimal one enables us to find
\begin{align}
\Gamma_2=Remainder.
\end{align} This together with $\Gamma_1=Remainder$ in \eqref{eq:estGamma1} implies $$D_3(n)=\Gamma_1+\Gamma_2=Remainder.$$ Put this into the $\partial_{t}z_n$-eqn
in \eqref{eq:ZnRough} to obtain the desired estimate \eqref{eq:ZNequation}.
\eqref{eq:betterZn} also helps us to improve the estimate \eqref{eq:estD2} for $D_2$
\begin{align}
D_2=Remainder.
\end{align} This together with \eqref{eq:estD1}, \eqref{eq:d1d2} implies for $\Omega$ in \eqref{eq:d1d2}
$$\Omega=D_1+D_2=Remainder.$$
Put this into \eqref{eq:LamGam} to obtain
the desired estimates \eqref{ExpanLambda}, \eqref{ExpanGamma} for $\dot\lambda$ and $\dot\gamma-\Upsilon.$
\end{proof}
The following result has been applied in \eqref{defA11} to show that the numerator is proportional to the denominator. Similar result can be found in \cite{GaWe}.
\begin{lemma}\label{coordinate-lemma}
For $|m|=|n|=1,$ and $m,\ n\in (\mathbb{Z}^{+}\cup\{0\})^{N}$
\begin{align}
\langle N_{m,n}^{Im},\ \phi^{\lambda}\rangle=\frac{1}{4i}(n-m)\cdot E(\lambda)\ [\langle m\cdot\xi,\ n\cdot \xi\rangle+\langle m\cdot \eta, \ n\cdot \eta\rangle].
\label{eq:UnexpectedFact}
\end{align}
\end{lemma}
\begin{proof}
We start with deriving an expression for $N_{m,n}^{Im},\ |m|=|n|=1.$
The explicit form of $\displaystyle\sum_{|m|+|n|=2}JN_{m,n}z^{m}\bar{z}^{n}$ in
\eqref{eq:SecondOrderTerm} implies that
\begin{align*}
\sum_{|m|+|n|=2}N^{Im}_{m,n}z^{m}\bar{z}^{n}=&2\phi^{\lambda}A_{1}B_{1}\\
=&\frac{1}{2i}\phi^{\lambda}(\displaystyle\sum_{n=1}^{N}z_{n}\xi_{n}+\sum_{n=1}^{N}\bar{z}_{n}\xi_{n})
(\sum_{m=1}^{N}z_{m}\eta_{m}-\sum_{m=1}^{N}\bar{z}_{m}\eta_{m}).
\end{align*}
Take the relevant terms to obtain
\begin{align}
N^{Im}_{m,n}=\frac{1}{2i}\phi^{\lambda}(\xi_{n}\eta_{m}-\xi_{m}\eta_{n}),
\end{align} here we used the notation
\begin{align}
\xi_n=n\cdot \xi,\ \eta_n=n\cdot \eta,\ \text{for}\ n\in (\mathbb{Z}^{+}\cup\{0\})^{N}\ \text{and}\ |n|=1.\label{eq:notationXiEta}
\end{align}
Hence the left hand side of \eqref{eq:UnexpectedFact} takes a new form,
\begin{align*}
\langle
N^{Im}_{m,n},\phi^{\lambda}\rangle=&\frac{1}{2i}\int
(\phi^{\lambda})^{2}(\xi_{n}\eta_{m}-\xi_{m}\eta_{n})\\
=&\frac{1}{4i} \big[\langle [L_{+}(\lambda)-L_{-}(\lambda)]\xi_n, \eta_{m}\rangle-\langle [L_{+}(\lambda)-L_{-}(\lambda)]\xi_m, \eta_{n}\rangle\big],
\end{align*} where we used the fact that $L_{+}(\lambda)-L_{-}(\lambda)=2(\phi^{\lambda})^2$, see \eqref{eq:defLLambda}.
Use the facts $L_{+}(\lambda)$ and $L_{-}(\lambda)$ are self-adjoint, and $$L_{-}(\lambda)\eta_{n}\ =\ E_n(\lambda)\xi_{n},\
\text{and}\ L_{+}(\lambda)\xi_{n}\ =\ E_n(\lambda)\eta_{n},\
n=1,2,\cdot\cdot\cdot,N,$$ in \eqref{eq:eigenf}, and hence the desired result, recall the notations in \eqref{eq:notationXiEta},
\begin{align}
\langle
N^{Im}_{m,n},\phi^{\lambda}\rangle=&\frac{1}{4i} (E_n(\lambda)-E_m(\lambda))[\langle \xi_m,\ \xi_n\rangle+\langle \eta_m,\ \eta_n\rangle]\nn\\
=&\frac{1}{4i}(n-m)\cdot E(\lambda)\ [\langle m\cdot\xi,\ n\cdot \xi\rangle+\langle m\cdot \eta, \ n\cdot \eta\rangle]
\end{align}
\end{proof}
\section{Proof of the Normal Form Equation \eqref{eq:detailedDescription}}\label{sec:NFT}
We expand the first two terms on the right hand side of the equations for $z_{n}$ in
\eqref{eq:ZNequation} to obtain
\begin{align}
&-\left< \sum_{|m|=2, |n|=1}JN_{m,n} z^{m}\bar{z}^{n},
\left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}\right)\right>
+
\frac{1}{2}\Upsilon\displaystyle\sum_{m=1}^{N}z_{m}\left<\left(
\begin{array}{lll}
-i\eta_{m}\\ \xi_{m} \end{array}\right),
\left(
\begin{array}{lll}
\eta_{k}\\
i\xi_{k}
\end{array}
\right)\right>\nn\\
&=\sum_{l=1}^{5}\Theta_{l}(k),
\end{align} where, recall the definitions of $JN_{m,n},\ |m|+|n|=3,$ from
\eqref{eq:JNm+n=3}, $\Theta_1(k)$ is defined as
\begin{align}
\Theta_{1}(k) :=-\left< \overline{X_1}R_{2,0},\left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}
\right)\right>=-\left< \sum_{|m|=2}R_{m,0}z^{m},\ (\overline{X_1})^* \left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}
\right)\right>,\label{eq:Theta1k}
\end{align}
where, recall the definition of $2\times 2$ matrix $X$ in \eqref{eq:XDef} and we divide it into two terms
$X=X_{1}+\overline{X_{1}}$ with $X_1$ defined as
\begin{align}
X_{1}:=\left(
\begin{array}{lll}
-i\phi^{\lambda}\ z\cdot\eta,& \phi^{\lambda}\ z\cdot\xi\\
-3 \phi^{\lambda } z\cdot\xi,&
i\phi^{\lambda}\ z\cdot\eta
\end{array}
\right),\label{eq:defX1}
\end{align}
\begin{align}
\Theta_{2}(k):=&-\left< X_{1}\left(
\begin{array}{lll}
\displaystyle\sum_{n=1}^{N}\sum_{|m|=|l|=1}P_{m,l}^{(n)} z^{m}\bar{z}^{l}\xi_{n}\\
\displaystyle\sum_{n=1}^{N}\sum_{|m|=|l|=1}Q_{m,l}^{(n)}z^{m}\bar{z}^{l}\eta_{n}
\end{array}
\right),\ \left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}
\right)\right>\nn\\
&-\left< \overline{X_{1}}\left(
\begin{array}{lll}
\displaystyle\sum_{|m|=2}z^{m}[\sum_{n=1}^{N}P_{m,0}^{(n)}\xi_{n}+A_{m,0}^{(1)}\partial_{\lambda}\phi^{\lambda}]\\
\displaystyle\sum_{|m|=2}z^{m}[\sum_{n=1}^{N}Q_{m,0}^{(n)}\eta_{n}+A_{m,0}^{(2)}\phi^{\lambda}]
\end{array}
\right), \left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}
\right)\right>,\label{smallDi}
\end{align}
\begin{align*}
\Theta_{3}(k):=&-\frac{1}{8}\left\langle\
((z\cdot\xi)^{2}-(z\cdot\eta)^{2})\left(
\begin{array}{lll}
i\bar{z}\cdot\eta\\ -\bar{z}\cdot\xi
\end{array}
\right), \left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array} \right)\ \right\rangle\\
&+\frac{1}{4}\left\langle\
(|z\cdot\xi|^2+|z\cdot\eta|^2)\left(
\begin{array}{lll}
i z\cdot\eta\\ z\cdot\xi
\end{array} \right), \left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array} \right)\right\rangle,
\end{align*}
\begin{align*}
\Theta_{4}(k):=\frac{1}{2}\ \Upsilon\left\langle\left(
\begin{array}{lll}
-i z\cdot\eta\\
z\cdot\xi
\end{array}
\right), \left(
\begin{array}{lll}
\eta_{k}\\
i\xi_{k}
\end{array}
\right)\right\rangle,
\nn\end{align*}
\begin{align*}
\Theta_{5}(k):=-\left\langle \sum_{|m|=|n|=1}R_{m,n}z^{m}\bar{z}^{n},\ X^{*}_{1}\left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}
\right)\right
\rangle,
\end{align*}
where, recall the definition of real function $\Upsilon$ from
\eqref{ExpanGamma}.
The result is the following
\begin{proposition}\label{prop:smallDivisor}
$Re\ \displaystyle\sum_{k=1}^{N}\bar{z}_k \Theta_1(k)$ can be decomposed into three terms,
\begin{align}
Re\ \sum_{k=1}^{N}\bar{z}_k \Theta_1(k)
=&- C (e_0-\lambda)\ \Gamma(z,\bar{z})\nn\\
&+(e_0-\lambda) \sum_{|m|=|n|=2}C_{m,n}(\lambda)\ E(\lambda)\cdot(m-n)\ z^{m}\bar{z}^{n}\label{eq:ReThet1}\\
&+|e_0-\lambda|^2\ \cO(|z|^4) \nn
\end{align} where $C$ is a positive constant, $\Gamma(z,\bar{z})$ is the positive term defined in Fermi-Golden-rule \eqref{Gammadef}, $C_{m,n}(\lambda)$ are uniformly bounded constants and recall that $e_0-\lambda>0,$ in Lemma \ref{LM:groundNon}.
For $Re\ \displaystyle\sum_{k=1}^{N}\bar{z}_k \Theta_2(k)$, there exist some constants $D_{m,n}(\lambda)$ such that
\begin{align}
Re\ \sum_{k=1}^{N}\bar{z}_k \Theta_2(k)
=&(e_0-\lambda) \sum_{|m|=|n|=2}D_{m,n}(\lambda)\ E(\lambda)\cdot(m-n)\ z^{m}\bar{z}^{n};\label{eq:ReThet2}
\end{align}
For $n=3,4,5,$ we have
\begin{align}
Re\ \sum_{k=1}^{N}\bar{z}_k \Theta_n (k)=0.\label{eq:ReThet3}
\end{align}
\end{proposition}
The proposition will be proved in subsequent subsections.
Next we prove the desired result \eqref{eq:detailedDescription}.\\
\textbf{Proof of \eqref{eq:detailedDescription}}
By the estimates above we have that
\begin{align}
\partial_{t}|z|^2=&-2C(e_0-\lambda)\ \Gamma(z,\bar{z})+(e_0-\lambda) \sum_{|m|=|n|=2} C_{m,n} \ E(\lambda)\cdot (m-n)z^{m}\bar{z}^n\nn\\
&+\cO((e_0-\lambda)^2|z|^4)+|z|Remainer,
\end{align} with $C_{m,n}$ being some constant. Here we can take $\displaystyle\sum_{|m|=|n|=2} C_{m,n} \ E(\lambda)\cdot (m-n)z^{m}\bar{z}^n$ to be real since it is in the equation for real parameter $|z|^2$ and $C\Gamma(z,\bar{z})$ is real. This implies $$\sum_{|m|=|n|=2} C_{m,n} \ E(\lambda)\cdot (m-n)z^{m}\bar{z}^n=\sum_{|m|=|n|=2} \overline{C_{m,n}} \ E(\lambda)\cdot (m-n)\bar{z}^{m} z^n,$$ and hence forces
\begin{align}
C_{m,n}=-\overline{C_{n,m}}.\label{eq:realIma}
\end{align}
By observing that $$E(\lambda)\cdot (m-n)z^{m}\bar{z}^n=i\partial_t z^{m}\bar{z}^n+|z|Remainder,$$ the fact that $(e_0-\lambda) \displaystyle\sum_{|m|=|n|=2}i C_{m,n} z^{m}\bar{z}^{n}$ is real implied by \eqref{eq:realIma}, we can define a new nonnegative parameter $\tilde{z}$ satisfying $\tilde{z}^2=|z|^2-(e_0-\lambda) \displaystyle\sum_{|m|=|n|=2}i C_{m,n} z^{m}\bar{z}^{n}$ such that
\begin{align}
\partial_{t} \tilde{z}^2= -2C(e_0-\lambda)\ \Gamma(z,\bar{z})+\cO((e_0-\lambda)^2|z|^4)+|z|Remainer
\end{align}
which is the desired estimate \eqref{eq:detailedDescription}.
\begin{flushright}
$\square$
\end{flushright}
Next we prove Proposition \ref{prop:smallDivisor}. In the proof the following results, from \cite{GaWe2011}, will be used. Recall that the function $\phi$ is the ground state for $-\Delta+V$ with eigenvalue $-e_0$, the functions $\xi^{lin}_{k}$, $k=1,2,\cdots,N,$ are neutral modes with eigenvalues $-e_k.$
\begin{lemma}
There exist constants $C_0,\ C_1\in\mathbb{R}$ such that in the space $\langle x\rangle^{-4}\mathcal{H}^{2}$
\begin{align}
\phi^{\lambda}=&C_0 (e_0-\lambda)^{\frac{1}{2}} \phi+\cO(|e_0-\lambda|^{\frac{3}{2}}),\nn\\
\partial_{\lambda}\phi^{\lambda}=&C_1 (e_0-\lambda)^{-\frac{1}{2}}\phi+\cO(|e_0-\lambda|^{\frac{1}{2}}),\label{eq:LambdaPhi2}\\
&\frac{1}{\langle \phi^{\lambda},
\partial_{\lambda}\phi^{\lambda}\rangle}\lesssim 1.\nn
\end{align}
For the neutral modes we have
\begin{align}\label{eq:asympto}
\|\langle x\rangle^{4}(\eta_{m}-\xi_{m}^{lin})\|_{\mathcal{H}^2},\ \|\langle x\rangle^{4}(\xi_{m}-\xi_{m}^{lin})\|_{\mathcal{H}^2},\ \|\langle x\rangle^{4}(\xi_{m}-\eta_{m})\|_{\mathcal{H}^2}=\cO(|e_0-\lambda|).
\end{align} Recall $P_{c}^{lin}$ is the orthogonal project onto the essential spectrum of $-\Delta+V$, and $P_{c}^{\lambda}$ of \eqref{Pcdef} is the Riesz projection onto the essential spectrum of $L(\lambda)$
\begin{align}\label{eq:projection}
P_{c}^{\lambda}=P_{c}^{lin}\left(
\begin{array}{ll}
1&0\\
0&1
\end{array}
\right)+\cO(|e_0-\lambda|).
\end{align}
\end{lemma}
\subsection{Proof of \eqref{eq:ReThet1}}
We start from the definition of $\Theta_1(k)$ in \eqref{eq:Theta1k}.
Compute directly to obtain
\begin{align*}
(\overline{X_1})^* \left(
\begin{array}{lll}
\eta_{k}\\
-i\xi_{k}
\end{array}
\right)=\left(
\begin{array}{ll}
-i\phi^{\lambda} z\cdot \eta\ \eta_k+3i \phi^{\lambda} z\cdot \xi\ \xi_k\\
\phi^{\lambda} z\cdot \xi\ \eta_k+\phi^{\lambda} z\cdot \eta\ \xi_k
\end{array}
\right).
\end{align*}
Hence
\begin{align}
Re \sum_{k}\bar{z}_k\Theta_{1}(k)=-\left< \sum_{|m|=2}R_{m,0}z^{m},\ \left(
\begin{array}{lll}
-i\phi^{\lambda} (z\cdot \eta)^2+3i \phi^{\lambda} (z\cdot \xi)^2\\
2\phi^{\lambda} z\cdot \xi\ z\cdot \eta
\end{array}
\right)\right>.\label{eq:theta1k}
\end{align}
We extract its main part by define
\begin{align}
D:=Re \sum_{|m|=2}&\big\langle z^{m}\big[(-\Delta+V+\lambda)J+i E(\lambda)\cdot m-0\big]^{-1} P_c \phi (\xi^{lin})^m \left(
\begin{array}{lll}
-i\\
1
\end{array}
\right),\ \times\nn\\
&\phi (z\cdot \xi^{lin})^2\left(
\begin{array}{lll}
i\\
1
\end{array}
\right)
\big\rangle.\label{eq:defD}
\end{align}
Here $(\xi^{lin})^{m}$ is defined as, for $m=(m_1,\ m_2,\ \cdots, \ m_N)\in (\mathbb{Z}^{+}\cup\{0\})^{N}$,
\begin{align*}
(\xi^{lin})^{m}:=(\xi_1^{lin})^{m_1}(\xi_2^{lin})^{m_2}\cdots (\xi_N^{lin})^{m_N}.
\end{align*}
The result is
\begin{lemma}\label{LM:simplify}
\begin{align}
Re \sum_{k=1}^{N}\bar{z}_k\Theta_{1}(k)&=-C(e_0-\lambda ) \ D+\cO(|e_0-\lambda |^2 |z|^4).
\end{align}
\end{lemma}
The lemma will be proved in Part \ref{Pa:simplify} below.
Next we study the term $D$. To facilitate our estimate we diagonalize the matrix operator in \eqref{eq:defD}. Define a unitary matrix $U$ by
\begin{align}
U\ :=\ \frac{1}{\sqrt{2}}\left(
\begin{array}{lll}
1&i\\
i&1
\end{array}
\right),\label{Udef}
\end{align} then we have that
$$J=i U \sigma_3 U^*,$$ with $\sigma_3$ being the third Pauli matrix.
Inset the identity $UU^*=U^* U=Id$ into appropriate places in the inner product of $D$ to obtain, recall the convention that $\langle f,\ g\rangle=\int f(x)\bar{g}(x)\ dx,$
\begin{align}
D
=&2\sum_{|m|=|n|=2}Re \langle
\big[i(-\Delta+V+\lambda- E(\lambda)\cdot m)+0\big]^{-1} P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}
\rangle \ z^{m}\bar{z}^{n}\nn\\
=&2 \sum_{|m|=|n|=2}Im\langle\big[-\Delta+V+\lambda- E(\lambda)\cdot m-i0\big]^{-1} P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle\ z^{m}\bar{z}^{n}.\label{eq:secondRes}
\end{align}
To cast the expression into a convenient form, we use the following two simple facts, for any functions $f,\ g$ and real constant $h$, $Im\langle f,\ g\rangle=\frac{1}{2i}[\langle f,\ g\rangle-\langle g,\ f\rangle]$ and $\langle f,\ (-\Delta+V-h-i0)^{-1}g\rangle=\langle (-\Delta+V-h+i0)^{-1}f,\ g\rangle.$ Compute directly to obtain
\begin{align}
D=&\frac{1}{i} \sum_{|m|=|n|=2} \langle L(m,n)\ P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle\ z^{m}\bar{z}^{n}.\label{express}
\end{align}
with $L(m,n)$ being a linear operator defined as
$$L(m,n):=
[-\Delta+V+\lambda- E(\lambda)\cdot m-i0]^{-1}-[-\Delta+V+\lambda- E(\lambda)\cdot n+i0]^{-1}
$$
In studying \eqref{express}, the main tool is a well known fact that, see e.g. \cite{RSIII}, for any constant $h>0,$
\begin{align}
&\langle \frac{1}{i} \big[ [-\Delta+V-h^2-i0]^{-1}-[-\Delta+V-h^2+i0]^{-1}\big]\ P_c f,\ g\rangle\nn\\
=& C h\int_{\mathbb{S}^2} \widehat{f}(h\sigma) \ \overline{\widehat{g}}(h\sigma)\ d\sigma \label{eq:measure}
\end{align} where $C$ is a positive constant, and the complex function $\widehat{f}$ is defined as,
\begin{align}
\widehat{f}(k):= \int_{\mathbb{R}^3} f(x)\ e(x, k)\ dx,
\end{align} hence $\widehat{f}(h\sigma)$ is $\widehat{f}$ restricted to the sphere $|k|=h\sigma,\ \sigma\in \mathbb{S}^2.$ Here the complex function $e:\ \mathbb{R}^3\times \mathbb{R}^3\rightarrow \mathbb{C}$ is defined as $$e(x,k):=[1+(-\Delta-|k|^2-i0)^{-1}V(x)]^{-1}\ e^{ix\cdot k}.$$
We continue to study \eqref{express}.
For the easiest cases
$
E(\lambda)\cdot m= E(\lambda)\cdot n,
$ apply \eqref{eq:measure} directly to obtain
\begin{align}
\frac{1}{i} \langle L(m,n) P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle
=C C_m \int_{\mathbb{S}^2} \widehat{\phi(\xi^{lin})^m}(C_m\sigma) \ \overline{\widehat{\phi(\xi^{lin})^n}}(C_m\sigma)\ d\sigma \label{eq:directFa}
\end{align} with $C_{m}\in \mathbb{C}$ defined as $$C_m:=\sqrt{E(\lambda)\cdot m-\lambda},$$ here $E(\lambda)\cdot m-\lambda$ is positive by the conditions that $2e_k<e_0$ and $E_k(\lambda)\approx e_0-e_k ,\ k=1,2,\cdots, N,$ and $\lambda\approx e_0.$ Recall that $m\in (\mathbb{Z}^{+}\cup\{0\})^{N}$ and $|m|=2.$
For the cases $ E(\lambda)\cdot m\not= E(\lambda)\cdot n, $ we claim, for some constant $C_{m,n}$
\begin{align}
\frac{1}{i} \langle L(m,n) P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle
=&C C_m^{\frac{1}{2}} C_n^{\frac{1}{2}} \int_{\mathbb{S}^2} \widehat{\phi(\xi^{lin})^m}(C_m\sigma) \ \overline{\widehat{\phi(\xi^{lin})^n}}(C_n\sigma)\ d\sigma\nn\\
&+C_{m,n} E(\lambda)\cdot (m-n).\label{eq:claim88}
\end{align}
If the claim holds, this together with \eqref{eq:directFa}, Lemma \ref{LM:simplify} and the fact $E_k(\lambda)=e_0-e_k+\cO(e_0-\lambda)$ by \eqref{eq:Enen}, implies the desired result \eqref{eq:ReThet1}.
What is left is to prove the claim \eqref{eq:claim88}. We start with decomposing
the left hand side into two parts
\begin{align}
\frac{1}{i} \langle L(m,n) P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle=A+B
\end{align} with
\begin{align*}
A:=&\frac{1}{i} \langle L(m,m) P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle,\\
B:=&\frac{1}{i} \langle [L(m,n)-L(m,m)] P_c\phi (\xi^{lin})^{m},\ \phi (\xi^{lin})^{n}\rangle.
\end{align*} By \eqref{eq:measure}, it is easy to see that
\begin{align}
A=&C C_m \int_{\mathbb{S}^2} \widehat{\phi(\xi^{lin})^m}(C_m\sigma) \ \overline{\widehat{\phi(\xi^{lin})^n}}(C_m\sigma)\ d\sigma\\
=&C C_m^{\frac{1}{2}} C_n^{\frac{1}{2}} \int_{\mathbb{S}^2} \widehat{\phi(\xi^{lin})^m}(C_m\sigma) \ \overline{\widehat{\phi(\xi^{lin})^n}}(C_n\sigma)\ d\sigma+C_{m,n} E(\lambda)\cdot (m-n),\nn
\end{align} where $C_{m,n}$ is a constant, and in the second step we use that the functions $\widehat{\phi(\xi^{lin})^m}$ and the scalar $C_m$ depend smoothly on $E(\lambda)\cdot m$ and $E(\lambda)\cdot n$.
For $B$, it is easy to see that for some constant $D_{m,n},$ $$B=D_{m,n} E(\lambda)\cdot (m-n).$$
Collecting the estimates above, we prove the claim \eqref{eq:claim88}.
Hence the proof is complete.
\subsubsection{Proof of Lemma \ref{LM:simplify}}\label{Pa:simplify}
We rewrite the expression in \eqref{eq:theta1k} as
\begin{align}
Re\sum_{k=1}^{N}\bar{z}_k \Theta_1(k)=-\left<
\sum_{|m|=2} R_{m,0}z^{m},\ A
\right>\label{eq:Aori}
\end{align} with the vector function $A$ defined as
\begin{align*}
A:=\left(
\begin{array}{lll}
-i\phi^{\lambda} (z\cdot \eta)^2+3i \phi^{\lambda} (z\cdot \xi)^2\\
2\phi^{\lambda} z\cdot \xi\ z\cdot \eta
\end{array}
\right).
\end{align*}
Apply the estimates of $\phi^{\lambda},$ $\xi$ and $\eta$ in
\eqref{eq:LambdaPhi2} and \eqref{eq:projection} to obtain
\begin{align}
A=C_1 (e_0-\lambda)^{\frac{1}{2}}\phi (z\cdot \xi^{lin})^2\left(
\begin{array}{lll}
i \\
1
\end{array}
\right)+\cO((e_0-\lambda)^{\frac{3}{2}}|z|^2),\label{eq:expressA}
\end{align} here the expansion is in the space $\langle x\rangle^{-4}\mathcal{L}^2$ for some $\epsilon_0>0,$ $C_1>0$ is a constant.
Now we turn to $R_{m,0}$, which is defined as
\begin{align*}
R_{m,0}=\big[ L(\lambda)+iE(\lambda)\cdot m-0\big]^{-1} P_c^{\lambda} JN_{m,0};
\end{align*} and for $JN_{m,0}$ we have, from \eqref{eq:SecondOrderTerm},
\begin{align}
\sum_{|m|=2}JN_{m,0} z^{m}=\sum_{|m|=2}\left(
\begin{array}{lll}
ImN_{m,0}\\
-ReN_{m,0}
\end{array}
\right)z^{m}=\frac{1}{4} \left(
\begin{array}{cc}
-2i \phi^{\lambda} z\cdot \xi \ z\cdot \eta\\
-3\phi^{\lambda} (z\cdot \xi)^2+\phi^{\lambda} (z\cdot \eta)^2
\end{array}
\right)\label{eq:JNm0}
\end{align}
Now we extract the main part of $R_{m,0}$ and $JN_{m,0}$ by applying the estimates of $\phi^{\lambda},$ $\xi$ and $\eta$ and $P_c$ in
\eqref{eq:LambdaPhi2} and \eqref{eq:projection}, and use that
\begin{align*}
\big[ L(\lambda)+iE(\lambda)\cdot m-0\big]^{-1}=\big[(-\Delta+V+\lambda)J+i E(\lambda)\cdot m-0\big]^{-1}+\cO(e_0-\lambda)
\end{align*}
to obtain
\begin{align}
R_{m,0}
=& C_2 (e_0-\lambda)^{\frac{1}{2}}\ z^{m}\big[(-\Delta+V+\lambda)J+i E(\lambda)\cdot m-0\big]^{-1} P_c \phi (\xi^{lin})^m \left(
\begin{array}{lll}
-i\\
1
\end{array}
\right)\nn\\
&+\cO((e_0-\lambda)^{\frac{3}{2}}) z^{m}
\label{eq:rmo}
\end{align}
where $C_2>0$ is a constant, the expansion is in the space $\langle x\rangle^{4}\mathcal{L}^2$.
Put this and \eqref{eq:expressA} into \eqref{eq:Aori} to obtain the desired result.
\begin{flushright}
$\square$
\end{flushright}
\subsection{Proof of \eqref{eq:ReThet2}}
To illustrate the ideas we consider part of it, namely
\begin{align}
\sum_{k}\bar{z}_k\tilde\Theta_2(k):=&\left< \overline{X_{1}}\left(
\begin{array}{lll}
\displaystyle\sum_{|m|=2}z^{m}\sum_{n=1}^{N}P_{m,0}^{(n)}\xi_{n}\\
\displaystyle\sum_{|m|=2}z^{m}\sum_{n=1}^{N}Q_{m,0}^{(n)}\eta_{n}
\end{array}
\right), \
\left(
\begin{array}{lll}
z\cdot \eta\\
-iz\cdot \xi
\end{array}
\right)\right>\nonumber.
\end{align} The other part is similar, hence omitted.
Compute directly to obtain
\begin{align}
&\sum_{k}\bar{z}_k\tilde\Theta_2(k)\nn\\
=
&\left< \left(
\begin{array}{lll}
\displaystyle\sum_{|m|=2}z^{m}\sum_{n=1}^{N}P_{m,0}^{(n)}\xi_{n}\\
\displaystyle\sum_{|m|=2}z^{m}\sum_{n=1}^{N}Q_{m,0}^{(n)}\eta_{n}
\end{array}
\right), \
(\overline{X_{1}})^*\left(
\begin{array}{lll}
z\cdot \eta\\
-iz\cdot \xi
\end{array}
\right)\right>\nonumber\\
=&4\left< \left(
\begin{array}{lll}
\displaystyle\sum_{|m|=2}z^{m}\sum_{n=1}^{N}P_{m,0}^{(n)}\xi_{n}\\
\displaystyle\sum_{|m|=2}z^{m}\sum_{n=1}^{N}Q_{m,0}^{(n)}\eta_{n}
\end{array}
\right), \
i\left(
\begin{array}{lll}
\displaystyle\sum_{|l|=2}ReN_{l,0}z^{l}\\
\displaystyle\sum_{|l|=2}ImN_{l,0}z^{l}
\end{array}
\right)\right>\nonumber\\
=& -4i \sum_{n=1}^{N}\sum_{|m|=2}\sum_{|l|=2} [ P_{m,0}^{(n)} z^{m} \ \langle \xi_n,\ ReN_{l,0}z^{l}\rangle+Q_{m,0}^{(n)} z^{m} \ \langle \eta_n,\ ImN_{l,0}z^{l}\rangle ]
\end{align}
where, in the second last step we used the definition of $ReN_{l,0}$ and $ImN_{l,0}$, $|l|=2,$ in \eqref{eq:JNm0}. Next, relate the definitions of $P_{m,0}^{(n)}$ and $Q_{m,0}^{(k)}$ in \eqref{eq:pq20and02} to $\langle \eta_n,\ ImN_{m,0}\rangle$ and $\langle \xi_n,\ ReN_{m,0}\rangle$, and then take the real part for $\sum_{k}\bar{z}_k\tilde\Theta_2(k)$ to obtain
\begin{align}
Re\ \sum_{k}\bar{z}_k\tilde\Theta_2(k)
=& 4\sum_{n=1}^{N} \sum_{|m|=|l|=2}Re\big[ [P_{m,0}^{(n)} \overline{Q_{l,0}^{(n)}} -Q_{m,0}^{(n)}\ \overline{P_{l,0}^{(n)}} ]\ l\cdot E(\lambda)\ z^{m}\bar{z}^{l}\big]\nonumber\\
=& 2\sum_{n=1}^{N} \sum_{|m|=|l|=2}Re\big[ [P_{m,0}^{(n)} \overline{Q_{l,0}^{(n)}} -Q_{m,0}^{(n)}\ \overline{P_{l,0}^{(n)}} ]\ l\cdot E(\lambda)\ z^{m}\bar{z}^{l}\big]\nonumber\\
&+2\sum_{n=1}^{N} \sum_{|m|=|l|=2}Re\big[ [\overline{P_{m,0}^{(n)}} Q_{l,0}^{(n)} -\overline{Q_{m,0}^{(n)}}\ P_{l,0}^{(n)} ]\ l\cdot E(\lambda)\ \bar{z}^{m} z^{l}\big]\nonumber\\
=&2\sum_{n=1}^{N} \sum_{|m|=|l|=2}Re\big[ [P_{m,0}^{(n)} \overline{Q_{l,0}^{(n)}}-Q_{m,0}^{(n)}\ \overline{P_{l,0}^{(n)}} ]\ (l-m)\cdot E(\lambda)\ z^{m}\bar{z}^{l} \big]\nn\\
=&\sum_{|m|=|l|=2} C_{m,l}\ (l-m)\cdot E(\lambda) z^{m}\bar{z}^{l}\label{TildeTheta}
\end{align} where, in the second step we used a simple fact that $Re\ a=Re\ \bar{a}$, for any parameter $a$, for the second term in the second step, we interchange the indices $m$ and $l$ to obtain the third step, and in the last step $C_{m,l}$ are constants naturally defined.
The proof is complete by observing that \eqref{TildeTheta} is the desired result.
\subsection{Proof of \eqref{eq:ReThet3}}
It is easy to see that for $j=3,4,$ $ \displaystyle\sum_{k=1}^{N}\bar{z}_{k}\Theta_{j}(k)$ are purely imaginary. Hence by taking the real part,
\begin{align}
Re\ \sum_{k=1}^{N}\bar{z}_{k}\Theta_{j}(k)=0.
\end{align}
Next
we turn to $\Theta_5.$ Use the definition of $X_1$ and compute directly to obtain
\begin{align*}
\sum_{k=1}^{N}\bar{z}_{k}\Theta_{5}(k)=-\left\langle \sum_{|m|=|n|=1}R_{m,n}z^{m}\bar{z}^{n},\ \left(
\begin{array}{lll}
i\phi^{\lambda}|z\cdot \eta|^2+3i\phi^{\lambda} |z\cdot \xi|^2\\
\phi^{\lambda}[ \bar{z}\cdot \xi \ z\cdot\eta- z\cdot \xi\cdot \bar{z}\cdot \eta]
\end{array}
\right) \right\rangle
\end{align*}
By noticing that $\displaystyle\sum_{|m|=|n|=1}R_{m,n}z^{m}\bar{z}^{n}$ is real by definition, and that the other vector function is purely imaginary, we have that the inner product is purely imaginary, hence
\begin{align}
Re\ \sum_{k=1}^{N}\bar{z}_{k}\Theta_{5}(k)=0.
\end{align}
Collect the estimates above to complete the proof.
\section{Proof of the Main Theorem \ref{THM:MainTheorem}}\label{ProveMain}
The proof is almost identical to the part in \cite{GaWe}, hence we only sketch it.
We begin by introducing a family of space-time norms, $ Z(T),\ {\cal R}_j(T)$, for measuring the decay of the $z(t)$ and $\vec{R}(t)$ for $0\le t\le T$, with $T$ arbitrary and large.
We then prove that this family of norms satisfy a set of coupled inequalities,
from which we can infer the desired large time asymptotic behavior.
Define
\begin{equation}
T_{0}:=|z(0)|^{-1},\label{Todef}
\end{equation}
where, recall that $|z(0)|$ is the initial amount of mass of neutral modes, see Theorem \ref{THM:MainTheorem}.
Now we define the controlling functions:
\begin{equation}\label{majorant}
\begin{array}{lll}
Z(T):=\displaystyle\max_{t\leq T}(T_{0}+t)^{\frac{1}{2}}|z(t)|,& &
\mathcal{R}_{1}(T):=\displaystyle\max_{t\leq
T}(T_{0}+t)\|\langle x\rangle^{-\nu} \vec{R}(t)\|_{\mathcal{H}^{3}},\\
\mathcal{R}_{2}(T):=\displaystyle\max_{t\leq
T}(T_{0}+t)\|\vec{R}(t)\|_{\infty},& &
\mathcal{R}_{3}(T):=\displaystyle\max_{t\leq
T}(T_{0}^{\frac{2}{3}}+t)^{\frac{7}{5}}\|\langle x\rangle^{-\nu}\tilde{R}(t)\|_{2},\\
\mathcal{R}_{4}(T):=\displaystyle\max_{t\leq
T}\|\vec{R}(t)\|_{\mathcal{H}^{3}},& &
\mathcal{R}_{5}(T):=\displaystyle\max_{t\leq
T}\frac{(T_{0}+t)^{\frac{1}{2}}}{\log(T_{0}+t)}\|\vec{R}(t)\|_{3}\ .\\
\end{array}
\end{equation}
To estimate $\mathcal{R}_{k}$, $k=1,2,3,5,$ or to control the dispersion $\vec{R}$, we use the propagator estimate, namely for any function $g\in \mathcal{L}^{1},$
\begin{align}
\|e^{tL(\lambda)}P_c^{\lambda}g\|_{\infty}\lesssim t^{-\frac{3}{2}}\|g\|_1.
\end{align} To estimate the decay of $|z|$, we define a positive constant $q$ by $$q^2=|z|^2+F(z,\bar{z})$$ where $F(z,\bar{z})$ is real and of order $|z|^4$, as stated in Statement C of Theorem \ref{GOLD:maintheorem}. By Statement C of Theorem \ref{GOLD:maintheorem}
\begin{align}
\frac{d}{dt}|q|^2\leq -C|q|^{4}+\cdots.
\end{align} If the omitted part is small, then we have $|q|^{2}=|z|^2\lesssim (1+t)^{-1}.$
The details are identical to the corresponding parts in \cite{GaWe}, and the proof is tedious. Hence we omit the detail.
The results are:
\begin{proposition}\label{prop:majorants}
\begin{align*}
\mathcal{R}_{1}\lesssim &T_{0}\|\langle x\rangle^{\nu}\vec{R}(0)\|_{\mathcal{H}^{2}}+\mathcal{R}_{4}\mathcal{R}_{2}+Z^{2}+T_{0}^{-\frac{1}{2}}[Z^{3}+Z\mathcal{R}_{1}+\mathcal{R}_{1}^{2}+\mathcal{R}_{2}^{2}].\\
\mathcal{R}_{2}\lesssim &T_{0}\|\vec{R}(0)\|_{1}+T_{0}\|\vec{R}(0)\|_{\mathcal{H}^{2}}+Z^{2}+\mathcal{R}_{4}^{2}\mathcal{R}_{2}+T_{0}^{-\frac{1}{2}}[Z^{3}+Z\mathcal{R}_{1}+\mathcal{R}_{1}^{2}].\\
\mathcal{R}_{3}\lesssim &
T_{0}\|\langle x\rangle^{\nu}\vec{R}(0)\|_{2}+T_{0}^{\frac{3}{2}}|z|^{2}(0)+T_{0}^{-\frac{1}{20}}
(Z^{3}
+Z\mathcal{R}_{3}+Z\mathcal{R}_{1}+\mathcal{R}_{5}^{3}+R_{2}^{2}\mathcal{R}_{4}).\\
\mathcal{R}_{4}^{2}\lesssim &
\|\vec{R}(0)\|^{2}_{\mathcal{H}^{2}}+T_{0}^{-1}[\mathcal{R}_{1}^{2}+Z^{2}\mathcal{R}_{1}+Z^{2}\mathcal{R}_{1}^{2}+\mathcal{R}_{4}^{2}\mathcal{R}_{2}^{2}].\\
\mathcal{R}_{5}\lesssim & T_{0}\|\vec{R}(0)\|_{1}+T_{0}\|\vec{R}(0)\|_{\mathcal{H}^{2}}+Z^{2}+T_{0}^{-\frac{1}{2}}[\mathcal{R}_{5}^{2}\mathcal{R}_{2}+Z^{4}+Z\mathcal{R}_{3}+\mathcal{R}_{1}^{2}+\mathcal{R}_{2}^{2}].\\
Z(T) \lesssim &1 + \frac{1}{T_0^{2\over5}}\ Z(T)\ \left(\ Z(T)+ \cR_1^2(T)+\cR_2^2(T)+Z(T)\cR_3(T)\ \right)
\end{align*}
\end{proposition}
\subsection{Proof of the Main Theorem ~\ref{THM:MainTheorem}}\label{subsec:proofsubmaintheorem}
Our goal is to prove that the functions $Z$ and $\mathcal{R}_{k},\ k=1,2,\cdots,5$, are uniformly bounded, using the estimates in Proposition \ref{prop:majorants}.
Define
$M(T):=\displaystyle\sum_{n=1}^{4}\mathcal{R}_{n}(T)$ and
\begin{equation}\label{defineS}
S:=T_{0}(\|\vec{R}(0)\|_{\mathcal{H}^{2}}+\|\langle x\rangle^{\nu}\vec{R}(0)\|_{2}+\|\vec{R}(0)\|_1)+\|\vec{R}(0)\|_{\mathcal{H}^2}+T_0^{\frac{3}{2}}|z_0|^2,
\end{equation} where, recall the definition of $T_{0}$ after
\eqref{majorant}. By the conditions on the initial condition \eqref{InitCond}
we have that $\mathcal{R}_{4}(0)$ is small, $M(0)$, $Z(0)$ and $S$ are
bounded. Then by the estimates in Proposition \ref{prop:majorants} we
obtain
\begin{align}
M(T)+Z(T)\leq \mu(S), \text{and}\ \mathcal{R}_{4}\ll 1,
\end{align} where
$\mu$ is a bounded function if $S$ bounded. This together with the definitions of $\mathcal{R}_k,\ k=1,2,\cdots,5,$ implies that
\begin{align}
\|\langle x\rangle^{-\nu}\vec{R}\|_{2},\ \|\vec{R}\|_{\infty}\leq
c(T_{0}+t)^{-1},\ |z(t)|\leq c(T_{0}+t)^{-\frac{1}{2}}\label{equa619a}
\end{align} which is part of Statements (A) and (C) in Theorem
~\ref{THM:MainTheorem}. The rest of (A), Equation
\eqref{eq:detailedDescription1}, is proved in
\eqref{eq:detailedDescription}.
\begin{flushright}
$\square$
\end{flushright} | 123,072 |
CORY COX GROUP FT. CAILI O’DOHERTY
Drummer Cory Cox is a native of Houston, TX. Inspired by his family’s passion for music, he studied privately and worked with the youth choir at his church at age eight. Developing an interest in music history, he enrolled into the arts program at Johnston Middle School and The High School for The Performing and Visual Arts. Working with various ensembles and mentors that introduced him to jazz and world music, he was inspired to pursue a music career that would take him around the world. While Cory was in high school, he was selected to become a member of the Texas Music Educators Association and the Gibson Baldwin Grammy Jazz Ensemble where he performed with the High School Grammy Jazz Choir.
Gaining exposure to various music programs around the country lead to Cory receiving a scholarship from Blue Note recording artist Jason Moran to attend college and he was selected as a 2006-08 Brubeck Institute Fellow at the University of the Pacific in Stockton, CA. As a Brubeck Fellow, he received ensemble coaching from mentors Dave Brubeck, Joe Gilman, Fred Hersch, Robert Glasper, Bob Hurst, Christian McBride, Jeff “Tain” Watts, Eric Harland, Freddie Hubbard, Nicholas Payton, Miguel Zenon and Joshua Redman. The Brubeck Institute Jazz Quintet was selected as the 2007 Down Beat Magazine best college ensemble. Cory credits the Brubeck institute as one the most insightful musical experiences.
After completing the two-year fellowship program, Cory moved to NYC to finish his undergrad studies at the New School University. Since moving to NY, Cory has performed and toured with the Dave Brubeck Quartet, Reggie Workman, the Ben Flocks Quartet, John Ellis, Joel Frahm, Javier Vercher, Marcus Strickland, Jimmy Owens, John Raymond, the Hironobu Saito Group, Thana Pavelic, Lorenzo Conte and many others throughout the U.S, Europe, Switzerland, Croatia and Japan. Cory currently teaches privately and lead ensemble workshops at Stanford Jazz in California and Litchfield Jazz in Connecticut. He credits Jesus, his family, Keith Sanders, Sebastian Whittaker, Craig Green, Conrad Johnson, Warren Sneed and his peers as his greatest inspiration. | 104,612 |
German bank needs €30m from State for low-rate loans plan
A key report on proposals for the model in Ireland is due, writes Fearghal O'Connor
German bank Sparkasse plans to establish a €30m pilot project in Westmeath and Fingal, with each area twinned for support with an individual not-for-profit regional bank in Germany, the Sunday Independent has learned.
But the plans hang on a key report into community banking models that is with Department of Finance officials and expected to be published within a fortnight, according to sources.
Supporters of the model argue that, because it does not need to make a profit for shareholders, it could charge low German-style interest rates on mortgages and business loans. Sparkasse would provide expertise and support but the €30m initial funding would need to come from State resources.
If the report is positive towards the Sparkasse-type model, there are plans to establish a stakeholder group to prepare a funding model and adapt it to Irish regulations and culture, said a source.
But according to sources close to the process there are still numerous hurdles, with an expectation that key government officials may be wary of the disruptive impact such a model could have on the local lending operations of pillar banks.
Similar fears around competition also exist within the credit union movement.
But some there see potential synergies that could allow credit unions greater access to SME and mortgage lending, it is understood. At least three Irish delegations may visit Germany over the coming months to see the banking model in action, including the Oireachtas finance and business enterprise committees, as well as Fingal officials and councillors.
"We're anxious to see the bank working in practice and to meet local government to hear how it works for them," said Fingal Fine Gael councillor Tom O'Leary, who has backed the plan.
German local government officials sit on the governance boards of regional Sparkasse banks, but a separate executive board made up of qualified individuals is tasked with day to day running.
O'Leary said he was aware of a lot of support for the plan within his own party and that the selection of Fingal - close to Taoiseach Leo Varadkar's political power base - as a pilot area was significant.
Westmeath Labour TD Willie Penrose - also a vocal advocate of the model - told a recent Oireachtas agriculture committee meeting that it could provide money to people "at reasonable prices".
"I imagine the bank representatives will tell me they are doing that," said Penrose.
"If that is the case, why is there hue and cry throughout the country? A well-to-do farmer dealing with one of the institutions before the committee asked me to keep highlighting the need for a Sparkasse model or a similar model in this country to give 'those boyos' - that is what he called the banks - a rub of competition.
"The farmer in question works extremely hard. He works 18 hours a day. He is working the flesh off his bones to meet his commitments. He believes the banks are charging too much."
Sunday Indo Business | 28,331 |
TITLE: Meeting of a symmetric random walk and a random walk on integers
QUESTION [2 upvotes]: Let $\{X_n\}$ be a symmetric random walk on integers and $\{Y_n\}$ be a random walk on integers with transition probabilities:
$$p(k,k+1)=p,~p(k,k-1)=q=1-p.$$
Suppose $\{X_n\}$ and $\{Y_n\}$ are independent. If $X_0=a>0,~Y_0=0$, evaluate the probability that $\{X_n\}$ and $\{Y_n\}$ will ever meet.
Attempt. Let $\{(X_n,Y_n)\}$ be the $2D$ random walk on
$\mathbb{Z}^2$ with transition probabilities:
$$p\big((k,m),(k\pm 1,m+1)\big)=\frac{p}{2},~p\big((k,m),(k\pm 1,m-1)\big)=\frac{q}{2}.$$
If $T=\inf\{n\geqslant 0: (X_n,Y_n)\in \Delta\}$ is the time the two random walks first meet and
$$h(k,m)=P\big((X_n,Y_n)\in \Delta|(X_0,Y_0)=(k,m)\big)$$ then we seek for $h(a,0).$ We know that:
$$h(k,m)=\frac{p}{2}\,h(k+1,m+1)+\frac{p}{2}\,h(k-1,m+1)
+\frac{1-p}{2}\,h(k+1,m-1)+\frac{1-p}{2}\,h(k-1,m-1)$$
and $h(m,m)=0$ for all integers $m,~k$. How can we procced to the solution of this recurrence equation?
REPLY [0 votes]: Following the guideline by @Rhys Steele, let's take the $1D$ random walk $Z_n=\frac{X_n-Y_n}{2}$, defined by the semi-differences of $X_n,\,Y_n$, with transition probabilities:
$$p(k,k+1)=\frac{q}{2},~~p(k,k-1)=\frac{p}{2},~~p(k,k)=\frac{1}{2}.$$
Under this modelling we are looking for $P(T_0<+\infty|Z_0=a/2),$ where $$T_0=\inf\{n\geqslant 0: Z_n=0\}$$ the first time $Z_n$'s hit $0$, in other wording, $X_n$'s and $Y_n$'s coincide. Obviously the desired probability equals $0$, if $a$ is odd. For the case $a$ is even, set $h(k)=P(T_0<+\infty|Z_0=a)$ and $h$ is the least non-negative solution of the equation:
$$h(k)=\frac{q}{2}\,h(k+1)+\frac{1}{2}\,h(k)+ \frac{p}{2}\,h(k-1)$$
equivalently:
$$q\,h(k+1)-h(k)+p\,h(k-1)=0,$$
along with $h(0)=0.$ The solution to the above problem is not unique, but under the condition that $h$ is the least non-negative solution we get the formula: $$P_{a/2}(T_0<+\infty)=\left \{\begin {array}{ll}
1&~,~~0<p \leqslant 1/2\\
\displaystyle \left(\frac{p}{1-p}\right)^{\frac{a}{2}}&~,~~1/2<p<1\\
\end{array}
\right..$$ | 89,310 |
Friends, Followers, and Papal Selfies: The Church and the New Media
Tuesday, February 11, 2014
7 p.m. • Donahue Auditorium
Dolan Center for Science and Technology
John Carroll University
Free and open to the public
Long before Twitter, smartphones, and YouTube, the Second Vatican Council issued its Decree on Social Communications (Inter Mirifica)— calling the church to embrace and employ the modern media in its evangelizing mission. Who could have imagined what the next 50 years would bring? The digital revolution has totally transformed our way of communicating. New forms of social media are reshaping the way we understand ourselves, associate with others, and engage the world. What does this mean for the church? And what is the role of the “New Media” in the “New Evangelization” called for by Pope Francis?
Vincent J. Miller, Ph.D.
Gudorf Chair in Catholic Theology and Culture at the University of Dayton
Author of Consuming Religion: Christian Faith and Practice in a Consumer Culture and currently working on a book about the impact of globalization on religious belief and communities.
Sponsored by:
- The Jack and Mary Jane Breen Chair in Catholic Systematic Theology
- Department of Theology and Religious Studies
- Tim Russert Department of Communication and Theatre Arts | 373,575 |
Employment and Volunteer Opportunities
SHINE Community Services is currently recruiting the following positions:
TREASURER
Are you looking to use your significant business or governance experience to benefit your community?
Shine Community Service’s mission is to contribute to the social, physical and emotional wellbeing of those requiring a range of care services including in-home services, centre based activities and social outings.
We are looking for a Treasurer to oversee and provide strong financial governance, leadership and advice. This volunteer position will work closely with the CEO and Board Members to:
Ensure financial governance, risk mitigation and financial robustness are maintained;
Closely monitor the financial performance of the organisation, working closely with the organisation’s staff;
Ensure that requirements and duties specified in the Constitution and other governing documents are met.
Based in Cottesloe, Shine has strong ties with the Shire of Peppermint Grove and the Towns of Cottesloe, Claremont and Mosman Park. Our funding arrangements are unique in this industry, and add a layer of interest and challenge to the role.
A strong understanding of how not-for-profit organisations operate and previous financial experience is essential (CPA / CA qualifications are preferred). Exposure to the aged care services or aligned industries is beneficial.
To register your interest as volunteer Treasurer, please email your application to
Queries may be emailed, or for a confidential discussion please call Alison Garton, Chair, on 9253 5555. | 115,480 |
\begin{document}
\maketitle
\setcounter{section}{0}
\begin{abstract}
We introduce the conception of matched pairs of $(H, \beta)$-Lie
algebras, construct an $(H, \beta)$-Lie algebra through them. We
prove that the cocycle twist of a matched pair of $(H, \beta)$-Lie
algebras can also be matched.
\par\smallskip
{\bf 2000 MSC:} 17B62, 18D35
\par\smallskip
{\bf Keywords:} $(H, \beta)$-Lie algebra, Matched Pair, Cocycle
Twist.
\end{abstract}
\section{Introduction and Preliminaries}
A generalized Lie algebra in the comodule category of a cotriangular
Hopf algebra which included Lie superalgebras and Lie color algebras
as special cases has been studied by many authors, see \cite{BFM01,
FM94} and the references therein.
On the other hand, there is a general theory of matched pairs of Lie
algebras which was introduced and studied by Majid in \cite{Ma90,
Ma95}. It says that we can construct a new Lie algebra through a
matched pair of Lie algebras.
In this note, we introduce the conception of matched pairs of $(H,
\beta)$-Lie algebras, construct an $(H, \beta)$-Lie algebra through
them. Furthermore, we prove that the cocycle twist of a matched
pair of $(H, \beta)$-Lie algebras can also be matched.
We now fix some notation. Let $H$ be a Hopf algebra, write the
comultiplication $\Delta: H\to H\ot H$ by $\Delta(h)=\sum h\di\ot
h\dii$. When $V$ is a left $H$-comodule with coaction $\rho: V\to
H\ot V$, we write $\rho(v)=\sum v\moi\ot v\moo$. We frequently omit
the summation sign in the following context.
A pair $(H, \beta)$ is called a cotriangular Hopf algebra, if $H$ is
a Hopf algebra, $\beta: H\ot H \to k$ is a convolution-invertible
bilinear map satisfying for all $ h, g, l\in H$,
(CT1)\quad $\beta(h\di, g\di)g\dii h\dii=\beta(h\dii, g\dii)h\di
g\di$;
(CT2)\quad$ \beta(h, gl)=\beta(h\di, g)\beta(h\dii, l)$;
(CT3)\quad$ \beta(hg, l)=\beta(g, l\di)\beta(h, l\dii)$;
(CT4)\quad$ \beta(h\di, g\di)\beta(g\dii,
h\dii)=\varepsilon(g)\varepsilon(h)$.
\noindent A map satisfying (CT2)--(CT4) is called a skew-symmetric
bicharacter. Throughout this note, we always assume $H$ is a
cotriangular Hopf algebra that is commutative and cocommutative.
A convolution invertible map $\si: H\ot H\to k$ is called a left
cocycle if for all $h, g, l\in H$,
$$\si(h\di, g\di)\si(h\dii g\dii, l)=\si(g\di, l\di)\si(h, g\dii l\dii),$$
and a right cocycle if
$$\si(h\di g\di, l)\si(h\dii, g\dii)=\si(h, g\di l\di)\si(g\dii, l\dii).$$
\begin{definition} Let $(H, \beta)$ be a cotriangular
Hopf algebra. An $(H, \beta)$-Lie algebra is a left $H$-comodule
$\cll$ together with a Lie bracket
$[,]: \cll \ot \cll \to \cll$ which is an $H$-comodule morphism satisfying, for
all $a, b, c\in \cll$
(1) $\beta$-anticommutativity:
$$ [a,b] =-\beta(a\moi, b\moi)[b\moo, a\moo],$$
(2) $\beta$-Jacobi identity:
$$[[a, b],c]+\beta(a\moi, b\moi c\moi)[[b\moo,c\moo],a\moo]+\beta(a\moi b\moi, c\moi)[[c\moo, a\moo], b\moo]=0.$$
\end{definition}
When $H=k\mathbb{Z}_2$, $\beta(x, y)=(-1)^{xy}$, for all $x, y\in
\mathbb{Z}_2$, this is exactly Lie superalgebra. When $H=kG$, where
$G$ is an abelian group with a bicharacter $\beta: G\times G\to
k^{*}$ such that $\beta(h, g)=\beta(g, h)^{-1}$ for all $h, g\in G$,
this is exactly Lie color algebra studied in \cite{CSO06, Sch79}.
\begin{example}\rm Let $A$ be a left $H$-comodule algebra. Define $[,
]_{\beta}$ to be $[a, b]_{\beta}:=ab- \sum \beta(a\moi, b\moi)$
$b\moo a\moo$. Then $(A, [, ]_{\beta})$ is an $(H, \beta)$-Lie
algebra and is denoted by $A_{\beta}$.
\end{example}
\begin{definition} Let $(H, \beta)$ be a cotriangular Hopf
algebra. An $(H, \beta)$-Lie coalgebra is a left $H$-comodule $\caa$
together with a Lie cobracket $ \delta: \caa \to \caa \ot \caa$
which is an $H$-comodule morphism satisfying,
(1) $\beta$-anticocommutativity:
$$ \delta(a) =-\beta(a\li\moi, a\lii\moi) a\lii\moo\ot a\li\moo,$$
(2) $\beta$-co-Jacobi identity:
\begin{eqnarray*}
a\li\li\ot a\li\lii \ot a\lii+\beta(a\li\li\moi a\li\lii\moi, a\lii\moi)a\lii \ot a\li\li\ot a\li\lii\\
+\beta(a\li\li\moi, a\li\lii\moi a\lii\moi)a\li\lii \ot a\lii \ot a\li\li=0.
\end{eqnarray*}
\noindent where we use the notion $\de(a)=\sum a\li\ot a\lii$ for
all $a\in \caa$.
\end{definition}
\begin{example}\rm Let $C$ be a left $H$-comodule coalgebra.
Define $\de_{\beta}:C\to C\ot C$ to be
$$\de_{\beta}(c)=\sum c\di\ot
c\dii- \beta(c\di\moi, c\dii\moi)c\dii\moo\ot c\di\moo.$$
Then $(C, \de_{\beta})$ is an $(H, \beta)$-Lie coalgebra and is denoted by
$(C_{\beta}, \de_{\beta})$.
\end{example}
\begin{proposition} Let $H$ be a Hopf algebra with a
skew-symmetric bicharacter $\beta: H\ot H\to k$, and suppose $\si:
H\ot H\to k$ is a left cocycle.
(a) Define $H_{\si}$ to be $H$ as a coalgebra, with
multiplication defined to be
$$h\cdot_{\si}l:=\si\inv(h\di, l\di)h\dii l\dii\si(h\diii, l\diii).$$
Then $H$ (with a suitable
antipode) is a Hopf algebra.
(b) Define the map $\beta\dsi: H\dsi\ot H\dsi\to k$ by, for all $h, l\in H$,
$$\beta\dsi(h, l):=\si\inv(l\di, h\di)\beta(h\dii, l\dii)\si(h\diii, l\diii).$$
If $(H,\beta)$ is cotriangular, then
$(H\dsi,\beta\dsi)$ is also cotriangular.
(c) If $A$ is a left $H$-comodule algebra, define $A\usi$ to be $A$
as a vector space and $H\dsi$-comodule, with multiplication given by:
$$a\cdot\usi b:=\si(a\moi, b\moi)a\moo b\moo.$$
Then $A\usi$ is an $H\dsi$-comodule algebra.
\end{proposition}
\begin{definition} \label{dfnlb} An $(H, \beta)$-Lie bialgebra
$\chh$ is a vector space equipped simultaneously with an $(H,
\beta)$-Lie algebra structure $(\chh, [,])$ and an $(H, \beta)$-Lie
coalgebra $(\chh, \delta)$ structure such that the following
compatibility condition is satisfied:
(LB):
\begin{eqnarray*}
\delta([a, b])&=& [a, b\li]\ot b\lii+ \beta(a\moi, b\li\moi) b\li\moo\ot[a\moo, b\lii]\\
&&+ a\li\ot [a\lii, b] +\beta(a\lii\moi, b\moi) [a\li, b\moo]\ot a\lii\moo.
\end{eqnarray*}
We denoted it by $(\chh, [,], \delta)$. \end{definition}
\section{Matched Pair of $(H, \beta)$-Lie Algebras}
Let $\caa, \chh$ be both $(H, \beta)$-Lie algebras. For $a, b\in \caa$, $h, g\in \chh$, denote maps $\trr :\chh \otimes \caa \to \caa$, $\trl :\chh \otimes \caa \to \chh$, by $ \trr (h \otimes a) = h \trr a$, $ \trl (h \otimes a) = h \trl a$. If $\chh$ is an $(H, \beta)$-Lie algebra and the map $\trr:\chh\ot \caa \to \caa$ satisfying
$$[h, g]\trr a=h\trr g\trr a-\beta(h\moi, g\moi)g\moo\trr h\moo\trr a ,$$
then $\caa$ is called a left $\chh$-module. Note that
when considering $(H, \beta)$-Lie algebras, all action maps must be
$H$-comodule maps. Thus for $h\in \chh, a\in \caa$, we have
$$\rho(h\trr a)=\sum h\moi a\moi\ot h\moo\trr a\moo.$$
If $\caa$ is an $\chh$-module Lie algebra, then
$$h\trr [a, b]=[h\trr a,b]+\beta(h\moi, a\moi)[ a\moo, h\moo\trr b]$$
and if $\caa $ is an
$\chh$-module Lie coalgebra, then
$$\de(h\trr a)=h\trr a\li\ot a\lii+\beta(h\moi, a\li\moi)a\li\moo\ot h\moo\trr a\lii.$$
\begin{definition} Let $\caa$ and $\chh$ be $(H,\beta)$-Lie algebras. If $\caa$ is a left $\chh$-module, $\chh$ is
a right $\caa$-module, and the following (BB1) and (BB2) hold, then
$(\caa, \chh)$ is called a\emph{ matched pair of $(H,\beta)$-Lie
algebras}.
(BB1):
$$h\trr [a, b]=[h\trr a, b]+\beta(h\moi, a\moi)[ a\moo, h\moo\trr
b]+(h\trl a)\trr b-\beta(a\moi, b\moi)(h\trl b\moo)\trr a\moo,$$
(BB2):
$$[h, g]\trl a=[h, g\trl a]+\beta(g\moi, a\moi)[h\trl a\moo,
g\moo]+h\trl (g\trr a)-\beta(h\moi, g\moi)g\moo\trl(h\moo\trr a).$$
\end{definition}
\begin{theorem}\label{th1}
If $(\caa, \chh)$ is a matched pair of $(H, \beta)$-Lie algebras,
then the double cross sum $\caa\bowtie \chh$ form an $(H, \beta)$-Lie
algebra which equals to $\caa\oplus \chh$ as linear space, but with Lie bracket
\begin{eqnarray*} [a\oplus h, b\oplus g]
&=&([a, b]+ h\trr b-\beta(g\moi, a\moi)g\moo\trr a\moo)\\
&&\oplus ([h, g]+h\trl b-\beta(g\moi, a\moi)g\moo\trl a\moo).
\end{eqnarray*}
\end{theorem}
\begin{proof} We show that the $\beta$-Jacobi identity holds for
$\caa \bowtie \chh$. By definition, $$[h, a]=h\trr a+h\trl a, [a,
h]=-\beta(a\moi, h\moi)a\moo\trr h\moo-\beta(a\moi, h\moi)a\moo\trl
h\moo.$$
Thus we have $$[[h, g], a]=[h, g]\trr a+[h, g]\trl a,$$
for alll $h, g\in \chh, a\in \caa$, and for the second item of $\beta$-Jacobi identity
\begin{eqnarray*}
&&\beta(h\moi g\moi, a\moi)[[a\moo, h\moo], g\moo]\\
&=&\beta(h\moii g\moii, a\moii)\beta(a\moi, h\moi)\beta((h\moo\trr
a\moo)\moi, g\moi)\\
&& g\moo\trr (h\moo\trr a\moo)\moo-\beta(h\moii g\moii,
a\moii)\beta(a\moi, h\moi)\\
&&[h\moo\trl a\moo, g\moo]+\beta(h\moii g\moii, a\moii)\beta(h\moi,
a\moi)\\
&&\beta((h\moo\trr a\moo)\moi, g\moi)g\moo\trl(h\moo\trr a\moo)\moo
\end{eqnarray*}
The right hand side is equal to:
\begin{eqnarray*}
&&\mbox{1st of RHS}\\
&=&\beta(h\moiii g\moii, a\moiii)\beta(a\moii,
h\moii)\beta(h\moi a\moi, g\moi)g\moo\trr h\moo\trr a\moo\\
&=&\beta(h\moiii, a\moiii)\beta(g\moiii, a\moiv)\beta(a\moii, h\moii)\\
&&\times\beta(h\moi, g\moi)\beta(a\moi, g\moii) g\moo\trr h\moo\trr a\moo\\
&=&\beta(g\moiii, a\moii)\beta(h\moi, g\moi)\beta(a\moi, g\moii) g\moo\trr h\moo\trr a\moo\\
&=&\beta(g\moii, a\moii)\beta(h\moi, g\moiii)\beta(a\moi, g\moi) g\moo\trr h\moo\trr a\moo\\
&=&\beta(h\moi, g\moi)g\moo\trr h\moo\trr a
\end{eqnarray*}
where we use the fact that $\trr: \chh\ot \caa\to \caa$ is a left
$H$-comodule map in the first equality, (CT2) and (CT3) for $\beta$
in second equality, the cocommutative of $H$ in the fourth equality
and (CT4) for $\beta$ in the fifth equality. Similarly,
$$\mbox{3rd of RHS} =\beta(h\moi, g\moi)g\moo\trl (h\moo\trr a).$$
and
\begin{eqnarray*}
\mbox{2ed of RHS} &=&-\beta(h\moii, a\moii)\beta(g\moii, a\moiii)\beta(a\moi, h\moi)[h\moo\trl a\moo, g\moo] \\
&=&-\beta(g\moi, a\moi)[h\trl a\moo, g\moo].
\end{eqnarray*}
As for the third item of $\beta$-Jacobi identity,
\begin{eqnarray*}
&&\beta(h\moi, g\moi a\moi)[[g\moo, a\moo], h\moo]\\
&=&-\beta(h\moii, g\moi a\moi)\beta((g\moo\trr a\moo)\moi,
h\moi)h\moo\trr(g\moo\trr a\moo)\moo\\
&&-\beta(h\moii, g\moi a\moi)\beta((g\moo\trr a\moo)\moi,
h\moi)h\moo\trl(g\moo\trr a\moo)\moo\\
&&+\beta(h\moi, g\moi a\moi)[g\moo\trl a\moo, h\moo]
\end{eqnarray*}
The right hand side is equal to
\begin{eqnarray*}
&&\mbox{1st of RHS}\\
&=&-\beta(h\moii, g\moii a\moii)\beta(g\moi a\moi, h\moi)h\moo\trr(g\moo\trr a\moo)\\
&=&-\beta(h\moiv, g\moii)\beta(h\moiii, a\moii)\beta(g\moi, h\moii)\\
&&\times\beta(a\moi, h\moi)h\moo\trr(g\moo\trr a\moo)\\
&=&-\beta(h\moiv, g\moii)\beta(h\moii, a\moii)\beta(g\moi, h\moiii)\\
&&\beta(a\moi, h\moi)h\moo\trr(g\moo\trr a\moo)\\
&=&-\beta(h\moii, g\moii)\beta(g\moi, h\moi)h\moo\trr(g\moo\trr
a\moo)\\
&=&-h\trr(g\trr a)
\end{eqnarray*}
Similarly,
$$\mbox{2ed of RHS} =-h\trl(g\trr a)$$
and
\begin{eqnarray*}
&&\mbox{3rd of RHS}=\\
&=&-\beta(h\moii, g\moi a\moi)\beta((g\moo\trl a\moo)\moi, h\moi)[h\moo, (g\moo\trl a\moo)\moo]\\
&=&-\beta(h\moii, g\moii a\moii)\beta(g\moi a\moi, h\moi)[h\moo, g\moo\trl a\moo]\\
&=&-\beta(h\moiv, g\moii)\beta(h\moiii, a\moii)\beta(g\moi, h\moi)\\
&&\times\beta(a\moi, h\moii)[h\moo, g\moo\trl a\moo]\\
&=&-\beta(h\moii, g\moii)\beta(h\moiii, a\moii)\beta(g\moi,h\moi)\\
&&\times\beta(a\moi, h\moiv)[h\moo, g\moo\trl a\moo]\\
&=&-\beta(h\moii, a\moii)\beta(a\moi, h\moi)[h\moo, g\moo\trl a\moo]\\
&=&-[h, g\trl a].
\end{eqnarray*}
Now by (BB2) and $\caa$ is a left $\chh$-module we have that the sum
of three item equals to zero. The other cases can be checked
similarly.\end{proof}
\begin{proposition} Assume that $\caa$ and $\chh$ are $(H,
\beta)$-Lie bialgebras, $(\caa, \chh)$ is a matched pair of $(H,
\beta)$-Lie algebras; $\caa$ is a left $\chh$-module Lie coalgebra;
$\chh$ is a right $\caa$-module $(H, \beta)$-Lie coalgebra. If
$(\id_{\chh}\ot \trl)(\delta_{\chh}\ot \id_{\caa})+(\trr
\ot\id_{\caa} )( \id_{\chh}\ot\delta_{\caa})=0$, i.e.
$\begin{array}{lc} (BB3):\ & \sum h\li\ot h\lii\trr a+\sum h\trl
a\li\ot a\lii=0,
\end{array}$
\noindent then $\caa\bowtie \chh$ becomes an $(H, \beta)$-Lie
bialgebra.
\end{proposition}
\begin{proof} The Lie algebra structure is as in theorem \ref{th1}.
The Lie cobracket is the one inherited from $\caa$ and $\chh$.
$\caa$ and $\chh$ are also $(H, \beta)$-Lie sub-bialgebras of
$\caa\bowtie \chh$. So we only check equation (LB) on $\caa\ot
\chh$. For $h\in \chh, a\in \caa$, $\de[h, a]=\de( h\trr a)+\de
(h\trl a)$, and by the ad-action on tenor product
\begin{eqnarray*}
&& h\trr \de(a)+\de(h)\trl a\\
&=& h\trr a\li\ot a\lii(1) +h\trl a\li\ot a\lii(2)+\beta(h\moi, a\li\moi)\\
&&a\li\moo\ot h\moo\trr a\lii(3)+ \beta(h\moi, a\li\moi)a\li\moo\ot h\moo\trl a\lii(4)\\
&&+h\li\ot h\lii \trl a(5) +h\li\ot h\lii \trr a(6) +\beta(h\lii\moi, a\moi)\\
&&h\li\trl a\moo\ot h\lii\moo(7)+\beta(h\lii\moi, a\moi)h\li\trr
a\moo\ot h\lii\moo(8)
\end{eqnarray*}
By (BB3), $(2)+(6)=0$, $(4)+(8)=0$. For the remaining four terms,
$\de( h\trr a)=(1)+(3)$ and $\de (h\trl a)=(5)+(7)$.
\end{proof}
\section{Cocycle Twists of Matched Pairs of $(H, \beta)$-Lie Algebras}
The cocycle twist $\caa\usi$ of an $(H, \beta)$-Lie algebra $\caa$
was introduced in \cite{BFM01}. It is an $(H, \beta\dsi)$-Lie
algebra with the map $[, ]\usi: \caa\ot \caa\to \caa$ given by
$$[a, b]\usi=\si(a\moi, b\moi)[a\moo, b\moo],$$
where $ \forall a, b\in \caa$.
If in addition, $\caa$ is a left $\chh$-module $\trr :\chh \otimes \caa \to \caa$, then we obtain that $\caa\usi$ is a
left $\chh\usi$-module $\trr\usi :\chh \otimes \caa \to \caa$,
$$h\trr\usi a=\si(h\moi, a\moi)h\moo\trr a\moo.$$
See \cite[Propostion 4.7]{BFM01}. Similarly, $\caa$ is a right $\chh$-module $\trl :\caa \otimes \chh \to \chh$, then we obtain a right $\caa\usi$-module $\trl\usi
:\chh \otimes \caa \to \chh$ by
$$h \trl\usi a=\si(h\moi,a\moi)h\moo\trl a\moo.$$
We now prove that the cocycle twist of a
matched pair of $(H, \beta)$-Lie algebras can also matched.
\begin{theorem}
If $(\caa, \chh)$ is a matched pair of $(H, \beta)$-Lie algebras,
then $(\caa\usi, \chh\usi)$ is a matched pair of $(H, \beta\dsi)$-Lie algebras.
Furthermore, their double cross sum $\caa\usi\bowtie \chh\usi$ form an $(H, \beta\dsi)$-Lie
algebra.
\end{theorem}
\begin{proof} Note that the bracket in $\caa\usi\bowtie
\chh\usi$ is given by
\begin{eqnarray*}
{[a, b]}\usi&=&\si(a\moi, b\moi)[a\moo, b\moo],\\
{[h, a]}\usi&=&\si(h\moi, a\moi)h\moo\trr a\moo+\si(h\moi,a\moi)h\moo\trl a\moo,\\
{[a, h]}\usi&=&-\si(h\moii,a\moii)\beta(h\moi, a\moi)h\moo\trr a\moo-\si(h\moii,a\moii)\beta(h\moi, a\moi)h\moo\trl a\moo.
\end{eqnarray*}
We check that the matched pair conditions (BB1) and (BB2) are valid on $(\caa\usi,
\chh\usi)$. We want to obtain that
\begin{eqnarray*}
h\trr\usi [a, b]\usi&=&[h\trr\usi a, b]\usi+\beta\dsi(h\moi, a\moi)[ a\moo, h\moo\trr\usi b]\usi\\
&&+(h\trl\usi a)\trr\usi b-\beta\dsi(a\moi, b\moi)(h\trl\usi b\moo)\trr\usi a\moo.
\end{eqnarray*} In fact,
\begin{eqnarray*} h\trr\usi [a, b]\usi &=&\si(h\moi, [a\moo,
b\moo]\moi)\si(a\moi,
b\moi)h\moo\trr [a\moo, b\moo]\moo\\
&=&\si(h\moi, a\moi b\moi)\si(a\moii, b\moii)h\moo\trr [a\moo,
b\moo]
\end{eqnarray*}
and \begin{eqnarray*} [h\trr\usi a, b]\usi &=& \si(h\moi,
a\moi)\si((h\moo\trr a\moo)\moi,
b\moi)[(h\moo\trr a\moo)\moo, b\moo]\\
&=& \si(h\moii, a\moii)\si(h\moi a\moi, b\moi)[h\moo\trr a\moo,
b\moo]
\end{eqnarray*}
Similarly, we get $ (h\trl\usi a)\trr\usi b = \si(h\moii,
a\moii)\si(h\moi a\moi, b\moi)(h\moo\trl a\moo)\trr b\moo$. Also,
\begin{eqnarray*}
&&\beta\dsi(h\moi, a\moi)[ a\moo, h\moo\trr\usi b]\usi\\
&=& \si(h\moiii, a\moiii)\beta(h\moii, a\moii)\si(a\moi, h\moi) [
a\moo, h\moo\trr\usi b]\usi\\
&=& \si(h\mov, a\moiv)\beta(h\moiv, a\moiii)\si(a\moii, h\moiii)\\
&&\si(h\moii, b\moii)\si(a\moi, h\moi b\moi)[a\moo, h\moo\trr b\moo]\\
&=& \si(h\mov, a\mov)\beta(h\moiv, a\moiv)\si\inv(a\moiii, h\moiii)\\
&&\si(a\moii, h\moii)\si(a\moi h\moi, b\moi)[a\moo, h\moo\trr b\moo]\\
&=& \si(h\moiii, a\moiii)\beta(h\moii, a\moii)\si(h\moi a\moi, b\moi)[a\moo, h\moo\trr b\moo]\\
&=& \si(h\moiii, a\moiii)\si(h\moii a\moii, b\moii)\beta(h\moi,
a\moi)[a\moo, h\moo\trr b\moo]
\end{eqnarray*}
Similarly, we get
\begin{eqnarray*}
\beta\dsi(a\moi, b\moi)(h\trl\usi b\moo)\trr\usi
a\moo&=&\si(h\moiii, a\moiii)\si(h\moii a\moii, b\moii)\\
&&\times\beta(h\moi,a\moi)(h\moo\trl b\moo)\trr a\moo.
\end{eqnarray*}
Now by the cocycle condition
of $\si$ and (BB1) on $(\caa, \chh)$, we get the result. Similar
argument can be performed for (BB2) on $(\caa\usi, \chh\usi)$.
\end{proof}
We now give the relationship between the $(H, \beta)$-Lie algebra
$\caa\usi\bowtie \chh\usi$ and $ (\caa\bowtie \chh)\usi$ by the
following theorem, the proof can easily be seen from their
construction so we omit it.
\begin{theorem}
If $(\caa, \chh)$ is a matched pair of $(H, \beta)$-Lie algebras,
then $\caa\usi\bowtie \chh\usi\cong (\caa\bowtie \chh)\usi$.
\end{theorem}
\section*{Acknowledgements}
The author would like to thank the referee for helpful comments and
suggestions. | 166,091 |
Peer dev, 2017-16
I was really stoked when Jerrod asked me to create the new logo for (In)disposable, later was renamed to Peer Dev., which originated as concept blog featuring people and their photos taken via disposable cameras. Jerrod was looking for a weathered and narrative-focused logo, something experimental with relatable elements of the midwest where this idea was born.
Final logo
Early iteration
Wallpaper pattern | 175,521 |
\begin{document}
\title{\LARGE Study of Robust Adaptive Power Allocation Techniques for Rate Splitting based MU-MIMO systems }
\author{A. R. Flores and R. C. de Lamare \vspace{-5.25em}
\thanks{This work was partially supported by the National Council for Scientific and Technological Development (CNPq), FAPERJ, FAPESP, and CGI.}\thanks{A. R. Flores is with the Centre for Telecommunications Studies, Pontifical
Catholic University of Rio de Janeiro, Rio de Janeiro 22451-900, Brazil (e-mail: [email protected]).}\thanks{R. C. de Lamare is with the Centre for Telecommunications Studies,
Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro 22451-900,
Brazil, and also with the Department of Electronic Engineering, University
of York, York YO10 5DD, U.K. (e-mail: [email protected]).}}
\maketitle
\begin{abstract}
Rate splitting (RS) systems can better deal with imperfect channel state information at the transmitter (CSIT) than conventional approaches. However, this requires an appropriate power allocation that often has a high computational complexity, which might be inadequate for practical and large systems. To this end, adaptive power allocation techniques can provide good performance with low computational cost. This work presents novel robust and adaptive power allocation technique for RS-based multiuser multiple-input multiple-output (MU-MIMO) systems. In particular, we develop a robust adaptive power allocation based on stochastic gradient learning and the minimization of the mean-square error between the transmitted symbols of the RS system and the received signal. The proposed robust power allocation strategy incorporates knowledge of the variance of the channel errors to deal with imperfect CSIT and adjust power levels in the presence of uncertainty. An analysis of the convexity and stability of the proposed power allocation algorithms is provided, together with a study of their computational complexity and theoretical bounds relating the power allocation strategies. Numerical results show that the sum-rate of an RS system with adaptive power allocation outperforms RS and conventional MU-MIMO systems under imperfect CSIT.
\end{abstract}
\begin{IEEEkeywords}
Rate splitting, multiuser MIMO, power allocation, ergodic sum rate.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}
\IEEEPARstart{W}{ireless} communications systems employing multiple antennas have the advantage of increasing the overall throughput without increasing the required bandwidth. For this reason, multiple-antenna systems are at the core of several wireless communications standards such as WiFi, Long Term Evolution (LTE) and the fifth generation (5G). However, such wireless systems suffer from multiuser interference (MUI). In order to mitigate MUI, transmit processing techniques have been employed in the downlink (DL), allowing accurate recovery of the data at the receivers. In general, a precoding technique maps the symbols containing the information to the transmit antennas so that the information arrives at the receiver with reduced levels of MUI. Due to its benefits, linear \cite{mmimo,Joham2005,wence,grbd,wlrbd,rmmse} and non-linear \cite{Zu2014,Peel2005,bbprec} precoding techniques have been extensively reported in the literature. The design of effective precoders demands very accurate channel state information at the transmitter (CSIT), which is an extremely difficult task to accomplish in actual wireless systems. Hence, the transmitter typically only has access to partial or imperfect CSIT. As a result, the precoder cannot handle MUI as expected, resulting in residual MUI at the receiver. This residual MUI can degrade heavily the performance of wireless systems since it scales with the transmit power employed at the base station (BS) \cite{Tse2005}.
\subsection{Prior and related work}
In this context, Rate Splitting (RS) has emerged as a novel approach that is capable of dealing with CSIT imperfection \cite{Clerckx2016} in an effective way. RS was initially proposed in \cite{Han1981} to deal with interference channels \cite{Carleial1978}, where independent transmitters send information to independent receivers \cite{Haghi2021}. Since then, several studies have found that RS outperforms conventional schemes such as conventional precoding in spatial division multiple access (SDMA), power-domain Non-Orthogonal Multiple Access (NOMA) \cite{Mao2018} and even dirty paper coding (DPC) \cite{Mao2020}. Interestingly, it turns out that RS constitutes a generalized framework which has as special cases other transmission schemes such SDMA, NOMA and multicasting \cite{Clerckx2020,Naser2020,Jaafar2020,Mao22}. The main advantage of RS is its capability to partially decode interference and partially treat interference as noise. To this end, RS splits the message of one or several users into a common message and a private message. The common message must be decoded by all the users that employ successive interference cancellation \cite{spa,mfsic,mbdf,bfidd,1bitidd}. On the other hand, the private messages are decoded only by their corresponding users.
RS schemes have been shown to enhance the performance of wireless communication systems. In \cite{Yang2013} RS was extended to the broadcast channel (BC) of multiple-input single-output (MISO) systems, where it was shown that RS provides gains in terms of Degrees-of-Freedom (DoF) with respect to conventional multiuser multiple-input multiple-output (MU-MIMO) schemes under imperfect CSIT. Later in \cite{Hao2017b}, the DoF of a MIMO BC and IC was characterized. RS has eventually been shown in \cite{Piovano2017} to achieve the optimal DoF region when considering imperfect CSIT, outperforming the DoF obtained by SDMA systems, which decays in the presence of imperfect CSIT.
Due to its benefits, several wireless communications deployments with RS have been studied. RS has been employed in MISO systems along with linear precoders \cite{Joudeh2016,Hao2015} in order to maximize the sum-rate performance under perfect and imperfect CSIT assumptions. Another approach has been presented in \cite{Joudeh2017} where the max-min fairness problem has been studied. In \cite{Hao2017} a K-cell MISO IC has been considered and the scheme known as topological RS presented. The topological RS scheme transmits multiple layers of common messages, so that the common messages are not decoded by all users but by groups of users. RS has been employed along with random vector quantization in \cite{Lu2018} to mitigate the effects of the imperfect CSIT caused by finite feedback. In \cite{Flores2020,rsthp}, RS with common stream combining techniques has been developed in order to exploit multiple antennas at the receiver and to improve the overall sum-rate performance. A successive null-space precoder, that employs null-space basis vectors to adjust the inter-user-interference at the receivers, is proposed in \cite{Krishnamoorthy2021}. The optimization of the precoders along with the transmission of multiple common streams was considered in \cite{Mishra2022}. In \cite{Li2020}, RS with joint decoding has been explored. The authors of \cite{Li2020} devised precoding algorithms for an arbitrary number of users along with a stream selection strategy to reduce the number of precoded signals.
Along with the design of the precoders, power allocation is also a fundamental part of RS-based systems. The benefits of RS are obtained only if an appropriate power allocation for the common stream is performed. However, the power allocation problem in MU-MIMO systems is an NP-hard problem \cite{Luo2008,Liu2011}, and the optimal solution can be found at the expense of an exponential growth in computational complexity. Therefore, suboptimal approaches that jointly optimize the precoder and the power allocation have been developed. Most works so far employ exhaustive search or complex optimization frameworks. These frameworks rely on the augmented WMMSE \cite{Mao2020,Maoinpress,JoudehClerckx2016,Kaulich2021,Mishra2022}, which is an extension of the WMMSE proposed in \cite{Christensen2008}. This approach requires also an alternating optimization, which further increases the computational complexity. A simplified approach can be found in \cite{Dai2016a}, where closed-form expressions for RS-based massive MIMO systems are derived. However, this suboptimal solution is more appropiate for massive MIMO deployments. The high complexity of most optimal approaches makes them impractical to implement in large or real-time systems. For this reason, there is a strong demand for cost-effective power allocation techniques for RS systems.
\subsection{Contributions}
In this paper, we present novel efficient robust and adaptive power
allocation techniques \cite{rapa} for RS-based MU-MIMO systems. In
particular, we develop a robust adaptive power allocation (APA-R)
strategy based on stochastic gradient learning
\cite{bertsekas,jidf,smtvb,smce} and the minimization of the
mean-square error (MSE) between the transmitted common and private
symbols of the RS system and the received signal. We incorporate
knowledge of the variance of the channel errors to cope with
imperfect CSIT and adjust power levels in the presence of
uncertainty. When the knowledge of the variance of the channel
errors is not exploited the proposed APA-R becomes the proposed
adaptive power allocation algorithm (APA). An analysis of the
convexity and stability of the proposed power allocation algorithms
along with a study of their computational complexity and theoretical
bounds relating the power allocation strategies are developed.
Numerical results show that the sum-rate of an RS system employing
adaptive power allocation outperforms conventional MU-MIMO systems
under imperfect CSIT assumption. The contributions of this work can
be summarized as:
\begin{itemize}
\item Cost-effective APA-R and APA techniques for power allocation are proposed based on stochastic gradient recursions and knowledge of the variance of the channel errors for RS-based and standard MU-MIMO systems.
\item An analysis of convexity and stability of the proposed power allocation techniques along with a bound on the MSE of APA and APA-R and a study of their computational complexity.
\item A simulation study of the proposed APA and APA-R, and the existing power allocation techniques for RS-based and standard MU-MIMO systems.
\end{itemize}
\subsection{Organization}
The rest of this paper is organized as follows. Section II describes the mathematical model of an RS-based MU-MIMO system. In Section III the proposed APA-R technique is presented, the proposed APA approach is obtained as a particular case and sum-rate expressions are derived. In Section IV, we present an analysis of convexity and stability of the proposed APA and APA-R techniques along with a bound on the MSE of APA and APA-R and a study of their computational complexity. Simulation results are illustrated and discussed in Section V. Finally, Section VI presents the conclusions of this work.
\subsection{Notation}
Column vectors are denoted by lowercase boldface letters. The vector $\mathbf{a}_{i,*}$ stands for the $i$th row of matrix $\mathbf{A}$. Matrices are denoted by uppercase boldface letters. Scalars are denoted by standard letters. The superscript $\left(\cdot\right)^{\text{T}}$ represents the transpose of a matrix, whereas the notation $\left(\cdot\right)^H$ stands for the conjugate transpose of a matrix. The operators $\lVert \cdot \rVert$, and $\mathbb{E}_x\left[\cdot\right]$ represent the Euclidean norm, and the expectation w.r.t the random variable $x$, respectively. The trace operator is given by $\text{tr}\left(\cdot\right)$. The Hadamard product is denoted by $\odot$. The operator $\text{diag}\left(\mathbf{a}\right)$ produces a diagonal matrix with the coefficients of $\mathbf{a}$ located in the main diagonal.
\section{System Model}
Let us consider the RS-based MU-MIMO system architecture depicted in Fig. \ref{System Model RS}, where the BS is equipped with $N_t$ antennas, serves $K$ users and the $k$th UE is equipped with $N_k$ antennas. Let us denote by $N_r$ the total number of receive antennas. Then, $N_r=\sum_{k=1}^K N_k$. For simplicity, the message intended for the $k$th user is split into a common message and a private message. Then, the messages are encoded and modulated. The transmitter sends one common stream and a total of $M$ private streams, with $M\leq N_r$. The set $\mathcal{M}_k$ contains the $M_k$ private streams, that are intended for the user $k$, where $M_k\leq N_k$. It follows that $M=\sum_{k^=1}^K M_k$.
\begin{figure}[htb!]
\begin{center}
\includegraphics[scale=0.45]{RS_system_model.eps}
\vspace{-1.0em}
\caption{RS MU-MIMO architecture.}
\label{System Model RS}
\end{center}
\vspace{-2em}
\end{figure}
The vector $\mathbf{s}^{\left(\text{RS}\right)}=\left[s_c,\mathbf{s}_p^{\text{T}}\right]^{\text{T}} \in \mathbb{C}^{M+1}$, which is assumed i.i.d. with zero mean and covariance matrix equal to the identity matrix, contains the information transmitted to all users, where $s_c$ is the common symbol and $\mathbf{s}_p=\left[\mathbf{s}_1^{\text{T}},\mathbf{s}_2^{\text{T}},\cdots,\mathbf{s}_K^{\text{T}}\right]^{\text{T}}$ contains the private symbols of all users. Specifically, the vector $\mathbf{s}_k \in \mathbb{C}^{M_k}$ contains the private streams intended for the $k$th user. The system is subject to a transmit power constraint given by $\mathbb{E}\left[\lVert\mathbf{x}^{\left(\text{RS}\right)}\rVert^2\right]\leq E_{tr}$, where $\mathbf{x}^{\left(\text{RS}\right)}\in \mathbb{C}^{N_t}$ is the transmitted vector and $E_{tr}$ denotes the total available power. The transmitted vector can be expressed as follows:
\begin{align}
\mathbf{x}^{\left(\text{RS}\right)}=&\mathbf{P}^{\left(\text{RS}\right)}\mathbf{A}^{\left(\text{RS}\right)}\mathbf{s}^{\left(\text{RS}\right)}=a_c s_c \mathbf{p}_c+\sum_{m=1}^{M}a_m s_m \mathbf{p}_m, \label{Transmit Signal}
\end{align}
where $\mathbf{A}^{\left(\text{RS}\right)}\in \mathbb{R}^{\left(M+1\right)\times \left(M+1\right)}$ represents the power allocation matrix and $\mathbf{P}^{\left(\text{RS}\right)}=[\mathbf{p}_c,\mathbf{p}_1,\cdots ,\mathbf{p}_K] \in \mathbb{C}^{N_t \times \left(M+1\right)}$ is used to precode the vector of symbols $\mathbf{s}^{\left(\text{RS}\right)}$. Specifically, $\mathbf{A}^{\text{RS}}=\text{diag}\left(\mathbf{a}^{\left(\text{RS}\right)}\right)$ and $\mathbf{a}^{\left(\text{RS}\right)}=\left[a_c, a_1,\cdots,a_M\right]^{\text{T}}\in \mathbb{R}^{M+1}$, where $a_c$ denotes the power allocated to the common stream and $a_k$ allocates power to the $k$th private stream. Without loss of generality, we assume that the columns of the precoders are normalized to have unit norm.
The model established leads us to the received signal described by
\begin{equation}
\mathbf{y}=\mathbf{H}\mathbf{P}^{\left(\text{RS}\right)}\text{diag}\left(\mathbf{a}^{\left(\text{RS}\right)}\right)\mathbf{s}^{\left(\text{RS}\right)}+\mathbf{n}, \label{Receive Signal Complete}
\end{equation}
where $\mathbf{n}=\left[\mathbf{n}_1^{\text{T}},\mathbf{n}_2^{\text{T}},\cdots,\mathbf{n}_K^{\text{T}}\right]^{\text{T}} \in \mathbb{C}^{N_r}$ represents the uncorrelated noise vector, which follows a complex normal distribution, i.e., $\mathbf{n}\sim \mathcal{CN}\left(\mathbf{0},\sigma_n^2\mathbf{I}\right)$. We assume that the noise and the symbols are uncorrelated, which is usually the case in real systems. The matrix $\mathbf{H}=\left[\mathbf{H}_1^{\text{T}},\mathbf{H}_2^{\text{T}},\cdots,\mathbf{H}_K^{\text{T}}\right]^{\text{T}}\in \mathbb{C}^{N_r\times N_t}$ denotes the channel between the BS and the user terminals. Specifically, $\mathbf{n}_k$ denotes the noise affecting the $k$th user and the matrix $\mathbf{H}_k\in \mathbb{C}^{N_k\times N_t}$ represents the channel between the BS and the $k$th user. The imperfections in the channel estimate are modelled by the random matrix $\tilde{\mathbf{H}}$. Each coefficient of $\tilde{\mathbf{H}}$ follows a Gaussian distribution with variance equal to $\sigma_{e,i}^2$ and $\mathbb{E}\left[\tilde{\mathbf{h}}_{i,*}^H\tilde{\mathbf{h}}_{i,*}\right]=\sigma_e^2\mathbf{I}\quad \forall i \in\left[1,N_r\right]$. Then, the channel matrix can be expressed as $\mathbf{H}=\hat{\mathbf{H}}+\tilde{\mathbf{H}}$, where the channel estimate is given by $\hat{\mathbf{H}}$. From \eqref{Receive Signal Complete} we can obtain the received signal of user $k$, which is given by
\begin{align}
\mathbf{y}_k=&a_c s_c \mathbf{H}_k\mathbf{p}_c+ \mathbf{H}_k\sum_{i\in \mathcal{M}_k}a_i s_i\mathbf{p}_i+\mathbf{H}_k\sum\limits_{\substack{l=1\\l \neq k}}^{K}\sum\limits_{j\in \mathcal{M}_l}a_j s_j\mathbf{p}_j+ \mathbf{n}_k.\label{Receive Signal per user}
\end{align}
Note that the RS architecture contains the conventional MU-MIMO as a special case where no message is split and therefore $a_c$ is set to zero. Then, the model boils down to the model of a conventional MU-MIMO system, where the received signal at the $k$th user is given by
\begin{equation}
\mathbf{y}_k=\mathbf{H}_k\sum_{i\in \mathcal{M}_k}a_i s_i\mathbf{p}_i+\mathbf{H}_k\sum\limits_{\substack{l=1\\l \neq k}}^{K}\sum\limits_{j\in \mathcal{M}_l}a_j s_j\mathbf{p}_j+\mathbf{n}_k\label{Receive Signal per user convetinoal MIMO}
\end{equation}
In what follows, we will focus on the development of power allocation techniques that can cost-effectively compute $a_c$ and $a_j, j = 1, \ldots, K$.
\section{Proposed Power Allocation Techniques}
In this section, we detail the proposed power allocation techniques. In particular, we start with the derivation of the ARA-R approach and then present the APA technique as a particular case of the APA-R approach.
\subsection{Robust Adaptive Power Allocation}\label{c5 section robust power allocation RS}
Here, a robust adaptive power allocation algorithm, denoted as APA-R, is developed to perform power allocation in the presence of imperfect CSIT. The key idea is to incorporate knowledge about the variance of the channel uncertainty \cite{locsme,okspme} into an adaptive recursion to allocate the power among the streams. The minimization of the MSE between the received signal and the transmitted symbols is adopted as the criterion to derive the algorithm.
Let us consider the model established in \eqref{Receive Signal Complete} and denote the fraction of power allocated to the common stream by the parameter $\delta$, i.e., $a_c^2=\delta E_{tr}$. It follows that the available power for the private streams is reduced to $\left(1-\delta\right)E_{tr}$. We remark that the length of $\mathbf{s}^{\left(\text{RS}\right)}$ is greater than that of $\mathbf{y}^{\left(\text{RS}\right)}$ since the common symbol is superimposed to the private symbols. Therefore, we consider the vector $\mathbf{y}'=\mathbf{T}\mathbf{y}^{\left(\text{RS}\right)}$, where $\mathbf{T}\in\mathbb{R}^{\left(M+1\right)\times M}$ is a transformation matrix employed to ensure that the dimensions of $\mathbf{s}^{\left(\text{RS}\right)}$ and $\mathbf{y}^{\left(\text{RS}\right)}$ match, and is given by
\begin{equation}
\mathbf{T}=\begin{bmatrix}
1 &1 &\cdots &1\\
1 &0 &\cdots &0\\
0 &1 &\cdots &0\\
\vdots &\vdots &\ddots &\vdots\\
0 &0 &\cdots &1
\end{bmatrix}.
\end{equation}
All the elements in the first row of matrix $\mathbf{T}$ are equal to one in order to take into account the common symbol obtained at all receivers. As a result we obtain the combined receive signal of all users. It follows that
\begin{equation}
\mathbf{y}'=\begin{bmatrix}
y_c\\
y_1\\
\vdots\\
y_M
\end{bmatrix}=\begin{bmatrix}
\sum_{i=1}^{M} y_i\\
y_1\\
\vdots\\
y_M
\end{bmatrix},
\end{equation}
where the received signal at the $i$th antenna is described by
\begin{equation}
y_i=a_c s_c\left(\hat{\mathbf{h}}_{i,*}+\tilde{\mathbf{h}}_{i,*}\right)\mathbf{p}_c+\sum_{j=1}^{M}a_j s_j \left(\hat{\mathbf{h}}_{i,*}+\tilde{\mathbf{h}}_{i,*}\right)\mathbf{p}_j+ n_i.
\end{equation}
Let us now consider the proposed robust power allocation problem for imperfect CSIT scenarios. By including the error of the channel estimate, the robust power allocation problem can be formulated as the constrained optimization given by
\begin{equation}
\begin{gathered}
\min_{\mathbf{a}} \mathbb{E}\left[\lVert\mathbf{s}^{\left(\text{RS}\right)}-\mathbf{y}'\left(\mathbf{H}\right)\rVert^2| \hat{\mathbf{H}}\right]\\
\text{s.t.}~~\text{tr}\left(\mathbf{P}^{\left(\text{RS}\right)}\text{diag}\left(\mathbf{a}^{\left(\text{RS}\right)}\odot \mathbf{a}^{\left(\text{RS}\right)}\right)\mathbf{P}^{\left(\text{RS}\right)^{H}}\right)=E_{tr},\label{Obejctive funtion robust}
\end{gathered}
\end{equation}
which can be solved by first relaxing the constraint, using an adaptive learning recursion and then enforcing the constraint.
In this work, we choose the MSE as the objective function due to its convex property and mathematical tractability, which help to find an appropriate solution through algorithms. The objective function is convex as illustrated by Fig. \ref{fig:MSECurve} and analytically shown in Section \ref{analysis}. In Fig. \ref{fig:MSECurve}, we plot the objective function using two precoders, namely the zero-forcing (ZF) and the matched filter (MF) precoders \cite{Joham2005}, where three private streams and one common stream are transmitted and the parameter $\delta$ varies with uniform power allocation across private streams.
\begin{figure}
\centering
\includegraphics[scale=0.4]{MSECurve.eps}
\vspace{-1.5em}\caption{Objective function considering a MU-MIMO system with $Nt=3$, $K=3$, and $\sigma_n^2=1$}\vspace{-1.5em}
\label{fig:MSECurve}
\end{figure}
To solve \eqref{Obejctive funtion robust} we need to expand the terms and evaluate the expected value. Let us consider that the square error is equal to $\varepsilon=\lVert \mathbf{s}^{\left(\text{RS}\right)}-\mathbf{y}' \rVert^2$. Then, the MSE is given by
\begin{align}
\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]=&-2a_c\sum_{i=1}^M\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}-2\sum_{j=1}^M a_j \Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}+2\sum_{j=1}^{M}a_j^2\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right)\nonumber\\
&+2a_c^2\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)+2\sum_{i=1}^{M-1}\sum_{q=i+1}^{M}\sum_{r=1}^M a_r^2\Re\left\{\hat{\phi}^{\left(i,r\right)^*}\hat{\phi}^{\left(q,r\right)}\right\}\nonumber\\
&+\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c^2\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+M\left(1+2\sigma_n^2\right)+1,\label{mean square error APA robust}
\end{align}
where $\hat{\phi}^{\left(i,c\right)}=\hat{\mathbf{h}}_{l,*}\mathbf{p}_c$ and $\hat{\phi}^{\left(i,l\right)}=\hat{\mathbf{h}}_{i,*}\mathbf{p}_l$ for all $i,l \in \left[1,M\right]$. The proof to obtain the last equation can be found in appendix \ref{Appendix MSE APA-R}. The partial derivatives of \eqref{mean square error APA robust} with respect to ${a}_c$ and ${a}_i$ are expressed by
\begin{align}
\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_c}=& 2\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}-2\sum_{i=1}^M\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}+4a_c\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right),\label{gradient robust apa ac}
\end{align}
\begin{align}
\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i}=&4a_i\sum_{l=1}^{M-1}\sum_{q=l+1}^{M} \Re\left[\hat{\phi}^{\left(l,i\right)^*}\hat{\phi}^{\left(q,i\right)}\right]-2 \Re{\left\{\hat{\phi}^{\left(i,i\right)}\right\}}+4a_i\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,i\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_i\rVert^2\right).\label{gradient robust apa ai}
\end{align}
The partial derivatives given by \eqref{gradient robust apa ac} and \eqref{gradient robust apa ai} represent the gradient of the MSE with respect to the power allocation coefficients. With the obtained gradient we can employ a gradient descent algorithm, which is an iterative procedure that finds a local minimum of a differentiable function. The key idea is to take small steps in the opposite direction of the gradient, since this is the direction of the steepest descent. Remark that the objective function used is convex and has no local minimum. Then, the recursions of the proposed APA-R technique are given by
\begin{align}
a_c\left[t+1\right]&=a_c\left[t\right]-\mu\frac{\partial\mathbb{E}\left[\lvert\varepsilon\rvert^2|\hat{\mathbf{H}}^{\text{T}}\right]}{\partial a_c},\nonumber\\
a_i\left[t+1\right]&=a_i\left[t\right]-\mu\frac{\partial\mathbb{E}\left[\lvert\varepsilon\rvert^2|\hat{\mathbf{H}}^{\text{T}}\right]}{\partial a_i},
\end{align}
where the parameter $\mu$ represents the learning rate of the adaptive algorithm. At each iteration, the power constraint is analyzed. Then, the coefficients are scaled with a power scaling factor by $\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]=\beta\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]$, where $\beta=\sqrt{\frac{1}{\textrm{tr}\left(\textrm{diag}\left(\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\odot \mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\right)\right)}}$ to ensure that the power constraint is satisfied. Algorithm \ref{algorithm RS APA} summarizes the proposed APA-R algorithm.
\begin{algorithm}[t!]
\normalsize
\SetAlgoLined
given $\hat{\mathbf{H}}$, $\mathbf{P}$, and $\mu$\;
$\mathbf{a}\left[1\right]=\mathbf{0}$\;
\For{$n=2$ \KwTo $I_t$}{
$\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_c}= 2\sum\limits_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}-2\sum\limits_{i=1}^M\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}+4a_c\left(\sum\limits_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)$\;
\vspace{0.1cm}
$\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i}=4a_i\sum\limits_{l=1}^{M-1}\sum\limits_{q=l+1}^{M} \Re\left[\hat{\phi}^{\left(l,i\right)^*}\hat{\phi}^{\left(q,i\right)}\right]-2 \Re{\left\{\hat{\phi}^{\left(i,i\right)}\right\}}+4a_i\left(\sum\limits_{l=1}^M\lvert\hat{\phi}^{\left(l,i\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_i\rVert^2\right)$\;
\vspace{0.1cm}
$a_c\left[n\right]=a_c\left[n-1\right]-\mu\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_c}$\;
\vspace{0.1cm}
$a_i\left[n\right]=a_i\left[n-1\right]-\mu\frac{\partial\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i}$\;
\vspace{0.1cm}
\If{$\textrm{\rm tr}\left(\textrm{\rm diag}\left(\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\odot \mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\right)\right)\neq 1$}{
\vspace{0.1cm}
$\beta=\sqrt{\frac{1}{\textrm{tr}\left(\textrm{diag}\left(\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\odot \mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\right)\right)}}$\;
\vspace{0.1cm}
$\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]=\beta\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]$\;
}
}
\caption{Robust Adaptive Power allocation}
\label{algorithm RS APA}
\end{algorithm}\vspace{-0.5em}
\subsection{Adaptive Power Allocation}
In this section, a simplified version of the proposed APA-R algorithm is derived. The main objective is to reduce the complexity of each recursion of the adaptive algorithm and avoid the load of computing the statistical parameters of the imperfect CSIT, while reaping the benefits of RS systems.
The power allocation problem can be reformulated as the constrained optimization problem given by
\begin{equation}
\begin{gathered}
\min_{\mathbf{a}} \mathbb{E}\left[\lVert\mathbf{s}^{\left(\text{RS}\right)}-\mathbf{y}'\rVert^2\right]\\
\text{s.t.}~~\text{tr}\left(\mathbf{P}^{\left(\text{RS}\right)}\text{diag}\left(\mathbf{a}^{\left(\text{RS}\right)}\odot \mathbf{a}^{\left(\text{RS}\right)}\right)\mathbf{P}^{\left(\text{RS}\right)^{H}}\right)=E_{tr},
\end{gathered} \label{Objective function apa}
\end{equation}
In this case, the MSE is equivalent to
\begin{align}
\mathbb{E}\left[\varepsilon\right]=&-2a_c\sum_{j=1}^{M}\Re\left\{\phi^{\left(j,c\right)}\right\}-2\sum_{l=1}^M a_l\Re\left\{\phi^{\left(l,l\right)}\right\}+2\left(\sum_{l=1}^M a_c^2\lvert\phi^{\left(l,c\right)}\rvert^2+\sum_{i=1}^M\sum_{j=1}^M a_j^2\lvert\phi^{\left(i,j\right)}\rvert^2\right)\nonumber\\
&+2\sum_{i=1}^{M-1}\sum_{q=i+1}^{M}\sum_{r=1}^M a_r^2\Re\left\{\phi^{\left(i,r\right)^*}\phi^{\left(q,r\right)}\right\}+\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c^2\phi^{\left(l,c\right)^*}\phi^{\left(j,c\right)}+M\left(1+2\sigma_n^2\right)+1, \label{mean square error APA RS}
\end{align}
where we considered that $\phi^{\left(i,c\right)}=\mathbf{h}_{l,*}\mathbf{p}_c$ and $\phi^{\left(i,l\right)}=\mathbf{h}_{i,*}\mathbf{p}_l$ for all $i,l \in \left[1,M\right]$. The proof to obtain \eqref{mean square error APA RS} can be found in appendix \ref{Appendix MSE APA}. Taking the partial derivatives of \eqref{mean square error APA RS} with respect to the coefficients of $\mathbf{a}^{\left(\text{RS}\right)}$ we arrive at
\begin{align}
\frac{\partial\mathbb{E}\left[\varepsilon\right]}{\partial a_c}&=4a_c \sum_{i=1}^{M}\lvert\phi^{\left(i,c\right)}\rvert^2+2a_c\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M \phi^{\left(l,c\right)^*}\phi^{\left(j,c\right)}-2\sum_{q=1}^{M}\Re\left[\phi^{\left(q,c\right)}\right],\label{gradient RS perfect CSIT common stream}\\
\frac{\partial\mathbb{E}\left[\varepsilon\right]}{\partial a_i}&=4a_i\sum_{j=1}^M \lvert\phi^{\left(j,i\right)}\rvert^2+4a_i\sum_{r=1}^{M-1}\sum_{q=r+1}^{M} \Re\left[\phi^{\left(r,i\right)^*}\phi^{\left(q,i\right)}\right]-2 a_i\Re\left[\phi^{\left(i,i\right)}\right],\label{gradient RS perfect CSIT private streams}
\end{align}
The power allocation coefficients are adapted using \eqref{gradient RS perfect CSIT common stream} and \eqref{gradient RS perfect CSIT private streams} in the following recursions:
\begin{align}
a_c\left[t+1\right]&=a_c\left[t\right]-\mu\frac{\partial\mathbb{E}\left[\varepsilon\right]}{\partial a_c}\nonumber\\
a_i\left[t+1\right]&=a_i\left[t\right]-\mu\frac{\partial\mathbb{E}\left[\varepsilon\right]}{\partial a_i}.\label{update equation for perfect CSIT}
\end{align}
\subsection{Sum-Rate Performance}
In this section, we derive closed-form expressions to compute the sum-rate performance of the proposed algorithms. Specifically, we employ the ergodic sum-rate (ESR) as the main performance metric. Before the computation of the ESR we need to find the average power of the received signal, which is given by
\begin{equation}
\mathbb{E}\left[\lvert y_k\rvert^2\right]=a_c^2\lvert \mathbf{h}_k^{\textrm{T}}\mathbf{p}_c\rvert^2+\sum_{i=1}^K a_i^2\lvert \mathbf{h}_k^{\textrm{T}}\mathbf{p}_i\rvert^2+\sigma_w^2.
\end{equation}
It follows that the instantaneous SINR while decoding the common symbol is given by
\begin{align}
\gamma_{c,k}&=\frac{a_c^2\lvert \mathbf{\hat{h}}_k^{\textrm{T}}\mathbf{p}_c\rvert^2}{\sum\limits_{i=1}^K a_i^2\lvert \mathbf{h}_k^{\textrm{T}}\mathbf{p}_i\rvert^2+\sigma_w^2}.\label{instantaneous SINR common rate}
\end{align}
Once the common symbol is decoded, we apply SIC to remove it from the received signal. Afterwards, we calculate the instantaneous SINR when decoding the $k$th private stream, which is given by
\begin{equation}
\gamma_k=\frac{a_k^2\lvert\mathbf{\hat{h}}_k^{\textrm{T}}\mathbf{p}_k\rvert^2}{\sum\limits_{\substack{i=1\\i\neq k}}^K a_i^2\lvert\mathbf{h}_k\mathbf{p}_i\rvert^2+\sigma_w^2}.\label{instantaneous SINR private rate}
\end{equation}
Considering Gaussian signaling, the instantaneous common rate can be found with the following equation:
\begin{equation}
R_{c,k}=\log_2\left(1+\gamma_{c,k}\right).\label{instantaneous common rate per user}
\end{equation}
The private rate of the $k$th stream is given by
\begin{equation}
R_{k}=\log_2\left(1+\gamma_{k}\right)\label{instantaneous private rate}
\end{equation}
Since imperfect CSIT is being considered, the instantaneous rates are not achievable. To that end, we employ the average sum rate (ASR) to average out the effect of the error in the channel estimate. The average common rate and the average private rate are given by
\begin{align}
\bar{R}_{c,k}&=\mathbb{E}\left[R_{c,k}|\mathbf{\hat{G}}\right] &
\bar{R}_{k}=\mathbb{E}\left[R_{k}|\mathbf{\hat{G}}\right],
\end{align}
respectively. With the ASR we can obtain the ergodic sum-rate (ESR), which quantifies the performance of the system over a large number of channel realizations and is given by
\begin{equation}
S_r=\min_{k}\mathbb{E}\left[\bar{R_{c,k}}\right]+\sum_{k=1}^K \mathbb{E}\left[\bar{R}_k\right],\label{system ergodic sum rate}
\end{equation}
\section{Analysis}
\label{analysis}
In this section, we carry out a convexity analysis and a statistical analysis of the proposed algorithms along with an assessment of their computational complexity in terms of floating point operations (FLOPs). Moreover, we derive a bound that establishes that the proposed APA-R algorithm is superior or comparable to the proposed APA algorithm.
\subsection{Convexity analysis}
In this section, we perform a convexity analysis of the optimization problem that gives rise to the proposed APA-R and APA algorithms. In order to establish convexity, we need to compute the second derivative of $\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]$ with respect to $a_c$ and $a_i$, and then check if it is greater than zero \cite{bertsekas}, i.e.,
\begin{equation}
\frac{\partial^2 \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_c \partial a_c} >0 ~{\rm and} ~ \frac{\partial^2 \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i \partial a_i} >0, ~ i=1,2, \ldots, K
\end{equation}
{Let us first compute $\frac{\partial^2 \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_c \partial a_c} $ using the results in \eqref{gradient robust apa ac}:
\begin{equation}
\begin{split}
\frac{\partial}{\partial a_c} \Bigg( \frac{\partial \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{ \partial a_c} \Bigg) & = \frac{\partial}{\partial a_c} \Bigg(2\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}-2\sum_{i=1}^M\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}+4a_c \left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2 \right) \Bigg) \\ & = 2\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M \hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+4\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right). \label{2ndderiv_ac}
\end{split}
\end{equation}
Now let us compute $\frac{\partial^2 \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_i \partial a_i} $ using the results in \eqref{gradient robust apa ai}:
\begin{equation}
\begin{split}
\frac{\partial}{\partial a_i} \Bigg( \frac{\partial \mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{ \partial a_i} \Bigg) & = \frac{\partial}{\partial a_i} \Bigg(4a_i\sum_{l=1}^{M-1}\sum_{q=l+1}^{M} \Re\left[\hat{\phi}^{\left(l,i\right)^*}\hat{\phi}^{\left(q,i\right)}\right]-2 \Re{\left\{\hat{\phi}^{\left(i,i\right)}\right\}}+4a_i\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,i\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_i\rVert^2\right) \Bigg) \\ & = 4\sum_{l=1}^{M-1}\sum_{q=l+1}^{M} \Re\left[\hat{\phi}^{\left(l,i\right)^*}\hat{\phi}^{\left(q,i\right)}\right]+4\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,i\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_i\rVert^2\right), \label{2ndderiv_ai}
\end{split}
\end{equation}
Since we have the sum of the strictly convex terms in \eqref{2ndderiv_ac} and \eqref{2ndderiv_ai} the objective function associated with APA-R is strictly convex \cite{bertsekas}. The power constraint is also strictly convex and only scales the powers to be adjusted. In the case of the APA algorithm, the objective function does not employ knowledge of the error variance $\sigma_{e,i}^2$ and remains strictly convex.
\subsection{Bound on the MSE of APA and APA-R}
Let us now show that the proposed APA-R power allocation produces a lower MSE than that of the proposed APA power allocation. The MSE obtained in \eqref{mean square error APA RS} assumes that the transmitter has perfect knowledge of the channel. Under such assumption the optimal coefficients $\mathbf{a}_{o}$ that minimize the error are found. However, under imperfect CSIT the transmitter is unaware of $\tilde{\mathbf{H}}$ and the adaptive algorithm performs the power allocation by employing $\hat{\mathbf{H}}$ instead of $\mathbf{H}$. This results in a power allocation given by $\hat{\mathbf{a}}^{\left(\text{APA}\right)}$ which originates an increase in the MSE obtained. It follows that
\begin{equation}
\varepsilon\left(\mathbf{H},\mathbf{a}_o\right)\leq\varepsilon\left(\mathbf{H},\hat{\mathbf{a}}^{\left(\text{APA}\right)}\right)
\end{equation}
On the other hand, the robust adaptive algorithm finds the optimal $\mathbf{a}_o^{\left(\text{APA-R}\right)}$ that minimizes $\mathbb{E}\left[\varepsilon\left(\mathbf{H},\mathbf{a}\right)\right|\hat{\mathbf{H}}]$ and therefore takes into account that only partial knowledge of the channel is available. Since the coefficients $\hat{\mathbf{a}}^{\left(\text{APA}\right)}$ and $\mathbf{a}_o^{\left(\text{APA-R}\right)}$ are different, we have
\begin{equation}
\mathbb{E}\left[\varepsilon\left(\mathbf{H,\mathbf{a}_o^{\left(\text{APA-R}\right)}}\right)|\hat{\mathbf{H}}\right]\leq\mathbb{E}\left[\varepsilon\left(\mathbf{H},\hat{\mathbf{a}}^{\left(\text{APA}\right)}\right)|\hat{\mathbf{H}}\right]
\end{equation}
Note that under perfect CSIT equation \eqref{mean square error APA robust} reduces to \eqref{mean square error APA RS}. In such circumstances $\mathbf{a}_o^{\left(\text{APA}\right)}=\mathbf{a}_o^{\left(\text{APA-R}\right)}$ and therefore both algorithms are equivalent. In the following, we evaluate the performance obtained by the proposed algorithms.
Specifically, we have that $\hat{\mathbf{a}}^{\left(\text{APA}\right)}=\mathbf{a}_o^{\left(\text{APA-R}\right)}+\mathbf{a}_e$, where $\mathbf{a}_e=\left[a_{c,e},a_{1,e},\cdots,a_{M,e}\right]^{\text{T}}$ is the error produced from the assumption that the BS has perfect CSIT.
Then, we have
\begin{align}
\mathbb{E}\left[\varepsilon^{\left(\text{APA}\right)}-\varepsilon\right.\left.^{\left(\text{APA-R}\right)}\right]&=-2a_{c,e}\sum_{i=1}^M\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}-2\sum_{j=1}^M a_{j,e} \Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}+\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_{c,e}^2\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}\nonumber\\
&+2a_{c,e}^2\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)+2\sum_{i=1}^{M-1}\sum_{q=i+1}^{M}\sum_{r=1}^M a_{r,e}^2\Re\left[\hat{\phi}^{\left(i,r\right)^*}\hat{\phi}^{\left(q,r\right)}\right]\nonumber\\
&+2\sum_{j=1}^{M}a_{j,e}^2\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right).
\end{align}
which is a positive quantity when $-2a_{c,e}\sum_{i=1}^M\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}<2a_{c,e}^2\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)$ and $-2\sum_{j=1}^M a_{j,e} \Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}<2\sum_{j=1}^{M}a_{j,e}^2\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right)$. The inequalities hold as long as $a_{c,e}\left[\sum_{i=1}^M\left(\Re\left\{\hat\phi^{\left(i,c\right)}\right\}\right)^2+\sum_{i=1}^M\left(\Im\left\{\hat\phi^{\left(i,c\right)}\right\}\right)^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right]>\sum_{i=1}^M\Re{\left\{\phi^{\left(i,c\right)}\right\}}$ and $\sum_{j=1}^{M}a_{j,e}\left[\sum_{l=1}^M\left(\Re\left\{\hat\phi^{\left(l,j\right)}\right\}\right)^2+\sum_{l=1}^M\left(\Im\left\{\hat\phi^{\left(l,j\right)}\right\}\right)^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right]>\sum_{j=1}^M\Re{\left\{\phi^{\left(j,j\right)}\right\}}$. As the error in the power allocation coefficients grows the left-hand side of the two last inequalities increases. This explains why the proposed APA-R performs better than the proposed APA algorithm.
\subsection{Statistical Analysis}
The performance of adaptive learning algorithms is usually measured in terms of its transient behavior and steady-state behaviour. These measurements provide information about the stability, the convergence rate, and the MSE achieved by the algorithm\cite{Yousef2001,Sayed2003}.
Let us consider the adaptive power allocation with the update equations given by \eqref{update equation for perfect CSIT}. Expanding the terms of \eqref{update equation for perfect CSIT} for the private streams, we get
\begin{align}
a_j\left[t+1\right]=&a_j\left[t\right]-4\mu a_j\left[t\right]\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+2\mu\Re\left\{\phi^{\left(j,j\right)}\right\}-4\mu a_j\left[t\right]\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}\Re\left\{\phi^{\left(q,j\right)^*}\phi^{\left(r,j\right)}\right\}.\label{update recursion coeff perfect csit}
\end{align}
Let us define the error between the estimate of the power coefficients and the optimal parameters as follows:
\begin{equation}
e_{a_j}\left[t\right]=a_j\left[t\right]-a_j^{\left(o\right)},\label{error optimal estimate}
\end{equation}
where $a_j^{\left(o\right)}$ represents the optimal allocation for the $j$th coefficient.
By subtracting \eqref{error optimal estimate} from \eqref{update recursion coeff perfect csit}, we obtain
\begin{align}
e_{a_j}\left[t+1\right]=&e_{a_j}\left[t\right]-4\mu a_j\left[t\right]\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+2\mu\Re\left\{\phi^{\left(j,j\right)}\right\}-4\mu a_j\left[t\right]\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}\Re\left\{\phi^{\left(q,j\right)^*}\phi^{\left(r,j\right)}\right\}\nonumber\\
=&e_{a_j}\left[t\right]+2\mu\Re\left\{\phi^{\left(j,j\right)}\right\}-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)a_j\left[t\right], \label{eq54}
\end{align}
where $f_{q,r}^{\left(j\right)}=\Re\left\{\left(\mathbf{h}_{q,*}\mathbf{p}_j\right)^*\left(\mathbf{h}_{r,*}\mathbf{p}_j\right)\right\}$. Expanding the terms in \eqref{eq54}, we get
\begin{align}
e_{a_j}\left[t+1\right]=&-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)a_j^{\left(o\right)}+e_{a_j}\left[t\right]+2\mu\Re\left\{\phi^{\left(j,j\right)}\right\}\nonumber\\
&-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)e_{a_j}\left[t\right].\nonumber
\end{align}
Rearranging the terms of the last equation, we obtain
\begin{align}
e_{a_j}\left[t+ \right.\left. 1\right]
&=\left\{1-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)\right\}e_{a_j}\left[t\right]\nonumber\\
&-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)a_j^{\left(o\right)}+2\mu\Re\left\{\phi^{\left(j,j\right)}\right\}.\label{eq55}
\end{align}
Equation \eqref{eq55} can be rewritten as follows
\begin{align}
e_{a_j}\left[t+ \right.\left. 1\right]
=&\left\{1-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)\right\}e_{a_j}\left[t\right]\nonumber\\
&+2\mu\left(\frac{\text{MSE}_{\textrm{min}}\left(a_j^{\left(o\right)}\right)}{a_j^{\left(o\right)}}-a_j^{\left(o\right)}\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right),\label{eq56}
\end{align}
where
\begin{align}
\text{MSE}_{\rm min}\left(a_j^{\left(o\right)}\right)=2 a_j^{\left(o\right)}\left(a_j^{\left(o\right)}\sum_{i=1}^{M} \lvert\phi^{\left(i,j\right)}\rvert^2-\Re\left\{\phi^{\left(j,j\right)}\right\}\right.\left.+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right).
\end{align}
Bu multiplying \eqref{eq56} by $e_a\left[t+1\right]$ and taking the expected value, we obtain
\begin{align}
\sigma_{e_{a_j}}^2\left[t\right.\left.1+\right]
=&\left\{1-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)\right\}^2\sigma_{e_{a_j}}^2\left[t\right]\nonumber\\
&+4\mu^2\left(\frac{\text{MSE}_{min}\left(a_j^{\left(o\right)}\right)}{a_j^{\left(o\right)}}-a_j^{\left(o\right)}\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)^2,
\end{align}
where we consider that $\mathbb{E}\left[e_{a_j}\left[i\right]\right]\approx \mathbf{0}$.The previous equation constitutes a geometric series with geometric ratio equal to $1-4\mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)$. Then, we have
\begin{equation}
\left\lvert 1-4 \mu\left(\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)\right\rvert<1
\end{equation}
Note that from the last equation the step size must fulfill
\begin{equation}
0<\mu_j<\frac{1}{2\lambda_j},
\end{equation}
with $\lambda_j=\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}$.
For the common power allocation coefficient we have the following recursion:
\begin{align}
a_c\left[t+1\right]=&a_c\left[t\right]-4\mu a_c\left[t\right]\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-2\mu\sum_{l=1}^{M}\Re\left\{\phi^{\left(l,c\right)}\right\}+2\mu a_c\left[t\right]f^{\left(c\right)},
\end{align}
where $f^{\left(c\right)}=\sum_{q=1}^{M}\sum_{\substack{r=1\\r\neq q}}^{M}\phi^{\left(q,c\right)^*}\phi^{\left(r,c\right)}$. The error with respect to the optimal power allocation of the common stream is given by
\begin{equation}
e_c\left[i\right]=a_c{\left[i\right]}-a_c^{\left(o\right)}.
\end{equation}
Following a similar procedure to the one employed for the private streams we arrive at
\begin{align}
e_c\left[t+1\right]=&\left\{1-4\mu\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-2\mu f^{\left(c\right)}\vphantom{\sum_{j=1}^{M}}\right\}e_c\left[t\right]-2\mu\left(2\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2+\mu f^{\left(c\right)}\right)a_c^{\left(o\right)}\nonumber\\
&-2\mu\sum_{l}^{M}\Re\left\{\phi^{\left(l,c\right)}\right\}.
\end{align}
Multiplying the previous equation by $e_c\left[t+1\right]$ and taking the expected value leads us to:
\begin{align}
\sigma_{e_c}^2\left[t+1\right]=&\left\{1-2\mu\left(2\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-f^{\left(c\right)}\right)\right\}\sigma_{e_c}^2\left[t\right]\nonumber\\
&-4\left(\frac{\text{MSE}_{\text{min}}\left(a_c^{\left(o\right)}\right)}{a_c^{\left(o\right)}}+a_c^{\left(o\right)}\sum_{j=1}^{M}\Re\left\{\phi^{\left(j,c\right)}\right\}\right)
\end{align}
It follows that the geometric ratio of the recursion is equal to $1-2\mu\left(2\sum\limits_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-f^{\left(c\right)}\right)$. Then, the step size must be in the following range:
\begin{equation}
0<\mu_c<\frac{1}{\lambda_c},
\end{equation}
where
\begin{equation}
\lambda_c=2\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-f^{\left(c\right)}
\end{equation}
The step-size of the algorithm must be less or equal to $\min\left(\mu_c,\mu_j\right)$ $\forall j \in \left\{1,2,\cdots,M\right\}$. The stability bounds provide useful information on the choice of the step sizes.
Let us now consider the APA-R algorithm. The \textit{a posteriori} error can be expressed as follows:
\begin{align}
e_{a_j}\left[t+1\right]=&e_{a_j}\left[t\right]+2\mu\Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}-4\mu\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right.\left.+\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)a_j\left[t\right].
\end{align}
Expanding the terms of the last equation, we get
\begin{align}
e_{a_j}\left[t+1\right]=&\left\{1-4\mu\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right.\right.\left.\left.+\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)\right\}e_{a_j}\left[t\right]\nonumber\\
&-4\mu\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right.\left.+\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)a_j^{\left(o\right)}+2\mu\Re\left\{\hat{\phi}^{\left(j,j\right)}\right\}.
\end{align}
The geometric ratio of the robust APA algorithm is given by $1-4\mu\left(\sum\limits_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right.$ $\left.+\sum\limits_{q=1}^{M-1}\sum\limits_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)$. Then, we have that
\begin{equation}
\left\lvert1-4\mu\left(\sum_{l=1}^{M}\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)\right\rvert<1
\end{equation}
Therefore, the step size must satisfy the following inequality:
\begin{equation}
0<\mu_j^{\left(\textrm{r}\right)}<\frac{1}{2\lambda^{\left(\textrm{r}\right)}_j},
\end{equation}
where $\lambda^{\left(\textrm{r}\right)}_j=\sum_{l=1}^{M}\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}$. Following a similar procedure for the power coefficient of the common stream lead us to
\begin{equation}
0<\mu_c^{\left(\textrm{r}\right)}<\frac{1}{\lambda^{\left(\textrm{r}\right)}_c},
\end{equation}
where
\begin{equation}
\lambda_c^{\left(\text{r}\right)}=2\sum_{j=1}^{M}\lvert\hat{\phi}^{\left(j,c\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2-f^{\left(c\right)}
\end{equation}
As in the previous case, the step size is chosen using $\min\left(\mu^{\left(\textrm{r}\right)}_c,\mu^{\left(\textrm{r}\right)}_j\right)$ $\forall j \in \left\{1,2,\cdots,M\right\}$. The variable $\lambda_c^{\left(\textrm{r}\right)}$ has an additional term given by $\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2$ when compared to $\lambda_c$ of the APA algorithm. In this sense, the upper bound of (70) is smaller than the bound in (65). In other words, the step size of APA-R takes smaller values than the step size of APA.
\subsection{Computational Complexity}
In this section the number of FLOPs performed by the proposed algorithms is computed. For this purpose let us consider the following results to simplify the analysis. Consider the complex vector $\mathbf{z}_1$ and $\mathbf{z}_2 \in \mathbb{C}^{n}$. Then, we have the following results:
\begin{itemize}
\item The product $\mathbf{z}_1^{\text{T}}\mathbf{z}_2$ requires $8n-2$ FLOPs.
\item The term $\lVert \mathbf{z}_1\rVert^2$ requires $7n-2$ FLOPs
\end{itemize}
The gradient in equation \eqref{gradient RS perfect CSIT private streams} requires the computation of three terms. The first term, which is given by $4 a_c \sum\limits_{i=1}^M\lvert\mathbf{h}_{i,*}\mathbf{p}_c\rvert^2$ needs a total of $8 N_tM +2M+1$ FLOPs. The evaluation of the second term results in $8N_tM+M$. For the last term we have a total of $\left(9M^2-9M+2\right)/2$. Considering a system where $N_t=N_r=M=n$ we have that the number of FLOPs required by the proposed adaptive algorithm is given by $\frac{41}{2}n^3+19n^2+\frac{5}{2}n+4$.
The computational complexity of the gradients in \eqref{gradient robust apa ac} and \eqref{gradient robust apa ai} can be computed in a similar manner. However, in this case we have an additional term given by $4 a_c M\sigma_{e,i}^2\lVert \mathbf{p}_i\rVert^2$, which requires a total of $7N_t+2$ FLOPs. Then, the robust adaptive algorithm requires a total of $\frac{41}{2}n^3+19n^2+\frac{19}{2}n+6$. It is important to mention that the computational complexity presented above represents the number of FLOPs of the adaptive algorithm per iteration.
In contrast, the optimal power allocation for the conventional SDMA system requires the exhaustive search over $\mathbf{A}$ with a fine grid. Given a system with $12$ streams and a grid step of $0.001$, the exhaustive search would require the evaluation of $5005000$ different power allocation matrices for each channel realization. In contrast, the adaptive approaches presented require only the computation of around $30$ iterations. Furthermore, the complexity of the exhaustive search for an RS system is even higher since the search is perform over $\mathbf{A}^{\left(\text{RS}\right)}$, which additionally contains the power allocated to the common stream.
Table \ref{Computational complexity power allocation} summarizes the computational complexity of the proposed algorithms employing the big $\mathcal{O}$ notation. In Table \ref{Computational complexity power allocation}, $I_o$ denotes the number of points of the grid given a step size, $I_w$ refers to the number of iterations of the alternating procedure and $I_a$ denotes the number of iterations for the adaptive algorithms. It is worth noting that $I_o>>I_a$. Moreover, the inner iterations employed in the WMMSE approach are much more demanding than the iterations of the proposed APA and APA-R algorithms. Fig \ref{Complexity} shows the computational complexity in terms of FLOPS assuming that the number of transmit antennas increases. The term CF represents the closed-form power allocation in \cite{Dai2016a}, which requires the inversion of an $N_t\times N_t$ matrix. The step of the grid was set to $0.01$ for the ES and the number of iterations to $30$ for the WMMSE, APA and APA-R approaches. Note that in practice the WMMSE iterates continuously until meeting a predefined accuracy. In general, it requires more than $30$ iterations. It is also important to point out that the cost per iteration of the adaptive approaches can be reduced after the first iteration. The precoders and the channel are fixed given a transmission block. Therefore, after the first iteration, we can avoid the multiplication of the precoder by the channel matrix in the update equation. In contrast, the WMMSE must perform the whole procedure again since the precoders are updated between iterations. This is illustrated in Fig. \ref{ComplexityPerIteration}.
\begin{table}[H]
\caption{Computational complexity of the power allocation algorithms.}
\begin{center}
\vspace{-.3cm}
\begin{tabular}{ p{4 cm} c}
\hline
\hline
Technique & $\mathcal{O}$\\
\hline
\rule{0pt}{3ex}
SDMA-ES & $\mathcal{O}\left(N_t I_o^2 M^3\right)$\\
\rule{0pt}{3ex}
WMMSE & $\mathcal{O}\left(I_w N_t M^3\right)$\\
\rule{0pt}{3ex}
RS-ES & $\mathcal{O}\left(N_t I_o^2 (M+1)^3\right)$\\
\rule{0pt}{3ex}
RS-APA & $\mathcal{O}\left(I_a N_t (M+1)^2\right)$\\
\rule{0pt}{3ex}
RS-APA-R & $\mathcal{O}\left(I_a N_t (M+1)^2\right)$\\
\rule{0pt}{3ex}
CF\cite{Dai2016a} & $\mathcal{O}\left( N_t^3\right)$\\
\hline\label{Computational complexity power allocation}
\end{tabular}
\end{center}
\end{table}
\vspace{-2em}
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=5.5cm, width=0.98\columnwidth]{Complexity_new.eps}
\caption{Number of FLOPs required by different power allocation algorithms considering a MU-MIMO system with an increasing number of antennas.}
\label{Complexity}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.45\textwidth}.
\includegraphics[height=5.6cm, width=0.98\columnwidth]{CostPerIteration.eps}
\caption{Number of FLOPs required per iteration considering a MU-MIMO system with $Nt=16$ and $M=16$.}
\label{ComplexityPerIteration}
\end{subfigure}
\caption{Computational Complexity}
\end{figure}
\section{Simulations}\label{c5 section simulations}
In this section, the performance of the proposed APA-R and APA algorithms is assessed against existing power allocation approaches, namely, the ES, the WMMSE \cite{JoudehClerckx2016}, the closed-form approach of \cite{Dai2016a}, the power allocation obtained directly from the precoders definition which is denoted here as random power allocation, and the uniform power allocation (UPA) approaches. Unless otherwise specified, we consider an RS MU-MIMO system where the BS is equipped with four antennas and transmits data to two users, each one equipped with two antennas. The inputs are statistically independent and follow a Gaussian distribution. A flat fading Rayleigh channel, which remains fixed during the transmission of a packet, is considered, we assume additive white Gaussian noise with zero mean and unit variance, and the SNR varies with $E_{tr}$. For all the experiments, the common precoder is computed by employing a SVD over the channel matrix, i.e. $\hat{\mathbf{H}}=\mathbf{S}\mathbf{\Psi}\mathbf{V}^H$. Then we set the common precoder equal to the first column of matrix $\mathbf{V}$, i.e., $\mathbf{p}_c=\mathbf{v}_1$.
In the first example, we consider the transmission under imperfect CSIT. The proposed APA-R algorithm is compared against the closed form expression from \cite{Dai2016a} and against ES with random and UPA algorithms. The first ES approach fixes a random power allocation for the private streams and then an exhaustive search is carried out to find the best power allocation for the common stream. The second scheme considers that the power is uniformly distributed among private streams and then performs an exhaustive search to find the optimum value for $a_c$. Fig. \ref{C5 Figure3} shows the performance obtained with a MF. Although ES obtains the best performance, it requires a great amount of resources in terms of computational complexity. Moreover, we can see that the closed-form power allocation does not allocate power to the common message in the low SNR regime. The reason for this behavior is that this power allocation scheme was derived for massive MIMO environments where the excess of transmit antennas gets rid of the residual interference at low SNR and no common message is required. As the SNR increases the residual interference becomes larger and the algorithm allocates power to the common stream.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.55]{C5_FMF_new.eps}
\vspace{-1.0em}
\caption{Sum-rate performance of RS-MF precoding scheme, $N_t=4$, $N_k=4$, $K=1$, and $\sigma_e^2=0.05$.}
\label{C5 Figure3}
\end{center}
\end{figure}
In the next example, the ZF precoder has been considered. The results illustrated in Fig. \ref{C5 Figure4} show that the techniques that perform ES, which are termed as RS-ZF-ES+Random and RS-ZF-ES-UPA, have the best performance. However, the APA and APA-R adaptive algorithms obtain a consistent gain when compared to the conventional ZF algorithm.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{C5_F4.eps}
\vspace{-1.0em}
\caption{Sum-rate performance of RS-ZF precoding scheme, $N_t=4$, $N_k=2$, $K=2$, and $\sigma_e^2=0.1$.}
\label{C5 Figure4}
\end{center}
\end{figure}
In Fig. \ref{C5_7_D} we employed the MMSE precoder. In this case, an exhaustive search was performed to obtain the best power allocation coefficients for all streams. The technique from [27] was also considered and is denoted in the figure as RS-WMMSE. We can observe that the best performance is obtained by ES. The proposed APA and APA-R algorithms obtain a consistent gain when compared to the conventional MMSE precoder. The performance is worse than that of ES and the RS-WMMSE, but the computational complexity is also much lower.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{C5_F7_E.eps}
\vspace{-1.0em}
\caption{Sum-rate performance of RS-MMSE precoding scheme, $N_t=4$, $N_k=2$, $K=2$, and $\sigma_e^2=0.2$.}
\label{C5_7_D}
\end{center}
\end{figure}
In the next example, we evaluate the performance of the proposed APA and APA-R techniques as the error in the channel estimate becomes larger. For this scenario, we consider a fixed SNR equal to 20 dB. Fig. \ref{VarErr} depicts the results of different power allocation techniques. The results show that the APA-R performs better than the APA as the variance of the error increases.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{VarErrV1.eps}
\vspace{-1.0em}
\caption{Sum-rate performance of RS-ZF precoding scheme, $N_t=4$, $N_k=2$, and $K=2$.}
\label{VarErr}
\end{center}
\end{figure}
Let us now consider the ESR obtained versus the number of iterations, which is shown in Fig. \ref{MSEperIteration}. The step size of the adaptive algorithms was set to $0.004$ and the SNR to $10$ dB. Fig. \ref{MSEperIteration} shows that APA and APA-R obtain better performance than WMMSE with few iterations, i.e., with reduced cost. Recall that the cost per iteration is much lower for the adaptive algorithms.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.55]{FigIterationsJournal10dB4.eps}
\vspace{-1.0em}
\caption{Sum-rate performance of RS-ZF precoding scheme, $N_t=4$, $N_k=1$,$K=4$, and $\sigma_e^2=0.2$.}
\label{MSEperIteration}
\end{center}
\end{figure}
In Fig. \ref{acPow} we can notice the power allocated to the common stream. For this simulation we consider the same setup as in the previous simulation. We can observe that the parameter $a_c$ increases with the SNR. In other words, as the MUI becomes more significant, more power is allocated to the common stream. We can also notice that the proposed APA-R algorithm allocates more power to the common stream than that of the APA algorithm.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{Power_ac.eps}
\vspace{-1.5em}
\caption{Power allocated to the common stream, $N_t=4$, $N_k=2$, $K=2$.}
\label{acPow}
\vspace{-1.5em}
\end{center}
\end{figure}
In the last example, we consider the ZF precoder in a MU-MIMO system where the BS is equipped with $24$ transmit antennas. The information is sent to $24$ users which are randomly distributed over the area of interest. Fig. \ref{C5 Figure6} shows the results obtained by employing the proposed APA and APA-R algorithms. Specifically, it can be noticed that the RS system equipped with APA-R can obtain a gain of up to $20 \%$ over that random allocation and up to $50 \%$ over that of a conventional MU-MIMO system with random allocation.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{C5_F6_B.eps}
\vspace{-1.5em}
\caption{Sum-rate performance of RS-ZF precoding scheme, $N_t=24$, $N_k=1$, $K=24$, and $\sigma_e^2=0.1$.}
\label{C5 Figure6}
\end{center}
\end{figure}
\vspace{-2.15em}
\section{Conclusion}
In this work, adaptive power allocation techniques have been developed for RS-MIMO and conventional MU-MIMO systems. Differently to optimal and WMMSE power allocation often employed for RS-based systems that are computationally very costly, the proposed APA and APA-R algorithms have low computational complexity and require fewer iterations for new transmission blocks, being suitable for practical systems. Numerical results have shown that the proposed power allocation algorithms, namely APA and APA-R, are not very far from the performance of exhaustive search with uniform power allocation. Furthermore, the proposed robust technique, i.e., APA-R, increases the robustness of the system against CSIT imperfections.
\appendices
\section{Proof of the MSE for the APA-R}\label{Appendix MSE APA-R}
In what follows the derivation of the MSE for the APA-R algorithm is detailed. Let us first expand the MSE, which is given by
\begin{align}
\mathbb{E}\left[\varepsilon\lvert\hat{\mathbf{H}}\right]=&\mathbb{E}\left[\left(\mathbf{s}^{\left(\text{RS}\right)}-\mathbf{y}'\right)^H\left(\mathbf{s}^{\left(\text{RS}\right)}-\mathbf{y}'\right)\lvert\hat{\mathbf{H}}\right]\nonumber\\
=&\underbrace{\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{s}^{\left(\text{RS}\right)}\lvert\hat{\mathbf{H}}\right]}_{T_1}-\underbrace{\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{y}'\lvert\hat{\mathbf{H}}\right]}_{T_2}-\underbrace{\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\text{RS}\right)}\lvert\hat{\mathbf{H}}\right]}_{T_3}+\underbrace{\mathbb{E}\left[\mathbf{y'}^H\mathbf{y}'\lvert\hat{\mathbf{H}}\right]}_{T_4}.\label{mean square error terms for RS APA Robust}
\end{align}
The first term of \eqref{mean square error terms for RS APA Robust} is independent from $\hat{\mathbf{H}}$ and can be reduced to the following expression:
\vspace{-2em}
\begin{align}
\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{s}^{\left(\text{RS}\right)}\right]=&\mathbb{E}\left[s_c^*s_c\right]+\mathbb{E}\left[s_1^*s_1\right]\cdots+\mathbb{E}\left[s_M^*s_M\right],=M+1.\label{c5 term1}
\end{align}
The second term given by $T_2$ requires the computation of the following term:
\begin{align}
\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{y}'=& s_c^* y_c+s_1^*y_1+\cdots+s_M^*y_{M},\nonumber\\
=&s_c^*\sum_{l=1}^{M}\left[a_c s_c\left(\hat{\mathbf{h}}_{l,*}+\tilde{\mathbf{h}}_{l,*}\right)\mathbf{p}_c+\sum_{j=1}^{M}a_j s_j \left(\hat{\mathbf{h}}_{l,*}+\tilde{\mathbf{h}}_{l,*}\right)\mathbf{p}_j+ n_l\right]\nonumber\\
&+s_1^*\left[a_c s_c \left(\hat{\mathbf{h}}_{1,*}+\tilde{\mathbf{h}}_{1,*}\right)\mathbf{p}_c+\sum_{l=1}^{M}a_l s_l\left(\hat{\mathbf{h}}_{1,*}+\tilde{\mathbf{h}}_{1,*}\right)\mathbf{p}_l+n_1\right]\nonumber\\
&+\cdots+s_M^*\left[a_c s_c \left(\hat{\mathbf{h}}_{M,*}+\tilde{\mathbf{h}}_{M,*}\right)\mathbf{p}_c+\sum_{l=1}^{M}a_l s_l\left(\hat{\mathbf{h}}_{M,*}+\tilde{\mathbf{h}}_{M,*}\right)\mathbf{p}_l+n_M\right].\label{c5 term2 new}
\end{align}
By evaluating the expected value of the different terms in \eqref{c5 term2 new} we get the following quantities:
\begin{align}
\mathbb{E}\left[s_i^*y_i|\hat{\mathbf{H}}\right]&=\mathbb{E}\left[s_i^*\left\{a_c s_c \left(\hat{\mathbf{h}}_{i,*}+\tilde{\mathbf{h}}_{i,*}\right)\mathbf{p}_c+\sum_{l=1}^{M}a_l s_l\left(\hat{\mathbf{h}}_{i,*}+\tilde{\mathbf{h}}_{i,*}\right)\mathbf{p}_l+n_i\right\}\lvert\hat{\mathbf{H}}\right],\nonumber\\
&=a_i\hat{\phi}^{\left(i,i\right)}.\\
\mathbb{E}\left[s_c^*y_c|\hat{\mathbf{H}}\right]&=\mathbb{E}\left[s_c^*\sum_{l=1}^{M}\left\{a_c s_c\left(\hat{\mathbf{h}}_{l,*}+\tilde{\mathbf{h}}_{l,*}\right)\mathbf{p}_c+\sum_{j=1}^{M}a_j s_j \left(\hat{\mathbf{h}}_{l,*}+\tilde{\mathbf{h}}_{l,*}\right)\mathbf{p}_j+ n_l\right\}\lvert\hat{\mathbf{H}}\right],\nonumber\\
&=a_c\sum_{i=1}^M \hat{\phi}^{\left(i,c\right)},
\end{align}where $\hat{\phi}^{\left(i,q\right)}=\hat{\mathbf{h}}_{i,*}\mathbf{p}_q$ and $\hat{\phi}^{\left(i,c\right)}=\hat{\mathbf{h}}_{i,*}\mathbf{p}_c$. These expressions allow us to compute $T_2$, which is expressed by
\begin{equation}
\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{y}'|\hat{\mathbf{H}}\right]=a_c\sum_{i=1}^M\hat{\phi}^{\left(i,c\right)}+\sum_{j=1}^{M}a_j\hat{\phi}^{\left(j,j\right)}.\label{c5 term 2 robust}
\end{equation}
The third term can be calculated in a similar manner and is given by
\begin{align}
\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\text{RS}\right)}\right.&\left.|\hat{\mathbf{H}}\right]=a_c\sum_{i=1}^M\hat{\phi}^{\left(i,c\right)^*}+\sum_{j=1}^{M}a_j\hat{\phi}^{\left(j,j\right)^*}.\label{c5 term 3 robust}
\end{align}
The last term of equation \eqref{mean square error terms for RS APA Robust} requires the computation of several quantities. Let us first consider the following quantity:
\begin{align}
\mathbf{y'}^H\mathbf{y}'=&y_c^* y_c+y_1^*y_1+\cdots+y_M^*y_M=\left(\sum_{l=1}^M y_l^*\right)\left(\sum_{j=1}^M y_j\right)+\sum_{i=1}^M y_i^*y_i.
\end{align}
Taking the expected value on the last equation results in
\begin{align}
\mathbb{E}\left[ \mathbf{y'}^H\mathbf{y}'|\hat{\mathbf{H}}\right]=&\sum_{l=1}^M\sum_{j=1}^M\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]+\sum_{i=1}^{M}\mathbb{E}\left[y_i^* y_i|\hat{\mathbf{H}}\right],\nonumber\\
=&\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]+2\sum_{i=1}^{M}\mathbb{E}\left[y_i^* y_i|\hat{\mathbf{H}}\right],\label{c5 term4 complete}
\end{align}
\begin{align}
\mathbb{E}\left[y_i^* y_i\lvert\hat{\mathbf{H}}\right]=&\mathbb{E}\left[\left\lvert a_c s_c \mathbf{h}_{i,*}\mathbf{p}_c+\sum_{q=1}^{M}a_q s_q \mathbf{h}_{i,*}\mathbf{p}_q+ n_i\right\rvert^2\rvert\hat{\mathbf{H}}\right],\nonumber\\
=&\mathbb{E}\left[a_c^2\lvert s_c\rvert^2\left\lvert\hat{\phi}^{\left(i,c\right)}+\tilde{\phi}^{\left(i,c\right)}\right\rvert^2+\sum_{q=1}^{M}a_q\lvert s_q\rvert^2\left\lvert\hat{\phi}^{\left(i,q\right)}+\tilde{\phi}^{\left(i,q\right)}\right\rvert^2+\lvert n_i\rvert^2 \lvert \hat{\mathbf{H}}\right],
\end{align}
where $\tilde{\phi}^{\left(i,c\right)}=\tilde{\mathbf{h}}_{i,*}\mathbf{p}_c$ and $\tilde{\phi}^{\left(i,q\right)}=\tilde{\mathbf{h}}_{i,*}\mathbf{p}_q$. Expanding the terms of the last equation results in
\begin{align}
\mathbb{E}\left[y_i^* y_i\lvert\hat{\mathbf{H}}\right]=&\mathbb{E}\left[\sum_{q=1}^M a_q^2\lvert s_q\rvert^2\left(\lvert\hat{\phi}^{\left(i,q\right)}\rvert^2+2\Re\left[\hat{\phi}^{\left(i,q\right)^*}\tilde{\phi}^{\left(i,q\right)}\right]+\lvert\tilde{\phi}^{\left(i,q\right)}\rvert^2\right)\lvert\hat{\mathbf{H}}\right]\nonumber\\
&+\mathbb{E}\left[a_c^2\lvert s_c\rvert^2\left(\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+2\Re\left[\hat{\phi}^{\left(i,c\right)^*}\tilde{\phi}^{\left(i,c\right)}\right]+\lvert\tilde{\phi}^{\left(i,c\right)}\rvert^2\right)|\hat{\mathbf{H}}\right]+\sigma_n^2.
\end{align}
Since the entries of $\tilde{\mathbf{h}}_{i,*}~~\forall i$ are uncorrelated with zero mean and also independent from $\mathbf{s}^{\left(\text{RS}\right)}$, we have
\begin{align}
\mathbb{E}\left[y_i^* y_i|\hat{\mathbf{H}}\right]=&a_c^2\left(\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+\mathbb{E}\left[\lvert\tilde{\phi}^{\left(i,c\right)}\rvert^2\lvert\hat{\mathbf{H}}\right]\right)+\sum_{q=1}^M a_q^2\left(\lvert\hat{\phi}^{\left(i,q\right)}\rvert^2+\mathbb{E}\left[\lvert\tilde{\phi}^{\left(i,q\right)}\rvert^2\lvert\hat{\mathbf{H}}\right]\right)+\sigma_n^2.\label{c5 term4 robust part1}
\end{align}
Note that $\lvert\tilde{\phi}^{\left(i,c\right)}\rvert^2$ and $\lvert\tilde{\phi}^{\left(i,q\right)}\rvert^2$ are independent from $\hat{\mathbf{H}}$. Thus, we get
\begin{align}
\mathbb{E}\left[\lvert\tilde{\phi}^{\left(i,c\right)}\rvert^2\right]=&\lvert p^{\left(c\right)}_1\rvert^2\mathbb{E}\left[\tilde{h}_{i,1}^*\tilde{h}_{i,1}\right]+\lvert p^{\left(c\right)}_2\rvert^2\mathbb{E}\left[\tilde{h}_{i,2}^*\tilde{h}_{i,2}\right]+\cdots+\lvert p^{\left(c\right)}_{N_t}\rvert^2\mathbb{E}\left[\tilde{h}_{i,N_t}^*\tilde{h}_{i,N_t}\right],\nonumber\\
=&\lvert p^{\left(c\right)}_1\rvert^2\sigma_{e,i}^2+\lvert p^{\left(c\right)}_2\rvert^2\sigma_{e,i}^2+\cdots\lvert p^{\left(c\right)}_{N_t}\rvert^2\sigma_{e,i}^2,\nonumber\\
=&\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2,
\end{align}
and similarly $\mathbb{E}\left[\lvert\tilde{\phi}^{\left(i,q\right)}\rvert^2\right]=\sigma_{e,i}^2\lVert\mathbf{p}_q\rVert^2.$
Then, \eqref{c5 term4 robust part1} turns into
\begin{align}
\mathbb{E}\left[ y_i^* y_i|\hat{\mathbf{H}}\right]=&a_c^2\left(\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)+\sum_{j=1}^{K}a_j^2\left(\lvert\hat{\phi}^{\left(i,j\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right)+\sigma_n^2.\label{c5 corr received signal same ant robust}
\end{align}
Let us now evaluate the expected value of $y_l^*y_j$ when $l\neq j$, which results in
\begin{align}
\mathbb{E}\left[y_l^* y_j\lvert\hat{\mathbf{H}}\right]=&\mathbb{E}\left[\left(a_c s_c \mathbf{h}_{l,*}\mathbf{p}_c+\sum_{q=1}^{M}a_q s_q \mathbf{h}_{l,*}\mathbf{p}_q+ n_l\right)^*\right.\times\left.\left(a_c s_c \mathbf{h}_{j,*}\mathbf{p}_c+\sum_{r=1}^{M}a_r s_r \mathbf{h}_{j,*}\mathbf{p}_r+ n_j\right)\lvert\hat{\mathbf{H}}\right],\nonumber\\
=&\sum_{q=1}^M a_q^2\mathbb{E}\left[\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}+ \hat{\phi}^{\left(l,q\right)^*}\tilde{\phi}^{\left(j,q\right)}+\tilde{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}+\tilde{\phi}^{\left(l,q\right)^*}\tilde{\phi}^{\left(j,q\right)}\lvert\hat{\mathbf{H}}\right].\nonumber\\
&+a_c^2\mathbb{E}\left[\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+\hat{\phi}^{\left(l,c\right)^*}\tilde{\phi}^{\left(j,c\right)}+\tilde{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+\tilde{\phi}^{\left(l,c\right)^*}\tilde{\phi}^{\left(j,c\right)}\lvert\hat{\mathbf{H}}\right].
\end{align}
Remark that $\tilde{\mathbf{h}}_l$ and $\tilde{\mathbf{h}}_j$ are independent $\forall~~l\neq j$ with zero mean. Thus, the last equation is reduced to
\begin{align}
\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]=&a_c^2\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+\sum_{q=1}^M a_q^2\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}.\label{c5 corr signal from different ant robust}
\end{align}
Equations \eqref{c5 corr received signal same ant robust} allow us to compute the second term of equation \eqref{c5 term4 complete}, which is given by
\begin{align}
\sum_{i=1}^M\mathbb{E}\left[y_i^*y_i|\right.\left.\hat{\mathbf{H}}\right]
=&\sum_{j=1}^{M}a_j^2\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e_i}^2\lVert\mathbf{p}_j\rVert^2\right)+a_c^2\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)+M\sigma_n^2.\label{c5 term 4 part 1 robust}
\end{align}
On the other hand, \eqref{c5 corr signal from different ant robust} allow us to obtain the first term of \eqref{c5 term4 complete}, which results in
\begin{equation}
\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]=\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\left(a_c^2\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+\sum_{q=1}^M a_q^2\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}\right)
\end{equation}
Applying the property $\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}+\hat{\phi}^{\left(l,q\right)}\hat{\phi}^{\left(j,q\right)^*}=2\Re\left[\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}\right]$, we can simplify half of the sums from the triple summation, i.e.,
\begin{equation}
\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\sum_{q=1}^Ma_q^2\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}=2\sum_{l=1}^{M-1}\sum_{j=i+1}^{M}\sum_{q=1}^M a_q^2\Re\left[\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}\right].\label{c5 triple summation explained new}
\end{equation}
It follows that
\begin{align}
\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]=&2\sum_{l=1}^{M-1}\sum_{j=i+1}^{M}\sum_{q=1}^M a_r^2\Re\left[\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}\right]+\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c^2\hat{\phi}^{\left(l,c\right)}\hat{\phi}^{\left(j,c\right)}.\label{c5 term4 part 2 robust}
\end{align}
By using \eqref{c5 term 4 part 1 robust} and \eqref{c5 term4 part 2 robust} in \eqref{c5 term4 complete} and then substituting \eqref{c5 term1} \eqref{c5 term 2 robust}, \eqref{c5 term 3 robust}, and \eqref{c5 term4 complete} in \eqref{mean square error terms for RS APA Robust} we can calculate the MSE, which is given by \eqref{mean square error APA robust}. This concludes the proof. \vspace{-1.5em}
\section{Proof of the MSE for the APA}\label{Appendix MSE APA}
Here, we describe in detail how to obtain the MSE employed in the APA algorithm. Let us first consider the MSE, which is given by
\begin{align}
\mathbb{E}\left[\varepsilon\right]=&{\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{s}^{\left(\text{RS}\right)}\right]}-{\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{y}'\right]}-{\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\text{RS}\right)}\right]}+{\mathbb{E}\left[\mathbf{y'}^H\mathbf{y}'\right]}.\label{mean square error terms for RS APA}
\end{align}
The first term of \eqref{mean square error terms for RS APA} is computed identically to \eqref{c5 term1}.
By taking the expected value of the second term in \eqref{mean square error terms for RS APA} and expanding the equation, we have
\begin{align}
\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{y}'\right]=&a_c\mathbb{E}\left[s_c^*s_c\right]\sum_{l=1}^M\mathbf{h}_{l,*}\mathbf{p}_c+\sum_{l=1}^M \mathbb{E}\left[s_c^*n_l\right]+ \sum_{l=1}^{M}\mathbf{h}_{l,*}\sum_{j=1}^{M}a_j \mathbb{E}\left[s_c^*s_j\right] \mathbf{p}_j\nonumber\\
&+a_c\sum_{l=1}^M\mathbb{E}\left[s_l^*s_c\right]\mathbf{h}_{l,*}\mathbf{p}_c+\sum_{q=1}^M\sum_{l=1}^{M}a_l \mathbb{E}\left[s_q^*s_l\right]\mathbf{h}_{q,*}\mathbf{p}_l+\sum_{l=1}^M\mathbb{E}\left[s_l^*n_l\right].\label{c5 term 2 full}
\end{align}
Since the symbols are uncorrelated, equation \eqref{c5 term 2 full} is reduced to
\begin{align}
\mathbb{E}\left[\mathbf{s}^{\left(\text{RS}\right)^H}\mathbf{y}'\right]=a_c\sum_{j=1}^{M}\mathbf{h}_{j,*}\mathbf{p}_c+\sum_{l=1}^M a_l \mathbf{h}_{l,*}\mathbf{p}_l.\label{c5 term2}
\end{align}
The third term of \eqref{mean square error terms for RS APA} can be computed in a similar way as the second term, which lead us to
\begin{align}
\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\text{RS}\right)}\right]=a_c\sum_{j=1}^{M}\left(\mathbf{h}_{l,*}\mathbf{p}_c\right)^{*}+\sum_{l=1}^M a_l \left(\mathbf{h}_{l,*}\mathbf{p}_l\right)^{*}.\label{c5 term3}
\end{align}
The last term of \eqref{mean square error terms for RS APA} is equal to
\begin{align}
\mathbb{E}\left[ \mathbf{y'}^H\mathbf{y}'\right]=\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\mathbb{E}\left[y_l^* y_j\right]+2\sum_{i=1}^{M}\mathbb{E}\left[y_i^* y_i\right],\label{c5 term4 APA complete}
\end{align}
Let us first compute the quantity given by
\begin{equation}
\mathbb{E}\left[y_i^* y_i\right]=a_c^2\lvert\mathbf{h}_{i,*}\mathbf{p}_c\rvert^2+\sum_{l=1}^M a_l^2\lvert\mathbf{h}_{i,*}\mathbf{p}_l\rvert^2+\sigma_n^2.\label{c5 term 1 of term4}
\end{equation}
Additionally, we know that
\begin{align}
\mathbb{E}\left[y_i^* y_j\right]=&\mathbb{E}\left[\left(a_c s_c \mathbf{h}_{i,*}\mathbf{p}_c+\sum_{q=1}^{M}a_q s_q \mathbf{h}_{i,*}\mathbf{p}_q+ n_i\right)^*\right.\times\left.\left(a_c s_c \mathbf{h}_{j,*}\mathbf{p}_c+\sum_{l=1}^{M}a_l s_l \mathbf{h}_{j,*}\mathbf{p}_l+ n_j\right)\right],\nonumber\\
=&a_c^2\left(\mathbf{h}_{i,*}\mathbf{p}_c\right)^*\left(\mathbf{h}_{j,*}\mathbf{p}_c\right)+\sum_{l=1}^Ma_l^2\left(\mathbf{h}_{i,*}\mathbf{p}_l\right)^*\left(\mathbf{h}_{j,*}\mathbf{p}_l\right),\label{c5 term 2 of term4}
\end{align}
for all $i\neq j$. From the last equation, we have
\begin{align}
\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\mathbb{E}\left[y_l^* y_j\right]=&\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\left[a_c^2\left(\mathbf{h}_{i,*}\mathbf{p}_c\right)^*\left(\mathbf{h}_{j,*}\mathbf{p}_c\right)+\sum_{l=1}^Ma_l^2\left(\mathbf{h}_{i,*}\mathbf{p}_l\right)^*\left(\mathbf{h}_{j,*}\mathbf{p}_l\right)\right],\nonumber\\
=&\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^Ma_c^2\left(\mathbf{h}_{i,*}\mathbf{p}_c\right)^*\left(\mathbf{h}_{j,*}\mathbf{p}_c\right)+\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\sum_{l=1}^Ma_l^2\left(\mathbf{h}_{i,*}\mathbf{p}_l\right)^*\left(\mathbf{h}_{j,*}\mathbf{p}_l\right),\nonumber\\
=&\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^Ma_c^2\phi^{\left(i,c\right)^*}\phi^{\left(j,c\right)}+\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\sum_{l=1}^Ma_l^2\phi^{\left(i,l\right)^*}\phi^{\left(j,l\right)},\label{c5 term 2 of term4 extended}
\end{align}
where we define $\phi^{\left(i,c\right)}=\mathbf{h}_{l,*}\mathbf{p}_c$ and $\phi^{\left(i,l\right)}=\mathbf{h}_{i,*}\mathbf{p}_l$ for all $i,l \in \left[1,M\right]$. Applying the property $\phi^{\left(i,l\right)^*}\phi^{\left(j,l\right)}+\phi^{\left(i,l\right)}\phi^{\left(j,l\right)^*}=2\Re\left[\phi^{\left(i,l\right)^*}\phi^{\left(j,l\right)}\right]$, we can simplify half of the sums from the triple summation, i.e.,
\begin{equation}
\sum_{i=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\sum_{l=1}^Ma_l^2\phi^{\left(i,l\right)^*}\phi^{\left(j,l\right)}=2\sum_{i=1}^{M-1}\sum_{q=i+1}^{M}\sum_{r=1}^M a_r^2\Re\left[\phi^{\left(i,r\right)^*}\phi^{\left(q,r\right)}\right].\label{c5 triple summation explained}
\end{equation}
The final step to obtain the last term of \eqref{mean square error terms for RS APA} is to employ \eqref{c5 term 1 of term4}, \eqref{c5 term 2 of term4 extended}, and \eqref{c5 triple summation explained} to compute the following quantities:
\begin{equation}
\sum_{i=1}^{M}\mathbb{E}\left[y_i^* y_i\right]=\sum_{l=1}^M a_c^2\lvert\mathbf{h}_{l,*}\mathbf{p}_c\rvert^2+\sum_{i=1}^M\sum_{j=1}^M a_j^2\lvert\mathbf{h}_{i,*}\mathbf{p}_j\rvert^2+M\sigma_n^2,\label{c5 term4 a}
\end{equation}
\begin{align}
\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M\mathbb{E}\left[y_l^* y_j\right]=&2\sum_{i=1}^{M-1}\sum_{q=i+1}^{M}\sum_{r=1}^M a_r^2\Re\left[\phi^{\left(i,r\right)^*}\phi^{\left(q,r\right)}\right]+\sum_{l=1}^M\sum\limits_{\substack{j=1\\j\neq l}}^M a_c^2\phi^{\left(l,c\right)^*}\phi^{\left(j,c\right)}.\label{c5 term4}
\end{align}
By using \eqref{c5 term4 a} and \eqref{c5 term4} in \eqref{c5 term4 APA complete} and then substituting \eqref{c5 term1}, \eqref{c5 term2}, \eqref{c5 term3}, \eqref{c5 term4 APA complete} in \eqref{mean square error terms for RS APA} we get the MSE in \eqref{mean square error APA RS}. \vspace{-1.5em}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi | 187,781 |
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TITLE: How do I show that the mean recurrence time for transient states is infinity?
QUESTION [0 upvotes]: The random variable $T_i$, the "Hitting Time of $i$" is defined to be the first $n$ such that $X_n=i$ given that $X_0=i$.
By the mean recurrence time of $T_i$, I mean the expected value of this random variable.
I wish to show that if $i$ is transient, then the expectation does not converge to any finite real number. While this, intuitively makes sense, I do not know how to formally prove this and any help is appreciated.
REPLY [2 votes]: Note that state $i$ is persistent iff,
$$ P(X_n = i \text{ for some } n \geq 1| X_0 = i) = 1$$
Each state is transient or persistent.
The hitting time of state $i$, $T_i$, is a random variable defined as the first time we visit state $i$:
$$ T_i = \min \{n | n \geq 1, X_n = i\} $$
where $T_i$ is defined as $\infty$ if this visit never happens.
We now show that $ P(T_i = \infty | X_0 = i) > 0 $ iff state $i$ is transient.
Then, the required result on the mean recurrence time follows, because the mean recurrence time $\mu_i$ is defined as:
$$\mu_i = E(T_i| X_0 = i) $$
Suppose state $i$ is transient. Then,
$$
\begin{align} P(T_i = \infty | X_0 = i) & = P(X_n \neq i \text{ for all } n \geq 1 | X_0 = i) \\
& = 1 - P(X_n = i \text{ for some } n \geq 1 | X_0 = i) \\
& > 1 - 1 = 0.
\end{align}
$$
Suppose state $i$ is persistent. Then,
$$
\begin{align} P(T_i = \infty | X_0 = i) & = P(X_n \neq i \text{ for all } n \geq 1 | X_0 = i) \\
& = 1 - P(X_n = i \text{ for some } n \geq 1 | X_0 = i) \\
& = 1 - 1 = 0.
\end{align}
$$
This shows both directions, and completes the proof. | 93,689 |
\section{Delay Analysis} \label{sec:Delay Analysis}
Different types of delay analysis have been studied in \cite{2009sundararajanfeedback,2012SadeghiDeliverydelayanalysis,2014Sadeghidynamic} for this
coding scheme in homogeneous networks. They considered both decoding delay and delivery delay. However, in such a system where the packets can be
used only if they are delivered, decoding delay is less important than the delivery delay; because it is possible that a receiver decodes a packet but it must wait
until it is actually delivered in order to the application layer.
In the literature, the delivery delay has been considered as the time between when a packet enters the transmission queue and its delivery to the application at each receiver \cite{2009sundararajanfeedback}. Using this definition of the delivery delay in heterogeneous networks, there may be a large difference between delivery times of a packet for different receivers. On the other hand, weak receivers left behind from the transmission queue, still seek for older packets to complete their delivery. Therefore, we study the delivery delay using a new definition, which is based on the time that each receiver waits for a packet after it is first requested by that receiver. Using this new definition, the delivery delay of the receivers is measured independently with respect to their capability of delivering packets. We believe that this new definition is more suitable for heterogeneous networks and moreover, it leads to a closed form for the delivery delay.
\begin{dfn}
The delivery delay of a packet for receiver \um{i} is the time between the first request of that packet and its delivery. The probability of having $T$ time slots delay for \um{i} is shown by $P^d_i(T)$.
\end{dfn}
\begin{propos}
Suppose that $d_{u_i}$ is the probability of delivery a packet by $U_i$ in each time slot. The probability of delivery delay for $U_i$ is given by:
\begin{equation}
P^d_i(T)=\begin{cases}
{d_{u_i}^2(1-d_{u_i})^{(T-1)}}/{R_i}&T>0,\\
1-\sum_{T=1}^{\infty}{P^d_i(T)}&T=0.
\end{cases}
\label{eq:delivery delay}
\end{equation}
\end{propos}
\begin{proof}
A packet is delivered to a receiver $U_i$ with the probability of $d_{u_i}$, then $U_i$ requests the next packet and it can receive that after $T>0$ time slots by the probability of $(1-d_{u_i})^{(T-1)}d_{u_i}$.
Therefor, the number of packets with delivery delay of $T>0$ is $t{d_{u_i}^2(1-d_{u_i})^{(T-1)}}$, and $P^d_i(T)$ is given by
${td_{u_i}^2(1-d_{u_i})^{(T-1)}}/{d_i(t)}$. By summation on $P^d_i(T)$ for $T>0$, all packets with non-zero delivery delay in \um{i} buffer are considered
and the probability of the rest of them is given by complement probability that is given in (\ref{eq:delivery delay}). These are the packets with zero delivery delay which have been decoded sooner than the previous packets. When the receiver needs them they have already been delivered and it is not needed to send request for them to the sender.
\end{proof}
To calculate $P^d_i(T)$, the probability of delivery is needed that could be complicated in some cases. For instance if there is more than one receiver in
group $\hh$, calculation of $d_{u_i}$ is rather complicated for receivers in this group. In the following, we compute $d_{u_i}$ for some special
cases and compare the results with simulations.
\subsection{No receiver in group $\hh$}\label{No receiver in group H}
In this case, the strongest receiver is always the leader and other ones receive their packets only via differential knowledges. Thus, according to Section \ref{Delivery rate} the probability of packet delivery is given by:
\begin{equation}
d_{u_i}=\begin{cases}
C_1&i=1,\\
D_i^1C_i&i>1.
\end{cases}
\label{NO_R_delay}
\end{equation}
For $U_1$ (the strongest receiver), $d_{u_1}$ is the same as channel capacity, because it is the strongest receiver and all packets in its buffer are assumed to be delivered. However, for the other receivers $d_{u_i}$ is different, since they also receive non-differential packets which affects the number of delivered packets and the delivery rate, while $d_{u_i}$ is the probability of receiving a requested packet and delivery of it at the same time.
Now $P^d_i(T)$ can be determined using (\ref{eq:delivery delay}).
\subsection{One receiver in $\hh$}\label{One receiver in H}
In this case, there are two leaders, the receiver in $\hh$, $U_1\equiv U_H$ and $U_2 \equiv U_L$. For $d_{u_i}$ we have:
\begin{equation}
\label{eq:du1}
d_{u_i}=\begin{cases}
\beta_HC_1=\lambda& i=1 \quad (U_1\equiv U_H),\\
(\beta_HD^H_L+\bar{\beta}_H)C_L&i=2 \quad (U_2 \equiv U_L),\\
(\beta_HD_i^H+\bar{\beta}_HD_i^L)C_i&i>2.
\end{cases}
\end{equation}
According to the delivery rate analysis in Section \ref{sec:Delivery Rate Analysis}, $\beta_H$ is the fraction of time that $\nn{1}$ is in the transmission queue, and $\uu{1}$ receives it with the probability of $C_1$.
So, because all packets received by $U_1$ are delivered, $d_{u_1}$ is given by (\ref{eq:du1}). On the other hand, the portion of time that
$U_L$ is the leader, is given by $\bar{\beta}_H$, and this receiver deliver packets via leader transmissions with the probability of $\bar{\beta}_HC_L$ and differential knowledge transmissions with the probability of $\beta_HD_L^HC_L$. Moreover, other receivers deliver
packets via differential knowledges from these two leaders by the probability given in (\ref{eq:du1}). Again, the delay distribution is given by
(\ref{eq:delivery delay}).
\subsection{More than one receiver in $\hh$}\label{More than one receiver in H}
When there is more than one receiver in $\hh$, all members of $\hh$ have a chance to be the leader and they receive differential knowledges from each other. Furthermore, using our analytical model, we can not calculate the probability of being the leader and differential knowledge for the receivers in $\hh$. However, if we had the leader and differential knowledge probabilities in $\hh$, then $d_{u_i}$ would be given by:
\begin{equation}
\label{eq:2inH}
d_{u_i}=\begin{cases}
\displaystyle{\sum_{\substack{k\\U_k \in \hh}}{\beta_kD_i^k}}C_i&U_i\in \hh,\\
(\beta_HD_i^H+\bar{\beta}_HD_i^L)C_i&U_i\in\LL.
\end{cases}
\end{equation}
Where $\beta_k$ is the probability of $U_k\in \hh$ be the leader and $D_i^k$ is the probability of differential knowledge for $U_i$ when $U_k$ is the leader (Note that $D_i^i=1$ which is correspond to the leader transmission for $U_i$). Because we cannot calculate the values of $\beta_k$s and $D_i^k$s for the receivers in $\hh$, to evaluate the accuracy of (\ref{eq:2inH}) , we extract these values from simulations and after calculating $d_{u_i}$ we put it in (\ref{eq:delivery delay}) and compare the results with simulation
(see Fig. \ref{pic:delay5_2}).
\subsection{Simulation results}
Here, the derived expressions for the delay distributions are compared with the values of simulation. Fig. \ref{pic:delay4_0} and Fig. \ref{pic:delay5_1} illustrate the simulation and calculation results for the cases of Section \ref{No receiver in group H} and \ref{One receiver in H} respectively. For these cases, the settings A and B of table \ref{tab:simconditions} have been used for simulation and (\ref{eq:delivery delay}), (\ref{NO_R_delay}) and (\ref{eq:du1}) have been used for calculation. As it is observed, the calculation shows perfect match with simulation for $U_1$ and $U_2$ and loses its accuracy for the other receivers in both settings due to the error in calculation of differential knowledge probability. It is noteworthy that $P_1^d(0)$ is zero, that shows $U_1$ did not receive non-differential packets, because in setting A, $U_1$ has been the leader in all time slots, and in setting B, $U_2$ could be the leader only when the requested packet of $U_1$ is not in the transmission queue. Moreover, for $U_1$ in both settings most of the packets are delivered with the delay of $T=1$ i.e. in the next time slot after requesting packets. However, for the other receivers $P_i^d(0)$ is maximum, since they receive most of their packets by non-differential transmissions from the leader. Although $P_i^d(0)$ is maximum, it does not mean that the receivers experience low delay. In order to compare the receivers in delay, we should look at the range of $T$ for each receiver. For instance, in setting A, $U_1$ has the range of $0\leq T\leq 6$ while for $U_2$ the range is $0 \leq T \leq 90$ (the ranges in figures are limited to have better illustration). The maximum value of $T$ increases for $U_3$ to $202$ and for the last receiver to $1100$. Furthermore, for $U_i$'s with $i>1$ the probability of delay has a slow decline for $T>0$ that shows the number of packets for each value of $T$ is close to each other.
The simulation results for Section \ref{More than one receiver in H} is depicted in Fig. \ref{pic:delay5_2} where setting C of table \ref{tab:simconditions}, (\ref{eq:delivery delay}) and (\ref{eq:2inH}) have been used for simulation and calculation. Note that in this case $P_1^d(0)$ is not zero, because the other members in $\hh$ could be the leader and all of the receivers in $\hh$ can receive differential knowledges and also non-differential transmissions from each other.
\begin{figure}[]
\centering
\begin{minipage}[]{0.4\textwidth}
\includegraphics[width=\textwidth]{delay4_0.pdf}
\caption{Simulation and analytical results for the probability of having $T$ time slots delivery delay ($P_i^d(T)$) of Setting A in table \ref{tab:simconditions}. To calculate the delivery delay, (\ref{NO_R_delay}) is used for $d_{u_i}$ and the delivery delay is given by (\ref{eq:delivery delay}).}
\label{pic:delay4_0}
\end{minipage}
\hfill
\begin{minipage}[]{0.4\textwidth}
\includegraphics[width=\textwidth]{delay5_1.pdf}
\caption{Simulation and analytical results for the probability of having $T$ time slots delivery delay ($P_i^d(T)$) of Setting B in table \ref{tab:simconditions}. To calculate the delivery delay, (\ref{eq:du1}) is used for $d_{u_i}$ and the delivery delay is given by (\ref{eq:delivery delay}).}
\label{pic:delay5_1}
\end{minipage}
\end{figure}
\begin{figure}[]
\centering
\begin{minipage}[]{0.4\textwidth}
\includegraphics[width=\textwidth]{delay5_2.pdf}
\caption{Simulation and analytical results for the probability of having $T$ time slots delivery delay ($P_i^d(T)$) of Setting C in table \ref{tab:simconditions}. To calculate the delivery delay, (\ref{eq:2inH}) is used for $d_{u_i}$ and the delivery delay is given by (\ref{eq:delivery delay}).}
\label{pic:delay5_2}
\end{minipage}
\hfill
\begin{minipage}[]{0.4\textwidth}
\includegraphics[width=\textwidth]{delay_expect.pdf}
\caption{Delay expectation of different receivers, calculation and simulation for the settings of table \ref{tab:simconditions}. For calculation (\ref{eq:delay expect}) is used.}
\label{pic:delay expect}
\end{minipage}
\end{figure}
Another parameter for comparing the delay of the receivers is the expected value of the delivery delay which is shown by $E_i\{T\}$ for receiver $U_i$. From (\ref{eq:delivery delay}), we have:
\begin{equation}
\label{eq:delay expect}
E_i\{T\}=\sum_{T=0}^{\infty}{TP_i^d(T)}=\frac{1}{R_i}.
\end{equation}
This is a reasonable result that the average delay of each receiver has an inverse relation to its delivery rate. In Fig. \ref{pic:delay expect}, the simulation and calculation of $E\{T\}$ have been depicted for settings of table \ref{tab:simconditions}. As the figure illustrates, by increasing the receiver index in each setting, the delivery delay and error margin increase. Since the expectation of delay is the inverse of the delivery rate, the error margin of delay expectation increases for weaker receivers because the delivery rate value of these receivers is small and a little error in calculation of it affects the delay expectation considerably.
Using this comparison, we conclude that the stronger receivers have less delay. | 133,256 |
\begin{document}
\title[Fractional nonlocal Ornstein--Uhlenbeck equation]{The fractional
nonlocal Ornstein--Uhlenbeck equation, \\ Gaussian symmetrization and regularity}
\begin{abstract}
For $0<s<1$, we consider the Dirichlet problem for the fractional nonlocal Ornstein--Uhlenbeck equation
$$\begin{cases}
(-\Delta+x\cdot\nabla)^su=f,&\hbox{in}~\Omega,\\
u=0,&\hbox{on}~\partial\Omega,
\end{cases}$$
where $\Omega$ is a possibly unbounded open subset of $\R^n$, $n\geq2$.
The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed.
We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions.
As consequences, novel $L^p$ and $L^p(\log L)^\alpha$
regularity estimates in terms of the datum $f$ are obtained
by comparing $u$ with half-space solutions.
\end{abstract}
\maketitle
\section{Introduction}
In the present paper we are interested in developing Gaussian symmetrization techniques and, as consequences, to
obtain novel $L^p$ and $L^p(\log L)^\alpha$ regularity estimates for solutions to
nonlocal equations driven by fractional powers of the Ornstein--Uhlenbeck (OU for short) operator subject to homogeneous
Dirichlet boundary conditions.
More precisely, we focus on problems of the form
\begin{equation}\label{Problema}
\begin{cases}
(-\Delta+x\cdot\nabla)^su=f,&\hbox{in}~\Omega,\\
u=0,&\hbox{on}~\partial\Omega,
\end{cases}\qquad\hbox{for}~0<s<1,
\end{equation}
where $\Omega$ is an open subset of
$\mathbb{R}^{n}$ with $\gamma(\Omega)<1$. Here $\gamma$ denotes the Gaussian measure on $\R^{n}$, see \eqref{eq:Gaussian density}.
Our problem \eqref{Problema} corresponds to a Markov process. Indeed,
there is a stochastic process $Y_t$ having as generator the fractional OU operator
\eqref{Problema} with homogeneous Dirichlet boundary condition.
The process can be obtained as follows.
We first kill an OU process $X_t$ at $\tau_\Omega$, the first exit time of $X_t$
from the domain $\Omega$. Let us denote the killed OU process by $X_t^\Omega$.
Then we subordinate the killed OU process $X_t^\Omega$ with an $s$-stable
subordinator $T_t$. Thus $Y_t=X^\Omega_{T_t}$ is the resulting process (see for instance \cite{Applebaum}).
As explained in \cite{Caffarelli-Stinga}, \eqref{Problema} also arises in the context of
nonlinear elasticity as the Signorini problem or the thin obstacle problem.
Nonlocal equations with fractional powers of the OU operator in $\Omega=\R^n$ have been
studied in the past. Indeed, a Harnack inequality for nonnegative solutions was proved in \cite{Stinga-Zhang}.
Fractional isoperimetric problems and semilinear equations in infinite
dimensions (Wiener space) have been considered in \cite{NPS} and \cite{NPS2}.
Fractional functional inequalities were recently analyzed in \cite{Caffarelli-Sire}.
The symmetrization techniques in elliptic and parabolic PDEs are nowadays very classical and efficient tools
to derive optimal \textit{a priori} estimates for solutions.
The investigation in such direction started with the fundamental paper by
H. Weinberger \cite{Wein62}, see also \cite{Maz}. The ideas were later fully formalized by
G. Talenti in \cite{Ta1} for the homogeneous Dirichlet problem
associated to a linear equation in divergence form with zero order term
on a bounded domain of $\R^{n}$.
In particular, \cite{Ta1} establishes a strong pointwise comparison between the Schwarz spherical rearrangement
of the solution $u(x)$ to the original problem, and the unique radial solution $v(|x|)$ of a suitable elliptic problem
defined on a ball having the same measure as the original domain and radial data.
In turn, this kind of result allows to obtain regularity estimates of solutions with optimal constants.
When dealing with parabolic equations, any form of pointwise comparison between the solution $u(x,t)$ of an
initial boundary value problem and the solution $v(|x|,t)$ of a related radial problem with respect to $x$
is in general no longer available. Indeed, in this case a weaker comparison result in the integral form, the so-called mass concentration comparison (or
comparison of concentrations), holds for all times $t>0$, see for instance \cite{Band2,Mossino}. For a detailed survey
on this theory we refer the interested reader to \cite{VAZSURV}.
Quite recently, symmetrization techniques have been successfully applied to a class of
fractional nonlocal equations. More precisely, results in terms of symmetrization were obtained
for equations driven by the fractional
Dirichlet Laplacian $$(-\Delta_D)^su=f,$$ and by the fractional Neumann
Laplacian $$(-\Delta_N)^su=f,$$ in bounded domains of $\R^n$, for $0<s<1$. These equations arise in
several important applications, see for example
\cite{Allen, Caffarelli-Stinga, Song-Vondracek, StingaVolz}. The fractional operators above are defined in terms of
the corresponding eigenfunction expansions. Then the characterization provided
by the extension problem of \cite{Stinga-Torrea}
via the Dirichlet-to-Neumann map for a (degenerate or singular) elliptic PDE
allows to treat the above-mentioned problems with local techniques
(we also refer the reader to \cite{Caffarelli-Silvestre}
for the fractional Laplacian on $\R^n$ and to \cite{Gale-Miana-Stinga} for the most general extension result
available, namely, for infinitesimal generators of integrated semigroups in Banach spaces).
This information was essential to start a program regarding the applications
of symmetrization in PDEs with fractional Laplacians. Indeed,
the first paper in such direction was the seminal work \cite{VolzDib} for the case of the fractional Dirichlet Laplacian.
Those ideas were extended and enriched with many other applications to nonlinear fractional parabolic
equations in \cite{VAZVOL3,VAZVOL1,VAZVOL2}.
When Neumann boundary conditions in fractional elliptic and parabolic problems are assumed,
the symmetrization tools applied to the extension problem
still lead to a comparison result, though of a different type, see \cite{VOLZNEUM}.
It is important to notice that all the comparison results in the nonlocal setting we just mentioned
are not pointwise in nature, but in the form of mass concentration comparison.
One motivation of such phenomenon relies on the fact that the symmetrization
argument applies on the extension problem with respect to the spatial variable $x$,
by freezing the extra extension variable $y>0$.
In other words, a comparison of the solution to the extension problem
is given in terms of the so-called Steiner symmetrization.
On the other hand, for elliptic equations involving the OU operator
$$\mathcal{L}=-\Delta+x\cdot\nabla,$$
the first comparison result through symmetrization, in the pointwise form, was obtained in \cite{Bet-Pster}.
The symmetrization has to take into account the natural variational structure of the OU operator.
Indeed, the Dirichlet problem for $\mathcal{L}$ is of the form
\begin{equation}\label{Problemalocale}
\begin{cases}
-\operatorname{div}(\varphi\nabla u)=f\varphi,&\hbox{in}~\Omega, \\
u=0,&\hbox{on}~\partial\Omega,
\end{cases}
\end{equation}
where $\varphi=\varphi(x)$ is the density of the Gaussian measure $d\gamma$ with respect to the Lebesgue measure:
\begin{equation}\label{eq:Gaussian density}
d\gamma(x)=\varphi(x)\,dx=(2\pi)^{-n/2}\exp(-|x|^2/2)\,dx,\quad\hbox{for}~x\in\mathbb{R}^{n}.
\end{equation}
The source term $f$ is then taken in the suitable class of weighted $L^p$ spaces.
Moreover, the meaningful case is when $\Omega$ is an unbounded open set. Here
we assume
$$\gamma(\Omega)<1.$$
Hence, the comparison result must be done through \emph{Gaussian symmetrization}
instead of the usual Schwarz symmetrization.
In this setting, one of the main tools in the proof is the
Gaussian isoperimetric inequality, which states that among all measurable subsets
of $\mathbb{R}^{n}$ with prescribed Gaussian measure, the half-space is the minimizer of the Gaussian perimeter.
It becomes rather intuitive to guess that the Schwarz spherical rearrangement of a function
(which is a special radial, decreasing function), appearing in the comparison results in the Lebesgue setting,
should now be replaced by the rearrangement with respect to the Gaussian measure.
The latter is a particular increasing function, depending only on one variable, defined in a half-space (see Subsection \ref{Gaussian rearrangements} for
definitions and related properties).
The authors of \cite{Bet-Pster} were able to apply this powerful machinery
to compare the solution $u$ (in the sense of rearrangement) to \eqref{Problemalocale}
with the solution $v$ to the problem
\begin{equation}\label{Problemalocalesymm}
\begin{cases}
-\operatorname{div}(\varphi\nabla v)=f^{\displaystyle\star}\varphi,&\hbox{in}~\Omega^{\displaystyle\star}\\
v=0,&\hbox{on}~\partial\Omega^{\displaystyle\star},
\end{cases}
\end{equation}
where $\Omega^{\displaystyle\star}$ is a half-space having the same Gaussian measure as $\Omega$
and $f^{\displaystyle\star}$ is the $n$-dimensional Gaussian rearrangement of $f$.
The solution $v$ to \eqref{Problemalocalesymm} (parallel to the classical case described in \cite{Ta1})
can be explicitly written, allowing the authors to derive the sharp a priori pointwise estimate
$$u^{\displaystyle\star}(x)\leq v(x),\quad\hbox{for}~x\in \Omega^{\displaystyle\star}.$$
This was the starting point to obtain regularity results for $u$ in Lorentz--Zygmund spaces.
Generalizations of this result for elliptic and parabolic problems involving
elliptic operators in divergence form which are degenerate with respect to the Gaussian measure
are contained in \cite{chiacchio,dFP}, see also references therein.
Our main concern is to get sharp estimates
for the solution $u$ to \eqref{Problema} by comparing it with the solution $\psi$ to the problem
\begin{equation}\label{problema simm}
\begin{cases}
\mathcal{L}^{s}\psi=f^{\displaystyle\star}, & \hbox{in}~\Omega^{\displaystyle\star},\\
\psi=0, & \hbox{on}~\partial\Omega^{\displaystyle\star}.
\end{cases}
\end{equation}
As our previous discussion evidences, \eqref{problema simm} is actually a one dimensional problem.
Our idea that yields the desired result reads as follows.
Using the main extension result of \cite{Stinga-Torrea} we can characterize the fractional OU operator
$\mathcal{L}^s$ in \eqref{Problema} as a suitable Dirichlet-to-Neumann map.
This allows us to obtain the solution $u$ to \eqref{Problema}
as the trace on $\Omega$ of the solution $w=w(x,y)$ of the following degenerate elliptic boundary value problem, which will be called the \emph{extension problem}
associated to \eqref{Problema}:
\begin{equation}\label{Problema estensione}
\begin{cases}
-\operatorname{div}(y^a\varphi(x)\nabla_{x,y}w)=0, &\hbox{in}~\mathcal{C}_{\Omega},\\
w=0,&\hbox{on}~\partial_{L}\mathcal{C}_{\Omega},\\
-\underset{y\rightarrow0^{+}}{\lim}y^aw_y=f,&\hbox{on}~\Omega.
\end{cases}
\end{equation}
Here
\begin{equation}\label{eq:a}
a:=1-2s\in(-1,1),
\end{equation}
while
\[
\mathcal{C}_{\Omega}:=\Omega\times(0,\infty)
\]
is the infinite cylinder of basis $\Omega$, and
$\partial_{L}\mathcal{C}_{\Omega}:=\partial\Omega\times[0,\infty)$ is its lateral boundary.
In a similar way, the solution $\psi$ to \eqref{problema simm} can be seen as
the trace over $\Omega^{\displaystyle\star}$ of the solution $v=v(x,y)$ to
\begin{equation}\label{Problema estensione sim}
\begin{cases}
-\operatorname{div}(y^a\varphi(x)\nabla_{x,y}v)=0, &\hbox{in}~\mathcal{C}_{\Omega}^{\displaystyle\star},\\
v=0,&\hbox{on}~\partial_{L}\mathcal{C}_{\Omega}^{\displaystyle\star},\\
-\underset{y\rightarrow0^{+}}{\lim}y^av_y=f^{\displaystyle\star},&\hbox{on}~\Omega^{\displaystyle\star},
\end{cases}
\end{equation}
where
\begin{equation}\label{eq:Steiner symmetrization}
\mathcal{C}_{\Omega}^{\displaystyle\star}:=\Omega^{\displaystyle\star}\times(0,\infty),
\end{equation}
and $\partial_{L}\mathcal{C}_{\Omega
}^{^{\displaystyle\star}}:=\partial\Omega^{\displaystyle\star}\times[0,\infty)$.
Therefore, the problem reduces to look for a \emph{mass concentration
comparison} between the solution $w$ to \eqref{Problema estensione} and the solution $v$ to
\eqref{Problema estensione sim}. More precisely, we prove that
\begin{equation}
\int_{0}^rw^{\circledast}(\sigma,y)\,d\sigma\leq \int_{0}^rv^{\circledast}(\sigma,y)\,d\sigma,
\quad\hbox{for all}~r\in[0,\gamma(\Omega)],\label{concineq}
\end{equation}
where, for all $y\geq0$, the functions $w^{\circledast}(\cdot,y)$ and
$v^{\circledast}(\cdot,y)$ are the one dimensional Gaussian rearrangements
of $w(\cdot,y)$ and $v(\cdot,y)$, respectively. The key role of this framework
is played by a novel second order derivation formula
for functions defined by integrals, see Corollary \ref{Secondderivform}, whose proof presents new nontrivial technical difficulties owed to the Gaussian
framework.
As a consequence, we will obtain $L^p$ and $L^p(\log L)^\alpha$ estimates for $u$ in terms of $f$.
The paper is organized as follows. Section \ref{Preliminari} contains
the preliminaries needed for the developments of our results.
In particular, we briefly describe some basic properties of the Gaussian measure and the
OU semigroup. Moreover, we carefully develop a full and self-contained analysis of the main
functional setting where problems \eqref{Problema} and \eqref{Problema estensione} are posed.
Section \ref{Preliminari} ends with the introduction of the basic definitions
and properties of symmetrization with respect to the Gaussian measure. In this regard, we will present the proof of the derivation formula stated in
Theorem \ref{Firstderivform}, whose consequence is the above-mentioned second order differentiation formula, see Corollary \ref{Secondderivform}.
Section \ref{Section:comparison} is entirely devoted to the proof
of the comparison \eqref{concineq}, that is, our main result Theorem \ref{primoteoremadiconfronto}.
In Section \ref{Section:regularity} we present our novel
Gaussian--Zygmund $L^p(\log L)^\alpha(\Omega,\gamma)$ and $L^p(\Omega,\gamma)$
regularity estimates for solutions $u$ in
terms of the datum $f$, see Theorem \ref{thm:integrability}.
More precisely, our main result (Theorem \ref{primoteoremadiconfronto})
is combined with $L^p(\log L)^\alpha$ regularity estimates of the solution $\psi$
to problem \eqref{problema simm}, which is obtained by using the explicit form
of $\psi$ in terms of the fractional integral $\mathcal{L}^{-s}(f^{\star})$ and the OU semigroup.
Finally, in the Appendix we shall use suitable estimates of the Mehler kernel
to exhibit a semigroup-based proof of the regularity estimates when the datum
$f$ belongs to the smaller Gaussian--Lebesgue space $L^p(\Omega,\gamma)$.
\section{Preliminaries, functional setting, and the second order derivation formula}\label{Preliminari}
In this section we recall the basic tools we
are going to use in the proof of our main comparison result,
Theorem \ref{primoteoremadiconfronto}, and its
consequences. First, we introduce some basics about Gaussian analysis and the OU semigroup.
Then the necessary functional
background to precise the fractional nonlocal equations \eqref{Problema} and \eqref{problema simm},
and their extension problems \eqref{Problema estensione} and \eqref{Problema estensione sim}
will be developed. Finally, after presenting
definitions and properties of rearrangement techniques in the Gaussian
framework, we will prove our novel
second order derivation formula, see Theorem \ref{Firstderivform}
and Corollary \ref{Secondderivform}.
\subsection{Gaussian analysis and the OU semigroup}
\subsubsection{Gaussian measure and isoperimetry}
Let $d\gamma$ be the $n$-dimensional normalized Gaussian measure on $\mathbb{R}^{n}$
defined in \eqref{eq:Gaussian density}. Let $\Omega$ be an open subset of $\R^n$, possibly unbounded.
We denote by $H^{1}(\Omega,\gamma)$ the Sobolev space
with respect to the Gaussian measure,
which is obtained as the completion of $C^{\infty}(\overline{\Omega})$ with respect to the norm
$$\|u\|_{H^{1}(\Omega,\gamma)}^2=\int_{\Omega}
u^{2}\,d\gamma(x)+\int_{\Omega}|\nabla u|^{2}\,d\gamma(x).$$
By $H^{1}_{0}(\Omega,\gamma)$ we denote
the closure of $C_c^{\infty}(\Omega)$ in the norm of $H^{1}(\Omega,\gamma)$.
The following Poincar\'{e} inequality holds (see for instance \cite{Eh2}):
if $\gamma(\Omega)<1$ then there exists a constant $C_\Omega>0$ such that
\begin{equation}
\int_{\Omega}| u|^{2}\,d\gamma(x)\leq C_{\Omega}\int_{\Omega}
|\nabla u|^{2}\,d\gamma(x),\quad\hbox{for all}~u\in H_{0}^{1}(\Omega,\gamma).
\label{Poincare}
\end{equation}
One of the main tools to prove the comparison result is the \emph{Gaussian
isoperimetric inequality}. Let us define the
perimeter with respect to Gaussian measure as
$$P(E)=\int_{\partial E}\varphi(x)\,d\mathcal{H}^{n-1}(x),$$
where $E$ is a set of locally finite perimeter and $\partial E$ denotes its reduced boundary. As usual, $\mathcal{H}^{n-1}$ denotes the
$(n-1)$-dimensional Hausdorff measure. It is well known (see \cite{Bo})
that among all measurable sets of $\mathbb{R}^{n}$ with prescribed Gaussian
measure, the half-spaces take the smallest perimeter. More precisely, we have
\begin{equation}
P(E)\geq\frac{1}{\sqrt{2\pi}}\exp\big(-[\Phi^{-1}(\gamma(E))]^2/2\big),
\label{dis isop}
\end{equation}
for all subsets $E\subset\mathbb{R}^{n}$, where, for $\lambda\in\mathbb{R}\cup\{-\infty,+\infty\}$, we set
\begin{equation}\label{Phi}
\Phi(\lambda):=\frac{1}{\sqrt{2\pi}}\int_{\lambda}^{\infty}e^{-r^2/2}\,dr.
\end{equation}
\subsubsection{The OU semigroup}
We recall some remarkable properties of the OU semigroup (see \cite{AS,bok} for further details)
which will turn out to be useful in the following.
The solution to the Cauchy problem
$$\begin{cases}
\rho_{t}+\mathcal{L}\rho=0,&\hbox{in}~\mathbb{R}^{n}\times(0,\infty), \\
\rho(x,0)=g(x),&\hbox{on}~\mathbb{R}^{n},
\end{cases}$$
is given by the OU semigroup
$$\rho(x,t)=e^{-t\mathcal{L}}g(x).$$
It is a classical fact that such a semigroup can be expressed in terms of a suitable integral kernel.
More precisely, if $g\in L^{p}(\mathbb{R}^{n},\gamma)$, for $1\leq p\leq\infty$, then
\begin{equation}\label{solutsemig}
e^{-t\mathcal{L}}g(x)=\int_{\mathbb{R}^{n}}M_{t}(x,y)g(y)\,d\gamma(y),\quad\hbox{for}~x\in\R^n,~t>0.
\end{equation}
Here $M_{t}(x,y)$ is the so-called Mehler kernel, which is defined by
\begin{equation}\label{eq:Mehler kernel}
M_{t}(x,y)=\frac{1}{(1-e^{-2t})^{n/2}}\exp\bigg(-\frac{e^{-2t}|x|^{2}-2e^{-t}\langle x,y
\rangle+e^{-2t}|y|^{2}}{2(1-e^{-2t})}\bigg).
\end{equation}
We recall that
\begin{equation}\label{M=1}
\int_{\R^n}M_{t}(x,y)\,d\gamma(y)=1,\quad\hbox{for all}~x\in\R^n,~t>0,
\end{equation}
and that if $g\in L^p(\R^n,\gamma)$, $1\leq p<\infty$, then
\begin{equation}\label{Stima Lp semigruppo}
\|e^{-t\mathcal{L}}g\|_{L^p(\R^n,\gamma)}
=\bigg\|\int_{\R^n}M_{t}(\cdot,y)g(y)\,d\gamma(y)\bigg\|_{L^{p}(\R^n,\gamma)}
\leq\left\Vert g\right\Vert _{L^{p}(\R^n,\gamma)}.
\end{equation}
It is standard to define the OU semigroup on a domain $\Omega$ of $\R^{n}$
subject to homogenous Dirichlet boundary conditions. Indeed, the solution to
the Cauchy--Dirichlet problem
\begin{equation} \label{p2}
\begin{cases}
\eta_{t}+\mathcal{L}\eta=0,&\hbox{in}~\Omega\times(0,\infty), \\
\eta(x,t)=0,&\hbox{on}~\partial\Omega\times[0,\infty), \\
\eta(x,0)=f(x),&\hbox{on}~\Omega,
\end{cases}
\end{equation}
is given by the semigroup generated by the OU in $\Omega$ with Dirichlet boundary conditions:
$$\eta(x,t)=e^{-t\mathcal{L}_{\Omega}}f(x).$$
It follows from standard parabolic regularity theory that $\eta$ is smooth in $\Omega\times(0,\infty)$.
Now, let us choose $\Omega=H$, where $H$ is the half-space $H:=\{x=(x_{1},x')\in\R^n :x_{1}>0,~x'\in\R^{n-1}\}$
and define
\begin{equation}\label{est1}
\widetilde{f}(x)=
\begin{cases}
f(x_{1},x^{\prime}),&\hbox{for}~x\in H,\\
-f(-x_{1},x^{\prime}),&\hbox{for}~x\in\R^n\setminus H.
\end{cases}
\end{equation}
Observe that for $1\leq p<\infty$ we have
\begin{equation}\label{norma prolungamento}
\|\widetilde{f}\|_{L^p(\R^n,\gamma)}=2\|f\|_{L^{p}(H,\gamma)}.
\end{equation}
It is not difficult to check (see for example \cite{Priola}) that in this case
the semigroup associated to \eqref{p2} is obtained as the restriction to $H$
of the OU semigroup on $\R^n$ applied to $\tilde{f}$, that is,
\begin{equation}\label{eta}
\eta(x,t)=e^{-t\mathcal{L}_H}f(x)=e^{-t\mathcal{L}}\widetilde{f}(x)\big|_H.
\end{equation}
Moreover, using the expression of the OU semigroup in terms of the Mehler kernel \eqref{solutsemig}
we see that the following explicit formula holds in dimension $n=1$:
\begin{equation}\label{eq:eta}
\eta(x,t)=\int_{0}^{\infty}\left[M_{t}(x,y)-M_{t}(x,-y)\right]f(y)\,d\gamma(y),
\quad\hbox{for all}~x>0,~t>0.
\end{equation}
\subsection{The fractional nonlocal OU equation and the extension problem}
We introduce now an appropriate functional setting, which is essential when dealing with problems \eqref{Problema} and
\eqref{Problema estensione}. In order to define the fractional powers
$\mathcal{L}^su$, $0<s<1$, we consider the sequence of eigenvalues
$0<\lambda_1\leq \lambda_2\leq\cdots\leq\lambda_k\nearrow\infty$
and the corresponding
orthonormal basis of Dirichlet eigenfunctions $\{\psi_{k}\}_{k\geq1}$ of
$\mathcal{L}$ in $L^{2}(\Omega,\gamma)$, see for example \cite{Betta-Chiacchio-Ferone}.
In other words, for every $k\geq1$, $\psi_{k}\in L^2(\Omega,\gamma)$
is a weak solution to the Dirichlet problem
$$\begin{cases}
-\operatorname{div}(\varphi\nabla\psi_{k})=\lambda_{k}\varphi\,\psi_{k},&\hbox{in}~\Omega,\\
\psi_{k}=0,&\hbox{on}~\partial\Omega.
\end{cases}$$
Now, let us define the Hilbert space
\[
\mathcal{H}^s(\Omega,\gamma)\equiv\operatorname{Dom}(\mathcal{L}^{s}
):=\Big\{u\in L^{2}(\Omega,\gamma):\sum_{k=1}^{\infty}\lambda_{k}
^s|\langle u,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^{2}<\infty\Big\},
\]
with scalar product
\[
\langle u,v\rangle_{\mathcal{H}^s(\Omega,\gamma)}:=\sum_{k=1}^{\infty}
\lambda_{k}^{s}\langle u,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}\langle
v,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}.
\]
Then the norm in $\mathcal{H}^s(\Omega,\gamma)$ is given by
\[
\Vert u\Vert_{\mathcal{H}^s(\Omega,\gamma)}^{2}=\sum_{k=1}^{\infty}\lambda
_{k}^{s}|\langle u,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^{2}.
\]
For $u\in\mathcal{H}^s(\Omega,\gamma)$, we define $\mathcal{L}^{s}u$ as the
element in the dual space $\big(\mathcal{H}^s(\Omega,\gamma)\big)^{\prime}$ through the formula
\[
\mathcal{L}^{s}u=\sum_{k=1}^{\infty}\lambda_{k}^{s}\langle u,\psi
_{k}\rangle_{L^{2}(\Omega,\gamma)}\psi_{k},\quad\hbox{in}~\big(\mathcal{H}^s(\Omega,\gamma)\big)^{\prime}.
\]
That is, for any function $v\in\mathcal{H}^s(\Omega,\gamma)$ we have
\[
\langle\mathcal{L}^su,v\rangle=\sum_{k=1}^{\infty}\lambda_{k}^{s}\langle
u,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}\langle v,\psi_{k}\rangle
_{L^{2}(\Omega,\gamma)}=\langle u,v\rangle_{\mathcal{H}^s(\Omega,\gamma)}.
\]
This identity can be rewritten as
\[
\langle\mathcal{L}^{s}u,v\rangle=\int_{\Omega}(\mathcal{L}^{s/2}
u)(\mathcal{L}^{s/2}v)\,dx,\quad\hbox{for every}~u,v\in\mathcal{H}^s(\Omega,\gamma),
\]
where $\mathcal{L}^{s/2}$ is defined by taking the power $s/2$ of the
eigenvalues $\lambda_{k}$.
\begin{remark}[The fractional OU operator is a nonlocal operator]
By using the method of semigroups as in \cite{Stinga-Torrea}, see also
\cite{Caffarelli-Stinga, StingaVolz, Stinga-Zhang}, it can be seen that the fractional
operator $\mathcal{L}^s$ is a nonlocal operator. Indeed, we have the semigroup and kernel formulas
\begin{align*}
\mathcal{L}^su(x) &=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\mathcal{L}_\Omega}u(x)
-u(x)\big)\,\frac{dt}{t^{1+s}} \\
&=\operatorname{PV}\int_{\Omega}\big(u(x)-u(y)\big)K_s(x,y)\,dy+u(x)B_s(x),
\end{align*}
where $\operatorname{PV}$ means that the integral is taken in the principal value sense. Here
$$e^{-t\mathcal{L}_\Omega}u(x)=\int_\Omega H_t(x,y)u(y)\,d\gamma(y),$$
is the semigroup generated by $\mathcal{L}$ in $\Omega$ with Dirichlet boundary conditions,
$H_t(x,y)$ is the corresponding heat kernel,
$$K_s(x,y)=\frac{1}{|\Gamma(-s)|}\int_0^\infty H_t(x,y)\,\frac{dt}{t^{1+s}},\quad x,y\in\Omega,$$
and
$$B_s(x)=\frac{1}{|\Gamma(-s)|}\int_0^\infty\big(1-e^{-t\mathcal{L}_\Omega}1(x)\big)\,\frac{dt}{t^{1+s}},
\quad x\in\Omega.$$
In the particular case of $\Omega=\R^n$, we have $H_t(x,y)=M_t(x,y)$, the Mehler kernel, and,
as a direct consequence of \eqref{M=1}, we see that $B_s(x)\equiv0$. Though this description
is important, we will not use it here. Instead, we will apply the extension technique.
\end{remark}
Recalling the notation in \eqref{eq:a}, we define the Sobolev energy space on the infinite cylinder $\mathcal{C}_{\Omega}$:
\[
H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)=\bigg\{ v\in
H_{\mathrm{loc}}^{1}(\mathcal{C}_{\Omega}):v=0~\hbox{on}~\partial_{L}
\mathcal{C}_{\Omega},~\iint_{\mathcal{C}_{\Omega}}y^a(v^{2}+|\nabla
_{x,y}v|^{2})\,d\gamma(x)\,dy<\infty\bigg\}.
\]
By the Gaussian Poincar\'{e} inequality \eqref{Poincare}, for each
$v\in H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ we have
\begin{align*}
\iint_{\mathcal{C}_{\Omega}}y^av^{2}\,d\gamma(x)\,dy & =\int_{0}^{\infty
}y^a\int_{\Omega}v^{2}\,d\gamma(x)\,dy\leq C_{\Omega}\int_{0}^{\infty}
y^a\int_{\Omega}|\nabla_{x}v|^{2}d\gamma(x)\,dy\\
& \leq C_{\Omega}\iint_{\mathcal{C}_{\Omega}}y^a|\nabla_{x,y}v|^{2}
\,d\gamma(x)\,dy.
\end{align*}
Thus we can equip the space $H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma
(x)\otimes y^ady)$ with the equivalent norm
\[
\Vert v\Vert_{H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)}
^{2}=\iint_{\mathcal{C}_{\Omega}}y^a|\nabla_{x,y}v|^{2}\,d\gamma(x)\,dy,
\]
which is actually the norm defined through the scalar product
\[
\langle v,w\rangle_{H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma
(x)\otimes y^ady)}=\iint_{\mathcal{C}_{\Omega}}y^a\,\nabla_{x,y}v\cdot \nabla_{x,y}w\,\,d\gamma(x)\,dy.
\]
Furthermore, since we can identify $H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$
with the space $H^{1}((0,\infty),y^ady;H_{0}^{1}(\Omega,\gamma))$,
we have that $H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ is a
Hilbert space.\\[0.5pt]
The following Theorem is a particular case of \cite[Theorem~1.1]{Stinga-Torrea},
see also \cite{Caffarelli-Stinga, Gale-Miana-Stinga, Stinga-Zhang}.
It provides the characterization of $\mathcal{L}^su$ as the Dirichlet-to-Neumann map
for a degenerate elliptic extension problem in the upper cylinder $\mathcal{C}_{\Omega}$, for any
$u\in\mathcal{H}^s(\Omega,\gamma)$. As the solution $w(x,y)$
is explicitly given by \eqref{trueextens} and \eqref{StingaTorreasemigr},
the proof is just a verification of the statements, see for example \cite{Stinga-Torrea,StingaVolz}.
\begin{theorem}[Extension problem]\label{extensth}
Let $u\in\mathcal{H}^s(\Omega,\gamma)$. Define
\begin{equation}
w(x,y)\equiv\mathcal{P}_y^su(x)=
\frac{2^{1-s}}{\Gamma(s)}\sum_{k=1}^{\infty}(\lambda_k^{1/2}y)^s\mathcal{K}_s(\lambda_k^{1/2}y)
\langle u,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}\psi_{k}(x),
\label{trueextens}
\end{equation}
for $y\geq0$, where $\mathcal{K}_s$ is the modified Bessel function of the second kind and order $0<s<1$.
Then $w\in H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ and it is the
unique weak solution to the extension problem
\begin{equation}
\begin{cases}
-\operatorname{div}_{x,y}(y^a\varphi(x)\nabla_{x,y}w)=0,&\hbox{in}~\mathcal{C}_{\Omega},\\
w=0, & \hbox{on}~\partial_{L}\mathcal{C}_{\Omega},\\
w(x,0)=u(x), & \hbox{on}~\Omega,
\end{cases}
\label{extensionproblem}
\end{equation}
that vanishes weakly as $y\to\infty$. More precisely,
\[
\iint_{\mathcal{C}_{\Omega}}y^a(\nabla_{x,y}w\cdot\nabla_{x,y}\xi)\,d\gamma
(x)\,dy=0,
\]
for all test functions $\xi\in H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma
(x)\otimes y^ady)$ with zero trace over $\Omega$, $\operatorname{tr}_{\Omega}
\xi=0$, and
\[
\lim_{y\rightarrow0^{+}}w(x,y)=u(x)
\]
in $L^{2}(\Omega,\gamma)$. Furthermore, the function $w$ is the unique
minimizer of the energy functional
\begin{equation}
\mathcal{F}(v)=\frac{1}{2}\iint_{\mathcal{C}_{\Omega}}y^a|\nabla_{x,y}
v|^{2}\,d\gamma(x)\,dy, \label{energufunct}
\end{equation}
over the set $\mathcal{U}=\left\{ v\in H_{0,L}^{1}(\mathcal{C}_{\Omega
},d\gamma(x)\otimes y^ady):\,\operatorname{tr}_{\Omega}v=u\right\} $.
We can also write
\begin{equation}
w(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_{0}^{\infty}e^{-y^{2}/(4t)}
e^{-t\mathcal{L}_{\Omega}}u(x)\,\frac{dt}{t^{1+s}}.\label{StingaTorreasemigr}
\end{equation}
Moreover,
$$-\lim_{y\rightarrow0^{+}}y^aw_{y}=c_s\mathcal{L}^su,\quad\hbox{in}~\big(\mathcal{H}^s
(\Omega,\gamma)\big)^{\prime},$$
where $c_s=\frac{\Gamma(1-s)}{4^{s-1/2}\Gamma(s)}>0$. Finally, the following energy identity holds:
\begin{equation}\label{importidentity}
\iint_{\mathcal{C}_{\Omega}}y^a|\nabla_{x,y}w|^{2}\,d\gamma(x)\,dy=
c_s\|\mathcal{L}^{s/2}u\|^{2}_{L^{2}(\Omega,\gamma)}.
\end{equation}
\end{theorem}
Theorem \ref{extensth} shows in particular that the domain
$\mathcal{H}^s(\Omega,\gamma)$ is contained in the range of the trace operator
on $H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ at $y=0$. The next Lemma
shows that actually these two spaces coincide.
\begin{lemma}[Trace inequality]\label{Identification}
We have
\[
\operatorname{tr}_{\Omega}(H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes
y^ady))=\mathcal{H}^s(\Omega,\gamma).
\]
Moreover, for all $v\in H^{1}_{0,L}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$,
\begin{equation}\label{trace}
\|\mathcal{L}^{s/2}v(x,0)\|_{L^{2}(\Omega,\varphi)}^2 \leq (2c_s)^{-1}
\iint_{\mathcal{C}_{\Omega}}
y^a|\nabla_{x,y}v|^{2}\,d\gamma(x)\,dy.
\end{equation}
In particular, equality holds in \eqref{trace} if $v=\mathcal{P}_y^s(\operatorname{tr}_{\Omega}v)(x)$,
(see \eqref{trueextens}).
\end{lemma}
\begin{proof}
Let $u=\tr_\Omega v$, for $v\in H_{0,L}^{1}(\mathcal{C}_{\Omega
},d\gamma(x)\otimes y^ady)$ and define the function $w$ as in \eqref{trueextens}.
It is readily checked that $w$ satisfies \eqref{extensionproblem}, so it minimizes the functional $\mathcal{F}$ in \eqref{energufunct}. Therefore, by \eqref{importidentity},
$\|\mathcal{L}^{s/2}u\|^{2}_{L^{2}(\Omega,\gamma)}\leq (c_s)^{-1}\Vert
v\Vert_{H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)}^{2}<\infty$,
that is, $u\in\mathcal{H}^s(\Omega,\gamma)$. Now \eqref{trace} is clear.
\end{proof}
\begin{proposition}[Compactness of the trace embedding]\label{compactness}We have
\[
\operatorname{tr}_{\Omega}(H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes
y^ady))\subset\subset L^{2}(\Omega,\gamma).
\]
\end{proposition}
\begin{proof}
We need to check that the trace operator $\operatorname{tr}_\Omega:
H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)\to L^{2}(\Omega,\gamma)$ is compact.
It is clear that $\operatorname{tr}_\Omega$ is continuous from
$H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ into $L^2(\Omega,\gamma)$
since \eqref{trace} holds. Similarly, the finite rank operators $T_j$, $j\geq1$, defined by
$$T_jv=\sum_{k=1}^j\langle v(\cdot,0),\psi_k\rangle_{L^2(\Omega,\gamma)}\psi_k,$$
are continuous from
$H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ into $L^{2}(\Omega,\gamma)$.
By using \eqref{trace} and the fact that $\lambda_k\nearrow\infty$, as $k\to\infty$, we see that,
if $v\in H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$,
\begin{align*}
\|T_jv-\tr_\Omega v\|_{L^2(\Omega,\gamma)}^2 &= \sum_{k=j+1}^\infty|\langle v(\cdot,0),\psi_k\rangle|^2 \\
&\leq \frac{1}{\lambda_{j+1}^s}\sum_{k=j+1}^\infty\lambda_k^s|\langle v(\cdot,0),\psi_k\rangle|^2
\leq \frac{1}{\lambda_{j+1}^s}\|v\|^2_{H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)}.
\end{align*}
Therefore $T_j$ converges to $\tr_\Omega$ in the operator norm, as $j\to\infty$, and $\tr_\Omega$ is compact.
\end{proof}
Using the previous preliminaries, it is natural to give the following definitions of weak solutions.
\begin{definition}[Weak solution of \eqref{Problema estensione}]\label{Defprobest}
Let $f\in L^{2}(\Omega,\gamma)$. We say that $w\in H^{1}_{0,L}
(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$ is a weak solution to the linear Dirichlet-Neumann extension problem \eqref{Problema estensione} if
\begin{equation}\iint_{\mathcal{C}_{\Omega}}y^a\nabla_{x,y}w\cdot\nabla_{x,y}v\,d\gamma
(x)\,dy=c_s^{-1}\int_{\Omega}f(x)v(x,0)\,d\gamma(x),\label{weakform}\end{equation}
for every $v\in H^{1}_{0,L}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$, where
$c_s>0$ is the constant appearing in Theorem \ref{extensth}.
\end{definition}
\begin{definition}[Weak solution of \eqref{Problema}]
If $w$ is the weak solution to \eqref{Problema estensione}, its trace $u:=w(\cdot,0)\in \mathcal{H}^{s}(\Omega,\gamma)$ on $\Omega$ will be called a weak
solution to
\eqref{Problema}.
\end{definition}
\begin{remark}
If we assume that $f$ is in the dual space $\mathcal{H}^{s}(\Omega,\gamma)^{\prime}$, it is clear that the right hand side
in \eqref{weakform} must be replaced by the dual product
$\langle f,v(\cdot,0)\rangle$. Then the (unique) solution $u$ to \eqref{Problema} will be again the trace over $\Omega$ of the unique solution $w$ to the
extension problem \eqref{Problema estensione}.
\end{remark}
The following is just a restatement of Theorem \ref{extensth}, see \cite[Theorem~1.1]{Stinga-Torrea} and
also \cite{Gale-Miana-Stinga}.
\begin{theorem}[Extension problem for negative powers]
\label{existweaksollin}
Given $f\in L^{2}(\Omega,\gamma)$, let $u\in\mathcal{H}^s(\Omega,\gamma)$
be the unique solution to problem \eqref{Problema}.
The solution $w$ (see \eqref{trueextens})
to the extension problem \eqref{extensionproblem} can be written as
\begin{equation}\label{eq:semigroup formula f}
\begin{aligned}
w(x,y) &= \frac{2^{1-s}}{\Gamma(s)}\sum_{k=1}^{\infty}(\lambda_k^{1/2}y)^s\mathcal{K}_s(\lambda_k^{1/2}y)
\frac{\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}}{\lambda_k^s}\psi_{k}(x) \\
&= \frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-y^{2}/(4t)}
e^{-t\mathcal{L}_{\Omega}}f(x)\,\frac{dt}{t^{1-s}}.
\end{aligned}
\end{equation}
In particular, this is the unique weak solution to \eqref{Problema estensione} and
\begin{equation}\label{eq:fractional integral}
w(x,0)=u(x)=\mathcal{L}^{-s}f(x)=\frac{1}{\Gamma(s)}\int_0^\infty e^{-t\mathcal{L}_\Omega}f(x)\,\frac{dt}{t^{1-s}}.
\end{equation}
\end{theorem}
The domain $\mathcal{H}^s(\Omega,\gamma)$ of the fractional nonlocal operator $\mathcal{L}^s$
can be characterized as a suitable interpolation space between two
Hilbert spaces. Indeed, using the abstract discrete version of the $J$-Theorem
(see for example the Appendix in \cite{SirBonfVaz}), it is straightforward to prove that
\begin{equation}
\mathcal{H}^s(\Omega,\gamma)=\left[H_{0}^{1}(\Omega,\gamma),L^{2}
(\Omega,\gamma)\right] _{1-s},\label{Gaussianintsp}
\end{equation}
where the space in the right hand side of \eqref{Gaussianintsp}
is the real interpolation space between
$H_{0}^{1}(\Omega,\gamma)$ and $L^{2}(\Omega,\gamma)$. Then
$\mathcal{H}^{1/2}(\Omega,\gamma)$ may be seen as the equivalent of the
Lions--Magenes space $H_{00}^{1/2}(\Omega)$ in the Gaussian setting.
\subsection{Gaussian rearrangements}\label{Gaussian rearrangements}
We give the notion of rearrangement with respect to the Gaussian measure. For extra details,
we refer the interested reader to the classical monographs \cite{Bennett} and \cite{CR}.
If $u$ is a measurable function in $\Omega$, we denote by
\begin{itemize}
\item $u^{\circledast}$ the one dimensional decreasing rearrangement of $u$ with respect
to the Gaussian measure (also called \emph{one dimensional Gaussian rearrangement of $u$}):
$$u^{\circledast}(r)=\inf\{t\geq0:\gamma_{u}(t)\leq r\},\quad r\in(0,\gamma(\Omega)],$$
where $\gamma_{u}(t)=\gamma(\{x\in\Omega:\left\vert u(x)\right\vert>t\})$ is the distribution function of $u$;
\item $u^{\displaystyle\star}$ the $n$-dimensional rearrangement of $u$ with respect the Gaussian measure:
$$u^{\displaystyle\star}(x)=u^{\circledast}\big(\Phi(x_{1})\big),\quad x\in\Omega^{\displaystyle\star},$$
where $\Omega^{\displaystyle\star}=\{x=( x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:x_{1}>\lambda\}$ is the
half-space such that $\gamma(\Omega^{\displaystyle\star})=\gamma(\Omega)$
and $\Phi$ is given by \eqref{Phi}.
\end{itemize}
By definition, $u^{\displaystyle\star}$ is a function which depends
only on the first variable $x_{1}$, it is increasing and its level sets are half-spaces. Moreover,
$u$, $u^{\circledast}$ and $u^{\displaystyle\star}$ have the same
distribution function. This implies that the Gaussian $L^{p}$ norm is invariant under these rearrangements:
$$\|u\|_{L^{p}(\Omega,\gamma)}=\|u^{\circledast}\|_{L^{p}(0,\gamma(\Omega))}=\|u^{{\displaystyle\star}}\|_{L^{p}(\Omega^{{\displaystyle\star}},\gamma)},\quad\hbox{for
any}~1\leq p\leq\infty.$$
If $u$ is defined on a half-space and $u=u^{\displaystyle\star}$ we sometimes say that $u$ is \emph{rearranged}. Furthermore,
if $u$ and $v$ are
measurable functions then the following Hardy-Littlewood inequality holds:
\begin{equation}
\int_{\Omega}|u(x)v(x)|\,d\gamma(x)\leq\int_{\Omega^{\displaystyle\star}}
u^{\displaystyle\star}(x)v^{\displaystyle\star}(x)\,d\gamma(x)
=\int_{0}^{\gamma(\Omega)}u^{\circledast}(r)v^{\circledast}(r)\,dr.
\label{Hardy-Litt}
\end{equation}
If $u$ is defined on $\Omega$, $v$ on $\Omega^{\displaystyle\star}$ and
the following estimate holds
\begin{equation}
\int_{0}^{\gamma(\Omega)}u^{\circledast}(r)\,dr\leq \int_{0}^{\gamma(\Omega)}v^{\circledast}(r)\,dr,\label{massconcent}
\end{equation}
the same inequality is called \emph{mass concentration inequality} (or \emph{comparison of mass concentration}). If $v=v^{\displaystyle\star}$ and
\eqref{massconcent} occurs, we also say that $u^{\displaystyle\star}$ is \emph{less concentrated} that $v$ and we write $u^{\displaystyle\star}\prec v$.
Moreover, \eqref{massconcent} implies that (see for instance \cite{Chong})
\[
\|u\|_{L^{p}(\Omega,\gamma)}\leq \|v\|_{L^{p}(\Omega^{\displaystyle\star},\gamma)},
\quad\hbox{for all}~1\leq p\leq\infty.
\]
We will often deal with two-variable functions
\begin{equation}\label{w}
w:(x,y)\in\mathcal{C}_{\Omega}=\Omega\times(0,\infty)\rightarrow w(x,y)\in{\mathbb{R}},
\end{equation}
which are measurable with respect to $x$.
In such a case it will be convenient to consider the
so-called \textit{Gaussian Steiner symmetrization} of $\mathcal{C}_{\Omega}$ with
respect to the variable $x$, namely, the set $\mathcal{C}_{\Omega}^{\displaystyle\star}$
as defined in \eqref{eq:Steiner symmetrization}.
In addition (see for instance \cite{chiacchio,Eh2}) we will denote by $\gamma_{w}(t,y)$
and $w^{\circledast}(r,y)$ the distribution
function and the one dimensional Gaussian decreasing rearrangements of \eqref{w}, with respect to $x$,
for each $y$ fixed. We will also define the function
$$w^{\displaystyle\star}(x,y)=w^{\circledast}\big(\Phi(x_{1}),y\big),$$
which is called the \emph{Gaussian Steiner symmetrization of $w$}, with respect to $x$, that is, with respect to the line $x=0$.
Clearly, for any fixed $y$, $w^{\displaystyle\star}(\cdot,y)$ is an increasing function depending only on $x_{1}$.
Now we recall a result that we will use in the proof of our main comparison result in Section
\ref{Section:comparison}.
\begin{proposition}[See {\cite[p.~255]{chiacchio}}]\label{prop chiacchio}
Consider the Cauchy--Dirichlet problem \eqref{p2} with $\Omega=\Omega^{\displaystyle\star}$.
If $f(x)=f^{\displaystyle\star}(x)$ for a.e. $x \in \Omega^{\displaystyle\star}$ and
$f^{\displaystyle\star}\in L^{2}(\Omega^{\displaystyle\star},\gamma)$, then the solution $\eta$ to
\eqref{p2} is such that $\eta(x,t)=\eta^{\displaystyle\star}(x,t)$, for a.e. $x\in\Omega^{\displaystyle\star}$
and for all $t\geq0$.
\end{proposition}
\subsection{The second order derivation formula}
It will be essential for us to be able to differentiate
with respect to the extra variable $y$ under the integral symbol in the expression
\[\int_{\{x:\,w(x,y)>w^{\circledast}(r,y)\}}\frac{\partial w}{\partial y}(x,y)\,d\gamma(x).\]
Equivalently, we need to derive the Gaussian version of the first and second order differentiation
formulas established for the Lebesgue measure in \cite{Al-Li-TRo, Band2, Ferone-Mercaldo, Mossino}. The first order differentiation formula can be stated
as follows:
\begin{proposition}[See \cite{chiacchio}, also \cite{simon}]
If $w\in H^{1}(0,T;L^{2}(\Omega,\gamma))$ is a nonnegative function,
for some $T>0$, then $w^{\circledast}\in H^{1}\big(0,T;L^{2}(0,\gamma(\Omega))\big)$.
In addition, if $\gamma(\{w(x,t)=w^{\circledast}(r,t)\})=0$ for a.e.
$(r,t)\in(0,\gamma(\Omega))\times(0,T)$, then the following derivation formula holds
\begin{equation}\label{Rakotoson derivation formula}
\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}
\frac{\partial}{\partial y}w(x,y)\,d\gamma(x)=\int_{0}^{r}
w^{\circledast}(\sigma,y)\,d\sigma.
\end{equation}
\end{proposition}
In order to prove our novel second order derivation formula,
we need the following version of the coarea formula (see \cite{Federer} and \cite[Theorem 11]{Haj}).
\begin{proposition}
\label{Prop coarea copy(1)}If $u\in W_{\mathrm{loc}}^{1,p}(\R^n)$, with $p>1$,
and $\psi:\R^n\to\R$ is a nonnegative measurable function, then there exists a representative of
$u$, denoted again by $u$, such that
\begin{equation}\label{coarea}
\int_{\R^n}\psi(x)|\nabla_{x}u|\,dx=\int_{-\infty}^{\infty}\bigg(\int_{\left\{x:\,u(x)=\tau\right\}}\psi(x)\,d\mathcal{H}^{n-1}(x)\bigg)\,d\tau.
\end{equation}
\end{proposition}
Now we present our new Gaussian derivation formulas, which are a nonstandard adaptation of the
derivation formula exhibited in \cite{Ferone-Mercaldo}.
\begin{theorem}\label{Firstderivform}
Let $0<\varepsilon<T<\infty$. Consider a nonnegative function
$$w=w(x,y)\in H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)\cap C^{1}(\Omega\times(\varepsilon,T)).$$
Suppose also that $w$ is $C^{1,\alpha}$ with respect to $y\in(\varepsilon,T)$,
for some $0<\alpha\leq1$, uniformly with respect to $x\in \Omega$.
Moreover, assume that $f(x,y)$ is a continuous function
on the cylinder $\mathcal{C}_{\Omega}$ such that $f\in H^{1}\normalcolor(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^a \,dy)$
and the function $f(x,y)\varphi(x)$
is Lipschitz with respect to $y\in( \varepsilon,T)$, uniformly with respect to $x\in$ $\Omega$.
Furthermore, suppose that
\begin{equation}
\gamma\Big(\big\{x\in\Omega:|\nabla_{x}w|=0,w(x,y)\in(0,\sup_{x}w(x,y))\big\}\Big)
=0,\quad\hbox{for all}~y\in(\varepsilon,T),
\label{cond aggiuntiva}
\end{equation}
and set
\[
H(t,y):=\int_{\left\{x:\,w(x,y)>t\right\}}f(x,y)\,d\gamma(x),
\]
for $t\in[0,\infty)$ and $y\in(\varepsilon,T)$. The following statements hold true.
\begin{enumerate}[$(i)$]
\item For any fixed $y\in(\varepsilon,T)$, $H(t,y)$ is differentiable with respect to $t$ for a.e. $t\geq0$ and
\begin{equation}
\frac{\partial}{\partial t}H(t,y)=-\int_{\left\{x:\,w(x,y)=t\right\}}
\frac{f(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x).
\label{t derivate}
\end{equation}
\item For any fixed $t\geq0$, $H(t,y)$ is differentiable with respect to $y$ and,
for a.e $y\in(\varepsilon,T)$,
\begin{equation}
\frac{\partial}{\partial y}H(t,y)=\displaystyle\int_{\left\{x:\,w(x,y)>t\right\}}
\frac{\partial}{\partial y}f(x,y)\,d\gamma(x)+
\displaystyle\int_{\left\{x:\,w(x,y)=t\right\}}\frac{\partial}{\partial y}w(x,y)\frac{f(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x).
\label{y derivate}
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
Let us first prove $(i)$. By the extension theorem (see for instance \cite{FEOPOST})
we can extend $w(\cdot,y)$ as a function in $H^1(\R^n)$, for a.e. $y>0$.
Condition \eqref{cond aggiuntiva} allows us to choose
$\psi(x)=\frac{f(x,y)}{|\nabla_xw|}\varphi(x)\chi_{\{w(x,y)>t\}}(x)$ and $u(x)=w(x,y)$
in the coarea formula \eqref{coarea} to get
$$\int_{\left\{x:\,w(x,y)>t\right\}}f(x,y)\,d\gamma(x)=\int_{t}^{\infty}\bigg(\int_{\left\{x:\,w(x,y)=\tau\right\}}
\frac{f(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)\bigg)\,d\tau,$$
for a.e. $t\geq0$. Thus \eqref{t derivate} follows.
Next we prove $(ii)$.
We observe that
\[
H(t,y)-H(t,\overline{y})=\triangle_{1}+\triangle_{2}+\triangle_{3},
\]
where
$$\triangle_{1}=\int_{\left\{x:\,w(x,\overline{y})>t\right\}}
[f(x,y)-f(x,\overline{y})]\,d\gamma(x),\quad
\triangle_{2}=\int_{\left\{x:\,w(x,y)>t\geq w(x,\overline{y})\right\}}f(x,y)\,d\gamma(x),$$
and
$$\triangle_{3}=-\int_{\left\{x:\,w(x,\overline{y})>t\geq w(x,y)\right\}}f(x,y)\,d\gamma(x).$$
Since $f(x,y)\varphi(x)$ is Lipschitz with respect
to $y$, uniformly in $x$, by Lebesgue's dominated convergence theorem we easily infer that
\begin{equation}
\underset{y\rightarrow\overline{y}}{\lim}\frac{\triangle_1}{y-\overline{y}}=\int_{\left\{x:\,w(x,\overline{y})>t\right\}}
\frac{\partial f}{\partial y}(x,\overline{y})\,d\gamma(x),
\label{DELTA 1}
\end{equation}
for a.e. $t$ and a.e. $\overline{y}\in(\varepsilon,T)$.
Let us next consider $\frac{\triangle_{2}}{y-\overline{y}}.$ We have
\begin{equation}
\frac{\triangle_{2}}{y-\overline{y}}=\frac{1}{y-\overline{y}}\displaystyle\int_{D_{1}}f(x,y)\,d\gamma(x)
+\frac{1}{y-\overline{y}}\int_{D_{2}}f(x,y)\,d\gamma(x),
\label{divisa}
\end{equation}
where
\[
D_{1}=\left\{x\in\Omega:w(x,y)>t\geq w(x,\overline{y}),\frac{\partial w
}{\partial y}(x,\overline{y})=0\right\},
\]
and
\[
D_{2}=\left\{x\in\Omega:w(x,y)>t\geq w(x,\overline{y}),\frac{\partial w
}{\partial y}(x,\overline{y})\neq0\right\}.
\]
We claim that
\begin{equation}
\lim_{y\rightarrow\overline{y}}\frac{1}{y-\overline{y}}\int_{D_{1}}f(x,y)\,d\gamma(x)=0,\quad
\hbox{for a.e.}~t\geq0.
\label{zero}
\end{equation}
Since $w(x,y)\in C^{1,\alpha}$ with respect to $y\in(\varepsilon,T)$, uniformly in $x\in\Omega$, we have
\begin{equation}
\left\vert \frac{\partial w}{\partial y}(x,y)-\frac{\partial w}{\partial
y}(x,\overline{y})\right\vert \leq c|y-\overline{y}|^{\alpha},\quad\hbox{for every}~x\in\Omega,
\label{holder continuity}
\end{equation}
for a constant $c>0$ independent on $x$, $y$ and $\overline{y}$.
Since for any
$x\in D_{1}$ we have $\frac{\partial}{\partial y}w(x,\overline{y})=0$, by \eqref{holder continuity} we easily find the uniform estimate
\[
\left\vert w(x,y)-w(x,\overline{y})\right\vert\leq
\int_{\overline{y}}^{y}\left|\frac{\partial}{\partial
z}w(x,z)\right|dz \leq c|y-\overline{y}|^{\alpha+1},\quad\hbox{for all}~x\in D_{1},
\]
which yields
\begin{equation}\label{1quotratio}
\left\vert \frac{1}{y-\overline{y}}\int_{D_{1}}f(x,y)\,d\gamma(x)\right\vert
\leq\frac{1}{\left\vert
y-\overline{y}\right\vert }\int_{\{x:\,t-c|y-\overline{y}|^{\alpha+1}<w(x,\overline{y})\leq t\}}
|f(x,y)|\,d\gamma(x).
\end{equation}
Let us set
\[
\Psi(t):=\int_{\left\{x:\,w(x,\overline{y})>t\right\}}\max_{y\in[\varepsilon,T]}|f(x,y)|\,d\gamma(x).
\]
Since $f\in L^{2}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^ady)$
and $f$ is continuous, by Fubini's theorem we have that
$$\int_{\Omega}|f(x,y)|\,d\gamma(x)<\infty,$$
for a.e. $y>0$, and $\Psi(t)<\infty$, for all $t\geq0$. Then \eqref{1quotratio} implies
\[
\left\vert \frac{1}{y-\overline{y}}\int_{D_{1}}f(x,y)\,d\gamma(x)\right\vert
\leq c\left\vert y-\overline{y}\right\vert ^{\alpha}\frac{\Psi\big(t-c\left\vert y-\overline
{y}\right\vert ^{\alpha+1}\big)-\Psi(t)}{c\left\vert y-\overline{y}\right\vert ^{\alpha+1}}.
\]
Since the function $\Psi$ it monotone, it is also differentiable almost everywhere and then \eqref{zero} holds.
Now let us evaluate the second term in \eqref{divisa}.
First we consider the case $y>\overline{y}.$
For $y$ sufficiently close to $\overline{y},$ we have
$$\frac{1}{y-\overline{y}}\int_{D_{2}}f(x,y)\,d\gamma(x)=\frac{1}{y-\overline{y}}\int_{D_{3}}f(x,y)\,d\gamma(x),$$
where $$D_{3}=\bigg\{x\in\Omega:w(x,y)>t\geq w(x,\overline{y}),\frac
{\partial w}{\partial y}(x,\overline{y})>0\bigg\}.$$
Let us set
\[
\Gamma_{t}=\left\{x\in\Omega:w(x,\overline{y})=t\right\} \cap\left\{
x\in\Omega:\frac{\partial w}{\partial y}(x,\overline{y})>0\right\} .
\]
In a neighborhood $B_{r}(\overline{x},\overline{y},t)$ of a point
$(\overline{x},\overline{y},t)\in\R^{n+2}$ with $\overline{x}\in\Gamma_{t}$, the equality $w(x,y)=t$
implicitly defines a function $y=v(x,t)$ such that $\overline{y}
=v(\overline{x},t)$ and $w(x,v(x,t))=t.$ Moreover for $y$ sufficiently close
to $\overline{y}$ we have
\[
D_{3}\cap B_{r}(\overline{x},\overline{y},t)=\left\{ x\in B_{r}(\overline
{x},\overline{y},t):\overline{y}<v(x,t)<y\right\} .
\]
Observe that the implicit function theorem gives $|\nabla_{x}v(x,t)|=|\nabla_{x}w(x,\overline{y})|/\frac{\partial w}{\partial y}
(x,\overline{y})$. Then using the coarea formula (\ref{coarea})
we have
\begin{equation}\label{delta 2}
\begin{aligned}
\underset{y\rightarrow\overline{y}^{+}}{\lim}\frac{1}{y-\overline{y}}\int_{D_{3}\cap B_{r}(\overline{x},\overline{y},t)}
&f(x,y)\,d\gamma(x)
=\underset{y\rightarrow\overline{y}^{+}}{\lim}\frac{1}{y-\overline{y}}\int_{\overline{y}}^{y}\int_{\left\{x:\,v(x,t)=s\right\}}
\frac{f(x,y)\,\varphi(x)}{|\nabla_{x}v|}\,d\mathcal{H}^{n-1}(x)\,ds\\
&=\int_{\{x\in B_{r}(\overline{x},\overline{y},t):v(x,t)=\overline{y}\}}\frac{f(x,\overline{y})}{\left\vert \nabla
_{x}v\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x)\\
&=\int_{\{x\in B_{r}(\overline{x},\overline{y},t):w(x,\overline{y})=t\}}\frac{\partial w}{\partial y}
(x,\overline{y})\frac{f(x,\overline{y})}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x).
\end{aligned}
\end{equation}
By \eqref{zero} and \eqref{delta 2} it follows that
\begin{equation}
\underset{y\rightarrow\overline{y}^{+}}{\lim}\frac{\triangle_{2}}
{y-\overline{y}}=\int_{\{x:\,w(x,\overline{y})=t,\frac{\partial w}{\partial
y}(x,\overline{y})>0\}}\frac{\partial}{\partial y}w(x,\overline
{y})\frac{f(x,\overline{y})}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x).
\label{i}
\end{equation}
By analogous arguments we obtain
\begin{equation}
\underset{y\rightarrow\overline{y}^{-}}{\lim}\frac{\triangle_{2}}
{y-\overline{y}}=\int_{\{x:\,w(x,\overline{y})=t,\frac{\partial w}{\partial
y}(x,\overline{y})<0\}}\frac{\partial w}{\partial y}(x,\overline
{y})\,\frac{f(x,\overline{y})}{\left\vert \nabla_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x).
\label{ii}
\end{equation}
In the same way we can prove the analogue of (\ref{i})\ and (\ref{ii}) with
$\triangle_{2}\ $replaced by $\triangle_{3}.$ Then
\begin{equation}
\underset{y\rightarrow\overline{y}}{\lim}\frac{\triangle_{2}+\triangle_{3}
}{y-\overline{y}}=\int_{\left\{ x:\,w(x,\overline{y})=t\right\} }
\frac{\partial w}{\partial y}(x,\overline{y})\,\frac{f(x,\overline{y}
)}{\left\vert \nabla_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x).
\label{DELTA 2 e3}
\end{equation}
Putting together \eqref{DELTA 1} and \eqref{DELTA 2 e3} we obtain assertion $(ii)$.
\end{proof}
By recalling that the rearrangement $w^{\circledast}$ of a function $w$ is the
generalized inverse function of the distribution function $\gamma_{w}$, and
applying the chain rule formula, we can prove our novel derivation formula.
\begin{corollary}[Gaussian second order derivation formula]\label{Secondderivform}
Under the assumptions of Theorem \ref{Firstderivform}, for a.e. $y\in(\varepsilon,T)$ the following
derivation formula holds:
\begin{multline}
\frac{\partial}{\partial y}\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}f(x,y)\,d\gamma(x)=
\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}\frac{\partial}{\partial y}f(x,y)\,d\gamma(x)\\
-\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{f(x,y)}{|\nabla_{x}w|}
\left[\frac{{\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}}
\frac{\frac{\partial}{\partial y}w(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)}
{{\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}}\frac{1}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)}
-\frac{\partial}{\partial y}w(x,y)\right]\varphi(x)d\mathcal{H}^{n-1}(x).\label{Der1}
\end{multline}
In particular, if $w(x,y)$ is $C^{1}$ and the functions $w(x,y)\varphi(x)$,
$\frac{\partial}{\partial y}w(x,y)\varphi(x)$ are Lipschitz in $y\in(\varepsilon,T)$,
uniformly with respect to $x\in\Omega$, we have
\begin{equation}
\begin{aligned}
&\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}\frac{\partial^{2}}{\partial y^{2}}w(x,y)\,d\gamma(x) \\
&=\frac{\partial^{2}}{\partial y^{2}}
\displaystyle\int_{0}^{r}
w^{\circledast}(\sigma,y)\,d\sigma-\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}
\frac{\big(\frac{\partial}{\partial y}w(x,y)\big)^{2}}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)\\
&\quad+\Bigg(\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}
\frac{\frac{\partial}{\partial y}w(x,y)}{|\nabla_{x}w|}\varphi(x)d\mathcal{H}^{n-1}(x)\Bigg)^{2}
\Bigg(\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}
\frac{\varphi(x)}{|\nabla_{x}w|}d\mathcal{H}^{n-1}(x)\Bigg)^{-1}.\label{Der2}
\end{aligned}
\end{equation}
\end{corollary}
\begin{proof}
In order to prove \eqref{Der1} we need to evaluate the $y$-derivative of
$H(t,y)$ when $t=w^{\circledast}(r,y).$ By a rearrangement property (see for example \cite{Bennett}) we have
\begin{equation}
\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}w(x,y)\,d\gamma(x)=\int_{0}^{s}w^{\circledast}(\sigma,y)\,d\sigma.\label{ww}
\end{equation}
Observe that by applying \eqref{coarea} it is not difficult to prove that
\begin{equation}
-\frac{\partial w^{\circledast}}{\partial r}=\Bigg(\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\varphi(x)}{|\nabla_{x}w|}\,d\mathcal{H}^{n-1}(x)\Bigg)^{-1}\label{derivatrearrang}.
\end{equation}
Now using \eqref{coarea}, \eqref{derivatrearrang}, \eqref{Rakotoson derivation formula} and the chain rule,
\begin{align}
\frac{\partial}{\partial y}w^{\circledast}(r,y)&=\frac{\partial}{\partial y}\left(\frac{\partial}{\partial
r}\int_{0}^{r}w^{\circledast}(\tau,y)d\tau\right)\nonumber\\
&=\frac{\partial}{\partial r}\left(\frac{\partial}{\partial y}\int_{0}^{r}w^{\circledast}(\tau,y)d\tau\right)
=\frac{\partial}{\partial r}\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}\frac{\partial w}{\partial y}d\gamma(x)\label{tt}\\
&=\frac{\partial}{\partial r}\int_{w^{\circledast}(r,y)}^{\infty}d\tau\int_{\left\{x:\,w(x,y)=\tau\right\}}\frac{\frac{\partial }{\partial
y}w(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)\nonumber\\
&=-\frac{\partial w^{\circledast}}{\partial r}\,\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\frac{\partial }{\partial
y}w(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)\nonumber\\
&=\frac{\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\frac{\partial}{\partial
y}w(x,y)}{\left\vert \nabla_{x}w\right\vert }\varphi(x)\,d
\mathcal{H}^{n-1}(x)}{\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{1}{\left\vert
\nabla_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x)\nonumber
}.
\end{align}
By (\ref{tt}), (\ref{t derivate}) and (\ref{y derivate}) we obtain
\begin{align}
&\frac{\partial}{\partial y}H(w^{\circledast}(r,y),y) =\left.
\frac{\partial}{\partial t}H(t,y)\right\vert _{t=w^{\circledast}(r,y)}
\frac{\partial}{\partial y}w^{\circledast}(r,y)+H_{y}(w^{\circledast}(r,y),y)\label{111}\\
& =-\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{f(x,y)}{\left\vert \nabla
_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x)\times\nonumber
\frac{\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\frac{\partial}{\partial
y}w(x,y)}{\left\vert \nabla_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x)}{\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{1}{\left\vert
\nabla_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}
(x)}\nonumber\\
&\quad+ \int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}\frac{\partial}{\partial y}f(x,y)\,d\gamma
(x)+\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\partial}{\partial y}
w(x,y)\frac{f(x,y)}{\left\vert \nabla_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x),\nonumber
\end{align}
which is \eqref{Der1}.
Now we are in position to prove \eqref{Der2}. Indeed,
by applying \eqref{111} with $f(x,y)=w_{y}(x,y)$ and \eqref{Rakotoson derivation formula},
we finally get
\begin{align*}
\frac{\partial^{2}}{\partial y^{2}}\int_{0}^{r}w^{\circledast}(\sigma,y)\,d\sigma
&=\frac{\partial}{\partial y}\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}\frac{\partial}{\partial y}w(x,y)\,d\gamma(x) \\
& =\int_{\left\{x:\,w(x,y)>w^{\circledast}(r,y)\right\}}\frac{\partial^{2}}{\partial y^{2}}w(x,y)\,d\gamma(x) \\
&\quad+\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\big(\frac{\partial}{\partial
y}w(x,y)\big)^{2}}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)\\
&\quad-\frac{
\bigg(\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(r,y)\right\}}\frac{\frac{\partial}{\partial
y}w(x,y)}{\|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x)\bigg)^{2}
}
{
\bigg(\displaystyle\int_{\left\{x:\,w(x,y)=w^{\circledast}(s,y)\right\}}\frac{1}{\left\vert \nabla
_{x}w\right\vert }\varphi(x)\,d\mathcal{H}^{n-1}(x)\bigg)
}.
\end{align*}
\end{proof}
\begin{remark}
The sum of the last two terms to the right-hand side of \eqref{Der2} is nonpositive, see \cite[Remark 2.8]{AlDiaz}.
\end{remark}
The following Lemma shows that we can actually apply the second order derivation formula
\eqref{Der2} to the solution $w$ to the extension problem \eqref{Problema estensione}, namely,
when $w=\mathcal{P}^s_yu$ is the extension of the solution
$u\in \mathcal{H}^s(\Omega,\gamma)$ to the linear problem \eqref{Problema}.
\begin{lemma}\label{lemma regolarita}
If $f\in L^{2}(\Omega,\gamma)$ then the second order derivation formula \eqref{Der2} can be applied to the solution $w$ to
problem \eqref{Problema estensione}.
\end{lemma}
\begin{proof}
Since $w\in C^\infty(\mathcal{C}_{\Omega})$,
by classical results on solutions of elliptic equations with analytic
coefficients (see for instance \cite{Hashimoto}),
$w$ is analytic. Hence condition \eqref{cond aggiuntiva} holds.
Next we have to show that the functions $w(x,y)\varphi
(x)$ and $\partial_{y}w(x,y)\varphi(x)$ are Lipschitz in $y\in(\varepsilon,T)$,
uniformly with respect to $x\in \Omega$. This follows because it is known that the solution
to the extension problem has the regularity $w\in C^\infty((0,\infty);\mathcal{H}^s(\Omega,\gamma))$,
see \cite{Gale-Miana-Stinga, Stinga-Torrea}.
For the sake of completeness, we also give a direct proof of this regularity result.
By Theorem \ref{existweaksollin} and using the well known identity
$\frac{d}{dt}\big(t^\nu\mathcal{K}_\nu(t)\big)=-t^\nu\mathcal{K}_{\nu-1}(t)$, for $\nu\in\R$, it follows that
\[
\partial_{y}w=-C_s\sum_{k=1}^{\infty}(\lambda_k^{1/2}y)^s\mathcal{K}_{s-1}(\lambda_k^{1/2}y)
\frac{\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}}{\lambda_k^{s-1/2}}\psi_{k}(x)
\]
and
\[
\partial_{yy}w=-C_s\sum_{k=1}^{\infty}\big[(\lambda_k^{1/2}y)^{s-1}\mathcal{K}_{s-1}(\lambda_k^{1/2}y)
-(\lambda_k^{1/2}y)^{s-1}\mathcal{K}_{s-2}(\lambda_k^{1/2}y)y\big]
\frac{\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}}{\lambda_k^{s-1}}\psi_{k}(x).
\]
Then, as $\mathcal{K}_\nu(t)\sim\sqrt{\frac{\pi}{2t}}e^{-t}$, as $t\to\infty$, and
$\mathcal{K}_\nu(t)\sim C_\nu t^{-\nu}$, as $t\to0$, we get
\begin{align*}
\int_{0}^{\infty}y^a\int_{\Omega}|\partial_{y}w|^{2}\,d\gamma(x)\,dy
&=C_s\sum_{k=1}^\infty\bigg[\int_0^\infty y^a|(\lambda_k^{1/2}y)^s\mathcal{K}_{s-1}(\lambda_k^{1/2}y)|^2\,dy\bigg]
\frac{|\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^2}{\lambda_k^{2s-1}} \\
&=C_s\sum_{k=1}^\infty\bigg[\int_0^\infty r|\mathcal{K}_{s-1}(r)|^2\,dr\bigg]
\frac{|\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^2}{\lambda_k^{s}}
\leq \frac{C_s}{\lambda_{1}^{s}}\|f\|^{2}_{L^{2}(\Omega,\gamma)},
\end{align*}
and $w(x,y)\varphi(x)$ is Lipschitz with respect to $y\in(0,\infty)$, uniformly in $x$.
On the other hand,
\begin{align*}
\int_{\e}^{\infty}&y^a\int_{\Omega}|\partial_{yy}w|^{2}\,d\gamma(x)\,dy \\
&\leq C_{s,\varepsilon}\sum_{k=1}^{\infty}\int_\e^\infty y^a|(\lambda_k^{1/2}y)^{s-1}[(\lambda_k^{1/2}y)^{-1/2}
e^{-\lambda_k^{1/2}y}(1+y)]|^2\,dy
\frac{|\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^2}{\lambda_k^{2s-2}}\\
&\leq C_{s,\varepsilon}\sum_{k=1}^{\infty}\int_\e^\infty y^{-2}e^{-2\lambda_k^{1/2}y}(1+y)^2\,dy
\frac{|\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^2}{\lambda_k^{s-1/2}}\\
&\leq C_{s,\varepsilon}\sum_{k=1}^{\infty}\int_\e^\infty e^{-2\lambda_k^{1/2}y}\,dy
\frac{|\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^2}{\lambda_k^{s-1/2}}\\
&= C_{s,\varepsilon}\sum_{k=1}^{\infty}\frac{e^{-2\e\lambda_{k}^{1/2}}}{\lambda_k^s}
|\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}|^{2}\leq
\frac{C_{s,\varepsilon}}{\lambda_1^s}\|f\|^{2}_{L^{2}(\Omega,\gamma)}.
\end{align*}
Hence $\partial_{y}w(x,y)\varphi(x)$ is Lipschitz
with respect to $y\in(\e,\infty)$, uniformly in $x\in\Omega$.
\end{proof}
\section{The comparison result}\label{Section:comparison}
With the previous results at hand, we are now in position to prove the main result of the paper.
\begin{theorem}[Comparison result]\label{primoteoremadiconfronto}
Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ with $\gamma(\Omega)<1$.
Let $u$ and $\psi$ be the weak solutions to \eqref{Problema} and \eqref{problema simm}, respectively,
with $f\in L^{2}(\Omega,\gamma)$. Then
\begin{equation}\int_{0}^{r}u^{\circledast}(\sigma)\,d\sigma\leq\int_{0}^{r}\psi^{\circledast}(\sigma)\,d\sigma,\label{confronto}
\quad\hbox{for all}~0\leq r\leq \gamma(\Omega),\end{equation}
that is,
\[
u^{\displaystyle\star}\prec \psi.
\]
\end{theorem}
\begin{proof}
By making the change of variables $y=(2s)z^{1/(2s)}$ (see \cite{Caffarelli-Silvestre}), we can write the extension problems
\eqref{Problema estensione} and \eqref{Problema estensione sim} as
\begin{equation}\label{Problema estensione z}
\begin{cases}
-\mathcal{L}w+z^{2-1/s}w_{zz}=0,&\hbox{in}~\mathcal{C}_{\Omega},\\
w=0,&\hbox{on}~\partial_{L}\mathcal{C}_{\Omega},\\
-\underset{z\rightarrow0^{+}}{\lim}w_z=d_sf,&\hbox{on}~\Omega.
\end{cases}
\end{equation}
and
\begin{equation}\label{Problema estensione sim z}
\begin{cases}
-\mathcal{L}v+z^{2-1/s}v_{zz}=0,&\hbox{in}~\mathcal{C}_{\Omega}^{\displaystyle\star},\\
v=0,&\hbox{on}~\partial_{L}\mathcal{C}_{\Omega}^{\displaystyle\star},\\
-\underset{z\rightarrow0^{+}}{\lim}v_z=d_sf^{\displaystyle\star},&\hbox{on}~\Omega^{\displaystyle\star},
\end{cases}
\end{equation}
for some explicit constant $d_s>0$, respectively.
Now, since $u$ is the trace on $\Omega$ of the
solution $w$ to \eqref{Problema estensione z} and $\psi$ is the trace on $\Omega^{\bigstar}$ of the solution $v$ to
\eqref{Problema estensione sim z}, the result will immediately
follow once we prove the concentration comparison inequality
\begin{equation}
\int_{0}^{r}w^{\circledast}(\sigma,z)\,d\sigma\leq\int_{0}^{r}v^{\circledast}(\sigma,z)\,d\sigma,
\quad\hbox{for all}~0\leq r\leq\gamma(\Omega),~\hbox{for any fixed}~z\geq0.
\label{concentrazione}
\end{equation}
We recall that $w$ is smooth for any $z>0$. For a fixed
$z>0$ and $t>0$, let
\[\varsigma_{h}^{z}(x):=
\begin{cases}
\mathrm{sign}\,w(x,z),&\hbox{if}~|w(x,z)|\geq t+h,\\ \medskip
\dfrac{|w(x,z)|-t}{h}\,\mathrm{sign}\,w,&\hbox{if}~t<|w(x,z)|<t+h,\\
0,&\hbox{otherwise}.
\end{cases}\]
By multiplying the first equation in \eqref{Problema estensione z} by $\varsigma_{h}^{z}(x)$
and integrating over $\Omega$ with respect to the Gaussian measure, we obtain
\begin{align*}
\frac{1}{h}\int_{\left\{x:\,t<|w(x,z)|<t+h\right\}}|\nabla_{x}w|^{2}\,d\gamma&-\frac{z^{2-1/s}}{h}
\int_{\left\{x:\,|w(x,z)|>t+h\right\}}\frac{\partial^{2}w}{\partial z^{2}}\,d\gamma\\
&-\frac{z^{2-1/s}}{h}\int_{\left\{x:\,t<|w(x,z)|<t+h\right\}}\frac{\partial^{2}w}{\partial
z^{2}}(|w|-t)\,\mathrm{sign}\,w\,d\gamma=0.
\end{align*}
Letting $h\rightarrow0$ we obtain
\begin{equation}
-\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\left\vert \nabla_{x}
w\right\vert ^{2}\,d\gamma(x)-z^{2-1/s}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\frac
{\partial^{2}w}{\partial z^{2}}\,d\gamma(x)=0. \label{sostituendo}
\end{equation}
\noindent On the other hand, the coarea formula \eqref{coarea} and the isoperimetric
inequality with respect to the Gaussian measure \eqref{dis isop} give
\begin{equation}
-\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\!\left\vert \nabla_x
w\right\vert \, d\gamma(x)\geq\int_{\partial\left\{ x:\,\left\vert
w(x,z)\right\vert >t\right\}^{\star}}\varphi(x)\,d\mathcal{H}^{n-1}
(x)=\frac{1}{\sqrt{2\pi}}\exp\Bigg(-\frac{\big[\Phi^{-1}\big(
\gamma_{w}(t)\big)\big]^{2}}{2}\Bigg) ,
\label{isope}
\end{equation}
where $\left\{x:\, \left\vert w(x,z)\right\vert >t\right\} ^{\displaystyle\star}$ is the
half-space having Gauss measure $\gamma_{w}(t)$.
By H\"{o}lder's inequality,
$$\frac{1}{h}\int_{\left\{x:\,t<|w(x,z)|<t+h\right\}}|\nabla_x w|\,d\gamma(x)\leq\bigg(\frac{1}{h}
\int_{\left\{x:\,t<|w(x,z)|<t+h\right\}}|\nabla_x w|^2\,d\gamma(x)\bigg)^{1/2}\bigg(\frac{1}{h}\int_{\left\{x:\,t<|w(x,z)|<t+h\right\}}\,d\gamma(x)\bigg)^{1/2},$$
for any $h>0$. Hence, by taking $h\to0$,
\[
-\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\left\vert \nabla_{x}
w\right\vert \ d\gamma(x)\leq\left( -\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}
\left\vert \nabla_{x}w\right\vert ^{2}\, d\gamma(x)\right)
^{1/2}\left( -\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}
\, d\gamma(x)\right) ^{1/2}.
\]
Then \eqref{isope} yields
\begin{equation}
-\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\left\vert \nabla_{x}
w\right\vert ^{2}\, d\gamma(x)\geq\frac{1}{2\pi}\left( -\gamma_{w}^{\prime
}(t)\right) ^{-1}\exp\!\left( -\left[ \Phi^{-1}\big(\gamma_{w}(t)\big) \right] ^{2}\right) . \label{isop+holder}
\end{equation}
By plugging \eqref{isop+holder} into \eqref{sostituendo} we have
$$-z^{2-1/s}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\frac{\partial^{2}w}{\partial z^{2}
}\ d\gamma(x)-\!\frac{1}{2\pi}\left( \gamma_{w}^{\prime}(t)\right)
^{-1}\exp\left( -\left[ \Phi^{-1}\big(\gamma_{w}(t)
\big)\right] ^{2}\right) \leq0.$$
Now we set
\[
W(r,y):=\int_{0}^{r}w^{\circledast}(\sigma,z)\,d\sigma.
\]
Using Lemma \ref{lemma regolarita} and the second order derivation formula (\ref{Der2})
we
find that $W$ verifies the following differential inequality
\begin{equation}
-z^{2-1/s}\frac{\partial^{2}W}{\partial z^{2}}-p(r)\frac{\partial^{2}W}{\partial r^{2}
}\leq0 \label{eq W}
\end{equation}
for a.e.~$(r,z)\in(0,\gamma(\Omega))\times(0,\infty)$, where $p(r)=\frac{1}{2\pi}\exp(-[\Phi^{-1}(r)]^{2})$.
Moreover, the first order derivation formula
\eqref{Rakotoson derivation formula} implies
\[
\frac{\partial W}{\partial z}(r,z)=\frac{\partial}{\partial z}\int
_{\{x:\,w(x,z)>w^{\circledast}(r,z)\}}w(x,z)\,d\gamma(x)=\int_{\{x:\,w(x,z)>w^{\circledast
}(r,z)\}}\frac{\partial}{\partial z}w(x,z)\,d\gamma(x).
\]
Then, by the Hardy--Littlewood inequality \eqref{Hardy-Litt}, we easily infer
\begin{align*}
\frac{\partial W}{\partial z}(r,0) & =\int_{\{x:\,w(x,0)>w^{\circledast}(r,0)\}}
\frac{\partial w}{\partial z}(x,0)\, d\gamma(x)=-d_s\int_{\{x:\,u(x)>u^{\circledast}
(r)\}}f(x)\, d\gamma(x)\\
& \geq-d_s\int_{0}^{r}f^{\circledast}(\sigma)\,d\sigma,\quad\hbox{for}~r\in(0,\gamma(\Omega)).
\end{align*}
Therefore $W$ satisfies the following boundary conditions
\begin{align*}
W(0,z)& =0,\text{ \ \ } z\in\left[ 0,\infty\right), \\
\frac{\partial W}{\partial r}(\gamma(\Omega),z) & =0,\text{ \ \ }
z\in\left[ 0,\infty\right), \\
\frac{\partial W}{\partial z}(r,0) & \geq-d_s\int_{0}^{r}f^{\circledast}
(\sigma)\,d\sigma,\text{ \ for }r\in(0,\gamma(\Omega)).
\end{align*}
Next let us turn our attention to problem \eqref{Problema estensione sim}. By Proposition \ref{prop chiacchio},
it follows that the function
$\displaystyle\eta(x,t):=\big(e^{-t(\mathcal{L}_{\Omega^{\displaystyle\star}})}f^{\displaystyle\star}\big)(x)$,
is rearranged with respect to $x$, that is, $\eta(x,t)=\eta^{\displaystyle\star}(x,t)$. Recall the semigroup formula \eqref{eq:semigroup formula f}:
\[
v(x,y)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-y^{2}/(4t)}
\eta(x,t)\,\frac{dt}{t^{1-s}}.
\]
It is then clear that (even after the change of variables $y=(2s)z^{1/(2s)}$)
$v$ is rearranged with respect to $x$ as well,
namely, $v(x,z)=v^{\displaystyle\star}(x,z)$. This implies that the level sets of
$v(\cdot,z)$ are half-spaces, which gives in turn that all the inequalities involved
in the symmetrization arguments for the solution $u$ we performed above
become equalities for $v$. Therefore, if
$$V(r,z):=\int_{0}^{r}v^{\circledast}(\sigma,z)\,d\sigma,$$
then
\begin{equation}
-z^{2-1/s}\frac{\partial^{2}V}{\partial z^{2}}-p(r)\frac{\partial^{2}
V}{\partial r^{2}}=0. \label{eq V}
\end{equation}
Regarding the boundary conditions, we have
\begin{align*}
\frac{\partial V}{\partial z}(r,0) & =-d_s\int_{\{x:\,\psi(x_{1})>\psi^{\circledast}(r)\}}f^{\bigstar}(x)\, d\gamma(x)\\
& =-d_s\int_{\Phi^{-1}(r)}^{\infty}f^{\circledast}(\Phi^{-1}(x_{1}))\, d\gamma(x)\\
& =-d_s\int_{0}^{r}f^{\circledast}(\sigma)\,d\sigma,\quad\hbox{for}~r\in(0,\gamma(\Omega)).
\end{align*}
Then $V$ satisfies:
\begin{align*}
V(0,z) & =0,\text{ \ \ } z\in[0,\infty), \\
\frac{\partial V}{\partial r}(\gamma(\Omega),z) & =0,\text{ \ \ }
z\in[0,\infty), \\
\frac{\partial V}{\partial z}(r,0) & =-d_s\int_{0}^{r}f^{\circledast}
(\sigma)\,d\sigma,\text{ \ for }r\in(0,\gamma(\Omega)).
\end{align*}
If we put $\displaystyle Z(r,z):=W(r,z)-V(r,z)=\int_{0}^{r}[w^{\circledast}(\sigma,z)-v^{\circledast}(\sigma,z)]\,d\sigma$,
then \eqref{eq W} and \eqref{eq V} imply that $Z$ is a subsolution to
\begin{equation}
-z^{2-1/s}\frac{\partial^{2}Z}{\partial z^{2}}-p(r)\frac{\partial^{2}
Z}{\partial r^{2}}\leq0,\label{subsolution}
\end{equation}
for a.e.~$(r,z)\in(0,\gamma(\Omega))\times(0,\infty)$,
together with the following boundary conditions
\begin{align}
Z(0,z) & =0,\text{ \ \ }z\in[0,\infty),\nonumber \\
\frac{\partial Z}{\partial r}(\gamma(\Omega),z) & =0,\text{ \ \ }\label{boundcond}
z\in[0,\infty), \\
\frac{\partial Z}{\partial z}(r,0) & \geq0,\text{ \ for }r\in(0,\gamma(\Omega)).\nonumber
\end{align}
Moreover, since $\|w(\cdot,z)\|_{L^{2}(\Omega,\gamma)},\|v(\cdot,z)\|_{L^{2}(\Omega^{\displaystyle\star},\gamma)}
\rightarrow0$, as $z\rightarrow\infty$, we have
$Z(r,z)\rightarrow 0$, as $z\rightarrow\infty$, uniformly in $r$. Now we
claim that $Z\leq 0$ in $[0,\gamma(\Omega)]\times[0,\infty)$.
Indeed, observe that \eqref{subsolution} can be rewritten as
\[
-p(r)^{-1}\frac{\partial^2Z}{\partial z^2}-z^{-2+1/s}
\frac{\partial^2Z}{\partial r^2}\leq0.
\]
Therefore, by multiplying both sides by $Z_{+}$, the positive part of $Z$, and integrating by parts over the strip $(0,\gamma(\Omega))\times(0,\infty)$, the boundary conditions
\eqref{boundcond}
and the fact that $Z(r,z)\rightarrow0$ as $z\rightarrow\infty$ imply
\begin{multline*}
\int_{0}^{\gamma(\Omega)}p(r)^{-1}\frac{\partial Z}{\partial z}(r,0)Z_{+}(r,0)\,dr
+\int_{0}^{\infty}\int_{0}^{\gamma(\Omega)}z^{-2+1/s}\bigg|\frac{\partial Z_{+}}{\partial r}\bigg|^{2}\,dr\,dz\\
+\int_{0}^{\infty}\int_{0}^{\gamma(\Omega)}
p(r)^{-1}\bigg|\frac{\partial Z_{+}}{\partial z}\bigg|^{2}\,dr\,dz\leq0,
\end{multline*}
namely,
\[
\int_{0}^{\infty}\int_{0}^{\gamma(\Omega)}z^{-2+1/s}\bigg|\frac{\partial Z_{+}}{\partial r}\bigg|^{2}\,dr\,dz
+\int_{0}^{\infty}\int_{0}^{\gamma(\Omega)}
p(r)^{-1}\bigg|\frac{\partial Z_{+}}{\partial z}\bigg|^{2}\,dr\,dz\leq0.
\]
Thus $Z_{+}\equiv0$ and the concentration comparison inequality \eqref{concentrazione} follows.
\end{proof}
\section{Regularity estimates}\label{Section:regularity}
We first introduce the Zygmund spaces, which appear naturally in the regularity scale for solutions to elliptic
equations with Gaussian measure in the local setting, see \cite{dFP}. We refer the reader to the monograph \cite{Bennett} for details about all the related properties we will use for our purposes.
\begin{definition}[Zygmund spaces]
Let $1\leq p<\infty$ and $\alpha\in\R$. The Zygmund space $L^p(\log L)^\alpha(\Omega,\gamma)$ is defined as the space of all measurable functions $u:\Omega\to\R$ such that
$$\int_\Omega\big[|u(x)|\log^\alpha(2+|u(x)|)\big]^p\,d\gamma(x)<\infty.$$
\end{definition}
If $\alpha=0$ the Zygmund space $L^p(\log L)^0(\Omega,\gamma)$ coincides with the weighted space $L^p(\Omega,\gamma)$. Moreover, if $p>q$ and
$\alpha,\beta\in\R$ then
$$L^p(\log L)^\alpha(\Omega,\gamma)\subset L^q(\log L)^\beta(\Omega,\gamma).$$
When $p=q$ and $\alpha >\beta$ one can prove that
\begin{equation}
L^p(\log L)^\alpha(\Omega,\gamma)\subset L^p(\log L)^\beta(\Omega,\gamma).\label{secondembed}
\end{equation}
\begin{remark}
The Zygmund space $L^p(\log L)^\alpha(\Omega,\gamma)$ can be equivalently defined as the space of measurable functions $u:\Omega\to\R$ such that the quantity (which is a quasi-norm in this space)
\begin{equation}
\bigg(\int_0^{\gamma(\Omega)}\big[(1-\log t)^\alpha
u^{\circledast}(t)\big]^p\,dt\bigg)^{1/p}\label{qnorm}
\end{equation}
is finite. Moreover, $L^p(\log L)^\alpha(\Omega,\gamma)$ is a Banach space when equipped with the norm
\begin{equation}
\|u\|_{L^p(\log L)^\alpha(\Omega,\gamma)}^p=\int_0^{\gamma(\Omega)}\big[(1-\log t)^\alpha
u^{\circledast\circledast}(t)\big]^p\,dt,\label{normavera}
\end{equation}
where
\[
u^{\circledast\circledast}(t):=\frac{1}{t}\int_{0}^{t}u^{\circledast}(\sigma)\,d\sigma.
\]
We stress that the quasi-norm \eqref{qnorm} is equivalent to the norm \eqref{normavera}, see \cite{Bennett} for more details.
\end{remark}
The main result of this section is the following regularity result for solutions to the fractional
nonlocal OU problem \eqref{Problema} in terms of the data $f$.
Notice that when $s=1$ we recover the corresponding regularity results
for the OU equation via Gaussian symmetrization contained in \cite{dFP}.
\begin{theorem}[Regularity estimates]\label{thm:integrability}
Let $\Omega$ be an open subset of $\mathbb{R}^{n}$, $n\geq2$, such that $\gamma(\Omega)\leq1/2$.
Fix $0<s<1$.
If $f\in L^{p}(\log L)^\alpha(\Omega,\gamma)$, where $\alpha\in\R$ for $2< p<\infty$, and $\alpha\geq-\frac{s}{2}$ for $p=2$, then the solution $u$ to
\eqref{Problema} belongs to
$L^{p}(\log L)^{\alpha+s}(\Omega,\gamma)$ and
$$\|u\|_{L^{p}(\log L)^{\alpha+s}(\Omega,\gamma)}\leq C\|f\|_{L^{p}(\log L)^\alpha(\Omega,\gamma)},$$
for a positive constant $C=C(n,p,\alpha,s,\gamma(\Omega))$ which is independent on $u$ and $f$.
\end{theorem}
In order to prove Theorem \ref{thm:integrability} we will first show that the space $\mathcal{H}^{s}(\Omega,\gamma)$ is embedded in the Zygmund space $L^2(\log L)^{s/2}(\Omega,\gamma)$. This will allow us to choose the datum
$f$ in the dual space $L^2(\log L)^{-s/2}(\Omega,\gamma)$ in problem \eqref{Problema}. In this way
Definition \ref{Defprobest}
will still make sense and $u=w(\cdot,0)$, where $w$ is the solution to
\eqref{Problema estensione}, will be the unique weak solution to problem
\eqref{Problema}. Towards this end we introduce the fractional Gaussian Sobolev space $H^{s}(\Omega,\gamma)$ as the real interpolation space defined by
\[
H^{s}(\Omega,\gamma)=\left[ H^{1}(\Omega,\gamma),L^{2}(\Omega
,\gamma)\right] _{1-s}.
\]
\begin{lemma}
For any $u\in\mathcal{H}^{s}(\Omega,\gamma)$ the following inequality holds
\begin{equation}
\int_{0}^{\gamma(\Omega)}[(1- \log r)^{s/2}u^{\circledast}(r)]^{2}\,dr\leq C\Vert u\Vert_{\mathcal{H}^{s}(\Omega
,\gamma)}^2 \label{emblorentz}
\end{equation}
where $C$ is a positive constant depending on $n,s$ and $\Omega$. In particular,
\[
\mathcal{H}^{s}(\Omega,\gamma)\hookrightarrow L^{2}(\log L)^{s/2}(\Omega,\gamma).
\]
\end{lemma}
\begin{proof}
Given any function $u\in\mathcal{H}^{s}(\Omega,\gamma)$ we consider the
extension $\widetilde{u}$ of $u$ by zero outside of $\Omega$. Since $\widetilde
{u}\in H^{s}(\mathbb{R}^{n},\gamma)$ and this last space
coincides with the Gaussian Besov space $B^{s}(\mathbb{R}^{n},\gamma)$ (see
\cite{Nikitin}), the embedding result contained in \cite[Theorem 23]{MaMi} yields
\begin{equation*}
\int_{0}^{1/2}[(1- \log r)^{s/2}u^{\circledast}(r)]^{2}\,dr\leq
C\Vert\widetilde{u}\Vert^{2}_{H^{s}(\mathbb{R}^{n},\gamma)},
\end{equation*}
for some constant $C>0$. A change of variable and the monotonicity of the decreasing rearrangement $u^{\circledast}$ lead to
\begin{equation}
\int_{0}^{1}[(1- \log r)^{s/2}u^{\circledast}(r)]^{2}\,dr
\leq 2 \int_{0}^{1/2}[(1- \log r)^{s/2}u^{\circledast}(r)]^{2}\,dr
\leq 2C\Vert\widetilde{u}\Vert^{2}_{H^{s}(\mathbb{R}^{n},\gamma)}.\label{emblorentz2}
\end{equation}
Now we observe that the Exact Interpolation Theorem (see \cite[Theorem 7.23]{AdamsFourn}) implies that
extending any function $u\in\mathcal{H}^{s}(\Omega,\gamma)$ by zero
outside of $\Omega$ defines a continuous extension map between $\mathcal{H}^{s}
(\Omega,\gamma)$ and $H^{s}(\mathbb{R}^{n},\gamma)$. Thus it follows that the norm at the right-hand side of
\eqref{emblorentz2} is bounded (up to a constant depending on $n,s$ and $\Omega$) by $\Vert u\Vert_{\mathcal{H}^{s}(\Omega,\gamma)}^2$ and the result follows.
\end{proof}
With these results at hand, we are able to show the generalization of the comparison result (Theorem \ref{primoteoremadiconfronto}) for $f$ in Zygmund spaces.
\begin{corollary}
Assume that $f\in L^2(\log L)^{-s/2}(\Omega,\gamma)$. Then Theorem \ref{primoteoremadiconfronto} still holds.
\end{corollary}
\begin{proof}
Let $f_{n}$ be a sequence of smooth function such that $f_{n}\rightarrow f$ strongly in
$L^{2}(\log L)^{-s/2}(\Omega,\gamma)$. Let $w_{n}$ be the unique
weak solution to problem \eqref{Problema estensione} with data $f_{n}$. By choosing $w_{n}$ as a test function in \eqref{weakform} we have
\begin{align*}
\iint_{\mathcal{C}_{\Omega}} y^{a}|\nabla_{x,y}w_{n}|^{2}\,d\gamma(x)\,dy
&= c_{s}^{-1}\int_{\Omega}f_{n}(x)w_{n}(x,0)\,d\gamma(x) \\
&\leq c_{s}^{-1}\|w_{n}(x,0)\|_{L^{2}(\log
L)^{s/2}(\Omega,\gamma)}\|f_{n}(x,0)\|_{L^{2}(\log L)^{-s/2}(\Omega,\gamma)}.
\end{align*}
Next we use \eqref{emblorentz} and the trace inequality \eqref{trace} to find
\[
\iint_{\mathcal{C}_{\Omega}} y^{a} |\nabla_{x,y}w_{n}|^{2}\,d\gamma(x)\,dy\leq C\|w_{n}\|_{H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes
y^{a}dy)}\,\|f_{n}(x,0)\|_{L^{2}(\log L)^{-s/2}(\Omega,\gamma)}.
\]
This allows us to extract a subsequence from $\left\{w_{n}\right\}$ (still labeled by $\left\{w_{n}\right\}$), such that
$w_{n}\rightharpoonup w$ weakly in $H_{0,L}^{1}(\mathcal{C}_{\Omega},d\gamma(x)\otimes y^{a}dy)$.
Then the compact embedding established in Proposition \ref{compactness} gives that,
up to a new subsequence, $w_{n}(\cdot,0)\rightarrow w(\cdot,0)$ strongly in $L^{2}(\Omega,\gamma)$.
Thus we can pass to the limit in the weak formulation \eqref{weakform} of $w_{n}$ and find that $w$ solves problem \eqref{Problema estensione}
corresponding to the data $f$. Thus $u:=w(\cdot,0)$ is the weak solution to problem \eqref{Problema}. In order to obtain the concentration inequality
\eqref{confronto}, we just observe that $f_{n}^{\displaystyle\star}$ approximates $f^{\displaystyle\star}$ in $L^{2}(\Omega^{\displaystyle\star},\gamma)$.
Then, if $\left\{w_{n}\right\}$ and $\left\{v_{n}\right\}$ are sequences of approximating solutions converging to $w$ and $v$ respectively, passing to the limit in the integral inequality
\begin{equation*}
\int_{0}^{s}w_{n}^{\circledast}(\sigma,0)\,d\sigma\leq\int_{0}^{s}v_{n}^{\circledast
}(\sigma,0)\,d\sigma,
\end{equation*}
we immediately get \eqref{confronto}.
\end{proof}
For the proof of Theorem \ref{thm:integrability} we need two further preliminary results, interesting in their own right. The following is a regularity
result for solutions of problems of the type \eqref{Problema} with \emph{rearranged} data, posed on the half-space $H$.
\begin{theorem}[Estimates for half-space solutions]\label{Regularity theorem}
Let $H=\{x\in\R^n:x_{1}>0\}$. Suppose that $h(x)=h^{\bigstar}(x)$, for all $x\in H$.
If $h\in L^{p}(\log L)^\alpha(H,\gamma)$ with $\alpha\in\R$ for $2<p<\infty$, and $\alpha\geq-\frac{s}{2}$ for $p=2$,
then the weak solution $\psi$ to
\begin{equation}\label{PPP}
\begin{cases}
\mathcal{L}^s\psi=h,&\hbox{in}~H,\\
\psi=0,&\hbox{on}~\partial H,
\end{cases}
\end{equation}
belongs to $L^{p}(\log L)^{\alpha+s}(H,\gamma)$ and
$$\|\psi\|_{L^{p}(\log L)^{\alpha+s}(H,\gamma)}\leq C\|h\|_{L^{p}(\log L)^\alpha(H,\gamma)},$$
for some constant $C=C(n,p,\alpha,s)>0$, which is independent on $\psi$ and $h$.
\end{theorem}
\begin{proof}
By \eqref{eq:fractional integral} and \eqref{est1}--\eqref{eta} we can write
$$\psi(x)=\frac{1}{\Gamma(s)}\int_0^\infty e^{-t(\mathcal{L}_H)}h(x)\,\frac{dt}{t^{1-s}}=
\frac{1}{\Gamma(s)}\int_0^\infty e^{-t\mathcal{L}}\widetilde{h}(x)\,\frac{dt}{t^{1-s}}=\mathcal{L}^{-s}\widetilde{h}(x).$$
Then the estimate follows from \cite[Theorem~5.7]{KM}.
\end{proof}
The next Lemma is a useful comparison principle for solutions of problems of the form \eqref{Problema} with rearranged data, having as a ground domain an
half-space of Gaussian measure larger than $1/2$.
\begin{lemma}[Comparison of half-space solutions]\label{semispazi}
Let $H_{\omega}=\{x\in\mathbf{\mathbb{R}}^{n}:x_{1}>\omega\}$, for some $\omega>0$.
Let $h\in L^{p}(\log L)^\alpha(H,\gamma)$ be a nonnegative function such that $h(x)=h^{\bigstar}(x)$ and let $\psi$ be the weak
solution to
$$\begin{cases}
\mathcal{L}^{s}\psi=h,& \hbox{in}~H_{\omega},\\
\psi=0,&\hbox{on}~\partial H_{\omega}.
\end{cases}$$
Then
$$\psi(x) \leq \zeta(x), \quad\hbox{for a.e.}~x\in H_{\omega},$$
where $\zeta$ is the weak solution to \eqref{PPP} with datum $\overline{h}$,
where $\overline{h}$ denotes the zero extension of $h$ in $H\setminus H_{\omega}$.
\end{lemma}
\begin{proof}
The function
\[
F(x,t):=e^{-t(\mathcal{L}_{H})}\overline{h}(x)-e^{-t(\mathcal{L}_{H_{\omega}})}h(x),
\]
solves the initial boundary value problem
$$\begin{cases}
\partial_tF=\Delta F-x\cdot\nabla F,&\hbox{in}~H_\omega\times(0,\infty), \\
F(x,t)\geq0,&\hbox{on}~\partial H_{\omega}\times(0,\infty), \\
F(x,0)=0,&\hbox{on}~H_{\omega}.\\
\end{cases}$$
Thus, by a standard maximum principle argument, $F\geq0$ in $H_{\omega}\times[0,\infty)$.
In other words,
$$e^{-t(\mathcal{L}_{H})}\overline{h}\geq e^{-t(\mathcal{L}_{H_{\omega}})}h,\quad
\hbox{for all}~x\in H_{\omega},~t\geq0.$$
Therefore, if $v$ and $\overline{v}$ denote the extensions as in
\eqref{StingaTorreasemigr} of $\psi$ and $\zeta$, respectively, then
$$\overline{v}(x,y)\geq v(x,y),\quad\hbox{for all}~x\in H_{\omega},~y\geq0.$$
The result follows by taking $y=0$ in this last inequality.
\end{proof}
Now we are finally able to present the proof of the regularity estimate, namely, Theorem \ref{thm:integrability}.
\begin{proof}[Proof of Theorem \ref{thm:integrability}]
Let $u$ be the weak solution to \eqref{Problema} defined in an open set $\Omega$
such that $\gamma(\Omega)\leq 1/2$, with corresponding datum $f$. By Theorem
\ref{primoteoremadiconfronto}, $u$ is less concentrated than the solution $\psi$ to \eqref{problema simm} defined in the half-space with the
same Gauss measure as $\Omega$ and datum $f^{\displaystyle\star}$. If $\gamma(\Omega)=1/2$ the assertion follows by Theorem \ref{Regularity theorem}. If
$\gamma(\Omega)<1/2$,
we first apply Lemma \ref{semispazi} to estimate $\psi$ in terms of the solution $\zeta$ to \eqref{PPP} defined in the
half-space $H=\{x\in\mathbf{\mathbb{R}}^{n}:x_{1}>0\}$ and having the extension of
$f^{\displaystyle\star}$ by zero to $H$ at the right-hand side.. Then Theorem \ref{Regularity theorem} allows us to conclude.
\end{proof}
\begin{remark}
We remark that other regularity results for problems involving fractional operators with bounded lower order terms,
but posed on bounded smooth domains, are contained in \cite{Grubb}.
\end{remark}
\section{Appendix: A semigroup method proof of the $L^p$ estimate}\label{Appendix}
For completeness and convenience of the reader, we give an alternative and more explicit proof of Theorem \ref{Regularity theorem} with $L^p$ data using the Mehler kernel to represent the inverse of the fractional
OU operator. Observe that such result is a particular case of Theorem \ref{Regularity theorem} since, when $f\in L^{p}(\Omega,\gamma)$, Theorem
\ref{Regularity theorem} and the embedding \eqref{secondembed} give $u\in L^{p}(\log L)^{s}(\Omega,\gamma)\subset L^{p}(\Omega,\gamma)$.
\begin{theorem}[Estimates for half-space solutions with $L^{p}$ data]\label{Regularity L^{p} theorem}
Let $H=\{x\in\R^n:x_{1}>0\}$. Suppose that $h(x)=h^{\bigstar}(x)$, for all $x\in H$. If $h\in L^{p}(H,\gamma)$, for $2\leq p<\infty$,
then the weak solution $\psi$ to \eqref{PPP} belongs to $L^{p}(H,\gamma)$ and
$$\|\psi\| _{L^{p}(H,\gamma)}\leq C\|h\|_{L^{p}(H,\gamma)},$$
for some constant $C=C(n,p,s)>0$, which is independent on $\psi$ and $h$.
\end{theorem}
\begin{proof}
The proof will be split in four steps.
\newline
\noindent\textbf{Step 1. The explicit solution via the semigroup kernel.}
By \eqref{eq:fractional integral}, and by using an abuse of notation, the solution $\psi$ to \eqref{PPP} can be written as
$$\psi(x)=\psi(x_{1})=\frac{1}{\Gamma(s)}
\int_0^\infty e^{-t(\mathcal{L}_H)}h(x)\,\frac{dt}{t^{1-s}}
=\int_{0}^{\infty}G(x_{1},y_{1})h(y_{1})\,d\gamma(y_{1}),$$
where (see \eqref{eq:eta})
$$G(x_{1},y_{1})=\frac{1}{\Gamma(s)}\int_{0}^{\infty}[M_{t}(x_{1},y_{1})-M_{t}(x_{1},-y_{1})]\,\frac{dt}{t^{1-s}}.$$
Next we write
\begin{equation}\label{G}
\begin{aligned}
G(x_{1},y_{1})&=\int_{0}^{c(p)}\cdots\,dt+\int_{c(p)}^{T(x_{1},y_{1})}\cdots\,dt+\int_{T(x_{1},y_{1})}^{\infty}\cdots\,dt \\
&=:G_{1}(x_{1},y_{1})+G_{2}(x_{1},y_{1})+G_{3}(x_{1},y_{1}),
\end{aligned}
\end{equation}
with $c(p)>1$ a suitable constant, and $T(x_{1},y_{1})=\max\{c(p),\log\left(
x_{1}^{2}+y_{1}^{2}\right) \}.$
It follows that
\begin{equation}\label{aaaa}
\|\psi\|_{L^{p}(H,\gamma)}^{p}
\leq\sum_{j=1}^3\int_{0}^{\infty}\left(\int_{0}^{\infty}G_{j}(x_{1},y_{1})h(y_{1})\,d\gamma(y_{1})\right)^{p}d\gamma(x_{1}).
\end{equation}
\noindent\textbf{Step 2. Estimate of the term $j=1$ in \eqref{aaaa}.} We observe that by
\eqref{Stima Lp semigruppo} and \eqref{norma prolungamento} we get
\begin{equation}\label{ssm}
\left\Vert \int_{-\infty}^{\infty}M_{t}(x_{1},y_{1})\widetilde
{h}(y_{1})\,d\gamma(y_{1})\right\Vert _{L^{p}(\R,\gamma)}
\leq\|\widetilde{h}\|_{L^{p}(\mathbb{R},\gamma)}
=2\left\Vert h\right\Vert _{L^{p}(H,\gamma)},
\end{equation}
where $\widetilde{h}$ is defined like in \eqref{est1}. Tonelli's theorem,
Minkowski's inequality and (\ref{ssm}) yield
\begin{align*}
& \left\Vert \int_{0}^{\infty}G_{1}(x_{1},y_{1})h(y_{1})\,d\gamma(y_{1})\right\Vert _{L^{p}(H,\gamma)}\\
& \leq c_s\int_{0}^{c(p)}\left\Vert \int_{0}^{\infty}\left[
M_{t}(x_{1},y_{1})-M_{t}(x_{1},-y_{1})\right]
h(y_{1})\,d\gamma(y_{1})\right\Vert _{L^{p}(H,\gamma)}\frac{dt}{t^{1-s}}\\
&=c_s\int_{0}^{c(p)}\left\Vert \int_{-\infty}^{\infty}M_{t}(x_{1},y_{1})\widetilde{h}(y_{1})\,d\gamma(y_{1})\right\Vert
_{L^{p}(\R,\gamma)}\frac{dt}{t^{1-s}} \\
&\leq 2c_s\left\Vert h\right\Vert _{L^{p}(H,\gamma)}\int_{0}^{c(p)}\frac{dt}{t^{1-s}}
=c_{s,p}\|h\|_{L^p(H,\gamma)}.
\end{align*}
\noindent\textbf{Step 3. Estimate of $G_{2}$ and $G_{3}$.} We prove that
$$\int_0^\infty\left(\int_0^\infty G_{j}^{p'}(x_{1},y_{1})\,d\gamma(y_{1})\right)^{p/p'}
d\gamma(x_{1})<\infty,\quad\hbox{for}~j=2,3.$$
By Jensen's inequality, it is enough to show that $G_{j}\in L^{p}(H\times H,\gamma\otimes\gamma)$, for $j=2,3$.
If $t>c(p)>1$ then $(1-e^{-2t})\sim1$ and $\left\vert M_{t}(x_{1},y_{1})\right\vert\leq
c\exp\left( 4e^{-t}\left\vert x_{1}\right\vert\left\vert y_{1}\right\vert \right)$, see \eqref{eq:Mehler kernel}.
It follows that
\begin{align*}
\left\vert G_{2}(x_{1},y_{1})\right\vert
&\leq c_s\int_{c(p)}^{T(x_{1},y_{1})}\vert M_{t}(x_{1},y_{1})-M_{t}(x_{1},-y_{1})\vert\,\frac{dt}{t^{1-s}} \\
& \leq \frac{c_s}{c(p)^{1-s}}\int_{c(p)}^{T(x_{1},y_{1})}\exp(4e^{-t}|x_{1}||y_{1}|)\,dt \\
& \leq \frac{c_s}{c(p)^{1-s}}\int_{c(p)}^{T(x_{1},y_{1})}\exp\left[2e^{-c(p)}(x_{1}^{2}+y_{1}^{2})\right]\,dt \\
& \leq\frac{c_s}{c(p)^{1-s}}\cdot\frac{T(x_{1},y_{1})}{\left( \varphi(x_{1})\right) ^{4e^{-c(p)}}
\left(\varphi(y_{1})\right) ^{4e^{-c(p)}}}=:\widetilde{G}_{2}(x_{1},y_{1}).
\end{align*}
We then get $\widetilde{G}_{2}(x_{1},y_{1})\in L^{p}(H\times H,\gamma\otimes\gamma)$
if we choose $4pe^{-c(p)}<1$, that is, if $c(p)>\max\{1,\log(4p)\}$. Moreover, by Taylor's formula
and using that $t>1$ and $e^{-t}(|x_1|^2+|y_1|^2)<1$,
\begin{align*}
M_{t}(x_{1},y_{1})-M_{t}(x_{1},-y_{1}) &\leq C_n
\Big|\exp\Big(\frac{e^{-t}\langle x_1,y_1\rangle}{1-e^{-2t}}\Big)-\exp\Big(-\frac{e^{-t}\langle x_1,y_1\rangle}{1-e^{-2t}}\Big)\Big| \\
&\leq Ce^{-t}|\langle x_1,y_1\rangle|\exp(ce^{-t}\langle x_1,y_1\rangle) \\
&\leq Ce^{-t}(|x_1|^2+|y_1|^2)\exp(ce^{-t}(|x_1|^2+|y_1|^2)) \\
&\leq Ce^{-t}(|x_1|^2+|y_1|^2).
\end{align*}
Then
\begin{align*}
\left\vert G_{3}(x_{1},y_{1})\right\vert & \leq C_s\int_{T(x_{1},y_{1})}^{\infty}
\left\vert M_{t}(x_{1},y_{1})-M_{t}(x_{1},-y_{1})\right\vert\,\frac{dt}{t^{1-s}}
\leq C_{n,s}\big(|x_1|^2+|y_1|^2\big)\int_{T(x_{1},y_{1})}^{+\infty}e^{-t}dt\\
& =C_{n,s}\big(\left\vert x_{1}\right\vert ^{2}+\left\vert y_{1}\right\vert^{2}\big)
e^{-T(x_{1},y_{1})}\leq C_{n,s}\in L^{p}(H \times H,\gamma\otimes\gamma).
\end{align*}
\noindent\textbf{Step 4. Estimates of the terms $j=2,3$ in \eqref{aaaa}.}
By H\"{o}lder's inequality and the estimates of Step 3, we get
\begin{align*}
&\int_{0}^{\infty}\left(\int_{0}^{\infty}G_{j}(x_{1},y_{1})h(y_{1})\,d\gamma(y_{1})\right)^{p}d\gamma(x_{1})\\
& \leq\int_{0}^{\infty}\left(\int_{0}^{\infty}G_{j}^{p'}(x_{1},y_{1})\,d\gamma(y_{1})\right)^{p/p'}\left(
\int_{0}^{\infty}|h(y_{1})|^p\,d\gamma(y_{1})\right)d\gamma(x_{1})\leq
c\left\Vert h\right\Vert _{L^{p}(H,\gamma)}^{p},
\end{align*}
for $j=2,3$ and for some positive constant $c=c(n,p,s)$.\\
Hence the desired result follows by collecting Steps 2 and 4 in estimate \eqref{aaaa}.
\end{proof}
\bigskip
\noindent\textbf{Acknowledgements.} Research partially supported by GNAMPA of INdAM, ``Programma
triennale della Ricerca dell'Universit\`{a} degli Studi di Napoli "Parthenope"
- Sostegno alla ricerca individuale 2015-2017" (Italy) and
by Grant MTM2015-66157-C2-1-P form Government of Spain. | 178,709 |
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TITLE: Reductions of Isogenous Elliptic Curves
QUESTION [0 upvotes]: I am looking for a full proof (references containing a full proof are more than welcome as well) of the following fact:
If $E$, $E'$ are $\mathbb Q$-isogenous elliptic curves over $\mathbb Q$, are their reductions mod $p$ $\mathbb F_p$-isogenous if $p$ is a prime of good reduction for the curves? The answer here was not quite complete; it did not explain why the reduced Tate modules were isomorphic. In addition, the last comment in that post says that one can ignore Tate modules and just reduce the isogeny mod $p$ to get the result, but I do not see why this is true.
REPLY [1 votes]: Let $f : E \to E'$ be an isogeny over $\mathbb{Q}$ and let $p$ be a prime of good reduction for $E, E'$.
One way to see that the reductions (of global minimal integral Weierstrass models) $\tilde{E}, \tilde{E'}$ are isogenous over $\mathbb F_p$ is to use Tate modules (which would allow to treat the case of general abelian varieties over number fields, using Tate's isogeny theorem).
The linked answer explains that for $\ell \neq p$, the map $V_{\ell}(f) : V_{\ell}(E) \to V_{\ell}(E')$ is a isomorphism of $G_{\mathbb{Q}}$-representations, with inverse $\frac{1}{\deg(f)} V_{\ell}(f^\vee)$.
If we denote by $r : E \to \tilde{E}$ the reduction mod $p$ map, then statement VII.3.1.b) in Silverman's book "Arithmetic of Elliptic curves" implies that the reduction map induces isomorphisms $E[m] \cong \tilde E[m]$ for all $m$ coprime to $p$, and thus we get an isomorphism $V_{\ell} r : V_{\ell} E \to V_{\ell}(\tilde E)$, and the same holds for $V_{\ell} r' : V_{\ell} E' \to V_{\ell}(\tilde E')$.
We may thus define an map $\overline{V_{\ell} f} : V_{\ell}(\tilde E) \to V_{\ell}(\tilde E')$ so that the following diagram commutes:
[The $T_\ell$ should be $V_\ell$ everywhere.]
Because all other 3 maps are isomorphisms, we see that $\overline{V_{\ell} f}$ is an isomorphism at least of $\mathbb Z_\ell$-modules. To conclude we need to see that $\overline{V_{\ell} f}$ is an isomorphism at least of $G_{\mathbb{F}_p}$-representations.
If we denote by
$D_p \cong
\mathrm{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)
\hookrightarrow
G_{\mathbb{Q}}$
the decomposition subgroup at $p$, we have a surjective morphism $\pi : D_p \twoheadrightarrow G_{\mathbb{F}_p}$.
Then the Galois-equivariance of $\overline{V_{\ell} f}$ follows from the one of $V_{\ell} f$ and $V_{\ell} r, V_{\ell} r'$.
Namely, we have
$$ V_{\ell} r ( \sigma \cdot t ) = \pi(\sigma) \cdot V_{\ell} r(t) $$
whenever $\sigma \in D_p, t \in V_{\ell} E$. This is because on the level of $P =(x,y) \in E[m] \subset E(\overline{\mathbb{Q}})$, we have
$$r(\sigma \cdot (x,y)) = \pi(\sigma) \cdot r(x,y)$$
since $\pi(\sigma) : \bar{x} \mapsto \overline{\sigma(x)}$ is well-defined ($D_p$ is the stabilizer of a prime above $p$ in $\mathcal{O}_{\overline{ \mathbb{Q} }}$).
Note that if $f$ is defined between the minimal integral Weierstrass models of $E,E'$ using a ratio of polynomials having coefficients in $\Bbb Z \setminus p \Bbb Z$, then there is a well-defined reduction $\tilde f : \tilde E \to \tilde E'$ which is a morphism of elliptic curves, and cannot be zero (otherwise $f$ would map all points of $E$ to points having some $p$ in the denominator in the $x,y$-coordinates), hence is an isogeny — so no need of Tate's theorem here in that case. | 154,866 |
\begin{document}
\title[Uniqueness sets]{Uniqueness sets for Fourier series}
\author[Vagharshakyan]{Ashot Vagharshakyan}
\address{Institute of Mathematics NAS}
\address{ of Armenia, Bagramian 24-b,}
\address{375019 Yerevan, Armenia}
\begin{abstract}
The paper discusses some uniqueness sets for Fourier series.
\end{abstract}
\maketitle
\section{Introduction}
In this paper the following problem is considered: to find
conditions on a set $E\in [-\pi,\,\pi]$ such that, if a function $f(x),\,-\pi<x<\pi,$ belongs to some space and its fourier
series converges to zero at each point of the set $E$, then $f(x)$
is identically zero.
The first nontrivial result, for trigonometric series, was
proved by G. Cantor and W. Young, see \cite{Bary}, p. 191.
\begin{theorem}
Let $c_k \to 0$ and for each point
$x\in [-\pi,\,\pi]\setminus F$
we have
\[
\lim_{n\to \infty}\sum_{k=-n}^nc_ke^{ikx}=0,
\]
where $F$ be a countable set. Then
$c_k=0,\,\,\,k\in Z$.
\end{theorem}
D. Menshov, see \cite{Bary}, p. 806, construct a nonzero
measure $\mu$ which has support of zero Lebesgue measure,
and its Fourier coefficients tend to zero. The partial
summs, of that Fourier series converges to zers out of
$supp(\mu)$.
\section{Auxliary definitions and results}
More information, about the following quantities, related with
Hausdorff's measures and capacities, one can find in \cite{Car},
pp. 13 - 46. For convenient of the reader, we give some definitions.
\begin{definition}
Let $0\leq h(x),\,\,0\leq x\leq 1$ be a nonnegative, increasing
function and $h(0)=0$. Let the subset
$E\subset \{z;\quad |z|=1\}$
be cover by the family of arcs
$\{S_k\}_{k=1}^{\infty}$,
i.e.
\begin{equation*}
E\subseteq \bigcup_{k=1}^{\infty}S_k.
\end{equation*}
Then we put
\begin{equation*}
M_{h}(E)=\inf \left(\sum_{k=1}^{\infty}h(|S_k|)\right),
\end{equation*}
where
$|S|$
is the length of the arc $S$ and the infimum is taken over all
families of cover.
\end{definition}
\begin{definition}
Let $0<\alpha<1$ and $E$ be baunded Borel set. The $C_{\alpha}(E)$-
capactiy of the set $E$ is defined by formula
\begin{equation*}
C_{\alpha}(E)=\left(\inf_{\mu\prec E}\int_{E}\int_{E}
\frac{d\mu(x)d\mu(y)}{|x-y|^{\alpha}}\right)^{-1},
\end{equation*}
where $\mu \prec E$ means, that $\mu$ is probality measure with support in $E$.
\end{definition}
For each $0<\alpha<1$ from Parseval's equality follows that there is a constant $M$ such that,
\begin{equation*}
\sum_{k=-\infty}^{\infty}|\hat{f}_{k}|^2|k|^{\alpha}\leq M\,\int_{-\pi}^{\pi}|f(x)|^2dx+M \,\int_{-\pi}^{\pi}\int_{\pi}^{\pi}
\frac{|f(x)-f(y)|^2}{|x-y|^{1+\alpha}}dxdy.
\end{equation*}
The following statement is a fragment of A. Zygmund's theorem, see \cite{K}, p.22. Let $g(-\pi)=g(\pi)$ and $g(x),\,-\pi\leq x\leq \pi]$ have a bounded variation. If
\begin{equation*}
|g(x)-g(y)|\leq M\,\cdot h(|x-y|).
\end{equation*}
Then, there is a constant $B$ such that the Fourier coefficients of the function $g(x)$ satisfy the inequalities
\begin{equation*}
\sum_{2^j\leq |k|<2^{j+1}}|\hat{g}_{k}|^2\leq
B\,2^{-j}h\left(\frac{\pi}{3}2^{1-j}\right).
\end{equation*}
\begin{definition}
Let us denote by $\Lambda(n)$ the known Mangold's function, equals
\begin{equation*}
\Lambda\left(p^n\right)=\ln p,
\end{equation*}
where $p$ is prime number and $n$ is natural number.
For other natural numbers $m$
\begin{equation*}
\Lambda\left(m\right)=0.
\end{equation*}
\end{definition}
It is known, that for an arbitrary
natural number $n$ the equality
\begin{equation*}
\ln n=\sum_{d|n}\Lambda(d)
\end{equation*}
holds, where the sum is taken over all positive divisors of $n$.
In the following theorem A. Broman, see \cite{B}, p. 851,
gived the characterization of close exeptional sets.
\begin{theorem}
Let $0<\alpha<1$ and
\[
\sum^{\infty}_{n=-\infty}\frac{|c_n|^2}{|n|^{\alpha}+1}<\infty.
\]
Let $F$ be close set and
\[
\lim_{r\to 1-0}\sum_{k=-\infty}^{\infty}r^{|k|}c_ke^{ikx}=0,
\]
for arbitrary $x\in [-\pi,~\pi]\setminus F$.
Then $c_k=0,\,\,\,k\in Z,$ if and only if
\[
C_{1-\alpha}(F)=0.
\]
\end{theorem}
A. Zygmund, see \cite{Z}, proved the following nontrivial result.
\begin{theorem}
Let $\epsilon>0$ and $\epsilon_n >0,\,\,n=1,2,\dots$ be an
arbitrary decreasing sequence, tending to zero. Let
$|c_n|\leq \epsilon_n,\,\,n=1,2,\dots$. There is a subset
$E\subseteq [-\pi,\,\,\pi]$ with measure, i. e. $m(E)>2\pi-\epsilon$,
such that, if for each point $x\in [-\pi,\,\,\pi]\setminus E$ we have
\begin{equation*}
\lim_{n\to \infty}\sum_{k=-n}^nc_ke^{ikx}=0,
\end{equation*}
then $c_k=0,\,\,\,k\in Z,$.
\end{theorem}
The proof of the following theorem one can find in \cite{Vag2}.
\begin{theorem}
Let $0\leq \alpha<1$ and
\begin{equation*}
\int^{\pi}_{-\pi}|f(x)|dx+\int^{\pi}_{-\pi}\int^{\pi}_{-\pi}
\frac{|f(x)-f(y)|}{|x-y|^{1+\alpha}}dx < \infty
\end{equation*}
Let $E\subset [-\pi,\,\pi]$ be a subset such that for almost
all points $x_0\in [-\pi,\,\pi]$ we have
\begin{equation*}
\sum_{n=1}^{\infty}2^{n(1-\alpha)}C_{1-\alpha}(E_n(x_0))=\infty,
\end{equation*}
where
\begin{equation*}
E_n(x_0)=\left\{x\in E;\,\,\frac{1}{2^{n+1}}
\leq |x-y|<\frac{1}{2^{n}}\right\}.
\end{equation*}
If
\begin{equation*}
\lim_{n\to \infty}\sum_{k=-n}^n\hat{f}_ke^{ikx}=0,\,\,x\in E
\end{equation*}
then $f(x)=0,\,\,x\in [-\pi,\,\,\pi]$.
\end{theorem}
In this paper we prove a new result of this type for
other classes of functions.
\section{New uniqueness result}
The following result, in different form,
one can find in the paper, see \cite{Vag1}.
\begin{theorem}
Let $f(-\pi)=f(\pi)$ be differetable function.
Then
\begin{equation*}
\frac{1}{\pi}\sum_{p\in P}\ln p
\left(\sum_{n=1}^{\infty}\left[\frac{1}{p^n}
\sum_{k=1}^{p^n}f\left(\frac{2\pi k}{p^n}\right)
-\hat{f}_0\right]\right)=
\sum_{n\neq 0, n=-\infty}^{\infty}\hat{f}_{n}\ln |n|.
\end{equation*}
\end{theorem}
\begin{proof}
We have
\begin{equation*}
\sum_{n=1}^{\infty}\Lambda(n)\left(\frac{1}{n}
\sum_{k=1}^{n}
\frac{1-|z|^2}{\left|1-z\exp \{-\frac{2\pi i k}{n}\}\right|^2}-1\right)=
\end{equation*}
\begin{equation*}
=\sum_{n=1}^{\infty}\Lambda(n)\frac{1}{n}
\sum_{k=1}^{n}\left(\sum_{j=-\infty}^{\infty}r^{|j|}
\exp\left\{ixj-\frac{2\pi i k j}{n}\right\}-1\right)=
\end{equation*}
\begin{equation*}
=\sum_{j=1}^{\infty}r^j \left( \sum_{j/n}\Lambda(n)\right)\cos(jx)=
\sum_{j=1}^{\infty}r^j \cos(jx)\ln j.
\end{equation*}
where $z=re^{ix},\,\,0\leq r<1$.Multiplying by the function $f(x)$ and after integrating we get
\begin{equation*}
\sum_{n=1}^{\infty}\Lambda(n)\left(\frac{1}{n}\sum_{k=1}^{n}\frac{1}{2\pi}
\int_{-\pi}^{\pi}
\frac{1-r^2}{\left|1-r\exp\{ix-\frac{2\pi i k}{n}\}\right|^2}f(x)dx
-\hat{f}_{0}\right)=
\end{equation*}
\begin{equation*}
=\frac{1}{2}\sum_{j\neq 0,\,j=-\infty}^{\infty}r^{|j|}\hat{f}_{j}\ln |j|.
\end{equation*}
Passing to the limit if
$r\to 1-0$
we get the required equality.
\end{proof}
{\bf Remark.} The getting result we can write in the form
\begin{equation*}
\frac{1}{\pi}\sum_{p\in P}\ln p
\left(\sum_{n=1}^{\infty}\left[\frac{1}{p^n}
\sum_{k=1}^{p^n}\delta\left(x-\frac{2\pi k}{p^n}\right)
-1\right]\right)=\sum_{n\neq 0,\,n=-\infty}^{\infty}
e^{inx}\ln n.
\end{equation*}
This formula is a geralization of Poisson's well known formula:
\begin{equation*}
\sum_{n=-\infty}^{\infty}\delta(x-2\pi n)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}
e^{inx},\,-\infty<x<\infty.
\end{equation*}
\begin{theorem}
Let $0<\alpha<1$ and a nonnegative function $x\leq h(x),\,\,0\leq x <1$ satisfy the condition
\begin{equation*}
\int_0^1 \frac{h(x)}{x^{2-\alpha}}\ln^2\frac{e}{x}dx<\infty.
\end{equation*}
Let for the function $f(x)$ we have
\begin{equation*}
\int_{-\pi}^{\pi}|f(x)|^2dx+\int_{-\pi}^{\pi}\int_{\pi}^{\pi}
\frac{|f(x)-f(y)|^2}{|x-y|^{1+\alpha}}dxdy<\infty.
\end{equation*}
Let there is a subset $E\subset [-\pi,\,\pi]$ such that:
1. $M_{h}(E)>0$,
2. if $x\in E$ then an arbitrary point
\begin{equation*}
x+\frac{2\pi k}{p^n},
\end{equation*}
where
$k\in Z,\,\, n\in N$
and
$p$
is prime number, by $mod (2\pi)$, belongs to the set $E$.
If for eavery $x\in E$ we have
\begin{equation*}
\lim_{r\to 1-0}\sum_{k=-\infty}^{\infty}r^{|k|}\hat{f}_{k}e^{ikx}=0,
\end{equation*}
then the function $f(x)$ is identicaly zero.
\end{theorem}
\begin{proof}
By O. Frostman's theorem, see \cite{Car}, p. 14, there is a
probability measure $d\mu$ such that $supp(d\mu)\subseteq E$,
\begin{equation*}
M_{h}(supp(\mu)) > 0.
\end{equation*}
and for each $0<\delta$ the inequality
\begin{equation*}
\int_{[x,\,x+\delta]}d\mu \leq A h(\delta)
\end{equation*}
hold. Let us assume at the points $0$ and $2\pi$ the function $\mu$ is
continuous and $\mu(0)+1=\mu(2\pi)$. Denote
\begin{equation*}
f(re^{ix})=\sum_{n=-\infty}^{\infty}r^{|n|}\hat{f}_ne^{inx}.
\end{equation*}
Then we have
\begin{equation*}
\frac{1}{\pi}\sum_{p\in P}\ln p
\left(\sum_{n=1}^{\infty}\left[\frac{1}{p^n}
\sum_{k=1}^{p^n}\int_E f\left(r\exp\left\{\frac{2\pi i k}{p^n}+ix\right\}\right)d\mu(x)
-\hat{f}_0\right]\right)=
\end{equation*}
\begin{equation*}
=\sum_{n=2}^{\infty}r^n\left[\hat{f}_n\int_Ee^{inx}d\mu(x)+
\hat{f}_{-n}\int_Ee^{-inx}d\mu(x)\right]\ln n=
\end{equation*}
\begin{equation*}
=2\pi i\sum_{n=2}^{\infty}
\left(\hat{f}_n\hat{g}_{-n}-\hat{f}_{-n}\hat{g}_{-n}\right)r^nn\ln n.
\end{equation*}
where
\begin{equation*}
g(x)=\mu(x)-\frac{x}{2\pi}.
\end{equation*}
Since the function $f(e^{ix})$ vanish on the set $E$ so, we have
\begin{equation*}
\left|\frac{1}{\pi}\sum_{p\in P} \hat{f}_0\ln p \right|\leq
2\sum_{n=2}^{\infty}(|\hat{f}_{-n}||\hat{g}_n|+|\hat{f}_{n}||\hat{g}_{-n}|)n\ln n.
\end{equation*}
Let us note
\begin{equation*}
\sum_{n=2}^{\infty}|\hat{f}_n||\hat{g}_{-n}|n\ln n\leq
\left(\sum_{n=1}^{\infty}|\hat{f}_n|^2n^{\alpha} \right)^{\frac{1}{2}}
\left(\sum_{n=1}^{\infty}|\hat{g}_{-n}|^2n^{2-\alpha}\ln^2 n \right)^{\frac{1}{2}}\leq
\end{equation*}
\begin{equation*}
\leq\left(\sum_{n=1}^{\infty}|\hat{f}_n|^2n^{\alpha} \right)^{\frac{1}{2}}
\left(\sum_{j=1}^{\infty}j^22^{(2-\alpha)j}
\sum_{n=2^j}^{2^{j+1}-1}|\hat{g}_{-n}|^2\right)^{\frac{1}{2}}\leq
\end{equation*}
\begin{equation*}
\leq M \left(\sum_{j=1}^{\infty} j^22^{(1-\alpha)j}h(2^{-j})\right)^{\frac{1}{2}}<\infty.
\end{equation*}
The inequality
\begin{equation*}
\left|\frac{1}{\pi}\sum_{p\in P}\hat{f}_0 \ln p \right|<\infty
\end{equation*}
valid only if
\begin{equation*}
\hat{f}_0=0.
\end{equation*}
Considering the functions $e^{inx}f(x),\,\,\,n\in Z$ we prove that $\hat{f}(n)=0,\,\,n\in Z.$
\end{proof} | 215,404 |
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Some believe these high premiums are a result of physicians having to defend against frivolous lawsuits. The Medical Insurance Exchange of California (MIEC) noted that 86% of claims filed against their insured Hawai‘i physicians are found to be without merit and result in no payment to the claimant. However, due to the costs of legal fees and court costs to defend such lawsuits, premiums still increase.
It is the widely held opinion that a medical tort reform bill would remedy this situation by putting caps on the awards for malpractice claims. Current Hawai‘i law has no limits on non-economic damages, which would include mental anguish or disfigurement. Often, awards for these claims are exorbitant and can be unpredictable. There exists a cap of $375,000 for pain and suffering however, the cap is worded such that it has little meaning in acting as an actual cap. Attorneys are able to bypass the current cap. The proposed legislation would limit non-economic damages to $250,000 in claims against physicians in specialty areas such as emergency medicine, neurology, Ob/Gyn, orthopedic and general surgery. The legislation also includes a proposed cap of $3 million for gross negligence awards.
Other states have passed medical malpractice reform legislation and have experienced dramatic increases in the quality of care. Texas, for example, implemented reform by limiting non-economic damages in medical liability cases to $250,000 in 2003. They’ve since seen increases in physicians coming to their state to practice and decreases in malpractice insurance rates, as well as, decreased lawsuit filings. Nevada, Mississippi, and Oklahoma have also passed legislation for caps on non-economic damages and have seen similar outcomes.
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TITLE: Help thinking combinatorics exercise
QUESTION [0 upvotes]: Be $m$ parallel straight lines, and in each, $n$ points are chosen.
$a)$ How many subsets of $4$ elements can be chosen with this points?
$b)$ How many of those subsets have as a maximum, two points from each straight line?
What I think is that there are $nm$ points, then, using $ mn \choose 4$ is the answer.
The second one I don't really know how to think it...
REPLY [1 votes]: I can't add to your comment because I don't have enough reputation, but I believe that your intuition is wrong for your most recent attempt.
$\binom{nm}2\binom{nm-2}2$ chooses 2 elements from the set and then 2 more elements from the same set of points. you can see that this actually simplifies to $\binom{nm}4$; constraining the solution set should not yield an increase in possible values
Think about how you can alter this equation such that you are limiting lines rather than points
Your most recent attempt:
$\binom{n}2\binom{m(n-1)}2+\binom{n}1\binom{n}1\binom{m(n-2)}2$
undercounts the first portion. Fixing the line you are choosing from, you are correct (careful with m's and n's) that you have
$\binom{n}2\binom{n(m-1)}2$. However, you also have $m$ ways to choose the line which you fix $\Rightarrow m\binom{n}2\binom{n(m-1)}2$ for [2,2] and [2,1,1]
For the number of ways to choose one line you have $\binom{mn}1\binom{n(m-1)}1\binom{n(m-2)}1\binom{n(m-3)}1 = n^4(m(m-1)(m-2)(m-3))$ for [1,1,1,1]
Note that the case in which we select [2, 1, 1] is contained in the first selection. Do you understand why? | 81,150 |
fans of Moon Knight were gifted with a pretty amazing finale that left us with a great set up for future stories between oscar isaac‘s Marc Spector, Steven Grant, and a new character to mix things up. In celebration of the show having a great run on Disney+, new Funko Pop! figures have been announced for the series, including May Calamawy‘s Layla El-Faouly and some of the Egyptian gods we got to meet along the way.
The Layla figure is a bit of a spoiler for the final episode of the limited series, but she’s so cute that it’s almost worth it. (If you haven’t seen the episode, don’t read any further, but the Layla pop is the cutest thing I’ve ever seen). We got to see Layla embody the Scarlet Scarab, and in doing so, she got a fun new look that should excite fans for her future, and the Funko of her new suit, complete with gilded scarab wings, is a great one to add to your collection.
We also got a look at some of the gods in the show, and there’s a new Khonshu Funko (who was voiced by F.Murray Abraham in the show), as well as an Ammit figure now, that we’ve seen the crocodile-headed goddess. Both of the otherworldly gods are Target exclusives, with Khonshu being a jumbo-sized Pop!.
For the Layla figure, she does seem to be specific to San Diego Comic-Con as Funko’s Instagram caption reads: “Funko San Diego Comic-Con 2022 Reveal: Pop! Marvel: Marvel Studios’ Moon Knight – Scarlet Scarab.” Does this mean we have to wait in line to get her from her at SDCC 2022? Yes, and I will, because this is truly a Funko that has me excited in a way I haven’t been for these figures in a while.
But also, the Gods are so cute too, which is an odd thing to say about Khonshu and Ammit). This isn’t the first Khonshu figure, but he’s the Temple of Khonshu Statue version of the character, and the caption for their Instagram announcement reads: “Make the Gods a part of your collection with Marvel Studios’ Moon Knight – Temple of Khonshu Statue and Ammit! Pre-orders will be available today across a variety of retailers!”
So why not add all three to your collection? Because if you have to pick only one, it’s going to be hard because they’re just so amazing. Moon Knight was an incredible addition to the Marvel Cinematic Universe and these Funkos are fit for the Gods (meaning they’re incredible!). Check out the full Instagram announcement below:
Did (Spoiler) Really Die in ‘Doctor Strange in the Multiverse of Madness’?
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Facebook quietly added a new mini basketball game to its mobile Messenger app this week as a treat for fans chatting about March Madness (or basketball in general).
To play the free-throw shooting mini-game, send the basketball emoji to a friend, then click on the ball in chat. In the game, players shoot the ball by flicking their finger in the direction they want to shoot, and try to hit as many consecutive baskets as possible. The game also gets incrementally more challenging as you score more baskets. According to The Next Web, the hoop starts to move after players hit 10 baskets. After 20 baskets, the hoop speeds up. It’s possible that it gets even faster at 30 baskets, but I definitely don’t have enough game to get that far, so I’ll let you confirm that for yourself.
When you exit the game, Facebook posts your high score as a system message in the chat, basically daring your friend(s) to do better. The game also shows the in-chat high score while you’re playing.
Related: Facebook Messenger for Mac desktop app spied in leaked photo
This is not the first easter egg game Facebook has slipped into Messenger. Users discovered in February that the Messenger team had programmed a basic chess program in the app, which you can turn on by typing “@FBChess” into chat.
Meanwhile, more and more companies are integrating with Messenger, expanding its functionality beyond simple chat. Last week Spotify rolled out sharing integration, which allows users to share song links directly through the app. Users are also be able to call a ride from Uber, book a hotel room at a Hyatt, or book a flight on airline KLM soon, based on previously announced partnerships.
Not every feature coming to Messenger is as useful as Uber or as fun as a basketball game. According to an internal document leaked in February, Facebook plans to give companies the ability to send targeted advertising directly to their customers, starting later this year. According to the document, users will be able to send ads to any user that’s previously contacted them through the platform. | 363,732 |
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Bottles of KonaRed cold-brew Kona blend coffee. | 274,052 |
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# UV Sanitizers are now used to decontaminate the "MASK" to reuse it. Link | 187,816 |
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NAIROBI, January 11, 2008 (CISA) -An association of Members of Parliament from the African Great Lakes Region will undertake a fact-finding mission on the post-election crisis in Kenya
The Great Lakes Parliamentary Forum on Peace -Amani Forum- said Thursday that a delegation from Burundi, Democratic Republic of Congo, Rwanda, Tanzania, Uganda and Zambia will be in Kenya from Jan. 13 to 21.
“The purpose of the mission will be to gain firsthand knowledge of the nature and consequences of post-election violence, to ascertain the causes of the post-election violence and to make recommendations on political and legal choices that can contribute to the resolution of election-related conflict in Kenya,” Amani Forum said.
The mission will be led by Rwandese parliamentarian Sheik Harelimana Abdul Karim, secretary general of AMANI Forum Rwanda.
“The parliamentarians will consult with various actors, including Members of Parliament and other leaders in Kenya. They will also interact with various civil society groups, peace organizations, religious, youth and elders.”
The delegation will visit Nairobi, Central Province, Rift Valley, Nyanza and Coast where many Kenyans have fled to escape violence.
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Tuesday, January 15, 2008
KENYA: Great Lakes Legislators Announce Probe Mission on Crisis | 195,112 |
\begin{document}
\maketitle
\begin{abstract}
In this paper we study K\"ahler manifolds that are strongly not relative to any projective K\"ahler manifold, i.e. those K\"ahler manifolds that do not share a K\"ahler submanifold with any projective K\"ahler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full K\"ahler immersion into the infinite dimensional complex projective space.
As application we get that the $1$-parameter families of Bergman--Hartogs and Fock--Bargmann--Hartogs domains are strongly not relative to projective K\"ahler manifolds.
\end{abstract}
\section{Introduction and statement of the main results}
According to \cite{relatives}, two K\"ahler manifolds are called {\em relatives} when they share a common K\"ahler submanifold, i.e. if a complex submanifold of one of them with the induced metric is biholomorphically isometric to a complex submanifold of the other one with the induced metric.
In his seminal paper \cite{calabi}, Calabi determined a criterion which characterizes K\"ahler manifolds admitting a K\"ahler immersion into finite or infinite dimensional complex space forms. The main tool he introduced is the {\em diastasis function} associated to a real analytic K\"ahler manifold, namely a particular K\"ahler potential characterized by being invariant under pull--back through a holomorphic map.
Thanks to this property, the diastasis plays a key role in studying when two K\"ahler manifolds are relatives.
In \cite{umehara} Umehara proved that two finite dimensional complex space forms with holomorphic sectional curvatures of different signs can not be relatives. Although, as firstly pointed out by Bochner in \cite{bochner}, when the ambient space is allowed to be infinite dimensional, the situation is different: any K\"ahler submanifold of the infinite dimensional flat space $\ell^2(\mathds{C})$ admits a K\"ahler immersion into the infinite dimensional complex projective space.
Umehara's work has been generalized in the recent paper by X. Cheng and A. J. Di Scala \cite{fubinirelatives}, where the authors state necessary and sufficient conditions for
complex space forms of finite dimension and different curvatures to not be relative to each others.
In \cite{relatives} A. J. Di Scala and A. Loi prove that a Hermitian symmetric space of noncompact type endowed with its Bergman metric is not relative to a projective K\"ahler manifold, i.e. a K\"ahler manifold which admit a local holomorphic and isometric (from now on {\em K\"ahler}) immersion into the {\em finite} dimensional complex projective space (see also \cite{huang} for the case of Hermitian symmetric spaces of noncompact type and Euclidean spaces), and their result has been generalized in \cite{mossa} to homogeneous bounded domains of $\mathds{C}^n$.
Throughout the paper, we say that a K\"ahler manifold is {\em projectively induced} when it admits a K\"ahler immersion into $\mathds{C}{\rm P}^{N\leq\infty}$. When we also specify that it is {\em infinite projectively induced}, we mean that the K\"ahler immersion is full into $\mathds{C}{\rm P}^{\infty}$.
In this paper we are interested in studying when a K\"ahler manifold $(M,g)$ is {\em strongly not relative} to any projective K\"ahler manifold, that is, when $(M,c\, g)$ is not relative to any projective K\"ahler manifold for any value of the constant $c>0$ multiplying the metric.
Our first result can be viewed as a generalization of the results in \cite{relatives, mossa} and can be stated as follows:
\begin{theor}\label{hbd}
Let $(M,g)$ be a K\"ahler manifold such that $(M,\beta g)$ is infinite projectively induced for any $\beta>\beta_0\geq 0$. If $(M,g)$ and $\mathds{C}{\rm P}^n$ are not relatives for any $n<\infty$, then $(M,g)$ is strongly not relative to any projective K\"ahler manifold.
\end{theor}
Observe that in general there are not reasons for a K\"ahler manifold which is not relative to another K\"ahler manifold to remain so when its metric is rescaled. For example, consider that the complex projective space $(\mathds{C}{\rm P}^2,c\,g_{FS})$ where $g_{FS}$ is the Fubini--Study metric, for $c=\frac23$ is not relative to $(\mathds{C}{\rm P}^2,g_{FS})$, while for positive integer values of $c$ it is (see \cite{fubinirelatives} for a proof).
In order to state our second result, consider a $d$-dimensional K\"ahler manifold $(M,g)$ which admits global coordinates $\{z_1,\dots, z_d\}$ and denote by $M_{j}$ the $1$-dimensional submanifold of $M$ defined by:
$$
M_j=\{ z\in M|\, z_1=\dots=z_{j-1}=z_{j+1}=\dots=z_d=0\}.
$$
When exists, a K\"ahler immersion $f\!:M\rightarrow \mathds{C}{\rm P}^\infty$ is said to be {\em transversally full} when for any $j=1,\dots, d$, the immersion restricted to $M_{j}$ is full into $\mathds{C}{\rm P}^\infty$.
\begin{theor}\label{trfull}
Let $(M, g)$ be a K\"ahler manifold infinite projectively induced through a transversally full map. If for any $\alpha\geq \alpha_0>0$, $(M,\alpha\, g)$ is infinite projectively induced then $(M,g)$ is strongly not relative to any projective K\"ahler manifold.
\end{theor}
As a consequence of Theorem \ref{hbd} and Theorem \ref{trfull} we get that the $1$-parameter families of Bergman--Hartogs and Fock--Bargmann--Hartogs domains, which we describe in Section \ref{bbh}, are strongly not relative to any projective K\"ahler manifold (see corollaries \ref{chrel} and \ref{fbhrel} below).\\
The paper consts of three more sections. In the first one we briefly recall the definition of diastasis function and its properties we need and in the second one we prove Theorem \ref{hbd} and Theorem \ref{trfull}. Finally, in the third and last section we apply our results to Bergman--Hartogs and Fock--Bargmann--Hartogs domains.
The author is very grateful to Prof. Andrea Loi for all the interesting discussions and comments that helped her to improve the contents and the exposition.
\section{Calabi's diastasis function}
Consider a real analytic K\"ahler manifold $(M,g)$ and let $\varphi\!: U\rightarrow \mathds{R}$ be a K\"ahler potential for $g$ defined on a coordinate neighborhood $U$ around a point $p\in M$. Consider the analytic extension $\tilde\varphi\!: W\rightarrow \mathds{R}$, $\tilde\varphi(z,\bar z)=\varphi(z)$, of $\varphi$ on a neighborhood $W$ of the diagonal in $U\times \bar U$. The {\em diastasis function} ${\rm D}(z,w)$ is defined by the formula:
\begin{equation}\label{diast}
{\rm D}(z,w):=\tilde\varphi(z,\bar z)+\tilde\varphi(w,\bar w)-\tilde\varphi(z,\bar w)-\tilde\varphi(w,\bar z).
\end{equation}
Observe that since:
$$
\frac{\partial^2}{\partial z\partial \bar z}{\rm D}(z,w)=\frac{\partial^2}{\partial z\partial \bar z}\tilde \varphi\left(z,\bar z\right)=\frac{\partial^2}{\partial z\partial \bar z}\varphi\left(z\right),
$$
once one of its two entries is fixed, the diastasis is a K\"ahler potential for $g$. We denote by ${\rm D}_0(z)$ the diastasis centered at the origin. The following theorem due to Calabi \cite{calabi}, expresses the diastasis' property which is fundamental for our purpose.
\begin{theor}[E. Calabi]\label{induceddiast}
Let $(M, g)$ and $(S,G)$ be K\"ahler manifolds and assume $G$ to be real analytic. Denote by $\omega$ and $\Omega$ the K\"ahler forms associated to $g$ and $G$ respectively. If there exists a holomorphic map $f\!:(M,g)\rightarrow (S,G)$ such that $f^*\Omega=\omega$, then the metric $g$ is real analytic. Further, denoted by ${\rm D}^M_p\!:U\rightarrow \mathds R$ and ${\rm D}^S_{f(p)}\!:V\rightarrow\mathds R$ the diastasis functions of $(M,g)$ and $(S,G)$ around $p$ and $f(p)$ respectively, we have ${\rm D}_{f(p)}^S\circ f={\rm D}^M_p$ on $f^{-1}(V)\cap U$.
\end{theor}
Consider the complex projective space $\mathds{C}{\rm P}^N_b$ of complex dimension $N\leq \infty$, with the Fubini-Study metric $g_{b}$ of holomorphic bisectional curvature $4b$ for $b>0$. When $b=1$ we denote by $g_{FS}$ and $\omega_{FS}$ the Fubini-Study metric and the Fubini-Study form respectively. Let $[Z_0,\dots,Z_N]$ be homogeneous coordinates,
$p=[1,0,\dots,0]$ and $U_0=\{Z_0\neq 0\}$. Define affine coordinates $z_1,\dots, z_N$ on $U_0$ by $z_j=Z_j/(\sqrt{b}Z_0)$. The diastasis on $U_0$ centered at the origin reads:
\begin{equation}\label{diastcp}
{\rm D}^b_0(z)=\frac{1}{b}\log\left(1+b\sum_{j=1}^N|z_j|^2\right).
\end{equation}
Due to Th. \ref{induceddiast} and the expression of $\mathds{C}{\rm P}^N_b$'s diastasis \eqref{diastcp}, if $f\!:S\rightarrow \mathds{C}{\rm P}^N_b$ is a holomorphic map, $f(z)=[f_0(z),f_1(z),\dots, f_N(z)]$, then the induced diastasis ${\rm D}^S_0$ in a neighborhood of a point $p\in S$ is given by:
$$
{\rm D}^S_0(z)=\frac{1}{b}\log\left(1+b\sum_{j=1}^N|f_j(z)|^2\right).
$$
Further, if the K\"ahler map $f$ is assumed to be {\em full}, i.e. the image $f(S)$ is not contained into any lower dimensional totally geodesic submanifold of $\mathds{C}{\rm P}_b^N$, then $f$ is univocally determined up to rigid motion of $\mathds{C}{\rm P}^N_b$ \cite[pp. 18]{calabi}:
\begin{theor}[Calabi's Rigidity]\label{local rigidityb}
If a neighborhood $V$ of a point $p$ admits a full K\"ahler immersion into $(\mathds{C}{\rm P}^N_b,g_b)$, then $N$ is univocally determined by the metric and the immersion is unique up to rigid motions of $(\mathds{C}{\rm P}^N_b,g_b)$.
\end{theor}
Observe that by Th. \ref{local rigidityb} above, a K\"ahler manifold which is infinite projectively induced does not admit a K\"ahler immersion into any finite dimensional complex projective space.
\section{Proof of Theorems \ref{hbd} and \ref{trfull}}
\begin{proof}[Proof of Theorem \ref{hbd}]
Observe first that due to Th. \ref{induceddiast} it is enough to prove that $(M, c\,g)$ is not relative to $\mathds{C}{\rm P}^n$ for any finite $n$ and any $c>0$. For any $c>0$, we can choose a positive integer $\alpha$ such that $c\alpha>\beta_0$. Denote by $\omega$ the K\"ahler form on $M$ associated to $g$. Let $F\!:M\rightarrow \mathds{C}{\rm P}^\infty$ be a full K\"ahler map such that $F^*\omega_{FS}=c\alpha\, \omega$. Then $\tilde F=F/\sqrt{\alpha}$ is a K\"ahler map of $(M,c\, g)$ into $ \mathds{C}{\rm P}^\infty_\alpha$.
Let $S$ be a common K\"ahler submanifold of $(M,c\,g)$ and $\mathds{C}{\rm P}^n$. Then by Th. \ref{induceddiast} for any $p\in S$ there exist a neighborhood $U$ and two holomorhic maps $f\!:U\rightarrow M$ and $h\!:U\rightarrow \mathds{C}{\rm P}^n$, such that $f^*(c\omega)|_U=( \tilde F\circ f)^*\omega_{FS}|_U=h^*\omega_{FS}|_U$.
Thus, by \eqref{diastcp} one has:
$$
\log\left(1+\sum_{j=1}^n|h_j|^2\right)=\frac1\alpha\log\left(1+\sum_{j=1}^\infty|(F\circ f)_j)|^2\right).
$$
i.e.:
\begin{equation}\label{contradiction}
\alpha\log\left(1+\sum_{j=1}^n|h_j|^2\right)=\log\left(1+\sum_{j=1}^\infty|(F\circ f)_j)|^2\right).
\end{equation}
Since $F\circ f$ is full and $\alpha$ is a positive integer, this last equality and Calabi rigidity Theorem \ref{local rigidityb} imply $n=\infty$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{trfull}]
Due to Th. \ref{hbd} and Th. \ref{induceddiast} we need only to prove that a if a K\"ahler manifold is infinite projectively induced through a transversally full immersion then it is not relative to $\mathds{C}{\rm P}^n$ for any $n$.
Assume that $S$ is a $1$-dimensional K\"ahler submanifold of both $\mathds{C}{\rm P}^n$ and $(M, g)$.
Then around each point $p\in S$ there exist an open neighborhood $U$ and two holomorphic maps $\psi\!:U\rightarrow \mathds{C}{\rm P}^n$ and $\varphi\!:U\rightarrow M$, $\varphi(\xi)=(\varphi_1(\xi),\dots,\varphi_d(\xi))$ where $\xi$ are coordinates on $U$, such that $\psi^*\omega_{FS}|_U=\varphi^*(c\omega)|_U$. Without loss of generality we can assume $\frac{\partial\varphi_1(\xi)}{\partial \xi}(0)\neq 0$.
Let $f\!:M\rightarrow \mathds{C}{\rm P}^\infty$ be a K\"ahler map from $(M, g)$ into $\mathds{C}{\rm P}^\infty$. Since by assumption $f$ is transversally full, $f=[f_0,\dots, f_j,\dots]$ contains for any $m=1,2,3,\dots$, a subsequence $\left\{f_{j_1},\dots, f_{j_m}\right\}$ of functions which restricted to $M_1$ are linearly independent.
The map $f\circ\varphi\!:U\rightarrow \mathds{C}{\rm P}^\infty$ is full, in fact $f|_{M_1}\circ \varphi$ is full since $\varphi_1(\xi)$ is not constant and for any $m=1,2,3,\dots$, $\left\{f_{j_1}(\varphi_1(\xi)),\dots, f_{j_m}(\varphi_1(\xi))\right\}$ is a subsequence of $\{f|_{M_1}\circ \varphi\}$ of linearly independent functions. Conclusion follows by Calabi's rigidity Theorem \ref{local rigidityb}.
\end{proof}
\section{Applications}\label{bbh}
Let $(\Omega, \beta g_B)$, $\beta >0$, denote a bounded domain of $\mathds{C}^d$ endowed with a positive multiple of its Bergman metric $g_B$. Recall that $g_B$ is the K\"ahler metric on $\Omega$ whose associated K\"ahler form $\omega_B$ is given by $\omega_B=\frac{i}{2}\partial\bar\partial\log \K(z,z)$, where $\K(z,z)$ is the reproducing kernel for the Hilbert space:
$$\mathcal{H} =\left\{\varphi\in{\rm hol}(\Omega),\ \int_\Omega |\varphi|^{2}\ \frac{\omega_0^d}{d!}<\infty\right\},$$
where $\omega_0=\frac{i}{2} \sum_{j=1}^d dz_j\wedge d\bar z_j$ is the standard K\"ahler form of $\mathds{C}^d$. It follows by \eqref{diast} that the diastasis function for $g_B$ is given by:
\begin{equation}\label{diastomega}
{\rm D}_0^\Omega(z)=\log\frac{\K(z,z)\K(0,0)}{|\K(z,0)|^2}.
\end{equation}
Observe that the Bergman metric $g_B$ admits a natural K\"ahler immersion into the infinite dimensional complex projective space (cfr. \cite{kodomain}). More precisely, if $\K(z,z)=\sum_{j=0}^\infty |\varphi_j(z)|^2$, the map:
\begin{equation}\label{bergmanimm}
\varphi\!:\Omega\rightarrow\mathds{C}{\rm P}^\infty,\quad \varphi=(\varphi_0,\dots, \varphi_j,\dots),
\end{equation}
is a K\"ahler immersion of $(\Omega, g_B)$ into $\mathds{C}{\rm P}^\infty$, for $\varphi^*g_{FS}=g_B$, as it follows by:
$$
\omega_B=\frac i2\partial\bar\partial\log(\K(z,z))=\frac i2\partial\bar\partial\log\left(\sum_{j=0}^\infty |\varphi_j(z)|^2\right)=\varphi^*\omega_{FS}.
$$
Further, such immersion is full since $\{\varphi_j\}$ is a basis for the Hilbert space $\mathcal H$ and a bounded domain does not admit a K\"ahler immersion into a finite dimensional complex projective space even when the metric is rescaled. Although the existence of a K\"ahler immersion of $(\Omega, \beta g_B)$ into $\mathds{C}{\rm P}^\infty$ is strictly related to the constant $\beta$ which multiplies the metric (see \cite{articwall} for the case when $\Omega$ is symmetric). In \cite{ishi} it is proven that the only homogeneous bounded domain which is projectively induced for all positive values of the constant multiplying the metric is a product of complex hyperbolic spaces.
Although, the property of being projectively induced for a large enough constant is not so unusual and the following holds \cite{loimossaber}:
\begin{theor}[A. Loi, R. Mossa]\label{loimossaimm}
Let $(\Omega,g)$ be a homogeneous bounded domain. Then, there exists $\alpha_0>0$ such that $(\Omega,\alpha g)$ is projectively induced for any $\alpha\geq\alpha_0>0$.
\end{theor}
Notice that it is an open question if the same statement holds dropping the homogeneous assumption.
Regarding the property of being relative to some projective K\"ahler manifold, we recall the following result due to A. J. Di Scala and A. Loi in \cite{relatives}, which plays a key role in the proof of Corollary \ref{chrel}.
\begin{theor}[A. J. Di Scala, A. Loi ]\label{loidiscalahbd}
A bounded domain of $\mathds{C}^n$ endowed with its Bergman metric and a projective K\"ahler manifold are not relatives.
\end{theor}
Observe that due to theorems \ref{loimossaimm} and \ref{loidiscalahbd}, Theorem \ref{hbd} implies that a bounded domain of $\mathds{C}^n$ endowed with its Bergman metric and a projective K\"ahler manifold are {\em strongly} not relatives. Althought, this result has been proven in a more general context by R. Mossa in \cite{mossa}, where he shows that a homogeneous bounded domain and a projective K\"ahler manifold are not relatives.\\
Let us now describe the family of Bergman--Hartogs domains.
For all positive real numbers $\mu$ a {\em Bergman-Hartogs domain} is defined by:
\begin{equation}\label{defm}
M_{\Omega}(\mu)=\left\{(z,w)\in \Omega\times\mathds{C},\ |w|^2<\tilde \K(z, z)^{-\mu}\right\},
\end{equation}
where $\tilde \K(z, z)=\frac{\K(z,z)\K(0,0)}{|\K(z,0)|^2}$ with $\K$ the Bergman kernel of $\Omega$.
Consider on $M_{\Omega}(\mu)$ the metric $g(\mu)$ whose associated K\"ahler form $\omega(\mu)$ can be described by the (globally defined) K\"ahler potential centered at the origin
\begin{equation}\label{diastM}
\Phi(z,w)=-\log(\tilde\K(z, z)^{-\mu}-|w|^2).
\end{equation}
The domain $\Omega$ is called the {\em base} of the Bergman--Hartogs domain
$M_{\Omega}(\mu)$ (one also says that
$M_{\Omega}(\mu)$ is based on $\Omega$). Observe that these domains include and are a natural generalization of Cartan--Hartogs domains which have been studied under several points of view (see e.g. \cite{fengtubalanced,berezinCH} and references therein). To the author knowledge, Bergman-Hartogs domains has been already considered in \cite{hao,hao2,hao3}.
In \cite{articwall} the author of the present paper jointly with A. Loi proved that when the base domain is symmetric $(M_{\Omega}(\mu),c\,g(\mu))$ admits a K\"ahler immersion into the infinite dimensional complex projective space if and only if $(\Omega, (c+m)\mu g_B)$ does for every integer $m\geq0$. As pointed out in \cite{hao}, a totally similar proof holds also when the base is a homogeneous bounded domain. This fact together with Theorem \ref{loimossaimm} proves that a Bergman--Hartogs domain $(M_{\Omega}(\mu),c\,g(\mu))$ is projectively induced for all large enough values of the constant $c$ multiplying the metric. Further, the immersion can be written explicitely as follows (cfr. \cite[Lemma 8]{balancedch}):
\begin{lemma}\label{chimm}
Let $\alpha$ be a positive real number such that the Bergman--Hartogs domain $(M_{\Omega}(\mu),\alpha\, g(\mu))$ is projectively induced. Then, the K\"ahler map $f$ from $(M_{\Omega}(\mu),\alpha\,g(\mu))$ into $\mathds{C}{\rm P}^\infty$, up to unitary transformation of $\mathds{C}{\rm P}^\infty$, is given by:
\begin{equation}\label{immf}
f=\left[ 1, s, h_{\mu\, \alpha},\dots,\sqrt{\frac{(m+ \alpha-1)!}{(\alpha-1)!m!}}h_{\mu(\alpha +m)}w^m,\dots\right],
\end{equation}
where $s=(s_1,\dots, s_m,\dots)$ with
$$s_m=\sqrt{\frac{(m+ \alpha-1)!}{(\alpha-1)!m!}}w^m,$$
and $h_k=(h_k^1,\dots,h_k^j,\dots)$ denotes the sequence of holomorphic maps on $\Omega$ such that the immersion $\tilde h_k=(1,h_k^1,\dots, h_k^j,\dots)$, $\tilde h_k\!:\Omega\rightarrow\mathds{C}{\rm P}^\infty$, satisfies $\tilde h_k^*\omega_{FS}=k \omega_B$, i.e.
\begin{equation}
1+\sum_{j=1}^{\infty}|h_k^j|^2=\tilde \K^{-k}.\nonumber
\end{equation}
\end{lemma}
\begin{proof}
The proof follows essentially that of \cite[Lemma 8]{balancedch} once considered that $\Phi(z,w)=-\log(\tilde \K(z, z)^{-\mu}-|w|^2)$ is the diastasis function for $(M_\Omega(\mu), g(\mu))$ as follows readily applying \eqref{diast}.
\end{proof}
Observe that such map is full, as can be easily seen for example by considering that for any $m=1,2,3,\dots,$ the subsequence $\{s_1,\dots, s_m\}$ is composed by linearly independent functions.
As a consequence of theorems \ref{hbd}, \ref{trfull}, \ref{loidiscalahbd} and Lemma \ref{chimm}, we get the following:
\begin{cor}\label{chrel}
For any $\mu>0$, a Bergman--Hartogs domain $(M_\Omega(\mu), g(\mu))$ is strongly not relative to any projective manifold.
\end{cor}
\begin{proof}
Observe first that due to Th. \ref{induceddiast} it is enough to prove that $(M_\Omega(\mu),\alpha g(\mu))$ is not relative to $\mathds{C}{\rm P}^n$ for any finite $n$. Further, by Th. \ref{hbd} and Th. \ref{loidiscalahbd}, a common submanifold $S$ of both $(M_\Omega(\mu),\alpha g(\mu))$ and $\mathds{C}{\rm P}^n$ is not contained into $(\Omega,\alpha g(\mu)|_\Omega)$, since $\alpha g(\mu)|_\Omega=\frac{\alpha\mu}\gamma g_B$ is a multiple of the Bergman metric on $\Omega$. Thus, due to arguments totally similar to those in the proof of Th. \ref{trfull}, it is enough to check that the K\"ahler immersion $f\!:M_\Omega(\mu)\rightarrow \mathds{C}{\rm P}^\infty$ is transversally full with respect to the $w$ coordinate. Conclusion follows then by \eqref{immf}.
\end{proof}
Finally, we describe what we need about the $1$-parameter family of Fock--Bargmann--Hartogs domains, referring the reader to \cite{fbh} and reference therein for details and further results. For any value of $\mu>0$, a Fock--Bargmann--Hartogs domain $D_{n,m}(\mu)$ is a strongly pseudoconvex, nonhomogeneous unbounded domains in $\mathds{C}^{n+m}$ with smooth real-analytic boundary, given by:
$$
D_{n,m}(\mu):=\{(z,w)\in \mathds{C}^{n+m}: ||w||^2< e^{-\mu||z||^2}\}.
$$
One can define a K\"ahler metric $\omega(\mu;\nu)$, $\nu>-1$ on $D_{n,m}(\mu)$ through the globally defined K\"ahler potential:
$$
\Phi(z,w):=\nu\mu||z||^2-\log(e^{-\mu||z||^2}-||w||^2).
$$
In \cite{fbh}, E. Bi, Z. Feng and Z. Tu prove that when $n=1$ and $\nu=-\frac{1}{m+1}$, the metric $\omega(\mu;\nu)$ is infinite projectively induced whenever it is rescaled by a big enough constant. More precisely they prove the following:
\begin{theor}[E. Bi, Z. Feng, Z. Tu]\label{fbhth}
The metric $\alpha g(\mu;\nu)$ on the Fock--Bargmann--Hartogs domain $D_{n,m}(\mu)$ is balanced if and only if $\alpha>m+n$, $n=1$, $\nu=-\frac1{m+1}$.
\end{theor}
Recall that a balanced K\"ahler metric is a particular projectively induced metric such that the immersion map is defined by a orthonormal basis of a weighted Hilbert space (see e.g. \cite{balancedch}).
In order to apply Th. \ref{trfull} to Fock--Bargmann--Hartogs domains we need the following lemma:
\begin{lemma}\label{focktr}
For any $\mu>0$ and any $\alpha>m+1$, a Fock--Bargmann--Hartogs domain $\left(D_{1,m}(\mu),\alpha\omega(\mu;-\frac1{m+1})\right)$ admits a transversally full K\"ahler immersion into $\mathds{C}{\rm P}^\infty$.
\end{lemma}
\begin{proof}
A K\"ahler immersion exists due to Th. \ref{fbhth}. In order to see that it is transversally full, observe that when $w_1=\dots= w_m=0$, $\alpha\omega(\mu;-\frac1{m+1})|_{M_1}$ is a multiple of the flat metric, and when only one $w_j$ is different from zero $\alpha\omega(\mu;-\frac1{m+1})|_{M_j}$ is a multiple of the hyperbolic metric.
\end{proof}
\begin{cor}\label{fbhrel}
For any $\mu>0$, a Fock--Bargmann--Hartogs domain $\left(D_{1,m}(\mu),\omega(\mu;-\frac1{m+1})\right)$ is strongly not relative to any projective manifold.
\end{cor}
\begin{proof}
If follows directly from Th. \ref{trfull} and Lemma \ref{focktr}.
\end{proof} | 82,586 |
\begin{document}
\begin{frontmatter}
\title{The property of the set of the real numbers generated by a Gelfond-Schneider operator and the countability of all real numbers}
\author[Slavica]{Slavica Vlahovic}
and
\author[Branislav]{Branislav Vlahovic\corauthref{cor}}
\ead{[email protected]}
\corauth[cor]{Corresponding author.}
\address[Slavica]{Gunduliceva 2, 44000 Sisak, Croatia}
\address[Branislav]{North Carolina Central University, Durham, NC 27707, USA}
\begin{abstract}
Considered will be properties of the set of real numbers $\Re$ generated by an operator that has form of an exponential function of Gelfond-Schneider type with rational arguments.
It will be shown that such created set has cardinal number equal to ${\aleph_0}^{\aleph_0}=c$. It will be also shown that the same set is countable. The implication of this contradiction to the countability of the set of real numbers will be discussed.
\end{abstract}
\begin{keyword}
denumerability \sep real numbers \sep countability \sep cardinal numbers
{\it MSC:} 11B05
\end{keyword}
\end{frontmatter}
\section{Introduction}
In 1900 D. Hilbert announced a list of twenty-three outstanding unsolved problems. The seventh problem was settled in 1934 by A. O. Gelfond and an independent proof by Th. Schneider in 1935. They proved that if $\alpha$ and $\beta$ are algebraic numbers with $\alpha \neq 0, \alpha \neq 1$, and if $\beta$ is not a real rational number, then any value of ${\alpha}^{\beta}$ is transcendental [1, 2].
For instance transcendental number is $2^{\sqrt{2}}$.
This can be written in the form of
\begin{equation}
{({2\over1})}^{[({2\over1})^{1\over2}]}={({m_{i1}\over n_{i1}})}^{[({{m_{i2}\over n_{i2}})}^{({m_{i3}\over n_{i3}})}]}
\label{gelement}
\end{equation}
where $m_i, n_i \in N$.
We can ask ourselves a following question which "class" of transcendental numbers can be presented this way? Or can any transcendental number be expressed in the form (\ref{gelement}). Answer is obvious; some transcendental numbers cannot be expressed this way, for instance number $e$ cannot be presented by
\begin{equation}
e={m_1\over n_1}^{[{m_2\over n_2}^{m_3\over n_3}]}
\label{e}
\end{equation}
because after taking logarithm from both sides
\begin{equation}
1={m_2\over n_2}^{m_3 \over n_3}ln{m_1\over n_1},
\label{le}
\end{equation}
and this cannot be, because $ln{m_1\over n_1}$ is always transcendental [3-5] for $m_1,n_1\in N$.
However, one can take more freedom and try to express the number $e$ in the form
\begin{equation}
{[{m_1\over n_1}^{m_2\over n_2}]}^{[{m_3\over n_3}^{m_4\over n_4}]}
\label{ec}
\end{equation}
or even more freedom and try to present the number $e$ in the form
${a_1}^{a_2}$, where both ${a_1}$ and ${a_2}$ can have the form (\ref{ec}).
Obviously, the argument such as shown in (\ref{le}), that number $e$ cannot be presented in such way, cannot be applied anymore since both ${a_1}$ and ${a_2}$ can be now transcendental numbers.
One can go even further (as it is done in [6]) and take much more freedom in generating the numbers or a set of numbers, which elements will be generated through a general element of the sequence that has the form:
\begin{equation}
{a_1}^{{a_2}^{{a_3}^{.^{.^{.^{{a_n}^{.^{.^{.}}}}}}}}}
\label{gsequence}
\end{equation}
where in (\ref{gsequence}) each element $a_i$ of bases and exponents has the
following form:
\begin{equation}
a_i={[({{m_{i1}\over n_{i1}})}^{({m_{i2}\over n_{i2}})}]}^{[({{m_{i3}\over n_{i3}})}^{({m_{i4}\over n_{i4}})}]}
\label{gelementn}
\end{equation}
where $m_{ij},n_{ij}\in N, i=1, 2, 3,...n, j = 1, 2, 3, 4$.
The question remains: which class of the transcendental numbers can be or can not be represented in this way? Can majority of the transcendental numbers be presented or can not be presented in this way? If some transcendental numbers can not be presented, is that set countable or not?
First let us note that the set of numbers generated through the operator (\ref{gsequence}) looks similar to the set of the real numbers. Such set does not have the first and last element, it has subset of all rational numbers, and it is dense everywhere in rational, algebraic and transcendental numbers. However, it may not be equal to the set of the real numbers since it is harder to prove that it is dense in Dedekind's sense, since this would require proof that it does not have holes, i.e. that all numbers can be represented in this way.
To avoid that difficulty, let as assume that some numbers can not be presented in this way and let us focus here only on estimating the number of the elements in such set, i.e. on determining the cardinality of such set of numbers.
\section{The cardinality of the generated set of numbers}
Let us generate the set of the real numbers through relation (\ref{gsequence}) where each base and exponent element $a_i$ has the form (\ref{gelementn}). The mechanism to generate the elements of the set is to write (\ref{gsequence}) for all possible combinations of arguments, with the sum of all bases and exponents equal to 2, 3, 4,... and so on. As the sum increases the number of the exponents will expand. The sample of such generated set with a procedure to avoid double counting of the same numbers is given in [6]. However, let us focus here on our main task which is to estimate the cardinality of such generated set when the process described above continues to infinity.
For each particular number the general element (\ref{gsequence}) that corresponds to that number will have a final number of the exponents $a_i$. However, since the process of generating new numbers continues to infinity there is no an upper limit for the number of the exponents $a_i$ that will be generated by the general element (\ref{gsequence}), which will also go to infinity. Each of the elements of $a_i$ will have $\aleph_0$ possible combinations. This is obvious, since for any arbitrary large value $n \in N$ which one could take for the number of combinations, that value will be exceeded in this described process. The same is true for the number of the exponents. The number of the exponents will also be $\aleph_0$, since again any arbitrary taken number that one could chose for the value of the number of the exponents (does not matter how large is the number) will be exceeded in the described process, which continues to infinity.
Therefore, the above described set will have ${\aleph_0}^{\aleph_0} = c$ elements which makes it equivalent in the cardinal number to the set of the real numbers. A one to one correspondence between such produced set and the set of natural numbers $N$ can be easily obtained by arranging the set elements by the sum of the exponents, as it is done for instance in [6].
We will not here proceed to discuss what could be wrong with the Cantor's famous diagonal proof of countability of the set of real numbers; some of the relevant remarks are done in [6, 7]. Let us note here that that proof could be wrong since it uses the method of induction which, as it is well known [8, 9, 10], can not be applied on the infinite sets. With that method one can only prove that a number created by the diagonal procedure can be different from any $n$ numbers in the set. The method can not prove that that number is different from any number in the assumed denumerable set, which has infinite number of the elements. So, one can move through that set using the diagonal procedure to higher positions numbers $n$ in the sequence, but can not go through all the set elements. At least it cannot be done by using the induction method.
\section{Conclusion}
It is proven that the set generated by the general element (\ref{gsequence}) has cardinal number equal to ${\aleph_0}^{\aleph_0} = c$. The same set is also denumerable, the elements can be ordered by the sum of the bases and exponents in (\ref{gsequence}). Therefore it is proven that the cardinality of the real and natural set of numbers are the same, i.e. that ${\aleph_0}^{\aleph_0}= c= \aleph_0$. | 2,899 |
\begin{document}
\section{Cofibration categories}
\label{cofibration-categories}
\begin{nul*}
Model categories are a framework
for ``large'' homotopy theories that admit all limits and colimits,
like the homotopy theory of all spaces.
However, we are also interested in ``small'' homotopy theories
like the homotopy theory of \emph{finite} spaces
(finite simplicial sets, or finite CW complexes).
Cofibration categories model finitely cocomplete homotopy theories
and functors between them that preserve finite colimits.
In this appendix we collect facts about cofibration categories
which will be used in the main text.
Our main aim is \cref{cc-unit-weq} which gives conditions
under which the unit of a relative adjunction between cofibration categories
is a weak equivalence.
\end{nul*}
\begin{definition}
\label{cofibration-category}
A \emph{cofibration category} is a category $C$
equipped with two classes of maps
called \emph{cofibrations} and \emph{weak equivalences}
satisfying the axioms below.
\begin{enumerate}
\item
$C$ has an initial object $\emptyset$,
and $\id_\emptyset$ is a cofibration.
\end{enumerate}
We call an object $A$ \emph{cofibrant}
if the unique map $\emptyset \to A$ is a cofibration.
\begin{enumerate}[resume]
\item
The weak equivalences of $C$ satisfy the two-out-of-three axiom,
and isomorphisms are weak equivalences.
\item
\label{pushout-cofibration}
The cofibrations of $C$ are closed under composition.
The pushout of a cofibration $j : A \to B$ with cofibrant domain
along a map $f : A \to A'$ to another cofibrant object exists,
and any such pushout is again a cofibration,
which is also a weak equivalence if $j$ is one.
\item
Every map with cofibrant domain
factors as a cofibration followed by a weak equivalence.
\item
$C$ satisfies the following conditions
which are equivalent given the preceding axioms
\cite[Theorem~7.2.7]{RB}:
\begin{itemize}
\item
A retract of a weak equivalence is a weak equivalence.
\item
The weak equivalences satisfy the two-out-of-six axiom:
if $gf$ and $hg$ are weak equivalences then so is $g$
(hence also $f$ and $h$).
\end{itemize}
\end{enumerate}
Condition (\textit{iii}) implies that
any isomorphism between cofibrant objects is a cofibration
(because it may be written as a pushout of $\id_\emptyset$).
We call a map of a cofibration category an \emph{acyclic cofibration}
if it is both a cofibration and a weak equivalence.
If $C$ and $D$ are cofibration categories,
then a functor $F : C \to D$ is \emph{exact}
if $F$ preserves cofibrations, acyclic cofibrations,
the initial object, and pushouts of cofibrations.
\end{definition}
\begin{nul}
There are a number of different notions of cofibration category
in the literature.
The one above is from \cite{Cis10}
(where it is called a ``categorie d\'erivable \`a droite'')
and \cite{RB}
(where it is called a ``precofibration category''),
except that we have built the saturation condition (\textit{v})
into the definition of a cofibration category:
this is the condition that ensures that
the weak equivalences of $C$ are precisely
the maps that become isomorphisms in the homotopy category.
Cofibration categories that appear in practice are saturated\rlap{.}
\footnote{
If you know an example of a non-saturated cofibration category,
please describe it at \cite{NonSatCofCat}!
}
We are mainly interested in cofibration categories
in which every object is cofibrant,
but allowing the possibility of non-cofibrant objects
will simplify some of the statements below.
\end{nul}
\begin{example}
\label{cc-of-model}
A model category $M$ has an ``underlying'' cofibration category
with the same underlying category, cofibrations, and weak equivalences.
In fact, this data determines the model category $M$
and so we may think of a model category as
a (very) special kind of cofibration category.
If $M$ and $N$ are model categories viewed as cofibration categories
then a functor $F : M \to N$ is a left Quillen functor
if and only if it is both exact and a left adjoint.
\end{example}
\begin{example}
If $C$ is a cofibration category
we write $C^\cof$ for the full subcategory of $C$ on the cofibrant objects.
Then $C^\cof$ is again a cofibration category:
the main point is that it inherits pushouts of cofibrations from $C$.
Note that normally $C^\cof$ will not have all pushouts, even if $C$ does.
\end{example}
\begin{example}
Write $\sSetfin$ for the category of finite simplicial sets.
We equip $\sSetfin$ with the structure of a cofibration category
by restricting the corresponding parts of
the model category structure on $\sSet$.
That is, a morphism $f : A \to B$ of $\sSetfin$ is:
\begin{itemize}
\item a cofibration if it is a monomorphism;
\item a weak equivalence if it is a weak equivalence in $\sSet$.
\end{itemize}
The axioms for a cofibration category follow from
the fact that $\sSet$ is a model category
with the exception of the last one, the factorization axiom.
For any map $f : A \to B$,
the mapping cylinder factorization
$A \to A \times \Delta^1 \amalg_A B \xrightarrow{\sim} B$
expresses $A$ as a cofibration followed by a weak equivalence,
with the intermediate object $A \times \Delta^1 \amalg_A B$
again a finite simplicial set.
\end{example}
\begin{example}
Write $\Topfin$ for the full subcategory of $\Top$
on those objects which are homeomorphic to finite cell complexes.
We equip $\Topfin$ with the structure of a cofibration category
by declaring a morphism $f : A \to B$ to be
\begin{itemize}
\item
a cofibration if it is (homeomorphic to)
the inclusion of a finite relative cell complex;
\item
a weak equivalence if it is a homotopy equivalence.
\end{itemize}
By Whitehead's theorem, the homotopy equivalences in $\Topfin$
are also those maps which are weak equivalences in $\Top$.
Again, the cofibration category axioms for $\Topfin$
are easily checked using the fact that $\Top$ is a model category
(under the standard Serre--Quillen model category structure)
together with the usual mapping cylinder construction.
This example can also be constructed by applying
the following result of Baues.
\end{example}
\begin{definition}
\label{cc-fibrant}
An object $X$ of a cofibration category is \emph{fibrant}
if for any acyclic cofibration $f : A \to B$ with cofibrant domain,
the induced map $\Hom(B, X) \xrightarrow{- \circ f} \Hom(A, X)$
is surjective.
\end{definition}
\begin{example}
Every object of $\Topfin$ is fibrant.
On the other hand, $\sSetfin$ has very few fibrant objects.
In particular, if $X$ is a fibrant finite simplicial set
then each homotopy group of $X$ is finite,
so $S^1 = \Delta^1/\partial \Delta^1$
cannot have a fibrant approximation in $\sSetfin$.
Suppose $M$ is a model category, $X$ is an object of $M$,
and either of the following two hypotheses holds.
\begin{enumerate}
\item
$M$ admits a class of generating acyclic cofibrations
with cofibrant domains.
\item
$X$ is cofibrant.
\end{enumerate}
Then $X$ is fibrant in $M$ as a cofibration category
(\cref{cc-of-model})
if and only if $X$ is fibrant in the usual sense
as an object of the model category $M$.
(Under the second hypothesis, to check that $X \to *$
has the right lifting property
with respect to an acyclic cofibration $i : A \to B$,
we first push forward $i$ to $X$,
making its domain cofibrant.)
\end{example}
\begin{proposition}[{\cite[Theorem I.3.3]{Baues}}]
\label{cc-of-icat}
Let $C$ be an \emph{$I$-category}, that is,
a category equipped with a class of cofibrations
and a ``cylinder'' functor $I : C \to C$
and natural transformations
$p : I \to \id_C$, $i_0 : \id_C \to I$ and $i_1 : \id_C \to I$
satisfying the following axioms:
\begin{enumerate}[label=(I\arabic*)]
\item
$pi_0 = pi_1 = \id$.
\item
$C$ has an initial object and pushouts of cofibrations,
and these colimits are preserved by the functor $I : C \to C$.
Pushouts of cofibrations are again cofibrations.
\item
The cofibrations are closed under composition
and for every object $X$, the map $\emptyset \to X$ is a cofibration.
Moreover, cofibrations have the \emph{homotopy extension property}:
for each cofibration $f : A \to B$, object $X$,
and $\varepsilon \in \{0, 1\}$,
any horizontal morphism
\[
\begin{tikzcd}
IA \amalg_{A,\varepsilon} B \ar[r] \ar[d] & X \\
IB \ar[ru, dashed]
\end{tikzcd}
\]
admits a lift as shown by the dotted arrow.
Here the vertical map is built from $i_\varepsilon : \id_C \to I$.
\item
For each cofibration $f : A \to B$,
the induced ``relative cylinder'' map $B \amalg_{A,0} IA \amalg_{A,1} B \to IB$
is a cofibration.
\item
For each object $X$, there exists an ``interchange'' map $T : IIX \to IIX$
such that $Ti_\varepsilon = Ii_\varepsilon$
and $T(Ii_\varepsilon) = i_\varepsilon$
for $\varepsilon \in \{0, 1\}$.
\end{enumerate}
Then
$C$ admits the structure of a cofibration category
with the same cofibrations,
and homotopy equivalences (defined using the cylinder functor $I$)
as the weak equivalences.
Moreover, every object of $C$ is both cofibrant and fibrant.
\end{proposition}
\begin{definition}
\label{ho-cat}
The \emph{homotopy category} $\Ho C$ of a category $C$ with weak equivalences $\mathcal{W}$
(such as a cofibration category or model category)
is the category $C[\mathcal{W}^{-1}]$ obtained by formally inverting
the weak equivalences of $C$.
\end{definition}
\begin{nul}
We recall the following basic facts about homotopy categories.
Functors that preserve weak equivalences
induce functors between homotopy categories.
For a cofibration category $C$,
the inclusion $C^\cof \to C$ induces
an equivalence of categories $\Ho C^\cof \to \Ho C$ \cite[Theorem 6.1.6]{RB}.
In particular, any object~$X$ has a cofibrant approximation $\tilde X$,
which can be obtained by factoring $\emptyset \to X$
into a cofibration $\emptyset \to \tilde X$
followed by a weak equivalence $\tilde X \to X$.
An exact functor $F : C \to D$ between cofibration categories
does not necessarily preserve all weak equivalences,
but it does preserve cofibrant objects
and weak equivalences between cofibrant objects
by ``Ken Brown's lemma''.
We define the \emph{(left) derived functor} of $F$
to be the functor $\lder F$ (defined up to unique natural isomorphism)
which fits in the square below.
\[
\begin{tikzcd}
\Ho C^\cof \ar[r, "\Ho F^\cof"] \ar[d, "\sim"'] &
\Ho D^\cof \ar[d, "\sim"] \\
\Ho C \ar[r, "\lder F"'] & \Ho D
\end{tikzcd}
\]
When every object of $C$ is cofibrant,
we simply take $\lder F = \Ho F$.
In general, the morphisms of $\Ho C$ are equivalence classes of
zigzags in which the backwards maps are weak equivalences.
However, in certain cases
we can give a more explicit description of the Hom-sets of $\Ho C$.
\end{nul}
\begin{definition}[{\cite[\S6.3]{RB}}]
Two maps $f_0 : A \to B$ and $f_1 : A \to B$
between cofibrant objects $A$ and $B$ of a cofibration category
are \emph{strictly left homotopic}
if there exists
a factorization $A \amalg A \to I \to A$ of the fold map $A \amalg A \to A$
into a cofibration followed by a weak equivalence
together with a map $H : I \to B$
such that the diagram below commutes.
\[
\begin{tikzcd}
A \amalg A \ar[r, "{\langle f_0, f_1 \rangle}"] \ar[d] & B \\
I \ar[ru, "H"']
\end{tikzcd}
\]
The maps $f_0$ and $f_1$ are \emph{left homotopic}
if they become strictly left homotopic
after postcomposition with some acyclic cofibration $B \to B'$.
\end{definition}
\begin{nul}
Left homotopic maps between cofibrant objects
become equal in the homotopy category.
In fact, the converse also holds \cite[Theorem 6.3.1]{RB}.
However, not every element of $\Hom_{\Ho C}(A, B)$
arises from a morphism in $C$ from $A$ to $B$.
When $A$ and $B$ are cofibrant,
each element of $\Hom_{\Ho C}(A, B)$ may be represented by a zigzag
of the form $[A \to B' \xleftarrow{\sim} B]$.
in which the map $B \xrightarrow{\sim} B'$
may be chosen to be an acyclic cofibration
\cite[Theorem 6.4.5]{RB}.
Unlike in a model category,
we generally cannot choose a single object~$\widehat B$
so that every element of $\Hom_{\Ho C}(A, B)$
is represented by a morphism from $A$ to $\widehat B$.
(Consider the fact that the Hom-sets of $\sSetfin$ are finite,
while $\Hom_{\Ho \sSetfin}(\partial \Delta^2, \partial \Delta^2) =
\Hom_{\Ho \sSet}(\partial \Delta^2, \partial \Delta^2) = \ZZ$, as we will see later.)
However, if $B$ happens to be fibrant,
then $\Hom_{\Ho C}(A, B)$ is given by the familiar description.
\end{nul}
\begin{proposition}
\label{cc-ho-cof-fib}
Suppose $A$ is cofibrant and $X$ is both cofibrant and fibrant.
Then strict left homotopy is an equivalence relation $\sim^{s\ell}$
on $\Hom_C(A, X)$
and the induced map $\Hom_C(A, X)/{\sim^{s\ell}} \to \Hom_{\Ho C}(A, X)$
is a bijection.
Furthermore, strict left homotopy may be detected
using any fixed cylinder object for $A$.
\end{proposition}
\begin{proof}
Under these hypotheses, any acyclic cofibration $X \to X'$ has a retraction,
so left homotopy and strict left homotopy of maps into $X$ agree.
Then the claim follows from
\cite[Proposition 7.3.2, Theorem 6.3.1 and Lemma 6.3.2]{RB}.
\end{proof}
\begin{example}
\label{ho-topfin-ff}
For any objects $A$ and $B$ of $\Topfin$,
the natural map $\Hom_{\Ho \Topfin}(A, B) \to \Hom_{\Ho \Top}(A, B)$
is a bijection:
both sides can be computed as homotopy classes of maps from $A$ to $B$
in the classical sense.
\end{example}
\begin{proposition}
\label{ho-ssetfin-ff}
The inclusion $\sSetfin \to \sSet$
induces a fully faithful functor $\Ho \sSetfin \to \Ho \sSet$.
\end{proposition}
\begin{proof}
This follows from \cite[Theorem 4.6]{BaSc2} and \cite[Proposition 6.1]{BaSc1}.
We give a quick sketch of the proof specialied to $\sSet$.
Let $B$ be a finite simplicial set.
While $B$ generally does not have a finite fibrant replacement,
we can find a sequence $B = B_0 \to B_1 \to B_2 \to \cdots$
of finite simplicial sets and acyclic cofibrations
such that $\widehat B = \colim_{i \in \NN} B_i$ is fibrant in $\sSet$
(for example by taking $B_i = \Ex^i B$).
Then if $A$ is a finite simplicial set,
every element $h$ of $\Hom_{\Ho \sSet}(A, B)$
is represented by an element of $\Hom_\sSet(A, \widehat B)$
and therefore by an element $g$ of $\Hom_\sSet(A, B_i)$ for some $i \in \NN$.
We send $h$ to the element $[A \xrightarrow{g} B_i \xleftarrow{\sim} B]$
of $\Hom_{\Ho \sSetfin}(A, B)$.
Then one calculates that
this map does not depend on the choice of $i$
or on the choice of representative $g$,
and provides an inverse to the natural map
$\Hom_{\Ho \sSetfin}(A, B) \to \Hom_{\Ho \sSet}(A, B)$.
\end{proof}
\begin{example}
\label{ssetfin-equiv-topfin}
Geometric realization $|{-}| : \sSet \to \Top$ is a left Quillen functor
and it restricts to an exact functor $|{-}| : \sSetfin \to \Topfin$.
We claim that the latter functor
induces an equivalence of homotopy categories.
Indeed, we have a commutative square
\[
\begin{tikzcd}
\Ho \sSetfin \ar[r, "\Ho |{-}|"] \ar[d] & \Ho \Topfin \ar[d] \\
\Ho \sSet \ar[r, "\Ho |{-}|"] & \Ho \Top
\end{tikzcd}
\]
in which the vertical morphisms are fully faithful functors
by \cref{ho-ssetfin-ff,ho-topfin-ff}
and the bottom functor is an equivalence.
Thus, it suffices to show that the top functor is essentially surjective.
This is a classical fact:
every finite cell complex
is homotopy equivalent to the geometric realization of a finite simplicial set.
\end{example}
\begin{nul*}
Next we describe a very useful criterion for detecting when
an exact functor induces an equivalence of homotopy categories,
due to Cisinski.
This criterion is the reason we included the saturation axiom
in the definition of a cofibration category.
\end{nul*}
\begin{definition}[{\cite[3.6]{Cis10}}]
An exact functor $F : C \to D$ between cofibration categories
satisfies the \emph{approximation property}
if:
\begin{enumerate}[label=(AP\arabic*)]
\item
When restricted to cofibrant objects, $F$ reflects weak equivalences.
\item
Suppose $A$ is a cofibrant object of $C$
and $f : FA \to B$ is a morphism to a cofibrant object of $D$.
Then there exists a morphism $u : A \to A'$ with $A'$ cofibrant
and a diagram
\[
\begin{tikzcd}
FA \ar[r, "f"] \ar[d, "Fu"'] & B \ar[d, "\sim"] \\
FA' \ar[r, "\sim"] & B'
\end{tikzcd}
\]
with $B'$ cofibrant and the marked morphisms weak equivalences in $D$.
\end{enumerate}
\end{definition}
\begin{proposition}[{\cite[Th\'eor\`eme 3.19]{Cis10}}]
Let $F : C \to D$ be an exact functor between cofibration categories.
Then $\lder F : \Ho C \to \Ho D$ is an equivalence of categories
if and only if $F$ satisfies the approximation property.
\end{proposition}
\begin{nul}
Let $C$ be a cofibration category and $Z$ a cofibrant object.
Then the slice category $C_{Z/}$ has an induced cofibration category structure
in which a morphism is a cofibration or a weak equivalence
if and only if its underlying map is one in $C$.
Note that normally $C_{Z/}$ will not have all objects cofibrant,
even if $C$ does.
When $C$ has a cofibrant terminal object $*$, we write $C_*$ for $C_{*/}$,
the category of pointed objects of $C$.
\end{nul}
\begin{proposition}
\label{cc-slice-equiv}
Let $F : C \to D$ be an exact functor between cofibration categories
which induces an equivalence of homotopy categories,
and let $Z$ be a cofibrant object of $C$.
Then the induced functor $F_{Z/} : C_{Z/} \to D_{FZ/}$ is also exact
and induces an equivalence of homotopy categories.
\end{proposition}
\begin{proof}
The verification that $F_{Z/}$ is exact is routine.
To show that it induces an equivalence of homotopy categories,
we check the approximation property.
Here, for an object $X$ or map $f$ of $C_{Z/}$,
we write $\underline{X}$ or $\underline{f}$
for the underlying object or map of $C$.
\begin{enumerate}[label=(AP\arabic*)]
\item
Suppose $f : A \to B$ is a map between cofibrant objects of $C_{Z/}$
such that $F_{Z/}(f)$ is a weak equivalence.
Then $\underline{A}$ and $\underline{B}$ are cofibrant objects of $C$
(because $Z$ is cofibrant,
and the maps $Z \to \underline{A}$ and $Z \to \underline{B}$
are cofibrations)
and $\underline{F_{Z/}(f)} = F\underline{f}$ is a weak equivalence.
Hence by (AP1) for $F$,
$\underline{f}$ and so also $f$ are weak equivalences.
\item
Similarly, suppose given cofibrant objects $A$ of $C_{Z/}$ and $B$ of $D_{FZ/}$
and a morphism $f : F_{Z/}A \to B$.
Unwinding the definitions, this corresponds to a commutative diagram
\[
\begin{tikzcd}[column sep=tiny]
& FZ \ar[ld, "Fa"'] \ar[rd, "b"] \\
F \underline{A} \ar[rr, "\underline{f}"] && \underline{B}
\end{tikzcd}
\]
in $D$, in which the maps $a$ and $b$ are cofibrations.
In particular, $\underline{A}$ and $\underline{B}$ are cofibrant
because $Z$ is.
So we may apply (AP2) for $F$ and enlarge the diagram to
\[
\begin{tikzcd}[column sep=tiny]
& FZ \ar[ld, "Fa"'] \ar[rd, "b"] \\
F \underline{A} \ar[rr, "\underline{f}"] \ar[d, "F\underline{u}"'] &&
\underline{B} \ar[d, "\sim"] \\
F \underline{A'} \ar[rr, "\sim"] && \underline{B'}
\end{tikzcd}
\]
for cofibrant objects $\underline{A'}$ and $\underline{B'}$
and a morphism $\underline{u} : \underline{A} \to \underline{A'}$ in $C$.
Using the factorization axiom as described in \cite[Scholie 3.7]{Cis10},
we may moreover assume that $\underline{u}$
and the map $\underline{B} \to \underline{B'}$
are cofibrations.
We may regard $\underline{u}$ and $\underline{B'}$
as underlying a morphism $u : A \to A'$ of $C_{Z/}$
and an object $B'$ of $D_{FZ/}$ respectively,
with the structural maps determined by the above diagram.
This translates back into a diagram
\[
\begin{tikzcd}
F_{Z/}A \ar[r, "f"] \ar[d, "F_{Z/}u"'] & B \ar[d, "\sim"] \\
F_{Z/}A' \ar[r, "\sim"] & B'
\end{tikzcd}
\]
of the form required for (AP2).
The objects $A'$ and $B'$ are cofibrant because
their structural maps $Z \to \underline{A'}$ and $FZ \to \underline{B'}$
are cofibrations.
\qedhere
\end{enumerate}
\end{proof}
\begin{remark}
The natural functor $\Ho(C_{Z/}) \to \Ho(C)_{Z/}$ is generally not faithful.
For example, the morphisms of $\Ho(\Top_{*/})$
are basepoint-preserving homotopy classes of basepoint-preserving maps,
while the morphisms of $\Ho(\Top)_{*/}$
are \emph{free} homotopy classes of maps that preserve the basepoint component.
\end{remark}
\begin{proposition}
\label{slice-fibrant}
Suppose $C$ is a cofibration category,
$Z$ is a cofibrant object of $C$
and $X$ is an object of $C_{Z/}$
whose underlying object $\underline{X}$ is fibrant in $C$.
Then $X$ is fibrant in $C_{Z/}$.
In particular, if every object of $C$ is fibrant,
the same is true of $C_{Z/}$.
\end{proposition}
\begin{proof}
We have to check that
for every cofibration $f : A \to B$ in $C_{Z/}$ with cofibrant domain,
any extension problem of the form below admits a lift.
\[
\begin{tikzcd}
A \ar[r] \ar[d, "f"'] & X \\
B \ar[ru, dashed]
\end{tikzcd}
\]
Unwinding the definitions, this is equivalent to finding a lift in the diagram
\[
\begin{tikzcd}[column sep=tiny]
& Z \ar[ld] \ar[rd] \\
\underline{A} \ar[rr] \ar[d, "\underline{f}"'] & & \underline{X} \\
\underline{B} \ar[rru, dashed]
\end{tikzcd}
\]
and $\underline{f}$ is also a cofibration with cofibrant domain,
so such a lift exists because $\underline{X}$ is fibrant.
\end{proof}
\begin{proposition}
\label{ho-ssetfinptd-ff}
The inclusion $\sSetfin_* \to \sSet_*$ induces
a fully faithful functor $\Ho \sSetfin_* \to \Ho \sSet_*$.
\end{proposition}
\begin{proof}
Consider the following diagram,
where all the functors are induced by either inclusions
or geometric realization.
\[
\begin{tikzcd}
& \Ho {(\Topfin_*)^\cof} \ar[d, "(1)"] \\
\Ho \sSetfin_* \ar[r, "(2)"] \ar[d, "(3)"'] &
\Ho \Topfin_* \ar[d, "(4)"] \\
\Ho \sSet_* \ar[r, "(5)"] & \Ho \Top_*
\end{tikzcd}
\]
We want to prove that the functor labeled (3) is fully faithful.
The horizontal functors are equivalences,
using \cref{cc-slice-equiv,ssetfin-equiv-topfin}.
Functor (1) is also an equivalence,
and then functor (4) is fully faithful
by the same argument as \cref{ho-topfin-ff}
(every object of $\Topfin_*$ is fibrant).
\end{proof}
\begin{nul}
Finally, we prove a specialized result
regarding the unit of a relative adjunction
between cofibration categories.
We apply it in the main text to promote
an exact functor inducing an equivalence on homotopy categories
to a Quillen equivalence between model categories.
We begin with a technical lemma.
It corresponds to the following fact about model categories:
if $F : C \rightleftarrows D : G$ is a Quillen adjunction
and $\lder F : \Ho C \to \Ho D$ is fully faithful,
then the ``derived unit'' of the adjunction is a weak equivalence.
In the setting of cofibration categories
we cannot generally expect an exact functor to have a right adjoint,
so we assume instead that $F$ is part of
an adjunction \emph{relative} to a functor $J : C \to C'$ (\cref{relative-adjoint}).
The conclusion is therefore weaker:
we can only deduce that the unit map
``looks like a weak equivalence''
to objects of $C'$ in the image of the functor $J$.
(Even for $C = \sSetfin$, $C' = \sSet$, $J$ the inclusion,
this is not by itself sufficient to conclude that
the unit map is actually a weak equivalence.)
\end{nul}
\begin{lemma}
Let $C$, $C'$ and $D$ be cofibration categories
and $F : C \to D$, $U : D \to C'$ an adjunction relative to $J : C \to C'$
(so there is an isomorphism
$\Hom_D(FA, X) \xrightarrow{\sim} \Hom_{C'}(JA, UX)$
natural in $A \in C$ and $X \in D$)
satisfying the following conditions:
\begin{enumerate}
\item
All objects of $C$ and of $C'$ are cofibrant.
\item
All objects of $D$ are fibrant,
as are their images under $U$.
\item
$J$ and $F$ are exact.
\item
$\lder J : \Ho C \to \Ho C'$ and $\lder F : \Ho C \to \Ho D$
are fully faithful.
\end{enumerate}
Then for any objects $A$ and $K$ of $C$,
the unit map $\eta_K : JK \to UFK$
(corresponding under the relative adjunction to $\id_{FK} : FK \to FK$)
induces an isomorphism
$(\eta_K)_* : \Hom_{\Ho C'}(JA, JK) \to \Hom_{\Ho C'}(JA, UFK)$.
\end{lemma}
\begin{proof}
First, let $A$ be an object of $C$ and $X$ a cofibrant object of $D$.
In the diagram
\[
\begin{tikzcd}
\Hom_D(FA, X) \ar[r, "\sim"] \ar[d] & \Hom_{C'}(JA, UX) \ar[d] \\
\Hom_{\Ho D}(FA, X) \ar[r, dashed, "\sim"] & \Hom_{\Ho C'}(JA, UX)
\end{tikzcd}
\]
each vertical map is the quotient by the strict left homotopy relation,
because all objects involved are cofibrant and $X$ and $UX$ are fibrant.
Moreover, choosing a cylinder object
$A \amalg A \to I \to A$ for $A$ in $C$,
we may detect these strict left homotopy relations
on the images of this cylinder object under $F$ and $J$.
Then an adjunction argument shows that
these two strict left homotopy relations correspond,
so the adjunction isomorphism descends to the homotopy category
as shown by the dotted arrow.
Now let $A$ and $K$ be objects of $C$.
We next claim the square
\[
\begin{tikzcd}
\Hom_{\Ho C}(A, K) \ar[r] \ar[d] & \Hom_{\Ho D}(FA, FK) \ar[d, dashed] \\
\Hom_{\Ho C'}(JA, JK) \ar[r, "(\eta_K)_*"] & \Hom_{\Ho C'}(JA, UFK)
\end{tikzcd}
\tag{$*$}
\]
commutes,
where the dotted arrow was constructed above, taking $X = FK$.
We can represent a general element of $\Hom_{\Ho C}(A, K)$
in the form $[A \to K' \xleftarrow{\sim} K]$
with the map $K \to K'$ an acyclic cofibration.
Its image in $\Hom_{\Ho C'}(JA, UFK)$ under the two solid arrows
is represented by the zigzag
$[JA \to JK' \xleftarrow{\sim} JK \xrightarrow{\eta_K} UFK]$,
while its image under the top map is $[FA \to FK' \xleftarrow{\sim} FK]$.
Since $FK \xrightarrow{\sim} FK'$ is an acyclic cofibration
and $FK$ is fibrant,
we can choose a retraction $r : FK' \to FK$,
so that the composition $FK \to FK' \xrightarrow{r} FK$ is the identity.
Then the homotopy class $[FA \to FK' \xleftarrow{\sim} FK]$
is represented by a map $FA \to FK' \xrightarrow{r} FK$ in $D$
which (by naturality) the adjunction isomorphism takes to the composition
$JA \to UFK' \xrightarrow{Ur} UFK$ in $C'$.
Hence, in the diagram
\[
\begin{tikzcd}
JA \ar[rd] & JK \ar[r, "\eta_K"] \ar[d, "\sim"'] & UFK \ar[d] \\
& JK' \ar[r, "\eta_{K'}"] & UFK' \ar[u, bend right, shift right, pos=.4, "Ur"']
\end{tikzcd}
\]
we need to check that the zigzags
\[
\begin{tikzcd}
JA \ar[rd] & JK \ar[r, "\eta_K"] \ar[d, "\sim"'] & UFK \\
& JK'
\end{tikzcd}
\]
and
\[
\begin{tikzcd}
JA \ar[rd] & & UFK \\
& JK' \ar[r, "\eta_{K'}"] & UFK' \ar[u, bend right, shift right, pos=.4, "Ur"']
\end{tikzcd}
\]
become equal in $\Ho C'$.
Indeed, we compute
\begin{align*}
& [JA \to JK' \xrightarrow{\eta_{K'}} UFK' \xrightarrow{Ur} UFK] \\
{}={} & [JA \to JK' \xleftarrow{\sim} JK \xrightarrow{\sim} JK'
\xrightarrow{\eta_{K'}} UFK' \xrightarrow{Ur} UFK] \\
{}={} & [JA \to JK' \xleftarrow{\sim} JK \xrightarrow{\eta_K} UFK
\to UFK' \xrightarrow{Ur} UFK] \\
{}={} & [JA \to JK' \xleftarrow{\sim} JK \xrightarrow{\eta_K} UFK].
\end{align*}
Finally, we conclude that $(\eta_K)_*$ is an isomorphism
because the other three maps in ($*$) are
(the top and left by the hypotheses on $J$ and $F$,
and the dotted one by the first part of the proof).
\end{proof}
\begin{nul}
Now we specialize to the following situation.
Let $J : \sSetfin \to \sSet$ be the inclusion,
let $D$ be a cofibration category
and suppose $|{-}| : \sSetfin \to D$ and $\Sing : D \to \sSet$
is an adjunction relative to $J$
satisfying the following conditions:
\begin{enumerate}
\item
Every object of $D$ is fibrant.
\item
$|{-}|$ is exact and induces an equivalence of homotopy categories.
\end{enumerate}
Then $\Sing X$ is also fibrant for every object $X$ of $D$,
because $\sSet$ has generating acyclic cofibrations
which are the images under $J$ of maps between cofibrant objects of $\sSetfin$.
Furthermore, we saw earlier that $J$ induces
a fully faithful functor $\lder J = \Ho J : \Ho \sSetfin \to \Ho \sSet$.
Therefore, the conditions for the lemma are satisfied.
Next, we pass to the pointed setting.
The inclusion $J_{*/} : \sSetfin_* \to \sSet_*$ is also an exact functor
which induces a fully faithful functor on homotopy categories
(\cref{ho-ssetfinptd-ff}),
and $|{-}|$ induces an exact functor
$|{-}|_{*/} : \sSetfin_* \to D_{|*|/}$
which again induces an equivalence of homotopy categories
(\cref{cc-slice-equiv}).
Define $\Sing' : D_{|{*}|/} \to \sSet_*$ by sending
$(|{*}| \xrightarrow{x} X)$ to
the pointed simplicial set given by the composition
$({*} \xrightarrow{\eta_*} \Sing |{*}| \xrightarrow{\Sing x} \Sing X)$.
One easily checks that $|{-}|_{*/}$ and $\Sing'$
form an adjunction relative to $J_{*/}$
whose unit map on an object $K$ of $\sSetfin_*$
has underlying map given by
the unit map of the original relative adjunction
on the underlying object of $K$.
Finally, by \cref{slice-fibrant}
every object of $D_{|{*}|/}$ is fibrant
and their images under $\Sing'$ are also fibrant.
Thus, the lemma also applies to this pointed situation.
\end{nul}
\begin{proposition}
\label{cc-unit-weq}
In the above situation,
the unit map $\eta_K : K \to \Sing |K|$ is a weak equivalence
for every object $K$ of $\sSetfin$.
\end{proposition}
\begin{proof}
We check that $\eta_K$ induces an isomorphism
on $\pi_0$ and on $\pi_n$ for every $n \ge 1$ and choice of basepoint of $K$.
Here, by $\pi_n$ of a general pointed simplicial set
we really mean the simplicial homotopy groups
of a functorial fibrant replacement in $\sSet_*$,
so that $\pi_n(-)$ is naturally isomorphic to
$\Hom_{\Ho \sSet_*}((\Delta^n/\partial \Delta^n, *), -)$.
The map on $\pi_0$ induced by $\eta_K$ is isomorphic to
\[
(\eta_K)_* : \Hom_{\Ho \sSet}(*, K) \to \Hom_{\Ho \sSet}(*, \Sing |K|)
\]
and so it is an isomorphism by the lemma.
For $\pi_n$, let $k \in K_0$ be a vertex of~$K$.
We must show that the induced map
$\pi_n(K, k) \to \pi_n(\Sing |K|, \eta_K(k))$
is an isomorphism.
Let $K_* = (K, k) \in \sSetfin_*$.
Then the pointed map $(K, k) \to (\Sing |K|, \eta_K(k))$
is the unit map $K_* \to \Sing' |K_*|_{*/}$ of
the relative adjunction between $|{-}|_{*/}$ and $\Sing'$.
By the lemma again,
$\Hom_{\Ho \sSet_*}((\Delta^n/\partial \Delta^n, *), -)$
takes this map to an isomorphism.
\end{proof}
\end{document} | 65,088 |
Liberia Football Association chairman Musa Bility has announced plans to stand for the presidency of Fifa, saying it is Africa’s time to lead world football.
The 48-year-old is the second person to declare his candidacy after former Brazil international Zico.
Bility said Africa is the largest voting bloc in Fifa and we must take the lead to bring football together. .
znbc
He’d be more corrupt than Blatter | 376,231 |
\begin{document}
\maketitle
\begin{abstract}
In this short note, we show that for any $\epsilon >0$ and $k<n^{0.5-\epsilon}$ the choice number of the Kneser graph $KG_{n,k}$ is $\Theta (n\log n)$.\end{abstract}
\section{Introduction}
Let $[n] = \{1,\ldots, n\}$ be the standard $n$-element set and, for a set $X$, let ${X\choose k}$ stand for all $k$-element subsets of $X$ ($k$-sets for short). For integers $n\ge 2k>0$, the Kneser graph $KG_{n,k}$ is a graph with the vertex set ${[n]\choose k}$ and the edge set that consists of all pairs of disjoint $k$-sets.
Recall that, for a graph $G$, the quantity $\chi(G)$ is the smallest number $s$ of colors such that there is a vertex coloring in $s$ colors in which the endpoints of each edge receive different colors (a {\it proper coloring}). The choice number $ch(G)$ is the smallest $s$ such that for any assignment of lists $S(v)$ to each vertex $v\in G$ with lists of size $s$ there is a proper coloring of the vertices of $G$ that uses the color from $S(v)$ for each $v$.
It is by now one of the classical results in combinatorics that $\chi(KG_{n,k}) = n-2k+2$. It was shown by Lov\'asz \cite{L}, answering the question by Kneser. In fact, the upper bound is easy: for each $1\le i\le n-2k+1,$ color in $i$ the sets with minimum element $i$. The remaining $k$-sets are subsets of $\{n-2k+2,\ldots, n\}$ and do not induce an edge in $KG_{n,k}.$ Thus, they can be colored in one color.
Lov\'asz' paper initiated the use of topological method in combinatorics. By now, different proofs \cite{Bar, Mat} of Lov\'asz' result are known; however, all of them rely on topological arguments. Rather quickly after B\'ar\'any's proof, Schrijver \cite{Sch} constructed vertex-critical subgraphs of Kneser graphs, that is, subgraphs with the same chromatic number by such that deletion of any vertex decreases the chromatic number. These subgraphs are induced subgraphs of $KG_{n,k}$ on the vertices that correspond to $k$-sets that do not contain two cyclically consecutive elements. Very recently, Kaiser and Stehl\'ik \cite{KS} constructed edge-critical subgraphs of Schrijver graphs. There, deletion of any edge decreases chromatic number. After a series of papers \cite{Kup1, AH, Kup2}, the second author and Kiselev \cite{KK} essentially determined the chromatic number of a random subgraph of $KG_{n,k},$ obtained by including each edge with probability $1/2.$ There are extensions of Lov\'asz' and Schrijver's results to hypergraphs \cite{AFL}.
In this note, we study the choice number of Kneser graphs.
\begin{thm}\label{thmub}
For any $n\ge 2k>0$ we have $ch(KG_{n,k}) \leq n \ln\frac{n}{k} + n.$
\end{thm}
It should be clear that $ch(KG_{n,k})\ge \chi(KG_{n,k})=n-2k+2$, and thus Theorem~\ref{thmub} implies that $ch(KG_{n,k}) = \theta (n)$ for $k = \theta (n)$.
The following result improves on this lower bound for relatively small $k$.
\begin{thm}\label{thm4}
Fix $s\ge 3.$ If $n$ is sufficiently large and $k\le n^{\frac 12-\frac 1s}$ then $ch(KG_{n,k})\ge \frac 1{2s^2} n\log n.$
\end{thm}
These two results leave open the following intriguing question.
\begin{prb}
Determine the asymptotics of $ch(KG_{n,k})$ for $\sqrt n\ll k\ll n$.
\end{prb}
\section{Proofs}
\begin{proof}[Proof of Theorem~\ref{thmub}]
We shall employ probabilistic method. Let $S(v)$ be the list of $m$ colors assigned to vertex $v$. Denote by $L$ the set of all colors assigned to at least $1$ vertex. In what follows, we slightly abuse notation and identify vertices of $KG_{n,k}$ and the corresponding $k$-sets.
Let us take a random map $f: L\to [n]$. Such a correspondence induces a coloring of $KG_{n,k}$ as follows.
We color a $k$-set $v$ in color $\gamma$, $\gamma\in S(v)$, if there is an element $i\in [n]$ such that, first, $i\in v$ and, second, $f(\gamma) = i.$ (If there are several such $\gamma$, then we use any of them.) It should be clear that such coloring, if it exists, is proper and respects the color lists. Indeed, if two sets share the same color $\gamma$, then they must share a common element $f(\gamma).$
The last part of the proof is to show that the probability that such a coloring exists is non-zero.
The probability that a vertex $v$ is not colored is at most $(1-\frac{k}{n})^{m}<e^{-m\frac{k}{n}}.$ Then, the probability that there is at least one vertex that is not colored is at most ${ n \choose k} \cdot e^{-m\frac{k}{n}}$. If this probability is strictly smaller than $1$, then with positive probability the opposite holds, and we have a proper coloring. Let us bound this last expression.
$${n \choose k} \cdot e^{-m\frac{k}{n}} < \left(\frac{ne}{k}\right)^k \cdot e^{-m\frac{k}{n}} = e^{k(\ln(\frac{ne}{k})-m\frac{k}{n})} < 1, $$ if $m > n \ln\frac{n}{k} + n.$
\end{proof}
\subsection{Proof of Theorem~\ref{thm4}}
We shall need the following structural result concerning intersecting families. Recall that a family $\ff\subset 2^{[n]}$ is {\it intersecting } if $F_1\cap F_2\ne \emptyset $ for any $F_1,F_2\in \ff.$
For a family $\ff$ and a set $S$, let us use the following notation: $$\ff(S):=\{F\in \ff: S\subset F\}.$$
\begin{thm}\label{thmintstruc} Consider an intersecting family $\ff \subset {[n]\choose k}$ and fix an integer $s\ge 2.$ Then either there exists a family $\G\subset {[n]\choose s}$, $|\G|\le k^s,$ such that $$\ff\subset \bigcup_{S\in \G}\ff(S),$$
or a set $I\subset [n]$ of size at most $s-1$ such that $$\ff\subset \bigcup_{i\in I}\ff(i).$$
\end{thm}
\begin{proof}
Recall that a {\it cover} of the family $\ff$ is a set $C$ with $F\cap C\ne \emptyset$ for any $F\in \ff.$ The {\it covering number} $\tau(\ff)$ is the minimum size of a cover of $\ff$.
If $\tau(\ff)\le s-1$ then simply take $I$ to be the smallest cover of $\ff.$
If $\tau(\ff)\ge s$ then we shall construct $\G$ using the following simple inductive argument for an intersecting $\ff,$ which is inspired by the paper of Erd\H os and Lov\' asz \cite{EL} (cf. also \cite{KK2}).
Take an arbitrary set $F\in \ff.$ Define $\G_1\subset {[n]\choose 1}$ as follows: $\G_1:=\{\{i\}: i\in F\}.$ Then $\ff\subset \cup_{i\in F}\ff(i)$ since $\ff$ is intersecting.
For each $1\le \ell<s$, let us show how to construct $\G_{\ell+1}$ from $\G_\ell$. Assume that we have a family $\G_\ell\subset {[n]\choose \ell}$ of at most $k^\ell$ sets such that $\ff\subset \cup_{G\in \G_{\ell}}\ff(G)$. For each set $G\in \G_\ell$, consider a set $F_G$ that is disjoint with $G$. Such a set must exist since $|G|<\tau(\ff)$. Put $G_{\ell+1}:=\{G\cup \{i\}: G\in \G_{\ell}, i\in F_G\}$. It should be clear that $|\G_{\ell+1}|\le k^{\ell+1}$ and that $$\ff\subset \bigcup_{G'\in \G_{\ell+1}}\ff(G').$$
Finally, we put $\G:=\G_s$.
\end{proof}
We shall also need the Tur\'an-type result for hypergraphs due to Katona, Nemeth and Simonovits \cite{KNS}. Recall that, for a hypergraph $H$, its {\it independence number} $\alpha(H)$ is the size of the largest subset of vertices that does not contain an edge of $H$.
\begin{thm}[\cite{KNS}]\label{thmtur}
If $\mathcal H\subset {X\choose s}$ is a hypergraph with $\alpha(H) = q$ then $$|\mathcal H|\ge {|X|\choose s}/{q\choose s}\ge \Big(\frac {|X|}{q}\Big)^s.$$
\end{thm}
\begin{proof}[Proof of Theorem~\ref{thm4}] We again employ the probabilistic method. For shorthand, put $u: = \frac 1{s^2}n\log n$.\footnote{We tacitly assume that $u$ is an integer.} Take a set of $u$ colors and correspond to each vertex of $KG_{n,k}$ a random subset of colors of size $u/2.$
Take an arbitrary independent set in $KG_{n,k}$ (i.e., an intersecting family in ${[n]\choose k}$) and fix an integer $s\ge 2.$ Using Theorem~\ref{thmintstruc}, we get that each such independent set is contained in one of the families from $\mathcal C$, where $\mathcal C$ consists of all families $\mathcal K$ of the following two forms:\footnote{Note that $\mathcal C$ is a family of families.}
\begin{itemize}
\item[type i:] all $k$-sets that intersect a fixed set $I(\mathcal K),$ $|I| =s-1$;
\item[type ii:] all $k$-sets that contain one of the $s$-sets from a family $\G(\mathcal K)\subset {[n]\choose s},$ $|\G(\mathcal K)|=k^s.$
\end{itemize}
Note that $|\mathcal C|\le n^{s-1}+{n\choose s}^{k^s}$. We say that a coloring $X =X_1\sqcup\ldots \sqcup X_m$ of a set $X$ {\it lies} in a cover $X =X_1'\cup\ldots \cup X_m'$ of the same set if $X_i\subset X_i'$ for each $i.$ Using this terminology, any possible partition of ${[n]\choose k}$ into $u$ independent sets lies in one of the
\begin{equation}\label{eqnumbercolor}|\mathcal C|^u = \Big({n\choose s}^{k^s}+n^{s-1}\Big)^{u}\le n^{sk^su}\end{equation}
covers, formed by a $u$-tuple of families from $\mathcal C$.
For a given cover $\mathcal K = \mathcal K_1\cup\ldots\cup \mathcal K_u$ from $\mathcal C,$ let us bound from above the probability of the event $A_{\mathcal K}$ that $KG_{n,k}$ can be colored in one of the colorings of ${[n]\choose k}$ that lie in $\mathcal K$ and that respects the lists assigned to the $k$-sets. For each $\ell\in [n]$, define $d_\ell = |\{j\in [u]: \mathcal K_j\text{ is of type i and }j\in I(\mathcal K_j)\}$. Then, clearly, $\sum_{i\in[n]} d_i \le (s-1)u$, and, using Markov inequality, we get that for any $0<\epsilon<1$ there is a set $W\subset [n]$ of $\epsilon n$ elements such that \begin{equation}\label{eqlowdeg}d_i\le (1-\epsilon)^{-1}(s-1)u/n\text{ for any }i\in W.\end{equation}
Next, define an $s$-graph
$$\mathcal H:=\big\{H\subset W: \mathcal K_j \text{ is of type ii and }H\in \G(\mathcal K_j)\big\}.$$
Note that $|\mathcal H|\le k^s u.$ Applying Theorem~\ref{thmtur}, we get that
$$\alpha(\mathcal H)\ge \frac{|W|}{|\mathcal H|^{1/s}}\ge \frac{\epsilon n}{ku^{1/s}}=:t.$$
Thus, there exists a set $Y\subset W$ of size $t$ that is independent in $\mathcal H.$
Take any $X\in {Y\choose k}$ and denote by $B_{X,\mathcal K}$ the event that $X$ cannot be colored via a coloring that lies in $\mathcal K$. As a vertex of $KG_{n,k},$ $X$ cannot be colored in color $j$ if $\mathcal K_j$ is of type ii, because $X$ is independent in $\mathcal H$. Next, it may be colored in $j$ with $\mathcal K_j$ of type i only if $j$ belongs to the randomly chosen subset of $u/2$ colors that was assigned to $X$. Recall that \eqref{eqlowdeg} holds. Denote $z:=(1-\epsilon)^{-1}k(s-1)u/n$ and note that $z=o(\sqrt n)$. Then the probability $p$ that $X$ cannot be colored using this coloring is at least
$$\frac{{u-\sum_{i\in X}d_i\choose u/2}}{{u\choose u/2}} \ge \frac{{u-z\choose u/2}}{{u\choose u/2}}=\prod_{j=0}^{z-1}\frac{\frac n2-i}{n-i}\ge 2^{-z}\Big(1-\frac{z}{n-z}\Big)^{z}=(1+o(1))2^{-z}.$$
Expanding the expressions for $z$ and $u$, we get that $$p = (1+o(1))2^{-z} = (1+o(1))n^{-(1-\epsilon)^{-1}k\frac {s-1}{s^2}}\ge n^{-\frac k{s+1}},$$ provided $\epsilon>0$ is sufficiently small.
Thus, $A_{\mathcal K} \subset \bigcap_{X\in {Y\choose k}}\bar B_{X,\mathcal K}$ and so
$$\Pr[A_{\mathcal K}] \le (1-p)^{{|Y|\choose k}}\le e^{-p{|Y|\choose k}}=e^{-p{t\choose k}}\le e^{-p(t/k)^k}.$$
If $3\le k\le 100$ then we put $s=3$. There are sufficiently large constants $C',C'', C$ depending on $k,\epsilon$ such that we have we have $$p(t/k)^k\ge C'n^{-\frac k{s+1}}t^k \ge C''n^{-\frac k{4}}\Big(\frac n {u^{1/3}}\Big)^k = Cn^{-\frac k{4}}\frac{n^{\frac {2k}{3}}}{\log^{k/3}n}
\ge n^{0.4k}\ge n^{1.2},$$ and so $${\rm P}[A_{\mathcal K}] \le e^{-n^{1.2}}.$$
If $100< k\le n^{\frac 12-\frac 1s}$ then we have $$p(t/k)^k\ge n^{-\frac k{s+1}}\Big(\frac{\epsilon n}{k^2u^{1/s}}\Big)^k = n^{-\frac k{s+1}}n^{\frac ks} \Big(\frac{\epsilon s^2}{\log n}\Big)^k\ge n^{k/s^3},$$
provided $n$ is sufficiently large, and so $${\rm P}[A_{\mathcal K}] \le e^{-n^{k/s^3}}.$$
We are now ready to conclude the proof. Denote by $U$ the event that there exists a proper coloring of $KG_{n,k}$ that respects the lists chosen as described in the beginning of the proof.
\begin{align*}
{\rm P}[U]\le \sum_{\mathcal K\in \mathcal C}{\rm P}[A_{\mathcal K}] \le n^{sk^su}\cdot \begin{cases}e^{-n^{1.2}},\ 3\le k\le 100\\ e^{-n^{k/s^3}}, k> 100\end{cases} <1,
\end{align*}
provided $n$ is sufficiently large. Thus, there exists a choice of lists of size $u/2$ each so that no proper coloring with such list is possible. This completes the proof of the theorem.\end{proof} | 119,712 |
TITLE: Helicity for massless particles
QUESTION [2 upvotes]: The little group for massless particles is $ISO(2)$, with the following Lie algebra: $$[A,B]=0, \; [J^3,A]=iB, \; [J^3,B]=-iA,$$ where $A,B$ generate translations and $J^3$ generates rotations. To obtain the massless projective representations of the Poincaré group, we first look at projective representations of the little group, thus $ISO(2)$, then use the method of induced representations, i.e. Mackey theory. How can one conclude, that the phases appearing in the projective representation are actually just signs, and not arbitrary phases at this point? Or conversely: how can one conclude that massless particles have integer or half integer helicity? For massive particles, integer or half integer spin is clear, because we look at projective representations of the little group $SO(3)$, which are in 1-1 correspondence with ordinary representations of its universal cover $SU(2)$, which are labelled by integer or half integer numbers. However, apparently the statement is true nevertheless even for massless particles, but I don't see why.
I've read something about this in Weinberg Vol. 1 p.89-90, but I am kind of confused by the statement made above equation (2.7.43), to which I was trying to find an alternative explanation.
How does Weinberg conclude that the loop given can be contracted to a point in the massless case?
REPLY [3 votes]: Abstractly the group $ISO(2)\supset SO(2)$ has fundamental group $\mathbb{Z}$.
However, the main point is that the massless little group is imbedded inside the restricted Lorentz group $SO^+(3,1)$, where we know for physical reasons that even winding number is contractable, and hence should be identified in the little group, corresponding to helicity $\in\mathbb{Z}/2$ irreps$^1$.
References:
S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; p. 89-90.
Wikipedia.
N. Straumann, arXiv:0809.4942; p. 7.
--
$^1$ Concerning the "continuous spin" representation, see Refs. 2 & 3. | 22,436 |
\begin{document}
\date{\today}
\title{Restricted extension of sparse partial edge colorings of complete graphs}
\author{
{\sl Carl Johan Casselgren}\footnote{Department of Mathematics, Link\"oping University, SE-581 83 Link\"oping, Sweden. \newline
{\it E-mail address:} [email protected] \, Casselgren was supported by a grant from the Swedish
Research Council (2017-05077)}
\and {\sl Lan Anh Pham }\footnote{Department of Mathematics,
Ume\aa\enskip University,
SE-901 87 Ume\aa, Sweden.
{\it E-mail address:} [email protected]
}
}
\maketitle
\bigskip
\noindent
{\bf Abstract.}
Given a partial edge coloring of a complete graph $K_n$ and lists of allowed
colors for the non-colored edges of $K_n$, can we extend the partial edge coloring
to a proper edge coloring of $K_n$ using only colors from the lists?
We prove that this question has a positive answer in the case when both the
partial edge coloring and the color lists satisfy certain sparsity conditions.
\bigskip
\noindent
\small{\emph{Keywords: Complete graph, Edge coloring, Precoloring extension,
List coloring}}
\section{Introduction}
An {\em edge precoloring} (or {\em partial edge coloring})
of a graph $G$ is a proper edge coloring of some
subset $E' \subseteq E(G)$; {\em a $t$-edge precoloring}
is such a coloring with $t$ colors.
A $t$-edge precoloring $\varphi$ is
{\em extendable} if there is a proper $t$-edge coloring $f$
such that $f(e) = \varphi(e)$ for any edge $e$ that is colored
under $\varphi$; $f$ is called an {\em extension}
of $\varphi$.
In general, the problem of extending a given edge precoloring
is an $\mathcal{NP}$-complete problem,
already for $3$-regular bipartite graphs \cite{Fiala}.
Questions on extending a partial edge coloring seems to have
been first considered for balanced complete bipartite graphs, and
these questions are usually referred to as the problem of
completing partial Latin squares.
In this form the problem appeared already in 1960, when Evans
\cite{Evans} stated his now classic conjecture that for
every positive integer $n$, if
$n-1$ edges in $K_{n,n}$ have been (properly) colored,
then the partial coloring can be extended to
a proper $n$-edge-coloring of $K_{n,n}$.
This conjecture was
solved for large
$n$ by H\"aggkvist \cite{Haggkvist78} and later for all $n$ by Smetaniuk
\cite{Smetaniuk},
and independently by Andersen and Hilton \cite{AndersenHilton}.
Similar questions have also been investigated for complete graphs
\cite{AndersenHilton2}.
Moreover, quite recently, Casselgren et al. \cite{CasselgrenMarkstromPham}
proved an analogue of this result
for hypercubes.
Generalizing this problem, Daykin and H\"aggkvist \cite{DH}
proved several results on
extending partial edge colorings of $K_{n,n}$, and they also conjectured that much denser partial colorings can be extended, as long as the
colored edges are spread out in a specific sense:
a partial $n$-edge coloring of $K_{n,n}$ is
{\em $\epsilon$-dense} if there are at most $\epsilon n$ colored edges
from $\{1,\dots,n\}$
at any vertex and each color in $\{1,\dots,n\}$
is used at most $\epsilon n$ times
in the partial coloring.
Daykin and H\"aggkvist \cite{DH} conjectured
that for every positive integer $n$,
every $\frac{1}{4}$-dense partial proper $n$-edge
coloring can be extended
to a proper $n$-edge coloring of $K_{n,n}$,
and proved a version of the conjecture for $\epsilon=o(1)$
(as $n \to \infty$) and $n$ divisible by 16.
Bartlett \cite{Bartlett} proved that this conjecture holds for
a fixed positive $\epsilon$, and recently a
different proof which improves the value of $\epsilon$ was
given in \cite{BKLOT}.
For general edge colorings of balanced complete
bipartite graphs, Dinitz conjectured,
and Galvin proved \cite{Galvin}, that if each edge of $K_{n,n}$
is given a list of $n$ colors, then there is a proper edge
coloring of $K_{n,n}$ with support in the lists.
Indeed, Galvin's result was a complete solution
of the well-known List Coloring Conjecture
for the case of bipartite multigraphs
(see e.g. \cite{haggkvist1992some} for more background on this conjecture and its relation
to the Dinitz' conjecture).
Motivated by the Dinitz' problem, H\"aggkvist \cite{Ha89}
introduced the notion of {\em $\beta n$-arrays},
which correspond to list assignments $L$ of forbidden colors for
$E(K_{n,n})$,
such that each edge $e$ of $K_{n,n}$ is assigned a list $L(e)$
of at most $\beta n$ forbidden colors from $\{1,\dots,n\}$,
and at every vertex $v$ each color is forbidden on
at most $\beta n$ edges incident to $v$;
we call
such a list assignment for $K_{n,n}$ {\em $\beta$-sparse}.
If $L$ is a list assignment for $E(K_{n,n})$, then
a proper $n$-edge coloring $\varphi$ of $K_{n,n}$ {\em avoids}
the list assignment $L$ if $\varphi(e) \notin L(e)$ for
every edge $e$ of $K_{n,n}$;
if such a coloring exists, then $L$ is {\em avoidable}.
H\"aggkvist conjectured that there exists a fixed
$\beta>0$, in fact also that $\beta=\frac{1}{3}$, such that
for every positive integer $n$,
every $\beta$-sparse list assignment for $K_{n,n}$ is avoidable.
That such a $\beta>0$ exists was proved
for
even $n$
by Andr\'en in her PhD thesis \cite{Andren2010latin},
and later for all
$n$ in \cite{AndrenCasselgrenOhman}.
Combining the notions of extending a sparse precoloring and
avoiding a sparse list assignment,
Andr\'en et al. \cite{AndrenCasselgrenOhman} proved that
there are constants $\alpha> 0$ and $\beta> 0$,
such that for every positive integer $n$,
every $\alpha$-dense partial edge
coloring of $K_{n,n}$ can be extended
to a proper $n$-edge-coloring avoiding any given $\beta$-sparse
list assignment $L$, provided that no edge $e$ is precolored by a color
that appears in $L(e)$. Quite recently, Casselgren et al.
obtained analogous results for hypercubes \cite{CasselgrenMarkstromPham2}.
In this paper, we consider the corresponding problem for complete graphs.
As mentioned above, edge precoloring extension problems have previously been
considered for complete graphs. The type of questions that we are
interested in here, however, seems to be a hitherto quite unexplored line
of research.
To state our main result, we need to introduce some terminology.
If $n$ is even, let $m=2n-1$ and if $n$ is odd, let $m=2n$.
A partial edge coloring of $K_{2n}$ or $K_{2n-1}$ with colors $1,\dots,m$ is
{\em $\alpha$-dense} if
\begin{itemize}
\item[(i)] every color appears on at most $\alpha m$ edges;
\item[(ii)] for every vertex $v$, at most $\alpha m$ edges incident to $v$ are precolored.
\end{itemize}
A list assignment $L$ for a complete graph $K_{2n}$ or $K_{2n-1}$
on the color set
$\{1,\dots,m\}$ is {\em $\beta$-sparse} if
\begin{itemize}
\item[(i)] $|L(e)| \leq \beta m$ for every edge;
\item[(ii)] for every vertex $v$, every color appears in the lists of at most $\beta m$ edges incident to $v$.
\end{itemize}
Our main result is the following.
\begin{theorem}
\label{mainth}
There are constants $\alpha> 0$ and $\beta> 0$ such that
for every positive integer $n$, if $\varphi$
is an $\alpha$-dense $m$-edge precoloring of $K_{2n}$, $L$ is a
$\beta$-sparse list assignment for $K_{2n}$ from the
color set $\{1,\dots,m\}$,
and $\varphi(e) \notin L(e)$ for every edge $e \in E(K_{2n})$,
then there is a proper
$m$-edge coloring of $K_{2n}$ which agrees with $\varphi$
on any precolored edge and which avoids $L$.
\end{theorem}
Since any complete graph $K_{2n-1}$ of odd order is a subgraph of $K_{2n}$,
Theorem \ref{mainth} holds for any $\alpha$-dense $m$-edge precoloring and
any $\beta$-sparse list assignment for $K_{2n-1}$.
Hence, we have the following. For an integer $p$, we
define
$t= 4r-1$ if $p=4r$ or $p=4r-1$, and $t= 4r-2$ if $p=4r-2$ or $p =4r-3$.
\begin{corollary}
\label{cor:main}
There are constants $\alpha> 0$ and $\beta> 0$ such that
for every positive integer $p$, if $\varphi$ is an $\alpha$-dense $t$-edge
precoloring of $K_p$, $L$ is a $\beta$-sparse list assignment
from the color set $\{1,\dots,t\}$, and $\varphi(e) \notin L(e)$ for every
edge $e \in E(K_{p})$, then there is a proper $t$-edge coloring of $K_p$
which agrees with $\varphi$ on any precolored edge and which avoids $L$.
\end{corollary}
Note that the number of colors in the preceding corollary is best possible
if $p \in \{4r, 4r-1\}$, while it remains an open question whether
$t=4r-2$ can be replaced by $t=4r-3$ if $p \in \{4r-2, 4r-3\}$.
The rest of the paper is devoted to the proof of Theorem
\ref{mainth}. The proof of this theorem is simlar to the proof of the
main result of \cite{AndrenCasselgrenMarkstrom}. However, we shall need to generalize
several tools from \cite{AndrenCasselgrenMarkstrom, Bartlett}
to the setting of complete graphs.
\section{Terminology, notation and proof outline}
Let $\{p_1,p_2,\dots,p_n, q_1,q_2,\dots,q_n\}$ be
$2n$ vertices of the complete graph $K_{2n}$
and $G_1$ ($G_2$) be the induced subgraph
of $K_{2n}$ on the vertex set
$\{p_1,p_2,\dots,p_n\}$ ($\{q_1,q_2,\dots,q_n\}$) and
$K_{n,n}$ be the complete bipartite graph
with the partite sets $\{p_1,p_2,\dots,p_n\}$ and $\{q_1,q_2,\dots,q_n\}$.
It is obvious that the complete graph $K_{2n}$
is the union of the complete bipartite
graph $K_{n,n}$ and the two copies $G_1$ and $G_2$ of the complete graph $K_n$.
For any proper edge coloring $h$ of $K_{2n}$, we denote by $h_K$
the restriction of this coloring to $K_{n,n}$; similarly,
$h_{G_1}$ and $h_{G_2}$ are the restrictions of $h$
to the subgraphs $G_1$ and $G_2$, respectively.
For a vertex $u \in V(K_{2n})$, we denote by $E_u$ the set of
edges with one endpoint being $u$,
and for a (partial) edge coloring $f$ of $K_{2n}$,
let $f(u)$ denote the set of colors on the edges in $E_u$ under $f$.
Let $\varphi$ be an $\alpha$-dense precoloring of $K_{2n}$.
Edges of $K_{2n}$ which are colored under $\varphi$,
are called {\em prescribed (with respect to $\varphi$)}.
For an edge coloring $h$ of $K_{2n}$,
an edge $e$ of $K_{2n}$ is called
{\em requested (under $h$ with respect to $\varphi$)}
if $h(e) = c$ and
$e$ is adjacent to an edge $e'$ such that $\varphi(e')=c$.
Consider a $\beta$-sparse list assignment $L$ for $K_{2n}$.
For an edge coloring $h$ of $K_{2n}$,
an edge $e$ of $K_{2n}$ is called a \textit{conflict edge (of $h$ with
respect to $L$)} if $h(e) \in L(e)$;
such edges are also referred to as just {\em conflicts}.
An \textit{allowed cycle (under $h$
with respect to $L$)} of $K_{2n}$ is a $4$-cycle
$\mathcal{C}=uvztu$ in $K_{2n}$ that is $2$-colored under $h$, and such that
interchanging colors on $\mathcal{C}$ yields a proper edge coloring
$h_1$ of $K_{2n}$ where none of $uv$, $vz$,
$zt$, $tu$ is a conflict edge.
We call such an interchange {\em a swap on $h$}, or a {\em swap on $\mathcal{C}$}.
\vspace{0.5cm}
\begin{enumerate}
\item[Step I.] Define a {\em standard $m$-edge coloring $h$}
of the complete graph $K_{2n}$. In particular, this coloring has the property
that ``most'' edges of $K_{n,n}$ are contained in a large number of
$2$-colored $4$-cycles.
\item[Step II.]
Given the standard $m$-edge coloring $h$
of $K_{2n}$,
from $h$ we construct a new
proper $m$-edge-coloring $h'$
that satisfies
certain sparsity conditions. These conditions shall be more precisely
articulated below.
\item[Step III.] From the precoloring $\varphi$ of $K_{2n}$, we
define a new edge precoloring $\varphi'$ such that an edge
$e$ of $K_{2n}$ is colored
under $\varphi'$ if and only if $e$ is colored under
$\varphi$ or $e$ is a conflict edge
of $h'$ with respect to $L$. We shall also require that
each of the colors in $\{1,\dots,m\}$ is used a bounded number of times
under $\varphi'$.
\item[Step IV.]
In this step we prove a series of lemmas which roughly implies that
for almost all pair of edges $e$ and $e'$ in $K_{2n}$, we can construct
a new edge coloring $h^T$ from $h'$ (or a coloring obtained from $h'$)
such that $h^T(e')=h'(e)$ by recoloring a ``small'' subgraph of $K_{2n}$.
\item[Step V.] Using the lemmas proved in the previous step,
we shall in this step from $h'$ construct a coloring $h_q$ of $K_{2n}$
that agrees with $\varphi'$ and which avoids $L$.
This is done iteratively by steps:
in each step we consider a prescribed edge $e$
of $K_{2n}$, such that $h'(e) \neq \varphi'(e)$, and
construct a subgraph $T_e$
of $K_{2n}$, such that performing a series of swaps on allowed
cycles, all edges of which are in $T_e$, we obtain a coloring
$h''_1$ where $h''_1(e) = \varphi'(e)$.
Hence, after completing this
iterative procedure we obtain a coloring that is an extension of $\varphi'$
(and thus $\varphi$), and which avoids $L$.
\end{enumerate}
In Step IV and V we shall generalize several tools from
\cite{AndrenCasselgrenMarkstrom, Bartlett} to the setting of complete graphs.
\section{Proof}
In this section we prove Theorem \ref{mainth}. In the proof we shall verify
that it is possible to perform Steps I-V described above
to obtain a proper $m$-edge-coloring of $K_{2n}$ that is an extension of $\varphi$
and which avoids $L$. This is done by proving some lemmas in each step.
The proof of Theorem \ref{mainth} involves a number of
functions and parameters:
$$\alpha, \beta, d, \epsilon, k, c(n), f(n)$$
and a number of inequalities that they must satisfy. For the reader's convenience,
explicit choices for which the proof holds are presented here:
$$\alpha = \frac{1}{1000000}, \quad \beta=\frac{1}{1000000}, \quad d= \frac{1}{200},
\quad
\epsilon =\frac{1}{50000},$$
$$k = \frac{1}{5000}, \quad c(n) = \left\lfloor\frac{n}{50000}\right\rfloor, \quad
f(n) = \left\lfloor\frac{n}{10000}\right\rfloor.$$
We shall also use the functions
$$c'(n) = c(n)/2, \quad H(n) = 9 \alpha m + 9 f(n) + 6 c(n) + 4dn, \quad
P(n)=dn+ \alpha m + f(n).$$
Furthermore, we shall assume that $n$ is large enough whenever necessary. Since the
proof contains a finite number of inequalities that are valid if $n$ is large
enough, say $n \geq N$,
this suffices for proving the theorem with $\alpha'$ and $\beta'$
in place of $\alpha$ and $\beta$, and where we set
$\alpha' = \min\{1/N, \alpha\}$
and $\beta' = \min\{1/N, \beta\}$.
We remark that since the numerical values of $\alpha$ and $\beta$ are not anywhere near
what we expect to be optimal, we have not put an effort into choosing optimal values
for these parameters. Since $K_{n,n}$ is a subgraph of $K_{2n}$, any upper bounds
on $\alpha$ and $\beta$ for the corresponding problem on
complete bipartite graphs are also valid in the setting of complete graphs;
see \cite{AndrenCasselgrenMarkstrom} for a more
elaborate discussion on this question.
Finally, for simplicity of notation,
we shall omit floor and celling signs whenever these are not crucial.
\begin{proof}[Proof of Theorem \ref{mainth}]
Let $\varphi$ be an $\alpha$-dense precoloring of $K_{2n}$, and let $L$
be a $\beta$-sparse list assignment for $K_{2n}$
such that $\varphi(e) \notin L(e)$ for every edge $e \in E(K_{2n})$.
\bigskip
\noindent
{\bf Step I:} Below we shall define the {\em standard $m$-edge coloring $h$}
of the complete graph $K_{2n}$ by
defining an $n$-edge
coloring for $K_{n,n}$ using the set of colors $\{1,2,\dots,n\}$ and
a $(m-n)$-edge
coloring for $G_1$ and $G_2$ using the set
of colors $\{n+1,\dots,m\}$.
Throughout this paper, we assume $x \mod k =k$ in the case when $x \mod k \equiv 0$.
Firstly, we define a proper $n$-edge coloring for $K_{n,n}$
using the set of colors $\{1,2,\dots,n\}$.
This coloring was used in \cite{AndrenCasselgrenOhman, AndrenCasselgrenMarkstrom,
Bartlett}, and
we shall give the explicit construction for the case when $n$ is even.
For the case $n$ is odd, one can modify the construction in the even case by swapping
on some $2$-colored $4$-cycles and using a transversal; the details are given in Lemma 2.1 in \cite{Bartlett}.
So suppose that $n=2r$.
For $1\leq i, j \leq n$, the standard coloring $h_K$ for $K_{n,n}$ is defined as follows.
\begin{equation}
h_K(p_iq_j) = \left\{ \begin{array}{llcl}
j-i +1 & \mod r & \mbox{for} & i,j \leq r,\\
i-j +1 & \mod r & \mbox{for} & i, j> r, \\
(j-i+1 &\mod r) +r & \mbox{for} & i \leq r, j>r,\\
(i-j+1 &\mod r) + r & \mbox{for} & i > r, j \leq r.
\end{array}\right.
\end{equation}
The following property of $h_K$ is fundamental for our proof.
If a $2$-colored $4$-cycle with colors $c_1$ and $c_2$
satisfies that
$$|\{c_1, c_2\} \cap \{1,\dots, r\}|=1$$
then $C$ is called a {\em strong} $2$-colored 4-cycle.
\begin{lemma}
\cite{AndrenCasselgrenOhman,AndrenCasselgrenMarkstrom, Bartlett}
Each edge in $K_{n,n}$ belongs to exactly $r$ distinct strong $2$-colored $4$-cycles
under $h_K$.
\end{lemma}
For the case when $n = 2r + 1$, we can construct an $n$-edge
coloring $h_K$ for $K_{n,n}$
such that all but at most $3n+7$ edges are in $\left \lfloor{\frac{n}{2}}\right \rfloor$ strong $2$-colored 4-cycles.
In particular, there is a vertex in $K_{n,n}$ where no edge belongs
to at least $\left \lfloor{\frac{n}{2}}\right \rfloor$ strong
$2$-colored $4$-cycles. The full proof appears in \cite{Bartlett} and therefore we omit the details here.
Secondly, let us define $(m-n)$-edge colorings of $G_1$ and $G_2$ using the set of colors $\{n+1,\dots,m\}$.
Suppose first that $n$ is odd, and recall that $m=2n$.
We define the colorings $h_{G_1}$ of $G_1$ and $h_{G_2}$ of $G_2$
by,
for $1\leq i, j \leq n$,
setting
$$h_{G_1}(p_ip_j)=h_{G_2}(q_iq_j)=(i+j \mod n) + n.$$
Assume now that $n$ is even, and recall that $m=2n-1$.
We define the colorings
$h_{G_1}$ of $G_1$ and $h_{G_2}$ of $G_2$ as follows:
\begin{itemize}
\item $h_{G_1}(p_ip_j)=h_{G_2}(q_iq_j)=(i+j \mod n-1) + n$ for $1\leq i, j \leq n-1$.
\item $h_{G_1}(p_ip_n)=h_{G_2}(q_iq_n)=(2i \mod n-1) + n$ for $1\leq i, j \leq n-1$.
\end{itemize}
It is straightforward to verify that $h_K$, $h_{G_1}$, $h_{G_2}$ are proper colorings.
Taken together, the colorings $h_K$, $h_{G_1}$, $h_{G_2}$ constitute the
standard $m$-edge coloring $h$ of $K_{2n}$.
\bigskip
\noindent
{\bf Step II:} Let $h$ be the $m$-edge coloring of $K_{2n}$ obtained in Step I, and let $\rho=(\rho_1,\rho_2)$
be a pair of permutations chosen
independently and uniformly at random from all $n!$ permutations of
the vertex labels of $G_1$ and $n!$ permutations of the vertex labels of $G_2$.
We permute the labels of the vertices with respect to the coloring of $h$,
while $\varphi$ is considered as a fixed partial coloring of $K_{2n}$.
Thus we can view a relabeling of the vertices in $G_1$ and $G_2$
with respect to $h$
(while
keeping colors of edges fixed)
as equivalent to defining a new proper edge coloring of $K_{2n}$ from $h$
by recoloring edges in $K_{2n}$. Hence,
we can
think of $\rho$ as being applied to the edge coloring $h$ of $K_{2n}$
thereby defining a new edge coloring of $K_{2n}$
(rather than permuting vertex labels).
Denote by $h'$ a random $m$-edge coloring obtained from $h$ by applying $\rho$ to $h$.
Note that if $u'=\rho(u)$ and $v'=\rho(v)$, then $h'(u'v')=h(uv)$.
\begin{lemma}
\label{alpha}
Suppose that
$\alpha, \beta, \epsilon$ are constants, and $c(n)$ and $c'(n)=c(n)/2$ are
functions of $n$,
such that
$n-1>2c(n)>4$ and
$$\Big( \dfrac{4\beta}{\epsilon - 4\beta}\Big)^{\epsilon - 4\beta}
\Big( \dfrac{1}{1 - 2\epsilon + 8\beta}\Big)^{1/2-\epsilon + 4\beta} <1,$$
$$\alpha, \beta < \dfrac{c(n)}{2(n-c(n))}
\Big(\dfrac{n - c(n)}{n}\Big)^{\frac{n}{c(n)}}, \, \text{ and }$$
$$\beta<\dfrac{c'(n)}{2(n-c'(n))}
\Big(\dfrac{n - c'(n)}{n}\Big)^{\frac{n}{c'(n)}}.$$
Then the probability that $h'$ fails the following
conditions tends to $0$ as $n \rightarrow \infty$.
\begin{itemize}
\item[(a)] All edges in $K_{n,n}$, except for $3n+7$, belong to at least
$\left \lfloor{\frac{n}{2}}\right \rfloor - \epsilon n$ allowed strong $2$-colored 4-cycles.
\item[(b)] Each vertex of $K_{n,n}$ is incident to at most $c'(n)$ conflict edges in $K_{n,n}$.
\item[(c)] For each color $c \in \{1,2,\dots,n\}$,
there are at most $c(n)$ edges in $K_{n,n}$
that are colored $c$ that are conflicts.
\item[(d)] For each color $c \in \{1,2,\dots,n\}$,
there are at most $c(n)$ edges in $K_{n,n}$
that are colored $c$ that are prescribed.
\item[(e)] For each pair of colors $c_1\in \{1,2,\dots,m\}$ and
$c_2 \in \{1,2,\dots,n\}$, there are at most $c(n)$ edges $e$ in $K_{n,n}$
with color $c_2$ such that $c_1 \in L(e)$.
\item[(f)] Each vertex of $G_1$ $(G_2)$ is incident to at most $c'(n)$ conflict edges in $G_1$ $(G_2)$.
\item[(g)] For each color $c \in \{n+1,n+2,\dots,m\}$,
there are at most $c(n)$ edges in $G_1$ $(G_2)$
that are colored $c$ that are conflicts.
\item[(h)] For each color $c \in \{n+1,n+2,\dots,m\}$,
there are at most $c(n)$ edges in $G_1$ $(G_2)$
that are colored $c$ that are prescribed.
\item[(i)] For each pair of colors $c_1\in \{1,2,\dots,m\}$ and
$c_2 \in \{n+1,n+2,\dots,m\}$,
there are at most $c(n)$ edges $e$ in $G_1$ $(G_2)$ with color $c_2$
such that $c_1 \in L(e)$.
\end{itemize}
\end{lemma}
Note that the conditions in the lemma imply that the following holds for
the coloring $h'$.
\begin{itemize}
\item[(a')] Each vertex of $K_{2n}$ is incident to at most $c(n)$ conflict edges;
\item[(b')] For each color $c \in \{1,2,\dots,m\}$,
there are at most $c(n)$ edges in $K_{2n}$
that are colored $c$ that are conflicts $($prescribed$)$;
\item[(c')] For each pair of colors $c_1, c_2 \in \{1,2,\dots,m\}$,
there are at most $c(n)$ edges $e$ in $K_{2n}$
with color $c_2$ such that $c_1 \in L(e)$.
\end{itemize}
For the proof of this lemma we shall use the following theorem,
see e.g. \cite{AndrenCasselgrenMarkstrom}.
\begin{theorem}
\label{balancedbipartite}
If $B$ is a balanced bipartite graph on $2n$ vertices and $d_1,\dots,d_n$
are the degrees of the vertices in one part of $B$, then the number of perfect matchings in $B$
is at most $\prod_{1 \leq i \leq n} (d_i!)^{1/d_i}$.
\end{theorem}
\begin{proof}[Proof of Lemma \ref{alpha}]
Let $\alpha'=2\alpha$ and $\beta'=2\beta$; then the
$\alpha$-dense precoloring $\varphi$ satisfies that
\begin{itemize}
\item[(I)] every color appears on at most $\alpha' n$ edges;
\item[(II)] for every vertex $v$, at most $\alpha' n$ edges incident with $v$ are precolored.
\end{itemize}
For the $\beta$-sparse list assignment $L$, we have
\begin{itemize}
\item[(III)] $|L(e)| \leq \beta' n$ for every edge of $K_{2n}$;
\item[(IV)] for every vertex $v$, every color appears in the lists
of at most $\beta' n$ edges incident to $v$.
\end{itemize}
By applying Lemmas 3.2, 3.3, 3.4 in \cite{AndrenCasselgrenMarkstrom}, we deduce
that
the probability that $h'$ fails conditions (a), (b), (c), (d) or (e) tends to $0$ as $n \rightarrow \infty$ if
$$\Big( \dfrac{2\beta'}{\epsilon - 2\beta'}\Big)^{\epsilon - 2\beta'} \Big( \dfrac{1}{1 - 2\epsilon + 4\beta'}\Big)^{1/2-\epsilon + 2\beta'} <1;$$
$$\alpha', \beta'<\dfrac{c(n)}{(n-c(n))} \Big(\dfrac{n - c(n)}{n}\Big)^{\frac{n}{c(n)}};
\beta'<\dfrac{c'(n)}{(n-c'(n))} \Big(\dfrac{n - c'(n)}{n}\Big)^{\frac{n}{c'(n)}}.$$
Since all of these inequalities are true, our remaining job is to prove that
the probability that
$h'$ fails conditions (f), (g), (h) or (i) tends to $0$ as $n \rightarrow \infty$.
\vspace{0.3cm}
\begin{itemize}
\item We first prove (f) for $G_1$. Given a vertex $u \in G_1$,
it is obvious that $|E_u \cap E(G_1)|=n-1$.
We estimate the number of choices for the
pair $\rho=(\rho_1,\rho_2)$ such that under $h'$ at least $c'(n)$ edges
in $E_u \cap E(G_1)$ are conflicts with $L$.
There are $n!$ ways of choosing the permutation $\rho_2$;
fix such a permutation $\rho_2$. Also, there are $n$ choices for
a vertex $u_0$ such that $\rho_1(u_0)=u$,
we fix such a vertex $u_0$.
Next, let $N$ be a subset of $E_{u_0} \cap E(G_1)$ such that $|N|=c'(n)$ and all edges in $N$ are mapped
to conflict edges by $\rho$, there are ${n-1} \choose c'(n)$ ways to chose $N$.
Next, let $B$ be a balanced bipartite graph defined as follows:
the parts of $B$ are $E_{u_0} \cap E(G_1)$
and $E_u \cap E(G_1)$ and there is an edge between $u_0 x \in E_{u_0} \cap E(G_1)$ and $uy \in E_{u} \cap E(G_1)$ if
\begin{itemize}
\item $u_0x \notin N$, or
\item $u_0x \in N$ and $h(u_0x) \in L(uy)$.
\end{itemize}
If $u_0x \notin N$, then the degree of $u_0x$ in $B$ is $n-1$.
If $u_0x \in N$, then the degree of $u_0x$ in $B$ is at most $\beta'n$ because the color $h(u_0x)$
occurs at most $\beta'n$ times in $E_u \cap E(G_1)$.
A perfect matching in $B$ corresponds to a choice of $\rho_1$ so that $u_0$ is mapped to $u$ and
all edges in $N$ are mapped to conflict edges under $h'$. By Theorem
\ref{balancedbipartite},
the number of perfect matchings in $B$ is at most
$$\big((\beta'n)!\big) ^{\frac{c'(n)}{\beta'n}} \big((n-1)!\big) ^{\frac{n-1-c'(n)}{n-1}}.$$
So the probability that $E_u \cap E(G_1)$ contains at
least $c'(n)$ conflicts with $L$ is at most
$$X=\dfrac{n! n {{n-1} \choose c'(n)} \big((\beta'n)!\big) ^{\frac{c'(n)}{\beta'n}}
\big((n-1)!\big) ^{\frac{n-1-c'(n)}{n-1}}}{(n!)^2}$$
$$=\dfrac{\big((\beta'n)!\big) ^{\frac{c'(n)}{\beta'n}} \big((n-1)!\big) ^{\frac{n-1-c'(n)}{n-1}}}{c'(n)! (n-1-c'(n))!}$$
Using Stirling's formula
$$n! =C_0n^{a_0} \Big(\frac{n}{e}\Big)^n$$
for some positive constants $C_0$ and $a_0$; we have:
$$X \leq Cn^a \Big( \dfrac{\beta'n}{c'(n)}\Big)^{c'(n)}
\Big(\dfrac{n-1}{n-1-c'(n)}\Big)^{(n-1-c'(n))}$$
$$=Cn^a \Big( \dfrac{\beta'n}{c'(n)}\Big)^{c'(n)} \Big(1+\dfrac{1}{(n-1-c'(n))/c'(n)}\Big)^{(n-1-c'(n))}$$
where $C$ and $a$ are some positive constants.
Since the function $f(x)=(1+\dfrac{1}{x})^{xy}$ is increasing for $x, y\geq 1$, we have
$$X<Cn^a \Big( \dfrac{\beta'n}{c'(n)}\Big)^{c'(n)} \Big(1+\dfrac{1}{(n-c'(n))/c'(n)}\Big)^{(n-c'(n))}$$
$$=Cn^a \Big( \dfrac{\beta'n}{c'(n)}\Big)^{c'(n)} \Big(\dfrac{n}{n-c'(n)}\Big)^{n-c'(n)}$$
$$=Cn^a \Big( \dfrac{\beta'(n-c'(n))}{c'(n)}\Big)^{c'(n)} \Big(\dfrac{n}{n-c'(n)}\Big)^{n}$$
Now, since $G_1$ has $n$ vertices, the probability that $h'$
fails condition (f) for $G_1$ is at most
$$Y=Cn^{a+1} \Big( \dfrac{\beta'(n-c'(n))}{c'(n)}\Big)^{c'(n)} \Big(\dfrac{n}{n-c'(n)}\Big)^{n}$$
Since $\beta'<\dfrac{c'(n)}{(n-c'(n))}
\Big(\dfrac{n - c'(n)}{n}\Big)^{\frac{n}{c'(n)}}$,
we have
$Y \rightarrow 0$ as $n \rightarrow \infty$; thus the probability that $h'$ fails condition (f) for $G_1$
tends to zero as $n \rightarrow \infty$.
That the probability that $h'$ fails condition (f) for $G_2$
tends to zero as $n \rightarrow \infty$ can be proved similarly.
\item Next, we prove (g) for $G_1$. Let $n_0=(n-1)/2$ if $n$ is odd and $n_0=n/2$ if $n$ is even.
The total number of edges in $G_1$ is $n(n-1)/2$. Since the total number of colors used to color
the edges of $G_1$ under $h$ is $n$ if $n$ is odd, and $n-1$ if $n$ is even,
there are exactly $n_0$ edges in $G_1$ that
are colored by a fixed color in $\{n+1, n+2, \dots, m\}$.
Let $c$ be a color in $\{n+1, n+2, \dots, m\}$ and
$P_c=\{x_1y_1, x_2y_2,\dots,x_{n_0}y_{n_0}\}$ be the set of edges
that are colored $c$ under $h$. We estimate the number of
choices for the
pair $\rho=(\rho_1,\rho_2)$ such that under $h'$
at least $c(n)$ edges in $G_1$ that are colored $c$
are conflicts with $L$.
There are $n!$ ways of choosing the permutation $\rho_2$;
fix such a permutation $\rho_2$.
\begin{itemize}
\item If $n$ is odd, there is only one vertex $u$ in $G_1$ that is
not contained in the set
$V'_c=\{x_1,x_2,\dots,x_{n_0}, y_1,y_2,\dots,y_{n_0}\}$; there are $n$ ways of
choosing a vertex $u_1$
such that $\rho_1(u)=u_1$, we fix such a vertex $u_1$.
Moreover, there
are $(n-1)\dots(n-n_0)$ ways of choosing
a set of vertices $\{x'_1,\dots,x'_{n_0}\}$ satisfying
that $\rho_1(x_i)=x'_i$, $i=1,\dots,n_0$; fix such a set
$\{x'_1,\dots,x'_{n_0}\}$ and let
$Q=V(G_1) \setminus \{x'_1,\dots,x'_{n_0}, u_1\}$.
Note that $n \big((n-1) \dots (n-n_0)\big) =n!/n_0!$ (since $n_0=(n-1)/2$ if $n$ is odd).
\item If $n$ is even, then
$V(G_1) = V'_c$, where the latter set is defined as above.
There are $n(n-1)\dots(n-n_0+1)$ ways of choosing
a set of vertices $\{x'_1,\dots,x'_{n_0}\}$
such that $\rho_1(x_i)=x'_i$, $i=1,\dots,n_0$; fix such a set
$\{x'_1,\dots,x'_{n_0}\}$ and let $Q=V(G_1) \setminus \{x'_1,\dots,x'_{n_0}\}$.
Note that $n(n-1)\dots(n-n_0+1) =n!/n_0!$.
\end{itemize}
Let $N$ be a subset of $P_c$ such that $|N|=c(n)$
and all edges in $N$ are mapped to conflict edges by $\rho$; there are
$n_0 \choose c(n)$ ways to chose $N$.
We now define a balanced bipartite graph $B$ as follows: the parts of $B$ are $P_c$
and $Q$ and there is an edge between $x_iy_i \in P_c$ and $y'_j \in Q$ if
\begin{itemize}
\item $x_iy_i \notin N$, or
\item $x_iy_i \in N$ and $h(x_iy_i)=c \in L(x'_iy'_j)$.
\end{itemize}
If $x_iy_i \notin N$, then the degree of $x_iy_i$ in $B$ is $n_0$.
If $x_iy_i \in N$, then the degree of $x_iy_i$ in $B$ is at most $\beta' n$ because the color $c$
occurs at most $\beta' n$ times in $E_{x'_i} \cap E(G_1)$.
A perfect matching in $B$ corresponds to a choice
of $\rho_1$ so that $\rho_1(x_i)=x'_i$, $i=1,\dots,n_0$ and
all edges in $N$ are mapped to conflict edges in $h'$. By Theorem
\ref{balancedbipartite},
the number of perfect matchings in $B$ is at most
$$\big((\beta' n)!\big) ^{\frac{c(n)}{\beta' n}} \big(n_0 !\big) ^{\frac{n_0-c(n)}{n_0}}.$$
So the probability that $P_c$ contains at least $c'(n)$ conflicts with $L$ is at most
$$X=\dfrac{n! (n!/n_0!) {n_0 \choose c(n)} \big((\beta' n)!\big) ^{\frac{c(n)}{\beta' n}} \big(n_0!\big) ^{\frac{n_0-c(n)}{n_0}}}{(n!)^2}$$
$$=\dfrac{\big((\beta' n)!\big) ^{\frac{c(n)}{\beta'n}}
\big(n_0!\big) ^{\frac{n_0-c(n)}{n_0}}}{c(n)! (n_0-c(n))!}$$
Using Stirling's formula and similar estimates as above we deduce
that
$$X \leq Cn^a \Big( \dfrac{\beta'n}{c(n)}\Big)^{c(n)} \Big(\dfrac{n_0}{n_0-c(n)}\Big)^{n_0-c(n)}$$
$$<Cn^a \Big( \dfrac{\beta'n}{c(n)}\Big)^{c(n)} \Big(\dfrac{n}{n-c(n)}\Big)^{n-c(n)}
=Cn^a \Big( \dfrac{\beta'(n-c(n))}{c(n)}\Big)^{c(n)} \Big(\dfrac{n}{n-c(n)}\Big)^{n}$$
where $C$ and $a$ are some positive constants.
Note that there are at most $n$ colors in $h_{G_1}$, thus the probability that $h'$ fails condition (g) for $G_1$ is at most
$$Y=Cn^{a+1} \Big( \dfrac{\beta'(n-c(n))}{c(n)}\Big)^{c(n)} \Big(\dfrac{n}{n-c(n)}\Big)^{n}.$$
Since $\beta'<\dfrac{c(n)}{(n-c(n))} \Big(\dfrac{n - c(n))}{n}\Big)^{\frac{n}{c(n)}}$, we have
$Y \rightarrow 0$ as $n \rightarrow \infty$; thus the probability that $h'$ fails condition (g) for $G_1$
tends to zero as $n \rightarrow \infty$.
That the probability that $h'$ fails condition (g) for $G_2$
tends to zero as $n \rightarrow \infty$ can be proved similarly.
\item The proof of (h) is almost identical to the proof of (g) except that
we use the property that at most $\alpha' n$ edges incident to any vertex are prescribed. We omit the details.
\item The proof of (i) is also almost identical to the proof of (g);
here one also has to use the
property that any fixed color $c_1$ appears in the lists of at most of $\beta' n$ edges incident to any given vertex;
additionally instead of having $n$ choices for a
color $c$ as in the proof of $(g)$, we will have $m(m-n)$ choices for a
pair of colors $(c_1,c_2)$. Here, as well, we omit the details.
\end{itemize}
\end{proof}
Lemma \ref{alpha} implies that there exists a pair of permutations
$\rho=(\rho_1,\rho_2)$ such that
if $h'$ is the proper $m$-edge coloring obtained from $h$ by applying $\rho$ to $h$ then $h'$
satisfies conditions (a)-(i) of Lemma \ref{alpha}.
\bigskip
\noindent
{\bf Step III:} Let $h'$ be the proper $m$-edge coloring satisfying conditions
(a)-(i) of Lemma \ref{alpha} obtained in the previous step.
We use the following lemma for extending $\varphi$ to a
proper $m$-edge precoloring $\varphi'$ of $K_{2n}$, such that an
edge $e$ of $K_{2n}$ is colored under $\varphi'$ if and only if
$e$ is precolored under $\varphi$ or $e$ is a conflict edge of
$h'$ with $L$.
\begin{lemma}
\label{gamma}
Let $\alpha, \beta$ be constants and $c(n),f(n)$ be functions of $n$ such that
$$m - \beta m -2 \alpha m - 2c(n) - \dfrac{2nc(n)}{f(n)} \geq 1.$$
There is a proper $m$-edge precoloring $\varphi'$ of $K_{2n}$ satisfying the following:
\begin{itemize}
\item[(a)] $\varphi'(uv)=\varphi(uv)$ for any edge $uv$ of $K_{2n}$ that is precolored under $\varphi$.
\item[(b)] For every conflict edge $uv$ of $h'$ that is not colored
under $\varphi$,
$uv$ is colored under $\varphi'$ and $\varphi'(uv) \notin L(uv)$.
\item[(c)] There are at most $\alpha m + c(n)$ prescribed edges at each vertex of $K_{2n}$ under $\varphi'$.
\item[(d)] There are at most $\alpha m +f(n)$ prescribed edges with color $i$, $i =1,\dots, m$, under $\varphi'$.
\end{itemize}
Furthermore, the edge coloring $h'$ of $K_{2n}$ and the precoloring
$\varphi'$ of $K_{2n}$ satisfy that
\begin{itemize}
\item[(e)] For each color $c \in \{1,2,\dots,n\}$, there are at most $2c(n)$ prescribed edges in $K_{n,n}$ with color $c$ under $h'$.
\item[(f)] For each color $c \in \{n+1,n+2,\dots,m\}$,
there are at most $2c(n)$ prescribed edges in $G_1$ $(G_2)$ with color $c$
under $h'$.
\end{itemize}
\end{lemma}
Note that the two conditions (e) and (f) imply that
\begin{itemize}
\item[(g)] For each color $c \in \{1,2,\dots,m\}$,
there are at most $2c(n)$ prescribed edges in $K_{2n}$ with color $c$ in $h'$.
\end{itemize}
\begin{proof}
We shall construct the coloring $\varphi'$ by assigning a color
to every conflict edge; this is done by iteratively constructing
an $m$-edge precoloring $\phi$ of the conflict edges of $K_{2n}$;
in each step we color a hitherto uncolored conflict edge, thereby
transforming a conflict edge to a prescribed edge.
A color $c$ is {\em $\phi$-overloaded} in $K_{2n}$ if $c$ appears on at least
$f(n)$ edges in $K_{2n}$ under $\phi$. Since each vertex of $K_{2n}$ is incident
with at most $c(n)$ conflict edges, the number of conflict edges in $K_{2n}$ is
at most $2nc(n)$; this implies that at most
$\dfrac{2nc(n)}{f(n)}$ colors are $\phi$-overloaded in $K_{2n}$.
Let $G$ be the subgraph of $K_{2n}$ induced by all conflict edges of $K_{2n}$.
Let us now construct the $m$-edge coloring $\phi$ of $G$.
We color the edges of $G$ by steps, and in each step we define a
list $\mathcal{L}(e)$ of allowed colors for
a hitherto uncolored edge $e =uv$ of $G$ by for
every color $c \in \{1,\dots, m\}$ including $c$ in $\mathcal{L}(e)$ if
\begin{itemize}
\item $c \notin L(uv)$,
\item $c$ does not appear in $\varphi(u)$ or $\varphi(v)$,
or on any previously colored edge of $G$ that is adjacent to $e$,
\item $c$ is not $\phi$-overloaded in $K_{2n}$.
\end{itemize}
Our goal is then to pick a color $\phi(e)$ from $\mathcal{L}(e)$ for $e$.
Given that this is possible for each edge of $G$,
this procedure clearly produces a $m$-edge-coloring $\phi$ of $G$, so that
$\phi$ and $\varphi$ taken together form a proper $m$-edge precoloring of $K_{2n}$.
Using the estimates above and the facts that $G$ has maximum degree $c(n)$,
and at most $\alpha m$ edges incident with any vertex $v$ of $K_{2m}$
are prescribed with respect to $\varphi$,
we have
$$\mathcal{L}(e) \geq m - \beta m -2 \alpha m - 2c(n) - \dfrac{2nc(n)}{f(n)}$$
for every edge $e$ of $G$ in the process of constructing $\phi$, and
by assumption $\mathcal{L}(e) \geq 1$.
Thus, we conclude that we can choose an allowed color for each conflict edge
so that the coloring $\phi$ satisfies the above conditions.
This implies that taking $\phi$ and $\varphi$ together we obtain
a proper $m$-precoloring $\varphi'$ of the edges of $K_{2n}$.
There are at most $\alpha m + c(n)$ prescribed edges at each vertex
of $K_{2n}$ under $\varphi'$
because the maximum degree of $G$ is $c(n)$. The fact that we do not use
$\phi$-overloaded
colors in $\phi$ implies that there are at most $\alpha m +f(n)$
prescribed edges with color $i$,
$i =1,\dots, m$, under $\varphi'$.
Let us next prove that the precoloring $\varphi'$ satisfies condition
(e). By the previous lemma, for each color $c \in \{1,2,\dots,n\}$,
there are at most $c(n)$ edges in $K_{n,n}$
that are colored $c$ that are prescribed under $\varphi$. Furthermore,
we have at most $c(n)$ edges in $K_{n,n}$
that are colored $c$ that are conflict; thus after transforming all
conflict edges to prescribed edges,
there are at most $2c(n)$ prescribed edges in $K_{n,n}$ with respect to
$\varphi'$ that are colored $c$ under $h'$.
A similar argument shows that condition (f) holds as well.
\end{proof}
\bigskip
\noindent
{\bf Step IV:} Let $h'$ be the $m$-edge coloring of
$K_{2n}$ obtained in Step II, and
suppose that $\hat h$
is a proper $m$-edge coloring of $K_{2n}$
obtained from $h'$ by performing a sequence of swaps.
We say that an edge $e$ in $K_{2n}$ is
{\em disturbed (in $\hat{h}$)} if $e$ appears in a swap
which is used for obtaining $\hat{h}$ from $h'$, or if $e$ is one of
the original at most $3n+7$ edges in $h'$ that do not belong
to at least $\left \lfloor{\frac{n}{2}}\right \rfloor - \epsilon n$
allowed strong $2$-colored 4-cycles in $h'$.
For a constant $d >0$, we say that a vertex $v$ or color $c$ is $d$-overloaded
if at least $d n$
edges which are incident to $v$ or colored $c$, respectively, are disturbed.
The following lemma is similar to Lemma $3.5$ and Lemma $3.6$ in
\cite{AndrenCasselgrenMarkstrom},
which are strengthened variants of Lemma 2.2. in \cite{Bartlett}. Thus, we shall skip the proof.
\begin{lemma}
\label{swapedgeinK}
Suppose that $h''$ is a proper $m$-edge coloring of $K_{2n}$ obtained from $h'$
by performing some sequence of swaps
on $h'$ and that at most $kn^2$ edges in $h''$ are disturbed
for some constant $k>0$. Suppose that
for each color $c$, at most $2c(n)+P(n)$ edges
with color $c$ under $h''$ are prescribed.
Moreover, let $\{t_1,\dots,t_a\}$ be a set of colors from $h''$.
If $$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n -
6d n - 5 \dfrac{k}{d}n - 4\alpha m - 8c(n) - 3a - 3\beta m - 2P(n)- 6 >0$$
then for any vertex $u_1$ of $G_1$ $(G_2)$ and all but at most
\begin{itemize}
\item $2 \dfrac{k}{d} n + \alpha m +c(n)+a$ choices of a vertex $u_2$ in
$G_2$ $(G_1)$,
such that $h''(u_1u_2) \in \{1,2,\dots,n\}$, and
\item $4 \dfrac{k}{d} n+ a+1+4c(n)+2\beta m + 2 \alpha m +2dn+P(n)$ choices of
a vertex $v_2$ in $G_2$ $(G_1)$, such that $h''(u_1v_2) \in \{1,2,\dots,n\}$,
\end{itemize}
there is a subgraph $T$ of $K_{n,n}$
and a proper $m$-edge coloring $h^T$ of $K_{2n}$, obtained from
$h''$ by performing a sequence of swaps on $4$-cycles in
$T$, that satisfies the following:
\begin{itemize}
\item the color of any edge of $T$ under $h''$ is not $d$-overloaded;
\item no edges that are prescribed (with respect to $\varphi'$) are in $T$;
\item $h''$ and $h^T$ differs on at most $16$ edges
\emph{(}i.e. $T$ contains at most $16$ edges\emph{)};
\item no edge with a color in $\{t_1,...,t_a\}$ under $h''$ is in $T$;
\item $h^T(u_1u_2)=h''(u_1v_2)$ and $h^T(u_1v_2)=h''(u_1u_2)$;
\item if there is a conflict of $h^T$ with respect to $L$, then this edge is also
a conflict of $h''$;
\item any edge in $G_1$ or $G_2$ that is requested under $h^T$ (with respect
to $\varphi'$) is also
requested under $h''$.
\end{itemize}
\end{lemma}
Lemma \ref{swapedgeinK} states that there are
many pairs of adjacent edges $e_x,e_y \in E(K_{n,n})$ satisfying that
$h''(e_x), h''(e_y) \in \{1,2,\dots,n\}$ such that we can exchange their colors
by recoloring a small subgraph of $K_{n,n}$.
When applying the preceding lemma, we shall
refer to $u_1u_2$ as the ``first edge'' and $u_1v_2$ as the ``second edge''.
Given an edge $e_x \in E(K_{n,n})$ such that $h''(e_x) \in \{1,2,\dots,n\}$,
the following lemma
is used for obtaining a coloring where an edge $e_y \in E(K_{n,n})$
adjacent to $e_x$ is colored $h''(e_x)$.
\begin{lemma}
\label{swapedgeinK1}
Suppose that $h''$ is a proper $m$-edge coloring of $K_{2n}$
obtained from $h'$ by performing some sequence of swaps
on $h'$ and that at most $kn^2$ edges in $h''$ are
disturbed for some constant $k>0$. Suppose that
for each color $c$, at most $2c(n)+P(n)$ edges with color
$c$ under $h''$ are prescribed,
and at most $H(n)$ edges with color $c$ are disturbed.
Moreover, let $\{t_1,\dots,t_a\}$ be a set of colors from $h''$.
If $$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n - 6d n -
5 \dfrac{k + 34/n^2}{d}n - 4\alpha m - 8c(n) - 3a - 3\beta m - 2P(n)- 6 >0$$
and
$$n - \Big(8 \dfrac{k + 34/n^2}{d} n + 2a + 3 + 8c(n) + 6\beta m +
4 \alpha m + 4dn+ 2P(n) + H(n)\Big)>0$$
then for any edge $u_1u_2$ of $K_{n,n}$ with
$$h''(u_1u_2)=c_1, \; c_1 \in \{1,2,\dots,n\}, \;
c_1 \notin \{t_1,\dots,t_a\}$$ and all but at most
$$4c(n) + P(n) + 2\beta m + 2\alpha m + 2a + 1 + 4 \dfrac{k+34/n^2}{d} n + H(n)$$
choices of a vertex $v_2$ satisfying that $u_1v_2 \in E(K_{n,n})$,
there is a subgraph $T$ of $K_{n,n}$ and a proper $m$-edge coloring
$h^T$ of $K_{2n}$, obtained from
$h''$ by performing a sequence of swaps on $4$-cycles in
$T$, that satisfies the following:
\begin{itemize}
\item except $c_1$, any color of an edge in $T$ under $h''$ is not $d$-overloaded;
\item except $u_1u_2$, no edge in $T$ is prescribed;
\item $h''$ and $h^T$ differs on at most $34$ edges \emph{(}i.e. $T$
contains at most $34$ edges\emph{)};
\item no edge with a color in $\{t_1,\dots,t_a\}$ under $h''$ is in $T$;
\item $h^T(u_1v_2)=h''(u_1u_2)=c_1$;
\item if there is a conflict of $h^T$ with $L$, then this edge is also
a conflict of $h''$ with $L$;
\item any edge in $G_1$ or $G_2$ that is requested under $h^T$ (with respect
to $\varphi'$) is also
requested under $h''$.
\end{itemize}
\end{lemma}
\begin{proof}
Without loss of generality, assume that $u_1 \in V(G_1)$;
this implies $u_2 \in V(G_2)$.
We choose $v_2 \in V(G_2)$ so that the following properties hold.
\begin{itemize}
\item The edge $v_1v_2$ in $K_{n,n}$ satisfying $h''(v_1v_2)=c_1$ is not
disturbed and not prescribed.
Since there are at most $2c(n)+P(n)$ prescribed edges
and at most $H(n)$ disturbed edges with color $c_1$ under $h''$,
and each such prescribed or disturbed edge of $K_{n,n}$ can be incident to at most
one vertex of $G_2$,
this eliminates at most $2c(n)+P(n)+H(n)$ choices.
\item The edge $u_1v_2$ and the edge $u_2v_1$
are both valid choices for the first edge in an application of
Lemma \ref{swapedgeinK}.
This eliminates at most
$$2\Big(2 \dfrac{k+34/n^2}{d} n + \alpha m +c(n)+a\Big)$$ choices.
The additive factor $34/n^2$ comes from the fact that performing
a sequence of swaps to transform $h''$ into $h^T$
will create at most $34$ additional disturbed edges.
\item $c_1 \notin L(u_1v_2) \cup L(u_2v_1)$ and $u_1u_2 \neq v_1v_2$.
This excludes at most $2\beta m + 1$ choices.
\end{itemize}
Thus we have at least
$$n - 4c(n)- P(n) - 2\beta m - 2\alpha m - 2a -1 - 4 \dfrac{k+34/n^2}{d} n - H(n)$$
choices for a vertex $v_2$ and an edge $v_1v_2$.
We note that this expression is greater than zero by assumption,
so we can indeed make the choice.
Next, we want to choose a color
$c_2 \in \{1,2,\dots,n\}$ such that the following properties hold.
\begin{itemize}
\item The edges $e_1$ and $e_2$ colored $c_2$ under $h''$
that are incident with $u_1$ and $u_2$, respectively,
are both valid choices for the second edge in
an application of Lemma \ref{swapedgeinK}; this eliminates at most
$$2\Big(4 \dfrac{k+34/n^2}{d} n+ a+1+4c(n)+2\beta m + 2 \alpha m +2dn+P(n)\Big)$$
choices.
Note that this condition implies that color $c_2$ is not $d$-overloaded.
\item $c_2 \neq c_1$ and $c_2 \notin L(u_1u_2) \cup L(v_1v_2)$.
This excludes at most $2\beta m+1$ choices.
\end{itemize}
Thus we have at least
$$n - \Big(8 \dfrac{k + 34/n^2}{d} n + 2a + 3 + 8c(n) +
6\beta m + 4 \alpha m + 4dn+ 2P(n)\Big)$$
choices. By assumption, this expression is greater than zero,
so we can indeed choose such color $c_2$. Now, since
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n - 6d n -
5 \dfrac{k+34/n^2}{d}n - 4\alpha m - 8c(n) - 3a - 3\beta m - 2P(n)- 6 >0,$$
we can apply Lemma \ref{swapedgeinK} two consecutive times
to exchange the color of $u_1v_2$ and $e_1$, and
similarly for $u_2v_1$ and $e_2$. Finally,
by swapping on the $2$-colored $4$-cycle
$u_1u_2v_1v_2u_1$, we get the proper coloring $h^T$
such that $h^T(u_1v_2)=h''(u_1u_2)=c_1$. Moreover,
since these swaps only involve edges
from $K_{n,n}$, they do not result in any ``new'' requested edges in
$G_1$ or $G_2$. Note that the same holds for conflict edges in $K_{2n}$.
Note that the subgraph $T$,
consisting of all edges used in the
swaps above,
contains two edges $u_1u_2$ and $v_1v_2$
and the additional edges needed for two applications
of Lemma \ref{swapedgeinK}; this implies that $T$ contains at
most $2+16 \times 2 =34$ edges. Furthermore,
except (possibly) $u_1u_2$, no edges in $T$ are prescribed; except $c_1$,
$T$ only contains edges with colors that are not $d$-overloaded.
Additionally, $T$ does not contain an edge with a color in $\{t_1,\dots,t_a\}$.
\end{proof}
As for Lemma \ref{swapedgeinK}, when applying Lemma \ref{swapedgeinK1},
we shall refer to $u_1u_2$ as the ``first edge'' and $u_1v_2$ as
the ``second edge''.
We use Lemma \ref{swapedgeinG} below for transforming a
coloring $h''$ into a coloring where
an edge $e_y \in E(K_{n,n})$ is colored by the color $h''(e_x)$
of an adjacent edge $e_x \in E(G_1)$ $(E(G_2))$, where
$h''(e_x) \in \{n+1,\dots,m\}$.
In applications of this lemma $u_1v_1$ will be referred to as the
``first edge'', and $u_1u_2$ as the ``second edge''.
\begin{lemma}
\label{swapedgeinG}
Suppose that $h''$ is a proper $m$-edge coloring of $K_{2n}$
obtained from $h'$ by performing some sequence of swaps
on $h'$ and that at most $kn^2$ edges in $h''$ are disturbed
for some constant $k>0$. Suppose further that
for each color $c$, at most $2c(n)+P(n)$
edges with color $c$ under $h''$ are prescribed,
and at most $H(n)$ edges with color $c$ are disturbed.
Moreover, let $\{t_1,\dots,t_a\}$ be a set of colors from $h''$.
If $$\left \lfloor{\frac{n}{2}}\right \rfloor -
2\epsilon n - 6d n - 5 \dfrac{k+34/n^2}{d}n - 4\alpha m -
8c(n) - 3a - 3\beta m - 2P(n)- 6 >0$$
and
$$n - (8 \dfrac{k+34/n^2}{d} n+ 2a+2+12c(n)+6\beta m +
8 \alpha m + 4dn + 2P(n) + 2H(n)\Big)>0$$
then for any edge $u_1v_1$ of $G_1$ $(G_2)$
with
$$h''(u_1v_1)=c_1, \; c_1 \in \{n+1,\dots,m\}, \;
c_1 \notin \{t_1,\dots,t_a\}$$ and all but at most
$$6c(n) + 2P(n) + 2\beta m + 2\alpha m + 2a + 1 + 4 \dfrac{k+34/n^2}{d} n + 2H(n)$$
choices of $u_2 \in V(G_2)$ $(V(G_1))$, there is a
subgraph $T$ of $K_{2n}$ and a proper $m$-edge coloring
$h^T$, obtained from $h''$ by performing a sequence of swaps on $4$-cycles in $T$,
that satisfies the following:
\begin{itemize}
\item except $c_1$, any color of an edge in $T$ under $h''$
is not $d$-overloaded;
\item except $u_1v_1$, no edge in $T$ is prescribed;
\item $h''$ and $h^T$ differs on at most $34$ edges
\emph{(}i.e. $T$ contains at most $34$ edges\emph{)};
\item no edge with a color in $\{t_1,\dots,t_a\}$ under $h''$ is in $T$;
\item $h^T(u_1u_2)=h''(u_1v_1)=c_1$;
\item if there is a conflict of $h^T$ with $L$,
then this edge is also a conflict of $h''$ with $L$;
\item any edge in $G_1$ or $G_2$ that is requested under $h^T$ (with respect
to $\varphi'$) is also
requested under $h''$.
\end{itemize}
\end{lemma}
\begin{proof}
Without loss of generality, assume that $u_1v_1 \in E(G_1)$. We choose
$u_2 \in V(G_2)$
such that the following properties hold.
\begin{itemize}
\item The edge $u_2v_2 \in E(G_2)$ satisfying $h''(u_2v_2)=c_1$
is not disturbed and not prescibed.
Since there are at most $2c(n)+P(n)$ prescribed edges
and at most $H(n)$ disturbed edges with color $c_1$ under $h''$;
and each prescribed or disturbed edge of $G_2$ can be
incident to at most two vertices of $G_2$,
this eliminates at most $2(2c(n)+P(n)+H(n))$ choices.
\item The edge $u_1u_2$ and $v_1v_2$ are both valid choices for the first
edge in an application of
Lemma \ref{swapedgeinK}. As in the proof of the preceding lemma,
this eliminates at most
$$2\Big(2 \dfrac{k+34/n^2}{d} n + \alpha m +c(n)+a\Big)$$ choices.
\item $c_1 \notin L(u_1u_2) \cup L(v_1v_2)$.
This excludes at most $2\beta m$ choices.
\end{itemize}
In the coloring $h'$, there are at least $n-1$ vertices in $G_2$ that
are incident with an edge of color $c_1$; thus we have at least
$$n - 1 - 6c(n)- 2P(n) - 2\beta m - 2\alpha m -
2a - 4 \dfrac{k+34/n^2}{d} n - 2H(n)$$
choices for $u_2$. We note that this expression is greater
than zero by assumption, so we can indeed make the choice.
Next, we want to choose a color $c_2 \in \{1,2,\dots,n\}$
(which implies $c_2 \neq c_1$) such that the following properties hold.
\begin{itemize}
\item The edges $e_1$ and $e_2$ colored $c_2$ under $h''$
that are incident with $u_1$ and $v_1$, respectively,
are both valid choices for the second edge
in
an application of Lemma \ref{swapedgeinK}; this eliminates at most
$$2\Big(4 \dfrac{k+34/n^2}{d} n+ a+1+4c(n)+2\beta m + 2 \alpha m +2dn+P(n)\Big)$$
choices.
\item $c_2 \notin L(u_1v_1) \cup L(u_2v_2)$.
This excludes at most $2\beta m$ choices.
\item $c_2 \notin \varphi'(u_1) \cup \varphi'(u_2) \cup \varphi'(v_1) \cup
\varphi'(v_2) \setminus \{\varphi'(u_1v_1), \varphi'(u_2v_2)\}$.
This condition is needed to ensure that performing a series
of swaps on $T$, does not result in a ``new'' requested edge in $G_1$ or $G_2$.
Since there are at most $\alpha m + c(n)$ prescribed
edges at each vertex of $K_{2n}$ under $\varphi'$,
this excludes at most $4(\alpha m+c(n))$ choices.
\end{itemize}
Thus we have at least
$$n - (8 \dfrac{k+34/n^2}{d} n+ 2a+2+12c(n)+6\beta m + 8 \alpha m +4dn+2P(n)\Big)$$
choices. By assumption, this expression is greater than zero,
so we can indeed choose such
color $c_2$. Now, since
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n -
6d n - 5 \dfrac{k+34/n^2}{d}n - 4\alpha m - 8c(n) - 3a - 3\beta m - 2P(n)- 6 >0,$$
we can apply Lemma \ref{swapedgeinK} two consecutive
times to exchange the colors of $u_1u_2$ and $e_1$, and
similarly for $v_1v_2$ and $e_2$. Finally, by swapping on the $2$-colored $4$-cycle
$u_1u_2v_2v_1u_1$, we get the proper coloring $h^T$ such that
$h^T(u_1u_2)=h''(u_1v_1)=c_1$.
Note that the subgraph $T$,
consisting of all edges used in the
swaps above,
contains two edges $u_1v_1$ and $u_2v_2$
and the additional edges needed for two applications of
Lemma \ref{swapedgeinK}; this
implies that $T$ uses at most $2+16 \times 2 =34$ edges.
Furthermore, except (possibly) $u_1v_1$, no edges in
$T$ are prescribed; except $c_1$,
$T$ only contains edges with colors that are not $d$-overloaded.
Additionally, $T$ does not contain an edge with a color in $\{t_1,\dots,t_a\}$.
\end{proof}
The following lemma is used for transforming the coloring $h''$
into a coloring where an edge $e_y \in E(K_{n,n})$
is colored by the color $h''(e_x)$ of an adjacent edge
$e_x \in E(K_{n,n})$, where $h''(e_x) \in \{n+1,\dots,m\}$.
When applying the lemma we shall refer to
$u_1u_2$ as the ``first edge'' and $u_1v_2$ as the ``second edge''.
\begin{lemma}
\label{swapedgeinKnew}
Suppose that $h''$ is a proper $m$-edge coloring of $K_{2n}$
obtained from $h'$ by performing some sequence of swaps
on $h'$ and that at most $kn^2$ edges in $h''$ are disturbed
for some constant $k>0$. Suppose further that
for each color $c$,
at most $2c(n)+P(n)$ edges with color $c$ under $h''$ are prescribed,
and at most $H(n)$ edges with color $c$ are disturbed.
Let $\{t_1,\dots,t_a\}$ be a set of colors from $h''$.
If $$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n -
6d n - 5 \dfrac{k+101/n^2}{d}n - 4\alpha m - 8c(n) - 3a - 3\beta m - 2P(n)- 6 >0$$
and
$$n - \Big(8 \dfrac{k+101/n^2}{d} n+ 2a+2+12c(n)+6\beta m +
8 \alpha m +4dn+2P(n) + 2H(n)\Big)>0$$
then for any edge $u_1u_2$ of $K_{n,n}$ with
$$h''(u_1u_2)=c_1, \; c_1 \in \{n+1,\dots,m\}, \; c_1 \notin \{t_1,\dots,t_a\}$$
and all but at most
$$5c(n) + 2P(n) + \alpha m + \beta m + a + 2 + 2\dfrac{k+67/n^2}{d} n + 2H(n)$$
choices of a vertex $v_2$ satisfying $u_1v_2 \in K_{n,n}$, there is a
subgraph $T$ of $K_{2n}$ and a proper $m$-edge coloring $h^T$,
obtained from $h''$ by performing a sequence of swaps on $4$-cycles in $T$,
that satisfies the following:
\begin{itemize}
\item except $c_1$, any color of an edge in $T$ under $h''$ is not $d$-overloaded;
\item except $u_1u_2$, no edge of $T$ is prescribed;
\item $h''$ and $h^T$ differs on at most $67$ edges
\emph{(}i.e. $T$ contains at most $67$ edges\emph{)};
\item no edge with a color in $\{t_1,\dots,t_a\}$ under $h''$ is in $T$;
\item $h^T(u_1v_2)=h''(u_1u_2)=c_1$;
\item if there is a conflict of $h^T$ with $L$,
then this edge is also a conflict of $h''$ with $L$;
\item any edge in $G_1$ or $G_2$ that is requested under $h^T$ (with respect
to $\varphi'$) is also
requested under $h''$.
\end{itemize}
\end{lemma}
\begin{proof}
Without loss of generality, assume that $u_1 \in V(G_1)$;
this implies $u_2 \in V(G_2)$. We choose $v_2 \in V(G_2)$ such that
the following properties hold.
\begin{itemize}
\item The edge $v_2x \in E(G_2)$ satisfying $h''(v_2x)=c_1$ is not disturbed.
As in the preceding lemma, this eliminates at most $2H(n)$ choices.
\item The edge $v_2x$ is not prescribed and $v_2 \neq u_2$.
This eliminates at most $2(2c(n)+P(n))+1$ choices.
\item The edge $u_1v_2$ is a valid choice for the
first edge
in an application of
Lemma \ref{swapedgeinK}. This eliminates at most
$2 \dfrac{k+67/n^2}{d} n + \alpha m +c(n)+a$ choices.
\item $L(u_1v_2)$ does not contain the color $c_1$. This eliminates at most $\beta m$ choices.
\end{itemize}
In the coloring $h'$,
there are at least $n-1$ vertices in $G_2$ that
are incident with an edge of color $c_1$; thus we have at least
$$n -1 - 5c(n)- 2P(n) - \alpha m - \beta m - a -1 - 2\dfrac{k+67/n^2}{d} n - 2H(n)$$
choices for $v_2$. Since this expression is greater than zero by assumption,
we can indeed make the choice.
Next, we want to choose a vertex $v_1 \in V(G_1)$ satisfying the following:
\begin{itemize}
\item The edge $v_2v_1$ is a valid choice for the second edge
in an application of Lemma \ref{swapedgeinG}.
This eliminates at most $$6c(n) + 2P(n) + 2\beta m + 2\alpha m +
2a + 1+ 4 \dfrac{k+(34+67)/n^2}{d} n + 2H(n)$$ choices.
\item The edge $u_2v_1$ is a valid choice for the first
edge
in an application of
Lemma \ref{swapedgeinK} and $v_1 \neq u_1$.
This eliminates at most $2 \dfrac{k+67/n^2}{d} n + \alpha m + c(n) + a +1$ choices.
\item $L(u_2v_1)$ does not contain the color $c_1$. This eliminates at most $\beta m$ choices.
\end{itemize}
Thus we have at least $$n - 7c(n)- 2P(n) - 3\alpha m - 3\beta m -
3a - 2 - 6\dfrac{k+101/n^2}{d} n - 2H(n)$$
choices for $v_1$. Since this expression is greater than zero by assumption,
we can indeed make the choice.
Finally, we want to choose a color $c_2 \in \{1,2,\dots,n\}$
(which implies $c_2 \neq c_1$) such that the following properties hold.
\begin{itemize}
\item The edges $e_1$ and $e_2$
colored $c_2$ under $h''$ that
are adjacent to $u_1$ and $u_2$, respectively,
are both valid choices for the second edge
in
an application of Lemma \ref{swapedgeinK}; this eliminates at most
$$2\Big(4 \dfrac{k+67/n^2}{d} n+ a+1+4c(n)+2\beta m + 2 \alpha m +2dn+P(n)\Big)$$
choices.
\item $c_2 \notin L(u_1u_2) \cup L(v_1v_2)$.
This excludes at most $2\beta m$ choices.
\end{itemize}
Thus we have at least
$$n - \Big(8 \dfrac{k+67/n^2}{d} n+ 2a+2+8c(n)+6\beta m +
4 \alpha m + 4dn + 2P(n)\Big)$$
choices. By assumption, this expression is greater than zero,
so we can indeed choose such
edges $e_1$ and $e_2$.
Now, since
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n - 6d n -
5 \dfrac{k+101/n^2}{d}n - 4\alpha m - 8c(n) - 3a - 3\beta m - 2P(n)- 6 >0$$
and
$$n - \Big(8 \dfrac{k+101/n^2}{d} n+ 2a+2+12c(n)+6\beta m +
8 \alpha m +4dn+2P(n) + 2H(n)\Big)>0,$$
we can apply Lemma \ref{swapedgeinK} two consecutive times to exchange the colors
of $u_1v_2$ and $e_1$, and
similarly for $u_2v_1$ and $e_2$.
We can thereafter apply Lemma \ref{swapedgeinG} to obtaing a coloring
where $v_1v_2$ is colored $c_1$.
Now, by swapping on the $2$-colored $4$-cycle
$u_1u_2v_1v_2u_1$, we get the proper coloring $h^T$ such
that $h^T(u_1v_2)=h''(u_1u_2)=c_1$.
Since the applications of Lemma \ref{swapedgeinK} and \ref{swapedgeinG}
do not result in any ``new'' requested edges in $G_1$ or $G_2$,
the transformations in this lemma
do not yield any ``new'' requested edges in $G_1$ or $G_2$;
the same holds for conflict edges in $K_{2n}$.
Note that the subgraph $T$,
consisting of all edges used in the
swaps above
contains an edge $u_1u_2$ and
all the additional edges needed for
applying Lemma \ref{swapedgeinK} twice and Lemma \ref{swapedgeinG} once;
this implies that $T$ contains at most $1+16 \times 2 +34=67$ edges.
Furthermore, except $u_1u_2$, no edges in $T$ are prescribed;
except $c_1$,
$T$ only contains edges with colors that are not $d$-overloaded.
Additionally, $T$ does not contain an edge with a color in
$\{t_1,\dots,t_a\}$.
\end{proof}
Given a color $c_1 \in \{1,2,\dots,n\}$, the final lemma in this
step is used for obtaining a coloring where an edge
in $G_1$ or $G_2$ is colored $c_1$. In applications of this
lemma we shall refer to $uv$ as the ``first edge''.
\begin{lemma}
\label{swapedgecolor}
Suppose that $h''$ is a proper $m$-edge coloring of $K_{2n}$ obtained
from $h'$ by performing some sequence of swaps
on $h'$ and that at most $kn^2$ edges in $h''$ are disturbed for
some constant $k>0$. Suppose further that
for each color $c$, at most $2c(n)+P(n)$ edges with color $c$ under $h''$ are
prescribed,
and at most $H(n)$ edges with color $c$ are disturbed.
Moreover, let $\{t_1,\dots,t_a\}$ be a set of colors from $h''$.
If $$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n -
6d n - 8 \dfrac{k+104/n^2}{d}n - 4\alpha m - 15c(n) - 4a - 6\beta m - 5P(n) - 2H(n) - 6 >0$$
then for any color $c_1 \in \{1,2,\dots,n\}$,
where $c_1 \notin \{t_1,\dots,t_a\}$,
there are at least
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 7c(n) - 3P(n) - dn - 2H(n)$$
choices of an edge $uv \in E(G_1)$ $(E(G_2))$, such that there is
a subgraph $T$ of $K_{2n}$ and a proper $m$-edge coloring $h^T$,
obtained from $h''$ by performing a sequence of swaps on $4$-cycles in $T$,
that satisfies the
following:
\begin{itemize}
\item except $c_1$, any color of an edge in $T$ under $h''$ is not
$d$-overloaded;
\item $T$ contains no prescribed edge;
\item $h''$ and $h^T$ differs on at most $70$ edges
\emph{(}i.e. $T$ contains at most $70$ edges\emph{)};
\item no edge with color in $\{t_1,\dots,t_a\}$ under $h''$ is in $T$;
\item $h^T(uv)=c_1$;
\item if there is a conflict of $h^T$ with $L$,
then this edge is also a conflict of $h''$ with $L$;
\item any edge in $G_1$ or $G_2$ that is requested under $h^T$ (with respect
to $\varphi'$) is also
requested under $h''$.
\end{itemize}
\end{lemma}
\begin{proof}
We will prove the lemma assuming $uv \in E(G_1)$;
the case when $uv \in E(G_2)$ is of course analogous.
Since at most $kn^2$ edges in $h''$ are disturbed, there are at most
$kn/d$ $d$-overloaded colors;
by assumption, $n -1 - kn/d - a>0$, so we can choose a color
$c_2 \in \{n+1,n+2,\dots,m\}$ such that $c_2 \notin \{t_1,\dots,t_a\}$
is not a $d$-overloaded color. Next, we choose an edge
$uv \in G_1$ satisfying $h''(uv)=c_2$
such that the following properties hold.
\begin{itemize}
\item The edge $uv$ is not prescribed.
Since there are at most $2c(n)+P(n)$ prescribed edges with
color $c_2$ in $h''$, this eliminates at most $2c(n)+P(n)$ choices.
\item The edge $uv$ is not disturbed and $c_1 \notin L(uv)$.
Since the color $c_2$ is not $d$-overloaded
and for each pair of colors $c_1,c_2 \in \{1,2,\dots,m\}$,
there are at most $c(n)$ edges $e$
in $K_{2n}$ with $h'(e) =c_2$ and $c_1 \in L(e)$
and at most $dn$ edges of color $c_2$ have been used in the
swaps for
transforming $h'$ to $h''$;
this eliminates at most $c(n)+dn$ choices.
\item $c_1 \notin \varphi'(u) \cup
\varphi'(v) \setminus \{\varphi'(uv) \}$.
This condition is needed to ensure that after performing the
swaps in this lemma,
$uv$ is not a requested edge in $G_1$.
Since there are at most $2c(n)+P(n)$ prescribed
edges with color $c_1$ in $h''$,
this excludes at most $2(2c(n)+P(n))$ choices.
\item The edges $e_1$ and $e_2$ colored $c_1$ under $h''$ that are incident
with $u$ and $v$, respectively,
are not disturbed. This condition implies that $e_1, e_2 \in K_{n,n}$
and this eliminates at most $2H(n)$ choices.
\end{itemize}
Under $h'$
there are $\left \lfloor{\frac{n}{2}}\right \rfloor$ edges in $G_2$ that
are colored $c_2$; thus we have at least
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 7c(n)- 3P(n) - dn- 2H(n)$$
choices for an edge $uv$. Since this expression is
greater than zero by assumption, we can indeed make the choice.
Next, we want to choose an edge $xy \in E(G_2)$ satisfying $h''(xy)=c_2$
such that the following properties hold.
\begin{itemize}
\item $c_1 \notin L(xy) \cup \varphi'(x) \cup
\varphi'(y) \setminus \{\varphi'(xy)\}$,
and the edge $xy$ is not prescibed and not disturbed.
As before, this eliminates at most $7c(n)+3P(n) + dn$ choices.
\item The edges $ux$ and $vy$ are both valid choices for the second
edge
in
an application of Lemma \ref{swapedgeinK1}.
This eliminates at most $$2\Big(4c(n) + P(n) + 2\beta m + 2\alpha m +
2a + 1 + 4 \dfrac{k+(34+70)/n^2}{d} n + H(n)\Big)$$ choices.
\item $c_2 \notin L(ux) \cup L(vy)$. This eliminates at most
$2\beta m$ choices.
\end{itemize}
Thus we have at least $$\left \lfloor{\frac{n}{2}}\right \rfloor -
\Big(15c(n) + 5P(n) + 4\alpha m + 6\beta m + dn + 4a +
2 + 8\dfrac{k+104/n^2}{d} n + 2H(n)\Big)$$
choices for $xy$. Since this expression is greater than zero by assumption,
we can indeed make the choice.
Now, since
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 2\epsilon n -
6d n - 8 \dfrac{k+104/n^2}{d}n - 4\alpha m - 15c(n) - 4a - 6\beta m - 5P(n) - 2H(n) - 6 >0$$
we can apply Lemma \ref{swapedgeinK1} two consecutive times to
obtain a coloring where $ux$ is colored $c_1$ and $vu$ is colored $c_1$.
Thereafter, finally, by swapping on the $2$-colored $4$-cycle
$uvyxu$, we get the proper coloring $h^T$ such that $h^T(uv)=h''(e_1)=c_1$.
Since the swaps used when applying Lemma \ref{swapedgeinK1}
do not result in any ``new'' requested edges in $G_1$ or $G_2$,
the transformations in this lemma do not yield any new requested edges
in $G_1$ or $G_2$; similarly for conflict edges of $K_{2n}$.
Note that the subgraph $T$, consisting of all edges used in the
swaps above,
contains two edges
$uv$ and $xy$ and all additional edges needed for applying
Lemma \ref{swapedgeinK1} twice;
this implies that $T$ contains at most $2+34 \times 2=70$ edges.
Furthermore, none of these edges in $T$ are prescribed; except $c_1$,
$T$ only contains edges with colors that are not $d$-overloaded.
Additionally, $T$ does not contain an edge with a color in $\{t_1,\dots,t_a\}$.
\end{proof}
{\bf Step V:} Let $\varphi'$ be the proper $m$-precoloring of $K_{2n}$
obtained in Step III and $h'$
be the $m$-edge coloring of $K_{2n}$ obtained in Step II.
In this step we shall from $h'$ construct a coloring $h_q$ of $K_{2n}$
that agrees with $\varphi$ and which avoids $L$.
This is done iteratively by steps:
in each step we consider a prescribed edge $e$
of $K_{2n}$, such that $h'(e) \neq \varphi'(e)$, and
perform a sequence of swaps on $2$-colored $4$-cycles to
obtain a coloring $h_e$ where
$e$ is colored $\varphi'(e)$.
In this process, special care is taken so that these swaps
do not result in
any new requested edges in $G_1$ or $G_2$;
in particular, this implies that every requested edge
with a color in $\{1,2,\dots,n\}$ is always in $K_{n,n}$
for any intermediate coloring of $K_{2n}$ that is constructed in
this iterative procedure.
We shall use the following lemma.
\begin{lemma}
\label{correctedge}
Suppose that $h''$ is a proper $m$-edge coloring of $K_{2n}$
obtained from $h'$ by performing some sequence of swaps
on $h'$ and that at most $kn^2$ edges in $h''$ are disturbed for
some constant $k>0$.
Suppose further that
\begin{itemize}
\item for each color $c$, at most $2c(n)+P(n)$ edges with
color $c$ under $h''$ are prescribed;
\item at most $H(n)$ edges with color $c$ are disturbed;
\item all requested edges with a color
from $\{1,2,\dots,n\}$ under $h''$ are in $K_{n,n}$.
\item if $e$ is a prescribed edge of $K_{n,n}$ that satisfies
$\varphi'(e) \neq h''(e)$, then $h''(e) \in \{1,\dots,n\}$.
\end{itemize}
Let $uv$ be an edge of $K_{2n}$ such that
$$h''(uv)=c_1, \;\; \varphi'(uv)=c_2, \;\; c_1 \neq c_2.$$
and set $$M=\left \lfloor{\frac{n}{2}}\right \rfloor -
\Big(2\epsilon n+24c(n) +
6dn + 9P(n) + 6\beta m + 4\alpha m + 10 + 8 \dfrac{k+(67+205)/n^2}{d} n +
6H(n)\Big)$$
If $M>0$, then there is a subgraph $T$ of $K_{2n}$
and a proper $m$-edge coloring $h^T$,
obtained from $h''$ by performing a sequence of swaps on $4$-cycles in $T$,
that satisfies the following:
\begin{itemize}
\item $h^T(uv)=c_2$;
\item $h''$ and $h^T$ differs on at most $205$ edges
\emph{(}i.e. $T$ contains at most $205$ edges\emph{)};
\item besides $uv$, $h''$ and $h^T$ disagree on at most $2$ prescribed edges;
\item if $h''$ and $h^T$ disagree on a prescribed edge $ab$
\emph{(}where $ab \neq uv$\emph{)}, then $ab$ is a requested edge,
$h^T(ab)$ is not $d$-overloaded and $h''(ab) \neq \varphi'(ab)$;
\item the subgraph $T$ contains at most three edges with color $c_1$ under $h''$,
and at most four edges with color $c_2$ under $h''$;
\item except $c_1$ and $c_2$, no colors of edges in $T$ (under $h''$)
are $d$-overloaded;
\item if there is a conflict of $h^T$ with $L$,
then this edge is also a conflict of $h''$ with $L$;
\item any edge in $G_1$ or $G_2$ that is requested under $h^T$ (with respect
to $\varphi'$) is also
requested under $h''$.
\end{itemize}
\end{lemma}
\begin{proof}
We shall contruct a subgraph $T$ of $K_{2n}$, and by performing a
sequence of swaps on $4$-cycles of $T$,
we shall obtain the coloring $h^T$ from $h''$,
where $h^T$ and $\varphi'$ agree on
the edge $uv$. We will accomplish this
by applying Lemmas \ref{swapedgeinK}, \ref{swapedgeinK1},
\ref{swapedgeinG}, \ref{swapedgeinKnew}, \ref{swapedgecolor},
and in our application of these lemmas,
we will avoid the colors $\{c_1,c_2\}$; so $a=2$.
Let $e_1$ and $e_2$ be the requested edges
incident with $u$ and $v$, respectively, satisfying that
$h''(e_1)= h''(e_2)=c_2$.
We shall consider four different cases.
\vspace{0.5cm}
\noindent
\textbf{Case \pmb{$1$}}. $uv \in E(K_{n,n})$ and $c_2 \in \{1,2,\dots,n\}$:
\medskip
\noindent
Since under $h''$, all requested edges with colors in
$\{1,2,\dots,n\}$ are in $K_{n,n}$, $e_1, e_2 \in E(K_{n,n})$.
Moreover, by assumption $c_2 \in \{1,\dots,n\}$,
so we can proceed as in the proof of Lemma $3.7$ in
\cite{AndrenCasselgrenMarkstrom} and use swaps on $4$-cycles,
all edges of which are contained in $K_{n,n}$, to obtain
a coloring $h^T$ where $h^T(uv)=c_2$.
Note also that this implies that every precolored edge $e$ of $K_{n,n}$
that satisfies $h''(e) \in \{1,\dots,n\}$, also satisfies
$h^T(e) \in \{1,\dots,n\}$.
The swaps needed for obtaining the required coloring will involve
at most $69$ edges, as described in proof of Lemma $3.7$ in
\cite{AndrenCasselgrenMarkstrom}.
The exact details of the transformation of the coloring $h''$ into $h^T$
can be found in \cite{AndrenCasselgrenMarkstrom},
so we omit them here.
\vspace{0.5cm}
\noindent
\textbf{Case \pmb{$2$}}. $uv \in E(K_{n,n})$ and $c_2 \in \{n+1,n+2,\dots,m\}$:
\medskip
\noindent
In this case, we will contruct a subgraph $T$ with at most $136$ edges.
Without loss of generality, we assume that $u \in V(G_1)$,
this implies $v \in V(G_2)$.
By assumption $c_1 \in \{1,\dots,n\}$;
we choose an edge $xy \in E(K_{n,n})$ ($x \in V(G_1)$ and $y \in V(G_2)$),
with $h''(xy)=c_1$
such that the following properties hold.
\begin{itemize}
\item The edge $xy$ is not disturbed and not prescribed and $c_2 \notin L(xy)$.
Since for each pair of colors $c_1,c_2 \in \{1,2,\dots,m\}$,
there are at most $c(n)$ edges $e$
in $K_{2n}$ with $h'(e) =c_1$ and $c_2 \in L(e)$,
and at most $H(n)$ edges of color $c_1$ have been used
in the swaps for transforming $h'$ into $h''$,
this eliminates at most $H(n)+2c(n)+P(n)+c(n)$ choices.
\item The vertex $x$ satisfies the following.
\begin{itemize}
\item If $e_2 \in E(G_2)$, then we choose $x$ such that
$vx$ is a valid choice for the second edge
in an application of Lemma \ref{swapedgeinG}.
This eliminates at most
$$6c(n) + 2P(n) + 2\beta m + 2\alpha m + 5 + 4 \dfrac{k+(34+136)/n^2}{d} n +
2H(n)$$ choices.
\item If $e_2 \in E(K_{n,n})$, then since
$h''(e_2)=c_2 \in \{n+1,n+2,\dots,m\}$, we choose $x$ such that $vx$
is a valid choice for the second edge
in an application of Lemma \ref{swapedgeinKnew}.
This eliminates at most $$5c(n) + 2P(n) + \alpha m + \beta m +
4 + 2\dfrac{k+(67+136)/n^2}{d} n + 2H(n)$$ choices.
So in both cases, this choosing process eliminates at most
$$6c(n) + 2P(n) + 2\beta m + 2\alpha m + 5 + 4 \dfrac{k+203/n^2}{d} n + 2H(n)$$ choices.
\end{itemize}
\item The vertex $y$ is chosen with same strategy as $x$.
Similarly, this eliminates at most
$$6c(n) + 2P(n) + 2\beta m + 2\alpha m + 5 + 4 \dfrac{k+203/n^2}{d} n + 2H(n)$$ choices.
\item $c_1 \notin L(uy) \cup L(vx)$. This excludes at most $2\beta m$ choices.
\end{itemize}
Thus we have at least $$n - \Big(15c(n) + 5P(n) + 6\beta m +
4\alpha m + 10 + 8 \dfrac{k+203/n^2}{d} n + 5H(n)\Big)$$
choices for an edge $xy$. Since this expression is greater than zero by assumption,
we can indeed make the choice.
Now, since $M>0$, we can apply Lemma \ref{swapedgeinG}
or Lemma \ref{swapedgeinKnew}
to obtain a coloring where $uy$ is colored $h''(e_1)$.
Similarly, we can apply Lemma \ref{swapedgeinG}
or Lemma \ref{swapedgeinKnew} to thereafter obtain a coloring
where $vx$ is colored $h''(vx)$. Next, by swapping on the $2$-colored $4$-cycle
$uvxyu$, we get the proper coloring $h^T$ such that $h^T(uv)=h''(e_1)=c_2$.
Since the swaps
from the applications of
Lemma \ref{swapedgeinG} and Lemma \ref{swapedgeinKnew}
do not result in any ``new'' requested edges in $G_1$ or $G_2$,
the swaps used in this case do not no yield
any new requested edges in $G_1$ or $G_2$; similarly for all conflict
edges of $K_{2n}$.
Here, the subgraph $T$ contains the edges $uv$ and $xy$ and all
additional edges used when applying the previous lemmas above;
in total there are at most $2+67 \times 2=136$ edges in $T$.
Note further that besides $uv$ the only edges of
$K_{2n}$ that might be prescribed and are used
in swaps for constructing $h^T$ are $e_1$ and $e_2$; this property
shall be used when applying Lemma \ref{correctedge}.
\vspace{0.5cm}
\noindent
\textbf{Case \pmb{$3$}}. $uv \in E(G_1)$ (or $uv \in E(G_2)$)
and $c_2 \in \{1,2,\dots,n\}$:
\medskip
\noindent
In this case, we shall construct a subgraph $T$ with at most $139$ edges.
Without loss of generality, we shall assume that $uv \in E(G_1)$. Moreover,
since all requested edges with a color in $\{1,2,\dots,n\}$ under $h''$ are
in $K_{n,n}$, $e_1, e_2 \in E(K_{n,n})$.
If $c_1 \in \{n+1,n+2,\dots,m\}$, then we choose an
edge $xy \in E(G_2)$ such that $h''(xy)=c_1$
and $xy$ is not prescribed or disturbed.
If $c_1 \in \{1,2,\dots,n\}$, then
we choose an edge $xy \in E(G_2)$ to be the
first edge in an application of Lemma \ref{swapedgecolor};
this choice implies that $xy$ is not prescribed and not disturbed.
So in both case we can have at least
$$\left \lfloor{\frac{n}{2}}\right \rfloor - 7c(n) -
3P(n) - dn - 2H(n)$$ choices for $xy$.
In addition to this, $xy$ also needs to satisfy the following:
\begin{itemize}
\item The edges $ux$ and $vy$ are valid choices for the second edge
in an application of Lemma \ref{swapedgeinK1}.
This eliminates at most $$2\Big(4c(n) + P(n) +
2\beta m + 2\alpha m + 5 + 4 \dfrac{k+(34+139)/n^2}{d} n + H(n)\Big)$$ choices.
\item $c_2 \notin L(xy) \cup \varphi'(x) \cup \varphi'(y)
\setminus \{\varphi'(xy)\}$.
This condition will imply that after performing all swaps in this case,
$xy$ is not a ``new'' requested edge of $G_2$.
Since there are at most $c(n)$ edges $e$
in $K_{2n}$ such that $h'(e) =c_1$ and $c_2 \in L(e)$,
and we have already excluded the choices for $xy$ which are disturbed,
this condition eliminates at most $c(n) + 2(2c(n)+P(n))$ choices.
\item $c_1 \notin L(ux) \cup L(vy)$. This excludes at most $2\beta m$ choices.
\end{itemize}
So in final, we have at least
$$\left \lfloor{\frac{n}{2}}\right \rfloor - \Big(20c(n) + dn + 7P(n) + 6\beta m + 4\alpha m + 10 + 8 \dfrac{k+173/n^2}{d} n + 4H(n)\Big)$$
choices for $xy$. Since this expression is greater than zero
by assumption, we can indeed make the choice.
Since $M>0$, firstly if $c_1 \in \{1,2,\dots,n\}$,
we can apply Lemma \ref{swapedgecolor} to obtain a coloring
where $xy$ is colored $c_1$.
Secondly, we can apply Lemma \ref{swapedgeinK1} twice to
obtain a coloring where $ux$ is colored $h''(e_1)$ and $vy$ is colored
$h''(e_2)$. Finally, by swapping on the $2$-colored $4$-cycle
$uvyxu$, we get the proper coloring $h^T$ such that $h^T(uv)=h''(e_1)=c_2$.
Note that this implies that $uv$ is not a requested edge under $h^T$.
More generally, since the swaps from the applications of
Lemma \ref{swapedgeinK1} and Lemma \ref{swapedgecolor}
do not result in any new requested edges in $G_1$ or $G_2$,
the swaps used in this case do not yield
any new requested edges in $G_1$ or $G_2$; similarly for conflict edges
in $K_{2n}$.
Here, the subgraph $T$ contains the
edges $uv$ and $xy$ and all the additional edges needed to apply the lemmas above
(if we need to apply Lemma \ref{swapedgecolor},
then $xy$ is included in the edges used when applying this lemma);
in total, $T$ uses at most $1+ 70 + 34 \times 2=139$ edges.
\vspace{0.5cm}
\noindent
\textbf{Case \pmb{$4$}}. $uv \in E(G_1)$ (or $uv \in E(G_2)$)
and $c_2 \in \{n+1,n+2,\dots,m\}$:
\medskip
\noindent
In this case we proceed similarly to Case $3$;
using the same setup as in Case 3, a slight difference between two cases
is that in Case $4$,
we can use Lemma \ref{swapedgeinG} or Lemma \ref{swapedgeinKnew} to
obtain a coloring where $ux$ is colored $h''(e_1)$
and $vy$ is colored $h''(e_2)$. Similar calculations as above yields that
we have at least
$$\left \lfloor{\frac{n}{2}}\right \rfloor - \Big(24c(n) + dn +
9P(n) + 6\beta m + 4\alpha m + 10 + 8 \dfrac{k+(67+205)/n^2}{d} n + 6H(n)\Big)$$
choices for $xy$; and this expression is greater than zero by assumption,
so we can indeed make the choice and perform the necessary swaps to
get the coloring $h^T$ satisfying
$h^T(uv)=h''(e_1)=c_2$. Here, $T$ contains at most $1+ 70 + 67 \times 2=205$ edges.
\bigskip
Finally, let us note that
in the first two cases, $T$ contains exactly two edges
with color $c_1$ under $h''$.
In the last two cases, $T$ contains exactly two edges
with color $c_1$ under $h''$ if we
do not have to apply Lemma \ref{swapedgecolor};
otherwise $T$ contains exactly three edges with color $c_1$ under $h''$.
Any application of Lemma \ref{swapedgeinK1}, \ref{swapedgeinG}
or \ref{swapedgeinKnew}
above uses at most two edges with color $c_2$ under $h''$, so
the subgraph $T$ contains at most four edges with color $c_2$ under $h''$.
Except $c_1$ and $c_2$, the subgraph $T$ only contains edges with
colors that are not $d$-overloaded.
\end{proof}
We will take care of every prescribed edge $e$ of $K_{2n}$
such that $h'(e) \neq \varphi'(e)$ by successively applying
Lemma \ref{correctedge}; using this lemma we can construct the
proper $m$-edge colorings of $K_{2n}$ $h_0=h'$,
$h_1$, $h_2$,
\dots, $h_q$, where $h_i$ is constructed from $h_{i-1}$ by an application of
Lemma \ref{correctedge} and $h_q$ is a extension
of $\varphi'$.
Since the number of prescribed edge at each vertex of $K_{2n}$
is at most $\alpha m+ c(n)$, the total number
of prescribed edges in $K_{2n}$ is at most $2n(\alpha m+c(n))$,
thus $q \leq 2n(\alpha m+c(n))$.
When we apply Lemma \ref{correctedge}, we first consider all prescribed edges $e$
in $K_{n,n}$ that satisfies $\varphi'(e) \in \{1,\dots, n\}$ (Case 1 in the lemma).
This is important, since otherwise we might recolor such edges by colors
from $\{n+1,\dots, m\}$, and are thereafter unable to apply Lemma \ref{correctedge}.
Thereafter we apply Lemma \ref{correctedge} to all prescribed edges $e$ of $K_{n,n}$
that satisfies $\varphi'(e) \in \{n+1, \dots,m\}$
(Case 2 in Lemma \ref{correctedge}).
Note that after performing all the swaps as described in the preceding paragraph,
we have not recolored any edge of $G_1$ or $G_2$.
Thus, if one of the requested edges $e_1$ and $e_2$
in Case 2 of the proof of Lemma \ref{correctedge}
is in $K_{n,n}$, then it has been used in a previous application of
Lemma \ref{correctedge} to a prescribed edge $e'$ of $K_{n,n}$
that satisfies $\varphi'(e') \in \{n+1, \dots,m\}$.
Moreover, since the only prescribed edges that are used in an application
of Lemma \ref{correctedge} is $uv$ and (possibly)
the requested edges $e_1$ and $e_2$,
it follows that every prescribed edge $e$ in $K_{n,n}$ that satisfies
$\varphi'(e) \in \{n+1,\dots, m\}$ is not recolored
by a color from $\{n+1,\dots,m\} \setminus \{\varphi'(e)\}$
in the process of applying Lemma \ref{correctedge} to the prescribed
edges $e$ of $K_{n,n}$ that satisfies $\varphi'(e) \in \{n+1, \dots,m\}$.
Thus, we may assume that $h''(e) \in \{1,\dots, n\}$ for any intermediate coloring
$h''$ and any precolored edge $e$ in $K_{n,n}$. Hence, we can perform
all the swaps as described in Case 2 in the proof of Lemma \ref{correctedge}.
Thereafter, we consider all prescribed edges of $G_1$ and $G_2$.
In an application of Lemma \ref{correctedge}
to obtain $h_i$ from $h_{i-1}$,
we use swaps involving at most three prescribed edges: the edge $uv$,
and the two adjacent requested edges
$e_1$ and $e_2$.
Since there are at most $2c(n)$ prescribed edges in $K_{2n}$
with any given color $c$ in $h'$,
there are at most $\alpha m+f(n)$ prescribed edges with
color $c$ under $\varphi'$, and
$h_i(e_1)$ and $h_i(e_2)$ are not $d$-overloaded colors
in the coloring $h_{i-1}$,
it follows that
for each $i=1,\dots,q$, there are at most
$2c(n)+ dn+ \alpha m+f(n)$ edges with color $c$ under $h_i$ that are prescribed.
Furthermore, each application of Lemma \ref{correctedge}
to a prescribed edge $uv$ with $h'(uv)=c$
constructs a subgraph $T$ with at most three edges with color
$c$ under $h'$; thus a color $c$ is used at most
$3\big(2c(n)+dn+\alpha m+f(n)\big)$ times in a subgraph $T$
where a prescribed edges has color $c$ in $h'$.
Moreover, there are at most $\alpha m+f(n)$ prescribed edges
with color $c$ under $\varphi'$, and a subgraph $T$ constructed
by an application of Lemma \ref{correctedge} uses at most four edges with color $c$.
Except for these edges, any other edges contained in a subgraph
created by an application of Lemma \ref{correctedge} are colored by
colors that are not $d$-overloaded.
Hence, at most
$$4\big(\alpha m+f(n)\big)+3\big(2c(n)+dn+\alpha m+
f(n)\big)+dn= 7\alpha m+7f(n)+6c(n)+4dn$$
distinct edges with color $c$ under $h'$ are used in swaps for
constructing $h_q$ from $h'$.
Let $H(n)=7\alpha m+7f(n)+6c(n)+4dn$, $P(n)=dn+\alpha m+f(n)$;
from the preceding paragraph we deduce that
as long as $kn^2 \geq 205 \times 2n(\alpha m+c(n))=410n(\alpha m+c(n))$
and
$$\left \lfloor{\frac{n}{2}}\right \rfloor - \Big(2\epsilon n + 24c(n) + 6dn + 9P(n) + 6\beta m + 4\alpha m + 10 + 8 \dfrac{k+272/n^2}{d} n + 6H(n)\Big)>0$$
$$c'(n)=c(n)/2; \; n-1>2c(n)>4; \;
\Big( \dfrac{4\beta}{\epsilon - 4\beta}\Big)^{\epsilon - 4\beta} \Big( \dfrac{1}{1 - 2\epsilon + 8\beta}\Big)^{1/2-\epsilon + 4\beta} <1$$
$$\alpha, \beta < \dfrac{c(n)}{2(n-c(n))} \Big(\dfrac{n - c(n)}{n}\Big)^{\frac{n}{c(n)}}; \;
\beta<\dfrac{c'(n)}{2(n-c'(n))} \Big(\dfrac{n - c'(n)}{n}\Big)^{\frac{n}{c'(n)}}$$
$$m - \beta m -2 \alpha m - 2c(n) - \dfrac{2nc(n)}{f(n)} \geq 1$$
for some constants $\alpha, \beta, \epsilon, k,d$
and functions $c(n)$, $f(n)$ of $n$,
we can apply Lemma \ref{alpha} to obtain $h'$,
Lemma \ref{gamma} to obtain $\varphi'$ and finally Lemma \ref{correctedge}
to obtain the coloring $h_q$ which is a completion
of $\varphi'$ that avoids $L$.
This completes the proof of Theorem \ref{mainth}.
\end{proof} | 136,920 |
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REAL PEOPLE, REALLY LAUGHING. For the joy of seeing laughter. Natural photos which capture people laughing....(NOT just smiling, acting silly or making funny faces). If you wish to post pics of funny faces, I suggest this group:
FUNNY FACES
Only 2 simple rules for this group:
1. They must be natural shots, natural people, naturally laughing
2. So we get a nice visual mix, only ONE photo posting at a time.
Now, let's share a laugh! ;)
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TITLE: Finite Field Question: Which of the followings are true?
QUESTION [1 upvotes]: I have the following True or False question that I am having trouble getting it correct. I've written down my thoughts on each choice. If anyone could verify my thoughts or tell me where I made a mistake, I would really appreciate.
Question: Which of the followings are true?
It is possible for an irreducible polynomial in $\mathbb{Q}[X]$ to be inseparable.
Every irreducible polynomial over a finite field $K$ is separable.
The polynomial $X^p - t\in\mathbb{F}_p (t)[X]$ over the finite field of rational functions $\mathbb{F}_p (t)$ is separable.
The field $\mathbb{F}_p(t^{1/p})$ for the polynomial $X^p - t$ has degree $p$ over $\mathbb{F}_p(t)$ and is a splitting field for $X^p - t$.
Let $K$ be the algebraic closure of $\mathbb{F}_p (t)$. Let $\mathbb{F}_p (t^{1/p})$ be a stem field of $X^p - t$. Then there are $p$ $\mathbb{F}_p (t)$-homomorphisms from $\mathbb{F}_p(t^{1/p})$ to $K$.
My thoughts:
False. Let $p\in\mathbb{Q}[X]$ be irreducible. Since $char(\mathbb{Q})=0$, we must have $\deg(\gcd(p,p'))\le\deg(p')<\deg(p)$. This implies that $\gcd(p,p')=1$, which implies that $p$ is separable.
My "guess" is False. We can take a polynomial $p$ such that $p'$ vanishes. This sounds plausible since we are in a finite field, and we can take polynomials in form of $X^{p^r}$. Then $p$ is inseparable. But I am not sure if we can get irreducible polynomials at the same time. I remember that $X^p - 1$ is irreducible but I forgot the proof.
False. Let $f = X^p -t$, then $f' = 0$ since $char(\mathbb{F}_p(t))=p$. Then $\gcd(f,f')=f\ne 1$. So $f$ has to be inseparable.
True. $t^{1/p}$ is a root of $f = X^p - t$ and $f$ is irreducible. So degree of extension is $p$. $\mathbb{F}_p(t^{1/p})$ is a splitting field because $X^p - t=(X-t^{1/p})^p$ due to Frobenius map.
I have no idea but I would guess "True" if I have to guess :(
REPLY [2 votes]: Corrections/comments/whatnot:
Your answer to part 2 is incorrect. The key is that when $K$ is a finite field of characteristic $p$, every one of its elements is also a $p$th power. That is to each $z\in K$ there exists a $y\in K$ such that $y^p=z$. If the derivative of $f(x)\in K[x]$ vanishes, then $f(x)=a_0+a_1x^p+a_2x^{2p}+\cdots+a_nx^{np}$ for some natural number $n$ and some elements $a_i\in K$. Selecting elements $b_i\in K$ such that $b_i^p=a_i$ we see, by the characteristic $p$ power law $(u+v)^p=u^p+v^p$, that
$$
f(x)=(b_0+b_1x+b_2x^2+\cdots+b_nx^n)^p.
$$
So if $f'=0$ then $f$ cannot be irreducible in $K[x]$.
Your answers to parts 1,3, and 4 are correct.
Your guess to part 5 is, however, incorrect. There is only a single such homomorphism. This is because the usual logic shows that such a homomorphism $\phi$ is fully determined, if we know $y=\phi(t^{1/p})\in K$. This element must satisfy the equation $y^p=t$ But the result of part 4 shows that there is only one such element $t\in K$. | 186,742 |
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The expense of spent ongoing happened to be installed changed internet dating January of assessment can that comes with the warmth. It is actually most high-priced to pay for the one-month registration even though the twelve-month registration ideal the cheapest entire per evaluation when it comes to price. At the time you sign up for a one-year account, gain extra six months included about the original account free-of-charge. When compared with almost every other internet dating internet sites, warmth. Even as a free of cost individual, feel free to use boards and so the easy messenger characteristic to speak with other free or superior users independently. You also get extra several months of program provided in the event you sign up to get reasonably limited manhood a person three or twelve period durations. You should also enjoy the sugars Daddie assessment. | 128,460 |
When I edit an contact on Android it is saved as a VCard3 even if it was retrieved as VCard4.
“davdroid: [syncadapter.ContactsSyncManager] Server advertises VCard/4 support: false”
EDIT: Using other clients (like Thunderbird+Cardbook Addon) I can create VCard4 contacts on the Radicale server.
Resulting in the following behavior: Using TB Cardbook to generate Contact results in a VCard4, then editing it with Davdroid results in a new VCard3.
Is this a bug of DAVdroid or Radicale? Can I circumvent it somehow?
I already posted that in the Radicale Bugtracker some time ago: | 366,559 |
TITLE: When the sum of all divisors of a natural number is odd
QUESTION [0 upvotes]: Prove that the sum of all divisors of a natural number $n$ is odd if and only if $n = 2^r \cdot k^2$ where $k$ and $r$ are natural numbers.
The first direction: if $k^2$ is an even number, we rewrite $2^r \cdot k^2$ as $2^{(r+m)} \cdot z$ where $z$ is a natural odd number. Now, the sum of all divisors of $2^{(r+m)}$ without $1$ is even, and the sum of of all divisors of $z$ is odd, so we get odd.
This is true?
To the other direction, I have no idea...
REPLY [1 votes]: A different approach:
Assume that the prime factorization of $n$ is $n=2^{a_0}p_1^{a_1}p_2^{a_2}\cdots p_s^{a_s}$ where $p_i$ are odd primes different with each other. Let's write $p_0=2$.
Any divisor of $n$ is of the form $p_0^{b_0}p_1^{b_1}p_2^{b_2}\cdots p_s^{b_s}$ with $0\leq b_i\leq a_i, \ \forall i$. Therefore the sum of the divisors of $n$ is $\sum p_0^{b_0}p_1^{b_1}p_2^{b_2}\cdots p_s^{b_s}$ where the sum range over all $b_i$ with $0\leq b_i\leq a_i, \ \forall i$.
Equivalently the sum can be written as a product:
$$\sum_{0\leq b_i\leq a_i}p_0^{b_0}p_1^{b_1}p_2^{b_2}\cdots p_s^{b_s}=\prod_{i=0}^s(1+p_i+p_i^2+\ldots+p_i^{a_i}).$$
Now the sum of the divisors of $n$ is odd iff $1+p_i+p_i^2+\ldots+p_i^{a_i}$ is odd for all $i=1,2,\ldots, s$. For $i=0, \ 1+p_i+p_i^2+\ldots+p_i^{a_i}$ is odd. For $i>0$ (i.e. $p_i$ odd) when is $1+p_i+p_i^2+\ldots+p_i^{a_i}$ odd?
If $n=2^rk^2$ what does this mean for the $a_i$? Conversely for what $a_i$, $n$ can be written as $2^rk^2?$ | 138,298 |
TITLE: Is there a temperature at which ice is denser than water?
QUESTION [14 upvotes]: Normally ice would float on water because its density is less compared to that of water as a liquid.
But is it possible that its density will increase due to a very low temperature or is ice in any case lighter than water?
REPLY [23 votes]: Ice can be denser than water for certain values of $P,T$. Look at these two pictures taken from here:
The darker areas in the second picture denotes areas of greater density. So you can clearly see that when pressure is increased, ice becomes denser than water along the coexistence line.
For example at $T=400$ K ice VII is clearly denser than water along the coexistence line ($P \simeq 2$ GPa).
Quoting from the page:
As pressure increases, the ice phases become denser. They achieve this by initially bending bonds, forming tighter ring or helical networks, and finally including greater amounts of network inter-penetration. This is particularly evident when comparing ice-five with the metastable ices (ice-four and ice-twelve) that may exist in its phase space.
At atmospheric pressure, $P_{atm} \simeq 100$ kPa $=10^5$ Pa, ordinary ice is always less dense than water.
Upadate: how to read the pictures
When I posted this answer, I may have taken for granted that everybody was able to read this kind of phase diagram, but since it looks like I was wrong, I will try to explain them better.
The first diagram shows the various phases of water as a function of the two parameters $P,T$. The first thing that must be noticed is that the pressure axis is logarithmic while the temperature axis is linear. This means that the plot is "compressed" in the vertical direction (you can see that the $T$ axis goes from $1$ to $800$ (almost 3 orders of magnitude) while the $P$ axis goes from $0.1$ to $10^{12}$ (13 orders of magnitude!)).
The black lines are coexistence lines. This means that along those lines two phases can coexist. If we want to compare water and ice, I think that the only meaningful way to do so is to compare them along the coexistence line, because only there it will be possible to have both of them. For example, you can see that ice VIII can never coexist with liquid water.
Our world is located at $P= 1$bar$\simeq 10^5$ Pa (red line):
You can in fact see that, at the red line, the solid-liquid transition is at $273$ K ($0$°C) and the liquid-vapor is at $373$ K ($100$°C) - as expected.
But things get different at different pressures. For example, at $10^6$ Pa ($10\times$atm.pressure), the liquid-vapor transition is at $450$K, and at $10^2$ Pa ($1/1000$ of atm.pressure) ice sublimates directly into vapor (there is no liquid state!).
Now, the density. You have to look at the second plot to see the density.
For example, let's take the $400$ K-$2\cdot10^9$Pa point (yellow arrow in the first plot). To see the density, look at the corresponding point in the second plot. You can see that the area corresponding to ice (ice VII) is darker than the area corresponding to water, so you can tell that ice is denser than water there, and so on.
If you take P=$10^5$ Pa (atm.pressure), you can see that ice is always less dense than water (lighter shading) there. | 92,430 |
It’s not often you find yourself excited to come to the office on a weekend, especially earlier than you usually arrive on weekdays. This past Saturday, June 25, was an exception, as we hosted our first public Brite Camp workshop on Python.
More than sixty people (a pretty even mix of Britelings and outside friends) showed up bright and early on Saturday to improve their coding skills. One-third of the crowd hadn’t ever programmed before; one-third had some understanding of basic programming concepts; the final one-third were developers who knew other languages and wanted to learn Python.
Bob, our Senior API Engineer, took the beginners through the basics of programming, with help from TAs Kiel and Steve, who walked around the class ensuring everyone understood what was going on. By the end of the day, the participants had a number guessing game to show their friends.
Senior Software Engineer, Nesan, took the intermediate class through all the beginner material and beyond. Realizing that he had a sharp audience, Nesan blew through the lesson plan in a matter of minutes, then focused the rest of the day on application of concepts. In the end, not only did participants have a more robust customized version of the number guessing game, they also had a voting app.
On the advanced end of things, Rome and his TA, Michael, took more experienced developers through the differences in Python, introduced them to Django for Web development, and helped them build their own URL shortener.
By the time 4:00 p.m. hit, we had to convince people that we were out of material for the day — they didn’t want to leave!
We couldn’t be happier with how the day went, and a post-event survey revealed the same from our participants. Several have emailed their teachers directly, some blogged about their experience, others provided us with feedback like:
“It was very down to Earth. And I was actually able to complete the challenges!! Great workshop. Would love to come again.”
“Fantastic! Great energy. Perfect pace. Teaching and helping was amazing. One of the best trainings I’ve ever been to.”
“Good presentation. Fun open format for projects that allowed us to mess around and be creative in our solutions.”
“The teachers were great and I loved the casual, friendly atmosphere!! “
“Great instruction, very engaging, and awesome attitude. Can’t wait for the next one. Very friendly all around. Everyone seemed to really want to do this, as opposed to just doing it because they said they would. “
“The instructors (especially Bob and Kiel) were awesome! Very patient and friendly. The orange t-shirts were hot. In all seriousness, I really did learn a lot in a short span of time. I felt comfortable asking lots of questions (which is critical for learning), and it was very rewarding to learn how to write a short program. I had a ton of fun!”
We’ll be sure to put on more public Brite Camps, and may even continue the Python curriculum in the coming months so folks can continue to learn! Stay tuned! | 47,703 |
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CodePlexProject Hosting for Open Source Software
Hello there, I really tried to search help and discussion but still don't know how to do that.
My documentation built to website and browsed e.g. in IE shows always in address bar only the URL to the Index.aspx e.g.
But your documentation on the web shows complete URL
Please, how can I force my website to show complete URL including stuff like
...Index.aspx?topic=html/b772e00e-1705-4062-adb6-774826ce6700.htm
Thank you for any help in advance and thank you for SHFB, Pavol
Currently, the website doesn't update the URL with the selected topic. It's probably possible by modifying TOC.js but I haven't looked into how it may be done.
Eric
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\begin{document}
\begin{abstract}
The recently discovered fourth class of Frobenius manifolds by Combe--Manin in \cite{CoMa} opened and highlighted new geometric domains to explore.
The guiding mantra of this article is to show the existence of hidden geometric aspects of the fourth Frobenius manifold, which turns out to be related to so-called {\it causality conditions.}
Firstly, it is proved that the fourth class of Frobenius manifolds is a Pseudo-Elliptic one.
Secondly, this manifold turns out to be a sub-manifold of a non-orientable Lorentzian projective manifold.
Thirdly, Maurer--Cartan structures for this manifold and hidden geometrical properties for this manifold are unraveled.
{\it In fine}, these investigations lead to the rather philosophical concept of causality condition, creating a bridge between the notion of causality coming from Lorentzian manifolds (originated in special relativity theory) and the one arising in probability and statistics.
\end{abstract}
\maketitle
\tableofcontents
\section*{Introduction}
The notion of Frobenius manifolds (resp. $F$-manifolds) is the fruit of fifty years of remarkable interaction between topology and quantum physics. This relation involves the most advanced and sophisticated ideas on each side, and lead to Topological Quantum Field Theory.
Three classes of Frobenius manifolds include quantum cohomology (topological sigma-models), unfolding spaces of singularities (Saito’s theory, Landau-Ginzburg models), and Barannikov--Kontsevich construction starting with the Dolbeault complex of a Calabi--Yau manifold and conjecturally producing the B-side of the mirror conjecture in arbitrary dimension \cite{Ma99}.
The recently discovered fourth class of Frobenius manifolds by Combe--Manin in \cite{CoMa} opened and highlighted new geometric domains to explore. The aim of this article is to show the existence of {\it hidden geometric aspects} of the fourth Frobenius manifold.
This class of Frobenius manifolds includes manifolds of probability distributions, given by a triple $(M,g,\nabla)$ where $M$ is a manifold, equipped with a Riemannian metric $g$ and a torsion free connection $\nabla$.
As it was shown in \cite{CoCoNen1,CoMa}, the $F$-manifolds in the sense of Combe--Manin \cite{CoMa} have an unexpected meaning from the point of view of algebraic geometry. To give an example, the tangent space to this manifold can be interpreted as a module over a certain unital associative, commutative rank 2 algebra. This algebra being a non division algebra, implies extremely rich non Euclidean geometrical properties and leads to unexpected developments, such as in \cite{CoMaMa1,CoMaMa2}.
\smallskip
These geometric investigations lead unexpectedly to the rather philosophical notion of {\it causality}. It turns out that, using our construction, a bridge between the notion of causality coming from Lorentzian manifolds given by S. W. Hawking and G. F. R Ellis in \cite{HaEl} and the notion of causality, arising in probability and statistics \cite{Sha96} can be established,.
\smallskip
According to B. Dubrovin (\cite{D96} and \cite{Ma99}), the main component of a Frobenius structure
on $M$ is a {\it (super)commutative, associative and bilinear over constants
multiplication $\circ : \cT_M \otimes \cT_M \to \cT_M$ on its tangent sheaf $\cT_M$}.
Additional parts of the structure in terms of which further restrictions upon $\circ$
might be given, are listed below:
\begin{list}{--}{}
\item A subsheaf of flat vector fields $\cT^f_M \subset \cT_M$ consisting of
tangent vectors flat in a certain affine structure.
\item A metric (nondegenerate symmetric quadratic form) $g: S^2(\cT_M)\to \cO_M$.
\item An identity $e$.
\item An Euler vector field $E$ .
\end{list}
We start, as above, with a family of data
\begin{equation}\label{E:FM}
(M; \quad \circ : \cT_M\otimes \cT_M \to \cT_M; \quad \cT_M^f \subset \cT_M; \quad
g : S^2(\cT_M) \to \cO_M),
\end{equation}
mostly omitting identity $e$ and Euler field $E$.
\vspace{3pt}
The main additional structure bridging these data together is a family of (local) {\it potentials} $\Phi$
(sections of $\cO_M$)
such that for any (local) flat tangent fields $X,Y,Z$ we have
\[
g(X\circ Y,Z) = g(X, Y\circ Z) =(XYZ)\Phi .
\]
If such a structure exists, then (super)commutativity and associativity of $\circ$
follows automatically, and we say that the family~\eqref{E:FM} defines a {\it Frobenius manifold.}
However, relying on the algebraic and geometric developments of \cite{CoCoNen1,CoMa}, it has been shown that we can consider this space of probability distributions as a manifold of 0-pairs i.e. the space of $(n-1)$-hyperplanes lying in a vector space of dimension $n$. The theory of $m$-pairs was introduced by E. Cartan in \cite{Ca31}, A.P Shirokov\, \cite{Sh87,Sh02} and A.P. Norden\, \cite{No76}. The interpretation of the fourth class of Frobenius manifold as a manifold of 0-pairs has been proved in \cite{CoMa} in Proposition 5.9.
\smallskip
Relying on this fundamental result, leads to proving that the fourth class of Frobenius manifolds are projective (non Euclidean) non orientable manifolds which have the structure of a Lorentzian manifold.
Throughout the paper we will prove these two following statements:
\begin{theorem-non A}
The class of the fourth Frobenius manifolds is a pseudo-Elliptic manifold of type $S^n_1$ equipped with the following {\it Maurer--Cartan structures}:
\[
\left\{\begin{aligned}
d{\bf r} &= \omega {\bf r} + \omega^sX_s,\\
dX_i &= \omega_i {\bf r} + \omega_i^jX_j.
\end{aligned}\right.\]
\end{theorem-non A}
\begin{theorem-non}
The fourth class of Frobenius manifolds satisfies the following geometric properties:
\begin{enumerate}
\item it is a projective non-orientable submanifold of a Lorentzian projective manifold;
\item it is equipped with a non Euclidean geometry;
\item this manifolds is uniquely determined by an orientable 2-sheeted cover;
\item it is decomposed into two domains, which are symmetric about a real (projective) Hermitian hyperquadric, being the symmetry hyperplane mirror.
\end{enumerate}
\end{theorem-non}
The method of the proof is essentially based on the Cartan's theory of $m$-pairs and Norden--Shirokov's normalization theory, which from an other point of view implies that Maurer--Cartan structure equations are used. The paper is decomposed as follows.
\medskip
\subsubsection{Plan of the paper}\
$\ast$ In section 1, we recall fundaments on manifolds of probability distributions, as well as differential manifolds defined over an associative, commutative algebra of finite dimension. Since it has been shown that the fourth Frobenius manifold is related to the geometry over the algebra of paracomplex numbers (see \cite{CoMa}), we focus in particular on the rank 2 algebra of paracomplex numbers.
\smallskip
$\ast$ In section 2, we develop the geometry of $m$-pairs, applied to the manifold of probability distributions. By $m$-pair, we mean a pair consisting of an $m$-plane and an $(n-m-1)$-plane in a space of dimension $n$. This relies on Cartan's $m$-pairs theory and Norden--Shirokov's normalisation theory. It leads to establishing a bridge between paracomplex geometry, projective geometry and the manifold of probability distributions.
\smallskip
$\ast$ In section 3, we show that the class of the fourth Frobenius manifold can be considered as pseudo-Elliptic manifolds. The link between section 3 and section 2 relies on a theorem of Rozenfeld\, (see section 4.1.1 in \cite{Roz97}). From this it follows that class of fourth Frobenius manifolds are Lorentzian. Maurer--Cartan structures for those manifolds are presented.
\smallskip
$\ast$ Section 4 considers the automorphism group of the manifold of probability distributions. We complete a theorem by Wolf \cite{Wo}. This theorem classifies a Riemannian symmetric space, the associated Lie group and its totally geodesic submanifolds. The theorem was stated only for the field of real, complex, quaternion, octonion numbers and was left open for algebras. We complete the result for the algebra of paracomplex numbers. In this section we prove that this specific manifold has a pair of totally geodesic submanifolds, which are isomorphic to the cartesian product of real projective spaces.
\smallskip
$\ast$ Section 5 is a conclusion and highlights the connection to causality problems. In particular, it raises questions of bridging causality notion investigated by Hawking, Ellis and the causality notion from probability and statistics.
\vfill\eject
\subsection*{List of notations}\
\vspace{5pt}
\begin{tabular}{l c l}
$A$ & & algebra of finite dimension \\
$A E^n$ & & affine space over the algebra $A$\\
$A\frak{M}^{m}$ & & $m$-module defined over algebra $A$\\
$\frak{A}$ & & spin factor algebra\\
$B_l^{n}(x,y)$ & & bilinear form of signature $(l,n-l)$\\
$\cC$ & & cone in $\mathcal{W}$ of (strictly) positive measures \\
$\fC$& & algebra of paracomplex numbers\\
$\fC E^m$ & & paracomplex affine space\\
$\fC K$ & & unitary paracomplex space\\
$\fC\cP^n$ & & projective paracomplex space\\
$E^{m}$ & & $m$-dimensional real linear space.\\
$\mathcal{I}$& & Ideal\\
$\frak{M}^{m}$ & & $m$-module \\
$R^n_l$ & & pseudo-Euclidean space of index $l$\\
$S^n_l$ & & pseudo-Elliptic space of index $l$\\
$\mathcal{W}$ & & linear space of signed measures \\
& & with bounded variations, vanishing on an ideal $\mathcal{I}$ \\
$X_m$ & & $m$-dimensional hypersurface in affine space\\
\end{tabular}
\medskip
\section{Overview on the paracomplex geometry}
In this section we present several building blocks to construct manifolds over a finite unital commutative associative algebra.
\subsection{Fifth Vinberg cone}
Consider $(X,\cF)$ the measure space, as defined in the annexe section. Let $\cI$ be an ideal of the $\sigma$--algebra $\cF$, on which measures vanish. Let $\mathcal{W}$ be a linear space of signed measures with bounded variations, vanishing on the ideal $\mathcal{I}$ of the $\sigma$-algebra $\mathcal{F}$. Let $\cC$ be a cone in $\mathcal{W}$ of (strictly) positive measures on the space $(X,\cF)$, vanishing only on an ideal $\cI$ of the $\sigma$--algebra $\cF.$ We have the following result, bridging the cone $\cC$ and the fifth Vinberg cone.
\begin{theorem}\label{T:+++}
The positive cone $\cC$, defined above, is a Vinberg cone, defined over the algebra of paracomplex numbers.
\end{theorem}
This follows from \cite{CoMa}.
\begin{remark}
The $n$-dimensional Vinberg cones are in bijection with (semi-simple) Jordan $n$-dimensional algebra.
Moreover, in the work of Vinberg, there were introduced left symmetric algebras on the convex homogeneous cones. This algebraic structure was given the name Vinberg algebras.
These algebras are also referred as {\it pre-Lie algebras}, in domains concerned by Hochschild cohomology problems \cite{G} and operads. In Gerstenhaber's works, the Lie bracket involved in the Gerstenhaber structure on the Hochschild cohomology comes from a pre-Lie algebra structure on the cochains.
\end{remark}
\smallskip
In this context a Vinberg cone $\cC\subset \cW$ is a non--empty subset, closed with respect to addition and multiplication by positive reals. A convex cone $\cC$ in a vector space $\cW$ with an inner product has a dual cone $\cC^*=\{a\in \cW:\, \forall b\in \cW,\, \langle a,b\rangle>0 \}$. The cone is self-dual when $\cC=\cC^*$. It is homogeneous when to any points $a,b\in \cC$ there is a real linear transformation $T:\cC\to \cC$ that restricts to a bijection $\cC\to \cC$ and satisfies $T(a)=b$. Moreover, the closure of $\cC$ should not contain a real linear subspace of positive dimension.
\smallskip
The automorphism group $\fG$ of this cone forms a real, solvable Lie subgroup of $GL_{n}(\R)$ and the action is simply transitive. The cone is invariant under this Lie group. One can easily pass to the corresponding Lie algebra. This Lie algebra inherits the property of being real and solvable. It splits into an abelian Lie algebra and a nilpotent one. One can establish a bijection between the Vinberg cone, the Lie group, the corresponding Lie algebra and the Vinberg algebra.
\smallskip
\subsection{Two dimensional unital algebras}
As a necessary tool towards the understanding of the cone $\cC$ and its related objects, we start with algebraic definitions. Namely, the definition of the rank 2 algebras, including the spin factor algebra.
Consider the unital and bi-dimensional algebra $\fA$, defined by following relations:
\[e_{i}\cdot e_{j}=\sum_{k}C^{k}_{ij}e_{k}\quad \text{with}\quad C^{k}_{ij}=C^{k}_{ji}.\]
The classification of 2-dimensional unital algebras, generated by generators $\{1, \e \}$, splits into three main classes:
\begin{equation}
\e^2=\left\{\begin{aligned}
1&\quad \text{in the paracomplex case, denoted:}\quad \fC\\
-1&\quad \text{in the complex case, denoted:} \quad \C \\
0 & \quad \text{in the dual case, denoted:} \quad C_0 \end{aligned}\right.
\end{equation}
\subsubsection{1. Paracomplex numbers} The algebra of paracomplex numbers given by $\langle 1,\e \, |\, \, \e^2=1\rangle$ can be defined, after a change of basis, by a pair of new generators such that:
\begin{equation}e_{-}= \frac{1-\e}{2},\quad e_{+}=\frac{1+\e}{2}.\end{equation}
These generators have the following relations:
\[ e_{-}\circ e_{-}=e_{-},\quad e_{+}\circ e_{+}=e_{+},\quad e_{-}\circ e_{+}=0,\]
\[e_{-}+ e_{+}=1,\quad e_{-}- e_{+}= \e.\]
we call this new basis a canonical basis. Notice that this new basis highlights the existence of a {\it pair of idempotents} i.e. $e_{-}^2=e_{-}$ and $e_{+}^2=e_{+}$. This algebra has 0 divisors, being different from 0: it is not a division ring. Paracomplex numbers can be associated to the coordinate ring $\fC=\R[x]/(x^2-1)$.
The structure constants $C^{k}_{ij}$ are:
\begin{equation}C_{11}^{1}=C_{12}^2=C_{22}^1=\, 1, \end{equation}
the other structure constants are null.
This semi-simple algebra is isomorphic to $\mathbb{R}\oplus \mathbb{R}$. It can be identified to $\mathbb{R}^{2}$, as a set, but not as an algebra.
Note that the algebra of paracomplex numbers is also known as the spin factor algebra (see p.4 in \cite{MC04}).
The conjugation operation is as follows. For a given paracomplex number $z=x+\e y$, we have its conjugated version written as $\overline{z}=x-\e y$.
\smallskip
\subsubsection{2. Complex numbers} For complex numbers, the algebra is given by $\{1,\e\, |\, \e^2=-1\}$. There is no pair of idempotents, contrarily to the paracomplex case.
The structure constants $C^{k}_{ij}$ are:
\[C_{11}^{1}=C_{12}^2=\, 1,C_{22}^1=\, -1 \]
the other structure constants are null.
\smallskip
\subsubsection{3. Dual numbers} For dual numbers, the algebra is given by $\{1,\e\, |\, \e^2=0\}$. The structure constants $C^{k}_{ij}$ are:
\[C_{11}^{1}=C_{12}^2=\, 1, \]
the other structure constants are null. This is a nilpotent algebra but not a division algebra.
\begin{remark}
Note that the algebra of paracomplex numbers is not a division algebra.
This has consequences on the geometry of the affine space defined over this algebra. However, note that neither paracomplex numbers nor complex numbers are nilpotent algebras.
\end{remark}
\subsection{Module over the spin factor algebra}
In this section, we rely on the following trilogy, connecting the following algebraic and geometric objects:
\[\Bigg\{n-\text{Algebra}\quad A \Bigg\} \leftrightarrow \Bigg\{m-\text{Module over algebra}\Bigg\} \leftrightarrow \Bigg\{mn-\text{Vector space} \Bigg\}.\]
In the same vein, a bigger step allows to relate the algebra $A$ to a manifold $M^{mn}$ over $A$ (it is discussed in the section \ref{S:CR}).
Let us denote by $A$ the real $n$-algebra, $\frak{M}^{m}$ the $m$-module and $E^{nm}$ the $nm$-dimensional vector space.
If $A$ is a real $n$-algebra with basis elements $e_a$, the linear space (or free module) $A\frak{M}^{m}$ admits a {\it real interpretation} in the space $\frak{M}^{mn}$.
In this interpretation, each vector $x = \{x^i\}$ in $A \frak{M}^{m}$, with coordinates $x^i = x^{ia}e_a$, is interpreted as the vector $\hat{x}=\{x^{ia}\}$ in $\frak{M}^{mn}$. If we replace in this definition the real linear space $\frak{M}^n$ by a linear space or free module $A\frak{M}^n$, we obtain the affine space $A E^n$ over the algebra $A$. In the space $A E^n$ the affine coordinates of points straight lines, planes, $m$-planes, and hyperplanes are defined in the same way as in $E^n$ (see \cite{Roz97}, section 2.1.2).
\smallskip
Let $\frak{A}$ be the spin factor algebra. Historically, the spin factor algebra on the space $\R\oplus\R^n$ for $n \geq 2$,
is equipped with the Jordan product on pairs ${\bf x} \bullet {\bf y} = (x_0y_0+ x\cdot y, x_0y+y_0x)$ where $\cdot$ denotes the usual dot product on $\R^n$ and ${\bf x}=(x_0,x)\in \R\oplus\R^n$. In its matricial representation version, the trace form---which is the usual matrix trace---is in this case $Tr({\bf x})=x_0$.
Construct the $m$-module over the spin factor algebra $\frak{A}\frak{M}^{m}$.
The affine representation of the algebra $\frak{A}$, or free module $\frak{A}\frak{M}^{m}$, admits a realisation in the real linear space $E^{2m}$. We develop this point of view below.
Let $E^{2m}$ be a $2m$-dimensional real linear space. {\it A paracomplex structure} on $E^{2m}$ is an endomorphism $\fK: E^{2m} \to E^{2m}$ such that $\fK^2=I$. The eigenspaces $E^{m}_+, E^{m}_-$ of $\fK$ with eigenvalues $1,-1$ respectively, have the same dimension. The pair $(E^{2m},\fK)$ will be called a {\it paracomplex vector space.} We define {\it the paracomplexification of $E^{2m}$} as $E^{2m}_\fC = E^{2m} \otimes_{\R} \fC$ and we extend $\fK$ to a $\fC$-linear endomorphism $\fK$ of $E^{2m}_\fC$.
\begin{lemma}[\cite{Roz97}]
Let $E^{2m}_\fC = E^{2m} \otimes_{\R} \fC$ be endowed with an involutive $\fC$-linear endomorphism $\fK$ of $E^{2m}_\fC$. Then, the space $E^{2m}_\fC$ is decomposed into the direct sum of a pair of $m$-dimensional subspaces $E^{m}_{+}$ and $E^{m}_{-}$ such that: \[E^{2m}_\fC=E^{m}_{+}\oplus E^{m}_{-},\] verifying:
\[
E^{m}_{+} = \{v\in E^{2m}_\fC \, |\, \fK v=\e v\}=\{v+\e\fK v\, |\, v \in E^{2m}_\fC\},
\]
\[
E^{m}_{-} = \{v\in E^{2m}_\fC |\, \fK v= -\e v\}=\{v-\e\fK v\, |\, v\in E^{2m}_\fC\}.
\]
\end{lemma}
\begin{remark}
This splitting also appears in the context of a space over complex numbers.
\end{remark}
\subsection{Paracomplex manifold}\label{S:CR}
We establish the final building block relating the unital real algebra $A$, the $A$-module and the manifold defined over $A$.
Let $y=f(x)$ be a (analytic) function, whose domain and range belong to a commutative algebra (i.e. $C_{jk}^h=C_{kj}^h$). We put $x=\sum_ix_ie_i,$ $y=\sum_iy_ie_i.$
From the generalized Cauchy--Riemann we have the following:
\begin{equation}\sum_h\frac{\partial y_i}{\partial x_h}C_{jk}^h=\sum_h\frac{\partial y_h}{\partial x_i}C_{hk}^j,\end{equation}
where $C_{jk}^h$ are the constant structures (see \cite{Sha66}).
\vskip.2cm
We restrict our attention to the case of paracomplex manifolds.
A {\it paracomplex manifold} is a real manifold $M$ endowed with a paracomplex structure $\fK$ that admits an atlas of paraholomorphic coordinates (which are functions with values in the algebra $\fC = \R + \e\R$ defined above), such that the transition functions are paraholomorphic.
Explicitly, this means the existence of local coordinates $(z_+^\alpha, z_-^\alpha),\, \alpha = 1\dots, m$ such that
paracomplex decomposition of the local tangent fields is of the form
\begin{equation}
T^{+}M=span \left\{ \frac{\partial}{\partial z_{+}^{\alpha}},\, \alpha =1,...,m\right\} ,
\end{equation}
\begin{equation}
T^{-}M=span \left\{\frac{\partial}{\partial z_{-}^{\alpha}}\, ,\, \alpha =1,...,m\right\} .
\end{equation}
Such coordinates are called {\it adapted coordinates} for the paracomplex structure $\fK$.
By abuse of notation, we write $\partial_z$ instead of $\frac{\partial}{\partial z^{\alpha}}$.
We associate with any adapted coordinate system $(z_{+}^{\alpha}, z_{-}^{\alpha})$ a paraholomorphic coordinate system $z^{\alpha}$ by
\begin{equation}
z^\alpha\, =\, \frac{z_{+}^{\alpha}+z_{-}^{\alpha}}{2} +\e\frac{z_{+}^{\alpha}-z_{-}^{\alpha}}{2}, \alpha=1,...,m .
\end{equation}
We define the paracomplex tangent bundle as the $\R$-tensor product $T^\fC M = TM \otimes \fC$ and we extend the endomorphism $\fK$ to a $\fC$-linear endomorphism of $T^\fC M$. For any $p \in M$, we have the following decomposition of $T_{p}^\fC M$:
\begin{equation}
T_p^\fC M=T_p^{1,0}M \oplus T_p^{0,1}M\,
\end{equation}
where
\begin{equation}
T_p^{1,0}M = \{v\in T_p^\fC M | \fK v=\e v\}=\{v+\e \fK v| v \in E^{2m}\} ,
\end{equation}
\begin{equation}
T_p^{0,1}M = \{v\in T_p^\fC M | \fK v= -\e v\}=\{v-\e \fK v|v\in E^{2m}\}
\end{equation}
are the eigenspaces of $\mathfrak{K}$ with eigenvalues $\pm \e$.
The following paracomplex vectors
\begin{equation}
\frac{\partial}{\partial z_{+}^{\alpha}}=\frac{1}{2}\left(\frac{\partial}{\partial x^{\alpha}} + \e\frac{\partial}{\partial y^{\alpha}}\right),\quad \frac{\partial}{\partial{z}_{-}^{\alpha}}=\frac{1}{2}\left(\frac{\partial}{\partial x^{\alpha}} - \e\frac{\partial}{\partial y^{\alpha}}\right)
\end{equation}
form a basis of the spaces $T_p^{1,0}M$ and $T_p^{0,1}M$.
This paragraph presents the final building block of the construction of the manifold over paracomplex numbers. Hence, now it is possible to present the proof of the Main Theorem A.
\section{First part of the proof of Theorem A}
In view of proving the Main Theorem A, we give a first part of the proof, by introducing the following construction of the paracomplex projective spaces and related notions.
\subsection{Construction of a paracomplex projective space}
Let us introduce a notion of projective spaces over paracomplex numbers $\fC\cP^n$. The notation $\fC E^n$ stands for paracomplex affine space.
Let $\R\cP^n$ be a real $n$- dimensional projective space. Any point of the $\R\cP^n$ space can be determined by a system of homogeneous coordinates $[X^0: X^1: ...: X^n]\in \R\cP^{n}$.
\smallskip
Points of $\fC\cP^n$ the projective paracomplex space are given by the homogeneous coordinates $[X^0: X^1:...:X^n]\in \fC\cP^n$, where $X^i$ are paracomplex numbers. In such a space $\fC\cP^n$ it is possible to define straight lines, planes and hyperplanes as in the real projective space $\R\cP^n$.
In that space, any point can be given by $n$ real numbers $x^1, x^2,..., x^n$ which are paracomplex numbers.
Exactly as in the case of real projective spaces, one defines the relation between the real affine space and the real projective space $\R\cP^n$ by considering that to any point $x^i$ one can attach homogeneous coordinate $X^i/X^0$, one can proceed similarly for the affine and projective paracomplex space. To any point of affine paracomplex space $\fC E^n$, given by coordinates $x^i$ (which are paracomplex numbers), there corresponds a homogeneous coordinate $X^i/X^0$ in the paracomplex projective space $\fC\cP^n$. As it is known, in the classical (real or complex) framework, the points at infinity in projective space are given by hyperplanes at infinity i.e. we have an equation of the type $X^0=0$. However, in the case of paracomplex projective spaces, there exist not only infinity hyperplanes but as well {\it special points}: the points corresponding to the {\it zero divisor}.
From another point of view, the paracomplex projective space $\fC\cP^n$ and the real projective space $\R\cP^n$ are directly related. The points of
$\fC\cP^n$ can be realized as pairs of points lying in the real projective space $\R\cP^n\times \R\cP^n$. Indeed, a given point $X^i$ of $\fC\cP^n$ is given by a pair of points in $\R\cP^n\times \R\cP^n$, where
\[X^i=x^i e_+ + y^ie_-,\] and these real projective points have for coordinates $x^i$ and $y^i.$
Let us consider a transformation of the coordinates $x^i$ and $y^i$ into the coordinates $kx^i$ and $ly^i.$ Then, the coordinates $x^i$ are replaced by
the coordinates $(ke_+ + le_{-})x^i.$ Under this transformation, a straight line is transformed into a pair of straight lines and the (hyper)planes are transformed into pairs of (hyper)planes, each of which lie respectively in a copy of $\R\cP^n$.
From these transformations, it follows that in the paracomplex projective space $\fC\cP^n$ one has the following property:
\smallskip
{\it Through two points can pass an infinite number of straight lines. Moreover, two straight lines of $\fC\cP^n$ in the same plane will intersect themselves in more than one point.}
\subsubsection{Projective Group of transformations}
In the $\fC\cP^n$ space one can define a group of collineations, correlations, resp. anti-collineation and anti-correlations.
The group of collineations is determined by the following system of equations:
\begin{equation}\label{E:sys}
k{X’}^i= \sum_j a^i_j X^j, \quad i,j= 1,2,\cdots,n.
\end{equation}
where $a^i_j$ are matrix entries, being paracomplex numbers, $k$ is an arbitrary paracomplex number.
The anti-collineation is obtained by taking the conjugation of all the $X'^j$ and $X^i$ in (\ref{E:sys}).
\medskip
\subsection{Some notions on $m$-pairs}
We have previously recalled rudiments of paracomplex geometry and showed the relation to the fourth Frobenius manifold.
In this section, we introduce the language of $m$-pairs. This allows us to show the connection to projective geometry and later on to the pseudo-Elliptic spaces.
\smallskip
We define $A\cP^{n}$ to be the $n$-dimensional projective space defined over an algebra $A$ (associative, commutative, unital of finite dimension). By abuse of notation and whenever the context is clear we will use simply the notation $\cP^{n}$. Let $X_{d}$ be a $d$-dimensional surface of the $n$-dimensional projective space $A\cP^{n}$, with $d\leq n$.
\begin{definition}[Normalized surface]
The surface $X_{d}$ is said to be normalized in the Norden sense, if at each point $p\in X_{d}$, are associated the two following hyperplanes:
\begin{enumerate}
\item Normal of first type, $P_{I}$, of dimension $n-d$, and intersecting the tangent $d$-plane $T_{p}X_{d}$ at a unique point $p \in X_{d}$.
\item Normal of second type, $P_{II}$, of dimension $d-1$, and included in the $d$-plane $T_{p}X_d$, not meeting the point $p$.
\end{enumerate}
\end{definition}
This decomposition expresses the duality of projective space. In particular, in the limit case, where $d=n$, then $P_{I}$ is reduced to the point $p$ and $P_{II}$ is the $(n-1)$-surface which does not contain the point $p$.
This property is nothing but the usual duality of projective space. Note that in this case, $X_{n}$ can be identified with the projective space $\cP^{n}$.
\begin{definition}\label{D:mpairs}
A pair consisting of an $m$-plane and an $(n-m-1)$-plane is called an $m$-pair.
\end{definition}
\smallskip
We establish a relation to Grassmannians. A Grassmannian $G(k,n)$ is a space that parametrizes all $k$-dimensional linear subspaces of the $n$-dimensional vector space.
In particular, the Grassmannian $G(1,n)$ is the space of lines through the origin in the vector space and is the same as the projective space $\cP^{n-1}$.
\smallskip
\begin{remark}
Each $m$-pair corresponds to a point of a Grassmannian of type $G(m,n)$. Reciprocally, every point in the Grassmannian manifold $G(m,n)$ defines an $m$-plane in $n$-space. Fibering these planes over the Grassmannian one arrives at the vector bundle, which generalizes the tautological bundle of a projective space. Similarly the $(n-m)$-dimensional orthogonal complements of these planes yield an orthogonal vector bundle.
\end{remark}
From~\cite{No58,Sh87}, for normalized surfaces associated to an $m$-pair space, the following properties holds:
\medskip
\begin{lemma}\label{L:pairs}
\
\begin{enumerate}
\item The space of $m$-pairs is a projective, differentiable manifold.
\item For any integer $m\geq 0$, a manifold of $m$-pairs contains 2 flat, affine and symmetric connections.
\end{enumerate}
\end{lemma}
Turning our attention to 0-pairs, an important key lemma relates 0-pairs, projective spaces and Grassmannians.
\subsection{Paracomplex projective geometry and the fourth Frobenius manifold}
We have the following:
\begin{lemma}[Key lemma]\label{R:1}
The space of $0$-pairs can be identified with an $(n-1)$-dimensional projective space. This is a Grassmannian space of type $G(1,n)$.
\end{lemma}
\begin{proof}
For $m=0$, a $0$-pair consists of a point and of an $(n-1)$-hyperplane. So, this amounts to considering the space of $(n-1)$-hyperplanes in an $n$-dimensional space. In other words, this is a Grassmannian of type $G(n-1,n)$. We have a (non-canonical) isomorphism of $G(n-1,n)$ and $G(1,n)$. This isomorphism of Grassmannians sends an $(n-1)$-dimensional subspace into its $1$-dimensional orthogonal complement. Since $G(1,n)$ is the same as the projective space $\cP^{n-1}$, therefore, we can identify 0-pairs to an $(n-1)$-dimensional projective space.
\end{proof}
This lemma plays an important role, in particular in relation to the next proposition.
\begin{proposition}\label{P:0-pairs}
Suppose that $(X,\cF)$ is a finite measurable set where the dimension of $X$ is $n+1$, and measures vanish only on an ideal $I$. Let $S$ be the space of probability distributions on $(X,\cF)$. Then, the space $S$ is a manifold of 0-pairs.
\end{proposition}
\begin{proof}
The statement corresponds to Proposition 5.9 in \cite{CoMa}.\end{proof}
We have introduced the previous part, in order to discuss the statistical manifolds. In particular, manifolds of probability distributions are related to the $m$-pairs in the following way:
\begin{corollary}\label{C:proj}
The fourth Frobenius manifold is identified with the paracomplex projective space $\fC\cP^{n}$.
\end{corollary}
\begin{lemma}\label{T:3}
The manifold of probability distributions $S$ has a pair of flat, affine, symmetric connections.
\end{lemma}
\begin{proof}
There are different ways of proving this. One possibility is that this follows from the calculation in \cite{BuNen1,BuNen2}.
Another, and more geometric, approach is to apply Lemma\, \ref{R:1} and Lemma\, \ref{L:pairs}.\end{proof}
\begin{remark}
In other words, the fibration is done with the algebra of 2 connections.
\end{remark}
Let us recall the following proposition:
\begin{proposition}\label{P:isome}
The space of $0$-pairs in the projective space is isometric to the hermitian projective space over the algebra of paracomplex numbers.
\end{proposition}
\begin{proof}
see e.g.~\cite{Roz97} section 4.4.5.
\end{proof}
Finally, from Proposition \ref{P:isome} and Proposition \ref{P:0-pairs} it follows that:
\begin{proposition}
The statistical manifold is isometric to the hermitian projective space over the algebra of paracomplex numbers.
\end{proposition}
As a last point bridging the statistical manifold and the paracomplex space, we have that:
\begin{lemma}
Suppose that $(X,\cF)$ is a finite measurable set where the dimension of $X$ is $n + 1$, and measures vanish only on an ideal $I$. The space $S$ of probability distributions on $(X,\cF)$ is isomorphic to the hermitian projective space over the cone $M_+(2, \C)$.
\end{lemma}
\begin{proof}
See Theorem 5.10 in \cite{CoMa}.
\end{proof}
\section{Second part of the proof of Theorem A}
In this section, Maurer--Cartan structures for the fourth Frobenius manifold are presented. These considerations arise from our next result bridging manifold of probability distributions and so-called {\it pseudo-Euclidean spaces.}
\subsection{Maurer--Cartan structures for the fourth Frobenius manifolds}
Pseudo-Euclidean spaces, denoted $R^n_l$, arise from the modification of one of the axioms of classical Euclidean spaces $\R^n$. The fifth axiom turns into the following:
{\it ``There are $l$ mutually orthogonal vectors $v_a$ with negative inner squares $v^2_a$ and $n-l$ mutually orthogonal vectors $v_u$ with positive inner squares $v^2_u$, and each vector $v_a$ is orthogonal to each vector $v_u$.''}
The space $R^n_l$ is a pseudometric space, and the integer $l$ is called the index of this space.
To establish the relation between manifold of probability distributions and pseudo-Euclidean spaces, we use Norden's normalisation theory \cite{No58,No76,Sh87}. More precisely, this evolves around {\it structural equations} of an affine connection space. The Norden method goes as follows.
Let $X_{m}$ be an $m$-dimensional surface. Equip it with coframes $\{\omega^i\}$ in an affine space $\R E^{n+1}$.
It is known that there exists an equivalence between an affine connection on $X_{m}$ and an infinitesimal connection in the principal bundle space of linear frames of the manifold $X_{m}$. So, we can establish the structural equations of the affine connection form in the following way:
\begin{equation} d\omega^{i}-\omega^s\wedge \omega_s^i=\Omega^i, \quad d\omega_j^{i}-\omega_j^s\wedge \omega_s^i=\Omega_j^i, \end{equation}
where $\omega_j^i$ are the connection forms and $\Omega^i, \Omega_j^i$ are respectively the torsion and curvature forms of the affine connection.
\smallskip
Consider a surface $X_m$ and its polar vector equation ${\bf r}={\bf r}(u^1,\dots, u^m)$. Let $M({\bf r})$ be a point on $X_m$ given by the intersection of the line passing through the origin and collinear to the vector ${\bf r}$.
Choose a framing of the surface $X_m$ given by $m$ framing vectors $\{{\bf e}_{i}\}_{i=1}^m$ at $M({\bf r})$ and belonging to the $(m + 1)$-dimensional subspace of $\R E^{n+1}$. Consider the normal space to the tangent space, generated by $n-m$ framing vectors, denoted ${\bf e}_\alpha$. The framings are defined as follows $\{{\bf r}, {\bf e}_i, {\bf e}_\alpha\}$ (notice that it includes the polar vector).
These framings verify the following classical system of Maurer--Cartan like equations:
\begin{equation}
\begin{aligned}
d{\bf r} &= \omega {\bf r} + \omega^s{\bf e}_s\\
d{\bf e}_i &= \omega_i {\bf r} + \omega_i^s{\bf e}_s+ \omega_i^\beta {\bf e}_\beta\\
d{\bf e}_\alpha &= \omega_\alpha {\bf r} + \omega_\alpha^s{\bf e}_s+ \omega_\alpha^\beta {\bf e}_\beta
\end{aligned}
\end{equation}
It was proved in \cite{Roz97}, section 4.1.1 and Theorem 4.3 and Theorem 4.1, that a paracomplex projective space is a pseudo-Euclidean manifold.
We now consider the geometry of the fourth Frobenius manifold and prove that:
\smallskip
\begin{theorem}\label{T:PE}
The manifold $S$ is a pseudo-Euclidean space $R^n_l$.
\end{theorem}
\begin{proof}
As was shown in Proposition \ref{P:0-pairs}, the fourth Frobenius manifold are defined by 0-pairs and therefore the covectors ${\bf e}_\alpha$ {\it do not} exist in our case.
We now apply the Norden method \cite{No76}. Let the polar vector {\bf r} be, in the manifold of probability distributions context, defined as the so-called affine canonical coordinates (\cite{Sh87}, p.146). Define, the coframe $\omega$ to be the affine connection component defined in \cite{Sh87}. The vectors ${\bf e}_i$ are nothing but the score vectors, which have been defined in the Appendix under the notation $X_j=\partial_j\ln \rho_{\theta}$, where $\rho_{\theta}$ is a probability distribution.
Then, the following system of equations is satisfied:
\begin{equation}\begin{aligned}
d{\bf r} &= \omega {\bf r} + \omega^sX_s\\
dX_i &= \omega_i {\bf r} + \omega_i^jX_j.
\end{aligned}\end{equation}
In this way we define {\it Maurer--Cartan structures} for the fourth Frobenius manifolds.
By Rozenfeld's theorem, a paracomplex projective space is a pseudo-Euclidean space. Thus, the manifold of probability distributions is a pseudo-Euclidean space and the statement is proven.
\end{proof}
\subsection{Pseudo-Ellipticity}
Let $R^n_l$, $0\geq l\geq n$, be a real coordinate $n$-dimensional vector space with bilinear form:
\begin{equation}
B^n_l(x,y)=-\sum_{i=1}^l x_iy_i +\sum_{j=l+1}^nx_jy_j.
\end{equation}
In virtue of Proposition \ref{P:0-pairs}, Lemma \ref{R:1}, and Lemma \ref{L:coco}, a manifold of probability distributions is a manifold of 0-pairs and dual to the section $\cH$ of the fifth Vinberg cone. So,
this implies, that the bilinear form is given by $B^n_1(x,y)=- x_1y_1 +\sum_{j=2}^nx_jy_j
$ and the the statement below follows.
\begin{proposition}
The fourth Frobenius manifold $S$ is a pseudo-Euclidean space $R^n_1$ of index one.
\end{proposition}
\begin{proof}
In order to show that the space has index 1, it was shown by Chentsov\cite{Ce3} that this manifold is geodesically convex and that the maximal submanifolds are totally geodesic. From Wolf's theorem \cite{Wo}, we have that the space $R^n_l$ is pseudo convex only if the index $l$ is
1.
\end{proof}
More precisely, we can refine our statement by stating that:
\begin{theorem}\label{P:pseudoell}
The manifold $S$ is a real pseudo-Elliptic space $S^n_1$ of index one.
\end{theorem}
\begin{proof}
The manifold $S$ equipped with a Riemannian metric $g$ is a Riemannian manifold $(S, g)$. In order to show that the manifold $S$ is a real pseudo-Elliptic space, it is sufficient to find the metric.
Bhattacharyya \cite{Ba} shows that the distance between two points in a manifold of probability distributions (i.e. distance between a pair probability distributions $P, P^*$), is given by:
\[d(P,P^*) = \int_{\Omega}\sqrt{\rho\rho^*}d\lambda,\]
where $\rho$ and $\rho^*$ are the Radon--Nikodym derivatives of $P$ and $P^*$ respectively w.r.t $\lambda$.
However, it was shown in \cite{BuCoNen99} (section 2 p.89) that the distance $d(P,P^*) $ is given by $\cos^2\omega= \int_{\Omega}\sqrt{\rho\rho^*}d\lambda$.
This coincides with the metric on the pseudo-Elliptic manifold:
\[cos^2\frac{\delta}{r}= \overline{XY,\alpha\beta}\]
where $X,Y$ are points, $\alpha,\beta$ are points at infinity and $\overline{XY,\alpha\beta}$ is the cross-ratio of these points. The real radius of the curvature is $r$. In particular, the interpretation regarding the manifold of probability distributions is as follows: the points $X, Y$ correspond to the probability distributions and $\alpha,\beta$ correspond to the probability distributions on the boundary of the cone.
This proves the statement about pseudo-Ellipticity.
\end{proof}
\section{Theorem B: the fourth Frobenius manifold is a Lorentzian manifold}
Now, we present the proof of the Main Theorem B.
\subsection{Lorentzian manifolds}
\begin{definition}\label{D:Lorentz}
A Lorentzian manifold is a pseudo Riemannian manifold which is equipped with an everywhere non-degenerate, smooth, symmetric metric tensor $g$ of signature $(1,n-1)$ i.e. such that the bilinear form verifies:
\[B^n_1(x,y)=- x_1y_1 +\sum_{j=2}^nx_jy_j.\]
\end{definition}
\begin{proposition}\label{P:Lor1}
A pseudo-Euclidean manifold in a pseudo-Euclidean space of index 1 is a Lorentzian manifold.
\end{proposition}
\begin{proof}
A pseudo-Euclidean space of index 1 $R^n_1$ has a bilinear from \[B^n_1(x,y)=- x_1y_1 +\sum_{j=2}^nx_jy_j
.\] Applying the Definition \ref{D:Lorentz}, we can say that a pseudo-Euclidean manifold of index 1 is a Lorentzian manifold.
\end{proof}
The cone of positive measures of bounded variations belongs to the fifth class (see \, \cite{CoMa}). This means we have a cone defined over the algebra of paracomplex numbers. This cone is also known in other branches as the time future like cone. In particular, we have that:
\begin{proposition}
The cone $\cC$ defined over the algebra of paracomplex numbers is a pseudo-Riemannian manifold $(M,g)$. This is a differentiable manifold $M$, equipped with an everywhere non-degenerate, smooth, symmetric metric tensor $g$ of signature $(1,n-1)$.
\end{proposition}
For a metric $g$, the signature $(1,n-1)$ implies that we have:
\[g=-dx_1^2+dx_2^2+\dots +dx_n^2.\]
In other words:
\begin{corollary}
The cone $\cC$ is a Lorentzian manifold.
\end{corollary}
As for the manifold of probability distributions, we have that:
\begin{proposition}\label{C:lo}
The manifold of probability distributions $S$ is a projective Lorentzian manifold.
\end{proposition}
\begin{proof}
A manifold of probability distributions is a pseudo-Elliptic space $S^n_1$, by Theorem\, \ref{P:pseudoell}. The bilinear form is given by: $B^n_1(x,y)=- x_1y_1 +\sum_{j=2}^nx_jy_j$. Applying Definition\, \ref{D:Lorentz}, the conclusion is straight forward.
\end{proof}
\subsection{The mirror symmetries of the fourth Frobenius manifold}
A central Hermitian hyperquadric form can be written generally as:
\[xQx+c=\sum_i \overline{x}^ix^i+c=0.\]
Those central Hermitian hyperquadrics are called Hermitian ellipsoids. Hermitian ellipsoids, in the paracomplex space $\fC E^n$, are homeomorphic to the topological product of $\R^n$ and a hypersphere in $\R^n$, denoted $S^{n-1}$. We call the hyperquadric given by the equation $xQx = 0$ an absolute hyperquadric of the pseudo-Elliptic space.
\begin{remark}
The equation $xQx = 0$ is the equation of an oval hyperquadric in $\cP^n$. If the vectors $x$ and $y$ represent points $X$ and $Y$ in this space, the vectors $Qx$ and $Qy$ can be regarded as covectors representing the hyperplanes polar to these points with respect to this oval hyperquadric.
\end{remark}
An $n$-dimensional non-Euclidean Riemann space can be determined as an $n$-dimensional projective space in which the distance between two points $x$ and $y$ is given by:
\[ \cos^2 \frac{\omega}{r}= \frac{(x,y)}{(x,x)\cdot (y,y)},\]
where $(x,y)$ is a bilinear form, $r$ and $\omega$ are real numbers.
However, in the case of a projective paracomplex space, the distance between points is given by a slightly different formula.
Historically, the idea to construct new type of non Euclidean space was given by C. Segre. Namely, he proposed to introduce new type of form and then to construct new types of spaces.
Let us introduce the notion of Hermitian form:
\[\{x,y\}=\overline{x}^0 y^0+ \overline{x}^1 y^1+...+\overline{x}^n y^n,\]
with property the that $\{x,y\}= \overline{\{y,x\}}$.
Therefore, the form $\{x,x\}$ is always real.
Now, we can introduce the distance $\omega$ in the paracomplex projective space as follows:
\[\cos^2 (\frac{\omega}{r}) = \frac{ \{x,y \}\cdot \{y,x\} }{\{x,x\}\cdot \{y,y\}},\]
where $r$ is a radius of the curvature of the space. This space is a so-called {\it unitary paracomplex space} $\fC K^n$.
\smallskip
For the convenience of the reader, we recall the definition of a unitary paracomplex space $\fC K^n$.
A vector space over the algebra of paracomplex numbers $\fC$ is a vector space endowed with an inner product $\{ \cdot, \cdot\}$ satisfying the following axioms:
\begin{enumerate}
\item $\{a,b\}=\overline{\{b,a\}}$
\item $\lambda\{a,b\}=\{\lambda a,b\}$, where $\lambda \in \fC$
\item$ \{a+b,c\}=\{a,c\}+ \{b,c\}$
\item If $a\neq 0$ then $\{a,a\}>0$.
\end{enumerate}
The collineation in the projective paracomplex space $\fC\cP^n$ which conserves the distances between two points we call the actions of the space. The matrices of these collineation are given by:
\[
\overline{a^0_i} a^0_j + \overline{a^1_i} a^1_j +...+\overline{a^n_i} a^n_j =\begin{cases}
1, &\text{if}\quad i= j,\\
0, &\text{if}\quad i\neq j.
\end{cases}\]
\begin{remark}
Notice that this is a discrete analog of the inner product defined for a paracomplex Hilbert space.
\end{remark}
In the projective paracomplex space one can easily see that there exists as well another action called anti-collineation as well conserving the distance between two points. Then the interpretation of the unitary paracomplex space $\fC K^n$ is the following one.
\begin{theorem}\label{T:mirr}
The statistical manifold can be defined as one of the two domains into which the hyperquadric $xQx = 0$ divides the paracomplex projective space $\cP^n$, where the distance between two points $X$ and $Y$ is given by: \[cos^2\frac{\delta}{r}= \overline{XY,\alpha\beta},\]
where $r$ is the curvature.
\end{theorem}
Here $\overline{XY,\alpha\beta}$ is the cross ratio of these points and their polar hyperplanes with respect to the hyperquadric $xQx =0$, with
$x$ being an arbitrary vector in the affine space, representing two points in $\fC\cP^n$.
\begin{proof}
This follows from the Theorem\, \ref{T:PE} and the Theorem 4.3, in \cite{Roz97}.
\end{proof}
\begin{remark}
Note that the elliptic space $S^n$ can be defined as the projective space $\cP^n$ in which we have a specific distance relying on the cross ratio $\overline{XY,\alpha\beta}$ where $X, Y$ are two points and $\alpha, \beta$ are their polar hyperplanes. This metric is defined as:
\[cos^2\frac{\delta}{r}=\overline{XY,\alpha\beta}\]
where $ \overline{XY,\alpha\beta}$ is the cross ratio of these points and their polar hyperplanes with respect to the imaginary hyperquadric $x^2 = 0$.
In this way, we achieve the first part of the proof of Main Theorem B.
\end{remark}
\section{Second part of the proof of Theorem B}
A Vinberg cone generates a class of Lie groups and of Lie algebras. There exists an automorphism group of the Vinberg cone (see section 1.1). Consider the positive cone $\cC$ of strictly positive measures on a space $(X,\mathcal{F})$, vanishing only on an ideal $I$ of the $\sigma$-algebra $\mathcal{F}$. In our case, the automorphism corresponds to the parallel transport, which we define below.
Let $\cW$ be the space of signed measures of bounded variations (i.e. signed measures whose total variation is bounded, vanishing only on an ideal $I$ of the $\sigma$-algebra $\mathcal{F}$). To any parallel transport $h$ in the covector space $\cW^{*}$ of the space $\cW$ of $\sigma$--finite measures, we associate
\[f\xrightarrow{h} f+h,\] an automorphism of the cone $\cC$:
\begin{equation}
\mu \xrightarrow{h} \nu,\, \text{ where }\, \frac{d\nu}{d\mu}(\omega)\, =\, \exp (h(\omega)),
\end{equation}
and $d\nu/d\mu$ is the Radon--Nikodym derivative of the measure $\nu$ w.r.t. the measure $\mu$.
This automorphism is a non--degenerate linear map of $\cW$ which leaves the cone invariant.
\smallskip
Denote by $\fG$ the group of all automorphisms $h$ such that $h\, =\, \ln\, \frac{d\nu}{d\mu}.$
The commutative subgroup of all ``translations'' of the cone $\cC$ is a simply transitive Lie group. To this Lie group $\fG$ the associated Lie algebra $\fg$ defines the derivation of the cone.
\smallskip
We focus on the geometry of $\cH$, defined in the Appendix. It has the {\it projective geometry} of a pencil of straight-lines of the cone $\cC$. The geodesics on the cone $\cC$ are the trajectories of 1-parameter subgroup of the group $\fG$ and can be written in the following way:
\begin{equation}\label{E:t}
f(\omega;s)=f(\omega;0)\exp\{s.h(\omega)\}.
\end{equation}
The geodesics on $\cH$ need to be logarithmic projections on $\cH$ of these trajectories. They are distinct from $f(\omega;s)$ by a normalization constant. In this way, a geodesic---crossing a given point $p(\omega; 0)$ in a given direction---can be determined by:
\begin{equation}\label{E:t}p(\omega;s)=\frac{1}{a(s)}p(\omega;0)\exp\{s.q(\omega)\},\quad \text{where} \quad a(s)=\int_{\Omega} \exp(s.q(\omega))p(\omega;0)d\mu.\end{equation}
By duality, we can obtain a similar approach to the geodesics of the manifold of probability distributions $S$, which are trajectories of 1-parameter subgroups.
\subsection{Real Interpretation of unitary paracomplex spaces $\fC K_n$.}
Recall that $\fC K_n$ is a paracomplex vector space on which an inner product of vectors is defined. In the $(2n+1)$-dimensional non Euclidean Riemann space $R^{2n+1}_l$, there exist families of so-called {\it paratactic} congruent straight lines. Each 0-pair is in on-to-one correspondence with a ray of our congruence. These families are $2n$ parametric families of straight lines along of which it is possible to apply a 1-parameter group of motions (translations along the congruent straight lines). These are so-called {\it paratactic transformations}.
That means, two straight lines of such a congruence have a 1-parameter set of common perpendiculars, being of the same length (and not just one perpendicular!). That is, the distance between these two straight lines is the same (we take the length of the perpendicular as the distance).
This type of congruence exists as well in the non-Euclidean pseudo-Riemannian space $R^{2n+1}_{n+1}$. In this space the bilinear form will be given by:
\[B^{2n+1}_n(x,y)= - x^0 y^0 - x^1 y^1 - ... - x^n y^n + x^{(n+1)} y^{(n+1)} +... + x^{(2n+1)} y^{(2n+1)}.\]
The unitary paracomplex space $\fC K_n$ is isomorphic to the non Euclidean pseudo-Riemannian space $R^{2n+1}_n$. This ends and gives a conclusion to discussions on statements appearing in Theorem B.
\subsection{An addition to Wolf's theorem}
In this paragraph, we prove an additional result to Wolf's classification theorem in \cite{Wo}. His result holds for the real, complex, octonionic and quaternonic number fields. However, there is a gap in what concerns algebras that are not fields: for example, paracomplex numbers. We remedy to this precise situation.
Let us go back to the automorphism group of the cone $\cC$.
\begin{lemma}
Let $\fG$ be the Lie group of automorphisms of the Vinberg cone $\cW$. Let $H$ be a subgroup of $\fG$ leaving the manifold $\cH$ invariant. Then,
$H$ is a Lie subgroup.
\end{lemma}
Let $\cH$ be a non-empty (compact) subspace of the Vinberg cone $\cW$. Consider the subgroup $H$ of $\fG\subset GL_n(\R)$ leaving this $\cH$ invariant. Then, it is known that this subgroup $H$ is compact.
It remains to apply the Cartan closed subgroup theorem: any closed subgroup $H$ of a Lie group $\fG$ is a Lie subgroup (and thus a submanifold) of $\fG$.
\begin{corollary}
$H$ is a compact Lie subgroup.
\end{corollary}
The manifold of probability distributions is isometric to a pseudo-Riemannian (projective) space. The group of motions is a simple Lie group of type $D_n$ (see \cite{Roz97}), which is associated to a special orthogonal group $SO(1,2n-1)$ group, and of real rank 1.
Therefore, we have that:
\begin{lemma}
The fourth Frobenius manifold $S$ can be considered as a compact symmetric space of rank 1.
\end{lemma}
\begin{proof} Indeed, this follows from the above arguments: it is a compact Lie group, the group of motions is a simple Lie group of type $D$. Since it is a projective space and of even dimension the only remaining possibility is to have a compact symmetric space of rank 1.
\end{proof}
\begin{theorem}
Consider the $2n$-dimensional manifold of probability distributions $S$. If $M$ is a totally geodesic submanifold of $S$ then it is a product of real projective spaces $\R\cP^r\times\R\cP^r$, where $1\leq r \leq 2n$.
\end{theorem}
\begin{proof}
The $2n$-dimensional manifold of probability distributions are by the statement (Lemma\, \ref{R:1} and Proposition\, \ref{P:0-pairs}) identified to a paracomplex projective space, which in turn is isomorphic to a pair of real projective spaces of real dimension $2n$. Now, we invoke the following argument of fixed point sets: for any isometry $f : M \to M$, the fixed point set is a totally geodesic submanifold of $M$.
Taking the isometry $f: \fC\cP^{2n}\to \fC\cP^{2n}$ in the paracomplex projective space $\fC\cP^{2n}$ such that $f: (a,b)\to (a,-b)$, where $a=(z_1,\dots, z_{r+1})$ and $b=(z_{r+2},\dots, z_{n+1})$, the set of points $(a,0)$ forms a fixed set under $f$. So, the fixed point set is $\fC\cP^r$. So, in particular, $\fC\mathbb{P}^r$ being isomorphic to $\R\cP^r\times \R\cP^r$ for $1\leq r \leq 2n$ it defines a totally geodesic manifold.
\end{proof}
\begin{remark}
This completes Wolf's theorem \cite{Wo}, which considers totally geodesic submanifolds for spaces defined over the fields $\R, \C, \mathbb{O},\mathbb{H}$. Here we define this for the algebra of paracomplex numbers.
\end{remark}
We have that the manifold of probability distributions is decomposed into a pair of totally geodesic submanifolds.
A submanifold $N$ of a Riemannian manifold $(M,g)$ is called totally geodesic if any geodesic on the submanifold $N$ with its induced Riemannian metric $g$ is also a geodesic on the Riemannian manifold $(M,g)$.
\begin{corollary}\label{C:sbmfd}
The $2n$-dimensional manifold of probability distributions $S$ has a pair of totally geodesic submanifolds, being real projective spaces.
\end{corollary}
Now, we prove the following result:
\begin{theorem}\label{T:pro}
The manifold $S$ is isomorphic to $\R\cP^n\times \R\cP^n$.
\end{theorem}
\begin{proof}
Indeed, by Proposition\, \ref{P:0-pairs}, we have that $S$ is a 0-pair. By Lemma \ref{R:1} a 0-pair is a projective space $P^n$. An $n$-dimensional paracomplex projective space is isomorphic to a cartesian product of $n-$dimensional real projective spaces $\R\cP^n\times \R\cP^n$ (see \cite{Roz97}). So, the manifold $S$ is isomorphic to the cartesian product $\R\cP^n\times \R\cP^n$.
\end{proof}
\begin{corollary}\label{}
The manifold $S$ is a projective variety.
\end{corollary}
\begin{proof}
The product of two projective varieties is a projective variety. Since $\R\cP^n$ are projective varieties, the statement follows directly.
\end{proof}
\begin{proposition}\label{P:Lo}
The manifold of probability distributions is a non-orientable Lorentzian manifold.
\end{proposition}
\begin{proof}
Let us apply Corollary\, \ref{C:lo} stating that the manifold of probability distributions is a Lorentzian manifold. The dimension of this manifold is even: it is a projective paracomplex space by Lemma \ref{R:1} and Proposition \ref{P:0-pairs}).
Theorem \ref{T:pro} states that the manifold $S$ is isomorphic to $\R\cP^{2m}\times \R\cP^{2m}$. By topological arguments, an even dimensional real projective space in non-orientable. Now, since the cartesian product of a pair of manifolds $M\times N$ is orientable iff the manifolds $M$ and $N$ are orientable, the conclusion is straight forward.
\end{proof}
We now prove the statement:
\begin{theorem}\label{T:nono}
The class of fourth Frobenius manifold is:
\begin{enumerate}
\item geodesically convex,
\item non-orientable,
\item non isochronous (time-sense not conserved) in the sense of Calabi--Markus (see \cite{CaMa}, for the exact terminology)
\end{enumerate}
even dimensional Lorentzian manifolds.
\end{theorem}
\begin{proof}
By Proposition \ref{P:Lo} we know that $S$ is a Lorentzian manifold.
The first (1) follows from the statement in Chensov \, \cite{Ch64}. For (2), we can use the knowledge developed and acquired above. Indeed, since $\R\cP^n$ is not orientable for $n$ even and applying the fact that a cartesian product of manifolds is orientable iff both manifolds are orientable it follows that since $S$ is of even dimension, $S$ is a non-orientable manifold. Now, applying the section 3 from \cite{CaMa} this implies that $S$ is geodesically convex and time-like non orientable.
\end{proof}
\begin{corollary}
The fourth Frobenius manifold is uniquely determined by an orientable 2-fold covering.
\end{corollary}
\begin{proof}
This is clear from elementary topology that we have $\R\cP^n\cong S^n/\sim$ where $\sim$ is the antipodal map. So, we have the fiber bundle
$S^n\to \R\cP^n$, with group $\Z_2$ of isometries $\pm I$. Another way of considering this to apply Section 3 of \cite{CaMa} and in particular the ingredients constituting the proof of Theorem 3 in \cite{CaMa}.
\end{proof}
\begin{proposition}\label{P:Pierce}
The space of probability distributions $S$ is a non-orientable Lorentzian manifold, decomposed into pseudo-Riemannian submanifolds, being symmetric to each other with respect to the Pierce mirror. This Pierce mirror being an Hermitian hyperquadric is an ellipsoid and it isomorphic to $\mathbb{R}^n\times S^{n-1}$. This is the mirror symmetry of the fourth Frobenius manifold.
\end{proposition}
\begin{proof}
The space of probability distributions $S$ is by Theorem \ref{T:nono} a non-orientable Lorentzian manifold.
Now, $S$ has a pair of pseudo-Riemannian submanifolds (by Collary \ref{C:sbmfd}).
Using Corollary \ref{C:proj} we have that $S$ is identified to a paracomplex projective space. Now, since the spin factor algebra has a pair of idempotents, this implies that there exists a Pierce mirror, inducing symmetries of the space.
We define the following morphism from the algebra of paracomplex numbers to $S^{2n}_1$. The Pierce mirror in the algebra corresponds to an involution in $S^{2n}_1$. The set of fixed points under this involution coincides with a hyperquadric.
More precisely, applying the Theorem\, \ref{T:mirr} (paragraph 4.2), it turns out that this hyperquadric is an Hermitian ellipsoid hyperquadric, and that this is the set of fixed points under the symmetry. Therefore, it is a mirror. Consider the distance from hyperquadric to both domains (totally geodesic submanifolds of the fourth Frobenius manifold). From the properties of the module over a paracomplex algebra and its real interpretation (see section 1.3), we see that the distances from this hyperplane to both domains are geometrically the same (see Theorem \ref{T:mirr} and Theorem 4.3 in \cite{Roz97}). Now, since both domains are isomorphic to each other, the hyperquadric is a reflection mirror.
This shows the statement.
\end{proof}
\section{Conclusion}
We have shown previously that the structure of the cone $\cC$ is a Lorenztian structure. Naturally, this leads to raising questions around {\it causality}, where the causality is interpreted here in the sense of S. W. Hawking $\&$ J. F. R. Ellis \cite{HaEl}.
\smallskip
Let us remark that for a space constructed from modules over an algebra, the principle of causality is no longer a hypothesis.
Such a space can be seen as a ``space-time'' {\it only if} we presuppose some {\it causality principle}. Indeed, in this way one can define a notion of time or {\it times}.
\smallskip
On the other side, we have a cone $\cC$ of measures of bounded variations. This is related to objects being central in machine learning and statistics.
Machine learners and statisticians have translated the philosophical idea of {\it causality} into a viable inferential tool \cite{Sha96}. The main question researchers posed over the past thirty years was the extent to which a change in a causal variable might influence changes in a collection of effect variables, on the basis of observing an uncontrolled idle system. Traditionally, causal inference methods rely on a prespecified set of problem variables and use tools from counterfactual analysis, structural equation models and graphical models.
\smallskip
To conclude, this leads to a defining a bridge, between causality as defined by S. W. Hawking $\&$ J. F. R. Ellis in \cite{HaEl} and causality defined for probability and statistics. Furthermore, this raises many questions and developments around these very active areas of research. | 214,714 |
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\begin{document}
\title[ ]{On integral models of Shimura varieties}
\author[G. Pappas]{G. Pappas}
\thanks{Partially supported by NSF grant DMS-1701619 and the Bell Companies Fellowship Fund through the Institute for Advanced Study}
\address{Dept. of
Mathematics\\
Michigan State
Univ.\\
E. Lansing\\
MI 48824\\
USA} \email{[email protected]}
\begin{abstract}
We show how to characterize integral models of Shimura varieties over places
of the reflex field where the level subgroup is parahoric by formulating a definition of a ``canonical" integral model. We then prove that in Hodge type cases and under a tameness hypothesis, the integral models constructed by the author and Kisin
in previous work are canonical and,
in particular, independent of choices.
\end{abstract}
\date{\today}
\maketitle
\tableofcontents
\def\ve{\bigskip}
\section*{Introduction}
In this paper, we show how to uniquely characterize integral models of Shimura varieties over some primes where non-smooth reduction is allowed. More specifically, we consider integral models over primes $p$ at which
the level subgroup is parahoric. Then, under some further assumptions, we provide a notion of a ``canonical" integral model.
At such primes, the Shimura varieties have integral models with complicated singularities (\cite{R}, \cite{R-Z}). This happens even for the most commonly used Shimura varieties with level structure, such as Siegel varieties, and it foils attempts to characterize the models by simple conditions. The main observation of this paper is that we can characterize these integral models by requiring that they support suitable versal ``$\Gg$-displays", i.e. filtered Frobenius modules with $\Gg$-structure, where $\Gg$ is the smooth integral $p$-adic group scheme which corresponds to the level subgroup. We then prove that these modules exist in most Hodge type cases treated by the author and Kisin in \cite{KP}. As a corollary, we show that these integral models of Shimura varieties with parahoric level structure, are independent of the choices made in their construction.
Let us first recall the story over ``good" places, i.e. over primes at which the level subgroup is hyperspecial. One expects that there is an integral model with smooth reduction at such primes. This expectation was first spelled out by Langlands in the 80's. Later, it was pointed out by Milne \cite{Milne} that one can uniquely characterize smooth integral models over the localization of the reflex field at such places by requiring that they satisfy a Neron-type extension property. Milne calls smooth integral models with this property ``canonical''. The natural integral models of Siegel Shimura varieties, at good primes, are smooth and satisfy the extension property. Therefore, they are canonical. In this case, the extension property follows by the Neron-Ogg-Shafarevich criterion and a purity result of Vasiu and Zink \cite{VasiuZink} about extending abelian schemes over codimension $\geq 2$ subschemes of smooth schemes. This argument extends to the very general class of Shimura varieties of abelian type at good primes, provided we can show there is a smooth integral model which is, roughly speaking, constructed using moduli of abelian varieties. This existence of such a canonical smooth integral model for Shimura varieties of abelian type at places over good primes was shown by Kisin (\cite{KisinJAMS}, see also earlier work of Vasiu \cite{VasiuAsian}).
The problem becomes considerably harder over other primes. Here, we are considering primes $p$ at which the level subgroup is parahoric. For the most part, we also require that the reductive group splits over a tamely ramified extension,
although our formulation is more general. Under these assumptions, models for Shimura varieties of abelian type, integral at places over such $p$, were constructed by Kisin and the author \cite{KP}.
This follows work of Rapoport and Zink \cite{R-Z}, of Rapoport and the author, and of many others, see \cite{Pappasicm}. The construction in
\cite{KP} uses certain simpler schemes, the ``local models" that depend only on the local Shimura data. Then, integral models for Shimura varieties of Hodge type are given by taking the normalization of the Zariski closure of a well-chosen embedding of the Shimura variety in a Siegel moduli scheme over the integers. More generally, models of Shimura varieties of abelian type are obtained from those of Hodge type by a quotient construction that uses Deligne's theory of connected Shimura varieties. All these integral models of Shimura varieties have the same \'etale local structure as the corresponding local models. However, the problem of characterizing them globally or showing that they are independent of choices was not addressed in \emph{loc. cit.}\footnote{with the exception of the very restricted result \cite[Prop. 4.6.28]{KP}.} Here, we give a broader notion of ``canonical" integral model and solve these problems when the varieties are of Hodge type.
Such a characterization was not known before, not even for general PEL type Shimura varieties.
Let us now explain these results more carefully.
Let $(G, X)$ be Shimura data \cite{DeligneCorvallis} with corresponding minuscule
cocharacter conjugacy class $\{\mu\}$ and reflex field $\eE$. For simplicity, we will assume that the center of $G$ does not contain a split $\R$-torus which is $\Q$-anisotropic. For an open compact subgroup $\eK\subset G(\bbA^f)$ of the finite adelic points of $G$, the Shimura variety
\[
{\rm Sh}_{\eK}(G, X)=G(\Q)\backslash (X\times G(\bbA_f)/\eK)
\]
has a canonical model over $\eE$.
Fix a prime $p$. Suppose $\eK=\eK_p\eK^p$, with $\eK_p\subset G(\Q_p)$ and
$\eK^p\subset G(\bbA^p_f)$, both compact open, with $\eK^p$ sufficiently small.
Assume that $p$ is odd and that:
1) The group $G$ splits over an tamely ramified extension of $\Q_p$.
2) The level $\eK_p$ is a \emph{parahoric} subgroup in the sense of Bruhat-Tits \cite{T}, i.e. $\eK_p$ is the connected stabilizer of a point in the (enlarged) building of $G(\Q_p)$. Then $\eK_p=\Gg(\Z_p)$, where $\Gg$ is a smooth connected affine group scheme over $\Z_p$ with $\Gg\otimes_{\Z_p}\Q_p=G_{\Q_p}$.
We will denote by ${\mathrm L}_\eK$ the pro-\'etale $\Gg(\Z_p)$-cover over $\Sh_{\eK}(G, X)$
given by the limit of $\Sh_{\eK'}(G, X)\to \Sh_{\eK}(G, X)$ where $\eK'=\eK'_p\eK^p\subset \eK=\eK_p\eK^p$, with $\eK'_p$ running over all compact open subgroups of $\eK_p$.
Now choose a place $v$ of $\eE$ over $p$. Let $\O_{\eE, (v)}$ be the localization of the ring of integers $\O_{\eE}$ at $v$.
Denote by $E$ the completion of $\eE$ at $v$, by $\O_E$ the integers of $E$ and fix an algebraic closure $k$ of the residue field $k_E$ of $E$. We can also consider $\{\mu\}$ as a conjugacy class of cocharacters which is defined over $E$.
We assume that the pair $(\Gg, \{\mu\})$ is of Hodge type, \emph{i.e.} there is a closed group scheme
immersion $\iota: \Gg\hookrightarrow \GL_n$ over $\Z_p$ such that $\iota(\mu)$ is conjugate to one of the standard minuscule cocharacters of $\GL_n$ and $\iota(\Gg)$ contains the scalars. In \cite{PZ}, (see also \cite{HPR}), we define the local model
\[
\Mloc=\Mloc(\Gg, \{\mu\}).
\]
This is a flat and projective $\O_E$-scheme with $\Gg$-action. Its generic fiber
is $G_E$-equivariantly isomorphic to the variety $X_\mu$ of parabolic subgroups of $G_E$ of type $\mu$,
and its special fiber is reduced. The map $\iota$ gives an equivariant closed immersion
\[
\iota_*: \Mloc\hookrightarrow {\rm Gr}(d, n)_{\O_E}
\]
in a Grassmannian, where $d$ is determined by $\iota(\mu)$.
We ask for $\O_{\eE, (v)}$-models $\SS_{\eK}=\SS_{\eK_p\eK^p}$ (schemes of finite type and flat over $\O_{\eE, (v)}$) of the Shimura variety $\Sh_{\eK}(G, X)$ which are normal. In addition, we require:
\begin{itemize}
\item[1)] For $\eK'^p\subset \eK^p$, there are finite \'etale morphisms
\[
\pi_{\eK'_p, \eK_p}: \SS_{\eK_p\eK'^p}\to \SS_{\eK_p\eK'^p}
\]
which extend the natural $\Sh_{\eK_p\eK'^p}(G, X)\to \Sh_{\eK_p\eK^p}(G, X)$.
\item[2)] The scheme $\SS_{\eK_p}=\varprojlim_{\eK^p}\SS_{\eK_p\eK^p}$ satisfies the ``extension property" for dvrs of mixed characteristic $(0, p)$, i.e. for any such dvr $R$
\[
\SS_{\eK_p}(R[1/p])=\SS_{\eK_p}(R).
\]
\item[3)] The $p$-adic formal schemes $\widehat \SS_{\eK}=\varprojlim_n \SS_\eK\otimes_{\O_{\eE, (v)}}\O_{\eE, (v)}/(p)^n$ support versal {$(\Gg, \Mloc)$-displays}
$
\Dd_\eK
$
which are {associated} to $\Lr_{\eK}$. We ask that these are compatible for varying $\eK^p$, i.e. that there are compatible isomorphisms
\[
\pi_{\eK'_p, \eK_p}^*\Dd_{\eK }\simeq \Dd_{\eK'}
\]
over the system of morphisms $\pi_{\eK'_p, \eK_p}$ of (1).
\end{itemize}
\smallskip
The notion of a $(\Gg, \Mloc)$-display is one of the central constructions of the paper;
we will explain this below. Our first main result is that, under the above assumptions, models $\SS_\eK$ which satisfy the above conditions are uniquely determined up to isomorphism. In particular, they only depend on the Shimura data $(G, X)$, $v$ and $\eK$. We call integral models $\SS_{\eK}$ which satisfy the above, \emph{canonical}. Our second main result is that, under some additional hypotheses,
the integral models of \cite{KP} support such $(\Gg, \Mloc)$-displays. Thus they are canonical and,
in particular, independent of choices in their construction.
We now explain the terms that appear in condition (3):
Suppose $R$ is a normal $p$-adically complete flat $\O_E$-algebra.
Denote by $W(R)$ the ring of ($p$-typical) Witt vectors with entries from $R$
and by $\phi: W(R)\to W(R)$ the Frobenius endomorphism.
A $(\Gg, \Mloc)$-display
$\calD=(\calP, q, \Psi)$ over $R$ consists of a $\Gg$-torsor $\calP$ over the ring of Witt vectors
$W(R)$, a $\Gg$-equivariant morphism
\[
q:\calP\otimes_{W(R)}R\to \Mloc,
\]
and a $\Gg$-isomorphism $\Psi: \calQ\xrightarrow{\sim} \calP $. Here, $\calQ$ is a $\Gg$-torsor over $W(R)$ constructed from $q$ (see Proposition \ref{propQconstr})
which comes together with an isomorphism of $G$-torsors
\[
\calQ[1/p]\xrightarrow{\simeq} \phi^*\calP[1/p]
\]
over $W(R)[1/p]$.
We can think of $\calQ$ as a ``modification of $\phi^*\calP$ along the divisor $p=0$,
bounded by $\Mloc$". This way, the notion of a $(\Gg, \Mloc)$-display resembles
that of a shutka and its mixed characteristic variants (\cite{Schber}). It is also a generalization of the notion of $(\Gg,\mu)$-display due to the author and B\"ultel \cite{BP} (this required the restrictive assumption that
$\Gg$ is reductive over $\Z_p$). By using Zink's Witt vector descent, we obtain
a straightforward extension of this notion from ${\rm Spf}(R)$ to non-affine $p$-adic formal schemes like $\widehat \SS_{\eK}$.
If $R$ is, in addition, a Noetherian complete local ring with perfect residue field, there is a similar notion of a {\sl Dieudonn\'e}
$(\Gg, \Mloc)$-display over $R$ for which all the above objects $ \calP$, $q$, $\Psi$,
are defined over Zink's variant $\hat W(R)$ of the Witt ring. Then, for every local Hodge embedding $\iota:\Gg\hookrightarrow \GL_n$, a Dieudonn\'e $(\Gg, \Mloc)$-display $\calD$ induces a classical Dieudonn\'e display over $R$. By Zink's theory \cite{Zink}, \cite{ZinkCFT}, this gives a $p$-divisible group over $R$.
In (3), we ask that $\Dd_\eK$ is ``associated" to ${\rm L}_{\eK}$. This definition is related to the notion of ``associated" used by Faltings, e.g. \cite{Fa1}. In our case, the definition also involves some constructions of (integral) $p$-adic Hodge theory (see Definition \ref{defass2}).
Finally, let us explain the term ``versal" in (3):
For $\bar x\in \SS_\eK(k)$, denote by $\hat R_{\bar x}$ the
completion of $\SS_\eK\otimes_{\O_{\eE, (v)}}\O_{\breve E}$ at $\bar x$, and set
\[
\calD_{\eK, \bar x}:= \Dd_\eK\otimes_{W(\O_{{\widehat\SS}_\eK})}W(\hat R_{\bar x})
\]
for the $(\Gg, \Mloc)$-display over $\hat R_{\bar x}$ obtained by base change. In view of the existence of
the isomorphisms above, we can omit the subscript $\eK$ and simply write $\calD_{\bar x}$.
The associated condition implies that $\calD_{\bar x}$ supports the structure of a Dieudonn\'e $(\Gg, \Mloc)$-display over $\hat R_{\bar x}$; in the introduction
we denote this also by $\calD_{\bar x}$.
We say that the $(\Gg, \Mloc)$-display $\Dd_\eK$ is versal if $\calD_{\bar x}$ is versal for all $\bar x\in \SS_\eK(k)$. This means that for all $\bar x\in \SS_\eK(k)$ and for a suitable choice of a section $s$ of the $\Gg$-torsor $\calP_{\bar x}\otimes_{ W(\hat R_{\bar x})}\hat R_{\bar x}\to \Spec(\hat R_{\bar x})$, the composition
\[
q\cdot s: \Spec(\hat R_{\bar x})\to \Mloc
\]
identifies $\hat R_{\bar x}$ with the corresponding completion of the local model $\Mloc$. In particular, this condition fixes the singularity of the integral model $\SS_{\eK}$ at $\bar x$.
Showing that, for given $(G, X)$, $v$ and $\eK$, integral models with versal $(\Gg, \Mloc)$-displays $\Dd_\eK$ exist, is quite involved. We use an intermediate notion, that of an ``associated system" $(\Lr_\eK, \{\calD_{\bar x}\}_{\bar x\in \SS_\eK(k)})$ (Definition \ref{defass}), in which $ \calD_{\bar x}$ are Dieudonn\'e $(\Gg, \Mloc)$-displays over the strict completions $\hat R_{\bar x}$, as before.
(A $(\Gg, \Mloc)$-display $\Dd_\eK$ which is associated to $\Lr_\eK$, gives such an associated system by base change, as above.) In fact, most of our constructions just use the associated system $(\Lr_\eK, \{\calD_{\bar x}\}_{\bar x\in \SS_\eK(k)})$. For example, we show that the existence of a versal associated system is enough to characterize the
integral model $\SS_\eK$ uniquely.
We show that $\Lr_{\eK}$ can be completed to a (unique up to isomorphism) associated system by employing comparison isomorphisms of integral $p$-adic Hodge theory that use work of Scholze and others \cite{Schber}, \cite{BMS}, and of Faltings \cite{Fa}, \cite{Fa1}; this works quite generally, i.e. with mild assumptions on the integral model. Proving that we have a versal system is harder since, by definition, versality imposes that the singularities
of the integral model $\SS_\eK$ agree with those of the local model $\Mloc$. We show this for Shimura data of Hodge type, which satisfy some mild additional conditions, by using the main result of \cite{KP}.
It follows that, under these assumptions, the integral models of \cite{KP} are also independent, up to isomorphism, on the various choices made in their construction.
Then, by employing some work of Hamacher and Kim we can also show that these models support a (global) versal $(\Gg, \Mloc)$-display $\Dd_\eK$ associated to $\Lr_\eK$. It then follows, that in this case, the integral models of \cite{KP} are canonical as per the definition above.
Our point of view fits with the well-established idea that most Shimura varieties should be moduli spaces of $G$-motives. As such, they should have (integral) canonical models. We can not make this precise yet. However, we consider the versal $\Gg$-display as the crystalline avatar of the universal $G$-motive and show that its existence is enough to characterize the integral model. In fact, there should be versions for other cohomology theories (see \cite{BMS}, \cite{SchICM}), that are also enough to characterize the models.
Let us now give an outline of the contents of the paper. In \S 1 we prove some preliminary facts about rings of Witt vectors and other $p$-adic period rings, that are used in the constructions. We continue with some more preliminaries on torsors in \S 2. In \S 3 we give the definition of (Dieudonn\'e) displays with $\Gg$-structure. In \S 4, we show how to construct, using the theory of Breuil-Kisin modules, such a display from a $\Gg(\Z_p)$-valued crystalline representation of a $p$-adic field. We also give some other similar constructions, for example a corresponding Breuil-Kisin-Fargues $\Gg$-module. In \S 5, we give the definition of an associated system and show that schemes that support suitable versal associated systems are (\'etale locally) determined by their generic fibers. We also show how to give an associated system starting from a $p$-divisible group whose Tate module carries suitable Galois invariant tensors. In \S 6, we apply this to Shimura varieties and show that systems
$(\SS_\eK, \Lr_\eK, \{\calD_{\bar x}\}_{\bar x\in \SS_\eK(k)})$
as above are unique. In \S 7, we prove that the integral models of Shimura varieties of Hodge type constructed in \cite{KP} carry versal associated systems and are therefore uniquely determined.
\subsection{Acknowledgements} We thank M. Rapoport and P. Scholze for useful suggestions and corrections, V. Drinfeld for interesting discussions,
and the IAS for support.
\subsection{Notations}
Throughout the paper $p$ is a prime and, as usual, we denote by $\Z_p$, $\Q_p$, the $p$-adic integers, resp. $p$-adic
numbers. We fix an algebraic closure $\bar\Q_p$ of $\Q_p$. If $F$ is a finite extension of $\Q_p$, we will
denote by $\O_F$ its ring of integers, by $k_F$ its residue field and by $\breve F$ the completion of the maximal unramified extension of $F$ in $\bar \Q_p$.
\vfill\eject
\section{Algebraic preliminaries}
\subsection{}
We begin with some preliminaries about rings of Witt vectors and other
$p$-adic period rings. The reader can skip the proofs at first reading and return
to them as needed.
\begin{para}
For a $\Z_{(p)}$-algebra $R$, we denote by
\[
W(R)=\{(r_1, r_2,\ldots , r_n, \ldots )\ |\ r_i\in R\}
\]
the ring of ($p$-typical) Witt vectors of $R$. Let $\phi: W(R)\to W(R)$ and $V: W(R)\to W(R)$
denote the Frobenius and Verschiebung, respectively. Let $I_R=V(W(R))\subset W(R)$ be the ideal of elements
with $r_1=0$. The projection $(r_1, r_2,\ldots )\mapsto r_1$ gives an isomorphism $W(R)/I_R\simeq R$. For $r\in R$, we set as usual
\[
[r]=(r, 0, 0, \ldots )\in W(R)
\]
for the Teichm\"uller representative.
Also denote by
\[
{\rm gh}: W(R)\rightarrow \prod\nolimits_{i\geq 1} R
\]
the Witt (``ghost'') coordinates ${\rm gh}=({\rm gh}_i)_i$. Recall,
\[
{\rm gh}_i((r_1, r_2, r_3, \ldots ))=r_1^{p^{i-1}}+pr_2^{p^{i-2}}+\cdots + p^{i-1}r_i.
\]
\end{para}
\begin{para}
Let $R$ be a complete Noetherian local ring with maximal ideal $\frakm_R$ and perfect residue field $k$
of characteristic $p$. Assume that $p\geq 3$. There is a splitting
$W(R)=W(k)\oplus W(\frakm_R)$ and following Zink (\cite{ZinkCFT}), we can consider the subring $\hat W(R)=W(k)\oplus \hat W(\frakm_R)\subset W(R)$, where $\hat W(\frakm_R)$ consists of those Witt vectors with $r_n\in \frakm_R$ for which the sequence $r_n$ converges to $0$ in the $\frakm_R$-topology of $R$. The subring $\hat W(R)$ is stable under $\phi$ and $V$. In this case, both $W(R)$ and $\hat W(R)$ are $p$-adically complete and separated local rings.
\end{para}
\begin{para}\label{Algcond0}
In what follows, we consider a $\Z_p$-algebra $R$ that satisfies:
\begin{itemize}
\item[(1)] $R$ is complete and separated for the adic topology given by a finitely generated ideal $\calA$ that contains $p$, and
\item[(2)] $R$ is formally of finite type over $W=W(k)$, where $k$ is a perfect field of characteristic $p$.
\end{itemize}
\end{para}
\subsection{}
Let $\frakM\subset W(R)$ be a maximal ideal with residue field $k'$.
We have $I_R\subset \frakM$, since $W(R)$ is $I_R$-adically complete and separated (\cite[Prop. 3]{Zink}). Let
$\frakm_R=\frakM/I_R\subset W(R)/I_R=R$ be the corresponding maximal ideal of $R$.
Suppose that $\hat R $ is the completion of $R$ at $\frakm$. Then $W(\hat R)$ is local henselian.
Denote by $W(R)^h_{\frakM}$ the henselization of the localization $W(R)_\frakM$.
\begin{lemma}\label{A1lemma} Assume, in addition to (1) and (2), that $R$ is an integral domain and $\Z_p$-flat.
Then, the natural homomorphism $R\to \hat R$ induces injections
\[
W(R)\subset W(R)^h_{\frakM}\subset W(\hat R).
\]
\end{lemma}
\begin{proof}
Since $R$ is $p$-torsion free,
\[
{\rm gh}: W(R)\hookrightarrow \prod\nolimits_{i\geq 1} R
\]
is an injective ring homomorphism. If $f=(f_1, f_2,\ldots )\not\in \frakM$, then $f_1\not\in \frakm$.
Since $p\in \frakm$, we have ${\rm gh}_i(f)\not\in\frakm$, for all $i$. In particular, ${\rm gh}_i(f)\neq 0$
and so since $R$ is a domain, $f$ is a not a zero divisor in $W(R)$. It follows that
$W(R)$ is a subring of $W(R)_\frakM$. We now consider $W(R)\subset W(\hat R)$. Notice that $f$ is invertible in $W(\hat R)$
since $\hat R$ is $p$-adically complete and $f_1$ is invertible in $\hat R$. Hence, we have an injection $W(R)_\frakM\hookrightarrow W(\hat R)$. The ring $W(\hat R)$ is local and henselian and $W(R)_\frakM\hookrightarrow W(\hat R)$ is a local ring homomorphism. It follows that the henselization
$W(R)^h_\frakM$ is contained in $W(\hat R)$.
\end{proof}
\quash{
\begin{lemma}\label{dejonglemma}
Assume, in addition to (1) and (2), that $R$ is a normal domain flat over $\Z_p$. Then, for all $n\geq 0$, including $n=\infty$, we have
\[
W_n(R)=\{f\in W_n(R)[1/p]\ |\ \forall F, \forall \xi: R\to \O_F, \ \xi(f) \in W_n(\O_F) \},
\]
where $F$ runs over all finite extensions of $W(k)[1/p]$ and $\xi$ over all $W(k)$-algebra homomorphisms.
\end{lemma}
\begin{proof}
Consider $f\in W_n(R)[1/p]$ so that $g=p^af\in W_n(R)$ for some $a\geq 0$.
Assume that $\xi(g)$ is divisible by $p^a$ in $W_n(\O_F)$, for all $\xi: R\to \O_F$.
We would like to show that $g$ is divisible by $p^a$ in $W_n(R)$. This will be the case when
certain universal polynomials in the ghost coordinates ${\rm gh}_i(g)$ which have coefficients in $\Z[1/p]$, take values in $R$. By \cite[Prop. 7.3.6]{deJongCrys} this is equivalent to asking that the same
polynomials in ${\rm gh}_i(\xi(g))$ take values in $\O_F$, for all $\xi$. This is true by our assumption.
\end{proof}
}
\subsection{}\label{Algcond}
For a $\Z_p$-algebra $R$ we consider the condition:
\begin{itemize}
\item[ (N)] $R$ is a \emph{normal} domain, is flat over $\Z_p$, and satisfies (1) and (2).
\end{itemize}
We say that $\Z_p$-algebra $R$ satisfies condition (CN) when, in addition to satisfying (N), $R$ is a (complete) local ring.
\begin{para}
Suppose $R$ satisfies (N). We start by recalling a useful statement shown by de Jong \cite[Prop. 7.3.6]{deJongCrys}, in a slightly different language.
\begin{prop}\label{dejong}
We have
\[
R=\{f\in R[1/p]\ |\ \forall F, \forall \xi: R\to \O_F, \ \xi(f) \in \O_F \},
\]
where $F$ runs over all finite extensions of $W(k)[1/p]$ and $\xi$ over all $W(k)$-algebra homomorphisms. \hfill $\square$
\end{prop}
\begin{cor}\label{dejonglemma}
We have
\[
W(R)= (W(R)[1/p])\cap\prod\nolimits_{\xi} W(\O_F)
\]
where $F$, $\xi$ are as above.
\end{cor}
\begin{proof}
Under our assumption, $W(R)$ is $\Z_p$-flat. Consider $f\in W(R)[1/p]$ so that $g=p^af\in W(R)$ for some $a\geq 0$.
Assume that $\xi(g)$ is divisible by $p^a$ in $W(\O_F)$, for all $\xi: R\to \O_F$.
We would like to show that $g$ is divisible by $p^a$ in $W(R)$. This will be the case when
certain universal polynomials in the ghost coordinates ${\rm gh}_i(g)$ which have coefficients in $\Z[1/p]$, take values in $R$. By Proposition \ref{dejong}, this is equivalent to asking that the same
polynomials in ${\rm gh}_i(\xi(g))$ take values in $\O_F$, for all $\xi$. This is true by our assumption.
\end{proof}
\begin{prop}\label{corInter2} Suppose $R$ satisfies (CN). We have
\[
W(R)\cap (\prod_{\xi: R\to \O_F}\hat W(\O_F))=\hat W(R).
\]
Here the product is over all finite extensions $F$ of $W(k)[1/p]$ and all $W(k)$-algebra homomorphisms $\xi: R\to \O_F$.
The intersection takes place in $\prod_{\xi: R\to \O_F} W(\O_F)$.
\end{prop}
\begin{proof}
This follows from the definitions and:
\begin{prop} Suppose that $(f_n)_n$ is a sequence of elements of the maximal ideal $\frakm_R$ such that, for every finite extension $F$ of $W(k)[1/p]$ and every $W(k)$-algebra homomorphism $\xi: R\to \O_F$, the sequence
$(\xi(f_n))_n$ converges to $0$ in the $p$-adic topology of $F$. Then $(f_n)_n$ converges to $0$ in the $\frakm_R$-topology.
\end{prop}
\begin{proof}
Under our assumption on $R$, there is a finite injective ring homomorphism
\[
\phi^*: R_0=W(k)\lps t_1,\ldots, t_r\rps\to R.
\]
We will use this to reduce the proof to the case $R=R_0$.
We start by showing that there is $d\geq 1$, such that every $f\in R$ satisfies an equation of the form
\begin{equation}\label{ff}
P(T; f)=T^d+a_1T^{d-1}+\cdots +a_{d-1}T+a_d=0, \qquad a_i=a_i(f)\in R_0.
\end{equation}
Indeed, let $d$ be the degree of the extension ${\rm Frac}(R)/{\rm Frac}(R_0)$ of fraction fields. Since both $R$ and $R_0$ are regular in codimension $1$, the morphism $\phi: \Spec(R)\to \Spec(R_0)$ is finite flat of degree $d$ over the complement $V=\Spec(R_0)-Z$ of some closed subscheme $Z$ of codimension $\geq 2$. We can obtain (\ref{ff}) by considering the action of the endomorphism $T$ given by $f$ on the $\O_{V}$-module $\phi_*(\O_{\phi^{-1}(V)})$ which is locally free of rank $d$. The coefficients $a_i$ are regular outside $Z$ and so they belong to $R_0$ (cf. \cite[Lemma 7.3.3]{deJongCrys}).
Assume that $(f_n)_n$ is a sequence of elements in $\frakm_R$ that satisfies the assumption of the proposition. Fix a finite Galois extension $F'$ of ${\rm Frac}(R_0)$ that contains ${\rm Frac}(R)$ and let $R'$ be the integral closure of $R_0$ in $F'$ so that $R_0\subset R\subset R'$.
For each $n$ we can write
\[
P(T; f_n)=\prod_{i=1}^d(T-f_{n, i})
\]
with $f_{n, i}\in R'$ and $f_{n, 0}=f_n$. The elements $f_{n, i}$ are Galois conjugates of $f_n$. We can now see that the assumption of the proposition
is satisfied for all the sequences $(f_{n, i})_n$ in $R'$. Therefore, it is also satisfied for the sequence of their symmetric functions $(a_i(f_n))_n$ in $R_0$, for each $i$.
Suppose now that we know
that the proposition is true for $R_0$. Then, we obtain that $a_i(f_n)$ converges to $0$ in the $\frakm_{R_0}$-topology.
This combined with (\ref{ff}) implies that $f_n^d$ converges to $0$ in the $\frak m_R$-topology. For $f\in R$, consider the sequence of ideals
\[
\cdots \subset I_{a+1}=(\frakm_R^{a+1}; f)\subset I_a=(\frakm_R^a; f)\subset \cdots
\]
of $R$. Krull's intersection theorem implies $\cap_{a=0}^{\infty}I_a=(0)$ and so, by Chevalley's lemma, the ideals $(I_a)_a$ also define the
$\frakm_{R}$-topology of $R$. Since $f_n^d\in \frakm_{R}^a$ implies $f_n^{d-1}\in (\frakm_R^a; f_n)$, we quickly obtain, by decreasing induction on $d$, that $f_n$ converges to $0$ in the $\frak m_R$-topology.
It remains to prove the proposition for the power series ring $R_0$:
Set $Y=\Spec(R_0)$, and let $h: \ti Y\to Y$ be the blowup of $Y$ at the maximal ideal $\frakm=(p, t_1,\ldots, t_r)$. The exceptional divisor $E$ can be identified with ${\mathbb P}(\frakm/\frakm^2)\cong {\mathbb P}^r_k$. Assume that $(f_n)_n$ is a sequence of elements in $\frakm$ that satisfies the assumption of the proposition but is such that $(f_n)_n$ does not converge to $0$ in the $\frakm$-topology. Then, by replacing $(f_n)_n$ by a subsequence, we can assume that, there is an integer $N\geq 1$, such that $f_n\in \frakm^N-\frakm^{N+1}$, for all $n$. Then the proper transform $Z(f_n)\subset \ti Y$ of $f_n=0$ intersects the exceptional divisor $E={\mathbb P}^r_k$ along a hypersurface $S_n\subset {\mathbb P}^r_k$ of degree $N$.
\begin{lemma} After replacing $(f_n)_n$ by a subsequence, we can find a $\bar k$-valued point $x$ of the exceptional divisor ${\mathbb P}^r_k$ which does not lie on any of the proper transforms $Z(f_n)$ of $f_n=0$, for all $n$.
\end{lemma}
\begin{proof} We argue by contradiction: Assume that for any given point $x\in {\mathbb P}^r(\bar k)$ and almost all $n$, $Z(f_n)$ contains $x$. Then, also for every finite set of points $A(m)=\{x_1,\ldots , x_m\}$, we have $A(m)\subset Z(f_n)$, for almost all $n$.
Since $Z(f_n)\cap {\mathbb P}^r$ is, for each $n$, a hypersurface of fixed degree $N$, when $m>>N$ this is not possible.
\end{proof}
\smallskip
We now extend $x$ given by the lemma,
to a point $\ti x\in \ti Y(\O_{F})=Y(\O_F)$, where $F$ is some finite extension of $
W(k)_\Q$. By assumption, $\ti x(f_n)\to 0$ in $F$. Since $\ti x$ misses the proper transform of $f_n=0$, this implies that the valuation of $h^*(f_n)$ along the exceptional
divisor grows without bound. This contradicts $f_n\not\in \frakm^{N+1}$.
\end{proof}\end{proof}
\end{para}
\subsection{}\label{appCompl}
We assume that $R$ satisfies (CN) with $k=\bar{\mathbb F}_p$.
Fix an algebraic closure $\overline{ F(R)}$ of the fraction field $F(R)={\rm Frac}(R)$. Denote by $\ti R$
the union of all finite normal $R$-algebras $R'$ such that:
\begin{itemize}
\item[1)] $R\subset R'\subset \overline{ F(R)}$, and
\item[2)] $R'[1/p]$ is finite \'etale over $R[1/p]$.
\end{itemize}
Note that all such $R'$ are local and complete. We will denote by $\bar R$ the integral closure
of $R$ in $\overline{ F(R)}$, so that $\bar R$ is the union of all $R'$ as in (1).
Let us set
\[
\Gamma_R={\rm Gal}(\ti R[1/p]/R[1/p]),
\]
which acts on $\ti R$.
Also denote by
\[
\ti R^\wedge=\varprojlim\nolimits_n \ti R/p^n\ti R,\quad \ti R^\wedge=\varprojlim\nolimits_n \bar R/p^n\bar R,
\]
the $p$-adic completions.
When $R=W=W(k)$, we denote ${\bar W}^\wedge={\ti W}^\wedge$ by $\O$.
\begin{prop}\label{prop131}
The natural maps $\bar R\to \bar R^\wedge$, $\ti R\to \ti R^\wedge$, are injections and induce
isomorphisms $\bar R/p^n\bar R\simeq \bar R^\wedge/p^n\bar R^\wedge$, $\ti R/p^n\ti R\simeq \ti R^\wedge/p^n\ti R^\wedge$,
for all $n\geq 1$.
\end{prop}
\begin{proof}
This is given by the argument in \cite[Prop. 2.0.3]{Brinon} which deals with the case of $\ti R$ and the case of $\bar R$ is similar.
\end{proof}
\begin{prop}\label{prop132} Let $S$ be $\ti R^\wedge$ or $\bar R^\wedge$.
a) $S$ is $p$-adically complete and separated and is flat over $\Z_p$.
b) $S$ is an integral perfectoid algebra (in the sense of \cite[3.2]{BMS}).
c) $S$ is local and strict henselian.
\end{prop}
\begin{proof}
Part (a) is also given by \cite[Prop. 2.0.3]{Brinon}.
Let us show (b) for $S=\ti R^\wedge$. The argument for $\bar R^\wedge$ is similar and actually simpler.
We see that $\ti R$ and so $S$, contains an element $\pi$ with $\pi^p=p$. Then $S$ is $\pi$-adically complete. Using
\cite[Lemma 3.10]{BMS}, it is now enough to show that the Frobenius $\phi: S/\pi S\to S/p S$ is an isomorphism and that $\pi$ is not a zero divisor in $S$. Since $S$ is $\Z_p$-flat, $\pi$ is not a zero divisor. By Proposition \ref{prop131}, we
have $\ti R/p\ti R\simeq S/pS$ and similarly $\ti R/\pi\ti R\simeq S/\pi S$. Hence, it is enough to show that
$\phi: \ti R/\pi \ti R\to \ti R/p \ti R$ is an isomorphism. Suppose now $x\in \ti R$ satisfies $x^p=p y$ with $y\in \ti R$. Then $(x/\pi)^p=y$ and since $\ti R$ is a union of normal domains, we have $x/\pi=z\in \ti R$. This shows injectivity. To show surjectivity, consider $a\in R'\subset \ti R$ and consider
\[
R''=R'[X]/(X^{p^2}-p X-a).
\]
This is a finite $R$-algebra. It is \'etale over $R'[1/p]$ since the derivative $p(pX^{p^2-1}-1)$ is a unit in $R''[1/p]$. Now there is $R''\to \overline{ F(R)}$ that extends $R'\subset \overline{ F(R)}$ and the image $b$ of $X$ in $\overline{ F(R)}$ is contained in a finite $R'$-algebra which is also \'etale over $R'[1/p]$. This gives $b\in \ti R$ with $b^{p^2}\equiv a\ {\rm mod\ } p\ti R$ which implies surjectivity.
For part (c), since $S=\ti R^\wedge$ is $p$-adically complete, it is enough to show that these properties are true for $S/pS\simeq \ti R/p\ti R$. We can see that $\ti R$ is both local and strict henselian, and then so is the quotient $\ti R/p\ti R$. The argument for $\bar R^\wedge$ is similar.
\end{proof}
\begin{thm}\label{FaltingsAlmost}
The action of $\Gamma_R$ on $\ti R$ extends to a
$p$-adically continuous action on $\ti R^\wedge$ and we have
\[
(\ti R^\wedge)^{\Gamma_R}=R.
\]
\end{thm}
\begin{proof}
By Faltings \cite{Fa} or \cite[Prop. 3.1.8]{Brinon}, we have
\[
R\subset (\ti R^\wedge)^{\Gamma_R}\subset R[1/p].
\]
Using this, we see that it remains to show that $\ti R^\wedge\cap R[1/p]=R$, with the intersection in $\ti R^\wedge[1/p]$.
Suppose $f\in \ti R^\wedge\cap R[1/p]$. By applying Proposition \ref{dejong}, we see that it is enough to show
that $\xi(f)\in \O_F$, for all $W$-algebra homomorphisms $\xi: R\to \O_F$ with $F$ a finite extension of
$W[1/p]$. Choose such a $\xi: R\to \O_F$. We can extend $\xi$ to $\bar \xi: \ti R\to \bar\O_F=\O_{\bar F}$ and then by $p$-adic completion to
\[
{\bar \xi}^\wedge: \ti R^\wedge\to \O.
\]
This gives ${\bar \xi}^\wedge: \ti R^\wedge[1/p]\to \O[1/p]$. But then
$\bar\xi^\wedge(f)\in F\cap \O=\O_F$.
\end{proof}
\smallskip
\subsection{}\label{appAinf}
In the rest of the section, we will restrict to the case $S=\ti R^\wedge$.
We will use the notations of \cite[\S 3]{BMS}. Consider the tilt
\[
S^\flat=\varprojlim\nolimits_\phi S/pS=\varprojlim\nolimits_\phi S
\]
and similarly for $\O^\flat$.
\begin{lemma}\label{localS}
The ring $S^\flat$ is local strict henselian with residue field $k$.
\end{lemma}
\begin{proof} As we have seen, the rings $S$ and $S/pS=\ti R/p\ti R$ are local and strict henselian with residue field $k$. Denote by $x\mapsto \bar x$
the map $S/pS\to k$.
The Frobenius $S/pS\to S/pS$ is surjective and, hence, $S^\flat\to S/pS$ is surjective. If
$
x=(x_0, x_1, x_2, \ldots )\in S^\flat
$
with $x_i\in S/pS$, $x_{i+1}^p=x_i$, has $x_0$ a unit, then all $x_i$ and also $x$ are units. Hence, $S^\flat$ is local with residue field $k$ and $(x_0, x_1, x_2, \ldots )\mapsto (\bar x_0, \bar x_1, \bar x_2, \ldots )$ is the residue field map $S^\flat\to k$.
Now consider $f(T)\in S^\flat[T]$ with a simple root $\kappa=(\kappa_0, \kappa_1, \kappa_2, \ldots )$ in $k$, with $\kappa_i=\sqrt[p^i]{\kappa_0}$. Since $S/pS$ is local henselian, the simple root $\kappa_i\in k$ of $f_i(T)$ lifts uniquely to a root $a_i\in S/pS$.
By uniqueness, we have $a_{i+1}^p=a_i$ and so $a=(a_0, a_1,\ldots )$ is a root in $S^\flat$ that lifts $\kappa$. Hence, $S^\flat$ satisfies Hensel's lemma.
\end{proof}
\begin{para}\label{thetanotations}
We set $A_{\rm inf}(S)=W(S^\flat)$ for Fontaine's ring. By \cite[Lemma 3.2]{BMS}, we have
\[
A_{\inf}(S)\cong\varprojlim\nolimits_{\phi} W_r(S).
\]
This gives corresponding homomorphisms
\[
\tilde \theta_r: A_\inf(S)\to W_r(S).
\]
We also have the standard homomorphism of $p$-adic Hodge theory
\[
\theta: A_\inf(S)\to S,
\]
given by
\[
\theta(\sum\nolimits_{n\geq 0} [x_n]p^n)=\sum\nolimits_{n\geq 0} x_n^{(n)}p^n.
\]
Here, we write $x=(x^{(0)}, x^{(1)},\ldots )\in S^\flat=\varprojlim\nolimits_\phi S$.
As in \cite[\S 3]{BMS}, the homomorphism $\theta$ lifts to
\[
\theta_\infty: A_\inf(S)\to W(S),
\]
given by
\[
\theta_\infty(\sum\nolimits_{n\geq 0} [x_n]p^n)=\sum\nolimits_{n\geq 0}[x^{(n)}_n]p^n.
\]
\end{para}
\begin{para}\label{par143}
In the following, $\alpha$ runs over all rings homomorphisms $\alpha: S\to \O$ which are obtained from some $W$-homomorphism
$ \ti R\to \ti W=\bar W$ by $p$-adic completion. There is a corresponding $A_\inf(S)\to A_\inf(\O)$ given by applying
the Witt vector functor to $\alpha^\flat: S^\flat\to \O^\flat$.
Note that if $F$ is a finite extension of $W[1/p]$, then any homomorphism $ \xi: R \to \O_F$ extends to such an $\alpha: S\to \O$.
\begin{lemma}\label{lemmainf0}
a) The homomorphism $A_\inf(S)\to \prod_{\alpha} A_\inf(\O)$ is injective.
b) The ring $A_\inf(S)$ is $p$-adically complete, local strict henselian and $\Z_p$-flat.
\end{lemma}
\begin{proof} We first note that $\ti R\to \prod_\alpha \bar W$ is injective and $\ti R\cap (\prod_\alpha p\bar W)=p\bar R$, as this easily follows by Proposition \ref{dejong} applied to the algebras $R'$. Therefore,
\[
S/pS=\ti R/p\ti R\subset \prod\nolimits_{\alpha} \bar W/p\bar W= \prod\nolimits_{\alpha}\O/p\O
\]
is injective. Hence, $S^\flat\to \prod_\alpha\O^\flat$ is injective and part (a) follows.
To show part (b), recall $A_\inf(S)=W(S^\flat)$.
The ring $S^\flat$ is perfect, and so $pW(S^\flat)=I_{S^\flat}$, $W(S^\flat)/pW(S^\flat)=S^\flat$. It follows that $A_\inf(S)$ is $p$-adically complete and that $p$ is not a zero divisor. Lemma \ref{localS} now implies that $A_\inf(S)$ is local and strict henselian. \end{proof}
\end{para}
\begin{para}
Now let us fix an embedding $\bar W\hookrightarrow \ti R$, which induces $\O\hookrightarrow S$.
Let
\[
\epsilon=(1, \zeta_p, \zeta_{p^2},\ldots )\in \O^\flat=\varprojlim\nolimits_\phi \O
\]
be a system of primitive $p$-th power roots of unity. Set
\[
\mu=[\epsilon]-1\in A_{\inf}(\O)\subset A_\inf(S).
\]
\begin{prop}\label{lemmainf1}
a) The element $\mu$ is not a zero divisor in $A_\inf(S)$.
b) Suppose that $f$ in $A_\inf(S)$ is such that, for every $\alpha: S\to \O$ obtained from $\ti R\to \bar W$
as above, $\alpha(f)$ is in the ideal
$(\mu)$ of $A_\inf(\O)$. Then, $f$ is in $(\mu)$.
\end{prop}
\begin{proof}
Part (a) follows from \cite[Prop. 3.17 (ii)]{BMS}. As in the proof of \emph{loc. cit.}, the ghost coordinate vectors of $\tilde\theta_r(\mu)$ are
\[
{\rm gh}(\tilde\theta_r(\mu))=(\zeta_{p^r}-1,\ldots, \zeta_p-1)\in S^r,
\]
and the result follows from this.
Let us show (b). Recall
\[
A_\inf(S)=\varprojlim\nolimits_{\phi} W_r(S)\subset \prod\nolimits_{r\geq 1} W_r(S); \ a\mapsto (\tilde \theta_r(a))_r.
\]
Now suppose that $\alpha(f)=\mu\cdot b_\alpha$, for $b_\alpha\in A_\inf(\O)$. Apply $\tilde \theta_r$ to obtain
\[
\alpha(\tilde \theta_r(f))=\tilde \theta_r(\alpha(f))=\tilde\theta_r(\mu)\cdot \tilde\theta_r(b_\alpha).
\]
This implies that, for all $i=1,\ldots, r$, and all $\alpha$,
\[
\zeta_{p^i}-1\ |\ \alpha({\rm gh}_i(\tilde \theta_r(f)))
\]
in $\O$. The same argument as in the proof of Lemma \ref{lemmainf0} (a) above, gives
\[
S/(\zeta_{p^i}-1)S\hookrightarrow \prod\nolimits_\alpha \O/(\zeta_{p^i}-1)\O.
\]
This implies that $\zeta_{p^i}-1$ (uniquely) divides ${\rm gh}_i(\tilde \theta_r(f))$ in $S$.
We claim that the quotients $g_{i, r}={\rm gh}_i(\tilde \theta_r(f))/(\zeta_{p^i}-1)$ in $S$ are the ghost coordinates
of an element $\gamma_r$ of $W_r(S)$, which is then the quotient $\tilde \theta_r(f)/\tilde \theta_r(\mu)$. To check this we have to show that certain universal polynomials in the $g_{i, r}$ with coefficients in $\Z[1/p]$ take values in $S$. This holds
after evaluating by $\alpha: S\to \O$ and so the same argument using Proposition \ref{dejong} as before, shows that it is true. It follows that, for all $r$, $\ti\theta_r(\mu)$ uniquely divides $\ti\theta_r(f)$ in $W_r(S)$ and, in fact,
\[
\ti\theta_r(f)=\ti\theta_r(\mu)\cdot \gamma_r.
\]
Applying $\phi$ gives
\[
\ti\theta_{r-1}(f)=\ti\theta_{r-1}(\mu)\cdot \phi(\gamma_r).
\]
in $W_{r-1}(S)$. Therefore, $\phi(\gamma_r)=\gamma_{r-1}$. Hence, there is $\gamma\in A_\inf(S)=\varprojlim\nolimits_{\phi} W_r(S)$ such that $\gamma_r=\ti \theta_r(\gamma)$. Then,
$
f=\mu\cdot \gamma.
$
\end{proof}
\begin{cor}\label{corinf}
a) We have
$
A_\inf(S)=(A_\inf(S)[1/\mu])\cap \prod_\alpha A_\inf(\O).
$
b) Suppose that
$M_1$ and $M_2$ are two finite free $A_\inf(S)$-modules with $M_1[1/\mu]=M_2[1/\mu]$, and such that
$\alpha^*M_1=\alpha^*M_2$ as $A_\inf(\O)$-submodules of $\alpha^*M_1[1/\mu]=\alpha^*M_2[1/\mu]$, for all $\alpha$. Then $M_1=M_2$.
\end{cor}
\begin{proof}
Part (a) follows directly from the previous proposition. Part (b) follows by applying (a) to the entries of the matrices expressing
a basis of $M_1$ as a combination of a basis of $M_2$, and vice versa.
\end{proof}
\end{para}
\begin{para}
As in \cite[\S 3]{BMS}, set $\xi=\mu/\phi^{-1}(\mu)$ which is a generator of the kernel of the homomorphism $\theta: A_\inf(S)\to S$.
Let $A_{\rm cris}(S)$ be the $p$-adic completion of the divided power envelope of $A_\inf(S)$ along $(\xi)$. By \cite[App. to XVII]{Schber}, the natural homomorphism
\[
A_\inf(S)\rightarrow A_{\rm cris}(S)
\]
is injective.
\begin{comment}
Set $B^+_{\rm cris}(S)=A_{\rm cris}(S)[1/p]$ and $B_{\rm cris}(S)=A_{\rm cris}(S)[1/(p\mu)]=A_{\rm cris}(S)[1/\mu]$.
\begin{lemma}\label{lemmainf2}
We have $A_\inf(S)\subset A_{\rm cris}(S)$ and
\[
A_{\rm inf}(S)\subset A_{\rm inf}(S)[1/(p\mu)]\subset A_{\rm cris}(S)[1/(p\mu)]=B_{\rm cris}(S).
\]
\end{lemma}
\begin{proof}
See \cite[App. to XVII]{Schber} for the first statement. The rest follows from this and the definitions since, by the above lemmas, $p$ and $\mu$ are not zero divisors in $A_\inf(S)$.
\end{proof}
\end{comment}
We also record:
\begin{lemma}\label{lemmainf3}
$A_\inf(S)^{\phi=1}=\Z_p$.
\end{lemma}
\begin{proof}
It is enough to show that $(S^\flat)^{\phi=1}={\mathbb F}_p$, i.e. that $(\ti R/p\ti R)^{\phi=1}=(S/pS)^{\phi=1}={\mathbb F}_p$.
Now argue as in \cite[6.2.19]{Brinon}:
Suppose $a\in \ti R$ is such that $a^p=a\, {\rm mod}\, p\ti R$. Then $a\in R'$, for some $R'/R$ finite normal with $R'[1/p]$ \'etale over $R[1/p]$ and $a^p=a\, {\rm mod}\, pR'$.
By Hensel's lemma for the $p$-adically complete $R'$,
the equation $x^p-x=0$ has a root $b$ in $R'$ which is congruent to $a\, {\rm mod}\, pR'$. But $R'$ is an integral domain, so any such root is one of the standard roots in $\Z_p$, so $b$ is in $\Z_p$ and $b=a\, {\rm mod}\, pR'$ in ${\mathbb F}_p$.
\end{proof}
\end{para}
\ve
\section{Shimura pairs and $\Gg$-torsors}
\subsection{}\label{ssShimura}
We first set some notation and define the notion of a local Hodge embedding.
\begin{para}
Let $G$ be a connected reductive algebraic group over $\Q_p$
and $\{\mu\}$ the $G(\bar \Q_p)$-conjugacy class of a
minuscule cocharacter $\mu: \Gm_{\bar\Q_p}\to G_{\bar \Q_p}$.
To such a pair $(G, \{\mu\})$, we associate:
\begin{itemize}
\item The reflex field $E\subset \bar\Q_p$. As usual, $E$ is the field of definition of the conjugacy class $\{\mu\}$ (i.e. the finite extension of $\Q_p$ which corresponds to the subgroup of $\sigma\in \Gal(\bar\Q_p/\Q_p)$ such that $\sigma(\mu)$ is $G(\bar\Q_p)$-conjugate to $\mu$.)
\item The $G$-homogeneous variety $X_\mu=X_\mu(G)$ of parabolic subgroups of $G$ of type $\mu$. This is a projective smooth $G$-variety defined over $E$ with $X_\mu(\bar\Q_p)=G(\bar\Q_p)/P_{\mu}(\bar\Q_p)$.
\end{itemize}
\end{para}
\begin{para}\label{par113}
We will consider a smooth connected affine group scheme $\Gg$
over $\Z_p$ with generic fiber $G$ and assume the following:
(H) There is a group scheme homomorphism
\[
\iota: \Gg\hookrightarrow \GL_n
\]
which is a closed immersion such that $\{\iota(\mu)\}$ is
conjugate to one of the standard minuscule cocharacters $\mu_d(a)={\rm diag}(a^{(d)}, 1^{(n-d)})$ of $\GL_n$ for some $1\leq d\leq n-1$, and $\iota(\Gg)$ contains the scalars $\Gm$.
We call such an $\iota$, a \emph{local Hodge embedding}.
The corresponding $\GL_n$-homogeneous space $X_{\mu_d}(\GL_n)$ is the Grassmannian ${\rm Gr}(d, n)$ of $d$-spaces in $\Q_p^n$. Under the assumption (H), $\iota$ gives an equivariant closed embedding $X_\mu\subset {\rm Gr}(d, n)_E$. Set $\Lambda=\Z_p^n$. The Grassmannian ${\rm Gr}(d, n) $ has a natural model over $\Z_p$ which we will denote by ${\rm Gr}(d, \Lambda)$.
Denote by $\M$ the Zariski closure of $X_\mu$ in ${\rm Gr}(d, \Lambda)_{\O_E}$.
It admits an action of $\Gg$ and a $\Gg$-equivariant closed immersion
\[
\iota_*: \M\hookrightarrow {\rm Gr}(d, \Lambda)_{\O_E}.
\]
\end{para}
\begin{para}
Denote by $\Lambda^\otimes =\oplus_{m, n\geq 0}\, \Lambda^{\otimes m}\otimes_{\Z_p}(\Lambda^\vee)^{\otimes n}$ the total tensor algebra of $\Lambda$, where $\Lambda^\vee={\rm Hom}_{\Z_p}(\Lambda, \Z_p)$.
By \cite[Prop. 1.3.2]{KisinJAMS}, see also \cite{DeligneLetter}, we can
realize $\Gg$ as the scheme theoretic fixer of a finite list of tensors $(s_a)\subset \Lambda^\otimes$: For any $\Z_p$-algebra
$R$ we have
\[
\Gg(R)=\{g\in \GL(\Lambda\otimes_{\Z_p}R)\ |\ g\cdot (s_a\otimes 1)=(s_a\otimes 1)\}.
\]
Since we assume that $\iota(\Gg)$ contains the scalars $\Gm$ and $a\in \Gm$ acts on $\Lambda^{\otimes m}\otimes_{\Z_p}(\Lambda^\vee)^{\otimes n}$ via $a^{m-n}$, we see that the $s_a$ are contained in the part of the tensor algebra with $n=m$.
In particular, we can assume that every tensor $s_a$ is given by a $\Z_p$-linear map $\Lambda^{\otimes n}\to \Lambda^{\otimes n}$, for some $n=n_a$.
\end{para}
\subsection{}
Let us now collect some general statements about $\Gg$-torsors.
We denote by ${\rm Rep}_{\Z_p}(\Gg)$ the exact tensor category of representations of $\Gg$ on finite free $\Z_p$-modules, i.e. of group scheme homomorphisms $\rho: \Gg\to \GL(\Lambda')$ with $\Lambda'$ a finite free $\Z_p$-module.
\begin{para}\label{gen12}
Let $A$ be a $\Z_p$-algebra. Set $S=\Spec(A)$. Suppose that
$T\to S$ is a $\Gg$-torsor. By definition, this means that $T$ supports a (left) $\Gg$-action $\Gg\times T\to T$ such that
$\Gg\times T \xrightarrow{\sim} T\times_S T $ given by $(g, t)\mapsto (gt, t)$ is an isomorphism, and $T\to S$ is faithfully flat. By descent, $T$ is affine, so $T=\Spec(B)$ with
$A\to B$ faithfully flat.
If $\rho: \Gg\to \GL(\Lambda')$ is in ${\rm Rep}_{\Z_p}(\Gg)$, we can consider the vector bundle over $S$ which is associated to $T$ and $\rho$:
\[
T(\rho)=T\times^\Gg_{\Spec(\Z_p)} \bbA({\Lambda'}) =(T\times_{\Spec(\Z_p)} \bbA({\Lambda'}))/\sim
\]
where $(g^{-1}t, \lambda)\sim (t, \rho(g)\lambda)$. Here, $\bbA({\Lambda'})$ is the affine space
$\Spec({\rm Symm}_{\Z_p}(\Lambda'^\vee))$ over $\Spec(\Z_p)$.
In what follows, we often abuse notation, and also denote
by $T(\rho)$ the corresponding $A$-module of global sections of the bundle $T(\rho)$.
By \cite{Broshi}, see also \cite[19.5.1]{Schber}, this construction gives an equivalence between the category of $\Gg$-torsors $T\to S$ and the category of exact tensor functors
\[
T: {\rm Rep}_{\Z_p}(\Gg)\to {\rm Bun}(S); \ \ \rho\mapsto T(\rho),
\]
into the category of vector bundles ${\rm Bun}(S)$ on $S$.
Assume now that $T$ is a $\Gg$-torsor and $\iota: \Gg\hookrightarrow \GL(\Lambda)$ is as in \S\ref{par113}.
\begin{prop}\label{torsorRep} The $A$-module $M=T(\iota)$ is locally free finite of rank $n$ and comes equipped with tensors $(m_a)\subset M^\otimes$ such that there is a $\Gg$-equivariant isomorphism
\[
T\simeq \underline{\rm Isom}_{(m_a), (s_a)}(T(\iota), \Lambda\otimes_{\Z_p}A).
\]
\end{prop}
\begin{proof} This is quite standard, see for example \cite[Cor. 1.3]{Broshi} for a similar statement. We sketch the argument: By the above, $M=T(\iota)$ is a locally free $A$-module of rank $n={\rm rank}_{\Z_p}(\Lambda)$. Since the construction of $T(\rho)$ commutes with tensor operations (i.e. $\rho\mapsto T(\rho)$ gives a tensor functor) we have
\[
M^\otimes\simeq T\times^\Gg_{\Spec(\Z_p)} \bbA( \Lambda^\otimes).
\]
We can think of $s_a\in \Lambda^\otimes$ as $\Gg$-invariant linear maps $s_a: \Z_p\to \Lambda^\otimes$ which give $1\times s_a: S=\Gg\backslash T\to M^\otimes$, i.e. tensors $m_a=1\times s_a\in M^\otimes$. Set
$T'= \underline{\rm Isom}_{(m_a), (s_a)}(M, \Lambda\otimes_{\Z_p}A)$ with its natural left $\Gg$-action.
The base change $T'\times_{S}T$ is equivariantly identified with $ \Gg\times T \simeq T\times_{S}T$ and the proof follows.
\end{proof}
\begin{Remark}\label{indIota}
{\rm
Suppose that $\iota':\Gg\hookrightarrow {\rm GL}(\Lambda')$ is another closed group scheme immersion that realizes $\Gg$ as the subgroup scheme that fixes $(s_b')\subset (\Lambda')^\otimes$. It follows that there is a $\Gg$-equivariant isomorphism
\[
\underline{\rm Isom}_{(m'_b), (s'_b)}(M', \Lambda'\otimes_{\Z_p}A)\xrightarrow{\sim} \underline{\rm Isom}_{(m_a), (s_a)}(M, \Lambda\otimes_{\Z_p}A).
\]}
\end{Remark}
For the following, we assume in addition that $A$ is local and henselian.
\begin{prop}\label{RaynaudGruson}
Suppose that $M$ is a finite locally free $A$-module and let $(m_a)\subset M^\otimes$.
Consider the $A$-scheme
\[
T= \underline{\rm Isom}_{(m_a), (s_a)}(M, \Lambda\otimes_{\Z_p}A)
\]
which supports a natural $\Gg$-action. Suppose that there exists a set of local $\Z_p$-algebra homomorphisms $\xi: A\to R_\xi$, with $\cap _{\xi}{\rm ker(\xi)}=(0)$, and such that, for every $\xi: A\to R_\xi$, the base change $\xi^*T:=T\times_{S}\Spec(R_\xi)$ is a $\Gg$-torsor over $\Spec(R_\xi)$.
Then, $T\to S$ is also a $\Gg$-torsor.
\end{prop}
\begin{proof}
The scheme $T$ is affine and $T\to S$ is of finite presentation. The essential difficulty is in showing that $T\to S$ is flat but under our assumptions, this follows from \cite[Thm. (4.1.2)]{RaynaudGruson}. The fiber of $T\to S$ over the closed point of $S$ is not empty, hence $T\to S$ is also faithfully flat. Now the base change $T\times_ST$ admits a tautological section which gives a
$\Gg$-isomorphism $T\times_S T\simeq \Gg\times T$. This completes the proof.
\end{proof}
\smallskip
\begin{cor}\label{CorRaynaudGruson}
Set $A=W(R)$, where $R$ satisfies (N) of \S\ref{Algcond}. Suppose that $M$ is a free locally finite $A$-module, and that $m_a$ and $T$ are as in the statement of Proposition \ref{RaynaudGruson}. Assume that for all $W(k)$-algebra homomorphisms $\ti x: R\to \O_F$, where $F$ runs over all finite extensions of $W(k)_\Q$, the pull-back $T\otimes_A W(\O_F)$ is a $\Gg$-torsor over $W(\O_F)$. Then $T$ is a $\Gg$-torsor over $A$.
\end{cor}
\begin{proof} We first show the statement when $R$ is in addition complete and local, i.e. it satisfies (CN). Then $W(R)$ is local henselian and the result follows from Proposition \ref{RaynaudGruson}
applied to the set of homomorphisms $\xi: A=W(R)\to R_\xi=W(\O_F)$
given as $\xi=W(\ti x)$.
We now deal with the general case. Under our assumptions, $A=W(R)$ is flat over $\Z_p$.
Let $\frakM\subset W(R)$ be a maximal ideal with residue field $k'$. We have $I_R\subset \frakM$, since $W(R)$ is $I_R$-adically complete and separated (\cite[Prop. 3]{Zink}). Let
$\frakm_R=\frakM/I_R\subset W(R)/I_R=R$ be the corresponding maximal ideal of $R$. Our assumptions on $R$ imply that the residue field $k'$ is a finite extension of $k$.
Suppose that $\hat R$ is the completion of $R\otimes_{W}W(\bar k')$ at $\frakm_R\otimes_{W}W(\bar k')$. Then $W(\hat R)$ is local and strictly henselian.
Denote by $W(R)^{\rm sh}_{\frakM}$ the strict henselization of the localization $W(R)_\frakM$.
By Lemma \ref{A1lemma} we have
\[
W(R)^{\rm sh}_{\frakM}\subset W(\hat R).
\]
We also have
\[
\hat R\subset \prod\nolimits_{\xi: R\to \O_F} \O_F
\]
where the product is over all $\xi: R\to \O_F$ that factor through $\hat R$.
By Proposition \ref{RaynaudGruson} applied to $R_\xi=W(\O_F)$, the base change $T\otimes_{W( R)}W(R)^{\rm sh}_\frakM$ is
a $\Gg$-torsor. By descent, so is the base change $T\otimes_{W(R)}W(R)_\frakM$ over
$W(R)_\frakM$.\quash{We are also given that $T[1/p]$ is a $G=\Gg[1/p]$-torsor.
Consider the ring $W(R)/pW(R)$; the natural map $W(R)/pW(R)\to R/pR$ is surjective
with nilpotent kernel $I_R/(pW(R)\cap I_R)=I_R/pI_R=I_R/I_R^2$ (see \cite[(3)]{Zink} for the last equality). The spectrum of $W(R)/pW(R)$ agrees with that of $R/pR$. Since $T[1/p]$ is flat over $W(R)[1/p]$, and}
Since this is true for all maximal ideals $\frakM\subset W(R)$, it follows that $T$ is flat over $W(R)$. The result now follows as in the proof of Proposition \ref{RaynaudGruson}.
\end{proof}
\begin{Remark}
{\rm When $R$ satisfies (CN), the corollary also holds with $W(R)$, $W(\O_F)$, replaced by $\hat W(R)$, $\hat W(\O_F)$ respectively.
}
\end{Remark}
\end{para}
\ve
\section{Displays with $\Gg$-structure}\label{s1}
In this section, we define $(\Gg, \M)$-displays and give some basic properties. We also define and study the notion of a versal $(\Gg, \M)$-display.
\subsection{}
This subsection contains the main construction needed for the definition of a $(\Gg, \M)$-display.
Suppose that $\Gg$ and $\calM$ are as in \S\ref{par113}.
\begin{para}
Suppose that $R$ satisfies (N). Set $A=W(R)$.
(If $R$ satisfies (CN) and $p\geq 3$, we can also consider $A=\hat W(R)$.)
Since $\phi(I_R)\subset p W(R)$ we have a ring homomorphism
\[
\bar\phi: R= W(R)/ I_R\xrightarrow{\phi} A/pA
\]
induced by the Frobenius $\phi: A\to A$.
\end{para}
\begin{para}\label{ss132}
Suppose that $\calF\subset \Lambda\otimes_{\Z_p} R$ corresponds to the image under $\iota_*$ of an $R$-valued point of $\M$.
Compose this with $\bar\phi: R\to A/pA$ to obtain an $A/pA$-submodule
\[
\bar\phi^* \calF\subset \Lambda\otimes_{\Z_p} A/pA
\]
which is locally an $A/pA$-direct summand. Let $U$ be the inverse image of $\bar\phi^* \F$ under the reduction
$\Lambda\otimes_{\Z_p}A\to \Lambda\otimes_{\Z_p}A/pA$ so that
\[
p\Lambda\otimes_{\Z_p}A\subset U\subset \Lambda\otimes_{\Z_p}A.
\]
We can write $\Lambda\otimes_{\Z_p}A=L\oplus T$, with $L$ and $T$ finite projective $A$-modules such that $\calF$ corresponds to $L\oplus I_RT$
under ${\rm id}_{\Lambda}\otimes {\rm gh}_1: \Lambda\otimes_{\Z_p}A\to \Lambda\otimes_{\Z_p}R$.
Then we also have
\[
U=\phi^*(L)\oplus p\phi^*(T)\subset \phi^*(\Lambda\otimes_{\Z_p}A)=\Lambda\otimes_{\Z_p}A.
\]
Notice that $U^\otimes\subset U^{\otimes}[1/p]=\Lambda^\otimes_A[1/p]$. We have
$s_a\otimes 1\in \Lambda^\otimes_A\subset \Lambda^\otimes_A[1/p]$.
\medskip
Set $D^\times=\Spec(W(k)\lps u\rps)-\{(p,u)\}$.
In what follows, we will also assume, without further mention, the following purity condition:
\begin{itemize}
\item[(P)] Every $\Gg$-torsor over $D^\times$ is trivial.
\end{itemize}
\begin{Remark}
{\rm The purity condition is known to hold for $k=\bar{\mathbb F}_p$ and all parahoric group schemes $\Gg$ with $G=\Gg\otimes_{\Z_p}\Q_p$ that splits over a tamely ramified extension of $\Q_p$ and has no factors of type $E_8$ (\cite[Prop. 1.4.3]{KP}). This result has been extended to many cases of wild ramification by Ansch\"utz \cite[Theorem 8.4]{An}; it is probably true for all parahoric group schemes. However, in general, it can fail for most smooth affine group schemes over $\Z_p$ with reductive generic fiber.
}
\end{Remark}
\begin{prop}\label{propQconstr}
a) The tensors $s_a\otimes 1 $ belong to $U^\otimes$.
b) The scheme of isomorphisms that respect the tensors
\[
\calQ:=\underline{\rm Isom}_{s_a\otimes 1 }(U, \Lambda_A)
\]
is a $\Gg$-torsor over $\Spec(A)$.
c) Since $U[1/p]=\Lambda_A[1/p]$, we have a trivialization
\[
\calQ[1/p]:=\calQ\otimes_{A}A[1/p]=G_{A[1/p]}.
\]
\end{prop}
\begin{proof} Parts (a) and (b) follows exactly as the corresponding statement of \cite[Cor. 3.2.11]{KP}
by using Proposition \ref{RaynaudGruson} and Corollary \ref{CorRaynaudGruson}. Alternatively, for part (a), we can use the natural direct sum decomposition of $U^\otimes=(\phi^*(L)\oplus p\phi^*(T))^\oplus$ as in the proof of Lemma \ref{Qconnection} below.
Part (c) is easy.
\end{proof}
\end{para}
\begin{para}\label{indpara}
For $R$ satisfying (N), we can consider the
set
\[
{\rm Gr}_{\Gg }(R)=\{(\calQ, \alpha)\}
\]
of isomorphism classes of pairs $(\calQ, \alpha)$ of a $\Gg$-torsor $\calQ$ over $W(R)$ with a trivialization $\alpha$ of $\calQ[1/p]$ over $W(R)[1/p]$. Note
that if, in addition, $R$ is complete local with algebraically closed residue field, then $W(R)$ is local strictly henselian, and
\[
{\rm Gr}_{\Gg }(R)\cong G(W(R)[1/p])/\Gg(W(R)).
\]
The construction $\calF\mapsto \bar\phi^*\calF\mapsto (\calQ, \alpha)$ above gives a functorial map
\[
\fraki_{\Gg,\mu}(R): \M(R)\to {\rm Gr}_{\Gg }(R)
\]
which is injective.
This map is independent of the embedding $\iota$. (As long as $\iota$ still gives $\M$ as the Zariski closure of $X_\mu$ in the integral Grassmannian.) To see this suppose that $\iota': \Gg\hookrightarrow \GL(\Lambda')$ is another local Hodge embedding which gives $\iota'_*: \M\hookrightarrow {\rm Gr}(d', \Lambda')_{\O_E}$. We can consider the product
\[
\iota''=\iota\times \iota': \Gg\xrightarrow{\Delta} \Gg\times \Gg\hookrightarrow \GL(\Lambda)\times \GL(\Lambda')\subset \GL(\Lambda\oplus \Lambda').
\]
This induces
\[
\iota''_*: \M\xrightarrow{\Delta }\M\times_{\O_E}\M\hookrightarrow {\rm Gr} (d, \Lambda)\times_{\O_E} {\rm Gr}(d', \Lambda')\subset {\rm Gr}(d+d', \Lambda\oplus \Lambda')_{\O_E}.
\]
By the construction, we have $\calF''=\calF\oplus \calF'$, $U''=U\oplus U'$, and the projections give $U''\to U$, $U'\to U$. These maps induce
$\Gg$-equivariant morphisms $\calQ\to \calQ''$ and $\calQ'\to \calQ''$ which are then isomorphisms
of $\Gg$-torsors.
\begin{Remark}\label{conjEmb}
{\rm a) The above applies to $\M=\Mloc$, where $\Mloc=\Mloc{(\Gg, \{\mu\})}$
are the local models of \cite{PZ}, when $p\nmid \pi_1(G_{\rm der}(\bar\Q_p))$
and there is a local Hodge embedding (H). (See \cite[\S 2.3]{KP}.)
b) We conjecture that the maps $\fraki_{\Gg,\mu}$ exist in greater generality: More precisely, we expect that there are always
canonical functorial injective maps
\[
\fraki_{\Gg,\mu}(R): \Mloc(R)\to {\rm Gr}_{\Gg }(R),
\]
for $R$ satisfying (N), and for the local models $\Mloc=\Mloc{(\Gg, \{\mu\})}$ associated to a pair $(\Gg, \{\mu\})$, where $\Gg$ is a tame parahoric group scheme and $\mu$ a minuscule coweight, as for example in \cite{HPR}.
The maps should be $\Gg$-equivariant in the sense that
\begin{equation}\label{conjphi}
\fraki_{\Gg,\mu}(g\cdot x)=\phi(g)\cdot \fraki_{\Gg,\mu}(x),
\end{equation}
where the action on the right hand side is on the trivialization $\alpha$.
In fact, the maps $ \fraki_{\Gg,\mu}$ should exist without the tameness hypotheses, for example for the restriction of scalars local models of Levin \cite{BLevin}, or for the local models conjectured to exist by Scholze \cite[Conjecture 21.4.1]{Schber}. The embedding $\fraki_{\Gg,\mu}$ could be interpreted as giving the formal completion $\hat{\rm M}^{\rm loc}$ as a subsheaf of a Witt-vector affine Grasmannian. A perfectoid version of such an embedding (i.e. for v-sheaves) is given in \cite[Chapter XXII]{Schber}.
}
\end{Remark}
\end{para}
\begin{para}\label{qMlocTor}
Suppose now that $\calP$ is a $\Gg$-torsor over $A$ given together with a $\Gg$-equivariant morphism
\[
q: \calP\otimes_{A}R\to \M.
\]
Then, by the previous construction together with descent,
we obtain a $\Gg$-torsor $\calQ$ together with an isomorphism of $G$-torsors
\[
\calQ[1/p]\xrightarrow{\simeq} \phi^*\calP[1/p]
\]
over $A[1/p]$.
This allows us to think of $\calQ$ as a ``modification of $\phi^*\calP$ along the divisor $p=0$,
bounded by $\M$".
\begin{para}
Denote by $\frak m_R$ the maximal ideal of $R$. Set $\fraka_R=\frakm_R^2+(p)$.
Observe that the Frobenius $\phi$ factors as
\[
W(R/\fraka_R)\to W(k)\xrightarrow{\phi} W(k)\to W(R/\fraka_R).
\]
\begin{lemma}\label{Qconnection}
There is a canonical isomorphism of $\Gg$-torsors
\begin{equation}\label{canonicaliso}
\calQ\otimes_{ W(R)}W(R/\fraka_R)\xrightarrow{\sim} \calQ_0\otimes_{W(k)}W(R/\fraka_R)
\end{equation}
where $\calQ_0:=\calQ\otimes_{W(R)}W(k)$.
\end{lemma}
\begin{proof}
We use a similar argument as in the proof of \cite[Lemma 3.1.9]{KP}.
Let us write $\Lambda\otimes_{\Z_p} W(R/\fraka_R)=L\oplus T$, with $L$ and $T$ free $W(R/\fraka_R)$-modules, such that $\F\otimes_R R/\fraka_R$ is given by $L$ modulo $I_{R/\fraka_R}$.
Then, as in \emph{loc.cit.}
\[
U\otimes_{W(R)}W(R/\fraka_R)=\phi^*(L)\oplus (p\otimes \phi^*(T))
\]
and $\phi^*(L)\simeq \phi^*(L_0)\otimes_{W(k)}W(R/\fraka_R)$, $\phi^*(T)\simeq \phi^*(T_0)\otimes_{W(k)}W(R/\fraka_R)$. Here,
we write $p\otimes -$ for $p\Z_p\otimes_{\Z_p} -$.
This gives the isomorphism
\begin{equation}\label{connU}
U\otimes_{W(R)}W(R/\fraka_R)\simeq U_0\otimes_{W(k)}W(R/\fraka_R),
\end{equation}
and it now remains to show that this respects the tensors $s_a$.
Assume that $s_a$ is given by $\Lambda^{\otimes n}\to\Lambda^{\otimes n}$, see \S\ref{par113}. Then, $s_a$ preserves the corresponding filtration on
\[
\Lambda^{\otimes n}\otimes_{\Z_p} W(R/\fraka_R)=(L\oplus T)^{\otimes n}=\oplus_{i=0} (L^{\otimes i}\otimes_{W(R/\fraka_R)} T^{\otimes (n-i)}).
\]
Hence, $s_a$ maps $L^{\otimes i}\otimes_{W(R/\fraka_R)} T^{\otimes (n-i)}$ to the sum $\oplus_{j\geq i} L^{\otimes j}\otimes_{W(R/\fraka_R)} T^{\otimes (n-j)}$.
So $s_a=\oplus_i s^i_{a}$, with
\[
s^i_{a}: L^{\otimes i}\otimes_{W(R/\fraka_R)} T^{\otimes (n-i)}\to \oplus_{j\geq i} L^{\otimes j}\otimes_{W(R/\fraka_R)} T^{\otimes (n-j)}.
\]
Now we have
\begin{eqnarray*}
U^{\otimes n}\otimes_{W(R)}W(R/\fraka_R)&=\bigoplus_{i=0} \phi^*(L)^{\otimes i}\otimes_{W(R/\fraka_R)} (p^{n-i}\Z_p\otimes_{\Z_p} \phi^*(T)^{\otimes (n-i)})\\
&=\bigoplus_{i=0}p^{n-i} \otimes (\phi^*(L^{\otimes i}\otimes_{W(R/\fraka_R)} T^{\otimes (n-i)})).\
\end{eqnarray*}
The corresponding endomorphism of $U^{\otimes n}\otimes_{W(R)}W(R/\fraka_R)$ is then given by
\[
\bigoplus\nolimits_{i=0}^n m_i\circ (p^{n-i}\otimes \phi^*(s^i_a)),
\]
where $m_i$ is
\[
\oplus_{j\geq i} (p^{n-i}\Z_p\hookrightarrow p^{n-j}\Z_p)\otimes_{\Z_p}{\rm id}_ {\phi^*(L^{\otimes j}\otimes_{W(R/\fraka_R)} T^{\otimes (n-j)})}.
\]
The isomorphism induced by (\ref{connU}) on $n$-th tensor powers, is obtained as a direct sum
of the isomorphisms
\[
p^{n-i} \otimes (\phi^*(L^{\otimes i}\otimes_{W(R/\fraka_R)} T^{\otimes (n-i)}))\xrightarrow{\sim}
p^{n-i} \otimes (\phi^*(L^{\otimes i}_0\otimes_{W(k)} T^{\otimes (n-i)}_0)).
\]
The result now follows.
\end{proof}
When $\calP$ is a $\Gg$-torsor over $\hat W(R)$, there are corresponding statements with $W(R)$ and $W(R/\fraka_R)$ replaced by $\hat W(R)$ and $\hat W(R/\fraka_R)$.
\end{para}
\begin{Remark}\label{rem139}
{\rm The pair $(\calQ, \calQ[1/p]\xrightarrow{\simeq} \phi^*\calP[1/p])$ only depends on $\calP$, $\M$ and $q$ and is independent of $\iota$; this follows from \S\ref{indpara}.}
\end{Remark}
\end{para}
\subsection{} We now give the definition of a $(\Gg, \M)$-display over $R$, where
$R$ satisfies (N)
of \S\ref{Algcond}.
\begin{Definition}
A $(\Gg, \M)$-display over $R$ is a triple $\calD=(\calP, q, \Psi)$ of:
\begin{itemize}
\item A $\Gg$-torsor $\calP$ over $W(R)$,
\item a $\Gg$-equivariant morphism $q:\calP\otimes_{W(R)}R\to \M$ over $\O_E$,
\item a $\Gg$-isomorphism $\Psi: \calQ\xrightarrow{\sim} \calP$ where $\calQ$ is the
$\Gg$-torsor over $W(R)$ induced by $q$ as in \ref{qMlocTor}.
\end{itemize}
\end{Definition}
\begin{para}\label{global}
Assume now that $\frakX$ is a $p$-adic formal scheme which is flat and formally of finite type
over ${\rm Spf}(\Z_p)$ and which is normal. By Zink's Witt vector descent \cite[\S1.3, Lemma 30]{Zink}, there is a sheaf of rings
$W(\O_{\frakX})$ over $ \frakX $ such that for every open affine formal subscheme ${\rm Spf}(R)\subset \frakX$, we have $\Gamma(\Spf(R), W(\O_{\frakX}))=W(R)$. It now makes sense to give the natural extension of the above definition:
A
$(\Gg, \M)$-display over $\frakX$ is a triple $\calD=(\calP, q, \Psi)$ with the data
$\calP$, $q$, $\Psi$ as above given over $W(\O_{\frakX})$.
\begin{Remark}\label{remark323}{\rm
Assume Scholze's conjecture \cite[Conj. 21.4.1]{Schber} on the existence
of local models $\Mloc=\Mloc{(\Gg, \{\mu\})}$ and also the conjecture of Remark \ref{conjEmb} (b). Then
we can easily extend the definition of a $(\Gg, \Mloc)$-display to all pairs
$(\Gg, \{\mu\})$ of a parahoric group scheme $\Gg$ over $\Z_p$ and a minuscule cocharacter $\mu$ of $\Gg_{\bar\Q_p}$.
}
\end{Remark}
\end{para}
In what follows, we assume until further notice that $R$ satisfies (CN).
Suppose that $\calD=(\calP, q, \Psi)$ is a $(\Gg, \M)$-display over $R$.
\begin{para} Recall that $(\calP, q)$ gives $\calQ$ and a $G$-isomorphism $\calQ[1/p]\cong \phi^*\calP[1/p]$.
Suppose we choose a section $s$ of $\calP$ over $W(R)$.
Denote by $\phi^*(s)$ the corresponding section of the base change $\phi^*\calP=\calP\otimes_{W(R),\phi}W(R)$.
Then, every section of $\calQ$ can be written as $g_\mu\cdot \phi^*(s)$ with $g_\mu\in G(W(R)[1/p])$. We have
\[
\calQ=\Gg\cdot g_\mu\cdot \phi^*(s)\simeq \Gg.
\]
Define $\Psi(1)\in \Gg(W(R))$ by
\[
\Psi(g_\mu\cdot \phi^*(s))=\Psi(1)\cdot s.
\]
The Frobenius of the $\Gg$-display is then the $G$-isomorphism
\[
F: \phi^*\calP[1/p]\xrightarrow{\sim}\calP[1/p]
\]
given by $p\cdot \Psi$.
We have
\[
F(\phi^*(s))=p\cdot g_\mu^{-1}\cdot \Psi(1) \cdot s.
\]
Changing the section $s$ to $s'=g\cdot s$
gives
\[
F(\phi^*(s'))=\phi(g)^{-1}\cdot (p\cdot g_\mu^{-1}\cdot \Psi(1))\cdot g^{-1} \cdot s'
\]
with $g\in \Gg(W(R))$.
\end{para}
\begin{para}
By Lemma \ref{Qconnection}, there is a natural $\Gg$-isomorphism
$\calQ\otimes_{W(R)}W(R/\fraka)\cong \calQ_0\otimes_{W(k)}W(R/\fraka)$.
\begin{Definition}
A section of $s$ of the $\Gg$-torsor $\calP$ is called rigid in the first order at $\frakm_R$ when, under this isomorphism,
\[
\Psi^{-1}(s)\, {\rm mod}\, W(\fraka)=\Psi^{-1}_0(s_0)\otimes 1,
\] where the subscript $0$ denotes reduction modulo $\frakm_R$.
\end{Definition}
Suppose now that $s$ is a section of $\calP$. Choose a section $g_\mu\cdot \phi^*(s)$ of $\calQ$
which is ``horizontal at $\frak m_R$", i.e. which respects the isomorphism
$\calQ\otimes_{W(R)}W(R/\fraka)\cong \calQ_0\otimes_{W(k)}W(R/\fraka)$. Then, $s$ is rigid in the first order
if the element $\Psi(1)\in \Gg(W(R))$, defined by
\[
\Psi(g_\mu\cdot \phi^*(s))=\Psi(1)\cdot s,
\]
is constant modulo $\fraka$, i.e. if
\[
\Psi(1)\, {\rm mod}\, W(\fraka)\in \Gg(W(k))\subset \Gg(W(R/\fraka)).
\]
\begin{Definition}\label{Def145}
A $(\Gg, \M)$-display $\calD$ over $R$ is \emph{versal}, if there is a section $s$ of $\calP$ which is rigid in the first order,
such that the composition $q\cdot s: \Spec(R)\to \M$ gives an isomorphism between $R$ and the corresponding completion of the local
ring of $\M\otimes_{\O_E}\O_{\breve E}$.
\end{Definition}
\begin{Remark}
{\rm a) If there is a section $s$ of $\calP$ which is rigid in the first order,
such that the composition $q\cdot s: \Spec(R)\to \M$ gives an isomorphism
on completions as above, the same conclusion is true for any other section
$s'$ of $\calP$ which is also rigid in the first order.
b) Using (a), we see that if $\calD$ is versal, and $s$ is any section of $\calP$ which is rigid in the first order, then the morphism
\[
q: \Gg\times \Spec(R)\xrightarrow{\buildrel s\over\sim}\calP\otimes_{W(R)}R \xrightarrow{\ q\ } \M
\] is formally smooth.
}
\end{Remark}
\begin{prop}\label{versalProp2}
Suppose that the $(\Gg, \calM)$-display $\calD=(\calP, q, \Psi)$ over $R$ is versal. Then $q: \calP\otimes_{W(R)}R\to \calM$ is formally smooth.
\end{prop}
\begin{proof}
Follows from (b) above.
\end{proof}
\end{para}
\begin{para}
We now assume that $p$ is odd.
\begin{Definition}
A Dieudonn\'e $(\Gg, \M)$-display over $R$ is a triple $\calD=(\calP, q, \Psi)$ of
a $\Gg$-torsor $\calP$ over $\hat W(R)$, a $\Gg$-equivariant morphism \[
q:\calP\otimes_{\hat W(R)}R\to \M
\] over $\O_E$, and
a $\Gg$-isomorphism $\Psi: \calQ\xrightarrow{\sim} \calP$ where $\calQ$ is the
$\Gg$-torsor over $\hat W(R)$ induced by $q$ as in \ref{qMlocTor}.
\end{Definition}
Note that a Dieudonn\'e $(\Gg, \M)$-display over $R$, produces a $(\Gg, \M)$-display over $R$
by base change along the inclusion $\hat W(R)\hookrightarrow W(R)$. Most of the notions defined above for $(\Gg, \M)$-displays, for example, the notion of versal, have obvious extensions for Dieudonn\'e $(\Gg, \M)$-displays. The obvious variant of Proposition \ref{versalProp2} for Dieudonn\'e displays holds.
\end{para}
\begin{para}
Suppose that $\calD=(\calP, q, \Psi)$ is a Dieudonn\'e $(\Gg, \M)$-display over $R$, and that $\iota: \Gg\hookrightarrow \GL(\Lambda)$ is a local Hodge embedding. We set
\[
M:=\calP(\iota),\quad M_1:=\calQ(\iota), \quad F^{\#}_1: \phi^*M_1\to \tilde M_1\xrightarrow{\Psi(\iota)} M
\]
As in \cite[Lemma 3.1.5]{KP}, the triple $(M, M_1, F_1)$ defines a Dieudonn\'e display over $R$, in the sense of Zink \cite{ZinkCFT}. We denote this Dieudonn\'e display by $\calD(\iota)$. By \cite{ZinkCFT}, there is a corresponding $p$-divisible group $\sG_R={\rm BT}(\calD(\iota))$ over $R$. It follows from \cite[Lemma 3.1.12]{KP}, that if $\calD$ is versal, then
$\sG_R$ is a versal deformation of its special fiber $\sG_R\times_R k$.
\end{para}
\ve
\section{Crystalline $\Gg$-representations}
In this section, we describe ``$\Gg$-versions'' of objects of integral $p$-adic Hodge theory
which can be associated to a $\Gg(\Z_p)$-valued crystalline representation.
\subsection{}\label{BKrep}
Fix $(G, \{\mu\})$ and $\Gg$ as in \S\ref{ssShimura}. Fix also a local Hodge embedding
\[
\iota:\Gg\hookrightarrow \GL(\Lambda)
\]
with $\M=\overline{\iota_*(X_\mu)}\subset {\rm Gr}(g,\Lambda)_{\O_E}$. Assume that $\Gg$ satisfies the purity condition (P) for $k=\bar{\mathbb F}_p$.
Let $F$ be a finite extension of $E$ or of $\breve E$ with residue field $k$. Let
\[
\rho: {\rm Gal}(\bar F/F)\to \Gg(\Z_p)
\]
be a Galois representation. We assume that $\iota\cdot \rho: {\rm Gal}(\bar F/F)\to \GL(\Lambda[1/p])$ is crystalline. We give three flavors of ``$\Gg$-versions'' of Frobenius modules
which can be associated to $\rho$ by integral $p$-adic Hodge theory.
\subsection{The Breuil-Kisin $\Gg$-module}
\begin{para}\label{BKtorsor}
Choose a uniformizer $\pi_F$ of $F$ and let $E(u)\in W(k)[u]$ be the Eisenstein polynomial
with $E(\pi_F)=0$. Choose also a compatible system of roots $\sqrt[p^n]{\pi_F}$ in $\bar F$.
The Breuil-Kisin $\Gg$-module attached to $\rho$, is
by definition, a pair $(\calP_{\rm BK}, \phi_{\calP_{\rm BK}})$ where
\begin{itemize}
\item[$\bullet$] $\calP_{\rm BK}$ is a $\Gg$-torsor over $\frakS=W(k)\lps u\rps$,
\item[$\bullet$] $\phi_{\calP_{\rm BK}}$ is an isomorphism of $\Gg$-torsors
\[
\phi_{\calP_{\rm BK}}: \phi^*\calP_{\rm BK}[1/E(u)]\xrightarrow{\sim } \calP_{\rm BK}[1/E(u)].
\]
\end{itemize}
It is constructed as follows. (It does depend on the choice of $\sqrt[p^n]{\pi_F}$, $n\geq 0$.)
As in the proof of \cite[Lemma 3.3.5]{KP}, we write $\O_{\calG}=\varinjlim_{i\in J} \Lambda_i$ with $\Lambda_i\subset \O_{\calG}$ of finite $\Z_p$-rank and $\calG$-stable. The Galois action on $\Lambda$ gives
actions on $\Lambda_i$ and on $\O_{\Gg}$. We apply the Breuil-Kisin functor
\[
\frakM: {\rm Rep}_{K}^{{\rm cris}, \circ}\to {\rm Mod}^\phi_{/\frakS}
\]
(see \cite[\S 1]{KisinJAMS}, \cite[Theorem 3.3.2]{KP} for notations and details of its properties.
This depends on the choice of $\sqrt[p^n]{\pi_F}$, $n\geq 0$, in $\bar F$).
Let
\[
\frakM(\O_{\calG}):=\varinjlim\nolimits_{i\in J} \frakM(\Lambda_i).
\]
By \cite[Theorem 3.3.2]{KP}, the composition of $\frakM$ with restriction to $D^\times$ is an exact faithful tensor functor.
Hence, we obtain that $\frakM(\O_{\calG})_{|D^\times}$ is a sheaf of algebras over $D^\times$ and that
\[
\calP^\times_{\rm BK}:=\underline{\Spec}(\frakM(\O_{\calG})_{|D^\times})
\]
is a $\calG$-torsor over $D^\times$. Using the purity assumption (P), we can extend $\calP^\times_{\rm BK}$ to a $\Gg$-torsor $\calP_{\rm BK}$
over $D=\Spec(W(k)\lps u\rps)$ as follows:
Let us consider the scheme
\[
\calP':= \underline {\rm Isom}_{\ti s_a, s_a\otimes 1}(\frakM(\Lambda), \Lambda\otimes_{\Z_p}W\lps u\rps)\subset \underline {\rm Hom}(\frakM(\Lambda), \Lambda\otimes_{\Z_p}W\lps u\rps)
\]
of isomorphisms taking
$\tilde s_a$ to $s_a\otimes 1$. Here, as in \emph{loc. cit.} $\tilde s_a\in \frakM(\Lambda)^\otimes$
are the tensors obtained by applying the functor $\frakM(-)$ to the Galois invariant tensors $s_{a}\in \Lambda^\otimes$.
By \cite[Lemma 3.3.5]{KP}, the scheme $\calP'$ is naturally a $\calG$-torsor over $D$, which, in fact, is trivial.
As in the proof of \cite[Lemma 3.3.5]{KP}, we see that there is a natural isomorphism
$\calP'_{|D^\times}\simeq \calP^\times_{\rm BK}$ as $\calG$-torsors over $D^\times$.
However, there is a bijection between sections of $\calP'$ over $W\lps u\rps$ and sections of $\calP'_{|D^\times }=\calP^\times_{\rm BK}$ over $D^\times$. Hence, we obtain that the
$\calG$-torsor $\calP^\times_{\rm BK}$ over $D^\times$ extends to a
$\calG$-torsor $\calP_{\rm BK} $ over $D$ which is, then, uniquely determined
and is independent of the choice of $\Lambda$.
The isomorphism $\phi_{\calP_{\rm BK}}$ comes directly from the construction.
Note here that we can view the Breuil-Kisin $\Gg$-module attached to $\rho$ as an exact tensor functor
\[
{\rm Rep}_{\Z_p}(\Gg)\to {\rm Mod}^\phi_{/\frakS}.
\]
\end{para}
\subsection{The Dieudonn\'e $\Gg$-display.}
\begin{para}\label{BKdisplay}
Assume here that $\iota\cdot \rho$ has Hodge-Tate weights $\leq 0$ and that in fact,
the deRham filtration on $D_{\rm dR}(\Lambda[1/p])$ is given by a $G$-cocharacter conjugate to $\mu$.
Then, there is also a
Dieudonn\'e $(\Gg, \M)$-display
\[
\calD_{\rho}=(\calP, q, \Psi)
\]
over $\O_F$ which is attached to $\rho$. This is constructed as follows:
Let $\frakS=W(k)\lps u\rps\to \hat W(\O_F)$ be the unique Frobenius equivariant map lifting the identity on $\O_F$ which is given by $u\mapsto [\pi_F]$. We set
\[
\calP:=\calP_{\rm BK}\otimes_{\frakS, \phi}\hat W(\O_F).
\]
To obtain the rest of the data of the Dieudonn\'e $\Gg$-display we proceed as follows:
Consider the Breuil-Kisin module $\frakM=\frakM(\Lambda)$ associated to $\Lambda$. It comes with the Frobenius $F: \phi^*\frakM\to \frakM$ and the cokernel $\frakM/F(\phi^*\frakM)$ is annihilated by $E(u)$. We can then write $\frakM=L\oplus T$, with $L$ and $T$ free $\frakS$-modules such that
\[
F(\phi^*\frakM)=L\oplus E(u)T.
\]
Write $\frakM_1\subset \phi^*\frakM$ for the largest $\frakS$-submodule such that $F(\frakM_1) \subset E(u)\frakM$.
Then $F(\frakM_1) = E(u)\frakM$.
The corresponding filtration
\[
\overline \frakM_1\subset \phi^*\frakM/E(u)\phi^*\frakM
\]
gives an $\O_F$-valued point of a Grassmannian which is in the closure of the
$G$-orbit of $\mu$, hence an $\O_F$-point of $\M$. This gives a $\Gg$-equivariant
morphism
\[
q: \phi^*\calP_{\rm BK}\otimes_\frakS \O_F\to \M.
\]
Since $\calP\otimes_{\hat W(\O_F)}\O_F=\phi^*\calP_{\rm BK}\otimes_{\frakS}\O_F$ we obtain
\[
q: \calP\otimes_{\hat W(\O_F)} \O_F\to \M.
\]
This gives $\calQ$ and $\Psi$ is then determined by $\phi_{\calP_{\rm BK}}$.
To give these more explicitly, set $M=\frakM\otimes_{\frakS, \phi}\hat W(\O_F)$.
We have
\[
\frakM_1\otimes_{\frakS}\hat W(\O_F)\subset M.
\]
Using that $\phi(E([\pi_F]))/p$ is a unit in $\hat W(\O_F)$, after applying $\phi$, we obtain a filtration
\[
p(\phi^*M)\subset \tilde M_1:=\phi^*(\frakM_1\otimes_{\frakS}\hat W(\O_F))\subset \phi^*M.
\]
As in the proof of \cite[Lemma 3.2.9]{KP} the tensors $\ti s_a\otimes 1=\phi^*(\ti s_a\otimes 1)$ lie in $\tilde M_1^\otimes$ and
\[
\calQ=\underline{\rm Isom}_{(\ti s_{a}\otimes 1), (s_a\otimes 1)}(\tilde M_1, \Lambda\otimes_{\Z_p}W(\O_F)).
\]
The divided Frobenius $F_1=p^{-1}F: \tilde M_1\xrightarrow{\sim} M$ gives the $\Gg$-isomorphism $\Psi:\calQ\xrightarrow{\sim}\calP$.
\end{para}
\subsection{The Breuil-Kisin-Fargues $\Gg$-module.}
Here, we use the notations of \S\ref{appCompl}, \S\ref{appAinf}. In particular, $\O$ is the $p$-adic completion of the integral closure $\bar\O_F$ of $\O_F$ in $\bar F$ and $\O^\flat$ is its tilt. For simplicity, set $A_\inf=A_\inf(\O)$.
\begin{para}\label{BKF}
By definition, a (finite free) Breuil-Kisin-Fargues (BKF) module over $A_\inf$ is a finite free $A_\inf$-module
$M$ together with an isomorphism
\[
\phi_M: (\phi^*M)[1/\phi(\xi)]\xrightarrow{\sim} M[1/\phi(\xi)]
\]
where $\xi$ is a generator of the kernel of $\theta$. (See \cite{Schber}, \cite{BMS}).
Similarly, a Breuil-Kisin-Fargues $\Gg$-module over $A_\inf$ is, by definition, a pair $(\calD_\inf, \phi_{\calD_{\inf}})$, where $\calD_\inf$ is a $\Gg$-torsor over $A_\inf$ and
\[
\phi_{\calD_\inf}: (\phi^*\calD_{\inf})[1/\phi(\xi)]\xrightarrow{\sim} \calD_{\inf}[1/\phi(\xi)].
\]
is a $\Gg$-equivariant isomorphism.
\end{para}
\begin{para}
Now fix a uniformizer $\pi=\pi_F$ of $F$ and also a compatible system of roots $\pi^{1/p^n}$, for $n\geq 1$, giving an element $\pi^\flat=(\pi, \pi^{1/p},\ldots ) \in \O^\flat$.
These choices define a $\phi$-equivariant homomorphism
\[
f : \frakS=W\lps u\rps\to A_\inf
\]
given by $u\mapsto [\pi^\flat]^p$ and which is the Frobenius on $W=W(k)$. By \cite[Proposition 4.32]{BMS}, the association
\[
\frakM\mapsto M=\frakM\otimes_{\frakS}A_\inf
\]
defines an exact tensor functor from Breuil-Kisin modules over $\frakS$ to Breuil-Kisin-Fargues (BKF) modules over $A_\inf$.
We can compose the above functor with the tensor exact functor
\[
{\rm Rep}_{\Z_p}(\Gg)\to {\rm Mod}^\phi_{/\frakS}
\]
given by the Breuil-Kisin $\Gg$-module $\calP_{\rm BK}$ over $D=\Spec(\frakS)$ of \S\ref{BKtorsor}. We obtain a tensor exact functor
\[
{\rm Rep}_{\Z_p}(\Gg)\to {\rm Mod}^\phi_{/A_\inf}
\]
to the category ${\rm Mod}^\phi_{/A_\inf}$ of finite free BKF modules over $A_\inf$.
This functor gives a $\Gg$-torsor $\calD_{\inf}$ over $A_{\inf}(\O)$ which admits a $\Gg$-equivariant isomorphism
\[
\phi_{\calD_\inf}: (\phi^*\calD_{\inf})[1/\phi(\xi)]\xrightarrow{\sim} \calD_{\inf}[1/\phi(\xi)].
\]
(Here, $\phi(\xi)=f(E(u))$ for $E(u)\in W\lps u\rps$ an Eisenstein polynomial for $\pi$.
The element $\xi$ generates the kernel of $\theta: A_\inf\to \O$.)
Hence, $(\calD_{\inf}, \phi_{\calD_\inf})$ is a Breuil-Kisin-Fargues $\Gg$-module
which is attached to $\rho$.
More explicitly, set
\[
M_\inf:=M_\inf(\Lambda)=\frakM(\Lambda)\otimes_\frakS A_\inf.
\]
The tensors $s_{a}\in \Lambda^\otimes$ induce $\phi$-invariant tensors $s_{a, \inf}\in M^\otimes_\inf$; these are the base changes of $\ti s_{a}\in \frakM(\Lambda)^\otimes$. We have
\[
\calD_{\inf}\cong \underline{\rm Isom}_{(s_{a, \inf}), (s_a\otimes 1)}(M_\inf, \Lambda\otimes_{\Z_p}A_{\inf}(\O))
\]
as $\Gg$-torsors.
\end{para}
\begin{para}
Assume that $\Lambda$, acted on by $\rho$, is isomorphic to the Galois representation on the linear dual
\[
T^\vee=T_p(\mathscr G^*)(-1)
\]
of the Tate module $T=T_p(\mathscr G)$ of a $p$-divisible group $\mathscr G$ over $\O_F$.
We have
\begin{equation}\label{dualinf}
M_\inf\cong M(\mathscr G^*)\{-1\}\cong {\rm Hom}_{A_\inf}(M(\mathscr G), A_\inf).
\end{equation}
Here, $M(\mathscr G)$, $M(\mathscr G^*)$ are the BKF modules associated (\cite[Theorem 17.5.2]{Schber}) to the base change over $\O$ of $\mathscr G$, resp. of the
Cartier dual $\mathscr G^*$ of the $p$-divisible group $\mathscr G$ and $\{-1\}$ denotes the Tate twist, as defined in \cite{BMS}. The second (duality) isomorphism in (\ref{dualinf}) can be shown by combining \cite[Prop. 5.2.8]{SW} with the constructions of \cite[\S 17]{Schber}.
The corresponding Frobenius $\phi_{M_\inf}$ satisfies
\[
\phi(\xi) M_\inf\subset \phi_{M_\inf}(M_\inf)\subset M_\inf.
\]
\end{para}
\begin{para}
Choose $p$-th power roots of unity giving $\epsilon=(1, \zeta_p, \zeta_{p^2}, \ldots )\in \O^\flat$ and set $\mu=[\epsilon]-1\in A_\inf$.
Let $\underline{\Q_p/\Z_p}$ be the constant $p$-divisible group. By \cite[theorem 17.5.2]{Schber}, there is a comparison map
\[
\Lambda^\vee\cong {\rm Hom}_\O(\mathscr G^*_\O, \underline {\Q_p/\Z_p}(1) )
\cong {\rm Hom}_{A_\inf, \phi}(M_\inf\{1\}, A_\inf\{1\} ).
\]
This induces the $\phi$-invariant
isomorphism
\[
\Lambda\otimes_{\Z_p} A_\inf[1/\mu]\cong M_\inf[1/\mu]\cong\frakM(\Lambda)\otimes_{\frakS} A_\inf[1/\mu].
\]
It follows from the constructions and \cite[4.26]{BMS} that under these isomorphisms the tensors $s_{a}\otimes 1$, $s_{a, \inf}$
and $\ti s_a$ correspond.
\end{para}
\begin{para}\label{BKFcompatible}
The constructions of the previous paragraphs are compatible in the following sense.
Assume that $\rho$ is as in the beginning of \S\ref{BKdisplay}; then $\Lambda$ is the linear dual of the Tate module of a $p$-divisible group over $\O_F$.
Fix $\pi=\pi_F$ of $F$ and a compatible system of roots $\pi^{1/p^n}$, for $n\geq 1$, giving $\pi^\flat=(\pi, \pi^{1/p},\ldots ) \in \O^\flat$ as above. Recall the homomorphism
\[
\theta_\infty: A_{\inf}(\O)=W(\O^\flat)\to W(\O), \quad \theta_\infty([(x^{(0)}, x^{(1)}, \ldots )])=[x^{(0)}].
\]
The diagram
$$
\begin{matrix}
\frakS & \xrightarrow{\ \ \ f\ \ \ } & A_{\inf}(\O)\\
\phi\downarrow\ \ \ \ &\ \ & \downarrow\theta_\infty\\
\hat W(\O_F)&\xrightarrow{\ \ \ \ \ \ \ \ } & W(\O), \\
\end{matrix}
$$
where the bottom horizontal map is given by the inclusion, commutes.
We then have isomorphisms of $\Gg$-torsors
\begin{equation}\label{compareEq}
\calP\otimes_{\hat W(\O_F)}W(\O)\simeq \calP_{\rm BK}\otimes_{\frakS, \phi} W(\O)\simeq \calD_\inf\otimes_{A_\inf(\O)}W(\O)
\end{equation}
which are compatible with the Frobenius structures: This can be seen by combining
results of \cite[\S 4]{BMS}, \cite[\S 17]{Schber}, and the above constructions.
Similarly, we can see that both the $(\Gg, \calM)$-display $\calD$ and the Breuil-Kisin-Fargues $\Gg$-module $(\calD_{\inf}, \phi_{\calD_\inf})$ are, up to a canonical isomorphism, independent of the choice of $\pi_F$ and its roots $\sqrt[p^n]{\pi_F} $ in $\bar F\subset C$.
\end{para}
\ve
\section{Associated systems}\label{s2}
Here, we define the notion of an associated system and give several results. The main result says, roughly, that any $\Gg(\Z_p)$ pro-\'etale local system which is given by the Tate module of a $p$-divisible group over a normal base with appropriate \'etale tensors, can be extended uniquely to an associated system (see Theorem \ref{Exists} for the precise statement).
We also show how to use the existence of \emph{versal} associated systems to compare formal completions of normal schemes with the same generic fiber (Proposition \ref{MainProp}). Finally, we show that the definition of associated is independent of the choice of the local Hodge embedding (Proposition \ref{indassociated}).
\subsection{}
We continue with the notations and assumptions of the previous section on $\Gg$ and $\calM$.
In particular, we assume that $\Gg$ satisfies the purity condition (P) for $k=\bar{\mathbb F}_p$.
Let us suppose that $\calX$ is a flat $\O_E$-scheme of finite type, which is normal
and has smooth generic fiber. Suppose that we are given a pro-\'etale $\Gg(\Z_p)$-local system $\Lr$ on $X=\calX[1/p]$, i.e. a Galois cover
of $X$ with group $\Gg(\Z_p)=\varprojlim\nolimits_n \Gg(\Z/p^n\Z)$.
\begin{para}\label{311} For any $\bar x\in \calX(k)$, let $\hat R_{\bar x}$ be the completion of the local ring $R_{\bar x}$ of $\breve \calX=\calX\otimes_{\O_E}{\O_{\breve E}}$ at
$\bar x$.
Suppose we have a Dieudonn\'e $(\Gg, \M)$-display $\calD_{\bar x}=(\calP_{\bar x}, q_{\bar x}, \Psi_{\bar x})$ over $\hat R_{\bar x}$.
In accordance with our notations in \S\ref{gen12}, we will denote by
\[
\calD_{\bar x}(\iota)=(M_{\bar x}, M_{1, \bar x}, F_{1, \bar x})
\]
the Dieudonn\'e display over $\hat R_{\bar x}$ induced from $\calD_{\bar x}$ using $\iota$
and the construction of \cite[3.1.5]{KP}.
By \cite{ZinkCFT}, there is a corresponding $p$-divisible group $\mathscr G(\bar x)$ over $\hat R_{\bar x}$,
of height equal to $n={\rm rank}_{\Z_p}(\Lambda)$, with
\[
\mathscr G(\bar x)^*={\rm BT}( \calD_{\bar x}(\iota)).
\]
Then, by \cite[Theorem B]{Lau}, we have a canonical isomorphism
\begin{equation}\label{Dieudonne}
\calD_{\bar x}(\iota)\cong{\DD}(\mathscr G(\bar x))(\hat W(\hat R_{\bar x}))
\end{equation}
of Dieudonn\'e displays, where on the right hand side, ${\DD}$ denotes the \emph{contravariant} Dieudonn\'e crystal.
\smallskip
Consider the following two conditions. The first is:
\medskip
A1) There is an isomorphism of $\Z_p$-local systems over $\hat R_{\bar x}[1/p]$ between the Tate module $T$ of the $p$-divisible group $\mathscr G({\bar x})={\rm BT}( \calD_{\bar x}(\iota))$ and the pull-back of $\Lr(\iota)^\vee$.
\medskip
Before we state the second condition, observe that, under (A1), for any $\ti x\in \calX(\O_F)$ that lifts $\bar x$, the Galois representation $\rho({x})$ obtained from $x^*\Lr$, is crystalline. By \cite[Theorem 3.3.2 (2)]{KP}, the isomorphism in (A1) induces an isomorphism
\begin{equation}\label{Dieudonne2}
\calD_{\rho(x)}(\iota)\cong\DD(\ti x^*\mathscr G(\bar x))(\hat W(\O_F)).
\end{equation}
Here, $\ti x^*\mathscr G(\bar x)$ is the $p$-divisible group over $\O_F$ obtained by base-changing $\mathscr G(\bar x)$ by $\ti x$ and $\calD_{\rho(x)}$ is the Dieudonn\'e $(\Gg, \calM)$-display attached to $\rho(x)$ by \S\ref{BKdisplay}.
\smallskip
We can now state the second condition:
\smallskip
A2) For every $\ti x\in \calX(\O_F)$ lifting $\bar x$, the isomorphism
\[
\calD_{\bar x}(\iota) \otimes_{\hat W(\hat R_{\bar x})}\hat W(\O_F)\xrightarrow{\sim} \calD_{\rho(x)}(\iota)
\]
of Dieudonn\'e displays over $\O_F$ induced by (\ref{Dieudonne}), (\ref{Dieudonne2}), and base change, gives an isomorphism of Dieudonn\'e $(\Gg,\M)$-displays
over $\O_F$ (so, in particular, respects the tensors).
\begin{Definition}\label{defass}
If (A1) and (A2) hold for $\bar x\in \calX(k)$, we say that $\Lr$ and $\calD_{\bar x}$ are \emph{associated}.
If (A1) and (A2) hold for all $\bar x\in \calX(k)$, we call $(\Lr, \{\calD_{\bar x}\}_{\bar x\in \calX(k)})$ an \emph{associated system}.
\end{Definition}
\begin{Definition}\label{versal}
The associated system $(\Lr, \{\calD_{\bar x}\}_{\bar x\in \X(k)})$ is \emph{versal} over $\X$,
if for every $\bar x\in \calX(k)$, $\calD_{\bar x}$ is versal in the sense of Definition \ref{Def145}.
\end{Definition}
The definition of ``associated'' uses the local Hodge embedding $\iota$ which we, for now, fix in our discussion. We will later show that it is independent of this choice, see Proposition \ref{indassociated}.
\begin{prop}\label{depL}
If $\Lr$ and $\calD_{\bar x}$, for $\bar x\in \calX(k)$, are associated,
then $\calD_{\bar x}$ is, up to isomorphism,
uniquely determined by $\Lr$.
\end{prop}
\begin{proof} Suppose that $\Lr$ and $\calD'_{\bar x}$ are also associated. Then $\mathscr G({\bar x})[1/p]\simeq \mathscr G'({\bar x})[1/p]$ as $p$-divisible groups over $\hat R_{\bar x}[1/p]$, since they both have the same Tate module which is given by the restriction of $\Lr(\iota)$ to $\hat R_{\bar x}[1/p]$. Tate's theorem applied to the normal Noetherian domain $\hat R_{\bar x}$, provides an isomorphism
$\alpha: \mathscr G({\bar x})\xrightarrow{\sim} \mathscr G'({\bar x})$. Therefore, we obtain an isomorphism
of Dieudonn\'e displays $\delta: \calD_{\bar x}(\iota)\xrightarrow{\sim} \calD'_{\bar x}(\iota)$.
This amounts to an isomorphism
\[
(M, M_1, F_1)\xrightarrow{\delta} (M', M'_1, F'_1).
\]
Here both $M$, $M'$ are free $\hat W(\hat R_{\bar x})$-modules of rank $n$. The
$(\Gg, \M)$-displays $\calD_{\bar x}$ and $\calD'_{\bar x}$ have $\Gg$-torsors $\calP$, $\calP'$
given by tensors $s_{a}\in M^{\otimes}$, $s'_{a}\in M'^{\otimes}$. We would like to show that $\delta: M\to M'$ lies in
the $\hat W(\hat R_{\bar x})$-valued points of
\[
\underline{ {\rm Hom}}_{(s_a), (s'_a)}(M, M')=\Spec(A/I)\hookrightarrow \underline{ {\rm Hom} }(M, M')= \Spec(A).
\]
Here, $A\simeq \hat W(\hat R_{\bar x})[(t_{ij})_{1\leq i, j\leq n}]$, non-canonically. Let us consider $f(t_{ij})\in I$. We would like to show that
$f(\delta_{ij})=0$ in $\hat W(\hat R_{\bar x})$, where $\delta_{ij}\in \hat W(\hat R_{\bar x})$ are the coordinates of the
$\hat W(\hat R_{\bar x})$-linear map $\delta$.
Condition (A2) implies that $(\ti x)^*(f(\delta_{ij}))=0$,
for all $\ti x$ lifting $\bar x$. This implies that $f(\delta_{ij})=0$, so $\delta$ respects the tensors. It now follows that
$\delta$ respects the rest of the data that give the $(\Gg, \M)$-displays
$\calD_{\bar x}$ and $\calD'_{\bar x}$.
\end{proof}
\subsection{}
Let $\Dd$ be a $(\Gg, \M)$-display over the $p$-adic
formal scheme $\frakX=\varprojlim_n \calX\otimes_{\O_E}\O_E/(p)^n$.
\begin{Definition}\label{defass2}
We say that the $(\Gg, \M)$-display $\Dd$ over $\frakX$ is associated to $\Lr$ if, for all
$\bar x\in \calX(k)$, there is
\begin{itemize}
\item a Dieudonn\'e $(\Gg, \M)$-display $\Dd_{\bar x}$
which is associated to $\Lr$,
\item
an isomorphism of $(\Gg, \M)$-displays
\[
\Dd_{\bar x}\otimes_{\hat W(R_x)}W(R_x)\simeq \Dd\otimes_{W(\O_{\frakX})}W(R_x).
\]
\end{itemize}
\end{Definition}
Note that, then, $(\Lr, \{\Dd_{\bar x}\}_{\bar x\in \X(k)})$ is an associated system.
\begin{Remark}
{\rm It would make sense to add to the Definition \ref{defass2} the following ``global" condition
on $\Dd $:
There is a convergent Frobenius $G$-isocrystal $\rm T$ over $\calX\otimes_{\O_E}k_E$ which is associated
(in the sense of \cite{Fa2}, see also \cite[Appendix A]{KMPS}) to the local system $\Lr\otimes_{\Z_p}\Q_p$ over $\calX[1/p]$, such that for each $n\geq 1$, the Frobenius $G$-torsor over $W_n(\O_{\frak X})[1/p]$ underlying $\Dd\otimes_{W(\O_{\frakX})}W_n(\O_{\frak X})[1/p]$ is given by the value ${\rm T}(W_n(\O_{\frak X}))$.
This is a natural condition which, with some more work, can be shown for the $(\Gg, \Mloc)$-displays $\Dd_\eK$ over the integral models $\SS_\eK$
of Shimura varieties that we construct in \S 6. (See \cite[Appendix A]{KMPS} for the construction of $\rm T$.) However, it is not needed for
our characterization of integral models.}
\end{Remark}
\begin{Definition}
We say that the $(\Gg, \Mloc)$-display $\Dd$ over $\frakX$ which is associated to $\Lr$, is versal over $\frakX$,
if the associated system $(\Lr, \{\Dd_{\bar x}\}_{\bar x\in \X(k)})$ is \emph{versal} over $\X$.
\end{Definition}
\subsection{}
Assume now that $\X$ and $\X'$ are two flat $\O_E$-schemes of finite type, normal with the \emph{same} smooth generic fiber $X=\X[1/p]=\X'[1/p]$. Suppose that $(\Lr, \{\calD_{\bar x}\}_{\bar x\in \X(k)})$ and $(\Lr', \{\calD'_{\bar x'}\}_{\bar x'\in \X'(k)})$ are versal associated systems on $\X$ and $\X'$ respectively, with $\Lr= \Lr'$ on $X$.
Denote by $\Y$ the normalization of the Zariski closure of the diagonal embedding of $X$ in the product $\X\times_{\Spec(\O_E)}\X'$. Denote by
\[
\X\xleftarrow{\ \pi } \Y\xrightarrow{\pi'}\X',
\]
the morphisms given by the two projections. For simplicity, we again set $\breve\Y=\Y\otimes_{\O_E}\O_{\breve E}$, $\breve \X=\X\otimes_{\O_E}\O_{\breve E}$, etc. For $\bar y\in \Y(k)$, set $\bar x=\pi(\bar y)$, $\bar x'=\pi'(\bar y)$.
\begin{prop}\label{MainProp}
a) We have
\[
\pi^* \calD_{\bar x} \simeq \pi'^* \calD'_{\bar x'}
\]
as Dieudonn\'e $(\Gg, \M)$-displays on the completion $\hat\O_{\breve\Y ,\bar y}$.
b) The morphism $\pi$ induces an isomorphism
\[
\pi^*: \hat\O_{\breve\X ,\bar x}\xrightarrow{\sim} \hat\O_{\breve\Y ,\bar y}
\] between the completions of $\breve \Y$ and $\breve \X$, at $\bar y$ and $\bar x$, respectively. Similarly, for $\pi'$.
\end{prop}
\begin{proof}
Part (a) follows by the argument in the proof of Proposition \ref{depL}.
Let us show (b).
For simplicity, set $R=\hat \O_{\breve \X, \bar x}$, $R'=\O_{\breve \X', \bar x'}$,
$R''=\O_{\breve \Y, \bar y}$.
By the construction of $\Y$, we have a local homomorphism
\[
R\hat\otimes_{\O_{\breve E}}R'\to R''
\]
which is finite. Write $R_1$ for its image:
\[
R\hat\otimes_{\O_{\breve E}}R' \twoheadrightarrow R_1\hookrightarrow R''
\]
Applying $\iota$ and the functor ${\rm BT}$ to $\calD_{\bar x}$, $\calD'_{\bar x}$, gives $p$-divisible groups $\mathscr G$, $\mathscr G'$ over $R$, $R'$ respectively. By (a) we have
\begin{equation}\label{IsoPdiv}
\pi^*\mathscr G\simeq \pi'^*\mathscr G'
\end{equation}
over $R''$. This isomorphism specializes to give $f_0: \bar x^*(\mathscr G)\xrightarrow{\sim} \bar x'^*(\mathscr G')$, an isomorphism of $p$-divisible groups over the field $k$.
Let us write $T=\Spf(U)$ for the base change to $\O_{\breve E}$ of the universal deformation space of a $p$-divisible group $\mathscr G_0$ over $k$ which is isomorphic to the $p$-divisible groups $\bar x^*(\mathscr G)$ and $\bar x'^*(\mathscr G')$ above, and fix such isomorphisms. This allows us to view $f_0$ as an isomorphism $f_0: \mathscr G_0\xrightarrow{\sim} \mathscr G_0$.
Set $\Spf(R)=S$, $\Spf(R')=S'$, $\Spf(R'')=S''$, and $Z=\Spf(R_1)$. By the versality condition on $\calD_{\bar x}$ and $\calD'_{\bar x}$,
$S$ and $S'$ can both be identified with closed formal subschemes of $T$ given by ideals $I$ and $I'$ of $U$, respectively. There is a closed formal subscheme $\Gamma$ of $T\hat\times_{\O_{\breve E}} T=\Spf(U\hat\otimes_{\O_{\breve E}}U)$ prorepresenting the subfunctor of pairs of deformations of $\mathscr G_0$ where $ f_0$ extends as an isomorphism. The subscheme $\Gamma$ is defined by the ideal generated by $(u\otimes 1-1\otimes f^*_0(u))$, $u\in U$, where $f^*_0: U\xrightarrow{\sim} U$ is the ``relabelling'' automorphism corresponding to $ f_0 $. By (\ref{IsoPdiv}), we have
that $Z\subset T\hat\times_{\O_{\breve E}} T$ is contained (scheme theoretically) in the ``intersection''
\[
\Gamma\cap (S\hat\times_{\O_{\breve E}} S')=\Spf(U\hat\otimes_{\O_{\breve E}} U/((u\otimes 1-1\otimes f^*_0(u))_{u\in U}, I\otimes U, U\otimes I').
\]
The projection makes this isomorphic to $S\cap {f^*_0}^{-1}(S')$, the formal spectrum of $R/J$, where we set $J:=f^*_0(I')R$. From
\[
\Spf(R_1)=Z\subset \Gamma\cap (S\hat\times_{\O_{\breve E}} S')\simeq \Spf(R/J)
\]
we have $\dim(R_1)\leq \dim(R/J)$. Since $R_1$ is integral of dimension equal to that of $R''$
and so of $R$, we have $\dim(R)=\dim(R_1)\leq \dim(R/J)\leq \dim(R)$. Since $R$ is an integral domain, this implies $J=(0)$ and that $R_1$, which is a quotient of $R/J$ of the same dimension, is also isomorphic to $R$. Since $R\simeq R_1\to R''$ is finite, and $R$, $R'$ and $R''$ are normal, the birational $R_1\to R''$ is an isomorphism; so is $R\to R''$ and, by symmetry, also $R'\to R''$.
\end{proof}
\end{para}
\subsection{}
Suppose that $\calX$, $\Lr$, and $\iota: \Gg\hookrightarrow \GL(\Lambda) $, are as in the beginning of Section \ref{s2}.
\begin{thm}\label{Exists}
Suppose that the \'etale local system $\Lr(\iota)^\vee$ is given by the Tate module of a $p$-divisible group $\sG$ over $\calX$.
Then $\Lr$ is part of a unique, up to isomorphism, associated system $(\Lr, \{\calD_{\bar x}\}_{\bar x\in \calX(k)})$, i.e. of a system which satisfies
(A1) and (A2) for $\iota$.
\end{thm}
\begin{proof}
The uniqueness part of the statement follows from Proposition \ref{depL}. Our task is to construct, for each $\bar x\in \calX(k)$, a $(\Gg, \calM)$-display $\calD_{\bar x}$ over the strict completion $R=\hat R_{\bar x}$ that satisfies (A1) and (A2). Let
\[
(M, M_1, F_1)
\]
be the Dieudonn\'e display obtained by the evaluation $M=\DD(\sG)(\hat W(R))$ of the (contravariant) Dieudonn\'e crystal of $\sG$ over $R$.
This gives a $(\GL_n, {\rm Gr}(d, \Lambda))$-display
$(\P_{\GL}, q_{\GL}, \Psi_{\GL})$.
We want to upgrade this to a $(\Gg, \calM)$-display, the main difficulty being the construction of appropriate tensors $s_a\in M^\otimes$. The construction occupies several paragraphs:
\begin{para}\label{332}
We recall the notations and results of \S\ref{appCompl}, \S\ref{appAinf}, for $R$.
In particular, we fix an algebraic closure $\overline{ F(R)}$ of the fraction field $F(R)$, we denote by $\bar R$ the integral closure of $R$ in $\overline{ F(R)}$ and by $\ti R$ the union of all finite
normal $R$-algebras $R'$ in $\overline{ F(R)}$ such that $R'[1/p]$ is \'etale over $R[1/p]$. Set $\bar R^\wedge$ and $\ti R^\wedge $ for their $p$-adic completions. For simplicity, we set
\[
S=\ti R^\wedge.
\]
Also, we set $\O$ for the $p$-adic completion of the integral closure $\bar\O_E$ of $\O_E$ in the algebraic closure $\bar E$.
By \S \ref{appCompl}, $\bar R^\wedge$, $S=\ti R^\wedge$, and $\O$, are integral perfectoid $\Z_p$-algebras in the sense of \cite[3.1]{BMS}, which are local Henselian and flat over $\Z_p$.
The Galois group
$\Gamma_R$ acts on $\ti R$, on $S$, and on $A_\inf(S)=W(S^\flat)$.
\end{para}
\begin{para}
Let $(M(\sG)=M(\mathscr G)(S), \phi_{M(\sG)})$ be the (finite free) Breuil-Kisin-Fargues module over $A_{\inf}(S)$ associated to the base change $\mathscr G_S$ of $\mathscr G$.
By \cite[Theorem 17.5.2]{Schber},
$M(\sG)(S)$ is the value of a functor which gives an equivalence between $p$-divisible groups over $S$ and finite projective BKF modules $(M, \phi_M)$ over $A_\inf(S)$ that satisfy
\[
M\subset \phi_{M}(M)\subset \frac{1}{\phi(\xi)}M.
\]
By \emph{loc. cit.}, the equivalence is functorial in $S$. Therefore, $M(\sG)(S)$ supports an action of $\Gamma_R$ which
commutes with $\phi_{M(\sG)(S)}$ and is semi-linear with respect to the action of $\Gamma_R$ on $A_\inf(S)$. By
\emph{loc. cit.}, we have
\begin{equation}
T={\rm Hom}_S(\underline{\Q_p/\Z_p}, \mathscr G_S)\xrightarrow{\sim} {\rm Hom}_{A_{\inf}(S), \phi}(A_{\inf}(S), M(\sG))=M(\sG)^{\phi_{M(\sG)}=1}.
\end{equation}
This gives the comparison homomorphism
\[
c: T\otimes_{\Z_p} A_{\inf}(S)\xrightarrow{\ } M(\sG)(S)
\]
which is $\phi$ and Galois equivariant.
Using the constructions in \cite[\S 17]{Schber} together with Lemma \ref{lemmainf0} and Proposition \ref{lemmainf1}, we see that $c$ is injective and gives
\[
T \otimes_{\Z_p}A_{\rm inf}(S)\subset M(\sG)(S)\subset T\otimes_{\Z_p}\frac{1}{\mu}A_{\rm inf}(S).
\]
Therefore, we obtain a ``comparison'' isomorphism
\begin{equation}\label{tensorInf}
T^{\otimes }\otimes_{\Z_p}A_{\rm inf}(S)[1/\mu]\xrightarrow{\sim} M(\sG)(S)^{\otimes}[1/\mu].
\end{equation}
\end{para}
\begin{para}\label{536}
Let us set:
\[
M_{\rm inf}(S):=M(\sG)(S)^\vee:={\rm Hom}_{A_\inf}(M(\sG)(S), A_\inf)\cong M(\sG^*)(S)\{-1\}.
\]
Let
\[
s_{a, \inf}\in M_\inf(S)^{\otimes}[1/\mu]
\]
be the tensors
which correspond to $s_{a}\in T^{\otimes}$ under (\ref{tensorInf}).
We have
\[
\phi_{M_\inf}(s_{a,\inf})=s_{a,\inf}.
\]
We can now construct a Breuil-Kisin-Fargues $\Gg$-module $(D_{\inf}(S), \phi_{D_{\inf}})$ over $A_\inf(S)$.
\begin{prop}\label{p537}
a) We have $s_{a, \inf}\in M_\inf(S)^\otimes$.
b) By (a), we can consider
the $\Gg$-scheme
\[
D_{\inf}(S):=\underline{\rm Isom}_{(s_{a, \inf}), (s_a\otimes 1)}(M_\inf(S), \Lambda\otimes_{\Z_p}A_\inf(S)).
\]
The scheme $D_{\inf}(S)$, with its natural $\Gg$-action, is a $\Gg$-torsor over $A_\inf(S)$.
c) There is a $\Gg$-equivariant isomorphism
\[
\phi_{D_{\inf}}: (\phi^*D_{\inf}(S))[1/\phi(\xi)]\xrightarrow{\sim}D_{\inf}(S)[1/\phi(\xi)],
\]
where $\xi$ is any generator of the kernel of $\theta: A_\inf(S)\to S$.
d) Suppose $\ti x: S\to \O$ extends a point $\ti x: R\to \bar\O_E$ which lifts $\bar x$.
Then the base change of $(D_{\inf}(S), \phi_{D_{\inf}})$ by $\ti x: S\to \O$ is isomorphic to the BKF $\Gg$-module over $A_\inf(\O)$ which is attached to $\ti x^*\Lr$ by \ref{BKF}.
\end{prop}
\begin{proof}
Consider $ \ti x: S\to \O$ as in (d). Let $(M(\sG)(\O), \phi_{M(\sG)(\O)})$ be the BKF module over $A_\inf(\O)$ associated to the
$p$-divisible group $\ti x^*\mathscr G$ over $\O$.
By functoriality under $S\to \O$ of the functor of \cite[Theorem 17.5.2]{Schber}, we have a canonical isomorphism
\[
M_\inf(S)\otimes_{A_\inf(S)}A_\inf(\O)\cong M_\inf(\O)=M(\sG)(\O)^\vee
\]
respecting the Frobenius structures.
\begin{lemma}\label{l538}
The pull-back
\[
\ti x^*(s_{a, \inf})\in M_\inf(S)^{\otimes}[1/\mu]\otimes_{A_\inf(S)}A_\inf(\O)=M_\inf(\O)^{\otimes}[1/\mu]
\]
lies in $M_\inf(\O)^{\otimes}$.
\end{lemma}
\begin{proof} The statement follows from \S \ref{BKF}, the functoriality under $S\to \O$ of the functor of \cite[Theorem 17.5.2]{Schber}, and the definition of $s_{a, \inf}$.
\end{proof}
\smallskip
Now we can proceed with the proof of the Proposition. Part (a) follows from the above Lemma and Lemma \ref{lemmainf1}.
By Lemma \ref{lemmainf0} and Proposition \ref{RaynaudGruson}, $D_{\inf}(S)$ is a $\Gg$-torsor, i.e. part (b) holds.
The identity $\phi_{M_\inf}(s_{a,\inf})=s_{a,\inf}$ holds in $M_\inf(S)^\otimes [1/\mu]$ and so also in $M_\inf(S)^\otimes$ since
\[
M_\inf(S)\subset M_\inf(S)[1/\mu]
\]
by Lemma \ref{lemmainf1}.
Therefore,
\[
\phi_{D_{\inf}}: (\phi^*D_{\inf}(S))[1/\phi(\xi)]\xrightarrow{\sim}D_{\inf}(S)[1/\phi(\xi)].
\]
is $\Gg$-equivariant, which is (c). Finally, (d) follows from the above and functoriality under $S\to \O$.
\end{proof}
\begin{Remark} {\rm
a) Using these constructions and the comparison
\[
T\otimes_{\Z_p}A_\inf(S)[1/\mu]\simeq M(\sG)(S)[1/\mu],
\]
we can see that the BKF $\Gg$-module $(D_\inf(S), \phi_{D_\inf})$ only depends, up to isomorphism, on $\Lr$.
Indeed, from \S\ref{BKF}, this statement is true when $S=\O$. In general,
the comparison isomorphism first implies that the $\Gg$-torsor $D_\inf(S)[1/\mu]$ depends, up to isomorphism, only on $\Lr$. Then, by considering
restriction along $\ti x: S\to\O$ and using Lemma \ref{lemmainf1} (b), we see that we can determine $D_\inf(S)$ and $\phi_{D_{\inf}}$
over $A_\inf(S)$.
b) As usual, we may think of $D_\inf(S)$ as an exact tensor functor
\[
{\rm Rep}_{\Z_p}(\Gg)\to {\rm Mod}^\phi_{/A_\inf(S)}.
\]
}
\end{Remark}
\end{para}
\begin{para}
We can now complete the proof of Theorem \ref{Exists}.
Recall that
\[
\theta_\infty: A_\inf(S)\to W(S)
\]
factors as a composition
\[
\theta_{\infty}: A_\inf(S)\to A_{\rm cris}(S)\to W(S).
\]
By, \cite[Theorem 17.5.2]{Schber}, the Frobenius module
\[
M_\inf(S)\otimes_{A_{\inf}(S)}A_{\cris}(S) =M(\sG^*)(S)\{-1\}\otimes_{A_{\inf}(S)}A_{\cris}(S)
\]
describes the covariant Dieudonn\'e module of the base change $\mathscr G^*_S$ evaluated at the divided power thickening $A_{\rm cris}(S)\to S$. By \cite[Theorem B]{Lau}, this evaluation of the Dieudonn\'e module is naturally isomorphic to $M\otimes_{\hat W(R) } W(S)$, with its Frobenius structure. Combining these now gives a natural isomorphism
\begin{equation}\label{combEq}
M\otimes_{\hat W(R) } W(S)\simeq M_\inf(S)\otimes_{A_{\inf}(S)} W(S)
\end{equation}
which is compatible with Frobenius and the action of $\Gamma_R$. We obtain
\[
M^\otimes\otimes_{\hat W(R) } W(S)\simeq M_\inf(S)^\otimes\otimes_{A_{\inf}(S)} W(S).
\]
Since the tensors $s_{a, \inf}\otimes 1\in M_\inf(S)^\otimes\otimes_{A_{\inf}(S)} W(S)$ are $\Gamma_R$-invariant,
we see that
\[
s_{a, \inf}\otimes 1\in (M^\otimes\otimes_{\hat W(R) } W(S))^{\Gamma_R}=M^\otimes\otimes_{\hat W(R)} (W(S))^{\Gamma_R}.
\]
By Theorem \ref{FaltingsAlmost}, $(W(S))^{\Gamma_R}=W(R)$. Therefore,
\begin{equation}\label{combEqTen}
s_{a, \inf }\otimes 1\in M^\otimes\otimes_{\hat W(R)} W(R).
\end{equation}
In fact, we also have:
\begin{prop}\label{Propinfres}
a) The tensors $\und s_a:=s_{a, \inf }\otimes 1$ lie in $M^\otimes$.
b) The identity
\begin{equation}\label{identityxi}
\ti x^*(\und s_a)=\ti s_{a,\ti x}
\end{equation}
holds in $ M^\otimes\otimes_{\hat W(R)}\hat W(\O_F)\cong M^\otimes_{\O_F}$.
\end{prop}
In the above, $(M_{\O_F}, M_{1, \O_F}, F_{1, \O_F})$ is the Dieudonn\'e display over $\O_F$ associated
by \S\ref{BKdisplay} to the Galois representation on $\Lambda$ given by $\ti x^*\Lr(\iota)$, and $\ti s_{a,\ti x}=\ti s_a\otimes 1$ are the corresponding tensors.
\begin{proof} Using the compatibility of the construction with pull-back along points $\ti x: R\to \O_F$ we first see that the identity (\ref{identityxi}) holds in the tensor product $ M^\otimes\otimes_{\hat W(R)} W(\O_F)$. However, the right hand side $\ti s_{a, \ti x}$ lies in the subset $M^\otimes\otimes_{\hat W(R)}\hat W(\O_F)$, and, hence, so is the left hand side $\ti x^*(\und s_a)$. Proposition \ref{corInter2} now implies (a), and (b) also follows.
\end{proof}
\smallskip
The tensors $\und s_a\in M^\otimes$ allow us to define
\[
\calP:=\underline{\rm Isom}_{\und s_a, s_a\otimes 1}(M, \Lambda\otimes_{\Z_p}\hat W(R)).
\]
By the above, $\xi^*\calP$ is isomorphic to the $\Gg$-torsor given in \S\ref{BKdisplay}. By Corollary \ref{CorRaynaudGruson} for $A=\hat W(R)$, $\calP$ is a $\Gg$-torsor over $\hat W(R)$. By definition, we have $\calP(\iota)=M$.
It remains to construct $q$ and $\Psi$.
\end{para}
The filtration $I_RM\subset M_1\subset M$ gives a filtration of $M/I_RM$:
\[
(0)\subset {\rm Fil}^1:=M_1/I_RM\subset {\rm Fil}^0:=M/I_RM.
\]
This induces a filtration ${\rm Fil}^{\otimes, 0}$ of $ (M/I_RM)^\otimes$ and we have
\[
\und s_a\in {\rm Fil}^{\otimes, 0}\subset (M/I_RM)^\otimes[1/p]
\]
since this is true at all $F$-valued points.
Hence, $q: \P_{\GL}\otimes_{\hat W(R)}R\to {\rm Gr}(d, \Lambda)$ restricted to $\P\otimes_{\hat W(R)}R\subset \P_{\GL}\otimes_{\hat W(R)}R$ lands in $X_\mu(G)$ on the generic fiber. Since $\M$ is the Zariski closure of $X_\mu(G)$ in ${\rm Gr}(d, \Lambda)_{\O_E}$, we obtain
\[
q: \P\otimes_{\hat W(R)}R\to \M.
\]
Recall that we use $q$ to define the $\Gg$-torsor $\calQ$. From the construction, we have a $\Gg$-equivariant closed immersion $\calQ\subset \calQ_{\GL}$.
Finally, let us give $\Psi$: We consider $\Psi_{\GL}:\calQ_{\GL}\xrightarrow{\sim} \calP_{\GL}$. We will check that this restricts to $\Psi: \calQ\xrightarrow{\sim}\calP$: For this, is enough to show that the map $\Psi: \tilde M_1\xrightarrow{\sim} M$ given by $\Psi_{\GL}$ preserves the tensors $\und s_a$.
This follows as in the proof of Proposition \ref{depL} by observing that this is the case after pulling back by all $\ti x: R\to\O_F$.
Indeed, $\Psi_{\O_F}$ preserves $\ti s_{a,\ti x}$ and so, by (\ref{identityxi}), $\ti x^*\Psi=\Psi_{\O_F}$ preserves $\ti x^*(\und s_a)$.
The above define the $(\Gg, \M)$-display $\calD_{\bar x}=(\P, q, \Psi)=(\P_{\bar x}, q_{\bar x}, \Psi_{\bar x})$. By its construction, $ \calD_{\bar x}$ satisfies (A1) and (A2). This completes the proof of Theorem \ref{Exists}.
\end{proof}
\begin{para}
In fact, the proof of the Theorem \ref{Exists} also gives:
\begin{prop}
There is an isomorphism of $\Gg$-torsors
\begin{equation}\label{combEq2}
\calP_{\bar x}\otimes_{\hat W(R)}W(S)\simeq D_{\inf}(S)\otimes_{A_\inf(S)}W(S),
\end{equation}
which is also compatible with the Frobenius structures.\hfill$\square$
\end{prop}
\end{para}
\subsection{} We show that the definition of ``associated" is independent of the choice of the local Hodge embedding. More precisely:
\begin{prop}\label{indassociated}
The notion of associated is independent of the choice of local Hodge embedding $\iota$, i.e. if the system $(\Lr, \{\calD_{\bar x}\}_{\bar x\in \calX(k)})$ satisfies the requirements of the definition of associated for $\iota$, it also satisfies them for any other local Hodge embedding $\iota'$ which produces
the same $\M$.
\end{prop}
\begin{proof}
Assume that $\Lr$ and $\calD_{\bar x}$ are associated for $\iota$. By Theorem \ref{Exists}, we can assume that
$\calD_{\bar x}$ is obtained from $\Lr$ and $\iota$ by the construction in its proof.
We will use the notations of \S\ref{311}, \S\ref{332}:
In particular, $R=\hat R_{\bar x}$ and $\sG=\sG(\bar x)$ is the $p$-divisible group over $R$ given by $\calD_{\bar x}(\iota)$.
By \cite{LauGalois}, the Tate module $T^*:=T_p(\mathscr G^*)(\bar R)$
can be identified with the kernel of
\[
F_{1 }-1: \hat M_1 \to \hat M =M \otimes_{\hat W(R)}\hat W({\bar R}).
\]
Here, we denote abusively by $\hat W(\bar R)$ the $p$-adic completion of
\[
\varinjlim_{L/F(R)}\hat W(R_L)
\]
where $L$ runs over finite extensions of $F(R)$ in $\overline {F(R)}$, and $R_L$ is the normalization of $R$ in $L$. There is a natural surjective homomorphism $\hat W(\bar R)\to \bar R^\wedge$.
In the above, we set
\[
\hat M_1={\rm ker}(\hat M\to (M/M_1)\otimes_R \bar R^\wedge).
\]
The isomorphism
\[
{\rm per}: T^*\xrightarrow{\sim} \ker( F_{1 }-1: \hat M_1 \to \hat M )
\]
induces the comparison homomorphism
\[
T^*\otimes_{\Z_p}\hat W(\bar R)\to M \otimes_{\hat W(R)}\hat W({\bar R}).
\]
Let us consider a second local Hodge embedding $\iota':\Gg\hookrightarrow \GL(\Lambda')$ which realizes $\Gg$ as the fixator of tensors $(s'_b)$. As before, we have a $p$-divisible group $\mathscr G'=\mathscr G'(\bar x)$ over $R=\hat R_{\bar x}$ given by $\calD_{\bar x}(\iota')=(M', M'_1, F'_1)$. Denote its Tate module by $T(\mathscr G')$. To show (A1) for $\iota'$, we have to compare $T(\mathscr G')$ with $T':=\Lr(\iota')^\vee=\Lambda'^\vee$.
Recall (\S\ref{536}), that we have a $\phi$-invariant isomorphism
\[
\Lambda \otimes_{\Z_p}A_{\rm inf}(S)[1/ \mu] \simeq M_\inf(S)[1/ \mu]
\]
that sends the tensors $s_{a}\otimes 1$ to $s_{a, \inf}$. By a standard Tannakian argument
we see that this gives an isomorphism
\begin{equation}\label{tannakaComp}
\Lambda'^\vee\otimes_{\Z_p}A_{\rm inf}(S)[1/\mu]\simeq M'_\inf(S)^\vee[1/\mu]
\end{equation}
where $M'_\inf(S)=D_{\inf}(S)(\iota')$ is obtained from the $\Gg$-torsor $D_{\inf}(S)$.
Since $\iota'$ is also a local Hodge embedding, we can see using
\cite[Theorem 17.5.2]{Schber}, that $M'_\inf(S)^\vee$ is the BKF module $M(\mathscr H)$ of some $p$-divisible group $\mathscr H$ over $S$.
Then, also
\[
M'_\inf(S)\simeq M(\mathscr H^*)\{-1\}.
\]
\begin{lemma}\label{twoBTs}
For all $\ti x: S\to \O$ obtained from $\ti x: \ti R\to \bar\O_E$, we have:
a) $\ti x^*\mathscr H\simeq \ti x^* \sG'$,
b) $T'\simeq T_p(\ti x^*\sG')$.
\end{lemma}
\begin{proof} Consider the Breuil-Kisin $\Gg$-module $(\calP_{\rm BK}, \phi_{\calP_{\rm BK}})$ associated to $\ti x^*\Lr$ by \S\ref{BKtorsor}. Then, $(\calP_{\rm BK}(\iota'), \phi_{\calP_{\rm BK}}(\iota'))$ gives a ``classical'' Breuil-Kisin module which corresponds to a $p$-divisible group $\sG'_{\ti x}$ over $\O_F$.
The construction of the Breuil-Kisin $\Gg$-module implies that $T'\simeq \ti x^*\Lr(\iota')^\vee$ is identified with the Tate module $T_p(\sG'_{\ti x})$. By the compatibility (\S\ref{BKFcompatible}) of the constructions
in \S\ref{BKF} and \S\ref{BKdisplay}, Proposition \ref{Propinfres}, and the fact \cite[Theorem 17.5.2]{Schber} that the functor $M(-)$
gives an equivalence of categories between $p$-divisible groups over $\O$ and (suitable) BKF modules over $A_\inf(\O)$, the base change of $\sG'_{\ti x}$ to $\O$ is isomorphic to both
$ \ti x^*\mathscr H$ and $\ti x^* \sG'$. This gives (a). In fact, since the Tate module of $\sG'_{\ti x}$ is identified with $T'$, we then obtain (b).
\end{proof}
\smallskip
From (\ref{tannakaComp}), we obtain an injection
\[
c': \Lambda'^\vee=T'\hookrightarrow M'_\inf(S)^\vee[1/\mu].
\]
\begin{lemma}\label{545}
The map $c'$ gives an isomorphism
\[
c': T'\xrightarrow{\ \sim } (M'_\inf(S)^\vee)^{\phi_{M'^\vee}=1} ={M(\mathscr H)}^{\phi_{M(\mathscr H)}=1}
\]
which identifies $T'$ with the Tate module of $\mathscr H$.
\end{lemma}
\begin{proof}
For all $\ti x: S\to \O$ given by $\ti x: \ti R\to \bar\O_E$, we consider the composition
\[
T'\hookrightarrow M'_\inf(S)^\vee[1/\mu] \to M'_\inf(\O)^\vee[1/\mu]
\]
where the second map is given by pull-back along $A_\inf(S)\to A_\inf(\O)$ and functoriality.
From Lemma \ref{twoBTs} and its proof, we see that this composition is identified with the comparison isomorphism for $\ti x^*\mathscr H$ and so its image is contained in $M'_\inf(\O)^\vee$, in fact in $(M'_\inf(\O)^\vee)^{\phi_{M'^\vee}=1}$. It follows from Corollary \ref{corinf} (a) that the image of $c'$ is contained in
$M'_\inf(S)^\vee$.
Hence, we obtain:
\[
c':T'= \Lambda'^\vee\hookrightarrow (M'_\inf(S)^\vee)^{\phi_{M'^\vee}=1}.
\]
Now, for all such $\ti x: S\to \O$, consider
\[
T'\hookrightarrow (M'_\inf(S)^\vee)^{\phi_{M'^\vee}=1}\to (M'_\inf(\O)^\vee)^{\phi_{M'^\vee}=1}.
\]
As above, this composition is identified with the comparison map
for $\ti x^*\mathscr H$ and is therefore an isomorphism. However,
\[
(M'_\inf(S)^\vee)^{\phi_{M'^\vee}=1}\simeq (M(\mathscr H))^{\phi_{M(\mathscr H)}=1}
\]
is the Tate module of $\mathscr H$, a finite free $\Z_p$-module of rank equal to
${\rm rank}_{\Z_p}(T')$. Therefore
\[
c': T'\xrightarrow{\ \sim } (M'_\inf(S)^\vee)^{\phi_{M'^\vee}=1} ={M(\mathscr H)}^{\phi_{M(\mathscr H)}=1}
\]
is an isomorphism and it identifies $T'$ with the Tate module of $\mathscr H$ as desired.
\end{proof}
\smallskip
The lemma and duality now gives that we also have
\begin{equation}\label{twedgeeq}
T'^\vee(1)\simeq {(M'_\inf(S)\{1\})}^{\phi_{M'\{1\}}=1}={M(\mathscr H^*)}^{\phi_{M(\mathscr H^*)}=1}
\end{equation}
with both sides identifying with the Tate module of $\mathscr H^*$.
\begin{lemma}
The natural homomorphism
\[
g: A_\inf(S)\xrightarrow{\theta_\infty} W(S)=W(\ti R^\wedge)\to W(\bar R^\wedge)
\]
factors through $ \hat W(\bar R)$ as a composition
\[
A_\inf(S)\hookrightarrow A_{\rm cris}(S)\to \hat W(\bar R)\to W(\bar R^\wedge).
\]
\end{lemma}
\begin{proof} The diagram
\[
\begin{matrix}
A_\inf(S) & \xrightarrow{\theta_\infty}& W(S) \\
\downarrow&&\downarrow \\
A_\inf(\bar R^\wedge)&\xrightarrow{\theta_\infty} &W(\bar R^\wedge)
\end{matrix}
\]
with vertical arrows given by $S\to \bar R^\wedge$, is commutative. Hence, the composition $g$ is equal to
\[
A_\inf(S)\to A_\inf(\bar R^\wedge)\xrightarrow{\theta_\infty} W(\bar R^\wedge).
\]
We want to show $\theta_\infty: A_\inf(\bar R^\wedge)\xrightarrow{\ } W(\bar R^\wedge)$ factors through $\hat W(\bar R)$.
We can argue as in the proof of \cite[Lemma 5.1]{LauGalois}: Note that, for each $n\geq 1$,
all elements $a$ of the kernel of $\hat W(\bar R)/p^n\to \bar R^\wedge/p\bar R^\wedge$ satisfy $a^{p^n}=0$. Therefore, by the universal property of the Witt vectors, this gives
\[
A_\inf(\bar R^\wedge)=W((\bar R^\wedge)^\flat )\to \hat W(\bar R)\to \bar R^\wedge/p\bar R^\wedge.
\]
This lifts $\theta_{\bar R^\wedge} : A_\inf(\bar R^\wedge) \to \bar R^\wedge$. Now since $\hat W(\bar R)\to \bar R^\wedge$ is a divided power extension of $p$-adic rings, the map $\theta_{\bar R^\wedge}$ factors
\[
A_\inf(\bar R^\wedge)\to A_{\rm cris}(\bar R^\wedge)\to \hat W(\bar R)\to \bar R^\wedge
\]
and using this we can conclude the proof.
\end{proof}
\smallskip
As in (\ref{combEq}), (\ref{combEq2}), we can use the above lemma to obtain an isomorphism
\begin{equation}\label{combEq3}
M'_\inf(S)\otimes_{A_{\inf}(S)}\hat W(\bar R)\simeq M'\otimes_{\hat W(R)}\hat W(\bar R)
\end{equation}
which respects the Frobenius and Galois structures. This combined with (\ref{twedgeeq}) gives
\[
T'^\vee(1)\simeq {(M'_\inf(S)\{1\})}^{\phi_{M'\{1\}}=1} \xrightarrow{ \ } \hat M'^{F_1=1}_1\simeq T(\mathscr G'^*)
\]
where both source and target are finite free $\Z_p$-modules of the same rank as that of $T'$.
By pulling back via all $\ti x: \bar R\to \bar \O_E$ and using Lemma \ref{twoBTs}, we see that this map is an isomorphism. Therefore, after taking duals, we have
\[
T'\simeq T(\mathscr G')
\]
which shows (A1).
For (A2), it is enough to show that the tensors $s'_b$ on $\calD_{\bar x}(\iota')$
restrict to the corresponding tensors on $\calD_{\rho(x)}(\iota')$. This follows from the above
construction and (\ref{combEq3}). \end{proof}
\medskip
\section{Canonical integral models}\label{sShimura}
\subsection{}
We now consider Shimura varieties and their arithmetic models. Under certain assumptions, we give a definition of a ``canonical'' integral model.
\begin{para}\label{31}
Let $\eG$ be a connected reductive group over $\mathbb Q$ and $X$ a conjugacy class of maps
of algebraic groups over $\RR$
$$ h: \mathbb S = \Res_{\C/\RR} \GG_m \rightarrow \eG_{\RR},$$
such that $(\eG,X)$ is a Shimura datum (\cite{DeligneCorvallis} \S 2.1.)
For any $\C$-algebra $R,$ we have $R\otimes_{\RR}\C = R \times c^*(R)$ where $c$
denotes complex conjugation, and we denote
by $\mu=\mu_h$ the cocharacter given on $R$-points by
$
R^\times \rightarrow (R \times c^*(R))^{\times} = (R\otimes_{\RR}\C)^\times = \mathbb S(R) \overset h \rightarrow \eG_{\C}(R).
$
Let $\AA_f$ denote the finite adeles over $\Q,$ and $\AA^p_f \subset \AA_f$ the subgroup
of adeles with trivial component at $p.$ Let $\eK = \eK_p\eK^p \subset \eG(\AA_f)$
where $\eK_p \subset \eG(\Q_p),$ and $\eK^p \subset \eG(\AA^p_f)$ are compact open subgroups.
If $\eK^p$ is sufficiently small then the Shimura variety
$$
\Sh_{\eK}(\eG,X)_{\C} = \eG(\Q) \backslash X \times \eG(\AA_f)/\eK
$$
has a natural structure of an algebraic variety over $\mathbb C$.
This has a canonical model $\Sh_{\eK}(\eG,X)$ over the reflex field; a number field ${\sf E} = {\sf E}(\eG,X)$ which is the minimal field
of definition of the conjugacy class of $\mu_h.$
We will always assume in the following that $\eK^p$ is sufficiently small; in particular, the quotient above exists as an algebraic variety.
Let $Z^s(G)$ be the maximal anisotropic rational subtorus of the center of $G$ that splits over $\R$.
In what follows, for simplicity, we will assume that $Z^s(G)$ is trivial.
Let $\Gg$ be a smooth connected affine flat group scheme over $\Z_p$ with generic fiber
$G_{\Q_p}$. We assume that $\Gg$ satisfies the purity condition (P) for $k=\bar{\mathbb F}_p$.
Now choose a place $v$ of $\eE$ over $p$, given by an embedding $\bar\Q\to \bar\Q_p$.
We denote by $E={\sf E}_v/\Q_p$ the local reflex field and by $\{\mu\}$ the
$G(\bar\Q_p)$-conjugacy class -which is defined over $E$- of the minuscule cocharacter $\mu_h$.
We denote by $\O_{\eE, (v)}$ the localization of the ring of integers $\O_{\eE}$ at $v$.
Fix a flat projective normal $\O_E$-scheme $\M$ with $\Gg$-action whose generic fiber is the homogeneous variety $X_\mu$. We call $(\Gg, \M)$ the \emph{local integral Shimura pair}.
Fix $\eK_p=\Gg(\Z_p)\subset G(\Q_p)$. We consider the system
of covers $\Sh_{\eK'}(G, X)\to \Sh_{\eK}(G, X)$ where $\eK'=\eK'_p\eK^p\subset \eK=\eK_p\eK^p$, with $\eK'_p$ running over all compact open subgroups of $\eK_p=\calG(\Z_p)$. This gives a pro-\'etale $\Gg(\Z_p)$-local system ${\rm L}_\eK$ over $\Sh_{\eK}(G, X)$.
\end{para}
\begin{para}\label{612}
Suppose now that $p>2$ and there is a local Hodge embedding
\[
\iota:\Gg \hookrightarrow \GL(\Lambda)
\]
with corresponding $\M=\overline {\iota_*(X_\mu)}\subset {\rm Gr}(d, \Lambda)_{\O_E}$. Then, we can consider Dieudonn\'e $(\Gg, \M)$-displays.
Suppose that for all sufficiently small $\eK^p$ we have $\OEv$-models $\SS_{\eK}=\SS_{\eK_p\eK^p}$ (schemes of finite type and flat over $\OEv$) of the Shimura variety $\Sh_{\eK}(G, X)$ which are normal. In addition, we require:
\begin{itemize}
\item[1)] For $\eK'^p\subset \eK^p$, there are finite \'etale morphisms
\[
\pi_{\eK'_p, \eK_p}: \SS_{\eK_p\eK'^p}\to \SS_{\eK_p\eK'^p}
\]
which extend the natural $\Sh_{\eK_p\eK'^p}(G, X)\to \Sh_{\eK_p\eK^p}(G, X)$.
\item[2)] The scheme $\SS_{\eK_p}=\varprojlim_{\eK^p}\SS_{\eK_p\eK^p}$ satisfies the ``extension property" for dvrs of mixed characteristic $(0, p)$:
\[
\SS_{\eK_p}(R[1/p])=\SS_{\eK_p}(R),
\]
for any such dvr $R$.
\item[3)] The $p$-adic formal schemes $\widehat \SS_{\eK}=\varprojlim_n \SS_\eK\otimes_{\O_{\eE, (v)}}\O_{\eE, (v)}/(p)^n$ support versal {$(\Gg, \M )$-displays}
$
\Dd_\eK
$
which are {associated} to $\Lr_{\eK}$. We ask that these are compatible for varying $\eK^p$, i.e. that there are compatible isomorphisms
\[
\pi_{\eK'_p, \eK_p}^*\Dd_{\eK }\simeq \Dd_{\eK'}.
\]
\end{itemize}
Instead of (3) we could also consider the alternative condition:
\begin{itemize}
\item[3*)] The schemes $\SS_{\eK}$ support versal associated systems
\[
\widehat\Dd_\eK:=({\rm L}_\eK, \{\calD_{\bar x}\}_{\bar x\in \SS_\eK(k)}),
\]
where $\calD_{\bar x}$ are Dieudonn\'e $(\Gg, \M)$-displays.
\end{itemize}
Note that (3) implies (3*); this follows from the definitions and Proposition \ref{depL}.
Theorem \ref{charThmA} below makes the following definition reasonable. Recall we fix the local integral Shimura pair $(\Gg, \M)$.
\begin{Definition}\label{canonical613}
A projective system of $\OEv$-models $\SS_{\eK} $ of the Shimura varieties $\Sh_{\eK}(G, X)$, for $\eK=\eK_p\eK^p$ with $\eK_p$ fixed as above, is \emph{canonical}, if the models are normal and satisfy the conditions (1), (2), (3) above.
\end{Definition}
We conjecture that, under the hypotheses above, such canonical models always exist.
In the next section, we show this for Shimura varieties of Hodge type at tame primes
of parahoric reduction with $\M=\Mloc$ the local models of \cite{PZ}, \cite{HPR}.
\begin{Remark}
{\rm
Note that, in our definition of canonical, we first make a choice for the $\O_E$-scheme $\M$. However, it is reasonable
to ask, beforehand, for a canonical choice of $\M$ which depends only on $\Gg$ and $\{\mu\}$ and use this in the definition. This canonical choice should be the ``local model''. At tame primes of parahoric reduction the local models $\Mloc$ of \cite{PZ}, (slightly modified, as in \cite{HPR}), are uniquely determined by $\Gg$ and $\{\mu\}$
\cite[Theorem 2.7, Remark 2.9]{HPR}; we use this choice in the next section.
In the general case when $\Gg$ is parahoric, Scholze conjectures \cite[Conjecture 21.4.1]{Schber} that there should always be a canonical choice
$\M={\mathbb M}^{\rm loc, flat}_{(\Gg, \mu)}$. He characterizes ${\mathbb M}^{\rm loc, flat}_{(\Gg, \mu)}$ uniquely in terms of $(\Gg, \{\mu\})$ by giving
its associated v-sheaf. We can then regard the integral models $\SS_{\eK}$ to be canonical only when they satisfy our conditions for this particular choice of $\M$. In any case, as it is shown in \cite{HPR}, $\Mloc={\mathbb M}^{\rm loc, flat}_{(\Gg, \mu)}$ in all the cases we consider in \S\ref{sHodge}. In fact, assuming also the truth of the Conjecture in Remark \ref{conjEmb}, we can make sense of $(\Gg, \Mloc)$-displays
and give a notion of a canonical integral model more generally (i.e. without any tameness or local Hodge type hypothesis).
}
\end{Remark}
\begin{thm}\label{charThmA} Fix $\eK_p=\Gg(\Z_p)$ as above. Suppose that $\SS_\eK$, $\SS'_{\eK}$ are $\OEv$-models of the Shimura variety ${\rm Sh}_{\eK}(G, X)$ for $\eK=\eK_p\eK^p$ that satisfy (1), (2) and (3*). Then there
are isomorphisms $\SS_\eK\simeq \SS'_{\eK}$
giving the identity on the generic fibers and which are compatible with the data in (1) and (3*).
\end{thm}
Since condition (3) implies (3*), this immediately gives:
\begin{cor}\label{charThm}
Fix $\eK_p=\Gg(\Z_p)$ as above. Suppose that $\SS_\eK$, $\SS'_{\eK}$ are canonical $\OEv$-models of the Shimura variety ${\rm Sh}_{\eK}(G, X)$ for $\eK=\eK_p\eK^p$. Then there are isomorphisms $\SS_\eK\simeq \SS'_{\eK}$
giving the identity on the generic fibers and which are compatible with the data in (1). \endproof
\end{cor}
\begin{proof} Let us denote by $\SS''_{\eK}$ the normalization of the Zariski closure of the diagonal embedding of ${\rm Sh}_{\eK}(G, X)$ in $\SS_{\eK}\times_{\OEv}\SS'_{\eK}$. This is a third $\OEv$-model of the Shimura variety ${\rm Sh}_{\eK}(G, X)$ which is also normal. We can easily see that $\SS''_{\eK}$, for varying $\eK^p$, come equipped with data as in (1) and that (2) is satisfied. Denote by
\[
\pi_{\eK}: \SS''_{\eK}\to \SS_{\eK}, \quad \pi'_{\eK}: \SS''_{\eK}\to \SS'_{\eK}
\]
the morphisms induced by the projections. Both of these morphisms are the identity on the generic fiber and so they are birational. Using condition (3*), we see that by Proposition \ref{MainProp}, $\pi_{\eK}$ and $\pi'_{\eK}$ give isomorphisms between the strict completions at geometric closed points of the special fiber. It follows that the fibers of $\pi_{\eK}$ and of $\pi'_{\eK}$ over all such points are zero-dimensional. Hence, $\pi_{\eK}$ and $\pi'_{\eK}$
are quasi-finite. The desired result now quickly follow from this, Zariski's main theorem and the following:
\begin{prop}\label{Proper}
The morphisms $\pi_{\eK}$ and $\pi_{\eK}'$ are proper.
\end{prop}
\begin{proof} It is enough to prove that $\pi_{\eK}$ is proper, the properness of $\pi_{\eK}'$ then given by symmetry. We can also base change to the strict completion $\O_{\breve E}$ of $\OEv$; for simplicity we will omit this base change from the notation. We apply the Nagata compactification theorem (\cite[Thm. 4.1]{Conrad}) to $\pi_{\eK}$.
This provides a proper morphism $\bar\pi_{\eK}: \calT\to \SS_{\eK}$
and an open immersion $j: \SS''_{\eK}\hookrightarrow \calT$ with $\pi_{\eK}=\bar\pi_{\eK}\cdot j$. By replacing $\calT$ by the scheme theoretic closure of $j$, we can assume that $j(\SS''_{\eK})$ is dense in $\calT$. Since $\pi_{\eK}[1/p]$ is an isomorphism and hence proper, $j[1/p]$ is also proper. Hence, $j[1/p]$ is an isomorphism as a proper open immersion with dense image. It follows that $\calT$ is flat over $\O_E$ and that
the ``boundary", $\calT-j(\SS''_{\eK})$, if non-empty, is supported on the special fiber
of $\calT\to \Spec(\O_E)$.
If $\calT-j(\SS''_{\eK})\neq \emptyset$, there is a $k$-valued point $\bar t$ of $\calT-j(\SS''_{\eK})$. By flatness,
$\bar t$ lifts to $\tilde t\in \calT(\O_F)$, for some finite extension $F/\breve E$. Set $t=x$ for the corresponding $F$-valued point of the Shimura variety $\calT[1/p]=\SS_{\eK}[1/p]=\SS''_{\eK}[1/p]$. This extends to $\tilde x:=\bar\pi_{\eK}(\tilde t) \in \SS_{\eK}(\O_F)$. Since $\O_F$ is strictly henselian, the point $\tilde x$ lifts to a point
\[
\ti z\in \SS_{\eK_p}(\O_F)=\varprojlim\nolimits_{\eK^p}\SS_{\eK_p\eK^p}(\O_F).
\]
By the dvr extension property for $\SS''_{\eK_p}$, this also gives a point $\ti z''
\in \SS''_{\eK_p}(\O_F)$. This maps to a point $\tilde x''\in \SS''_{\eK}(\O_F)$ which agrees
with $x\in \SS_{\eK}(F)$ on the generic fiber. Since $\bar\pi_{\eK}: \calT\to \SS_{\eK}$ is separated, this implies that $\bar t$ lies on $j(\SS''_{\eK})$, which is a contradiction. We conclude that $j$ is an isomorphism and so $\pi_{\eK}$ is proper.
\end{proof}
\end{proof}
\end{para}
\section{Shimura varieties at tame parahoric primes}\label{sHodge}
\subsection{}
We now concentrate our attention to Shimura varieties of Hodge type at primes where the level is parahoric (\cite{KP}).
\begin{para}
Fix a $\Q$-vector space $V$ with a perfect alternating pairing $\psi.$
For any $\Q$-algebra $R,$ we write $V_R = V\otimes_{\Q}R.$
Let $\GSp = \GSp(V,\psi)$ be the corresponding group of symplectic similitudes, and let $S^{\pm}$ be the Siegel
double space, defined as the set of maps $h: \mathbb S \rightarrow \GSp_{\RR}$ such that
\begin{enumerate}
\item The $\C^\times$-action on $V_{\RR}$ gives rise to a Hodge structure
\[
V_{\C} \iso V^{-1,0} \oplus V^{0,-1}
\] of type $(-1,0), (0,-1)$.
\item $(x,y) \mapsto \psi(x, h(i)y)$ is (positive or negative) definite on $V_{\RR}.$
\end{enumerate}
\smallskip
\end{para}
\begin{para}\label{assumptions}
Let $(G, X)$ be a Shimura datum and $\eK=\eK_p\eK^p\subset G(\AA_f)$ with $\eK_p\subset G(\Q_p)$ and $\eK^p\subset G(\AA_f^p)$ as above, where $p$ is an odd prime. We assume:
\begin{itemize}
\item[1)] $(G, X)$ is of Hodge type: There is a symplectic faithful representation
$\rho: G\hookrightarrow {\rm {GSp}}(V, \psi)$ inducing an embedding of Shimura data
\[
(G, X)\hookrightarrow ({\rm {GSp}}(V, \psi), S^{\pm}).
\]
\item[2)] $G$ splits over a tamely ramified extension of $\Q_p$.
\item[3)] $\eK_p=\calG(\Z_p)$, where $\calG$ is the Bruhat-Tits stabilizer group scheme $\calG_x$ of a point $x$ in the extended Bruhat-Tits building of $G(\Q_p)$ and $\calG$ is connected, \emph{i.e.} we have
$\calG=\calG_x=\calG_x^\circ$.
\item[4)] $p\nmid \pi_1(G_{\rm der}(\bar\Q_p))$.
\end{itemize}
Note that by \cite[Prop. 1.4.3]{KP}, the group scheme $\Gg$ satisfies the purity condition (P)
of \S \ref{ss132}.
We now fix a place $v$ of the reflex field ${\sf E}$ over $p$ and let $E={\sf E}_v/\Q_p$ and $\{\mu\}$ be as in \S\ref{sShimura} above.
Associated to $(G_{\Q_p}, \{\mu\})$ and $x$, we have the local model
\[
\Mloc=\Mloc_{x}(G, \{\mu\}).
\]
This is a flat and projective scheme over the ring of integers $\O_E$ which supports an action of
$\calG_{\O_E}$.
(Here, we can also use the slight modification of the definition of \cite{PZ} given by \cite{HPR}.
Then, $\Mloc$ has reduced special fiber and is normal, even when $p$ divides the order
of $\pi_1(G_{\rm der}(\bar\Q_p))$.)
It is shown in \cite{HPR}, that $\Mloc_{x}(G, \{\mu\})$
depends only on ${(\calG, \{\mu\})}$, up to $\calG_{\O_E}$-equivariant isomorphism.
In \cite[2.3.1, 2.3.15, 2.3.16]{KP}, it is shown that under the assumptions (1)-(4) above, there is a (possibly different) Hodge embedding
\[
\iota: (G, X)\hookrightarrow ({\rm {GSp}}(V, \psi), S^\pm)
\] and a $\Z_p$-lattice $\Lambda\subset V_{\Q_p}$
such that $\Lambda\subset \Lambda^\vee$ and
\begin{itemize}
\item[a)] There is a group scheme homomorphism
\[
\iota: \calG\hookrightarrow {\rm GL}(\Lambda )
\]
which is a closed immersion and extends
\[
G_{\Q_p}\hookrightarrow {\rm {GSp}}(V_{\Q_p}, \psi_{\Q_p})\subset {\rm {GL}}(V_{\Q_p}).
\]
\item[b)] There is a corresponding equivariant closed immersion
\[
\iota_*: \Mloc_{x}(G, \{\mu\})\hookrightarrow {\rm {Gr}}(g, \Lambda)_{\O_E}.
\]
\end{itemize}
Here, $\dim_{\Q_p}(V)=2g$ and ${\rm {Gr}}(g, \Lambda)$ is the Grassmannian over $\Z_p$.
\end{para}
\begin{para}
Let $V_{\Z_{(p)}} = \Lambda \cap V,$ and fix a $\Z$-lattice $V_{\Z} \subset V$ such that $V_{\Z}\otimes_{\Z}\Z_{(p)} = V_{\Z_{(p)}}$
and $V_{\Z} \subset V_{\Z}^\vee.$
Consider the Zariski closure $G_{\Z_{(p)}}$ of $\eG$ in $\GL(V_{\Z_{(p)}})$; then $G_{\Z_{(p)}}\otimes_{\Z_{(p)}}\Z_p\cong \Gg$. Fix a collection of tensors $(s_{a}) \subset V_{\Z_{(p)}}^\otimes$
whose stabilizer is $G_{\Z_{(p)}}.$
This is possible by \cite[Lemma 1.3.2]{KisinJAMS} and \cite{DeligneLetter}.
Set $\eK_p = \calG(\Z_p),$ and $\eK^\flat_p = \mathrm{GSp}(V_{\Q_p})\cap \GL(\Lambda).$ We set $\eK = \eK_p\eK^p$ and similarly for $\eK^\flat.$
By \cite[Lemma 2.1.2]{KisinJAMS}, for any compact open subgroup $\eK^p \subset G(\AA^p_f)$ there exists
$\eK^{\flat p} \subset \GSp(\AA^p_f)$ such that $\iota$ induces an embedding over $\sf E$
$$
\Sh_{\eK}(\eG,X) \hookrightarrow \Sh_{\eK^\flat}(\GSp(V, \psi), S^{\pm})\otimes_\Q{\sf E}.
$$
The choice of lattice $V_{\Z}$ gives an interpretation of the Shimura variety $\Sh_{\eK^\flat}(\GSp, S^{\pm})$ as a moduli scheme of
polarized abelian varieties with $\eK^{\flat p}$-level structure, and hence to an integral model $\calA_{\eK^\flat}=\SSh_{\eK^
\flat}(\GSp, S^{\pm})$ over $\Z_{(p)}$
(see \cite{KisinJAMS}, \cite{KP}).
We denote by $\SSh^-_{\eK}(\eG,X)$ the (reduced) closure of $\Sh_{\eK}(\eG,X)$ in the $\OEv$-scheme
$\SSh_{\eK^\flat}(\GSp, S^{\pm})\otimes_{\Z_{(p)}}\OEv,$ and by $\SSh_{\eK}(\eG,X),$ the normalization of the closure $\SSh_{\eK}(\eG,X)^-.$
For simplicity, we set
\[
\SSh_{\eK}:=\SSh_{\eK}(\eG,X)
\]
when there is no danger of confusion.
\begin{thm}\label{exAssSystem}
Assume that $p$ is odd and that the Shimura data $(G, X)$ and the level subgroup $\eK$ satisfy the assumptions (1)-(4) of \S \ref{assumptions}. Then, the $\OEv$-models $\SS_{\eK}(G, X)$ support versal associated systems
\[
\widehat\Dd_\eK=({\rm L}_\eK, \{\calD_{\bar x}\}_{\bar x\in \SS_{\eK}(k)}),
\]
where $\calD_{\bar x}$ are Dieudonn\'e $(\Gg, \Mloc)$-displays.
\end{thm}
\begin{proof} Recall the pro-\'etale $\Gg(\Z_p)$-local system ${\rm L}_\eK$ over $\Sh_{\eK}(G, X)$ given as in \S \ref{31} above. Let $h: A \rightarrow \SS_{\eK}$ denote the restriction of the universal abelian scheme via $
\SS_{\eK}\rightarrow \SSh_{\eK^\flat}(\GSp, S^{\pm})
$. Then the $\Z_p$-local system $\Lr_\eK(\iota)$ is isomorphic to the $\Z_p$-dual of the local system given by the
Tate module of the $p$-divisible group $A[p^\infty]$ of the universal abelian scheme over $\SS_{\eK}$. The tensors
$s_a\in \Lambda^\otimes$ give corresponding global sections $s_{a, \et}$ of $\Lr_\eK(\iota)^\otimes$ over
$\Sh_{\eK}(G, X)$. Theorem \ref{Exists} implies that ${\rm L}_\eK$ extends to an associated system $({\rm L}_\eK, \{\calD_{\bar x}\}_{\bar x\in \SS_{\eK}(k)})$, where $\calD_{\bar x}$ are Dieudonn\'e $(\Gg, \Mloc)$-displays.
It remains to show:
\begin{prop}\label{versalProp}
For every $\bar x\in \SS_{\eK}(k)$, the Dieudonn\'e $(\Gg, \Mloc)$-display $\calD_{\bar x}$ over
$R=\hat\O_{\breve \SS_\eK, \bar x}$ is versal.
\end{prop}
\begin{proof} Set $\calD_{\bar x}=(\calP, q, \Psi)$.
Choose a section $s$ of $\calP$ over $\hat W(R)$ which is rigid in the first order.
Then the corresponding section $\Spec(\hat W(R))\to \calP\subset \calP_{\GL}$ is rigid in the first order for
the $\GL$-display $(\P_{\GL}, q_{\GL}, \Psi_{\GL})$ induced by $\iota$ and $\calD_{\bar x}$. We have a morphism
\[
q\cdot s: \Spec(R)\to \Mloc\subset {\rm Gr}(g,\Lambda)_{\O_{\breve E}}.
\]
We also have the natural morphism
\[
i: \Spec(R)=\Spec(\hat\O_{\breve \SS_\eK, \bar x})\to \calA_{\eK^\flat}\otimes_{\Z_p}\O_{\breve E}.
\]
By the main result of \cite{KP}, $i$ induces a surjection between the cotangent spaces at $i(\bar x)$ and $\bar x$. By \cite[(3.1.12), (3.2.12)]{KP}, since
$s$ is rigid for the $\GL$-display, the morphism $\Spec(R)\to \Mloc\subset {\rm Gr}(g,\Lambda)_{\O_E}$
also induces a surjection on cotangent spaces. We obtain that $\Spec(R)\to \Mloc$ also induces
a surjection
\[
\hat\O_{{\breve {\rm M}}^{\rm loc}, \bar y}\to R=\hat\O_{\breve\SS_\eK, \bar x}
\]
where $\bar y=(q\cdot s)(\bar x)$.
This surjection between
complete local normal rings of the same dimension has to be an isomorphism. This completes our proof.
\end{proof}
\end{proof}
\end{para}
By combining Theorems \ref{exAssSystem} and \ref{charThm} we now obtain:
\begin{thm}\label{indThm}
Assume that $p$ is odd and that the Shimura data $(G, X)$ and the level subgroup $\eK$ satisfy the assumptions (1)-(4) of \S \ref{assumptions}. Suppose $v$ is a place of $\eE$ over $p$. Then the $\O_{\eE, (v)}$-scheme $\SSh_{\eK}(G, X)$ of \cite{KP} is independent of the choices of Hodge embedding $\rho: (G, X)\hookrightarrow ({\rm {GSp}}(V, \psi), S^\pm)$, lattice $V_{\Z}\subset V $ and tensors $(s_a)$, used in its construction. \hfill $\square$
\end{thm}
\subsection{} Finally, we show:
\begin{thm}\label{thmLast}
Assume that $p$ is odd and that the Shimura data $(G, X)$ and the level subgroup $\eK$ satisfy the assumptions (1)-(4) of \S \ref{assumptions}. Then, the $\OEv$-models $\SS_{\eK}(G, X)$ of \cite{KP} are canonical, in the sense of Definition \ref{canonical613} and for $\M=\Mloc$.
\end{thm}
\begin{proof} We set
$
\widehat\SS_\eK=\varprojlim_n \SS_\eK\otimes_{\O_{\eE, (v)}}\otimes \O_{\eE, (v)}/(p)^n$ for the formal scheme obtained as the
$p$-adic completion of $\SS_\eK$. The Dieudonn\'e crystal $\DD_\eK:=\DD(A[p^\infty])(W(\O_{\widehat\SS_\eK}))$ of the universal $p$-divisible group over $\SS_\eK$ gives a $\GL$-display over $\widehat\SS_\eK$. By work of Hamacher and Kim \cite[3.3]{HamaKim}, there are Frobenius invariant tensors $s_{a, \rm univ}\in \DD_\eK^\otimes$ which have the following property: For every $\bar x\in \SS_\eK(k)$, the base change isomorphism
\begin{equation}\label{lastbc}
\DD_\eK\otimes_{W(\O_{\widehat\SS_\eK})}W(\hat R_{\bar x})\simeq M_{\bar x} \otimes_{\hat W(\hat R_x)}W(\hat R_{\bar x})
\end{equation}
maps $s_{a, \rm univ}$ to $\und s_a\otimes 1$. Here, we write $\calD_{\bar x}(\iota)=(M_{\bar x}, M_{1, \bar x }, F_{1, \bar x })$
and we recall that $\und s_a\in M^{\otimes}_{\bar x}$ are the tensors which are associated to $s_{a,\et}\in \Lr_{\eK}(\iota)^\otimes$ and are given by the $\Gg$-torsor $\calP_{\bar x}$ of the
Dieudonn\'e $(\Gg, \Mloc)$-display $\calD_{\bar x}=(\calP_{\bar x}, q_{\bar x}, \Psi_{\bar x})$.
We can now use this to give a $(\Gg, \Mloc)$-display $\Dd_\eK=(\calP_\eK, q_\eK, \Psi_\eK)$ over $\widehat\SS_\eK$ as follows:
We first set
\[
\calP_\eK=\underline{\rm Isom}_{(s_{a,\rm univ}), (s_a\otimes 1)}( \DD_\eK, \Lambda\otimes_{\Z_p}W(\O_{\widehat\SS_\eK})).
\]
Consider an open affine formal subscheme ${\rm Spf}(R)\subset \widehat\SS_\eK$. Then $R$ satisfies condition (N) of \S\ref{Algcond}. We can now see, using Corollary \ref{CorRaynaudGruson} and the fact that (\ref{lastbc}) above respects the tensors, that the restriction of $\calP_{\eK}$ over $W(R)$ is a $\Gg$-torsor. It remains to give $q_\eK$ and $\Psi_\eK$. Recall that, under our assumptions, \cite[Theorem 4.2.7]{KP} gives a (``local model") diagram
\[
\SS_{\eK}\xleftarrow{} \widetilde\SS_\eK\xrightarrow{q_\eK} \Mloc
\]
in which the left arrow is a $\Gg$-torsor and the right arrow is smooth and $\Gg$-equivariant. The $\Gg$-torsor
$\widetilde\SS_\eK\to \SS_\eK$ is given as
\[
\widetilde\SS_\eK:=\underline{\rm Isom}_{(s_{a,\rm DR}), (s_a)}( {\rm H}^1_{\rm DR}(A), \Lambda\otimes_{\Z_p}\O_{ \SS_\eK}).
\]
Since by \cite[Cor. 3.3.4]{HamaKim} the comparison
\[
\DD_\eK\otimes_{W(\O_{\widehat\SS_\eK})}\O_{\widehat\SS_\eK}\cong
\rH^1_{\rm DR}(A)
\]
takes $s_a\otimes 1 $ to $s_{a, \rm DR}$, we have
\[
\calP_\eK\otimes_{W(\O_{\widehat\SS_\eK})}\O_{\widehat\SS_\eK}\cong \widetilde\SS_\eK\xrightarrow{ q_\eK} \Mloc
\]
which gives the desired $q_\eK$. Finally, we can give $\Psi_\eK$ using the Frobenius structure on $\DD_\eK$ (since this respects the tensors). Then
$\DD_\eK$ gives the desired $(\Gg, \Mloc)$-display which satisfies the requirements of \S \ref{612}. The result follows.
\end{proof}
\bigskip | 54,090 |
Model Naomi Campbell was hospitalized in Brazil for the removal of a small cyst, her rep confirmed Tuesday.
.”
Gynecologist Jose Aristodemo Pinotti told the Agencia Estado news service that Cambell had emergency laparoscopic abdominal surgery. “I cannot reveal what Naomi had, nor how serious her condition was, but I can say I operated on her yesterday and that she is completely cured,” Pinotti said.
Sirio Libanes Hospital said Campbell was admitted Sunday night and that she was also being cared for by David Uip, one of Brazil’s leading experts in the treatment of infectious diseases. | 328,800 |
Award Winning Um Salon Beauty Hair Salon
Acclaimed Designer Illustrates The Um Salon Beauty Hair Salon
The lead designer of the award winning project UM Salon - Beauty Hair Salon by Acclaimed Designer. .
Award Winning Um Salon Beauty Hair Salon Images:
End of the article, click here to go back to previous page, or alternatively go back to Worlddesigndays Blog homepage. | 9,609 |
We want our entire church to join us for an exciting home huddle. We will answer your questions and celebrate what God is doing in the life of our church and our plans to “Imagine More’ for our future. Pick a single date that works for you and come hear about the exciting future of our church! Sunday Oct 6th 4-5:30PM (Brent & Jeanine Wakefield Home) Monday Oct. 7th 7-8:30PM (Brian & Miranda Stewart Home) Tuesday Oct. 8th 7-8:30PM (Roger Haserot Home) Wednesday Oct 9th 7-8:30PM (Justin & Nicole Kenyon Home) Thursday Oct. 10th 7-8:30PM (Alonzo & Nancy Becerra Home) NO HOME HUDDLE ON FRIDAY Saturday Oct. 12th 9-10:30AM (Pastor Shawn & Patty Robinson) | 130,229 |
TITLE: The box has minimum surface area
QUESTION [6 upvotes]: Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube.
Could you give me some hints what we are supposed to do??
$$$$
EDIT:
Having found that for $z=\frac{V}{xy}$ the function $A_{\star}(x, y)=A(x, y, \frac{V}{xy})$ has its minimum at $(\sqrt[3]{V}, \sqrt[3]{V})$, how do we conclude that the box is a cube??
We have that $x=y$. Shouldn't we have $x=y=z$ to have a cube??
REPLY [0 votes]: What part is not clear? The symmetry automatically pulls you into other two situations cyclically.
Let us take Lagrange Multiplier (as others have also done).
I take the unified Lagrangian combining object and constraint functions of
Volume and Area together.
I also choose it such that in $ V - A \cdot \lambda \tag{1}$ $\lambda$ would be
physically a linear dimension for a side of a rectangular parallelepiped,except
for a constant factor.
$ x y z - ( x y + y z + z x) \lambda \tag{2}$
Partial differentiation with respect to x gives $ y z - ( y+z) \lambda =0,\, \dfrac1y + \dfrac1z= \dfrac{1}{\lambda} \tag{3}$
Remember that when number of independent variables are more than 2, partial differentiation should be done with respect to each variable.
So similarly by cyclic symmetry, $ z x -( z + x)\lambda =0 , \,\dfrac1z + \dfrac1x= \dfrac{1}{\lambda} \tag{4} $
and
$ xy -( x+y)\lambda =0 , \,\dfrac1x + \dfrac1y= \dfrac{1}{\lambda} \tag{5} $
Summming up the three and halving,
$ \dfrac1x + \dfrac1y + \dfrac1z= \dfrac{3}{2\lambda} \tag{6} $
Subtracting from this the second part of $ (3), (4), (5)$ we get
$ \dfrac1x = \dfrac{1}{2 \lambda} \tag{7}, x = 2 \lambda $
that gives you
$ x = y = z = 2 \lambda = a , $ say.
So finally $ V = a^3$ and $ A = 6 a^2.$ | 189,345 |
With the new year approaching, I'm thinking of what will be left behind in 2009, as well as the fresh start offered by 2010. It seems particularly appropriate then to focus on endings and/or beginnings for our stretch.
Leave me a note about your poem and I'll post the results here later this week.
Leave me a note about your poem and I'll post the results here later this week.
Left Behind: 2009.
c2009 by Jane Yolen,
A Song for New Year's Eve.
c2009.
c2009 by K. Thomas Slesarik
SOMEDAY.
I am LOVING these poems. Thanks.
Jane
I also am enjoying these poems as they make me want to try writing a free verse poem. Thanks
Ken
A New Year Begins
Like a field of fresh fallen snow
a new year sparkles
with possibility
unblemished, unspoiled, unbroken—
a magical moment
gone too soon.
**on the beginning of winter...**
FIRST SNOW AT THE NEW HOUSE
(Hope the format holds! and Happy New Year, everyone!
Julie
***
A STORY FOR THE NEW YEAR.
Forgot to say wow! nice work this week! Thanks, as usual, Tricia.
Been meaning to say, I'm glad you've joined in, Steven--I really like your poems.
And Julie, nice line about the "familiar adjectives and prepositions."
Jane, I'm still trying to figure out how to leave my lust for chocolate behind!
Thank you Tricia! And Happy New Year to all the wonderful poets who gather here.
VIRGIN EMBRACE
A new year
beckons
with arms
spread
amply
inviting me
into
its virgin
embrace.
© Carol Weis | 83,213 |
\section{Introduction}
\gianluca{This space is for general comments: 1) we have to squeeze the doc, as virginialake produces useless blank space.;3) title of def/prop + consistency, 5) no $\bc$-definability, we can simply speak of $\bc$-membership; 6) left-right implication' rather than `left-right direction'}
\anupam{recall: threads are, by definition, maximal, so their start point is determined. They are also not necessarily infinite (e.g. if they end at a weakening.}
\anupam{whenever we use `elementary', I am trying not to qualify space or time.}
\anupam{on $\nbc$: perhaps soundness for $\felementary$ is an instance of that for Leivant's system based on higher-type predicative recursion, under a suitable translation. To mention.}
Formal proofs are traditionally seen as finite mathematical objects modelling logical or mathematical reasoning. Non-wellfounded (but finitely branching) proofs represent a generalization of the notion of formal proof to an infinitary setting. In non-wellfounded proof theory, special attention is devoted to \emph{regular} (or \emph{circular}) proofs, i.e.~those non-wellfounded proofs having only finitely many distinct sub-proofs. Regular proofs can be turned into finite structures called \emph{cycle normal forms}, usually given as finite trees with additional ‘backpointers’. However, regular proofs admit fallacious reasoning. To recover consistency, a standard solution is based on the introduction of non-local correctness criteria, e.g.~\emph{progressiveness}, typically checked by B\"{u}chi automata on infinite words.
Regular proofs, and their corresponding cycle normal forms, have been employed to reason about modal $\mu$-calculus and fixed-point logics~\cite{niwinski1996games,dax2006proof}, induction and coinduction~\cite{brotherston2011sequent}, Kleene algebra~\cite{das2017cut,DP18}, linear logic~\cite{baelde2016infinitary}, arithmetic~\cite{das2018logical}, system $\tgodel$~\cite{Kuperberg-Pous21, Das2021}, and continuous cut-elimination~\cite{mints1978finite, fortier2013cuts}. In particular, \cite{Kuperberg-Pous21} and \cite{Das2021} investigate the computational aspects of regular proofs. Due to their coinductive nature, non-wellfounded proofs are able to define any number-theoretic (partial) function. Definability can be then restricted by combining regularity and progressiveness: the former rules out uncomputable functions, while the latter recovers termination.
Little is known, however, about the complexity-theoretic aspects of regular proofs. Kuperberg, Pinault and Pous~\cite{kuperberg2019cyclic} introduced a circular proof system based on Kleene algebras and seen as a computational machinery for recognising languages. The system and its affine version capture, respectively, the regular languages and the languages accepted in logarithmic time by random-access Turing machines. Their work does not consider the cut rule, which allows for a proof-theoretical description of function composition and is the key ingredient to express recursion in a cyclic-theoretic setting.
The present paper aims at bridging the gap between regular proofs and Implicit Computational Complexity (ICC), a branch of computational complexity studying machine-free languages and calculi able to capture a given complexity class without relying on explicit resource bounds. Our starting point is Bellantoni and Cook's function algebra $\bc$ characterising the polynomial time computable functions ($\fptime$) using \emph{safe recursion}. Functions of $\bc$ have shape $f(x_1,...,x_n;y_{1},...,y_{m})$, where the semicolon separates the \emph{normal} arguments $x_1,...,x_n$ from the \emph{safe} arguments $y_{1},...,y_{m}$. The idea behind safe recursion is that only normal arguments can be used as recursive parameters, while recursive calls can only appear in safe position. This prevents recursive calls becoming recursive parameters of other previously defined functions. In the spirit of the Curry-Howard paradigm, which identifies proofs and programs, $\bc$ can be alternatively designed as a sequent calculus proof-system where safe recursion is introduced by a specific inference rule $\saferec$. In this system, a two-sorted function $f(x_1,...,x_n;y_{1},...,y_{m})$ is represented by a derivation of the sequent $\lists{n}{\sn}{\sn}, \lists{m}{\n}{\n}\seqar \n$ where $\n$ is the ground type for natural numbers and $\sn$ is its modal version.
Starting from $\bc$ we shall consider the non-wellfounded proofs, or \emph{coderivations}, generated by the rules of the subsystem $\bcnorec\dfn \bc \setminus \saferec$. The circular proof system $\ncbc$ is then obtained by considering the regular and progressing coderivations of $\bcnorec$ which satisfy a further criterion, called \emph{safety}. On the one hand, regularity and progressiveness ensure that coderivations of $\ncbc$ define total computable functions; on the other hand, the latter criterion ensures that the recursion mechanisms defined in this infinitary setting are safe, i.e.~the recursive call of a function is never the recursive parameter of the step function.
Despite $\ncbc$ having only ground types, it is able to define safe recursion schemes that nest recursive calls, a property that typically arises in higher-order recursion. This is in fact a peculiar feature of regular proofs extensively studied by Das in~\cite{Das2021}, who has shown that the number-theoretic functions definable by type level $n$ proofs of a circular version of system $\tgodel$ are exactly those ones definable by type level $n+1$ proofs of $\tgodel$.
In the setting of ICC, Hofmann~\cite{Hofmann97} and Leivant~\cite{Leivant99} observed that the capability of nesting recursive calls by higher-order safe recursion mechanisms can be used to characterise the elementary time functions ($\felementary$). In particular, in~\cite{Hofmann97} Hofmann presents the type system $\slr$ (Safe Linear Recursion) as a higher-order version of $\bc$. This system has been shown to capture $\fptime$ once a linearity restriction on the higher-order ‘safe' recursion operator is imposed, which prevents duplication of the recursive calls, and hence their nesting.
Following~\cite{Hofmann97}, we introduce a linearity requirement for $\ncbc$ that is able to control the interplay between loops and the cut rule, called \emph{left-leaning criterion}. The resulting circular proof system is called $\cbc$. Intuitively, $\cbc$ can be seen as a circular version of $\bc$, while $\ncbc$ can be seen as the circular version of a new function algebra, called $\nbc$, which generalises $\bc$ by permitting nested versions of the safe recursion scheme. In particular, in order to define the nesting of recursive calls, the function algebra for $\ncbc$ crucially requires the introduction of ‘auxiliary functions', i.e.~oracles.
The main results of our paper are the following:
\begin{itemize}
\item a function is definable in $\ncbc$ iff it is $\nbc$ iff it is in $\felementary$;
\item a function is definable in $\cbc$ iff it is in $\fptime$;
\end{itemize}
First, we define the function algebras $\nbcpp$ and $\bcpp$, respectively obtained by extending $\nbc$ and $\bc$ with a safe recursion mechanism based on permutation of prefixes. Then we show that $\nbc$ and $\nbcpp$ capture precisely the elementary time computable functions, while $\bcpp$ captures the polytime ones. Completeness for $\ncbc$ can be achieved by showing how to represent the functions of $\nbc$. In a similar way, we represent the functions of $\bc$ in $\cbc$. Soundness is subtler, as it relies on a translation of coderivations of $\ncbc$ and $\cbc$, which are essentially coinductive objects, into the inductively defined functions of $\nbcpp$ and $\bcpp$, respectively.
\subsection*{Overview of results}
The paper is structured as follows. Section~\ref{sec:preliminaries} discusses $\bc$ as a proof system. In Section~\ref{sec:two-tiered-circular-systems-on-notation} we present the non-wellfounded proof system $\bcnorec$ and its semantics. We then analyse some proof-theoretical conditions able to restrict the set of definable functions of $\bcnorec$.
This leads us to the circular proof systems $\ncbc$ and $\cbc$. In Section~\ref{sec:some-variants} we present the function algebras $\nbc$, $\bcpp$ and $\nbcpp$, which implement various safe recursion schemes. Section~\ref{sec:characterization-results-for-function-algebras} shows that $\bcpp$ captures $\fptime$ (\Cref{cor:bcpp-bc-fptime}) and that both $\nbc$ and $\nbcpp$ capture $\felementary$ (Corollary~\ref{cor:nb-elementary-characterization}). These results require a Bounding Lemma (Lemma~\ref{lem:boundinglemma}) and the encoding of the elementary time functions into $\nbc$ (Theorem~\ref{thm:elementary-in-nbc}). In Section~\ref{sec:completeness} we show that any function definable in $\bc$ is also definable in $\cbc$ (Theorem~\ref{thm:bc-in-cbc}), and that any function definable in $\nbc$ is also definable in $\ncbc$ (Theorem~\ref{thm:nbc-in-ncbc}). In Section~\ref{sec:translation} we present a translation of $\ncbc$ into $\nbcpp$ that maps $\cbc$ coderivations into $\bcpp$ functions (Lemma~\ref{lem:translation}).
The main results of this section are displayed in Figure~\ref{fig:picture-main-results}. Many proofs are reported in the appendices.
\begin{figure}[t]
\centering
\begin{tikzcd}[column sep=large]
&&& \felementary \arrow[rrrd,bend left=10, leftarrow, pos=0.3, "\text{Theorem}~\ref{thm:fp-soundness}"]&&& \\
\nbc \arrow[rrr, "\text{Theorem}~\ref{thm:nbc-in-ncbc}"]\arrow[rrru,bend left=10, leftarrow, pos=0.7, "\text{Theorem}~\ref{thm:elementary-in-nbc}"] && &\ncbc
\arrow[rrr, "\text{Lemma}~\ref{lem:translation}"] & && \nbcpp \\
\bc \arrow[rrr, "\text{Theorem}~\ref{thm:bc-in-cbc}"]\arrow[rrrd,bend right=10, leftrightarrow, dashed, pos=0.7, swap, "\text{\cite{BellantoniCook}}"] &&& \cbc
\arrow[rrr, "\text{Lemma}~\ref{lem:translation}"]& && \bcpp \\
&&& \fptime \arrow[rrru,bend right=10, leftarrow, pos=0.3, swap, "\text{Theorem}~\ref{thm:fp-soundness}"]&& &
\end{tikzcd}
\caption{Summary of the main results of the paper, where $\rightarrow$ indicates an inclusion ($\subseteq$) of function classes.
}
\label{fig:picture-main-results}
\end{figure} | 116,250 |
Doctor Who available on iTunes
In time for Christmas, and for the first time, Doctor Who fans will be able to download series 1 – 4 from the iTunes Store in the UK.
From BBC Worldwide, the programmes will become available throughout December, with one series being released each week from the 2 December, and Series 4 becoming available on the 23 December. | 5,379 |
...An Taoiseach, Charles Haughey TD, is pictured leaving the Pro-Cathedral after the mass.Add to Cart
- Filename
- T3-053.jpg
- Copyrighted to SKP & Associates Ltd trading as Lensmen & Associates, Lensmen Photographic Agency, Lensmen Photographic Archive and Irish Photo Archive..COPYRIGHT AND RELATED RIGHTS ACT, 2000..Under the Copyright and Related Rights Act, 2000 the copyright
- Image Size
- 533x379 / 94.4KB
- image library Irish heritage Irish culture Irish photos news photos old print Irish history historical photos Charles Haughey TD Ireland history pictures Irish society black and white black and white pictures Dail Eireann photographs of Ireland antique photos Irish images President Hillery 1980s Irish pictures photography print Republic of Ireland vintage photos Irish gifts Irish people film photography the eighties old pictures Pro-Cathedral Dublin images of Ireland black and white prints picture library Irish old photos sepia
- Contained in galleries
- Search ALL, Search ALL, 1989 - Mass For The 26th Dáil. (T3).
| 148,373 |
TITLE: Understanding G.H Hardy's Proof for Infinitely Many Rational Numbers
QUESTION [0 upvotes]: Taken from A Course of Pure Mathematics:
Irrational Numbers: If the reader will mark off on the line all the points corresponding to the rational numbers whose denominators are $1,2,3\ldots$ in succession, he will readily convince himself that he can cover the line with rational points as closely as he likes. We can state this more precisely as follows: if we take any segment $BC$ on $A$, we can find as many rational points as we please on $BC$.
Suppose, for example, that $BC$ falls within the segment $A_1A_2$. It is evident that if we choose a positive integer $k$ so that$$k.BC>1\tag1$$
And divide $A_1A_2$ into $k$ equal parts, then at least one of the points of division (say $P$) must fall inside $BC$, without coinciding with either $B$ or $C$. For if this were not so, $BC$ would be entirely included in one of the $k$ parts into which $A_1A_2$ has been divided, which contradicts the supposition $(1)$. But $P$ obviously corresponds to a rational number whose denominator is $k$. Thus at least one rational point $P$ lies between $B$ and $C$. But when we can find another such point $Q$ between $B$ and $P$, another between $B$ and $Q$, and so on indefinitely; i.e., as we asserted above, we can find as many as we please. We may express this by saying that $BC$ includes infinitely many rational points.
Question: What does the passage mean by this? I can't follow through with the proof, can you explain it to me?
I tried drawing a diagram with horizontal line $A_1A_2$ and starting off with an example, such as when $BC=1/3$ and $k=4$. However, no matter what, I just can't follow through.
The English is confusing me.
REPLY [2 votes]: If you divide $A_1A_2$ (which is assumed of unit length) into $k=4$ equal parts,then each part has length $\frac14$, hence is shorter than $BC$. It follows that $BC$ cannot fit between consecutive partitioning points, hence must overlap at least two (consecutive) of the $k$ parts at least partly. The partition point between these parts is in the interior of $BC$. | 70,138 |
Less process improvements
The fresh new suggestions felt contained in this report work with bolder a method to slow down the pain from how does Illinois cash quick cash work college student debt. But you will find less change to your processes and you may build from payment which could and assist consumers by simply making it more straightforward to supply masters or remain on payment preparations. Among those choices are talked about lower than.
Consumers already into IDR need to go due to a yearly records process to re-apply. That is an unneeded nightmare for everybody in it. In the event the individuals aren’t reapproved in the long run, they truly are knocked out-of IDR and just have outstanding desire capitalized. Servicers, at the same time, need to spend your time searching for and verifying records to possess consumers whoever commission situation has already been managed. That will take time out of contacting a whole lot more disturb consumers.
Obtaining and you may existence for the Public service Loan Forgiveness is going to be a period of time-ingesting process that includes taking paperwork closed of the borrower’s boss
In lieu of annual reapplication, borrowers should be able to authorize the Irs so you can immediately share their upgraded economic suggestions using their tax returns each year. Performing this will allow repayments in order to instantly to switch and prevent the newest dependence on most borrowers to re-apply annually.
You can find tall arguments regarding the whether defaulting all consumers into the IDR are a good idea because of issues about forcing borrowers so you’re able to spend even when they can’t spend the money for IDR fee, certainly one of other problems. However, IDR will likely be a lot more of an automated tool having consumers who will be if not poised to go into default. That would involve granting new Irs the capacity to express monetary details about one debtor that is 180 or even more months delinquent thus their servicer can enroll them during the IDR. This would keep consumers that have a great $0 commission regarding default without run their region, when you’re servicers could potentially give less fee for others.
One challenge with taking troubled borrowers onto IDR would be the fact men and women agreements is more difficult to sign up for than other repayment choice particularly a great forbearance. A debtor which simply would like to stop costs into the a forbearance is going to do therefore by asking for you to definitely on the internet or higher the phone. At the same time, a debtor who wants to explore IDR must over documents and you can present income studies, except if they thinking-approve which they lack people income. Even though it is crucial that you link IDR costs in order to accurate money pointers, individuals shall be permitted to vocally provide this type of research inturn to have a temporary sixty-time acceptance to own IDR. Borrowers’ money was established you to definitely matter for 2 days, going for time for you to deliver the real paperwork necessary to stay for the plan.
This new education loan series experience already some punitive with regards to of how it normally garnish wages, seize taxation refunds, and take a portion of Social Security monitors. Concurrently, the new wide variety obtained from garnishment can also be bigger than what a borrower for the IDR carry out spend. For example, the typical payment toward IDR is determined on 10 percent out of discretionary income. By comparison, salary garnishment takes as much as 15 per cent away from disposable spend. 56 The fresh wage garnishment system is to getting fairer so you can borrowers of the just taking the exact same express of money as an IDR commission. It should will also get entry to income tax investigation just to influence the size of a family for calculating that it percentage number. If at all possible, the machine should imagine an easy way to ensure it is quantity accumulated compliment of garnishment in order to matter on the forgiveness into IDR.
Rather than signing large numbers of personal PSLF versions, employers need the capacity to mass approve qualification because of their staff. By way of example, immediately following a manager must signal a good PSLF function getting an excellent borrower, they might into the further years simply posting a page on servicer list all the anyone he’s got official prior to now who are however functioning in the company. This will reduce the load on the companies, because they don’t have to signal personal forms, and also support much easier handling.
Recent Comments | 143,477 |
Subsets and Splits