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The symbols $\triangle$, $\square$, $\diamond$, $\clubsuit$ represent four different integers from 1 to 9. Using the equations below, what is the value of $\square$ ? \begin{align*}
\triangle + \square &= \clubsuit \\
\triangle + \triangle &= \diamond +\diamond + \diamond + \diamond + \diamond \\
\triangle + \triangle &= \clubsuit + \diamond.
\end{align*}
For simplicity, replace the triangle with the letter $a$, the square with the letter $b$, the diamond with the letter $c$, and the club with the letter $d$. The three given equations become \begin{align*}
a+b&=d\\
2a&=5c\\
2a&=c+d
\end{align*} We want to find the value of $b$. We can substitute the second equation into the third equation to eliminate $a$, to get $5c=c+d \Rightarrow 4c=d$. Since $a$, $b$, $c$, and $d$ are all integers from 1 to 9, we know that $d$ must be either 4 or 8 and $c$ correspondingly either 1 or 2. The first case, $c=1$ and $d=4$, does not work because plugging those two values into the third given equation gives $2a=5$, which is impossible if $a$ is an integer. Thus, $c=2$ and $d=8$. Plugging these values into the third given equation to solve for $a$, we have $2a=2+8\Rightarrow a=5$. Plugging $a=5$ and $d=8$ into the first equation to solve for $b$, we have $5+b=8 \Rightarrow b=3$. Thus, the value of the square is $\boxed{3}$. | Math Dataset |
Pascale Charpin 1,, and Jie Peng 2,
INRIA, 2 rue Simone Iff, Paris, France
Mathematics and Science College of Shanghai Normal University, Shanghai, China
* Corresponding author: Pascale Charpin
Received October 2018 Revised January 2019 Published June 2019
The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper, we consider more generally the codes associated with functions that have differential uniformity at least $ 4 $. We emphasize, for such a function $ F $, the role of codewords of weight $ 3 $ and $ 4 $ and of some cosets of its associated code $ C_F $. We give some properties on codes associated with differential uniformity exactly $ 4 $. We obtain lower bounds and upper bounds for the numbers of codewords of weight less than $ 5 $ of the codes $ C_F $. We show that the nonlinearity of $ F $ decreases when these numbers increase. We obtain a precise expression to compute these numbers, when $ F $ is a plateaued or a differentially two-valued function. As an application, we propose a method to construct differentially $ 4 $-uniform functions, with a large number of $ 2 $-to-$ 1 $ derivatives, from APN functions.
Keywords: Vectorial function, power function, derivative, Boolean function, linear code, coset of code, plateaued function, bent functions, differential uniformity, differentially two-valued function, Walsh spectrum.
Mathematics Subject Classification: Primary: 94A60, 11T71; Secondary: 14G50.
Citation: Pascale Charpin, Jie Peng. Differential uniformity and the associated codes of cryptographic functions. Advances in Mathematics of Communications, 2019, 13 (4) : 579-600. doi: 10.3934/amc.2019036
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Pascale Charpin Jie Peng | CommonCrawl |
Nizar Touzi
Nizar Touzi (born 1968 in Tunisia) is a Tunisian-French mathematician. He is a professor of applied mathematics at École polytechnique. His research focuses on analysis, statistics and algebra. He is being known for publications on optimization and stochastic control.
Nizar Touzi
Touzi in Oberwolfach
Born1968
Tunisia
NationalityFrench-Tunisian
Alma materUniversité Paris IX - Dauphine
Scientific career
FieldsMathematics
ThesisModèles à volatilité stochastique : arbitrage, équilibre et inférence statistique (1993)
Doctoral advisorÉric Michel Renault
Education
Touzi completed his PhD in Applied Mathematics at the Paris Dauphine University under Éric Michel Renault in January 1994. He began his post-doctoral studies at the University of Chicago, doing such from October 1993 to May 1994. After this, he had an HDR at his alma mater, Paris Dauphine University, in January 1999.[1]
Career
Touzi began his academic career as an assistant professor at this same institution in September 1994. He worked there for five years before becoming a professor of applied mathematics at Pantheone-Sorbonne University in Paris in September 1999.[1]
Touzi’s most cited work, Applications of Malliavin Calculus to Monte Carlo Methods in Finance,[2] was published right before this career change in August 1999. In 2001, Touzi transitioned to the Center for Research in Economics and Statistics to continue teaching applied mathematics. Along with teaching, he also co-led the Finance and Insurance Laboratory at CREST. Between 2001 and 2005, Touzi was an invited professor at multiple institutions, including the University of British Columbia, Princeton University, and the Center for Interuniversity Research and Analysis of Organizations.
In September 2005, Touzi accepted a new position as the Chair in Mathematical Finance at the Tanaka Business School of Imperial College London. He worked at the Tanaka Business School for almost a year before holding his most recent and current position as a professor of applied mathematics at École polytechnique. He was also the head of the Department of Applied Mathematics at École polytechnique from September 2014 to August 2017.[1]
Research
Touzi's most cited paper, Applications of Malliavin Calculus to Monte Carlo methods in finance, co-authored by Eric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux and Pierre-Louis Lions, describes an original probabilistic method to compute option contract Greeks: delta, gamma, theta, and vega. The method is derived from the formula for integration-by-parts and uses principles from Malliavin calculus. Their approach, when computed on standard European option contracts and compared to results yielded from the Monte Carlo method, happens to be more efficient. This paper had a significant impact in the world of mathematical finance, as previous option contract pricing models were based around the Black-Scholes model and Monte Carlo simulations.
Awards
• Best Young Researcher in Finance Award 2007 of the Europlace Institute of Finance.[3]
• The University of Toronto Dean’s Distinguished Visitor Chair, Fields Institute, April-June 2010.[3]
• Invited Session Speaker at the International Congress of Mathematics, August 2010, Hyderabad (India).[3]
• ERC Advanced Grant 2012.[3]
• French Academy of Science Bachelier Prize 2012.[3]
• Oxford University Visiting Man Chair, One month within the period January July 2014.[3]
• Minerva Lectures at Columbia University, October 2013, New York.[3]
• Monetary Authority of Singapore (MAS) Visiting Professor, National University of Singapore, January-February 2018, Singapore.[3]
• Van Eenam, Butcher, and Butcher Financial/Actuarial Faculty Lecture, the University of Michigan, Department of Mathematics, April 2018, Ann Arbor.[3]
References
1. Touzi, Nizar. "Nizar Touzi CV" (PDF). Retrieved December 6, 2022.
2. Fournié, E.; Lasry, JM.; Lebuchoux, J.; et, al. (1999). "Applications of Malliavin calculus to Monte Carlo methods in finance". Finance and Stochastics. 3 (4): 391–412. doi:10.1007/s007800050068. S2CID 6683178.
3. "Nizar Touzi: H-index & awards - academic profile". Research.com. Retrieved December 5, 2022.
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| Wikipedia |
Home O Level Forces & Turning Effect Of Forces How To Add Forces (O Level)
How To Add Forces (O Level)
January 10, 2020 November 17, 2014 by Mini Physics
Show/Hide Sub-topics (Forces & Turning Effect Of Forces | O Level)
What is a force?
Addition Of Forces
Balanced force
Unbalanced force
Turning Effect
Moment Of A Force
Rotational Equilibrium
Since force is a vector quantity, forces can be represented by an arrow diagram.
The magnitude of the force is represented by the length of the arrow
The direction of the force is represented by the direction in which the arrow is pointed.
Resultant Force
Resultant force is the combination of 2 or more forces.
The effect on a body produced by 2 or more forces acting on it will be the same as that produced by their resultant force.
Hence, resultant force is used to simplify force diagrams – it is easier to deal with one resultant force than multiple forces.
Adding Forces in Same Direction
The above figure shows two forces – 10 N and 20 N acting on a car. The resultant force will be 30 N to the right, which is obtained by adding the two forces numerically.
In the general case, where there is $F_{1}$ to $F_{n}$ acting on an object in the same direction,
$$F_{\text{resultant}} = F_{1} + F_{2} + … + F_{n}$$
Adding Forces in Opposite Direction
When the applied forces are in the opposite direction, the resultant force is dependent on the magnitude of the forces. Using the first car in the above figure, there are two forces 20 N and 40 N acting on it. The resultant force will be 20 N as $F_{\text{resultant}} = 40 \, – 20 = 20 \, \text{N}$ to the right.
When the two forces are same in magnitude but different in direction, the resultant force will be 0 (as seen above)
Slightly more advanced trick:
Taking rightward as positive, the forces acting on the car will be -20 N (Negative because the force is to the left) and + 40 N.
Using the addition of forces, $F_{\text{resultant}} = (\, – 20) + 40 = + 20 \, \text{N}$, which is 20 N to the right.
Worked Example
Three forces of 3 N, 1.5 N and 2 N are acting on an object, as shown in the picture below.
What is the resultant due to the three forces?
From the diagram above and taking rightward as positive, the resultant force is given by:
$$\begin{aligned} F_{\text{resultant}} &=3-1.5-2 \\ &=-0.5 \text{ N} \end{aligned}$$
Forces at an angle to each other (Drawing force diagrams)
This will be more complicated than the previous two cases. We can use the parallelogram law of vector addition to find the resultant force.
Consider the above diagram, we are given two forces – A and B. We will shift B to match up with A as seen below. Ensure that force A and B are drawn to scale (e.g. 1 cm to 5 N)
After shifting B, we draw two more lines to make it a parallelogram. The resultant force will just be the red line in the diagram below. You can measure the diagonal (A + B line) to find the resultant force.
Bonus: Can you work out what happens if the two forces are at $90^{\circ}$ to each other?
Ans: The parallelogram is now a square and you can just use Pythagoras' Theorem to find the magnitude of the resultant force. E.g.
Using the Pythagoras' Theorem is another method of finding resultant force from the addition from two vectors.
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Very much helpful and interesting it is very easy to under stand ilke it very much | CommonCrawl |
\begin{document}
\begin{abstract}
As suggested by the title, this paper is a survey of recent results
and questions on the collection of computably enumerable sets under
inclusion. This is not a broad survey but one focused on the
author's and a few others' current research.
\end{abstract}
\maketitle
There are many equivalent ways to definite a computably enumerable or
c.e.\ set. The one that we prefer is the domain of a Turing machine
or the set of balls accepted by a Turing machine. Perhaps this
definition is the main reason that this paper is included in this
volume and the corresponding talk in the ``Incomputable'' conference.
The c.e.\ sets are also the sets which are $\Sigma^0_1$ definable in
arithmetic.
There is a computable or effective listing, $\{ M_e | e \in \omega\}$,
of all Turing machines. This gives us a listing of all c.e.\ sets,
$x$ in $W_e$ at stage $s$ iff $M_e$ with input $x$ accepts by stage
$s$. This enumeration of all c.e.\ sets is very dynamic. We can
think of balls $x$ as flowing from one c.e.\ set into another. Since
they are sets, we can partially order them by inclusion, $\subseteq$
and consider them as model, $\mathcal{E} = \langle \{ W_e | e \in
\omega\}, \subseteq\rangle$. All sets (not just c.e.\ sets) are
partially ordered by Turing reducibility, where $A \leq_T B$ iff
there is a Turing machine that can compute $A$ given an oracle for
$B$.
Broadly, our goal is to study the structure $\mathcal{E}$ and learn
what we can about the interactions between definability (in the
language of inclusion $\subseteq$), the dynamic properties of c.e.\
sets and their Turing degrees. A very rich relationship between
these three notions has been discovered over the years. We cannot
hope to completely cover this history in this short paper. But, we
hope that we will cover enough of it to show the reader that the
interplay between these three notions on c.e.\ sets is, and will
continue to be, an very interesting subject of research.
We are assuming that the reader has a background in computability
theory as found in the first few chapters of \citet{Soare:87}. All
unknown notation also follows \cite{Soare:87}.
\section{Friedberg Splits}
The first result in this vein was \citet{Friedberg:58}, every
noncomputable c.e.\ set has a Friedberg split. Let us first
understand the result then explore why we feel this result relates to
the interplay of definability, Turing degrees and dynamic properties
of c.e.\ sets.
\begin{definition}
$A_0 \sqcup A_1 = A$ is a \emph{Friedberg split} of $A$ iff, for all
$W$ (all sets in this paper are always c.e.), if $W-A$ is not a
c.e.\ set neither are $W-A_i$. \end{definition}
The following definition depends on the chosen enumeration of all c.e. sets. We use the enumeration given to us in the second paragraph of this paper, $x \in W_{e,s}$ iff $M_e$ with input $x$ accepts by stage $s$, but with the convention that if $x \in W_{e,s}$ then $e,x < s$ and, for all stages $s$, there is at most one pair $e, x$ where $x$ enters $W_e$ at stage $s$. Some details on how we can effectively achieve this type of enumeration can be found in \citet[Exercise I.3.11]{Soare:87}. Moreover, when given a c.e.\ set, we are given the index of this c.e.\ set in terms of our enumeration of all c.e.\ sets. At times we will have to appeal to Kleene's Recursion Theorem to get this index.
\begin{definition}
For c.e.\ sets $A = W_e$ and $B=W_i$, $$A \backslash B = \{ x |
\exists s [x \in (W_{e,s}-W_{i,s})]\}$$ and $A\searrow B = A
\backslash B \cap B$. \end{definition}
By the above definition, $A\backslash B$ is a c.e.\ set. $A \backslash B$ is the set of balls that enter $A$ before they enter $B$. If $x \in A \backslash B$ then $x$ may or may not enter $B$ and if $x$ does enters $B$, it only does so after $x$ enters $A$ (in terms of our enumeration). Since the intersection of two c.e.\ sets is c.e, $A \searrow B$ is a c.e.\ set. $A\searrow B $ is the c.e.\ set of balls $x$ that first enter $A$ and then enter $B$ (under the above enumeration).
Note that $W\backslash A = (W-A) \sqcup (W\searrow A)$ ($\sqcup$ is the disjoint union). Since $W\backslash A$ is a c.e.\ set, if $W-A$ is not a c.e.\ set then $W\searrow A$ must be infinite. (This happens for all enumerations.) Hence infinitely many balls from $W$ must flow into $A$.
\begin{lemma}[Friedberg]
Assume $A= A_0 \sqcup A_1$, and, for all $e$, if $W_e \searrow A$ is
infinite then both $W_e \searrow A_0$ and $W_e \searrow A_1$ are
infinite. Then $A_0 \sqcup A_1$ is a Friedberg split of $A$.
Moreover if $A$ is not computable neither are $A_0$ and $A_1$. \end{lemma}
\begin{proof}
Assume that $W-A$ is not a c.e.\ set but $X= W-A_0$ is a c.e.\ set.
$X-A = W-A$ is not a c.e.\ set. So $X\searrow A$ is infinite and
therefore $X \searrow A_0$ is infinite. Contradiction.
If $A_0$ is computable then $X= \overline{A}_0$ is a c.e.\ set and
if $A$ is not computable then $X-A$ cannot be a c.e.\ set. So use
the same reasoning as above to show $X \searrow A_0$ is infinite for
a contradiction. \end{proof}
Friedberg more or less invented the priority method to split every c.e.\ set into two disjoint c.e.\ sets while meeting the hypothesis of the above lemma. The main idea of Friedberg's construction is when a ball $x$ enters $A$ at stage $s$ to add it to one of $A_0$ or $A_1$ but which set $x$ enters is determined by priority. Let \begin{equation}\tag*{$\mathcal{P}_{e,i,k}$}
\label{eq:1}
\text{if }W_e \searrow A \text{ is infinite then } |W_e \searrow A_i
| \geq k. \end{equation}
We say $x$ meets $\mathcal{P}_{e,i,k}$ at stage $s$ if $|W_e \searrow A_i | < k$ by stage $s-1$ and if we add $x$ to $A_i$ then $|W_e
\searrow A_i | \geq k$ by stage $s$. Find the highest $\langle e, i, k \rangle$ that $x$ can meet and add $x$ to $A_i$ at stage $s$. It is not hard to show that all the $\mathcal{P}_{e,i,k}$ are meet.
It is clear that the existence of a Friedberg split is very dynamic. Let's see why it is also a definable property. But, first, we need to understand what we can say about $\mathcal{E}$ with inclusion. We are not going to go through the details but we can define union, intersection, disjoint union, the empty set and the whole set. We can say that a set is complemented. A very early result shows that if $A$ and $\overline{A}$ are both c.e.\ then $A$ is computable. So it is definable if a c.e.\ set is computable.
Inside every computable set we can repeat the construction of the halting set. So a c.e.\ set $X$ is finite iff every subset of $X$ is computable. Hence $W-A$ is a c.e.\ set iff there is a c.e. set $X$ disjoint from $A$ such that $W \cup A = X \sqcup A$. So saying that $A_0 \sqcup A_1 =A$ is a Friedberg split and $A$ is not computable is definable.
Friedberg's result answers a question of Myhill, ``Is every non-recursive, recursively enumerable set the union of two disjoint non-recursive, recursively enumerable sets?'' The question of Myhill was asked in print in the Journal of Symbolic Logic in June 1956, Volume 21, Number 2 on page 215 in the ``Problems'' section of the JSL. This question was the eighth problem appearing in this section. The question about the existence of maximal sets, also answered by Friedberg, was ninth. This author does not know how many questions were asked or when this section was dropped. Myhill also reviewed \citet{Friedberg:58} for the AMS, but the review left no clues why he asked the question in the first place.
The big question in computability theory in the 1950's was ``Does there exist an incomplete noncomputable c.e. set''? \citet{Kleene.Post:54} showed that there are a pair of incomparable Turing degrees below $\bf{0'}$. We feel that after Kleene-Post, Myhill's question is very natural. So we can claim that the existence of a Friedberg split for every c.e.\ set $A$ fits into our theme, the interplay of definability, dynamic properties and Turing degree on the c.e.\ sets.
\subsubsection{Recent Work and Questions on Friedberg Splits}
Given a c.e.\ set one can uniformly find a Friedberg split. It is known that there are other types of splits. One wonders if any of these non-Friedberg splits can be done uniformly. It is also known that for some c.e.\ sets the only nontrivial splits ($A = A_0 \sqcup A_1$ and the $A_0$ and $A_1$ are not computable) are Friedberg. So one cannot hope to get a uniform procedure which always provides a nontrivial non-Friedberg split of every noncomputable c.e.\ set. But it would be nice to find a computable function $f(e) = \langle e_0, e_1 \rangle$ such that, for all $e$, if $W_e$ is noncomputable then $W_{e_0} \sqcup W_{e_1} = W_e$ is a nontrivial split of $W_e$ and, for every c.e.\ set $A$, if $A$ has a nontrivial non-Friedberg split and $A = W_e$ (so $W_e$ is any enumeration of $A$), and then $W_{e_0} \sqcup W_{e_1} = W_e$ is a nontrivial non-Friedberg split. So, if $A$ has a nontrivial non-Friedberg split and $W_e$ is any enumeration of $A$, $f$ always gives out a nontrivial non-Friedberg split. In work yet to appear, the author has shown that such a computable $f$ cannot exist.
Let $\mathcal{P}$ be a property in $\mathcal{E}$. We say that $A$ is \emph{hemi}-$\mathcal{P}$ iff there are c.e.\ sets $B$ and $C$ such that $A \sqcup B = C$ and $C$ has $\mathcal{P}$. We can also define \emph{Friedberg}-$\mathcal{P}$ iff there are c.e.\ sets $B$ and $C$ such that $A \sqcup B = C$ is a Friedberg split and $C$ has $\mathcal{P}$. If $\mathcal{P}$ is definable then \emph{hemi}-$\mathcal{P}$ and \emph{Friedberg}-$\mathcal{P}$ are also definable. One can get lots of mileage from the \emph{hemi}-$\mathcal{P}$, see \citet{Downey.Stob:92} and \citet{Downey.Stob:93}. Most of these results are about properties $\mathcal{P}$ where every nontrivial split of a set with $\mathcal{P}$ is Friedberg. We feel that one should be using \emph{Friedberg}-$\mathcal{P}$ rather than \emph{hemi}-$\mathcal{P}$. To that end we ask the following:
\begin{question}
Is there a definable $\mathcal{P}$ such that the Friedberg splits
are a proper subclass of the nontrivial splits?
\end{question}
We feel that the Friedberg splits are very special and they should
not be able to always cover all the nontrivial splits of every
definable property.
\section{All orbits nice? No!}
As we mentioned earlier, Friedberg also constructed a maximal set
answering another question of Myhill. A maximal set, $M$, is a c.e.\
set such that for every superset $X$ either $X=^* M$ ($=^*$ is equal
modulo finite) or $W =^* \omega$. Being maximal is definable.
Friedberg's construction of a maximal set is very dynamic.
\citet{Martin:66*1} showed that all maximal sets must be high. A
further result of \citet{Martin:66*1} shows a c.e.\ degree is high
iff it contains a maximal set. A remarkable result of Soare
\cite{Soare:74} shows that the maximal sets form an orbit, even an
orbit under automorphisms computable from $\bf 0''$ or
$\Delta^0_3$-automorphisms.
The result of Soare gives rise to the question are all orbits as nice
as the orbit of the maximal sets? We can go more into the formality
of the question but that was dealt with already in another survey
paper, \citet*{MR2395047}. To tell if two c.e. sets, $A$ and $B$,
are in the same orbit, it is enough to show if there is an
automorphism $\Phi$ of $\mathcal{E}$ taking the one to the other,
$\Phi(A) = B$ (we write this as $A$ is \emph{automorphic} to $B$).
Hence it is $\Sigma^1_1$ to tell if two sets are in the same orbit.
The following theorem says that is the best that we can do and hence
not all orbits are as nice as the orbits of maximal sets. The
theorem has a number of interesting corollaries.
\begin{theorem}[\citet*{MR2425182}]
\label{sw} There is a c.e.\ set $A$ such that the index set $\{i :
W_i \approx A\}$ is $\Sigma^1_1$-complete. \end{theorem}
\begin{corollary}[\citet{MR2425182}]
Not all orbits are elementarily definable; there is no arithmetic
description of all orbits of $\mathcal{E}$. \end{corollary}
\begin{corollary}[\citet{MR2425182}]
The Scott rank of $\mathcal{E}$ is $\wock +1$. \end{corollary}
\begin{theorem}[\citet{MR2425182}]\label{sec:maincor}
For all finite $\alpha > 8$ there is a properly $\Delta^0_{\alpha}$
orbit. \end{theorem}
These results were completely explored in the survey, \cite{MR2395047}. So we will focus on some more recent work. In the work leading to the above theorems, Cholak and Harrington also showed that:
\begin{theorem}[\cite{MR2366962}]
Two simple sets are automorphic iff they are $\Delta^0_6$
automorphic. A set $A$ is \emph{simple} iff for every (c.e.) set
$B$ if $A \cap B$ is empty then $B$ is finite. \end{theorem}
Recently Harrington improved this result to show:
\begin{theorem}[Harrington 2012, private email]
The complexity of the $\mathcal{L}_{\omega_1,\omega}$ formula
describing the orbit of any simple set is very low (close to 6).
\end{theorem}
That leads us to make the following conjecture:
\begin{conjecture}
We can build the above orbits in Theorem~\ref{sec:maincor} to
have complexity close to $\alpha$ in terms of the
$\mathcal{L}_{\omega_1,\omega}$ formula describing the orbit.
\end{conjecture}
\section{Complete Sets}
Perhaps the biggest questions on the c.e. sets are the following:
\begin{question}[Completeness]
Which c.e.\ sets are automorphic to complete sets?
\end{question}
Motivation for this question dates back to Post. Post was trying to
use properties of the complement of a c.e.\ set to show that the set
was not complete. In the structure $\mathcal{E}$ all the sets in the
same orbit have the same definable properties.
By \citet{Harrington.Soare:98}, \cite{Harrington.Soare:91}, and
\cite{Harrington.Soare:96}, we know that not every c.e.\ set is
automorphic to a complete set and, furthermore, there is a dichotomy
between the ``prompt'' sets and the ``tardy'' (nonprompt) sets with
the ``prompt'' sets being automorphic to complete sets. We will
explore this dichotomy in more detail, but more definitions are
needed:
\begin{definition}
$X = (W_{e_1} - W_{e_2}) \cup (W_{e_3} - W_{e_4}) \cup \ldots
(W_{e_{2n-1}} - W_{e_{2n}})$ iff $X$ is $2n$-c.e.\ and $X$ is
$2n+1$-c.e.\ iff $X = Y \cup W_e$, where $Y$ is $2n$-c.e. \end{definition}
\begin{definition}
Let $X^n_e$ be the $e$th $n$-c.e.\ set. $A$ is \emph{almost
prompt} iff there is a computable nondecreasing function $p(s)$
such that for all $e$ and $n$ $ \text{if } X^n_e = \overline{A}
\text{ then } (\exists x) (\exists s) [ x \in X^n_{e,s} \text{
and } x \in A_{p(s)}]$.
\end{definition}
\begin{theorem}[\citet{Harrington.Soare:96}]
Each almost prompt sets are automorphic to some complete set. \end{theorem}
\begin{definition}
$D$ is \emph{$2$-tardy} iff for every computable nondecreasing
function $p(s)$ there is an $e$ such that $ X^2_e = \overline{D}$
and $(\forall x) (\forall s) [\text{if }x \in X^2_{e,s}$ then $ x
\not\in D_{p(s)}]$ \end{definition}
\begin{theorem}[\citet{Harrington.Soare:91}]
There are $\mathcal{E}$ definable properties $Q(D)$ and $P(D,C)$ such that
\begin{enumerate}
\item $Q(D)$ implies that $D$ is $2$-tardy and hence the orbit of
$D$ does not contain a complete set.
\item for $D$, if there is a $C$ such that $P(D,C)$ and $D$ is
$2$-tardy then $Q(D)$ (and $D$ is high).
\end{enumerate} \end{theorem}
The $2$-tardy sets are not almost prompt and the fact they are not almost prompt is witnessed by $e=2$. It would be nice if the above theorem implied that being $2$-tardy was definable. But it says with an extra definable condition being $2$-tardy is definable.
\citet{Harrington.Soare:91} ask if each $3$-tardy set is computable by some $2$-tardy set. They also ask if all low$_2$ simple sets are almost prompt (this is the case if $A$ is low). With Gerdes and Lange, Cholak answered these negatively:
\begin{theorem}[\citet*{MR2926283}]\label{sec:work-tardy-sets-1}
There exists a properly $3$-tardy $B$ such that there is no
$2$-tardy $A$ such that $B \leq_T A$. Moreover, $B$ can be built
below any prompt degree. \end{theorem}
\begin{theorem}[\citet*{MR2926283}]
There is a low$_2$, simple, $2$-tardy set. \end{theorem}
Moreover, with Gerdes and Lange, Cholak showed that there are definable (first-order) properties $Q_n(A)$ such that if $Q_n(A)$ then $A$ is $n$-tardy and there is a properly $n$-tardy set $A$ such that $Q_n(A)$ holds. Thus the collection of all c.e.\ sets not automorphic to a complete set breaks up into infinitely many orbits.
But, even with the work above, the main question about completeness and a few others remain open. These open questions are of a more degree-theoretic flavor. The main still open questions are:
\begin{question}[Completeness]
Which c.e.\ sets are automorphic to complete sets? \end{question}
\begin{question}[Cone Avoidance]\label{sec:complete-sets}
Given an incomplete c.e.\ degree $\mathbf{d}$ and an incomplete
c.e.\ set $A$, is there an $\Ahat$ automorphic to $A$ such that
$\mathbf{d} \not\leq_T \Ahat$? \end{question}
It is unclear whether these questions have concrete answers. Thus the following seems reasonable.
\begin{question}
Are these arithmetical questions? \end{question}
Let us consider how we might approach these questions. One possible attempt would be to modify the proof of Theorem~\ref{sw} to add degree-theoretic concerns. Since the coding comes from how $A$ interacts with the sets disjoint from it, we should have reasonable degree-theoretic control over $A$. The best we have been able to do so far is alter Theorem~\ref{sw} so that the set constructed has hemimaximal degree and everything in its orbit also has hemimaximal degree. However, what is open is whether the orbit of any set constructed via Theorem~\ref{sw} must contain a representative of every hemimaximal degree or only have hemimaximal degrees. If the infinite join of hemimaximal degrees is hemimaximal then the degrees of the sets in these orbits only contain the hemimaximal degrees. But, it is open whether the infinite join of hemimaximal degrees is hemimaximal.
\subsection{Tardy Sets}
As mentioned above, there are some recent results on $n$-tardy and very tardy sets (a set is very tardy iff it is not almost prompt). But there are several open questions related to this work. For example, is there a (first-order) property $Q_\infty$ so that if $Q_\infty(A)$ holds, then $A$ is very tardy (or $n$-tardy, for some $n$). Could we define $Q_\infty$ such that $Q_n(A)\implies Q_\infty(A)$? How do hemi-$Q$ and $Q_3$ compare? But the big open questions here are the following:
\begin{question}
Is the set $B$ constructed in Theorem~\ref{sec:work-tardy-sets-1}
automorphic to a complete set? If not, does $Q_3(B)$ hold? \end{question}
It would be very interesting if both of the above questions have a negative answer.
Not a lot about the degree theoretic properties of the $n$-tardies is known. The main question here is whether Theorem~\ref{sec:work-tardy-sets-1} can be improved to $n$ other than $2$.
\begin{question}
For which $n$ are there $n+1$ tardies which are not computed by
$n$-tardies? \end{question}
But there are many other approachable questions. For example, how do the following sets of degrees compare: \begin{itemize} \item the hemimaximal degrees? \item the tardy degrees? \item for each $n$, $\{\mathbf{d} : $ there is an $n$-tardy $D$ such
that $\mathbf{d} \leq_T D\}$? \item $\{\mathbf{d} : $ there is a $2$-tardy
$D$ such that $Q(D)$ and $\mathbf{d} \leq_T D\}$? \item $\{\mathbf{d} : $ there is an $A \in \mathbf{d}$ which is not
automorphic to a complete set$\}$? \end{itemize} Does every almost prompt set compute a $3$-tardy? Or a very tardy? \citet{Harrington.Soare:98} show there is a maximal $2$-tardy set. So there are $2$-tardy sets which are automorphic to complete sets. Is there a nonhigh, nonhemimaximal, $2$-tardy set which is automorphic to a complete set?
\subsection{Cone Avoidance, Question~\ref{sec:complete-sets}}
The above prompt vs.\ tardy dichotomy gives rise to a reasonable way to address Question~\ref{sec:complete-sets}. An old result of Cholak \cite{mr95f:03064} and, independently, Harrington and Soare \cite{Harrington.Soare:96}, says that every c.e.\ set is automorphic to a high set. Hence, a positive answer to both the following questions would answer the cone avoidance question but not the completeness question. These questions seem reasonable as we know how to work with high degrees and automorphisms, see \cite{mr95f:03064},
\begin{question}
Let $A$ be incomplete. If the orbit of $A$ contains a set of high
prompt degree, must the orbit of $A$ contain a set from all high
prompt degrees? \end{question}
\begin{question}
If the orbit of $A$ contains a set of high tardy degree, must the
orbit of $A$ contain a set from all high tardy degrees? \end{question}
Similarly we know how to work with prompt degrees and automorphisms, see \citet*{mr92j:03039} and \citet{Harrington.Soare:96}. We should be able to combine the two. No one has yet explored how to work with automorphisms and tardy degrees.
\section{$\mathcal{D}$-Maximal Sets}
In the above sections we have mentioned maximal and hemimaximal sets several times. It turns out that maximal and hemimaximal sets are both $\mathcal{D}$-maximal.
\begin{definition}
$\mathcal{D}(A) = \{ B : \exists W ( B \subseteq A \cup W \text{ and
} W \cap A = \emptyset)\}$ under inclusion. Let
$\mathcal{E}_{\mathcal{D}(A)}$ be $\mathcal{E}$ modulo $\mathcal{D}(A)$. \end{definition}
$\mathcal{D}(A)$ is the ideal of c.e.\ sets of the form $\tilde{A} \sqcup \tilde{D}$ where $\tilde{A} \subseteq A$ and $\tilde{D} \cap A = \emptyset$.
\begin{definition}
$A$ is \emph{$\mathcal{D}$-hhsimple} iff $\mathcal{E}_{\mathcal{D}(A)}$ is a
$\Sigma^0_3$ Boolean algebra. $A$ is \emph{$\mathcal{D}$-maximal}
iff $\mathcal{E}_{\mathcal{D}(A)}$ is the trivial Boolean algebra iff for all
c.e.\ sets $B$ there is a c.e. set, $D$, disjoint from $A$, such
that either $B \subset A \cup D$ or $B \cup D \cup A = \omega$. \end{definition}
Maximal sets and hemimaximal sets are $\mathcal{D}$-maximal. Plus, there are many other examples of $\mathcal{D}$-maximal sets. In fact, with the exception of the creative sets, all known elementary definable orbits are orbits of $\mathcal{D}$-maximal sets. In the lead up to Theorem~\ref{sw}, Cholak and Harrington were able to show:
\begin{theorem}[\cite{mr2004f:03077}]
If $A$ is $\mathcal{D}$-hhsimple and $A$ and $\Ahat$ are in the
same orbit then $\mathcal{E}_{\mathcal{D}(A)} \cong_{\Delta^0_3}
\mathcal{E}_{\mathcal{D}(\Ahat)}$. \end{theorem}
So it is an arithmetic question to ask if the orbit of a $\mathcal{D}$-maximal set contains a complete set. But the question remains does the orbit of every $\mathcal{D}$-maximal set contain a complete set? It was hoped that the structural properties of $\mathcal{D}$-maximal sets would be sufficient to allow us to answer this question.
Cholak, Gerdes, and Lange \cite{pub2} have completed a classification of all $\mathcal{D}$-maximal sets. The idea is to look at how $\mathcal{D}(A)$ is generated. For example, for a hemimaximal set $A_0$, $\mathcal{D}(A_0)$ is generated by $A_1$, where $A_0 \sqcup A_1$ is maximal. There are ten different ways that $\mathcal{D}(A)$ can be generated. Seven were previously known and all these orbits contain complete and incomplete sets. Work from \citet{MR1264963} shows that these seven types are not enough to provide a characterization of all $\mathcal{D}$-maximal sets. Cholak, Gerdes, and Lange construct three more types and show that these ten types provide a characterization of all $\mathcal{D}$-maximal sets. We have constructed three new types of $\mathcal{D}$-maximal sets; for example, a $\mathcal{D}$-maximal set where $\mathcal{D}(A)$ is generated by infinitely many not disjoint c.e\ sets. We show these three types plus another split into infinitely many different orbits. We can build examples of these sets which are incomplete or complete. But, it is open if each such orbit contains a complete set. So, the structural properties of $\mathcal{D}$-maximal sets was not enough to determine if each $\mathcal{D}$-maximal set is automorphic to a complete set.
It is possible that one could provide a similar characterization of the $\mathcal{D}$-hhsimple sets. One should fix a $\Sigma^0_3$ Boolean algebra, $\mathcal{B}$, and characterize the $\mathcal{D}$-hhsimple sets, $A$, where $\mathcal{E}_{\mathcal{D}(A)} \cong \mathcal{B}$. It would be surprising if, for some $\mathcal{B}$, the characterization would allow us to determine if every orbit of these sets contains a complete set.
\section{Lowness}
Following his result that the maximal sets form an orbit, Soare\cite{Soare:82} showed that the low sets resemble computable sets. A set $A$ is low$_n$ iff $\mathbf{0}^{(n)} \equiv_T A^{(n)}$. We know that noncomputable low sets cannot have a computable set in their orbit, so, the best that Soare was able to do is the following:
\begin{definition}
$\mathcal{L}(A)$ are the c.e.\ supersets of $A$ under inclusion.
$\mathcal{F}$ is the filter of finite sets. $\mathcal{L}^*(A)$ is
$\mathcal{L}(A)$ modulo $\mathcal{F}$. \end{definition}
\begin{theorem}[\citet{Soare:82}]
If $A$ is low then $\mathcal{L}^*(A) \approx
\mathcal{L}^*(\emptyset)$. \end{theorem}
In 1990, Soare conjectured that this can be improved to low$_2$. Since then there have been a number of related results but this conjecture remains open. To move forward some definitions are needed:
\begin{definition}
$A$ is \emph{semilow} iff $\{ i | W_i \cap A \neq \emptyset \}$ is
computable from $\mathbf{0'}$. $A$ is \emph{semilow$_{1.5}$} iff
$\{ i | W_i \cap A \text{ is finite}\} \leq_1 \mathbf{0''}$. $A$ is
\emph{semilow$_2$} iff $\{ i | W_i \cap A \text{ is finite}\}$ is
computable from $\mathbf{0''}$. \end{definition}
Semilow implies semilow$_{1.5}$ implies semilow$_2$, if $A$ is low then $\overline{A}$ is semilow, and low$_2$ implies semilow$_2$ (details can be found in \citet{Maass:83} and \citet{mr95f:03064}). \citet{Soare:82} actually showed that if $\overline{A}$ is semilow then $\mathcal{L}^*(A) \approx \mathcal{L}^*(\emptyset)$. \citet{Maass:83} improved this to when $\overline{A}$ is semilow$_{1.5}$.
In Maass's proof semilow$_{1.5}$ness is used in two ways: A c.e.\ set, $W$, is \emph{well-resided outside $A$} iff $W \cap \overline{A}$ is infinite. Semilow$_{1.5}$ makes determining which sets are well-resided outside $A$ a $\Pi^0_2$ question. The second use of semilow$_{1.5}$ was to capture finitely many elements of $W \cap \overline{A}$. For that Maass showed that semilow$_{1.5}$ implies the \emph{outer splitting property}:
\begin{definition}
$A$ has the \emph{outer splitting property} iff there are computable
functions $f, h$ such that, for all $e$, $W_e = W_{f(e)} \sqcup
W_{h(e)}$, $W_{f(e)} \cap \overline{A}$ is finite, and if $W_e \cap
\overline{A}$ is infinite then $W_{f(e)} \cap \overline{A}$ is
nonempty. \end{definition}
Cholak used these ideas to show that:
\begin{theorem}[\citet{mr95f:03064})]
If $A$ has the outer splitting property and $\overline{A}$ is
semilow$_2$ then $\mathcal{L}^*(A) \approx
\mathcal{L}^*(\emptyset)$. \end{theorem}
It is known that there is a low$_2$ set which does not have the outer splitting property, see \citet*[Theorem~4.6]{Downey:2013wq}. So to prove that if $A$ is low$_2$ then $\mathcal{L}^*(A) \approx \mathcal{L}^*(\emptyset)$ will need a different technique. However, \citet{Lachlan:68} showed that every low$_2$ set has a maximal superset using the technique of \emph{true stages}. Perhaps the true stages technique can be used to show Soare's conjecture.
Recently there has been a result of Epstein.
\begin{theorem}[\citet{epstein:10} and \cite{MR3003266}]
There is a properly low$_2$ degree $\mathbf{d}$ such that if $A
\leq_T \mathbf{d}$ then $A$ is automorphic to a low set.
\end{theorem}
Epstein's result shows that there is no collection of c.e.\ sets
which is closed under automorphisms and contains at least one set of
every nonlow degree. Related results were discussed in
\citet{mr2001k:03085}.
This theorem does have a nice yet unmentioned corollary: The
collection of all sets $A$ such that $\overline{A}$ is semilow (these
sets are called \emph{speedable}) is not definable. By
\citet*[Theorem~4.5]{Downey:2013wq}, every nonlow c.e. degree
contains a set $A$ such that $\overline{A}$ is not semilow$_{1.5}$
and hence not semilow. So there is such a set $A$ in $\mathbf{d}$.
$A$ is automorphic to a low set $\Ahat$. Since $\Ahat$ is low,
$\overline{\Ahat}$ is semilow.
Esptein's result leads us wonder if the above results can be improved
as follows:
\begin{conjecture}[Soare]
Every semilow set is (effectively) automorphic to a low set.
\end{conjecture}
\begin{conjecture}[Cholak and Epstein]
Every set $A$ such that $A$ has the outer splitting property and
$\overline{A}$ is semilow$_2$ is automorphic to a low$_2$ set.
\end{conjecture}
Cholak and Epstein are currently working on a proof of the latter
conjecture and some related results. Hopefully, a draft will be
available soon.
\end{document} | arXiv |
British Journal of Nutrition
Volume 110 Issue 7
Nutritional disturbance in acid...
Core reader
Nutritional disturbance in acid–base balance and osteoporosis: a hypothesis that disregards the essential homeostatic role of the kidney
Review of acid–base homeostasis
Is there a reason to question the traditional, accepted approach to analyse acid–base chemistry?
Nutrition and acid–base balance
Claude Bernard's nutrition experiments in rabbits
Origin of the hypothesis involving bone as regulator of acid–base balance
From severe renal metabolic acidosis to the hypothesis of 'latent' acidosis of nutritional origin
Bone alkali store and overestimated proton retention in chronic renal failure
Extrapolation from severe metabolic acidosis in rodents to the putative nutritional protein origin of osteoporosis in the general human population
Evaluation of the acid and alkali nutritional load
Mathematical model to estimate the potential renal acid load of foods
Relationship between bone health and the acid or alkaline load of the diet
Reviews of studies dealing with the dietary acid load hypothesis of osteoporosis
Randomised clinical trials with potassium alkali in postmenopausal women
Phosphate intake and calcium balance
Age decline in renal function and osteoporosis: are they causally related?
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British Journal of Nutrition, Volume 110, Issue 7
14 October 2013 , pp. 1168-1177
Jean-Philippe Bonjour (a1)
(a1)
Division of Bone Diseases, Geneva University Hospitals and Faculty of Medicine, Rue Gabrielle-Perret-Gentil, CH-1211Geneva 14, Switzerland
Copyright: © The Author 2013
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
DOI: https://doi.org/10.1017/S0007114513000962
Published online by Cambridge University Press: 04 April 2013
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The nutritional acid load hypothesis of osteoporosis is reviewed from its historical origin to most recent studies with particular attention to the essential but overlooked role of the kidney in acid–base homeostasis. This hypothesis posits that foods associated with an increased urinary acid excretion are deleterious for the skeleton, leading to osteoporosis and enhanced fragility fracture risk. Conversely, foods generating neutral or alkaline urine would favour bone growth and Ca balance, prevent bone loss and reduce osteoporotic fracture risk. This theory currently influences nutrition research, dietary recommendations and the marketing of alkaline salt products or medications meant to optimise bone health and prevent osteoporosis. It stemmed from classic investigations in patients suffering from chronic kidney diseases (CKD) conducted in the 1960s. Accordingly, in CKD, bone mineral mobilisation would serve as a buffer system to acid accumulation. This interpretation was later questioned on both theoretical and experimental grounds. Notwithstanding this questionable role of bone mineral in systemic acid–base equilibrium, not only in CKD but even more in the absence of renal impairment, it is postulated that, in healthy individuals, foods, particularly those containing animal protein, would induce 'latent' acidosis and result, in the long run, in osteoporosis. Thus, a questionable interpretation of data from patients with CKD and the subsequent extrapolation to healthy subjects converted a hypothesis into nutritional recommendations for the prevention of osteoporosis. In a historical perspective, the present review dissects out speculation from experimental facts and emphasises the essential role of the renal tubule in systemic acid–base and Ca homeostasis.
'It is no exaggeration to say that the composition of the body fluids is determined not by what the mouth takes in but by what the kidneys keep: they are the master chemists of our internal environment. When, among other duties, they excrete the ashes of our body fires, or remove from the blood the infinite variety of foreign substances that are constantly absorbed from our indiscriminate gastrointestinal tracts, these excretory operations are incidental to the major task of keeping our internal environment in an ideal, balanced state.'
Homer W. Smith (From Fish to Philosopher)(1)
The hypothesis suggesting that a diet increasing the urinary excretion of acid ion (proton = H+) could be a risk factor for osteoporosis was proposed more than 40 years ago(2). Conversely, the contention that a diet rich in alkaline or basic (OH−) functions would be beneficial to bone health continues to generate a substantial scientific interest. The recurring resurgence of this interest is relayed in the general population by various mass media spreading the belief of small but very active groups of opponents to the use of any animal products(3). The same view can be expressed by the scientific community via analyses or meta-analyses of studies which suggest that certain nutrients, particularly animal protein, or foods such as meat or dairy products, by virtue of their supposed 'acidogenic' properties, may increase the risk of osteoporosis. At the same time, or in response to these suggestions appearing in the scientific literature, there is growing interest in miraculous benefits claimed for so-called 'alkalinogenic' diets or nutritional products, such as those proposed on numerous websites. Consuming 'alkalis' will bring about a number of benefits, expanding from hair loss treatment to the prevention of cancers, infections, allergies, obesity, 'all types of rheumatism' and, ultimately, osteoporosis, the subject of the present review.
The keen interest in alkali has also found followers among certain anthropologists who argue that the contemporary diet, when compared with that which prevailed before the Neolithic period, has led to osteoporosis together with other diseases linked to the modern way of life, several being hypothetically caused by nutrition-induced metabolic acidosis(4).
One may be surprised by this keen interest in alkalis and the associated fear of acid, forgetting that the skin or the oesophagus tolerates caustic soda (Na+OH−) as poorly as hydrochloric acid (H+Cl−). Yet, basic physiology shows that our bodies are equipped with several systems capable of neutralising or generating protons, such as the bicarbonate–CO2 buffer:
$$\begin{eqnarray} H^{ + } + HCO_{3}^{ - }\leftrightarrow H_{2}CO_{3}\leftrightarrow H_{2}O + CO_{2} \end{eqnarray}$$
This system enables very effective neutralisation of the excess of H+ ions by moving this reaction to the right and therefore increasing the production of CO2, which, in physiological conditions, is easily eliminated via the respiratory tract(5). In addition to this pulmonary mechanism, the renal tubular system is extremely well equipped to maintain the acid–base balance of the extracellular compartment by modulating the reabsorption of bicarbonate and the secretion of protons. These processes are linked to buffer systems able to eliminate the excess of H+ ions produced by cellular metabolism, without substantially lowering urinary pH(5). The main urinary buffer systems are:
$$\begin{eqnarray} (i)\,HPO_{4}^{2 - }\leftrightarrow H_{2}PO_{4}^{1\cdot 8 - }\,(divalent\,phosphates\leftrightarrow monovalent\,phosphates) \end{eqnarray}$$
$$\begin{eqnarray} (ii)\,NH_{3}\leftrightarrow NH_{4}^{ + }\,(ammonia\leftrightarrow ammonium) \end{eqnarray}$$
The composition of the extracellular fluid in which the cells of the body exert their specific functions must deviate towards neither the acid nor the alkaline side. Measurable deviations are due to pathological disturbances that affect primarily the digestive tract, intermediary metabolism, the pulmonary system or renal functions. The four classic disturbances of acid–base balance with clinically significant consequences are, on the one hand, acidosis and alkalosis of metabolic origin and, on the other hand, acidosis and alkalosis of respiratory origin(5, 6). Furthermore, deviations from an extracellular pH of 7·35 can be corrected or attenuated by both the capacity of chemical buffers and the physiological regulation at the respiratory and renal tubular levels. The mobilisation of such compensatory mechanisms is expressed by changes in the distribution of buffer system components. These basic concepts are essential for the understanding of the relationship between nutrition and bone health.
As discussed below, the notion of latent acidosis(7), as well as the relationship between ageing, renal functional decline and blood acid–base composition(8), have been suggested to be causally related to the increased prevalence of osteoporosis in the elderly population. However, alterations in blood pH, [HCO3−] and/or pCO2 have not been documented in relation to changes in the foods or nutrients purported to cause osteoporosis in otherwise healthy individuals(9, 10).
The traditional, accepted bicarbonate-centred formulation of acid–base interpretation was questioned about 25 years ago by Stewart(11), who promoted the so-called 'strong ion difference' (SID) approach. According to the mathematical model from which this theory was worked out, the components of the volatile bicarbonate–CO2 buffer system (CO2, HCO3−, H2CO2 and CO32 −) were dependent variables of the difference in the net charges of fixed cations and anions fully dissociated in solution. Thus, according to Stewart(11, 12), the strong ion difference [Na+] − [Cl−] or SID would be a determinant of [H+]. However, 30 years after Stewart(11, 12), Kurtz et al. (13) thoroughly analysed the physico-chemical, physiological and clinical aspects of Stewart's theory when compared with the traditional, accepted bicarbonate-centred approach. In this very comprehensive review(13) it was underscored that Stewart's theory(11, 12) reintroduced the confusion in the acid–base literature that existed from the beginning of the twentieth century and had prevailed until the early 1950s. During that period, clinical chemists considered Na+ as a base and Cl− as an acid(13). Such a consideration entirely disregarded the key position of H+ in acid–base reactions. This misconception in clinical acid–base chemistry was dispelled in the mid-late 1950s by Relman(14) and Christiensen(15), whose 'prescient analysis foreshadows in some sense the current issues in the literature as they relate to the Stewart framework'(13). Furthermore, the bicarbonate-centred approach utilising the Henderson–Hasselbach equation is a mechanistic formulation that reflects the underlying acid–base situation(13). It remains the most reliable and used method for physiologists and clinicians to assess acid–base chemistry in human blood(13). Therefore, it is inaccurate to claim an absence of consensus as to how to assess acid–base balance by referring primarily to the SID and bicarbonate-centred approaches without emphasising the most cogent arguments developed by Kurtz et al. (13). Adherents to the notion of diet-induced acidosis as an essential mechanism for the high prevalence of osteoporosis in the Western world suggest that if no change is observed, this does not mean there is none. However, in order to support the diet-induced acidosis hypothesis of osteoporosis, it would seem necessary to objectively measure whether diet alters blood acid–base equilibrium, and, particularly, whether such alteration can be found in association with bone fragility.
In the presence of one of the above-mentioned acid–base balance disturbances, foods, depending on their nutrient composition, can either slightly accentuate or ameliorate a pathological condition. However, in the absence of such pathologies, food components trigger neither extracellular fluid acidosis nor alkalosis.
Any influence of nutritional origin that slightly disrupts the acid–base equilibrium is at once corrected by biochemical buffering systems operating in both the extracellular and intracellular compartments. Then, as indicated above, come into play the homeostatic systems involved in the regulation of pulmonary ventilation and urinary acid excretion via modulation of the renal tubular reabsorption or 'reclamation' of filtered bicarbonates and of proton secretion(5).
Over the last two decades, tremendous progress has been achieved in understanding the cellular and molecular mechanisms involved in renal tubular acidification (see for reviews Weiner & Hamm(16), Hamm et al. (17), Koeppen(18) and Weiner & Verlander(19)). Nevertheless, the fundamental concepts elucidated several decades ago on the overall renal control of extracellular proton homeostasis remain valid.
Homeostasis is defined as the stabilisation of the various physiological constants of the 'internal environment'. It has played an essential part in the evolution of life, from the most elementary unicellular organism to Homo sapiens, both in its phylogenetic and ontogenetic trajectories. Bearing in mind the capacity of physiological systems to adapt in response to environmental changes, homeostasis provides a scientific explanation for the basic mechanism of biological evolution(1).
Homeostasis includes the maintenance of a constant extracellular concentration of protons. Extracellular levels of other ions such as Na, K, Ca and inorganic phosphate are also barely affected by fluctuations in their respective nutritional intakes, unless their variations are very large in quantity and extend over prolonged periods.
That diet alters urinary acidity had already been demonstrated in the nineteenth century by Bernard(20) in his fundamental experiments on rabbits. By substituting cold boiled beef for their usual dietary regimen (consisting essentially of grass), cloudy, alkaline urine became clear and acidic, like the urine of carnivores(20). For this eminent physiologist, whose major contribution was to the elucidation of the homeostasis of the internal environment, these experiments carried out on rabbits represented a particularly cogent example of functional adaptation to environmental variations(20). The urinary acidity changes observed in response to food substitution are particularly relevant to the considerations discussed below.
A century after Bernard's(20) observations in rabbits, Relman and his colleagues(21–24) in Boston carried out a series of classical experiments with the objective of establishing, via quantitative data, the vital role of the kidney in acid–base balance. First, in healthy human subjects, i.e. those with normal renal function, Relman et al. (21) demonstrated that acid urinary excretion perfectly counterbalanced the net production of non-volatile acid. These experiments showed that the regulation mechanisms for the proton balance were indeed functioning. They signified that, in the absence of renal insufficiency, there was no argument for the involvement of organs other than the kidney in the maintenance of the homeostasis of non-volatile acids.
They then applied their technique to patients suffering from acidosis through chronic renal insufficiency(22, 24). In these studies, carried out on a small number of patients with a pathologically decreased but stable serum level of bicarbonates, their method of calculation indicated a positive balance of protons(22, 24). This led to the hypothesis that the quantity of acid retained in the body, indirectly estimated and not measured, was neutralised by the release of bicarbonates by the dissolution of bone mineral(22, 24). (Bone mineral is not pure hydroxyapatite. The apatite crystals contain impurities, most notably carbonate (CO32 −) in place of the phosphate group. The concentration of carbonate (4–6 %) makes bone mineral similar to a carbonate apatite. Other documented substitutions are K, Mg, Sr and Na in place of the Ca ions, and Cl and F in place of the hydroxyl groups. These impurities reduce the crystallinity and solubility of the apatite(25).)
In order to document this hypothetical bone mobilisation of bicarbonates, the Relman team carried out an initial study on five normal subjects(23). The administration of large doses of NH4Cl, drastically decreasing the blood level of bicarbonates from 26·5 to 18·8 mEq/l, was associated with a negative Ca balance, attributed to the mobilisation of calcium carbonate of skeletal origin(23). This interpretation was therefore based on the measurement of a decreased but stable level of bicarbonates, whereas during the same period, the estimate of acid balance indicated a progressive accumulation of protons(23). The negative Ca balance was due to an increase in urinary losses, the intestinal Ca absorption being unchanged(23). The change in the rate of urinary Ca excretion was therefore interpreted as a consequence of the mobilisation of Ca from the bones, associated with the release of buffer substances due to the dissolution of bone mineral in the presence of severe metabolic acidosis(23). The authors did not consider the possibility that the mobilisation of bone Ca might be secondary to an effect on the renal tubular reabsorption of Ca. In several subsequent studies, it turned out that acidosis is a factor that considerably inhibits the tubular reabsorption of Ca(26). Consequently, the mobilisation of bone Ca observed in these earlier experiments(23) may therefore actually represent a secondary phenomenon, compensating for the tendency towards hypocalcaemia rather than being the cause of the negative Ca balance(26).
In a subsequent study from the same group, Ca balance was determined in eight patients suffering from severe renal insufficiency(24). In the majority of these patients, there were signs of osteodystrophy including generalised skeletal demineralisation and radiological evidence of secondary hyperparathyroidism as expressed at the phalanges by the presence of sub-periosteal resorption(24). Administration of NaHCO3, causing an increase in the concentration of serum bicarbonates from 18·7 to 27·4 mEq/l and thereby correcting metabolic acidosis, was associated with modest improvement in the negative Ca balance, from − 5·3 to − 1·5 mEq/d. Moreover, this correction was essentially due to a decrease in the faecal excretion of Ca, the urinary excretion being considerably reduced in these patients(24).
The three above-mentioned studies, two conducted in patients with chronic renal insufficiency(22, 24) and one in normal subjects rendered severely acidotic through the administration of NH4Cl(23), are the basis of the hypothesis that bone mineral plays an important part in whole-body acid–base balance. This role would rely on the mobilisation of alkaline ions from the bone, thereby offsetting the excess of acid. This hypothesis is still being considered as a well-established scientific fact. The putative bone buffer mobilisation would be operational not only in the case of severe renal insufficiency, but also in the absence of any pathology, affecting the respiratory and/or renal regulatory systems involved in the maintenance of acid–base balance.
According to this hypothesis, the 'Western diet', in particular, would be a risk factor for osteoporosis, as it may supply an excess of protons that the pulmonary and renal systems would no longer be in a position to eliminate and which, therefore, would require the mobilisation of calcium bicarbonate from the bone tissue. However, it has been demonstrated that, in subjects in good health, blood pH and serum level of bicarbonates are not altered following dietary manipulations that induce alterations in urinary proton excretion, such as quantitative variations in the protein intake or qualitative differences in the diet, when comparing omnivorous and vegetarian subjects(27–29). In the absence of studies demonstrating the existence of an acid–base imbalance in the extracellular fluid, the notion of a 'latent' metabolic acidosis state has been put forward(7). This expression appears to be a misuse of language. The term 'latent' in a medical context is used to describe a state during which a clearly identified pathological disturbance or a pathogenic agent of a disease is detectable but remains inactive. A good example is the varicella zoster virus that remains latent after the initial bout of chicken pox has ended. When the virus becomes reactivated, usually several decades later, it causes herpes zoster. However, this phenomenon does not apply to the putative relationship between metabolic acidosis, the incriminated state of nutritional origin and osteoporosis. Therefore, the systemic acidosis of pure dietary origin remains a hypothesis that has not been scientifically demonstrated but which, in a certain number of publications (see below), is considered to be a proven pathophysiological mechanism leading to osteoporosis.
Even the hypothesis that bone is very important in maintaining stable serum HCO3− in established chronic metabolic acidosis has been challenged on the grounds of both theory and experimental data(30–32).
Even if one admits that in the experiments conducted in patients suffering from acidosis due to chronic renal insufficiency(22, 24), the stability of low serum bicarbonates would be the consequence of some alkali mobilisation from an endogenous source, the origin cannot be the bone mineral(30–32). Indeed, the quantity of buffering substances released from the bone would be largely insufficient to neutralise the acid assumed to have accumulated in the course of years when chronic renal insufficiency has been developing(30–32). It was estimated that about 50 % of bone mineral would have to be dissolved over approximately 1·8 years in order to achieve such an acid neutralisation(30–32). In other words, calculation based on the total Ca and alkali content in the skeleton indicates that with a supposed proton retention of 12–19 mEq daily in chronic renal acidosis(22, 24), it would take 3·6 years for the bone alkali store to be exhausted in order to buffer this amount of acid(32).
Thus, a quantitative estimate of the bone alkali content rules out that mobilisation of apatite mineral would be implicated in the maintenance of the low serum level of bicarbonates observed in the metabolic acidosis of chronic renal insufficiency.
A re-evaluation of the various components of acid–base balance(32, 33) made highly questionable the hypothesis that bone alkali mobilisation is an important process in maintaining a stable low level of serum bicarbonate in chronic metabolic acidosis(30–32). Important technical progress has made possible the determination of the net gastrointestinal absorption of alkali, applying a method that avoids imprecise measurements of the quantities consumed and excreted in the faeces(30–32). With the use of this technique, as well as taking into account the urinary excretion of organic cations and anions (see below), the acid–base balance appeared to be neutral in end-stage renal disease patients(33, 34). Consequently, with no excess of protons to be neutralised, there was no reason to invoke the mobilisation of alkali from the bone tissue in chronic renal insufficiency with stable metabolic acidosis.
Thus, a technical error, corresponding either to an underestimate of the net quantity of acid excreted, or to an overestimate of the net acid production, has perpetuated the incorrect concept that bone mineral plays a substantial part in acid–base balance in patients suffering from chronic renal acidosis. This incorrect concept does not mean that the acidosis generated by severe chronic kidney disease would not contribute to renal osteodystrophy. Nevertheless, other mechanisms probably play a more important part than acidosis per se in the deterioration of bone integrity in the case of severe chronic renal failure (for a review, see Hruska & Mathew(35)).
The effects of metabolic acidosis on the skeleton were examined both in vitro and in vivo in animal experiments(36–40). The results of these studies have been interpreted as supporting the hypothesis of an acid-buffering role of bone mineral. They are considered as experimental evidence in favour of the putative causal relationship between the so-called 'Western diet' and the prevalence of osteoporosis in the general population(4, 7, 41–44).
Furthermore, these observations, whether on isolated bone cells or on rodents(36, 37, 38–40), taken together with the fact that food intake modifies the degree of acidification of urine, as already demonstrated by Bernard(20) in the mid-nineteenth century, provided the rationale for exploring whether there would be a possible relationship between protein intake and osteoporosis, and, particularly, whether protein from animal v. vegetable sources would be more detrimental to bone health. To this end, many epidemiological studies have been published in the course of the last 16 years(45–56). Several of these reports appear to present some methodological flaws. Examples include the following: the age of the included subjects (varying between 35 and 74 years); the absence of an analytical distinction between sex; the inclusion of both pre-menopausal and postmenopausal women; the scarce or rather poor estimation of physical activity; the non-appreciation of the risk of falls; the variable levels of protein intake, often with average consumption above the recommended nutritional intake, therefore limiting the impact of protein malnutrition. In such disparate clinical conditions, it seems questionable to draw a synthesis from these studies by calculating an average relative risk with regard to the development of low bone mineral density (BMD)/content and/or fracture risk.
Furthermore, in some reports testing the a priori hypothesis that acidic urinary excretion (particularly when positively related to protein intakes) would reflect metabolic acidosis and thereby should be associated with poor bone health, the data were a posteriori equivocally handled in favour of the postulated assumption. Thus, when the whole cohort did not show any associated relationship, further analysis focused on subgroups as computed by cross-tabulation combining highest protein with lowest Ca intakes(53), or on subjects with a history of fracture exclusively(54), or still on participants with high, but not with low urinary acid excretion(57).
Starting from the hypothesis that the quantity of residual acid in the diet would influence the bone integrity of subjects otherwise in good health, several methods were proposed, based on studies conducted in the context of chronic renal insufficiency. First, it should be specified that measuring the pH of foods does not reflect the acid or alkali load they provide to the body. For example, orange juice has a low pH, by virtue of its high citric acid content, whereas once it has been ingested, it adds an alkali load to the body. Sulphurous amino acids (R-S) are neutral, but add acid loads once they have been metabolised, the reaction being:
$$\begin{eqnarray} R\hyphen S\rightarrow CO_{2} + urea + H_{2}SO_{4} \end{eqnarray}$$
Foods contain numerous chemical substances. Their absorption depends not only on the type of substances ingested, but also on interactions with gastric acid and other nutrients in simultaneously ingested foods. Therefore, it is almost impossible to predict the impact of food ingestion on the regulation of acid–base balance(32).
Moreover, since the intestinal absorption of the acid or alkali loads of food is incomplete, it is still necessary to be able to measure their quantity when excreted in the faeces. Taking into account both the experimental and analytical difficulties associated with such measurements, a simplified method has been developed and validated among subjects with chronic renal acidosis(31, 32–34, 58). According to this method, in the steady state, the total amount of inorganic cations $$(Na^{ + } + K^{ + } + Ca^{2 + } + Mg^{2 + }) $$ minus the total amount of anions $$(Cl^{ - } + P^{1\cdot 8 - }) $$ measured in the urine over 24 h can be used to estimate the net gastrointestinal absorption of alkalis. This measurement has the advantage of also including any other source of alkalis translocated into the extracellular environment, hypothetically including those from the bone tissue(31).
The principle according to which, at the steady state, the quantity of electrolytes excreted in the urine equals their quantity absorbed by the intestine has led to the development of mathematical models in order to estimate the relationship between food intake and net renal acid excretion (NAE)(59). NAE includes the daily urinary excretion of both inorganic and organic acids. This measurement provides an estimate of net endogenous acid production (NEAP)(60). The analytical difficulty relating to the measurement of urinary organic acids (OA), which include citric, lactic, oxalic, malic and succinic acids, as well as glutamic and aspartic amino acids, has been circumvented by an estimate derived from the body surface. The equation used is:
$$\begin{eqnarray} OA\,(mEq/d) = body\,surface\times (41/1\cdot 73), \end{eqnarray}$$
in which the value 41 corresponds to the median daily urinary excretion of OA for an average body surface of 1·73 m2 among subjects in good health(60, 61). This anthropometric estimate of OA is included in the calculation of the potential renal acid load (PRAL) of foods(62).
This calculation avoids the direct measurement of NAE, which is already an indirect measurement in itself of the NEAP. The PRAL can be estimated relatively easily from dietary studies, using weekly diaries or regular questionnaires, in which the quantities ingested are analysed according to nutritional composition tables. The nutrients taken into account for the PRAL calculation are: (phosphorus+protein) − (K+Ca+Mg).
The estimate of endogenous acid production has been further simplified by considering only protein and K intakes(63). An analysis of about twenty different diets followed by 141 subjects aged 17–73 years showed a coefficient of correlation (R 2) of 0·36 (P= 0·006) with a positive slope between protein intake and renal net acid excretion (RNAE, taken as a NEAP index), whereas it was 0·14, with a negative slope, for K intake(63). By the regression of the protein:K ratio, the R 2 became 0·72 (P< 0·001)(63). The use of this simple ratio estimates the acid load of foods according to the following equation(63):
$$\begin{eqnarray} RNAE\,(mEq/d) = - 10\cdot 2 + 54\cdot 5\,(protein\,(g/d)/K\,(mEq/d)). \end{eqnarray}$$
Physiologically, the meaning of the protein:K ratio remains obscure. Indeed, K per se cannot be considered as an alkalinising ion, since hyperkalaemic states are usually the generator of acidosis and not of metabolic alkalosis(6).
Of note, the development of a tool enabling the estimation of the PRAL of foods was aimed at modifying the urinary pH by dietetic means, particularly in the context of preventing recurrent urinary lithiasis(62). Thus, taking into account the differences in pH-dependent mineral solubility, the nutritional approach for preventing the recurrence of calcium phosphate or uric acid lithiasis, for example, has consisted in promoting acidification or alkalinisation of urine, respectively (see for a review Grases et al. (64) and Moe et al. (65)).
Over the last two decades, several reports have considered the relationship between the Ca economy and bone metabolism and K intake from foods or from the administration of potassium bicarbonate or citrate salts(42, 66–75). In the context of osteoporosis, human intervention studies have been designed to test whether the administration of alkalinising salts may favourably affect Ca and bone metabolism and therefore eventually be developed as anti-osteoporotic therapy(66, 67, 69–75). The results obtained by the end of relatively short time interventions suggested that taking alkalinising salts may transiently reduce bone turnover markers, and/or increase the balance of bone health, and thus lead to '…tipping the scales in favour of potassium-rich, bicarbonate-rich foods'(42). However, prolonged randomised studies did not confirm such a positive influence on Ca economy and bone loss prevention(72, 73). Decreased intestinal Ca absorption can explain reduced calciuria (UCa), with K salts yielding no significant net change in Ca balance(70, 73). Furthermore, in terms of skeletal health, in a 2-year randomised placebo-controlled trial in healthy postmenopausal women aged 55–65 years, potassium citrate administered in two doses (moderate: 18·5 mEq/d and high: 55·5 mEq/d) had no persistent effect on biochemical markers of bone remodelling measured at regular intervals. In line with this negative assessment, the reduction in areal BMD observed at the end of the intervention did not slow down, despite an increase in urinary pH and excretion of K in the course of 2 years of treatment(72). In this trial, the consumption of additional fruits and vegetables (+300 g/d) increasing the urinary excretion of K neither reduced bone turnover nor prevented areal BMD decline when compared with the placebo group(72). As reported in short-term studies, a temporary reduction in bone markers was observed 4–6 weeks after the start of the treatment(72). In other words, the classical study supporting the 'benefits' of nutritional alkalinisation for bone health(66) was not confirmed by a long-term clinical trial, not only measuring bone remodelling, but also bone loss following the menopause, at two skeletal sites of extreme importance in the risk of osteoporotic fractures – spine and proximal femur(72).
Despite this negative evidence from a well-designed clinical trial(72) and long-term preclinical investigations showing no relationship between urinary acid excretion and either bone status (density and strength) or remodelling(76), the idea that taking bicarbonates or alkaline K salts would be beneficial to the Ca economy and might result in better bone health and thereby prevent osteoporotic fractures continues to generate reports aimed at demonstrating such a therapeutical possibility(43, 57, 74, 77).
In the context of osteoporosis prevention in postmenopausal women and the elderly, modification of dietary habits could be plausible so long as long-term efficacy can be clearly demonstrated. In contrast, the daily consumption of alkaline salt preparations over several decades appears to be hazardous in the absence of an evaluation of possible long-term toxicity. For example, the risk of enhancing vascular calcifications cannot be ruled out, particularly when alkaline salts are combined with Ca and vitamin D supplementation. In the study by Jehle et al. (71), the lumbar spine BMD difference without a consistent change in bone remodelling markers between the potassium citrate and potassium chloride groups could, as suggested by the authors, be fully attributed to the enhanced non-cellular matrix mineralisation and thus be largely independent of bone cell-mediated events. Whether such Ca deposition in soft tissues, resulting from the consumption of alkaline salt supplements and an increased supply of Ca–vitamin D, could also occur in the cardiovascular system(78, 79) is unclear and is a risk that could overbalance the small and inconsistent benefit over placebo observed on bone integrity with alkaline supplements after 1 or 2 years of intervention(71, 72, 80).
Recent reports have not sustained the existence of a pathophysiological mechanism linking the consumption of some nutrients, particularly animal protein, to the induction of a biologically significant metabolic acidosis that would result in a negative Ca balance, bone loss and eventually osteoporotic fracture.
A first meta-analysis including twenty-five clinical trials, and adhering to rigorous pre-defined quality criteria, focused on the association between NAE and UCa(81). The analysed trials consisted in nutritional treatment and were carried out on healthy subjects in order to test the effect of either two types of food (meat v. soya), or certain nutrients (quantity of protein or dairy protein v. soya protein), or even acidifying (NH4Cl) or alkalinising (citrate, sodium bicarbonate or K) salt supplements. A significant linear relationship was found between net acid excretion and Ca excretion for both acidic and alkaline urine(81). Whether this increase in UCa when associated with net acid excretion would correspond to a decrease in Ca balance was examined in another meta-analysis(82). The included studies had all employed stringent methods to measure Ca balance and bone metabolism in relation to changes in NAE(82). The treatments were carried out on adult subjects in good health and consisted of modifications of protein intake, in terms of quantity or quality(82). Despite an increase in UCa in response to the nutritional treatment, Ca balance, as well as bone resorption evaluated by measuring the type I collagen N-telopeptide, did not show any correlation with the acid load of the dietary regimens tested(82). This meta-analysis did not suggest that protein-induced UCa associated with increased NAE would exert a negative impact on bone health, leading to osteoporosis in the long term. Therefore, it does not argue in favour of the theory advocating alkaline diets.
Furthermore, two other recent original reports did not sustain the hypothesis that a high dietary acid load might be detrimental to bone integrity. In the Framingham Osteoporosis Study, dietary acid load, estimated by the NEAP and PRAL, was not associated with BMD at any skeletal sites among 1069 'Original' and 2919 'Offspring' cohort participants(83). A possible exception was in older men with a trend between the NEAP and the femoral neck but not lumbar spine BMD, whereas no association was found with PRAL(83). Moreover, there was no interaction between either the NEAP or PRAL and total Ca intake(83). Thus, this study did not support the hypothesis that a high dietary acid load combined with a relatively low Ca intake might accelerate bone loss and increase the risk of fragility fracture(83). Another report was quite consistent with the detailed analysis of the data from the two Framingham generation cohorts(83). Indeed, no apparent relationship was found between urinary pH or urinary acid excretion and either the change in lumbar or femoral BMD or in the incidence of fractures after 5 years of monitoring including approximately 6800 person-years (age at baseline: approximately 59 years; female sex: 70 %) in a prospective investigation(9).
Another recent and comprehensive review reported on a systematic search of the published literature for randomised intervention trials, prospective cohort studies and meta-analysis of the acid-ash or acid–base hypothesis in relation to bone-related outcomes. In these studies, the dietary acid load was altered, or an alkaline diet or alkaline salts were provided to healthy human adults(10). The objective of this systematic review was to evaluate the relationship between the dietary acid load and osteoporosis using Hill's epidemiological criteria of causality(84). It was concluded that a causal association between the dietary acid load and osteoporotic bone disease is not supported by evidence, nor that an alkaline diet favourably influences bone health(10). Furthermore, assuming that fruit and vegetables are beneficial to bone health, such a positive influence would be mediated by mechanisms other than those related to their alkalinising potential, as experimentally demonstrated several years ago(85).
The bone data from two independent long-term randomised clinical trials testing K alkali supplements against placebo in healthy postmenopausal women(69, 72) have been analysed in one single publication(80). This analysis clearly shows, after 2 years of intervention, that K alkali treatment does not alter BMD changes at both lumbar spine and hip levels and has no effect on markers of bone resorption(80). Therefore, the previously reported long-term persistence of the urine Ca-lowering effect of potassium bicarbonate(69) was not associated with a significant benefit in terms of postmenopausal osteoporosis prevention(80). Likewise, both the greater spinal or hip BMD and the lower bone resorption markers, which were found to be associated with reduced estimates of NEAP and higher dietary K intakes in cross-sectional population studies of pre- and postmenopausal women(86, 87), were not confirmed in long-term randomised trials(72, 80). When compared with the null finding of these two trials(72, 80), a report, still in press(77), describes a positive effect of potassium citrate associated with supplements of calcium carbonate and vitamin D3 on BMD. This effect, recorded in a 2-year randomised trial carried out in healthy, elderly men and women studied together, remains to be mechanistically explained since it was observed, as in the two above-mentioned studies(72, 80), in the absence of any persistent reduction in bone resorption markers(77).
The dietary acid load hypothesis also postulates that increasing the urinary excretion of phosphate, considered as an 'acidic' ion, enhances UCa and contributes to the loss and fragility of bones with ageing(59, 88, 89). In sharp contrast with this hypothesis but in full agreement with physiological notions on the phosphate–Ca interaction(90), analysis of twelve human studies indicated that higher phosphate intakes were associated with decreased UCa and improved Ca balance(91).
It can be argued that the age-related decline in renal function, with its associated trend towards metabolic acidosis, would be sufficiently important to accelerate bone resorption while reducing bone formation(8), and thus could eventually explain the increased incidence of osteoporotic fractures with ageing. According to this putative pathophysiological mechanism, it would be justified to treat age-related osteoporosis by potassium bicarbonate administration or by appropriate modifications of the net dietary acid–base load(8, 66). However, there is no evidence that elderly patients with established osteoporosis, as documented by either spine or hip BMD T-score ≤ − 2·5 or by one prevalent vertebral fracture, have a lower glomerular filtration rate and more severe metabolic acidosis(92) compared with age- and sex-matched non-osteoporotic subjects(93, 94). Furthermore, in the National Health and Nutrition Examination Survey (NHANES) III population, a much larger number of subjects have osteoporosis/osteopenia(95) rather than a low glomerular filtration rate(93) or metabolic acidosis(94). In the analysis of the NHANES III survey, BMD was not found to be diminished by mild or moderate renal insufficiency(96). In fact, renal function itself was not independently associated with BMD, after taking into account sex, age and body weight(96). Furthermore, in this large survey, changes in serum bicarbonate were not apparent until chronic renal insufficiency, as estimated by the Cockcroft–Gault creatinine clearance, was ≤ 20 ml/min(97). Taken together, these results do not support the notion that age-related metabolic acidosis that would result from the deterioration of renal function could be pathophysiologically implicated in the marked increase in the prevalence of osteoporosis observed with ageing in the general population.
It is a well-established biological fact that the degree of urinary acidity varies according to the type of consumed foods. In the middle of the nineteenth century, Bernard(20) considered this variation to be an example of physiological control in the internal environment. A century later, experiments carried out among patients suffering from severe metabolic acidosis caused by renal insufficiency, or among healthy subjects made acidotic by administering NH4Cl, suggested the involvement of bone tissue in maintaining the acid–base balance. This hypothesis was later refuted on the basis of both theoretical and experimental arguments. Despite this rebuttal, the hypothesis was put forward that bone could play a buffering role, with the consideration that nutrients, particularly animal proteins with their acid load, could be a major cause of osteoporosis. Several recent human studies have shown that there is no relationship between nutritionally induced variations of urinary acid excretion and Ca balance, bone metabolism and the risk of osteoporotic fractures. Variations in human diets across a plausible range of intakes have been shown to have no effect on blood pH. Consistent with this lack of a mechanistic basis, long-term studies of alkalinising diets have shown no effect on the age-related change in bone fragility. Consequently, advocating the consumption of alkalinising foods or supplements and/or removing animal protein from the human diet is not justified by the evidence accumulated over the last several decades.
The author is grateful to Professor Robert P. Heaney, Creighton University, USA, for reading and providing helpful comments on the manuscript. The author received no financial support for writing the present review. There is no conflict of interest to disclose.
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\author{A. Stoimenow\footnotemark[1]\\[2mm] \small Ludwig-Maximilians University Munich, Mathematics\\ \small Institute, Theresienstra\ss e 39, 80333 M\"unchen, Germany,\\ \small e-mail: {\tt [email protected]},\\ \small WWW\footnotemark[2]\enspace: {\hbox{\tt http://www.informatik.hu-berlin.de/ \raisebox{-0.8ex}{\tt\raisebox{-0.8ex}{\tt\old@tl{}}{}}stoimeno}} }
{\def\fnsymbol{footnote}{\fnsymbol{footnote}} \footnotetext[1]{Supported by a DFG postdoc grant.} \footnotetext[2]{All my papers, including those referenced here, are available on my webpage or by sending me an inquiry to the email address I specified above. Hence, \em{please}, do not complain about some paper being non-available before trying these two options. Thank you.} }
\title{\large\bf \uppercase{The crossing number and maximal bridge length of a knot diagram}\\[4mm] {\it\small This is a preprint. I would be grateful for any comments and corrections!}}
\date{\large Current version: \today\ \ \ First version: \makedate{15}{2}{1999}}
\maketitle
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{\let\@noitemerr\relax \vskip-2.7em\kern0pt\begin{abstract} \noindent{\bf Abstract.}\enspace We give examples showing that Kidwell's inequality for the maximal degree of the Brandt-Lickorish-Millett-Ho polynomial is in general not sharp.\\ {\it Keywords:} Brandt-Lickorish-Millett-Ho polynomial, unknotting number, crossing number, bridge length\\ {\it AMS subject classification:} 57M25 \end{abstract} }
{\parskip0.2mm\tableofcontents}
\section{Introduction}
The $Q$ (or absolute) polynomial is a polynomial invariant in one variable $z$ of links (and in particular knots) in $S^3$ without orientation defined by being 1 on the unknot and the relation \begin{eqn}\label{Qrel} A_{1}+A_{-1}=z(A_{0}+A_{\infty})\,, \end{eqn} where $A_i$ are the $Q$ polynomials of links $K_i$ and $K_i$ ($i\in{\Bbb Z} \cup\{\infty\}$) possess diagrams equal except in one room, where an $i$-tangle (in the Conway \cite{Conway} sense) is inserted, see figure \reference{figtan}.
\begin{figure}
\caption{The Conway tangles.}
\label{figtan}
\end{figure}
It has been discovered in 1985 independently by Brandt, Lickorish and Millett \cite{BLM} and Ho \cite{Ho}. Several months after its discovery, Kauffman \cite{Kauffman} found a 2-variable polynomial $F(a,z)$, specializing to $Q$ by setting $a=1$.
In \cite{Kidwell}, Kidwell found a nice inequality for the maximal degree of the $Q$ polynomial.
\begin{theorem} (Kidwell) Let $D$ be a diagram of a knot (or link) $K$. Then \begin{eqn}\label{Qineq} \max\deg Q(K)\,\le\,c(D)-d(D)\,, \end{eqn} where $c(D)$ is the crossing number of $D$ and $d(D)$ its maximal bridge length, i.~e., the maximal number of consecutive crossing over- or underpasses. Moreover, if $D$ is alternating (i.~e. $d(D)=1$) and prime, then equality holds in \eqref{Qineq}. \end{theorem}
In \cite[problem 4, p.~560]{Morton} he asked whether \eqref{Qineq} always becomes equality when minimizing the r.h.s. over all diagrams $D$ of $K$. From the theorem it follows that this is true for alternating knots and also for those non-alternating knots $K$, where $\max\deg Q(K)=c(K)-2$ (here $c(K)$ denotes the crossing number of $K$). All non-alternating knots in Rolfsen's tables \cite{Rolfsen} have this property except for one~-- the Perko knot $10_{161}$ (and its obversed duplication $10_{162}$), where $\max\deg Q=6$. Hence, as quoted by Kidwell, this knot became a promising candidate for strict inequality in \eqref{Qineq}. To express ourselves more easily, we define
\begin{defi} Call a knot $K$ $Q$-maximal, if \eqref{Qineq} with the r.h.s. minimized over all diagrams $D$ of $K$ becomes equality. \end{defi}
The aim of this note is to show that indeed the Perko knot is not $Q$-maximal. We give several modifications of our arguments and examples showing how they can be applied to exhibit non-$Q$-maximality.
\section{\label{S2}Plane curves}
We start by some discussion on plane curves.
\begin{defi} A non-closed plane curve is a $C^1$ map $\gm\,:\,[0,1]\,\to\,{\Bbb R}^2$ with $\gm(0)\ne \gm(1)$ and only transverse self-intersections. $\gm$ carries a natural orientation. \end{defi}
\begin{exam} Here are some plane curves: \[ \@ifnextchar[{\@epsfs}{\@@epsfs}[6mm]{t-curve0}\quad \@ifnextchar[{\@epsfs}{\@@epsfs}[1.5cm]{t-curve1}\quad \@ifnextchar[{\@epsfs}{\@@epsfs}[1.5cm]{t-curve2} \] \end{exam}
In the following, whenever talking of plane curves we mean non-closed ones with orientation unless otherwise stated. However, in some cases it is possible to forget about orientation if it is irrelevant. It is convenient to identify $\gm$ with $\gm([0,1])$ wherever this causes no confusion. Whenever we want to emphasize that a line segment in a local picture starts with an endpoint, the endpoint will be depicted as a thickened dot.
\begin{defi} The crossing number $c(\gm)$ of a curve $\gm$ is the number of self-intersections (crossings). The curve $\gm$ with $c(\gm)=0$ is called trivial. \end{defi}
\begin{defi} We call a non-closed curve $\tl\gm$ similar to $\gm$ (and denote it by $\tl\gm\sim\gm$) if $\tl\gm(0)=\gm(0)$, $\tl\gm(1)=\gm(1)$ and $\tl\gm$ intersects $\gm$ only transversely. The distance $d(\gm)$ of $\gm$ call the number $\min\{\,\#(\gm\cap\tl\gm)\,:\,\tl\gm\sim\gm\,\}-2$ (the `$-2$' provided to ignore the coincidence of start- end endpoint). A curve $\tl\gm$ realizing this minimum is called minimal similar to $\gm$. Such $\tl\gm$ can be chosen to have no self-intersections. \end{defi}
\begin{exam} The curves $\@ifnextchar[{\@epsfs}{\@@epsfs}[2mm]{t-curve0}$ and $\@ifnextchar[{\@epsfs}{\@@epsfs}[0.6cm]{t-curve1}$ have $d=0$, while $d\left(\@ifnextchar[{\@epsfs}{\@@epsfs}[0.6cm]{t-curve2} \right)=1$. \end{exam}
\begin{defi} We call a plane curve $\gm$ composite, if there is a \em{closed} plane curve $\gm'$ (with no self-intersections) such that $\gm'$ intersects $\gm$ in exactly one point, transversely, and in both components of ${\Bbb R}^2\sm\gm'$ there are crossings of $\gm$. In this case $\gm'$ separates $\gm$ into two parts $\gm_1$ and $\gm_2$, which we call components of $\gm$. We write $\gm=\gm_1\#\gm_2$. Conversely, this can be used to define the operation `$\#$' (connected sum) of $\gm_1$ and $\gm_2$, wherever $\gm_1(1)$ or $\gm_2(0)$ are in the closure of the unbounded component of their complements. We call $\gm$ prime, if it is not composite. \end{defi}
\begin{exam} \[ \begin{array}{*{10}c} \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1} & \,=\, & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1.1} & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1} & \,=\, & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2.1} \\ \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1-} & \,=\, & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1.1-} & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2} & \,=\, & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2.2} \\ \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2} & \,=\, & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve1.2} & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2-} & \,=\, & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2.2-} \\ & & & & & \@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2-}&\#&\@ifnextchar[{\@epsfs}{\@@epsfs}[1.2cm]{t-curve2} & \,=\, & \mbox{\Large ???} \end{array} \] \end{exam} This example visualizes that the connected sum in general depends on the orientation of the summands and their order.
It is clear that the crossing number is additive under connected sum and it's a little exercise to verify that the distance is as well.
The path we are going to follow starts with the following
\begin{exer}\label{x1} Verify that the complement of a curve $\gm$ has $c(\gm)+1$ connected components, and conclude from that that $d(\gm)\le c(\gm)$. \end{exer}
\noindent Hint: One way to show that is to observe it when $\gm$ is trivial, to prove that you can obtain any $\gm$ from the trivial one by the four local moves
\begin{equation}\label{Reimov} \begin{array}{*6c} \diag{6mm}{1.5}{1.5} {\pictranslate{0.25 0}{
\braid{0.5 0.5}{1 1}
} \piccurve{0.25 1}{0.25 1.25}{0.45 1.5}{0.75 1.5} \pictranslate{1.5 0}{\picscale{-1 1}{
\piccurve{0.25 1}{0.25 1.25}{0.45 1.5}{0.75 1.5}
}} }\quad&\llra&\quad \diag{6mm}{1.5}{1.3} {\pictranslate{0.25 0}{ \picline{0 0}{0 0.8} \picline{1 0}{1 0.8} \piccirclearc{0.5 0.8}{0.5}{0 180} }}&\qquad \diag{6mm}{1}{2}{ \braid{0.5 0.5}{1 1} \braid{0.5 1.5}{1 1} }\quad&\llra&\quad \diag{6mm}{1}{2}{ \picline{0 0}{0 2} \picline{1 0}{1 2} } \\[8mm] \diag{6mm}{2.5}{1.5}{ \picline{0 0.4}{2.5 0.4} \picline{2 1.5}{0.5 0} \picline{2 0}{0.5 1.5} }\quad&\llra&\quad \diag{6mm}{2.5}{1.5}{ \picline{0 1.1}{2.5 1.1} \picline{2 1.5}{0.5 0} \picline{2 0}{0.5 1.5} }&\qquad \diag{6mm}{2}{2}{
\picline{1 2}{1 0}
\picline{0 1}{2 1}
\ptat{0 1} }\quad&\llra&\quad \diag{6mm}{2}{2}{
\picline{0 2}{0 0}
\picline{1 1}{2 1}
\ptat{1 1} } \end{array} \end{equation}
and to trace how the number of components and $c(\gm)$ change under these moves.
Our first aim is to improve slightly the inequality of the exercise.
\begin{lemma}\label{l1} $d(\gm)\le\max(2,c(\gm)-2)$ if $\gm$ prime. \end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Consider the first (and analogously last) crossing of $\gm$ (that is, the crossings passed as first and last by $\gm$). Denote by letters the connected components of the complement near these crossings: \[ \diag{1cm}{2}{2}{
\picline{1 2}{1 0}
\picvecline{0 1}{0.7 1}
\picline{0.7 1}{2 1}
\ptat{2 1}
\picputtext{1.5 1.5}{$a'$}
\picputtext{1.5 0.5}{$a'$}
\picputtext{0.5 1.5}{$b'$}
\picputtext{0.5 0.5}{$c'$} }\qquad \diag{1cm}{2}{2}{
\picline{1 2}{1 0}
\picvecline{0 1}{2 1}
\ptat{0 1}
\picputtext{1.5 1.5}{$b$}
\picputtext{1.5 0.5}{$c$}
\picputtext{0.5 1.5}{$a$}
\picputtext{0.5 0.5}{$a$} }\] First note that $a\not\in\{b,c\}$, else if w.l.o.g. $a=c$ there would be a closed curve $\gm'$ like \[ \diag{1cm}{2}{3}{
\pictranslate{0 1}{
\picline{1 2}{1 0}
\picvecline{0 1}{2 1}
\ptat{0 1}
\picputtext{1.5 0.5}{$a$}
\picputtext{0.5 1.5}{$a$}
\picputtext{0.5 0.5}{$a$}
}
\piclinedash{0.2 0.1}{0.25}
\piccircle{1 1}{0.6}{}
\picputtext{1.7 0.5}{$\gm'$} } \] intersecting $\gm$ in only one point and either $d(\gm)=0$ or $\gm$ is composite.
Then note that $b\ne c$, because else there would be a $\gm'$ like \[ \diag{1cm}{3}{2}{
\picline{1 2}{1 0}
\picvecline{0 1}{2 1}
\ptat{0 1}
\picputtext{1.5 1.5}{$b$}
\picputtext{1.5 0.5}{$b$}
\picputtext{0.5 1.5}{$a$}
\picputtext{0.5 0.5}{$a$}
\piclinedash{0.2 0.1}{0.25}
\piccircle{2 1}{0.6}{} } \] and the 2 curve segments could not be connected.
Therefore, $b\ne c$ and a minimal curve $\tl\gm\sim\gm$ would not need to pass through one of $b$ and $c$. The same holds for the last crossing of $\gm$. Hence we avoid $\tl\gm$ passing through at least two components of the complement of $\gm$, unless $\{b,c\} \cap\{b',c'\}\ne\vn$, but then $d(\gm)\le 2$.
\@mt{\Box}
For the Perko knot we need to work a little harder.
\begin{lemma}\label{l2} $d(\gm)\le\max(3,c(\gm)-3)$ if $\gm$ prime. \end{lemma}
\begin{defi} An isolated crossing of $\gm$ is a crossing $p$ such that there is a closed curve $\gm'$ with $\gm\cap\gm'=\{p\}$ and $\gm'$ intersects transversely both strands of $\gm$ intersecting at $p$. \end{defi}
\@ifnextchar[{\@proof}{\@proof[\unskip]}[of lemma] If $\gm$ has an isolated crossing, then one of the components of ${\Bbb R}^2\sm\gm$ has both and the other one has no one of the endpoints of $\gm$. Removing the part of $\gm$ in latter component and smoothing $\gm$ near $p$ reduces $c(\gm)$, but not $d(\gm)$, hence we may (say, by induction on $c(\gm)$) assume that $\gm$ has no isolated crossing.
Now consider a crossing of $\gm$ which is neither the first nor the last and denote the components near it by $l$, $m$, $n$ and $o$. \begin{eqn}\label{crossing} \diag{1cm}{2}{2}{
\picline{1 2}{1 0}
\picline{0 1}{2 1}
\picputtext{1.5 1.5}{$m$}
\picputtext{1.5 0.5}{$o$}
\picputtext{0.5 1.5}{$l$}
\picputtext{0.5 0.5}{$n$} } \end{eqn} Call 2 components neighbored if the intersection of the closures of their fragments in \eqref{crossing} is a line, and opposite if it is just the crossing itself.
By primality of $\gm$ any two neighbored components are distinct and by non-isolatedness of the crossing so are any two opposite components.
Hence $l$, $m$, $n$ and $o$ are pairwise distinct. Now call $b$, $b'$ the components which were found not to be passed by a minimal similar curve $\gm'$ to $\gm$ by the proof of lemma \reference{l1} and $a$, $a'$ the components denoted so in the same proof.
Then distinguish some cases.
\begin{caselist} \case\label{c1} No one of $b$, $b'$ is among $l$, $m$, $n$ and $o$. As $\gm'$ certainly does not pass through one of $l$, $m$, $n$ and $o$ you have a third component not passed by $\gm'$ and you are done.
\case Exactly one of $b$, $b'$, say $b$, is among $l$, $m$, $n$ and $o$. You would be done as in case \reference{c1} unless $\gm'$ does not pass only through $b$. Then you have a picture like this: \[ \diag{1cm}{3}{3}{
\pictranslate{0.5 0.5}{
\picline{1 2}{1 0}
\picline{0 1}{2 1}
\picputtext{1.5 1.5}{$m$}
\picputtext{1.5 0.5}{$b$}
\picputtext{0.5 1.5}{$l$}
\picputtext{0.5 0.5}{$n$}
}
\piclinedash{0.2 0.1}{0.25}
\picstroke{
\piccurve{3 3}{2.5 2.5}{1.5 2.5}{1.3 1.9}
\picveccurveto{1 1.5}{1 0.8}{0 0}
}
\picputtext{2.7 2.6}{$\gm'$} } \] Then $\gm'$ passes through $m$ and $n$, w.l.o.g. first through $m$ and then through $n$. But then $\gm'$ is not minimal because passing through $a$, $c$ and $m$ (and possible further components between $c$ and $m$) before passing through $n$ could be replaced by just passing through $a$ and $b$ to arrive to $n$. By this contradiction you are done here.
\case Both $b$, $b'$ are among $l$, $m$, $n$ and $o$. If $b$ and $b'$ are neighbored, then $d(\gm)\le 3$. \end{caselist}
Therefore, by this case distinction you are done unless at any crossing of $\gm$ except the first and the last one $b$ and $b'$ participate as opposite components. In particular $b$ participates as a neighboring component at any crossing of $\gm$ except possibly the last one. But then one can see that $\gm$ must look like \[ \diag{1cm}{6}{6}{
\pictranslate{3 3}{
\picmultigraphics[rt]{6}{360 7 :}{
\picline{2 23.5 polar}{2 75 polar}
\picarcangle{2 23.5 polar}{3 50 polar}{2 75 polar}{0.2}
}
\picline{2 23.5 polar}{2.5 420 360 7 : - polar}
\picline{2 -27.5 polar}{2.5 420 360 7 : - N polar}
\picline{2 23.5 polar}{2 -27.5 polar}
\ptat{2.5 420 360 7 : - polar}
\ptat{2.5 420 360 7 : - N polar}
\picputtext{0 0}{$b$}
\picputtext{-1.5 2.5}{$b'$}
} }\,. \] To see this, start with \[ \diag{5mm}{6}{6}{
\pictranslate{3 3}{
\picmultigraphics[rt]{7}{360 7 :}{
\picline{2.8 10 polar}{2.8 75 polar}
}
\picputtext{0 0}{$b$}
} } \] and then observe that there is only one way to reconnect the outer arcs not creating crossings (except possibly the last one) and having $b'$ as specified, and moreover it works only if the number of crossings is odd.
But for such a curve $d(\gm)=0$.
\@mt{\Box}
\section{Non-$Q$-maximal knots}
Now we are prepared to exhibit the Perko knot as non-$Q$-maximal.
\begin{theorem} If $D$ is a prime diagram of a knot $K$ of $c(D)$ crossings with a bridge of length $l=c(D)-k$, and $D$ has minimal crossing number among all such diagrams for fixed $k$, then $l\le \max(3,k-3)$, hence $c(D)\le k+\max(3,k-3)$. \end{theorem}
From this we have the desired example:
\begin{exam} If $10_{161}$ were $Q$-maximal, then we could pose $k=6$ in the theorem and would obtain a 9 crossing diagram of the knot, which does not exist. Hence $10_{161}$ is not $Q$-maximal. \end{exam}
\@ifnextchar[{\@proof}{\@proof[\unskip]}[of theorem] This is basically lemma \reference{l2}. Consider $\gm'$ to be the part of $D$ consisting of the maximal (length) bridge and $\gm$ consisting of the rest of (the solid line of) $D$ with signs of all crossings ignored. Then the freedom to move the bridge corresponds to the freedom to move $\gm'$.
\@mt{\Box}
Clearly, for many phenomena Rolfsen's tables up to 10 crossings are very limited. Verifying the list of non-alternating knots of at most 15 crossings provided by Thistlethwaite (see \cite{HTW}), I found 189 15 crossing knots for which $\max\deg Q\le 8$, and hence for which we would be done showing non-$Q$-maximality already with lemma \reference{l1} (or even exercise \reference{x1}). The most striking examples are the knots $15_{119574}$ and $15_{119873}$, where $\max\deg Q=4$ (although for both $\max\deg_z F(a,z)=11$, the coefficients of the 7 highest powers of $z$ cancel when setting $a=1$).
There are several ways how the theorem can be modified.
\begin{theorem} If $D$ is a diagram of a knot $K$ of $c(D)$ crossings with a bridge of length $l=c(D)-k$, then $u(K)\le \br{k/2}$, where $u(K)$ denotes the unknotting number of $K$. \end{theorem}
\@ifnextchar[{\@proof}{\@proof[\unskip]} By switching at most half of the crossings in $D$ not involved in the maximal bridge, the remaining part $\gm$ of the plane curve (this time \em{with} signs of the crossings) can be layered, i.~e., any crossing is passed the first time as over- and then as under- crossing or vice versa. But reinstalling the bridge to a layered $\gm$ gives a layered, and hence unknotted, diagram.
\@mt{\Box}
\begin{corr}\label{c2} If $u(K)>\br{\max\deg Q(K)/2}$, then $K$ is not $Q$-maximal.
\@mt{\Box} \end{corr}
Unfortunately, this corollary does not work to show non-$Q$-maximality of Perko's knot. Verifying both hand-sides of the inequality (using that the unknotting number of $10_{161}$ is 3, see \cite{pos,Kawamura, Tanaka}), we find that we just have equality. And that equality does not suffice is seen, e.~g., from all 8 closed positive braid knots in Rolfsen's tables (see \cite{Cromwell,Busk}) and more generally from the $(2,n)$-torus knots for $n$ odd.
For knots of $>10$ crossings unknotting numbers are not tabulated (anywhere I know of) and a general machinery does not exist to compute them, hence when wanting to extend the search space for examples applicable to corollary \reference{c2}, it makes sense to replace the unknotting number by lower bounds for it, which can be computed straightforwardly. I tried two such bounds. First we have the signature $\sg$.
\begin{corr}
If $|\sg(K)|>\max\deg Q(K)$, then $K$ is not $Q$-maximal.
\@mt{\Box} \end{corr}
Clearly, replacing $Q$ by lower bounds for it makes the condition more and more restrictive. However, when checking the above mentioned list of $189$ knots, I found that at least one of them satisfied strict inequality. It is $15_{166028}$, where $\sg=8$ and $\max\deg Q=7$.
\begin{figure}
\caption{Three non-$Q$-maximal knots.}
\label{fignm}
\end{figure}
Another possibility is to minorate $u(K)$ by the bound coming from the $Q$ polynomial itself.
\begin{corr}\label{c3} If $2\log_{-3}Q(-1)>\max\deg Q(K)$, then $K$ is not $Q$-maximal.
\@mt{\Box} \end{corr}
\begin{rem} The negative logarithm base may disturb the reader because such logarithms are usually not defined. But by work of Sakuma, Murakami, Nakanishi (see Theorem 8.4.8 (2) of \cite{Kawauchi}) and Lickorish and Millett \cite{LickMil} $Q(-1)$ is always a(n integral) power of $-3$ and this one it is referred to by this expression. \end{rem}
The inequality in corollary \reference{c3} looks rather bizarre. First, the inequality $u(K)\ge \log_{-3}Q(-1)$ is in general much less sharp than the one with the signature and secondly, the inequality in corollary \reference{c3} requires the coefficients of $Q$ to be of an average magnitude which grows exponentially with $\max\deg Q$. Thus, non-surprisingly, my quest for applicable examples among the non-alternating 15 and 16 crossing knots ended with no success in this case.
\begin{question} Is there a knot $K$ with $2\log_{-3}Q(-1)>\max\deg Q(K)$? \end{question}
I nevertheless gave the above inequality, because it is self-contained w.r.t. $Q$ and would decide about non-$Q$-maximality from $Q$ itself (without knowing anything else about the knot) and hence is, in some sense, also beautiful.
\section{A question on plane curves}
The machinery of the dependence of $d(\gm)$ on $c(\gm)$ we developed just as far as necessary for our knot theoretical context, but possibly it is also interesting in its own right.
\begin{question} Which is the best upper bound for $d(\gm)$ in terms of $c(\gm)$, i.~e. a function $f:\,{\Bbb N}\to{\Bbb N}$ such that for any $\gm$ we have $f(c(\gm))\ge d(\gm)$? \end{question}
We proved that for \em{prime} $\gm$ we can choose $f(n):=\max(3,n-3)$. Turning back to our example $\@ifnextchar[{\@epsfs}{\@@epsfs}[6mm]{t-curve2}$, where $c=2$ and $d=1$, and applying connected sums we find that we cannot choose $f(n)$ better than $\br{n/2}$ (similar prime examples as \[ \@ifnextchar[{\@epsfs}{\@@epsfs}[17mm]{t-curve3} \] exist as well). But possibly this indeed is the best upper bound. Unfortunately, proving it seems a matter of further tricky labour as in \S\reference{S2}.
{\small
}
\end{document} | arXiv |
Does the (relativistic) mass change? Why?
I learned recently that when an object moves with a velocity comparable to the velocity of light the (relativistic) mass changes. How does this alteration take place?
special-relativity speed-of-light mass reference-frames
Emilio Pisanty
BIJU SAHABIJU SAHA
$\begingroup$ I'm not comfortable enough on the topic of Special Relativity to give a full answer, but what you're referring to is not (necessarily) what is actually going on. When introducing the concept of an upper limit on velocity, many like to use the interpretation that the 'mass' of the object in motion is changing. Once you've studied the topic in a bit more depth, you see rather that it is better to talk about the relationship between energy and momentum, which does not require any mysterious changing masses. $\endgroup$ – Daniel Blay Aug 12 '12 at 11:50
$\begingroup$ duplicated by physics.stackexchange.com/q/71772 $\endgroup$ – Ben Crowell Sep 1 '13 at 14:22
$\begingroup$ This paper gives an approach I hadn't seen before: Sonego and Pin, "Deriving relativistic momentum and energy," arxiv.org/abs/physics/0402024 $\endgroup$ – Ben Crowell Sep 6 '13 at 23:25
In relativistic mechanics, there is a conserved quantity, relativistic momentum:
$\vec p = \gamma m \vec v$
$\gamma = \dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
where m is the invariant mass or less precisely, the rest mass.
Now, one interpretation is to identify $\gamma m$ as the relativistic mass, a speed dependent mass. But this is actually unnatural as it leads to the notion of directionally dependent inertia; objects having more inertia along the direction of motion.
In fact, it is more natural to identify $\gamma \vec v$ as the spatial components of a four-vector, the four-velocity $\mathbf{U}$.
Then, the four-momentum is just $m\mathbf{U}$ with spatial components $\vec p$:
$m\mathbf U = (\gamma m c, \gamma m \vec v)$
Alfred CentauriAlfred Centauri
$\begingroup$ +1 Good answer. Note that other quantities, such as temperature, can be anisotropic (directionally dependent); or at least working physicists in such disciplines as near-Earth space physics find it useful to talk of such things when dealing e.g. with the motions of ions in plasma bounded by the Earth's magnetic field. Nevertheless, the fact that it is simpler to speak of the four-velocity than an anisotropic mass is motivation enough to abandon the notion of relativistic mass. $\endgroup$ – Niel de Beaudrap Aug 12 '12 at 13:31
$\begingroup$ @NieldeBeaudrap, good comment and I'm considering editing my answer to address it. $\endgroup$ – Alfred Centauri Aug 12 '12 at 21:45
$\begingroup$ Not a good answer--- the relativistic mass is independent of direction, it's only when you take it out of the derivative, and interpret m as the ratio of F to a, rather than as the ratio of p to m, that the direction dependent business starts. So you shouldn't say that $\gamma m$ is a directional mass, or a transverse mass, because the generalization is of $p=mv$ not $F=ma$. Otherwise fine. $\endgroup$ – Ron Maimon Aug 13 '12 at 4:31
$\begingroup$ @RonMaimon, I didn't write "$\gamma m$ is a directional mass", I wrote "it leads to the notion of directionally dependent inertia". $\endgroup$ – Alfred Centauri Aug 13 '12 at 11:01
$\begingroup$ This is helpful because it points out to the OP that his/her question was phrased in obsolete terminology. However, it doesn't fundamentally address the question. The question is asking for why there is a factor of $\gamma$ involved, regardless of whether or not we group the factors of $p=(m\gamma)v$ and refer to $m\gamma$ as mass. $\endgroup$ – Ben Crowell Sep 1 '13 at 16:36
Here is a (still rather long) sketch of Einstein's original development of the relativistic kinetic energy, from his celebrated 1905 paper "On the Electrodynamics of Moving Bodies" linked to in Ben Crowell's answer. This answer's approach is also motivated by a paper linked by dmckee in chat.
Having noted the inconsistency of Maxwell's electrodynamics with standard Newtonian mechanics, Einstein offers up his principles for a new dynamics:
the dynamical laws should be the same in two coordinate systems in uniform relative motion (so there's no preferred system of "absolute rest")
the constancy of the speed of light $c$.
Note that the first is actually already satisfied by Newtonian dynamics; it's the second principle that's revolutionary.
From these principles he develops the Lorentz transformation, from which in turn flow (among many other things):
time dilation: a moving clock appears to run slow. From the point of view of a reference frame in which a clock, moving with velocity $v$, marks a time interval $\Delta \tau$, the reference frame's clock system records a longer interval $\Delta t$: $$ \Delta t = \gamma \Delta \tau \quad \text{ where } \gamma = \frac{1}{\sqrt{1-\left( \frac{v}{c} \right)^2}} $$ (I have to note here that Einstein actually uses the symbol $\beta$ for what we call $\gamma$; since $\beta$ now means something completely different, this change in notation causes me no end of confusion. I wonder when the change occurred?)
the velocity addition formula, which for co-linear velocities $v$ and $w$ gives a resultant $V$: $$ V = \Phi(v,w) = \frac{v + w}{1 +\frac{vw}{c^2}} $$ If $w << v$, we can write (in the limit $w \rightarrow 0$):
$$ V = v + dv = v + \phi(v) w \quad \text{ where } \phi(v) = \left. \frac{\partial \Phi}{\partial w} \right|_{w=0} = \frac{1}{\gamma ^2}$$
Note that Newtonian dynamics is recovered by setting $\phi=1$, which amounts to $\gamma=1$.
Einstein next demonstrates that Maxwell's equations already satisfy both his principles, and that the electric field component $E$ in the direction of motion of a moving frame is the same in both moving and stationary frames.
With all that as preparation, it's time for the main event: consider a charge $q$ accelerated from rest by a uniform electric field $E$ across a potential energy difference $W=qEl$. By conservation of energy, the final kinetic energy $T$ of the charge will be $T=W$.
At any point in its motion, where the particle's instantaneous velocity is $v$ in the lab frame, one can establish a co-moving reference frame in which Newton's law applies (instantaneously) for the change in velocity $dw$ (in that frame): $$ dw = \frac{q}{m} E d \tau $$
Transforming this expression to the lab frame using the above results, one finds: $$ \frac{dv}{\phi(v)} = \frac{q}{m} E \frac{dt}{\gamma} \quad \text{ or } \quad dv = \frac{1}{\gamma^3} \frac{qE}{m} dt$$
Since the rate of energy change (the power) is: $$ \frac{dT}{dt} = -\frac{dW}{dt} = \frac{d}{dt} (qEx) = qEv $$ we find, for the kinetic energy of the accelerated charge: $$ T = \int_0^{t_f} qEv dt = m \int_0^{v_f} \gamma^3 v dv = mc^2 (\gamma - 1)$$ where the $\gamma$ in the result is evaluated at the final velocity $v_f$.
Note that in the Newtonian limit $\gamma=1$, the integral evaluates to the familiar $\frac{1}{2} mv_f^2$.
Art BrownArt Brown
$\begingroup$ Nice answer, +1. However, I've never been satisfied with the logical justification of the step in Einstein's original derivation where he assumes that the work-kinetic energy theorem holds without modification in relativity. $\endgroup$ – Ben Crowell Sep 3 '13 at 15:54
$\begingroup$ @Ben You said something similar to me the other day too, and while I commend the search for deeper meaning when-even and where-ever, this one is puzzling me a little bit. Do you know of a deeper reason for introducing the work-energy theorem in Newtonian mechanics that "well, it turns out to be useful"? $\endgroup$ – dmckee♦ Sep 11 '13 at 14:32
$\begingroup$ @dmckee: In Newtonian mechanics the work-energy theorem is a theorem, which can be proved from Newton's laws. That's completely different logically from assuming that the it remains valid without any change in form when we generalize to SR. $\endgroup$ – Ben Crowell Sep 11 '13 at 15:50
$\begingroup$ @Ben it is trivial to show that $\Delta W = \frac{1}{2} m \Delta (v^2)$, but that isn't the magic. The magic is that this is a useful thing to know. That the energy feeds into a conservation rule. Without going the Noterian route, I find it hard to justify that the energy should matter without just going forward (show that it comes up in conservative fields and other places so that we can solve problems). Maybe that's just a sign of ignorance on my part, but that is why it doesn't bother me to start computing things with the theorem and seeing if it turns out to be useful in relativity. $\endgroup$ – dmckee♦ Sep 11 '13 at 16:22
$\begingroup$ @dmckee: In the broadest context, conservation of energy is indeed "magic" in the sense that we just play with it, patch it up as needed, add new forms of energy, etc., as needed in order to make it a valid law. But in a more restricted context, e.g., Newtonian gravity, conservation of energy is a theorem that follows from Newton's laws, and the work-kinetic energy theorem is a part of the machinery of that theorem. The full dynamics of SR was found purely deductively before any expts were available to confirm any aspect of it. If some portion of that deduction is fallacious, that's a flaw. $\endgroup$ – Ben Crowell Sep 11 '13 at 22:39
Let's start by assuming the postulates of special relativity given in Einstein 1905a. One of these is that $c$ is the same in all frames of reference. There are really two things we would like to do: (1) prove that the usual formulas from Newtonian mechanics no longer give a usable description of dynamics, and (2) find out how to modify those formulas.
Task #1 is pretty straightforward. For example, suppose we have an elastic, one-dimensional collision between objects $M$ and $m$, with $M \gg m$, in a frame of reference where $m$ is initially at rest and $M$ has initial velocity $v$. If we assume the Newtonian expressions for momentum and kinetic energy, then the result of such a collision is that $m$'s final velocity is $v'=2v$. In the case where $v=c/2$, this would cause $m$ to fly off at $v'=c$. But this contradicts Einstein's second postulate, because if it's possible for material objects to move at $c$, then it's possible for observers to move at $c$, but then in such an observer's frame of reference, a ray of light could be moving at zero speed.
We can also see qualitatively from this argument that inertia must increase at speeds comparable to $c$. For consistency with the postulates of relativity, the actual result of this collision must be $v'<c$. The mass $m$ is acting as though it has more than the expected resistance to the change in its state of motion. There are two equivalent ways of stating this: (a) we can say that $m$ increases with speed, or (b) we can modify the equations for energy and momentum while considering $m$ to be a constant. It doesn't fundamentally matter whether we choose a or b; it just amounts to reshuffling a certain correction factor in certain equations. Up until about 1950, a was more popular, but these days all physicists use b.
So now we have task #2, which is to quantitatively fix up the dynamical formulas in Newtonian mechanics so that they are relativistically correct. There are a lot of different ways to do this. The route Einstein originally took was to demonstrate equivalence of mass and energy (Einstein 1905b). The paper is pretty readable, but if you really want to continue with this approach and develop a full treatment of momentum, in my opinion it gets a little cumbersome. A more modern approach, demonstrated in Einstein 1935, is to think in terms of four-vectors. This approach allows for a pretty compact derivation, at the expense of some abstraction.
The kinematical consequences of the postulates in Einstein 1905a are summarized by the Lorentz transformation, which converts the time and space coordinates of an event $(t,x,y,z)$ into coordinates $(t',x',y',z')$ in another frame that is in motion relative to the first at a velocity $v$. It's not my purpose to rederive the Lorentz transformation here, so I'll just appeal to its properties as needed. This makes it natural to start talking about vectors $\textbf{r}$ and $\textbf{r}'$ in four dimensions. These are called four-vectors. We really have to throw away the old notion of a three-vector, because a three-vector like $(x,y,z)$ doesn't have any well-defined transformation properties; we can't tell what it would look like in another frame without knowing $t$.
Just as Newtonian mechanics has uniform rules for operating on displacement vectors, force vectors, momentum vectors, etc., we expect that the Lorentz transformation will be applicable to all the corresponding objects in relativity. You can take this as a postulate if you like.
The fundamental laws of physics are conservation laws, such as conservation of momentum. The above considerations tell us that in order to generalize conservation of momentum to relativity, we're going to have to make a four-vector out of the Newtonian three-momentum. If the law is reexpressed in terms of a four-vector, then the equation will automatically be valid regardless of what frame we're in, since both sides of the equation will transform identically.
The Lorentz transformation of a zero vector is always zero. This means that the momentum four-vector of a material object can't equal zero in the object's rest frame, since then it would be zero in all other frames as well. So for an object of mass $m$, let its momentum four-vector in its rest frame be $(f(m),0,0,0)$, where $f$ is some function that we need to determine, and $f$ can depend only on $m$ since there is no other property of the object that can be dynamically relevant here. Since conservation laws are additive, $f$ has to be $f(m)=km$ for some universal constant $k$. In sensible relativistic units where $c=1$, $k$ is unitless. Since we want $\textbf{p}=m\textbf{v}$ to hold for four-vectors so as to recover the appropriate Newtonian limit for massive bodies, and since $v_t=1$ in that limit, we need $k=1$.
Transforming this momentum four-vector into some other frame, we find that its timelike component is no longer $m$. It equals $m$ plus an expression whose low-velocity limit is the kinetic energy. We interpret this expression as the relativistic kinetic energy. We no longer have separate conservation of mass, only conservation of mass-plus-energy or "mass-energy," $E$.
The Lorentz transformation always preserves the norm of a vector $\textbf{r}$, defined by $r_t^2-r_x^2-r_y^2-r_z^2$. For a body of mass $m$, the norm of the momentum four-vector will always be $m^2$, regardless of what frame we're in. The result is
$$ m^2=E^2-p^2 \qquad ,$$
which is valid for both massive and massless particles. In the $m \ne 0$ case, one can then prove that $p=m\gamma v$. The mass $m$ is constant, which is the modern convention. In school textbooks that are still stuck in the 1940's, $m\gamma$ is referred to as the relativistic mass, $m$ as the rest mass.
Einstein, "On the electrodynamics of moving bodies," 1905; English translation at http://fourmilab.ch/etexts/einstein/specrel/www/
Einstein, "Does the inertia of a body depend upon its energy-content?," 1905; English translation at http://fourmilab.ch/etexts/einstein/E_mc2/www/
Einstein, "Elementary derivation of the equivalence of mass and energy," Bull. Amer. Math. Soc. 41 (1935), 223-230, http://www.ams.org/journals/bull/1935-41-04/S0002-9904-1935-06046-X/home.html
In Newtonian physics the mass of a particle of matter does not change . It is defined by
$F=ma$ , where $F$ is the force necessary to apply to this specific mass $m$ in order to accelerate it by an acceleration $a$.
When velocities approach the velocity of light, experiments have told us that the higher the velocity of the particle the more force must be applied for the same acceleration $a$.
The theory of special relativity addresses this behavior , and it has been validated again and again by experiments. From the link:
To an observer who is not accelerating, it appears as though the object's inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is in fact observed in particle accelerators, where each charged particle is accelerated by the electromagnetic force.
One can find the formula of the mass change in the above link.
Now there is no other answer to "why", then "because that is the way nature behaves".
anna vanna v
$\begingroup$ I updated the question slightly. $\endgroup$ – Qmechanic♦ Aug 12 '12 at 11:59
$\begingroup$ Why the downvote? $\endgroup$ – Abhimanyu Pallavi Sudhir Sep 1 '13 at 4:19
$\begingroup$ Now there is no other answer to "why", then "because that is the way nature behaves". Not true. The relativistic behavior of inertia and momentum follows logically from the postulates of special relativity. $\endgroup$ – Ben Crowell Sep 1 '13 at 16:34
$\begingroup$ @BenCrowell special relativity and its postulates was fitted to the fact that experimental data showed such a behavior. It is not the data that follows special relativity, special relativity is the mathematical description of what we have observed/measured in nature, a shorthand for all that data. $\endgroup$ – anna v Sep 1 '13 at 16:38
$\begingroup$ @BenCrowell The beauty of a successful/validated theory is that it hangs together consistently describing the data and predicting new behaviors. It would be invalidated if one datum was not fitted/predicted. It is trite, but a theory can never be proven correct, it is validated by data; it can be falsified by one wrong datum, i.e. behavior of nature. $\endgroup$ – anna v Sep 1 '13 at 16:44
when an object moves with a velocity comparable to the velocity of light the (relativistic) mass changes [...]
This premise appears mistaken.
When and while some specific object which is identified by some specific (intrinsic, proper, invariant) mass $m$ moves with some specific constant speed $v$ (relative to a suitable system of participants who are capable of evaluating this speed of this object, in comparison to the speed of light in vacuum $c_0$)
then the so-called "relativistic mass" of this object, in this trial, $m / \sqrt{ 1 - (v/c_0)^2 }$, does not change, but remains constant as well.
Instead, different trials may be considered, in which the same specific object of specific (intrinsic, proper, invariant) mass $m$ moves with different speeds such that its "relativistic mass" consequently differs from trial to trial.
(The history of the notion "relativistic mass" and its limited utility in comparison to the notion of "(intrinsic, proper, invariant) mass" has already been addressed in other answers.)
Additionally to Alfred Centauri's answer I can say, that mass
$m_{rel}=\gamma m$
is AUTOMATICALLY implies directional inertia, since it is not constant.
Any non-constant mass causes a propulsion force.
From definition of force
$F=\frac{dp}{dt}=\frac{dm_{rel}v}{dt}=v\frac{dm_{rel}}{dt}+m_{rel}\frac{dv}{dt}$
$F_{prop}=v\frac{dm_{rel}}{dt}$
It is zero for constant mass, but not zero for non-constant.
This is essential part of mechanics, and is used in rocket engineering.
DimsDims
Whether or not it's gaining mass depends on how one defines mass. The "relativistic mass" is simply the total energy, with a factor of c squared thrown in.
I recall reading that Einstein himself was against the idea of such a concept, and is quoted as saying that the only mass one should consider is the "rest mass". I would have to agree.
Relativistic mass is basically something that is only used when scientists are explaining things to non-scientists. The reason you can't reach the speed of light is not because an object is inherently changing, it's because of the relationship between relative speed and energy.
Bear in mind also, that, if we use relativistic mass, we are no longer justified in saying light is massless. Light has relativistic mass, because it has energy.
anotherguyanotherguy
This is a simple way to understand mass. We know Light (and EM in general) does not have a rest mass. But a standing wave or a trapped beam of light between two mirrors (as in lasers), does have a rest mass. So we conclude that rest mass is nothing more than a trapped/arrested momentum, or energy if you like since the two are derivable from each other. The trapping can be done either by walls like mirrors or a cavity wall, or can be done by trapping in a circular motion (without the need for walls). Thus we say here that mass and relativistic mass must be equivalent to an un-trapped momentum, whereas a rest mass is equivalent to a trapped/arrested momentum. Many examples confirm this picture. First you have the annihilation and creation experiments where rest mass becomes pure energy flux and visa versa. Then you have the case where the mass of a proton being much larger than the total rest masses of the internal constituents- as the excess rest mass comes from the huge momentum of constituents moving at relativistic speeds. By the same logic, the electron itself must be no more than a trapped momentum. This of course goes well with our knowledge of the electron.. as having a real mechanical spin (as per Einstein-de-Hass experiment), and also having an internal clock- the Zitterbewegung that is also related to the spin. Then you have the case of a double trapping.. where you have rest masses with high speeds trapped in a larger structure- as in the case of the nucleus. This again makes the mass of the nucleus larger than the sum of the rest masses of the internal components. But such difference starts becoming less and less as the structure becomes large. A hot matter for example, has a larger rest mass than a cold one, but the difference can't be measured- being very small. We also note that here we agree with Einstein formula E=m c^2, since in a standing wave you have double the kinetic energy.. that is E=2* .5 m v^2=m v^2=mc^2 as the speed in our case is that of light.
RiadRiad
Mass in physics is a mathematical construct, and mass of an object approaching $\infty $ as the speed of an object approaches $ c $ is a mathematical consequence of the postulates of Special Relativity.
ConstantineConstantine
$\begingroup$ The OP knows that it follows from the postulates of SR, but wants to know how. This doesn't address the question. $\endgroup$ – Ben Crowell Sep 1 '13 at 14:51
protected by Qmechanic♦ Sep 24 '13 at 18:44
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Maria Angela Ardinghelli
Maria Angela Ardinghelli (1730–1825) was an Italian translator, mathematician, physicist and noble,[1] mostly known as the Italian translator of the works of Stephen Hales, a Newtonian physiologist. She translated two of his works; Haemastaticks and Vegetable Staticks. Aside from Ardinghelli's historical invisibility, she managed to remain relevant without being shunned into social isolation or derision by sharing her works with specific audiences.
Maria Angela Ardinghelli
Portrait medallion by Jean-Jacques Caffieri (1755). Archives de l’Académie des Sciences, Paris. © Académie des Sciences—Institut de France
Born1730 (1730)
Naples
Died1825 (aged 94–95)
NationalityItalian
Known forExpert in mathematical physics, Italian translation of the works of Stephen Hales
Scientific career
FieldsMathematics, Physics
PatronsJean-Antoine Nollet
InfluencesPietro Della Torre and Vito Caravelli
Influenced
• Jean-Antoine Nollet
• Alexis Claude Clairaut
Background
Maria Angela Ardinghelli was born in Naples (Kingdom of Naples) into a noble family of Florentine origin. Having lost her brother during their childhood, Maria Angela thus became an only child. Her father turned to educating her, and by the age of fourteen she was fluent in Latin.[2] She studied philosophy and physical-mathematical sciences under the physicist and mathematician Giovanni Maria Della Torre and Vito Caravelli. She also studied English and French.
Ardinghelli was neither an aristocrat nor a member of ascendant middle class. Her family was from Florence, described as “one of the most distinguished and ancient of Italy”, in the sixteenth century. When the Medici family climbed into power in Tuscany the Ardinghelli family fled Tuscany for Naples.
As was obligatory for the aristocratic women of the time, Maria Angela was a literate poet and Latinist, as well as expert of mathematical physics. She belonged to the circle of the prince of Tarsia, founded in 1747, which, in intellectual circles in Naples, had the strongest association to Newton, experimental physics and electricity. The library and the laboratory of Tarsia was to be of much use to her.
Ardinghelli never wanted to leave Naples. She made it clear that she would never leave her family, rejecting marriage with French architect Julien Leroy and the possibility of becoming the scientific tutor to the royal princesses at Versailles. She stayed in Naples where she hosted many conversazioni as meeting points for traveling naturalist and corresponding with the Paris Academy of Science.
Maria Angela Ardinghelli had acted as an informal correspondent for the Paris Academy of Sciences. She had connected the scientific communities of Naples and France. When Maria Angela reached the apex of her popularity she devised a few strategies to maintain her anonymity, which she succeeded at. In spite of Ardinghelli's historical invisibility, she selectively chose from her works what she wanted visible to specific audiences in order to protect herself from social isolation.
Ardinghelli and Nollet
As a correspondent and member of the Paris Academy of Sciences Maria Angela was catapulted to fame by abbé Jean-Antoine Nollet. Nollet met Ardinghelli at conversazioni, hosted by her in Naples during his journey through Italy in 1749. Nollet, an acclaimed celebrity, published a volume on electricity in which he needed to defend his theories against those of Benjamin Franklin. Nollet wrote nine letters to nine different savants distinguished in the field of physics. The first letter was to Ardinghelli. In the letter he writes about her translation of Hales's Haemastaticks and writes: “very virtuous young lady, who in a short time has made a lot of progress in the field of physics.” This public declaration of esteem made Ardinghelli well known.
Accomplishments
Expert in mathematical physics, Ardinghelli's fame is mainly due to the translation of key works of the English physicist Stephen Hales Haemastaticks and Vegetable Staticks. She also performed scientific experiments inspired by Hales works. She was identified as an informal correspondent and cultural mediator for foreign scientist and naturalist traveling to Italy. Being a mediator opened a door and put her in the position to meet Jean-Antoine Nollet, whom appointed her to be an informal correspondent for the Paris Academy of Sciences. Working for the Paris Academy of Sciences had her connection the scientific communities of France and Naples. In Maria Angela's translations, she broadened herself to more than just the footnotes that typical translators confined themselves to. She opened herself in the dedication and in the "To the Reader" sections of her translations. In these sections, she opened herself up to the members of higher classes. She corresponded with leading scientists of the time, including, to name a few, the mathematician and astronomer and physicist Alexis Claude Clairaut and Jean-Antoine Nollet.
References
1. Ogilvie, Marilyn Bailey (1986). Women in science : antiquity through the nineteenth century : a biographical dictionary with annotated bibliography (3 print. ed.). Cambridge, Mass.: MIT Press. ISBN 978-0-262-15031-6.
2. Bertucci, Paola (June 2013). The Invisible Woman (PDF). The University of Chicago Press. p. 231.
• http://scienzaa2voci.unibo.it/biografie/67-ardinghelli-maria-angela
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| Wikipedia |
OSSperf – a lightweight solution for the performance evaluation of object-based cloud storage services
Christian Baun1,
Henry-Norbert Cocos1 &
Rosa-Maria Spanou1
This paper describes the development and implementation of a lightweight software solution, called OSSperf, which can be used to investigate the performance of object-based public cloud storage services like Amazon S3, Google Storage and Microsoft Azure Blob Storage, as well as private cloud re-implementations. Furthermore, this paper presents an explanation of the output of the tool and some lessons learned during the implementation.
A form of cloud services, which belong to the Infrastructure as a Service (IaaS) delivery model, is the group of object-based storage services. Examples for public cloud offerings, which belong to this kind of services, are the Amazon Simple Storage Service (S3) [1], the Google Cloud Storage (GCS) [2] and Microsoft Azure Blob Storage [3]. Furthermore, several free private cloud solutions exist, which re-implement the S3-functionality. Examples for service solutions, which implement the S3 API and are licensed according to a free software license, are Eucalyptus Walrus [4, 5], Ceph [6, 7], Nimbus Cumulus [8, 9], Fake S3 [10], Minio [11], Riak Cloud Storage [12], S3ninja [13], S3rver [14] and Scality S3 [15]. OpenStack Swift [16], which is the object storage component of the IaaS solution, provides a similar functionality, but implements the Swift API, which is quite similar to the S3 API.
In this work, different approaches and already existing solutions for the performance evaluation of object-based storage services are evaluated and a new solution, called OSSperf is developed. This new benchmark testing solution is intended to be lightweight, which means it causes only little effort for installation and usage. It shall support a wide number of different public and private cloud services with their different APIs. Additionally, it shall analyze the performance of the most important features of object-based storage services and in order to simulate scenarios of different degrees of utilization, the tool must provide a parallel operation mode for the upload and download of objects.
This paper is organized as follows. "Related work" section contains a discussion of related work and explains the motivation for the development of OSSperf.
In "Development and implementation of OSSperf" section, the design and implementation of the software and its functioning are discussed. In addition, this section provides a description of some challenges, we were facing during the development and implementation process.
"Analyzing the benchmark" section presents some benchmark results of OSSperf and explains how to analyze them.
Finally, "Conclusions" section presents conclusions and directions for future work.
Analyzing the performance of object-based storage services is a task, which has been addressed by several other works. First works on this topic were carried out and published shortly after the availability of the Amazon S3 service offering in 2006 (for the United States) and 2007 (for Europe), that became a blueprint for a large number of commercial competitors or free re-implementations. In the literature, several works cover the topic of measuring the performance of object-based storage services with the S3 interface and with the Microsoft Azure Blob Storage interface.
Garfinkel [17] evaluated in 2007 the throughput, which the Amazon S3 service offering can deliver via HTTP GET requests with objects of different sizes over several days from several locations by using a self-written client. The tool was implemented in C++ and used libcurl [18] for the interaction with the storage service. The focus of this work is the download performance of Amazon S3. Other operations like the upload performance are not investigated. Unfortunately, this tool has never been released by the author and the work does not investigate the performance and functionality of private cloud scenarios.
Palankar et al. [19] evaluated in 2008 the ability of Amazon S3 to provide storage to large-scale science projects from the perspective of cost, availability and performance. Among others, the authors evaluated the throughput (HTTP GET operations) which S3 can deliver in single-node and multi-node scenarios. They also measured the performance from different remote locations. No used tool has been released by the authors and the work does not mention the performance and functionality of private cloud scenarios.
Li et al. [20] analyzed the performance of the four public cloud service offerings Amazon S3, Microsoft Azure Blob Storage and Rackspace Cloud Files with their self developed Java software solution CloudCmp [21] for objects of 1 kB and 10 MB in size. The authors among others compare the scalability of the mentioned blob services by sending multiple concurrent operations and were able to make bottlenecks visible when uploading or downloading multiple objects of 10 MB in size.
Calder et al. [22] measured in 2011 the throughput (HTTP GET and HTTP PUT operations) of the Microsoft Azure Blob Storage service for objects of 1 kB and 4 MB in size. Unfortunately, the authors do not describe which tools they did use for their work and the authors do not mention the performance and functionality of further public cloud offerings or of private cloud scenarios.
Zheng et al. [23, 24] described in 2012 and 2013 the Cloud Object Storage Benchmark – COSBench [25], which is able to measure the performance of different object-based storage services. It is developed in Java and provides a web-based user interface and helpful documentation for users and developers. The tool is licensed according to the Apache 2.0 license and it supports the S3 API and the Swift API, but not the API of the Microsoft Azure Blob Storage service. COSBench can simulate different sorts of workload. It is among others possible to specify the number of workers, which interact with the storage service, and the read/write ratio of the access operations. This way, the software implements a parallel operation mode. The complexity of COSBench is also a drawback, because the installation and configuration require much effort.
Bessani et al. [26] analyzed in 2013 among others the required time to upload and download objects of 100 kB, 1 MB, and 10 MB in size from Amazon S3, Microsoft Azure Blob Storage and Rackspace Cloud Files from clients that were located on different parts of the globe. In their work, the authors describe DepSky [27], a software solution that can be used to create a virtual storage cloud by using a combination of diverse cloud service offerings in order to achieve better levels of availability, security, privacy and prevent the situation of a vendor lock-in. While the DepSky software has been released to the public by the authors, they did not publish a tool to carry out performance measurements of storage services so far.
McFarland [28] implemented in 2013 in the programming language Python two applications, called S3-perf, which make use of the boto [29] library to measure the download and upload data rate of the Amazon S3 service offering for different file object sizes. Those solutions offer only little functionality. They do only provide the option to measure the required time to execute upload and download operations sequentially. Parallel operations are not supported and also fundamental operations other than the upload and download objects are not considered. Furthermore, the solution does not support the Swift API or the API of the Microsoft Azure Blob Storage service and the software license is unclear.
Jens Hadlich implemented in 2015 an utility, called the S3 Performance Test Tool [30] in Java, which can be used to measure the required time to create and erase buckets, as well as for uploading and downloading files into Amazon S3 or S3-compatible object storage systems. The tool allows to upload and download objects in parallel. It is free software and licensed according to the MIT License. A drawback of the S3 Performance Test Tool is that it does not support the Swift API or the Microsoft Azure Blob Storage API and it does not calculate the achieved bandwidth during upload and download operations.
Land [31] analyzed in 2015 the performance of the public cloud object-based storage services Amazon S3, Google Cloud Storage and Microsoft Azure Blob Storage with files of different sizes by using the command line tools of the service providers and by mounting buckets of the services as file systems in user-space. Attaching the storage of an object-based storage service via a file systems in user-space driver is a quite special application, which is hardly comparable to the typical sort of interaction with such services. Additionally, this work does not analyze the performance and functionality of private cloud scenarios. The author uploaded for his work a file of 100 MB in size and a local git repository with several small files into the evaluated storage services and afterwards erased these files, but in contrast to our work, he did not investigate the performance of the single storage service related operations in detail.
Bjornson [32] measured in 2015 the latency - time to first byte (TTFB) and the throughput of the storage service offerings Amazon S3, Google Cloud Storage, and Microsoft Azure Blob Storage for objects ranging from 16 KB to 32 MB. The author used the Object Storage Benchmark, which is a part of the PerfKit Benchmarker [33] test suite. This tool is free software and licensed according to the Apache 2.0 license. The software supports not only multiple storage service interfaces, but also sequential and parallel operations. The author discovered, that the different public cloud services have different performance characteristics, which depend heavily on the object size. Thus, it is important for users and administrators to benchmark storage services with objects, that are comparable in size with the objects their web applications use. The work does not consider the typical storage service related operations in detail and it does not analyze the performance and functionality of private cloud scenarios. The PerfKit Benchmarker suite is a set of different benchmark applications to investigate different performance aspects of cloud Infrastructure services. The complexity of this collection is also a drawback, because the configuration and proper handling requires much effort.
In contrast to the related works in this section (see Table 1), we wanted to develop and implement a lightweight solution to analyze the performance of the most important storage service operations and not only of upload and download operations or of the latency. Furthermore we wanted to develop a tool which can interact not only with the Amazon S3 API, but also with the Swift API and the API of the Microsoft Azure Blob Storage service. Last but not least, the development of the tool aimed to create a solution which can be deployed with minimum effort and is simple to use.
Table 1 Software solutions to analyze the performance of object-based storage services
Development and implementation of OSSperf
The analysis of the existing benchmark testing solutions in "Related work" section resulted in the development of a new tool. The preconditions of developing a lightweight solution, which causes only little effort for installation and usage, led to the development of a bash script. In order to simplify the development and implementation task, already existing command line tools were selected to carry out all interaction with the used storage services. In practice, the users and administrators of storage services have already installed some of these command line tools and are trained in working with at least one of them.
Users of OSSperf have the freedom to choose between these command line tools for the interaction with the used storage services:
az [34]. A python client for the Azure Blob Storage API.
gsutil [35]. A python client for the Google Cloud Storage service offering, which uses the S3 API.
mc [36]. The so called Minio Client. It is developed in the programming language go and can interact with storage services that (re-)implement the API of the S3.
s3cmd [37]. A python client, that can interact with storage services that (re-)implement the API of the S3.
swift [38]. A python client for the Swift API.
Access to the storage services need to be configured in the way the above mentioned command line tools expect it.
Per default, if no other client software is specified via a command line parameter, OSSperf will try to use s3cmd for the interaction with the desired storage service. Once a storage service is registered in the configuration file of s3cmd, OSSperf can interact with the S3-compatible REST API of the service.
REST is an architectural-style that relies on HTTP methods like GET or PUT. S3-compatible services use PUT to receive the list of buckets that are assigned to a user account, or a list of objects inside a bucket, or an object itself. Buckets and objects are created with PUT and DELETE is used to erase buckets and objects. POST can be used to upload objects and HEAD is used to retrieve meta-data from an account, bucket or object. Uploading files into S3-compatible services is done via POST or PUT directly from the client of the user. Table 2 gives an overview of methods used to interact with S3-compatible storage services [39].
Table 2 Description of the HTTP methods with request-URIs that are used to interact with storage services
If the Swift API or the Microsoft Azure Blob Storage API shall be used, OSSperf does not use the command line tool s3cmd, but the Swift client (swift) or the Azure client (az). In contrast to s3cmd, the swift tool uses no configuration file with user credentials. Users of swift need to specify the endpoint of the used storage service as well as username and password inside the environment variables ST_AUTH, ST_USER and ST_KEY. Users of az need to specify the username and password inside the environment variables AZURE_STORAGE_ACCOUNT and AZURE_STORAGE_ACCESS_KEY.
Users of OSSperf have the freedom to use the Minio Client (mc) or the python client for the Google Cloud Storage (gsutil), instead of s3cmd, for the interaction with storage services that implement the S3-API. In this case, it is necessary to specify the connection parameters of the storage service as the first step in the appropriate configuration files of theses clients. Afterwards, the alternative clients can be set via command line parameter and used by OSSperf.
One of the aims during the development of OSSperf was to create a tool, which tests the performance of the most common used bucket- and object-related operations. These are in detail the creation and erasure of buckets and the upload, download and erasure of objects, as well as the operation, which returns a list of the objects, that are assigned to a specific bucket. These operations are in other words the CRUD actions (see Table 3), which are the four basic functions of persistent storage and are mostly mentioned in the field of SQL databases and therefore mapped to basic SQL statements, but they can also be mapped to HTTP methods, which are the foundation of the interaction with object-based storage services.
Table 3 The CRUD actions and their matching HTTP methods
To investigate the performance of the CRUD actions, when doing a performance evaluation with OSSperf, the tool executes these six steps for a specific number of objects of a specific size:
Upload one or more objects into this bucket
Fetch the list of objects inside the bucket
Download the objects from the bucket
Erase the objects inside the bucket
Erase the bucket
The time, which is required to carry out these operations is individually measured and can be used to analyze the performance of these commonly used storage service operations.
Users of OSSperf have the freedom to specify the number of files via command-line parameters, which shall be created, uploaded and downloaded, as well as their individual size. The files are created via the command line tool dd and contain data from the pseudorandom number generator behind the special file /dev/random.
In order to be able to simulate different load scenarios, the OSSperf tool supports the parallel transfer of objects, as well as requesting delete operations in parallel. If the parallel flag is set, the steps 2, 4 and 5 are executed in parallel by using the command line tool parallel. The steps 1, 3 and 6 can not in principle be executed in parallel.
Setting the bucketname, that OSSperf shall use, can also be done via command line parameter. The default bucketname is ossperf-testbucket, but in case the performance of a public cloud service offering shall be investigated, it may be required for the user to specify another name for the bucket, because the buckets inside a storage service need to have unique names.
A common assumption, when using storage services, which implement the same API, is that all operations cause identical results. But during the development of OSSperf, some challenges caused by non-matching service behavior emerged. One issue is the encoding of the bucket names. In order to be conforming to the DNS requirements, bucket names should not contain capital letters. To comply with this rule, the free private cloud storage service solutions Minio, Riak CS, S3rver, and Scality S3, do not accept bucket names with upper-case letters.
Other services like Nimbus Cumulus and S3ninja only accept bucket names, which are encoded entirely in upper case letters. The service offerings Amazon S3 and Google Cloud Storage, as well as the private cloud solution Fake S3 are more generous in this case and accept bucket names, which are written in lower case and upper-case letters.
In order to be compatible with the different storage service solutions and offerings, OSSperf allows the user to specify the encoding of the bucket name with a command line parameter.
Every time, OSSperf is executed, the tool prints out a line of data, which informs the user about the date (column 1) and time (column 2) when the execution was finished, the number of created objects (column 3), the size of the single objects in bytes (column 4), as well as the required time in seconds to execute the single steps 1–6 (columns 5–10). Column number 11 contains the sum of all time values, which is calculated by using the command line tool bc. The final columns 12 and 13 present the bandwidth B U during the upload (step 2) and the bandwidth B D during the download (step 4) operations. These values are also calculated with the command line tool bc with Eqs. 1 and 2, where N is the number of files, S is the size of the single files, T U is the required time to upload and T D the required time to download the files. Figure 1 contains five lines of output data, generated by OSSperf. These lines are a part of the output data, which were used to generate Fig. 3 and Table 4.
$$\begin{array}{@{}rcl@{}} \frac{S ~{\left[Byte\right]} \times N \times 8}{T_{U}~\text{[s]}}\ /\ 1000\ /\ 1000\ = B_{U}~\left[\text{Mbit/s}\right] \end{array} $$
Example of output data of OSSperf
Functioning of the object storage interface
Required time to upload and download sets of 10 files into a multi-node storage service implemented with Minio
Table 4 Bandwidth during upload and download when using a multi-node storage service implemented with Minio in a private context
$$\begin{array}{@{}rcl@{}} \frac{S ~{\left[Byte\right]} \times N \times 8}{T_{D}~\text{[s]}}\ /\ 1000\ /\ 1000\ = B_{D}~\left[\text{Mbit/s}\right] \end{array} $$
The structure of the output simplifies the analysis of the performance measurements by using command line tools like sed, awk and gnuplot.
Analyzing the benchmark
An example for the presentation of benchmark results, gained with OSSperf, present Fig. 3 and Table 4. For this scenario, the free storage service solution Minio [11] was installed in an 8-node deployment on a cluster of Raspberry Pi 2 single board computers [40, 41]. The client, which executed OSSperf, was connected via Ethernet with the same network switch as the cluster nodes. Each Raspberry Pi node has a 100 MBit/s Ethernet interface.
The gained data shows that the parallel upload of files is beneficial for objects of all sizes, but the download of objects <128 kB in size requires a longer time in parallel transfer compared to sequential mode. With a growing object size, the service performs better in parallel transfer compared with sequential mode.
The effect, that the bandwidth gets better with a growing file size, is caused by the protocol overhead1 of the object-based storage services (see Fig. 2). This overhead exists for file transmissions of any size and its portion of the potential throughput shrinks with a growing file size. The overhead is caused among others by theses characteristics:
For upload and download operations, one communication partner needs to wrap the files into HTTP messages, TCP segments, IP packages and Ethernet frames and the other communication partner needs to unwrap them.
The transmission of the HTTP messages via a computer network requires time (= network latency).
Each upload operation requires the client to wait for a reply of the server, that indicates the request was successful.
One of the reasons, why the upload performance in Fig. 3 and Table 4 is slower compared to the download performance, may be the effort to store files in minio multi-node deployments. Minio uses erasure code and checksums [42] to increase the availability of the stored data, which causes much effort on the single board computers, which were used in this scenario.
Another example of benchmark results, gained with OSSperf, present Fig. 4 and Table 5. In this scenario, OSSperf was used to investigate the performance of the Amazon S3 service offering. Also when using S3, the upload performance is slower, compared with the download performance. The measured bandwidth during upload is close to the maximum data rate, the used internet service provider offers for upstream. Equal to the private cloud scenario from Fig. 3 and Table 4, the parallel upload of the ten files is never faster compared with the sequential upload. And again, the parallel upload is only beneficial when downloading large files. In this scenario, the turnaround point is when downloading objects ≥512 kB in size.
Required time to upload and download sets of 10 files into the Amazon S3 public cloud storage service offering
Table 5 Bandwidth during upload and download when using the public cloud storage service offering Amazon S3
When investigating the performance of a public cloud storage service like Amazon S3, it must always be kept in mind, that the performance of this service may vary over the time of the day and that it is among others influenced by the network path between client and service, as well as the current network utilization.
The development of OSSperf resulted in a solution to evaluate the performance of object-based cloud storage services, that is more lightweight than COSBench or the PerfKit Benchmarker test suite, because its installation and configuration does not require much effort and it has only few dependencies. OSSperf only requires the bash command line interpreter, md5sum for the creation and validation of the checksums, bc to carry out the calculations, parallel to implement the parallel mode and command line clients like az, gsutil, mc, s3cmd and swift for the interaction with the storage services.
In contrast to the works of Garfinkel and of Palankar et al., which both only consider HTTP GET requests, OSSperf carries out typical storage service operations and evaluates the time, which is required to execute these operations.
OSSperf supports the APIs of several different public and private cloud services by using clients that can interact via the S3-API, the Swift-API and the Azure Blob Storage API. This is a huge benefit in contrast to the other analyzed solutions, because the S3 Performance Test Tool, S3-perf, the work of Calder et al., the work of Garfinkel and the work of Palankar et al. support only a single API. CloudCmp, COSBench and the PerfKit Benchmarker test suite support only two APIs.
In contrast to S3-perf, the work of Calder et al., the work of Garfinkel and the work of Palankar et al., OSSperf is able to operate in sequential and parallel mode to create load situations of different sort.
The scope of functions sets OSSperf apart from the other solutions, we investigated in this work (see Table 1). With the performance information of OSSperf, users and administrators of storage services can find out which operations are most time consuming. This information is essential to optimize applications, which use storage services, as well as for optimizing storage service deployments.
OSSperf is free software and licensed under the terms of the GPLv3. The source code and documentation can be found inside the Git repository: https://github.com/christianbaun/ossperf/
Next steps are the implementation for support of further command line clients to interact with object-based storage services like Rackspace Cloud Files [43] in order to allow a more detailed investigation of the way, the different clients influence the performance.
1 The network-level efficiency of object-based cloud storages is better compared with storage services like Google Drive [44], Microsoft OneDrive [45], Dropbox [46], Box [47] or Apple iCloud [48] – such services are sometimes called Personal Cloud Storage – because they all implement on top of object-based storage services an additional synchronization protocol in order to keep the data in a consistent state between client site and the service provide [49–51].
Application programming interface
AWS:
Blob:
Binary large object
CRUD:
Create, read, update and delete
DNS:
GCS:
GPL:
Hypertext transfer protocol
kB:
Kilobyte
MB:
REST:
Representational state transfer
S3:
Simple storage service
TCP:
Transmission control protocol
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Baun C, Cocos HN, Spanou RM (2017) Performance Aspects of Object-based Storage Services on Single Board Computers. Open J Cloud Comput (OJCC) 4(1):1–16.
Minio Erasure Code Quickstart Guide. https://github.com/minio/minio/tree/master/docs/erasure.
Rackspace Cloud Files. https://www.rackspace.com/cloud/files/.
Google Drive. https://www.google.com/drive/.
Microsoft OneDrive.https://onedrive.live.com.
Dropbox. https://www.dropbox.com.
Box. https://www.box.com.
iCloud. https://www.icloud.com.
Li Z, Jin C, Xu T, Wilson C, Liu Y, Cheng L, Liu Y, Dai Y, Zhang ZL (2014) Towards Network-level Efficiency for Cloud Storage Services In: Proceedings of the 2014 ACM Conference on Internet Measurement Conference. IMC '14, 115–128.
Drago I, Bocchi E, Mellia M, Slatman H, Pras A (2013) Benchmarking personal cloud storage In: Proceedings of the 2013 ACM Conference on Internet Measurement Conference. IMC '13, 205–212.
Drago I, Mellia M, M. Munafo M, Sperotto A, Sadre R, Pras A (2012) Inside Dropbox: Understanding Personal Cloud Storage Services In: Proceedings of the 2012 ACM Internet Measurement Conference. IMC '12, 481–494.
OSSperf. https://github.com/christianbaun/ossperf.
Many thanks to Katrin Baun for her assistance in improving the quality of this paper.
This work was funded by the Hessian Ministry for Science and the Arts ('Hessisches Ministerium für Wissenschaft und Kunst') in the framework of research for practice ('Forschung für die Praxis') and by the Faculty of Computer Science and Engineering Science of the Frankfurt University of Applied Sciences in the framework of 'Innovationsfonds Forschung' (IFOFO).
The source code and documentation of the OSSperf software is stored inside the Git repository: https://github.com/christianbaun/ossperf/.
Dr. Christian Baun is a Professor at the Faculty of Computer Science and Engineering of the Frankfurt University of Applied Sciences in Frankfurt am Main, Germany. He earned his Diploma degree in Informatik (Computer Science) in 2005 and his Master degree in 2006 from the Mannheim University of Applied Sciences. In 2011, he earned his Doctor degree from the University of Hamburg. He is author of several books, articles and research papers. His research interest includes operating systems, distributed systems and computer networks.
Henry-Norbert Cocos studies computer science at the Frankfurt University of Applied Sciences. His research interest includes distributed systems and single board computers. Currently, he constructs a 256 node cluster of Raspberry Pi 3 nodes which shall be used to analyze different parallel computation tasks. For this work, he analyzes which administration tasks need to be carried out during the deployment and operation phase and how these tasks can be automated.
Rosa-Maria Spanou studies computer science at the Frankfurt University of Applied Sciences. Her research interest includes distributed systems and single board computers. Currently, she constructs and analyzes different multi-node object-based cloud storage solutions.
Frankfurt University of Applied Sciences, Frankfurt am Main, 60318, Germany
Christian Baun
, Henry-Norbert Cocos
& Rosa-Maria Spanou
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CB and HNC carried out the literature review. CB and RMS developed the initial concept of the OSSperf benchmark solution and CB did most of the implementation. CB drafted the manuscript. HNC and RMS jointly provided useful remarks and critically reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Christian Baun.
Baun, C., Cocos, H. & Spanou, R. OSSperf – a lightweight solution for the performance evaluation of object-based cloud storage services. J Cloud Comp 6, 24 (2017) doi:10.1186/s13677-017-0096-x
DOI: https://doi.org/10.1186/s13677-017-0096-x | CommonCrawl |
What does Stephen Hawking mean by 'an infinite universe'?
In the recent $100m search for extra terrestrial life project project, Stephen Hawking is quoted in the following way:
"We believe that life arose spontaneously on Earth," Hawking said at Monday's news conference, "So in an infinite universe, there must be other occurrences of life."
My understanding is that scientists believe that the universe isn't infinite - there there is a finite number of stars, it's continuously expanding but has a finite size at a given time.
What's Stephen Hawking refering to here?
universe cosmology
HDE 226868♦
dwjohnstondwjohnston
Due to the finite speed of light, and the finite age of the Universe, only a portion of it is observable. When people talk about "the size of the Universe", "the number of stars in the Universe", etc., they usually refer to the observable Universe, i.e. the sphere in which we are centered, and which has a radius given by the distance light has been able to travel in the 13.8 billion years since the Big Bang. Note that since the Universe is expanding, this radius is more than 13.8 billion light-years. In fact it's 46.3 billion light-years.
Observations indicate that, on large scales (i.e. above roughly half a billion light-years), the Universe is homogeneous (the same everywhere) and isotropic (the same in all directions). Assuming that this is indeed true is known as the cosmological principle. If the rest of the Universe follows this principle, then there are three possible overall "versions" of universes that we can live in. We call these versions "flat", "closed", and "open". Whereas a globally closed universe would have a finite extent, globally flat or open universes must be infinitely large.
The observable Universe is, within measuring uncertainties but too a very high precision, flat (e.g. Planck Collaboration et al. 2016). Hence, we might think that the whole Universe is, in fact, infinite. But sort of like standing in a large forest with limited visibility doesn't tell you whether the forest is just larger than you can see, or if it's infinitely large, we can't with our current theories and observations know whether the Universe is finite or infinite.
pelapela
29.7k7575 silver badges106106 bronze badges
$\begingroup$ Well, no. All the observations indicate is that the chunk we see is flat and homogeneous. There is no possible observation that could positively conclude that the universe is infinite, for obvious reasons. There's always the question of what may be far beyond the current limits of observation. $\endgroup$
– Florin Andrei
$\begingroup$ Yes, you're right, of course. I do like to think that the rest of the Universe is similar to what we see, though. $\Omega_\mathrm{tot}$ is so close to unity, and in the past is has been tremendously more close to unity. But you're right, the only thing we can say with some certainty is that the Universe is much larger than what we can see. $\endgroup$
– pela
I believe his meaning is being misinterpreted as a claim that the universe actually is infinite. If someone says "in an infinite universe", they are not asserting that the universe is indeed infinite (as no scientist could ever claim to know), they are simply saying "if the universe is infinite." It is merely a case from which to begin the discussion. For example, if you were having a discussion about whether to pass a seatbelt law, you might start with "in a world where people always make the safer choice in everything they do, we would not need seatbelt laws." That would not be a claim we live in such a world, it is merely setting a kind of baseline for going on to talk about what is the actual situation.
Ken GKen G
A relativistic universe which is expanding faster than light (like ours) is effectively infinite for all practical purposes.
Also due to its relativistic nature and faster than light expansion, even if you assume it's not infinite at some given moment, it still doesn't have any edges or borders - for you. You're in some random place in it, the maximum speed of any possible interaction is speed of light, and the universe is expanding faster than light - then anything that happens at some (real or imaginary) "edge" is outside your realm of existence. It is effectively infinite - for you.
Give it enough time and it would grow as big as you want. Give it asymptotically infinite time and it will grow asymptotically infinite.
As to what the "edge" might be, see Eternal Inflation. This is a model in modern cosmology where local bubbles like ours have stopped inflating (they still expand, but not at the tremendous rate of the initial inflationary phase); however, inflation continues forever outside the bubbles and at the edges. Therefore the bubbles keep growing indefinitely at faster than light speeds. Because speed of light is a limit for any interaction inside the bubbles, for any internal observer each bubble is effectively infinite for any practical purpose.
Be aware that there is no proof that this is actually the case, but this is a model that fits well what we now observe.
EDIT: TBH, I'm not even sure this is a question for StackExchange. It's very open-ended, and we don't really have the conclusive answers. All we can say for now is that the universe appears to be infinite for all practical purposes, but we can't know for sure. So often scientists like Hawking just simplify the language and refer to the universe as "infinite" without any of the qualifiers that would be required in a strict context.
I don't think there's any single, final answer here.
Florin AndreiFlorin Andrei
$\begingroup$ But isn't the fundamental assumption behind Hawking's statement that there is an infinite number of planets? How does "effectively infinite" address that? $\endgroup$
– Pete Becker
$\begingroup$ It doesn't. There's a fundamental difference between "absurdly humongous beyond imagination" and "infinite". But I think most astronomer/physicists think that the Universe is in fact infinite, lest it would conflict with the cosmological principle, which we wouldn't like. $\endgroup$
$\begingroup$ Also, +1, but note that the term "expanding faster than the speed of light" necessitates a a chosen distance scale, since expansion rate is proportional to distance. The part of the Universe that is closer to us than roughly $10^{10}$ lightyears recede at speeds lower than that of light. $\endgroup$
$\begingroup$ @PeteBecker In a relativistic universe (like ours), there is an upper limit to the speed of any interaction. Nothing can reach arbitrarily far away, arbitrarily quickly. That being the case, there is no practical difference between an universe infinite-right-now, and an universe which is expanding faster than light over very large scales. In the Eternal Inflation model, the bubble doesn't even have an edge - the "edge" is the place/time where/when the universe is young, perpetually created, eternally inflating. An observer there would simply see a younger universe, that's the only difference. $\endgroup$
Is the Universe infinite?
If the Universe is infinite, why isn't it of infinite density?
What suggests the cyclic nature of the universe? Proposed by Stephen Hawking
Does a flat universe mean not finite and circular? | CommonCrawl |
Climate change and trend analysis of temperature: the case of Addis Ababa, Ethiopia
Zinabu Assefa Alemu ORCID: orcid.org/0000-0003-0693-85961 &
Michael O. Dioha2
This paper presents the trend analysis of temperature and the effect of climate variation in the city of Addis Ababa, Ethiopia. The paper seeks to provide up-to-date information for the better management of climate change in the city. The analysis is based on the temperature difference in the city over two stations—Bole and Entoto. The overall purpose of this study is to investigate the possible trend of temperature variation as well as the effect of climate change in the study area.
The Mann-Kendall (MK) trend test and Sen's slope estimate were employed to find the nature of the temperature trend and significance level in the city.
It was found that the MK2/MK3 statistic (Z) value for minimum, maximum and average temperatures for Bole station are 6.21/5.99, 2.49/2.6, and 6.09/6.14 respectively. The positive Kendall's Z value shows an upward trend and implies an increasing trend over time. This indicates a significant increase in the trend at a 5% level of significance since the significance level (alpha) is greater than the computed p-value (0.05 > p-values (0.0001)). Whereas for Entoto station, the MK1 statistic (Z) results are 1.64 for minimum, while the MK2/MK3 static (Z) are 0.71/0.65 for the maximum, and 0.17/1.04 for average temperature, and this positive value shows an indicator of an increasing trend. However, the increase is not significant at the 5% significant level since the computed p-value is larger than the significant level (alpha = 0.05).
There is a tendency of temperature increments in Bole station. This could be due to the influence of climate change which can lead to weather extremes in the capital city. Therefore, the study recommends that the variability of temperature needs further monitoring technique, and there is a need to consider the increasing temperature trend to minimize its effects on human health.
Climate change has become one of the most essential concerns in the field of sustainable development, and its impacts (rising of sea levels, melting of polar ice caps, wild bush fires, intense droughts etc.) can be felt in different parts of the globe (Dioha and Kumar 2020; Ali et al. 2013). The warming of our planet due to the emission of greenhouse gases is now unquestionable; and over the last century, the CO2 atmospheric concentration has increased significantly and has, in turn, induced the average global temperature to increase by 0.74 °C as compared with the preindustrial era (UNFCCC 2007). The high temperature in urban areas affects mostly health, economy, leisure activities, and wellbeing of urban residents. Thermal stress caused by warming highly affects the health of vulnerable peoples (Tan et al. 2010; Patz et al. 2005). Developing countries are mostly affected by climate change, and Ethiopia is an example of the most vulnerable countries (Cherie and Fentaw 2015).
The intensity and frequency of extremes can be easily changed by climate change and the changes in climate extremes and their impacts on a variety of physical and biological systems examined by the Intergovernmental Panel on Climate Change (IPCC) and their effects can also contribute to global warming (IPCC 2007). Many factors such as the expansion of cities, and fast population growth rate along with migration from rural to urban areas pose a major challenge for city planners and also contributes to increasing climate change (WHO and UNICEF 2006; Alemu and Dioha 2020). Using various General Circulation Models, Feyissa et al. (2018) suggested future climatic changes and argued that a rise in temperature will exacerbate the urban heat highland effects in warm seasons and an increase in precipitation. Some environmental harms such as high temperature and extreme rainfall, which results in flooding in Addis Ababa, could be signals of climate change (Birhanu et al. 2016). Also, the city temperature is mostly affected by anthropogenic activities along with climate change.
The Mann–Kendall (MK) (non-parametric) test is usually used to detect an upward trend or downward (i.e. monotonic trends) in a series of hydrological data (climate data) and environmental data. The null hypothesis for this test indicates no trend, whereas the alternative hypothesis indicates a trend in the two-sided test or a one-sided test as an upward trend or downward trend (Pohlert 2020). The Sen's estimator is another non-parametric method used for the trend analysis of hydroclimate data set. It is also used to identify the trend magnitude. Hence, this test computes the linear rate of change (slope) and the intercept as shown in Sen's method (Sen 1968). The MK test is widely documented in various literature, as a powerful trend test for effective analysis of seasonal and annual trends in environmental data, hydrological data (climate data), and this test is preferred over other tests because of its applicability in time-series data, which does not follow the statistical distribution.
There are numerous examples of MK trend test applications such as Asfaw et al. (2018) who used the MK test for the detection of trends in time series analysis and the result revealed that inter-annual and intra-annual variability of rainfall as well as the severity index value for Palmer drought shows that the trend for the number of drought years was increasing. Another study also employed a non-parametric MK test and Sen's slope estimates to test the trend of each extreme temperature and rainfall indices as well as their statistical significance in the Western Tigray, Ethiopia (Berhane et al. 2020). Similarly, the trend analysis of temperature in Gombe state, Nigeria was analyzed using the MK trend test and Sen's estimator to decide the nature of the temperature trend and significance level. The study found that average and maximum temperatures revealed positive Kendall's statistics (Z) (Alhaji et al. 2018). In a different study, Yadav et al. (2014) used the MK test and the Sen's Slope for the analysis of both trends and slope magnitude. The results indicated that in all thirteen areas of Uttarakhand (India), the trends of temperature and precipitation are increasing in some months, whereas in some other months the trends were decreasing. Getachew (2018) used the MK trend test for the analysis of rainfall and temperature trends in the south Gonder zone (Ethiopia). The study found that a statistically significant increase in Nefas Mocha and Addis Zemen for mean annual temperature. Kuriqi et al. (2020) applied the MK methodology to validate findings from Sen's slope trend analysis in a study on the seasonality shift and streamflow flow variability trends in India.
Furthermore, the MK test and the Sen's estimator test has been applied to examine the significant trend of rainfall, temperature, and runoff in the Rangoon watershed in Dadeldhura district of Nepal. The result revealed that there were warming trends in the study area (Pal et al. 2017). In contrast, Machida et al. (2013) studied whether the MK test is an effective methodology for detecting software aging from traces of computer system metrics. But, the MK test result shows it is not a powerful trend test. The authors' experimental study showed that the use of MK trend test in detecting software aging is highly exposed in creating false positives (Machida et al. 2013). Other studies have applied the MK test for the assessment of spatial and temporal trends such as in Northern Iran (Biazar and Ferdosi 2020) and in Kansas, USA (Anandhi 2013). Despite the various application of the MK trend test in different parts of the world, studies analyzing the non-parametric MK test is commonly employed to detect monotonic trends in a series of environmental data, climate data or hydrological data. But some limited studies such as Machida et al (2013) showed that the MK test is not a powerful trend test for software aging and the variation in this result and other studies is because of differences in study variables/materials.
However, the MK test is a non-parametric (distribution-free) test which is used to analyze time-series data for consistently monotonic trends. These non-parametric methods have several benefits such as the handling of missing data, the requirement of few assumptions, and the data distribution independence (Öztopal and Sen 2016; Wu and Qian 2017; Kisi 2015). Nevertheless, the major disadvantage of the method is the influence of autocorrelation in data on its test significance. Several modifications in the MK test have been proposed by different authors to remove the influence of autocorrelation done with varied techniques and one of the most common tests is corrected for bias before pre-whitening (Malik et al. 2019; Sanikhani et al. 2018; Su et al. 2006). The MK test is mostly chosen for the analysis of climatic data since its measurement does not follow the normal distribution. Thus, the present study has employed the MK trend test and Sen's slope estimate to understand the nature of the temperature trend and significance level in the study area. Hence, the current study is conducted based on the temperature variation in the city of Addis Ababa over two stations—Bole and Entoto. The historic temperature used for Bole station is from 1983 to 2016 and the Entoto station from 1989 to 2016. In addition to this, the selection of this station is also classified based on geographical variation and the altitude differences.
The overall objective of this study is to investigate the trend of temperature in Addis Ababa City by using the Mann–Kendall trend test and Sen's slope estimate as well as to look at the effect of climate change in the study area. The result of this study (i.e. temperature trends and their descriptive statistics) will help city planners in forecasting weather variations. It will also support universal health coverage by predicting the situation/seasons in order to control seasonal disease outbreaks. In terms of contribution to the existing literature, this study introduces one of the earliest case studies in this subject matter for Ethiopia and the findings will be useful in mitigating the adverse impacts of climate change in the country. Also, the analytical framework presented here can be employed by other researchers to study temperature variations in other regions of the world. While this paper is crafted with a local case study, the results will also be useful for international literature.
The rest of this paper is structured thus: Sect. 2 explains the research methods used in the study which incorporates the study area, data quality control, different MK Tests, Sen's Slope estimator, ITA analysis, data collection, and processing, as well as data analysis tools. Section 3 describes the results and brief discussion, while the general study conclusions are presented in Sect. 4.
We employed the Mann–Kendall (MK) trend test and Sen's slope estimate to examine the nature of the temperature trend and significance level in the study area. Figure 1 shows the general study methodological framework.
General methodological approach
Description of the study area
Addis Ababa is the capital city of Ethiopia and it is found in the heart of the country surrounded by Oromia which is geographically located at longitude 38° 44′ E and latitude of 9° 1′ N. According to the 2007 census, the city have a total population of 2,739,551 inhabitants. Addis Ababa comprises 6 zones and 28 woredas. Addis Ababa covers an area of about 540 Km2 and it lies between 2,200 to 2,500 m above sea level. The city lies at the foot of the 3,000 m high Entoto Mountains and the mountain Entoto is located in Gullele Sub City (within Addis Ababa city Administration). Furthermore, the lowest and the highest annual average temperature of the city is between 9.89 and 24.64 °C (FDRE 2018; CSA 2007). Figure 2 shows a map of the study area.
Map of the study area
Data quality control
The quality of the data was visually and statistically assessed. Visually, the temperature data were checked and detected for outlier and missing data to avoid erroneous/typing error data that can cause changes in the final result. Whereas, the MK test method was checked and tested statistically with the trend free pre-whiting process and the variance correction approaches before applying the test. The trend free pre-whiting process was proposed to remove the serial correlation from the data before applying the trend test (Yue et al. 2002; Hamed 2009). Likewise, to overcome the limitation of the occurrence of serial autocorrelation in time series, the variance correction procedure was applied as proposed by (Hamed and Rao 1998).
Mann–Kendall test (MK1)
MK trend test is a non-parametric test used to identify a trend in a series. It is also used to determine whether a time series has a monotonic upward or downward trend. The non-parametric MK test is commonly employed to detect monotonic trends in a series of environmental data, hydrological data, or climate data. The null hypothesis (H0) shows no trend in the series and the data which come from an independent population are identically distributed. The alternative hypothesis, Ha, indicates that the data follow a monotonic trend (i.e. negative, non-null, or positive trend). There are two benefits of using this test. First, it does not require the data to be normally distributed since the test is non-parametric (distribution-free test) and second, the test has low sensitivity to abrupt breaks due to inhomogeneous time series. The data values are evaluated as an order time series and all subsequent data values are likened from each data value. The time series x1, x2, x3… xn represents n data points.
The MK test statistic (S) is calculated as follows:
$${\text{S}} = \sum\nolimits_{{{\text{i}} = 1}}^{{{\text{n}} - 1}} {\sum\nolimits_{{{\text{j}} = {\text{i}} + 1}}^{{\text{n}}} {sign({\text{Xj}} - {\text{Xi}})}}$$
$$\mathrm{sgn}(\mathrm{x})=\left\{\begin{array}{c} 1 \quad{\text{if}\quad{\text{ x}>0}}\\ 0\quad{\text{if}\quad{\text{ x}=0}}\\ -1 \quad{\text{if}\quad{\text{ x}<0}}\end{array}\right.$$
Note that if S > 0, then later observations in the time series tend to be larger than those that appear earlier in the time series and it is an indicator of an increasing trend, while the reverse is true if S < 0 and this indicates a decreasing trend.
The mean of S is E[S] = 0 and the variance \(({\upsigma }^{2}\)) of S is given by
$${\upsigma }^{2} =\frac{1}{18}\left\{\mathrm{ n}\left(\mathrm{n}-1\right)\left(2\mathrm{n}+5\right)-{\sum }_{\mathrm{j}=1}^{\mathrm{p}}\mathrm{tj}(\mathrm{tj}-1)(2\mathrm{tj}+5)\right\}$$
where p is the number of the tied groups in the data set and tj is the number of data points in the jth tied group. The statistic S is approximately normally distributed provided that the following Z-transformation is employed:
$$\mathrm{Z}=\left\{\begin{array}{c} \frac{\mathrm{S}-1}{\sqrt{{\upsigma }^{2} }} \quad{{\text {if} \,\text{s}>0}}\\ 0 \quad{{\text {if}\, \text{s}=0}}\\ \frac{\mathrm{S}+1}{\sqrt{{\upsigma }^{2} }} \quad{{\text{if}\, \text{s}<0}}\end{array}\right.$$
A normal approximation test that could be used for datasets with more than 10 values was described, provided there are not many tied values within the data set. If there is no monotonic trend (the null hypothesis), then for time series with more than ten elements, z ∼ N (0, 1), i.e. z has a standard normal distribution. The probability density function for a normal distribution with a mean of 0 and a standard deviation of 1 is given by the following equation:
$$\mathrm{f }(\mathrm{z})\hspace{0.17em}=\hspace{0.17em}\frac{1}{\sqrt{2\uppi }}{\mathrm{e}}^{\frac{{-\mathrm{z}}^{2}}{2}}$$
The statistic S is closely related to Kendall's as given by:
$$\uptau = \frac{\mathrm{S}}{\mathrm{D}}$$
$${\mathrm{D}= \left[ \frac{1}{2}\mathrm{n}(\mathrm{n}-1)-\frac{1}{2}{\sum }_{\mathrm{j}=1}^{\mathrm{p}}\mathrm{tj}(\mathrm{tj}-1)\right]}^{1/2}{\left[\frac{1}{2}\mathrm{n}(\mathrm{n}-1)\right]}^{1/2}$$
where p is the number of the tied groups in the data set and tj is the number of data points in the jth tied group.
All the above procedures used to compute the Mann–Kendall Trend test were collected and referenced from (Zaiontz 2020; Kendall 1975; Pohlert 2020).
Mann–Kendall test with trend-free pre-whitening (MK2)
Hamed (2009) recommended that there will be a decrease or an increase in S value when autocorrelation is positive or negative which is underestimated or overestimated by the original variance V(S). Therefore, when trend analysis is conducted for this data using MK1, it will show positive or negative trends when there is no trend. Hence, the trend free pre-whiting process (TFPW) was proposed by Hamed (2009) and the proposed pre-whitening technique in which the slope and lag-1 serial correlation coefficient are simultaneously estimated. The lag-1 serial correlation coefficient is then corrected for bias before pre-whitening. Finally, the lag-1 serial correlation components are removed from the series before applying the trend test. The following steps are used to determine trend analysis using the MK2 test. Calculate the lag-1 (k = 1) autocorrelation coefficient (r1) using:
$$\mathrm{r}1 = \frac{\frac{1}{\mathrm{n}-\mathrm{k}}\sum_{i=1}^{n-k}\left(\mathrm{ Xi }- \stackrel{-}{\mathrm{X }}\right) \left(\mathrm{ Xi}+\mathrm{k }- \stackrel{-}{\mathrm{X }}\right)}{\frac{1}{\mathrm{n}} \sum_{i=1}^{n}{ (\mathrm{ X }- \stackrel{-}{\mathrm{X }})}^{2}}$$
If the condition \(\frac{-1-1.96 \sqrt{n-2}}{\mathrm{n}-1}\) ≤ r1 ≤ \(\frac{-1+1.96 \sqrt{n-2}}{\mathrm{n}-1}\) is satisfied, then the series is assumed to be independent at a 5% significance level and there is no need for pre-whitening. Otherwise, pre-whitening is required for the series before applying the MK1 test.
Equation (9) is used to remove the trend in time series data to get detrended time series.
$$\upbeta ={\text{median}}\left[ \frac{\mathrm{Xj}-\mathrm{Xi}}{\mathrm{j}-\mathrm{i}}\right]\quad {{\text {for all}}\quad{{\text{i < j}}}}.$$
Equation (8) is used to calculate lag-1 autocorrelations for detrended time series given by Xi. Using Eq. (11), remove the lag-1 autoregressive component (AR (1)) from the detrended series to get a residual series.
Yet again, (β * i) value is added to the residual series as follows;
Thus, the MK test is applied to the blended series Yi to determine the significance of the trend.
Mann-Kendall test with variance correction (MK3)
Sometimes, removing lag-1 autocorrelation is not enough for many hydrological time-series datasets. To overcome the limitation of the presence of serial autocorrelation in time-series, a correction procedure was proposed by (Hamed and Rao 1998). First, the corrected variance S is calculated by Eq. (13), where V (S) is the variance of the MK1 and CF is the correction factor due to the existence of serial correlation in the data.
$${\text{Corrected}}\, {\text{variance}}\, {\text{S}} ({\text{V}}*\left( {\text{S}} \right)) = {\text{CF}} \times {\text{V}} \left( {\text{S}} \right)$$
$$\mathrm{CF }=1+\frac{2}{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{\sum }_{\mathrm{k}=1}^{\mathrm{n}-1}(\mathrm{n}-\mathrm{k})(\mathrm{n}-\mathrm{k}-1)(\mathrm{n}-\mathrm{k}-2){r}_{k}^{R}$$
where rRk is lag-ranked serial correlation, while n is the total number of observations.
The advantage of the MK3 test over the MK2 test is that it includes all possible serial correlations (lag-k) in the time series, while MK2 only considers the lag-1 serial correlation (Yue et al. 2002).
Sen's Slope estimator
Sen's estimator is another non-parametric test used to identify a trend in a series as well as it shows the magnitude of the trend. The Sen's slop estimate requires at least 10 values in a time series. This test computes both the slope (i.e. linear rate of change) and intercepts according to Sen's method (Sen 1968). Likewise, as Drápela and Drápelová (2011) described that the linear model can be calculated as follows:
$$\mathrm{f}\left(\mathrm{x}\right)=\text{Qx}+\text{B}$$
where Q is the slope, B is constant. According to Pohlert (2020), initially, a set of linear slopes is calculated as follows (Eq. 16):
$$\mathrm{Qi }= \frac{\mathrm{Xj}-\mathrm{Xk}}{\mathrm{j}-\mathrm{k}}\quad\quad{{\text{for j }} = {\text{ 1}},{\text{ 2}},{\text{ 3}} \ldots {\text{ N}}}$$
where Q is the slope, X denotes the variable, n is the number of data, and j, k are indices where j > k. The slope is estimated for each observation and the corresponding intercept is also the median of all intercepts. Median is computed from N observations of the slope to estimate the Sen's Slope estimator (Eq. 17):
$$\text{Q}=\left[\begin{array}{cc}\mathrm{Q}\frac{\mathrm{N}+1}{2} & \mathrm{if \,N \,is\, odd}\\ \frac{1}{2}\left(\mathrm{Q}\frac{\mathrm{N}}{2}+\mathrm{Q}\frac{\mathrm{N}+1}{2}\right)&\mathrm{if\, N \,is\, even}\end{array}\right]$$
$$\text{N}=\frac{\mathrm{n}(\mathrm{n}-1)}{2}$$
where N is the Slope observations and n is the values of Xk in the time series.
According to Mondal et al. (2012), when the N Slope observations are shown as Odd, the Sen's Estimator is computed as Qmed = (N + 1)/2 and for Even times of observations the Slope estimate as Qmed = [(N/2) + ((N + 1)/2)]/2. The two-sided test is carried out at 100(1 – α) % of the confidence interval to obtain the true slope for the non-parametric tests in the series.
The positive slope Qi obtained shows an increasing/upward trend whereas the negative slope Qi obtained shows a decreasing/downward trend. But, if the slope is zero there is no trend other than things remain the same. To obtain an estimate of B (constant) in Eq. (8), the n values of differences xi—Qti is calculated. The median of these values gives an estimate of constant B. The estimates for the constant B of lines of the 99% and 95% confidence intervals are calculated by a similar procedure (Pohlert 2020; MAKESENS 2002).
Innovative trend analysis (ITA) method
The Innovative trend analysis method was proposed by Şen (2011) for the detection of trends in time series. In this method, data are equally divided into two segments between the first dates to the last date. Both segments are arranged in ascending order and presented in the X- and Y-axis. The first segment is presented in the horizontal axis (x-axis) while the second segment is presented in the vertical axis (y-axis) in the Cartesian coordinate system. A bisector line at 1:1 (450) line divides the diagram into two equal triangles. If the data points lay on the 1:1 line, there is no trend in the data. If the data points exist in the top triangle, it is indicative of a positive trend (increasing trend). If the data lies in the bottom triangle, it indicates a negative trend (decreasing trend) in the data (Zhang et al. 2008; Şen 2011). The Innovative Trend Analysis (ITA) of different temperatures graphs/plots for both stations were investigated through RStudio (i.e. package used 'trendchange::innovtrend (X)') as developed by Şen (2011).
The descriptive statistics table provides summary information on the binning input variables. The descriptive procedure displays univariate summary statistics for several variables in a single table. Statistics include the sample size (observations), mean, minimum, maximum, variance, standard deviation, and the number of cases with valid values. Values for Minimum and Maximum correspond to the lowest and highest categories of the factor variable. Whereas, the mean (is computed as the sum of all data values Xi, divided by the sample size n:
$$\text{Mean} \left( {\bar{X}} \right) = \sum\nolimits_{{i = 1}}^{n} {\frac{{Xi}}{n}}$$
The sample variance is the classical measures of spread. Like the mean, they are strongly influenced by outlying values. Both the variance and standard deviation are measures of variability in a population. Variance is the average squared deviations from the arithmetic mean and the standard deviation is the square root of the variance. Thus, the variance is nothing but the square of the standard deviation, i.e.,
$$\text{Variance}({\upsigma }^{2})=\frac{\sum {(\mathrm{ X }- \stackrel{-}{\mathrm{X }})}^{2}}{(\mathrm{n}-1)}$$
$$\text{Standard Deviation}\left(\upsigma \right)=\sqrt{{\upsigma }^{2}}$$
All the above procedures used to compute the descriptive statistics were collected and referenced from (Gupta 2007; Helsel and Hirsch 2002).
Data collection and processing
The daily climatic data for minimum and maximum temperatures were obtained from the National Meteorological Agency (NMA). The historic temperature was collected from two stations which are Bole station from 1983 to 2016 (NY = 34) and Entoto station from 1989 to 2016 (NY = 28). The overall research data for this study were collected based on secondary data sources to address the goals of the study. The data was used to analyses the temperature trend of Addis Ababa city.
MK Trend Test and Sen's Slope estimator were used to study the trend analysis of temperature. The Mann–Kendall trend test results such as MK stat(s), Kendall's tau, test statistics (Z), and P-value as well as the Sen's slope Q were computed using XRealStats, XLSTAT 2020, and RStudio (i.e. documentation for package 'modifiedmk' version 1.5.0). MAKESENS version 1.0 was used for the graph of Sen's estimate whereas the descriptive statistical techniques such as minimum, maximum, mean, standard deviation, variance, and also average annual temperatures graph were computed using Microsoft excel. The analyzed data was used to detect the trend of climate change.
Mann-Kendall test result
Trend analysis of Temperature for Addis Ababa City was done with 34 years of temperature data from Bole station (1983–2016) along with 28 years of temperature data from Entoto station (1989–2016). MK test and Sen's Slope estimator has been used to determine the temperature trend. Figure 3a–d shows the graph of minimum, maximum, and average temperature, in addition to the comparison of the temperatures for Bole station, whereas Fig. 4a–d shows the graph of minimum, maximum, and average temperature, as well as the comparison of the temperatures for Entoto station.
(a) Plot of minimum temperature, average temperature, and maximum temperature from 1983 to 2016 for Bole, (b) Plot of minimum temperature from 1983 to 2016 for Bole, (c) Plot of maximum temperature from 1983 to 2016 for Bole, (d) Plot of average temperature from 1983 to 2016 for Bole
(a) Plot of minimum temperature, average temperature, and maximum temperature from 1989—2016 for Entoto, (b) Plot of minimum temperature from 1989–2016 for Entoto, (c) Plot of maximum temperature from 1989 to 2016 for Entoto, (d) Plot of average temperature from 1989 to 2016 for Entoto station
From the MK test result, it was found that the Z value of MK2/MK3 for minimum, maximum, and average temperatures for Bole station are 6.21/5.99, 2.49/2.6, and 6.09/6.14 respectively, as stated in (Table 1). The positive Kendall's Z value shows an upward trend and also implies an increasing trend over time. This indicates that there is a significant increase in the trend at a 5% level of significance since the p-value is less than the significant level alpha (0.05) (Table 1). Whereas, for the Entoto station, the test statistic (Z) value of MK1 for minimum temperatures is 1.64, and the test statistic (Z) value of MK2/MK3 for maximum and average temperatures are 0.71/0.65, and 0.17/1.04 respectively, as displayed in (Table 1) and the positive value indicates an increasing trend but not significant at 5% significant level since the p-value is greater than the significant level alpha = 0.05 (Table 1). However, the minimum temperature for the Entoto station used the original MK test without using the modified MK test since the criteria stated in Eq. (8) is satisfied and thus no need of pre-whitening test before applying the MK test. Therefore, we simply use the result of MK1 without applying the serial correlation test.
Table 1 Trend analysis of temperature using MK1/MK2/MK3 test for Bole and Entoto stations
The result obtained in this study agrees with the findings of an earlier study by Getachew (2018), whose results revealed that for maximum temperature, an increasing trend analysis is found to be statistically significant as the computed p-value (i.e. 0.03) is lower than the significance level (alpha = 0.05) and the researcher rejects the null hypothesis and accepts the alternative hypothesis for Addis Zemen station at south Gonder zone. Similarly, in Ethiopia, a study conducted by Johannes and Mebratu (2009) shows that over the past five decades the temperature has been increasing annually at a rate of 0.2 °C. Conversely, the increasing trend of minimum temperature for Addis Zemen station is statistically insignificant as the computed p-value (i.e. 0.284) is greater than the significance level (alpha = 0.05) and thus the researcher cannot reject the null hypothesis (Getachew 2018). Table 1 shows the MK trend test result. The annual temperature (i.e. minimum, maximum, and average temperature) for Bole station shows a positive trend and statistically significant because the computed p-value is lower than the significance level alpha = 0.05, and one can accept the alternative hypothesis and reject the null hypothesis. On the other hand, the trend analysis of annual temperature (i.e. minimum, maximum, and average temperature) for the Entoto station shows an increasing trend but not statistically significant, thus, we can accept the null hypothesis as the computed p-value is greater than the significant level (alpha = 0.05).
Sen's estimate and computed data
The simple non-parametric procedure developed by (Zaiontz 2020) was used to estimate the slopes (change per unit time) present in the trend and the Sen's estimate graph/figure was computed by (MAKESENS 2002). The positive sign indicates the increasing slope, and the negative sign implies the decreasing slope, whereas, the zero slope shows no trend in the data for the study period and things remain the same. The Sen's slope estimates as shown in Table 1 and Fig. 5a–c for minimum, maximum, and average temperature from 1983 to 2016 for Bole station respectively shows an increasing trend and this agrees with the MK statistic (Z) result of positive values. It was found that the Z value of MK2/MK3 for minimum, maximum, and average temperatures for Bole station is 6.21/5.99, 2.49/2.6, and 6.09/6.14 respectively (Table 1). The positive Kendall's Z value shows an upward trend and also implies an increasing trend over time. This indicates that there is a significant increase in the trend at a 5% level of significance since the computed p-value is less than the significant level alpha (0.05) (Table 1).
(a) Sen's slope of minimum temperature from 1983–2016 for Bole, (b) Sen's slope of maximum temperature from 1983 to 2016 for Bole, (c) Sen's slope of average temperature from 1983–2016 for Bole
The Sen's slope estimates as shown in Table 1 and Fig. 6a–c for minimum, maximum, and average temperature from 1989 to 2016 for Entoto station respectively, depicts an increasing trend and the Sen's slope agrees with the MK1 statistic (Z) result of positive values of 1.64 for minimum temperature, whereas the MK2/MK3 statistic (Z) result of positive values of 0.71/0.65 for maximum and 0.17/1.04 for average temperature and this shows an indicator of an increasing trend. However, the increasing trend is not significant at 5% significant level since the computed p-value is greater than the significant level (Table 1).
(a) Sen's slope of minimum temperature from 1989–2016 for Entoto, (b) Sen's slope of maximum temperature from 1989–2016 for Entoto, (c) Sen's slope of average temperature from 1989–2016 for Entoto station
Furthermore, the result found in this present study for minimum, maximum, and average temperature for Entoto station and Bole station are different even though the stations exist in the capital city and this dissimilarity occurs because of geographical variation. Bole station exists inside the capital city and many constructions and other transport facilities take place due to this there is a temperature increment, whereas, the Entoto station exists near the National park and mountains and because of this, the temperature is almost stable. The Sen's Slope estimator displays that there is a tendency of temperature increments in Bole station. Thus, the increasing trend of temperature due to climate change and other factors can lead to weather extremes in the capital city (FDRE 2018; Fig. 5a–c, Fig. 6a–c).
Innovative trend analysis (ITA) method and computed figures
The simple non-parametric procedure was used to estimate the graph/figure through Rstudio using the package 'trendchange::innovtrend (X)' (Şen 2011). A bisector line at 1:1 straight line divides the diagram into two equal triangles. If the data points lay on the 1:1 line, there is no trend in the data. If the data points exist in the top triangle, it is indicative of an increasing trend. If the data lies in the bottom triangle, it indicates a decreasing trend in the data.
The Innovative Trend Analysis in Fig. 7a–c for minimum, maximum, and average temperatures from 1983 to 2016 for Bole station was computed. So, the data points exist in the top triangle and this shows an increasing trend and this strongly agrees with the MK2/MK3 result of positive values. It was found that the Z value of MK2/MK3 for minimum, maximum, and average temperatures for Bole station is 6.21/5.99, 2.49/2.6, and 6.09/6.14 respectively. The positive Kendall's Z value shows an upward trend and also implies an increasing trend over time. This indicates that there is a significant increase in the trend at a 5% level of significance since the computed p-value is less than the significant level (alpha = 0.05) (Table 1).
(a) Plot of ITA for minimum temperature from 1983–2016 for Bole Station, (b) Plot of ITA for maximum temperature from 1983–2016 for Bole Station, (c) Plot of ITA for average temperature from 1983–2016 for Bole Station, (d) Plot of ITA for minimum temperature from 1989–2016 for Entoto Station, (e) Plot of ITA for maximum temperature from 1989–2016 for Entoto Station, (f) Plot of ITA for average temperature from 1989–2016 for Entoto Station
The Innovative Trend Analysis in Fig. 7d–f for minimum, maximum, and average temperatures from 1989 to 2016 for the Entoto station was computed. So, the data points lay on 1:1 line and this shows no trend in the data and this result agrees with the MK1 result of positive values for minimum temperature and MK2/MK3 result of positive values for maximum, and average temperature. It was found that the Z value of MK1 for minimum temperature for the Entoto station is 1.64 whereas the Z value of MK2/MK3 for maximum and average temperatures for the Entoto station is 0.71/0.65, and 0.17/1.04 respectively. The positive Kendall's Z value shows an indicator of an increasing trend. But, the increasing trend is not significant at 5% significant level since the computed p-value is greater than the significant level alpha = 0.05. The Innovative Trend Analysis (ITA) Method was used for a further checkup to compare with the result of the MK1/MK2/MK3 and Sens slope estimate for the trend and significant test (Table 1; Fig. 7a-c, Fig. 7d-f).
Descriptive statistics of annual average temperature
Table 2 shows the minimum, maximum, and average temperature among the two stations. The average annual minimum temperature ranges from 7.66 °C to 11.61 °C, and from 4.14 °C to 10.02 °C for Bole and Entoto stations respectively. The average annual maximum temperature ranges from 22.65 °C to 24.52 °C for Bole station whereas, for Entoto station, it ranges from 16.18 °C to 19.67 °C. The average annual average temperature ranges between 15.20 °C to 17.87 °C, and between 10.7 °C to 14.64 °C for Bole and Entoto stations respectively.
Table 2 Descriptive statistics of annual average temperature
The result obtained in this study agrees with the findings of an earlier study, and whose results revealed that the mean annual maximum temperature ranges from 18.3 °C to 26.3 °C in Nefas Mewcha and Mekane Eyesus, while the mean annual minimum temperature ranges from 7.82 °C to 11.57 °C for Nefas Mewcha and Addis Zemen stations at south Gonder zone (Getachew 2018). Likewise, in this current study, the mean annual minimum temperature ranges from 8.56 °C to 9.82 °C for Entoto and Bole station whereas the mean annual maximum temperature ranges from 18.25 °C to 23.52 °C for Entoto and Bole stations, as well as the mean annual average temperature, ranges from 13.40 °C to 16.67 °C for Entoto and Bole station (Table 2). Conversely (Getachew 2018), the mean annual maximum temperature ranges between 26.9 °C and 32.2 °C for Addis Zemen stations at the south Gonder zone and the result disagrees with the present study of the mean annual the maximum temperature for Entoto and Bole station and this dissimilarity happens as a result of topographic variation and geographical location of the station.
From the study, it can be concluded that the trend analysis of annual temperature for Bole station shows a positive trend and statistical significance. As the computed p-value is lower than the alpha (significance level), one should reject the null hypothesis and accept the alternative hypothesis. On the other hand, the trend analysis of annual temperature for the Entoto station shows an increasing trend but not statistically significant. Hence, one cannot reject the null hypothesis, H0 as the computed p-value is greater than the significant level of alpha (0.05). Furthermore, the study showed that both the Mann–Kendall trend test and Sen's Slope estimator reveals that there is a tendency of temperature increase in the study area. Thus, the increasing trend of temperature due to climate change and other factors can lead to weather extremes in the capital city.
CSA:
Central Statistical Agency
Correction Factor
FDRE:
Federal Democratic Republic of Ethiopia
H0:
Alternative hypothesis
ITA:
Innovative Trend Analysis
Mann Kendall
NMA:
National Meteorological Agencies
NY:
TFPW:
Trend Free Pre-Whiting Process
UNFCCC:
UNICEF:
United Nations International Children's Emergency Fund
: Normalized test statistics
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We want to express our greatest appreciation to National Meteorological Agencies (NMA) for providing necessary data. The opinion expressed herein are the authors' own and do not necessarily express the view of NMA.
No funding has been received for this study.
Ethiopian Public Health Institute, P.O.Box: 1242, Addis Ababa, Ethiopia
Zinabu Assefa Alemu
Department of Energy and Environment, TERI School of Advanced Studies, 10 Institutional Area, Vasant Kunj, New Delhi, 110 070, India
Michael O. Dioha
ZA performed the study design, statistical analysis of results, data interpretation, and writing the manuscript. MO Conceptualization, draft review and edit of the manuscript. All authors read and approved the final manuscript.
Correspondence to Zinabu Assefa Alemu.
Ethics approval and consent to participant
Consent to publication
The authors have no competing interest to declare.
Alemu, Z.A., Dioha, M.O. Climate change and trend analysis of temperature: the case of Addis Ababa, Ethiopia. Environ Syst Res 9, 27 (2020). https://doi.org/10.1186/s40068-020-00190-5
Mann-kendall test
Sen's slope | CommonCrawl |
\begin{definition}[Definition:Normed Division Algebra]
Let $\left({A_F, \oplus}\right)$ be a unitary algebra where $A_F$ is a vector space over a field $F$.
Then $\left({A_F, \oplus}\right)$ is a normed divison algebra iff $A_F$ is a normed vector space such that:
:$\forall a, b \in A_F: \left \Vert{a \oplus b}\right \Vert = \left \Vert{a}\right \Vert \left \Vert{b}\right \Vert$
where $\left \Vert{a}\right \Vert$ denotes the norm of $a$.
\end{definition} | ProofWiki |
\begin{document}
\title[Small Prime Powers in the Fibonacci Sequence ]
{\Large{Small Prime Powers in the Fibonacci Sequence }} \author{J. Mc Laughlin} \address{Mathematics Department\\
University of Illinois \\
Champaign - Urbana, Illinois 61820} \email{[email protected]} \keywords{Fibonacci, binary recurrences} \subjclass{Primary:11B39, Secondary:11B37} \date{December,13,2000} \begin{abstract} It is shown that there are no non-trivial fifth-, seventh-, eleventh-, thirteenth- or seventeenth powers in the Fibonacci sequence.
For eleventh, thirteenth- and seventeenth powers an alternative
(to the usual exhaustive check of products of powers of fundamental units) method is used to overcome the problem of having a large number of independent units and relatively high bounds on their exponents.
It is envisaged that the same method can be used to decide the question of the existence of higher small prime powers in the Fibonacci sequence and that the method can be applied to
other binary recurrence sequences. The alternative method
mentioned may have wider applications. \end{abstract}
\maketitle
\section{Introduction } The Fibonacci sequence \{$F_{n}$\}$_{n=0}^{\infty}$ is defined by setting $F_{0} = 0, F_{1} = 1$ and, for $n \geq 2$, by setting $F_{n} = F_{n-1} + F_{n-2}$ .
Cohn~\cite{C64} and Wylie~\cite{W64} proved independently, by elementary means, that the only squares in the Fibonacci sequence are $F_{0}=0,F_{1}= F_{2} = 1$ and $F_{12} = 144$.
In~\cite{LF70}, London and Finkelstein used previous results on solutions to two diophantine equations to show that the only cubes in the Fibonacci sequence are $F_{0}=0$, $F_{1}= F_{2} = 1$ and $F_{6} = 8$. In \cite{LW81}, Lagarias and Weisser gave a complete determination of all Fibonacci numbers of the form $2^{a}3^{b}x^{3}$.
In~\cite{P83}, Peth\H{o} used linear forms in logarithms together with a computer search using congruence considerations to give an alternative proof of London and Finkelstein's result.
Peth\H{o} (\cite{P82}) and Shorey and Stewart (\cite{SS83}) proved independently that there are only finitely many perfect powers in any non-trivial binary recurrence sequence.
In~\cite{P81} Peth\H{o} states that if $F_{m} = x^{q}$, for some positive integers $x$ and $q$, then $q<10^{98}$.
In the same paper he also states that he used the same method that he used in~\cite{P83} to show that the only fifth powers in the Fibonacci sequence are $F_{0}=0$ and $F_{1}= F_{2} =1$.
In this paper the method outlined by Peth\H{o} in \cite{P83} is used as a starting point and then linear forms in logarithms together with the LLL algorithm are used to reprove the result for fifth powers and to prove that the only seventh-, eleventh-, thirteenth- or seventeenth powers in the Fibonacci sequence are $F_{0}=0$ and $F_{1}= F_{2} =1$. An alternative method (to the usual exhaustive check of products of powers of fundamental units) is used to complete the search in the case of eleventh, thirteenth- and seventeenth powers.
It is envisaged that the same method can be used to decide the question of the existence of higher small prime powers in the Fibonacci sequence and that the method can be applied to search for prime powers in other binary reurrence sequences. The alternative method
mentioned above may have wider applications.
Computations were performed using {\itshape Magma\/}, {\itshape Mathematica\/} and {\itshape Pari-gp\/} and were carried out on Sun ultra 5- and Sun ultra 10 computers.
\section{Elementary Considerations}
It is easy to show that {\allowdisplaybreaks \begin{align*} &\phantom{as}\\ F_{m+1}^{2}-F_{m+1}F_{m}-F_{m}^2 &= (-1)^{m}.\\ &\phantom{as} \end{align*} } Peth\H{o}, in~\cite{P83}, gives the following lemma: \begin{lemma} Let $q \geq 3$ be a positive integer. If $F_{m} = x^{q}$, for some positive integer $x$, then $m=0,1,2,6$ or $\exists$ a prime $p$,
$p|m$, such that $F_{p} = x_{1}^{q}$, for some positive integer $x_{1}$. \end{lemma} Hence it can be assumed that $m$ is an odd prime. Suppose $F_{m}=x^{n}$ and $F_{m+1} = y$, for some positive integers $x,y$ and some prime $n \geq 5$. Then $y^{2}-yx^{n}-x^{2n}+1=0$, and regarding this equation as a quadratic in $y$ and looking at its discriminant, it follows that $5x^{2n}-4=z^{2}$, for some positive integer $z$. Clearly $z=5v \pm 1$, for some positive integer $v$ and thus that {\allowdisplaybreaks \begin{align}\label{eq:2} &\phantom{as}\notag\\ x^{2n}=(2v)^{2}+(v\pm1)^2.\\ &\phantom{as}\notag \end{align} } It is clear that $(x,v)=1$ and it is not difficult to show that $x$ must be odd and $v$ must be even. Looking at~\eqref{eq:2} in $\mathbb{Z}[\,i\,]$, where $i = \sqrt{-1}$, it can further be shown that $v\pm1+2iv$ has to be an $n$th power and that there exists integers $A$ and $B_{1}$ such that {\allowdisplaybreaks \begin{align}\label{eq:3} &\phantom{as}\notag\\ &v\pm1+2iv = (A+B_{1}i)^{n} \Longrightarrow x^{2} = A^{2}+B_{1}^{2}.\\ &\phantom{as}\notag \end{align} }
$A$ has to be odd and $B_{1}$ has to be even ($=2B$, say). Suppose $n=2m+1$. Comparing real and imaginary parts of the first equation in \eqref{eq:3}, it follows that {\allowdisplaybreaks \begin{align}\label{eq:4} &\phantom{as}\notag\\ \pm1=B^{n}\sum_{j=0}^{m} (-1)^j2^{2j}\left(\binom{n}{2j}\left(\frac{A}{B}\right)^{n-2j}- \binom{n}{2j+1}\left(\frac{A}{B}\right)^{n-2j-1}\right).
\end{align} } Let {\allowdisplaybreaks \begin{align}\label{fn} f_{n}(x)= \sum_{j=0}^{m}(-1)^j2^{2j}\left(\binom{n}{2j}x^{n-2j}- \binom{n}{2j+1}x^{n-2j-1}\right). \end{align} } Assume $f_{n}(x)$ is irreducible (which can easily be proved for the cases examined: $ n = 5, 7, 11, 13$ and $17$) and let the roots be denoted $\theta_{1}, \cdots, \theta_{n}$. Let $\theta_{i}$ denote anyone of these roots.
From \eqref{eq:4} and \eqref{fn} we have that \begin{equation*} \pm1 =\prod_{k=1}^{n}\left(A-\theta_{k}B\right) = N_{K/Q}\left(A-\theta_{i}B\right). \end{equation*} Hence $A-\theta_{i}B$ is a unit in $\mathbb{Q}(\theta_{i})$. For the cases examined
it turns out that $f_{n}(x)$ has all its roots real and that
the rank of the group of units is $n-1$. \footnote{It may be easy to show that $f_{n}(x)$ is irreducible for all primes $n$ and, using Sturm's Theorem, that $f_{n}(x)$ has all real roots for every prime $n$, in which case the rank of the unit group is $n-1$ for all prime $n$. However, this is not examined here.}
Denote a set of fundamental units in $\mathbb{Q}(\theta_{i})$ by $\epsilon_{1}^{(i)}, \cdots, \epsilon_{n-1}^{(i)}$ and let $\beta_{i}:=A-\theta_{i}B$. Then there exists integers $u_{1},\cdots, u_{n-1}$ such that {\allowdisplaybreaks \begin{align}\label{eq:6} \beta_{i} = \pm \prod_{k=1}^{n-1}\epsilon_{k}^{(i)u_{k}}. \end{align} }
Let $U = \max_{1\leq k \leq n-1}|u_{k}|$. We next find an initial bound on $U$.
\section{Finding an initial bound on the exponents of the fundamental units}
Suppose $j$ is such that
$|\beta_{j}|= \min_{1\leq i \leq n}|\beta_{i}|$ and let $K= \mathbb{Q}(\theta_{j})$. Define the following numbers: {\allowdisplaybreaks \begin{align}\label{ceqs}
c_{1} &= \min_{i,\,\,i \ne j}|\theta_{j} - \theta_{i}|, &c_{2} = \max_{r \ne s \ne t \ne r}
\left|\frac{\theta_{t}-\theta_{r}}{\theta_{t}-\theta_{s}}\right|,\\ &\phantom{as} &\phantom{as} \notag\\ c_{2a} &= \max_{(s,t),\,\,\,j \ne s \ne t \ne j}
\left|\frac{\theta_{j}-\theta_{s}}{\theta_{j}-\theta_{t}}\right|,
&c_{3} = \max_{i,\,\,i \ne j}|\theta_{i} - \theta_{j}|. \phantom{asdd}\notag\\ &\phantom{as} &\phantom{as} \notag \end{align} } It is also assumed that {\allowdisplaybreaks \begin{align}\label{beq1}
|B| \geq \max \left\{4, \sqrt[\uproot{2}n]{2c_{2}}\frac{2}{c_{1}}\right\}.\\ &\phantom{as}\notag \end{align} } For $i \ne j$, we have that {\allowdisplaybreaks \begin{align}\label{bet1}
|B||\theta_{i} - \theta_{j}|
&\leq |\beta_{j}|+|\beta_{i}|
\leq 2|\beta_{i}| \Longrightarrow
\displaystyle{\frac{ |B|c_{1}}{2} \leq|\beta_{i} |}\\ &\phantom{as}\notag \end{align} } This implies that {\allowdisplaybreaks \begin{align}\label{bet2}
|\beta_{j} | &= \displaystyle{\frac{1}{\prod_{i=1,i \ne j}^{n}|\beta_{i} |}
\leq \left(\frac{2}{|B|c_{1}}\right)^{n-1}}. \end{align} } Similarly, it follows from \eqref{beq1} that
\begin{align}\label{bet3} &\phantom{as}\notag\\
|\beta_{i}|\leq |B||\theta_{i} - \theta_{j}| + |\beta_{j} |
\leq \displaystyle{
|B|\left(c_{3}+ \frac{c_{1}}{4c_{2}}\right)}=c_{4}|B|, \\&\phantom{as}\notag \end{align}
where\,\,$c_{4}=\displaystyle{c_{3}+c_{1}/(4c_{2})}$. Using \eqref{bet1}, \eqref{beq1} and \eqref{bet3}, it follows that
\begin{align}\label{bet4} &\phantom{as}\notag\\
|\log|\beta_{i} || \leq c_{5}\log|B|, \\&\phantom{as}\notag \end{align}
where
$c_{5} = \displaystyle{ \max \left\{1+|\log( c_{1}/2)|/ \log 4, 1+ |\log c_{4}| /\log 4\,\, \right\}}$.
Let $\{ i_{k}: k=1, \cdots, n-1\}$ denote the set $\{1,2,\cdots, n\} \setminus \{j\}$. From \eqref{eq:6}, {\allowdisplaybreaks \begin{align*} \left( \begin{matrix}
\log|\beta_{i_{1}}| \\ \vdots \\
\log|\beta_{i_{n-1}}| \end{matrix} \right) &= \left( \begin{matrix}
\log|\epsilon_{1}^{(i_{1})}| & \dots & \log|\epsilon_{n-1}^{(i_{1})}| \\ \vdots & \ddots & \vdots \\
\log|\epsilon_{1}^{(i_{n-1})}| & \dots & \log|\epsilon_{n-1}^{(i_{n-1})}| \end{matrix} \right) \left( \begin{matrix} u_{1}\\ \vdots\\ u_{n-1} \end{matrix} \right)\\ &\phantom{as}\\ &:= M\left( \begin{matrix} u_{1}\\ \vdots\\ u_{n-1} \end{matrix} \right), \\ &\phantom{as}\\ \Longrightarrow \left( \begin{matrix} u_{1}\\ \vdots\\ u_{n-1} \end{matrix} \right) &=M^{-1}\left( \begin{matrix}
\log|\beta_{i_{1}}| \\ \vdots \\
\log|\beta_{i_{n-1}}| \end{matrix} \right).\\ &\phantom{as} \end{align*} } Suppose $M^{-1}:=\left(m_{r,s}\right),\,\, 1\leq r\leq n-1,\,\,\, 1\leq s\leq n-1$. Then {\allowdisplaybreaks \begin{align*} &\phantom{as}\\ U
&\leq \max_{1 \leq r \leq n-1}\{|m_{r,1}|+\cdots+|m_{r,n-1}|\} \times
\max_{1 \leq t \leq n-1}\{|\log|\beta_{i_{t}}||\} \leq c_{6}\log|B|,\\ &\phantom{as} \end{align*} } where $c_{6}=
\max_{1 \leq r \leq n-1}\{|m_{r,1}|+\cdots+|m_{r,n-1}|\}\times c_{5}$. Thus {\allowdisplaybreaks \begin{equation}\label{eq:8}
\exp{(U/c_{6})} \leq |B|. \end{equation} } Let {\allowdisplaybreaks \begin{align}\label{lam}
\Lambda &=
\log\left|\frac{\theta_{j}-\theta_{k}}{\theta_{j}-\theta_{l}}\right|+ \sum_{r=1}^{n-1}u_{r}
\log\left|\frac{\epsilon_{r}^{(l)}}{\epsilon_{r}^{(k)}}\right|
=\log\left|\frac{\theta_{j}-\theta_{k}}{\theta_{j}-\theta_{l}}
\frac{A-\theta_{l}B}{A-\theta_{k}B}\right|. \end{align} }
By Siegel's identity, {\allowdisplaybreaks \begin{align}\label{sieg} &\phantom{as}\notag \\
\left|\frac{\theta_{l}-\theta_{k}}{\theta_{l}-\theta_{j}}
\frac{A-\theta_{j}B}{A-\theta_{k}B}\right|&=
\left|\frac{\theta_{j}-\theta_{k}}{\theta_{j}-\theta_{l}}
\frac{A-\theta_{l}B}{A-\theta_{k}B} - 1\right|.\\ &\phantom{as}\notag \end{align} } From the definition of the $\beta_{i}$'s, \eqref{ceqs}, \eqref{bet1} and \eqref{bet2}, it follows that {\allowdisplaybreaks \begin{align}\label{ineq1} &\phantom{as}\notag\\
&\left|\frac{\theta_{l}-\theta_{k}}{\theta_{l}-\theta_{j}}
\frac{A-\theta_{j}B}{A-\theta_{k}B}\right|
\leq c_{2}\left(\frac{2}{|B|c_{1}}\right)^{n}<\frac{1}{2}\\ &\phantom{as}\notag \end{align} } From \eqref{sieg} and\eqref{ineq1}, it follows that {\allowdisplaybreaks \begin{align}\label{ineq2} &\frac{\theta_{j}-\theta_{k}}{\theta_{j}-\theta_{l}} \frac{A-\theta_{l}B}{A-\theta_{k}B} >\frac{1}{2}>0\\ &\phantom{as}\notag\\ &\Longrightarrow \Lambda = \log \frac{\theta_{j}-\theta_{k}}{\theta_{j}-\theta_{l}} \frac{A-\theta_{l}B}{A-\theta_{k}B}\\ &\phantom{as}\notag\\
&\Longrightarrow |\Lambda|<2c_{2}\left(\frac{2}{|B|c_{1}}\right)^{n}.\\ &\phantom{as}\notag \end{align} } The last inequality follows, in the case $\Lambda>0$,
from \eqref{ineq1}, \eqref{sieg} and the fact that $e^{x}-1 > x$, for $x>0$. In the case $\Lambda < 0$, we also use \eqref{ineq2}. Note that $\Lambda \not = 0$, or else the right side of \eqref{sieg} is zero, implying, on the left side of \eqref{sieg}, that either $\theta_{j}$ is rational, or $\theta_{l}=\theta_{k}$. However, both of these are impossible, since $\theta_{j}$ is algebraic of degree $n$, and $f_{n}(x)$ has distinct roots.
Combining this last inequality for $|\Lambda|$ with~\eqref{eq:8} it follows that {\allowdisplaybreaks \begin{equation}\label{eq:10} \frac{c_{1}^n}{2^{n+1}c_{2}} \exp{(nU/c_{6})}
<\frac{1}{|\Lambda|}. \end{equation} } Next, the following theorem of Baker and W\H{u}stholz (\cite{BW93}) gives an upper bound on
$1/|\Lambda|$\,\,: \begin{theorem} Denote by $\alpha\sb{1},\cdots,\alpha\sb{n}$ algebraic numbers, not $0$ or $1$, by $\log\alpha\sb{1}$, $\cdots$, $\log\alpha\sb{n}$ determinations of their logarithms, by $d$ the degree over ${Q}$ of the number field ${Q}(\alpha\sb{1},\cdots,\alpha\sb{n})$ and by $b\sb{1},\cdots,b\sb{n}$ rational integers, not all $0$ and let $B=\max\{\vert b\sb{1}\vert ,\cdots,\vert b\sb{n}\vert ,e\sp{1/d}\}$.
Define $\log A\sb{i}= \max\{h(\alpha\sb{i}),(1/d)\vert \log\alpha\sb{i}\vert ,1/d\}$ ($1\le i\le n$), where $h(\alpha)$ denotes the absolute logarithmic Weil height of $\alpha$.
Assuming
the number $\Lambda=b\sb{1}\log\alpha\sb{1}+\cdots+b\sb{n}\log\alpha\sb{n}$ does not vanish, then {\allowdisplaybreaks \begin{align*} \vert \Lambda\vert \ge\exp\{-C(n,d)\log A\sb{1}\cdots\log A\sb{n}\log B\}, \end{align*} }
where $C(n,d)=18(n+1)!n\sp{n+1}(32d)\sp{n+2}\log(2nd)$. \end{theorem} Here {\allowdisplaybreaks \begin{align*} h(\alpha)=\frac{1}{[\mathbb{Q}(\alpha):\mathbb{Q}]}
\log \left|\, a_{0}\,\prod_{r=1}^{s}\max\{1,|\alpha^{i}|\}\right|, \end{align*} } where the minimal polynomial of $\alpha$ has leading coefficient $a_{0}$ and $\alpha = \alpha^{1},\cdots,\alpha^{s}$ are the conjugates of $\alpha$.
In our application, it can be seen from \eqref{lam} that
$n$ has the same meaning as previously, that
$b_{1} = 1$, that $\alpha_{1}=
\left|(\theta_{j}-\theta_{k})/(\theta_{j}-\theta_{l})\right|$, and, for $j=2,\cdots,n$, that $b_{j} = u_{j-1}$ and
$\alpha_{j}=\left|\epsilon_{j-1}^{(l)}/ \epsilon_{j-1}^{(k)}\right|$.
Let $\gamma_{i} = \epsilon_{i}^{(l)}/ \epsilon_{i}^{(k)}$, with conjugates $\gamma_{i}=\gamma_{i}^{1},\cdots,\gamma_{i}^{s}$. Since $\gamma_{i}$ is a unit its minimal polynomial has its leading coefficient $a_{0} = 1$ and since $f_{n}(x)$ has all real roots,
$|\gamma_{i}^{r}|= \pm \gamma_{i}^{r}$. {\allowdisplaybreaks \begin{align*}
\displaystyle{|\gamma_{i}^{r}| \leq \eta_{i} :=
\frac{\max\{|\epsilon_{i}^{(r)}|: 1 \leq r \leq n\}}
{\min\{|\epsilon_{i}^{(r)}|: 1 \leq r \leq n\}}}
\Longrightarrow h(\gamma_{i}) \leq \log \eta_{i}.
\end{align*} } Let $\delta_{jkl} =\displaystyle{(\theta_{j}-\theta_{k})/(\theta_{j}-\theta_{l})}$ and suppose the minimum polynomial of $\delta_{jkl}$ is $g(x)$. The conjugates of $\delta_{jkl}$ are bounded by $c_{2a}$. For small primes $n$ the Galois group associated to the polynomial $f_{n}(x)$ can be determined using a computer Algebra system like Magma. Let $p(x)=\prod(x-\delta_{rst})(\theta_{r}-\theta_{t})$ where the product is taken over all conjugates $\delta_{rst}$ of $\delta_{jkl}$.
$p(x)\in \mathbb{Z}[\,x\,]$ and $g(x)\,|\,p(x)$. For low values of $n$, $p(x)$ can be calculated numerically and $g(x)$ and thus $d_{g} =$ degree$(g(x))$ and $a_{0}$ can be determined explicitly. In fact, for the cases examined, $d_{g}= n(n - 1)$ and $a_{0}=2^{2(n-1)}$. Then {\allowdisplaybreaks \begin{align*} h(\delta_{jkl}) \leq \frac{\log 2^{2(n-1)}}{ n(n - 1)} + \log c_{2a}. \end{align*} } {\allowdisplaybreaks \begin{align*}
\left[\mathbb{Q}\left( \displaystyle{\delta_{jkl}, \epsilon_{1}^{(l)}/ \epsilon_{1}^{(k)}, \cdots, \epsilon_{i}^{(l)}/ \epsilon_{i}^{(k)} }\right):\mathbb{Q}\right] &\leq
[\mathbb{Q}(\theta_{1},\cdots,\theta_{n}):\mathbb{Q}] =:D. \\&\phantom{as} \end{align*} } For the values of $n$ examined, $D = n(n-1)$.
Let $C(n,D)$ be as defined in the theorem. Then {\allowdisplaybreaks \begin{align*}
|\Lambda|&>\exp\left(-C(n,D)\prod_{i}\log \eta_{i} \left(\frac{\log 2^{2(n-1)}}{ n(n - 1)} + \log c_{2a}\right) \log U\right)\\ &=\exp(-c_{7}\log U) = U^{-c_{7}},\notag \end{align*} } where $c_{7}= C(n,D)\prod_{i}\log \eta_{i} (\frac{\log 2^{2(n-1)}}{ n(n - 1)} + \log c_{2a})$. Combining this inequality with \eqref{eq:10} it follows that \begin{equation*} \frac{c_{1}^n}{2^{n+1}c_{2}} \exp{(nU/c_{6})}<U^{c_{7}}. \end{equation*}
If it is assumed that $|U| \geq 4$ then \[\frac{U}{\log U} \leq \frac{c_{6}c_{7}}{n}+
\frac{c_{6}\left|\log \displaystyle{\frac{c_{1}^n}{2^{n+1}c_{2}}}\right|} {n\log 4 } . \] Thus an upper bound can be found for $U$. Denote this upper bound by $K_{3}$. This bound is generally too large to enable the remaining cases to be tested so it is next reduced, using the LLL algorithm.
\section{Reducing the bound } To reduce the bound a version of the LLL algorithm is applied, as outlined in the paper of Tzanakis and De Weger~\cite{TD89}. Using the notation of their paper:
Let \begin{equation}\label{eq:14a} \Lambda = \delta + a_{1}\mu_{1}+ \cdots + a_{q}\mu_{q}, \end{equation} where the $a_{i}$'s are integers and the $\mu$'s and $\delta$ are real,
with $\delta \ne 0$. Let $A = \max_{1 \leq i \leq q}|a_{i}|$. $K_{1},K_{2}$ and $K_{3}$ are positive numbers satisfying \begin{equation}\label{eq:14}
|\Lambda|<K_{1}\exp (-K_{2}A), A<K_{3}. \end{equation} Choose $c_{0}= \sigma_{1}K_{3}^{q}$ where $\sigma_{1}>1$. Consider the lattice $\Gamma$ associated with the matrix \begin{equation*} \mathcal{A}= \left( \begin{matrix} 1 &0& \dots & 0 &0\\ 0&1&\dots&0&0\\ \vdots &\vdots& \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & 0 \\ \text{[}c_{0}\mu_{1}] &[c_{0}\mu_{2}] & \dots & [c_{0}\mu_{q-1}] & [c_{0}\mu_{q}] \end{matrix} \right) \end{equation*} Find a reduced basis $\textbf{$b_{1}$},\dots,\textbf{$b_{q}$}$ for this basis and let $\mathcal{B}$ be the matrix associated with this basis. \[ \text{Let }\textbf{x}=\left( \begin{matrix} 0 \\ \vdots \\ 0\\ \text{[}-c_{0}\delta] \end{matrix} \right) \in \mathbb{Z}^{q}\text{ and let }\textbf{s}= \left( \begin{matrix} s_{1} \\ \vdots \\ s_{q-1}\\ s_{q} \end{matrix} \right) = \mathcal{B}^{-1}\textbf{x}.
\] Let $||y||$ denote the distance from $y$ to the nearest integer. \begin{proposition} Let $i^{*} = max\{i:1 \leq i \leq q$ and $s_{i} \notin \mathbb{Z}\}$. If \begin{equation*}
2^{-(q-1)/2}||s_{i^{*}}|||\textbf{$b_{1}$}| \geq\sqrt{\left(4q^{2}+3q-\frac{3}{4}\right)}K_{3}, \end{equation*} Then every solution of~\eqref{eq:14a} satisfying~\eqref{eq:14} satisfies \begin{equation*} A<\frac{1}{K_{2}}\log \left( \frac{c_{0} K_{1}}{q K_{3}}\right). \end{equation*} \end{proposition} From \eqref{eq:10}, this proposition can be applied with $a_{r} = u_{r}$, for $1 \leq r \leq n-1$, and \begin{align*} &q=n-1,& &A=U,& &K_{1}= \frac{2^{n+1}c_{2}}{c_{1}^n},& \\ &K_{2}= \frac{n}{c_{6}},& &\delta =
\log\left|\frac{\theta_{j}-\theta_{k}}{\theta_{j}-\theta_{l}}\right|,& &\mu_{r}=
\log\left|\frac{\epsilon_{r}^{(l)}}{\epsilon_{r}^{(k)}}\right|.& \end{align*} Once a new lower bound is found the proposition is then applied again by now setting $K_{3}=1/K_{2}\log \left( c_{0} K_{1}/(q K_{3})\right)$
and this is repeated until the bound is reduced as far as possible.
\section{Completing the Search}
Once $K_{3}$ has been reduced as much as the LLL algorithm will allow,
a computer search is done of products of powers of the fundamental units, with these powers bounded by this final value of $K_{3}$, as on the right hand side of~\eqref{eq:6}, to see if any of these products have the form of the left side of~\eqref{eq:6}. However, see later for the cases $n=11$,$13$ and $17$.
Remark: These calculations have to be carried out for each value of $j$ -- in other words all of the $c_{i}$'s, apart from $c_{2}$ are dependent on the choice of $j$.
These theoretical results are now applied for $n=5,7,11,13$ and $17$.
\section{Small prime powers in the Fibonacci sequence} Remark: As noted above, most of the $c_{i}$ depend on the choice of $j$ and hence are given as vectors (the first component being the value got by letting $j = 1$ and so on). For a fixed $j$, $k$ is chosen to be $j+1 \mod{n}$ and $l$ is chosen to be $j+2 \mod{n}$. In what follows, $\theta$ denotes a root of $f_{n}(x)$.
1)The case $n=5$: $D=20$ and \[f_{5}(x)=-16 + 80x + 40x^2 - 40x^3 - 5x^4 + x^5. \] The zeroes of $f_{5}(x)$ are \[ \{ -4.64105,-1.1869,0.185992,1.75785,8.88411\}. \] A set of fundamental units in $\mathbb{Q}(\theta)$ is {\allowdisplaybreaks \begin{align*}
&\{1/16\theta^3 + 3/16\theta^2 - 1/4\theta - 1/4, \\
& 3/32\theta^4 - 27/32\theta^3 - 1/4\theta^2 + 25/8\theta - 1/2,\\
& 5/16\theta^4 - 9/8\theta^3 - 223/16\theta^2 - 15/2\theta + 33/4, \\
& 1/32\theta^4 - 13/32\theta^3 - 1/8\theta^2 + 87/8\theta - 2\}. \end{align*} } {\allowdisplaybreaks \begin{align*}
& c_{1} = \{3.4541, 1.3728, 1.3728, 1.5718, 7.1262 \},\\
&c_{2} = 7.3356,\\
&c_{2a} = \{3.9156, 7.3356, 6.3356, 4.5336, 1.8979 \},\\
&c_{3} = \{ 13.5251, 10.0710, 8.6981, 7.1262, 13.5251 \},\\
&c_{5} = \{2.8850, 2.6694, 2.5642, 2.4219, 2.8916 \},\\
&c_{6} = \{1.8086, 1.6734, 1.4252, 1.3461, 1.4617 \},\\
&c_{7} = \{1.7353\times10^{32}, 2.3987\times10^{32},
2.2438\times10^{32}, 1.89018\times10^{32}, \\ &\phantom{asfdsfsdafsadfasfsdfsadf} 9.70057\times10^{31} \},\\
&K_{1} = \{0.954799, 96.2577, 96.2577, 48.9276, 0.0255452\}\text{ and } \\
&K_{2} = \{2.76452, 2.98782, 3.50828, 3.71432, 3.42053\}. \end{align*} } Initially, $K_{3} = \{10^{34},10^{34},10^{34},10^{34},10^{34}\}$,
and eventually $K_{3}$ $=$ $\{11,12,$ $10,10,8\}$. Finally, a check on products of powers of fundamental units, with the power being bounded in absolute value by $12$ shows that there are no fifth powers in the Fibonacci sequence other than the trivial ones.
2)The case $n=7$: \, $D=42$ and \[f_{7}(x) =64 - 448 x - 336 x^2 + 560 x^3 + 140 x^4 - 84 x^5 - 7 x^6 + x^7. \] The zeroes of $f_{7}(x)$ are \[ \{ -6.68663,-2.19286,-0.804777,0.132665,1.13197,2.88015,
12.5395\}. \]
A set of fundamental units for $\mathbb{Q}(\theta)$ is {\allowdisplaybreaks \begin{align*} \{ &5/512\theta^6 - 33/512\theta^5 - 211/256\theta^4 + 59/64\theta^3 + 31/8\theta^2 - 73/32\theta + 5/16,\\
& 1/512\theta^6 - 7/512\theta^5 - 11/64\theta^4 + 25/64\theta^3 + 27/32\theta^2 - 39/32\theta + 1/4, \\
& 3/512\theta^6 - 37/512\theta^5 - 13/64\theta^4 + 177/64\theta^3 - 171/32\theta^2 + 87/32\theta + 1/4,\\ & 5/256\theta^6 - 7/64\theta^5 - 433/256\theta^4 - 9/32\theta^3 + 69/32\theta^2 + 9/8\theta - 1/16,\\ & 7/256\theta^5 - 33/256\theta^4 - 81/32\theta^3 - 75/32\theta^2 + 87/16\theta + 11/16, \\
& 1/256\theta^6 - 7/256\theta^5 - 5/16\theta^4 + 11/16\theta^3 + 39/16\theta^2 - 21/16\theta - 2\}. \end{align*} } {\allowdisplaybreaks \begin{align*} &c_{1}=\{4.49377, 1.38808, 0.937441, 0.937441, 0.999309, 1.74818, 9.65932\},\\ &c_{2}= 14.2348,\\ &c_{2a}=\{4.27839, 10.6135, 14.2348, 13.2348, 11.4154, 5.52537, 1.99042\},\\ &c_{3}=\{19.2261, 14.7323, 13.3443, 12.4068, 11.4075, 9.65932, 19.2261\},\\ &c_{5}=\{3.13545, 2.94165, 2.86996, 2.81749, 2.75706, 2.63825, 3.13883\},\\ &c_{6}=\{2.05127, 2.13505, 2.08302, 2.04265, 1.8492, 1.61462, 2.10145\},\\ &c_{7}=\{1.03176 \times 10^{47}, 1.59932 \times 10^{47}, 1.78271 \times 10^{47},
1.7372 \times 10^{47},\\ &1.6448 \times 10^{47}, 1.19154 \times 10^{39},
5.53721 \times 10^{39}\}\\ &K_{1} = \{0.0984719, 367.017, 5727.71, 5727.71, 3661.77, 73.0283, 0.000464476\},\\ &K_{2}= \{3.41253, 3.27862, 3.36051, 3.42691, 3.78542, 4.33539, 3.33104\} \end{align*} } $K_{3}$ is initially
$\{10^{49},10^{49},10^{49},10^{49}, 10^{49},10^{49},10^{49}\}$ and eventually $\{16, 17,$ $ 17, 18,
17, 15, 16\}$. A check on products of powers of fundamental units, with the power being bounded in absolute value by $18$ produces no such products which are linear in $\theta$ and thus
shows that there are no seventh powers in the Fibonacci sequence other than the trivial ones .
Remark: A set of fundamental units for the cases $n=11,13$ and $17$ are not included here because they are so large. However, they can be found in Appendix I. They can also be easily generated in GP/PARI. For $n=11$, the following code produces a set of fundamental units:\footnote{ Guillaume Hanrot pointed out to me that my use of the GP/PARI \emph{bnfinit} command in a preprint version of this paper (where I used it without the flag ``1'') did not necessarily produce a system of fundamental units. Thus it is necessary to a little more to show, for each $n$ in question, that the system of units I used, $\{\epsilon_{i}\}_{i=1}^{n-1}$, is indeed a fundamental system. Let the system of fundamental units generated by \emph{bnfinit}($f_{n},1$) in GP/PARI, version 2.1.0, be denoted by $\{\alpha_{i}\}_{i=1}^{n-1}$. For the cases $n=5$ and $n=7$, this set equals $\{\epsilon_{i}\}_{i=1}^{n-1}$. For $n=11$, \[ \{\epsilon_{i} \}_{i=1}^{10} = \left \{ \alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{8}, \alpha_{9},\alpha_{7},\frac{\alpha_{2}\alpha_{6}\alpha_{7}} {\alpha_{3}\alpha_{4}},\alpha_{10} \right \}. \] For $n=13$, \[ \{\epsilon_{i} \}_{i=1}^{12} = \left \{ \alpha_{8},\alpha_{5},\alpha_{2},\alpha_{1},\alpha_{3},\alpha_{4}, -\frac{\alpha_{3}\alpha_{4}} {\alpha_{6}}, \alpha_{7}, \frac{1} {\alpha_{10}}, -\alpha_{4}\alpha_{9}, \frac{\alpha_{1}\alpha_{5}\alpha_{6}\alpha_{11}} {\alpha_{3}\alpha_{4}\alpha_{7}}, \alpha_{12} \right \}. \] For $n=17$, \begin{multline*} \{\epsilon_{i} \}_{i=1}^{16} = \biggm \{ \frac{\alpha_{1}^{2}\alpha_{2}\alpha_{3}^{2}\alpha_{6}\alpha_{7}^{2} \alpha_{11}\alpha_{16}} {\alpha_{4}^{2}\alpha_{5}\alpha_{8}^{2}\alpha_{9}\alpha_{12}}, \alpha_{2},\alpha_{4},\alpha_{3},\alpha_{5},\alpha_{6},\alpha_{7}, \alpha_{8},\alpha_{9},\alpha_{12},\\ \frac{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{11}} {\alpha_{4}}, \alpha_{13},\alpha_{10},\alpha_{14}, -\frac{\alpha_{2}} {\alpha_{10}\alpha_{15}}, -\frac{1} {\alpha_{1}} \biggm \}. \end{multline*} Each of these sets is clearly also a set of fundamental units.
}
\begin{align*} \{f&= Pol(1024 - 11264x - 14080x^2 + 42240x^3 + 21120x^4\\& - 29568x^5 - 7392x^6 + 5280x^7 + 660x^8 - 220x^9 - 11x^{10} + x^{11},x);\\ &bnfinit(f,1)[8][5] \} \end{align*}
3)The case $n=11$: \, $D=110$ and \begin{multline*} f_{11}(x) =1024 - 11264\,x - 14080\,x^2 + 42240\,x^3 + 21120\,x^4 -
29568\,x^5\\ - 7392\,x^6 + 5280\,x^7 + 660\,x^8 -
220\,x^9 - 11\,x^{10} + x^{11}. \end{multline*} The zeroes of $f_{11}(x)$ are \begin{multline*} \{ -10.6902,\,-3.93193,\,-2.12056,\,-1.16928,\,-0.496752,\,0.0843495,\\
0.680024,\,1.40783,\,2.51488,\,4.91791,\,19.8037\}. \end{multline*}
\allowdisplaybreaks{ \begin{multline*} c_{1}=\{6.75826,\, 1.81137,\, 0.951281,\, 0.672528,\, 0.581101,\, 0.581101,\\
0.595674, \, 0.727806,\,1.10705,\, 2.40303,\, 14.8858\}. \end{multline*} $c_{2}=34.9345$. \begin{multline*} c_{2a}=\{4.5121,\,13.1037,\, 23.0471,\, 31.1853,\, 34.9345,\, 33.9345,\, 32.1043,\\ 25.2758,\,
15.6171,\, 6.49517,\, 2.04852 \}. \end{multline*} \begin{multline*} c_{3}=\{30.4939,\, 23.7356,\, 21.9243,\, 20.973,\, 20.3005,\, 19.7194,\, 19.1237,\\ 18.3959,\, 17.2888,\, 15.6081,\, 30.4939 \}. \end{multline*} \begin{multline*} c_{5}=\{3.46637,\, 3.28489,\, 3.22745,\, 3.1954,\, 3.17187,\, 3.15092,\, 3.12881,\\ 3.10086,\, 3.05622,\, 2.98291,\, 3.46774 \}. \end{multline*} \begin{multline*} c_{6}=\{3.03767,\, 2.38601,\, 3.34699,\, 3.39081,\, 2.34158,\, 3.44318,\, 3.15266,\\ 3.38847,\, 2.68879,\, 2.40831,\, 3.06079 \}. \end{multline*} \begin{multline*} c_{7}=\{2.8731\times{10}^{78},\,4.7491\times{10}^{78},\,5.7427\times{10}^{78},\,
6.2748\times{10}^{78},\\ 6.4746\times{10}^{78},\, 6.4235\times{10}^{78},\,
6.326\times{10}^{78},\,5.9051\times{10}^{78},\,\\5.0579\times{10}^{78},\,
3.5141\times{10}^{78},\, 1.4836\times{10}^{78} \} \end{multline*} \begin{multline*} K_{1} = \{0.0001,\,207.753,\,247867.,\,1.1241\times{10}^7,\,5.6085\times{10}^7,\,
5.6085\times{10}^7, \\ 4.271\times{10}^7,\, 4.7146\times{10}^6,\,46750.2,\,9.2739,\,1.7994\times{10}^{-8}š\}. \end{multline*} \begin{multline*} K_{2}= \{3.6212,\, 4.61021,\, 3.28653,\, 3.24406,\, 4.69768,\, 3.19472,\, 3.48911,\,\\
3.2463,\, 4.09106,\, 4.56752,\, 3.59385\}. \end{multline*} } Initially, \begin{multline*} K_{3}= \{ \,{10}^{81},\,\,\,{10}^{81},\,{10}^{81},\,{10}^{81},
\,{10}^{81},\,{10}^{81},\,\,\,{10}^{81},\,{10}^{81},
\,{10}^{81},\,{10}^{81},\,\, 10^{80}\},
\end{multline*} and finally $K_{3}= \{32,\, 27,\, 41,\, 43,\, 29,\, 47,\, 40,\, 43,\, 32,\, 26,\, 28\}$.
Checking all products of powers of ten independent units and with these powers being bounded absolutely by $47$
would take some time but there is a much quicker way, which is now described. The same method is applied for $n=13$ and $17$.
Let $p = \sum_{i=0}^{10}a_{i}\theta^{i}$, $q = \sum_{i=0}^{10}b_{i}\theta^{i}$ be two numbers in $\mathbb{Q}(\theta)$. Then $p \times q = \sum_{i=0}^{10}c_{i}\theta^{i}$,
for some $c_{i} \in \mathbb{Q}$ and if the $a_{i}$ are bounded in absolute value by $K_{a}$ and the $b_{i}$ are bounded in absolute value by $K_{b}$ then a bound for the $c_{i}$ can be found in terms of $K_{a}$ and $K_{b}$. In fact the $c_{i}$ are bounded by $16564181057933828K_{a}K_{b}$. Another way to see this is to regard $p$ and $q$ as polynomials in $\theta$ and reduce their product modulo $f_{11}(\theta)$. Let $M = 16564181057933828$.
Next, pick out the coefficient $v$ that is largest in absolute value in the following list of powers of the fundamental units:
$\{\epsilon_{r}^{i}: 1 \leq r \leq 10, -47 \leq i \leq 47 \}$, regarding the members of this set as polynomials in $\theta$ (This involves checking $950$ units rather than $95^{10}$). We have {\allowdisplaybreaks \begin{multline*} v=1/512 \times (20107468130152762104958655475357868066478593\\ 506162563506545 02987105724151326017006680926209502\\ 499326050998267664485654586806568806547). \end{multline*} } Thus the coefficient of any power of $\theta$ in any expression of the form $\prod_{r=1}^{10}\epsilon_{r}^{i_{r}}$ where $-47 \leq i_{r} \leq 47$ is less than $M^{9}v^{10}$. In particular, $\beta_{i} = A + B\theta_{i}$ must have $A$ and $B$ less than $M^{9}v^{10}$ and thus, from~\eqref{eq:3}, $x \leq \sqrt{5} M^{9}v^{10}$ and thus $F_{m} = x^{11} \leq (\sqrt{5} M^{9}v^{10})^{11} \backsimeq 7.7943\times 10^{15864}$. Since $F_{m} \backsimeq ((1+\sqrt{5})/2)^{m}/\sqrt{5} \Longrightarrow m \leq 75913$. The odd terms in the Fibonacci sequence up to $F_{75913}$ can be checked directly to see if any are eleventh powers and no non-trivial eleventh powers are found.
More efficiently, one can use the standard trick that
if an eleventh power exists, it must be an eleventh power residue modulo every prime. Choose, say, ten primes $p_{1}, \cdots, p_{10} \equiv 1 (\mod{11})$ and calculate the eleventh power residues in each case. Using the fact that $F_{i+3} = 3F_{i+1} - F_{i-1}$ (it being necessary to consider $F_{j}$ for $j$ odd) and working modulo each of the ten primes in parallel, it is a matter of seconds to check up $j=75913$ for eleventh powers (by checking if $F_{j} \mod{p_{i}}$ is an eleventh power residue for $p_{i}$, for each $i$).
4)The case $n=13$: \, $D=156$ and \begin{multline*} f_{13}(x) =-4096 + 53248\,x + 79872\,x^2 - 292864\,x^3\\ - 183040\,x^4 +
329472\,x^5 + 109824\,x^6 - 109824\,x^7\\ - 20592\,x^8 +
11440\,x^9 + 1144\,x^{10} - 312\,x^{11} - 13\,x^{12} +
x^{13}. \end{multline*} The roots of $f_{13}(x)$ are \begin{multline*} \{ -12.6754,\,-4.75486,\,-2.68723,\,-1.64838,\,-0.960337,\,-0.41792,\\
0.0713607,\,0.569323,\,1.14244,\,1.90337,\,3.13069,\,5.90001,\,
23.4269\}. \end{multline*} \begin{multline*} c_{1}=\{7.92054,\, 2.06763,\, 1.03885,\, 0.688043,\, 0.542417,\, 0.48928,\\ 0.48928,\, 0.497963,\, 0.573112,\, 0.760936,\, 1.22732,\, 2.76932,\,17.52696209\,\}. \end{multline*} $c_{2}= 48.7346$. \begin{multline*} c_{2a}=\{4.55807,\, 13.63,\, 25.1375,\, 36.4444,\, 44.9604,\, 48.7346,\, 47.7346,\\ 45.9023,\,
38.8833,\, 28.2856,\, 16.537,\, 6.70758,\, 2.05982 17.52\}. \end{multline*} \begin{multline*} c_{3}=\{36.1023,\, 28.1818,\, 26.1142,\, 25.0753,\, 24.3873,\, 23.8449,\, 23.3556,\\ 22.8576,\, 22.2845,\, 21.5236,\, 20.2963,\, 18.5754,\, 36.1023.52\}. \end{multline*} \begin{multline*} c_{5}=\{3.58782,\, 3.40862,\, 3.35353,\, 3.3242,\, 3.30411,\, 3.28788,\, 3.27293,\\ 3.25738,\,
3.23908,\, 3.21405,\, 3.17179,\, 3.10821,\, 3.588823.52\}. \end{multline*} \begin{multline*} c_{6}=\{5.2067,\, 3.88525,\, 4.86669,\, 3.30557,\, 3.62425,\, 3.62461,\, 3.28336,\\ 3.98813,\,
2.97576,\, 3.41762,\, 3.86076,\, 3.09418,\, 4.097533.52\}. \end{multline*} \begin{multline*} c_{7}=\{1.30674\times{10}^{95},\,2.18839\times{10}^{95},\,2.68104\times{10}^{95},\,
2.97999\times{10}^{95},\, \\3.14901\times{10}^{95},\,3.21389\times{10}^{95},\,
3.1972\times{10}^{95},\,3.1657\times{10}^{95},\\ 3.03213\times{10}^{95},\,
2.77601\times{10}^{95},\,2.34399\times{10}^{95},\,\\1.6177\times{10}^{95},\,
6.67449\times{10}^{94}\}. \end{multline*} \begin{multline*} K_{1} = \{1.65365\times{10}^{-6},\,63.2569,\,486473.,\,1.03099\times{10}^8,\, 2.26948\times{10}^9, \\ 8.66973\times{10}^9,\,8.66973\times{10}^9,\,
6.89763\times{10}^9,\,1.10954\times{10}^9,\\ 2.78439\times{10}^7,\, 55693.3,\, 1.41715,\, 5.421\times{10}^{-11}\}. \end{multline*} \begin{multline*} K_{2}= \{2.49678,\, 3.34599,\, 2.67122,\, 3.93275,\, 3.58695,\, 3.5866,\, 3.95935,\\ 3.25967,\,
4.36864,\, 3.80382,\, 3.36721,\, 4.20144,\, 3.17265 \}. \end{multline*} Initially, \begin{multline*} K_{3} =\{{10}^{98},\, {10}^{98},\, {10}^{98},\, {10}^{98},\,
{10}^{98},\, {10}^{98},\, {10}^{98},\, {10}^{98},\, \\
{10}^{98},\, {10}^{98},\, {10}^{98},\, {10}^{97},\,
{10}^{97}\} \end{multline*} and eventually $K_{3}=\{53,\, 45,\, 58,\, 49,\, 55,\, 55,\, 43,\, 52,\, 44,\, 50,\, 55,\, 41,\, 47\}$.
With the same notation that was used for the case $n=11$, {\allowdisplaybreaks \begin{multline*} v=1/4096 \times (93158647867090656840416856127516852294230637148702\\ 851086532124807140957259454209260273172314431029910278429059765\\ 08393206322152405473550058771766196947352038793187460444181), \end{multline*} } $M=316357820342343521286$, and so $F_{m} \leq (\sqrt{5}M^{11}v^{12})^{13} \backsimeq
5.4892\times 10^{29199}$ and so $m \leq 139720$. A check shows that $F_{j}$ is not a thirteenth power for $3 \leq j \leq 139720$.
5)The case $n=17$: \,$D=272$ and \begin{multline*} f_{17}(x) =-65536 + 1114112\,x + 2228224\,x^2 - 11141120\,x^3 -
9748480\,x^4\\ + 25346048\,x^5 + 12673024\,x^6 -
19914752\,x^7 - 6223360\,x^8\\
+ 6223360\,x^9 +
1244672\,x^{10} - 792064\,x^{11} - 99008\,x^{12} +\\
38080\,x^{13} + 2720\,x^{14} - 544\,x^{15} - 17\,x^{16} +
x^{17}. \end{multline*} The zeroes of $f_{17}(x)$ are \begin{multline*} \{ -16.6323,\,-6.36449,\,-3.75619,\,-2.50343,\,-1.72576,\,-1.16412,\\
-0.712712,\,-0.317684,\,0.0545603,\,0.430621,\,0.838223,\,1.31512,\\
1.92569,\,2.80429,\,4.30707,\,7.83505,\,30.6661\}. \end{multline*} {\allowdisplaybreaks \begin{multline*} c_{1}=\{10.2678,\,2.60829,\,1.25276,\,0.777669,\,0.561638,\,0.451412,\\
0.395027,\,0.372245,\, 0.372245,\,0.376061,\,0.407602,\,0.476899,\,
0.61057,\,\\ 0.878601,\,1.50278,\, 3.52799,\,22.831 \}, \end{multline*} $c_{2}= 83.2349$. \begin{multline*} c_{2a}=\{4.60646,\,14.1973,\,27.4771,\,42.6525,\,57.6738,\,70.5125,\,79.4345,\\
83.2349,\, 82.2349,\,80.4005,\,73.1789,\,61.5455,\,47.0714,\,31.7115,\\
17.5402,\,6.93523,\,2.07167 \}. \end{multline*} \begin{multline*} c_{3}=\{47.2984,\,37.0306,\,34.4223,\,33.1695,\,32.3918,\,31.8302,\,31.3788,\\
30.9838,\, 30.6115,\,30.2355,\,29.8279,\,29.351,\,28.7404,\,27.8618,\\
26.359,\,24.4674,\,47.298467 \}. \end{multline*} \begin{multline*} c_{5}=\{3.78233,\,3.60548,\,3.55271,\,3.52594,\,3.50882,\,3.49619,\,3.48589,\\
3.47675,\,3.46803,\,3.45911,\,3.44932,\,3.4377,\,3.42255,\,3.40018,\\
3.36024,\,3.30671,\,3.782967 \}. \end{multline*} \begin{multline*} c_{6}=\{6.96297,\,6.95734,\,5.89133,\,6.24564,\,4.71335,\,4.94999,\,5.6139,\\
7.93478,\,7.87795,\,4.98754,\,6.01398,\,7.84567,\,6.87894,\,5.26067,\\
5.08113,\,5.13209,\,8.364577 \}. \end{multline*} \begin{multline*} c_{7}=\{2.15293\times{10}^{126},\,3.65902\times{10}^{126},\,
4.54254\times{10}^{126},\,5.13093\times{10}^{126},\\
5.53464\times{10}^{126},\,5.80357\times{10}^{126},\,
5.96299\times{10}^{126},\,6.02552\times{10}^{126},\\
6.00935\times{10}^{126},\,5.97916\times{10}^{126},\,
5.85324\times{10}^{126},\,5.62158\times{10}^{126},\\
5.26283\times{10}^{126},\, 4.73432\times{10}^{126},\,
3.94195\times{10}^{126},\,2.7004\times{10}^{126},\\
1.08369\times{10}^{126} \}. \end{multline*} \begin{multline*} K_{1} = \{1.3922\times{10}^{-10},\,1.82303,\,473234.,\,1.56794\times{10}^9,\,
3.9635\times{10}^{11},\\ 1.62592\times{10}^{13},\,1.57104\times{10}^{14},\,
4.3128\times{10}^{14},\,4.3128\times{10}^{14},\,3.62626\times{10}^{14},\\
9.22206\times{10}^{13},\,6.39148\times{10}^{12},\,9.57958\times{10}^{10},\,
1.96963\times{10}^8,\,21460.7,\\ 0.01073,\,1.75352\times{10}^{-16} \}. \end{multline*} \begin{multline*} K_{2}= \{2.44149,\,2.44346,\,2.88559,\,2.7219,\,3.60678,\,3.43435,\,3.0282,\,\\
2.14247,\,2.15792,\, 3.40849,\,2.82675,\,2.1668,\,2.47131,\,3.23152,\,
3.34572,\,\\3.31249,\,2.03238 \}. \end{multline*} } Initially, \begin{multline*} K_{3}= \{10^{134},\,\,10^{134},\,10^{134},\,10^{134},\,10^{134},\,10^{134}, 10^{134},10^{134}, 10^{134},\,10^{134},\,\\ 10^{134},\,10^{134},\,10^{134},\,10^{134},\,10^{134},\,10^{134},\, 10^{133} \} \end{multline*}
and eventually \begin{multline*} K_{3}=\{
93,\,103,\, 91,\, 100,\, 76,\, 81,\, 93,\, 135,\, 134,\, 82,\, 102,\, \\ 132,\, 113,\, 83,\, 77,\, 73,\, 106
\}. \end{multline*} With the same notation as above $M=416654165624561667592653373446$ and \allowdisplaybreaks{ \begin{multline*} v=1/16384\times (3001857560961454376246370500531976342749510086525\\3 7165233797532993409765024146699104795999297502467819521882253175\\87 914591257285898763774317774494225802733903917259383140801108235\\422 8188011341225855205786331925792324752396626091769176991951013\\90211 864167053047355724708811474915491143119130389255089524641379\\747951 73794783022459203950545394990641318064587830393038066236424\\9440973 7496287203469936364043185785741160332733990397733872532456\\24091789 0154747), \end{multline*} } $F_{m} \leq (\sqrt{5}M^{15}v^{16})^{17}$ $\backsimeq 3.2504 \times 10^{128942}$ and thus $m \leq 616986$. A check shows that $F_{j}$ is not a seventeenth power in the range $3 \leq j \leq 616986$.
\section{Conclusion} The same method could be used to extend the results about prime powers in other binary recurrence sequences, in particular the Lucas sequence. It is possible the alternative method
used to overcome the problems of have a large number of independent units and a large bound on their exponents may be applied in other situations.
\emph{Acknowledgements:} I wish to thank Guillaume Hanrot for several helpful comments. He drew my attention to his paper \cite{H00}, in which he describes a method for solving Thue equations without the full unit group, a method can be applied to the problem of finding small prime powers in the Fibonacci sequence. He also points out that, once the LLL reduction of the bound on the powers of the units has been completed, that there several methods for shortening the final check of remaining possible cases (See \cite{BH96}).
Finally, he pointed out that the ``thueinit'' and ``thue'' commands in PARI/GP can be used to solve the associated Thue equation, when the degree is small.
\section{Appendix I : Fundamental Units in $\mathbb{Q}(\theta)$ for the cases $n=11,13$ and $17$.} For the case $n=11$ a set of fundamental units in the associated field is the following: {\allowdisplaybreaks \begin{multline*} \biggm\{
\frac{209}{256} - \frac{2945\,x}{256} +
\frac{83\,x^2}{8} + \frac{7449\,x^3}{512} -
\frac{21179\,x^4}{2048} - \frac{4375\,x^5}{1024} +
\frac{4257\,x^6}{2048} + \frac{2711\,x^7}{8192} - \\
\frac{5987\,x^8}{65536} - \frac{323\,x^9}{65536} +
\frac{7\,x^{10}}{16384},\,\,\,\,
\frac{421}{256} + \frac{1643\,x}{512} -
\frac{1343\,x^2}{512} - \frac{3605\,x^3}{512} - \\
\frac{5311\,x^4}{2048}
+ \frac{4537\,x^5}{4096} +
\frac{2571\,x^6}{4096} - \frac{77\,x^7}{8192} -
\frac{1631\,x^8}{65536} - \frac{101\,x^9}{131072} +
\frac{13\,x^{10}}{131072},\\
- \frac{3}{128} + \frac{207\,x}{256} -
\frac{271\,x^2}{256} - \frac{39\,x^3}{256} +
\frac{801\,x^4}{1024} - \frac{715\,x^5}{2048} -
\frac{27\,x^6}{2048} + \frac{113\,x^7}{4096} - \\
\frac{71\,x^8}{32768}
- \frac{17\,x^9}{65536} +
\frac{x^{10}}{65536},\,\,\,\,- \frac{81}{256} +
\frac{1147\,x}{256} + \frac{1309\,x^2}{256} -
\frac{2417\,x^3}{256} - \frac{5511\,x^4}{2048} + \\
\frac{4209\,x^5}{2048} + \frac{1089\,x^6}{4096} -
\frac{369\,x^7}{4096} - \frac{297\,x^8}{65536} +
\frac{27\,x^9}{65536},\,\,\,\,
\frac{147}{128} + \frac{47\,x}{128} +
\frac{1111\,x^2}{256} -\\ \frac{2577\,x^3}{256} -
\frac{803\,x^4}{256} + \frac{5\,x^5}{2} +
\frac{1353\,x^6}{4096} - \frac{463\,x^7}{4096} -
\frac{187\,x^8}{32768} + \frac{17\,x^9}{32768},\\
- \frac{31}{4} - \frac{2423\,x}{256} +
\frac{2271\,x^2}{128} + \frac{315\,x^3}{256} -
\frac{21377\,x^4}{1024} - \frac{4845\,x^5}{2048} +
\frac{16773\,x^6}{4096} + \frac{1491\,x^7}{4096}\\ -
\frac{2779\,x^8}{16384} - \frac{503\,x^9}{65536} +
\frac{49\,x^{10}}{65536},\,\,\,\,
\frac{17}{16} - \frac{909\,x}{256} -
\frac{23\,x^2}{128} + \frac{2757\,x^3}{512} -
\frac{737\,x^4}{1024}\\ - \frac{1705\,x^5}{1024} -
\frac{55\,x^6}{4096} + \frac{659\,x^7}{8192} -
\frac{7\,x^8}{16384} - \frac{31\,x^9}{65536} +
\frac{x^{10}}{65536},\,\,\,\, -\frac{501}{256} +
\frac{2155\,x}{512} +\\ \frac{13771\,x^2}{512} +
\frac{2787\,x^3}{256} - \frac{49915\,x^4}{2048} -
\frac{23013\,x^5}{4096} + \frac{9541\,x^6}{2048} +
\frac{1145\,x^7}{2048} - \frac{12909\,x^8}{65536} - \\
\frac{1273\,x^9}{131072} + \frac{117\,x^{10}}{131072},\,\,\,\,
\frac{113}{256} + \frac{4569\,x}{256} +
\frac{2559\,x^2}{64} - \frac{927\,x^3}{512} -
\frac{61735\,x^4}{2048} - \frac{1721\,x^5}{512} + \\
\frac{5429\,x^6}{1024} + \frac{3943\,x^7}{8192} -
\frac{14059\,x^8}{65536} - \frac{641\,x^9}{65536} +
\frac{31\,x^{10}}{32768},\\
\frac{625}{256} - \frac{7401\,x}{256} +
\frac{1651\,x^2}{256} + \frac{3747\,x^3}{128} -
\frac{19629\,x^4}{2048} - \frac{14289\,x^5}{2048} + \\
\frac{8857\,x^6}{4096} + \frac{439\,x^7}{1024} -
\frac{6255\,x^8}{65536} - \frac{365\,x^9}{65536} +
\frac{15\,x^{10}}{32768} \biggm\}. \end{multline*} }
For $n=13$ a set of fundamental units in the associated field is:
{\allowdisplaybreaks \normalsize{ \begin{multline*} \biggm\{\frac{23}{1024} - \frac{10109\theta}{2048} -
\frac{44003\theta^2}{4096} + \frac{124839\theta^3}{8192} +
\frac{136209\theta^4}{4096} - \frac{20289\theta^5}{16384} -
\frac{433515\theta^6}{32768}\\ - \frac{61481\,\theta^7}{65536} +
\frac{369007\,\theta^8}{262144} + \frac{49647\,\theta^9}{524288} -
\frac{39319\,\theta^{10}}{1048576} -
\frac{2917\,\theta^{11}}{2097152} + \frac{243\,\theta^{12}}{2097152}
,\\ -\frac{3727}{1024} +
\frac{7181\,\theta}{128} - \frac{216211\,\theta^2}{4096} -
\frac{121847\,\theta^3}{1024} + \frac{655867\,\theta^4}{8192} +
\frac{119945\,\theta^5}{2048} - \frac{991747\,\theta^6}{32768}\\ -
\frac{37475\,\theta^7}{4096} + \frac{888013\,\theta^8}{262144} +
\frac{7157\,\theta^9}{16384} -
\frac{101367\,\theta^{10}}{1048576} -
\frac{1149\,\theta^{11}}{262144} + \frac{335\,\theta^{12}}{1048576},\\
\frac{1407}{1024} - \frac{2625\,\theta}{256} -
\frac{160297\,\theta^2}{4096} + \frac{2013\,\theta^3}{2048} +
\frac{395753\,\theta^4}{8192} + \frac{6531\,\theta^5}{1024} -
\frac{524081\,\theta^6}{32768}\\ - \frac{31605\,\theta^7}{16384} +
\frac{427491\,\theta^8}{262144} + \frac{8689\,\theta^9}{65536} -
\frac{45573\,\theta^{10}}{1048576} -
\frac{891\,\theta^{11}}{524288} + \frac{143\,\theta^{12}}{1048576},\\
\frac{195}{256} - \frac{15321\,\theta}{2048} -
\frac{58735\,\theta^2}{2048} + \frac{5855\,\theta^3}{8192} +
\frac{289521\,\theta^4}{8192} + \frac{76579\,\theta^5}{16384} -
\frac{191731\,\theta^6}{16384}\\ - \frac{92625\,\theta^7}{65536} +
\frac{19531\,\theta^8}{16384} + \frac{50851\,\theta^9}{524288} -
\frac{16651\,\theta^{10}}{524288} -
\frac{2605\,\theta^{11}}{2097152} + \frac{209\,\theta^{12}}{2097152}
,\\ \frac{1}{16} - \frac{639\,\theta}{512} -
\frac{175\,\theta^2}{64} + \frac{3547\,\theta^3}{2048} +
\frac{8821\,\theta^4}{2048} - \frac{2439\,\theta^5}{4096} -
\frac{243\,\theta^6}{128}\\ - \frac{1129\,\theta^7}{16384} +
\frac{3441\,\theta^8}{16384} + \frac{1581\,\theta^9}{131072} -
\frac{23\,\theta^{10}}{4096} - \frac{105\,\theta^{11}}{524288} +
\frac{9\,\theta^{12}}{524288},\\
\frac{243}{1024} - \frac{2135\,\theta}{1024} -
\frac{22321\,\theta^2}{4096} + \frac{64117\,\theta^3}{4096} +
\frac{53391\,\theta^4}{8192} - \frac{60195\,\theta^5}{8192} -
\frac{48321\,\theta^6}{32768}\\ + \frac{27909\,\theta^7}{32768} +
\frac{22815\,\theta^8}{262144} - \frac{6291\,\theta^9}{262144} -
\frac{1053\,\theta^{10}}{1048576} + \frac{81\,\theta^{11}}{1048576},\\
\frac{243}{1024} - \frac{1659\,\theta}{512} -
\frac{26341\,\theta^2}{4096} + \frac{5907\,\theta^3}{512} +
\frac{149841\,\theta^4}{8192} - \frac{10071\,\theta^5}{4096} -
\frac{229029\,\theta^6}{32768}\\ - \frac{525\,\theta^7}{2048} +
\frac{190759\,\theta^8}{262144} + \frac{5441\,\theta^9}{131072} -
\frac{20001\,\theta^{10}}{1048576} -
\frac{89\,\theta^{11}}{131072} + \frac{61\,\theta^{12}}{1048576},\\
- \frac{593}{512} + \frac{19249\,\theta}{1024} -
\frac{51101\,\theta^2}{2048} - \frac{123517\,\theta^3}{4096} +
\frac{66827\,\theta^4}{2048} + \frac{124229\,\theta^5}{8192} -
\frac{185417\,\theta^6}{16384}\\ - \frac{85397\,\theta^7}{32768} +
\frac{159199\,\theta^8}{131072} + \frac{35493\,\theta^9}{262144} -
\frac{17713\,\theta^{10}}{524288} -
\frac{1529\,\theta^{11}}{1048576} + \frac{115\,\theta^{12}}{1048576}
,\\ \frac{797}{1024} - \frac{19295\,\theta}{2048} -
\frac{93697\,\theta^2}{4096} + \frac{419113\,\theta^3}{8192} +
\frac{307313\,\theta^4}{4096} - \frac{249571\,\theta^5}{16384} -
\frac{912353\,\theta^6}{32768}\\ - \frac{8383\,\theta^7}{65536} +
\frac{736037\,\theta^8}{262144} + \frac{67797\,\theta^9}{524288} -
\frac{75437\,\theta^{10}}{1048576} -
\frac{5115\,\theta^{11}}{2097152} + \frac{453\,\theta^{12}}{2097152}
,\\ - \frac{2287}{1024} -
\frac{31689\,\theta}{256} - \frac{334863\,\theta^2}{4096} +
\frac{308411\,\theta^3}{2048} + \frac{751519\,\theta^4}{8192} -
\frac{11207\,\theta^5}{256} -\\ \frac{878031\,\theta^6}{32768} +
\frac{50093\,\theta^7}{16384} + \frac{632093\,\theta^8}{262144} +
\frac{813\,\theta^9}{65536} - \frac{59971\,\theta^{10}}{1048576} -
\frac{813\,\theta^{11}}{524288} + \frac{169\,\theta^{12}}{1048576},\\
\frac{27}{512} + \frac{2813\,\theta}{1024} -
\frac{90129\,\theta^2}{2048} - \frac{427171\,\theta^3}{4096} +
\frac{155347\,\theta^4}{1024} + \frac{639577\,\theta^5}{8192} -
\frac{1076753\,\theta^6}{16384}\\ - \frac{494651\,\theta^7}{32768} +
\frac{983883\,\theta^8}{131072} + \frac{216065\,\theta^9}{262144} -
\frac{111149\,\theta^{10}}{524288} -
\frac{9527\,\theta^{11}}{1048576} + \frac{721\,\theta^{12}}{1048576}
,\\ - \frac{303}{512} + \frac{909\,\theta}{128} +
\frac{4857\,\theta^2}{2048} - \frac{5137\,\theta^3}{128} +
\frac{71615\,\theta^4}{4096} + \frac{24109\,\theta^5}{1024} -
\frac{146907\,\theta^6}{16384}\\ - \frac{7933\,\theta^7}{2048} +
\frac{136005\,\theta^8}{131072} + \frac{5593\,\theta^9}{32768} -
\frac{16083\,\theta^{10}}{524288} - \frac{25\,\theta^{11}}{16384} +
\frac{55\,\theta^{12}}{524288}\biggm\}. \end{multline*} } } Finally, for the case $n=17$ a set of fundamental units in the associated field is: {\allowdisplaybreaks \normalsize{ \begin{multline*} \biggm\{ \frac{1669}{8192} + \frac{1013441\,\theta}{32768} -
\frac{831099\,\theta^2}{16384} -
\frac{15553837\,\theta^3}{131072} +
\frac{9705537\,\theta^4}{65536} +
\frac{70985341\,\theta^5}{524288}\\ -
\frac{33065551\,\theta^6}{262144} -
\frac{124377673\,\theta^7}{2097152} +
\frac{21555899\,\theta^8}{524288} +
\frac{90616595\,\theta^9}{8388608} -
\frac{22540321\,\theta^{10}}{4194304}\\ -
\frac{26767543\,\theta^{11}}{33554432} +
\frac{4417175\,\theta^{12}}{16777216} +
\frac{2772335\,\theta^{13}}{134217728} -
\frac{256021\,\theta^{14}}{67108864} -
\frac{66067\,\theta^{15}}{536870912} + \\
\frac{3807\,\theta^{16}}{536870912},\\
- \frac{3381}{16384} +
\frac{1443\,\theta}{512} - \frac{1104995\,\theta^2}{65536} -
\frac{596807\,\theta^3}{32768} +
\frac{12341635\,\theta^4}{262144} +
\frac{1742777\,\theta^5}{65536}\\ -
\frac{39959399\,\theta^6}{1048576} -
\frac{6905549\,\theta^7}{524288} +
\frac{50345477\,\theta^8}{4194304} +
\frac{1358191\,\theta^9}{524288} -
\frac{25774521\,\theta^{10}}{16777216}\\ -
\frac{1694321\,\theta^{11}}{8388608} +
\frac{4980657\,\theta^{12}}{67108864} +
\frac{91457\,\theta^{13}}{16777216} -
\frac{285757\,\theta^{14}}{268435456} -
\frac{4507\,\theta^{15}}{134217728} + \\
\frac{527\,\theta^{16}}{268435456},\\
\frac{13}{64} - \frac{11119\,\theta}{4096} -
\frac{5847\,\theta^2}{512} + \frac{359545\,\theta^3}{16384} +
\frac{899457\,\theta^4}{16384} - \frac{810279\,\theta^5}{65536} - \\
\frac{1690541\,\theta^6}{32768} - \frac{148823\,\theta^7}{262144} +
\frac{4347743\,\theta^8}{262144} +
\frac{1161099\,\theta^9}{1048576} -
\frac{544193\,\theta^{10}}{262144} - \\
\frac{616077\,\theta^{11}}{4194304} +
\frac{406671\,\theta^{12}}{4194304} +
\frac{87947\,\theta^{13}}{16777216} -
\frac{11269\,\theta^{14}}{8388608} -
\frac{2581\,\theta^{15}}{67108864} + \\
\frac{161\,\theta^{16}}{67108864},\\
\frac{3177}{4096} - \frac{37351\,\theta}{8192} -
\frac{260767\,\theta^2}{16384} + \frac{854595\,\theta^3}{32768} +
\frac{5165539\,\theta^4}{65536} - \frac{160099\,\theta^5}{131072} - \\
\frac{17391923\,\theta^6}{262144} -
\frac{5968849\,\theta^7}{524288} +
\frac{18808385\,\theta^8}{1048576} +
\frac{6236715\,\theta^9}{2097152} -
\frac{8469197\,\theta^{10}}{4194304} - \\
\frac{2025815\,\theta^{11}}{8388608} +
\frac{1527105\,\theta^{12}}{16777216} +
\frac{217439\,\theta^{13}}{33554432} -
\frac{85281\,\theta^{14}}{67108864} -
\frac{5339\,\theta^{15}}{134217728} + \\
\frac{313\,\theta^{16}}{134217728},\\
\frac{35177}{4096} + \frac{87353\,\theta}{8192} -
\frac{1685455\,\theta^2}{16384} - \frac{1286573\,\theta^3}{32768} +
\frac{18808867\,\theta^4}{65536} +
\frac{10349213\,\theta^5}{131072}\\ -
\frac{62922435\,\theta^6}{262144} -
\frac{26769697\,\theta^7}{524288} +
\frac{79984865\,\theta^8}{1048576} +
\frac{25086155\,\theta^9}{2097152} -
\frac{40730397\,\theta^{10}}{4194304}\\ -
\frac{8793575\,\theta^{11}}{8388608} +
\frac{7786305\,\theta^{12}}{16777216} +
\frac{1029535\,\theta^{13}}{33554432} -
\frac{441361\,\theta^{14}}{67108864} -
\frac{26891\,\theta^{15}}{134217728} + \\
\frac{1609\,\theta^{16}}{134217728},\\
\frac{8353}{16384} + \frac{82469\,\theta}{32768} -
\frac{334519\,\theta^2}{65536} - \frac{2389817\,\theta^3}{131072} +
\frac{4045291\,\theta^4}{262144} +
\frac{16465649\,\theta^5}{524288} - \\
\frac{21174147\,\theta^6}{1048576} -
\frac{41242981\,\theta^7}{2097152} +
\frac{41290081\,\theta^8}{4194304} +
\frac{33247615\,\theta^9}{8388608} -
\frac{25999301\,\theta^{10}}{16777216} -\\
\frac{9888859\,\theta^{11}}{33554432} +
\frac{5556713\,\theta^{12}}{67108864} +
\frac{993323\,\theta^{13}}{134217728} -
\frac{335425\,\theta^{14}}{268435456} -
\frac{22599\,\theta^{15}}{536870912} + \\
\frac{1269\,\theta^{16}}{536870912},\\
\frac{17529}{8192} - \frac{199385\,\theta}{16384} -
\frac{2570675\,\theta^2}{32768} - \frac{1099275\,\theta^3}{65536} +
\frac{22040999\,\theta^4}{131072} +
\frac{14557163\,\theta^5}{262144} - \\
\frac{67044783\,\theta^6}{524288} -
\frac{35241615\,\theta^7}{1048576} +
\frac{82863837\,\theta^8}{2097152} +
\frac{30445541\,\theta^9}{4194304} -
\frac{41990713\,\theta^{10}}{8388608} - \\
\frac{10034929\,\theta^{11}}{16777216} +
\frac{8051613\,\theta^{12}}{33554432} +
\frac{1124425\,\theta^{13}}{67108864} -
\frac{458917\,\theta^{14}}{134217728} -
\frac{28469\,\theta^{15}}{268435456} + \\
\frac{1683\,\theta^{16}}{268435456},\\
-\frac{15439}{16384} +
\frac{7457\,\theta}{4096} - \frac{772229\,\theta^2}{65536} -
\frac{225567\,\theta^3}{4096} + \frac{19970281\,\theta^4}{262144} +
\frac{5384407\,\theta^5}{65536} - \\
\frac{83962553\,\theta^6}{1048576} -
\frac{5203139\,\theta^7}{131072} +
\frac{118428135\,\theta^8}{4194304} +
\frac{7938383\,\theta^9}{1048576} -
\frac{63912463\,\theta^{10}}{16777216} - \\
\frac{149561\,\theta^{11}}{262144} +
\frac{12691267\,\theta^{12}}{67108864} +
\frac{249817\,\theta^{13}}{16777216} -
\frac{739187\,\theta^{14}}{268435456} -
\frac{2983\,\theta^{15}}{33554432} + \\
\frac{1375\,\theta^{16}}{268435456},\\
\frac{37669}{16384} - \frac{782207\,\theta}{32768} -
\frac{3931563\,\theta^2}{65536} +
\frac{29154715\,\theta^3}{131072} +
\frac{38848887\,\theta^4}{262144} -
\frac{120152419\,\theta^5}{524288} - \\
\frac{105613751\,\theta^6}{1048576} +
\frac{158737183\,\theta^7}{2097152} +
\frac{108365877\,\theta^8}{4194304} -
\frac{77583405\,\theta^9}{8388608} -
\frac{44555329\,\theta^{10}}{16777216} + \\
\frac{12880225\,\theta^{11}}{33554432} +
\frac{6875085\,\theta^{12}}{67108864} -
\frac{438065\,\theta^{13}}{134217728} -
\frac{313501\,\theta^{14}}{268435456} -
\frac{7771\,\theta^{15}}{536870912} + \\
\frac{913\,\theta^{16}}{536870912},\\
\frac{26149}{8192} - \frac{89167\,\theta}{32768} -
\frac{84531\,\theta^2}{2048} + \frac{2581855\,\theta^3}{131072} +
\frac{6955001\,\theta^4}{65536} -
\frac{13565819\,\theta^5}{524288} - \\
\frac{9794367\,\theta^6}{131072} +
\frac{15629499\,\theta^7}{2097152} +
\frac{5367431\,\theta^8}{262144} -
\frac{1891597\,\theta^9}{8388608} -
\frac{2435655\,\theta^{10}}{1048576} - \\
\frac{2428259\,\theta^{11}}{33554432} +
\frac{1707267\,\theta^{12}}{16777216} +
\frac{541463\,\theta^{13}}{134217728} -
\frac{45335\,\theta^{14}}{33554432} -
\frac{19063\,\theta^{15}}{536870912} + \\
\frac{1259\,\theta^{16}}{536870912},\\
-\frac{91977}{8192} +
\frac{530479\,\theta}{16384} + \frac{729419\,\theta^2}{32768} -
\frac{6980111\,\theta^3}{65536} +
\frac{1869453\,\theta^4}{131072} +
\frac{27472683\,\theta^5}{262144} - \\
\frac{18142681\,\theta^6}{524288} -
\frac{42686171\,\theta^7}{1048576} +
\frac{31187031\,\theta^8}{2097152} +
\frac{28182221\,\theta^9}{4194304} -
\frac{18441695\,\theta^{10}}{8388608} - \\
\frac{7619597\,\theta^{11}}{16777216} +
\frac{3855551\,\theta^{12}}{33554432} +
\frac{725657\,\theta^{13}}{67108864} -
\frac{232019\,\theta^{14}}{134217728} -
\frac{15945\,\theta^{15}}{268435456} + \\
\frac{883\,\theta^{16}}{268435456},\\
- \frac{7709}{16384} +
\frac{71959\,\theta}{8192} - \frac{663207\,\theta^2}{65536} -
\frac{282043\,\theta^3}{8192} + \frac{9339035\,\theta^4}{262144} +
\frac{5176181\,\theta^5}{131072} - \\
\frac{34425699\,\theta^6}{1048576} -
\frac{4589823\,\theta^7}{262144} +
\frac{46980309\,\theta^8}{4194304} +
\frac{6714145\,\theta^9}{2097152} -
\frac{25183029\,\theta^{10}}{16777216} - \\
\frac{61627\,\theta^{11}}{262144} +
\frac{5002729\,\theta^{12}}{67108864} +
\frac{202099\,\theta^{13}}{33554432} -
\frac{292097\,\theta^{14}}{268435456} -
\frac{2379\,\theta^{15}}{67108864} + \\
\frac{545\,\theta^{16}}{268435456},\\
\frac{310213}{16384} + \frac{379055\theta}{32768} -
\frac{10426491\theta^2}{65536} -
\frac{10145735\theta^3}{131072} +
\frac{89578131\theta^4}{262144} +
\frac{62976811\theta^5}{524288} - \\
\frac{273648223\theta^6}{1048576} -
\frac{136720787\theta^7}{2097152} +
\frac{336238489\theta^8}{4194304} +
\frac{116558733\theta^9}{8388608} -
\frac{169272961\theta^{10}}{16777216} - \\
\frac{38784309\,\theta^{11}}{33554432} +
\frac{32283889\,\theta^{12}}{67108864} +
\frac{4403545\,\theta^{13}}{134217728} -
\frac{1832805\,\theta^{14}}{268435456} -
\frac{112817\,\theta^{15}}{536870912} + \\
\frac{6703\,\theta^{16}}{536870912},\\
- \frac{3375}{8192} +
\frac{59225\,\theta}{4096} + \frac{2751061\,\theta^2}{32768} +
\frac{3266747\,\theta^3}{32768} -
\frac{12989039\,\theta^4}{131072} - \frac{222733\,\theta^5}{1024} - \\
\frac{24349275\,\theta^6}{524288} +
\frac{41990993\,\theta^7}{524288} +
\frac{71338227\,\theta^8}{2097152} -
\frac{9003067\,\theta^9}{1048576} -
\frac{43379513\,\theta^{10}}{8388608} + \\
\frac{1560157\,\theta^{11}}{8388608} +
\frac{8421299\,\theta^{12}}{33554432} +
\frac{43097\,\theta^{13}}{8388608} -
\frac{454593\,\theta^{14}}{134217728} -
\frac{10569\,\theta^{15}}{134217728} +
\frac{383\,\theta^{16}}{67108864},\\
\frac{17285}{16384} - \frac{465327\,\theta}{32768} -
\frac{4632383\,\theta^2}{65536} + \frac{899987\,\theta^3}{131072} +
\frac{48004271\,\theta^4}{262144} +
\frac{15671757\,\theta^5}{524288} - \\
\frac{152102419\,\theta^6}{1048576} -
\frac{52147849\,\theta^7}{2097152} +
\frac{187354341\,\theta^8}{4194304} +
\frac{51864899\,\theta^9}{8388608} -
\frac{93440845\,\theta^{10}}{16777216} - \\
\frac{18753527\,\theta^{11}}{33554432} +
\frac{17634053\,\theta^{12}}{67108864} +
\frac{2246239\,\theta^{13}}{134217728} -
\frac{992033\,\theta^{14}}{268435456} -
\frac{59731\,\theta^{15}}{536870912} + \\
\frac{3601\,\theta^{16}}{536870912},\\
\frac{3503}{8192} - \frac{76383\,\theta}{8192} -
\frac{770045\,\theta^2}{32768} + \frac{733515\,\theta^3}{16384} +
\frac{4622523\,\theta^4}{131072} -
\frac{5753349\,\theta^5}{131072} -\\
\frac{8650317\theta^6}{524288} + \frac{472931\theta^7}{32768} +
\frac{6723545\theta^8}{2097152} -
\frac{3876569\theta^9}{2097152} -
\frac{2151455\theta^{10}}{8388608} +
\frac{369863\theta^{11}}{4194304} + \\
\frac{246705\,\theta^{12}}{33554432} -
\frac{41315\,\theta^{13}}{33554432} -
\frac{7063\,\theta^{14}}{134217728} +
\frac{67\,\theta^{15}}{33554432} + \frac{3\,\theta^{16}}{134217728}\biggm\}. \end{multline*} } }
\end{document} | arXiv |
Journal of the Korean Mathematical Society (대한수학회지)
Korean Mathematical Society (대한수학회)
DOI QR Code
CONJUGACY SEPARABILITY OF CERTAIN GENERALIZED FREE PRODUCTS OF NILPOTENT GROUPS
Kim, Goansu ;
Tang, C.Y.
Received : 2011.07.04
https://doi.org/10.4134/JKMS.2013.50.4.813
Citation PDF KSCI
It is known that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroups are cyclic or central in both factor groups. However, those generalized free products may not be conjugacy separable when the amalgamated subgroup is a direct product of two infinite cycles. In this paper we show that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroup is ${\langle}h{\rangle}{\times}D$, where D is in the center of both factors.
generalized free products;residually finite;conjugacy separable;nilpotent groups
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November 2015 , Volume 126, Issue 1–2, pp 25–56 | Cite as
Global distribution and surface activity of macromolecules in offline simulations of marine organic chemistry
Oluwaseun O. Ogunro
Susannah M. Burrows
Amanda A. Frossard
Forrest Hoffman
Robert T. Letscher
J. Keith Moore
Lynn M. Russell
Shanlin Wang
Oliver W. Wingenter
First Online: 13 October 2015
Organic macromolecules constitute a high percentage of remote sea spray. They enter the atmosphere through adsorption onto bubbles followed by bursting at the ocean surface, and go on to influence the chemistry of the fine mode aerosol. We present a global estimate of mixed-layer macromolecular distributions, driven by offline marine systems model output. The approach permits estimation of oceanic concentrations and bubble film surface coverages for several classes of organic compound. Mixed layer levels are computed from the output of a global ocean ecodynamics model by relating the macromolecules to standard biogeochemical tracers. Steady state is assumed for labile forms, and for longer-lived components we rely on ratios to existing transported variables. Adsorption is then represented through conventional Langmuir isotherms, with equilibria deduced from laboratory analogs. Open water concentrations locally exceed one micromolar carbon for the total of proteins, polysaccharides and refractory heteropolycondensates. The shorter-lived lipids remain confined to regions of strong biological activity. Results are evaluated against available measurements for all compound types, and agreement is generally well within an order of magnitude. Global distributions are further estimated for both fractional coverage of bubble films at the air–water interface and the two-dimensional concentration excess. Overall, we show that macromolecular mapping provides a novel tool for the comprehension of oceanic surfactant patterns. These results may prove useful in planning field experiments and assessing the potential response of surface chemical behaviors to global change.
Organic macromolecules Mixed layer distributions Langmuir adsorption Fractional coverage Bubble films Air–water interface
Responsible Editor: Mark Brush.
Organic carbon is now regularly reported as a ubiquitous natural constituent during observations of the sub-micron marine aerosol (O'Dowd et al. 2004; Russell et al. 2010; Lapina et al. 2011). Macromolecules of biological origin can be conveyed vertically through the ocean mixed layer to the sea surface microlayer by several mechanisms: upwelling in the general circulation, attachment to air bubbles, and association with buoyant exopolymeric particles (Cunliffe et al. 2011; Wurl et al. 2011). Bubble bursting at the ocean surface produces nascent sea spray aerosol droplets; this is a major process capable of transferring organic matter from ocean water into the marine boundary layer (Russell et al. 2010; Cunliffe et al. 2011). Emitted aerosol particles containing long-chain carbon molecules can contribute significantly to the atmospheric particle population and affect the concentrations of cloud condensation nuclei (CCN; Meskhidze et al. 2011). The CCN in turn influence cloud formation and development, with impacts on climate through modification of the albedo.
Aerosol processes currently represent a poorly-understood component of the global biogeochemical system, leading to major uncertainties in radiative forcing as presented in recent climate assessments (Solomon et al. 2007). Natural emissions lie at the heart of this issue, including sources from marine biogenic carbon which extend over much of the planet. Total aerosol forcing has shifted significantly across the globe since the industrial revolution, but the largest uncertainties are associated with the quantification of naturally occurring particles (Carslaw et al. 2013). The oceanic contribution is dominant in terms of coverage area. Fluxes of seawater organics into the remote aerosol system may evolve significantly over the next few decades (O'Dowd et al. 2004; Meskhidze et al. 2011). Changes must be considered both in local mass transfer rates and the regional influence of biogeography. The prediction of future distributions of macromolecules and surfactants will require simultaneous consideration of biological, chemical and physical mechanisms affecting the introduction and enrichment of organic matter.
The fraction of organic material emitted to the atmosphere in fine mode sea spray has recently been parameterized for global atmospheric models as a function of the satellite-observed ocean chlorophyll-a concentration (Vignati et al. 2010; Lapina et al. 2011; Meskhidze et al. 2011). The approach is sometimes augmented by a dependence on wind speed (Gantt et al. 2011) or particle size (Long et al. 2011). In these representations, chlorophyll acts as a proxy for the marine ecodynamic processes that are more fundamentally responsible for the generation of high molecular weight substances. Here we present preliminary tests of an alternative technique, which accounts more completely for macromolecular chemical resolution within the sea (Benner 2002) and then utilizes the distributions to predict the organic mass composition in fine-mode sea spray aerosol (Burrows et al. 2014). The following quantities are explored as offline carbon additions to a global marine biogeochemical model: proteins, polysaccharides, lipids, certain aminosugar derivatives and especially peptidoglycan, heteropolycondensate byproducts in the mixed layer, and the humics of central water masses.
We work from previously simulated values of primary production, phytoplankton distributions and generic dissolved organic matter. Our base model is the Biogeochemistry-Ecosystem-Circulation package for ecological dynamics (Moore et al. 2004). This code runs as a standard module inside the Parallel Ocean Program and it is also a subcomponent of the Community Earth System Model. Our aim is to develop and test surface ocean biogeochemical schemes capable of speciating dissolved organic carbon into macromolecular and polymeric subclasses. Attempts are also made to deal with heterogeneous molecular recombination processes and oligomeric decay products, in at least a preliminary fashion. To indicate zones of potential influence for the individual organic compounds, we additionally estimate and map Langmuir adsorption-driven surface coverage. The results can be applied conceptually to any of the several air–water interface types residing near the atmospheric boundary, including wave-generated bubble films and the global sea surface microlayer. The resolved concentrations that we generate are evaluated against several specific types of chemical oceanographic measurement. Finally, interpretations are offered for the simulated global patterns, both in terms of surfactant coverage and also the chemically resolved carbon excess.
Chemical oceanographers first divided the large scale dissolved organic fraction into its macromolecular constituency in roughly the 1970s. Consideration of the potential surfactant effects began at about the same time. As might be expected, structures known to exist inside phytoplanktonic and microbial cells were quickly identified as fresh release forms (Lee and Bada 1977; Gagosian and Stuermer 1977). Organic films were simultaneously documented at the multiple water–air interface types (Garrett 1967; Hoffman and Duce 1976). Analytical techniques demanded by such studies were quite challenging, since diverse functional groups had to be distinguished and/or separated from the heterogeneous ionic medium that is seawater. Nonetheless progress has been steady -contrast the reviews by Liss (1975) and then Liss et al. (1997) as an instructive set of examples. Highlights during this period of growth in the laboratory and seagoing effort have included the elucidation of bubble surface chemistry (Blanchard 1975, 1989), recognition of the importance of polymeric colloidal/gel processing (Wells and Goldberg 1993; Chin et al. 1998), and transfer effects of films on sea-air gas fluxes (Frew et al. 1990; Tsai and Liu 2003).
It should be clear from this brief history that major concepts underlying the present work have been explored by a range of research groups operating over several decades. But planetary scale marine systems models have only been widely available since just prior to the turn of the century. Tracer lists are now long enough to begin distinguishing the dissolved organics in an idealized manner. Initially the perspective of the overall carbon cycle has most often been adopted, with discrimination mainly according to reactivity or lability (e.g. Moore et al. 2004; Hansell et al. 2012). Our contribution is to attempt a functionally realistic representation of individual macromolecules, so that surfactant properties can be intercompared. We achieve this working from a family of intermediate complexity ecodynamics models already incorporating a subset of reactive organic precursors (Moore et al. 2002, 2004).
General and oceanographic surface film literatures have been mined to provide for the required theoretical information. A brief list of sample references might include Ter Minassian-Seraga (1956) on surfactant kinetics; Adamson (1960) offering a classic textbook approach to the two dimensional physical chemistry; Somorjai (1972) which is a complementary text but stronger in some ways and particularly with regard to adsorption enthalpy and temperature dependence; Liss (1975) which is among the first comprehensive reviews in the marine microlayer literature and nicely introduces the concept of ocean surface pressure; Barger and Means (1985) as an illustrative first attempt to develop interfacial equations of state from a coastal point of view. Laboratory investigations have also been scanned over the last few decades for relevant equilibrium data, and useful examples may be found in the Graham and Phillips (1979) study of stock proteins functioning as surfactants; Adamson and Gast (1997) reviewing some of the more modern adsorption research; Baeza et al. (2005) on interactions in the equilibrated multicomponent state; Moore et al. (2008) focusing on coastal and Gulf Stream humics; and finally Brzozowska et al. (2012) in which the kinetic concepts of Ter Minassian-Seraga (1956) are updated for natural salt water systems.
Results of this preparation process can be summarized through equations describing first the distributions and then surfactant behaviors of the various marine lipid and polymer classes. What follows is in fact a strong condensation from earlier presentations by our own group (Elliott et al. 2014; Burrows et al. 2014). In these publications we emphasized respectively, the time evolution of surface film structures for a few key ecological provinces and then the potential mechanisms of adsorptive transfer from bubbles into the aerosol. In the current work, we extend the effort by outlining a global marine mixed layer or chemical oceanographic viewpoint for the first time. It is also the case that previous approaches relied on the simplest possible molecular proxies. Mainly we restricted ourselves to familiar, pure macromolecules and avoided any mixed functionalities. Here the aminosugars and heterogeneous oligomers/polymers are treated in somewhat more detail.
Methods: assumptions and equations
Our strategy is to blend a contemporary marine biogeochemical knowledge base with fundamental aspects of surfactant theory. Through this combination we seek to better define the properties of a major environmental interface—the boundary between sea and air, which is nominally unbroken but complicated by the formation of bubble and atmospheric particle plumes (Liss 1975; Blanchard 1989; Russell et al. 2010). We build from the output of a well-established ocean systems model (Moore et al. 2002, 2004; Elliott et al. 2011; Wang et al. 2015). This baseline code utilizes a simulated fluid dynamic general circulation to transport resources into the sunlit euphotic zone. There they support fixation of carbon from dissolved inorganic to particulate forms including pigments. Distributions for upwelling nutrients plus those of the carbonates and other small molecules are verifiable by soundings gathered into rich large-scale data sets (e.g. World Ocean Atlas and others at nodc.noaa.gov). Phytoplanktonic chlorophyll has been monitored in surface waters for several decades via remote sensing (Longhurst 1998; oceancolor.gsfc.nasa.gov), while detailed trophic dynamics including zooplankton counts can be compared with about half a dozen time series obtained at oceanographic stations (Moore et al. 2002).
Given this strong backdrop of metrics and data comparison, reliance on a biogeochemical model involves at least two primary assumptions -fidelity in under-sampled surface ecosystems, and also for tracer distributions in underlying water. Effectively, we apply the validated global driver code as a sophisticated interpolation tool. Elemental cycles are extended into lesser-known mixed layer situations or else downward through the column, to depths from which refractory organics sometimes convect (Dittmar and Kattner 2003; Moore et al. 2004). Remote sensing has only begun to distinguish major classes of the biota, so additionally we depend on computations for local taxonomy. This is of greatest importance for chitin, and certain other amino polysaccharides generated mainly by the prokaryotes (Benner 2002; Benner and Kaiser 2003; Aluwihare et al. 2005; Kaiser and Benner 2009). Populations of the producers themselves may be highly variable, so that cross-checking of the resulting concentrations is essential.
Our physical chemical approach is by contrast quite generic. The surfactant studies we draw upon have deep historical roots going back more than a century. Both textbooks and recent environmental studies cite Langmuir–Blodgett trough experiments of the early nineteen hundreds as a baseline (Adamson and Gast 1997; Svenningsson et al. 2006; Tuckermann 2007; Fuentes et al. 2010). The familiar approximation of an idealized monolayer must be made in order to translate laboratory results to global ocean interfaces -dissolved material must accumulate along atmospheric boundaries in a manner described at least in part by linear saturation (Elliott et al. 2014). Liquid phase surface tension measurements are then readily interpretable. The concepts recur regularly in the marine microlayer literature (Liss 1975; Barger and Means 1985; Frew et al. 1990; Kujawinsky et al. 2002) and also concerning the behavior of organics in tropospheric aerosol droplets (Blanchard 1975, 1989; Svenningsson et al. 2006; Meskhidze et al. 2011; Burrows et al. 2014). It thus seems logical to use Langmuir theory as a chemical bridge from ocean to atmosphere. Monolayer behavior will of course break down near the point of full coverage due to intermolecular interactions, and at high two dimensional pressures as the condensed phase collapses (Adamson 1960; Graham and Phillips 1979). But our hypothesis is that nonetheless, textbook surface chemistry should capture the basics of adsorptive equilibration along with intermolecular site competition.
In the equation set to follow, low level theory is presented incorporating all these approximations. We deal first with macromolecular reactive transport in the global modeling context, then with the tendency for organics to adhere to bubble fields. Coherent mixed layer distributions are defined for a subset of marine surfactant proxies, working from biogeochemical simulations of the broader reservoir of dissolved organic carbon. Simple Langmuir style computation is applied with thermochemical parameters held constant, even in the regime approaching or surpassing total coverage. Both the biogeochemical and surfactant models are of necessity incomplete, but we maintain that they nonetheless constitute appropriate starting points. Implications of our major assumptions are explored further after a full exposition of results, in the concluding discussion section.
Organic compound distribution estimates are built up primarily from the global marine ecodynamics framework known as BEC, standing for Biogeochemistry-Ecosystem-Circulation (Moore et al. 2004; Letscher et al. 2015; Wang et al. 2015). This code simulates multielement flow through several trophic levels in the upper ocean, with limitations by the usual nutrient types and light availability. Competitions are included for about half a dozen different phytoplankton classes. Consumers are represented as a single agglomerate zooplankton weighted between micro- and mesoscale forms depending on the balance of food items. Heterotrophic bacteria are represented only as a time constant acting on dissolved organics in order to remineralize them. Thus the microbial recyclers are not carried in BEC explicitly. Since they are actually critical to the production of certain combined polysaccharide-proteins, we adapt a convenient parameterization from earlier work on sulfur biogenic aerosol sources (Elliott 2009). All ecodynamics are embedded in the Parallel Ocean Program or POP in order to provide a transport context or in other words, global current and mixing fields (Moore et al. 2004). Emphasis in the original code family is placed upon primary production and traditional inorganic carbonate chemistry. The relevant parameters, source-sink terms and ordinary differential equations are described in the main appendix of Moore et al. (2002). Critical routings for the dissolved organics are developed further in Moore et al. (2004).
BEC ecodynamic tracer fields are imported into our calculations directly from the most recent versions, incorporating improvements which mainly involve phytoplankton-specific parameter settings (Wang et al. 2015). Our treatment of the organics involves streamlined parallel kinetics operating conceptually on high molecular weight chains portrayed as individual forms of carbon. We do not attempt to conserve the other major atom types. Relevant high molecular weight compounds are released primarily by the disruption of autotrophic cells (Wakeham et al. 1997; Kujawinsky et al. 2002), with exudation appended in some cases (Benner 2002; Elliott et al. 2014). Thus injection patterns are determined by a combination of primary and secondary production inherited from the driver program. But the various lipids and polymers are all given distinct, functionally specific residence times. In most cases these refer to disappearance, with only a nominal allowance for interchange among polymer types or the accumulation of refractories (Carlson 2002; Nagata et al. 2003).
The notation here derives from our ice algal equation set (Elliott et al. 2012, 2015), but it is tailored to the processing of POP history files. A vector of macromolecular carbon quantities C mac or individually C i is generated from BEC autotrophic carbon concentrations C phyto (C p ) through a short series of manipulations. The phytoplanktonic index is positioned as a superscript because it can be dropped as soon as total injections are collected. An intermediate step is apportionment of the long chain compounds within producer cells, based on a commonly accepted global average breakdown (Parsons et al. 1984; Wakeham et al. 1997; Benner 2002). Values are very consistent across the literature at 60, 20 and 20 percent for the major, fresh organic types protein, polysaccharide and lipid (f for i = 1–3). A disruption rate constant subscripted by disr is estimated from the local BEC mortality and zooplankton levels, so that macromolecular spill rate patterns can be computed. Together these stages in the processing may be summarized as
$$ C_{i}^{p} = f_{i} C^{p} ;\quad Spill_{i} = \mathop \sum \limits_{p} k_{disr}^{p} C_{i}^{p} $$
Model and mapping calculations are performed in units of μM carbon, with ecodynamic flow carried in μM/d in all cases. For the first three compound types rough time constants are next selected, characterizing the extent to which material is transported from its point of origin. A significant fraction of the lipidic mass tends to be short lived relative to horizontal mixing (Kattner et al. 1983; Parrish et al. 1992). The aliphatics are therefore handled at local steady state. Some polysaccharides by contrast survive on the order of months and are transported outward to the regional scale (Wakeham et al. 1997; Goldberg et al. 2009). BEC semilabile dissolved organics constitute an overlapping tracer set (Moore et al. 2004) and so we assign an adjustable but major portion to act as a surrogate for the carbohydrates. The proteins are often classified as intermediate in stability (Wakeham et al. 1997) and in this early simulation we represent them merely as an average of the other two macromolecules. In order of actual computation then,
$$ C_{lip} = Spill_{lip} \tau_{lip} ;\;C_{poly} = f_{semi} C_{semi} \left( {BEC} \right); \;C_{prot} = \left( {C_{lip} + C_{poly} } \right)/2 $$
Here indices become abbreviations in the interest of readability, and our choices should be self-explanatory (prot, poly, lip). Chemical structures which are independent of phytoplanktonic injection are dealt with as special cases. For example peptidoglycan is relatively short lived as a combined polymer (Nagata et al. 2003) so that once again a steady state approximation is reasonable. But heterotrophic bacteria are assumed to comprise the major source, and their distributions have to be estimated from earlier sulfur cycle simulations (Elliott 2009 shown as E09). A significant fraction of net carbon cycling through the microbial loop can be assigned, since the polymer is a major component of bacterial cell walls (Benner and Kaiser 2003). The proportion f pept is therefore varied around a central value of ten percent to achieve agreement with in situ data. Hence an additional relationship is required:
$$ C_{pept} = f_{pept} Loop_{micr} (E09)\tau_{pept} $$
The quantity Loop is just another form of elemental flow. Heteropolycondensates are computed by difference relative to the global field of total dissolved organics (Hansell et al. 2012). This is accomplished by subtracting the sum of results 1 through 3 from the contemporary total carbon distribution. Implicitly then, mixed functionality compounds represent the accumulation of dissolved detritus (Benner 2002). Humic acid is estimated based on the Arctic central layer concentrations provided by Dittmar and Kattner (2003), which are likely representative of water masses below the thermocline (Benner 2002; Hansell et al. 2012). Zones where winter vertical convection penetrates deeply are identified from the three dimensional general circulation model output. All of the time constants and concepts employed in the above equations are condensed for reference in Table 1.
Schematic mechanism for the production and transformation of specified macromolecular categories, in mixed and central layers of the global ocean
AA, Hetpc
Nagata et al. (2003), Tanoue (1992)
Sugars, Hetpc
Goldberg et al. (2009), Hansell et al. (2012)
CH in Hetpc
Parrish et al. (1992)
Chitosan, lysates
Aluwihare et al. (2005), Benner and Kaiser (2003)
Peptidoglycan
Benner and Kaiser (2003), Nagata et al. (2003)
Heteropoly
Humics
Benner (2002), Hansell et al. (2012), Wells (2002)
Humic Acid
Dittmar and Kattner (2003)
AA the marine amino acids, CH alkyl functionality, Hetpc or Heteropoly heteropolycondensates comprising a processed, dissolved organic carbon background
Surface interactions of the marine macromolecules distributed by Eqs. 1–3 are determined according to, and represented by, idealized two dimensional chemistry on the Gibbs phase plane (Adamson 1960; Somorjai 1972). A thermochemical system coupling Langmuir monolayer formation with the surface tension parameterization of Szyszkowski is adopted as the primary model. Its potential for interpreting marine surfactant behaviors is apparent from widespread application to environmental organics including lipids, freshly released biomacromolecules, oxidation products and hydrolytic derivatives (Graham and Phillips 1979; Svenningsson et al. 2006; Tuckermann 2007; Moore et al. 2008). Selected experimental data are shown in Table 2, then translated into mapping information for the planetary scale interfaces lying between sea and air. The latter include not only the perimeter of wave driven bubbles, but also the global microlayer and even the nascent spray aerosol early in its evolution. Our intention however is to work first and foremost from a bottom up or chemical oceanographic perspective. The reader is referred to Elliott et al. (2014) plus Burrows et al. (2014) for detailed treatments extending into the atmospheric boundary layer.
Surfactant behaviors for macromolecular model compounds
Model Species
C 1/2
Γmax
Lysozyme
~10−4
Graham and Phillips (1979)
Alginate
>10−1
Babak et al. (2000)
<10−6
Brzozowska et al. (2012), Christodoulou and Rosano (1968)
Kumar (2000)
Benner and Kaiser (2003), Damodaran and Razumovsky (2003)
Moore et al. (2008)
Riverine Fulvic
Svenningsson et al. (2006)
Half saturation concentrations for Langmuir isothermal surface coverage are given in molar units (C 1/2), and the associated maximum excess or two dimensional concentration is expressed for convenience as carbon atoms per square angstrom
GAGP gum Arabic glycoprotein, present along with the gum polysaccharide as an impurity. Heteropoly see Table 1
Thermochemical reference forms are as follows: The biomacromolecules and sea salt act as solutes, the solvent water surrounds them, and a subset of organics resides at the interface. The latter potentially equilibrate as far as the gas phase. Given bulk liquid and gaseous mixtures of j components, total moles and the Gibbs free energy are both readily expressible as sums. But the mass of surfactants is explicitly accounted such that rearrangement isolates the quantities of interest on the left hand side. They are superscripted just below with the word "Surface".
$$ n_{j}^{l} + n_{j}^{g} + n_{j}^{Surface} = n_{j} ; \quad G^{l} + G^{g} + G^{Surface} = G $$
$$ n_{j}^{Surface} = n_{j} - \left( { n_{j}^{l} + n_{j}^{g} } \right); \quad G^{Surface} = G - \left( {G^{l} + G^{g} } \right) $$
A differential change in free energy is given by
$$ dG = - SdT + Vdp + \sigma dA + \mathop \sum \limits_{j} \mu_{j} dn_{j} $$
where σ is the surface tension in units, for example, of N/m or J/m2 and other symbols have their usual meanings. The surface energy contribution is additive in 6 upon the standard textbook dG. Under local conditions in the marine aqueous system, temperature may be considered constant. Furthermore the volume of the dividing plane is zero by definition (Adamson 1960; Tuckermann 2007). Since all chemical potentials must be in balance at equilibrium, the expression for surface free energy change simplifies and is readily integrated.
$$ \mu_{j}^{Surface} = \mu_{j}^{l} = \mu_{j}^{g} $$
$$ dG^{Surface} = \sigma dA + \mathop \sum \limits_{j} \mu_{j} dn_{j}^{Surface} ; \;G^{Surface} = \sigma A + \mathop \sum \limits_{j} \mu_{j} n_{j}^{Surface} $$
The tension is a quantity often measured by surface chemists and in fact it is involved in most of the Table 2 determinations. Working from 4 to 8 the value of \( \sigma \) can now be understood as a function of composition. The product rule and Gibbs–Duhem-type arguments provide the key relationship, after matching with earlier expressions. If we exclude all surfactants but the macromolecules, subscript j reduces to i since these are the only substances with a two dimensional presence.
$$ dG^{Surface} = \sigma dA + Ad\sigma + \mathop \sum \limits_{i} n_{i}^{Surface} d\mu_{i} + \mathop \sum \limits_{i} \mu_{i} dn_{i}^{Surface} $$
$$ Ad\sigma = - \mathop \sum \limits_{i} n_{i}^{Surface} d\mu_{i} $$
At this point we focus for purposes of clarity on a single unnamed organic which may be of oceanographic interest. The summation is no longer needed and moles are converted into an analog of concentration, Γ = n/A. Typically this quantity is referred to as the "excess", because it is not accounted for in the aqueous phase. It will be computed and plotted once the relevant Langmuir equilibria and competitions are fully developed. In the present work we alternately apply units of carbon atoms per square angstrom and moles per square meter, in order to visualize the surfactant situation at the microscopic scale or else in the context of standard seawater salinity.
For any dilute solute, chemical potential is related to the logarithm of concentration C. In all the critical cases here, we will refer to local levels of the dissolved carbon atoms from Eqs. 1–3. Various useful forms of the Gibbs surface equation are:
$$ d\sigma = - \varGamma d\mu ;d\mu = RTd\ln C; \quad d\sigma /dC = - \varGamma RT/C $$
In weak solutions, a linear approximation may be invoked and macromolecules necessarily exert a lowering influence on the interfacial tension. Surface pressure Π is typically defined as the difference in σ from pure to water-solute situations.
$$ \sigma = \sigma^{*} + - const.C;\quad \varPi = const.C; \quad d\sigma /dC = - const.; \quad \varPi = \varGamma RT $$
Expression 12 is sometimes referred to as the "two dimensional ideal gas law" since back-substitution of the excess yields ΠA = nRT (Liss 1975; Barger and Means 1985). In principle, some combination of 11 with the simple "law" might be used to model the behavior of hypothetical marine surfactants. But the idealized relationships often break down under ambient conditions. They are most appropriate to point solutes/adsorbers lacking structure and molecular interaction (Graham and Phillips 1979; Tuckermann 2007; Elliott et al. 2014).
The Gibbs equation combines conveniently with results from adsorption kinetics to yield a fit to the tension curve in the region of growing surfactant influence (Somorjai 1972). Rates of site occupation and desorption are given by k a C, where the proportionality constant has units of per (concentration time), and k d in the reverse direction which is first order. Fractional coverage θ is then a function of the ratio between the two, and this in turn may be thought of as an adsorptive equilibrium constant. Its reciprocal is just the concentration at half coverage C 1/2. We rely heavily on half saturation as a reference point while organizing laboratory surfactant data.
$$ K = k_{a} /k_{d} ; \quad \theta = KC/\left( {1 + KC} \right); \quad C_{1/2} = K^{ - 1} $$
An assymptote for monolayer coverage or excess Γ max may also be assumed, although even this is not a given as polymers reconfigure, stack and react (Adamson 1960; Graham and Phillips 1979). A particularly compact mathematical form is then attained. Substitution and integration yield what is often called the Langmuir-Szyszkowski equation, after major contributors working in the early era.
$$ \varGamma = \varGamma_{\hbox{max} } \theta = \varGamma_{\hbox{max} } KC/\left( {1 + KC} \right) $$
$$ d\sigma = - \varGamma RTdC/C; \quad d\sigma = - \varGamma_{\hbox{max} } KRT\left( {dC/\left( {1 + KC} \right)} \right) $$
$$ \varPi = \varGamma_{\hbox{max} } RT\ln \left( {1 + KC} \right) $$
The two dimensional equilibrium constants represented in Table 2 were obtained by fitting surface tension or pressure data to the forms in 11–16, with one exception. Stearic acid is sufficiently insoluble in seawater that a kinetic dissolution approach had to be applied (Ter Minassian-Seraga 1956; Brzozowska et al. 2012). Note that in the table, reciprocal equilibria are expressed in pure molar units as opposed to the model/simulation standard of μM. Values for limiting areal concentrations are often reported in mg/m2 but we convert to our own dimensions for visualization and then oceanographic application. The Γmax may also be derived by curve fitting but are measurable directly, by weighing out a particular sample then generating the spread film on a flat water surface (Adamson 1960; Graham and Phillips 1979; Damodaran and Razumovsky 2003).
When the distinctions "i" are maintained and propagated beyond Eqs. 9–10 in order to represent multiple adsorbing agents, competition for the available interfacial surface area enters in (Somorjai 1972; Tuckermann 2007; Burrows et al. 2014). Multicomponent coverage can be described by the chemically resolved forms
$$ \theta_{k} = K_{k} C_{k} /\left( {1 + \mathop \sum \limits_{i} K_{i} C_{i} } \right); \quad \theta_{total} = \mathop \sum \limits_{k} \theta_{k} ; \quad \varGamma_{k} = \varGamma_{\hbox{max} ,k} \theta_{k} $$
where the final subscript k is introduced to specify a single compound type. This is merely to distinguish intercomparisons among our marine organics from the overall components j of the hypothetical aqueous-to-gas reference state in 4 and 5. The derivation once again involves kinetic approach to (and desorption from) the water–air interface. It is sometimes provided in detail in physical chemistry source books (e.g. Laidler 1965). Ultimately the metrics collected in 17 are computed globally, to assess tradeoffs between selected biomacromolecules functioning as surfactants. But it should be clear from our development that at the level of chemical thermodynamics, the methods are quite approximate.
Distributing the macromolecules
Concentration mapping exercises were conducted offline relative to geocycling output from the BEC model running inside of the Community Earth System Model (Moore et al. 2004). Our calculations build on on a recent time-slice simulation of the marine methane cycle (Elliott et al. 2011) by utilizing major biotic and seawater composition outputs in an analysis and plotting script. Biogeochemical relationships adopted for the present work have been summarized in Eqs. 1 through 3 with time constants provide in Table 1. Our offline model is written in the NOAA Ferret scripting language, and it is available from the authors on request. More detailed descriptions appear in companion manuscripts emphasizing a few well studied ecosystems plus an atmospheric viewpoint involving the estimation of aerosol organic fractions (Elliott et al. 2014; Burrows et al. 2014). The complete global mechanism has been published as a Los Alamos COSIM technical report (Climate Ocean Sea Ice Modeling).
The preexisting external model partitions total chlorophyll into the major phytoplanktonic classes: small autotrophs (0.2–2 µm), diazotrophs, coccolithophores, diatoms and high-latitude Phaeocystis. In the present work, we additionally represented intermediate competitors including the chlorophytes in order to extend the C p vector, along with cyanobacteria and heterotrophic bacteria in some tests (Gregg et al. 2003; Elliott 2009). These additional types may act as distinct specialist producers of certain surface active polymers. In some regional ecosystems, fibers and external cell barriers are synthesized as long rigid strands or even chemical grids. Thus we also introduced chitin and peptigoglycan as carbon sources associated with the cell perimeter. To achieve the biological resolution required for estimation of these polymers, two further steps were taken. Small autotrophs were apportioned post hoc using an alternate ecodynamic framework (Gregg et al. 2003), and a steady-state concentration of consumer microbes was computed according to the algorithm previously developed for dimethyl sulfide.
Polysaccharides may be considered relatively long-lived among ocean organics (Goldberg et al. 2011), and so their distributions were represented as a constant fraction f semi of the moderately or semi-labile dissolved organic carbon imported from BEC. Many lipids lie at the opposite, reactive end of the macromolecular spectrum (Kattner et al. 1983). They are removed from the water column by heterotrophic bacteria sufficiently rapidly that their concentrations can be represented at local steady state. Proteins are often of intermediate stability, so they are computed in the current work simply as a weighted average of the previous two classes (Wakeham et al. 1997). Our approach can thus be thought of as a conceptual parallel steady state, but simultaneously it is tailored to the necessities of an offline mapping analysis. The balance between actual and indirect kinetics is dictated by the equations in 2. For the polymeric categories carbohydrate and protein which are closely linked to dynamic output, we assume only implicitly that release is a byproduct of phytoplanktonic cell breakage. Hence the values are entirely consistent with processing in the parent routines (grazing, viral attack or senescence; Moore et al. 2004). Arguments in the dissolved organic literature often suggest that cell disruption is a major source channel for such fresh biomacromolecules (Kujawinsky et al. 2002; Kuznetsova et al. 2004; Kaiser and Benner 2009). Lipid production rates on the other hand were calculated fully explicitly, based on fixed BEC zooplankton and grazing fields. The carbon proportion of fatty acids and sterols was therefore estimated more directly, at the global average for phytoplanktonic composition determined by f lip . Photochemical production is known for certain species in the organic spectrum including aliphatics (Benner 2002), but it is not considered at this point in our research.
We further assessed diatomaceous and zooplanktonic (copepod) sources for the structural material chitin, a common acetylated aminopolysaccharide. It is produced both for purposes of protective coverage and as a component of fibrils. Distributions were essentially determined by habitat and succession. It may be concluded, however, that this particular decay series is unlikely to support adsorptive coverage of water-to-air interfaces at the global scale. Chitin is a very insoluble polymer (Bhosle et al. 1998; Kumar 2000), and source-receptor logic applied to large scale monosugar observations indicates that solute release is weak (Benner and Kaiser 2003). Essentially, we argue from an empirical standpoint that this macromolecule is present at only trace levels across the surface ocean, at least in its dissolved form. By contrast peptidoglycan and its byproduct oligomers are reported as common components of open seawater (Benner and Kaiser 2003; Nagata et al. 2003; Aluwihare et al. 2005). In this instance, dissolved concentrations were estimated as a fraction of carbon flow through heterotrophic bacterial consumption channels (the microbial loop). Yields were adjusted to match both the reported concentrations and reactivity (Eq. 3, Nagata et al. 2003). Prokaryotic cyanophytes may also be producers of peptidoglycan, but in the interest of simplicity we defer their simulation here.
Organics injected by cell disruption as protein, polysaccharide and lipid are degraded in the ocean via random enzymatic and photochemical processes, into a massive but poorly known reservoir which is functionally mixed and polymeric (Benner 2002). We refer to such byproducts here collectively as the heteropolycondensates (Wells 2002). The material may be visualized as a combination of monomers, substituents and functional groups derived from the multiple source pools (Malcolm 1990). But it has been mapped solely as a difference quantity, relative to either independent or BEC estimates of the dissolved carbon total. The full complement of mixed layer macromolecules including proteins through heterogeneous polymers may further be incorporated at high latitudes into several modes of deep water formation. Through dark reactions at depth, conversion takes place into material that is essentially recalcitrant to chemical breakdown. Lifetimes may be comparable to residence times in the abyss (Dittmar and Kattner 2003), and in some cases they exceed thousands of years.
The deep carbon returns to the surface annually due to convective mixing in the winter hemisphere. Upward transport is estimated here for zones of high-latitude turnover based on numerical mixed-layer depths output along with the POP general circulation. Maximization tends to occur in an extremely geographically variable manner. We assumed complete vertical dilution of all constituents from the lower mixing limit at several hundred meters all the way to the surface. Results were then checked against climatological seasonal cycles and penetration depths compiled in an authoritative biogeography of the sea (Longhurst 1998). Humic acid concentrations were set equal to Arctic measurement data, presumed here to be representative of global high latitudes (Dittmar and Kattner 2003). Since we deduce central ocean humate from observations, source distributions are included in principle. Nonidentified, extrahumic amphiphiles were not simulated.
References in the final column of Table 1 are included as a guide to more comprehensive materials, for biogeochemical channels and time scales dictating the distributions of all macromolecular categories: generalized proteins (Tanoue 1992), carbohydrates comprising a portion of the traditional semi-labile dissolved organic carbon (Goldberg et al. 2009; Hansell et al. 2012) and the relatively short-lived lipids (Parrish et al. 1992). Aminosugar reactions in the mixed layer include hydrolysis leading to de-acetylation, the loss of amide linkages and decay of any substituent peptide appendages (Benner and Kaiser 2003; Nagata et al. 2003; Aluwihare et al. 2005). Like other processes under consideration, these are parameterized and hence treated in aggregate. More detailed structural definitions for the heteropolycondensate tracer are available in the literature regarding refractory and colloidal carbon (Wells 2002; Dittmar and Kattner 2003).
Adsorption in the model
Surfactants dissolved in the upper few meters of the ocean will adsorb readily to any air–water phase boundaries. Hence they are expected to coat the exterior of wave-generated bubbles and also liquid (aqueous) sections of the sea-air interface (Cunliffe et al. 2011). From these starting points, macromolecules may be emitted rapidly to the atmosphere following film breakage at the upper limit of the water column (O'Dowd et al. 2004; Russell et al. 2010). Per our final equation set 17, standard multispecies Langmuir adsorption provides approximate coverages while multiplication by the laboratory maximum excess allows for conversion to areal concentrations. We represent surfactant processes in this simple way along all global air–water interfaces, i.e. the ocean surface microlayer and cavity interiors. Rising bubbles transport high molecular weight organics vertically, and additional material is attached at the end of the upward trajectory during passage into the atmosphere (Burrows et al. 2014). Along these pathways, relative amounts of the various macromolecules should be conveyed conservatively into nascent spray particles since timescales are short and no chemical reactions are postulated. Thus our maps are also relevant to the period prior to the onset of atmospheric photochemical degradation and formation of secondary organic aerosol.
Surfactant data are presented in Table 2 for a series of proxy compounds corresponding to the selected list of macromolecules. Most quantities were extracted directly from laboratory experiments, drawing upon the Gibbs-Langmuir-Szyszkowski relationships of Eqs. 11–16. Denatured proteins are represented by the hydrolytic catalyst lysozyme (Graham and Phillips 1979). This particular structure is actually of terrestrial origin. But its adsorptive behaviors should largely be independent of any intracellular function. Polysaccharides may be very insoluble when hydrophobic substituents are attached. We adopt the synthetic analog alginate as an initial proxy for the class of pure polysaccharides (Babak et al. 2000). Lipids may be well represented either by long chain fatty acids or sterols (Kattner et al. 1983; Loh et al. 2008). Here we assign the role to stearate (Christodoulou and Rosano 1968; Brzozowska et al. 2012), with the option to switch to cholesterol.
Chitin, one of the most common polysaccharides in the ocean, is an example of high insolubility (Bhosle et al. 1998; Kumar 2000). Although hypothetical derivatives might prove to be adsorptive, we momentarily discount the contribution of this particular carbon decay line on a mass flow basis. The proteo-substituted amino sugars may be more surface active due to hydrophobic side group effects. In the absence of data, glycoproteins extracted from a well-known food stabilizer (gum arabic) are cited as a peptidoglycan analog (Damodaran and Razumovsky 2003). Heteropolycondensates and humics are discussed in the environmental chemistry literature primarily as coastal samples. Gulf Stream and commercial fulvics are taken to be representative (Svenningsson et al. 2006; Moore et al. 2008). Half saturations are listed in the table as C 1/2 for conceptual convenience, but in fact they can also be thought of as reciprocal equilibrium constants K −1 following 13. As in the case of other parameters, coverage levels were derived assuming ideal dilute surfactant behavior. And all equilibria are uncertain by perhaps an order of magnitude, since real macromolecules exhibit significant deviations from the assumptions of Gibbs theory (Graham and Phillips 1979; Baeza et al. 2005; Shogren and Biresaw 2007).
Upward transport on the buoyant global bubble field will necessarily be modulated by the surface excess. This is just the density of material associated with unit adsorption area, as defined in the logic leading to Eqs. 17. For some substances of interest, the maximum is about one when expressed in units of atoms per square angstrom (Christodoulou and Rosano 1968; Graham and Phillips 1979). This value constitutes a useful conceptual starting point, because one side of the reference area is about the length of a carbon–carbon bond. But in several cases we recommend a range of areal concentrations, and these deserve additional comment. In controlled experiments, model heterogeneous polymers tend to form irregular and extremely complex structures spanning the water interface. Chains or filaments of hydrophiles may be suspended below a few hydrophobic constituents lining the surface (Babak et al. 2000; Damodaran and Razumovsky 2003; Moore et al. 2008). Stacked configurations could also be present in the marine environment, and so they are included here for potential sensitivity testing.
Results: concentration distributions
The concentration calculations are presented as a series of plots for the chosen macromolecular carbon types, each displaying panels from the four seasons. Monthly average distributions are shown for central months beginning with February representing winter, and arranged clockwise from upper left. Marine macromolecular compounds are discussed in the usual order, with protein, polysaccharide, lipid, peptidoglycan, heteropolycondensate products and finally the deep humics organized as Figs. 1, 2, 3, 4, 5, 6. We begin our analysis in all cases with austral summer. Ring ecosystems of the Southern Ocean are the most extensive and intense on the planet (Longhurst 1998). But winter deep convection is more pronounced and visually interesting in the Arctic regime than it is in the Antarctic. Our interpretations move equator-ward within the February plots, and then advance in time through the annual cycle. Typically, the northern hemispheric results follow similar trends that are merely delayed by 6 months.
Global surface ocean protein distributions estimated by our offline mapping procedure, central months of the four seasons. Units are micromolar (µM) dissolved carbon
Global surface ocean polysaccharide distributions estimated by our offline mapping procedure, central months of the four seasons. Units are micromolar (µM) dissolved carbon
Global surface ocean lipid distributions estimated by our offline mapping procedure, central months of the four seasons. Units are micromolar (µM) dissolved carbon
Global surface ocean peptidoglycan distributions estimated by our offline mapping procedure, central months of the four seasons. Units are micromolar (µM) dissolved carbon
Global surface ocean heteropolycondensate distributions estimated by our offline mapping procedure, central months of the four seasons. Units are micromolar (µM) dissolved carbon
Global surface ocean humic distributions estimated by our offline mapping procedure, central months of the four seasons. Units are micromolar (µM) dissolved carbon
Seasonal protein distributions are displayed in Fig. 1. As expected, patterns mimic those of primary production or generalized marine biological activity. The ecodynamic sequence of phytoplanktonic growth followed by cell disruption is primarily responsible for the release of fresh bioorganic molecules. Proteins are thought to be relatively short lived (Tanoue 1992; Goldberg et al. 2009; Hansell et al. 2012), so that their concentrations are a relatively small fraction of total dissolved organic carbon. Values in the central gyres fall in the μM range, but since the Langmuir adsorption reference point is small (Table 2), contributions to the surface chemistry may still be expected in the baseline configuration. Extremes in the Sunda-Arafura shelf province (Indonesia to New Guinea; Longhurst 1998) can be attributed to stagnation and a corresponding buildup of semilabile material in the parent model (Moore et al. 2004). Proceeding through the seasons, a shift in protein steady state follows the path of the sun northward then returning. This fundamental cycle is common to several of the compounds studied here. An exception will be refractory carbon sourced from the abyss.
Patterns are similar for the polysaccharides as shown in Fig. 2, but concentrations are several times higher locally (note the change in color scale). These are realistic relative levels (Benner 2002; Hansell et al. 2012), and they reflect in part the differential lifetimes from Table 1. Maximal concentrations of the pure carbohydrate class amount to tens of μM. This result is nominally consistent with measurements of neutral sugars conducted at key sites such as the major oceanographic stations (Goldberg et al. 2009, 2011), since they tend to be located within or at the edges of oligotrophic gyres. In certain areas, it is likely that serious overestimation of polysaccharide concentrations is occurring. The Arafura shelf again stands out, but maxima are also evident in the Humboldt Current extending all the way to coastal Peru. Eastern basin upwelling zones are relatively understudied for their organic chemistry in the Southern Hemisphere, but fractions approaching half of all dissolved organic carbon must be considered suspect. Similar concerns apply to the Canary Islands ecosystem during northern summer and in the far eastern Pacific off Central America. This potential problem can be traced to our oversimplification of reactive transport, since we currently describe polysaccharides as a constant proportion of BEC semilabiles (Moore et al. 2004). It is likely the issues can be resolved when the macromolecules are simulated online within the general circulation.
Lipids are treated as a highly labile fraction here, so their concentrations are associated with primary production. The Falkland Islands and Subantarctic Ring zones are particularly prominent in the February panel of Fig. 3. Significant activity is also predicted for peripheral embayments surrounding the Antarctic continent and extending along the Palmer Peninsula. This effect may partially be explained by the coincidence of our monthly average plotting strategy with the mid-summer peak in polar biological activity (Longhurst 1998). The ice domain and receding pack edge, however, are often sites for strong blooms, whether for the diatoms or competing Phaeocystis. In the Antarctic polar provincial ecosystems, biological activity accelerates as ice coverage melts back toward the continent and the day lengthens moving into summer. Only negligible concentrations of lipid material are simulated at low latitudes in the global open ocean, but all members of this organic class are strong surfactants (Brzozowska et al. 2012), so that our exercise will ultimately lead to significant net coverage estimates in many areas. In northern hemispheric summer, the bloom fans out around and below the Arctic pack. Algal concentrations are probably overpredicted in this circumstance, but it is interesting to note that the lipids closely follow.
Our initial attempt to represent the amino polysaccharides adopts the bacterial cell wall component peptidoglycan as a model (Fig. 4). Production of this external structural compound was computed in proportion to heterotrophic biomass density, and the latter was taken in turn from a parameterization developed to consume organic sulfur (Elliott 2009). Peptide substituent arms of the polymer are thought to degrade in only a few days, but the core oligosaccharide chain lasts much longer (Nagata et al. 2003). Still, we simulate the distribution as a local steady state, and in most areas of the global surface ocean this should serve as an adequate entry point. The production rate was adjusted to produce an average concentration of about one μM in keeping with the few available measurements (Benner and Kaiser 2003). But the microbial input was limited to ten percent of total carbon flow (Eq. 3). Concentration distributions of peptidoglycan are the most uniform among all compounds dealt with here. This is because the source is linked to recycling as opposed to primary production. The relationship we derive is approximately proportional to the square root of phytoplanktonic biomass (Parsons et al. 1984). Indications are that the aminosugars will congregate in zones of stronger upwelling. Plus once again the annual cycle is driven in all areas by season and solar angle. Like the lipids, aminated polymers tend to track their ecodynamic forcings below the Arctic pack during the melt (August). In this case however, the driver organisms are prokaryotic.
Heteropolycondensates will ultimately be treated in marine organic chemistry simulations as true recombination products. Here, we estimate their distributions mainly by invoking mass balance and difference relative to total dissolved organic carbon (Hansell et al. 2012). We have also compared results with dynamic simulations of real surface refractories, conducted online inside the BEC code but only recently available. The two approaches lead to consistent patterns, and the heterogeneous carbon distribution shown in Fig. 5 is representative. Areas of high concentration tend to be anticorrelated with those of the fresh macromolecules discussed above, because the heterogeneous material is produced exclusively by carbon degradation. But the molecules build to high levels of tens μM, approaching the total of dissolved organic carbon concentration within the central gyres. This is because refractory residence times assumed in the model exceed surface ocean horizontal mixing periods. Accumulation in the gyres is thus the rule for this material. Stark minima in some locations should be discounted as an artifact of the differencing procedure. We believe this to be the case between Indonesia and Australia, and probably also in the Eastern South Pacific and the Atlantic Canary Islands province. The effect is likely real in the high Arctic, however, since there it is attributable to deep convective mixing.
Winter overturning acts as a major source of humic acid in the final concentration distributions presented in Fig. 6. In the Northern Hemisphere, deep water penetration is focused in the Labrador and GIN Seas (Greenland-Iceland-Norway; Longhurst 1998). Here, our offline model dilutes the remnant mixed layer and then partially replaces surface organics with deep humic substance (Benner 2002; Dittmar and Kattner 2003). Like the heteropolycondensates, humic acids are of mixed functionality. But they must still be considered chemically distinct, due to long isolation from photochemical processing. The abyssal carbon is stirred to the surface, and there it displaces any constituents remaining from the peak of summer biological activity. Humics do not appear to be strongly surface active (Svenningsson et al. 2006), but the high North Atlantic is especially windy in the wintertime and aerosol effects may persist and propagate into early spring (O'Dowd et al. 2004). Concentrations of upwelled carbon at the surface approach the levels assigned in the present work for central water layers (i.e. depths of 200 meters). The value is set at 50 μM (Dittmar and Kattner 2003). In the corresponding August plot, it is apparent that deep convection is less extensive in the Southern Ocean. Circumpolar current systems block high-salinity water masses which might otherwise be injected toward the pole from lower latitudes by the several western boundary currents (Longhurst 1998). POP results suggest that the strongest activity occurs along the southern frontal regime of the Tasman Sea.
Data comparisons
The maps of Figs. 1, 2, 3, 4, 5, 6 can now be checked against regional scale in situ observations across the global ocean. Serious caveats apply to such comparisons and must be mentioned to begin. Mainly the difficulties are attributable to complexities of the wet chemistry. Methods sections in the measurement papers of Table 3 reveal that samples may be subjected to: freezing, drying, desalination, ultrafiltration, extraction, and acidification. In some instances multiples steps from this list are interwoven into intricate laboratory procedures. The concentration information may only be deducible indirectly, from chemical behaviors recorded at various experimental stages. Polymers may be partially or almost entirely hydrolyzed, reduced or oxidized and organic derivitization is often required to achieve spectroscopic detectability. Rough analytical treatment can cause the breakup of combined heterogeneous forms to yield fragments which are artifacts, amphiphiles or both (Borch and Kirchman 1997; Witter and Luther 2002; Aluwihare et al. 2005; Van Mooy and Fredricks 2010). Still, the level of agreement among diverse methods in the table indicates a certain amount of fidelity. Recently introduced and improved techniques can be down-selected without changing our overall conclusions (e.g. Kaiser and Benner 2009; Goldberg et al. 2009, 2011). A closer examination of the links between experiment, structure and surface activity is definitely warranted. For the moment, however, we take the conservative standpoint that the tabulated data serve at the very least as a strong guide to organic functionality.
Comparison of simulated mixed layer concentrations with selected, chemically resolved data for the marine macromolecules
BPLR
Benner and Kaiser (2003)
PEQD
(Equatorial)
Kaiser and Benner (2009)
NPTG
Loh et al. (2008)
0.1−3
Gagosian et al. (1982)
Marty et al. (1979)
NECS
0.1−0.3
Kattner et al. (1983)
NWCS
1−5
Delmas et al. (1984)
GFST
Kuznetsova et al. (2004)
Lee and Bada (1977)
Dittmar et al. (2001)
(Tropical)
Aluwihare et al. (2005)
1−10
Goldberg et al. (2009)
SPSG
Goldberg et al. 2011)
Borch and Kirchman (1997)
Mykelstad et al. (1997)
APLR
Witter and Luther (2002)
10−20
Pakulski and Benner (1994)
Bhosle et al. (1998)
McCarthy et al. (1998)
Case studies are listed roughly in order of increasing measured levels, as an aid to the eye. Sites are classified according to the Longhurst ecogeographical system, with abbreviations defined in the text. Concentrations are given in units of micromolar (μM) carbon
In order to support a relatively complete comparison, we identified measurements spanning the entire historical period discussed in the background section at the top of the text (Lee and Bada 1977 to Goldberg et al. 2011). Collected data are ordered by increasing global average concentration, so that the progression is regular and a quick evaluation of model fidelity is possible. Thus the compound order shown in Tables 1 and 2, which is based on ecodynamic mass flow considerations (Parsons et al. 1984; Benner 2002), is approximately reversed. Observations are presented beginning with peptidoglycan then moving toward the high concentration mixed refractories. Locations are specified using the Longhurst biogeographic province system (Longhurst 1998). The lettered sets of interest here are APLR (the Austral Polar province along the periphery of the Antarctic continent), BPLR (Atlantic Boreal Polar, covering much of the Arctic Ocean along with the east coast of Greenland), GFST (the Gulf Stream), MONS ((Monsoonal regime of the Indian Ocean), NAST (North Atlantic Subtropical gyre, the western half of which is almost synonymous with the Sargasso Sea), NECS (Northeast Atlantic Coastal Shelves but corresponding closely with the North Sea), NPTG (the vast North Pacific Tropical Gyre, but the majority of points were obtained in the vicinity of the Hawaiian Islands), NWCS (the Northwest Atlantic Coastal Shelves province, which is highly biologically active and includes the Canadian Maritimes), PEQD (the Pacific Equatorial Divergence extending from the Galapagos westward to about the dateline), SARC (the Atlantic Subarctic zone encompassing Britain, Scandinavia and the Barents Sea), and finally SPSG (the South Pacific Subtropical Gyre transected by cruises in several cases but with key samples collected near the Humboldt Current). Note that the distribution of ecogeography is skewed heavily toward the Northern Hemisphere, as might be expected for a difficult and evolving set of ambient analytical chemical measurements.
Concentrations for open water observations of prokaryotic structural material appear at the top of Table 3. They seem quite realistic. Peptidoglycan and its oligomeric decay products are produced ubiquitously (Benner and Kaiser 2003; Nagata et al. 2003), so they are expected to maintain a fairly uniform global distribution. Overpredictions in the high Arctic are likely traceable to a general ice domain productivity bias, as discussed in the results section just above. Quoted measurements actually represent the sum of hydrolysable, monomeric amino sugar constituents in many cases. Thus a certain amount of the original environmental material may have gone undetected. Here, we incorporate only the heterotrophic source for peptidoglycan. Photosynthetic bacteria will surely add to the concentrations. Our mapped estimates of this prokaryotic byproduct may therefore constitute underestimates at several levels. But the mechanism entails sufficient flexibility that it can be adjusted as necessary in the future.
With the exception of a single comparison for the North Pacific (Loh et al. 2008), baseline lipid concentrations are consistently low by an order of magnitude or more relative to the data sets summarized in Table 3. Here it seems likely that the chosen observations represent overestimates. A few concentration values are available from aerosol-oriented studies conducted in less productive tropical waters, for a variety of fatty acid chain lengths (Marty et al. 1979; Gagosian et al. 1982). But some of these levels are actually calculated indirectly based on remote air samples rather than seawater determinations. For example, a local ocean-to-atmosphere emission model was developed to budget the carbonaceous aerosol at isolated Enewetak Atoll (Gagosian et al. 1982). Overall the spread in observed oceanic lipid concentrations is exceptionally broad. The Canadian maritime biogeochemical province, represented in multiple cases by μM of the lipids or more, actually spans heterogeneous coastal, slope, and open water regimes. The cited investigations average data collected in shore or else over the continental shelf (Delmas et al. 1984; Parrish et al. 1992). Hence they probably reflect enhanced primary production.
Starting with the proteins our offline mapping procedure begins to overestimate the molar concentrations relative to available data. However, the baseline scheme for this particular organic class involves only a zeroth order weighting of semi-labile and shorter lived macromolecular distributions. In future work, we plan to conduct dynamic simulations online, inside of ocean circulation coding. The polysaccharide concentrations compare reasonably well to observations except at the highest concentrations, but this in fact is where the analytical chemistry is at its most challenging. A mix of total carbohydrates and neutral sugars is actually represented, as discriminated by hydrolytic, spectroscopic derivative, redox and chromatographic techniques of variable internal reproducibility. Our mixed polymeric material concentrations also agree well with observations. But since this field is generated approximately by working backward from different sorts of refractory carbon climatology, the outcome is expected. Problems should be anticipated for all carbohydrate material in high productivity coastal waters and inland seas, since they are currently under sampled and difficult to model. Indonesia and the Humboldt Current offer prominent examples but no data sets appear to be available.
Measurements for the individual macromolecules are regionally sparse, but collected at the global scale they are nonetheless sufficient for a preliminary statistical analysis. A nonparametric approach is presented as a series of box diagrams in Fig. 7. Selected observations extracted from the range of Table 3 studies are visualized in a quartile format. Dissolved concentrations are separated into low and high carbon subsets since the spread among compounds is broad. Outliers are separated by whiskers and depicted explicitly where they become relevant. All conclusions drawn from our biogeographic arguments are confirmed in the figure. Simulated peptidoglycan concentrations constitute slight overpredictions, but the analytical chemistry certainly leaves room for average measured concentrations to rise. Lipid simulations fall below the interquartile range of available data, but some major studies are biased coastally. The global protein discrepancy is small and readily adjustable since in this case our modeling really constitutes a weighting procedure. Sugar calculations are reasonable excluding a few extremes which may again be attributable to complexities of the experiments. Heteropolycondensate values are in the dozens of micromolar but show relative agreement, as expected from our heavily data-driven approach. In several important cases in the plots of 7, estimated and observed bars overlap, lending confidence to the details of our method. Only the lipid discrepancy is large and it may well reflect skewed sampling.
Global box plot comparisons of modeled versus measured concentration, for a selection of data from the references of Table 3. Central quartiles and outlier boundaries are provided so that dispersion and skewness of the underlying distributions are apparent in a nonparametric sense. Abbreviations: Pept –peptidoglycan. Prot –proteins. Poly –polysaccharides. Hetpc see Table 1
Overall, it can be concluded from Table 3 and Fig. 7 that our mapping procedure achieves a reasonable level of fidelity relative to the dissolved organic carbon measurement literature. All the major compounds under consideration are represented in a simple offline fashion to within an order of magnitude of their empirical concentrations, wherever convenient data are at hand for evaluation. The lipids are a possible exception but this may be due primarily to coastal offsets. A combination of dynamic computation with enhanced observation and interpretation is likely to reduce the gap. We infer that the marine organic chemical simulations developed in the present work are of sufficient quality that we can now move forward, to an exploration of global scale Langmuir adsorption. This is accomplished by combining information in the tables and figures as they have been presented so far, according to logic laid out in the methods section.
Results: surface activity
Surfactant kinetics are often considered to be rapid relative to timescales for bubble transport through the upper mixed layer. This has partly been a matter of convenience, but the conclusion is consistent with a large body of laboratory evidence. Even in small tank experiments implying short and artificial trajectories, surface saturation is often demonstrable (Hoffman and Duce 1976; Blanchard 1989). In a recent study of interfacial buildup by the marine organics, Fuentes et al. (2010) obtained equilibration times much faster than bubble return to the atmosphere. The molecules under study consisted of typical phytoplanktonic exudates. The authors offer a Gibbs-Langmuir-Szyszkowski based analysis of the process, incorporating realistic macromolecular diffusion rates and bubble boundary layer theory. They conclude that for typical marine situations, full surface coverage is achieved in milliseconds. In fact the researchers go so far as to state that it is effectively instantaneous.
There are undoubtedly circumstances in the global regime under which restricted bubble penetration and slow polymer diffusion rates lead to significant undersaturation. But for the moment and in common with many preceding groups, we assume standard Langmuir equilibrium as a startup expedient. Coverage and carbon mass are then readily calculated from the relationships in 17. In Figs. 8 and 9 we couple the mixed layer distributions presented earlier with settings compiled in Table 2. Normalized concentrations K i C i are inserted into the competitive multispecies isotherm yielding coverage and excess for individual components k. Surfactant contributions to the global monolayer are then distributed across the surface ocean based on our parameterized biogeochemistry. From the fractional surface coverage and the maximum areal concentration of each species we compute the actual excess, i.e., an estimate of carbon burden per unit interfacial area of the bubble or microlayer. For convenience in comparison with film and aerosol salt content, the unit of choice is now moles/m2.
Fractional Langmuir coverage of air–water interfaces in the upper ocean, for a baseline run defined by Table 2. Only the February result is shown, but other seasons follow annual trends from 1 through 6. The quantity is dimensionless and is presented as the base 10 logarithm
Carbon mass excess on air–water interfaces of the upper ocean, for a baseline run defined by Table 2. Only the February result is shown, but other seasons follow annual trends from 1 through 6. Units are moles/m2 but the color bar is logarithmic
Surfactant mass is readily interpretable in the context of ocean interfacial chemistry through a series of orientation calculations. First of all, we convert the lower limit excess data of Table 2 from the microscopically intuitive value one per square angstrom to an equivalent level of 2 × 10−4 mol carbon/m2. This facilitates a calculation of critical "column" concentrations, provided for comparison with the amount of salt in a bursting bubble film. Rising bubbles may acquire additional organic layers during their transition through the ocean–atmosphere interface (Elliott et al. 2014; Burrows et al. 2014). It is clear from Table 2 data that maximum excess is a strong function of molecular structure. Values are often many times greater than a single C per bond length. Complex stacking configurations are regularly reported in experimental studies (e.g. Graham and Phillips 1979; Shogren and Biresaw 2007; Damodaran and Razumovsky 2003; Babak et al. 2000). Therefore we adopt the round figure spread 1–100 × 10−4 mol/m2 as a reference range. Potential carbon densities for the marine macromolecules are completely bracketed in two dimensions, under conditions of full or maximum surface coverage.
Breaking bubbles and the nascent spray are thought to be similar in thickness at about one-tenth micron (Gong 2003; O'Dowd and De Leeuw 2007; Modini et al. 2013; Burrows et al. 2014). The spectrum of sea spray flux peaks near this value, so the original film may be thinner—surface tension reshaping follows closely upon bursting once the material enters the atmosphere. The average salinity of the ocean is about 35 psu, so that the NaCl concentration can be fixed at 0.5 molar for our purposes (Longhurst 1998). This translates to a reference column salt burden of 5 × 10−5 mol/m2 over a nominal distance of 10−7 m. Thus for saturated air–water interfaces, carbon in the bio-macromolecules would be quite capable of competing on a molar basis with sodium or chloride during transport into the atmospheric boundary layer (Elliott et al. 2014; Burrows et al. 2014). This result is corroborated in many recent marine aerosol measurement studies (O'Dowd et al. 2004; O'Dowd and De Leeuw 2007; Russell et al. 2010; Lapina et al. 2011; Cunliffe et al. 2011). Our goal is to offer more detailed resolution in the chemical oceanographic sense, through the various numerical mapping techniques.
The previous tables and figures already suggest indirectly that two-dimensional surface fractions may be large over much of the global ocean. For example, compare protein and lipid maxima with their respective half saturations in Table 2 (converting molar equilibria to μM). Normalization of these concentrations as in C/C 1/2 (Eq. 13) gives a quick indication of the approach to unit coverage in a Langmuir monolayer (Adamson 1960; Adamson and Gast 1997; Elliott et al. 2014). Since both ambient levels and half saturation values are in the vicinity of μM, large proportions are plausible. To demonstrate this visually and geographically, we present the standard Langmuir theta parameter (fractional coverage) as calculated for the three most adsorptive carbon classes (proteins, lipids, peptidoglycan) and also for the massive reservoir of mixed polymers (heteropolycondensates). Global distributions are shown for this suite of compounds in Fig. 8, for the month of February. In the other months/seasons patterns merely progress according to the solar cycle as shown in the plots of 1-6. Competitive multispecies coverage in the monolayer is expressed as a base 10 logarithm in order to accommodate the wide range which must be simulated.
Figure 8 is taken as a baseline result and major inferences are as follows. In this particular simulation, proteins approach two-dimensional saturation over large sections of the ocean environment, as indicated by light red to red on the log color scale. Lipids also achieve high fractional coverage, but only in regions of intense biological activity. In either case, our reference points regarding film column burden suggest that the enrichment of organics through Langmuir adsorption may well be sufficient to explain their large contribution to the fine-mode aerosol. Protein coverage extends into the central basin gyres, since this class is treated partly as a transportable semi-labile. By contrast the shorter-lived lipids are computed assuming steady state with phytoplanktonic cell disruption, so that they are confined to immediate source regions. Heteropolycondensate coverages tend to surpass those of peptidoglycan. This is despite a lower baseline half-saturation concentration in the latter case. Hence the dominance of heteropolycondensates is attributable to their much higher concentrations in the ocean, resulting from the well-known accumulation of refractory carbon at middle latitudes (Fig. 5; Benner 2002; Hansell et al. 2012).
Carbon excess or absolute two-dimensional concentration is summarized for the Northern Hemispheric winter period in Fig. 9. Values are presented on a log scale and now they can be directly compared with the sodium content of an idealized breaking seawater film: 5 × 10−5 mol/m2 assuming thickness of 0.1 micron and treating the system as a homogeneous slab. Protein and lipid-derived carbon are again the dominant groups, and they are potentially salt competitive on a carbon-molar basis in areas of strong primary production. The relative ranking of the macromolecular groups remains in the order from Fig. 8, since water column concentration distributions are unaltered and the two dimensional maxima do not differ dramatically. For this baseline simulation, we assigned an intermediate maximum excess of 10 wherever Table 2 data fell in the range 1–100 and 1 wherever the value fell between 1 and 10 atoms per square angstrom (parentheticals in the main table field).
Several authors on this paper are involved in organic functional group analysis of marine aerosol composition. Filter samples have been collected during ship-board experiments, and artificially-generated marine particles were produced by creating bursting bubbles in natural seawater during research cruises (AF, LR). Fourier transform infrared spectroscopy (FTIR) was used to determine amounts of the major bonding structures. A key finding has been that carbon-hydrogen moieties (meaning alkanes) and the hydroxyl (generally in carbohydrate-like structures) are present in roughly equal proportions, whether in clean marine particles or bubbling analogs (Russell et al. 2010; Frossard et al. 2014). Proteins are the dominant surfactant species in our baseline run, but on average they contain only a low ratio of OH to CH. For example, among the most abundant oceanic amino acids only serine has the hydroxyl attached to its R substituent (Benner 2002; Parsons et al. 1984). The implication of such comparisons is that additional sources of carbohydrate may be required in the model. For example, the relative adsorption tendency of the polysaccharides may be underestimated in the runs presented so far. Or there may be complementary mechanisms of sea-air transfer involving colloids and gels (Cunliffe et al. 2011; Wurl et al. 2011). In addition to the fresh and conceptually pure polysaccharides of Table 1, both peptidoglycan and the heteropolycondensates are carbohydrate-rich, so that they could play into any alternate mechanisms
Spectroscopic analysis can also be extended beyond alkyl versus hydroxyl to include a wide range of organic functional groups. Amines and amides have been reported in some regions, and these would in fact be consistent with the high protein excess obtained here in our Langmuir scenario (Russell et al. 2010; Frossard et al. 2014). The actual molar ratios of major functional groups in seawater organics are presented in Table 4 as derived from standard analog compound structures. Bonds are counted according to the methods of Russell et al. (2010) so that the values are FTIR relevant. It becomes clear that saccharides and mixed polymers must be involved in spray generation in at least some combination. But additionally, amino acids arising from the proteins are implicated as a major source of nitrogen atoms. Lipids provide a likely explanation for CH (alkane) richness in a given bloom regime, but proteins and amino sugars cannot be excluded as contributors. The situation is complex, but it is probably amenable to a more detailed source-receptor type analysis. We believe this could serve as a secondary application of the mapping procedures introduced here.
Selected functional relationships from the ocean macromolecules considered here as potential adsorbing agents
CH/C
OH/C
Am/C
Benner (2002); Parsons et al. (1984)
Parsons et al. (1984); Goldberg et al. (2009)
Brzozowska et al. (2012)
Peptide arms
Nagata et al. (2003); Parsons et al. (1984)
Benner (2002); Malcolm (1990)
Composition is characterized by the molar ratio of each group to total carbon atoms in a given molecular structure. Values in the table are condensed from a data base maintained by the authors. It is referred to as GELOMOLD, for Global Excel Listing Of Marine Organic and Ligand Data
CH and OH individual carbon-hydrogen or hydroxyl bonds. Am either amide or amine. 95/5 percentages of carbon present as polysaccharide versus protein in the mixed condensate
An overestimation of protein bubble coverage could of course be partially attributable to biases in the underlying concentration distribution (Table 3). But uncertainties in the Langmuir coefficients are much greater (references in Table 2). Thus we will conclude this section with a brief sensitivity study focusing on the half saturation parameter. The reference compound lysozyme that was initially adopted to represent marine protein has a C 1/2 of 10−4 molar. This particular proxy choice is of course terrestrial, and it was necessitated by a lack of laboratory measurements for ambient marine polymers. Lysozyme is globular, and denaturing should relax the structure of any enzyme released freely into seawater. However, our most important protein reference also contains data for casein (Graham and Phillips 1979). We thus experimented with a smaller half concentration of 10−5. Continuing the pattern, we also reduced the C 1/2 associated with heteropolycondensate carbon to the level 10−2, since proteinaceous impurities are likely to enhance adsorptivity (Damodaran and Razumovsky 2003). Results for the sensitivity test phase of our calculations are shown in Figs. 10 and 11, which are patterned after 8 and 9 respectively in terms of compound order and seasonality.
Fractional Langmuir coverage of air–water interfaces in the upper ocean, sensitivity test with half saturations decreased 10x for proteins and the heterogeneous polymers. Only the February result is shown, but other seasons follow annual trends from 1 through 6. The quantity is dimensionless and is presented as the base 10 logarithm
Carbon mass excess on air–water interfaces of the upper ocean, sensitivity test with half saturations decreased 10x for proteins and the heterogeneous polymers. Only the February result is shown, but other seasons follow annual trends from 1 through 6. Units are moles/m2 but the color bar is logarithmic
Given the altered half saturations, carbon coverage attributable to the protein source is increased considerably over the entire global ocean. As expected, the heteropolycondensate contribution also increases. This second run naturally moves net surface fractions and masses in the same direction for the protein and processed macromolecular reservoirs. Such changes may bring the model somewhat closer to agreement with the FTIR measurements of artificially-generated marine aerosol particles. The increase in mixed condensate implies a higher relative contribution from OH over large areas, and especially in the subtropics. Lipids remain locally dominant in Figs. 10 and 11, but their influence is now more tightly confined to productive ring ecosystems of the Southern Ocean (Longhurst 1998). This result is consistent with a correlation of alkyl to ocean chlorophyll, as observed in the spectroscopic analyses of Frossard et al. (2014).
Preliminary sensitivity test results presented in Figs. 10 and 11 touch upon only two of the potential means by which model-data agreement might be improved here. In addition to these cases, a series of simulations was performed in order to explore other mechanisms for increasing the concentration of OH functional groups toward the observed range. For instance the baseline analog for polysaccharides was replaced with gum arabic in one set of runs (Damodaran and Razumovsky 2003). The half saturation concentration for this well-known food additive is about an order of magnitude less than that of alginate. In another test, the equilibrium constant for peptidoglycan was reduced by a factor of ten. We also represented the high latitude humics as amphiphiles, giving them a C 1/2 of 10−5 in several different computations. All these alternative scenarios provide possible ways of increasing the hydroxyl in marine aerosols.
True polymeric surfactants will behave in a non-ideal manner, with surface coverage as a function of aqueous concentration deviating significantly from Langmuir adsorption. The irregularities may result from subtleties of the two dimensional packing structure, or else conformation changes as film pressure increases. Under these circumstances the excess maximum becomes a variable quantity (e.g. Graham and Phillips 1979). Moreover, the macromolecules under consideration are often incorporated into larger colloidal particles in the surface ocean. These in turn may be efficiently collected by pure mechanical impaction on rising bubbles. Additionally, some marine particulates are capable of rising gradually to the surface due to their own buoyancy (Cunliffe et al. 2011). Polymers and gels will interact across the marine size spectrum, opening a variety of multiphase transport pathways (Wells 2002). Our view is that the classic Langmuir approach to surface activity allows for construction of a semi-quantitative but conceptually powerful foundation, from which these other issues may be discussed and investigated.
Summary and discussion
The sea-air transfer of marine macromolecules influences background aerosol composition over the majority of the planetary surface (O'Dowd et al. 2004; Russell et al. 2010; Cunliffe et al. 2011; Lapina et al. 2011). Uncertainties in biogenic emissions propagate into the chemistry of nascent spray, and could ultimately impact global climate through CCN fields operating on cloud albedo (Meskhidze et al. 2011; Carslaw et al. 2013). Effects may be felt on atmospheric particle size, hygroscopicity, vapor uptake and reactivity. Phytoplanktonic ecodynamics are widely considered to be a major driver of primary organic aerosol sources (O'Dowd et al. 2004; Meskhidze et al. 2011; Gantt et al. 2011). Chlorophyll-a correlation methods have often been invoked initially in order to represent organic distributions in the water column. In the present work we explore an option based on more complete biogeochemistry. Dynamics of the source macromolecules are represented at the level of internal mixed layer processing across the entire surface ocean. Concentration distributions are estimated for a chemically resolved set of long-chain organics and polymers. The patterns analyzed are constructed for the organic compounds through offline mapping techniques, using both steady state and proxy tracer methods as needed (Eqs. 1–3; Table 1; Figs. 1, 2, 3, 4, 5, 6).
Since sea spray is generated through bubble breaking at the surface, the suite of marine macromolecules is treated as a collection of ideal isothermal Langmuir adsorbers (Eqs. 13–17). Surfactant equilibrium parameters are derived from appropriate laboratory studies, and assessed for their significance in the open ocean context (Table 2; Figs. 8, 9, 10, 11). Fractional coverage of the air–water interface is estimated as a competitive, multispecies process for the entire global ocean–atmosphere interfacial system. The surface fractions are then converted into densities expressed as the carbon excess, a measure of the areal organic burden. This value is compared with the mass of salt contained in idealized bubble films and thus also in the sea spray aerosol. The polymer and surfactant chemistry relationships involved have been discussed in the literature over a period of several decades (e.g. Liss 1975; Liss et al. 1997; review sections in Elliott et al. 2014 or Burrows et al. 2014). Our main contribution in the present work is to apply modern Earth System model output and mapping tools to the problem (Gregg et al. 2003; Moore et al. 2004; Elliott 2009; Elliott et al. 2011). We develop estimates for the biogeographic distributions of multiple organic classes based on a state-of-the-art marine ecodynamics package. Relative mass contributions are then computed for surfactant films forming at the global air–water interface. Results may be taken as chemically and biogeographically resolved indicators of input to the organic portion of the nascent aerosol.
Our primary motivation in this research is to identify regional variations in ocean carbon composition which may transfer to the atmosphere. A Langmuir mechanism is adapted to this task because it constitutes a reasonable and tractable starting point. We begin by identifying six classes of high molecular weight organic compounds for simulation: aggregated proteins, polysaccharides, and lipids along with the rigid structural polymer chitin, prokaryotic cell wall derivatives such as peptidoglycan and its oligomeric decay products, the dominant highly processed refractories of the central gyres, and finally, deep water humic acid (Table 1). In the offline maps, protein, polysaccharide and heteropolycondensate reservoirs accumulate concentrations of tens μM carbon in the open surface sea, but regional distinctions are significant. By contrast, the lipids and peptidoglycan maximize in zones of high biological activity because they are relatively labile, and peaks are only of order μM. Peptidoglycan contours extend into the central gyres because this molecule is produced in our representation by the heterotrophic bacteria. Mixed polymers are computed from the difference between climatological refractory DOC and the remaining surface organic molecules, but it is clear that they accumulate at middle latitudes. We assume humics are transported upward from deeper ocean layers, and this occurs exclusively during deep winter convection.
In addition to simulating distributions of the macromolecules, we also estimate both fractional surfactant coverages and surface excess (two dimensional) concentrations for each class of compound. Results should be applicable to a variety of air–water interfaces in the mixed layer, e.g. both the internal surfaces of wave-generated bubbles and the ocean–atmosphere interfacial microlayer (Russell et al. 2010; Cunliffe et al. 2011). Our Langmuir chemical calculations are presented only for the month of February, since the effects of seasonal cycling are readily apparent from annual mass distributions. In a baseline model configuration, proteins are globally dominant and they appear capable of saturating upper ocean surfaces over some of biogeographic space and time. This is due to a combination of high concentrations and the laboratory adsorptivity, but of course both quantities are moderately uncertain. The lipids are strongly hydrophobic and they too can support near unit coverage, but influence is limited to areas of high primary productivity. These extend along subtropical and subpolar fronts, eastern basin upwellings, the equator, and even into the winter hemisphere in some ecological provinces. Peptidoglycan remains greatly undersaturated in the sense of surface activity in our baseline run, since both concentrations and the adsorptive tendency are intermediate. Mixed refractory products are competitive with lipids in the central basins where their relative concentrations are greatest. The baseline calculations take into account nonideal monolayer configurations for aminosugar and polysaccharide structures, reflecting the potential for balling and clumping of their carbohydrate chains (Damodaran and Razumovsky 2003; Shogren and Biresaw 2007).
The Langmuir parameters adopted here are mainly estimated based on the behavior of analog compounds in the laboratory (Table 2). Both equilibrium half-saturation levels and the maximum excess carbon concentration are highly uncertain when applied as global averages. Indications are that real values could differ by an order of magnitude or more. A primary sensitivity test was therefore conducted, focusing on our estimates of the adsorption equilibrium constants (Eqs. 13 and 17). In a set of simulations moving beyond the baseline, a direct relationship was demonstrated for the half saturation concentrations with fractional coverage. New choices of adsorptive proxy species increased the regional dominance of proteins and the proportion of carbohydrate in our simulated coatings. We were thus able to raise the profile of saccharide hydroxyl functionalities in the blend of molecular structures lining air–water surfaces. We show that this brings the modeled-mapped results closer to experimental values, obtained during ship-based aerosol/bubble sampling with spectroscopic detection (Russell et al. 2010; Frossard et al. 2014).
The runs described here were performed entirely offline in the computational sense, by importing results from a well-established marine systems model into a convenient analysis and graphics package (Moore et al. 2004; NOAA Ferret). Steady state kinetics or approximate distributions are implied for all tracers involved. It is likely that results will vary somewhat when the opportunity arises to repeat our research in the context of real chemical transport dynamics. In more complete calculations, we intend to include a broader variety of model compounds including lipoproteins, lipopolysaccharides, phosphorylated organics, plus intermediate and mixed oligomers (Parsons et al. 1984; Benner 2002). Statistical approaches may then be required to organize an increasing number of conversion rate and surfactant parameter estimates. We are aware that the equilibria lying at the core of the approach are very much approximate. In many cases data were selected and interpreted from room temperature, low ionic strength laboratory studies. Salt concentrations approach unit molarity across the global surface ocean, and polar temperatures drop well below the freezing point of pure water (Parsons et al. 1984; Longhurst 1998). Thermochemical adjustments are thus called for, in order to incorporate realistic temperature and salt effects. The Langmuir model assumes idealized surfactant behaviors such as a fixed maximum excess, and the absence of chemical interactions in two dimensions (Adamson 1960; Somorjai 1972; Adamson and Gast 1997). Actual surfactants display reconfigurations, which may dramatically alter the holding capacity of a phase boundary. Two-dimensional acid base relationships and reactive chemistry can also occur along aqueous interfaces (Kanicky and Shah 2002). Electrostatic interactions may enhance the effective thickness of the adsorbed material (Parra-Barraza et al. 2005). At least some of these challenges may be susceptible in the near term to a careful blend of theoretical and laboratory investigations. Detailed extensions of our work may prove crucial in extreme environments such as polar and ice (brine) biogeochemical systems.
The discussion as presented thus far necessarily involves a lengthy catalog of uncertainties, and so perhaps it is useful to recap at this stage. Errors in our approach may derive from several sources, and all are reflected in the parameters underlying a macromolecular chemistry-transport-surfactant formulation (equations plus Tables 1 and 2). Although global organics may prove critical to comprehension of the marine aerosol (O'Dowd et al. 2004; Russell et al. 2010), processes controlling their distribution and behavior remain very poorly known. Injection schemes, carbon proportions and removal rates adopted preliminarily here are all highly idealized (Parsons et al. 1984; Hansell et al. 2012; Table 1). Biomass fractions are represented as globally averaged carbon percentages rounded in the tens place. Mechanisms are streamlined and lifetimes are merely best estimates to the nearest power of ten in units of days. Physical chemistry is simulated via simple polymers which are often missing organic functionalities or combinations thereof. Higher fidelity representatives should be sought, from among biomacromolecular permutations which are nearly endless. Thus the list of tabled compounds will become steadily longer, though of course complexity of the real marine environment can never be approached. Even fundamental dependencies of the core chemical equilibria on temperature or salinity have been subverted for the moment (Adamson 1960; Somorjai 1972; Graham and Phillips 1979; Elliott et al. 2014).
And although we may have succeeded in mapping certain prominent bonding structures to within an order of magnitude (Table 3 and Fig. 7), error bars are much greater for their surfactant properties including coverage and excess (Table 2). This conclusion can be drawn from our own survey of the surface chemical literature, with order of magnitude biases likely in many cases (e.g. contrast the protein classes of Graham and Phillips 1979). Langmuir coverage and carbon mass per area lie at the heart of our arguments, and so it might be concluded that the fundamental logic itself is uncertain. But by the same token our conceptual/theoretical grounding is firm, whether viewed from the standpoint of marine organic chemistry or surfactant physics (Liss 1975; Adamson and Gast 1997; Liss et al. 1997; Benner 2002; Elliott et al. 2014). Hence the arguments and results merit at least initial consideration. All of our statements regarding potential biases can clearly be qualified –this particular systems-level distribution and mapping exercise is the first of its kind, so that further research will definitely improve the situation.
Perhaps the most important discrepancy in our research is revealed by comparison with the organic functional groups identified in spectroscopic data (Russell et al. 2010; Frossard et al. 2014). FTIR analyses of atmospheric marine aerosol particles, artificially generated analogs, and seawater organic matter all indicate that carbohydrates with the expected large fractions of hydroxyl functionality must be prominent components. While there is also evidence for the presence of organic nitrogen in the marine atmosphere (Mopper and Zika 1987; Russell et al. 2010), the majority of material seems to be saccharidic in nature. We begin to approach this result by optimizing the chosen equilibrium values, but other resolutions are possible as well. It is conceivable that polysaccharides form small particles in the water column which are simply swept to the level of the microlayer and above via bubble impaction (Wells 2002; Wurl et al. 2011). Carbohydrate structures are known gelling agents and in fact have a buoyant rising component of their own, associated with low internal density (Cunliffe et al. 2011). The inability of our preferred, baseline parameter set to completely explain observed functional breakdowns is unsurprising. It may merely demonstrate that higher fidelity half-concentrations need to be incorporated. But it is also likely that mechanical lifting of the polymer field must be parameterized, or that multiple layer effects need to be incorporated. A portion of the dissolved polysaccharide is acidic (Hung et al. 2001), so that the abundant divalent cations of seawater may act as bridging ions to attach the sugars as cooperative layers. In future computations, numerical connections could be made with the rising bubble field, buoyant particle types and multilevel isotherms to augment vertical transport through the water column.
These and other additional topics will be investigated in second-generation simulations already underway in our group. We are currently planning for the direct coupling of organic chemistry into the ocean circulation components of generalized Earth System Models. The goal will be to provide realistic sources of biogenic aerosol that are simultaneously dynamic and responsive to climate feedbacks. The concentration fields we describe will likely shift significantly in the era of global warming -as the ocean stratifies, as areas of primary production shift poleward, and as growing seasons lengthen toward the poles. We are hopeful in the meantime that mapped estimates of macromolecular distributions will prove useful to the contemporary aerosol community as a research guide. A sense of regional and global availability is afforded for chemically resolved ocean carbon. For instance, mapped information regarding the relative abundance of surfactants may be helpful in the design of ship and aircraft based experiments.
Participants at Los Alamos National Laboratory and New Mexico Tech thank the U.S. Department of Energy SciDAC program (Scientific Discovery for Advanced Computing), and specifically its ACES4BGC project (Applying Computationally Efficient Schemes for Biogeochemical Cycles). SMB was supported by the Office of Science Biological and Environmental Research division of the U.S. Department of Energy, as part of an Earth System Modeling Program. Additionally, contributions by AAF and LMR were supported by NSF grants OCE-1129580 and AGS-1360645. JKM and RTL acknowledge support from the DOE Office of Biological and Environmental Research, Grant ER65358. Validation exercises were conducted as part of the DOE marine biogeochemistry Benchmarking and Feedbacks effort.
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© The Author(s) 2015
1.New Mexico Institute of Mining and TechnologySocorroUSA
2.Pacific Northwest National LaboratoryRichlandUSA
3.Los Alamos National LaboratoryLos AlamosUSA
4.Scripps Institution of OceanographyLa JollaUSA
5.University of CaliforniaBerkeleyUSA
6.Oak Ridge National LaboratoryOak RidgeUSA
7.University of CaliforniaIrvineUSA
Ogunro, O.O., Burrows, S.M., Elliott, S. et al. Biogeochemistry (2015) 126: 25. https://doi.org/10.1007/s10533-015-0136-x
Accepted 28 August 2015
First Online 13 October 2015
DOI https://doi.org/10.1007/s10533-015-0136-x
Online ISSN 1573-515X | CommonCrawl |
Determinants of underweight, stunting and wasting among schoolchildren
Mekides Wolde1,
Yifru Berhan2 &
Alemzewed Chala1
BMC Public Health volume 15, Article number: 8 (2015) Cite this article
The cause of under-nutrition in schoolchildren is complex and varying from region to region. However, identifying the cause is the basic step for nutritional intervention programs.
School based cross-sectional survey was conducted among 450 schoolchildren aged 7-14 years, using multi-stage sampling techniques in Dale Woreda, southern Ethiopia.
A structured questionnaire and 24-hour recall methods were administered to determine the sociodemographic and dietary intake of participants. Stool microscopic examination was done. Weight and height were measured using a standard calibrated scale. Odds ratio generated from logistic regression was used to determine the strength of variables association.
Older age group (10-14 vs. 7-9) (AOR = 3.4; 95% CI, 1.7-6.6) and having Trichuris Trichura infection (AOR = 3.9; 95% CI, 1.4 -11.6) increased the risk of being stunted. Children whose mothers have completed primary education are less likely to be stunted than children whose mothers do not have formal education (AOR = 0.3; 95% CI, 0.2-0.8).
Having large family size (AOR = 3.3; 95% CI, 1.4-7.9) and inadequate intake of carbohydrate (AOR = 3.1; 95% CI, 1.4-6.8) were independent predictors of wasting. Children whose mothers completed primary education are less likely to be underweight (AOR = 0.3; 95% CI, 0.1-0.9). Children live in food insecure households are more likely to be stunted, under-weight and wasted than children live in food secure households (AOR = 2.5; 95%, 1-5.6; AOR = 3.9; 95% CI, 1.2-12.0; AOR = 4.8; 95% CI, 1.7-13.6;).
Household food insecurity, low maternal education and infection with Trichuris trichura were some of the major factors contributing to under-nutrition in the study area.
Under- nutrition among school-age children is a common problem in developing nations. It may turn out from a broad range of aspects like prenatal under-nutrition, deficiencies of macro and micronutrient, infection and possibly socioeconomic conditions [1].
United nation for children's fund [2] reported that more than 200 million school-age children were stunted by the year 2000.In the same report, it was pointed out that the proportion of stunted schoolchildren with impaired physical and mental development will grow up to 1 billion by the year 2020 unless a tangible action is undertaken.
The economic cost of under-nutrition is highly substantial. According to the World Health Organization, underweight is the single largest risk factor contributing to the global burden of disease in the developing world. It leads to nearly 15 percent of the total disability-adjusted life years (DALY) losses in countries with high children mortality [3]. It is also proved that 1% loss in adult height occurred due to childhood stunting which in return associate with 1.4% loss in productivity [4]. Moreover; poor nutrition and health among children contribute to the general inefficiency of education systems worldwide. Varies researches have shown that improved nutrition and health lead to better performance, fewer repeated classes and reduced dropout rates [5].
In Ethiopia, particularly in the study area, there is scarcity of information on the determinants of under-nutrition among schoolchildren. Moreover, schoolchildren are at high risk of nutritional deficiency and their nutritional status is poorly documented. Because most of the studies in this country have been conducted on under five children. We surmise that identifying the contributing factors for under-nutrition among schoolchildren is the basic step to set a sustainable and effective nutritional intervention in the study area. Thus, the present study was designed to assess determinants of under-nutrition among schoolchildren aged 7-14 years.
This school based cross-sectional survey was conducted among randomly selected schoolchildren aged 7-14 years old from randomly selected three schools in Dale
Woreda. Dale woreda is found in the Sidama zone of Southern Nations and Nationalitities Peoples' Regional State of Ethiopia. The woreda is located about 326 Km South from Addis Ababa, capital city of Ethiopia. The Woreda has a total area of 28,444 hectares; total population of 222,068 with 37,027 households. The Woreda is characterized by 1% high land and 99% mid-altitude agro-ecologies and produces a variety of crops and livestock. The area is known for its coffee production. At the time of the research, the Woreda had 36 kebeles, 7 health centers, 29 health posts and 1 hospital.
The study was conducted at three schools in October 2012 and 450 students were sampled using single population proportion formula:
$$ \mathrm{n}=\left[\frac{{\left({\mathrm{Z}}_1-\frac{\upalpha}{2}\right)}^2\uprho \left(1-\uprho \right)}{{\mathrm{d}}_2}\right] $$
Where: Z = Standard normal variable at 95% confidence level (1.96)
P = Anticipated proportion, (23.1%)
d = 0.05 (5% margin of error)
To minimize errors arising from the likelihood of non-compliance, 10% of the sample size was added. Finally, a design effect (DE) of 1.5 is used to minimize bias arising from not using simple random sampling technique.
A multi-stage sampling technique was used to select a representative sample of schoolchildren from the study area. The list of schools existing in Dale Woreda was obtained from Woreda's education office, and three schools (Debub Kegie Millennium, Soyama and Degara) were selected using lottery method. List of all students aged 7-14 years and whose grade 1-8 was obtained from each school head. A total of 1397, 1648 and 1550 schoolchildren were found in Debub kegie millennium, Soyama and Degara schools, respectively. The number of schoolchildren aged 7-14 years from each school and grade included in the study was determined by using proportional allocation (PPS) to size. Finally, systematic sampling system was applied to select the study participants from each grade and sex using the respective class rosters for 2011/2012 academic year as the sample frame; the sampling interval for the systematic sampling system was calculated by dividing the total number of students in the roster to the specific students to be selected from each class (Figure 1).
Schematic presentation of sampling procedure, Dale Woreda, 2012.
A standardized questionnaire was developed based on known and hypothesized risk factors. Household food insecurity status was assessed using the nine Food and Nutrition Technical Assistance (FANTA) scale guideline questions [6]. The questionnaire was constructed in English and translated into Sidama (local language). The questionnaire was pretested among 45 schoolchildren (10% of the sample size). The socio-demographic and dietary interviews were done at the household level.
Parents and children whose age is 8 years and above were respondents. An oriented health extension worker recorded observations on physical situations for each child in the school setting. The data collectors all had education levels of grade 10 or above and were native to the study area.
After data collection had been completed, some of the variables were categorized for analysis purpose. Categorized variables include: age of a child which is grouped into two (7-9 years and 10 and above years), family size grouped into three: small, medium and large, maternal educational status grouped into three: no formal education, primary education completed and secondary education and above, household food insecurity status grouped into four: food secure, mild food insecure, moderate food insecure and severe food insecure households.
For the anthropometric data digital portable calibrated SECA weighing scale was used to measure the weight of studied children. The children were weighed wearing lightly clothed, without shoes and with empty pockets. The calibrated SECA balance scale has intervals/sensitivity of 0.1 kg with a capacity of 130 kg.
Height was measured to the nearest 0.1 cm precision and length up to 2 meters using the same device that has a scale and sliding headpiece. Weighing scale was calibrated to zero before taking every measurement. Clinical examination of study participants was done for variables such as the presence or absence of clean clothing, dental caries, trimmed fingernails, presence of dirt on fingernails, wearing shoes, paleness on the conjunctiva and palms, edema and for any more body abnormalities by the principal investigators and an assistant nurse. All the observations were recorded in the recording format.
Stool samples were collected and preserved using WHO standard operating procedures for the parasitological examination of feces [7]. Accordingly, study participants provided labeled plastic cups with serial numbers, soft tissue paper, and clean wooden applicator stick. And they were instructed to bring 2 g or about a thumb of fresh stool sample of their own. Students who brought inadequate stool sample were re-instructed to bring another sample. Samples collected from study participants were preserved in a tube containing 10% formalin in 0.85% saline, and were transported to Hawassa university microbiology laboratory for microscopic examination following WHO standard operating procedures [7]. Ten percent (10%) of the total fecal specimen (450) was re-checked in the laboratory by a senior laboratory technologist.
Dietary assessments were conducted at the household level and the respondents were children's parents and children themselves. Trained data collectors filled the 24 hour recall dietary history with specific probes, to help the respondent remember all foods consumed throughout the day. To account for any day of the week effects on food and/or nutrient intake, weekends, weekdays and market days were proportionately represented in the survey.
Ethical clearance for this study was obtained from the institutional review board of Hawassa University. We obtained a written informed parental consent. The students' privacy during the interview, stool collection, and anthropometric measurements were maintained and data obtained from them was kept confidential.
All statistical analyses were done using SPSS for windows version 16 statistical package. Data entry for anthropometric indices was made using WHO Anthro plus version 1.0.4. The nutrient and energy content of foods consumed by the index child in the preceding 24-hours was calculated using the Ethiopian food composition table [8] and micro soft excel.
All continuous data were checked for normality using the kolmogorov-smirnove test. Descriptive statistical tests were applied to indicate the prevalence of under-nutrition as frequencies and percentages. To test the presence as well as strength of association between under nutrition and factors, binary and multivariate logistic regression tests were used. Variables used for the regression model include: sex of a child, age of a child, maternal educational status, family size, family monthly income, infection with ascariasis only, hookworm only, trichuriasis only, household food security status, and dietary intake interms of energy, carbohydrate, protein, Fe, Zn and Vit A.
Factors showed significant association in the univariate analysis and factors identified in different literatures as predictors were entered into the multivariate analysis.
Under-nutrition was defined for a child, who had less than-2 z-scores (SD) from the NCHS median reference population values [9].
Operational definitions
Adequate intake: intake of the individual is equal or above the estimated average intake (EAR) for nutrients which have EAR or intake equal or above the recommended daily allowances (RDA) for nutrients which do not have EAR.
Helminthes: are parasites which are multicellular, bilaterally symmetrical worms having three germ layers, transmitted through contact with fecally contaminated soil or water.
Helminthic infections positive: direct microscopic evidence of one or more helminthic parasites.
Personal hygiene indicators: These indicators are: responses of participants to questions such as hand washing practices before meal, hand washing practices after use of latrine, trimmed finger nails, neatness of clothing, use of detergent to wash hands and frequency of body bath were considered. Score 0 for no practice, 1 for some times and 2 for mostly practiced.
Socio-demographic characteristics
Out of the total sample, 445 (98.8%) were eligible for analysis. The mean (SD) age of the study participants was 10.7 (±2.0) years. The majority of this study participants, 425 (95.5%) and 442 (99.3%), were protestant and Sidama by religion and ethnicity, respectively. Farming was the main source of economy for approximately 90% of the households. Two hundred thirty five (52.8%) and 100(22.5%) of the study participants' mothers and fathers had never attended a formal education, respectively. The studied households had median family size of six. Most of the households, 426 (95%), were headed by the father. Two hundred fifty two (56.6%) of the participants reported a family monthly income of about 10.0 USD or below per month. The access to mass media was 44.7% (Table 1).
Table 1 Socio-demographic characteristics of studied children in the age range of 7-14 years in Sothern Ethiopia
Water, environmental and personal hygiene conditions
Tap water was the main source of drinking water for approximately half of the studied households, 234 (52.6%). Three hundred fifty-nine (80.7%) of the respondents never used a treatment for their drinking water.
Two hundred thirty eight (53.5%) had a pit latrine with slab. It was also found that 180 (40.4%) of the households disposed their solid wastes in an open field.
This study has revealed that 182 (40.9%) of the respondents used soap or ash most of the time to wash their hands. One hundred ninety four (56.4%) of the studied children had trimmed their fingernails, but 130 (29.2%) did not wash their hands before meals. It was also observed that 139 (31.2%) of the participants did not wear shoes on the date of data collection.
Under nutrition
Anthropometric measurements of weight and height of children were done. Accordingly, stunting was found to be a common nutritional problem, in which 25.6% of studied children were found to be below-2SD while 10.3% of the children were severely stunted. Based on the BMI for age status, 64 (14%) of studied children had wasting while, forty two (19%) of studied children had underweight (Table 2).
Table 2 Prevalence of under nutrition of studied children aged 7-14 years (HAZ and BMI) and aged 7-9 years (WAZ) in Southern Ethiopia
Intestinal Helminthiasis
The overall prevalence of helminthes infection was 286(64.3%) while soil transmitted helminthes infections namely: Ascaris lumbericoides, Hookworm and Trichuris trichiura infection were more common in the studied children. Two hundred fifty six (57.6%) of children were infected by Ascaris lumbericoides. Infection with more than one helminthes (mixed infection) was found in 33 (7.4%) of studied children.
Determinants of underweight, stunting and wasting
Binary logistic regression was carried out to determine factors associated with under-nutrition. Infection with Trichuris trichura, sociodemographic characteristics, dietary intake and household food insecurity status were major factors associated with under-nutrition.
According to the binary logistic regression model maternal education status (AOR = 0.3; 95% CI, 0.1-0.9) and household food insecurity (AOR = 3.9; 95% CI, 1.2-12) were independently associated with the child's low weight for age status (Table 3). Multivariate analysis of risk factors for stunting revealed that the odds of being stunted increased 4 times for children who had Trichuris trichura infection than children who do not have infection.
Table 3 Multivariate analyses of risk factors for under-weight in Southern Ethiopia
Age of a child (AOR = 3.4; 95% CI, 1.7-6.6), household food insecurity status (AOR = 4.1; 95% CI, 1.6-10) and maternal education (AOR = 0.3; 95% CI, 0.2-0.8) were found to be independent predictor of the height for age status. Furthermore, children with poor personal hygiene score were more likely to be stunted (AOR = 6.9; 95% CI, 2.8-17) (Table 4).
Table 4 Multivariate analyses of risk factors for stunting in Southern Ethiopia
Living in a food insecure household (AOR = 4.8; 95% CI, 1.7-13.6), inadequate intake of carbohydrate (AOR = 3.1; 95% CI, 1.4-6.8) and having a large family size (AOR = 3.3; 95% CI, 1.4-7.9) are identified as major risk factors for wasting (Table 5).
Table 5 Multivariate analysis of risk factors for wasting in Southern Ethiopia
The nutritional status of children in this study showed that under-nutrition was prevalent as compared to the WHO/2007 international reference standards [10]. In this study, multiple factors are associated with under-nutrition which was similarly explained by a study done in Brazil [11]. This study found that children live in food insecure households are more likely to be stunted, underweight and wasted. Which is supported by a study done by Belachew and his colleagues [12]. Ali and his colleagues also found that severe household food insecurity was significantly associated with underweight in Ethiopia [13].
A study done by Donna and his associates among Hispanic children stated that food insecurity was negatively associated with children's BMI for age [14]. However, a study done in USA revealed that there is no significant association between household food insecurity status and nutritional status of schoolchildren [15]. The fact that the problem of food insecurity is low in the USA may be the reason for the absence of association. Reports from different organizations like the World Bank statistics documented that, children who live in households lacking access to sufficient food are more likely to predispose to poor nutrition and health related problems than children from food secure households [16].
According to the current study, children whose mothers have never attended a formal education were more likely to be stunted and underweight as compared to children whose mothers had formal education. This is in line with a study done by Joshi and his colleagues and Emina and his associates [17,18]. Similarly, a study done in Nigeria, Pakistan and Pune revealed that low maternal education is one of the risk factor for stunting in the studied children [19-21].
In the present study, infection with Trichuris trichiura was significantly associated with stunting. This is due to chronic nature of Trichuris trichiura infection; in which it can stay in a human gut for more than three years [22,23]. Similar to this finding, other studies conducted elsewhere among schoolchildren found that infection with Trichuris trichiura is associated with stunting [24-26]. However, a study done in different areas of Ethiopia reported that there is no significant association between Trichuris trichiura infection and stunting [27-29].
In the current study it is found that as age of a child increase the likely hood of a child to be stunted will be increased this might be due to the fact that stunting is a chronic nutritional problem in which, once a child is stunted it might be difficult to revise in the late childhood. In this study, having large family size was found to be a risk factor for wasting. This is in line with a study done in North West Ethiopia [30]. Similarly, a study reported by Dona and his associates and Babar and his colleagues showed that the larger the size of family the poorer the nutritional status of the children would be seen [14,20].
According to the present study, none of the macronutrients or micronutrients were found significantly associated with under-weight or stunting. The findings are consistent with a study done in Vietnam [31]. However, a study done in Meghalaya/north east India found that the average energy intake was significantly lower in children with stunting than in children who had normal height for age status [32].
In the current study carbohydrate intake of children in the preceding 24 hours was found to be an independent factor for children's BMI for age status. In contrary to this, a study done in Iran revealed that there was no significant association between carbohydrate intake and wasting [33]. The presence of association between carbohydrate intake and wasting in the current study may be because wasting is an acute form of malnutrition in which a recent inadequate intake of food may affect the BMI for age status. The explanation for the absence of association between dietary intake and under-weight or stunting in the current study may be, because stunting is a chronic nutritional problem. In addition to this, a single 24 hr data may not show the usual intake of participants.
The school based cross sectional nature of this study might lead to miss children who did not get a chance to attend school and it also limited us to not determine the direction of association. A single 24 hour recall dietary data might not reflect the usual intake of participants. This study did not assess puberty which may affect the nutritional requirement and nutritional status of studied children.
In the current study, under-nutrition in the schoolchildren has multiple factors; Low maternal education and household food insecurity status were the independent factors for under-weight. Having large family size, inadequate carbohydrate intake and household food insecurity status were independent predictors for low BMI for age status. Having Trichuris trichura infection, living in a food insecure household, poor maternal education and children aged 10-14 years were risk factors for stunting.
No significant association was found between any of helminthic infections and under-weight or wasting.
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This study was sponsored by Engine/USAID project, Ethiopia. We are grateful to the study participants and data collectors.
We declare that this research article is our original work and all sources of materials used for article have been dully acknowledged. We solemnly declare that this article is not submitted to any other institution anywhere for the award of any academic degree, diploma or certificate.
Hawassa University, College of Agriculture, Hawassa, Ethiopia
Mekides Wolde & Alemzewed Chala
Hawassa University, College of Medicine and Health Sciences, Hawassa, Ethiopia
Yifru Berhan
Mekides Wolde
Alemzewed Chala
Correspondence to Yifru Berhan.
Authors listed have contributed sufficiently to the project to be included as authors, MW carried out design of the project. MW also performed statistical analysis and interpretation of data as well as revision of the manuscript; YB has been involved in design and analysis of the project. YB has also contributed in drafting the manuscript; and has given final approval of the version to be published. AC participated in data analysis, interpretation and revision of manuscript. All authors read and approved the final manuscript.
Wolde, M., Berhan, Y. & Chala, A. Determinants of underweight, stunting and wasting among schoolchildren. BMC Public Health 15, 8 (2015). https://doi.org/10.1186/s12889-014-1337-2
Crossectional
Determinant
Schoolchildren
Under-nutrition | CommonCrawl |
\begin{document}
\title{$f$-cohomology and motives over number rings} \author{Jakob Scholbach \footnote{Universit{\"a}t M{\"u}nster, Mathematisches Institut, Einsteinstr. 62, D-48149 M{\"u}nster, Germany, \href{mailto:[email protected]}{[email protected]} }}
\maketitle
\begin{abstract} This paper is concerned with an interpretation of $f$-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of $f$-cohomology of mixed motives $M_\eta$ over a number field $F$---one via ramification conditions on $\ell$-adic realizations, another one via the $K$-theory of proper regular models---both agree with motivic cohomology of $\eta_{!*} M_\eta[1]$. Here $\eta_{!*}$ is constructed by a limiting process in terms of intermediate extension functors $j_{!*}$ defined in analogy to perverse sheaves. \end{abstract}
The aim of this paper is to give an interpretation of $f$-cohomology in terms of motives over number rings. The notion of $f$-cohomology goes back to Beilinson who used it to formulate a conjecture about special $L$-values \cite{Beilinson:Higher,Beilinson:Notes}. The most classical example is what is now called $\mathrm{H}^1_f(F, \mathbf{1}(1))$, $f$-cohomology of $\mathbf{1}(1)$, the motive of a number field $F$, twisted by one. This group is $\mathcal O_F^{\times} {\otimes}_\mathbb{Z} \mathbb{Q}$, as opposed to the full motivic cohomology $\mathrm{H}^1(F, \mathbf{1}(1)) = F^{\times} {\otimes} \mathbb{Q}$. Together with the Dirichlet regulator, it explains the residue of the Dedekind zeta function $\zeta_F(s)$ at $s=1$. This idea has been generalized in many steps and many ways, for example to the notion of Selmer complexes \cite{Nekovar:Selmer}. This work is concerned with the $f$-cohomology of a mixed motive $M_\eta$ over $F$. There are two independent yet conjecturally equivalent ways to define $\mathrm{H}^1_f(F, M_\eta) \subset \mathrm{H}^1(F, M_\eta)$. We interpret the two definitions of $f$-cohomology as motivic cohomology of suitable motives over ${\mathcal{O}_F}$. This idea is due to Huber.
There are two approaches to $\mathrm{H}^1_f(M_\eta)$. The first is due to Beilinson \cite[Remark 4.0.1.b]{Beilinson:Height}, Bloch and Kato \cite[Conj. 5.3.]{BlochKato} and Fontaine \cite{Fontaine:Valeurs,FPR}. It is given by picking elements in motivic cohomology acted on by the local Galois groups in a prescribed way (\refde{H1flocal}, \refde{H1fglobal}, \refde{H1fEllAdic}). The second definition of $\mathrm{H}^1_f(M_\eta)$, due to Beilinson \cite[Section 8]{Beilinson:Notes}, applies to $M_\eta = \mathrm{h}^{i-1}(X_\eta)(n)$, with $X_\eta$ smooth and projective over $F$, $i - 2n < 0$. It is given by the image of $K$-theory of a regular proper model $X$ of $X_\eta$ (\refde{H1fImage}). Such a model may not exist, but there is a unique meaningful extension of this definition to all Chow motives over $F$ due to Scholl \cite{Scholl:Integral}.
Our main results (Theorems \ref{theo_charH1fmotivic}, \ref{theo_summaryH1f}, \ref{theo_Scholl}) show that both definitions of $\mathrm{H}^1_f(M_\eta)$ agree with $\mathrm{H}^0(\eta_{!*} \mathrm{h}^{i-1}(X_\eta, n)[1])$. Here $\eta_{!*}$ is a functor that attaches to any suitable mixed motive over $F$ one over ${\mathcal{O}_F}$. It is defined by a limiting process using the intermediate extension $j_{!*}$ familiar from perverse sheaves \cite{BBD} along open immersions $j: U \rightarrow {\Spec} {\OF}$. Even to formulate such a definition, one has to rely on profound conjectures, namely the existence of mixed motives over (open subschemes of) ${\Spec} {\OF}$. The proof of the main theorems also requires us to assume a number of properties related to weights of motives.
We point out that previously Jannsen and Scholl have shown the agreement of these two notions (in the case $M_\eta = h^{i}(X_\eta)(n)$, $X_\eta / F$ smooth and proper) under weaker hypotheses than the ones considered here \cite{Scholl:IntegralII}. Also Scholl unconditionally proved the agreement for products of smooth projective curves over $F$ (\textit{op.~cit.}). Our motivation for this studying and employing this stronger set of assumptions about motives lies in an application to special $L$-values conjectures \cite{Scholbach:specialL}. Very briefly, Beilinson's conjecture concerning special $L$-values for mixed motives $M_\eta$ over $\mathbb{Q}$ has $f$-cohomology as motivic input. $L$-functions of such motives can be generalized to motives over $\mathbb{Z}$ such that the classical $L$-function of $M_\eta$ agrees with the $L$-function (over $\mathbb{Z}$) of $\eta_{!*} M_\eta[1]$. Thereby the $L$-function and the motivic data in Beilinson's conjecture belong to the same motive over $\mathbb{Z}$, thus giving content to a more general conjecture about special $L$-values for motives over $\mathbb{Z}$. In this light it is noteworthy that $\mathrm{H}^0(\eta_{!*} \mathrm{h}^{2n-1}(X_\eta, n)[1])$ identifies with the group that occurs in the part of Beilinson's conjecture that describes special values at the central point.
The contents of the paper are as follows: \refsect{axiomaticdescription} is the basis of the remainder; it lists a number of axioms on triangulated categories of motives. Such categories $\DM_\gm(S)$ have been constructed by Voevodsky \cite{Voevodsky:TCM} and Hanamura \cite{Hanamura:Motives1} (over fields) and Levine \cite{Levine:MixedMotives} (over bases $S$ over a field). The various approaches are known to be (anti-)equivalent, at least for rational coefficients \cite[Section VI.2.5]{Levine:MixedMotives}, \cite[Section 4]{Bondarko:Differential}. Over more general bases $S$, the category $\category {DM}(S)$ has been constructed by Ivorra \cite{Ivorra:Realisation} and Cisinski and D\'eglise \cite{CisinskiDeglise:Triangulated}. We sum up the properties of this construction by specifying a number of axioms concerning triangulated categories of motives that will be used in the sequel. They are concerned with the ``core'' behavior of $\category {DM}(S)$, that is: functoriality, compacity, the monoidal structure and the relation to algebraic $K$-theory, as well as localization, purity, base-change and resolution of singularities. We work with motives with rational coefficients only, since this is sufficient for all our purposes. We use a contravariant notation for motives, that is to say the functor that maps any scheme $X$ to its motive $\operatorname{M}(X)$ shall be contravariant. This is in line with most pre-Voevodsky papers.
\refsect{realizations} is a very brief reminder on realizations. The existence of various realizations, due to Huber and Ivorra \cite{Huber:Mixed, Huber:Realization, Ivorra:Realisation}, is pinning down the intuition that motives should be universal among (reasonable) cohomology theories.
After \refsect{perverse}, a brief intermezzo on perverse $\ell$-adic sheaves over ${\mathcal{O}_F}$, \refsect{conjprop} spells out a number of conjectural properties (also called axioms in the sequel) of $\DM_\gm(S)$, where $S$ is either a finite field ${\mathbb{F}_{\pp}}$, a number field $F$ or a number ring ${\mathcal{O}_F}$. The first group of these properties centers around the existence of a category of mixed motives $\category {MM}(S)$, which is to be the heart of the so-called motivic $t$-structure. The link between mixed motives over ${\mathcal{O}_F}$ and ${\mathbb{F}_{\pp}}$ or $F$ is axiomatized by mimicking the exactness properties familiar from perverse sheaves (\refax{exactness}). A key requirement on mixed motives is that the realization functors on motives should be exact (\refax{tstructureRealizations}). For the $\ell$-adic realization over ${\Spec} {\OF}[1/\ell]$, this requires a notion of perverse sheaves over that base (\refsect{perverse}). Another important conjectural facet of mixed motives are weights. Weights are an additional structure encountered in both Hodge structures and $\ell$-adic cohomology of algebraic varieties over finite fields, both due to Deligne \cite{Deligne:Hodge3, Deligne:Weil2}. They are important in that morphisms between Hodge structures or $\ell$-adic cohomology groups are known to be strictly compatible with weights, moreover, they are respected to a certain extent by smooth maps and proper maps. It is commonly assumed that this should be the case for mixed motives, too. We show in a separate work that the $t$-structure axioms and the needed weight properties hold in the triangulated subcategory $\category{DATM}({\mathcal{O}_F}) \subset \DM_\gm({\mathcal{O}_F})$ of Artin-Tate motives (as far as they are applicable) \cite{Scholbach:Mixed}.
The remaining two sections assume the validity of the axiomatic framework set up so far. The first key notion in \refsect{motivesOF} is the intermediate extension $j_{!*} M$ of a mixed motive $M$ along some open embedding $j$ inside ${\Spec} {\OF}$. This is done as in the case of perverse sheaves, due to Beilinson, Bernstein and Deligne \cite{BBD}. Quite generally, much of this paper is built on the idea that the abstract properties of mixed perverse sheaves (should) give a good model for mixed motives over number rings. Next we develop a notion of smooth motives, which is an analog of lisse \'etale sheaves. This is needed to use a limiting technique to get the extension functor $\eta_{!*}$ that extends motives over $F$ to ones over ${\mathcal{O}_F}$. Finally, we apply the axiom on the exactness of $\ell$-adic realization to show that intermediate extensions commute with the realization functors. This will be a stepstone in a separate work on $L$-functions of motives \cite{Scholbach:specialL}.
\refsect{fcohomology} gives the comparison theorems on $f$-cohomology mentioned above. The two definitions of $f$-cohomology being quite different, the proofs of the comparison statements are different, too: the first is essentially based on the Hochschild-Serre spectral sequence. The crystalline case of that definition of $f$-cohomology is disregarded throughout. The second proof is a purely formal, if occasionally intricate bookkeeping of cohomological degrees and weights.
The problem of finding a motivic interpretation of terms such as $\mathrm{H}^1_f(M_\eta)$ underlying the formulation of Beilinson's conjecture has been studied by Scholl \cite{Scholl:Remarks, Scholl:Integral, Scholl:IntegralII}, who develops an abelian category $\category {MM}(F / {\mathcal{O}_F})$ of mixed motives over ${\mathcal{O}_F}$ by taking mixed motives over $F$, and imposing additional non-ramification conditions. Conjecturally, the group $\operatorname{Ext}^i_{\category {MM}(F / {\mathcal{O}_F})} (\mathbf{1}, \mathrm{h}^i(X_\eta, n))$ for $X_\eta / F$ smooth and projective, $i = 0$, $1$, agrees with what amounts to $\mathrm{H}^{i-1}(\eta_{!*} \mathrm{h}^{2n-1}(X_\eta, n)[1])$.
No originality is claimed for Sections \ref{sect_axiomaticdescription}, \ref{sect_realizations}, and \ref{sect_conjprop}, except perhaps for the formulation of the relation of mixed motives over ${\mathcal{O}_F}$ and $F$ and the residue fields ${\mathbb{F}_{\pp}}$, which however is a natural and immediate translation of the theory of perverse sheaves. I would like to thank Denis-Charles Cisinski and Fr\'ed\'eric D\'eglise for communicating to me their work on $\DM_\gm(S)$ over general bases \cite{CisinskiDeglise:Triangulated} and Baptiste Morin for explaining me a point in \'etale cohomology. Most of all, I gratefully acknowledge Annette Huber's advice in writing my thesis, of which this paper is a part.
\section{Geometric motives} \mylabel{sect_axiomaticdescription}
Throughout this paper, $F$ is a number field, ${\mathcal{O}_F}$ its ring of integers, $\mathfrak{p}$ stands for a place of $F$. For finite places, the residue field is denoted ${\mathbb{F}_{\pp}}$. By scheme we mean a Noetherian separated scheme. Actually, it suffices to think of schemes of finite type over one of the rings just mentioned. In this section $S$ denotes a fixed base scheme.
This section is setting up a number of axioms describing a triangulated category $\DM_\gm(S)$ of geometric motives over $S$. They will be used throughout this work. As pointed out in the introduction, the material of this section is due to Cisinski and D\'eglise \cite{CisinskiDeglise:Triangulated}, who build such a category of motives using Ayoub's base change formalism \cite{Ayoub:Six1}.
\axio \mylabel{axio_motiviccomplexes} (Motivic complexes and functoriality) \begin{itemize} \item There is a triangulated $\mathbb{Q}$-linear category $\category {DM}(S)$\index{DM@$\category {DM}(-)$}. It is called category of \Def{motivic complexes} over $S$ (with rational coefficients). It has all limits and colimits. \item (Tensor structure) The category $\category {DM}(S)$ is a triangulated symmetric monoidal category (see e.g. \cite[Part 2, II.2.1.3]{Levine:MixedMotives}). Tensor products commute with direct sums. The unit of the tensor structure is denoted $\mathbf{1}_S$ or $\mathbf{1}$\index{1@$\mathbf{1}$}. Also, there are internal $\mathrm{Hom}$-objects in $\category {DM}$, denoted $\underline{\Hom}$. The \Def{dual} $M^\vee$ of an object $M \in \category {DM}(S)$ is defined by $M^\vee := \underline{\Hom}(M, \mathbf{1})$. \item For any map $f: X \rightarrow Y$ of schemes, there are pairs of adjoint functors \eqn \mylabel{eqn_adjstar} f^*: \category {DM}(Y) \leftrightarrows \category {DM}(X) : f_* \xeqn such that $f^* \mathbf{1}_Y = \mathbf{1}_X$ and, if $f$ is quasi-projective, $$f_!: \category {DM}(X) \leftrightarrows \category {DM}(Y): f^!.$$ \end{itemize} \xaxio
The existence of $f_!$ and $f^!$ is restricted to quasi-projective maps since the abstract construction of these functors in Ayoub's work \cite[Section 1.6.5]{Ayoub:Six1}, on which Cisinski's and D\'eglise's construction of motives over general bases \cite{CisinskiDeglise:Triangulated} relies, has a similar restriction.
Recall that an object $X$ in a triangulated category $\mathcal T$ closed under arbitrary direct sums is \Def{compact} if $\mathrm{Hom}(X, -)$ commutes with direct sums. The subcategory of $\mathcal T$ of compact objects is triangulated and closed under direct summands (a.k.a. a \Def{thick subcategory}) \cite[Lemma 4.2.4]{Neeman:TC}. The category $\mathcal T$ is called \Def{compactly generated} if the smallest triangulated subcategory closed under arbitrary sums containing the compact objects is the whole category $\mathcal T$.
\axio \mylabel{axio_compact} (Compact objects)
The motive $\mathbf{1} \in \category {DM}(S)$ is compact.
The functors $f^*$ and $f^!$, whenever defined, and ${\otimes}$ and $\underline{\Hom}$ preserve compact objects. The same is true for $f_*$ and $f_!$ if $f$ is of finite type.
The canonical map $M \rightarrow (M^\vee)^\vee$ is an isomorphism for any compact object $M$.
\xaxio
\defi \mylabel{defi_motive}
The subcategory of compact objects of $\category {DM}(S)$ is denoted $\DM_\gm(S)$\index{DMgm@$\DM_\gm(-)$} and called the category of \Def{geometric motives} over $S$.
For any map $f: X \rightarrow S$ of finite type, the object $\operatorname{M}_S(X) := \operatorname{M}(X) := f_* f^* \mathbf{1} \in \DM_\gm(S)$ is called the \Def{motive} of $X$ over $S$. By adjunction, $\operatorname{M}$ is a contravariant functor from schemes of finite type over $S$ to $\DM_\gm(S)$. For any quasi-projective $f: X \rightarrow S$, the \emph{motive with compact support}\index{motive!with compact support} of $X$, $\M_{\mathrm c}(X)$, is defined as $f_! f^* \mathbf{1} \in \DM_\gm(S)$.
The smallest thick subcategory of $\category {DM}(S)$ containing the image of $\operatorname{M}$ is denoted ${\DMgmbeff}(S)$ and called the category of \Def{effective geometric motives}. The closure of that subcategory under all direct sums is called the category of \Def{effective motives}, ${\DM_\mathrm{eff}}(S)$.
\xdefi
\axio \mylabel{axio_monoidal} (Tensor product vs. fiber product) The functor $\operatorname{M}$ is an additive tensor functor, i.e., maps disjoint unions of schemes over $S$ to direct sums and fiber products of schemes over $S$ to tensor products in $\DM_\gm(S)$. \xaxio
\axio \mylabel{axio_compactlygenerated} (Compact generation) The categories $\category {DM}(S)$ and ${\DM_\mathrm{eff}}(S)$ are compactly generated. \xaxio
The category $\category {DM}(S)$, being closed under countable direct sums is pseudo-abelian \cite[Lemma II.2.2.4.8.1]{Levine:MixedMotives}, i.e., it contains kernels of projectors. In particular, the projector $\operatorname{M}(\P_S) \rightarrow \operatorname{M}(S) \rightarrow \operatorname{M}(\P_S)$ has a kernel $K$ (the first map is induced by the projection onto the base, the second map stems from the rational point $0 \in \P_S$). The object $$\mathbf{1}(-1) := K[2],$$ is called \Def{Tate object} or Tate motive. The resulting decomposition $\operatorname{M}(\P_S) = \mathbf{1} \oplus \mathbf{1}(-1)[-2]$ implies $\mathbf{1}(-1) \in {\DMgmbeff}(S)$.
\axio \mylabel{axio_TateCancel} (Cancellation and Effectivity)
In $\DM_\gm(S)$ (and thus in $\category {DM}(S)$), the Tate object $\mathbf{1}(-1)$ has a tensor-inverse denoted $\mathbf{1}(1)$\index{1(1)@$\mathbf{1}(1)$}. For any $M \in \category {DM}(S)$, $n \in \mathbb{Z}$, set $M(n) := M {\otimes} \mathbf{1}(1)^{{\otimes} n}$. Then there is a canonical isomorphism called \Def{cancellation} isomorphism ($n \in \mathbb{Z}$, $M, N \in \category {DM}(S)$): $$\mathrm{Hom}_{\category {DM}(S)}(M, N) \cong \mathrm{Hom}_{\category {DM}(S)}(M(n), N(n)).$$
The smallest tensor subcategory of $\DM_\gm(S)$ that contains ${\DMgmbeff}(S)$ and $\mathbf{1}(1)$ is $\DM_\gm(S)$. In other words, $\DM_\gm(S)$ is obtained from ${\DMgmbeff}(S)$ by tensor-inverting $\mathbf{1}(-1)$.
\xaxio
\defi \mylabel{defi_motiviccoho} Let $M$ be any geometric motive over $S$. We write $\mathrm{H}^i(M) := \mathrm{H}^i(S, M) := \mathrm{Hom}_{\category {DM}(S)}(\mathbf{1}, M[i])$. For $M = \operatorname{M}(X)(n)$ for any $X$ over $S$ we also write $\mathrm{H}^i(X, n) := \mathrm{H}^i(\operatorname{M}(X)(n)) = \mathrm{Hom}_{\DM_\gm(S)} (\mathbf{1}, \operatorname{M}(X)(n)[i]) \stackrel{\text{\refeq{adjstar}}}= \mathrm{Hom}_{\DM_\gm(X)} (\mathbf{1}, \mathbf{1}(n)[i])$. This is called \Def{motivic cohomology} of $M$ and $X$, respectively. \xdefi
\axio \mylabel{axio_RelationKTheory} (Motivic cohomology vs. K-theory)\index{motivic cohomology!Relation to $K$-theory} For any regular scheme $X$, there is an isomorphism $\mathrm{H}^i(X, n) \cong K_{2n-i}(X)^{(n)}_\mathbb{Q}$, where the right hand term denotes the Adams eigenspace of algebraic $K$-theory tensored with $\mathbb{Q}$ \cite{Quillen:HAK}. \xaxio
This is a key property of motives, since algebraic $K$-theory is a universal cohomology theory in the sense that Chern characters map from algebraic $K$-theory to any other (reasonable) cohomology theory of algebraic varieties \cite{Gillet:RR}. For $S$ a perfect field, this axiom is given by \cite[Prop. 4.2.9]{Voevodsky:TCM} and its non-effective analogue. See also \cite[Theorem I.III.3.6.12.]{Levine:MixedMotives}.
Recall Grothendieck's category of pure motives $\category {M}_\sim(K)$ with respect to an adequate equivalence relation $\sim$, see e.g.\ \cite[Section 4]{Andre:Motifs}. For rational equivalence they are also called Chow motives, since, for any smooth projective variety $X$ over a field $K$, \eqn \mylabel{eqn_ChowM} \mathrm{Hom}_{\category {M}_\mathrm{rat}(K)}(\mathbf{1}(-n), h(X)) = \mathrm{CH}^n(X), \xeqn where $h(X)$ denotes the Chow motive of $X$ and the right hand term is the \Def{Chow group} of cycles of codimension $n$ in $X$. This way, the above axiom models the fact \cite[2.1.4]{Voevodsky:TCM} that Chow motives are a full subcategory of $\DM_\gm(K)$. Under the embedding $\category {M}_\mathrm{rat}(K) \subset \DM_\gm(K)$, $h(X, n)$ maps to $\operatorname{M}(X)(n)[2n]$.
\bem We do not need to assume \emph{expressis verbis} homotopy invariance (i.e., $\mathbf{1} \stackrel \cong \rightarrow pr_* pr^* \mathbf{1} \in \DM_\gm(S)$ for $pr: S {\times} \mathbb{A}^1 \rightarrow S$) nor the projective bundle formula \cite[Prop. 3.5.1]{Voevodsky:TCM}. (Note, however, that $K'$-theory does have such properties.) \xbem
\axio \mylabel{axio_localization} (Localization) Let $i: Z \rightarrow S$ be any closed immersion and $j: V \rightarrow S$ the open complement. The adjointness maps give rise to the following distinguished triangles in $\category {DM}(S)$: $$j_! j^* \rightarrow \mathrm{id} \rightarrow i_* i^*,$$ $$i_* i^! \rightarrow \mathrm{id} \rightarrow j_* j^*.$$ (In particular, $f_* f^* \cong \mathrm{id}$, where $f: X_\mathrm{red} \rightarrow X$ denotes the canonical map of the reduced subscheme structure.) In addition, one has $j^* j_* = \mathrm{id}$ and $i^* i_* = \mathrm{id}$, equivalently $j^* i_* = i^* j_! = 0$. \xaxio
\axio \mylabel{axio_purity} (Purity and base change) \begin{itemize} \item For any quasi-projective map $f$, there is a functorial transformation of functors $f_! \rightarrow f_*$. It is an isomorphism if $f$ is projective. \item (\Def{Relative purity}): If $f$ is quasi-projective and smooth of constant relative dimension $d$, there is a functorial (in $f$) isomorphism $f^! \cong f^*(d)[2d]$. \item (\Def{Absolute purity}): If $i : Z \rightarrow U$ is a closed immersion of codimension $c$ of two regular schemes $Z$ and $U$, there is a natural isomorphism $i^! \mathbf{1} \cong \mathbf{1}(-c)[-2c]$. \item (\Def{Base change}): For any two quasi-projective maps $f$ and $g$ let $f'$ and $g'$ denote the pullback maps: \eqn \label{eqn_basechange} \xymatrix { X' {\times}_X Y \ar[r]^{g'} \ar[d]^{f'} & Y \ar[d]^f \\ X' \ar[r]^g & X } \xeqn Then there is canonical isomorphism of functors $$f^* g_! \stackrel{\cong}{{\longrightarrow}} g'_! f'^*.$$ \end{itemize} \xaxio
This axiom is proven by Cisinski \& D\'eglise using Ayoub's general base change formalism. See in particular \cite[1.4.11, 12]{Ayoub:Six1} for the construction of the base change map. See also \cite[Theorem I.I.2.4.9]{Levine:MixedMotives} for a similar statement in Levine's category of motives.
\defi \mylabel{defi_Verdierdual} Let $f: S \rightarrow {\Spec} {\Z}$ be the structural map. Assume $f$ is quasi-projective. Then $D(M) := \underline{\Hom}(M, f^! \mathbf{1}(1)[2])$ is called \Def{Verdier dual} of $M$. \xdefi
By the preceding axioms, $D$ induces a contravariant endofunctor of $\DM_\gm(S)$. The shift and twist in the definition is motivated as follows: given some complex analytic space $X$, the Verdier dual of a sheaf $\mathcal F$ on $X$ is defined by $$D(\mathcal F) := \underline{\RHom}_{\category {D}(\Shv{}{X})}(\mathcal F, f^! \mathbb{Z}),$$ where $f$ denotes the projection to a point, see e.g.\ \cite[Ch. VI]{Iversen:Cohomology}. When $X$ is smooth of dimension $d$, one has $f^! \mathbb{Z} = f^* \mathbb{Z} (d)[2d] = \mathbb{Z}(d)[2d]$. A similar fact holds for $\ell$-adic sheaves (see e.g.\ \cite[Section II.7-8]{KiehlWeissauer}). The above definition mimics this situation insofar as ${\Spec} {\Z}$ is seen as an analogue of a smooth affine curve.
Let us give a number of consequences of the preceding axioms, in particular purity, base change and localization: in \refeq{basechange}, suppose that $f$ is smooth and $g: X' \rightarrow X$ is a codimension one closed immersion between regular schemes. Then there is a canonical isomorphism \eqn \mylabel{eqn_purityExample} g^! \operatorname{M}_X(Y) = \operatorname{M}_{X'} (X' {\times}_X Y)(-1)[-2]. \xeqn
Let $Z \subset X$ be a closed immersion of quasiprojective schemes over $S$. Then there is a distinguished triangle of motives with compact support $$\M_{\mathrm c}(Z) \rightarrow \M_{\mathrm c}(X) \rightarrow \M_{\mathrm c}(X \backslash Z).$$
Let $S$ be a scheme of equidimension $d$ such that the structural map $f: S \rightarrow {\Spec} {\Z}$ factors as $$S \stackrel j {\longrightarrow} S' \stackrel i {\longrightarrow} \mathbb A^n_\mathbb{Z} \text{ or } \P[n]_\mathbb{Z} \stackrel p {\longrightarrow} {\Spec} {\Z},$$ where $j$ is an open immersion into a regular scheme $S'$, $i$ is a closed immersion and $p$ is the projection map. Then $f^! \mathbf{1} = \mathbf{1} (d-1)[2d-2]$, as one sees by applying relative purity to $p$ and to $j$, and absolute purity to $i$. In particular, the Verdier duality functor on any open subscheme $S$ of ${\Spec} {\OF}$ is given by $D_{\DM_\gm(S)} (?) = \underline{\Hom}(?, \mathbf{1}(1)[2])$ while on $\DM_\gm({\mathbb{F}_{\pp}})$ it is given by $\underline{\Hom}(?, \mathbf{1}) = ?^\vee$.
\axio \mylabel{axio_Verdierdual} (Verdier dual) The Verdier dual functor $D$ exchanges ``$!$'' and ``$*$'' throughout, e.g., there are natural isomorphisms $D (f^! M) \cong f^* D(M)$ for any quasi-projective map $f: X \rightarrow Y$ and $M \in \category {DM}(Y)$ and similarly with $f_!$ and $f_*$. \xaxio
\lemm \mylabel{lemm_reflexivity} Let $S$ be such that $f^! \mathbf{1} = f^* \mathbf{1}(d)[2d]$ for some integer $d$, where $f: S \rightarrow {\Spec} {\Z}$ is the structural map. For example, $S$ might be regular and affine or projective over $\mathbb{Z}$ (see above), or smooth over ${\Spec} {\Z}$ (purity). Then, for any compact object $M \in \DM_\gm(S)$, the canonical map $M \rightarrow D(D(M))$ is an isomorphism. This will be referred to as \Def{reflexivity} of Verdier duality. \xlemm \pr By \refax{compactlygenerated}, it suffices to check it for $M = \pi_* \pi^* \mathbf{1}$, where $\pi: X \rightarrow S$ is some map of finite type. In this case it follows for adjointness reasons and the assumption.
\end{proof}
\axio \mylabel{axio_resolution} (Resolution of singularities) Let $K$ be a field. As a triangulated additive tensor category (i.e., closed under triangles, arbitrary direct sums and tensor product), $\category {DM}(K)$ is generated by $\mathbf{1}(-1)$ and all $\operatorname{M}(X)$, where $X / K$ is a smooth projective variety.
When $S$ is an open subscheme of ${\Spec} {\OF}$, the generators of $\category {DM}(S)$ are $\mathbf{1}(-1)$, ${i_\mathfrak{p}}_* \operatorname{M}(X_\mathfrak{p})$, and $\operatorname{M}(X)$, instead, where $X_\mathfrak{p}$ is any projective and smooth variety over ${\mathbb{F}_{\pp}}$, $i_\mathfrak{p}$ denotes the immersion of any closed point ${\mathbb{F}_{\pp}}$ of $S$, and $X$ is any regular, flat projective scheme over ${\mathcal{O}_F}$. \xaxio
Consequently, the subcategories of compact objects $\DM_\gm(-)$ are generated as a thick tensor subcategory by the mentioned objects. In Voevodsky's theory of motives over a field of characteristic zero, this is \cite[Section 4.1]{Voevodsky:TCM}. This uses Hironaka's resolution of singularities. Over a field of positive characteristic and number rings, one has to use de Jong's resolution result, see \cite[Lemma B.4]{HuberKahn:Slice}.
We also need a limit property of the generic point. Let $S$ be an open subscheme of ${\Spec} {\OF}$, let $\eta: {\Spec} {F} \rightarrow S$ be the generic point.
\axio \mylabel{axio_generic} (Generic point) Let $M$ be any geometric motive over $S$. The natural maps $j_* j^* M \rightarrow \eta_* \eta^* M$ give rise to an isomorphism $\varinjlim j_* j^* M = \eta_* \eta^* M$, where the colimit is over all open subschemes $j : S' \rightarrow S$. It induces a distinguished triangle in $\category {DM}(S)$ \eqn \mylabel{eqn_localgeneric} \oplus_{\mathfrak{p} \in S} {i_\mathfrak{p}}_* i_\mathfrak{p}^! M \rightarrow \mathrm{id} \rightarrow \eta_* \eta^* M, \xeqn where the sum runs over all closed points $\mathfrak{p} \in S$ and $i_\mathfrak{p}$ is the closed immersion. \xaxio
\section{Realizations}\mylabel{sect_realizations}
One of the main interests in motives lies in the fact that they are explaining (or are supposed to explain) common phenomena in various cohomology theories. These cohomology functors are commonly referred to as \Def{realization} functors. They typically have the form $\DM_\gm(S) \rightarrow \category {D}^\mathrm{b}(\mathcal{C})$, where $\mathcal{C}$ is an abelian category whose objects are amenable with the methods of (linear) algebra, such as finite-dimensional vector spaces or finite-dimensional continuous group representations or constructible sheaves.
For example, let $\ell$ be a prime and let $S$ be either a field of characteristic different from $\ell$ or a scheme of finite type over ${\Spec} {\OF}[1/\ell]$. The $\ell$-adic cohomology maps any scheme $X$ of finite type over $S$ to $${\R} {\Gamma}_\ell (X) := \mathrm{R} \pi_* \pi^* {\mathbb{Q}_{{\ell}}} \in \category {D}^\mathrm{b}_c \left( S, {\mathbb{Q}_{{\ell}}} \right),$$ where $\pi : X \rightarrow S$ is the structural map and the right hand category denotes the ``derived'' category of constructible ${\mathbb{Q}_{{\ell}}}$-sheaves on $S$ (committing the standard abuse of notation, see e.g.\ \cite[II.6., II.7.]{KiehlWeissauer}). This functor factors over the \emph{$\ell$-adic realization functor}\index{l-adic realization@$\ell$-adic realization} (\cite[p. 772]{Huber:Realization}, \cite{Ivorra:Realisation}) ${\R} {\Gamma}_\ell : \DM_\gm(S) \rightarrow \category {D}^\mathrm{b}_c (S, {\mathbb{Q}_{{\ell}}})$. When $S$ is of finite type over ${\mathbb{F}_{\pp}}$, the realization functor actually maps to $\category {D}^\mathrm{b}_{c, m}(S, \ol {\mathbb{Q}_{{\ell}}})$, the full subcategory of complexes $C$ in $\category {D}^\mathrm{b}_c (S, \ol {\mathbb{Q}_{{\ell}}})$ such that all $\mathrm{H}^n(C)$ are mixed sheaves \cite[1.2]{Deligne:Weil2}.
Further realization functors include Betti, de Rham and Hodge realization. See e.g. \cite[2.3.5]{Huber:Realization}. The following axiom says (in particular) that the $\ell$-adic realization of $\operatorname{M}(X)$ does give the $\ell$-adic cohomology groups.
\axio \mylabel{axio_realfunct} (Functoriality and realizations) The $\ell$-adic realization functor commutes with the six Grothendieck functors $f_*$, $f_!$, $f^!$, $f^*$, ${\otimes}$ and $\underline{\Hom}$ (where applicable). For example, for any map $f: S' \rightarrow S$ and any geometric motive $M$ over $S'$: $$(f_* M)_\ell = f_* (M_\ell).$$ \xaxio
\section{Interlude: Perverse sheaves over number rings} \mylabel{sect_perverse} This section is devoted to a modest extension of $\ell$-adic perverse sheaves \cite{BBD} to the situation where the base $S$ is an open subscheme of ${\Spec} {\OF}[1/\ell]$. It is needed to formulate \refax{tstructureRealizations} for the $\ell$-adic realization of motives over number rings. This section may be considered a reformulation in ``perverse language'' of the well-known duality for cohomology of the inertia group \cite[Dualit\'e]{SGA4-1/2}. In a nutshell, the theory of perverse sheaves on varieties over ${\mathbb{F}_q}$ stakes on relative purity, that is $f^! {\mathbb{Z}_{{\ell}}} = f^* {\mathbb{Z}_{{\ell}}} (n)[2n]$ for a smooth map $f$ of relative dimension $n$. The analogous identity for a closed immersion $i: {\Spec} {\Fpp} \rightarrow S$ reads \eqn \mylabel{eqn_absolutepurity} i^! {\mathbb{Z}_{{\ell}}} = i^* {\mathbb{Z}_{{\ell}}} (-1)[-2]. \xeqn It is a reformulation of well-known cohomological properties of the inertia group: $\mathrm{H}^1(I_\mathfrak{p}, V) = (V(-1))_{I_\mathfrak{p}}$ for any $\ell$-adic module with continuous $I_\mathfrak{p}$-action ($\mathfrak{p} \nmid \ell$). All higher group cohomologies of $I_\mathfrak{p}$ vanish.
Let $\category {D}^\mathrm{b}(S, {\mathbb{Z}_{{\ell}}})$ be the bounded ``derived'' category of ${\mathbb{Z}_{{\ell}}}$-sheaves on $S$ as constructed by Ekedahl \cite{Ekedahl:Adic}. All following constructions can be done for ${\mathbb{Q}_{{\ell}}}$ instead of ${\mathbb{Z}_{{\ell}}}$, as well. We keep writing $j_*$ for the total derived functor, commonly denoted $\mathrm{R} j_*$ etc. However, $\mathrm{R}^n j_*$ etc.\ keep their original meaning.
As in \textit{loc.~cit.}, see especially [2.2.10, 2.1.2, 2.1.3, 1.4.10]\footnote{In the sequel, any reference in brackets refers to \cite{BBD}.}, one first defines a notion of stratification, and secondly obtains a $t$-structure on the subcategory $\category {D}^\mathrm{b}_{(\Sigma, L)}(S, {\mathbb{Z}_{{\ell}}})$ that are constructible with respect to a given stratification $\Sigma = \{ \Sigma_i \}$ and a set $L$ of irreducible lisse sheaves on the strata. Thirdly, one takes the ``limit'' over the stratifications. The union of all $\category {D}^\mathrm{b}_{(\Sigma, L)}(S, {\mathbb{Z}_{{\ell}}})$ is the ``derived'' category $\category {D}^\mathrm{b}_c(S, {\mathbb{Z}_{{\ell}}})$ of constructible sheaves. In order to extend the $t$-structure on the subcategories to one on $\category {D}^\mathrm{b}_c(S, {\mathbb{Z}_{{\ell}}})$, one has to check that the inclusion $\category {D}^\mathrm{b}_{(\Sigma', L')}(S, {\mathbb{Z}_{{\ell}}}) \rightarrow \category {D}^\mathrm{b}_{(\Sigma, L)}(S, {\mathbb{Z}_{{\ell}}})$ is $t$-exact for any refinement of stratifications. Here we employ a different argument. The proof of [2.1.14, 2.2.11] relies on relative purity for $\ell$-adic sheaves \cite[Exp. XVI, 3.7]{SGA4:3}. As in the proof of [2.1.14] we have to check the following: let $\Sigma_i \stackrel{a}\rightarrow \Sigma'_i \stackrel b {\longrightarrow} S$ be the inclusions of some strata and let $C \in \category {D}^{\mathrm{b}, \geq 0}_{(\Sigma', L')} (S, {\mathbb{Z}_{{\ell}}})$. Then $C \in \category {D}^{\mathrm{b}, \geq 0}_{(\Sigma, L)} (S, {\mathbb{Z}_{{\ell}}})$. We can assume $\dim \Sigma_i = 0$, $\dim \Sigma'_i = 1$, since all other cases are clear. Thus, $b$ is an open immersion. We may also assume for notational simplicity that $\Sigma_i = {\Spec} {\Fpp}$. Let $j$ be the complementary open immersion to $a$. By definition, $\mathrm{H}^n b^! C = b^! \mathrm{H}^n C = b^* \mathrm{H}^n C$ is locally constant and vanishes for $n < -1$. In the parlance of Galois modules this means that, viewed as a $\pi_1(\Sigma'_i)$-representation, the action of the inertia group $I_\mathfrak{p} \subset \pi_1(\Sigma'_i)$ on that sheaf is trivial. Thus $$a^! \mathrm{H}^n b^* C = a^* (\mathrm{R}^1 j_* j^* \mathrm{H}^n b^* C) [-2] = \mathrm{H}^1 (I_\mathfrak{p}, \mathrm{H}^n b^* C)[-2] = a^* \mathrm{H}^n b^* C (-1)[-2].$$ (We have used $\mathfrak{p} \nmid \ell$ at this point.) The spectral sequence $$\mathrm{H}^{p-2} a^* \mathrm{H}^q b^! C (-1) = \mathrm{H}^p a^! \mathrm{H}^q b^! C \Rightarrow \mathrm{H}^n a^! b^! C$$ is such that the left hand term vanishes for $p \neq 2$ since $a^*$ is exact w.r.t.~the standard $t$-structure. It also vanishes for $q < -1$ by the above. Hence the right hand term vanishes for $n = p + q < 1$. A fortiori it vanishes for $n < -\dim {\mathbb{F}_{\pp}} = 0$.
Objects in the heart of this $t$-structure on $\category {D}^\mathrm{b}_c(S, {\mathbb{Z}_{{\ell}}})$ are called \Def{perverse sheaves} on $S$. For example ${\mathbb{Z}_{{\ell}}}[1]$ and $i_* {\mathbb{Z}_{{\ell}}}$ for any immersion $i$ of a closed point are perverse sheaves on $S$. The \Def{Verdier dual} of any $\mathcal{C} \in \category {D}^\mathrm{b}_c(S, {\mathbb{Z}_{{\ell}}})$ is defined by $D(C) := \underline{\Hom} (C, {\mathbb{Z}_{{\ell}}}(1)[2])$. As above, we have dropped ``$\mathrm{R}$'' from the notation, so that this $\underline{\Hom}$ means what is usually denoted $\underline {\RHom}$.
\lemm Let $j: S' \rightarrow S$ be an open immersion and $i: Z \rightarrow S$ a closed immersion. Let $\eta: {\Spec} {F} \rightarrow S$ be the generic point. Then $j_*$, $j_!$, $i_*$, $\eta^*[-1]$, $j^*$ and $D$ are $t$-exact, while $i^*$ ($i^!$) is of cohomological amplitude $[-1, 0]$ ($[0, 1]$), in particular right-exact (left- exact, respectively). Finally, the $t$-structure on $\category {D}^\mathrm{b}_c(S, {\mathbb{Z}_{{\ell}}})$ is non-degenerate \cite[p. 32]{BBD}. \xlemm
\pr The only non-formal statement is the exactness of $j_*$. The corresponding precursor result [4.1.10] is a reformulation of \cite[Th. 3.1., Exp. XIV]{SGA4:2}, which says for any affine map $j: X \rightarrow Y$ over schemes over a field $K$, and any (honest) sheaf $\mathcal F$ which is torsion (prime to $\operatorname{char} K$) $$d(\mathrm{R}^q j_* \mathcal F)) \leq d(\mathcal F) - q$$ where $d(\mathcal G) := \sup \{ \dim \ol{\{x\}} , \mathcal G_{\ol x} \neq 0 \}$ for any sheaf $\mathcal G$. In our situation, we are given a locally constant sheaf $\mathcal F$ on $S'$ whose torsion is prime to all characteristics of $S$. The conclusion of the theorem also holds for $j$, as follows from the cohomological dimension of $I_\mathfrak{p}$, which is one.
\end{proof}
Let $\mathcal F$ be any perverse sheaf on $S'$. Following [1.4.22], let the \Def{intermediate extension} $j_{!*} \mathcal F$ be the image of the map $j_! \mathcal F \rightarrow j_* \mathcal F$ of perverse sheaves on $S$. As in [2.1.11] one sees that it can be calculated in terms of the good truncation with respect to the standard $t$-structure: $j_{!*} \mathcal F = \tau_{\leq -1}^{can} j_* \mathcal F.$ If $\mathcal F = \mathcal G[1]$, where $\mathcal G$ is a lisse (honest) sheaf on $S'$, this gives $(\mathrm{R}^0 j_* \mathcal G)[1]$.
\section{Mixed motives}\mylabel{sect_conjprop} Throughout this section, let $S = {\Spec} {F}$ or ${\Spec} {\Fpp}$ or an open subscheme of ${\Spec} {\OF}$.\\
This section formulates a number of axioms concerning weights and the motivic $t$-structure on triangulated categories of motives over $S$. In contrast to the axioms listed in \refsect{axiomaticdescription}, the axioms mentioned in this section are wide open, so it might be more appropriate to call them conjectures instead.
\axio \mylabel{axio_cohomdim} (Motivic $t$-structure and cohomological dimension)
The category of geometric motives $\DM_\gm (S)$ has a non-degenerate $t$-structure \cite[Def. 1.3.1]{BBD} called \Def{motivic $t$-structure}. Its heart is denoted $\category {MM}(S)$\index{MM@$\category {MM}(-)$}. Objects of $\category {MM}(S)$ are called \Def{mixed motives} over $S$.
For any $M \in \DM_\gm(S)$, there are $a$, $b \in \mathbb{Z}$ such that $\tau_{\leq a} M = \tau_{\geq b} M = 0$. Here and in the sequel, $\tau_{\leq -}$ and $\tau_{\geq -}$ denote the truncation functors with respect to the motivic $t$-structure.
The \Def{cohomological dimension} of $\DM_\gm({\mathbb{F}_{\pp}})$ and $\DM_\gm(F)$ is 0 and 1, respectively, in the sense that $$\mathrm{Hom}_{\category {DM}({\mathbb{F}_{\pp}})}(M, N[i]) = 0$$ for all mixed motives $M, N$ over ${\mathbb{F}_{\pp}}$ and $i > 0$ and similarly for mixed motives over $F$ and $i>1$. (For $i < 0$ the term vanishes by the $t$-structure axioms.)
The $t$-structures are such that over $S = \mathrm{Spec}\text{ } F$ or ${\Spec} {\Fpp}$, $\mathbf{1} \in \category {MM}(S)$, while for an open subscheme $S \subset {\Spec} {\OF}$, $\mathbf{1}[1] \in \category {MM}(S)$.
\xaxio
The existence of the motivic $t$-structure on $\DM_\gm(K)$ satisfying the axioms listed in this section is part of the general motivic conjectural framework, see e.g.\ \cite[App. A]{Beilinson:Height}, \cite[Ch. 21]{Andre:Motifs}. The idea of building a triangulated category of motives and descending to mixed motives by means of a $t$-structure is due to Deligne. The existence of a motivic $t$-structure on $\DM_\gm(K)$ is only known in low dimensions: the subcategory of Artin motives, i.e., motives of zero-dimensional varieties, carries such a $t$-structure \cite[Section 3.4.]{Voevodsky:TCM}. By \textit{loc.~cit.}, \cite{Orgogozo:Isomotifs}, the subcategory of $\DM_\gm(K)$ generated by motives of smooth varieties of dimension $\leq 1$ is equivalent to the bounded derived category of 1-motives \cite[Section 10]{Deligne:Hodge3} up to isogeny. Finally, if $K$ is a field satisfying the Beilinson-Soul\'e vanishing conjecture, such as a finite field or a number field, the category of Artin-Tate motives over $K$ enjoys a motivic $t$-structure \cite{Levine:TateMotives, Wildeshaus:ATM}. The results on Artin-Tate motives are generalized to bases $S$ which are open subschemes of ${\Spec} {\OF}$ in \cite{Scholbach:Mixed}.
The conjecture about the cohomological dimension is due to Beilinson. A (fairly weak) evidence for this conjecture is the cohomological dimension of Tate motives over $F$ and ${\mathbb{F}_{\pp}}$, which is one and zero, respectively. This follows from vanishing properties of $K$-theory of these fields.
The normalization in the last item is merely a matter of bookkeeping, but is motivated by similar shifts in perverse sheaves (\refsect{perverse}). The existence of a motivic $t$-structure is not expected to hold for motives with integral coefficients.
We do not (need to) assume that the canonical functor $\category {D}^\mathrm{b}(\category {MM}(S)) \rightarrow \DM_\gm(S)$ is an equivalence of categories or, equivalently \cite[Lemma 1.4.]{Beilinson:Derived}, $\operatorname{Ext}^i_{\category {MM}(S)} (A, B) = \mathrm{Hom}_{\DM_\gm(S)} (A, B[i])$ for all mixed motives $A$ and $B$.
\axio \mylabel{axio_exactness} (Exactness properties) Let $S \subset {\Spec} {\OF}$ be an open subscheme, let $i: {\Spec} {\Fpp} \rightarrow {\Spec} {\OF}$ be a closed point, $j: U \rightarrow S$ an open immersion and $\eta: \mathrm{Spec}\text{ } F \rightarrow S$ the generic point.
Then $j^* = j^!$, $\eta^*[-1]$, $i_*$, $j_*$ and $j_!$ are exact with respect to the motivic $t$-structures on the involved categories of geometric motives. Further, $i^*$ is right-exact, more precisely it maps objects in cohomological degree $0$ to degrees $[-1, 0]$. Dually, $i^!$ has cohomological amplitude $[0, 1]$. Verdier duality $D$ is ``anti-exact'', i.e., maps objects in positive degrees to ones in negative degrees and vice versa. \xaxio
The axiom is motivated by the same exactness properties in the situation of perverse sheaves over ${\Spec} {\OF}[1/\ell]$ (\refsect{perverse}). The corresponding exactness properties of the above functors on Artin-Tate motives, where the motivic $t$-structure is available, are established in \cite{Scholbach:Mixed}.
\defi The cohomology functor with respect to the motivic $t$-structure on $\DM_\gm(S)$ is denoted ${\p\mathrm{H}}^*$. For any scheme $X / S$, we write $$\mathrm{h}^i(X, n) := {\p\mathrm{H}}^i \operatorname{M}_S(X)(n).$$ \xdefi
\axio \mylabel{axio_numhom} Let $X_\eta / F$ be any smooth projective variety. Then numerical equivalence and homological equivalence (with respect to any Weil cohomology) agree on $X_\eta$. \xaxio
Let either $S$ be a field and let $C$ stand for the $\ell$-adic realization (in case $\text{char } S \neq \ell$), Betti, de Rham or absolute Hodge realization or let $S \subset {\Spec} {\OF}[1/\ell]$ be an open subscheme and let $C$ be the $\ell$-adic realization. We write ${\R} {\Gamma}_C : \DM_\gm(S) \rightarrow \category {D}^\mathrm{b}(\mathcal{C})$ for the realization functor, where $\category {D}^\mathrm{b}(\mathcal{C})$ is understood as a placeholder of the target category of $C$. (We abuse the notation insofar as that target category is not a derived category in the strict sense when $C$ is the $\ell$-adic realization.) For all realizations over a field, this category is endowed with the usual $t$-structure on the derived category of an exact category, e.g.\ on $\category {D}^\mathrm{b}_c(K, {\mathbb{Q}_{{\ell}}})$ for $\ell$-adic realization. When $C$ is the $\ell$-adic realization over an open subscheme $S$ of ${\Spec} {\OF}[1/\ell]$, we take the perverse $t$-structure on $\category {D}^\mathrm{b}_c(S, {\mathbb{Q}_{{\ell}}})$ defined in \refsect{perverse}. Using this, we have the following axiom:
\axio \mylabel{axio_tstructureRealizations} (Exactness of realization functors) Realization functors ${\R} {\Gamma}_C$ are exact with respect to the motivic $t$-structure on $\DM_\gm(S)$. Equivalently, as the $t$-structure on $\category {D}^\mathrm{b}(\mathcal{C})$ is non-degenerate, ${\R} {\Gamma}_C({\p\mathrm{H}}^0 M) = {\p\mathrm{H}}^0 {\R} {\Gamma}_C (M)$ for any geometric motive $M$ over $S$. On the left, ${\p\mathrm{H}}^0$ denotes the cohomology functor belonging to the motivic $t$-structure on $\DM_\gm(S)$, while on the right hand side it is the one belonging to the $t$-structure on $\category {D}^\mathrm{b}(\mathcal{C})$. \xaxio
This axiom is, if fairly loosely, motivated by a similar fact in the theory of mixed Hodge modules: let $X$ be any complex algebraic variety. Then, under the faithful ``forgetful functor'' from the derived category of mixed Hodge modules to the derived category of constructible sheaves with rational coefficients $$\category {D}^\mathrm{b} (\category{MHM}(X)) \rightarrow \category {D}^\mathrm{b}_c (X, \mathbb{Q})$$ the category $\category{MHM}(X)$ corresponds to perverse sheaves on $X$.
Recall that in an abelian category $\mathcal{C}$, a morphism $f: (X, W^*) \rightarrow (Y, W^*)$ between filtered objects is called \Def{strict} if $f(W^n X) = f(X) \cap W^n Y$ for all $n$.
\axio \mylabel{axio_weight} (Weights)
Any mixed motive $M$ over $S$ has a functorial finite exhaustive separated filtration $W_* M$ called \Def{weight filtration}, i.e., a sequence of subobjects in the abelian category $\category {MM}(S)$ $$0 = W_a M \subset W_{a+1} M \subset \dots \subset W_b M = M.$$ Any morphism between mixed motives is strict with respect to the weight filtration.
Tensoring any motive with $\mathbf{1}(n)$ shifts its weights by $-2n$.
Let ${\R} {\Gamma}_C : \DM_\gm(S) \rightarrow \category {D}^\mathrm{b}(\mathcal{C})$ be any realization functor that has a notion of weights (such as the $\ell$-adic realization when $S = {\Spec} {\Fpp}$ or the Hodge realization when $S = \mathrm{Spec}\text{ } \mathbb{Q}$). Then $$\operatorname{gr}_n^W {\R} {\Gamma}_C (M) = {\R} {\Gamma}_C (\operatorname{gr}_n^W M)$$ for any mixed motive $M$ over $S$.
\xaxio
\defi
For any $M \in \category {MM}(S)$, we write $\operatorname{wt}(M)$ for the (finite) set of integers $n$ such that $\operatorname{gr}^W_n M \neq 0$. For $M \in \DM_\gm(S)$, define\index{wt@$\operatorname{wt}(-)$} $\operatorname{wt}(M) := \cup_{i \in \mathbb{Z}} \operatorname{wt}({\p\mathrm{H}}^i(M)) -i$.
\xdefi
\axio \mylabel{axio_respectweights} (Preservation of weights) Let $f: X \rightarrow S$ be a quasi-projective map. Then the functors $f_! f^*$ preserve negativity of weights, i.e., given a geometric motive $M$ over $S$ with weights $\leq 0$, $f_! f^* M$ also has weights $\leq 0$. Dually, $f_* f^!$ preserves positive weights.
In the particular case $S \subset {\Spec} {\OF}$ (open), let $j: U \rightarrow S$ and $\eta : {\Spec} {F} \rightarrow S$ be an open immersion into $S$ and the generic point of $S$, respectively. Let $i: {\Spec} {\Fpp} \rightarrow S$ be a closed point. Then, $i^*$ and $j_!$ preserve negativity of weights and dually for $i^!$ and $j_*$. Finally, $j^*$ and $\eta^*$ both preserve both positivity and negativity of weights. \xaxio
The preceding weight axioms are motivated by the very same properties of $\ell$-adic perverse sheaves on schemes over $\mathbb{C}$ or finite fields \cite[5.1.14]{BBD}, number fields \cite{Huber:Perverse} as well as Hodge structures \cite[Th. 8.2.4]{Deligne:Hodge3} and Hodge modules (see \cite[Chapter 14.1]{PetersSteenbrink} for a synopsis). In these settings, actually $f_!$ and $f^*$ preserve negative weights, but we do not need weights for motives over more general bases than the ones above. The weight formalism we require is stronger than the one provided by the differential-graded interpretation of $\DM_\gm$ over a field \cite{Bondarko:Differential} or \cite[6.7.4]{BeilinsonVologodsky}.
\bem Over $S = {\Spec} {\OF}$, we actually only use the following weight properties: for any $M \in \DM_\gm(S)$, the interval $\operatorname{wt}(M)$ containing the weights of $M$ satisfies the following two properties: first, it is compatible under functoriality as in \ref{axio_respectweights} and, second, $j_{!*}$ preserves weights of pure smooth motives. (See Definitions \ref{defi_intermediateextension}, \ref{defi_genericallysmooth} for these two notions and the proof of \refth{summaryH1f}.) \xbem
\bsp \mylabel{bsp_weightTate} For any projective (smooth) scheme $X$ of finite type over $S$, the weights of $\mathrm{h}^i(X)(n)$ are $\leq i - 2n$ ($\geq i-2n$, respectively). \xbsp
\axio \mylabel{axio_mixedpure} (Mixed vs. pure motives) For any field $K$, the subcategory of pure objects in $\category {MM}(K)$ identifies with $\category {M}_\mathrm{num}(K)$, the category of numerical pure motives over $K$. \xaxio
By \refax{cohomdim}, there is an exact sequence $$0 \rightarrow \mathrm{H}^1(\mathrm{h}^{2n-1}(X_\eta, n)) \rightarrow \mathrm{H}^{2n}(X_\eta, n) \rightarrow \mathrm{H}^0(\mathrm{h}^{2n}(X_\eta, n)) \rightarrow 0.$$ By Axioms \ref{axio_numhom} and \ref{axio_mixedpure}, it reads \eqn \mylabel{eqn_morphismsChow} 0 \rightarrow \mathrm{CH}^n(X_\eta)_{\mathbb{Q},\mathrm{hom}} \rightarrow \mathrm{CH}^n(X_\eta)_\mathbb{Q} \rightarrow \mathrm{CH}^n(X_\eta)_\mathbb{Q} / \mathrm{hom} \rightarrow 0. \xeqn Here $\mathrm{CH}^m(X_\eta)_{\mathbb{Q},\mathrm{hom}}$ and \index{CHmmodulohom@$\mathrm{CH}^m(-)_\mathbb{Q} / \mathrm{hom}$} $\mathrm{CH}^m (X_\eta)_\mathbb{Q} / \mathrm{hom}$ are by definition the kernel and the image (seen as a quotient of the Chow group) of the cycle class map from the $m$-th Chow group to $\ell$-adic cohomology of $X_\eta$, $\mathrm{CH}^m (X_\eta)_\mathbb{Q} \rightarrow \mathrm{H}^{2m}(X_\eta, {\mathbb{Q}_{{\ell}}}(m))$ \cite[VI.9]{Milne:EC}.
As a consequence of the weight filtration, every mixed motive is obtained in finitely many steps by taking extensions of motives in $\category {M}_\mathrm{num}(K)$. Recall also that for any $X / {\mathbb{F}_q}$ which is smooth and projective the spectral sequence $$\operatorname{Ext}^p_{\category {MM}({\mathbb{F}_q})}(\mathbf{1}, \mathrm{h}^q(X)) \Rightarrow \mathrm{Hom}_{\DM_\gm({\mathbb{F}_q})}(\mathbf{1}, \operatorname{M}(X)[p+q])$$ degenerates by \refax{cohomdim} and yields an agreement \eqn \mylabel{eqn_numrat} \mathrm{CH}^q(X) / \mathrm{num} = \mathrm{Hom}_{\category {M}_\mathrm{num}({\mathbb{F}_q})}(\mathbf{1}, \mathrm{h}_\mathrm{num}^q(X)) \stackrel{\text{\ref{axio_mixedpure}}}= \mathrm{Hom}_{\category {MM}({\mathbb{F}_q})}(\mathbf{1}, \mathrm{h}^q(X)) = \mathrm{CH}^q(X), \xeqn i.e., the agreement of rational and numerical equivalence (and thus, of all adequate equivalence relations).
\bem \mylabel{bem_hardLefschetz} Recall that the agreement of numerical and homological equivalence on all smooth projective varieties over $F$ implies the motivic hard Lefschetz \cite[5.4.2.1]{Andre:Motifs}: for such a variety $X_\eta / F$ of constant dimension $d_\eta$, let $i \leq d_\eta$ and $a$ any integer. Then, taking the $(d_\eta - i)$-fold cup product with the cycle class of a hyperplane section with respect to an embedding of $X_\eta$ into some projective space over $F$ yields an isomorphism (``hard Lefschetz isomorphism'') \eqn \mylabel{eqn_hardLefschetz} \mathrm{h}^i(X_\eta, a) \stackrel{\cong}{{\longrightarrow}} \mathrm{h}^{2d_\eta - i}(X_\eta, d_\eta - i + a). \xeqn The hard Lefschetz is known to imply a non-canonical decomposition \cite{Deligne:Decompositions} $$\operatorname{M}(X_\eta) \cong \bigoplus \mathrm{h}^n(X_\eta)[-n].$$ \xbem
We need to assume the following generalization of this. It will be used in \refle{smooth}, which in turn is crucial in \refsect{fcohomology}. Note that the index shift in the second part is due to the normalization in \refax{cohomdim}: for $S = {\Spec} {\OF}$ and a closed point $i$ as above, take for example $X = S$, $\operatorname{M}(S) = \mathbf{1} = \mathrm{h}^{1}(S)[-1]$ (sic) and $i^* \operatorname{M}(S) = \mathbf{1}_{{\mathbb{F}_{\pp}}} = \mathrm{h}^0(\mathrm{Spec}\text{ } {\mathbb{F}_{\pp}})$.
\axio \mylabel{axio_smoothprojective} (Decomposition of smooth projective varieties)
Let $X / S$ be smooth and projective. In $\DM_\gm(S)$, there is a non-canonical isomorphism $$\phi_X : \operatorname{M}(X) \cong \bigoplus_n \mathrm{h}^n(X)[-n].$$ For open subschemes $S \subset {\Spec} {\OF}$, this isomorphism is compatible with pullbacks along all closed points $i: {\Spec} {\Fpp} \rightarrow S$ in the following sense: let $X_\mathfrak{p}$ be the fiber of $X$ over ${\mathbb{F}_{\pp}}$, and let $\psi$ be the isomorphism making the following diagram commutative. Its left hand isomorphism is an instance of base change. $$\xymatrix{ i^* \operatorname{M}(X) \ar[r]^(.4){i^* \phi_X} \ar[d]^\cong & \oplus_n i^* \mathrm{h}^n(X)[-n] \ar[d]^\psi \\ \operatorname{M}(X_\mathfrak{p}) \ar[r]^(.4){\phi_{X_\mathfrak{p}}} & \oplus_m \mathrm{h}^m(X_\mathfrak{p})[-m]} $$ Then $\psi$ respects the direct summands, i.e., induces isomorphisms $$i^* \mathrm{h}^n(X)[-n] \cong \mathrm{h}^{n-1}(X_\mathfrak{p})[-n+1].$$ \xaxio
\section{Motives over number rings}\mylabel{sect_motivesOF}
In the following sections we assume the axioms of Sections \ref{sect_axiomaticdescription}, \ref{sect_realizations}, and \ref{sect_conjprop}. Unless explicitly mentioned otherwise, let $S$ be an open subscheme of ${\Spec} {\OF}$, let $i: {\Spec} {\Fpp} \rightarrow {\Spec} {\OF}$ be a closed point, $j: S' \rightarrow S$ an open subscheme and $\eta: \mathrm{Spec}\text{ } F \rightarrow S$ the generic point.
This section derives a number of basic results about motives over $S$ from the axioms spelled out above. We define and study the \emph{intermediate extension} $j_{!*}: \category {MM}(S') \rightarrow \category {MM}(S)$ in analogy to perverse sheaves (\refde{intermediateextension}). An ``explicit'' set of generators of $\DM_\gm(S)$ (\refsa{generatorsoverOF}) is obtained using $j_{!*}$. We introduce a notion of \emph{smooth motives} (\refde{genericallysmooth}), which should be thought of as analogs of lisse sheaves. Using this notion, we extend the intermediate extension to a functor $\eta_{!*}$ spreading out motives over $F$ with a certain smoothness property to motives over $S$, cf.\ \refde{genericintermediateextension}. This functor will be the main technical tool in dealing with $f$-cohomology in \refsect{fcohomology}. In \refle{excreal} we express the $\ell$-adic realization of motives of the form $j_{!*} M$ in sheaf-theoretic language.
\subsection{Cohomological dimension}
The following is an immediate consequence of \refax{exactness}: \lemm \mylabel{lemm_eta} For any scheme $X$ over $S$ we have $\eta^* [-1] \mathrm{h}^i(X,n) = \mathrm{h}^{i-1}(X {\times}_S F, n)$. \xlemm
The following lemma parallels (and is a consequence of) \refax{cohomdim}.
\lemm \mylabel{lemm_cohomdimOF} The cohomological dimension of $\DM_\gm(S)$ is two, that is to say, for any two mixed motives $M, N$ over $S$, $$\mathrm{Hom}_{\DM_\gm(S)}(M, N[i]) = 0$$
for all $i > 2$. In particular $\mathrm{H}^{i}(M)$ vanishes for $|i|>1$. \xlemm \pr Apply $\mathrm{Hom}(M,-)$ to the localization triangle $\oplus_{\mathfrak{p} \in S} {i_\mathfrak{p}}_* i_\mathfrak{p}^! N \rightarrow N \rightarrow \eta_* \eta^* N$ of \refax{generic}, where $i_\mathfrak{p}$ are the immersions of the closed points of $S$. The terms adjacent to $\mathrm{Hom}(M, N[i])$ are $\mathrm{Hom}(M, \oplus_{\mathfrak{p}} {i_\mathfrak{p}}_* i_\mathfrak{p}^! N[i]) = \oplus_{\mathfrak{p}} \mathrm{Hom}(i_\mathfrak{p}^* M, i_\mathfrak{p}^! N[i])$ (as $M$ is compact) and $\mathrm{Hom}(M, \eta_* \eta^* N[i]) = \mathrm{Hom}(\eta^* M, \eta^* N[i])$. The latter term vanishes for $i>1$ since $\eta^*[-1]$ is exact and the cohomological dimension of $\DM_\gm(F)$ is one.
To deal with the former term, we have to take into account that $i_\mathfrak{p}^!$ and $i_\mathfrak{p}^*$ are not $t$-exact, but of cohomological amplitude $[0, 1]$ and $[-1, 0]$, respectively. By decomposing $i_\mathfrak{p}^! N$ into its ${\p\mathrm{H}}^{1}$- and ${\p\mathrm{H}}^0$-part and similarly with $i_\mathfrak{p}^* M$ and using that the cohomological dimension of $\DM_\gm({\mathbb{F}_{\pp}})$ is zero, the term vanishes for $i > 2$. Using general $t$-structure properties, the second claim is a particular case of the first one. \end{proof}
\subsection{Intermediate extension} \defi (Motivic analog of \cite[Def. 1.4.22]{BBD}) \mylabel{defi_intermediateextension} The \Def{intermediate extension} $j_{!*}$\index{j@$j_{"!*}$} of some mixed motive $M$ over $S'$ is defined as $$j_{!*} M := \mathrm{im} (j_! M \rightarrow j_* M).$$ The image is taken in the abelian category $\category {MM}(S)$, using the exactness of $j_!$ and $j_*$, \refax{exactness}. \xdefi
\bem \mylabel{bem_triangles} Let $i: Z \rightarrow S$ be the complement of $j$. The localization triangles (\refax{localization}) and cohomological amplitude of $i^*$ (\refax{exactness}) yield short exact sequences in $\category {MM}(S)$ \eqn \mylabel{eqn_local1} 0 \rightarrow i_* {\p\mathrm{H}}^{-1} i^* j_* M \rightarrow j_! M \rightarrow j_{!*} M \rightarrow 0, \xeqn \eqn \mylabel{eqn_local2} 0 \rightarrow j_{!*} M \rightarrow j_* M \rightarrow i_* {\p\mathrm{H}}^{0} i^* j_* M \rightarrow 0. \xeqn These triangles are the same as for perverse sheaves in the situation that the analog of \refax{exactness}, \cite[4.1.10]{BBD}, is applicable. \xbem
\lemm \mylabel{lemm_properties} \begin{itemize} \item Given any mixed motive $M$ over $S'$, $j_{!*} M$ is, up to a unique isomorphism, the unique mixed extension of $M$ (i.e., an object $X$ in $\category {MM}(S)$ such that $j^* X = M$) not having nonzero subobjects or quotients of the form $i_* N$, where $i: Z \rightarrow S$ is the closed complement of $j$ and $N$ is a mixed motive on $Z$. \item For any two composable open immersions $j_1$ and $j_2$ we have ${j_1}_{!*} \circ {j_2}_{!*} = ({j_1} \circ j_2)_{!*}$. \item $j_{!*}$ commutes with duals, i.e., $D(j_{!*} -) \cong j_{!*} D(-)$.
\end{itemize} \xlemm \pr The proofs of the same facts for perverse sheaves \cite[Cor. 1.4.25, 2.1.7.1]{BBD} carry over to this setting. The first statement easily implies the last one.
\end{proof}
The following proposition makes precise the intuition that any motive $M$ over $S$ should be reconstructed by its generic fiber (over $F$) and a finite number of special fibers (over various ${\mathbb{F}_{\pp}}$).
\satz \mylabel{satz_generatorsoverOF} As a thick subcategory of $\category {DM}(S)$, $\DM_\gm(S)$ is generated by motives of the form \begin{itemize} \item $i_* \operatorname{M}(X_\mathfrak{p})(m)$, where $X_\mathfrak{p} / {\mathbb{F}_{\pp}}$ is smooth projective, $m \in \mathbb{Z}$ and $i: {\Spec} {\Fpp} \rightarrow S$ is any closed point and \item $j_{!*} j^* \mathrm{h}^k(X, m)$, where $X$ is regular, flat and projective over $S$ with smooth generic fiber, and $j: S' \rightarrow S$ is such that $X {\times}_S S'$ is smooth over $S'$ and $k$ and $m$ are arbitrary. \end{itemize} \xsatz \pr Let $\mathcal D \subset \DM_\gm(S)$ be the thick category generated by the objects in the statement. By resolution of singularities over $S$ (\refax{resolution}), $\DM_\gm(S)$ is the thick subcategory of $\category {DM}(S)$ generated by objects $i_* \operatorname{M}(X_\mathfrak{p})(m)$ and $\operatorname{M}(X)(m)$, where $X_\mathfrak{p}$ and $X$ are as in the statement and $m \in \mathbb{Z}$.
It is therefore sufficient to see $M := \operatorname{M}(X) \in \mathcal D$. Let $j : S' \rightarrow S$ be such that $X_{S'}$ is smooth over $S'$. By \ref{axio_localization} it is enough to show $j_* j^* M \in \mathcal D$, since motives over finite fields are already covered. Applying the truncations with respect to the motivic $t$-structure to $j_* j^* M$ and exactness of $j_*$, $j^*$ (\refax{exactness}) shows that we may deal with $j_* j^* \mathrm{h}^k(X, m)$ for all $k$ instead of $j_* j^* M$. (Only finitely many $k$ yield a nonzero term by \refax{cohomdim}.) By \refbe{triangles}, there is a short exact sequence of mixed motives $$0 \rightarrow j_{!*} j^* \mathrm{h}^k(X, m) \rightarrow j_* j^* \mathrm{h}^k(X, m) \rightarrow i_* {\p\mathrm{H}}^0 i^* j_* j^* \mathrm{h}^k(X, m) \rightarrow 0.$$ Here $i$ is the complement of $j$. The left and right hand motives are in $\mathcal D$, hence so is the middle one. \end{proof}
\subsection{Smooth motives} \mylabel{sect_smoothmotives} The notion of smooth motives (a neologism leaning on lisse sheaves) is a technical stepstone for the definition of the generic intermediate extension $\eta_{!*}$, cf.\ \refde{genericintermediateextension}. Roughly speaking, smoothness for mixed motives $M$ means that $i^* M $ and $i^! M$ do not intermingle in the sense that their cohomological degrees are disjoint.
\defi \mylabel{defi_genericallysmooth} Let $M$ be a geometric motive over $S$. It is called \Def{smooth}\index{motive!smooth} if for any closed point $i: {\Spec} {\Fpp} \rightarrow S$ there is an isomorphism $$i^! M \cong i^* M (-1)[-2].$$ $M$ is called \Def{generically smooth}\index{motive!generically smooth} if there is an open (non-empty) immersion $j: S' \rightarrow S$ such that $j^* M$ is smooth. \xdefi
Let $X / S$ be a scheme with smooth generic fiber $X_\eta$. Then $\operatorname{M}_S(X)$ is a generically smooth motive.
The isomorphism in \refde{genericallysmooth} is not required to be canonical in any sense. Therefore, the subcategory of smooth motives is \emph{not} triangulated in $\DM_\gm(S)$.
\lemm \mylabel{lemm_basic} Let $M$ be a smooth mixed motive over $S$. Let $i: Z \rightarrow S$ be proper closed subscheme, let $j: S' \rightarrow S$ be its complement. Then $i^! M = ({\p\mathrm{H}}^1 i^! M)[-1]$ and dually $i^* M = ({\p\mathrm{H}}^{-1} i^* M)[1]$. \xlemm \pr By assumption $i^! M \cong i^* M(-1)[-2]$. By \refax{exactness}, the left hand side of that isomorphism is concentrated in degrees $[0, 1]$. The right hand side is in degrees $[1, 2]$. This shows $i^! M = {\p\mathrm{H}}^1 (i^! M)[-1]$ by \refax{cohomdim} and similarly for $i^* M$. \end{proof}
The following is the key relation of smooth motives and the intermediate extension. Note the similarity with \refle{locconst}.
\lemm \mylabel{lemm_smooth_and_j} Let $M$ be a smooth mixed motive over $S$. Then $M$ is canonically isomorphic to $j_{!*} j^* M$. \xlemm \pr Let $i : Z \rightarrow S$ be the complement of $j$. Given any $i_* N \subset M$ with $N \in \category {MM}(Z)$, we apply the left-exact functor $i^!$ and see $N \subset {\p\mathrm{H}}^0(i^! M) \stackrel{\ref{lemm_basic}}= 0$. Quotients of $M$ of the form $i_* N$ are treated dually. We now invoke \refle{properties}. \end{proof}
\lemm \mylabel{lemm_smooth} Let $X$ be any smooth projective scheme over $S$. Set $M := \operatorname{M}(X)$. Then all $\mathrm{h}^n X = {\p\mathrm{H}}^n M$ are smooth. \xlemm \pr Let $f_{m,n}$ be the $(m,n)$-component of the bottom isomorphism making the following commutative: $$ \xymatrix{ i^! M \ar[r]^{\cong, \text{ see \refeq{purityExample}}} \ar[d]^{\cong, \ref{axio_smoothprojective}} & i^* M(-1)[-2] \ar[d]^{\cong, \ref{axio_smoothprojective}} \\ \oplus_m A_m := \oplus i^! ({\p\mathrm{H}}^m M)[-m] \ar[r]^{\cong} & \oplus_n B_n := \oplus i^* ({\p\mathrm{H}}^n M)(-1)[-n-2]. }$$ We claim $f_{m,n} = 0$ for all $m \neq n$, from which the lemma follows. By \refax{smoothprojective} we have $B_n \cong \mathrm{h}^{n-1}(X_\mathfrak{p})[-n-1](-1)$. Using this and the reflexivity of the Verdier dual functor, we obtain an isomorphism $A_m \cong ({\p\mathrm{H}}^{m+1} i^! M)[-1-m]$. Hence $B_n$ is concentrated in cohomological degree $n+1$, while $A_m$ is in degree $n+2$. (The a priori bounds of \refax{exactness} would be $[m, m+1]$ and $[n+1, n+2]$, respectively.) As the cohomological dimension of motives over ${\mathbb{F}_{\pp}}$ is zero (\refax{cohomdim}), the only way for $f_{m,n} \neq 0$ is $m=n$.
\end{proof}
\subsection{Generic intermediate extension} \lemm \mylabel{lemm_spreadout} (Spreading out morphisms) Given two geometric motives $M$ and $M'$ over $S$ together with a map $\phi_\eta: \eta^* M \rightarrow \eta^* M'$, there is an open subscheme $j: S' \subset S$ and a map $\phi_{S'}: j^* M \rightarrow j^* M'$ which extends $\phi_\eta$. Any two such extensions agree when restricted to a possibly smaller open subscheme. In particular, if $\phi_\eta$ is an isomorphism, then $\phi_{S'}$ is an isomorphism for sufficiently small $S'$. \xlemm \pr The adjunction map $M \rightarrow \eta_* \eta^* M$ and $\eta_* \phi_\eta$ give a map $M \rightarrow \eta_* \eta^* M'$, hence by \refeq{localgeneric} a map $M \rightarrow \oplus_\mathfrak{p} {i_\mathfrak{p}}_* i_\mathfrak{p}^! M'[1]$. As $M$ is compact, it factors over a finite sum $\oplus_{\mathfrak{p} \in T} {i_\mathfrak{p}}_* i_\mathfrak{p}^! M'[1]$. Let $j: S' \rightarrow S$ be the complement of the points in $T$. The map $M \rightarrow \eta_* \eta^*M'$ factors over ${j}_* j^* M'$ and gives a map $j^* M \rightarrow j^* M'$ which extends $\phi_\eta$. The first claim is shown.
For the unicity of the extension, we may assume that $\phi_\eta$ is zero, and show that $\phi_{S'}$ is zero for some suitable $S'$. This is the same argument as before: the map $M \rightarrow {j}_* j^* M'$ constructed in the previous step factors over $\oplus_{\mathfrak{p} \in S'} {i_\mathfrak{p}}_* i_\mathfrak{p}^! M'$, since $M \rightarrow \eta_* \eta^* M'$ is zero. By compacity of $M$, only finitely many primes in the sum contribute to the map, discarding these yields the claim.
If $\phi_\eta$ is an isomorphism, $\psi_\eta := \phi_\eta^{-1}$ can be extended to some $\psi_{S'}$. As both $\phi_{S'} \circ \psi_{S'}$ and $\mathrm{id}_{S'}$ extend $\mathrm{id}_\eta$, they agree on some possibly smaller open subscheme of $S$ and similarly with $\psi_{S'} \circ \phi_{S'}$.
\end{proof}
\bem The lemma shows the full faithfulness of the functor $$\varinjlim_{S' \subset S} \DM_\gm(S') \stackrel{\eta^*}{{\longrightarrow}} \DM_\gm(F).$$ Its essential surjectivity is a consequence of \refax{compactlygenerated}, so we have an equivalence. However, we will stick to the more basic language of colimits in $\category {DM}(S)$ instead of colimits of the categories of geometric motives. \xbem
\defi \mylabel{defi_genericintermediateextension} Let $M_\eta \in \DM_\gm(F)$ be a motive such that there exists a generically smooth mixed motive $M$ over $S$ (\refde{genericallysmooth}) with $\eta^* M \cong M_\eta$. Then the \Def{generic intermediate extension} $\eta_{!*} M_\eta$ is defined as $$\eta_{!*} M_\eta := j_{!*} j^* M$$ where $j: S' \rightarrow S$ is an open immersion such that $j^* M$ is smooth. \xdefi
This is independent of the choices of $j$ and $M$ (Lemmas \ref{lemm_smooth_and_j}, \ref{lemm_spreadout}) and functorial (\ref{lemm_spreadout}). For a mixed, non-smooth motive $M$, there need not be a map $j_{!*} j^* M \rightarrow M$. Therefore, $\varinjlim j_{!*} j^* M$ does not make sense unless there is some smoothness constraint on $M_\eta$.
\subsection{Intermediate extension and $\ell$-adic realization} \mylabel{sect_intermediateell} This subsection deals with the interplay of the (generic) intermediate extension functor on mixed motives and the $\ell$-adic realization. In this subsection, $S$ is an open subscheme of ${\Spec} {\OF}[1/\ell]$. The following lemma is well-known.
\lemm \mylabel{lemm_locconst} Let $\mathcal F$ be an \'etale (honest) locally constant sheaf on $S$. Let $\eta: {\Spec} {F} \rightarrow {\Spec} {\OF}[1/\ell]$ be the generic point. Then the canonical map $\mathcal F \rightarrow \mathrm{R}^0 \eta_* \eta^* \mathcal F$ is an isomorphism. \xlemm
\lemm \mylabel{lemm_excreal} Let $M$ be a mixed motive over $S'$. Let $j: S' \rightarrow S$ be an open immersion. Then $$(j_{!*} M)_\ell = j_{!*} (M_\ell).$$ Let $i$ be the complementary closed immersion to $j: S' \rightarrow S$ and let $\eta'$ and $\eta$ be the generic point of $S'$ and $S$, respectively. If $M$ is additionally smooth, one has $$(i^* j_{!*} M)_\ell = i^* j_{!*} M_\ell = i^* (\mathrm{R}^0 \eta_* \eta'^* M_\ell[-1])[1].$$
\xlemm
To clarify the statement, note that the $\ell$-adic realization of $M$ is a perverse sheaf on $S'$ by \refax{tstructureRealizations}. Thus, $j_{!*}$ (\refsect{perverse}) can be applied to it.
\pr The first statement follows from \refax{realfunct}, the definition of $j_{!*}$ and the exactness of ${\R} {\Gamma}_\ell$ (\refax{tstructureRealizations}).
Let now $M$ be mixed and smooth over $S'$. As $M_\ell$ is a perverse sheaf by \ref{axio_tstructureRealizations}, there is an open immersion $j' : S'' \rightarrow S'$ such that $j'^* M_\ell[-1]$ is a locally constant (honest) sheaf on $S''$. As $M$ is smooth we know from Lemmas \ref{lemm_properties} and \ref{lemm_smooth_and_j} $$i^* j_{!*} M = i^* (j \circ j')_{!*} j'^* M.$$ By the interpretation of $(j \circ j')_{!*}$ in terms of $\mathrm{R}^0 (j \circ j')_*$ (\refsect{perverse}) we have $$(i^* j_{!*} M)_\ell = i^* j_{!*} M_\ell = i^* (\mathrm{R}^0 (j \circ j')_* j'^* M_\ell[-1])[1] \stackrel{\text{\ref{lemm_locconst}}}= i^* (\mathrm{R}^0 \eta_* \eta'^* M_\ell[-1])[1].$$ \end{proof}
\section{$f$-cohomology} \mylabel{sect_fcohomology}
\subsection{$f$-cohomology via non-ramification} Let $F$ be a number field. For any place $\mathfrak{p}$ of $F$, let $F_\mathfrak{p}$ be the completion, $G_\mathfrak{p}$ the local Galois group. For finite places, $I_\mathfrak{p}$ denotes the inertia group. For brevity, we will usually write $\mathrm{H}^*(M)$ for $\mathrm{H}^*(S, M)$, where $M$ is any motive over some base $S$.
\defi \cite[Section 3]{BlochKato} \mylabel{defi_H1flocal} Let $V$ be a finite-dimensional $\ell$-adic vector space, endowed with a continuous action of $G_\mathfrak{p}$, where $\mathfrak{p}$ is a finite place of $F$ not over $\ell$. Set $$\mathrm{H}^i_f(F_\mathfrak{p}, V) := \left\{ \begin{array}{ll}
\mathrm{H}^0 (F_\mathfrak{p}, V) & i=0\\ \ker \mathrm{H}^1(F_\mathfrak{p}, V) \rightarrow \mathrm{H}^1(I_\mathfrak{p}, V) & i=1 \\ 0 & \text{else.}
\end{array} \right.$$
\xdefi
\bem If $\mathfrak{p}$ lies over $\ell$, the definition is completed by $\mathrm{H}^1_f(F_\mathfrak{p}, V) := \ker \mathrm{H}^1(F_\mathfrak{p}, V) \rightarrow \mathrm{H}^1(F_\mathfrak{p}, B_{crys} {\otimes} V)$, where $B_{crys}$ denotes the ring of $\mathfrak{p}$-adic periods \cite{FontaineMessing}. We will disregard this case throughout. \xbem
\lemm \mylabel{lemm_charH1f} Let $\eta_\mathfrak{p}: \mathrm{Spec}\text{ } F_\mathfrak{p} \rightarrow \mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}$ be the generic point of the completion of ${\mathcal{O}_F}$ at $\mathfrak{p}$. Using the above notation, for $\mathfrak{p}$ not over $\ell$, there is a canonical isomorphism $\mathrm{H}^1_f (F_\mathfrak{p}, V) \cong \mathrm{H}^1 ({\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^0 {\eta_\mathfrak{p}}_{*} V)$. (The right hand side denotes $\ell$-adic cohomology over ${\mathcal{O}_{F_\mathfrak{p}}}$.) \xlemm \pr For any $\ell^n$-torsion sheaf $\mathcal F$ on $F_\mathfrak{p}$ we write $A(\mathcal F) := \ker \mathrm{H}^1(F_\mathfrak{p}, \mathcal F) \rightarrow \mathrm{H}^1(I_\mathfrak{p}, \mathcal F)$. The ${\mathbb{Q}_{{\ell}}}$-sheaf $V$ is, by definition, of the form $U {\otimes}_{\mathbb{Z}_{{\ell}}} {\mathbb{Q}_{{\ell}}}$, where $U = (U_n)_n$ is a projective system of $\mathbb{Z} / \ell^n$-sheaves. By definition $$\mathrm{H}^1(F_\mathfrak{p}, V) = \varprojlim_{n \in \mathbb{N}} \mathrm{H}^1(F_\mathfrak{p}, U_n) {\otimes} {\mathbb{Q}_{{\ell}}}$$ and similarly for $\mathrm{H}^1(I_\mathfrak{p}, V)$. Both $\varprojlim_n$ and $- {\otimes}_{\mathbb{Z}_{{\ell}}} {\mathbb{Q}_{{\ell}}}$ are left-exact functors, so one has $$\mathrm{H}^1_f(F_\mathfrak{p}, V) = \left (\varprojlim_n A(U_n) \right) {\otimes} {\mathbb{Q}_{{\ell}}}.$$ Thus it is sufficient to show $A(U) = \mathrm{H}^1({\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^0 {\eta_\mathfrak{p}}_{*} U)$ for any $\ell^n$-torsion sheaf $U$ over $F_\mathfrak{p}$.
Recall the description of \'etale sheaves on ${\mathcal{O}_{F_\mathfrak{p}}}$ from \cite[II.3.12, II.3.16]{Milne:EC}. Let $i: {\Spec} {\Fpp} \rightarrow \mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}$ be the closed point. As ${\mathcal{O}_{F_\mathfrak{p}}}$ is a henselian ring \cite[Prop. I.4.5]{Milne:EC}, for any sheaf $\mathcal F$ on $\mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}$, the global sections depend only on the special fiber and $$ \Gamma_{\mathrm{Spec}\text{ } F_\mathfrak{p}} = \Gamma_{\mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}} \circ ({\eta_\mathfrak{p}}_{*}) = \Gamma_{\mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}} \circ (i_* i^* {\eta_\mathfrak{p}}_{*}). $$
These functors can be interpreted using group cohomology: $\Gamma_{\mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}} \circ i_* = \Gamma_{{\mathbb{F}_{\pp}}}$ and $(-)^{I_\mathfrak{p}} = i^* {\eta_\mathfrak{p}}_{*}$ (\textit{loc.~cit.}). The Hochschild-Serre spectral sequence for $(-)^{G_\mathfrak{p}} = (-)^{\mathrm{Gal}({\mathbb{F}_{\pp}}) } \circ (-)^{I_\mathfrak{p}}$ can be translated to $$ \mathrm{H}^p ( \mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}, i_* i^* \mathrm{R}^q {\eta_\mathfrak{p}}_* U) \Rightarrow \mathrm{H}^n (F_\mathfrak{p}, U).$$ In addition we have the Leray spectral sequence $$ \mathrm{H}^p ( \mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^q {\eta_\mathfrak{p}}_* U) \Rightarrow \mathrm{H}^n (F_\mathfrak{p}, U). $$ The exact sequence of low degrees of the Hochschild-Serre sequence maps to the sequence below: $$ \xymatrix { 0 \ar[r] & \mathrm{H}^1(\mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^0 {\eta_\mathfrak{p}}_{*} U) \ar[r] & \mathrm{H}^1(F_\mathfrak{p}, U) \ar[r] \ar[d]^= & \mathrm{H}^0(\mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^1 {\eta_\mathfrak{p}}_* U) \ar[d] \\ 0 \ar[r] & A(U) \ar[r] & \mathrm{H}^1(F_\mathfrak{p}, U) \ar[r] & \mathrm{H}^1(I_\mathfrak{p}, U) } $$ As $\mathrm{H}^0(\mathrm{Gal}({\mathbb{F}_{\pp}}), \mathrm{H}^1(I_\mathfrak{p}, U)) \subset \mathrm{H}^1(I_\mathfrak{p}, U)$ and $\Gamma_{\mathcal{O}_{F_\mathfrak{p}}} = \Gamma_{\mathcal{O}_{F_\mathfrak{p}}} \circ i_* i^*$, the right hand map is injective, therefore there is a unique isomorphism between the left hand terms making the diagram commutative.
\end{proof}
In order to proceed to a global level, the following definition is done:
\defi \mylabel{defi_H1fglobal} \cite[II.1.3]{FPR} Given an $\ell$-adic continuous representation $V$ of $G = \mathrm{Gal}(F)$, define $\mathrm{H}^i_f(F, V)$\index{H1fF@$\mathrm{H}^1_f(F, -)$} to be such that the following diagram is cartesian. In the lower row, $V$ is considered a $G_\mathfrak{p} = \mathrm{Gal}(F_\mathfrak{p})$-module by restriction. \begin{displaymath} \xymatrix { \mathrm{H}^i_f(F, V) \ar[r] \ar[d] & \mathrm{H}^i(F, V) \ar[d] \\ \prod \mathrm{H}^i_f(F_\mathfrak{p}, V) \ar[r] & \prod \mathrm{H}^i(F_\mathfrak{p}, V) } \end{displaymath}
The product ranges over all finite places $\mathfrak{p}$ of $F$. We define $\mathrm{H}^i_{f,\backslash \operatorname{crys}}(F, V)$\index{H1flF@$\mathrm{H}^i_{f,\backslash \ell}(F, V)$} similarly, except that in the lower row of the preceding diagram only places $\mathfrak{p}$ that do not lie over $\ell$ occur. \xdefi
\lemm \mylabel{lemm_charH1fglobal} Let $V$ be an $\ell$-adic \'etale sheaf on $\mathrm{Spec}\text{ } F$. Then there is a natural isomorphism $$\mathrm{H}^i_{f,\backslash \operatorname{crys}}(F, V) \cong \mathrm{H}^1 ({\mathcal{O}_F}[1/\ell], \mathrm{R}^0 {\eta}_{*} V).$$ \xlemm \pr By the same argument as in the previous proof, we may assume that $V$ is a sheaf of $\mathbb{Z} / \ell^n$-modules, since the isomorphism we are going to establish is natural in $V$ and $$\mathrm{H}^i_{f,\backslash \operatorname{crys}}(F, V) = \ker \mathrm{H}^i(F, V) \rightarrow \prod_{\mathfrak{p} \nmid \ell} \left ( \mathrm{H}^i(F_\mathfrak{p}, V) / \mathrm{H}^i_f(F_\mathfrak{p}, V) \right).$$ Consider the following cartesian diagram ($\mathfrak{p} \nmid \ell$) $$\xymatrix{ \mathrm{Spec}\text{ } F_\mathfrak{p} \ar[r]^{\eta_\mathfrak{p}} \ar[d]^b & \mathrm{Spec}\text{ } {\mathcal{O}_{F_\mathfrak{p}}} \ar[d]^a & \mathrm{Spec}\text{ } {\mathbb{F}_{\pp}} \ar[l]^{i_\mathfrak{p}} \ar[d]^= \\ \mathrm{Spec}\text{ } F \ar[r]^{\eta} & \mathrm{Spec}\text{ } {\mathcal{O}_F}[1/\ell] & \mathrm{Spec}\text{ } {\mathbb{F}_{\pp}} \ar[l]^{i} } $$ In the derived category of $\mathbb{Z} / \ell^n$-sheaves on ${\Spec} {\OF}[1/\ell]$, there is a triangle $\mathrm{R}^0 \eta_* V \rightarrow \mathrm{R} \eta_* V \rightarrow \mathrm{R}^1 \eta_*[-1] V$. Likewise,
$\mathrm{R}^0 {\eta_\mathfrak{p}}_* b^* V \rightarrow {\mathrm{R} \eta_\mathfrak{p}}_* b^* V \rightarrow \mathrm{R}^1 {\eta_\mathfrak{p}}_* b^* V[-1]$. (We have used $\mathfrak{p} \nmid \ell$, since the inertia group has cohomological dimension bigger than one for $\mathfrak{p} | \ell$.) This yields exact horizontal sequences, the vertical maps are adjunction maps
\scriptsize $$ \xymatrix { 0 \ar[r] & \mathrm{H}^1(\mathrm{Spec}\text{ } {\mathcal{O}_F}[1/\ell], {\eta}_{*} V) \ar[r] \ar[d] & \mathrm{H}^1(F, V) \ar[r] \ar[d] & \mathrm{H}^0(\mathrm{Spec}\text{ } {\mathcal{O}_F}[1/\ell], \mathrm{R}^1 \eta_* V) \ar[d]^\alpha \\ 0 \ar[r] & \prod_{\mathfrak{p} \nmid \ell} \mathrm{H}^1({\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^0 {\eta_\mathfrak{p}}_* b^* V) \ar[r] & \prod_{\mathfrak{p} \nmid \ell} \mathrm{H}^1(F_\mathfrak{p}, b^* V) \ar[r] & \prod_{\mathfrak{p} \nmid \ell} \mathrm{H}^0({\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^1 {\eta_\mathfrak{p}}_* b^* V) } $$ \normalsize We will show that $\alpha$ is injective. Hence, the left square is cartesian and by definition and \refle{charH1f} the claim is shown. Indeed, $\alpha$ factors as $$\mathrm{H}^0({\mathcal{O}_F}[1/\ell], \mathrm{R}^1 \eta_* V) \subset \prod_{\mathfrak{p} \nmid \ell} \mathrm{H}^0({\mathbb{F}_{\pp}}, i_\mathfrak{p}^* \mathrm{R}^1 \eta_* V) \rightarrow \prod_{\mathfrak{p} \nmid \ell} \mathrm{H}^0({\mathcal{O}_{F_\mathfrak{p}}}, \mathrm{R}^1 {\eta_\mathfrak{p}}_* b^* V) \left ( \stackrel \cong = \prod_{\mathfrak{p} \nmid \ell} \mathrm{H}^0 ({\mathbb{F}_{\pp}}, i_\mathfrak{p}^* \mathrm{R}^1 {\eta_\mathfrak{p}}_* b^* V) \right).$$ using $i^* \mathrm{R}^1 \eta_* V = i_\mathfrak{p}^* a^* \mathrm{R}^1 \eta_* V = i_\mathfrak{p}^* \mathrm{R}^1 {\eta_\mathfrak{p}}_* b^* V$. \end{proof}
\defi \cite[Remark 4.0.1.b]{Beilinson:Height}, \cite[Conj. 5.3]{BlochKato}, \cite[Section 6.5]{Fontaine:Valeurs}, \cite[III.3.1.3]{FPR} \mylabel{defi_H1fEllAdic} Let $M_\eta$ be a mixed motive over $F$. Let, similarly to \refde{H1fglobal}, $\mathrm{H}^i_f (M_\eta)$ be defined such that the following diagram, in which the bottom products are taken over all primes $\ell$, is cartesian. As usual, ${M_\eta}_\ell$ is the $\ell$-adic realization, seen as a $G$-module.
\begin{displaymath} \xymatrix { \mathrm{H}^i_f(F, {M_\eta}) \ar[r] \ar[d] & \mathrm{H}^i(F, {M_\eta}) \ar[d] \\ \prod_\ell \mathrm{H}^i_f(F, {M_\eta}_\ell) \ar[r] & \prod_\ell \mathrm{H}^i(F, {M_\eta}_\ell) } \end{displaymath}
Again, to rid ourselves from crystalline questions at $\mathfrak{p} | \ell$, we define $\mathrm{H}^i_{f,\backslash \operatorname{crys}}(F, {M_\eta})$ by replacing $\prod_\ell \mathrm{H}^i_f(F, {M_\eta}_\ell)$ in the bottom row by $\prod_\ell \mathrm{H}^i_{f,\backslash \operatorname{crys}}(F, {M_\eta}_\ell)$. \xdefi
We are now going to exhibit an interpretation of $f$-cohomology thus defined in terms of the generic intermediate extension $\eta_{!*}$. Recall that we are assuming in this section the axioms of Sections \ref{sect_axiomaticdescription}, \ref{sect_realizations}, and \ref{sect_conjprop}. Mixed motives are needed to even define $\eta_{!*}$. Moreover, for the comparison result, we need to assume the following conjecture.
\lemm \mylabel{lemm_injectivity} Let $N$ be any mixed motive over ${\mathbb{F}_{\pp}}$. The $\ell$-adic realization map $\mathrm{H}^0({\mathbb{F}_{\pp}}, N) \rightarrow \mathrm{H}^0({\mathbb{F}_{\pp}}, N_\ell) := N_\ell^{\mathrm{Gal}({\mathbb{F}_{\pp}})}$ is injective. \xlemm \pr By the strictness of the weight filtration, the canonical maps $$\mathrm{H}^0 (\operatorname{gr}_0^W N) \leftarrow \mathrm{H}^0(W_0 N) \rightarrow \mathrm{H}^0(N)$$ are both isomorphisms. Moreover, the $\ell$-adic realization functor commutes with $\operatorname{gr}_0^W$ by \refax{weight}, so that we can replace $N$ by $\operatorname{gr}_0^W$ and assume that $N$ is pure of weight $0$. In view of our assumptions on motives, cf.\ \refeq{numrat}, all adequate equivalence relations agree, so that we may regard $N$ as a pure motive with respect to any adequate equivalence relation. As the injectivity is stable under taking direct summands, we may assume $N = h(X, n)$ for $X$ smooth and projective over ${\mathbb{F}_{\pp}}$, by definition of pure motives and \refax{mixedpure}. The left hand side is given by $\mathrm{CH}^n(X)$, so the map is injective by \refeq{numrat}. \end{proof}
\theo \mylabel{theo_charH1fmotivic} Let $M$ be a generically smooth mixed motive over ${\mathcal{O}_F}$ (\refde{genericallysmooth}). Set $\eta^* M [-1] =: M_\eta$. There is a natural isomorphism $$\mathrm{H}^0({\mathcal{O}_F}, \eta_{!*} \eta^* M) \stackrel{\cong}{{\longrightarrow}} \mathrm{H}^1_{f, \backslash \operatorname{crys}}(F, M_\eta).$$ \xtheo \pr Notice that $\eta_{!*} \eta^* M$ is well-defined by the assumptions. We want to show that there is a cartesian commutative diagram $$ \xymatrix { \mathrm{H}^0(\eta_{!*} \eta^* M) \ar[r] \ar@{.>}[d] & \mathrm{H}^0(\eta_* \eta^* M) = \mathrm{H}^1(M_\eta) \ar[d] \\ \prod_\ell \mathrm{H}^1_{f,\backslash \ell}(F, {M_\eta}_\ell) \ar[r] & \prod_\ell \mathrm{H}^1(F, {M_\eta}_\ell) } $$ Let $j: U \rightarrow {\Spec} {\OF}$ be any open immersion such that $j^* M$ is smooth. We have $\eta_{!*} \eta^* M = j_{!*} j^* M$. The left hand term of the exact sequence $$\oplus_{\mathfrak{p} \in U} \mathrm{H}^0({i_\mathfrak{p}}_* i_\mathfrak{p}^! M) \rightarrow \mathrm{H}^0(j_* j^* M) \rightarrow \mathrm{H}^0(\eta_* \eta^* M) \rightarrow \oplus_{\mathfrak{p} \in U} \mathrm{H}^1({i_\mathfrak{p}}_* i_\mathfrak{p}^! M)$$ induced by \refeq{localgeneric} vanishes as $i_\mathfrak{p}^! M$ is concentrated in cohomological degree $1$ for $\mathfrak{p} \in U$ (\refle{basic}). Any $a \in \mathrm{H}^0(\eta^* M)$ maps to a finite sub-sum of $\oplus_{\mathfrak{p} \in {\Spec} {\OF}} \mathrm{H}^1({i_\mathfrak{p}}_* i_\mathfrak{p}^! M)$, so letting $j$ be the open complement of these points, $a$ lies in (the image of) $\mathrm{H}^0(j_* j^* M)$: $$\mathrm{H}^0(\eta^* M) = \varinjlim_{j: U \rightarrow {\Spec} {\OF} \atop j^* M \text{ smooth}} \mathrm{H}^0(j_* j^* M).$$
By \refle{prepinjectivity} below, the map $\mathrm{H}^0 (j_{!*} j^* M) \rightarrow \mathrm{H}^0(j_* j^* M) \rightarrow \mathrm{H}^0(\eta^*M)$ is injective. Therefore, taking the colimit over all $U$ such that $M|_U$ is smooth, the exact localization sequence $$0 \rightarrow \mathrm{H}^0(j_{!*} j^* M) \rightarrow \mathrm{H}^0(j_* j^* M) \rightarrow \oplus_{\mathfrak{p} \notin U} \mathrm{H}^0 ({\p\mathrm{H}}^0 i_\mathfrak{p}^* j_* j^* M)$$ stemming from \refeq{local2} gives $$0 \rightarrow \mathrm{H}^0(j_{!*} j^* M) \rightarrow \mathrm{H}^0(\eta_* \eta^* M) \rightarrow \oplus_{\mathfrak{p}} \mathrm{H}^0 ({\p\mathrm{H}}^0 i_\mathfrak{p}^* {j_\mathfrak{p}}_* j_\mathfrak{p}^* M).$$ Here $j_\mathfrak{p}$ is the complementary open immersion to $i_\mathfrak{p}$ and the direct sum is over all (finite) places $\mathfrak{p}$ of ${\mathcal{O}_F}$. We have $i_\mathfrak{p}^* \eta_* \eta^* M = i_\mathfrak{p}^* {j_\mathfrak{p}}_* j_\mathfrak{p}^* M$, so the top sequence in the following commutative diagram is exact: \scriptsize \eqn \mylabel{eqn_diagramreal} \xymatrix{ 0 \ar[r] & \mathrm{H}^0(j_{!*} j^* M) \ar[r] \ar[d] & \mathrm{H}^1(M_\eta) \ar[r] \ar[d] &
\oplus_{\mathfrak{p}} \mathrm{H}^0({\p\mathrm{H}}^0 i_\mathfrak{p}^* \eta_* \eta^* M) \ar[d] \\ 0 \ar[r] & \prod_\ell \mathrm{H}^0((j_\ell^* j_{!*} j^* M)_\ell) \ar[r] & \prod_\ell \mathrm{H}^1({M_\eta}_\ell) \ar[r] & \prod_\ell \oplus_{\mathfrak{p} \nmid \ell} \mathrm{H}^0(({\p\mathrm{H}}^0 i_\mathfrak{p}^* \eta_* \eta^* M)_\ell) } \xeqn \normalsize The lower row denotes $\ell$-adic cohomology over ${\mathcal{O}_F}[1/\ell]$, $F$, and the various ${\mathbb{F}_{\pp}}$, respectively. Moreover, $j_\ell: {\Spec} {\OF}[1/\ell] \rightarrow {\Spec} {\OF}$ is the open immersion. The remainder of the proof consists in the following steps: we show that the diagram is commutative, that the second row is exact, identify its lower leftmost term and show that the rightmost vertical map is injective. This implies that the left square is cartesian, hence the theorem follows.
We write $\iota$ and $\iota_\ell$ for the open immersions of $U \cap {\Spec} {\OF}[1/\ell]$ into ${\Spec} {\OF}[1/\ell]$ and $U$, respectively.
By \refle{excreal} and the exactness of $j_\ell^*$ we have $$(j_\ell^* j_{!*} j^* M)_\ell = (\iota_{!*} \iota^* j_\ell^* M)_\ell = \iota_{!*} \iota^* (j_\ell^* M)_\ell.$$ Thus \refeq{diagramreal} is commutative since every term at the bottom just involves the $\ell$-adic realization of the motive above it, restricted to ${\Spec} {\OF}[1/\ell]$.
The exactness of the bottom row is shown separately for each $\ell$, so $\ell$ is fixed for this argument. By the characterization just mentioned, $\iota_{!*} \iota^* (j_\ell^* M)_\ell$ does not change when shrinking $U$, since $j_{!*} j^* M$ is independent of $U$ (as soon as $M$ is smooth over $U$). On the other hand, by the exactness of the $\ell$-adic realization functor (\refax{tstructureRealizations}) $(j_\ell^* M)_\ell$ is a perverse sheaf on ${\Spec} {\OF}[1/\ell]$, so is a locally constant sheaf (shifted into degree $-1$) on a suitable small open subscheme. Hence we may assume that $\iota^* (j_\ell^* M)_\ell$ is a locally constant sheaf in degree $-1$. By \refsect{perverse}, $ \iota_{!*} \iota^* (j_\ell^* M)_\ell = (\mathrm{R}^0 \iota_* \iota^* (j_\ell^* M)_\ell[-1])[+1]$, so the lower row is the exact cohomology sequence belonging to the distinguished triangle of sheaves on ${\Spec} {\OF}[1/\ell]$ $$\mathrm{R}^0 \eta_\ell {}_* (M_\eta)_\ell \rightarrow \mathrm{R} \eta_\ell {}_* (M_\eta)_\ell \rightarrow (\mathrm{R}^1 \eta_\ell {}_* (M_\eta)_\ell)[-1].$$ Here $\eta_\ell: {\Spec} {F} \rightarrow {\Spec} {\OF}[1/\ell]$ is the generic point. As is well-known, there is an isomorphism \eqn \mylabel{eqn_pfiso} D := \mathrm{R}^1 \eta_\ell {}_* \eta_\ell^* A \stackrel \cong \rightarrow \bigoplus_{\mathfrak{p} \nmid \ell} {i_\mathfrak{p}}_* i_\mathfrak{p}^* \mathrm{R}^1 \eta_\ell {}_* \eta_\ell^* A =: \bigoplus B_\mathfrak{p} \xeqn for any generically locally constant constructible $\ell$-adic sheaf $A$, such as $M_\ell[-1]$. Indeed, the adjunction map $a: D \rightarrow \prod_{\mathfrak{p} \nmid \ell} B_\mathfrak{p}$ factors over the direct sum: note that $(\oplus B_\mathfrak{p}) / \ell^n = \oplus (B_\mathfrak{p} / \ell^n)$ and likewise with the product. Then $$\mathrm{Hom}(D, \oplus B_\mathfrak{p}) = \varprojlim_n \mathrm{Hom}(D/\ell^n, \oplus (B_\mathfrak{p} / \ell^n)) \subset \varprojlim_n \mathrm{Hom}(D/\ell^n, \prod (B_\mathfrak{p} / \ell^n))$$ and to see that $a$ lies in the left hand subgroup, it is enough to consider the $\mathbb{Z}/\ell^n$-sheaves $D / \ell^n$ etc. The corresponding map $\mathrm{H}^1(\mathrm{Gal}(F), A / \ell^n) \rightarrow \prod \mathrm{H}^1 (I_\mathfrak{p}, A / \ell^n)$ (Galois cohomology of the inertia groups) factors over the direct sum, since the left hand term agrees with $\mathrm{H}^1(\mathrm{Gal}(F' / F), A)$ for some finite extension $F'/F$. This uses that $A / \ell^n$ is constructible. The extension $F' / F$ is ramified in finitely many places (only), so the claimed factorization follows. This implies \refeq{pfiso} and thus the exactness of the lower row of \refeq{diagramreal}. By \refle{charH1fglobal} and \refle{locconst}, the factors in the lower left-hand term of \refeq{diagramreal} agree with $\mathrm{H}^1_{f, \backslash \operatorname{crys}}(F, \eta^* M_\ell[-1])$.
To show that the rightmost vertical map of \refeq{diagramreal} is an injection, let $a = (a_\mathfrak{p})_{\mathfrak{p} \in {\Spec} {\OF}}$ be a nonzero element of the rightmost upper term. Only finitely many $a_\mathfrak{p}$ are nonzero. Pick some $\ell$ not lying under any of these prime ideals $\mathfrak{p}$. Then the image of $a$ in $\oplus_{\mathfrak{p} \nmid \ell} \mathrm{H}^0(({\p\mathrm{H}}^0 i_\mathfrak{p}^* \eta_* \eta^* M)_\ell)$ is nonzero by \refle{injectivity}. \end{proof}
\lemm \mylabel{lemm_prepinjectivity} Let $M$ be a mixed motive over $S$ such that $j^* M$ is smooth for some open immersion $j: U \rightarrow S$. Then both maps $\mathrm{H}^0 (j_{!*} j^* M) \rightarrow \mathrm{H}^0(j_* j^* M) \rightarrow \mathrm{H}^0(\eta^*M)$ are injective. \xlemm \pr Indeed the kernels are $\mathrm{H}^{-1}({\p\mathrm{H}}^0 i^* j_* j^* M) = 0$ and $\oplus_{\mathfrak{p} \in U} \mathrm{H}^0(i_\mathfrak{p}^! M)$, which vanishes since $i_\mathfrak{p}^! M$ sits in cohomological degree $+1$, for $M$ is smooth around $\mathfrak{p} \in U$ (\refle{basic}). \end{proof}
\subsection{$f$-cohomology via $K$-theory of regular models} \mylabel{sect_fcohomologyII}
\defi \mylabel{defi_H1fImage} Let $X_\eta$ be a smooth and projective variety over $F$. Let $X / {\mathcal{O}_F}$ be any projective model, i.e., $X {\times}_{\mathcal{O}_F} F = X_\eta$. Then we define $$\mathrm{H}^i(X_\eta, n)_{\mathcal{O}_F} := \operatorname{im} (\mathrm{H}^i(X, n) \rightarrow \mathrm{H}^i(X_\eta, n)).$$
\xdefi
Recall that we are assuming the axioms of Sections \ref{sect_axiomaticdescription}, \ref{sect_realizations}, and \ref{sect_conjprop}; the full force of mixed motives will be made use of in the sequel.
\theo \mylabel{theo_summaryH1f} The above is well-defined, i.e., independent of the choice of the model $X$. More precisely we have natural isomorphisms: $$\mathrm{H}^0(\eta_{!*} \mathrm{h}^{i-1}(X_\eta, n)[1]) = \left \{ \begin{array}{cl} \mathrm{H}^{i}(X_\eta, n)_{{\mathcal{O}_F}} & i<2n \\ \mathrm{CH}^n(X_\eta)_{\mathbb{Q},\mathrm{hom}} & i=2n \end{array} \right.$$ Moreover $$\mathrm{H}^{-1}(\eta_{!*} \mathrm{h}^{i-1}(X_\eta, n)[1]) = \mathrm{H}^0 (\mathrm{h}^{i-1}(X_\eta, n)).$$ \xtheo
When $X$ is regular, the definition and the statement are due to Beilinson \cite[Lemma 8.3.1]{Beilinson:Notes}. In this case one has $\mathrm{H}^{i}(X_\eta, n)_{{\mathcal{O}_F}} = \operatorname{im} K'_{2j-i}(X)_\mathbb{Q}^{(j)} \rightarrow K'_{2j-i}(X_\eta)_\mathbb{Q}^{(j)}$, but that expression does in general depend on the choice of the model \cite{deJeu:Appendix, deJeu:Further}. An extension of Beilinson's definition to all Chow motives over $F$ due to Scholl is discussed in the theorem below. We first provide a preparatory lemma.
\lemm \mylabel{lemm_excimage}
Let $M \in \category {MM}({\Spec} {\OF})$ be a mixed, generically smooth motive with strictly negative weights (\refde{genericallysmooth}). Let $j: U \rightarrow {\Spec} {\OF}$ be an open non-empty immersion such that $M|_U$ is smooth. The natural map $j_{!*} j^* M \rightarrow \eta_* \eta^* M$ gives rise to an isomorphism $$\mathrm{H}^0(j_{!*} j^* M) = \mathrm{im} \left ( \mathrm{H}^0(M) \rightarrow \mathrm{H}^0(\eta_* \eta^* M) \right).$$ \xlemm
\pr By \refle{prepinjectivity}, $\mathrm{H}^0(j_* j^* M) \rightarrow \mathrm{H}^0(\eta_* \eta^* M)$ is injective. Hence it suffices to show $\mathrm{H}^0(j_{!*} j^*M) = \operatorname{im} (\mathrm{H}^0 M \rightarrow \mathrm{H}^0 (j_* j^* M))$. Let $i$ be the complement of $j$. From \refeq{local1}, \refeq{local2}, we have a commutative exact diagram \scriptsize $$ \xymatrix{ & \mathrm{H}^0(j_! j^* M) \ar[r]^\alpha \ar@{->>}[d] & \mathrm{H}^0(M) \ar[r] \ar[d] & \mathrm{H}^0(i_* i^* M) \\ 0 = \mathrm{H}^{-1}(i_* {\p\mathrm{H}}^0 i^* j_* j^* M) \ar[r] & \mathrm{H}^0(j_{!*}j^* M) \ar@{>->}[r] \ar[d] & \mathrm{H}^0(j_* j^* M) \\ & \mathrm{H}^{1}(i_* {\p\mathrm{H}}^{-1} i^* j_* j^* M) =0 } $$ \normalsize The indicated vanishings are because of $t$-structure reasons and \refax{cohomdim}, respectively. It remains to show that $\alpha$ is surjective. As $i^* M$ is concentrated in cohomological degrees $[-1, 0]$ (\refax{exactness}), there is an exact sequence $$0 = \mathrm{H}^1({\p\mathrm{H}}^{-1}i^* M) \rightarrow \mathrm{H}^0(i^* M) \rightarrow \mathrm{H}^0({\p\mathrm{H}}^0 i^* M).$$ However $\mathrm{H}^0({\p\mathrm{H}}^0 i^* M)=0$ as $i^*$ preserves negative weights (\refax{respectweights}) and by strictness of the weight filtration and compatibility with the $t$-structure (\refax{weight}). \end{proof}
\pr Let $j: U \rightarrow {\Spec} {\OF}$ be an open nonempty immersion (which exists by smoothness of $X_\eta$) such that $X_U$ is smooth over $U$. By definition of $\eta_{!*}$ and Lemmas \ref{lemm_eta} and \ref{lemm_smooth}, the left hand term in the theorem agrees with $\mathrm{H}^0(j_{!*} \mathrm{h}^i (X_U, n))$. In the sequel, we write $M := \mathrm{h}^{i}(X, n)$ and $M_\eta := \eta^* [-1] M = \mathrm{h}^{i-1}(X_\eta, n)$.
We first do the case $i < 2n$. The spectral sequences $$\mathrm{H}^a(\mathrm{h}^b (X, n)) \Rightarrow \mathrm{H}^{a+b}(X, n), \, \mathrm{H}^a(\mathrm{h}^b (X_\eta, n)) \Rightarrow \mathrm{H}^{a+b}(X_\eta, n)$$ resulting from repeatedly applying truncation functors of the motivic $t$-structure converge since the cohomological dimension is finite (\refax{cohomdim} over $F$, \refle{cohomdimOF} over ${\mathcal{O}_F}$). By \refle{cohomdimOF}, $\mathrm{H}^i(-)$, applied to mixed motives over ${\mathcal{O}_F}$, is non-zero for $i \in \{-1, 0, 1\}$ only. We thus have to consider two exact sequences. The exact functor $\eta^* [-1]$ maps to similar exact sequences for motivic cohomology over $F$ (the indices work out properly, see \refle{eta}):
\eqn \mylabel{eqn_ss1} \xymatrix{ 0 \ar[r] & K \ar[r] \ar[d] & \mathrm{H}^{i}(X,n) \ar[r] \ar[d] & \mathrm{H}^{-1}(\mathrm{h}^{i+1}(X, n)) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & K_\eta \ar[r] & \mathrm{H}^{i}(X_\eta, n) \ar[r] & \mathrm{H}^0(\mathrm{h}^{i}(X_\eta, n)) \stackrel{\ref{axio_weight}, \ref{bsp_weightTate}}= 0 \ar[r] & 0 } \xeqn
\eqn \mylabel{eqn_ss2} \xymatrix{ 0 \ar[r] & \mathrm{H}^1(\mathrm{h}^{i-1} (X, n)) \ar[r] \ar[d] & K \ar[r] \ar[d] & \mathrm{H}^0 (M) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \mathrm{H}^2(\mathrm{h}^{i-2}(X_\eta, n)) \stackrel{\ref{axio_cohomdim}}= 0 \ar[r] & K_\eta \ar[r] & \mathrm{H}^1(M_\eta) \ar[r] & 0 } \xeqn Here, $K$ and $K_\eta$ are certain $E_3$-terms of the spectral sequences above. The rightmost vertical map in \refeq{ss1} is injective as one sees by combining \refeq{localgeneric} with the left-exactness of $i_\mathfrak{p}^!$. Hence \eqnarr \mathrm{H}^{i}(X_\eta, n)_{{\mathcal{O}_F}} & = & \operatorname{im} (\mathrm{H}^{i}(X, n) \rightarrow \mathrm{H}^i(X_\eta, n)) = \operatorname{im} (K \rightarrow K_\eta) \\ & = & \operatorname{im} (\mathrm{H}^0 (M) \rightarrow \mathrm{H}^1(M_\eta)) \xeqnarr The motive $M = \mathrm{h}^{i}(X, n)$ is a generically smooth (mixed) motive by \refle{smooth}. (Recall that this uses the decomposition axiom \ref{axio_smoothprojective} for smooth projective varieties.) By \refbs{weightTate}, its weights are strictly negative. Thus \refle{excimage} applies and the case $i < 2n$ is shown.
We now do the case $i = 2n$. The motive $j^* M$ is pure of weight zero (\refbs{weightTate}), hence by strictness of the weight filtration for motives over ${\mathcal{O}_F}$ and \refeq{local1}, \refeq{local2} the same is true for $E := j_{!*} j^* M$. (This is an avatar of \cite[Cor. 5.3.2]{BBD}.) Thus ${\p\mathrm{H}}^{1} i^! E$ has strictly positive weights because of \refax{respectweights} and the compatibility of weights and the motivic $t$-structure, i.e., $\operatorname{wt} {\p\mathrm{H}}^{1} (-) \subset \operatorname{wt} (-) + 1$. Therefore $\mathrm{H}^0({\p\mathrm{H}}^{1} i^! E) = 0$. Here $i$ is any closed immersion. The localization triangle \refeq{localgeneric} yields $$\mathrm{H}^0 (E) \stackrel{\alpha}\rightarrow \mathrm{H}^0(\eta_* \eta^* E) \stackrel{\text{\refeq{morphismsChow}}}= \mathrm{CH}^n(X_\eta)_{\mathbb{Q}, \mathrm{hom}} \rightarrow \oplus_\mathfrak{p} \mathrm{H}^1(i_\mathfrak{p}^! E) = \oplus \mathrm{H}^0({\p\mathrm{H}}^1(i^! E)) = 0.$$ Therefore, $\alpha$ is surjective. The injectivity of $\alpha$ is \refle{prepinjectivity}.
To calculate $\mathrm{H}^{-1}(\eta_{!*} M_\eta [1])$, let $j: U \rightarrow {\Spec} {\OF}$ be as above. The natural map $\mathrm{H}^{-1}({\Spec} {\OF}, j_{!*} j^* M) \rightarrow \mathrm{H}^{-1}(U, j^* M)$ is an isomorphism by the exact cohomology sequence belonging to \refeq{local2}. Thus we have to show $$\mathrm{H}^{-1}({\Spec} {\OF}, j_* j^* M) = \mathrm{H}^{-1}({\Spec} {\OF}, \eta_* \eta^* M).$$ This follows from the localization axiom \ref{axio_localization} and $i_\mathfrak{p}^! M$ being in cohomological degree $+1$ for all points $\mathfrak{p}$ in $U$ (\refle{basic}), so that $\mathrm{H}^0({\mathbb{F}_{\pp}}, i_\mathfrak{p}^! M) = \mathrm{H}^{-1}({\mathbb{F}_{\pp}}, i_\mathfrak{p}^! M) = 0$. \end{proof}
By a theorem of Scholl \cite[Thm. 1.1.6]{Scholl:Integral}, there is a unique functorial and additive (i.e., converting finite disjoint unions into direct sums) way to extend the definition of $\mathrm{H}^i(X_\eta, n)_{\mathcal{O}_F}$ as the image of the $K$-theory of a \emph{regular} proper flat model (\refde{H1fImage}) to all Chow motives over $F$, in particular to ones of smooth projective varieties $X_\eta / F$ that do not possess a regular proper model $X$. The following theorem compares this definition with the one via intermediate extensions.
\theo \mylabel{theo_Scholl} Let $h_\eta$ be a direct summand in the category of Chow motives of $h(X_\eta, n)$ where $X_\eta / F$ is smooth projective. Let $i \in \mathbb{Z}$ be such that $i -2n <0$. Let $\iota: \category {M}_\mathrm{rat}(F) \rightarrow \DM_\gm(F)$ be the embedding. Then, the group $$\mathrm{H}^i(h_\eta)_{\mathcal{O}_F} := \mathrm{H}^0 (\eta_{!*} ( {\p\mathrm{H}}^{i-2n-1} (\iota (h_\eta)) [1])).$$ is well-defined and agrees with the aforementioned definition by Scholl. \xtheo \pr Recall $\iota (h(X_\eta, n)) = \operatorname{M}(X_\eta, n)[2n] \in \DM_\gm(F)$. We first check that the group is well-defined: let $X / {\mathcal{O}_F}$ be a projective model of $X_\eta$. By \refle{spreadout}, there is some model $M \in \category {MM}({\mathcal{O}_F})$ of ${\p\mathrm{H}}^{i-2n-1} \iota (h_\eta)[1]$ and an open subscheme $U$ of ${\Spec} {\OF}$ such that $M$ is a direct summand of ${\p\mathrm{H}}^{i-1} \operatorname{M}(X)(n)$ and such that $X {\times} U$ is smooth over $U$. Then $\mathrm{h}^{i-1} (X, n)$ is a smooth motive when restricted to $U$ (\refle{smooth}). Hence so is $M$. Thus $\eta_{!*}$ can be applied to $({\p\mathrm{H}}^{i-2n-1} \iota (h_\eta)) [1]$.
The assignment $h_\eta \mapsto \mathrm{H}^0 (\eta_{!*} ({\p\mathrm{H}}^{i-2n-1} \iota (h_\eta)) [1])$ is functorial and additive and $h(X_\eta)(n)$ maps to $$\mathrm{H}^0 (\eta_{!*} ({\p\mathrm{H}}^{i-1} \operatorname{M}(X_\eta, n)) [1]) \stackrel{\text{\ref{theo_summaryH1f}}} = \mathrm{H}^i(X_\eta, n)_{\mathcal{O}_F}.$$ Thus the two definitions agree by Scholl's theorem. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Realizing compactly generated pseudo-groups of dimension one} \begin{abstract} Many compactly generated pseudo-groups of local transformations on $1$-manifolds are realizable as the transverse dynamic of a foliation of codimension $1$ on a compact manifold of dimension $3$ or $4$. \end{abstract} \keywords{Pseudo-group, compactly generated, foliation, taut, holonomy pseudo-group, realization of pseudo-groups}
\maketitle
\begin{center} Ga\"{e}l Meigniez \footnote{Partially supported by the Japan Society for Promoting Science, No.\ L03514} \end{center} \date{\today}
After C. Ehresmann, \cite{ehr1954},
given a foliation $\mbox{$\mathcal F$}$ of codimension $q$ on a compact manifold $M$~, its transverse dynamic
is represented by its \emph{holonomy pseudo-group} of local transformations on any exhaustive transversal $T$~. The inverse problem has been raised by A. Haefliger: given a pseudo-group
of local transformations of some manifold of dimension $q$~, realize it, if possible, as the dynamic of some foliation of codimension $q$ on some compact manifold. The difficulty here lies in the compactness. More precisely, Haefliger discovered a necessary condition: the pseudo-group must be \emph{compactly generated} \cite{hae1985}\cite{hae2002}.
He asked if this condition is sufficient.
The present paper intends to study the case $q=1$~.
A counterexample is known: there exists a compactly generated pseudo-group of local transformations of the line, which is not realizable. It contains a \emph{paradoxical Reeb component:} a full subpseudo-group equivalent to the holonomy of a Reeb component, but whose boundary orbit has some complicated isotropy group on the exterior side \cite{mei2010}.
The object of the
present paper is, on the contrary, to give a \emph{positive} answer to Haefliger's question for many pseudo-groups of dimension one.
Recall that a codimension $1$ foliation is (topologically) \emph{taut} if through every point there passes a transverse loop, or a transverse path with extremities on $\partial M$ (we refer e.g. to \cite{can2000} for the elements on foliations) . Equivalently, the foliation has no \emph{dead end component.} These notions are easily translated for pseudo-groups: one has the notions of tautness and of dead end components for a pseudo-group of dimension $1$ (see paragraph \ref{tautness_sbs} below). For example, every pseudo-group of dimension $1$ without closed orbit is taut. It turns out that, to realize a given compactly generated pseudo-group
of dimension $1$, the extra necessary and/or sufficient conditions that we find, bear on the isotropy groups of the closed orbits bounding the dead end components, if any.
We also pay attention to the dimension of the realization. Of course,
a pseudo-group which is realized by some foliation $\mbox{$\mathcal F$}$ on some manifold $M$~,
is also realized by the pullback of $\mbox{$\mathcal F$}$ into $M\times{\bf S}^1$~. One can ask to realize a pseudo-group, if possible, in the smallest possible dimension. The dynamics of the foliations on surfaces being very restrictive, the dimension $3$ will be in general the first candidate.
There is a well-known constraint specific to dimension $3$ in the nontaut case. Namely, remember that for elementary Euler characteristic reasons, in every compact foliated $3$-manifold which is not taut, every leaf bounding a dead end component is a $2$-torus or an annulus (S. Goodman) \cite{can2000}.
This phenomenon has a counterpart
in the holonomy pseudo-group: for every
orbit bounding a dead end component, its isotropy group is commutative of
rank at most two.
\medbreak A few precisions must be given before the results.
Equivalence: the pseudo-groups must be considered up to an equivalence called \emph{Haefliger equivalence}. Given
two different exhaustive transversals for a same foliated manifold, the two holonomy pseudo-groups are Haefliger-equivalent \cite{hae1985}\cite{hae1988}\cite{hae2002}.
A foliated manifold is said to \emph{realize} a pseudo-group $G$ if its holonomy pseudo-group on any exhaustive transversal is Haefliger-equivalent to $G$~.
Differentiability: The given pseudo-group being of class $C^r$~, $0\le r\le\infty$~, the realizing foliation will be $C^{\infty, r}$, that is, globally $C^r$ and tangentially smooth \cite{can2000}. We also consider the pseudo-groups of class $PL$~: the realizing foliations will be $C^{\infty, PL}$.
Orientation: for simplicity, all pseudo-groups are understood \emph{orientable,} that is, orientation-preserving. All foliations are understood tangentially orientable and transversely orientable.
Boundaries: by a "foliated manifold", we understand a manifold $M$ with a smooth boundary (maybe empty), endowed with a foliation $\mbox{$\mathcal F$}$ such that each connected component of $\partial M$ is either a leaf of $\mbox{$\mathcal F$}$ or transverse to $\mbox{$\mathcal F$}$~.
So, $\partial M$ splits into a tangential boundary $\partial_\parallel M$~, which is seen in the holonomy pseudo-group, and a transverse boundary $\partial_\pitchfork M$~, which is not. However, in the realization problem, the choice of allowing a transverse boundary or not, only affects the dimension of the realization. For, if $G$ is realized by some foliation $\mbox{$\mathcal F$}$ on some manifold $M$~, with some transverse boundary components, then it is also realized, without transverse boundary components, by the pullback of $\mbox{$\mathcal F$}$ in a manifold of one more dimension, namely: $$(M\times{\bf S}^1)\cup_{(\partial_\pitchfork M\times{\bf S}^1)} (\partial_\pitchfork M\times {\bf D}^2)$$
\begin{TA} Every pseudo-group of dimension $1$ which is \emph{compactly generated} and \emph{taut}, is realized by some foliated compact $3$-manifold, without transverse boundary. \end{TA}
Essentially, our method is that the pseudo-group is first easily realized as the dynamic of a Morse-singular foliation on a compact $3$-manifold. The singularities are of Morse indices $1$ and $2$~, in equal number. Then, thanks to tautness, from every singularity of index $2$ there is a positively transverse path to some (distant) singularity of index $1$~. Thanks
to some geometric manifestation of
compact generation, the pair is cancelled, \emph{not} in Morse's way, but rather by the means of an elementary surgery of index $2$ performed on the manifold, without changing the dynamic of the foliation.
An analogous construction can also be made inside a given foliated manifold; this leads to the following dimension reduction.
\begin{pro}\label{reduction_pro} Let $(M,\mbox{$\mathcal F$})$ be a compact manifold of dimension $n\ge 4$, endowed with a foliation of codimension $1$ which is topologically taut.
Then, there is a proper compact
submanifold $M'\subset M$ of dimension $n-1$ transverse to $\mbox{$\mathcal F$}$, such that: \begin{itemize} \item Every leaf of $\mbox{$\mathcal F$}$ meets $M'$; \item Every two points of $M'$ which lie on the same leaf of $\mbox{$\mathcal F$}$,
lie on the same leaf of $\mbox{$\mathcal F$}\vert M'$. \end{itemize} \end{pro}
In particular, the holonomy pseudo-group of $\mbox{$\mathcal F$}\vert M'$ is Haefliger-equivalent to the holonomy pseudo-group of $\mbox{$\mathcal F$}$. Here, \emph{proper} means that $\partial_\pitchfork M'=\emptyset$ and that $\partial_\parallel M'=M'\cap\partial_\parallel M$. Of course, from proposition \ref{reduction_pro}, it follows by induction on $n$, that $M$ contains a proper compact
submanifold \emph{of dimension $3$} transverse to $\mbox{$\mathcal F$}$, with the same two properties.
Recently, Martinez Torres, Del Pino and Presas have obtained by very different means a similar result in the particular case where $M$ admits a global closed $2$-form inducing a symplectic form on every leaf \cite{mar2014}. I thank Fran Presas for pointing out to me the general problem. \medbreak
We also get a \emph{characterization} of the dynamics of all foliations, taut or not, on compact $3$-manifolds.
\begin{TB}\label{B_thm} Let $G$ be a pseudo-group of dimension $1$~. Then, the following two properties are equivalent. \begin{enumerate} \item $G$ is realizable by some foliated compact $3$-manifold
(possibly with a transverse boundary);
\item $G$ is compactly generated; and for every orbit of $G$ in the boundary of every dead end component, its isotropy group is commutative of rank at most $2$~. \end{enumerate} \end{TB}
As a basic but fundamental
example, the nontaut pseudo-group of local transformations of the real line generated by two homotheties $t\mapsto\lambda t$~, $t\mapsto\mu t$~, with $\lambda$, $\mu>0$ and $\log \mu/\log\lambda\notin{\bf Q}$~, verifies the properties of theorem B, and is \emph{not} realizable by any foliated compact $3$-manifold without boundary --- see paragraph \ref{homothety_sbs} below. We know no simple, necessary and sufficient
conditions for realizing a nontaut pseudo-group on a compact $3$-manifold without transverse boundary.
More generally, skipping the condition of rank at most $2$:
\begin{TC} Let $G$ be a pseudo-group of dimension $1$ which is {compactly generated} and such that, for every orbit in the boundary of every dead end component, its isotropy group is commutative. Then, $G$ is realizable by some foliated compact orientable $4$-manifold, without transverse boundary. \end{TC}
\begin{cor} Every pseudo-group of dimension $1$ and of class $PL$ which is compactly generated, is realizable (in dimension $4$).
\end{cor}
There remain several open questions between these positive results and the negative result of \cite{mei2010}.
Regarding the isotropy groups of the orbits bounding the dead end components, the zoology of the groups that always allow a realization in high dimension, remains obscure (and may be intractable).
Consider a pseudo-group $G$ of dimension $1$ which is compactly generated. If $G$ is real analytic, is it necessarily realizable? If $G$ is realizable, is it necessarily realizable in dimension $4$~?
Also, beyond the realization problem itself, one can ask for a more universal property. Call $G$ \emph{universally realizable} if there is a system of foliated compact manifolds, each realizing $G$~, and of foliation-preserving embeddings, whose inductive limit is a Haefliger classifying space for $G$~. One can prove (not tackled in the present paper) that every compactly generated pseudo-group of dimension one and class $PL$ is universally realizable. What if we change $PL$ for "taut"? for "real analytic"?
The problem is that the method of the present paper, essentially
the cancellation of a pair of distant singularities of indices $1$ and $n-1$ in a Morse-singular foliation on a $n$-manifold by an elementary surgery without changing the dynamic, is very specific to these indices, and we know no equivalent e.g. for a pair of singularities of index $2$ in ambient dimension $4$~. \medbreak
\section{Preliminaries on pseudo-groups} In the first two paragraphs \ref{pghe_sbs} and \ref{compact_generation_sbs}, we recall concepts and facts about Haefliger equivalence and compact generation, in a form that fits our purposes. The material here is essentially due to Haefliger. In the third paragraph \ref{tautness_sbs}, we translate into the frame of pseudo-groups of dimension $1$, the notion of topological tautness, which is classical in the frame of foliations of codimension $1$. \medbreak
In ${\bf R}^n$~, one writes
${\bf D}^n$ the compact unit ball; ${\bf S}^{n-1}$ its boundary;
$*$ the basepoint $(1,0,\dots,0)\in{\bf S}^{n-1}$~; and $dx_n$ the foliation
$x_n=constant$~. ``Smooth'' means $C^\infty$.
\subsection{Pseudo-groups and Haefliger equivalences}\label{pghe_sbs}
An arbitrary differentiability class is understood. Let $T$~, $T'$ be manifolds of the same dimension, not necessarily compact. Smooth boundaries are allowed.
A \emph{local transformation} from $T$ to $T'$ is a diffeomorphism $\gamma$ between two nonempty, topologically open subsets $Dom(\gamma)\subset T$~, $Im(\gamma)\subset T'$~. Note that the boundary is necessarily invariant~:$$Dom(\gamma)\cap\partial T=\gamma{^{-1}}( \partial T' )$$Given also a local transformation $\gamma'$ from $T'$ to $T''$~, the composite $\gamma'\gamma$ is defined whenever $Im(\gamma)$ meets $Dom(\gamma')$ (an inclusion is not necessary), and one has~:$$Dom(\gamma'\gamma)=\gamma{^{-1}}(Dom(\gamma'))$$ Given two sets of local transformations $A$~, $B$~, as usual, $AB$ denotes the set of the composites of all composable pairs $\alpha\beta$~,
where $\alpha\in A$ and $\beta\in B$~. Also, $1_U$ denotes the identity map of the set $U$~.
\begin{dfn}\cite{veb1931} A \emph{pseudo-group} on a manifold $T$ is a set $G$ of local self-transformations of $T$ such that~: \begin{enumerate} \item For every nonempty, topologically open $U\subset T$~, the identity map $1_U$ belongs to $G$~; \item $GG=G{^{-1}}=G$~; \item For every local self-transformation $\gamma$ of $T$~, if $Dom(\gamma)$ admits an open cover $(U_i)$
such that every restriction $\gamma\vert U_i$ belongs to $G$~, then $\gamma$ belongs to $G$~. \end{enumerate} \end{dfn}
Then, by (1) and (2), $G$ is also stable by restrictions: if $\gamma$ belongs to $G$ and if $U\subset Dom(\gamma)$ is nonempty open, then $\gamma\vert U$ belongs to $G$~.
Example 1. Every set $S$ of local self-transformations of $T$ is contained in a smallest pseudo-group $<S>$ containing $S$~, called the pseudo-group \emph{generated} by $S$~. A local transformation $\gamma$ of $T$ belongs to $<S>$ if and only if, in a neighborhood of every point in its domain, $\gamma$ splits as a composite $\sigma_\ell\dots\sigma_1$~, with $\ell\ge 0$ and $\sigma_1$~,\dots, $\sigma_\ell\in S\cup S{^{-1}}$~.
Example 2. Given a pseudo-group $(G,T)$~, and a nonempty open subset $U\subset T$~, one has on $U$ a \emph{restricted} pseudo-group $G\vert U:=1_UG1_U$~:
the set of the elements of $G$ whose domains and images are both contained in $U$~.
Example 3. More generally, given a pseudo-group $(G,T)$~, a manifold $T'$~, and a set $F$ of local transformations from $T'$ to $T$~,
one has on $T'$ a \emph{pullback} pseudo-group $F^*(G):=<F{^{-1}} GF>$~. \medbreak Under a pseudo-group $(G,T)$~, every point $t\in T$ has~: \begin{enumerate} \item An \emph{orbit} $G(t)$~: the set of the images $\gamma(t)$ through
the local transformations
$\gamma\in G$ defined at $t$~; \item An \emph{isotropy group} $G_t$~:
the group of the germs at $t$ of the
local transformations $\gamma\in G$ defined at $t$ and fixing $t$~. \end{enumerate}
Call an open subset $T'\subset T$ \emph{exhaustive} if $T'$ meets every orbit. Call the pseudo-group $G$ \emph{cocompact} if $T$ admits a relatively compact exhaustive open subset. Call the pseudo-group $G$ \emph{connected} if every two points of $T$ are linked by a finite sequence of points of $T$~,
of which every two consecutive ones lie in the same orbit or in the same connected component of $T$~. Obviously, every pseudo-group splits as a disjoint sum of connected ones. \medbreak Let $(M,\mbox{$\mathcal F$})$ be a manifold foliated in codimension $q$~. A smooth boundary is allowed, in which case each connected component of $\partial M$ must be tangent to $\mbox{$\mathcal F$}$ or transverse to $\mbox{$\mathcal F$}$~. One writes $\partial_\parallel M$ the union of the tangential components. By a \emph{transversal,} one means a $q$-manifold $T$ immersed into $M$ transversely to $\mbox{$\mathcal F$}$~, not necessarily compact, and such that $\partial T=T\cap\partial_\parallel M$~. One calls $T$ \emph{exhaustive} (or \emph{total}) if it meets every leaf.
\begin{dfn}\label{holonomy_dfn}\cite{ehr1954} The \emph{holonomy pseudo-group} $Hol(\mbox{$\mathcal F$},T)$ of a foliation $\mbox{$\mathcal F$}$ on an exhaustive transversal
$T$ is the pseudo-group \emph{generated} by the local transformations $\gamma$ of $T$ for which there exists a map $$f_\gamma: [0,1]\times Dom(\gamma)\to M$$ such that~: \begin{itemize} \item $f_\gamma\pitchfork\mbox{$\mathcal F$}$ and $f_\gamma^*\mbox{$\mathcal F$}$ is the slice foliation on $[0,1]\times Dom(\gamma)$, whose leaves are the $[0,1]\times t$'s ($t\in Dom(\gamma)$); \item $f_\gamma(0,t)=t$ and $f_\gamma(1,t)=\gamma(t)$~,
for every $t\in Dom(\gamma)$~.
\end{itemize} \end{dfn}
We may call $f_\gamma$ a \emph{fence} associated to $\gamma$. This holonomy pseudo-group does represent the dynamic of the foliation:
there is a one-to-one correspondence $L\mapsto L\cap T$ between the leaves of $\mbox{$\mathcal F$}$ and the orbits of $Hol(\mbox{$\mathcal F$},T)$~; a topologically closed orbit corresponds to a topologically closed leaf; the isotropy group of $Hol(\mbox{$\mathcal F$},T)$ at any point
is isomorphic with the holonomy group of the corresponding leaf; etc. \begin{dfn}\label{equivalence_dfn}\cite{hae1984} A \emph{Haefliger equivalence} between two pseudo-groups $(G_i,T_i)$ ($i=0,1$) is a pseudo-group $G$ on the disjoint union $T_0\sqcup T_1$~, such that
$G\vert T_i=G_i$ ($i=0,1$) and that every orbit of $G$ meets both $T_0$ and $T_1$~. \end{dfn}
Example 1. The two holonomy pseudo-groups of a same foliation on two exhaustive transversals are Haefliger equivalent.
Example 2. The restriction of a pseudo-group $(G,T)$ to any exhaustive open subset of $T$ is Haefliger-equivalent to $(G,T)$~.
Example 3. More generally, let $(G,T)$ be a pseudo-group, and let $F$ be
a set of local transformations from $T'$ to $T$~. Assume that: \begin{enumerate} \item $FF{^{-1}}\subset G$~; \item $\cup_{\phi\in F}Dom(\phi)=T'$~; \item $\cup_{\phi\in F}Im(\phi)$ is $G$-exhaustive in $T$~. \end{enumerate} Then, the pseudo-group $<F\cup G>$ on $T\sqcup T'$ is a Haefliger equivalence between $(G,T)$ and $(F^*(G), T')$~.
\medbreak The Haefliger equivalence is actually an equivalence relation between pseudo-groups. Given two Haefliger equivalences: $G$ between $(G_0,T_0)$ and $(G_1,T_1)$, and
$G'$ between $(G_1,T_1)$ and $(G_2,T_2)$, one forms the pseudo-group $<G\cup G'>$ on $T_0\sqcup T_1\sqcup T_2$~. Then, $<G\cup G'>\vert(T_0\sqcup T_2)$ is a
Haefliger equivalence between $(G_0,T_0)$ and $(G_2,T_2)$~.
Every Haefliger equivalence induces a one-to-one correspondence between the orbit spaces
$T_i/G_i$ ($i=0,1)$~. A closed orbit corresponds to a closed orbit. The isotropy groups at points on corresponding orbits are isomorphic.
\subsection{Compact generation}\label{compact_generation_sbs}
Let $(G,T)$ be a pseudo-group. We say that
$\gamma\in G$ is ($G$-) {\it extendable} if there exists some $\bar\gamma\in G$
such that $Dom(\gamma)$ is contained \emph{and relatively compact}
in $Dom(\bar\gamma)$, and that $\gamma=\bar\gamma\vert Dom(\gamma)$.
The composite of two extendable elements is also extendable. The inverse of an extendable element is also extendable.
\begin{dfn}\label{CG_dfn} {\rm (Haefliger)\cite{hae1985}} A pseudo-group $(G,T)$ is \emph{compactly generated} if
there are an exhaustive, relatively compact, open subset $T'\subset T$~, and finitely many elements of $G\vert T'$ which are $G$-{extendable}, and which generate $G\vert T'$~. \end{dfn}
\begin{pro}{\rm (Haefliger)\cite{hae1985}\cite{hae2002}}\label{invariance_pro} Compact generation is invariant by Haefliger equivalence. \end{pro} \begin{pro}{\rm (Haefliger)\cite{hae1985}\cite{hae2002}}
The holonomy pseudo-group of every foliated compact manifold is compactly generated. \end{pro} We shall also use the following fact, which amounts to say that the choice of $T'$ is arbitrary. \begin{lem}\label{transversal_change_lem}{\cite{hae1985}\cite{hae2002}} Let $(G,T)$ be a {compactly generated} pseudo-group, and $T''\subset T$ be \emph{any}
exhaustive, relatively compact, open subset. Then there are finitely many elements of $G\vert T''$ that are {extendable} in $G$~, and that generate $G\vert T''$~. \end{lem}
Note: pseudo-groups vs. groupoids. The above definition of compact generation, although it may look strange at first look, is relevant;
in particular because it is preserved through Haefliger equivalences.
N. Raimbaud has shown that compact generation has a somewhat more natural generalization in the frame of topological groupoids. Write $\Gamma(T)$ the topological groupoid of the germs of local transformations of $T$~. Let $G$ be any pseudo-group on $T$~. Then, the set $\Gamma$ of germs $[g]_t$~, for all $g\in G$ and all $t\in Dom(g)$~, is in $\Gamma(T)$ an open subgroupoid whose space of objects is the all of $T$~. It is easily verified that
one gets this way a bijection between the set of pseudo-groups on $T$ and the set of open subgroupoids in $\Gamma(T)$ whose space of objects is $T$~.
The pseudo-group $G$ is compactly generated if and only if the topological groupoid $\Gamma$
contains an exhaustive, relatively compact, open subset, which generates a full subgroupoid \cite{rai2009}.
\subsection{Tautness for pseudo-groups of dimension $1$}\label{tautness_sbs} We now consider a pseudo-group $(G,T)$ \emph{of dimension $1$~,} that is, $\dim T=1$~; and \emph{oriented,} that is, $T$ is oriented and $G$ is orientation-preserving. From now on, all pseudo-groups will be understood of dimension $1$ and oriented.
By a \emph{positive arc} $[t,t']$ of origin $t$ and extremity $t'$~, we mean an orientation-preserving embedding of the interval $[0,1]$ into $T$ sending $0$ to $t$ and $1$ to $t'$~.
A \emph{positive chain} is a finite sequence of positive arcs, such that the extremity of each (but the last) lies on the same orbit as the origin of the next. A \emph{positive loop} is a positive chain such that the extremity of the last arc lies on the same orbit at the origin of the first.
\begin{dfn} A pseudo-group $(G,T)$ of dimension 1 is \emph{taut} if every point of $T$ lies either on a positive chain whose origin and extremity belong to $\partial T$~, or on a positive loop. \end{dfn} \begin{pro}\label{taut_pro} Let $(G,T)$ be a cocompact pseudo-group of dimension 1. Then, $(G,T)$
is taut if and only if it is Haefliger-equivalent to
some pseudo-group $(G',T')$ such that $T'$ is a finite disjoint
union of compact intervals and circles. \end{pro}
\begin{proof} One first easily verifies that tautness is invariant by Haefliger equivalence. "If" follows.
Conversely, given a taut cocompact pseudo-group $(G,T)$~, by cocompactness there is a finite family $C$ of positive chains, each being a loop or having extremities on $\partial T$~, such that every orbit of $G$ meets on at least one of them.
Consider one of these chains $c=([t_i,t'_i])$ ($0\le i\le\ell(c)$) which is not a loop: its origin $t_0$ and extremity $t'_{\ell(c)}$ lie on $\partial T$~. For every $1\le i\le\ell(c)$~, one has $t_{i}=g_i(t'_{i-1})$ for some $g_i\in G$ whose domain and image are small. Let $$U_0:=[t_0,t'_0]\cup Dom(g_{1})$$ $$U_{\ell(c)}:=Im(g_{\ell(c)})\cup[t_{\ell(c)},t'_{\ell(c)}]$$ and for each
$1\le i\le{\ell(c)}-1$~, let $$U_i:=Im(g_i)\cup[t_i,t'_i]\cup Dom(g_{i+1})$$
One makes an abstract copy $U'_i$ of each $U_i$~. Write
$f_{c,i}:U'_i\to U_i$ for the identity.
These abstract copies are glued together by means of the $g_i$'s into a single compact segment $T'_c$~. Thus, $T'_c$ has an atlas of maps which are
local transformations $f_{c,i}$ ($0\le i\le\ell(c)$) from $T'_c$ to $T$~, such that every change of maps $g_i=f_{c,i}f_{c,i-1}{^{-1}}$ belongs to $G$~. The images of the $f_{c,i}$'s cover the chain $c$~.
In the same way, for every $c\in C$ which is a loop, one makes a circle
$T'_c$ together with an atlas $f_{c,i}$ ($0\le i\le\ell(c)$) of maps
which are local transformations from $T'_c$ to $T$~, such that every change of maps belongs to $G$~. The images of the maps cover the chain $c$~.
Let $T'$ be the disjoint union of the $T'_c$'s, for $c\in C$~. By the example 3 above after the definition \ref{equivalence_dfn}, $G$ is Haefliger-equivalent to the pseudo-group $F^*(G)$ of local transformations of $T'$~. \end{proof}
In case $(G,T)$ is connected, one can be more precise (left as an exercise): \begin{pro}\label{connected_taut_pro} Let $(G,T)$ be a connected, cocompact pseudo-group of dimension 1. Then, $(G,T)$
is taut if and only if it is Haefliger-equivalent to
some pseudo-group $(G',T')$ such that $T'$ is either a finite disjoint
union of compact intervals,
or a single circle. \end{pro}
Our last lemma has no relation to tautness. For a compactly generated pseudo-group of dimension one, one can give a more precise form to the generating system defining compact generation:
\begin{lem}\label{intervals_lem} Let $(G,T)$ be a compactly generated pseudo-group of dimension $1$~. Then~: \begin{enumerate} \item There is a $G$-exhaustive, open, relatively compact $T'\subset T$
which
has finitely many connected components;
\item For every $T'$ as above, $G\vert T'$ admits a finite set of $G$-extendable generators whose domains and images
are intervals. \end{enumerate} \end{lem} \begin{proof} (1) The pseudo-group $G$~, being compactly generated, is in particular cocompact: there is a compact $K\subset T$ meeting every orbit. Being compact, $K$ meets only finitely many connected components $T_i$ of $T$~. For each $i$~, let $T'_i\subset T_i$ be relatively compact, open, connected, and contain $K\cap T_i$~. Then, $T':=\cup_iT'_i$ is $G$-exhaustive, open, relatively compact, and has finitely many connected components.
(2) By the lemma \ref{transversal_change_lem}, $G\vert T'$ admits a finite set $(g_i)$ $(i=1,\dots,p)$ of $G$-extendable generators. For each $1\le i\le p$~, let $\bar g_i$ be a $G$-extension of $g_i$~. Let $U_i\subset Dom(\bar g_i)$ be open, relatively compact, contain $Dom(g_i)$~, and have finitely many connected components. Then, $U_i\cap T'$ has finitely many connected components. Each of these components is either an interval or a circle. In the second case, we cover this circle by two open intervals. We get a cover of $U_i\cap T'$ by a finite family $(I_{j})$ ($j\in J_i$) of intervals open and relatively compact in $Dom(\bar g_i)$ .
The finite family $(\bar g_i\vert I_j)$ $(1\le i\le p, j\in J_i)$ is $G$-extendable and generates $G\vert T'$~. \end{proof}
\section{Proof of theorem A and of proposition \ref{reduction_pro}}\label{A_section} \def{\mathcal R}{{\mathcal R}}
\subsection{Proof of theorem A} We are given a taut, compactly generated pseudo-group $(G,T)$ of dimension $1$ and class $C^r$, $0\le r\le\infty$, or $PL$ . We have to realize $G$ as the holonomy pseudo-group of some foliated compact $3$-manifold. By proposition \ref{taut_pro}, we can assume that $T$ is compact: a finite disjoint union of compact intervals and circles. By lemma \ref{intervals_lem} applied to $T'=T$~, the pseudo-group $G$ admits a finite system $g_1$~,\dots, $g_p$
of $G$-extendable generators whose domains and images are intervals.
The proof uses Morse-singular foliations. It would be natural to define them as the Haefliger structures whose singularities are quadratic, but this would lead to irrelevant technicalities. A simpler concept will do.
\begin{dfn}\label{morse_dfn}
A \emph{Morse foliation} $\mbox{$\mathcal F$}$ on a smooth $n$-manifold $M$ is a foliation of codimension one and class $C^{\infty,r}$ on the complement of finitely many singular points, such that on some open neighborhood of each, $\mbox{$\mathcal F$}$ is conjugate to the level hypersurfaces of some nondegenerate quadratic form on some neighborhood of $0$ in ${\bf R}^n$~. The conjugation must be $C^0$~; it must be smooth except maybe at the singular point. \end{dfn}
We write $Sing(\mbox{$\mathcal F$})\subset M$ for the finite set of singularities. Note that $\mbox{$\mathcal F$}$ is smooth on some neighborhood of $Sing(\mbox{$\mathcal F$})$, minus $Sing(\mbox{$\mathcal F$})$. The holonomy pseudo-group of $\mbox{$\mathcal F$}$ is defined,
on any exhaustive transversal disjoint from the singularities, as the holonomy pseudo-group of the regular foliation $\mbox{$\mathcal F$}\vert(M\setminus Sing(\mbox{$\mathcal F$}))$~.
We shall first realize $(G,T)$ as the holonomy pseudo-group of a Morse foliation on a compact $3$-manifold. Then, compact generation will allow us to perform a surgery on this manifold and regularize the foliation, without changing its transverse structure.
\smallbreak To fix ideas, at first we assume that $T$ is without boundary: that is, a finite disjoint union of circles.
Let $M_0:={\bf S}^2\times T$ and let $\mbox{$\mathcal F$}_0$ be the foliation of $M_0$ by $2$-spheres: its holonomy pseudo-group on the exhaustive transversal $\ast\times T$ is the trivial pseudo-group. Write ${\rm pr}_2:M_0\to T$ for the second projection.
For every $1\le i\le p$~, write $(u_i,u'_i)\subset T$ for the open interval that is the domain of $g_i$~,
and write $(v_i,v'_i)$ the image of $g_i$~. Fix some extension $\bar g_i\in G$~.
Choose two embeddings $e_i:{\bf D}^3\to M_0$ and $f_i:{\bf D}^3\to M_0$ such that \begin{enumerate} \item $e_i({\bf D}^3)$ and $f_i({\bf D}^3)$ are disjoint from each other and from $\ast\times T$~; \item ${\rm pr}_2(e_i({\bf D}^3))=[u_i,u'_i]$ and ${\rm pr}_2(f_i({\bf D}^3))=[v_i,v'_i]$~; \item $e_i^*\mbox{$\mathcal F$}_0=f_i^*\mbox{$\mathcal F$}_0$ is the trivial foliation $dx_3\vert{\bf D}^3$~; \item ${\rm pr}_2\circ f_i=\bar g_i\circ{\rm pr}_2\circ e_i$~. \end{enumerate}
We perform on $M_0$ an elementary surgery of index $1$ by cutting the interiors of $e_i({\bf D}^3)$ and of $f_i({\bf D}^3)$, and by pasting their boundary 2-spheres. The points $e_i(x)$ and $f_i(x)$ are pasted, for every $x\in\partial{\bf D}^3$~.
We perform such a surgery on $M_0$ for every $1\le i\le p$~, choosing of course the embeddings $e_i$~, $f_i$ two by two disjoint. Let $M_1$ be the resulting manifold.
Obviously, $\mbox{$\mathcal F$}_0$ induces on $M_1$ a Morse foliation $\mbox{$\mathcal F$}_1$~, with $2p$ singularities, one at every point $s_i:=e_i(0,0,-1)=f_i(0,0,-1)$~, of Morse index $1$~; and one at every point $s'_i:=e_i(0,0,+1)=f_i(0,0,+1)$~, of Morse index $2$~. It is easy and standard to endow $M_1$ with a smooth structure, such that $\mbox{$\mathcal F$}_1$ is of class $C^{\infty,r}$, and smooth in a neighborhood of the singularities, minus the singularities.
By (4), the holonomy of $\mbox{$\mathcal F$}_1$ on the transversal $*\times T\cong T$ is generated by the local transformations $g_i$~. That is, it coincides with $G$~.
Up to now, we have not used fully the fact that $G$ is compactly generated. Now, we point a consequence of this fact, which is actually its geometric translation.
Consider in general
some Morse foliation $\mbox{$\mathcal X$}$ on some $3$-manifold $X$~, and some singularity $s$ of index $1$~. On some neighborhood of $s$~, the Morse foliation $\mbox{$\mathcal X$}$ admits the first integral $Q:=-x_0^2+x_1^2+x_2^2$ in some continuous local coordinates $x_0$~, $x_1$~, $x_2$, smooth except maybe at the singularity. The two components of the singular cone at $s$~, namely $Q{^{-1}}(0)\cap\{x_0<0\}$ and $Q{^{-1}}(0)\cap \{x_0>0\}$~, may either belong to the same leaf of the regular foliation $\mbox{$\mathcal X$}\vert(X\setminus Sing(\mbox{$\mathcal X$}))$~, or not. If they do, then there is a loop $\lambda:[0,1]\to X$ such that \begin{itemize} \item $\lambda(0)=\lambda(1)=s$~; \item $\lambda$ is tangential to $\mbox{$\mathcal X$}$~; \item $\lambda(t)\notin Sing(\mbox{$\mathcal X$})$ for every $0<t<1$~; \item $x_0(\lambda(t))\le 0$ (resp. $\ge 0$) for every $t$ close enough to $0$ (resp. $1$). \end{itemize}
Such a loop has a holonomy germ $h(\lambda)$ on the pseudo-transversal arc $x_0=x_1=0$~, $x_2\ge 0$~. This is the germ at $0$ of some homeomorphism of the nonnegative half-line. \begin{dfn}\label{levitt_dfn} If moreover
the holonomy $h(\lambda)$ is the identity, then we call $\lambda$ a \emph{Levitt loop} for $\mbox{$\mathcal X$}$ at $s$~. \end{dfn}
In the same way, at every singularity of index $2$~, the Morse foliation $\mbox{$\mathcal X$}$ admits the first integral $Q':=x_0^2-x_1^2-x_2^2$ in some local coordinates $x_0$~, $x_1$~, $x_2$~. The notion of a Levitt loop is defined symmetrically by reversing the transverse orientation of $\mbox{$\mathcal X$}$~.
\begin{lem}\label{levitt_lem} The Morse foliation $\mbox{$\mathcal F$}_1$ admits a Levitt loop at every singularity. \end{lem}
\begin{proof} Consider e.g. a singularity $s_i$ of index $1$~. In $M_0$~, let $a$ be a path from the point $(*,u_i)$ to the point $e_i(0,0,-1)$ in the sphere ${\bf S}^2\times u_i$~; and let $b$ be a path from $(*,v_i)$ to $f_i(0,0,-1)$ in the sphere ${\bf S}^2\times v_i$~. Then, in $M_1$~,
the path $ab{^{-1}}$ is tangential to $\mbox{$\mathcal F$}_1$ and passes through $s_i$~. Obviously, the holonomy $h(ab{^{-1}})$ of $\mbox{$\mathcal F$}_1$
on $*\times T\cong T$ along this path is well-defined on the right-hand side of $u_i$~. That is, $h(ab{^{-1}})$ is a germ of homeomorphism of $T$ from some interval $[u_i,u_i+\epsilon)\subset T$ onto some interval $[v_i,v_i+\eta)\subset T$~.
By properties (2) through (4) above,$$h(ab{^{-1}})=g_i\vert[u_i,u_i+\epsilon).$$
On the other hand, recall that $\bar g_i\in G$~. Since $G$ is the holonomy pseudo-group of $\mbox{$\mathcal F$}_1$ on $*\times T$~, there is in $M\setminus Sing(\mbox{$\mathcal F$}_1)$~, a path $c$ from $u_i$ to $v_i$~, tangential to $\mbox{$\mathcal F$}_1$~, and whose holonomy on $*\times T$ is the germ of $\bar g_i$ at $u_i$~. Then, $\lambda:=a{^{-1}} cb$ is a Levitt loop at $s_i$~. \end{proof}
To simplify the argument in the rest of the construction, it is convenient
(although in fact not necessary) that $\mbox{$\mathcal F$}_1$ admit at each singularity a \emph{simple} Levitt loop. We can get this extra property as follows. Let $s$ be a singularity of $\mbox{$\mathcal F$}_1$~, let $\lambda$ be a Levitt loop for $\mbox{$\mathcal F$}_1$ at $s$~, and let
$L$ be the leaf singular at $s$~, containing $\lambda$~. After a generic perturbation of $\lambda$ in $L$~, the loop $\lambda$ is immersed and self-transverse in $L$~. Let $x$ be a self-intersection point of $\lambda$~. Since $\mbox{$\mathcal F$}_1$ is taut, there passes through $x$ an embedded transverse circle $C\subset M_1$~, disjoint from $*\times T$~.
We perform a surgery on $M_1$~, cutting a small tubular neighborhood $N\cong{\bf D}^2\times{\bf S}^1$
of $C$ in $M_1$~, in which $\mbox{$\mathcal F$}_1$ is the foliation by the ${\bf D}^2\times t$'s; and we glue $\Sigma\times{\bf S}^1$~, where $\Sigma$ is the compact connected orientable surface of genus $1$ bounded by one circle, foliated by the $\Sigma\times t$'s. The holonomy pseudo-group of the foliation $\mbox{$\mathcal F$}_1$ on $*\times T$ is not changed. After the surgery, there is at $s$ a Levitt loop with one less self-intersection. Of the two pieces of $\lambda$ that passed through $x$~, now one passes in the new handle and is disjoint from the other.
After a finite number of such surgeries, for every $1\le i\le p$~, the Morse foliation $\mbox{$\mathcal F$}_1$ admits at the singularity $s_i$ (resp. $s'_i$) a simple Levitt loop $\lambda_i$ (resp. $\lambda'_i$).
Fix some $1\le i\le p$~. We shall somewhat cancel the pair of
singularities $s_i$ and $s'_i$ of $\mbox{$\mathcal F$}_1$~, at the price of a surgery on $M_1$~, without changing the
transverse structure of $\mbox{$\mathcal F$}_1$~.
First, we use fully the fact that $G$ is taut: there is a path $p_i:[0,1]\to M_1$ from $p_i(0)=s'_i$ to $p_i(1)=s_i$~, and
positively transverse to $\mbox{$\mathcal F$}_1$ except at its endpoints.
The geometry is as follows (figure \ref{one_fig}). \begin{figure}\label{one_fig}
\end{figure}
Let $Q(x_1,x_2,x_3)$ be a quadratic form
of Morse index $1$ with respect to some local system of coordinates at $s_i$~, which is a local first integral for $\mbox{$\mathcal F$}_1$~. Then, $p_i$ arrives at $s_i$ by one of the two components of the cone $Q<0$~. Reversing if necessary the orientation of $\lambda_i$, one can arrange that $\lambda_i$ quits $s_i$ in the boundary of the same half cone.
Symmetrically,
let $Q'(x_1,x_2,x_3)$ be a quadratic form of Morse index $2$ with respect to some local system of coordinates at $s'_i$~, which is a local first integral for $\mbox{$\mathcal F$}_1$~. Then,
$p_i$ quits $s'_i$ by one of the two components
of the cone $Q'>0$~.
Reversing if necessary the orientation of $\lambda'_i$, one can arrange that $\lambda'_i$ arrives at $s'_i$ in the boundary of the same half cone.
We shall perform a surgery on $M_1$~, and modify $\mbox{$\mathcal F$}_1$~, in an arbitrarily small neighborhood of $\lambda'_i\cup p_i\cup\lambda_i$~, to cancel the singularities $s_i$~, $s'_i$~, without changing the holonomy pseudo-group of the foliation.
To this aim, the composed path $\lambda'_ip_i\lambda_i$ (that is, $\lambda'_i$ followed by $p_i$ followed by $\lambda_i$) is homotoped, relatively to its endpoints,
into some path $q_i$ also positively transverse to $\mbox{$\mathcal F$}_1$~, except at its endpoints $q_i(0)=s'_i$ and $q_i(1)=s_i$~. The homotopy consists in pushing the two tangential Levitt loops to some nearby, positively transverse paths, and in rounding the two corners; it is $C^0$-small.
Notice that $p_i$ and
$q_i$ arrive at $s_i$ by the two opposite components of the cone $Q<0$~.
Symmetrically,
$p_i$ and $q_i$ quit $s'_i$ by the two opposite components
of the cone $Q'>0$~.
By construction, for a convenient choice of the parametrization $t\mapsto q_i(t)$, the transverse path $q_i$ is \emph{$\mbox{$\mathcal F$}_1$-equivalent} to $p_i$~, that is, the diffeomorphism $p_i(t)\mapsto q_i(t)$ belongs to the holonomy pseudo-group of $\mbox{$\mathcal F$}_1$ on the union of the two transversal open arcs $p_i\cup q_i\setminus \{s_i,s'_i\}$~.
After a small, generic perturbation of $p_i$ and $q_i$ relative to their endpoints $s_i$, $s'_i$, we arrange that $p_i$ and $q_i$ are two embeddings of the interval into $M_1$~; that they are disjoint, but at their endpoints; and also that they are disjoint from $p_j$~, $q_j$ for every $j\neq i$~, and also disjoint from the transversal $*\times T$~.
Recall (definition \ref{morse_dfn}) that $\mbox{$\mathcal F$}_1$ is smooth in a neighborhood of $s_i$ and $s'_i$ (but maybe at $s_i$ and $s'_i$).
After a $C^r$-small perturbation of $\mbox{$\mathcal F$}_1$ in some small neighborhood of $p_i$ and $q_i$, relative to some small neighborhoods of $s_i$ and $s'_i$, the foliation $\mbox{$\mathcal F$}_1$ is smooth in some neighborhood of $p_i\cup q_i$ (but maybe at $s_i$ and $s'_i$). \medbreak Now, we shall perform on $M_1$ an elementary surgery of index $2$ along every embedded circle $p_i\cup q_i$ ($1\le i\le p$)
(figure \ref{two_fig}). \begin{figure}\label{two_fig}
\end{figure}
That is, we cut some small tubular neighborhood $N_i\cong{\bf S}^1\times{\bf D}^2$ of $p_i\cup q_i$~, and we paste ${\bf D}^2\times{\bf S}^1$
(here the choice of the framing is irrelevant). We shall
obtain
a closed $3$-manifold $M$~. We shall, for a convenient choice of the $N_i$'s~, extend the foliation $\mbox{$\mathcal F$}_1\vert(M_1\setminus\cup_i N_i)$ to $M$~, as a (regular) foliation, still admitting $*\times T$ as an exhaustive transversal, and whose holonomy pseudo-group on $*\times T$ will still be $G$~.
To this end, first notice that, by definition \ref{levitt_dfn}, and since $\lambda_i$ (resp. $\lambda'_i$) is a simple loop, there is some small open neighborhood $U_i$ (resp. $U'_i$) of $\lambda_i$ (resp. $\lambda'_i$) in $M_1$~, such that the foliation $\mbox{$\mathcal F$}_1$ admits in $U_i\setminus s_i$ (resp. $U'_i\setminus s'_i$)
a first integral $F_i$ (resp. $F'_i$)
whose level sets are connected.
Precisely, for every $t<F_i(s_i)$ (resp. $t>F'_i(s'_i)$), the level set $F_i{^{-1}}(t)$ (resp. $F'_i{^{-1}}(t)$) is
an open disk. For every $t>F_i(s_i)$ (resp. $t<F'_i(s'_i)$), the level set $F_i{^{-1}}(t)$ (resp. $F'_i{^{-1}}(t)$) is the connected orientable open surface of genus one with one end.
Choose a compact $3$-ball $B_i\subset U_i$ containing $s_i$ and
such that $F_i\vert B_i$ is
topologically conjugate to a quadratic form $Q$ of signature $-++$~, with three different eigenvalues, on the unit ball. Choose a compact $3$-ball
$B'_i\subset U'_i$ containing $s'_i$ and such that $F'_i\vert B'_i$ is
topologically conjugate to a quadratic form $Q'$ of signature $--+$~, with three different eigenvalues,
on the unit ball.
Choose some tubular neighborhood $S_i$
of the circle $p_i\cup q_i$~,
so thin that $S_i\cap\partial B_i$ (resp. $S_i\cap\partial B'_i$) is
contained in the cone $Q<0$ (resp. $Q'>0$)~, and such that $\mbox{$\mathcal F$}_1\vert S_i$
is a foliation by disks, except on the intersections
of $S_i$ with $Q\ge 0$ and with $Q'\le 0$~.
Define $N_i:=B_i\cup B'_i\cup S_i$~.
We can arrange that $N_i$ is a smooth solid torus.
Then, after reparametrizing the values of $F_i$ and of $F'_i$~,
they obviously extend to a function $F''_i$ on $N_i\setminus\{s_i,s'_i\}$ as follows.
\begin{enumerate}
\item $F''_i$ is a first integral for $\mbox{$\mathcal F$}_1$ on $N_i\setminus\{s_i,s'_i\}$~;
\item $F''_i$ coincides with $F_i$ on $B_i$ and with $F'_i$ on $B'_i$~;
\item $F''_i\vert\partial N_i$ has exactly eight Morse critical points: two minima
and two critical points of index $1$ on $\partial B'_i$~, two critical points of index $1$ and two maxima
on $\partial B_i$~;
\item The values of $F''_i$ at these critical points are respectively $-2,-2,$ $-1,-1,$ $1,1,$ $2,2$~; \item The sign of the tangency between $F''_i$ and $\partial N_i$ at each critical point is as follows: the descending gradient of $F''_i$ exits $N_i$ at the four critical points on $\partial B'_i$~, and enters $N_i$ at the four critical points on $\partial B_i$~; \item One has $F''_i(p_i(u))=F''_i(q_i(u))$ for every $u\in[0,1]$~. \end{enumerate}
On the other hand, in the handle $H_i:={\bf D}^2\times{\bf S}^1$~, one has the function $h:=x_2(1+y_1^2)$~, where ${\bf D}^2\subset{\bf R}^2$ (resp. ${\bf S}^1\subset{\bf R}^2$) is defined by $x_1^2+x_2^2\le 1$ (resp. $y_1^2+y_2^2=1$). In $H_i$, the function $h$ has no critical point. On $\partial H_i$, by (3), (4) and elementary Morse theory, $h\vert\partial H_i$ is smoothly conjugate to $F''_i\vert\partial N_i$~. We attach $H_i$ to $M\setminus Int(N_i)$ so that the functions $F''_i$ and $h$ coincide on $\partial N_i\cong\partial H_i$~.
We extend $\mbox{$\mathcal F$}_1$ inside $H_i$ as the foliation defined by $h$~.
By (5), the sign of the tangency between $h$ and $\partial H_i$ at each singularity is the same as the sign of the tangency between $F''_i$ and $\partial N_i$~.
So, the resulting foliation is regular.
Having done this for every pair of singularity $s_i$~, $s'_i$~, $i=1,\dots, p$~, we get a regular foliation $\mbox{$\mathcal F$}$ on a closed $3$-manifold $M$~.
We claim that $\mbox{$\mathcal F$}$ admits $*\times T$ as an exhaustive transversal, and has the same holonomy pseudo-group $G$ as $\mbox{$\mathcal F$}_1$ on $*\times T$~. Obviously, $\mbox{$\mathcal F$}$ has no leaf contained in any $H_i$~. So, the claim
amounts to verify the following. Let $\gamma:[0,1]\to M_1$ (resp. $M$) be a path tangential to $\mbox{$\mathcal F$}_1$ (resp. $\mbox{$\mathcal F$}$) and whose endpoints belong to $M_1\setminus\cup_i Int(N_i)$~. Then, there is a path $\gamma':[0,1]\to M$ (resp. $M_1$) tangential to $\mbox{$\mathcal F$}$ (resp. $\mbox{$\mathcal F$}_1$) with the same endpoints, and such that the holonomy of $\mbox{$\mathcal F$}_1$ (resp. $\mbox{$\mathcal F$}$) along $\gamma$ is the same as the holonomy of $\mbox{$\mathcal F$}$ (resp. $\mbox{$\mathcal F$}_1$) along $\gamma'$~.
We can assume that $\gamma$ is contained in some
$N_i$ (resp. $H_i$), with endpoints on $\partial N_i=\partial H_i$~. Let $t:=F''_i(\gamma(0))=h(\gamma(0)) =F''_i(\gamma(1))=h(\gamma(1))$~.
First, consider the case where $\gamma$ is contained in $N_i$ and tangential to $\mbox{$\mathcal F$}_1$~. There are three subcases, depending on $t$.
First subcase: $F''_i(s'_i)<t<F''_i(s_i)$~. Then, the level set $F''_i{^{-1}}(t)$ is the disjoint union of two disks, so $\gamma$ has the same endpoints as some path $\gamma'$ contained in $\partial F''_i{^{-1}}(t)$~, and we are done.
Second subcase: $F''_i(s_i)\le t<2$~. Then, consider the level set $F_i{^{-1}}(t)\subset U_i$~. Obviously, the intersection of this level set with $U_i\setminus Int(B_i)$ is connected: a pair of pants when $t<1$, a pair of pants when $t>1$, and it is also connected when $t=1$. So, $\gamma$ has the same endpoints as some path $\gamma'$ contained in this intersection, and we are done. (If the endpoints of $\gamma$ do not lie on the same connected component of the boundary of the annulus $F_i{^{-1}}(t)\cap B_i$~, then the path $\gamma'$ will be close to the Levitt loop $\lambda_i$).
The third and last subcase $-2<t\le F''_i(s'_i)$ is symmetric to the second.
Now, consider the second case, where $\gamma$ is contained in $H_i$ and tangential to $\mbox{$\mathcal F$}$~.
In the subcases $-2<t<-1$ and $1<t<2$~, the level set $h{^{-1}}(t)$ is the disjoint union of two disks. Thus,
$\gamma(0), \gamma(1)$ are also the endpoints of some path $\gamma'$ contained in $\partial(h{^{-1}}(t))$~, and we are done. The like holds for $t=-2, -1, 1$ or $2$.
In the subcase $-1<t<1$, the level set $h{^{-1}}(t)$ is an annulus. If
$\gamma(0), \gamma(1)$ belong to a same component of $\partial(h{^{-1}}(t))$~, we are done. In the remaining sub-subcase, $\gamma(0), \gamma(1)$ belong to the two different circle components of $\partial(h{^{-1}}(t))$~. By (6), these two circles are also the boundaries of the two disk leaves of $\mbox{$\mathcal F$}_1\vert S_i$ through $p_i(u)$ and $q_i(u)$~, for some $u\in(0,1)$~. Now, recall that the diffeomorphism $p_i(u)\mapsto q_i(u)$ between the transversals $p_i$ and $q_i$ belongs to the holonomy pseudo-group of $\mbox{$\mathcal F$}_1$ on $p_i\cup q_i$~. In other words, there is a path $\gamma':[0,1]\to M_1$ tangential to $\mbox{$\mathcal F$}_1$ with the same endpoints as $\gamma$~, and such that the holonomy of $\mbox{$\mathcal F$}$ along $\gamma$ is the same as the holonomy of
$\mbox{$\mathcal F$}_1$ along $\gamma'$~.
Theorem A is proved in the case of a pseudo-group $(G,T)$ without boundary. \medbreak
Now, let us prove theorem A for a taut, compactly generated
pseudo-group $(G,T)$
such that $T$ has a boundary. One can assume that
$(G,T)$ is connected. Thus, one is reduced to the case where $T$
is a finite disjoint union of compact intervals (proposition \ref{connected_taut_pro}).
The construction is much the same as in the case without boundary. We stress the few differences.
We start from the manifold $M_0:={\bf S}^2\times T$~. For some of the generators $g_i$~, their domains and images meet the boundary, i.e. they are semi-open intervals. Consider for example a $g_i$ whose domain meets the positive boundary $\partial_+T$ (the boundary points where the
tangent vectors which are positive with respect to the orientation of $T$ , exit from $T$). That is, $Dom(g_i)=(u_i,u'_i]$ and $Im(g_i)=(v_i,v'_i]$ and $Dom(g_i)\cap\partial T=u'_i$ and $Im(g_i)\cap\partial T=v'_i$~.
Such a generator will be introduced in the holonomy of the foliation by performing, somewhat, a \emph{half} elementary surgery of index $1$ on the manifold $M_0$~. Put for every $n$~: $${2\mun}{\bf D}^n:=\{(x_1,\dots,x_n)\in{\bf R}^n\ \vert\ x_1^2+\dots+x_n^2\le 1,
x_n\le 0\} $$
Its boundary splits as the union of ${\bf D}^{n-1}$ (the subset defined in ${2\mun}{\bf D}^n$ by $x_n=0$) and ${2\mun}{\bf S}^{n-1}$
(the subset defined in ${2\mun}{\bf D}^n$ by $x_1^2+\dots+x_n^2=1$)~. Fix some extension $\bar g_i\in G$~.
Choose two embeddings $e_i:{2\mun}{\bf D}^3\to M_0$ and $f_i:{2\mun}{\bf D}^3\to M_0$ such that \begin{enumerate} \item $e_i{^{-1}}(\partial M_0)=f_i{^{-1}}(\partial M_0) ={\bf D}^2$~; \item $e_i({2\mun}{\bf D}^3)$ and $f_i({2\mun}{\bf D}^3)$ are disjoint from each other and from $T\times\ast$~; \item ${\rm pr}_2(e_i({2\mun}{\bf D}^3))=[u_i,u'_i]$ and ${\rm pr}_2(f_i({2\mun}{\bf D}^3))=[v_i,v'_i]$~; \item $e_i^*\mbox{$\mathcal F$}_0=f_i^*\mbox{$\mathcal F$}_0$ is the trivial foliation $dx_3$ on ${2\mun}{\bf D}^3$~; \item ${\rm pr}_2\circ f_i=\bar g_i\circ{\rm pr}_2\circ e_i$~. \end{enumerate}
We perform on $M_0$ a surgery by cutting
$e_i({2\mun}{\bf D}^3\setminus{2\mun}{\bf S}^2)$ and $f_i({2\mun}{\bf D}^3\setminus{2\mun}{\bf S}^2)$ and by pasting
$e_i({2\mun}{\bf S}^2)$ with $f_i({2\mun}{\bf S}^2)$~. The points $e_i(x)$ and $f_i(x)$ are pasted, for every $x\in{2\mun}{\bf S}^2$~.
This surgery produces a single singularity
$s_i:=e_i(0,0,-1)=f_i(0,0,-1)$~, of Morse index $1$~.
The case of a generator $g_i$ whose domain meets $\partial_-T$ is of course symmetric.
After performing a surgery for every generator, we get a resulting compact manifold
$M_1$~, and a Morse foliation $\mbox{$\mathcal F$}_1$ induced on $M_1$ by $\mbox{$\mathcal F$}_0$~, with some singularities of indices $1$ and $2$~. The boundary of $M_1$ is the disjoint union of two closed connected surfaces $\partial_-M_1$~, $\partial_+M_1$~, both
tangential to $\mbox{$\mathcal F$}_1$. At every point of
$\partial_-M_1$ (resp. $\partial_+M_1$)~, the tangent vectors positively transverse to $\mbox{$\mathcal F$}_1$ enter into (resp. exit from) $M_1$~. The holonomy pseudo-group of $\mbox{$\mathcal F$}_1$ on $*\times T$ coincides with $G$~.
These singularities are eliminated one after the other (\emph{not} by pairs). Let us eliminate e.g. a singularity $s_i$ of index $1$~.
On the one hand, by tautness, there is a path $p_i$~, positively transverse to $\mbox{$\mathcal F$}_1$ but at $s_i$~, from $p_i(0)\in \partial_-M_1$ to $p_i(1)=s_i$~.
On the other hand, by compact generation, $\mbox{$\mathcal F$}_1$ admits a Levitt loop $\lambda_i$ at $s_i$~. We can arrange that $\lambda_i$ is a simple loop: if it has a transverse self-intersection $x$~, then, by tautness, through $x$ there passes an arc $A$ embedded in $M_1$~, positively transverse to $\mbox{$\mathcal F$}_1$~, and whose endpoints lie on $\partial M_1$~. We perform a surgery on $M_1$ along $A$~, cutting a small tubular neighborhood $\cong{\bf D}^2\times[0,1]$ and pasting $\Sigma\times[0,1]$ (recall that $\Sigma$ is the disk endowed with a handle: see the paragraph below the proof of lemma \ref{levitt_lem}). The holonomy pseudo-group of the foliation is not changed. After the surgery, $s_i$ admits a Levitt loop with one less self-intersection.
The composed path $p_i\lambda_i$ is homotoped to a path $q_i$ positively transverse to $\mbox{$\mathcal F$}_1$~, arriving at $s_i$ through the component of the cone $Q<0$ opposite to that of $p_i$~; and $q_i$ is $\mbox{$\mathcal F$}_1$-equivalent to $p_i$~. During the homotopy, the extremity endpoint $s_i$ is fixed, but the origin endpoint moves in $\partial_-M_1$~. One arranges that $p_i\cap q_i=s_i$~.
The singularity $s_i$ is eliminated by, somewhat, a \emph{half} elementary surgery of index $2$ along the arc $p_i\cup q_i$ : one cuts a small tubular neighborhood of this arc, $N_i\cong[0,1]\times{\bf D}^2$~, such that
$N_i\cap\partial M_1\cong\{0,1\}\times{\bf D}^2$~; and one pastes ${2\mun}{\bf D}^2\times{\bf S}^1$ foliated by the restricted function $h\vert({2\mun}{\bf D}^2\times{\bf S}^1)$~. Every arc $({2\mun}{\bf S}^1)\times\theta\in\partial({2\mun}{\bf D}^2)\times{\bf S}^1$ is identified with $[0,1]\times\theta\in[0,1]\times\partial{\bf D}^2$~.
The details are just like in the case without boundary.
\subsection{Proof of proposition \ref{reduction_pro}} We are now given a compact manifold $M$ of dimension $n\ge 4$ endowed with a codimension $1$, taut foliation $\mbox{$\mathcal F$}$; and we have to find in $M$ a proper hypersurface $M'$ transverse to $\mbox{$\mathcal F$}$, such that the inclusion induces a bijection between the spaces of leaves
$M'/(\mbox{$\mathcal F$}\vert M')$ and $M/\mbox{$\mathcal F$}$.
To fix ideas, we consider only the case where $M$ is closed connected and where $\mbox{$\mathcal F$}$ is smooth.
Endow $M$ with an auxiliary Riemannian metric.
Write $\rho(\mbox{$\mathcal F$})$ the infimum of the injectivity radii of the leaves.
Fix a positive length $\delta<\rho(\mbox{$\mathcal F$})/4$ so small that the following \emph{tracking} property holds for every leaf $L$ of $\mbox{$\mathcal F$}$ and for every locally finite, $\delta$-dense subset $A\subset L$ (in the sense that every point of $L$ is at distance \emph{less than $\delta$} from some point of $A$).
For every shortest geodesic segment $[a,b]$ whose endpoints lie in $A$ and whose length is less than $\rho(\mbox{$\mathcal F$})/2$, there exists in $A$ a finite sequence $a_0=a$, \dots, $a_\ell=b$, such that $d(a_{i-1},a_{i})<2\delta$ ($1\le i\le\ell$), and that the shortest geodesic segments $[a_0,a_1]$, \dots, $[a_{\ell-1},a_\ell]$, $[b,a]$ form a simple loop bounding a $2$-disk embedded in $L$.
Choose a circle $T$ embedded into
$M$ transversely to $\mbox{$\mathcal F$}$, and such that $T\cap L$ is $\delta$-dense in every leaf $L$.
Write $G$ the holonomy pseudo-group of $\mbox{$\mathcal F$}$
on $T$. For any $g\in G$ and $r>0$, say that $g$ is \emph{$r$-short} if for every $t\in Dom(g)$, the distance from $t$ to $g(t)$ in the leaf of $\mbox{$\mathcal F$}$ through $t$ is less than $r$. At every point of $T$, one has only finitely many $2\delta$-short germs of local transformations of $T$ belonging to $G$. Thus, one has
a finite family $g_1,\dots, g_p\in G$ such that
every domain $Dom(g_i)$ is an interval $(t_i,t'_i)\subset T$~; and that every $2\delta$-short germ in $G$
is the germ of some $g_i$ at some point of its domain. Moreover, one can arrange
that $g_1$, \dots, $g_p$ are $(\rho(\mbox{$\mathcal F$})/2)$-short and
$G$-extendable; and that
the leaves through the endpoints $t_i, t'_i$
are two by two distinct. One writes $\hat g_i$ the extension of $g_i$ to the compact interval $[t_i,t'_i]$~.
The family $g_1$, \dots, $g_p$ generates $G$. Indeed, since $\delta<\rho(\mbox{$\mathcal F$})/2$, for every leaf $L$, the fundamental groupoid of the pair
$(L,L\cap T)$ is generated by the geodesic segments of length less than $2\delta$
whose enpoints lie in $L\cap T$.
One can arrange moreover that to each $\hat g_i$ is associated a fence (recall
definition \ref{holonomy_dfn}) $f_i$, such that the
image rectangles $Im(f_1)=f_1([0,1]\times[t_1,t'_1])$,
\dots, $Im(f_p)=f_p([0,1]\times[t_p,t'_p])$ are two by two disjoint in $M$. Indeed,
one first has the fences composed by the tangential shortest geodesic segments
$[t,\hat g_i(t)]$ ($t\in[t_i,t'_i]$). Since the leaves are of dimension $n-1\ge 3$,
after a fine enough
subdivision of the domains of the $g_i$'s into smaller subintervals, and after
a small generic perturbation of the arcs $[t,g_i(t)]$
relative to their endpoints, the image rectangles are two by two disjoint. Let
$s_i:=f_p(1/2,t_i)$ (resp. $s'_i:=f_p(1/2,t'_i)$) be the middle of the lower (resp. upper)
edge of each fence.
\medbreak
The rest of the proof of proposition \ref{reduction_pro} is alike the proof of theorem A, except that
the construction is made inside $(M,\mbox{$\mathcal F$})$. The dimension of the construction
here is $n-1$, rather than $3$ as it was in theorem A, but this does not make
any substantial difference. Here is a sketch.
Let $K:=T\cup Im(f_1)\cup\dots\cup Im(f_p)$, a $2$-complex embedded into $M$.
One has in $M$ a small compact neighborhood $\Omega$ of
$K\setminus\{s_1,\dots,s_p,s'_1,\dots,s'_p\}$
whose smooth boundary $M_1:=\partial\Omega$ is much like in the proof of theorem $A$.
Precisely, $s_1,\dots,s_p,s'_1,\dots,s'_p\in M_1$~; and
$M_1$ is transverse to $\mbox{$\mathcal F$}$ but at each $s_i$ (resp. $s'_i$), where $\mbox{$\mathcal F$}_1:=\mbox{$\mathcal F$}\vert M_1$ has a Morse singularity of index
$1$ (resp. $n-2$). One has in $M_1$ a circle $*\times T$ close and parallel to $T$,
transverse to $\mbox{$\mathcal F$}_1$, and meeting
every leaf of $\mbox{$\mathcal F$}_1$, such that the holonomy of $\mbox{$\mathcal F$}_1$ on
$\ast\times T\cong T$ is generated by $g_1$, \dots, $g_p$. That is, it coincides
with $G$.
Now, we use the tracking property to find some convenient Levitt loops. Consider any $s_i$~. In the leaf $L_i$ of $\mbox{$\mathcal F$}$ through $s_i$, the shortest geodesic segment $[t_i,\hat g_i(t_i)]$ has length less than $\rho(\mbox{$\mathcal F$})/2$, thus it is tracked by a piecewise geodesic path
$a_0=t_i$, \dots, $a_\ell=\hat g_i(t_i)$, such that $d(a_{i-1},a_{i})<2\delta$ ($1\le i\le\ell$); and $[a_0,a_1]$, \dots, $[a_{\ell-1},a_\ell]$, $[\hat g_i(t_i),t_i]$ form a simple loop $\lambda_{\rm geod}$ bounding a $2$-disk embedded in $L_i$. Close to $\lambda_{\rm geod}$~, one has a loop $\lambda_K$ in $L_i\cap K$~. Close to $\lambda_K$~, one has a loop $\lambda_i$ in $L_i\cap M_1$~, passing through $s_i$~. Obviously, $\lambda_i$ is a Levitt loop for $\mbox{$\mathcal F$}_1$ at $s_i$~.
If the fences $f_1$~,\dots, $f_n$ have been
taken close enough to the geodesic ones, and if the neighborhood $\Omega$ of $K$ has been taken thin enough, then $\lambda_i$ is also simple, and bounds also a disk $\Delta_i$ embedded in $L_i$~.
The like holds at every $s'_i$~, and yields a simple Levitt loop
$\lambda'_i$ bounding a disk $\Delta'_i$ embedded in the leaf of $\mbox{$\mathcal F$}$ through $s'_i$~.
The leaves of $\mbox{$\mathcal F$}$ through the singularities of $\mbox{$\mathcal F$}_1$ being two by two distinct, the disks are two by two disjoint.
Like in the proof of theorem $A$, we have in $M_1$ a simple path $p_i$
from $s'_i$ to $s_i$, positively
transverse to $\mbox{$\mathcal F$}_1$ but at its endpoints. The composed path $\lambda'_ip_i
\lambda_i$ is perturbated in $M_1$, relative to his endpoints,
into some simple path $q_i$ transverse to $\mbox{$\mathcal F$}_1$ and disjoint from $p_i$,
but at its endpoints. The union of the
disks $\Delta_i$ and $\Delta'_i$ with a thin strip is perturbated into
a $2$-disk $\Delta''_i$ (rather obvious on figure \ref{one_fig}) such that
\begin{itemize}
\item $\Delta''_i$ is embedded into $M$;
\item $\partial\Delta''_i=\Delta''_i\cap M_1=p_i\cup q_i$;
\item $\Delta''_i$ is transverse to $\mbox{$\mathcal F$}$ and $\mbox{$\mathcal F$}\vert\Delta''_i$
is the foliation of the $2$-disk by parallel straight segments.
\end{itemize}
The hypersurface $M'\subset M$ is built from $M_1$
by cutting, for every $1\le i\le p$,
a small tubular neighborhood of $p_i\cup q_i$ in $M_1$, diffeomorphic
with ${\bf S}^1\times{\bf D}^{n-2}$,
and pasting the boundary of the sphere bundle normal to $\Delta''_i$
in $M$, diffeomorphic to ${\bf D}^2\times{\bf S}^{n-3}$.
\section{proof of theorems B and C}\label{BC_section}
\subsection{Examples: realizing the homothety pseudo-groups}\label{homothety_sbs}
First, we discuss the realization of some elementary but fundamental examples:
the homothety pseudo-groups. They constitute the most simple nontaut,
compactly generated pseudo-groups.
Given some positive real numbers $\lambda_1$~, \dots, $\lambda_r$~,
let $G(\lambda_1,\dots,\lambda_r)$ be the pseudo-group of local transformations of the real line generated by the
homotheties $t\mapsto\lambda_1 t$~, \dots, $t\mapsto\lambda_r t$~.
We assume that
the family $\log\lambda_1$~, \dots, $\log\lambda_r$
is of linear rank $r$ over ${\bf Q}$~.
For $r=1$~, the pseudo-group
$G(\lambda_1)$ has two obvious realizations of interest.
The first is
on the annulus $A:={\bf S}^1\times[0,1]$~.
The compact leaf is
${\bf S}^1\times (1/2)$;
the other leaves are transverse to $\partial A$ and spiral towards ${\bf S}^1\times (1/2)$.
The second realization is on $\partial(A\times{\bf D}^2)\cong{\bf S}^2\times{\bf S}^1$~.
The compact leaf is a $2$-torus, and splits ${\bf S}^2\times{\bf S}^1$ into
two Reeb components.
On the contrary, $G(\lambda_1)$ is \emph{not} realizable on $T^2$. Indeed,
the foliation would be transversely oriented and have a single compact leaf,
whose linear holonomy would be nontrivial, a contradiction.
The case $r=2$ is analogous. The torus $T^2$ is endowed with the angle coordinates $x, y$~. One realizes
$G(\lambda_1,\lambda_2)$ on $V:=T^2\times[0,1]$ by a foliation $\mbox{$\mathcal F$}(\lambda_1,\lambda_2)$
transverse to both boundary tori,
where its trace is the linear irrational foliation $dx\log\lambda_1+dy\log\lambda_2=0$~. The torus $T^2\times(1/2)$ is a compact leaf; the other leaves spiral towards it.
Notice that $G(\lambda_1,\lambda_2)$ is \emph{not} realizable by any foliation $\mbox{$\mathcal F$}$ on any closed, orientable $3$-manifold $M$~. For, by contradiction,
$\mbox{$\mathcal F$}$ would have a unique compact leaf $L$ diffeomorphic to $T^2$~, along which $M$
would split into two compact $3$-manifolds $M'$~, $M''$~. On ${\bf R}\setminus 0$~, the differential $1$-form $dt/t$ is invariant by $G(\lambda_1,\lambda_2)$~. There would correspond on $M\setminus L$ a nonsingular closed $1$-form $\omega$ of rank $r=2$~, such that $\mbox{$\mathcal F$}\vert(M\setminus L)=\ker\omega$~.
In $H^1(M';{\bf R})$~,
the de Rham cohomology class $[\omega]$
decomposes as $(\log\lambda_1)e_1+(\log\lambda_2)e_2$~, with $e_1,e_2\in H^1(
M';{\bf Z})$~.
The restriction $[\omega]\vert L\in H^1(L;{\bf R})$ is of rank $2$~,
being the class of the linear holonomy of $\mbox{$\mathcal F$}$ along $L$~. Thus, $e_1\vert L$ and $e_2\vert L$ are not ${\bf Q}$-colinear
in $H^1(L;{\bf Q})$~.
Since $L$ is a $2$-torus, $(e_1\vert L)\wedge(e_2\vert L)\neq 0$ in
$H^2(L;{\bf Z})$~. In other words, $e_1\wedge e_2\in H^2(M';{\bf Z})$ is nonnull
on the fundamental class of $\partial M'$~. This contradicts Stokes theorem,
$M'$ being an orientable compact $3$-manifold.
One can ask if things would turn better if one dropped the condition that the realization be a \emph{tangentially orientable} foliation. It is not difficult to see that the answer is negative: $G(\lambda_1,\lambda_2)$ is also {not} realizable by any foliation $\mbox{$\mathcal F$}$~ , even not orientable, on any closed $3$-manifold $M$~. This is left as an exercise. I thank the referee for pointing out a mistake at this point
in the first version of this paper.
For \emph{every} $r\ge 2$~, the pseudo-group $G(\lambda_1,\dots,\lambda_r)$ is
realizable on a closed orientable $4$-manifold. Indeed, in a first place,
for each $2\le i\le r$~,
just as above,
realize $G(\lambda_1,\lambda_i)$ by a foliation $\mbox{$\mathcal F$}(\lambda_1,\lambda_i)$ on $V:=T^2\times[0,1]$~.
So, $G(\lambda_1,\lambda_i)$ is also realized by the pullback $\mbox{$\mathcal F$}_i$
of $\mbox{$\mathcal F$}(\lambda_1,\lambda_i)$ in the $4$-manifold
$$M_i:=\partial(V\times{\bf D}^2)
\cong T^2\times{\bf S}^2$$
The compact leaf $L_i$ of $\mbox{$\mathcal F$}_i$ is the $3$-torus $T^2\times{\bf S}^1$.
For each $i=3,\dots,r$, in $L_2$ and in $L_i$, we pick some embedded circle $C_i\subset L_2$ (resp. $C'_i\subset L_i$)
parallel to the first circle factor: the holonomy of $\mbox{$\mathcal F$}_2$ (resp. $\mbox{$\mathcal F$}_i$) along $C_i$
(resp. $C'_i$) is the germ of $t\mapsto\lambda_1t$
at $0$~. We arrange that $C_3$, \dots, $C_r$ are two by two disjoint. The loop $C_i$ (resp. $C'_i$) has in $M_2$ (resp. $M_i$) a small tubular neighborhood $N_i$ (resp. $N'_i$) $\cong{\bf D}^3\times{\bf S}^1$~, on
the boundary of which $\mbox{$\mathcal F$}_2$ (resp. $\mbox{$\mathcal F$}_i$) traces a foliation composed of two Reeb components,
realizing $G(\lambda_1)$~.
We cut from $M_2$~, \dots, $M_r$ the interiors of $N_3$~, \dots, $N_r$~,
$N'_3$~, \dots, $N'_r$~. We paste every $\partial N_i$ with $\partial N'_i$~,
such that $\mbox{$\mathcal F$}_2\vert\partial N_i$ matches $\mbox{$\mathcal F$}_i\vert\partial N'_i$~. We get
a closed $4$-manifold with a foliation realizing $G(\lambda_1,
\dots,\lambda_r)$~.
\medbreak
The realization of pseudo-groups of homotheties \emph{with boundary} is much alike:
let $2{^{-1}} G(\lambda_1,\dots,\lambda_r)$ be the pseudo-group of local transformations of the \emph{half}-line ${\bf R}_{\ge 0}$ generated by some family of
homotheties $t\mapsto\lambda_1 t$~, \dots, $t\mapsto\lambda_r t$~,
of rank $r$~. Each of the above realizations of $G(\lambda_1,\dots,\lambda_r)$
splits along its unique compact leaf into two realizations of
$2{^{-1}} G(\lambda_1,\dots,\lambda_r)$.
\subsection{Novikov decomposition for pseudo-groups, and hinges}
Let $(G,T)$ be a compactly generated pseudo-group of dimension $1$~.
We consider the closed orbits (the orbits topologically closed in $T$~).
\begin{lem}\label{closed_orbits_lem}
The union of the closed orbits is topologically closed in $T$~.
\end{lem}
\begin{proof} We know no proof for this fact
in the pseudo-group frame.
To prove it, we realize the pseudo-group, as in section \ref{A_section}, by
a Morse foliation $\mbox{$\mathcal F$}$
on a compact manifold $M$~. Since the homology of $M\setminus Sing(\mbox{$\mathcal F$})$
is of finite rank, Haefliger's argument \cite{hae1962} applies and shows that the union of the closed leaves is closed.
\end{proof}
We call a closed orbit \emph{isolated} (resp. \emph{left isolated})
(resp. \emph{right isolated}) if it admits some neighborhood
(resp. left neighborhood) (resp. right neighborhood) in $T$
which meets no other closed orbit.
By a \emph{component} of $(G,T)$ , one means
a submanifold $T'\subset T$ of dimension $1$, topologically closed in $T$,
and saturated for $G$~.
By an \emph{$I$-bundle} (resp. an \emph{${\bf S}^1$-bundle})
we mean the pseudo-group generated by
a finite number $r$ of \emph{global} diffeomorphisms on the compact interval
(resp. the circle). It is of course realized on some compact $3$-manifold
fibred over some closed surface (``suspension''). Every pseudo-group
Haefliger-equivalent to an $I$-bundle (resp. an {${\bf S}^1$-bundle})
is also called an $I$-bundle (resp. an {${\bf S}^1$-bundle}).
The smallest possible $r$ is the \emph{rank} of the $I$-bundle
(resp. ${\bf S}^1$-bundle).
Any closed orbit whose isotropy group has infinitely many fix points
bounds an $I$-bundle. Precisely,
\begin{lem}\label{accumulation_lem}
Let $G(t)\subset T$ be a closed orbit,
and let $h_1$, \dots, $h_r$ be elements of $G$ whose germs at $t$ generate the isotropy
group $G_t$. Assume that $h_1$, \dots, $h_r$ admit a sequence $(t_n)$ of
common fix points other than $t$, decreasing (resp. increasing) to $t$.
Put $I_n:=[t,t_n]$ (resp. $[t_n,t]$).
Then, for every $n$ large enough,
\begin{itemize}
\item The restricted pseudo-group $(G\vert I_n,I_n)$ is generated by
$h_1\vert I_n$, \dots, $h_r\vert I_n$;
\item The $G$-saturation of $I_n$
is an $I$-bundle component of $(G,T)$, Haefliger-equivalent to $(G\vert I_n,I_n)$.
\end{itemize}
\end{lem}
\begin{proof}
One first reduces oneself to the case where $G(t)=\{t\}$, as follows. Let $U:=(T\setminus G(t))\cup{t}$. By Baire's theorem, every closed orbit is discrete. So, $U$ is open in $T$ and meets every orbit. We change $(G,T)$ for $(G\vert U,U)$, which is also compactly generated by proposition \ref{invariance_pro}.
So, we assume that $G(t)=\{t\}$.
Since $G$ is compactly generated,
one has a topologically open, relatively compact $T'\subset T$ meeting
every $G$-orbit (in particular $t\in T'$) such that $G\vert T'$
admits a system of generators $g_1$~, \dots, $g_p$ which are $G$-extendable.
Let $\bar g_1$~, \dots, $\bar g_p\in G$ be some extensions.
If
$t$ lies in the topological boundary of $Dom(g_i)$ with respect to $T$~,
then we can avoid
this by changing $g_i$ for $\bar g_i\vert(Dom(g_i)\cup(t-\epsilon,t+\epsilon'))$~, where $(t-\epsilon,t+\epsilon')$
is relatively compact in $Dom(\bar g_i)\cap T'$~.
The like holds for $Im(g_i)$~.
Thus, after permuting the generators, for some $0\le q\le p$~, the point
$t
$ belongs to the domains and to the images of $g_1$~, \dots, $g_q$~;
but $t$
does not belong, nor is adherent, to the domains nor to the images of
$g_{q+1}$~, \dots, $g_p$~.
Also, restricting the domain of each $h_i$, we arrange
that $h_i\in G\vert T'$ and that $h_i$ is $G$-extendable.
Then, we can add the family $(h_i)$ to the family
of generators $(g_i)$. So, we can assume
that $r\le q$ and that $g_1=h_1$, \dots, $g_r=h_r$.
Then, for every $r+1\le i\le q$~, the generator $g_i$ coincides with some
composite of $g_1$, \dots, $g_r$ on some small compact neighborhood $N_i$ of $t$.
We can change $g_i$ for $g_i\vert(Dom(g_i)\setminus N_i)$.
Finally, we have obtained a family of generators $g_1$, \dots, $g_p$ for $G\vert T'$,
such that
$t$ belongs to the domains and to the images of $g_1=h_1$~, \dots, $g_r=h_r$~;
but that $t$
does not belong, nor is adherent, to the domains nor to the images of
$g_{r+1}$~, \dots, $g_p$~.
For every $n$ large enough,
$I_n$ is contained in $T'$ and in the domains of $g_1$, \dots, $g_r$;
and $I_n$ is
invariant by $g_1=h_1$, \dots, $g_r=h_r$; and $I_n$
is disjoint from the supports of $g_{r+1}$, \dots, $g_p$. Thus, $I_n$
is saturated for $G$, and $G\vert I_n$ is generated by $g_1\vert I_n$, \dots,
$g_r\vert I_n$. The interval $I_n$ is an $I$-bundle component of $G$.
\end{proof}
We call an orbit \emph{essential} (with respect to $(G,T)$)
if it meets no transverse positive loop
and no transverse positive chain whose both endpoints lie on $\partial T$ . Every essential
orbit is closed (obviously). In the union of the closed orbits,
the union of the essential orbits is topologically closed (obviously) and open
(by lemma \ref{accumulation_lem}).
We call an $I$-bundle component
of $G$ \emph{essential} (with respect to $(G,T)$) if its boundary orbits are essential
with respect to $(G,T)$. Then, every
closed orbit interior to this $I$-bundle is also essential with respect to $(G,T)$.
The ``Novikov decomposition'' is well-known for foliations on compact manifolds.
Every compact connected manifold endowed with a foliation of codimension one,
either is an ${\bf S}^1$-bundle, or splits along finitely many compact leaves bounding some
dead end components, into compact components, such that each component is
an $I$-bundle, or its interior is topologically taut. For compactly generated
pseudo-groups, one has an analogous
decomposition (exercise):
\begin{pro}\label{novikov_pro} (Novikov decomposition)
Let $(G,T)$ be a connected, compactly generated
pseudo-group of dimension $1$. Assume that $(G,T)$ is not an ${\bf S}^1$-bundle.
Then, $T$ splits, along finitely many essential orbits,
into finitely many components
$T_i$,
such that for each $i$:
(a) the component $(G\vert T_i,T_i)$ is an essential $I$-bundle,
or
(b)
the interior of the component, $(G\vert Int(T_i), Int(T_i))$, is taut.
\end{pro}
Novikov decompositions are functorial with respect to Haefliger
equivalences: given a Haefliger
equivalence between two pseudo-groups, to every Novikov decomposition of the
one corresponds naturally a Novikov decomposition of the other.
We shall not use this decomposition under this form, nor prove it in general. We need, under the hypotheses of theorems B and C, the more precise form \ref{splitting_pro} below.
\medbreak
From now on, we assume moreover that in the
compactly generated, $1$-dimensional pseudo-group $(G,T)$, every essential orbit is \emph{commutative,}
that is, its isotropy group is commutative. By the \emph{essential rank} of $(G,T)$,
we mean the supremum of the ranks of the isotropy groups of the essential orbits.
The proof of theorems B and C somewhat consists in realizing independently every
component of some Novikov decomposition, and pasting these realizations together.
The interior of every component falling to (b) in proposition \ref{novikov_pro}, is realized on a closed
$3$-manifold, thanks to theorem A and to the following
\begin{lem}\label{complement_lem}
Let $(G,T)$ be a compactly generated pseudo-group of dimension $1$~.
Let $G(t_0)\subset T$ be an isolated closed orbit, whose isotropy group is commutative.
Then, the subpseudo-group $G\vert(T\setminus G(t_0))$ is also compactly generated.
\end{lem}
\begin{proof}
We treat the case where the orbit $G(t_0)$ is contained in $\partial_-T$~. Of course,
the case where it is contained in $\partial_+T$ is symmetric; and
the case where
it is contained in $Int(T)$
is much alike.
Since $G$ is compactly generated,
one has a topologically open, relatively compact $T'\subset T$ meeting
every $G$-orbit such that $G\vert T'$
admits a system of generators $g_1$~, \dots, $g_p$ which are $G$-extendable.
Just as in the proof of lemma \ref{accumulation_lem},
one can arrange that $G(t_0)=\{t_0\}$~ (in particular, $t_0\in T'$); and that, for some $0\le r\le p$~, the point
$t_0$ belongs to the domains and to the images of $g_1$~, \dots, $g_r$~;
and that $t_0$
does not belong, nor is adherent, to the domains nor to the images of
$g_{r+1}$~, \dots, $g_p$~.
Since $t_0$ is isolated as a closed orbit
of $G$~, one has $r\ge 1$~.
We can arrange moreover, to simplify notations, that the family $(g_i)$
is symmetric: the inverse of every $g_i$ is some $g_j$~.
The isotropy group of $t_0$ being commutative,
there is a $u_0>t_0$ so close to $t_0$ that
\begin{enumerate}
\item For every $r+1\le i\le p$~, the interval $[t_0,u_0]$ does not meet $Dom(g_i)$~;
\item For every $1\le i,j\le r$ and every $t\in[t_0,u_0]$~,
one has $t\in Dom(g_i)$ and $g_i(t)\in Dom(g_j)$ and $g_ig_j(t)=g_jg_i(t)$~.
\end{enumerate}
Put $T'':=T'\setminus[t_0,u_0]$ and $G_0:=G\vert(T\setminus t_0)$~.
We shall show that
every orbit of $G_0$ meets $T''$~, and that the pseudo-group
$G_0\vert T''$ is generated by $g_1\vert T''$~, \dots, $g_p\vert T''$~.
Every $g_i\vert T''$ being $G_0$-extendable, it will follow that $G_0$ is
compactly generated.
To this end, define by induction two sequences $u_n\in[t_0,u_0]$ and $1\le i(n)\le r$~, such that $u_{n+1}:=g_{i(n)}(u_{n})$ is the minimum of $g_1(u_{n})$~, \dots, $g_r(u_n)$~.
Because $t_0$ is isolated as a closed orbit
of $G$~, there is no common fixed point for $g_1$~, \dots, $g_r$ in the interval
$(t_0,u_0]$~. Thus, $(u_n)$ decreases to $t_0$~. Also, for every $n\ge 0$~: $$g_{i(n)}{^{-1}}((u_{n+1},u_0])\subset(u_n,u_0]\cup T''\ \ \ (*)$$ In particular, every orbit of $G_0$ meets $T''$~.
Consider the germ $[g]_t$ of some $g\in G_0$ at some point $t\in Dom(g)$
such that $t\in T''$ and $g(t)\in T''$~. Since the $g_i$'s generate $G\vert T'$~,
this germ can be decomposed as a word $w$ in the germs of the generators: $$[g]_t=[g_{j(\ell)}]_{t(\ell-1)}\dots[g_{j(1)}]_{t(0)}$$
where $1\le j(1), \dots, j(\ell)\le p$~, where $t(0)=t$~,
and where for every $0\le k\le\ell$
one has $t(k):=g_{j(k)}\circ\dots\circ g_{j(1)}(t)\in T'$~.
We call the finite sequence $t(0)$~, \dots, $t(\ell)$ the \emph{trace} of $w$~.
We have to prove that $[g]_t$ admits also a second such decomposition, whose
trace is moreover contained in $T''$~.
We make a double induction: on the smallest integer $n\ge 0$ such that the trace
of $w$
is disjoint from $[t_0,u_{n}]$~, and, if $n\ge 1$~, on the number of $k$'s
for which $t_k\in(u_{n},u_{n-1}]$~.
Assume that $n\ge 1$~. Let $1\le k\le\ell-1 $ be an index for which $t_{k}\in(u_n,u_{n-1}]$~.
Consider the word
$$w':=g_{j(\ell)}
\dots g_{j(k+2)}
g_{i(n-1)}
g_{j(k+1)}
g_{j(k)}
g_{i(n-1)}{^{-1}}
g_{j(k-1)}
\dots g_{j(1)}
$$
By the property (2) above applied at the point $t_k$ to the pair
$g_{i(n-1)}{^{-1}}, g_{j(k+1)}$ and to the pair $g_{i(n-1)}{^{-1}}, g_{j(k)}{^{-1}}$~,
the composite $w'$ is defined at $t$~, and $w'$ has the same germ at $t$ as $w$~.
The trace of $w'$ at $t$ is the same as the trace of $w$~,
except that $t(k)$ has been changed for the three points
${g_{i(n-1)}{^{-1}}(t(k-1))}$~, ${g_{i(n-1)}{^{-1}}(t(k))}$ and ${g_{i(n-1)}{^{-1}}(t(k+1))}$~. By ($\ast$),
none of the three lies in $[t_0,u_{n-1}]$~. The induction is complete.
\end{proof}
The pasting of the realizations of the Novikov components
will be a little
delicate. The following
notion allows us to take in account,
with every commutative closed orbit, its isotropy group;
and with every commutative $I$-bundle,
the holonomy of its boundary orbits on the exterior side.
\begin{dfn}\label{thick_dfn}
We call a pseudo-group $(\Gamma,\Omega)$ of dimension $1$
a \emph{hinge} if $\Omega$ is an interval, either open, or compact, or semi-open;
and
if there exist a $\Gamma$-invariant compact
interval $[a,b]\subset\Omega$, with $a\le b$, and a system of generators
$\gamma_1$~, \dots, $\gamma_r$ for $\Gamma$, such that
\begin{enumerate} \item The domains and the images of $\gamma_1$~,\dots, $\gamma_r$ are intervals containing $[a,b]$~; \item For every $\gamma,\eta\in\Gamma$~, one has $\gamma\eta=\eta\gamma$ and
$\gamma{^{-1}}\eta=\eta\gamma{^{-1}}$ and $\gamma{^{-1}}\eta{^{-1}}=\eta{^{-1}}\gamma{^{-1}}$ wherever both composites are defined;
\item Every neighborhood of $[a,b]$ in $\Omega$ meets every orbit of $\Gamma$~.
\end{enumerate}
\end{dfn}
We call $[a,b]$ the \emph{core}.
Write $\partial\Omega$ the boundary of $\Omega$ as a manifold, that is,
the boundary points of $\Omega$ belonging to $\Omega$, if any. By (3), every such boundary point copincides with $a$ or $b$~.
The hinge is \emph{degenerate} if $a=b$, in which case $a=b$ is the (unique) closed
orbit of $(\Gamma, \Omega)$. The hinge is \emph{nondegenerate} if $a<b$, in which case $[a,b]$ is the (maximal) $I$-bundle component of $(\Gamma, \Omega)$.
The smallest possible $r$ is the \emph{rank} of $(\Gamma,\Omega)$~.
\begin{lem}\label{no_fix_point_lem} Let $(\Gamma,\Omega)$ be a hinge. Then, there exists a local transformation in $\Gamma$ whose domain contains the core, and which is fix-point free outside the core. \end{lem}
\begin{proof} i) In case $\Omega$ coincides with the core $[a,b]$~, there is nothing to prove. \medbreak ii) Consider the case $\partial\Omega=a$~.
Let
$\gamma_1$~, \dots, $\gamma_r$ be as in definition \ref{thick_dfn}. Write $\gamma_{r+i}=\gamma_i{^{-1}}$ ($1\le i\le r$)~.
Fix $u_0>b$ so close to $b$~, that $u_0$ belongs to the domains of $\gamma_1$~, \dots, $\gamma_{2r}$~. Define by induction two sequences $u_n\in(b,u_0]$ and $1\le i(n)\le 2r$~, such that $u_{n+1}:=\gamma_{i(n)}(u_{n})$ is the minimum of $\gamma_1(u_{n})$~, \dots, $\gamma_{2r}(u_n)$~. \medbreak Claim: $\gamma:=\gamma_{i(0)}$ is fix-point free in $(b,u_0]$~.
Indeed, consider any $t\in(b,u_0]$~. By property (3) of definition \ref{thick_dfn},
there is no common fixed point for $\gamma_1$~, \dots, $\gamma_r$ in the interval
$(b,u_0]$~. Thus, $(u_n)$ decreases to $b$~.
By construction, the two following properties are obvious, for every $n\ge 1$ and every $0\le i\le 2r$~: $$\gamma_{i(n)}{^{-1}}((u_{n+1},u_n])\subset(u_n,u_{n-1}]\ \ (A)$$ $$\gamma_i(b,u_{n}]\subset (b,u_{n-1}]\ \ (B)$$
Let $N$ be the integer such that $t\in(u_{N+1},u_N]$~. By (A), the composite $\alpha:=\gamma_{i(1)}{^{-1}}\dots\gamma_{i(N)}{^{-1}}$ is defined at $t$, and $\alpha(t)\in(u_1,u_0]$~. Consequently, $\gamma(\alpha(t))<\alpha(t)$~. If $N=0$~, this means that $\gamma(t)<t$~, and we are done. So, assume $N\ge 1$~. Then, $\alpha$ is defined on the whole interval $(b,u_{N-1})$~, which contains $t$ and $\gamma(t)$ (by B).
On the other hand, by (B), for every $k=0,\dots,N$~, the composite $$w_k:=\gamma_{i(1)}{^{-1}}\dots\gamma_{i(k)}{^{-1}}\gamma\gamma_{i(k+1)}{^{-1}}\dots\gamma_{i(N)}{^{-1}}$$ is defined at $t$~. By property (2) of definition \ref{thick_dfn}~ , one has $w_0(t)=w_1(t)$~,\dots, $w_{N-1}(t)=w_N(t)$~. So, $\alpha(\gamma(t))=\gamma(\alpha(t))<\alpha(t)$~. Applying $\alpha{^{-1}}$~, we get $\gamma(t)<t$~. The claim is proved. \medbreak
iii) The case $\partial\Omega=b$ is of course symmetric to ii). \medbreak iv) In the remaining case $\partial\Omega=\emptyset$, we need a little more argument to get a local transformation which is fix-point free \emph{in the same time} on the left of $a$, and on the right of $b$~.
Like in case iii), one makes a composite $\gamma$ of the generators $\gamma_i$'s, defined on some neighborhood of $[a,b]$~, and such that $\gamma(t)<t$ for every $t>b$ in the domain of $\gamma$~. Symmetrically, one makes a composite $\delta$ of the generators $\gamma_i$'s, defined on some neighborhood of $[a,b]$~, and such that $\delta(t)>t$ for every $t<a$ in the domain of $\delta$~. Clearly, by property (2), $\gamma\delta=\delta\gamma$ on some small neighborhood $[s_0,t_0]$ of $[a,b]$~.
If $\gamma$ (resp. $\delta$) is fix-point free on $[s_0,a)$ (resp. on $(b,t_0]$)~, then $\gamma$ (resp. $\delta$) restricted to $(s_0,t_0)$ works. So, disminushing the interval $[s_0,t_0]$ if necessary, we can assume that $\gamma(s_0)=s_0$ and $\delta(t_0)=t_0$~. Then, $\gamma\delta$ restricted to $(s_0,t_0)$
works. Indeed, let $t_n:=\gamma^n(t_0)$ and $s_n:=\delta^n(s_0)$~. Then, the sequence $t_n$ decreases to $b$~, the sequence $s_n$ increases to $a$~, and $\gamma(s_n)=s_n$~, and $\delta(t_n)=t_n$~, and $\gamma\delta(t_n)=t_{n+1}$~, and $\gamma\delta(s_n)=s_{n+1}$~. So, $\gamma\delta$ is fix-point free in $[s_0,a)$ and in $(b,t_0]$~.
\end{proof}
Every hinge is easily realized:
\begin{lem}\label{commutative_realization_lem}
Let $(\Gamma,\Omega)$ be a hinge of rank $r\ge 1$.
\begin{enumerate}
\item If $r\le 2$, the hinge $(\Gamma,\Omega)$ is realized on $T^2\times[0,1]$, with $2-\sharp(\partial\Omega)$ transverse boundary components;
\item For every $r$, the hinge $(\Gamma,\Omega)$ is realized on some compact $4$-manifold, without
transverse boundary component.
\end{enumerate}
\end{lem}
\begin{proof}
The realization is much like in the particular
case of the homothety pseudo-groups,
seen above at paragraph \ref{homothety_sbs}.
Consider, to fix ideas, the case where $\partial\Omega=\emptyset$~.
(1) Let us assume that $r= 2$,
and let us realize $(\Gamma,\Omega)$ on $T^2\times[0,1]$~. Recall that $[a,b]\subset\Omega$ is the core (definition \ref{thick_dfn}).
The suspension of $\gamma_1$ and $\gamma_2$ over $T^2$
provides, in $T^2\times\Omega$~, a foliation $\mbox{$\mathcal F$}$ on some open neighborhood $U$ of $T^2\times[a,b]$~. By property (3) of definition \ref{thick_dfn}, the local transformations $\gamma_1$ and $\gamma_2$ have
no common fix point outside $[a,b]$~.
Consequently, one has an embedding of $T^2\times[0,1]$ into $U$
containing
$T^2\times[a,b]$ in its interior, and meeting every leaf of $\mbox{$\mathcal F$}$~,
and such that $T^2\times 0$ and $T^2\times 1$
are embedded transversely to $\mbox{$\mathcal F$}$~. It is easily verified that $\mbox{$\mathcal F$}\vert(T^2\times[0,1])$
realizes $(\Gamma,\Omega)$~.
(2) Now, let $r$ be any integer $\ge 2$~. After lemma \ref{no_fix_point_lem}, we can assume that $\gamma_1$ is fix-point free outside $[a,b]$~. Also, by a restriction of $\Omega$ which amounts to a Haefliger equivalence of the hinge pseudo-group, one arranges that $\Omega=Dom(\gamma_1)\cup Im(\gamma_1)$~. Then, for each $2\le i\le r$~, just as in case (1), one
realizes the pseudo-group $\Gamma_i:=<\gamma_1,\gamma_i>$ on $\Omega$ by a foliation $\mbox{$\mathcal F$}^3_i$ on $V:=T^2\times[0,1]$~.
So, $\Gamma_i$ is also realized by the pullback $\mbox{$\mathcal F$}^4_i$
of $\mbox{$\mathcal F$}^3_i$ in $M_i:=\partial(V\times{\bf D}^2)
\cong T^2\times{\bf S}^2$~. The foliation $\mbox{$\mathcal F$}^4_i$ contains a ``core'' $I$-bundle
$B_i\cong T^3\times[a,b]$.
For each $i=3,\dots,r$, in $B_2$ (resp. $B_i$), we pick some embedded annulus $A_i:=C_ i\times[a,b]\subset B_2$
(resp. $A'_i:=C'_ i\times[a,b]\subset B_i$), where $C_i$ (resp. $C'_i$) is a circle embedded in $T^3$ and
parallel to the first circle factor. The foliation $\mbox{$\mathcal F$}^4_2\vert A_i$ (resp. $\mbox{$\mathcal F$}^4_i\vert A'_i$) is the suspension of $\gamma_1\vert[a,b]$~;
the holonomy of $\mbox{$\mathcal F$}^4_2$ (resp. $\mbox{$\mathcal F$}^4_i$) along $C_i\times a$
(resp. $C'_i\times a$) is the germ of $\gamma_1$
at $a$~; the holonomy of $\mbox{$\mathcal F$}^4_2$ (resp. $\mbox{$\mathcal F$}^4_i$) along $C_i\times b$
(resp. $C'_i\times b$) is the germ of $\gamma_1$
at $b$~.
We arrange that $C_3$, \dots, $C_r$ are two by two disjoint in $T^3$~.
The annulus $A_i$ (resp. $A'_i$) has in $M_2$ (resp. $M_i$) a small tubular neighborhood $N_i$ (resp. $N'_i$) $\cong{\bf D}^3\times{\bf S}^1$~, on
the boundary of which $\mbox{$\mathcal F$}^4_2$ (resp. $\mbox{$\mathcal F$}^4_i$) traces a foliation composed of an $I$-bundle and two Reeb components,
realizing $(<\gamma_1>,\Omega)$~.
We cut from $M_2$~, \dots, $M_r$ the interiors of $N_3$~, \dots, $N_r$~,
$N'_3$~, \dots, $N'_r$~. We paste every $\partial N_i$ with $\partial N'_i$~,
such that $\mbox{$\mathcal F$}^4_2\vert\partial N_i$ matches $\mbox{$\mathcal F$}^4_i\vert\partial N'_i$~. We get
a closed $4$-manifold with a foliation realizing $(\Gamma,\Omega)$~. \end{proof}
\begin{pro} Let $(G,T)$ be a compactly generated pseudo-group of dimension $1$, in which every essential orbit is commutative.
Then, after a Haefliger equivalence, $T$ splits as a disjoint union $$T=T_0\sqcup\Omega_1\sqcup\dots\sqcup\Omega_\ell$$ such that \begin{enumerate}\label{splitting_pro} \item $T_0$ is a finite disjoint union of circles and compact intervals; \item Each $\Omega_k$ ($1\le k\le\ell$) is the domain of a hinge $\Gamma_k\subset G$ whose rank is at most the essential rank of $(G,T)$; \item Each core $[a_k,b_k]\subset\Omega_k$ is $G$-saturated; \item For every $t\in\Omega_k\setminus[a_k,b_k]$
($1\le k\le\ell$),
the orbit $G(t)$ meets $T_0$. \end{enumerate} \end{pro}
We begin to prove proposition \ref{splitting_pro}.
By a \emph{subpseudo-group} in $(G,T)$~,
we mean a pseudo-group $(\Gamma,\Omega)$ such that $\Omega\subset T$ is topologically open,
and that $\Gamma\subset G\vert\Omega$~.
\begin{dfn}\label{faithful_dfn}
Let $(\Gamma,\Omega)\subset(G,T)$ be a hinge subpseudo-group.
Let $[a,b]\subset\Omega$ be its core.
(a) Assume that
$(\Gamma,\Omega)$ is
degenerate $(a=b)$. We call the hinge subpseudo-group
\emph{faithful} if $G(a)$ is closed in $T$ and if $\Gamma_{a}=G_a$ (isotropy groups).
(b) Assume that
$(\Gamma,\Omega)$ is
nondegenerate $(a\neq b)$. We call the hinge subpseudo-group
\emph{faithful}
if the $G$-saturation of $[a,b]$ is a component of $(G,T)$,
if $\Gamma\vert[a,b]=G\vert[a,b]$,
if $\Gamma_a=G_a$, and if $\Gamma_b=G_b$.
\end{dfn}
In case (b),
the $G$-saturation of $[a,b]$ is necessarily an $I$-bundle component of $G$.
The notion of subpseudo-group is \emph{not} functorial with respect
to the Haefliger equivalences. The following notion solves this difficulty.
\begin{dfn}\label{extension_dfn} Given two pseudo-groups $(G,T)$ and $(\Gamma,\Omega)$, an \emph{extension}
of $(G,T)$ by $(\Gamma,\Omega)$ is a pseudo-group $\bar G$ on the disjoint
union $\bar T:=T\sqcup\Omega$ such that
\begin{itemize}
\item $T$ is exhaustive for $\bar G$;
\item $G=\bar G\vert T$;
\item $\Gamma\subset\bar G$.
\end{itemize}
\end{dfn}
In particular, $(\bar G,\bar T)$ is Haefliger-equivalent to $(G,T)$, and $(\Gamma,\Omega)$ is a subpseudo-group of $(\bar G,\bar T)$.
For example, given two pseudo-groups $(G,T)$, $(\Gamma,\Omega)$
and given a Haefliger equivalence $(\bar\Gamma,\Omega\sqcup \Omega_0)$
between $(\Gamma,\Omega)$ and some subpseudo-group $(\Gamma_0,\Omega_0)\subset (G,T)$,
one has a natural extension of $(G,T)$ by $(\Gamma,\Omega)$: namely,
$\bar G$ is the pseudo-group on $T\sqcup\Omega$ generated by
$G\cup\bar\Gamma$.
An extension $(\bar G,\bar T)$ of a pseudo-group $(G,T)$ by a hinge
$(\Gamma,\Omega)$
is called \emph{faithful} if $(\Gamma,\Omega)\subset(\bar G,\bar T)$ is faithful;
\emph{essential} if its core is an essential orbit or
an essential $I$-bundle in $(\bar G,\bar T)$.
\begin{lem}\label{local_hinge_extension_lem}
For every $t\in T$ such that $G(t)$ is essential,
there is an essential faithful extension $(\bar G,\bar T)$ of $(G,T)$ by a hinge $(\Gamma,\Omega)$ s.t:
\begin{itemize}
\item The rank of the hinge $(\Gamma,\Omega)$ is at most the essential rank of $(G,T)$;
\item The core of $(\Gamma,\Omega)$
meets $\bar G(t)$;
\item The core of $(\Gamma,\Omega)$
meets also every essential orbit of $\bar G$ close enough to
$\bar G(t)$.
\end{itemize} \end{lem}
\begin{proof}
First case: $G(t)$ is not contained in any
$I$-bundle component of $(G,T)$ of rank $0$.
In this case, we shall actually find a
faithful hinge subpseudo-group in $(G,T)$ whose core
meets $G(t)$ and every neighboring essential orbit.
Let $r:=\mbox{$\mbox{\rm rank}$}(G_t)$ and choose $h_1$, \dots, $h_r\in G$
such that their germs at $t$
are a basis of $G_t$. Let $\Omega$ be a small interval containing $t$,
topologically open in $T$, and contained in
the intersection of the domains and of the images of $h_1$, \dots, $h_r$.
Put $\gamma_i:=h_i\vert(\Omega\cap h_i{^{-1}}(\Omega))$ ($i=1,\dots, r$)
and $\Gamma:=<\gamma_1,\dots,\gamma_r>$.
For $\Omega$ small enough, the properties (1) and (2) of definition \ref{thick_dfn} are fulfilled for every small enough, $\Gamma$-invariant interval $[a,b]$~.
First subcase: $G(t)$ is isolated (recall the vocabulary that follows lemma \ref{closed_orbits_lem} above).
Put $a=b:=t$.
For $\Omega$ small enough, by lemma
\ref{accumulation_lem}, $h_1,\dots, h_r$ have no common fix point in $\Omega$. In consequence, for every $t'\in\Omega\setminus\{t\}$, there is an $i$ for which one of the four following properties holds: $t'\in Dom(\gamma_i)$ and $t<\gamma_i(t')<t'$, or $t'\in Dom(\gamma_i{^{-1}})$ and $t<\gamma_i{^{-1}}(t')<t'$, or $t'\in Dom(\gamma_i)$ and $t'<\gamma_i(t')<t$, or $t'\in Dom(\gamma_i{^{-1}})$ and $t'<\gamma_i{^{-1}}(t')<t$. The property (3) of definition \ref{thick_dfn} follows.
Second subcase: $G(t)$ is not isolated from either side. In that subcase, by lemma \ref{accumulation_lem},
we can shorten $\Omega$ to arrange that moreover none of the endpoints of $\Omega$ is a fix point common to $h_1$, \dots, $h_r$. Let $a$ and $b$ be the smallest and the largest fix points common to
$h_1$, \dots, $h_r$
in $\Omega$. Then, $a<t<b$. For every $t'\in\Omega\setminus[a,b]$, there is an $i$ for which
one of the four following properties holds: $t'\in Dom(\gamma_i)$ and $b<\gamma_i(t')<t'$, or $t'\in Dom(\gamma_i{^{-1}})$ and $b<\gamma_i{^{-1}}(t')<t'$, or $t'\in Dom(\gamma_i)$ and $t'<\gamma_i(t')<a$, or $t'\in Dom(\gamma_i{^{-1}})$ and $t'<\gamma_i{^{-1}}(t')<a$. The property (3) of definition \ref{thick_dfn} follows.
Third subcase: $G(t)$ is isolated from exactly one side. The argument is similar to the first two subcases.
Second case: $G(t)$ is contained in an $I$-bundle component $C\subset T$ of rank $0$. That is, $C$ is a $1$-manifold, topologically closed in $T$,
and $G\vert C$ is Haefliger-equivalent to the trivial pseudo-group
on the interval $[0,1]\subset{\bf R}$. In other words, one has an orientation-preserving
map $f:C\to[0,1]$ which is etale (that is, a local diffeomorphism); and the Haefliger equivalence is nothing but the pseudo-group
on the disjoint union $C\sqcup[0,1]$ generated by the set of the local sections of $f$. The boundary $\partial C$ is made of of two orbits
$\partial_-C=G(t_0)$ and $\partial_+C=G(t_1)$.
We can assume that $C$ is
maximal among the $I$-bundle components of rank $0$.
Assume also, to fix ideas, that $C$ is interior to $T$ (the other cases being alike and simpler).
Thus, the isotropy group of $G$ at $t_0$ (resp. $t_1$) is nontrivial on the left (resp. right).
Pick some small open interval $(u_0,v_0)\subset T$ containing $t_0$ and whose intersection with $C$ is $[t_0,v_0)$; and
pick some small open interval $(u_1,v_1)\subset T$ containing $t_1$ and whose intersection with $C$ is $(u_1,t_1]$. Take the intervals so small that $f(v_0)<f(u_1)$. One extends $f\vert[t_0,v_0)$ into a diffeomorphism $f_0$ from the interval $(u_0,v_0)$ onto the interval $(-\infty,f(v_0))$. The choice among the extensions is arbitrary. Similarly, one extends $f\vert(u_1,t_1]$ into a diffeomorphism $f_1$ from the interval $(u_1,v_1)$ onto the interval $(f(u_1),+\infty)$. Let $T'$ be the disjoint union $T\sqcup{\bf R}$. Let $G'$ be the pseudo-group on $T'$ generated by $G$, $f$, $f_0$, and $f_1$. Obviously, $T$ is exhaustive in $(G',T')$, and $G=G'\vert T$, and $G'\vert[0,1]$ is the trivial pseudo-group on $[0,1]$, and the orbit $G'(t)$ meets $[0,1]$ at $f(t)$.
Let $r:=\max(\mbox{$\mbox{\rm rank}$}(G'_0),\mbox{$\mbox{\rm rank}$}(G'_1))$. One immediately
makes $h_1$, \dots, $h_r\in G'\vert{\bf R}$ whose domains and images
contain $[0,1]$, which are the identity on $(0,1)$, whose germs at $0$ generate
$G'_0$, and whose germs at $1$ generate $G'_1$.
Let $\Omega\subset{\bf R}$ be an open interval containing $[0,1]$, and contained in
the intersection of the domains and of the images of $h_1$, \dots, $h_r$.
For $\Omega$ small enough, the property (2) of definition \ref{thick_dfn} is fulfilled.
By lemma \ref{accumulation_lem},
we can moreover shorten $\Omega$ to arrange that
none of its endpoints is a fix point common to $h_1$, \dots, $h_r$.
Put $\gamma_i:=h_i\vert(\Omega\cap h_i{^{-1}}(\Omega))$ ($i=1,\dots, r$)
and $\Gamma:=<\gamma_1,\dots,\gamma_r>$. Let $a$ and $b$ be the smallest and the largest fix points common to
$h_1$, \dots, $h_r$
in $\Omega$. The property (3) of definition \ref{thick_dfn} is fulfilled. The pseudo-group $(G'\vert(T\sqcup\Omega),T\sqcup\Omega)$ is a faithful extension of $(G,T)$ by the hinge $(\Gamma,\Omega)$.
\end{proof}
\begin{proof}[Proof of proposition \ref{splitting_pro}] The pseudo-group $(G,T)$ being cocompact, and the union of the essential leaves being topologically closed in $T$, one has a compact $K\subset T$
whose $G$-saturation coincides with this union. By lemma \ref{local_hinge_extension_lem}, every point of $K$ has a
neighborhood in $K$ whose orbits meet the core of the hinge after one essential, faithful hinge
extension, whose rank is at most the essential rank of $(G,T)$. One extracts a finite subcover. There corresponds a finite sequence of essential faithful extensions by hinges $(\Gamma_k,\Omega_k)$ ($1\le k\le\ell$),
whose ranks are at most the essential rank of $(G,T)$.
Let $(\bar G,\bar T)$ be the resulting global extension of $(G,T)$; let $[a_k,b_k]$ be the core of $(\Gamma_k,\Omega_k)$; and let $C_k\subset \bar T$ be the $\bar G$-saturation of
$[a_k,b_k]$. It is easy to arrange that $C_1$,\dots, $C_\ell$ are two by two disjoint. A closed orbit of $\bar G$ is contained in $C_1\cup\dots\cup C_\ell$ iff it is essential.
Consequently, the pseudo-group $(\bar G\vert U=G\vert U,U)$ is taut, where
$$U:= T\setminus((C_1\cup\dots\cup C_\ell)\cap T)$$ Also, the topological closure $\bar U$ of $U$ in $T$ being a component of $(G,T)$, the restricted pseudo-group $(G\vert\bar U,\bar U)$ is compactly generated. By lemma \ref{complement_lem}, $(G\vert U,U)$ is also compactly generated. By proposition \ref{taut_pro}, $(G\vert U,U)$ is Haefliger-equivalent to some pseudo-group $(G_0,T_0)$
on a finite disjoint union $T_0$ of compact intervals and circles. By the example that follows the definition \ref{extension_dfn} above, we get an extension $(\tilde G,\tilde T)$ of $(\bar G,\bar T)$ by $(G_0,T_0)$. One has $\tilde T=\bar T\sqcup T_0$. Let $$\tilde T':=T_0\sqcup\Omega_1\dots\sqcup\Omega_\ell\subset\tilde T$$ By construction, $\tilde T'$ is exhaustive in $(\tilde G, \tilde T)$. We change $(G,T)$ for $(\tilde G\vert\tilde T',\tilde T')$. The properties of proposition \ref{splitting_pro} are fulfilled.
\end{proof}
\subsection{End of the proofs of theorems B and C}
Let, as before, $(G,T)$ be a compactly generated pseudo-group of dimension $1$, in which every essential orbit is commutative. Our task is to realize $(G,T)$, in dimension $3$ if possible, and $4$ if not.
Without loss of generality, $(G,T)$ is under the form described by
proposition \ref{splitting_pro}. We shall first realize separately $(G\vert T_0,T_0)$, $(\Gamma_1,\Omega_1)$, \dots, $(\Gamma_\ell,\Omega_\ell)$; and then perform some surgeries along some loops in the realizations, transverse to the foliations. It is convenient to begin with somewhat introducing these loops into the pseudo-group.
For each $k$, if
$a_k\notin\partial_-\Omega_k$
(resp. $b_k\notin\partial_+\Omega_k$), write $\Omega_k^-$
(resp. $\Omega_k^+$) the connected component of $\Omega_k\setminus[a_k,b_k]$ on the left of $a_k$ (resp. on the right of $b_k$).
\begin{lem}\label{interval_lem} a) In case $a_k\notin\partial_-\Omega_k$, there exist in $\Omega_k^-$
two points $a'_k<a''_k<a_k$, and $\phi_k\in\Gamma_k$, such that
i) The interval $(a'_k,a''_k)$ is exhaustive for $\Gamma_k\vert\Omega_k^-$;
ii) $[a'_k,a''_k]\subset Dom(\phi_k)\cap Im(\phi_k)$;
iii) $\phi_k(t)>t$ for every $t\in[a'_k,a''_k]$;
iv) $\phi_k(a'_k)<a''_k$.
b) Symmetrically, in case $b_k\notin\partial_+\Omega_k$,
there exist in $\Omega_k^+$
two points $b_k<b'_k<b''_k$, and $\psi_k\in\Gamma_k$, such that
i) The interval $(b'_k,b''_k)$ is exhaustive for $\Gamma_k\vert\Omega_k^+$;
ii) $[b'_k,b''_k]\subset Dom(\psi_k)\cap Im(\psi_k)$;
iii) $\psi_k(t)>t$ for every $t\in[b'_k,b''_k]$;
iv) $\psi_k(b'_k)<b''_k$. \end{lem} \begin{proof}[Proof of a)] Recall $\gamma_1$, \dots, $\gamma_r$ of definition \ref{thick_dfn}. Choose
$a'_k<a_k$, so close to $a_k$ that it belongs to the domain and to the image
of $\gamma_i$,
for every $1\le i\le r$.
Let $\gamma_j^{\epsilon_j}(a'_k)$
be the maximum of the values $\gamma_i(a'_k)$, $\gamma_i{^{-1}}(a'_k)$ ($1\le i\le r$).
Put $\phi_k:=\gamma_j^{\epsilon_j}$. Choose $a''_k$ in the interval $(\gamma_j^{\epsilon_j}(a'_k),a_k)$, so close to $\gamma_j^{\epsilon_j}(a'_k)$ that iii) holds. The properties i), ii) and iv) are obvious. \end{proof}
For each $k=1$, \dots, $\ell$, it follows from ii), iii) and iv) that,
in case $a_k\notin\partial_-\Omega_k$ (resp. $b_k\notin\partial_+\Omega_k$),
the subpseudo-group of $(\Gamma_k,\Omega_k)$ generated by $\phi_k\vert(a'_k,a''_k)$ (resp. $\psi_k\vert(b'_k,b''_k)$) is Haefliger-equivalent to the trivial pseudo-group on the circle. In case $\partial\Omega_k=\emptyset$ (resp. $\{a_k\}$) (resp. $\{b_k\}$) (resp. $\{a_k,b_k\}$), by the example following the definition \ref{extension_dfn},
we get an extension $(\hat\Gamma_k,\hat\Omega_k)$
of the hinge $(\Gamma_k,\Omega_k)$ by the trivial pseudo-group on the disjoint union of two circles $S_k^-\sqcup S_k^+$ (resp. one circle $S_k^+$) (resp. one circle $S_k^-$) (resp. $\emptyset$).
In other words, we have an extension $(\hat G,\hat T)$ of $(G,T)$ by the trivial pseudo-group on the disjoint union $S$ of all the $S_k^\pm$'s ($1\le k\le\ell$). In particular, $\hat T=T\sqcup S$. Write $\hat T_0:=T_0\sqcup S\subset\hat T$ and $\hat G_0:=\hat G\vert\hat T_0$. Also write $$A:=[a_1,b_1]\cup \dots \cup [a_\ell,b_\ell]$$
\begin{lem}\label{generation_lem}
The pseudo-group $\hat G$ on $\hat T$ is generated by $\hat
G_0$, $\hat\Gamma_1$, \dots, and $\hat\Gamma_\ell$.
\end{lem}
\begin{proof}
We have to verify that the germ $[g]_t$ of every $g\in\hat G$ at
every $t\in Dom(g)$, is generated by $\hat G_0$ and the $\hat\Gamma_k$'s.
If $t\in\Omega_k\setminus[a_k,b_k]$, then (lemma
\ref{interval_lem}, i)) there is
some $\gamma\in\Gamma_k$ such that $\gamma(t)\in (a'_k,a''_k)$ or
$\gamma(t)\in (b'_k,b''_k)$, and thus some $\hat\gamma\in\hat\Gamma_k$
such that $\hat\gamma(t)\in S_k^\pm$.
We are thus reduced to the case $t\in\hat T_0\cup A$. Symmetrically, one can assume also that $g(t)\in\hat T_0\cup A$.
By proposition \ref{splitting_pro}, (3), either $t, g(t)\in\hat T_0$ (and thus $[g]_t\in
\hat G_0$) or $t, g(t)\in [a_k,b_k]$ for some $1\le k\le\ell$. In that second case,
the extension of $(G,T)$ by $(\Gamma_k,\Omega_k)$ being faithful,
$g\in\Gamma_k$. \end{proof}
\begin{proof}[Proof of theorem B] We have to prove that (2) implies (1).
Start from a pseudo-group $(\hat G,\hat T)$
as in lemma \ref{generation_lem}, Haefliger-equivalent to $(G,T)$.
On the one hand, the restriction of $\hat G$ to $\hat T\setminus Int(A)$, being a component
of $(\hat G,\hat T)$,
is compactly generated by lemma \ref{complement_lem}. Since $\hat T_0\subset \hat T\setminus Int(A)$
is exhaustive, $(\hat G_0,\hat T_0)$ is compactly generated.
This pseudo-group is also taut, $\hat T_0$ being a disjoint union of
circles and compact intervals. By theorem A, $(\hat G_0,\hat T_0)$
is realized by a foliated compact $3$-manifold $(M_0,\mbox{$\mathcal F$}_0)$,
without transverse boundary. More precisely, from the proof of theorem A,
$\hat T_0$ is
embedded into $M_0$ as an exhaustive transversal to $\mbox{$\mathcal F$}_0$,
and $\hat G_0$ is the holonomy pseudo-group of
$\mbox{$\mathcal F$}_0$ on $\hat T_0$. One takes off from $M_0$ a small open tubular
neighborhood $N_0$ of $S$, such that $\mbox{$\mathcal F$}_0\vert\partial N_0$ is
the trivial foliation by $2$-spheres.
On the other hand, for each $k=1$, \dots, $\ell$, one realizes $(\Gamma_k,
\Omega_k)$ by a foliation $\mbox{$\mathcal F$}_k$ on $M_k:=T^2\times[0,1]$ (lemma
\ref{commutative_realization_lem}). Obviously, $\mbox{$\mathcal F$}_k$ admits transverse loops
corresponding to $S_k^\pm$, in the sense that $\hat\Omega_k$
embeds into $M_k$ as an exhaustive transversal to $\mbox{$\mathcal F$}_k$,
and $\hat\Gamma_k$ is the holonomy pseudo-group of
$\mbox{$\mathcal F$}_k$ on $\hat\Omega_k$.
One takes off from $M_k$ a small open tubular
neighborhood $N_k$ of $S_k^\pm$, such that $\mbox{$\mathcal F$}_k\vert\partial N_k$ is
the trivial foliation by $2$-spheres.
One pastes $\sqcup_{1\le k\le\ell}\partial N_k\cong{\bf S}^2\times S$ with $\partial N_0
\cong{\bf S}^2\times S$,
with respect to the identity of $S$. One gets a foliation $\mbox{$\mathcal F$}$ on $$M_0\cup_{{\bf S}^2\times S}(M_1
\sqcup\dots\sqcup M_\ell)$$ whose holonomy on the exhaustive transversal
$T$ coincides with $G$, by lemma \ref{generation_lem}.
\end{proof}
\begin{proof}[Proof of theorem C] The same as for (2) implies (1) in theorem B,
but instead of the foliated $3$-manifold $(M_0,\mbox{$\mathcal F$}_0)$, we use the foliated
$4$-manifold $(M_0\times{\bf S}^1, pr_1^*(\mbox{$\mathcal F$}_0))$; and instead of $T^2\times[0,1]$,
we use a $4$-dimensional realization of $(\Gamma_k,\Omega_k)$
(lemma \ref{commutative_realization_lem}).
\end{proof}
\noindent Universit\'e de Bretagne Sud
\noindent Universit\'e Europ\'eenne de Bretagne
\noindent Laboratoire de Math\'ematiques de Bretagne Atlantique, UMR 6205
\noindent Postal address: UBS, LMBA, B\^atiment Yves Coppens, Tohannic
\noindent B.P. 573
\noindent F-56019 VANNES CEDEX, France
\noindent [email protected]
\end{document} | arXiv |
\begin{document}
\title{On the bipartite graph packing problem} \begin{abstract} The graph packing problem is a well-known area in graph theory. We consider a bipartite version and give almost tight conditions on the packability of two bipartite sequences. \end{abstract}
\keywords{graph packing, bipartite, degree sequence}
\section{Notation}
We consider only simple graphs. Throughout the paper we use common graph theory notations: $d_G(v)$ (or briefly, if $G$ is understood from the context, $d(v)$) is the degree of $v$ in $G$, and $\Delta(G)$ is the maximal and $\delta(G)$ is the minimal degree of $G$, and $e(X,Y)$ is the number of edges between $X$ and $Y$ for $X\cap Y = \emptyset$. For any function $f$ on $V$ let $f(X)=\sum\limits_{v\in X}f(v)$ for every $X\subseteq V$. $\pi(G)$ is the degree sequence of $G$.
\section{Introduction}
Let $G$ and $H$ be two graphs on $n$ vertices. We say that $G$ and $H$ \emph{pack} if and only if $K_n$ contains edge-disjoint copies of $G$ and $H$ as subgraphs.
The graph packing problem can be formulated as an embedding problem, too. $G$ and $H$ pack if and only if $H$ is a subgraph of $\overline{G}$ ($H\subseteq \overline G$).
A classical result is the theorem of Sauer and Spencer.
\begin{theorem}[Sauer, Spencer \cite{sauer}] Let $G_1$ and $G_2$ be graphs on $n$ vertices with maximum degrees $\Delta_1$ and $\Delta_2$, respectively. If $\Delta_1\Delta_2<\frac{n}{2}$, then $G_1$ and $G_2$ pack. \label{0:1} \end{theorem}
Many questions in graph theory can be formulated as special packing problems, see \cite{kierstead}. The main topic of the paper is a type of these packing questions, which is called degree sequence packing to be defined in the next section. Some results in this field are similar to that of Sauer and Spencer (\autoref{0:1}).
The structure of the paper is as it follows. First, we define the degree sequence packing problem, and survey some results. Next, we state and prove our main result and also show that it is tight. In particular, we improve a bound given by Diemunsch et al.~\cite{ferrara} Finally, we consider some corollaries of our main theorem.
\section{Degree sequence packing}
\subsection{Graphic sequence packing}
Let $\pi = (d_1,\ldots,d_n)$ be a graphic sequence, which means that there is a simple graph $G$ with vertices $\{v_1,\ldots,v_n\}$ such that $d(v_i) = d_i$. We say that $G$ \emph{represents} $\pi$.
Havel \cite{havel} and Hakimi \cite{hakimi} gave a characterization of graphic sequences.
\begin{theorem} [Hakimi \cite{hakimi}] Let $\pi = \{a_1,\ldots,a_n\}$ be a sequence of integers such that $n-1\geq a_1\geq \cdots\geq a_n\geq 0$. Then $\pi$ is graphic if and only if by deleting any term $a_i$ and subtracting 1 from the first $a_i$ terms the remaining list is also graphic. \end{theorem} Kleitman and Wang \cite{kleitman} extended this result to directed graphs.
Two graphic sequences $\pi_1$ and $\pi_2$ \textit{pack} if there are graphs $G_1$ and $G_2$ representing $\pi_1$ and $\pi_2$, respectively, such that $G_1$ and $G_2$ pack. Obviously, the order does not matter.
There is an alternative definition to the packability of two graphic sequences. $\pi_1$ and $\pi_2$ \textit{pack with a fixed order} if there are graphs $G=(V,E_1)$ and $H=(V,E_2)$ with $V(\{v_1,\ldots,v_n\})$ such that $d_G(v_i)=\pi_1(i)$ and $d_H(v_i)=\pi_2(i)$ for all $i=1,\ldots,n$.
A detailed study of degree sequence packing we refer to Chapter 3 of Seacrest's PhD Thesis \cite{seacrest}.
One of the first results in (unordered or fixed order) degree sequence packing is the Lov\'asz--Kundu Theorem \cite{lovasz2, kundu}.
\begin{theorem}[Kundu \cite{kundu}] A graphic sequence $\pi = (d_1,\ldots,d_n)$ has a realization containing a $k$-regular subgraph if and only if $\pi - k = (d_1-k,\ldots,d_n-k)$ is graphic. \end{theorem} Though we use the first definition, we give a result for the latter. Let $\Delta_i = \Delta(\pi_i)$ the largest degree and $\delta_i = \delta(\pi_i)$ the smallest degree of $\pi_i$ for $i=1,2$.
Busch et al.~\cite{busch} gave a condition for the packability of two graphic sequences with a fixed order. By $\pi_1+\pi_2$ they mean the vector sum of (the ordered) $\pi_1$ and $\pi_2$.
\begin{theorem}[Busch et al. \cite{busch}] Let $\pi_1$ and $\pi_2$ be graphic sequences of length $n$ with $\Delta = \Delta(\pi_1 + \pi_2)$ and $\delta = \delta(\pi_1+\pi_2)$. If $\Delta\leq \sqrt{2\delta n} - (\delta-1)$, then $\pi_1$ and $\pi_2$ pack with a fixed oreder. When $\delta = 1$, strict inequality is required. \end{theorem} Diemunsch et al.~\cite{ferrara} showed a condition for (unordered) graphic sequences.
\begin{theorem}[Diemunsch et al. \cite{ferrara}]\label{0:2} Let $\pi_1$ and $\pi_2$ be graphic sequences of length $n$ with $\Delta_2 \geq\Delta_1$ and $\delta_1\geq 1$.
If \begin{equation} \left\{\begin{array}{rl} (\Delta_2+1)(\Delta_1+\delta_1) \leq \delta_1 n + 1,&when~\Delta_2+2\geq \Delta_1+\delta_1,~and\\ \dfrac{(\Delta_2+1+\Delta_1+\delta_1)^2}{4}\leq \delta_1n+1,&when~\Delta_2+2< \Delta_1+\delta_1, \end{array}\right. \end{equation} then $\pi_1$ and $\pi_2$ pack.
\end{theorem}
\subsection{Bipartite packing}
We study the bipartite packing problem as it is formulated by Catlin \cite{catlin}, Hajnal and Szegedy \cite{hajnal} and was used by Hajnal for proving deep results in complexity theory of decision trees \cite{hajnal2}.
Let $G_1 = (A,B;E_1)$ and $G_2 = (S,T;E_2)$ bipartite graphs with $|A|=|S|=m$ and $|B|=|T|=n$. They pack in the bipartite sense (i.e. they have a \textit{bipartite packing}) if there are edge-disjoint copies of $G_1$ and $G_2$ in $K_{m,n}$.
Let us define the \textit{bigraphic sequence packing problem}. We say that a sequence $\pi = (a_1,\ldots,a_m,b_1,\ldots,b_n)$ is \emph{bigraphic}, if $\pi$ is the degree sequence of a bipartite graph $G$ with vertex class sizes $m$ and $n$, respectively \cite{west}.
Two bigraphic sequences $\pi_1$ and $\pi_2$ without a fixed order \textit{pack}, if there are edge-disjoint bipartite graphs $G_1$ and $G_2$ with degree sequences $\pi_1$ and $\pi_2$, respectively, such that $G_1$ and $G_2$ pack in the bipartite sense.
Similarly to general graphic sequences, we can also define the packing with a fixed order.
Diemunsch et al.~\cite{ferrara} show the following for bigraphic sequences:
\begin{theorem}[Diemunsch et al. \cite{ferrara}] \label{0:3} Let $\pi_1$ and $\pi_2$ be bigraphic sequences with classes of size $r$ and $s$. Let $\Delta_1\leq \Delta_2$ and $\delta_1\geq 1$. If \begin{equation} \Delta_1\Delta_2\leq\frac{r+s}{4}, \end{equation}
then $\pi_1$ and $\pi_2$ pack. \end{theorem} The following lemma, formulated by Gale \cite{gale} and Ryser \cite{ryser}, will be useful. We present the lemma in the form as discussed in Lov\'asz, Exercise 16 of Chapter 7 \cite{lovasz}.
\begin{lemma}[Lov\'asz \cite{lovasz}] \label{4}
Let $G$ be a bipartite graph and $\pi$ a bigraphic sequence on $(A, B)$. \begin{equation}
\pi(X) \leq e_G(X,Y) + \pi(\overline{Y})~~~\forall X\subseteq A,~\forall Y\subseteq B, \end{equation}
then $\pi$ can be embedded into $G$ with a fixed order. \end{lemma} For more results in this field, we refer the reader to the monography on factor theory of Yu and Liu \cite{yu}.
\section{Main result}
\begin{theorem} \label{1}
For every $\varepsilon\in(0,\frac{1}{2})$ there is an $n_0=n_0(\varepsilon)$ such that if $n>n_0$, and $G(A,B)$ and $H(S,T)$ are bipartite graphs with $|A|=|B|=|S|=|T|=n$ and the following conditions hold, then $H\subseteq G$.
\begin{enumerate} \setlength\itemindent{15ex}
\item[\emph{Condition 1:}] \label{1:1} $d_G(x) > \(\frac{1}{2}+\varepsilon\)n$ holds for all $x\in A\cup B$
\item[\emph{Condition 2:}] $d_H(x) < \frac{\varepsilon^4}{100}\frac{n}{\log n}$ holds for all $x\in S$,
\item[\emph{Condition 3:}] $d_H(y)=1$ holds for all $y\in T$. \end{enumerate} \end{theorem} We prove \autoref{1} in the next section. First we indicate why we have the bounds in Conditions 1 and 2.
Condition 1 of \autoref{1} is necessary. Suppose that $\frac{n}{2}-1 < d_G(x)$. That allows $G = K_{\frac{n}{2}+1,\frac{n}{2}-1}\cup K_{\frac{n}{2}-1,\frac{n}{2}+1}$. For all $\varepsilon>0$ there is an $n_0$ such that if $n>n_0$ degrees are higher than $\left(\frac{1}{2}-\varepsilon\right)n$, but there is no perfect matching (i.e. 1-factor) in the graph.
Condition 2 is necessary as well. To show it, we give an example. Let $G = G(n,n,p)$ a random bipartite graph with $p>0.5$ and vertex class sizes of $n$. Let $H(S,T)$ be the following bipartite graph: each vertex in $T$ has degree $1$. In $S$ all vertices have degree 0, except $\frac{\log n}{c}$ vertices with degree $\frac{cn}{\log n}$. The graph $H$ cannot be embedded into $G$, which follows from the example of Koml\'os et al. \cite{komlos}
Before proving \autoref{1} we compare our main theorem with the previous results.
\begin{remark} There are graphs which can be packed using \autoref{1}, but not with \autoref{0:1}. \end{remark}
Indeed, $\Delta_1 >\frac{n}{2} $ and we can choose $\Delta_2 >1$. Thus, $\Delta_1\Delta_2>\frac{n}{2}$. However, with \autoref{1} we can pack $G$ and $H$.
\begin{remark} There are graphs which can be packed using \autoref{1}, but not with \autoref{0:2}. \end{remark}
Let $\pi_1 = \pi(H)$ and $\pi_2 = \pi(\overline{G})$.
$\delta_1 = 1$ and $\Delta_1\leq \frac{n}{100\log n}$.
If $\Delta_2 \approx \frac{n}{2}$, then $\Delta_2+2 \geq \Delta_1+\delta_1$.
Furthermore, \begin{equation} (\Delta_2+1)(\Delta_1+\delta_1) \approx \frac{n}{2}\cdot \frac{n}{c\log n} \gg n. \end{equation} Although the conditions of \autoref{0:2} are not satisfied, $\pi_1$ and $\pi_2$ still pack.
\begin{remark} There are graphs which can be packed using \autoref{1}, but not with \autoref{0:3}. \end{remark}
Let $\pi_1 = \pi(H)$ and $\pi_2 = \pi(\overline G)$, as above. The conditions of \autoref{0:3} are not satisfied, however, \autoref{1} gives a packing of them.
As it is transparent, our main theorem can guarantee packings in cases, that were far beyond reach by the previous tecniques.
\section{Proof}
We formulate the key technical result for the proof of \autoref{1} in the following lemma.
\begin{lemma}
\label{5}
Let $\varepsilon$ and $c$ such that in \autoref{1}. Let $G$ and $H$ be bipartite graphs with classes $Z$ and $W$ of sizes $z$ and $n$, respectively, where $z>\frac{2}{\varepsilon}$.
Suppose that
\begin{enumerate}
\item $d_G(x)>\(\frac{1}{2}+\varepsilon\)n$ for all $x\in Z$ and
\item $d_G(y)>\(\frac{1}{2}+\frac{\varepsilon}{2}\)z$ for all $y\in W$.
\newcounter{enumTemp}
\setcounter{enumTemp}{\theenumi}
\end{enumerate}
Assuming
\begin{enumerate}
\setcounter{enumi}{\theenumTemp}
\item There is an $M\in \mathbb{N}$ and with $\delta\leq \frac{\varepsilon}{10}$ we have
$$M\leq d_H(x)\leq M(1+\delta)~\forall x\in Z,$$
and
\item $d_H(y) = 1~\forall y\in W$.
\end{enumerate}
Then there is an embedding of $H$ into $G$. \end{lemma}
\begin{proof}
We show that the conditions of \autoref{4} are satisfied.
Let $X\subseteq Z$, $Y\subseteq W$. We have five cases to consider depending on the size of $X$ and $Y$.
In all cases we will use the obvious inequality $Mz\leq n$, as $d_H(X) = d_H(Y)$. For sake of simplicity, we use $e(X,Y)=e_G(X,Y)$. \begin{enumerate}[(a)] \item
$|X|\leq\frac{z}{2(1+\delta)}$ and $|Y|\leq\frac{n}{2}$.
We have
\begin{equation}
d_H(X)\leq M(1+\delta)|X| \leq M(1+\delta)\frac{z}{2(1+\delta)} = \frac{Mz}{2}\leq \frac{n}{2} \leq |\overline{Y}|=d_H\(\overline{Y}\).
\end{equation}
\item $|X|\leq\frac{z}{2(1+\delta)}$ and $|Y|>\frac{n}{2}$.
Let $\phi = \frac{|Y|}{n}-\frac{1}{2}$, so $|Y| = \(\frac{1}{2}+\phi\)n$. Obviously, $0\leq\phi\leq\frac{1}{2}$.
Therefore, $d_H(\overline Y)=|\overline Y| = \(\frac{1}{2}-\phi\)n$.
Since $d_H(X)\leq \frac{n}{2}$, as we have seen above, furthermore,
\begin{equation}
e(X,Y)\geq (\varepsilon + \phi)n|X|\geq (\varepsilon+ \phi)n,
\end{equation}
we obtain $d_H(X)\leq d_H(\overline Y)+e_G(X,Y)$.
\item $\frac{z}{2}\geq|X|>\frac{z}{2(1+\delta)}$ and $|Y|\leq\frac{n}{2}$.
\label{case_c}
Let $\psi = \frac{|X|}{z}-\frac{1}{2(1+\delta)}$, hence, $|X| = \(\frac{1}{2(1+\delta)}+\psi\)z$. Let $\psi_0 = \frac{\delta}{2(1+\delta)}=\frac{1}{2}-\frac{1}{2(1+\delta)}$, so $\psi\leq \psi_0$. This means that $|X|=\(\frac{1}{2}-\psi_0+\psi\)z$.
As $0<\delta\leq\frac{\varepsilon}{10}$, we have $\psi_0<\frac{\delta}{2}\leq \frac{\varepsilon}{20}$.
Let $\phi = \frac{1}{2}-\frac{|Y|}{n}$, so $|Y| = \(\frac{1}{2}-\phi\)n$. As $|Y|\leq\frac{n}{2}$, this gives $0\leq \phi\leq \frac{1}{2}$.
\begin{enumerate}[(1)]
\item $d_H(\overline Y)=|\overline Y| = n\(\frac{1}{2}+\phi\)$
\item As above, $d_H(X)\leq M(1+\delta)|X| = Mz(1+\delta)\(\frac{1}{2(1+\delta)}+\psi\)\leq n(1+\delta)\(\frac{1}{2(1+\delta)}+\psi\)$.
\item We claim that $e(X,Y)\geq |Y|\(\frac{\varepsilon}{2}-\psi_0+\psi\)z$. Indeed, the number of neighbours of a vertex $y\in Y$ in $X$ is at least $\(\frac{\varepsilon}{2}+\psi-\psi_0\)z$, considering the degree bounds of $W$ in $H$.
\end{enumerate}
We show $d_H(X)\leq e(X,Y) + d_H(\overline Y)$.
It follows from
\begin{equation}
n(1+\delta)\(\frac{1}{2(1+\delta)} + \psi\) \leq n\(\frac{1}{2}-\phi\)\(\frac{\varepsilon}{2}-\psi_0+\psi\)z + n\(\frac{1}{2}+\phi\).
\end{equation}
This is equivalent to
\begin{equation}
\psi + \delta\psi\leq z\(\frac{1}{2}-\phi\)\(\frac{\varepsilon}{2}+\psi-\psi_0\) + \phi.
\label{eq1}
\end{equation}
The left hand side of \eqref{eq1} is at most $\psi_0 + \delta\psi_0 \leq \frac{\delta}{2}+\frac{\delta^2}{2}\leq \delta$, as $\delta\leq\varepsilon\leq\frac{1}{2}$.
If $\phi > \delta$, \eqref{eq1} holds, since $\frac{\varepsilon}{2}+\psi-\psi_0\geq 0$, using $\psi_0\leq\frac{\varepsilon}{20}$.
Otherwise, if $\phi\leq \delta$, the right hand side of \eqref{eq1} is
\begin{equation}
z\(\frac{1}{2}-\phi\)\(\frac{\varepsilon}{2}+\psi-\psi_0\) \geq \(\frac{1}{2}-\delta\)\(\frac{\varepsilon}{2}-\frac
{\delta}{2}\)z.
\end{equation}
We also have
\begin{equation}
\frac{\varepsilon}{4}+\frac{\delta^2}{2}-\frac{\delta\varepsilon}{2}-\frac{\delta}{4} > \delta,
\end{equation}
since
\begin{equation}
\varepsilon + 2\delta^2-2\delta\varepsilon > \varepsilon - 2\frac{\varepsilon^2}{10}>\frac{\varepsilon}{2}>5\delta,
\end{equation}
using $\delta\leq \frac{\varepsilon}{10}$.
This completes the proof of this case.
\item $|X|>\frac{z}{2}$ and $|Y|\leq\frac{n}{2}$.
We have
\begin{enumerate}[(1)]
\item $d_H(X)= d_H(Z)-d_H(\overline X) = n - d_H(\overline X) \leq n - M|\overline X|,$
\item $d_H(\overline Y) = n- |Y|$ and
\item $e(X,Y) \geq |Y|\(|X|-\frac{z}{2}+\frac{\varepsilon z}{2}\)$, using to the degree bound on $Y$.
\end{enumerate}
All we have to check is whether
\begin{equation}
n - M|\overline X| \leq n - |Y| + |Y|\(|X|-\frac{z}{2} + \frac{\varepsilon z}{2}\)
\end{equation}
It is equivalent to
\begin{equation}
0 \leq |Y|\(|X|-\frac{z}{2}+\frac{\varepsilon z}{2}-1\)+M\(z-|X|\)
\label{eq4}
\end{equation}
\eqref{eq4} has to be true for any $Y$ and $M$. Specially, with $|Y|=M=1$, \eqref{eq4} has the following form:
\begin{equation}
0\leq |X| - \frac{z}{2} + \frac{\varepsilon z}{2}-1 + z - |X| = \frac{z}{2}+\frac{\varepsilon z}{2}-1.
\label{eq5}
\end{equation}
\eqref{eq5} is true if $z\geq 2$.
If $z = 1$, then $Z=\{v\}$ is only one vertex, which is connected to each vertex in $W$. In this case, \autoref{5} is obviously true.
\item $|X|>\frac{z}{2(1+\delta)}$ and $|Y|>\frac{n}{2}$.
Let $\psi = \frac{|X|}{z}-\frac{1}{2(1+\delta)}$, hence, $|X| =z\(\frac{1}{2(1+\delta)}+\psi\)$. Let $\psi_0 = \frac{\delta}{2(1+\delta)}$, as it was defined in Case \eqref{case_c}. Again, $\psi_0\leq\frac{\delta}{2}$. We have $0\leq\psi\leq\frac{1}{2}+\psi_0\leq \frac{1+\delta}{2}$.
Let $\phi = \frac{|Y|}{n}-\frac{1}{2}$, hence, $|Y| = n\(\frac{1}{2}+\phi\)$.
We have
\begin{enumerate}[(1)]
\item $d_H(X)\leq zM(1+\delta)\(\frac{1}{2(1+\delta)}+\psi\)\leq n (1+\delta)\(\frac{1}{2(1+\delta)}+\psi\),$
\item $d_H(\overline Y) = n\(\frac{1}{2}-\phi\)$ and
\item $e(X,Y)\geq z\(\frac{1}{2(1+\delta)}+\psi\)(\phi+\varepsilon)n.$
\end{enumerate}
From the above it is sufficient to show that
\begin{equation}
n(1+\delta)\(\frac{1}{2(1+\delta)}+\psi\)\leq n\(\frac{1}{2}-\phi\) + z\(\frac{1}{2(1+\delta)+\psi}\)(\phi+\varepsilon)n.
\end{equation}
It is equivalent to
\begin{equation}
\psi(1+\delta)\leq -\phi + z\(\frac{1}{2(1+\delta)}+\psi\)(\phi + \varepsilon).
\label{eq2}
\end{equation}
Using $\psi\leq\frac{1+\delta}{2}$ and $\delta\leq\frac{\varepsilon}{10}$, the left hand side of \eqref{eq2} is at most
\begin{equation}
\frac{1+\delta}{2}(1+\delta) = \frac{1}{2}+\delta+\frac{\delta^2}{2}\leq \frac{1}{2}+\frac{\varepsilon}{10}+\frac{\varepsilon^2}{200}\leq\frac{1}{2}+\frac{1}{10}=\frac{3}{5},
\end{equation}
as $\varepsilon\leq\frac{1}{2}$.
The right hand side of \eqref{eq2} is
\begin{equation}
\phi\frac{z-2(1+\delta)}{2(1+\delta)} + \frac{z}{2(1+\delta)}\varepsilon + z\psi(\phi+\varepsilon)
\label{eq3}
\end{equation}
The first and the last term of \eqref{eq3} is always positive. (We use that $z>3$.) Therefore, \eqref{eq3} is at least $\frac{z}{2(1+\delta)}\varepsilon$.
It is enough to show that
\begin{equation}
\frac{3}{5} \leq \frac{z}{2(1+\delta)}\varepsilon.
\end{equation}
This is true indeed, since $\varepsilon>\frac{2}{z}$ and $\delta\leq\frac{\varepsilon}{10}\leq\frac{1}{20}$.
We have proved what was desired.
\end{enumerate}
\end{proof}
\begin{proof} (\autoref{1}) First, form a partition $C_0,C_1,\ldots,C_k$ of $S$ in the graph $H$. For $i>0$ let $u\in C_i$ if and only if $\frac{\varepsilon^4}{100}\frac{n}{\log n}\cdot\frac{1}{(1+\delta)^{i-1}}\geq d_H(u)>\frac{\varepsilon^4}{100}\frac{n}{\log n}\cdot\frac{1}{(1+\delta)^{i}}$ with $\delta = \frac{\varepsilon}{10}$. Let $C_0$ be the class of the isolated points in $S$. Note that the number of partition classes, $k$ is $\log_{1+\delta}n = \log_{1 + \frac{\varepsilon}{10}} n =\frac{\log n}{\log \(1 + \frac{\varepsilon}{10}\)} = c\log n$.
Now, we embed the partition of $S$ into $A$. Take a random ordering of the vertices in $A$. The first $|C_1|$ vertices of $A$ form $A_1$, the vertices $|C_1|+1,\ldots,|C_1|+|C_2|$ form $A_2$ etc., while $C_0$ maps to the last $|C_0|$ vertices. Obviously, $C_0$ can be always embedded.
We say that a partition class $C_i$ is \emph{small} if $|C_i|\leq \frac{16}{\varepsilon^2}\log n$.
We claim that the total size of the neighbourhood in $B$ of small classes is at most $\frac{\varepsilon n}{4}$.
The size of the neighbourhood of $C_i$ is at most \begin{equation} \frac{\varepsilon^4}{100}\frac{n}{\log n}\cdot \frac{1}{(1+\delta)^{i-1}}\cdot\frac{16}{\varepsilon^2}\log n. \end{equation} If we sum up, we have that the total size of the neighbourhood of small classes is at most
\begin{align} \sum_{i=1}^{k}\frac{\varepsilon^4}{100}\frac{n}{\log n}\cdot \frac{1}{(1+\delta)^{i-1}}\cdot\frac{16}{\varepsilon^2}\log n = \frac{4}{25}\varepsilon^2n\sum_{i=0}^{k-1}\frac{1}{(1+\delta)^i}\leq \nonumber\\ \leq\frac{4}{25}\varepsilon^2n\frac{1+\delta}{\delta}\leq\frac{4}{25}\varepsilon^2n\frac{3/20}{\varepsilon/10}\leq \frac{\varepsilon n}{4}.\phantom{mmmmmmmmmmm} \end{align} The vertices of the small classes can be dealt with using a greedy method: if $v_i$ is in a small class, choose randomly $d_H(v_i)$ of its neighbours, and fix these edges. After we are ready with them, the degrees of the vertices of $B$ are still larger than $\(\frac{1}{2}+\frac{\varepsilon}{2}\)n$.
Continue with the large classes. Reindex the large classes $D_1,\ldots,D_\ell$ and form a random partition $E_1,\ldots,E_\ell$ of the unused vertices in $B$ such that $|E_i| = \sum\limits_{u\in D_i} d_H(u)$. We will consider the pairs $(D_i,E_i)$.
We will show that the conditions of \autoref{5} are satisfied for $(D_i,E_i)$.
For this, we will use the Azuma--Hoeffding inequality.
We have to show that for any $i$ every vertex $y\in E_i$ has at least $\(\frac{1}{2}+\frac{\varepsilon}{4}\)z$ neighbours in $D_i$ and every vertex $x\in D_i$ has at least $\(\frac{1}{2}+\frac{\varepsilon}{2}\)z$ in $E_i$.
Then we apply \autoref{5} with $\frac{\varepsilon}{2}$ instead of $\varepsilon$, and we have an embedding in each pair $(D_i,E_i)$, which gives an embedding of $H$ into $G$.
Let $|D_i| = z$. We know $z > \frac{16}{\varepsilon^2} \log n$, as $D_i$ is large.
Build a martingale $\mathcal{Z} = \mathcal{Z}_0,\mathcal{Z}_1,\ldots,\mathcal{Z}_z$. Consider a random ordering $v_1,\ldots,v_z$ of the vertices in $Z$. Let $X_i=1$ if $v_i$ is a neighbour of $y$, otherwise, let $X_i = 0$. Let $\mathcal{Z}_i = \sum\limits_{j=1}^i X_j$, and let $\mathcal{Z}_0 = 0$. This chain $\mathcal{Z}_i$ is a martingale indeed with martingale differences $X_i\leq 1$, which is not hard to verify.
According to the Azuma--Hoeffding inequality \cite{azuma, hoeffding} we have the following lemma:
\begin{lemma}[Azuma \cite{azuma}] If $\mathcal{Z}$ is a martingale with martingale differences $1$, then for any $j$ and $t$ the following holds: \label{azulemma} \begin{equation} \mathbb{P}\(\mathcal{Z}_j\geq\mathbb{E}\mathcal{Z}_j-t\)\geq 1 - e^{-\frac{t^2}{2j}}. \end{equation} \end{lemma}
The conditional expected value $\mathbb{E}(\mathcal{Z}_z|\mathcal{Z}_0)$ is $\mathbb{E}\mathcal{Z}_z = \(\frac{1}{2}+\frac{3\varepsilon}{4}\)z$.
\autoref{azulemma} shows that
\begin{equation} \mathbb{P}\(\mathcal{Z}_z\geq\(\frac{1}{2}+\frac{\varepsilon}{2}\)z\) \geq 1 - e^{-\frac{\varepsilon^2z^2/4}{2z}} = 1-e^{-\varepsilon^2 z / 8}. \label{azueq} \end{equation} We say that a vertex $v\in E_i$ is \textbf{bad}, if it has less than $\(\frac{1}{2}+\frac{\varepsilon}{2}\)z$ neighbours in $D_i$. \autoref{azulemma} means that a vertex $v$ is bad with probability at most $e^{-\varepsilon^2 z/8}$. As we have $n$ vertices in $B$, the probability of the event that any vertex is $C$-bad is less than \begin{equation} n\cdot e^{-\varepsilon^2z/8}<\frac{1}{n}, \end{equation} as $z>\frac{16}{\varepsilon^2}\log n$.
Then we have that with probability $1-\frac{1}{n}$ no vertex in $E_i$ is bad. Thus, Condition (ii) of \autoref{5} is satisfied with probability 1 for any pair $(D_i,E_i)$.
Using \autoref{azulemma}, we can also show that each $x\in D_i$ has at least $\(\frac{1}{2}+\frac{\varepsilon}{2}\)|E_i|$ neighbours in $E_i$ with probability 1.
Thus, the conditions of \autoref{5} are satisfied, and we can embed $H$ into $G$. The proof of \autoref{1} is finished. \end{proof}
\begin{koszi} I would like to thank my supervisor, B\'ela Csaba his patient help, without whom this paper would not have been written. I also express my gratitude to P\'eter L. Erd\H os and to P\'eter Hajnal thoroughly for reviewing and correcting the paper. This work was supported by T\'AMOP-4.2.2.B-15/1/KONV-2015-0006 \end{koszi}
\end{document} | arXiv |
Tesseractic honeycomb
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.
Tesseractic honeycomb
Perspective projection of a 3x3x3x3 red-blue chessboard.
TypeRegular 4-space honeycomb
Uniform 4-honeycomb
FamilyHypercubic honeycomb
Schläfli symbols{4,3,3,4}
t0,4{4,3,3,4}
{4,3,31,1}
{4,4}(2)
{4,3,4}×{∞}
{4,4}×{∞}(2)
{∞}(4)
Coxeter-Dynkin diagrams
4-face type{4,3,3}
Cell type{4,3}
Face type{4}
Edge figure{3,4}
(octahedron)
Vertex figure{3,3,4}
(16-cell)
Coxeter groups${\tilde {C}}_{4}$, [4,3,3,4]
${\tilde {B}}_{4}$, [4,3,31,1]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive
Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex.
It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are Self-dual.
Coordinates
Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).
Sphere packing
Like all regular hypercubic honeycombs, the tesseractic honeycomb corresponds to a sphere packing of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic honeycomb of 4 dimensions, vertex-centered 3-spheres and cell-inscribed 3-spheres will both fit at once, forming the unique regular body-centered cubic lattice of equal-sized spheres (in any number of dimensions). Since the tesseract is radially equilateral, there is exactly enough space in the hole between the 16 vertex-centered 3-spheres for another edge-length-diameter 3-sphere. (This 4-dimensional body centered cubic lattice is actually the union of two tesseractic honeycombs, in dual positions.)
This is the same densest known regular 3-sphere packing, with kissing number 24, that is also seen in the other two regular tessellations of 4-space, the 16-cell honeycomb and the 24-cell-honeycomb. Each tesseract-inscribed 3-sphere kisses a surrounding shell of 24 3-spheres, 16 at the vertices of the tesseract and 8 inscribed in the adjacent tesseracts. These 24 kissing points are the vertices of a 24-cell of radius (and edge length) 1/2.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3,3,4}. Another form has two alternating tesseract facets (like a checkerboard) with Schläfli symbol {4,3,31,1}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {∞}4. One can be made by stericating another.
Related polytopes and tessellations
The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
C4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1
1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13
[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×6
14, 15, 16, 17
The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.
B4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]: ×1
5, 6, 7, 8
<[4,3,31,1]>:
↔[4,3,3,4]
↔
×2
9, 10, 11, 12, 13, 14,
(10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3]
↔
↔
×3
1, 2, 3, 4
[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×12
20, 21, 22, 23
The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not).
The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.
The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb.
Birectified tesseractic honeycomb
A birectified tesseractic honeycomb, , contains all rectified 16-cell (24-cell) facets and is the Voronoi tessellation of the D4* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{4}$×2, [[4,3,3,4]] symmetry, alternately colored from ${\tilde {C}}_{4}$, [4,3,3,4] symmetry, three colors from ${\tilde {B}}_{4}$, [4,3,31,1] symmetry, and 4 colors from ${\tilde {D}}_{4}$, [31,1,1,1] symmetry.
See also
Regular and uniform honeycombs in 4-space:
• 16-cell honeycomb
• 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) - Model 1
• Klitzing, Richard. "4D Euclidean tesselations". x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o *d4x, x∞o o3o3o *d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o - test - O1
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
\begin{definition}[Definition:Graham's Number]
Let $3 \uparrow 3$ denote Knuth's notation for powers:
:$3 \uparrow 3 := 3^3$
Further, let:
:$3 \uparrow \uparrow 3 := 3 \uparrow \paren {3 \uparrow 3} = 3^{\paren {3^3} }$
and so define:
:$3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_n 3 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} \paren {3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} 3}$
Thus, for example:
:$3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow \paren {3 \uparrow \uparrow \uparrow 3}$
Let:
: $n_1 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{3 \uparrow \uparrow \uparrow \uparrow 3} 3$
That is, a total of $3 \uparrow \uparrow \uparrow \uparrow 3$ instances of $\uparrow$.
Similarly, let:
:$n_2 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_1} 3$
In general for $m \ge 2$:
:$n_m := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{m - 1} } 3$
and so specifically:
:$n_{63} := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{62} } 3$
It is this $n_{63}$ which is defined as '''Graham's number'''.
{{NamedforDef|Ronald Lewis Graham|cat = Graham}}
\end{definition} | ProofWiki |
Small dodecahemicosacron
In geometry, the small dodecahemicosacron is the dual of the small dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the great dodecahemicosacron.
Small dodecahemicosacron
TypeStar polyhedron
Face—
ElementsF = 30, E = 60
V = 22 (χ = −8)
Symmetry groupIh, [5,3], *532
Index referencesDU62
dual polyhedronSmall dodecahemicosahedron
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.[1] In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
Since the small dodecahemicosahedron has ten hexagonal faces passing through the model center, it can be seen as having ten vertices at infinity.
See also
• Hemi-icosahedron - The ten vertices at infinity correspond directionally to the 10 vertices of this abstract polyhedron.
References
1. (Wenninger 2003, p. 101)
• Wenninger, Magnus (2003) [1983], Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (Page 101, Duals of the (nine) hemipolyhedra)
External links
• Weisstein, Eric W. "Small dodecahemicosacron". MathWorld.
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
| Wikipedia |
Threshold dynamics of a bacillary dysentery model with seasonal fluctuation
Zhenguo Bai and Yicang Zhou
A bacillary dysentery model with seasonal fluctuation is formulated and studied. The basic reproductive number $\mathcal {R}_0$ is introduced to investigate the disease dynamics in seasonal fluctuation environments. It is shown that there exists only the disease-free periodic solution which is globally asymptotically stable if $\mathcal {R}_0<1$, and there exists a positive periodic solution if $\mathcal {R}_0>1$. $\mathcal {R}_0$ is a threshold parameter, its magnitude determines the extinction or the persistence of the disease. Parameters in the model are estimated on the basis of bacillary dysentery epidemic data. Numerical simulations have been carried out to describe the transmission process of bacillary dysentery in China.
Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model withseasonal fluctuation. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 1-14. doi: 10.3934/dcdsb.2011.15.1.
A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity
Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli and Giuseppe Tomassetti
We consider a system of partial differential equations which describes anti-plane shear in the context of a strain-gradient theory of plasticity proposed by Gurtin in [6]. The problem couples a fully nonlinear degenerate parabolic system and an elliptic equation. It features two types of degeneracies: the first one is caused by the nonlinear structure, the second one by the dependence of the principal part on twice the curl of a planar vector field. Furthermore, the elliptic equation depends on the divergence of such vector field - which is not controlled by twice the curl - and the boundary conditions suggested in [6] are of mixed type.
To overcome the latter complications we use a suitable, time-dependent representation of a divergence-free vector field which plays the role of the elastic stress. To handle the nonlinearities, by a suitable reformulation of the problem we transform the original system into one satisfying a monotonicity property which is more "robust" than the gradient flow structure inherited as an intrinsic feature of the mechanical model. These two insights make it possible to prove existence and uniqueness of a solution to the original system.
Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Giuseppe Tomassetti. A nonlocal and fully nonlineardegenerate parabolic system from strain-gradient plasticity. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 15-43. doi: 10.3934/dcdsb.2011.15.15.
Permeation flows in cholesteric liquid crystal polymers under oscillatory shear
Zhenlu Cui and Qi Wang
We investigate the permeation flow of cholesteric liquid crystal polymers (CLCPs) subject to a small amplitude oscillatory shear using a tensor theory developed by the authors [8]. We model the material system by the Stokes hydrodynamic equations coupled with the orientational dynamics. At low frequencies, the steady permeation modes are recovered and the director rotates in phase with the applied shear. At high frequencies, the out of phase component dominates the dynamics. The asymptotic formulas for the loss modulus ($G''$) and storage modulus ($G^{'}$) are obtained at both low and high frequencies. In the low frequency limit, both the loss modulus and the storage modulus are shown to exhibit a classical frequency $\omega$ dependence ($G^{''} \propto \omega$, $G^{'} \propto \omega^2$ ) with the proportionality of order $O(Er)$ and $O(q)$, respectively, where $\frac{2\pi}{q}$ defines the pitch of the chiral liquid crystal and $Er$ is the Ericksen number of the liquid crystal polymer system. The magnitudes of dimensionless complex flow rate and complex viscosity are calculated. They are shown to have two Newtonian plateaus at low and high frequencies while a power-law response at intermediate frequencies.
Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 45-60. doi: 10.3934/dcdsb.2011.15.45.
Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays
Yoichi Enatsu, Yukihiko Nakata and Yoshiaki Muroya
In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number $R_0$ and the distributed delays do not influence the global dynamics of the model.
Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 61-74. doi: 10.3934/dcdsb.2011.15.61.
Numerical simulations of diffusion in cellular flows at high Péclet numbers
Yuliya Gorb, Dukjin Nam and Alexei Novikov
We study numerically the solutions of the steady advection-diffu-sion problem in bounded domains with prescribed boundary conditions when the Péclet number Pe is large. We approximate the solution at high, but finite Péclet numbers by the solution to a certain asymptotic problem in the limit Pe $\to \infty$. The asymptotic problem is a system of coupled 1-dimensional heat equations on the graph of streamline-separatrices of the cellular flow, that was developed in [21]. This asymptotic model is implemented numerically using a finite volume scheme with exponential grids. We conclude that the asymptotic model provides for a good approximation of the solutions of the steady advection-diffusion problem at large Péclet numbers, and even when Pe is not too large.
Yuliya Gorb, Dukjin Nam, Alexei Novikov. Numerical simulations of diffusion in cellular flows at high P\u00E9clet numbers. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 75-92. doi: 10.3934/dcdsb.2011.15.75.
Bifurcations of an SIRS epidemic model with nonlinear incidence rate
Zhixing Hu, Ping Bi, Wanbiao Ma and Shigui Ruan
The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence $\beta SI^p/(1+\alpha I^q)$. The existence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical behavior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifurcation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.
Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinearincidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 93-112. doi: 10.3934/dcdsb.2011.15.93.
Allee effects in an iteroparous host population and in host-parasitoid interactions
Sophia R.-J. Jang
We investigate a stage-structured model of an iteroparous population with two age classes. The population is assumed to exhibit Allee effects through reproduction. The asymptotic dynamics of the model depend on the maximal reproductive number of the population. The population may persist if the maximal reproductive number is greater than one. There exists a population threshold in terms of the unstable interior equilibrium. The host population will become extinct if its initial distribution lies below the threshold and the host population can persist indefinitely if its initial distribution lies above the threshold. In addition, if the unstable equilibrium is a saddle point and the system has no $2$-cycles, then the stable manifold of the saddle point provides the Allee threshold for the host. Based on this host population system, we construct a host-parasitoid model to study the impact of Allee effects upon the population interaction. The parasitoid population may drive the host to below the Allee threshold so that both populations become extinct. On the other hand, under some conditions on the parameters, the host-parasitoid system may possess an interior equilibrium and the populations may coexist as an interior equilibrium.
Sophia R.-J. Jang. Allee effects in an iteroparous host population and inhost-parasitoid interactions. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 113-135. doi: 10.3934/dcdsb.2011.15.113.
The initial layer for Rayleigh problem
Hung-Wen Kuo
Rayleigh's problem of an infinite flat plate set into uniform motion impulsively in its own plane is studied by using the BKW model, the linearized Boltzmann equation and the full Boltzmann equation, respectively. The purpose is to study the gas motion under the diffuse reflection boundary condition. For a small impulsive velocity (small Mach number) and short time, the flow behaves like a free molecule flow. Our analysis is based on certain pointwise estimates for the solution of the problem and flow velocity.
Hung-Wen Kuo. The initial layer for Rayleigh problem. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 137-170. doi: 10.3934/dcdsb.2011.15.137.
Traveling wave solutions for Lotka-Volterra system re-visited
Anthony W. Leung, Xiaojie Hou and Wei Feng
Using a new method of monotone iteration of a pair of smooth lower- and upper-solutions, the traveling wave solutions of the classical Lotka-Volterra system are shown to exist for a family of wave speeds. Such constructed upper and lower solution pair enables us to derive the explicit value of the minimal (critical) wave speed as well as the asymptotic decay/growth rates of the wave solutions at infinities. Furthermore, the traveling wave corresponding to each wave speed is unique up to a translation of the origin. The stability of the traveling wave solutions with non-critical wave speed is also studied by spectral analysis of a linearized operator in exponentially weighted Banach spaces.
Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 171-196. doi: 10.3934/dcdsb.2011.15.171.
Approximate tracking of periodic references in a class of bilinear systems via stable inversion
Josep M. Olm and Xavier Ros-Oton
This article deals with the tracking control of periodic references in single-input single-output bilinear systems using a stable inversion-based approach. Assuming solvability of the exact tracking problem and asymptotic stability of the nominal error system, the study focuses on the output behavior when the control scheme uses a periodic approximation of the nominal feedforward input signal $u_d$. The investigation shows that this results in a periodic, asymptotically stable output; moreover, a sequence of periodic control inputs $u_n$ uniformly convergent to $u_d$ produce a sequence of output responses that, in turn, converge uniformly to the output reference. It is also shown that, for a special class of bilinear systems, the internal dynamics equation can be approximately solved by an iterative procedure that provides of closed-form analytic expressions uniformly convergent to its exact solution. Then, robustness in the face of bounded parametric disturbances/uncertainties is achievable through dynamic compensation. The theoretical analysis is applied to nonminimum phase switched power converters.
Josep M. Olm, Xavier Ros-Oton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 197-215. doi: 10.3934/dcdsb.2011.15.197.
On spatiotemporal pattern formation in a diffusive bimolecular model
Rui Peng and Fengqi Yi
This paper continues the analysis on a bimolecular autocatalytic reaction-diffusion model with saturation law. An improved result of steady state bifurcation is derived and the effect of various parameters on spatiotemporal patterns is discussed. Our analysis provides a better understanding on the rich spatiotemporal patterns. Some numerical simulations are performed to support the theoretical conclusions.
Rui Peng, Fengqi Yi. On spatiotemporal patternformation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 217-230. doi: 10.3934/dcdsb.2011.15.217.
Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems
Jianhe Shen, Shuhui Chen and Kechang Lin
A semi-analytical procedure for studying stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous dynamical systems is developed. This procedure is based mainly on the incremental harmonic balance (IHB) method. It is composed of three key steps, namely, the determination of limit cycles by IHB method, the calculation of transition matrix by precise integration (PI) algorithm and the discrimination of limit cycle stability by Floquet theory. As an application, the procedure is used to investigate the dynamics of the limit cycle of a three-dimensional nonlinear autonomous system. The symmetry-breaking bifurcation, the first and the second period-doubling bifurcations of the limit cycle are identified. The critical parameter values corresponding to these bifurcations are calculated. The phase portraits and bifurcation points agree well with those of direct numerical integrations by using Runge-Kutta method.
Jianhe Shen, Shuhui Chen, Kechang Lin. Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 231-254. doi: 10.3934/dcdsb.2011.15.231.
Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition
Li-Li Wan and Chun-Lei Tang
The existence and multiplicity of homoclinic orbits for a class of the second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \forall t \in \mathbb{R}$, are obtained via the concentration-compactness principle and the fountain theorem respectively, where $W(t, x)$ is superquadratic and need not satisfy the (AR) condition with respect to the second variable $ x\in\mathbb{R}^{N}$.
Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (<i>AR<\/i>) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 255-271. doi: 10.3934/dcdsb.2011.15.255.
Global convergence of a predator-prey model with stage structure and spatio-temporal delay
Rui Xu
In this paper, a predator-prey model with stage structure for the predator and a spatio-temporal delay describing the gestation period of the predator under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states is established. Sufficient conditions are derived for the global attractiveness of the positive steady state and the global stability of the semi-trivial steady state of the proposed problem by using the method of upper-lower solutions and its associated monotone iteration scheme. Numerical simulations are carried out to illustrate the main results.
Rui Xu. Global convergence of a predator-prey model with stage structure and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 273-291. doi: 10.3934/dcdsb.2011.15.273.
Analysis of a delayed free boundary problem for tumor growth
Shihe Xu
In this paper we study a delayed free boundary problem for the growth of tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the process of proliferation is delayed compared to apoptosis. By $L^p$ theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the asymptotic behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a dormant state as $t\rightarrow\infty.$
Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 293-308. doi: 10.3934/dcdsb.2011.15.293.
Traveling waves for models of phase transitions of solids driven by configurational forces
Shuichi Kawashima and Peicheng Zhu
This article is concerned with the existence of traveling wave solutions, including standing waves, to some models based on configurational forces, describing respectively the diffusionless phase transitions of solid materials, e.g., Steel, and phase transitions due to interface motion by interface diffusion, e.g., Sintering. These models were proposed by Alber and Zhu in [3]. We consider both the order-parameter-conserved case and the non-conserved one, under suitable assumptions. Also we compare our results with the corresponding ones for the Allen-Cahn and the Cahn-Hilliard equations coupled with linear elasticity, which are models for diffusion-dominated phase transitions in elastic solids.
Shuichi Kawashima, Peicheng Zhu. Traveling waves for models of phase transitions of solids driven byconfigurational forces. Discrete & Continuous Dynamical Systems - B, 2011, 15(1): 309-323. doi: 10.3934/dcdsb.2011.15.309. | CommonCrawl |
\begin{definition}[Definition:Diagonal Relation]
Let $S$ be a set.
The '''diagonal relation on $S$''' is a relation $\Delta_S$ on $S$ such that:
:$\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$
Alternatively:
:$\Delta_S = \set {\tuple {x, y}: x, y \in S: x = y}$
\end{definition} | ProofWiki |
Technical feasibility of decrypting https by replacing the computer's PRNG
Intel has an on-chip RdRand function which supposedly bypasses the normally used entropy pool for /dev/urandom and directly injects output. Now rumors are going on that Intel works together with the NSA... and knowing that PRNGs are important for cryptography is enough to get this news spreading.
I personally don't believe this is true, so this is entirely hypothetical: Let's assume that indeed RdRand does what news says it does and that it indeed outputs randomness into a place where applications and libraries would look for cryptographically secure randomness.
How feasible is it that the chip's manufacturer can predict the output of this PRNG when it passed tests from the people applying the use of this RdRand instruction in kernels?
If the chip's manufacturer can predict the output of the PRNG to some extent, how feasible is it that they can decrypt any https traffic between two systems using their chips? (Or anything else requiring randomness, https is only an example.)
My reason for asking: http://cryptome.org/2013/07/intel-bed-nsa.htm
As said, I don't believe everything written here, but I find it very interesting to discuss the possibility technically.
tls randomness pseudo-random-generator cryptographic-hardware backdoors
LucLuc
$\begingroup$ I just removed all comments because they were not related to the topic of the question, or about cryptography at all. Please constrain yourself to clarifications and similar comments about the question. $\endgroup$ – Paŭlo Ebermann Jul 15 '13 at 18:33
$\begingroup$ This question appears to be off-topic because it is about Hardware backdooring of a system. Probably belongs elsewhere. $\endgroup$ – minar Jul 18 '13 at 6:13
$\begingroup$ Steve Blank thinks it's feasible, in fact, it sounds like he'd be surprised if there weren't a back door: steveblank.com/2013/07/15/… $\endgroup$ – Kinnard Hockenhull Jul 23 '13 at 3:00
1 - How feasible is it that the chip's manufacturer can predict the output of this PRNG when it passed tests from the people applying the use of this RdRand instruction in kernels?
A strong stream cipher's output is random and unpredictable to anyone not knowing the key. See where this is heading? Just because something looks random doesn't mean it's random.
2 - If the chip's manufacturer can predict the output of the PRNG to some extent, how feasible is it that they can decrypt any https traffic between two systems using their chips? (Or anything else requiring randomness, https is only an example.)
If you can predict the PRNG you can basically predict the secrets used for key exchange, and from that deduce the shared secret. Then you can simply decrypt the communication.
orlporlp
$\begingroup$ I think it wouldn't be that hard for hardware experts to reverse-engineer the RdRand algorithm so as to establish whether it is a legitimate TRNG or is doing something strange like generating some kind of keystream to introduce a backdoor (publishing their research, of course). Though again, as said in the question's link, entropy pool poisoning is rather difficult to exploit. $\endgroup$ – Thomas Jul 14 '13 at 4:29
$\begingroup$ @Thomas I heard that Linux uses RdRand directly in some places instead of just mixing it into the pool. In that case you don't need to poison the entropy pool. $\endgroup$ – CodesInChaos Jul 14 '13 at 8:53
$\begingroup$ Hmm I think I get it. I was thinking "how'd you make it seem random while knowing the output" and an encryption algorithm is the obvious answer. But then how do you know how far into the output stream the PRNG is? Well you don't really need to, deducing that is much faster (just reproduce the output stream and test everything) than cracking true randomness. Use the chip's serial + a static salt as first input, then make the PRNG's state persistent between reboots. That could probably totally work. Waiting for other answers, but I'll accept soon if none are added. Thanks for your answer! $\endgroup$ – Luc Jul 14 '13 at 13:16
$\begingroup$ Can this be tested? Is there a way to know if this is true? Can some test suite be written which can be run on our machines to test if RdRand function is doing something nefarious? $\endgroup$ – notthetup Jul 14 '13 at 15:06
$\begingroup$ @notthetup Yes and no. It might be possible that hardware experts are capable of opening the chip and study the circuit and deduce from that what RdRand does, but this is extremely hard to do due to the small scale (not to mention expensive). But if we assume Intel does have nefarious purposes with RdRand and built it around a cipher of which they know the key, then no we can't detect it from a test suite, if the used cipher is strong. A strong cipher is explicitly designed to be indistinguishable from random noise - in fact, it's considered to be broken if it is. $\endgroup$ – orlp Jul 14 '13 at 16:51
Have you heard of the strange story of Dual_EC_DRBG? A random number generator suggested and endorsed by the government that exhibits some very suspicious properties.
http://www.schneier.com/blog/archives/2007/11/the_strange_sto.html
From that article:
This is how it works: There are a bunch of constants -- fixed numbers -- in the standard used to define the algorithm's elliptic curve. These constants are listed in Appendix A of the NIST publication, but nowhere is it explained where they came from.
What Shumow and Ferguson showed is that these numbers have a relationship with a second, secret set of numbers that can act as a kind of skeleton key. If you know the secret numbers, you can predict the output of the random-number generator after collecting just 32 bytes of its output. To put that in real terms, you only need to monitor one TLS internet encryption connection in order to crack the security of that protocol. If you know the secret numbers, you can completely break any instantiation of Dual_EC_DRBG.
So the short answer is yes, it is possible to create a random number generating algorithm that has exploitable weaknesses, in particular weakness that only the creator of the algorithm may be able to exploit.
CodesInChaos
Joshua KoldenJoshua Kolden
$\begingroup$ This attack reminds me of the attack on the Netscape browser's PRNG back in the 1990s: cs.berkeley.edu/~daw/papers/ddj-netscape.html Today, people better understand the need for a secure PRNG, so it should be better tested. But all it would take is some espionage and skullduggery, and a weak algorithm could be dropped into the chip after testing. Unless a production chip is retested, it would go unnoticed. It's like the old joke: how do you really know when a random number generator is broken? $\endgroup$ – John Deters Jul 15 '13 at 22:03
$\begingroup$ Also it's worth noting that a similar method has the added property that even the chip manufacture would not necessarily know they where introducing a back door. In their eyes they are just using a government approved algorithm, and would naturally assume it is secure. $\endgroup$ – Joshua Kolden Jul 16 '13 at 2:10
$\begingroup$ Dual_EC_DRBG wasn't exploitable because of some weakness only the author knew how to exploit. In fact, as soon as it was released people concluded that its very design was such that kleptography was possible. $\endgroup$ – forest Jan 9 '19 at 11:21
As nightcracker correctly stated, any strong cryptographic PRNG will produce a stream of numbers that pass statistical tests.
However, the manufacturer has some constraints:
Independent tests will be performed on multiple processors that are set up in an identical manner, so each processor must produce different outputs.
Any given processor must produce a different output stream on each power up.
A simple scheme would be to use the processor serial number as an input to the PRNG to ensure different processors had different outputs, and have an undisclosed non-volatile register (e.g. a power-on counter) to ensure each boot was different.
A scheme such as this would probably resist any attempts at analysis using only its outputs: a standard cryptographic PRNG with a global secret (common across all processors), processor ID and power on counter as inputs. At this point, a large scale surveillance infrastructure on observing a new user would only have a space of a few millions of possible processor IDs, plus a few hundreds or thousands of possible boot counts. This could all be easily precomputed too, so would be very practical to hook into a surveillance infrastructure with today's computing power. (Once a user's processor ID and boot count have been identified once, it is of course much easier to keep track of this than have to do a full search each time).
However, the odds are that Intel aren't betting their international sales solely on another fab not having the inclination to open up their chip and check this (e.g. ARM would have a strong incentive to identify such foul play). Update: but they could be compelled by the government to put such a back door in whether it is in their commercial interests or not. Update 2: They, or their fab, could also use stealthy dopant-level modifications to make it extremely hard to detect the modifications, even by someone with Intel-like capabilities (see the first case study, Chapter 3 in the referenced paper).
I'm not an expert in microprocessor hardware, so can't comment on techniques that might introduce biases or other predictable features without being detected. One possible backdoor might be to severely constrain the next requested output from RdRand only after performing a computation such as would be needed to verify the authenticity of a certificate signed by one of a set of long-lived root CAs (perhaps China's CNNIC would be a useful candidate?).
Being able to predict that the output of RdRand is within a searchable subset of possible outputs doesn't alone mean an attacker could break the system - it depends how that output is used. For example if the consuming application uses it as just another optional input to its entropy pool, then being able to predict that input means the user is no better than without RdRand, but equally is not worse off.
CodesInChaos points out that Linux has used RdRand directly at times; Intel are also encouraging direct use of the instruction. So it is not unreasonable to imagine a browser or other TLS client that uses output from RdRand as its sole source of entropy. If this is the case then an observer who can predict the output from RdRand can indeed compromise your security.
Most cryptosystems fail if the entropy input can be predicted, including SSL/TLS.
To pick a couple of examples in use by popular websites from the many possible TLS key exchange options:
My TLS connection to gmail currently uses Ephemeral Elliptic Curve Diffie-Hellman (ECDHE; I believe this is Google's default these days if your browser supports it). If an observer can enumerate the possible random numbers used by my browser, then the observer knows my secret key $d$, so can compute the shared secret $x_k$ by calculating $x_k = dQ$, where Q is the Google server's ephemeral public key (and vice versa if the observer can predict Google's secret key). $x_k$ is used as the premaster secret - the secret from which all other secrets are derived, so obtaining it breaks all of the assurances that TLS aims to provide.
Your TLS connection to Wikipedia uses an RSA based key exchange, as that's all they support (e.g. mine is currently TLS_RSA_WITH_RC4_128_SHA). With these ciphersuites, the premaster secret is generated by the client using its random number generator, and sent to the server. Being able to predict the random number directly gives an attacker the secrets they need.
MichaelMichael
$\begingroup$ Useful information, thanks for your answer! And you mention a very legitimate question: What's in it for Intel besides becoming good mates with the NSA? Merely being good mates doesn't bring them much profit. Yet they could still say it's secure crypto, and it is as long as nobody knows the initial state. They might just get away with it in the corporate world. But I'm just guessing really. $\endgroup$ – Luc Jul 14 '13 at 22:54
$\begingroup$ @Luc, can you say "government contracts"? Sure, I knew you could. $\endgroup$ – John Deters Jul 15 '13 at 22:37
I am the designer of the random number generator that is behind the Intel RdRand instruction.
It isn't. We cannot. It passes the tests because it is a cryptographically secure random number generator being fed by a 2.5Gbps TRNG source.
The manufacturer would not intercept traffic at this point. The plaintext is present on the system. The attacker would attack the place in the system that the plaintext resides. E.G. in the network stack where the encryption/decryption of the link cipher takes place, or in the key establishment code. This is more the realm of traditional software vulnerability attacks. There's no need to pull off the very difficult task of injecting known non-random numbers into an RNG and trying to make sure it gets used on the right cycle by the right instruction in the right bit of the software such that you can reverse engineer the key.
David JohnstonDavid Johnston
$\begingroup$ Just playing devils advocate: A cryptographically secure RNG being fed about no input (or the current timestamp + device serial number or such) would also pass the same tests, wouldn't it? $\endgroup$ – Paŭlo Ebermann Sep 10 '13 at 19:19
$\begingroup$ @PaŭloEbermann Yep, it would. A backdoored RNG would be as simple as combining a very weak (say 32-bit) random value with a master key. Without knowledge of the master key, the output seems completely random and is unbreakable. With knowledge of the master key, the output has a keyspace of only $2^{32}$. There would be no way to tell based on the output only if that were the case. $\endgroup$ – forest Dec 4 '18 at 11:21
There are two ways I can see for the RNG to be cooked. (For the record, I don't see any reason at all to suspect this of Intel, but I also think prudent cryptographic design requires us to think through what would happen if our RNG were flawed or backdoored.)
First, your RNG could not have enough entropy. That's what got the Netscape RNG many years ago, and also what is apparently behind all those RSA keys with shared prime factors that Lenstra et al and Henninger et al found a couple years ago. And it's what got the Taiwanese smart cards that have been in the news more recently. If you don't get enough entropy, then you will not get any security. You can imagine the Intel RNG having some kind of flaw or intentional weakness where it never gets more than, say, 40 bits of entropy, and then generates outputs using CTR-DRBG with AES (from SP 900-90).
Second, your RNG could have an actual trapdoor or an unintentional weakness. That's what's alleged to have with the Dual EC DRBG in SP 800-90. If an attacker knows the relationship between P and Q, and sees one full DRBG output, he can recover the future state of the DRBG and predict every future output. You can imagine something like this--the Intel RNG uses CTR-DRBG, but if it wired all but 40 bits of the key to some known values, then the outputs would pass statistical tests, but would be weak to an attacker who knew those bits.
In either case, if you used the RNG outputs directly, you would be vulnerable. In the first case, anything you do with the RNG outputs that doesn't add in some other entropy will be vulnerable, since the attacker can just guess the entropy and then generate everything himself. In the second case, you'd need to give the attacker a little output to run his attack on, but (depending on details of the backdoor) a couple IVs for the encryption algorithm might be enough to leak the secret.
The best way to avoid both of these potential attacks is to combine the Intel RNG outputs with OS-collected entropy. There are several ways to do this, but I think the best one is something you can find in SP 800-90--you can seed an RNG, and then keep generating outputs with prediction resistance.
a. Use /dev/random to get at least 128 bits of entropy, concatenate that with a few outputs from RDRAND, and use the result to seed a software instance of CTR-DRBG using AES.
b. Each time you get a request to generate some bits of output, you do the following:
(i) Get an output from RDRAND.
(ii) Reseed the CTR-DRBG instance using that output as the new seed material. (The previous state's entropy is preserved by reseeding, so this can't weaken what you already had.)
(iii)Generate your outputs.
If /dev/random gives you 128 bits of entropy and CTR-DRBG is secure, then you could let your attacker choose every value you get from RDRAND, and he couldn't make your random numbers any less secure.
If RDRAND is good, then the attacker could choose the bits you get from /dev/random and your random numbers would still be secure.
Alternatively, you could simply XOR RDRAND outputs with /dev/urandom outputs. It's easy to see that this can't be any weaker than the stronger of the two, as long as neither one is able to predict the other's values. That's discussed in the draft SP 800-90C.
Disclaimer: I'm one of the authors of the 800-90 standards, and the designer of CTR-DRBG.
$\begingroup$ "you could let your attacker choose every value you get from RDRAND, and he couldn't make your random numbers any less secure" is not true because the CPU can see your /dev/random pool. See DJB's blog.cr.yp.to/20140205-entropy.html $\endgroup$ – Navin Mar 14 '17 at 18:24
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Attack of the Middle Square Weyl Sequence PRNG | CommonCrawl |
\begin{document}
\title{Derivational modal logics with the difference modality}
\abstract{In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, $T_1$-spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, $\mathbf R^n~ (n\ge 2)$. We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.}
\section{Introduction} \label{sec:intro} Topological modal logic was initiated by the works of A. Tarski and J.C.C. McKinsey in the 1940s. They were first to consider both topological interpretations of the diamond modality: one as closure, and another as derivative.
Their studies of closure modal logics were rather detailed and profound. In particular, in the fundamental paper \cite{MT} they have shown that the logic of any metric separable dense-in-itself space is ${\bf S4}$. This remarkable result also demonstrates a relative weakness of the closure operator to distinguish between interesting topological properties.
The derivational interpretation gives more expressive power. For example, the real line can be distinguished from the real plane (the observation made by K. Kuratowski as early as in 1920s, cf. \cite{K22}); the real line can be distinguished from the rational line \cite{Sh90}; $T_0$ and $T_D$ separation axioms become expressible \cite{BEG}, \cite{E1}. However, in \cite{MT} McKinsey and Tarski only gave basic definitions for derivational modal logics and put several problems that were solved much later.
The derivational semantics also has its limitations (for example, it is still impossible to distinguish ${\bf R}^2$ from ${\bf R}^3$). Further increase of expressive power can be provided by the well-known methods of adding universal or difference modalities \cite{GorPas}, \cite{GarGor}. In the context of topological semantics this approach also has proved fruitful --- for example, connectedness is expressible in modal logic with the closure and the universal modality \cite{Sh99}, and the $T_1$ separation axiom in modal logic with the closure and the difference modality \cite{kudinov_2005}.
Until the early 1990s, when the connections between topological modal logic and Computer Science were established, the interest in that subject was moderate. Leo Esakia was one of the enthusiasts of modal logical approach to topology, and he was probably the first to appreciate the role of the derivational modality, in particular, in modal logics of provability \cite{E}. Another strong motivation for further studies of derivational modal logics (`d-logics') were the axiomatization problems left open in \cite{MT}.\footnote{The early works of the second author in this field were greatly influenced by Leo Esakia.} In recent years d-logics have been studied rather intensively, a brief summary of results can be found in section 3 below.
In this chapter the first thorough investigation is provided for logics in the most expressive language in this context\footnote{Some other kinds of topomodal logics arise when we deal with topological spaces with additional structures, e.g. spaces with two topologies, spaces with a homeomorphism etc. (cf. \cite{H}).}, namely the derivational modal logics with the difference modality (`dd-logics'). It unifies earlier studies by the first author in closure modal logics with the difference modality (`cd-logics') and by the second author in d-logics.
The diagram in section 12 compares the expressive power of different kinds of topomodal logics. Our conjecture is that dd-logics are strictly more expressive than the others, but it is still an open question if the dd-language is stronger than the cd-language. Speaking informally, it is more convenient --- for example, the Kuratowski's axiom for ${\bf R}^2$ (Definition 9.1) is expressible in cd-logic as well, but in a more complicated form \cite{kudinov_2006}.
We show that still in many cases properties of dd-logics are similar to those of d-logics: finite axiomatizability, decidability and the finite model property (fmp). Besides specific results characterizing logics of some particular spaces, our goal was to propose some general methods. In fact, nowadays in topomodal logic there are many technical proofs, but few general methods. In this chapter we propose only two simplifying novelties --- dd-morphisms (section 6) and the Glueing lemma \ref{Glue}, but we hope that much more can be done in this direction, cf. the recent paper \cite{Hodkinson}.
In more detail, the plan of the chapter is as follows. Preliminary sections 2--4 include standard definitions and basic facts about modal logics and their semantics. Some general completeness results for dd-logics can be found in sections 5, 7. In section 5 we show that every extension of the minimal logic $\lgc{K4^\circ D^+}$ by variable-free axioms is topologically complete. In section 8 we prove the same for extensions of $\lgc{DT_1}$ (the logic of dense-in-themselves $T_1$-spaces); the proof is based on a construction of d-morphisms from the recent paper \cite{BLB}.
In section 6 we consider validity-preserving maps from topological to Kripke frames (d-morphisms and dd-morphisms) and prove a modified version of McKinsey--Tarski's lemma on dissectable spaces. In section 7 we prove that $\lgc{DT_1}$ is complete w.r.t. an arbitrary zero-dimensional dense-in-itself separable metric space by the method from \cite{Sh90}, \cite{Sh00}.
Sections 8--10 study the axiom of connectedness $AC$ and Kuratowski's axiom $Ku$ related to local 1-componency. In particular we prove that the logic $\lgc{DT_1CK}$ with both these axioms has the fmp. This is a refinement of an earlier result \cite{Sh90}, \cite{Sh00} on the fmp of the d-logic $\lgc{D4}+Ku$ (the new proof uses a simpler construction).
Section 11 contains our central result: $\lgc{DT_1CK}$ is the dd-logic of ${\bf R}^n$ for $n>1$. The proof uses an inductive construction of dd-morphisms onto finite frames of the corresponding logic, and it combines methods from \cite{Sh90}, \cite{Sh00}, \cite{kudinov_2006}, with an essential improvement motivated by \cite{LB2} and based on the Glueing lemma.
The final section discusses some further directions and open questions. The Appendix contains technical details of some proofs.
\section{Basic notions} \label{sec:bas} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})}
The material of this section is quite standard, and most of it can be found in \cite{CZ97}. We consider {\em $n$-modal (propositional) formulas} constructed from a countable set of propositional variables $PV$ and the connectives $\bot$, $\rightarrow$, $\square_1,\ldots,\square_n$. The derived connectives are $\wedge,\;\vee,\;\neg,\;\top,\;{\leftrightarrow},\;\Diamond_1,\ldots,\Diamond_n$. A formula without occurrences of propositional variables is called {\em closed}.
A {\em (normal) $n$-modal logic } is a set of modal formulas containing the classical tautologies, the axioms $\square_i(p\rightarrow q)\rightarrow (\square_i p \rightarrow\square_i q)$ and closed under the standard inference rules: Modus Ponens ($A,~A\rightarrow B/B$), Necessitation ($A/\square_i A$), and Substitution $(A(p_j)/A(B))$.
To be more specific, we use the terms `($\square_1,\ldots,\square_n$)-modal formula' and `($\square_1,\ldots,\square_n$)-modal logic'.
${\bf K}_n$ denotes the minimal $n$-modal logic (and ${\bf K}={\bf K}_1$). An $n$-modal logic containing a certain $n$-modal logic $\mathbf{\Lambda}$ is called an \emph{extension} of $\mathbf{\Lambda}$, or a \emph{$\mathbf{\Lambda}$-logic}. The minimal $\mathbf{\Lambda}$-logic containing a set of $n$-modal formulas $\Gamma$ is denoted by $\mathbf{\Lambda}+\Gamma$. In particular, $${\bf K4}:={\bf K}+\square p\rightarrow\square\square p,~ {\bf S4}:={\bf K4}+\square p\rightarrow p, ~{\bf D4}:={\bf K4}+\Diamond\top, $$ $$ {\bf K4}^\circ := {\bf wK4}:= {\bf K} + p\wedge \square p\rightarrow \square \square p. $$
The {\em fusion} $L_1*L_2$ of modal logics $L_1,~L_2$ with distinct modalities is the smallest modal logic in the joined language containing $L_1\cup L_2$.
A {\em (normal) $n$-modal algebra} is a Boolean algebra with extra $n$ unary operations preserving ${\bf 1}$ (the unit) and distributing over $\cap$; they are often denoted by $\square_1,\ldots,\square_n$, in the same way as the modal connectives. A {\em valuation} in a modal algebra $\mathfrak{A}$ is a set-theoretic map $\theta: PV \longrightarrow \mathfrak{A}$. It extends to all $n$-modal formulas by induction: $$\theta(\bot)={\varnothing},~\theta(A\rightarrow B)=-\theta(A)\cup\theta(B),~ \theta(\square_i A)=\square_i\theta(A).$$ A formula $A$ is {\em true in} $\mathfrak{A}$ (in symbols: $\mathfrak{A}\vDash A$) if $\theta(A)=\textbf{1}$ for any valuation $\theta$. The set ${{\bf L}}(\mathfrak{A})$ of all $n$-modal formulas true in an $n$-modal algebra $\mathfrak{A}$ is an $n$-modal logic called the {\em logic of $\mathfrak{A}$}.
An $n$-modal {\em Kripke frame } is a tuple $ F=(W,R_1,\ldots,R_n)$, where $W$ is a nonempty set (of worlds), $R_i$ are binary relations on $W$. We often write $x\in F$ instead of $x\in W$. In this chapter (except for Section 2) all 1-modal frames are assumed to be transitive. The associated $n$-modal algebra $MA(F)$ is $2^W$ (the Boolean algebra of all subsets of $W$) with the operations $\square_1,\ldots,\square_n$ such that $\square_i V= \{x\mid R_i(x)\subseteq V\}$ for any $V\subseteq W$.
A {\em valuation} in $F$ is the same as in $MA(F)$, i.e., this is a map from $PV$ to ${\cal P}(W)$ (the power set of $W$). A {\em (Kripke) model} over $F$ is a pair $M=(F,\theta)$, where $\theta$ is a valuation in $F$. The notation $M,x\vDash A$ means $x\in\theta(A)$, which is also read as `$A$ is \emph{true in $M$ at} $x$'. A (modal) formula $A$ is {\em true in} $M$ (in symbols: $M\vDash A$) if $A$ is true in $M$ at all worlds. A formula $A$ is called {\em valid in} a Kripke frame $F$ (in symbols: $F\vDash A$) if $A$ is true in all Kripke models over $F$; this is obviously equivalent to $MA(F)\vDash A$.
The {\em modal logic} ${{\bf L}}(F)$ of a Kripke frame $F$ is the set of all modal formulas valid in $F$, i.e., ${{\bf L}}(MA(F))$. For a class of $n$-modal frames ${\cal C}$, the {\em modal logic} of ${\cal C}$ (or \emph{the modal logic determined} by ${\cal C}$) is ${{\bf L}}({\cal C}):=\bigcap\{{{\bf L}}(F)\mid F\in{\cal C}\}$. Logics determined by classes of Kripke frames are called {\em Kripke complete}. An $n$-modal frame validating an $n$-modal logic $\mathbf{\Lambda}$ is called a {\em $\mathbf{\Lambda}$-frame}. A modal logic has the {\em finite model property (fmp)} if it is determined by some class of finite frames.
It is well known that $(W,R)\vDash{\bf K4}$ iff $R$ is transitive; $(W,R)\vDash{\bf S4}$ iff $R$ is reflexive transitive (a {\em quasi-order}).
A {\em cluster} in a transitive frame $(W,R)$ is an equivalence class under the relation $\sim_R:=(R \cap R^{-1}) \cup I_W$, where $I_W$ is the equality relation on $W$. A {\em degenerate cluster} is an irreflexive singleton. A cluster that is a reflexive singleton, is called {\em trivial}, or {\em simple}. A {\em chain} is a frame $(W,R)$ with $R$ transitive, antisymmetric and linear, i.e., it satisfies $\forall x\forall y~ (xRy\vee yRx\vee x=y)$. A point $x\in W$ is {\em strictly (R-)minimal} if $R^{-1}(x)={\varnothing}$.
A {\em subframe} of a frame $F=(W, R_1,\ldots,R_n)$ obtained by restriction to $V\subseteq W$, is
$F|V:=(V,R_1|V,\ldots,R_n|V)$. Then for any Kripke model $M=(F,\theta)$ we have a {\em submodel}
$M|V:=(F|V,\theta|V)$, where $(\theta|V)(q):=\theta(q)\cap V$ for each $q\in PV$. If $R_i(V)\subseteq V$ for any
$i$, the subframe $F|V$ and the submodel $M|V$ are called {\em generated}.
The {\em union} of subframes $F_j=F|W_j,~j\in J$ is the subframe
$\bigcup\limits_{j\in J}F_j:=F|\bigcup\limits_{j\in J}W_j$.
A {\em generated subframe (cone) with the root $x$} is
$F^x:=F|R^*(x)$, where $R^*$ is the reflexive transitive closure of $R_1\cup\ldots\cup R_n$; so for a transitive frame $(W,R)$,~$R^*=R\cup I_W$ is the reflexive closure of $R$ (which is also denoted by $\overline{R}$). A frame $F$ is called {\em rooted} with the root $u$ if $F=F^u$. Similarly we define a cone $M^x$ of a Kripke model $M$.
Every finite rooted transitive frame $F=(W,R)$ can be presented as the union $(F|C)\cup F^{x_1}\cup\ldots\cup F^{x_m}$ ($m\geq 0$), where $C$ is the root cluster, $x_i$ are its successors (i.e., $x_i\not\in C,~\overline{R}^{-1}(x_i)=\sim_R(x_i)\cup C$). If $C$ is non-degenerate, the frame
$F|C$ is $(C,C^2)$, which we usually denote just by $C$. If $C=\{a\}$ is degenerate, $ F|C$ is $(\{a\},{\varnothing})$, which we denote by $\breve{a}$.
Let us fix the propositional language (and the number $n$) until the end of this section. \begin{lem}\label{gl} {\em (Generation Lemma)} \begin{enumerate} \item ${{\bf L}}(F)=\bigcap\{{{\bf L}}(F^x) \mid x\in F\}$. \item If $F$ is a generated subframe of $G$, then ${{\bf L}}(G)\subseteq {{\bf L}}(F)$. \item If $M$ is a generated submodel of $N$, then for any formula $A$ for any $x$ in $M$ \[ N,x\vDash A\;\mbox{iff}\; M,x\vDash A. \] \end{enumerate} \end{lem}
\begin {lem}\label{root} For any Kripke complete modal logic $\mathbf{\Lambda}$, \[ \mathbf{\Lambda}={{\bf L}}(\mbox{all }\mathbf{\Lambda}\mbox{-frames})={{\bf L}}(\mbox{all rooted }\mathbf{\Lambda}\mbox{-frames}). \] \end{lem}
A {\em p-morphism } from a frame $(W,R_1,\ldots,R_n)$ onto a frame $ (W^\prime,R_1^\prime,\ldots,R_n^\prime)$ is a surjective map $f:W\longrightarrow W^\prime $ satisfying the following conditions (for any $i$): \begin{enumerate} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})} \item $\forall x\forall y~ (xR_i y\Rightarrow f(x) R_i'f(y))$ (monotonicity); \item $\forall x\forall z~ (f(x)R_i' z\Rightarrow\exists y
(f(y) = z ~\&~ xR_iy))$ (the lift property). \end{enumerate} If $xR_iy$ and $f(x)R_i'f(y)$, we say that $xR_iy$ {\em lifts} $f(x)R_i'f(y)$.
Note that (1)$~\&~$(2) is equivalent to $$\forall x~f(R_i(x))=R_i'(f(x)).$$ $f:\; F\twoheadrightarrow F'$ denotes that $f$ is a p-morphism from $F$ onto $F^\prime $.
Every set-theoretic map $f:W\longrightarrow W'$ gives rise to the dual morphism of Boolean algebras $2^f:2^{W'}\longrightarrow 2^W$ sending every subset $V\subseteq W'$ to its inverse image $f^{-1}(V)\subseteq W$.
\begin {lem}\label{p1} {\em (P-morphism Lemma)}~ \renewcommand{(\arabic{enumi})}{(\arabic{enumi})} \begin{enumerate} \item $f:\; F\twoheadrightarrow F'$ iff $2^f$ is an embedding of $MA(F')$ in $MA(F)$. \item $f:\; F\twoheadrightarrow F'$ implies ${\text{\bf {\em L}}}(F)\subseteq{\text{\bf {\em L}}}(F').$ \item If $f:\; F\twoheadrightarrow F'$, then $F\vDash A\Leftrightarrow F'\vDash A$ for any closed formula $A$. \end{enumerate} \end{lem}
In proofs of the fmp in this chapter we will use the well-known filtration method \cite{CZ97}. Let us recall the construction we need.
Let $\Psi$ be a set of modal formulas closed under subformulas. For a Kripke model $M = (F,\varphi )$ over a frame $F = (W,R_1,\ldots,R_n)$, there is the equivalence relation on $W$ $$ x \equiv_\Psi y \Longleftrightarrow \forall A\in\Psi (M,x \vDash A \Leftrightarrow M,y \vDash A).$$ Put $W^{\prime}:=W/\equiv_\Psi;~~x^\sim\,:=\; \equiv_\Psi(x)$ (the equivalence class of $x$),\\ $\varphi^\prime (q) :=\{x^\sim\mid x\in \varphi (q)\}$ for $q \in PV\cap\Psi$ (and let $\varphi'(q)$ be arbitrary for $q \in PV-\Psi$).
\begin{lem}\label{L26}{\em(Filtration Lemma)} Under the above assumptions, consider the relations ${\underline{R}}_i, R'_i $ on $W^\prime$ such that $$a{\underline{R}}_ib \hbox{ iff } \exists x\in a~ \exists y\in b~ xR_iy ,$$ \[ R'_i= \begin{cases} \mbox{the transitive closure of }{\underline{R}}_i & \mbox{ if }R_i\mbox{ is transitive,}\nonumber\\ {\underline{R}}_i & \mbox{otherwise.}\nonumber
\end{cases} \] Put $M^\prime := (W^\prime, R'_1,\ldots,R'_n, \varphi^\prime)$. Then for any $x\in W, ~A\in\Psi $ : $$M,x \vDash A\mbox{ iff }M',x^\sim \vDash A.$$ \end{lem}
\begin{defi} An {\em $m$-formula} is a modal formula in propositional variables $\{ p_1,\dots,p_m\}$. For a modal logic $\mathbf{\Lambda}$ we define the {\em $m$-weak} (or {\em $m$-restricted) canonical frame $F_{\mathbf{\Lambda}\lceil m} := (W, R_1,\ldots, R_n)$ and canonical model} $M_{\mathbf{\Lambda}\lceil m}:= (F_{\mathbf{\Lambda}\lceil m}, \varphi)$, where $W$ is the set of all maximal $\mathbf{\Lambda}$-consistent sets of $m$-formulas, $
x R_i y \mbox{ iff for any }m\mbox{-formula } A$\\ $(\square_i A\in x \Rightarrow A\in y), $ \[ \varphi(p_i): = \begin{cases} \{ x\mid p_i\in x\}& \mbox{ if }i\leq m,\nonumber\\ {\varnothing}& \mbox{ if }i> m. \nonumber
\end{cases} \] $\mathbf{\Lambda}$ is called {\em weakly canonical} if $ F_{\mathbf{\Lambda}\lceil m}\vDash\mathbf{\Lambda}$ for any finite $m$. \end{defi}
\begin{propo}\label{cm} For any $m$-formula $A$ and a modal logic $\mathbf{\Lambda}$ \begin{enumerate} \item $M_{\mathbf{\Lambda}\lceil m},x \vDash A \mbox{ iff }A\in x$; \item $M_{\mathbf{\Lambda}\lceil m}\vDash A \mbox{ iff }A\in\mathbf{\Lambda}$; \item if $\mathbf{\Lambda}$ is weakly canonical, then it is Kripke complete. \end{enumerate} \end{propo}
\begin{coro}\label{cm1} If for any $m$-formula $A$, $M_{\mathbf{\Lambda}\lceil m},x\vDash A{\Leftrightarrow} M_{\mathbf{\Lambda}\lceil m},y\vDash A$, then $x=y$. \end{coro}
\begin{defi} A cluster $C$ in a transitive frame $(W,R)$ is called \emph{maximal} if \mbox{$\overline{R}(C) = C$.} \end{defi}
\begin{lem}\label{L82}\label{L84} Let $ F_{\mathbf{\Lambda}\lceil m} =(W,R_1,\ldots,R_n)$ and suppose $\mathbf{\Lambda}\vdash\square_1p\rightarrow\square_1\square_1p $ (i.e., $R_1$ is transitive). Then every generated subframe of $(W,R_1)$ contains a maximal cluster. \end{lem}
The proof is based on the fact that the general Kripke frame corresponding to a canonical model is descriptive; cf. \cite{CZ97}, \cite{F85} for further details\footnote{For the 1-modal case this lemma has been known as folklore since the 1970s; the second author learned it from Leo Esakia in 1975.}.
\section{Derivational modal logics} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})} We denote topological spaces by $\mathfrak{X}, \mathfrak{Y},\ldots$ and the corresponding sets by $X,Y,\dots$.\footnote{Sometimes we neglect this difference.} The interior operation in a space $\mathfrak X$ is denoted by ${\bf I}_X$ and the closure operation by ${\bf C}_X$, but we often omit the subscript $X$. A set $S$ is a {\em neighbourhood} of a point $x$ if $x\in {\bf I} S$; then $S-\{x\}$ is called a {\em punctured neighbourhood} of $x$.
\begin{defi} Let $\mathfrak X$ be a topological space, $V\subseteq X$. A point $x\in X$ is said to be {\em limit} for $V$ if $x\in{\bf C}(V-\{x\})$; a non-limit point of $V$ is called {\em isolated}.
The {\em derived set of $V$} (denoted by ${\bf d} V$, or by ${\bf d}_X V$) is the set of all limit points of $V$. The unary operation $V \mapsto {\bf d} V$ on ${\cal P}(X)$ is called {\em the derivation} (in $\mathfrak X$).
A set without isolated points is called {\em dense-in-itself}. \end{defi}
\begin{lem}\label{dY}\cite{K66} For a subspace $\mathcal Y\subseteq \mathfrak X$ and $V\subseteq X$ ${\bf d}_Y(V\cap Y)={\bf d}_X(V\cap Y)\cap Y$; if $Y$ is open, then ${\bf d}_Y(V\cap Y)={\bf d}_XV\cap Y$. \end{lem}
\begin{defi} The \emph{derivational algebra of a topological space $\mathfrak X$} is $DA(X):=(2^X, {\bf \tilde{d}} )$, where $2^X$ is the Boolean algebra of all subsets of $X$, ${\bf \tilde{d}} V := -{\bf d} (-V)$\footnote{There is no common notation for this operation; some authors use $\mathbf{\tau}$.}. The \emph{closure algebra of a space $\mathfrak X$} is $CA(\mathfrak X):=(2^X,{\bf I} )$. \end{defi}
\begin{rem}\rm In \cite{MT} the derivational algebra of $\mathfrak X$ is defined as $(2^X,{\bf d} )$, and the closure algebra as $(2^X,{\bf C} )$, but here we adopt equivalent dual definitions. \end{rem}
It is well known that $CA(\mathfrak X), ~DA(\mathfrak X)$ are modal algebras, $CA(\mathfrak X)\vDash {\bf S4}$ and $DA(\mathfrak X) \vDash {\bf K4}^\circ$ (the latter is due to Esakia).
Every Kripke ${\bf S4}$-frame $F=(W,R)$ is associated with a topological space $N(F)$ on $W$, with the {\em Alexandrov} (or {\em right}) {\em topology} $\{V\subseteq W\mid R(V)\subseteq V\}$. In $N(F)$ we have ${\bf C} V=R^{-1}(V), ~{\bf I} V=\{x\mid R(x)\subseteq V\}$; thus $MA(F)=CA(N(F))$.
\begin{defi} A modal formula $A$ is called {\em d-valid in a topological space} $\mathfrak X$ (in symbols, $\mathfrak X\vDash^d A$) if it is true in the algebra $DA(\mathfrak X)$. The logic ${{\bf L}}(DA(\mathfrak X))$ is called the {\em derivational modal logic} (or the {\em d-logic}) of $\mathfrak X$ and denoted by ${{\bf L}}{\bf d} (\mathfrak X)$.
A formula $A$ is called {\em c-valid} in $\mathfrak X$ (in symbols, $\mathfrak X\vDash^c A$) if it is true in $CA(\mathfrak X)$. ${{\bf L}}{\bf c} (\mathfrak X):={{\bf L}}(CA(\mathfrak X))$ is called the {\em closure modal logic,} or the \emph{c-logic} of $\mathfrak X$. \end{defi}
\begin{defi} For a class of topological spaces ${\cal C}$ we also define the {\em d-logic} ${{\bf L}}{\bf d}({\cal C}):=\bigcap\{{{\bf L}}{\bf d}(\mathfrak X)\mid \mathfrak X\in{\cal C}\}$ and the {\em c-logic} ${{\bf L}}{\bf c}({\cal C}):=\bigcap\{{{\bf L}}{\bf c}(\mathfrak X)\mid \mathfrak X\in{\cal C}\}$. Logics of this form are called {\em d-complete} (respectively, {\em c-complete} ). \end{defi}
\begin{defi} A {\em valuation} in a topological space $\mathfrak X$ is a map $\varphi : PV \longrightarrow {\cal P}(\mathfrak X)$. Then $(\mathfrak X,\varphi)$ is called a {\em topological model} over $\mathfrak X$. \end{defi}
So valuations in $\mathfrak X$, $CA(\mathfrak X)$, and $DA(\mathfrak X)$ are the same. Every valuation $\varphi$ can be prolonged to all formulas in two ways, according either to $CA(\mathfrak X)$ or $DA(\mathfrak X)$. The corresponding maps are denoted respectively by $\varphi_c$ or $\varphi_d$. Thus \begin{align*} \varphi_d (\square A) &= {\bf \tilde{d}}\varphi_d (A),~& \varphi_d (\Diamond A) &= {\bf d}\varphi_d (A),\\ \varphi_c (\square A) &= {\bf I}\varphi_c (A),~& \varphi_c (\Diamond A) &= {\bf C}\varphi_c (A). \end{align*}
A formula $A$ is called {\em d-true} (respectively, {\em c-true}) in $(\mathfrak X,\varphi)$ if $\varphi_d ( A) =X$ (respectively, $\varphi_c ( A) =X$ ). So $A$ is d-valid in $\mathfrak X$ iff $A$ is d-true in every topological model over $\mathfrak X$, similarly for c-validity.
\begin{defi} A modal formula $A$ is called \emph{d-true at a point} $x$ in a topological model $(\mathfrak X,\varphi)$ if $x\in\varphi_d (A)$. \end{defi}
Instead of $x\in\varphi_d(A)$, we write $x \vDash^d A$ if the model is clear from the context. Similarly we define the c-truth at a point and use the corresponding notation.
From the definitions we obtain \begin{lem} For a topological model over a space $\mathfrak X$ \begin{itemize} \item $x \vDash^d \square A$ iff $\exists U\ni x~ (U$ is open in $\mathfrak X~ \&~ \forall y\in U-\{ x\}~ y \vDash^d A)$; \item $x \vDash^d \Diamond A$ iff $\forall U\ni x~ (U$ is open in $\mathfrak X~ \Rightarrow~ \exists y\in U-\{ x\}~ y \vDash^d A)$. \end{itemize} \end{lem}
\begin{defi}\label{lt1} A \emph{local $T_1$-space} (or a {\em $T_D$-space} \cite{AT62}) is a topological space, in which every point is {\em locally closed}, i.e, closed in some neighbourhood. \end{defi}
Note that a point $x$ in an Alexandrov space $N(W,R)$ is closed iff it is minimal (i.e., $R^{-1}(x)=\{x\}$); $x$ is locally closed iff $R(x)\cap R^{-1}(x)=\set{x}$. Thus $N(F)$ is local $T_1$ iff $F$ is a poset.
\begin{lem}\label{L310}\cite{E1} For a topological space $\mathfrak X$ \begin{enumerate} \item $\mathfrak X \vDash^d {\bf K4}\mbox{ iff } \mathfrak X\mbox{ is local }T_1;$ \item $\mathfrak X \vDash^d \Diamond\top\mbox{ iff } \mathfrak X\mbox{ is dense-in-itself}.$ \end{enumerate} \end{lem}
\begin{defi} A Kripke frame (W,R) is called {\em weakly transitive} if $R\circ R\subseteq \overline{R}$. \end{defi} It is obvious that the weak transitivity of $R$ is equivalent to the transitivity of $\overline{R}$.
\begin{propo}\label{P32}\cite{E1}~ (1) $(W,R) \vDash {\bf K4}^\circ\mbox{ iff } (W,R)\mbox{ is weakly transitive};$\\ (2) ${\bf K4}^\circ$ is Kripke-complete. \end{propo}
\begin{lem}\label{L33}\cite{E1}~ (1) Let $F = (W,R)$ be a Kripke ${\bf S4}$-frame, and let $R^\circ := R-I_W$, $F^\circ := (W,R^\circ)$. Then ${\bf Ld}(N(F)) = {{\bf L}}(F^\circ)$.\\ (2) Let $F=(W,R)$ be a weakly transitive irreflexive Kripke frame, and let $\overline{F}=: (W, \overline{R})$ be its reflexive closure. Then $ {{\bf L}}{\bf d} (N(\overline{F}))={{\bf L}}(F) $.\\ (3) If $\mathbf{\Lambda}={{\bf L}}({\cal C})$, for some class ${\cal C}$ of weakly transitive irreflexive Kripke frames, then $\mathbf{\Lambda}$ is d-complete. \end{lem} \begin{proof} (1) Note that $R^\circ(x)$ is the smallest punctured neighbourhood of $x$ in the space $N(F)$. So the inductive d-truth definition in a topological model $(N(F),\varphi)$ coincides with the inductive truthdefinition in the Kripke model $(W,R^\circ,\varphi)$.
(2) Readily follows from (1), since $\overline{R}$ is transitive and $(\overline{R})^\circ=R$.
(3) Follows from (2). {\hspace*{\fill} } \end{proof}
\begin{defi} For a 1-modal formula $A$ we define $A^\sharp$ as the formula obtained by replacing every occurrence of every subformula $\square B$ with $\overline{\square}B:=\square B\wedge B$.
For a 1-modal logic $\mathbf{\Lambda}$ its {\em reflexive fragment} is $^\sharp\mathbf{\Lambda}:=\{A\mid\mathbf{\Lambda}\vdash A^\sharp\}$. \end{defi}
\begin{propo}\label{cldl}\cite{BEG}~ (1) If $\mathbf{\Lambda}$ is a ${\bf K4}^\circ$-logic, then $^\sharp\mathbf{\Lambda}$ is an ${\bf S4}$-logic. \\ (2) For any topological space $X$, ${\bf Lc}(\mathfrak X)= \,^\sharp{\bf Ld}(\mathfrak X)$, \\ (3) For any weakly transitive Kripke frame $F$, ${\bf L}(\overline{F})=\, ^\sharp{\bf L}(F)$. \end{propo}
\begin{proof} (1) It is clear that for a weakly transitive $\mathbf{\Lambda}$, $\overline{\square}$ satisfies the axioms of ${\bf S4}$, so $^\sharp\mathbf{\Lambda}$ contains these axioms. Since $^\sharp$ distributes over implication, it follows that $^\sharp\mathbf{\Lambda}$ is closed under Modus Ponens. For the substitution closedness, note that for any variable $p$ and formulas $A,B$ $([B/p]A)^\sharp=[B^\sharp/p] A^\sharp$; thus $A\in\, ^\sharp\mathbf{\Lambda}$ implies $[ B/p]A\in\, ^\sharp\mathbf{\Lambda}$.Finally, since $(\square A)^\sharp= \overline{\square}\,A^\sharp$, it is clear that $A\in\, ^\sharp\mathbf{\Lambda}$ only if $\square A\in\,^\sharp\mathbf{\Lambda}$.
(2) By definitions, $${\bf Lc}(X)\vdash A \;\mbox{iff}\; CA(X)\vDash A,$$ $$^\sharp{\bf Ld}(X)\vdash A \;\mbox{iff}\; {\bf Ld}(X)\vdash A^\sharp\;\mbox{iff}\; DA(X)\vDash A^\sharp.$$ Let us show that that $ CA(X){\,\not\mo\,} A$ iff $ DA(X){\,\not\mo\,} A^\sharp$. In fact, consider a topological model $(X,\varphi)$. We claim that \[ \varphi_c(B)=\varphi_d(B^\sharp)\eqno(*) \] for any formula $B$. This is easily checked by induction, the crucial case is when $B=\square B_1$; then by definitions and the induction hypothesis we have: \[ \varphi_c(B)={\bf I}\varphi_c(B_1)={\bf I}\varphi_d(B_1^\sharp)= \boxdot\varphi_d(B_1^\sharp)\cap \varphi_d(B_1^\sharp)=\varphi_d(\overline{\square}\,B_1^\sharp)= \varphi_d(B^\sharp). \] The claim (*) implies that $\varphi_c(A)\neq X$ iff $\varphi_d(A^\sharp)\neq X$ as required.
(3) On the one hand, $${\bf L}(\overline{F})={{\bf L}}(MA(\overline{F}))= {\bf L} (CA(N(\overline{F}))= {\bf Lc}(N(\overline{F})).$$ On the other hand, by Lemma \ref{L33}(2), \[ {\bf L}(F)= {\bf Ld}(N(\overline{F})), \] and we can apply (2) to $N(F)$.{\hspace*{\fill} } \end{proof}
Let us give some examples of d-complete logics. \begin{enumerate} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})} \item ${\bf Ld}(\mbox{all ~topological ~spaces})= {\bf K4}^\circ$. This was proved by L. Esakia in the 1970s and published in \cite{E1}. \item ${\bf Ld}(\mbox{all ~local~}T_1\mbox{-spaces})= {\bf K4}$. This is also a result from \cite{E1}.
\item ${\bf Ld}(\mbox{all }T_0\mbox{-spaces})= {\bf K4}^\circ + p \land \Diamond (q \land \Diamond p) \to \Diamond p \lor \Diamond(q \land \Diamond q)$. This result is from \cite{BEG2011}.
\item L. Esakia \cite{E} also proved that G\"odel - L\"ob logic ${\bf GL} := {\bf K} + \square (\square p\supset p)\supset \square p$ is the derivational logic of the class of all topological scattered spaces (a space is {\em scattered} if each its nonempty subset has an isolated point). \item The papers \cite{A87}, \cite{A88}, \cite{Bl} give a complete description of d-logics of ordinals with the interval topology: ${\bf Ld}(\alpha)$ is either ${\bf GL}$ (if $\alpha\geq\omega^\omega$), or ${\bf GL}+ \square^n \bot$ (if $\omega^{n-1}\leq\alpha<\omega^n$). In particular, ${\bf Ver} := {\bf K} + \square \bot$ is the d-logic of any finite ordinal (and of any discrete space). \item The well-known ``difference logic" \cite{S80}, \cite{DR93} ${\bf DL} := {\bf K4}^\circ + \Diamond \square p \supset p,$ is determined by Kripke frames with the difference relation: ${\bf DL} = {{\bf L}}(\{(W,\neq_W)\mid W\neq{\varnothing}\}),$ where $\neq_W:=W^2-I_W$; hence by \ref{cldl}, ${\bf DL}$ is the d-logic of the class of all trivial topological spaces. However, for any particular trivial space $\mathfrak X$, ${\bf Ld}(\mathfrak X)\neq {\bf DL}$. Moreover, ${\bf Ld}(\mathfrak X)$ is not finitely axiomatizable for any infinite trivial $\mathfrak X$\cite{KudShap}; this surprising result is easily proved by a standard technique using Jankov formulas (cf. \cite{Kudinov08}). \item In \cite{Sh00} it was proved that ${\bf Ld}\mbox{(all 0-dimensional separable metric spaces)}={\bf K4}.$ All these spaces are embeddable in ${\bf R}$ \cite{K66}. \item In \cite{Sh00} it was also proved that for any dense-in itself separable metric space $\mathfrak X$, ${\bf Ld}(\mathfrak X)={\bf D4}$; this was a generalization of an earlier proof \cite{Sh90} for $\mathfrak X={\bf Q}$. A more elegant proof for ${\bf Q}$ is in \cite{LB1}. \item Every extension of ${\bf K4}$ by a set of closed axioms is a d-logic of some subspace of ${\bf Q}$ \cite{BLB}. This gives us a continuum of d-logics of countable metric spaces.
\item In \cite{Sh90} ${\bf Ld}({\bf R}^2)$ was axiomatized and it was also proved that the d-logics of ${\bf R}^n$ for $n\geq 2$ coincide. We will simplify and extend that proof in the present chapter. \item $\mathbf{Ld}({\bf R})$ was described in \cite{Sh00}; for a simpler completeness proof cf. \cite{LB2}. \item $\mathbf{Ld} (\mbox{all Stone spaces})= {\bf K4}$ and $\mathbf{Ld} (\mbox{all weakly scattered Stone spaces})= {\bf K4} + \Diamond \top \to \Diamond \Box \perp$, cf. \cite{BEG2010}.
\item d-logics of special types of spaces were studied in \cite{BEG}, \cite{LB1}. They include submaximal, perfectly disconnected, maximal, weakly scattered and some others. \end{enumerate}
However, not all extensions of ${\bf K4}^\circ$ are d-complete. In fact, the formula $p\supset \Diamond p$ never can be d-valid, because ${\bf d} Y = {\varnothing}$ for a singleton $Y$. So every extension of ${\bf S4}$ is d-incomplete, and thus Kripke completeness does not imply d-completeness.
\begin{propo}\label{P42} Let $F=(\omega^*,\prec)$ be the ``standard irreflexive transitive tree", where $\omega^*$ is the set of all finite sequences in $\omega$; $\alpha\prec\beta$ iff $\alpha$ is a proper initial segment of $\beta$. Then $${\bf D4} = {{\bf L}}(F)={{\bf L}}{\bf d} (N(\overline{F})) = {{\bf L}}{\bf d} ({\cal D}),$$ where ${\cal D}$ denotes the class of all dense-in-themselves local $T_1$-spaces. \end{propo} \begin{proof} The first equality is well known \cite{VB83}; the second one holds by \ref{L33}. By \ref{L310}, ${\bf D4}$ is d-valid exactly in spaces from ${\cal D}$. So $N(\overline{F})\in{\cal D}$, ${\bf D4}\subseteq{{\bf L}}{\bf d} ({\cal D}) $, and the third equality follows.{\hspace*{\fill} } \end{proof}
\section{Adding the universal modality and the difference modality} \label{sec:basics} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})}
Recall that the {\em universal modality} $[\forall]$ and the {\em difference modality} $[\neq]$ correspond to Kripke frames with the universal and the difference relation. So (under a valuation in a set $W$) these modalities are interpreted in the standard way: \begin{align*} x\vDash[\forall]A &\;\mbox{iff}\;~ \forall y\in W~y\vDash A;~ &x\vDash[\neq]A &\;\mbox{iff}\; \forall y\in W~(y\neq x \Rightarrow y\vDash A). \end{align*}
The corresponding dual modalities are denoted by $\langle\exists\rangle$ and ${\langle\ne\rangle}$.
\begin{defi} For a $[\forall]$-modal formula $A$ we define the $[\neq]$-modal formula $A^u$ by induction: \begin{equation*} A^u:=A\mbox{ for }A\mbox{ atomic},\ \
(A\supset B)^u:=A^u\supset B^u,\ \ ([\forall]B)^u :=[\neq]B^u\wedge B^u. \end{equation*} \end{defi}
We can consider 2-modal topological logics obtained from ${\bf Lc}(\mathfrak X)$ or ${\bf Ld}(\mathfrak X)$ by adding the universal or the difference modality\footnote{So we extend the definitions of the d-truth or the c-truth by adding the item for $[\forall]$ or $[\neq]$.}. Thus for a topological space $\mathfrak X$ we obtain four 2-modal logics : ${\bf Lc}_\forall(\mathfrak X)$ (the {\em closure universal (cu-) logic}), ${\bf Ld}_\forall(\mathfrak X)$ (the {\em derivational universal (du-) logic}), ${\bf Lc}_{\neq}(\mathfrak X)$ (the {\em closure difference (cd-) logic}), ${\bf Ld}_{\neq}(\mathfrak X)$ (the {\em derivational difference (dd-) logic}). Similar notations (${\bf Lc}_\forall({\cal C})$ etc.) are used for logics of a class of spaces ${\cal C}$, and respectively we can define four kinds of topological completeness (cu-, du-, cd-, dd-) for 2-modal logics.
cd-logics were first studied in \cite{Gab01},
cu-logics in \cite{Sh99}, du-logics in \cite{LB2}, but dd-logics have never been addressed so far.
For a $\square$-modal logic ${{\bf L}}$ we define the 2-modal logics \begin{align*} {{\bf L}}{\bf D} &:={{\bf L}}*{\bf D}{{\bf L}}+[\neq]p\wedge p\rightarrow\square p, ~ &{{\bf L}}{\bf D}^+ &:={{\bf L}}*{\bf D}{{\bf L}}+[\neq]p\rightarrow\square p, \\ {{\bf L}}{\bf U}&:={{\bf L}}*{\bf S5}+[\forall]p\rightarrow\square p. \end{align*} Here we suppose that ${\bf S5}$ is formulated in the language with $[\forall]$ and ${\bf D}{{\bf L}}$ in the language with $[\neq]$. The following is checked easily: \begin{lem}\label{L41} For any topological space $\mathfrak X$, \[ {\bf Lc}_\forall(\mathfrak X)\supseteq {\bf S4}{\bf U},\quad {\bf Ld}_\forall(\mathfrak X)\supseteq{\bf K4}^\circ{\bf U},\quad {\bf Lc}_{\neq}(\mathfrak X)\supseteq {\bf S4}{\bf D},\quad {\bf Ld}_{\neq}(\mathfrak X) \supseteq{\bf K4}^\circ{\bf D}^+. \] \end{lem}
\begin{defi} For a 1-modal Kripke frame $F=(W,R)$ we define 2-modal frames $F_\forall:=(F,W^2),~ F_{\neq}:=(F,\neq_W)$ and modal logics ${{\bf L}}_\forall(F):={{\bf L}}(F_\forall),~ {{\bf L}}_{\neq}(F):={{\bf L}}(F_{\neq})$.
\end{defi}
Sahlqvist theorem \cite{CZ97} implies \begin{propo}~ The logics ${\bf S4}{\bf U},~ {\bf K4}^\circ{\bf U}, ~ {\bf S4}{\bf D},~{\bf K4}^\circ{\bf D}^+ $ are Kripke complete. \end{propo} Using the first-order equivalents of the modal axioms for these logics (in particular, Proposition \ref{P32}) we obtain
\begin{lem}\label{L46}~ For a rooted Kripke frame $G=(W,R,S)$ \begin{enumerate} \item $ G\vDash{\bf S4}{\bf U}\;\mbox{iff}\; R\mbox{ is a quasi-order }\&~S=W^2, $ \item $ G\vDash{\bf K4}^\circ{\bf U} \;\mbox{iff}\; R\mbox{ is weakly transitive }\&~S=W^2, $ \item $ G\vDash{\bf S4}{\bf D} \;\mbox{iff}\; R\mbox{ is a quasi-order }\&~\overline{S}=W^2, $ \item $ G\vDash{\bf K4}^\circ{\bf D}^+\;\mbox{iff}\; R\mbox{ is weakly transitive }\&~\overline{S}=W^2\&~R\subseteq S. $ \end{enumerate} Also note that $\overline{S}=W^2\;\mbox{iff}\;\neq_W\subseteq S$. \end{lem}
\begin{defi}\label{F0} A rooted Kripke $\lgc{K4^\circ D}^+$-frame described by Lemma \ref{L46} (4) is called {\em basic}. The class of these frames is denoted by $\mathfrak{F}_0$. \end{defi}
Next, we easily obtain the 2-modal analogue to Lemma \ref{L33}.
\begin{lem}\label{L47}~ (1) Let $F$ be an ${\bf S4}$-frame. Then \[ {\bf Ld}_{\neq}(N(F))={{\bf L}}_{\neq}(F^\circ),~ {\bf Ld}_\forall(N(F))={{\bf L}}_\forall(F^\circ). \] (2) Let $F$ be a weakly transitive irreflexive Kripke frame. Then \[ {\bf Ld}_{\neq}(N(\overline{F}))={{\bf L}}_{\neq}(F),~ {\bf Ld}_\forall(N(\overline{F}))={{\bf L}}_\forall(F). \] (3) Let ${\cal C}$ be a class of weakly transitive irreflexive Kripke 1-frames. Then ${{\bf L}}_{\neq}({\cal C})$ is dd-complete, ${{\bf L}}_{\forall}({\cal C})$ is du-complete. \end{lem} Let us extend the translations $(-)^\sharp$, $(-)^u$ to 2-modal formulas.
\begin{defi} $(-)^u$ translates $(\square, [\forall])$-modal formulas to $(\square, [\neq])$-modal formulas so that $([\forall]B)^u =[\neq]B^u\wedge B^u$ and $(-)^u$ distributes over the other connectives.
Similarly, $(-)^\sharp$ translates $(\square,[\neq])$-modal formulas and $(\square,[\forall])$-modal formulas to formulas of the same kind, so that $(\square\,B)^\sharp =\square\,B^\sharp\wedge B^\sharp$ and $(-)^\sharp$ distributes over the other connectives. \begin{align*} ^u\mathbf{\Lambda}&:=\{A\mid A^u\in\mathbf{\Lambda}\}\mbox{ for a }(\square,[\forall])\mbox{-modal logic } \mathbf{\Lambda}\mbox{ (the {\em universal fragment}),}\\ ^\sharp\mathbf{\Lambda}&:=\{A\mid A^\sharp\in\mathbf{\Lambda}\}\mbox{ for a }(\square,[\neq])\mbox{- or a } (\square,[\forall])\mbox{-modal }\mathbf{\Lambda}\mbox{ (the {\em reflexive fragment}),}\\ \,^\sharp\phantom{}^u\mathbf{\Lambda}&:= \,^\sharp(^u\mathbf{\Lambda})\mbox{ for a }(\square,[\neq])\mbox{-modal } \mathbf{\Lambda}\mbox{ (the {\em reflexive universal fragment}).} \end{align*} \end{defi}
\begin{propo}\label{P49}~ (1) The map $\mathbf{\Lambda}\mapsto\,^\sharp\mathbf{\Lambda}$ sends $\lgc{K4^\circ D^+}$-logics to $\lgc{S4U}$-logics.\\ (2) The map $\mathbf{\Lambda}\mapsto\,^u\mathbf{\Lambda}$ sends $\lgc{K4^\circ D^+}$-logics to $\lgc{ K4^\circ U}$-logics and $\lgc{S4D}$-logics to $\lgc{S4U}$-logics. \\(3) The map $\mathbf{\Lambda}\mapsto\,^\sharp\phantom{}^u\mathbf{\Lambda}$ sends $\lgc{K4^\circ D^+}$-logics to $\lgc{S4U}$-logics. \\(4) For a topological space $\mathfrak X$ \[ {{\bf L}}{\bf c}_{\neq}(\mathfrak X)=\,^\sharp{{\bf L}}{\bf d}_{\neq}(\mathfrak X),~ {{\bf L}}{\bf d}_{\forall}(\mathfrak X)=\,^u{{\bf L}}{\bf d}_{\neq}(\mathfrak X),~ {{\bf L}}{\bf c}_{\forall}(\mathfrak X)=\,^u{{\bf L}}{\bf c}_{\neq}(\mathfrak X)={} ^\sharp{{\bf L}}{\bf d}_{\forall}(\mathfrak X). \] \\(5) For a weakly transitive Kripke frame $F$ \[ {{\bf L}}_{\neq}(\overline{F})=\,^\sharp{{\bf L}}_{\neq}(F),~ ~{{\bf L}}_{\forall}(F)=\,^u{{\bf L}}_{\neq}(F), ~{{\bf L}}_{\forall}(\overline{F})=\,^\sharp{{\bf L}}_{\forall}(F)= \,^\sharp\,^u{{\bf L}}_{\neq}(F). \] \end{propo}
Proposition \ref{P49}\,(4) implies that dd-logics are the most expressive of all kinds of the logics we consider.
\begin{coro} If ${{\bf L}}{\bf d}_{\neq}(\mathfrak X)= {{\bf L}}{\bf d}_{\neq}(\mathfrak Y)$ for spaces $\mathfrak X,\mathfrak Y$, then all the other logics (du-, cu-, cd-, d-, c-) of these spaces coincide. \end{coro}
Let \[
AT_1:= {[\ne]} p \rightarrow {[\ne]} \square p, \quad AC:= \left[\forall\right](\square p \lor \square\neg p) \rightarrow \left[\forall\right] p \lor \left[\forall\right] \lnot p.
\]
\begin{propo}\label{P412} For a topological space $\mathfrak X$
\begin{enumerate} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})} \item $\mathfrak X \models^d \Diamond\top$ iff $\mathfrak X$ is dense-in-itself; \item $\mathfrak X \models^d AT_1$ iff $\mathfrak X \models^c AT_1$ iff $\mathfrak X$ is a $T_1$-space; \item $\mathfrak X\vDash^d AC^{\sharp}$ iff $\mathfrak X \models^c AC$ iff $\mathfrak X$ is connected.
\end{enumerate} \end{propo}
\begin{proof}
(1) and the first equivalence in (2) are trivial. The first equivalence in (3) follows from \ref{P49}(4). The remaining ones are checked easily, cf. \cite{kudinov_2006}, \cite{Sh99}.{\hspace*{\fill} } \end{proof}
For a $\square$-modal logic $\lgc{L}$ put \[
\begin{array}{rcl} \lgc{LD^+T_1} := \lgc{LD^+} + AT_1, &\qquad& \lgc{LD^+T_1C} := \lgc{LD^+} + AT_1 + AC^{\sharp u}.
\end{array} \] Also put \[ \lgc{KT_1}:=\lgc{K4D^+T_1},~\lgc{DT_1}:=\lgc{D4D^+T_1}, ~\lgc{DT_1C}:=\lgc{D4D^+T_1C}. \]
\begin{propo}\cite{kudinov_2006}\label{pr:DS_AT1} If $F = (W, R, R_D)$ is basic, then \mbox{$F \models AT_1$} iff all $R_D$-irreflexive points are strictly $R$-minimal iff $R_D \circ R \subseteq R_D$. \end{propo}
\begin{rem}\rm Density-in-itself is expressible in cd-logic and dd-logic by the formula $DS:= {[\ne]} p \supset \Diamond p$, So for any space $\mathfrak X$, $\mathfrak X\vDash^c DS\;\mbox{iff}\;\mathfrak X\vDash^d DS\;\mbox{iff}\; \mathfrak X\vDash^d\Diamond\top$.
It is known that $DS$ axiomatizes dense-in-themselves spaces in cd-logic \cite{kudinov_2006}. However, in dd-logic this axiom is insufficient: ${\bf Ld}_{\neq}$(all dense-in-themselves spaces) $= \lgc{D4^\circ D^+}=\lgc{K4^\circ D^+}+\Diamond\top$, and it is {\em stronger} than $\lgc{K4^\circ D^+}+DS$. (To see the latter, consider a singleton Kripke frame, which is $R_D$-reflexive, but $R$-irreflexive.) Therefore $\lgc{K4^\circ D^+}+DS$ is dd-incomplete. \end{rem}
\begin{rem}\rm Every $T_1$-space is a local $T_1$-space, so the dd-logic of all $T_1$-spaces contains $\square p\rightarrow\square\square p$. However, $\lgc{K4^{\circ}D^+T_1}\,{\not\vdash}\,\square p\rightarrow\square\square p$. In fact, consider a 2-point frame $F:=(W,\neq_W,W^2)$. It is clear that $F\vDash\lgc{K4^{\circ}D^+}$. Also $F\vDash AT_1$, by Proposition \ref{pr:DS_AT1}, but $F{\,\not\mo\,}\square p\rightarrow\square\square p$, since $\neq_W$ is not transitive.
It follows that $\lgc{K4^{\circ}D^+T_1}$ is dd-incomplete; $T_1$-spaces are actually axiomatized by ${\bf KT_1}$ (Corollary \ref{C713}). \end{rem}
\newcommand{\mathbf{Lc_\forall}}{\mathbf{Lc_\forall}} \newcommand{\mathbf{Lc_{\ne}}}{\mathbf{Lc_{\ne}}} \newcommand{\mathbf{Ld_\forall}}{\mathbf{Ld_\forall}} \newcommand{\mathbf{Ld_{\ne}}}{\mathbf{Ld_{\ne}}}
Let us give some examples of du-, cu- and cd-complete logics. \begin{enumerate} \item $\mathbf{Lc_\forall}(\hbox{all spaces}) = \lgc{S4U}$.
\item $\mathbf{Lc_\forall}(\hbox{all connected spaces}) = \mathbf{Lc_\forall}(\mathbf R^n) = \lgc{S4U} + AC$ for any $n\ge 1$ \cite{Sh99}\footnote{The paper \cite{Sh99} contains a stronger claim: $\mathbf{Lc_\forall}(\mathfrak X)=\lgc{S4U} + AC$ for any connected dense-in-itself separable metric $\mathfrak X$. However, recently we found a gap in the proof of Lemma 17 from that paper. Now we state the main result only for the case ${\mathfrak X}=\mathbf R^n$; a proof can be obtained by applying the methods of the present Chapter, but we are planning to publish it separately.}
\item $\mathbf{Ld_\forall}(\hbox{all spaces}) = \lgc{S4D}$ \cite{DR93}.
\item $\mathbf{Lc_{\ne}}(\mathfrak X) = \lgc{S4DT_1+DS}$, where $\mathfrak X$ is a 0-dimensional separable metric space \cite{kudinov_2006}.
\item $\mathbf{Lc_{\ne}}({\bf R}^n)$ for any $n\ge 2$ is finitely axiomatized in \cite{kudinov_2005}; all these logics coincide.
\item $\mathbf{Ld_\forall}({\bf R})$ is finitely axiomatized in \cite{LB2}. \end{enumerate}
\section{dd-completeness of $\lgc{K4^\circ D^+}$ and some of its extensions} This section contains some simple arguments showing that there are many dd-complete bimodal logics.
All formulas and logics in this section are $(\square,{[\ne]})$-modal. An arbitrary Kripke frame for $(\square,{[\ne]})$-formulas is often denoted by $(W,R,R_D)$.
\begin{lem}\label{L35}~ (1) Every weakly transitive Kripke 1-frame is a p-morphic image of some irreflexive weakly transitive Kripke 1-frame. \\ (2) Every rooted $\lgc{K4^\circ D^+}$-frame is a p-morphic image of some $R$- and $R_D$-irreflexive rooted $\lgc{K4^\circ D^+}$-frame. \end{lem} \begin{proof} (1) Cf. \cite{E1}.
(2) Similar to the proof of (1). For $F=(W,R,R_D)\in \mathfrak{F}_0$ put \[ W_r:=\{a\mid aR_Da\},~ W_i=W-W_r, \quad \tilde{W}:= W_i\cup (W_r \times \{ 0,1\}). \] Then we define the relation $\tilde{R}$ on $\tilde{W}$ such that \begin{align*}
(b,j) \tilde{R}a &\mbox{ ~~iff ~~ } bRa,& a \tilde{R} (b,j) &\mbox{ ~~iff ~~ } aRb,\\ (b,j) \tilde{R} (b',k) &\mbox{ ~~iff ~~ } bRb'\ \&\ b\ne b' \vee b= b'\ \&\ j\ne k,& a\tilde{R} a' &\mbox{ ~~iff ~~ } aRa'. \end{align*}
Here $a,a'\in W_i;~ b,b'\in W_r; ~j,k\in\{ 0,1\}$. So we duplicate all $R_D$-reflexive points making them irreflexive (under both relations). It follows that $\tilde{F}:=(\tilde{W},\tilde{R},\neq_{\tilde{W}})\in \mathfrak{F}_0$ and $\tilde{R}$ is irreflexive; the map $f:\tilde{W} \to W$ sending $(b,j)$ to $b$ and $a$ to itself (for $b\in W_r, ~ a\in W_i$) is a p-morphism $\tilde{F}\twoheadrightarrow F$. {\hspace*{\fill} } \end{proof}
\begin{propo}\label{P53} Let $\Gamma$ be a set of closed 2-modal formulas, $\mathbf{\Lambda}:= \lgc{K4^\circ D^+}+\Gamma$. Then \begin{enumerate} \item $\mathbf{\Lambda}$ is Kripke complete. \item $\mathbf{\Lambda}$ is dd-complete. \end{enumerate} \end{propo}
\begin{proof} (1) $\lgc{K4^\circ D^+}$ is axiomatized by Sahlqvist formulas. One can easily check that (in the minimal modal logic) every closed formula is equivalent to a positive formula, so we can apply Sahlqvist theorem.
(2) Suppose $A\not\in\mathbf{\Lambda}$. By (1) and the Generation lemma there exists a rooted Kripke 2-frame $F$ such that $F\vDash L$ and $F{\,\not\mo\,} A$. Then by Lemma \ref{L35}, for some irreflexive weakly transitive 1-frame $G=(W,R)$ there is a p-morphism $(G,\neq_W)\twoheadrightarrow F$. By the p-morphism lemma $(G,\neq_W){\,\not\mo\,} A$ and $(G,\neq_W)\vDash\mathbf{\Lambda}$ (since $\Gamma$ consists of closed formulas). Hence by Lemma \ref{L47}, $\mathbf{\Lambda}\subseteq\lgc{Ld}_{\neq}(N(\overline{G}))$, $A\not\in\lgc{Ld}_{\neq}(N(\overline{G}))$. {\hspace*{\fill} } \end{proof}
\begin{rem}\rm Using Proposition \ref{P53} and the construction from \cite{BLB} one can prove that there is a continuum of dd-complete logics. Such a claim is rather weak, because Proposition \ref{P53} deals only with Alexandrov spaces. In section 7 we will show how to construct many dd-complete logics of metric spaces. \end{rem}
\section{d-morphisms and dd-morphisms; extended McKinsey - Tarski's Lemma} In this section we recall the notion of a d-morphism (a validity-preserving map for d-logics) and introduce dd-morphisms, the analogues of d-morphisms for dd-logics. This is the main technical tool in the present chapter. Two basic lemmas are proved here, an analogue of McKinsey--Tarski's lemma on dissectability for d-morphisms and the Glueing lemma.
The original McKinsey--Tarski's lemma \cite{MT} states the existence of a c-morphism (cf. Remark \ref{R61} ) from an arbitrary separable dense-in-itself metric space onto a certain quasi-tree of depth 2. The separability condition is actually redundant \cite[Ch.~3]{RS} (note that the latter proof is quite different from \cite{MT}\footnote{\label{foot:7}Recently P. Kremer \cite{Kremer} has showed that ${\bf S4}$ is {\em strongly complete} w.r.t. any dense-in-itself metric space. His proof uses much of the construction from \cite{RS}.}). But c-morphisms preserve validity only for c-logics, and unfortunately, the constructions by McKinsey--Tarski and Rasiowa--Sikorski cannot be used for d-morphisms. So we need another construction to prove a stronger form of McKinsey--Tarski's lemma.
\begin{defi}\label{D51} Let $\mathfrak X$ be a topological space, $F=(W,R)$ a transitive Kripke frame. A map $f: X\longrightarrow W$ is called a {\em d-morphism} from $\mathfrak X$ to $F$ if $f$ is open and continuous as a map $\mathfrak X\longrightarrow N(\overline{F})$ and also satisfies \begin{align*} \mbox{r-density}:&~~\forall w\in W( wRw\Rightarrow f^{-1} (w)\subseteq {\bf d} f^{-1} (w)),\\ \mbox{i-discreteness}:&~~\forall w\in W( \neg wRw\Rightarrow f^{-1} (w)\cap {\bf d} f^{-1} (w)={\varnothing}). \end{align*} If $f$ is surjective, we write $f: \mathfrak X\twoheadrightarrow^d F$. \end{defi}
\begin{propo}\label{P52}\cite{BEG}~ (1) $f$ is a d-morphism from $\mathfrak X$ to $F$ iff $2^f$ is a homomorphism from $MA(F)$ to $DA(\mathfrak X)$.\\ (2) If $f: \mathfrak X\twoheadrightarrow^d F$, then ${{\bf L}}{\bf d} (\mathfrak X) \subseteq {{\bf L}}(F)$. \end{propo}
\begin{coro}\cite{Sh90}\label{L52} A map $f$ from a topological space $\mathfrak X$ to a finite transitive Kripke frame $F$ is a d-morphism iff $$\forall w\in W~ {\bf d} f^{-1} (w) = f^{-1}(R^{-1}(w)).$$
\end{coro}
\begin{proof} $2^f$ preserves Boolean operations. It is a homomorphism of modal algebras iff it preserves diamonds, i.e., iff for any $V\subseteq W$, $$f^{-1}(R^{-1}(V))={\bf d} f^{-1}(V).$$ Inverse images and ${\bf d}$ distribute over finite unions, so the above equality holds for any (finite) $V$ iff it holds for singletons, i.e., \[\hspace{3.8cm}\ f^{-1}(R^{-1}(w))= {\bf d} f^{-1}(w).\qedhere \] \end{proof}
\begin{rem}\label{R61}\rm For a space $\mathfrak X$ and a Kripke ${\bf S4}$-frame $F=(W,R)$ one can also define a {\em c-morphism} $\mathfrak X\longrightarrow F$ just as an open and continuous map $f: \mathfrak X\longrightarrow N(F)$. So every d-morphism to an ${\bf S4}$-frame is a c-morphism. It is well known \cite{RS} that $f: X\longrightarrow W$ is a c-morphism iff $2^f$ is a homomorphism $MA(F) \longrightarrow CA(\mathfrak X)$. Again for a finite $F$ this is equivalent to $$ \forall w\in W~{\bf C} f^{-1}(w) = f^{-1}(R^{-1}(w)). $$ \end{rem}
\begin{lem}\label{Restlem}
If $f:\mathfrak X\twoheadrightarrow^d F$ for a finite frame $F$ and $\mathcal Y\subseteq\mathfrak X$ is an open subspace, then $f|Y$ is a d-morphism. \end{lem}
\begin{proof}
We apply Proposition \ref{P52}. Note that $f|Y$ is the composition $f \cdot j$, where
$j:Y\hookrightarrow X$ is the inclusion map. Then $2^{ f|Y}=2^j\cdot 2^f$. Since $2^f$ is a homomorphism $MA(F)\longrightarrow DA(\mathfrak X)$, it remains to show that $2^j$ is a homomorphism $DA(\mathfrak X)\longrightarrow DA(\mathcal Y)$, i.e., it preserves the derivation: $j^{-1}({\bf d} V)={\bf d}_Yj ^{-1}(V)$, or ${\bf d} V\cap Y={\bf d}_Y(V\cap Y)$, which follows from \ref{dY}. {\hspace*{\fill} } \end{proof}
\begin{defi}\label{D53} A set $\gamma$ of subsets of a topological space $\mathfrak X$ is called \emph{dense} at $x\in X$ if every neighbourhood of $x$ contains a member of $\gamma$. \end{defi}
\begin{propo}\label{P54} For $m>0,~ l>0$ let $\Phi_{ml}$ be a ``quasi-tree" of height 2, with singleton maximal clusters and An
$m$-element root cluster (Fig. \ref{fig:Phiml}). For $l=0, ~m>0, ~ \Phi_{ml}$ denotes an $m$-element cluster.
Let $\mathfrak X$ be a dense-in-itself separable metric space, $B\subset X$ a closed nowhere dense set. Then there exists a d-morphism $g: \mathfrak X\twoheadrightarrow^d \Phi_{ml}$ with the following properties: \begin{enumerate} \item $B \subseteq g^{-1}(b_1)$; \item every $g^{-1}(a_i)$ (for $i\leq l$ ) is a union of a set $\alpha_i$ of disjoint open balls, which is dense at any point of $g^{-1}(\{ b_1,...,b_m\}$). \end{enumerate} \end{propo} \begin{proof} Let $X_1,\dots, X_n,\dots$ be a countable base of $\mathfrak X$ consisting of open balls. We construct sets $A_{ik}, ~B_{jk}$ for $1\leq i\leq l, ~1\leq j\leq m, ~k\in\omega$, with the following properties: \begin{enumerate} \renewcommand{(\arabic{enumi})}{(\arabic{enumi})} \item $A_{ik}$ is the union of a finite set $\alpha_{ik}$ of nonempty open balls whose closures are disjoint; \item ${\bf C} A_{ik} \cap {\bf C} A_{i'k} = {\varnothing}$ for $i\not= i'$ ; \item $\alpha_{ik} \subseteq \alpha_{i,k+1};~A_{ik} \subseteq A_{i,k+1}$; \item $B_{jk}$ is finite; \item $B_{jk} \subseteq B_{j,k+1};$ \item $A_{ik} \cap B_{jk} = {\varnothing} ;$ \item $X_{k+1} \subseteq\bigcup\limits_{i=1}^l A_{ik}\Rightarrow \alpha_{i,k+1} = \alpha_{ik}, ~B_{j,k+1} = B_{jk};$ \item if $X_{k+1}\not\subseteq \bigcup\limits_{i=1}^l A_{ik}$, there are closed nontrivial balls $P_1,\dots,P_l$ such that for any $i$, $j$ $$P_i \subseteq X_{k+1}-A_{ik},~ \alpha_{i,k+1} = \alpha_{ik} \cup \{ {\bf I} P_i\},~ (B_{j,k+1} - B_{jk}) \cap X_{k+1}\not= {\varnothing}; $$ \item $A_{ik} \subseteq X-B$; \item $B_{jk} \subseteq X-B$; \item $j\not= j'\Rightarrow B_{j'k} \cap B_{jk} = {\varnothing}$ . \end{enumerate} We carry out both the construction and the proof by induction on $k$.
Let $k=0$. $(X-B)$ is infinite, since it is nonempty and open in a dense-in-itself $\mathfrak X$. Take distinct points $v_1,\dots,v_l \not\in B$ and disjoint closed nontrivial balls $ Z_1,\dots,Z_l \subset X-B$ with centres at $v_1,\dots,v_l$ respectively (see Fig.\ref{fig:McT-constraction}).
\begin{figure}
\caption{Case k = 0}
\label{fig:McT-constraction}
\end{figure}
Put
$$\alpha_{i0}: =\{ {\bf I} Z_i\}; ~A_{i0}: = {\bf I} Z_i;$$ then $Z_i = {\bf C} A_{i0}$. As above, since $(X-B) -\bigcup\limits^l_{i=1}Z_i$ is nonempty and open, it is infinite. Pick distinct $w_1,\dots,w_m\in X-B$ and put $B_{j0}: = \{ w_j\}$. Then the required properties hold for $k=0$.
At the induction step we construct $A_{i,k+1}, B_{j,k+1}$. Put $Y_k :=\bigcup\limits ^l_{i=1}A_{ik}$ and consider two cases.\\ (a) $X_{k+1}\subseteq Y_k$. Then put: $$\alpha_{i,k+1}: = \alpha_{ik};~ A_{i,k+1}: = A_{ik}; ~B_{j,k+1}: = B_{jk}.$$ (b) $X_{k+1}\not\subseteq Y_k$. Then $X_{k+1} \not\subseteq {\bf C} Y_k$. In fact, $X_{k+1} \subseteq {\bf C} Y_k$ implies $X_{k+1} \subseteq {\bf I} {\bf C} Y_k=Y_k$, since $X_{k+1}$ is open and by (1) and (2). So we put $$W_0 := X_{k+1} - {\bf C} Y_k -\bigcup^m_{j=1}B_{jk},~ W := W_0 - B.$$ Since $(X_{k+1} - {\bf C} Y_k )$ is nonempty and open and every $B_{jk}$ is finite by (4), $W_0$ is also open and nonempty (by the density of $\mathfrak X$). By the assumption of \ref{P54}, $B$ is closed, and thus $W$ is open.
$W$ is also nonempty. In fact, otherwise $W_0 \subseteq B$, and then $W_0 \subseteq {\bf I} B={\varnothing}$ (since $B$ is nowhere dense by the assumption of \ref{P54}).
Now we argue similarly to the case $k=0$. Take disjoint closed nontrivial balls $P_1,\dots,P_l\subset W$. Then $W - \bigcup\limits^l_{i=1}P_i$ is infinite, so we pick distinct $b_{1,k+1},\dots, b_{m,k+1}$ in this set and put $$B_{j,k+1}: = B_{jk} \cup \{ b_{j,k+1}\},~ \alpha_{i,k+1}: = \alpha_{ik} \cup \{ {\bf I} P_i\},~ A_{i,k+1}: = A_{ik} \cup {\bf I} P_i.$$ In the case (a) all the required properties hold for $(k+1)$ by the construction.
In the case (b) we have to check only (1), (2), (6), (8)--(11).
(8) holds, since by construction we have \begin{align*} P_i\subset W\subset& X_{k+1}-{\bf C} Y_k\subset X_{k+1}-A_{ik};\\ b_{j,k+1}\in W \subseteq& X_{k+1},~ b_{j,k+1}\in (B_{j,k+1}-B_{jk}). \end{align*}
(1): From IH it is clear that $\alpha_{i,k+1}$ is a finite set of open balls and their closures are disjoint; note that $P_i \cap {\bf C} A_{ik} = {\varnothing}$, since $P_i \subseteq W \subseteq -{\bf C} A_{ik}.$
(2): We have \begin{align*} &{\bf C} A_{i,k+1} \cap {\bf C} A_{i^\prime,k+1} = ({\bf C} A_{ik} \cup P_i) \cap ({\bf C} A_{i^\prime k}\cup P_{i^\prime}) =\\ =&({\bf C} A_{ik} \cap {\bf C} A_{i^\prime k}) \cup ({\bf C} A_{ik} \cap P_{i'}) \cup ({\bf C} A_{i^\prime k} \cap P_i) \cup (P_i \cap P_{i'})
={\bf C} A_{ik} \cap {\bf C} A _{i^\prime k} = {\varnothing} \end{align*} by IH and by the construction; note that
$P_i, P_i' \subseteq W \subseteq -{\bf C} Y_k$.
(6): We have $$ A_{i,k+1} \cap B_{j,k+1} = (A_{ik} \cap B_{jk} ) \cup ({\bf I} P_i \cap \set{b_{j,k+1}}) \cup ( A_{ik} \cap \set{ b_{j,k+1}}) \cup ({\bf I} P_i \cap B_{jk}) = {\varnothing} $$ by IH and since $b_{j,k+1}\not\in P_i,~ b_{j,k+1}\in W \subseteq X-Y_k$, $P_i \subset W \subseteq X-B_{jk}$ .
(9): We have $A_{i,k+1} = A_{ik} \cup {\bf I} P_i \subseteq -B,$ since $A_{ik} \subseteq -B$ by IH, and $P_i \subset W \subseteq -B$ by the construction.
Likewise, (10) follows from $B_{jk} \subseteq -B$ and $b_{j,k+1}\in W \subseteq -B$.
To check (11), assume $j \not= j'$. We have $B_{j',k+1} \cap B_{j,k+1} = B_{j'k} \cap B_{jk}$, since $b_{j^\prime,k+1} \not= b_{j,k+1} , ~ b_{j,k+1}\in W \subseteq -B_{j^\prime k}$ and $b_{j^\prime,k+1}\in W\subseteq -B_{jk}$. Then apply IH.
Therefore the required sets $A_{ik}, B_{jk}$ are constructed. Now put $$\alpha_i: =\bigcup_k \alpha_{ik},~ A_i:= \bigcup \alpha_i=\bigcup_k A_{ik}, ~ B_j: =\bigcup_k B_{jk},$$ $$ B^\prime _1: = X - (\bigcup_i A_i \cup \bigcup_j B_j),$$ and define a map $g: X\longrightarrow \Phi_{ml}$ as follows: \[ g(x):=\left\{ \begin{array}{ll} a_i & \mbox{if } x\in A_i, \\ b_j & \mbox{if } x\in B_j,~j\neq 1,\\ b_1 & \mbox{otherwise (i.e., for } x\in B^\prime _1). \\ \end{array} \right.
\] By (2), (3), (5), (6), (11), $g$ is well defined; by (9), (10) $B \subseteq g^{-1}(b_1)$.
To prove that $g$ is a d-morphism, we check some other properties. $$\leqno(12)\quad X - \bigcup\limits^l_{i=1}A_i \subseteq {\bf d} B_j.$$
In fact, take an arbitrary $x\not\in \bigcup\limits^l_{i=1}A_i $ and show that $x\in {\bf d} B_j$, i.\/e., $$\leqno(13)\quad(U-\{ x\}) \cap B_j \not= {\varnothing} . $$ for any neighbourhood $U$ of $x$. First assume that $x\not\in B_j$. Take a basic open $X_{k+1}$ such that $x\in X_{k+1} \subseteq U$. Then \mbox{$X_{k+1}\not\subseteq \bigcup\limits^l_{i=1}A_i$,} and (8) implies $B_{j,k+1} \cap X_{k+1} \not= {\varnothing} .$ Thus $B_j \cap U \not= {\varnothing}$. So we obtain (13).
Suppose $x\in B_j$; then $x\in B_{jk}$ for some $k$. Since $\mathfrak X$ is dense-in-itself and $\set{X_1,\,X_2,\dots}$ is its open base, $\setdef[X_{s+1}]{s\geq k}$ is also an open base (note that every ball in $\mathfrak X$ contains a smaller ball). So $x\in X_{s+1}\subseteq U$ for some $s\ge k$. Since $x\not\in \bigcup\limits^l_{i=1}A_i $, we have $X_{s+1}\not\subseteq\bigcup\limits^l_{i=1}A_i$, and so $(B_{j,s+1}-B_{js}) \cap X_{s+1} \not= {\varnothing}$ by (8); thus $(B_j-B_{js}) \cap U \not= {\varnothing}$. Now $x\in B_{jk} \subseteq B_{js}$ implies (13).
$$\leqno(14)\qquad {\bf d} B_j \subseteq X -\bigcup\limits^l_{i=1}A_i.$$ In fact, $B_j\subseteq -A_i$, by (3), (5), (6). So ${\bf d} B_j \subseteq {\bf d} (-A_i) \subseteq -A_i$, since $A_i $ is open.
Similarly we obtain $$\leqno(15)\qquad {\bf d} B^\prime _1\subseteq X -\bigcup\limits^l_{i=1}A_i, \qquad {\bf d} A_i \subseteq X- \bigcup\limits_{r\not=i}A_r.$$ Also note that $$\leqno(16)\qquad A_i \subseteq {\bf d} A_i,$$ since $A_i$ is open, $\mathfrak X$ is dense-in-itself. Similary to (12) we have $$\leqno(17)\qquad \alpha_i \mbox{ is dense at every point of }B_j, B'_1\ \mbox{ (and thus $B_j,\ B'_1 \subseteq {\bf d} A_i$}).$$
To conclude that $g$ is a d-morphism, note that $$g^{-1}(a_i) = A_i, ~g^{-1}(b_j) = B_j~ (\mbox{for }j\not= 1),~ g^{-1}(b_1) = B^\prime _1,$$ and so by (15), (16), (17) \begin{align*} {\bf d} g^{-1}(a_i) &= {\bf d} A_i = X - \bigcup\limits_{r\not=i}A_r = g^{-1}(R^{- 1}(a_i)), \end{align*} and by (12), (14), (15) \begin{align*} {\bf d} g^{-1}(b_j) &= {\bf d} B_j = X - \bigcup\limits^l_{i=1}A_i = g^{-1}(R^{- 1}(b_j))\ \ \mbox{(for $j\ne 1$)},\\ \qquad\qquad\qquad {\bf d} g^{-1}(b_1) &= {\bf d} B^\prime _1= X - \bigcup\limits^l_{i=1}A_i = g^{-1}(R^{- 1}(b_1)). \qedhere \end{align*} \end{proof} \begin{figure}
\caption{Frame $\Phi_{ml}$.}
\label{fig:Phiml}
\end{figure}
For the proof see Appendix.
\begin{lem}\label{refroot} Assume that \begin{enumerate} \item $\mathfrak X$ is a dense-in-itself separable metric space, \item $B\subset X$ is closed nowhere dense, \item $F = C\cup F_1\cup \dots\cup F_l$ is a $\lgc{D4}$-frame, where $C = \{ b_1,\dots,b_m\} $ is a non-degenerate root cluster, $F_1,\dots,F_l$ are the subframes generated by the successors of $C$, \item for any nonempty open ball $U$ in $\mathfrak X$, for any $i\in\{1,\ldots,l\}$ there exists a d-morphism $f_i^U:U\twoheadrightarrow^d F_i$. \end{enumerate} Then there exists $f:\mathfrak X\twoheadrightarrow^d F$ such that $f(B)=\{b_1\}$. \end{lem} \begin{proof} First, we construct $g: \mathfrak X\twoheadrightarrow^d \Phi_{ml}$ according to Proposition \ref{P54}. Then $B\subseteq g^{-1}(b_1)$ and $A_i = g^{-1}(a_i)$ is the union of a set $\alpha_i$ of disjoint open balls. Then put \begin{equation}\label{Eq:fPhiml} f(x):=\left\{ \begin{array}{ll} g(x) & \mbox{if } g(x)\in C, \\ f_i^U(x)& \mbox{if } x\in U,~U\in\alpha_i.\\ \end{array} \right. \end{equation}
Since $g$ and $f_i^U$ are surjective, the same holds for $f$. So let us show $${\bf d} f^{-1}(a) = f^{-1}(R^{-1}(a))$$ ($R$ is the accessibility relation on $F$). First suppose $a\in C$. Then (since $g$ is a d-morphism) $${\bf d} f^{-1}(a) = {\bf d} g^{-1}(a) = g^{-1}(C) = f^{-1}(C) = f^{-1}(R^{-1}(a)). $$ Now suppose $a \notin C$, $I = \setdef[i]{a \in F_i}$, and let $R_i$ be the accessibility relation on $F_i$. We have: \begin{align*} f^{-1}(a) =&\bigcup\limits_{i\in I}\bigcup\limits_{U\in\alpha_i}(f_i^U)^{-1}(a),& R^{-1}(a) =& C \cup \bigcup\limits_{i\in I} R_i^{-1}(a), \end{align*} and so $$ f^{-1}(R^{-1}(a)) =
g^{-1}(C) \cup \bigcup\limits_{i\in I}\bigcup\limits_{U\in\alpha_i} (f_i^U)^{-1}(R_i^{-1}(a)).
$$ Since $f_i^U$ is a d-morphism, \begin{equation}\label{eq:L69:2} f^{-1}(R^{-1}(a)) = g^{-1}(C) \cup\bigcup\limits_{i\in I}\bigcup\limits_{U\in\alpha_i} {\bf d}_U((f_i^U)^{-1}(a))
\subseteq g^{-1}(C) \cup {\bf d} f^{-1}(a). \end{equation}
Let us show that \begin{equation}\label{eq:L69:3} g^{-1}(C) \subseteq {\bf d} f^{-1}(a).
\end{equation}
In fact, let $x\in g^{-1}(C)$. Since $\alpha_i$ is dense at $x$, every neighbourhood of $x$ contains some $U\in\alpha_i$. Since $f^U_i$ is surjective, $f(u) = f^U_i (u) = a$ for some $u\in U$. Therefore, $x\in{\bf d} f^{-1}(a)$.
(\ref{eq:L69:2}) and (\ref{eq:L69:3}) imply $f^{-1}(R^{-1}(a)) \subseteq {\bf d} f^{-1}(a)$. Let us prove the converse: \begin{equation}\label{eq:L69:5} {\bf d} f^{-1}(a) \subseteq f^{-1}(R^{-1}(a)). \end{equation}
We have $A_j \cap f^{-1}(a)={\varnothing}$ for $j\notin I$ and $A_j$ is open, hence $A_j \cap {\bf d} f^{-1}(a)={\varnothing}$. Thus ${\bf d} f^{-1}(a) \subseteq g^{-1}(C) \cup A_i$. Now $g^{-1}(C)\subseteq f^{-1}(R^{-1}(a))$ by (\ref{eq:L69:2}), so it remains to show that for any $i\in I$ \begin{equation}\label{eq:L69:7} {\bf d} f^{-1}(a)\cap A_i\subseteq f^{-1}(R^{-1}(a)). \end{equation}
To check this, consider any $x\in{\bf d} f^{-1}(a)\cap A_i$. Then $x\in U$ for some $U\in \alpha_i$, and thus by \ref{dY} and (\ref{eq:L69:2}) $x\in{\bf d} f^{-1}(a)\cap U={\bf d}_U(f^{-1}(a)\cap U)={\bf d}_U(f^U_i)^{-1}(a)\subseteq f^{-1}(R^{-1}(a))$. This implies (\ref{eq:L69:7}) and completes the proof of (\ref{eq:L69:5}). {\hspace*{\fill} } \end{proof}
Recall that $\partial$ denotes the boundary of a set in a topological space: $\partial A := {\bf C} A - {\bf I} A$.
\begin{lem}\label{Glue}{\em (Glueing lemma)} Let $\mathfrak X$ be a local $T_1$-space satisfying
\noindent (a) $X=X_1\cup Y\cup X_2$ for closed nonempty subsets $X_1,Y,X_2$ such that \begin{itemize} \item $X_1\cap X_2=X_1\cap{\bf I} Y= X_2\cap{\bf I} Y ={\varnothing}$, \item $\partial X_1\cup\partial X_2=\partial Y$, \item ${\bf d}{\bf I} Y=Y$ (i.e., $Y$ is regular and dense in-itself). \end{itemize} or
(b) $X=X_1\cup X_2$ is a nontrivial closed partition.
Let $F=(W,R)$ be a finite ${\bf K4}$-frame, $F_1=(W_1,R_1),~ F_2=(W_2,R_2)$ its generated subframes such that $W=W_1\cup W_2$ and suppose there are d-morphisms $f_i:\mathfrak X_i\twoheadrightarrow^d F_i$, $i=1,\,2$, where $\mathfrak X_i$ is the subspace of $\mathfrak X$ corresponding to $X_i$.
In the case (a) we also assume that $F_1, F_2$ have a common maximal cluster $C$, $f_i(\partial X_i)\subseteq R^{-1}(C)$ for $i=1,2$ and there is $g:{\bf I} Y\twoheadrightarrow^d C$ (where $C$ is regarded as a frame with the universal relation, ${\bf I} Y$ as a subspace of $\mathfrak X$). Then $f_1\cup f_2\cup g:\mathfrak X\twoheadrightarrow^d F$ in the case (a), $f_1\cup f_2:\mathfrak X\twoheadrightarrow^d F$ in the case
(b).\footnote{$f_1\cup f_2$ is the map $f$ such that $f|X_i=f_i$; similarly for $f_1\cup f_2\cup g$.} \end{lem}
\begin{figure}
\caption{Case (a)}
\label{fig:glueing_lemma}
\end{figure}
\begin{proof} Let $f:= f_1\cup f_2\cup g $ (or $f:= f_1\cup f_2$), $ F_i=(W_i,R_i)$, ${\bf d}:={\bf d}_X,~{\bf d}_i:={\bf d}_{X_i}$. For $w\in W$ there are four options.
(1) $w\in W_1-W_2$. Then ${\bf d} f^{-1}(w)={\bf d} f_1^{-1}(w)={\bf d}_1 f_1^{-1}(w) = f_1^{-1}(R_1^{-1}(w))$ (since $X_1$ is closed and $f_1$ is a d-morphism). It remains to note that $R_1^{-1}(w)=R^{-1}(w)\subseteq W_1-W_2$, and thus $f_1^{-1}(R_1^{-1}(w))= f^{-1}(R^{-1}(w)).$
(2) $w\in W_2-W_1$. Similar to the case (1).
(3) $w\in (W_1\cap W_2)-C$ in the case (a) or $w\in W_1\cap W_2$ in the case (b). Then $f^{-1}(w)=f_1^{-1}(w)\cup f_2^{-1}(w)$, so similarly to (1), \[ {\bf d} f^{-1}(w)= {\bf d}_1 f_1^{-1}(w)\cup {\bf d}_2 f_2^{-1}(w) = f_1^{-1}(R_1^{-1}(w))\cup f_2^{-1}(R_2^{-1}(w))= f^{-1}(R^{-1}(w)). \]
(4) $w\in C$ in case (a). First note that ${\bf d} g^{-1}(w)=Y$. In fact, $g$ is a d-morphism onto the cluster $C$, so ${\bf d}_{{\bf I} Y} g^{-1}(w)= g^{-1}(C)={\bf I} Y$. Hence ${\bf I} Y\subseteq {\bf d} g^{-1}(w )\subseteq {\bf d}{\bf I} Y=Y$, and thus $$Y={\bf d}{\bf I} Y\subseteq{\bf d}{\bf d} g^{-1}(w ) \subseteq {\bf d} g^{-1}(w )$$ by \ref{L310}(2).
Next, since $X_1$, $X_2$ are closed and $f_1$, $f_2$ are d-morphisms we have
\begin{align*} {\bf d} f^{-1}(w) &= {\bf d} f_1^{-1}(w)\cup {\bf d} f_2^{-1}(w)\cup {\bf d} g^{-1}(w) = {\bf d}_1 f_1^{-1}(w)\cup {\bf d}_2 f_2^{-1}(w)\cup Y =&\\ &= f_1^{-1}(R_1^{-1}(w))\cup f_2^{-1}(R_2^{-1}(w))\cup Y= f^{-1}(R^{-1}(w)). &\qedhere \end{align*}
\end{proof}
The case (b) of the previous lemma can be generalized as follows. \begin{lem}\label{L67} Suppose a topological space $\mathfrak X$ is the disjoint union of open subspaces: $\mathfrak X=\bigsqcup\limits_{i\in I}\mathfrak X_i$. Suppose a Kripke ${\bf K4}$-frame $F$ is the union of its generated subframes: $F=\bigcup\limits_{i\in I}F_i$ and suppose $f_i:\mathfrak X_i\twoheadrightarrow^{d}F_i$. Then $\bigcup\limits_{i\in I}f_i:\mathfrak X\twoheadrightarrow^{d}F$. \end{lem}
\begin{defi}\label{def_pmorphism}
Let $\mathfrak X$ be a topological space,
$F= (W, R, R_D)$ be a frame. Then a surjective map $f: X \longrightarrow W$ is called a
{\em dd-morphism}
(in symbols, $f:\mathfrak X \twoheadrightarrow^{dd} F$) if \begin{enumerate}
\item $f:\mathfrak X \pmor^d{} (W, R)$ is a d-morphism ;
\item $f:(X, \ne_X) \twoheadrightarrow (W, R_D)$ is a p-morphism of Kripke frames. \end{enumerate} \end{defi}
\begin{lem}\label{lem_pmorphism}
If $f:\mathfrak X \twoheadrightarrow^{dd} F$, then ${{\bf L}}{\bf d}_{\neq}(\mathfrak X) \subseteq {{\bf L}}(F)$ and for any closed 2-modal $A$ \[ \mathfrak X\vDash A\Leftrightarrow F\vDash A. \] \end{lem}
\begin{proof} Similar to \ref{P52} and \ref{p1}. {\hspace*{\fill} } \end{proof}
\begin{defi}
A set-theoretic map $f:X\longrightarrow Y$ is called {\em n-fold at} $y\in Y$ if $|f^{-1}(y)|=n$;\footnote{$|\ldots|$ denotes the cardinality.} $f$ is called {\em manifold at} $y$ if it $n$-fold for some $n>1$. \end{defi}
\begin{propo}\label{lem:dmortodDmor}~
(1) Let $G = (X, \ne_X)$, $F = (W, S)$ be Kripke frames such that $\overline{S}=W^2$, and let $f : X \longrightarrow W$ be a surjective function. Then $$f : G \twoheadrightarrow F \quad\;\mbox{iff}\;\quad f\mbox{ is manifold exactly at }S\mbox{-reflexive points of }F.$$ (2) Let $\mathfrak X$ be a $T_1$-space, $F= (W, R, R_D)$ a rooted $\lgc{KT_1}$-frame, $f:\mathfrak X\twoheadrightarrow^d(W,R)$. Then $f:\mathfrak X\twoheadrightarrow^{dd}F$ iff for any strictly $R$-minimal $v$ \[ v R_D v\Leftrightarrow f\mbox{ is manifold at }v. \] (3) If $\mathfrak X$ is a $T_1$-space, $f: \mathfrak X \pmor^d{} F=(W, R)$ and $R^{-1}(w)\neq{\varnothing}$ for any $w\in W$, then $f:\mathfrak X \twoheadrightarrow^{dd} F_{\forall}$, where $F_{\forall}:= (W, R, W^2)$.
\end{propo}
\begin{proof} (1) Note that $f$ is a p-morphism iff for any $x\in X$ \[ f(X-\{x\})=S(f(x))= \begin{cases} W & {\rm if~}f(x)Sf(x),\nonumber\\ W-\{f(x)\} & {\rm otherwise.}\nonumber
\end{cases} \]
(2) By (1), $f:\mathfrak X\twoheadrightarrow^{dd}F$ iff \[ \forall v\in W(v R_D v\Leftrightarrow \abs{f^{-1}(v)} > 1). \] The latter equivalence holds whenever $ R^{-1}(v)\neq{\varnothing}$. In fact, then by Corollary \ref{L52}, ${\bf d} f^{-1} (v) = f^{-1}(R^{-1}(v)) \neq{\varnothing}$, and thus $f^{-1}(v)$ is not a singleton (since $\mathfrak X$ is a $T_1$-space). $ R^{-1}(v)\neq{\varnothing}$ also implies $vR_Dv$, by Proposition \ref{pr:DS_AT1}.
(3) follows from (2). {\hspace*{\fill} } \end{proof}
After we have proved the main technical results, in the next sections we will study dd-logics of specific spaces.
\section{$\lgc{D4}$ and $\lgc{DT_1}$ as logics of zero-dimensional dense-in-themselves spaces}
In this section we will prove the d-completeness of $\lgc{D4}$ and dd-completeness of $\lgc{DT_1}$ w.r.t. zero-dimensional spaces. The proof follows rather easily from the previous section and an additional technical fact (Proposition \ref{P61}) similar to the McKinsey--Tarski lemma.
Recall that a (nonempty) topological space $\mathfrak X$ is called {\em zero-dimensional} if clopen sets constitute its open base \cite{A77}. Zero-dimensional $T_1$-spaces with a countable base are subspaces of the Cantor discontinuum, or of the set of irrationals \cite{K66}.
\begin{lem}\label{L71} Let $\mathfrak X$ be a zero-dimensional dense-in-itself Hausdorff space. Then for any $n$ there exists a nontrivial open partition $\mathfrak X=\mathfrak X_1\sqcup\ldots\sqcup\mathfrak X_n$, in which every $\mathfrak X_i$ is also a zero-dimensional dense-in-itself Hausdorff space. \end{lem}
\begin{proof} It is sufficient to prove the claim for $n=2$ and then apply induction. A dense-in-itself space cannot be a singleton, so there are two different points $x,y\in X$. Since $\mathfrak X$ is $T_1$ and zero-dimensional, there exists a clopen $U$ such that $x\in U,~y\not\in U$. So $X=U\cup (X-U)$ is a nontrivial open partition. The Hausdorff property, density-in-itself, zero-dimensionality are inherited for open subspaces. {\hspace*{\fill}} \end{proof}
\begin{propo}\label{P61} Let $\mathfrak X$ be a zero-dimensional dense-in-itself metric space, $y\in X$. Let $\Psi_l$ be the frame consisting of an irreflexive root $b$ and its reflexive successors $a_0,\dots,a_{l-1}$ (Fig. \ref{fig:Phiml_irreflexive_root}).
\begin{figure}
\caption{Frame $\Psi_l$.}
\label{fig:Phiml_irreflexive_root}
\end{figure}
Then there exists $f: \mathfrak X\twoheadrightarrow^d \Psi_l$ such that $f(y) = b$ and for every $i$ there is an open partition of $f^{-1}(a_i)$, which is dense at y. \end{propo} \begin{proof} Let $O(a,r) := \setdef[x\in X]{\rho(a,x)<r}$, where $\rho$ is the distance in $\mathfrak X$.
There exist clopen sets $Y_0, Y_1, \dots$ such that $$\set{y}\subset\dots\subset Y_{n+1} \subset Y_n \subset\dots Y_1 \subset Y_0 = X$$ \ and $Y_n \subseteq O(y, 1/n)$ for $n >0$.
These $Y_n$ can be easily constructed by induction. Then $$\bigcap\limits_n Y_n =\set{y}, \ \hbox{and}\ X-\{ y\} =\bigsqcup\limits_n X_n, $$ where $X_n = Y_n-Y_{n+1}$. Note that the $X_n$ are nonempty and open, $X_n \subseteq O(y, 1/n)$ for $n>0$.
Now define a map $f: X\longrightarrow\Psi_l$ as follows: \[ f(x)=\left\{ \begin{array}{ll} a_{r(n)} & \mbox{if } x\in X_n; \\ b & \mbox{if } x=y,\\ \end{array} \right.
\] where $r(n)$ is the remainder of dividing $n$ by $l$; it is clear that $f$ is surjective.
Let us show that for any $x$,
\[ x\in {\bf d} f^{-1}(u) \mbox{ iff }f(x)Ru. \eqno(*) \]
(i) Assume that $u=a_j$. Then
$f^{-1}(u)=\bigcup\limits_n X_{nl+j}$, and $$f(x)Ru \mbox{ iff }(f(x)=b \mbox{ or } f(x)=u).$$ To prove `if' in (*), consider two cases.
1. Suppose $f(x)=u, ~x\in X_{nl+j}$. Since $X_{nl+j}$ is nonempty and open, it is dense-in-itself, and thus $x\in {\bf d} X_{nl+j}\subseteq {\bf d} f^{-1}(u)$.
2. Suppose $f(x)=b$, i.e. $x=y$. Then $x\in {\bf d} f^{-1}(u)$, since $X_{nl+j}\subseteq O(y, 1/n)$.
The previous argument also shows that $\{ X_{nl+j}\mid n\geq 0\}$ is an open partition of $f^{-1}(a_j)$, which is dense at $y$.
To prove `only if', suppose $f(x)Ru$ is not true. Then $f(x) = a_k$ for some $k \not= j$, and so for some $n$, $x\in X_n$, $X_n\cap f^{-1}(u)={\varnothing}$. Since $X_n$ is open, $x\not\in {\bf d} f^{-1}(u)$.
(ii) Assume that $u = b$. Then $f^{-1}(u) = \{ y\}$, and so ${\bf d} f^{-1}(u) = {\varnothing} = f^{- 1}(R^{-1}(u))$. {\hspace*{\fill} }\end{proof}
\begin{propo}\label{P64} Let $\mathfrak X$ be a zero-dimensional dense-in-itself separable metric space, $F$ a finite rooted ${\bf D4}$-frame. Then there exists a d-morphism $\mathfrak X\twoheadrightarrow^dF$, which is 1-fold at the root of $F$ if this root is irreflexive. \end{propo} \begin{proof} By induction on the size of $F$.
(i) If $F$ is a finite cluster, the claim follows from Proposition \ref{P54}.
(ii) If $F = C\cup F_1\cup \dots\cup F_l$, where $C = \{ b_1,\dots,b_m\} $ is a non-degenerate root cluster, $F_1,\dots,F_l$ are the subframes generated by the successors of $C$, we can apply Lemma \ref{refroot}. In fact, every open ball $U$ in $\mathfrak X$ is zero-dimensional and dense-in-itself.
(iii) Suppose $F=\breve{b}\cup F_0\cup\dots\cup F_{l-1}$, where $b$ is an irreflexive root of $F$, $F_i$ are the subframes generated by the successors of $b$. There exists $g: X\twoheadrightarrow^d \Psi_l$ by \ref{P61}, with an arbitrary
$y\in X$. Then $g^{-1}(a_i)$ is a union of a set $\alpha_i$ of disjoint open sets, and $\alpha_i$ is dense at $y$. If $U\in\alpha_i$, then by IH, there exists $f^U_i: U\twoheadrightarrow^d F_i$. Put \[ f(x)=\left\{ \begin{array}{ll} b & \mbox{ if } x=y; \\ f_i^U(x)& \mbox{ if } x\in U,~U\in\alpha_i.\\ \end{array} \right.
\] Then similarly to Lemma \ref{refroot} it follows that $f: X\twoheadrightarrow^d F$.
Finally note that if the root of $F$ is irreflexive, the first step of the construction is case (iii), so the preimage of the root is a singleton. {\hspace*{\fill} } \end{proof} \begin{theorem}\label{T65} If $\mathfrak X$ is a zero-dimensional dense-in-itself separable metric space, then ${{\bf L}}{\bf d} (\mathfrak X)={\bf D4}$. \end{theorem} \begin{proof} By Propositions \ref{P64} and \ref{P52} ${{\bf L}}{\bf d} (\mathfrak X) \subseteq {{\bf L}} (F)$ for any finite rooted ${\bf D4}$-frame $F$, thus ${{\bf L}}{\bf d} (\mathfrak X) \subseteq {\bf D4}$, since ${\bf D4}$ has the fmp. By Lemma \ref{L310} ${\bf D4}\subseteq{{\bf L}}{\bf d} (\mathfrak X)$ . {\hspace*{\fill} }\end{proof}
\begin{lem}\label{P741} Let $\mathfrak X$ be a zero-dimensional dense-in-itself separable metric space, $F$ a finite ${\bf D4}$-frame. Then there exists a d-morphism $\mathfrak X\twoheadrightarrow^d F$, which is 1-fold at all strictly minimal points. \end{lem}
\begin{proof}
$F = F_1\cup\ldots\cup F_n$ for different finite rooted ${\bf D4}$-frames $F_i$. By Lemma \ref{L71}, $\mathfrak X=\mathfrak X_1\sqcup\ldots\sqcup\mathfrak X_n$ for zero-dimensional dense-in-themselves subspaces $\mathfrak X_i$, which are also metric and separable. By Proposition \ref{P64}, we construct $f_i:\mathfrak X_i\twoheadrightarrow^d F_i$. Then by Lemma \ref{L67}, $\bigcup\limits_{i=1}^nf_i:\mathfrak X\twoheadrightarrow^d F$. Every strictly minimal point of $F$ is an irreflexive root of a unique $F_i$, so its preimage is a singleton. {\hspace*{\fill} } \end{proof}
\begin{propo}\label{P:dd-morph_zero_dem} Let $\mathfrak X$ be a zero-dimensional dense-in-itself separable metric space, $F \in \mathfrak{F}_0$ a finite $\lgc{DT_1}$-frame. Then there exists a dd-morphism $\mathfrak X\twoheadrightarrow^{dd}F$. \end{propo} \begin{proof} We slightly modify the proof of the previous lemma. Let $F=(W,R,R_D)$, $G=(W,R)$. Then $G = G_1\cup\ldots\cup G_n$ for different cones $G_i$. We call $G_i$ {\em special} if its root is strictly $R$-minimal and $R_D$-reflexive. We may assume that exactly $G_1,\ldots,G_m$ are special. Then we count them twice and present $G$ as $G_1\cup G'_1\cup\ldots\cup G_m\cup G'_m\cup G_{m+1}\cup\ldots\cup G_n$, where $G'_i=G_i$ for $i\leq m$ (or as $G_1\cup G'_1\cup\ldots\cup G_m\cup G'_m$ if $m=n$).
Now we can argue as in the proof of Lemma \ref{P741}. By Lemma \ref{L71}, $\mathfrak X=\mathfrak X_1\sqcup\mathfrak X'_1\sqcup\ldots\sqcup\mathfrak X_m\sqcup \mathfrak X'_m\sqcup \mathfrak X_{m+1}\sqcup\ldots\sqcup \mathfrak X_n$ for zero-dimensional dense-in-itself separable metric $\mathfrak X_i, \mathfrak X'_i$. By Proposition \ref{P64}, we construct the maps $f_i:\mathfrak X_i\twoheadrightarrow^d G_i$, $f'_i:\mathfrak X'_i\twoheadrightarrow^d G'_i$, which are 1-fold at irreflexive roots; hence by Lemma \ref{L67}, $f:\mathfrak X\twoheadrightarrow^d G$ for $f:= \bigcup\limits_{i=1}^nf_i\cup \bigcup\limits_{i=1}^mf'_i $.
Every strictly minimal point $a \in G$ is an irreflexive root of a unique $G_i$. If $a$ is $R_D$-irreflexive, then $G_i$ is not special, so $f^{-1}(a)=f_i^{-1}(a)$ is a singleton. If $a$ is $R_D$-reflexive, then $G_i$ is special, so $f^{-1}(a)= f_i^{-1}(a)\cup (f'_i)^{-1}(a)$, and thus $f$ is 2-fold at $a$. Therefore, $f:\mathfrak X\twoheadrightarrow^{dd} F$ by Proposition \ref{lem:dmortodDmor}. {\hspace*{\fill} } \end{proof}
\begin{lem}\label{LF1}
Let $M = (W, R, R_D, \varphi)$ be a rooted Kripke model over a basic frame\footnote{Basic frames were defined in Section 4.} validating $ AT_1$, $\Psi$ a set of 2-modal formulas closed under subformulas. Let $M'=(W',R',R_D',\theta')$ be a filtration of $M$ through $\Psi$ described in Lemma \ref{L26}\footnote{Recall that $R^\prime$ is the transitive closure of ${\underline{R}}$, $R_D'=\underline{R_D}$.}. Then the frame $(W',R',R_D')$ is also basic and validates $AT_1$. \end{lem}
\begin{proof} In fact, $R'$ is transitive by definition. For any two different $a, b\in W'$ we have $a R_D'b$, since $xR_Dy$ for any $x\in a,~y\in b$ (as $F\in\mathfrak{F}_0$).
Next, note that if $a$ is $ R_D'$-irreflexive, then $a=\{x\}$ for some $R_D$-irreflexive $x$. In this case, since $(W, R, R_D)\vDash AT_1$, there is no $y$ such that $yRx$ (Proposition \ref{pr:DS_AT1}), hence $(R')^{-1}(a)={\varnothing}$, and thus $(W',R',R_D')\vDash AT_1$.
Finally, $R'\subseteq R_D'$. In fact, all different points in $F'$ are $ R_D'$-related, so it remains to show that every $ R_D'$-irreflexive point is $R'$-irreflexive. As noted above, such a point is a singleton class $x^\sim=\{x\}$, where $x$ is $R_D$-irreflexive. Then $x$ is $R$-minimal, so in $W'$ there is no loop of the form $x^\sim{\underline{R}} x_1{\underline{R}}\ldots{\underline{R}} x^\sim$, and thus $x^\sim$ is $R'$-irreflexive. {\hspace*{\fill} } \end{proof}
By a standard argument Lemma \ref{LF1} implies
\begin{theorem}\label{Tfmp} Every logic of the form $\lgc{KT_1}+A$, where $A$ is a closed 2-modal formula, has the finite model property. \end{theorem}
\begin{proof}
Let $L$ be such a logic and suppose $L\,{\not\vdash}\, B$. By Proposition \ref{P53} $L$ is Kripke complete,
so by the Generation lemma there is a rooted Kripke frame $F=(W,R,R_D)$ such that $F\vDash L,~F{\,\not\mo\,} B$.
Then $F$ is basic by definition. Let $M=(F,\theta)$ be a Kripke model over $F$ refuting $B$. Let $\Psi$
be the set of all subformulas of $A$ or $B$, and let us construct the filtration
$M'=(W',R',R_D',\theta')$ of $M$ through $\Psi$ as in Lemmas \ref{L26}(2) and \ref{LF1}. By the previous
lemma, $F':= (W',R',R_D')\vDash\lgc{KT_1}$.
By the Filtration lemma, $M'{\,\not\mo\,} B$. By the same lemma, the truth of $A$ is preserved in $M'$, so $F'\vDash A$, since $A$ is closed. Therefore, $F'\vDash L$.
\end{proof}
\begin{theorem}\label{thm:DLogic_zerospace} Let $\mathfrak X$ be a zero-dimensional dense-in-itself separable metric space. Then ${{\bf L}}{\bf d}_{\neq} (\mathfrak X)=\lgc{DT_1}$. \end{theorem} \begin{proof} For any finite $\lgc{DT_1}$-frame $F$ we have ${{\bf L}}{\bf d}_{\neq} (\mathfrak X) \subseteq {{\bf L}} (F)$ by Proposition \ref{P:dd-morph_zero_dem} and Lemma \ref{lem_pmorphism}. By the previous theorem, $\lgc{DT_1}$ has the fmp, so ${{\bf L}}{\bf d}_{\neq} (\mathfrak X) \subseteq \lgc{DT_1}$. Since $\mathfrak X\vDash^d \lgc{DT_1}$ (Proposition \ref{P412}), it follows that ${{\bf L}}{\bf d}_{\neq} (\mathfrak X) = \lgc{DT_1}$. {\hspace*{\fill} } \end{proof}
\begin{propo}{\cite[Lemma 3.1]{BLB}}\label{prop:QtoF} Every countable\footnote{In this chapter, as well as in \cite{BLB}, `countable' means `of cardinality at most $\aleph_0$'.} rooted $\lgc{K4}$-frame is a d-morphic image of a subspace of ${\bf Q}$. \end{propo}
To apply this proposition to the language with the difference modality, we need to examine the preimage of the root for the constructed morphism. Fortunately, in the proof of Proposition \ref{prop:QtoF} in \cite{BLB} the preimage of a root $r$ is a singleton iff $r$ is irreflexive.
\begin{lem}\label{prop:QtoF_1fold}
Let $F$ be a countable $\lgc{K4}$-frame. Then there exists a d-morphism from a subspace of ${\bf Q}$ onto $F$, which is 1-fold at all strictly minimal points. \end{lem} \begin{proof} Similar to Lemma \ref{P741}. We can present $F$ as a countable union of different cones $\bigcup\limits_{i\in I}F_i$ and ${\bf Q}$ as a disjoint union $\bigsqcup\limits_{i\in I}\mathfrak X_i$ of spaces homeomorphic to ${\bf Q}$. By Proposition \ref{prop:QtoF} (and the remark after it), for each $i$ there exists $f_i: \mathcal Y_i \pmor^d{} F_i$ for some subspace $\mathcal Y_i \subseteq \mathfrak X_i$ such that $f_i$ is 1-fold at the root $r_i$ of $F_i$ if $r_i$ is irreflexive. Now by Lemma \ref{L67} $f:= \bigcup\limits_{i\in I}f_i: \bigsqcup\limits_{i\in I}\mathcal Y_i \twoheadrightarrow^{d}F$, and $f$ is 1-fold at all strictly minimal points of $F$ (i.e., the irreflexive $r_i$) --- since every $r_i$ belongs only to $F_i$, so $f^{-1}(r_i)=f_i^{-1}(r_i)$. {\hspace*{\fill} } \end{proof}
\begin{propo}\label{prop:subsetQtoF_ddmorph}
Let $F$ be a countable $\lgc{KT_1}$-frame. Then there exists a dd-morphism from a subspace of ${\bf Q}$ onto $F$. \end{propo}
\begin{proof} Similar to Proposition \ref{P:dd-morph_zero_dem}. If $F = (W, R, R_D)$, the frame $G=(W,R)$ is a countable union of different cones. There are two types of cones: non-special $G_i~ (i\in I)$ and special (with strictly $R$-minimal and $R_D$-reflexive roots) $H_j~ (j\in J)$: \[ G = \bigcup\limits_{i \in I} G_i \cup \bigcup\limits_{j \in J} H_j. \] Then we duplicate all special cones \[ G = \bigcup\limits_{i \in I} G_i \cup \bigcup\limits_{j \in J} H_j \cup \bigcup\limits_{j \in J} H'_j \] and as in the proof of \ref{prop:QtoF_1fold}, construct $f: \bigsqcup\limits_{i\in I}\mathcal Y_i\sqcup\bigsqcup\limits_{j\in J}\mathcal Z_j\sqcup\bigsqcup\limits_{j\in J}\mathcal Z'_j \twoheadrightarrow^{d}F$. This map is 1-fold exactly at all $R_D$-irreflexive points, so it is a dd-morphism onto $F$. {\hspace*{\fill} } \end{proof}
\begin{coro}\label{C713}
$\mathbf{Ld}_{\ne}(\mbox{all $T_1$-spaces}) = \lgc{KT_1}$. \end{coro}
\begin{proof} Note that $\lgc{KT_1}$ is complete w.r.t.~countable frames and every subspace of ${\bf Q}$ is $T_1$. {\hspace*{\fill} } \end{proof}
\begin{propo}\label{P714}
Let $\mathbf{\Lambda} = \lgc{KT_1} + \Gamma$ be a consistent logic, where $\Gamma$ is a set of closed formulas. Then $\mathbf{\Lambda}$ is dd-complete w.r.t.\/ subspaces of ${\bf Q}$. \end{propo} \begin{proof} Since every closed formula is canonical, $\mathbf{\Lambda}$ is Kripke complete. So for every formula $A \notin \mathbf{\Lambda}$ there is a frame $F_A$ such that $F_A \models \mathbf{\Lambda}$ and $F_A \nvDash A$. By Proposition \ref{prop:subsetQtoF_ddmorph}, there is a subspace $\mathfrak X_A \subseteq {\bf Q}$ and $f_A: \mathfrak X_A \twoheadrightarrow^{dd} F_A$. Then $\mathfrak X_A{\,\not\mo\,} A,~\mathfrak X_A\vDash\mathbf{\Lambda}$ by Lemma \ref{lem_pmorphism}. Therefore ${{\bf L}}{\bf d}_{\neq} (\mathcal K) = \mathbf{\Lambda}$ for $\mathcal K := \setdef[\mathfrak X_A]{A \notin \mathbf{\Lambda}}$. {\hspace*{\fill} } \end{proof}
\begin{rem}\rm A logic of the form described in Proposition \ref{P714} is dd-complete w.r.t. a set of subspaces of ${\bf Q}$. This set may be non-equivalent to a single subspace. For example, there is no subspace $\mathfrak X\subseteq{\bf Q}$ such that $\lgc{KT_1}={{\bf L}}{\bf d}_{\neq}(\mathfrak X) $. In fact, consider \[ A:=[\neq]\square\bot\wedge\square\bot. \] Then $A$ is satisfiable in $\mathfrak X$ iff $\mathfrak X\vDash^d A$ iff $\mathfrak X$ is discrete. So $A$ is consistent in $\lgc{KT_1}$. Now if $\lgc{KT_1}={{\bf L}}{\bf d}_{\neq}(\mathfrak X)$, then $A$ must be satisfiable in $\mathfrak X$, hence $\mathfrak X\vDash^d A$; but $\lgc{KT_1}\not\vdash A$, and so we have a contradiction. \end{rem}
\section{Connectedness}
Connectedness was the first example of a property expressible in cu-logic, but not in c-logic. The corresponding connectedness axiom from \cite{Sh99} will be essential for our further studies. In this section we show that it is weakly canonical, i.e., valid in weak canonical frames --- a fact not mentioned in \cite{Sh99}.
\begin{lem}\label{L80}\cite{Sh99} A topological space $\mathfrak X$ is connected iff $\mathfrak X\vDash^c AC$, where \[ AC :=[\forall](\square p\vee\square\neg p)\to [\forall] p\vee[\forall]\neg p. \] \end{lem}
For the case of Alexandrov topology there is an equivalent definition of connectedness in relational terms. \begin{defi}\label{D75} For a transitive Kripke frame $F = (W,R)$ we define the {\em comparability} relation $R^\pm:=R\cup R^{-1}\cup I_W$.
$F$ is called {\em connected} if the transitive closure of $R^\pm$ is universal. A subset $V\subseteq W$ is called {\em connected} in $F$ if the frame $F|V$ is connected.
A 2-modal frame $(W,R,S)$ is called \emph{(R)-connected} if $(W,R)$ is connected. \end{defi} Thus $F$ is connected iff every two points $x,y$ can be connected by a {\em non-oriented path} (which we call just a {\em path}), a sequence of points $x_0x_1\ldots x_n$ such that $x=x_0 R^\pm x_1\ldots R^\pm x_n=y$.
From \cite{Sh99} and Proposition \ref{P49} we obtain
\begin{lem}\label{L801} (1) For an ${\bf S4}$-frame $F$, the associated space $N(F)$ is connected iff $F$ is connected.\\ (2) For a ${\bf K4}$-frame $F$, $F_{\forall}\models AC^{\sharp u}$ iff $F$ is connected. \end{lem}
\begin{lem}\label{mcl} Let $M=(W,R,R_D,\theta)$ be a rooted generated submodel of m-weak canonical model for a modal logic $\mathbf{\Lambda}\supseteq{\bf K4D^+}$. Then \begin{enumerate} \item Every $R$-cluster in $M$ is finite of cardinality at most $2^m$. \item $(W,R)$ has finitely many $R$-maximal clusters.
\item For each $R$-maximal cluster $C$ in $M$ there exists an $m$-formula $\beta(C)$ such that: $$\forall x\in M~ (M,x \vDash \beta(C)\Leftrightarrow x\in\overline{R}^{-1} (C)).$$ \end{enumerate} \end{lem}
The proof is similar to \cite[Section 8.6]{CZ97}.
\begin{lem}\label{lem_KripkeCompletAC} Every rooted generated subframe of a weak canonical frame for a logic $\mathbf{\Lambda}\supseteq{\bf K4D^+} + AC^{\sharp u} $ is connected. \end{lem}
\begin{proof} Let $M$ be a weak canonical model for $\mathbf{\Lambda}$, $M_0$ its rooted generated submodel with the frame $F=(W,R,R_D)$, and suppose $F$ is disconnected. Then there exists a nonempty proper clopen subset $V$ in the space $N(W,\overline{R})$. Let $\Delta$ be the set of all $R$-maximal clusters in $V$ and put \[
B := \bigvee\limits_{C \in \Delta} \beta(C). \] Then $B$ defines $V$ in $M_0$, i.e., $V=\overline{R}^{-1}(\bigcup{\Delta})$. In fact, $\bigcup{\Delta}\subseteq V$ implies $\overline{R}^{-1}(\bigcup{\Delta}) \subseteq V $, since $V$ is closed. The other way round, $V\subseteq\overline{R}^{-1}(\bigcup{\Delta})$, since for any $v\in V$, $\overline{R}(v)$ contains an $R$-maximal cluster $C\in{\Delta}$, and $\overline{R}(v)\subseteq V$ as $V$ is open.
So $w \models B$ for any $w \in V$, and since $V$ is open, $w \models \overline{\square} B$. By the same reason, $w\models \overline{\square}\lnot B$ for any $w \not \in V $ . Hence \[ M_0 \models \left[\forall\right] (\overline{\square} B \lor \overline{\square} \lnot B). \] By Proposition \ref{cm} all substitution instances of $AC$ are true in $M_0$. So we have \[ M_0 \models \left[\forall\right] (\overline{\square} B \lor \overline{\square} \lnot B) \rightarrow \left[\forall\right] B \lor \left[\forall\right] \lnot B, \] and thus \[ M_0 \models \left[\forall\right] B \lor \left[\forall\right] \lnot B. \] This contradicts the fact that $V$ is a nonempty proper subset of $W$. {\hspace*{\fill} } \end{proof}
In d-logic instead of connectedness we can express some its local versions; they will be considered in the next section.
\section{Kuratowski formula and local 1-componency} In this section we briefly study Kuratowski formula distinguishing ${\bf R}$ from ${\bf R}^2$ in d-logic. Here the main proofs are similar to the previous section, so most of the details are left to the reader.
\begin{defi} We define {\em Kuratowski formula} as \[ Ku:= \quad \square (\overline{\square} p \vee \overline{\square} \neg p) \rightarrow \square p \vee \square \neg p. \] \end{defi}
The spaces validating $Ku$ are characterized as follows \cite{LB2}.
\begin{lem}\label{lem:th32_from_LB2} For a topological space $\mathfrak X$, $\mathfrak X\vDash^d Ku$ iff
\noindent for any $x\in X$ and any open neighbourhood $U$ of $x$, if $U-\{x\}$ is a disjoint union $V_1\cup V_2$ of sets open in the subspace $U-\{x\}$, then there exists a neighbourhood\footnote{In \cite{LB2} neighbourhoods are supposed open, but this does not matter here, since every neighbourhood contains an open neighbourhood.} $V \subseteq U$ of $x$ such that $V-\{x\} \subseteq V_1$ or $V-\{x\} \subseteq V_2$. \end{lem}
\begin{defi} A topological space $\mathfrak X$ is called {\em locally connected} if every neighbourhood of any point $x$ contains a connected neighbourhood of $x$. Similarly, $\mathfrak X$ is called {\em locally 1-component} if every punctured neighbourhood of any point $x$ contains a connected punctured neighbourhood of $x$. \end{defi}
It is well known \cite{A77} that in a locally connected space every neighbourhood $U$ of any point $x$ contains a connected {\em open} neighbourhood of $x$ (e.g. the connected component of $x$ in ${\bf I} U$).
\begin{lem}\label{LK1} If $\mathfrak X$ is locally 1-component, then $\mathfrak X\vDash^d Ku$. \end{lem}
The proof is straightforward, and we leave it to the reader.
\begin{lem}\label{LK2}~
(1) Every space d-validating $Ku$ has the following \emph{non-splitting property}:
(NSP) If an open set $U$ is connected, $x\in U$ and $U-\{x\}$ is open, then $U-\{x\}$ is connected.\\ (2) Suppose $\mathfrak X$ is locally connected and local $T_1$. Then (NSP) holds in $\mathfrak X$ iff $\mathfrak X$ is locally 1-component iff $\mathfrak X\vDash^d Ku$.
\end{lem}
\begin{proof} (1) We assume $\mathfrak X\vDash^d Ku$ and check (NSP). Suppose $U$ is open and connected, $U^\circ:=U-\{x\}$ is open, and consider a partition $U^\circ=U_1\cup U_2$ for open $U_1,U_2$. By \ref{lem:th32_from_LB2} there exists an open $V\subseteq U$ containing $x$ such that $V\subseteq\{x\}\cup U_1$ or $V\subseteq\{x\}\cup U_2$. Consider the first option (the second one is similar). We have a partition \[ U=(\{x\}\cup U_1)\cup U_2, \] and $\{x\}\cup U_1= V\cup U_1$, so $\{x\}\cup U_1$ is open. Hence by connectedness, $U= \{x\}\cup U_1$, i.e., $U^\circ = U_1$. Therefore, $U^\circ $ is connected.
(2) It suffices to show that (NSP) implies the local 1-componency. Consider $x\in X$ and its neighbourhood $U_1$. Since $\mathfrak X$ is local $T_1$, $U_1$ contains an open neighborhood $U_2$, in which $x$ is closed, i.e., ${\bf C}\{x\}\cap U_2=\{x\}$. By the local connectedness, $U_2$ contains a connected open neighbourhood $U_3$, and again ${\bf C}\{x\}\cap U_3=\{x\}$; thus $U_3-\{x\}$ is open. Eventually, $U_3-\{x\}$ is connected, by (NSP). {\hspace*{\fill} } \end{proof}
\begin{rem}\rm The {\em ($n$-th)
generalized Kuratowski formula} is the following formula in variables $p_0,\dots,p_n$ $$Ku_n := \square \bigvee\limits^n_{k=0}\overline{\square} Q_k \rightarrow \bigvee\limits^n_{k=0}\square\neg Q_k,$$ where $Q_k := p_k \wedge\bigwedge\limits_{j\not= k}\neg p_j$.
The formula $Ku_1$ is related to the equality found by Kuratowski \cite{K22}: $$(*)\quad{\bf d} ((x \cap {\bf d} (-x)) \cup (-x \cap {\bf d} x)) = {\bf d} x \cap {\bf d} (-x),$$ which holds in every algebra $DA({\bf R}^n)$ for $n>1$, but not in $DA({\bf R})$. This equality corresponds to the modal formula $$Ku^\prime:=\quad \Diamond ((p \wedge \Diamond \neg p) \vee (\neg p \wedge \Diamond p)) {\leftrightarrow}\Diamond p \wedge \Diamond \neg p,$$ and one can show that ${\bf D4} + Ku^\prime = {\bf D4} + Ku_1={\bf D4} + Ku.$ \end{rem}
\begin{rem}\rm The class of spaces validating $Ku_n$ is described in \cite{LB2}. In particular, it is valid in all locally $n$-component spaces defined as follows.
A neighbourhood $U$ of a point $x$ in a topological space is called {\em $n$-component at $x$} if the punctured neighbourhood $U-\set{x}$ has at most $n$ connected components. A topological space is called {\em locally $n$-component} if the $n$-component neighbourhoods at each of its point constitute a local base (i.e., every neighbourhood contains an $n$-component neighbourhood). \end{rem}
\begin{lem}\label{L77}\cite{LB2} For a transitive Kripke frame $(W,R)$
$(W,R) \vDash Ku$ iff for any $R$-irreflexive $x$, the subset $R(x)$ is connected (in the sense of Definition \ref{D75}). \end{lem}
\begin{theorem}\label{th:D4KU1_KripCompl} The logics ${\bf K4}+Ku$, ${\bf D4}+Ku$ are weakly canonical, and thus Kripke complete. \end{theorem}
A proof of \ref{th:D4KU1_KripCompl} based on Lemma \ref{L77} and a 1-modal version of Lemma \ref{mcl} is straightforward, cf. \cite{Sh90} or \cite{LB2} (the latter paper proves the same for $Ku_n$).
Hence we obtain \begin{theorem} The logic $\lgc{DT_1K}:=\lgc{DT_1}+Ku$ is weakly canonical, and thus Kripke complete. \end{theorem}
\begin{proof}
(Sketch.) For the axiom $Ku$ the argument from the proof of \ref{th:D4KU1_KripCompl} is still valid due to definability of all maximal clusters (Lemma \ref{mcl}). The remaining axioms are Sahlqvist formulas. {\hspace*{\fill} }\end{proof}
\begin{theorem}\label{th:DT_1CK_KripCompl} The logic $\lgc{DT_1CK}:=\lgc{DT_1K}+AC^{\sharp u}$ is weakly canonical, and thus Kripke complete. \end{theorem}
\begin{proof} We can apply the previous theorem and Lemma \ref{lem_KripkeCompletAC}. {\hspace*{\fill} }\end{proof}
Completeness theorems from this section can be refined: in the next section we will prove the fmp for the logics considered above.
\section{The finite model property of ${\bf D4K}$, $\lgc{DT_1K}$, and $\lgc{DT_1CK}$}
For the logic ${\bf D4}+Ku$ the first proof of the fmp was given in \cite{Sh90}. Another proof (also for ${\bf D4}+Ku_n$) was proposed by M. Zakharyaschev \cite{Z}; it is based on a general and powerful method.
In this section we give a simplified version of the proof from \cite{Sh90}. It is based on a standard filtration method, and the same method is also applicable to 2-modal logics $\lgc{DT_1K}$, $\lgc{DT_1CK}$.
\begin{theorem}\label{thm:fmp_DT_1CK} The logics $\lgc{DT_1K}$ and $\lgc{DT_1CK}$ have the finite model property. \end{theorem}
\begin{proof} Let $\mathbf{\Lambda}$ be one of these logics. Consider an $m$-formula $A\not\in\Lambda$. Take a generated submodel $M = (W, R, R_D, \varphi)$ of the $m$-restricted canonical model of $\Lambda$ such that $M,u{\,\not\mo\,} A$ for some $u$. As we know, its frame is basic and its $R$-maximal clusters are definable (Lemma \ref{mcl}).
Put \begin{align*} \Psi_0 &:= \setdef[\beta(C)]{C\ \hbox{is an $R$-maximal cluster in $M$}}, \\ \Psi_1 &:= \set{A} \cup \setdef[\overline{\square}\gamma]{\gamma\ \hbox{is a Boolean combination of formulas from $\Psi_0$}}, \\ \Psi &:= \hbox{the closure of $\Psi_1$ under subformulas.} \end{align*}
The set $\Psi$ is obviously finite up to equivalence in $\mathbf{\Lambda}$.
Take the filtration $M' = (W', R', R'_D, \varphi')$ of $M$ through $\Psi$ as in Lemma \ref{LF1}. By that lemma, $F':=(W', R', R'_D)\vDash {\bf KT_1}$. The seriality of $R'$ easily follows from the seriality of $R$.
Next, if $\mathbf{\Lambda}=\lgc{DT_1CK}$, the frame $(W, R, R_D)$ is connected by Lemma \ref{lem_KripkeCompletAC}. So for any $x,y \in W$ there is an $R$-path from $x$ to $y$. $aRb$ implies $a^\sim R'b^\sim$, so there is an $R'$-path from $x^\sim$ to $y^\sim$ in $F'$. Therefore $F'\vDash AC^{\sharp u}$. It remains to show that $F'\vDash Ku$. Consider an $R'$-irreflexive point $x^\sim \in W'$ and assume that $R' (x^\sim )$ is disconnected. Let $V$ be a nonempty proper connected component of $R'(x^\sim)$. Consider \begin{align*} \Delta &:= \setdef[C]{\exists y (y^\sim \in V \;\&\; C\subseteq R(y) \;\&\; C \hbox{ is an $R$-maximal cluster in $M$)}}; \\ B &:= \bigvee\limits_{C \in \Delta} \beta (C), \end{align*} where $\beta (C)$ is from Lemma \ref{mcl}. Note that \[ \leqno(1)\quad z\in C~\&~ C\in{\Delta}\Rightarrow z^\sim \in V. \] In fact, if $C\in{\Delta}$, then for some $y^\sim \in V$ we have $yRz$; hence $y^\sim R'z^\sim$, so $z^\sim \in V$, by the connectedness of $V$.
Let us show that for any $y^\sim \in R'(x^\sim)$ \[ \leqno(2)\quad M', y^\sim \models B \ \ \hbox{iff}\ \ M, y \models B \ \ \hbox{iff}\ \ y^\sim \in V, \] i.e., $B$ defines $V$ in $ R'(x^\sim)$.
The first equivalence holds by the Filtration Lemma, since $B \in \Psi_1$.
Let us prove the second equivalence. To show `if', suppose $y^\sim\in V$. By Lemma \ref{L82}, in the restricted canonical model there is a maximal cluster $C$ $R$-accessible from $y$; then $M, y \models \beta(C)$. We have $C \in \Delta$, and thus $M, y \models B$.
To show `only if', suppose $y^\sim \not\in V$, but $M, y \models B$. Then $M,y \models \beta (C)$, for some $C \in \Delta$, hence $C \subseteq R(y)$, i.e., $yRz$ for some (and for all) $z\in C$; so it follows that $y^\sim R' z^\sim$. Thus $y^\sim$ and $z^\sim$ are in the same connected component of $R'(x^\sim)$, which implies $z^\sim \not\in V$. However, $z^\sim \in V$ by (1), leading to a contradiction.
By Proposition \ref{cm} all substitution instances of $Ku$ are true in $M$. So \[ M\vDash Ku(B):=\square (\overline{\square} B \lor \overline{\square}\lnot B) \to \square B \lor \Box \lnot B. \]
Consider an arbitrary $y\in R(x)$. Then for any $z\in R(y)$, $y^\sim$ and $z^\sim$ are in the same connected component of $R'(x^\sim)$. Thus $y^\sim$ and $z^\sim$ are both either in $V$ or not in $V$, and so by (2), both of them satisfy either $B$ or $\lnot B$. Hence $M, y\models \overline{\square} B \lor \overline{\square}\lnot B$. Therefore, $x$ satisfies the premise of $Ku(B)$. Consequently, $x$ must satisfy the conclusion of $Ku(B)$. Thus $M,x \models \square B$ or $M,x \models \square \lnot B$. Since $\square B, \square \lnot B \in\Psi_1$, the Filtration Lemma implies $M', x^\sim\vDash \square B$ or $M', x^\sim\vDash \square\neg B$. Eventually by (2), $V=R'(x^\sim)$ or $V={\varnothing}$, which contradicts the assumption about $V$.
To conclude the proof, note that $A\in \Psi$, so by the Filtration Lemma $M',u^\sim\nvDash A$. As we have proved, $F' \models \mathbf{\Lambda}$. Therefore $\mathbf{\Lambda}$ has the fmp. {\hspace*{\fill} }\end{proof}
\begin{theorem}\label{T101} The logic ${\bf D4K}$ has the finite model property. \end{theorem}
\begin{proof} Use the argument from the proof of \ref{thm:fmp_DT_1CK} without the second relation. {\hspace*{\fill} }\end{proof}
Thanks to the fmp, we have a convenient class of Kripke frames for the logic $\lgc{DT_1CK}$. This will allow us to prove the topological completeness result in the next section.
\section{The dd-logic of $\mathbf R^n$, $n\ge 2$.}
This section contains the main result of the Chapter. The proof is based on the fmp theorem from the previous section and a technical construction of a dd-morphism presented in the Appendix.
In this section $\norm{\cdot}$ denotes the standard norm in $\mathbf R^n$, i.e. for $x\in \mathbf R^n$ \[ \norm{x} = \sqrt{x_1^2+\ldots+x_n^2}. \]
We begin with some simple observations on connectedness. For a path $\alpha = w_0 w_1 \ldots w_{n}$ in a $\lgc{K4}$-frame $(W,R)$ we use the notation $ \overline{R}(\alpha) := \bigcup\limits_{i=0}^n \overline{R} (w_{i})$. A path $\alpha$ is called {\em global} (in $F$) if $\overline{R} (\alpha)=W$.
\begin{lem}\label{globpath} Let $F = (W, R)$ be a finite connected ${\bf K4}$-frame, $w,v\in W$. Then there exists a global path from $w$ to $v$. \end{lem}
\begin{proof} In fact, in the finite connected graph $(W, R^\pm)$ the vertices $w,v$ can be connected by a path visiting all the vertices (perhaps, several times). {\hspace*{\fill} } \end{proof}
\begin{lem}\label{LPF0} Let $F = (W, R, R_D)$ be a finite rooted $\lgc{DT_1CK}$-frame. Then the set of all $R_D$-reflexive points in $F$ is connected. \end{lem}
\begin{proof} Let $x,y$ be two $R_D$-reflexive points. Since $(W,R)$ is connected, there exists a path connecting $x$ and $y$. Consider such a path $\alpha$ with the minimal number $n$ of $R_D$-irreflexive points, and let us show that $n=0$.
Suppose not. Take an $R_D$-irreflexive point $z$ in $\alpha$; then $\alpha=x\ldots uzv\ldots y$, for some $u,v$, and it is clear that $zRu$, $zRv$, since $z$ is strictly $R$-minimal. By Lemma \ref{L77}, $R(z)$ is connected, so $u$, $v$ can be connected by a path $\beta$ in $R(z)$. Thus in $\alpha$ we can replace the part $uzv$ with $\beta$, and the combined path $x\ldots\beta\ldots y$ contains $(n-1)$ $R_D$-irreflexive points, which contradicts the minimality of $n$. {\hspace*{\fill} } \end{proof}
\begin{lem}\label{lem:pathinaFrame} Let $F = (W, R, R_D)$ be a finite rooted $\lgc{DT_1CK}$-frame and let $w', w'' \in W$ be $R_D$-reflexive. Then there is a global path $\alpha = w_0 \ldots w_n$ in $(W,R)$ such that $w' = w_0$, $w_n = w''$ and all $R_D$-irreflexive points occur only once in $\alpha$. \end{lem}
\begin{proof}
Let $\set{u_1, \,\ldots, \;u_k}$ be the $R_D$-irreflexive points. By connectedness there exists paths $\alpha_0$, \ldots, $\alpha_{k}$ respectively from $w'$ to $u_1$, from $u_1$ to $u_2$, \ldots, from $u_k$ to $w''$.
By Lemma \ref{LPF0}, the set $W' := W - \set{u_1, \ldots, u_k}$ is connected.
Hence we may assume that each $\alpha_i$ does not contain $R_D$-irreflexive points except its ends. Also there exists a loop $\beta$ in $F':=F|W'$ from $w''$ to $w''$ such that \hbox{$W - \bigcup\limits_{i=1}^{k-1} \overline{R}(\alpha_i) \subseteq \overline{R}(\beta).$} \begin{figure}
\caption{Path $\alpha$.}
\label{fig:PathAlpha}
\end{figure} Then we can define $\alpha$ as the joined path $\alpha_0\ldots \alpha_k \beta$, (Fig. \ref{fig:PathAlpha}). {\hspace*{\fill} }\end{proof}
\begin{propo}\label{pr:dd-morphismRntoF} For a finite rooted $\lgc{DT_1CK}$-frame $F = (W, R, R_D)$ and $R$-reflexive points $w', w'' \in W$, the following holds. \begin{description}
\item[(a)] If $X=\set{ x\in {\bf R}^n \mid ||x|| \le r }$, $n \ge 2$, then there exists $f: X \twoheadrightarrow^{dd} F$ such that $f(\partial X) = \set{w'}$; \item[(b)] If $0\leq r_1<r_2$ and \begin{align*}
X& = \setdef[x\in {\bf R}^n]{r_1\leq ||x||\leq r_2 }, \\
Y'& =\setdef[x\in {\bf R}^n]{||x||=r_1 },\ Y'' =\setdef[x\in {\bf R}^n]{||x||=r_2 }, \end{align*} then there exists $f: X \twoheadrightarrow^{dd} F$ such that
$f(Y') = \set{w'}$, $f(Y'') = \set{w''}$. \end{description} \end{propo}
\begin{proof} By induction on $\abs{W}$. Let us prove (a) first. There are five cases:
\textbf{(a1)} $W = R(b)$ (and hence $b R b$) and $b = w'$. Then there exists $f:X\twoheadrightarrow^d (W,R)$. In fact, let $C$ be the cluster of $b$ (as a subframe of $(W,R)$). Then $(W,R)=C$ or $(W,R)=C\cup F_1\cup\ldots\cup F_l$, where the $F_i$ are generated by the successors of $C$. If $(W,R)=C$, we apply Proposition \ref{P54}; otherwise we apply Lemma \ref{refroot} and IH.
By \ref{pr:DS_AT1} it follows that $R_D$ is universal. And so by \ref{lem:dmortodDmor}(3) $f$ is a dd-morphism.
\textbf{(a2)} $W = R(b)$ and not $w'Rb$. We may assume that $r=3$. Put \[
X_1 := \set{x \mid ||x|| \le 1},~
Y := \set{ x \mid 1\le ||x|| \le 2 },~
X_2 := \set{ x \mid 2 \le ||x|| \le 3 }. \]
By the case (a1), there is $f_1: X_1 \twoheadrightarrow^{dd} F$ with $f_1(\partial X_1) = \set{b}$. Let $C$ be a maximal cluster in $R(w')$. By \ref{P54} there is $g: {\bf I} Y \twoheadrightarrow^d C$. Since $R(w')\neq W$, we can apply IH to the frame $F':=F^{w'}_{\forall}$ and construct a dd-morphism $f_2: X_2 \twoheadrightarrow^{dd} F'$ with $f_2(\partial X_2) = \set{w'}$. Now since $f_i(\partial X_i)\subseteq R^{-1}(C)$, the Glueing lemma \ref{Glue} is applicable. Thus $f:X\twoheadrightarrow^d F$ for $f:=f_1\cup f_2\cup g$ (See Fig. \ref{fig:dd-morphism_a23}, Case (a2)). Note that $\partial X\subset \partial X_2$, so $f(\partial X)=f_2(\partial X)=\{w'\}$.
As in the case (a1), $f$ is a dd-morphism by \ref{lem:dmortodDmor}.
\textbf{(a3)} $(W,R)$ is not rooted. By Lemma \ref{lem:pathinaFrame} there is a global path $\alpha$ in $F$ with a single occurrence of every $R_D$-irreflexive point. We may assume that $ \alpha =
b_0 c_0 b_1 c_1\ldots c_{m-1} b_m$, $b_m = w'$ and for any $i<m$ $c_i \in C_i \subseteq R(b_i) \cap R(b_{i+1})$, where $C_i$ is an $R$-maximal cluster. Such a path is called \emph{reduced}. For $0 \le j \le m$ we put $F_j := F|\overline{R}(b_j)$.
Since $(W,R)$ is not rooted, each $F_j$ is of smaller size than $F$, so we can apply the induction hypothesis to $F_j$. We may assume that
$$X = \set{ x \mid ||x||\leq 2m+1}, \ Y=\set{ x \mid ||x||=2m+1}.$$ Then put \[
X_i := \set{ x \mid ||x|| \leq i+1 }\mbox{ for }0 \le i \le 2m, \] \[ Y_i := \partial X_i,~ {\Delta} _i := {\bf C} (X_{i} -X_{i-1})\mbox{ for }0 \le i \le 2m. \] By IH and Proposition \ref{P54} there exist \begin{align*} f_0:& X_0 \twoheadrightarrow^{dd} F_0 \hbox{ such that } f_0(Y_0) = \{c_0\},\\ f_{2j}:& {\Delta} _{2j} \twoheadrightarrow^{dd}F_{j} \hbox{ such that } f_{2j}(Y_{2j}) = \{c_j\}, \ f_{2j}(Y_{2j-1}) = \{c_{j-1}\}\ \mbox{ for }1 \le j \le m,\\ f_{2j-1}:& \Phi {\Delta}_{2j+1} \pmor^d{} C_j \mbox{ for }0 \le j \le m-1. \end{align*}
\begin{figure}
\caption{dd-morphism $f$}
\label{fig:dd-morphism_a23}
\end{figure}
One can check that $f:X\twoheadrightarrow^{dd} F$ for $f:=\bigcup\limits_{j=0}^{2m}f_j$ (Fig. \ref{fig:dd-morphism_a23}).
\textbf{(a4)} $W = \overline{R}(b)$, $\lnot b R_D b$ (and so $\lnot b R b$). We may assume that
$$X = \set{ x \mid ||x||\leq 2}, \ Y=\set{ x \mid ||x||=2}.$$ Then similar to case (a3) put \[X_0 := X, ~Y_0 := Y, ~
X_i := \set{ x \mid ||x|| \leq \frac 1i },~ Y_i := \partial X_i,~ {\Delta} _i := {\bf C} (X_{i} -X_{i+1}), ~ (i>0). \]
Consider the frame $F' := F|W'$, where $W' = W - \set{b}$. Note that $w'\in W'$, since $w'Rw'$, by the assumption of \ref{pr:dd-morphismRntoF}. By Lemma \ref{L77} $F'$ is connected, and thus $F'\vDash \lgc{DT_1CK}$. By Lemma \ref{lem:pathinaFrame} there is a reduced global path $\alpha = a_1 \ldots a_m$ in $F'$ such that $ a_1 = w'$. Let \[ \gamma = a_1 a_2 \ldots a_{m-1} a_m a_{m-1} \ldots a_2 a_1 a_2 \ldots \] be an infinite path shuttling back and forth through $\alpha$. Rename the points in $\gamma$: \begin{equation}
\gamma = b_0 c_0 b_1 c_1 \ldots b_m c_m b_{m+1} \ldots \end{equation}
Again as in the case (a3) we put $F_j := F|\overline{R}(b_j)$, and assume that $c_j \in C_j$ and $C_j$ is an $R$-maximal cluster. By IH there exist \begin{align*} f_0: & {\Delta} _0 \twoheadrightarrow^{dd} F_0 \hbox{ such that } f_0(Y_0) = \{b_0\} = \{w'\}, \ f_1(Y_1) =\{ c_0\},\\ f_{2j}: & {\Delta} _{2j} \twoheadrightarrow^{dd}F_{j} \hbox{ such that } f_{2j}(Y_{2j}) = \{c_{j-1}\}, \ f_{2j}(Y_{2j+1}) = \{c_{j}\}\hbox{ for }j>0, \end{align*} and by Proposition \ref{P54} there exist $f_{2j+1}: \Phi {\Delta} _{2j+1} \pmor^d{} C_j$. Put \[ f(x) := \left\{ \begin{array}{ll} b &\hbox{ if } x = {\bf 0},\\ f_{2j}(x)&\hbox{ if } x \in \Delta_{2j},\\ f_{2j+1}(x)&\hbox{ if } x \in \Phi\Delta_{2j+1}, \end{array} \right. \] One can check that $f$ is d-morphic (Fig. \ref{fig:dd-morphism_a45}).
\textbf{(a5)} $W = \overline{R}(b)$, $\lnot b R b$ and $b R_D b$. Then $R_D$ is universal, $w'\ne b$. Put \[
X' := \set{x \mid ||x|| < 1}, ~
X_4 := \set{ x \mid 1 \le ||x|| \le 2}, \] and let $X_1,X_2$ be two disjoint closed balls in $X'$, $X_3 := X'- X_1 - X_2$.
Let $C$ be a maximal cluster in $R(w')$,
$F' := F|R(w')$. Then there exist: \begin{align*}
f_i:& X_i \twoheadrightarrow^d (W,R) \mbox{ for }i=1,2\mbox{ such that } \ f_i(\partial X_i) = \set{w'}, \hbox{ by the case (a4),}\\
f_3:& X_3 \twoheadrightarrow^d C, \hbox{ by Proposition \ref{P54},}\\ f_4:& X_4 \twoheadrightarrow^{dd} F' \mbox{ such that } \ f_4(\partial X_4) = \set{w'}, \hbox{ by the induction hypothesis.} \end{align*}
\begin{figure}
\caption{dd-morphism $f$}
\label{fig:dd-morphism_a45}
\end{figure}
Put $f:=f_1\cup f_2\cup f_3\cup f_4$ (Fig. \ref{fig:dd-morphism_a45}). Then $f(\partial \mathfrak X) = \set{w'}$.
By Lemma \ref{Glue} (b) $f_1\cup f_2:X_1\cup X_2\twoheadrightarrow^d F$, and hence $f:X\twoheadrightarrow^d F$ by Lemma \ref{Glue} (a).
$f$ is manifold at $b$, thus it is a dd-morphism by \ref{lem_pmorphism}.
Now we prove (b). There are three cases.
\textbf{(b1)} $w' = w'' = b$ and $W = R(b)$. The argument is the same as in the case (a1), using Proposition \ref{P54}, Lemma \ref{refroot}, the induction hypothesis, and Proposition \ref{lem:dmortodDmor}.
\textbf{(b2)} $w' = w'' = b$, but $W \neq R(b)$. Consider a maximal cluster $C \subseteq R (b)$. Since all spherical shells for different $r_1$ and $r_2$ are homeomorphic, we assume that $r_1 = 1$, $r_2 = 4$. Consider the sets \begin{equation*}
X_1 := \set{x \mid 1 \le ||x|| \le 2},\ \
X' := \set{x \mid 2 < ||x|| < 3},\ \
X_3 := \set{x \mid 3 \le ||x|| \le 4}, \end{equation*}
and let $X_0\subset X'$ be a closed ball, $X_2 := X' - X_0$. Let $F' :=F| R (b)$. There exist \begin{align*}
f_1:& X_1 \twoheadrightarrow^{dd} F'\mbox{ such that } f_1(\partial X_1) = \set{b}, \hbox{ by the case (b1)},\\
f_2:& X_2 \twoheadrightarrow^d C, \hbox{ by Proposition \ref{P54}},\\
f_3:& X_3 \twoheadrightarrow^{dd} F'\mbox{ such that } \ f_3(\partial X_3) = \set{b}, \hbox{ by the case (b1)},\\
f_0:& X_0 \twoheadrightarrow^{dd} F\mbox{ such that } \ f_4(\partial X_0) = \set{b},
\hbox{ by the statement (a) for }F. \end{align*} One can check that $f:X\twoheadrightarrow^{dd} F$ for $f:=f_0\cup f_1\cup f_2\cup f_3$.
\textbf{(b3)} $w' \ne w''$ and for some $b\in W$, $W = R(b)$, so $F$ has an $R$-reflexive root. Let \[
F_1 := F|R(w'), ~ F_2 := F|R(w''), \] and let $C_i$ be an $R$-maximal cluster in $F_i$ for $i\in \set{1,2}$.
We assume that $r_1 = 1$, $r_2 = 6$ and consider the sets \begin{align*}
X_i &:= \set{x \mid i \le ||x|| \le i+1}, \ i \in \set{1, \ldots, 5}. \end{align*}
By the case (b1) and Proposition \ref{P54} we have
\begin{align*}
f_1:&~ X_1 \twoheadrightarrow^{dd} F_1 \mbox{ such that } f_1(\partial X_1) = \set{w'}, &f_2:&~ \Phi X_2 \twoheadrightarrow^d C_1,\\
f_3:&~ X_3 \twoheadrightarrow^{dd} F\mbox{ such that } \ f_3(\partial X_3) = \set{b}, &f_4:&~ \Phi X_4 \twoheadrightarrow^d C_2, \\
f_5:&~ X_5 \twoheadrightarrow^{dd} F_2 \mbox{ such that } f_1(\partial X_5) = \set{w''}. \end{align*}
\begin{figure}
\caption{dd-morphism $f$}
\label{fig:dd-morphism_b23}
\end{figure}
One can check that $f:X\twoheadrightarrow^{dd} F$ for $f:= \bigcup\limits_{i=1}^5 f_i$ (Fig. \ref{fig:dd-morphism_b23}, Case (b3)).
\textbf{(b4)} $w' \ne w''$ and $W \ne R(b)$ for any $b\in W$. By Lemma \ref{LPF0} there is a reduced path $\alpha = b_0 c_0 b_1 \ldots c_{m-1} b_m$ from $b_0 = w'$ to $b_m = w''$ that does not contain $R_D$-irreflexive points, $c_i \in C_i$, where $C_i$ is an $R$-maximal cluster. We may also assume that \begin{equation}\label{eq:Rbi}
\overline{R}(b_i) \ne W, \hbox{ for any $i\in \set{1, \ldots, m-1}$}. \end{equation} In fact, if the frame $(W,R)$ is not rooted, then (\ref{eq:Rbi}) obviously holds. If $(W,R)$ is rooted, then its root $r$ is irreflexive and by Lemma \ref{L77}, $R(r)$ is connected, so there exists a path $\alpha$ in $R(r)$ satisfying (\ref{eq:Rbi}).
Put \[ F_0 := F,~
F_j := F|R(b_j), 1 \le j \le m. \]
Assuming that $r_1 = 1$, $r_2 = 2m+1$ we define \begin{align*}
X_i &:= \setdef{||x|| \leq i +1 },~ Y_i := \partial X_i~ (\mbox{for }0 \le i \le 2m+1),\\ {\Delta} _i &:= {\bf C} (X_{i+1} -X_{i}) \ (\mbox{for }0\le i\le 2m). \end{align*}
\begin{figure}
\caption{dd-morphism $f$, case (b4)}
\label{fig:dd-morphism_b4}
\end{figure}
By the cases (b2), (b1), Proposition \ref{P54}, and the induction hypothesis there exist \begin{align*} &f_0: {\Delta}_0 \twoheadrightarrow^{dd} F=F_0 \hbox{ such that } f_0(Y_0) = f_0(Y_1) = \set{w'}; \\ &f_{2j}: {\Delta} _{2j} \twoheadrightarrow^{dd} F_{j} \hbox{ such that } f_{2j}(Y_{2j+1}) = \set{c_{j}}, \ f_{2j}(Y_{2j}) = \{c_{j-1}\}\ (1 \le j \le m);\\ &f_{2j-1}: \Phi {\Delta} _{2j-1} \twoheadrightarrow^d C_{j-1} \ (1 \le j \le m),\\ &f_{2m}: {\Delta} _{2m} \twoheadrightarrow^{dd} F_m \hbox{ such that } f_{2m}(Y_{2m}) = \set{c_{m}}, \ f_{2m}(Y_{2m+1}) = \set{w''}. \end{align*}
We claim that $f:X\twoheadrightarrow^{dd} F$ for $f:= \bigcup\limits_{i=0}^{2m} f_i$ (Fig. \ref{fig:dd-morphism_b4}). First, we prove by induction using Lemma \ref{Glue} (see previous cases) that $f$ is a d-morphism. Note that $f(Y')= f(Y_0) = \set{w'}$ and $f(Y'')= f(Y_{2m+1}) = \set{w''}$.
Second, there are no $R_D$-irreflexive points in $\alpha$, so all preimages of $R_D$-irreflexive points are in $\Delta_0$; since $f_0$ is a dd-morphism, $f$ is 1-fold at any $R_D$-irreflexive point and manifold at all the others. Thus $f$ is a dd-morphism by Proposition \ref{lem:dmortodDmor}. \end{proof}
\begin{theorem}\label{T125}
For $n \ge 2$, the dd-logic of ${\bf R}^n$ is $\lgc{DT_1CK}$. \end{theorem}
\begin{proof} Since ${\bf R}^n$ is a locally 1-component connected dense-in-itself metric space, ${\bf R}^n \models^d \lgc{DT_1CK}$.
Now consider a formula $A \notin \lgc{DT_1CK}$. Due to the fmp (Theorem \ref{thm:fmp_DT_1CK}) there exists a finite rooted Kripke frame $F = (W, R, R_D) \vDash \lgc{DT_1CK}$ such that $F \nvDash A$. By Proposition $\ref{pr:dd-morphismRntoF}$ there exists $f:{\bf R}^n \twoheadrightarrow^{dd} F$. Hence ${\bf R}^n \nvDash^d A$ by Lemma \ref{lem_pmorphism}. {\hspace*{\fill} }\end{proof}
\section{Concluding remarks} \textbf{Hybrid logics.} Logics with the difference modality are closely related to hybrid logics. The paper \cite{Litak06} describes a validity-preserving translation from the language with the topological and the difference modalities into the hybrid language with the topological modality, nominals and the universal modality.
Apparently a similar translation exists for dd-logics considered in our chapter. There may be an additional option --- to use `local nominals', propositional constants that may be true not in a single point, but in a discrete set. Perhaps one can also consider `one-dimensional nominals' naming `lines' or `curves' in the main topological space; there may be many other similar options.
\textbf{Definability.} Among several types of topological modal logics considered in this chapter dd-logics are the most expressive. The correlation between all the types are shown in Fig. \ref{fig:languages}.
A language ${\cal L}_1$ is \emph{reducible} to ${\cal L}_2$ (${\cal L}_1 \le {\cal L}_2$) if every ${\cal L}_1$-definable class of spaces is ${\cal L}_2$-definable; ${\cal L}_1<{\cal L}_2$ if ${\cal L}_1\le {\cal L}_2$ and ${\cal L}_2 \nleq {\cal L}_1$. The non-strict reductions 1--7 in Fig. \ref{fig:languages} are rather obvious. Let us explain, why 1--6 are strict.
\begin{figure}
\caption{Correlation between topomodal languages.}
\label{fig:languages}
\end{figure}
The relations 1 and 2 are strict, since the c-logics of ${\bf R}$ and ${\bf Q}$ coincide \cite{MT}, while the cu- and d-logics are different \cite{Sh99, E1}.
The relation 3 is strict, since in d-logic without the universal modality we cannot express connectedness (this follows from \cite{E1}). The relations 4 and 6 are strict, since the cu-logics of ${\bf R}$ and ${\bf R}^2$ are the same \cite{Sh99}, while the cd- and du-logics are different \cite{Gab01, LB2}.
In cd- and dd-logic we can express \emph{global 1-componency}: the formula \[ {[\ne]} (\overline{\square} p \lor \overline{\square} \lnot p) \to {[\ne]} p \lor {[\ne]} \lnot p \] is c-valid in a space $\mathfrak X$ iff the complement of any point in $\mathfrak X$ is connected. So we can distinguish the line $\mathbf R$ and the circle ${\bf S^1}$. In du- (and cu-) logic this is impossible, since there is a local homemorphism $f(t) = e^{it}$ from $\mathbf R$ onto $S^1$. It follows that the relation 5 is strict. Our conjecture is that the relation 7 is strict as well.
\textbf{Axiomatization.} There are several open questions about axiomatization and completeness of certain dd-logics.
1. The first group of questions is about the logic of ${\bf R}$. On the one hand, in \cite{Kudinov08} it was proved that $\mathbf{Lc}_{\ne}({\bf R})$ is not finitely axiomatizable. Probably, the same method can be applied to $\mathbf{Ld}_{\ne}({\bf R})$. On the other hand, $\mathbf{Lc}_{\ne}({\bf R})$ has the fmp \cite{Kud11}, and we hope that the same holds for the dd-logic. The decidability of $\mathbf{Ld}_{\ne}({\bf R})$ follows from \cite{BG}, since this logic is a fragment of the universal monadic theory of ${\bf R}$; and by a result from \cite{Rey} it is PSPACE-complete. However, constructing an explicit infinite axiomatization of $\mathbf{Lc}_{\ne}({\bf R})$ or $\mathbf{Ld}_{\ne}({\bf R})$ might be a serious technical problem.
2. A `natural' semantical characterization of the logic $\lgc{DT_1C}+Ku_2$ (which is a proper sublogic of $\mathbf{Ld}_{\ne}({\bf R})$) is not quite clear. Our conjecture is that it is complete w.r.t. 2-dimensional cell complexes, or more exactly, adjunction spaces obtained from finite sets of 2-dimensional discs and 1-dimensional segments.
3. We do not know any syntactic description of dd-logics of 1-dimensional cell complexes (i.e., unions of finitely many segments in ${\bf R}^3$ that may have only endpoints as common).
Their properties are probably similar to those of $\mathbf{Ld}_{\ne}({\bf R})$.
4. It may be interesting to study topological modal logics with the graded difference modalities ${[\ne]}_n A$ with the following semantics: $x\models {[\ne]}_n A$ iff there are at least $n$ points $y\ne x$ such that $y \models A$.
5. The papers \cite{MT} and \cite{Kremer} prove completeness and strong completeness of ${\bf S4}$ w.r.t.~any dense-in-itself metric space. The corresponding result for d-logics is completeness of $\mathbf{D4}$ w.r.t. an arbitrary dense-in-itself separable metric space. Is separabilty essential here? Does strong completeness hold in this case? Similar questions make sense for dd-logics.
6. \cite{Gab01} presents a 2-modal formula cd-valid exactly in $T_0$-spaces. However, the cd-logic (and the dd-logic) of the class of $T_0$-spaces is still unknown. Note that the d-logic of this class has been axiomatized in \cite{BEG2011}; probably the same technique is applicable to cd- and dd-logics.
7. In footnote 7 we have mentioned that there is a gap in the paper \cite{Sh99}. Still we can prove that for any connected, locally connected metric space $\mathfrak X$ such that the boundary of any ball is nowhere dense, ${{\bf L}}{\bf c}_\forall(\mathfrak X)=\mathbf{S4U} + AC$. But for an arbitrary connected metric space $\mathfrak X$ we do not even know if ${{\bf L}}{\bf c}_\forall(\mathfrak X)$ is finitely axiomatizable.
8. Is it possible to characterize finitely axiomatizable dd-logics that are complete w.r.t.~Hausdorff spaces? metric spaces? Does there exist a dd-logic complete w.r.t. Hausdorff spaces, but incomplete w.r.t. metric spaces?
9. Suppose we have a c-complete modal logic $L$, and let ${\cal K}$ be the class of all topological spaces where $L$ is valid. Is it always true that ${\bf Lc}_\forall({\cal K})=LU$? and ${\bf Lc}_{\ne}({\cal K})=LD$? Similar questions can be formulated for d-complete modal logics and their du- and dd-extensions.
10. An interesting topic not addressed in this chapter is the complexity of topomodal logics. In particular, the complexity is unknown for the d-logic (and the dd-logic) of ${\bf R}^n$ ($n>1$).
We would like to thank anonymous referee who helped us improve the first version of the manuscript.
The work on this chapter was supported by RFBR grants 11-01-00281-a, 11-01-00958-a, 11-01-93107-CNRS-a and the Russian President's grant NSh-5593.2012.1.
\end{document} | arXiv |
mimi tsuruga
math ◇ data ◇ elasticsearch
Topology of Data
Topological Software
Simplicial Homology
January 31, 2013 / matsguru
I started writing a post about Homology Algorithms but realized that some discussion about homology should come first. And for those of you who already know what homology is, we should at least be on the same page in terms of language and notation. Besides, there's more than one way to think about homology.
The way that I understand homology—the way I describe it here—is influenced almost entirely by Carsten Lange. He helped me understand the basics of algebraic topology so that I can read about discrete Morse theory without feeling like a total oaf. This also means that we are thinking about homology with the eventual goal of computing (i.e., with a computer) the homology of some explicit, finite discrete topological space. In particular, we will work with simplicial complexes.
References used include the classics, Munkres and Hatcher, and for a speedy introduction, Zomorodian's "Topology for Computing". (And, as usual, a ton of help has been provided by my advisors Frank and Bruno.)
And before I begin, let me just say: Big ups to Carsten and all dedicated educators like him who, for countless hours, over many days and weeks, sit together with students one-on-one until they have the confidence to seek knowledge on their own.
Homology in topology
The big question in topology is whether or not two spaces are homeomorphic. To say that two spaces are homeomorphic, we need only find any homeomorphism between them. But to say that two spaces are not homeomorphic, we would have to show that no function between them is a homeomorphism. One way to handle this latter (infinite) case is by using topological invariants.
A topological invariant is a property of a space that can be used to distinguish topological spaces. For example, say we have two sacks labeled $A$ and $B$. We can't see inside them, but we can tell that sack $A$ has one item in it while sack $B$ has two. Clearly the contents of sack $A$ and sack $B$ cannot be the same.
However, if $A$ and $B$ both have one item each, we can't tell whether those two items are of the same type. Perhaps one is a steak and the other a rice crispy treat—clearly very different objects, both nutritionally and topologically. (Can you tell I missed lunch?)
The topological invariant described in these examples is connectedness, which is only useful when the number of connected components is different. But if they have the same number of connected components, we can't say definitively whether the objects are of the same topological type. And that's how it works with all topological invariants.
Definition (topological invariant) Given two topological spaces $K_1$ and $K_2$, a map $\tau$ is a topological invariant if whenever $\tau(K_1) \neq \tau(K_2)$, then $K_1 \ncong K_2$. But if $\tau(K_1) = \tau(K_2)$, then $K_1 \overset{?}{\cong} K_2$ and we don't know whether they are homeomorphic or not.
Homology groups are topological invariants. We take some geometric information about the topological space, form some algebraic object (specifically, a group) containing that information, then apply some machinery from group theory to say something about the original space.
And if two topological spaces give rise to homology groups that are not isomorphic, then those two spaces are not homeomorphic.
Definition of Homology
Let's jump right in with a definition, then explain it in detail using an example.
Definition (homology group) For a topological space $K$, we associate a chain complex $\mathcal{C}(K)$ that encodes information about $K$. This $\mathcal{C}(K)$ is a sequence of free abelian groups $C_p$ connected by homomorphisms $\partial_p: C_p \to C_{p-1}$. \begin{align*}0 \overset{\partial_{d+1}}{\longrightarrow} C_d \overset{\partial_{d}}{\longrightarrow} C_{d-1} \overset{\partial_{d-1}}{\longrightarrow} \cdots \overset{\partial_{2}}{\longrightarrow} C_1 \overset{\partial_{1}}{\longrightarrow} C_0 \overset{\partial_{0}}{\longrightarrow} 0\end{align*} Then the $p$-th homology group is defined as \begin{align*}H_p(K):= \text{ker}(\partial_p) / \text{im}(\partial_{p+1}).\end{align*}
That's a lot of terms we have yet to define.
Let's start at the beginning with our input space $K$. We can assume that $K$ is a simplicial complex since we'll only be dealing with simplicial complexes and they're just nice to handle. So $K$ is a simplicial complex of dimension $d$. This means that for any face $\sigma \in K$, its dimension is at most $|\sigma| \le d$.
Figure 1: Simplicial complex $K$ with vertices labeled 1,2,3,4.
In Figure 1, we have our simplicial complex $K$. I've labeled the vertices so it's easier for me to refer to the many parts of $K$. Our complex $K$ consists of
one 2-cell: $f_1 = [2 \, 3\, 4]$;
five 1-cells: $e_1=[1 \, 2], e_2=[1 \, 4], e_3=[2 \, 3], e_4=[2 \, 4], e_5=[3 \, 4]$; and
four 0-cells: $v_1=[1], v_2=[2], v_3=[3], v_4=[4]$.
This complex $K$ is of dimension $d=2$.
The group $C_p$ is a group of something called $p$-chains. Recall that a group is a set with a binary operation (in our case, addition) that is (i) associative, (ii) has an identity element, and (iii) has inverses. And when we refer to any group, instead of listing all the elements of the group all the time, it's much more economical to talk about groups with respect to its generators. For the generators of $C_p$, we have to go back to $K$.
The $p$ in $C_p$ refers to the dimension of cells in $K$. A 1-chain of $K$, for example, is some strange abstract object that is generated by all the 1-cells of $K$, that is, we "add" up a bunch of 1-cells. So $e_1 + 2 e_5$ is an example of a 1-chain. It has no obvious geometric meaning (though, apparently, some people would disagree). If it helps, you can think of $C_p$ as a vectorspace whose basis is the set of $p$-cells of $K$ and the scalars are in $\mathbb{Z}$.
Figure 2: Relabeled and oriented.
Notice that the labeling of the vertices from Figure 1 gives us a nice built-in orientation, i.e., ordering, for each cell, see Figure 2. For example, edge $e_1 = [1 \, 2]$ so the orientation we give this edge is from $v_1$ to $v_2$ (because $1<2$). This orientation will help us define the additive inverse of a 1-cell. So $-e_1$ is what you would expect it to be; it's just $e_1$ oriented in the opposite direction, from $v_2$ to $v_1$.
Now the definition states that $C_p$ is not just any old group. It is a free Abelian group. This is not as bad as it sounds. It just means, again, that we can just go back to thinking of $C_p$ like it's a vectorspace.
So $C_p$ is free, which just means that every $p$-chain in $C_p$ can be uniquely written as $\sum_i n_i \sigma_i^p$, where $n_i \in \mathbb{Z}$ and $\sigma_p$ are $p$-cells. This idea should be deeply ingrained into your understanding of basis vectors in a vectorspace.
And $C_p$ is also Abelian. So our addition is commutative $\sigma + \tau = \tau + \sigma$ and that also let's us keep thinking about this vectorspace idea.
So now that we know what the $C_p$ are, let's talk about the homomorphisms $\partial_p$ between them. These homomorphisms have a special name; they are called boundary maps. This is where the geometry and combinatorics fit together very nicely. Let's go back to our Figure 2.
The boundary of some $p$-cell $\sigma^p=[v_0 \, v_1 \, \dots \, v_p]$ is a $(p-1)$-chain with the alternating sum $$\partial_p(\sigma) = \sum_{i=0}^{p} (-1)^i [v_0 \, v_1 \, \cdots \, \hat{v_i} \, \cdots \, v_p ]$$ where $\hat{v_i}$ means you leave that vertex out.
For example, $\partial_1(e_1) = v_2 – v_1$ or $\partial_2(f_1) = \partial_2([2 \, 3 \, 4]) = [3 \, 4] – [2 \, 4] + [2 \, 3] = e_3 + e_5 – e_4$. This corresponds nicely with the figure. Just follow along the blue arrows on the boundary of $f_1$ from $v_2$ in a counter clockwise direction.
The boundary function works similarly for all dimensions $p$. For the higher dimensions, just keep in mind what I said earlier about orientation being determined by the labels. The orientation for an edge is easy to visualize, but not so much for, say, a 6-simplex. Everything works out nicely combinatorially with the above definition.
And because $C_p$ is a free abelian group, we see that the boundary map $\partial_p$ is a homomorphism. Notice also that $\partial_p \circ \partial_{p+1} = 0$. Whenever this happens (to any sequence of Abelian groups with homomorphisms between them with this property), we call such a sequence a chain complex.
Geometry of homology
Let's introduce a few more important terms.
The set $B_p = \text{im}(\partial_{p+1})$ is the set of boundaries and the set $Z_p = \text{ker}(\partial_p)$ is the set of cycles (which, in German, is "Zyklus").
So a cycle is any $p$-chain that sums up to zero. This is a very combinatorial/algebraic definition for a word that sounds very geometric. Intuitively, when you hear the word "cycle", you might think of some path that closes, maybe like a loop. It's very important, however, to recognize that "cycle" and "loop" (from homotopy theory) are very very different objects.
Let me go off on a short tangent here. Have you noticed that the word "curve" is different depending on who you talk to? For a physicist, a curve is a trajectory–a function $\gamma: [0,1] \to X$ for some space $X$. For a geometer, however, a curve is a subset of a space. Merely the image $\text{im}(\gamma)$. So it doesn't matter whether a mosquito flies around in a circle 7 times or just once; it still traces just a single circle.
Analogously, a cycle doesn't care about how you "go around the cycle" as long as the algebraic sum of the corresponding $(p-1)$-chains in the cycle adds up to zero.
And remember: since we have an Abelian group, the order of the addition (or "direction" of the cycle) doesn't matter at all. For a loop, however, it matters whether you go around the loop clockwise or counterclockwise.
Let's look again at the composition $\partial_p \circ \partial_{p+1} = 0$. It says, geometrically, that the boundary of $(p+1)$-chains are cycles. So the set of boundaries $B_p$ is contained in the set of cycles $Z_p$, i.e., $B_p \subseteq Z_p$.
Not only that, because $\partial_{p+1}$ is a homomorphism, $B_p$ is even a subgroup of $Z_p$. Geometrically speaking, the quotient group $H_p = Z_p / B_p$ means we partition the set of cycles $Z_p$ to those that bound and those that don't. This means we can isolate cycles that do not bound cells. That is, we can find holes!
Let's go back to our example and think about $H_1 (K)$. The "simplest" 1-chains that make a loop are $e_1 + e_4 – e_2 = \ell_1$ and $e_3 + e_5 – e_4=\ell_2$. All other loops are some linear combination of $\ell_1, \ell_2$. For example $e_1 + e_3 + e_5 – e_2 + (e_4 – e_4) = \ell_1 + \ell_2$. So $Z_1$ is generated by $\ell_1, \ell_2$, i.e., $Z_1 = \{c\, \ell_1 + d\, \ell_2 \mid c,d \in \mathbb{Z} \} \cong \mathbb{Z}^2$. But we also know that $\ell_2 = \partial_{2} (f_1)$ and since $f_1$ is the only 2-cell in $K$, we have $C_2 = \{ c \,f_1 \mid c \in \mathbb{Z} \}$ and therefore $B_1 = \text{im}(\partial_{2}) = \{ c \, \ell_2 \mid c \in \mathbb{Z}\} \cong \mathbb{Z}$.
So $H_1 = Z_1/B_1 \cong \mathbb{Z}^2 / \mathbb{Z} \cong \mathbb{Z}$.
Using homology
Now that the homology group has been defined, we should know what it means. There is an important theorem in group theory that we should recall at this point. It's so important that it is a fundamental theorem.
Theorem (Fundamental Theorem of Finitely Generated Abelian Groups) Every finitely generated Abelian group $G$ is isomorphic to a direct product \begin{align*}G \cong H \times T, \end{align*} where $H = \mathbb{Z} \times \cdots \times \mathbb{Z} = \mathbb{Z}^{\beta}$ is a free Abelian subgroup of $G$ having finite rank $\beta$ and $T = \mathbb{Z}_{t_1} \times \mathbb{Z}_{t_2} \times \cdots \times \mathbb{Z}_{t_k}$ consists of finite cyclic groups of rank $t_1, \dots, t_k$.
The $T$ is called the torsion subgroup of $G$; the $\beta$ is unique and called the Betti number of $G$; and the $t_1, \dots, t_k$ are also unique and called the torsion coefficients of $G$.
This fundamental theorem says that the homology group $H_p$ (which is a quotient group of a finitely generated Abelian group over a finitely generated Abelian group and therefore itself a finitely generated Abelian group) will always be isomorphic to something that looks like \begin{align*}H_p \cong \mathbb{Z}_{t_1} \times \cdots \times \mathbb{Z}_{t_k} \times \mathbb{Z}^{\beta}\end{align*} a finite direct sum of (finite and infinite) cyclic groups. So now all topological spaces can be nicely categorized by their homology groups according to their Betti numbers and torsion coefficients.
The homology groups $H_p$ are of dimension $p$ and each of them have an associated Betti number $\beta_p$. Since we figured out for our example that $H_1 (K) \cong \mathbb{Z} = \mathbb{Z}^1$, our Betti number $\beta_1 = 1$.
These Betti numbers have a wonderful geometric meaning. This number $\beta_p$ counts the number of $p$-dimensional holes in $K$. Ah, yes. We finally get to something meaningful and even useful!
Betti numbers can count, as in our example, the number of holes.
Since $\beta_1(K)=1$, this means $K$ has one 1-dimensional hole. And there is indeed a hole in $K$ (because there is no face $[1 \, 2 \, 4]$).
But what is a "hole"?
Honestly, I don't know how to draw a good picture of a general $p$-dimensional hole (though examples of specific cases are easy). Thinking of holes like in homotopy theory–as the empty spaces where $p$-spheres $S^p$ cannot be shrunk down to a point–is a very nice and intuitive notion. But there are some spaces that, like the Poincare sphere for example, have $H_1 = 0$ and $\pi_1 \neq 0$. So it's not enough to think about holes as the holes we see when working with fundamental groups.
We saw from the definition that homology groups are equivalence classes of cycles, of those that bound and those that don't. So we collect all the "simplest" cycles and check whether they are the boundary of any faces. Any cycle that does not have a corresponding face which it bounds must bound a hole.
It turns out that counting holes has all kinds of industry applications. It can be used to analyze the structure of solid materials or proteins. It can be used in image processing or pattern recognition. It can even be used in robotics and sensor networks.
So homology algorithms have been developed to help these industry people to count holes. And now we know exactly how to write such an algorithm, right?
The significance of adipra
November 28, 2012 / matsguru
I realize that anyone reading this post doesn't know what adipra is. How can you? I made it up.
The sphere I constructed most recently was an idea of Karim Adiprasito. While building it, I needed to name it so that I can refer to it in my code. So I temporarily named it adipra and that's what I'm going to call it here.
Actually, in this post, I don't want to talk about what adipra is. Not yet. Let's talk just about what makes adipra special. For anyone who's interested, all the specifics can be found here.
We need a little language to get started.
Def A combinatorial d-manifold is a triangulated d-manifold whose vertex links are PL spheres.
Let's assume we know what a triangulated d-manifold means without hashing out the details here. If you need to, you can think of it as a simplicial complex.
The star of a vertex v in a triangulated d-manifold T is the collection of facets of T that contain v. The link of a vertex v is like the star of v, but take out all the v's. For example, let's take a simple hexagon with a vertex in the center.
Example of star and link of vertex
Let T=$\{[0\, 1 \,2],[0\, 1\, 6],[0\, 2\, 3],[0\, 3\, 4],[0\, 4\, 5],[0\, 5\, 6]\}$, then \begin{align*}\text{star}(3,T) &=\{[0\, 2 \,3],[0\, 3\, 4]\} \text{ (green triangles)}\\ \text{link}(3,T)&=\{[0\, 2],[0\, 4]\} \text{ (red edges)} \end{align*}
That's simple enough to understand even in higher dimensions. Just remember we're always doing things combinatorially.
Next is the PL sphere. I said earlier that we can assume we know what a triangulated d-manifold is. What we're actually talking about is a triangulated PL d-manifold. A PL-sphere is a PL manifold that is bistellarly equivalent to the boundary of a d-simplex. I'll write in more detail what bistellarly equivalent means in a later post. For now, just think of it as the discrete version of being homeomorphic.
Putting all that together, we understand a combinatorial d-manifold to be a triangulated manifold, that is, some simplicial complex-like thing where we require that each of its verticies is sort of covered by a ball. Naturally, the next question to ask is: are there triangulated manifolds that are not combinatorial?
For d=2,3, all triangulations (of the d-sphere) are combinatorial. For d=2, the vertex links should be homeomorphic to $S^1$ or bistellarly equivalent to the triangle (boundary of a 2-simplex). Similarly, for d=3, vertex links are $S^2$. For d=4, all triangulated 4-manifolds are also combinatorial. This result is due to Perelman. The vertex links are 3-spheres, which, as you know, is what Perelman worked on. [Remember that PL=DIFF in dim 4.] But it falls apart for d$\ge 5$ as there are non-PL triangulations of the d-sphere.
Ok, so here's a spoiler about adipra: it's a non-PL triangulation of the 5-sphere. But that's not all.
There's a nice theorem by Robin Forman, the father of discrete Morse theory.
Theorem (Forman) Every combinatorial d-manifold that admits a discrete Morse function with exactly two critical cells is a combinatorial d-sphere.
Actually, this is not really Forman's theorem. This is a theorem by Whitehead which Forman reformulated using his language of discrete Morse theory. This is the original theorem.
Theorem (Whitehead) Any collapsible combinatorial d-manifold is a combinatorial d-ball.
What Forman did is take a sphere, take out one cell, call that guy a critical cell, then collapse the rest of it down (using Whitehead's theorem) to get the other critical cell. Thus you have two critical cells.
So the question you ask next is: can you have non-combinatorial spheres with 2 critical cells?
Yes, actually, you can! Karim showed that you can have a non-PL/non-combinatorial triangulation of the 5-sphere that has a discrete Morse function with exactly 2 critical cells! And then I built an explicit example (with Karim's instructions) and called it adipra.
KolKom 2012
November 26, 2012 / matsguru / 1 Comment
My first conference talk since starting Phase II was at KolKom 2012, the colloquium on combinatorics in Berlin. But I don't know anything about combinatorics or optimization. What am I doing at a combinatorics conference!?
Actually, I don't know much about anything. But it's not hard to make me sound like I know about some things. The title of my talk was "constructing complicated spheres as test examples for homology algorithms". Yes, algorithms! That's the key! It was Frank's idea to add the word "algorithms" to my title so that I can at least pretend to belong to this conference. It made for a very long title, but at least my abstract was accepted.
You can look at my slides, but I'll give a short overview here.
We begin with a definition of homology and motivate some applications for homology algorithms. Betti numbers, in particular, are interesting for industrial applications. To compute the Betti numbers of some given simplicial complex, we will be dealing with matrices. We want to find the Smith Normal Form of these matrices. But the bigger the complex, the bigger the matrix, and the longer it takes to find the Smith normal form. So we throw the complex into a preprocess to reduce the size of the input. This preprocess uses discrete Morse theory.
Morse theory is used to distinguish topological spaces in differential topology. Forman came up with a nice discrete counterpart that conveniently preserves many of the main results from smooth Morse theory including the Morse inequalities. The Morse inequality $\beta_i \le c_i \; \forall i$ means that the Betti number $\beta_i$ is bounded above by $c_i$, the number of critical points of dimension $i$. Since our original goal was to compute Betti numbers, this result will definitely come in handy.
The critical points $c_i$ are the critical points of some discrete Morse function. These discrete Morse functions can be interpreted as a sort of collapse. By collapsing big complexes, we can reduce big complexes (that have big matrices for which we need to find the Smith normal form) to smaller complexes.
Ideally, we want to find the discrete Morse function with the lowest $c_i$'s. But computing the optimal discrete Morse vector, a vector $c = (c_0,c_1,c_2, \dots)$, is $\mathcal{NP}$-hard! So Lutz and Benedetti have come up with the idea of using random discrete Morse theory. Surprisingly, their algorithm is able to find the optimal discrete Morse vector most (but not all) of the time. To get a better idea of when they're better able to find the optimum, they came up with a complicatedness factor which measures how hard it is to find that optimum.
That's where I come in. I constructed some complicated simplicial complexes. They are being used to test some homology algorithms that are currently under development.
The first of my complicated complexes is the Akbulut-Kirby sphere, one of a family of Cappell-Shaneson spheres. These spheres have an interesting history. The AK-sphere, in particular, was one thought to be an exotic sphere. Exotic spheres are spheres that are homeomorphic, but not diffeomorphic to a standard sphere. In dimension $4$, this would mean it could be a counterexample to the smooth Poincar\'e conjecture, or SPC4 for short. Unfortunately, some years after the AK-sphere was proposed, it was shown to be standard after all. You can learn more about it here.
So I built an AK-sphere. Actually, I wrote code that can build any of the whole family of the CS-spheres. Before talking about what we did, let's start with how these spheres are built. To understand the construction, you have to first accept two simple ideas:
The boundary of a solid ball is a sphere.
That's not too hard to accept, which is why I said it first even though it's the last step.
Given a donut, if you fill the "hole" (to make it whole, HA!), it will become a solid ball.
That's also not too difficult to see. You can imagine the filling to be something like a solid cylinder that you "glue" onto the inside part of the donut.
Now for some language. This donut is actually a ball with a handle, which is known as a 1-handle. The filling for the hole is called a 2-handle. And we can specify how the 2-handle is glued in using an attaching map.
So to build the AK-sphere here's what we do:
Take a 5-ball.
Attach two 1-handles. Let's call one of them $x$ and the other $y$ so we can tell them apart.
Glue in two 2-handles to close the "holes" but use attaching maps that have the following presentation: \begin{align*} xyx&= yxy & x^r = y^{r-1}\end{align*}This means that the attaching map goes (on the boundary of the 5-ball) along $x$ first, then $y$, then $x$, then $y$ in the reverse direction, then $x$ in reverse, and finally $y$ in reverse. Similarly with $x^r = y^{r-1}$. [Note: For the AK-sphere, $r = 5$. By varying $r$ you get the family of CS-spheres.]
Take its boundary.
Since the 2-handles have closed the holes created by the 1-handles, what we have left after step 3 is again a 5-ball, though a bit twisted. So its boundary in step 4 should be a sphere.
To construct these spheres, we decided to go in a different order. It would be hard to imagine the attaching maps in dimension 5. So we went down a smidge to dimension 3. And instead of drawing the paths of the attaching maps on/in a ball, we built the ball around the paths.
As you might imagine, these paths come out a bit knotted. But by bringing them up in dimension, we can "untangle" it because there's more space in higher dimensions. The way that we go up in dimension is by "crossing by $I$" where $I$ is the usual unit interval. Crossing by $I$ has the convenient side effect that everything you "crossed" ends up on the boundary of the resulting complex.
So we start with these two paths (some triangulated torii). We fill in the space between them with cleverly placed tetrahedra so that it ends up being a ball with two 1-handles. We now have a genus 2 simplicial complex, where two sets of tetrahedra are labelled as the path representing $xyx = yxy$ and the other $x^r = x^{r-1}$. Cross $I$, cross $I$ so that we are in dimension 5 with the two paths on the boundary of this ball with two 1-handles. Glue in the (5-dimensional) 2-handles (whatever that means). Take the boundary. Done!
The experiments we've run on them so far have shown some promise. The complexes are not too large (with f-vector$=(496,7990,27020,32540,13016)$) so the experiments don't take too long to run. They've already been used by Konstantin Mischaikow and Vidit Nanda to improve their on-going Perseus algorithm. [edit: In the original post, I mistakenly referenced CHomP, which has been static for some time.]
The other complex I constructed is Mazur's 4- manifold. We do a similar thing, but this time there's only one path to deal with (that is, only one 1-handle to close up with one 2-handle). The results from this one has some nice topological properties. I'll get to that in a later post once we're ready to publish. | CommonCrawl |
\begin{document}
\title[Shape theorem for an epidemic model in $d\ge 3$]{A shape theorem for an epidemic model in dimension $d\ge 3$}
\author{E. D. Andjel} \author{N. Chabot} \author{E. Saada}
\address{Aix-Marseille Universit\'e,
CNRS, Centrale Marseille, I2M, UMR 7373,\newline
Technop\^ole Ch\^ateau-Gombert, 39 rue Fr\'ed\'eric Joliot-Curie,\newline 13453 Marseille Cedex 13, France.\newline Visiting IMPA, Rio de Janeiro, Brasil.} \email{[email protected]}
\thanks{E. D. Andjel was partially supported by PICS no. 5470, by CNRS and by CAPES.} \address{Lyc\'ee Lalande,\newline 16 rue du Lyc\'ee,\newline 01000 Bourg en Bresse, France.} \email{[email protected]}
\address{CNRS, UMR 8145, MAP5, Universit\'e Paris Descartes, Sorbonne Paris Cit\'e,\newline 45 rue des Saints-P\`eres,\newline 75270 Paris cedex 06, France.} \email{[email protected]} \urladdr{\url{http://www.math-info.univ-paris5.fr/\~esaada/}} \thanks{E. Saada was partially supported by PICS no. 5470}
\subjclass[2000]{60K35, 82C22.} \keywords{Shape theorem, epidemic model, first passage locally dependent percolation,
dynamic renormalization.}
\begin{abstract}
We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${\mathbb Z}^d,d\ge 3$, when there is no extinction of the infection. For this, we derive percolation estimates (using dynamic renormalization techniques) for a locally dependent random graph in correspondence with the epidemic model. \end{abstract}
\maketitle
\section{Introduction}\label{sec:intro}
\citet{MR0496851,MR0480208} has introduced a stochastic spatial
epidemic model on ${\mathbb Z}^d$ called ``general
epidemic model'', describing the evolution of individuals submitted to infection by contact contamination of infected neighbors. More precisely, on each site of ${\mathbb Z}^d$ there is an individual who can be healthy, infected, or immune. At time 0, there is an infected individual at the origin, and all other sites are occupied by healthy individuals. Each infected individual emits germs according to a Poisson process, it stays infected for a random time, then it recovers and becomes immune to further infection. A germ emitted from $x\in{\mathbb Z}^d$ goes to one of the neighbors $y\in{\mathbb Z}^d$ of $x$ chosen at random. If the individual at $y$ is healthy then it becomes infected and begins to emit germs; if this individual is infected or immune, nothing happens. The germ emission processes and the durations of infections of different individuals are mutually independent.\\ \\
After Mollison's papers, this epidemic model has given rise to many studies, and other models that are variations of this ``SIR'' (Susceptible-Infected-Recovered) structure have been introduced. A first direction to study such models is whether the different states asymptotically survive or not, according to the values of the involved parameters (e.g. the infection and recovery rates). A second direction
is the obtention of a shape theorem for the asymptotic behavior of infected individuals, when there is no extinction of the infection (throughout this paper, ``extinction'' is understood as ``extinction of the infection''). \\ \\
\citet{K} proved that for $d=1$, extinction is almost sure for the general epidemic model.
\citet{MR0675138} has studied the threshold behavior of this model in dimension $d\ge 2$. He proved that the process has a critical infection rate below which extinction is almost certain, and above which there is survival, thus closing this question. His work (as well as the following ones on this model) is based on the analysis of an oriented percolation model, that he calls a ``locally dependent random graph'', in correspondence with the epidemic model. See also the related paper \citet{MR0766826}.\\ \\
In the general epidemic model on ${\mathbb Z}^2$, when there is no extinction, \citet{MR0978353} have derived a shape theorem for the set of infected or immune individuals
when the contamination rule is nearest neighbor,
and the durations of infection are positive with a positive probability. A second moment is required for those durations only to localize the infected but not immune individuals within the shape obtained. This result was extended to a finite range contamination rule by \citet{MR1245289}. The proofs in \citet{MR0978353,MR1245289} are based on the correspondence with the locally dependent random graph;
they refer to \citet{MR0624685}, which deals with first passage percolation (see also \citealp{MR0876084}),
including the possibility of infinite passage times. They rely on circuits to delimit and control open paths. This technique cannot be used for dimension greater than $2$.\\ \\
There was no investigation of the shape theorem for the general epidemic model in higher dimensions, until \citet[unpublished]{C} proved it for a nearest neighbor contamination rule in dimension $d\ge 3$,
with the restriction to deterministic durations of infection: in that case the oriented percolation model is comparable to a non-oriented Bernoulli percolation model (as noticed in \citealp{MR0675138}, the case with constant durations of infection in the epidemic model is the only one where the edges are independent
in the percolation model).
Analyzing the epidemic model for $d\ge 3$ required heavier techniques than before: \citet{C} used results from \citet{MR1404543} and \citet{MR1068308} for non-oriented Bernoulli percolation to derive, for the percolation model,
exponential estimates in the subcritical case on the one hand, and estimates using percolation on slabs on the other hand. To apply those results to the epidemic model required to find an alternative, in the percolation model,
to the neighborhoods (for points in ${\mathbb Z}^2$) delimited by circuits of \citet{MR0978353}. \citet{C} introduced new types of random neighborhoods characterized by the properties of the percolation model in dimension $d\ge 3$. \\ \\
In the present work, we complete the derivation of the shape theorem for the set of infected or immune individuals in the general epidemic model with a nearest neighbor contamination rule in dimension $d\ge 3$, by proving it for random durations of infection, which are positive with a positive probability. There, the comparison with non oriented percolation done in \citet{C} is no longer valid, and we have to deal with an oriented dependent percolation model, with possibly infinite passage times.
Our approach consists in adapting the dynamic renormalization techniques
of \citet{MR1068308} without calling on \citet{MR1404543}. This simplifies the paper, but we obtain sub-exponential estimates (which suffice for our purposes), instead of exponential estimates as in the paper \citet{C}.
With this in hand, it is then possible to catch hold of the skeleton of the latter:
We take advantage of the random neighborhoods introduced there
(they turn out to be still valid in our setting) to derive the shape theorem. Similarly to \citet{MR0978353}, we require a moment of order $d$ of the durations of infection only to localize the infected but not immune individuals within the shape obtained. \\ \\
Let us mention two recent works, \citet{CT} and \citet{MR2923190}, on shape theorems for (or related to) first passage (non dependent) percolation on ${\mathbb Z}^d$ with various assumptions on the passage times, for which the approach in \citet{MR0876084} is extended. \\ \\
Our paper is organized as follows. In Section \ref{sec:model} we define the general epidemic model, the locally dependent random graph, we explicit their link, and we state the shape theorem (Theorem \ref{th:shape}). Section \ref{sec:appliquer_GM} is devoted to the necessary percolation estimates on the locally dependent random graph needed for Theorem \ref{th:shape}. We prove the latter in Section \ref{sec:Nicolas}, thanks to an analysis of the travel times for the epidemic. In Appendix \ref{sec:appendix}, we prove all the results of Section \ref{sec:appliquer_GM} requiring dynamical renormalisation techniques.
\section{The set-up: definitions and results}\label{sec:model}
Let $d\ge 3$. The epidemic model on ${\mathbb Z}^d$ is represented by a Markov process $(\eta_t)_{t\ge 0}$ of state space $\Omega=\{0,i,1\}^{{\mathbb Z}^d}$. The value $\eta_t(x)\in\{0,i,1\}$ is the state of the individual located at site $x$ at time $t$: state 1 if the individual is healthy (but not immune), state $i$ if it is infected, or state 0 if it is immune. We will shorten this in ``site $x$ is healthy, infected or immune''.
We assume that at time 0, the origin $o=(0,\ldots,0)$ is the only infected site while all other sites are healthy. That is, the initial configuration $\eta_0$ is given by
\begin{equation}\label{eq:IC} \eta_0(o)=i,\qquad\forall\, z\not= o, \,\eta_0(z)=1. \end{equation}
We now describe how the epidemic propagates, then we introduce a related locally dependent oriented bond percolation model on ${\mathbb Z}^d$, and finally we link the two models. We assume that all the processes and random variables we deal with are defined on a common probability space, whose probability is denoted by $P$, and the corresponding expectation by $E$. \par
For $x=(x_1,\ldots,x_d)\in{\mathbb Z}^d, y=(y_1,\ldots,y_d)\in{\mathbb Z}^d$,
$\| x-y\|_1=\sum_{i=1}^d |x_i-y_i|$ denotes the $l^1$ norm of $x-y$,
and we write $x\sim y$ if $x,y$ are neighbors, that is $\| x-y\|_1=1$.
Let $(T_x, e(x,y): x,y \in {\mathbb Z}^d, x\sim y)$ be independent random variables such that \par \noindent 1) the $T_x$'s are nonnegative with a common distribution satisfying $P(T_x=0)<1$;\par \noindent 2) the $e(x,y)$'s are exponentially distributed with a parameter $\lambda>0$. \\
We stress that the only assumption on the $T_x$'s is that their distribution is not a Dirac mass on 0. They could be infinite, or without any finite moment.
We define
\begin{equation}\label{def:open-closed-bonds} X(x,y)= \begin{cases} 1 &\text{if $e(x,y)<T_x$;}\\ 0 &\text{otherwise.} \end{cases} \end{equation}\\
In the epidemic model, for a given infected individual $x$, $T_x$ denotes the amount of time $x$ stays infected; during this time of infection, $x$ emits germs according to a Poisson process of parameter $2d\lambda$; when $T_x$ is over, $x$ recovers and its state becomes 0 forever. An emitted germ from $x$ at some time $t$ reaches $y$ (say), one of the $2d$ neighbors of $x$, uniformly. If this neighbor $y$
is in state 1 at time $t^-$, it immediately changes to state $i$ at time $t$,
from $t$ begins the duration of infection $T_y$, and $y$ begins to emit germs according to the same rule as $x$ did; if this neighbor $y$ is in state 0 or $i$ at time $t^-$, nothing happens. \\ \\
In the percolation model, for $x,y\in{\mathbb Z}^d, x\sim y$, the oriented bond $(x,y)$ is said to be \textit{open with passage time} $e(x,y)$ (abbreviated $\lambda$\textit{-open}, or \textit{open} when the parameter is fixed) if $X(x,y)=1$ and \textit{closed} (with infinite passage time) if $X(x,y)=0$. As in \citet{MR0675138}, we call this oriented percolation model
a {\sl locally dependent random graph}. Indeed the fact that {\sl any} of the bonds exiting from site $x$ is open depends on the r.v. $T_x$.\par
For $x,y\in{\mathbb Z}^d$ (not necessarily neighbors), ``$x\to y$'' means that there exists (at least) {\sl an open path} from $x$ to $y$, that is a path of open oriented bonds,
$\Gamma_{x,y}=(z_0=x,z_1,\ldots,z_n=y)$. \par
If $x\to y,\,x\not=y$, we define the \textit{passage time on} $\Gamma_{x,y}$ to be (see \eqref{def:open-closed-bonds})
\begin{equation}\label{eq:passage-time-on-Gamma} \overline{\tau}(\Gamma_{x,y})=\sum_{j=0}^{n-1}e(z_j,z_{j+1}) \end{equation}
and, if $x=y$, we put $\overline{\tau}(\Gamma_{x,x})=0$.\par \noindent
We then define the \textit{travel time from $x$ to} $y$ to be
\begin{equation}\label{eq:passage-time-from-x-to-y} \tau(x,y)= \begin{cases} \displaystyle{\inf_{\{\Gamma_{x,y}\}}}\,\overline{\tau}(\Gamma_{x,y})
&\text{if $x\not=y,x\to y$,}\\ 0 &\text{if $x=y$,}\\ +\infty &\text{otherwise.} \end{cases} \end{equation}
where the infimum is over all possible open paths from $x$ to $y$. \\ \\
Coming back to the epidemic model, note that when the initial configuration is $\eta_0$ defined in \eqref{eq:IC}, for a given site $z$, $\tau(o,z)$ is the duration for the infection to propagate from $o$ to $z$, changing successively the values on all the sites of the involved path $\Gamma_{o,z}$ from 1 to $i$.\\ \\
To link the two models, we define, for $t\ge 0$,
\begin{eqnarray}\label{eq:immune_at_t} \Xi_t &= \{x\in{\mathbb Z}^d: x \mbox{ is immune at time } t\} &= \{x\in{\mathbb Z}^d: \eta_t(x)=0\};\\ \label{eq:infected_at_t} \Upsilon_t &= \{x\in{\mathbb Z}^d: x \mbox{ is infected at time } t\} &= \{x\in{\mathbb Z}^d: \eta_t(x)=i\}. \end{eqnarray}
We have, for $z\in{\mathbb Z}^d,\,t\ge 0$,
\begin{equation}\label{eq:xi-zeta-tau}
z\in\Upsilon_t\cup \Xi_t \quad\hbox{ if and only if }\quad {\tau}(o,z)\le t. \end{equation}
Indeed, ${\tau}(o,z)\le t$ means that the infection has reached site $z$ before time $t$, so that site $z$ is either still infected or already immune at time $t$, that is $z\in\Upsilon_t\cup \Xi_t$. Conversely, if $z$ is infected or immune at time $t$, it means that it has already been infected.\\ \\
In the epidemic model, we denote by $C_o^{\rm out}$ the set of sites that will ever become infected, that is
\begin{equation}\label{eq:C_o}
C_o^{\rm out}=\{x\in{\mathbb Z}^d:\exists\, t\ge 0, \eta_t(x)=i|\eta_0(o)=i,\forall\, z\not= o, \eta_0(z)=1\}. \end{equation}
Then, by \eqref{eq:xi-zeta-tau}, $C_o^{\rm out}$ is the set of sites that can be reached from the origin following an open path in the percolation model. See also \citet[(1.2)]{MR0978353}, \citet[p. 322]{MR0496851} and \citet[Lemma 3.1]{MR0675138}. \\ \\
More generally, in the percolation model, for each $x\in {\mathbb Z}^d$ we define the \textit{ingoing and outgoing clusters to and from} $x$ to be
\begin{equation}\label{def:clusters_of_x} C_x^{\rm in} =\{y\in {\mathbb Z}^d:y\rightarrow x\},\qquad C_x^{\rm out} =\{y\in {\mathbb Z}^d:x\rightarrow y\}, \end{equation}
and the corresponding critical values to be
\begin{equation}\label{def:critical_of_x} \lambda_c^{\rm in}= \inf \{\lambda: P(\vert C_x^{\rm in}\vert =+\infty)>0\},\quad \lambda_c^{\rm out} = \inf \{\lambda: P(\vert C_x^{\rm out}\vert= +\infty)>0\}, \end{equation}
where
$|A|$ denotes the cardinality of a set $A$. \par
In Section \ref{sec:appliquer_GM}, we will first prove the following proposition
about these critical values.
\begin{proposition}\label{cor:lambda-i=lambda-o}
We have $\lambda_c^{\rm in}=\lambda_c^{\rm out}.$ This common value will be denoted by $\lambda_c=\lambda_c({\mathbb Z}^d)$. \end{proposition}
Assuming that $\lambda>\lambda_c$, the most important part of our work in Section \ref{sec:appliquer_GM} will then be, thanks to dynamic renormalization techniques, to analyze for the percolation model percolation on slabs in Theorem \ref{prop:clusters_rentrant-sortant_infinis}, and, through a succession of lemmas, to establish in Proposition \ref{lem:ap2} subexponential estimates for the length of the shortest path between two points $x$ and $y$ given that $x\to y$. This will imply (see Remark \ref{rk:csq-ap2}) uniqueness of the infinite cluster of sites connected to $+\infty$. Proposition \ref{lem:ap2} contains the crucial properties we will need on the percolation model to derive our main result, the shape theorem, that we now state.
\begin{theorem}\label{th:shape} Assume $\lambda>\lambda_c$, and the initial configuration of the epidemic model $(\eta_t)_{t\ge 0}$ to be given by \eqref{eq:IC}. Then there exists a convex subset $D\subset {\mathbb R}^d$ such that, for all $\varepsilon>0$ we have, for $t$ large enough
\begin{equation}\label{eq:shape} \Bigl((1-\varepsilon)tD \cap C_o^{\rm out} \Bigr)\subset \Bigl( \Xi_t \cup \Upsilon_t\Bigr) \subset\Bigl( (1+\varepsilon)tD \cap C_o^{\rm out} \Bigr)\mbox{ a.s.}
\end{equation}
and if $E(T_o^d)<\infty$ we also have
\begin{equation}\label{eq:couronne}
\Upsilon_t \subset\Bigl( (1+\varepsilon)tD \setminus (1-\varepsilon)tD \Bigr) \mbox{ a.s. for}\ t \mbox{ large enough.} \end{equation}
\end{theorem}
In other words, the epidemic's progression follows linearly the boundary of a convex set.
Note that a moment assumption on $T_o$ is only required to localize the infected individuals, and not for the first part of the theorem, for which there is no assumption on the distribution of $T_o$. It is also remarkable that the fact that $T_o$ could be either very small or very large with respect to the exponential variables $e(x,y)$ does not play any role. \\ \\
We prove Theorem \ref{th:shape} in Section \ref{sec:Nicolas}. For this, we follow some of the fundamental steps of \citet{MR0978353}, but since in dimensions three or higher, circuits are not useful as in dimension 2, we had to find other methods of proofs. \\ \\
By \eqref{eq:xi-zeta-tau}, we have to analyze travel times to prove Theorem \ref{th:shape}.
On the percolation model, we first construct, in Section \ref{subsec:widetildeC},
for each site $z\in{\mathbb Z}^d$ a random neighborhood ${\mathcal V}(z)$ in such a way that two neighborhoods are always connected by open paths (these neighborhoods have to be different from those delimited by circuits of \citealp{MR0978353}). For $z,y\in{\mathbb Z}^d$, we show that the travel time $\tau(z,y)$
is `comparable' (in a sense precised in Lemma \ref{lem:comparaison-approximation}) to the travel time ${\widehat\tau}(z,y)$ to go from ${\mathcal V}(z)$ to ${\mathcal V}(y)$.
Then we approximate the travel time between sites by a subadditive process, and we derive (in Theorem \ref{th:radial-limits} and Section \ref{subsec:extending_mu}) a radial limit $\mu(x)$ (for all $x$), which is asymptotically the linear growth speed of the epidemic in direction $x$. In Theorem \ref{th:widehat_t-serie} we control how ${\widehat\tau}(o,\cdot)$ grows. Finally we prove in Theorem \ref{th:At-encadre} an asymptotic shape theorem for $\widehat\tau(o,\cdot)$, from which we deduce Theorem \ref{th:shape}.
\section{Percolation estimates}\label{sec:appliquer_GM}
In this section we collect some results concerning the locally dependent random graph,
given by the random variables $(X(x,y),x,y\in {\mathbb Z}^d)$
introduced in \eqref{def:open-closed-bonds}.
Our goal is to derive subexponential estimates
in Proposition \ref{lem:ap2}.
\begin{remark}\label{rk:indepdtvectors} Although the r.v.'s $(X(x,y),x,y\in {\mathbb Z}^d)$ are not independent, if we denote by $({\rm e}_1,\ldots,{\rm e}_d)$ the canonical basis of ${\mathbb Z}^d$, then the random vectors $\{X(x,x+{\rm e}_1),\ldots,X(x,x+{\rm e}_d),X(x,x-{\rm e}_1),\ldots,X(x,x-{\rm e}_d): x\in {\mathbb Z}^d\}$ (in which each component depends on $T_x$) are i.i.d., since two different vectors for $z,y\in{\mathbb Z}^d$ depend respectively on $T_z$ and $T_y$ which are independent. This small dependence forces us to explain why and how some results known for independent percolation remain valid in this context.
\end{remark}
\begin{remark}\label{rk:FKG}
The function $X(x,y)$ is increasing in the independent random variables $T_x$ and $-e(x,y)$. It then follows
that the r.v.'s $(X(x,y): x,y \in {\mathbb Z}^d, y\sim x)$ satisfy the following property:\\
\newline {\rm (FKG)} Let $U$ and $V$ be bounded measurable increasing functions of the random variables $(X(x_j,y_j): x_j,y_j \in {\mathbb Z}^d, y_j\sim x_j, j\in {\mathbb N})$, then $E(UV)\ge E(U)E(V)$. \\
\newline For the proof of this property, we refer to \citet[Lemma (2.1)]{MR0978353}
with the help of \citet[Lemma 4.1 and its Corollary]{MR0115221}
if $U$ and $V$ depend on a finite number of variables
$X(x_j,y_j)$, and to \citet[Chapter 2]{MR1707339} to take the limit
for an infinite number of variables.
\\
\newline We will use this property
in the proofs of Theorem \ref{prop:clusters_rentrant-sortant_infinis},
Lemma \ref{lem:APp} and Lemma \ref{lem:k_x-exp_tail}
below for $U,V$ indicator functions involving open paths without loops,
thus we will speak of increasing events rather than increasing functions.
\end{remark}
For $n\in{\mathbb N}\setminus\{0\}$, let $B(n)=[-n,n]^d$, let $\partial B(n)$ denote the \textit{inner vertex boundary} of $B(n)$, that is
\begin{equation}\label{innervertexboundary} \partial B(n)= \{x\in{\mathbb Z}^d:x\in B(n), x\sim y \mbox{ for some } y\notin B(n)\}; \end{equation}
and, for $x\in{\mathbb R}^d$, $B_x(n)=x+B(n)$.
For $A,R\subset{\mathbb Z}^d$, ``$A\rightarrow R$'' means that there exists an open path $\Gamma_{x,y}$ from some $x\in A$ to some $y\in R$.
\begin{theorem}\label{th:outgoing-decay}
(i) Suppose $\lambda <{\lambda}_c^{\rm out}$, then there exists $\beta_{\rm out}>0$ such that for all $n>0$, $P(o\rightarrow \partial B(n))\leq \exp (-\beta_{\rm out} n).$
\noindent (ii) Suppose $\lambda <{\lambda}_c^{\rm in}$, then there exists $\beta_{\rm in}>0$ such that for all $n>0$, $P(\partial B(n) \rightarrow o)\leq \exp (-\beta_{\rm in} n).$ \end{theorem}
Theorem \ref{th:outgoing-decay}\textit{(i)} is a special case of \citet[Theorem (3.1)]{MR1624925}, whose proof can be adapted to obtain Theorem \ref{th:outgoing-decay}\textit{(ii)}. It is worth noting that in the context of our paper, by Remark \ref{rk:FKG}, \citet[Theorem (3.1)]{MR1624925} can be proved using the BK inequality instead of the Reimer inequality
(see \citealp[Theorems (2.12), (2.19)]{MR1707339}).
Theorem \ref{th:outgoing-decay} yields Proposition \ref{cor:lambda-i=lambda-o}: \\
\begin{proof}{Proposition}{cor:lambda-i=lambda-o}
Suppose $\lambda <{\lambda}_c^{\rm in}$. Then by translation invariance and Theorem \ref{th:outgoing-decay}\textit{(ii)} we have that for any $x\in \partial B(n)$, $P(o\rightarrow x)\leq \exp (-\beta_{\rm in} n)$. Adding over all points of $\partial B(n)$ we get $P(o\rightarrow \partial B(n))\leq K'n^{d-1} \exp (-\beta_{\rm in} n)$ for some constant $K'$, which implies that $\lim_{n\to +\infty} P(o\rightarrow \partial B(n))=0$. Therefore $\lambda\leq \lambda_c^{\rm out}$ and $\lambda_c^{\rm in}\leq \lambda_c^{\rm out}$. The other inequality is obtained similarly.
\end{proof}
\mbox{}\\ \\
{}From now on, we assume $\lambda>\lambda_c({\mathbb Z}^d)$ and define the following events:
For $x,y \in{\mathbb Z}^d,A\subset{\mathbb Z}^d$,
\newline \textit{(i)} The event $\{ x\rightarrow y \ \mbox {within }\ A \}$ consists of all points in our probability space for which there exists an open path
$\Gamma_{x,y}=(x_0=x,x_1,\ldots,x_n=y)$ from $x$ to $y$ such that
$x_j\in A$ for all $j\in\{0,\ldots,n-1\}$.
Note that the end point $y$ may not belong to $A$.
\newline \textit{(ii)} The event $\{x\to y \hbox{ outside } A\}$ consists of all points in our probability space for which
there exists an open path $\Gamma_{x,y}=(x_0=x,x_1,\ldots,x_n=y)$ from $x$ to $y$ such that none of the $x_j$'s ($j\in\{0,\ldots,n\}$) belongs to $A$.
\begin{definition}\label{def:within-outside} For $x\in{\mathbb Z}^d,A\subset{\mathbb Z}^d$ let
\begin{eqnarray*}\label{eq:C_xA} C_x^{\rm in}(A) &=&\{y\in A:y\rightarrow x\ \hbox{ within }A \}\qquad \mbox{and}\cr C_x^{\rm out}(A) &=&\{y\in A:x\rightarrow y\ \hbox{ within }A\}. \end{eqnarray*}
\end{definition}
Note that by this definition $C_x^{\rm in}(A)\subset A$ and $C_x^{\rm out}(A)\subset A$.\\ \\
The rest of this section relies heavily on the techniques of \citet{MR1068308} or \citet[Chapter 7]{MR1707339}. We assume the reader familiar with them. We postpone to Appendix \ref{sec:appendix} the proofs of Theorem \ref{prop:clusters_rentrant-sortant_infinis} and Lemma \ref{lem:connections} below, which require a thoughtful adaptation of \citet[Chapter 7]{MR1707339} for our context of dependent percolation. Nonetheless, it is possible to go directly to Section \ref{sec:Nicolas}, where these techniques are no longer used, assuming that Proposition \ref{lem:ap2} holds.\par
Next theorem is crucial, it states that there is percolation on slabs.
\begin{theorem}\label{prop:clusters_rentrant-sortant_infinis}
Assume $\lambda>\lambda_c$. For any $k\in{\mathbb N}\setminus\{0\}$, let $S_k=\{0,1,\dots,k\}\times{\mathbb Z}^{d-1}$ denote the slab of thickness $k$ containing $o$. Then for $k$ large enough we have
$$ \inf_{x\in S_k}P(\vert C_x^{\rm in}(S_k)\vert =+\infty)>0, \quad \mbox{ and }\quad \inf_{x\in S_k}P(\vert C_x^{\rm out}(S_k)\vert =+\infty)>0. $$
\end{theorem}
We introduce now some notation about the shortest path between two points $x$ and $y$ such that $x\to y$.
\begin{notation}\label{not:extvertbound-Dxy}
(a) For $A\subset{\mathbb Z}^d$
we define the {\sl exterior vertex boundary} of $A$ as:
\begin{equation}\label{exteriorvertexboundary} \Delta_v A= \{x\in{\mathbb Z}^d:x\notin A, x\sim y \mbox{ for some } y\in A\}. \end{equation}
(b) If $x\rightarrow y$ let $D(x,y)$ be the smallest number of bonds required to build an open path from $x$ to $y$
(hence in this path there is no loop, and the $D(x,y)$ bonds are distinct).
If $x\not\rightarrow y$, we put $D(x,y)=+\infty$.\par
\noindent (c) For $A\subset{\mathbb Z}^d$, $x\in A,y\in \Delta_v A$, ``$D(x,y)<m \mbox{ within }\ A$''
means that there is an open path $\Gamma_{x,y}$
using less than $m$ bonds from $x$ to $y$ whose sites are all in $A$ except $y$.
\end{notation}
The end of this section provides some upper bounds for the tail of the conditional distribution of $D(x,y)$ given the event $\{x\rightarrow y\}$. We derive Proposition \ref{lem:ap2}, required in Section \ref{sec:Nicolas}, thanks to Lemmas \ref{lem:connections}, \ref{lem:reaching faces}, \ref{lem:APp}. These estimates are not optimal and better results could be obtained by a thoughtful adaptation of the methods of \citet{MR1404543}.
Instead of getting exponential decays in $\|x-y\|_1$ (or in $n$) we get exponential decays in
$\|x-y\|_1^{1/d}$ (or in $n^{1/d}$). We have adopted this approach because those weaker results suffice for our purposes and are simpler to obtain, thus making our proof much easier to follow: it is possible to read our work knowing only \citet{MR1068308} and not \citet{MR1404543}. Next lemma is inspired by \citet[Section 5\textit{(f)} p. 454]{MR1068308}.
\begin{lemma}\label{lem:connections}
Assume $\lambda>\lambda_c$. There exist $\delta>0$, $k\in {\mathbb N}\setminus\{0\}$ and ${\rm C}_1={\rm C}_1(k)>0$ such that
\noindent (i) $\forall n>0,\ x\in B(n+k)\setminus B(n),\ y\in (B(n+k)\setminus B(n)) \cup \Delta_v(B(n+k)\setminus B(n) ) $ we have :
$$P(x\rightarrow y \ \mbox{ within }\ B(n+k)\setminus B(n))>\delta. $$
(ii) Let for $(n,m)\in{\mathbb Z}^2$ with $n<m$, and for $\ell\ge 0$,
\begin{eqnarray} A(n,m,\ell)&=&\{z: -k+n\leq z_1<n, -\infty<z_2\leq \ell+k\}\cup\cr && \{z:-k+n\leq z_1\leq m+k, \ell<z_2\leq \ell+k\}\cup \cr &&\{z:m<z_1\leq m+k, -\infty<z_2\leq \ell+k\}.\label{eq:A_nm} \end{eqnarray}
$\forall n<m,\ \forall x \in A(n,m,0),\forall y\in A(n,m,0)\cup \Delta_v A(n,m,0)$, we have:
$$P(D(x,y)<{\rm C}_1(\| x-y\|_1 +(-x_2)^+ +(-y_2)^+)\ \mbox{ within } A(n,m,0))>\delta.$$
\end{lemma}
We again introduce some notation, to decompose in Lemma \ref{lem:reaching faces} a path from the center of a box to its boundary through hyperplanes.
\begin{notation}\label{not:HnJnGn} Let $k$ be given by Lemma \ref{lem:connections} and let $x$ and $y$ be points in ${\mathbb Z}^d$. For $\ell\in {\mathbb Z}$ let $H_\ell=\{z\in {\mathbb Z}^d:z_1=\ell\}$ and define the events, for $n\in{\mathbb N}$,
\begin{eqnarray*}
J_n&=&\{x\rightarrow H_{x_1-1-jk}\ \mbox{within}\ B_x(nk),
j=0,\dots, \lfloor n/2\rfloor\}\cap \cr &&\quad\{ H_{y_1+1+jk}\rightarrow y\ \mbox{within}\ B_y(nk), j=0,\dots, \lfloor n/2\rfloor\},\cr G_n&=&\{x\rightarrow \partial B_x(nk), \ \partial B_y(nk)\rightarrow y\}, \end{eqnarray*}
where, for any $a\in{\mathbb R}$, $\lfloor a\rfloor$ denotes the greatest integer not greater than $a$. \end{notation}
\begin{lemma}\label{lem:reaching faces}
Assume $\lambda>\lambda_c$. Let $k$ be given by Lemma \ref{lem:connections} and let $x,y$ be points in ${\mathbb Z}^d$. Then, for $n\in {\mathbb N}\setminus\{0\}$ there exists $\beta>0$ such that
\[ P(J_n\vert G_n)\ge 1-\exp(-\beta n). \]
\end{lemma}
\begin{proof}{Lemma}{lem:reaching faces}
By translation invariance we may assume that $x$ is the origin. We start showing that for some constant $\beta'>0$ and all $n$
\begin{eqnarray} && P(o\rightarrow H_{-1-jk}\ \mbox{within}\ B(nk),
j=0,\dots, \lfloor n/2\rfloor {\vert} o\rightarrow \partial B(nk))\cr && \ge 1-\exp(-\beta' n).\label{inter} \end{eqnarray}
For this we first observe that
\begin{eqnarray*} &&\{o\rightarrow H_{-1-jk}\ \mbox{within}\ B(nk) \mbox{ for some }\lfloor n/2\rfloor \leq j\leq n \} \\ &&\subset \{ o\rightarrow H_{-1-jk}\ \mbox{within}\ B(nk),
j=0,\dots, \lfloor n/2\rfloor \}.
\end{eqnarray*}
Hence \eqref{inter} follows from
\begin{eqnarray*}\label{inter'} && P(o\rightarrow H_{-1-jk} \mbox{ within } B(nk) \mbox{ for some }\lfloor n/2\rfloor \leq j\leq n {\vert} o\rightarrow \partial B(nk))\cr &&\ge 1-\exp(-\beta' n), \end{eqnarray*}
which is a consequence of Lemma \ref{lem:connections}\textit{(i)}. Since $P(\partial B_y(kn)\rightarrow y)$ is bounded below as $n$ goes to infinity, \eqref{inter} implies that
$$P(o\rightarrow H_{-1-jk}\ \mbox{ within}\ B(nk),
j=0,\dots, \lfloor n/2\rfloor {\vert} G_n )$$
converges to $1$ exponentially fast. Similarly one proves that
$$ P(H_{y_1+1+jk}\rightarrow y\ \mbox{within}\ B_y(nk), j=0,\dots, \lfloor n/2\rfloor \vert G_n) $$
converges to $1$ exponentially fast, and the lemma follows. \end{proof}
\mbox{}\\ \\
In Lemma \ref{lem:APp} below we prove a chemical distance bound that will be used later on to derive in Remark \ref{rk:csq-ap2}, through Proposition \ref{lem:ap2}, the uniqueness of the infinite cluster of sites connected to $+\infty$. The main technique is to construct an open path in a ring after independent attempts thanks on the one hand to Lemma \ref{lem:reaching faces} whose $J_n$'s enable to get disjoint slabs, and on the other hand to Lemma \ref{lem:connections}\textit{(ii)} once we find the appropriate ring.
\begin{lemma}\label{lem:APp}
Assume $\lambda>\lambda_c$. Let $k$ be given by Lemma \ref{lem:connections}, and let $G_n$ be as in Lemma \ref{lem:reaching faces}. Then, there exist constants ${\rm C}_2$, ${\rm C}_3$ and $\alpha_2>0$ such that, for all $x,y \in {\mathbb Z}^d$,
$n\in {\mathbb N}\setminus\{0\}$, we have
$$P(D(x,y)> {\rm C}_2 \|x-y\|_1+{\rm C}_3(nk)^d \vert \ G_n)\leq \exp(-\alpha_2 n). $$
\end{lemma}
\begin{proof}{Lemma}{lem:APp}
Again, by translation invariance we may assume that $x$ is the origin
and without loss of generality, we also assume that $y_1>0$ and $y_2\ge 0$. By Lemma \ref{lem:reaching faces} it suffices to show that
$$P(D(o,y)> {\rm C}_2 \|y\|_1+{\rm C}_3(nk)^d \vert \ J_n)$$
decays exponentially in $n$.
For $0\le j\le \lfloor n/2\rfloor$, let (see \eqref{eq:A_nm})
$A_j=A(-jk,y_1+jk,y_2+jk)$.
\begin{figure}
\caption{the event $W_3$}
\label{fig:dessin_lemme3.4-2}
\end{figure}
Note that the sets $A_0,\dots, A_{\lfloor n/2\rfloor}$ are disjoint. Figure 1
should help the reader to visualize them. Our aim is to
find paths from $o$ to $y$ through independent attempts, which will enable to use Lemma \ref{lem:connections}\textit{(ii)} in each set $A_j$. This is why we have first replaced $G_n$ by $J_n$ to condition with.\par
On the event $J_n$, we can reach from the origin each of the sets $A_i$ by means of an open path contained in $B(nk)$ and from each of these sets we can reach $y$ by means of an open path contained in $ B_y(nk)$. Hence, on $J_{n}$ for each $j\in \{0,\dots,\lfloor n/2\rfloor\}$ there exist a random point $U_j\in B(nk)\cap A_j$ and an open path from $o$ to $U_j$ such that all its sites except $U_j$ are in $B(nk)\cap (\cap_{\ell=j}^{\lfloor n/2\rfloor}A_\ell^c)$. If there are many possible values of $U_j$ we choose the first one in some arbitrary deterministic order. Similarly, there is a random point $V_j\in B_y(nk)\cap \Delta_v A_j$ and an open path from $V_j$ to $y$ with all its sites in $B_y(nk)\cap (\cap_{\ell=j}^{\lfloor n/2\rfloor}A_\ell^c)$. Let $u^j$ and $v^j$ be possible values of $U_j$ and $V_j$ respectively. Then let $\rm C_1$ be as in Lemma \ref{lem:connections} and define
\begin{eqnarray*}
F_j(u^j,v^j)&=&\{U_j=u^j,V_j=v^j\}, \cr E_j(u^j,v^j)&=&\{D(u^j,v^j)< {\rm C_1}(\| u^j-v^j\|_1+\vert u^j_2\vert +\vert v^j_2\vert)\ \mbox{within}\ A_j\}\ \mbox{and} \cr W_j&=&\cup_{u^j,v^j}\left(F_j(u^j,v^j)\cap E_j(u^j,v^j)\right), \end{eqnarray*}
where the union is over all possible values of $U_j$ and $V_j$. Now we define a subset of ${\mathbb Z}^d$
\begin{equation}
R_j=\Big(B(nk)\cup B_y(nk)
\cup( A_0\cup\dots \cup A_{j-1})\Big)\cap \Big( A_j^c\cap \dots \cap A_{n-1}^c\Big), \end{equation}
and we denote by $\sigma_j$ the $\sigma$-algebra generated by $\{T_x,e(x,y):x \in R_j,x\sim y\}$. Then, noting that $\bold{1}_{F_j(u^j,v^j)}\Pi_{\ell=0}^{j-1}\bold{1}_{W_\ell^c} $ is $\sigma_j$-measurable, write for $j=1,\dots, \lfloor n/2\rfloor$:
\begin{eqnarray} && P\left(W_j\cap J_n \cap( \cap_{\ell=0}^{j-1}W_\ell^c)\right)=\sum_{u^j,v^j}E\left(\bold{1}_{F_j(u^j,v^j)}\mathbf{1}_{E_j(u^j,v^j)} \bold{1}_{J_n}(\Pi_{\ell=0}^{j-1}\bold{1}_{W_\ell^c})\right)\nonumber\\ &=& \sum_{u^j,v^j}E\left(\bold{1}_{F_j(u^j,v^j)}(\Pi_{\ell=0}^{j-1}\bold{1}_{W_\ell^c}) E( \bold{1}_{J_n} \bold{1}_{E_j(u^j,v^j)}\vert \sigma_j)\right)\nonumber\\ &\ge& \sum_{u^j,v^j}P(E_j(u^j,v^j))E \left( \bold{1}_{F_j(u^j,v^j)}(\Pi_{\ell=0}^{j-1}\bold{1}_{W_\ell^c}) E( \bold{1}_{J_n} \vert \sigma_j)\right)\nonumber\end{eqnarray} \begin{eqnarray}&=&\sum_{u^j,v^j}P(E_j(u^j,v^j)) E\left( \bold{1}_{F_j(u^j,v^j)}(\Pi_{\ell=0}^{j-1}\bold{1}_{W_\ell^c}) \bold{1}_{J_n} \right)\nonumber\\ &\ge& \delta \sum_{u^j,v^j}P\left(F_j(u^j,v^j)\cap J_n \cap(\cap_{\ell=0}^{j-1}W_\ell^c)\right) =\delta P\left(J_n\cap(\cap_{\ell=0}^{j-1}W_\ell^c)\right), \label{last-en-plus} \end{eqnarray}
where the sums are over all possible values of $U_j$ and $V_j$, the first inequality follows from Remark \ref{rk:FKG} since both $J_n$ and $E_j(u^j,v^j)$ are increasing events, and from the fact that $E_j(u^j,v^j)$ is independent of $\sigma_j$;
the second inequality follows from Lemma \ref{lem:connections}\textit{(ii)} and the last equality from the fact that $J_n$ is contained in the union of the $F_j(u^j,v^j)$'s which are disjoint. We rewrite \eqref{last-en-plus} as
\begin{eqnarray*} P\left(J_n\cap(\cap _{\ell=0}^{j}W_\ell^c)\right)&\le& (1-\delta)P\left(W_j\cap J_n\cap(\cap _{\ell=0}^{j-1}W_\ell^c)\right)\cr &\le& (1-\delta)P\left(J_n\cap(\cap _{\ell=0}^{j-1}W_\ell^c)\right) \end{eqnarray*}
Now, proceeding by induction on $j$ one gets:
$$P\left(J_n\cap(\cap _{\ell=0}^{\lfloor n/2\rfloor-1}W_\ell^c)\right) \le (1-\delta)^{\lfloor n/2\rfloor}P(J_n).$$
Since we can choose ${\rm C}_2$ and ${\rm C}_3$ in such a way that the event
$\{D(o,y)> {\rm C}_2 \|y\|_1+{\rm C}_3(nk)^d \}$ does not occur if any of the $W_i$'s occurs, the lemma follows.
\end{proof}
\mbox{}\\ \\
Next proposition concludes this section.
\begin{proposition}\label{lem:ap2}
Assume $\lambda>\lambda_c$.
(i) Let ${\rm C}_2$ be as in Lemma \ref{lem:APp}. Then, there exists $\alpha_3>0$ such that
for all $x,y\in{\mathbb Z}^d,n\in{\mathbb N}$, we have
$$P(D(x,y)\geq {\rm C}_2\| x-y\|_1+n^d\vert x\rightarrow y)\leq \exp(-\alpha_3 n);$$
(ii) $P(x\rightarrow y \vert \ \vert C_x^{\rm out} \vert =+\infty, \ \vert C_y^{\rm in} \vert =+\infty )=1$. \end{proposition}
\begin{proof}{Proposition}{lem:ap2}
\textit{(i)} Modifying the constant $\alpha_2$, the statement of Lemma \ref{lem:APp} above holds for ${\rm C}_3=1/k^d$. \par
\textit{(ii)} We have that $\{x\to \infty\hbox{ and } \infty \to y \}=\cap_n G_n$. Hence for all $k$,
\begin{eqnarray*} P(D(x,y)=+\infty, x\to \infty\hbox{ and } \infty \to y ) &\leq& P(D(x,y)=+\infty, G_k) \\ &\leq& P(D(x,y)=+\infty \vert G_k), \end {eqnarray*}
which converges to 0 when $k$ goes to infinity by Lemma \ref{lem:APp}. We thus have $P(D(x,y)=+\infty \vert x\to \infty\hbox{ and } \infty \to y )=0$.
\end{proof}
\section{ The shape theorem }\label{sec:Nicolas}
In the percolation model, let $C^\infty$ be the cluster of sites connected to $\infty$:
\begin{equation}\label{eq:widetildeC} C^\infty= \{x\in{\mathbb Z}^d: x\to \infty \hbox{ and } \infty\to x\}. \end{equation}
\begin{remark}\label{rk:csq-ap2} As a consequence of Proposition \ref{lem:ap2}\textit{(ii)}, $C^\infty$ is a connected set: if two sites $x,y$ of ${\mathbb Z}^d$ belong to $C^\infty$, then $x\to y$ and $y\to x$. \end{remark}
\subsection{Neighborhoods in $C^\infty$}\label{subsec:widetildeC}
In this subsection, we construct neighborhoods ${\mathcal V}(\cdot)$ of sites in ${\mathbb Z}^d$.\par
We first deal separately with finite clusters, which will have no influence on the asymptotic shape of the epidemic. We will include them in the neighborhoods ${\mathcal V}(\cdot)$ of sites we construct.
\begin{definition}\label{def:racines} For $x\in{\mathbb Z}^d$, let
\[ \begin{cases} R_x^{\rm out}=\{y\in{\mathbb Z}^d: x\to y \hbox{ outside } C^\infty \}
&\text{(outgoing root from $x$);}\\ R_x^{\rm in}=\{y\in{\mathbb Z}^d: y\to x \hbox{ outside } C^\infty \}
&\text{(incoming root to $x$).} \end{cases}
\]
\end{definition}
In particular $x$ belongs to $R_x^{\rm out}$ and $R_x^{\rm in}$ if and only if $x\notin{C^\infty}$. Otherwise $R_x^{\rm out}$ and $R_x^{\rm in}$ are empty. By next lemma, the distribution of the radius of $R_o^{\rm out}\cup R_o^{\rm in}$ decreases exponentially.
\begin{lemma}\label{lem:exp_decay_R} There exists $\sigma_1=\sigma_1(\lambda,d)>0$ such that, for all $n\in{\mathbb N}$, $$P\left((R_o^{\rm out}\cup R_o^{\rm in})\cap \partial B(n)\neq\emptyset\right)\le \exp(- \sigma_1 n).$$ \end{lemma}
\begin{proof}{Lemma}{lem:exp_decay_R}
For $n\in{\mathbb N}\setminus\{0\}$, $R_o^{\rm out}\cap \partial B(2n)\neq\emptyset$ means that there exists an open path $o\to \partial B(2n)$ outside $C^\infty$. This implies that there exists $x\in \partial B(n)$ satisfying $o\to x\to \partial B(2n)$ outside $C^\infty$. Similarly, $R_o^{\rm in}\cap \partial B(2n)\neq\emptyset$ implies that there exists $x\in \partial B(n)$ satisfying $\partial B(2n)\to x\to o$ outside $C^\infty$. Then for such a point, either the cluster $C_x^{\rm out}$ or the cluster $C_x^{\rm in}$ is finite, and has a radius larger than or equal to $n$.
Relying on Proposition \ref{p1},\textit{b)} in Appendix \ref{sec:appendix}, we can follow the proof of \citet[Theorems (8.18), (8.21)]{MR1707339} to get the existence of $\sigma_0=\sigma_0(\lambda,d)>0$ such that:
\begin{equation}\label{eq:analogue_G-thm8.21} \begin{cases}
P(C_x^{\rm out} \cap \partial B_x(n)\neq\emptyset, |C_x^{\rm out}| <+\infty) \le \exp(- \sigma_0 n);\\
P(C_x^{\rm in} \cap \partial B_x(n)\neq\emptyset, |C_x^{\rm in}| <+\infty) \le \exp(- \sigma_0 n). \end{cases}
\end{equation}
Hence
\begin{eqnarray*} P\left((R_o^{\rm out}\cup R_o^{\rm in})\cap \partial B(2n)\neq\emptyset\right) &\le& P\left(R_o^{\rm out}\cap \partial B(2n)\neq\emptyset\right)\cr && +P\left(R_o^{\rm in}\cap \partial B(2n)\neq\emptyset\right)\cr
&\le& 2\sum_{x\in\partial B(n)} P(|C_x^{\rm out}| <+\infty, x\to \partial B_x(n))\cr &&
+2\sum_{x\in\partial B(n)} P(|C_x^{\rm in}| <+\infty, \partial B_x(n)\to x)\cr
& \le & 4|\partial B(n)| \exp(- \sigma_0 n) \end{eqnarray*}
which induces the result.
\end{proof}
\mbox{}\\ \\
To define the neighborhood ${\mathcal V}(x)$ on $C^\infty$ of a site $x$, we introduce the smallest box whose interior contains $R_x^{\rm out}$ and $R_x^{\rm in}$, which contains elements of $C^\infty$, and is such that two elements of $C^\infty$ in this box are connected by an open path which does not exit from a little larger box. For this last condition, which will enable to bound the
travel time through ${\mathcal V}(x)$, we use the parameter ${\rm C}_2$ obtained in Lemma \ref{lem:APp}.
\begin{definition}\label{def:k_x} Let ${\rm C}'={\rm C_2}d+2$. Let $\kappa(x)$ be the smallest $l\in{\mathbb N}\setminus\{0\}$ such that
\[ \begin{cases} (i)\,\,\,\,\, \partial B_x(l) \cap \left(R_x^{\rm out}\cup R_x^{\rm in}\right)=\emptyset;\\ (ii)\,\,\, B_x(l) \cap C^\infty \not= \emptyset;\\ (iii)\, \forall\, (y,z) \in (B_x(l) \cap C^\infty)^2,\,y\to z \hbox{ within } B_x({\rm C}'l). \end{cases}
\]
\end{definition}
\begin{remark}\label{rk:(i)} By \textit{(i)} above, $R_x^{\rm out}\cup R_x^{\rm in}\subset B_x(\kappa(x))$. \end{remark}
In the next lemma, we bound the probability a box of size $n$ does not admit properties \textit{(i)--(iii)} above, that is, we prove that the random variable $\kappa(x)$ has a sub-exponential tail.
\begin{lemma}\label{lem:k_x-exp_tail}
There exists a constant $\sigma=\sigma(\lambda,d)>0$ such that, for any $n\in{\mathbb N}$,
\[ P(\kappa(x)\ge n)\le \exp(-\sigma n^{1/d}). \]
\end{lemma}
\begin{proof}{Lemma}{lem:k_x-exp_tail}
We show that the probability that any of the 3 conditions in Definition \ref{def:k_x} is not achieved for $n$ decreases exponentially in $n^{1/d}$: \par
\noindent \textit{(i)} By translation invariance, we have by Lemma \ref{lem:exp_decay_R},
\begin{equation}\label{eq:non_i} P\left( \partial B_x(n)\cap\left(R_x^{\rm out}\cup R_x^{\rm in}\right) \not=\emptyset\right) \le \exp(-\sigma_1 n). \end{equation}
\noindent \textit{(ii)} There exist $k\in{\mathbb N}$,
$\sigma_2=\sigma_2(\lambda,d)>0$ such that for any $n\in{\mathbb N}$,
\begin{equation}\label{eq:non_ii} P(B_x(n) \cap C^\infty=\emptyset) \le \exp(-\sigma_2\lfloor n/(k+1)\rfloor). \end{equation}
Indeed, let $k=k(\lambda,d)$ be large enough for the conclusions
of Theorem \ref{prop:clusters_rentrant-sortant_infinis} to hold on the slab $S_k$. Then we have
\begin{eqnarray*} & P(B_x(n) \cap C^\infty=\emptyset) \le P(\forall\,z\in\{ x+ j{\rm e}_1,0\le j\le n\},z\notin C^\infty)\cr &= P(\forall\,z\in\{ x+ j{\rm e}_1,0\le j\le n\},C_z^{\rm in} \hbox{ or } C_z^{\rm out} \hbox{ is finite}) \end{eqnarray*}
We denote by $S_k(l)=\{l(k+1),\cdots,(l+1)(k+1)-1\}\times{\mathbb Z}^{d-1}$ for $l\ge 0$ the slab of thickness $k$ to which $z$ belongs. If $C_z^{\rm in}$ (or $C_z^{\rm out}$) is finite, so is $C_z^{\rm in}(S_k(l))$ (or $C_z^{\rm out}(S_k(l))$). Because $\{ \vert C_z^{\rm in}(S_k(l))\vert =+\infty \}$ and $\{ \vert C_z^{\rm out}(S_k(l))\vert =+\infty \}$ are increasing events
it follows from Theorem \ref{prop:clusters_rentrant-sortant_infinis} and
the FKG inequality (see
Remark \ref{rk:FKG}) that
\begin{eqnarray} &\inf_{u\in S_k(l)}P(\vert C_u^{\rm in}(S_k(l))\vert =\vert C_u^{\rm out}(S_k(l))\vert =+\infty)\cr &\ge \inf_{u\in S_k(l)}\left(P(\vert C_u^{\rm in}(S_k(l))\vert=+\infty) P(\vert C_u^{\rm out}(S_k(l))\vert =+\infty)\right)>0.\label{eq:bis-infinf} \end{eqnarray}
Since events occurring in two different slabs are independent, we have
\begin{eqnarray*} && P(\forall\,z\in\{ x+ j{\rm e}_1,0\le j\le n\},z\notin C^\infty)\cr
&\le & P(\forall\,l\ge 0,\forall\,z\in\{ x+ j{\rm e}_1,0\le j\le n\}\cap S_k(l), \cr
&&\qquad C_z^{\rm in}(S_k(l)) \hbox{ or } C_z^{\rm out}(S_k(l)) \hbox{ is finite})\cr
&\le & \left(P(\forall\,z\in\{ j{\rm e}_1,0\le j\le k\},\right.\cr
&&\left.\qquad C_z^{\rm in}(S_k(0)) \hbox{ or } C_z^{\rm out}(S_k(0))
\hbox{ is finite}\right)^{\lfloor n/(k+1)\rfloor}\cr
&\le & \exp(-\sigma_2 \lfloor n/(k+1)\rfloor)
\end{eqnarray*}
with $\sigma_2=\sigma_2(\lambda,d)>0$, independent of $n$, because, for $z_0=\lfloor k/2\rfloor{\rm e}_1$, using \eqref{eq:bis-infinf} we have
\begin{eqnarray*} & P(\exists\,z\in\{ x+ j{\rm e}_1,0\le j\le k\},
|C_z^{\rm in}(S_k(0))|=|C_z^{\rm out}(S_k(0))|=+\infty)\cr
\ge & P(|C_{z_0}^{\rm in}(S_k(0))| = | C_{z_0}^{\rm out}(S_k(0))|=+\infty) >0. \end{eqnarray*}
\noindent \textit{(iii)} There exists $ \sigma_3=\sigma_3(\lambda,d)>0$ such that
\begin{eqnarray}\label{eq:sigma_3} & P\left(\exists\,
(y,z) \in (B_x(n) \cap C^\infty)^2,\,y\not\to z \hbox{ within }(B_x({\rm C}'n)\right)\cr
& \le \exp (- \sigma_3 n^{1/d}). \end{eqnarray}
Indeed, if no open path from $y$ to $z$ (both in $B_x(n)\cap C^\infty$) is contained in $B_x({\rm C}'n)$, then $D(y,z)\ge 2({\rm C}'-1)n$. Given our choice of ${\rm C}'$
this implies that $D(y,z)\ge {\rm C_2}\|y-z\|_1 +n$. Therefore \eqref{eq:sigma_3} follows from Proposition \ref{lem:ap2}\textit{(i)}.
\end{proof}
\mbox{} \\ \\
We define the (site) neighborhood in $C^\infty$ of $x$ by
\begin{equation}\label{def:calV-x} {\mathcal V}(x)=B_x(\kappa(x))\cap C^\infty. \end{equation}
\begin{remark}\label{rk:calV-a-2-points}
(a) By Definition \ref{def:k_x}\textit{(ii)}, ${\mathcal V}(x)\not=\emptyset$. \par \noindent (b) By Definition \ref{def:k_x}\textit{(iii)}, for all $y,z$ in ${\mathcal V}(x)$, there exists at least one open path from $y$ to $z$, denoted by $\Gamma^*_{y,z}$, contained in
$B_x({\rm C}'\kappa(x))$. If there are several such paths we
choose the first one according to some deterministic order. \end{remark}
We finally define an ``edge'' neighborhood $\overline\Gamma(x)$ of $x$:
\begin{eqnarray}\label{def:barGamma-x} \overline\Gamma(x)&=&\{(y',z')\subset B_x(\kappa(x)), (y',z')\hbox{ open}\}\cup\cr &&\qquad\{(y',z')\in\Gamma^*_{y,z},y,z\in {\mathcal V}(x)\}. \end{eqnarray}
Those neighborhoods satisfy
\begin{equation}\label{eq:borner_calV-x_et_Gamma-x} {\mathcal V}(x)\subset B_x(\kappa(x));\qquad\overline\Gamma(x)\subset B_x({\rm C}'\kappa(x)). \end{equation}
\subsection{ Travel times and radial limits}\label{subsec:Radial-limits}
We now come back to the spatial epidemic model.
In this subsection, we estimate the time needed by the epidemic
to cover $C^\infty$, taking advantage of the analysis of paths
in the percolation model done in Section \ref{sec:appliquer_GM}. We first define an approximation for the passage time of the epidemic, then we prove the existence of radial limits for this approximation and for the epidemic.
We will follow for this the spirit of the construction in \citet{MR0978353}. \\ \\
By analogy with \citet{MR0624685,MR0978353} (although neighborhoods in our context are defined differently), we define, for $x,y\in{\mathbb Z}^d$, the travel time from ${\mathcal V}(x)$ to ${\mathcal V}(y)$ and the time spent around $x$ to be (remember \eqref{eq:passage-time-from-x-to-y})
\begin{eqnarray}\label{def:approx-passage-time}
\widehat{\tau}(x,y)&=&\displaystyle{\inf_{x'\in{\mathcal V}(x), y'\in{\mathcal V}(y)} \tau(x',y')}
;\\\label{def:2-approx-passage-time}
u(x)&=&
\begin{cases} \displaystyle{\sum_{(y',z')\in\overline\Gamma(x)} \tau(y',z')}
&\text{if $\overline\Gamma(x)\not=\emptyset$,}\\ 0 &\text{otherwise.} \end{cases}
\end{eqnarray}
By Remarks \ref{rk:csq-ap2}, \ref{rk:calV-a-2-points}\textit{(a)}, $\widehat{\tau}(x,y)$ is finite.
If ${\mathcal V}(x)\cap{\mathcal V}(y)\not=\emptyset$, then $\widehat{\tau}(x,y)=0$.\\ \\
We now show that if $y\in C_x^{\rm out}\setminus R_x^{\rm out}$, $\widehat{\tau}(x,y)$ approximates $\tau(x,y)$.
\begin{lemma}\label{lem:comparaison-approximation} For $x\in{\mathbb Z}^d$, if $y\in C_x^{\rm out}\setminus R_x^{\rm out}$, we have
\begin{equation}\label{eq:comparaison-approximation} \widehat{\tau}(x,y) \le \tau(x,y)\le u(x)+\widehat{\tau}(x,y)+u(y). \end{equation}
\end{lemma}
\begin{proof}{Lemma}{lem:comparaison-approximation}
Let $\Gamma_{x,y}$ be an open path
from $x$ to $y$ such that $\tau(x,y)=\overline{\tau}(\Gamma_{x,y})$.
Since $y\notin R_x^{\rm out}$ this path must intersect $C^\infty$.
Let $c_1$ and $c_2$ be the first and last points we
encounter in $C^\infty$ when moving from $x$ to $y$ along $\Gamma_{x,y}$. By Definition \ref{def:k_x}\textit{(i)}, $c_1 \in{\mathcal V}(x)$ and $c_2 \in{\mathcal V}(y)$: indeed (for instance for $c_1$), either $x\in{C^\infty}$ and $c_1=x$, or the point $a\in\partial B_x(\kappa(x))\cap\Gamma_{x,y}$ does not belong to $R_x^{\rm out}$ and $c_1$ is the first point on $\Gamma_{x,y}$ between $x$ and $a$; we might have $c_1=c_2$, if ${\mathcal V}(x)\cap{\mathcal V}(y)\not=\emptyset$. We have, denoting by $\vee$ the concatenation of paths,
\[ \Gamma_{x,y}=\Gamma_{x,c_1}\vee\Gamma_{c_1,c_2}\vee\Gamma_{c_2,y} \]
where
$\Gamma_{x,c_1}$ (resp. $\Gamma_{c_2,y}$) is an open path from $x$ to $c_1$ contained in $B_x(\kappa(x))$ (resp. from $c_2$ to $y$ contained in $B_y(\kappa(y))$) and $\Gamma_{c_1,c_2}$ is an open path from $c_1$ to $c_2$. We then obtain the first inequality of \eqref{eq:comparaison-approximation} since:
$$\widehat{\tau} (x,y)\le \overline{\tau}(\Gamma_{c_1,c_2})\le \overline{\tau}(\Gamma_{x,y})=\tau(x,y).$$
To prove the second inequality of \eqref{eq:comparaison-approximation}, let $\Gamma_{d_1,d_2}$ be an open path from $d_1 \in{\mathcal V}(x)$ to $d_2\in{\mathcal V}(y)$
such that $\overline{\tau}(\Gamma_{d_1,d_2})=\widehat{\tau}(x,y)$. Since the open paths $\Gamma_{x,c_1}$ from $x$ to $c_1$ and $\Gamma^*_{c_1,d_1}$ (which exists by Remark \ref{rk:calV-a-2-points}\textit{(b)}) from $c_1$ to $d_1$ have edges in $\overline{\Gamma}(x)$ (see \eqref{def:barGamma-x}), the open path $\Gamma_{x,d_1}=\Gamma_{x,c_1}\vee\Gamma^*_{c_1,d_1}$ from $x$ to $d_1$ satisfies $\overline{\tau}(\Gamma_{x,d_1})\le u(x)$. Similarly, there is an open path $\Gamma_{d_2,y}$ from $d_2$ to $y$ such that $\overline{\tau}(\Gamma_{d_2,y})\le u(y)$. We conclude with $$\tau(x,y)\le \overline{\tau}(\Gamma_{x,d_1})+\overline{\tau}(\Gamma_{d_1,d_2}) +\overline{\tau}(\Gamma_{d_2,y})\le u(x)+\widehat{\tau}(x,y)+u(y).$$
\end{proof}
\mbox{}\\ \\
We now prove that $\widehat{\tau}(.,.)$ is almost subadditive, which will enable us later on in Theorem \ref{th:radial-limits} to appeal to Kingman's Theorem.
\begin{lemma}\label{lem:sous-additif}
For all $x,y,z\in{\mathbb Z}^d$, we have the \rm{subadditivity property}
\begin{equation}\label{eq:sous-additif} \widehat{\tau}(x,z)\le\widehat{\tau}(x,y)+u(y)+\widehat{\tau}(y,z). \end{equation}
\end{lemma}
\begin{proof}{Lemma}{lem:sous-additif}
Let $\Gamma_{a,b}$ be an open path from $a\in{\mathcal V}(x)$ to $b\in{\mathcal V}(y)$ such that
$\widehat{\tau}(x,y)=\overline{\tau}(\Gamma_{a,b})$.
Similarly, let $\Gamma_{c,d}$ be an open path from $c\in {\mathcal V}(y)$ to $d\in{\mathcal V}(z)$ such that
$\widehat{\tau}(y,z)=\overline{\tau}(\Gamma_{c,d})$ (we
might have $a=b$, $c=d$ or $b=c$). Since both $b$ and $c$
are in ${\mathcal V}(y)$ there exists an open path
$\Gamma^*_{b,c}$ from $b$ to $c$ such that
$\overline{\tau}(\Gamma^*_{b,c})\le u(y)$
(see Remark \ref{rk:calV-a-2-points}\textit{(b)} and \eqref{def:barGamma-x}).
The lemma then follows since the concatenation of these three paths is
an open path from a point of ${\mathcal V}(x)$ to a point of ${\mathcal V}(z)$ and
$$\widehat{\tau}(x,z)\le \overline{\tau}(\Gamma_{a,b}) + \overline{\tau}(\Gamma^*_{b,c})+\overline{\tau}(\Gamma_{c,d})\le \widehat{\tau}(x,y) +u(y)+\widehat{\tau}(y,z).$$
\end{proof}
\mbox{}\\ \\
We introduce a new notation, for the length of the shortest path between two neighborhoods. For $x,y\in {\mathbb Z}^d$, let
\begin{equation}\label{eq:barD} \overline{D}(x,y)= \inf_{x'\in {\mathcal V}(x),y'\in {\mathcal V}(y)} D(x',y'). \end{equation}
Note that unlike $D(x,y)$, $\overline{D}(x,y)$
is always finite. Next proposition corresponds
to Proposition \ref{lem:ap2}\textit{(i)}
for $\overline{D}(x,y)$ instead of $D(x,y)$.
It will be used in
Lemma \ref{lem:regularite-approx-tau} which follows.
\begin{proposition}\label{lem:bar-D} There exist constants ${\rm C}_4$ and $\alpha_4>0$ such that
$$P(\overline{D}(x,y)\ge {\rm C_4}\|x-y\|_1 +n)\le \exp(-\alpha_4 n^{1/d}), \qquad \forall\, x,y \in {\mathbb Z}^d,n\in {\mathbb N}.$$
\end{proposition}
\begin{proof}{Proposition}{lem:bar-D}
Let ${\rm C_2}$ be as in Lemma \ref{lem:APp} and Proposition \ref{lem:ap2}. Then
\begin{eqnarray*}
&&P(\overline{D}(x,y)\ge {\rm C_2}\|x-y\|_1 +(2d+1){\rm C_2}n)\cr\le&& P(\kappa(x)>n)+P(\kappa(y)>n)\cr
&+&P(\overline{D}(x,y)\ge {\rm C_2}\|x-y\|_1 +(2d+1){\rm C_2}n,\kappa(x)\le n, \kappa(y)\le n)\cr \le&& P(\kappa(x)>n)+P(\kappa(y)>n)\cr
&+&\sum_{x'\in B_x(n),y'\in B_y(n)}P(D(x',y')\ge {\rm C_2}\|x-y\|_1 +(2d+1){\rm C_2}n, x'\rightarrow y' )\cr
\le&& P(\kappa(x)>n)+P(\kappa(y)>n)\cr
&+&\sum_{x'\in B_x(n),y'\in B_y(n)}P(D(x',y')\ge {\rm C_2}\|x'-y'\|_1 +{\rm C_2}n, x'\rightarrow y'). \end{eqnarray*}
The result follows from Proposition \ref{lem:ap2} and Lemma \ref{lem:k_x-exp_tail}. \end{proof}
\mbox{}\\ \\
Of course, the random variables $u(x)$ and $\widehat{\tau}(x,y)$ are almost surely finite. But we will need later on repeatedly a better control of their size, provided by our next lemma.
\begin{lemma}\label{lem:regularite-approx-tau} For all $x,y\in{\mathbb Z}^d$, $r\in{\mathbb N}\setminus\{0\}$, $u(x)$ and $\widehat{\tau}(x,y)$ have a finite $r$-th moment. \end{lemma}
\begin{proof}{Lemma}{lem:regularite-approx-tau}
By Lemma \ref{lem:k_x-exp_tail}, $u(x)$ is bounded above by a sum of passage times $e(y,z)$ with $y$ and $z$ in the box $B_x(Y)$, where $Y$ is a random variable whose moments are all finite.
By Lemmas \ref{lem:k_x-exp_tail} and \ref{lem:bar-D}
the same happens to $\widehat{\tau}(x,y)$
(if $x'\in {\mathcal V}(x),y'\in {\mathcal V}(y)$ are the sites that achieve
$\overline{D}(x,y)$, then $\widehat{\tau}(x,y)\le {\tau}(x',y')$).
Therefore it suffices to show that if
$(X_i,i\in {\mathbb N})$ is a sequence of i.i.d. random variables and
$N$ is a random variable taking values in ${\mathbb N}$, then the
moments of $\sum_{i=1}^N X_i$
are all finite if it is the case for both the $X_i$'s and $N$.
To prove this write:
\begin{eqnarray*} E(\vert \sum_{i=1}^N X_i\vert^r)&= &\sum_{n=1}^{\infty}E(\vert X_1+\dots+X_n\vert^r {\bf 1}_{\{N=n\}})\cr &\leq&\sum_{n=1}^{\infty}[E(\vert X_1+\dots+X_n\vert^{2r})P(N=n)]^{1/2} \cr &\leq&\sum_{n=1}^{\infty}[E(\vert X_1\vert +\dots+\vert X_n\vert)^{2r}P(N=n)]^{1/2}\cr &\leq&\sum_{n=1}^{\infty} [n^{2r}{\rm C}_{2r}P(N=n)]^ {1/2}
\end{eqnarray*}
where the second line comes from Cauchy-Schwartz' inequality, the factor $n^{2r}$ counts the number of terms in the development of $(\vert X_1\vert +\dots+\vert X_n\vert)^{2r}$ and the constant ${\rm C}_{2r}$ depends on the distribution of the $X_i$'s. As $N$ has all its moments finite $P(N=n)$ decreases faster than $n^{-2r-4}$ and the sum is finite.
\end{proof}
\mbox{}\\ \\
We now construct a process $(\vartheta_\cdot)$ which will be subadditive in every direction, and will have a.s., by Kingman's Theorem, a radial limit denoted by $\mu$. We will then check that $\widehat\tau(o,\cdot)$ also has, in every direction, the same radial limit, and we will extend this conclusion to $\tau(o,\cdot)$ on the set $C_o^{\rm out}$ of sites that have ever been infected. Hence we first prove
\begin{theorem}\label{th:radial-limits} For all $z\in{\mathbb Z}^d$, there exists $\mu(z)\in{\mathbb R}^+$ such that almost surely
\begin{eqnarray}\label{eq:lim-widehat_tau} &&\lim_{n\to +\infty}\frac {\widehat{\tau}(o,nz)}n =\mu(z) \qquad\hbox{ and }\\ \label{eq:radial-limit_2} &&\lim_{n\to +\infty}\left[\frac{\tau (o,nz)}n - \mu(z)\right]{\bf 1}_{\{nz\in C_o^{\rm out}\}}=0. \end{eqnarray}
\end{theorem}
\begin{proof}{Theorem}{th:radial-limits}
\textit{(i)} For all $z\in{\mathbb Z}^d$, $(m,n)\in{\mathbb N}^2$, let
\begin{equation}\label{eq:vartheta_theta} \vartheta_z(m,n)=\widehat{\tau}(mz,nz)+u(nz). \end{equation}
The process $(\vartheta_z(m,n))_{(m,n)\in{\mathbb N}^2}$ satisfies the hypotheses of Kingman' subadditive ergodic theorem (see \citealp[Theorem VI.2.6]{MR2108619}) by \eqref{eq:sous-additif}. Hence (noticing also that $\vartheta_z(0,n)=\vartheta_{nz}(0,1)$) there exists $\mu(z)\in{\mathbb R}^+$ such that
\begin{eqnarray}\label{eq:lim-vartheta_theta} \lim_{n\to +\infty}\frac 1n \vartheta_z(0,n) &=&\lim_{n\to +\infty} E\left(\frac{\vartheta_z(0,n)}{n}\right) =\lim_{n\to +\infty} E\left(\frac{\vartheta_{nz}(0,1)}{n}\right)\cr &=&\inf_{n\in{\mathbb N}} E\left(\frac{\vartheta_z(0,n)}{n}\right) =\inf_{n\in{\mathbb N}} E\left(\frac{\vartheta_{nz}(0,1)}{n}\right) =\mu(z)\, \end{eqnarray}
a.s. and in $L^1$. Since the random variables $(u(z):z\in {\mathbb Z}^d)$ are identically distributed, it follows from Lemma \ref{lem:regularite-approx-tau} and Chebychev's inequality that \newline $\sum_{n=0}^\infty P(u(nz)>n\varepsilon)<+\infty$
for all $\varepsilon>0$, so that by Borel-Cantelli's Lemma
\begin{equation}\label{eq:lim-u} \lim_{n\to +\infty}\frac {u(nz)}n =0,\,\mbox{ a.s.} \end{equation}
Thus by \eqref{eq:vartheta_theta}, \eqref{eq:lim-vartheta_theta}, \eqref{eq:lim-u}
we have \eqref{eq:lim-widehat_tau} for all $z\in{\mathbb Z}^d$.\\ \\
\textit{(ii)} Since $ R_o^{\rm out}$ is a.s. finite, if $nz\in C_o^{\rm out}$, then $nz\in C_o^{\rm out}\setminus R_o^{\rm out}$ for $n$ large enough. Hence, from
Lemma \ref{lem:comparaison-approximation}, for $n$ large enough we have
\[
\left|\displaystyle{\frac{\tau(o,nz)}n}-\mu(z)\right|{\bf 1}_{\{nz\in C_o^{\rm out}\setminus R_o^{\rm out}\}}
\le \displaystyle{\frac{u(o) + u(nz)}n}+\left|\displaystyle{\frac{\widehat{\tau}(o,nz)}n}-\mu(z)\right| \]
and we conclude that \eqref{eq:radial-limit_2} is satisfied by \eqref{eq:lim-u} and \eqref{eq:lim-widehat_tau}. \end{proof}
\subsection{Extending $\mu$}\label{subsec:extending_mu}
We have proved the existence of a linear propagation speed in every direction of ${\mathbb Z}^d$. Now, to derive an asymptotic shape result, in particular for the approximating travel times $(\widehat\tau(x,y),x,y\in{\mathbb Z}^d)$, we need to
extend $\mu$ from ${\mathbb Z}^d$ to
a Lipschitz, convex and homogeneous function on ${\mathbb R}^d$. The asymptotic shape of the epidemic will be given by the convex set $D$ defined in \eqref{def:D} below. As a first step, we prove properties of $\mu$ on ${\mathbb Z}^d$.
\begin{lemma}\label{ajoute1}
The function $\mu$ satisfies the following properties for all $x,y\in {\mathbb Z}^d$, $k \in {\mathbb N}$:
\newline (i) $\displaystyle{\mu(x)=\lim_{n\to+\infty} E\left(\frac{\widehat \tau (o,nx)}{n}\right)}$,\newline (ii) $\mu(x+y)\leq \mu(x)+\mu(y)$,\newline (iii) $\mu(x)=\mu(-x)$,\newline
(iv) $\mu({\rm e}_i)=\mu({\rm e}_\ell),\,\forall i,\ell\in \{1,\dots,d\}$,\newline
(v) $\mu(kx)=k\mu(x)$,\newline
(vi) $\mu(x)\leq \mu({\rm e}_1)\|x\|_1$. \end{lemma}
\begin{proof}{Lemma}{ajoute1} Since $\vartheta_x(0,n)=\widehat{\tau}(o,nx)+u(nx)$, part \textit{(i)} follows from \eqref{eq:lim-vartheta_theta} and \eqref{eq:lim-u}.
To prove part \textit{(ii)} write:
\begin{eqnarray*} &&\mu(x+y)=\lim_{n\to+\infty} E\left(\frac{\widehat \tau (o,n(x+y))}{n}\right)\\ &\leq& \lim_{n\to+\infty} E\left(\frac{\widehat \tau (o,nx)}{n}\right) + \lim_{n\to+\infty} E\left(\frac{\widehat \tau (nx,n(x+y))}{n}\right) +\lim_{n\to+\infty} E\left(\frac{u(nx)}{n}\right)\\ &=& \lim_{n\to+\infty} E\left(\frac{\widehat \tau (o,nx)}{n}\right) + \lim_{n\to+\infty} E\left(\frac{\widehat \tau (nx,n(x+y))}{n}\right)\\ &=&\mu(x)+\mu(y), \end{eqnarray*}
where the first equality follows from part \textit{(i)}, the inequality from \eqref{eq:sous-additif}, the second equality from \eqref{eq:lim-u}
and the third one from part \textit{(i)} and translation invariance of
$\widehat \tau$. Parts \textit{(iii)}--\textit{(iv)} follow immediately from
part \textit{(i)} and the corresponding properties of $\widehat \tau(o,x)$. To prove part \textit{(v)} write:
$$\mu(kx) =\lim_{n\to +\infty} E\left(\frac{\vartheta_{nkx}(0,1)}{n}\right) =k\lim_{n\to +\infty} E\left(\frac{\vartheta_{nkx}(0,1)}{nk}\right) =k\mu(x),$$
where the first and third equalities follow from \eqref{eq:lim-vartheta_theta}. Finally, part \textit{(vi)} follows from parts \textit{(ii)--(iv)}. \end{proof}
\mbox{}\\ \\
Next corollary extends Lemma \ref{ajoute1}\textit{(iii)}--\textit{(iv)}.
\begin{corollary}\label{cor:per}
For any permutation $\sigma$ of $\{1,\cdots,d\}$, any $y=(y_1, y_2,\cdots,y_d)\in{\mathbb Z}^d$ and any choice of the signs $\pm$,
\[ \mu(\pm y_{\sigma(1)}, \pm y_{\sigma(2)},\cdots,\pm y_{\sigma(d)})=\mu(y_1, y_2,\cdots,y_d). \]
\end{corollary}
\begin{proof}{Corollary}{cor:per} Clearly $\widehat \tau (o,(y_1,y_2,\dots,y_d))$ has the same distribution as $\widehat \tau(o,(\pm y_{\sigma(1)}, \pm y_{\sigma(2)},\cdots,\pm y_{\sigma(d)}))$ for any choice of the signs and any permutation $\sigma$, hence the corollary follows
from Lemma \ref{ajoute1}\textit{(i)}. \end{proof}
\begin{lemma}\label{ajoute2} Let $\gamma^*=\mu({\rm e}_1)$. Then $\gamma^*$ is a Lipschitz constant for $\mu$. For all $u,v\in {\mathbb Z}^d$ we have
$$\vert \mu(u)-\mu(v)\vert \leq \gamma^* \|u-v\|_1. $$ \end{lemma}
\begin{proof}{Lemma}{ajoute2}
Let $y=u-v,\,x=v$. We have
$$\mu(u)-\mu(v)=\mu(x+y)-\mu(x)\leq \mu(y)=\mu(u-v)\leq \mu({\rm e}_1)\|u-v\|_1,$$
where the inequalities follow from Lemma \ref{ajoute1}\textit{(ii)} and \textit{(vi)}. Similarly, taking $x=u,\,y=v-x$ gives
$$\mu(v)-\mu(u)\leq \mu({\rm e}_1)\|v-u\|_1,$$
and the lemma follows. \end{proof}
\mbox{}\\ \\
\noindent
In a second step, we extend $\mu$ to ${\mathbb R}^d$ and we introduce the set $D$.
\begin{proposition}\label{lem:definir_phi} There exists an extension of $\mu$ to ${\mathbb R}^d$, which is Lipschitz with Lipschitz constant $\gamma^*$ given by Lemma \ref{ajoute2}, convex and homogeneous on ${\mathbb R}^d$. Moreover, $\mu(x) = 0$ if and only if $x=o$ and the set
\begin{equation}\label{def:D} D=\{x\in{\mathbb R}^d:\mu(x)\le 1\} \end{equation}
is convex, bounded and contains an open ball centered at $o$.
\end{proposition}
\begin{proof}{Proposition}{lem:definir_phi} We start by extending $\mu$ to ${\mathbb Q}^d$. For $x\in{\mathbb Q}^d\setminus\{o\}$ let
\begin{eqnarray}\label{eq:Nx} N_x&=&\min\{k\ge 1,k\in {\mathbb N} :kx\in{\mathbb Z}^d\}\quad \hbox{ and}\\ \label{def:bis-mu-x}\mu(x)&=&\frac{\mu(N_x x)}{N_x}.\end{eqnarray}
We now prove that this extension is homogeneous: let $\alpha\in {\mathbb Q}$ be positive and let $x\in {\mathbb Q}^d,\,x\neq o$. Then, there exist $k_1,k_2\in {\mathbb N}$ multiples of $N_x$ and $N_{\alpha x}$ respectively, such that $k_1x,k_2\alpha x\in {\mathbb Z}^d$ and $k_1x=k_2\alpha x$. Write
\begin{eqnarray*} \mu(\alpha x) =\frac{\mu(N_{\alpha x}\alpha x )}{N_{\alpha x}}=\frac{\mu(k_2 \alpha x)}{k_2} =\frac{\mu(k_1 x)}{k_2}=\frac{k_1 }{k_2}\frac{\mu(k_1 x)}{k_1}=\alpha \frac{\mu(N_x x)}{N_x}=\alpha \mu(x), \end{eqnarray*}
using \eqref{def:bis-mu-x} for the first equality, Lemma \ref{ajoute1}\textit{(v)} for the second and fifth ones.\par \noindent
To prove that $\mu$ is Lipschitz on ${\mathbb Q}^d$, let $x,y\in {\mathbb Q}^d\setminus\{o\}$. Then,
\begin{eqnarray*}
\left\vert \mu(x)-\mu(y)\right\vert&=&\left\vert \frac{\mu(N_x x)}{N_x}-\frac{\mu(N_y y)}{N_y}\right\vert =\left\vert \frac{\mu(N_yN_x x)}{N_yN_x}-\frac{\mu(N_xN_y y)}{N_xN_y}\right\vert\\ &=& \frac{\vert \mu(N_xN_y x)-\mu(N_xN_y y)\vert}{N_xN_y}\\
&\leq & \frac{\gamma^* \|N_xN_y x-N_xN_y y\|_1}{N_xN_y}=\gamma^* \| x- y\|_1, \end{eqnarray*}
using Lemma \ref{ajoute1}\textit{(v)} for the second equality and Lemma \ref{ajoute2} for the inequality.\\ \\
To prove that $\mu$ is convex on ${\mathbb Q}^d$, take $x,y \in {\mathbb Q}^d$ and $\alpha \in {\mathbb Q} \cap (0,1)$. Then let $k_1,k_2$ be elements in ${\mathbb N}$ such that $k_1\alpha \in {\mathbb N}$, $k_2 x \in {\mathbb Z}^d$, $k_2 y \in {\mathbb Z}^d$ and write:
\begin{eqnarray*} &&\mu (\alpha x+(1-\alpha)y)=\lim_{n \to+\infty}E\Big( \frac{\widehat \tau(o,n\alpha x+n(1-\alpha)y)}{n}\Big)\cr &=&\lim_{n \to+\infty} E\Big(\frac {\widehat \tau(o,nk_1\alpha k_2x+nk_1(1-\alpha)k_2y)}{nk_1k_2}\Big)\cr &\leq&\lim_{n \to+\infty} E\Big(\frac {\widehat \tau(o,nk_1\alpha k_2x)+\widehat \tau (o,nk_1(1-\alpha)k_2y)+ u(nk_1\alpha k_2x)}{nk_1k_2}\Big)\cr &=&\lim_{n \to+\infty} E\Big(\frac {\widehat \tau(o,nk_1\alpha k_2x)+\widehat \tau (o,nk_1(1-\alpha)k_2y)}{nk_1k_2}\Big)\cr &=&\frac{\mu(k_1k_2\alpha x)+\mu(k_1k_2(1-\alpha)y)}{k_1k_2}\cr &=&\alpha \mu(x)+(1-\alpha)\mu(y), \end{eqnarray*}
where the first equality follows from Lemma \ref{ajoute1}\textit{(i)}, the inequality from Lemma \ref{lem:sous-additif}, the third equality
from \eqref{eq:lim-u},
the fourth from Lemma \ref{ajoute1}\textit{(i)} and the last one from the homogeneity of $\mu$ on ${\mathbb Q}^d$.\\ \\
Because $\mu$ is homogeneous, Lipschitz and convex on ${\mathbb Q}^d$, we can extend $\mu$ by continuity to ${\mathbb R}^d$.\\ \\
To prove that $\mu(x)>0$ if $x\neq o$ we argue by contradiction: assume $\mu(x)=0$ and without loss of generality that $x=(x_1,\dots,x_d)$ with $x_1\neq 0$. First note that since $\mu$ is Lipschitz and homogeneous, the conclusion of Corollary \ref{cor:per} also holds for any $(x_1,\dots,x_d)\in {\mathbb R}^d$, then write
\begin{eqnarray*} \mu(2x_1,0,\cdots,0)&=&\mu(2x_1,0,\cdots,0)- \mu(x_1,x_2,\cdots,x_d)\\ &\le&\mu(x_1,-x_2,\cdots,-x_d)=0, \end{eqnarray*}
using Lemma \ref{ajoute1}\textit{(ii)} for the inequality, and Corollary \ref{cor:per} with the assumption $\mu(x) = 0$ for the last equality. Then since $\mu$ is homogeneous we get $\mu({\rm e}_1)=0$. However, considering a standard first passage percolation model with passage times $e(z,y)$ and adding a `tilde' to quantities associated to this model, we have $\widetilde\tau(o,z)\leq \tau(o,z)$ a.s. for all $z\in {\mathbb Z}^d$. Since by \citet[Theorem (2.18)]{MR0876084},
$$\lim_{n\to+\infty} \widetilde \tau (o,n{\rm e}_1)=\widetilde \mu ({\rm e}_1),$$
it follows from \eqref{eq:radial-limit_2} that $\widetilde\mu({\rm e}_1)\leq \mu({\rm e}_1)=0$.
But from \citet[Theorems (1.7) and (1.15)]{MR0876084} we get $\widetilde\mu({\rm e}_1)>0$, thus reaching a contradiction.\\ \\
The convexity of $\mu$ implies that $D$ is convex. We prove by contradiction that $D$ contains an open ball centered at $o$: otherwise, there exists a sequence $(x_n)_{n\in{\mathbb N}}$ such that $x_n\notin D,\,\lim_{n\to+\infty}x_n=0$; therefore on the one hand $\mu(x_n)>1$, and on the other hand $\lim_{n\to+\infty}\mu(x_n)=0$ because $\mu(o)=0$ and $\mu$ is continuous, hence a contradiction.\\ \\
Finally we argue again by contradiction to prove that the set $D$ is bounded:
otherwise there would exist a sequence $(y_n)_{n\in{\mathbb N}}$ with $y_n\in D$ and $\|y_n\|_1>n$. Then $x_n=y_n/\|y_n\|_1$ satisfies $\|x_n\|_1=1$, and, since $\mu$ is homogeneous,
$\lim_{n\to+\infty}\mu(x_n)=0$. By compactness $(x_n)_{n\in{\mathbb N}}$
has a converging subsequence to some $x$ such that $\mu(x)=0$ with $\|x\|_1=1$;
since we have already proved there is no such $x$ we get a contradiction.
\end{proof}
\subsection{Behavior of $\widehat \tau$}\label{subsec:behavior-widehat_tau}
Our next result establishes how $\widehat \tau(o,z)$ grows for $z\in{\mathbb Z}^d$.
\begin{theorem}\label{th:widehat_t-serie} There exist $K=K(\lambda,d)>0$ and $\alpha>0$ such that
\begin{eqnarray*}
P(\widehat\tau(o,z)>K\|z\|_\infty)&\leq& \exp(-\alpha(\|z\|_\infty^{1/d}),\ \ \forall \ z\in {\mathbb Z}^d,\cr P(\widehat\tau(o,z)>K(\|z\|_\infty+n))&\leq& \exp(-\alpha n^{1/d}),\ \ \forall \ z\in {\mathbb Z}^d,n\in {\mathbb N},\cr
\sum_{z\in{\mathbb Z}^d} P({\widehat\tau}(o,z)>K\|z\|_\infty)&<&+\infty. \end{eqnarray*}
\end{theorem}
\begin{proof}{Theorem}{th:widehat_t-serie}
Let $K\ge 0, z\in{\mathbb Z}^d$ and let
$\mathcal B=B(o,(\|z\|_\infty+n)/4)\times B(z,(\|z\|_\infty+n)/4)$. Then write:
\begin{eqnarray}\label{eq:calcul1-widehat_t-serie}
&&P({\widehat\tau}(o,z)>K(\|z\|_\infty+n))\cr
&\le & P(4\kappa(z)>\|z\|_\infty+n)+ P(4\kappa(o)>\|z\|_\infty+n)
+ P(A) \end{eqnarray}
where
\begin{eqnarray}\nonumber A&=&\{{\widehat\tau}(o,z)>K(\|z\|_\infty +n), 4\kappa(z)
\le \|z\|_\infty +n, 4\kappa(o) \le \|z\|_\infty +n\}\\\label{eq:calcul2-widehat_t-serie}
&\subset& \displaystyle{\cup_{(x,y)\in \mathcal B}\{ x\to y, \tau(x,y)>K(\|z\|_\infty+n)\}}.
\end{eqnarray}
Note that if $(x,y)\in \mathcal B$ we have
\begin{eqnarray}\label{eq:x-y and z}
\|z\|_\infty-n&\le& 2\|x-y\|_\infty\le 3\|z\|_\infty+n\quad\mbox{ and}\cr 3(\|z\|_\infty+n)&=& 3\|z\|_\infty+n+2n\ge 2(\|x-y\|_\infty+n). \end{eqnarray}
{}From \eqref{eq:calcul2-widehat_t-serie}, \eqref{eq:x-y and z}, for ${\rm C}_2$ given in Proposition \ref{lem:ap2}, we get:
\begin{eqnarray}\label{eq:calcul5-widehat_t-serie} && P(A) \le
\sum_{x\in B(o,(\|z\|_\infty+n)/4)}\ \ \sum_{y\in B(z,(\|z\|_\infty+n)/4)} \cr
&&\Big(P( 3\tau(x,y)> 2K(\|x-y\|_\infty +n), D(x,y)<({\rm C}_2+1)(\|x-y\|_1+n))\cr
&&\quad+P(x\to y, D(x,y)\geq ({\rm C}_2+1)(\|x-y\|_1+n ))\Big). \end{eqnarray}
It now follows from Proposition \ref{lem:ap2}\textit{(i)} that we have, for some $\alpha_5>0$,
\begin{eqnarray}\label{eq:calcul6-widehat_t-serie}
&&P(x\to y, D(x,y)\geq ({\rm C}_2+1)(\|x-y\|_1+n ))\cr
&\le& \exp(-\alpha_5(\|x-y\|_1+n)^{1/d})\cr
&\le& \exp(-\alpha_5(\|x-y\|_\infty+n)^{1/d}). \end{eqnarray}
Then, taking $K$ large enough, by large deviation results for exponential variables, we also have, for some $\alpha_6>0$,
\begin{eqnarray}\label{eq:calcul7-widehat_t-serie}
&& P( 3\tau(x,y)>2K(\|x-y\|_\infty +n), D(x,y)<({\rm C}_2+1)(\|x-y\|_1+n)) \cr
&\le& P( 3\tau(x,y)>2K(\|x-y\|_\infty +n), D(x,y)<({\rm C}_2+1)d(\|x-y\|_\infty+n))\cr
&\le& \exp(-\alpha_6(\|x-y\|_\infty+n)). \end{eqnarray}
Hence, from \eqref{eq:x-y and z}--\eqref{eq:calcul7-widehat_t-serie}, for some constants $R$ and $\alpha_7>0$ we have:
\begin{eqnarray*}\label{eq:calcul9-widehat_t-serie}
&&P(A) \le R (\|z\|_\infty+n)^{2d}\exp(-\alpha_7 (\|z\|_\infty+n)^{1/d}), \end{eqnarray*}
which gives, by modifying the constants,
\begin{eqnarray}\label{eq:calcul10-widehat_t-serie}
&&P(A) \le R' \exp(-\alpha_8 (\|z\|_\infty+n)^{1/d}). \end{eqnarray}
The theorem's statements now follow from \eqref{eq:calcul9-widehat_t-serie}, \eqref{eq:calcul1-widehat_t-serie} and Lemma \ref{lem:k_x-exp_tail}.
\end{proof}
\subsection{Asymptotic shape for $\widehat \tau$}\label{subsec:shape-widehat_t}
Next theorem is the last necessary step to prove the shape theorem.
\begin{theorem}\label{th:At-encadre} Let $\varepsilon>0$, and
${\widehat A}_t=\{z\in{\mathbb Z}^d:{\widehat\tau}(o,z)\le t\}.$
Then, a.s. for $t$ large enough, for $D$ defined in \eqref{def:D},
\begin{equation}\label{eq:At-encadre} (1-\varepsilon)tD\cap{\mathbb Z}^d\subset{\widehat A}_t\subset(1+\varepsilon)tD\cap{\mathbb Z}^d. \end{equation}
\end{theorem}
In the sequel $K$ and $\alpha$ are fixed constants satisfying the conclusions of Theorem \ref{th:widehat_t-serie}, $\gamma^*$ is the Lipschitz constant of $\mu$ (see Lemma \ref{ajoute2}) and $N_x$ was defined in \eqref{eq:Nx} for any $x\in {\mathbb Q}^d\setminus \{o\}$.
To prove Theorem \ref{th:At-encadre} we need the two following lemmas.
\begin{lemma}\label{lem:covering B_0} Let $\rho>0$ and let $\delta \le \rho/(2K)$. Then, for all $x\in {\mathbb Q}^d\setminus \{o\}$,
\begin{eqnarray}\label{eq:f1} \sum_{k>0}P(\sup_{z\in B_{kN_xx}(\delta kN_x)\cap{\mathbb Z}^d}\widehat \tau(kN_xx,z)\ge kN_x\rho)&<&\infty,\\ \label{eq:f2} \sum_{k>0}P(\sup_{z\in B_{kN_xx}(\delta kN_x)\cap{\mathbb Z}^d}\widehat \tau(z,kN_xx)\ge kN_x\rho)&<&\infty. \end{eqnarray}
\end{lemma}
\begin{proof}{Lemma}{lem:covering B_0}
We derive only \eqref{eq:f1}, since the proof of \eqref{eq:f2} is analogous. By translation invariance
$$P(\sup_{z\in B_{kN_xx}(\delta kN_x)\cap {\mathbb Z}^d}\widehat \tau(kN_xx,z)\ge kN_x\rho) =P(\sup_{z\in B(\delta kN_x)\cap {\mathbb Z}^d}\widehat \tau(o,z)\ge kN_x\rho).$$
Hence it suffices to show that
$$\sum_{k>0} P(\sup_{z\in B(\delta kN_x)\cap {\mathbb Z}^d}\widehat \tau(o,z)\ge kN_x\rho)<\infty.$$
Let $k>0,z\in B(\delta kN_x)\cap{\mathbb Z}^d$. By Theorem \ref{th:widehat_t-serie}
we have:
\begin{eqnarray*}
P(\widehat \tau (o,z)\ge kN_x\rho)&\le& P(\widehat \tau (o,z)\ge K\|z\|_\infty +kN_x\rho/2)\cr &\le& \exp\Big(-\alpha \Big\lfloor \frac{kN_x\rho}{2K}\Big\rfloor^{1/d}\Big). \end{eqnarray*}
Therefore, for some constant ${\rm C}$,
\begin{eqnarray*} \sum_{k>0}P(\sup_{z\in B(\delta kN_x)\cap{\mathbb Z}^d}\widehat \tau(o,z)\ge kN_x\rho) &\le& \sum_{k>0} {\rm C} (\delta kN_x)^d \exp\Big(-\alpha \Big\lfloor \frac{kN_x\rho}{2K}\Big\rfloor^{1/d}\Big)\cr &<&\infty. \end{eqnarray*}
\end{proof}
\mbox{}\\ \\
For $x\in{\mathbb Q}^d\setminus\{o\}$, $\delta>0$, we define the cone associated to $x$ of amplitude $\delta$ as
\begin{equation}\label{eq:cone_de_x} C_x(\delta)={\mathbb Z}^d\cap \Big(\cup_{t\ge 0}B_{tx}(\delta t)\Big). \end{equation}
\begin{lemma}\label{lem:included-cone}
Let $x\in {\mathbb Q}^d\setminus \{o\}$. Then for any $0<\delta' <\delta$
the set $C_x(\delta')\setminus \cup_{k\ge 0} B_{kN_x x}(\delta k N_x)$ is finite. \end{lemma}
\begin{proof}{Lemma}{lem:included-cone} Let
$$t_0=\frac{N_x\|x\|_1}{\delta-\delta'}.$$
Since ${\mathbb Z}^d\cap \Big(\cup_{t\ge 0}^{t_0}B_{tx}(\delta' t)\Big)$ is finite, it suffices to show that
$$\cup_{t\ge t_0}^{\infty}B_{tx}(\delta' t) \subset \cup_{k\ge 0} B_{kN_x x}(\delta k N_x).$$
To prove this, pick $z\in B_{t_1x}(\delta' t_1)$ for some $t_1\ge t_0$. Let $k_0=\inf\{i\in {\mathbb N}: iN_x\ge t_1\}$. Hence $0\le k_0N_x-t_1<N_x$, and
\begin{eqnarray*}
\|z-k_0N_xx\|_1&\le&\|z-t_1x\|_1+|t_1-k_0N_x|\|x\|_1\\
&<& \|z-t_1x\|_1+N_x\|x\|_1
\le \delta't_1 +N_x\|x\|_1\\ &=&\delta't_1+(\delta-\delta')t_0\leq \delta t_1\leq \delta k_0 N_x. \end{eqnarray*}
Therefore $z\in B_{k_0 N_x x}(\delta k_0 N_x)$ and the lemma is proved. \end{proof}
\mbox{}\\ \\
In the next proof we use that since the
Lipschitz constant of $\mu$ is $\gamma^*$
for the norm $\|.\|_1$ (by Proposition \ref{lem:definir_phi}),
it is $\gamma=\gamma^*d$ for the norm $\|.\|_\infty$.\par
\begin{proof}{Theorem}{th:At-encadre}
Fix $\varepsilon\in (0,1)$ and let $\rho,\delta $ and $\iota$ be three small positive parameters such that $\delta \le \rho/(2K)$, whose values will be determined later. The set ${\mathcal Y}=\{x\in {\mathbb Q}^d:1-2\iota <\mu(x)<1-\iota\}$ is a ring between two balls with the same center but with a different radius, because by Proposition \ref{lem:definir_phi},
$\mu$ is homogeneous and positive except that $\mu(o)=0$. Hence the (compact) closure of ${\mathcal Y}$, which is recovered by balls of the same radius centered on the rational points of ${\mathcal Y}$, is in fact covered by a finite number of such balls. Thus there exists a finite subset $Y$ of ${\mathcal Y}$
such that ${\mathbb Z}^d \subset \cup_{x\in Y}C_x(\delta/2)$ (if the balls recover the ring, the cones associated to them recover the whole space). Hence, to prove the first inclusion of \eqref{eq:At-encadre} it suffices to show that for any $x\in Y$ and any sequences $(t_n)_{n>0}$ and $(z_n)_{n>0}$ such that $t_n \uparrow \infty$ in ${\mathbb R}^+$,
$z_n\in C_x(\delta/2)\cap{\mathbb Z}^d$ with $\|z_n\|_\infty\ge n$ and
$\mu(z_n)\le (1-\varepsilon) t_n$, we have $\widehat \tau (o,z_n)\le t_n$
a.s. for $n$ sufficiently large. So, let $(t_n)_{n>0}$ and $(z_n)_{n>0}$ be such sequences. Using Lemma \ref{lem:included-cone}
(taking a subsequence if necessary) let $k_n\in {\mathbb N} $ be such that $z_n\in B_{k_nN_xx}( \delta k_nN_x)$, hence $k_n\ge {\rm C}n$ for some constant ${\rm C}$. Since
$\mu$ is Lipschitz, write
$$k_n N_x (1-2\iota)\le \mu(k_nN_xx)\le \mu(z_n) +\gamma \delta k_n N_x\le (1-\varepsilon)t_n+\gamma \delta k_n N_x, $$
so that
\begin{equation}\label{eq:this inequality} k_nN_x\le \Big(\frac{1-\varepsilon}{1-2\iota -\gamma \delta}\Big)t_n. \end{equation}
It now follows from \eqref{eq:this inequality} and the subadditivity property \eqref{eq:sous-additif} of $\widehat \tau$ that:
$$\frac{\widehat \tau(o,z_n)}{t_n}\le \Big(\frac{1-\varepsilon}{1-2\iota -\gamma \delta}\Big)\Big(\frac{\widehat \tau(o,k_nN_xx)}{k_nN_x} +\frac{u(k_nN_xx)}{k_nN_x} +\frac{\widehat \tau(k_nN_xx,z_n)}{k_nN_x}\Big).$$
Therefore, by Theorem \ref{th:radial-limits}, Lemma \ref{lem:regularite-approx-tau} (the variables $u(.)$ are identically distributed, and $k_n\ge {\rm C}n$), Lemmas \ref{lem:definir_phi} and \ref{lem:covering B_0} we obtain:
$$\limsup_{n\to +\infty} \frac{\widehat \tau(o,z_n)}{t_n}\le \Big(\frac{1-\varepsilon}{1-2\iota -\gamma \delta}\Big)\Big(\mu(x) +\rho \Big)\qquad \mbox{a.s.}$$
Since $x\in Y$ this implies:
$$\limsup_{n\to +\infty} \frac{\widehat \tau(o,z_n)}{t_n} \le \Big(\frac{1-\varepsilon}{1-2\iota -\gamma \delta}\Big)\Big(1-\iota+\rho \Big)\qquad \mbox{a.s.}$$
Taking $\iota$, $\rho$ and $\delta$ small enough, the right hand side is strictly less than $1$ which proves that $\widehat \tau (o,z_n)\le t_n$ a.s. for $n$ sufficiently large.
Similarly, to prove the second inclusion of \eqref{eq:At-encadre} it suffices to show that for any $x\in Y$ and any sequences $t_n \uparrow \infty$ in ${\mathbb R}^+$ and
$z_n$ in $C_x(\delta/2)\cap{\mathbb Z}^d$ such that $\mu(z_n)\ge (1+\varepsilon) t_n$ we have
$\widehat \tau (o,z_n)> t_n$ a.s. for $n$ sufficiently large. As before, taking subsequences if necessary,
we let $(t_n)_{n>0}$ and $(z_n)_{n>0}$ be such sequences, and $k_n\in {\mathbb N} $ be such that $z_n\in B_{k_nN_xx}( \delta k_nN_x)$. Proceeding then as for the first inclusion, we get:
$$k_nN_x(1-\iota)\ge \mu(k_nN_xx)\ge \mu(z_n)-\gamma \delta k_nN_x \ge (1+\varepsilon)t_n -\gamma \delta k_nN_x,$$
$$k_nN_x\ge \Big( \frac{1+\varepsilon}{1-\iota +\gamma \delta}\Big)t_n,$$
$$\frac{\widehat \tau(o,z_n)}{t_n}\ge \Big(\frac{1+\varepsilon}{1-\iota +\gamma \delta}\Big) \Big( \frac{\widehat \tau (o,k_nN_xx)}{k_nN_x} -\frac{u(z_n)}{k_nN_x}-\frac{\widehat \tau (z_n,k_nN_xx)}{k_nN_x}\Big),$$
and
\begin{eqnarray*} \liminf_{n\to +\infty} \frac{\widehat \tau(o,z_n)}{t_n} &\ge& \Big(\frac{1+\varepsilon}{1-\iota +\gamma \delta}\Big) \Big(\mu(x) -\rho\Big)\qquad \mbox{a.s.}\cr &\ge&\Big(\frac{1+\varepsilon}{1-\iota +\gamma \delta}\Big) \Big(1-2\iota-\rho\Big)\qquad \mbox{a.s.} \end{eqnarray*}
Now, taking $\iota$, $\rho$ and $\delta$ small enough, the right hand side is strictly bigger than $1$ and the second inclusion of \eqref{eq:At-encadre} is proved.
\end{proof}
\subsection{ Asymptotic shape for the epidemic }\label{subsec:shape-thm}
We can now prove our main result, the shape theorem.
\begin{proof}{Theorem}{th:shape}
Let $\varepsilon>0$ be given. \par
\noindent \textit{(i)} We first show that the infection grows at least linearly as $t$ goes to infinity, that is,
\begin{equation}\label{eq:ne-stagne-pas1} P\Big(\big(\Upsilon_t\cup \Xi_t\big) \supset \big((1-\varepsilon)tD\cap C_o^{\rm out} \big)\mbox{ for all $t$ large enough} \Big)=1. \end{equation}
Since $R_o^{\rm out}$ is finite a.s. this will follow from:
\begin{equation}\label{eq:ne-stagne-pas2} P\Big(\big(\Upsilon_t\cup \Xi_t \big)\supset \big((1-\varepsilon)tD\cap (C_o^{\rm out} \setminus R_o^{\rm out})\big)\mbox{ for all $t$ large enough} \Big)=1. \end{equation}
By Theorem \ref{th:At-encadre}, if $0<a<b$ then $atD\cap {\mathbb Z}^d \subset \widehat A_{bt}$ a.s. for $t$ large enough. Hence, for $t$ large enough
$z\in (1-\varepsilon)tD\cap (C_o^{\rm out} \setminus R_o^{\rm out})$ implies
\begin{equation}\label{eq:tau_z-borne}
{\widehat\tau}(o,z)\le (1-\varepsilon/2)t\mbox{ a.s.} \end{equation}
and by Lemma \ref{lem:comparaison-approximation}, ${\tau}(o,z)\le(1-\varepsilon/2)t+u(o)+u(z)$. Since $u(o)< +\infty$ a.s. we have $u(o)<(\varepsilon/4)t$ a.s. for $t$ large enough. Hence,
by \eqref{eq:xi-zeta-tau}, \eqref{eq:ne-stagne-pas2} will follow if we show that
$\sup_{z \in tD}u(z)\leq (\varepsilon/4) t\,$ a.s. for $t$ large enough, which is implied by $\sup_{z \in (n+1)D}u(z)\leq (\varepsilon/4)n\,$ a.s. for $n=\lfloor t \rfloor$. By Proposition \ref{lem:definir_phi}, $D$ is bounded, hence the number of points in $(n+1)D$ with coordinates in ${\mathbb Z}$ is less than ${\rm C_5}(n+1)^d$ for some constant ${\rm C_5}$. Then write
\begin{eqnarray*} P\left(\sup_{z \in (n+1)D}u(z)\geq \frac{\varepsilon n}4\right)&\leq& {\rm C_5}(n+1)^d P\left(u(o)\geq \frac{\varepsilon n}4\right)\cr&\leq& {\rm C_5}(n+1)^d \frac{4^{d+2}}{(\varepsilon n)^{d+2}}E(u(o)^{d+2}). \end{eqnarray*}
Thus, by Lemma \ref{lem:regularite-approx-tau}, $\sum_{n\in{\mathbb N}} P(\sup_{z \in (n+1)D}u(z)\geq \varepsilon n/4)<\infty$, and \eqref{eq:ne-stagne-pas2} follows from Borel-Cantelli's Lemma.\\ \\
\textit{(ii)} Next we show that
\begin{equation}\label{eq:ne-deborde-pas} P\Big(\big(\Upsilon_t\cup \Xi_t\big) \subset\big( (1+\varepsilon)tD\cap C_o^{\rm out}\big) \mbox{ for all $t$ large enough} \Big)=1. \end{equation}
If $z$ belongs to $\Xi_t$ or $\Upsilon_t$, then by \eqref{eq:xi-zeta-tau} and Lemma \ref{lem:comparaison-approximation}, ${\widehat\tau}(o,z)\le t$ for $z\in C_o^{\rm out}\setminus R_o^{\rm out}$, which implies $z\in(1+\varepsilon)tD$ for $t$ large enough by Theorem \ref{th:At-encadre}. Since $R_o^{\rm out}$ is finite \eqref{eq:ne-deborde-pas} follows.\par
Putting together \eqref{eq:ne-stagne-pas1} and \eqref{eq:ne-deborde-pas} yields \eqref{eq:shape}.\\ \\
\textit{(iii)} Finally, assuming $E(\vert T_z\vert^d)<\infty$, we show that
\begin{equation}\label{eq:brule-rapidement} P(\Upsilon_t\cap(1-\varepsilon)tD=\emptyset \mbox{ for $t$ large enough})=1. \end{equation}
Let $z\in(1-\varepsilon)tD\cap C_o^{\rm out}$, then, by \eqref{eq:xi-zeta-tau}, \eqref{eq:ne-stagne-pas1} and the same reasoning as for \eqref{eq:tau_z-borne}, we have $\tau(o,z)\le (1-\varepsilon/2)t$ if $t$ is large enough. Hence, \eqref{eq:brule-rapidement} will follow if we show that $T_z\ge (\varepsilon/2)\tau(o,z)$ occurs only for a finite number of $z$'s. Indeed otherwise $T_z\le (\varepsilon/2)(1-\varepsilon/2)t$ so that $\tau(o,z)+T_z<t$ if $t$ is large enough: it means that the infection has reached site $z$ and the time of infection from $z$ is over before time $t$, hence $z$ has recovered by time $t$, that is $z\in\Xi_t,z\notin\Upsilon_t$. \par
But
for $\delta=(2(1+\varepsilon)\sup_{x\in D}\|x\|_\infty)^{-1}$ (by Proposition \ref{lem:definir_phi}, $D$ is bounded),
we have $\tau(o,z) \ge \delta \|z\|_\infty$ except for a finite number of $z$'s.
Because if $z$ satisfies $\tau(o,z) < \delta \|z\|_\infty$, then by
\eqref{eq:xi-zeta-tau} and \eqref{eq:ne-deborde-pas}, for $\delta \|z\|_\infty$ larger than some $t_0$, we have
$z\in\big(\Upsilon_{\delta \|z\|_\infty}\cup \Xi_{\delta \|z\|_\infty}\big)
\subset (1+\varepsilon)\delta \|z\|_\infty D$, hence the contradiction $\|z\|_\infty\le \|z\|_\infty/2$. \par
Therefore, it suffices to show that for any $\delta'>0$ the event
$\{T_z\ge \delta'\|z\|_\infty\}$ can only occur for a finite number of $z$'s.
This will follow from Borel-Cantelli's Lemma once we prove that $\sum_{z\in{\mathbb Z}^d} P(T_z\ge \delta' \|z\|_\infty)<\infty$. To do so we write, since the $T_z$'s are identically distributed:
$$\sum_{z\in{\mathbb Z}^d} P(T_z\ge \delta' \|z\|_\infty)
=\sum_{n\in{\mathbb N}}\sum_{z:\|z\|_\infty=n}P(T_z\ge \delta' n) \le c\sum_{n\in{\mathbb N}} n^{d-1}P(T_o\ge \delta' n)$$
for some constant $c$, and this last series converges because $T_o$ has a finite moment of order $d$.
Putting together \eqref{eq:shape} and \eqref{eq:brule-rapidement} yields \eqref{eq:couronne}.
\end{proof}
\begin{appendix}\section{}\label{sec:appendix}
In this appendix we prove Theorem \ref{prop:clusters_rentrant-sortant_infinis}, Lemma \ref{lem:connections} and \eqref{eq:analogue_G-thm8.21} in the proof of Lemma \ref{lem:exp_decay_R}.
These proofs rely on dynamic renormalisation techniques introduced in \citet{MR1124831}. In applying these techniques we follow \citet[Chapter 7]{MR1707339} and \citet{MR1068308}, but we introduce some modifications.
In particular by considering some larger boxes we avoid using the sprinkling technique
more than once on any given bond. Because of this we only need to consider two different
values of the infection parameter and we do not need to introduce
the updating functions of \citet{MR1707339}.
To simplify the notation we write the proofs for $d=3$, but their generalizations to higher dimensions presents no problems.\\ \\
We introduce parameters whose values will be settled in Lemma \ref{lsuppl} below. We fix $\lambda'>\lambda_c$ and adopt the terminology of \citet[Chapter 7]{MR1707339}. Nonetheless, we might change names of constants if this creates confusions with the rest of our paper. In the sequel $n,m$ and $N$ are positive integers such that
\begin{equation}\label{mnN} 2m<n\qquad\hbox{ and }\qquad N=n+m+1. \end{equation}
We consider our percolation model on the slab ${\mathbb Z}^2 \times [-3N,3N]$.
Recall that we denote by $({\rm e}_1,{\rm e}_2,{\rm e}_3)$ the canonical basis of ${\mathbb Z}^3$. For $x=(x_1,x_2,x_3)\in {\mathbb Z}^3$ and $k\in {\mathbb N}$ such that $-3N+k\leq x_3\leq 3N-k$ we recall that $B(k)=[-k,k]^3$ and $B_x(k)=x+[-k,k]^3$. We divide the face $F(n)=\{x: x\in\partial B(n), x_1=n\}$ of $\partial B(n)$ in 4 quadrants:
\begin{eqnarray*} T^{+,+}(n)&=&\{x: x\in\partial B(n), x_1=n, x_2\geq 0, x_3\geq 0\},\\ T^{+,-}(n)&=&\{x: x\in\partial B(n), x_1=n, x_2\geq 0, x_3\leq 0\},\\ T^{-,+}(n)&=&\{x: x\in\partial B(n), x_1=n, x_2\leq 0, x_3\geq 0\},\\ T^{-,-}(n)&=&\{x: x\in\partial B(n), x_1=n, x_2\leq 0, x_3\leq 0\}. \end{eqnarray*}
and, for any choice of $(i,j)\in \{+,-\}^2$, we define a box
of thickness $2m+1$ composed of translates of the corresponding quadrant, by
\begin{equation}\label{eq:def-Tijmn}
T^{i,j}(m,n)=\cup_{\ell=1}^{2m+1}\{\ell {\rm e}_1+T^{i,j}(n)\}.
\end{equation}
For any of the above sets a subindex as $y$ means we translate it by $y$.
\begin{definition} A {\rm seed} is a translate of $B(m)$ such that all its edges are $\lambda'$-open. \end{definition}
We will be looking for oriented open paths starting in a seed inside $B(2N)$, and: either \textit{(a)} contained in the union of boxes $B(3N)\cup B_{6N{\rm e}_1}(3N)$ and reaching a seed inside $B_{6N{\rm e}_1}(2N)\cap B_{8N{\rm e}_1}(2N)$;
or \textit{(b)} contained in the union of boxes $B(3N)\cup B_{6N{\rm e}_2}(3N)$ and reaching a seed inside $B_{6N{\rm e}_2}(2N)\cap B_{8N{\rm e}_2}(2N)$. We will construct those paths in Lemma \ref{l3} below. \\ \\
An important tool in \citet[Chapter 7]{MR1707339} is the \textit{sprinkling technique}, which enables some bonds, that would be closed otherwise, to be independently open with a probability larger than some $\varepsilon'>0$. We therefore need to find a way to proceed similarly, in spite of the fact that we work with dependent percolation. For this, it is convenient to define the processes for different values of the rate of propagation $\widetilde\lambda$ on our common probability space and compare them.
Let $(e_1(x,y),x,y\in{\mathbb Z}^3)$ be a collection of independent
exponential r.v.'s with parameter $1$. Then let $e_{\widetilde\lambda}(x,y)={\widetilde\lambda}^{-1}e_1(x,y)$, and
\begin{equation}\label{def:sprinkling_open-closed-bonds} X_{\widetilde\lambda}(x,y)= \begin{cases} 1 &\text{if $e_{\widetilde\lambda}(x,y)<T_x$;}\\ 0 &\text{otherwise.} \end{cases} \end{equation}
We recall that the event $\{T_x>e_{\widetilde\lambda}(x,y) \}$ occurs if and only if the oriented bond $(x,y)$ is $\widetilde\lambda$-open. \\ \\
The following lemma implies that given $\lambda>\delta_1>0$,
there exists $\iota>0$ such that for any $\widetilde\lambda$ such that $\widetilde\lambda+\delta_1<\lambda$ the random field $\{X_{\widetilde\lambda +\delta_1}(u,v):u,v \in {\mathbb Z}^3\}$ is stochastically above the random field $\{\max \{X_{\widetilde\lambda }(u,v),Y(u,v)\}:u,v \in {\mathbb Z}^3\}$ where the random variables $Y(u,v)$ are i.i.d. Bernoulli with parameter $\iota$ and are independent of the random variables $X_{\widetilde\lambda}(u,v)$. This lemma justifies the use of the sprinkling technique
needed to prove Lemmas \ref{l2} and \ref{l3} below.
\begin{lemma}\label{lem:sprinkling}
Assume $\lambda>\delta_1>0$. There exists $\iota>0$ such that for any $\widetilde\lambda>0$ such that $\widetilde\lambda+\delta_1<\lambda$, and any $x,y\in {\mathbb Z}^3,$ with $y\sim x$,
\begin{eqnarray}\nonumber P(X_{\widetilde\lambda+\delta_1 } (x,y)=1&\vert& X_{\widetilde\lambda+\delta_1}(u,v),\, u,v\in {\mathbb Z}^3,\, u\sim v, (u,v)\not=(x,y);\\\nonumber &&X_{\widetilde\lambda}(u,v),\, u,v\in {\mathbb Z}^3,\, u\sim v)>\iota\ \ \ a.s. \end{eqnarray}
\end{lemma}
\begin{proof}{Lemma}{lem:sprinkling}
Recall that $X_{\widetilde\lambda+\delta_1}(x,y)$ is independent of the random variables $X_{\widetilde\lambda+\delta_1}(u,v),X_{\widetilde\lambda}(u,v),u\neq x,v\sim u$, hence it suffices to show that
\begin{eqnarray}\label{(3)bis} P(X_{\widetilde\lambda+\delta_1}(x,y)=1&\vert& X_{\widetilde\lambda+\delta_1}(x,z),\ z\sim x,
z\neq y;\nonumber\\ && X_{\widetilde\lambda}(x,z),z\sim x)>
\iota \ \ \ a.s.
\end{eqnarray}
Since we are now conditioning on a finite number of random variables taking only values $0$ and $1$,
\eqref{(3)bis} will follow from:
\begin{eqnarray} P(X_{\widetilde\lambda+\delta_1}(x,y)=1&\vert& X_{\widetilde\lambda+\delta_1}(x,z)=a_z, z\sim x, z\neq y;\nonumber\\ &&
X_{\widetilde\lambda}(x,z)=b_z,z\sim x)> \iota\label{ex-lem3.6} \end{eqnarray}
for all choices $a_z$ and $b_z$ in $\{0,1\}$ with $b_z\leq a_z$.\\ \\
To prove \eqref{ex-lem3.6}, we denote by ${\mathcal N}_{x}$ the union of all partitions of the set $\{z\in {\mathbb Z}^3:\,z\sim x\}$ of neighbors of $x$ into three disjoint sets called ${\mathcal N}_{x}^1,{\mathcal N}_{x}^0,{\mathcal N}_{x}^{0,1}$ such that $y\in{\mathcal N}_{x}^1\cup{\mathcal N}_{x}^{0,1}$. Using the inequality $P(A\vert B)\geq P(A\cap B)$ for two events $A,B$, and taking an arbitrary partition in ${\mathcal N}_{x}$ gives
\begin{eqnarray*} &&P(X_{\widetilde\lambda+\delta_1}(x,y)=1\,\vert\, X_{\widetilde\lambda+\delta_1}(x,u)=0,\, \forall u\in{\mathcal N}_{x}^0,\\ &&\quad X_{\widetilde\lambda}(x,v)=1,\, \forall v\in{\mathcal N}_{x}^1 ; X_{\widetilde\lambda}(x,w)=0, X_{\widetilde\lambda+\delta_1}(x,w)=1,\, \forall w\in{\mathcal N}_{x}^{0,1})\\ &&\ge P(X_{\widetilde\lambda+\delta_1}(x,y)=1,\, X_{\widetilde\lambda+\delta_1}(x,u)=0,\, \forall u\in{\mathcal N}_{x}^0,\\ &&\quad X_{\widetilde\lambda}(x,v)=1,\, \forall v\in{\mathcal N}_{x}^1 ; X_{\widetilde\lambda}(x,w)=0, X_{\widetilde\lambda+\delta_1}(x,w)=1,\, \forall w\in{\mathcal N}_{x}^{0,1}).
\end{eqnarray*}
Let $a>0$ be such that $P(T_{x} \in [a, a +\gamma ])>0$ for all $\gamma >0$, and let $\delta_2\in (0,a)$ be such that $b$ defined by $$b:=\frac{(a+\delta_2)\widetilde\lambda}{\widetilde\lambda+\delta_1}$$ satisfies $b< a$. On the event
\begin{eqnarray}\nonumber &&\{T_{x} \in [a,a+\delta_2/2), e_{\widetilde\lambda} ({x},w) \in [a+\delta_2/2, a+\delta_2),\, \forall w\in{\mathcal N}_{x}^{0,1},\\\nonumber &&\,e_{\widetilde\lambda} ({x},v) \in [a-\delta_2/2, a),\, \forall v\in{\mathcal N}_{x}^1,\\\label{eq:onthisevent} &&e_{\widetilde\lambda+\delta_1} ({x},u) \in [a+\delta_2/2, a+\delta_2),\, \forall u\in{\mathcal N}_{x}^0\}, \end{eqnarray}
for all sites $v$ such that $v\in{\mathcal N}_{x}^1$ we have $e_{\widetilde\lambda}(x,v)<T_{x}$ hence $X_{\lambda'}(x,v)=1$; for all sites $u$ such that $u\in{\mathcal N}_{x}^0$ we have $T_{x}<e_{\lambda'+\delta_1}(x,u)$ hence $X_{\widetilde\lambda+\delta_1}(x,u)=0$; and for all sites $w$ such that $w\in{\mathcal N}_{x}^{0,1}$ we have $T_{x}< e_{\widetilde\lambda}(x,w)$ and
\[e_{\widetilde\lambda+\delta}(x,w)= \frac{\widetilde\lambda}{\widetilde\lambda+\delta_1} e_{\widetilde\lambda}(x,w)
\leq \frac{\widetilde\lambda}{\widetilde\lambda+\delta_1}(a+\delta_2)=b<a \leq T_{x}\]
hence $X_{\widetilde\lambda}(u,w)=0$ and $X_{\widetilde\lambda+\delta_1}(u,w)= 1$. \par
Since the probability $p({\mathcal N}_{x}^0,{\mathcal N}_{x}^1,{\mathcal N}_{x}^{0,1})$ of the event \eqref{eq:onthisevent} is strictly positive,
we conclude the proof of \eqref{ex-lem3.6} by taking
$$\iota=\inf_{{\mathcal N}_{x}}p({\mathcal N}_{x}^0,{\mathcal N}_{x}^1,{\mathcal N}_{x}^{0,1}).$$
\end{proof}
\mbox{}\\ \\
It is in view of this sprinkling procedure that we chose a propagation rate $\lambda'>\lambda_c$.
As in \citet[Section 7.2]{MR1707339}, we go on with two key geometrical lemmas. The first one, Lemma \ref{l1}, corresponds to \citet[Lemma 7.9]{MR1707339}, with a very similar proof that we omit consequently. The second one, Lemma \ref{l2}, corresponds to \citet[Lemma 7.17]{MR1707339}, that it generalizes in view of its applications for Theorem \ref{prop:clusters_rentrant-sortant_infinis} and Lemma \ref{lem:connections}.
\begin{lemma}\label{l1} If $\lambda_c<\lambda'$ and $\eta>0$, then there exist integers $m=m(\lambda',\eta)$ and $n=n(\lambda',\eta)$
satisfying \eqref{mnN} and such that
\begin{eqnarray*} &&P\big(\mbox{there exists a }\lambda' \mbox{-open path in }B(n)\cup T^{i,j}(m,n)\mbox{ from }B(m)\\ &&\mbox{to a seed contained in } T^{i,j}(m,n)\big)>1-\eta, \end{eqnarray*}
for any choice of $(i,j)\in\{+,-\}^2$. \end{lemma}
\begin{notation}\label{sigma-fields} Given a subset $V$ of ${\mathbb Z}^3$, $\delta>0$ and $x\in V$, we let $\sigma(x,V,\lambda',\delta)$ be the $\sigma$-algebra generated by the indicator functions of the following collection of events:
\begin{equation}\label{eq:coll-evt} \{T_y>e_{\lambda'}(y,z): y\in V, z\sim y \}\cup \{T_y>e_{\lambda'+\delta}(y,z):y\in V\cap B_x(n)^c,z\sim y \}. \end{equation}
Note that when $V\cap B_x(n)^c=\emptyset$, $\sigma(x,V,\lambda',\delta)$ is simply the $\sigma$-algebra generated by the indicator functions of $\{T_y>e_{\lambda'}(y,z): y\in V, z\sim y \}$, which we will denote by $\sigma(V,\lambda')$. \end{notation}
For $A$ a subset of ${\mathbb Z}^3$, recall from \eqref{exteriorvertexboundary} that $\Delta_v A$ denotes the exterior vertex boundary of $A$.
\begin{lemma}\label{l2}
If $\lambda_c< \lambda'$ and $\epsilon,\delta >0$, there exists $m=m(\lambda',\epsilon,\delta)$ and $n=n(\lambda',\epsilon,\delta)$ satisfying \eqref{mnN}
and with the following property: \newline For any choice of $(i,j)\in\{+,-\}^2$, any $x\in {\mathbb Z}^3$, any set $L\subset{\mathbb Z}^3$ such that
\begin{equation}\label{eq:Lsuchthat} B_x(m)\subset L\subset{\mathbb Z}^3\setminus T^{i,j}_x(m,n) \end{equation}
and for any $\sigma(x,L,\lambda',\delta)$-measurable
event $H$ of strictly positive probability we have:
\begin{equation}\label{eq:cl-l2} P(G^{i,j}\vert H)\geq 1-\epsilon , \end{equation}
where
\begin{eqnarray*} G^{i,j}&=&\big\{\mbox{there exists a path contained in } B_x(n)\cup T^{i,j}_x(m,n) \mbox{ going from }L\\ && \mbox{ to a seed contained in } T^{i,j}_x(m,n) \mbox{ and such that}\\ &&\mbox{its first edge } (u,v) \mbox{ with }u\in L,v\in\Delta_v L \mbox{ is }(\lambda'+\delta) \mbox{-open} \\ && \mbox{and all}\mbox{ its other edges are } \lambda'\mbox{-open}\big\}. \end{eqnarray*}
\end{lemma}
Note that since a $\lambda'$-open edge is also $(\lambda'+\delta)$-open, all the involved edges
in the path in $G^{i,j}$ are $(\lambda'+\delta)$-open, but
we do not know if the first edge of this path is $\lambda'$-open.
\begin{figure}
\caption{In the open path, the first edge going out of $L$ is $(\lambda'+\delta)$-open, and all other edges are $\lambda'$-open.}
\label{fig:dessin_lemme1.5}
\end{figure}
\begin{proof}{Lemma}{l2}
Given $A$ a subset of ${\mathbb Z}^3$ and a subset $C$ of $\Delta_v A$, let
\begin{equation}\label{def:AbuildrelC} \{A \buildrel{\lambda'+\delta}\over{\Rightarrow}C\} \end{equation}
be the event that at least one of the bonds going from $A$ to $C$ is $(\lambda'+\delta)$-open. Note that this is a stronger condition than to have a $(\lambda'+\delta)$-open path from $A$ to $C$. \par
Let $\alpha$ be the probability that any given bond is $\lambda'$-open.\par
Since the model is invariant under translations and 90 degree rotations, it suffices to show the lemma when $x$ is the origin and $(i,j)=(+,+)$. We will hence drop $x,i$ and $j$ from the notation.
Let
\begin{eqnarray} &&V(L)=\big\{ z\in \Delta_v(L\cap B(n)): \mbox{ there exists a }\lambda'\mbox{-open path contained in }\nonumber\\ &&B(n)\cup T(m,n)\setminus L\mbox{ going from }z \mbox{ to a seed contained in }T(m,n)\big\}. \label{def:V(L)}
\end{eqnarray}
Here it is understood that there always is a $\lambda'$-open path from a point to itself. Therefore any $ z\in \Delta_v(L\cap B(n)) $ contained in a seed in $T(m,n)$ is in $V(L)$. Note also that since the path is contained in $B(n)\cup T(m,n)$, $V(L)$ is a subset of $B(n)\cup T(m,n)$. Now write
\begin{equation}\label{def-G} G=\cup_{K}\big(\{L \buildrel{\lambda'+\delta}\over{\Rightarrow}K\}\cap\{V(L)=K\}\big) \end{equation}
where the union is over all possible values of $V(L)$.\par
Our next step is to show that if $m$ and $n$ are properly chosen, the set $V(L)$ is large with probability close to $1$. Let $k$ be a positive integer. Note that if $V(L)$ has at most $k$ elements, by the FKG inequality (see Remark \ref{rk:FKG}), the probability that all the bonds entering $V(L)$ are $\lambda'$-closed is at least $(1-\alpha)^{6k}$ (recall that $\alpha$ is the probability that any given bond is $\lambda'$-open, and that we are working in ${\mathbb Z}^3$). All the $\lambda'$-open paths going from $L$ to a seed contained in $T(m,n)$ have to pass through a point in $V(L)$. Hence if all the bonds entering $V(L)$ are $\lambda'$-closed, such a path does not exist. Thus we have
\begin{eqnarray} && P\big(\mbox{there exists a }\lambda'\mbox{-open path contained in }B(n)\cup T(m,n)\setminus L \nonumber\\ &&\quad\mbox{ going from } L \mbox{ to a seed contained in }T(m,n)\,\vert\, \,\vert V(L)\vert\leq k\big)\nonumber\\ &&\leq 1-(1-\alpha)^{6k}.\label{(1)}
\end{eqnarray}
But according to Lemma \ref{l1} there exist $m$ and $n$ such that $2m<n$ and
\begin{eqnarray*} && P\big(\mbox{there exists a }\lambda'\mbox{-open path contained in } B(n)\cup T(m,n) \\ &&\quad\mbox{ going from }B(m) \mbox{ to a seed contained in }T(m,n)\big)
\end{eqnarray*}
is as close to $1$ as we wish.
This implies that
\begin{eqnarray} && P\big(\mbox{there exists a }\lambda'\mbox{-open path contained (except its initial point) in}\cr &&\quad B(n)\cup T(m,n)\setminus L\mbox{ going from }L\cr&&\quad \mbox{ to a seed contained in }T(m,n)\big) \label{csq-lem-1.1}
\end{eqnarray}
is as close to $1$ as we wish uniformly in $L$'s such that
\begin{equation}\label{such-sets-L} B(m)\subset L \subset {\mathbb Z}^3\setminus T(m,n). \end{equation}
The probability \eqref{csq-lem-1.1} is equal to
\begin{eqnarray} && P\big(\mbox{there exists a }\lambda'\mbox{-open path contained (except its initial point) in }\nonumber\\ &&\quad B(n)\cup T(m,n)\setminus L \mbox{ going from }L\mbox{ to a seed contained in }\nonumber\\ &&\quad T(m,n)\,\vert\, \vert V(L)\vert \leq k) P(\vert V(L)\vert \leq k\big)\nonumber\\ &&+ P\big(\mbox{there exists a }\lambda'\mbox{-open path contained (except its initial point) in }\nonumber\\ &&\quad B(n)\cup T(m,n)\setminus L \mbox{ going from }L\mbox{ to a seed contained in }\nonumber\\ &&\quad T(m,n)\, \vert\, \vert V(L)\vert > k\big) P( \vert V(L)\vert > k)\nonumber\\ &&\leq \big(1-(1-\alpha)^{6k}\big) P( \vert V(L)\vert \leq k)+P(\vert V(L)\vert > k)\nonumber\\ &&=1-(1-\alpha)^{6k} P( \vert V(L)\vert \leq k),\label{detail(2)}
\end{eqnarray}
where the inequality comes from \eqref{(1)}. For this upper bound to be close to 1, $P( \vert V(L)\vert \leq k)$ has to be small.
Hence, it follows from \eqref{(1)}--\eqref{detail(2)} that
for any $\epsilon_0>0$ and any $k\in {\mathbb N}$,
we can choose $m$ and $n$ with $2m<n$ in such a way that
\begin{equation}\label{(2)} P(\vert V(L)\vert \leq k)\leq \epsilon_0 \end{equation}
for all sets $L$ satisfying \eqref{such-sets-L}. \par
Let $K$ be a subset of $ \Delta_v(L\cap B(n)) $.
We will now provide a lower bound to $P(L
\buildrel{\lambda'+\delta}\over{ \Rightarrow}K\, \vert\, H)$
which depends on the cardinality of $K$ but is independent of $H$.
Suppose $K$ has at least $6r$ elements.
Each point $u\in K$ has a neighbor $v\in L\cap B(n) $, that we associate to $u$.
But since each point of ${\mathbb Z}^3$ has 6 nearest neighbors,
$v$ could be a neighbor of up to 6 points of $K$, to which it could have been associated.
Then, there exist distinct $x_1,\dots,x_r \in L\cap B(n) $ and distinct
$y_1,\dots,y_r \in K $
such that $x_i\sim y_i$ for $i=1,\dots,r$. \par
Since $x_i\in L\cap B(n)$ and $H$ is $\sigma(o,L,\lambda',\delta)$-measurable, by Lemma \ref{lem:sprinkling}, for some $\iota >0$ we have
\begin{equation}\label{(3)} P(X_{\lambda'+\delta}(x_i,y_i)=0\,\vert\, X_{\lambda'+\delta}(x_j,y_j)=0,j=1,\dots i-1;H)<1-\iota \end{equation}
for $i=1,\dots,r$. It now follows from \eqref{(3)} and an inductive argument that
\[ P(X_{\lambda'+\delta}(x_i,y_i)=0,\, i=1,\dots,r\,\vert\, H)<(1-\iota)^r. \]
Hence for all $K\subset \Delta_v(L\cap B(n))$ such that $\vert K\vert \geq 6r$ we have:
\begin{equation}\label{(4)} P(L \buildrel{\lambda'+\delta}\over{ \Rightarrow}K \,\vert\, H)\geq 1-(1-\iota)^r.
\end{equation}
Now write:
\begin{eqnarray*} P(G, V(L)=K\,\vert\, H)&=&P(L \buildrel{\lambda'+\delta}\over{ \Rightarrow}K,V(L)=K\,\vert\, H)\\ &=&P(L \buildrel{\lambda'+\delta}\over{ \Rightarrow}K\,\vert\, V(L)=K, H)P(V(L)=K\,\vert\, H).
\end{eqnarray*}
But the event $\{V(L)=K\}$ is measurable with respect to the $\sigma$-algebra
generated by the random variables $(T_x,e_{\lambda'}(x,y):x\notin L)$ while both
$\{L
\buildrel{\lambda'+\delta}\over{ \Rightarrow}K \}$ and $H$ are measurable with
respect to the $\sigma$-algebra generated by the random variables
$(T_x,e_{\lambda'}(x,y),e_{\lambda'+\delta}(x,y):x\in L)$.
Therefore $\{V(L)=K\}$ is independent of the pair of events
$H,\{L
\buildrel{\lambda'+\delta}\over{ \Rightarrow}K \}$, so that
\[ P(G, V(L)=K\,\vert\, H)=P(L \buildrel{\lambda'+\delta}\over{ \Rightarrow}K\,\vert\, H)P(V(L)=K) \]
Then summing up over all sets $K$ such that $\vert K\vert\geq 6r$,
it follows from \eqref{def-G} and \eqref{(4)} that
\[ P(G\vert H)\geq (1-(1-\iota)^r) P(\vert V(L)\vert\geq 6r). \]
To complete the proof of Lemma \ref{l2} first pick $r$ such that
$(1-\iota)^r<{\epsilon}/{2}$ and then use \eqref{(2)} to pick $m$ and $n$ such that $P(\vert V(L)\vert\geq 6r)\geq 1-{\epsilon}/{2}$. \end{proof}
\mbox{}\\
\begin{notation}\label{def:barTx}
For a given $x\in{\mathbb Z}^3$ and $i=1,2,3$, $H_x^i$ will denote the hyperplane perpendicular to
${\rm e}_i$ passing through $x$. Before stating the next lemma where $x\in B(2N-m)$, we define $\overline T_x(m,n)$ to be the thickened box built from the quadrant opposite to the one $x$ belongs to in the face $H_x^1\cap B(2N-m)$, that is
\begin{eqnarray*} T^{+,+}_x(m,n) &\mbox{ if }& x_2\leq 0 \mbox { and }x_3\leq 0,\\ T^{+,-}_x(m,n) &\mbox{ if }& x_2\leq 0 \mbox { and }x_3> 0,\\ T^{-,+}_x(m,n) &\mbox{ if }& x_2> 0 \mbox { and }x_3\leq 0,\\ T^{-,-}_x(m,n) &\mbox{ if }& x_2> 0 \mbox { and }x_3> 0. \end{eqnarray*}
\end{notation}
Thanks to Lemma \ref{l2}, in the following lemma we construct open paths starting in a seed inside $B(2N)$, and reaching either a seed inside $B_{6N{\rm e}_1}(2N)\cap B_{8N{\rm e}_1}(2N)$ or
inside $B_{6N{\rm e}_2}(2N)\cap B_{8N{\rm e}_2}(2N)$. For $i\in\{1,2\}$, the successive seeds in these open paths will have centers belonging to the
hyperplanes $H^i_{x+N{\rm e}_i}, H^i_{x+2N{\rm e}_i},
H^i_{x+3N{\rm e}_i}$, $H^i_{x+4N{\rm e}_i},\ldots$;
these successive seeds will be respectively contained in
$B_{N{\rm e}_i}(2N)$, $B_{2N{\rm e}_i}(2N),
B_{3N{\rm e}_i}(2N), B_{4N{\rm e}_i}(2N),\ldots$,
and we
will stop as soon as we will get a seed in $B_{8N{\rm e}_i}(2N)$. This construction will use a \textit{steering procedure} in which, at each stage, the choice of a seed in $\overline T_.(m,n)$ compensates from an earlier deviation.
\begin{lemma}\label{l3} Given $\lambda'>\lambda_c$, for any $\epsilon,\delta >0$ there exist $n=n(\lambda',\epsilon,\delta),m=m(\lambda',\epsilon,\delta)$ and $N$ satisfying \eqref{mnN}, such that for any $x\in B(2N-m)$,
\begin{eqnarray*} P(C_x^i) \geq 1-8\epsilon, \qquad \mbox{ for } i\in\{1,2\} \end{eqnarray*}
where
\begin{eqnarray*} C_x^i&=&\big\{ \mbox{there exists a seed }B_{y}(m) \mbox{ contained in }B_{8N{\rm e}_i}(2N), \mbox{ with }y_i\le 8N \\ && \mbox{ and a path contained in }B(3N) \cup B_{6N{\rm e}_i}(3N)\\ &&\mbox{ from } B_x(m) \mbox{ to } B_{y}(m) \mbox { whose edges are all }(\lambda'+\delta)\mbox{-open and }\\ &&\mbox{those which are to the right of (resp. above)
the hyperplane }\\ && H^i_{y-N{\rm e}_i} \mbox{ when $i=1$ (resp. $i=2$) are }\lambda'\mbox{-open}\big\} \end{eqnarray*}
\end{lemma}
\begin{figure}
\caption{Event $C_x^1$}
\label{fig:dessin_lemme1.7}
\end{figure}
\begin{proof}{Lemma}{l3} We consider $C_x^1$. Since the model is invariant under $90$ degree rotations, the proof will also be valid for $C_x^2$.
Let $V_1$ be the set of vertices of all paths starting at $B_x(m)$
and contained in $B_x(n)\cup \overline T_x(m,n)$ whose first edge is $(\lambda'+\delta)$-open and all the other edges are $\lambda'$-open.
Let
$$A^1_x=\{V_1\mbox{ contains a seed in } \overline T_x(m,n)\}.$$
Note that since $x\in B(2N-m)$ a path contained in $B_x(n)\cup \overline T_x(m,n)$ is also contained in $B(3N)$. Note also that the center of a seed in $\overline T_x(m,n) $ belongs to
$H^1_{x+N{\rm e}_1}$, and that by our definition of $\overline T_x(m,n)$ (in Notation \ref{def:barTx})
this seed is contained in $B_{N{\rm e}_1}(2N)$. Thanks to Lemma \ref{l2} with $L=B_x(m)$ and $H$ the whole probability space (that is without conditioning), there exist $n,m$ such that
\begin{equation}\label{PA1} P(A^1_x)>1-\epsilon. \end{equation}
If $A^1_x$ occurs, of all the seeds in $\overline T_x(m,n)\cap V_1$
we choose one according to some arbitrary deterministic order.
We now define a random variable $Z_1$ as follows: on $A^1_x$, $Z_1$
is the center of the chosen seed and on $(A^1_x)^c$,
$Z_1=\Delta$ where $\Delta$ is an extra point we add to ${\mathbb Z}^3$.
Note that on $A^1_x$, $Z_1$ takes values in the hyperplane
$H^1_{x+N{\rm e}_1}$. The random variable $Z_1$ is a function of $V_1$
which we denote by $F_1$. We now wish to give a lower bound to the
conditional probability given $\{Z_1=z_1\}$ with
$z_1\neq \Delta $ that there is a path contained in
$B_{z_1}(n)\cup \overline T_{z_1}(m,n)$ from
$V_1$ to a seed in $\overline T_{z_1}(m,n)$ and having the following properties:\par
\textit{(i)} all its bonds are $(\lambda'+\delta)$-open; \par
\textit{(ii)} all its bonds which are to the right of $H^1_{x+(N+m){\rm e}_1}$ are $\lambda'$-open. \par\noindent
Therefore to obtain the lower bound we let $L_1$ be a value of $V_1$ containing
a seed in $\overline T_x(m,n)$ and consider the event:
\begin{eqnarray*} A^1(x,L_1)&=&\big\{\mbox{there exist }v_1\in L_1\cap B_{F_1(L_1)}(n) \mbox{ and
a path}\\ && \mbox{from }v_1 \mbox{ to a seed in } \overline T_{F_1(L_1)}(m,n),
\mbox{contained in } \\ && B_{F_1(L_1)}(n)\cup \overline T_{F_1(L_1)}(m,n) \mbox{ whose edges are all }(\lambda'+\delta) \mbox{-open}\\ && \mbox{and those to the right of } H^ 1_{x+(N+m){\rm e}_1} \mbox{ are } \lambda'\mbox{-open}\big\}. \end{eqnarray*}
The event $\{V_1=L_1\}$ is $\sigma(L_1,\lambda')$-measurable (recall Notation \ref{sigma-fields}), hence it follows from Lemma \ref{l2} that
\begin{equation}\label{PA1cond} P(A^1(x,L_1)\vert V_1=L_1)\geq 1-\epsilon. \end{equation}
Let $V_2$ be the set of vertices of all the paths
with the following properties: \par
\textit{(i)} they start from $B_x(m)$; \par
\textit{(ii)} they are contained in $B(3N)\cup B_{N{\rm e}_1}(3N)$ and lie entirely to the left of $H^1_{x+(2N+m){\rm e}_1}$; \par
\textit{(iii)} all their edges are $(\lambda'+\delta)$-open and those to the right of $H^ 1_{x+(N+m){\rm e}_1}$ are $ \lambda' $-open. \par
We also define the event
$$A^2_x=\{V_2 \mbox{ contains a seed centered in } H^1_{x+2N{\rm e}_1} \cap B_{2N{\rm e}_1}(2N-m)\}.$$
Noting that $A^2_x$ contains $A^1(x,L_1) \cap \{V_1=L_1\}$ for any $L_1$ containing a seed in $\overline T_x(m,n)$, summing over all such $L_1$'s we get by \eqref{PA1} and \eqref{PA1cond}
\begin{eqnarray} P(A^2_x)&\geq& \sum_{L_1} P(A^1(x,L_1)\vert V_1=L_1)P(V_1=L_1)\cr &\geq& \sum_{L_1} (1-\epsilon)P(V_1=L_1)=(1-\epsilon)P(A^1_x)\cr \label{eq:PA2-l3} &\geq&(1-\epsilon)^2 \geq 1-2\epsilon.
\end{eqnarray}
Now we define a random variable $Z_2$ as follows: on the event $A^2_x$ among the seeds contained in $V_2$ and centered in $H^1_{x+2N{\rm e}_1} \cap B_{2N{\rm e}_1}(2N-m)$ we choose one according to some arbitrary deterministic order and we let $Z_2 $ be its center. On $(A^2_x)^c$ we let $Z_2=\Delta $. Thus, $Z_2$ is a function $F_2$ of $V_2$. As before we let $L_2$ be a possible value of $V_2$ containing a seed centered in $H^1_{x+2N{\rm e}_1} \cap B_{2N{\rm e}_1}(2N-m)$ and consider the event:
\begin{eqnarray*} A^2(x,L_2)&=&\big\{ \mbox{there exist }v_2\in L_2\cap B_{F_2(L_2)}(n) \mbox{ and a path from }v_2 \mbox{ to a} \\ &&\mbox{seed in } \overline T_{F_2(L_2)}(m,n),\mbox{ contained in } B_{F_2(L_2)}(n)\cup \overline T_{F_2(L_2)}(m,n)\\ && \mbox{whose edges are all }(\lambda'+\delta) \mbox{-open and those to the right of }\\ &&H^ 1_{x+(2N+m){\rm e}_1} \mbox{ are } \lambda'\mbox{-open}\big\}. \end{eqnarray*}
The event $\{V_2=L_2\}$ is $\sigma(F_2(L_2),L_2,\lambda',\delta)$-measurable, hence
it follows from Lemma \ref{l2} that
$$P(A^2(x,L_2)\vert V_2=L_2)\geq 1-\epsilon. $$
We now let $V_3$ be the set of vertices belonging to all the paths with the following properties: \par
\textit{(i)} they start from $B_x(m)$; \par
\textit{(ii)} they are contained in $B(3N)\cup B_{N{\rm e}_1}(3N)$ and lie entirely to the left of $H^1_{x+(3N+m){\rm e}_1}$; \par
\textit{(iii)} all their edges are $(\lambda'+\delta)$-open and those to the right of $H^ 1_{x+(2N+m){\rm e}_1}$ are $ \lambda' $-open. \par
We also define the event:
$$A^3_x=\{V_3 \mbox{ contains a seed centered in } H^1_{x+3N{\rm e}_1} \cap B_{3N{\rm e}_1}(2N-m)\}.$$
Since $ A^3_x$ contains $A^2(x,L_2) \cap \{V_2=L_2\}$ we can argue as before and get:
$$P(A^3_x)\geq 1-3\epsilon .$$
The argument is then repeated until we reach a seed in $B_{8N{\rm e}_1}(2N)$. The total number of steps needed is at most $8$. Since at each step the probability is reduced by $\epsilon$, the lemma is proved. \end{proof}
\mbox{}\\ \\
Then define
\begin{equation}\label{eq:Cx} C_x=C^1_x\cap C^2_x. \end{equation}
{}From Lemma \ref{l3} we get:
\begin{corollary}\label{c1} Given $\lambda'>\lambda_c$, for any $\epsilon,\delta >0$ there exist $n,m,N$ satisfying \eqref{mnN} and such that for any $x\in B(2N-m)$ we have $$P(C_x)\geq 1-16\epsilon.$$ \end{corollary}
Next lemma fixes the values of all the parameters introduced up to now.
\begin{lemma}\label{lsuppl} Assume $\lambda >\lambda_c$. Then, there exist constants $m, N , K$ and $\iota>0$ such that for all $k$,
\begin{eqnarray*} &&P(\mbox{there exists a }\lambda\mbox{-open path contained in }\\ &&[-3N,(3+8k)N]\times [-3N,(3+8k)N]\times [-3N,3N] \\ &&\mbox{from }B(m)\mbox{ to a seed in } B_{8Nk{\rm e}_1+8Nk{\rm e}_2}(2N)\\ &&\mbox{whose number of edges is at most }2Kk) \geq \iota. \end{eqnarray*}
\end{lemma}
\begin{proof}{Lemma}{lsuppl} We first fix $\epsilon>0$ small enough for the two dimensional oriented site percolation of parameter
$1-16\epsilon$ to be supercritical. Then we take $\lambda'> \lambda_c$ and $\delta >0$ such that $\lambda'+\delta<\lambda$. Finally for those values of $\epsilon$, $\delta$ and $\lambda'$ we fix $n$, $m$ and $N=n+m+1$ satisfying \eqref{mnN} and such that the conclusion of Corollary \ref{c1} is valid. \\ \\
We create a two dimensional oriented site percolation on $({\mathbb Z}_+)^2$ associated to the percolation model we already have. We will refer to this model as the ``renormalized model'', while the percolation model we already had on ${\mathbb Z}^3$ will be referred to as the ``original model''. On the renormalized model all the paths are oriented upwards and towards the right; moreover, two subsequent sites of a path are at euclidean distance $1$. We now explain the way in which these models are associated. In the renormalized model site $(0,0)$ is always considered open, site $(0,1)$ is open (closed) if $C^1_0$ occurs (does not occur) in the original model. Similarly, $(1,0)$ is open (closed) if $C^2_0$ occurs (does not occur) in the original model. Note that although the states of these last two sites $(0,1)$ and $(1,0)$ are dependent, by Corollary \ref{c1} they are both open with probability at least $1-16\epsilon$. We then proceed recursively as follows.\\ \\
At the $n$-th step we will look at the points in $\{(x,y)\in {\mathbb Z}^2_+: x+y=n-1\}$ which have been reached in the renormalized model from
$(0,0)$ following open paths and order them according to their second coordinates.
We start from the point having the lowest second coordinate.
Assume it is $(x_1, n-1-x_1)$. This point was reached from either
$(x_1-1, n-1-x_1)$ or $(x_1, n-2-x_1)$. In the first case, in the original model
a seed is reached in the left portion of $B_{8Nx_1{\rm e}_1+8N(n-1-x_1){\rm e}_2}(2N)$ (remember the description given before the statement of Lemma \ref{l3}).
Let $z_1$ be the center of this seed. If $C^1_{z_1}$ occurs
(does not occur) in the original model we say that site $(x_1+1, n-1-x_1)$ in the renormalized model
is open (closed). And if $C^2_{z_1}$ occurs (does not occur) in the original model we say that site $(x_1, n-x_1)$ is open (closed). Note that since
$z_1$ is in the left portion of $B_{8Nx_1{\rm e}_1+8N(n-1-x_1){\rm e}_2}(2N)$, when we attempt to move upwards, the first seed we are seeking is centered to the right of $z_1$ due to our steering procedure, thus avoiding regions where we have already used $(\lambda'+\delta)$-open edges. In the second case, the seed reached in the original model (we again denote its center by $z_1$) is in the lowest portion of $B_{8Nx_1{\rm e}_1+8N(n-1-x_1){\rm e}_2}(2N)$ and when we want
to establish if $C^1_{z_1}$ occurs we will be looking for paths reaching a seed whose center
is above $z_1$.
We then move to the second point in
$\{(x,y)\in {\mathbb Z}^2_+: x+y=n-1\}$ which has been reached in the renormalized model from
$(0,0)$ following open paths. Let $(x_2,n-1-x_2)$ be that point and let $z_2$ be the
center of the seed located inside $B_{8Nx_2{\rm e}_1+8N(n-1-x_2){\rm e}_2}(2N)$
which was reached in the original model following open paths starting at $B(m)$.
Two different cases arise: either $x_2=x_1-1$ or $x_2<x_1-1$. In the first case the point $(x_2+1,n-1-x_2)=(x_1,n-x_1)$
has already been declared open or closed and remains in that state.
Then, we declare $(x_2,n-x_2)$ open (closed) if $C^2_{z_2}$ occurs (does not occur) in the original model.
In the second case (when $x_2<x_1-1$) we declare $(x_2+1,n-1-x_2)$ open (closed) if
$C^1_{z_2}$ occurs (does not occur) in the original model and we declare $(x_2,n-x_2)$
open (closed) if $C^2_{z_2}$ occurs (does not occur) in the original model.
Then we go on.\\ \\
We now note that for all $n$ each site examined in the set $\{(x,y):x+y=n\}$ has probability bigger than $1-8\epsilon$ of being open and that such sites are dependent at most by pairs. This implies, as explained in the following lines, that the open cluster of the origin is stochastically above the open cluster of an independent oriented site percolation model of parameter $1-16\epsilon$. \\ \\
For this, we again proceed by induction on $n$. We denote by $a_1,a_2,\ldots,a_k$ the points in the open cluster of the origin that belong to $\{(x,y)\in {\mathbb Z}^2_+: x+y=n\}$. We assume that they are ordered according to their second coordinates. Point $a_1$ has two neighbors $b_1,b_2$ on $\{(x,y)\in {\mathbb Z}^2_+: x+y=n+1\}$. They are both open with probability at least $1-16 \epsilon$, which is stochastically larger than if they were both independently open with probability $1-16 \epsilon$. In other words, if a random vector $(Y_1,Y_2)$ with coordinates taking values in $\{0,1\}$ is such that $ P(Y_1=Y_2=1)\geq 1-16 \epsilon$, then the vector $(Y_1,Y_2)$ is stochastically larger than the vector $(X_1,X_2)$ where $X_1$ and $X_2$ are independent Bernoulli r.v.'s of parameter $1-16 \epsilon$. Going on, if $a_2 =a_1+(-1,1)$, then we just have to consider the point $b_3=a_2+(0,1)$, because $a_2+(1,0)$ has already been examined. This point $b_3$ will be open with probability at least $1-8\epsilon$ independently of what happened with $b_1$ and $b_2$. Otherwise if $a_2$ is more distant from $a_1$ we have to examine $b_3= a_2+(1,0)$ and $b_4=a_2+(1,0)$: they will both be open with probability at least $1-16 \epsilon$ independently of what happened with $b_1$ and $b_2$, and so on.
In the end, to each examined point on $\{(x,y)\in {\mathbb Z}^2_+: x+y=n+1\}$ is attached a r.v. with value 1 if it is open and 0 if it is closed. The r.v.'s thus obtained are stochastically larger than a sequence of independent Bernoulli r.v.'s of parameter $ 1-16 \epsilon$.
\\ \\
Thus, for our choice of $\epsilon$ the renormalized model is supercritical and there exists a constant
$\iota >0$ such that $P((0,0)\rightarrow (k,k))\geq \iota$ for all $k\in {\mathbb N}$. Note also that the existence of an open oriented path from $(0,0)$ to $(k,k)$ (which has length $2k$) in the renormalized model implies the existence of a $(\lambda'+\delta)$-open path in the original model from $B(m)$ to some seed in $B_{8Nk{\rm e}_1+8Nk{\rm e}_2}(2N)$ whose number of edges is bounded above by $2Kk$ where $K$ is some constant that depends on $N$ but not on $k$. Indeed suppose that the point following $(0,0)$ in the path of
the renormalized model is $(1,0)$. This means that there exists an open path in the original model
from a seed in $B(2N)$ to a seed in $B_{8Ne_1}(2N)$. This last path is not oriented, but being contained in $B(3N) \cup B_{6N{\rm e}_1}(3N)$, it uses only edges in this set. The total number of edges in the latter is a function of $N$ which does not depend on $k$, that we denote by $K(N)$. Hence the derived open path in the original model from a seed in $B(2N)$ to a seed in $B_{8Nk{\rm e}_1+8Nk{\rm e}_2}(2N)$ has a number of edges bounded by $2K(N)k$. \end{proof}
\mbox{}\\ \\
For our next result we define the boxes: $$\overline{B}_{i,j}=B_{(3+8i)N{\rm e}_1+(3+8j)N{\rm e}_2}(2N)$$ where $i$ and $j$ are non-negative integers.
\begin{corollary}\label{C1}
Assume $\lambda >\lambda_c$. Let $N$ be as in the conclusion of Lemma \ref{lsuppl}. Then, there exist $\iota'>0$ and $K'\in {\mathbb N}$ such that: For any $k\in {\mathbb N}$ and any $0\leq i_1,i_2,j_1,j_2\leq k$ we have
\begin{eqnarray*} &&P(\mbox{there exists a }\lambda\mbox{-open path contained in }\\ &&[0,(6+8k)N]\times [0,(6+8k)N]\times [-3N,3N] \mbox{ from }\overline{B}_{i_1,i_2}\mbox{ to } \\ && \overline{B}_{j_1,j_2} \mbox{ whose number of edges is at most } 2K'(\vert i_1-j_1\vert
+\vert i_2-j_2\vert)) \geq \iota'. \end{eqnarray*}
\end{corollary}
\begin{proof}{Corollary}{C1}
We wish to join $\overline{B}_{i_1,i_2}$ to $\overline{B}_{j_1,j_2}$. Lemma \ref{lsuppl} enables to go from a box to another one along a diagonal direction issued from that box. Hence applying Lemma \ref{lsuppl} we get
\begin{eqnarray*} &&P(\mbox{there exists a }\lambda\mbox{-open path contained in }\\ &&[0,(6+8k)N]\times [0,(6+8k)N]\times [-3N,3N] \mbox{ from }\overline{B}_{i_1,i_2} \mbox{ to } \\ && \overline{B}_{i_1+r,i_2+r} \mbox{ whose number of edges is at most } 2K r) \geq \iota \end{eqnarray*}
for all $r \in {\mathbb N}$ such that $i_1+r, i_2+r \leq k$. Since the percolation model is invariant under 90 degree rotations, the same inequality holds if instead of adding $(r,r)$ to $(i_1,i_2)$ we add $(r,-r)$,$(-r,r)$ or $(-r,-r)$. That is, instead of going in one direction of one diagonal issued from $(i_1,i_2)$, we may take this diagonal in the other direction, or one direction of the other diagonal issued from $(i_1,i_2)$. This depends on the relative positions of $(i_1,i_2)$ and $(j_1,j_2)$ within the square $[0,k]\times[0,k]$ to which they both belong. More precisely, from $(i_1,i_2)$ and from $(j_1,j_2)$
is issued a diagonal, and those two diagonals intersect within $[0,k]\times[0,k]$. If this intersection point has integer coordinates, it can be written $(i_1+r_1,i_2+r_2)$ as well as $(j_1+\ell_1,j_2+\ell_2)$, with
$r_1=r_2$ or $r_1=-r_2$ (depending on which diagonal issued from $(i_1,i_2)$ was used), and with $\ell_1=\ell_2$ or $\ell_1=-\ell_2$ similarly. If this intersection point does not have integer coordinates, on each of the involved diagonals there is one point with integer coordinates, with those two points at distance 1, of the form $(i_1+r_1,i_2+r_2)$ and $(j_1+\ell_1, j_2+\ell_2)$, always with $r_1=r_2$ or $r_1=-r_2$, and $\ell_1=\ell_2$ or $\ell_1=-\ell_2$. To summarize, there exist integers $r_1,r_2,\ell_1,\ell_2$ with the following properties
\begin{enumerate} \item $r_2=r_1$ or $r_2=-r_1$ and $\ell_2=\ell_1$ or $\ell_2=-\ell_1$; \item $0\leq i_1+r_1,i_2+r_2,j_1+\ell_1,j_2+\ell_2\leq k$; \item either $\vert i_1+r_1-(j_1+\ell_1)\vert+\vert i_2+r_2-(j_2+\ell_2)\vert=0$ \newline or
$\vert i_1+r_1-(j_1+\ell_1)\vert+\vert i_2+r_2-(j_2+\ell_2)\vert=1$; \item $\vert r_1\vert +\vert \ell_1\vert \leq \vert i_1-j_1\vert +\vert i_2-j_2\vert$. \end{enumerate}
\begin{figure}
\caption{Two possible cases}
\label{fig:dessin_corollary1.10}
\end{figure}
The corollary now follows from Lemma \ref{lsuppl}, the FKG inequality (see Remark \ref{rk:FKG}) and the fact that the distance from a point in $\overline{B}_{i_1+r_1,i_2+r_2}$ to a point in $\overline{B}_{j_1+\ell_1,j_2+\ell_2}$ is bounded above by $20N$. \end{proof}
\mbox{}\\ \\
\begin{proposition}\label{p1} Suppose $\lambda >\lambda_c$. Then there exist constants $C, N$ and $\delta_1 >0$ such that
a) for all $M\geq 6N$, $x,y\in [0,M]\times [0,M]\times [-3N,3N]$,
\begin{eqnarray*} &&P\big(\mbox{there exists an open path from } x\mbox{ to }y\mbox{ contained in }\\
&&[0,M]\times [0,M]\times[-3N,3N]\mbox { with at most }C\|x-y\|_1 \mbox{ edges}\big)\geq \delta_1 \end{eqnarray*}
b) The original model is supercritical in a slab on thickness $k=6N$.
\end{proposition}
\begin{proof}{Proposition}{p1}
It follows from Lemma \ref{lsuppl} that the probability of having an open path of length $n$
starting in $B(m)$ and contained in the slab $ {\mathbb Z} \times {\mathbb Z}\times [-3N,3N] $ does not converge to $0$ as $n$ goes to infinity. This proves part\textit{ b)}. To prove part \textit{a)} consider the boxes $B_{(3N+8Ni){\rm e}_1+(3N+8Nj){\rm e}_2}$ with $0\leq i,j\leq (\frac{M}{N}-6)\frac{1}{8}$.
Then, note that for any point in $[0,M]\times [0,M]\times [-3N,3N]$ there is such a box at distance at most $12N$. The result now follows from this, the FKG inequality (see Remark \ref{rk:FKG}) and Corollary \ref{C1}.
\end{proof}
\mbox{}\\ \\
We have now all the ingredients for the proofs of Theorem \ref{prop:clusters_rentrant-sortant_infinis}, Lemma \ref{lem:connections}, and \eqref{eq:analogue_G-thm8.21} of Lemma \ref{lem:exp_decay_R}.\\ \\
\noindent
\textit{Proof of Theorem \ref{prop:clusters_rentrant-sortant_infinis}}.
Let $x$ and $y$ be two points in a slab of thickness $6N$. By Proposition \ref{p1}, the probability to have an open path from $x$ to $y$ in the slab is larger than $\delta_1$. Therefore the probability for the outgoing cluster from $x$ in the slab to be
infinite, as well as the probability for the incoming cluster to $y$ in the slab to be infinite, is at least $\delta_1$.
Note that Proposition \ref{p1} gives more precise information, since it restricts the involved open paths to a part of the slab, and gives an upper bound on the lengths of the paths.
\mbox{$\square$}
\mbox{}\\ \\
\noindent
\textit{Proof of Lemma \ref{lem:connections}}. For two points $x$ and $y$, the idea to build an open path from $x$ to $y$ is to combine paths in different slabs using in each one Proposition \ref{p1},\textit{a)}.\\ \\
\textit{(i)} Let $\delta_1,M$ and $C$ be given by Proposition \ref{p1}, and let $k\ge M$. For $n>0$, let $x=(x_1,x_2,x_3)\in B_{n+k}\setminus B_n, \ y=(y_1,y_2,y_3)\in (B_{n+k}\setminus B_n)\cup \Delta_v(B_{n+k}\setminus B_n )$. Assume for instance that $x_1<-n$, $n<y_1$, $-n<x_2<n$ and $-n<y_2<n$. Let $u,v\in B_{n+k}\setminus B_n$ with $-n<u_1,n<u_2$ and $n<v_1,v_2$. By Proposition \ref{p1},\textit{a)} there exist with a probability larger than $\delta_1$ an open path from $x$ to $u$, as well as from $u$ to $v$ and from $v$ to $y$. By FKG inequality (see Remark \ref{rk:FKG}) there exists therefore with a probability larger than $\delta_1^3$ an open path from $x$
to $y$. Since this particular case gives the maximal distance between $x$ and $y$, $\delta=\delta_1^3$ enables us to conclude.\\ \\
\textit{(ii)} Let $n<m,\ x \in A(n,m,0),y\in A(n,m,0)\cup \Delta_v A(n,m,0)$. We proceed similarly to \textit{(i)}. Assume for instance that $x_1<n,\,x_2<0$ and $m<y_1,\,y_2<0$. Let $u,v\in A(n,m,0)$ be such that $u_1<n,\,0<u_2$ and $m<v_1,\,0<v_2$. By Proposition \ref{p1},\textit{a)} there exist with a probability larger than $\delta_1$ an open path from $x$ to $u$, as well as from $u$ to $v$ and from $v$ to $y$. We conclude with $\delta=\delta_1^3$ and $C_1=C$.\\ \\
Note that we have to add $(-x_2)^+ +(-y_2)^+$ in part \textit{(ii)} of the lemma because if
$x\in \{z: -k+n\leq z_1<n, -\infty<z_2\leq 0\}$ and $y\in \{z:m<z_1\leq m+k, -\infty<z_2\leq 0\}$, to move from $x$ to $y$ staying in $A(n,m,0)$ we need to reach first the set $ \{z:-k+n\leq z_1\leq m+k, 0<z_2\leq k\}$ (i.e. to increase the second coordinate until it is positive).
\mbox{$\square$}
\mbox{}\\ \\
\noindent
\textit{Proof of \eqref{eq:analogue_G-thm8.21} of Lemma \ref{lem:exp_decay_R}}. Relying on Proposition \ref{p1},\textit{b)}, we can follow the proof of \citet[Theorems (8.18), (8.21)]{MR1707339} to derive \eqref{eq:analogue_G-thm8.21}.
\mbox{$\square$}
\end{appendix}
\mbox{}\\ \\
\noindent{\bf Acknowledgements.} We thank Geoffrey Grimmett for useful discussions.
We thank referees for helpful comments and suggestions. This work was initiated during the semester ``Interacting Particle Systems, Statistical Mechanics and Probability Theory'' at CEB, IHP (Paris), whose hospitality is acknowledged. Part of this paper was written while E.A. was visiting IMPA, Rio de Janeiro and thanks are given for the hospitality encountered there.
\end{document} | arXiv |
\begin{document}
\title[The index of compact constant mean curvature surfaces] {Lower bounds for the index of compact constant mean curvature surfaces in $\mathbb R^{3}$ and $\mathbb S^{3}$}
\author[Cavalcante]{Marcos P. Cavalcante}
\address{ \newline Instituto de Matem\'atica \newline Universidade Federal de Alagoas (UFAL) \newline Campus A. C. Sim\~oes, BR 104 - Norte, Km 97, 57072-970. \newline Macei\'o - AL -Brazil} \email{[email protected]}
\thanks{The first author was partially supported by CNPq-Brazil.}
\author[de Oliveira]{Darlan F. de Oliveira} \address{ \newline Departamento de Ci\^encias Exatas \newline Universidade Estadual de Feira de Santana (UEFS) \newline Avenida Transnordestina, S/N, Novo Horizonte, 44036-900
\newline Feira de Santana - BA - Brazil} \email{[email protected]}
\subjclass[2010]{53C42, 49Q10, 35P15.}
\date{\today}
\keywords{constant mean curvature surfaces, Morse index, spectrum.}
\begin{abstract} {Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$ or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded from below by a linear function of the genus. As a by-product we obtain a comparison theorem between the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.. } \end{abstract}
\maketitle
\section{Introduction}
Let $\overline M^{3}$ be a complete Riemannian 3-manifold and let $M\subset \overline M^{3} $ be a compact surface immersed in $\bar M^3$. It is well known that $M$ is a \emph{minimal surface} if it is a critical point of the area functional, and $M$ is a \emph{constant non zero mean curvature (CMC) surface} if it is a critical point of the area functional for those variations that preserve the enclosed volume. When the ambient space is the Euclidean three-space, it means that minimal surfaces are the mathematical models of soap films while constant mean curvature surfaces are those of soap bubbles.
The stability properties of minimal and CMC surfaces are given by the study of the second variation of the area functional. In order to give a precise definition let us assume that $M$ is closed and two-sided, and denote by $N$ a unit normal vector field along $M$. Given $u\in C^\infty(M)$, a smooth function on $M$ and considering variations given by $V:=uN$, the second variation formula is given by the quadratic form
\[
Q(u,u):=\int_M\|\nabla u\|^2 -(\overline {\ric}(N)+\|A\|^2)u^2 \,dM,
\] where $\overline {\ric}(N)$ is the Ricci curvature of $\overline M$ in the direction of
$N$ and $\|A\|^2$ stands for the square norm of the second fundamental form of $M$ (see Section \ref{preliminaries} for details). For CMC sufaces the requirement that variations preserve the enclosed volume is equivalent to consider functions satisfying $\int_Mu\,dM=0$. For future reference we will denote by $\mathcal F$ the space of smooth functions satisfying this property.
The \emph{Morse index} of a minimal surface $M$ is defined as the maximal dimension of a vector subspace of $C^\infty(M)$ where $Q$ is negative defined and will be denoted by $\ind(M)$. If $M$ is a CMC surface, we define the \emph{weak index} $\indw(M)$ as the maximal dimension of a vector subspace of $\mathcal F$ where $Q$ is negative defined.
The index is always finite when $M$ is compact and if the index is zero, we say that the surface is \emph{stable}. In fact, the index indicates the number of directions whose variations decrease area. If $M$ is non compact, we can extend the notion of index by taking a limit of the indices of an exhaustion on $M$. Finally, we recall that all these concepts can be given in higher dimensions as well.
A classical theorem proved independently by do Carmo and Peng \cite{dCP}, Fischer-Colbrie and Schoen \cite{FCS} and Pogorelov \cite{P} asserts that \emph{a complete stable minimal surface in $\field{R}^3$ is a flat plane}.
If $M$ is a complete minimal surface with finite index immersed in a manifold $\overline M$ with nonnegative scalar curvature, Fischer-Colbrie \cite{FC} and Gulliver \cite{G}, also independently, proved that $M$ is conformally diffeomorphic to a compact Riemann surface with genus $g$ and punctured at finitely many points $p_1,\ldots, p_r$, corresponding to the ends of $M$. In \cite{R}, using harmonic $1$-forms to construct test functions, Ros proved that the index of a minimal surface immersed in $\field{R}^3$ or in quotients of $\field{R}^3$ is bounded from below by a linear function of its genus, namely he proved that $\ind(M)\geq 2g/3 $ if $M$ is orientable and $\ind(M)\geq g/3 $ if $M$ is nonorientable. In the case of oriented minimal surfaces $M\subset \field{R}^3$, Chodosh and M\'aximo \cite{CM}, improved Ros' ideas and proved that $Ind(M)\geq \frac 2 3 (g+r) - 1$.
For closed minimal hypersurfaces in the unit sphere $M\subset\mathbb S^{n+1}$, Savo \cite{Sa} proved that the Morse index is bounded from below by a linear function of its first Betti number (the genus in dimension 2) of $M$.
This result was generalized recently by Ambrozio, Carlotto and Sharp in \cite{ACS} and by Mendes and Radesh in \cite{MR} for a wide class of ambient manifolds with positive curvature. These results provide a partial answer for a conjecture of Marques and Neves which asserts that \emph{the index of a compact minimal hypersurface immersed in ambient spaces with positive Ricci curvature is bounded from below by a linear function of the first Betti number} (see \cite{Ma} and \cite{N}).
Also recently, Chao Li proved in \cite{Li} that the index and the nullity (the dimension of the subspace of solutions to $J=0$) of a complete minimal hypersurface with finite total curvature in the Euclidean space is bounded from below by a linear function of the number of ends and its first Betti number.
In the case of CMC surfaces, Barbosa, Do Carmo and Eschenburg \cite{BdCE} proved that \emph{geodesic spheres are the only compact constant mean curvature hypersurfaces in space forms that are stables}, that is, whose weak index is zero. On the other hand, there are few results in the literature about index estimates of other examples of CMC hypersufaces. To the best of the authors' knowledge, estimates were given by Lima-Sousa Neto-Rossman \cite{LSR} and Rossman \cite{Ro} for CMC tori in $\field{R}^3$, by Perdomo-Brasil \cite{PB} for CMC hypersurfaces in the sphere, but in terms of the dimension, and by Rossman-Sultana \cite{RS} and Ca\~nete \cite{Ca} for CMC tori in $\mathbb S ^{3}.$
The purpose of the present paper is to generalize Ros and Savos's estimates to obtain lower bounds for the week stability index of compact CMC surfaces immersed in either $\mathbb R^{3}$ or in $\mathbb S^{3}$ in terms of the genus. To do that we apply Ros' ideas \cite{R} making use of harmonic 1-forms to construct test functions related to the topology of the surface.
We point out that the similar ideas were used by Palmer \cite{Pa} to obtain a lower bound for the index of the energy functional and also by Torralbo and Urbano in \cite{TU} to classify stable CMC spheres in homogeneous spaces.
This paper is organized as follows. Section \ref{r} is devoted to state the main results of the paper. In Section \ref{preliminaries} we present the precise definition of index
we discuss some known examples of CMC surfaces and they indices. In Section \ref{test} we present some auxiliares results that will be used in Section \ref{proofs}, a section dedicated to the proofs of the theorems.
\section{Results}\label{r}
In this section we present the precise statements of our results. For simplicity let us denote by $\bar{M}_c^{3}$ the space form of constant curvature $c\in \{0,1\}$, that is, the $3$-dimensional unit sphere $\mathbb S^{3}$ for $c=1$ and the $3$-dimensional Euclidean space $\mathbb R^{3}$ for $c=0$.
Our main theorem read as follows.
\begin{theorem}\label{ind} Let $M^2$ be a compact constant mean curvature surface with genus $g$ immersed in $\bar M_c^{3}.$ Then, \[ \indw(M)\geq\frac{g}{3+c}. \] \end{theorem}
Note that if $M$ is stable CMC surface in $\bar M_c^{3}$, then Theorem \ref{ind} implies that $M$ is a topological sphere and by Chern-Hopf Theorem \cite{C,H} it is a round sphere. So we recover Barbosa-do Carmo-Eschenburg Theorem \cite{BdCE} in dimension $n=2$.
\begin{remark} It is a natural question to wonder whether there is a lower bound for the weak index of compact CMC hypersurfaces in $\mathbb S^{n+1}$ or in $\field{R}^{n+1}$, $n\geq 3$, in terms of the first Betti number. \end{remark}
As a by product of the technique, we obtain a comparison between the eigenvalues $ \lambda_{\alpha}^J$ of the Jacobi operator of a CMC surface immersed in $\bar M_c^{3}$ and the eigenvalues $\lambda_{\beta}^{\Delta}$ of the Hodge Laplacian acting on $1$-forms, in the same spirit as Savo did for minimal hypersurfaces in $\mathbb S^{n+1}$.
\begin{theorem}\label{esp} Let $M^2$ be a compact immersed surface in $\bar M_c^{3}$ with constant mean curvature $H$. Then for all positive integers $\alpha$ we have \begin{eqnarray*} \lambda_{\alpha}^J\leq -2(c+H^2)+\lambda_{m(\alpha)}^{\Delta}, \end{eqnarray*} where $m(\alpha)>2(3+c)(\alpha-1).$ \end{theorem}
\begin{remark} Bingqing Ma and Guangyue Huang obtained in \cite{MH} a similar inequality as in Theorem \ref{esp} for CMC hypersurfaces in $\mathbb S^{n+1}$, but involving the norm of the second fundamental form of $M$. They comparison theorem does not imply index estimates for CMC hypersurfaces. \end{remark}
\begin{remark} If $M$ is a compact minimal hypersurface in $\mathbb S^{n+1}$ which is not totally geodesic, then $-n$ is an eigenvalue of $J$ with multiplicity at least $n+2$ (see \cite{U}, \cite{EL}, \cite{Sa}), in particular its index is at least $n+3$, since the first eigenvalue is simple. Savo \cite{Sa} used this fact to improve his estimates for the index of minimal hypersurfaces in the sphere. We would like to point out here that Perdomo and Brasil proved in \cite{PB} that if $M$ is a compact CMC hypersurface in $\mathbb S^{n+1}$ which is not totally umbilical, then $\indw(M)\geq n+1$, however the negative eigenvalues are not explicit and thus we cannot use it to improve our estimates. \end{remark}
\section{Preliminaries}\label{preliminaries}
In this section we will considerer hypersurfaces in any dimension.
\subsection{The index of constant mean curvature hypersurfaces}
Let $\bar M^{n+1}$ be a Riemannian manifold and let $\psi:M^n\rightarrow\bar M^{n+1}$ be an immersed two sided compact hypersurface without boundary. We consider in $M$ the Riemannian metric $g$ induced by $\psi$. Let $\nabla$ and $\bar{\nabla}$ be the Levi-Civita connections on $M$ and $\bar M$, respectively. Fixed a unit normal vector field $N$ along $M$, we will denote by $A$ its associated shape operator, that is, \begin{equation*} AX=-\bar{\nabla}_X N \ \ \mbox{for all} \ X\in TM. \end{equation*}
The mean curvature function of $M$ is then defined as $H=(1/n)tr A$. It is well known that every smooth function $u\in C^{\infty}(M)$ induces a normal variation
$\psi_t:M^n\rightarrow \bar{M}^{n+1}$ given by \[ \psi_t(x)=exp_{\psi(x)}(tu(x)N_x), \] where $exp$ denotes the exponencial map in $\bar{M}^{n+1}.$ Since $M$ is closed and $\psi_0=\psi,$ there exists $\epsilon>0$ such that \begin{equation*} M_{u,t}=\{exp_{\psi(x)}(tu(x)N);x\in M\} \end{equation*} are immersed hypersurfaces for all $t\in(-\epsilon,\epsilon).$ We can consider the area functional $ \mathcal A_u:(-\epsilon,\epsilon)\rightarrow\mathbb{R} $ which is given by \begin{equation*} \mathcal A_u(t)=\int_MdM_{u,t}, \end{equation*} where $dM_{u,t}$ is the $n$-dimensional area element of the metric induced on $M$ by $\psi_t.$ The first variation formula for the area is given by \begin{equation*} \mathcal A'_u(0)=-n\int_MuHdM. \end{equation*} As a direct consequence, minimal hypersurfaces are characterized as critical points of the area functional, while constant mean curvature (CMC) hypersurfaces are the critical points of the area functional restricted to variations that preserves volume, that is, $\displaystyle\int_Mu\,dM=0.$ For such critical points, the second variation of the area functional is given by the following quadratic form \[
\mathcal A''_u(0)=\int_M\|\nabla u\|^2 -(\overline {\ric}(N)+\|A\|^2)u^2 \,dM. \]
Here $\|A\|^2=tr(A^2)$ is the Hilbert-Schmidt norm of $A$ and $\overline{\textrm{Ric}}(N)$ denotes the Ricci curvature of $\bar M$ in the direction of $N$. Integrating by parts we can write \[ \mathcal A''_u(0)=\int_MuJu \,dM. \]
where $J=\Delta-\ric(N)-\|A\|^2$ is the so called Jacobi operator or stability operator of $M$. We also point out that we are using the geometric definition of the Laplace-Beltrami operator, that is, $\Delta u=\delta du$ where $\delta w=-tr \nabla w$ for $w\in \Omega^1(M).$
The index of a CMC hypersurface $M$ is denoted by $\indw(M)$ and defined as the maximum dimension of any subspace $V$ of \[ \displaystyle\mathcal{F}=\big\{u\in C^{\infty}(M); \int_Mu=0\big\} \] on which $\mathcal{A}''_u(0)$ is negative definite. In other words, $\indw(M)$ is the number of negative eigenvalues of $J$, which is necessarily finite for closed hypersurfaces. In the special case that $\bar M^{3}$ has constant sectional curvature $c$, the Jacobi operator reads as \begin{equation*}\label{jeq}
J=\Delta-\|A\|^2-2c, \end{equation*} and this form will be used in the rest of the paper.
\subsection{Examples of compact CMC hypersurfaces and their indices} The most simple examples of closed CMC surfaces in the Euclidean space are the geodesic spheres, which are the only stables ones \cite{BdC}. In 1986, Wente \cite{W} constructed the first examples of CMC tori in $\field{R}^3$ solving a question posed by Hopf in \cite{H}. After that, all CMC tori in space forms were classified in a series of works (see \cite{Ab}, \cite{Bo}, \cite{PS} and \cite{UY}) and their indices were estimated by Lima, Sousa Neto and Rosmann in \cite{LSR} and by Rosmann in \cite{Ro}. One can summarizing they results as follows \begin{quote} {\it The index of a CMC torus in $\field{R}^3$ is at least 8 and there are CMC tori with arbitrarily large index.} \end{quote} Many examples of compact CMC surfaces with genus $g\ge 2$ were constructed by Kapouleas in \cite{Ka1, Ka2, Ka3} using the gluing method.
When the ambient space is the round sphere $\mathbb S^{n+1}$ many examples of CMC hypersurfaces are known. Again, geodesic spheres are the only stable CMC hypersurfaces \cite{BdCE}. The next important family is given by the CMC Clifford tori, including the case $H=0$.
In fact, it was proved by S. Brendle \cite{B} that that Clifford tori are the only minimal embedded tori in $\mathbb S^3$, confirming longstanding conjecture of H. Lawson. It is proved by B. Andrews and H. Li \cite{AL} that all CMC embedded tori in $\mathbb S^3$ are rotational surfaces, confirming longstanding conjectures of Pinkall-Sterling. We also note that Andrews and Li \cite{AL} gave a complete classifications of all CMC embedded tori in $\mathbb S^3$. It follows from the work of Simons \cite{S} that any compact minimal hypersurface not totally geodesic in $\mathbb S^{n+1}$ has index $\ind(M)\geq n+3$ and it is also well known that the minimal Clifford torus has index $n+3$. It is a natural problem to classify the minimal hypersurfaces $M\subset \mathbb S^{n+1}$ with $\ind(M)=n+3$. This problem was solved when $n=2$ by Urbano in \cite{U}, showing that the minimal Clifford tori are the only minimal surface in $\mathbb S^3$ whose index is $5$. This problem is still open in higher dimensions.
In the case $H\neq 0$, Perdomo and Brasil \cite{PB} proved that if $M\subset \mathbb S^{n+1}$ is a compact CMC hypersurface not totally umbilical, then $\indw(M)\geq n+1$ (see also \cite{A} for a nice survey). Recently, Alias and Piccione \cite{AP} showed the existence of infinite sequences of isometric embeddings of tori with constant mean curvature in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus. In the same spirit as above, the index of CMC tori of revolution in $\mathbb S^3$ were estimated by Rossman and Sultana in \cite{RS} and by Ca\~nete in \cite{Ca}. On the other hand, higher genus CMC surfaces in $\mathbb S^3$ were constructed by Butscher-Pacard \cite{BP} (see also \cite{B1,B2} for examples higher dimensions), but no index estimates is known.
\section{Test functions based on coordinates of vector fields} \label{test}
In this section we will consider CMC immersed surfaces $\psi:M\to\bar{M}_c^{3}$. In the spherical case, $c=1$, we will also consider the unit normal vector field $\nu=-\psi$ along $\mathbb{S}^{3}$, such that the second fundamental form of the inclusion map, $\bar{\psi}:\mathbb{S}^{3}\rightarrow \mathbb{R}^{4}$, is the identity. That is, $D_X\nu=-X$ where $D$ denotes the Levi-Civita connection in the Euclidean space. It follows immediately that
\begin{eqnarray*}\label{conexao} D_Y X-\nabla_Y X=\left\langle X,Y\right\rangle\nu+\left\langle AX,Y\right\rangle N,
\mbox{for all} \ X,Y\in TM \end{eqnarray*} for $M$ immersed in $\mathbb{S}^{3}$ and
\begin{eqnarray*}\label{conexao_r} D_Y X-\nabla_Y X=\left\langle AX,Y\right\rangle N, \ \mbox{for all} \ X,Y\in TM \end{eqnarray*} for $M$ immersed in $\mathbb{R}^{3}.$
Fixed an orthonormal basis $\mathcal{E}=\{\bar E_1,\dots,\bar E_{3+c}\}$ of parallel vector fields on $\mathbb{R}^{3+c}$ we will denote by \[{E_i}:=\bar E_i-\langle \bar E_i,N\rangle N-c\langle \bar E_i,\nu\rangle\nu \] the vector fields given by the orthogonal projection of $\bar E_i$ on $TM$. For the fixed basis $\mathcal{E}$ we will consider the smooth support functions $f_i, g_i:M\to\field{R}$ given by \[ f_i=\left\langle \bar{E_i},\nu\right\rangle \quad \textrm{ and } \quad g_i=\left\langle \bar{E_i},N\right\rangle, \] for $1\leq i\leq 3+c$.
Let $\xi\in TM$ be a smooth vector field on $M$ and let $\omega$ denote its dual $1$-form, that is $\xi = \omega^\#$. Inspired in the works of Ros and Savo we will use the coordinates of $\xi\in TM$ as test functions. They are given by \[ w_i:=\langle E_i,\xi\rangle, \quad 1\leq i\leq 3+c. \]
Let us denote by $\nabla^{*}\nabla$ the rough Laplacian acting on vector fields and by $\Delta$ the Hodge Laplacian acting on $1$-forms. They are defined, respectively, by \[ \nabla^{*}\nabla\xi=-\trace \nabla^2\xi \quad \mbox{and}\quad \Delta\omega=d\delta\omega+\delta d\omega, \] where $d$ is the exterior differential and $\delta=-\star d\star$ is the formal adjoint of $d$ with respect to the canonical $L^2$-inner product on $1$-forms induced by the Riemannian metric of $M$. We define the Laplacian of the vector fields $\xi$ as being $\Delta\xi = (\Delta\omega)^{\#}$ and so these Laplacians are related by the well known Bochner formula \begin{equation}\label{bochner} \Delta \xi=\nabla^{*}\nabla\xi+K\xi, \end{equation} where $K$ is the Gauss curvature of $M$.
In order to compute the Jacobi operator of $w_i$ we need the following lemma. \begin{lemma}\label{lapwi}Let $M^{2}$ be an orientable CMC surface immersed in $\bar M_c^3.$ Then, using the above notation we have \[
\Delta w_i=(\|A\|^2-4H^2)w_i+2H\langle AE_i,\xi\rangle-2g_i\langle A,\nabla \xi\rangle+2cf_i\diver\xi+\langle E_i,\Delta \xi\rangle, \] for $1\leq i\leq 3+c$. \end{lemma} \proof Fixed a point $p\in M$, we consider a local orthonormal frame $\{e_1,e_2\}$ on $M$ which is geodesic at $p.$ A direct computation shows that $\nabla_{e_\ell}E_i=g_iAe_\ell+cf_ie_\ell,$ $\ell=1,2.$ Thus, using Einstein summation notation, we get \begin{eqnarray*} \Delta w_i &=&-e_\ell e_\ell\langle E_i,\xi\rangle\\
&=&-e_\ell(\langle \nabla _{e_\ell} E_i,\xi\rangle+\langle E_i, \nabla_{e_\ell}\xi\rangle)\\
&=&-\langle\nabla_{e_\ell}\nabla_{e_\ell}E_i,\xi\rangle-2\langle \nabla_{e_\ell} E_i,\nabla_{e_\ell}\xi\rangle-
\langle E_i, \nabla_{e_\ell}\nabla_{e_\ell}\xi\rangle\\
&=&-\langle \nabla_{e_\ell}(g_iAe_\ell+cf_ie_\ell),\xi\rangle-2\langle g_iAe_\ell +
cf_ie_\ell,\nabla_{e_\ell}\xi\rangle+\langle E_i,\nabla^*\nabla \xi\rangle\\
&=&\langle AE_i,e_\ell\rangle\langle Ae_\ell,\xi\rangle-g_i\langle (\nabla_{e_\ell}A)e_\ell,\xi\rangle+c\langle E_i,e_\ell\rangle\langle e_\ell,\xi\rangle\\
&&-2g_i\langle A,\nabla \xi\rangle+2cf_i\diver \xi+\langle E_i,\nabla^*\nabla \xi\rangle\\
&=&\langle A^2E_i,\xi\rangle-2g_i\langle A,\nabla \xi\rangle+2cf_i\diver\xi
+\langle E_i,\Delta \xi\rangle\\&&-2H^2\langle E_i,\xi\rangle+\frac{\|A\|^2}{2}\langle E_i,\xi\rangle,
\end{eqnarray*} where in the last equality we used the Bochner equation (\ref{bochner}) and the Gauss equation in the form below: \[
K=c+2H^2-\frac{1}{2}\|A\|^2. \] To conclude the proof we note that shape operator $A$ satisfies the following equation \[
A^2=\frac{1}{2}(\|A\|^2-4H^2)I_2+2HA. \] \endproof
We end this section noting that the coordinates of harmonic vector fields are admissible functions to compute the index of CMC surfaces. Moreover we have:
\begin{lemma}\label{wi} If $\xi\in TM$ is a harmonic vector field, then $w_i= \langle E_i, \xi\rangle$ and $\bar w_i=\langle E_i,\star\xi\rangle$ satisfy \[ \int_M w_i =\int_M \bar w_i =0
\] for $1\leq i\leq 3+c.$ \end{lemma} \proof If $\xi$ is harmonic we have $\diver\xi=0$ and therefore \[ \int_M w_i=-\int_M\langle \nabla f_i,\xi\rangle=-\int_Mf_i\diver\xi=0. \] Now, recall that, in an orthonormal basis $\{e_1,e_2\}$ of $TM$, the Hodge star operator is defined by \[ \star e_1=e_2,\star e_2=-e_1. \] Since $\Delta$ comutes with $\star$ it follows that $\star\xi$ is also a harmonic vector field on $M$ and this concludes the proof. \endproof
\section{Proofs of the theorems}\label{proofs}
For simplicity we will present the proofs in the case of CMC surfaces in the unit sphere $\mathbb S^3$. The case of CMC surfaces in $\mathbb R^3$ follows the same steps.
Let us denote by $\mathcal{L}^{\Delta}_m$ the vector space given by the direct sum of the eigenspaces generated by $\xi_1, \xi_2, \ldots, \xi_m$, the first $m$ eigenfunctions of the Hodge Laplacian $\Delta$ and let us denote by $\mathcal{H}^1(M)$ the vector space of the harmonic vector fields on $M$. Notice that $\dim \mathcal{H}^1(M) =2g$, where $g$ is the genus of surface $M$, and $\mathcal{H}^1(M)\subset \mathcal{L}^{\Delta}_m$ .
\subsection{Proof of Theorem \ref{esp}}
Since $J$ is an elliptic self-adjoint operator, it admits a sequence of eigenvalues diverging to infinity, \[ \lambda_1^{J}\leq \lambda_2^{J} \leq\cdots\leq \lambda_k^{J} \leq\cdots \]
Fix an orthonormal basis $\{\phi_1,\phi_2,\ldots\}$ of $C^{\infty}(M)$ given by eigenfunctions of the Jacobi operator, that is, $J\phi_i=\lambda_i^J\phi_i$. We denote by $\mathcal{J}^{p} :=\langle \phi_1,\cdots,\phi_{p}\rangle^{\bot}$ the linear space orthogonal to the first $p$ eigenfunctions of the Jacobi operator.
Initially, we look for vector fields $\xi\in \mathcal{L}^{\Delta}_m$ such that the functions $w_i,\bar w_i\in \mathcal{J}^{\alpha-1}$, for some $\alpha\in \mathbb N$ and $i\in\{1,\dots, 4\}.$ In other words, we have a system with $8(\alpha-1)$ homogenous linear equations in the variable $\xi$ \begin{equation}\label{sys} \int_Mw_i\phi_k=\int_M\bar w_i\phi_k=0,\quad 1\leq i\leq 4 \quad \mbox{and} \quad 1\leq k\leq \alpha-1. \end{equation} Therefore, if $m(\alpha)=\dim \ \mathcal{L}^{\Delta}_m>8(\alpha-1),$ then the system (\ref{sys}) has at least a non trivial solution $\xi\in \mathcal{L}^{\Delta}_m$ such that $w_i,\bar w_i\in \mathcal{J}^{\alpha-1}$ for all $1\leq i\leq 4.$ By the min-max principle we have \begin{eqnarray*} \lambda_{\alpha}^J\leq\frac{\int_Mw_iJw_i}{\int_Mw_i^2}\quad \mbox{and}\quad \lambda_{\alpha}^J\leq\frac{\int_M\bar w_iJ\bar w_i}{\int_M\bar w_i^2}. \end{eqnarray*} Now, using Lemma \ref{lapwi} we get
{\setlength\arraycolsep{1pt} \begin{eqnarray*} \lambda_{\alpha}^J\int_Mw_i^2&\leq &-(2+4H^2)\int_Mw_i^2+2H\int_M\langle E_i,A\xi\rangle w_i\\ && +\int_M\langle E_i,\Delta \xi\rangle w_i-2\int_Mg_i\langle A,\nabla \xi\rangle w_i+2\int_Mf_i\delta\xi w_i. \end{eqnarray*}}
Summing upon $i=1,\ldots, 4$ we obtain {\setlength\arraycolsep{1pt} \begin{eqnarray*}
\lambda_{\alpha}^J\int_M\|\xi\|^2&\leq &-(2+4H^2)\int_M\|\xi\|^2+2H\int_M\langle A\xi,\xi\rangle +\int_M\langle\Delta \xi,\xi\rangle. \end{eqnarray*}}
Analogously, we do the same to the test functions $\bar w_i$: {\setlength\arraycolsep{1pt}
\begin{eqnarray*}
\lambda_{\alpha}^J\int_M\|\xi\|^2&\leq &-(2+4H^2)\int_M\|\xi\|^2+2H\int_M\langle A\star\xi,\star\xi\rangle +\int_M\langle\Delta \star\xi,\star\xi\rangle. \end{eqnarray*}}
Summing these last two inequalities we have {\setlength\arraycolsep{0pt}
\begin{eqnarray}\label{lambda}
\lambda_{\alpha}^J\int_M\|\xi\|^2&\leq &-(2+4H^2)\int_M\|\xi\|^2+H\int_M\langle A\xi,\xi\rangle+\langle A\star\xi,\star\xi\rangle\\
&&+\frac{1}{2}\int_M(\langle\Delta \xi,\xi\rangle+\langle\Delta \star\xi,\star\xi\rangle) \nonumber.
\end{eqnarray}} Now, we observe that
\begin{equation}\label{AA}
\langle A\xi,\xi\rangle+\langle A\star\xi,\star\xi\rangle=2H\|\xi\|^2 \end{equation} for any $\xi\in TM$.
If $\xi\in \mathcal{L}^{\Delta}_m$ we can $\xi=\alpha_i\xi_i$ and then
\begin{eqnarray}\label{xi} \int_M\langle\Delta \star\xi,\star\xi\rangle&=&\int_M\langle\Delta \xi,\xi\rangle\\ \nonumber
&=&\lambda_i^{\Delta}\int_M\alpha_i\alpha_k \langle\xi_i,\xi_k\rangle\\ \nonumber
&\leq&\lambda_{m(\alpha)}\int_M\|\xi\|^2. \nonumber
\end{eqnarray} Plugging (\ref{AA}) and (\ref{xi}) into (\ref{lambda}) we obtain \[ \lambda_{\alpha}^J\leq -2(1+H^2)+\lambda_{m(\alpha)}^{\Delta} \] where $m(\alpha)>8(\alpha-1)$.
\subsection{Proof of Theorem \ref{ind}}
We start as in proof of Theorem \ref{esp} but now choosing an orthonormal basis $\{\phi_1,\phi_2,\cdots\}$ of the space $\mathcal{F}=\big\{u\in C^{\infty}(M); \int_Mu=0\big\}$ given by eigenfunctions of the Jacobi operator. We already know from Lemma \ref{wi} that for any $\xi\in\mathcal{H}^1(M)$ the test functions $w_i,\bar w_i\in\mathcal{F}.$
Consider then vector fields $\xi\in \mathcal{H}^1(M)$ such that the test functions $w_i,\bar w_i\in \mathcal{J}^{\alpha-1}$, for some $\alpha\in \mathbb N$ and $i\in\{1,\dots, 4\}.$
As before, if $\dim \mathcal{H}^1(M)=2g>8(\alpha-1),$ then the system (\ref{sys}) has at least a non trivial solution $\xi\in \mathcal{H}^1(M)$.
Following the same steps as above, we use Lemma \ref{lapwi} but now for the harmonic vector fields $\xi$ and its dual $\star \xi$. We obtain
{\setlength\arraycolsep{1pt}
\begin{eqnarray*}
\lambda_{\alpha}^J\int_M\|\xi\|^2&\leq &-2(1+H^2)\int_M\|\xi\|^2.
\end{eqnarray*}}
Hence, we conclude that $\lambda_{\alpha}^J< 0$ and then $\indw (M)\geq \alpha.$ Since $\alpha$ can be chosen as the largest integer such that $2g>8(\alpha-1)$ we get \[ \indw(M)\geq \frac{g}{4}. \]
\end{document} | arXiv |
\begin{document}
\title{Algebraic Geometric Comparison of Probability Distributions}
\author{\name Franz J.~Kir\'{a}ly \email [email protected] \\
\addr Machine Learning Group, Computer Science \\
Berlin Institute of Technology (TU Berlin) \\
Franklinstr.~28/29, 10587 Berlin, Germany\\
and Discrete Geometry Group, Institute of Mathematics, FU Berlin \\
\AND
\name Paul von B\"unau \email [email protected]
\AND
\name Frank C.~Meinecke \email [email protected] \\
\addr Machine Learning Group, Computer Science \\
Berlin Institute of Technology (TU Berlin) \\
Franklinstr.~28/29, 10587 Berlin, Germany\\
\AND
\name Duncan A.~J.~Blythe \email [email protected] \\
\addr Machine Learning Group, Computer Science \\
Berlin Institute of Technology (TU Berlin) \\
Franklinstr.~28/29, 10587 Berlin, Germany\\
and Bernstein Center for Computational Neuroscience (BCCN), Berlin \\
\AND
\name Klaus-Robert Müller \email [email protected] \\
\addr Machine Learning Group, Computer Science \\
Berlin Institute of Technology (TU Berlin) \\
Franklinstr.~28/29, 10587 Berlin, Germany\\
and IPAM, UCLA, Los Angeles, USA
}
\editor{The editor}
\maketitle
\begin{abstract} We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identifiability criterion.
\end{abstract}
\begin{keywords} Computational algebraic geometry, Approximate algebra, Unsupervised Learning \end{keywords}
\section{Introduction} \label{sec:Intro} Comparing high dimensional probability distributions is a general problem in machine learning, which occurs in two-sample testing (e.g.~\cite{Hotelling31,Gretton07akernel}), projection pursuit (e.g.~\cite{Fried74}), dimensionality reduction and feature selection (e.g.~\cite{Tor03Feature}). Under mild assumptions, probability densities are uniquely determined by their cumulants which are naturally interpreted as coefficients of homogeneous multivariate polynomials. Representing probability densities in terms of cumulants is a standard technique in learning algorithms. For example, in Fisher Discriminant Analysis~\citep{Fisher36}, the class conditional distributions are approximated by their first two cumulants.
In this paper, we take this viewpoint further and work explicitly with polynomials. That is, we treat estimated cumulants not as constants in an objective function, but as objects that we manipulate algebraically in order to find the optimal solution. As an example, we consider the problem of finding the linear subspace on which several probability distributions are identical: given $D$-variate random variables $X_1, \ldots, X_m$, we want to find the linear map $P \in \R^{d \times D}$ such that the projected random variables have the same probability distribution, \begin{align*}
P X_1 \sim \cdots \sim P X_m . \end{align*} This amounts to finding the directions on which all projected cumulants agree. For the first cumulant, the mean, the projection is readily available as the solution of a set of linear equations. For higher order cumulants, we need to solve polynomial equations of higher degree. We present the first algorithm that solves this problem explicitly for arbitrary degree, and show how algebraic geometry can be applied to prove properties about it.
\begin{figure}
\caption{
Illustration of the optimization approach. The left panel shows the contour plots
of three sample covariance matrices. The black line is the true one-dimensional subspace
on which the projected variances are exactly equal, the magenta line corresponds
to a local minimum of the objective function. The right panel shows the value of
the objective function over all possible one-dimensional subspaces, parameterized
by the angle $\alpha$ to the horizontal axis; the angles corresponding to the global minimum
and the local minimum are indicated by black and magenta lines respectively.
}
\label{fig:ml_optim}
\end{figure}
To clarify the gist of our approach, let us consider a stylized example. In order to solve a learning problem, the conventional approach in machine learning is to formulate an objective function, e.g.~the log likelihood of the data or the empirical risk. Instead of minimizing an objective function that involves the polynomials, we consider the polynomials as \textit{objects in their own right} and then solve the problem by algebraic manipulations. The advantage of the algebraic approach is that it captures the inherent structure of the problem, which is in general difficult to model in an optimization approach. In other words, the algebraic approach actually \textit{solves} the problem, whereas optimization \textit{searches} the space of possible solutions guided by an objective function that is minimal at the desired solution, but can give poor directions outside of the neighborhood around its global minimum. Let us consider the problem where we would like to find the direction $v \in \RR^2$ on which several sample covariance matrices $\Sigma_1, \ldots, \Sigma_m \subset \R^{2 \times 2}$ are equal. The usual ansatz would be to formulate an optimization problem such as \begin{align} \label{eq:objfun}
v^* = \argmin_{\| v \| = 1} \sum_{1 \le i,j \le m} \left( v^\top \Sigma_i v - v^\top \Sigma_j v \right) ^2 . \end{align} This objective function measures the deviation from equality for all pairs of covariance matrices; it is zero if and only if all projected covariances are equal and positive otherwise. Figure~\ref{fig:ml_optim} shows an example with three covariance matrices (left panel) and the value of the objective function for all possible projections $v = \begin{bmatrix} \cos(\alpha) & \sin(\alpha) \end{bmatrix}^\top$. The solution to this non-convex optimization problem can be found using a gradient-based search procedure, which may terminate in one of the local minima (e.g.~the magenta line in Figure~\ref{fig:ml_optim}) depending on the initialization.
However, the natural representation of this problem is not in terms of an objective function, but rather a system of equations to be solved for $v$, namely \begin{align} \label{eq:simple_example}
v^\top \Sigma_1 v = \cdots = v^\top \Sigma_m v . \end{align} In fact, by going from an algebraic description of the set of solutions to a formulation as an optimization problem in Equation~\ref{eq:objfun}, we lose important structure. In the case where there is an exact solution, it can be attained explicitly with algebraic manipulations. However, when we estimate a covariance matrix from finite or noisy samples, there exists no exact solution in general. Therefore we present an algorithm which combines the statistical treatment of uncertainty in the coefficients of polynomials with the exactness of algebraic computations to obtain a consistent estimator for $v$ that is computationally efficient.
Note that this approach is not limited to this particular learning task. In fact, it is applicable whenever a set of solutions can be described in terms of a set of polynomial equations, which is a rather general setting. For example, we could use a similar strategy to find a subspace on which the projected probability distribution has another property that can be described in terms of cumulants, e.g.~independence between variables. Moreover, an algebraic approach may also be useful in solving certain optimization problems, as the set of extrema of a polynomial objective function can be described by the vanishing set of its gradient. The algebraic viewpoint also allows a novel interpretation of algorithms operating in the feature space associated with the polynomial kernel. We would therefore argue that methods from computational algebra and algebraic geometry are useful for the wider machine learning community.
\begin{figure}
\caption{
Representation of the problem: the left panel shows sample covariance
matrices $\Sigma_1$ and $\Sigma_2$ with the desired projection $v$.
In the middle panel, this projection is defined as
the solution to a quadratic polynomial. This polynomial is
embedded in the vector space of coefficients spanned by the monomials
$X^2, Y^2$ and $X Y$ shown in the right panel.
}
\label{fig:alg_setting}
\end{figure}
Let us first of all explain the representation over which we compute. We will proceed in the three steps illustrated in Figure~\ref{fig:alg_setting}, from the geometric interpretation of sample covariance matrices in data space (left panel), to the quadratic equation defining the projection $v$ (middle panel), to the representation of the quadratic equation as a coefficient vector (right panel). To start with, we consider the Equation~\ref{eq:simple_example} as a set of homogeneous quadratic equations defined by \begin{align} \label{eq:homo_simple_example}
v^\top ( \Sigma_i - \Sigma_j ) v = 0 \; \; \forall \, 1 \le i,j \le m, \end{align} where we interpret the components of $v$ as variables, $v = \begin{bmatrix} X & Y \end{bmatrix}^\top$. The solution to these equations is the direction in $\R^2$ on which the projected variance is equal over all covariance matrices. Each of these equations corresponds to a quadratic polynomial in the variables $X$ and $Y$, \begin{align} \label{eq:polyintro}
q_{ij} & = v^\top (\Sigma_i - \Sigma_j) v \nonumber \\
& = v^\top \begin{bmatrix}
a_{11} & a_{12} \\ a_{21} & a_{22}
\end{bmatrix} v \nonumber \\
& = a_{11} X^2 + ( a_{12} + a_{21}) X Y + a_{22} Y^2, \end{align} which we embed into the vector space of coefficients. The coordinate axis are the monomials $\{ X^2, X Y, Y^2 \}$, i.e.~the three independent entries in the Gram matrix $(\Sigma_i - \Sigma_j)$. That is, the polynomial in Equation~\ref{eq:polyintro} becomes the coefficient vector \begin{align*}
\vec{q}_{ij} = \begin{bmatrix} a_{11} & a_{12}+a_{21} & a_{22} \end{bmatrix}^\top . \end{align*} The motivation for the vector space interpretation is that every linear combination of the Equations~\ref{eq:homo_simple_example} is also a characterization of the set of solutions: this will allow us to find a particular set of equations by linear combination, from which we can directly obtain the solution. Note, however, that the vector space representation does not give us all equations which can be used to describe the solution: we can also multiply with arbitrary polynomials. However, for the algorithm that we present here, linear combinations of polynomials are sufficient.
\begin{figure}
\caption{
Illustration of the algebraic algorithm. The left panel shows the vector space
of coefficients where the polynomials corresponding to the
Equations~\ref{eq:homo_simple_example} are considered as elements of the vector space
shown as red points. The middle
panel shows the approximate $2$-dimensional subspace (blue surface) onto which we project the
polynomials. The right panel shows the one-dimensional intersection (orange line) of the approximate
subspace with the plane spanned by spanned by $\{X Y, Y^2\}$. This subspace is spanned by the polynomial
$Y ( \alpha X + \beta Y )$, so we can divide by the variable $Y$.
}
\label{fig:alg_approach}
\end{figure}
Figure~\ref{fig:alg_approach} illustrates how the algebraic algorithm works in the vector space of coefficients. The polynomials $\mathcal{Q} = \{ q_{ij} \}_{i,j=1}^n$ span a space of constraints which defines the set of solutions. The next step is to find a polynomial of a certain form that immediately reveals the solution. One of these sets is the linear subspace spanned by the monomials $\{X Y, Y^2\}$: any polynomial in this span is divisible by $Y$. Our goal is now to find a polynomial which is contained in both this subspace and the span of $\mathcal{Q}$. Under mild assumptions, one can always find a polynomial of this form, and it corresponds to an equation \begin{align} \label{eq:solution_span}
Y ( \alpha X + \beta Y ) = 0 . \end{align} Since this polynomial is in the span of $\mathcal{Q},$ our solution $v$ has to be a zero of this particular polynomial: $v_2 ( \alpha v_1 + \beta v_2) = 0$. Moreover, we can assume\footnote{This is a consequence of the generative model for the observed polynomials which is introduced in Section~\ref{sec:alg_prob-poly}. In essence, we use the fact that our polynomials have no special property (apart from the existence of a solution) with probability one. } that $v_2 \neq 0$, so that we can divide out the variable $Y$ to get the linear factor $( \alpha X + \beta Y )$, \begin{align*}
0 = \alpha X + \beta Y = \begin{bmatrix} \alpha & \beta \end{bmatrix} v . \end{align*} Hence $v = \begin{bmatrix} -\beta & \alpha \end{bmatrix}^\top$ is the solution up to arbitrary scaling, which corresponds to the one-dimensional subspace in Figure~\ref{fig:alg_approach} (orange line, right panel). A more detailed treatment of this example can also be found in Appendix~\ref{app-example}.
In the case where there exists a direction $v$ on which the projected covariances are exactly equal, the linear subspace spanned by the set of polynomials $\mathcal{Q}$ has dimension two, which corresponds to the degrees of freedom of possible covariance matrices that have fixed projection on one direction. However, since in practice covariance matrices are estimated from finite and noisy samples, the polynomials $\mathcal{Q}$ usually span the whole space, which means that there exists only a trivial solution $v = 0$. This is the case for the polynomials pictured in the left panel of Figure~\ref{fig:alg_approach}. Thus, in order to obtain an approximate solution, we first determine the approximate two-dimensional span of $\mathcal{Q}$ using a standard least squares method as illustrated in the middle panel. We can then find the intersection of the approximate two-dimensional span of $\mathcal{Q}$ with the plane spanned by the monomials $\{X Y, Y^2\}$. As we have seen in Equation~\ref{eq:solution_span}, the polynomials in this span provide us with a unique solution for $v$ up to scaling, corresponding to the fact that the intersection has dimension one (see the right panel of Figure~\ref{fig:alg_approach}). Alternatively, we could have found the one-dimensional intersection with the span of $\{ X Y, X^2 \}$ and divided out the variable $X$. In fact, in the final algorithm we will find all such intersections and combine the solutions in order to increase the accuracy. Note that we have found this solution by solving a simple least-squares problem (second step, middle panel of Figure~\ref{fig:alg_approach}). In contrast, the optimization approach (Figure~\ref{fig:ml_optim}) can require a large number of iterations and may converge to a local minimum. A more detailed example of the algebraic algorithm can be found in Appendix~\ref{app-example}.
\begin{figure}
\caption{
The left panel shows two sample covariance matrices in the plane, along with
a direction on which they are equal. In the right panel, a third (green) covariance
matrix does not have the same projected variance on the black direction.
}
\label{fig:spurious_example}
\end{figure}
The algebraic framework does not only allow us to construct efficient algorithms for working with probability distributions, it also offers powerful tools to prove properties of algorithms that operate with cumulants. For example, we can answer the following central question: how many distinct data sets do we need such that the subspace with identical probability distributions becomes uniquely identifiable? This depends on the number of dimensions and the cumulants that we consider. Figure~\ref{fig:spurious_example} illustrates the case where we are given only the second order moment in two dimensions. Unless $\Sigma_1 - \Sigma_2$ is indefinite, there \textit{always} exists a direction on which two covariance matrices in two dimensions are equal (left panel of Figure~\ref{fig:spurious_example}) --- irrespective of whether the probability distributions are actually equal. We therefore need at least three covariance matrices (see right panel), or to consider other cumulants as well. We derive a tight criterion on the necessary number of data sets depending on the dimension and the cumulants under consideration. The proof hinges on viewing the cumulants as polynomials in the algebraic geometry framework: the polynomials that define the sought-after projection (e.g.~Equations~\ref{eq:homo_simple_example}) generate an ideal in the polynomial ring which corresponds to an algebraic set that contains all possible solutions. We can then show how many independent polynomials are necessary so that the dimension of the linear part of the algebraic set has smaller dimension in the generic case. We conjecture that these proof techniques are also applicable to other scenarios where we aim to identify a property of a probability distribution from its cumulants using algebraic methods.
Our work is not the first that applies geometric or algebraic methods to Machine Learning or statistics: for example, methods from group theory have already found their application in machine learning, e.g.~\cite{Ris07Skew,RisBor08Skew}; there are also algebraic methods estimating structured manifold models for data points as in \cite{GPCA05} which are strongly related to polynomial kernel PCA --- a method which can itself be interpreted as a way of finding an approximate vanishing set.
The field of Information Geometry interprets parameter spaces of probability distributions as differentiable manifolds and studies them from an information-theoretical point of view (see for example the standard book by \cite{Ama00}), with recent interpretations and improvements stemming from the field of algebraic geometry by \cite{Wat09}. There is also the nascent field of algebraic statistics which studies the parameter spaces of mainly discrete random variables in terms of commutative algebra and algebraic geometry, see the recent overviews by \cite[chapter 8]{Stu02} and \cite{Stu10} or the book by \cite{Gib10} which also focuses on the interplay between information geometry and algebraic statistics. These approaches have in common that the algebraic and geometric concepts arise naturally when considering distributions in parameter space.
Given samples from a probability distribution, we may also consider algebraic structures in the data space. Since the data are uncertain, the algebraic objects will also come with an inherent uncertainty, unlike the exact manifolds in the case when we have an a-priori family of probability distributions. Coping with uncertainties is one of the main interests of the emerging fields of approximative and numerical commutative algebra, see the book by \cite{Ste04} for an overview on numerical methods in algebra, and \citep{KrePouRob09} for recent developments in approximate techniques on noisy data. There exists a wide range of methods; however, to our knowledge, the link between approximate algebra and the representation of probability distributions in terms of their cumulants has not been studied yet.
The remainder of this paper is organized as follows: in the next Section~\ref{sec:alg_prob}, we introduce the algebraic view of probability distribution, rephrase our problem in terms of this framework and investigate its identifiability. The algorithm for the exact case is presented in Section~\ref{sec:exact}, followed by the approximate version in Section~\ref{sec:approx}. The results of our numerical simulations and a comparison against Stationary Subspace Analysis (SSA)~\citep{PRL:SSA:2009} can be found in Section~\ref{sec:sims}. In the last Section~\ref{sec:concl}, we discuss our findings and point to future directions. The appendix contains an example and proof details.
\section{The Algebraic View on Probability Distributions} \label{sec:alg_prob}
In this section we introduce the algebraic framework for dealing with probability distributions. This requires basic concepts from complex algebraic geometry. A comprehensive introduction to algebraic geometry with a view to computation can be found in the book~\citep{Cox}. In particular, we recommend to go through the Chapters~1 and 4.
In this section, we demonstrate the algebraic viewpoint of probability distributions on the application that we study in this paper: finding the linear subspace on which probability distributions are equal. \begin{Prob}\label{Prob:orig} Let $X_1, \ldots, X_{m}$ be a set of $D$-variate random variables, having smooth densities. Find all linear maps $P\in \R^{d\times D}$ such that the transformed random variables have the same distribution, $$P X_1 \sim \cdots \sim P X_m .$$ \end{Prob} In the first part of this section, we show how this problem can be formulated algebraically. We will first of all review the relationship between the probability density function and its cumulants, before we translate the cumulants into algebraic objects. Then we introduce the theoretical underpinnings for the statistical treatment of polynomials arising from estimated cumulants and prove conditions on identifiability for the problem addressed in this paper.
\subsection{From Probability Distributions to Polynomials} \label{sec:alg_prob-poly} The probability distribution of every smooth real random variable $X$ can be fully characterized in terms of its \textit{cumulants}, which are the tensor coefficients of the cumulant generating function. This representation has the advantage that each cumulant provides a compact description of certain aspects of the probability density function. \begin{Def} Let $X$ be a $D$-variate random variable. Then by $\cumu_n(X)\in \R^{D^{(\times n)}}$ we denote the $n$-th cumulant, which is a real tensor of degree $n$. \end{Def}
Let us introduce a useful shorthand notation for linearly transforming tensors. \begin{Def} Let $A\in \CC^{d\times D}$ be a matrix. For a tensor $T\in \R^{D^{(\times n)}}$ (i.e.~a real tensor $T$ of degree $n$ of dimension $D^n=D\cdot D\cdot \ldots\cdot D$) we will denote by $A\circ T$ the application of $A$ to $T$ along all tensor dimensions, i.e. $$\left(A\circ T\right)_{i_1\dots i_n}=\sum_{j_1=1}^D\dots \sum_{j_n=1}^D A_{i_{1}j_{1}}\cdot\ldots\cdot A_{i_{n}j_{n}}T_{j_1\dots j_n}.$$ \end{Def}
The cumulants of a linearly transformed random variable are the multilinearly transformed cumulants, which is a convenient property when one is looking for a certain linear subspace. \begin{Prop} Let $X$ be a real $D$-dimensional random variable and let $A \in \R^{d \times D}$ be a matrix. Then the cumulants of the transformed random variable $A X$ are the transformed cumulants, \begin{align*}
\cumu_n(A X) = A \circ \cumu_n(X). \end{align*} \end{Prop}
We now want to formulate our problem in terms of cumulants. First of all, note that $P X_i \sim P X_j$ if and only if $v X_i\sim v X_j$ for all row vectors $v\in \lspan P^\top.$ \begin{Prob} Find all $d$-dimensional linear subspaces in the set of vectors \begin{align*}
S & = \{ v \in \R^D \; \left| \; v^\top X_1 \sim \cdots \sim v^\top X_m \} \right. \\
& = \{v\in \R^D \; \left| \; v^\top \circ \cumu_n (X_i)=v^\top \circ \cumu_n (X_j),\; n\in\N, 1 \le i,j\le m\} \right. . \end{align*} \end{Prob}
Note that we are looking for linear subspaces in $S$, but that $S$ itself is not a vector space in general. Apart from the fact that $S$ is homogeneous, i.e.~$\lambda S = S$ for all $\lambda \in \R$, there is no additional structure that we utilize.
For the sake of clarity, in the remainder of this paper we restrict ourselves to the first two cumulants. Note, however, that one of the strengths of the algebraic framework is that the generalization to arbitrary degree is straightforward; throughout this paper, we indicate the necessary changes and differences. Thus, from now on, we denote the first two cumulants by $\mu_i=\cumu_1(X_i)$ and $\Sigma_i=\cumu_2(X_i)$ respectively for all $1 \leq i \leq m$. Moreover, without loss of generality, we can shift the mean vectors and choose a basis such that the random variable $X_m$ has zero mean and unit covariance. Thus we arrive at the following formulation. \begin{Prob} Find all $d$-dimensional linear subspaces in
$$S=\{v\in \R^D \; | \; v^\top (\Sigma_i-I)v=0,\; v^\top \mu_i=0 , \; 1 \le i \le (m-1)\}.$$ \end{Prob} Note that $S$ is the set of solutions to $m-1$ quadratic and $m-1$ linear equations in $D$ variables. Now it is only a formal step to arrive in the framework of algebraic geometry: let us think of the left hand side of each of the quadratic and linear equations as polynomials $q_1, \ldots, q_{m-1}$ and $f_1, \ldots, f_{m-1}$ in the variables $T_1, \ldots, T_D$ respectively, \begin{align*}
q_i = \begin{bmatrix} T_1 \cdots T_D \end{bmatrix} \circ ( \Sigma_i - I )
\hspace{0.3cm} \text{ and } \hspace{0.3cm} f_i = \begin{bmatrix} T_1 \cdots T_D \end{bmatrix} \circ \mu_i, \end{align*} which are elements of the polynomial ring over the complex numbers in $D$ variables, $\C[T_1,\dots, T_D]$. Note that in the introduction we have used $X$ and $Y$ to denote the variables in the polynomials, we will now switch to $T_1, \ldots, T_D$ in order to avoid confusion with random variables. Thus $S$ can be rewritten in terms of polynomials, \begin{align*}
S = \left\{ v \in \R^D \; \left| \; q_i(v) = f_i(v) = 0 \; \forall \, 1 \leq i \leq m-1 \right\} \right., \end{align*} which means that $S$ is an algebraic set. In the following, we will consider the corresponding complex vanishing set \begin{align*}
S &=\VS(q_1, \ldots, q_{m-1}, f_1,\dots, f_{m-1})\\
&:= \left\{ v \in \C^D \; \left| \; q_i(v) = f_i(v) = 0 \; \forall \, 1 \leq i \leq m-1 \right\} \subseteq \C^D \right. \end{align*} and keep in mind that eventually we will be interested in the real part of $S$. Working over the complex numbers simplifies the theory and creates no algorithmic difficulties: when we start with real cumulant polynomials, the solution will always be real. Finally, we can translate our problem into the language of algebraic geometry. \begin{Prob}\label{Prob:Alg} Find all $d$-dimensional linear subspaces in the algebraic set $$S = \VS(q_1, \ldots, q_{m-1}, f_1,\dots, f_{m-1}).$$ \end{Prob}
So far, this problem formulation does not include the assumption that a solution exists. In order to prove properties about the problem and algorithms for solving it we need to assume that there exist a $d$-dimensional linear subspace $S' \subset S$. That is, we need to formulate a \textit{generative model} for our observed polynomials $q_1, \ldots, q_{m-1}, f_1,\dots, f_{m-1}$. To that end, we introduce the concept of a \textit{generic} polynomial, for a technical definition see Appendix~B. Intuitively, a generic polynomial is a continuous, polynomial valued random variable which almost surely has no algebraic properties except for those that are logically implied by the conditions on it. An algebraic property is an event in the probability space of polynomials which is defined by the common vanishing of a set of polynomial equations in the coefficients. For example, the property that a quadratic polynomial is a square of linear polynomial is an algebraic property, since it is described by the vanishing of the discriminants. In the context of Problem~\ref{Prob:Alg}, we will consider the observed polynomials as generic conditioned on the algebraic property that they vanish on a fixed $d$-dimensional linear subspace $S'$.
One way to obtain generic polynomials is to replace coefficients with e.g.~Gaussian random variables. For example, a generic homogeneous quadric $q \in \C[T_1, T_2]$ is given by \begin{align*}
q = Z_{11} T^2_1 + Z_{12} T_1 T_2 + Z_{22} T^2_2, \end{align*} where the coefficients $Z_{ij} \sim \Gauss(\mu_{ij}, \sigma_{ij})$ are independent Gaussian random variables with arbitrary parameters. Apart from being homogeneous, there is no condition on $q$. If we want to add the condition that $q$ vanishes on the linear space defined by $T_1 = 0$, we would instead consider \begin{align*}
q = Z_{11} T^2_1 + Z_{12} T_1 T_2 . \end{align*} A more detailed treatment of the concept of genericity, how it is linked to probabilistic sampling, and a comparison with the classical definitions of genericity can be found in Appendix~\ref{sec:gendef}.
We are now ready to reformulate the genericity conditions on the random variables $X_1,\dots, X_m$ in the above framework. Namely, we have assumed that the $X_i$ are general under the condition that they agree in the first two cumulants when projected onto some linear subspace $S'$. Rephrased for the cumulants, Problems~\ref{Prob:orig} and \ref{Prob:Alg} become well-posed and can be formulated as follows. \begin{Prob}\label{Prob:Alg_gen} Let $S'$ be an unknown $d$-dimensional linear subspace in $\C^D$. Assume that $f_1,\dots, f_{m-1}$ are generic homogenous linear polynomials, and $q_1,\dots, q_{m-1}$ are generic homogenous quadratic polynomials, all vanishing on $S'.$ Find all $d$-dimensional linear subspaces in the algebraic set $$S=\VS(q_1, \ldots, q_{m-1}, f_1,\dots, f_{m-1}).$$ \end{Prob} As we have defined ``generic'' as an implicit ``almost sure'' statement, we are in fact looking for an algorithm which gives the correct answer with probability one under our model assumptions. Intuitively, $S'$ should be also the only $d$-dimensional linear subspace in $S$, which is not immediately guaranteed from the problem description. Indeed this is true if $m$ is large enough, which is the topic of the next section.
\subsection{Identifiability} \label{sec:alg_prob-ident} In the last subsection, we have seen how to reformulate our initial Problem~\ref{Prob:orig} about comparison of cumulants as the completely algebraic Problem~\ref{Prob:Alg_gen}. We can also reformulate identifiability of the true solution in the original problem in an algebraic way: identifiability in Problem~\ref{Prob:orig} means that the projection $P$ can be uniquely computed from the probability distributions. Following the same reasoning we used to arrive at the algebraic formulation in Problem~\ref{Prob:Alg_gen}, one concludes that identifiability is equivalent to the fact that there exists a unique linear subspace in $S$.
Since identifiability is now a completely algebraic statement, it can be treated also in algebraic terms. In Appendix~\ref{app-generic}, we give an algebraic geometric criterion for identifiability of the stationary subspace; we will sketch its derivation in the following.
The main ingredient is the fact that, intuitively spoken, every generic polynomials carries one degree of freedom in terms of dimension, as for example the following result on generic vector spaces shows:
\begin{Prop}\label{Prop:GenVec-main} Let $\calP$ be an algebraic property such that the polynomials with property $\mathcal{P}$ form a vector space $V$. Let $f_1,\dots, f_n\in \C[T_1,\dots T_D]$ be generic polynomials satisfying $\mathcal{P}$. Then $$\rk \lspan (f_1,\dots, f_n)=\min (n, \dim V).$$ \end{Prop} \begin{proof} This is Proposition \ref{Prop:GenVec} in the appendix. \end{proof}
On the other hand, if the polynomials act as constraints, one can prove that each one reduces the degrees of freedom in the solution by one:
\begin{Prop}\label{Thm:genintcont} Let $Z$ be a sub-vector space of $\C^D$. Let $f_1,\dots, f_n$ be generic homogenous polynomials in $D$ variables (of fixed, but arbitrary degree each), vanishing on $Z$. Then for their common vanishing set $\VS (f_1,\dots, f_n)=\{x\in\C^D\mid f_i(x)=0\;\forall i\}$, one can write $$\VS (f_1,\dots, f_n) = Z \cup U,$$ where $U$ is an algebraic set with $$\dim U\le \max (D-n,\; 0).$$ \end{Prop} \begin{proof} This follows from Corollary \ref{Cor:KrullHt-Hom} in the appendix. \end{proof} Proposition \ref{Thm:genintcont} can now be directly applied to Problem~\ref{Prob:Alg_gen}. It implies that $S=S'$ if $2(m-1)\ge D+1$, and that $S'$ is the maximal dimensional component of $S$ if $2(m-1) \ge D-d+1$. That is, if we start with $m$ random variables, then $S'$ can be identified uniquely if $$2(m-1) \ge D-d+1$$ with classical algorithms from computational algebraic geometry in the noiseless case. \begin{Thm}\label{Thm:ident} Let $X_1,\dots, X_m$ be random variables. Assume there exists a projection $P\in \R^{d\times D}$ such that the first two cumulants of all $P X_1, \ldots, P X_m$ agree and the cumulants are generic under those conditions. Then the projection $P$ is identifiable from the first two cumulants alone if $$m \ge \frac{D-d+1}{2}+1.$$ \end{Thm} \begin{proof} This is a direct consequence of Proposition \ref{Prop:ident_ex} in the appendix, applied to the reformulation given in Problem \ref{Prob:Alg_gen}. It is obtained by applying Proposition \ref{Thm:genintcont} to the generic forms vanishing on the fixed linear subspace $S'$, and using that $S'$ can be identified in $S$ if it is the biggest dimensional part. \end{proof} We have seen that identifiability means that there is an algorithm to compute $P$ uniquely when the cumulants are known, resp.~to compute a unique $S$ from the polynomials $f_i,q_i$. It is not difficult to see that an algorithm doing this can be made into a consistent estimator when the cumulants are sample estimates. We will give an algorithm of this type in the following parts of the paper.
\section{An Algorithm for the Exact Case} \label{sec:exact} In this section we present an algorithm for solving Problem~\ref{Prob:Alg_gen}, under the assumption that the cumulants are known exactly. We will first fix notation and introduce important algebraic concepts. In the previous section, we derived in Problem~\ref{Prob:Alg_gen} an algebraic formulation of our task: given generic quadratic polynomials $q_1,\dots, q_{m-1}$ and linear polynomials $f_1,\dots, f_{m-1}$, vanishing on a unknown linear subspace $S'$ of $\C^D$, find $S'$ as the unique $d$-dimensional linear subspace in the algebraic set $\VS(q_1, \ldots, q_{m-1}, f_1,\dots, f_{m-1})$. First of all, note that the linear equations $f_i$ can easily be removed from the problem: instead of looking at $\C^D$, we can consider the linear subspace defined by the $f_i$, and examine the algebraic set $\VS(q'_1, \ldots, q'_{m-1}),$ where $q'_i$ are polynomials in $D-m+1$ variables which we obtain by substituting ${m-1}$ variables. So the problem we need to examine is in fact the modified problem where we have only quadratic polynomials. Secondly, we will assume that ${m-1}\ge D$. Then, from Proposition~\ref{Thm:genintcont}, we know that $S=S'$ and Problem~\ref{Prob:Alg_gen} becomes the following. \begin{Prob}\label{Prob:Alg_SSA_naive} Let $S$ be an unknown $d$-dimensional subspace of $\C^D$. Given ${m-1} \geq D$ generic homogenous quadratic polynomials $q_1,\dots, q_{m-1}$ vanishing on $S$, find the $d$-dimensional linear subspace $$S=\VS(q_1, \ldots, q_{m-1}).$$ \end{Prob} Of course, we have to say what we mean by \textit{finding} the solution. By assumption, the quadratic polynomials already fully describe the linear space $S$. However, since $S$ is a linear space, we want a basis for $S$, consisting of $d$ linearly independent vectors in $\C^D$. Or, equivalently, we want to find linearly independent linear forms $\ls_1,\dots, \ls_{D-d}$ such that $\ell_i(x)=0$ for all $x\in S.$ The latter is the correct description of the solution in algebraic terms. We now show how to reformulate this in the right language, following the algebra-geometry duality. The algebraic set $S$ corresponds to an ideal in the polynomial ring $C[T_1,\dots, T_D]$. \begin{Not} We denote the polynomial ring $\C[T_1,\dots, T_D]$ by $R$. The ideal of $S$ is an ideal in $R$, and we denote it by by $\idS=\Id (S).$ Since $S$ is a linear space, there exists a linear generating set $\ls_1, \ldots, \ls_{D-d}$ of $\idS$ which we will fix in the following. \end{Not} We can now relate the Problem~\ref{Prob:Alg_SSA_naive} to a classical problem in algebraic geometry. \begin{Prob}\label{Prob:Alg_SSA} Let $m > D$ and $q_1,\dots, q_{m-1}$ be generic homogenous quadratic polynomials vanishing on a linear $d$-dimensional subspace $S\subseteq \C^D$. Then find a linear basis for the radical ideal \begin{align*}
\sqrt{\langle q_1, \ldots, q_{m-1}\rangle} = \Id(\VS(q_1, \ldots, q_{m-1}))=\Id (S). \end{align*} \end{Prob} The first equality follows from Hilbert's Nullstellensatz. This also shows that solving the problem is in fact a question of computing a radical of an ideal. Computing the radical of an ideal is a classical problem in computational algebraic geometry, which is known to be difficult (for a more detailed discussion see Section~\ref{sec:exact-prwork}). However, if we assume ${m-1}\ge D(D+1)/2 - d(d+1)/2$, we can dramatically reduce the computational cost and it is straightforward to derive an approximate solution. In this case, the $q_i$ generate the vector space of homogenous quadratic polynomials which vanish on $S$, which we will denote by $\idS_2.$ That this is indeed the case, follows from Proposition \ref{Prop:GenVec-main}, and we have $\dim \idS_2 = D(D+1)/2 - d(d+1)/2,$ as we will calculate in Remark \ref{Rem:gensm}.
Before we continue with solving the problem, we will need to introduce several concepts and abbreviating notations. First we introduce notation to denote sub-vector spaces which contain polynomials of certain degrees. \begin{Not} Let $\mathcal{I}$ be a sub-$\mathbb{C}$-vector space of $R$, i.e.~$\mathcal{I}=R$, or $\mathcal{I}$ is some ideal of $R$, e.g.~$\mathcal{I}=\idS.$ We denote the sub-$\mathbb{C}$-vector space of homogenous polynomials of degree $k$ in $\mathcal{I}$ by $\mathcal{I}_k$ (in commutative algebra, this is standard notation for homogenously generated $R$-modules). \end{Not} For example, the homogenous polynomials of degree $2$ vanishing on $S$ form exactly the vector space $\idS_2$. Moreover, for any $\mathcal{I}$, the equation $\mathcal{I}_k=\mathcal{I}\cap R_k$ holds. The vector spaces $R_2$ and $\idS_2$ will be the central objects in the following chapters. As we have seen, their dimension is given in terms of triangular numbers, for which we introduce some notation:
\begin{Not} We will denote the $n$-th triangular number by $\Delta (n) = \frac{n(n+1)}{2}$. \end{Not}
The last notational ingredient will capture the structure which is imposed on $R_k$ by the orthogonal decomposition $\C^D = S \oplus S^\perp.$ \begin{Not} Let $S^\perp$ be the orthogonal complement of $S$. Denote its ideal by $\idN = \Id\left(S^\perp\right)$. \end{Not} \begin{Rem}\label{Rem:calcidS} As $\idN$ and $\idS$ are homogenously generated in degree one, we have the calculation rules \begin{align*} \idS_{k+1}=\idS_k\cdot R_1\quad\mbox{and}\quad \idN_{k+1}=\idN_k\cdot R_1,\\ (\idS_1)^k=(\idS^k)_k\quad\mbox{and}\quad (\idN_1)^k=(\idN^k)_k \end{align*} where $\cdot$ is the symmetrized tensor or outer product of vector spaces (these rules are canonically induced by the so-called graded structure of $R$-modules). In terms of ideals, the above decomposition translates to $$R_1=\idS_1\oplus \idN_1.$$ Using the above rules and the binomial formula for ideals, this induces an orthogonal decomposition \begin{align*} R_2=&R_1\cdot R_1=(\idS_1\oplus \idN_1)\cdot (\idS_1\oplus \idN_1)= (\idS_1)^2\oplus (\idS_1\cdot\idN_1) \oplus (\idN_1)^2\\ & = \idS_1\cdot(\idS_1\oplus\idN_1) \oplus (\idN^2)_2 = \idS_1\cdot R_1 \oplus (\idN^2)_2 = \idS_2 \oplus (\idN^2)_2 \end{align*} (and similar decompositions for the higher degree polynomials $R_k$). \end{Rem}
The tensor products above can be directly translated to products of ideals, as the vector spaces above are each generated in a single degree (e.g.~$\idS^k, \idN^k$, are generated homogenously in degree $k$). To express this, we will define an ideal which corresponds to $R_1$: \begin{Not} We denote the ideal of $R$ generated by all monomials of degree $1$ by $\idM = \langle T_1,\dots, T_D \rangle $. \end{Not} Note that ideal $\idM$ is generated by all elements in $R_1.$ Moreover, we have $\idM_k=R_k$ for all $k\ge 1$. Using $\idM$, one can directly translate products of vector spaces involving some $R_k$ into products of ideals: \begin{Rem}\label{Rem:calcidM} The equality of vector spaces $$\idS_{k}=\idS_1\cdot (R_1)^{k-1}$$ translates to the equality of ideals $$\idS\cap \idM^k= \idS\cdot \idM^{k-1},$$ since both the left and right sides are homogenously generated in degree $k$. \end{Rem}
\subsection{The Algorithm} \begin{table}[h] \begin{center}
\begin{tabular}{l|l}
$S \subset \C^D$ & $d$-dimensional projection space \\
$R=\C [T_1,\dots T_D]$ & Polynomial ring over $\C$ in $D$ variables \\
$R_k$ & $\C$-vector space of homogenous $k$-forms in $T_1, \ldots, T_D$\\
$\Delta (n) =\frac{n(n+1)}{2}$ & $n$-th triangular number \\
$\idS=\langle\ls_1,\dots, \ls_{D-d}\rangle = \Id(S) $ & The ideal of $S$, generated by linear polynomials $\ls_i$\\
$\idS_k = R_k\cap \idS $ & $\C$-vector space of homogenous $k$-forms vanishing on $S$\\
$\idN = \Id(S^\perp)$ & The ideal of $S^\perp$\\
$\idN_k = R_k\cap \idN $ & $\C$-vector space of homogenous $k$-forms vanishing on $S^\perp$\\
$\idM=\langle T_1,\dots, T_D\rangle$ & The ideal of the origin in $\C^D$ \end{tabular} \caption{
Notation and important definitions
\label{table:notation} } \end{center} \end{table} In this section we present an algorithm for solving Problem~\ref{Prob:Alg_SSA}, the computation of the radical of the ideal ${\langle q_1,\dots, q_{m-1}\rangle}$ under the assumption that $$m\ge \Delta(D)-\Delta(d)+1.$$ Under those conditions, as we will prove in Remark \ref{Rem:gensm} (iii), we have that $$ \langle q_1,\dots, q_{m-1}\rangle =\idS_2.$$ Using the notations previously defined, one can therefore infer that solving Problem~\ref{Prob:Alg_SSA} is equivalent to computing the radical $\idS=\sqrt{\idS\cdot \idM}$ in the sense of obtaining a linear generating set for $\idS$, or equivalent to finding a basis for $\idS_1$ when $\idS_2$ is given in an arbitrary basis. $\idS_2$ contains the complete information given by the covariance matrices and $\idS_1$ gives an explicit linear description of the space of projections under which the random variables $X_1, \ldots, X_m$ agree.
\begin{algorithm}[h] \caption{\label{alg:exact_covonly} The \textit{input} consists of the quadratic forms $q_1, \ldots, q_{m-1} \in R$, generating $\idS_2,$ and the dimension $d$; the \textit{output} is the linear generating set $\ls_1, \ldots, \ls_{D-d}$ for $\idS_1$.
}
\begin{algorithmic}[1]
\State Let $\pi \gets (1 \, 2 \, \cdots \, D)$ be a transitive permutation of the variable indices $\{ 1, \ldots, D \}$
\State Let $Q \gets \begin{bmatrix} q_1 & \cdots & q_{m-1} \end{bmatrix}^\top$ be the $((m-1) \times \nD)$-matrix
of coefficient vectors, where every row corresponds to a polynomial and every column to a monomial $T_i T_j$.
\For {$k=1, \ldots, D-d$} \label{al:for}
\State
\begin{minipage}[t]{14cm}
Order the columns of $Q$ according to the lexicographical ordering of monomials $T_i T_j$
with variable indices permuted by $\pi^k$, i.e.~the ordering of the columns is given
by the relation $\succ$ as
\begin{align*}
T_{\pi^k(1)}^2 & \succ T_{\pi^k(1)}T_{\pi^k(2)} \succ
T_{\pi^k(1)}T_{\pi^k(3)}\succ \dots \succ T_{\pi^k(1)}T_{\pi^k(D)} \succ
T_{\pi^k(2)}^2\\
&\succ T_{\pi^k(2)}T_{\pi^k(3)}\succ \dots \succ T_{\pi^k(D-1)}^2 \succ
T_{\pi^k(D-1)}T_{\pi^k(D)}\succ T_{\pi^k(D)}^2
\end{align*}
\end{minipage}
\State Transform $Q$ into upper triangular form $Q'$ using Gaussian elimination
\State
\begin{minipage}[t]{14cm}
The last non-zero row of $Q'$ is a polynomial $T_{\pi^k(D)} \ell$, where $\ell$
is a linear form in $\idS$, and we set $\ls_k \gets \ell$\label{alg:exact-crux}
\end{minipage}
\EndFor \end{algorithmic} \end{algorithm}
Algorithm~\ref{alg:exact_covonly} shows the procedure in pseudo-code; a summary of the notation defined in the previous section can be found in Table~\ref{table:notation}. The algorithm has polynomial complexity in the dimension $d$ of the linear subspace $S$. \begin{Rem}\label{Rem:compAlgI} Algorithm~\ref{alg:exact_covonly} has average and worst case complexity \begin{align*}
O\left( (\nD-\nd)^2\nD \right), \end{align*} In particular, if $d$ is not considered as parameter of the algorithm, the average and the worst case complexity is $O(D^6).$ On the other hand, if $\nD-\nd$ is considered a fixed parameter, then Algorithm 1 has average and worst case complexity $O(D^2).$ \end{Rem} \begin{proof} This follows from the complexities of the elementary operations: upper triangularization of a generic matrix of rank $r$ with $m$ columns matrix needs $O(r^2m)$ operations. We first perform triangularization of a rank $\nD-\nd$ matrix with $\nD$ columns. The permutations can be obtained efficiently by bringing $Q$ in row-echelon form and then performing row operations. Operations for extracting the linear forms and comparisons with respect to the monomial ordering are negligible. Thus the overall operation complexity to calculate $\fraks_1$ is $O((\nD-\nd)^2 \nD).$
Note that the difference between worst- and average case lies at most in the coefficients, since the inputs are generic and the complexity only depends on the parameter $D$ and not on the $q_i$. Thus, with probability $1,$ exactly the worst-case-complexity is attained. \end{proof}
There are two crucial facts which need to be verified for correctness of this algorithm. Namely, there are implicit claims made in Line~\ref{alg:exact-crux} of Algorithm~\ref{alg:exact_covonly}: First, it is claimed that the last non-zero row of $Q'$ corresponds to a polynomial which factors into certain linear forms. Second, it is claimed that the $\ell$ obtained in step~\ref{alg:exact-crux} generate $\fraks$ resp.~$\fraks_1$. The proofs of these non-trivial claims can be found in Proposition~\ref{Prop:Alg1corr} in the next subsection.
Dealing with additional linear forms $f_1,\dots, f_{m-1},$ is possible by way of a slight modification of the algorithm. Because the $f_i$ are linear forms, they are generators of $\fraks.$ We may assume that the $f_i$ are linearly independent. By performing Gaussian elimination before the execution of Algorithm~\ref{alg:exact_covonly}, we may reduce the number of variables by ${m-1}$, thus having to deal with new quadratic forms in $D-m+1$ instead of $D$ variables. Also, the dimension of the space of projections is reduced to $\min(d-m+1, -1).$ Setting $D'=D-m+1$ and $d'=\min (d-m+1,-1)$ and considering the quadratic forms $q_i$ with Gaussian eliminated variables, Algorithm~\ref{alg:exact_covonly} can be applied to the quadratic forms to find the remaining generators for $\idS_1.$ In particular, if $m-1\ge d,$ then there is no need for considering the quadratic forms, since $d$ linearly independent linear forms already suffice to determine the solution.
We can also incorporate forms of higher degree corresponding to higher order cumulants. For this, we start with $\idS_k,$ where $k$ is the degree of the homogenous polynomials we get from the cumulant tensors of higher degree. Supposing we start with enough cumulants, we may assume that we have a basis of $\idS_k.$ Performing Gaussian elimination on this basis with respect to the lexicographical order, we obtain in the last row a form of type $T_{\pi^k(D)}^{k-1}\ell,$ where $\ell$ is a linear form. Doing this for $D-d$ permutations again yields a basis for $\idS_1.$
Moreover, slight algebraic modifications of this strategy also allow to consider data from cumulants of different degree simultaneously, and to reduce the number of needed polynomials to $O(D)$; however, due to its technicality, this is beyond the scope of the paper. We sketch the idea: In the general case, one starts with an ideal $$\calI=\langle f_1,\dots, f_m\rangle,$$ homogenously generated in arbitrary degrees. such that $\sqrt{\calI}=\fraks.$ Proposition~\label{Prop:dehom-rad-generic} in the appendix implies that this happens whenever $m\ge D+1.$ One then proves that due to the genericity of the $f_i,$ there exists an $N$ such that $$\calI_N=\fraks_N,$$ which means that $\fraks_1$ can again be obtained by calculating the saturation of the ideal $\calI$. When fixing the degrees of the $f_i$, we will have $N=O(D)$ with a relatively small constant (for all $f_i$ quadratic, this even becomes $N=O(\sqrt{D})$). So algorithmically, one would first calculate $\calI_N=\fraks_N,$ which then may be used to compute $\fraks_1$ and thus $\fraks$ analogously to the case $N=2$, as described above.
\subsection{Proof of correctness} \label{sec:exact-correct}
In order to prove the correctness of Algorithm~\ref{alg:exact_covonly}, we need to prove the following three statements. \begin{Prop}\label{Prop:Alg1corr} For Algorithm~\ref{alg:exact_covonly} it holds that\\ \itboxx{i} $Q$ is of rank $\nD-\nd$.\\ \itboxx{ii} The last column of $Q$ in step 6 is of the claimed form.\\ \itboxx{iii} The $\ls_1, \ldots, \ls_{D-d}$ generate $\idS_1$. \end{Prop} \begin{proof} This proposition will be proved successively in the following: (i) will follow from Remark~\ref{Rem:gensm} (iii); (ii) will be proved in Lemma~\ref{Lem:Xell}; and (iii) will be proved in Proposition~\ref{Prop:gens}. \end{proof} Let us first of all make some observations about the structure of the vector space $\idS_2$ in which we compute. It is the vector space of polynomials of homogenous degree $2$ vanishing on $S$. On the other hand, we are looking for a basis $\ls_1,\dots, \ls_{D-d}$ of $\idS_1$. The following remark will relate both vector spaces: \begin{Rem}\label{Rem:gensm} The following statements hold:\\ \itboxx{i} $\idS_2$ is generated by the polynomials $\ls_iT_j, 1\le i\le D-d, 1\le j\le D, .$ \\ \itboxx{ii} $\dim_\C \idS_2=\nD-\nd$\\ \itboxx{iii} Let $q_1, \ldots, q_m$ with $m\ge \nD-\nd$ be generic homogenous quadratic polynomials in $\fraks$. Then $\langle q_1,\dots,q_m \rangle=\idS_2.$ \end{Rem} \begin{proof} (i)~In Remark~\ref{Rem:calcidS}, we have concluded that $\idS_2=\idS_1\cdot R_1.$ Thus the product vector space $\idS_2$ is generated by a product basis of $\idS_1$ and $R_1$. Since $T_j,1\le j\le D$ is a basis for $R_1$, and $\ls_i,1\le i\le D-d$ is a basis for $\idS_1$, the statement holds. (ii)~In Remark~\ref{Rem:calcidM}, we have seen that $R_2=\idS_2\oplus (\idN_1)^2,$ thus $\dim \idS_2= \dim R_2 - \dim (\idN_1)^2.$ The vector space $R_2$ is minimally generated by the monomials of degree $2$ in $T_1,\dots T_D$, whose number is $\nD$. Similarly, $(\idN_1)^2$ is minimally generated by the monomials of degree $2$ in the variables $\ls'_1,\dots, \ls'_d$ that form the dual basis to the $\ls_i$. Their number is $\nd$, so the statement follows. (iii)~As the $q_i$ are homogenous of degree two and vanish on $S$, they are elements in $\idS_2.$ Due to (ii), we can apply Proposition~\ref{Prop:GenVec-main} to conclude that they generate $\idS_2$ as vector space. \end{proof}
Now we continue to prove the remaining claims. \begin{Lem}\label{Lem:Xell} In Algorithm~\ref{alg:exact_covonly} the $(\nD-\nd)$-th row of $Q'$ (the upper triangular form of $Q$) corresponds to a $2$-form $T_{\pi(D)}\ell$ with a linear polynomial $\ell\in \idS_1$. \end{Lem} \begin{proof} Note that every homogenous polynomial of degree $k$ is canonically an element of the vector space $R_k$ in the monomial basis given by the $T_i$. Thus it makes sense to speak about the coefficients of $T_i$ for an $1$-form resp.~the coefficients of $T_iT_j$ of a $2$-form.
Also, without loss of generality, we can take the trivial permutation $\pi=\id$, since the proof
will not depend on the chosen lexicographical ordering and thus will be naturally invariant under permutations of variables. First we remark: since $S$ is a generic $d$-dimensional linear subspace of $\C^D$, any linear form in $\idS_1$ will have at least $d+1$ non-vanishing coefficients in the $T_i.$ On the other hand, by displaying the generators $\ls_i, 1\le i\le D-d$ in $\idS_1$ in reduced row echelon form with respect to the $T_i$-basis, one sees that one can choose all the $\ls_i$ in fact with exactly $d+1$ non-vanishing coefficients in the $T_i$ such that no nontrivial linear combination of the $\ls_i$ has less then $d+1$ non-vanishing coefficients. In particular, one can choose the $\ls_i$ such that the biggest (w.r.t.~the lexicographical order) monomial with non-vanishing coefficient of $\ls_i$ is $T_i$.
Remark~\ref{Rem:gensm} (i) states that $\idS_2$ is generated by $$\ls_iT_j, 1\le i\le D-d, 1\le j\le D.$$ Together with our above reasoning, this implies the following.
{\bf Fact 1:} There exist linear forms $\ls_i,1\le i\le D-d$ such that: the $2$-forms $\ls_iT_j$ generate $\idS_2,$ and the biggest monomial of $\ls_iT_j$ with non-vanishing coefficient under the lexicographical ordering is $T_iT_j.$ By Remark~\ref{Rem:gensm} (ii), the last row of the upper triangular form $Q'$ is a polynomial which has zero coefficients for all monomials possibly except the $\nd+1$ smallest, $$T_{D-d}T_D,T_{D-d+1}^2, T_{D-d+1}T_{D-d+2},\dots, T_{D-1}T_D,T_D^2.$$ On the other hand, it is guaranteed by our genericity assumption that the biggest of those terms is indeed non-vanishing, which implies the following.
{\bf Fact 2:} The biggest monomial of the last row with non-vanishing coefficient (w.r.t~the lexicographical order) is that of $T_{D-d}T_D.$
Combining Facts 1 and 2, we can now infer that the last row must be a scalar multiple of $\ls_{D-d}T_D$: since the last row corresponds to an element of $\idS_2,$ it must be a linear combination of the $\ls_iT_j.$ By Fact 1, every contribution of an $\ls_iT_j, (i,j)\neq (D-d,D)$ would add a non-vanishing coefficient lexicographically bigger than $T_{D-d}T_D$ which cannot cancel. So, by Fact 2, $T_D$ divides the last row of the upper triangular form of $Q,$ which then must be $T_D \ls_{D-d}$ or a multiple thereof. Also we have that $\ls_{D-d}\in\fraks$ by definition. \end{proof} It remains to be shown that by permutation of the variables we can find a basis for $\idS_1$. \begin{Prop}\label{Prop:gens} The $\ell_1,\dots,\ell_{D-d}$ generate $\idS_1$ as vector space and thus $\idS$ as ideal. \end{Prop} \begin{proof} Recall that $\pi^i$ was the permutation to obtain $\ell_i.$ As we have seen in the proof of Lemma~\ref{Lem:Xell}, $\ell_i$ is a linear form which has non-zero coefficients only for the $d+1$ coefficients $T_{\pi^i(D-d)},\dots, T_{\pi^i(D)}.$ Thus $\ell_i$ has a non-zero coefficient where all the $\ell_j,j<i$ have a zero coefficient, and thus $\ell_i$ is linearly independent from the $\ell_j,j<i.$ In particular, it follows that the $\ell_i$ are linearly independent in $R_1$. On the other hand, they are contained in the $D-d$-dimensional sub-$\C$-vector space $\idS_1$ and are thus a basis of $\idS_1$, and also a generating set for the ideal $\idS.$ \end{proof} Note that all of these proofs generalize to $k$-forms. For example, one calculates that $$\dim_\CC \idS_k= {D+k-1 \choose k} - {d+k-1 \choose k},$$ and the triangularization strategy yields a last row which corresponds to $T_{\pi(D)}^{k-1}\ell$ with a linear polynomial $\ell\in \idS_1$
\subsection{Relation to Previous Work in Computational Algebraic Geometry} \label{sec:exact-prwork}
In this section, we discuss how the algebraic formulation of the cumulant comparison problem given in Problem~\ref{Prob:Alg_SSA} relates to the classical problems in computational algebraic geometry.
Problem~\ref{Prob:Alg_SSA} confronts us with the following task: given polynomials $q_1,\dots, q_{m-1}$ with special properties, compute a linear generating set for the radical ideal $$\sqrt{\langle q_1, \ldots, q_{m-1}\rangle}=\Id(\VS (q_1,\dots, q_{m-1})).$$ Computing the radical of an ideal is a classical task in computational algebraic geometry, so our problem is a special case of radical computation of ideals, which in turn can be viewed as an instance of primary decomposition of ideals, see \cite[4.7]{Cox}.
While it has been known for long time that there exist constructive algorithms to calculate the radical of a given ideal in polynomial rings \cite{Her26}, only in the recent decades there have been algorithms feasible for implementation in modern computer algebra systems. The best known algorithms are those of \cite{GiaTraZac88Gro}, implemented in AXIOM and REDUCE, the algorithm of \cite{EisHunVas92Dir}, implemented in Macaulay 2, the algorithm of \cite{CabConCar97Yet}, currently implemented in CoCoA, and the algorithm of \cite{KriLog91Alg} and its modification by \cite{Lap06Alg}, available in SINGULAR.
All of these algorithms have two points in common. First of all, these algorithms have computational worst case complexities which are doubly exponential in the square of the number of variables of the given polynomial ring, see \cite[section 4.]{Lap06Alg}.
Although the worst case complexities may not be approached for the problem setting described in the current paper, these off-the-shelf algorithms do not take into account the specific properties of the ideals in question.
On the other hand, Algorithm~\ref{alg:exact_covonly} can be seen as a homogenous version of the well-known Buchberger algorithm to find a Groebner basis of the dehomogenization of $\fraks$ with respect to a degree-first order. Namely, due to our strong assumptions on $m$, or as is shown in Proposition~\ref{Prop:dehom-rad-generic} in the appendix for a more general case, the homogenous saturations of the ideal $\langle q_1,\dots, q_{m-1}\rangle = \frakm\cdot \fraks$ and the ideal $\fraks$ coincide. In particular, the dehomogenizations of the $q_i$ constitute a generating set for the dehomogenization of $\fraks$. The Buchberger algorithm now finds a reduced Groebner basis of $\fraks$ which consists of exactly $D-d$ linear polynomials. Their homogenizations then constitute a basis of homogenous linear forms of $\fraks$ itself. It can be checked that the first elimination steps which the Buchberger algorithm performs for the dehomogenizations of the $q_i$ correspond directly to the elimination steps in Algorithm~\ref{alg:exact_covonly} for their homogenous versions. So our algorithm performs similarly to the Buchberger algorithm in a noiseless setting, since both algorithms compute a reduced Groebner basis in the chosen coordinate system.
However, in our setting which stems from real data, there is a second point which is more grave and makes the use of off-the-shelf algorithms impossible: the computability of an exact result completely relies on the assumption that the ideals given as input are exactly known, i.e.~a generating set of polynomials is exactly known. This is not a problem in classical computational algebra; however, when dealing with polynomials obtained from real data, the polynomials come not only with numerical error, but in fact with statistical uncertainty. In general, the classical algorithms are unable to find any solution when confronted even with minimal noise on the otherwise exact polynomials. Namely, when we deal with a system of equations for which over-determination is possible, any perturbed system will be over-determined and thus have no solution. For example, the exact intersection of $N>D+1$ linear subspaces in complex $D$-space is always empty when they are sampled with uncertainty; this is a direct consequence of Proposition~\ref{Thm:genintcont}, when using the assumption that the noise is generic. However, if all those hyperplanes are nearly the same, then the result of a meaningful approximate algorithm should be a hyperplane close to all input hyperplanes instead of the empty set.
Before we continue, we would like to stress a conceptual point in approaching uncertainty. First, as in classical numerics, one can think of the input as theoretically exact, but with fixed error $\varepsilon$ and then derive bounds on the output error in terms of this $\varepsilon$ and analyze their asymptotics. We will refer to this approach as {\it numerical uncertainty}, as opposed to {\it statistical uncertainty}, which is a view more common to statistics and machine learning, as it is more natural for noisy data. Here, the error is considered as inherently probabilistic due to small sample effects or noise fluctuation, and algorithms may be analyzed for their statistical properties, independent of whether they are themselves deterministic or stochastic. The statistical view on uncertainty is the one the reader should have in mind when reading this paper.
Parts of the algebra community have been committed to the numerical viewpoint on uncertain polynomials: the problem of numerical uncertainty is for example extensively addressed in Stetter's standard book on numerical algebra \citep{Ste04}. The main difficulties and innovations stem from the fact that standard methods from algebra like the application of Groebner bases are numerically unstable, see \cite[chapter 4.1-2]{Ste04}.
Recently, the algebraic geometry community has developed an increasing interest in solving algebraic problems arising from the consideration of real world data. The algorithms in this area are more motivated to perform well on the data, some authors start to adapt a statistical viewpoint on uncertainty, while the influence of the numerical view is still dominant. As a distinction, the authors describe the field as approximate algebra instead of numerical algebra. Recent developments in this sense can be found for example in \citep{Hel06} or the book of \cite{KrePouRob09}. We will refer to this viewpoint as the statistical view in order to avoid confusion with other meanings of approximate.
Interestingly, there are significant similarities on the methodological side. Namely, in computational algebra, algorithms often compute primarily over vector spaces, which arise for example as spaces of polynomials with certain properties. Here, numerical linear algebra can provide many techniques of enforcing numerical stability, see the pioneering paper of \cite{Cor95}. Since then, many algorithms in numerical and approximate algebra utilize linear optimization to estimate vector spaces of polynomials. In particular, least-squares-approximations of rank or kernel are canonical concepts in both numerical and approximate algebra.
However, to the best of our knowledge, there is to date no algorithm which computes an ``approximate'' (or ``numerical'') radical of an ideal, or an approximate saturation, and also none in our special case. In the next section, we will use estimation techniques from linear algebra to convert Algorithm~\ref{alg:exact_covonly} into an algorithm which can cope with the inherent statistical uncertainty of the estimation problem.
\section{Approximate Algebraic Geometry on Real Data} \label{sec:approx} \label{sec:assa-appr}
In this section we show how algebraic computations can be applied to polynomials with inexact coefficients obtained from estimated cumulants on finite samples. Note that our method for computing the approximate radical is not specific to the problem studied in this paper.
The reason why we cannot directly apply our algorithm for the exact case to estimated polynomials is that it relies on the assumption that there exists an exact solution, such that the projected cumulants are equal, i.e.~we can find a projection $P$ such that the equalities \begin{align*}
P \Sigma_1 P^\top = \cdots = P \Sigma_m P^\top \hspace{0.3cm} \text{and} \hspace{0.3cm}
P \mu_1 = \cdots = P \mu_m \end{align*} hold exactly. However, when the elements of $\Sigma_1, \ldots, \Sigma_m$ and $\mu_1, \ldots, \mu_m$ are subject to random fluctuations or noise, there exists no projection that yields exactly the same random variables.
In algebraic terms, working with inexact polynomials means that the joint vanishing set of $q_1, \ldots, q_{m-1}$ and $f_1, \ldots, f_{m-1}$ consists only of the origin $0 \in \C^D$ so that the ideal becomes trivial: $$\langle q_1,\dots, q_{m-1}, f_1,\dots, f_{m-1}\rangle = \frakm.$$ Thus, in order to find a meaningful solution, we need to compute the radical approximately.
In the exact algorithm, we are looking for a polynomial of the form $T_D\ell$ vanishing on $S$, which is also a $\CC$-linear combination of the quadratic forms $q_i.$ The algorithm is based on an explicit way to do so which works since the $q_i$ are generic and sufficient in number. So one could proceed to adapt this algorithm to the approximate case by performing the same operations as in the exact case and then taking the $(\nD-\nd)$-th row, setting coefficients not divisible by $T_D$ to zero, and then dividing out $T_D$ to get a linear form. This strategy performs fairly well for small dimensions $D$ and converges to the correct solution, albeit slowly.
Instead of computing one particular linear generator as in the exact case, it is advisable to utilize as much information as possible in order to obtain better accuracy. The least-squares-optimal way to approximate a linear space of known dimension is to use singular value decomposition (SVD): with this method, we may directly eliminate the most insignificant directions in coefficient space which are due to fluctuations in the input.
To that end, we first define an approximation of an arbitrary matrix by a matrix of fixed rank. \begin{Def} Let $A \in \C^{m \times n}$ with singular value decomposition $A = U D V^*,$ where $D =\diag (\sigma_1,\dots, \sigma_p)\in \C^{p \times p}$ is a diagonal matrix with ordered singular values on the diagonal, \begin{align*}
|\sigma_{1}| \geq |\sigma_{2}| \geq \cdots \geq |\sigma_{p}| \geq 0. \end{align*} For $k\le p,$ let $D'=\diag (\sigma_1,\dots, \sigma_k, 0,\dots, 0).$ Then the matrix $A' = U D' V^*$ is called {\it rank $k$ approximation} of $A.$ The null space, left null space, row span, column span of $A'$ will be called {\it rank $k$ approximate null space, left null space, row span, column span} of $A.$ \end{Def} For example, if $u_1, \ldots, u_p$ and $v_1, \ldots, v_p$ are the columns of $U$ and $V$ respectively, the rank $k$ approximate left null space of $A$ is spanned by the rows of the matrix \begin{align*}
L = \begin{bmatrix}
u_{p-k+1} & \cdots &
u_p
\end{bmatrix}^\top , \end{align*} and the rank $k$ approximate row span of $A$ is spanned by the rows of the matrix \begin{align*}
S = \begin{bmatrix}
v_{1} & \cdots &
v_p
\end{bmatrix}^\top . \end{align*} We will call those matrices the {\it approximate left null space matrix} resp.~the {approximate row span matrix \it} of rank $k$ associated to $A.$ The approximate matrices are the optimal approximations of rank $k$ with respect to the least-squares error.
We can now use these concepts to obtain an approximative version of Algorithm~\ref{alg:exact_covonly}. Instead of searching for a single element of the form $T_D\ell,$ we estimate the vector space of all such elements via singular value decomposition --- note that this is exactly the vector space $\left(\langle T_D\rangle \cdot \fraks\right)_2$, i.e.~the vector space of all homogenous polynomials of degree two which are divisible by $T_D$. Also note that the choice of the linear form $T_D$ is irrelevant, i.e.\ we may replace $T_D$ above by any variable or even linear form. As a trade-off between accuracy and runtime, we additionally estimate the vector spaces $\left(\langle T_D\rangle \cdot \fraks\right)_2$ for all $1\le i\le D$, and then least-squares average the putative results for $\fraks$ to obtain a final estimator for $\fraks$ and thus the desired space of projections.
\begin{algorithm}[h] \caption{\label{alg:approx_covonly} The \textit{input} consists of noisy quadratic forms $q_1, \ldots, q_{m-1} \in \C[T_1, \ldots, T_D]$, and the dimension $d$; the \textit{output} is an approximate linear generating set $\ls_1, \ldots, \ls_{D-d}$ for the ideal $\idS$. }
\begin{algorithmic}[1]
\State Let $Q \gets \begin{bmatrix} q_1 & \cdots & q_{m-1} \end{bmatrix}^\top$ be the $(m-1 \times \nD)$-matrix
of coefficient vectors, where every row corresponds to a polynomial and every column to a monomial $T_i T_j$
in arbitrary order.
\For{$i=1,\ldots,D$}
\State
\begin{minipage}[t]{14cm}
Let $Q_i$ be the $((m-1) \times \nD - D)$-sub-matrix of $Q$ obtained by removing all
columns corresponding to monomials divisible by $T_i$
\end{minipage}
\State
\begin{minipage}[t]{14cm}
Compute the approximate left null space matrix $L_i$ of $Q_i$ of rank $(m-1) -\nD+ \nd +D-d$
\end{minipage}
\State Compute the approximate row span matrix $L'_i$ of $L_i Q$ of rank $D-d$
\State
\begin{minipage}[t]{14cm}
Let $L''_i$ be the $(D-d \times D)$-matrix obtained from $L'_i$ by removing all columns corresponding to
monomials not divisible by $T_i$
\end{minipage}
\EndFor
\State Let $L$ be the $(D(D-d)\times D)$-matrix obtained by vertical concatenation of $L''_1, \ldots, L''_D$
\State Compute the approximate row span matrix
$A = \begin{bmatrix} a_1 & \cdots & a_{D-d} \end{bmatrix}^\top$ of $L$ of rank $D-d$
and let
$\ls_i = \begin{bmatrix} T_1 & \cdots & T_D \end{bmatrix} a_i $ for all $1 \leq i \leq D-d$.
\end{algorithmic} \end{algorithm} We explain the logic behind the single steps: In the first step, we start with the same matrix $Q$ as in Algorithm 1. Instead of bringing $Q$ into triangular form with respect to the term order $T_1 \prec\dots\prec T_D,$ we compute the left kernel space row matrix $S_i$ of the monomials not divisible by $T_i$. Its left image $L_i=S_i Q$ is a matrix whose row space generates the space of possible last rows after bringing $Q$ into triangular form in an arbitrary coordinate system. In the next step, we perform PCA to estimate a basis for the so-obtained vector space of quadratic forms of type $T_i$ times linear form, and extract a basis for the vector space of linear forms estimated via $L_i.$ Now we can put together all $L_i$ and again perform PCA to obtain a more exact and numerically more estimate for the projection in the last step. The rank of the matrices after PCA is always chosen to match the correct ranks in the exact case.
Note that Algorithm~\ref{alg:approx_covonly} is a consistent estimator for the correct space of projections if the covariances are sample estimates. Let us first clarify in which sense consistent is meant here: If each covariance matrix is estimated from a sample of size $N$ or greater, and $N$ goes to infinity, then the estimate of the projection converges in probability to the true projection. The reason why Algorithm~\ref{alg:approx_covonly} gives a consistent estimator in this sense is elementary: covariance matrices can be estimated consistently, and so can their differences, the polynomials $q_i$. Moreover, the algorithm can be regarded as an almost continuous function in the polynomials $q_i$; so convergence in probability to the true projection and thus consistency follows from the continuous mapping theorem.
The runtime complexity of Algorithm~\ref{alg:approx_covonly} is $O(D^6)$ as for Algorithm~\ref{alg:exact_covonly}. For this note that calculating the singular value decomposition of an $m\times n$-matrix is $O(mn\max (m,n)).$
If we want to consider $k$-forms instead of $2$-forms, we can use the same strategies as above to numerically stabilize the exact algorithm. In the second step, one might want to consider all sub-matrices $Q_M$ of $Q$ obtained by removing all columns corresponding to monomials divisible by some degree $(k-1)$ monomial $M$ and perform the for-loop over all such monomials or a selection of them. Considering $D$ monomials or more gives again a consistent estimator for the projection. Similarly, these methods allow us to numerically stabilize versions with reduced epoch requirements and simultaneous consideration of different degrees.
\section{Numerical Evaluation} \label{sec:sims}
In this section we evaluate the performance of the algebraic algorithm on synthetic data in various settings. In order to contrast the algebraic approach with an optimization-based method (cf.~Figure~\ref{fig:ml_optim}), we compare with the Stationary Subspace Analysis (SSA) algorithm~\citep{PRL:SSA:2009}, which solves a similar problem in the context of time series analysis. To date, SSA has been successfully applied in the context of
biomedical data analysis~\citep{BunMeiSchMul10Finding},
domain adaptation~\citep{HarKawWasBun10SSA}, change-point detection~\citep{BunMeiSchMul10Boosting} and computer vision~\citep{MeiBunKawMul09Learning}.
\subsection{Stationary Subspace Analysis} \label{sec:sims-SSA}
Stationary Subspace Analysis~\citep{PRL:SSA:2009, MulBunMeiKirMul11SSAToolbox} factorizes an observed time series according to a linear model into underlying stationary and non-stationary sources. The observed time series $x(t) \in \R^D$ is assumed to be generated as a linear mixture of stationary sources $s^\s(t) \in \R^d$ and non-stationary sources $s^\n(t) \in \R^{D-d}$,
\begin{align}
x(t) = A s(t) = \begin{bmatrix} A^{\s} & A^{\n} \end{bmatrix}
\begin{bmatrix} s^{\s}(t) \\ s^{\n}(t) \end{bmatrix} , \label{eq:mixing_model} \end{align} with a time-constant mixing matrix $A$. The underlying sources $s(t)$ are not assumed to be independent or uncorrelated.
The aim of SSA is to invert this mixing model given only samples from $x(t)$. The true mixing matrix $A$ is not identifiable~\citep{PRL:SSA:2009}; only the projection $P \in \R^{d \times D}$ to the stationary sources can be estimated from the mixed signals $x(t)$, up to arbitrary linear transformation of its image. The estimated stationary sources are given by $\hat{s}^\s(t) = P x(t)$, i.e.~the projection $P$ eliminates all non-stationary contributions: $P A^\n = 0$.
The SSA algorithms~\citep{PRL:SSA:2009,HarKawWasBun10SSA} are based on the following definition of stationarity: a time series $X_t$ is considered stationary if its mean and covariance is constant over time, i.e.~$\E[X_{t_1}] = \E[X_{t_2}]$ and $\E[X_{t_1} X_{t_1}^\top] = \E[X_{t_2} X_{t_2}^\top]$ for all pairs of time points $t_1, t_2 \in \N$. Following this concept of stationarity, the projection $P$ is found by minimizing the difference between the first two moments of the estimated stationary sources $\hat{s}^\s(t)$ across epochs of the times series. To that end, the samples from $x(t)$ are divided into $m$ non-overlapping epochs of equal size, corresponding to the index sets $\mathcal{T}_1, \ldots, \mathcal{T}_{m}$, from which the mean and the covariance matrix is estimated for all epochs $1 \leq i \leq m$, \begin{align*}
\emu_i = \frac{1}{|\mathcal{T}_i|} \sum_{t \in \mathcal{T}_i} x(t) \hspace{0.5cm} \text{ and } \hspace{0.5cm}
\esi_i = \frac{1}{|\mathcal{T}_i|-1} \sum_{t \in \mathcal{T}_i} \left( x(t)-\emu_i \right)\left( x(t)-\emu_i \right)^\top . \end{align*} Given a projection $P$, the mean and the covariance of the estimated stationary sources in the $i$-th epoch are given by $\emu^\s_i = P \emu_i$ and $\esi^\s_i = P \esi_i P^\top$ respectively. Without loss of generality (by centering and whitening\footnote{A whitening transformation is a basis transformation $W$ that sets the sample covariance matrix to the identity. It can be obtained from the sample covariance matrix $\hat{\Sigma}$ as $W = \hat{\Sigma}^{-\frac{1}{2}}$} the average epoch) we can assume that $\hat{s}^\s(t)$ has zero mean and unit covariance.
The objective function of the SSA algorithm~\citep{PRL:SSA:2009} minimizes the sum of the differences between each epoch and the standard normal distribution, measured by the Kullback-Leibler divergence $\KLD$ between Gaussians: the projection $P^*$ is found as the solution to the optimization problem, \begin{align*}
P^* & =
\argmin_{P P^\top = I} \; \sum_{i=1}^{m} \KLD \Big[ \Gauss(\emu^\s_i,\esi^\s_i) \; \Big|\Big| \; \Gauss(0,I) \Big] \\
& = \argmin_{P P^\top = I} \; \sum_{i=1}^{m} \left(
- \log\det\esi^\s_i
+ (\hat{\mu}^\s_i)^\top \emu^\s_i \right), \end{align*} which is non-convex and solved using an iterative gradient-based procedure.
This SSA algorithm considers a problem that is closely related to the one addressed in this paper, because the underlying definition of stationarity does not consider the time structure. In essence, the $m$ epochs are modeled as $m$ random variables $X_1, \ldots, X_{m}$ for which we want to find a projection $P$ such that the projected probability distributions $P X_1, \ldots, P X_{m}$ are equal, up to the first two moments. This problem statement is equivalent to the task that we solve algebraically.
\subsection{Results} \label{sec:sims-exps}
In our simulations, we investigate the influence of the noise level and the number of dimensions on the performance and the runtime of our algebraic algorithm and the SSA algorithm. We measure the performance using the subspace angle between the true and the estimated space of projections $S$.
The setup of the synthetic data is as follows: we fix the total number of dimensions to $D=10$ and vary the dimension $d$ of the subspace with equal probability distribution from one to nine. We also fix the number of random variables to $m=110$. For each trial of the simulation, we need to choose a random basis for the two subspaces $\R^D = S \oplus S^\perp$, and for each random variable, we need to choose a covariance matrix that is identical only on $S$. Moreover, for each random variable, we need to choose a positive definite disturbance matrix (with given noise level $\sigma$), which is added to the covariance matrix to simulate the effect of finite or noisy samples.
The elements of the basis vectors for $S$ and $S^\perp$ are drawn uniformly from the interval $(-1, 1)$. The covariance matrix of each epoch $1 \leq i \leq m$ is obtained from Cholesky factors with random entries drawn uniformly from $(-1, 1)$, where the first $d$ rows remain fixed across epochs. This yields noise-free covariance matrices $C_1, \ldots, C_m \in \R^{D \times D}$ where the first $(d \times d)$-block is identical. Now for each $C_i$, we generate a random disturbance matrix $E_i$ to obtain the final covariance matrix $$C'_i = C_i + E_i .$$ The disturbance matrix $E_i$ is determined as $
E_i = V_i D_i V_i^\top $ where $V_i$ is a random orthogonal matrix, obtained as the matrix exponential of an antisymmetric matrix with random elements and $D_i$ is a diagonal matrix of eigenvalues. The noise level $\sigma$ is the log-determinant of the disturbance matrix $E_i$. Thus the eigenvalues of $D_i$ are normalized such that $$
\frac{1}{10} \sum_{i=1}^{10} \log D_{ii} = \sigma . $$ In the final step of the data generation, we transform the disturbed covariance matrices $C'_1, \ldots, C'_m$ into the random basis to obtain the cumulants $\Sigma_1, \ldots, \Sigma_m$ which are the input to our algorithm.
\begin{figure}
\caption{
Comparison of the algebraic algorithm and the SSA algorithm. Each panel shows
the median error of the two algorithms (vertical axis) for varying numbers of
stationary sources in ten dimensions (horizontal axis). The noise level increases
from the left to the right panel; the error bars extend from the 25\% to the 75\%
quantile estimated over 2000 random realizations of the data set.
}
\label{fig:alg1}
\end{figure}
The first set of results is shown in Figure~\ref{fig:alg1}. With increasing noise levels (from left to right panel) both algorithms become worse. For low noise levels, the algebraic method yields significantly better results than the optimization-based approach, over all dimensionalities. For medium and high-noise levels, this situation is reversed.
\begin{figure*}
\caption{
The left panel shows a comparison of the algebraic method and the SSA algorithm
over varying noise levels (five stationary sources in ten dimensions), the two curves
show the median log error. The right panel shows a comparison of the runtime for varying
numbers of stationary sources. The error bars extend from the 25\% to
the 75\% quantile estimated over 2000 random realizations of the data set.
}
\label{fig:alg2}
\end{figure*}
In the left panel of Figure~\ref{fig:alg2}, we see that the error level of the algebraic algorithm decreases with the noise level, converging to the exact solution when the noise tends to zero. In contrast, the error of original SSA decreases with noise level, reaching a minimum error baseline which it cannot fall below. In particular, the algebraic method significantly outperforms SSA for low noise levels, whereas SSA is better for high noise. However, when noise is too high, none of the two algorithms can find the correct solution. In the right panel of Figure~\ref{fig:alg2}, we see that the algebraic method is significantly faster than SSA.
\section{Conclusion} \label{sec:concl}
In this paper we have shown how a learning problem formulated in terms of cumulants of probability distributions can be addressed in the framework of computational algebraic geometry. As an example, we have demonstrated this viewpoint on the problem of finding a linear map $P \in \R^{d \times D}$ such that a set of projected random variables $X_1, \ldots, X_m \in \R^D$ have the same distribution, \begin{align*}
P X_1 \sim \cdots \sim P X_m . \end{align*} To that end, we have introduced the theoretical groundwork for an algebraic treatment of inexact cumulants estimated from data: the concept of polynomials that are \textit{generic} up to a certain property that we aim to recover from data. In particular, we have shown how we can find an approximate exact solution to this problem using algebraic manipulation of cumulants estimated on samples drawn from $X_1, \ldots, X_m$. Therefore we have introduced the notion of computing an \textit{approximate saturation} of an ideal that is optimal in a least-squares sense. Moreover, using the algebraic problem formulation in terms of generic polynomials, we have presented compact proofs for a condition on the identifiability of the true solution.
In essence, instead of searching the surface of a non-convex objective function involving the cumulants, the algebraic algorithm directly finds the solution by manipulating cumulant polynomials --- which is the more natural representation of the problem. This viewpoint is not only theoretically appealing, but conveys practical advantages that we demonstrate in a numerical comparison to Stationary Subspace Analysis \citep{PRL:SSA:2009}: the computational cost is significantly lower and the error converges to zero as the noise level goes to zero. However, the algebraic algorithm requires $m \geq \nD$ random variables with distinct distributions, which is quadratic in the number of dimensions $D$. This is due to the fact that the algebraic algorithm represents the cumulant polynomials in the vector space of coefficients. Consequently, the algorithm is confined to linearly combining the polynomials which describe the solution. However, the set of solutions is also invariant under multiplication of polynomials and polynomial division, i.e.~the algorithm does not utilize all information contained in the polynomial equations. We conjecture that we can construct a more efficient algorithm, if we also multiply and divide polynomials.
The theoretical and algorithmic techniques introduced in this paper can be applied to other scenarios in machine learning, including the following examples. \begin{itemize}
\item \textbf{Finding properties of probability distributions.} Any inference problem that
can be formulated in terms of polynomials, in principle,
amenable our algebraic approach; incorporating polynomial constraints is also straightforward.
\item \textbf{Approximate solutions to polynomial equations.} In machine learning,
the problem of solving polynomial equations can e.g.~occur in the context of finding
the solution to a constrained nonlinear optimization problem by means of setting the gradient to
zero.
\item \textbf{Conditions for identifiability.} Whenever a machine learning problem can be formulated
in terms of polynomials, identifiability of its generative model can also be phrased in terms of algebraic geometry,
where a wealth of proof techniques stands at disposition.
\end{itemize}
We argue for a cross-fertilization of approximate computational algebra and machine learning: the former can benefit from the wealth of techniques for dealing with uncertainty and noisy data; the machine learning community may find a novel framework for representing learning problems that can be solved efficiently using symbolic manipulation.
\acks{We thank Marius Kloft and Jan Saputra M\"uller for valuable discussions. We are particularly grateful to Gert-Martin Greuel for his insightful remarks. We also thank Andreas Ziehe for proofreading the manuscript. This work has been supported by the Bernstein Cooperation (German Federal Ministry of Education and Science), Förderkennzeichen 01~GQ~0711, and the Mathematisches Forschungsinstitut Oberwolfach (MFO). A preprint version of this manuscript has appeared as part of the Oberwolfach Preprint series \citep{Kir11}. }
\appendix
\section{An example} \label{app-example} In this section, we will show by using a concrete example how the Algorithms~\ref{alg:exact_covonly} and \ref{alg:approx_covonly} work. The setup will be the similar to the example presented in the introduction. We will use the notation introduced in Section~\ref{sec:exact}. \begin{Ex}\rm In this example, let us consider the simplest non-trivial case: Two random variables $X_1,X_2$ in $\R^2$ such that there is exactly one direction $w\in \R^2$ such that $w^\top X_1=w^\top X_2$. I.e.~the total number of dimensions is $D=2$, the dimension of the set of projections is $d=1$. As in the beginning of Section~\ref{sec:exact}, we may assume that $\R^2=S\oplus S^\perp$ is an orthogonal sum of a one-dimensional space of projections $S$ and its orthogonal complement $S^\perp$. In particular, $S^\perp$ is given as the linear span of a single vector, say $\begin{bmatrix} \alpha & \beta \end{bmatrix}^\top$. The space $S$ is also the linear span of the vector $\begin{bmatrix} \beta & -\alpha \end{bmatrix}^\top.$
Now we partition the sample into $D(D+1)/2-d(d+1)/2=2$ epochs (this is the lower bound needed by Proposition~\ref{Prop:Alg1corr}). From the two epochs we can estimate two covariance matrices $\hat{\Sigma}_1,\hat{\Sigma}_2$. Suppose we have \begin{align}
\hat{\Sigma}_1 & = \begin{bmatrix}
a_{11} & a_{12} \\ a_{21} & a_{22}
\end{bmatrix} . \end{align} From this matrices, we can now obtain a polynomial \begin{align}
q_1 & = w^\top (\hat{\Sigma}_1 - I) w \nonumber \\
& = w^\top \begin{bmatrix}
a_{11}-1 & a_{12} \\ a_{21} & a_{22}-1
\end{bmatrix} w \nonumber \\
& = (a_{11}-1) T_1^2 + ( a_{12} + a_{21}) T_1 T_2 + (a_{22}-1) T_2^2, \end{align} where $w=\begin{bmatrix} T_1 & T_2 \end{bmatrix}^\top.$ Similarly, we obtain a polynomial $q_2$ as the Gram polynomial of $\hat{\Sigma}_2-I.$
First we now illustrate how Algorithm~\ref{alg:exact_covonly}, which works with homogenous exact polynomials, can determine the vector space $S$ from these polynomials. For this, we assume that the estimated polynomials are exact; we will discuss the approximate case later. We can also write $q_1$ and $q_2$ in coefficient expansion: \begin{align*} q_1&=q_{11}T_1^2+q_{12}T_1T_2+q_{13}T_2^2\\ q_2&=q_{21}T_1^2+q_{22}T_1T_2+q_{23}T_2^2. \end{align*} We can also write this formally in the $(2\times 3)$ coefficient matrix $Q=(q_{ij})_{ij},$ where the polynomials can be reconstructed as the entries in the vector $$Q\cdot \begin{bmatrix} T_1^2 & T_1T_2 & T_2^2 \end{bmatrix}^\top.$$ Algorithm~\ref{alg:exact_covonly} now calculates the upper triangular form of this matrix. For polynomials, this is equivalent to calculating the last row \begin{align*}
&q_{21} q_1 -q_{11} q_2\\ &=[q_{21}q_{12}-q_{11}q_{22}]T_1T_2+[q_{21}q_{13}-q_{11}q_{23}]T_2^2. \end{align*} Then we divide out $T_2$ and obtain $$P=[q_{21}q_{12}-q_{11}q_{22}]T_1+[q_{21}q_{13}-q_{11}q_{23}]T_2.$$ The algorithm now identifies $S^\perp$ as the vector space spanned by the vector $$\begin{bmatrix} \alpha & \beta \end{bmatrix}^\top =\begin{bmatrix}q_{21}q_{12}-q_{11}q_{22} & q_{21}q_{13}-q_{11}q_{23}\end{bmatrix}^\top.$$ This already finishes the calculation given by Algorithm~\ref{alg:exact_covonly}, as we now explicitly know the solution $$\begin{bmatrix} \alpha & \beta \end{bmatrix}^\top$$ To understand why this strategy works, we need to have a look at the input. Namely, one has to note that $q_1$ and $q_2$ are generic homogenous polynomials of degree $2$, vanishing on $S$. That is, we will have $q_i(x)=0$ for $i=1,2$ and all points $x\in S.$ It is not difficult to see that every polynomial fulfilling this condition has to be of the form $$(\alpha T_1+\beta T_2)(a T_1 +bT_2)$$ for some $a,b\in \C;$ i.e.~a multiple of the equation defining $S$. However we may not know this factorization a priori, in particular we are in general agnostic as to the correct values of $\alpha$ and $\beta$. They have to be reconstructed from the $q_i$ via an algorithm. Nonetheless, a correct solution exists, so we may write \begin{align*} q_1&=(\alpha T_1+\beta T_2)(a_1 X +b_1 T_2)\\ q_2&=(\alpha T_1+\beta T_2)(a_2 X +b_2 T_2), \end{align*} with $a_i,b_i$ generic, without knowing the exact values a priori. If we now compare to the above expansion in the $q_{ij},$ we obtain the linear system of equations \begin{align*} q_{i1}&=\alpha a_i\\ q_{i2}&=\alpha b_i+ \beta a_i\\ q_{i3}&=\beta b_i \end{align*} for $i=1,2$, from which we may reconstruct the $a_i,b_i$ and thus $\alpha$ and $\beta$. However, a more elegant and general way of getting to the solution is to bring the matrix $Q$ as above into triangular form. Namely, by assumption, the last row of this triangular form corresponds to the polynomial $P$ which vanishes on $S$. Using the same reasoning as above, the polynomial $P$ has to be a multiple of $(\alpha T_1+\beta T_2)$. To check the correctness of the solution, we substitute the $q_{ij}$ in the expansion of $P$ for $a_i,b_i$, and obtain \begin{align*} P=&[q_{21}q_{12}-q_{11}q_{22}]T_1T_2+[q_{21}q_{13}-q_{11}q_{23}]T_2^2\\ =&[\alpha a_2 (\alpha b_1+\beta a_1)-\alpha a_1 (\alpha b_2+\beta a_2)]T_1T_2+[\alpha a_2 \beta b_1 -\alpha a_1 \beta b_2]T_2^2\\ =&[\alpha^2 a_2 b_1-\alpha^2 a_1 b_2]T_1T_2+[\alpha\beta a_2 b_1 -\alpha\beta a_1b_2]T_2^2\\ =&(\alpha T_1 + \beta T_2) \alpha [a_2b_1-a_1b_2] T_2. \end{align*} This is $(\alpha T_1+\beta T_2)$ times $T_2$ up to a scalar multiple - from the coefficients of the form $P$, we may thus directly reconstruct the vector $\begin{bmatrix} \alpha & \beta \end{bmatrix}$ up to a common factor and thus obtain a representation for $S,$ since the calculation of these coefficients did not depend on a priori knowledge about $S.$
If the estimation of the $\hat{\Sigma}_i$ and thus of the $q_i$ is now endowed with noise, and we have more than two epochs and polynomials, Algorithm~\ref{alg:approx_covonly} provides the possibility to perform this calculation approximately. Namely, Algorithm~\ref{alg:approx_covonly} finds a linear combination of the $q_i$ which is approximately of the form $T_D\ell$ with a linear form $\ell$ in the variables $T_1,T_2$. The Young-Eckart Theorem guarantees that we obtain a consistent and least-squares-optimal estimator for $P$, similarly to the exact case. The reader is invited to check this by hand as an exercise. \end{Ex}
Now the observant reader may object that we may have simply obtained the linear form $(\alpha T_1+\beta T_2)$ and thus $S$ directly from factoring $q_1$ and $q_2$ and taking the unique common factor. This is true, but this strategy can only be applied in the very special case $D-d=1.$ To illustrate the additional difficulties in the general case, we repeat the above example for $D=4$ and $d=2$ for the exact case: \begin{Ex}\rm In this example, we need already $D(D+1)/2-d(d+1)/2=7$ polynomials $q_1,\dots, q_7$ to solve the problem with Algorithm~\ref{alg:exact_covonly}. As above, we can write \begin{align*}
q_i=&q_{i1}T_1^2+q_{i2}T_1T_2+q_{i3}T_1T_3+q_{i4}T_1T_4+q_{i5}T_2^2\\ &+q_{i6}T_2T_3+q_{i7}T_2T_4+q_{i8}T_3^2+q_{i9}T_3T_4+q_{i,10}T_4^2\\ \end{align*} for $i=1,\dots, 7$, and again we can write this in a $(7\times 10)$ coefficient matrix $Q=(q_{ij})_{ij}$. In Algorithm~\ref{alg:exact_covonly}, this matrix is brought into triangular form. The last row of this triangular matrix will thus correspond to a polynomial of the form $$P =p_7T_2T_4+p_{8}T_3^2+p_{9}T_3T_4+p_{10}T_4^2$$ A polynomial of this form is not divisible by $T_4$ in general. However, Proposition~\ref{Prop:Alg1corr} guarantees us that the coefficient $p_8$ is always zero due to our assumptions. So we can divide out $T_4$ to obtain a linear form $$p_7T_2+p_{9}T_3+p_{10}T_4.$$ This is one equation defining the linear space $S$. One obtains another equation in the variables $T_1,T_2,T_3$ if one, for example, inverts the numbering of the variables $1-2-3-4$ to $4-3-2-1$. Two equations suffice to describe $S$, and so Algorithm~\ref{alg:exact_covonly} yields the correct solution.
As in the example before, it can be checked by hand that the coefficient $p_7$ indeed vanishes, and the obtained linear equations define the linear subspace $S$. For this, one has to use the classical result from algebraic geometry that every $q_i$ can be written as $$q_i=\ell_1 P_1+\ell_2 P_2,$$ where the $\ell_i$ are fixed but arbitrary linear forms defining $S$ as their common zero set, and the $P_i$ are some linear forms determined by $q_i$ and the $\ell_i$ (this is for example a direct consequence of Hilbert's Nullstellensatz). Caution is advised as the equations involved become very lengthy - while not too complex - already in this simple example. So the reader may want to check only that the coefficient $p_8$ vanishes as claimed.
\end{Ex}
\section{Algebraic Geometry of Genericity} \label{app-generic}
In the paper, we have reformulated a problem of comparing probability distributions in algebraic terms. For the problem to be well-defined, we need the concept of genericity for the cumulants. The solution can then be determined as an ideal generated by generic homogenous polynomials vanishing on a linear subspace. In this supplement, we will extensively describe this property which we call genericity and derive some simple consequences.
Since genericity is an algebraic-geometric concept, knowledge about basic algebraic geometry will be required for an understanding of this section. In particular, the reader should be at least familiar with the following concepts before reading this section: Polynomial rings, ideals, radicals, factor rings, algebraic sets, algebra-geometry correspondence (including Hilbert's Nullstellensatz), primary decomposition, height resp.~dimension theory in rings. A good introduction into the necessary framework can be found in the book of \cite{Cox}.
\label{sec:Alg-Generic} \subsection{Definition of genericity} \label{sec:gendef} In the algebraic setting of the paper, we would like to calculate the radical of an ideal $$\calI=\langle q_1,\dots, q_{m-1}, f_1,\dots, f_{m-1}\rangle.$$ This ideal $\calI$ is of a special kind: its generators are random, and are only subject to the constraints that they vanish on the linear subspace $S$ to which we project, and that they are homogenous of fixed degree. In order to derive meaningful results on how $\calI$ relates to $S$, or on the solvability of the problem, we need to model this kind of randomness.
In this section, we introduce a concept called genericity. Informally, a generic situation is a situation without pathological degeneracies. In our case, it is reasonable to believe that apart from the conditions of homogeneity and the vanishing on $S$, there are no additional degeneracies in the choice of the generators. So, informally spoken, the ideal $\calI$ is generated by generic homogenous elements vanishing on $S.$ This section is devoted to developing a formal theory in order to address such generic situations efficiently.
The concept of genericity is already widely used in theoretical computer science, combinatorics or discrete mathematics; there, it is however often defined inexactly or not at all, or it is only given as an ad-hoc definition for the particular problem. On the other hand, genericity is a classical concept in algebraic geometry, in particular in the theory of moduli. The interpretation of generic properties as probability-one-properties is also a known concept in applied algebraic geometry, e.g.~algebraic statistics. However, the application of probability distributions and genericity to the setting of generic ideals, in particular in the context of conditional probabilities, are original to the best of our knowledge, though not being the first one to involve generic resp.~general polynomials, see \citep{Iar84}. Generic polynomials and ideals have been also studied in \citep{Fro94}. A collection of results on generic polynomials and ideals which partly overlap with ours may also be found in the recent paper \citep{Par10}.
Before continuing to the definitions, let us explain what genericity should mean. Intuitively, generic objects are objects without unexpected pathologies or degeneracies. For example, if one studies say $n$ lines in the real plane, one wants to exclude pathological cases where lines lie on each other or where many lines intersect in one point. Having those cases excluded means examining the ``generic'' case, i.e. the case where there are $n(n+1)/2$ intersections, $n(n+1)$ line segments and so forth. Or when one has $n$ points in the plane, one wants to exclude the pathological cases where for example there are three affinely dependent points, or where there are more sophisticated algebraic dependencies between the points which one wants to exclude, depending on the problem.
In the points example, it is straightforward how one can define genericity in terms of sampling from a probability distribution: one could draw the points under a suitable continuous probability distribution from real two-space. Then, saying that the points are ``generic'' just amounts to examine properties which are true with probability one for the $n$ points. Affine dependencies for example would then occur with probability zero and are automatically excluded from our interest. One can generalize this idea to the lines example: one can parameterize the lines by a parameter space, which in this case is two-dimensional (slope and ordinate), and then sample lines uniformly distributed in this space (one has of course to make clear what this means). For example, lines lying on each other or more than two lines intersecting at a point would occur with probability zero, since the part of parameter space for this situation would have measure zero under the given probability distribution.
When we work with polynomials and ideals, the situation gets a bit more complicated, but the idea is the same. Polynomials are uniquely determined by their coefficients, so they can naturally be considered as objects in the vector space of their coefficients. Similarly, an ideal can be specified by giving the coefficients of some set of generators. Let us make this more explicit: suppose first we have given a single polynomial $f\in \C[X_1,\dots X_D]$ of degree $k$.
In multi-index notation, we can write this polynomial as a finite sum $$f=\sum_{\alpha\in \NN^D}c_\alpha X^\alpha\,\quad \mbox{with}\;c_\alpha\in \C.$$
This means that the possible choices for $f$ can be parameterized by the ${D+k \choose k}$ coefficients $c_I$ with $\|I\|_1\le k.$ Thus polynomials of degree $k$ with complex coefficients can be parameterized by complex ${D+k \choose k}$-space.
Algebraic sets can be similarly parameterized by parameterizing the generators of the corresponding ideal. However, this correspondence is highly non-unique, as different generators may give rise to the same zero set. While the parameter space can be made unique by dividing out redundancies, which gives rise to the Hilbert scheme, we will instead use the redundant, though pragmatic characterization in terms of a finite dimensional vector space over $\C$ of the correct dimension.
We will now fix notation for the parameter space of polynomials and endow it with algebraic structure. The extension to ideals will then be derived later. Let us write $\calM_k$ for complex ${D+k \choose k}$-space (we assume $D$ as fixed), interpreting it as a parameter space for the polynomials of degree $k$ as shown above. Since the parameter space $\calM_k$ is isomorphic to complex ${D+k\choose k}$-space, we may speak about algebraic sets in $\calM_k$. Also, $\calM_k$ carries the complex topology induced by the topology on $\R^{2k}$ and by topological isomorphy the Lebesgue measure; thus it also makes sense to speak about probability distributions and random variables on $\calM_k.$ This dual interpretation will be the main ingredient in our definition of genericity, and will allow us to relate algebraic results on genericity to the probabilistic setting in the applications. As $\calM_k$ is a topological space, we may view any algebraic set in $\calM_k$ as an event if we randomly choose a polynomial in $\calM_k$: \begin{Def} Let $X$ be a random variable with values in $\calM_k$. Then an event for $X$ is called {\it algebraic event} or {\it algebraic property} if the corresponding event set in $\calM_k$ is an algebraic set. It is called {\it irreducible} if the corresponding event set in $\calM_k$ is an irreducible algebraic set. \end{Def}
If an event $A$ is irreducible, this means that if we write $A$ as the event ``$A_1$ and $A_2$'', for algebraic events $A_1,A_2$, then $A=A_1$, or $A=A_2.$ We now give some examples for algebraic properties.
\begin{Ex}\rm\label{Ex:algevts} The following events on $\calM_k$ are algebraic: \begin{enumerate}
\item The sure event.
\item The empty event.
\item The polynomial is of degree $n$ or less.
\item The polynomial vanishes on a prescribed algebraic set.
\item The polynomial is contained in a prescribed ideal.
\item The polynomial is homogenous.
\item The polynomial is a square.
\item The polynomial is reducible. \end{enumerate} Properties 1-5 are additionally irreducible.
We now show how to prove these claims: 1-2 are clear, we first prove that properties 3-5 are algebraic and irreducible. By definition, it suffices to prove that the subset of $\calM_k$ corresponding to those polynomials is an irreducible algebraic set. We claim: in any of those cases, the subset in question is moreover a linear subspace, and thus algebraic and irreducible. This can be easily verified by checking directly that if $f_1,f_2$ fulfill the property in question, then $f_1+\alpha f_2$ also fulfills the property.
Property 6 is algebraic, since it can be described as the disjunction of the properties ``The polynomial is homogenous and of degree $n$'' for all $n\le k.$ Those single properties can be described by linear subspaces of $\calM_k$ as above, thus property 6 is parameterized by the union of those linear subspaces. In general, these are orthogonal, so property 6 is not irreducible.
Property 7 is algebraic, as we can check it through the vanishing of a system of generalized discriminant polynomials. One can show that it is also irreducible since the subset of $\calM_k$ in question corresponds to the image of a Veronese map (homogenization to degree $k$ is a strategy); however, since we will not need such a result, we do not prove it here.
Property 8 is algebraic, since factorization can also be checked by sets of equations. One has to be careful here though, since those equations depend on the degrees of the factors. For example, a polynomial of degree $4$ may factor into two polynomials of degree $1$ and $3$, or in two polynomials of degree $2$ each. Since in general each possible combination defines different sets of equations and thus different algebraic subsets of $\calM_k$, property 8 is in general not irreducible (for $k\le 3$ it is). \end{Ex}
The idea defining a choice of polynomial as generic follows the intuition of the affirmed non-sequitur: a generic, resp.~generically chosen polynomial should not fulfill any algebraic property. A generic polynomial, having a particular simple (i.e.~irreducible) algebraic property, should not fulfill any other algebraic property which is not logically implied by the first one. Here, algebraic properties are regarded as the natural model for restrictive and degenerate conditions, while their logical negations are consequently interpreted as generic, as we have seen in Example~\ref{Ex:algevts}. These considerations naturally lead to the following definition of genericity in a probabilistic context:
\begin{Def}\label{Def:genrand} Let $X$ be a random variable with values in $\calM_k$. Then $X$ is called {\it generic}, if for any irreducible algebraic events $A,B,$ the following holds:
The conditional probability $P_X(A|B)$ exists and vanishes if and only if $B$ does not imply $A$. \end{Def} In particular, $B$ may also be the sure event.
Note that without giving a further explication, the conditional probability $P_X(A|B)$ is not well-defined, since we condition on the event $B$ which has probability zero. There is also no unique way of remedying this, as for example the Borel-Kolmogorov paradox shows. In section~\ref{sec:genalt}, we will discuss the technical notion which we adopt to ensure well-definedness.
Intuitively, our definition means that an event has probability zero to occur unless it is logically implied by the assumptions. That is, degenerate dependencies between events do not occur.
For example, non-degenerate multivariate Gaussian distributions or Gaussian mixture distributions on $\calM_k$ are generic distributions. More general, any positive continuous probability distribution which can be approximated by Gaussian mixtures is generic (see Example~\ref{Ex:gen-nongen}). Thus we argue that non-generic random variables are very pathological cases. Note however, that our intention is primarily not to analyze the behavior of particular fixed generic random variables (this is part of classical statistics). Instead, we want to infer statements which follow not from the particular structure of the probability function, but solely from the fact that it is generic, as these statements are intrinsically implied by the conditional postulate in Definition~\ref{Def:genrand} alone. We will discuss the definition of genericity and its implications in more detail in section~\ref{sec:genalt}.
With this definition, we can introduce the terminology of a generic object: it is a generic random variable which is object-valued.
\begin{Def} We call a generic random variable with values in $\calM_k$ a generic polynomial of degree $k.$ When the degree $k$ is arbitrary, but fixed (and still $\ge 1$), we will say that $f$ is a generic polynomial, or that $f$ is generic, if it is clear from the context that $f$ is a polynomial. If the degree $k$ is zero, we will analogously say that $f$ is a generic constant.\\
We call a set of constants or polynomials $f_1,\dots, f_m$ generic if they are generic and independent.\\
We call an ideal generic if it is generated by a set of $m$ generic polynomials.\\
We call an algebraic set generic if it is the vanishing set of a generic ideal.\\
Let $\calP$ be an algebraic property on a polynomial, a set of polynomials, an ideal, or an algebraic set (e.g.~homogenous, contained in an ideal et.). We will call a polynomial, a set of polynomials, or an ideal, a {\it generic} $\calP$ polynomial, set, or ideal, if it the conditional of a generic random variable with respect to $\calP$.\\
If $\calA$ is a statement about an object (polynomial, ideal etc), and $\calP$ an algebraic property, we will say briefly ``A generic $\calP$ object is $\calA$'' instead of saying ``A generic $\calP$ object is $\calA$ with probability one''. \end{Def}
Note that formally, these objects are all polynomial, ideal, algebraic set etc -valued random variables. By convention, when we state something about a generic object, this will be an implicit probability-one statement. For example, when we say\\
``A generic green ideal is blue'',\\
this is an abbreviation for the by definition equivalent but more lengthy statement\\
``Let $f_1,\dots, f_m$ be independent generic random variables with values in $\calM_{k_1},\dots,\calM_{k_m}.$ If the ideal $\langle f_1,\dots, f_m\rangle$ is green, then with probability one, it is also blue - this statement is independent of the choice of the $k_i$ and the choice of which particular generic random variables we use to sample.\\
On the other hand, we will use the verb ``generic'' also as a qualifier for ``constituting generic distribution''. So for example, when we say\\
``The Z of a generic red polynomial is a generic yellow polynomial'',\\
this is an abbreviation of the statement\\
``Let $X$ be a generic random variable on $\calM_k,$ let $X'$ be the yellow conditional of $X$. Then the Z of $X'$ is the red conditional of some generic random variable - in particular this statement is independent of the choice of $k$ and the choice of $X$.''\\
It is important to note that the respective random variables will not be made explicit in the following subsections, since the statements will rely only on its property of being generic, and not on its particular structure which goes beyond being generic.\\
As an application of these concepts, we may now formulate the problem of comparing cumulants in terms of generic algebra:
\begin{Prob}\label{Prob:SSA-alg} Let $\fraks=\Id(S)$, where $S$ is an unknown $d$-dimensional subspace of $\C^D$. Let $$\calI=\langle f_1,\dots, f_m \rangle$$ with $f_i\in \fraks$ generic of fixed degree each (in our case, one and two), such that $\sqrt{\calI}=\fraks.$
Then determine a reduced Groebner basis (or another simple generating system) for $\fraks.$ \end{Prob}
As we will see, genericity is the right concept to model random sampling of polynomials, as we will derive special properties of the ideal $\calI$ which follow from the genericity of the $f_i$.
\subsection{Zero-measure conditionals, and relation to other types of genericity}\label{sec:genalt} In this section, se will discuss the definition of genericity in Definition~\ref{Def:genrand} and ensure its well-definedness. Then we will invoke alternative definitions for genericity and show their relation to our probabilistic intuitive approach from section~\ref{sec:gendef}. As this section contains technical details and is not necessary for understanding the rest of the appendix, the reader may opt to skip it.
An important concept in our definition of genericity in Definition~\ref{Def:genrand} is the conditional probability $P_X(A|B)$. As $B$ is an algebraic set, its probability $P_X(B)$ is zero, so the Bayesian definition of conditional cannot apply. There are several ways to make it well-defined; in the following, we explain the Definition of conditional we use in Definition~\ref{Def:genrand}. The definition of conditional we use is one which is also often applied in this context. \begin{Rem}\label{Rem:genmeas} Let $X$ be a real random variable (e.g.~with values in $\calM_k$) with probability measure $\mu$. If $\mu$ is absolutely continuous, then by the theorem of Radon-Nikodym, there is a unique continuous density $p$ such that $$\mu(U)=\int_U p\, d\lambda$$ for any Borel-measurable set $U$ and the Lebesgue measure $\lambda$. If we assume that $p$ is a continuous function, it is unique, so we may define a restricted measure $\mu_B$ on the event set of $B$ by setting $$\nu(U)=\int_U p\, dH,$$
for Borel subsets of $U$ and the Hausdorff measure $H$ on $B$. If $\nu(B)$ is finite and non-zero, i.e.~$\nu$ is absolutely continuous with respect to $H$, then it can be renormalized to yield a conditional probability measure $\mu(.)|_B=\nu(.)/\nu(B).$ The conditional probability $P_X(A|B)$ has then to be understood as
$$P_X(A|B)=\int_{B}\mathbbm{1} (A\cap B)\,d\mu\mid_B,$$ whose existence in particular implies that the Lebesgue integrals $\nu (B)$ are all finite and non-zero. \end{Rem}
As stated, we adopt this as the definition of conditional probability for algebraic sets $A$ and $B$. It is important to note that we have made implicit assumptions on the random variable $X$ by using the conditionals $P_X(A|B)$ in Remark~\ref{Rem:genmeas} (and especially by assuming that they exist): namely, the existence of a continuous density function and existence, finiteness, and non-vanishing of the Lebesgue integrals. Similarly, by stating Definition~\ref{Def:genrand} for genericity, we have made similar assumptions on the generic random variable $X$, which can be summarized as follows:
\begin{Ass}\label{Ass:gen} $X$ is an absolutely continuous random variable with continuous density function $p$, and for every algebraic event $B$, the Lebesgue integrals $$\int_B p\, dH,$$ where $H$ is the Hausdorff measure on $B$, are non-zero and finite. \end{Ass}
This assumption implies the existence of all conditional probabilities $P_X(A|B)$ in Definition~\ref{Def:genrand}, and are also necessary in the sense that they are needed for the conditionals to be well-defined. On the other hand, if those assumptions are fulfilled for a random variable, it is automatically generic:
\begin{Rem}\label{Prop:gen=cont} Let $X$ be a $\calM_k$-valued random variable, fulfilling the Assumptions in~\ref{Ass:gen}. Then, the probability density function of $X$ is strictly positive. Moreover, $X$ is a generic random variable. \end{Rem} \begin{proof} Let $X$ be a $\calM_k$-valued random variable fulfilling the Assumptions in~\ref{Ass:gen}. Let $p$ be its continuous probability density function.
We first show positivity: If $X$ would not be strictly positive, then $p$ would have a zero, say $x$. Taking $B=\{x\},$ the integral $\int_B p\, dH$ vanishes, contradicting the assumption.
Now we prove genericity, i.e.~that for arbitrary irreducible algebraic properties $A,B$ such that $B$ does not imply $A$, the conditional probability $P_X(A|B)$ vanishes. Since $B$ does not imply $A$, the algebraic set defined by $B$ is not contained in $A$. Moreover, as $B$ and $A$ are irreducible and algebraic, $A\cap B$ is also of positive codimension in $B$. Now by assumption, $X$ has a positive continuous probability density function $f$ which by assumption restricts to a probability density on $B$, being also positive and continuous. Thus the integral
$$P_X(A|B)=\int_B \mathbbm{1}_A f(x)\, dH,$$ where $H$ is the Hausdorff measure on $B$, exists. Moreover, it is zero, as we have derived that $A$ has positive codimension in $B$. \end{proof}
This means that already under mild assumptions, which merely ensure well-definedness of the statement in the Definition~\ref{Def:genrand} of genericity, random variables are generic. The strongest of the comparably mild assumptions are the convergence of the conditional integrals, which allow us to renormalize the conditionals for all algebraic events. In the following example, a generic and a non-generic probability distribution are presented.
\begin{Ex}\rm\label{Ex:gen-nongen} Gaussian distributions and Gaussian mixture distributions are generic, since for any algebraic set $B$, we have $$\int_B \mathbbm{1}_{\mathcal{B}(t)}\, dH = O(t^{\dim B}),$$
where $\mathcal{B}(t)=\{x\in\mathbb{R}^n\;;\; \|x\|<t\}$ is the open disc with radius $t.$ Note that this particular bound is false in general and may grow arbitrarily large when we omit $B$ being algebraic, even if $B$ is a smooth manifold. Thus $P_X(A|B)$ is bounded from above by an integral (or a sum) of the type $$\int_{0}^\infty\exp(-t^2)t^a\;dt\quad\mbox{with}\; a\in\mathbb{N}$$ which is known to be finite.
Furthermore, sums of generic distributions are again generic; also, one can infer that any continuous probability density dominated by the distribution of a generic density defines again a generic distribution.
An example of a non-generic but smooth distribution is given by the density function $$p(x,y)=\frac{1}{\mathcal{N}}e^{-x^4y^4}$$ where $\mathcal{N}$ is some normalizing factor. While $p$ is integrable on $\mathbb{R}^2,$ its restriction to the coordinate axes $x=0$ and $y=0$ is constant and thus not integrable. \end{Ex}
Now we will examine different known concepts of genericity and relate them briefly to the one we have adopted.
A definition of genericity in combinatorics and geometry which can be encountered in different variations is that there exist no degenerate interpolating functions between the objects:
\begin{Def}\label{Def:gencomb} Let $P_1,\dots, P_m$ be points in the vector space $\mathbb{C}^n$. Then $P_1,\dots, P_m$ are general position (or generic, general) if no $n+1$ points lie on a hyperplane. Or, in a stronger version: for any $d\in\mathbb{N}$, no (possibly inhomogenous) polynomial of degree $d$ vanishes on ${n+d \choose d}+1$ different $P_i$. \end{Def} As $\calM_k$ is a finite dimensional $\mathbb{C}$-vector space, this definition is in principle applicable to our situation. However, this definition is deterministic, as the $P_i$ are fixed and no random variables, and thus preferable when making deterministic statements. Note that the stronger definition is equivalent to postulating general position for the points $P_1,\dots, P_m$ in any polynomial kernel feature space.
Since not lying on a hyperplane (or on a hypersurface of degree $d$) in $\mathbb{C}^n$ is a non-trivial algebraic property for any point which is added beyond the $n$-th (resp.~the ${n+d\choose d}$-th) point $P_i$ (interpreted as polynomial in $\calM_k$), our definition of genericity implies general position. This means that generic polynomials $f_1,\dots, f_m\in\calM_k$ (almost surely) have the deterministic property of being in general position as stated in Definition~\ref{Def:gencomb}. A converse is not true for two reasons: first, the $P_i$ are fixed and no random variables. Second, even if one would define genericity in terms of random variables such that the hyperplane (resp.~hypersurface) conditions are never fulfilled, there are no statements made on conditionals or algebraic properties other than containment in a hyperplane, also Lebesgue zero sets are not excluded from occuring with positive probability.
Another example where genericity classically occurs is algebraic geometry, where it is defined rather general for moduli spaces. While the exact definition may depend on the situation or the particular moduli space in question, and is also not completely consistent, in most cases, genericity is defined as follows: general, or generic, properties are properties which hold on a Zariski-open subset of an (irreducible) variety, while very generic properties hold on a countable intersection of Zariski-open subsets (which are thus paradoxically ''less'' generic than general resp.~generic properties in the algebraic sense, as any general resp.~generic property is very generic, but the converse is not necessarily true). In our special situation, which is the affine parameter space of tuples of polynomials, these definitions can be rephrased as follows: \begin{Def}\label{Def:gencomb} Let $B\subseteq\mathbb{C}^k$ be an irreducible algebraic set, let $P=(f_1,\dots, f_m)$ be a tuple of polynomials, viewed as a point in the parameter space $B.$ Then a statement resp.~property $A$ of $P$ is called very generic if it holds on the complement of some countable union of algebraic sets in $B.$ A statement resp.~property $A$ of $P$ is called general (or generic) if it holds on the complement of some finite union of algebraic sets in $B.$ \end{Def} This definition is more or less equivalent to our own; however, our definition adds the practical interpretation of generic/very generic/general properties being true with probability one, while their negations are subsequently true with probability zero. In more detail, the correspondence is as follows:
If we restrict ourselves only to algebraic properties $A$, it is equivalent to say that the property $A$ is very generic, or general for the $P$ in $B$, and to say with our original definition that a generic $P$ fulfilling $B$ is also $A$; since if $A$ is by assumption an algebraic property, it is both an algebraic set and a complement of a finite (countable) union of algebraic sets in an irreducible algebraic set, so $A$ must be equal to an irreducible component of $B$; since $B$ is irreducible, this implies equality of $A$ and $B$. On the other hand, if $A$ is an algebraic property, it is equivalent to say that the property not-$A$ is very generic, or general for the $P$ in $B$, and to say with our original definition that a generic $P$ fulfilling $B$ is not $A$ - this corresponds intuitively to the probability-zero condition $P(A|B)=0$ which states that non-generic cases do not occur. Note that by assumption, not-$A$ is then always the complement of a finite union of algebraic sets.
\subsection{Arithmetic of generic polynomials} In this subsection, we study how generic polynomials behave under classical operations in rings and ideals. This will become important later when we study generic polynomials and ideals.\\
To introduce the reader to our notation of genericity, and since we will use the presented facts and similar notations implicitly later, we prove the following
\begin{Lem}\label{Lem:Gen-arith} Let $f\in\C [X_1,\dots, X_D]$ be generic of degrees $k.$ Then:\\ \itboxx{i} The product $\alpha f$ is generic of degree $k$ for any fixed $\alpha\in\C\setminus \{\0\}.$\\ \itboxx{ii} The sum $f + g$ is generic of degree $k$ for any $g\in \C [X_1,\dots, X_D]$ of degree $k$ or smaller.\\ \itboxx{iii} The sum $f + g$ is generic of degree $k$ for any generic $g\in \C [X_1,\dots, X_D]$ of degree $k$ or smaller. \end{Lem} \begin{proof} (i) is clear since the coefficients of $g_1$ are multiplied only by a constant. (ii) follows directly from the definitions since adding a constant $g$ only shifts the coefficients without changing genericity. (iii) follows since $f,g$ are independently sampled: if there were algebraic dependencies between the coefficients of $f+g$, then either $f$ or $g$ was not generic, or the $f,g$ are not independent, which both would be a contradiction to the assumption. \end{proof}
Recall again what this Lemma means: for example, Lemma~\ref{Lem:Gen-arith} (i) does not say, as one could think:\\
``Let $X$ be a generic random variable with values in the vector space of degree $k$ polynomials. Then $X=\alpha X$ for any $\alpha\in \C\setminus \{0\}.$''\\
The correct translation of Lemma~\ref{Lem:Gen-arith} (i) is:\\
``Let $X$ be a generic random variable with values in the vector space of degree $k$ polynomials. Then $X'=\alpha X$ for any fixed $\alpha\in \C\setminus \{0\}$ is a generic random variable with values in the vector space of degree $k$ polynomials''\\
The other statements in Lemma~\ref{Lem:Gen-arith} have to be interpreted similarly.\\
The following remark states how genericity translates through dehomogenization: \begin{Lem}\label{Lem:dehom} Let $f\in\C [X_1,\dots, X_D]$ be a generic homogenous polynomial of degree $d.$ \\ Then the dehomogenization $f(X_1,\dots, X_{D-1},1)$ is a generic polynomial of degree $d$ in the polynomial ring $\C [X_1,\dots, X_{D-1}].$\\
Similarly, let $\fraks\idof \C [X_1,\dots, X_D]$ be a generic homogenous ideal. Let $f\in \fraks$ be a generic homogenous polynomial of degree $d.$ \\ Then the dehomogenization $f(X_1,\dots, X_{D-1},1)$ is a generic polynomial of degree $d$ in the dehomogenization of $\fraks.$ \end{Lem} \begin{proof} For the first statement, it suffices to note that the coefficients of a homogenous polynomial of degree $d$ in the variables $X_1,\dots, X_D$ are in bijection with the coefficients of a polynomial of degree $d$ in the variables $X_1,\dots, X_{D-1}$ by dehomogenization. For the second part, recall that the dehomogenization of $\fraks$ consists exactly of the dehomogenizations of elements in $\fraks.$ In particular, note that the homogenous elements of $\fraks$ of degree $d$ are in bijection to the elements of degree $d$ in the dehomogenization of $\fraks$. The claims then follows from the definition of genericity. \end{proof}
\subsection{Generic spans and generic height theorem} In this subsection, we will derive the first results on generic ideals. We will derive an statement about spans of generic polynomials, and generic versions of Krull's principal ideal and height theorems which will be the main tool in controlling the structure of generic ideals. This has immediate applications for the cumulant comparison problem.
Now we present the first result which can be easily formulated in terms of genericity:
\begin{Prop}\label{Prop:GenVec} Let $P$ be an algebraic property such that the polynomials with property $P$ form a vector space $V$. Let $f_1,\dots, f_m\in \C[X_1,\dots X_D]$ be generic polynomials satisfying $P.$ Then $$\rk \lspan (f_1,\dots, f_m)=\min (m, \dim V).$$ \end{Prop} \begin{proof} It suffices to prove: if $i\le M,$ then $f_i$ is linearly independent from $f_1,\dots f_{i-1}$ with probability one. Assuming the contrary would mean that for some $i$, we have $$f_i=\sum_{k=0}^{i-1}f_k c_k\quad\mbox{for some}\; c_k\in \CC,$$ thus giving several equations on the coefficients of $f_i.$ But these are fulfilled with probability zero by the genericity assumption, so the claim follows. \end{proof}
This may be seen as a straightforward generalization of the statement: the span of $n$ generic points in $\C^D$ has dimension $\min (n,D).$
We now proceed to another nontrivial result which will now allow us to formulate a generic version of Krull's principal ideal theorem:
\begin{Prop}\label{Prop:NoZero} Let $Z\subseteq \C^D$ be a non-empty algebraic set, let $f\in \C[X_1,\dots X_D]$ generic. Then $f$ is no zero divisor in $\calO(Z)=\C[X_1,\dots X_D]/\Id(Z).$ \end{Prop} \begin{proof} We claim: being a zero divisor in $\calO(Z)$ is an irreducible algebraic property. We will prove that the zero divisors in $\calO(Z)$ form a linear subspace of $\calM_k,$ and linear spaces are irreducible.\\
For this, one checks that sums and scalar multiples of zero divisors are also zero divisors: if $g_1,g_2$ are zero divisors, there must exist $h_1,h_2$ such that $g_1h_1=g_2h_2=0.$ Now for any $\alpha\in \C,$ we have that $$(g_1+\alpha g_2) (h_1h_2)=(g_1h_1)h_2 + (g_2h_2)\alpha h_1= 0.$$ This proves that $(g_1+\alpha g_2)$ is also a zero divisor, proving that the zero divisors form a linear subspace and thus an irreducible algebraic property.
To apply the genericity assumption to argue that this event occurs with probability zero, we must exclude the possibility that being a zero divisor is trivial, i.e.~always the case. This is equivalent to proving that the linear subspace has positive codimension, which is true if and only if there exists a non-zero divisor in $\calO(Z).$ But a non-zero divisor always exists since we have assumed $Z$ is non-empty: thus $\Id(Z)$ is a proper ideal, and $\calO(Z)$ contains $\C,$ which contains a non-zero divisor, e.g.~the one.
So by the genericity assumption, the event that $f$ is a zero divisor occurs with probability zero, i.e.~a generic $f$ is not a zero divisor. Note that this does not depend on the degree of $f.$ \end{proof} Note that this result is already known, compare Conjecture B in \citep{Par10}.
A straightforward generalization using the same proof technique is given by the following \begin{Cor}\label{Cor:NoZero} Let $\calI\idof \C[X_1,\dots, X_D]$, let $P$ be a non-trivial algebraic property. Let $f\in \C[X_1,\dots X_D]$ be a generic polynomial with property $P$. If one can write $f=f'+c$, where $f'$ is a generic polynomial subject to some property $P'$, and $c$ is a generic constant, then $f$ is no zero divisor in $\C[X_1,\dots, X_D]/\calI.$ \end{Cor} \begin{proof} First note that $f$ is a zero divisor in $\C[X_1,\dots, X_D]/\calI$ if and only if $f$ is a zero divisor in $\C[X_1,\dots, X_D]/\sqrt{\calI}.$ This allows us to reduce to the case that $\calI=\Id (Z)$ for some algebraic set $Z\subseteq \C^D.$
Now, as in the proof of Proposition~\ref{Prop:NoZero}, we see that being a zero divisor in $\calO(Z)$ is an irreducible algebraic property and corresponds to a linear subspace of $\calM_k$, where $k=\deg f.$ The zero divisors with property $P$ are thus contained in this linear subspace. Now let $f$ be generic with property $P$ as above. By assumption, we may write $f=f'+c.$ But $c$ is (generically) no zero divisor, so $f$ is also not a zero divisor, since the zero divisors form a linear subspace of $\calM_k.$ Thus $f$ is no zero divisor. This proves the claim. \end{proof}
Note that Proposition~\ref{Prop:NoZero} is actually a special case of Corollary~\ref{Cor:NoZero}, since we can write any generic polynomial $f$ as $f'+c$, where $f'$ is generic of the same degree, and $c$ is a generic constant.
The major tool to deal with the dimension of generic intersections is Krull's principal ideal theorem: \begin{Thm}[Krull's principal ideal theorem]\label{Thm:KrullPI} Let $R$ be a commutative ring with unit, let $f\in R$ be non-zero and non-invertible. Then $$\htid \langle f\rangle\le 1,$$ with equality if and only if $f$ is not a zero divisor in $R$. \end{Thm} The reader unfamiliar with height theory may take $$\htid \calI = \codim \VS(\calI)$$ as the definition for the height of an ideal (caveat: codimension has to be taken in $R$).
Reformulated geometrically for our situation, Krull's principal ideal theorem implies: \begin{Cor}\label{Cor:KrullPI-geom} Let $Z$ be a non-empty algebraic set in $\C^D.$Then $$\codim (Z\cap \VS(f))\le \codim Z+1.$$ \end{Cor} \begin{proof} Apply Krull's principal ideal theorem to the ring $R=\calO(Z)=\C [X_1,\dots, X_D]/\Id(Z).$ \end{proof} Together with Proposition~\ref{Prop:NoZero}, one gets a generic version of Krull's principal ideal theorem:
\begin{Thm}[Generic principal ideal theorem]\label{Thm:KrullGenPI} Let $Z$ be a non-empty algebraic set in $\C^D$, let $R=\calO (Z),$ and let $f\in \C[X_1,\dots, X_D]$ be generic. Then we have $$\htid \langle f\rangle = 1.$$ \end{Thm} In its geometric formulation, we obtain the following result. \begin{Cor}\label{Cor:KrullPI} Consider an algebraic set $Z\subseteq \C^D,$ and the algebraic set $\VS(f)$ for some generic $f\in \C [X_1,\dots, X_D].$ Then $$\codim (Z\cap \VS (f))=\min (\codim Z + 1,\; D+1).$$ \end{Cor} \begin{proof} This is just a direct reformulation of Theorem~\ref{Thm:KrullGenPI} in the vein of Corollary~\ref{Cor:KrullPI-geom}. The only additional thing that has to be checked is the case where $\codim Z = D+1,$ which means that $Z$ is the empty set. In this case, the equality is straightforward. \end{proof}
The generic version of the principal ideal theorem straightforwardly generalizes to a generic version of Krull's height theorem. We first mention the original version: \begin{Thm}[Krull's height theorem]\label{Thm:KrullHt} Let $R$ be a commutative ring with unit, let $\calI=\langle f_1,\dots, f_m \rangle \idof R$ be an ideal. Then $$\htid \calI \le m,$$ with equality if and only if $f_1,\dots, f_m$ is an $R$-regular sequence, i.e.~$f_i$ is not invertible and not a zero divisor in the ring $R/\langle f_1,\dots, f_{i-1}\rangle$ for all $i$. \end{Thm} The generic version can be derived directly from the generic principal ideal theorem:
\begin{Thm}[Generic height theorem]\label{Thm:KrullGenHt} Let $Z$ be an algebraic set in $\C^D,$ let $\calI=\langle f_1,\dots, f_m\rangle$ be a generic ideal in $\C [X_1,\dots, X_D].$ Then $$\htid (\Id(Z)+\calI) = \min (\codim Z + m,\; D+1).$$ \end{Thm} \begin{proof} We will write $R=\calO(Z)$ for abbreviation.
First assume $m\le D+1-\codim Z.$ It suffices to show that $f_1,\dots, f_m$ forms an $R$-regular sequence, then apply Krull's height theorem. In Proposition~\ref{Prop:NoZero}, we have proved that $f_i$ is not a zero divisor in the ring $\calO(Z\cap\VS(f_1,\dots, f_{i-1}))$ (note that the latter ring is nonzero by Krull's height theorem). By Hilbert's Nullstellensatz, this is the same as the ring $R/\sqrt{\langle f_1,\dots, f_{i-1}\rangle}.$ But by the definition of radical, this implies that $f_i$ is no zero divisor in the ring $R/\langle f_1,\dots, f_{i-1}\rangle,$ since if $f_i\cdot h=0$ in the first ring, we have $$(f_i\cdot h)^N=f_i\cdot (f_i^{N-1}h^N)=0$$ in the second. Thus the $f_i$ form an $R$-regular sequence, proving the theorem for the case $m\le D+1-\codim Z.$
If now $m> k:=D+1-\codim Z,$ the above reasoning shows that the radical of $\Id(Z)+\langle f_1,\dots, f_k\rangle$ is the module $\langle 1\rangle,$ which means that those are equal. Thus $$\Id(Z)+\langle f_1,\dots, f_k\rangle=\Id(Z)+\langle f_1,\dots, f_m\rangle=\langle 1\rangle,$$ proving the theorem.
Note that we could have proved the generic height theorem also directly from the generic principal ideal theorem by induction. \end{proof}
Again, we give the geometric interpretation of Krull's height theorem:
\begin{Cor}\label{Cor:genint} Let $Z_1$ be an algebraic set in $\C^D$, let $Z_2$ be a generic algebraic set in $\C^D$. Then one has $$\codim (Z_1\cap Z_2)=\min (\codim Z_1+\codim Z_2,\; D+1).$$ \end{Cor} \begin{proof} This follows directly from two applications of the generic height theorem~\ref{Thm:KrullGenHt}: first for $Z=\C^D$ and $Z_2=\VS(\calI)$, showing that $\codim Z_2$ is equal to the number $m$ of generators of $\calI;$ then, for $Z=Z_1$ and $Z_2=\VS(\calI),$ and substituting $m=\codim Z_2.$ \end{proof}
We can now immediately formulate a homogenous version of Proposition~\ref{Cor:genint}:
\begin{Cor}\label{Cor:genintproj} Let $Z_1$ be a homogenous algebraic set in $\C^D$, let $Z_2$ be a generic homogenous algebraic set in $\C^D$. Then one has $$\codim (Z_1\cap Z_2)=\min (\codim Z_1+\codim Z_2,\; D).$$ \end{Cor} \begin{proof} Note that homogenization and dehomogenization of a non-empty algebraic set do not change its codimension, and homogenous algebraic sets always contain the origin. Also, one has to note that by Lemma~\ref{Lem:dehom}, the dehomogenization of $Z_2$ is a generic algebraic set in $\C^{D-1}.$ \end{proof}
Finally, using Corollary~\ref{Cor:NoZero}, we want to give a more technical variant of the generic height theorem, which will be of use in later proofs. First, we introduce some abbreviating notations: \begin{Def} Let $f\in \C[X_1,\dots X_D]$ be a generic polynomial with property $P$. If one can write $f=f'+c$, where $f'$ is a generic polynomial subject to some property $P'$, and $c$ is a generic constant, we say that $f$ has {\it independent constant term}. If $c$ is generic and independent with respect to some collection of generic objects, we say that $f$ has independent constant term with respect to that collection. \end{Def} In this terminology, Corollary~\ref{Cor:NoZero} rephrases as: a generic polynomial with independent constant term is no zero divisor. Using this, we can now formulate the corresponding variant of the generic height theorem:
\begin{Lem}\label{Lem:KrullGenHt} Let $Z$ be an algebraic set in $\C^D.$ Let $f_1,\dots, f_m\in\C[X_1,\dots, X_D]$ be generic, possibly subject to some algebraic properties, such that $f_i$ has independent constant term with respect to $Z$ and $f_1,\dots, f_{i-1}.$ Then $$\htid (\Id(Z)+\calI) = \min (\codim Z + m,\; D+1).$$ \end{Lem} \begin{proof} Using Corollary~\ref{Cor:NoZero}, one obtains that $f_i$ is no zero divisor modulo $\Id(Z)+\langle f_1,\dots, f_{i+1}\rangle.$ Using Krull's height theorem yields the claim. \end{proof}
\subsection{Generic ideals} The generic height theorem~\ref{Thm:KrullGenHt} has allowed us to make statements about the structure of ideals generated by generic elements without constraints. However, the ideal $\calI$ in our the cumulant comparison problem is generic subject to constraints: namely, its generators are contained in a prescribed ideal, and they are homogenous. In this subsection, we will use the theory developed so far to study generic ideals and generic ideals subject to some algebraic properties, e.g.~generic ideals contained in other ideals. We will use these results to derive an identifiability result on the marginalization problem which has been derived already less rigourously in the supplementary material of \citep{PRL:SSA:2009} for the special case of Stationary Subspace Analysis.
\begin{Prop}\label{Prop:dehom-rad-generic} Let $\fraks \idof \C [X_1,\dots, X_D]$ be an ideal, having an H-basis $g_1,\dots, g_n$. Let $$\calI=\langle f_1,\dots, f_m\rangle,\quad m\ge \max(D+1, n)$$ with generic $f_i\in \fraks$ such that $$\deg f_i\ge \max_j \left(\deg g_j\right)\quad \mbox{for all}\; 1\le i\le m.$$ Then $\calI=\fraks.$ \end{Prop} \begin{proof} First note that since the $g_i$ form a degree-first Groebner basis, a generic $f\in \fraks$ is of the form $$f=\sum_{k=1}^n g_kh_k\quad\mbox{with generic}\;h_k,$$ where the degrees of the $h_k$ are appropriately chosen, i.e. $\deg h_k\le \deg f - \deg g_k$.
So we may write $$f_i=\sum_{k=1}^n g_kh_{ki}\quad\mbox{with generic}\;h_{ki},$$ where the $h_{ki}$ are generic with appropriate degrees, and independently chosen. We may also assume that the $f_i$ are ordered increasingly by degree.\\
To prove the statement, it suffices to show that $g_j\in \calI$ for all $j$. Now the height theorem~\ref{Thm:KrullGenHt} implies that $$\langle h_{11},\dots h_{1m}\rangle=\langle 1\rangle,$$ since the $h_{ki}$ were independently generic, and $m\ge D+1.$ In particular, there exist polynomials $s_1,\dots, s_m$ such that $$\sum_{i=1}^m s_i h_{1i}=1.$$ Thus we have that \begin{align*} \sum_{i=1}^m s_i f_i = \sum_{i=1}^m s_i \sum_{k=1}^n g_kh_{ki}= \sum_{k=1}^n g_k \sum_{i=1}^m s_ih_{ki}\\ =g_1+ \sum_{k=2}^n g_k \sum_{i=1}^m s_ih_{ki}=:g_1+ \sum_{k=2}^n g_k h'_k. \end{align*} Subtracting a suitable multiple of this element from the $f_1,\dots, f_m,$ we obtain $$f'_i=\sum_{k=2}^n g_k(h_{ki}-h_{1i}h'_k)=:\sum_{k=2}^n g_k h'_{ki}.$$ We may now consider $h_{1i}h'_k$ as fixed, while the $h_{ki}$ are generic. In particular, the $h'_{ki}$ have independent constant term, and using Lemma~\ref{Lem:KrullGenHt}, we may conclude that $$\langle h'_{21},\dots, h'_{2m} \rangle=\langle 1\rangle,$$ allowing us to find an element of the form $$g_2+\sum_{k=3}^n g_k \cdot\dots$$ in $\calI$. Iterating this strategy by repeatedly applying Lemma~\ref{Lem:KrullGenHt}, we see that $g_k$ is contained in $\calI,$ because the ideals $\calI$ and $\fraks$ have same height. Since the numbering for the $g_j$ was arbitrary, we have proved that $g_j\in \calI$, and thus the proposition. \end{proof} The following example shows that we may not take the degrees of the $f_i$ completely arbitrary in the proposition, i.e.~the condition on the degrees is necessary: \begin{Ex}\rm Keep the notations of Proposition~\ref{Prop:dehom-rad-generic}. Let $\fraks=\langle X_2-X_1^2, X_3\rangle,$ and $f_i\in \fraks$ generic of degree one. Then $$\langle f_1,\dots, f_m\rangle = \langle X_3\rangle.$$ This example can be generalized to yield arbitrarily bad results if the condition on the degrees is not fulfilled.
However note that when $\fraks$ is generated by linear forms, as in the marginalization problem, the condition on the degrees vanishes. \end{Ex}
We may use Proposition~\ref{Prop:dehom-rad-generic} also in another way to derive a more detailed version of the generic height theorem for constrained ideals: \begin{Prop}\label{Prop:KrullHt-algset} Let $V$ be a fixed $d$-codimensional algebraic set in $\C^D.$ Assume that there exist $d$ generators $g_1,\dots, g_d$ for $\Id(V).$ Let $f_1,\dots, f_m$ be generic forms in $\Id (V)$ such that $\deg f_i\ge \deg g_i$ for $1\le i\le \min (m,d)$. Then we can write $\VS (f_1,\dots, f_m)=V\cup U$ with $U$ an algebraic set of $$\codim U\ge\min (m,\;D+1),$$ the equality being strict for $m < \codim V.$ \end{Prop} \begin{proof} If $m\ge D+1$, this is just a direct consequence of Proposition~\ref{Prop:dehom-rad-generic}.
First assume $m = d.$ Consider the image of the situation modulo $X_{m},\dots, X_D.$ This corresponds to looking at the situation $$\VS (f_1,\dots, f_m)\cap H\subseteq H\cong \C^{m-1},$$ where $H$ is the linear subspace given by $X_m=\dots = X_D=0.$ Since the coordinate system was generic, the images of the $f_i$ will be generic, and we have by Proposition~\ref{Prop:dehom-rad-generic} that $\VS (f_1,\dots, f_m)\cap H = V\cap H.$ Also, the $H$ can be regarded as a generic linear subspace, thus by Corollary~\ref{Cor:genint}, we see that $\VS (f_1,\dots, f_m)$ consists of $V$ and possibly components of equal or higher codimension. This proves the claim for $m = \codim V.$
Now we prove the case $m\ge d.$ We may assume that $m=D+1$ and then prove the statement for the sets $\VS (f_1,\dots, f_i), d\le i\le m.$ By the Lasker-Noether-Theorem, we may write $$\VS (f_1,\dots, f_d)= V \cup Z_1 \cup\dots \cup Z_N$$ for finitely many irreducible components $Z_j$ with $\codim Z_j\ge \codim V.$ Proposition~\ref{Prop:dehom-rad-generic} now states that $$\VS (f_1,\dots, f_m)=V.$$ For $i\ge d,$ write now $$Z_{ji}=Z_j\cap \VS (f_1,\dots, f_i)= Z_j\cap \VS (f_{d+1},\dots, f_i).$$ With this, we have the equalities \begin{align*} \VS (f_1,\dots, f_i)&= \VS (f_1,\dots, f_d)\cap \VS (f_{d+1},\dots, f_i)\\ &= V \cup (Z_1\cap \VS (f_{d+1},\dots, f_i))\cup\dots \cup (Z_N\cap \VS (f_{d+1},\dots, f_i))\\ &= V\cup Z_{1i}\cup\dots\cup Z_{Ni}. \end{align*} for $i\ge d.$ Thus, reformulated, Proposition~\ref{Prop:dehom-rad-generic} states that $Z_{jm}=\varnothing$ for any $j$. We can now infer by Krull's principal ideal theorem~\ref{Thm:KrullPI} that $$\codim Z_{ji}\le \codim Z_{j,i-1}+1$$ for any $i,j$. But since $\codim Z_{jm}=D+1,$ and $\codim Z_{jd}\ge d,$ we thus may infer that $\codim Z_{ji}\ge i$ for any $d\le i\le m.$ Thus we may write $$\VS (f_1,\dots, f_i)=V\cup U\quad\mbox{with}\;U=Z_{1i}\cup\dots\cup Z_{Ni}$$ with $\codim U\ge i,$ which proves the claim for $m\ge \codim V.$
The case $m < \codim V$ can be proved again similarly by Krull's principal ideal theorem~\ref{Thm:KrullPI}: it states that the codimension of $\VS (f_1,\dots, f_i)$ increases at most by one with each $i$, and we have seen above that it is equal to $\codim V$ for $i=\codim V.$ Thus the codimension of $\VS (f_1,\dots, f_i)$ must have been $i$ for every $i\le \codim V.$ This yields the claim. \end{proof} Note that depending on $V$ and the degrees of the $f_i,$ it may happen that even in the generic case, the equality in Proposition~\ref{Prop:KrullHt-algset} is not strict for $m\ge \codim V$: \begin{Ex}\rm Let $V$ be a generic linear subspace of dimension $d$ in $\C^D,$ let $f_1,\dots, f_m\in \Id(V)$ be generic with degree one. Then $\VS (f_1,\dots, f_m)$ is a generic linear subspace of dimension $\max (D-m, d)$ containing $V.$ In particular, if $m\ge D-d,$ then $\VS (f_1,\dots, f_m)=V.$ In this example, $U= \VS(f_1,\dots, f_m)$, if $m < \codim V,$ with codimension $m$, and $U=\varnothing$, if $m\ge \codim V,$ with codimension $D+1.$
Similarly, one may construct generic examples with arbitrary behavior for $\codim U$ when $m\ge \codim V,$ by choosing $V$ and the degrees of $f_i$ appropriately. \end{Ex}
Similarly as in the geometric version for the height theorem, we may derive the following geometric interpretation of this result: \begin{Cor} Let $V\subseteq Z_1$ be fixed algebraic sets in $\C^D$. Let $Z_2$ be a generic algebraic set in $\C^D$ containing $V.$ Then $$\codim (Z_1\cap Z_2 \setminus V)\ge \min (\codim (Z_1 \setminus V) + \codim (Z_2 \setminus V),\; D+1).$$ \end{Cor} Informally, we have derived a height theorem type result for algebraic sets under the constraint that they contain another prescribed algebraic set $V$. \\
We also want to give a homogenous version of Proposition~\ref{Prop:KrullHt-algset}, since the ideals in the paper are generated by homogenous forms: \begin{Cor}\label{Cor:KrullHt-Hom} Let $V$ be a fixed homogenous algebraic set in $\C^D$. Let $f_1,\dots, f_m$ be generic homogenous forms in $\Id (V),$ satisfying the degree condition as in Proposition \ref{Prop:KrullHt-algset}. Then $\VS (f_1,\dots, f_m)=V+ U$ with $U$ an algebraic set fulfilling $$\codim U\ge \min (m,\;D).$$ In particular, if $m> D,$ then $\VS (f_1,\dots, f_m)=V.$ Also, the maximal dimensional part of $\VS (f_1,\dots, f_m)$ equals $V$ if and only if $m > D- \dim V.$ \end{Cor} \begin{proof} This follows immediately by dehomogenizing, applying Proposition~\ref{Prop:KrullHt-algset}, and homogenizing again. \end{proof} From this Corollary, we now can directly derive a statement on the necessary number of epochs for the identifiability of the projection making several random variables appear identical. For the convenience of the reader, we recall the setting and then explain what identifiability means. The problem we consider in the main part of the paper can be described as follows:
\begin{Prob} Let $X_1,\dots, X_{m}$ be smooth random variables, let $$q_i=[T_1,\dots, T_D]\circ \left(\kappa_2(X_i)-\kappa_2(X_{m})\right),\;1\le i\le {m-1}$$ and $$f_i=[T_1,\dots, T_D]\circ\left( \kappa_1(X_i)-\kappa_1(X_{m})\right),\;1\le i\le {m-1}$$ be the corresponding polynomials in the formal variables $T_1,\dots, T_D$. What can one say about the set $$S'=\VS ( q_1, \ldots, q_{m-1}, f_1,\dots, f_{m-1} ).$$ \end{Prob} If there is a linear subspace $S$ on which the cumulants agree, then the $q_i, f_i$ vanish on $S$. If we assume that this happens generically, the problem reformulates to \begin{Prob} Let $S$ be a $d$-dimensional linear subspace of $\C^D$, let $\fraks=\Id(S),$ and let $f_1,\dots, f_{N}$ be generic homogenous quadratic or linear polynomials in $\fraks$. How does $S'=\VS ( f_1,\dots, f_{N} )$ relate to $S$?. \end{Prob}
Before giving bounds on the identifiability, we first begin with a direct consequence of Corollary~\ref{Cor:KrullHt-Hom}: \begin{Rem}\label{Rem:cumpoly} The highest dimensional part of $S'=\VS (f_1,\dots, f_N)$ is $S$ if and only if $$ N > D - d.$$ \end{Rem} For this, remark that $\Id(S)$ is generated in degree one, and thus the degree condition in Corollary \ref{Cor:KrullHt-Hom} becomes empty.
We can now also get an identifiability result for $S$:
\begin{Prop}\label{Prop:ident_ex} Let $f_1,\dots, f_{N}$ be generic homogenous polynomials of degree one or two, vanishing on a linear space $S$ of dimension $d>0$. Then $S$ is identifiable from the $f_i$ alone if $$N \ge D-d+1.$$ Moreover, if all $f_i$ are quadrics, then $S$ is identifiable from the $f_i$ alone only if $$N\ge 2.$$ \end{Prop} \begin{proof} Note that the $f_1,\dots, f_N$ are generic polynomials contained in $\fraks:=\Id(S)$.
First assume $N \ge D-d+1.$ We prove that $S$ is identifiable: using Corollary~\ref{Cor:KrullHt-Hom}, one sees now that the common vanishing set of the $f_i$ is $S$ up to possible additional components of dimension $d-1$ or less. I.e.~the radical of the ideal generated by the $f_i$ has a prime decomposition $$\sqrt{\langle f_1,\dots, f_{N}\rangle} = \fraks\cap \frakp_1\cap \dots \cap\frakp_k,$$ where the $\frakp_i$ are of dimension $d-1$ or less, while $\fraks$ has dimension $d$. So one can use one of the existing algorithms calculating primary decomposition to identify $\fraks$ as the unique component of the highest dimensional part, which proves identifiability if $N \ge D-d+1$.
Now we prove the only if part: assume that $N=1$, i.e.~we have only a single $f_1$. Since $f_1$ is generic with the property of vanishing on $S$, we have $$f_1=\sum_{i=1}^{D-d}g_ih_i,$$ where $g_1,\dots, g_{D-d}$ is some homogenous linear generating set for $\Id (S),$ and $h_1,\dots, h_{D-d}$ are generic homogenous linear forms. Thus, the zero set $\VS(f_1)$ also contains the linear space $S'=\VS (h_1,\dots, h_{D-d})$ which is a generic $d$-dimensional linear space in $\mathbb{C}^D$ and thus different from $S$; no algorithm can decide whether $S$ or $S'$ is the correct solution, so $S$ is not identifiable. \end{proof}
Note that there is no obvious reason for the lower bound $N \ge D-d+1$ given in Proposition~\ref{Prop:ident_ex} to be strict. While it is most probably the best possible bound which is in $D$ and $d$, in general it may happen that $S$ can be reconstructed from the ideal $\langle f_1,\dots, f_{N}\rangle$ directly. The reason for this is that a generic homogenous variety of high enough degree and dimension does not need to contain a linear subspace of fixed dimension $d$ in general.
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\end{document} | arXiv |
Hessian automatic differentiation
In applied mathematics, Hessian automatic differentiation are techniques based on automatic differentiation (AD) that calculate the second derivative of an $n$-dimensional function, known as the Hessian matrix.
When examining a function in a neighborhood of a point, one can discard many complicated global aspects of the function and accurately approximate it with simpler functions. The quadratic approximation is the best-fitting quadratic in the neighborhood of a point, and is frequently used in engineering and science. To calculate the quadratic approximation, one must first calculate its gradient and Hessian matrix.
Let $f:\mathbb {R} ^{n}\rightarrow \mathbb {R} $, for each $x\in \mathbb {R} ^{n}$ the Hessian matrix $H(x)\in \mathbb {R} ^{n\times n}$ is the second order derivative and is a symmetric matrix.
Reverse Hessian-vector products
For a given $u\in \mathbb {R} ^{n}$, this method efficiently calculates the Hessian-vector product $H(x)u$. Thus can be used to calculate the entire Hessian by calculating $H(x)e_{i}$, for $i=1,\ldots ,n$.[1]
The method works by first using forward AD to perform $f(x)\rightarrow u^{T}\nabla f(x)$, subsequently the method then calculates the gradient of $u^{T}\nabla f(x)$ using Reverse AD to yield $\nabla \left(u\cdot \nabla f(x)\right)=u^{T}H(x)=(H(x)u)^{T}$. Both of these two steps come at a time cost proportional to evaluating the function, thus the entire Hessian can be evaluated at a cost proportional to n evaluations of the function.
Reverse Hessian: Edge_Pushing
An algorithm that calculates the entire Hessian with one forward and one reverse sweep of the computational graph is Edge_Pushing. Edge_Pushing is the result of applying the reverse gradient to the computational graph of the gradient. Naturally, this graph has n output nodes, thus in a sense one has to apply the reverse gradient method to each outgoing node. Edge_Pushing does this by taking into account overlapping calculations.[2]
The algorithm's input is the computational graph of the function. After a preceding forward sweep where all intermediate values in the computational graph are calculated, the algorithm initiates a reverse sweep of the graph. Upon encountering a node that has a corresponding nonlinear elemental function, a new nonlinear edge is created between the node's predecessors indicating there is nonlinear interaction between them. See the example figure on the right. Appended to this nonlinear edge is an edge weight that is the second-order partial derivative of the nonlinear node in relation to its predecessors. This nonlinear edge is subsequently pushed down to further predecessors in such a way that when it reaches the independent nodes, its edge weight is the second-order partial derivative of the two independent nodes it connects.[2]
Graph colouring techniques for Hessians
The graph colouring techniques explore sparsity patterns of the Hessian matrix and cheap Hessian vector products to obtain the entire matrix. Thus these techniques are suited for large, sparse matrices. The general strategy of any such colouring technique is as follows.
1. Obtain the global sparsity pattern of $H$
2. Apply a graph colouring algorithm that allows us to compact the sparsity structure.
3. For each desired point $x\in \mathbb {R} ^{n}$ calculate numeric entries of the compact matrix.
4. Recover the Hessian matrix from the compact matrix.
Steps one and two need only be carried out once, and tend to be costly. When one wants to calculate the Hessian at numerous points (such as in an optimization routine), steps 3 and 4 are repeated.
As an example, the figure on the left shows the sparsity pattern of the Hessian matrix where the columns have been appropriately coloured in such a way to allow columns of the same colour to be merged without incurring in a collision between elements.
There are a number of colouring techniques, each with a specific recovery technique. For a comprehensive survey, see.[3] There have been successful numerical results of such methods.[4]
References
1. Bruce Christianson. Automatic Hessians by Reverse Accumulation, http://imajna.oxfordjournals.org/content/12/2/135.abstract.
2. R. Gower, M. Mello. A new framework for the computation of Hessians. In: Optimization Methods and Software. doi: http://www.tandfonline.com/doi/full/10.1080/10556788.2011.580098.
3. A. H. Gebremedhin, A. Tarafdar, A. Pothen, and A. Walther. Efficient Computation of Sparse Hessians Using Coloring and Automatic Differentiation". In: INFORMS J. on Computing 21.2 (2009), pp. 209-223. doi:10.1287/ijoc.1080.0286
4. A. Walther. Computing sparse Hessians with automatic differentiation". In: ACM Trans. Math. Softw. 34.1 (2008), pp. 1-15. ISSN 0098-3500. doi: http://doi.acm.org/10.1145/1322436.1322439.
| Wikipedia |
Masha has a quadratic of the form $x^2+bx+1$, where $b$ is a specific positive number. Using her knowledge of how to complete the square, Masha is able to rewrite this quadratic in the form $(x+m)^2-63$. What is $b$?
The expansion of $(x+m)^2-63$ is $x^2+2mx+m^2-63$, which has a constant term of $m^2-63$. This constant term must be equal to the constant term of the original quadratic, so $m^2-63 = 1$, which yields the possibilities $m=8$ and $m=-8$.
If $m=8$, then $(x+m)^2-63 = x^2+16x+1$. If $m=-8$, then $(x+m)^2-63 = x^2-16x+1$. Of these two possibilities, only the first conforms to our information that $b$ was a positive number. So, the original quadratic was $x^2+16x+1$, giving $b=\boxed{16}$. | Math Dataset |
Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;[1][2] equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.[3][4][5][6]
The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom,[7][8] before the development of the Schrödinger equation. However, this approach is rarely used today.
In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system.[9] The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere,[10] so that the whole problem is symmetric under certain rotations of the four-dimensional space.[11] This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.[12]
The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector,[13][14] the Runge–Lenz vector[15] and the Lenz vector.[8] Ironically, none of those scientists discovered it.[15] The LRL vector has been re-discovered and re-formulated several times;[15] for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics.[2][14][16] Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.[17][18][19]
Context
A single particle moving under any conservative central force has at least four constants of motion: the total energy E and the three Cartesian components of the angular momentum vector L with respect to the center of force.[20][21] The particle's orbit is confined to the plane defined by the particle's initial momentum p (or, equivalently, its velocity v) and the vector r between the particle and the center of force[20][21] (see Figure 1). This plane of motion is perpendicular to the constant angular momentum vector L = r × p; this may be expressed mathematically by the vector dot product equation r ⋅ L = 0. Given its mathematical definition below, the Laplace–Runge–Lenz vector (LRL vector) A is always perpendicular to the constant angular momentum vector L for all central forces (A ⋅ L = 0). Therefore, A always lies in the plane of motion. As shown below, A points from the center of force to the periapsis of the motion, the point of closest approach, and its length is proportional to the eccentricity of the orbit.[1]
The LRL vector A is constant in length and direction, but only for an inverse-square central force.[1] For other central forces, the vector A is not constant, but changes in both length and direction. If the central force is approximately an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction.[14] A generalized conserved LRL vector ${\mathcal {A}}$ can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form.[18][19]
The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector. Thus, the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum.
History of rediscovery
The LRL vector A is a constant of motion of the Kepler problem, and is useful in describing astronomical orbits, such as the motion of planets and binary stars. Nevertheless, it has never been well known among physicists, possibly because it is less intuitive than momentum and angular momentum. Consequently, it has been rediscovered independently several times over the last three centuries.[15]
Jakob Hermann was the first to show that A is conserved for a special case of the inverse-square central force,[22] and worked out its connection to the eccentricity of the orbital ellipse. Hermann's work was generalized to its modern form by Johann Bernoulli in 1710.[23] At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of A, deriving it analytically, rather than geometrically.[24] In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below,[16] using it to show that the momentum vector p moves on a circle for motion under an inverse-square central force (Figure 3).[12]
At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis.[25] Gibbs' derivation was used as an example by Carl Runge in a popular German textbook on vectors,[26] which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom.[27] In 1926, Wolfgang Pauli used the LRL vector to derive the energy levels of the hydrogen atom using the matrix mechanics formulation of quantum mechanics,[7] after which it became known mainly as the Runge–Lenz vector.[15]
Mathematical definition
An inverse-square central force acting on a single particle is described by the equation
$\mathbf {F} (r)=-{\frac {k}{r^{2}}}\mathbf {\hat {r}} ;$ ;}
The corresponding potential energy is given by $V(r)=-k/r$. The constant parameter k describes the strength of the central force; it is equal to G⋅M⋅m for gravitational and −ke⋅Q⋅q for electrostatic forces. The force is attractive if k > 0 and repulsive if k < 0.
The LRL vector A is defined mathematically by the formula[1]
$\mathbf {A} =\mathbf {p} \times \mathbf {L} -mk\mathbf {\hat {r}} ,$
where
• m is the mass of the point particle moving under the central force,
• p is its momentum vector,
• L = r × p is its angular momentum vector,
• r is the position vector of the particle (Figure 1),
• $\mathbf {\hat {r}} $ is the corresponding unit vector, i.e., $\mathbf {\hat {r}} ={\frac {\mathbf {r} }{r}}$, and
• r is the magnitude of r, the distance of the mass from the center of force.
The SI units of the LRL vector are joule-kilogram-meter (J⋅kg⋅m). This follows because the units of p and L are kg⋅m/s and J⋅s, respectively. This agrees with the units of m (kg) and of k (N⋅m2).
This definition of the LRL vector A pertains to a single point particle of mass m moving under the action of a fixed force. However, the same definition may be extended to two-body problems such as the Kepler problem, by taking m as the reduced mass of the two bodies and r as the vector between the two bodies.
Since the assumed force is conservative, the total energy E is a constant of motion,
$E={\frac {p^{2}}{2m}}-{\frac {k}{r}}={\frac {1}{2}}mv^{2}-{\frac {k}{r}}.$
The assumed force is also a central force. Hence, the angular momentum vector L is also conserved and defines the plane in which the particle travels. The LRL vector A is perpendicular to the angular momentum vector L because both p × L and r are perpendicular to L. It follows that A lies in the plane of motion.
Alternative formulations for the same constant of motion may be defined, typically by scaling the vector with constants, such as the mass m, the force parameter k or the angular momentum L.[15] The most common variant is to divide A by mk, which yields the eccentricity vector,[2][16] a dimensionless vector along the semi-major axis whose modulus equals the eccentricity of the conic:
$\mathbf {e} ={\frac {\mathbf {A} }{mk}}={\frac {1}{mk}}(\mathbf {p} \times \mathbf {L} )-\mathbf {\hat {r}} .$
An equivalent formulation[14] multiplies this eccentricity vector by the major semiaxis a, giving the resulting vector the units of length. Yet another formulation[28] divides A by $L^{2}$, yielding an equivalent conserved quantity with units of inverse length, a quantity that appears in the solution of the Kepler problem
$u\equiv {\frac {1}{r}}={\frac {km}{L^{2}}}+{\frac {A}{L^{2}}}\cos \theta $
where $\theta $ is the angle between A and the position vector r. Further alternative formulations are given below.
Derivation of the Kepler orbits
The shape and orientation of the orbits can be determined from the LRL vector as follows.[1] Taking the dot product of A with the position vector r gives the equation
$\mathbf {A} \cdot \mathbf {r} =A\cdot r\cdot \cos \theta =\mathbf {r} \cdot \left(\mathbf {p} \times \mathbf {L} \right)-mkr,$
where θ is the angle between r and A (Figure 2). Permuting the scalar triple product yields
$\mathbf {r} \cdot \left(\mathbf {p} \times \mathbf {L} \right)=\left(\mathbf {r} \times \mathbf {p} \right)\cdot \mathbf {L} =\mathbf {L} \cdot \mathbf {L} =L^{2}$
Rearranging yields the solution for the Kepler equation
${\frac {1}{r}}={\frac {mk}{L^{2}}}+{\frac {A}{L^{2}}}\cos \theta $
This corresponds to the formula for a conic section of eccentricity e
${\frac {1}{r}}=C\cdot \left(1+e\cdot \cos \theta \right)$
where the eccentricity $e={\frac {A}{\left|mk\right|}}\geq 0$ and C is a constant.[1]
Taking the dot product of A with itself yields an equation involving the total energy E,[1]
$A^{2}=m^{2}k^{2}+2mEL^{2},$
which may be rewritten in terms of the eccentricity,[1]
$e^{2}=1+{\frac {2L^{2}}{mk^{2}}}E.$
Thus, if the energy E is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"[1]), the eccentricity is greater than one and the orbit is a hyperbola.[1] Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola.[1] In all cases, the direction of A lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.[1]
Circular momentum hodographs
The conservation of the LRL vector A and angular momentum vector L is useful in showing that the momentum vector p moves on a circle under an inverse-square central force.[12][15]
Taking the dot product of
$mk{\hat {\mathbf {r} }}=\mathbf {p} \times \mathbf {L} -\mathbf {A} $
with itself yields
$(mk)^{2}=A^{2}+p^{2}L^{2}+2\mathbf {L} \cdot (\mathbf {p} \times \mathbf {A} ).$
Further choosing L along the z-axis, and the major semiaxis as the x-axis, yields the locus equation for p,
$p_{x}^{2}+\left(p_{y}-{\frac {A}{L}}\right)^{2}=\left({\frac {mk}{L}}\right)^{2}.$
In other words, the momentum vector p is confined to a circle of radius mk/L = L/ℓ centered on (0, A/L).[29] For bounded orbits, the eccentricity e corresponds to the cosine of the angle η shown in Figure 3. For unbounded orbits, we have $A>mk$ and so the circle does not intersect the $p_{x}$-axis.
In the degenerate limit of circular orbits, and thus vanishing A, the circle centers at the origin (0,0). For brevity, it is also useful to introduce the variable $ p_{0}={\sqrt {2m|E|}}$.
This circular hodograph is useful in illustrating the symmetry of the Kepler problem.
Constants of motion and superintegrability
The seven scalar quantities E, A and L (being vectors, the latter two contribute three conserved quantities each) are related by two equations, A ⋅ L = 0 and A2 = m2k2 + 2 mEL2, giving five independent constants of motion. (Since the magnitude of A, hence the eccentricity e of the orbit, can be determined from the total angular momentum L and the energy E, only the direction of A is conserved independently; moreover, since A must be perpendicular to L, it contributes only one additional conserved quantity.)
This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since the initial time is not determined by a constant of motion. The resulting 1-dimensional orbit in 6-dimensional phase space is thus completely specified.
A mechanical system with d degrees of freedom can have at most 2d − 1 constants of motion, since there are 2d initial conditions and the initial time cannot be determined by a constant of motion. A system with more than d constants of motion is called superintegrable and a system with 2d − 1 constants is called maximally superintegrable.[30] Since the solution of the Hamilton–Jacobi equation in one coordinate system can yield only d constants of motion, superintegrable systems must be separable in more than one coordinate system.[31] The Kepler problem is maximally superintegrable, since it has three degrees of freedom (d = 3) and five independent constant of motion; its Hamilton–Jacobi equation is separable in both spherical coordinates and parabolic coordinates,[17] as described below.
Maximally superintegrable systems follow closed, one-dimensional orbits in phase space, since the orbit is the intersection of the phase-space isosurfaces of their constants of motion. Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross products of all of these gradients. As a result, all superintegrable systems are automatically describable by Nambu mechanics,[32] alternatively, and equivalently, to Hamiltonian mechanics.
Maximally superintegrable systems can be quantized using commutation relations, as illustrated below.[33] Nevertheless, equivalently, they are also quantized in the Nambu framework, such as this classical Kepler problem into the quantum hydrogen atom.[34]
Evolution under perturbed potentials
The Laplace–Runge–Lenz vector A is conserved only for a perfect inverse-square central force. In most practical problems such as planetary motion, however, the interaction potential energy between two bodies is not exactly an inverse square law, but may include an additional central force, a so-called perturbation described by a potential energy h(r). In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow apsidal precession of the orbit.
By assumption, the perturbing potential h(r) is a conservative central force, which implies that the total energy E and angular momentum vector L are conserved. Thus, the motion still lies in a plane perpendicular to L and the magnitude A is conserved, from the equation A2 = m2k2 + 2mEL2. The perturbation potential h(r) may be any sort of function, but should be significantly weaker than the main inverse-square force between the two bodies.
The rate at which the LRL vector rotates provides information about the perturbing potential h(r). Using canonical perturbation theory and action-angle coordinates, it is straightforward to show[1] that A rotates at a rate of,
${\begin{aligned}{\frac {\partial }{\partial L}}\langle h(r)\rangle &={\frac {\partial }{\partial L}}\left\{{\frac {1}{T}}\int _{0}^{T}h(r)\,dt\right\}\\[1em]&={\frac {\partial }{\partial L}}\left\{{\frac {m}{L^{2}}}\int _{0}^{2\pi }r^{2}h(r)\,d\theta \right\},\end{aligned}}$
where T is the orbital period, and the identity L dt = m r2 dθ was used to convert the time integral into an angular integral (Figure 5). The expression in angular brackets, ⟨h(r)⟩, represents the perturbing potential, but averaged over one full period; that is, averaged over one full passage of the body around its orbit. Mathematically, this time average corresponds to the following quantity in curly braces. This averaging helps to suppress fluctuations in the rate of rotation.
This approach was used to help verify Einstein's theory of general relativity, which adds a small effective inverse-cubic perturbation to the normal Newtonian gravitational potential,[35]
$h(r)={\frac {kL^{2}}{m^{2}c^{2}}}\left({\frac {1}{r^{3}}}\right).$
Inserting this function into the integral and using the equation
${\frac {1}{r}}={\frac {mk}{L^{2}}}\left(1+{\frac {A}{mk}}\cos \theta \right)$
to express r in terms of θ, the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be[35]
${\frac {6\pi k^{2}}{TL^{2}c^{2}}},$
which closely matches the observed anomalous precession of Mercury[36] and binary pulsars.[37] This agreement with experiment is strong evidence for general relativity.[38][39]
Poisson brackets
The unscaled functions
The algebraic structure of the problem is, as explained in later sections, SO(4)/Z2 ~ SO(3) × SO(3).[11] The three components Li of the angular momentum vector L have the Poisson brackets[1]
$\{L_{i},L_{j}\}=\sum _{s=1}^{3}\varepsilon _{ijs}L_{s},$
where i=1,2,3 and εijs is the fully antisymmetric tensor, i.e., the Levi-Civita symbol; the summation index s is used here to avoid confusion with the force parameter k defined above. Then since the LRL vector A transforms like a vector, we have the following Poisson bracket relations between A and L:[40]
$\{A_{i},L_{j}\}=\sum _{s=1}^{3}\varepsilon _{ijs}A_{s}.$
Finally, the Poisson bracket relations between the different components of A are as follows:[41]
$\{A_{i},A_{j}\}=-2mH\sum _{s=1}^{3}\varepsilon _{ijs}L_{s},$
where $H$ is the Hamiltonian. Note that the span of the components of A and the components of L is not closed under Poisson brackets, because of the factor of $H$ on the right-hand side of this last relation.
Finally, since both L and A are constants of motion, we have
$\{A_{i},H\}=\{L_{i},H\}=0.$
The Poisson brackets will be extended to quantum mechanical commutation relations in the next section and to Lie brackets in a following section.
The scaled functions
As noted below, a scaled Laplace–Runge–Lenz vector D may be defined with the same units as angular momentum by dividing A by $ p_{0}={\sqrt {2m|H|}}$. Since D still transforms like a vector, the Poisson brackets of D with the angular momentum vector L can then be written in a similar form[11][8]
$\{D_{i},L_{j}\}=\sum _{s=1}^{3}\varepsilon _{ijs}D_{s}.$
The Poisson brackets of D with itself depend on the sign of H, i.e., on whether the energy is negative (producing closed, elliptical orbits under an inverse-square central force) or positive (producing open, hyperbolic orbits under an inverse-square central force). For negative energies—i.e., for bound systems—the Poisson brackets are[42]
$\{D_{i},D_{j}\}=\sum _{s=1}^{3}\varepsilon _{ijs}L_{s}.$
We may now appreciate the motivation for the chosen scaling of D: With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of L and the three components of D forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to so(4), the Lie algebra of the 4-dimensional rotation group SO(4).[43]
By contrast, for positive energy, the Poisson brackets have the opposite sign,
$\{D_{i},D_{j}\}=-\sum _{s=1}^{3}\varepsilon _{ijs}L_{s}.$
In this case, the Lie algebra is isomorphic to so(3,1).
The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the square root of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where $|H|=H$) and negative values (where $|H|=-H$).
Laplace-Runge-Lenz operator for the hydrogen atom in momentum space
Scaled Laplace-Runge-Lenz operator in the momentum space was found recently in.[44][45] Formula for the operator is simplier than in the position space:
${\hat {\mathbf {A} }}_{\mathbf {p} }=\imath ({\hat {l}}_{\mathbf {p} }+1)\mathbf {p} -{\frac {(p^{2}+1)}{2}}\imath \mathbf {\nabla } _{\mathbf {p} },$
where "degree operator"
${\hat {l}}_{\mathbf {p} }=(\mathbf {p} \mathbf {\nabla } _{\mathbf {p} })$
multiplies a homogeneous polynomial by its degree.
Casimir invariants and the energy levels
The Casimir invariants for negative energies are
${\begin{aligned}C_{1}&=\mathbf {D} \cdot \mathbf {D} +\mathbf {L} \cdot \mathbf {L} ={\frac {mk^{2}}{2|E|}},\\C_{2}&=\mathbf {D} \cdot \mathbf {L} =0,\end{aligned}}$
and have vanishing Poisson brackets with all components of D and L,
$\{C_{1},L_{i}\}=\{C_{1},D_{i}\}=\{C_{2},L_{i}\}=\{C_{2},D_{i}\}=0.$
C2 is trivially zero, since the two vectors are always perpendicular.
However, the other invariant, C1, is non-trivial and depends only on m, k and E. Upon canonical quantization, this invariant allows the energy levels of hydrogen-like atoms to be derived using only quantum mechanical canonical commutation relations, instead of the conventional solution of the Schrödinger equation.[8][43] This derivation is discussed in detail in the next section.
Quantum mechanics of the hydrogen atom
Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical operators is specified by the Poisson bracket of the corresponding classical variables, multiplied by iħ.[46]
By carrying out this quantization and calculating the eigenvalues of the C1 Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum.[7] This elegant 1926 derivation was obtained before the development of the Schrödinger equation.[47]
A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of p and L must be defined carefully.[8] Typically, the operators for the Cartesian components As are defined using a symmetrized (Hermitian) product,
$A_{s}=-mk{\hat {r}}_{s}+{\frac {1}{2}}\sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{sij}(p_{i}\ell _{j}+\ell _{j}p_{i}),$
Once this is done, one can show that the quantum LRL operators satisfy commutations relations exactly analogous to the Poisson bracket relations in the previous section—just replacing the Poisson bracket with $1/(i\hbar )$ times the commutator.[48][49]
From these operators, additional ladder operators for L can be defined,
${\begin{aligned}J_{0}&=A_{3},\\J_{\pm 1}&=\mp {\tfrac {1}{\sqrt {2}}}\left(A_{1}\pm iA_{2}\right).\end{aligned}}$
These further connect different eigenstates of L2, so different spin multiplets, among themselves.
A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined,
$C_{1}=-{\frac {mk^{2}}{2\hbar ^{2}}}H^{-1}-I,$
where H−1 is the inverse of the Hamiltonian energy operator, and I is the identity operator.
Applying these ladder operators to the eigenstates |ℓmn〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, C1, are seen to be quantized, n2 − 1. Importantly, by dint of the vanishing of C2, they are independent of the ℓ and m quantum numbers, making the energy levels degenerate.[8]
Hence, the energy levels are given by
$E_{n}=-{\frac {mk^{2}}{2\hbar ^{2}n^{2}}},$
which coincides with the Rydberg formula for hydrogen-like atoms (Figure 6). The additional symmetry operators A have connected the different ℓ multiplets among themselves, for a given energy (and C1), dictating n2 states at each level. In effect, they have enlarged the angular momentum group SO(3) to SO(4)/Z2 ~ SO(3) × SO(3).[50]
Conservation and symmetry
The conservation of the LRL vector corresponds to a subtle symmetry of the system. In classical mechanics, symmetries are continuous operations that map one orbit onto another without changing the energy of the system; in quantum mechanics, symmetries are continuous operations that "mix" electronic orbitals of the same energy, i.e., degenerate energy levels. A conserved quantity is usually associated with such symmetries.[1] For example, every central force is symmetric under the rotation group SO(3), leading to the conservation of the angular momentum L. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the spherical harmonics of the same quantum number ℓ without changing the energy.
The symmetry for the inverse-square central force is higher and more subtle. The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers ℓ and m. The symmetry is more subtle, however, because the symmetry operation must take place in a higher-dimensional space; such symmetries are often called "hidden symmetries".[51]
Classically, the higher symmetry of the Kepler problem allows for continuous alterations of the orbits that preserve energy but not angular momentum; expressed another way, orbits of the same energy but different angular momentum (eccentricity) can be transformed continuously into one another. Quantum mechanically, this corresponds to mixing orbitals that differ in the ℓ and m quantum numbers, such as the s(ℓ = 0) and p(ℓ = 1) atomic orbitals. Such mixing cannot be done with ordinary three-dimensional translations or rotations, but is equivalent to a rotation in a higher dimension.
For negative energies – i.e., for bound systems – the higher symmetry group is SO(4), which preserves the length of four-dimensional vectors
$|\mathbf {e} |^{2}=e_{1}^{2}+e_{2}^{2}+e_{3}^{2}+e_{4}^{2}.$
In 1935, Vladimir Fock showed that the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a three-dimensional unit sphere in four-dimensional space.[10] Specifically, Fock showed that the Schrödinger wavefunction in the momentum space for the Kepler problem was the stereographic projection of the spherical harmonics on the sphere. Rotation of the sphere and re-projection results in a continuous mapping of the elliptical orbits without changing the energy, an SO(4) symmetry sometimes known as Fock symmetry;[52] quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number n. Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector L and the scaled LRL vector A formed the Lie algebra for SO(4).[11][42] Simply put, the six quantities A and L correspond to the six conserved angular momenta in four dimensions, associated with the six possible simple rotations in that space (there are six ways of choosing two axes from four). This conclusion does not imply that our universe is a three-dimensional sphere; it merely means that this particular physics problem (the two-body problem for inverse-square central forces) is mathematically equivalent to a free particle on a three-dimensional sphere.
For positive energies – i.e., for unbound, "scattered" systems – the higher symmetry group is SO(3,1), which preserves the Minkowski length of 4-vectors
$ds^{2}=e_{1}^{2}+e_{2}^{2}+e_{3}^{2}-e_{4}^{2}.$
Both the negative- and positive-energy cases were considered by Fock[10] and Bargmann[11] and have been reviewed encyclopedically by Bander and Itzykson.[53][54]
The orbits of central-force systems – and those of the Kepler problem in particular – are also symmetric under reflection. Therefore, the SO(3), SO(4) and SO(3,1) groups cited above are not the full symmetry groups of their orbits; the full groups are O(3), O(4), and O(3,1), respectively. Nevertheless, only the connected subgroups, SO(3), SO(4), and SO+(3,1), are needed to demonstrate the conservation of the angular momentum and LRL vectors; the reflection symmetry is irrelevant for conservation, which may be derived from the Lie algebra of the group.
Rotational symmetry in four dimensions
The connection between the Kepler problem and four-dimensional rotational symmetry SO(4) can be readily visualized.[53][55][56] Let the four-dimensional Cartesian coordinates be denoted (w, x, y, z) where (x, y, z) represent the Cartesian coordinates of the normal position vector r. The three-dimensional momentum vector p is associated with a four-dimensional vector ${\boldsymbol {\eta }}$ on a three-dimensional unit sphere
${\begin{aligned}{\boldsymbol {\eta }}&={\frac {p^{2}-p_{0}^{2}}{p^{2}+p_{0}^{2}}}\mathbf {\hat {w}} +{\frac {2p_{0}}{p^{2}+p_{0}^{2}}}\mathbf {p} \\[1em]&={\frac {mk-rp_{0}^{2}}{mk}}\mathbf {\hat {w}} +{\frac {rp_{0}}{mk}}\mathbf {p} ,\end{aligned}}$
where $\mathbf {\hat {w}} $ is the unit vector along the new w axis. The transformation mapping p to η can be uniquely inverted; for example, the x component of the momentum equals
$p_{x}=p_{0}{\frac {\eta _{x}}{1-\eta _{w}}},$
and similarly for py and pz. In other words, the three-dimensional vector p is a stereographic projection of the four-dimensional ${\boldsymbol {\eta }}$ vector, scaled by p0 (Figure 8).
Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the z axis is aligned with the angular momentum vector L and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the y axis. Since the motion is planar, and p and L are perpendicular, pz = ηz = 0 and attention may be restricted to the three-dimensional vector ${\boldsymbol {\eta }}=(\eta _{w},\eta _{x},\eta _{y})$. The family of Apollonian circles of momentum hodographs (Figure 7) correspond to a family of great circles on the three-dimensional ${\boldsymbol {\eta }}$ sphere, all of which intersect the ηx axis at the two foci ηx = ±1, corresponding to the momentum hodograph foci at px = ±p0. These great circles are related by a simple rotation about the ηx-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; however, such a rotation is orthogonal to the usual three-dimensional rotations, since it transforms the fourth dimension ηw. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.
An elegant action-angle variables solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates ${\boldsymbol {\eta }}$ in favor of elliptic cylindrical coordinates (χ, ψ, φ)[57]
${\begin{aligned}\eta _{w}&=\operatorname {cn} \chi \operatorname {cn} \psi ,\\[1ex]\eta _{x}&=\operatorname {sn} \chi \operatorname {dn} \psi \cos \phi ,\\[1ex]\eta _{y}&=\operatorname {sn} \chi \operatorname {dn} \psi \sin \phi ,\\[1ex]\eta _{z}&=\operatorname {dn} \chi \operatorname {sn} \psi ,\end{aligned}}$
where sn, cn and dn are Jacobi's elliptic functions.
Generalizations to other potentials and relativity
The Laplace–Runge–Lenz vector can also be generalized to identify conserved quantities that apply to other situations.
In the presence of a uniform electric field E, the generalized Laplace–Runge–Lenz vector ${\mathcal {A}}$ is[17][58]
${\mathcal {A}}=\mathbf {A} +{\frac {mq}{2}}\left[\left(\mathbf {r} \times \mathbf {E} \right)\times \mathbf {r} \right],$
where q is the charge of the orbiting particle. Although ${\mathcal {A}}$ is not conserved, it gives rise to a conserved quantity, namely ${\mathcal {A}}\cdot \mathbf {E} $.
Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as[18]
${\mathcal {A}}=\left({\frac {\partial \xi }{\partial u}}\right)\left(\mathbf {p} \times \mathbf {L} \right)+\left[\xi -u\left({\frac {\partial \xi }{\partial u}}\right)\right]L^{2}\mathbf {\hat {r}} ,$
where u = 1/r and ξ = cos θ, with the angle θ defined by
$\theta =L\int ^{u}{\frac {du}{\sqrt {m^{2}c^{2}(\gamma ^{2}-1)-L^{2}u^{2}}}},$
and γ is the Lorentz factor. As before, we may obtain a conserved binormal vector B by taking the cross product with the conserved angular momentum vector
${\mathcal {B}}=\mathbf {L} \times {\mathcal {A}}.$
These two vectors may likewise be combined into a conserved dyadic tensor W,
${\mathcal {W}}=\alpha {\mathcal {A}}\otimes {\mathcal {A}}+\beta \,{\mathcal {B}}\otimes {\mathcal {B}}.$
In illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated.[18] Since the force is central,
$\mathbf {F} (r)=-k\mathbf {r} ,$
the angular momentum vector is conserved and the motion lies in a plane.
The conserved dyadic tensor can be written in a simple form
${\mathcal {W}}={\frac {1}{2m}}\mathbf {p} \otimes \mathbf {p} +{\frac {k}{2}}\,\mathbf {r} \otimes \mathbf {r} ,$
although p and r are not necessarily perpendicular.
The corresponding Runge–Lenz vector is more complicated,
${\mathcal {A}}={\frac {1}{\sqrt {mr^{2}\omega _{0}A-mr^{2}E+L^{2}}}}\left\{\left(\mathbf {p} \times \mathbf {L} \right)+\left(mr\omega _{0}A-mrE\right)\mathbf {\hat {r}} \right\},$
where
$\omega _{0}={\sqrt {\frac {k}{m}}}$
is the natural oscillation frequency, and
$A=(E^{2}-\omega ^{2}L^{2})^{1/2}/\omega .$
Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problems
The following are arguments showing that the LRL vector is conserved under central forces that obey an inverse-square law.
Direct proof of conservation
A central force $\mathbf {F} $ acting on the particle is
$\mathbf {F} ={\frac {d\mathbf {p} }{dt}}=f(r){\frac {\mathbf {r} }{r}}=f(r)\mathbf {\hat {r}} $
for some function $f(r)$ of the radius $r$. Since the angular momentum $\mathbf {L} =\mathbf {r} \times \mathbf {p} $ is conserved under central forces, $ {\frac {d}{dt}}\mathbf {L} =0$ and
${\frac {d}{dt}}\left(\mathbf {p} \times \mathbf {L} \right)={\frac {d\mathbf {p} }{dt}}\times \mathbf {L} =f(r)\mathbf {\hat {r}} \times \left(\mathbf {r} \times m{\frac {d\mathbf {r} }{dt}}\right)=f(r){\frac {m}{r}}\left[\mathbf {r} \left(\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}\right)-r^{2}{\frac {d\mathbf {r} }{dt}}\right],$
where the momentum $ \mathbf {p} =m{\frac {d\mathbf {r} }{dt}}$ and where the triple cross product has been simplified using Lagrange's formula
$\mathbf {r} \times \left(\mathbf {r} \times {\frac {d\mathbf {r} }{dt}}\right)=\mathbf {r} \left(\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}\right)-r^{2}{\frac {d\mathbf {r} }{dt}}.$
The identity
${\frac {d}{dt}}\left(\mathbf {r} \cdot \mathbf {r} \right)=2\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}={\frac {d}{dt}}(r^{2})=2r{\frac {dr}{dt}}$
yields the equation
${\frac {d}{dt}}\left(\mathbf {p} \times \mathbf {L} \right)=-mf(r)r^{2}\left[{\frac {1}{r}}{\frac {d\mathbf {r} }{dt}}-{\frac {\mathbf {r} }{r^{2}}}{\frac {dr}{dt}}\right]=-mf(r)r^{2}{\frac {d}{dt}}\left({\frac {\mathbf {r} }{r}}\right).$
For the special case of an inverse-square central force $ f(r)={\frac {-k}{r^{2}}}$, this equals
${\frac {d}{dt}}\left(\mathbf {p} \times \mathbf {L} \right)=mk{\frac {d}{dt}}\left({\frac {\mathbf {r} }{r}}\right)={\frac {d}{dt}}(mk\mathbf {\hat {r}} ).$
Therefore, A is conserved for inverse-square central forces[59]
${\frac {d}{dt}}\mathbf {A} ={\frac {d}{dt}}\left(\mathbf {p} \times \mathbf {L} \right)-{\frac {d}{dt}}\left(mk\mathbf {\hat {r}} \right)=\mathbf {0} .$
A shorter proof is obtained by using the relation of angular momentum to angular velocity, $\mathbf {L} =mr^{2}{\boldsymbol {\omega }}$, which holds for a particle traveling in a plane perpendicular to $\mathbf {L} $. Specifying to inverse-square central forces, the time derivative of $\mathbf {p} \times \mathbf {L} $ is
${\frac {d}{dt}}\mathbf {p} \times \mathbf {L} =\left({\frac {-k}{r^{2}}}\mathbf {\hat {r}} \right)\times \left(mr^{2}{\boldsymbol {\omega }}\right)=mk\,{\boldsymbol {\omega }}\times \mathbf {\hat {r}} =mk\,{\frac {d}{dt}}\mathbf {\hat {r}} ,$
where the last equality holds because a unit vector can only change by rotation, and ${\boldsymbol {\omega }}\times \mathbf {\hat {r}} $ is the orbital velocity of the rotating vector. Thus, A is seen to be a difference of two vectors with equal time derivatives.
As described elsewhere in this article, this LRL vector A is a special case of a general conserved vector ${\mathcal {A}}$ that can be defined for all central forces.[18][19] However, since most central forces do not produce closed orbits (see Bertrand's theorem), the analogous vector ${\mathcal {A}}$ rarely has a simple definition and is generally a multivalued function of the angle θ between r and ${\mathcal {A}}$.
Hamilton–Jacobi equation in parabolic coordinates
The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates (ξ, η), which are defined by the equations
${\begin{aligned}\xi &=r+x,\\\eta &=r-x,\end{aligned}}$
where r represents the radius in the plane of the orbit
$r={\sqrt {x^{2}+y^{2}}}.$
The inversion of these coordinates is
${\begin{aligned}x&={\tfrac {1}{2}}(\xi -\eta ),\\y&={\sqrt {\xi \eta }},\end{aligned}}$
Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations[17][60]
${\begin{aligned}2\xi p_{\xi }^{2}-mk-mE\xi &=-\Gamma ,\\2\eta p_{\eta }^{2}-mk-mE\eta &=\Gamma ,\end{aligned}}$
where Γ is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta px and py shows that Γ is equivalent to the LRL vector
$\Gamma =p_{y}(xp_{y}-yp_{x})-mk{\frac {x}{r}}=A_{x}.$
Noether's theorem
The connection between the rotational symmetry described above and the conservation of the LRL vector can be made quantitative by way of Noether's theorem. This theorem, which is used for finding constants of motion, states that any infinitesimal variation of the generalized coordinates of a physical system
$\delta q_{i}=\varepsilon g_{i}(\mathbf {q} ,\mathbf {\dot {q}} ,t)$
that causes the Lagrangian to vary to first order by a total time derivative
$\delta L=\varepsilon {\frac {d}{dt}}G(\mathbf {q} ,t)$
corresponds to a conserved quantity Γ
$\Gamma =-G+\sum _{i}g_{i}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right).$
In particular, the conserved LRL vector component As corresponds to the variation in the coordinates[61]
$\delta _{s}x_{i}={\frac {\varepsilon }{2}}\left[2p_{i}x_{s}-x_{i}p_{s}-\delta _{is}\left(\mathbf {r} \cdot \mathbf {p} \right)\right],$
where i equals 1, 2 and 3, with xi and pi being the i-th components of the position and momentum vectors r and p, respectively; as usual, δis represents the Kronecker delta. The resulting first-order change in the Lagrangian is
$\delta L={\frac {1}{2}}\varepsilon mk{\frac {d}{dt}}\left({\frac {x_{s}}{r}}\right).$
Substitution into the general formula for the conserved quantity Γ yields the conserved component As of the LRL vector,
$A_{s}=\left[p^{2}x_{s}-p_{s}\ \left(\mathbf {r} \cdot \mathbf {p} \right)\right]-mk\left({\frac {x_{s}}{r}}\right)=\left[\mathbf {p} \times \left(\mathbf {r} \times \mathbf {p} \right)\right]_{s}-mk\left({\frac {x_{s}}{r}}\right).$
Lie transformation
The Noether theorem derivation of the conservation of the LRL vector A is elegant, but has one drawback: the coordinate variation δxi involves not only the position r, but also the momentum p or, equivalently, the velocity v.[62] This drawback may be eliminated by instead deriving the conservation of A using an approach pioneered by Sophus Lie.[63][64] Specifically, one may define a Lie transformation[51] in which the coordinates r and the time t are scaled by different powers of a parameter λ (Figure 9),
$t\rightarrow \lambda ^{3}t,\qquad \mathbf {r} \rightarrow \lambda ^{2}\mathbf {r} ,\qquad \mathbf {p} \rightarrow {\frac {1}{\lambda }}\mathbf {p} .$
This transformation changes the total angular momentum L and energy E,
$L\rightarrow \lambda L,\qquad E\rightarrow {\frac {1}{\lambda ^{2}}}E,$
but preserves their product EL2. Therefore, the eccentricity e and the magnitude A are preserved, as may be seen from the equation for A2
$A^{2}=m^{2}k^{2}e^{2}=m^{2}k^{2}+2mEL^{2}.$
The direction of A is preserved as well, since the semiaxes are not altered by a global scaling. This transformation also preserves Kepler's third law, namely, that the semiaxis a and the period T form a constant T2/a3.
Alternative scalings, symbols and formulations
Unlike the momentum and angular momentum vectors p and L, there is no universally accepted definition of the Laplace–Runge–Lenz vector; several different scaling factors and symbols are used in the scientific literature. The most common definition is given above, but another common alternative is to divide by the quantity mk to obtain a dimensionless conserved eccentricity vector
$\mathbf {e} ={\frac {1}{mk}}\left(\mathbf {p} \times \mathbf {L} \right)-\mathbf {\hat {r}} ={\frac {m}{k}}\left(\mathbf {v} \times \left(\mathbf {r} \times \mathbf {v} \right)\right)-\mathbf {\hat {r}} ,$
where v is the velocity vector. This scaled vector e has the same direction as A and its magnitude equals the eccentricity of the orbit, and thus vanishes for circular orbits.
Other scaled versions are also possible, e.g., by dividing A by m alone
$\mathbf {M} =\mathbf {v} \times \mathbf {L} -k\mathbf {\hat {r}} ,$
or by p0
$\mathbf {D} ={\frac {\mathbf {A} }{p_{0}}}={\frac {1}{\sqrt {2m|E|}}}\left\{\mathbf {p} \times \mathbf {L} -mk\mathbf {\hat {r}} \right\},$
which has the same units as the angular momentum vector L.
In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by −1. Other common symbols for the LRL vector include a, R, F, J and V. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.
An alternative conserved vector is the binormal vector B studied by William Rowan Hamilton,[16]
$\mathbf {B} =\mathbf {p} -\left({\frac {mk}{L^{2}r}}\right)\ \left(\mathbf {L} \times \mathbf {r} \right),$
which is conserved and points along the minor semiaxis of the ellipse. (It is not defined for vanishing eccentricity.)
The LRL vector A = B × L is the cross product of B and L (Figure 4). On the momentum hodograph in the relevant section above, B is readily seen to connect the origin of momenta with the center of the circular hodograph, and to possess magnitude A/L. At perihelion, it points in the direction of the momentum.
The vector B is denoted as "binormal" since it is perpendicular to both A and L. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.
The two conserved vectors, A and B can be combined to form a conserved dyadic tensor W,[18]
$\mathbf {W} =\alpha \mathbf {A} \otimes \mathbf {A} +\beta \,\mathbf {B} \otimes \mathbf {B} ,$
where α and β are arbitrary scaling constants and $\otimes $ represents the tensor product (which is not related to the vector cross product, despite their similar symbol). Written in explicit components, this equation reads
$W_{ij}=\alpha A_{i}A_{j}+\beta B_{i}B_{j}.$
Being perpendicular to each another, the vectors A and B can be viewed as the principal axes of the conserved tensor W, i.e., its scaled eigenvectors. W is perpendicular to L ,
$\mathbf {L} \cdot \mathbf {W} =\alpha \left(\mathbf {L} \cdot \mathbf {A} \right)\mathbf {A} +\beta \left(\mathbf {L} \cdot \mathbf {B} \right)\mathbf {B} =0,$
since A and B are both perpendicular to L as well, L ⋅ A = L ⋅ B = 0.
More directly, this equation reads, in explicit components,
$\left(\mathbf {L} \cdot \mathbf {W} \right)_{j}=\alpha \left(\sum _{i=1}^{3}L_{i}A_{i}\right)A_{j}+\beta \left(\sum _{i=1}^{3}L_{i}B_{i}\right)B_{j}=0.$
See also
• Astrodynamics
• Orbit
• Eccentricity vector
• Orbital elements
• Bertrand's theorem
• Binet equation
• Two-body problem
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Further reading
• Baez, John (2008). "The Kepler Problem Revisited: The Laplace–Runge–Lenz Vector" (PDF). Retrieved 2021-05-31.
• Baez, John (2003). "Mysteries of the gravitational 2-body problem". Archived from the original on 2008-10-21. Retrieved 2004-12-11.
• Baez, John (2018). "Mysteries of the gravitational 2-body problem". Retrieved 2021-05-31. Updated version of previous source.
• D'Eliseo, M. M. (2007). "The first-order orbital equation". American Journal of Physics. 75 (4): 352–355. Bibcode:2007AmJPh..75..352D. doi:10.1119/1.2432126.
• Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Bibcode:2013qtm..book.....H, ISBN 978-1461471158.
• Leach, P. G. L.; G. P. Flessas (2003). "Generalisations of the Laplace–Runge–Lenz vector". J. Nonlinear Math. Phys. 10 (3): 340–423. arXiv:math-ph/0403028. Bibcode:2003JNMP...10..340L. doi:10.2991/jnmp.2003.10.3.6. S2CID 73707398.
| Wikipedia |
September 22, 2020 AEST
Long-Distance Person Travel: A Cluster-Based Approach
Mina Hassanvand,
cluster analysis fuzzy c-means k-means agglomerative hierarchical clustering travel behaviour travel modelling long-distance person travel
Photo by frank mckenna on Unsplash
Hassanvand, Mina. 2020. "Long-Distance Person Travel: A Cluster-Based Approach." Findings, September. https://doi.org/10.32866/001c.17291.
Table 1: Clustering Steps
Figure 1: Count of Trips by Distance and Canadian Nights Away for 2017 (TSRC) 4167 (AB) to (AB) Trips
Table 2: List of Variables
Table 3: Clusters Found in the 2017 (TSRC) (AB) to (AB) Trips at 68% Confidence Level
Figure 2: 3D Representation of 10 Clusters Across a few Key Dimensions: Household (HH) Members on the Trip, Travelling Group Size, Total Nights Away from home while on Trip, One-way Distance from Home in km, Trips Frequency, Total Amount of Spending while on Trip in 2017 $CAD
Figure 3: 2017 (TSRC) (AB) to (AB) Trips (AHC) Ward's Dendrogram
Many long-distance person trips (LDPT) modelling efforts fail to accurately represent trips using traditional segmentation approaches. Thus, a clustering approach was used herein to segment an intra-provincial trips data set. The trips' segments found were short economical getaways (36%), same-day shopping (16%), personal business (14%), visiting friends/relatives (10%), business/casino trips (10%), young adults playing team sports (6%), same-day trips of snow/festival loving young families with kids (3%), costly cottage/camping trips (3%), seniors with medical appointments (2%), and multiple city visitors (1%). The existence of clusters and associated activities shows what segmentation approaches modern models should follow.
The analysis of travel demand often includes the partitioning of the demand into "market segments" seeking to separate these influences into groups. As demonstrated in (Travel Demand Modelling 2016; Ben-Akiva and Lerman 1987; Limtanakool, Dijst, and Schwanen 2006; Bhat 1997b; Koppelman and Sethi 2000; Mandel, Gaudry, and Rothengatter 1997; Larse 2010; Wardman, Toner, and Whelan 1997; LaMondia, Bhat, and Hensher 2008; Carlsson 1999; Hensher 1991; Morrison and Winston 1985; Hassanvand 2020), the development of such models includes substantial effort that encompasses enormous revalidation work, which has been reduced with the aid of advanced computer technology. However, there still exists a gap in model design that arise from failure to segment markets.
This paper defines travel alternatives made by market segments with the aim of optimizing the selection of descriptive variables and strengthening the explanatory power of the model. Nearly all long-distance person trips (LDPT) models (Federal Highway Administration 2015; Golob 2001; Kizielewicz et al. 2017; Bhat 1997a; Golob and Hensher 1998; Badoe and Miller 1998; Lieberman et al. 2001) developed in various countries are based not on empirical procedures but rather by educated guesses to describe the travel market variables. This research transforms the approach to traditional segmentations using computer science approaches for network-based data that stem from fuzzy logic (Zadeh 1965). Such approaches are based on grouping of data points by examining their proximity (e.g. Euclidean distance) to one another. This is essential as LDPT is not merely the longer version of short-distance daily trips. While fuzzy-neuro models have been used in transit and some short-distance models (Kumar, Sarkar, and Madhu 2013; Sarkar 2012; Tharwat 2014; Roxas 2016; Yaldi et al, n.d.; Gite 2013), they have not been used in LDPT – excluding goods movement, trucking, or air travel.
Clustering is a statistical tool used in pattern recognition and machine learning to find similar groups in seemingly dissimilar network-based datasets (e.g. transport data). Objects in a cluster/class share many characteristics but are very dissimilar to objects not belonging to that particular cluster (Punj and Stewart 1983). In most of the classification works (Milligan 1996; Posse 1998; Everitt, Landau, and Leese 2001), considerable number of algorithms belong to two major types of clustering used here namely Hierarchical and Partitional. The former is based on finding clusters hierarchy using a criterion and producing a dendrogram. The latter is partitioning the data based on minimization of an objective function such as the squared error function (Kaufman and Rousseeuw 1990; Bezdek 1974):
\(J\mathbf{=}\sum_{\mathbf{j = 1}}^{\mathbf{k}}\mathbf{\ }\sum_{\mathbf{i = 1}}^{\mathbf{n}}\left\| \mathbf{x}_{\mathbf{i}}^{\left( \mathbf{j} \right)}\mathbf{-}\mathbf{c}_{\mathbf{j}} \right\|^{\mathbf{2}}\hspace{40mm}(1)\)
Where \(\left\| \mathbf{x}_{\mathbf{i}}^{\left( \mathbf{j} \right)}\mathbf{-}\mathbf{c}_{\mathbf{j}} \right\|^{\mathbf{2}}\) is the distance between a data item \(\mathbf{x}_{\mathbf{i}}^{\left( \mathbf{j} \right)}\) and a centre point \(\mathbf{c}_{\mathbf{j}}\).
One type of partitional clustering, with steps shown in Table 1, is called the k-means approach as a specific form of the more general fuzzy c-means clustering that minimizes a similar objective function [34]:
Table 1:Clustering Steps
The uniqueness of this work lies partly in the essential three-step cluster validity checks which are often ignored in many clustering themed studies (Dunn 1974; Zaki and Meira, Jr 2014):
Cluster tendency checks: which is a measure of clusterability of a data set considering that algorithms such as k-mean unquestioningly find some clusters in a data set regardless. Thus, to ensure the data is actually clusterable, one must examine it for its clustering tendency using indices such as the Hopkins statistics prior to any clustering practices.
Cluster stability checks: is the practice of clustering randomly generated data sets out of the original data and data belonging to other years/locations in order to examine if the resulting clusters are persistent and show up each time. Also, clustering the data set using fuzzy c-means provides an additional check on the existence/lack of potential outliers and acts as a precautionary measure against model-dependency of results.
Cluster validity: consists of three tests namely External (one-way ANOVA, Post-hoc Bonferroni, and Logistic Regression), Internal (Beta-CV index), and Relative (Elbow method). Other tests include variables' correlation checks, F-tests, Grubb's test of outliers, Ward's (AHC) dendrogram analyses and stopping rules comparison of a large Duda-Hart Je(2)/Je(1) index with a small Pseudo T-tests and a large Calinski-Harabasz Pseudo-F indices for detection of number of clusters (Everitt, Landau, and Leese 2001).
The publicly available standardized and weighted 2017 Travel Survey of Residents of Canada (TSRC) data set – including 14064 Province of Alberta (AB) residents, 4167 AB to AB trips, and 6128 nights travelled – is examined and compared with 2016 and 2015 data. TSRC is a supplement of the Canadian Labour Force Survey (LFS) (Statistics Canada 2017b, 2017a) after which TSRC questions are asked of a random 18+ household (HH) member regarding any one-way 40+ km trips from home finished in the previous month (same-day/overnight) plus any overnight trips ended two months before regardless of distance. Figure 1 shows trip counts by distance/purpose followed by variables' list in Table 2. Analysis is based on 100 variables with minimal correlations from socio-demographic factors to places visited and 37 different activities divided into same-day/overnight.
Figure 1:Count of Trips by Distance and Canadian Nights Away for 2017 (TSRC) 4167 (AB) to (AB) Trips
Table 2:List of Variables
Table 3 describes 10 clusters found in the 2017 data set followed by Figure 2 which is a 3D representation of clusters' center points across some of the most important dimensions (for brevity). Figure 3 represents a dendrogram of classes hierarchy found through (AHC) clustering.
Table 3:Clusters Found in the 2017 (TSRC) (AB) to (AB) Trips at 68% Confidence Level
Figure 2:3D Representation of 10 Clusters Across a few Key Dimensions: Household (HH) Members on the Trip, Travelling Group Size, Total Nights Away from home while on Trip, One-way Distance from Home in km, Trips Frequency, Total Amount of Spending while on Trip in 2017 $CAD
Figure 3:2017 (TSRC) (AB) to (AB) Trips (AHC) Ward's Dendrogram
Examinations revealed the data possess a natural structure with 10 clusters at 68% confidence level. Such results are consistent with other literature findings for LD trips (Future Foundation 2015; Birley and Westhead 1990; Mooi and Sarstedt 2011). For example, trips done for pleasure have consistently been found to belong to mostly the top two categories of LDPT. The second largest cluster is representative of individual adults from the same HH who travel in smaller groups with no kids. Their purpose is mainly same-day trips of shopping with moderate levels of spending with activities such as walking. This finding is novel and could be a characteristic of the Province of Alberta, in that malls and shopping centres such as Banff, Lake Louise, West Edmonton mall, or other shopping avenues are also long-distance traveller attractors. The existence of such clusters demonstrates how traditional LDPT trips segmentation through "guessing variables and rechecking" are obsolete and would need to be enhanced using comprehensive clustering approaches targeted for network-based data to better represent the overall LDPT market while relying less on conjecture and assumptions.
Submitted: September 21, 2020 AEST
Accepted: September 21, 2020 AEST
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Calculate: {\text{begin}array l x+y-3z=-13 } 5x+4y+z=17-3x+y-4z=-34\text{end}array .
Expression: $\left\{\begin{array} { l } x+y-3z=-13 \\ 5x+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$
Solve the equation for $x$
$\left\{\begin{array} { l } x=-13-y+3z \\ 5x+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$
Substitute the given value of $x$ into the equation $5x+4y+z=17$
$\left\{\begin{array} { l } 5\left( -13-y+3z \right)+4y+z=17 \\ -3x+y-4z=-34\end{array} \right.$
Substitute the given value of $x$ into the equation $-3x+y-4z=-34$
$\left\{\begin{array} { l } 5\left( -13-y+3z \right)+4y+z=17 \\ -3\left( -13-y+3z \right)+y-4z=-34\end{array} \right.$
$\left\{\begin{array} { l } -y+16z=82 \\ -3\left( -13-y+3z \right)+y-4z=-34\end{array} \right.$
$\left\{\begin{array} { l } -y+16z=82 \\ 4y-13z=-73\end{array} \right.$
Multiply both sides of the equation by $4$
$\left\{\begin{array} { l } -4y+64z=328 \\ 4y-13z=-73\end{array} \right.$
Sum the equations vertically to eliminate at least one variable
$51z=255$
Divide both sides of the equation by $51$
$z=5$
Substitute the given value of $z$ into the equation $4y-13z=-73$
$4y-13 \times 5=-73$
Solve the equation for $y$
$y=-2$
Substitute the given values of $\begin{array} { l }y,& z\end{array}$ into the equation $x=-13-y+3z$
$x=-13-\left( -2 \right)+3 \times 5$
$x=4$
The possible solution of the system is the ordered triple $\left( x, y, z\right)$
$\left( x, y, z\right)=\left( 4, -2, 5\right)$
Check if the given ordered triple is a solution of the system of equations
$\left\{\begin{array} { l } 4+\left( -2 \right)-3 \times 5=-13 \\ 5 \times 4+4 \times \left( -2 \right)+5=17 \\ -3 \times 4+\left( -2 \right)-4 \times 5=-34\end{array} \right.$
Simplify the equalities
$\left\{\begin{array} { l } -13=-13 \\ 17=17 \\ -34=-34\end{array} \right.$
Since all of the equalities are true, the ordered triple is the solution of the system
Solve for: ((n+1) !)/((n-2) !)
Evaluate: 4(3y-1)=8y
Solve for: (2+5) / 6 * 2
Calculate: (2-(-4)^2+2)/((2-4)^2)
Solve for: (d)/(dx) (sin(x)^2)
Solve for: (x-225)/(25)=1.2816
Evaluate: (/(1) 2) (/(4) 5)
Evaluate: (-14xy^{-3})/(28x^{-7)y^{-3}} * (-5x^{-1}y^{-2})/((-y)^2)
Calculate: y=-x^2-8x-7
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\begin{document}
\title[Polynomial and Multilinear Hardy--Littlewood Inequalities]{Polynomial and Multilinear Hardy--Littlewood Inequalities: Analytical and Numerical Approaches} \author[Campos]{J. Campos} \address{Departamento de Ci\^{e}ncias Exatas\\ \indent Universidade Federal da Para\'{\i}ba \\ \indent 58.297-000 - Rio Tinto, Brazil.} \email{[email protected]} \author[Cavalcante]{W. Cavalcante} \address{Departamento de Matem\'{a}tica\\ \indent Universidade Federal de Pernambuco\\ \indent50.740-560 - Recife, Brazil.} \email{[email protected]} \author[F\'{a}varo]{V. F\'{a}varo} \address{Faculdade de Matem\'{a}tica\\ \indent Universidade Federal de Uberl\^{a}ndia\\ \indent 38.400-902 - Uberl\^{a}ndia, Brazil.} \email{[email protected]} \author[N\'{u}\~{n}ez]{D. N\'{u}\~{n}ez-Alarc\'{o}n} \address{Departamento de Matem\'{a}tica\\ \indent Universidade Federal de Pernambuco\\ \indent 50.740-560 - Recife, Brazil.} \email{[email protected]} \author[Pellegrino]{D. Pellegrino} \address{Departamento de Matem\'{a}tica \\ \indent Universidade Federal da Para\'{\i}ba \\ \indent 58.051-900 - Jo\~{a}o Pessoa, Brazil.} \email{[email protected] and [email protected]} \author[Serrano]{D. M. Serrano-Rodr\'iguez} \address{Departamento de Matem\'{a}tica\\ \indent Universidade Federal de Pernambuco\\ \indent 50.740-560 - Recife, Brazil.} \email{[email protected]} \thanks{J. Campos was supported by a CAPES Postdoctoral scholarship. V. V. F \'{ }avaro was supported by FAPEMIG Grants CEX-APQ-01409-12, PPM-00086-14 and CNPq Grants 482515/2013-9, 307517/2014-4. W. Cavalcante is supported by Capes. D. N\'{u}\~{n}ez and D.M. Serrano were supported by CNPq Grant 461797/2014-3. D. Pellegrino was supported by CNPq} \subjclass[2010]{46G25, 47L22, 47H60.} \keywords{Absolutely summing operators; Hardy--Littlewood inequality; Bohnenblust--Hille inequality.} \maketitle
\begin{abstract} We investigate the growth of the polynomial and multilinear Hardy--Littlewood inequalities. Analytical and numerical approaches are performed and, in particular, among other results, we show that a simple application of the best known constants of the Clarkson inequality improves a recent result of Araujo et al. We also obtain the optimal constants of the generalized Hardy--Littlewood inequality in some special cases. \end{abstract}
\section{Introduction}
The investigation of polynomials and multilinear operators acting on Banach spaces is a fruitful topic of investigation that dates back to the $30^{\prime }s$ (see, for instance \cite{bh, hardy, LLL} and, for recent papers, \cite{rueda, bayy, botelho, dimant, garcia} among many others).
Let $\mathbb{K}$ be the real or complex scalar field, and$\,n\geq 1$ be a positive integer. In 1930 Littlewood proved his well-known $4/3$ inequality to solve a problem posed by P.J. Daniell (see \cite{LLL}). The Littlewood's $ 4/3$ inequality asserts that \begin{equation*} \left( \sum\limits_{i,j=1}^{n}\left\vert T(e_{i},e_{j})\right\vert ^{\frac{4 }{3}}\right) ^{\frac{3}{4}}\leq \sqrt{2}\left\Vert T\right\Vert \end{equation*} for all positive integers $n$ and every continuous bilinear form $ T:c_{0}\times c_{0}\rightarrow \mathbb{K}$, where\linebreak $\Vert T\Vert
:=\sup_{z^{(1)},z^{(2)}\in B_{c_{0}}}|T(z^{(1)},z^{(2)})|$. The exponent $ 4/3 $ is optimal and in the case $\mathbb{K}=\mathbb{R}$ the optimality of the constant $\sqrt{2}$ is also known (see \cite{diniz}). Soon afterwards this inequality was generalized by Hardy and Littlewood (\cite{hardy}, 1934) for bilinear forms on $\ell _{p}$ and, in 1982 Praciano-Pereira (\cite{pra}) extended the result of Hardy and Littlewood to $m$-linear forms on $\ell _{p} $. Another generalization of the Hardy--Littlewood inequalities for $m$ -linear forms was obtained by Dimant and Sevilla-Peris, and will be treated in Remark \ref{988}.
The Hardy--Littlewood inequalities for $m$-linear forms is the following result:
\textbf{Theorem (Hardy--Littlewood/Praciano-Pereira).} Let $m\geq 2$ be a positive integer. For $p\geq 2m,$ there is a constant $C_{\mathbb{K} ,m,p}\geq 1$ such that \begin{equation*} \left( \sum_{i_{1},...,i_{m}=1}^{n}\left\vert T(e_{i_{1}},\ldots ,e_{i_{m}})\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp} }\leq C_{\mathbb{K},m,p}\left\Vert T\right\Vert , \end{equation*}
for all positive integers $n$ and all $m$-linear forms $T:\ell _{p}^{n}\times \cdots \times \ell _{p}^{n}\rightarrow \mathbb{K}$.
The exponent $\frac{2mp}{mp+p-2m}$ is optimal and $\Vert T\Vert
:=\sup_{z^{(1)},...,z^{(m)}\in B_{\ell _{p}^{n}}}|T(z^{(1)},...,z^{(m)})|\,$ . In the limiting case ($p=\infty ,$ considering, of course $f(\infty ):=\lim_{p\rightarrow \infty }f(p)$ regardless of the function $f$), we recover the classical multilinear Bohnenblust--Hille inequality (see \cite {bh}). The original upper estimate for $C_{\mathbb{K},m,p}$ is $2^{\frac{m-1 }{2}}$. Recently, in some papers (see \cite{adv22, ap, ooo}), this estimate was improved for all $m$ and $p$ with the only exception of the case $C_{ \mathbb{R},m,2m}$.
The precise behavior of the growth of the optimal constants $C_{\mathbb{K} ,m,p}$ is still unknown (some partial results can be found in \cite{laa, adv22, ap}).
Up to now, the best known lower estimates for $C_{\mathbb{R},m,p}$ are always smaller than $2$ and again the more critical situation is when $p=2m$ , where the lower estimates presented in \cite{laa} are more difficult to obtain and not explicitly stated for the case $p=2m$.
In view of the special role played by the constants $C_{\mathbb{R},m,2m}$ and since this case is a kind of dual version of the classical Bohnenblust--Hille inequality (see details in Section \ref{890}), in the Sections 3 and 4 we investigate this critical case and obtain quite better lower estimates. Our approach has two novelties: a new class of multilinear forms, not investigated before in similar context, and a new numerical approach in this framework. As it will be clear along the paper the new family of multilinear forms introduced in this paper is more effective to obtain good lower estimates for the Hardy--Littlewood inequality.
In Section \ref{ggee} we investigate the generalized Hardy--Littlewood inequality. Our approach provides new lower bounds for this inequality. As a consequence of our results, in Theorem \ref{optimal} we obtain optimal constants for some cases of three-linear forms.
In Section \ref{ii1} we investigate the polynomial Hardy--Littlewood inequality. The approaches of Sections \ref{ggee} and \ref{ii1} are entirely analytic and do not depend on computation assistance.
\section{ The multilinear Hardy--Littlewood inequality\label{890}}
From now on, if $p\in \left( 1,2\right) $, $p^{\ast }$ is the extended real number such that $\frac{1}{p}+\frac{1}{p^{\ast }}=1.$ Also, $E^{\prime }$ denotes the topological dual of a Banach space $E$. By $\mathcal{L}\left( ^{m}E;F\right) $ we denote the Banach space of all (bounded) $m$-linear operators $U:E\times \cdots \times E\rightarrow F$, with $E$, $F~$Banach spaces over $\mathbb{K}$. For $1\leq s\leq r<\infty ,\,U\in \mathcal{L} \left( ^{m}E;F\right) $ is called \emph{multiple $(r,s)$-summing}, if there exists a constant $C>0$ such that \begin{equation*} \left( \sum_{i_{1},\dots ,i_{m}=1}^{n}\left\Vert U\left( x_{i_{1}},\dots ,x_{i_{m}}\right) \right\Vert _{F}^{r}\right) ^{\frac{1}{r}}\leq C\left\Vert U\right\Vert \prod_{k=1}^{m}\left\Vert \left( x_{i_{k}}\right) _{i_{k}=1}^{n}\right\Vert _{w,s} \end{equation*} for all finite choice of vectors $x_{i_{k}}\in E,\,1\leq i_{k}\leq n,\,1\leq k\leq m$, where \begin{equation*}
\Vert \left( x_{i}\right) _{i=1}^{n}\Vert _{w,s}:=\sup_{\Vert \varphi \Vert _{E^{\prime }}\leq 1}\left( \sum_{i=1}^{n}|\varphi (x_{i})|^{s}\right) ^{ \frac{1}{s}}. \end{equation*} The vector space of all multiple $(r,s)$-summing operators in $\mathcal{L} \left( ^{m}E;F\right) $ is denoted by $\Pi _{(r,s)}\left( ^{m}E;F\right) $. For more details of the theory of multiple summing operators theory see \cite {matos, santosp, per}.
In the terminology of the multiple summing operators, it is well known (see, for instance, \cite[Section 5]{dimant}) that the Hardy--Littlewood/Praciano-Pereira inequality is equivalent to the equality
\begin{equation*} \Pi _{\left( \frac{2mp}{mp+p-2m};p^{\ast }\right) }(^{m}E;\mathbb{K})= \mathcal{L}(^{m}E;\mathbb{K}). \end{equation*} In other words, if $m\geq 2$ and $p\geq 2m$, then there is a constant $C_{ \mathbb{K},m,p}\geq 1$ such that \begin{equation*} \left( \sum_{i_{1},...,i_{m}=1}^{n}\left\vert T(x_{i_{1}},\ldots ,x_{i_{m}})\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp} }\leq C_{\mathbb{K},m,p}\left\Vert T\right\Vert \prod\limits_{k=1}^{m}\left\Vert \left( x_{i_{k}}\right) _{i_{k}=1}^{n}\right\Vert _{w,p^{\ast }} \end{equation*} for all $m$-linear forms $T:E\times \cdots \times E\rightarrow \mathbb{K}$, for all finite choice of vectors $x_{i_{k}}\in E,\,1\leq i_{k}\leq n,\,1\leq k\leq m$.
As mentioned in the introduction, the case $p=2m$ in the Hardy--Littlewood inequality is specially interesting. In this case we have very few information on the constants involved, and moreover, this case is a kind of dual version of the Bohnenblust--Hille inequality, in the sense that in the pair of parameters $\left( \frac{2mp}{mp+p-2m};p^{\ast }\right) $, each case has a coordinate which is kept constant (in reverse location). More specifically, in the terminology of the multiple summing operators, the Bohnenblust--Hille inequality asserts that \begin{equation*} \Pi _{\left( \frac{2m}{m+1};1\right) }(^{m}E;\mathbb{K})=\mathcal{L}(^{m}E; \mathbb{K}) \end{equation*} for all Banach spaces $E$. On the other hand, when $p=2m$, the Hardy--Littlewood inequality is equivalent to \begin{equation*} \Pi _{\left( 2;\frac{2m}{2m-1}\right) }(^{m}E;\mathbb{K})=\mathcal{L}(^{m}E; \mathbb{K}) \end{equation*} for all Banach spaces $E$.
Up to now the best known upper estimates for the constants $\left( C_{ \mathbb{R},m,p}\right) _{m=1}^{\infty }$ can be found in \cite[page 1887]{ap} and \cite{ooo}. The updated results on the lower bounds for these constants are:
\begin{itemize} \item $C_{\mathbb{R},m,p}\geq 2^{\frac{mp+2m-2m^{2}-p}{mp}}$ for $p>2m$ and $ C_{\mathbb{R},m,p}>1$ for $p=2m$ (see \cite{laa}); \end{itemize}
From now on $p^{\ast }$ denotes the conjugate number of $p$. In this section we find an overlooked (and simple) connection between the Clarkson's inequalities and the Hardy--Littlewood's constants which helps to find analytical lower estimates (without the use of a computational aid) for these constants.
\begin{theorem} \label{popkhtr} Let $m\geq 2$ and $p\geq 2m$. The optimal constants of the Hardy-Littlewood inequalities satisfies \begin{equation*} C_{\mathbb{R},m,p}\geq \frac{2^{\frac{2mp+2m-p-2m^{2}}{mp}}}{\sup_{x\in \lbrack 0,1]}\frac{((1+x)^{p^{\ast }}+(1-x)^{p^{\ast }})^{\frac{1}{p^{\ast }} }}{{(1+x^{p})^{1/p}}}}. \end{equation*} \end{theorem}
\begin{proof} For a given Banach space $E$ we know that $\Psi :\mathcal{L}\left( ^{2}E; \mathbb{R}\right) \rightarrow \mathcal{L}\left( E;E^{\ast }\right) $ given by $\Psi (T)(x)(y)=T\left( x,y\right) $ is an isometric isomorphism. For $ E=\ell _{p}^{2}$ and using the characterization of the dual of $\ell _{p}^{2} $, we conclude that for the bilinear form \begin{equation*} \begin{array}{ccccl} T_{2,p} & : & \ell _{p}^{2}\times \ell _{p}^{2} & \rightarrow & \mathbb{R} \\ & & ((x_{i}^{(1)}),(x_{i}^{(2)})) & \mapsto & x_{1}^{(1)}x_{1}^{(2)}+x_{1}^{(1)}x_{2}^{(2)}+x_{2}^{(1)}x_{1}^{(2)}-x_{2}^{(1)}x_{2}^{(2)}, \end{array} \end{equation*} we have \begin{equation*} \begin{array}{ccccl} \Psi (T_{2,p}) & : & \ell _{p}^{2} & \rightarrow & \ell _{p^{\ast }}^{2} \\ & & (x_{i}) & \mapsto & (x_{1}+x_{2},x_{1}-x_{2}). \end{array} \end{equation*} Since $p\geq 2m$ and $m\geq 2$, using the best constants from the Clarkson's inequality in the real case (see \cite[Theorem 2.1]{mali}) we know the norm of the linear operator $\Psi (T_{2,p})$ (and consequently the norm of the bilinear form $T_{2,p}$), i.e., \begin{equation*} \Vert T_{2,p}\Vert =\left\Vert \Psi (T_{2,p})\right\Vert =\sup_{x\in \lbrack 0,1]}\frac{((1+x)^{p^{\ast }}+(1-x)^{p^{\ast }})^{\frac{1}{p^{\ast }}}}{ (1+x^{p})^{1/p}}. \end{equation*} Now, as in \cite{laa}, we define inductively \begin{equation*} \begin{array}{rccl} T_{m,p}: & \ell _{p}^{2^{m-1}}\times \cdots \times \ell _{p}^{2^{m-1}} & \rightarrow & \mathbb{R} \\ & (x^{(1)},...,x^{(m)}) & \mapsto & (x_{1}^{(m)}+x_{2}^{(m)})T_{m-1,p}(x^{(1)},...,x^{(m)}) \\ & & & +(x_{1}^{(m)}-x_{2}^{(m)})T_{m-1,p}(B^{2^{m-1}}(x^{(1)}),...,B^{2}(x^{(m-1)})), \end{array} \end{equation*} where $x^{(k)}=(x_{j}^{(k)})_{j=1}^{{2^{m-1}}}\in \ell _{p}^{2^{m-1}}$, $ 1\leq k\leq m$, and $B$ is the backward shift operator in $\ell _{p}^{2^{m-1}}$ and, again as in \cite{laa}, we conclude that \begin{align*}
|T_{m,p}(x^{(1)},...,x^{(m)})|& \leq
|x_{1}^{(m)}+x_{2}^{(m)}||T_{m-1,p}(x^{(1)},...,x^{(m)})| \\ &
+|x_{1}^{(m)}-x_{2}^{(m)}||T_{m-1,p}(B^{2^{m-1}}(x^{(1)}),B^{2^{m-2}}(x^{(2)}),...,B^{2}(x^{(m-1)}))| \\ & \leq \Vert T_{m-1,p}\Vert
(|x_{1}^{(m)}+x_{2}^{(m)}|+|x_{1}^{(m)}-x_{2}^{(m)}|) \\ & \leq 2\Vert T_{m-1,p}\Vert \Vert x^{(m)}\Vert _{p}, \end{align*} i.e., \begin{equation*} \Vert T_{m,p}\Vert \leq 2^{m-2}\Vert T_{2,p}\Vert . \end{equation*} \end{proof}
Now we have \begin{equation*} (4^{m-1})^{\frac{mp+p-2m}{2mp}}=\left( \sum_{j_{1},...,j_{m}=1}^{2^{m-1}}\left\vert T_{m,p}(e_{j_{1}},...,e_{j_{m}})\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{ \frac{mp+p-2m}{2mp}}\leq C_{\mathbb{R},m,p}2^{m-2}\Vert T_{2,p}\Vert \end{equation*} and thus \begin{equation*} C_{\mathbb{R},m,p}\geq \frac{(4^{m-1})^{\frac{mp+p-2m}{2mp}}}{2^{m-2}\Vert T_{2,p}\Vert }=\frac{2^{2\left( m-1\right) \left( \frac{mp+p-2m}{2mp}\right) -\left( m-2\right) }}{\sup_{x\in \lbrack 0,1]}\frac{\left( \left( 1+x\right) ^{p^{\ast }}+\left( 1-x\right) ^{p^{\ast }}\right) ^{1/p^{\ast }}}{\left( 1+x^{p}\right) ^{1/p}}} \end{equation*}
When $m=2,$ using estimates of \cite[page 1369]{mali}, note that \begin{align*} C_{\mathbb{R},2,4}& \geq \frac{2}{\sqrt{3}}>1.1546 \\ C_{\mathbb{R},2,8}& \geq \frac{2^{\frac{5}{4}}}{1.892}>1.2570 \\ C_{\mathbb{R},2,p}& \geq \frac{2^{\frac{2mp+2m-p-2m^{2}}{mp}}}{1.9836}>1.3591 \text{ for }p=1+\log _{9/10}1/19 \\ C_{\mathbb{R},2,p}& \geq \frac{2^{\frac{2mp+2m-p-2m^{2}}{mp}}}{1.9999}>1.4105 \text{ for }p=1+\log _{99/100}1/199. \end{align*} Using the old estimates of \cite{laa} for $p>2m$ (i.e., $C_{\mathbb{R} ,m,p}\geq 2^{\frac{mp+2m-2m^{2}-p}{mp}}$) we can easily verify that the old estimates are worse. Also, in the old estimates we have no closed formula for the case $p=2m.$
\begin{remark} One may try to use the complex Clarkson's inequalities to obtain nontrivial lower bounds for the constants of the complex Hardy-Littlewood inequality. But, this is not effective, since we just get trivial lower bounds, i.e., $1$ . \end{remark}
\begin{remark}[The case $m<p<2m$] \label{988}There is also a version of Hardy--Littlewood's inequality for $ m<p<2m,$ due to Dimant and Sevilla-Peris (\cite{dimant} and the forthcoming Section 6). In this case, the optimal exponent is $\frac{p}{p-m}$ and we still denote the optimal constant for this inequality by $C_{\mathbb{K},m,p}$ . The best information we have so far for the lower estimates for the constant $C_{\mathbb{R},m,p}$ are trivial, that is, \begin{equation*} 1\leq C_{\mathbb{R},m,p}\leq (\sqrt{2})^{m-1}. \end{equation*} Similarly to the argument used in the proof of the Theorem \ref{popkhtr}, we can also provide a closed formula (which depends on $p$) for the lower bounds of $C_{\mathbb{R},m,p}$, but in this case, we do not always have nontrivial information. More precisely, we prove that \begin{equation*} C_{\mathbb{R},m,p}\geq \frac{2^{\frac{mp+2m-2m^{2}}{p}}}{\sup_{x\in \lbrack 0,1]}\frac{((1+x)^{p^{\ast }}+(1-x)^{p^{\ast }})^{\frac{1}{p^{\ast }}}}{ (1+x^{p})^{\frac{1}{p}}}}. \end{equation*} It is important to mention this case because, for suitable choices of $p$, we get nontrivial lower estimates for $C_{\mathbb{R},m,p}$. For instance, \begin{equation*} C_{\mathbb{R},2,7/2}\geq 1.104,\ \ C_{\mathbb{R},3,28/5}\geq 1.025,\ \ \text{ and}\ \ C_{\mathbb{R},100,199999/1000}\geq 1.003. \end{equation*} This leads us to question the following: Would also be the optimal constants of the Hardy--Littlewood inequality for $m<p<2m$ strictly greater than $1$? \end{remark}
\section{First numerical estimates (using well-known multilinear forms)}
Since the publication of \cite{diniz}, the family of $m$-linear forms $T_{m}:\ell _{\infty }\times \cdots \times \ell _{\infty }$ defined inductively by \begin{equation} T_{2}(x,y)=x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}, \label{t2} \end{equation} \begin{align} T_{3}(x,y,z)& =(z_{1}+z_{2})\left( x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}\right) \label{t3} \\ & +(z_{1}-z_{2})\left( x_{3}y_{3}+x_{3}y_{4}+x_{4}y_{3}-x_{4}y_{4}\right) , \notag \end{align} \begin{align} T_{4}(x,y,z,w)& =\left( w_{1}+w_{2}\right) \left( \begin{array}{c} (z_{1}+z_{2})\left( x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}\right) \\ +(z_{1}-z_{2})\left( x_{3}y_{3}+x_{3}y_{4}+x_{4}y_{3}-x_{4}y_{4}\right) \end{array} \right) \label{t4} \\ & +\left( w_{1}-w_{2}\right) \left( \begin{array}{c} (z_{3}+z_{4})\left( x_{5}y_{5}+x_{5}y_{6}+x_{6}y_{5}-x_{6}y_{6}\right) \\ +(z_{3}-z_{4})\left( x_{7}y_{7}+x_{7}y_{8}+x_{8}y_{7}-x_{8}y_{8}\right) \end{array} \right) \notag \end{align} and so on, have been used to find lower estimates for Bohnenblust--Hille and related inequalities (se also \cite{ooo}). In the context of the Hardy--Littlewood inequalities we also have good results, but in the next section we invent different multilinear forms that, in our context, provide better estimates.
The numerical issue involved to obtain our estimates is the calculus of $
\Vert T_{m}\Vert $ when $\ell _{\infty }$ is replaced by $\ell _{p}$ (in this case we write $T_{m,p}$ instead of $T_{m}$)$.$ This task refers to a typical nonlinear optimization problem subject to restrictions. Namely, we want to find a global maximum of $|T_{m,2m}(x^{(1)},\ldots ,x^{(m)})|$ with $ x^{(i)}\in B_{\ell _{2m}}$, $i=1,\ldots ,m$ for the operators (\ref{t2}), ( \ref{t3}), (\ref{t4}), etc.
To perform this computer-aided calculus we use a couple of software: multi-paradigm numerical computing environment called MATLAB (MATrix LABoratory) (see \cite{gilat}) to specify the problem and a software library for large-scale nonlinear optimization called Interior Point to solve it. Mathematical details of the algorithm used by interior-point can be found in several publications (see for instance \cite{Byrd,Byrd2,Waltz}).
As the interior-point algorithm is designed to find local solutions for a given optimization problem starting from a initial data, it is necessary to find all local solutions (all maxima) and take the greatest of them. This can be done taking a reasonable distribution of starting points throughout the domain of the operator.
Performing these calculations for $T_{m,2m}$, we obtain \begin{equation}
\begin{tabular}{|c|cl|} \hline
& \multicolumn{1}{|c|}{} & \\
$C_{\mathbb{R},2,4}>$ & $\frac{2}{1.74}$ & \multicolumn{1}{|l|}{$>1.149$} \\
& & \multicolumn{1}{|l|}{} \\
$C_{\mathbb{R},3,6}>$ & $\frac{4}{3.29}$ & \multicolumn{1}{|l|}{$>1.215$} \\
& \multicolumn{1}{|c|}{} & \multicolumn{1}{|l|}{} \\
$C_{\mathbb{R},4,8}>$ & \multicolumn{1}{|c|}{$\frac{8}{6.40}$} & $>1.250$ \\
& \multicolumn{1}{|c|}{} & \\
$C_{\mathbb{R},5,10}>$ & \multicolumn{1}{|c|}{$\frac{16}{12.60}$} & $>1.269$ \\
& \multicolumn{1}{|c|}{} & \\
$C_{\mathbb{R},6,12}>$ & \multicolumn{1}{|c|}{$\frac{32}{25.00}$} & $>1.280$ \\
& \multicolumn{1}{|c|}{} & \\
$C_{\mathbb{R},7,14}>$ & \multicolumn{1}{|c|}{$\frac{64}{49.47}$} & $>1.293$ \\
& \multicolumn{1}{|c|}{} & \\
$C_{\mathbb{R},8,16}>$ & \multicolumn{1}{|c|}{$\frac{128}{98.36}$} & $>1.301$ \\
& \multicolumn{1}{|c|}{} & \\
$C_{\mathbb{R},9,18}>$ & \multicolumn{1}{|c|}{$\frac{256}{195.81}$} & $ >1.\,\allowbreak 307.$ \\
& \multicolumn{1}{|c|}{} & \\ \hline \end{tabular} \label{dell99} \end{equation}
\section{New multilinear forms and better estimates\label{ujp}}
Up to now the best known multilinear forms to use in order to find lower bounds for the Bohnenblust--Hille and Hardy--Littlewood inequalities were those defined in (\ref{t2}), (\ref{t3}), (\ref{t4}) and so on. Now we show that for $m=4,8,16,...$ we get better estimates using slightly different multilinear forms and numerical computation. Define \begin{equation*} \widetilde{T}_{2}(x,y)=x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}, \end{equation*} \begin{align*} \widetilde{T}_{4}(x,y,z,w)& =\left( x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}\right) \left( z_{1}w_{1}+z_{1}w_{2}+z_{2}w_{1}-z_{2}w_{2}\right) \\ & +\left( x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}\right) \left( z_{3}w_{3}+z_{3}w_{4}+z_{4}w_{3}-z_{4}w_{4}\right) \\ & +\left( x_{3}y_{3}+x_{3}y_{4}+x_{4}y_{3}-x_{4}y_{4}\right) \left( z_{1}w_{1}+z_{1}w_{2}+z_{2}w_{1}-z_{2}w_{2}\right) \\ & -\left( x_{3}y_{3}+x_{3}y_{4}+x_{4}y_{3}-x_{4}y_{4}\right) \left( z_{3}w_{3}+z_{3}w_{4}+z_{4}w_{3}-z_{4}w_{4}\right) , \end{align*}
\begin{eqnarray*} \widetilde{T}_{8}(x,y,z,w,r,s,t,u) &=&\widetilde{T}_{4}(x,y,z,w)\widetilde{T} _{4}(r,s,t,u) \\ &&+\widetilde{T}_{4}(x,y,z,w)\widetilde{T}_{4}(B^{4}\left( r\right) ,B^{4}\left( s\right) ,B^{4}\left( t\right) ,B^{4}\left( u\right) ) \\ &&+\widetilde{T}_{4}(B^{4}(x),B^{4}(y),B^{4}(z),B^{4}(w))\widetilde{T} _{4}(r,s,t,u) \\ &&-\widetilde{T}_{4}(B^{4}(x),B^{4}(y),B^{4}(z),B^{4}(w))\widetilde{T} _{4}(B^{4}\left( r\right) ,B^{4}\left( s\right) ,B^{4}\left( t\right) ,B^{4}\left( u\right) ), \end{eqnarray*} and so on (recall that $B^{4}$ is the shift operator, as defined before). Using $\widetilde{T}_{4},\widetilde{T}_{8},$ etc, we obtain \begin{equation}
\begin{tabular}{|c|c|l|} \hline & & \\ $C_{\mathbb{R},4,8}>$ & $\frac{2^{3}}{6.20}$ & $>1.290$ \\ & & \\ $C_{\mathbb{R},8,16}>$ & $\frac{2^{7}}{91.48}$ & $>1.399$ \\ & & \\ $C_{\mathbb{R},16,32}>$ & $\frac{2^{15}}{22137.70}$ & $>1.\,\allowbreak 480,$ \\ & & \\ \hline \end{tabular} \label{t44} \end{equation}
and this procedure seems clearly better than the former.
\section{On the generalized Hardy--Littlewood inequality\label{ggee}}
The main goal in this section is to provide optimal constants for some cases of three-linear forms in the recently extended version of the Hardy--Littlewood inequality, presented in \cite{n, dimant}:
\textbf{Theorem (Generalized Hardy--Littlewood inequality).} If\textit{\ }$ m\geq 2$ is a positive integer,\linebreak $2m\leq p\leq \infty $ \textit{and} $\mathbf{q}:=(q_{1},...,q_{m})\in \left[ \frac{p}{p-m},2\right] ^{m}$ are \textit{such that} \begin{equation} \frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}=\frac{mp+p-2m}{2p}, \label{ppoo} \end{equation} t\textit{hen there exists a constant} $C_{m,p,\mathbf{q}}^{\mathbb{K}}\geq 1$ \textit{such that} \begin{equation} \left( \sum_{j_{1}=1}^{n}\left( \sum_{j_{2}=1}^{n}\left( \cdots \left( \sum_{j_{m}=1}^{n}\left\vert T(e_{j_{1}},...,e_{j_{m}})\right\vert ^{q_{m}}\right) ^{\frac{q_{m-1}}{q_{m}}}\cdots \right) ^{\frac{q_{2}}{q_{3}} }\right) ^{\frac{q_{1}}{q_{2}}}\right) ^{\frac{1}{q_{1}}}\leq C_{m,p,\mathbf{ q}}^{\mathbb{K}}\left\Vert T\right\Vert \label{g33} \end{equation} \textit{for all continuous }$m$\textit{--linear forms }$T:\ell _{p}^{n}\times \cdots \times \ell _{p}^{n}\rightarrow \mathbb{K}$\textit{\ and all positive integers }$n.$\textit{\ }
The case $p=\infty $ recovers the so called generalized Bohnenblust--Hille inequality (see \cite{alb}) and when $p=\infty $ and $ q_{1}=\cdots =q_{m}=\frac{2m}{m+1}$ we recover the classical Bohnenblust--Hille inequality. The optimal constants $C_{m,p,\mathbf{q}}^{ \mathbb{K}}$ are known in very few cases, namely
(i) $p=\infty $ and $(q_{1},...,q_{m})=\left( 1,2,...,2\right) $. In this case (see \cite[Theorem 2.1]{ooo}) $C_{m,\infty ,\mathbf{q}}^{\mathbb{R} }=\left( \sqrt{2}\right) ^{m-1}$ for all $m\geq 2;$
(ii) \ $\left( m,p\right) =\left( 2,\infty \right) $ and no restriction on $ q_{1},q_{2}.$ In this case (see \cite[Theorem 6.3]{alb}) $C_{2,\infty , \mathbf{q}}^{\mathbb{R}}=\sqrt{2}.$
In these two cases these optimal constants are obtained by using special multilinear forms to find lower bounds that match exactly with the known upper bounds of the constants. This approach seems to be not effective in other cases, but we do not know if the reason is a fault of the method or a weakness of our estimates of upper bounds (i.e, maybe the known upper bounds are not good enough). Using this technique it was recently shown in \cite[ Theorem 2.3]{ooo} that for a constant $\alpha \in \lbrack 1,2]$ and a multiple exponent $\mathbf{q}=(\alpha ,\frac{2\alpha m-2\alpha }{\alpha m-2+\alpha },...,\frac{2\alpha m-2\alpha }{\alpha m-2+\alpha }),$ we have \begin{equation} C_{m,\infty ,\mathbf{q}}^{\mathbb{R}}\geq 2^{\frac{2m-\alpha m-4+3\alpha }{ 2\alpha }}. \label{jfapel} \end{equation}
By using the Minkowski inequality, we obtain that for $\mathbf{q}=(\frac{ 2\alpha m-2\alpha }{\alpha m-2+\alpha },...,\frac{2\alpha m-2\alpha }{\alpha m-2+\alpha },\alpha )$, with $\alpha >\frac{2m}{m+1}$ the estimate (\ref {jfapel}) gives us
\begin{equation*} C_{m,\infty ,\mathbf{q}}^{\mathbb{R}}\geq 2^{\frac{2m-\alpha m-4+3\alpha }{ 2\alpha}}. \end{equation*}
In this section, we show that for a constant $\alpha \in \lbrack 1,2]$ and a $\mathbf{q}=(\frac{2\alpha m-2\alpha }{\alpha m-2+\alpha },...,\frac{2\alpha m-2\alpha }{\alpha m-2+\alpha },\alpha )$ we have \begin{equation} C_{m,\infty ,\mathbf{q}}^{\mathbb{R}}\geq 2^{\frac{3\alpha m-2m-5\alpha +4}{ 2\alpha \left( m-1\right) }}. \label{thispel} \end{equation}
For $\alpha >\frac{2m}{m+1}$ the new estimate is strictly bigger (and thus better) than the previous.
When $m=3$ and $\alpha =2$ we obtain $C_{3,\infty ,\mathbf{q}}^{\mathbb{R} }\geq 2^{3/4}$ and since we already know (see \cite[Lemma 2.1]{adv22}) that $ C_{3,\infty ,\mathbf{q}}^{\mathbb{R}}\leq 2^{3/4}$, we conclude that the optimal constant is $2^{3/4}.$
The result proved here is:
\begin{proposition} Let $\alpha \in \lbrack 1,2]$ be a constant and $\mathbf{q}=(\beta _{m},...,\beta _{m},\alpha )$ be a multiple exponent of the generalized Bohnenblust--Hille inequality for real scalars. Then \begin{equation*} C_{m,\infty ,\mathbf{q}}^{\mathbb{R}}\geq 2^{\frac{3\alpha m-2m-5\alpha +4}{ 2\alpha \left( m-1\right) }}. \end{equation*} \end{proposition}
\begin{proof} The $m$-linear operators that we will use are defined inductively as in (\ref {t2}), (\ref{t3}) and (\ref{t4}) . Since \begin{align*} {\left( \frac{\left( 2^{m-1}\right) ^{2}}{2}2^{\frac{1}{\alpha }\beta _{m}}\right) ^{\frac{1}{\beta _{m}}}}& {=}\left( 2^{2m-3}2^{\frac{1}{\alpha } \beta _{m}}\right) ^{\frac{1}{\beta _{m}}}=\left( \sum\limits_{i_{1},\ldots ,i_{m-1}=1}^{2^{m-1}}\left( \sum\limits_{i_{m}=1}^{2}\left\vert T_{m}(e_{i_{^{1}}},...,e_{im})\right\vert ^{\alpha }\right) ^{\frac{1}{ \alpha }\beta _{m}}\right) ^{\frac{1}{\beta _{m}}} \\ & \leq C_{m,\infty ,\mathbf{q}}^{\mathbb{R}}\left\Vert T_{m}\right\Vert \end{align*} and $\beta _{m}=\frac{2\alpha m-2\alpha }{\alpha m-2+\alpha }$ we conclude that \begin{equation*} C_{m,\infty ,\mathbf{q}}^{\mathbb{R}}\geq \frac{\left( 2^{2m-3}\left( 2\right) ^{\frac{1}{\alpha }\beta _{m}}\right) ^{\frac{1}{\beta _{m}}}}{ 2^{m-1}}=2^{\frac{3\alpha m-2m-5\alpha +4}{2\alpha \left( m-1\right) }}. \end{equation*} \end{proof}
\begin{theorem} \label{optimal} The optimal constant of the generalized Bohnenblust--Hille inequality for $m=3$ and $\mathbf{q}=(4/3,4/3,2)$ or $\mathbf{q} =(4/3,8/5,8/5)$ or $\mathbf{q}=(4/3,2,4/3)$ is $C_{3,\infty ,\mathbf{q}}^{ \mathbb{R}}=2^{3/4}$. \end{theorem}
\begin{proof} From \cite[Lemma 2.1]{adv22} we obtain, for $m$ and $\mathbf{q}$ satisfying the hypotheses of the theorem, the estimate \begin{equation*} C_{3,\infty ,\mathbf{q}}^{\mathbb{R}}\leq 2^{3/4}. \end{equation*} Using (\ref{thispel}) we prove that for $\mathbf{q}=(4/3,4/3,2)$ we have \begin{equation*} C_{3,\infty ,\mathbf{q}}^{\mathbb{R}}\geq 2^{3/4} \end{equation*}
and, finally, using (\ref{jfapel}) we show that \ for $\mathbf{q} =(4/3,8/5,8/5)$ we have \begin{equation*} C_{3,\infty ,\mathbf{q}}^{\mathbb{R}}\geq 2^{3/4}. \end{equation*} On the other hand, using the operator $T_{3}$ (see (\ref{t3})) we have \begin{equation*} 2^{\frac{11}{4}}=\left( \sum\limits_{i_{1}=1}^{4}\left( \sum\limits_{i_{2}=1}^{4}\left( \sum\limits_{i_{3}=1}^{2}\left\vert T_{3}(e_{i_{^{1}}},e_{i_{2}},e_{i_{3}})\right\vert ^{4/3}\right) ^{\frac{3}{2 }}\right) ^{\frac{2}{3}}\right) ^{\frac{3}{4}}\text{,} \end{equation*} and thus for $\mathbf{q}=(4/3,2,4/3)$ we get \begin{equation*} C_{3,\infty ,\mathbf{q}}^{\mathbb{R}}\geq \frac{2^{\frac{11}{4}}}{2^{2}}=2^{ \frac{3}{4}}\text{.} \end{equation*} \end{proof}
\section{The polynomial Hardy--Littlewood inequality\label{ii1}}
Let $E$ be a real or complex Banach space and $m$ be a positive integer and let $\mathbb{K}$ be the real or complex scalar field. A map $P:E\rightarrow \mathbb{K}$ is a homogeneous polynomial on $E$ of degree $m$ if there exists a symmetric $m$-linear form $L$ on $E^{m}$ such that $P(x)=L(x,\ldots ,x)$ for all $x\in E$. We denote by ${\mathcal{P}}(^{m}E)$ the space of continuous $m$-homogeneous polynomials on $E$ endowed with the usual norm \begin{equation*}
\Vert P\Vert :=\sup \{|P(x)|:\left\Vert x\right\Vert =1\}. \end{equation*} Observe that an $m$-homogeneous polynomial in ${\mathbb{K}}^{n}$ can be written as \begin{equation*} P(x)={\sum\limits_{\left\vert \alpha \right\vert =m}}a_{\alpha }x^{\alpha }, \end{equation*}
where $x=(x_{1},\ldots ,x_{n})\in {\mathbb{K}}^{n}$, $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in ({\mathbb{N}}\cup \{0\})^{n}$, $|\alpha
|=\alpha _{1}+\cdots +\alpha _{n}$ and $x^{\alpha }=x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}$. We denote \begin{equation*} \left\vert P\right\vert _{p}:=\left( {\sum\limits_{\left\vert \alpha \right\vert =m}}\left\vert a_{\alpha }\right\vert ^{p}\right) ^{1/p} \end{equation*} and \begin{equation*} \left\vert P\right\vert _{\infty }:=\max \left\vert a_{\alpha }\right\vert . \end{equation*}
The polynomial Hardy--Littlewood inequality is:
\textbf{Theorem (Polynomial Hardy--Littlewood inequality).} For $m<p\leq \infty $ there is a constant $D_{\mathbb{K},m,p}\geq 1$ such that
\begin{equation} \begin{tabular}{clll} $\left( {\sum\limits_{\left\vert \alpha \right\vert =m}}\left\vert a_{\alpha }\right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}$ & $\leq $ & $D_{ \mathbb{K},m,p}\left\Vert P\right\Vert $, & $\text{if }m<p\leq 2m,$ \\ & & & \\ $\left( {\sum\limits_{\left\vert \alpha \right\vert =m}}\left\vert a_{\alpha }\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}$ & $\leq $ & $D_{\mathbb{K},m,p}\left\Vert P\right\Vert \text{,}$ & $\text{if }p\geq 2m$ \end{tabular} \label{6544} \end{equation} for all positive integers $n$ and all $m$-homogeneous polynomials $P:\ell _{p}^{n}\rightarrow \mathbb{K}$ given by \begin{equation*} P(x)={\sum\limits_{\left\vert \alpha \right\vert =m}}a_{\alpha }x^{\alpha }. \end{equation*}
This is a consequence of the multilinear Hardy--Littlewood inequality, previously described, and the following inequality also known as Hardy--Littlewood inequality \cite{dimant}:
\textbf{Theorem (Hardy--Littlewood/Dimant--Sevilla-Peris).} For $m<p\leq 2m$ , there is a constant $C_{\mathbb{K},m,p}\geq 1$ such that \begin{equation*} \left( \sum_{i_{1},...,i_{m}=1}^{n}\left\vert T(e_{i_{1}},\ldots ,e_{i_{m}})\right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq C_{ \mathbb{K},m,p}\left\Vert T\right\Vert \end{equation*} for all positive integers $n$ and all $m$-linear forms $T:\ell _{p}^{n}\times \cdots \times \ell _{p}^{n}\rightarrow \mathbb{K}$.
Above, the exponent $\frac{p}{p-m}$ is optimal and therefore in (\ref{6544}) both exponents $\frac{p}{p-m}$ and $\frac{2mp}{mp+p-2m}$ are optimal. The case $p=\infty $ in the appropriate inequality of (\ref{6544}), is the classical polynomial Bohnenblust--Hille inequality (see \cite{bh}).
From now on $D_{\mathbb{K},m,p}$ denotes the optimal constants satisfying ( \ref{6544}). As in the multilinear case, the precise behaviour of the growth of the constants $D_{\mathbb{K},m,p}$ is still unknown (partial results can be found in \cite{campos, nunez1}). For instance, in \cite[Theorem 3.1] {campos} it is proved that for $p\geq 2m$ we have
\begin{equation*} D_{\mathbb{R},m,p}\geq \left( \sqrt[16]{2}\right) ^{m}. \end{equation*}
When $p=\infty $ we know that (see \cite{bohr, campos1}) \begin{eqnarray*} \lim \sup_{m}D_{\mathbb{R},m,\infty }^{1/m} &=&2; \\ \lim \sup_{m}D_{\mathbb{C},m,\infty }^{1/m} &=&1. \end{eqnarray*}
It will be convenient to define $H_{1}=\left\{ \left( p,m\right) \in \mathbb{ R}\times \mathbb{N}:m<p<2m\right\} $ and $H_{2}=\left\{ \left( p,m\right) \in \mathbb{R}\times \mathbb{N}:p\geq 2m\right\} $ with any total order. The main results of this section are the following:
\begin{lemma} \label{the:Q_2k} Let $j=1,2.$ Then \begin{equation*} \lim \sup_{H_{j}}D_{\mathbb{R},m,p}^{1/m}\geq 2. \end{equation*} \end{lemma}
\begin{proof} Consider the sequence of norm-one $j$-homogeneous polynomials $Q_{j}:\ell _{p}\rightarrow \mathbb{R}$ defined recursively by \begin{align*} Q_{2}(x_{1},x_{2})& =x_{1}^{2}-x_{2}^{2}, \\ Q_{2^{m}}(x_{1},\ldots ,x_{2^{m}})& =Q_{2^{m-1}}(x_{1},\ldots ,x_{2^{m-1}})^{2}-Q_{2^{m-1}}(x_{2^{m-1}+1},\ldots ,x_{2^{m}})^{2}. \end{align*} From the proof of \cite[Theorem 3.1]{campos1}, we known that \begin{equation}
|Q_{2^{m}}^{n}|_{\infty }\geq \left( \frac{2^{n}}{n+1}\right) ^{2^{m}-1} \label{eq_m} \end{equation} for every natural number $n,m$.
\noindent Next, since for every homogeneous polynomial $P$ we obviously have \begin{equation*}
|P|_{p}\geq |P|_{\infty }, \end{equation*} from \eqref{eq_m} we conclude that \begin{equation*} D_{{\mathbb{R}},n2^{m},p}\geq \left( \frac{2^{n}}{n+1}\right) ^{2^{m}-1}. \end{equation*} Note that
\begin{equation*} D_{\mathbb{R},n2^{m},p}^{1/n2^{m}}\geq \left( \left( \frac{2^{n}}{n+1} \right) ^{2^{m}-1}\right) ^{\frac{1}{n2^{m}}}=\left( \frac{2^{n}}{n+1} \right) ^{\frac{2^{m}-1}{n2^{m}}} \end{equation*} and making $m\rightarrow \infty $ we have \begin{equation*} \left( \frac{2^{n}}{n+1}\right) ^{\frac{2^{m}-1}{n2^{m}}}\rightarrow \frac{2 }{\left( n+1\right) ^{1/n}} \end{equation*} and now making $n\rightarrow \infty $ we have \begin{equation*} \frac{2}{\left( n+1\right) ^{1/n}}\rightarrow 2. \end{equation*} \end{proof}
From now on we write \begin{eqnarray*} \rho \left( p,m\right) &=&\frac{p}{p-m}\text{ if }m<p\leq 2m, \\ \rho \left( p,m\right) &=&\frac{2mp}{mp+p-2m}\text{ if }p\geq 2m. \end{eqnarray*}
Now we prove the theorem:
\begin{theorem} Let $j=1,2.$ At least one of the following two sentences hold true: \end{theorem}
(a) $\lim \sup_{H_{j}}D_{\mathbb{R},m,p}^{1/m}=2.$
(b) $\lim \sup_{H_{j}}D_{\mathbb{C},m,p}^{1/m}>1.$
\begin{proof} Suppose that (a) is not true for some $j$. So (using the previous result) we would have\linebreak $\lim \sup_{H_{j}}D_{\mathbb{R},m,p}^{1/m}>\left( 2+\varepsilon \right) >2.$ Therefore, for each $k\in \mathbb{N}$ there is $ n_{k}\in \mathbb{N}$, $\left( p_{_{k}},m_{k}\right) $ $\in $ $H_{j}$ and a $ m_{k}$-homogeneous polynomial $P_{m_{k}}:\ell _{p_{k}}^{n_{k}}\rightarrow \mathbb{R}$ such that \begin{equation*} \left( {\sum\limits_{\left\vert \alpha \right\vert =m_{k}}}\left\vert a_{\alpha }\right\vert ^{\rho \left( p_{_{k}},m_{k}\right) }\right) ^{\frac{1 }{\rho \left( p_{_{k}},m_{k}\right) }}\leq D_{\mathbb{R},m_{k},p_{k}}\left \Vert P_{m_{k}}\right\Vert , \end{equation*} with \begin{equation*} D_{\mathbb{R},m_{k},p_{k}}>\left( 2+\varepsilon \right) ^{m_{k}}. \end{equation*} Considering the complexification of $P_{m_{k}}$ we know that \begin{equation*} \left\Vert \left( P_{m_{k}}\right) _{\mathbb{C}}\right\Vert \leq 2^{m_{k}-1}\left\Vert P_{m_{k}}\right\Vert \end{equation*} and now looking for the complex polynomials $\left( P_{m_{k}}\right) _{ \mathbb{C}}$ we would have \begin{eqnarray*} \left( {\sum\limits_{\left\vert \alpha \right\vert =m_{k}}}\left\vert a_{\alpha }\right\vert ^{\rho \left( p_{k},m_{k}\right) }\right) ^{\frac{1}{ \rho \left( p_{k},m_{k}\right) }} &\leq &D_{\mathbb{C},m_{k},p_{k}}\left \Vert \left( P_{m_{k}}\right) _{\mathbb{C}}\right\Vert \\ &\leq &D_{\mathbb{C},m_{k},p_{k}}2^{m_{k}-1}\left\Vert P_{m_{k}}\right\Vert \end{eqnarray*} and thus \begin{equation*} D_{\mathbb{R},m_{k},p_{k}}\leq D_{\mathbb{C},m_{k},p_{k}}2^{m_{k}-1}, \end{equation*} i.e.,
\ \begin{equation*} D_{\mathbb{R},m_{k},p_{k}}^{1/m_{k}}\leq D_{\mathbb{C} ,m_{k},p_{k}}^{1/m_{k}}2^{\frac{m_{k}-1}{m_{k}}}\leq 2D_{\mathbb{C} ,m_{k},p_{k}}^{1/m_{k}}. \end{equation*} Now, since \begin{equation*} D_{\mathbb{R},m_{k},p_{k}}^{1/m_{k}}>2+\varepsilon \end{equation*} we conclude that \begin{equation*} D_{\mathbb{C},m_{k},p_{k}}^{1/m_{k}}>1+\frac{\varepsilon }{2}>1 \end{equation*} for all $k$, and thus \begin{equation*} \lim \sup_{H_{j}}D_{\mathbb{C},m,p}^{1/m}>1. \end{equation*} Reciprocally, if (b) is not true for some $j$, then $\lim \sup_{H_{j}}D_{ \mathbb{C},m,p}^{1/m}=1$ and thus $\lim \sup_{H_{j}}D_{\mathbb{R} ,m,p}^{1/m}\leq 2$ and from the previous lemma we conclude that \begin{equation*} \lim \sup_{H_{j}}D_{\mathbb{R},m,p}^{1/m}=2. \end{equation*} \end{proof}
\begin{verbatim}
\end{verbatim}
\textbf{Acknowledgement.} This preprint is the result of the union and improvement of two arXiv preprints of some of its authors: arXiv 1504.04207 by D.\ Pellegrino and arXiv 1503.00618. The preprint arXiv 1504.04207, as now is incorporated to the present paper, does not exist as an independent paper anymore. It is just part of the present new paper.
\end{document} | arXiv |
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean.
Definition
Given a series $\{s_{n}\}$, the Riesz mean of the series is defined by
$s^{\delta }(\lambda )=\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }s_{n}$
Sometimes, a generalized Riesz mean is defined as
$R_{n}={\frac {1}{\lambda _{n}}}\sum _{k=0}^{n}(\lambda _{k}-\lambda _{k-1})^{\delta }s_{k}$
Here, the $\lambda _{n}$ are a sequence with $\lambda _{n}\to \infty $ and with $\lambda _{n+1}/\lambda _{n}\to 1$ as $n\to \infty $. Other than this, the $\lambda _{n}$ are taken as arbitrary.
Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of $s_{n}=\sum _{k=0}^{n}a_{k}$ for some sequence $\{a_{k}\}$. Typically, a sequence is summable when the limit $\lim _{n\to \infty }R_{n}$ exists, or the limit $\lim _{\delta \to 1,\lambda \to \infty }s^{\delta }(\lambda )$ exists, although the precise summability theorems in question often impose additional conditions.
Special cases
Let $a_{n}=1$ for all $n$. Then
$\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}\zeta (s)\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{n}b_{n}\lambda ^{-n}.$
Here, one must take $c>1$; $\Gamma (s)$ is the Gamma function and $\zeta (s)$ is the Riemann zeta function. The power series
$\sum _{n}b_{n}\lambda ^{-n}$
can be shown to be convergent for $\lambda >1$. Note that the integral is of the form of an inverse Mellin transform.
Another interesting case connected with number theory arises by taking $a_{n}=\Lambda (n)$ where $\Lambda (n)$ is the Von Mangoldt function. Then
$\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }\Lambda (n)=-{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{\rho }{\frac {\Gamma (1+\delta )\Gamma (\rho )}{\Gamma (1+\delta +\rho )}}+\sum _{n}c_{n}\lambda ^{-n}.$
Again, one must take c > 1. The sum over ρ is the sum over the zeroes of the Riemann zeta function, and
$\sum _{n}c_{n}\lambda ^{-n}\,$
is convergent for λ > 1.
The integrals that occur here are similar to the Nörlund–Rice integral; very roughly, they can be connected to that integral via Perron's formula.
References
• ^ M. Riesz, Comptes Rendus, 12 June 1911
• ^ Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes" (PDF). Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942. Archived from the original (PDF) on 7 February 2012.
• Volkov, I.I. (2001) [1994], "Riesz summation method", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
\begin{document}
\title{On nonlinear instability of liquid Lane-Emden stars}
\author{Zeming Hao
\and Shuang Miao}
\maketitle
\begin{abstract}
We establish a dynamical nonlinear instability of liquid Lane-Emden stars in $\mathbb R^{3}$ whose adiabatic exponents take values in $[1,\frac43)$. Our proof relies on a priori estimates for the free boundary problem of a compressible self-gravitating liquid, as well as a quantitative analysis of the competition between the fastest linear growing mode and the source.
\end{abstract}
\section{Introduction}\label{1}
A classical model for Newtonian stars is given by the compressible Euler-Poisson system, as an idealized self-gravitating gas or liquid surrounded by vacuum, kept together by self-consistent gravitational force. In this paper, we consider the compressible Euler-Poisson equations for a perfect fluid with no heat conduction and no viscosity. It is given by
\begin{align}\label{main eq}
\begin{split}
&\rho\left( u_{t}+u\cdot\nabla u\right) =-\nabla P
-\rho\nabla\psi,\quad\quad\quad \ in \ \mathcal{B}(t) \\
&\rho_{t}+\mathrm{div}\,(\rho u)=0 ,\quad \quad\quad\quad\quad\quad\quad\ \ in \ \mathcal B(t)
\end{split}
\end{align} here \(u\) is the fluid velocity, \(\rho\) is the fluid density, $P$ is the fluid pressure, and \(\psi\) is the gravitational potential. All these quantities are functions of spacetime variable $(x,t)$. \(\psi\) is defined as \begin{align}\label{potential}
\psi(t,x):=-\int_{\mathcal{B}(t)}\frac{\rho(t,y)}{|x-y|}dy, \end{align} so that with\(\chi_{\mathcal{B}(t)}\) denoting the characteristic function of \(\mathcal{B}(t)\) \begin{align*} \Delta\psi(t,x)=4\pi \rho\chi_{\mathcal{B}(t)}. \end{align*} The fluid is initially supported on a compact domain \(\mathcal{B}_{0}\subset\mathbb{R}^{3}\). The domain $\mathcal B(t)$ occupied by fluid does not have a fixed shape and as the system evolves \(\mathcal{B}(t)\) changes in time. We consider a free boundary problem such that there is no surface tension on $\partial\mathcal B(t)$ and any fluid particle on the boundary always stays on the boundary in the evolution. Therefore the following boundary conditions hold:
\begin{align}\label{boundary conditions}
P=0\quad \textrm{on}\quad \partial\mathcal{B}(t),\quad \partial_{t}+u\cdot\nabla\in \,T(t,\partial\mathcal{B}(t)). \end{align} There are four equations (the scalar continuity equation plus the vector momentum equation) but five unknown functions (\(\rho,P\) and \(u\)) in \eqref{main eq}. To close the system, we need to specify an equation of state. The model we consider is that of a barotropic fluid where the pressure is a function of the fluid density only and is expressed through the following \emph{polytropic} equation of state \begin{align}\label{eq of state} P=K\rho^{\gamma}-C, \end{align} for some constants \(K,C>0\) and \(1\le\gamma\le 2\). The fact that the model \eqref{main eq}-\eqref{boundary conditions} describes the motion of a \emph{liquid} is reflected by the boundary condition $P=0$ on $\partial\mathcal B(t)$ and the equation of state \eqref{eq of state}. In particular \eqref{eq of state} implies that on $\partial\mathcal B(t)$ the fluid density is a positive constant $\rho_{0}>0$. Without loss of generality, we can take \(K=C=1\). As we shall see, the constant \(\gamma\), called the adiabatic index, plays a crucial role in our analysis.
The Euler-Poisson system \eqref{main eq}-\eqref{boundary conditions} has steady-state solutions called Lane-Emden stars. In the case of radial symmetry, the steady-state equation becomes an ODE. Lam \cite{L} has shown that liquid Lane-Emden stars are linearly unstable when $1\le\gamma<\frac43$ and the density at the center is large enough. We shall study the nonlinear dynamical behavior of the Lane-Emden stars in this regime. \subsection{History and background} The Euler-Poisson system with the polytropic equation of state has played a crucial role in astrophysics. It is used to model stellar structure and evolution by Chandrasekhar \cite{Chan}, Shapiro, and Teukolsky \cite{ST}. When the pressure and the gravity are perfectly balanced, the Euler-Poisson system with the polytropic equation of state has an important class of time-independent steady state solutions known as the Lane-Emden stars. For \(\gamma=\frac{4}{3}\), astrophysicists Goldreich and Weber \cite{GW} found a special class of expanding and collapsing solutions that model stellar collapse and expansion such as supernova expansion. To understand the physical interest corresponding to these special solutions, it is important to have an existence theory for the Euler and Euler-Poisson equation.
In the gaseous case, where the fluid density vanishes on the boundary, the local existence for the free boundary problem of the Euler-Poisson system \eqref{main eq}-\eqref{boundary conditions} (or Euler system for which the self-gravity is not present) has been extensively studied, see \cite{CS,JMas,IT,GL} and references therein. In the liquid case, the corresponding local existence for the Euler-Poisson system was proved in \cite{GLL} (see also \cite{Lind05,Trak09} for the Euler system, and \cite{Oliy,MSW} for relativistic liquid), where the proof relies on delicate energy estimates and optimal elliptic estimates in terms of boundary regularity. The local well-posedness theory provides a solid foundation for studying the stability of Lane-Emden stars.
There has been development on the problem of gaseous Lane–Emden stars. The linear stability was studied in \cite{Lin,JMak}, in which the authors show that Lane–Emden stars in \(\mathbb{R}^{3}\) are linearly unstable when \(1<\gamma<\frac{4}{3}\) and linearly stable when \(\frac{4}{3}\le\gamma <2\). Jang \cite{Jang08,Jang14} prove the full nonlinear, dynamical instability of the steady profile for \(\frac{6}{5}\le\gamma<\frac{4}{3}\) using a bootstrap argument and nonlinear weighted energy estimates. But the nonlinear stability in the range \(\frac{4}{3}<\gamma <2\) has not been fully understood and remains open. By the linear stability, under the assumption that a global-in-time solution exists, nonlinear stability for \(\frac{4}{3}<\gamma <2\) was established by Rein in \cite{Rein} using a variational approach based on the fact that the steady states are minimizers of an energy functional. See other approaches in \cite{LS09,LS08}. In the critical case \(\gamma=\frac{4}{3}\) the Lane-Emden star is nonlinearly unstable despite the conditional linear stability - in fact, Deng, Liu, Yang, and Yao \cite{DLYY} show that the energy of a steady state is zero and any small perturbation can make the energy positive and cause part of the gas go to infinity. Despite the above advances for gaseous stars, only recently a few results have been established for the stability problem of Lane–Emden stars for a liquid.
Assuming spherical symmetry, so that \(\vec{u}(x,t)=u(r,t)\hat{r}\) where \(\hat{r}=\frac{x}{|x|}\), the continuity equation in \(\eqref{main eq}\) becomes \begin{align} D_{t}\rho+\rho\frac{\partial_{r}(r^{2}u)}{r^{2}}=\partial_{t}\rho+u\partial_{r}\rho+\rho\frac{\partial_{r}(r^{2}u)}{r^{2}}=0. \end{align} The momentum equation reads \begin{align}\label{momentum eq SS} D_{t}u+\partial_{r}\psi+\frac{1}{\rho}\partial_{r}P=\partial_{t}u+u\partial_{r}u+\partial_{r}\psi+\frac{1}{\rho}\partial_{r}P=0. \end{align} The gravitational potential satisfies the Poisson equation \begin{align}\label{Poisson eq SS} 4\pi\rho=\Delta\psi=\frac{1}{r^{2}}\partial_{r}(r^{2}\partial_{r}\psi). \end{align} We put \(\eqref{Poisson eq SS}\) into the momentum equation \(\eqref{momentum eq SS}\) to get \begin{align} \partial_{t}u+u\partial_{r}u+\frac{4\pi}{r^{2}}\int_{0}^{r}s^{2}\rho(s)ds+\frac{1}{\rho}\partial_{r}p=0. \end{align} Now we write the Euler-Poisson system \(\eqref{main eq}\) in Lagrangian coordinates. Let \(\eta(y,t)\) be the radial position of the fluid particle at time \(t\) so that \begin{align*} \partial_{t}\eta=u\circ\eta \ \ \ \ \text{with} \ \ \ \ \eta(y,0)=\eta_{0}(y). \end{align*} Here \(\eta_{0}\) is not necessarily the identity map and depend on the initial density profile and in fact. Then we have the Lagrangian variables \begin{align*} &\upsilon=u\circ\eta \ \ (Lagrangian\ velocity)\\ &f=\rho\circ\eta\ \ (Lagrangian\ density)\\ &\varphi=\psi\circ\eta\ \ (Lagrangian\ potential)\\ &J=\frac{\eta^{2}}{y^{2}}\partial_{y}\eta\ \ (Jacobian\ determinant\ (renormalized)). \end{align*} So the continuity equation in Lagrangian coordinate is \begin{align*} \partial_{t}f+f\frac{\partial_{t}J}{J}=0\ \ \ \ and\ so \ \ \ \ \partial_{t}\log(fJ)=0\ \ \Rightarrow \ \ fJ=f_{0}J_{0}. \end{align*} And the momentum equation becomes \begin{align*} \partial_{t}^{2}\eta+\frac{4\pi}{\eta^{2}}\int_{0}^{y}s^{2}(f_{0}J_{0})ds+\frac{1}{f_{0}J_{0}}\frac{\eta^{2}}{y^{2}}\partial_{y}\left( f_{0}J_{0}\frac{3y^{2}}{\partial_{y}\eta^{3}}\right) ^{\gamma}=0. \end{align*} Then we consider the linearized equation of Euler-Poisson system \(\eqref{main eq}-\eqref{boundary conditions}\) for perturbation. Assume \(\bar{\rho}\) is a steady state solution. For a small perturbation \(\rho_{0}=\bar{\rho}+\varepsilon\) and \(u_{0}=\upsilon\), we can pick \(\eta_{0}\) such that \(f_{0}J_{0}=\bar{\rho}\). Let \(\eta(y, t)=y(1+\zeta(y,t))\) and \(\sigma=\log\frac{\rho}{\bar{\rho}}\), we obtain the linearized equation for the perturbation \begin{align}\label{linearized eq SS} \left\{ \begin{aligned} &\partial_{t}\upsilon+\frac{4}{\bar{\rho}}\zeta\partial_{y}\bar{\rho}^{\gamma}-\gamma\frac{1}{\bar{\rho}}\partial_{y}(\bar{\rho}^{\gamma}(3\zeta+y\partial_{y}\zeta))=0\\ &\partial_{t}\sigma+\partial_{y}\upsilon+\frac{2\upsilon}{y}=0\\ &\partial_{t}\zeta-\frac{\upsilon}{y}=0 \end{aligned} \right. \end{align} Lam \cite{L} shows that liquid Lane-Emden stars are linearly stable when \(\frac{4}{3}\le\gamma\le2\), or \(1\le\gamma<\frac{4}{3}\) for stars with small central density. It is also shown in \cite{L} that the above linearized equation admits a growing mode solution of the form \(\zeta(y,t)=e^{\lambda t}\chi(y)\) with \(\lambda>0\) when \(\gamma\in[1,\frac43)\) and the star possesses a large central density. The linear stability of the gaseous Lane-Emden stars does not depend on its central density. However in the liquid case, the stability of the Lane-Emden stars does depend on its central density, see \cite{L} and the references (for instance \cite{hadvzic2021stability,hadvzic2021turning}) therein. The following lemma (see \cite{guo}) is crucial for us to obtain nonlinear instability. \begin{lemma}\label{1.4.}
Assume that \(L\) is a linear operator on a Banach space \(X\) with norm \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert\), and \(e^{tL}\) generates a strongly continuous semigroup on \(X\) such that
\begin{align}
\arrowvert\arrowvert e^{tL}\arrowvert\arrowvert_{(X,X)}\le C_{L}e^{\lambda t}
\end{align}
for some \(C_{L}\) and \(\lambda> 0\). Assume a nonlinear operator \(N(y)\) on \(X\) and another norm \(\arrowvert\arrowvert\arrowvert\cdot\arrowvert\arrowvert\arrowvert\), and constant \(C_{N}\), such that
\begin{align}
\arrowvert\arrowvert N(y)\arrowvert\arrowvert\le C_{N}\arrowvert\arrowvert\arrowvert y\arrowvert\arrowvert\arrowvert^{2}
\end{align}
for all \(y\in X\) and \(\arrowvert\arrowvert\arrowvert y\arrowvert\arrowvert\arrowvert <\infty\). Assume for any solution \(y(t)\) to the equation
\begin{align}
y'=Ly+N(y)
\end{align}
with \(\arrowvert\arrowvert\arrowvert y(t)\arrowvert\arrowvert\arrowvert\le \sigma\), there exist \(C_{0},C_{\sigma}>0\) such that for any small \(\epsilon>0\), there exists \(C_{\epsilon}>0\) such that the following sharp energy estimate holds:
\begin{align}\label{1.15}
\arrowvert\arrowvert\arrowvert y(t)\arrowvert\arrowvert\arrowvert\le C_{0}\arrowvert\arrowvert\arrowvert y(0)\arrowvert\arrowvert\arrowvert+\int_{0}^{T}\epsilon\arrowvert\arrowvert\arrowvert y(s)\arrowvert\arrowvert\arrowvert+C_{\sigma}\arrowvert\arrowvert\arrowvert y(s)\arrowvert\arrowvert\arrowvert^{\frac{3}{2}}+C_{\epsilon}\arrowvert\arrowvert y(s)\arrowvert\arrowvert ds.
\end{align}
Consider a family of initial data \(y^{\delta}(0)=\delta y_{0}\) with \(\arrowvert\arrowvert y_{0}\arrowvert\arrowvert=1\) and \(\arrowvert\arrowvert\arrowvert y_{0}\arrowvert\arrowvert\arrowvert<\infty\) and let \(\theta_{0}\) be a sufficiently small (fixed) number. Then there exists some constant \(C>0\) such that if
\begin{align}
0\le t\le T^{\delta}\equiv\frac{1}{\lambda}\log\frac{\theta_{0}}{\delta}
\end{align}
we have
\begin{align}\label{1.17}
\arrowvert\arrowvert y(t)-\delta e^{Lt}y_{0}\arrowvert\arrowvert\le C[\arrowvert\arrowvert\arrowvert y_{0}\arrowvert\arrowvert\arrowvert^{2}+1]\delta^{2}e^{2\lambda t}.
\end{align}
In particular, if there exists a constant \(C_{p}\) such that \(\arrowvert\arrowvert e^{Lt}y_{0}\arrowvert\arrowvert\ge C_{p}e^{\lambda t}\) then at the escape time
\begin{align}\label{1.18}
\arrowvert\arrowvert y(T^{\delta})\arrowvert\arrowvert\ge\tau_{0}>0,
\end{align}
where \(\tau_{0}\) depends explicitly on \(C_{L},C_{N},C_{0},C_{\sigma},C_{p},\lambda,y_{0},\sigma\) and is independent of \(\delta\). \end{lemma} \begin{remark}
\(y(t)\) represents the perturbation away from the
trivial steady state and \(\lambda\) is the fastest growing mode of the linear operator \(L\). It is worth noting that The small constant \(\epsilon\) only needs to be less than, say, \(\frac{\lambda}{4}\). \end{remark} \begin{remark}
In our application, \(\arrowvert\arrowvert\arrowvert\cdot\arrowvert\arrowvert\arrowvert\) represents higher order Sobolev norms (including internal and boundary norms). \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert\) represents \(L^{2}\) norm in the fluid interior. \(\arrowvert\arrowvert\arrowvert\cdot\arrowvert\arrowvert\arrowvert\) is a stronger norm than \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert\) and the assumption \(\arrowvert\arrowvert N(y)\arrowvert\arrowvert\le C_{N}\arrowvert\arrowvert\arrowvert y\arrowvert\arrowvert\arrowvert^{2}\) means that the nonlinearity is not bounded in the weaker norm \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert\). This is a key analytical
difficulty in many instability problems. \end{remark} Lemma \(\ref{1.4.}\) provides a general framework to study nonlinear instability, see other approaches \cite{VF,ChSu} for study of nonlinear instability in free boundary problems. The most delicate part of applying this framework is that the higher order Sobolev norm \(\arrowvert\arrowvert\arrowvert\cdot\arrowvert\arrowvert\arrowvert\) does not create a faster growth rate than the weaker counterpart \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert\), over the time scale of \(0\le t\le T^{\delta}\). In other words, roughly speaking, the stronger norm \(\arrowvert\arrowvert\arrowvert\cdot\arrowvert\arrowvert\arrowvert\) is controlled reversely by the weaker norm \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert\). This is the key to close the instability argument. \subsection{Statement of the main result} In this paper we prove the nonlinear instability for the liquid Lane-Emden stars. To state the main theorem, we consider the Euler-Poisson system \eqref{main eq}-\eqref{boundary conditions} in a perturbative form around the equilibrium state \(\bar{\rho}\) (Let \(\rho=\bar{\rho}+\tilde{\rho}\), then the unknowns of the system \eqref{main eq}-\eqref{boundary conditions} become \((u,\tilde{\rho})\)). Our main result establishes the full nonlinear dynamical instability of the Lane-Emden equilibrium. In the statement of the following theorem, for any \(\delta>0\) and \(\theta_{0}>\delta\), we define \begin{align} T^{\delta}\equiv\frac{1}{\sqrt{\mu_{0}}}\log\frac{\theta_{0}}{\delta}, \end{align} where \(\sqrt{\mu_{0}}\) is the sharp linear growth rate. \begin{theorem}\label{main th}
Assume that \(1\le\gamma<\frac{4}{3}\) and the stars have large central densities. For any sufficiently small \(\delta>0\), there exists a family of initial data \((u^{\delta}(0),\tilde{\rho}^{\delta}(0))=\delta(u_{0},\tilde{\rho}_{0})\) and \(T^{\delta}>0\) such that the perturbed solutions \((u^{\delta}(t),\tilde{\rho}^{\delta}(t))\) to the Euler-Poisson system \eqref{main eq}-\eqref{boundary conditions} for \(t\in[0,T^{\delta}]\) satisfy
\begin{align*}
\arrowvert\arrowvert(u^{\delta}(T^{\delta}),\tilde{\rho}^{\delta}(T^{\delta}))\arrowvert\arrowvert_{L^{2}}\ge \tau_{0}>0,
\end{align*}
where \(\tau_{0}\) is independent of \(\delta\). \end{theorem}
\begin{remark}
The above result shows that no matter how small the amplitude of initial perturbed
data is taken to be, we can find a solution such that the corresponding energy escapes at a time \(T^{\delta}\): there is no stabilization of the system. We conclude from this that liquid Lane-Emden stars with large central density for \(1\le\gamma<\frac{4}{3}\) are nonlinearly unstable.
\end{remark} \begin{remark}
We note that the nonlinear dynamics of any general perturbation is dominated by the fastest linear growing mode \(\sqrt{\mu_{0}}\) up to the time scale of \(T^{\delta}\). This implies that the nonlinear instability is essentially driven by the linear instability. The hypothesis of Theorem \(\ref{main th}\) guarantees the occurrence of the linear instability and a fast linear growth rate. \end{remark} \subsection{Main ideas for the proof}
Assume the exact solution \(y_{\ast}\) of Euler-Poisson system \(\eqref{main eq}-\eqref{boundary conditions}\) is written as \begin{align*}
y_{\ast}=\bar{y}+y_{L}+y_{e}, \end{align*} where \(\bar{y}\) is the steady state, \(y_{L}\) is the solution of linearized equation and \(y_{e}\) is the error. Based on linear instability, we have \(\Arrowvert y_{L}\Arrowvert\sim\delta e^{\lambda t}\), where \(\delta\) is the magnitude of the initial perturbation. Suppose there exists small enough \(\epsilon>0\) and constants \(C_{\sigma},C_{\epsilon}>0\) such that the perturbation solution \(y:=y_{L}+y_{e}\) satisfies the following sharp energy estimate \begin{align*}
\arrowvert\arrowvert\arrowvert y(t)\arrowvert\arrowvert\arrowvert\le\delta+\int_{0}^{t}\epsilon\lambda \arrowvert\arrowvert\arrowvert y(s)\arrowvert\arrowvert\arrowvert+C_{\sigma}\arrowvert\arrowvert\arrowvert y(s)\arrowvert\arrowvert\arrowvert^{\frac{3}{2}}+C_{\epsilon}\arrowvert\arrowvert y(s)\arrowvert\arrowvert ds. \end{align*} Then according to Lemma \(\ref{1.4.}\), we have \(\Arrowvert y\Arrowvert\sim\frac{1}{2}\delta e^{\lambda t}\), which means linear solution \(y_{L}\) dominate the nonlinear correction \(y_{e}\). Therefore at the escape time \(t=T^{\delta}\), we shall see \begin{align*}
\Arrowvert y(T^{\delta})\Arrowvert\gtrsim\frac{1}{2}\delta e^{\lambda T^{\delta}}=\frac{1}{2}\theta_{0}, \end{align*} where \(\theta_{0}\) is independent of \(\delta\), and instability happen.
Therefore an appropriate a priori estimate for the perturbed solution is the key to prove the nonlinear instability. To obtain such an estimate, we follow the approach in \cite{MSW}. We start by deriving quasilinear system from compressible Euler-Poisson system \eqref{main eq}-\eqref{boundary conditions}. By introducing an acoustical metric $g$ and its corresponding wave operator $\Box_{g}$ (see \eqref{2.4}), we obtain a coupled quasilinear system for fluid velocity \(u\) and logarithmic density perturbation variable \(\varepsilon\) (see \(\eqref{2.5},\eqref{2.15},\eqref{2.20}\) and \(\eqref{2.24}\)): \begin{equation}\label{1.19} \left\{ \begin{aligned} &\left(B^{2}+a\nabla_{n}^{\left( g\right) }\right)u^{i}=-c_{s}^{2}\nabla_{i} B\bm{\uprho}-B\nabla_{i}\psi\quad \textrm{on}\ \partial\mathcal B(t),\\ &\square_{g}u^{i}=-\left(1+c_{s}^{-1}c'_{s}\right) \left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}u^{i}+B\nabla_{i}\psi-2c_{s}^{-1}c'_{s}(B\bm{\uprho})\nabla_{i}\psi\quad \textrm{in}\ \mathcal B(t), \end{aligned} \right. \end{equation} and \begin{equation}\label{1.20} \left\{ \begin{aligned} \square_{g}\varepsilon=&-3c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}\varepsilon+\mathscr{Q}\quad\textrm{in}\ \mathcal B(t),\quad\quad\varepsilon,\ B\varepsilon=0\ \ \textrm{on}\ \ \partial\mathcal B(t), \\ \square_{g}B\varepsilon=&-(1+3c_{s}^{-1}c'_{s})\left( g^{-1}\right) ^{\alpha\beta}\left( \partial_{\alpha}\bm{\uprho}\right) \left( \partial_{\beta}B\varepsilon\right)-2c_{s}c'_{s}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}B\varepsilon\right)+\mathscr{P}\quad \textrm{in}\ \mathcal B(t). \end{aligned} \right. \end{equation} Here $\mathscr{Q}, \mathscr{P}$ are lower order terms whose precise expressions are given in \eqref{lower order terms}. $\bm{\uprho}$ is the logarithmic density defined to be $\log\frac{\rho}{\rho_{0}}$. $B$ is the vectorfield $\partial_{t}+u^{a}\partial_{a}$ and $c_{s}$ is the sound speed defined by $c_{s}:=\sqrt{\frac{dP}{d\rho}}$. In \cite{MSW} a similar quasilinear system was shown to be hyperbolic type, and a local-wellposedness result of free boundary hard phase fluids was obtained. For the fluid velocity equations \(\eqref{1.19}\), We multiply the boundary equation and interior equation by $\frac{1}{a}(Bu)$, \(Bu\) respectively and integrate. By observing the signs of the boundary terms, we obtain the following energy in Lemma \(\ref{3.2.}\) \begin{align*}
\int_{\mathcal B(t)}\left| \partial_{t,x}u\right| ^{2}dx+\int_{\partial\mathcal B(t)}\frac{1}{a}\left| Bu\right| ^{2}dS. \end{align*} For the wave equation for \(B\varepsilon\) with Dirichlet boundary conditions, we choose a suitable multiplier consisting of an appropriate linear combination of \(B\) and the normal \(n\) to $\partial\mathcal B(t)$, and apply integration by parts in Lemma \(\ref{3.3.}\). The energy functional for \(\eqref{1.20}\) controls \begin{align*}
\int_{\mathcal B(t)}\left| \partial_{t,x}B\varepsilon\right| ^{2}dx+\int_{0}^{t}\int_{\partial\mathcal B(\tau)}\left| \partial_{t,x}B\varepsilon\right| ^{2}dSd\tau. \end{align*} To control higher order Sobolev norms, we commute \(B^{k}\) with \(\eqref{1.19}\) and \(\eqref{1.20}\) (see \(\eqref{3.28}-\eqref{3.30}\)). Then we use elliptic estimates and the corresponding wave equation to control the higher order Sobolev norms in terms of the $L^{2}$-norms of $B^{k}$-derivatives (see details in Proposition \ref{5.2.}). For the most challenge gravitational potential in the source, we apply Hilbert transform and techniques in Clifford analysis to convert the derivative of \(\nabla \psi\) of any order into a function equal to zero on the boundary, which is treated in Section \(\ref{4}\). Then the energy functional \(\eqref{5.3}\) can control the \(L^{2}\) norm of the gravitational terms by using elliptic estimates repeatedly and controlling commutator errors. For other applications of Clifford analysis in fluid free boundary problems, see \cite{wu1999well, MS,ZZ}.
To close the a priori estimate, we need to control the source carefully, which we divide into three types: \emph{the top order linear source}, \emph{the non-top order linear source} and \emph{the nonlinear source}. According to Lemma \ref{1.4.} and Sobolev embedding, the nonlinear source is almost in good forms. The details are given in Section \ref{5}. For the linear source, we need to prove that their coefficients can be controlled by the fastest linear growth mode \(\mu_{0}\). By the standard Sobolev interpolation inequality, which helps us to transfer large coefficients of strong norm to above \(L^{2}\) norm, we can control the non-top order linear terms. Finally for the top order linear terms, as we shall show in Proposition \(\ref{5.1.}\), the coefficient \(C_{1}(\bar{\rho})\) in front of these terms is in form of \begin{align} C_{1}(\bar{\rho})\sim\left( c_{s}^{2}(\bar{\bm{\uprho}})+c_{s}(\bar{\bm{\uprho}})c'_{s}(\bar{\bm{\uprho}})\right)\nabla\bar{\bm{\uprho}} \sim\gamma\bar{\rho}^{\gamma-1}\nabla\bar{\bm{\uprho}}\lesssim-\frac{1}{\bar{\rho}}\partial_{y}\bar{\rho}^{\gamma}. \end{align} Observing the influence of the adiabatic index \(\gamma\) on the properties of the steady-state solution, we divide \(\gamma\) into three different intervals. Applying the basic relations \(\eqref{5.49}, \eqref{5.55}\) and asymptotic properties \(\eqref{333}\) satisfied by the steady-state solution, we prove that the coefficient \(C_{1}(\bar{\rho})\) can be dominated by the fastest linear growth mode \(\mu_{0}\) in all three intervals when the density of the star center is large enough. The details for treating the linear source are given in Section \ref{6}. Combined with the previous discussion and the a priori estimate \(\eqref{5.5}\), we finally prove that the liquid Lane-Emden stars are nonlinearly unstable.
\subsection{Outline of the paper} In Section \(\ref{2}\) we derive the quasilinear equations for fluid velocity \(u\) and logarithmic density perturbation variable \(\varepsilon\). In Section \(\ref{3}\), we derive the basic energy inequalities and higher order equations. In Section \(\ref{4}\), we introduce Hilbert transform and describe how to apply them in the a priori estimates. Based on these, in Section \(\ref{5}\) we prove a sharp nonlinear energy estimate (see \ref{5.5}), and the integral terms of the right-hand side of \(\eqref{5.5}\) contain the three types of sources (i.e. the top order linear terms, the non-top order linear terms and the nonlinear terms). In Section \(\ref{6}\), we show that the coefficients of the linear terms on the right-hand side of \(\eqref{5.5}\) can be dominated by the fastest linear growth mode \(\mu_{0}\), which in fact proves our main result Theorem \(\ref{main th}\) by using a bootstrap argument.
\subsection*{Acknowledgment} The authors would like to thank Qingtang Su and Yanlin Wang for stimulating discussions on this project. This work was supported by National Key R\& D Program of China 2021YFA1001700, NSFC grant 12071360, and the Fundamental Research Funds for the Central Universities in China.
\section{Quasilinear equations for compressible Euler-Poisson system}\label{2} We start by recalling the notation \begin{align*} \bm{\uprho}=\log(\rho/\rho_{0}), \ \ B=\partial_{t}+u^{a}\partial_{a},\ \ c_{s}^{2}=\frac{dP}{d\rho}, \end{align*} where the constant $\rho_{0}>0$ is the boundary value of $\rho$. Then the compressible Euler-Poisson system \(\eqref{main eq}\) write \begin{align}\label{2.1} B\bm{\uprho}=&-\mathrm{div}\, u,\\ \label{2.2} Bu^{i}=&-c_{s}^{2}\nabla_{i}\bm{\uprho}-\nabla_{i}\psi. \end{align} Let \(X:\mathbb{R}\times\mathcal B(0)\to \mathcal B\) be the Lagrangian parametrization of \(\mathcal B=\mathcal B(t)\). We can express the logarithmic density as the sum of the steady state and the perturbation \begin{align*} \bm{\uprho}(x,t)=\bar{\bm{\uprho}}(X^{-1}(x,t))+\varepsilon(x,t),\ \ \ \ where\ \ \ \ \bar{\bm{\uprho}}=\log\bar{\rho} \end{align*} By a slight abuse of notation, we often write \(\bar{\bm{\uprho}}(x,t)\) instead of \(\bar{\bm{\uprho}}(X^{-1}(x,t))\), therefore we have \begin{align}\label{2.3} B\bar{\bm{\uprho}}=0. \end{align} In order to obtain quasilinear equations, we introduce the following acoustical metric and the corresponding covariant wave operator \begin{align}\label{2.4} \begin{split} &g:=-dt\otimes dt+c_{s}^{-2}\sum_{a=1}^{3}\left(dx^{a}-u^{a}dt\right)\otimes \left(dx^{a}-u^{a}dt\right),\\ &g^{-1}:=-B\otimes B+c_{s}^{2}\sum_{a=1}^{3}\partial_{a}\otimes\partial_{a},\\
&\square_{g}:=\frac{1}{\sqrt{|\det g|}}\partial_{\alpha} \left\{\sqrt{|\det g|}(g^{-1})^{\alpha\beta}\partial_{\beta}\right\}. \end{split} \end{align} It is straightforward to verify that \(g^{-1}\) is the matrix inverse of \(g\), and in this case the vectorfield \(B\) is timelike, future-directed, orthogonal to \(\mathcal B_{t}\), and with unit-length.
\begin{lemma}[\(\textbf{Quasilinear equation on the free boundary}\)]
Given the system \(\eqref{2.1}-\eqref{2.2}\), we have the following equation on the boundary:
\begin{align}\label{2.5}
\left(B^{2}+a\nabla_{n}^{\left( g\right) }\right)u^{i}=-c_{s}^{2}\nabla_{i}B\varepsilon-B\nabla_{i}\psi,\quad \textrm{on}\quad \partial\mathcal B(t).
\end{align}
Here \(a=\sqrt{\partial_{\alpha}\bm{\uprho}\partial^{\alpha}\bm{\uprho}}\).
\begin{proof}
According to \eqref{boundary conditions},we get
\begin{align}\label{2.6}
B\bm{\uprho}=0,\quad \textrm{on}\quad \partial\mathcal B(t).
\end{align}
Taking a \(B\) derivative to the momentum equation \eqref{2.2}, using \(\eqref{2.6}\) and restricting it to the boundary we get
\begin{align}\label{2.7}
B^{2}u^{i}+c_{s}^{2}B\nabla_{i} \bm{\uprho}=-B\nabla_{i}\psi.
\end{align}
Computing the commutator \(\left[B,\nabla_{i}\right]\bm{\uprho}=B\nabla_{i} \bm{\uprho}-\nabla_{i} B\bm{\uprho}=-\nabla \bm{\uprho}\cdot\nabla u^{i} \) yields
\begin{align}\label{2.8}
B^{2}u^{i}-c_{s}^{2}\nabla \bm{\uprho}\cdot\nabla u^{i}=-\nabla_{i} B\bm{\uprho}-B\nabla_{i}\psi.
\end{align}
We now simplify the term \(\left( c_{s}^{2}\nabla \bm{\uprho}\cdot\nabla u^{i}\right) \). According to the expression of \(g^{-1}\) ,we have
\(\partial^{a}=-B^{a}B+c_{s}^{2}\partial_{a}\) and \(\partial^{0}=-B\). Using \(\eqref{2.6}\) we get
\begin{align*}
\partial^{a}\bm{\uprho}=c_{s}^{2}\partial_{a}\bm{\uprho},\quad \partial^{0}\bm{\uprho}=0, \quad \textrm{on}\quad \partial\mathcal B(t),
\end{align*}
and hence
\begin{align*}
c_{s}^{2}\nabla \bm{\uprho}\cdot\nabla u^{i}=\partial^{\alpha}\bm{\uprho}\partial_{\alpha}u^{i}=\partial_{\alpha}\bm{\uprho}\partial^{\alpha}u^{i}.
\end{align*}
We use \(\nabla^{\left( g\right) }\) for the (spacetime) gradient (with respect to \(g\)). Let \(n\) be the unit outward pointing (spacetime) normal to \(\partial\mathcal B(t)\). Since \(\bm{\uprho}=0\ \textrm{on}\ \partial\mathcal B(t),\ -\nabla^{\left( g\right) } \bm{\uprho}=an\) on the free boundary \(\partial\mathcal B(t)\), with \(a=\sqrt{\partial_{\alpha}\bm{\uprho}\partial^{\alpha}\bm{\uprho}}\). According to \(\eqref{2.3}\) and simple calculations, we get
\begin{align}\label{2.9}
\left(B^{2}+a\nabla^{\left( g\right)}_{n}\right)u^{i}=-c_{s}^{2}\nabla_{i} B\varepsilon-B\nabla_{i}\psi,
\end{align}
which completes the proof of \(\eqref{2.5}\).
\end{proof} \end{lemma}
\begin{lemma}[\(\square_{g}\) \(\textbf{relative to the Cartesian coordinates}\)]
The covariant wave operator \(\square_{g}\) acts on scalar functions \(\phi\) via the following identity:
\begin{align}\label{2.11}
\square_{g}\phi=-BB\phi+c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\phi+2c_{s}^{-1}c'_{s}\left(B\varepsilon\right)B\phi-\left(\partial_{a}u^{a}\right)B\phi-c_{s}^{-1}c'_{s}\left( g^{-1}\right)^{\alpha\beta}\left(\partial_{\alpha}\bm{\uprho}\right) \partial_{\beta}\phi.
\end{align}
\begin{proof}
It is straightforward to compute using equations \(\eqref{2.4}\) that relative to Cartesian coordinates, we have
\begin{align}\label{2.12}
\det g=-c_{s}^{-6}
\end{align}
and hence
\begin{align}\label{2.13}
\sqrt{|\det g|}g^{-1}=-c_{s}^{-3}B\otimes B+c_{s}^{-1}\sum_{a=1}^{3}\partial_{a}\otimes\partial_{a}.
\end{align}
Using\(\eqref{2.4}\), \(\eqref{2.12}\) and \(\eqref{2.13}\), we compute that
\begin{align}\label{2.14}
\begin{split}
\square_{g}\phi=&-c_{s}^{3}\left(B^{\, \alpha}\partial_{\alpha}\left( c_{s}^{-3}\right) \right) B^{\,\beta}\partial_{\beta}\phi-\left( \partial_{\alpha}B^{\,\alpha}\right) B^{\,\beta}\partial_{\beta}\phi-\left( B^{\,\alpha}\partial_{\alpha}B^{\,\beta}\right) \partial_{\beta}\phi\\
&-B^{\,\alpha}B^{\,\beta}\partial_{\alpha}\partial_{\beta}\phi+c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\phi-c_{s}c'_{s}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \partial_{b}\phi.
\end{split}
\end{align}
Finally, from \(\eqref{2.14}\), the expression for \(B\), the expression for \(g^{-1}\), and simple calculations, we arrive at \(\eqref{2.11}\).
\end{proof} \end{lemma} We now establish equation for \(u\). \begin{lemma}[\(\textbf{Wave equation for}\) \(u\)]
The compressible Euler-Poisson equations \(\eqref{2.1}-\eqref{2.2}\) imply the following covariant wave equation for the scalar-valued function \(u^{i}\):
\begin{align}\label{2.15}
\begin{split}
\square_{g}u^{i}=-\left(1+c_{s}^{-1}c'_{s}\right) \left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}u^{i}+B\nabla_{i}\psi-2c_{s}^{-1}c'_{s}(B\varepsilon)\nabla_{i}\psi.
\end{split}
\end{align}
\begin{proof}
First, we use \(\eqref{2.11}\) with \(\phi=u^{i}\) to deduce
\begin{align}\label{2.16}
\square_{g}u^{i}=-BBu^{i}+c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}u^{i}+2c_{s}^{-1}c'_{s}(B\varepsilon) Bu^{i}-\left( \partial_{a}u^{a}\right) Bu^{i}-c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}u^{i}.
\end{align}
Next, we use \(\eqref{2.1}-\eqref{2.2}\) to compute that
\begin{align}\label{2.17}
\begin{split}
BBu^{i}=&-c_{s}^{2}\delta^{ia}B\partial_{a}\bm{\uprho}-2c_{s}c'_{s}B\bm{\uprho}\delta^{ia}\partial_{a}\bm{\uprho}-B\nabla_{i}\psi\\
=&-c_{s}^{2}\delta^{ia}\partial_{a}\left( B\bm{\uprho}\right) +c_{s}^{2}\delta^{ia}\partial_{a}u^{b}\partial_{b}\bm{\uprho}-2c_{s}c'_{s}B\bm{\uprho}\delta^{ia}\partial_{a}\bm{\uprho}-B\nabla_{i}\psi\\
=&c_{s}^{2}\delta^{ia}\delta_{c}^{b}\partial_{a}\left( \partial_{b}u^{c}\right) +c_{s}^{2}\delta^{ia}\partial_{a}u^{b}\partial_{b}\bm{\uprho}-2c_{s}c'_{s}B\bm{\uprho}\delta^{ia}\partial_{a}\bm{\uprho}-B\nabla_{i}\psi\\
=&c_{s}^{2}\delta^{bc}\partial_{b}\partial_{c}u^{i}+c_{s}^{2}\delta^{ia}\partial_{c}\left( \partial_{a}u^{c}-\partial_{c}u^{a}\right) +c_{s}^{2}\delta^{ia}\partial_{a}u^{b}\partial_{b}\bm{\uprho}-2c_{s}c'_{s}B\bm{\uprho}\delta^{ia}\partial_{a}\bm{\uprho}-B\nabla_{i}\psi\\
=&c_{s}^{2}\delta^{bc}\partial_{b}\partial_{c}u^{i}+c_{s}^{2}\delta^{ia}\partial_{c}\left( \partial_{a}u^{c}-\partial_{c}u^{a}\right)+c_{s}^{2}\left( \partial_{i}u^{b}-\partial_{b}u^{i}\right)\partial_{b}\bm{\uprho}\\
&+c_{s}^{2}\delta^{ab}\partial_{a}u^{i}\partial_{b}\bm{\uprho}-2c_{s}c'_{s}B\bm{\uprho}\delta^{ia}\partial_{a}\bm{\uprho}-B\nabla_{i}\psi\\
=&c_{s}^{2}\delta^{bc}\partial_{b}\partial_{c}u^{i}+c_{s}^{2}\delta^{ab}\partial_{a}u^{i}\partial_{b}\bm{\uprho}-2c_{s}c'_{s}B\bm{\uprho}\delta^{ia}\partial_{a}\bm{\uprho}-B\nabla_{i}\psi,
\end{split}
\end{align}
because \(\textrm{curl}u=0\).
Next, substituting the RHS \(\eqref{2.17}\) for the term \(-BBu^{i}\) on RHS \(\eqref{2.16}\), we arrive at
\begin{align}\label{2.18}
\begin{split}
\square_{g}u^{i}=\left\{-c_{s}^{2}\delta^{ab}\left( \partial_{b}\bm{\uprho}\right) \partial_{a}u^{i}-\left( \partial_{a}u^{a}\right) Bu^{i} \right\}-c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}u^{i}+B\nabla_{i}\psi-2c_{s}^{-1}c'_{s}(B\bm{\uprho})\nabla_{i}\psi.
\end{split}
\end{align}
To handle the terms \(\left\{\cdot\right\}\) in \(\eqref{2.18}\), we use \(\eqref{2.1}\),\(\eqref{2.2}\) and \(\eqref{2.4}\) to obtain
\begin{align}\label{2.19}
-c_{s}^{2}\delta^{ab}\left( \partial_{b}\bm{\uprho}\right) \partial_{a}u^{i}-\left( \partial_{a}u^{a}\right) Bu^{i}=-c_{s}^{2}\delta^{ab}\left( \partial_{b}\bm{\uprho}\right) \partial_{a}u^{i}+\left( B\bm{\uprho}\right) Bu^{i}=-\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}u^{i}.
\end{align}
Finally, substituting \(\eqref{2.19}\) into \(\eqref{2.18}\), we conclude the desired equation \(\eqref{2.15}\).
\end{proof} \end{lemma} We now derive equation for \(\varepsilon\). \begin{lemma}[\(\textbf{Wave equation for}\) \(\varepsilon\)]
The compressible Euler-Poisson equations \(\eqref{2.1}-\eqref{2.2}\) imply the following covariant wave equation for the logarithmic density perturbation variable \(\varepsilon\):
\begin{align}\label{2.20}
\begin{split}
\square_{g}\varepsilon=&-3c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}\varepsilon+2\sum_{1\le a<b\le 3}\left\{\partial_{a}u^{a}\partial_{b}u^{b}-\partial_{a}u^{b}\partial_{b}u^{a}\right\}\\
&-c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\bar{\bm{\uprho}}-2c_{s}^{2}c'_{s}\delta^{ab}(\partial_{a}\bm{\uprho})(\partial_{b}\bar{\bm{\uprho}})-\Delta\psi.
\end{split}
\end{align}
\begin{proof}
First,using \(\eqref{2.11}\) with \(\phi=\varepsilon\) and equation \(\eqref{2.1}\), we compute that
\begin{align}\label{2.21}
\square_{g}\varepsilon=-BB\varepsilon+c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\varepsilon+2c_{s}^{-1}c'_{s}(B\varepsilon) (B\varepsilon)+\left( \partial_{a}u^{a}\right) ^{2}-c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}\varepsilon.
\end{align}
Next, we use \(\eqref{2.1}-\eqref{2.2}\) to compute that
\begin{align}\label{2.22}
\begin{split}
BB\varepsilon=&-\partial_{a}\left( Bu^{a}\right) +\left( \partial_{a}u^{b}\right) \partial_{b}u^{a}\\
=&c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\bm{\uprho}+\delta^{ab}\left( \partial_{a}c_{s}^{2}\right) \partial_{b}\bm{\uprho}+\left( \partial_{a}u^{b}\right) \partial_{b}u^{a}+\Delta\psi\\
=&c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\bm{\uprho}+2c_{s}c'_{s}\delta^{ab}\partial_{a}\bm{\uprho}\partial_{b}\bm{\uprho}+\left( \partial_{a}u^{b}\right) \partial_{b}u^{a}+\Delta\psi.
\end{split}
\end{align}
Finally, using \(\eqref{2.22}\) to substitute for the term \(-BB\bm{\uprho}\) on RHS \(\eqref{2.21}\) and using the identities
\begin{align}\label{2.23}
\left( \partial_{a}u^{a}\right) ^{2}-\left( \partial_{a}u^{b}\right) \partial_{b}u^{a}=2\sum_{1\le a<b\le 3}\left\{\partial_{a}u^{a}\partial_{b}u^{b}-\partial_{a}u^{b}\partial_{b}u^{a}\right\}
\end{align}
and \(B\bm{\uprho}B\varepsilon-c_{s}^{2}\delta^{ab}\partial_{a}\bm{\uprho}\partial_{b}\varepsilon=-\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}\varepsilon\), , we arrive at the desired expression \(\eqref{2.20}\).
\end{proof} \end{lemma} It's worth noting that there is no zeroth order term on RHS \(\eqref{2.20}\) because \(\bar{\bm{\uprho}}\) satisfies the steady equation. We now establish equation for \(B\varepsilon\). \begin{lemma}[\(\textbf{Wave equation for}\) \(B\varepsilon\)]
The compressible Euler-Poisson equations \(\eqref{2.1}-\eqref{2.2}\) imply the following covariant wave equation for \(B\varepsilon\):
\begin{align}\label{2.24}
\begin{split}
\square_{g}B\varepsilon=& -(1+3c_{s}^{-1}c'_{s})\left( g^{-1}\right) ^{\alpha\beta}\left( \partial_{\alpha}\bm{\uprho}\right) \left( \partial_{\beta}B\varepsilon\right)-2c_{s}c'_{s}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}B\varepsilon\right)+2c_{s}^{2}\left( \partial_{a}\partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)\\
&-2c_{s}c'_{s}\delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\partial_{b}\bm{\uprho}\right)+2c_{s}^{2}\left( \partial_{a}\partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)-2\left( c'_{s}c'_{s}+c_{s}c'\!'_{s}\right) \delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right)\\
&+8c_{s}c'_{s}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)-B\Delta\psi +2\left( \partial_{a}\partial_{b}\psi\right) \left( \partial_{a}u^{b}\right)
+2\left( \partial_{a}u^{b}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{c}u^{a}\right).
\end{split}
\end{align}
\begin{proof}
First,using \(\eqref{2.11}\) with \(\phi=B\varepsilon\) to deduce
\begin{align}\label{2.25}
\square_{g}B\varepsilon=-BBB\varepsilon+c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}B\varepsilon+2c_{s}^{-1}c'_{s}\left( B\varepsilon\right) \left( BB\varepsilon\right) -\left( \partial_{a}u^{a}\right) BB\varepsilon-c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\left( \partial_{\alpha}\bm{\uprho}\right) \left( \partial_{\beta}B\varepsilon\right) .
\end{align}
Next, commuting equation \(\eqref{2.22}\) with the operator \(B\), we get
\begin{align}\label{2.26}
\begin{split}
BBB\varepsilon=&2c_{s}c'_{s}\delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\partial_{b}\bm{\uprho}\right) +c_{s}^{2}\delta^{ab}\left( B\partial_{a}\partial_{b}\bm{\uprho}\right) +2\left( c'_{s}c'_{s}+c_{s}c'\!'_{s}\right) \delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right) \\
&+B\Delta\psi+\left\{4c_{s}c'_{s}\delta^{ab}\left( B\partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right) +2\left( B\partial_{a}u^{b}\right) \left( \partial_{b}u^{a}\right) \right\}.
\end{split}
\end{align}
Next, using the identity
\begin{align}\label{2.27}
\partial_{a}B\phi=B\partial_{a}\phi+\left( \partial_{a}u^{c}\right) \left( \partial_{c}\phi\right)
\end{align}
with \(\phi=\bm{\uprho}\) and \(\phi=u^{b}\) respectively, we deduce the terms \(\left\{\cdot\right\}\) in \(\eqref{2.26}\) and obtain
\begin{align}\label{2.28}
\begin{split}
\left\{\cdot\right\}=&4c_{s}c'_{s}\delta^{ab}\left( \partial_{a}B\varepsilon\right) \left( \partial_{b}\bm{\uprho}\right) -4c_{s}c'_{s}\delta^{ab}\left( \partial_{a}u^{c}\right) \left( \partial_{c}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right)+2\left( \partial_{a}Bu^{b}\right) \left( \partial_{b}u^{a}\right) -2\left( \partial_{a}u^{b}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{c}u^{a}\right)
\end{split}
\end{align}
Next, using the identity
\begin{align}\label{2.29}
\partial_{a}\partial_{b}B\varepsilon=B\partial_{a}\partial_{b}\bm{\uprho}+\left( \partial_{a}\partial_{b}u^{c}\right) \left( \partial_{c}\bm{\uprho}\right)+\left( \partial_{a}u^{c}\right) \left( \partial_{b}\partial_{c}\bm{\uprho}\right)+\left( \partial_{b}u^{c}\right) \left( \partial_{a}\partial_{c}\bm{\uprho}\right),
\end{align}
we get
\begin{align}\label{2.30}
\begin{split}
c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}B\varepsilon=&c_{s}^{2}\delta^{ab}\left( B\partial_{a}\partial_{b}\bm{\uprho}\right) +c_{s}^{2}\delta^{ab}\left( \partial_{a}\partial_{b}u^{c}\right) \left( \partial_{c}\bm{\uprho}\right) +2c_{s}^{2}\delta^{ab}\left( \partial_{a}u^{c}\right) \left( \partial_{b}\partial_{c}\bm{\uprho}\right) \\
=&c_{s}^{2}\delta^{ab}\left( B\partial_{a}\partial_{b}\bm{\uprho}\right) +c_{s}^{2}\left( \Delta u^{c}\right) \left( \partial_{c}\bm{\uprho}\right) +2c_{s}^{2}\left( \partial_{a}u^{b}\right) \left( \partial_{a}\partial_{b}\bm{\uprho}\right).
\end{split}
\end{align}
Next, we use \(\eqref{2.25},\eqref{2.26},\eqref{2.28}\) and \(\eqref{2.30}\) to derive the equation
\begin{align}\label{2.31}
\begin{split}
\square_{g}B\varepsilon=& -c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\left( \partial_{\alpha}\bm{\uprho}\right) \left( \partial_{\beta}B\varepsilon\right)-4c_{s}c'_{s}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}B\varepsilon\right)+c_{s}^{2}\left( \triangle u^{c}\right) \left( \partial_{c}\bm{\uprho}\right)\\
&-2c_{s}c'_{s}\delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\partial_{b}\bm{\uprho}\right)+2c_{s}^{2}\left( \partial_{a}\partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)-2\left( c'_{s}c'_{s}+c_{s}c'\!'_{s}\right) \delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right)\\
&+4c_{s}c'_{s}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)-B\Delta\psi \\
&-2\left( \partial_{a}Bu^{b}\right) \left( \partial_{b}u^{a}\right)+2\left( \partial_{a}u^{b}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{c}u^{a}\right)+2c_{s}^{-1}c'_{s}\left( B\varepsilon\right) \left( BB\varepsilon\right) +\left( B\varepsilon\right) \left( BB\varepsilon\right) .
\end{split}
\end{align}
Finally, noticing the identity
\begin{align}\label{2.32}
\begin{split}
-2\left( \partial_{a}Bu^{b}\right) \left( \partial_{b}u^{a}\right)=
2c_{s}^{2}\left( \partial_{a}\partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right) +4c_{s}c'_{s}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)+2\left( \partial_{a}\partial_{b}\psi\right) \left( \partial_{a}u^{b}\right) ,
\end{split}
\end{align}
according to \(\textrm{curl}(\textrm{curl}u)=\nabla\left( \nabla\cdot u\right) -\Delta u=0\), and \(\left( B\bm{\uprho}\right) \left( BB\varepsilon\right) -c_{s}^{2}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}B\varepsilon\right) =-\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}B\varepsilon\), we arrive at the desired expression \(\eqref{2.24}\).
\end{proof} \end{lemma} In conclusion, we reformulate the compressible Euler-Poisson equation in the following quasilinear form: \begin{equation}\label{2.33} \left\{ \begin{aligned} &\left(B^{2}+a\nabla_{n}^{\left( g\right) }\right)u^{i}=-c_{s}^{2}\nabla_{i} B\bm{\uprho}-B\nabla_{i}\psi\quad \textrm{on}\ \partial\mathcal B(t),\\ &\square_{g}u^{i}=-\left(1+c_{s}^{-1}c'_{s}\right) \left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}u^{i}+B\nabla_{i}\psi-2c_{s}^{-1}c'_{s}(B\bm{\uprho})\nabla_{i}\psi\quad \textrm{in}\ \mathcal B(t), \end{aligned} \right. \end{equation} and \begin{equation}\label{2.34} \left\{ \begin{aligned} \square_{g}\varepsilon=&-3c_{s}^{-1}c'_{s}\left( g^{-1}\right) ^{\alpha\beta}\partial_{\alpha}\bm{\uprho}\partial_{\beta}\varepsilon+\mathscr{Q}\quad\textrm{in}\ \mathcal B(t),\quad\quad\varepsilon,\ B\varepsilon=0\ \ \textrm{on}\ \ \partial\mathcal B(t), \\ \square_{g}B\varepsilon=&-(1+3c_{s}^{-1}c'_{s})\left( g^{-1}\right) ^{\alpha\beta}\left( \partial_{\alpha}\bm{\uprho}\right) \left( \partial_{\beta}B\varepsilon\right)-2c_{s}c'_{s}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}B\varepsilon\right)+\mathscr{P}\quad \textrm{in}\ \mathcal B(t), \end{aligned} \right. \end{equation} where \(a\), \(\mathscr{Q}\) and \(\mathscr{P}\) are defined by \begin{align}\label{lower order terms}
\begin{split} a:=&\sqrt{\partial_{\alpha}\bm{\uprho}\partial^{\alpha}\bm{\uprho}},\\ \mathscr{Q}:=&-c_{s}^{2}\delta^{ab}\partial_{a}\partial_{b}\bar{\bm{\uprho}}-2c_{s}^{2}c'_{s}\delta^{ab}(\partial_{a}\bm{\uprho})(\partial_{b}\bar{\bm{\uprho}})-\Delta\psi+2\sum_{1\le a<b\le 3}\left\{\partial_{a}u^{a}\partial_{b}u^{b}-\partial_{a}u^{b}\partial_{b}u^{a}\right\}\\ \mathscr{P}:=&-2c_{s}c'_{s}\delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\partial_{b}\bm{\uprho}\right)+2c_{s}^{2}\left( \partial_{a}\partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right)-2\left( c'_{s}c'_{s}+c_{s}c'\!'_{s}\right) \delta^{ab}\left( B\varepsilon\right) \left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right)-B\Delta\psi\\ &+8c_{s}c'_{s}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right) +2\left( \partial_{a}\partial_{b}\psi\right) \left( \partial_{a}u^{b}\right) +2c_{s}^{2}\left( \partial_{a}\partial_{b}\bm{\uprho}\right) \left( \partial_{a}u^{b}\right) +2\left( \partial_{a}u^{b}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{c}u^{a}\right). \end{split} \end{align} We write the right hand side of equation \(\eqref{2.33}\) and \(\eqref{2.34}\) as the sum of the main linear terms and remainder, which \(\mathscr{Q}\) and \(\mathscr{P}\) include the lower order linear and nonlinear terms. In the next section we will discuss the necessary analytic tools to resolve above quasilinear system.
\section{Energy inequality and higher order equations}\label{3} In this section we first consider the following model \begin{equation}\label{3.1} \left\{ \begin{aligned} &\left(B^{2}+an\right)\phi=f\\ &\square_{g}\phi=g \end{aligned} \right. \end{equation} here we use the notation \(n\phi\) for \(\nabla_{n}^{\left( g\right) }\phi\) and \(a\) is as in \(\eqref{2.5}\). In subsection \(\ref{222}\) we use \(\nabla\) for \(\nabla^{\left( g\right) }\).
To obtain an energy inequality for the wave operator \(\square_{g}\), we start as usual by choosing a vector field \(X\) (the multiplier) and writing \begin{align}\label{3.2} \left( \square_{g}\phi\right) \left( X\phi\right) =\mathrm{div}\, P +q. \end{align} Here \(P\) is an appropriate field whose coefficients are quadratic forms in the components of \(\nabla\phi\), and \(q\) is a quadratic form in these components with variable coefficients. Then by integrating \(\left( \square_{g}\phi\right) \left( X\phi\right) \) in some spacetime domain \(\mathcal{D}\), and using the Stokes formula, we can obtain boundary terms \begin{align*} \int_{\partial\mathcal{D}}\left\langle P,N\right\rangle dv, \end{align*} where \(N\) is unit outside normal, which yield the energy of \(\phi\). We shall applying the above energy identity to our setting. \subsection{Energy inequality}\label{222} The energy-momentum tensor \(Q\) is a symmetric 2-tensor defined by \begin{align}\label{Q}
Q\left( X,Y\right) =\left( X\phi\right) \left( Y\phi\right) -\frac{1}{2}\left\langle X,Y \right\rangle |\nabla\phi|^{2},\quad Q_{\alpha\beta}=\left( \partial_{\alpha}\phi\right)\left( \partial_{\beta}\phi\right)-\frac{1}{2}g_{\alpha\beta}|\nabla\phi|^{2}, \end{align}
where \(|\nabla\phi|^{2}=\partial_{\alpha}\phi\partial^{\alpha}\phi\), and the deformation tensor of a given vector field \(X\) is the symmetric 2-tensor \(^{\left( X\right)}\pi \) defined by \begin{align} ^{\left( X\right)}\pi\left( Y,Z\right) =\left\langle D_{Y}X,Z\right\rangle +\left\langle D_{Z}X,Y\right\rangle ,\quad ^{\left( X\right)}\pi_{\alpha\beta}=D_{\alpha}X_{\beta}+D_{\beta}X_{\alpha}. \end{align} Next, we introduce a key formula. \begin{lemma}
Let \(\phi\) be a given \(C^{2}\) function and \(Q\) be the associated energy–momentum tensor. Let \(X\) be a vector field, and set \(P_{\alpha}=Q_{\alpha\beta}X^{\beta}=\left( X\phi\right) \left( \partial_{\alpha}\phi\right) -\frac{1}{2}X_{\alpha}|\nabla\phi|^{2}\). Then
\begin{align}\label{key formula}
\mathrm{div}\, P =\left( \square_{g}\phi\right)\left( X\phi\right) +\frac{1}{2}Q^{\alpha\beta\left( X\right) }\pi_{\alpha\beta}.
\end{align} \end{lemma} To compute \(^{\left( X\right)}\pi\), we utilize the following formula \begin{align}\label{pi} ^{\left( X\right)}\pi^{\alpha\beta}=\partial^{\alpha}\left( X^{\beta}\right) +\partial^{\beta}\left( X^{\alpha}\right) -X\left( g^{\alpha\beta}\right) . \end{align}
Now, using \(\eqref{Q}\), \(\eqref{key formula}\) and \(\eqref{pi}\), we compute that \begin{align}\label{divP} \begin{split}
\mathrm{div}\, P =&\left( \square_{g}\phi\right)\left( X\phi\right) +\frac{1}{2}\left[ \left( \partial_{\alpha}\phi\right) \left( \partial_{\beta}\phi\right) -\frac{1}{2}g_{\alpha\beta}|\nabla\phi|^{2}\right] \left[ \partial^{\alpha}\left( X^{\beta}\right) +\partial^{\beta}\left( X^{\alpha}\right) -X\left( g^{\alpha\beta}\right)\right] \\
=&\left( \square_{g}\phi\right)\left( X\phi\right)+\partial^{\alpha}\left( X^{\beta}\right) \left( \partial_{\alpha}\phi\right) \left( \partial_{\beta}\phi\right) -\frac{1}{2}X\left( g^{\alpha\beta}\right) \left( \partial_{\alpha}\phi\right) \left( \partial_{\beta}\phi\right) -\frac{1}{2}\left( \partial_{\alpha}X^{\alpha}\right) |\nabla\phi|^{2}+\frac{1}{4}g_{\alpha\beta}X\left( g^{\alpha\beta}\right) |\nabla\phi|^{2}. \end{split} \end{align} Noticing the identity \begin{align}\label{Xnabla2}
|g|^{-\frac{1}{2}}X\left( |g|^{\frac{1}{2}} \right)=-\frac{1}{2}g_{\alpha\beta}X\left( g^{\alpha\beta}\right), \end{align} we arrive at the following identity \begin{align}\label{key identity}
\left( \square_{g}\phi\right)\left( X\phi\right)=\mathrm{div}\, P+\frac{1}{2}\left( \partial_{\alpha}X^{\alpha}\right) |\nabla\phi|^{2}+\frac{1}{2}X\left(g^{\alpha\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) -\partial^{\alpha}\left(X^{\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right)+\frac{1}{2}|g|^{-\frac{1}{2}}X\left( |g|^{\frac{1}{2}} \right) |\nabla\phi|^{2} , \end{align}
where \(|g|=|\det g|\). Then we establish main energy identity for \(\eqref{3.1}\). \begin{lemma}[\(\textbf{Energy identity}\)]\label{3.2.}
Suppose \(\phi\) satisfies \(\eqref{3.1}\), then the following energy identity holds:
\begin{align}\label{energy}
\begin{split}
&\int_{\mathcal B(T)}\left( \left( B\phi\right) ^{2}+\frac{1}{2}|\nabla\phi|^{2}\right) dx+\int_{\partial\mathcal B(T)}\frac{1}{2a}\left( B\phi\right) ^{2}dS\\
&=\int_{\mathcal B(0)}\left( \left( B\phi\right) ^{2}+\frac{1}{2}|\nabla\phi|^{2}\right) dx+\int_{\partial\mathcal B(0)}\frac{1}{2a}\left( B\phi\right) ^{2}dS\\
&+\int_{0}^{T}\int_{\partial\mathcal B(t)}\frac{1}{a}f\left( B\phi\right) dSdt-\int_{0}^{T}\int_{\mathcal B(t)}g\left( B\phi\right) dxdt-\int_{0}^{T}\int_{\partial\mathcal B(t)}\frac{1}{2a^{2}}\left( Ba\right) \left( B\phi\right) ^{2}dSdt\\
&+\int_{0}^{T}\int_{\partial\mathcal B(t)}\frac{\mathrm{div}\,\mkern-17.5mu\slash\ \ B}{2a}\left( B\phi\right) ^{2}dSdt+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}B\left(g^{\alpha\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) dxdt\\
&-\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}\left( B\bm{\uprho}\right) |\nabla\phi|^{2}dxdt-\int_{0}^{T}\int_{\mathcal B(t)}\partial^{\alpha}\left(B^{\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) dxdt+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}|g|^{-\frac{1}{2}}B\left( |g|^{\frac{1}{2}} \right) |\nabla\phi|^{2}dxdt,
\end{split}
\end{align}
where \(\mathrm{div}\,\mkern-17.5mu\slash\ \) denotes the divergence operator on \(\partial\mathcal B\). \end{lemma} \begin{proof}
Multiplying the first equation in \(\eqref{3.1}\) by \(\frac{1}{a}\left( B\phi\right) \) we get
\begin{align}\label{1/a B phi}
\frac{1}{2}B\left[ \frac{1}{a}\left( B\phi\right) ^{2}\right] +\left( n\phi\right) \left( B\phi\right) =\frac{1}{a}f\left( B\phi\right) -\frac{1}{2a^{2}}\left( Ba\right) \left( B\phi\right) ^{2}.
\end{align}
we have
\begin{align}\label{div}
\int_{0}^{T}\int_{\partial\mathcal B(t)}\left( \frac{1}{2}B\left( \frac{1}{a}\left( B\phi\right) ^{2}\right) +\frac{\mathrm{div}\,\mkern-17.5mu\slash\ \ B}{2a}\left( B\phi\right) ^{2}\right) dSdt=\int_{\partial\mathcal B(T)}\frac{1}{2a}\left( B\phi\right) ^{2}dS-\int_{\partial\mathcal B(0)}\frac{1}{2a}\left( B\phi\right) ^{2}dS.
\end{align}
Integrating \(\eqref{1/a B phi}\) over \(\partial\mathcal B=\cup_{t\in[0,T]}\partial\mathcal B(t)\) and using \(\eqref{div}\) we get
\begin{align}\label{energy1}
\begin{split}
&\int_{\partial\mathcal B(T)}\frac{1}{2a}\left( B\phi\right) ^{2}dS+\int_{0}^{T}\int_{\partial\mathcal B(t)}\left( n\phi\right) \left( B\phi\right)dSdt\\
&=\int_{\partial\mathcal B(0)}\frac{1}{2a}\left( B\phi\right) ^{2}dS+\int_{0}^{T}\int_{\partial\mathcal B(t)}\frac{1}{a}f\left( B\phi\right) dSdt\\
&-\int_{0}^{T}\int_{\partial\mathcal B(t)}\frac{1}{2a^{2}}\left( Ba\right) \left( B\phi\right) ^{2}dSdt+\int_{0}^{T}\int_{\partial\mathcal B(t)}\frac{\mathrm{div}\,\mkern-17.5mu\slash\ \ B}{2a}\left( B\phi\right) ^{2}dSdt.
\end{split}
\end{align}
To treat the second term on the left, we integrate \(\eqref{key identity}\) with \(X=B\) over \(\cup_{t\in[0,T]}\mathcal B(t)\). Using the fact that \(B\) is tangent to \(\partial\mathcal B\) and Stokes formula, we get
\begin{align}\label{energy2}
\begin{split}
&\int_{\mathcal B(T)}\left( \left( B\phi\right) ^{2}+\frac{1}{2}|\nabla\phi|^{2}\right) dx-\int_{0}^{T}\int_{\partial\mathcal B(t)}\left( n\phi\right) \left( B\phi\right)dSdt\\
&=\int_{\mathcal B(0)}\left( \left( B\phi\right) ^{2}+\frac{1}{2}|\nabla\phi|^{2}\right) dx-\int_{0}^{T}\int_{\mathcal B(t)}g\left( B\phi\right) dxdt+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}B\left(g^{\alpha\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) dxdt\\
&-\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}\left( B\bm{\uprho}\right) |\nabla\phi|^{2}dxdt-\int_{0}^{T}\int_{\mathcal B(t)}\partial^{\alpha}\left(B^{\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) dxdt+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}|g|^{-\frac{1}{2}}B\left( |g|^{\frac{1}{2}} \right) |\nabla\phi|^{2}dxdt.
\end{split}
\end{align}
The lemma follows by adding \(\eqref{energy2}\) to \(\eqref{energy1}\). \end{proof} Because the vectorfield \(B\) is timelike and future-directed, according to the positivity of energy-momentum tensor, the first term on the left of \(\eqref{energy}\) satisfies \begin{align*}
\left( B\phi\right) ^{2}+\frac{1}{2}|\nabla\phi|^{2}\gtrsim\left| \partial_{t,x}\phi\right|^{2}. \end{align*} For the steady-state solution to \eqref{main eq}-\eqref{boundary conditions}, we have the following \emph{Taylor sign condition} \begin{align}\label{Taylor sign} \nabla_{\mathcal{N}}p\le-c_{0}<0,\quad on \quad \partial\mathcal B ,\quad\quad where\quad \nabla_{\mathcal{N}}=\mathcal{N}^{a}\partial_{a}. \end{align} Since the solution we shall construct in this paper is a small perturbation of the steady-state solution up to time $T^{\delta}$, provided that $\theta_{0}$ is sufficiently small, the condition \eqref{Taylor sign} holds also for the perturbed solution. Therefore \(a\) must be positive and the left hand side of the energy identity \(\eqref{energy}\) controls \begin{align*}
\int_{\partial\mathcal B(T)}\frac{1}{a}\left| B\phi\right| ^{2}dS+\int_{\mathcal B(T)}\left| \partial_{t,x}\phi\right| ^{2}dx. \end{align*} Next, we give two energy estimates for the wave operator: \begin{lemma}\label{3.3.}
There is a (future-directed and timelike) vectorfield \(Q\) such that for any \(\phi\) which is constant on \(\partial\mathcal B\),
\begin{align}\label{estimate1}
\begin{split}
&\int_{\mathcal B\left( T\right) }\left| \partial_{t,x}\phi\right|^{2}dx+\int_{0}^{T}\int_{\partial\mathcal B\left( t\right)}\left| \partial_{t,x}\phi\right|^{2}dSdt\\
&\lesssim\int_{\mathcal B\left( 0\right) }\left| \partial_{t,x}\phi\right|^{2}dx+\left| \int_{0}^{T}\int_{\mathcal B\left( t\right) } \left( \square_{g}\phi\right) \left( Q\phi\right) dxdt\right| \\
&+\int_{0}^{T}\int_{\mathcal B\left( t\right) }\left| \frac{1}{2}\left( \partial_{\alpha}Q^{\alpha}\right) |\nabla\phi|^{2}+\frac{1}{2}Q\left(g^{\alpha\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) -\partial^{\alpha}\left(Q^{\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right)+\frac{1}{2}|g|^{-\frac{1}{2}}Q\left( |g|^{\frac{1}{2}} \right) |\nabla\phi|^{2} \right| dxdt.
\end{split}
\end{align} \end{lemma} \begin{proof}
Integrating \(\eqref{key identity}\) with \(X=Q\) over \(\cup_{t\in[0,T]}\mathcal B(t)\), we get
\begin{align}\label{energyQ}
\begin{split}
&\int_{\mathcal B(T)}\left( \left( Q\phi\right) \left( B\phi\right) +\frac{1}{2}Q^{0}|\nabla\phi|^{2}\right) dx-\int_{0}^{T}\int_{\partial\mathcal B(t)}n_{\alpha}\left( \left( Q\phi\right) \left( \partial^{\alpha}\phi\right) -\frac{1}{2}Q^{\alpha}|\nabla\phi|^{2}\right) dSdt\\
=&\int_{\mathcal B(0)}\left( \left( Q\phi\right) \left( B\phi\right) +\frac{1}{2}Q^{0}|\nabla\phi|^{2}\right) dx-\int_{0}^{T}\int_{\mathcal B(t)}\left( \square_{g}\phi\right) \left( Q\phi\right) dxdt+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}Q\left(g^{\alpha\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) dxdt\\
&+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}\left(\partial_{\alpha}Q^{\alpha}\right) |\nabla\phi|^{2}dxdt-\int_{0}^{T}\int_{\mathcal B(t)}\partial^{\alpha}\left(Q^{\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) dxdt+\int_{0}^{T}\int_{\mathcal B(t)}\frac{1}{2}|g|^{-\frac{1}{2}}Q\left( |g|^{\frac{1}{2}} \right) |\nabla\phi|^{2}dxdt.
\end{split}
\end{align}
Since \(\phi\) is constant on \(\partial\mathcal B\) , \(\partial_{\alpha}\phi\partial^{\alpha}\phi=\pm\sqrt{\partial_{\alpha}\phi\partial^{\alpha}\phi}\left( n\phi\right)\). We get
\begin{align}
\partial_{\alpha}\phi\partial^{\alpha}\phi=\left( n\phi\right) ^{2}.
\end{align}
and
\begin{align}
n_{\alpha}\left( \left( Q\phi\right) \left( \partial^{\alpha}\phi\right) -\frac{1}{2}Q^{\alpha}\left( \partial_{\beta}\phi\right) \left( \partial^{\beta}\phi\right)\right)=\frac{1}{2}Q^{n}\left( n\phi\right) ^{2}
\end{align}
on \(\partial\mathcal B\), where \(Q^{n}=\left\langle Q,n\right\rangle\). Therefore letting \(Q\) be a future-directed timelike vectorfield with \(Q^{n}<0\) in \(\eqref{energyQ}\) (for instance \(Q= \nu B-n\) for some large \(\nu\)) we arrive at \(\eqref{estimate1}\). \end{proof} This energy estimate can be used for the second equation in \(\eqref{2.34}\). \begin{lemma}\label{3.4.}
There exists a (future-directed and timelike) vectorfield \(Q\) such that for any function \(\phi\),
\begin{align}\label{estimate2}
\begin{split}
&\int_{\mathcal B\left( T\right) }\left| \partial_{t,x}\phi\right|^{2}dx+\int_{0}^{T}\int_{\partial\mathcal B\left( t\right)}\left| \partial_{t,x}\phi\right|^{2}dSdt\\
&\lesssim\int_{\mathcal B\left( 0\right) }\left| \partial_{t,x}\phi\right|^{2}dx+\left| \int_{0}^{T}\int_{\mathcal B\left( t\right) } \left( \square_{g}\phi\right) \left( Q\phi\right) dxdt\right| +\int_{0}^{T}\int_{\partial\mathcal B\left( t\right) }\left( \left( n\phi\right) ^{2}+\left( B\phi\right) ^{2}\right) dSdt\\
&+\int_{0}^{T}\int_{\mathcal B\left( t\right) }\left| \frac{1}{2}\left( \partial_{\alpha}Q^{\alpha}\right) |\nabla\phi|^{2}+\frac{1}{2}Q\left(g^{\alpha\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right) -\partial^{\alpha}\left(Q^{\beta} \right)\left(\partial_{\alpha}\phi \right)\left(\partial_{\beta}\phi \right)+\frac{1}{2}|g|^{-\frac{1}{2}}Q\left( |g|^{\frac{1}{2}} \right) |\nabla\phi|^{2} \right| dxdt.
\end{split}
\end{align} \end{lemma} \begin{proof}
We choose \(Q^{n}=\left\langle Q,n\right\rangle>0\). For instance, let \(Q=\nu B+n\) with \(\nu>0\) chosen so that \(Q\) is future-directed and timelike. Then on \(\partial\mathcal B\),
\begin{align*}
\begin{split}
&-n_{\alpha}\left( \left( Q\phi\right) \left( \partial^{\alpha}\phi\right) -\frac{1}{2}Q^{\alpha}\left( \partial_{\beta}\phi\right) \left( \partial^{\beta}\phi\right)\right)\\
=&-\left(\nu B\phi\right) \left( n\phi\right) -\left( n\phi\right) ^{2}+\frac{1}{2}\partial_{\alpha}\phi\partial^{\alpha}\phi
\ge c_{1}\left| \partial_{t,x}\phi\right|^{2}-c_{2}\left( \left( n\phi\right) ^{2}+\left( B\phi\right) ^{2}\right)
\end{split}
\end{align*}
for some constants \(c_{1}, c_{2}>0\) depending only on \(B\). The lemma follows by \(\eqref{energyQ}\) with \(Q=\nu B+n\). \end{proof} This energy estimate can be used for controlling arbitrary derivatives of an arbitrary function on the boundary in terms of the normal and material derivatives. \begin{lemma}\label{3.5.}
For any \(\phi\),
\begin{align}\label{3.20}
c_{s}^{2}\Delta\phi=\square_{g}\phi+BB\phi-\left( 3c_{s}^{-1}c'_{s}+1\right) B\bm{\uprho}B\phi+c_{s}c'_{s}\delta^{ab}\partial_{a}\bm{\uprho}\partial_{b}\phi.
\end{align} \end{lemma} \begin{proof} This is a direct consequence of \(\eqref{2.11}\). \end{proof}
In order to apply Lemma \(\ref{3.5.}\), we introduce the following standard elliptic estimates, whose proof can be found in \cite{Taylor-book1}: \begin{lemma}\label{3.6.}
For any \(t>0,k=0,1\), we have
\begin{align}
\Arrowvert\phi\Arrowvert_{H^{k+1}\left( \mathcal B_{t}\right) }\lesssim\Arrowvert\Delta\phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }+\Arrowvert\phi\Arrowvert_{H^{k+\frac{1}{2}}\left( \partial\mathcal B_{t}\right) },
\end{align}
and
\begin{align}
\Arrowvert\phi\Arrowvert_{H^{2}\left( \mathcal B_{t}\right) }\lesssim\Arrowvert\Delta\phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }+\Arrowvert N\phi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) },
\end{align}
where \(N\) is a transversal vectorfield to \(\partial\mathcal B_{t}\subseteq\mathcal B_{t}\), and where the implicit constants depend on \(\mathcal B_{t}\). \end{lemma}
\subsection{Higher order equations}Here we derive the higher order versions of \(\eqref{2.33}\) and \(\eqref{2.34}\). We give some important commutator identities, which are valid for any \(\phi\). \begin{align}\label{commutator1} &[B,\partial_{a}]\phi=-\left( \partial_{a}u^{b}\right) \left( \partial_{b}\phi\right) .\\\label{commutator2} &[B,\partial_{a}\partial_{b}]\phi=-\left( \partial_{a}u^{c}\right) \left( \partial_{b}\partial_{c}\phi\right) -\left( \partial_{b}u^{c}\right) \left( \partial_{a}\partial_{c}\phi\right) -\left( \partial_{a}\partial_{b}u^{c}\right) \left( \partial_{c}\phi\right) \\ &[B,\Delta]\phi=-2\delta^{ab}\left( \partial_{a}u^{c}\right) \left( \partial_{b}\partial_{c}\phi\right) +\delta^{ab}\left(\partial_{a}B\bm{\uprho}\right) \left( \partial_{b}\phi\right) . \end{align} \begin{align} \begin{split} [B,B^{2}-c_{s}^{2}\delta^{ab}\partial_{a}\bm{\uprho}\partial_{b}]\phi=&c_{s}^{2}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{c}\phi\right) +c_{s}^{2}\delta^{ab}\left( \partial_{c}\bm{\uprho}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{a}\phi\right)\\ &-c_{s}^{2}\delta^{ab}\left( \partial_{a}B\bm{\uprho}\right) \left( \partial_{b}\phi\right), \quad \textrm{on}\quad \partial\mathcal B . \end{split} \end{align} \begin{align} \begin{split} [B,\square_{g}]\phi=&-c_{s}^{2}\delta^{ab}\left( \partial_{a}\partial_{b}u^{c}\right) \left( \partial_{c}\phi\right) -2c_{s}^{2}\delta^{ab}\left( \partial_{a}u^{c}\right) \left( \partial_{b}\partial_{c}\phi\right) +2c_{s}c'_{s}B\bm{\uprho}\delta^{ab}\partial_{a}\partial_{b}\phi\\ &+3\left( c_{s}^{-1}c''_{s}-c_{s}^{-2}c'_{s}c'_{s}\right) \left( B\bm{\uprho}\right) ^{2}B\phi+\left( 1+3c_{s}^{-1}c'_{s}\right) BB\bm{\uprho}B\phi\\ &-\left( c'_{s}c'_{s}+c_{s}c''_{s}\right) B\bm{\uprho}\delta^{ab}\partial_{a}\bm{\uprho}\partial_{b}\phi-c_{s}c'_{s}\delta^{ab}\left( \partial_{a}B\bm{\uprho}\right) \left( \partial_{b}\phi\right) \\ \label{3.27} &+c_{s}c'_{s}\delta^{ab}\left( \partial_{a}\bm{\uprho}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{c}\phi\right)+c_{s}c'_{s}\delta^{ab}\left( \partial_{c}\bm{\uprho}\right) \left( \partial_{b}u^{c}\right) \left( \partial_{a}\phi\right). \end{split} \end{align} The above identities can be obtained by direct calculation. Applying \(\eqref{commutator1}-\eqref{3.27}\) we can calculate the higher order versions of \(\eqref{2.33}\) and \(\eqref{2.34}\), which we record in the following lemmas.
\begin{lemma}\label{3.7.}
For any \(k\ge 0\)
\begin{align}\label{3.28}
\left( B^{2}+an\right) B^{k}u=-c_{s}^{2}\nabla B^{k+1}\varepsilon+F_{k}
\end{align}
where \(F_{k}\) is a linear combination (coefficients are related to \(c_{s}\)) of terms of the forms\\
\begin{enumerate}[\quad (1)]
\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla B^{k_{m+1}}\bm{\uprho}\right)\), where \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k-1\).\label{k1}
\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla B^{k_{m+1}}B\bm{\uprho}\right)\), where \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k-1\).\label{k2}
\item \(\left( B^{k+1}\nabla\psi\right) \)\label{k3}
\end{enumerate} \end{lemma} \begin{proof}
We proceed inductively. For $k=0$ the statement already contained in the first equation in \eqref{2.33}.
\begin{align*}
B\left( anB^{j}u\right)=B\left( -c_{s}^{2}\delta^{ab}\partial_{a}\bm{\uprho}\partial_{b}B^{j}u\right) =&anB^{j+1}u-c_{s}^{2}\delta^{ab}\left( \partial_{a}B\bm{\uprho}\right) \left( \partial_{b}B^{j}u\right)\\
&+c_{s}^{2}\delta^{ab}\left( \partial_{b}u^{c}\right) \left( \left( \partial_{a}\bm{\uprho}\right) \left( \partial_{c}B^{j}u\right) +\left( \partial_{c}\bm{\uprho}\right) \left( \partial_{a}B^{j}u\right) \right) ,
\end{align*}
so \([B,B^{2}+an]B^{j}u\) has the right form. Next, in view of \(\eqref{commutator1}\), \(B\) applied to the terms in (1), (2) and (3) with \(k\) replaced by \(j\), as well as \(\nabla B^{j+1}\varepsilon\), also has the desired form. \end{proof} \begin{lemma}\label{3.8.}
For any \(k\ge 0\)
\begin{align}\label{3.29}
\square_{g}B^{k}u=G_{k}
\end{align}
where \(G_{k}\) is a linear combination (coefficients are related to \(c_{s}\) and its derivatives) of terms of the forms\\
\begin{enumerate}[\quad (1)]
\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla B^{k_{m+1}}\bm{\uprho}\right)\), where \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k\).\label{j1}
\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla^{(2)}B^{k_{m+1}}u\right) \), where \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k-1\).\label{j2}
\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right) \left( B^{k_{m+1}}\nabla\psi\right) \), where \(k_{1}+\cdot\cdot\cdot+k_{m}\le k\) and \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k+1\)\label{j3}
\end{enumerate} \end{lemma} \begin{proof}
Again we proceed inductively. For \(k=0\) the statement already contained in the second equation in \(\eqref{2.33}\). Assume it holds for \(k=j\) and let us prove it for \(k=j+1\). By computing, \([B,\square_{g}]B^{j}u\) has the right form contained in (1) and (2). Similarly, \(B\) applied to above forms have the desired forms by \(\eqref{commutator1}\) and \(\eqref{commutator2}\). \end{proof} \begin{lemma}\label{3.9.}
For any \(k\ge 0\)
\begin{align}\label{3.30}
\square_{g}B^{k+1}\varepsilon=H_{k}
\end{align}
where \(H_{k}\) is a linear combination (coefficients are related to \(c_{s}\) and its derivatives) of terms of the forms\\
\begin{enumerate}[\quad (1)]
\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla B^{k_{m+1}}\bm{\uprho}\right) \left( \nabla B^{k_{m+2}}B\bm{\uprho}\right)\), where \(k_{1}+\cdot\cdot\cdot+k_{m}\le k-1\) and \(k_{1}+\cdot\cdot\cdot+k_{m+2}\le k\).
\label{i1}\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla B^{k_{m+1}}\bm{\uprho}\right) \left( \nabla B^{k_{m+2}}\bm{\uprho}\right)\), where \(k_{1}+\cdot\cdot\cdot+k_{m+2}\le k\).
\label{i2}\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla^{\left( 2\right) } B^{k_{m+1}}\bm{\uprho}\right) \), where \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k\).
\label{i3}\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left( \nabla B^{k_{m+1}}\bm{\uprho}\right) \left( \nabla^{\left( 2\right) } B^{k_{m+2}}u\right) \), where \(k_{1}+\cdot\cdot\cdot+k_{m+2}\le k-1\).
\label{i4}\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\), where \(k_{1}+\cdot\cdot\cdot+k_{m}\le k\).
\label{i5}\item \(\left( \nabla B^{k_{1}}u\right)...\left( \nabla B^{k_{m}}u\right)\left(B^{k_{m+1}}\nabla\nabla\psi \right) \), where \(k_{1}+\cdot\cdot\cdot+k_{m+1}\le k\). \label{i6}
\item \(\left( B^{k+1}\Delta\psi\right) \).\label{i7}
\end{enumerate} \end{lemma} \begin{proof}
For \(k=0\) the statement already contained in the second equation in \(\eqref{2.34}\). Assume it holds for \(k=j\) and let us prove it for \(k=j+1\). By computing, \([B,\square_{g}]B^{j+1}\varepsilon\) has the right form. Similarly, \(B\) applied to above forms have the desired forms by \(\eqref{commutator1}\) and \(\eqref{commutator2}\). \end{proof} Compared to the system studied in \cite{MSW}, the gravitational potential \(\psi\) on the right-hand side of equation \(\eqref{3.28}-\eqref{3.30}\) will also affect the energy estimate. We shall use the Hilbert transform to control its contribution.
\section{Analytic tools}\label{4} In this section we first recall basic algebraic properties of the Clifford algebra \(\mathcal{C}=Cl_{0,2}(\mathbb{R})\). The algebra \(\mathcal C\) is the associative algebra generated by the four basis elements \((1,e_{1},e_{2},e_{3})\) over \(\mathbb R\), satisfying the relatins \begin{align*} 1e_{i}=e_{i},\quad e_{i}e_{j}=-e_{j}e_{i},\quad i\ne j,\ i,j=1,2,3,\quad e_{1}e_{2}=e_{3},\quad e_{i}^{2}=-1,\ i=1,2,3. \end{align*} Every element \(a\in\mathcal C\) has a unique representation \(a=a^{0}+\sum_{i=1}^{3}a^{i}e_{i}\). We identify the real numbers \(\mathbb R\) with Clifford numbers using the relation \(a\mapsto a1\), and identify vectors in \(\mathbb R^{3}\) with Clifford vectors using the relation \(\vec{a}\mapsto a^{i}e_{i}\). The Clifford differentiation operator \(\mathcal D\), acting on Clifford algebra-valued functions, is defines as \begin{align*} \mathcal D=\sum_{i=1}^{3}\partial_{x^{i}}e_{i}, \end{align*} where \(x=(x^{1},x^{2},x^{3})\) are the usual rectangular coordinates in \(\mathbb R^{3}\). Let \(\Omega\) be a \(C^{2}\), bounded, and simply-connected domain in \(\mathbb R^{3}\) with boundary \(\Sigma\). We say that a function \(f\) defined on \(\Omega\) is Clifford analytic, if \(\mathcal D f=0\). It is straightforward to verify that a vector-valued function \(f\) is Clifford analytic is equivalent to \(f\) being curl and divergence free. For the convenience of following discussion, we introduce the Lagrangian parametrization of the surface. Let \(\xi:\mathbb R\times S_{R}\to\partial\mathcal B\) be the Lagrangian parametrization of \(\partial\mathcal B=\partial\mathcal B(t)\), satisfying \(\xi(0,p)=p\) for all \(p\) in \(S_{R}\) and \begin{align} \xi_{t}(t,p)=u(t,\xi(t,p)). \end{align} \(n(t,p)={{\bf n}}(t,\xi(t,p))\) denote the exterior unit normal to \(\partial\mathcal B(t)\). In arbitrary (orientation preserving) local coordinates \((\alpha,\beta)\) on \(S_{R}\) we have \begin{align}
n=\frac{N}{|N|},\quad where \quad N=\xi_{\alpha}\times\xi_{\beta}. \end{align} If \({\bf f}:\mathcal B \to \mathbb R\) is a (possibly time-dependent) differentiable function, and \(f={{\bf f}}\circ\xi\), then by a slight abuse of notation we write \begin{align} \nabla f=\left( \nabla{{\bf f}}\right)\circ\xi,\quad df=\left(d{{\bf f}} \right) \circ\xi, \end{align} where \(d\) denotes the exterior differentiation operators on \(\partial\mathcal B\). With this notation, and using the fact that \(N=\xi_{\alpha}\times\xi_{\beta}\) \begin{align}
n\times\nabla f:=\left( {{\bf n}}\times\nabla{{\bf f}}\right) \circ\xi=\frac{\xi_{\beta}f_{\alpha}-\xi_{\alpha}f_{\beta}}{|N|}. \end{align} For a Clifford algebra-valued function \(f\) (possibly time-dependent) on \(\Sigma\) we define the Hilbert transform of \(f\) as \begin{align} H_{\Sigma}f(\xi)=p.v.\int_{\Sigma}K(\xi'-\xi)n(\xi')f(\xi')dS(\xi'),\quad \xi\in\Sigma, \end{align} where \begin{align}
K(x):=-\frac{1}{2\pi}\frac{x}{|x|^{3}},\quad\quad x\in \mathbb R^{3}, \end{align} and \(K(\xi'-\xi)n(\xi')f(\xi')\) are usual Clifford product. Then we introduce an important property of Hilbert transform described in the following Lemma. \begin{lemma}
\cite{clifford}\cite{wu1999well}If \(f\) is the restriction
to \(\Sigma\) of a Clifford analytic function \({\bf f}\) defined in a neighborhood of \(\Omega\), then \(f(\xi)=H_{\Sigma}f(\xi)\). Similarly, if \(f\) is the restriction
to \(\Sigma\) of a Clifford analytic function \({\bf f}\) defined in a neighborhood of \(\Omega^{c}\), then \(f(\xi)=-H_{\Sigma}f(\xi)\). Finally the operator \(H_{\Sigma}:L^{2}(\Sigma,dS)\to L^{2}(\Sigma,dS)\) are bounded and linear. \end{lemma} In our application, we express the gravitational term on $\partial\mathcal B(t)$ in terms of Hilbert transform (see \cite{MS}). By the boundedness of the operator \(H_{\Sigma}\), we can control the gravitational term on the boundary. We achieve this process through the following Lemma.
\begin{lemma}\label{4.2.}
Suppose the functions \(\phi:\mathbb R^{3}\to\mathbb R\) and \(\Phi:\overline{\mathcal B}\to\mathbb R\) satisfy
\begin{equation}\label{4.7}
\Delta \phi=\left\{
\begin{aligned}
&\Delta \Phi \quad \quad in\ \mathcal B(t),\\
&0\quad\quad \ \ \ in \ \overline{\mathcal{B} (t)} ^{c}
\end{aligned}
\right.
\end{equation}
Then
\begin{equation}
\nabla \phi=\frac{1}{2}(I-H_{\partial\mathcal B_{t}})\nabla\Phi,\quad on\ \partial\mathcal B_{t}.
\end{equation} \end{lemma} \begin{proof}
According to \(\eqref{4.7}\), we have
\begin{align}
\nabla\cdot\left( \nabla\phi-\nabla\Phi\right) =0\quad and \quad \nabla\times\left( \nabla\phi-\nabla\Phi\right) =0,\quad\quad in\ \mathcal B(t).
\end{align}
This indicates that \((\nabla\phi-\nabla\Phi)\) is a Clifford analytic function in \(\mathcal B(t)\) and hence
\begin{align}
\begin{split}
&H_{\partial\mathcal B_{t}}\left( \nabla\phi-\nabla\Phi \right) =\nabla\phi-\nabla\Phi\\
\Rightarrow \quad \quad &(I-H_{\partial\mathcal B_{t}})\nabla\phi=(I-H_{\partial\mathcal B_{t}})\nabla\Phi.
\end{split}
\end{align}
Similarly since \(\nabla\phi\) is curl and divergence-free outside of \(\mathcal B(t)\),
we get
\begin{align}
(I+H_{\partial\mathcal B_{t}})\nabla\phi=0.
\end{align}
The desired result follows because
\begin{align}
\nabla\phi=\frac{1}{2}(I-H_{\partial\mathcal B_{t}})\nabla\phi+\frac{1}{2}(I+H_{\partial\mathcal B_{t}})\nabla\phi=\frac{1}{2}(I-H_{\partial\mathcal B_{t}})\nabla\Phi.
\end{align} \end{proof} This theorem allows us to use the Hilbert transform to estimate \(\Arrowvert\nabla\psi\Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right) }\). In the application we usually take \(\Phi=0\) on \(\partial\mathcal B(t)\). \begin{lemma}
Given the Euler-Poisson system \(\eqref{main eq}-\eqref{boundary conditions}\), we have the following equation for the gravitational potential:
\begin{align}
\partial_{t}\psi=\nabla\cdot\int_{\mathcal B(t)}\frac{\rho u}{|x-y|}dy
\end{align} \end{lemma} \begin{proof}
According to the definition of the gravitational potential \(\psi\), we have
\begin{align}
\begin{split}
\partial_{t}\psi&=-\frac{\partial}{\partial t}\int_{\mathcal B(t)}\frac{\rho}{|x-y|}dy\\
&=-\int_{\mathcal B(t)}\frac{\partial_{t}\rho}{|x-y|}+\mathrm{div}\,\left( \frac{\rho u}{|x-y|}\right) dy\\
&=-\int_{\mathcal B(t)}\frac{\partial_{t}\rho}{|x-y|}+\frac{\rho\mathrm{div}\, u}{|x-y|}+\frac{u\cdot\rho}{|x-y|}+\rho u\cdot\nabla_{y}\left(\frac{1}{|x-y|} \right) dy.
\end{split}
\end{align}
Using the momentum equation in \(\eqref{main eq}\), the first three terms under the integral cancel. Therefore
\begin{align}
\partial_{t}\psi=-\int_{\mathcal B(t)}\rho u\cdot\nabla_{y}\left(\frac{1}{|x-y|} \right) dy=\int_{\mathcal B(t)}\rho u\cdot\nabla_{x}\left(\frac{1}{|x-y|} \right) dy=\nabla\cdot\int_{\mathcal B(t)}\frac{\rho u}{|x-y|} dy.
\end{align} \end{proof} The above result allows us to estimate \(\Arrowvert\nabla_{t}\psi\Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right) }\) using Lemma \(\ref{4.2.}\) hence \(\Arrowvert B^{k}\psi\Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right) }\) as well.
\begin{proposition}\label{4.4.}
Suppose the bootstrap assumptions \(\eqref{5.4}\) hold. Then the following estimates hold for any function \(f\) defined on \(\partial\mathcal B\) and any \(k\le l\):
\begin{align}
\Arrowvert[B^{k},H_{\partial\mathcal B_{t}}]f\Arrowvert_{L^{2}(\partial\mathcal B_{t})}\lesssim\sum_{i+j\le k-1, j\le 10}(\Arrowvert B^{i}u\Arrowvert_{H^{1}(\partial\mathcal B_{t})}\Arrowvert \nabla B^{j}f\Arrowvert_{L^{\infty}(\partial\mathcal B_{t})}+\Arrowvert B^{j}u\Arrowvert_{W^{2,\infty}(\partial\mathcal B_{t})}\Arrowvert B^{i}f\Arrowvert_{L^{2}(\partial\mathcal B_{t})}).
\end{align} \end{proposition} Before stating the proof the Proposition \(\eqref{4.4.}\) we record some important Lemmas. For any two Clifford algebra-valued functions \(f\) and \(g\) define \begin{align}
Q(f,g):=\frac{1}{|N|}\left( f_{\alpha}g_{\beta}-f_{\beta}g_{\alpha}\right), \end{align}
here \(|N|=|\xi_{\alpha}\times\xi_{\beta}|\) which makes \(Q(f,g)\) coordinate-invariant. If \(f\) and \(g\) are vector-valued we also define \begin{align*}
\vec{Q}(f,g):=\frac{1}{|N|}(f_{\alpha}\times g_{\beta}-f_{\beta}\times g_{\alpha}). \end{align*} \begin{lemma}\label{4.5.}
If \(f,g\) and \(h\) are scalar-valued then
\begin{align}\label{4.18}
\int_{\partial\mathcal B(t)}Q(f,g)hdS=-\int_{\partial\mathcal B(t)}fQ(h,g)dS.
\end{align} \end{lemma} \begin{proof}
In the scalar case we have
\begin{align}
Q(f,g)dS=(f_{\alpha}g_{\beta}-f_{\beta}g_{\alpha})d\alpha\wedge d\beta=df\wedge dg,
\end{align}
where \(d\) denotes the exterior differentiation operator on \(\partial\mathcal B(t)\). The identity \(\eqref{4.18}\) follows by Stokes' Theorem
\begin{align}
\begin{split}
0=\int_{\partial\mathcal B(t)}d(fhdg)&=\int_{\partial\mathcal B(t)}hdf\wedge dg+\int_{\partial\mathcal B(t)}fdh\wedge dg\\
&=\int_{\partial\mathcal B(t)}Q(f,g)hdS+\int_{\partial\mathcal B(t)}Q(f,g)hdS.
\end{split}
\end{align} \end{proof} \begin{lemma}
Let \(f\) be a Clifford algebra-valued function. Then
\begin{align}\label{4.21}
[\partial_{t}^{k},H_{\Sigma}]f\sim\sum_{i+j+p+l\le k-1}\int_{\Sigma}\partial_{t}^{i}K(\xi'-\xi)\partial_{t}^{j}(\xi_{t}-\xi'_{t})Q\left( \partial_{t}^{p}\xi',\partial_{t}^{l}f'\right)dS'.
\end{align} Here ``$\sim$" means we drop the numeric constant coefficients in front of the integrals in the sum. \end{lemma} \begin{proof}
We first prove the following commutator formulas for the Hilbert transform
\begin{align}\label{4.22}
[\partial_{t},H_{\Sigma}]f=\int_{\Sigma}K(\xi'-\xi)(\xi_{t}-\xi'_{t})Q\left( \xi',f'\right)dS'.
\end{align}
Suppose \(\eta\in \mathbb R^{3}\) are arbitrary vectors and \(\xi\neq\xi'\), and let \(K=K(\xi'-\xi)\). Then in local coordinates \((\alpha,\beta)\) on \(\Sigma\), we have
\begin{align}\label{4.23}
-(\eta\cdot\nabla)K(\xi'_{\alpha'}\times\xi'_{\beta'})+(\xi'_{\alpha'}\cdot\nabla)K(\eta\times\xi'_{\beta'})+(\xi'_{\beta'}\cdot\nabla)K(\xi'_{\alpha'}\times\eta)=0.
\end{align}
Now let’s prove \(\eqref{4.22}\). By definition, we have
\begin{align}\label{4.24}
\begin{split}
[\partial_{t},H_{\Sigma}]f&=\partial_{t}(H_{\Sigma}f)-H_{\Sigma}(\partial_{t}f)\\
&=\int\int\partial_{t}\left(K(\xi'-\xi)(\xi'_{\alpha'}\times\xi'_{\beta'}) \right) (f'-f)d\alpha'd\beta'\\
&=\int\int\partial_{t}\left(K(\xi'-\xi)\right) (\xi'_{\alpha'}\times\xi'_{\beta'}) (f'-f)d\alpha'd\beta'\\
&+\int\int K(\xi'-\xi)(\xi'_{t\alpha'}\times\xi'_{\beta'}+\xi'_{\alpha'}\times\xi'_{t\beta'})(f'-f)d\alpha'd\beta'
\end{split}
\end{align}
Notice that
\begin{align}
\partial_{t}(K(\xi'-\xi))=((\xi'_{t}-\xi_{t})\cdot\nabla)K(\xi'-\xi),\quad \partial_{\alpha'}K(\xi'-\xi)=(\xi'_{\alpha'}\cdot\nabla)K(\xi'-\xi)
\end{align}
and
\begin{align}
\partial_{\beta'}K(\xi'-\xi)=(\xi'_{\beta'}\cdot\nabla)K(\xi'-\xi).
\end{align}
In \(\eqref{4.23}\) we take \(\eta=\xi'_{t}-\xi_{t}\) and apply to \(\eqref{4.24}\). We get
\begin{align}\label{4.27}
\begin{split}
[\partial_{t},H_{\Sigma}]f&=\int\int{\partial_{\alpha'}K((\xi'_{t}-\xi_{t})\times\xi'_{\beta'})+\partial_{\beta'}K(\xi'_{\alpha'}\times(\xi'_{t}-\xi_{t}))}(f'-f)d\alpha'd\beta'\\
&+\int\int K(\xi'-\xi)(\xi'_{t\alpha'}\times\xi'_{\beta'}+\xi'_{\alpha'}\times\xi'_{t\beta'})(f'-f)d\alpha'd\beta'.
\end{split}
\end{align}
Applying integration by parts to the first term on the right hand side of \(\eqref{4.27}\), we obtain
\begin{align}
\begin{split}
[\partial_{t},H_{\Sigma}]f&=-\int\int K(\xi'-\xi)\left((\xi'_{t}-\xi_{t})\times\xi'_{\beta'}f'_{\alpha'}+\xi'_{\alpha'}\times(\xi'_{t}-\xi_{t})f'_{\beta'} \right) d\alpha'd\beta'\\
&-\int\int K(\xi'-\xi)(\xi'_{t\alpha'}\times\xi'_{\beta'}+\xi'_{\alpha'}\times\xi'_{t\beta'})(f'-f)d\alpha'd\beta'\\
&+\int\int K(\xi'-\xi)(\xi'_{t\alpha'}\times\xi'_{\beta'}+\xi'_{\alpha'}\times\xi'_{t\beta'})(f'-f)d\alpha'd\beta'\\
&=\int\int K(\xi'-\xi) ((\xi'_{t}-\xi_{t})\times(\xi'_{\beta'}f'_{\alpha'}-\xi'_{\alpha'}f'_{\beta'}))d\alpha'd\beta'\\
&=\int_{\Sigma}K(\xi'-\xi)(\xi_{t}-\xi'_{t})Q\left( \xi',f'\right)dS'.
\end{split}
\end{align}
Finally the estimate \([\partial_{t}^{k},H_{\Sigma}]f\) follows by writing
\begin{align}
[\partial_{t}^{k},H_{\Sigma}]f=[\partial_{t},H_{\Sigma}]\partial_{t}^{k-1}f+\partial_{t}[\partial_{t},H_{\Sigma}]\partial_{t}^{k-2}f+\dots+\partial_{t}^{k-1}[\partial_{t},H_{\Sigma}]f.
\end{align} \end{proof} We next turn to estimates on singular integral operators. Let \(J:S_{R}\to \mathbb R_{k}\), \(F:\mathbb R^{k}\to \mathbb R\), \(A:S_{R}\to \mathbb R\) be smooth functions. We want to estimate singular integrals of the following forms: \begin{align}\label{4.30}
C_{1}f(p):=p.v.\int_{S_{R}}F\left(\frac{J(p)-J(q)}{|p-q|} \right) \frac{\Pi^{N}_{i=1}(A_{i}(p)-A_{i}(q))}{|p-q|^{N+2}}f(q)dS(q), \end{align} where \(dS\) denotes the surface measure on \(S_{R}\), and where we assume that the kernel \begin{align}
k_{1}(p,q)=F\left(\frac{J(p)-J(q)}{|p-q|} \right) \frac{\Pi^{N}_{i=1}(A_{i}(p)-A_{i}(q))}{|p-q|^{N+2}} \end{align} is odd, that is, \(k_{1}(p,q)=-k_{1}(q,p)\). \begin{lemma}\label{4.7.}
\cite{wu2011global}With the same notation as \(\eqref{4.30}\), we have
\begin{align}\label{4.32}
\Arrowvert C_{1}f\Arrowvert_{L^{2}(S_{R})}\le C \prod_{i=1}^{N}\left( \Arrowvert\nabla\mkern-10.5mu\slash A_{i} \Arrowvert_{L^{\infty}(S_{R})}+R^{-1}\Arrowvert A_{i} \Arrowvert_{L^{\infty}(S_{R})}\right)\Arrowvert f\Arrowvert_{L^{2}(S_{R})},
\end{align}
and
\begin{align}\label{4.33}
\begin{split}
\Arrowvert C_{1}f\Arrowvert_{L^{2}(S_{R})}\le&C\left(\Arrowvert\nabla\mkern-10.5mu\slash A_{1} \Arrowvert_{L^{2}(S_{R})}+R^{-1}\Arrowvert A_{1} \Arrowvert_{L^{2}(S_{R})} \right) \\
&\times\prod_{i=2}^{N}\left(\Arrowvert\nabla\mkern-10.5mu\slash A_{i} \Arrowvert_{L^{\infty}(S_{R})}+R^{-1}\Arrowvert A_{i} \Arrowvert_{L^{\infty}(S_{R})} \right) \Arrowvert f\Arrowvert_{L^{\infty}(S_{R})},
\end{split}
\end{align}
where \(\nabla\mkern-10.5mu\slash\) denotes the covariant differentiation operator with respect
to the standard metric on \(S_{R}\) and the constants depend on \(F\), \(\Arrowvert\nabla\mkern-10.5mu\slash J\Arrowvert_{L^{\infty}}\). \end{lemma}
\begin{proof}[Proof of Proposition \(\eqref{4.4.}\)]
We consider \(\Arrowvert [\partial_{t}^{k},H_{\partial\mathcal B_{t}}]f\Arrowvert_{L^{2}(S_{R})}\) in three limiting cases. First we notice the components of the term
\begin{align}\label{4.34}
\int_{\partial\mathcal B_{t}}\partial_{t}^{k-1}K(\xi'-\xi)(\xi_{t}-\xi'_{t})Q\left(\xi',f' \right) dS'
\end{align}
satisfy condition of Lemma \(\ref{4.7.}\) by the appropriate transformation. Therefore \(\eqref{4.34}\) can be regarded as lower order terms according to \(\eqref{4.32}\). For the components of the term
\begin{align}\label{4.35}
\int_{\partial\mathcal B_{t}}K(\xi'-\xi)\partial_{t}^{k-1}(\xi_{t}-\xi'_{t})Q\left(\xi',f' \right) dS',
\end{align}
we use Lemma \(\ref{4.7.}\) to bound the \(L^{2}(S_{R})\) norms of \(\eqref{4.34}\) by the right-hand side of \(\eqref{4.33}\). So
\begin{align}\label{4.36}
\left| \left| \int_{\partial\mathcal B_{t}}K(\xi'-\xi)\partial_{t}^{k-1}(\xi_{t}-\xi'_{t})Q\left(\xi',f' \right) dS'\right| \right| _{L^{2}(S_{R})}\lesssim(\Arrowvert \nabla\mkern-10.5mu\slash \partial_{t}^{k-1}u\Arrowvert_{L^{2}(S_{R})}+R^{-1}\Arrowvert \partial_{t}^{k-1}u\Arrowvert_{L^{2}(S_{R})})\Arrowvert \nabla\mkern-10.5mu\slash f\Arrowvert_{L^{\infty}(S_{R})}.
\end{align}
For the term
\begin{align}\label{4.37}
\int_{\partial\mathcal B_{t}}K(\xi'-\xi)(\xi_{t}-\xi'_{t})Q\left(\xi',\partial_{t}^{k-1}f' \right) dS',
\end{align}
We rewrite the term \(\eqref{4.37}\) as
\begin{align}
\begin{split}
\int_{\partial\mathcal B_{t}}K^{i}(\xi'-\xi)(\xi^{j}_{t}-(\xi'_{t})^{j})Q\left((\xi')^{p},\partial_{t}^{k-1}f' \right) dS'{ e_{i}e_{j}e_{p}}:=A_{ijp}{ e_{i}e_{j}e_{p}},
\end{split}
\end{align}
suppose \(f\) is scalar function. For any \(i,j,p=1,2,3\), let \((\xi')^{i}-\xi^{i}=(\zeta')^{i}\), \(\xi^{j}_{t}-(\xi'_{t})^{j}=(\eta')^{j}\), \(\partial_{t}^{k-1}f'=\tilde{f'}\)
and using integration-by-parts formula of Lemma \(\ref{4.5.}\) in the componentwise, we get
\begin{align}\label{4.39}
\begin{split}
&A_{ijp}\sim\int_{\partial\mathcal B_{t}}\frac{(\zeta')^{i}(\eta')^{j}}{|\zeta|^{3}}Q((\xi')^{p},\tilde{f'})\\
=&\int_{\partial\mathcal B_{t}}Q\left( (\xi')^{p},\frac{(\zeta')^{i}(\eta')^{j}}{|\zeta|^{3}}\right)\tilde{f'}dS' \\
=&\int_{\partial\mathcal B_{t}}\frac{(\zeta')^{i}}{|\zeta|^{3}}\tilde{f'}Q((\xi')^{p},(\eta')^{j})dS'+\int_{\partial\mathcal B_{t}}\frac{(\eta')^{j}}{|\zeta|^{3}}\tilde{f'}Q((\xi')^{p},(\zeta')^{i})dS'+\int_{\partial\mathcal B_{t}}\frac{(\zeta')^{i}(\eta')^{j}\zeta}{|\zeta|^{5}}\cdot\tilde{f'}Q((\xi')^{p},\zeta)dS'\\
\lesssim&(\Arrowvert \nabla\mkern-10.5mu\slash u\Arrowvert_{L^{\infty}(S_{R})}+R^{-1}\Arrowvert u\Arrowvert_{L^{\infty}(S_{R})})\Arrowvert\partial_{t}^{k-1}f\Arrowvert_{L^{2}(S_{R})}.
\end{split}
\end{align}
The last estimate follows from \(\eqref{4.32}\). All other cases can be considered combinations of \(\eqref{4.34}\), \(\eqref{4.35}\) and \(\eqref{4.37}\). Based on above discussion, combining \(\eqref{4.21}\), \(\eqref{4.36}\) and \(\eqref{4.39}\) , we finally obtain
\begin{align}
\begin{split}
\Arrowvert [\partial_{t}^{k},H_{\partial\mathcal B_{t}}]f\Arrowvert_{L^{2}(S_{R})}
\lesssim&\sum_{i+j\le k-1, j\le 10}(\Arrowvert\partial_{t}^{i}u\Arrowvert_{H^{1}(S_{R})}\Arrowvert \nabla\mkern-10.5mu\slash \partial_{t}^{j}f\Arrowvert_{L^{\infty}(S_{R})}+\Arrowvert\partial_{t}^{j}u\Arrowvert_{W^{2,\infty}(S_{R})}\Arrowvert \partial_{t}^{i}f\Arrowvert_{L^{2}(S_{R})})\\
\lesssim&\sum_{i+j\le k-1, j\le 10}(\Arrowvert B^{i}u\Arrowvert_{H^{1}(\partial\mathcal B_{t})}\Arrowvert \nabla B^{j}f\Arrowvert_{L^{\infty}(\partial\mathcal B_{t})}+\Arrowvert B^{j}u\Arrowvert_{W^{2,\infty}(\partial\mathcal B_{t})}\Arrowvert B^{i}f\Arrowvert_{L^{2}(\partial\mathcal B_{t})}).
\end{split}
\end{align}
which completes the proof of Proposition \(\eqref{4.4.}\).
\end{proof}
\section{Priori estimate}\label{5}
For any function \(\phi\) we define the energies \begin{align} \begin{split}
&E[\phi,t]:=\int_{\mathcal B(t)}|\partial_{t,x}\phi|^{2}dx+\int_{\partial\mathcal B(t)}\frac{1}{a}|B\phi|^{2}dS,\\
&\underline{E}[\phi,t]:=\int_{\mathcal B(t)}|\partial_{t,x}\phi|^{2}dx. \end{split} \end{align} Higher order energies are defined as \begin{align} E_{j}[\phi,t]=E[B^{j}\phi,t],\quad E_{\le k}[\phi,t]=\sum_{j=0}^{k}E_{j}[\phi,t],\quad\underline{E}_{j}[\phi,t]=\underline{E}[B^{j}\phi,t],\quad\underline{E}_{\le k}[\phi,t]=\sum_{j=0}^{k}\underline{E}_{j}[\phi,t]. \end{align} To simplify notation we introduce the unified energy \begin{align}\label{5.3} {\mathscr{E}}_{l}(t):=\sum_{2j+k\le l+2}\left( \Arrowvert B^{k}u\Arrowvert^{2}_{H^{j}\left( \mathcal B_{t}\right) }+ \Arrowvert B^{k+1}\varepsilon\Arrowvert^{2}_{H^{j}\left( \mathcal B_{t}\right) } \right)+\underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]. \end{align} Our goal in this section is to prove the following a priori estimate.
\begin{proposition}\label{5.1.}
Suppose \(u,\varepsilon\) is a solution to \(\eqref{2.33}-\eqref{2.34}\) with
\begin{align}\label{5.4}
{\mathscr{E}}_{l}(t)\le C_{l},\quad |\nabla^{(m)}X(\cdot,t)|\le C_{X},\quad|J(t)-1|\le \frac12,\quad\left| \tilde{J}(t)-1\right| \le \frac12,
\end{align}
for some constants \(C_{l},C_{X},C_{J},C_{\tilde{J}}>0\) and \(l\) sufficiently large satisfying \(0\le m\ll l\), where \(J(t)\) and \(\tilde{J}(t)\) are the Jacobian of the Lagrangian coordinate transformation from \(\mathcal B(0)\) to \(\mathcal B(t)\) and \(\partial\mathcal B(0)\) to \(\partial\mathcal B(t)\) respectively.
Then we have
\begin{align}\label{5.5}
{\mathscr{E}}_{l}(t)\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds,
\end{align}
for some positive constants \(C_{0},C_{1}(\bar{\rho}), C_{2}(\bar{\rho}),C_{h}\) and \(t\in[0,T]\). \end{proposition} \begin{remark}
Combined with the discussion in Section \(\ref{1}\), the estimate \(\eqref{5.5}\) and \(\eqref{6.1}\), which we shall prove in the next section, actually show our main conclusion Theorem \(\ref{main th}\), that liquid Lane-Emden stars are nonlinearly unstable, where \({\mathscr{E}}_{l}\) corresponds to a stronger energy norm \(\arrowvert\arrowvert\arrowvert\cdot\arrowvert\arrowvert\arrowvert^{2}\) and \(\Arrowvert\varepsilon,u\Arrowvert^{2}_{L^{2}}\) corresponds to the weaker norm \(\arrowvert\arrowvert\cdot\arrowvert\arrowvert^{2}\). \end{remark} \begin{remark}
The bootstrap assumptions \(\eqref{5.4}\) implies that the norm in the Lagrangian coordinate system is equivalent to the norm in the Cartesian coordinate system, so Proposition \(\ref{5.1.}\) still holds in the Lagrangian coordinate system. It is worth noting that when the conclusion \(\eqref{5.5}\) holds, we can close the bootstrap assumptions using the fundamental theorem of calculus. \end{remark}
To prove Proposition \(\ref{5.1.}\) we need to show that higher order energies \(\underline{E}_{\le l+1}[\varepsilon,t]\) and \(E_{\le l}[u,t]\) give pointwise control on lower order derivatives of \(u\) and \(\varepsilon\) and \(L^{2}\) control of lower order Sobolev norms of \(u,\varepsilon\). The result is stated in the following proposition.
\begin{proposition}\label{5.2.}
Under the assumptions of Proposition \(\eqref{5.1.}\), for any \(t\in[0,T]\), we have
\begin{align}\label{5.6}
\begin{split}
&\sum_{k+2p\le l+2}\Arrowvert\partial_{t,x}^{p}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }+ \sum_{k+2p\le l+2}\Arrowvert\partial_{t,x}^{p}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right)} \\
&\lesssim \underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2p\le l+2}\Arrowvert\partial_{t,x}^{p}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2p\le l+2}\Arrowvert\partial_{t,x}^{p}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau.
\end{split}
\end{align}
The implicit coefficient in this estimate depends on the constants of the bootstrap assumption \(\eqref{5.4}\). \end{proposition}
Before we give the proof of the proposition, we need some preparation. First, we introduce some notations: \begin{align} \nabla_{i}:=\partial_{i},\quad\underline{n}:=\frac{(\partial_{1}\bm{\uprho},\partial_{2}\bm{\uprho},\partial_{3}\bm{\uprho})}{\sqrt{\sum_{i=1}^{3}(\partial_{i}\bm{\uprho})^{2}}},\quad\underline{n}_{i}:=\delta_{ij}\underline{n}^{j},\quad \nabla\mkern-10.5mu\slash_{i}:=\partial_{i}-\underline{n}_{i}\underline{n}^{j}\partial_{j}. \end{align} Note that \(\nabla\mkern-10.5mu\slash_{i},i=1,2,3\) are defined globally, are tangential to \(\partial\mathcal B_{t}\), and span \(T\partial\mathcal B_{t}\). \begin{lemma}\label{5.5.}
For any smooth function \(\phi\), the following estimate holds:
\begin{align}\label{4.12}
\begin{split}
\Arrowvert\phi\Arrowvert_{H^{j}\left( \mathcal B_{t}\right)}\lesssim\Arrowvert\phi\Arrowvert_{H^{j-1}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla^{(j-2)}\Delta\phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}
+\Arrowvert[\nabla\mkern-10.5mu\slash,\nabla^{(j-2)}]\phi\Arrowvert_{H^{1}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla\mkern-10.5mu\slash\phi\Arrowvert_{H^{j-1}\left( \mathcal B_{t}\right)}.
\end{split}
\end{align} \end{lemma} \begin{proof}
Using the first estimate in Lemma \(\ref{3.6.}\) and trace theorem
\begin{align*}
\Arrowvert\phi\Arrowvert_{H^{j}\left( \mathcal B_{t}\right)}\lesssim&\Arrowvert\phi\Arrowvert_{H^{j-1}\left( \mathcal B_{t}\right)}+\Arrowvert\Delta\nabla^{(j-2)}\phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla\mkern-10.5mu\slash\nabla^{(j-2)}\phi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}+\Arrowvert\nabla^{(j-2)}\phi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}\\
\lesssim&\Arrowvert\phi\Arrowvert_{H^{j-1}\left( \mathcal B_{t}\right)}+\Arrowvert\Delta\nabla^{(j-2)}\phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla\mkern-10.5mu\slash\nabla^{(j-2)}\phi\Arrowvert_{H^{1}\left( \mathcal B_{t}\right)}.
\end{align*}
The desired estimate follows after commuting various operators. \end{proof} \begin{lemma}\label{5.6.}
Under the bootstrap assumption \(\eqref{5.4}\), we have
\begin{align}
\Arrowvert\nabla^{a}B^{k}u\Arrowvert_{L^{\infty}\left( \mathcal B_{t}\right)}+ \Arrowvert\nabla^{a}B^{k+1}\varepsilon\Arrowvert_{L^{\infty}\left( \mathcal B_{t}\right)}\lesssim {\mathscr{E}}_{l}^{\frac{1}{2}}(t),\quad\quad 0\le a \le p-2,\quad k\le l-2p-2, t\in[0,T].
\end{align} \end{lemma} \begin{proof}
This follows from the Sobolev embedding \(H^{2}(\mathcal B_{t})\hookrightarrow L^{\infty}(\mathcal B_{t})\). \end{proof} \begin{lemma}\label{5.7.}
Under the bootstrap assumption \(\eqref{5.4}\), if \(2a+k \le l+1, t\in[0,T]\), then
\begin{align}
\Arrowvert\nabla^{a}B^{k}u\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+ \Arrowvert\nabla^{a}B^{k+1}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}\lesssim \Arrowvert\nabla^{a}B^{k}u\Arrowvert_{L^{2}\left( \mathcal B_{0}\right)}+ \Arrowvert\nabla^{a}B^{k+1}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}^{\frac{1}{2}}(s)ds.
\end{align}
The implicit coefficient in this estimate depends on the constants of the bootstrap assumption \(\eqref{5.4}\). \end{lemma} \begin{proof}
We rcall the Lagrangian parametrization of \(X\), that is
\begin{align*}
\partial_{\tau}X(\tau,y)=Bu(X(\tau,y)),\quad\quad X(0,y)=y.
\end{align*}
If \(p_{t}\) is a point on \(\mathcal B_{t}\), we let \(p_{0}\) be the point on \(\mathcal B_{0}\) such that \(X(t,p_{0})=p_{t}\). For any function \(\phi\)
\begin{align}
\phi(p_{t})-\phi(p_{0})=\int_{0}^{t}B\phi(p_{\tau})d\tau.
\end{align}
According to the bootstrap assumption \(\ref{5.4}\), we can bound the Jacobian of the Lagrangian coordinate transformation from \(\mathcal B_{0}\) to \(\mathcal B_{t}\), therefore we get
\begin{align*}
\Arrowvert\phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}\lesssim \Arrowvert\phi\Arrowvert_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}\Arrowvert B\phi\Arrowvert_{L^{2}\left( \mathcal B_{s}\right)}ds.
\end{align*}
We apply this estimate to \(\phi=\nabla^{a}B^{k}u\) as well \(\phi=\nabla^{a}B^{k+1}\varepsilon\). Then as long as \(2a+k\le l+1\), we have
\begin{align*}
\Arrowvert B\phi\Arrowvert_{L^{2}\left( \mathcal B_{s}\right)}\lesssim {\mathscr{E}}_{l}^{\frac{1}{2}}(s),
\end{align*}
which completes the proof of Lemma \(\ref{5.7.}\). \end{proof}
\begin{proof}[Proof of Proposition \(\eqref{5.2.}\)]
Note that we only need to consider \(\partial_{x}^{p}B^{k}u\). Indeed, using induction on the order of \(\partial_{t}\), for \(\partial_{t}\partial_{x}^{p-1}B^{k}u\), we have
\begin{align}
\begin{split}
\partial_{t}\partial_{x}^{p-1}B^{k}u=&B\partial_{x}^{p-1}B^{k}u-u^{a}\partial_{a}\partial_{x}^{p-1}B^{k}u\\
=&[B,\partial_{x}^{p-1}]B^{k}u+\partial_{x}^{p-1}B^{k+1}u-u^{a}\partial_{a}\partial_{x}^{p-1}B^{k}u.
\end{split}
\end{align}
If we can estimate \(\partial_{t}^{p'}\partial_{x}^{p-p'}B^{k}u\), for \(\partial_{t}^{p'+1}\partial_{x}^{p-p'-1}B^{k}u\), we have
\begin{align}
\partial_{t}^{p'+1}\partial_{x}^{p-p'-1}B^{k}u=\partial_{t}\partial_{t}^{p'}\partial_{x}^{p-p'-1}B^{k}u.
\end{align}
The induction argument follows exactly the same way as we treat the case when \(p'=0\). The argument for \(\varepsilon\) is the same. Turning to \(\partial_{x}^{p}B^{k}u\), we will use an induction argument on \(p\). When \(p=1\), the result follows
directly by definition. Now we assume that the estimate holds for index less or equal to \(1\le p \le \frac{M+2}{2}-1\), that is,
\begin{align}\label{5.14} \begin{split} &\sum_{q\le p}\sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }+ \sum_{q\le p}\sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right)} \\ &\lesssim \underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau. \end{split} \end{align} and prove the estimates for \(p+1\), that is,
\begin{align}\label{5.15} \begin{split} &\sum_{k\le l-2p}\Arrowvert\partial_{t,x}^{p+1}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }+ \sum_{k\le l-2p}\Arrowvert\partial_{t,x}^{p+1}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right)} \\ &\lesssim \underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau. \end{split} \end{align} We start with the estimate for \(\Arrowvert\nabla_{x}^{p+1}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) } \) and in fact first estimate \(\Arrowvert\nabla_{x}^{(2)}\nabla\mkern-10.5mu\slash^{(p-1)}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }\). To apply Lemma \(\eqref{3.5.}\) to \(\phi:=\nabla\mkern-10.5mu\slash^{p-1}B^{k+1}\varepsilon\) we need to estimate \(\Arrowvert\Delta\nabla\mkern-10.5mu\slash^{(p-1)}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right)}\). Using the notation of Lemma \(\eqref{3.9.}\), we have \begin{align}\label{5.16} \begin{split} \Delta\nabla\mkern-10.5mu\slash^{p-1}B^{k+1}\varepsilon\sim&\nabla\mkern-10.5mu\slash^{p-1}H_{k}+[\nabla\mkern-10.5mu\slash^{p-1},\square_{g}]B^{k+1}\varepsilon+\left( \nabla\bm{\uprho}\right) \left( \nabla\nabla\mkern-10.5mu\slash^{p-1}B^{k+1}\varepsilon\right) \\ &+\nabla\nabla\mkern-10.5mu\slash^{p-1}B^{k+2}\varepsilon+\nabla[B,\nabla\mkern-10.5mu\slash^{p-1}]B^{k+1}\varepsilon. \end{split} \end{align} Except for \(\nabla\mkern-10.5mu\slash^{p-1}H_{k}\) the \(L^{2}\left( \mathcal B_{t}\right)\) norms of all the terms on the right-hand side of \(\eqref{5.16}\) are bounded by the right-hand side of \(\eqref{5.15}\) using the induction hypothesis \(\eqref{5.14}\). Here for the terms where derivatives hit the coefficients of \(\nabla\mkern-10.5mu\slash\) it suffices to observe that these coefficients are functions of \(\nabla\bm{\uprho}\). Next we investigate the structure of \(\nabla\mkern-10.5mu\slash^{p-1}H_{k}\). In view of Lemma \(\eqref{3.9.}\), the remainder terms in \(\nabla\mkern-10.5mu\slash^{p-1}H_{k}\) are \begin{align} \nabla^{p+1}B^{k}\varepsilon,\ \ \nabla^{p+1}B^{k-1}u \ \ and \ \ \nabla^{p-1}B^{k}\nabla\nabla\psi. \end{align} The \(L^{2}\left( \mathcal B_{t}\right)\) norm of all other term appearing in \(\nabla\mkern-10.5mu\slash^{p-1}H_{k}\) can be bounded by the right-hand side of \(\eqref{5.15}\) using the induction hypothesis \(\eqref{5.14}\). For the first two terms above, since \(k\le l-2p\) we can use Lemma \(\ref{5.7.}\) to bound the \(L^{2}\left( \mathcal B_{t}\right)\) norms of these terms by the right-hand side of \(\eqref{5.15}\) as well. For the last term, we can use Elliptic estimate and Hilbert transform (which is introduced in the Section \(\eqref{4}\)) frequently to bound its \(L^{2}\left( \mathcal B_{t}\right)\) norms.
Next we turn to the gravity term. In view of Lemma \(\eqref{3.6.}\),we have \begin{align}\label{5.18} \begin{split} \Arrowvert\nabla^{p-1}B^{k}\nabla\nabla\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\lesssim&\Arrowvert\nabla\nabla\nabla^{p-1}B^{k}\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }+\Arrowvert\nabla^{p-1}[B^{k},\nabla\nabla]\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\\ \lesssim&\Arrowvert\nabla^{p-1}B^{k}\psi\Arrowvert_{H^{\frac{3}{2}}\left( \partial\mathcal B_{t}\right)}+\Arrowvert\nabla^{p-1}[\Delta,B^{k}]\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}\\ &+\Arrowvert\nabla^{p-1}B^{k}\Delta\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla^{p-1}[B^{k},\nabla\nabla]\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\\ :=&\Arrowvert\nabla^{p-1}B^{k}\psi\Arrowvert_{H^{\frac{3}{2}}\left( \partial\mathcal B_{t}\right)}+R_{1} \end{split} \end{align} We focus on the boundary term, \begin{align}\label{5.19} \begin{split} \Arrowvert\nabla^{p-1}B^{k}\psi\Arrowvert_{H^{\frac{3}{2}}\left( \partial\mathcal B_{t}\right)}&\lesssim\Arrowvert\nabla\nabla^{p-1}B^{k}\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}+\Arrowvert\nabla^{p-1}B^{k}\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}\\ &\lesssim\Arrowvert B^{k}\nabla\nabla^{p-1}\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}+\Arrowvert [B^{k},\nabla\nabla^{p-1}]\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}+\Arrowvert\nabla^{p-1}B^{k}\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right)}\\ &\lesssim\Arrowvert B^{k}\nabla\nabla^{p}\psi\Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}+R_{2}. \end{split} \end{align} Let function \(\Phi\) satisfy \(\Delta\Phi=4\pi G\nabla^{p}\rho,\ in \ \mathcal B(t),\ \ \Phi=0,\ on \ \partial\mathcal B(t) \), then \begin{align}\label{5.20} \begin{split} \Arrowvert B^{k}\nabla\nabla^{p}\psi\Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}&=\frac{1}{2}\Arrowvert B^{k}\left( I-H_{\partial\mathcal B_{t}}\right)\nabla\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}\\ &\lesssim\Arrowvert H_{\partial\mathcal B_{t}} B^{k}\nabla\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}+\Arrowvert [B^{k},H_{\partial\mathcal B_{t}}]\nabla\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}. \end{split} \end{align} For the first term above, we have \begin{align}\label{5.21} \begin{split} \Arrowvert H_{\partial\mathcal B_{t}} B^{k}\nabla\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}&\lesssim\Arrowvert B^{k}\nabla\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}\\ &\lesssim\Arrowvert \nabla B^{k}\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}+\Arrowvert [B^{k},\nabla]\Phi \Arrowvert_{L^{2}\left( \partial\mathcal B_{t}\right)}\\ &\lesssim\Arrowvert B^{k}\Phi \Arrowvert_{H^{2}\left( \mathcal B_{t}\right)}+\Arrowvert [B^{k},\nabla]\Phi \Arrowvert_{H^{1}\left( \mathcal B_{t}\right)}\\ &\lesssim\Arrowvert B^{k}\Delta\Phi \Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert [B^{k},\Delta]\Phi \Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert [B^{k},\nabla]\Phi \Arrowvert_{H^{1}\left( \mathcal B_{t}\right)}\\ &\lesssim\Arrowvert \nabla^{p}B^{k}\rho \Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert [B^{k},\nabla^{p}]\rho \Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+R_{3}. \end{split} \end{align} The first two terms on the right-hand side of \(\eqref{5.21}\) are bounded by the right-hand side of \(\eqref{5.15}\) using the induction hypothesis \(\eqref{5.14}\). According to trace theorem and commutator identities, the top order terms about \(u,\varepsilon\) in \(R_{1},R_{2}\) and \(R_{3}\) can be bounded as well. For the remainder terms in \(R_{1},R_{3}\) \begin{align}\label{5.22} \sum_{1\le i\le 2}\sum_{j\le k-1}\Arrowvert\nabla^{p-1}\nabla^{i}B^{j}\psi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}\quad and \quad \sum_{1\le i\le 2}\sum_{j\le k-1}\Arrowvert\nabla^{p-1}\nabla^{i}B^{j}\Phi\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}, \end{align} we can use elliptic estimates repeatedly to control. For the remainder terms in \(R_{2}\), we can choose the appropriate function \(\Phi\) to bound them in the same way as in the treatment of \(B^{k}\nabla\nabla^{p}\psi\) in \(\eqref{5.20}\). The fact that \([B^{k},H_{\partial\mathcal B_{t}}]\nabla\Phi\) are the lower order terms and some remaining details are proved in Section \(\ref{4}\). Based on this discussion, Using \(\eqref{5.16}\) and Lemma \(\ref{3.5.}\), for any \(k\le l-2p\) we obtain \begin{align}\label{5.23} \begin{split} &\Arrowvert\nabla^{(2)}\nabla\mkern-10.5mu\slash^{p-1}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }\\ \lesssim&\underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau. \end{split} \end{align} Next we apply Lemma \(\ref{5.5.}\) to \(\phi:=\nabla\nabla\mkern-10.5mu\slash^{p-2}B^{k+1}\varepsilon\) to get \begin{align}\label{5.24} \begin{split} \Arrowvert\nabla\nabla\mkern-10.5mu\slash^{p-2}B^{k+1}\varepsilon\Arrowvert_{H^{2}\left( \mathcal B_{t}\right)}\lesssim&\Arrowvert\nabla\nabla\mkern-10.5mu\slash^{p-2}\Delta B^{k+1}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert[\nabla\nabla\mkern-10.5mu\slash^{p-2},\Delta]B^{k+1}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla\mkern-10.5mu\slash^{p-2}B^{k+1}\varepsilon\Arrowvert_{H^{2}\left( \mathcal B_{t}\right)}\\ &+\Arrowvert[\nabla,\nabla\mkern-10.5mu\slash]\nabla\mkern-10.5mu\slash^{p-2}B^{k+1}\varepsilon\Arrowvert_{H^{1}\left( \mathcal B_{t}\right)}+\Arrowvert\nabla\mkern-10.5mu\slash^{p-1}B^{k+1}\varepsilon\Arrowvert_{H^{2}\left( \mathcal B_{t}\right)}. \end{split} \end{align} By \(\eqref{5.23}\) and the arguments leading to it, all the terms on the right-hand side of \(\eqref{5.24}\) except \begin{align*} \Arrowvert\nabla\nabla\mkern-10.5mu\slash^{p-2}\Delta B^{k+1}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)} \end{align*} are bounded by the right-hand side of \(\eqref{5.15}\). The term \(\Arrowvert\nabla\nabla\mkern-10.5mu\slash^{p-2}\Delta B^{k+1}\bm{\uprho}\Arrowvert_{L^{2}\left( \mathcal B_{t}\right)}\) is bounded in the same way as in the treatment of \(\nabla\mkern-10.5mu\slash^{p-1}H_{k}\) above. Summarizing we have obtained \begin{align}\label{5.25} \begin{split} &\Arrowvert\nabla^{(3)}\nabla\mkern-10.5mu\slash^{p-2}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }\\ \lesssim&\underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau. \end{split} \end{align} Repeating the argument inductively for \(\phi:=\nabla^{2}\nabla\mkern-10.5mu\slash^{p-3}B^{k+1}\varepsilon,\nabla^{3}\nabla\mkern-10.5mu\slash^{p-4}B^{k+1}\varepsilon,\dots\) we finally obtain \begin{align}\label{5.26} \begin{split} &\Arrowvert\nabla^{p}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }\\ \lesssim&\underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau. \end{split} \end{align} Next we use the second estimate in Lemma \(\ref{3.6.}\) to estimate \(\Arrowvert\nabla^{p+1}B^{k}u\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) },\ k+2p+2\le l+2\), under the induction hypothesis \(\eqref{5.14}\). The second estimate in Lemma \(\ref{3.6.}\) gives \begin{align}\label{5.27} \Arrowvert\nabla^{p+1}B^{k}u\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\lesssim\Arrowvert\Delta\nabla^{p-1}B^{k}u\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }+ \Arrowvert\delta^{ab}\left(\nabla_{a}\bm{\uprho} \right) \nabla_{b}\left( \nabla^{p-1}B^{k}u \right) \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }. \end{align} The term \(\Delta\nabla^{p-1}B^{k}u\) has the similar structure to the corresponding term in \(\eqref{5.16}\) and can be handled using similar considerations, so we concentrate on the boundary contribution \(\Arrowvert\delta^{ab}\left(\nabla_{a}\bm{\uprho} \right) \nabla_{b}\left( \nabla^{p-1}B^{k}u \right) \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\). Using the trace theorem and Lemma \(\ref{3.7.}\), we have \begin{align} \begin{split} &\Arrowvert\delta^{ab}\left(\nabla_{a}\bm{\uprho} \right) \nabla_{b}\left( \nabla^{p-1}B^{k}u \right) \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\\ \lesssim&\Arrowvert[\delta^{ab}\left(\nabla_{a}\bm{\uprho} \right) \nabla_{b},\nabla^{p-1}]B^{k}u \Arrowvert_{H^{1}\left( \mathcal B_{t}\right) }+\Arrowvert\nabla^{p-1}B^{k+2}u \Arrowvert_{H^{1}\left( \mathcal B_{t}\right) }+\Arrowvert\nabla^{p}B^{k+1}\varepsilon \Arrowvert_{H^{1}\left( \mathcal B_{t}\right) }+\Arrowvert\nabla^{p-1}F_{k} \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }. \end{split} \end{align} Except for the last term \(\Arrowvert\nabla^{p-1}F_{k} \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\), all other terms on the right above are bounded by the right-hand side of \(\eqref{5.15}\) using the induction hypothesis \(\eqref{5.14}\) and \(\eqref{5.26}\). For \(\Arrowvert\nabla^{p-1}F_{k} \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\), in view of Lemma \(\ref{3.7.}\) the highest order terms are \begin{align*} \Arrowvert \nabla^{p+1}B^{k-1}u\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) },\quad \Arrowvert \nabla^{p+1}B^{k}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\quad and \quad \Arrowvert \nabla^{p-1}B^{k+1}\nabla\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }. \end{align*} The term \(\Arrowvert \nabla^{p+1}B^{k}\varepsilon\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\) was already bounded in \(\eqref{5.26}\), and \(\Arrowvert \nabla^{p+1}B^{k-1}u\Arrowvert_{L^{2}\left( \mathcal B_{t}\right) }\) can be handled using Lemma \(\ref{5.7.}\). Noticing the following result \begin{align} \Arrowvert \nabla^{p-1}B^{k+1}\nabla\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\sim \Arrowvert \nabla^{p-1}B^{k}\nabla_{t,x}\nabla\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }. \end{align} Therefore we can bound \(\Arrowvert \nabla^{p-1}B^{k+1}\nabla\psi\Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\) in the same way as in the treatment of \(\eqref{5.19}-\eqref{5.22}\). Putting everything together we have proved that \begin{align}\label{5.30} \begin{split} &\Arrowvert\delta^{ab}\left(\nabla_{a}\bm{\uprho} \right) \nabla_{b}\left( \nabla^{p-1}B^{k}u \right) \Arrowvert_{H^{\frac{1}{2}}\left( \partial\mathcal B_{t}\right) }\\ &\lesssim\underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau. \end{split} \end{align} Combining \(\eqref{5.27}\) and \(\eqref{5.30}\), we finally obtain \begin{align}\label{5.31} \begin{split} &\Arrowvert\nabla^{p}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{t}\right) }\\ \lesssim&\underline{E}_{\le l+1}[\varepsilon,t]+E_{\le l}[u,t]+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k}u\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right) }+ \sum_{k+2q\le l+2}\Arrowvert\partial_{t,x}^{q}B^{k+1}\varepsilon\Arrowvert^{2}_{L^{2}\left( \mathcal B_{0}\right)}+\int_{0}^{t}{\mathscr{E}}_{l}(\tau)d\tau, \end{split} \end{align} which completes the proof of Proposition \(\ref{5.2.}\). \end{proof} The following we only need to show that \(\underline{E}_{\le l+1}[\varepsilon,t]\) and \(E_{\le l}[u,t]\) are bounded by the right-hand side of \(\eqref{5.5}\). we will complete this work by using energy estimates \(\eqref{energy}\), \(\eqref{estimate1}\) and \(\eqref{estimate2}\). In Lemma \(\ref{3.3.}\), the main term of the right-hand side of \(\eqref{estimate1}\) is \begin{align*}
\left| \int_{0}^{t}\int_{\mathcal B\left( s\right) } \left( \square_{g}\phi\right) \left( Q\phi\right) dxds\right|, \end{align*} and the ramainder terms are naturally controlled by the right-hand side \(\eqref{5.5}\). For the above term with \(\phi=B^{k+1}\varepsilon\), we need consider all possible case of \(\square_{g}B^{k+1}\varepsilon\) stated in Lemma \(\ref{3.9.}\). Similarly in Lemma \(\ref{3.2.}\), for the right-hand side of \(\eqref{energy}\) with \(\phi=B^{k}u\), what we need to controll is \begin{align*}
\left| \int_{0}^{t}\int_{\partial\mathcal B(\tau)}F_{k}\left( B^{k+1}u\right) dSd\tau\right| +\left| \int_{0}^{t}\int_{\mathcal B(s)}G_{k}\left( B^{k+1}u\right) dxds\right| . \end{align*} To simplify notation we define as \begin{align*} \mathcal E_{k}(t):=\underline{E}_{\le l+1}[\varepsilon,t]+ E_{\le l}[u,t]. \end{align*} We first consider two special cases intorduced in the following Lemma. \begin{lemma}\label{5.8.}
Suppose the bootstrap assumption \(\eqref{5.4}\) hold. Then there exist some positive constants \(C_{0}\), \(C_{1}(\bar{\rho})\), \(C_{2}(\bar{\rho})\), \(C_{h}\) for any \(k\le l\) and \(t\in[0,T]\), we have
\begin{align}\label{5.32}
\int_{0}^{t}\int_{\mathcal B(\tau)}|\nabla^{2}B^{k}\varepsilon|^{2}dxd\tau\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds.
\end{align} \end{lemma} \begin{proof}
We proceed inductively. For \(k=0\) this follows for instance by \(\eqref{3.20}\) and \(\eqref{2.20}\) using the elliptic estimates. Iductively, suppose the statement of the lemma holds for \(k\le j-1\le l-1\) and let us prove it for \(k=j\). Applying Lemma \(\ref{3.5.}\) and elliptic estimates with \(\phi=B^{j}\varepsilon\), we need to bound the \(\Arrowvert H_{j-1}\Arrowvert^{2} _{L^{2}(\mathcal B_{s})}\), with \(H_{j-1}\) as in Lemma \(\ref{3.9.}\). The contribution from line \(\eqref{i1},\eqref{i2},\eqref{i3},\eqref{i4},\eqref{i5}\) and \(\eqref{i7}\) in Lemma \(\ref{3.9.}\) can be controlled by the energy \({\mathscr{E}}_{l}\) (the top order terms are \(\nabla^{2}B^{k-2}u\ and\ \nabla^{2}B^{k-1}\varepsilon\)). Then we consider the contribution of line \(\eqref{i6}\) and focus on \(\Arrowvert B^{k}\nabla\nabla\psi\Arrowvert^{2}_{L^{2}(\mathcal B_{s})}\) in particular. Using similar processing methods like \(\eqref{5.18}-\eqref{5.22}\), \(\Arrowvert B^{k-1}\nabla\nabla\psi\Arrowvert^{2}_{L^{2}(\mathcal B_{s})}\) can be bounded by
\begin{align}
\sum_{i\le k-2}\int_{\mathcal B(s)}|\nabla^{2}B^{i}u|^{2}dx+\sum_{i\le k-1}\int_{\mathcal B(s)}\left(|\nabla B^{i}\varepsilon|^{2} +|\nabla B^{i}u|^{2}\right) dx,
\end{align}
which completes the proof of Lemma \(\ref{5.8.}\). \end{proof} The next lemma allows us to estimate the \(L^{2}(\partial\mathcal B_{s})\) norm of \(\nabla B^{k-1}u\). \begin{lemma}\label{5.9.}
Suppose the bootstrap assumption \(\eqref{5.4}\) hold. Then there exist some positive constants \(C_{0}\), \(C_{1}(\bar{\rho})\), \(C_{2}(\bar{\rho})\), \(C_{h}\) for any \(k\le l-1\) and \(t\in[0,T]\), we have
\begin{align}\label{5.34}
\int_{0}^{t}\int_{\partial\mathcal B(\tau)}|\nabla B^{k}u|^{2}dSd\tau\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds.
\end{align} \end{lemma} \begin{proof}
For \(k=0\) this follows for instance from the trace theorem. Proceeding inductively, we assume \(\eqref{5.34}\) holds for \(k\le j-1\le l-2\) and prove it for \(k=j\). Applying Lemma \(\ref{3.4.}\) with \(\phi=B^{j}u\), what term we need to process is
\begin{align*}
\int_{0}^{t}\int_{\mathcal B(s)}|G_{j}|^{2}dxds+\int_{0}^{t}\int_{\partial\mathcal B(\tau)}|nB^{j}u|^{2}dSd\tau,
\end{align*}
and the raminder terms are bounded by the right-hand side of \(\eqref{5.34}\). For the contribution \(nB^{j}u\) on \(\mathcal B\) we apply Lemma \(\ref{3.7.}\) to write
\begin{align}\label{5.35}
-nB^{j}u=\frac{1}{a}BB^{j+1}u+\frac{1}{a}\nabla B^{j+1}\varepsilon-\frac{1}{a}F_{j}.
\end{align}
The first two terms on the right-hand side of \(\eqref{5.35}\) are bounded by using the trace theorem and Lemma \(\ref{5.8.}\). The contributin of \(F_{j}\) can be bounded by the right-hand side of \(\eqref{5.34}\) using the Hilbert transform and the elliptic estimates. Finally we consider the contribution of \(G_{j}\). The line \(\eqref{j1}\) and \(\eqref{j2}\) in Lemma \(\ref{3.8.}\) are bounded by
\begin{align}\label{5.36}
\sum_{i\le j}\int_{0}^{t}\int_{\mathcal B(s)}|\nabla B^{i}\varepsilon|^{2}dxds+\sum_{i\le j}\int_{0}^{t}\int_{\mathcal B(s)}|\nabla B^{i}u|^{2}dxds+\sum_{i\le j-1}\int_{0}^{t}\int_{\mathcal B(s)}|\nabla^{2} B^{i}u|^{2}dxds,
\end{align}
hence by the right-hand side of \(\eqref{5.34}\). For the term \(B^{j+1}\nabla\psi\) from line \(\eqref{j3}\) in \(G_{j}\), we actually consider
\begin{align}
\int_{0}^{t}\int_{\mathcal B(s)}|B^{j}\nabla_{t,x}\nabla\psi |^{2}dxds,
\end{align}
which is bounded by \(\eqref{5.34}\) in the same way as in the treatment of \(\eqref{5.18}-\eqref{5.22}\). \end{proof} We turn to the proof of Proposition \(\ref{5.1.}\) \begin{proof}[Proof of Proposition \(\eqref{5.1.}\)]
We proceed inductively on \(k\) to show the pirori estimete \(\eqref{5.5}\), that is, there exist some positive constants \(C_{0}\), \(C_{1}(\bar{\rho})\), \(C_{2}(\bar{\rho})\), \(C_{h}\) for any \(j\le l\) and \(t\in[0,T]\)
\begin{align}\label{5.38}
\underline{E}_{\le j+1}[\varepsilon,t]+E_{\le j}[u,t]\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds.
\end{align}
For small enough \(j\), the estimate \(\eqref{5.38}\) is clearly true. Now we assume that \(\eqref{5.38}\) holds for \(j\le k-1\le l-1\) and show it for \(j=k\).\\
{\bf Step\ 1:} First we show that
\begin{align}\label{5.39}
\underline{E}_{\le k+1}[\varepsilon,t]\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds.
\end{align}
As we have discussed before, the method is to apply energy estimate \(\eqref{estimate1}\) with \(\phi=B^{k+1}\varepsilon\) to equation \(\eqref{3.30}\), which can be done because \(B^{k+1}\varepsilon=0\) on \(\partial\mathcal B\). It is obvious that the main term of the right-hand side of \(\eqref{estimate1}\) is
\begin{align*}
\left| \int_{0}^{t}\int_{\mathcal B\left( s\right) } H_{k} \left( QB^{k+1}\varepsilon\right) dxds\right|,
\end{align*}
where \(H_{k}\) is as in Lemma \(\ref{3.9.}\). If \(H_{k}\) is of the form \(\eqref{i1},\eqref{i2},\eqref{i3},\eqref{i5}\) and \(\eqref{i7}\) in Lemma \(\ref{3.9.}\), then we can use Cauchy-Schwarz to bound above contribution by
\begin{align*}
\sum_{i\le k}\int_{0}^{t}\int_{\mathcal B(s)}|\nabla B^{i}u|^{2}dxds+\sum_{i\le k+1}\int_{0}^{t}\int_{\mathcal B(s)}|\nabla B^{i}\varepsilon|^{2}dxds+\sum_{i\le k}\int_{0}^{t}\int_{\mathcal B(s)}|\nabla^{2} B^{i}\varepsilon|^{2}dxds.
\end{align*}
According to Lemma \(\ref{5.8.}\) the above three terms are bounded by the right-hand side of \(\eqref{5.39}\). If \(H_{k}\) is of the form \(\eqref{i6}\) in Lemma \(\ref{3.9.}\) we use a similar approach like \(\eqref{5.18}-\eqref{5.22}\). It is worth noting that we should use Lemma \(\eqref{3.6.}\) first to bound \(\Arrowvert B^{k}\nabla\psi\Arrowvert_{H^{1}\left( \mathcal B_{s}\right) }\). Because we can bound \(\Arrowvert\nabla B^{k-1}u\Arrowvert_{L^{2}\left( \partial\mathcal B_{s}\right) }\) rather than \(\Arrowvert\nabla^{2}B^{k-1}u\Arrowvert_{L^{2}\left( \mathcal B_{s}\right) }\) according to Lemma \(\eqref{5.9.}\). Finally for the contribution from line \(\eqref{i4}\) in Lemma \(\ref{3.9.}\) we use a few integration by parts. We treat the most difficult case when \(k_{m+2}=k-1\), and write the resulting expression as
\begin{align}
\tilde{F}^{ab}\nabla_{a}\nabla_{b}B^{k-1}u=\tilde{F}^{ab}g_{a\beta}g^{\alpha\beta}\nabla_{\alpha}\nabla_{b}B^{k-1}u:=F^{\alpha b}\nabla_{\alpha}\nabla_{b}B^{k-1}u.
\end{align}
Replacing \(H_{k}\) by this expression and making some simple calculations, we have
\begin{align}\label{5.41}
\begin{split}
(F^{\alpha b}\nabla_{\alpha}\nabla_{b}B^{k-1}u)(QB^{k+1}\varepsilon)=&\nabla_{\lambda}[(F^{\lambda b}\nabla_{b}B^{k-1}u)(QB^{k+1}\varepsilon)-(F^{\alpha b}\nabla_{b}B^{k-1}u)(B^{\lambda}Q\nabla_{\alpha}B^{k}\varepsilon)]\\
&-(\nabla_{\lambda}F^{\lambda b})(\nabla_{b}B^{k-1}u)(QB^{k+1}\varepsilon)-(F^{\lambda b}\nabla_{b}B^{k-1}u)(\nabla_{\lambda}Q^{\alpha})(\nabla_{\alpha}B^{k+1}\varepsilon)\\
&+(BF^{\alpha b})(\nabla_{b}B^{k-1}u)(Q\nabla_{\alpha}B^{k}\varepsilon)+(F^{\alpha b}\nabla_{b}B^{k}u)(Q\nabla_{\alpha}B^{k}\varepsilon)\\
&-(\nabla_{b}B^{\lambda})(F^{\alpha b}\nabla_{\lambda}B^{k-1}u)(Q\nabla_{\alpha}B^{k}\varepsilon)+(F^{\alpha b}\nabla_{b}B^{k-1}u)(BQ^{\lambda})(\nabla_{\lambda}\nabla_{\alpha}B^{k}\varepsilon)\\
&+(F^{\alpha b}\nabla_{b}B^{k-1}u)(Q^{\lambda}[B,\nabla_{\lambda}\nabla_{\alpha}]B^{k}\varepsilon)+(F^{\alpha b}\nabla_{b}B^{k-1}u)(\nabla_{\lambda}B^{\lambda})(Q\nabla_{\alpha}B^{k}\varepsilon).
\end{split}
\end{align}
Except for the first line, the other terms can be bounded by the same arguments as above. For the first line we integrate by parts. The resulting on \(\mathcal B_{t}\) involve at most one top order term, hence these terms can be bounded by using Cauchy-Schwarz inequality with a small constant and using the induction hypothesis. Finally, since \(B\) is tangential to \(\partial\mathcal B\), the integration of boundary term can be bounded by Cauchy-Schwarz inequality with a small constant as well. This finishes the proof of \(\eqref{5.39}\).\\
{\bf Step\ 2:} Here we show that
\begin{align}\label{5.42}
\left| \int_{0}^{t}\int_{\mathcal B(s)}G_{k}\left( B^{k+1}u\right) dxds\right|\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds.
\end{align}
where \(G_{k}\) is as in Lemma \(\ref{3.8.}\). The form \(\eqref{j1}\) and \(\eqref{j3}\) can be bounded in the same way as in Step 1 above. For the form \(\eqref{j2}\) of \(G_{k}\), we also treat the hardest case when \(k_{m+1}=k-1\), and express \(G_{k}\) as
\begin{align*}
\tilde{G}^{ab}\nabla_{a}\nabla_{b}B^{j-1}u=\tilde{G}^{ab}g_{a\beta}g^{\alpha\beta}\nabla_{\alpha}\nabla_{b}B^{j-1}u:=G^{\alpha b}\nabla_{\alpha}\nabla_{b}B^{j-1}u.
\end{align*}
Then we have
\begin{align}\label{5.43}
\begin{split}
(G^{\alpha b}\nabla_{\alpha}\nabla_{b}B^{j-1}u)(B^{j+1}u)=&\nabla_{\lambda}[(G^{\lambda b}\nabla_{b}B^{j-1}u)(B^{j+1}u)-(G^{\alpha b}\nabla_{b}B^{j-1}u)(B^{\lambda}\nabla_{\alpha}B^{j}u)]\\
&-(\nabla_{\alpha}G^{\alpha b})(\nabla_{b}B^{j-1}u)(B^{j+1}u)+(BG^{\alpha b})(\nabla_{b}B^{j-1}u)(\nabla_{\alpha}B^{j}u)\\
&+(G^{\alpha b}\nabla_{b}B^{j}u)(\nabla_{\alpha}B^{j}u)-(G^{\alpha b}\nabla_{\lambda}B^{j-1}u)(\nabla_{b}B^{\lambda})(\nabla_{\alpha}B^{j}u)\\
&+(G^{\alpha b}\nabla_{b}B^{j-1}u)(\nabla_{\lambda}B^{\lambda})(\nabla_{\alpha}B^{j}u)-(G^{\alpha b}\nabla_{b}B^{j-1}u)(\nabla_{\alpha}B^{\lambda})(\nabla_{\lambda}B^{j}u).
\end{split}
\end{align}
Integrating \(\eqref{5.43}\) over \(\cup_{s\in[0,t]}\mathcal B(s)\), the contribution of the right-hand side of \(\eqref{5.43}\) can be bounded by using integration by parts and Cauchy-Schwarz inequality as well as in the treatment of \(H_{k}\). This completes the proof of \(\eqref{5.42}\).\\
{\bf Step\ 3:} Then we show that
\begin{align}\label{5.44}
E_{\le k}[u,t]\le C_{0}{\mathscr{E}}_{l}(0)+\int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s)+C_{h}{\mathscr{E}}_{l}^{\frac{3}{2}}(s)ds.
\end{align}
We apply the Lemma \(\ref{3.2.}\) with \(\phi=B^{k}u\). we note that except for the integration of boundary term
\begin{align}
\int_{0}^{t}\int_{\partial\mathcal B(\tau)}F_{k}\left(B^{k+1}u\right)dSd\tau ,
\end{align}
all other terms on the right-hand side of \(\eqref{energy}\) are bounded by using Lemma \(\ref{estimate1}\) and \(\eqref{5.42}\). The contribution of the form \(\eqref{k1}\) and \(\eqref{k2}\) in Lemma \(\ref{3.7.}\) are bounded by using trace theorem, \(\eqref{5.32}\), \(\eqref{5.34}\) and induction hypothesis. For the terms of the form \(\eqref{k3}\), we consider the most difficult case
\begin{align}
\int_{0}^{t}\int_{\partial\mathcal B(\tau)}|B^{k}\nabla_{t,x}\nabla\psi|^{2}dSd\tau,
\end{align}
which is bounded in the same way as in the treatment of \(\eqref{5.20}-\eqref{5.22}\).Note that \(\eqref{5.6}\), \(\eqref{5.39}\) and \(\eqref{5.44}\) complete the proof of the Proposition \(\ref{5.1.}\).\\ \end{proof}
\section{Nonlinear instability}\label{6} Here we describe how to close the bootstrap argument in Lemma \(\ref{1.4.}\) from linear instability to nonlinear instability. As discussed before, we know that the emergence of nonlinear instability requires the stronger norm to be controlled reversely by the weaker norm, that is, we need to prove that there exists small enough \(\epsilon>0\) and constant \(C_{\epsilon}>0\), such that \begin{align}\label{6.1} \int_{0}^{t}C_{1}(\bar{\rho}){\mathscr{E}}_{l}(s)+C_{2}(\bar{\rho})\mathcal E_{l-1}(s) ds\le\int_{0}^{t}\epsilon\mu_{0}{\mathscr{E}}_{l}(s)+C_{\epsilon}\left(\Arrowvert \varepsilon \Arrowvert^{2}_{L^{2}(\mathcal B_{s})}+\Arrowvert u \Arrowvert^{2}_{L^{2}(\mathcal B_{s})} \right) ds. \end{align}
We recall the conclusion of reference \cite{L}, when \(1\le\gamma<\frac{4}{3}\) and stars with large central density, liquid Lane-Emden stars are linearly unstable, and the fastest linear growth mode \(\mu_{0}\) in increases with the increase of center density (that is, \(\mu_{0}\) is related to steady state \(\bar{\rho}\)). Observing the structure of \(F_{k},G_{k},H_{k}\), we notice that the coefficients of linear term with the hightest derivative of \(\varepsilon\) and \(u\) are dependent on the steady state \(\bar{\rho}\), and roughly \(C_{1}(\bar{\rho})\) increases as the steady state \(\bar{\rho}\) increases. Therefore, we need to obtain the exact magnitude relationship between the linear term coefficient \(C_{1}(\bar{\rho})\) and the linear growth mode \(\mu_{0}\), which is the main goal of this section. In \cite{L}, the author transformed \(\eqref{linearized eq SS}\) into a Sturm-Liouville type equation with Robin type boundary condition, and then proved the existence of the fastest linear growth mode \(\mu_{0}\). We outline the results here. By setting \(\zeta(y,t)=e^{\lambda t}\chi(y)\) we rewrite \(\eqref{linearized eq SS}\) as \begin{align}\label{5.48} \begin{split} L\chi:=-\gamma\partial_{y}(\bar{\rho}^{\gamma}y^{4}\partial_{y}\chi)+(4-3\gamma)y^{3}\chi\partial_{y}\bar{\rho}^{\gamma}=&-\lambda^{2}y^{4}\bar{\rho}\chi \\ 3\chi(R)+R\partial_{y}\chi(R)=&0. \end{split} \end{align} Given \(\chi\in C^{2}([0,R])\) satisfying the boundary condition of \(\eqref{5.48}\), we use integration by parts \begin{align}\label{111} \langle L\chi,\chi\rangle=&\int_{0}^{R}\gamma\bar{\rho}^{\gamma}y^{4}(\partial_{y}\chi)^{2}+(4-3\gamma)y^{3}\chi^{2}\partial_{y}\bar{\rho}^{\gamma}dy+3\gamma R^{3}\chi(R)^{2}, \end{align} where \(\left\langle \cdot\right\rangle \) represent the usual \(L^{2}\) inner product. Let \(H^{1}_{r}(B_{R}(\mathbb R^{5}))\) denote the subspace of spherically symmetric functions in \(H^{1}(B_{R}(\mathbb R^{5}))\). We can conclude \begin{align}\label{5.49} \inf_{\Arrowvert\chi\Arrowvert_{y^{4}\bar{\rho}}=1}\langle L\chi,\chi\rangle:=\mu_{\ast}=\inf[\mu:\exists\chi\ne 0 s.t. L\chi=\mu y^{4}\bar{\rho}\chi], \end{align} where \(\Arrowvert\chi\Arrowvert^{2}_{\omega}=\langle \chi,\omega\chi\rangle \). Moreover there exists \(\chi_{\ast}\in H^{1}_{r}(B_{R}(\mathbb R^{5}))\) that allows the infimum to be reached. This shows that if there exist \(\chi\) such that \(\langle L\chi,\chi\rangle <0\), then by \(\eqref{5.49}\) there exist \(-\mu_{0}<0\) and \(\chi_{\ast}\) such that \(L\chi_{\ast}=-\mu_{0}y^{4}\bar{\rho}\chi_{\ast}\). The following we need to estimate the magnitude of \(\mu_{0}\). We know that the family of gaseous steady states are self-similar, so that the family is given by \(\bar{\rho}_{\kappa}(y)=\kappa\bar{\rho}_{\ast}(\kappa^{1-\gamma/2}y)\) where \(\bar{\rho}_{\ast}\) is a steady state. The corresponding liquid star has \(R_{\kappa}=\kappa^{-(1-\gamma/2)}\bar{\rho}^{-1}_{\ast}(1/\kappa)\). We deal with three cases individually.\\ {\bf Case\ 1:} \(\frac{6}{5}<\gamma<\frac{4}{3}\)
With \(\eqref{111}\), we have \begin{align} \begin{split} \langle L_{\kappa}\chi,\chi\rangle=&\int_{0}^{R_{\kappa}}\gamma\bar{\rho}_{\kappa}^{\gamma}y^{4}(\partial_{y}\chi)^{2}+(4-3\gamma)y^{3}\chi^{2}\partial_{y}\bar{\rho}_{\kappa}^{\gamma}dy+3\gamma R_{\kappa}^{3}\chi(R_{\kappa})^{2}\\ =&R_{\kappa}^{3}k^{\gamma}\int_{0}^{1}\gamma\bar{\rho}_{\ast}(\bar{\rho}_{\ast}^{-1}(1/\kappa)z)^{\gamma}z^{d+1}(\partial_{z}\tilde{\chi})^{2}+(4-3\gamma)z^{d}\bar{\rho}_{\ast}^{-1}(1/\kappa)\tilde{\chi}^{2}(\bar{\rho}_{\ast}^{\gamma})'(\bar{\rho}_{\ast}^{-1}(1/\kappa)z)dz+3\gamma R_{\kappa}^{3}\chi(R_{\kappa})^{2}. \end{split} \end{align} When \(\gamma>\frac{6}{5}\), the gaseous steady state \(\bar{\rho}_{\ast}\) has compact support. Then \(\bar{\rho}_{\ast}^{-1}(1/\kappa)\to\bar{\rho}^{-1}_{\ast}(0)=R_{\ast}\) as \(\kappa\to\infty\). According to dominated convergence we have \begin{align}\label{5.51} \begin{split} \langle L_{\kappa}1,1\rangle=&(4-3\gamma)R_{\kappa}^{3}\kappa^{\gamma}\bar{\rho}_{\ast}^{-1}(1/\kappa)\int_{0}^{1}z^{d}(\bar{\rho}_{\ast}^{\gamma})'(\bar{\rho}_{\ast}^{-1}(1/\kappa)z)dz+3\gamma R_{\kappa}^{3}\\ \sim&-\kappa^{\frac{5}{2}\gamma-3}\to-\infty\quad\quad as\quad\quad \kappa\to\infty, \end{split} \end{align} and \begin{align}\label{5.52} \langle 1,y^{4}\bar{\rho}_{\kappa}1\rangle=\int_{0}^{R_{\kappa}}y^{4}\bar{\rho}_{\kappa}(y)dy=R_{\kappa}^{5}\kappa\int_{0}^{1}z^{4}\bar{\rho}_{\ast}(\bar{\rho}_{\ast}^{-1}(1/\kappa)z)dz\sim\kappa^{\frac{5}{2}-4}. \end{align} Combining \(\eqref{5.49}\), \(\eqref{5.51}\) and \(\eqref{5.52}\), we have \(\mu_{0}\sim\kappa\) when \(\kappa\) is large enough. Then we turn to consider the magnitude of \(C_{1}(\bar{\rho}_{\kappa})\). According to equations \(\eqref{2.33}\) and \(\eqref{2.34}\), we can obtain \begin{align}\label{5.54} C_{1}(\bar{\rho}_{\kappa})\sim\left( c_{s}^{2}(\bar{\bm{\uprho}}_{\kappa})+c_{s}(\bar{\bm{\uprho}}_{\kappa})c'_{s}(\bar{\bm{\uprho}}_{\kappa})\right)\nabla\bar{\bm{\uprho}}_{\kappa} \sim\gamma\bar{\rho}_{\kappa}^{\gamma-1}\nabla\bar{\bm{\uprho}}_{\kappa}\lesssim-\frac{1}{\bar{\rho}_{\kappa}}\partial_{y}\bar{\rho}^{\gamma}_{\kappa}. \end{align} In spherical symmetry, the steady state satisfies equation \begin{align}\label{5.55} \frac{4\pi}{y^{2}}\int_{0}^{y}s^{2}\bar{\rho}(s)ds+\frac{1}{\bar{\rho}}\partial_{y}\bar{\rho}^{\gamma}=0. \end{align} Let \(y=\nu R_{\kappa}\), where \(0<\nu\le 1\), we have \begin{align}
|C_{1}(\bar{\rho}_{\kappa})|\lesssim\frac{1}{\nu^{2}R_{\kappa}^{2}}\int_{0}^{\nu R_{\kappa}}s^{2}\bar{\rho}_{\kappa}(s)ds=\nu R_{\kappa}\kappa\int_{0}^{1}z^{2}\bar{\rho}_{\ast}(\bar{\rho}_{\ast}^{-1}(1/\kappa)\nu z)dz\lesssim\kappa^{\frac{\gamma}{2}}. \end{align}
Since \(\frac{\gamma}{2}<1\), there exists small enough \(\epsilon>0\) such that \(|C_{1}(\bar{\rho}_{\kappa})|\le \epsilon\mu_{0}\) when \(\kappa\) is large enough. \\ {\bf Case\ 2:} \(\gamma=\frac{6}{5}\)
From the explicit formula which has been proven in \cite{heinzle2002finiteness}, we have \begin{align}\label{22} \begin{split} \bar{\rho}_{\kappa}(y)=&\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}y^{2}\right) ^{-\frac{5}{2}}\\ \partial_{y}\bar{\rho}_{\kappa}^{\gamma}(y)=&-\frac{4\pi}{3}\kappa^{\frac{2}{5}}y\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}y^{2}\right) ^{-4}\\ R_{\kappa}=&\frac{3}{\sqrt{2\pi}}\kappa^{-\frac{2}{5}}\left( \kappa^{\frac{2}{5}}-1\right) ^{\frac{1}{2}}. \end{split} \end{align} From \(\eqref{111}\) we have \begin{align} \begin{split} \langle L_{\kappa}1,1\rangle=&\int_{0}^{R_{\kappa}}(4-3\gamma)y^{3}\partial_{y}\bar{\rho}_{\kappa}^{\gamma}dy+3\gamma R_{\kappa}^{3}\\ =&-\frac{4\pi}{3}(4-3\gamma)\kappa^{\frac{2}{5}}\int_{0}^{R{\kappa}}y^{4}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}y^{2}\right) ^{-4}dy+3\gamma R_{\kappa}^{3}\\ =&-\frac{4\pi}{3}(4-3\gamma)\kappa^{-\frac{3}{5}}\int_{0}^{\kappa^{\frac{1}{5}}R{\kappa}}z^{4}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}z^{2}\right) ^{-4}dz+3\gamma R_{\kappa}^{3}\\ =&-3(4-3\gamma)\left( \frac{9}{2\pi}\right) ^{\frac{3}{2}}\kappa^{-\frac{3}{5}}\int_{\kappa^{-\frac{2}{5}}}^{\kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}R_{\kappa}^{2}}\frac{\left( s-\kappa^{-\frac{2}{5}}\right) ^{\frac{3}{2}}}{s^{4}}ds+3\gamma R_{\kappa}^{3}\\ \le&-3(4-3\gamma)\left( \frac{9}{4\pi}\right) ^{\frac{3}{2}}\kappa^{-\frac{3}{5}}\int_{2\kappa^{-\frac{2}{5}}}^{\kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}R_{\kappa}^{2}}s^{-\frac{5}{2}}ds+3\gamma R_{\kappa}^{3}\\ =&-2(4-3\gamma)\left( \frac{9}{4\pi}\right) ^{\frac{3}{2}}\kappa^{-\frac{3}{5}}\left( \left( 2\kappa^{-\frac{2}{5}}\right) ^{-\frac{3}{2}}-\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}R_{\kappa}^{2}\right) ^{-\frac{3}{2}}\right) +3\gamma R_{\kappa}^{3}\\ \to&-2(4-3\gamma)\left( \frac{9}{8\pi}\right) ^{\frac{3}{2}}\quad\quad as\quad\quad \kappa\to\infty, \end{split} \end{align} and \begin{align*} \begin{split} \langle 1,y^{4}\bar{\rho}_{\kappa}1\rangle=&\int_{0}^{R_{\kappa}}y^{4}\bar{\rho}_{\kappa}(y)dy =\int_{0}^{R_{\kappa}}y^{4}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}y^{2}\right) ^{-\frac{5}{2}}dy\\ =&\kappa^{-1}\int_{0}^{\kappa^{\frac{1}{5}}R_{\kappa}}z^{4}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}z^{2}\right) ^{-\frac{5}{2}}dz =\frac{9}{4\pi}\left( \frac{9}{2\pi}\right) ^{\frac{3}{2}}\kappa^{-1}\int_{\kappa^{-\frac{2}{5}}}^{\kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}R_{\kappa}^{2}}\frac{\left( s-\kappa^{-\frac{2}{5}}\right) ^{\frac{3}{2}}}{s^{\frac{5}{2}}}ds\\ \lesssim&\kappa^{-1}\int_{\kappa^{-\frac{2}{5}}}^{\kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}R_{\kappa}^{2}}s^{-1}ds=\kappa^{-1}\left( \log\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}R_{\kappa}^{2}\right) -\log\kappa^{-\frac{2}{5}}\right) \lesssim\frac{\log\kappa}{\kappa}. \end{split} \end{align*} According to \(\eqref{5.49}\), we know \(\mu_{0}\sim \frac{\kappa}{\log\kappa}\) when \(\kappa\) is large enough. As discussed in case 1, we consider the relationship between the coefficient \(C_{1}(\bar{\rho}_{\kappa})\) and \(\kappa\). Using the explicit formula \(\eqref{22}\) and \(\eqref{5.54}\) we have \begin{align*}
\left| C_{1}(\bar{\rho}_{\kappa})\right| \lesssim- \frac{1}{\bar{\rho}_{\kappa}}\partial_{y}\bar{\rho}_{\kappa}^{\gamma} =\frac{4\pi}{3}\kappa^{\frac{2}{5}}y\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}y^{2}\right) ^{-\frac{3}{2}}\le\max_{0\le\nu\le 1}\frac{4\pi}{3}\kappa^{\frac{2}{5}}\nu R_{\kappa}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\right) ^{-\frac{3}{2}}. \end{align*} Naturally we define the function \begin{align*} f(\nu):=\kappa^{\frac{2}{5}}\nu R_{\kappa}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\right) ^{-\frac{3}{2}} \end{align*} It is straightforward to calculate \begin{align*} f'(\nu)=\kappa^{\frac{2}{5}}R_{\kappa}\frac{\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\right) ^{\frac{3}{2}}-\frac{2\pi}{3}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\right) ^{\frac{1}{2}}}{\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\right) ^{3}}=\kappa^{\frac{2}{5}}R_{\kappa}\frac{\kappa^{-\frac{2}{5}}-\frac{4\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}}{\left( \kappa^{-\frac{2}{5}}+\frac{2\pi}{9}\kappa^{\frac{2}{5}}\nu^{2}R_{\kappa}^{2}\right) ^{\frac{5}{2}}} \end{align*} and the maximum is attained when \(\nu=\frac{3}{2\sqrt{\pi}}\kappa^{-\frac{2}{5}}R_{\kappa}^{-1}\). Therefore we have \begin{align*}
\max\left|C_{1}(\bar{\rho}_{\kappa}) \right| \lesssim f(\frac{3}{2\sqrt{\pi}}\kappa^{-\frac{2}{5}}R_{\kappa}^{-1})\lesssim\kappa^{\frac{3}{5}} \end{align*}
Since \(\kappa^{\frac{3}{5}}\ll\frac{\kappa}{\log\kappa}\) when \(\kappa\) is large enough, there exists small enough \(\epsilon>0\) such that \(|C_{1}(\bar{\rho}_{\kappa})|\le \epsilon\mu_{0}\).\\ {\bf Case\ 3:} \(1\le\gamma<\frac{6}{5}\)\\ Let \(\bar{\rho}_{\ast}\) be a gaseous steady state and \(\bar{\rho}_{\kappa}(r)=\kappa\bar{\rho}_{\ast}(\kappa^{1-\gamma/2}r)\). Then there exist \(c>0\) such that \begin{align}\label{333} \begin{split}
\left| r^{\frac{2}{2-\gamma}}\bar{\rho}_{\kappa}(r)-\left(\frac{1}{2\pi}\frac{\gamma(4-3\gamma)}{(2-\gamma)^{2}}
\right)^{\frac{1}{2-\gamma}} \right| =&\left| r^{\frac{2}{2-\gamma}}\bar{\rho}_{\kappa}(r)-\upsilon^{\ast}_{1}\right| \lesssim\left( \kappa^{1-\frac{\gamma}{2}}r\right) ^{-c}\\
\left| r^{\frac{2}{2-\gamma}-3}\bar{m}_{\kappa}(r)-\frac{2\gamma}{2-\gamma}\left(\frac{1}{2\pi}\frac{\gamma(4-3\gamma)}{(2-\gamma)^{2}}
\right)^{\frac{\gamma-1}{2-\gamma}}\right| =&\left|r^{\frac{2}{2-\gamma}-3}\bar{m}_{\kappa}(r)-\upsilon^{\ast}_{2}\right| \lesssim \left( \kappa^{1-\frac{\gamma}{2}}r\right) ^{-c}\\ R_{\kappa}\to R_{\infty}:=(\upsilon^{\ast}_{1})^{1-\frac{\gamma}{2}}=&\left(\frac{1}{2\pi}\frac{\gamma(4-3\gamma)}{(2-\gamma)^{2}} \right)^{\frac{1}{2}}\quad\quad as\quad\quad \kappa\to\infty, \end{split} \end{align} where \begin{align*} \bar{m}_{\kappa}(r)=4\pi\int_{0}^{r}s^{2}\bar{\rho}_{\kappa}(s)ds. \end{align*} These are known results (see \cite{L}). Based on the above formula we first consider the bound on the coefficient \(C_{1}(\bar{\rho}_{\kappa})\). From \(\eqref{5.55}\), we have \begin{align*}
\left| C_{1}(\bar{\rho}_{\kappa})\right| \lesssim- \frac{1}{\bar{\rho}_{\kappa}}\partial_{y}\bar{\rho}_{\kappa}^{\gamma}=\frac{4\pi}{y^{2}}\int_{0}^{y}s^{2}\bar{\rho}_{\kappa}(s)ds \end{align*} Let \(y=\nu R_{\kappa}\). when \(0\le \nu\le \kappa^{-(1-\frac{\gamma}{2})}\) we have \begin{align*}
\left| C_{1}(\bar{\rho}_{\kappa})\right| \lesssim&\frac{1}{\nu^{2} R_{\kappa}^{2}}\int_{0}^{\nu R_{\kappa}}s^{2}\kappa\bar{\rho}_{\ast}(\kappa^{1-\frac{\gamma}{2}}s)ds =\nu R_{\kappa}\int_{0}^{1}z^{2}\kappa\bar{\rho}_{\ast}(\kappa^{1-\frac{\gamma}{2}}\nu R_{\kappa}z)dz\\ \lesssim&\kappa^{-(1-\frac{\gamma}{2})}\kappa R_{\kappa}\int_{0}^{1}z^{2}\Arrowvert\bar{\rho}_{\ast}\Arrowvert_{\infty}dz \lesssim \kappa^{\frac{\gamma}{2}} \end{align*} When \(\kappa^{-(1-\frac{\gamma}{2})}\le\nu\le 1\) we have \begin{align*}
\left| C_{1}(\bar{\rho}_{\kappa})\right| \lesssim\frac{1}{\nu^{2} R_{\kappa}^{2}}\int_{0}^{\nu R_{\kappa}}s^{2}\kappa\bar{\rho}_{\ast}(\kappa^{1-\frac{\gamma}{2}}s)ds= \frac{1}{\nu^{2} R_{\kappa}^{2}}\int_{0}^{\nu R_{\kappa}\kappa^{1-\frac{\gamma}{2}}}\kappa\kappa^{-3(1-\frac{\gamma}{2})}z^{2}\bar{\rho}_{\ast}(z)dz. \end{align*} Fix some large \(M>0\) and divide the region of integration into two parts. Then we have \begin{align*}
\left| C_{1}(\bar{\rho}_{\kappa})\right| \lesssim&\frac{\kappa^{1-3(1-\frac{\gamma}{2})}}{\nu^{2} R_{\kappa}^{2}}\int_{M}^{\nu R_{\kappa}\kappa^{1-\frac{\gamma}{2}}}\bar{\rho}_{\ast}(z)z^{2}dz+\frac{\kappa^{1-3(1-\frac{\gamma}{2})}}{\nu^{2} R_{\kappa}^{2}}\int_{0}^{M}\bar{\rho}_{\ast}(z)z^{2}dz\\ \lesssim&\frac{\kappa^{1-3(1-\frac{\gamma}{2})}}{\nu^{2} R_{\kappa}^{2}}\int_{M}^{\nu R_{\kappa}\kappa^{1-\frac{\gamma}{2}}}z^{2-\frac{2}{2-\gamma}}dz+\frac{\kappa^{1-3(1-\frac{\gamma}{2})}}{\nu^{2} R_{\kappa}^{2}}\int_{0}^{M}\Arrowvert\bar{\rho}_{\ast}\Arrowvert_{\infty}z^{2}dz\\ \lesssim&\frac{\kappa^{1-3(1-\frac{\gamma}{2})}}{\nu^{2} R_{\kappa}^{2}}\left( \nu R_{\kappa}\kappa^{1-\frac{\gamma}{2}}\right) ^{3-\frac{2}{2-\gamma}}+\frac{\kappa^{1-3(1-\frac{\gamma}{2})}}{\nu^{2} R_{\kappa}^{2}}\int_{0}^{M}\Arrowvert\bar{\rho}_{\ast}\Arrowvert_{\infty}z^{2}dz\\ \lesssim&\nu^{1-\frac{2}{2-\gamma}}\kappa^{1-3(1-\frac{\gamma}{2})+(1-\frac{\gamma}{2})(3-\frac{2}{2-\gamma})}+\nu^{-2}\kappa^{1-3(1-\frac{\gamma}{2})}\lesssim\kappa^{\frac{\gamma}{2}} \end{align*} According to previous discussion, as long as the linear growth mode \(\mu_{0}\) increase in magnitude with respect to \(\kappa\) is larger than \(\frac{\gamma}{2}\), we will finish the proof. From \(\eqref{111}\) we have \begin{align*} \langle L_{\kappa}\chi,\chi\rangle=&\int_{0}^{R_{\kappa}}\gamma\bar{\rho}_{\kappa}^{\gamma}y^{4}(\partial_{y}\chi)^{2}+(4-3\gamma)y^{3}\chi^{2}\partial_{y}\bar{\rho}_{\kappa}^{\gamma}dy+3\gamma R_{\kappa}^{3}\chi(R_{\kappa})^{2}\\ =&\int_{0}^{R_{\kappa}}\gamma\bar{\rho}_{\kappa}^{\gamma}y^{4}(\partial_{y}\chi)^{2}-(4-3\gamma)y\chi^{2}\bar{m}_{\kappa}\bar{\rho}_{\kappa}dy+3\gamma R_{\kappa}^{3}\chi(R_{\kappa})^{2}. \end{align*} Fix \(\nu>0\), and suppose \begin{align}\label{444}
\chi(y)=\left\{ \begin{aligned} &\nu^{-a}\kappa^{a(1-\frac{\gamma}{2})},\quad\quad 0\le y\le \nu\kappa^{-(1-\frac{\gamma}{2})}\\ &y^{-a},\quad\quad\nu\kappa^{-(1-\frac{\gamma}{2})}\le y\le R_{\kappa} \end{aligned} \right., \end{align} we have \begin{align}\label{5.60} \begin{split} \langle L_{\kappa}\chi,\chi\rangle\le&\int_{\nu\kappa^{-(1-\frac{\gamma}{2})}}^{R_{\kappa}}\gamma\bar{\rho}_{\kappa}^{\gamma}y^{4}(\partial_{y}\chi)^{2}-(4-3\gamma)y\chi^{2}\bar{m}_{\kappa}\bar{\rho}_{\kappa}dy+3\gamma R_{\kappa}^{3-2a}\\ \le&\int_{\nu\kappa^{-(1-\frac{\gamma}{2})}}^{R_{\kappa}}\gamma\left( 1+\epsilon_{1}\right) ^{\gamma}\left( \upsilon^{\ast}_{1}\right) ^{\gamma}y^{5-\frac{2+\gamma}{2-\gamma}}\left( \partial_{y}\chi\right) ^{2}-\frac{2\gamma(4-3\gamma)}{2-\gamma}\left( 1-\epsilon_{2}\right) ^{2}\left( \upsilon^{\ast}_{1}\right) ^{\gamma}y^{4-\frac{4}{2-\gamma}}\chi^{2}dy+3\gamma R_{\kappa}^{3-2a}\\ =&\gamma\left( \upsilon^{\ast}_{1}\right) ^{\gamma}\left( a^{2}\left( 1+\epsilon_{1}\right) ^{\gamma}-2\left( 1-\epsilon_{2}\right) ^{2}\left( 3-\frac{2}{2-\gamma}\right) \right) \int_{\nu\kappa^{-(1-\frac{\gamma}{2})}}^{R_{\kappa}}y^{3-\frac{2+\gamma}{2-\gamma}-2a}dy, \end{split} \end{align} and \begin{align*} \langle\chi,y^{4}\bar{\rho}_{\kappa}\chi\rangle=\int_{0}^{R_{\kappa}}y^{4}\bar{\rho}_{\kappa}\chi^{2}dy=\int_{\kappa^{-(1-\frac{\gamma}{2})}}^{R_{\kappa}}y^{4-2a}\bar{\rho}_{\kappa}dy+\int_{0}^{\kappa^{-(1-\frac{\gamma}{2})}}y^{4}\bar{\rho}_{\kappa}\kappa^{a(1-\frac{\gamma}{2})}dy:=A+B. \end{align*} We estimate the two parts respectively. With \(\eqref{333}\), we have \begin{align}\label{5.61} A\lesssim\int_{\kappa^{-(1-\frac{\gamma}{2})}}^{R_{\kappa}}y^{4-2a-\frac{2}{2-\gamma}}dy \end{align} and \begin{align}\label{5.62} B=\int_{0}^{\kappa^{-(1-\frac{\gamma}{2})}}y^{4}\kappa^{1+a(1-\frac{\gamma}{2})}\bar{\rho}_{\ast}(\kappa^{1-\frac{\gamma}{2}}y)dy\le\kappa^{1+a(1-\frac{\gamma}{2})}\Arrowvert\bar{\rho}_{\ast}\Arrowvert_{\infty}\int_{0}^{\kappa^{-(1-\frac{\gamma}{2})}}y^{4}dy\lesssim \kappa^{1+(a-5)(1-\frac{\gamma}{2})}. \end{align} Combining \(\eqref{5.60}\), \(\eqref{5.61}\) and \(\eqref{5.62}\), if there exists some constant \(a>0\) such that \begin{align}\label{5.63} \left\{ \begin{aligned} &-(1-\frac{\gamma}{2})(4-\frac{2+\gamma}{2-\gamma}-2a)>\frac{\gamma}{2}\\ &a^{2}\left( 1+\epsilon_{1}\right) ^{\gamma}-2\left( 1-\epsilon_{2}\right) ^{2}\left( 3-\frac{2}{2-\gamma}\right)<0\\ &4-\frac{2}{2-\gamma}-2a\ge-1\\ &1+(a-5)(1-\frac{\gamma}{2})\le 0 \end{aligned} \right. \end{align}
hold, then it means that \(\mu_{0}\) has a larger magnitude than \(\kappa\). we can take large enough \(\kappa\) such that \(|C_{1}(\bar{\rho}_{\kappa})|\le \epsilon\mu_{0}\) for small enough \(\epsilon>0\). By simplifying \(\eqref{5.63}\), we only need to find the constant \(a>0\) that satisfies \begin{align*} 2-\frac{1}{2-\gamma}<a<\sqrt{\frac{(1-\epsilon_{2})^{2}}{(1+\epsilon_{1})^{\gamma}} \left( 6-\frac{4}{2-\gamma}\right) }. \end{align*} It is obvious that there exists a constant \(a>0\) satisfying the above inequality when \(\epsilon_{1}\) and \(\epsilon_{2}\) are small enough (equivalent to picking sufficiently large \(\nu\) in \(\eqref{444}\)). This completes the argument.
For the non-top order terms on the left-hand side of \(\eqref{6.1}\), it is straightforward to be bounded by applying Sobolev interpolation inequality (the reason is that the coefficient \(C_{2}(\bar{\rho}_{\kappa})\) may have a faster growth rate than \(\mu_{0}\) as \(\kappa\to\infty\)). The following we roughly illustrate the idea. \begin{align}\label{5.56}
\int_{0}^{t}\mathcal E_{l-1}ds=\sum_{j=0}^{l-1}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}B^{j}u|^{2}dxds+\sum_{j=0}^{l-1}\int_{0}^{t}\int_{\partial\mathcal B(\tau)}|B^{j+1}u|^{2}dSd\tau+\sum_{j=0}^{l}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}B^{j}\varepsilon|^{2}dxds. \end{align} The first term on the right-hand side of \(\eqref{5.56}\) can be written as \begin{align}\label{5.57} \begin{split}
&\sum_{j=0}^{l-1}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}B^{j}u|^{2}dxds\\
\le&\theta_{1}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}B^{l}u|^{2}dxds+C_{\theta_{1}}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}u|^{2}dxds\\
\le&\theta_{1}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}B^{l}u|^{2}dxds+\theta_{2}C_{\theta_{1}}\int_{0}^{t}\int_{\mathcal B(s)}|\partial_{t,x}^{2}u|^{2}dxds+C_{\theta_{1}}C_{\theta_{2}}\int_{0}^{t}\int_{\mathcal B(s)}|u|^{2}dxds, \end{split} \end{align} where we apply Sobolev interpolation inequality to time norm and space norm of Lagrange coordinates respectively (in this case, \(B\sim\partial_{t}\) and \(\partial_{x}\sim\nabla\)). For some large \(\kappa\), we take \(\theta_{1},\theta_{2}\) small enough such that \begin{align} C_{2}(\bar{\rho})(\theta_{1}+\theta_{2}C_{\theta_{1}})\lesssim\epsilon\mu_{0}. \end{align} Then the terms on the right-hand side of \(\eqref{5.57}\) can be bounded. The remainder terms on the the right-hand side of \(\eqref{5.56}\) is bounded by the trace theorem and the same way. Combined with the previous discussion, we have completed the proof of \(\eqref{6.1}\).
\centerline{\scshape Zeming Hao}
{\footnotesize
\centerline{School of Mathematics and Statistics, Wuhan University}
\centerline{Wuhan, Hubei 430072, China}
\centerline{\email{[email protected]}} }
\centerline{\scshape Shuang Miao}
{\footnotesize
\centerline{School of Mathematics and Statistics, Wuhan University}
\centerline{Wuhan, Hubei 430072, China}
\centerline{\email{[email protected]}} }
\end{document} | arXiv |
Effect of transcutaneous acupoint electrical stimulation on propofol sedation: an electroencephalogram analysis of patients undergoing pituitary adenomas resection
Xing Liu1,
Jing Wang2,
Baoguo Wang1,
Ying Hua Wang3,4,
Qinglei Teng1,
Jiaqing Yan5,
Shuangyan Wang1 &
You Wan6
Transcutaneous acupoint electrical stimulation (TAES) as a needleless acupuncture has the same effect like traditional manual acupuncture. The combination of TAES and anesthesia has been proved valid in enhancing the anesthetic effects but its mechanisms are still not clear.
In this study, we investigated the effect of TAES on anesthesia with an electroencephalogram (EEG) oscillation analysis on surgery patients anesthetized with propofol, a widely-used anesthetic in clinical practice. EEG was continuously recorded during light and deep propofol sedation (target-controlled infusion set at 1.0 and 3.0 μg/mL) in ten surgery patients with pituitary tumor excision. Each concentration of propofol was maintained for 6 min and TAES was given at 2–4 min. The changes in EEG power spectrum at different frequency bands (delta, theta, alpha, beta, and gamma) and the coherence of different EEG channels were analyzed.
Our result showed that, after TAES application, the EEG power increased at alpha and beta bands in light sedation of propofol, but reduced at delta and beta bands in deep propofol sedation (p < 0.001). In addition, the EEG oscillation analysis showed an enhancement of synchronization at low frequencies and a decline in synchronization at high frequencies between different EEG channels in either light or deep propofol sedation.
Our study showed evidence suggested that TAES may have different effects on propofol under light and deep sedation. TAES could enhance the sedative effect of propofol at low concentration but reduce the sedative effect of propofol at high concentration.
Transcutaneous acupoint electric stimulation (TAES), or "needleless acupuncture", is an easy and non-invasive alternative to needle-based electro-acupuncture (EA). It combines the advantages of both acupuncture and transcutaneous electrical nerve stimulation by pasting electrode pads on the acupoints instead of piercing the skin with needles. Several studies indicated that intraoperative TAES could enhance the sedative effect of propofol, a widely-used sedative anesthetic [1–3], In addition, it could reduce the opioids consumption and the incidence of anesthesia-related side-effects, while improve the quality of recovery from anesthesia [4, 5]. Our recent study also suggested that TAES may exert analgesic effect, and the sufentanil consumption was significantly reduced during craniotomy [6]. Electroacupuncture stimulation at ST36 and PC6 has been reported to significantly deepen the sedation level of general anaesthesia [7]. These results suggested that acupuncture and related techniques may have both analgesic and sedative effects. Nevertheless, the combination of TAES and anesthetic has been reported to be benefit, but it is still unclear how acupuncture works in propofol-induced deep or light sedation.
Electroencephalography (EEG) is a sensitive method for measuring the brain activities and is widely applied to monitor the depth of anesthesia. Previous researches showed that activity of EEG changed under anesthesia, power of alpha frequency band (8–12 Hz), especially the frontal alpha, was increased under propofol [8, 9] and beta, gamma frequency band was varies [8]. In addition, disrupted coherence of EEG activity was considered as the leading underlying mechanism of anesthesia [10, 11]. Evidence indicates that TAES or acupuncture could induce the changes of EEG activity. During acupuncture, activity of alpha and theta oscillations of EEG in human being increased [12]. After TAES, the power of theta frequency band was decreased [13]. These results suggested that TAES may modulate the activity and coherence of EEG to improve the sedition under anesthesia [13, 14]. In the present study, we investigated the effect of TAES on propofol anesthesia with an EEG oscillation analysis in patients undergoing pituitary adenoma resection.
Patient selection and clinical procedures
This study was approved by the ethics review board of the Beijing Sanbo Brain Hospital (2013121101) and registered in the Chinese Clinical Trial Registry (registration number: ChiCTR-TRC-13004051). All participants provided their written informed consent and consent to publish the individual and identifiable patient details before being enrolled in this study. Inclusion criteria were as follows: (1) The age of patients should range from 18 to 65 years; (2) Patients without gender limited; (3) The Body Mass Index (BMI) of patients should range from 18 to 30 Kg/m2; (4) Patients meet the standard of American Society of Anesthesiology (ASA), Physical Status matain ASA I-II; (5) All patients signed their written informed consent. Exclusion criteria included: (1) Patients had a history of needleless acupuncture within 6 months; (2) Patients in lactation or pregnant; (3) Patients involved in other clinical trial within nearly 4 weeks; (4) Patients took sedatives and analgesics for a long-term, and have been addicted or alcoholics; (6) Patients with extreme anxiety fear, non-cooperation or communication barriers during the test. A total of ten patients scheduled for pituitary tumor excision and met with all above criteria were enrolled in this study during October 2013 to June 2014. All participants received standard pre-anesthesia assessments, and were tested with normal hearing and urine toxicology to exclude other potential factors, which might interact with propofol or confound the EEG adversely.
Participants were fasted for at least 8 h before the procedure. The heart rate of patients was monitored with an electrocardiogram, oxygen saturation through pulse oximetry, respiration and expiratory carbon dioxide with capnography, and blood pressure through non-invasive cuff to ensure the patients' safety. There were at least three anesthesiologists involved in each study: one was in charge of the medical management of the subject during the study, the second handled the propofol administration, and the third accomplished EEG recording. When the patient became apneic, the first anesthesiologist assisted breathing with bag/mask ventilation. A phenylephrine infusion was applied to maintain mean arterial pressure above the specific level determined from the patients' baseline measurement.
Experimental design and procedure
The experiment paradigm was shown in Fig. 1. Before propofol infusion was started, we recorded the EEG for about 5 min as a baseline when the patient was kept in a conscious, eye-closed and calm situations, this phase was named as phase 0. Then we used a computer-controlled infusion to achieve propofol target effect-site concentrations of 1 μg/mL and then up to 3 μg/mL. Propofol was administered as target-controlled infusion (TCI) based on the pharmacokinetic model by Marsh et al. [15], and the target plasma concentration of sufentanil was set at a certain value (1 or 3 μg/mL) during the whole anesthesia. The concentration level on each target effect-site was maintained for 6 min, then divided each 6 min into three phases (2 min/phase): phase 1–3 (1 μg/mL propofol), and phase 4–6 (3 μg/mL propofol), respectively. TAES were applied in the phase two and phase five.
Experimental paradigm. A computer-controlled infusion method was used to achieve propofol target effect-site concentrations of 1 and 3 μg/mL. The target effect-site concentration level was measured for 6 min and then increased to 3 μg/mL. Every 6-min observation period was divided into three phases, and each phase lasted for 2 min. TAES was administrated at the phases two and five, respectively
TAES
TAES was applied to the acupoints of Hégǔ (LI 4), Wàiguān (TE 5), Zúsānlǐ (ST 36) and Qiūxū (GB 40) on the left side of patient in phase two and five, respectively. The stimulation was applied by the HANS acupoint nerve stimulator (HANS 200A, Nanjing Jisheng Medical Technology Co., Ltd., Nanjing, China) with a dense-disperse frequency of 2/100 Hz (alternated once every 3 s; 0.6 ms at 2 Hz and 0.2 ms at 100 Hz). EEG was recorded in the whole procedure. We defined the whole procedure into three states: state one represents the basal state including phase 0; state two represents low-concentration of propofol (1 μg/mL) including phases one, two, and three (phase 1 is the basal state of phases 2 and 3); state three represents high-concentrationμg/mL of propofol (3 μg/mL) including phases four, five, six (phase 4 is the basal state of phases 5 and 6).
EEG recordings
Scalp EEG electrodes were positioned at Fp1, Fp2, Fz3, F4, C3, C4, Cz, P3, P4, O1, O2, F7, F8, T3, T4, T5, and T6 (EEG, international 10–20 system); all channels were referenced to A1, A2 (bilateral Mastoid). Electrode impedances were reduced to below 5 kΩ prior to data collection. EEG signals were collected using the Nicolet One EEG-64 device (Nicolet Corp., USA) with a sampling frequency of 1024 Hz. The signals were band-passed at 1.6–45 Hz to avoid baseline drift and high frequency noise.
Power spectrum analysis
The power spectrum of EEG signals was estimated with a customized procedure as our previous reports [16, 17]. Considering the EEG spectrum should be relatively stable during the short time of each phase (2 min), we used the following method to reduce the abnormal variance in the power spectrum:
EEG data were segmented into epochs of 12 s;
For each epoch, (1) Perform the Morlet wavelet transform with the wavelet central angle frequency of 6 (ω0 = 6) at frequency band 2.0 Hz to 45 Hz, with a frequency resolution of 0.5 Hz; (2) EEG could have little chance of sudden change during anesthesia. Therefore, we reasonably treat the abrupt change in EEG energy as induced by artifacts. A common fluctuation range of wavelet energy at a particular frequency is within 1 uV. So in this study 1 uV is chosen as the threshold for removing artifacts. Then for each frequency, outliers in corresponding wavelet coefficients which has a standard deviation (SD) larger than 1 μV was removed with a threshold of mean ± SD; (3) For each frequency, repeat step 2.2 with remaining coefficient, until the SD is less than 1 μV, or removed values exceeds a ratio of 20 %;
For each frequency, the mean coefficients of all epochs are considered as the power of that frequency.
EEG coherence analysis
We estimated the magnitude squared coherence between each pair of channels using Welch's overlapped averaged phaseogram method [18]. The method was described as follows:
For two time series x(n) and y(n), estimate the power spectral density (PSD) by the discrete Fourier transform
$$ {P}_{xx}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \sum_{m=-\infty}^{\infty }}{R}_{xx}(m){e}^{-j\omega m} $$
$$ {P}_{yy}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \sum_{m=-\infty}^{\infty }}{R}_{yy}(m){e}^{-j\omega m} $$
Then the magnitude-squared coherence between the two signals is given by
$$ {C}_{xy}\left(\omega \right)=\frac{{\left|{P}_{xx}\left(\omega \right){P}_{yy}^{*}\left(\omega \right)\right|}^2}{P_{xx}\left(\omega \right){P}_{yy}\left(\omega \right)} $$
where \( * \) denotes the conjugate of a complex number.
$$ \left(\omega \right)=\frac{1}{2\pi }{\displaystyle \sum_{m=-\infty}^{\infty }}{R}_{xx}(m){e}^{-j\omega m} $$
where * denotes the conjugate of a complex number.
We calculated the coherence index of every channel before and after TAES at each band.
Statistical analysis was performed using SAS 9.0 software (SAS Institute Inc., Cary, NC, USA). The power spectral data before and after TAES were analyzed by paired-sample t-test. And the power spectral data from different concentration level of propofol were analyzed by One-Way ANOVA. The further analysis used Dunnett-t test, and phase 0 as a control group. Paired-sample t-test was also applied to analyze the coherence index before and after TAES at each band for the same channel. A p value of < 0.05 was considered to be statistically significant.
Effect of propofol on the EEG power
To investigate the effect of propofol on the ongoing brain activities, we calculated the averaged absolute spontaneous EEG powers at each frequency band for each recording session (Fig. 2). One-way ANOVA analysis showed significant difference at delta (F(2, 158), = 187.411, p < 0.001), theta (F(2, 158) = 130.379, p < 0.001), alpha (F(2, 158) = 742.112, p < 0.001), and beta (F(2, 158) = 243.857, p < 0.001) bands, but not at gamma band (F(2, 158) = 0.528, p = 0.59). Further analysis with Dunnett t-test revealed that the power of both alpha and beta frequency oscillations increased significantly at phase one compared with phase 0 (p < 0.001), while the power of other bands showed no significant change. In addition, the powers of all of alpha, beta, delta, and theta bands increased significantly at phase four (p < 0.001), while the power of gamma band showed no significant changes compared with phase 0.
Effect of propofol concentration on the EEG power at different frequency bands. a. The EEG power spectrum at each frequency band. b. Topoplot of EEG power at each frequency band. Phase 0: before propofol infusion, phase 1: 1 μg/mL propofol, phase 4: 3 μg/mL propofol. Data were color-coded and plotted at the corresponding position on the scalp. ** p < 0.001, and * p < 0.05 vs the phase 0
Furthermore, we analyzed the power changes of different bands in each channel of different phases. Compared with the powers of each band in the corresponding channel of phase 0, the power of alpha band in channels O2 and P4 increased significantly (alpha band: F(2, 8) = 121.778, p <0.05), and the power of beta band in all channels increased significantly in phase 1 (F(2,8) = 46.388, p < 0.001,). And the power of delta, theta, alpha, and beta bands in all channels increased significantly in phase four (delta band: F(2, 8) = 187.411, p < 0.001; theta band: F(2, 8) = 130.379, p < 0.001; alpha band: F(2,8) = 742.112, p < 0.001; beta band: F(2, 8) = 243.857, p < 0.001).
Effect of TAES on EEG power at different concentrations of propofol
We compared and analyzed the averaged absolute power changes of EEG at each frequency band in different phases. Figure 3 showed the effect of TAES on the changes of EEG power in different frequency bands at different concentrations of propofol, we can see an increase at low frequency bands in deep propofol sedation. To observe the main effects of TAES on the propofol-induced EEG power changes, we summarize the EEG power changes in phase one and three and phase four and six, respectively (Fig. 4a and b). Compared with those in phase one, the powers of alpha and beta bands increased significantly (t = 7.324, p < 0.001; t = 9.302, p < 0.001) at phase three, whereas the powers of the other bands did not show any significant changes. Compared with those in phase four, the powers of delta and beta bands in phase six showed significant decrease (t = 7.819, p < 0.001; t = 17.312, p < 0.001), whereas the powers of the other bands did not show any significant changes.
Effect of TAES on the changes of EEG power induced at different propofol concentrations. a. EEG power spectrum at propofol concentration of 1 μg/mL; b. EEG power spectrum at propofol concentration of 3 μg/mL. In phase two and phase five, TAES were applied. ** p < 0.001 vs the phase one or four
Topoplot of EEG power at different frequency bands during the different phases. a: Topoplot of EEG power at the propofol concentration of 1 μg/mL; b: Topoplot of EEG power at the propofol concentration of 3 μg/mL TAES was applied in phase two and phase five. Data were color-coded and plotted at the corresponding position on the scalp
Furthermore, we analyzed the power changes of each band in different phases at each channel (Fig. 5). Compared with that in phase one, the power of alpha band in phase three was among 12 of 16 channels, and the power of beta band was among eight of 16 channels, both of them increased significantly (p < 0.001, t = 9.457), but the power of other bands did not show any significant changes. Compared with that in phase four, the power of beta band in all channels decreased significantly in phase six, except for channel Fp2. The power of theta band in channels O1 and O2, and the power of delta bands in channel T3, T6 and C4 increased significantly (p < 0.001, t = 7.331). The power of other bands did not show any significant changes in phase 6. The channels that TAES had significant influences on the EEG power at low- and high-concentrations of propofol were summarized in Table 1.
Effect of TAES on the changes of EEG power spectra at each individual channel at different propofol concentrations. a: EEG power spectra at propofol concentration of 1 μg/mL; b: EEG power spectra at propofol concentration of 3 μg/mL. In each plot, the averaged absolute EEG power before TAES (red line), during TAES (green line), and after TAES (blue line) is plotted at each electrode
Table 1 Channels with significant change (p < 0.05) at different frequency bands, induced by TEAS at two propofol concentrations
Effects of TAES on EEG coherence among different channels at different concentrations of propofol
We used magnitude squared coherence to estimate the correlation index between different EEG channels in each band before and after TAES at propofol target effect-site concentrations of 1 μg/mL and 3 μg/mL. Then we used paired-t test to analyze the changes of the correlation indices. The discrepancy and the extent of coherence changes of different channels in each band at different propofol concentrations were presented in Fig. 6. The following effects can been seen : Firstly, the synchronization between each pair of channels increased in low-frequency oscillations (delta, theta and alpha), but the synchronism in high-frequency oscillations (beta and gamma) decreased at 1 μg/mL propofol. The synchronization increased in theta and alpha bands at 3 μg/mL propofol, while decreased in beta band, and did not show any significant changes in the rest of the bands; Secondly, the synchronization in ipsilateral hemisphere was stronger than that in bilateral hemispheres at 1 μg/mL propofol. However, it did not show any significant discrepancies between two hemispheres; Thirdly, the synchronization in right hemisphere was stronger than that in left hemisphere at 3 μg/mL propofol.
The effect of TAES on synchronization among EEG channels at different frequency bands at different propofol concentrations. a: Synchronization among EEG channels at propofol concentration of 1 μg/mL; b: Synchronization among EEG channels at propofol concentration of 3 μg/mL. Red nodes indicate electrodes projected onto the cortex. Lines between nodes indicate a significant change in synchronization between the two channels before and after TAES. The color of lines represents the strength of synchronization. The red color indicates an increase in synchronization, and the blue indictaed a decrease
We adopted a self-control design to compare the changes of brain oscillations before and after TAES in patients undergoing pituitary adenoma resection. The individual difference of EEG data was large. The self-control design can minimize the interference of the individual difference to the results. We used a computer-controlled infusion to achieve stable propofol target effect-site concentrations of 1 μg/mL and 3 μg/mL before and after TAES, respectively. Bonhomme et al. reported that the objects become slightly drowsy at 0.5 μg/mL propofol, inarticulate and sluggish at 1.5 μg/mL, and irresponsive at all at 3.5 μg/mL [19]. Purdon et al. identified two states, one where subjects had a nonzero probability of response to auditory stimulated another where subjects were unconscious with a zero probability of response [9]. Then, some researchers defined upper two states as propofol-induced unconsciousness trough-max (TM) and peak-max (PM), respectively [20]. Furthermore, Akeju's team found that the neurophysiological signatures were stably maintained over changing propofol effect-site concentrations: approximately 1 to 2 μg/mL for TM and approximately 3 to 5 μg/mL for PM [20]. Thus, we defined 1 μg/mL and 3 μg/mL propofol as light and deep sedation respectively in present study. It not only made it easy to investigate the effect of TAES in a stable neurophysiological state, but also reduced the dose of propofol used for the research.
Brain oscillation at low- and high-concentrations of propofol
Compared with baseline, the power of alpha and beta oscillations had a significant increase at 1 μg/mL propofol, and power of delta, theta, alpha, and beta significantly increased at 3 μg/mL propofol. Numerous studies have investigated the effect of propofol on EEG, and propofol might exhibit anesthetic effect by potentiating GABAA receptors. The effects on macroscopic dynamics were noticeable in the EEG, which contained several stereotyped patterns during maintenance of propofol anesthesia. These patterns were like that powers in (0.5–4 Hz) delta range increase in light anesthetic level [21, 22]; with the increasing concentration, an alpha (~10 Hz) rhythm [23–25] is coherent across frontal cortex; powers in alpha range then became smaller and theta or delta powers become dominant in deeper levels. With further deeper levels, burst suppression, an alternation between bursts of high-voltage activity and phases of flat EEG was lasting for several seconds [26, 27]. Some researchers investigated the change of EEG power at the loss of consciousness and sedation induced by propofol, a significant increase of beta and alpha bands were observed at sedation of propofol, corresponding to 15–25 Hz and 12–15 Hz, respectively. Additionally, they noticed an enhancement in delta, alpha, and theta power were noticed during propofol induced loss of consciousness [22]. Our results were not completely concordant with pervious study. The reason might be the different propofol concentration used that leading to the different level of sedation, or TAES might influence the brain oscillation, since all patients accepted TAES before the increase in popofol concentration.
TAES modulation on brain oscillation in light and deep propofol sedation
The validity of TAES in anesthesia was controversial. Some researchers argued that acupuncture was "only" a placebo procedure based on sensory input. However, we observed significant changes of the ongoing power spectra in different frequency oscillations ranging from delta to beta band except for the gamma band.
It is known that alpha oscillation mainly serves as a top-down controlled inhibitory mechanism. Beta oscillation may be involved in the maintenance of the current sensorimotor or cognitive state, and the extraordinary enhancement of beta oscillation may result in an abnormal persistence of the current situation and a deterioration of flexible behaviors and cognitive controls [28]. Theta oscillation serves as an essential network-level role in hippocampal learning and memory. For example, theta oscillations promote plasticity [29] and support memory processes requiring interregional signal integration [30–32]. Conversely, suppression of the theta rhythm impairs learning and memory [33–35]. Delta-band oscillation is more often seen to be related to deep-sleep states and compromise of neuronal function [36]. The latter findings support the belief that low-frequency oscillations might actually influence in active processing [37]. For our result, the brain oscillations induced by TAES in light and deep propofol sedation were different. The power increase in light propofol sedation following TAES intervention occurred at alpha and beta bands, while reduction of power at delta and beta bands was in deep propofol sedation.
As mentioned before, the alpha power has become a reverse measure of activation [38–40]. The beta oscillation might be related to deterioration of cognitive control. The power increases of alpha and beta bands after TAES in propofol sedation might indicate that TAES could strengthen the sedation effect of propofol. The power of delta and beta oscillation was significantly induced after TAES, especially for beta oscillation at 3 μg/mL propofol. The decrease in beta oscillation occurred at all channels. Elevated endogenous GABA levels could cause the elevation of beta power [41, 42]. Electro acupuncture may induce release of endogenous endomorphins that activate μ opioid receptors in GABA-nergic neurons to suppress the release of GABA [43]. Taken together, it is conceivable that the decreased power of beta oscillations in our results might reflect the inhibition of GABAergic interneurons by TAES.
More importantly, we found an enhancement of synchronization at low frequency and a decline in synchronization at high frequency between different channels after TAES, and at different propofol concentrations. Synchronous rhythms represent a vital mechanism for expressing temporal coordination of neural activity in the brain wide network. Coherent oscillations are generated by many generally neuronal synchrony. It may contribute to well-timed coordination and communication between neural populations simultaneously engaged in a cognitive process. It is well known that slow waves oscillations are the electrophysiological correlate of millions of neurons switching between up and down states. The large slow waves may link to decreases in effective connectivity, which presence of widespread cortical disability between up and down states during early NREM sleep [44, 45]. The high frequency oscillations like beta and gamma may play an important role in integrating the unity of conscious perception [46]. It has been accepted that low-frequency oscillations might be involved in the integration of information across widely spatial distribution of neural assemblies and high-frequency oscillations distributed over a more limited topographic area. In our study, we found that the synchronization of low frequency (delta, theta) oscillation occurred widely across brain areas, while the coherence of high frequency (beta, gamma) occurred within more limited areas. Taken together, we speculated that TAES exerted antinociceptive effect by modulation of the power and coherence between different channels, which disturbed the cortex excitability and effective connectivity.
The deficiency of this article and the future directions of research
There are still some limitations in this view, including limited samples. In the future studies, we will enlarge the sample size to solve this problem. Moreover, there is a washout phase in the self-control design. Although we did not find the conclusive evidence about corporation effect of TAES for 2 min, the post effect of TAES should be taken into consideration when analyze and discuss the EEG changes at 3 μg/mL propofol.
Changes in EEG signature induced by TAES under light or deep sedation were different. TAES might strengthen the effect of propofol during light sedation, whereas it might have an antagonism to propofol in deep sedation. TAES may exert antinociceptive effect by modulating the power and coherence between different channels, which disturbed the cortex excitability and effective connectivity. However, it is difficult for us to simply come to the conclusion that whether TAES is beneficial or harmful in propofol anesthesia, and a large cohort studies are still needed to further clarify the potential mechanisms of TAES.
TEAS:
Transcutaneous acupoint electrical stimulation
EGG:
EA:
American society of anesthesiology
Peak-max
Trough-max
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This research was supported by grants from the National Basic Research Program of the Ministry of Science and Technology of China (2013CB531905), the National Natural Science Foundation of China (81230023 and 81221002).
Department of Anesthesiology, Beijing Sanbo Brain Hospital, Capital Medical University, Beijing, 100093, China
Xing Liu, Baoguo Wang, Qinglei Teng & Shuangyan Wang
Department of Neurobiology, Capital Medical University, Beijing, 100069, China
Center for Collaboration and Innovation in Brain and Learning Sciences, Beijing Normal University, Beijing, 100875, China
Ying Hua Wang
State Key Laboratory of Cognitive Neuroscience and Learning & IDG/Mc Govern Institute for Brain Research, Beijing Normal University, Beijing, 100875, China
Institute of Electrical Engineering, Yanshan University, Qinhuangdao, 066004, China
Jiaqing Yan
Neuroscience Research Institute, Key Lab for Neuroscience, Peking University Health Science Center, Beijing, 100191, China
You Wan
Xing Liu
Baoguo Wang
Qinglei Teng
Shuangyan Wang
Correspondence to Baoguo Wang or You Wan.
The authors have no conflicts of interest to declare.
Xing Liu, Qing Lei Teng, and Shang Yan Wang collected data and Xing Liu draft the manuscript, Jing Wang reviewed the data collection, analysis and manuscript, Ying Hua Wang and Jia Qing Yan draw the figures, analyzed data and performed statistics, YW and GBW designed the study, analyzed the data and finalized the manuscript. All authors read and approved the final manuscript.
Liu, X., Wang, J., Wang, B. et al. Effect of transcutaneous acupoint electrical stimulation on propofol sedation: an electroencephalogram analysis of patients undergoing pituitary adenomas resection. BMC Complement Altern Med 16, 33 (2016). https://doi.org/10.1186/s12906-016-1008-1
Transcutaneous acupoint electrical stimulation (TAES)
Electroencephalogram (EGG)
Power spectrum | CommonCrawl |
Anomalous birefringence and enhanced magneto-optical effects in epsilon-near-zero metamaterials based on nanorods' arrays
I. A. Kolmychek, A. R. Pomozov, V. B. Novikov, A. P. Leontiev, K. S. Napolskii, and T. V. Murzina
I. A. Kolmychek,* A. R. Pomozov, V. B. Novikov, A. P. Leontiev, K. S. Napolskii, and T. V. Murzina
M.V. Lomonosov Moscow State University, 119991, Leninskie Gory, Moscow, Russia
*Corresponding author: [email protected]
V. B. Novikov https://orcid.org/0000-0002-6368-9672
K. S. Napolskii https://orcid.org/0000-0002-9353-9114
I Kolmychek
A Pomozov
V Novikov
A Leontiev
K Napolskii
T Murzina
I. A. Kolmychek, A. R. Pomozov, V. B. Novikov, A. P. Leontiev, K. S. Napolskii, and T. V. Murzina, "Anomalous birefringence and enhanced magneto-optical effects in epsilon-near-zero metamaterials based on nanorods' arrays," Opt. Express 27, 32069-32074 (2019)
Nanophotonics, Metamaterials, and Photonic Crystals
Faraday effect
Raman scattering
Spontaneous emission
Thin films
Original Manuscript: July 11, 2019
Revised Manuscript: September 10, 2019
Manuscript Accepted: October 7, 2019
Hyperbolic metamaterials based on the ordered arrays of metal nanorods in a dielectric media are of great interest owing to new optical effects appearing in such artificial media. Here we study the effects in the polarization state of light passing through a nanocomposite material consisting of Au nanorods in porous alumina and a similar structure supplemented by a nanolayer of ferromagnetic nickel. We demonstrate that close to the epsilon-near-zero dispersion point, under the transition to the hyperbolic dispersion region, the nanocomposites reveal anomalously high modulation of the polarization state of light, which appears as polarization plane rotation and ellipticity changes of probing radiation with a zero ellipticity. This effect is applied for the giant enhancement of the Faraday effect in a continuous ferromagnetic film staying in contact with hyperbolic material. These findings open a path for the design of polarization state control by using hyperbolic metamaterials.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Magneto-optical effects in hyperbolic metamaterials
I. A. Kolmychek, A. R. Pomozov, A. P. Leontiev, K. S. Napolskii, and T. V. Murzina
Optical characterization of epsilon-near-zero, epsilon-near-pole, and hyperbolic response in nanowire metamaterials
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Optica 3(3) 339-346 (2016)
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New J. Phys. (1)
Phys. Rev. B (2)
Fig. 1. (a) Scheme of the "HMM$+$Ni film" sample and the experimental setup for the Faraday effect measurements; (b) transmission spectra of the sample for different angles of incidence; (c) calculated spectra of the effective components Re$\epsilon _{\perp }$ (black line), Re$\epsilon _{\parallel }$ (red line). Shaded areas in (b), (c) correspond to the hyperbolic dispersion spectral regions.
Fig. 2. The data obtained from the ellipsometry measurements: (a) the complex Jones matrix element T$_1$ of the transmitted light for $\theta =$30$^{\circ }$; (b), (c), (d) the wavelength dependencies of the characteristics of the polarization state of the transmitted beam calculated using the ellipsometry data for $\phi =10^{\circ }$. Angles of incidence are indicated on the panels.
Fig. 3. (a) Angular-wavelength spectrum of $\rho$ in Faraday geometry for the p-polarized fundamental beam, the values of the $\rho$ in percents correspond to the colour scale; (b) cross-sections of the angular-wavelength spectrum for $\theta = 0,30$, and $45^{\circ }$; (c) Faraday rotation spectrum of the Ni film (black curve, right axis) and corresponding rotation of the axis of the polarization ellipse $\alpha$ in the "HMM$+$Ni film" (red curve, left axis), $\theta =45^{\circ }$.
Fig. 4. Calculated angular-wavelength spectra of the (a) $Re(q_z)/(\omega /c)$ and (b) $Im(q_z)/(\omega /c)$ values for TE and TM modes in HMM; bottom map in (a) is $Re(\Delta q_z)/(\omega /c)$. | CommonCrawl |
The global dynamical complexity of the human brain network
Xerxes D. Arsiwalla ORCID: orcid.org/0000-0003-1485-18531 &
Paul F. M. J. Verschure1,2
How much information do large brain networks integrate as a whole over the sum of their parts? Can the dynamical complexity of such networks be globally quantified in an information-theoretic way and be meaningfully coupled to brain function? Recently, measures of dynamical complexity such as integrated information have been proposed. However, problems related to the normalization and Bell number of partitions associated to these measures make these approaches computationally infeasible for large-scale brain networks. Our goal in this work is to address this problem. Our formulation of network integrated information is based on the Kullback-Leibler divergence between the multivariate distribution on the set of network states versus the corresponding factorized distribution over its parts. We find that implementing the maximum information partition optimizes computations. These methods are well-suited for large networks with linear stochastic dynamics. We compute the integrated information for both, the system's attractor states, as well as non-stationary dynamical states of the network. We then apply this formalism to brain networks to compute the integrated information for the human brain's connectome. Compared to a randomly re-wired network, we find that the specific topology of the brain generates greater information complexity.
From a computational neuroscience perspective, the brain is oftentimes abstracted as a complex information processing network, that integrates sensory inputs from multiple modalities in order to generate action and cognition. In this paper, we ask a much simpler question: viewing the brain as a dynamical network of neural masses, how can one compute the information integrated by such networks in the course of dynamical transitions from one state to another? A possible approach, among others, is to look at information-theoretic complexity measures that seek to quantify information generated by all causal sub-processes in such a network. One candidate measure for global dynamical complexity is integrated information, usually denoted as Φ. It was first introduced in neuroscience as a complexity measure for neural networks, and by extension, as a possible correlate of consciousness itself (Tononi et al. 1994). It is defined as the quantity of information generated by a network as a whole, due to its causal dynamical interactions, and one that is over and above the information generated independently by the disjoint sum of its parts. As a complexity measure, Φ seeks to operationalize the intuition that complexity arises from simultaneous integration and differentiation of the network's structural and dynamical properties. As such, the interplay of integration and differentiation in a network's dynamics is hypothesized to generate information that is highly diversified yet integrated, thereby creating patterns of high complexity. The aim of this paper is to develop mathematical tools for computing integrated information (analytically when possible, otherwise numerically) for large networks. We then apply this framework to the large-scale structural connectivity network of the human brain.
Let us begin with a brief review of the rich history of this field. The earliest proposals defining integrated information were made in the pioneering work of (Tononi 2004; Tononi and Sporns 2003; Tononi et al. 1994). Since then, considerable progress has been made towards development of a normative theory as well as applications of integrated information (Arsiwalla and Verschure 2013; 2016a; Balduzzi and Tononi 2008; 2009; Barrett and Seth 2011; Krohn and Ostwald 2016; Mediano et al. 2016; Oizumi et al. 2014; Tononi 2012). Similar information-based approaches have also been successfully applied to many-body problems in other domains, such as, for the problem of estimating microstates of statistical mechanical ensembles (Arsiwalla 2009). In fact, there are now several candidate measures of integrated information such as neural complexity (Tononi et al. 1994), causal density (Seth 2005), Φ from integrated information theory: IIT 1.0, 2.0 & 3.0 (Balduzzi and Tononi 2008; Oizumi et al. 2014; Tononi 2004), stochastic interaction (Ay 2015; Wennekers and Ay 2005), empirical Φ (Barrett and Seth 2011) and synergistic Φ (Griffith 2014; Griffith and Koch 2014), plus several variations of these (see Tegmark (2016) for an overview). Table 1 summarizes these measures along with corresponding information metrics upon which they have been based.
Table 1 Candidate measures of integrated information shown alongside information metrics used in their respective formulations
Many of the above measures have been useful in different domains of validity. However, applications to realistic data and in particular to large-scale networks have proven computationally challenging. With a focus on developing computational tools, we discuss three of the above measures in more detail. The measure of (Balduzzi and Tononi 2008) has been quite useful for discrete-state, deterministic, Markovian systems with the maximum entropy distribution. On the other hand, the measure of (Barrett and Seth 2011) has been applied to continuous-state, stochastic, non-Markovian systems and in principle, admits dynamics with any empirical distribution (although in practice, it is easier to use assuming Gaussian distributions). The formulation in (Barrett and Seth 2011) is based on mutual information, whereas (Balduzzi and Tononi 2008) uses a measure based on the Kullback-Leibler divergence. Note however, that in some cases the measure of (Barrett and Seth 2011) can take negative values and that complicates its interpretation. The Kullback-Leibler based definition computes the information generated during state transitions and as we shall see remains positive in the regime of stable dynamics. This makes it easier to interpret as an integrated information measure. Both measures (Balduzzi and Tononi 2008; Barrett and Seth 2011) make use of a normalization scheme in their formulations. Normalization inadvertently introduces ambiguities in computations. The normalization is actually used for the purpose of determining the partition of the network that minimizes the integrated information, but a normalization dependent choice of partition ends up influencing the value and interpretation of Φ. An alternate measure based on the Earth Mover's distance was proposed in (Oizumi et al. 2014). This does away with the normalization problem (though the current version is not formulated for continuous-state variables). However, the formulation of (Oizumi et al. 2014) lies outside the scope of standard information theory and is still very difficult for performing computations on large networks.
A comment on network partitions is relevant at this point. The three measures of Φ discussed above, make use of what is called the minimum information partition or minimum information bi-partition (MIP/MIB). The issue with this partitioning is that it leads to a combinatorial explosion in the number of configurations to be evaluated when working with networks having a large number of nodes. As a result, application of the above measures to compute integrated information of very large networks remains challenging, particularly for the scale of networks obtained from neuroimaging data. On the other hand, in earlier work (Arsiwalla and Verschure 2013), we have introduced a formulation of integrated information that overcomes both, the normalization and combinatorial problem by using a different partitioning of the network called the maximum information partition (MaxIP), which opens the prospect of large-scale applications. However, the formulation in (Arsiwalla and Verschure 2013) was only applicable for uncorrelated node dynamics, which may not be realistic enough for many biological systems.
In this paper, we seek to go beyond (Arsiwalla and Verschure 2013; 2016a; 2016b), starting with an extension of the formalism to include node correlations and also non-stationarity. In order to do that, we solve the discrete-time Lyapunov equation, the solution of which, is then used to get fully analytic expressions for Φ with network correlations. We consider networks with linear stochastic dynamics, which generate multivariate time-series signals. Furthermore, our networks are plastic, in the sense that connection weights are scalable using a global coupling parameter. We compute Φ as a function of this coupling. We also extend our framework to include non-stationary dynamics. This gives us Φ as a function of time, computed through the temporal evolution of the system. The stationary solution yields Φ at the fixed-point attractor, whereas the non-stationary solution leads to Φ elsewhere in the phase space of the system.
As proof of principle, we apply our formulation to the structural connectivity network of white matter fiber tracts in the human cerebral cortex, obtained from diffusion spectrum imaging (Hagmann et al. 2008; Honey et al. 2009). This network has 998 nodes, representing neuronal populations. The edges are weighted fiber counts between populations. Implementing stochastic Gaussian dynamics on this network, we determine stationary solutions to the dynamical system from which we compute the information integrated in bits. To contrast with a null-model, we randomly re-wire the original network and repeat the computation. The original network scores higher on integrated information for all allowed couplings in the stationary as well as non-stationary regime.
Stochastic integrated information
Mathematical formulation
We consider networks with linear stochastic dynamics. The state of each node is given by a random variable pertaining to a given probability distribution. These variables may either be discrete-valued or continuous. However, for many biological applications, Gaussian distributed, continuous-valued state variables are fairly reasonable abstractions (for example, aggregate neural population firing rate, EEG or fMRI signals). The state of the network X t at time t is taken as a multivariate Gaussian variable with distribution \(\phantom {\dot {i}\!}\mathbf {P}_{\mathbf {X}_{\mathbf {t}}} (\mathbf {x}_{\mathbf {t}}) \). x t denotes an instantiation of X t with components \({{x_{t}^{i}}}\) (i going from 1 to n, n being the number of nodes). When the network makes a transition from an initial state X 0 to a state X 1 at time t=1, observing the final state generates information about the system's initial state. The information generated equals the reduction in uncertainty regarding the initial state X 0 . This is given by the conditional entropy H(X 0 |X 1 ). In order to extract that part of the information generated by the system as a whole, over and above that generated individually by its parts, one computes the relative conditional entropy given by the Kullback-Leibler divergence of the conditional distribution \(\mathbf {P}_{\mathbf {X}_{\mathbf {0}} | \mathbf {X}_{\mathbf {1}} = \mathbf {x}^{\prime }} (x) \) of the system with respect to the joint conditional distributions \(\prod _{k=1}^{r} \mathbf {P}_{\mathbf {M}^{\mathbf {k}}_{\mathbf {0}} | {\mathbf {M}^{\mathbf {k}}_{\mathbf {1}} = \mathbf {m}^{\prime }}} \) of its non-overlapping sub-systems demarcated with respect to a partition \({\mathcal {P}}_{r}\) of the system into r distinct sub-systems. Denoting this as \({\Phi _{{\mathcal {P}}_{r}}}\), we have
$$\begin{array}{@{}rcl@{}} {\Phi_{\mathcal{P}_{r}}} \left(\mathbf{X}_{\mathbf{0}} \rightarrow \mathbf{X}_{\mathbf{1}} = \mathbf{x}^{\prime}\right) = \, D_{KL} \left({\mathbf{P}_{{\mathbf{X}}_{\mathbf{0}} | \mathbf{X}_{\mathbf{1}} = \mathbf{x}^{\prime}}} \left|{\vphantom{\mathbf{P}_{{\mathbf{X}}_{\mathbf{0}}}}}\right| \prod\limits_{k=1}^{r} {\mathbf{P}_{{\mathbf{M}^{\mathbf{k}}_{\mathbf{0}}} | {\mathbf{M}^{\mathbf{k}}_{\mathbf{1}}} = \mathbf{m}^{\prime}}} \right) \end{array} $$
where for an r partitioned system, the state variable X 0 can be decomposed as a direct sum of state variables of the sub-systems
$$\begin{array}{@{}rcl@{}} {\mathbf{X}_{\mathbf{0}} = {\mathbf{M}_{\mathbf{0}}^{\mathbf{1}}} \oplus {\mathbf{M}_{\mathbf{0}}^{\mathbf{2}}} \oplus \cdots \oplus {\mathbf{M}_{\mathbf{0}}^{\mathbf{r}}} = \bigoplus_{\mathbf{k} = \mathbf{1}}^{\mathbf{r}} {\mathbf{M}_{\mathbf{0}}^{\mathbf{k}}} } \end{array} $$
and similarly, X 1 decomposes as
For stochastic systems, it is useful to work with a measure that is independent of any specific instantiation of the final state x ′. So we average with respect to final states to obtain an expectation value from Eq. (1). After some algebra, we get
$$ \left< \Phi \right>_{\mathcal{P}_{r}} ({\mathbf{X}_{\mathbf{0}} \rightarrow \mathbf{X}_{\mathbf{1}}}) = - {\mathbf{H} (\mathbf{X}_{\mathbf{0}} | \mathbf{X}_{\mathbf{1}})} + \sum\limits_{k=1}^{r} {\mathbf{H} \left({\mathbf{M}^{\mathbf{k}}_{\mathbf{0}}} | {\mathbf{M}^{\mathbf{k}}_{\mathbf{1}}}\right) } $$
This is our definition of integrated information, which we use in the rest of this paper. Note that the measure described in (Balduzzi and Tononi 2008) is not applicable to networks with stochastic dynamics. They do use Eq. (1) as their definition but endow their nodes with discrete states. On the other hand, (Barrett and Seth 2011) uses a different definition of integrated information, where conditional entropies as in Eq. (4) are replaced by conditional mutual information. This definition only matches the definition of Eq. (1) in special cases but not in general for any distribution. From an information theory perspective, the Kullback-Leibler divergence offers a principled way of comparing probability distributions, hence we follow that approach in formulating our measure in Eq. (4).
The state variable at each time t=0 and t=1 follows a multivariate Gaussian distribution
$$ {\mathbf{X}_{\mathbf{0}} \sim \mathcal{N} \left(\bar{\mathbf{x}}_{\mathbf{0}}, \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}})\right) } \qquad {\mathbf{X}_{\mathbf{1}} \sim \mathcal{N}} \left({\bar{\mathbf{x}}_{\mathbf{1}}, \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{1}})} \right) $$
The generative model for this system is equivalent to a multi-variate auto-regressive process (Barrett et al. 2010)
$$ {\mathbf{X}_{\mathbf{1}} = \mathcal{A} \; \mathbf{X}_{\mathbf{0}} + \mathbf{E}_{\mathbf{1}} } $$
where \(\mathcal {A}\) is the weighted adjacency matrix of the network and E 1 is Gaussian noise. Next, taking the mean and covariance respectively on both sides of this equation, while holding the residual independent of the regression variables, yields
$$\begin{array}{@{}rcl@{}} {\bar{\mathbf{x}}_{\mathbf{1}} = \mathcal{A} \; \bar{\mathbf{x}}_{\mathbf{0}} } \quad \qquad {\boldsymbol{\Sigma}(\mathbf{X}_{\mathbf{1}}) = \mathcal{A} \; \boldsymbol{\Sigma}(\mathbf{X}_{\mathbf{0}}) \; \mathcal{A}^{\mathbf{T}} + \boldsymbol{\Sigma}(\mathbf{E}) } \end{array} $$
In the absence of any external inputs, stationary solutions of a stochastic linear dynamical system as in Eq. (6) are fluctuations about the origin. Therefore, we can shift coordinates to set the means \({\bar {\mathbf {x}}_{\mathbf {0}}}\) and consequently \(\bar {\mathbf {x}}_{\mathbf {1}}\) to the zero. The second equality in Eq. (7) is the discrete-time Lyapunov equation and its solution will give us the covariance matrix of the state variables.
The conditional entropy for a multivariate Gaussian variable was computed in (Barrett and Seth 2011)
$$ {\mathbf{H} (\mathbf{X}_{\mathbf{0}} | \mathbf{X}_{\mathbf{1}})} = \frac{1}{2} n \log (2 \pi e) - \frac{1}{2} \log \left[ \det {\boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}} | \mathbf{X}_{\mathbf{1}})} \right] $$
which is fully specified by the conditional covariance matrix. Inserting this in Eq. (4) yields
$$ \left< \Phi \right>_{\mathcal{P}_{r}} ({\mathbf{X}_{\mathbf{0}} \rightarrow \mathbf{X}_{\mathbf{1}}}) = \frac{1}{2} \log \left[ \frac{\prod_{\mathbf{k} = 1}^{r} \det {\boldsymbol{\Sigma} \left({\mathbf{M}^{\mathbf{k}}_{\mathbf{0}}} | {\mathbf{M}^{\mathbf{k}}_{\mathbf{1}}}\right)} }{\det {\boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}} | \mathbf{X}_{\mathbf{1}})} } \right] $$
Now, in order to compute the conditional covariance matrix we make use of the identity (proof of this identity for the Gaussian case was demonstrated in (Barrett et al. 2010))
$$ {\boldsymbol{\Sigma} (\mathbf{X} | \mathbf{Y}) = \boldsymbol{\Sigma}(\mathbf{X}) - \boldsymbol{\Sigma} (\mathbf{X}, \mathbf{Y}) \boldsymbol{\Sigma} (\mathbf{Y})^{-\mathbf{1}} \boldsymbol{\Sigma} (\mathbf{X}, \mathbf{Y})^{\mathbf{T}} } $$
The appropriate covariance we will need to insert in this expression is
$$ {\boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}}, \mathbf{X}_{\mathbf{1}}) \equiv \left< \left(\mathbf{X}_{\mathbf{0}} - \bar{\mathbf{x}}_{\mathbf{0}} \right) \left(\mathbf{X}_{\mathbf{1}} - \bar{\mathbf{x}}_{\mathbf{1}} \right)^{\mathbf{T}} \right> = \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}}) \, \mathcal{A}^{\mathbf{T}} } $$
which gives for the conditional covariance
$$\begin{array}{@{}rcl@{}} {\boldsymbol{\Sigma} \left(\mathbf{X}_{\mathbf{0}} | \mathbf{X}_{\mathbf{1}}\right) = \boldsymbol{\Sigma}\left(\mathbf{X}_{\mathbf{0}}\right) - \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}}) \, \mathcal{A}^{\mathbf{T}} \, \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{1}})^{-\mathbf{1}} \mathcal{A} \; \Sigma (\mathbf{X}_{\mathbf{0}})^{\mathbf{T}} } \end{array} $$
And similarly for the sub-systems
$$\begin{array}{@{}rcl@{}} {\boldsymbol{\Sigma} \left({\mathbf{M}^{\mathbf{k}}_{\mathbf{0}}} | {\mathbf{M}^{\mathbf{k}}_{\mathbf{1}}}\right)} = {\boldsymbol{\Sigma}\left({\mathbf{M}_{\mathbf{0}}^{\mathbf{k}}}\right)} - {\boldsymbol{\Sigma}\left({\mathbf{M}_{\mathbf{0}}^{\mathbf{k}}}\right) \, {\mathcal{A}^{\mathbf{T}}} \big{|}_{\mathbf{k}} \, { \boldsymbol{\Sigma}\left({\mathbf{M}_{\mathbf{1}}^{\mathbf{k}}}\right)}^{-\mathbf{1}} \mathcal{A} \big{|}_{\mathbf{k}} \, {\boldsymbol{\Sigma} \left({\mathbf{M}_{\mathbf{0}}^{\mathbf{k}}}\right)}^{\mathbf{T}}} \end{array} $$
where k indexes the partition such that \(\mathbf {{M_{0}^{k}}}\) denotes the k th sub-system at t=0 and \( \mathcal {A} \big {|}_{k}\) denotes the restriction of the adjacency matrix to the k th sub-network.
Further, for linear multi-variate systems, a unique fixed point always exists. We try to find stable stationary solutions of the dynamical system. In that regime, the multi-variate probability distribution of states approaches stationarity and the covariance matrix converges, such that
$$\begin{array}{@{}rcl@{}} {\boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{1}}) = \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}})} \end{array} $$
t=0 and t=1 refer to time-points taken after the system converges to the fixed point. Then the discrete-time Lyapunov equations can be solved iteratively for the stable covariance matrix Σ(X t ). For networks with symmetric adjacency matrix and independent Gaussian noise, the solution takes a particularly simple form
$$\begin{array}{@{}rcl@{}} {\boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{t}}) = \left(\mathbf{1} - \mathcal{A}^{\mathbf{2}} \right)^{-\mathbf{1}} \boldsymbol{\Sigma}(\mathbf{E}) } \end{array} $$
and for the parts, we have
$$\begin{array}{@{}rcl@{}} {\boldsymbol{\Sigma}({\mathbf{M}_{\mathbf{0}}^{\mathbf{k}}}) = \boldsymbol{\Sigma} (\mathbf{X}_{\mathbf{0}}) \big{|}_{\mathbf{k}} } \end{array} $$
given by the restriction of the full covariance matrix on the k th sub-network. Note that Eq. (16) is not the same as Eq. (15) on the restricted adjacency matrix as that would mean that the sub-network has been explicitly severed from the rest of the system. Indeed, Eq. (16) is precisely the covariance of the sub-network while it is still part of the network and <Φ> yields the integrated and differentiated information of the whole network that is greater than the sum of these connected parts. Inserting Eqs. (12), (13), (15) and (16) into Eq. (9) yields <Φ> as a function of network weights for symmetric and correlated networks. For the case of asymmetric weights, the entries of the covariance matrix cannot be explicitly expressed as a matrix equation. However, they may still be solved by Jordan decomposition of both sides of the Lyapunov equation.
The maximum information partition
Following (Arsiwalla and Verschure 2013; Edlund et al. 2011), the maximum information partition (MaxIP) is defined as the partition of the system into its irreducible parts. This is the finest partition and is unique as there is only one way to combinatorially reduce a system into all of its sub-units. This partition can directly be found by construction and does not require a normalization scheme for sampling through the space of multi-partitions in order to search for the one that either maximizes or minimizes the integrated information. Consequently, the resulting value of <Φ> computed using the MaxIP is free from normalization dependencies.
Moreover, the MaxIP also helps reduce computational cost. This can be seen as follows. Prescriptions using the MIP/MIB are typically evaluated for a large class of network bi-partitions, whereas the MaxIP is uniquely defined. The number of bi-partitions of a set of n elements is given by the sum of binomial coefficients \(\sum _{p = 1}^{[n/2]} \,^{n}C_{p}\), where n C p =n!/p! (n−p)! with n!=n×(n−1)×⋯×1 and [n/2] denotes the nearest integer less that or equal to n/2. Among all possible bi-partitions, MIP/MIB prescriptions usually restrict to those that divide the system into approximately equal parts. This still leaves us with n C [n/2] configurations for which <Φ> has to be computed. Table 2 summarizes how this number scales with network size from a single node to a million nodes.
Table 2 Scaling of network configurations upon computing Φ using the MIP/MIB versus using the MaxIP for networks with n nodes
Another interesting feature of the MaxIP is that <Φ> computed using this partition in fact accounts for the maximum amount of information that the network can integrate compared to any other bi-, tri- or multi-partition of the system. This is due to the fact that this partition cannot be decomposed further. Every other partition will be coarser than the MaxIP and will therefore have at least some of its parts as composites of the irreducible units in the MaxIP. As these composites integrate more information than its own irreducible units, subtracting the information of a composite (when treating the composite as a part) from the information of the whole system will always produce a smaller <Φ> than that obtained by subtracting the information of each irreducible unit of the network from that of the whole network. Therefore <Φ> computed using the MaxIP is the maximum possible integrated information of the system compared to <Φ> computed using any other partition of the network. In that sense, unlike the MIP or MIB, the MaxIP in fact captures the complete information integrated by the network and is therefore a more natural choice for quantifying whole versus parts.
Analytic solutions for <Φ>
Now that we have a rigorous analytic formulation of integrated information, let us first demonstrate examples of computations performed using artificial networks. In Fig. 1 we consider two artificial networks. For these cases, we want to compute the exact analytic solution for <Φ>. Each of these networks have 8 dimensional adjacency matrices with bi-directional weights (though our analysis does not depend on that and works as well with directed graphs). We want to compute <Φ> as a function of network weights, which we keep as free parameters. However, in order to constrain the space of parameters, we shall set all weights to a single parameter, the global coupling strength g. This gives us <Φ> as an analytic function of g.
Graphs of two artificial networks, (A) and (B)
<Φ> for attractor states
We first compute <Φ> in the stationary regime, that is, when the system has converged to its fixed-point attractor state. The results for the two networks labeled a and b respectively are shown in Eqs.(17), (18) respectively. These are computed for a single time-step, corresponding to the stable stationary solution of the system.
$$\begin{array}{@{}rcl@{}} \left< \Phi \right>_{A} &=& \frac{1}{2} \log \frac{\left(1-43 g^{2} \right)^{8} }{ \left(1-50 g^{2}+49 g^{4} \right)^{8}} \end{array} $$
$$\begin{array}{@{}rcl@{}} \left< \Phi \right>_{B} &=& \frac{1}{2} \log \frac{B_{1} \cdot B_{2} \cdot B_{3} \cdot B_{4} \cdot B_{5}}{\left(-1+g^{2} \right)^{4} \left(1-8 g^{2}+4 g^{4} \right)^{6} \left(1-17 g^{2}+72 g^{4}-64 g^{6}+16 g^{8} \right)^{8}} \end{array} $$
$$\begin{array}{@{}rcl@{}} B_{1} &=& \left(1-15 g^{2}+56 g^{4}-56 g^{6}+16 g^{8} \right) \\ B_{2} &=& \left(1-15 g^{2}+54 g^{4}-54 g^{6}+16 g^{8} \right) \\ B_{3} &=& \left(1-22 g^{2}+159 g^{4}-426 g^{6}+336 g^{8}-80 g^{10} \right)^{2} \\ B_{4} &=& \left(1-21 g^{2}+147 g^{4}-401 g^{6}+374 g^{8}-136 g^{10}+16 g^{12} \right)^{2} \\ B_{5} &=& \left(1-23 g^{2}+183 g^{4}-612 g^{6}+835 g^{8}-526 g^{10}+152 g^{12}-16 g^{14} \right)^{2} \end{array} $$
Note that the mathematical framework described above is in no way limited by the size of the network and thus, in principle, can be applied to networks of any size, to yield exact results. The only practical difficulty would be in the form of available computing hardware resources. Hence, for very large data networks, such as those from brain imaging, numerical computations of <Φ> would be more practical to perform.
<Φ> for non-stationary dynamics
The mathematical formulation developed above can also be used to compute <Φ> at non-stationary points in the solution space of networks with linear stochastic dynamics. We show this explicitly for Network B in Fig. 1. We compute <Φ> for the complete temporal evolution of the system starting from an initial condition at t=0 until the system stabilizes at the fixed point attractor. In the non-stationary case, Eqs. (14), (15) and (16) no longer hold. However, everything up to and including Eq. (13) are valid. Hence, the covariance matrix Σ(X t ) is simply computed recursively following Eq. (7). Subsequently, Σ(X t ) and <Φ> are both computed for each time-point t.
Figure 2 shows the multivariate time-series signals generated by Network B for two different coupling strengths g. The critical value of g for this network is 0.3023, at which the dynamics becomes unstable. For g≤0.3023, the system converges to the fixed-point at the origin. In Fig. 3 we plot temporal profiles of <Φ> for both the above values of g, which shows increasing integrated information for stronger coupled networks.
Simulated time-series data for Network B following the generative model in Eq. (6). The plot on the left shows simulated data corresponding to network coupling strength g=0.3000 and variance of noise σ=1. The plot on the right refers to data from the same network with coupling g=0.3022 and the same noise amplitude. Each plot shows 8 time-series profiles, corresponding to the 8 nodes of the network (note that several of these profiles intersect or overlap with each other, hence in the above plots they appear to be clustered together). The time-series for each node is shown in a different color and the color scheme is the same for the plot on the left and the one on the right. Stability of the system is guaranteed until the critical coupling at g=0.3023. Closer to the critical point, the system takes longer to converge to the fixed-point attractor at x = 0
Temporal profiles of <Φ> for Network B corresponding to the two coupling strengths used in Fig. 2. <Φ> saturates as the system approaches the stable attractor with greater integrated information for dynamics closer to the critical point
Application to brain connectomics
The framework described above, provides us with all the mathematical tools to compute how much information is integrated in bits in a single time-step, by a large network with linear stochastic (Gaussian) dynamics. We apply the above formulation to the whole brain structural connectivity network of the human cerebral cortex, using data published in the seminal work of (Hagmann et al. 2008; Honey et al. 2009). This data is acquired from high-resolution T1-weighted diffusion spectrum imaging (DSI). The data preprocessing pipeline, as described in (Hagmann et al. 2008), involves white and gray matter segmentation from the T1 images, followed by parcellation into 66 anatomical regions and subsequently 998 individual regions of interest (ROIs) based on Talairach coordinates. After that, whole brain tractography is performed to obtain estimates of axonal trajectories across the entire white matter. From this, connection weights between pairs of ROIs are determined, resulting in a weighted network of structural connectivity across the entire brain. We have displayed the data in matrix form as a 998 dimensional matrix on the left-hand side of Fig. 4. The 998 voxels (ROIs) represent nodes of the network. Each node is physically a population of neurons. The edges are weighted fiber counts between populations. Additionally, we include a global coupling variable g, multiplying the entire matrix, that can be used to tune the overall strength of the weights.
Left: Connectivity matrix of human cerebral white matter. Right: Randomized version of the same matrix, preserving network weights. The data consists of white matter fiber tracts from 998 cortical voxels. The connectivity matrix on the left is a weighted matrix with the color-bar (in the middle) indicating connection strengths. The randomized matrix on the right is obtained by randomly shuffling positions of weights from the connectivity matrix
To simulate brain dynamics, one may chose from among a variety of possible models, discussed in (Arsiwalla et al. 2013, 2015a,b; Galán 2008). To run these simulations, one may use customizable tools such as those described in (Betella et al. 2014a,b; 2013; Omedas et al. 2014). The simplest model among the ones mentioned above is the linear stochastic Wilson-Cowen model. In fact, it can be seen from (Galán 2008) that Eq. (6) is precisely a special case of the discrete-time limit of the linear stochastic Wilson-Cowen model. That is what we use here. The brain's state of spontaneous activity or resting-state is usually identified as the attractor state of these models. This corresponds to finding stable stationary solutions of the system. This is precisely the regime in which we compute <Φ> in bits as a function of the coupling g. The results are shown in the red profile in Fig. 5. Further, in order to contrast this result with a null model, we also rewired the edges of the connectome network randomly, while preserving the magnitude of the weights. This generates the randomized data matrix shown on the right-hand side of Fig. 4. We also compute <Φ> for this matrix. The resulting profile is the blue curve in Fig. 5. For extremely small couplings, the two networks are indistinguishable on <Φ> scores, however, as g grows, the architecture of the brain's network turns out to perform better at integrating information than its randomized counterpart.
<Φ> as a function of global coupling strength g. <Φ> for the data (shown as red points) and for the randomized network (shown as blue points). Stationary solutions exist up to g = 1.49, the critical point of the data network
While Fig. 5 depicts the fixed-point behavior of <Φ> as a function of g, in Fig. 6 we show the full time-course of <Φ> for both the connectome network as well as its randomized counterpart at a specific value of the coupling g=1.488. The non-stationary behavior is computed using linearized dynamics as discussed above. Asymptotic values of <Φ> in Fig. 6 converge to those in Fig. 5 at g=1.488. Once again we find that the connectomic network completely dominates its randomized counterpart in the quantity of information it integrates and this difference only gets more pronounced upon approaching the attractor state. Note that for a more thorough comparison, one might also want to check the above against an entire distribution of random networks. However, the main point of this paper is to demonstrate a systematic computation of how much information a realistic large network integrates. Functionally, what this corresponds to in terms of brain function or disease is an interesting question by itself. A possible approach towards addressing those issues would be to perform computations as the one demonstrated above for a large repertoire of neuroimaging studies ranging from task-based paradigms to disease states and use that to calibrate brain functional states on a scale of information complexity. Another question on which there is still no consensus concerns consciousness (Arsiwalla et al. 2016a). While it is generally agreed that information integration is a necessary component of phenomenological consciousness, by itself, it may not be sufficient (Arsiwalla et al. 2016b, Verschure 2016).
A comparison of temporal profiles of <Φ> for the brain connectome network versus its randomized counterpart, both computed at a fixed coupling g=1.488. The asymptotes of these profiles match the stationary values of <Φ> in Fig. 5 for the given coupling
In this paper, we have demonstrated a computational framework, built on a rich body of earlier work on information-theoretic complexity measures and applied it to compute the integrated information of large networks endowed with linear stochastic dynamics. Integrated information is interesting as a global measure of a system's dynamical complexity. Whereas local complexity measures such as Granger causality, transfer entropy or synergistic mutual information have been very successful at quantifying local information processes of complex systems (Wibral et al. 2014), global measures such as integrated information serve to complement local measures and give insights on the system's collective behavior.
Earlier attempts to compute integrated information have been limited to relatively small networks. This was mainly due to normalization ambiguities and explosive combinatorics associated with bi-partitions used therein. Instead, what we find is that the finest partitioning of the system solves all these problems and opens the window of applicability to large-scale networks. In particular, we apply our formulation to the human brain connectome network. This network is constructed from white matter tractography data from the human cerebral cortex and consists of 998 nodes with about 28,000 symmetric and weighted connections between them (Hagmann et al. 2008; Honey et al. 2009). Using a discrete-time linear stochastic neuronal population model to generate the dynamics of neural activity on this network, we compute the integrated information of this dynamical system during state transitions for both, stationary as well as non-stationary dynamics. For the linearized system, the stationary solution corresponds to the network's resting state attractor. The computed integrated information depends on both, the structural anatomy as well as the network's dynamical operating point, that is, the value of the global coupling g.
We see potentially useful applications of our information-based measures for other types of physiological data as well, for example, tracing studies or detailed microscopic connectivity data. As for neuroimaging studies, information-based methods offer a useful way to quantify complexity of brain functions. The clinical utility of our measure would be in identifying information-based differences between healthy subjects and patients of neurodegenerative diseases. Just as we identified a transitionary phase after which an anatomical network strongly differs in information integration and differentiation from a randomly rewired network, similar comparative analysis for patients compared to healthy controls might provide a quantification of the extent of the disorder and even provide an analytic way to suggest diagnostic surgical rewiring to restore network processing.
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This work has been supported by the European Research Council's CDAC project: "The Role of Consciousness in Adaptive Behavior: A Combined Empirical, Computational and Robot based Approach" (ERC-2013- ADG 341196).
All authors contributed to this work. Both authors read and approved the final manuscript.
The authors declare that they have no competing interests. Both authors read and approved the final manuscript.
Synthetic Perceptive Emotive and Cognitive Systems (SPECS) Lab, Center of Autonomous Systems and Neurorobotics, Universitat Pompeu Fabra, Barcelona, Spain
Xerxes D. Arsiwalla
& Paul F. M. J. Verschure
Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
Paul F. M. J. Verschure
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Correspondence to Xerxes D. Arsiwalla.
Arsiwalla, X.D., Verschure, P.F. The global dynamical complexity of the human brain network. Appl Netw Sci 1, 16 (2016). https://doi.org/10.1007/s41109-016-0018-8
Received: 16 July 2016
Brain networks
Neural dynamics
Complexity measures | CommonCrawl |
\begin{document}
\title{A unified approach to embeddings of a line in 3-space}
\begin{abstract} While the general question of whether every closed embedding of an affine line in affine \(3\)-space can be rectified remains open, there have been several partial results proved by several different means. We provide a new approach, namely constructing (strongly) residual coordinates, that allows us to give new proofs of all known partial results, and in particular generalize the results of Bhatwadekar-Roy and Kuroda on embeddings of the form \((t^n,t^m,t^l+t)\). \end{abstract}
\section{Introduction}
Let \(k\) be an algebraically closed field of characteristic zero throughout. The embedding problem, one of the central problems in affine algebraic geometry, asks whether every closed embedding \(\mathbb{A}^m \hookrightarrow \mathbb{A}^n\) is equivalent to the standard embedding (such embeddings are called {\em rectifiable}). In general, this is a very difficult problem that has led to several fruitful avenues of research. If \(n \geq 2m+2\), then every embedding is rectifiable due to a general result of Srinivas \cite{Srinivas}. Two remaining cases have attracted the most attention: the case where \(n=m+1\), and the case where \(m=1\). In the former case, it is often referred to as the Abhyankar-Sathaye embedding conjecture, and can be reformulated as asking whether every hyperplane in \(\mathbb{A}^n\) is a coordinate; as this is not the focus of this paper, we simply suggest the papers \cite{Sathaye,RS,WrightCancel,Kaliman} to the interested reader, and note that it remains open for \(n \geq 3\).
In this paper, we are interested in the \(m=1\) case, i.e. embeddings \(\mathbb{A} \hookrightarrow \mathbb{A}^n\). Note that the result of Srinivas \cite{Srinivas} provides an affirmative answer in the case \(n \geq 4\). In the case \(n=2\), an affirmative answer was provided by Abyhankar and Moh \cite{AM}, and independently, Suzuki \cite{Suzuki}. Thus, in the \(m=1\) case, the only remaining open case is embeddings \(\mathbb{A} \hookrightarrow \mathbb{A}^3\), so we direct our attention there.
Abhyankar \cite{Abhyankar} conjectured (among other things) that the family of embeddings \((t^n, t^{n+1}, t^{n+2}+t)\) are not rectifiable. Craighero \cite{Craighero1, Craighero2} showed for \(n=3\) and \(n=4\), these embeddings are, in fact, rectifiable; but it remains an open question for \(n \geq 5\). Later, Bhatwadekar and Roy \cite{BR} considered the more general family of embeddings \((t^{n}, t^m, t^l+t)\), and showed that many of these are rectifiable. More recently, Kuroda \cite{Kuroda} showed that another subset of this family are rectifiable. These results are stated precisely in Section \ref{sec:unified} below.
All of these papers use different approaches. In \cite{Craighero1}, Craighero explicitly writes down the rectifying automorphism, while in \cite{Craighero2} he explicitly computes a polynomial, and as this polynomial is linear in a variable, appeals to a well-known result of Sathaye \cite{Sathaye} to see it can be extended to a rectifying automorphism. Bhatwadekar and Roy \cite{BR} make use of the Dedekind conductor, while Kuroda \cite{Kuroda} constructs rectifying automorphisms from exponentials of locally nilpotent derivations similar to the Nagata map.
In this paper, we take a single approach that we apply to recover all of these results and extend some of them. To broadly explain our approach, first note that an embedding \(\mathbb{A} \hookrightarrow \mathbb{A}^3\) corresponds to a surjection of polynomial rings \(k[x,y,z] \rightarrow k[t]\). It is well-known that an embedding is rectifiable if and only if there is a coordinate of \(k[x,y,z]\) that maps to \(t\). Our approach is to construct (strongly) residual coordinates that map to \(t\), and then use a criterion of \cite{StronglyResidual} to see that they are coordinates (see Corollary \ref{cor:criterion} below).
In the next section, we review some standard definitions and results from the field, and then describe our general approach. Then in Section \ref{sec:unified}, we precisely state and give new proofs of the results of \cite{Craighero1, Craighero2, BR, Kuroda}; in particular, we also prove generalizations of the results of Bhatwadekar and Roy \cite{BR} and Kuroda \cite{Kuroda} in Theorems \ref{thm:BRgeneral} and \ref{thm:Kurodageneral}, respectively.
\section{Preliminaries}\label{sec:preliminaries}
Let us begin by quickly recalling some fairly standard definitions; more detailed explanations can be found in \cite{Arno} or \cite{Wright}. \begin{itemize} \item We use \({\rm GA}_n(k)\) to denote the general automorphism group \(\Aut _k \mathbb{A} ^n\). This group is anti-isomorphic to the automorphism group of the polynomial ring \(k^{[n]}\). \item \({\rm EA}_n(k)\) denotes the subgroup of \({\rm GA}_n(k)\) generated by elementary automorphisms, i.e. those fixing \(n-1\) variables. \item A polynomial \(f \in k^{[n]}\) is called a {\em coordinate} (or {\em variable} by some authors) if there exist \(f_2,\ldots,f_n\) such that \((f,f_2,\ldots,f_n) \in {\rm GA}_n(k)\). \item An embedding \(\phi:\mathbb{A} \hookrightarrow \mathbb{A}^3\) is called {\em rectifiable} if there exists \(\theta \in {\rm GA}_3(k)\) such that \(\theta \phi = (t,0,0)\). \item For any morphism \(\phi:\mathbb{A}^m \rightarrow \mathbb{A}^n\), we use \(\phi ^*\) to denote the corresponding map \(\phi^*:k^{[n]} \rightarrow k^{[m]}\). \end{itemize}
The following is a well-known algebraic characterization of rectifiable embeddings. \begin{lemma}\label{lem:coordinate} Let \(\phi: \mathbb{A} \hookrightarrow \mathbb{A}^n\) be an embedding. If \(f \in k^{[n]}\) is a coordinate with \(\phi^*(f)=t\), then \(\phi\) is rectifiable. \end{lemma} \begin{proof} Let \(\theta=(f,f_2,\ldots,f_n) \in {\rm GA}_n(k)\). Since \(\phi^*(f)=t\), we have \(\theta \phi=(t,g_2(t),\ldots,g_n(t))\) for some polynomials \(g_2, \ldots, g_n \in k[t]\). Let \(\psi = (x_1,x_2-g_2(x_1),\ldots,x_n-g_n(x_1)) \in {\rm GA}_n(k)\). Then \(\psi \theta \phi = (t,0,\ldots,0)\), so \(\phi\) is rectifiable. \end{proof}
This lemma underlies our approach in this paper: given an embedding, we aim to construct a coordinate that the embedding maps to \(t\). To construct coordinates, we rely up on the idea of (strongly) residual coordinates.
\subsection{Strongly residual coordinates and associated embeddings}
\begin{definition}A polynomial \(f \in k[x]^{[n]}\) is called a {\em residual coordinate} if it becomes a coordinate modulo \(x-c\) for each \(c \in k\). A polynomial is called a {\em strongly residual coordinate} if it becomes a coordinate modulo \(x\), and also a coordinate over \(k[x,x^{-1}]\). \end{definition}
Note that strongly residual coordinates are residual coordinates. A special case of the Dolgachev-Weisfeiler conjecture asserts that all residual coordinates are coordinates. This remains open for \(n \geq 3\), with the V\'en\'ereau polynomial \(y+x(xz-y(yu+z^2))\) being a famous example of a (strongly) residual coordinate whose status as a coordinate remains unresolved\footnotemark.
\footnotetext{In fact, the V\'en\'ereau polynomial is also a hyperplane, meaning it is a potential counterexample to the Abhyankar-Sathaye conjecture as well. We refer the reader to \cite{KVZ,Vtype,bivariables} for further reading on the V\'en\'ereau polynomial.}
To illustrate our methods, let us consider the well known construction of the Nagata automorphism. Let \(\alpha = (x,y,z-\frac{y^2}{x}) \in {\rm GA}_2(k[x,x^{-1}])\), and set \(\beta = (x,y+x^2z,z) \in {\rm GA}_2(k[x])\). It is straightforward to compute that \[\alpha ^{-1} \beta \alpha = (x,y+x(xz-y^2), z+2y(xz-y^2)+x(xz-y^2)^2).\] This resulting element of \({\rm GA}_2(k[x])\) is known as the Nagata automorphism; after many years, it was famously shown \cite{SU} to not be generated by elementary and affine elements of \({\rm GA}_3(k)\). Since it is a conjugation of an elementary automorphism (over \(k[x,x^{-1}]\)), it can also be obtained as an exponential of a locally nilpotent derivation, namely \((xz-y^2)\left(x\frac{\partial}{\partial y}+2y\frac{\partial}{\partial z}\right)\); Kuroda exploits this fact in \cite{Kuroda}. However, if we instead consider \(\alpha _0 = (x,y,z-\frac{y^2}{x^2})\) and \(\beta _0 = (x,y+x^3z,z)\), then \(\alpha _0 ^{-1} \beta_0 \alpha _0 \notin {\rm GA}_2(k[x])\); however, letting \(\gamma = (x,y,z+\frac{2y^3}{x})\), then a straightforward computation verifies that \(\gamma \alpha _0 ^{-1} \beta _0 \alpha _0 \in {\rm GA}_2(k[x])\). Note that since we are no longer simply conjugating, it is not an exponential of a locally nilpotent derivation.
We now generalize this approach; the following is a special case of Theorem 13 of \cite{StronglyResidual}, but we also present a short direct proof here. \begin{theorem}\label{thm:cleardenominators} Let \(\alpha, \beta \in {\rm EA}_2(k[x,x^{-1}])\) be of the form \(\alpha =(x,y,z+x^{-l}P(y))\) for some \(P \in k[x][y]\) and \(\beta = (x,y+x^a(x^lz)^b,z)\) for some \(a,b \in \mathbb{N}\). Then there exists \(R \in k[x][y]\) such that setting \(\gamma = (x,y,z+x^{-l}R(y))\), then \(\gamma \beta \alpha \in {\rm GA}_2(k[x])\). Moreover, if \(P(y) \in (y^c)\) for some \(c \in \mathbb{N}\), then \(R(y)\in (y^c)\). \end{theorem}
\begin{proof} We begin by proving the following: \begin{claim} Let \(F \in k[x]^{[2]}\) and \(G \in k[x]^{[1]}\), and suppose \[\phi = (x,y+x^a(x^lz+P(y))^b,z+x^azF(x^lz,y)+x^{-l}G(y)) \in {\rm GA}_2(k[x,x^{-1}]).\] Then setting \(\theta = (x,y,z-x^{-l}G(y)) \in {\rm EA}_2(k[x,x^{-1}])\), there exist \(\tilde{F} \in k[x]^{[2]}\) and \(\tilde{G} \in k[x]^{[1]}\) such that \[\theta \phi = (x,y+x^a(x^lz+P(y))^b,z+x^az\tilde{F}(x^lz,y)+x^{-l+a}\tilde{G}(y)) \in {\rm GA}_2(k[x,x^{-1}]).\] Moreover, \(\tilde{G}(y) \in (y^c)k[x,y]\) as well. \end{claim} \begin{proof} This follows from direct computation. Note that \(\theta\) fixes \(x\) and \(y\), so we need only compute \[(\theta \phi)^*(z) = z+x^azF(x^lz,y)+x^{-l}\left(G(y) - G(y+x^a(x^lz+P(y))^b)\right).\] The key observation is to apply Taylor's formula to obtain
\[G(y)-G(y+x^a(x^lz+P(y))^b) = -\sum _{i=1} \frac{1}{i!}G^{(i)}(y)(x^a(x^lz+P(y))^b)^i.\] Applying a binomial expansion to \((x^lz+P(y))^{bi}\), we see that there exists \(\tilde{F}_0 \in k[x]^{[2]}\) such that \[G(y)-G(y+x^a(x^lz+P(y))^b) = -\sum _{i=1} \frac{1}{i!}G^{(i)}(y)x^{ai}(P(y))^{bi} + x^{l+a}z\tilde{F}_0(x^lz,y). \] Then we observe that setting \begin{align*} \tilde{G}(y)=-\sum _{i=1} \frac{1}{i!}G^{(i)}(y)x^{a(i-1)}(P(y))^{bi} &&\text{and}&& \tilde{F}(x^lz,y)=F(x^lz,y)+\tilde{F}_0(x^lz,y),\end{align*} we obtain the desired result.
\end{proof}
Now, beginning with \(\beta \alpha\) and repeatedly applying the claim, we may inductively produce a sequence \(\theta _0, \ldots, \theta _r\) with \(\theta _i = (x,y,z-x^{-l+ai}G_i(y))\) (with each \(G_i(y) \in (y^c)\)) such that \[\theta _r \cdots \theta _0 \beta \alpha = (x,y+x^a(x^lz+P(y))^b,z+x^az\tilde{F_r}(x^lz,y)+x^{-l+a(r+1)}\tilde{G_r}(y))\] for some \(\tilde{F_r}(x^lz,y), \tilde{G_r}(x^lz,y) \in k[x][x^lz,y]\). Let \(r\) be minimal so that \(a(r+1) \geq l\), and set \(\gamma = \theta _r \cdots \theta _0\). Then letting \(R(y) = -\sum _{i=0} ^r x^{ai}G_i(y) \in (y^c)\), we see \(\gamma = (x,y,z+x^{-l}R(y))\) with \(\gamma \beta \alpha \in {\rm GA}_2(k[x,x^{-1}])\) and \(\gamma \beta \alpha \in (k[x,y,z])^3\). It is well known (e.g. Proposition 1.1.7 in \cite{Arno}) that this implies that \(\gamma \beta \alpha \in {\rm GA}_2(k[x])\) as required. \end{proof}
\begin{corollary}\label{cor:criterion} Let \(\phi: \mathbb{A} \hookrightarrow \mathbb{A}^3\) be an embedding. Suppose \(\alpha, \beta \in {\rm EA}_2(k[x,x^{-1}])\) are of the form \(\alpha =(x,y,z+x^{-l}P(y))\) and \(\beta = (x,y+xQ(x^lz),z)\) for some \(P,Q \in k[x]^{[1]}\), and that \(\phi^* \alpha^* \beta^*(y)=t\). Then \(\phi\) is rectifiable. \end{corollary} \begin{proof} Apply the previous theorem to produce \(\gamma = (x,y,z+x^{-l}R(y))\) such that \(\gamma \beta \alpha \in {\rm GA}_2(k[x])\). Let \(f=(\gamma \beta \alpha)^*(y)\), so \(f\) is a coordinate. Since \(\gamma^*(y)=y\), we have \(t=\phi^*\alpha^*\beta^*(y) = \phi^*\alpha^*\beta^*\gamma^*(y)=\phi^*(\gamma \beta \alpha)^*(y)=\phi^*(f)\). Lemma \ref{lem:coordinate} then completes the proof. \end{proof}
\begin{remark} An elementary example of how this corollary can be used can be found in Theorem \ref{thm:Craighero1}. \end{remark} \subsection{A useful combinatorial lemma} Here we prove a combinatorial lemma that we use in the proof of Lemma \ref{lem:cd} below; the reader may prefer to skip the proof for now, and proceed to Section \ref{sec:unified}. This is used only in the proof of Lemma \ref{lem:cd} below. For an integer \(m\), let us write \(\delta _m = \begin{cases} 1 & m \text{ is even} \\ 0 & m \text{ is odd}\end{cases}\). \begin{lemma}\label{lem:combinatorial} Let \(m \in \mathbb{N}\). Then there exist \(\alpha _0, \ldots, \alpha _\floor{\frac{m}{2}}, \beta \in k\) such that, in \(k[s]\), \[ \sum _{i=0} ^\floor{ \frac{m}{2}} \alpha _i s^i(1+s)^{m-2i} = 1+ \beta \delta _m s^{\frac{m}{2}}+s^m .\] \end{lemma}
\begin{proof} First, suppose \(m=2k+1\) is odd. We induct on \(k\), with the \(k=0\) case being trivial. For \(k>0\), we first expand \((1+s)^{2k+1}\) noting the symmetry of coefficients:
\begin{equation*} (1+s)^{2k+1} = 1+s^{2k+1} + \sum _{j=1} ^{k} {2k+1 \choose j} (s^j + s^{2k+1-j}) = 1+ s^{2k+1} + \sum _{j=1} ^{k} {2k+1 \choose j}s^j (1 + s^{2k+1-2j}). \end{equation*}
For \(1 \leq j \leq k\), by the induction hypothesis we choose \(\alpha _{j,0},\ldots,\alpha _{k-j}\) such that \[1+s^{2k+1-2j} =\sum _{i=0} ^{k-j} \alpha _{j,i} s^i (1+s)^{2k+1-2j-2i}.\] Then \begin{align*} (1+s)^{2k+1} &=
1+ s^{2k+1} + \sum _{j=1} ^{k} {2k+1 \choose j}s^j \sum _{i=0} ^{k-j} \alpha _{j,i} s^i (1+s)^{2k+1-2j-2i}. \end{align*}
Letting \(\alpha _{0,i} = -{2k+1 \choose i}\) for \(0 \leq i \leq k\), we then have
\begin{align*} 1+s^{2k+1}&= -\sum _{j=0} ^{k} \sum _{i=0} ^{k-j}{2k+1 \choose j} \alpha _{j,i} s^{i+j} (1+s)^{2k+1-2(i+j)} \\ &= \sum _{r=0} ^k s^{r} (1+s)^{2k+1-2r} \left(-\sum _{j=0} ^{r} {2k+1 \choose j} \alpha _{j,r-j}\right) . \end{align*}
Now, suppose \(m=2k\) is even. We again induct on \(k\), with \(k=0\) being trivial. For \(k>0\) we proceed similarly by expanding \((1+s)^{2k+1}\), noting however that there is an odd number of terms now.
\begin{equation*} (1+s)^{2k} = 1+{2k \choose k } s^{k}+s^{2k} + \sum _{j=1} ^{k-1} {2k \choose j} (s^j + s^{2k-j}) = 1+ {2k \choose k} s^{k}+s^{2k} + \sum _{j=1} ^{k-1} {2k \choose j}s^j (1 + s^{2k-2j}) \end{equation*}
For \(1 \leq j \leq k-1\), by the induction hypothesis we choose \(\alpha _{j,0},\ldots,\alpha _{k-j}, \beta _j\) such that \[1+s^{2k-2j} = \beta _j s^{k-j}+\sum _{i=0} ^{k-j} \alpha _{j,i} s^i (1+s)^{2k-2j-2i}.\] Then \begin{align*} (1+s)^{2k} &=
1+ {2k \choose k} s^{k}+s^{2k} + \sum _{j=1} ^{k-1} {2k \choose j}s^j \left( \beta _j s^{k-j}+\sum _{i=0} ^{k-j} \alpha _{j,i} s^i (1+s)^{2k-2j-2i} \right) \\ &=1+ s^{k} \left({2k \choose k}+\sum _{j=1} ^{k-1} {2k \choose j} \beta _j\right)+s^{2k} +\sum _{j=1} ^{k-1} \sum _{i=0} ^{k-j}{2k \choose j} \alpha _{j,i} s^{i+j} (1+s)^{2k-2(i+j)} \end{align*}
Now, set \(\alpha _{0,i} = -{2k \choose i}\) for \(0 \leq i \leq k\), and set \(\beta = {2k \choose k}+\sum _{j=1} ^{k-1} {2k \choose j} \beta _j\). Then
\begin{align*} 1+\beta s^k + s^{2k} &= -\sum _{j=0} ^{k} \sum _{i=0} ^{k-j}{2k \choose j} \alpha _{j,i} s^{i+j} (1+s)^{2k-2(i+j)}\\
&=\sum _{r=0} ^k s^{r} (1+s)^{2k-2r} \left(-\sum _{j=0} ^{r} {2k \choose j} \alpha _{j,r-j}\right) . \end{align*}
\end{proof}
\section{A unified approach to known results on embedding}\label{sec:unified}
In this section we summarize all results known to us on embeddings \(\mathbb{A} \hookrightarrow \mathbb{A}^3\), and show how they can all be proved via Corollary \ref{cor:criterion}. For convenience, we will adopt the notation \(X=\phi^*(x)\), \(Y=\phi^*(y)\), and \(Z=\phi^*(z)\). So for example, when considering the Abhyankar embeddings \((t^n, t^{n+1}, t^{n+2}+t)\), we will write \(X=t^n\), \(Y=t^{n+1}\), and \(Z=t^{n+2}+t\). As an elementary example, consider the \(n=2\) Abhyankar embedding \((t^2, t^3, t^4+t)\). It is straightforward to see that \(Z-X^2=t\), and the polynomial \(z-x^2\) is a coordinate (of the elementary automorphism \((x,y,z-x^2)\)), so by Lemma \ref{lem:coordinate} the embedding \((t^2,t^3,t^4+t)\) is rectifiable.
\subsection{Craighero's results} The \(n=3\) and \(n=4\) cases of Abhyankar's conjecture were resolved by Craighero \cite{Craighero1,Craighero2}; we are able to give short proofs here.
\begin{theorem}[Craighero]\label{thm:Craighero1} The embedding \(\phi = (t^3,t^4,t^5+t)\) is rectifiable. \end{theorem} \begin{proof} We begin by computing \[Y-\frac{Z^2}{X^2}+2 = t^4-\frac{t^{10}+2t^6+t^2}{t^6} +2 =-t^{-4}.\] Let \(\alpha = \left(x,y-\frac{z^2}{x^2}+2,z\right)\) and \(\beta = (x,y,z+x(x^2y))\). Then we compute \begin{align*} \phi^*\alpha^*\beta^*(z) &= \phi^*\alpha^*(z+x(x^2y)) \\ &= \phi^*\left(z+x^3\left(y-\frac{z^2}{x^2}+2\right)\right) \\ &= Z+X^3\left(Y-\frac{Z^2}{X^2}+2\right) \\ &= (t^5+t)+t^{9}(-t^{-4}) \\ &= t. \end{align*} Thus, by Corollary \ref{cor:criterion}, the embedding is rectifiable.
\end{proof}
\begin{theorem}[Craighero]\label{thm:Craighero2} The embedding \(\phi = (t^4,t^5,t^6+t)\) is rectifiable. \end{theorem} \begin{proof} We first observe that \(Z^2-X^3=2t^7+t^2\). Next, letting \(a,b \in k\) be arbitrary for the moment, we compute \begin{align*} Y+a\frac{Z^3}{X^2}+bXZ -3a &= t^5+\frac{a\left(t^{18}+3t^{13}+3t^8+t^3\right)}{t^8}+b(t^{10}+t^5) -3a \\ &= (a+b)t^{10}+(1+3a+b)t^5+at^{-5}. \end{align*} Setting \(a=-\frac{1}{2}\) and \(b=\frac{1}{2}\) causes these first two coefficients to vanish, so we see \[Y-\frac{1}{2}\frac{Z^3}{X^2}+\frac{1}{2}XZ+\frac{3}{2} = -\frac{1}{2}t^{-5}. \]
Thus, \[\left(Y-\frac{1}{2}\frac{Z^3}{X^2}+\frac{1}{2}XZ+\frac{3}{2} \right) +\frac{1}{4}\frac{Z^2-X^3}{X^3}=-\frac{1}{2}t^{-5}+\frac{1}{4}\frac{2t^7+t^2}{t^{12}} =\frac{1}{4}t^{-10}.\]
Now, let \begin{align*} \alpha &= \left(x, y-\frac{1}{2}\frac{z^3}{x^2}+\frac{1}{2}xz+\frac{3}{2} +\frac{1}{4}\frac{z^2-x^3}{x^3},z\right) \\ \beta &= (x,y,z+x(-4x^3y)) \end{align*}
Then we see \begin{align*} \phi^*\alpha^*\beta^*(z) &= Z-4X^4\left(Y-\frac{1}{2}\frac{Z^3}{X^2}+\frac{1}{2}XZ+\frac{3}{2} +\frac{1}{4}\frac{Z^2-X^3}{X^3}\right) \\ &= t^{6}+t-4(t^{16})\left(-\frac{1}{4}t^{-10}\right) \\ &= t. \end{align*} Thus \(\phi\) is rectifiable by Corollary \ref{cor:criterion}.
\end{proof}
\subsection{Generalizing Bhatwadekar-Roy's results} The next results were due to Bhatwadekar and Roy \cite{BR}, who studied embeddings of the form \((t^n, t^{an+1}, t+t^l)\) for positive integers \(a,n\) and \(l>n\). This more general class includes the Abhyankar embeddings mentioned above. Interstingly, in positive characteristic they were able to show that all such embeddings are rectifiable; in the characteristic zero case we are interested in, they obtained two positive results: when \(l\equiv -1 \pmod n\), and when \(n=4\). We first generalize the former in Theorem \ref{thm:BRgeneral}, and then provide a new proof of the latter in Theorem \ref{thm:BR2}.
\begin{theorem}\label{thm:BRgeneral} Let \(m,n \in \mathbb{N}\) be coprime positive integers. Write \(m \equiv c \pmod n\) for some \(0<c<n\), and set \(d=n-c\). Write \(\lambda_1 c=\mu_1 n -1\) and \(\lambda _2 d = \mu _2 n -1\) for \(\lambda _1, \lambda _2 \in \mathbb{N}\) and minimal \(\mu_1, \mu_2 \in \mathbb{N}\). If \(b > \min\{\mu_1, \mu_2\}\), then the embedding \(\phi = (t^n, t^{m}, t+t^{bn-1})\) is rectifiable. \end{theorem}
\begin{remark} When \(c=1\), we obtain one of the results of Bhatwadekar and Roy in \cite{BR}. \end{remark}
\begin{example} Let \(n=5\) and \(m=7\), so \(c=2\), in which case \(\mu_1=1\). Then we see the embedding \((t^5,t^7,t+t^9)\) is rectifiable. \end{example}
More generally, we have \begin{corollary} Let \(n \in \mathbb{N}\) be odd. Then for any \(b>1\), the embedding \((t^n,t^{n+2}, t+t^{bn-1})\) is rectifiable. \end{corollary} \begin{proof} Note that \(c=2\), so since \(n\) is odd, \(n-1\) is even and thus \(\mu_1=1\). \end{proof}
In order to prove Theorem \ref{thm:BRgeneral}, we first prove the following lemma. \begin{lemma}\label{lem:cd} Suppose \(\phi=(t^n, t^m, t^{bn-1})\) is an embedding. Then for any \(r \in \mathbb{N}\) there exits \(p \in k[x,z]\) such that \(\phi^*(p)=t^r+t^{(bn-1)r}\). \end{lemma} \begin{proof} We begin by applying Lemma \ref{lem:combinatorial}. Recall that \(\delta _r = \begin{cases} 1 & r \text{ is even} \\ 0 & r \text{ is odd}\end{cases}\); then by Lemma \ref{lem:combinatorial}, there exist \(\alpha _0, \ldots, \alpha _{ \floor{\frac{r}{2}}}, \beta \in k\) such that, for any \(s \in k[t]\), \begin{equation}\label{eq:s}
\sum _{i=0} ^\floor{ \frac{r}{2}} \alpha _i s^i(1+s)^{r-2i} = 1+ \beta \delta _{r} s^{\frac{r}{2}}+s^{r} .
\end{equation} Now we set \(\displaystyle p= \sum _{i=0} ^\floor{\frac{r}{2}} \alpha _i x^{bi}z^{r-2i} -\beta \delta _{r} x^{b \frac{r}{2}} \in k[x,z]\), and we compute \begin{align*} \phi^*(p)&= \sum _{i=0} ^\floor{\frac{r}{2}} \alpha _i (t^n)^{bi}(t+t^{bn-1})^{r-2i} - \beta \delta _{r} (t^n)^{b \frac{r}{2}} \\ &= \sum _{i=0} ^\floor{\frac{r}{2}} \alpha _i t^{nbi+r-2i}(1+t^{bn-2})^{r-2i} - \beta \delta _{r} t^{bn \frac{r}{2}} \\ &= t^{r}\left(\sum _{i=0} ^\floor{\frac{r}{2}} \alpha _i \left(t^{bn-2}\right)^i(1+t^{bn-2})^{r-2i} - \beta \delta _{r} \left(t^{bn-2}\right)^{\frac{r}{2}}\right) \end{align*} Substituting \(s=t^{bn-2}\) into \eqref{eq:s} above, we obtain \[\phi^*(p)=t^{r}\left(1+\left(t^{bn-2}\right)^{r}\right)=t^{r}+t^{(bn-1)r}.\] \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:BRgeneral}] Write \(m=an+c\) for some \(a \in \mathbb{N}\). We begin by supposing \(a \geq bd-1\), so \(a+1=bd+h\) for some \(h \in \mathbb{N}\). In this case, we can write \(m=(a+1)n-d=(bn-1)d+hn\). By Lemma \ref{lem:cd}, there exists \(p \in k[x,z]\) such that \(\phi^*(p)=t^d+t^{(bn-1)d}\). Then, setting \(\theta = (x,-y+x^hp(x,z),z) \in {\rm GA}_3(k)\), we see that \(\theta \phi = (t^n, t^{hn+d},t+t^{bn-1})\), and this is rectifiable if and only if \(\phi\) is rectifiable. Note that since \(b\geq 2\), \(hn+d < (bn-1)d+hn=m\). Thus, by repeating this process we may continue until we have \(a < bd-1\).
So now it suffices to assume \(a <bd-1\). We proceed in two cases; first, we suppose \(b > \mu _2\). By Lemma \ref{lem:cd}, there exists \(p \in k[x,z]\) such that \(\phi^*(p)=t^d+t^{(bn-1)d}\). We then set \(\alpha = \left(x,y-\frac{p(x,z)}{x^{bd-a-1}},z\right) \) and \(\beta = \left( x,y,z+ x^{b-\mu _2}\left(x^{bd-a-1}y\right)^{\lambda _2} \right)\), and compute
\begin{align*} \phi^* \alpha^* \beta^*(z) &= Z+ X^{b-\mu _2}(X^{bd-a-1}Y-p(X,Z))^{\lambda _2} \\ &= t+t^{bn-1} + \left(t^n\right)^{b-\mu _2} \left( (t^n)^{bd-a-1} t^{an+c} - \left( t^{d}+t^{(bn-1)d}\right) \right)^{\lambda _2} \\ &=t+t^{bn-1}+t^{n(b-\mu _2)}\left(t^{bdn-n+c} - t^{d}-t^{(bn-1)d} \right)^{\lambda _2} \\ &= t+t^{bn-1} +t^{nb-\lambda _2 d - 1}(-t^{\lambda _2 d}) \\ &= t. \end{align*} Then by Corollary \ref{cor:criterion}, \(\phi\) is rectifiable.
Now, we must consider the case \(b > \mu _1\). Then by Lemma \ref{lem:cd}, find \(q \in k[x,z]\) such that \(\phi^*(q)=t^c+t^{(bn-1)c}\), and as above, \(p \in k[x,z]\) such that \(\phi^*(p)=t^d+t^{(bn-1)d}\).
Now set \(\alpha = \left(x,y-\frac{p(x,z)}{x^{bd-a-1}}+\frac{q}{x^{bc+bd-a-2}},z\right) \) and \(\beta = \left( x,y,z- x^{b-\mu _1}\left(x^{bc+bd-a-2}y\right)^{\lambda _1} \right)\), and compute \begin{align*} \phi^* \alpha^* \beta^*(z) &= Z- X^{b-\mu _1}(X^{bc+bd-a-2}Y-X^{bc-1}p(X,Z)+q(X,Z))^{\lambda _1} \\ &= t+t^{bn-1} - \left(t^n\right)^{b-\mu _1} \left( (t^n)^{bc+bd-a-2} t^{an+c} -(t^n)^{bc-1} \left( t^{d}+t^{(bn-1)d}\right)+\left(t^c+t^{(bn-1)c}\right) \right)^{\lambda _1} \\ &=t+t^{bn-1}-t^{n(b-\mu _1)}\left(t^{bcn+bdn-2n+c} - t^{bcn-n+d}-t^{bnc-n+(bn-1)d}+t^c+t^{(bn-1)c} \right)^{\lambda _1} \\ &= t+t^{bn-1} -t^{nb-\lambda _1 c - 1}(t^{\lambda _1 c}) \\ &= t. \end{align*} Then again by Corollary \ref{cor:criterion}, \(\phi\) is rectifiable. \end{proof}
We next give a new proof of the other result of \cite{BR}. \begin{theorem}[Bhatwadekar-Roy] \label{thm:BR2} \(\phi = (t^4, t^{4a+1}, t^m+t)\) is rectifiable for any \(a,m \in \mathbb{N}\). \end{theorem} \begin{proof} We divide the proof into four cases, based on the residue of \(m\) modulo \(4\). Three cases are relatively straightforward, while the case \(m \equiv 2 \pmod 4\) generalizes our proof of Theorem \ref{thm:Craighero2} above.
\noindent \textbf{Case 1: \(m \equiv 1 \pmod 4\).} Write \(m=4k+1\) for some \(k \in \mathbb{N}\). Without loss of generality, we may assume \(a < k\); for if \(a \geq k\), we may apply the map \((x,-y+x^{a-k}z,z)\) to produce the embedding \((t^4, t^{4(a-k)+1},t^m+t)\), and repeat until \(a <k\). But in this case, observe that \(Z-X^{k-a}Y=t\), and \(z-x^{k-a}y\) is a coordinate.
\noindent \textbf{Case 2: \(m \equiv 2 \pmod 4\).} Write \(m=4k+2\) for some \(k \in \mathbb{N}\). This is the hardest case, but proceeds in the same way as our proof of Theorem \ref{thm:Craighero2}. First, we observe that \[Z^2-X^{2k+1}=2t^{4k+3}+t^2.\]
\begin{align*} Y+c\frac{Z^3}{X^{2k+1-a}}+dX^{a}Z-3cX^{a-k} &=(c+d)t^{4a+4k+2}+(1+3c+d)t^{4a+1}+c\frac{1}{t^{8k-4a+1}} \end{align*} Setting \(c=-\frac{1}{2}\) and \(d=\frac{1}{2}\) causes these first two coefficients to vanish, so we see \[Y-\frac{1}{2}\frac{Z^3}{X^{2k+1-a}}+\frac{1}{2}X^aZ+\frac{3}{2}X^{a-k} = -\frac{1}{2}\frac{1}{t^{8k-4a+1}} \]
Thus \[Y-\frac{1}{2}\frac{Z^3}{X^{2k+1-a}}+\frac{1}{2}X^aZ+\frac{3}{2}X^{a-k} +\frac{1}{4} \frac{Z^2-X^{2k+1}}{X^{3k-a+1}} = \frac{1}{4}\frac{1}{t^{12k-4a+2}} \]
Now, let \begin{align*} \alpha &= \left(x, \left(y-\frac{1}{2}\frac{z^3}{x^{2k+1-a}}+\frac{1}{2}x^az+\frac{3}{2}x^{a-k} \right) +\frac{1}{4}\frac{z^2-x^3}{x^{3k-a+1}},z\right) \end{align*}
Note that if \(a > 3k+1\), then \(\alpha \in {\rm EA}_3(k)\). And in this case, we have \(12k-4a+2=4(3k-a+1)-2<0\); so then we have \(\alpha \phi= (t^4, \frac{1}{4}t^{4a-(12k+2)},t^{m}+t) \), with \(4a-(12k+2)< 4a+1\). Now, if \( k\geq a-3k-1\), we can set \(\beta = (x,y,z-4x^{4k-a+1}y) \in {\rm EA}_3(k)\) , and compute \( \phi^*\alpha^*\beta^*(z) = t\), and as \(\beta \alpha \in {\rm GA}_3(k)\), \(\phi\) is rectifiable by Lemma \ref{lem:coordinate}. If instead \(k < a-3k-1\), we set \(\beta = (x,-4y+x^{a-4k-1}z,z) \in {\rm GA}_3(k)\), and compute that \(\beta \alpha \phi = (t^4, t^{4(a-4k-1)+1},t^m+t)\).
This process can be repeated, so we may now proceed assuming without loss of generality that \(a \leq 3k+1\). We construct \(\alpha\) as above (but now, \(\alpha \in {\rm EA}_2(k[x,x^{-1}])\)), and further define \begin{align*} \beta &= \left(x,y,z+x^{k}(-4x^{3k-a+1}y)\right) \end{align*}
Then we see \begin{align*} \phi^*\alpha^*\beta^*(z) &= Z-4X^{4k-a+1}\left(Y-\frac{1}{2}\frac{Z^3}{X^{2k+1-a}}+\frac{1}{2}X^aZ+\frac{3}{2}X^{a-k} +\frac{1}{4}\frac{Z^2-X^{2k+1}}{X^{3k-a+1}}\right) \\ &= t^{4k+2}+t-4(t^{16k-4a+4})\left(\frac{1}{4}t^{-12k+4a-2}\right) \\ &= t. \end{align*} Thus \(\phi\) is rectifiable by Corollary \ref{cor:criterion}.
\noindent \textbf{Case 3: \(m \equiv 3 \pmod 4\).} Write \(m=4k+3\) for some \(k \in \mathbb{N}\). Without loss of generality, we may assume \(a \leq 3k+2\); for if \(a >3k+2\), we set \(\alpha = (x,-y+x^{a-2-3k}z^3-3x^{a-2k-1}z,z) \in {\rm GA}_3(k)\) and compute \begin{align*} \phi^*\alpha^*(y) &= -Y+X^{a-2-3k}Z^3-3X^{a-2k-1}Z \\ &=-t^{4a+1}+t^{4(a-2-3k)}(t^{4k+3}+t)^3 -3t^{4a-8k-4}(t^{4k+3}+t)\\ &= -t^{4a+1}+t^{4a-8-12k}\left(t^{12k+9}+3t^{8k+7}+3t^{4k+5}+t^3\right) - 3t^{4a-4k-1}-3t^{4a-8k-3} \\ &= t^{4a-12k-5} \end{align*}
This process can be repeated until \(a \leq 3k+2\).
Now, assuming \(a \leq 3k+2\), we set \(\alpha = \left(x,y-\frac{z^3}{x^{3k+2-a}}-3x^{a-2k-1}z,z\right)\) and \(\beta = (x,y,z+x^k(-x^{3k+2-a}y))\),
and compute \begin{align*} \phi^*\alpha^*\beta^*(z) &= Z-X^K\left(Z^3-X^{3k-a+2}Y-3X^{k+1}Z\right) \\ &=t+t^{4k+3}-t^{4k}\left( (t^{4k+3}+t)^3-t^{4(3k-a+2)}(t^{4a+1})-3t^{4k+4}(t^{4k+3}+t)\right) \\ &= t+t^{4k+3}-t^{4k}\left(t^{12k+9}+3t^{8k+7}+3t^{4k+5}+t^3 - t^{12k+9}-3t^{8k+7}-3t^{4k+5}\right) \\ &= t+t^{4k+3}-t^{4k}\left(t^3 \right) \\ &= t. \end{align*} Thus \(\phi\) is rectifiable by Corollary \ref{cor:criterion}.
\noindent \textbf{Case 4: \(m \equiv 0 \pmod 4\).} In this case, \(m=4k\) for some \(k \in \mathbb{N}\), so \(Z-X^k=t\), and \(z-x^k\) is a coordinate. \end{proof}
\subsection{Generalizing Kuroda's result}
This section is devoted to proving the following theorem. \begin{theorem}\label{thm:Kurodageneral} Let \(n,a,c,l,s \in \mathbb{N}\) such that \(cl<a\). Then the embedding \((t^n, t^{an+c}, t+t^{(an+c)s-ln})\) is rectifiable. \end{theorem} \begin{remark}Kuroda \cite{Kuroda} proved the special case of \(c=1\) and \(a=2l+m\) for some nonnegative integer \(m\) (in which case the assumption \(cl<a\) is satisfied automatically). \end{remark} \begin{proof}[Proof of Theorem \ref{thm:Kurodageneral}]
Let \(\alpha = \left(x,y,z-\frac{y^s}{x^l}\right)\) and \(\beta = \left(x,y-x^{a-cl}(x^lz)^c,z)\right) \)
We compute \begin{align*} \phi^*\alpha^*\beta^*(y) &= Y-X^{a-cl}\left(X^lZ-Y^s\right)^c \\ &=t^{an+c}-t^{n(a-cl)}\left(t^{ln}(t+t^{(an+c)s-ln})-t^{(an+c)s}\right)^c \\ &=t^{an+c}-t^{n(a-cl)}\left(t^{ln+1}\right)^c \\ &= t^{an+c}-t^{n(a-cl)+c(ln+1)} \\ &=0 \end{align*} It is also easy to see that \(\phi^*\alpha^*\beta^*(z)=t\), so we have \(\beta \alpha \phi = (t^n,0,t)\). Now, from Theorem \ref{thm:cleardenominators}, we can produce \(\gamma = (x,y,z+\frac{R(y)}{x^l})\) such that \(\gamma \beta \alpha \in {\rm GA}_2(k[x])\), and \(R(y) \in (y^s)k[x,y]\), so that \((\gamma \beta \alpha \phi)^*(z)=(\beta \alpha \phi)^*(z+\frac{R(y)}{x^l})=t\). Then letting \(f=(\gamma \beta \alpha)^*(z)\), we have \(f\) is a coordinate with \(\phi^*(f)=t\), so \(\phi\) is rectifiable by Lemma \ref{lem:coordinate}. \end{proof}
Letting \(c=1,a=3,l=2\), we obtain the following: \begin{corollary}For any \(n \in \mathbb{N}\), the embedding \(\left(t^n, t^{3n+1}, t^{4n+2}+t\right)\) is rectifiable. \end{corollary}
Letting \(c=2,a=5,l=2\), we obtain the following: \begin{corollary}For any \(n \in \mathbb{N}\), the embedding \(\left(t^n, t^{5n+2}, t^{8n+4}+t\right)\) is rectifiable. \end{corollary}
\end{document} | arXiv |
Kim Thibault
College Math, Calculus, Complex Number
Let us take a moment to ponder how truly bizarre the Laplace transform is.
You put in a sine and get an oddly simple, arbitrary-looking fraction. Why do we suddenly have squares?
You look at the table of common Laplace transforms to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results.
Or so we thought when we first encountered the cursive $\mathcal{L}$ in school.
What does the Laplace transform do, really?
Some Preliminary Examples
Looking Inside the Laplace Transform of Sine
Diverging Functions: What the Laplace Transform is for
A Transform of Unfathomable Power
At a high level, Laplace transform is an integral transform mostly encountered in differential equations — in electrical engineering for instance — where electric circuits are represented as differential equations.
In fact, it takes a time-domain function, where $t$ is the variable, and outputs a frequency-domain function, where $s$ is the variable. Definition-wise, Laplace transform takes a function of real variable $f(t)$ (defined for all $t \ge 0$) to a function of complex variable $F(s)$ as follows:
\[\mathcal{L}\{f(t)\} = \int_0^{\infty} f(t) e^{-st} \, dt = F(s) \]
What fate awaits simple functions as they enter the Laplace transform?
Take the simplest function: the constant function $f(t)=1$. In this case, putting $1$ in the transform yields $1/s$, which means that we went from a constant to a variable-dependent function.
(Odd but not too worrying. After all, we've seen $1/x$ integrating to $\ln x$ in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.)
Let us take it up a notch, with the linear function $f(t) = t$. After the transformation, it is turned into $1/s^2$, which means that we went from $1 \to 1/s$ to $t \to 1/s^2$. A pattern begins to emerge.
Now what about $f(t)=t^n$? With this simple power function, we end up with: \[ \mathcal{L}\{ t^n \} = \frac{n!}{s^{n+1}}\] So there was a factorial in $\mathcal{L}\{t\}$ all along, hidden by the fact that $1! = 1$. What else is the transform hiding?
Here, a glance at a table of common Laplace transforms would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly — which brings us back to the sine function.
Let us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a complex exponential — an indirect result of the celebrated Euler's formula:\[e^{it} = \cos t + i \sin t\]In fact, a sine is often expressed in terms of exponentials for ease of calculation, so if we apply that to the function $f(t) = \sin (at)$, we would get: \[ \sin(at) = \frac{e^{iat}-e^{-iat}}{2i} \]Thus the Laplace transform of $\sin(at)$ then becomes:
\[ \mathcal{L}\{\sin(at)\} = \frac{1}{2i} \int\limits_0^{\infty} (e^{iat}-e^{-iat}) e^{-st} \, dt \]which means that we have a product of exponentials. Distributing the terms, we get:
\[ \mathcal{L}\{\sin(at)\} = \frac{1}{2i} \int\limits_0^{\infty} e^{iat-st}-e^{-iat-st} \, dt \]
Here, factoring the $t$ in the exponents yields:
\[ \mathcal{L}\{\sin(at)\} = \frac{1}{2i} \int\limits_0^{\infty} e^{(ia-s)t}-e^{(-ia-s)t} \, dt \]and since $\mathrm{Re}(s) \gt 0$ by assumption, we can proceed with the integration from $0$ to $\infty$ as usual:
\[ \mathcal{L}\{\sin(at)\} = \left.\frac{e^{(ia-s)t}}{2i (ia-s)}\right|_0^{\infty}-\left.\frac{e^{(-ia-s)t}}{2i (-ia-s)}\right|_0^{\infty} \]
Let us simplify further. Distributing the $i$ inside the parentheses, we get:
\[ \mathcal{L}\{\sin(at)\} = \left.\frac{e^{(ia-s)t}}{2(-a-is)}\right|_0^{\infty}-\left.\frac{e^{(-ia-s)t}}{2(a-is)}\right|_0^{\infty} \]By evaluating the $t$ at the boundaries, we get:
\[ \mathcal{L}\{\sin(at)\} = \left( \frac{e^{(ia-s) \cdot \infty}}{2(-a-is)}-\frac{e^{(ia-s) \cdot 0}}{2 (-a-is)}\right)-\left(\frac{e^{(-ia-s)\cdot\infty}}{2(a-is)}-\frac{e^{(-ia-s)\cdot 0}}{2(a-is)}\right) \]And because $\mathrm{Re}(s) > 0$ by assumption, both $e^{(ia-s) \cdot \infty}$ and $e^{(-ia-s)\cdot\infty}$ oscillate to $0$ (i.e., vanish at infinity), after which we are then left with:\[ \mathcal{L}\{\sin(at)\} = \frac{1}{-2(-a-is)} + \frac{1}{2(a-is)} \]Once there, merging the fractions together would yield:\begin{align*} \mathcal{L}\{\sin(at)\} & = \frac{2(a-is)-2(-a-is)}{-4 (a-is)(-a-is)} \\ & = \frac{2a-2is + 2a+2is}{4 (a^2 + isa-isa + s^2)} \\ & = \frac{4a}{4(a^2 + s^2)} \\ & = \frac{a}{a^2 + s^2} \end{align*}which shows that after Laplace transform, a sine is turned into a more tractable geometric function. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: \[ \mathcal{L}\{\cos (at)\} = \frac{s}{a^2 + s^2} \qquad (\mathrm{Re}(s) > 0) \] But then, one might argue "Why do we need to transform trigonometric functions like this when we can just integrate them?"
What if we throw a wrench in there by introducing a diverging function, say, $f(t)=e^{at}$? As it turns out, the Laplace transform of the exponential $e^{at}$ is actually deceptively simple: \begin{align*} \mathcal{L}\{e^{at}\} & = \int_0^{\infty} e^{at}e^{-st} \, dt \\ & = \int_0^{\infty} e^{(a-s)t} \, dt \end{align*}Here, we see that so long as $\mathrm{Re}(s) \gt a$, we would get that: \begin{align*} \int_0^{\infty} e^{(a-s)t} \, dt & = \left. \frac{e^{(a-s)t}}{a-s} \right|_0^{\infty} \\ & = 0-\frac{1}{a-s} \\ & = \frac{1}{s-a} \end{align*} That is, as long as $\mathrm{Re}(s) > a$, the Laplace transform of $e^{at}$ is a simple $1/(s-a)$. Here's a video version of the derivation for the record.
On the other hand, if we mix the exponential $e^{at}$ with the power function $t^n$, we would then have: \[ \mathcal{L}\{t^n e^{at}\} = \int\limits_0^{\infty} t^n e^{at} e^{-st} \, dt \] which, after a bit of recursion and integration by parts, would become:\[ \frac{n!}{(s-a)^{n+1}} \]Here, notice how the transforms of exponential and power function are both represented in the expression, with the factorial $n!$, the $1/(s-a)$ fraction, and the $n + 1$ exponent.
In fact, it turns out that we can integrate any function with the Laplace transform, as long as it does not diverge faster than the $e^{at}$ exponential. In the tables of Laplace transforms, you might have noticed the $\mathrm{Re}(s) \gt a$ condition. That is what the condition is alluding to.
However, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the derivatives as well. In goes $f^{(n)}(t)$. Something happens. Then out goes:\[ s^n \mathcal{L}\{f(t)\}-\sum_{r=0}^{n-1} s^{n-1-r} f^{(r)}(0) \]For example, when $n=2$, we have that:\[ \mathcal{L}\{f^{\prime\prime}(t)\} = s^2 \mathcal{L}\{f(t)\}-sf(0)-f'(0) \]In addition to the derivatives, the $\mathcal{L}$ can also process some integrals: the integral sine, cosine and exponential, as well as the error function — to name a few.
But that's not all. There is also the inverse Laplace transform, which takes a frequency-domain function and renders a time-domain function.
In fact, performing the transform from time to frequency and back once introduces a factor of $1/2\pi$. Sometimes, you'll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of $1/\sqrt{2\pi}$.
This is as if the Kraken could restitute the boat intact — but only for a factor of $1/2\pi$.
The Laplace transform, even after all those years, never ceases to bring us awe with its power. Here's a table summarizing the transforms we've discussed thus far:
Laplace Transform
$1$ $\dfrac{1}{s}$
$t$ $\dfrac{1}{s^2}$
$t^n$ $\dfrac{n!}{s^{n+1}}$
$e^{at}$ $\dfrac{1}{s-a}$
$\sin(at)$ $\dfrac{a}{a^2+s^2}$
$\cos(at)$ $\dfrac{s}{a^2+s^2}$
$t^n e^{at}$ $\dfrac{n!}{(s-a)^{n+1}}$
$f^{(2)}(t)$ $\displaystyle s^2 \mathcal{L}\{f(t)\}-sf(0)-f'(0)$
$f^{(n)}(t)$ $\displaystyle s^n \mathcal{L}\{f(t)\}-\sum_{r=0}^{n-1} s^{n-1-r} f^{(r)}(0)$
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Kim Thibault is an incorrigible polymath. After a Ph.D. in Physics, she did applied research in machine learning for audio, then a stint in programming, to finally become an author and scientific translator. She occasionally solves differential equations as a hobby. Her blog can be found at kimthibault.mystrikingly.com/blog and her professional profile at linkedin.com/in/kimthibaultphd.
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Identification of positive selection in genes is greatly improved by using experimentally informed site-specific models
Jesse D. Bloom ORCID: orcid.org/0000-0003-1267-34081
Biology Direct volume 12, Article number: 1 (2017) Cite this article
Sites of positive selection are identified by comparing observed evolutionary patterns to those expected under a null model for evolution in the absence of such selection. For protein-coding genes, the most common null model is that nonsynonymous and synonymous mutations fix at equal rates; this unrealistic model has limited power to detect many interesting forms of selection.
I describe a new approach that uses a null model based on experimental measurements of a gene's site-specific amino-acid preferences generated by deep mutational scanning in the lab. This null model makes it possible to identify both diversifying selection for repeated amino-acid change and differential selection for mutations to amino acids that are unexpected given the measurements made in the lab. I show that this approach identifies sites of adaptive substitutions in four genes (lactamase, Gal4, influenza nucleoprotein, and influenza hemagglutinin) far better than a comparable method that simply compares the rates of nonsynonymous and synonymous substitutions.
As rapid increases in biological data enable increasingly nuanced descriptions of the constraints on individual protein sites, approaches like the one here can improve our ability to identify many interesting forms of selection in natural sequences.
This article was reviewed by Sebastian Maurer-Stroh, Olivier Tenaillon, and Tal Pupko. All three reviewers are members of the Biology Direct editorial board.
An important goal of biology is to identify genetic modifications that have led to evolutionarily significant changes in phenotype. In the case of protein-coding genes, this means identifying mutations that were fixed by selection to alter properties such as the activity of enzymes or the antigenicity of viral proteins.
This goal is challenging because not all mutations that fix do so because they confer beneficial phenotypic effects that are selected by evolution. Sometimes mutations fix because they adaptively alter phenotype, but mutations also fix due to forces such as genetic drift or hitchhiking. Therefore, it is difficult to examine gene sequences and pinpoint specific mutations that have changed evolutionarily relevant phenotypes. As Zuckerkandl and Pauling [1] noted a half-century ago:
[Many] substitutions may lead to relatively little functional change, whereas at other times the replacement of one single amino acid residue by another may lead to a radical functional change... It is the type rather than number of amino acid substitutions that is decisive.
Unfortunately, Zuckerkandl and Pauling [1] did not provide a prescription for determining the "type" of substitution that leads to phenotypic change, and such a prescription remains elusive decades later.
Because it is difficult to determine a priori which substitutions have altered relevant phenotypes, methods have been devised that compare homologous sequences to identify sites where mutations have been positively selected by evolution. The basic strategy is to formulate a null model for evolution, and then identify sites that have evolved in ways incompatible with this model. If the null model adequately describes evolution in the absence of selection for phenotypic change, then sites that deviate from the model are ones where mutations have been selected because they alter evolutionarily relevant phenotypes.
For protein-coding genes, the most widely used methods for identifying specific sites of positive selection are built around the null model that nonsynonymous and synonymous mutations should fix at equal rates. These methods estimate the rates of fixation of nonsynonymous (dN) and synonymous (dS) mutations at each codon site r [2–6]. The ratio d N/d S at r is taken as a measure of selection. If the ratio is clearly >1 then pressure for phenotypic change is favoring fixation of protein-altering nonsynonymous mutations, and the site is under diversifying selection. If the ratio is clearly <1 then nonsynoymous mutations are being purged to prevent phenotypic change, and the site is under purifying selection.
Although d N/d S methods are tremendously useful (the leading software implementations HyPhy and PAML have each been cited thousands of times [7, 8]), their underlying null model is clearly oversimplified. A random nonsynonymous mutation completely inactivates the typical protein ≈40% of the time [9]. So unsurprisingly, most genes have many sites with d N/d S<1. This finding is often of little biological value, since researchers frequently already know that the gene they are studying is under some type of protein-level constraint.
Perhaps more importantly, d N/d S methods also can fail to identify sites that have fixed adaptive mutations. For instance, T-cells drive fixation of immune-escape mutations in influenza – but because the relevant sites are under strong constraint, d N/d S remains <1 and the relative increase in nonsynonymous substitution rate is only apparent in comparison to homologs not subject to immune selection [10]. Therefore, even positive selection for adaptive mutations can fail to elevate d N/d S>1 at functionally constrained sites.
The limitations of simply comparing the rates of fixation of nonsynonymous and synonymous mutations have become especially glaring in light of deep mutational scanning experiments. These experiments, which subject libraries of mutant genes to selection in the lab and query the fate of each mutation by deep sequencing [11, 12], can measure the preference of each site in a protein for each amino acid [13]. A clear result is that sites vary wildly in their amino-acid preferences. Some sites are relatively unconstrained and prefer all amino acids roughly equally; for these sites, simply testing for d N/d S>1 is a reasonable approach for identifying positive selection. But most sites strongly prefer one or a few amino acids, so positive selection would not necessarily be expected to elevate d N/d S>1 for these sites.
As an example, Fig. 1 shows the amino-acid preferences of five sites in TEM-1 β-lactamase as measured by the deep mutational scanning of Stiffler et al [14]. Mutations at three of these sites confer antibiotic or inhibitor resistance in β-lactamases [15]. Inspection of Fig. 1 shows that the two sites not implicated in resistance have evolved in ways that seem roughly compatible with their amino-acid preferences measured in the lab: site 201 tolerates many amino acids in the lab and is moderately variable in nature, while site 242 strongly prefers glycine in the lab and is conserved at that identity in nature. But the three sites involved in resistance have evolved in ways that seem to deviate from their amino-acid preferences measured in the lab: site 238 substitutes from the lab-preferred glycine to the less preferred serine, site 240 repeatedly substitutes to lysine despite not strongly preferring this amino acid in the lab, and site 244 substitutes from the lab-preferred arginine to several less preferred amino acids. So given the experimentally measured preferences, it is fairly apparent that the sites where mutations contribute to antibiotic resistance are evolving in ways that deviate from the preferences measured in the lab. But as Fig. 1 shows, a d N/d S method fails to find any site with d N/d S>1 at a false-discovery rate (FDR) of 0.05. As this example shows, a null model that fails to account for site-specific amino-acid preferences can overlook sites that fix adaptive mutations.
Different sites are expected to evolve differently, but d N/d S methods ignore this fact and so have limited power to detect positive selection. a The amino-acid preferences of five sites in TEM-1 β-lactamase as measured by deep mutational scanning (using the data measured with the highest concentration of ampicillin in [14]; letter heights are proportional to amino-acid preferences). Three sites experience mutations that confer extended-spectrum antibiotic or inhibitor resistance [15]. The two sites not involved in resistance are evolving in a way that seems roughly compatible with the experimentally measured amino-acid preferences, while the three sites implicated in resistance are evolving in ways that clearly deviate from the preferences (for instance, site 238 mutates from highly preferred glycine to the very low preference amino-acid serine). b A standard d N/d S model (the M0 variant [4] of the Goldman-Yang model [23], abbreviated GY94) assumes all sites evolve under uniform constraints. When this model is used to fit a site-specific d N/d S, no sites are deemed under diversifying selection (d N/d S>1) at a FDR of 0.05 for testing all sites, although the non-resistance site 242 is deemed under purifying selection (d N/d S<1). The violin plot shows the distribution of P-values for sites having d N/d S> or <1. All sites below the bottom dotted blue line are deemed to have d N/d S<1 at an FDR of 0.05. No sites have d N/d S>1 at this FDR, so the top dotted blue line indicate the P-value that would be needed for a site to have d N/d S>1 at a significance level of 0.05 using a Bonferroni correction. A full analysis of all sites and further details are later in the paper. See Additional file 16 for subtleties about amino-acid preferences versus equilibrium frequencies
Here I describe how the limitations of d N/d S methods illustrated in Fig. 1 can be overcome by defining selection relative to a null model established by experimentally measured site-specific amino-acid preferences. This more nuanced null model can be used to identify sites of diversifying selection for unusually rapid amino-acid change via a statistically principled extension to standard d N/d S methods. The more nuanced null model can also be used to heuristically identify sites of differential selection for unexpected amino acids. Both of these strategies ultimately seek to identify sites that are evolving differently in nature than expected from constraints measured in the lab. Although the lab measurements are undoubtedly imperfect proxies for actual selective constraints in nature, they provide a better model for evolution in nature than phylogenetic substitution models commonly used to identify positive selection in nature. I demonstrate that this is the case by analyzing four genes, and showing that the experimentally informed methods greatly outperform a standard d N/d S method at identifying sites of antibiotic-resistance and immune-escape mutations. As deep mutational scanning data become more widespread, approaches like the one here can enhance our ability to identify sites of biologically interesting selection.
An evolutionary null model informed by experimentally measured amino-acid preferences
To remedy the limitations of d N/d S methods illustrated in Fig. 1, we formulate a description of how sites should evolve if selection in nature matches the constraints measured by deep mutational scanning in the lab. This description consists of a set of site-specific experimentally informed codon models (ExpCM). The ExpCM used here are similar but not identical to those in [16, 17]. Specifically, they differ from the model in [17] by inclusion of an ω parameter representing the relative rate of nonsynonymous to synonymous substitutions, and by handling the nucleotide mutation terms via an HKY85-style [18] formalism rather than the formalism in [17].
Deep mutational scanning experiments provide direct measurements of the preference π r,a of each site r for each amino acid a (for details of how these preferences can be obtained from the experimental data, see [13]). These preferences are normalized so \(\sum _{a} \pi _{r,a} = 1\). We use the preferences to define an ExpCM for each site. As is typical for phylogenetic substitution models, each ExpCM is a reversible stochastic matrix giving the rates of substitution between codons. The rate P r,x y from codon x to y at site r is written in mutation-selection form as
$$ P_{r,xy} = Q_{xy} \times F_{r,xy} $$
where Q xy represents the rate of mutation from x to y and F r,x y represents the selection on this mutation. The mutation terms are identical across sites, but the selection terms are site-specific.
The mutation terms Q xy are given by a HKY85 model [18], and consist of a transition-transversion ratio κ and four nucleotide parameters ϕ A , ϕ C , ϕ G , and ϕ T that sum to one. These ϕ parameters give the expected nucleotide composition in the absence of selection on amino acids; the actual nucleotide frequencies are also influenced by the selection (for this reason, the ϕ terms cannot simply be equated with the empirical alignment frequencies). The mutation term is:
$$ \begin{aligned} Q_{xy} = \left\{\begin{array}{ll} 0 & \text{\textit{x} and \textit{y} differ by \(> 1\) nucleotide,} \\ \phi_{w} & \text{\textit{x} can be converted to \textit{y} by transversion to \textit{w},} \\ \kappa \times \phi_{w} & \text{\textit{x} can be converted to \textit{y} by transition to \textit{w}.} \\ \end{array}\right. \end{aligned} $$
The site-specific amino-acid preferences π r,a enter the model via the selection terms F r,x y . Let A(x) denote the amino acid encoded by codon x, let β be the stringency parameter described in [17], and let ω be a gene-wide relative rate of fixation of nonsynonymous to synonymous mutations after accounting for the amino-acid preferences. Then:
$$ \begin{aligned} F_{r,xy} =\left\{ \begin{array}{ll} 1 & \text{if}\, \mathcal{A}(x) = \mathcal{A}(y) \\ \omega & \text{if}\, \mathcal{A}(x) \ne \mathcal{A}(y)\, \text{and}\, \pi_{r,\mathcal{A}(x)} = \pi_{r,\mathcal{A}(y)} \\ \omega \times \frac{\ln\left[\frac{\left(\pi_{r,\mathcal{A}(y)}\right)^{\beta}}{\left(\pi_{r,\mathcal{A}(x)}\right)^{\beta}}\right]}{1 - \frac{\left(\pi_{r,\mathcal{A}(x)}\right)^{\beta}}{\left(\pi_{r,\mathcal{A}(y)}\right)^{\beta}}} & \text{otherwise.} \end{array}\right. \end{aligned} $$
The functional form relating F r,x y to π r,a for nonsynonymous mutations is that derived by Halpern and Bruno [19] under certain (probably unrealistic) assumptions about the evolutionary process and the relationship between the preferences and amino-acid fitnesses (see also [20–22]). Relative to the equation of Halpern and Bruno [19], Eq. 3 removes terms related to mutation (these are captured by Q xy ) and corrects a typographical error in the denominator. The stringency parameter β is >1 if natural selection favors high-preference amino acids with greater stringency than the experiments used to measure π r,a , and is <1 if it favors them with less stringency. Under the assumptions of Halpern and Bruno [19], β is related to effective population size. Note that if β=0, then the substitution model defined by Eq. 1 reduces to a F1X4 version of the M0 variant [4] of the Goldman-Yang [23] model. The ω parameter indicates if there is a retardation (ω<1) or acceleration (ω>1) in the rate of fixation of nonsynonymous mutations relative to synonymous mutations after accounting for the preferences. In [17], it is shown that a model of the form defined by P r,x y is reversible and has stationary state
$$ p_{r,x} = \frac{\left(\pi_{r,\operatorname{A}(x)}\right)^{\beta} \times \phi_{x_{1}} \times \phi_{x_{2}} \times \phi_{x_{3}}}{\sum_{y} \left(\pi_{r,\operatorname{A}(y)}\right)^{\beta} \times \phi_{y_{1}} \times \phi_{y_{2}} \times \phi_{y_{3}}} $$
where x 1, x 2, and x 3 are the nucleotides at positions 1, 2, and 3 of codon x.
The ExpCM can be used to calculate the likelihood of a phylogenetic tree and an alignment of genes using the algorithm of Felsenstein [24], which implicitly assumes that sites evolve independently. The set of ExpCM for a given gene have six free parameters: ω, β, κ, and three of the ϕ's. The π r,a values are not free parameters, since they are specified a priori from experimental data. The values of the six free parameters are fit by maximum likelihood.
Overall, the ExpCM describe how sites evolve if selection in nature is concordant with the amino-acid preferences measured in the lab.
Identifying sites of diversifying selection
Having established a null model for how a gene should evolve if selection adheres to the constraints measured in the lab, we next want to identify sites that deviate from this model. Such sites are likely targets of additional selection. One such form of selection is diversifying selection for amino-acid change, as occurs at viral epitopes under continual pressure to escape newly generated immunity.
To detect diversifying selection, we use an approach analogous the fixed effects likelihood (FEL) method [5, 25, 26]. After fixing the tree and model parameters to their maximum likelihood values for the entire sequence, for each site r we fit a synonymous rate μ r and a parameter ω r corresponding to the nonsynonymous rate relative to the synonymous rate by replacing Eq. 3 with
$$ \begin{aligned} F_{r,xy} =\left\{ \begin{array}{lll} \mu_{r} & \text{if}\, \mathcal{A}(x) = \mathcal{A}(y) \\ \mu_{r} \times \omega_{r} & \text{if}\, \mathcal{A}(x) \ne \mathcal{A}(y) \text{and}\, \pi_{r,\mathcal{A}(x)} = \pi_{r,\mathcal{A}(y)} \\ \mu_{r} \times \omega_{r} \times \frac{\ln\left[\frac{\left(\pi_{r,\mathcal{A}(y)}\right)^{\beta}}{\left(\pi_{r,\mathcal{A}(x)}\right)^{\beta}}\right]}{1 - \frac{\left(\pi_{r,\mathcal{A}(x)}\right)^{\beta}}{\left(\pi_{r,\mathcal{A}(y)}\right)^{\beta}}} & \text{otherwise.} \end{array}\right. \end{aligned} $$
and optimizing with respect μ r and ω r . The reason that we fit μ r as well as ω r is to accommodate synonymous rate variation among sites; this can be important for the reasons described in [27]. The null hypothesis is that ω r =1. Following [5], we compute a P-value for rejecting this null hypothesis by using a \({\chi _{1}^{2}}\) test to compare the likelihood when fitting both μ r and ω r to that when fitting only μ r and fixing ω r =1. The key statistic is not ω r itself, but rather the difference in log likelihood (the likelihood ratio) from which we compute the P-value for rejecting the null hypothesis of ω=1 in favor of ω r >1 or ω r <1. The former case implies diversifying selection, while the latter case indicates a selective constraint on amino-acid change that is not adequately captured by the preferences. To account for the fact that a different test is performed for each site, we control the FDR using the Benjamini-Hochberg procedure [28]. As demonstrated below, this approach has excellent power to pinpoint sites like 238 and 244 in Fig. 1, which fix multiple nonsynonymous mutations despite being under strong functional constraint.
Identifying sites of differential selection
Some interesting forms of selection do not cause sites to change repeatedly, but rather lead them to substitute to amino acids that are unexpected given the amino-acid preferences measured in the lab. Such sites are under differential selection to fix mutations different from those expected if selection in nature parallels that in the lab.
To detect differential selection, we compare the preferences measured in the lab to those that optimally describe evolution in nature. We again begin by fixing the tree and model parameters to their maximum likelihood values determined over the whole gene. We then examine the effect of allowing the preferences at each site to differ from the values measured in the lab. Specifically, denote the preferences that optimally describe evolution in nature as \(\hat {\pi }_{r,a}\), with \(\sum _{a} \hat {\pi }_{r,a} = 1\). Denote the differential preference Δ π r,a for amino-acid a at site r as the difference between \(\hat {\pi }_{r,a}\) and the experimentally measured preferences rescaled by the stringency parameter: \(\Delta \pi _{r,a} = \hat {\pi }_{r,a} -\frac {\left (\pi _{r,a}\right)^{\beta }}{\sum _{a'} \left (\pi _{r,a'}\right)^{\beta }}\). If we redefine Eq. 3 by replacing (π r,a )β with \(\hat {\pi }_{r,a}\) as in
$$ F_{r,xy} \,=\,\left\{\!\! \begin{array}{ll} 1 & \text{if}\, \mathcal{A}(x) =\! \mathcal{A}(y) \\ \omega & \text{if}\, \mathcal{A}(x) \ne\! \mathcal{A}(y) \text{and}\, \hat{\pi}_{r,\mathcal{A}(x)} \,=\, \hat{\pi}_{r,\mathcal{A}(y)} \\ \omega \times\! \frac{\ln\left[\frac{\hat{\pi}_{r,\mathcal{A}(y)}}{\hat{\pi}_{r,\mathcal{A}(x)}}\right]}{1 - \frac{\hat{\pi}_{r,\mathcal{A}(x)}}{\hat{\pi}_{r,\mathcal{A}(y)}}} & \text{otherwise,} \end{array}\right. $$
then we can determine the preferences that optimally describe natural evolution by optimizing with respect to \(\hat {\pi }_{r,a}\) after fixing the tree and model parameters to their maximum likelihood values for the entire sequence. However, unconstrained optimization of Eq. 6 will overfit the data [29, 30]. We therefore instead optimize the product of Eq. 6 and an Eq. that regularizes the Δ π r,a values by biasing them towards zero:
$$ \Pr\left(\left\{\hat{\pi}_{r,a}\right\} \mid \left\{\pi_{r,a}\right\}, \beta\right) = \prod_{a} \left(\frac{1}{1 + C_{1} \times \left(\Delta\pi_{r,a}\right)^{2}}\right)^{C_{2}} $$
where C 1 and C 2 determine how strongly \(\hat {\pi }_{r,a}\) is biased towards the experimentally measured preferences. Here I use C 1=150 and C 2=0.5; Eq. 7 is illustrated in Additional file 1. Effectively, this equation biases the estimated values towards the prior expectation from the deep mutational scanning, although the equation is not a true prior as we are using a maximum-likelihood rather than a Bayesian approach. Note that while the underlying rationale for regularizing the Δ π r,a values is clear, the regularization implemented by Eq. 7 was chosen heuristically with the rationale that the marginal cost of shifting Δ π r,a away from zero should initially be steep but then flatten somewhat, corresponding to the intuition that most sites have little differential selection, but some have a lot. However, a more statistically principled method for assessing the support for non-zero Δ π r,a values is an important area for future work.
A differential preference of Δ π r,a >0 implies that natural evolution favors amino-acid a at site r more than expected, whereas Δ π r,a <0 implies that evolution disfavors this amino acid. The total differential selection at r is quantified as half the absolute sum of the differential preferences, \(\frac {1}{2} \sum _{a} \left |\Delta \pi _{r,a}\right |\); this quantity ranges from zero to one. As demonstrated below, this approach has excellent power to pinpoint sites like 238 and 240 in Fig. 1, which fix mutations to unexpected amino acids. However, I emphasize that this test for differential selection is heuristic, and does not incorporate formal statistical significance testing.
Choice of four genes to test approaches to identify sites of selection
To test the approaches for detecting selection described above, I selected four genes: the DNA-binding domain of yeast Gal4, β-lactamase, the nucleoprotein (NP) of human influenza, and the hemagglutinin (HA) of human seasonal H1N1 influenza. Previous deep mutational scanning studies have measured the effects of all mutations to these genes [14, 31–33], enabling calculation of their site-specific amino-acid preferences. For β-lactamase there are actually two deep mutational scanning datasets: one from Firnberg et al [34] and a more recent one from Stiffler et al [14]. As will be shown below, a likelihood-based model comparison shows that the latter of these two datasets provides a better description of β-lactamase evolution in nature, and so for that reason this is the β-lactamase deep mutational scanning dataset used in the current study. For each gene, I assembled an alignment of homologs for evolutionary analysis (Table 1).
Table 1 The four genes analyzed in this study
A great deal is known about the pressures that have shaped the evolution of all four genes. Gal4 performs a function that is conserved among homologs from widely diverged species, and does not appear to be changing phenotypically [35, 36]. However, the other three genes are undergoing adaptive evolution: β-lactamases evolve resistance to new antibiotics and inhibitors [15, 37], while NP and HA evolve to escape the immune response in humans [10, 38–40]. These genes therefore provide an excellent test case. Gal4 is a "negative control": no sites in this gene should be identified as under selection to fix adaptive mutations. But an effective approach for identifying positive selection should pinpoint the sites of drug-resistance and immune-escape mutations in the other three genes.
Experimentally informed site-specific models are vastly better descriptors of evolution
Our basic assumption is that site-specific ExpCM are a better null model for evolution than the non-site-specific models used by d N/d S methods. Prior work has shown that experimentally informed site-specific models similar to the ExpCM defined here greatly outperform non-site-specific models [16, 17, 32, 33]. To confirm this result for the ExpCM and genes here, I compared the ExpCM to the several variants [4] of the Goldman-Yang style models [23] (denoted as GY94) commonly used by d N/d S methods. I used F3X4 equilibrium frequencies for GY94, with the nine F3X4 parameters estimated by maximum likelihood. These equilibrium frequencies are not site-specific; this is the major difference between GY94 and ExpCM (Fig. 1).
To compare the models and perform the other analyses in this paper, I developed the software package phydms (phylogenetics informed by deep mutational scanning; https://github.com/jbloomlab/phydms). This software interfaces with and extends Bio++ [41, 42] to enable analyses with both ExpCM and GY94 models. The analyses described in this paper use phydms version 1.2.3.
I used phydms to infer a maximum-likelihood phylogenetic tree for each gene using GY94 with a single gene-wide d N/d S ratio (the M0 model in [4]). After fixing the tree topology to that estimated using GY94 M0, I re-optimized the branch lengths and model parameters by maximum likelihood for four additional models. The first is GY94 M3 [4], in which the likelihood for each site is a linear combination of those under three different d N/d S values, with these values and their weights shared across the whole alignment and optimized by maximum likelihood. The second is ExpCM. The third is ExpCM with the amino-acid preferences averaged across sites – this averaging makes the model non-site-specific, but captures any gene-wide trends in the deep mutational scanning data. The final is ExpCM with the amino-acid preferences randomized among sites – this model is still site-specific, but the site-specific parameters are no longer associated with the actual site for which they were measured.
I compared these models using Akaike Information Criteria (AIC) [43], which measures model fit penalized by the number of free parameters. Table 2 shows that ExpCM describe the evolution of all four genes far better than any other model. This table also shows that for β-lactamase, the new Stiffler et al [14] deep mutational scanning dataset informs ExpCM that are superior to those informed by the older Firnberg et al [34] deep mutational scanning dataset, although ExpCM informed by either dataset are vastly superior to any GY94 models. The huge superiority of ExpCM over the GY94 models is because ExpCM capture site-specific evolutionary constraints, as demonstrated by the fact that ExpCM in which preferences are averaged across sites are comparable to GY94. The poor performance of the randomized ExpCM is because a site-specific model only helps if the experimentally measured preferences are assigned to the correct sites. Indeed, Table 2 shows that randomly assigned site-specific preferences are so detrimental that they are nearly completely flattened by fitting a stringency parameter β that is close to zero, effectively making the randomized ExpCM non-site-specific. Overall, Table 2 confirms previous work [16, 17, 32, 33, 44] showing that experimentally informed site-specific models provide vastly improved descriptions of evolution.
Table 2 Site-specific ExpCM are vastly better than GY94 or ExpCM with preferences averaged or randomized across sites
Another informative comparison is between the d N/d S of GY94 and the ω of ExpCM. ExpCM can represent protein-level constraint either via the site-specific amino-acid preferences or by shrinking ω to <1. In contrast, GY94 can only represent constraint by shrinking d N/d S even if the actual selection is for preferred amino acids at each site rather than against amino-acid change per se [45]. Table 2 shows that the ExpCM ω is always greater than the GY94 d N/d S. This effect is most striking for β-lactamase: while GY94 suggests selection against amino-acid change per se by fitting d N/d S=0.3, ExpCM indicate that this selection is actually accounted for by the site-specific amino-acid preferences by fitting ω=1. For the other three genes, the ExpCM ω is <1 indicating that the site-specific amino-acid preferences don't capture all constraints, but the ExpCM ω is still always substantially greater than the GY94 d N/d S.
The ExpCM stringency parameter β also provides useful information. Recall that β>1 means that natural evolution selects for preferred amino acids with greater stringency than the deep mutational scanning. Table 2 shows that for both influenza genes (NP and HA), the stringency of natural selection exceeds that of the deep mutational scanning, indicating that the selection in experiments in [32] and [33] was not as rigorous as selection in nature. For β-lactamase, the stringency of natural evolution is approximately equal to that of the deep mutational scanning, providing a second indication (along with the fitting of ω≈1) that the experiments in [14] did an excellent job of capturing the constraints on β-lactamases in nature. Only for Gal4 is β<1: either the selections in [31] were more stringent than natural selection, or the measured preferences are not completely representative of those in nature and so β is fit to <1 to somewhat flatten these preferences.
The stringency-rescaled amino-acid preferences are in Fig. 2 and Additional files 2, 3 and 4. These figures reveal remarkable variation in constraint among sites, explaining why ExpCM better describe evolution than non-site-specific models. Overall, the results in this section verify that ExpCM offer a better evolutionary null model, and so motivate their use in identifying diversifying and differential selection.
Site-specific amino-acid preferences for β-lactamase. The height of each letter is proportional to the preference for that amino acid at that site, and letters are colored by amino-acid hydrophobicity. These are the preferences experimentally measured in [14] for TEM-1 β-lactamase under selection with 2.5 mg/ml ampicillin, re-scaled by the stringency parameter β= 1.01 from Table 2. The re-scaling is done so that if the experimentally measured preference for amino-acid a at site r is π r,a , then the rescaled preference is proportional to (π r,a )β. The β-lactamase sequence is numbered using the Ambler scheme [82], meaning that residue numbers 239 and 253 are skipped. Comparable data for Gal4, NP, and HA are shown in Additional files 2, 3 and 4, respectively
Experimentally informed site-specific models better detect diversifying selection
I used the ExpCM to identify sites of diversifying selection for amino-acid change. This was done by using phydms to fit ω r and a synonymous rate for each site r via Eq. 5, fixing all other parameters at their optimized values. To compare to a standard d N/d S method, I also fit a d N/d S ratio and synonymous rate for each site using GY94 with all other parameters fixed to the values optimized under GY94 M3 (equivalent to the fixed effects likelihood or FEL method as implemented in [5]).
Figure 3 a shows that ExpCM have much greater power to identify diversifying selection than the GY94 d N/d S method. For Gal4, GY94 finds many sites with d N/d S<1, but no sites with d N/d S>1 at an FDR of 0.05. As discussed in the Introduction, identifying sites with d N/d S<1 points to the naivety of the GY94 null model rather than unexpected biology, since any reasonable researcher would have already expected Gal4's protein sequence to be under evolutionary constraint. The more plausible ExpCM null model finds that all sites in Gal4 are evolving as expected from the measurements in the lab (for no sites does it reject the null hypothesis ω r =1). For the other three genes, GY94 again finds that there are many sites with d N/d S<1 while failing to identify any sites with d N/d S>1 at an FDR of 0.05 – despite the fact that there is clear evidence that all three genes fix drug-resistance or immune-escape mutations. In contrast, the more realistic ExpCM find sites of diversifying selection for all three genes: there are three sites with ω r >1 in β-lactamase, four in NP, and two in HA.
The experimentally informed models (ExpCM) identify many sites of diversifying or differential selection that are missed by a standard d N/d S analysis (GY94). a The violin plots show the distribution of P-values that a site is under diversifying selection for (positive numbers) or against (negative numbers) amino-acid change (ω r indicates both the ExpCM parameter in Eq. 5 and the GY94 d N/d S ratio). The portion of the distribution above / below the dotted blue lines contains all sites for which there is support for rejecting the null hypothesis ω r =1 at a FDR of 0.05. When there are no sites with support at this FDR, the dotted blue lines indicate the P-value that would be needed for a site to have ω r >1 or <1 at a significance level of 0.05 using a Bonferroni correction. The d N/d S method identifies many sites of purifying selection, but fails to find any sites of selection for amino-acid change. The ExpCM model already accounts for basic functional constraints and so doesn't identify any sites with ω r <1, but does identify sites of diversifying selection in all genes except Gal4 (which is not thought to evolve under pressure for phenotypic change). b The violin plots shown the distribution of differential selection at each site inferred with the ExpCM. Since Gal4 is not under selection for phenotypic change, I defined a heuristic threshold at 2-times the Gal4 maximum value of 0.27. At this threshold, sites of differential selection are identified for all three other genes. The legend labels all sites under diversifying or differential selection. This analysis was performed using phydms; Additional file 17 shows that similar results are obtained if the d N/d S analysis is instead performed using HyPhy [7]
To statistically validate the ExpCM approach for identifying diversifying selection, I used pyvolve [46] to simulate alignments of NP under ExpCM informed by the experimentally measured preferences and using the tree inferred for the actual NP sequences. In each simulation, I randomly selected five sites and placed them under diversifying with ω r values ranging from 5 to 30. I then analyzed the simulated alignments for diversifying using the ExpCM and the FEL-like GY94 d N/d S method. As shown in Additional file 5, ExpCM consistently outperformed GY94 at identifying the simulated sites of diversifying selection. Additional file 5 also shows that the Benjamini-Hochberg procedure [28] effectively controlled the false discovery rate. These simulations demonstrate the statistical soundness of the ExpCM approach for identifying diversifying selection.
Both the FEL-like GY94 d N/d S method and the ExpCM used for the analysis in Fig. 3 a test for diversifying selection across the phylogeny. But in many cases, diversifying selection is episodic. Therefore, d N/d S methods have been extended to identify sites under diversifying selection in only some lineages [6, 47–49]. I used one of these methods, MEME [6], to test for episodic diversifying selection. Additional file 6 shows that MEME identifies one site of diversifying selection each in β-lactamase and NP, and no sites in HA or Gal4. This makes MEME more powerful than the FEL-like GY94 method but still less powerful than ExpCM. However, MEME and ExpCM outperform the FEL-like GY94 method for orthogonal reasons: MEME is superior because it can identify episodic selection, whereas ExpCM are superior because they account for functional constraints on individual sites. In principle, it should be possible to merge ExpCM with methods to identify episodic diversifying selection.
A variety of other d N/d S methods have also been developed. The most prominent other class includes so-called "random effects" methods that use an empirical Bayesian approach to share information about the distribution of d N/d S across sites [2, 50–52]. The relative pros and cons of "random effects" methods versus the "fixed effects" methods used in this paper remain an area of active discussion [5, 53]. It is beyond the scope of the current study to compare these two classes of methods. Here I simply note that as with the test for episodic selection described in the previous paragraph, ExpCM substitution models could in principle also be incorporated into the "random effects" framework, since the essential differences between "random effects" and "fixed effects" methods are due to how parameters are handled rather than the substitution model itself.
Overall, the results in this section show that ExpCM are better at identifying diversifying selection than several standard d N/d S methods. The reason for this superiority is that the ExpCM account for variation in the inherent constraints on different sites, and so have greater power to recognize when a functionally constrained site is changing more rapidly than expected.
Experimentally informed site-specific models enable detection of differential selection
ExpCM also enable identification of differential selection for unexpected amino acids. I used phydms to estimate the differential preference Δ π r,a of each site r for each amino-acid a by optimizing the product of Eq. 6 and Eq. 7 after fixing all other parameters. The differential selection at each site r was quantified as \(\frac {1}{2} \sum _{a} \left |\Delta \pi _{r,a}\right |\), which can range from zero to one.
Figure 3 b shows the distribution of site-specific differential selection. As expected, no sites in Gal4 are under strong differential selection. But for each of the other genes, a small subset of sites are under strong differential selection. I heuristically classified differential selection as "significant" if it exceeded 2-times the maximum value for Gal4. At this threshold, there are seven sites of differential selection in β-lactamase, nine in NP, and three in HA. So overall, Fig. 3 b suggests that most sites are evolving as expected in all four genes, but a small fraction of sites are under differential selection in β-lactamase, NP, and HA due to their roles in drug resistance or immune escape. This result is concordant from what we expect given biological knowledge about the selection pressures on these genes. Note that similarly reasonable results are not obtained using the non-phylogenetic Kullback-Leibler divergence to measure differences between amino-acid frequencies in nature and the experimentally measured amino-acid preferences (Additional file 7). This fact emphasizes the importance of examining evidence for diversifying selection in a phylogenetic framework rather than analyzing them using statistical approaches that treat them as independent samples from some underlying ensemble.
A more detailed portrayal of the diversifying selection at each site is in Fig. 4 and Additional files 8, 9, and 10. For each site, these images display the evidence for diversifying selection, the strength of differential selection, and the differential preference for each amino acid at sites under non-negligible differential selection.
Site-specific selection on β-lactamase inferred with experimentally informed models. The height of each letter above/below the black center line is proportional to the differential selection for/against that amino acid at that site relative to what is expected from the amino-acid preferences in Fig. 2. The overlay bar shows the evidence for diversifying selection at each site, which is manifested by strong evidence for a ratio ω r of nonsynonymous to synonymous substitution rates that is higher (red) or lower (blue) than expected from the amino-acid preferences. The β-lactamase sequence is numbered using the Ambler scheme [82], meaning that residue numbers 239 and 253 are skipped. Comparable data for Gal4, NP, and HA are shown in Additional files 8, 9, and 10, respectively
There are sites in β-lactamase, NP, and HA that are under both diversifying and differential selection, but there are also sites that are only under one of these forms of selection (Fig. 3). These findings make sense: often, pressure for amino-acid change will drive multiple substitutions to non-preferred amino-acid identities, leaving traces of both types of selection. But sometimes, a relatively unconstrained site substitutes to a variety of different amino acids, leading to diversifying but not differential selection. In other cases, a site fixes just one or a few substitutions to a non-preferred amino acid that confers some enduring phenotypic benefit, leading to differential but not diversifying selection.
The identified sites of selection are consistent with existing biological knowledge
The ExpCM identified sites of differential and diversifying selection in all three genes that are undergoing adaptive evolution (β-lactamase, NP, and HA), while GY94 identified no sites with d N/d S<1 in any of the genes. But before concluding that this result indicates the superiority of the ExpCM, we must answer the following question: are the identified sites actually the locations of substitutions that have altered evolutionarily relevant phenotypes? To answer this question, I examined the literature on drug resistance in β-lactamases and immune escape by NP and HA (Table 3).
Table 3 At most sites of selection identified using ExpCM, mutations affect drug resistance or immune escape
For β-lactamases, [15] reports 18 sites at which mutations known to affect resistance are observed in clinical isolates. The ExpCM identify 9 sites of selection; 6 of these 9 sites are among the 18 known sites of resistance mutations (Table 3). There are 263 residues in the mature β-lactamase protein, so we can reject the possibility that the identified sites are not associated with resistance mutations (P=10−6, Fisher's exact test). So for β-lactamase, the ExpCM mostly identify sites that have been independently shown to affect drug resistance.
NP is under immune selection to escape T cells [10, 38] and probably also antibodies [54, 55]. The ExpCM identify 10 sites of selection. I searched the literature and found reports that 8 of these 10 sites are relevant to immune escape (Table 3). So for NP, the ExpCM mostly identify sites that have been independently shown to affect immunogenicity.
HA is under immune selection to escape antibodies. Caton et al [40] used antibodies to map escape mutations in H1 HA. A reasonable definition of the antigenic portion of HA is the set of sites identified in [40] plus any sites in three-dimensional contact with these sites (a contact is defined as a C α −C α distance ≤6Å in PDB 1RVX). Using this definition, 86 of the 509 sites in the HA ectodomain are in the antigenic portion of the molecule. The ExpCM identify 3 sites of selection, all of which are in the antigenic portion of HA. We can reject the possibility that these identified sites are not associated with the antigenic portion of the molecule (P=0.005, Fisher's exact test). So for HA, the ExpCM identify sites that have been independently shown to affect immunogenicity.
Overall, these results show that sites of selection identified by ExpCM are indeed the locations of substitutions that alter evolutionarily relevant phenotypes. For a concrete illustration of sites of adaptive substitutions that are identified by ExpCM but not by a d N/d S method, Fig. 5 shows the results of the ExpCM analysis of the five example sites in β-lactamase discussed in the Introduction and Fig. 1. Three of these five sites experience substitutions that affect resistance, but a d N/d S method fails to flag any of them as under diversifying selection (d N/d S>1) since it doesn't account for site-specific constraints (Fig. 1). Figure 5 shows that ExpCM correctly identify all three resistance sites as under diversifying or differential selection, while finding that the non-resistance sites are evolving as expected. Visual inspection of the two figures provides an intuitive explanation of why accounting for site-specific amino-acid preferences makes ExpCM so much more powerful at identifying sites of selection to alter evolutionarily relevant phenotypes.
The experimentally informed models (ExpCM) correctly identify the three β-lactamase sites in Fig. 1 that contribute to drug resistance. Figure 1 showed five sites in β-lactamase, three of which (238, 240, and 244) experience substitutions that contribute to drug resistance. However, a d N/d S analysis (GY94) fails to identify any of these sites as under diversifying selection (d N/d S>1) at a FDR of 0.05 for testing all sites (dotted blue lines). In contrast, ExpCM correctly determine that the three resistance sites are under diversifying (238 and 244) or differential (238 and 240) selection, and that the two non-resistance sites (201 and 242) are evolving as expected. ExpCM outperform the d N/d S method because they implement a null model that accounts for the site-specific amino-acid preferences shown in Fig. 1; for instance, this null model is not surprised that site 242 remains fixed at the highly preferred amino-acid R, but does find it noteworthy that site 240 substitutes to K multiple times even though that is not a particularly preferred amino acid
I have described an approach that uses experimentally informed models to identify sites of biologically interesting selection in protein-coding genes. This approach asks the following question: Is a site evolving differently in nature than expected from constraints measured in the lab? In contrast, traditional d N/d S methods simply ask: Is a site evolving non-neutrally? The former question is sometimes more informative than the latter. It is by now abundantly clear that most protein residues are under some type of constraint, so finding that a site evolves non-neutrally is often unsurprising. Instead, we want to identify sites of substitutions that have altered evolutionarily relevant phenotypes. As demonstrated here, experimentally informed models have much greater power to identify such sites. The improvement is remarkable: while a d N/d S method fails to find any sites of adaptive evolution in the genes examined, experimentally informed models identify 22 sites of diversifying or differential selection, most of which fix mutations that have been independently shown to affect drug resistance or immunogenicity.
What accounts for the improved power of the experimentally informed site-specific models? As vividly illustrated by the deep mutational scanning studies that provide the data used here (Fig. 2 and Additional files 2, 3, and 4), there is vast variation in the constraints on sites within a protein. Therefore, the significance that we should ascribe to a substitution depends on where it occurs: several changes at an unconstrained site may be unremarkable, but a single substitution away from a preferred amino acid at a constrained site probably reflects some powerful selective force. Whereas d N/d S methods treat all substitutions equally, the models used here evaluate the significance of each substitution in the context of the experimentally measured amino-acid preferences of the site at which it occurs.
Does this reliance on experimental measurements make the approach less objective? At first glance, the fact that d N/d S methods are uncontaminated by messy experiments feels reassuring. In contrast, experimentally informed models are dependent on all the subjective decisions associated with experimental design and interpretation. In addition, experiments in the lab may fail to fully capture all the selection pressures operating in nature. But in truth, experimentally informed models simply make explicit something that is already true: we define positive selection with respect to a null model for evolution in the absence of this selection. At least for the genes examined here, sites of known adaptive mutations are better identified by leveraging imperfect experiments that capture many of the constraints on natural evolution than by objectively testing the implausible null hypothesis that every site is evolving neutrally.
An assumption of experimentally informed site-specific models is that amino-acid preferences are conserved among the homologs under analysis. At first glance this assumption seems tenuous – epistasis can shift the effects of mutations as a gene evolves [56–58]. But it is rare for epistatic shifts to be large enough to undermine the advantage of site-specific models: this fact is demonstrated by direct experiments [32, 59, 60], the observation that parallel viral lineages tend to substitute to the same preferred amino acids at each site [61], and the empirical superiority of site-specific models in fitting phylogenies of diverged homologs (Table 2, [17, 32]). Therefore, epistasis does not subvert the basic advantage of a model informed by site-specific amino-acid preferences.
Of course, experimentally informed site-specific models require measurement of amino-acid preferences. However, advances in deep mutational scanning will make this requirement less and less of an impediment [11, 12]. In a fitting twist, one of the pioneers of deep mutational scanning [11] was also the first to sequence a gene from influenza [62, 63]. At the time, sequencing the homologous gene from thousands of other viral strains must have seemed unimaginable – a few decades later, for this study I had to subsample the ≫105 publicly available influenza sequences down to a manageable number. The core techniques of deep mutational scanning – sequencing and gene/genome engineering – are improving at a similar pace, so coming years will see measurement of the amino-acid preferences of many more genes.
Another possibility is to use non-experimental strategies to inform site-specific models like the one here. One strategy is to predict site-specific constraints from higher-level properties such as solvent accessibility [64–66] or via molecular simulation [67–70]. It remains unclear whether such non-experimental strategies can predict site-specific amino-acid preferences with sufficient accuracy to inform substitution models that can match the ExpCM used here. Another strategy is to infer preferences from naturally occurring sequences [30, 71–75]. If care is taken to avoid the over-fitting that could accompany inferring preferences from the same naturally occurring sequences that are being analyzed for selection, then this might be a viable approach. Indeed, while the current paper was under review, Rodrigue and Lartillot published an elegant study that implements an approach along these lines [76]. But I suggest that direct measurement of amino-acid preferences via deep mutational scanning may well prove the best solution in many cases: after all, biology is full of properties that are challenging to predict or infer, but are now routinely measured in high-throughput.
Overall, I have described a new approach that leverages high-throughput experimental data to identify sites of selection in protein-coding genes. This approach clearly outperforms a standard implementation of the widely used d N/d S strategy, however there is much room for improvement. The utility of the d N/d S strategy has been enhanced by innovations that have made it possible to do things like test for selection only along certain branches [6, 49], utilize Bayesian approaches to share information across sites [2, 50–52], better incorporate synonymous rate variation [77], and more rapidly perform the computational analyses [52, 78]. Most of these innovations could also be used in combination with the experimentally informed models described here. Methodological improvements of this sort, coupled with growing amounts of deep mutational scanning data, could make experimentally informed models an increasingly powerful tool to identify genotypic changes that have altered phenotypes of interest.
Software implementing the analyses
The algorithms described in this paper are implemented in the phydms software package, which is available at https://github.com/jbloomlab/phydms. This package is written in Python, and uses cython to interface with and extend Bio++ (http://biopp.univ-montp2.fr/, [41, 42]) for the likelihood calculations. Special thanks to Laurent Guéguen and Julien Dutheil for generously making the cutting-edge version of Bio++ available and providing assistance in its use. The software uses dms_tools (https://github.com/jbloomlab/dms_tools, [13]) and weblogo (http://weblogo.threeplusone.com/, [79]) for visualizing the results. The analyses in this paper used phydms version 1.2.3.
Amino-acid preferences for the four proteins
The amino-acid preferences were taken from previously published deep mutational scanning experiments. For NP, the preferences were taken from [32], using the average of the measurements for the two NP variants. For HA, the preferences were taken from [33]. For β-lactamase, [14] provides "relative fitness" scores, which are log10 enrichment ratios. I used the scores for the selections on 2.5 mg/ml of ampicillin (the highest concentration), averaging the scores for the two replicates. Following the definition in [13] of the preferences as the normalized enrichment ratios, the preferences π r,a are calculated from the relative fitness scores S r,a so that \(\pi _{r,a} \propto \max \left (10^{S_{r,a}}, 10^{-4}\right)\) and \(1 = \sum _{a} \pi _{r,a}\). For Gal4, [31] provides "effect scores", which are the log2 of the enrichment ratios. The preferences are calculated from the effect scores E r,a so that \(\pi _{r,a} \propto \max \left (2^{E_{r,a}}, 2 \times 10^{-4}\right)\) and \(1 = \sum _{a} \pi _{r,a}\). A few effect scores are missing from [31], so these scores are set to the average for all mutations for which scores are provided. The formulas to convert the β-lactamase and Gal4 scores to preferences include the max operators to avoid estimating preferences of zero; the minimal allowable values specified by the second argument to these operators are my guess of the lowest frequency that would have been reliably observed in each experiment.
For the comparison of the two different deep mutational scanning datasets for β-lactamase shown in Table 2, the measurements from the Firnberg et al [34] deep mutational scanning were converted into site-specific amino-acid preferences as described in [17].
Alignments of naturally occurring sequences for each protein
For NP, the sequence alignment was constructed by extracting all post-1950 full-length NPs in the Influenza Virus Resource [80] that are descended in purely human lineages from the 1918 virus (H1N1 from 1950–1957 and 1977–2008, H2N2 from 1957–1968, and H3N2 from 1968–2015), and retaining just two sequences per-subtype per-year to yield a manageable alignment. The rationale for using only post-1950 sequences is that most viruses isolated before then were passaged extensively in the lab prior to sequencing. For HA, the alignment was constructed by extracting all post-1950 sequences in the human seasonal H1N1 lineage (H1N1 from 1950–1957 and 1977–2008), and retaining just four sequences per year to yield a manageable alignment. For β-lactamase, the alignment consists of the TEM and SHV β-lactamases used in [17]. For Gal4, a set of homologs was obtained by performing a tblastn search of the Gal4 DNA-binding domain used by [31] against wgs (limiting by saccharomyceta (taxid:716545)) and chromosomes for hits with E≤0.01, and retaining only sequences that aligned to the Gal4 DNA-binding domain with ≥ 70% protein identity and ≤ 5% gaps. For all genes, alignments were made pairwise to the sequence used for the deep mutational scanning with EMBOSS needle [81], and sites were purged if they were gapped in that sequence.
Sequence numbering
In the figures and tables, the residues in NP are numbered sequentially beginning with one at the N-terminal methionine. The residues in HA are numbered using the H3 numbering scheme (the one used in PDB 4HMG), and the site-specific selection analysis is performed only for the residues in HA ectodomain (residues present in PDB 4HMG). The residues in β-lactamase are numbered using the Ambler scheme [82]. The residues in Gal4 are numbered using the scheme in [31].
The software package that implements the algorithms described in this paper is available at https://github.com/jbloomlab/phydms. The analyses were performed using version 1.2.3 of the phydms software. Data and scripts to perform the specific analyses are provided as Additional files 11, 12, 13, 14 and 15.
Reviewer Report 1: Sebastian Maurer-Stroh, Bioinformatics Institute (BII), A*STAR, Singapore
Reviewer summary –
Interesting well conceived approach.
Author response: Thank you for the kind words.
Reviewer recommendations to author –
This is an interesting approach to overcome simplifications of dN/dS site selection models by using site-specific experimental data from deep mutational scanning. As beautifully detailed and desirable this sounds, one should not forget that the experimental setup is detrimental for the types or aspects of protein function that can actually be investigated which directly influences the range of obtainable interpretations. For example, influenza hemagglutinin has multiple roles to fulfill on top of antigenic drift such as pH-dependent conformational changes and receptor binding. Similarly, functional roles of the nucleoprotein are not only thermal stability and immune response evasion but also RNA packing and sub-cellular shuttling. Also beta-lactamases will mutate differently under different pressures from different antibiotics or in competition with other bacteria. The difficulty of the experimental setup to represent the full complexity of natural selection pressures is not always just a limiting factor but looking only at some aspects of function at any one time allows elegantly gauging details of specifically targeted evolutionary forces at play. The notion of the critical influence of the experimental setup is mentioned in the discussion but would be good to be included also in the introduction.
Author response: This is an important point. I have elaborated the paragraph in the Discussion that describes how experiments in the lab will sometimes fail to fully capture selection in nature (This is the paragraph beginning, "Does this reliance on experimental measurements make the approach less objective?" I have also added mention of this point in the Introduction by emphasizing that "lab measurements are undoubtedly imperfect proxies for actual selective constraints in nature."
The reviewer also makes excellent points regarding influenza hemagglutinin in particular. Although I do not go into these points in the current manuscript (which focuses more on the general approach than the details of HA), the reviewer's intuition is validated by recent work for my group specifically focusing on HA [ 44 ] which found that the experimentally informed models identify both sites of actual positive selection from immunity and sites subject to lab-specific selection pressures related to proteolytic activation of HA. However, despite these caveats, I think that the current manuscript clearly demonstrates that site-specific models informed by imperfect experiments are superior to the much more unrealistic standard non-site-specific models.
The formalism of the approach is well developed and intuitively makes sense but the practical result for hemagglutinin left me a bit wanting. Certainly the identified sites for HA in Table 3 are important but they seem only a small subset of such sites that can be identified with other methods (e.g. SLAC from HyPhy package over naturally occurring sequences finds dozens that can be rationalized to make sense through overlap with known epitopes etc). Could it simply be that the, in some cases, used heuristic Gal4-based thresholding is too conservative and considering less stringent criteria would find more of the presumably true sites?
Author response: I think the relative paucity of sites identified in HA is due to the fact that the analysis focuses on seasonal H1 HA rather than H3 HA. For instance, I ran the H1 HA alignment used in this paper through SLAC as implemented in the DataMonkey web interface to HyPhy (data not shown). The SLAC analysis only identified two sites of positive selection for the H1 alignment. I would expect that all approaches would identify more sites in H3 HA, since human H3N2 influenza undergoes more rapid antigenic drift than human seasonal H1N1 influenza [ 83 ]. Such an analysis will be possible once deep mutational scanning data are available for an H3 HA.
By the way, the criterion of Caton epitope residues plus everything within 6A does includes a lot of structurally buried residues. Maybe an additional surface accessibility criterion to enrich for direct epitope candidates may be justifiable here? If I am not wrong, HA 225 (in H3 numbering in Table 3) is a classical host/passage specificity position in H1 context and it is good to be highlighted by the new approach but its potentially broader functional importance on receptor binding should also be mentioned and referenced accordingly.
Author response: These are both good points. The three sites of selection listed in the table are at least partially surface-exposed. As the reviewer points out, some of the 89 sites within 6Å are buried, and so are probably not true antigenic sites. Accounting for this fact would deflate the denominator in the Fisher's exact test that we use to test the significance that we are identifying true antigenic sites, and so further improve the P-value for supporting the validity of our ExpCM method. However, I prefer to be conservative and keep all 89 sites in the denominator, since in some cases mutations at buried sites may still introduce slight conformational changes or N-linked glycosylation motifs that escape antibodies.
The point about HA site 225 in receptor-binding is a good one. I have added a line in the table that emphasizes that mutations at site 225 are implicated in both host adaptation and lab passaging adaptation via changing receptor binding, and have cited the following relevant references: [ 84 – 87 ].
The following additional points are meant to stimulate further thoughts for future work: Empirical average (neither site- nor protein-specific) amino acid substitution tables have been derived en masse since the early works of Dayhoff (PAM, JTT, BLOSUM,...). Picking one of the most popular, BLOSUM62, how similar or different is it for the studied proteins' ExpCM results?
Author response: Good question. Empirical amino-acid substitution matrices themselves cannot be directly substituted for codon substitution models. But there are a variety of empirical codon substitution models, which combine empirical amino-acid substitution models with codon substitution models. One such set of models are Kosiol et al 2007 models [ 88 ]. In prior work [ 16 , 17 , 33 ] I have compared these Kosiol 2007 models to the various forms of the Goldman-Yang style models used here, as well as earlier versions of the ExpCM. As described in that prior work, the Kosiol 2007 models in general were not substantially better (and were often actually worse) than the Goldman-Yang models in terms of phylogenetic fit. Therefore, it appears that an empirical model that tries to account for amino-acid substitutions in a way that is NOT site-specific does not lead to substantial improvements. This is probably because protein-level constraints are highly site-specific, and cannot effectively be modeled in an "average" across sites.
Classical substitution matrices are traditionally derived from globular regions of proteins forming 3D structures but un- or dynamically structured N- or C-terminal stretches are also under selection pressure for targeting motifs and other constraints. An unbiased but complete scanning method may be equally applicable also in non-globular regions and pinpoint critical sites often neglected by earlier approaches?
Author response: This is another good question. As the reviewer suggests, I would expect that perhaps the site-specific amino-acid preferences for unstructured protein domains to be quite a bit different than for globular proteins. To my knowledge, no one has yet performed deep mutational scanning on an unstructured protein domain. But once such experiments are done, as the reviewer suggests, it would be very interesting to test whether such experiments could inform substitution models.
On the complexity of adaptive mutations in the substrate binding pocket of beta-lactamases, I found it curious that antibiotics resistance genes in microbiomes of an un-contacted Amazonian tribe had the capacity to also neutralize synthetic man-made antibiotics they have never been exposed to (http://www.sciencemag.org/news/2015/04/resistance-antibiotics-found-isolated-amazonian-tribe). This highlights plasticity of the natural repertoire of substrate binding pocket residues to accommodate a broad range of unknown substrates directly or with few mutations.
Author response: This is an interesting observation. As more deep mutational scanning data sets become available, it will be interesting to compare the inherent plasticity of different active sites.
Adaptive mutations are of great importance not just in the context of pathogens but it would be interesting to also apply deep scanning and ExpCM on key genes in human diseases (P53, KRas, EGFR, …)
Author response: This is a great suggestion. Some recent studies by other groups have already started to move in this direction; see for instance [ 89 , 90 ]. These studies may have the potential to aid in the prospective identification of disease-causing human mutations.
Last but not least, the manuscript and suppl. material with code links are commendably complete descriptions of the work.
Author response: Thanks! Hopefully the availability of the code and data will help enable others to extend and improve the approaches described in this manuscript.
Additional responses from reviewer after reading the revised version. The quoted text indicates the author's comments in the revision:
"I think the relative paucity of sites identified in HA is due to the fact that the analysis focuses on seasonal H1 HA rather than H3 HA." Indeed, could be true.
"... keep all 89 sites in the denominator, since in some cases mutations at buried sites may still introduce slight conformational changes or N-linked glycosylation motifs that escape antibodies." Ok to keep all 89 sites for this paper but remove in the response the comment on buried N-glycosylation sites. The latter most commonly are not buried due to the simple necessity of access for the modifying enzyme machinery [91].
Author response: The reviewer is correct that the N-linked glycans themselves are not buried. I had meant that in some cases the Ser/Thr in the Asn-Xaa-Ser/Thr glycosylation motif might be buried, but admittedly this is probably a rare event.
"As described in that prior work, the Kosiol 2007 models in general were not substantially better (and were often actually worse)... " Sure, I did not mean that they would be better in performance but more that it might be interesting to study trends in observed differences to possibly improve them with some extra rules e.g. something that would filter out less reliable sites where differences are always high. In other words, some amino acid substitution pairs may be more site-specific than others? In any case, partially addressed before and possible extension for future work.
Author response: I agree that this is an interesting area for future work.
"To my knowledge, no one has yet performed deep mutational scanning on an unstructured protein domain. But once such experiments are done, as the reviewer suggests, it would be very interesting to test whether such experiments could inform substitution models." Most proteins are not fully structured but typically feature flexible N- and C-termini as well as often only partially structured longer loop regions. One way to define these unstructured regions is by looking for unresolved residues in crystal structures despite being part of the used sequence. These are easy to see when looking at the sequence tab of PDB files online. In fact for the H1N1 HA deep scan, it seem the author has deep scanning data for ∼18 unstructured residues in the N-terminus and 60 in the C-terminus [44]. Surprisingly there seems to be quite some constrained sites in the C-term here which also may point to functional importance as motifs or partial or conditional structure.
Author response: This is a good idea – it would be interesting to specifically look at unstructured regions in proteins that have already been studied by deep mutational scanning. Such an analysis is beyond the scope of the current study, but is an interesting topic for future work. As the reviewer notes, the conservation at some sites in the C-terminus of HA is compatible with the fact that parts of the transmembrane domain and cytoplasmic tail are important for virion formation, such as via interactions between HA's cytoplasmic tail and the matrix protein.
Reviewer Report 2: Olivier Tenaillon, INSERM, France
In his manuscript entitled "Identification of positive selection in genes is greatly improved by using experimentally informed site specific models", Jesse Bloom propose to use quantitative information based in deep mutational scanning experiments to detect selection in phylogenies. In previous articles, he proposed to use such information to improve the phylogenetic reconstruction, in the present one he extends the approach to detection of selection, the rational being that a better underlying model allows a finer detection of selection, and a site specific model gives more power to detect local effects. He applies his method to 4 genes, one in which no selection is expected and 3 in which there are target sites for selection. The results suggest a better detection of sites under selection. I really appreciated the approach used and have just minor comments.
Author response: Thank you for the nice summary and kind words about the manuscript.
The method relies on the use of deep mutational scanning experiments, but does not mention how good and precise these experiments have to be. For instance, the Stiffler et al experiments [14] on beta-lactamases are done after 3 generations of growth and give mostly a growth, no growth information (actually Firnberg and Ostermeier data [34] would have been more appropriate as they provide a much finer resolution). Indeed, in that paper the distribution of fitness is almost completely bimodal for mutation effects. These experiments are much less costly than others that will do deep scanning with much more time points (or concentrations) and therefore with higher fitness resolution for the mutants. So how important is the precision of the experimental data? Would a binary fit for each amino acid mutation work as well? This is important for two reasons: first it can define somehow that price required to get a good signal with mutational scanning. Second, if the data are binary, then mutation prediction approach may be relevant. In a recent paper, Figliuzzi et al (MBE, 2016 [75] that should at least be cited along side with Hopf in arRxiv [74]), Martin Weigt's group showed that the DCA and Independent model based on protein alignment were providing a good prediction of mutation effects produced in experiments especially on grow no-grow kind of data. If the improvement of the present approach is not very sensitive to the quality of the experimental data, then it would gain incredibly in usage if predictions from pfam alignment such as the ones done by DCA were used rather than costly experiments.
Author response: These are all great points.
The first question is how to choose the which deep mutational scanning dataset to use to inform the substitution models. As the reviewer points out, there are currently two deep mutational scanning datasets for beta-lactamase: the one by Stiffler et al used in the current manuscript [ 14 ], and an earlier dataset by Firnberg and Ostermeier [ 34 ]. In prior work [ 17 ], I have shown that the Firnberg dataset also improves phylogenetic fit. But the initial version of this manuscript only used the newer Stiffler data set. So how do we know which is better? We can compare how well different deep mutational scanning datasets actually describe the constraints on natural evolution using maximum-likelihood phylogenetics via AIC, exactly as is traditionally done to compare substitution models [ 43 ]. Specifically, we can perform phylogenetic fitting of ExpCM informed by each dataset to see which one yields a higher likelihood of the actual natural sequences. The new Table 2 now includes analyses with ExpCM informed by each deep mutational scanning dataset. As can be seen from this table, ExpCM informed by the Stiffler dataset describe the natural evolution of β -lactamase better than ExpCM informed by the Firnberg dataset ( Δ A I C = 204). Therefore, by the criterium typically used to compare substitution models, the Stiffler dataset is superior. Note however that either dataset informs ExpCM that are clearly better than standard GY94-type models.
The foregoing analyses do not provide a basis for concluding why the Stiffler dataset is superior to the Firnberg one. As the reviewer notes, one difference is that the more extended selection in the Stiffler et al experiments leads to more binary measurements. But the differences could also be due to reasons that are more technical than biological. For instance, Stiffler et al perform two full biological replicates of their deep mutational scanning, and I have used the average of the two replicates – this averaging presumably reduces experimental noise. In contrast, Firnberg et al did not perform replicates of their experiment, so perhaps there is more noise that has not been averaged away. Consistent with this idea, analyses of other genes have shown that averaging across experimental replicates of deep mutational scanning typically improves ExpCM [ 44 ], presumably by reducing the effects of measurement noise.
Thanks for pointing out the Figliuzzi et al [ 75 ] study that predicts mutational effects from sequence alignments. I have added mention of this study to the paragraph in the Discussion that addresses whether site-specific amino-acid preferences could be computationally inferred from natural alignments rather than measured experimentally (this is the paragraph beginning "Another possibility is to use non-experimental strategies to inform site-specific models like the one here."). The short answer is that I do not know whether computational methods like those used in Figliuzzi et al [ 75 ] could be used in place of deep mutational scanning – but certainly I agree that this would greatly expand the utility of approaches like the one that I describe in the current manuscript. One caveat about inferring the preferences from natural sequence alignments is that care must be taken to avoid over-fitting the data, as the preferences would then come from the same alignment that is being analyzed phylogenetically – in my current manuscript, the preferences are from a separate dataset (the deep mutational scanning) from the natural sequence alignment. However, it may be possible to infer the preferences without overfitting – see for instance a paper by Rodrigue and Lartillot [ 76 ] that was published while the current manuscript was under review. Certainly I hope that the current manuscript will help inspire future work to see if the site-specific amino-acid preferences can also be obtained in other "cheaper" ways than deep mutational scanning – although I would note that deep mutational scanning itself is also getting progressively cheaper.
The differential selection is interesting but not as intuitive than the diversifying one. The experiments being made in the lab, they may lack some facets of selection. So the test will tell us if sites are significantly different from the selection in the lab. However, we can not, in many cases, know whether this is a true mark of selection in the wild or a limited power of the experimental setting to provide a good model.
Author response: This is a good point. I have added text to the Discussion that emphasizes that the diversifying selection test looks for differences between selection in nature and what is expected given measurements in the lab. I have emphasized why this will sometimes (but not always) be informative for identifying mutations of biological interest.
In the different sets of genes studied here the difference of selection between laboratory and other experiments is relevant: lack of immune system, or lack of new antibiotic, but how general can that be? It could be worth discussing briefly that issue, to give some intuition to future users about the meaning of the signal they may get.
This is a good point. I have added text that describes how the tests are especially useful when we know that there are selection pressures (such as immunity or drug resistance) that are present in nature but not in the lab. Similar situations where there are known external pressures in nature but not in the lab will occur sometimes (as in the case of the influenza genes and β -lactamase), but not in other cases (such as Gal4).
It would be appropriate to plot the trees of each gene alignment that are used for inference and present the state of the candidate mutations.
Given the large number of candidate mutations, it is not feasible to make trees that display the states of each of the relevant sites for all genes. However, I have included the phylogenetic trees in the relevant Additional files so that those can be opened in a program such as FigTree to examine the trees and map mutations to the branches.
Minor issues –
Are all mutations with a signal reported in the violin graphs?
Author response: Yes, in the violin plots, the points indicate all mutations with a signal of either differential or diversifying selection.
Shouldn't "beta-lactamase" be used throughout the paper rather than "lactamase"?
Author response: Yes. In the revised version, I have made sure to fully write out " β -lactamase" rather than sometimes just saying "lactamase."
Reviewer Report 3: Tal Pupko, Tel Aviv University, Israel
Dr. Bloom is pushing forward an innovative idea: to integrate data from deep mutational scanning to improve the performance of the challenging task of identifying positively selected sites. To this end, he proposes a novel codon model that explicitly integrates such data within its parameters. I enjoyed the new concept, and I was convinced by the benefit of integrating such experimental data to improve dN/dS methods. I have some comments and suggestions to make the manuscript more accurate and informative.
Author response: Thank you for the nice summary of the manuscript and the kind words.
All comments (major and minor) in the order they appear in the manuscript.
I felt that the first sentence is phrased in a non-scientific language. It is written that an important goal is to "identify genetic modifications that have led to interesting changes in phenotype." Who decides what is interesting and what is not? I would rephrase to states that scientists want a better map between generic modifications and phenotypic variation.
Author response: I have changed the word "interesting" to "evolutionarily significant," which seems less subjective. However, I think some level of subjectivity is inherent in studying phenotypic changes. The researcher defines what is considered a phenotype that is worthy of study: for instance, in influenza virology we generally consider mutations that alter immunogenicity or host tropism to be "important," and in the study of bacterial antibiotic resistance genes we typically consider as "important" mutations that enhance resistance to new drugs. But our choice to focus on those phenotypes is somewhat subjective. The approach in the current manuscript identifies sites that are evolving differently in nature than expected from experiments in the lab – but the choice to compare natural evolution to the "null model" of experiments in the lab is subjective, and is guided by the idea that pressures present in nature but absent in the lab are often relevant to phenotypes we consider "important" (for instance, immunogenicity for influenza, or extended-spectrum drug resistance for lactamase). I have elaborated on this point in the Discussion in the paragraph beginning "Does this reliance on experimental measurements make the approach less objective?"
In page 2, it is written "for protein-coding genes, the most widely used methods for identifying specific sites of selection are built around the null model that non-synonymous and synonymous mutations should fix at equal rates." I think this is inaccurate. Most biochemists interested to find purifying selective forces acting on their protein of interest do not use dN/dS methods. Instead, they use tools such as Consurf, which explicitly account for the physiochemical nature of the amino acids. Codon models are almost only used when explicitly searching for positive selection.
Author response: This is a good point. I have changed "specific sites of selection" to "specific sites of positive selection."
Page 2, change "amino-acid mutation" to "non-synonymous mutation."
Author response: Thanks for catching this inconsistency in word usage, I have made this change.
Page 2, it is claimed "detecting purifying selection as manifested by dN/dS < 1 points more to the naivety of the null model than unexpected biology". As stated above, from a biochemical perspective it is highly important to know which sites are highly conserved and which ones are not. Such information is used, for example, for predicting which sites are buried and which are exposed to the solvent, which mutations are likely to cause diseases, and when the molecular mechanism of an enzyme is elucidated. Thus, when a codon model predicts and quantifies sites as being evolved under dN/dS < 1, this points to the fact that the model genuinely captures variation in purifying selective forces among amino acid sites. It does not point for a naivety of the model. Further, for dN/dS < 1, there is not a null model and an alternative model (which is not the case when searching for positive selection), so it is not clear what "null model" is in this statement.
Author response: These are good points. I have simply removed the referenced sentence altogether, since it is unclear for the reason that reviewer notes. Specifically, the reviewer is correct that (depending on the question at hand), finding dN/dS <1 may be important (for instance, it is important for identifying disease-causing mutations, but not for finding viral immune escape mutations). However, it is true that many methods (such as FEL, FUBAR) for analyzing site-specific selection test both the alternative model dN/dS <1 and the alternative model dN/dS >1 against the null model dN/dS = 1, so for these methods there is a null model when testing dN/dS <1.
Regarding the paragraph starting with "perhaps more importantly, dN/dS methods also have limited power to identify sites that have fixed adaptive mutations": the term "fixed adaptive mutations" should be explained. Further, it is claimed that dN/dS methods have limited power, but only one example is provided (ref 10). As it is stated, the claim is not supported.
Author response: I have changed the text from "also have limited power to identify" to "also can fail to identify." This avoids a blanket statement that dN/dS methods lack power, since as the reviewer points out, I only cite a single example. However, I think that example justifies the statement that dN/dS methods can fail. And of course, the results in the current manuscript provide many more examples of sites of immune-escape or drug resistance mutations that are under positive selection but are not identified by a standard dN/dS method but are identified by the ExpCM.
It is written that "the limitation of the null model that assumes equal rates of fixation of non-synonymous and synonymous mutations have become... ". The standard codon models assume omega varies over sites according to a beta distribution (sometimes, a gamma distribution is assumed). By doing so, they assume that for most sites, the fixation rate of non-synonymous mutations is lower than the rate of synonymous mutations. Hence this statement in this sentence is inaccurate.
Author response: I have re-written the text to read: "The limitations of simply comparing the rates of fixation of nonsynonymous and synonymous mutations have become especially glaring in light of deep mutational scanning experiments." This statement along with the remainder of the paragraph effectively captures the key point that when sites are under very different levels of inherent constraint, a method that does not assign a different expectation of the expected constraint to each site will have difficulty identifying positive selection at constrained sites.
In the last paragraph of the introduction it is written "But most sites strongly prefer one or a few amino acids; dN/dS methods do not offer a plausible null model for these sites". This is again, inaccurate. There were many efforts to include amino acid preference with codon models. See for example (1) "An Empirical Codon Model for Protein Sequence Evolution", a paper from the group of Nick Goldman; (2) "Empirical codon substitution matrix", from the group of Gaston Gonnet; (3) "A Combined Empirical and Mechanistic Codon Model" from my own group; (4) A book chapter about empirical and semi empirical codon evolutionary models in the book "Codon Evolution: Mechanisms and Models" edited by David Liberles.
Author response: I have re-worded the sentence in question. However, none of the references mentioned by the reviewer include site-specific constraints. They do treat different nonsynonymous substitutions differently, but this treatment is the same across sites (with the possible incorporation of a distributed rate parameter). Therefore, the stationary state of these models is homogeneous across sites (a rate parameter does not alter the model's stationary state since it is simply a constant multiplying the transition matrix). The key difference of the ExpCM used here is that the treatment of each nonsynonymous substitution depends on the site, and so each site has a different stationary state. The re-wording of the sentence should better emphasize the key distinction.
I had difficulties to understand figure 1A. To the best of my understanding, a comparison is made between amino acid preferences as measured by the deep mutations scanning of Stiffler et al. to the amino acid preferences in "nature". However, it is not clear how the amino acid preferences in nature were computed. In addition, in Stiffler et al. several deep mutations scanning experiments were conducted. Which one is presented and why? It should also be better explained in which sites positive selection is expected, what is the "real" omega, what is the inferred omega of PAML.
Author response: I have clarified the figure. I have added text to explain that the preferences shown in the figure are for the measurements from deep mutational scanning. None of the preferences are taken from natural sequence data – instead, there is just a comparison with which mutations are common in the naturally occurring sequences. I have added text to the legend explain that the Stiffler data is from the experiments with the highest concentration of ampicillin (this was previously explained only in the Methods). The violin plots show the P-value for ω >1 for each site computed using the FEL method; these P-values are shown rather than the ω value itself because site-specific estimates of ω are known to be numerically unreliable and so most methods focus on estimating the P-value (or posterior probability) of ω >1 rather than the numerical value of ω itself. The top column of text explains which sites are implicated in extended-spectrum antibiotic resistance; these are the ones that might reasonably be posited to be under positive selection.
The methods are compared only to the "Goldman-Yang model" from 1994. In 1994 Goldman and Yang were the first codon model published, back to back with a paper by Muse and Gaut. I would suggest to use codon models that are used now, e.g., the M8-M8A model. Also, I am not sure that the real codon model proposed in Goldman and Yang (1994) was used. GY94, as stated in that paper in equation 3 includes explicit consideration of the amino acid type. Maybe the Muse and Gaut (1994) codon model was used instead? In light of these comments, I suggest that more details are provided for this figure so that readers can be convinced that a problem with standard codon models exists.
Author response: This is a good point. The original Goldman-Yang paper [ 23 ] includes the possibility of weighting substitutions by amino-acid similarity ( d ij terms in their notation). In subsequent work [ 4 ], Yang and Goldman largely abandoned these weightings (i.e., made all d ij ) terms equal, and then defined various variants of these models (e.g., M0, M3, M8, etc). However, the literature commonly refers to all these model variants as "Goldman-Yang" style models, even though the reviewer is correct that they do not contain the weightings in the original Goldman-Yang paper. To clarify this, I have explicitly indicated that I have used specific M variants of the Goldman-Yang style models (e.g., M0) as defined in [ 4 ]. As far as the M8 model, I have chosen to instead use the M3 model for this paper. Like the M8 model, the M3 model allows multiple categories of ω . In earlier work using similar models [ 17 ], I have shown that the M3 and M8/M8a models give comparable performance.
Figure 1, the P values are corrected for multiple testing using FDR. But in the legend it is written that Bonferroni correction is used. Maybe this should be better clarified?
Author response: The tests were performed using an FDR. But in the case where there are no sites that are significant at an FDR of 0.05, the blue line indicates the P-value that would be needed by a single site to be significant with P =0.05 using a Bonferroni correction. This is equivalent to the FDR cutoff for just one site, since FDR and Bonferroni are identical when there is just one significant site. I have added text to clarify this.
In the last paragraph of the introduction it is claimed that the goal is to detect sites under "differential selection for unexpected amino acids". Is this identical with the goal of "detecting sites evolving under positive selection"? There are many other works that aim to detect selection shifts (e.g., the extensive literature on covarion models). This is not the same as to detect positive selection.
Author response: This is a valid point, although as the results in the manuscript show, in many cases the sites of differential selection turn out to be sites of adaptive mutations. I have added a sentence in the last paragraph of the Introduction emphasizing that this strategy "seeks to identify sites that are evolving differently in nature than expected from constraints measured in the lab." As I think the subsequent results show, in many cases these sites turn out to be ones that have fixed immune-escape or drug-resistance mutations that would typically be envisioned as having arisen from positive selection for adaptation.
The first part of the results is dedicated to a description of the ExpCM model. It is written: "The ExpCM used here are similar but not identical to those in [16,17]". However, the differences are not explicitly stated nor are the reasons for changing the model. I suggest making this statement more explicit.
Author response: Good suggestion. I have clarified in the text how the ExpCM differ. They differ by including the ω term, and by using a slightly different model (an HKY85 model) for handling the nucleotide mutation rates.
In Equation 5, variability in the synonymous rate among sites is included. Why not to include it already in the null model, i.e., Equation 3 (see also "Towards realistic codon models: among site variability and dependency of synonymous and non-synonymous rates." [77])? Also, when comparing to the standard model, how can one know the contribution of adding the data from the deep mutation scanning versus the contribution to power stemming from adding a component of synonymous variation over sites? At any rate, a more elaborate way to test for deviation from the null model, would be to generate an alternative model for all sites that would allow omega to vary across sites. Then to estimate, for example, the posterior expectation of omega for each site. Such an approach would allow for example to account for uncertainty in model parameters, by adding a BEB (Bayes Empirical Bayesian) component.
Author response: The synonymous rate variation is not included in the gene-wide model, but is included in the site-specific fitting to test for diversifying selection. Specifically, when fitting Equation 5, the null model is to fit just μ r (synonymous rate) and fix ω r =1, while the alternative model is to fit both ω r and μ r , so this ensures that any improvement in site-specific estimation is not due to the synonymous rate. The reason I have taken this approach is that it is used in the FEL approaches (and many other standard approaches) to which the comparisons are made (see [ 27 ]). Therefore, all the comparisons between the null and alternative models of both the ExpCM and more standard GY94-style models handle synonymous rate variation comparably, ensuring an apples-to-apples comparison.
The reviewer is correct that real biological processes might involve synonymous rate variation as well. This possibility is nicely discussed in the reviewer's own paper on the topic [ 77 ]. Therefore, in the concluding paragraph of the Discussion, I have cited [ 77 ] and added mention of how better incorporating synonymous rate variation might be one possible way to extend/improve ExpCM. Note however that the pros and cons of incorporating synonymous rate variation remain a topic of active debate [ 92 ], although I tend to side with the reviewer [ 77 ] and others [ 27 ] that incorporating such variation is beneficial.
I agree that using an empirical Bayes approach is an alternative framework to try, although again the relative pros and cons of these so-called "random effects" methods versus their "fixed effects" alternatives remains a topic of active debate [ 5,53,93 ]. I discuss this issue and the possibility of extending ExpCM to an empirical Bayes framework in the Results paragraph beginning "A variety of other d N/d S methods have also been developed."
Below equation 5 it is written that the key statistic is not omega itself, but rather the P-value. I don't think the P value is the statistic, but rather, the log likelihood ratio.
Author response: I have clarified this point by stating that the key statistic is the difference in log likelihoods (the likelihood ratio), from which a P-value can be computed.
The balancing term introduced in equation 7 seems to be equivalent to assuming a specific prior distribution over the amino acid distributions. However, the connection with a prior distribution is very implicit. I suggest moving to a Bayesian approach and thus making this prior assumption explicit. If this is not feasible in the current version of the manuscript, please consider stating this link to an implicit prior.
Author response: In a Bayesian approach, this equation would be the equivalent of a prior. However, since the current manuscript uses a maximum-likelihood approach, the equation is better thought of as a regularization term, since we are not actually sampling from the posterior established by the likelihood and prior, but rather simply maximizing the likelihood subject to the regularization established by Equation 7. I have added a sentence making this link between regularization and a Bayesian prior. Given the current computational implementation, it is not straightforward to move the analysis to a Bayesian approach. But as mentioned in the response two before this one, this is an interesting area for future work, and one that I discuss in the manuscript.
When comparing the power of the ExpCM method to "GY94", it seems to me that there is also a difference in the false positive rate.
Author response: This is true. The ExpCM has a false-discovery rate that is close to what is expected given the FDR of 0.05, while the GY94 has a lower false-discovery rate but also a much higher false negative rate.
Minor comments -
Consider reducing the number of additional files and move some info into the main text.
Author response: I admit there are a lot of additional files. However, for both an earlier version of this manuscript submitted elsewhere and the first version I posted on bioRxiv, I received exactly the opposite complaint that there were too many figures that would be better moved to additional files! So I think I am going to keep it as is, knowing that in the final published version (which will have working links) it will be much easier for the reader to access the additional files.
AIC:
Akaike Information Criterion
ExpCM:
Experimentally informed codon model
GY94:
Goldman Yang 1994 substitution model
Hemagglutinin
NP:
Nucleoprotein
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Tremendous thanks to Laurent Guéguen and Julien Dutheil for developing the Bio++ libraries, generously making the cutting-edge version of this software freely available, and providing assistance in its use. Thanks to Erick Matsen and Sergei Kosakovsky-Pond for helpful comments. This work was supported by the NIGMS of the NIH under grant R01GM102198.
This work was supported by an NIH/NIGMS grant (R01GM102198) and a Pew Scholars grant to JDB. The funders had no role in study design, data collection and interpretation, or the decision to submit the work.
All custom scripts and data are available in Additional files 11, 12, 13, 14 and 15. The software is available at https://github.com/jbloomlab/phydms.
The contact information for JDB is: Division of Basic Sciences and Computational Biology Program, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N, Seattle, WA 98109. E-mail: [email protected].
The author declares that he has no competing interests.
Division of Basic Sciences and Computational Biology Program, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N, Seattle, 98109, WA, USA
Jesse D. Bloom
Correspondence to Jesse D. Bloom.
Graph of the function used to regularize the Δ π r,a values when inferring differential selection. The log of the regularization defined by Eq. 7 is a sum of terms like this taken over all differential preferences at a site. This regularization has the property that the marginal cost of shifting Δ π r,a away from zero is initially steep but then flattens somewhat as Δ π r,a becomes large. This corresponds to the intuition that most sites will be evolving as expected (and so have Δ π r,a ∼0), but a few sites might be under strong differential selection. This plot uses C 1=150 and C 2=0.5. (PDF 284 kb)
Site-specific amino-acid preferences for Gal4. Shown are the preferences experimentally measured by [31] for the DNA-binding domain of yeast Gal4, re-scaled by the stringency parameter β= 0.82 from Table 2. (PDF 25 kb)
Site-specific amino-acid preferences for NP. Site-specific amino-acid preferences for influenza NP. Shown are the preferences experimentally reported in [32] for the average of the measurements on the A/PR/8/1934 and A/Aichi/2/1968 strains, re-scaled by the stringency parameter β=2.43 from Table 2. (PDF 68 kb)
Site-specific amino-acid preferences for HA. Shown are the preferences experimentally measured by [33] for influenza HA (A/WSN/1933, H1N1 strain), re-scaled by the stringency parameter β= 1.61 from Table 2. The residues are numbered according to the H3 numbering scheme (the one used in PDB 4HMG), and data are only shown for sites in the HA ectodomain (residues present in the crystal structure in PDB 4HMG). (PDF 100 kb)
Simulations validate the statistical approach used to identify diversifying selection. Using the actual ExpCM parameters for NP in Table 2 except fixing ω=1 for all sites except for those selected to be simulated under diversifying selection, I used pyvolve [46] to simulate 40 alignments along the tree inferred from the actual NP sequences. For each simulation, I randomly selected 5 sites to place under diversifying selection, with ω r values ranging from 1 (no diversifying selection) to 30 (very strong diversifying selection). I then analyzed the data using phydms in the same way that the actual data were analyzed. Sites were called as being under significant diversifying selection using the false discovery rates (FDRs) indicated in the figure. The top panel shows that ExpCM greatly outperformed the FEL-like GY94 method at identifying true positives. The bottom panel shows that the Benjamini-Hocbherg [28] procedure effectively controls the fraction of false discoveries among the sites called as being under diversifying selection using ExpCM. The Benjamini-Hochberg procedure may be slightly too conservative for ExpCM (for every value of ω r the actual rate of false discoveries is slightly below the FDR), but the differences seem modest. The computer code to perform these simulations is in Additional file 17. (PDF 150 kb)
This figure is same as Fig. 3 a but also includes an analysis with MEME [6] as implemented in HyPhy [7]. MEME reports the P-value that a site has d N/d S>1 on at least some branches of the tree. As can be seen from this figure, MEME is somewhat more powerful than the GY94-based FEL approach, presumably because some sites are only under episodic diversifying selection. While the GY94-based FEL approach identifies no sites of diversifying selection, MEME identifies one site of diversifying selection in β-lactamase and one site in NP. However, MEME still identifies fewer sites for all genes than the ExpCM. (PDF 287 kb)
This figure shows the distribution over sites of the Kullback-Leibler divergence of the experimentally measured amino-acid preferences from the alignment frequencies. Note that the Kullback-Leibler divergence does not take phylogeny into account, and so will be confounded the incomplete sampling of potentially tolerated amino acids by natural evolution. The distribution of per-site Kullback-Leibler divergences shown here lacks the biologically sensible features of the differential selection computed in a phylogenetic framework and shown in Fig. 3 b. For instance, Gal4 has many sites with very high Kullback-Leibler divergence even though on biological grounds we expect it to be evolving mostly in the absence of positive selection. In contrast, β-lactamase and NP tend to have lower Kullback-Leibler divergence even though we know that they evolve under selection for adaptive mutations that confer drug resistance or immune escape. The biologically unreasonable distribution of Kullback-Leibler divergences shown in this plot are probably due to the failure of the Kullback-Leibler divergence to account for phylogeny, which may in turn make the results highly sensitive to uneven phylogenetic sampling and differences in the total sequence divergence spanned by the alignments (see Table 1). The Kullback-Leibler divergence was computed using logarithms taken to the base two. (PDF 86 kb)
Site-specific selection on Gal4 inferred with the experimentally informed models. This figure is equivalent to Fig. 4 but for Gal4. (PDF 252 kb)
Site-specific selection on NP inferred with the experimentally informed models. This figure is equivalent to Fig. 4 but for NP. (PDF 278 kb)
Additional file 10
Site-specific selection on HA inferred with the experimentally informed models. This figure is equivalent to Fig. 4 but for HA. (PDF 282 kb)
The data and code for running the analysis for Gal4. This is a 7-Zip file containing an iPython notebook and the relevant data files. (7z 1260 kb)
The data and code for running the analysis for lactamase. This is a 7-Zip file containing an iPython notebook and the relevant data files. (7z 2692 kb)
The data and code for running the analysis for NP. This is a 7-Zip file containing an iPython notebook and the relevant data files. (7Z 2910 kb)
The data and code for running the analysis for HA. This is a 7-Zip file containing an iPython notebook and the relevant data files. (7Z 3000 kb)
The data and code for running the pyvolve simulations. This is a 7-Zip file containing an iPython notebook and the relevant data files. (7Z 7450 kb)
Clarification of subtleties in the relationship between amino-acid preferences and substitution model equilibrium frequencies. Figure 1 shows the experimentally measured amino-acid preferences and the equilibrium frequencies of the GY94 model. The equilibrium frequencies of the experimentally informed codon models (ExpCM) are given by Eq. 4, and are similar but not identical to the preferences: the ExpCM equilibrium frequencies are also influenced by the unequal number of codons per amino acid, nucleotide mutation biases, and the stringency parameter β. The equilibrium frequencies of the GY94 model already account for the codon/mutation factors. To clarify these distinctions, this figure shows the preferences and equilibrium frequencies of the ExpCM model, and the "all-equal" amino-acid preferences that would lead to the equilibrium frequencies of the GY94 model if the nucleotide frequency parameters in that model are construed as representing mutation-level rather than selection-level processes. Note that the logo plots show the amino-acid frequencies implied by the equilibrium codon frequencies (i.e. the sum of the frequencies of all encoding codons for each amino acid). (PDF 76 kb)
The results of the d N/d S analysis are qualitatively similar when using HyPhy rather than phydms. This figure shows the same data as that in Fig. 3 a, but also includes the results of a d N/d S analysis using the fixed effects likelihood (FEL) method implemented in HyPhy [7]. The results are not identical to the phydms GY94 results because the HyPhy implementation differs slightly from the phydms implementation: HyPhy performs the d N/d S analysis using the substitution model of [102] rather than GY94, and infers a neighbor-joining tree with a nucleotide substitution model rather than a maximum-likelihood tree using a codon model. Nonetheless, the results of the HyPhy FEL analysis are highly similar to those of the phydms GY94 analysis, both in terms of the overall distribution of results and in terms of the values for the specific indicated sites. The point markers represent the same sites as in Fig. 3. (PDF 236 kb)
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Bloom, J.D. Identification of positive selection in genes is greatly improved by using experimentally informed site-specific models. Biol Direct 12, 1 (2017). https://0-doi-org.brum.beds.ac.uk/10.1186/s13062-016-0172-z
DOI: https://0-doi-org.brum.beds.ac.uk/10.1186/s13062-016-0172-z
Deep mutational scanning
Substitution model
Diversifying selection
dN/dS | CommonCrawl |
Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate
DCDS-B Home
Global behavior of delay differential equations model of HIV infection with apoptosis
January 2016, 21(1): 121-131. doi: 10.3934/dcdsb.2016.21.121
Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities
Jackson Itikawa 1, and Jaume Llibre 2,
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
Received April 2015 Revised June 2015 Published November 2015
We classify the global phase portraits in the Poincaré disc of the differential systems $\dot{x}=-y+xf(x,y),$ $\dot{y}=x+yf(x,y)$, where $f(x,y)$ is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in [9] completes the classification of the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.
Keywords: Poincaré disk, uniform isochronous center, Polynomial vector field, quartic polynomial differential system., phase portrait.
Mathematics Subject Classification: Primary: 34C05, 34C2.
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Jackson Itikawa Jaume Llibre | CommonCrawl |
\begin{document}
\maketitle
\begin{abstract} Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In this paper, we study monomial GAPN functions for odd characteristic. In particular, we give all monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. \end{abstract}
{\footnotesize {\it Keywords:} APN function, GAPN function, exceptional exponent, Gold function, inverse permutation, EA-equivalent, algebraic degree,
finite field }
{\footnotesize {\it 2010 MSC:} 94A60, 05B25 }
\section{Introduction}
Let $F = \mathbb{F}_{p^n}$ be a finite field of characteristic $p$. A function $f \colon F \to F$ is an almost perfect nonlinear (APN) function if \begin{align*}
N_f (a, b) \coloneqq \# \Set{x \in F | D_{a}f(x) \coloneqq f(x+a) - f (x) =b} \leq 2 \end{align*} for all $a \in F^\times$ and $b \in F$. When $p=2$, such functions have useful properties and applications in cryptography, finite geometries and so on. On the other hand, APN functions for odd characteristic have quite different properties from the even characteristic case. In \cite{GAPN1}, the definition of APN functions for odd characteristic was modified to satisfy some similar properties of APN functions for even characteristic. In fact, the author defined a generalized almost perfect nonlinear (GAPN) functions as follows (see \cite[Definition 1.1]{GAPN1}): A function $f \colon F \to F$ is a GAPN function if \begin{align*}
\tilde{N}_f (a, b) \coloneqq \# \Set{x \in F | \tilde{D}_{a}f(x) \coloneqq \sum_{i \in \mathbb{F}_{p}}f(x+ia) =b} \leq p \end{align*} for all $a \in F^\times$ and $b \in F$. Note that when $p=2$, GAPN functions coincide with APN functions.
In addition, a few examples of GAPN functions on $F = \mathbb{F}_{p^n}$ was constructed. For example, \begin{itemize} \item the inverse permutation \begin{align*} f \colon F \longrightarrow F, \ \ x \longmapsto x^{p^n - 2} \end{align*} is a GAPN function of algebraic degree $n (p-1) - 1$ if $p$ is odd, and \item the generalized Gold function \begin{align*} f \colon F \longrightarrow F, \ \ x \longmapsto x^{p^i + p - 1}. \end{align*} is a GAPN function of algebraic degree $p$ if $\gcd (i,n) = 1$.
\end{itemize} Here see \cite[Section 2]{GAPN1} for the algebraic degree, and see Table 1 below for the Gold functions.
In this paper, we study monomial GAPN functions for odd characteristic. Note that monomial APN functions for even characteristic have been studied by many researchers. The following Table \ref{monomial APN} is a complete list, up to CCZ-equivalence, of known monomial APN functions for even characteristic, where $d^{\circ} (f_d)$ is the algebraic degree of $f_d$. We will give a generalization of Welch functions (see sub-subsection \ref{generalized Welch}). \begin{table}[h] \begin{center} \caption{Known monomial APN functions $f_d (x) = x^d$ on $\mathbb{F}_{2^n}$} \label{monomial APN}
\begin{tabular}{|l|c|c|c|c|} \hline & Exponents $d$ & Conditions & $d^{\circ} (f_d)$ & References \\ \hline \hline Gold functions & $2^i+1$ & $\gcd (i,n) = 1$ & $2$ & \cite{Gold1968maximal} \cite{Nyberg1994differentially} \\ \hline Kasami functions & $2^{2i} - 2^i+1$ & $\gcd (i,n) = 1$ & $i+1$ & \cite{JW93} \cite{Kasami71} \\ \hline Welch functions & $2^t+3$ & $n = 2t + 1$ & $3$ & \cite{Dobbertin99Welch} \\ \hline Niho functions & $2^t + 2^{\frac{t}{2}}-1$, $t$ is even & $n=2t+1$ & $\frac{t+2}{2}$ & \cite{Dobbertin99Niho} \\
& $2^t + 2^{\frac{3t+1}{2}}-1$, $t$ is odd & $n=2t+1$ & $t+1$ & \\ \hline Inverse function & $2^{2t}-1$ & $n=2t+1$ & $n-1$ & \cite{BD94} \cite{Nyberg1994differentially} \\ \hline Dobbertin functions & $2^{4t}+2^{3t}+2^{2t}+2^t - 1$ & $n=5t$ & $t+3$ & \cite{Dobbertin01} \\ \hline
\end{tabular} \end{center} \end{table}
Every function $f \colon F \to F$ can be represented uniquely as a polynomial function $f(x) = \sum_{d=0}^{p^n - 1} c_d x^d \in F [x]$. We can extend $f$ to an extension field of $F$ by using this unique polynomial formula. Then we can define a generalization of exceptional APN functions as follows (see \cite{AMR10} and \cite{HM2011} for exceptional APN functions): \begin{definition} \begin{enumerate} \item[(1)] A function $f \colon F \to F$ is \textbf{$p$-exceptional} if $f$ is a GAPN function on $F$ and is also GAPN function on infinitely many extension fields of $F$. \item[(2)] The exponent $d$ is $p$-\textbf{exceptional} if $f(x) = x^d$ is a GAPN function on infinitely many extension fields of $\mathbb{F}_{p}$. \end{enumerate} \end{definition} Note that $2$-exceptional exponents are so-called exceptional exponents. For any $p$, generalized Gold functions are $p$-exceptional clearly. When $p=2$, Kasami functions are also $2$-exceptional functions. In addition, the following Theorem was conjectured by Dillon \cite{Dillon02} and was proved by Hernando and McGuire \cite{HM2011}:
\begin{theorem} When $p=2$, the only $2$-exceptional monomial APN functions are the Gold and Kasami functions. In other words, the only $2$-exponential exponents are the Gold and Kasami numbers. \end{theorem}
In this paper, we will give all monomial GAPN functions of algebraic degree $p$ or $n (p-1) - 1$ (see Subsection \ref{minimum algebraic degree} and \ref{maximum algebraic degree}, respectively). Note that if $f \colon \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ is a GAPN function, then we have $p \leq d^{\circ} (f) \leq n (p-1) - 1$ (see Proposition \ref{Prop: basic result}). In addition, we will show that when $p \geq 3$, any monomial function of algebraic degree $p$ is $p$-exceptional on some extension field of $\mathbb{F}_p$ (see Proposition \ref{main1}). Moreover for odd prime $p$, we will give a conjecture for the existences of GAPN functions $f$ on $\mathbb{F}_{p^n}$ with $p < d^{\circ} (f) < n (p-1) - 1$, and $p$-exceptional exponents (see Conjecture \ref{conj 1}).
\section{Monomial GAPN functions}
In this section, we mainly assume that $p$ is an odd prime. Let $F = \mathbb{F}_{p^n}$ be a finite field of characteristic $p$. Let $f_d$ be a monomial function \begin{align*} f_d \colon F \longrightarrow F, \ \ x \longmapsto x^d. \end{align*} For any $x \in F$, we have that $x^{p^n} = x$, so we may assume that $d < p^n$. Then the exponent $d$ has the $p$-adic expansion $d = \sum_{s=0}^{n-1} d_s p^s$. Let $w_p (d)$ denote the sum of the coefficients $d = \sum_{s=0}^{n-1} d_s$, and we call it the $p$-weight of $d$. Clearly, we have $w_p (d) \leq n (p-1)$. For any monomial function $f_d \colon F \to F$, the algebraic degree $d^{\circ} (f_d)$ of $f_d$ coincides with the $p$-weight of $d$ (see \cite[Section 2]{GAPN1} for more details): \begin{align*} d^{\circ} (f_d) = w_p (d) \leq n (p-1). \end{align*}
\begin{proposition} \label{Prop: basic result} On the above notations, \begin{enumerate} \item[(1)] if $d^{\circ} (f_d) < p$, then $f_d$ is not a GAPN function on $F$, and \item[(2)] if $d^{\circ} (f_d)$ is even, then $f_d$ is not a GAPN function on $F$. \end{enumerate} In particular, for any monomial GAPN function $f_d$, we have $p \leq d^{\circ} (f_d) \leq n (p-1) - 1$. \end{proposition}
\begin{proof} (1) is clear from Proposition 2.12 in \cite{GAPN1}. We prove (2). Assume that $d^{\circ} (f_d)$ is even. Then $d$ is even. Hence if the equation $\tilde{D}_a f_d (x) = b$ has a solution $x_0$, then any point in $\left( x_0 + a \mathbb{F}_p \right) \cup \left( - x_0 + a \mathbb{F}_p \right)$ is also a solution of the equation. Hence $\tilde{N}_{f_d} (a, b) \geq 2p$, and hence $f_d$ is not a GAPN function. \end{proof}
In the following, we will give all GAPN functions on $F = \mathbb{F}_{p^n}$ with algebraic degree $p$ or $n (p-1) - 1$.
\subsection{Monomial GAPN functions on $\mathbb{F}_{p^n}$ with minimum algebraic degree} \label{minimum algebraic degree}
Let $f_d$ be a monomial function on $F = \mathbb{F}_{p^n}$ with $d^{\circ} (f_d) = p$. Then we have that \begin{align*} d = p^{i_1} + p^{i_2} + \cdots + p^{i_p} \ \ \mbox{with} \ \ i_1 \leq i_2 \leq \cdots \leq i_p.
\end{align*} Hence we have that \begin{align*} d = p^{i_i} d', \ \ \mbox{where} \ \ d' = 1 + p^{i_2 - i_1} + \cdots + p^{i_p - i_1}, \end{align*} and hence we obtain $f_d = f_{d'} \circ {\rm Fb}_{p^{i_1}}$, where ${\rm Fb}_{p^{i_1}}$ is the Frobenius isomorphism ${\rm Fb}_{p^{i_1}} (x) = x^{p^{i_1}}$. In particular, $f_d$ and $f_{d'}$ are EA-equivalence (see \cite[Section 2]{GAPN1}), and hence $f_d$ is a GAPN function if and only if $f_{d'}$ is a GAPN function by Proposition 2.1 in \cite{GAPN1}. Therefore we may assume that $i_1 = 0$, that is, we may assume that \begin{align}\label{p-exceptional} d = 1 + p^{i_2} + \cdots + p^{i_p} \ \ \mbox{with} \ \ 0 \leq i_2 \leq \cdots \leq i_p \ \ \mbox{and} \ \ (i_2, \dots, i_p) \ne (0, \dots, 0). \end{align} Then we can write \begin{align} \label{alpha} d = \sum_{s=0}^{n-1} \alpha_s p^s \ \ \ (\alpha_s \in \mathbb{F}_p, \ \alpha_0 + \cdots + \alpha_{n-1} = p). \end{align} We define the polynomial $D(X) \in \mathbb{F}_p [X]$ as follows: \begin{align} \label{polynomial D} D(X) \coloneqq \sum_{s=0}^{n-1} \alpha _s X^s \left( = 1 + X^{i_2} + \cdots + X^{i_p} \right) \in \mathbb{F}_{p} [X]. \end{align} On the above notations, we have the following criterion: \begin{theorem} \label{Thm: criterion}
$f_d$ is a GAPN function on $F$ if and only if $D (\beta) \ne 0$ for any $\beta \in \overline{\mathbb{F}_p} \setminus \{ 1\}$ such that $\beta^n = 1$,
where $\overline{\mathbb{F}_p}$ is the algebraic closure of $\mathbb{F}_p$. \end{theorem}
\begin{proof}
By the proof of Lemme 3.3 in \cite{GAPN1}, we obtain that \begin{align*} \varphi_d (x) \coloneqq \tilde{D}_1 f_d (x) = x + x^{p^{i_2}} + \cdots + x^{p^{i_{p}}} \ \ (x \in F), \end{align*} and hence $\varphi_d \colon F \to F$ is $\mathbb{F}_p$-linear. Thus $\dim_{\mathbb{F}_p} {\rm Ker} \left( \varphi_d \right) = n - \dim_{\mathbb{F}_p} {\rm Im} \left( \varphi_d \right) $. Then by Lemma 3.3, (iii) in \cite{GAPN1}, $f_d$ is a GAPN function on $F$ if and only if \begin{align*}
{\rm Ker} \left( \varphi_d \right) = \Set{ x \in F | x + x^{p^{i_2}} + \cdots + x^{p^{i_{p}}} = 0 } = \mathbb{F}_p . \end{align*}
This is equivalent to that \begin{align} \label{dim im} \dim_{\mathbb{F}_p} {\rm Im} \left( \varphi_d \right) = n - 1, \ \mbox{that is}, \ \# {\rm Im} \left( \varphi _d \right) = p^{n-1} . \end{align} On the other hand, by Theorem 2.5 in \cite{CCMPT2013}, we have that $\# {\rm Im} \left( \varphi _d \right) = p^{{\rm rk} (M_d)}$. Here $M_d$ is the $n \times n $ matrix defined by \begin{align*} M_d = \left[ \begin{array}{cccc} \alpha_0 & \alpha_{n-1} & \cdots & \alpha_1 \\ \alpha_1 & \alpha_0 & \cdots & \alpha_2 \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{n-1} & \alpha_{n-2} & \cdots& \alpha_0 \\ \end{array} \right], \end{align*} where $\alpha _0$, $\dots$, $\alpha_{n-1}$ are defined by (\ref{alpha}).
Hence (\ref{dim im}) is equivalent to that ${\rm rk} (M_d) = n - 1$. Let $I_n$ be the identity matrix of size $n$. Then we have that \begin{align*} \det \left( \mu I_n - M_d \right) &= \det \left[ \begin{array}{cccc} \mu - \alpha_0 & - \alpha_{n-1} & \cdots & - \alpha_1 \\ -\alpha_1 & \mu - \alpha_0 & \cdots & -\alpha_2 \\ \vdots & \vdots & \ddots & \vdots \\ -\alpha_{n-1} & -\alpha_{n-2} & \cdots& \mu - \alpha_0 \\ \end{array} \right] \\ &= \prod_{\beta^n=1} \left( \mu - \alpha_0 - \alpha_1 \beta - \dots - \alpha_{n-1} \beta ^{n-1} \right) = \mu \prod_{\beta^n=1, \beta \ne 1} \left( \mu - D (\beta) \right), \end{align*} and hence the eigenvalues of $M_d$ are $0$ and $D (\beta)$ ($\beta \in \overline{\mathbb{F}_p} \setminus \{ 1\}$, $\beta^n = 1$). Therefore ${\rm rk} (M_d) = n - 1$ if and only if $D (\beta) \ne 0$ for any $\beta \in \overline{\mathbb{F}_p} \setminus \{ 1\}$ with $\beta^n = 1$. \end{proof}
Generalized Gold functions $f \colon \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$, $f(x) = x^{p^i + p-1}$ are GAPN functions if $\gcd (i, n) =1$. In particular, they are $p$-exceptional clearly. More generally, we obtain the following Proposition:
\begin{proposition} \label{main1}
Any exponent $d$ given by (\ref{p-exceptional}) is a $p$-exceptional exponent. \end{proposition}
\begin{proof} It follows immediately from the following Lemma \ref{key lemma}. \end{proof}
\begin{lemma} \label{key lemma} \begin{enumerate} \item[(i)] For any exponent $d$ given by (\ref{p-exceptional}), there exists $n \in \mathbb{N}$ with $d < p^n$ such that $f_d$ is a monomial GAPN function of algebraic degree $p$ on $\mathbb{F}_{p^n}$. \item[(ii)] Any monomial GAPN function of algebraic degree $p$ on $\mathbb{F}_{p^n}$ for some $n$ is $p$-exceptional. \end{enumerate}
\end{lemma}
\begin{proof} We first prove (i). Let $d$ be an exponent given by (\ref{p-exceptional}). By Theorem \ref{Thm: criterion}, it is sufficient to show that there exists $n \in \mathbb{N}$ with $d < p^n$ such that \begin{align} \label{want (i)}
\Set{ \beta \in \overline{\mathbb{F}_{p}} | D(\beta) = 0 }
\cap \Set{ \gamma \in \overline{\mathbb{F}_{p}} | \gamma ^n = 1} = \{ 1 \}, \end{align} where $D(X) \in \mathbb{F}_{p} [X]$ is defined by (\ref{polynomial D}). Note that the polynomial $D(X)$ depends only on exponent $d$. Then the set
$\Set{ \beta \in \overline{\mathbb{F}_{p}} | D(\beta) = 0 }$ is a finite set. Let $\beta_1$, $\dots$, $\beta_m \in
\Set{ \beta \in \overline{\mathbb{F}_{p}} | D(\beta) = 0 } \setminus \{ 1 \}$ be all elements which have finite orders and let \begin{align*}
n_j \coloneqq \min \Set{ N \in \mathbb{N} | \beta_j^N = 1 } (> 1). \end{align*} Then there exist $n \in \mathbb{N}$ with $d < p^n$ such that $ \gcd (n, n_j) = 1$ ($j = 1$, $\dots$, $m$).
Then we have $\beta_j \not \in \Set{ \gamma \in \overline{\mathbb{F}_{p}} | \gamma ^n = 1}$ ($j=1$, $\dots$, $m$) and hence we obtain (\ref{want (i)}).
Next we prove (ii). Let $f_d$ be a monomial GAPN function of algebraic degree $p$ on $\mathbb{F}_{p^n}$ for some $n$. By theorem \ref{Thm: criterion}, we have that \begin{align} \label{intersection}
\Set{ \beta \in \overline{\mathbb{F}_{p}} | D(\beta) = 0 }
\cap \Set{ \gamma \in \overline{\mathbb{F}_{p}} | \gamma ^n = 1} = \{ 1 \}. \end{align} Similarly to above, let $\beta_1$, $\dots$, $\beta_m \in
\Set{ \beta \in \overline{\mathbb{F}_{p}} | D(\beta) = 0 } \setminus \{ 1 \}$
be all elements which have finite orders and let $n_j \coloneqq \min \Set{ N \in \mathbb{N} | \beta_j^N = 1 }$ ($> 1$). By (\ref{intersection}),
we obtain that $\beta_j \not \in \Set{ \gamma \in \overline{\mathbb{F}_{p}} | \gamma ^n = 1}$, that is, $n$ is not divisible by $n_j$ for each $j=1$, $\dots$, $m$. Then for any prime $q$ such that $\gcd (n_j , q) = 1$ ($j=1$, $\dots$, $m$), the number $q n$ is not divisible by $n_j$ ($j=1$, $\dots$, $m$), that is, $
\beta_j \not \in \Set{ \gamma \in \overline{\mathbb{F}_{p}} | \gamma ^{qn} = 1} $ ($j=1$, $\dots$, $m$). Hence we get \begin{align*}
\Set{ \beta \in \overline{\mathbb{F}_{p}} | D(\beta) = 0 }
\cap \Set{ \gamma \in \overline{\mathbb{F}_{p}} | \gamma ^{qn} = 1} = \{ 1 \}, \end{align*} and hence $f_d$ is also GAPN function on $\mathbb{F}_{p^{qn}}$, which is an extension field of $\mathbb{F}_{p^n}$. Since there exist infinitely many such prime numbers, $f_d$ is $p$-exceptional. \end{proof}
\subsubsection{Example: a generalization of Welch functions} \label{generalized Welch}
If $n$ is odd, then the function defined by \begin{align*} f (x) = x^{2^t + 3}, \ \ \ t = \frac{n-1}{2} \end{align*} is APN on $\mathbb{F}_{2^n}$ (see \cite{Dobbertin99Welch}). Such functions are called the Welch functions (see Table \ref{monomial APN}). Here we construct a generalization of Welch functions.
\begin{proposition}
Let \begin{align*} d = p^t + p + 1, \ \ \ t = \left\{ \begin{array}{cl} \frac{n-1}{2} & (\mbox{$n$ is odd}), \\ \frac{n}{2} & (\mbox{$n$ is even}). \\ \end{array} \right. \end{align*} Then $f_d$ is a GAPN function of $\mathbb{F}_{p^n}$ if and only if $p = 2$ and $n$ is odd, or $p=3$. \end{proposition}
\begin{proof} When $p=2$, the function $f_d$ is the Welch function if and only if $n$ is odd.
Since $d^{\circ} (f_d) = 3$, by Proposition \ref{Prop: basic result}, (ii), the function $f_d$ is not a GAPN function on $\mathbb{F}_{p^n}$ when $p \geq 5$. Let $p=3$. We prove that $f_d$ is a GAPN function on $\mathbb{F}_{3^n}$. Since $d^{\circ} (f_d) = 3$, by Theorem \ref{Thm: criterion}, it is sufficient to show that \begin{align} \label{GWelch} D (\beta) = 1 + \beta + \beta^t \ne 0 \ \ \mbox{for any $\beta \in \overline{\mathbb{F}_p} \setminus \{ 1 \}$ with $\beta^n=1$}. \end{align} Assume that $D (\beta) = 0$ for some $\beta \in \overline{\mathbb{F}_p} \setminus \{ 1 \}$ with $\beta^n=1$. If $n$ is even, then we have \begin{align*} 1 = \beta^n = \beta^{2t} = (-1-\beta)^2 = 1 - \beta + \beta^2, \ \ \mbox{that, is, } \ \ \beta (\beta - 1) = 0, \end{align*} which is absurd. If $n$ is odd, then we have \begin{align*} 1 = \beta^n = \beta^{2 t + 1} = \beta \left( -1 - \beta \right)^2 = \beta - \beta ^2 + \beta ^3 , \ \ \mbox{that, is}, \ \ \beta^2 + 1 = \beta (\beta^2 + 1). \end{align*} Since $\beta \ne 1$, we get $\beta^2 = -1$. Hence we have \begin{align*} 0 = 1+ \beta + \beta^t = \left\{ \begin{array}{c l} \beta-1 & (t \equiv 0 \mod 4), \\ -\beta+1 & (t \equiv 1 \mod 4), \\ \beta & (t \equiv 2 \mod 4), \\ 1 & (t \equiv 3 \mod 4). \\ \end{array} \right. \end{align*} In any case, they are contradictions. Therefore we obtain (\ref{GWelch}). \end{proof}
\subsection{Monomial GAPN functions on $\mathbb{F}_{p^n}$ with maximum algebraic degree} \label{maximum algebraic degree}
For odd prime $p$, the inverse permutation \begin{align*} f_{p^n-2} \colon F \longrightarrow F, \ \ x \longmapsto x^{p^n-2} \end{align*} is a GAPN function (see \cite[Section 3]{GAPN1}). Note that when $p=2$ the inverse function is APN if and only if $n$ is odd. Then $p^n-2$ has the $p$-adic expansion \begin{align*} p^n - 2 = (p-1) \left( 1 + p + \cdots + p^{n-1} \right)- 1 = (p-2) + (p-1) \left( p + \cdots + p^{n-1} \right). \end{align*} Hence $d^{\circ} (f_{p^n-2}) = n (p-1) - 1$. More generally, we have the following proposition:
\begin{proposition} Any monomial function on $F = \mathbb{F}_{p^n}$ with algebraic degree $n (p-1) - 1$ is EA-equivalent to the inverse permutation.
In particular, it is a GAPN function on $F$.
\end{proposition} \begin{proof} Since $w_p (d) = n (p-1) - 1$, the exponent $d$ is given by \begin{align*} (p-1) (1 + p + \cdots + p^{n-1}) - p^j = p^n - p^j - 1, \end{align*} for some $j \in \left\{ 0, 1, \dots, n-1 \right\}$. Hence any monomial function on $F$ with algebraic degree $n (p-1) - 1$ is given by \begin{align*} f_{(j)} \colon F \longrightarrow F, \ \ x \longmapsto x^{p^n - p^j -1}
\end{align*} for some $j \in \left\{ 0, 1, \dots, n-1 \right\}$.
Let ${\rm Fb}_{(j)}$ be a Frobenius isomorphism \begin{align*} {\rm Fb}_{(j)} \colon F \longrightarrow F, \ \ x \longmapsto x^{p^{n-j}}. \end{align*} Then we have \begin{align*} \left( f_{(j)} \circ {\rm Fb}_{(j)} \right) (x) = \left( x^{p^{n-j}} \right)^{p^n - p^j - 1}
= x^{-1} = f_{p^n-2} (x), \end{align*} where we put $0^{-1} := 0$. Hence $f_{(j)}$ is EA-equivalent to the inverse function $f_{p^n-2}$. By Proposition 2.1 in \cite{GAPN1}, $f_{(j)}$ is a GAPN function on $F$.
\end{proof}
\subsection{The other monomial GAPN functions on $\mathbb{F}_{p^n}$}
By simple computations, we can show that when $p=3$ and $n \in \left\{ 6, 7, 8 \right\}$, there are no monomial GAPN functions $f_d$ on $\mathbb{F}_{3^n}$ with $3 < d^{\circ} (f_d) < 2 n - 1$. More generally, we give the following conjecture:
\begin{conjecture} \label{conj 1} Let $p$ be an odd prime. For sufficiently large $n$, there are no monomial GAPN functions $f_d$ on $\mathbb{F}_{p^n}$ with $p < d^{\circ} (f_d) < n (p-1) - 1$.
In particular,
the only $p$-exceptional exponents are given by (\ref{p-exceptional}). \end{conjecture}
\begin{table}[h]
\begin{tabular}{l} Masamichi Kuroda \\%& \hspace*{10mm}& Shuhei Tsujie \\ Department of Mathematics \\%& & Department of Mathematics \\ Hokkaido University \\%& & Hokkaido University \\ Sapporo 060-0810 \\%& & Sapporo 060-0810\\ Japan \\%& & Japan \\ [email protected] \\%& & [email protected] \end{tabular} \end{table}
\end{document} | arXiv |
Friday, September 30, 2016 ... / /
Barry Barish deserves a LIGO Nobel prize, too
It's not certain at all that next Tuesday, the Nobel prize in physics will be given to the people associated with LIGO, the (double) L-shaped experiment that announced the detection of Einstein's gravitational waves in February 2016. Recall that the waves were actually detected on September 14th, 2015, a year ago, just four months after an interview where someone said that the detection would take place "within five years". Sometimes things are slower than expected but indeed, sometimes they are faster, too.
Despite the uncertainty about the 2016 Nobel, the LIGO possibility is reasonably likely by now. Rumors indicate that the January 31st deadline for the nominations didn't turn out to be a fatal obstacle for the LIGO-related candidates.
In February, I was afraid that there could be some politically correct folks who would want to reward the current leaders of the experiment – basically random politically chosen hires. But thankfully or hopefully, it seems that the probability of this scenario has decreased and the actual fathers of the LIGO success – which made the decisive steps decades ago – are more likely to win.
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In 61 days, a Slovak billionaire will monitor every single Czech cash transfer in real time
Millions of stupid and jealous Czech sheep embrace the new "1984"
Unless something unexpected happens – and I pray that it will – the first batch of 50,000 of Czech businesses, mainly restaurants and hotels etc., will be obliged to immediately report every single payment from a consumer to the ministry of finance led by the Slovak-born food industry billionaire, media mogul, a VIP ex-member of the communist party, and a former communist snitch Andrej Babiš (net worth over $3 billion).
The consumer gets a receipt and he or she – an amateur snitch – will be able to send the receipt's ID to a server of the ministry and verify that the payment has been reported by the business. A motivation is that he may win a lottery for the amateur snitches. Andrej Babiš's former career of a snitch is seen in every aspect of this sick system. If the payment hasn't been reported, the businessman will immediately face existentially threatening fines and other punishments that the minister himself may decide about – or forgive. The law defining the EET things is a classic "rubber law" that may be bent by the executive power. It's a similar kind of a law that made Adolf Hitler the Führer.
This system is meant to guarantee that the taxes from that payment – every payment – will be sent to the government. Andrej Babiš, a member of the very bottom of the Czechoslovak moral cesspool who would have been executed in late 1989 if we hadn't decided to make our revolution in the "velvet" way (for example, their dirty family disinherited a relative who "dared to emigrate" from the communist Czechoslovakia, to emphasize how deeply into the communist leaders' aßes they are willing to climb in order to keep their undeserved advantages), and a guy who already owns most of the largest newspapers, will have access to all the information about every single cash payment to every business on the territory of Czechia, at least after all businesses are included into the system in a coming year or so.
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Thursday, September 29, 2016 ... / /
Aspects of the Indian-Pakistani (so far) miniwar
The British Empire has been in charge of the British India for some time. In 1947, that territory declared their independence and new countries, Pakistan and India, were created. Pakistan is some 98% Islamic. India is mostly Hinduist (Buddhism is below 1% these days) and only 15% Islamist (Christianity is over 2%, the third largest religion there). However, you may see that India is still the by far more diverse country among the two.
I would surely say that India is the more "politically Western" country among the two. You could say that it's "ironic" given the Pakistani prime minister Nawaz Sharif's Western suit and "Aryan" skin color on the picture above with the visually darker and more folklore-dressed Indian prime minister Modi. But India's fights against the Islamic terrorists basically coincide with the logic of similar fights that sensible Western countries have to wage.
There have been three conflicts between India and Pakistan. Most of the conflicts are linked to the most disputed part of the border, inside Kashmir. Kashmir is a territory in the Northern part of the Pakistani-Indian border, a cool region adjacent to the Himalayas. Both countries claim all of it. In practice, it's divided to two similarly large parts by the de facto (but internationally unrecognized) border, the so-called "Line of Control" (LoC). That's where the newest tension is concentrated, too.
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Rainer Weiss' birthday: from Slovakia to circuits, vinyl in Manhattan to LIGO
Along with Kip Thorne and Ronald Drever, Rainer Weiss is one of the most likely "triplet" that can share the Nobel prize in physics next Tuesday. Weiss' key contribution already occurred in 1967 – see the history of LIGO – when he began to construct a laser interferometer and published a text pointing out its usefulness.
WVXU, a BBC-linked news source, just released a fun biography:
A physicist who proved Einstein right started by tinkering with the family record player
Aside from fundamental physics, one of the additional reasons why this biography may be relevant on this blog are his family's links to Czechoslovakia.
Other texts on similar topics: Czechoslovakia, experiments, LIGO, politics, science and society, stringy quantum gravity
Wednesday, September 28, 2016 ... / /
92% in unhealthy air? Another example of a boy who cries wolf
By Václav Klaus, Czech ex-president
Today in the morning, my smartphone beeped and informed me about the reports that "an overwhelming majority of the inhabitants of the planet, namely 92 percent, is living at places where the air pollution surpasses the limits defined by the World Health Organization".
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Tuesday, September 27, 2016 ... / /
Most Czech viewers: Trump won 1st debate
Donald and Hillary met in the
first presidential debate (90 minutes of video)
at Hofstra University, New York. Lester Holt was the moderator. The host had the full control over the questions, the audience – partly students – was expected to remain silent and not to use cameras and phones.
Hillary said "Hey Donald!" and he shook her hand, apparently confident in his immunity against pneumonia and other contagious diseases. Quite generally, I am sure it's right to say that they behaved in a much more friendly way towards each other than their voters. ;-) Concerning similar formalities and speaking strategies, Trump was attempting to interrupt Hillary more than 20 times but only succeeded once. She didn't try to interrupt him, with three failed exceptions.
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Sunday, September 25, 2016 ... / /
Civil casualties in Aleppo are sad but negligible
I just listened to a rant by Samantha Power, the current U.S. ambassador to the U.N.
Well, what a hateful woman – when it comes to anything that has a relationship with Russia. She has nothing to do with the America that we used to love and that was inspiring us. One of the reasons I would love Trump to win is that he could end this absolutely insane anti-Russian hysteria in the U.S. Among other things, he could help to fire this particular insufferable female talking head.
But it's not just Samantha Power. Boris Johnson talks about Russian war crimes in Syria while The Telegraph shocks us with the Aleppo horror. From that paper, you may learn what has actually happened.
In a hugely intense bombing of the anti-Assad forces in Aleppo, an operation masterminded by the Kremlin and Assad, "dozens" of civilians have been killed. That's sad. (Media close to the Kremlin dispute even these dozens of death but let me assume that these sad reports are true.) But is that unexpected? Is that a lot?
Other texts on similar topics: Middle East, politics, Russia
Saturday, September 24, 2016 ... / /
Orbán wants to build a Hong Kong for Arabs and blacks in Libya
Hungary's prime minister Viktor Orbán keeps on presenting creative proposals that actually make sense.
Hungarian Prime Minister says EU should set up refugee city in Libya
He wants to grab a piece of land outside the EU – most specifically, he mentioned a place in Libya – and build a large enough camp that should host a really, really large number of migrants or self-described refugees. I chose the term "Hong Kong" because there could be a million people over there; the terms "Liberland" or "Dachau" look too small for Orbán's project.
Note that Hong Kong was liberated from the Japanese overlords in 1945 by a combination of Chinese and British troops. The latter players were enough to bring the place under the British control which meant a huge economic advantage. As previously negotiated, Hong Kong returned under the control of the People's Republic of China in 1997. So far, they haven't destroyed the place – because at least in the economic sense, the mainland China has largely embraced capitalism by itself.
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NASA-sponsored article makes millions of Ophiuchus-born women hysterical
Every five years or so (see 2007 and 2011), I write a blog post about the 13rd zodiac sign, the Serpentarius (the Greek name Ophiuchus is preferred by many these days but not by me) – the wearer of the snakes – in which I was born, much like everyone whose birthday is between November 29th and December 17th or so (more dates).
Fall 2016 just began and it was inevitable that someone makes sure that this insight shocks millions of people, especially women. And it's here. See the recent Ophiuchus articles on Google News.
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Europe has a chance to be "out" when Paris comes to force
The Paris agreement is a recent meaningless remake of the 1997 meaningless Kyoto treaty that, like the predecessor, tries to "fight against the climate change". Err once, err twice...
If you remember the Kyoto treaty, Al Gore signed it but the U.S. has never ratified it, and neither has Australia. Canada later withdrew from that pact. I think that the impact of this absence of the U.S. on the production of carbon dioxide – let alone the climate – was non-existent. In fact, the U.S. saw a greater decrease of "CO2 produced per dollar or capita" than the average Kyoto signatory. But because the U.S. stayed out, the American climate alarmists couldn't show their muscles as aggressively as their European counterparts.
Even though conservative Americans love to imagine that their nation is always more conservative than the European nations, I think that it doesn't apply to the current U.S. administration that is more left-wing than most European governments. This has many manifestations but one of them concerns the climate hysteria.
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Media downgrade Roger Penrose to an invisible appendix of crank Lee Smolin
A week ago, Roger Penrose released his new book Fashion, Faith, and Fantasy. We already knew that he was writing a book of this name in 2009 and Penrose was actually giving lectures with this name in 2006 or earlier. So you may say that this book is something that Roger Penrose – and he's far from an average man – has been working on for something like one decade. Also, the book has almost 200 figures which are freely available.
I wrote the clearest description of the book that is now out in 2014. Is the demand for this product of a decade-long effort by a famous thinker appropriate? Before we turn to this question, let me remind you about the content of the book or the meaning of the words in the title.
The three words, fashion, faith, and fantasy, primarily refer to string theory, quantum mechanics, and inflation, respectively. Roger Penrose has some problems with all these three things – and others. So he invents slogans to dismiss all these three important theories. String theory is a bubble, quantum mechanics is a religious cult, and inflationary cosmology is a result of folks on drugs who see pink elephants around. (Penrose's explanations are less concise and less colorful than mine, he's no Motl.)
As I have discussed in previous blog posts, his negative opinions on all these three theories are fundamentally wrong.
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Thomson Reuters: Nobel for Cohen, LIGO 3, or control theory
We got used to the predictions of the Nobel prize winners by Thomson Reuters. The awards will be announced between October 3rd and 10th. The predictions are in the article
Web of Science Predicts 2016 Nobel Prize Winners
Let us spend less time with the disciplines different than physics.
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Nanopoulos' and pals' model is back to conquer the throne
Once upon a time, there was an evil witch-and-bitch named Cernette whose mass was \(750\GeV\) and who wanted to become the queen instead of the beloved king.
Fortunately, that witch-and-bitch has been killed and what we're experiencing is
The Return of the King: No-Scale \({\mathcal F}\)-\(SU(5)\),
Li, Maxin, and Nanopoulous point out. It's great news that the would-be \(750\GeV\) particle has been liquidated. They revisited the predictions of their class of F-theory-based, grand unified, no-scale models and found some consequences that they surprisingly couldn't have told us about in the previous 10 papers and that we should be happy about, anyway.
Al Jazeera attempts a terrorist attack against the Czech gambling industry
"No stronger retrograde force exists in the world. Far from being moribund, Mohammedanism is a militant and proselytising faith," Winston Churchill famously pointed out.
We could have seen another example just five hours ago when Al Jazeera, a hybrid of written Mohammedanism and modern Western left-wing brainwashing outlets, picked a particular target, the Czech gambling industry:
Czech Republic: A dangerous gambling addiction
The author, an American living in Burma ($5,000 is the GDP per capita, PPP), described Czechia as a decaying society where 110,000 gamblers (over 1%) should probably be stored in a psychiatric asylum. (The number 110,000 was picked from some random government documents and compared with numbers from other governments – which obviously use completely different methods so he was comparing apples with oranges.)
He admitted that these industries were overregulated during communism (which doesn't mean that gambling was absent: the totto-lotto ["Sportka" existed during socialism] and betting on sports was alive and reasonably well ["Sazka" was the large company that did this business already during socialism], while avoiding the efficiency of capitalism) but he described the results of freedom in this business as catastrophic.
Slot machines, quizomats – machines that test the encyclopedic knowledge or IQ, betting on sports, and other things were all included in his picture of the Armageddon.
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Czech presidents would pick Trump
There are various people in the Europe – and even in Czechia – who have endorsed Hillary Clinton for the U.S. president. Well, even though the late Václav Havel could be one of these people if he were around, the Czech presidents who are alive beg to differ.
President Emeritus Václav Klaus believes (and so do his aides) that Hillary's reign would be a continuation of the ongoing tragic drift towards the PC post-democracy. He thinks that Trump is a natural political animal who is currently playing the role of a campaigner and who will behave differently, more responsibly, once he sits in the White House. However, Klaus often says that "unfortunately, Hillary will probably win".
Well, I actually think that Trump's victory is more likely.
Today, the current Czech president Miloš Zeman, the founder of the modern social democracy in Czechia, was interviewed by iDNES TV and its boss Jaroslav Plesl, a journalist owned by the billionaire Andrej Babiš.
The first half of the 15-minute interview is dedicated to the Czech regional elections (in October 2016), the Czech economy, budget deficits etc. The elections are less important than the parliamentary elections. Those things are totally boring for 98% of TRF voters. Let me jump to the foreign policy questions.
Monday, September 19, 2016 ... / /
Anti-string crackpots being emulated by a critic of macroeconomics
While only a few thousand people in the world – about one part per million – have some decent idea about what string theory is, the term "string theory" has undoubtedly penetrated the mass culture. The technical name of the theory of everything is being used to promote concerts, "string theory" is used in the title of books about tennis, and visual arts have lots of "string theory" in them, too.
But the penetration is so deep that even the self-evidently marginal figures such as the anti-string crackpots have inspired some followers in totally different parts of the human activity. In particular, five days ago, a man named Paul Romer wrote a 26-page-long rant named
The Trouble With Macroeconomics
See also Stv.tv and Power Line Blog for third parties' perspective.
If you think that the title is surprisingly similar to the title of a book against physics, "The Trouble With Physics", well, you are right. It's no coincidence. Building on the example of the notorious anti-physics jihadist named Lee Smolin, Paul Romer attacks most of macroeconomics and what it's been doing since the 1970s.
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When the beating of your heart echoes the beating of the Trumps
Most of the time, Hillary Clinton isn't working too hard to help the Trump campaign. In fact, sometimes it looks like she wants to harm Trump and maybe even return to the White House by the front entrance. But she has made an exception when she finally picked the anthem for the Trump campaign.
I had to listen to the song twice in order for me to find the melody catchy and thrice to almost safely remember it.
The anthem is a key music from "Les Deplorables" and it is called "When the Beating of Your Heart Echoes the Beating of the Drumpfs".
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String theory lives its first, exciting life
Gross, Dijkgraaf mostly join the sources of deluded anti-string vitriol
Just like the Czech ex-president has said that the Left has definitively won the war against the Right for any foreseeable future, I think it's true that the haters of modern theoretical physics have definitively won the war for the newspapers and the bulk of the information sources.
The Quanta Magazine is funded by the Simons Foundation. Among the owners of the media addressing non-experts, Jim Simons is as close to the high-energy theoretical physics research community as you can get. But the journalists are independent etc. and the atmosphere among the physics writers is bad so no one could prevent the creation of an unfortunate text
The Strange Second Life of String Theory
by Ms K.C. Cole. The text is a mixed, and I would say mostly negative, package of various sentiments concerning the state of string theory. Using various words, the report about an alleged "failure of string theory" is repeated about 30 times in that article. It has become nearly mandatory for journalists to write this spectacular lie to basically every new popular text about string theory. Only journalists who have some morality avoid this lie – and there aren't too many.
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Bratislava EU summit
A summit about the EU future takes place in the Slovak capital today. The location was determined by the ongoing Slovak EU presidency. Because they should focus on a "more distant" future, the British leaders will be absent for the first time. That's why the discussions on Brexit won't take place – or may turn out to be inconsequential.
Nominal EU President Donald Tusk urges a brutally honest assessment while the BBC is certain that the meeting will be a waste of time. Well, I also guess that they won't solve anything meaningfully but I still think that a meeting organized by the Slovak government has greater chances than those in Brussels and elsewhere. The Visegrád Group may present some reform plan for the EU (a "cultural counterrevolution") – a plan trying to make the EU less integrated, among other things. We will see whether a unified V4 voice exists and has a chance to be heard. Czech PM Sobotka would like to focus the summit on the "fight against illegal immigration and terrorism".
The Slovak presidential palace
In the morning, delegations arrive to the Bratislava Castle. The folks should talk for some two or three hours over there. The Slovak government insisted on a different arrangement than the talks in Brussels usually have; that includes some small rooms for discussions of couples or small groups. I surely think it's a good idea for the Slovaks – and other individual organizers – to bring some of their own approach. They also emphasize that the participants will be served proper Slovak wine instead of some French plonk. :-)
Noncommutativity and observer dependence of QM are morally equivalent
I want to add a playful "research" twist to my explanations of the foundations of quantum mechanics.
In many blog posts, I argued that the need to express the facts about Nature relatively to an observer – the "subjective" character of knowledge – is a defining property of quantum mechanics. Quantum mechanics is the only known consistent framework that makes it impossible to assume (like classical physics did) that all observations may be derived as consequences of some shared objective data (data independent of an observer). This property of quantum mechanics will be referred to as its observer dependence.
However, most recently 2 weeks ago, I also argued that all novelties of quantum mechanics may be reduced to the nonzero commutators. After all, Planck's constant \(\hbar\) is the constant that is approximated by \(\hbar=0\) in classical physics (or the classical limits of quantum theory) which means that it quantifies the deviations from classical physics. At the same moment, all commutators of observables that exist classically as well as quantum mechanically are proportional to \(\hbar\) in the quantum theory. The nonzero commutators are equivalent to what we will call noncommutativity of quantum mechanics.
OK, these seem like two different "defining" principles of quantum mechanics. Which of them is right? Needless to say, you may say that both principles are right and must be embraced to construct the correct framework. However, I will try to defend a prettier statement. These two principles aren't really independent. To some extent, they are equivalent!
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Sarkozy is a climate skeptic
TheLocal.fr brought us some surprising news:
Sarkozy comes out of the closet as a climate skeptic
Between 2007 and 2012, Sarkozy was the president of France and we could often see him as a member of a powerful European couple along with Angela Merkel. I think that during Sarkozy's times, France was viewed as a more important part of the couple than it is now. The effect of a switch to Sarkozy seems self-evident – although some unspectacular evolution of the French economy relatively to the German one could have contributed, too.
His fresh statements to AFP make it clear that he is no lukewarmer.
Climate has been changing for four billion years. Sahara has become a desert, it isn't because of industry. You need to be as arrogant as men are to believe we changed the climate.
Yup, if you are as non-arrogant as women are :-), except for Catherine Hayhoe or the Latin American female crackpot on top of the UNFCCC whose name I have forgotten (update: something like Figueres), you know that global warming is a pile of šit.
He seems to be as skeptical as you or me. The climate has been changing for quite some time. And desertification of Sahara – which, by the way, I also consider a vastly bigger issue than the change of the global temperature by a degree – wasn't caused by men.
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An interview with Václav Klaus
About the final victory of the Left, redefinition of the main political themes, Merkelism, and U.S. elections
An interview with the Czech ex-president was recorded by PragueTV less than two weeks ago and I found it interesting enough to translate those 23 minutes for you.
See the original "Prague Café" interview, a video in Czech.
Via klaus.cz
Moderator, Petr Žantovský:
Welcome to the Prague Café. Today, we have moved the program to slightly atypical spaces, namely those of the Institute of Václav Klaus. The reason is simple. Our guest today is the former president Václav Klaus. Good afternoon.
Klaus:
Good afternoon. Well, I just hope that you haven't brought the spirit of the Prague Café [=PC intellectual elites of Prague] into these sacred spaces of our institute.
A new paper ruling out many non-local realist theories
As explained 893 times on this blog, quantum mechanics has replaced, refused, and retired a basic assumption of classical physics, realism, while it has kept the principles of relativity, the Lorentz invariance, and the locality that follows from them.
So we have theories of nearly everything – quantum field theories – and a likely theory of everything – string theory – that predict that the behavior of physics objects in the Minkowski space is perfectly local and Lorentz-invariant but the required correct description cannot be realist. Instead, quantum mechanics is a set of laws of Nature that must be applied on a set of intrinsically subjective observations by an observer – and the precise description of a process generally does depend on whom we consider an observer and what are his observations, on answers that must be given and "inserted" to the quantum mechanical black box before the final calculations of quantum mechanics are made.
Quantum mechanics has been disputed by folks like Einstein, de Broglie, and Schrödinger from the beginning. However, the particular set of facts in the previous two paragraphs began to be obfuscated with the works of John Bell who proved a correct theorem – but started to draw completely incorrect consequences out of it and encouraged many other people to do the same.
The theorem shows that "local realist theories" cannot agree with the observations because those sometimes show greater correlations than the correlations obeying Bell's inequalities, an inevitable consequence of "local realist theories". So far so good. But Bell incorrectly suggested – and many mindless, totally deluded people repeated – that the resolution is that the world is realist but non-local.
A great scholarly talk on history of PC
I thought that because of my Soviet bloc background, years at Harvard, and many years I dedicated to this issue, I knew a lot about the PC, its history, its genesis etc. So I couldn't learn anything really new, right? Today, I was led to the four-months-old talk by
Stephanie Maier from Americans for Prosperity: The Sordid History of Political Correctness
and it was an eye-opener. (Hat tip: Ron Paul.)
Other texts on similar topics: education, freedom vs PC, politics
Vote-swapping is immoral, ineffective, misguided, and evidence that all leftist activists are filth
I do remember the year 2000. One of the games that accompanied the Gore-Bush presidential elections was the swapping of the votes between different leftists. A Ralph Nader voter in a swing state was viewed as damaging for Gore. So a Gore voter in a different state found him, promised to vote for Nader instead, and the Nader fan had the duty to vote for Gore, after all.
In that way, Gore was supposed to win the swing states where every vote mattered. On the other hand, Nader got the same number of votes in the whole U.S. You may remember that these groups of leftists ultimately didn't succeed, Gore lost, and Nader became pretty much irrelevant, too. But could they have succeeded? Hasn't the vote-swapping worked in the opposite direction than they intended?
Scott Aaronson celebrates this "wonderful" idea in 2016, too. Gore has been replaced by Hillary and Nader has been replaced by candidates such as Jill Stein.
The Ninth Circuit ruled that vote-swapping is legal. Let's use it to stop Trump
People like Aaronson are bigots and technologists of power. They don't discuss politics, their understanding of the political questions – especially subtle ones – is extremely superficial. But they want to be disciplined servants of their extreme left-wing ideology and invent the right methods to make this ideology conquer the world.
Once people like Aaronson take over a country, they become secretaries responsible for the logistics – deciding e.g. how to get the deplorables to a Gulag with a limited number of buses etc.
Hillary, pneumonia, and leftists' dishonesty and irresponsibility
Hillary fainted during an event commemorating the 15th anniversary of the 9/11 Muslim attacks on America (and my PhD defense). Soon afterwards, her doctor revealed that she was diagnosed with pneumonia on Friday. It's my feeling that the doctor had been asked to remain silent and she did something that the Hillary campaign didn't want.
For the first time, Hillary's health was acknowledged to be a serious issue. Her two-day trip to California was cancelled. And the leaders of the Democratic National Council have called for an emergency meeting that is already considering – right now – Hillary's replacement. The replacement hypothetically could be Sanders but probably won't be. Biden and Mooch (whoever the latter is supposed to be) are mentioned as serious contenders; Monica Lewinski is the most frequently mentioned replacement for Hillary among the pranksters.
She hasn't been replaced yet. But especially in combination with some previous pronouncements and events, the recent events are disturbing for numerous reasons.
My mathematics: an innocent song that wouldn't be PC in the U.S.
I was listening to some young Czech musicians' remakes of various songs.
One of the singing girls who impressed me as musicians was Naty [Natalia] Hrychová, with her three-year-old The Fifth (Pátá, a remake of Downtown – it sounds similar; the title means "Five O'Clock" and the song is about the moment when classes are finally over), created to match the key and rhythm of the same song by Zuzana Norisová from The Rebels, a retro-movie (yes, the actors star as much younger students). Naty attended a rural basic school in a village or town I've never heard of.
So I accidentally looked at her newer songs. And two weeks ago, she posted "My Mathematics" song two weeks ago – it has over 320,000 views (and over 1700 comments) by now.
I didn't instantly fall in love with the song (it got much better after a month!) but the lyrics is all about mathematics. More importantly, I want to convey the observation that despite all the pro-EU and climate-change and other politically correct brainwashing, often sponsored by the bureaucrats in the EU, certain things such as feminism haven't crippled the Czech schools yet.
Other texts on similar topics: education, freedom vs PC, music
A male chemical engineer attacked for daring to explain why water boils at low pressure
A gang of teenagers in Toulon, the French riviera, has attacked and broken the nose and similar facial bones of two men who accompanied women in shorts – because, according to the gang, the women in shorts were "whores". The gangsters must have failed to notice that there are millions of women in shorts on the French riviera.
Well, this gang almost certainly consisted of boys educated in an uncivilized society following the rules of the Middle Ages so you can't be surprised too much. When such folks get the space to terrorize the public for such crazy reasons – such as wearing the shorts – they will do so.
But don't we have similar mob attacks in the West? Well, we do. They often lead to different outcomes than broken noses but the mob mentality and irrationality are completely analogous to those of the Toulon attackers.
Other texts on similar topics: astronomy, experiments, freedom vs PC, science and society
Was Mother Teresa a good woman? All of us get brainwashed about something
Mother Teresa (1910-1997) has been awarded sainthood by the appropriate Catholic apparatchiks. Just a decade ago, I would agree that she was one of the holiest people in the 20th century and so on. You can hear this statement everywhere so it must probably be true – like the statement that the Greenland is the world's largest island. Now, my opinions are closer to those of Florin M.
This episode of "Penn and Teller's Bullšit" – well, the video only shows 1/3 of an episode that focused on Mother Teresa – was an eye-opener for me when I watched it some 6 years ago. Christopher Hitchens was their most favorite "critic" and he is the #1 source for Florin, too.
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Operation Ronce: is French army ready to re-conquer Islamic no-go zones?
Yesterday, our glorious confederate EU leaders have decided to kickstart the greatest "humanitarian" gesture in the EU history, in their efforts to at least infinitesimally please their new great friend Erdogan.
All the illegals who are currently in Turkey and plan to continue their illegal trip to the EU have been donated free debit cards with €0.348 billion on them. The money should make it easier for them to buy boats, get to Greece or Turkey, or take a flight to Western Europe, and neutralize some cops or other Europeans who would dare to try to enforce the law.
Fighters in Daesh were given gift cards to stay in any EU hotel of their choice and those who have already murdered some infidels have received life-long permits valid in all EU brothels including the headquarters of the European Commission. OK, I made the previous sentence up but even the second paragraph is just stunning. The EU has officially turned into one of the most generous sponsors of terrorism in the world.
Meanwhile, provocative writer Éric Zemmour – whose parents were Algerian Jews – has been interviewed by RTL, a French TV. This interview optimistically suggests that the freedom of expression isn't dead in France yet. I guess that such an interview couldn't appear at the most influential German TV stations anymore.
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Weapons of math destruction are helpful tools
Cathy O'Neill has worked as a scholar at Columbia before she became a "quant" for D.E. Shaw in Fall 2007, making sure that the company, the financial system, and the derivatives in particular would work flawlessly in the following year.
With these achievements, she is back with a book, willing to share her wisdom. I won't link to particular reviews and interviews because I don't think that there are innovative ideas in individual articles in that group. Instead, see e.g. Google News and Google.
Other texts on similar topics: computers, mathematics, science and society
iPhone 7 wireless buds: when tech progress focuses on impractical changes
After several years, I watched the full Apple product event last night. I was impressed by the technological achievements but the show hasn't made me "dream" about the new iPhone 7 or Apple Watch 2 etc.
Let me start with a review of the event. Before Tim Cook, the Apple CEO, arrived, we saw a funny exchange he had with the taxi driver. It looked like a video that was filmed just before Cook entered the hall in San Francisco.
Cook and various people were giving their talks. A billion of iPhones have been sold, that's why they are everywhere. Super Mario (Run) has finally been brought to iPhones. The game's father, my uncle Šigeru Mija-Motl, presented the new game in half-English, half-Japanese.
Other people were telling us about the projects to teach kids how to code (in Swift) using an iPad app or what was that. Another woman showed that Apple's counterpart of the Microsoft Office will now enable the real-time collaboration of many users that create a presentation or another file. Apple Watch 2 was waterproof, GPS-enhanced, and a hike enhanced by the product was sketched. Apple Watch Nike Plus could be useful for runners, a Nike guy argued.
This watch will tease the owners – it's sunny outside or Sunday and you really need to go running, Joe is ahead of you etc. – a reason why I surely won't buy such an annoying, arrogant watch. ;-)
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Chen-Ning Yang against Chinese colliders
The plans to build the world's new greatest collider in China have many prominent supporters – including Shing-Tung Yau, Nima Arkani-Hamed, David Gross, Edward Witten – but SixthTone and South China Morning Post just informed us about a very prominent foe: Chen-Ning Yang, the more famous part of Lee-Yang and Yang-Mills.
He is about 94 years old now but his brain is very active and his influence may even be enough to kill the project(s).
The criticism is mainly framed as a criticism of CEPC (Circular Electron-Positron Collider), a 50-70-kilometer-long [by circumference] lepton accelerator. But I guess that if the relevant people decided to build another hadron machine in China, and recall that SPPC (Super Proton-Proton Collider) is supposed to be located in the same tunnel, his criticism would be about the same. In other words, Yang is against all big Chinese colliders. If you have time, read these 403 pages on the CEPC-SPPC project. Yang may arguably make all this work futile by spitting a few milliliters of saliva.
He wrote his essay for a Chinese newspaper 3 days ago,
China shouldn't build big colliders today (autom. EN; orig. CN)
The journalists frame this opinion as an exchange with Shing-Tung Yau who famously co-wrote a pro-Chinese-collider book.
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Euro-koruna cap may collapse already in 2016
The previous text on this topic of the Czech currency cap was posted on July 26th.
So far, I am largely unprepared for the change (also owning several stock funds denominated in other currencies, without any insurance of the Forex risks) but wealthy and money-savvy TRF readers may do better. As Reuters reported yesterday, the shorting of the EUR-CZK currency pair may be the currency trade of 2017, according to ING (and surely others).
Except that once all the people become familiar with the opportunity, events may speed up and the change may already materialize in 2016.
For an older example of the same idea, the separation of the Czech and Slovak currencies in early 1993 was also accelerated once the Slovaks began to realize that their Slovak savings may deteriorate in the case of the currency split, and that's why they moved the cash to the Czech banks and Slovak banks were running out of cash, thus forcing to do the split quickly. Czechoslovakia was only split on January 1st and the plan was to keep the currency union except that the aforementioned pressures from the self-fulfilling prophesy forced the folks to (flawlessly) separate the currencies already on February 8th, just some 37 days later.
What's going on now?
See a server with all Czech and Slovak banknotes between 1918 and 1993.
Czechoslovaks and especially Czechs have been highly conservative with their currencies and savings. The conservativeness has many implications. Unlike other post-communist countries, we have always used primarily our currency for trading, saving, loans etc. Also, the inflation rate and personal and public debt level were much lower than in Hungary but even in less extreme former communist countries.
Other texts on similar topics: Czechoslovakia, Europe, markets, politics
A response to a Jordanian U.N. bureaucrat
A Muslim calling himself "the U.N. high commissioner for human rights" has prepared a rant against the Western politicians who oppose the Islamization of Europe and North America. He gave the speech at the Hague yesterday.
A full transcript is available and will appear below, too. Let me respond to that diatribe.
Dear Friends, I wish to address this short statement to Mr Geert Wilders, his acolytes, indeed to all those like him – the populists, demagogues and political fantasists.
Geert Wilders is a Dutch lawmaker with no executive power right now, even though his Party for Freedom has a significant chance to win the next elections in 2017. It is weird for a U.N. official to single out an innocent Dutch citizen as a target of his personal attacks. Should we understand it as a new U.N. policy to bully innocent yet politically inconvenient individuals in Europe?
Perhaps, the fact that the speech was given at the Hague is supposed to be an explanation why the U.N. official targeted a Dutch politician and tried to intervene into the internal Dutch domestic affairs. However, the U.N. official should only see the Hague as a neutral place that hosts some of the international organizations similar to the organizations that employs the official himself. This location doesn't give him the moral right to intervene into the Dutch domestic political affairs.
The Dutch internal politics is not your business, Mr Hussein, just like the internal politics of Kuwait was not the business of your Iraqi namesake. You may object that there is a difference because Saddam has already been executed while you are still walking and talking. But the difference between the two of you could be just 10 years.
Other texts on similar topics: Europe, Middle East, politics, religion
Europhys News: 9/11 saw "controlled demolitions"
Europhysics News is a journal published by the European Physical Society. It sounds like a probable counterpart of the American Physical Society. Well, I have some problems with this organization, APS, but I still believe that their physics journal wouldn't publish what the EPS physics journal did.
In their
September 2016 issue of Europhys News,
there is an article on pages 23-28 (out of 36) claiming that the collapse of the skyscrapers on 9/11/2001 had to be due to a controlled demolition. Holy cow.
Almost exactly 10 years ago, I discussed a preprint incorrectly (as I showed) saying that the collapse seemed too fast. Here we go again.
Other texts on similar topics: Europe, experiments, science and society
Energiewende increased electricity prices by 40% since 2007
Thank God, my homeland hasn't been fully occupied by Merkel's Germany yet which is why my compatriots may write and read what is happening in the world, including Germany, and – among other things – how their government has f*cked up the energy industry.
A translation of an article from yesterday.
"E" for Energiewende. Germans gave a new word for billions to the world
(The word Energiewende means something like The Energy Breakthrough – I guess this is how it would be called in the U.S.)
Germans are boasting about their path towards a green energy industry. Hundreds of thousands of new jobs have been created. However, the bill is being paid by the German households that have to face the electricity prices higher by 40%.
Other texts on similar topics: Europe, politics
AfD beats Merkel's CDU in Northern DDR state elections
Mecklenburg-Western Pomerania is the state (Bundesland) covering the Northernmost 25% strip of former East Germany.
Pomerania itself is a region belonging to Germany and Poland these days; the name "Pomerania" (PL: Pomorze, CZ: Pomořany) approximately means "Area By the Sea" in Slavic languages. The equally named 7th century West Slavic tribe of Pomeranians has modern descendants in Poland, the Kashubians. Most of the Bundesland is Mecklenburg in the West, a historical land speaking Low German (Mecklenburg is a distorted Mikilenburg, a big castle; it was Veligrad in old Slavic languages and it is Megalopolis in modern Latin). Their union is the poorest Bundesland in Germany. It probably harbors the smallest number of Muslims as well – but the Bundesland is among those that understand that a big Muslim community is a problem.
On August 24th, Czech ex-president Václav Klaus visited Schwerin, the capital of this Bundesland, and gave a speech (DE) to support the Alternative für Deutschland. I think it's fortunate that not many Czechs are reading these things because they would probably be easy targets of parodies.
If I add just a little bit of the parody flavor, he said that he had really, really never visited the part of DDR North of Berlin before. Why would I be visiting such a boring flat place without mountains? How could I enjoy right-wing sports such as skiing over here? Or should I swim in your cold Baltic Sea? Funnily enough, as a kid, with my family, I have spent a week in the Rügen Island (CZ: ostrov Rujána). It was fun. Cold hardy right-wing boys such as your humble correspondent don't have a problem to swim in cold water. I've tried an Austrian lake close to 10 °C, too. ;-)
Serious neutrinoless double beta-decay experiment cools down
Data collection to begin in early 2017
The main topic of my term paper in a 1998 Rutgers Glennys Farrar course was the question "Are neutrinos Majorana or Dirac?". I found the neutrino oscillations more important which is why I internalized that topic more deeply – although it was supposed to be reserved by a classmate of mine (and for some Canadian and Japanese guys who got a recent Nobel prize for the experiments). At any rate, the question I was assigned may be experimentally answered soon. Or not. (You may also want to see a similarly old term paper on the Milky Way at the galactic center.)
Neutrinos are spin-1/2 fermions. Their masses may arise just like the masses of electrons or positrons. In that case, we need a full Dirac spinor, two 2-component spinors, distinct particles and antiparticles (neutrinos and antineutrinos), and everything about the mass works just like in the case of the electrons and positrons. The Dirac mass terms are schematically\[
{\mathcal L}_{\rm Dirac} = m\bar\Psi \Psi = m \epsilon^{AB} \eta_A \chi_B + {\rm h.c.}
\] If neutrinos were Dirac particles in this sense, it would mean that right-handed neutrinos and left-handed antineutrinos do exist, after all – just like the observed left-handed neutrinos and right-handed antineutrinos. They would just be decoupled i.e. hard to be created.
Other texts on similar topics: experiments, string vacua and phenomenology
Obama's "ratification" of the Paris climate treaty is a joke
I was asked to promote an article about Clexit (climate exit) at Breitbart. Well, I don't see how a website like mine could be effectively helping to increase the traffic at Breitbart that has far more visitors. It feels like the repetitive e-mails I am receiving from Donald Trump saying that he needs my money. Doesn't he have enough of his money and wouldn't it be more natural for him to send some money to me? ;-)
But as you can see, I did my best, included a hyperlink, and wrote a controversial description that will make some reader click at it. :-)
Orange countries are the current self-described "ratified signatories" of the Paris agreement.
Also, even though I am subscribed under Clexit, I don't see how the article by our "founding secretary of Clexit" exceeds or differs from thousands of other review articles about the climate issue that have been written in the last decade or two, except for his somewhat ludicrous title. Well, some of them were much deeper than this one. So apologies but with my lack of faith that this article makes any difference, I will probably not be an effective servant of the "founding secretary". For his age, his behavior looks childish to me.
Other texts on similar topics: climate, Kyoto, politics
\(3.5\keV\) line claimed to come from bare sulfur nuclei
The ET radio signal at \(11\,{\rm GHz}\) was due to a Soviet satellite, LUX and others have found no dark matter directly, the LHC hasn't proven any deviation from the Standard Model whatsoever, and Tracy Slatyer now believes that the seemingly exciting Fermi bubbles arrive from some boring pulsars.
Signs of any progress in physics through the experiment are being carefully stopped by Mother Nature. She is telling everyone: Stop with these ludicrous experiments and start to work on string theory seriously. ;-) What other anomalies get killed these days?
In two 2014 articles
Signal of neutrino dark matter (February, by Adam Falkowski)
Controversy about the \(3.5\keV\) line (August),
you could have learned about some tentative astronomical observations of X-rays with energy \(3.5\keV\), sometimes attributed to a \(7\keV\) sterile neutrino dark matter or something else that was equally new.
Other texts on similar topics: astronomy, experiments, string vacua and phenomenology
Billionaire Babiš in hot water after gypsy concentration camp comments
If I ignore his war against the small entrepreneurs and his self-evident conflicts of interest, I think that the Slovak-born food industry billionaire Andrej Babiš – widely considered as the most powerful Czech politician at this moment (and often painted as the next Führer) – is doing a good job as the finance minister (but so did many predecessors).
For example, the Czech government budget shows the record surplus over CZK 80 billion after August; it seems very likely to me that the whole year will end in a surplus. Obviously, I don't think that it's mostly Babiš's achievement but I don't want to discuss these things here.
Off-topic: Obama, Kerry, and their equally racist f*cked-up friends have imposed sanctions on [Czech company] Škoda JS [nuclear machinery] Pilsen, once a part of the Škoda Holding and now the main sponsor of the great ice-hockey team in my hometown, HC Škoda Pilsen (which has several North American players, a Native American 2nd-U.S-army-imitating logo, and a post-NHL owner) – because of some Russian owners. This gets rather personal. Do these megajerks in the U.S. administration also want to damage the ice-hockey team, using the fallacy by association? I prefer if you die quickly, you unAmerican Kerries.
However, it's his (and his voters') general totalitarian mindset – so perfectly compatible with the persuasive claims that he was a communist secret agent codenamed Bureš – that is so terribly troubling.
A minister (the most SJW-like minister Jiří Dienstbier Jr) and an emeritus opposition leader (Karel Schwarzenberg, an aristocrat) demand his resignation after some potentially controversial statements about the gypsies and work that he made in Northern Bohemia yesterday. This main part of the Sudetenland was mostly successfully repopulated after the 1945 expulsion of the ethnic Germans. When things are added, the Romani people represent a significant fraction of the population today and there are a few well-known gypsy ghettos in Northern Bohemia.
Colorado Springs instructors ban discussions on the climate, even in students' leisure time
The College Fix, a right-wing online newspaper on U.S. education, has brought us an incredible story (hat tip: Climate Depot, see also Newz Sentinel):
Professors tell students: Drop class if you dispute man-made climate change
Note that University of Notre Dame is a private Catholic university in Indiana. I thought that this apparently wasn't enough to save the students from the extreme far left brainwashing – but Bill Z. has corrected my misunderstanding of the location.
As Kate Hardiman, a pretty student at Notre Dame, describes, three women teach a bizarre class at University of Colorado in Colorado Springs (the main campus is in Boulder, however, and I spent a month over there in 1999 at TASI) named Medical Humanities in the Digital Age. If you click at the link, you will see that the students are being indoctrinated about the climate apocalypse, alleged bad health effects of fracking, asked to quantify their carbon footprint, and they're supposed to imagine that they are looking at the evil white male imperialists from the viewpoint of the Native Americans, not to mention lots of similar things.
According to the content, it is a typical far left extremist indoctrination course. You may also notice that the name of the course doesn't describe the content well.
Other texts on similar topics: climate, education, freedom vs PC
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Rapid photocatalytic degradation of phenol from water using composite nanofibers under UV
Alaa Mohamed ORCID: orcid.org/0000-0002-1439-86171,2,
Samy Yousef3,4,
Walaa S. Nasser5,
T. A. Osman6,
Alexander Knebel1,
Elvia P. Valadez Sánchez1 &
Tawheed Hashem1,7
The removal of phenol from aqueous solution via photocatalytic degradation has been recognized as an environmentally friendly technique for generating clean water. The composite nanofibers containing PAN polymer, CNT, and TiO2 NPs were successfully prepared via electrospinning method. The prepared photocatalyst is characterized by SEM, XRD, and Raman spectroscopy. Different parameters are studied such as catalyst amount, the effect of pH, phenol concentration, photodegradation mechanism, flow rate, and stability of the composite nanofiber to evaluate the highest efficiency of the photocatalyst.
The composite nanofibers showed the highest photodegradation performance for the removal of phenol using UV light within 7 min. The pH has a major effect on the photodegradation of phenol with its maximum performance being at pH 5.
Given the stability and flexibility of the composite nanofibers, their use in a dynamic filtration is possible and can be even reused after several cycles.
The contamination of water resources is a serious environmental concern. The water quality has been constantly decreasing due to the existence of different pollutants such as dyes, synthetic hormones, phenol, and pharmaceuticals [1, 2]. Phenol and phenolic compounds are one of the most persistent toxic organic pollutants discharged in wastewater effluents, which are resistant to environmental degradation through chemical, biological, and photolytic processes [3, 4]. The major sources of phenol and phenolic compounds are discharges of chemical process industries such as pulp and paper, pharmaceutical, agrochemical, petrochemical, and pesticide production [5]. The accumulation of these pollutants can have adverse effects on human health due to their toxicity, their ability to disrupt the endocrine system and their carcinogenic behavior [6]. Therefore, the improper handling and disposal of these toxic carcinogenic compounds pose a significant risk to the environment and the ecosystem [7, 8]. In order to deal with such pollutants, conventional adsorption technologies, such as biological degradation, chemical oxidation, adsorption, and photocatalysis are used for removal procedures [9, 10].
In recent years, photocatalysis has become an alternative for the removal of organic pollutants, including the treatment of phenolic wastewater s due to its economic, efficient, and green feature [11, 12]. Photocatalysis has many advantages including its use in UV, visible light radiation, and no waste because of its complete mineralization at the end of disintegration [13]. In addition, organic pollutants decompose in the presence of a wideband gap semiconductor which can promote reactions in the existence of UV light without being consumed in the entire reaction.
TiO2 is one of the most active catalysts among the semiconductors for the treatment of organic and contaminated components, due to its photosensitivity, low price, biological activity, and chemical stability [14, 15]. In practical applications, it is necessary to recover the remaining TiO2 nanoparticles in the photocatalytic reaction solution. In addition, carbon nanotube (CNT) has received a lot of attention as adsorbent due to their large specific surface area, small size, high porosity, and light mass density [16, 17]. The combining of TiO2 with CNTs has shown a major effect that can improve the overall performance of the photocatalytic process [15, 18]. TiO2 is actually an n-type semiconductor, however, in the existence of CNTs, photogenerated electrons can move freely towards to the surface of CNTs surface, which may have a lower Fermi level and leave too much valence band holes in the TiO2 to migrate to the surface and react [19, 20]. Also, CNTs can provide TiO2 with space restriction and a larger contact area, which leads to faster observed redox reaction rate [21]. Further, the application of anodic potentials to irradiated TiO2/CNTs composite can be further improved [22, 23]. The uniformity of the oxide coating and the physical properties of the composite materials vary depending on the manufacturing processes. In this regard, composite nanofibers have been used to immobilize TiO2/CNTs on the nanofibers [24, 25].
The hybrid photocatalysis and nanofiber membrane combines the advantages of both, membrane filtration and photocatalytic degradation. This approach allows the possibility of creating composite nanofibers membranes with superior removal efficiency and selectivity that lead to the development of a new water treatment solution [26, 27]. Given its facility of control and environmental sustainability, electrospinning is a versatile and effective process for the production of composite nanofibers with diameters from nanometers to submicrometers and characterized by their low cost, ease of use, and unique properties [28]. Electrospun PAN can be a promising carrier for immobilized catalytic materials since electrospun nanofibers based on PAN are hydrophobic, have a low density and are flexible in operation, thereby ensure easy floating on liquid or fixing at the desired location in the reactors [29, 30]. The above considerations form the basis of our further work. Herein, we investigate the utilization of a PAN-CNT/TiO2-NH2 nanofiber membrane in the application of organic wastewater decontamination. In this paper, the properties of PAN-CNT/TiO2-NH2 composite nanofibers by SEM, XRD, Raman spectra, and UV–Vis were investigated. The effect of catalyst dosage, contact time, pH solution, photodegradation mechanism, flow rate, and phenol concentration on the photocatalytic degradation of phenol under UV light are presented.
Analytical-grade chemicals used in this work were purchased from Sigma-Aldrich Sweden Ltd. PAN (MW = 150,000), Aeroxide TiO2, 3-aminopropyltriethoxysilane, glutaraldehyde, polyacrylonitrile, dimethylformamide (DMF), sodium hydroxide and hydrochloric acid were used in the experimental studies. The variation of pH in the feed solutions was maintained by adding 0.1 M HCl and 0.1 M NaOH. MWCNTs were synthesized following the procedure described in Yousef et.al [31]. Phenol (99.0%) was used to make standards and aqueous solutions for photocatalytic reactions.
Material preparation
10 wt. % of PAN was dissolved in DMF and stirred for 5 h. The, 1 wt. % of CNT was added to the PAN/DMF solution. The mixed solution was stirred for an additional 24 h and used for nanofiber preparation. For electrospun nanofibers, 1 mL/h flow rate and 25 kV was required. The modified surfaces of TiO2 NPs were described in our previous work. A mixture of 0.5 g of TiO2 and 10 mL deionized water were mixed to facilitate the adsorption of the hydroxyl group. The TiO2 were dispersed in 100 ml of toluene via ultrasonication for 30 min. Then 3 ml of the silane was added to the solution. The suspension was refluxed at 110 °C for 24 h creating NH2 functional groups on the titanium dioxide. The nanofibers were then immersed in a crosslinking medium composed of 100 mL distilled water and 2.5 wt% GA and then shaken for 24 h.
The surface morphology of the composite nanofibers were characterized by field emission scanning electron microscope (FE-SEM, S-5000, Hitachi, Japan) using an acceleration voltage of 10 kV. The crystal structure of the samples was determined via X-ray diffraction (XRD) patterns, which was recorded with a Bruker D8-Advance using Cu-Ka radiation at a step time of 0.2 s per point in the range of 20–80°. Raman spectra (Thermo scientific, DXR) were recorded between 0 and 2000 cm−1 with 514 nm wavelength. The UV–Vis absorption spectrum of the phenol samples were recorded after irradiation in a wavelength range of 200–500 nm with a UV–visible spectrophotometer (Lambda 35, PerkinElmer).
Photocatalysis experiments
The photocatalytic degradation experiments of phenol were carried out under dynamic conditions of a flow rate of 7 ml/min using ultraviolet (UV) radiation as shown in Fig. 1. The photoreactor is a cylindrical pyrex-glass column with a capacity of 250 ml, an inner diameter of 20 mm, and a height of 300 mm. A UV-A lamp (315–400 nm) was used as a UV light source with an intensity of 100 W. The nanofiber mat (3 × 6 cm, 50 mg) was immersed in the phenol solution with the concentration of 10 mg/L before each experiment. The suspension was kept in the dark for 30 min to achieve the equilibrium on the surfaces of the composite nanofibers. After that, the sample was exposed to UV light; reference samples were collected for analysis. At a given interval time, 3 mL suspension was collected and analyzed using a UV–visible spectrophotometer. All experiments were duplicated to assure the consistency and reproducibility of the results. The effect of initial phenol concentration (10–100 mg/L) and pH values from (2–9) were examined. The photodegradation efficiency of phenol can be expressed as follows:
$$\mathrm{Photodegradation efficiency }(\mathrm{\%}) =\frac{{C}_{i}-{C}_{o}}{{C}_{i}}\bullet 100,$$
where Ci and C0 (mg/L) are the initial and equilibrium concentration of phenol. The filtration performance of the composite nanofiber membranes was also characterized. After pre-wetting the nanofibers mat with DI water for 1 h, the membrane was fixed in the photoreactor. The pure water flux was calculated at various flow rates as follows:
$$\mathrm{Pure water flux }\left(\mathrm{J}0\right)= \frac{{m}_{p}}{{t\bullet {\rho }_{w}\bullet A}_{m}},$$
where mP is the mass of permeate (g), t is the filtration time, Am is the area of the composite membrane (cm2), and ρw is the density of water (g/cm3).
Schematic diagram of photoreactor system
To confirm the successful crosslinking process, preparation, and immobilization of NPs loaded on the nanofiber, SEM, XRD, and Raman spectra were carried out. Figure 2a and b shows the surface morphologies of the PAN-CNT and PAN-CNT/TiO2-NH2 composite nanofibers. It was clearly observed that the composite nanofiber membrane was smooth and uniform and crosslinked well with the NPs on the top surface of the nanofiber. The XRD analysis of PAN-CNT/TiO2-NH2 composite nanofiber is presented in Fig. 2c, proving the highly crystalline character of the TiO2 and CNTs nanoparticles. In addition, Fig. 2d shows the Raman spectra of the PAN, CNT, TiO2, and PAN-CNT/TiO2-NH2 composite nanofibers. The result shows that the composite nanofibers contain all the characteristic peaks of PAN, CNT, and TiO2, confirming the successful fabrication of the composite nanofiber membrane [32, 33].
a SEM image of PAN-CNT nanofiber, b PAN-CNT/TiO2-NH2 composite nanofiber, c XRD analysis, and d Raman spectra of the PAN-CNT/TiO2-NH2 composite nanofibers
Photocatalytic performance of composites nanofibers
To maximize the degradation efficiency, catalyst dose was found to be the potential catalyst for the degradation of phenol [34]. In order to explore its effective concentration, reactions were carried out with different catalyst quantities (TiO2-NH2) ranging from 5 to 30 mg and the amount of CNT fixed at 3 mg at 10 mg/L concentrations of phenol. At higher TiO2-NH2 the catalytic performance of the composite nanofibers is much better for the same photocatalytic time as shown in Fig. 3. The 20 mg catalyst showed the best photodegradation efficiency in 7 min. This behavior is due to the increase in the number of active sites and the high surface area of the nanofiber [35]. Therefore, the increase in electron–hole pairs on the catalyst surface leads to higher amounts of reactive hydroxyl radicals, which can be attributed to better degradation. The increased loading with CNT/TiO2-NH2 provides more binding sites for substrate molecules to adsorb on the catalyst surface [36].
Effect of catalyst amount on the photodegradation of phenol (10 mg/L phenol, time 7 min, and pH 7)
Furthermore, the pH of the system exerts a profound influence on adsorption and absorption due to its influence on the surface properties of the adsorbent and ionization/dissociation of the ion [37]. In this study, the pH value range between 2 and 9 was explored, as shown in Fig. 4. Apparently, after 7 min of UV irradiation, the degradation efficiency at pH 5 is about 99.2% and drops to 85% at pH 9. The photodegradation efficiency at pH 2 is slightly lower than at pH 6, which is due to the potential dissociation of PAN-CNT/TiO2-NH2 composite nanofiber and the Cl− ions that generated through pH adjustment compete with phenol for adsorption, thereby reducing adsorption and photocatalytic performance under strongly acidic conditions [38]; whereas, at high alkaline, the photodegradation performance is highly lower. The variation in the photodegradation efficiency with pH can be attributed to the strong electrostatic interaction between phenol and TiO2 at a neutral state [39].
Effect of pH on the photodegradation of phenol (10 mg/L phenol)
The suitable condition for the maximum degradation of 10 mg/L phenol was finalized by optimizing various reaction parameters. The effect of phenol concentration on degradation efficiency was performed by increasing the concentration from 10 mg/L to 100 mg/L (Fig. 5). The result shows that at pH 5 the photodegradation efficiency decreased with increasing phenol concentration from 10 to 100 mg/L. The photodegradation efficiency at 100 mg/L was 60% in less than 10 min. This behavior may be attributed to the fact that due to the increased phenol concentration, the energy of the UV light is intercepted before reacting with the photocatalyst.
Effect of phenol concentration on the photodegradation of phenol
Figure 6 shows the individual components of the composite nanofiber to evaluate the photodegradation mechanism of phenol. The photocatalytic degradation efficiency of PAN/TiO2-NH2 and PAN-CNT was 79 and 45%, respectively. These results indicate that the photodegradation performance of phenol over the PAN-CNT/TiO2-NH2 composite nanofiber is more than PAN/TiO2-NH2 which can be attributed to that CNTs play an important role in improving the photodegradation efficiency and the increasing the number of active sites and the adsorption strength of each active site. The photodegradation mechanism of phenol using the composite nanofiber depends on the active species OH which is the key to the degradation process of organic substrates. To initiate the photoreaction, TiO2 is irradiated with energy equal or higher than its band gap. The photogenerated electrons are raised to the conduction band (CB) and photogenerated holes remain in the valence band (VB) [45]. The positively charged electron hole (h +) reacts with water to form (•OH) as shown in Fig. 7. Meanwhile the excited electron reduces an oxygen molecule to (O2•−) for degrading phenol to small molecules [46]. In addition, several researchers proposed that photocatalyst attack the hydroxyl radicals on the phenyl ring, resulting in the formation of catechol, then further oxidizing to oxalic acid and mineralizing to CO2 and H2O.
Effect material components on the photodegradation of phenol (10 mg/L, pH 5)
Mechanism of phenol degradation on the composite nanofiber membrane
The reusability and stability performance of photocatalytic composite nanofibers is highly attractive for practical applications. Therefore, the photocatalytic degradation was performed for 3 cycles of photodegradation at 10 mg/L and pH 5 for 15 min under UV irradiation. Figure 8 shows its reusability and stability performance. The result showed that no significant decrease could be observed in the catalytic performance of the composite nanofiber membrane after three recycle experiments. This reveals that the catalyst is stable for more than 3 cycles and can be easily recycled for further use, which gives it a great potential in practical water treatment.
Stability of the PAN-CNT/TiO2-NH2 composite nanofiber during phenol degradation
The pure water flux of the composite nanofiber membrane was tested as a function of the flow rate as shown in Fig. 9. The results show that the flux increases linearly with the flow rate. In general, the high flux of the composite nanofiber membrane is strongly desired as it consumes less energy and has high throughput which are the most important advantages pursued.
Photodegradation efficiency and flux as a function of flow rate using the PAN-CNT/TiO2-NH2 composite nanofiber membrane
In summary, PAN-CNT/TiO2-NH2 composite nanofibers were successfully prepared via electrospinning processes for photocatalytic degradation of phenol by UV light. The composite nanofibers have stable performance and high photodegradation for phenol under UV light; nearly 99% within 7 min. This is attributed to the large surface area of the composite nanofiber which leads to significant improvement in light absorption, thereby achieving accelerated photodegradation and improving the photodegradation efficiency. The best result was achieved at the optimal condition of 20 mg catalyst, 10 mg/L phenol in neutral pH which is 5, and 7 min. In addition, the composite nanofibers are stable and reusable more than three times, which is an attractive performance for practical applications. Therefore, the present photocatalytic composite nanofiber membrane is an attractive option for large-scale environmental purification. Furthermore, compared to conventional nano-sized powder photocatalytic materials, it can be easily separated from the filtration system after photocatalytic reaction and reused more easily.
TiO2 :
CNT:
Carbon nanotube
PAN:
Polyacrylonitrile
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T. Hashem acknowledges support through the Cluster "3DMM2O" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany`s Excellence Strategy – 2082/1 – 390761711.
Open Access funding enabled and organized by Projekt DEAL. Cluster "3DMM2O" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany`s Excellence Strategy – 2082/1 – 390761711.
Institute of Functional Interfaces (IFG), Karlsruhe Institute of Technology (KIT), Hermann-von Helmholtz-Platz 1, 76344, Eggenstein-Leopoldshafen, Germany
Alaa Mohamed, Alexander Knebel, Elvia P. Valadez Sánchez & Tawheed Hashem
Department of Mechatronics, Fifth Settlement, Canadian International College, New Cairo, Egypt
Alaa Mohamed
Department of Production Engineering, Faculty of Mechanical Engineering and Design, Kaunas University of Technology, 51424, Kaunas, Lithuania
Samy Yousef
Department of Materials Science, Physical and Chemical Properties of Materials, South Ural State University (National Research University), Lenin Prospect 76, 454080, Chelyabinsk, Russia
Research Institute of Medical Entomology, Giza, 12611, Egypt
Walaa S. Nasser
Mechanical Design and Production Engineering Department, Cairo University, Giza, 12613, Egypt
T. A. Osman
Institute of Physics and Technology, International X-Ray Optics Lab, National Research Tomsk Polytechnic University (TPU), 30 Lenin ave, Tomsk, 634050, Russia
Tawheed Hashem
Alexander Knebel
Elvia P. Valadez Sánchez
AM: conceptualization, methodology, investigation, writing—original draft, writing—review and editing. SY: review and editing. WN: writing—original draft, analysis. TO, AK, EV, TH: review and editing. All authors read and approved the final manuscript.
Correspondence to Alaa Mohamed or Tawheed Hashem.
Mohamed, A., Yousef, S., Nasser, W.S. et al. Rapid photocatalytic degradation of phenol from water using composite nanofibers under UV. Environ Sci Eur 32, 160 (2020). https://doi.org/10.1186/s12302-020-00436-0
Composite nanofibers
Photodegradation
Electrospinning | CommonCrawl |
Richard W. Cottle
Richard W. Cottle (29 June 1934) is an American mathematician. He was a professor of Management Science and Engineering at Stanford University, starting as an Acting Assistant Professor of Industrial Engineering in 1966 and retiring in 2005. He is notable for his work on mathematical programming/optimization, “Nonlinear programs”, the proposal of the linear complementarity problem, and the general field of operations research.
Richard W. Cottle
Born29 June 1934
Chicago, Illinois
NationalityAmerican
Alma materHarvard College, University of California at Berkeley
Life and career
Early life and family
Richard W. Cottle was born in Chicago on 29 June 1934 to Charles and Rachel Cottle. He started his elementary education in the neighboring village of Oak Park, Illinois and graduated from Oak Park-River Forest High School. After that, admitted to Harvard, Cottle began by studying government (political science) and taking premedical courses. After the first semester, he changed his major to mathematics in which he earned his bachelor's (cum laude) and master's degrees. Around 1958, he became interested in teaching secondary-level mathematics. He joined the Mathematics Department at the Middlesex School in Concord, Massachusetts where he spent two years. Midway through the latter period, he married his wife, Suzanne.[1]
Career[2][3]
While teaching at Middlesex School, he applied and was admitted to the PhD program in mathematics at the University of California at Berkeley, with the intention of focusing on geometry. Meanwhile, he also received an offer from the Radiation Laboratory at Berkeley as a part-time computer programmer. Through that work, some of which involved linear and quadratic programming, he became aware of the work of George Dantzig and Philip Wolfe. Soon thereafter he became a member of Dantzig's team at UC Berkeley Operations Research Center (ORC). There he had the opportunity to investigate quadratic and convex programming. This developed into his doctoral dissertation under the guidance of Dantzig and Edmund Eisenberg. Cottle's first research contribution, "Symmetric Dual Quadratic Programs," was published in 1963. This was soon generalized in the joint paper "Symmetric Dual Nonlinear Programs," co-authored with Dantzig and Eisenberg. This led to the consideration of what is called a "composite problem," the first-order optimality conditions for symmetric dual programs. This in turn, was named "the fundamental problem" and still later (in a more general context) "the complementarity problem." A special case of this, called "the linear complementarity problem",[4] is a major part of Cottle's research output. Also in 1963, he was a summer consultant at the RAND Corporation working under the supervision of Philip Wolfe. This resulted in the RAND Memo, RM-3858-PR, "A Theorem of Fritz John in Mathematical Programming."
In 1964, upon completion of his doctorate at Berkeley, he worked for Bell Telephone Laboratories in Holmdel, New Jersey. In 1965, he was invited to visit Stanford's OR Program, and in 1966, he became an Acting Assistant Professor of Industrial Engineering at Stanford. The next year he became an Assistant Professor in Stanford's new Department of Operations Research. He became an Associate Professor in 1969 and Full Professor in 1973. He chaired the department from 1990 to 1996. During 39 years on the active faculty at Stanford he had over 30 leadership roles in national and international conferences. He served on the editorial board of 8 scholarly journals, and was Editor-in-Chief of the journal, Mathematical Programming. He served as the associate chair of the Engineering-Economic Systems & Operations Research Department (EES & OR) after the merger of the two departments. In 2000, EES & OR merged again, this time with the Industrial Engineering & Engineering Management Department to form Management Science and Engineering (MS&E). During his sabbatical year at Harvard and MIT (1970-1971), he wrote “Manifestations of the Schur Complement’’, one of his most cited papers. In 1974, he started working on “The Linear Complementarity Problem,” one his most noted publications. In the mid 1980s, two of his former students, Jong-Shi Pang and Richard E. Stone, joined him as co-authors of this book which was published in 1992. “The Linear Complementarity Problem” won the Frederick W. Lanchester Prize of the Institute for Operations Research and the Management Sciences (INFORMS) in 1994. “The Linear Complementarity Problem” was republished by the Society for Industrial and Applied Mathematics in the series “Classics in Applied Mathematics series” in 2009. During 1978–1979, he spent a sabbatical year at the University of Bonn and the University of Cologne. There he wrote the paper “Observations on a Class of Nasty Linear Complementarity Problems’’ which relates the celebrated Klee-Minty result on the exponential time behavior of the simplex method of linear programming with the same sort of behavior in Lemke's algorithm for the LCP and hamiltonian paths on the n-cube with the binary Gray code representation of the integers from 0 to 2^n - 1. Also during this time he solved the problem of minimally triangulating the n-cube for n = 4 and worked with Mark Broadie to solve a restricted case for n = 5. In 2006 he was appointed a fellow of INFORMS[5] and in 2018 received the Saul I. Gass Expository Writing Award.
Contributions
Linear complementarity Problem
Cottle is best known for his extensive publications on the Linear Complementarity Problem (LCP). This work includes analytical studies, algorithms, and the interaction of matrix theory and linear inequality theory with the LCP. Much of this is an outgrowth of his doctoral dissertation supervised by George Dantzig, with whom he collaborated in some of his earliest papers. The leading example is "Complementary pivot theory of mathematical programming," published in 1968.
Definitions
The standard form of the LCP is a mapping:
$f:R^{n}\rightarrow R^{n}\mathrm {\textvisiblespace} f(x)=q+Mx$ (1)
Given $f$, find a vector $x\in R^{n}$ , such that $x_{i}\geq 0$, $f_{i}(x)\geq 0$ and $x_{i}f_{i}(x)=0$, for $i=1,2,...,n$
Because the affine mapping f is specified by vector and matrix, the problem is ordinarily denoted LCP(q, M) or sometimes just (q, M). A system of the form (1) in which f is not affine is called a nonlinear complementarity problem and is denoted NCP($f$). The notation CP($f$) is meant to cover both cases."[6]
Polyhedral sets having a least element
According to a paper by Cottle and Veinott: "For a fixed m $\times $ n matrix A, we consider the family of polyhedral sets $X_{b}=\{x|Ax\geq b\},\mathrm {\textvisiblespace} b\in R_{m}$ , and prove a theorem characterizing, in terms of A, the circumstances under which every nonempty X_b has a least element. In the special case where A contains all the rows of an n $\times $ n identity matrix, the conditions are equivalent to A^T being Leontief.[7]
Publications and others
Publications and Professional Activities
This list has been retrieved from the website.[8]
• Richard W. Cottle: On "Pre-historic" Linear Programming and the Figure of the Earth. J. Optimization Theory and Applications 175(1): 255-277 (2017)
• Ilan Adler, Richard W. Cottle, Jong-Shi Pang: Some LCPs solvable in strongly polynomial time with Lemke's algorithm. Math. Program. 160(1-2): 477-493 (2016)
• Richard W. Cottle: A field guide to the matrix classes found in the literature of the linear complementarity problem. J. Global Optimization 46(4): 571-580 (2010)
• Richard W. Cottle: A brief history of the International Symposia on Mathematical Programming. Math. Program. 125(2): 207-233 (2010)
• Richard W. Cottle: Linear Complementarity Problem. Encyclopedia of Optimization 2009: 1873-1878
• Richard W. Cottle, Ingram Olkin: Closed-form solution of a maximization problem. J. Global Optimization 42(4): 609-617 (2008)
• Richard W. Cottle: Book Review. Optimization Methods and Software 23(5): 821-825 (2008)
• Richard W. Cottle: George B. Dantzig: a legendary life in mathematical programming. Math. Program. 105(1): 1-8 (2006)
• Ilan Adler, Richard W. Cottle, Sushil Verma: Sufficient matrices belong to L. Math. Program. 106(2): 391-401 (2006)
• Richard W. Cottle: George B. Dantzig: Operations Research Icon. Operations Research 53(6): 892-898 (2005)
• Richard W. Cottle: Quartic Barriers. Comp. Opt. and Appl. 12(1-3): 81-105 (1999)
• Richard W. Cottle: Linear Programs and Related Problems (Evar D. Nering and Albert W. Tucker). SIAM Review 36(4): 666-668 (1994)
• Richard W. Cottle: The Principal Pivoting Method Revisited. Math. Program. 48: 369-385 (1990)
• Muhamed Aganagic, Richard W. Cottle: A constructive characterization of Qo-matrices with nonnegative principal minors. Math. Program. 37(2): 223-231 (1987)
• Mark Broadie, Richard W. Cottle: A note on triangulating the 5-cube. Discrete Mathematics 52(1): 39-49 (1984)
• Richard W. Cottle, Richard E. Stone: On the uniqueness of solutions to linear complementarity problems. Math. Program. 27(2): 191-213 (1983)
• Richard W. Cottle: Minimal triangulation of the 4-cube. Discrete Mathematics 40(1): 25-29 (1982)
• Richard W. Cottle: Observations on a class of nasty linear complementarity problems. Discrete Applied Mathematics 2(2): 89-111 (1980)
• Yow-Yieh Chang, Richard W. Cottle: Least-index resolution of degeneracy in quadratic programming. Math. Program. 18(1): 127-137 (1980)
• Richard W. Cottle: The journal. Math. Program. 19(1): 1-2 (1980)
• Richard W. Cottle: Completely- matrices. Math. Program. 19(1): 347-351 (1980)
• Muhamed Aganagic, Richard W. Cottle: A note on Q-matrices. Math. Program. 16(1): 374-377 (1979)
• Richard W. Cottle, Jong-Shi Pang: A Least-Element Theory of Solving Linear Complementarity Problems as Linear Programs. Math. Oper. Res. 3(2): 155-170 (1978)
• Richard W. Cottle: Three remarks about two papers on quadratic forms. Zeitschr. für OR 19(3): 123-124 (1975)
• Richard W. Cottle: Book reviews. Math. Program. 4(3): 349-350 (1973)
• Richard W. Cottle: Monotone solutions of the parametric linear complementarity problem. Math. Program. 3(1): 210-224 (1972)
• Richard W. Cottle, Jacques A. Ferland: On pseudo-convex functions of nonnegative variables. Math. Program. 1(1): 95-101 (1971)
• Richard W. Cottle: Letter to the Editor - On the Convexity of Quadratic Forms Over Convex Sets. Operations Research 15(1): 170-172 (1967)
Membership
1. International Linear Algebra Society 1989–2005.
2. Gesellschaft für Mathematik, Ökonomie, und Operations Research 1984–1998
3. Mathematical Programming Society 1970
4. INFORMS 1995
5. The Institute of Management Sciences 1967–1995
6. Operations Research Society of America 1962–1995
7. Society for Industrial and Applied Mathematics 1966
8. Mathematical Association of America 1958-2017
9. American Mathematical Society 1958
Further reading
R. W. Cottle and G. B. Dantzig. Complementary pivot theory of mathematical programming. Linear Algebra and its Applications, 1:103-125, 1968
References
1. "Cottle, Richard W." purl.stanford.edu. Retrieved 2018-11-09.
2. "Cottle, Richard W." purl.stanford.edu. Retrieved 2018-11-09.
3. INFORMS. "Cottle, Richard W." INFORMS. Retrieved 2018-11-09.
4. Cottle, Richard W. (2008), "Linear Complementarity Problem", Encyclopedia of Optimization, Springer US, pp. 1873–1878, doi:10.1007/978-0-387-74759-0_333, ISBN 9780387747583
5. Fellows: Alphabetical List, Institute for Operations Research and the Management Sciences, retrieved 2019-10-09
6. Cottle, Richard W. (2008), "Linear Complementarity Problem", Encyclopedia of Optimization, Springer US, pp. 1873–1878, doi:10.1007/978-0-387-74759-0_333, ISBN 9780387747583
7. Cottle, Richard W.; Veinott, Arthur F. (December 1972). "Polyhedral sets having a least element". Mathematical Programming. 3–3 (1): 238–249. doi:10.1007/bf01584992. ISSN 0025-5610. S2CID 34876749.
8. "dblp: Richard W. Cottle". dblp.uni-trier.de. Retrieved 2018-10-19.
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| Wikipedia |
\begin{document}
\title{Dynamical Zeta Functions and Kummer Congruences}
\author{J. Arias de Reyna}
\address{Department of Mathematical Analysis, University of Seville, Seville, Spain}
\curraddr{Facultad de Matem‡ticas, Universidad de Sevilla, Apdo.~1160, 41080-Sevilla, Spain} \email{[email protected]}
\thanks{This research was supported by Grant BFM2000-0514.}
\subjclass{Primary 11B68, 37C30; Secondary 11B37, 37B10}
\date{June 26, 2003 }
\keywords{Kummer congruences, Bernoulli numbers, Euler numbers, integer sequences, zeta function}
\begin{abstract} We establish a connection between the coefficients of Artin-Mazur zeta-functions and Kummer congruences. \par This allows to settle positively the question of the existence of a map $T\colon X\to X$ such that the number of fixed
points of $T^n$ are $|E_{2n}|$, where $E_{2n}$ are the Euler numbers. Also we solve a problem of Gabcke related to the coefficients of Riemann-Siegel formula. \end{abstract}
\maketitle
\def\mathbf{N}{\mathbf{N}} \def\mathbf{Z}{\mathbf{Z}} \def\operatorname{Fix}{\operatorname{Fix}} \def\operatorname{sgn}{\operatorname{sgn}} \def\operatorname{Tr}{\operatorname{Tr}}
\section*{Introduction}
In this paper we establish a connection between two important topics the Artin-Mazur zeta function and Kummer's congruences. Some connection between Kummer's congruences and periodic points are pointed in the paper by Everest, van der Poorten, Puri and Ward \cite{E}.
Inspired by the Hasse-Weil zeta function of an algebraic variety over a finite field, Artin and Mazur \cite{AM} defined the Artin-Mazur zeta function for an arbitrary map $T\colon X\to X$ of a topological space $X$: $$Z(T;x):=\exp\left(\sum_{n=1}^\infty\frac{\operatorname{Fix} T^n}{n} x^n\right).$$ Where $\operatorname{Fix} T^n$ is the number of isolated fixed points of $T^n$.
Manning \cite{M} proved the rationality of the Artin-Mazur zeta function for diffeomorphisms of a smooth compact manifold satisfying Smale axiom $A$.
Following \cite{P}, call a sequence $a=(a_n)_{n\ge1}$ of non-negative integers \emph{ realizable} if there is a set $X$ and a map $T\colon X\to X$ such that $a_n$ is the number of fixed points of $T^n$.
We must notice that in \cite{P} it is proved that if $(a_{n})$ is realizable, then there exists a compact space $X$ and a homeomorphism $T\colon X\to X$, such that $a_{n}=\operatorname{Fix} T^n$.
Puri and Ward \cite{PW} proved that a sequence of non-negative integers
$(a_{n})_{n\ge1}$ is realizable if and only if $\sum_{d\mid n}\mu(n/d)a_d$ is non negative and divisible by $n$ for all $n\ge1$. Here $\mu(n)$ denotes the well known Mšbius function (see \cite{A}), defined by $\mu(n)=(-1)^k$ if $n$ is a product of $k$ different prime numbers, and $\mu(n)=0$ if $n$ is not squarefree.
We shall delete the positivity condition, so we shall say that the sequence of integers $(a_n)_{n=1}^\infty$ is \emph{pre-realizable} if $\sum_{d\mid n}\mu(n/d)a_d$ is divisible by $n$ for every natural number.
In 1851 Kummer \cite{Km} discovered what we call Kummer's
congruences for Bernoulli numbers, (see the book by Nielsen \cite{N}).
Carlitz \cite{C} extended these
congruences to the generalized Bernoulli numbers of Leopoldt. Some
restrictions of Carlitz's results has been removed by the work of
Fresnel \cite{F}. These congruences are important for the definition
of the $p$-adic $L$-functions.
We establish a connection between these concepts that we can
formulate as in the following theorem.
\begin{theorem} Let $(a_n)$ be a sequence that satisfies Kummer congruences for every rational prime, then for every natural number $b$ the sequence $(a_{b+n})_{n=1}^\infty$ is pre-realizable. \end{theorem}
This theorem allow us to solve a problem posed by Gabcke \cite{G}. This is connected with the Riemann-Siegel formula. In the investigation of the zeta function of Riemann it is important to compute the values of this function $\zeta(1/2+it)$ at points on the critical line with $t$ very high. Riemann found a very convenient formula for these computations, yet he does not publish anything about this formula. In 1932 C.~L.~Siegel was able to recover it from Riemann's nachlass. Now this formula is known as the Riemann-Siegel formula.
To obtain the terms of this formula play a role certain numbers $\lambda_n$ that can be defined by a recurrence relation \begin{equation*} \begin{split} \lambda_0&=1,\\
(n+1)\lambda_{n+1}&=\sum_{k=0}^n2^{4k+1}|E_{2k+2}|\lambda_{n-k}. \end{split} \end{equation*} Here $E_{2n}$ denotes Euler numbers defined by \begin{equation*} \frac{1}{\cosh x}=\sum_{n=0}^\infty \frac{E_n}{n!}x^n. \end{equation*} Hence $E_{2n+1}=0$ for $n\ge0$ and $$E_0=1,\quad E_2=-1,\quad E_4=5,\quad E_6=-61,\quad E_8=1\,385,\quad \dots$$
Gabcke \cite{G} observed that the first six numbers $\lambda_n$ are integers and conjectured that this is so for all of them. Gabcke also considers analogous sequences $(\varrho_n)$ and $(\mu_n)$. Although he does not mention it, the same motivations for his conjecture also supports that these too are integers sequences. We prove all these conjectures. The proof of these assertions was the first motivation of this paper.
In \cite{PW} Puri and Ward ask if the sequence
$(|E_{2n}|)_{n\ge1}$ is realizable. As we will see the solution of Gabcke's problem is connected with this one. We shall prove that in fact it is realizable.
Notations: When $p$ is a prime number and $m$ an integer we shall put $p^\alpha\parallel m$ to indicate that $p^\alpha$ is the greatest power of $p$ that divides $m$. We indicate this relation also by $\nu_p(m)=\alpha$. We shall put $n\perp m$ to say that $n$ and $m$ are relatively prime.
\section{Dynamical Zeta Function}
\begin{theorem}\label{th:beene}
Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers and
define the sequence $(b_n)_{n=1}^\infty$ by
\begin{equation}\label{eq:defb}
nb_n=\sum_{d\mid n}\mu(n/d)a_d.
\end{equation}
Then we have the equality between formal power series
$$ \prod_{n=1}^\infty(1-x^n)^{-b_n}=
\exp\Bigl(\sum_{n=1}^\infty\frac{a_n}n\Bigr).$$ \end{theorem}
\begin{proof}
By the well known Mšbius inversion formula the relation
(\ref{eq:defb}) is
equivalent to
\begin{equation}\label{E:Mobius}
a_n=\sum_{d\mid n} d b_d,
\end{equation} therefore we have the following equalities between formal power series \begin{eqnarray} \log\prod_{n=1}^\infty(1-x^n)^{-b_n}&=&-\sum_{n=1}^\infty b_n\log(1-x^n)= \sum_{n=1}^\infty\sum_{k=1}^\infty b_n\frac{x^{nk}}{k}\nonumber\\ &=& \sum_{m=1}^\infty\frac{x^m}{m} \Bigl(\sum_{n\mid m}nb_n\Bigr)= \sum_{m=1}^\infty\frac{x^m}{m} a_m.\nonumber \end{eqnarray} And this is equivalent to the equality we want to prove. \end{proof}
\begin{theorem}\label{th:Aene}
Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers and
define the sequence $(A_n)_{n=0}^\infty$ by the recurrence
relation
\begin{equation*}\begin{split}
A_{0}&=1,\\
(n+1)A_{n+1}&= \sum_{k=0}^n A_{n-k} a_{k+1},\qquad n\ge0.
\end{split}\end{equation*}
Then we have the equality between formal power series
$$\sum_{n=0}^\infty A_n x^n=
\exp\Bigl(\sum_{n=1}^\infty\frac{a_n}{n}x^n\Bigr).$$ \end{theorem}
\begin{proof}
First we have the equality between formal power series
$$\sum_{n=1}^\infty n A_n x^{n-1}=\Bigl(\sum_{n=0}^\infty A_n x^n\Bigr)
\Bigl(\sum_{n=1}^\infty a_n x^{n-1}\Bigr)$$
because by the hypothesis the coefficient of $x^n$ is equal in both members.
Since $A_0=1$ integrating formally give us
$$\log\Bigl(\sum_{n=0}^\infty A_n x^n\Bigr)=\sum_{n=1}^\infty \frac{a_n}{n}
x^n.$$
That is equivalent to the relation we wanted to prove. \end{proof}
The following theorem gives various equivalent conditions for a sequence of integers
$(a_n)_{n=1}^\infty$ to be pre-realizable.
\begin{theorem} \label{T:equiv} Given a sequence $(a_n)_{n\ge1}$ of integers, the following conditions are equivalent: \begin{enumerate} \item[(a)] The numbers $(b_n)_{n\ge 1}$ defined by $$n b_n=\sum_{d\mid n} \mu(n/d) a_d$$ are integers for every $n\in\mathbf{N}$.
\item[(b)] The numbers $(A_n)_{n\ge0}$ defined by \begin{equation}\label{eq:defA} \begin{split} A_{0}&=1,\\ (n+1)A_{n+1}&= \sum_{k=0}^n A_{n-k} a_{k+1},\qquad n\ge0 \end{split}\end{equation} are integers for every $n\ge0$.
\item[(c)] For every prime number $p$ and natural numbers $n$, $\alpha$ with $p\perp n$ we have $$a_{np^\alpha}\equiv a_{n p^{\alpha-1}}\pmod{p^\alpha}.$$ \end{enumerate} \end{theorem}
\begin{proof} First we prove the equivalence of (a) and (b).
(a)$\Longrightarrow$(b). Assume (a). By the definition of the $(b_n)$ and
Theorem \ref{th:beene} we have $$\prod_{n=1}^\infty(1-x^n)^{-b_n}=
\exp\Bigl(\sum_{n=1}^\infty\frac{a_n}n x^n\Bigr).$$ and by the condition (a) of the theorem the $b_n$ are integers. Let $(A_n)_{n=0}^\infty$ the numbers defined by \ref{eq:defA}, we have to show that they are integers. By Theorem \ref{th:Aene} these numbers satisfies the relation
$$\sum_{n=0}^\infty A_n x^n=
\exp\Bigl(\sum_{n=1}^\infty\frac{a_n}n x^n\Bigr).$$ Thus we have $$\sum_{n=0}^\infty A_n x^n=\prod_{n=1}^\infty(1-x^n)^{-b_n}.$$ Expanding this product, since the $b_n$ are integers, we get that the $A_n$ are also integers. Hence we have that (a) implies (b).
(b)$\Longrightarrow$(a). Now, by hypothesis the numbers $(A_n)_{n\ge0}$ are integers. We can determine inductively a unique sequence of integers $c_n$ such that $$\sum_{n=0}^\infty A_n x^n=\prod_{n=1}^\infty (1-x^n)^{-c_n}.$$
In the first step observe that the coefficients of $x$ in both members must be the same, hence $$A_1=c_1. $$
Then observe that $$(1-x)^{c_1} \Bigl(\sum_{n=0}^\infty A_n x_n\Bigr)=1+ \sum_{n=2}^\infty A^{(2)}_n x^n,$$ where the numbers $A^{(2)}_n$ are integers.
Assume by induction that we have determined integers $c_j$, for $j=1$, $2$, \dots $n-1$ such that $$\prod_{j=1}^{n-1}(1-x^j)^{c_j}\Bigl(\sum_{n=0}^\infty A_n x^n\Bigr)= 1+\sum_{k=n}^\infty A^{(n)}_k x^k.$$ Then the $A^{(n)}_k$ are integers and we can define $c_n=A_n^{(n)}$, that satisfies the induction hypothesis. Now we have $$\prod_{j=1}^\infty (1-x^j)^{c_j}\Bigl(\sum_{n=0}^\infty A_n x^n\Bigr)=1.$$ By the hypothesis and Theorem \ref{th:Aene} $$\sum_{n=0}^\infty A_n x^n= \exp\left(\sum_{n=1}^\infty\frac{a_n}{n} x^n\right).$$ Therefore $$\prod_{n=1}^\infty (1-x^n)^{-c_n}=\sum_{n=0}^\infty A_n x^n= \exp\left(\sum_{n=1}^\infty\frac{a_n}{n} x^n\right).$$ Now take logarithms in both members to obtain $$\sum_{n=1}^\infty c_n\sum_{k=1}^\infty \frac{x^{kn}}k= \sum_{n=1}^\infty\frac{a_n}{n} x^n.$$ Reasoning as in the proof of Theorem \ref{th:beene} we get $$a_m=\sum_{n\mid m} n c_n.$$ Therefore by the Mšbius inversion formula $c_n=b_n$ the numbers defined on condition (a), and by construction these numbers $c_n$ are integers. Thus we have proved (a).
(a)$\Longrightarrow$(c). We know that condition (a) is equivalent to the existence of integers $b_n$ that satisfy the equation (\ref{E:Mobius}).
Assume that $p$ is a prime number and $n$ and $\alpha$ natural numbers such that $p\perp n$. Then $$a_{np^{\alpha}}=\sum_{d\mid np^\alpha} d b_d= \sum_{k=0}^\alpha\sum_{d\mid n} d p^k b_{d p^k}.$$ Analogously $$a_{np^{\alpha-1}}=\sum_{k=0}^{\alpha-1}\sum_{d\mid n} d p^k b_{d p^k}.$$ Therefore $$a_{np^{\alpha}}-a_{np^{\alpha-1}}=\sum_{d\mid n} d p^\alpha b_{d p^\alpha} \equiv0\pmod{p^{\alpha}}.$$
(c)$\Longrightarrow$(a). Let $n$ be an integer. We have to show that $$\sum_{d\mid n} \mu(n/d) a_d$$ is divisible by $n$. Let $p^\alpha\parallel n$, with $\alpha\ge1$,
then $n=p^\alpha m$ with $p\perp m$.
Since $\mu(k)\ne0$ only when $k$ is squarefree, we get \begin{multline*} \sum_{d\mid n} \mu(n/d) a_d=\sum_{d\mid m} \mu(m/d) a_{d p^\alpha }- \sum_{d\mid m}\mu(m/d) a_{d p^{\alpha-1}}\\= \sum_{d\mid m}\mu(m/d)\bigl( a_{d p^\alpha } -a_{d p^{\alpha-1}}\bigr) \equiv 0\pmod{p^{\alpha}}. \end{multline*} The sum is divisible for every primary divisor of $n$, and therefore divisible by $n$. \end{proof}
\section{Kummer congruences}
In 1851 Kummer \cite{Km} proved the following theorem:
\begin{theorem}[Kummer]
Let $p$ be a prime number. Assume that
$$\sum_{n=0}^\infty a_n\frac{x^n}{n!}=\sum_{k=0}^\infty
c_k(e^{bx}-e^{ax})^k,$$
where $a$, $b$ and the $c_k$ are integral $\pmod p$. Then the $a_n$
are integers
$\pmod p$ and for $e\ge 1$, $n\ge1$, $m\ge0$, and
$p^{e-1}(p-1)\mid w$ we have
\begin{equation}\label{E:Kummer}
\sum_{s=0}^n(-1)^s\binom{n}{s}a_{m+sw}\equiv0
\pmod{(p^m,p^{ne})}.
\end{equation} \end{theorem}
The congruences (\ref{E:Kummer}) are usually called Kummer congruences. We shall say that the sequence $(a_n)$ satisfies Kummer congruences if we have (\ref{E:Kummer}) for every prime number $p$. By Kummer theorem these sequences exist, but we are interested in some particular sequences.
\begin{theorem}\label{T:Euler}
The sequence $(E_{2n})_{n=1}^\infty$ satisfies Kummer congruences.
\end{theorem}
\begin{proof}
Since
$$\frac1{\cosh
x}=\frac2{e^x+e^{-x}}=\frac{2}{2+(e^{x/2}-e^{-x/2})^2},$$
Kummer theorem proves that $(E_n)_{n=1}^{\infty}$ satisfies
Kummer congruence (\ref{E:Kummer}) for every odd prime number $p$.
Therefore the sequence $(E_{2n})_{n=1}^\infty$ satisfies
these congruences for every odd prime number $p$. This reasoning
can be found in Kummer \cite{Km}.
The above procedure does not give the case $p=2$, but
Fresnel \cite{F} has extended Kummer congruences for Euler
number, as we see in the following lines.
Let $\chi$ the function from $\mathbf{Z}$ to $\{-1,0,1\}$, such that
$\chi(n)=0$ if $n$ is even, $\chi(n)=1$ if $n\equiv1\pmod4$ and
$\chi(n)=-1$ if $n\equiv3\pmod4$, then the generalized Bernoulli
numbers associated to this character, ---see \cite{F} for
details---
are related to Euler numbers as
$$\frac{B^n(\chi)}{n}=-\frac{E_{n-1}}{2}.$$
In \cite{F} p.~319 we found that, when $2^e\parallel w$, with $e\ge1$
$$\sum_{s=0}^n(-1)^s\binom{n}{s}\frac{B_{m+sw}(\chi)}{m+sw}\equiv0
\pmod{(2^{n(e+2)}, 2^{m-1})}.$$
With a change of notation this is equivalent to
$$\sum_{s=0}^n(-1)^s\binom{n}{s}\frac{E_{2m+sw}}{2}\equiv0
\pmod{(2^{n(e+2)}, 2^{2m})}.$$
Obviously this implies that for $2^{e-1}\mid w$ we have
$$\sum_{s=0}^n(-1)^s\binom{n}{s}E_{2(m+sw)}\equiv0
\pmod{(2^{ne},2^m)}.$$ \end{proof}
The above theorem is a model of many more interesting examples. In Carlitz \cite{C}, it is proved that if $\chi$ is a primitive character $\bmod f$, and $f$ is divisible by at least two distinct rational primes, then $B^n(\chi)/n$ is an algebraic integer and $$\sum_{s=0}^n(-1)^s\binom{n}{s}\frac{B^{n+1+sw}(\chi)}{n+1+sw}\equiv0 \pmod{(p^n,p^{e n})},$$ if $p^{e-1}(p-1)\mid w$.
Thus the sequence $(a_n)_{n=1}^\infty$, with $a_n=B^{n+1}(\chi)/(n+1)$ satisfies Kummer congruences, if the character $\chi$ is real. When $\chi$ is complex the sequence defined by $a_n=\operatorname{Tr}(B^{n+1}(\chi)/(n+1))$ satisfies Kummer congruences.
The sequences that satisfies Kummer congruences are pre-realizable, as we will see in the following theorem.
\begin{theorem}\label{T:KummerK}
Let $(a_n)_{n=1}^\infty $ a sequence that satisfies Kummer
congruences. Then
$$a_{b+np^\alpha}\equiv a_{b+np^{\alpha-1}}\pmod{p^\alpha},$$
for every natural numbers $b$, $n$, $\alpha$ and prime number $p$
such that $p\perp n$.
That is to say that if $(a_n)$ satisfies Kummer congruences then
for every natural number $b$,
the sequence
$(a_{b+n})_{n=1}^\infty$ is pre-realizable.
\end{theorem}
\begin{proof}
By (\ref{E:Kummer}), with $n=1$ we have
$$a_{m+p^{e-1}(p-1)}\equiv a_m\pmod{(p^m, p^e)}.$$
Therefore, for every natural number $k$, and assuming $m\ge e$
$$a_{m+kp^{e-1}(p-1)}\equiv a_m\pmod{ p^e}.$$
Now take $m=b+np^{\alpha-1}$, $k=n$ and $e=\alpha$. If
$b+np^{\alpha-1}\ge \alpha$, we get
$$a_{b+np^{\alpha-1}+np^{\alpha-1}(p-1)}\equiv
a_{b+np^{\alpha-1}}\pmod{p^\alpha}.$$
Since
$p^{\alpha-1}\ge\alpha$ for $p$ prime and $\alpha\ge1$, the
condition is satisfied and we get
$$a_{b+np^\alpha}\equiv
a_{b+np^{\alpha-1}}\pmod{p^\alpha}.$$ \end{proof}
\section{Euler numbers as numbers of fixed points}
We are now in position to solve the problem posed by Puri and Ward in \cite{PW},
they ask if the sequence $(|E_{2n}|)_{n=1}^\infty$ is realizable. We shall show that this is true.
\begin{theorem}\label{th:Ereal}
There exists a map $T\colon X\to X$, such that
$$|E_{2n}|=\operatorname{Fix} T^n.$$ \end{theorem}
\begin{proof}
First we show that $|E_{2n}|$ is a pre-realizable sequence.
By Theorem \ref{T:Euler} the sequence $(E_{2n})_{n=1}^\infty$
satisfies Kummer congruences. Thus by Theorem \ref{T:KummerK},
for $p\perp m$,
$$E_{2m p^{\alpha}}\equiv E_{2m p^{\alpha-1}}\pmod{p^\alpha}.$$
Therefore,
$$|E_{2m p^{\alpha}}|\equiv |E_{2m p^{\alpha-1}}|\pmod{p^\alpha}.$$ By Theorem \ref{T:equiv} it follows that the numbers $b_n$, defined by
$$n b_n=\sum_{d\mid n}\mu(n/d)|E_{2d}|,$$ are integers.
Now we must show that the numbers $b_n$ are non negative. To this end we observe that
$$nb_n\ge |E_{2n}|-\sum_{d=1}^{n/2}|E_{2d}|.$$ Now we apply the well known formula $$1\le
\Bigl(\frac{\pi}{2}\Bigr)^{2d+1}\frac{|E_{2d}|}{(2d)!}=2\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{2d+1}}\le 2.$$ Thus \begin{eqnarray} n b_n&\ge& (2n)!\Bigl(\frac{2}{\pi}\Bigr)^{2n+1}-2\sum_{d=1}^{n/2} (2d)!\Bigl(\frac{2}{\pi}\Bigr)^{2d+1}\nonumber\\ &\ge&(2n)!\Bigl\{ \Bigl(\frac{2}{\pi}\Bigr)^{2n+1}-2\frac{(n)!}{(2n)!} \sum_{d=1}^\infty\Bigl(\frac{2}{\pi}\Bigr)^{2d+1}\Bigr\}.\nonumber \end{eqnarray} We can compute the last sum and we get $0.433\dots$, therefore $$nb_n\ge (2n)!\Bigl\{ \Bigl(\frac{2}{\pi}\Bigr)^{2n+1}-\frac{(n)!}{(2n)!}\Bigr\}.$$ This is positive for $n\ge2$, and we have $b_1=1\ge0$. \end{proof}
The first values of these three sequences in this case are the following: \begin{center}
\begin{tabular}[c]{|c|c|c|c|c|c|c|c|c|c|} \hline $a_n$ & & 1 & 5 & 61 & 1385 & 50521 & 2702765 & 199360981 & \dots \\ \hline $b_n$ & & 1 & 2 & 20 & 345 & 10104 & 450450 & 28480140 & \dots \\ \hline $A_n$ & 1 & 3 & 23 & 371 & 10515 & 461869 & 28969177 & 2454072147 &
\dots \\ \hline \end{tabular} \end{center}
\section{Solution of Gabcke problem}
By Theorem \ref{T:equiv} the assertion of Gabcke, ---that the numbers $\lambda_n$ are integers---, is
equivalent to say that the sequence $a_{k}=2^{4k-3}|E_{2k}|$ is pre-realizable. We shall show that in fact it is realizable.
If $(a_n)$ and $(a'_n)$ are realizable, then the sequence $(a_na'_n)$ is also realizable. In fact given $T\colon X\to X$ and $T'\colon Y\to Y$ such that $a_n=\operatorname{Fix} T^n$ and $a'_n=\operatorname{Fix} T'{}^n$, then it is easy to see that $T\times T'\colon X\times Y\to X\times Y$ satisfies $a_na'_n=\operatorname{Fix}(T\times T')$.
Therefore by Theorem \ref{th:Ereal} what we need is to prove that the sequence $2^{4n-3}$ is realizable. This follows from the following theorem.
\begin{theorem}
Let $a$ and $b\in\mathbf{N}$ such that $b\mid a$ and for every prime number $p\mid a$
$p\mid (a/b)$. Then the sequence $a^n/b$ is realizable. \end{theorem}
\begin{proof}
By the result of Puri and Ward we must show that the sequence $a^n/b$ is
pre-realizable and that the corresponding $b_n$ are non-negative
integers.
First let $p\perp a$ be a prime number, and let $n\perp p$ and
$\alpha$ be natural numbers. We must show that
$$a^{np^\alpha}/b\equiv a^{np^{\alpha-1}}/b\pmod {p^\alpha}.$$
Since $b\perp p$, this is equivalent to
$$a^{np^\alpha}\equiv a^{np^{\alpha-1}}\pmod {p^\alpha}.$$
Now for $\alpha=1$ this is Fermat's little theorem, and for a
general $\alpha$ it follows, by induction, from the fact that
for $\alpha\ge1$ if $a\equiv b\bmod {p^{\alpha}}$, then
$a^p\equiv b^p\bmod{p^{\alpha+1}}$.
Now if $p\mid a$, assume that $p^r\parallel a$ and $p^s\parallel b$.
By hypothesis we have $r\ge s+1$.
We have to show that
$$a^{np^\alpha}/b\equiv a^{np^{\alpha-1}}/b\pmod {p^\alpha},$$
where $p\perp n$ and $\alpha\ge1$. But the two numbers are
divisible by $p^{rnp^{\alpha-1}-s}$. All we have to show is that
$rnp^{\alpha-1}\ge s+ \alpha$. We can assume that $n=1$.
For $\alpha=1$ this is $r\ge
s+1$ that is true by hypothesis. For other values of $\alpha$,
$\alpha\ge2$ and we
have
$$rp^{\alpha-1}=r(p^{\alpha-1}-1)+r\ge (\alpha-1)+(s+1).$$
Now we define the numbers $b_n$ by
$$nb_n=\sum_{d\mid n} \mu(n/d) a^d/b.$$
By the previous reasoning we know the $b_n$ are integers.
If $a=1$, it is easy to see that $b_1=1$ and $b_n=0$ for $n>1$. In
other case $a\ge2$ and
we have
$$nb_n\ge \frac1b \Bigl( a^n-\sum_{d=1}^{n/2} a^d\Bigr).$$
This is easyly seen to be non-negative. \end{proof}
\begin{corollary}\label{th:cor} The sequence $(a_n)_{n=1}^\infty$, where $a_n=2^{4n-3}$ is realizable. \end{corollary}
The three sequences associated to this realizable sequence are \begin{center}
\begin{tabular}[c]{|c|c|c|c|c|c|c|c|c|c|c|} \hline $a_n$ & & 2 & 32 & 512 & 8192 & 131072 & 2097152 & 33554432 & 536870912 & \dots \\ \hline $b_n$ & & 2 & 15 & 170 & 2040 & 26214 & 349435 & 4793490 & 67107840 & \dots \\ \hline $A_n$ & 1 & 2 & 18 & 204 & 2550 & 33660 & 460020 & 6440280 & 91773990 & \dots \\ \hline \end{tabular} \end{center}
Now we are in position to prove Gabcke's conjecture. \begin{theorem}
Let $\lambda_n$ the numbers defined by
\begin{equation}\label{E:deflambda}
\begin{split}
\lambda_0&=1,\\
(n+1)\lambda_n&= \sum_{k=0}^n 2^{4k+1}|E_{2k+2}|\lambda_{n-k},
\qquad (n\ge0),
\end{split}
\end{equation} $\varrho_n$ those defined by
\begin{equation}\label{E:defvarrho}
\begin{split}
\varrho_0&=-1,\\
(n+1)\varrho_n&= -\sum_{k=0}^n 2^{4k+1}|E_{2k+2}|\varrho_{n-k},
\qquad (n\ge0),
\end{split}
\end{equation} and finally let $\mu_n=(\lambda_n+\varrho_n)/2$. All those numbers are integers.
\end{theorem}
\begin{proof}
By Theorem \ref{th:Ereal}, and Corollary \ref{th:cor} the
sequences $(|E_{2n}|)_{n=1}^\infty$ and $(2^{4n-3})_{n=1}^\infty$
are realizable. Since the product of two realizable sequences is
realizable, the sequence
$(2^{4n-3}|E_{2n}|)_{n=1}^\infty$ is realizable. Therefore it
satisfies condition (a) of Theorem \ref{T:equiv}. So it satisfies
condition (b), but this is precisely that the numbers $\lambda_n$
are integers.
Now condition (c) of the same Theorem gives us that with
$a_n=2^{4n-3}|E_{2n}|$ we have for every prime number $p$ and
natural numbers $n\perp p$ and $\alpha$ that
$$a_{np^\alpha}\equiv a_{n p^{\alpha-1}}\pmod {p^\alpha}.$$
Thus the same congruences are satisfied by the numbers $a'_n=-a_n$.
Once again Theorem \ref{th:Ereal} says that the numbers $a'_n$
satisfies condition (b). This is the same as saying that the
numbers $A'_n$ defined by
\begin{minipage}{1cm}\begin{eqnarray}\end{eqnarray}\end{minipage}
\begin{minipage}{10cm}
\begin{eqnarray*}
A'_0&=&1\\
(n+1)A'_n&=& -\sum_{k=0}^n 2^{4k+1}|E_{2k+2}|A'_{n-k}
\qquad (n\ge0)
\end{eqnarray*}\end{minipage}
\noindent are integers. But it is easyly seen that
$\varrho_n=-A'_n$.
The affirmation about the numbers $\mu_n$ follows from the fact that $\lambda_n\equiv\varrho_n\pmod{2}$. That we prove in Theorem \ref{T:oh!}.
\end{proof}
The following theorem is well known. I give a proof for
completeness.
\begin{theorem} Let $s(n)$ be the sum of the digits of the binary representation of $n$, then $$s(n)=n-\sum_{j=1}^\infty \Bigl\lfloor\frac{n}{2^j}\Bigr\rfloor.$$ \end{theorem}
\begin{proof} Let the binary representation of $n$ be of type $\cdots 0\overbrace{11\cdots1}^{\text{$k$ times}}$, with $k\ge0$, then $n+1=\cdots 1\overbrace{00\cdots0}^{\text{$k$ times}}$. Therefore $$s(n)-k=s(n+1)-1.$$ Also $k=\nu_2(n+1)$ the exponent of $2$ in the prime factorization of $n+1$.
Thus we have proved that for every integer $n\ge0$ \begin{equation}\label{E:referencia} s(n+1)+\nu_2(n+1)=s(n)+1. \end{equation} We add this equalities for $n=0$, $1$, \dots, $n-1$ to get $$s(n)+\sum_{k=1}^n \nu_2(k)=n.$$
It is easily checked that $$\sum_{k=1}^n \nu_2(k)=\sum_{j=1}^\infty \Bigl\lfloor\frac{n}{2^j}\Bigr\rfloor.$$ \end{proof}
\begin{theorem} \label{T:oh!} The numbers $\lambda_n$ and $\varrho_n$ defined by Equations (\ref{E:deflambda}) and (\ref{E:defvarrho}) satisfy $$\nu_2(\lambda_n)=\nu_2(\varrho_n)=s(n).$$ \end{theorem}
\begin{proof} First consider the sequence $\lambda_n$. Clearly the theorem is true for the first $\lambda_n$ which are $$\lambda_0=1,\quad \lambda_1=2, \quad \lambda_3=82,\quad \lambda_4= 10572.$$ Since Euler numbers $E_{2k}$ are odd, from the definition of $\lambda_n$ it follows that \begin{equation}\label{E:induccion} \nu_2(n+1)+\nu_2(\lambda_{n+1})=\nu_2\Bigl(
\sum_{k=0}^n 2^{4k+1}|E_{2k+2}|\lambda_{n-k}\Bigr). \end{equation} By induction the terms of this sum are exactly divided by the powers of $2$ of exponents $$1+s(n),\quad 5+ s(n-1),\quad 9+s(n-2),\quad \dots \quad (4n+1)+s(0)$$ This is a strictly increasing sequence, since $$s(n)-s(n-1)=1-\nu_2(n)<4.$$
Hence from (\ref{E:induccion}) we get $$\nu_2(n+1)+\nu_2(\lambda_{n+1})=1+s(n).$$ By (\ref{E:referencia}) $$\nu_2(\lambda_{n+1})= s(n)-\nu_2(n+1)+1=s(n+1).$$
The same proof applies to the sequence $(\varrho_n)$. \end{proof}
\section{Examples}
We give here some examples of numbers satisfying our Theorem \ref{T:equiv}. First consider the case of the numbers of Gabcke $A_n=\lambda_n$. The first terms of the associated sequences are given by the following table. \begin{center}
\begin{tabular}[c]{|c|c|c|c|c|c|c|c|c|} \hline $a_n$ & & 2 & 160 & 31232 & 11345920 & 947622146676 & 957663025230936 & \dots \\ \hline $b_n$ & & 2 & 79 & 10410 & 2836440 & 1324377702 & 944684832315 & \dots \\ \hline $\lambda_n$ & 1 & 2 & 82 & 10572 & 2860662 & 1330910844 & 947622146676 & \dots \\ \hline \end{tabular} \end{center}
We can give arbitrarily a sequence of integers $(b_n)$ and obtain sequences $(a_n)$ and $(A_n)$ that automatically satisfy our theorems. We give two simple examples.
With $b_n=1$ for every $n$, we get $a_n =\sigma(n)$. \begin{center}
\begin{tabular}[c]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline $a_n$ & & 1 & 3 & 4 & 7 & 6 & 12 & 8 & 15 & 13 & 18 & 12 & 28 & \dots \\ \hline $b_n$ & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \dots \\ \hline $A_n$ & 1 & 1 & 2 & 3 & 5 & 7 & 11 & 15 & 22 & 30 & 42 & 56 & 77 & \dots \\ \hline \end{tabular} \end{center}
With $b_n=-24$, the numbers $A_n$ are given by Ramanujan's $\tau$ function
$A_n=\tau(n+1)$. \begin{center}
\begin{tabular}[c]{|c|c|c|c|c|c|c|c|c|c|c|} \hline $a_n$ & & -24 & -72 & -96 & -168 & -144 & -288 & -192 & -360 & \dots \\ \hline $b_n$ & & -24 & -24 & -24 & -24 & -24 & -24 & -24 & -24 & \dots \\ \hline $A_n$ & 1 & -24 & 252 & -1472 & 4830 & -6048 & -16744 & 84480 & -113643 & \dots \\ \hline \end{tabular} \end{center}
Finally let $(T_n)$ be the tangent numbers with the notation of \cite{K}. We have $T_{2n}=0$ $$T_{2n+1}=(-1)^{n} \frac {4^{n+1}(4^{n+1}-1)B_{2n+2}}{2n+2}.$$ It can be proved that $a_n=(-1)^nT_{2n+1}$ satisfies Kummer congruences. It follows that the sequence $(T_{2n+1})$ is realizable, in this case the three sequences are \begin{center}
\begin{tabular}[c]{|c|c|c|c|c|c|c|c|c|c|} \hline $a_n$ & & 2 & 16 & 272 & 7936 & 353792 & 22368256 & 1903757312 & \dots \\ \hline $b_n$ & & 2 & 7 & 90 & 1980 & 70758 & 3727995 & 271965330 & \dots \\ \hline $A_n$ & 1 & 2 & 10 & 108 & 2214 & 75708 & 3895236 & 280356120 & \dots \\ \hline \end{tabular} \end{center}
\end{document} | arXiv |
Result and discussion
RANdom SAmple Consensus (RANSAC) algorithm for material-informatics: application to photovoltaic solar cells
Omer Kaspi1, 2,
Abraham Yosipof3Email author and
Hanoch Senderowitz2Email authorView ORCID ID profile
Published: 6 June 2017
An important aspect of chemoinformatics and material-informatics is the usage of machine learning algorithms to build Quantitative Structure Activity Relationship (QSAR) models. The RANdom SAmple Consensus (RANSAC) algorithm is a predictive modeling tool widely used in the image processing field for cleaning datasets from noise. RANSAC could be used as a "one stop shop" algorithm for developing and validating QSAR models, performing outlier removal, descriptors selection, model development and predictions for test set samples using applicability domain. For "future" predictions (i.e., for samples not included in the original test set) RANSAC provides a statistical estimate for the probability of obtaining reliable predictions, i.e., predictions within a pre-defined number of standard deviations from the true values. In this work we describe the first application of RNASAC in material informatics, focusing on the analysis of solar cells. We demonstrate that for three datasets representing different metal oxide (MO) based solar cell libraries RANSAC-derived models select descriptors previously shown to correlate with key photovoltaic properties and lead to good predictive statistics for these properties. These models were subsequently used to predict the properties of virtual solar cells libraries highlighting interesting dependencies of PV properties on MO compositions.
RANSAC
Material-informatics
QSAR
Material informatics is a rapidly developing field engaged with the application of informatics principles to materials science in order to assist in the discovery and development of new materials [1–5]. Developments in material informatics take advantage of the vast empirical and computational information on structures and properties of materials available in multiple databases such as MatWeb (http://www.matweb.com/) which includes properties for over 115,000 materials and MatDat (https://www.matdat.com/) which includes over 1000 datasets of materials, to name but a few. [6–10] Turning this large volume of information into knowledge could be performed in multiple ways using multiple data mining procedures. As an example, AFLOW [6] (http://aflowlib.org/) is a database of density functional theory (DFT) calculations performed on more than 1.5 million materials with known crystal structures. Isayev et al. [5]. used this database to introduce the term "material cartography" for representing a library of materials as a network. The resulting network was subsequently mined using various machine learning methods in search for materials with interesting properties.
A pre-requisite to any data mining procedure is a data curation stage [11]. Data curation is important for two main reasons: (1) Publically available data sets may contain multiple errors; (2) even a small number of errors may compromise the quality of QSAR models [11]. For example, Olah et al. [12, 13] have shown an error rate as high as 8% in the WOMBAT database and Young et al. [14] have recorded error rates between 0.1 and 3.4% in a variety of databases. More recently, Isayev et al. [5] have demonstrated several errors in the AFLOW database including duplicate compounds and incorrect extraction of literature data. In general, data curation involves steps like the removal of duplicates, compounds with wrong Lewis structures, compounds for which descriptors could not be calculated, and in case of experimentally measured data the removal of compounds which suffer from errors caused by the measurement process.
Due to the sheer size of material databases, data curation cannot be performed manually but rather requires a computational workflow. Indeed several such workflows have been reported in the literature [11, 15, 16]. However, even a stringent curation workflow cannot clean a database from noise that often accompanies experimental data. The presence of noise might mask the information that the data hold, thereby compromising data interpretation, model generation and decisions making.
In general, noise could be classified as either internal or external. Internal noise is inherent to the measurement process of the data, affects all data points, and is assumed to be distributed normally. In contrast, external noise results from sources exterior to the system due to an error in the measurement itself or from extreme behavior that does not match the overall behavior of the majority of samples. While all samples experience internal noise, some may also experience (greater) external noise and could therefore be regarded as outlier samples. Thus, an outlier is an observation on the dataset, which appears to be inconsistent with the rest of the data [17].
Important aspects of data mining in material informatics are database searching, similarity searches, and the usage of machine learning algorithms for pattern recognition and derivation of predictive models [18, 19]. Multiple terms have been used to describe such models including Quantitative Structure Activity/Property Relationship (QSAR/QSPR) models [20, 21], Quantitative Materials Structure–Property Relationships (QMSPR) models [5], and Quantitative Nanostructure Activity Relationship (QNAR) models [22]. All models attempt to correlate specific activities (or properties) for a set of materials with (calculated or measured) molecular descriptors by means of a mathematical model. Such models should both provide scientific insight into the problem in hand as well as allow for the prediction of the results of future experiments. An important characteristic of QSAR models is therefore their predictive power. However the presence of outliers (i.e., noise) may bias the dataset to the point of compromising the ability of machine learning algorithms to build predictive models. Consequently, a common practice of QSAR modeling is the prior removal of outlying samples prior to model generation [23]. Accordingly, several methods for the removal of outliers were reported in the literature [24–29].
Two more aspects of machine learning algorithms which critically affect performances are the selection of specific descriptors that best correlate with the activity under study from the initial pool of descriptors and the definition (and application) of the model's applicability domain, namely, the region in material space in which the model is expected to give accurate predictions. Multiple descriptors selection (i.e. feature selection) methods have been developed including filter methods, wrapper methods and embedded methods [30]. Similarly several algorithms for the definition of applicability domains have been reported [31].
Most QSAR studies treat the removal of outliers, the selection of descriptors and the definition of applicability domain as separate stages within a QSAR workflow, often using different tools for each task [11, 20, 32, 33]. Thus, there is an interest in presenting a "one stop shop" algorithm for the performance of all tasks. The advantages of such an algorithm are the potential prevention of errors resulting from interfaces between different components as well as easier accessibility, in particular by non-experts. In contrast a "one stop shop" algorithm is by its nature non-modular, offering minimal flexibility in the modeling process.
With this in mind we present in this work the adaptation, implementation, and the first application of the RANdom SAmple Consensus (RANSAC) method [34] to the field of material-informatics by deriving predictive models for key photovoltaic properties of solar cells. RANSAC is a modeling tool widely used in the Image Processing field [34–36] primarily for image noise filtration. The algorithm produces and validates a linear QSAR model based on the Minimum Least Square (LMS) method by (1) filtering noisy samples (i.e., outliers), (2) selecting the best features (i.e., descriptors), (3) deriving a QSAR model from training set samples and (4) predicting the activity of test set samples while invoking the concept of applicability domain, all in a single process without the need of complementary processes. For prediction of samples not in the original test set (i.e., samples for which no activity data are available), RANSAC provides a statistical estimate for the probability of obtaining reliable predictions, i.e., predictions within a pre-defined number of standard deviations from the true values. These characteristics make RANSAC an appealing addition to the arsenal of tools available for the derivation of predictive QSAR models.
As a first application, we chose to test the performances of RANSAC in the important field of solar cells which emerge as one of the main resources for clean energy. Briefly, a typical solar cell (photovoltaic device) operates by: (1) Generation of charge carriers (electrons and holes) following the absorption of photons; (2) Separation of the photo-generated charge carriers via charge selective contact(s); (3) Collection of the photo-generated charge carriers at an external circuit resulting in electricity.
In particular we focus our attention on solar cells entirely composed of metal oxides (MOs). Such cells possess many favorable properties including natural abundance of the constituting materials, ease of fabrication and long time stability. However, such cells do not demonstrate sufficient efficiency in converting sunlight to electricity thereby requiring the development of new cells potentially composed of new MOs or MO combinations [37]. Such developments could be facilitated by the development of QSAR models to predict key solar cells properties such as current, voltage, and quantum efficiency. Yet despite their importance only few QSAR studies were reported on solar cells [38–40] and even fewer on MO-based solar cells [41].
MO-based solar cells are often produced using combinatorial techniques resulting in solar cell libraries [37, 42]. Following fabrication, the libraries are subjected to medium throughput measurements to characterize their composition/structure as well as their photovoltaic (PV) properties. Due to the technical challenges involved in both fabrication and characterization, the resulting libraries often contain noisy data [42] making them ideal candidates for the RANSAC algorithm. The main objective of the present study is therefore to establish the usefulness of the RANSAC algorithm in cleaning and analyzing datasets of solar cells libraries and predicting their PV properties. For this purpose, we used three recently published datasets experimentally-derived from two different solar cells libraries. The first library is a \(TiO_{2} |Cu_{2} O\) library reported by Pavan et al. [43]. The library consists of two datasets, one with Ag back contacts and the other with Ag|Cu back contacts. The second library is a \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library reported by Majhi et al. [15]. The two libraries comprised of \(TiO_{2} |Cu_{2} O\) based solar cells were previously modeled using k nearest neighbors (kNN) and genetic algorithm allowing for a facile comparison between the performances of the different algorithms. The third library (\(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3} )\) was previously analyzed using visualization methods [36]. We demonstrate that the RANSAC algorithm filters the sample space from noisy data (i.e., outliers), automatically selects descriptors previous shown to correlate with key PV properties and generates models with good predictive statistics for these properties.
RANSAC overview
RANdom SAmple Consensus (RANSAC) [34] is a method for deriving a model based on linear regression, performed on input data that may include noisy samples (both internal and external noise). The basic assumption of the algorithm is that the measured activity (\(Y_{measured} (\bar{x}))\) depends on a set of noise-free variables (e.g., descriptors; \(\bar{x}\)) and on noise added to them; Eq. (1).
$$Y_{measured} \left( {\bar{x}} \right) = Y_{{noise{-}free}} \left( {\bar{x}} \right) + N$$
where \(Y_{{noise{-}free}} (\bar{x})\) is the expected activity in a noise-free environment and \(N\) is a random internal noise. RANSAC assumes that the internal noise obeys the homoscedastic assumption, namely, that it has a constant distribution across all activity values. Using this assumption, boundaries could be set to form a "strip" that classifies the samples as either affected by internal noise only (model-compatible samples residing within the "strip") or such that are affected both by internal and by external noise (model-incompatible samples residing outside the "strip"). Importantly, these boundaries should be a priori provided to the algorithm, based on the system's characteristics and are expressed as the distance, in number of standard deviations (n), from the model (see below and Fig. 1).
A possible RANSAC output where the desired model is of the first power (i.e., straight line). The algorithm assumes that due to internal noise, samples will not be exactly on the model but within a normal distribution around it. Conceptually, this variance forms a "strip" where all samples that lay within its boundaries are influenced by internal noise only. Samples within a "strip" are defined as model-compatible. Samples outside the "strip" are defined as model-incompatible
Mathematically, the following definition applies [Eq. (2)]:
$$\begin{array}{*{20}l} {if\,\frac{{Y_{measured} \left( {\bar{x}} \right) - Y_{calculated} \left( {\bar{x}} \right)}}{\sigma } > n} \hfill & {Model{-}Incompatible\,Sample} \hfill \\ {else} \hfill & {Model{-}Compatible\,Sample} \hfill \\ \end{array}$$
where \(Y_{calculated} \left( {\bar{x}} \right)\) is the calculated activity (see below), σ is the standard deviation of the sample and n is the width of the "strip" (in units of σ). Operationally, RANSAC incorporates the following stages (Fig. 2): (1) Model construction: randomly select a subsample from the dataset and fit to it a linear curve using linear regressions Least Mean Squares (LMS). (2) Model scoring: classify all samples as either model-compatible or model incompatible (based on the a priori provided "strip" width). (3) Iterative phase: repeat steps (1) and (2) to build multiple models each based on other randomly selected subsamples. For each model count the number of model-compatible and model-incompatible samples (4) Model selection: select the model with the largest number of model-compatible samples, calculate LMS, discard model-incompatible samples (i.e., outliers) and calculate LMS again. This model will be used for subsequent predictions.
Description of the RANSAC algorithm as used for model construction
Model construction
For RANSAC to build a model, it must first draw a subsample from all the samples used for model training (i.e., training set) and use it to construct a regression line. For a single observation the model takes the form of Eq. (3):
$${\text{y}}_{\text{i}} = w_{0} x_{i0} + {\text{w}}_{1} x_{i1} + {\text{w}}_{2} x_{i1}^{2} + \cdots + {\text{w}}_{\text{p}} x_{i1}^{p} + {\text{w}}_{{{\text{p}} + 1}} x_{i2} + {\text{w}}_{{{\text{p}} + 2}} x_{i2}^{2} + \cdots + w_{p *d} x_{id}^{p}$$
Where y is the dependent variable, \(\bar{x}\) is the vector of the independent variables (i.e., descriptors), i denotes sample i, p is the power of the best fit curve, d is the dimensionality of the model (i.e., number of descriptors) and \(\bar{W}\) is a vector holding the weights calculated using the linear regression. Note that \(\bar{W}\) may have zero values for one or more input descriptors meaning, that these descriptors were not selected by the model. For multiple samples, the matrix form is used [Eq. (4)]:
$$\bar{Y}_{calculated} = X\bar{W}$$
$$X = \left( {\begin{array}{*{20}c} {x_{10} } & \cdots & {x_{1d}^{p} } \\ \vdots & \ddots & \vdots \\ {x_{i0} } & \cdots & {x_{id}^{p} } \\ \end{array} } \right),\quad \bar{W} = \left[ {\begin{array}{*{20}c} {w_{0} } \\ \vdots \\ {w_{d *p} } \\ \end{array} } \right],\quad {\bar{\text{Y}}}_{\text{calculated}} = \left[ {\begin{array}{*{20}c} {y_{0} } \\ \vdots \\ {y_{d} } \\ \end{array} } \right]$$
$$\bar{W} = \left( {X^{T} X} \right)^{ - 1} X^{T} \bar{Y}$$
The size of the subsample drawn by RANSAC should match the power (p) of the desired equation [Eq. (3)]. For example, for an equation with p = 3, a subsample of size 4 should be drawn.
Model scoring
The basic assumption underlying the RANSAC algorithm is that the set of samples (expressed as data points) could be approximated by a model of a certain dimensionality (d), where each dimension is represented by a descriptor raised up to a maximum power (p) allowed for the model. If this assumption holds true, then one would expect to have most dataset points residing within a "strip" of a given width around the best fit curve calculated for a subsample (i.e., model compatible samples). The "strip" could be used for several purposes: (1) Scoring models by counting the number of dataset points residing within their boundaries (the larger the number, the better the model). Models are scored based on the entire training set and not only on the drawn sub-sample used for their construction. (2) Identifying outliers by observing training set samples residing outside the "strip's" boundaries. (3) Defining the "strip" as the model's applicability domain for test set predictions. RANSAC scores a model based on the number of model-compatible samples from within the training set.
Iterative phase and select highest scoring model
RANSAC is an iterative algorithm that requires many repetitions of the model construction and scoring phases (i.e., iterations) in order to obtain the best model. Furthermore, the number of the required iterations depends on the size of the dataset with larger datasets requiring more iterations. At each iteration, the algorithm counts the number of model-compatible samples and outputs the weights vector (\(\bar{W}\)) that corresponds to the highest ranked model (i.e., model with the highest score). For this model the LMS error is calculated both before and after the removal of outliers (i.e., model-incompatible samples). It is important to note that the size of the "strip" (which ultimately determines the number of model compatible samples) may vary between libraries and should be specifically chosen for each library.
The best model emerging from the iterative phase is used for predictions. For test set samples, their known activities allow to classify them as either within or outside the model's applicability domain (i.e., either within or outside the "strip"). The percentage of within-"strip" samples provides an estimate for the percentage of "correct" (i.e., within the predefined number of standard deviations (n) from the true value) predictions for "future" samples, that is samples with unknown activities. RANSAC does not feature an inherent applicability domain for individual samples although a descriptors based applicability domain approach could of course be used [31].
For all RANSAC's applications described in this work the following parameters were used: The number of iterations was set to 105 to derive a polynomial equation of the 5th power. The size of the "strip" (i.e. the models' boundaries) was set to be ±1 standard deviation around \(\bar{Y}_{measured}\) derived from the training set. The algorithm was coded in MATLAB version R2014a.
Metal-oxide solar cells library
The basic assembly of MO solar cell library includes (see Fig. 3): (1) a transparent conducting oxide (TCO) coated on a glass, typically in the form of fluorine doped tin oxide (FTO); (2) a window layer, which is a wide band-gap n-type semiconductor (typically TiO2); (3) a light absorbing layer (absorber); (4) Metal back contact; (5) Metal frame (front contact) soldered directly onto the FTO.
A schematic representation of the PV solar cells libraries. a \(TiO_{2} |Cu_{2} O\) library (with Ag and Ag|Cu back contacts), b \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library
\(TiO_{2} |Cu_{2} O\) library (Fig. 3a)
An experimental library of solar cells was obtained from Pavan et al. [43]. This library was generated on precut glass coated with fluorine doped tin oxide (FTO) substrates onto which a TiO2 window layer with a linear gradient was deposited, followed by an absorber layer of Cu2O. Inserting two different grids of 13 × 13 = 169 back-contacts, namely, silver only (Ag) and silver and copper (Ag|Cu) deposited one after the other, lead to two sub-libraries (datasets) each consisting of 169 cells. In this work we omitted the non-photovoltaic cells leaving a total of 162 and 166 cells for the Ag and Ag|Cu back contact data base respectively.
\(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library (Fig. 3b)
This library was constructed in a manner roughly similar to the \(TiO_{2} |Cu_{2} O\) libraries, with the same window layer (TiO2) but different target metal oxide for the absorber layer (Co3O4) and also included a third recombination layer (MoO3). On top of the MoO3 layer a 13 × 13 grid of Au back contacts was placed, thus forming a library of 169 cells. The library was characterized by the varying thicknesses of the TiO2, Co3O4, MoO3 layers. For this library 19 cells were removed due to lack of photovoltaic activities (thus 150 cells remained).
Library characterization
Each solar cell was characterized by its material descriptors (independent variables) and experimentally measured photovoltaic activities (dependent variables). Material descriptors included the thickness of the window layer (\(d_{{TiO_{2} }}\)), the thickness of the absorber layers (\(d_{{Cu_{2} o}} \,{\text{and}}\,d_{{Co_{3} O_{4} }}\)), the thickness of the recombination layer (\(d_{{MoO_{3} }}\)), the thickness ratio between the absorber layer and the sum of the absorber and window layers (ratio), the thickness ratio between the absorber layer and the sum of the absorber and the recombination layers (ratio_AR), and the band gap of absorber layer (BGP). The band gap is the energy difference (in electron volts) between the top of the valence band and the bottom of the conduction band. Overall, for the \(TiO_{2} |Cu_{2} O\) libraries four descriptors were consider namely: \(d_{{TiO_{2} }} , d_{{Cu_{2} o}} ,ratio,\) and BGP, and for the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library five descriptors were consider namely: \(d_{{TiO_{2} }} , d_{{Co_{3} O_{4} }} , d_{{MoO_{3} }} ,\) \(ratio,\) and \(ratio\_AR\). Tables 1 and 2 present the range values for each of the descriptors.
Descriptor ranges for the \(TiO_{2} |Cu_{2} O\) library (with Ag and Ag|Cu back contacts)
\(TiO_{2} |Cu_{2} O\) (both libraries)
\(d_{{TiO_{2} }}\) (nm)
70.0–311.5
\(d_{{Cu_{2} o}}\) (nm)
BGP (eV)
Descriptor ranges for the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library
\(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\)
\(d_{{Co_{3} O_{4} }}\) (nm)
\(d_{{MoO_{3} }}\) (nm)
Ratio_AR
In this work we focused on three experimentally measured PV activities (dependent variables, end points): (1) the short circuit photocurrent density (J SC ) which is the current density through the solar cell when the voltage across the cell is zero. (2) The open circuit voltage (Voc) which is the maximum voltage available from a solar cell. This voltage occurs at an open circuit. (3) The internal quantum efficiency (IQE) which reflects the charge separation and collection efficiencies of a device and is calculated by Eq. (6) where \(J_{max}\) is the maximum theoretical calculated photocurrent. The distributions of the three PV activities are represented by boxplots in Fig. 4 and their ranges are given in Table 3.
Boxplots of the three PV activities (J SC , Voc, and IQE). a–c The three PV activities distribution for the \(TiO_{2} |Cu_{2} O\) library (with Ag and Ag|Cu back contacts). d–f The three PV activities distribution for the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library. The boxplots show the median values (solid horizontal line), 50th percentile values (box outline), the lower and upper quartile (whiskers, vertical lines), and outlier values (open circles)
Activity ranges for the three libraries
\(TiO_{2} |Cu_{2} O\) (Ag)
\(TiO_{2} |Cu_{2} O\) (Ag|Cu)
J SC \(\left( {\upmu{\text{A/cm}}^{2} } \right)\)
V OC \(\left( {\text{V}} \right)\)
IQE (%)
$$IQE = \frac{Jsc}{{J_{max} }}$$
Model fitting and statistical parameters
The datasets were divided into training and validation (test) sets using a recently published representativeness algorithm [44]. Subsets selected by this algorithm were previously employed as external validation sets in QSAR modeling [24, 25, 41, 44]. Each dataset was divided into a training set composed of 80% of the original dataset (130, 134 and 120 cells for the \(TiO_{2} |Cu_{2} O\) with Ag back contacts, \(TiO_{2} |Cu_{2} O\) with Ag|Cu back contacts and \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) datasets, respectively) and a test set containing the remaining cells (32 samples for the \(TiO_{2} |Cu_{2} O\) with Ag and Ag|Cu back contact and 30 samples for the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) dataset). The \(TiO_{2} |Cu_{2} O\) libraries with Ag and Ag|Cu Back Contacts were previously modeled by Yosipof et al. [41]. For the purpose of comparison, the training and test sets described above, were made identical to those described by Yosipof et al. [41].
To evaluate the RANSAC model performances on the training set we used \(Q_{train}^{2}\) as expressed by Eq. (7). The RANSAC algorithm excludes samples from the training set-based error calculation if residing outside the model's boundaries (e.g., "strip"). Thus the model's error is derived without these samples. This is analogous to outlier removal. Below we therefore report two \(Q_{train}^{2}\) estimates, the first based on all samples and the second based on samples surviving RANSAC's inherent outlier removal.
The performances of the RANSAC algorithm on the test set (\(Q_{ext}^{2}\)) were calculated in a similar manner [Eq. (8)]. Similarly to outlier removal, the "strip" calculated by the RANSAC algorithm was used to evaluate the applicability domain (AD) of the resulting model. Accordingly, two estimates of \(Q_{ext}^{2}\) were calculated one pertaining to the entire test set and one, for that portion of the test set which resided within the model's applicability domain.
$$Q_{train}^{2} = 1 - \frac{{\mathop \sum \nolimits \left( {Y_{measured,train} - Y_{predicted,train} } \right)^{2} }}{{\mathop \sum \nolimits \left( {Y_{measured,train} - \bar{Y}_{measured,train} } \right)^{2} }}$$
$$Q_{ext}^{2} = 1 - \frac{{\mathop \sum \nolimits \left( {Y_{measured,test} - Y_{predicted, test} } \right)^{2} }}{{\mathop \sum \nolimits \left( {Y_{measured,test} - \bar{Y}_{measured,test} } \right)^{2} }}$$
where \(Y_{measured}\) is the experimental result, \(Y_{predicted}\) is the predicted value and \(\bar{Y}_{measured}\) is the mean of the experimental results over training set samples.
In addition, we used the R2 (squared correlation coefficient) between the predicted (\(Y_{predicted}\)) and the experimental (\(Y_{measured }\)) data for both training and test set.
Finally, to assess model significance and to rule-out chance correlation, Y-randomization procedure was applied to all models.
Performances of RANSAC-derived models
The RANSC algorithm was applied to the three datasets described above. For each dataset, three models were derived to describe their photovoltaic (PV) properties (J SC , V OC and IQE). Table 4 presents the number of training set and test set samples found to reside within the model's "strip" (i.e., model-compatible samples). Model-incompatible samples in the training and test sets are referred to as outliers and outside of the model's AD, respectively. As can clearly be seen, the vast majority (≥85%) of the samples are included within the "strip" for both the training and test sets. This suggests that (1) predictive models could likely be derived for this dataset and (2) the model described by the "strip" forming curve approximates most of the training set and test set samples to within one standard deviation (the pre-defined "strip" width; see Methods section) from their experimental values. One could therefore propose that the majority of future samples will be similarly predicted. However, in two cases the number of model compatible cells was below the 85% threshold (the V OC models for the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library with 84 and 80% of model compatible cells for the training and test sets, respectively), indicating higher variance for this property in this dataset in comparison with the other properties/datasets. In accord with this observation, the performances of the V OC model from the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library were exceptionally poor (Table 5). This model was therefore excluded from the analysis reported below.
Number of model-compatible samples for the three datasets based on the RANSAC models
IQE
# Model—compatible training samples
125/130 (96%)
# Model—compatible test samples
32/32 (100%)
120/120 (100%)
RANSC model performance for the three datasets
\(Q_{train}^{2} (R^{2} )\)
\(Q_{train}^{2}\) \((R^{2} )\) (no outliers)
\(Q_{ext}^{2} (R^{2} )\)
\(Q_{ext}^{2}\) (AD) \((R^{2} )\)
J SC
V OC
−0.06 (0.03)
A comparison of model coverage, based on test set samples, between RANSAC and kNN models
RANSAC coverage (%)
kNN coverage* (%)
*The data for kNN were taken from Table 5 in Ref. [41]
Overall, the RANSAC algorithm led to models with good statistical parameters (Table 5) for training set samples for J SC (\(Q_{train}^{2}\) between 0.74 and 0.77), Voc (\(Q_{train}^{2}\) between 0.57 and 0.62 excluding the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library; see above) and IQE (\(Q_{train}^{2}\) between 0.71 and 0.85). Upon the removal of outliers, the statistical parameters for all models improved with the largest improvement being obtained for V OC (\(Q_{train}^{2}\) between 0.78–0.82, 0.65–0.73 (excluding the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\)) and 0.78–0.85 for J SC , V OC and IQE, respectively).
The performances of the RANSAC models on the test set samples followed a trend similar to that observed for the training set. Thus, for all test sets, \(Q_{ext}^{2}\) was found to be between 0.69–0.82, 0.62–0.80, and 0.69–0.79 for J SC , V OC (excluding the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3} library\)) and IQE, respectively. Similar results were obtained for R2 values between the predicted and the actual activities (Table 5). As expected for datasets devoid of significant activity cliffs, when considering only samples within the models' applicability domains, these numbers improved to 0.82–0.87 and 0.79–0.83 for J SC , and IQE, respectively. For V OC of the \(TiO_{2} |Cu_{2} O\) (Ag) library, no test set samples were filtered by the applicability domain leading to no change in model performances (\(Q_{ext}^{2}\) = 0.80). However for this property a significant increase in the \(TiO_{2} |Cu_{2} O\) (Ag|Cu) library upon the removal of only two samples was observed (\(Q_{ext}^{2}\) = 0.62 and 0.73 without and with the model's AD, respectively).
Figures 5 and 6 present predicted versus experimentally measured values for all three PV properties considered in this work across the three datasets following outlier removal for training set samples and considering only samples within the models applicability domains for the test set.
Predicted versus experimental PV properties for train set samples following the removal of outliers. a–c J SC , V OC and IQE for the TiO2|Cu2O library with Ag back contacts, d–f J SC , V OC and IQE for the TiO2|Cu2O library with Ag|Cu back contacts, g–i J SC , V OC and IQE for the TiO2|Co3O4|MoO3 library
Predicted versus experimental PV properties for test set samples residing with the models applicability domains. a–c J SC , V OC and IQE for the TiO2|Cu2O library with Ag back contacts, d–f J SC , V OC and IQE for the TiO2|Cu2O library with Ag|Cu back contacts, g–i J SC , V OC and IQE for the TiO2|Co3O4|MoO3 library
Finally, Y-randomization procedure was applied to all models and no statistically significant models were derived.
Two of the above described datasets [\(TiO_{2} |Cu_{2} O\) (Ag) and \(TiO_{2} |Cu_{2} O\) (Ag|Cu)] were previously modeled by Yosipof et al. [41] using kNN and a Genetic Programming (GP) approach, thereby allowing for a direct comparison between the performances of the resulting models (the results of kNN and GP models from Yosipof et al. [41] are presented in Table 7). GP produced models with \(Q_{ext}^{2}\) values between 0.74–0.76, 0.50–0.78 and 0.72 for J SC , V OC and IQE respectively. The corresponding numbers obtained by RANSAC are \(Q_{ext}^{2}\) = 0.69–0.76, 0.62–0.80, and 0.69–0.78 for J SC , V OC and IQE, respectively, with no AD and \(Q_{ext}^{2}\) = 0.84–0.87, 0.73–0.80 and 0.82–0.83 for J SC , V OC and IQE, respectively, with AD. These results suggest that the performances of the RANSAC models are similar to those of the GP with no consideration of the AD and provide significant improvement upon the application of AD. Of note, there is no inherent definition of AD in the GP method. For kNN, \(Q_{ext}^{2}\) was reported to be 0.89–0.92, 0.56–0.89, and 0.87–0.91 for J SC , V OC and IQE, respectively, with no AD and \(Q_{ext}^{2}\) 0.88–0.92, 0.55–0.89 and 0.87–0.89 for J SC , V OC and IQE, respectively, with AD. Thus, kNN provides models with higher prediction statistics than RANSAC in particular when the AD is not considered. However, the performances of RANSAC approach those of kNN upon the introduction of the AD. Moreover, the test set coverage provided by RANSAC is generally higher than that provided by kNN (Table 6). Finally, in contrast with kNN, RANSAC provides a model in the form of a QSAR equation which enhances model interpretability.
kNN and GP models performance retrieved from Yosipof et al. [41]
\(Q_{train}^{2}\)
\(Q_{ext}^{2}\) \((R^{2} )\) (AD)
RANSAC as a feature selection tool
Table 8 presents the model equations produced by RANSAC for the different PV properties of the three datasets.
RANSAC derived models for different PV properties
PV Property
\(J_{SC} \,\left( {\upmu{\text{A/cm}}^{ 2} } \right) = - 7.9 \times 10^{ - 5} d_{{TiO_{2} }}^{3} + 6.32 \times 10^{ - 7} d_{{TiO_{2} }}^{4} - 1.3 \times 10^{ - 9} d_{{TiO_{2} }}^{5} + 7.27 \times 10^{ - 6} d_{{Cu_{2} o}}^{3} - 1.8 \times 10^{ - 8} d_{{Cu_{2} o}}^{4} + 1.47 \times 10^{ - 11} d_{{Cu_{2} o}}^{5}\)
\(V_{OC} \,\left( {\text{V}} \right) = 4.44 \times 10^{ - 8} d_{{TiO_{2} }}^{3} - 2 \times 10^{ - 10} d_{{TiO_{2} }}^{4} + 2.12 \times 10^{ - 13} d_{{TiO_{2} }}^{5} + 1.76 \times 10^{ - 8} d_{{Cu_{2} o}}^{3} - 5.7 \times 10^{ - 11} d_{{Cu_{2} o}}^{4} + 4.87 \times 10^{ - 14} d_{{Cu_{2} o}}^{5}\)
\({\text{IQE}}\,\left( {\% } \right) = - 9.6 \times 10^{ - 8} d_{{TiO_{2} }}^{3} + 1.43 \times 10^{ - 9} d_{{TiO_{2} }}^{4} - 3.7 \times 10^{ - 12} d_{{TiO_{2} }}^{5} + 3.39 \times 10^{ - 8} d_{{Cu_{2} o}}^{3} - 9.2 \times 10^{ - 11} d_{{Cu_{2} o}}^{4} + 8.52 \times 10^{ - 14} d_{{Cu_{2} o}}^{5}\)
\(J_{SC} \,\left( {\upmu{\text{A/cm}}^{ 2} } \right) = 1.69 \times 10^{ - 5} d_{{TiO_{2} }}^{3} - 8.5 \times 10^{ - 8} d_{{TiO_{2} }}^{4} + 1.13 \times 10^{ - 10} d_{{TiO_{2} }}^{5} + 3.15 \times 10^{ - 6} d_{{Cu_{2} o}}^{3} - 1.9 \times 10^{ - 9} d_{{Cu_{2} o}}^{4} - 1.7 \times 10^{ - 12} d_{{Cu_{2} o}}^{5}\)
\(V_{OC} \,\left( {\text{V}} \right) = 9.31 \times 10^{ - 9} d_{{TiO_{2} }}^{3} + 4.01 \times 10^{ - 11} d_{{TiO_{2} }}^{4} - 2 \times 10^{ - 13} d_{{TiO_{2} }}^{5} + 2.04 \times 10^{ - 8} d_{{Cu_{2} o}}^{3} - 6.9 \times 10^{ - 11} d_{{Cu_{2} o}}^{4} + 6.18 \times 10^{ - 14} d_{{Cu_{2} o}}^{5}\)
\({\text{IQE}}\,\left( {\% } \right) = - 9.47 \times 10^{ - 7} d_{{TiO_{2} }}^{3} + 7.2 \times 10^{ - 9} d_{{TiO_{2} }}^{4} - 1.4 \times 10^{ - 11} d_{{TiO_{2} }}^{5} + 1.2 \times 10^{ - 7} d_{{Cu_{2} o}}^{3} - 3.6 \times 10^{ - 10} d_{{Cu_{2} o}}^{4} + 3.08 \times 10^{ - 13} d_{{Cu_{2} o}}^{5}\)
\(J_{SC} \,\left( {\upmu{\text{A/cm}}^{ 2} } \right) = - 1.5 \times 10^{ - 8} d_{{Co_{3} O_{4} }}^{4} + 6.09 \times 10^{ - 11} d_{{Co_{3} O_{4} }}^{5} + 3.74 \times 10^{ - 6} d_{{MoO_{3} }}^{4} - 4.5 \times 10^{ - 8} d_{{MoO_{3} }}^{5} + 5.17 \times 10^{ - 9} d_{{TiO_{2} }}^{4} - 1.3 \times 10^{ - 11} d_{{TiO_{2} }}^{5}\)
\(V_{OC} \,\left( {\text{V}} \right) = 2.08 \times 10^{ - 10} d_{{Co_{3} O_{4} }}^{4} - 6.8 \times 10^{ - 13} d_{{Co_{3} O_{4} }}^{5} + 4.42 \times 10^{ - 7} d_{{MoO_{3} }}^{4} - 6.7 \times 10^{ - 9} d_{{MoO_{3} }}^{5} - 4 \times 10^{ - 11} d_{{TiO_{2} }}^{4} + 7.35 \times 10^{ - 14} d_{{TiO_{2} }}^{5}\)
\({\text{IQE}}\,\left( {\% } \right) = - 3.9 \times 10^{ - 7} d_{{Co_{3} O_{4} }}^{3} + 3.26 \times 10^{ - 9} d_{{Co_{3} O_{4} }}^{4} - 7.2 \times 10^{ - 12} d_{{Co_{3} O_{4} }}^{5} - 3.9 \times 10^{ - 11} d_{{MoO_{3} }}^{5} + 1.52 \times 10^{ - 10} d_{{TiO_{2} }}^{4} - 3.9 \times 10^{ - 13} d_{{TiO_{2} }}^{5}\)
For both \(TiO_{2} |Cu_{2} O\) datasets it is evident that while four descriptors were evaluated by RANSAC, only two were picked by the algorithm as predictors of photovoltaic activities. Importantly, these two descriptors give rise to six terms in the resulting QSAR equations due to their power form. Thus, RANSAC "expands" the small number of final descriptors by using them in multiple forms. A potential drawback of the resulting models is therefore reduced interpretability of terms including "high power" descriptors. The \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) dataset was characterized by five descriptors and only three were selected by RANSAC leading to models with six terms (Table 8).
The features selected by the RANSAC algorithm could be compared with those selected by the kNN and GP models reported by Yosipof et al. [41]. As can be deduced from Table 9, all methods selected the same descriptors for the \(TiO_{2} |Cu_{2} O\) (Ag) library while kNN replaced \(d_{{TiO_{2} }}\) by the ratio descriptor for the \(TiO_{2} |Cu_{2} O\) (Ag|Cu) library. While GP sometimes selected a smaller number of "base descriptors", it compensated for this smaller number by incorporating these descriptors in more complex mathematical equations. In contrast, the RANSAC algorithm is limited to simple polynomial equation (to the 5th power in this study).
Featured selected for the \(TiO_{2} |Cu_{2} O\) libraries by the various methods
\(d_{{Cu_{2} o}} ,d_{{TiO_{2} }}\)
\(d_{{Cu_{2} o}}\)
\(d_{{Cu_{2} o}}\), Ratio
RANSAC derived virtual cells
RANSAC derived models could be used to predict PV properties of virtual solar cell libraries. These predictions could serve two purposes: (1) identify trends related to the dependence of PV properties on descriptors values, which are not easily discernible from the resulting equations. (2) Provide a theoretical basis for and guide future experiments.
\(TiO_{2} |Cu_{2} O\) (Ag) and \(TiO_{2} |Cu_{2} O\) (Ag|Cu) virtual libraries
The original \(TiO_{2} |Cu_{2} O\) (Ag) and \(TiO_{2} |Cu_{2} O\) (Ag|Cu) libraries were of identical compositions with \(d_{{TiO_{2} }}\) between 70 and 311.5 nm and \(d_{{Cu_{2} O}}\) between 249 and 596 nm. The virtual cell should cover these ranges and expand upon them to allow RANSAC-based extrapolations. With this in mind, thickness values for the different layers were selected to be between 200 and 700 nm and between 40 and 400 nm for the Cu2O and TiO2 layers, respectively, where each range was divided into 100 bins (a total of 10,000 cells per virtual library). These specific ranges were selected following several iterations designed to find the model's limits, beyond which the results would not be physically meaningful (i.e., have negative PV values). Next, the PV properties (J SC , V OC , IQE) of each cell were predicted using the RANSAC models presented in Table 8. The results of these predictions are presented in Fig. 7 and demonstrate a few trends: (1) all PV activities primarily depend on the thickness of the Cu2O layer rather than on the thickness of the TiO2 layer. This trend was noted by Pavan et al. [43]. but only for J SC . (2) J SC presents a marked increase for Cu2O thicknesses above 500 nm (where J SC equals \(200\frac{\mu A}{{cm^{2} }}\)) as seen in Fig. 7a, d. Similar trends (yet with less sharp transitions) are also seen for IQE and V OC (Fig. 7b, e and c, f, respectively). Interestingly, Cu2O thicknesses above 500 nm where hardly explored by the original library. (3) The nature of the back contact (Ag vs. Ag|Cu) has the largest effect on the dependence of J SC on the thickness of the Cu2O layer (compare Fig. 7a, d) which is followed by V OC (compare Fig. 7b, e). In contrast, the dependence of IQE on the thickness of the Cu2O layer is the least affected by the back contact (compare Fig. 7c, f). (4) Certain combinations of \(d_{{TiO_{2} }}\) and \(d_{{Cu_{2} O}}\) are predicted to have both high J SC and V OC values. These trends are largely in accord with previous conclusions on these systems deduced from experiments and other data mining approaches [41].
Virtual cells based on the \(TiO_{2} |Cu_{2} O\) with Ag back contacts [a J SC (μA/cm2); b V OC (V); c IQE (%)] and \(TiO_{2} |Cu_{2} O\) With Ag|Cu Back Contacts [d J SC (μA/cm2); e V OC (V); f IQE (%)] solar cells libraries. The white regions are outside of the models' applicability domain
\(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) virtual library
In a similar manner, another virtual library was constructed for the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) MOs composition. In the original library, the thicknesses of the different layers ranged from 259 to 355, nm, from 30.7 to 245 nm and from 38.9 to 61.8 for TiO2, Co3O4 and MoO3, respectively. In the virtual library, these ranges were increased to 30–500 and 40–100 nm for the Co3O4 layer and MoO3 layer, respectively (50 bins for each range) while the TiO2 layer was kept at a constant value of 340 nm. This led to a virtual library consisting of 2500 cells.
For this particular library, Koushik et al. [15] showed that IQE is mainly affected by the thickness of both the \(Co_{3} O_{4}\) and \(MoO_{3}\) layers. This conclusion was further supported by a computational analysis [36]. Figure 8 shows that RANSAC's prediction is in line with this proposition (i.e., to achieve relatively high IQE values, the thickness of the \(Co_{3} O_{4}\) layer must be low, smaller than 150 nm and this property is also influenced by the thickness of the \(MoO_{3}\) layer). In addition, RANSAC's models point to an inherent problem in producing solar cells with both high J SC and IQE values for this MOs combination since the former seems to yield maximum value at \(Co_{3} O_{4}\) layer thickness at the 500 nm region while the latter, yields its global maxima at the 30 nm region. Finally, Fig. 8 suggests possible combinations for additional experiments that may lead to high IQE values, for example small thicknesses of both \(Co_{3} O_{4}\) and \(MoO_{3}\) layers.
Virtual cells based on the \(TiO_{2} \left| {Co_{3} O_{4} } \right|MoO_{3}\) library [a J SC (μA/cm2); b V OC (V); c IQE (%)]. The white regions are outside of the models' applicability domain
To the best of our knowledge, this is the first application of the RANSAC algorithm in materials-informatics and certainly for the analysis of solar cells libraries. Overall, RANSAC demonstrated a promising ability to develop predictive models for key PV properties across multiple libraries. The statistical parameters of the resulting models favorably compare with results obtained from genetic programing and kNN-derived models. Furthermore, the trends observed either from the models in their equation form or from the virtual cells are in agreement with previous findings [43, 45].
The performances of RANSAC together with the ability to use it as a "one stop shop" for model derivation and validation makes the algorithm an appealing additional to the arsenal of modeling tools in chemo- and material-informatics. This opens new opportunities for understanding the factors controlling the properties of materials and for the design of new materials with improved properties. Clearly, the applications of RANSAC (as well as of all other data mining tools) should be conducted in close collaboration with experimentalists to provide physics/chemistry based explanation to the observed trends and to capitalize on the results. We expect that the RANSAC algorithm will find multiple usages in chemoinformatics and materials-informatics researches.
All authors conceived, designed, wrote, read and approved the final manuscript.
The authors acknowledge financial support from the Israeli National Nanotechnology Initiative (INNI, FTA project).
The libraries used in this article as well as any supporting tools will be provided upon request from the authors.
This work was supported by the Israeli National Nanotechnology Initiative (INNI, FTA project).
Department of Systems Engineering, Afeka – Tel-Aviv Academic College of Engineering, Tel-Aviv, Israel
Department of Chemistry, Bar-Ilan University, 5290002 Ramat-Gan, Israel
Faculty of Business Administration, College of Law & Business, 26 Ben Gurion Street, Ramat-Gan, P.O. Box 852, 5110801 Bnei Brak, Israel
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Primitive notion
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms.[1] Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem).
For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them,[2] their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both".[3]
Details
Alfred Tarski explained the role of primitive notions as follows:[4]
When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously. The sentence which determines the meaning of a term in this way is called a DEFINITION,...
An inevitable regress to primitive notions in the theory of knowledge was explained by Gilbert de B. Robinson:
To a non-mathematician it often comes as a surprise that it is impossible to define explicitly all the terms which are used. This is not a superficial problem but lies at the root of all knowledge; it is necessary to begin somewhere, and to make progress one must clearly state those elements and relations which are undefined and those properties which are taken for granted.[5]
Examples
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics:
• Set theory: The concept of the set is an example of a primitive notion. As Mary Tiles writes:[6] [The] 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive, undefined, term. As evidence, she quotes Felix Hausdorff: "A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit."
• Naive set theory: The empty set is a primitive notion. To assert that it exists would be an implicit axiom.
• Peano arithmetic: The successor function and the number zero are primitive notions. Since Peano arithmetic is useful in regards to properties of the numbers, the objects that the primitive notions represent may not strictly matter.[7]
• Axiomatic systems: The primitive notions will depend upon the set of axioms chosen for the system. Alessandro Padoa discussed this selection at the International Congress of Philosophy in Paris in 1900.[8] The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."[9]
• Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness, and incidence.
• Euclidean geometry: Under Peano's axiom system the primitive notions are point, segment, and motion.
Russell's primitives
In his book on philosophy of mathematics, The Principles of Mathematics Bertrand Russell used these notions: For the class-calculus (set theory) he used relations, taking set membership as a primitive notion. To establish sets he also requires propositional functions as primitive, as well as the phrase "such that" as used in set builder notation. (pp 18,9) Regarding relations, Russell takes as primitive notions the converse relation and complementary relation of a given xRy. Furthermore, logical products of relations and relative products of relations are primitive. (p 25) As for denotation of objects by description, Russell acknowledges that a primitive notion is involved. (p 27) The thesis of Russell’s book is "Pure mathematics uses only a few notions, and these are logical constants." (p xxi)
See also
• Axiomatic set theory
• Foundations of geometry
• Foundations of mathematics
• Logical atomism
• Logical constant
• Mathematical logic
• Notion (philosophy)
• Natural semantic metalanguage
References
1. More generally, in a formal system, rules restrict the use of primitive notions. See e.g. MU puzzle for a non-logical formal system.
2. Euclid (300 B.C.) still gave definitions in his Elements, like "A line is breadthless length".
3. This axiom can be formalized in predicate logic as "∀x1,x2∈P. ∃y∈L. C(y,x1) ∧ C(y,x2)", where P, L, and C denotes the set of points, of lines, and the "contains" relation, respectively.
4. Alfred Tarski (1946) Introduction to Logic and the Methodology of the Deductive Sciences, p. 118, Oxford University Press.
5. Gilbert de B. Robinson (1959) Foundations of Geometry, 4th ed., p. 8, University of Toronto Press
6. Mary Tiles (2004) The Philosophy of Set Theory, p. 99
7. Phil Scott (2008). "Mechanising Hilbert's Foundations of Geometry in Isabelle (see ref 16, re: Hilbert's take)". CiteSeerX 10.1.1.218.9262. {{cite web}}: Missing or empty |url= (help)
8. Alessandro Padoa (1900) "Logical introduction to any deductive theory" in Jean van Heijenoort (1967) A Source Book in Mathematical Logic, 1879–1931, Harvard University Press 118–23
9. Haack, Susan (1978), Philosophy of Logics, Cambridge University Press, p. 245, ISBN 9780521293297
| Wikipedia |
Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system
Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies
June 2018, 38(6): 2879-2910. doi: 10.3934/dcds.2018124
Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces
Qunyi Bie 1,, , Haibo Cui 2, , Qiru Wang 3, and Zheng-An Yao 3,
College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
* Corresponding author: [email protected]
Received July 2017 Published April 2018
Fund Project: Research Supported by the NNSF of China (Grant Nos.11271379, 11271381, 11671406, 11601164 and 11701325), the National Basic Research Program of China (973 Program) (Grant No. 2010CB808002), the Natural Science Foundation of Fujian Province of China (Grant Nos. 2016J05010 and 2017J05007) and the Scientific Research Funds of Huaqiao University (Grant No.15BS201).
The present paper is devoted to the compressible nematic liquid crystal flow in the whole space $ \mathbb{R}^N\,(N≥ 2)$. Here we concentrate on the incompressible limit in the $ L^p$ type critical Besov spaces setting. We first establish the existence of global solutions in the framework of $ L^p$ type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.
Keywords: Liquid crystal flow, $ L^p$ type critical Besov spaces, global existence, incompressible limit.
Mathematics Subject Classification: Primary: 35Q35, 76N10; Secondary: 35B40.
Citation: Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
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Qunyi Bie Haibo Cui Qiru Wang Zheng-An Yao
\begin{document}$ L^p$\end{document} type critical Besov spaces" readonly="readonly"> | CommonCrawl |
Extracting multiple layers of social networks through a 7-month survey using a wearable device: a case study from a farming community in Japan
Masashi Komori ORCID: orcid.org/0000-0001-7951-89261,
Kosuke Takemura2,
Yukihisa Minoura3,
Atsuhiko Uchida4,
Rino Iida4,
Aya Seike4 &
Yukiko Uchida4
Journal of Computational Social Science volume 5, pages 1069–1094 (2022)Cite this article
As individuals are susceptible to social influences from those to whom they are connected, structures of social networks have been an important research subject in social sciences. However, quantifying these structures in real life has been comparatively more difficult. One reason is data collection methods—how to assess elusive social contacts (e.g., unintended brief contacts in a coffee room); however, recent studies have overcome this difficulty using wearable devices. Another reason relates to the multi-layered nature of social relations—individuals are often embedded in multiple networks that are overlapping and complicatedly interwoven. A novel method to disentangle such complexity is needed. Here, we propose a new method to detect multiple latent subnetworks behind interpersonal contacts. We collected data of proximities among residents in a Japanese farming community for 7 months using wearable devices which detect other devices nearby via Bluetooth communication. We performed non-negative matrix factorization (NMF) on the proximity log sequences and extracted five latent subnetworks. One of the subnetworks represented social relations regarding farming activities, and another subnetwork captured the patterns of social contacts taking place in a community hall, which played the role of a "hub" of diverse residents within the community. We also found that the eigenvector centrality score in the farming-related network was positively associated with self-reported pro-community attitude, while the centrality score regarding the community hall was associated with increased self-reported health.
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Human behaviors sometimes show high complexity— person who behaved nicely in one moment may express hostility in another moment, sometimes appearing to be a different person altogether [21]. The accumulated wisdom of social psychology tells us how humans are susceptible to social influences (e.g., [2, 7, 34]) and thereby behave in a context-dependent way. A person's behavior can vary from context to context because one's behavior is influenced by whom one is with (e.g., [27]). Furthermore, people are generally embedded in multiple networks in their daily lives. Boissevain [5] argues that people participate in several activity fields and each activity field can be regarded as a social subnetwork consisting of a set of people who potentially share a common relationship (Fig. 1). Even in a relationship between two people, it is not possible to characterize a person by a single role. Imagine a small social group of people in a given region who are mutually acquainted with each other. There exist several functional subnetworks within the group that have specific roles. Many of the group's members participate in multiple subnetworks and may have different roles within each subnetwork. They are exposed to social influences sourced from different subnetworks and are faced with different obligations stemming from different social roles across these subnetworks. This diversity of social contexts can lead to the complexity of people's behaviors. A relation between people who interact with each other in different positions and roles in different networks is termed multiplex or many-stranded. This kind of overlap is more common in small and isolated societies and communities [5]. To disentangle these complex behaviors, it is essential to have methods that assess the networks embedded within the social group and the roles members have within each subnetwork. In the current study, we measured interpersonal contact (proximity logs) between residents in a rural area over a long period of time using electronic devices. We aim to develop a methodology for capturing a comprehensive view of a social group by discovering the multiple subnetworks that exist in a group, based on the results of long-term proximity measurement data matrix factorization.
The problem of finding unknown subnetworks of a community has been an age-old struggle faced by many researchers. In anthropological studies, participant observations have been used to comprehensively observe the social networks of small social groups [5, 30]. However, such methods may be difficult for a study of a community of a few hundred residents. In that case, the most common survey method today is to have participants report on their own social networks in questionnaires based on recall or recognition [41]. However, the data reported by participants in these types of surveys are heavily biased and inaccurate at both the dyadic and structural levels, when compared to objective observational records, and cannot be a substitute for observational data [3, 4, 11, 28]. To conduct more reliable research, we need a method for objectively observing interpersonal contacts.
One commonly used method of objectively assessing social networks, is through Email and SNS communication logs [13, 24, 26]. However, communication often happens outside such electronic media, and face-to-face verbal and nonverbal communication also plays a strong role in maintaining social relationships. In many instances (e.g., in rural areas with many elderly people), electronic communication logs may be insufficient for revealing social networks, and a method of measuring face-to-face interpersonal interactions is needed.
One recent solution is to use wearable devices. These allow the assessment of face-to-face interactions with high temporal resolution over long periods of time [8, 12, 44]. For example, the MIT Reality Mining Project uses a device named "sociometric badges" to record interpersonal interactions between employees in a work environment over time [40]. "Sociometric badges" are devices capable of collecting multi-channel log data, including physical proximity. The strength of person-to-person proximity measured by this wearable device has been demonstrated to be linked to the subjective quality of communication [32, 42]. Using wearable devices for social network research thus allows for long-term observation, which is also useful for extracting more stable social networks than short-term observation [15].
Social network studies using wearable devices have primarily targeted interpersonal relationships in the work environment, such as within business organizations [32, 44]. However, till date, no social network survey using wearable devices has been conducted on elderly people living in rural areas. The potential value of such a wearable device is more likely to be realized in a field where research participants can move freely, rather than in a temporally and spatially confined area (such as a workplace environment). As such, this method was especially appropriate for our study, which tracked the daily interpersonal contact history of participants residing within a rural community.
Participants carried a small smartphone from morning to evening, without restrictions on time and area, so that we can collect data of diverse social interactions, not limited to specific types of interactions (e.g., business conversations).
Though data obtained from wearable devices can provide clues to reveal the structure of the community, analysis is still a dogmatic premise. Network clustering is a conventional method for revealing the structure of a social group from the records of social ties, by estimating the sub-groups to which a group's members belong from the topography of the social connection data. Various methodologies of network clustering have been proposed [14], that ultimately serve to assign members to one or more sub-groups. However, these network clustering methods are based on cross-sectional data of social ties at a single time-point, and therefore cannot account for temporal changes in connections obtained longitudinally by wearable devices.
In this study, we adopted a different approach to extract social networks, which is based on cluster analysis, due to necessity of integrating high (temporal) resolution data logging of interpersonal interactions. These were measured over a long period of time with wearable devices. Similar to the contact-tracing applications used for COVID-19 management in some countries, the wearable devices model human interaction by communicating wirelessly with other devices within an immediate vicinity and store the information as a proximity log. The log data contain richer social information than general social network data, yielding information, for example, on who was with whom and for how long, and who was present and absent at the same time on a given occasion. To take advantage of these benefits (or using wearable devices), our study adopts an approach that resembles factor analyses. The subnetworks within a community are considered as latent common factors that cannot be directly observed, and interpersonal contact history is considered as an observation of the latent network. We also consider the social network of the entire community to be the superposition of these latent subnetworks. That is, the problem of finding unknown subnetworks of a community can be replaced by extracting common factors from interpersonal contact history.
By applying factor extraction method to the entire interpersonal contact histories of a community, it is possible to find potential common factors, or subnetworks. It is necessary to consider that the factors of these histories are non-negative, i.e., equal to 0, if no contact occurs and positive if there is contact. We, therefore, applied non-negative matrix factorization (NMF) to interpersonal contact history in this study. NMF is one of the methods to decompose matrices that have 0 or positive values [9, 25], and it has been widely applied to identify structures of various types of data, such as images, documents, genes, as well as time series data of acoustic signals. Thus, NMF is optimal for understanding the structure of social networks from interpersonal contact histories. This study attempts to discover unknown subnetworks within a community by decomposing the interpersonal contact history data into weighted rows of ties and network activity sequences using NMF. The social network created from the weighted row of ties corresponds to a set of participants who are connected to each other at the same time (we call this a subnetwork), and the network activity sequence corresponds to the activity levels of the subnetwork. One of the advantages of this method is that the latent networks can be extracted without depending on specific events or locations.
We collected proximity logs from wearable devices for 7 months. The study was conducted in a local community in Kyoto Prefecture, Japan. Data of physical proximity among community residents were analyzed by NMF to extract multiple latent subnetworks. To examine the validity of our new methodology, temporal characteristics of these latent subnetworks (e.g., pattern of temporal changes of physical proximity among community members) were examined through comparisons with qualitative data we collected through interviews with the community leader (e.g., event schedules in the community).
We also administered a questionnaire survey to the same residents to measure their subjective health and attitude toward the community (e.g., attachment to the community and trust toward other members). Several studies have suggested that pro-community attitude of residents is important for a community as it promotes the community's crime control [35, 37] and disaster prevention [16, 31]. Through the correlations between these variables (subjective health and pro-community attitude) and the positions of residents in latent subnetworks (i.e., centrality scores), we examined (1) whether our new method can extract relevant indices in predicting important variables in the fields of social science and public health, and (2) which subnetworks are (and which subnetworks are not) relevant to these variables.
Our study site was a local Japanese agricultural community. Though human communities have complex overlapping multiple networks, agricultural communities are generally outstanding in this aspect. One characteristic of such communities is that people work near their homes (short or non-existent commuting times). This proximity of workplaces and residences results in co-existence of several different types of social networks in a single community, where a resident may participate in multiple networks while playing different social roles across networks. For example, one resident, Mr. A, a rice-crop farmer, would participate in a network for agricultural infrastructure maintenance with other farmers in the community. He may also participate in another network for the neighborhood watch, and perhaps an association (network) for homeowners, and a network for private social gatherings among friends. These different networks can co-exist in the same community in a complex intertwined way. The strength of Mr. A's connection with Ms. B can be different across different networks (e.g., Mr. A and Ms. B could be close in infrastructure maintenance network, but perhaps distant in the social gathering network). As our purpose was to develop a methodology to extract latent networks from a complex accumulation of multiple interconnected networks, a local agricultural community provided an excellent context to test our idea.
Another (related) helpful characteristic of a localized agricultural community for our purpose is in its physical distance from other communities. In urbanized areas, communities and groups are generally located close to each other. In local areas, communities are relatively disconnected and traffic between them is less frequent. As a result, social relationships are relatively exclusive of individuals outside the community. Therefore, significant relationships of a resident are generally complete within their community. This completeness of networks is desirable when we try to capture influences of networks on one's life outcomes. If an individual has many important connections with outsiders, investigating the network structures within a community does little to help us with identifying important networks for that individual.
Schematic illustration of social multiplexity based on [5]. Individuals have a variety of roles in society
Study site, a landscape and b the community hall
Study site and participants
The study site was a local community (a farming village) located in a northern rural area of Kyoto Prefecture, Japan (Fig. 2). The geographical area of the community is approximately 5.0 km\(^2\) [36], and the total population of the community is 840, consisting of 318 households [10], for a population density of 168 persons/km\(^2\). The percentage of elderly people (aged 65 or older) is 30.5% [10]. This community is a part of a farming area, where 11.5% of the households are engaged in farming (mainly paddy cultivation) [29, 36]. To illustrate the extent to which this community is sparsely populated and small, we compared it with another community: in downtown Kyoto City (Nakagyo Ward), the population density is 14,756 persons/km\(^2\) (i.e., approximately 88 times as dense as the studied community), the percentage of elderly people is 24.7%, and the percentage of individuals engaging in farming is only 0.06% [36]. Thus, when compared with this urbanized area, our studied community is less populated, more aged, and more devoted to farming. Within the accessible area for the studied community's residents, there is only one elementary school and one junior high school, and no senior high school or university. Therefore, residents had to leave the community at least temporarily if they intended to pursue higher education.
As typical of rural Japanese communities, this community has several seasonal events. This includes a New Year's Day celebration at a local shrine, a summer evening festival, an autumn festival, and other community-based rituals (e.g., a community gathering to make a sacred rope devoted for the local shrine at the end of a year).
Prior to the study, we conducted two pilot tests to examine the communication infrastructure (the 3G line) at the study site. We also checked the functionality of the devices and applications, and their usability for participants (especially for elders, who were often unfamiliar with using smartphones). The first pilot test (April–May 2017) was conducted with 10 participants. Following this, we upgraded the application and created a user manual to improve usability for participants. The second test (August–September 2017) was conducted with 18 participants.
After the two pilot tests (late October 2017), we started a campaign to recruit participants for the main study. Following the advice of the community leader, we distributed flyers that targeted several local groups, such as a sports team, the neighborhood watch, a social group of elderly people, and so on. The community leader helped us approach diverse range of groups so that our study would cover a broad range of people in the community. We also recruited participants at a community event (a local festival) where many locals attended. The study was explained to the residents as one that investigated people's behavior and health. Participants would carry two small devices (wristwatch-type activity tracker and smartphone) with them and would receive the activity tracker (Go: Withings) as compensation for their participation when the data collection was completed. They were also informed that health guidance by a social worker and a doctor would be provided to participants who requested it, based on the daily step and sleep data obtained from the activity trackers and a self-report health questionnaire. Those who agreed to participate in the study received the devices, a consent form, the self-report health questionnaire (this included items on gender, age, and body size), and the user manual of the devices. The consent form and the health questionnaire were returned to the project team via post. For some participants, a DVD of a short video clip that explained how to use the devices was provided. In total, it took approximately 1 month for a sufficient number of devices to reach the community (late November 2017).
We recruited a total of 90 community residents, who received one device pair each. From this pool, we analyzed data from 58 participants, whose log data confirmed that they had been carrying the devices continuously for more than 2 months. Table 1) shows characteristics of the participants.
The study was approved by the Institutional Review Board at Kyoto University. All participants gave their informed consent. The participants provided their residential addresses to receive feedback by mail. Such identifying information was, however, accessible only to one project member who could not access the data obtained by wearable devices.
While the study commenced on November 1, 2017, it took 1 month for a sufficient number of devices to reach the community, so for this analysis, the study period was set from December 1, 2017 to June 30, 2018.
Each participant carried two devices with them: a wristwatch-type activity tracker and smartphone, of which the latter was the primary device for the current study. The smartphone (BL-01: BIGLOBE; height: 41 mm; width: 47 mm; thickness: 16 mm) was equipped with Android 4.2.2 and Bluetooth 4.0 (Class 2), and the maximum range of Bluetooth communication was 10 m (Fig. 3).
A custom-made application was installed on this terminal. This application recorded the MAC addresses of nearby Bluetooth devices and the time of detection, and the Bluetooth antenna was refreshed every 5 min. This made it possible to record the MAC addresses of other devices that were nearby (within 10 m) at 5-min intervals while the participants were out. With this application, the recorded MAC addresses and time data were uploaded to an online storage once a day via the 3G line. The activity meter data were also uploaded via the 3G network.
Participants were instructed to always carry the terminal and instructed to connect the device to a charger when they returned home. The terminal was set to restart automatically when disconnected from the charger.
To help the participants use the device properly (and to motivate them to bring the devices with them), we occasionally contacted them. In December 2017 (i.e., the first month of the data collection), we contacted participants (via telephone) whose data had not been uploaded for three consecutive days or more, and repeated explanations on how to use the device. In February 2018 (i.e., the third month of the data collection), members of the project team visited the community and gave each participant an interim report of daily step data, sleep data, and data of self-report health questionnaire along with health guidance by a social worker and a doctor. On this occasion, maintenance of the devices (e.g., battery change, application update) was carried out when necessary. We also gave the residents a brief lecture on health. Participants who could not meet the project team at this occasion received the interim report via mail. In April 2018, we provided spare devices to the community leader, in case there were some participants who needed them.
Wearable device (smartphone)
Self-report questionnaire on community-related attitudes
In the month of May, 2018 (the sixth month of the data collection), the participants received a paper-and-pencil questionnaire via mail (or through the community leader). The questionnaire included demographic items (e.g., occupation, educational background, and marital status) and two sets of self-report items to measure participants' attitudes, as well as their perceptions on their community life. The first set consisted of items to assess several aspects of a participant's positive attitude toward the community, such as community attachment, cooperative behavior toward the community, and trust toward community members (see Table 2 for the items). The items were from a series of large-scale social surveys that some of the current authors had conducted (for related studies, see [17, 18, 39] ).Footnote 1 The second set was designed to measure participants' openness, or attitudes toward new ideas and new people coming from outside of the community (see Table 3 for the items). These items were also from the same series of large-scale social surveys. For both sets of items, response options were on 5-point scales, with options ranging from 1 (strongly disagree) to 5 (strongly agree). In addition, the questionnaire also included an item to measure subjective health ("How would you rate your health at the present time?") [23] and happiness ("How would you rate your current level of happiness?") [1]. For these two items, response options were on 11-point scales, with options ranging from 0 (very bad/very unhappy) to 10 (very good/very happy). The questionnaire was completed anonymously and then returned to the project team via mail directly or through the community leader (the anonymity of the responses was maintained as the questionnaire was placed in an envelope and sealed).
Table 1 Sample characteristics
Qualitative data
After the primary data collection, the project team visited the community to collect qualitative data on community activities (e.g., a festival, gatherings/meetings of the aged club, activities of a farming group) that occurred during the study period. We interviewed the community leader, the leader of the aged club, and a community hall staff.
Factorization of proximity log data
During the survey period, the average number of times that one participant's device detected another's was 9.26 times/day, in which the device scanning was performed every 5 min. In this study, we assumed that when one device detected another device nearby, there was social contact between the owners of those devices. For all combinations of participants, the levels of social contacts every 30 min were scored based on the total number of device detections in the 30 min (48 epochs/day), resulting in 10,176 epochs (I) for the survey interval (212 days). The number of combinations (J) of all the participants were \({}_{58}\mathrm {C}_{2}\), as \(N=58\).
Let \({\mathbf {Y}}\) be \(I\times J\) social contact matrix, where the element \(y_{ij}\) represents the number of social contacts in a combination of two participants at epoch i. We assume that the social contacts reflect the sum of the activities of \(K(K\ll I,J)\) latent social networks with different configurations (Fig. 4). This study aims to find the latent networks in the target community by decomposing the social contact matrix \({\mathbf {Y}}\) into the product of the basis matrix \({\mathbf {H}}\) representing the time series of network activities and the coefficient matrix \({\mathbf {U}}\) corresponding to the levels of connections between two participants:
$$\begin{aligned} {\mathbf {Y}} \simeq \mathbf {HU}. \end{aligned}$$
The dimensions of the factorized matrices \({\mathbf {H}}\) and \({\mathbf {U}}\) are \(I\times K\) and \(K\times J\) respectively (Fig. 5). The element \(h_{ik}\) of \(I\times K\) basis matrix \({\mathbf {H}}\) can be regarded as the activity of the latent network k at epoch i. Thus, the matrix \({\mathbf {H}}\) shows the time series changes of the activity levels of the latent networks. The element \(u_{kj}\) of \(K\times J\) coefficient matrix \({\mathbf {U}}\) describes the degree of connectivity between each participant in latent network k.
Latent subnetworks and observed network
Dimensionality reduction with non-negative matrix factorization (NMF): the matrix \({\mathbf {Y}}\) is represented by the smaller matrices \({\mathbf {H}}\) and \({\mathbf {U}}\)
All the elements of social contacts matrix \({\mathbf {Y}}\) are non-negative values, by requirement. Moreover, the basis matrix \({\mathbf {H}}\) and the coefficient matrix \({\mathbf {U}}\) should consist of non-negative elements, because it is natural to think that latent social networks have additive effects on social contact rather than subtractive effects. Thus, in this study, we utilized non-negative matrix factorization (NMF) to find latent networks in the community.
NMF attempts to find an approximate factorization for \(\widehat{{\mathbf {Y}}}\simeq {\mathbf {Y}}\) that minimizes the distance D between \(\widehat{{\mathbf {Y}}}\) and \({\mathbf {Y}}\). In this study, we consider NMF in which the distance D is measured by Euclidean distance between the matrices. The function \(D_{\text {EU}}\) to be minimized is given by
$$\begin{aligned} D_{\text {EU}}({\mathbf {Y}},\widehat{{\mathbf {Y}}})=\parallel {\mathbf {Y}}-\mathbf {HU}\parallel ^{2}_{F}=\sum _{ij}\left( y_{ij}-\left( hu\right) _{ij}\right) ^{2}, \end{aligned}$$
where \(\parallel \cdot \parallel _{F}\) denotes the Frobenius norm, and \(y_{ij}\simeq (hu)_{ij}=\sum _{k=1}^{K}h_{jk}u_{jk}\) is subject to the constraints of \(h_{i\alpha },u_{\alpha j}\ge 0\), where \(0\le i\le I,0\le k \le K,0\le j \le J\). All computations were done within R using the package NMF [19]. The optimal number of ranks K was determined to be five based on the cophenetic correlation coefficient and the residual sum of squares, as well as interpretability (see below for our interpretation of each subnetwork).
To elucidate the characteristics of each factor, we examine the basis matrix \({\mathbf {H}}\) and the coefficient matrix \({\mathbf {U}}\). Each column of the basis matrix \({\mathbf {H}}\) corresponds to a time series of each factor, which represents the activity level of the network associated with each factor (48 epochs/day). Meanwhile, the coefficient matrix \({\mathbf {U}}\) represents the strength of participants' connections with each other associated with a particular factor. We reconstructed each factor and its corresponding network from the coefficient matrix \({\mathbf {U}}\) as an undirected graph (Fig. 6). Almost all the participants in our surveyed community were acquainted with each other. In evaluating the characteristics of individuals in such a small community, it is necessary to consider the importance of neighboring individuals. Therefore, we used eigenvector centrality to measure centrality [6]. Eigenvector centrality measures the importance of a node by considering the importance of its neighbors. It assigns a relative score to every node in the network based on the assumption that a connection to a high-scoring node will contribute more to that node's score than an equivalent connection to a low-scoring node. Centrality score (eigenvector) of each participant was calculated for each subnetwork; they were log-transformed to be used in later analyses (distributions of those log-transformed centrality scores are shown by Fig. S1 in the Supplementary Material). The temporal patterns of change in the activity level of each potential subnetwork, corresponding to the basis matrix \({\mathbf {H}}\), is shown in Figs. 7 and 8. Figure 7 covers the entire study period (December 1, 2017 to June 30, 2018) for all the subnetworks. On the other hand, Fig. 8 shows time activity levels of three subnetworks that had unique patterns (discussed below) for more focused periods of time. High activity levels reflect the high proportion of members of each subnetwork participating in the activity. As shown in Fig. 7, many epochs of factors 1 and 5 have high activity levels, suggesting that these factors are associated with public events. On the other hand, factors 2 and 4 show low activity levels, indicating that they are mainly related to personal contacts.
The social network corresponding to each factor. Each dot represents a participant. The darker the color, the higher the age
The time series of social activity levels corresponding to each factor. The bottom graph combines the graphs of all factors. The vertical axis represents the coefficients of the basis matrix \({\mathbf {H}}\). The higher the value, the more contact among the members in the subnetwork at the epoch
Excerpt from the base matrix \({\mathbf {H}}\), which represents the change in activity level over the month for the networks corresponding to the first, third and fifth factors. The higher the value, the more contacts among the members in the subnetwork at the epoch. Each label indicates the point when various activities took place, as revealed by the interviews. Factor 1 was associated with rice farming activities. Factor 3 was associated with activities at the community hall. Factor 5 was found to be related to various activities including community promotion activities
The first factor (Factor 1) showed higher levels of activities in April (Fig. 7). According to interviews with the community leader, collaborative community activities related to rice farming, such as sowing rice seeds, transplanting rice seedlings, rice field maintenance, and weeding, occurred during the period of high activity in April. This work was mainly performed by members of a farming group in the community, and their activity schedule corresponded to the time series pattern of interpersonal contacts in Factor 1 (Fig. 8). This suggests that Factor 1 is a component related to the collaborative work of agriculture. Further, Fig. 6 shows that Factor 1 was linked to the network of relatively older participants, which is consistent with the fact that the members of the farming group were relatively old. Figure 6 also suggests that ties were not equally distributed among participants—some were densely connected while others were not connected in this subnetwork. This consistent pattern was also found from the distribution of centrality scores (Fig. S1), which shows a negatively skewed distribution. Thus, in this latent subnetwork, there was a divide among the participants in terms of the degree of connectedness with others. This is consistent with our interpretation that this factor is related to agricultural activities (e.g., transplanting rice seedlings), in that activities related to this subnetwork required some (but not all) residents' cooperation.
The second factor (Factor 2) appeared to reflect diurnal activity, and we interpret it as representative of daily interactions with family. Figure 6 shows that the network associated with Factor 2 was composed of a small number of combinations of the participants. Similarly, Fig. S1 (histogram of log-transformed centrality score) shows that a large part of participants (approximately 40%) were located at the median (= − 6) of this distribution, suggesting that many participants were connected with some other participants to the similar degree. These characteristics of this factor are consistent with our interpretation that Factor 2 was related to interpersonal contacts among family members living together.
The third factor (Factor 3) was also associated with everyday activities (Figs. 7 and 8). A notable characteristic of this factor is that it had low-level but long-period activities, unlike the spike-like patterns shown in Factor 1. Generally, the activities started in the morning and ceased around the evening (before night), and often showed short-period reductions around noon. The level of activities was generally low, suggesting that only a few people were there at one time. Though there were some days that had higher levels of activities (e.g., March 12, 15 and 22), the interview data suggested that community activities (e.g., a social gathering of the elders' club) occurred at the community hall on these days (Fig. 8). These patterns suggest that Factor 3 reflected interpersonal contacts among people who visited the community hall for different reasons. From the interview, the community hall functioned as a gathering point for the residents, with several different kinds of activities taking place in the hall (Fig. 2b). On weekdays, one staff member was continually stationed in the hall (from morning to evening) even without any scheduled special activities, and assisted residents who visited the hall. Therefore, we concluded that the community hall served as "hub" that connected residents directly or indirectly, for various reasons (e.g., to participate in a club activity, to meet the hall staff, to get archived documents about the community history). Given that the residents differed in the frequency of visiting the community hall, the number of chances to interact with other residents at the community hall was also different.
The fourth factor (Factor 4) was hard to interpret. Higher activity levels were observed on January 8 and June 2, 2018 (Fig. 7), but it was unclear what kind of contact these were; we could not discern any related activities from the interviews conducted. Moreover, lower activity levels were observed during the months of April and May. The centrality distribution score (Fig. S1) was not skewed, unlike Factors 1 and 5. There was no clear peak (mode) in the distribution. At this point, we are hesitant to interpret this factor. The subnetwork of Factor 4 (Fig. 6) shows that there are several small groups that are not strongly connected to each other. It suggests that there were several gatherings whose activity level decreased due to unknown factors during the same period in April and May. The activity levels of Factor 1 indicate that farming activities were more active during April and May. This agreement implies that farming activities might have suppressed the activities of the subnetwork of Factor 4.
The fifth factor (Factor 5) is associated with the network which saw involvement from many community members across a wide range of ages (Fig. 6), though age was positively associated with centrality score in this factor (\(r = 0.33\); see Supplementary Material). Figure S1 suggests that despite the presence of some participants with low centrality scores, the majority of participants had similarly high scores for centrality. High activity levels (Figs. 7 and 8) coincided with community festivals, meetings of local non-profit groups, bazaars, community gardening activities, and settlement promotion activities (inferred from interviews with the community leader). This suggested that Factor 5 was a component closely connected to community promotion activities, which was organized by the community promotion committee and attended by many residents.
Another major activity was seen on February 7 in all the five factors (Fig. 7). This was the day the project team visited the community to conduct interviews with the participants and provided healthcare information.
Table 4 shows descriptive statistics of the centrality scores of the latent networks as well as the self-report scales. Table 5 shows the correlations between them. All the centrality scores of latent networks were positively correlated with each other. Among them, the strongest correlation was found between Factor 1 (activities of farming group) and Factor 5 (community promotion activities), suggesting that residents who were located at the center of the network for farming activities were also located at the center of the network for community promotion activities.
Reliability of self-report scales
The internal consistency of items for measuring positive attitude toward the community (pro-community attitude) is shown in Table 2. We used a principal component analysis (PCA) to assess the internal consistency of the nine items for measuring pro-community attitude and the five items for openness, separately. For pro-community attitude, two items had low factor loadings (see Table 2) and thus were excluded. The remaining seven items showed sufficiently high Cronbach's coefficient alpha (0.88) and McDonald's coefficient omega (0.91). "Pro-community attitude" was computed by averaging seven items. See footnoteFootnote 2 for validation checks of this composite measure. The item "I participate in community activities (e.g., meeting and events)," which was not included in the pro-community attitude score, was also used in the analyses below as a separate item measuring "participation in community activities."
For openness, one item with low factor loading was excluded (Table 3) and the remaining four items showed acceptable Cronbach's coefficient alpha (0.67) and McDonald's coefficient omega (0.80). These were averaged to provide a measure of "openness". See footnoteFootnote 3 for validation checks of this composite measure.
Table 2 Internal consistency of items for measuring pro-community attitude
Table 3 Internal consistency of items for measuring openness
Table 4 Descriptive statistics
Table 5 Correlations between the log-transformed eigenvector centrality scores and self-report scales
Correlations between centrality scores in latent networks and self-report scales
Among the self-report scales, pro-community attitude was positively correlated with participation in community activities (Table 5). Pro-community attitude was also positively correlated with happiness but only marginally. Participation in community activities also had weak (marginally significant) positive correlation with openness and subjective health. Openness and subjective health were positively correlated with each other. Finally, subjective health was positively correlated with happiness.
Our primary focus was toward understanding the relationship between pro-community attitude and centrality scores of latent networks. Centrality in the Factor 1 network (activities of farming group) was positively associated with pro-community attitude. Factor 2, which presumably reflected network among family members, did not have any significant correlation with the self-report scales. Centrality in the Factor 3 network (contacts at the community hall) was positively associated with subjective health. Centrality in the Factor 4 network, an unknown network, was also positively correlated with pro-community attitude. Centrality in the Factor 5 network (community promotion activities) did not have any correlation with the self-report scales. Participation in community activities (self-report) did not correlate with any centrality scores of latent networks. We will discuss this in "Discussion".
With the development of IoT technology, the process of obtaining detailed data on people's spatial proximity over time has become increasingly accessible. Social network surveys using such IoT technologies have higher ecological validity than surveys using paper questions or interviews, because they capture the real day-to-day behavior displayed by survey participants and is free of response and recall biases. However, in communities with complex interpersonal relationships, where multiple social networks overlap, it remains a challenge to discover interpretable social networks from large amounts of long-term, digitally recorded data. This may be partially due to the insufficient use of temporal information in social network studies using wearable devices. In this study, we aimed to extract latent social subnetworks in a local community by factorizing the time series log data matrix of the spatio-temporal proximity using NMF, and to evaluate the interpretability of the extracted latent network. We conducted a 7-month study using wearable devices in a farming community in Japan. This dataset provides rich information on changes of interpersonal contacts—not only micro-scale changes (i.e., changes within 30 min), but also macro-scale (seasonal) changes. Seasonal differences are especially important given that several activities in farming communities are season dependent. In addition, our dataset itself is an important contribution, as it consists of a wide range of age groups in a (relatively) isolated population, and are generally harder to access for researchers beyond university students or crowdsourced workers who are commonly used in this discipline.
We extracted five latent subnetworks from proximity logs. The proximity logs were decomposed into a basis matrix (corresponding to the temporal activity patterns of each subnetwork) and a weight matrix (corresponding to each tie between the members). We found that the extracted subnetworks showed reasonable and interpretable temporal patterns, suggesting the validity of our method. For example, Factor 1 showed a time series pattern of social contacts that tracked the activities of the farming group. Factor 1 showed high levels of activities around early April, where farming activities are generally busy. At a more fine-grained scale, high levels of activities were observed exactly on days when the farming group's activities (e.g., transplanting rice seedlings) were carried out. The other subnetworks had unique characteristics, and some of them showed interpretable patterns (e.g., Factor 3 seemed associated with contacts among a wide range of residents at the community hall).
We also found that the extracted subnetworks provided useful information in predicting important variables in the fields of social science (pro-community attitude) and public health (self-rated health). We measured these variables by the self-report scales and examined which subnetworks are (and which ones are not) relevant to these variables. As a result, we found that the centrality score of Factor 1 was positively associated with scores of pro-community attitude. That is, individuals at the center of the farming-related social network were more likely to be involved in reciprocal/cooperative relationships in the community than other individuals. This finding is consistent with previous studies showing that farming is connected to several collective activities in communities such as collective works to maintain shared facilities (e.g., [39]).
Interestingly, the farming-related network was not the only one sustaining reciprocal/cooperative relationships in the community. The centrality score of Factor 4 was also associated with pro-community attitude, suggesting that this unknown subnetwork is essential for cooperative relationships in the community. Yet, the time series pattern of this subnetwork did not correspond to the dates of community activity that we learned from the interview with the community leader and others. One interpretation could be that Factor 4 perhaps reflects more casual gatherings of informal groups and random encounters, which may play important roles to maintain cooperative relationships [33]. From another perspective, it may be that a person's high frequency of random contacts (rather than appointed ones) with others reflects how deeply their daily activities are intertwined with those of other residents (and thereby tended to show higher pro-community attitude). Either way, this unexpected association suggests that our method helps reveal social networks that are hidden yet play important roles in communities.
However, not all the subnetworks were related to pro-community attitude. For example, the centrality score of Factor 2 (family contacts) was not correlated with pro-community attitude. In addition, unexpectedly, the self-report measure of participation in community activities did not correlate with any of the centrality scores (Table 5) including Factor 1 (activities of the farming group). If the farming group plays central roles for reciprocal/cooperative relationships in the community (as seen in the correlations between pro-community attitude and Factor 1 centrality score), those with high centrality in Factor 1 should show the greater tendency to participate in community activities than others. One possibility has to do with a ceiling effect for the item of participation in community activities. The item might assess light commitment to the community (e.g., dropping by a community festival) rather than more heavy commitment (e.g., involving in the festival as a staff). In fact, the median of this item was relatively high (4.00 in the five-point scale ranging from 1 to 5) compared to pro-community attitude (median = 3.64). If that was the case, even though members of the farming group committed more deeply to participation in community activities than other residents, the current self-report item for participation in community activities failed to capture such a difference. This implies that we need an item asking how deeply one engages (rather than asking about participation) in community activities if researchers are interested in individual differences in the involvement in agricultural communities, where participation in community activities is generally high compared to other types of communities (e.g., [39]).
Self-rated health was positively correlated with Factor 3. Factor 3 presumably reflected a wide range of interpersonal contacts occurring at the community hall. The community hall played the role of a "hub" in this community. Different groups in the community visited the hall either regularly, or on an ad-hoc basis. For example, the elderly group had regular meetings at the hall. A group of women regularly gathered at the hall to conduct exercise sessions, and several people gathered there for occasional drinking sessions. The hall might be a place to connect different groups and diverse residents from the community. Then, Factor 3, which traced interpersonal contacts at the community hall, might be a network covering a wider range of social relationships in the community than the other subnetworks. If so, it is understandable that a person who cannot engage in even such a network may have a health issue that prevents them from commuting to the hall. Taken together, the findings suggest that our method can extract a social network that helps us identify individuals who may be of poor health.
First, the current study showed that spatio-temporal proximity data over a long time recorded by wearable devices is useful to detect meaningful structure of social networks. This is important given that self-report methods of interpersonal contacts have a non-negligible limitation that it cannot capture unnoticed/unmemorable interpersonal contacts [3, 4, 28]. Such contacts can still be a significant source of social influence on human minds [20] and thus need to be quantified. Using wearable devices for a long period can be a solution for this methodological issue. In fact, the Factor 4 subnetwork, which was seemingly representative of hidden social networks, was linked to pro-community attitude.
Second, the current study provides a novel method to extract complex and multi-layered social network structures. By factorizing proximity data, we were able to extract multiple (and mutually overlapping) latent subnetworks. This method is especially useful when researchers try to unravel complex interwoven ties among community members. People are often embedded in multiple social networks simultaneously (e.g., a researcher may be involved in multiple collaborative research projects, while teaching multiple courses, on top of working as a committee member for the university administration), and these networks are somewhat overlapping (e.g., one of their colleagues in the committee is also a member of the research project). By capturing such different types of networks simultaneously, we can examine what kind of networks (e.g., casual network, formal network) can be a channel for the transmission of various types of social influences (e.g., [7]).
Third, the current study provides a new perspective on social networks. Traditionally, studies have developed methods to classify individuals into clusters (e.g., [14]) and assess networks among them. In the current study, we proposed a new method to classify ties (or proximity) into latent subnetworks. This approach is based on a perspective to view proximities as observations that reflect latent structures behind of them. By viewing interpersonal contacts in this way, the multi-layered nature of social relationships in the real world can be targeted in empirical investigations. Under this view, it is only natural that individuals are embedded in multiple (and possibly overlapping) subnetworks and are sometimes forced to play different social roles across various contexts. As different social roles sometimes place conflicting obligations on an individual, the multi-layered nature of social networks is an important research theme regarding social stress that people face in their daily lives.
Limitations and future directions
First, our sample size was not large, and our efforts to recruit participants did not necessarily cover the entire community. We largely relied on the community leader's direct and indirect connections to recruit participants. This limitation comes from our decision to conduct the study in a farming community, in which social networks likely overlap and are interwoven complexly. As smartphones were not so common in such a community at the time of the survey, we needed to distribute the devices and asked the participants to bring them every day. Given that the devices were not necessarily familiar to the participants, this was not an easy request to accept. Therefore, we had to rely on the community leader, as the most influential person in the community, to recruit participants. Future studies with a larger sample and wider range of participants in a target community are needed. Alternatively, given the difficulty to collect network data over a long period of time, each study can be replicated at a smaller level with a relatively smaller sample size. By accumulating data from such studies, researchers can perform a meta-analysis, which would help overcome any problem caused by a small sample size and help conduct more fine-grained analyses as well (e.g., examination of possible moderating effects by gender). If accumulated data come from different types of communities, meta-analysis would also enable the examination of the generalizability of the findings across different contexts. The current study can be a part of such accumulation.
Second, on a related note, we could not ensure that participants brought the devices with them every time they left the house. When not physically with their devices, any encounters with other participants would be missed. One solution could have been to use participants' own smartphones, but with the low rates of smartphone ownership, this was another limitation of our decision to conduct the study in a farming community. For this study, a farming community provided an ideal circumstance for our research question (that is, complexly overlapping subnetworks), but future studies in more urbanized settings would be useful to examine the applicability of our method to different settings.
Third, like most factor analyses, deciding on the number of factors (subnetworks) was a challenge. In our case, the number of factors was determined not only by referring to the cophenetic coefficients and the residual sum of squares, but also by considering the interpretability of qualitative data obtained through the interview. Yet, it is still difficult to narrow down interpretations (e.g., we cannot be sure if "contacts at the community hall" really occurred at the community hall). In future research, the utilization of GPS location data may provide useful information for determining the appropriate number of factors. This study only measures physical proximity data and does not measure communication via PC or smartphone. Therefore, we were unable to examine whether or not there exists a subnetwork that is based on electronic communication, though the existence of such a subnetwork is possible. In future research, it should be important to examine the differences and interactions between social networks based on electronic communication and networks based on face-to-face communication.
Finally, to measure participants' health, we relied on self-reported questionnaires. As there would be an issue of reference group effect [22], the results should be interpreted with caution (e.g., elderly people might rate their health in comparison with other elderly people, not younger people). Future research should consider using biological markers to assess participants' health.
Human society often comprises several, multi-layered, complex social networks. To understand interpersonal behavior, we must first disentangle such complexity and extract interpretable subnetworks. To this end, the current study proposed a new method using NMF. This method successfully extracted five subnetworks from a 7-month survey of a farming community of Japan, that used wearable devices to track instances of social interaction. The extracted subnetworks helped predict individual differences within the community along the levels of pro-community attitude and health. The study contributes to the literature by adding a new method and a new perspective to comprehend face-to-face social interactions and structures of latent social networks that explain these interactions.
The datasets generated during the current study are not publicly available owing to privacy issues. However, they are available from the corresponding author upon reasonable request.
The series of large-scale social surveys were conducted in more than 500 communities mainly located in the western part of Japan. All surveys used paper-and-pencil questionnaires, and most of them were delivered via mail to residents in the target communities. The communities were diverse, including urban, farming, and fishing communities, scattered across different geographical areas of Japan. The surveys were conducted as part of a multiple-purpose project. Therefore, each survey needed to measure many concepts, and hence could not have many items for each one. As such, the project team developed items to measure several different aspects of participants' community life by a small number of items. The items we used in the current study came from the surveys of this project. Some of them were created by modifying items from standardized scales (e.g., the generalized trust scale of Yamagishi and Yamagishi [43]). We chose items to assess several aspects of a participant's positive attitude toward the community and examined their combinations (as described in "Results"). We took the same approach for the openness measure as well.
Two surveys from the aforementioned large-scale project (see footnote 1) can provide data to examine the validity of this composite measure of our pro-community attitude. One survey ("survey 1") collected data from 6409 individuals from 533 communities and used the same six items for pro-community attitude as the current study (the item "I try to always follow the established rules of the community" was not used in this survey). The other survey ("survey 2") collected data from 1066 individuals from 91 communities and used the same seven items for pro-community attitude as the current study. These two datasets showed that the internal reliability of this composite measure was high (\(\alpha s = 0.77\) and 0.81 in surveys 1 and 2, respectively). In addition, these two datasets showed that the composite measure of pro-community attitude had reasonable correlations with other variables. For example, the pro-community attitude score was positively correlated with how long a participant had lived in the community (r = 0.25, p < 0.001 in both surveys 1 and 2). Participation in collective activities in the community (e.g., maintenance work on public facilities, disaster-prevention group activities; see [39]) was also positively correlated with the pro-community attitude score (\(rs = 0.30\) and \(0.31, ps < 0.001\) in surveys 1 and 2, respectively). These findings are supportive of our approach combining the seven items to create a measure of pro-community attitude.
The same two surveys (see footnote 2) can also provide data to examine the validity of this composite measure of openness. Survey 1 used the same four items for openness as the current study. Survey 2 used the same three items for openness as the current study (the item "I would be happy if a person from another country settled in my community" was not used in survey 2). These two datasets showed that the internal reliability of this composite measure was not necessarily high as in the current study (\(\alpha s = 0.59\) and 0.58 in surveys 1 and 2, respectively). Yet, the composite measure of openness had reasonable correlations in these datasets. The surveys had two items from a scale of innovation-promotive behavior [38]. The two items were originally designed to measure a behavioral tendency to make proposals for a company (in surveys 1 and 2, the items were modified to fit in the context of local communities). The composite measure of openness was positively correlated with this active behavioral tendency (\(rs = 0.22\) and \(0.38, ps < 0.001\) in surveys 1 and 2, respectively).
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We thank Katsuhiko Kawaguchi, Kazuma Higashida, and Maki Higashida for their great support of our research in the community. We also thank everyone in the community for their kind help with our research. We thank Sachie Kaneko for her support of building connections with community members, and Kongmeng Liew for his helpful comments on earlier versions of this manuscript. This research was supported by the JST-RISTEX project on 'Designing a sustainable society through intergenerational co-creation'.
This research was supported by the JST-RISTEX project on 'Designing a sustainable society through intergenerational co-creation'.
Osaka Electro-Communication University, Neyagawa, Japan
Masashi Komori
Shiga University, Hikone, Japan
Kosuke Takemura
Bukkyo University, Kyoto, Japan
Yukihisa Minoura
Kyoto University, Kyoto, Japan
Atsuhiko Uchida, Rino Iida, Aya Seike & Yukiko Uchida
Atsuhiko Uchida
Rino Iida
Aya Seike
Yukiko Uchida
MK, KT, and YU designed the study; MK, YM, AU, RI, and AS executed the study; MK, YM, and AU analyzed the data; MK, KT, and MY prepared the manuscript.
Correspondence to Masashi Komori.
The study was approved by the Institutional Review Board at Kyoto University (27-P-29).
Consent to participate
All the participants provided their informed consent.
Below is the link to the electronic supplementary material.
Supplementary file1 (PDF 201 KB)
Komori, M., Takemura, K., Minoura, Y. et al. Extracting multiple layers of social networks through a 7-month survey using a wearable device: a case study from a farming community in Japan. J Comput Soc Sc 5, 1069–1094 (2022). https://doi.org/10.1007/s42001-022-00162-y
Issue Date: May 2022
Farming community
Non-negative matrix factorization | CommonCrawl |
\begin{document}
\title{Conjunction of Conditional Events and T-norms} \begin{abstract} We study the relationship between a notion of conjunction among conditional events, introduced in recent papers, and the notion of Frank t-norm. By examining different cases, in the setting of coherence, we show each time that the conjunction coincides with a suitable Frank t-norm. In particular, the conjunction may coincide with the Product t-norm, the Minimum t-norm, and Lukasiewicz t-norm. We show by a counterexample, that the prevision assessments obtained by Lukasiewicz t-norm may be not coherent. Then, we give some conditions of coherence when using Lukasiewicz t-norm. \keywords{Coherence, Conditional Event, Conjunction, Frank t-norm.} \end{abstract} \section{Introduction} In this paper we use the coherence-based approach to probability of de Finetti (\cite{biazzo00,biazzo05,coletti02,CoSV13,CoSV15,defi36,definetti74,gilio02,gilio12ijar,gilio16,gilio13,PfSa19}). We use a notion of conjunction which, differently from other authors, is defined as a suitable conditional random quantity with values in the unit interval (see, e.g. \cite{GiSa13c,GiSa13a,GiSa14,GiSa19,SPOG18}).
We study the relationship between our notion of conjunction and the notion of Frank t-norm. For some aspects which relate probability and Frank t-norm see, e.g., \cite{Tasso12,coletti14FSS,coletti04,Dubois86,Flaminio18,Navara05}. We show that, under the hypothesis of logical independence, if the prevision assessments involved with the conjunction $(A|H) \wedge (B|K)$
of two conditional events are coherent, then the prevision of the conjunction coincides, for a suitable $\lambda \in [0,+\infty]$, with the Frank t-norm $T_\lambda(x,y)$, where $x=P(A|H), y=P(B|K)$. Moreover, $(A|H) \wedge (B|K)= T_\lambda(A|H,B|K)$. Then, we consider the case $A=B$, by determining the set of all coherent assessment $(x,y,z)$ on $\{A|H,A|K, (A|H) \wedge (A|K)\}$. We show that, under coherence, it holds that $(A|H) \wedge (A|K)= T_\lambda(A|H,A|K)$, where $\lambda \in [0,1]$. We also study the particular case where $A=B$ and $HK=\emptyset$. Then, we consider conjunctions of three conditional events and we show that to make prevision assignments by means of the Product t-norm, or the Minimum t-norm, is coherent. Finally, we examine the Lukasiewicz t-norm and we show by a counterexample that coherence is in general not assured. We give some conditions for coherence when the prevision assessments are made by using the Lukasiewicz t-norm.
\section{Preliminary Notions and Results}
In our approach, given two events $A$ and $H$, with $H \neq \emptyset$, the conditional event $A|H$ is looked at as a three-valued logical entity which is true, or false, or void, according to whether $AH$ is true, or $\widebar{A}H$ is true, or $\widebar{H}$ is true. We observe that the conditional probability and/or conditional prevision values are assessed in the setting of coherence-based probabilistic approach.
In numerical terms $A|H$ assumes one of the values $1$, or $0$, or $x$, where $x=P(A|H)$ represents the assessed degree of belief on $A|H$. Then, $A|H=AH+x\widebar{H}$.
Given a family $\mathcal F = \{X_1|H_1,\ldots,X_n|H_n\}$, for each $i \in \{1,\ldots,n\}$ we denote by $\{x_{i1}, \ldots,x_{ir_i}\}$ the set of possible values of $X_i$ when $H_i$ is true; then, for each $i$ and $j = 1, \ldots, r_i$, we set $A_{ij} = (X_i = x_{ij})$.
We set $C_0 = \widebar{H}_1 \cdots \widebar{H}_n$ (it may be $C_0 = \emptyset$); moreover, we denote by $C_1, \ldots, C_m$ the constituents contained in $H_1\vee \cdots \vee H_n$. Hence $\bigwedge_{i=1}^n(A_{i1} \vee \cdots \vee A_{ir_i} \vee \widebar{H}_i) = \bigvee_{h = 0}^m C_h$. With each $C_h,\, h \in \{1,\ldots,m\}$, we associate a vector $Q_h=(q_{h1},\ldots,q_{hn})$, where $q_{hi}=x_{ij}$ if $C_h \subseteq A_{ij},\, j=1,\ldots,r_i$, while $q_{hi}=\mu_i$ if $C_h \subseteq \widebar{H}_i$; with $C_0$ it is associated $Q_0=\mathcal M = (\mu_1,\ldots,\mu_n)$. Denoting by $\mathcal I$ the convex hull of $Q_1, \ldots, Q_m$, the condition $\mathcal M\in \mathcal I$ amounts to the existence of a vector $(\lambda_1,\ldots,\lambda_m)$ such that: $ \sum_{h=1}^m \lambda_h Q_h = \mathcal M \,,\; \sum_{h=1}^m \lambda_h = 1 \,,\; \lambda_h \geq 0 \,,\; \forall \, h$; in other words, $\mathcal M\in \mathcal I$ is equivalent to the solvability of the system $(\Sigma)$, associated with $(\mathcal F,\mathcal M)$,
\begin{equation}\label{SYST-SIGMA} (\Sigma) \quad \begin{array}{ll} \sum_{h=1}^m \lambda_h q_{hi} = \mu_i \,,\; i \in\{1,\ldots,n\} \,, \sum_{h=1}^m \lambda_h = 1,\;\;\lambda_h \geq 0 \,,\; \,h \in\{1,\ldots,m\}\,. \end{array} \end{equation} Given the assessment $\mathcal M =(\mu_1,\ldots,\mu_n)$ on $\mathcal F =
\{X_1|H_1,\ldots,X_n|H_n\}$, let $S$ be the set of solutions $\Lambda = (\lambda_1, \ldots,\lambda_m)$ of system $(\Sigma)$. We point out that the solvability of system $(\Sigma)$ is a necessary (but not sufficient) condition for coherence of $\mathcal M$ on $\mathcal F$. When $(\Sigma)$ is solvable, that is $S \neq \emptyset$, we define: \begin{equation}\label{EQ:I0} \begin{array}{ll} I_0 = \{i : \max_{\Lambda \in S} \sum_{h:C_h\subseteq H_i}\lambda_h= 0\},\;
\mathcal F_0 = \{X_i|H_i \,, i \in I_0\},\;\; \mathcal M_0 = (\mu_i ,\, i \in I_0)\,. \end{array} \end{equation}
For what concerns the probabilistic meaning of $I_0$, it holds that $i\in I_0$ if and only if the (unique) coherent extension of $\mathcal M$ to $H_i|(\bigvee_{j=1}^nH_j)$ is zero. Then, the following theorem can be proved (\cite[Theorem 3]{BiGS08}) \begin{theorem}\label{CNES-PREV-I_0-INT}{\rm [{\em Operative characterization of coherence}]
A conditional prevision assessment ${\mathcal M} = (\mu_1,\ldots,\mu_n)$ on
the family $\mathcal F = \{X_1|H_1,\ldots,X_n|H_n\}$ is coherent if
and only if the following conditions are satisfied: \\
(i) the system $(\Sigma)$ defined in (\ref{SYST-SIGMA}) is solvable; (ii) if $I_0 \neq \emptyset$, then $\mathcal M_0$ is coherent. } \end{theorem} Coherence can be related to proper scoring rules (\cite{BiGS12,GiSa11a,LSA12,LSA15,LaSA18}).
\begin{definition}\label{CONJUNCTION}Given any pair of conditional events $A|H$ and $B|K$, with $P(A|H)=x$ and $P(B|K)=y$, their conjunction
is the conditional random quantity $(A|H)\wedge(B|K)$, with $\mathbb{P}[(A|H)\wedge(B|K)]=z$, defined as \begin{equation}\label{EQ:CONJUNCTION}
(A|H)\wedge(B|K) =\left\{\begin{array}{ll} 1, &\mbox{if $AHBK$ is true,}\\ 0, &\mbox{if $\widebar{A}H\vee \widebar{B}K$ is true,}\\ x, &\mbox{if $\widebar{H}BK$ is true,}\\ y, &\mbox{if $AH\widebar{K}$ is true,}\\ z, &\mbox{if $\widebar{H}\,\widebar{K}$ is true}. \end{array} \right. \end{equation} \end{definition}
In betting terms, the prevision $z$ represents the amount you agree to pay, with the proviso that you will receive the quantity $(A|H)\wedge(B|K)$. Different approaches to compounded conditionals, not based on coherence, have been developed by other authors (see, e.g., \cite{Kauf09,mcgee89}).
We recall a result which shows that Fr\'echet-Hoeffding bounds still hold for the conjunction of conditional events (\cite[Theorem~7]{GiSa14}). \begin{theorem}\label{THM:FRECHET}{\rm
Given any coherent assessment $(x,y)$ on $\{A|H, B|K\}$, with $A,H,B$, $K$ logically independent, $H\neq \emptyset, K\neq \emptyset$, the extension $z = \mathbb{P}[(A|H) \wedge (B|K)]$ is coherent if and only if the following Fr\'echet-Hoeffding bounds are satisfied:
\begin{equation}\label{LOW-UPPER}
\max\{x+y-1,0\} = z' \; \leq \; z \; \leq \; z'' = \min\{x,y\} \,.
\end{equation} }\end{theorem} \begin{remark}
From Theorem \ref{THM:FRECHET}, as the assessment $(x,y)$ on $\{A|H,B|K\}$ is coherent for every $(x,y)\in[0,1]^2$, the set $\Pi$ of coherent assessments $(x,y,z)$ on $\{A|H,B|K,(A|H)\wedge(B|K)\}$ is \begin{equation}\label{EQ:PI2} \begin{small} \Pi=\{(x,y,z): (x,y)\in[0,1]^2, \max\{x+y-1,0\} \leq z\leq \min\{x,y\} \end{small} \}. \end{equation} The set $\Pi$ is the tetrahedron with vertices the points $(1,1,1)$, $(1,0,0)$, $(0,1,0)$, $(0,0,0)$. For other definition of conjunctions, where the conjunction is a conditional event, some results on lower and upper bounds have been given in \cite{SUM2018S}. \end{remark}
\begin{definition}\label{DEF:CONGn} Let be given $n$ conditional events $E_1|H_1,\ldots,E_n|H_n$. For each subset $S$, with $\emptyset\neq S \subseteq \{1,\ldots,n\}$, let $x_{S}$ be a prevision assessment on $\bigwedge_{i\in S} (E_i|H_i)$. The conjunction $\mathcal{C}_{1\cdots n}=(E_1|H_1) \wedge \cdots \wedge (E_n|H_n)$ is defined as \begin{equation}\label{EQ:CF} \begin{array}{lll} \mathcal{C}_{1\cdots n}= \left\{ \begin{array}{llll} 1, &\mbox{ if } \bigwedge_{i=1}^n E_iH_i, \mbox{ is true} \\ 0, &\mbox{ if } \bigvee_{i=1}^n \widebar{E}_iH_i, \mbox{ is true}, \\ x_{S}, &\mbox{ if } \bigwedge_{i\in S} \widebar{H}_i\bigwedge_{i\notin S} E_i{H}_i\, \mbox{ is true}, \; \emptyset \neq S\subseteq \{1,2\ldots,n\}. \end{array} \right. \end{array} \end{equation} \end{definition}
In particular, $\mathcal{C}_1=E_1|H_1$; moreover, for $\mathcal{S}=\{i_1,\ldots,i_k\}\subseteq \{1,\ldots,n\}$, the conjunction $\bigwedge_{i\in S} (E_i|H_i)$ is denoted by $\mathcal{C}_{i_1\cdots i_k}$ and $x_{\mathcal{S}}$ is also denoted by $x_{i_1\cdots i_k}$.
Moreover, if $\mathcal{S}=\{i_1,\ldots,i_k\}\subseteq \{1,\ldots,n\}$, the conjunction $\bigwedge_{i\in S} (E_i|H_i)$ is denoted by $\mathcal{C}_{i_1\cdots i_k}$ and $x_{\mathcal{S}}$ is also denoted by $x_{i_1\cdots i_k}$. In the betting framework, you agree to pay $x_{1\cdots n}=\mathbb{P}( \mathcal{C}_{1\cdots n})$ with the proviso that you will receive: $1$, if all conditional events are true; $0$, if at least one of the conditional events is false; the prevision of the conjunction of that conditional events which are void, otherwise. The operation of conjunction is associative and commutative. We observe that, based on Definition \ref{DEF:CONGn}, when $n=3$ we obtain \begin{equation}\label{EQ:CONJUNCTION3} \begin{small} \begin{array}{lll} \mathcal{C}_{123} =\left\{ \begin{array}{llll} 1, &\mbox{ if } E_1H_1E_2H_2E_3H_3 \mbox{ is true},\\ 0, &\mbox{ if } \widebar{E}_1H_1 \vee \widebar{E}_2H_2 \vee \widebar{E}_3H_3 \mbox{ is true},\\ x_1,& \mbox{ if } \widebar{H}_1E_2H_2E_3H_3 \mbox{ is true},\\ x_2,& \mbox{ if } \widebar{H}_2E_1H_1E_3H_3 \mbox{ is true},\\ x_3, &\mbox{ if } \widebar{H}_3E_1H_1E_2H_2 \mbox{ is true}, \\ x_{12}, &\mbox{ if } \widebar{H}_1\widebar{H}_2E_3H_3 \mbox{ is true}, \\ x_{13}, &\mbox{ if } \widebar{H}_1\widebar{H}_3E_2H_2 \mbox{ is true}, \\ x_{23}, &\mbox{ if } \widebar{H}_2\widebar{H}_3E_1H_1 \mbox{ is true}, \\ x_{123}, &\mbox{ if } \widebar{H}_1\widebar{H}_2\widebar{H}_3 \mbox{ is true}. \\ \end{array} \right. \end{array} \end{small} \end{equation}
We recall the following result (\cite[Theorem 15]{GiSa19}). \begin{theorem}\label{THM:PIFOR3} Assume that the events $E_1, E_2, E_3, H_1, H_2, H_3$ are logically independent, with $H_1\neq \emptyset, H_2\neq \emptyset, H_3\neq \emptyset$. Then, the set $\Pi$ of all coherent assessments $\mathcal{M}=(x_1,x_2,x_3,x_{12},x_{13},x_{23},x_{123})$ on $\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}, \mathcal{C}_{123}\}$ is the set of points $(x_1,x_2,x_3,x_{12},x_{13},x_{23},x_{123})$ which satisfy the following conditions \begin{equation} \small \label{EQ:SYSTEMPISTATEMENT} \left\{ \begin{array}{l} (x_1,x_2,x_3)\in[0,1]^3,\\ \max\{x_1+x_2-1,x_{13}+x_{23}-x_3,0\}\leq x_{12}\leq \min\{x_1,x_2\},\\ \max\{x_1+x_3-1,x_{12}+x_{23}-x_2,0\}\leq x_{13}\leq \min\{x_1,x_3\},\\ \max\{x_2+x_3-1,x_{12}+x_{13}-x_1,0\}\leq x_{23}\leq \min\{x_2,x_3\},\\ 1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}\geq 0,\\
x_{123}\geq \max\{0,x_{12}+x_{13}-x_1,x_{12}+x_{23}-x_2,x_{13}+x_{23}-x_3\},\\ x_{123}\leq \min\{x_{12},x_{13},x_{23},1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}\}. \end{array} \right. \end{equation} \end{theorem} \begin{remark}\label{REM:INEQPI} As shown in (\ref{EQ:SYSTEMPISTATEMENT}), the coherence of $(x_1,x_2,x_3,x_{12},x_{13},x_{23},x_{123})$ amounts to the condition \begin{equation}\label{EQ:INEQPI} \begin{array}{ll} \max\{0,x_{12}+x_{13}-x_1,x_{12}+x_{23}-x_2,x_{13}+x_{23}-x_3\}\,\;\leq \;x_{123}\;\leq \\ \leq\;\; \min\{x_{12},x_{13},x_{23},1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}\}. \end{array} \end{equation} Then, in particular,
the extension $x_{123}$ on $\mathcal{C}_{123}$ is coherent if and only if $x_{123}\in[x_{123}',x_{123}'']$, where
$x_{123}'=\max\{0,x_{12}+x_{13}-x_1,x_{12}+x_{23}-x_2,x_{13}+x_{23}-x_3\}$, $ x_{123}''= \min\{x_{12},x_{13},x_{23},1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}\}.$
\end{remark} Then, by Theorem \ref{THM:PIFOR3} it follows \cite[Corollary 1]{GiSa19} \begin{corollary}\label{COR:PIFOR3} For any coherent assessment $(x_1,x_2,x_3,x_{12},x_{13},x_{23})$ on $\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}\}$
the extension $x_{123}$ on $\mathcal{C}_{123}$ is coherent if and only if $x_{123}\in[x_{123}',x_{123}'']$, where \begin{equation}\label{EQ:INECOR} \begin{array}{ll} x_{123}'=\max\{0,x_{12}+x_{13}-x_1,x_{12}+x_{23}-x_2,x_{13}+x_{23}-x_3\},\\ x_{123}''= \min\{x_{12},x_{13},x_{23},1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}\}. \end{array} \end{equation} \end{corollary} We recall that in case of logical dependencies, the set of all coherent assessments may be smaller than that one associated with the case of logical independence. However (see \cite[Theorem 16]{GiSa19}) the set of coherent assessments is the same when $H_1=H_2=H_3=H$ (where possibly $H=\Omega$; see also \cite[p. 232]{Joe97}) and a corollary similar to Corollary \ref{COR:PIFOR3} also holds in this case. For a similar result based on copulas see \cite{Dura08}.
\section{Representation by Frank t-norms for $(A|H)\wedge(B|K)$} We recall that for every $\lambda \in[0,+\infty]$ the Frank t-norm $T_{\lambda}:[0,1]^2\rightarrow [0,1]$ with parameter $\lambda$ is defined as \begin{equation}\label{EQ:FRANK} \begin{small} T_{\lambda}(u,v)=\left\{\begin{array}{ll} T_{M}(u,v)=\min\{u,v\}, & \text{ if } \lambda=0,\\ T_{P}(u,v)=uv, & \text{ if } \lambda=1,\\ T_{L}(u,v)=\max\{u+v-1,0\}, & \text{ if } \lambda=+\infty,\\ \log_{\lambda}(1+\frac{(\lambda^u-1)(\lambda^v-1)}{\lambda-1}), & \text{ otherwise}. \end{array}\right. \end{small} \end{equation} We recall that $T_{\lambda}$ is continuous with respect to $\lambda$; moreover, for every $\lambda \in[0,+\infty]$, it holds that $T_{L}(u,v)\leq T_{\lambda}(u,v) \leq T_{M}(u,v)$, for every $(u,v)\in[0,1]^2$ (see, e.g., \cite{KlMP00},\cite{KlMe05}). In the next result we study the relation between our notion of conjunction and t-norms. \begin{theorem}\label{THM_TNORM2}
Let us consider the conjunction $(A|H)\wedge(B|K)$, with $A, B, H, K$ logically independent and with $P(A|H)=x$, $P(B|K)=y$. Moreover, given any $\lambda \in[0,+\infty]$, let $T_{\lambda}$ be the Frank t-norm with parameter $\lambda$. Then, the assessment
$z=T_{\lambda}(x,y)$ on $(A|H)\wedge(B|K)$ is a coherent extension of $(x,y)$ on $\{A|H,B|K\}$; moreover $(A|H)\wedge(B|K)=T_{\lambda}(A|H, B|K)$. Conversely, given any coherent extension $z=\mathbb{P}[(A|H)\wedge(B|K)]$ of $(x,y)$, there exists $\lambda \in[0,+\infty]$ such that $z=T_{\lambda}(x,y)$. \end{theorem} \begin{proof}
We observe that from Theorem \ref{THM:FRECHET}, for any given $\lambda$, the assessment $z=T_{\lambda}(x,y)$ is a coherent extension of $(x,y)$ on $\{A|H,B|K\}$. Moreover, from (\ref{EQ:FRANK}) it holds that $T_{\lambda}(1,1)=1$, $T_{\lambda}(u,0)=T_{\lambda}(0,v)=0$, $T_{\lambda}(u,1)=u$, $T_{\lambda}(1,v)=v$. Hence, \begin{equation}\label{EQ:FRANKBIS} \begin{small}
T_{\lambda}(A|H,B|K) =\left\{\begin{array}{ll}
1, &\mbox{ if $AHBK$ is true,}\\
0, &\mbox{ if $\widebar{A}H$ is true or $\widebar{B}K$ is true,}\\
x, &\mbox{ if $\widebar{H}BK$ is true,}\\
y, &\mbox{ if $\widebar{K}AH$ is true,}\\
T_{\lambda}(x,y), &\mbox{ if $\widebar{H}\,\widebar{K}$ is true}, \end{array} \right. \end{small} \end{equation} and, if we choose $z=T_{\lambda}(x,y)$,
from (\ref{EQ:CONJUNCTION}) and (\ref{EQ:FRANKBIS}) it follows that $(A|H)\wedge(B|K)=T_{\lambda}(A|H, B|K)$. \\ Conversely, given any coherent extension $z$ of $(x,y)$, there exists $\lambda$ such that $z=T_{\lambda}(x,y)$. Indeed, if $z=\min\{x,y\}$, then $\lambda=0$; if $z=\max\{x+y-1,0\}$, then $\lambda=+\infty$; if $\max\{x+y-1,0\}<z<\min\{x,y\} $, then by continuity of $T_{\lambda}$ with respect to $\lambda$ it holds that $z=T_{\lambda}(x,y)$
for some $\lambda \in\,]0,\infty[$ (for instance, if $z=xy$, then $z=T_1(x,y)$) and hence $(A|H)\wedge(B|K)=T_{\lambda}(A|H, B|K)$. \qed \end{proof} \begin{remark}
As we can see from (\ref{EQ:CONJUNCTION}) and Theorem \ref{THM_TNORM2}, in case of logically independent events, if the assessed values $x,y,z$ are such that $z=T_{\lambda}(x,y)$ for a given $\lambda$, then the conjunction $(A|H)\wedge (B|K)=T_{\lambda}(A|H,B|K)$. For instance, if $z=T_1(x,y)=xy$, then
$(A|H)\wedge (B|K)=T_{1}(A|H,B|K)=(A|H)\cdot (B|K)$. Conversely, if $(A|H)\wedge (B|K)=T_{\lambda}(A|H,B|K)$ for a given $\lambda$, then $z=T_{\lambda}(x,y)$. Then, the set $\Pi$ given in (\ref{EQ:PI2}) can be written as $\Pi=\{(x,y,z): (x,y)\in[0,1]^2, z=T_{\lambda}(x,y), \lambda \in [0,+\infty]\}.$ \end{remark}
\section{Conjunction of $(A|H)$ and $(A|K)$}
In this section we examine the conjunction of two conditional events in the particular case when $A=B$, that is $(A|H)\wedge (A|K)$. By setting $P(A|H)=x$, $P(A|K)=y$ and $\mathbb{P}[(A|H)\wedge (A|K)]=z$, it holds that \[
(A|H)\wedge (A|K)=AHK+x\widebar{H}AK+y\widebar{K}AH+z\widebar{H}\,\widebar{K}\in\{1,0,x,y,z\}. \]
\begin{theorem}\label{THM:A=B}
Let $A, H, K$ be three logically independent events, with $H\neq \emptyset$, $K\neq \emptyset$. The set $\Pi$ of all coherent assessments $(x,y,z)$ on the family $\mathcal F=\{A|H,A|K,(A|H)\wedge (A|K)\}$ is given by
\begin{equation}\label{EQ:PIA=B}
\Pi=\{(x,y,z): (x,y)\in[0,1]^2, T_P(x,y)= xy\leq z\leq \min\{x,y\}=T_{M}(x,y)\}.
\end{equation} \end{theorem} \begin{proof}
Let $\mathcal M=(x,y,z)$ be a prevision assessment on $\mathcal F$.
The constituents associated with the pair $(\mathcal F,\mathcal M)$ and contained in $H \vee K$ are:
$ C_1=AHK$, $C_2=\widebar{A}HK$, $C_3=\widebar{A}\widebar{H}K$, $C_4=\widebar{A}H\widebar{K}$,
$C_5=A\widebar{H}K$, $C_6=AH\widebar{K}$.
The associated points $Q_h$'s are $Q_1=(1,1,1), Q_2=(0,0,0), Q_3=(x,0,0), Q_4=(0,y,0), Q_5=(x,1,x), Q_6=(1,y,y)$. With the further constituent $C_0=\widebar{H}\widebar{K}$ it is associated the point $Q_0=\mathcal{M}=(x,y,z)$.
Considering the convex hull $\mathcal I$ (see Figure \ref{FIG:IEA1}) of $Q_1, \ldots, Q_6$, a necessary condition for the coherence of the prevision assessment $\mathcal M=(x,y,z)$ on $\mathcal F$ is that $\mathcal M \in \mathcal I$, that is the following system must be solvable
\[
(\Sigma) \left\{
\begin{array}{l}
\lambda_1+x\lambda_3+x\lambda_5+\lambda_6=x,\;\;
\lambda_1+y\lambda_4+\lambda_5+y\lambda_6=y,\;\;
\lambda_1+x\lambda_5+y\lambda_6=z,\\
\sum_{h=1}^6\lambda_h=1,\;\;
\lambda_h\geq 0,\; h=1,\ldots,6.
\end{array}
\right.
\]
First of all, we observe that solvability of $(\Sigma)$ requires that $z\leq x$ and $z\leq y$, that is $z\leq \min\{x,y\}$. We now verify that $(x,y,z)$, with $(x,y)\in[0,1]^2$ and $z=\min\{x,y\}$, is coherent. We distinguish two cases: $(i)$ $x\leq y$ and $(ii)$ $x> y$. \\
Case $(i)$.
In this case $z=\min\{x,y\}=x$. If $y=0$
the system $(\Sigma)$ becomes
\[
\begin{array}{l}
\lambda_1+\lambda_6=0,\;\;
\lambda_1+\lambda_5=0,\;\;
\lambda_1=0,\;
\lambda_2+\lambda_3+\lambda_4=1,\;\;
\lambda_h\geq 0,\;\; h=1,\ldots,6.
\end{array}
\]
which is clearly solvable. In particular there exist solutions with $\lambda_2>0,\lambda_3>0, \lambda_4>0$, by
Theorem \ref{CNES-PREV-I_0-INT},
as the set $I_0$ is empty the solvability of $(\Sigma)$ is sufficient for coherence of the assessment $(0,0,0)$.
If $y>0$ the system $(\Sigma)$ is solvable and a solution is $ \Lambda=(\lambda_1,\ldots, \lambda_6)=(x,\frac{x(1-y)}{y},0,\frac{y-x}{y},0,0)$.
We observe that, if $x>0$, then $\lambda_1>0$ and $I_0=\emptyset$ because $\mathcal{C}_1=HK\subseteq H\vee K$, so that $\mathcal M=(x,y,x)$ is coherent. If $x=0$ (and hence $z=0$), then $\lambda_4=1$ and $I_0\subseteq \{2\}$. Then, as the sub-assessment $P(A|K)=y$ is coherent, it follows that the assessment $\mathcal M=(0,y,0)$ is coherent too.\\
Case $(ii)$. The system is solvable and a solution is $
\Lambda=(\lambda_1,\ldots, \lambda_6)=(y,\frac{y(1-x)}{x},\frac{x-y}{x},0,0,0).$
We observe that, if $y>0$, then $\lambda_1>0$ and $I_0=\emptyset$ because $\mathcal{C}_1=HK\subseteq H\vee K$, so that $\mathcal M=(x,y,y)$ is coherent. If $y=0$ (and hence $z=0$), then $\lambda_3=1$ and $I_0\subseteq \{1\}$. Then, as the sub-assessment $P(A|H)=x$ is coherent, it follows that the assessment $\mathcal M=(x,0,0)$ is coherent too.
Thus, for every $(x,y)\in[0,1]^2$, the assessment $(x,y,\min\{x,y\})$ is coherent and, as $z\leq \min\{x,y\}$, the upper bound on $z$ is $\min\{x,y\}=T_M(x,y)$. \\
We now verify that $(x,y,xy)$, with $(x,y)\in[0,1]^2$ is coherent; moreover we will show that $(x,y,z)$, with $z<xy$, is not coherent, in other words the lower bound for $z$ is $xy$.
First of all, we observe that $\mathcal M=(1-x)Q_4+xQ_6$, so that a solution of $(\Sigma)$ is $\Lambda_1=(0,0,0,1-x,0,x)$.
Moreover, $\mathcal M=(1-y)Q_3+yQ_5$, so that another solution is $\Lambda_2=(0,0,1-y,0,y,0)$. Then $
\Lambda=\frac{\Lambda_1+\Lambda_2}{2}=(0,0,\frac{1-y}{2},\frac{1-x}{2},\frac{y}{2},\frac{x}{2}) $
is a solution of $(\Sigma)$ such that $I_0=\emptyset$. Thus the assessment $(x,y,xy)$ is coherent for every $(x,y)\in[0,1]^2$.
In order to verify that $xy$ is the lower bound on $z$ we observe that the points $Q_3,Q_4,Q_5,Q_6$ belong to a plane $\pi$ of equation:
$yX+xY-Z=xy$, where $X,Y,Z$ are the axis' coordinates. Now, by considering the function $f(X,Y,Z)= yX+xY-Z$, we observe that for each constant $k$ the equation $f(X,Y,Z)=k$ represents a plane which is parallel to $\pi$ and coincides with $\pi$ when $k=xy$. We also observe that
$f(Q_1)=f(1,1,1)=x+y-1=T_L(x,y)\leq xy=T_P(x,y)$,
$f(Q_2)=f(0,0,0)=0 \leq xy=T_P(x,y)$, and
$f(Q_3)=f(Q_4)=f(Q_5)=f(Q_6)= xy=T_P(x,y)$.
Then, for every $\mathcal P=\sum_{h=1}^6\lambda_hQ_h$, with $\lambda_h\geq 0$ and $\sum_{h=1}^6\lambda_h=1$, that is $\mathcal P\in \mathcal I$, it holds that $
f(\mathcal P)=f\big(\sum_{h=1}^6\lambda_hQ_h\big)=\sum_{h=1}^6\lambda_hf(Q_h)\leq xy. $
On the other hand, given any $a>0$, by considering $\mathcal P=(x,y,xy-a)$ it holds that $
f(\mathcal P)=f(x,y,xy-a)=xy+xy-xy+a= xy+a>xy.
$
Therefore, for any given $a>0$ the assessment $(x,y,xy-a)$ is not coherent because $(x,y,xy-a)\notin \mathcal I$. Then, the lower bound on $z$ is $xy=T_{P}(x,y)$. Finally, the set of all coherent assessments $(x,y,z)$ on $\mathcal F$ is the set $\Pi$ in
(\ref{EQ:PIA=B}).
\qed \end{proof}
\begin{figure}
\caption{Convex hull $\mathcal I$ of the points $Q_1, Q_2,Q_3, Q_4, Q_5,Q_6$.
$\mathcal M'=(x,y,z'), \mathcal M''=(x,y,z'')$, where $(x,y)\in[0,1]^2$, $z'=xy$, $z''=\min\{x,y\}$. In the figure the numerical values are: $x=0.35$, $y=0.45$, $z'=0.1575$, and $z''=0.35$.}
\label{FIG:IEA1}
\end{figure} Based on Theorem \ref{THM:A=B}, we can give an analogous version for the Theorem \ref{THM_TNORM2} (when $A=B$). \begin{theorem}\label{THM_TNORM2A=B}
Let us consider the conjunction $(A|H)\wedge(A|K)$, with $A, H, K$ logically independent and with $P(A|H)=x$, $P(A|K)=y$.
Moreover, given any $\lambda \in[0,1]$, let $T_{\lambda}$ be the Frank t-norm with parameter $\lambda$. Then, the assessment
$z=T_{\lambda}(x,y)$ on $(A|H)\wedge(A|K)$ is a coherent extension of $(x,y)$ on $\{A|H,A|K\}$; moreover $(A|H)\wedge(A|K)=T_{\lambda}(A|H, A|K)$. Conversely, given any coherent extension $z=\mathbb{P}[(A|H)\wedge(A|K)]$ of $(x,y)$, there exists $\lambda \in[0,1]$ such that $z=T_{\lambda}(x,y)$. \end{theorem} The next result follows from Theorem \ref{THM:A=B} when $H$, $K$ are incompatible. \begin{theorem}\label{THM:SETPROD}
Let $A, H, K$ be three events, with $A$ logically independent from both $H$ and $K$, with $H\neq \emptyset$, $K\neq \emptyset$, $HK=\emptyset$. The set $\Pi$ of all coherent assessments $(x,y,z)$ on the family $\mathcal F=\{A|H,A|K,(A|H)\wedge (A|K)\}$ is given by $\Pi=\{(x,y,z): (x,y)\in[0,1]^2, z= xy=T_P(x,y)\}.$ \end{theorem} \begin{proof}
We observe that
\[
\begin{small}
(A|H)\wedge (A|K)=\left\{\begin{array}{ll}
0, &\mbox{ if } \widebar{A}\widebar{H}K \vee \widebar{A}H\widebar{K} \mbox{ is true,}\\
x, &\mbox{ if } \widebar{H}AK \mbox{ is true,}\\
y, &\mbox{ if } AH\widebar{K}\mbox{ is true,}\\
z, &\mbox{ if } \widebar{H}\widebar{K} \mbox{ is true.}\\
\end{array}
\right.
\end{small}
\] Moreover, as $HK=\emptyset$, the points $Q_h$'s are $(x,0,0), (0,y,0), (x,1,x), (1,y,y)$, which coincide with the points $Q_3,\ldots, Q_6$ of the case $HK\neq \emptyset$. Then, as shown in the proof of Theorem \ref{THM:A=B}, the condition $\mathcal M=(x,y,z)$ belongs to the convex hull of $(x,0,0), (0,y,0), (x,1,x), (1,y,y)$ amounts to the condition $z=xy$. \qed \end{proof} \begin{remark} From Theorem \ref{THM:SETPROD}, when $HK=\emptyset$ it holds that $
(A|H)\wedge (A|K)=(A|H)\cdot (A|K)=T_P(A|H,A|K), $
where $x=P(A|H)$ and $y=P(A|K)$. \end{remark}
\section{Further Results on Frank t-norms} In this section we give some results which concern Frank t-norms and the family $\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}, \mathcal{C}_{123}\}$. We recall that, given any t-norm $T(x_1,x_2)$ it holds that $T(x_1,x_2,x_3)=T(T(x_1,x_2),x_3)$. \subsection{On the Product t-norm} \begin{theorem}\label{THM:PROD} Assume that the events $E_1, E_2, E_3, H_1, H_2, H_3$ are logically independent, with $H_1\neq \emptyset, H_2\neq \emptyset, H_3\neq \emptyset$. If the assessment $\mathcal{M}=(x_1,x_2,x_3,x_{12},x_{13},x_{23},x_{123})$ on $\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}, \mathcal{C}_{123}\}$ is such that $(x_1,x_2,x_3)\in[0,1]^3$, $x_{ij}=T_{1}(x_i,x_j)=x_ix_j$, $i\neq j$, and $x_{123}=T_{1}(x_1,x_2,x_3)=x_1x_2x_3$, then $\mathcal M$ is coherent. Moreover, $\mathcal{C}_{ij}=T_{1}(\mathcal{C}_i,\mathcal{C}_j)=\mathcal{C}_i\mathcal{C}_j$, $i\neq j$, and $\mathcal{C}_{123}=T_{1}(\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3)=\mathcal{C}_{1}\mathcal{C}_{2}\mathcal{C}_{3}$. \end{theorem} \begin{proof} From Remark \ref{REM:INEQPI}, the coherence of $\mathcal M$ amounts to the inequalities in (\ref{EQ:INEQPI}). As $x_{ij}=T_{1}(x_i,x_j)=x_ix_j$, $i\neq j$, and $x_{123}=T_{1}(x_1,x_2,x_3)=x_1x_2x_3$, the inequalities (\ref{EQ:INEQPI}) become \begin{equation} \begin{array}{ll} \max\{0,x_1(x_2+x_3-1),x_{2}(x_1+x_3-1),x_3(x_1+x_2-1)\}\,\;\leq \;x_{1}x_2x_3\;\leq \\ \leq\;\; \min\{x_{1}x_2,x_{1}x_3,x_{2}x_3,(1-x_1)(1-x_2)(1-x_3)+x_1x_2x_3\}. \end{array} \end{equation} Thus, by recalling that $x_i+x_j-1\leq x_ix_j$, the inequalities are satisfied and hence $\mathcal M$ is coherent. Moreover, from (\ref{EQ:CONJUNCTION}) and (\ref{EQ:CONJUNCTION3}) it follows that $ \mathcal{C}_{ij}=T_{1}(\mathcal{C}_i,\mathcal{C}_j)=\mathcal{C}_i\mathcal{C}_j$, $i\neq j$, and $\mathcal{C}_{123}=T_{1}(\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3)=\mathcal{C}_{1}\mathcal{C}_{2}\mathcal{C}_{3}$.\qed \end{proof} \subsection{On the Minimum t-norm} \begin{theorem}\label{THM:MIN}
Assume that the events $E_1, E_2, E_3, H_1, H_2, H_3$ are logically independent, with $H_1\neq \emptyset, H_2\neq \emptyset, H_3\neq \emptyset$.
If the assessment $\mathcal{M}=(x_1,x_2,x_3,x_{12},x_{13},x_{23},x_{123})$ on
$\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}, \mathcal{C}_{123}\}$ is such that $(x_1,x_2,x_3)\in[0,1]^3$, $x_{ij}=T_{M}(x_i,x_j)=\min\{x_i,x_j\}$, $i\neq j$, and $x_{123}=T_{M}(x_1,x_2,x_3)=\min\{x_1,x_2,x_3\}$, then
$\mathcal M$ is coherent. Moreover, $\mathcal{C}_{ij}=T_{M}(\mathcal{C}_i,\mathcal{C}_j)=\min\{\mathcal{C}_i,\mathcal{C}_j\}$, $i\neq j$, and $\mathcal{C}_{123}=T_{M}(\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3)=\min\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}\}$. \end{theorem} \begin{proof}
From Remark \ref{REM:INEQPI}, the coherence of $\mathcal M$ amounts to the inequalities in (\ref{EQ:INEQPI}). Without loss of generality, we assume that $x_1\leq x_2\leq x_3$. Then
$x_{12}=T_{M}(x_1,x_2)=x_1$,
$x_{13}=T_{M}(x_1,x_3)=x_1$,
$x_{23}=T_{M}(x_2,x_3)=x_2$, and $x_{123}=T_{M}(x_1,x_2,x_3)=x_1$.
The inequalities (\ref{EQ:INEQPI}) become
\begin{equation}\label{EQ:MIN} \begin{array}{ll} \max\{0,x_{1},x_{1}+x_2-x_3\}=x_1\,\;\leq \;x_{1}\;\leq x_1= \min\{x_{1},x_{2},1-x_3+x_{1}\}. \end{array} \end{equation}
Thus, the inequalities are satisfied and hence $\mathcal M$ is coherent. Moreover, from (\ref{EQ:CONJUNCTION}) and (\ref{EQ:CONJUNCTION3}) it follows that $\mathcal{C}_{ij}=T_{M}(\mathcal{C}_i,\mathcal{C}_j)=\min\{\mathcal{C}_i,\mathcal{C}_j\}$, $i\neq j$, and $\mathcal{C}_{123}=T_{M}(\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3)=\min\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}\}$.
\qed \end{proof} \begin{remark} As we can see from
$(\ref{EQ:MIN})$ and Corollary \ref{COR:PIFOR3}, the assessment $x_{123}=\min\{x_1,x_2,x_3\}$ is the unique coherent extension on $\mathcal{C}_{123}$ of the assessment $
(x_1,x_2,x_3,\min\{x_1,x_2\},\min\{x_1,x_3\},\min\{x_2,x_3\})$ on
$\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}\}$. \\ We also notice that, if $\mathcal{C}_1\leq \mathcal{C}_2\leq \mathcal{C}_3$, then $\mathcal{C}_{12}=\mathcal{C}_1$, $\mathcal{C}_{13}=\mathcal{C}_1$, $\mathcal{C}_{23}=\mathcal{C}_2$, and $\mathcal{C}_{123}=\mathcal{C}_1$. Moreover, $x_{12}=x_1$,
$x_{13}=x_1$,
$x_{23}=x_2$, and $x_{123}=x_1$. \end{remark}
\subsection{On Lukasiewicz t-norm} We observe that in general the results of Theorems \ref{THM:PROD} and \ref{THM:MIN} do not hold for the Lukasiewicz t-norm (and hence for any given Frank t-norm), as shown in the example below. We recall that $T_L(x_1,x_2,x_3)=\max\{x_1+x_2+x_3-2,0\}$. \begin{example} The assessment $(x_1,x_2,x_3,T_L(x_1,x_2),T_L(x_1,x_3),T_L(x_2,x_3)$, $T_L(x_1,x_2,x_3))$ on the family $\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}, \mathcal{C}_{123}\}$, with $(x_1,x_2,x_3)=(0.5,0.6,0.7)$ is not coherent. Indeed, by observing that $T_L(x_1,x_2)=0.1$ $T_L(x_1,x_3)=0.2$, $T_L(x_2,x_3)=0.3$, and $T_L(x_1,x_2,x_3)=0$, formula (\ref{EQ:INEQPI}) becomes $ \max\{0, 0.1+0.2-0.5,0.1+0.3-0.6,0.2+0.3-0.7\}\,\;\leq \;0\; \leq\;\; \min\{0.1,0.2,0.3,1-0.5-0.6-0.7+0.1+0.2+0.3\}$,
that is: $ \max\{0, -0.2\}\,\;\leq \;0\;\leq \min\{0.1,0.2,0.3,-0.2\}; $ thus the inequalities are not satisfied and the assessment is not coherent. \end{example} More in general we have \begin{theorem}\label{THM:LUK} The assessment $(x_1,x_2,x_3,T_L(x_1,x_2),T_L(x_1,x_3),T_L(x_2,x_3))$ on the family $\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}\}$, with $T_L(x_1,x_2)>0$, ${T_L(x_1,x_3)>0}$, ${T_L(x_2,x_3)>0}$ is coherent if and only if
$x_1+x_2+x_3-2\geq 0$. Moreover, when $x_1+x_2+x_3-2\geq 0$ the unique coherent extension $x_{123}$ on $\mathcal{C}_{123}$ is $x_{123}=T_L(x_1,x_2,x_3)$. \end{theorem} \begin{proof} We distinguish two cases: $(i)$ $x_1+x_2+x_3-2< 0$; $(ii)$ $x_1+x_2+x_3-2\geq 0$.\\ Case $(i)$. From (\ref{EQ:SYSTEMPISTATEMENT}) the inequality $1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}\geq 0$ is not satisfied because $ 1-x_1-x_2-x_3+x_{12}+x_{13}+x_{23}=x_{1}+x_2+x_3-2<0. $ Therefore the assessment is not coherent.\\ Case $(ii)$. We set $x_{123}=T_L(x_1,x_2,x_3)=x_1+x_2+x_3-2$. Then, by observing that $0<x_i+x_j-1\leq x_1+x_2+x_3-2$, $i\neq j$, formula (\ref{EQ:INEQPI}) becomes $\max\{0,x_{1}+x_2+x_3-2\}\,\;\leq \;x_{1}+x_2+x_3-2\; \leq\;\; \min\{x_{1}+x_2-1,x_{1}+x_3-1,x_{2}+x_3-1,x_1+x_2+x_3-2\}$,
that is: $ \;x_{1}+x_2+x_3-2\;\leq \;x_{1}+x_2+x_3-2\;\leq \;x_{1}+x_2+x_3-2. $ Thus, the inequalities are satisfied and the
assessment $
(x_1,x_2,x_3,T_L(x_1,x_2),T_L(x_1,x_3),T_L(x_2,x_3), T_L(x_1,x_2,x_3))$ on $\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23},\mathcal{C}_{123}\}$
is coherent and the sub-assessment
$
(x_1,x_2,x_3,T_L(x_1,x_2),T_L(x_1,x_3),T_L(x_2,x_3))
$ on
$\mathcal F$ is coherent too.
\qed \end{proof} A result related with Theorem \ref{THM:LUK} is given below. \begin{theorem}
If the assessment $(x_1,x_2,x_3,T_L(x_1,x_2),T_L(x_1,x_3),T_L(x_2,x_3)$, $T_L(x_1,x_2,x_3))$ on the family $\mathcal F=\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}, \mathcal{C}_{12}, \mathcal{C}_{13}, \mathcal{C}_{23}, \mathcal{C}_{123} \}$, is such that $T_L(x_1,x_2,x_3)>0$, then the assessment is coherent. \end{theorem} \begin{proof}
We observe that $T_L(x_1,x_2,x_3)=x_1+x_2+x_3-2>0$; then
$x_i>0$, $i=1,2,3$, and $0<x_i+x_j-1\leq x_1+x_2+x_3-2$, $i\neq j$.
Then formula (\ref{EQ:INEQPI})) becomes:
$\;\;\max\{0,x_{1}+x_2+x_3-2\}\,\;\leq \;x_{1}+x_2+x_3-2\;\leq\\
\leq\;\;
\min\{x_{1}+x_2-1,x_{1}+x_3-1,x_{2}+x_3-1,x_1+x_2+x_3-2\},$
that is: \\ $x_{1}+x_2+x_3-2\;\leq \;x_{1}+x_2+x_3-2\;\leq \;x_{1}+x_2+x_3-2.$ \\ Thus, the inequalities are satisfied and the assessment is coherent. \qed \end{proof} \section{Conclusions} We have studied the relationship between the notions of conjunction and of Frank t-norms.
We have shown that, under logical independence of events and coherence of prevision assessments, for a suitable $\lambda \in [0,+\infty]$ it holds that $\mathbb{P}((A|H) \wedge (B|K))= T_\lambda(x,y)$ and $(A|H) \wedge (B|K)= T_\lambda(A|H,B|K)$. Then, we have considered the case $A=B$, by determining the set of all coherent assessment $(x,y,z)$ on $(A|H,B|K,(A|H) \wedge (A|K))$. We have shown that, under coherence, for a suitable $\lambda \in [0,1]$ it holds that
$(A|H) \wedge (A|K)= T_\lambda(A|H,A|K)$. We have also studied the particular case where $A=B$ and $HK=\emptyset$. Then, we have considered the conjunction of three conditional events and we have shown that the prevision assessments produced by the Product t-norm, or the Minimum t-norm, are coherent. Finally, we have examined the Lukasiewicz t-norm and we have shown, by a counterexample, that coherence in general is not assured. We have given some conditions for coherence when the prevision assessments are based on the Lukasiewicz t-norm. Future work should concern the deepening and generalization of the results of this paper. \\ \ \\ {\bf Acknowledgments}. We thank three anonymous referees for their useful comments.
\end{document} | arXiv |
\begin{document}
\renewcommand{***}{***}
\thispagestyle{empty}
\ArticleName{Infinitesimal Poisson Algebras and Linearization \\ of Hamiltonian Systems} \ShortArticleName{Infinitesimal Poisson Algebra}
\Author{D. Garc\'ia-Beltr\'an~${}^{1^\dag}$, J. C. Ru\'iz-Pantale\'on~${}^{2^\ddag}\,$ and Yu. Vorobiev~${}^{3^{\S}}$}
\AuthorNameForHeading{Garc\'ia-Beltr\'an, Ru\'iz-Pantale\'on, Vorobiev}
\Address{${\,}^{\dag}$~\,CONACyT Research-Fellow, Departamento de Matem\'aticas, Universidad de Sonora, M\'exico} \Address{${\,}^{\ddag}$~Instituto de Matem\'aticas, Universidad Nacional Aut\'onoma de M\'exico, M\'exico} \Address{${\,}^{\S}$~\,Departamento de Matem\'aticas, Universidad de Sonora, M\'exico}
\EmailD{$^{1}$\href{mailto:email@address}{jcpanta\,@im.unam.mx},
$^{2}$\href{mailto:email@address}{dennise.garcia\,@unison.mx},
$^{3}$\href{mailto:email@address}{yurimv\,@guaymas.uson.mx}}
\Abstract{Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides a suitable framework for the study of the Hamiltonization problem for the linearized dynamics along Poisson submanifolds.}
\Keywords{Poisson algebra, Poisson submanifold, Hamiltonian system, linearization, contravariant derivative.}
\Classification{53D17, 37J05, 53C05}
\section{Introduction}
In this paper, we describe a class of Poisson algebras which appear in the context of infinitesimal geometry of Poisson submanifolds, known also as first class constraints \cite{We83, M.C.Marle2000, Zambon}. One of our motivations is to provide a suitable framework for a non-intrinsic Hamiltonian formulation of linearized Hamiltonian dynamics along Poisson submanifolds of nonzero dimension. This question can be viewed as a part of a general Hamiltonization problem for projectable dynamics on fibered manifolds studied in various situations in \cite{Vor04, Vor05, Vorob05, MaRR-91, DavVor-08}. The main feature of our case is that we have to state the Hamiltonization problem in a class of Poisson algebras which do not define any Poisson structures, in general. This situation is related with the problem of the construction of first order approximations of Poisson structures around Poisson submanifolds \cite{Marcut12, Marcut14} which is only well-studied in the case of symplectic leaves \cite{Vor2001, Vor04}.
Let $S$ be an embedded Poisson submanifold of a Poisson manifold \ec{(M,\{,\}_{M})}. Then, for every \,\ec{H \in \Cinf{M}},\, the Hamiltonian vector field \ec{X_{H}} on $M$ is tangent to $S$ and hence can be linearized along $S$. The linearized procedure for \ec{X_{H}} at $S$ leads to a linear vector field \,\ec{\mathrm{var}_{S}X_{H} \in \X{\mathrm{lin}}(E)}\, on the normal bundle of $S$ defined as a quotient vector bundle \,\ec{E=\mathsf{T}_{S}M/\mathsf{T}{S}}.\, In the zero-dimensional case, when \,\ec{S=\{q\}}\, is a \emph{singular point} of the Poisson structure on $M$, the linear vector field \ec{\mathrm{var}_{S}X_{H}} is Hamiltonian relative to the induced Lie-Poisson bracket on \,\ec{E=\mathsf{T}_{q}M}.\, If \,\ec{\dim S > 0},\, then the linearized dynamical model associated to \ec{\mathrm{var}_{S}X_{H}}, called a \textit{first variation system}, does not inherit any natural Hamiltonian structure from the original Hamiltonian system.
This fact gives rise to the so-called Hamiltonization problem for \ec{\mathrm{var}_{S}X_{H}} which is formulated in a class of Poisson algebras on the space of fiberwise affine functions \ec{\Cinf{\mathrm{aff}}(E)} on $E$. In general, this setting can not be extended to the level of Poisson structures on $E$, because of the following observation due to I. M\u{a}rcut \cite{Marcut12}: a first-order local model for the Poisson structure around the Poisson submanifold $S$ does not always exists. For example, a linearized Poisson model exists in the special case when $S$ is a symplectic leaf \cite{Vor2001}.
By using the infinitesimal data of the Poisson submanifold $S$, we introduce a family of Poisson algebras on \ec{\Cinf{\mathrm{aff}}(E)} whose Lie brackets \ec{\{,\}^{\mathscr{L}}} are parameterized by transversals $\mathscr{L}$ of $S$, that is, by subbundles of \ec{\mathsf{T}_{S}M} complementary to $\mathsf{T}{S}$. These algebras are called \emph{infinitesimal Poisson algebras} and, in fact, are independent of $\mathscr{L}$ modulo isomorphisms. For every $\mathscr{L}$, the first variation system defines a derivation of the corresponding Poisson algebra. We derive the following criterion for the existence of a Hamiltonian structure for the first variation system of \ec{X_{H}} relative to the underlying class of Poisson algebras.
\begin{criterion}\label{criterion} If the flow of the Hamiltonian vector field \ec{X_{H}} admits an invariant transversal \,\ec{\mathscr{L} \subset \mathsf{T}_{S}M}\, of the Poisson submanifold $S$,
\begin{equation}\label{I1}
\big(\mathrm{d}_{q}\mathrm{Fl}_{X_{H}}^{t}\big)(\mathscr{L}_{q}) \,=\, \mathscr{L}_{\mathrm{Fl}_{X_{H}}^{t}(q)}, \quad \forall\,q \in S,
\end{equation} then the first variation system \ec{\mathrm{var}_{S}X_{H}} is a Hamiltonian derivation of the corresponding infinitesimal Poisson algebra,
\begin{equation*}
\dlie{\mathrm{var}_{S}X_{H}}(\cdot) \,=\, \{\phi_{H},\cdot\}^{\mathscr{L}},
\end{equation*} for a certain \,\ec{\phi_{H} \in \Cinf{\mathrm{aff}}(E)}.\, The converse is also true. \end{criterion}
In the case, when $S$ is a symplectic leaf, this criterion is valid in a class of Poisson structures around $S$, called coupling Poisson structures \cite{Vor04, Vorob05}. Here, we also give an application of this result to the linearization of Hamiltonian group actions at $S$. An interesting question is to extend such a criterion to general Poisson submanifolds using, for example, an approach developed in \cite{FloresPantaYura}, results of \cite{Marcut12, Marcut14} and the recent unpublished results on the existence of local models by R. Fernandes and I. Marcut (available at \href{http://www.unige.ch/math/folks/nikolaev/assets/files/GP-20200409-RuiFernandes.pdf}{http://www.unige.ch/math/folks/nikolaev/assets/files/GP-20200409-RuiFernandes.pdf}).
The paper is organized as follows. In Section 2, we recall the definitions of Poisson submanifolds and their infinitesimal data. In Section 3, we describe a class of infinitesimal Poisson algebras on the space of fiberwise affine functions \ec{C_{\operatorname{aff}}^{\infty}(E)} and formulate a result on the first order approximation of the original Poisson algebra around a Poisson submanifold. In Section 4, we show that a factorization of the Jacobi identity for the infinitesimal Poisson algebras leads to their parametrization by means of the so-called Poisson triples involving contravariant derivatives. In Section 5, we give a proof of the first order approximation result which is based on a correspondence between the Poisson triples and the transversal subbundles over a Poisson submanifold. In Section 6, we recall a linearization procedure for dynamical systems at an invariant submanifold which gives a class of projectable vector fields on the normal bundle determining the first variation systems. Section 7 is devoted to the Hamiltonization problem for first variation systems over a Poisson submanifold. First, we derive a geometric criterion for the existence of Hamiltonian structures and then, give its analytic version formulated as the solvability condition of an associated linear nonhomogeneous differential equation. Finally, in Section 8, we apply the Hamiltonization criterion to the construction of linearized models for Hamiltonian group actions around symplectic leaves.
\section{Preliminaries}\label{preliminaries}
Here, we recall some facts about Poisson submanifolds; for more details see \cite{We83, M.C.Marle2000, Zambon}.
Let \ec{(M,\Pi)} be a Poisson manifold equipped with a Poisson bivector field \,\ec{\Pi \in \Gamma\wedge^{2}\mathsf{T}{M}}\, and the Poisson bracket
\begin{equation*}
\{f,g\}_{M} \,=\, \Pi(\mathrm{d}{f},\mathrm{d}{g}), \quad f,g \in \Cinf{M}.
\end{equation*} An (immersed) submanifold \,\ec{\iota: S \hookrightarrow M}\, is said to be a \emph{Poisson submanifold} of $M$ if the \emph{Poisson bivector field $\Pi$ is tangent} to $S$:
\begin{equation}\label{Po1}
\Pi_{q} \in \wedge^{2}\mathsf{T}_{q}S, \quad \forall\, q \in S.
\end{equation} This means that $S$ inherits a (unique) Poisson structure \,\ec{\Pi_{S} \in \Gamma\wedge^{2}\mathsf{T}{S}}\, such that the inclusion $\iota$ is a Poisson map. The corresponding Poisson bracket is denoted by
\begin{equation*}
\big\{\bar{f},\bar{g}\big\}_{S} \,:=\, \Pi_{S}\big(\mathrm{d}{\bar{f}},\mathrm{d}{\bar{g}}\big), \quad \bar{f},\bar{g} \in \Cinf{S}.
\end{equation*}
There are several equivalent characterizations of when a submanifold is Poisson. Consider the induced bundle morphism \,\ec{\Pi^{\natural}:\mathsf{T}^{\ast}M \rightarrow \mathsf{T}{M}}\, defined by \,\ec{\alpha \mapsto \Pi^{\natural}(\alpha) := \mathbf{i}_{\alpha}\Pi},\, and denote by \ec{{\mathsf{T}{S}}^{\circ}} the annihilator of $\mathsf{T}{S}$. Then, condition (\ref{Po1}) can be reformulated in one of the following ways:
\begin{equation}\label{Po2}
\Pi^{\natural}\big({\mathsf{T}{S}}^{\circ}\big) = \{0\} \qquad \text{or} \qquad \Pi^{\natural}\big(\mathsf{T}_{S}^{\ast}M\big) \,\subseteq\, \mathsf{T}{S}.
\end{equation}
This implies that every Hamiltonian vector field \,\ec{X_{H} = \Pi^{\natural}\mathrm{d}{H}}\, is \emph{tangent} to $S$. Moreover, if $S$ is an embedded submanifold, then the first condition in (\ref{Po2}) is equivalent to the following: the vanishing ideal \ec{I(S) = \left\{f \in \Cinf{M} \,|\, f|_{S} = 0\right\}} is also an \emph{ideal in the Lie algebra} \ec{(\Cinf{M},\{,\}_{M})}.
Symplectic leaves are the simplest type of Poisson submanifolds. If $S$ is a symplectic leaf of $\Pi$ (i.e., a maximal integral manifold of the characteristic foliation), then \,\ec{\Pi^{\natural}(\mathsf{T}_{S}^{\ast}M) = \mathsf{T}{S}}.\, In this case, the Poisson tensor \ec{\Pi_{S}} is \emph{nondegenerate} and defines a symplectic form \ec{\omega_{S}} on $S$,
\begin{equation}\label{SS}
\omega_{S}^{\flat} \,=\, -\big(\Pi_{S}^{\natural}\big)^{-1}.
\end{equation}
In general, a Poisson submanifold $S$ is the union of open subsets of the symplectic leaves of $\Pi$.
Now, consider the \emph{cotangent Lie algebroid} of the Poisson manifold \ec{(M,\Pi)}:
\begin{equation}\label{cotangentalgebroid}
A \,:=\, \left(\mathsf{T}^{\ast}M, [,]_{A}, \Pi^{\natural}: \mathsf{T}^{\ast}M \rightarrow \mathsf{T}{M} \right),
\end{equation} where
\begin{equation*}
[\alpha,\beta]_{A} \,:=\, \mathbf{i}_{\Pi^{\natural}(\alpha)}\mathrm{d}\beta \,-\, \mathbf{i}_{\Pi^{\natural}(\beta)} \mathrm{d}\alpha \,-\, \mathrm{d}\langle \alpha,\Pi^{\natural}(\beta) \rangle
\end{equation*} is the Lie bracket for 1-forms on $M$.
The key property is that the cotangent Lie algebroid $A$ \eqref{cotangentalgebroid} admits a natural restriction to the Poisson submanifold $S$ in the sense that there exists a Lie algebroid \ec{A_{S}} over $S$,
\begin{equation*}
A_{S} \,:=\, \left(\mathsf{T}_{S}^{\ast}M, [,]_{A_{S}}, \Pi^{\natural}|_{S}:\mathsf{T}_{S}^{\ast}M \rightarrow \mathsf{T}{S} \right),
\end{equation*} such that the restriction map \,\ec{\Gamma\,\mathsf{T}^{\ast}M \rightarrow \Gamma\,\mathsf{T}_{S}^{\ast}M}\, is a \emph{Lie algebra homomorphism}. Here, the restrictions of the Lie bracket and the anchor are well-defined because of the property that the Poisson tensor $\Pi$ is tangent to $S$.
We observe that there exists a short exact sequence of Lie algebroids
\begin{equation*}
0 \longrightarrow {\mathsf{T}{S}}^{\circ} \longrightarrow A_{S} \longrightarrow \mathsf{T}^{\ast}S \longrightarrow 0,
\end{equation*} where \ec{\mathsf{T}^{\ast}S} is the cotangent Lie algebroid of \ec{(S,\Pi_{S})} and \ec{{\mathsf{T}{S}}^{\circ}} is a Lie algebroid with zero anchor. The last fact is a consequence of property (\ref{Po2}) which reads as
\begin{equation*}
{\mathsf{T}{S}}^{\circ} \,\subseteq\, \ker\big(\Pi^{\natural}|_{S}\big).
\end{equation*} It follows also that the annihilator \ec{{\mathsf{T}{S}}^{\circ}} is an \emph{ideal} in \ec{A_{S}}.
So, follow \cite{IKV-98, Marcut12, Marcut14}; by the \emph{infinitesimal data} of the Poisson submanifold $S$ we will mean the restricted Lie algebroid \ec{A_{S}}. In the case when $S$ is a symplectic leaf, \ec{A_{S}} is a transitive Lie algebroid \cite{Mac, Vor04, DVY-12}.
\section{Infinitesimal Poisson Algebras}\label{infinitesimal Poisson algebras}
Suppose we start with an embedded Poisson submanifold \ec{(S,\Pi_{S})} of a Poisson manifold \ec{(M,\Pi)}. By using the infinitesimal data of $S$, our point is to construct a Poisson algebra \ec{P_{1}} which gives a \emph{first-order approximation} to the original one
\begin{equation}\label{P1}
P = \big(\Cinf{M},\cdot,\{,\}_{M}\big)
\end{equation} in some natural sense.
Consider the normal bundle of $S$
\begin{equation*}
E \,:=\, \mathsf{T}_{S}M \,/\, \mathsf{T}{S}, \qquad \pi:E \longrightarrow S,
\end{equation*} and the co-normal (dual) bundle \,\ec{E^{\ast} \rightarrow S}.\, Denote by
\begin{equation}\label{EcProyCoc}
\nu: \mathsf{T}_{S}M \longrightarrow E
\end{equation} the quotient projection.
Consider a $\Cinf{S}$-module \emph{of fiberwise affine} $\Cinf{}$-\emph{functions} on $E$:
\begin{equation*}
\Cinf{\mathrm{aff}}(E) \,:=\, \pi^{\ast}\Cinf{S} \oplus \Cinf{\mathrm{lin}}(E) \,\simeq\, \Cinf{S} \oplus \Gamma{E^{\ast}}.
\end{equation*} So, every element \,\ec{\phi \in \Cinf{\mathrm{aff}}(E)}\, is represented as
\begin{equation*}
\phi \,=\, \pi^{\ast}f + \ell_{\eta} \,\simeq\, f \oplus \eta,
\end{equation*} where \,\ec{f \in \Cinf{S}}\, and \,\ec{\eta \in \Gamma{E^{\ast}}}.\, Here we use the canonical identification \,\ec{\ell:\Gamma{E^{\ast}} \rightarrow \Cinf{\mathrm{lin}}(E)}\, given by \,\ec{\ell_{\eta}(z) = \langle \eta_{\pi(z)},z \rangle},\, for \,\ec{z \in E}.\, First, we remark that \ec{\Cinf{\mathrm{aff}}(E)} is a \emph{commutative algebra} with ``infinitesimal'' multiplication
\begin{equation}\label{LB2}
\phi_{1} \cdot \phi_{2} \,=\, \pi^{\ast}(f_{1}f_{2}) \,+\, \ell_{(f_{1}\eta_{2} + f_{2}\eta_{1})}
\end{equation} or, equivalently,
\begin{equation}\label{LB1}
(f_{1} \oplus \eta_{1}) \cdot (f_{2} \oplus \eta_{2}) \,=\, f_{1}f_{2} \oplus (f_{1}\eta_{2} + f_{2}\eta_{1}).
\end{equation} Let \,\ec{\iota_{0}:S \hookrightarrow E}\, be the zero section of the normal bundle. Then, we have the canonical splitting
\begin{equation}\label{CS1}
\mathsf{T}_{S}E \,=\, \mathsf{T}{S} \oplus E,
\end{equation} and the projection \,\ec{\mathsf{T}_{S}E \rightarrow E}\, along $\mathsf{T}{S}$ whose adjoint gives a vector bundle morphism \,\ec{E^{\ast} \rightarrow \mathsf{T}_{S}^{\ast}E}.\, On the other hand, we have the dual decomposition of (\ref{CS1})
\begin{equation}\label{CS2}
\mathsf{T}_{S}^{\ast}E \,=\, E^{\circ} \oplus {\mathsf{T}{S}}^{\circ},
\end{equation} and the projection \,\ec{\mathrm{pr}:\mathsf{T}_{S}^{\ast}E \rightarrow {\mathsf{T}{S}}^{\circ}}\, along \ec{E^{\circ}}. Then, decomposition (\ref{CS2}) induces the vector bundle isomorphism \,\ec{\chi: E^{\ast} \rightarrow {\mathsf{T}{S}}^{\circ} \varhookrightarrow \mathsf{T}_{S}^{\ast}E}.\, Now, we define a linearization map
\begin{equation*}
\mathrm{Aff}: \Cinf{E} \longrightarrow \Cinf{\mathrm{aff}}(E), \qquad F \,\longmapsto\, \mathrm{Aff}(F) \,=\, \pi^{\ast}f + \ell_{\eta},
\end{equation*}
with \,\ec{f = \iota_{0}^{\ast}F}\, and \,\ec{\eta = \chi^{-1} \circ \mathrm{pr}(\mathrm{d}{F}|_{S})}.\, Here \,\ec{\mathrm{d}{F}|_{S} \in \Gamma\,\mathsf{T}_{S}^{\ast}E}\, is the restricted differential of \,\ec{F \in \Cinf{E}}.\, It is easy to see that $\mathrm{Aff}$ is a homomorphism of commutative algebras.
Now, consider the $\Cinf{S}$-module of \emph{fiberwise linear functions} \ec{\Cinf{\mathrm{lin}}(E)} and the $\Cinf{S}$-module isomorphism
\begin{equation*}
\Cinf{\mathrm{lin}}(E) \,\overset{\ell^{-1}}{\longrightarrow}\, \Gamma{E^{\ast}} \,\overset{\chi}{\longrightarrow}\, \Gamma\,{\mathsf{T}{S}}^{\circ}.
\end{equation*} Then, the bracket on the Lie algebroid \ec{A_{S}} induces an \emph{intrinsic Lie algebra structure} on \ec{\Cinf{\mathrm{lin}}(E)}:
\begin{equation*}
\{\varphi_{1},\varphi_{2}\}^{\operatorname{lin}} \,:=\, \ell\circ\chi^{-1}\left(\big[\chi \circ \ell^{-1}(\varphi_{1}),\chi \circ \ell^{-1}(\varphi_{2})\big]_{A_{S}}\right).
\end{equation*} This bracket together with trivial (zero) multiplication on \ec{\Cinf{\mathrm{lin}}(E)} defines a Poisson algebra structure.
It is useful also to given an alternative description of \ec{\Cinf{\mathrm{lin}}(E)}. Indeed, for any \,\ec{\eta_{1},\eta_{2} \in \Gamma{E^{\ast}}}\, define the bracket
\begin{equation}\label{FLB}
[\eta_{1},\eta_{2}]_{E^{\ast}} \,=\, \chi^{-1}\big(\,[\chi(\eta_{1}),\chi(\eta_{2})]_{A_{S}}\big),
\end{equation} which is $\Cinf{S}$-bilinear. This follows from \eqref{Po2}. Therefore, the co-normal bundle \ec{E^{\ast}} over $S$ inherits from \ec{[,]_{A_{S}}} a fiberwise Lie bracket \,\ec{S \ni q \mapsto [,]_{E_{q}^{\ast}}}\, smoothly varying with \,\ec{q \in S}.\, In other hand, the co-normal bundle \ec{E^{\ast}} is a \emph{bundle of Lie algebras} (not necessarily locally trivial). Moreover, this gives rise to a \emph{Lie-Poisson structure} (a \emph{vertical Lie-Poisson tensor}) on $E$.
\begin{example} If $S$ is a symplectic leaf, then the bundle of Lie algebras \ec{(E^{\ast},[,]_{E^{\ast}})} is \emph{locally trivial} and the corresponding typical fiber is called the \emph{isotropy algebra} of the leaf. \end{example}
So, taking into account that we have two intrinsic Poisson algebras $\Cinf{S}$ and \ec{\Cinf{\mathrm{lin}}(E)} associated with the Poisson submanifold $S$, we arrive at the following definition.
\begin{definition} By an \emph{infinitesimal Poisson algebra (IPA)} we mean a Poisson algebra
\begin{equation}\label{AF1}
\big(\Cinf{\mathrm{aff}}(E )= \pi^{\ast}\Cinf{S} \oplus \Cinf{\mathrm{lin}}(E),\cdot,\{,\}^{\mathrm{aff}}\,\big),
\end{equation} which consists of the commutative algebra \ec{(\Cinf{\mathrm{aff}}(E),\cdot)} in (\ref{LB2}) and a Lie bracket \ec{\{,\}^{\mathrm{aff}}} on \ec{\Cinf{\mathrm{aff}}(E)} satisfying the conditions:
\begin{itemize}
\item [(a)] the natural projection \,\ec{\Cinf{\mathrm{aff}}(E) \rightarrow \Cinf{S}}\, is a Poisson algebra homomorphism,
\item [(b)] for any \,\ec{\varphi_{1},\varphi_{2} \in \Cinf{\mathrm{lin}}(E)},\, we have
\begin{equation*}
\{0 \oplus \varphi_{1},0 \oplus \varphi_{2}\}^{\mathrm{aff}} \,=\, 0 \oplus \{\varphi_{1},\varphi_{1}\}^{\mathrm{lin}}.
\end{equation*}
\end{itemize} \end{definition}
Observe that for any infinitesimal Poisson algebra, we have an short exact sequence of Poisson algebras
\begin{equation*}
0 \longrightarrow \Cinf{\mathrm{lin}}(E) \varlonghookrightarrow \Cinf{\mathrm{aff}}(E) \longrightarrow \Cinf{S} \longrightarrow 0,
\end{equation*} where \ec{\Cinf{\mathrm{lin}}(E)} is an ideal.
To end this section we give a positive answer to the question on the existence of a first order approximation of the Poisson algebra (\ref{P1}) around an embedded Poisson submanifold.
By an exponential map we mean a diffeomorphism \,\ec{\mathbf{e}:E \rightarrow M}\, from the total space of the normal bundle onto a neighborhood of $S$ in $M$ which is identical on $S$, \,\ec{\mathbf{e}|_{S} = \mathrm{id}_{S}},\, and such that the composition
\begin{equation*}
E_{q} \,\varlonghookrightarrow\, \mathsf{T}_{q}E \,\overset{\mathrm{d}_{q}\mathbf{e}}{\longrightarrow}\, \mathsf{T}_{q}M \,\overset{\nu_{q}}{\longrightarrow}\, E_{q}
\end{equation*} is the \emph{identity map} of the fiber \,\ec{E_{q}=\pi^{-1}(q)}\, over \,\ec{q \in S}.\, An exponential map always exists \cite{LibMar-87}.
\begin{theorem}\label{thmIPA} For every (embedded) Poisson submanifold \,\ec{S \subset M}\, and an exponential map \,\ec{\mathbf{e}:E \rightarrow M},\, there exists an infinitesimal Poisson algebra \,\ec{P_{1} = (\Cinf{\mathrm{aff}}(E),\cdot,\{,\}^{\mathrm{aff}})},\, which is a first order approximation to \,\ec{P = (\Cinf{M},\cdot,\{,\}_{M})}\, around the zero section \,\ec{S \hookrightarrow E},\, in the sense that
\begin{equation}\label{AP1}
\{\phi_{1} \circ \mathbf{e}^{-1},\phi_{2} \circ \mathbf{e}^{-1}\}_{M} \circ \mathbf{e} \,=\, \{\phi_{1},\phi_{2}\}^{\mathrm{aff}} + \mathscr{O}_{2},
\end{equation} for all \,\ec{\phi_{1},\phi_{2} \in \Cinf{\mathrm{aff}}(E)}. \end{theorem}
Observe that condition (\ref{AP1}) can be reformulated as follows: the mapping
\begin{equation}\label{AP2}
\operatorname{Aff} \circ\, \mathbf{e}^{\ast}: \Cinf{M} \rightarrow \Cinf{\mathrm{aff}}(E)
\end{equation} is a Poisson algebra homomorphism.
The proof of this fact will be given in the next sections.
\section{Poisson Triples}\label{Poisson triples}
Here, we describe a structure of infinitesimal Poisson algebras by using the notion of a \emph{contravariant derivative} on a vector bundle over a Poisson manifold introduced in \cite{Va90} (see also \cite{Va94, Fer2000}).
Consider the co-normal bundle \ec{E^{\ast}} over the Poisson submanifold \,\ec{S \subset M}.\, Recall that a contravariant derivative $\mathscr{D}$ on \ec{E^{\ast}} consists of $\R{}$-linear operators \,\ec{\mathscr{D}_{\alpha}:\Gamma{E^{\ast}} \rightarrow \Gamma{E^{\ast}}}\, which are $\Cinf{S}$-linear in \,\ec{\alpha \in \Gamma\,\mathsf{T}^{\ast}S}\, and satisfy the Leibniz-type rule
\begin{equation*}
\mathscr{D}_{\alpha}(f\eta) \,=\, f\mathscr{D}_{\alpha}(\eta) \,+\, \big(\dlie{\Pi_{S}^{\natural}(\alpha)}f\big)\eta.
\end{equation*} for \,\ec{f \in \Cinf{S}}, \,\ec{\eta \in \Gamma{E^{\ast}}}.\, The curvature \ec{\mathrm{Curv}^{\mathscr{D}}} of $\mathscr{D}$ is defined as
\begin{equation*}
\mathrm{Curv}^{\mathscr{D}}(\alpha_{1},\alpha_{2}) \,:=\, [\mathscr{D}_{\alpha_{1}},\mathscr{D}_{\alpha_{2}}] - \mathscr{D}_{[\alpha_{1},\alpha_{2}]_{\mathsf{T}^{\ast}S}}.
\end{equation*} Here, \ec{[,]_{\mathsf{T}^{\ast}S}} denotes the Lie bracket for 1-forms on the Poisson manifold \ec{(S,\Pi_{S})}.
\begin{remark} Every covariant derivative (linear connection) \,\ec{\nabla:\Gamma\,\mathsf{T}{S} \times \Gamma{E^{\ast}} \rightarrow \Gamma{E^{\ast}}}\, induces a contravariant derivative $\mathscr{D}$ which is defined as
\begin{equation}\label{CC1}
\mathscr{D}_{\alpha} = \nabla_{\Pi_{S}^{\natural}(\alpha)},
\end{equation} and satisfies the following property:
\begin{equation}\label{CC2}
\Pi_{S}^{\natural}(\alpha) = 0 \quad \Longrightarrow \quad \mathscr{D}_{\alpha} =\, 0.
\end{equation} In general, condition (\ref{CC2}) does not imply the existence of a covariant derivative satisfying (\ref{CC1}) (for more details, see \cite{Fer2000}). \end{remark}
Now, suppose we are given a triple \ec{\big([,]_{E^{\ast}},\mathscr{D},\mathscr{K}\big)} consisting of
\begin{itemize}
\item the fiberwise Lie algebra bracket \ec{[,]_{E^{\ast}}} on \ec{E^{\ast}} given by (\ref{FLB}),
\item a contravariant derivative \,\ec{\mathscr{D}:\Gamma\,\mathsf{T}^{\ast}S \times \Gamma{E^{\ast}} \rightarrow \Gamma{E^{\ast}}}\, on the co-normal bundle \ec{E^{\ast}} over the Poisson manifold \ec{(S,\Pi_{S})},
\item a $\Cinf{S}$-bilinear antisymmetric mapping \,\ec{\mathscr{K}: \Gamma\,\mathsf{T}^{\ast}S \times \Gamma\,\mathsf{T}^{\ast}S \rightarrow \Gamma{E^{\ast}}}.
\end{itemize}
Assume that the triple \ec{\big([,]_{E^{\ast}},\mathscr{D},\mathscr{K}\big)} satisfies the following conditions:
\begin{align}
& \hspace{0.88cm} [\mathscr{D}_{\alpha},\mathrm{ad}_{\eta}] \,=\, \mathrm{ad}_{\mathscr{D}_{\alpha}\eta}, \label{Ja1} \\[0.15cm]
& \hspace{0.88cm} \mathrm{Curv}^{\mathscr{D}}(\alpha,\beta) \,=\, \mathrm{ad}_{\mathscr{K}(\alpha,\beta)}, \label{Ja3} \\[0.15cm]
& \underset{(\alpha,\beta,\gamma)}{\mathfrak{S}} \mathscr{D}_{\alpha}\mathscr{K}(\beta,\gamma) \,+\, \mathscr{K}(\alpha,[\beta,\gamma]_{\mathsf{T}^{\ast}S}) \,=\, 0, \label{Ja2}
\end{align} for all \,\ec{\alpha,\beta,\gamma \in \Gamma\,\mathsf{T}^{\ast}S}, \,\ec{\eta \in \Gamma{E^{\ast}}}.\, Here, \,\ec{\mathrm{ad}_{\eta}(\cdot) := [\eta,\cdot]_{E^{\ast}}}.
\begin{definition} A setup \ec{\big([,]_{E^{\ast}},\mathscr{D},\mathscr{K}\big)} satisfying (\ref{Ja1})-(\ref{Ja2}) is said to be a \emph{Poisson triple} of a Poisson submanifold \ec{(S,\Pi_{S})} in \ec{(M,\Pi)}. \end{definition}
Here we arrive at the basic fact.
\begin{lemma}\label{LemaTripleToAlg} Every Poisson triple \ec{\big([,]_{E^{\ast}},\mathscr{D},\mathscr{K}\big)} of a Poisson submanifold \ec{S \subset M} induces an infinitesimal Poisson algebra \,\ec{(\Cinf{\mathrm{aff}}(E) \simeq \Cinf{S} \oplus \Gamma{E^{\ast}},\cdot,\{,\}^{\mathrm{aff}})}\, with multiplication (\ref{LB1}) and the Lie bracket given by
\begin{equation}\label{PLB1}
\{f_{1} \oplus \eta_{1},f_{2} \oplus \eta_{2}\}^{\mathrm{aff}} \,:=\, \{f_{1},f_{2}\}_{S} \oplus \big( \mathscr{D}_{\mathrm{d}{f_{1}}}\eta_{2} - \mathscr{D}_{\mathrm{d}{f_{2}}}\eta_{1} + [\eta_{1},\eta_{2}]_{E^{\ast}} + \mathscr{K}(\mathrm{d}{f_{1}},\mathrm{d}{f_{2}}) \big).
\end{equation} \end{lemma}
The proof of this fact is a direct verification that conditions (\ref{Ja1})-(\ref{Ja2}) give a factorization of the Jacobi identity for bracket (\ref{PLB1}).
Using formula (\ref{PLB1}), one can show that the converse is also true; that is, each infinitesimal Poisson algebra induces a Poisson triple.
\begin{corollary}\label{corIPAPT} There is a one-to-one correspondence between infinitesimal Poisson algebras and Poisson triples. \end{corollary}
\begin{example} Consider a Poisson triple \ec{\big([,]_{E^{\ast}},\mathscr{D},\mathscr{K}\big)} in the case when the fiberwise Lie algebra on \ec{E^{\ast}} is abelian and the contravariant derivative is flat, \,\ec{[,]_{E^{\ast}} \equiv 0}\, and \,\ec{\mathscr{K} = 0}.\, Then, $\mathscr{D}$ is related with the notion of a Poisson module (see \cite{Bursztyn}) and defines the Lie bracket of the form
\begin{equation*}
\{f_{1} \oplus \eta_{1},f_{2} \oplus \eta_{2}\}^{\mathrm{aff}} \,=\, \{f_{1},f_{2}\}_{S} \oplus \left(\mathscr{D}_{\mathrm{d}{f_{1}}} \eta_{2} - \mathscr{D}_{\mathrm{d}{f_{2}}}\eta_{1}\right).
\end{equation*} \end{example}
\begin{remark} The notion of Poisson triples can be generalize to the more general situation, starting with a module over an abstract Poisson algebra. One can extend Corollary \ref{corIPAPT} to this case by using the correspondence between Poisson algebras and Lie algebroids \cite{Marcut12, DVY-12, DVYu-15}. \end{remark}
\section{Existence of Infinitesimal Poisson Algebra}\label{existence of infinitesimal Poisson algebra}
In this section, we prove the existence of an infinitesimal Poisson algebra structure on the commutative algebra \ec{\Cinf{\mathrm{aff}}(E)} of fiberwise affine functions on the normal bundle $E$ of an embedded Poisson submanifold \ec{(S,\Pi_{S})} in a Poisson manifold \ec{(M,\Pi)}. According to Lemma \ref{LemaTripleToAlg}, it suffices to show that there exists a Poisson triple of $S$.
Pick a splitting
\begin{equation}\label{ND1}
\mathsf{T}_{S}M \,=\, \mathsf{T}{S} \oplus \mathscr{L},
\end{equation} where \,\ec{\mathscr{L} \subset \mathsf{T}_{S}M}\, is a subbundle complementary to $\mathsf{T}{S}$, called a \emph{transversal} of $S$. Consider also the dual decomposition
\begin{equation}\label{ND2}
\mathsf{T}_{S}^{\ast}M \,=\, \mathscr{L}^{\circ} \oplus {\mathsf{T}{S}}^{\circ},
\end{equation} and the quotient projection \,\ec{\nu:\mathsf{T}_{S}M \rightarrow E}\, (\ref{EcProyCoc}). Then, the image of the adjoint morphism \,\ec{\nu^{\ast}:E^{\ast} \rightarrow \mathsf{T}_{S}^{\ast}M}\, is \,\ec{\nu^{\ast}(E^{\ast}) = {\mathsf{T}{S}}^{\circ} \varhookrightarrow \mathsf{T}_{S}^{\ast}M}\, and hence \ec{\nu^{\ast}} gives a vector bundle isomorphism between \ec{E^{\ast}} and \ec{{\mathsf{T}{S}}^{\circ}}. Moreover, decomposition (\ref{ND2}) induces the vector bundle isomorphism \,\ec{\tau_{\mathscr{L}}: \mathsf{T}^{\ast}S \rightarrow \mathscr{L}^{\circ}}.
Denote by \,\ec{\varrho_{\mathscr{L}}:\mathsf{T}_{S}^{\ast}M \rightarrow {\mathsf{T}{S}}^{\circ}}\, the projection along \ec{\mathscr{L}^{\circ}} according to the decomposition (\ref{ND2}).
\begin{lemma}\label{LemaTransversalTriple} Every transversal $\mathscr{L}$ of $S$ induces a Poisson triple
\begin{equation}\label{TR1}
\big(\,[,]_{E^{\ast}},\mathscr{D} = \mathscr{D}^{\mathscr{L}},\mathscr{K} = \mathscr{K}^{\mathscr{L}}\big),
\end{equation} where the contravariant derivative $\mathscr{D}$ and tensor filed $\mathscr{K}$ are given by
\begin{equation}\label{SU1}
\nu^{\ast}(\mathscr{D}_{\alpha}\eta) \,:=\, [\tau_{\mathscr{L}}(\alpha),\nu^{\ast}(\eta)]_{A_{S}},
\end{equation} and
\begin{equation}\label{SU2}
\nu^{\ast}\big(\mathscr{K}(\alpha,\beta)\big) \,:=\, \varrho_{_{\mathscr{L}}}\big([\tau_{\mathscr{L}}(\alpha),\tau_{\mathscr{L}}(\beta)]_{A_{S}}\big),
\end{equation} for all \,\ec{\alpha,\beta \in \Gamma\mathsf{T}^{\ast}S}\, and \,\ec{\eta \in \Gamma{E^{\ast}}}. \end{lemma} \begin{proof} Taking into account that \,\ec{{\mathsf{T}{S}}^{\circ} \subset \mathsf{T}_{S}^{\ast}M}\, is an ideal relative to the Lie bracket \ec{[,]_{A_{S}}}, we get that under the $\mathscr{L}$-dependent identification
\begin{equation}\label{ID1}
\tau_{\mathscr{L}} \oplus \nu^{\ast}: \mathsf{T}^{\ast}S \oplus E^{\ast} \,\longrightarrow\, \mathscr{L}^{\circ} \oplus {\mathsf{T}{S}}^{\circ} = \mathsf{T}_{S}^{\ast}M,
\end{equation} the triple (\ref{TR1}) transforms to the following one
\begin{equation}\label{Tr2}
\big([,]_{{\mathsf{T}{S}}^{\circ}}, \mathscr{D}', \mathscr{K}'\big),
\end{equation} where \,\ec{\mathscr{D}':\Gamma\mathscr{L}^{\circ} \times \Gamma{\mathsf{T}{S}}^{\circ} \rightarrow \Gamma{\mathsf{T}{S}}^{\circ}}\, is a contravariant derivative on the vector bundle \ec{{\mathsf{T}{S}}^{\circ}} given by \,\ec{\mathscr{D}_{\alpha}'\zeta = [\alpha',\zeta]_{A_{S}}},\, for all \,\ec{\alpha' =\tau^{-1}_{\mathscr{L}}(\alpha) \in \mathsf{T}^{\ast}S, \alpha\in \mathscr{L}^{\circ}}\, and \,\ec{\zeta \in {\mathsf{T}{S}}^{\circ}}.\, Moreover, the fiberwise Lie bracket \ec{[,]_{{{\mathsf{T}{S}}^{\circ}}}} and the tensor field $\mathscr{K}'$ take the form
\begin{equation*}
[\zeta_{1},\zeta_{2}]_{{\mathsf{T}{S}}^{\circ}} \,=\, [\zeta_{1},\zeta_{2}]_{A_{S}}, \qquad \mathscr{K}'(\alpha',\beta') \,=\, \varrho_{_{\mathscr{L}}}\big([\alpha',\beta']_{A_{S}} \big).
\end{equation*} By using identification (\ref{ID1}), one can show that the factorization of the Jacobi identity for the bracket \ec{[,]_{A_{S}}} just leads to the relations like (\ref{Ja1})-(\ref{Ja2}) for triple (\ref{Tr2}). So, this implies that the original triple (\ref{TR1}) is Poisson. \end{proof}
Combining the above results, we arrive at the following result on the parametrization of infinitesimal Poisson algebras.
\begin{proposition} Every transversal $\mathscr{L}$ in (\ref{ND1}) induces an infinitesimal Poisson algebra \,\ec{P_{1}^{\mathscr{L}} = (\Cinf{\mathrm{aff}}(E),\cdot, \newline \{,\}^{\mathscr{L}} )},\, where the Lie bracket \ec{\{,\}^{\mathscr{L}}} is defined by formula (\ref{PLB1}) involving the Poisson triple \ec{([,]_{E^{\ast}},\mathscr{D}^{\mathscr{L}},\mathscr{K}^{\mathscr{L}})} (\ref{TR1}). Moreover, the algebra \ec{P_{1}^{\mathscr{L}}} is independent of $\mathscr{L}$ up to isomorphism. \end{proposition} \begin{proof} The first assertion follows from Lemma \ref{LemaTripleToAlg} and Lemma \ref{LemaTransversalTriple}. Next, fixing a transversal $\mathscr{L}$ of $S$, we observe that any another transversal $\widetilde{\mathscr{L}}$, \,\ec{\mathsf{T}_{S}M = \mathsf{T}{S} \oplus \widetilde{\mathscr{L}}}\, is represented as follows
\begin{equation}\label{F1}
\widetilde{\mathscr{L}} \,=\, \{w+\delta(w) \,|\, w \in \mathscr{L}\},
\end{equation} where \,\ec{\delta:\mathscr{L} \rightarrow \mathsf{T}{S}}\, is a vector bundle morphism. On the contrary, for a given $\mathscr{L}$, an arbitrary vector bundle morphism $\delta$ from $\mathscr{L}$ to $\mathsf{T}{S}$ induces a transversal $\widetilde{\mathscr{L}}$ by formula (\ref{F1}). Therefore, we have the following transition rule for the contravariant derivatives \,\ec{\mathscr{D} = \mathscr{D}^{\mathscr{L}}}\, and \,\ec{\widetilde{\mathscr{D}} = \mathscr{D}^{\widetilde{\mathscr{L}}}}\, associated with two transversals $\mathscr{L}$ and $\widetilde{\mathscr{L}}$ of $S$:
\begin{equation}\label{RU1}
\widetilde{\mathscr{D}}_{\alpha} = \mathscr{D}_{\alpha} + \mathrm{ad}_{\mu(\alpha)}.
\end{equation} Here \,\ec{\mu:\mathsf{T}^{\ast}S \rightarrow E^{\ast}}\, is a vector bundle morphism of the form
\begin{equation}\label{RU3}
\mu \,=\, -\big(\nu|_{\mathscr{L}}\big)^{\ast - 1} \circ \delta^{\ast}.
\end{equation} Moreover, for tensor fields \,\ec{\mathscr{K} = \mathscr{K}^{\mathscr{L}}}\, and \,\ec{\widetilde{\mathscr{K}} = \mathscr{K}^{\widetilde{\mathscr{L}}}},\, we also have
\begin{equation}\label{RU2}
\widetilde{\mathscr{K}}(\alpha,\beta) \,=\, \mathscr{K}(\alpha,\beta) \,+\, \mathscr{D}_{\alpha}\mu(\beta) \,-\, \mathscr{D}_{\beta}\mu(\alpha) \mu\big([\alpha,\beta]_{\mathsf{T}^{\ast}S}\big) \,+\, [\mu(\alpha),\mu(\beta)]_{E^{\ast}}.
\end{equation} Finally, by using transition rules (\ref{RU1}), (\ref{RU2}) and by direct computations, we verify that the transformation \,\ec{f \oplus \eta \mapsto f \oplus (\eta + \mu(\mathrm{d}{f}))}\, gives an isomorphism between Poisson algebras \ec{P_{1}^{\mathscr{L}}} and \ec{P_{1}^{\widetilde{\mathscr{L}}}}. \end{proof}
To complete the proof of Theorem \ref{thmIPA}, we observe that for a given exponential map \,\ec{\mathbf{e}:E \rightarrow M},\, the algebra \ec{P_{1}^{\mathscr{L}}} gives a first order approximation to the original one \,\ec{P=\Cinf{M}},\, in the sense of (\ref{AP1}), under the following choice of $\mathscr{L}$:
\begin{equation}\label{CC}
\mathscr{L}_{q} = \big(\mathrm{d}_{q}\mathbf{e}\big)\big(E_{q}\big), \quad \forall\,q \in S.
\end{equation}
\begin{remark}\label{remark:cociente} As was observed in \cite{Marcut12}, the infinitesimal data of $S$ intrinsically induce the Poisson algebra \,\ec{\Cinf{M}/I^{2}(S)}.\, One can show that \,\ec{P^{\mathscr{L}}_{1}}\, is isomorphic to this Poisson algebra. \end{remark}
\section{The Linearization Procedure along Submanifolds}\label{the linearization procedure along submanifolds}
Here, we describe a general linearization procedure for vector fields at invariant submanifolds (see, also \cite{MaRR-91}).
Let $M$ be a $\Cinf{}$ manifold $M$ and \,\ec{S \subset M}\, be an embedded submanifold. Suppose that we are given a vector field $X$ on $M$ which is \emph{tangent} to $S$, \,\ec{X_{q} \in \mathsf{T}_{q}S}, for all \,\ec{q \in M};\, and hence its flow \ec{\mathrm{Fl}_{X}^{t}} leaves $S$ \emph{invariant}. The Lie algebra of such vector fields is denoted by $\X{S}(M)$.
Consider the normal bundle \,\ec{E=\mathsf{T}_{S}M/\mathsf{T}{S}}\, of $S$ with canonical projection \,\ec{\pi:E \rightarrow S}.\, Denote by \ec{\X{\mathrm{lin}}(E)} the Lie algebra of \emph{linear vector fields} on $E$. Each element $V$ of \ec{\X{\mathrm{lin}}(E)} is characterized by the properties: $V$ descends under $\pi$ to a vector field $v$ on $S$, and the Lie derivative $\dlie{V}$ leaves invariant the subspace \ec{\Cinf{\mathrm{lin}}(E)}.
Then, for every linear vector field \,\ec{V \in \X{\mathrm{lin}}(E)},\, the Lie derivative \,\ec{\dlie{V}:\Cinf{\mathrm{aff}}(E) \rightarrow \Cinf{\mathrm{aff}}(E)}\, induces a \emph{derivation} of the commutative algebra \ec{\Cinf{\mathrm{aff}}(E)} with multiplication (\ref{LB2}). It is clear that $\dlie{V}$ leaves invariant the components \ec{\pi^{\ast}\Cinf{S}} and \ec{\Cinf{\mathrm{lin}}(E)} in decomposition (\ref{AF1}).
Denote by \,\ec{\rho_{\varepsilon}:E \rightarrow E}\, the dilation, that is, the fiberwise multiplication on $E$ by a factor \,\ec{\varepsilon > 0}.\, Fix an exponential map \,\ec{\mathbf{e}:E \rightarrow M}\, from the total space onto a neighborhood of $S$ in $M$. Since \,\ec{\mathbf{e}|_{S} = \mathrm{id}_{S}},\, the pullback vector field \ec{\mathbf{e}^{\ast}X} is tangent to the zero section \,\ec{S \subset E}\, and its restriction to $S$ is just the restriction \,\ec{v := X|_{S}}\, of $X$ to $S$.
Denote \,\ec{\mathbf{e}_{\varepsilon} := \mathbf{e} \circ \rho_{\varepsilon}}.\, Then, one can show that the following limit
\begin{equation*}\label{DL1}
\mathrm{var}_{S}X \,:=\, \lim_{\varepsilon \rightarrow 0}\mathbf{e}_{\varepsilon}^{\ast}X \,\in\, \X{\mathrm{lin}}(E)
\end{equation*}
exists and gives a \emph{linear vector field} on $E$ which descends to the restriction \,\ec{v = X|_{S}}, \,\ec{\mathrm{d}\pi \circ \mathrm{var}_{S}X = v \circ \pi},\, and is independent of the choice of an exponential map \ec{\mathbf{e}}. It is clear that the zero section \,\ec{S \hookrightarrow E}\, is an invariant submanifold of the vector field \ec{\mathrm{var}_{S}X} whose restriction to $S$ is just $v$.
The linear dynamical system \ec{(E,\mathrm{var}_{S}X,S)} on the normal bundle $E$ is called the \emph{first variation system} of the vector field $X$ over an invariant submanifold \,\ec{S \subset M}.
Observe that the linear vector field \ec{\mathrm{var}_{S}X} gives a $0$th-\emph{order approximation} to $X$ around the submanifold $S$, in the sense that \,\ec{\mathbf{e}_{\varepsilon}^{\ast}X = \mathrm{var}_{S}X + \mathscr{O}(\varepsilon)}\, as \,\ec{\varepsilon \rightarrow 0}.
Indeed, fix a transversal \,\ec{\mathscr{L} \subset \mathsf{T}_{S}M}\, of $S$ in (\ref{ND1}) and consider the canonical decomposition (\ref{CS1}). Pick an exponential map \,\ec{\mathbf{e}:E \rightarrow M}\, satisfying the compatibility condition (\ref{CC}). Then, we have the expansion
\begin{equation}\label{FVS3}
\mathbf{e}_{\varepsilon}^{\ast}X \,=\, \mathrm{var}_{S}(X) \,+\, \varepsilon\,\mathscr{T} \,+\, \mathscr{O}(\varepsilon^{2}),
\end{equation} where the vector field $\mathscr{T}$ on $E$ is uniquely determined by the choice of a transversal $\mathscr{L}$ in (\ref{ND1}) modulo vertical vector fields on $E$, that is, by elements of \,\ec{\X{V}(E) = \Gamma\,\mathrm{Ver}(E)}.\, Here, \,\ec{\mathrm{Ver}(E) = \ker{\mathrm{d}\pi}}\, is the vertical subbundle of $E$. The image of the vector field $\mathscr{T}$ in (\ref{FVS3}) under the natural projection \,\ec{\X{E} \rightarrow \X{E}/\X{V}(E)}\, is called the \emph{dynamical torsion} of the vector $X$ relative to a transversal $\mathscr{L}$ to the invariant submanifold $S$ and denoted by \ec{\mathrm{tor}_{S}(X,\mathscr{L})}.
Therefore, the first variation system \ec{(E,\mathrm{var}_{S}X,S)} gives a natural \emph{linearized model} for the original dynamical system \ec{(M,X,S)}.
It is also useful to give a coordinate representation for the linearized model. Let \,\ec{(x,y)=(x^{i},y^{a})}\, be a coordinate system on $E$, where \ec{(x^{i})} are coordinates on $S$ and \ec{(y^{a})} are coordinates along the fibers with respect to a basis \ec{(e_{a})} of local sections of $E$. Then,
\begin{equation}\label{FVS8}
v \,=\, v^{i}(x)\frac{\partial}{\partial x^{i}}, \qquad \mathbf{e}^{\ast}X \,=\, X^{i}(x,y)\frac{\partial}{\partial x^{i}} \,+\, X^{a}(x,y)\frac{\partial}{\partial y^{a}},
\end{equation} with \,\ec{X^{i}(x,0) = v^{i}(x)}, \,\ec{X^{a}(x,0)=0}.\, So, we have
\begin{equation*}
\mathrm{var}_{S}X \,=\, v^{i}(x)\frac{\partial}{\partial x^{i}} \,+\, \frac{\partial X^{a}}{\partial y^{b}}\bigg|_{(x,0)}y^{b}\frac{\partial}{\partial y^{a}},
\end{equation*} and
\begin{equation*}
\mathscr{T} \,=\, \frac{\partial X^{i}}{\partial y^{a}}\bigg|_{(x,0)}y^{a}\frac{\partial}{\partial x^{i}} \,+\, \frac{1}{2}\frac{\partial^{2}X^{a}}{\partial y^{b}\partial y^{c}}\bigg|_{(x,0)}y^{b}y^{c}\frac{\partial}{\partial y^{a}}.
\end{equation*} Therefore, locally, the dynamical torsion is represented as
\begin{equation}\label{TOR}
\mathrm{tor}_{S}(X,\mathscr{L}) \,=\, \frac{\partial X^{i}}{\partial y^{a}}\bigg|_{(x,0)}y^{a}\frac{\partial}{\partial x^{i}}.
\end{equation}
Recall that a \emph{transversal} $\mathscr{L}$ of $S$ is said to be $X$-\emph{invariant}, if the subbundle \,\ec{\mathscr{L} \subset \mathsf{T}_{S}M}\, is invariant under the differential of the flow $X$ (condition (\ref{I1})).
The vanishing of the dynamical torsion has the following meaning.
\begin{lemma}\label{LemmaTransversal} A transversal $\mathscr{L}$ of $S$ is $X$-invariant if and only if
\begin{equation}\label{FVS12}
\mathrm{tor}_{S}(X,\mathscr{L}) \,=\, 0.
\end{equation} \end{lemma} \begin{proof} Fixing an exponential map $\mathbf{e}$ satisfying condition (\ref{CC}), let us consider the pull-back vector field \ec{\mathbf{e}^{\ast}X} on $E$. Then, the $X$-invariance of the transversal $\mathscr{L}$ is equivalent to the invariance of the splitting \,\ec{\mathsf{T}_{S}E=\mathsf{T}{S} \oplus E}\, with respect to the flow of \ec{\mathbf{e}^{\ast}X}. In infinitesimal terms, the \ec{\mathbf{e}^{\ast}X}-invariance of the subbundle $E$ of \ec{\mathsf{T}_{S}E} is expressed as follows
\begin{equation}\label{FVS13}
[\mathbf{e}^{\ast}X,Y]_{q} \,\in\, E_{q} \subset \mathsf{T}_{q}E,
\end{equation} for any \,\ec{q \in S}\, and \,\ec{Y \in \X{V}(E)}.\, Taking \,\ec{Y = \frac{\partial}{\partial y^{b}}}\, and by using (\ref{FVS8}), we get
\begin{equation*}
\big[\mathbf{e}^{\ast}X,\tfrac{\partial}{\partial y^{b}}\big] \,=\, -\left(\frac{\partial X^{i}}{\partial y^{b}}{(x,y)}\frac{\partial}{\partial x^{i}} + \frac{\partial X^{a}}{\partial y^{b}}{(x,y)}\frac{\partial}{\partial y^{a}}\right).
\end{equation*}
It follows that, in local terms, condition (\ref{FVS13}) reads \,\ec{{\partial X^{i}}/{\partial y^{b}}\,|_{(x,0)} = 0},\, for \,\ec{b=1,\ldots, \dim S}.\, Comparing this with (\ref{TOR}), we prove (\ref{FVS12}). \end{proof}
We conclude this section with the following observation on the symmetry properties of the linearized dynamics over $S$. It follows from (\ref{FVS3}) that the correspondence
\begin{equation}\label{FVS15}
\X{S}(M) \ni X \,\longmapsto\, \mathrm{var}_{S}X \in \X{\mathrm{lin}}(E)
\end{equation} is a \emph{Lie algebra homomorphism}, \,\ec{\mathrm{var}_{S}[X_{1},X_{2}] = [\mathrm{var}_{S}X_{1},\mathrm{var}_{S}X_{2}]}.
In context of the symmetries of a given vector field $X$ and its first variation system, we have the following consequence: the image under the homomorphism (\ref{FVS15}) of the Lie algebra of vector fields on $M$ which are tangent to $S$ and commute with $X$ belongs to the Lie algebra of linear vector fields on $E$ commuting with \ec{\mathrm{var}_{S}X}.
Moreover, we have the following fact. For every \,\ec{H \in \Cinf{M}},\, denote by \,\ec{H_{\mathscr{L}}^{\mathrm{aff}} \in \Cinf{\mathrm{aff}}(E)}\, its \emph{first-order approximation} around $S$, defined by means of homomorphism (\ref{AP2}),
\begin{equation}\label{LH1}
H_{\mathscr{L}}^{\mathrm{aff}} \,:=\, \mathrm{Aff}(H \circ \mathbf{e}) \,=\, \pi^{\ast}h + \ell_{\eta^{\mathscr{L}}} \,=\, F^{(0)} + F_{\mathscr{L}}^{(1)}.
\end{equation}
Here, \,\ec{h = H|_{S}},
\begin{equation}\label{LH2}
\eta^{\mathscr{L}} =\, \chi^{-1} \circ \mathrm{pr}\big(\mathrm{d}(H \circ \mathbf{e})|_{S}\big),
\end{equation} and an exponential map \,\ec{\mathbf{e}:E \rightarrow M}\, is compatible with a given transversal $\mathscr{L}$ by condition (\ref{CC}).
\begin{lemma} Let \,\ec{F \in \Cinf{M}}\, be a first integral of a vector field \,\ec{X \in \X{S}(M)}.\, Suppose that a transversal $\mathscr{L}$ is $X$-invariant. Then, the fiberwise affine function \ec{F_{\mathscr{L}}^{\mathrm{aff}}} is a first integral of the first variation system \ec{\mathrm{var}_{S}X},
\begin{equation}\label{FI2}
\dlie{\mathrm{var}_{S}X}F^{(0)} \,=\, 0 \qquad \text{and} \qquad \dlie{\mathrm{var}_{S}X}F_{\mathscr{L}}^{(1)} \,=\, 0.
\end{equation} \end{lemma} \begin{proof} The equality \,\ec{\dlie{X}F=0}\, implies that
\begin{equation}\label{FVS19}
\dlie{\mathbf{e}_{\varepsilon}^{\ast}X}\big(\mathbf{e}_{\varepsilon}^{\ast}F\big) \,=\, 0.
\end{equation}
In particular, \,\ec{F^{(0)}=\pi^{\ast}(\iota_{S}^{\ast}F)}\, is a first integral of the restriction \,\ec{v=X|_{S}}.\, On the other hand, by decomposition (\ref{FVS3}) we get
\begin{equation}\label{FVS20}
\dlie{\mathbf{e}_{\varepsilon}^{\ast}X}(\mathbf{e}_{\varepsilon}^{\ast}F) \,=\, \pi^{\ast}\dlie{v}(\iota_{S}^{\ast}F) \,+\, \varepsilon\big(\dlie{\mathrm{var}_{S}X}F_{\mathscr{L}}^{(1)} + \dlie{\mathscr{T}}F^{(0)}\big) \,+\, \mathscr{O}(\varepsilon^{2}).
\end{equation} The $X$-invariance of the transversal $\mathscr{L}$ is equivalent to condition (\ref{FVS12}). This means that the vector field $\mathscr{T}$ is vertical and hence \,\ec{\dlie{\mathscr{T}}(\pi^{\ast}f)=0},\, for any \,\ec{f \in \Cinf{S}}.\, Then, (\ref{FI2}) follows from (\ref{FVS19}), (\ref{FVS20}). \end{proof}
\section{The Hamiltonization Problem}\label{the hamiltonization problem}
As we mentioned above, the linearization of Hamiltonian dynamics at invariant submanifolds may destroy the Hamiltonian property. This feature of the linearization procedure gives rise to the Hamiltonization problem for linearized models around invariant (Poisson) submanifolds. We study this problem in the class of infinitesimal Poisson algebras described in the previous sections.
Let \ec{(S,\Pi_{S})} be an embedded Poisson submanifold of a Poisson manifold \ec{(M,\Pi)}. Let \,\ec{X_{H}=\mathbf{i}_{\mathrm{d}{H}}\Pi}\, be a Hamiltonian vector field on $M$ of a function \,\ec{H \in \Cinf{M}}.\, Then, \ec{X_{H}} is \emph{tangent} to $S$ and its restriction \,\ec{v_{h} = X_{H}|_{S}}\, is a Hamiltonian vector field on \ec{(S,\Pi_{S})}, \,\ec{v_{h}=\mathbf{i}_{\mathrm{d}{h}}\Pi_{S}}\, with \,\ec{h=H|_{S}}.
Consider the first variation system \ec{\mathrm{var}_{S}X_{H}} on the normal bundle $E$ of $S$.
To describe the properties of \ec{\mathrm{var}_{S}X_{H}}, let us fix a transversal $\mathscr{L}$ of $S$ and pick an exponential map \,\ec{\mathbf{e}:E \rightarrow M}\, satisfying (\ref{FVS20}). Then, by Theorem \ref{thmIPA} and Corollary \ref{corIPAPT}, we have the infinitesimal Poisson algebra \ec{(\Cinf{\mathrm{aff}}(E),\cdot,\{,\}^{\mathscr{L}})} associated with a Poisson triple \ec{\big([,]_{E^{\ast}},\mathscr{D}^{\mathscr{L}},\mathscr{K}^{\mathscr{L}}\big)}.
\begin{lemma} The first variation system of \ec{X_{H}} over $S$ is a derivation of the infinitesimal Poisson algebra \ec{(\Cinf{\mathrm{aff}}(E),\cdot,\{,\}^{\mathscr{L}})}, \,\ec{\mathrm{var}_{S}X_{H} \in \mathrm{Der}(\Cinf{\mathrm{aff}}(E))}. \end{lemma}
The next question is to find out under which conditions for the transversal $\mathscr{L}$, the derivation \ec{\mathrm{var}_{S}X_{H}} is Hamiltonian relative to \ec{\{,\}^{\mathscr{L}}}. We formulate the following criterion for the existence of a Hamiltonian structure for the first variation system.
\begin{theorem}\label{TeoHamPoissAlg} The first variation system \ec{\mathrm{var}_{S}X_{H}} is a Hamiltonian derivation of the infinitesimal Poisson algebra \ec{(\Cinf{\mathrm{aff}}(E),\cdot,\{,\}^{\mathscr{L}})} if and only if the transversal $\mathscr{L}$ to the Poisson submanifold $S$ is \ec{X_{H}}-invariant. In this case, \ec{\mathrm{var}_{S}X_{H}} is Hamiltonian relative to the coupling Lie bracket \ec{\{,\}^{\mathscr{L}}} (\ref{PLB1}) on \ec{\Cinf{\mathrm{aff}}(E)} associated to the Poisson triple \ec{\big([,]_{E^{\ast}},\mathscr{D}^{\mathscr{L}},\mathscr{K}^{\mathscr{L}}\big)} and the fiberwise affine function \ec{H_{\mathscr{L}}^{\mathrm{aff}}} in (\ref{LH1}),
\begin{equation}\label{HC1}
\dlie{\mathrm{var}_{S}X_{H}}\phi \,=\, \{H_{\mathscr{L}}^{\mathrm{aff}},\phi\}^{\mathscr{L}}, \quad \forall\,\phi \in \Cinf{\mathrm{aff}}(E).
\end{equation} Moreover, if \,\ec{F \in \Cinf{M}},\, is a first integral of the Hamiltonian system \ec{X_{H}}, then its first order approximation \ec{F_{\mathscr{L}}^{\mathrm{aff}}} is a Poisson commuting first integral of \ec{\mathrm{var}_{S}X_{H}}, \,\ec{\big\{H_{\mathscr{L}}^{\mathrm{aff}},F_{\mathscr{L}}^{\mathrm{aff}}\big\}^{\mathscr{L}} = 0}. \end{theorem}
As consequence of this theorem, we derive Criterion 1.1.
\begin{corollary} The existence of a Hamiltonian structure for the first variation system \ec{\mathrm{var}_{S}X_{H}} is provided by the existence of an invariant splitting (\ref{ND1}) for the original Hamiltonian system. \end{corollary}
We prove Theorem \ref{TeoHamPoissAlg} in few steps.
Given an arbitrary transversal $\mathscr{L}$ and an exponential map $\mathbf{e}$ satisfying condition (\ref{CC}), consider the contravariant derivative \,\ec{\mathscr{D}=\mathscr{D}^{\mathscr{L}}}\, and define the horizontal lift \ec{\mathrm{hor}_{\alpha}^{\mathscr{D}}} of a 1-form \,\ec{\alpha \in \Gamma\,\mathsf{T}^{\ast}S}\, as a linear vector field on $E$ given by \,\ec{\dlie{\mathrm{hor}_{\alpha}^{\mathscr{D}}}\ell_{\eta} = \ell_{\mathscr{D}_{\alpha}\eta}},\, for \,\ec{\eta \in \Gamma{E^{\ast}}}.\, In particular, for \,\ec{\alpha=\mathrm{d}{f}},\, the horizontal lift \ec{\mathrm{hor}_{\mathrm{d}{f}}^{\mathscr{D}}} descends to the Hamiltonian vector field \ec{v_{f}} on $S$. Moreover, consider a vertical bivector field \,\ec{\Lambda \in \Gamma\wedge^{2}\mathrm{Ver}(E)}\, which is fiberwise Lie-Poisson structure associated to the Lie bracket \ec{[\eta_{1},\eta_{2}]_{E^{\ast}}}, \,\ec{\Lambda(\mathrm{d}\ell_{\eta_{1}},\mathrm{d}\ell_{\eta_{2}}) = \ell_{[\eta_{1},\eta_{2}]_{E^{\ast}}}},\, for any \,\ec{\eta_{1},\eta_{2} \in \Gamma{E^{\ast}}}.
\begin{lemma} The first variation system admits the following $\mathscr{L}$-dependent decomposition into horizontal and vertical components
\begin{equation}\label{HF1}
\mathrm{var}_{S}X_{H} \,=\, \mathrm{hor}_{\mathrm{d}{h}}^{\mathscr{D}^{\mathscr{L}}} +\, \mathbf{i}_{\mathrm{d}\ell_{\eta^{\mathscr{L}}}}\Lambda,
\end{equation}
where \,\ec{h = H|_{S}}\, and \,\ec{\eta^{\mathscr{L}} \in \Gamma{E^{\ast}}}\, is defined by (\ref{LH2}). \end{lemma}
Therefore, formula (\ref{HF1}) shows that under a fixed transversal $\mathscr{L}$ of $S$, the first variation system \ec{\mathrm{var}_{S}X_{H}} is uniquely determined by the element \,\ec{h \oplus \eta \in \Cinf{S} \oplus \Gamma{E^{\ast}}},\, which is given in local coordinates as
\begin{equation*}
\eta^{\mathcal{L}} =\, \eta_{a}^{\mathscr{L}}e^{a}, \qquad \eta_{a}^{\mathscr{L}}(x) \,:=\, \frac{\partial(H \circ \mathbf{e})}{\partial y^{a}}{(x,0)},
\end{equation*} where \ec{(e^{a})} is the dual basis of local sections of \ec{E^{\ast}}.
\begin{lemma}\label{LemaIntrisicTorsion} The derivation \ec{\mathrm{var}_{S}X_{H}} is Hamiltonian relative to the Lie bracket \ec{\{,\}^{\mathscr{L}}} and function \ec{H_{\mathscr{L}}^{\mathrm{aff}}}, that is, condition (\ref{HC1}) holds, if and only if the element \ec{h \oplus \eta^{\mathscr{L}}} satisfies the equation
\begin{equation}\label{HSFVS32}
\mathbf{i}_{\mathrm{d}{h}}\mathscr{K}^{\mathscr{L}} - \mathscr{D}^{\mathscr{L}}\eta^{\mathscr{L}} \,=\, 0.
\end{equation} \end{lemma}
This fact follows from the representation (\ref{HF1}) and definition of the Lie bracket \ec{\{,\}^{\mathscr{L}}}.
Now, let us derive a formula for the torsion term in decomposition (\ref{FVS3}) of \ec{\mathbf{e}_{\varepsilon}^{\ast}X_{H}}. In coordinates \,\ec{(x,y) = (x^{i},y^{a})},\, we have
\begin{equation*}
\mathscr{D}_{\mathrm{d}{x^{j}}}^{\mathscr{L}}\big(\eta_{a}e^{a}\big) \,=\, \left(\mathscr{D}_{b}^{ja}\eta_{a} + \psi^{ji}\frac{\partial\eta_{b}}{\partial x^{i}}\right)e^{b}, \qquad \mathscr{K}^{\mathscr{L}}(\mathrm{d} x^{i},\mathrm{d} x^{j}) \,=\, \mathscr{K}_{a}^{ij}e^{a},
\end{equation*} where \,\ec{\Pi_{S}=\frac{1}{2}\psi^{ij}(x)\frac{\partial}{\partial x^{i}} \wedge \frac{\partial}{\partial x^{j}}}\, is the Poisson tensor on $S$. Moreover, by using these relations and definitions (\ref{SU1}), (\ref{SU2}), for the Poisson tensor \ec{\mathbf{e}_{\varepsilon}^{\ast}\Pi} on $E$, we have the following expansions of the pairwise Poisson brackets:
\begin{align}
\{x^{i},x^{j}\}_{E} \,&=\, \psi^{ij}(x) + \varepsilon\,\mathscr{K}_{a}^{ij}(x)y^{a} + \mathscr{O}(\varepsilon^{2}), \label{EcRelat1}\\
\{x^{i},y^{a}\}_{E} \,&=\, \varepsilon\,\mathscr{D}_{b}^{ia}(x)y^{b} + \mathscr{O}(\varepsilon^{2}), \label{EcRelat2} \\
\{y^{a},y^{b}\}_{E} \,&=\, \tfrac{1}{\varepsilon}\lambda_{c}^{ab}(x)y^{c} + \mathscr{O}(1). \label{EcRelat3}
\end{align} By these relations, we compute the term of order $\varepsilon$ in the expansion of \,\ec{\mathbf{e}_{\varepsilon}^{\ast}X_{H} = (\mathbf{e}_{\varepsilon}^{\ast}\Pi)^{\natural}\mathrm{d}(H \circ \mathbf{e}_{\varepsilon})}:
\begin{equation*}
\mathrm{tor}_{S}(X_{H},\mathscr{L}) \,=\, \left( \frac{\partial h}{\partial x^{i}}\mathscr{K}_{b}^{ij} - \psi^{ji}\frac{\partial\eta_{b}}{\partial x^{i}} - \mathscr{D}_{b}^{ja}\eta_{a}\right)y^{b}\frac{\partial}{\partial x^{j}}.
\end{equation*} It follows from here that condition (\ref{HSFVS32}) means that \,\ec{\mathrm{tor}_{S}(X_{H},\mathscr{L})=0}\, and hence by Lemma \ref{LemmaTransversal} it is equivalent to the \ec{X_{H}}-invariance of the transversal $\mathscr{L}$. Applying Lemma \ref{LemaIntrisicTorsion} ends the proof of Theorem \ref{TeoHamPoissAlg}
\begin{example}\label{exm:contraexam} Consider the Lie-Poisson bracket on \,\ec{\operatorname{e}^{\ast}(3)=\mathbb{R}^{6}=\mathbb{R}_{w}^{3}\times\mathbb{R}_{z}^{3}}:
\begin{equation*}
\{w^{i},w^{j}\} \,=\, \epsilon^{ijk}w_{k}, \qquad \{w^{i},z^{j}\} \,=\, \epsilon^{ijk}z_{k}, \qquad \{z_{i},z_{j}\} \,=\, 0.
\end{equation*} The $3$-dimensional submanifold \,\ec{S = \big\{z=0\big\} = \mathbb{R}_{w}^{3}\times\{0\}}\, is a Poisson submanifold where the rank of the Poisson tensor takes values $2$ or $0$. For the transversal $\mathscr{L}$ generated by \ec{{\partial}/{\partial z^{a}}}, \,\ec{a=1,2,3};\, we choose a tubular neighborhood $U$ of $S$ as \,\ec{U = S \times \mathbb{R}_{z}^{3}}\, equipped with coordinates \ec{x=w} and \ec{y=z}. Then, by using relations (\ref{EcRelat1})-(\ref{EcRelat3}), we compute \,\ec{\psi^{ij}(x) = \epsilon^{ijk}x^{k}}\, and the corresponding Poisson triple \,\ec{\mathcal{\mathscr{D}}_{b}^{ia} = \epsilon^{iab}, \mathcal{\mathscr{K}}_{a}^{ij} = 0, \lambda_{c}^{ab} = 0}.\, So, the contravariant derivative $\mathscr{D}$ is flat and the fiberwise Lie algebra is abelian. Moreover, one can show that, in this case, condition (\ref{CC2}) does not hold and hence $\mathscr{D}$ can not be generated by a linear connection in the sense of (\ref{CC1}). \end{example}
\begin{remark} Algebraically, Theorem \ref{TeoHamPoissAlg} is based on the following arguments. As we have mentioned in Remark 5.3, for a given transversal $\mathscr{L}$, the infinitesimal Poisson algebra \,\ec{P^{\mathscr{L}}_{1} = (\Cinf{S} \oplus \Gamma{E^{\ast}}, \cdot ,\{,\}^{\mathscr{L}})} is naturally identified with the quotient Poisson algebra \ec{\Cinf{M} / I^{2}(S)}. Every vector field $X$ on $M$ tangent to $S$ induces a derivation \ec{X^{(2)}} of \ec{\Cinf{M} / I^{2}(S)} because it preserves \ec{I^{2}(S)}. In the case when \,\ec{X=X_{H}}, it holds that \ec{X_{H}^{(2)}} is the Hamiltonian derivation of the element \,\ec{H + I^{2}(S) \in \Cinf{M} / I^{2}(S)}.\, Under the above identification, the derivation \ec{X_{H}^{(2)}} has two components: one that is diagonal acting on \ec{\Cinf{S} \oplus \Gamma{E^{\ast}}}, and one that sends \ec{\Cinf{S}} to \ec{\Gamma{E^{\ast}}} and is induced by the torsion \ec{\mathrm{tor}_{S}(X,\mathscr{L})}. Then, \ec{X_{H}^{(2)}} coincides with \ec{\mathrm{var}_{S}X} if and only if the torsion vanishes. Therefore, the torsionless condition implies that, under the identification \,\ec{H + I^{2}(S) = H^{\mathrm{aff}} = h \oplus \eta^{\mathscr{L}}},\, the derivation \ec{\mathrm{var}_{S}X} is Hamiltonian relative to \ec{H^{\mathrm{aff}}}. \end{remark}
It is useful to reformulate the criterion in Theorem \ref{TeoHamPoissAlg}, as the solvability condition of a global differential equation associated with the infinitesimal data of the submanifold $S$.
By (\ref{HSFVS32}) and the transition rules (\ref{F1}), (\ref{RU1}), (\ref{RU2}), we derive the following criterion.
\begin{proposition} Fix a transversal $\mathscr{L}$ and consider the element \ec{h \oplus \eta^{\mathscr{L}}} representing the first variation system \ec{\mathrm{var}_{S}X_{H}}. If the morphism \,\ec{\mu:\mathsf{T}^{\ast}S \rightarrow E^{\ast}}\, satisfies the equation
\begin{equation}\label{EcHamVar}
\big(\mathbf{i}_{\mathrm{d}{h}} \circ \mathscr{D}^{\mathscr{L}} + \mathrm{ad}_{\eta} + \mathscr{D}^{\mathscr{L}} \circ \mathbf{i}_{\mathrm{d}{h}}\big)(\mu) \,=\, \mathscr{D}^{\mathscr{L}}\eta \,-\, \mathbf{i}_{\mathrm{d}{h}}\mathscr{K}^{\mathscr{L}},
\end{equation} then \ec{\mathrm{var}_{S}X_{H}} is a Hamiltonian derivation with respect to the Poisson bracket \ec{\{,\}^{\mathscr{\widetilde{L}}}} associated to the transversal given by \,\ec{\widetilde{\mathscr{L}} = (\mathrm{id}+\delta)(\mathscr{L})},\, where a vector bundle morphism \,\ec{\delta:\mathscr{L} \rightarrow \mathsf{T}{S}}\, is defined in (\ref{RU3}). The corresponding Hamiltonian is given by \,\ec{H_{\widetilde{\mathscr{L}}}^{\operatorname{aff}} = \pi^{\ast}h + \ell_{(\eta^{\mathscr{L}}-\mu(v_{h}))}}. \end{proposition}
Taking into account the relation
\begin{equation*}
\big(\mathscr{D}^{\mathscr{L}}\mu\big)(\alpha_{1},\alpha_{2}) \,=\, \mathscr{D}_{\alpha_{1}}^{\mathscr{L}}\mu(\alpha_{2}) \,-\, \mathscr{D}_{\alpha_{2}}^{\mathscr{L}}\mu(\alpha_{1}) \,-\, \mu\big([\alpha_{1},\alpha_{2}]_{\mathsf{T}^{\ast}S}\big),
\end{equation*} for \,\ec{\alpha_{1},\alpha_{2} \in \Gamma\,\mathsf{T}^{\ast}S},\, we represent equation (\ref{EcHamVar}) for $\mu$ in the intrinsic form
\begin{equation}\label{1}
\mathscr{D}_{\mathrm{d}{h}}^{\mathscr{L}} \circ \mu \,-\, \mu \circ \mathrm{L}_{v_{h}} +\, \mathrm{ad}_{\eta} \circ \mu \,=\, \mathscr{D}^{\mathscr{L}}\eta^{\mathscr{L}} \,-\, \mathbf{i}_{\mathrm{d}{h}}\mathscr{K}^{\mathscr{L}}.
\end{equation} Locally, this equation can be rewritten in terms of (local) vector fields \,\ec{\mu_{a} = \mu_{a}^{i}(x)\frac{\partial}{\partial x^{i}}}\, on $S$ as follows
\begin{equation}\label{2}
[v_{h},\mu_{b}] \,+\, \big(\mathbf{i}_{\mathrm{d}{h}}\mathscr{D}_{b}^{a} - \lambda_{b}^{ac}\eta_{c}\big)\mu_{a} \,=\, -\, \Pi_{S}^{\natural}\mathrm{d}_{S}\eta_{b} \,+\, \eta_{a}\mathscr{D}_{b}^{a} \,-\, \mathbf{i}_{\mathrm{d}{h}}\mathscr{K}_{b},
\end{equation} where \,\ec{\mathscr{D}_{b}^{a}=\mathscr{D}_{b}^{ia}\frac{\partial}{\partial x^{i}}}\, and \,\ec{\mathscr{K}_{b} = \frac{1}{2}\mathscr{K}_{b}^{ij}\frac{\partial}{\partial x^{i}} \wedge \frac{\partial}{\partial x^{j}}}.\, If the normal bundle of $S$ is trivial, then one can think of equations (\ref{2}) as a global matrix representation of (\ref{1}).
Finally, consider the case when a given contravariant derivative \,\ec{\mathscr{D} = \mathscr{D}^{\mathscr{L}}}\, admits representation (\ref{CC1}) for a certain covariant derivative \,\ec{\nabla:\Gamma\,\mathsf{T}{S} \times \Gamma{E^{\ast}} \rightarrow \Gamma{E^{\ast}}}.\, Assume also that there exists a vector valued 2-form \,\ec{\mathscr{R}\in\Omega^{2}(S;E^{\ast})}\, such that the tensor field \,\ec{\mathscr{K}=\mathscr{K}^{\mathscr{L}}}\, is represented as
\begin{equation*}
\mathscr{K}(\alpha_{1},\alpha_{2}) \,=\, \mathscr{R}\big(\Pi_{S}^{\natural}\alpha_{1},\Pi_{S}^{\natural}\alpha_{2}\big),
\end{equation*} for \,\ec{\alpha_{1},\alpha_{2}\in\Gamma\,\mathsf{T}^{\ast}S}.\, Then, we have the following covariant version of equation (\ref{1}).
\begin{proposition} If a vector valued 1-form \,\ec{\vartheta \in \Omega^{1}(S;E^{\ast})}\, satisfies the equation
\begin{equation}\label{4}
\nabla_{v_{h}}\vartheta \,-\, \vartheta \circ \mathrm{L}_{v_{h}} + [\eta^{\mathscr{L}},\vartheta]_{E^{\ast}} \,=\, \nabla\eta^{\mathscr{L}} - \mathbf{i}_{v_{h}}\mathscr{R},
\end{equation} then \,\ec{\mu = \vartheta \circ \Pi_{S}^{\natural}}\, is a solution to (\ref{1}). \end{proposition}
Therefore, under above assumptions, the solvability of (\ref{4}) gives a sufficient condition for the Hamiltonization of the first variation system in the class of infinitesimal Poisson algebras.
In the case when $S$ is a symplectic leaf, the Poisson tensor \ec{\Pi_{S}} is nondegenerate and the solvability conditions for (\ref{1}) and (\ref{4}) are equivalent. The solvability of (\ref{4}) guaranties the existence of a Hamiltonian structure for \ec{\mathrm{var}_{S}X_{H}} in the class of coupling Poisson structures on $E$ \cite{Vor2001, Vor04}.
\section{The Case of a Symplectic Leaf}\label{the case of a symplectic leaf}
Let \ec{(S,\omega_{S})} be an embedded symplectic leaf of \ec{(M,\Pi)}. So, the Poisson tensor \ec{\Pi_{S}} is nondegenerate and induces the symplectic form \ec{\omega_{S}} on $E$ defined by (\ref{SS}). As we mentioned above, in this case the Hamiltonization criterion for the first variation system \ec{\mathrm{var}_{S}X_{H}} can be formulated in a class of Poisson structures \cite{Vor05, Vor04}. First, we observe that contravariant derivative \ec{\mathscr{D}^{\mathscr{L}}} induces a covariant derivative \,\ec{\nabla = \nabla^{\mathscr{L}}}\, on \ec{E^{\ast}} given by (\ref{CC1}). Then, the adjoint derivative \ec{(\nabla^{\mathscr{L}})^{\ast}} is a \emph{linear Poisson connection} on the normal bundle \ec{(E,\Lambda)}. Introducing the following antisymmetric mapping \,\ec{\sigma^{\mathscr{L}}:\Gamma\,\mathsf{T}{M} \times \Gamma\,\mathsf{T}{M} \rightarrow \Cinf{\mathrm{aff}}(E)},
\begin{equation}\label{EcSigmaL}
\sigma^{\mathscr{L}}(u_{1},u_{2}) \,:=\, \omega_{S}(u_{1},u_{2}) \,+\, \ell \circ \mathscr{K}^{\mathscr{L}}\big((\Pi_{S}^{\natural})^{-1}u_{1},(\Pi_{S}^{\natural})^{-1}u_{2}\big),
\end{equation} we arrive at the following fact \cite{Vor2001}: in a neighborhood of the zero section \,\ec{S \hookrightarrow E},\, every transversal $\mathscr{L}$ induces a Poisson tensor \ec{\Pi^{\mathscr{L}}} defined as a \emph{coupling Poisson structure} associated with the geometric data \ec{((\nabla^{\mathscr{L}})^{\ast},\sigma^{\mathscr{L}},\Lambda)}.
Remark that in general, the coupling Lie bracket \ec{\{,\}^{\mathscr{L}}} gives only a first-order approximation to the coupling Poisson structure \,\ec{\Pi^{\mathscr{L}} = \Pi^{\mathscr{L}}_{H} + \Lambda}\, in the sense that (see also \cite{Vor2001, Vor04})
\begin{equation*}
\Pi^{\mathscr{L}}(\mathrm{d}\phi_{1},\mathrm{d}\phi_{2}) \,=\, \{\phi_{1},\phi_{2}\}^{\mathscr{L}} + \mathscr{O}_{2}.
\end{equation*} Here, \ec{\Pi^{\mathscr{L}}_{H}} is the \ec{(\nabla^{\mathscr{L}})^{\ast}}-horizontal part uniquely defined by \ec{\sigma^{\mathscr{L}}}. One can show that the remainder in this equality vanishes if the zero curvature condition holds, \,\ec{\mathscr{K}^{\mathscr{L}} \equiv 0}.\, In this case, the Lie bracket \ec{\{,\}^{\mathscr{L}}} is canonically extended to a Poisson structure defined around the leaf $S$.
So, in the symplectic case, we have the following version of Theorem \ref{TeoHamPoissAlg}. \cite{Vor05}.
\begin{theorem}\label{TeoVarHam} If a transversal $\mathscr{L}$ is \ec{X_{H}}-invariant, then \ec{\mathrm{var}_{S}X_{H}} is a \emph{Hamiltonian vector field} on $E$ relative to the coupling Poisson structure \ec{\Pi^{\mathscr{L}}} and the affine function \ec{H_{\mathscr{L}}^{\mathrm{aff}}},
\begin{equation}\label{EcVarXH}
\mathrm{var}_{S}X_{H} \,=\, \mathbf{i}_{\mathrm{d}{H}_{\mathscr{L}}^{\mathrm{aff}}}\Pi^{\mathscr{L}}.
\end{equation} \end{theorem} \begin{proof} Consider the coupling Poisson tensor \ec{\Pi^{\mathscr{L}}} associated to the data \,\ec{(\nabla^{\ast} = (\nabla^{\mathscr{L}})^{\ast}, \sigma = \sigma^{\mathscr{L}}, \Lambda)},
\begin{equation*}
\Pi^{\mathscr{L}} = -\tfrac{1}{2}\sigma^{ij}\,\mathrm{hor}_{i}^{\nabla^{\ast}} \wedge \mathrm{hor}_{j}^{\nabla^{\ast}} + \Lambda, \qquad i,j = 1, \ldots, m.
\end{equation*} Here, \,\ec{\sigma^{is}\sigma_{sj} = \delta^{i}_{j}}, and \ec{\sigma_{ij}} are the components of the coupling form $\sigma$. Then, using the representation (\ref{HF1}) for \ec{\mathrm{var}_{S}X_{H}} and the relationship (\ref{EcSigmaL}) between \ec{\sigma^{\mathscr{L}}} and \ec{\mathscr{K}^{\mathscr{L}}}, by direct computation, we verify that condition (\ref{EcVarXH}) for \,\ec{H_{\mathscr{L}}^{\mathrm{aff}} = \pi^{\ast}h + \ell_{\eta}}\, is just equivalent to the equation (\ref{HSFVS32}) for \ec{h \oplus \eta^{\mathscr{L}}}. This fact together with Theorem \ref{TeoHamPoissAlg} and Lemma \ref{LemaIntrisicTorsion} ends the proof of the theorem. \end{proof}
Finally, we formulate the following consequence of this result for the existence of linearized models of Hamiltonian group actions. Let \,\ec{\Phi:G \times M \rightarrow M}\, be a \emph{canonical action} of a connected Lie group $G$ on a Poisson manifold \ec{(M,\Pi)}, with a momentum map \,\ec{\mathrm{J}:M \rightarrow \mathfrak{g}^{\ast}},
\begin{equation*}
X_{a}\big|_{m} \,=\,\frac{\mathrm{d}}{\mathrm{d} t}\bigg|_{t=0}\big[\Phi_{\exp(ta)}(m)\big] \,=\, \Pi^{\natural}\mathrm{d}\mathrm{J}_{a}\big|_{m}, \quad \forall\,a \in \mathfrak{g}.
\end{equation*} Then, the $G$-action leaves invariant a given (embedded) symplectic leaf \,\ec{S \subset M}\, and hence on the normal bundle \,\ec{\pi:E \rightarrow S},\, there exists an induced linearized $G$-action \,\ec{\varphi_{g}:E \rightarrow E}\, defined by
\begin{equation*}
\big(\nu_{g \cdot m}\big)(\mathrm{d}_{m}\Phi_{g}) \,=\, \varphi_{g} \cdot \nu_{m}, \quad m \in S,
\end{equation*} where \,\ec{\nu:\mathsf{T}_{S}M \rightarrow E}\, is the quotient projection.
\begin{theorem}\label{TeoAction} If the $G$-action is proper, then there exists a $G$-invariant transversal \,\ec{\mathscr{L} \subset \mathsf{T}_{S}M}\, of $S$, and in a $G$-invariant neighborhood of $S$ in $E$, the \emph{linearized} $G$-action $\varphi$ is \emph{canonical} relative to the coupling Poisson structure \ec{\Pi^{\mathscr{L}}} with fiberwise affine momentum map \,\ec{\mathrm{j}:E \rightarrow \mathfrak{g}^{\ast}}:
\begin{equation*}
\mathrm{var}_{S}X_{a} \,=\, \frac{\mathrm{d}}{\mathrm{d} t}\bigg|_{t=0}\big[\varphi_{\exp(ta)}\big] \,=\, \Pi^{\mathscr{L}}\mathrm{d}\,\mathrm{j}_{a},
\end{equation*} where \,\ec{\mathrm{j}_{a}=\mathrm{Aff}(\mathrm{J}_{a} \circ \mathbf{e}) \in \Cinf{\mathrm{aff}}(E)}. \end{theorem}
The proof follows from Theorem \ref{TeoVarHam} and the fact \cite{DuKo00}: each proper action of a Lie group $G$ admits a $G$-invariant Riemannian metric on $M$. Then, a $G$-invariant transversal $\mathscr{L}$ is defined as the orthogonal complement to $\mathsf{T}{S}$ in \ec{\mathsf{T}_{S}M}.
Notice that the assertion of Theorem \ref{TeoAction} is true when the Lie group $G$ is compact, since in this case, the action is proper.
\end{document} | arXiv |
Srinivasan Natarajan
Volume 111 Issue 5 October 1999 pp 627-637 Inorganic And Analytical
A synthetic iron phosphate mineral, spheniscidite, [NH4]+[Fe2(OH)(H2O)(PO4)2]−H2O, exhibiting reversible dehydration
Amitava Choudhury Srinivasan Natarajan
Spheniscidite, a synthetic iron phosphate mineral has been synthesized by hydrothermal methods. The material is isotypic with another iron phosphate mineral, leucophosphite. Spheniscidite crystallizes in the monoclinic spacegroupP21/n. (a=9·845(1),b=9·771(3),c=9·897(1),β=102·9°,V=928·5(1),Z=4,M=372·2,dcalc=2·02 g cm−3 andR=0·02). The structure consists of a network of FeO6 octahedra vertex-linked with PO4 tetrahedra forming 8-membered one-dimensional channels in which the NH4+ ions and H2O molecules are located. The material exhibits reversible dehydration and good adsorption behaviour. Magnetic susceptibility measurements indicate that the solid orders antiferromagnetically.
Inorganic-organic hybrid framework solids
Recent developments in the area of hybrid structures are overviewed with special emphasis on iron phosphate-oxalate materials. The structure of the iron phosphate-oxalates consists of iron phosphate chains or layers that are connected by oxalate moieties completing the architecture. The compounds exhibit interesting magnetic properties originating from the super-exchange interactions that are predominantly anti-ferromagnetic, involving the iron phosphates and the oxalate moieties. One of the materials,IV, also exhibits interesting adsorptive properties reminiscent of aluminosilicate zeolites. The aluminum phosphate-oxalate,VII, indicates that hybrid structures can be formed with zeolite architecture.
Synthesis of a 4-membered ring zinc phosphate monomer and its condensation self-assembly into an open-framework structure
S Neeraj Srinivasan Natarajan C N R Rao
Hydrothermal synthesis and structure of framework cobalt phosphates
Amitava Choudhury Srinivasan Natarajan C N R Rao
Synthesis and characterization of submicron-sized mesoporous aluminosilicate spheres
Gautam Gundiah M Eswaramoorthy S Neeraj Srinivasan Natarajan C N R Rao
Mesoporous aluminosilicate spheres of 0.3–0.4 Μm diameter, with different Si/Al ratios, have been prepared by surfactant templating. Surface area of these materials is in the 510–970 m2 g-1 range and pore diameter in the 15–20 å range.
Volume 115 Issue 5-6 October 2003 pp 573-586
A two-dimensional yttrium phthalate coordination polymer, [Y4(H2O)2(C8H4O4)6]∞, exhibiting different coordination geometries
A Thirumurugan Srinivasan Natarajan
A hydrothermal reaction of a mixture of Y(NO3)3, 1,2-benzenedicarboxylic acid (1,2-BDC) and NaOH gives rise to a new yttrium phthalate coordination polymer, [Y4H2O2C8H4O4)6]∞,I. The Y ions inI are present in four different coordination environments with respect to the oxygen atoms (CN6 = octahedral, CN7 = pentagonal bipyramid, CN8 = dodecahedron and CN9 = capped square anti-prism). The oxygen atoms of the 1,2-BDC are fully deprotonated, and show variations in their connectivity with Y atoms. The Y atoms themselves are connected through their vertices forming infinite Y-O-Y one-dimensional chains. The Y-O-Ychains are cross-linked by the 1,2-BDC anions forming a corrugated layer structure. The layers are supported by favourableπ…π interactions between the benzene rings of the 1,2-BDC anions. The variations in the coordination environment of the Y atoms and the presence of Y-O-Y interactions along with the favourableπ…π interactions between the benzene rings from different layers are noteworthy structural features. Crystal data: triclinic, space group =P−1 (no. 2),a = 12.6669 (2),b = 13.8538 (2),c = 16.0289 Å,α = 75.20 (1),β = 69.012 (1),γ= 65.529 (1)°,V = 2371.28 (7) Å3,Dcalc = 1.922 g cm−1, μ(MoKα) = 4.943 mm−1. A total of 9745 reflections collected and merged to give 6566 unique reflections (Rint = 0.0292) of which 5252 withI>2σ(I) were considered to be observed. FinalR2 = 0.0339,wR2 = 0.0724 andS = 1.036 were obtained for 704 parameters.
Volume 116 Issue 2 March 2004 pp 65-69
A lanthanum pyromellitate coordination polymer with three-dimensional structure
S V Ganesan Srinivasan Natarajan
A new three-dimensional metal-organic coordination polymer, [La2(H2O)2(H2BTEC)(BTEC)], 1, was hydrothermally synthesized and characterized by single crystal X-ray diffraction. The three-dimensional framework is built up from La2O16 dimers connected by carboxylate anions. The polymer exhibits strong photoluminescence at room temperature with the main emission band at 390 nm (λex = 338 nm). Crystal data: triclinic, space group P(−1),a = 6.4486(3),b = 9.4525(5),c = 9.6238(5) Å, α= 88.24(1), β = 74.67(2), γ= 76.76(1)°,V = 550.45(5)
Volume 117 Issue 3 May 2005 pp 219-226
Hydrothermal synthesis and structure of [(C4N2H12)3][P2Mo5O23]·H2O and [(C3N2H12)3][P2Mo5O23]·4H2O
Two new compounds, [(C4N2H12)3][P2Mo5O23]·H2O,I, and [(C3N2H12)3][P2Mo5O23]4H2O,II, have been prepared employing hydrothermal methods in the presence of aliphatic organic amine molecules. Both the compounds possess the same polyoxoanion, pentamolybdatobisphosphate, (P2Mo5O23)6-. The anions consist of a ring of five MoO6 distorted octahedra with four edge connections and one corner connection. The phosphate groups cap the pentamolybdate ring anion on either side. The anion is stabilized by strong hydrogen bonds involving the hydrogen atoms of the amine molecules and the oxygen atoms of the polyoxoanion and water molecules. Crystal data:I, monoclinic, space group = P21/n (no. 14), mol. wt. = 1192.1,a = 9.4180(1),b = 18.1972(3),c = 19.4509(1) å, Β = 103.722(1)‡,V = 3238.37(7) å3, Z = 4;II, triclinic, space group = P1 (no. 2), mol. wt. = 1210.1,a 9.5617(9),b = 13.3393(12),c = 13.7637(12) å, α = 88.735(1), Β = 75.68(1), γ = 87.484(2)‡,V = 1699.2(3) å3.
Volume 118 Issue 1 January 2006 pp 57-65
Synthesis, structure and magnetic properties of the polyoxovanadate cluster [Zn2(NH2(CH2)2NH2)5][{Zn(NH2(CH2)2NH2)2}2{V18O42(H2O)}].xH2O (x ∼ 12), possessing a layered structure].xH2O (x ∼ 12), possessing a layered structure
Srinivasan Natarajan K S Narayan Swapan K Pati
A hydrothermal reaction of a mixture of ZnCl2, V2O5, ethylenediamine and water gave rise to a layered poly oxovanadate material [Zn2(NH2(CH2)2NH2)5][{Zn(NH2(CH2)2NH2)2}5{V18O42(H2O)}].xH2O (x ∼ 12) (I) consisting of [V18O42]12− clusters. These clusters, with all the vanadium ions in the +4 state, are connected together through Zn(NH2(CH2)2NH2)2 linkers forming a two-dimensional structure. The layers are also separated by distorted trigonal bipyramidal [Zn2(NH2(CH2)2NH2)5] complexes. The structure, thus, presents a dual role for the Zn-ethyl-enediamine complex. The magnetic susceptibility studies indicate that the interactions between the V centres inI are predominantly antiferromagnetic in nature and the compound shows highly frustrated behaviour. The magnetic properties are compared to the theoretical calculations based on the Heisenberg model, in addition to correlating to the structure. Crystal data for the complexes are presented.
Volume 118 Issue 6 November 2006 pp 525-536
The use of hydrothermal methods in the synthesis of novel open-framework materials
Srinivasan Natarajan Sukhendu Manual Partha Mahata Vandavasi Koteswara Rao Padmini Ramaswamy Abhishek Banerjee Avijit Kumar Paul K V Ramya
The preparation of inorganic compounds, exhibiting open-framework structures, by hydrothermal methods has been presented. To illustrate the efficacy of this approach, few select examples encompassing a wide variety and diversity in the structures have been provided. In all the cases, good quality single crystals were obtained, which were used for the elucidation of the structure. In the first example, simple inorganic network compounds based on phosphite and arsenate are described. In the second example, inorganic-organic hybrid compounds involving phosphite/arsenate along with oxalate units are presented. In the third example, new coordination polymers with interesting structures are given. The examples presented are representative of the type and variety of compounds one can prepare by careful choice of the reaction conditions.
Magnetic behaviour in metal-organic frameworks — Some recent examples
Partha Mahata Debajit Sarma Srinivasan Natarajan
The article describes the synthesis, structure and magnetic investigations of a series of metal-organic framework compounds formed with Mn+2 and Ni+2 ions. The structures, determined using the single crystal X-ray diffraction, indicated that the structures possess two- and three-dimensional structures with magnetically active dimers, tetramers, chains, two-dimensional layers connected by polycarboxylic acids. These compounds provide good examples for the investigations of magnetic behaviour. Magnetic studies have been carried out using SQUID magnetometer in the range of 2-300 K and the behaviour indicates a predominant anti-ferromagnetic interactions, which appears to differ based on the M-O-C-O-M and/or the M-O-M (M = metal ions) linkages. Thus, compounds with carboxylate (Mn- O-C-O-Mn) connected ones, [C3N2H5][Mn(H2O){C6H3(COO)3\}], I, [{Mn(H2O)3}{C12H8O(COO)2}]$\cdots$H2O, II, [{Mn(H2O)}{C12H8O(COO)2}], III, show simple anti-ferromagnetic behaviour. The compounds with Mn-O/OH-Mn connected dimer and tetramer units in [NaMn{C6H3(COO)3}], IV, [Mn2($\mu_3$-OH) (H2O)2{C6H3(COO)3}]$\cdot$2H2O, V, show canted-antiferromagnetic and anti-ferromagnetic behaviour, respectively. The presence of infinite one-dimensional -Ni-OH-Ni- chains in the compound, [NiR_2R(HR_2RO)($\mu_3$-OH)2(C8H5NO4], VI, gives rise to ferromagnet-like behaviour at low temperatures. The compounds, [Mn3{C6H3(COO)3}$2], VII and [{Mn(OH)}2{C12H8O(COO)2}], VIII, have two-dimensional infinite -Mn-O/OH-Mn- layers with triangular magnetic lattices, which resemble the Kagome and brucite-like layer. The magnetic studies indicated canted-antiferromagnetic behaviour in both the cases. Variable temperature EPR and theoretical magnetic modelling studies have been carried out on selected compounds to probe the nature of the magnetic species and their interactions with them.
Volume 122 Issue 5 September 2010 pp 771-785
Effect of metal ion doping on the photocatalytic activity of aluminophosphates
Avijit Kumar Paul Manikanda Prabu Giridhar Madras Srinivasan Natarajan
The metal ions (Ti+4, Mg+2, Zn+2 and Co+2) have been substituted in place of Al$^{+3}$ in aluminophosphates (AlPOs). These compounds were used for the first time as possible photocatalysts for the degradation of organic dyes. Among the doped AlPOs, ZnAlPO-5, CoAlPO-5, MgAlPO-11, 18 and 36 did not show any photocatalytic activity. MgAlPO-5 showed photocatalytic activity and different loading of Mg (4, 8, 12 atom % of Mg) were investigated. The activity can be enhanced by the increasing of concentration of the doped metal ions. TiAlPO-5 (4, 8, 12 atom % of Ti) showed the highest photocatalytic activity among all the compounds and its activity was compared to that of Degussa P25 (TiO2). The activity of photocatalysts was correlated with the diffuse reflectance and photoluminescence spectra.
Volume 124 Issue 2 March 2012 pp 339-353
The relevance of metal organic frameworks (MOFs) in inorganic materials chemistry
Srinivasan Natarajan Partha Mahata Debajit Sarma
The metal organic frameworks (MOFs) have evolved to be an important family and a corner stone for research in the area of inorganic chemistry. The progress made since 2000 has attracted researchers from other disciplines to actively engage themselves in this area. This cooperative synergy of different scientific believes have provided important edge and spread to the chemistry of metal-organic frameworks. The ease of synthesis coupled with the observation of properties in the areas of catalysis, sorption, separation, luminescence, bioactivity, magnetism, etc., are a proof of this synergism. In this article, we present the recent developments in this area.
Volume 126 Issue 5 September 2014 pp 1477-1491 Special issue on Chemical Crystallography
A Reactive Intermediate, [Ni5(C6H4N3)6(CO)4], in the Formation of Nonameric Clusters of Nickel, [Ni9(C6H4N3)12(CO)6] and [Ni9(C6H4N3)12(CO)6].2(C3H7NO)
Subhradeep Mistry Srinivasan Natarajan
Three new molecular compounds, [Ni5(bta)6(CO)4], I, [Ni9(bta)12(CO)6], II, [Ni9(bta)12(CO)6].2(C3H7NO), III, (bta = benzotriazole) were prepared employing solvothermal reactions. Of these, I have pentanuclear nickel, whereas II and III have nonanuclear nickel species. The structures are formed by the connectivity between the nickel and benzotriazole giving rise to the 5- and 9-membered nickel clusters. The structures are stabilised by extensive $\pi \ldots \pi$ and C-H$\ldots \pi$ interactions. Compound II and III are solvotamorphs as they have the same 9-membered nickel clusters and have different solvent molecules. To the best of our knowledge, the compounds I-III represent the first examples of the same transition element existing in two distinct coordination environment in this class of compounds. The studies reveal that compound I is reactive and could be an intermediate in the preparation of II and III. Thermal studies indicate that the compounds are stable upto 350°C and at higher temperatures (∼800°C) the compounds decompose into NiO.Magnetic studies reveal that II is anti-ferromagnetic. | CommonCrawl |
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xaktly | Virology
Structures of viruses
The virology pages are organized like this:
What is a virus?
Review of genomes, RNA, DNA and replication
Viral genomes
Attachment & entry
Viral replication
Basic immunology
The plaque assay
The genome of a virus – its business end, consisting of RNA or DNA – is susceptible to being broken up and rendered useless by host defenses and other environmental factors. It must, therefore, be protected if a virus is to infect a host cell successfully and co-opt the host machinery to make copies of itself.
All viruses enclose their genetic material in a capsid (Latin root = "case") made of protein. Sometimes capsids are coated with a phospholipid membrane derived from host cell, and some viruses surround the genome with an inner capsid, a nucleocapsid.
All viral capsids are composed of many identical copies of just a few small-to-average sized proteins (20-50 KDa, see pro tip below). Those proteins, modules that can sometimes form more than one kind of quaternary structure complex, are the repeating building blocks of the amazing and diverse virus structures we can observe.
Capsid proteins are often named with the prefix "VP," for "viral protein" or "virion protein." The name VP20 would be a viral protein of mass 20 KiloDaltons (KDa), approximately.
Virus structures have to be held together tightly enough so that they can protect the enclosed genetic material, but not so tightly that they won't be able to open up and release that material (or sometimes let host-cell proteins in) when it's time. It's a tricky balance. The resulting protein structures are called metastable, implying that they lie on the cusp of an energetic divide, and could as easily remain together as fall apart.
We can break the structures of viruses down into two main categories, based mostly on the symmetry of their construction from protein subunits, and one catch-all category, the "complex structures."
We'll use those three categories. Viruses are further classified as to whether the genetic material is contained inside a single capsid, inner and outer capsids, and/or whether they receive a membrane envelope (outer coating). Viruses are referred to as being enveloped or non-enveloped.
Here are just a few examples (electron micrographs) of the main types of virus particles
The Ebola virus is a rod-shaped virus that tends to twist a bit at one end. The rod is formed from many copies of a single viral protein that assemble in a helical pattern, forming a hollow tube. Ebola viruses are roughly 80 nm (800 Å) in diameter and they can exceed 1000 nm long.
The Corona virus is an icosahedral (roughly soccer-ball shaped) virus with protein "spikes" located at the vertices. Several copies of one or more distinct viral proteins assemble to form a highly-symmetrical closed form – an icosahedron. Coronavirus capsids are about 50 nm (500 Å) in diameter.
The structure of the Lambda phage (a phage is a virus that infects bacteria) is a complex structure consisting of a rod-like helical section, an icosahedral "head" and several smaller structures like the protein bits that look like legs. The diameter of the icosahedral "head" is about 55 nm and the "tail" is about 150 nm long.
Pro tip:
In biology, the atomic mass unit (amu) is called the Dalton (Da). The mass of carbon is 12 amu or 12 Da. The average mass of the 20 amino acids (aa) is 110 Da, so a 10 KDa (10,000 Da) protein consists of about
$$\frac{10,000 \, Da}{110 \, Da/aa} = 91 \, \text{amino acids}$$
Think of it as about 90 amino acids for every 10 KDa of mass.
Three structural types, ± envelopes
The three classes of virus structure are shown – just schematically – here. Helical packing of many copies of (usually) just one protein forms a cyclindrical capsid that encloses the viral genome, either loosely or by forming close contacts with the DNA or RNA. Spherical particles have icosahedral symmetry, sometimes with the geometry of a soccer ball. Complex viruses can be combinations of icosahedral and helical construction, and sometimes something quite different. We'll go through each of these below.
Simple modules → complex structures
Viruses are efficient. The complete capsid structure of a typical virus is composed of no more than just two or three proteins, and often just one, arranged symmetrically so as to build a hollow structure that is strong enough to contain and protect the traveling genome, yet sufficiently weakly-bound that the capsid can open to release the genetic material when needed.
While the primary structure of a protein is completely determined by covalent bonds, the 2˚, 3˚ and 4˚ structures depend on weaker intermolecular attractions, such as hydrogen bonds and charge-charge interactions. These weaker interactions can be quite specific, binding two proteins or parts of one protein in a very specific conformation. Because they are weaker than covalent bonds, they can be relatively easily broken as needed.
A good paradigm for these weak interactions is the base-pairing between the two strands of the DNA double helix: The helices are quite spefically matched to one another through H-bonding interactions, but bound weakly enough that they can be separated for transcription and copying.
The only thing needed for a virus particle capsid to assemble is the presence of its constituent proteins. That is, if all parts of a viral capsid are present in a cell or a solution, viral capsids (virus-like particles or VLPs) will spontaneously self-assemble.
The arrangement of constituent proteins in virus capsids follow simple symmetry rules.
Helical viruses
Helical virus capsids are composed of hollow cyclinders formed from a repeated protein unit(s) that link to neighboring proteins in a spiraling pattern. The genetic material is usually wound in a helical pattern inside the cylindrical capsid. Here's a schematic depiction of how it looks.
Example: Tobacco mosaic virus (TMV)
The tobacco mosaic virus (TMV) is a nice example of a non-enveloped helical virus. TMV was the first virus ever isolated, so we know a lot about it. TMV causes the leaves of tobacco plants to have a mottled appearance, and therefore affects its commercial value.
A leaf on a TMV-infected tobacco plant shows mottled coloring. A healthy leaf is a continuous green color. Image: Clemson University - USDA Cooperative Extension Slide Series
Electron micrographs of TMV show the virus particles to be cylindrical, and of uniform diameter and length. These virus particles are rigid, but helical viruses need not be completely rigid in form, as we shall see.
Image: Public domain
The cylindrical capsid of TMV is composed of 2130 copies of a single coat protein called CP (for "coat protein" – I know, shocking). Here is an artistic representation of that capsid. The proteins are in blue (identical, but alternately shaded so that you can differentiate neighboring molecules), and the ssRNA strand, comprising 6400 bases, is in red. It folds in a helix in close contact with the inner ends of the capsid proteins.
Source: Protein data bank
A ribbon representation of the structure of the 158 amino-acid (so that's about 1.75 KDa) coat protein (CP) of TMV is shown below. Two molecules are shown to give an idea of their packing in the capsid. Each chain is covalently-linked, and consists of a group of four alpha helices with a fifth (on the left) directed toward the inside of the capsid. Three RNA bases form weak attachments to every CP protein. The attractive interactions between any two CP molecules are due to weaker intermolecular forces such as hydrogen bonds (H-bonds).
Example: Ebola virus
Ebola virus, which causes a severe hemorrhagic fever in humans, is an enveloped helical virus that forms often-bent tubes containing its RNA genome. Note the similarities between this electron micrograph of ebola viruses and the one above.
Here is a schematic view of the structure of Ebola.
The RNA genome is coated with two capsid proteins called N and VP30. These inner capsids are sometimes referred to as nucleocapsids because they (1) coat the nucleic acid polymers directly and (2) are the innermost of two capsid layers. The outer capsid of Ebola is formed from copies of two proteins, VP24 and VP40, the latter being most abundant and mainly responsible for the structure of the capsid. VP24 plays a role in inhibiting several cellular virus defenses. The outer capsid receives a membrane coating from the host, and displays copies of a glycoprotein along its surface. The Ebola capsid also encloses its own RNA polymerase (RNAp) and a multi-purpose helper protein called VP35.
All of these example viruses will spontaneously self assemble in solution if we can arrange for all of the protein and genetic components to come together artificailly in the lab. We can also make empty (of genetic material) virus particles called VLPs, virus-like particles, that contain no genetic material. VLPs have been used in this way as cell-specific delivery systems in living organisms, like humans. VLPs are a target of a lot of research. They have been used to create vaccines and to attempt various gene therapies.
The icosahedron and icosahedral symmetry
The icosahedron is one of the Platonic solids, objects identified by Plato which are regular, closed polyhedrons with congruent, regular (all sides congruent) polygon faces, with the same number of faces meeting at each vertex. The five platonic solids are
tetrahedron (four triangular faces)
cube (six square faces)
octahedron (eight triangular faces)
dodecahedron (twelve pentagonal faces)
icosahedron (twenty triangular faces)
The figures above illustrate the symmetry elements of a simple icosahedron. Icosahedrons have 30 two-fold axes of symmetry (red, one for each edge), 20 three-fold axes of symmetry (green, one for each face), and 12 five-fold axes of symmetry (blue, one for each vertex).
While many virus capsids have icosahedral symmetry, their appearance is nearly spherical. That's not a contradiction. When we refer to icosahedral symmetry, we refer to the symmetry of construction of the capsid, not necessarily its shape.
The soccer ball is an example of a truncated icosahedron, one in which there is one kind of vertex (three sides meet) and two kinds of faces: pentagons and hexagons. In order for an icosahedral solid to be able to close, it must include five pentagons, the black faces of the soccer ball.
The covalent molecule C60, a major component of soot, forms as a truncated icosahedron. This symmetry is ubiquitous in nature.
An object need not have an icosahedral shape to have icosahedral symmetry.
Arrangement of viral proteins on an icosahedron
Source: Swiss Institute of Bionformatics
Icosahedral packing of three VPs is illustrated schematically here. VP1, VP2 and VP3 form a triangle, and 24 such triangles form the complete icosahedron. In the figures below, the structures of several viruses are shown, some in ribbon diagrams and some as space-filling models. Each was determined from detailed structural measurements. See if you can identify triangular, pentagonal and hexagonal subunits in each.
The rotavirus causes much of the childhood diarrhea around the globe. In countries without the means to sufficiently clean water, or which lack the means to rehydrate sufferers with electrolyte-containing solutions, infected children often die. Two vaccines (as of 2021) are now available. If they can be distributed, many young lives may be saved.
The Dengue virus causes Dengue fever, which can be dangerous upon the first infection. If a human is infected a second time, the chances of survival are currently drastically reduced. Dengue is an arbovirus, one that is transmitted by an arthropod insect, in this case, the mosquito Aedes aegypti. See if you can identify triangular and pentagonal substructures in the image.
The Hepatitis B virus infects cells of the liver and can lead to cirrhosis (hardening of the organ through scarring), liver cancer and other kinds of liver failure. A small set of proteins forms a fairly complex virus structure.
Norwalk virus
The Norwalk virus is a norovirus. It is causes gastroenteritis – generally vomiting, diarrhea and dehydration. The disease is uncomfortable but usually resolves within a couple of days. Norovirus infections are second to colds (caused by rhinovirus or coronavirus) in the US.
Make your own virus model
You can make your own icosahedral virus model. Click here to download a .pdf file that you can print, cut out with scissors and tape or glue together. It's a good exercise to get a better feel for the symmetry of icosahedral virus particles.
Complex structures
Complex structures can consist of helical and icosohedral elements or they can take some other shape. Here is a brief tour of a couple of interesting complex viruses.
Lambda phage
A phage is a virus that infects a bacterium. An electron micrograph of the lambda phage, which infects the Escherichia coli (E. coli) bacterium, was shown above. Here is a schematic diagram of the phage. It acts like a sort of syringe to inject its DNA genome into its host.
The Rabies virus
The rabies virus is a bullet-shaped virus consisting of nucleocapsid that surrounds a single-stranded -RNA genome. Glycoproteins project through the membrane derived from host cells.
Source: modified from the figure in Abraham, et al., Int. J. Current Microbiology & Appl. Sci., 6(12):2064-2085 (2017).
Rabies affects the central nervous system of humans. As of 2021, there is no cure for an infected person who is showing symptoms of rabies. Caught immediately, there is antibody treatment and a vaccine than can stem the infection during its incubation period. In the US, rabies affects only a handful of people per year.
Protein structure
There are four levels of protein structure:
1˚ or primary structure is the ordering of the amino acids along the polypeptide chain.
2˚ or secondary structure is the formation of alpha helices, beta sheets and other structure that depends upon the sequence.
3˚ or tertiary structure is specific interactions between secondary structure elements to form the completed fold of a protein.
4˚ or quaternary structure is specific attractive interaction between one or more protein sub-units (identical or different) to form a stable complex of proteins.
Virome
The virome revers to the collection of all viral genomes, or even just viruses, that live within an organism, a set of organisms or a region on Earth. It is used rather loosely in context.
We can refer, for example, to the global virome (all viruses on Earth) or the human virome, the viruses that infect humans.
(par' · uh · dime)
A paradigm is a typical example or commonly-used model of some concept. A bunny rabbit is a paradigm of warm, fuzzy animals. | CommonCrawl |
\begin{document}
\title{Shortcuts to adiabatic rotation of a two-ion chain}
\author{Ander Tobalina} \address{Departament of Physical Chemistry, University of the Basque Country (UPV/EHU), Apdo. 644, 48080 Bilbao, Spain}
\author{Juan Gonzalo Muga} \address{Departament of Physical Chemistry, University of the Basque Country (UPV/EHU), Apdo. 644, 48080 Bilbao, Spain}
\author{Ion Lizuain} \address{Department of Applied Mathematics, University of the Basque Country UPV/EHU, Donostia-San Sebastian, Spain}
\author{Mikel Palmero} \ead{[email protected]} \address{Department of Applied Physics, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain}
\begin{abstract}
We inverse engineer fast rotations of a linear trap with two ions for a predetermined rotation angle and time, avoiding final excitation. Different approaches are analyzed and compared when the ions are of the same species or of different species.
The separability into dynamical normal modes for equal ions in a common harmonic trap, or for different ions in non-harmonic traps with up to quartic terms allows for simpler computations of the rotation protocols. For non-separable scenarios, in particular for different ions in a harmonic trap, rotation protocols are also found using more costly numerical optimisations.
\end{abstract}
\section{Introduction}
Trapped ions stand out as a flexible architecture to control internal and/or motional states and dynamics
for fundamental research of quantum phenomena and technological applications. Pure motional control without internal state transitions is in particular crucial in proposals of two-qubit gates, see e.g. \cite{Palmero2017}, or interferometry \cite{Campbell2017,Martinez-Garaot2018,Rodriguez-Prieto2020}, as well as to scale up the number of ions for quantum information processing \cite{Kielpinski2002,Rowe2002,Reichle2006,Home2009,Roos2012,Monroe2013,Kaushal2020,Wan2020}. The toolbox of basic operations induced by controlling the voltage of electrodes in different Paul trap configurations or detuned laser fields includes transport, expansions and compressions, separation and merging of ion chains, and rotations, the latter being the central topic of this work.
Specific motivations to implement rotations are: reordering an ion chain (to scale up quantum information processing or to locate cooling ions at appropriate positions) \cite{Splatt2009,Kaufmann2017}; rotation sensing \cite{Campbell2017}; different simulations (e.g. of black holes \cite{horstmann2010} or diatomic molecules \cite{Urban2019}); probing the exchange phase of quantum statistics
\cite{roos2017}; or sorting ions according to charge and mass \cite{Masuda2015}.
Trap rotations, to impart some angular momentum to an ion or ion chain, or to reorient the longitudinal axis of the trap, have been implemented in experiments with improving accuracy \cite{Splatt2009,Urban2019,Kaufmann2017,vanMourik2020}, and investigated theoretically \cite{Palmero2016,Lizuain2017}.
Motional control operations, and rotations in particular, need in most applications to be fast, relative to adiabatic dynamics, but also gentle, avoiding final excitations, two requirements met with shortcut to adiabaticity (STA) driving protocols \cite{Guery2019}. There are different STA techniques but, for trapped ion driving, STA invariant-based inverse engineering has proven useful \cite{Palmero2017, Martinez-Garaot2018, Rodriguez-Prieto2020, Lizuain2017, Torrontegui2011, Palmero2013, Palmero2014, Lu2014, Lu2015, Chen2010, Palmero2015a, Palmero2015, Sagesser2020}, also to design trap rotations for a single ion \cite{Palmero2016}.
In this paper we extend to a two-ion chain the design of STA 1D-trap rotations done in reference \cite{Palmero2016}. Our aim is to inverse engineer the rotation angle to implement a fast process, free from final excitations. The work in reference \cite{Palmero2016} was indeed presented as a preliminary step towards the more complex scenario of the chain rotation, which allows for different, and surely more relevant applications, in particular reordering. Engineering the two-ion rotation also entails non-trivial technical complications due to the increase in the number of equations to be solved, and also because, for some configurations, in particular for two different species in a harmonic trap, there is not in general a point-transformation that provides independent dynamical normal modes \cite{Lizuain2017}\footnote{``Dynamical normal modes" generalise regular (static) normal modes. They are independent concerted motions represented by harmonic oscillators with time-dependent parameters \cite{Lizuain2017, Palmero2014}, generally with a time-dependent oscillation frequency.}. Inverse engineering is much easier -to describe the motion and with respect to computational time- for independent modes than for a system which is not separable by point-transformations.\footnote{Separability by non-point transformations is possible in principle but it is considerably more involved in terms of its interpretation and practical use. Its application to inverse engineer one-particle rotations in anysotropic traps was explored in \cite{Lizuain2019} under some strong restrictions in process timing and rotation speed.}
\begin{figure}
\caption{(Color online) Schematic representation of the rotation of two ions confined along a line rotated by an angle $\theta_f$ in a time time $t_f$.}
\label{fig7_1}
\end{figure}
We introduce now the basic model. We opt for a cavalier, idealised modelling where the trap is assumed for simplicity to be tightly confined in the radial direction, as depicted in figure \ref{fig7_1}, i.e., we leave aside peculiarities of the experimental settings, such as micromotion effects and detailed electrode configurations, that may vary significantly among different traps. Our solutions will therefore be guiding starting points for a realistic implementation
\cite{vanMourik2020, Kaufmann2018}.
The trapping line rotates in a horizontal plane in a time $t_f$ up to a predetermined final angle, $\theta_f= \pi$ in all examples. {We first find the classical Hamiltonian from the corresponding Lagrangian and then quantise the result.
Let $s_i$, $i=1,2$, denote the points on the line where each ion lays. $s_i$ may take positive and negative values. The Cartesian (lab frame) components of a trajectory $s_i(t)$ are $x_i=x_i(s,t)$, $y_i=y_i(s,t)$,
\begin{equation} x_i = s_i\cos (\theta), \quad y_i=s_i\sin (\theta), \end{equation}
where $\theta=\theta (t)$ is the rotation angle.} For two different ions in a common trap potential $f(s_i)$ the Lagrangian is (we have considered that the magnetic interaction between the two moving charges can be safely neglected, see Appendix A)
\begin{eqnarray} \label{Lagrangian_2diffion} L = \sum_{i=1,2}\left[\frac{m_i}{2} \dot{s}_i^2-f(s_i)+ \frac{m_i{\dot\theta}^2}{2}s_i^2\right]-\frac{C_c}{s_2-s_1}, \end{eqnarray}
with corresponding Hamiltonian
\begin{eqnarray} H&=&\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V, \label{Hgeneral}\\ V&=&\sum_{i=1,2}\left[f(s_i)-\frac{m_i}{2}{\dot{\theta}}^2 s_i^2 \right]+\frac{C_c}{s_2-s_1}. \label{pot} \end{eqnarray}
In the Coulomb repulsion term
$C_c={e^2}/{(4\pi\epsilon_0)}$, where $\epsilon_0$ is the vacuum permittivity, and $e$ the electric charge of the electron.
The equilibrium positions $\{s_i^{(0)}\}$ of the ions are found by solving the set of equations $\{{\partial V}/{\partial s_i}=0\}$. Since different external traps may be considered, the following equations are for a generic $V$, the results for the simple harmonic trap are given later in section \ref{htrap}.
We define the equilibrium distance between ions as
\begin{equation} d=s_2^{(0)}-s_1^{(0)}
\end{equation}
and expand $V$ around the equilibrium positions, keeping terms up to second order. Using mass-weighted coordinates $\tilde s_i=\sqrt{m_i} s_i$ and momenta $\tilde p_i=p_i / \sqrt{m_i}$, $H$ is simplified to the quadratic form
\begin{equation} \label{general}
H=\frac{\tilde p_1^2}{2} + \frac{\tilde p_2^2}{2}
\!+\!(\tilde s_1-s_1^{(0)},\tilde s_2-s_2^{(0)}) {{\bm{\mathsf{ v}}}}
\left(\begin{array}{l}
\tilde{s}_1-s_1^{(0)}\\
\tilde{s}_2-s_2^{(0)}
\end{array}\right), \end{equation}
where the matrix $\bm{\mathsf v}$ has elements ${\sf v}_{ij}=\frac{1}{\sqrt{m_im_j}}\left.\frac{\partial^2 V}{\partial s_i\partial s_j}\right|_{\{s_i,s_j\}=\{s_i^{(0)},s_j^{(0)}\}}$.
\section{Diagonalisation and dynamical normal modes: setting the equations}
We may try to decouple the dynamics by diagonalising ${\bm{\mathsf{v}}}$. As explained in reference \cite{Lizuain2017}, moving to a frame defined by the eigenvectors of ${\bm{\mathsf{v}}}$ leads, after a classical canonical transformation or, equivalently, quantum unitary transformation, to the following effective Hamiltonian \cite{Lizuain2017}
\begin{eqnarray}
H'=\sum_{\nu=\pm}\left[\frac{p_\nu^2}{2}+\frac{\Omega_\nu^2}{2}\left(s_\nu+\frac{\dot p_{0\nu}}{\Omega_\nu^2}\right)^{\!\!2}\right]\!\!-\!\dot \mu (s_-p_+-s_+p_-), \label{effective} \end{eqnarray}
where $\mu$ is the tilting angle of the potential in configuration space defined by the relation
\begin{eqnarray} \label{decoupling_cond} \tan 2\mu=\frac{2v_{12}}{v_{11}-v_{22}},
\end{eqnarray}
and the momentum shifts
\begin{eqnarray}
p_{0+}&=&\dot s_1^{(0)} \sqrt{m_1}\cos\mu +\dot s_2^{(0)}\sqrt{m_2}\sin\mu,
\\
p_{0-}&=&-\dot s_1^{(0)} \sqrt{m_1}\sin\mu +\dot s_2^{(0)} \sqrt{m_2}\cos\mu,
\end{eqnarray}
have been defined.
The coordinates that diagonalise ${\bm{\mathsf{v}}}$ are
\begin{eqnarray}
\!\!\!\!\!\!\!s_+&=&\sqrt{m_1} (s_1\!-\! s_1^{(0)}) \cos\mu\! +\! \sqrt{m_2} (s_2\!-\!s_2^{(0)})\sin\mu,\\ \!\!\!\!\!\!\!s_-&=&-\sqrt{m_1} (s_1\!-\!s_1^{(0)}) \sin\mu\! +\! \sqrt{m_2} (s_2\!-\!s_2^{(0)}) \cos\mu,
\end{eqnarray}
with conjugate momenta
\begin{eqnarray}
p_+&=&\frac{\cos\mu}{\sqrt{m_1}}p_1 + \frac{\sin\mu}{\sqrt{m_2}}p_2,\\ p_-&=&-\frac{\sin\mu}{\sqrt{m_1}}p_1 + \frac{\cos\mu}{\sqrt{m_2}}p_2.
\end{eqnarray}
The squares of the frequencies are
\begin{eqnarray}
\Omega_+^2&=&v_{11}\cos^2\mu+v_{22}\sin^2\mu+v_{12}\sin{2\mu},
\nonumber\\
\Omega_-^2&=&v_{11}\sin^2\mu+v_{22}\cos^2\mu-v_{12}\sin{2\mu}.
\label{omegas} \end{eqnarray}
$s_\pm$ describe independent, dynamical normal modes whenever $\mu$ is time independent, see equation (\ref{effective}). In a quantum scenario this means that any wave-function dynamics can be decomposed in terms of the dynamics of two independent harmonic oscillators with time-dependent parameters. Different scenarios to achieve this decoupling are considered in the following.
\subsection{Results for the harmonic trap\label{htrap}}
For the common harmonic external potential with spring constant $k$, $f(s_i)={k}s_i^2/2$, which is the only configuration considered hereafter in the main text, the potential (\ref{pot}) takes the form
\begin{eqnarray} V&=&\frac{1}{2}u_1 s_1^2+\frac{1}{2}u_2 s_2^2+\frac{C_c}{s_2-s_1}, \label{Hamiltonian_2ionrot} \end{eqnarray}
where \begin{equation} u_i= m_i(\omega_i^2 -\dot{\theta}^2), \quad m_i\omega_i^2=k. \label{spco} \end{equation}
The $u_i$ are effective spring constants affected by the rotation speed. Unless $m_1=m_2$, they are different for both ions. With this $V$ we find the explicit relations
\begin{eqnarray}
s_i^{(0)}&=&-\left[\frac{C_c u_j^2}{u_i (u_i+u_j)^2}\right]^{\!1/3}\!, i\ne j,
\nonumber\\
d&=&\left[\frac{C_c(u_1+u_2)}{u_1u_2}\right]^{1/3},
\nonumber\\
{\bm{\mathsf v}}&=&\left(\begin{array}{lr}
\frac{\frac{2 C_c}{d^3}+u_1}{m_1} & -\frac{2 C_c}{d^3\sqrt{m_1 m_2} } \\
-\frac{2 C_c}{d^3\sqrt{m_1 m_2}} & \frac{\frac{2 C_c}{d^3}+u_2}{m_2}
\end{array}\right),
\nonumber\\
\label{decoupling_condh}
\tan 2\mu&=&\frac{4 C_c \sqrt{m_1m_2}}{(m_1-m_2) \left(2 C_c+d^3 k\right)}. \end{eqnarray}
\section{Equal ions}
If the ions are equal, $m_1=m_2=m$, the tilting angle takes the constant value $\mu=-\pi/4$. The decoupling condition is therefore identically satisfied at all times.
Also, $u_1=u_2=u=m\omega^2$, with
\begin{equation} \omega^2=\omega_0^2-\dot\theta^2 \label{ome_ef} \end{equation}
and $\omega_0$ constant. The angular velocity of the rotation $\dot{\theta}(t)$ could be negative at some intervals, whereas $\omega^2$ may also be positive or negative.
The equilibrium positions are simplified to
\begin{eqnarray} s_2^{(0)}=-s_1^{(0)}=\frac{x_0}{2}\quad\textrm{with} \quad x_0=\left(\frac{2Cc}{m\omega^2}\right)^{1/3}, \end{eqnarray}
which are symmetrical with respect to the trap centre $s=0$. The decoupled, effective Hamiltonian is therefore
\begin{eqnarray}
H''=\sum_{\nu=\pm}\left[\frac{p_\nu^2}{2}+\frac{\Omega_\nu^2}{2}\left(s_\nu+\frac{\dot p_{0\nu}}{\Omega_\nu^2}\right)^2\right], \label{18} \end{eqnarray}
with
\begin{eqnarray}
s_\pm&=&\sqrt{\frac{m}{2}}\left[\left(s_1+\frac{x_0}{2}\right)\mp\left(s_2-\frac{x_0}{2}\right)\right], \nonumber\\ p_{0+}&=&- \sqrt{\frac{m}{2}} \dot x_0,
\quad p_{0-}=0, \nonumber\\ \Omega_+^2&=&3\omega^2,\quad \Omega_-^2=\omega^2. \label{OmegaNM}
\end{eqnarray}
We consider rotation protocols with a smooth behavior of $\theta$ at the boundary times $t_b=0,t_f$,
\begin{eqnarray} \theta(0)&=&0,\quad \theta(t_f)=\theta_f, \label{cond1_2ion}\\ \dot{\theta}(t_b)&=&\ddot{\theta}(t_b)=0. \label{cond2_2ion} \end{eqnarray}
These conditions imply that
\begin{eqnarray} \omega(t_b)&=&\omega_0, \\ \dot\omega(t_b)&=&\ddot\omega(t_b)=\dot{p}_{0\nu}(t_b)=0. \label{cond3_2ion} \end{eqnarray}
\begin{figure*}
\caption{(Color online) Two equal ions. Exact energy excess (final minus initial energy) starting from the ground state and with dynamics driven by the full potential (\ref{Hamiltonian_2ionrot}) according to equation (\ref{controlparameter}) for the parameters $c_{3-6}$ that minimise the excitation in the normal modes. (a) represents this excitation in a linear scale and (b) in a logarithmic scale. Dotted blue line: protocol using all 4 free parameters in the ansatz for $\theta(t)$; Short-dashed green line: only 3 free parameters, i.e., $c_6=0$; Long-dashed black line: only 2 free parameters, i.e., $c_5=c_6=0$; Dash-dotted orange line: only one free parameter, i.e., $c_4=c_5=c_6=0$; Solid red line fixes: $c_{3-6}=0$ so there is no optimisation. The evolution was done for two $^{40}$Ca$^+$ ions, with an external trap frequency $\omega_0/(2\pi)=1.41$ MHz and a total rotation angle $\theta_f=\pi$. }
\label{fig7_2}
\end{figure*}
The two independent harmonic oscillators expand or compress through the time dependence of $\Omega_\nu$ and experiment a ``transport'', in $s_\nu$ space, along $\dot{p}_{0\nu}/\Omega_\nu^2$.
The Hamiltonian~(\ref{18}) has a dynamical invariant \cite{Lewis1969}
\begin{equation} I = \sum_{\nu=\pm} \frac{1}{2}[b_\nu(p_\nu-\dot{\alpha}_\nu)-\dot{b}_\nu(s_\nu-\alpha_\nu)]^2 + \frac{1}{2}\Omega_{0\nu}^2\left(\frac{s_\nu-\alpha_\nu}{b_\nu}\right)^2, \end{equation}
where $\Omega_{0\pm}=\Omega_\pm(0)$, and $b_\pm$ (scaling factors of the normal mode wavefunctions) and $\alpha_\pm$ (reference classical trajectories for each oscillator) are auxiliary functions that have to satisfy, respectively, the Ermakov and Newton equations,
\begin{eqnarray} \label{auxiliaryequations} \ddot{b}_\pm+\Omega_\pm^2 b_\pm &=& \frac{\Omega_{0\pm}^2}{b_\pm^3}, \\ \ddot{\alpha}_\pm+\Omega_\pm^2\alpha_\pm &=& \dot{p}_{0\pm}. \end{eqnarray}
The time-dependent Schr\" odinger equation can be solved by superposing, with constant coefficients, elementary solutions which are also eigenstates of the invariant, with the (``Lewis-Riesenfeld'') phase adjusted to be also solutions of the Schr\"odinger equation \cite{Torrontegui2011},
\begin{equation}
|\psi_{n\pm}''\rangle = e^{\frac{i}{\hbar}\left[\frac{\dot{b}_\pm s_\pm^2}{2b_\pm}+(\dot{\alpha}_\pm b_\pm-\alpha_\pm\dot{b}_\pm)\frac{s_\pm}{b_\pm}\right]}\frac{1}{\sqrt{b_\pm}}\Phi_n(\sigma_\pm), \label{elementary} \end{equation}
where $\sigma_\pm=\frac{s_\pm-\alpha_\pm}{b_\pm}$ and $\Phi_n$ are the eigenfunctions for the static harmonic oscillators with frequencies $\Omega_{0,\pm}$. The average energies for the $n$th elementary solution of each mode can be calculated \cite{Palmero2015a, Palmero2015},
\begin{eqnarray} \label{EnergyNM}
E''_{n\pm} &=& \langle \psi''_{n\pm}|H''|\psi''_{n\pm}\rangle\nonumber\\
&=& \frac{(2n+1)\hbar}{4\Omega_{0\pm}}\left(\dot{b}^2_\pm+\Omega_\pm^2b_\pm^2+\frac{\Omega_{0\pm}^2}{b_\pm^2}\right)
+ \frac{1}{2}\dot{\alpha}_\pm^2+\frac{1}{2}\Omega_\pm^2\left(\alpha_\pm-\frac{\dot{p}_{0\pm}}{\Omega_{\pm}^2}\right)^2. \end{eqnarray}
As $\dot{p}_{0\nu}(t_f)=0$, the final values are minimised when the only contribution is due to the eigenenergies for the oscillators, with
\begin{eqnarray} b_\pm(t_f)&=&1,\alpha(t_f)=\dot{\alpha}(t_f)=\dot{b}_\pm(t_f)=0. \label{bco}
\end{eqnarray}
\subsection{Inverse engineering} \label{equalionssec}
Imposing commutativity between Hamiltonian and invariant at initial $t=0$ and final times $t=t_f$, the invariant drives the initial eigenstates of $H$ to corresponding final eigenstates along the elementary solutions (\ref{elementary}), although there could be diabatic excitations at intermediate times, when the commutation between Hamiltonian and invariant is not guaranteed. By inspection of equation (\ref{elementary}), commutativity at the boundary times is achieved if the conditions in equation (\ref{bco}) are satisfied,
which occur automatically when the final energies (\ref{EnergyNM}) are minimised.
To inverse engineer the rotation we proceed similarly to reference \cite{Palmero2016}, with an ansatz for $\theta(t)$ that satisfies boundary conditions~(\ref{cond1_2ion}) and~(\ref{cond2_2ion}) with some free parameters. We use up to 4 free parameters,
\begin{eqnarray} \label{controlparameter} \theta (t) &=& \frac{1}{16}(32c_3+80c_4+144c_5+224c_6-9\theta_f)\cos\left(\frac{\pi t}{t_f}\right) \nonumber\\
&-& \frac{1}{16}(48c_3 + 96 c_4 + 160 c_5 + 240 c_6 - \theta_f)\cos\left(\frac{3\pi t}{t_f}\right) \nonumber\\
&+& c_3\cos\left(\frac{5 \pi t}{t_f}\right) + c_4\cos\left(\frac{7 \pi t}{t_f}\right) + c_5 \cos\left(\frac{9 \pi t}{t_f}\right) + c_6 \cos\left(\frac{11 \pi t}{t_f}\right) + \frac{\theta_f}{2}.\nonumber\\ \end{eqnarray}
This gives an expression of $\dot{\theta}$, from which we find $\omega$ in equation~(\ref{ome_ef}). We introduce $\omega$ in~(\ref{OmegaNM}) to get the normal mode angular frequencies $\Omega_\pm$ needed in the Ermakov equation~(\ref{auxiliaryequations}). For a given set of values of these parameters we solve the ``direct problem'' (Ermakov and Newton equations) with initial conditions
\begin{eqnarray} b_\pm(0)=1, \dot{b}_\pm(0)=0, \nonumber\\ \alpha_\pm(0)=\dot{\alpha}_\pm(0)=0, \end{eqnarray}
and compute easily the final energies with equation (\ref{EnergyNM}). The values of the parameters are varied with a subroutine that minimises the sum of the final mode energies~(\ref{EnergyNM}) (we use the MatLab `fminsearch' and $n=0$ but note that the optimal final values of $b(t_f)$, $\alpha(t_f)$ and their derivatives would minimise the energies for any $n$). The excess energy found with the optimal parameters for the normal modes is negligible in the range of final times depicted in figure \ref{fig7_2}.
\begin{figure}
\caption{(Color online) Evolution of the control parameter $\theta(t)$ for different final times when designed using all 4 free parameters. Dashed black line: $t_f=1$ $\mu$s, and optimisation parameters $c_{3-6}=(5.134,-5.360,59.577,91.234)\times 10^{-4}$; Solid blue line: $t_f=2$ $\mu$s, and optimisation parameters $c_{3-6}=(3.093,0.971,3.386,-6.036)\times 10^{-4}$; Dotted red line: $t_f=3$ $\mu$s, and optimisation parameters $c_{3-6}=(1.400,-0.270,0.182,-0.117)\times 10^{-4}$. Other parameters as in figure \ref{fig7_2}. }
\label{fig7_3}
\end{figure}
Once the free parameters are defined such that the design of $\theta$ minimises the excitation energy of the normal modes, we perform the quantum evolution driven by the full Hamiltonian with (\ref{Hamiltonian_2ionrot}) to check the performance of the designed protocol. We use the ``Split-Operator Method'', and the initial ground state is found performing an evolution in imaginary time. Figure \ref{fig7_2} shows the final excitation, i.e., the excess energy with respect to the initial energy after performing the evolution with the full Hamiltonian (\ref{Hgeneral}) using the potential~(\ref{Hamiltonian_2ionrot}). In figure \ref{fig7_2} (a) this excitation is depicted in a linear scale, and in figure \ref{fig7_2} (b) in a logarithmic scale. The results improve significantly by using more optimisation parameters. Even when using a single optimising parameter, the results are clearly better than the protocol without free parameters. Figure \ref{fig7_3} shows some examples of the rotation protocols with 4 parameters for different rotation times.
\begin{figure}
\caption{(Color online) Two different ions. Exact energy excess (final minus initial energy) when the initial ground state is driven by the full potential~(\ref{Hamiltonian_2ionrot}) according to equation~(\ref{controlparameter}) for the parameters $c_{3-6}$ that minimise this excitation. Dotted blue line: protocol using all 4 free parameters in the ansatz for $\theta(t)$; Short-dashed green line: only 3 free parameters, i.e., $c_6=0$; Long-dashed black line: only 2 free parameters, i.e., $c_5=c_6=0$; Dash-dotted orange line: only one free parameter, i.e., $c_4=c_5=c_6=0$; Solid red line fixes: $c_{3-6}=0$ so there is no optimisation. The evolution was done for a $^{40}$Ca$^+$ and a $^9$Be$^+$ ion, with an external trap frequency for the Ca ion of $\omega_1/(2\pi)=1.41$ MHz and a total rotation angle $\theta_f=\pi$. }
\label{fig7_4}
\end{figure}
\section{Two different ions}
Let us first explore some possible manipulations to make the modes separable when the ions are different. From the expression of $\tan 2\mu$ in equation (\ref{decoupling_condh}), $d^3 k$ should be constant. If the only parameter that depends on time is $d$, this condition cannot be satisfied. But if $k$ is allowed to be a time dependent controllable parameter, it would be in principle possible. If we set the constant as $B$ then, from equation (\ref{decoupling_condh}), the relation
\begin{eqnarray}
\tan 2\mu=\frac{4C_c\sqrt{m_1m_2}}{(m_1-m_2)(2C_c+B)},
\end{eqnarray}
fixes $\mu$ to have independent dynamical modes. Using the expressions for $d$ and the $u_i$, this condition may be satisfied for two values of $\dot\theta^2=a_lk$ for each $k$, $l=1,2$, where the $a_l$ are two constants.
The proportionality between $\dot{\theta}^2$ and $k$, however, is problematic. If we wish to approach $\dot\theta=0$ smoothly at the time boundaries, then $k\to 0$ there, which implies a vanishing trapping potential and $d\to\infty$.
A way out is explored in Appendix B making use of a more complex external trap potential with linear and quartic terms added, as in reference \cite{Sagesser2020}. In the main text we stay within the harmonic trap configuration with constant $k$ and renounce to separate the modes. Thus a different, pragmatic strategy is adopted, minimising the excitation energy directly to find the rotation protocol.
\begin{figure}
\caption{(Color online) Equilibrium (dashed lines) and dynamical (solid lines) positions of the ions versus time: $s_1^{(0)}$ and $s_1$ (Calcium ion, blue lines); $s_2^{(0)}$ and $s_2$ (Berilium ion, black lines), for a final time $t_f=1$ $\mu$s and for the optimising parameters $c_{3-6}=(1.757,1.824,1.120,-0.234)\times 10^{-2}$ with the protocol in equation~(\ref{controlparameter}). The initial state is the ground state. The $s_i$ are average positions from the quantum dynamics. }
\label{fig7_5}
\end{figure}
We use the same ansatz for the parameter control $\theta$ as in equation~(\ref{controlparameter})
and solve the full (quantum) dynamics for the potential~(\ref{Hamiltonian_2ionrot}) to find the final excess energy for specific values of the free parameters $c_{3-6}$. Then, as in section \ref{equalionssec}, we minimise the excess energy letting the MATLAB subroutine `fminsearch' find the optimal parameters.
In figure \ref{fig7_4} we depict this final excitation, optimising the result using from 1 to 4 free parameters for the $\theta$, and compare it with the results for no free parameters. This direct minimisation provides even better results than the indirect one based on the normal mode energy in section \ref{equalionssec}. The best protocol (4 optimising parameters) gives an excitation below 0.1 quanta at a final time $t_f=0.56$ $\mu$s.
The price to pay though, is that the computational time required increases dramatically, as we have to solve the full dynamics of the system at every iteration of the shooting method we use to optimise, whereas in the method based on normal modes we only needed to solve four ordinary differential equations at each iteration. Figure \ref{fig7_5} shows the equilibrium and dynamical positions of both ions during the evolution for $t_f=1$ $\mu$s. The trajectories are not symmetric since the two ions experience different effective spring constants, see equation~(\ref{spco}).
\section{Discussion}
We have designed protocols to rotate a linear trap containing two ions, without final excitation.
For two equal ions in a rotating, rigid harmonic trap, there are uncoupled dynamical normal modes. The separation facilitates inverse engineering since it is only necessary to solve ordinary differential equations for independent variables to minimise the final energy. These Ermakov and Newton equations are for the auxiliary functions in the invariants associated with the uncoupled Hamiltonians. Following this method and for a given ansatz for the rotation angle and for some allowed final excitation threshold, process-time lower limits are met due to the eventual failure of the small oscillation regime for very rapid rotations. Faster processes can be achieved by increasing the number of parameters in the ansatz. For two different ions in a harmonic trap, this method is not possible as the modes are coupled for a rigid trap, or can be uncoupled for a non-rigid trap but only for impractical boundary conditions for the trap. Instead we used direct optimisation of the rotation ansatz parameters with the full Hamiltonian. This direct approach is efficient with respect to the lower time limits but the computational effort is much more demanding.
A natural extension of this work would be considering different boundary conditions, for example a final rotating trap with $\dot\theta(t_f)\ne 0$, as in reference \cite{Urban2019}, to transfer an angular momentum to the chain. Another possible future extension would be adding noises and perturbations to make the protocols robust with respect to them. Finally, specific protocols could be designed to simultaneously rotate longer chains of ions, although it is possible to sequentially rotate them in groups of 2 using the protocols designed here.
\appendix
\section{Magnetic force vs electric force\label{magnetic}}
Two charged particles moving in a direction perpendicular to the direction in which they are aligned experience a magnetic force, with magnitude
\begin{equation} F_{mag}=\frac{\mu_0}{4\pi}\frac{e^2}{r^2}|\vec{v}_1\times (\vec{v}_2\times\hat{r})|, \end{equation}
where
$\mu_0$ is the permeability constant, $\vec{v}_i$ the velocity vectors of each ion, $\vec{r}=\vec{s}_2-\vec{s}_1$ the position vector of ion 2 with reference to ion 1, and $\hat{r}=\vec{r}/r$. The Coulomb interaction, which is the only one considered so far, gives a force of magnitude
\begin{equation} F_{el}=\frac{C_c}{{r}^2}. \end{equation}
The ratio of these two forces is, using $\mu_0\epsilon_0=c^{-2}$, where $c$ is the speed of light,
\begin{equation} R=\frac{F_{mag}}{F_{el}}=\frac{ |\vec{v}_1\times (\vec{v}_2\times\hat{r})| }{ c^2 }. \end{equation}
With $|\vec{v}_1|,|\vec{v}_2|\approx \frac{r}{2}\dot{\theta}$
we get
\begin{equation} R\approx \frac{r^2\dot{\theta}^2}{4c^2}. \end{equation}
For the protocols designed in the main text, the maximum values during the simulations at the represented times are $\dot{\theta}_{max}=5\times10^6$ s$^{-1}$ and $r_{max}=5.5\times 10^{-6}$ m so
the magnetic interaction is negligible with respect to the electric force.
\section{Rotation of two different species ions based on dynamical normal modes\label{dyndecoupling}}
\begin{figure}
\caption{ (a) Normal mode excitation $\Delta E=E(t_f)-E(0)$ in units of the initial energy $E_0\equiv E(0)$ for different final times. The protocol rotates a $^{40}$Ca$^+$ and a $^9$Be$^+$ ion in a double well potential with $m_1\omega_1^2=m_2\omega_2^2=-4.7$ pN/m and $\beta=0.52$ mN/m$^3$. The initial state is a product of the ground states of each normal mode and thus, the energy of the system is computed as $E=E''_{0+}+E''_{0-}$, see equation~(\ref{EnergyNM}). The solid red line represents a non-optimised protocol; blue dotted and black dashed and lines represent optimised protocols using one and two parameters respectively. (b) Initial potential configuration and (c) the required $\gamma(t)$, see equation~(\ref{dt}), for the protocol with two optimisation parameters ($c_3=0.0059$ and $c_4=0.0285$) and $t_f=1$ $\mu$s. (d) Corresponding evolution of the equilibrium positions, whose initial value is also represented in (b). The blue solid line is for $^{40}$Ca$^+$ and the green dashed line for $^9$Be$^+$. }
\label{Exc_gamma_V0}
\end{figure}
If the matrix ${\bm{\mathsf{v}}}$ is time dependent, the normal modes get decoupled if ${\mathsf{v}}_{11}={\mathsf{v}}_{22}$, see equation (\ref{decoupling_cond}), i.e.,
\begin{equation} \label{sepcon}
\frac 1 m_1 \frac{\partial^2 V}{\partial s_1^2}\bigg|_{s_1^{(0)}} =\frac 1 m_2\frac{\partial^2 V}{\partial s_2^2}\bigg|_{s_2^{(0)}}. \end{equation}
As explained in the main text, the rotation of different ions trapped by a rigid harmonic potential cannot be described in general in terms of dynamical normal modes. For a non-rigid one there is a formal solution which does not lead to practically useful boundary conditions. Here we consider different confining potentials that obey equation~(\ref{sepcon}), and thus allow us to inverse engineer the rotation using the Lewis-Riesenfeld family of invariants.
We use for the equilibrium positions the parametrisation $s_1^{(0)}=s_0-d/2$ and $s_2^{(0)}=s_0+d/2$, where $s_0$ is the middle point between them.
Specifically we consider a tilted double well potential, which combines a repulsive harmonic potential with the confinement provided by the quartic term and a linear term \cite{Sagesser2020},
\begin{equation} \label{quarticpot} V = \gamma(t) (s_1+s_2) + \frac 1 2 u_1(t) s_1^2+\frac 1 2 u_2(t) s_2^2 + \beta (s_1^4+s_2^4)+ \frac {C_c}{s_2-s_1}. \end{equation}
This gives the potential matrix
\begin{equation}
{\bm{\mathsf{v}}}=
\left(\begin{array}{lr}
\frac{\frac{2 C_c}{d^3}+u_1+12\left(-\frac{d}{2}+s_0\right)^{\!2}\beta}{m_1} & -\frac{2 C_c}{d^3\sqrt{m_1 m_2}}\\
-\frac{2 C_c}{d^3\sqrt{m_1 m_2}} & \frac{\frac{2 C_c}{d^3}+u_2+12\left(\frac{d}{2}+s_0\right)^{\!2}\beta}{m_2}
\end{array}\right). \nonumber
\end{equation}
The main-text equations from equation (\ref{effective}) to (\ref{omegas}) are still valid here. We assume that the controllable parameters are the linear potential and the rotation speed. Equation~(\ref{sepcon}) is satisfied whenever $d$ obeys
\begin{eqnarray} \label{eqd} &\,&
\Big\{ A({m_1}-{m_2}) ({u_1}-{u_2}) +d^2\left[6 A \beta ({m_1} + m_2) + ({m_1}-{m_2}) ({u_1}-{u_2})^2\right] \nonumber\\ &+& 24 \beta {C_c} d ({m_1}-{m_2}) +12 \beta ^2 d^6 ({m_1}-{m_2}) \Big\}=0, \end{eqnarray}
where we have defined
\begin{equation} A=\sqrt{d^3\! \left[24 \beta {C_c}\!-\!12 \beta^{2}
d^5\!-\!12 \beta d^3 ({u_1}\!+\!{u_2})\!+\!d ({u_1}\!-\!{u_2})^2\right]}. \end{equation}
The force $\gamma(t)$ that would produce the desired evolution for $d$ is
\begin{eqnarray} \gamma &=& \frac 1 {108 \beta ^2 d^6} \Big\{18 \beta {C_c} d^2 ({u_2}-{u_1})-24 \beta ^2 d^5 A - d ({u_1}-{u_2})^2 A \nonumber\\ &-& 6 \beta {C_c} A +36 \beta ^2 d^7 ({u_1}-{u_2}) + d^3 \left[-6 \beta A ( {u_1} + u_2) -({u_1}-{u_2})^3\right] \Big\}, \label{dt} \end{eqnarray}
and the corresponding evolution for the middle point between the ions is
\begin{equation} s_0=\frac{A +d^2 ({u_1}-{u_2})}{12 \beta d^3}. \label{mpdw} \end{equation}
The frequencies of the normal modes $\Omega_\pm$ can be analytically expressed in terms of $d$, the parameters that define the potential ($u_1$, $u_2$ and $\beta$) and the masses $m_1$ and $m_2$, but they are too lengthy to be reproduced here. Provided that equation~(\ref{eqd}) is satisfied, the rotation of the potential in equation~(\ref{quarticpot}) is governed by an uncoupled Hamiltonian of the form (\ref{18}), with the corresponding frequencies $\Omega_\pm$ and momentum shifts that read
\begin{equation} p_{0\pm}=\frac{1}{\sqrt{2}} \left[(\sqrt{m_1} \pm \sqrt{m_2}) \dot s + (\sqrt{m_2} \mp \sqrt{m_1}) \frac{\dot d}{2} \right]. \end{equation}
From here on the procedure to design the protocol is similar to the one explained in section \ref{equalionssec}. We start from the same ansatz for $\theta(t)$, see equation~(\ref{controlparameter}), which satisfies the boundary conditions~(\ref{cond1_2ion}) and (\ref{cond2_2ion}) by design, and search for the values of the free parameters that minimise the final excitation. Decoupling the dynamics of the system into independent dynamical normal modes, however, is more demanding here than for equal ions. We compute the necessary force, see equation~(\ref{dt}), and equilibrium positions, see Eqs.~(\ref{eqd}) and (\ref{mpdw}), for each test value of the free parameters in $\theta(t)$.
Figure \ref{Exc_gamma_V0}(a) shows that, for a rotation of a $^{40}$Ca$^+$ and a $^9$Be$^+$ ion chain, any of the protocols produce no excitations in the normal modes for processes as fast as $0.4$ $\mu$s.
It also illustrates the improvement of the results by increasing the number of free parameters for $\theta(t)$.
Normal mode excitation is an approximation of the exact excitation, nevertheless, our results suggest that performing the rotation with the double well may provide excitationless protocols at short time scales.
Figure \ref{Exc_gamma_V0}(b) and (c) depict, respectively, the initial potential and the required force $\gamma(t)$ for a specific rotation protocol using the tilted double well potential in equation~(\ref{quarticpot}). Notice that even the lowest value of the force, at boundary times, produces a considerable bias with little to none barrier potential between the two wells. Despite this, each equilibrium position, whose evolution is depicted in figure \ref{Exc_gamma_V0}(c), initially lays in its own well. This unusual potential shape would be the price to pay for mode separability. We note that a potential bias may be imposed or cancelled using STA methods as well \cite{Martinez-Garaot2015}.
\end{document} | arXiv |
\begin{document}
\title{Height estimates for Killing graphs} \author{Debora Impera} \address{Dipartimento di Matematica e Applicazioni\\ Universit\`a di Milano--Bicocca Via Cozzi 53\\ I-20125 Milano, ITALY} \email{[email protected]} \author{Jorge H. de Lira} \address{Departamento de Matem\'atica\\Universidade Federal do Cear\'a-UFC\\60455-760 Fortaleza, CE, Brazil.} \email{[email protected]} \author{Stefano Pigola} \address{Sezione di Matematica - DiSAT\\ Universit\'a dell'Insubria - Como\\ via Valleggio 11\\ I-22100 Como, Italy} \email{[email protected]} \author{Alberto G. Setti} \address{Sezione di Matematica - DiSAT\\ Universit\'a dell'Insubria - Como\\ via Valleggio 11\\ I-22100 Como, Italy} \email{[email protected]} \date{\today} \maketitle
\begin{abstract} The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height function in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled. \end{abstract}
\section*{Introduction and main results}
Let $\left(M, \langle \cdot, \cdot \rangle_M\right) $ be a complete, $(n+1) $-dimensional Riemannian manifold endowed with a complete Killing vector field $Y$ whose orthogonal distribution has constant rank $n$ and it is integrable. Let $(P, \langle \cdot, \cdot \rangle_P)$ be an integral leaf of that distribution equipped with its induced complete Riemannian metric $\langle \cdot, \cdot \rangle_P$. The flow $\vartheta: P \times \mathbb{R} \rightarrow M$ generated by $Y$ is an isometry between $M$ and the warped product $P\times_{e^{-\psi}}\mathbb{R} $ with metric \[ \langle \cdot, \cdot \rangle_{M}= \langle \cdot, \cdot \rangle_P+e^{-2\psi}\mathrm{d} s \otimes \mathrm{d} s \]
where $s$ is the flow parameter and $\psi = - \log |Y|$.
Let $\Omega\subset P$ be a possibly unbounded domain with regular boundary $\partial\Omega\neq\emptyset$. The {\it Killing graph} of a smooth function $u:\bar{\Omega}\rightarrow \mathbb{R} $ is the hypersurface $\Sigma\subset M$ parametrized by the map \[ X(x)=\vartheta(x,u(x)), \quad x\in \bar\Omega. \] Obviously, if $Y$ is a parallel vector field, then $M$ is isometric with the Riemannian product $P\times\mathbb{R} $ and the notion of a Killing graph reduces to that of a usual vertical graph.
The above terminology, together with some existence results, was first introduced by M. Dajczer, P.A. Hinojosa, and J.H. de Lira in \cite{DHL-CalcVar}. Since then, Killing graphs have become the subject of a systematic investigation both in order to understand their geometry and as a tool to study different problems such as the existence of solutions of the asymptotic Plateau problem in certain symmetric spaces; \cite{CR-Asian, Ri-preprint}.
The aim of this paper is to obtain quantitative height estimates for a smooth Killing graph \[ \Sigma=\mathrm{Graph}_{\bar \Omega}\left( u\right) \hookrightarrow M=P\times_{e^{-\psi}}\mathbb{R} \] parametrized over (the closure of) a possibly unbounded domain $\Omega\subset P$ and whose smooth boundary $\partial\Sigma\neq\emptyset$ is contained in the totally geodesic slice $P\times\{0\}$ of $M$.
When $\psi (x) \equiv \mathrm{const}$ and the ambient manifold is the Riemannian product $P\times\mathbb{R} $, it is well understood that quantitative a-priori estimates can be deduced by assuming that the mean curvature $H$ of the graph is constant (\textit{CMC graphs} for short). For bounded domains into the Euclidean plane $P=\mathbb{R} ^{2}$ this was first observed in seminal papers by E. Heinze, \cite{He}, and J. Serrin, \cite{Se}. More precisely, assume that $H>0$ with respect to the\textit{ downward pointing Gauss map} $\mathcal{N}$. Then, $\Sigma$ is confined into the slab $\mathbb{R} ^{2}\times \lbrack0,1/H]$ regardless of the size of the domain $\Omega$. This type of estimates has been recently extended to unbounded domains $\Omega \subset\mathbb{R} ^{2}$ by A. Ros and H. Rosenberg, \cite{RoRo-AJM}. Their technique, which is based on smooth convergence of CMC surfaces, requires strongly that the base leaf $P=\mathbb{R} ^{2}$ is homogeneous and cannot be trivially adapted to general manifolds. In the case of a generic base manifold $P$, and maintaining the assumption that the CMC vertical graph $\Sigma$ is parametrized over a bounded domain, the corresponding height estimates have been obtained by D. Hoffman, J.H. de Lira and H. Rosenberg, \cite{HLR-TAMS}, J.A. Aledo, J.M. Espinar and J.A. Galvez, \cite{AEG-Illinois}, L. Al\'\i as and M. Dajczer, \cite{AD-PEMS} etc. The geometry of $P$ enters the game in the form of curvature conditions, namely, the Ricci curvature of \ $P$ cannot be too much negative when compared with $H$. In particular, for non-negatively Ricci curved bases the height estimates hold with respect to any choice of $H>0$. In the very recent \cite{IPS-Crelle}, the boundedness assumption on the domain $\Omega$ has been replaced by a quadratic volume growth condition, thus obtaining a complete extension of the Ros-Rosenberg result to any complete manifold $P$ with non-negative sectional curvature and dimensions $n\leq4$. The restriction on the dimension is due to the fact that, up to now, it is not known whether CMC graphs over non-negatively curved manifolds $P$ are necessarily contained in a vertical slab. Granted this, the desired estimates can be obtained.
In a slightly different perspective, qualitative bounds of the height of CMC vertical graphs on bounded domains have been obtained by J. Spruck, \cite{S-PAMQ}. It is worth to observe that his technique, based on Serrin-type gradient estimates and Harnack inequalities, is robust enough to give a-priori bounds even in the case where the mean curvature is non-constant. Actually, it works even for Killing graphs up to using the work by M. Dajczer and J.H. de Lira, \cite{DL-Poincare}; see also \cite{DLR-JAM}. However, in the Killing setting, the problem of obtaining quantitative bounds both on bounded and\ on unbounded domains remained open.
Due to the structure of the ambient space, it is reasonable to expect that an a-priori estimate for the height function of a CMC Killing graph is sensitive of the deformation function ${\psi}$. In fact, since the length element of the fibre $\left\{ x\right\} \times\mathbb{R} $ is weighted by the factor $e^{-\psi(x)}$, a reasonable pointwise bound should be of the form \[
0 \leq e^{-\psi(x)}u(x) \lesssim \frac{1}{|H|}. \] Actually, the same weight $e^{-\psi(x)}$ appears also in the expression of the volume element of $\Sigma$ thus suggesting the existence of an intriguing interplay between Killing graph and smooth metric measure spaces (also called {\it weighted manifolds}). Since this interplay represents the leading idea of the entire paper we are going to take a closer look at how it arises.
In view of the fact that we are considering graphical hypersurfaces, weighted structures should appear both at the level of the base manifold $P$, where $\Sigma$ is parametrised, and at the level of the ambient space $M$, where $\Sigma$ is realised. In fact, these two weighted contexts will interact in the formulation of the main result. To begin with, we note that the induced metric on $\Sigma = \mathrm{Graph}_{\bar \Omega}(u)$ is given by \begin{equation} \langle \cdot, \cdot \rangle_{\Sigma} = \langle \cdot, \cdot \rangle_P+e^{-2\psi}\mathrm{d} u\otimes \mathrm{d} u.\label{1st} \end{equation} Thus, the corresponding Riemannian volume element $\mathrm{d} \Sigma$ has the expression \begin{equation}\label{volumelement} \mathrm{d}\Sigma= W e^{-\psi}\mathrm{d} P, \end{equation}
where $W= \sqrt{e^{2\psi} +|\nabla^Pu|^2}$ and $\mathrm{d} P$ is the volume element of $P$. As alluded to above, the special form of \eqref{volumelement}, when compared with the case of a product ambient space, suggests to switch the viewpoint from that of the Riemannian manifold $(P,\langle \cdot, \cdot \rangle_{P})$ to that of the smooth metric measure space \[ P_{\psi}:= (P,\langle \cdot, \cdot \rangle_{P},\mathrm{d} P_{\psi}) \] where we are using the standard notation \[ \mathrm{d} P_{\psi} = e^{-\psi} \mathrm{d} P. \] In particular, \[ \mathrm{d} \Sigma = W \mathrm{d} P_{\psi} \] and we are naturally led to investigate to what extent the geometry of $\Sigma$ is influenced by the geometry of the weighted space $P_{\psi}$. As we shall see momentarily, the geometry of $P_{\psi}$ will enter the game in the form of a growth condition on the weighted volume of its geodesic balls $B^{P}_{R}(o)$: \[ \mathrm{vol}_{\psi} (B^{P}_{R}(o)) = \int_{B^{P}_{R}(o)} \mathrm{d} P_{\psi}. \] In a different direction, we observe that the smooth metric measure space structure of $P_{\psi}$ extends to the whole ambient space up to identifying $\psi : P \to \mathbb{R}$ with the function $\bar \psi : P \times_{e^{-\psi}} \mathbb{R} \to \mathbb{R}$ given by \[ \bar \psi (x,s) = \psi (x). \] With a slight abuse of notation, we write \begin{equation}
M_{\psi} := (M, \langle \cdot, \cdot \rangle_{M}, \mathrm{d}_{\psi}M) = (P \times_{e^{-\psi}} \mathbb{R} , e^{-\psi} \mathrm{d} M) \end{equation} and we can consider the original Killing graph as an hypersurface \[ \Sigma = \mathrm{Graph}_{\bar \Omega}(u) \hookrightarrow M_{\psi}. \] Previous works on classical height estimates for CMC graphs show that the relevant geometry of the ambient space is subsumed to a condition on its Ricci tensor. Thus, if we think of realizing $\Sigma$ inside $M_{\psi}$ we can expect that height estimates need a condition on its Bakry-\'Emery Ricci tensor defined by \[ \mathrm{Ric}^{M}_{\psi} = \mathrm{Ric}^{M} + \mathrm{Hess}^{M}(\psi). \] We shall come back to this later on. Following \cite{DHL-CalcVar}, we now orient $\Sigma$ using the {\it upward pointing} unit normal \begin{equation} \label{N}
\mathcal{N} = \frac{e^{2\psi} Y - \vartheta_* \nabla^P u}{\sqrt{e^{2\psi}+|\nabla^P u|^2}}=\frac{1}{W}\big(e^{2\psi} Y - \vartheta_* \nabla^P u\big). \end{equation} Note that $\mathcal{N}$ is upward pointing in the sense that \begin{equation}\label{NY} \langle \mathcal{N}, Y\rangle_{M}=\frac{1}{W} >0. \end{equation} Let $H:\Omega\subseteq P \to \mathbb{R} $ be the corresponding mean curvature function. The weighted $n$-volume associated to (the restriction of) $\psi$ (to $\Sigma$) is defined by \begin{equation} \label{Apsi} \mathcal{A}_{\psi}[\Sigma] := \int_\Sigma \mathrm{d} \Sigma_{\psi}. \end{equation} We are not concerned with the convergence of the integral. Given a compactly supported variational vector field $Z$ along $\Sigma$ the first variation formula reads \begin{equation} \delta_Z \mathcal{A}_{\psi} = \int_\Sigma \left(\mathrm{div} ^\Sigma Z - \langle {\nabla}^{M} \psi, Z\rangle_{M}\right) \mathrm{d} \Sigma_{\psi}. \end{equation} In particular, if $Z = v \mathcal{N}$ for some $v\in C_{c}^\infty(\Sigma)$ we have \begin{equation} \delta_Z \mathcal{A}_{\psi} = \int_\Sigma \left(-nH - \langle \nabla^{M} \psi, \mathcal{N}\rangle_{M}\right) v \, \mathrm{d} \Sigma_{\psi}= -n\int_\Sigma H_{\psi} v \, \mathrm{d} \Sigma_\psi, \end{equation} where, using the definition proposed by M. Gromov, \cite{Gro}, \begin{equation} \label{Hpsi} H_{\psi} = H + \frac1n \langle {\nabla}^{M} \psi, \mathcal{N}\rangle_{M} \end{equation} is the {\it $\psi$-weighted mean curvature} of $\Sigma$.
The way we have followed to introduce the weighted structure on the ambient space $M$ may look the most natural: it is trivially compatible with the weighted structure of the base space $P_{\psi}$ and with the weighted height function of $\Sigma$. Moreover, the weight $\psi$ appears in the volume element of $\Sigma$. However, it is worth to note that this is not the only ``natural'' choice. This becomes clear as soon as we express the mean curvature (and its modified version) of $\Sigma = \mathrm{Graph}_{\Omega}(u)$ in the classical form of a capillarity equation. Indeed, it is shown in \cite{ DL-Poincare, DHL-CalcVar} that \begin{equation}\label{capillary-2} \mathrm{div} ^P\Big(\frac{\nabla^P u}{W}\Big) =n H_{-\psi}. \end{equation} which, in view of \eqref{Hpsi}, is completely equivalent to \begin{equation}\label{capillary-3} \mathrm{div} _{\psi}^P\left(\frac{\nabla^P u}{W} \right)= nH. \end{equation} Here, we are using the standard notation for the {\it weighted divergence} in $P_{\psi}$: \[ \mathrm{div}^{P}_{\psi}X \, = e^{\psi} \mathrm{div} (e^{-\psi} X) = \mathrm{div}^{P} X - \langle X, \nabla \psi\rangle_{P}. \] Thus, with respect to the capillarity type equation \eqref{capillary-2}, the natural and relevant weighted structure on $M$ arises from the weight $-\psi$ and we might be led to consider $\Sigma$ as an hypersurface in $M_{-\psi}$. On the other hand, the original choice $\psi$ fits very well into the capillarity type equation \eqref{capillary-3}. Both these weighted structures are relevant and a choice has to be made. To give an idea of this kind of duality between $\psi$ and $-\psi$ structures, we extend, in the setting of Killing graphs, the classical relation between the mean curvature of the graph and the isoperimetric properties of the parametrization domain. This is the content of the following weighted versions of a result by E. Heinz, \cite{He2}, S.S. Chern, \cite{Ch}, H. Flanders, \cite{Fl}, and I. Salavessa \cite{Salavessa-PAMS}. Define the ``standard'' and the``weighted'' Cheeger constants of a domain $\Omega$ by, respectively, \begin{equation}\nonumber \mathfrak{b}(\Omega) = \inf_{D} \frac{\mathrm{vol}(\partial D)}{\mathrm{vol}(D)}, \quad \mathfrak{b}_\psi (\Omega) = \inf_{D} \frac{\mathrm{vol}_\psi(\partial D)}{\mathrm{vol}_\psi(D)}, \end{equation} where $D$ is a bounded subdomain with compact closure in $\Omega$ and with regular boundary $\partial D \not= \emptyset$. \begin{proposition*} Let $\Sigma = \mathrm{Graph}_{\Omega}(u) \hookrightarrow P \times_{e^{-\psi}} \mathbb{R}$ be an $n$-dimensional Killing graph defined over the domain $\Omega\subset P$. Then, the mean curvature $H$ of $\Sigma$, and its weighted version $H_{-\psi}$, satisfy the following inequalities: \begin{equation}
n\inf_\Omega |H_{-\psi}|\le \mathfrak{b}(\Omega), \quad n\inf_\Omega |H|\le \mathfrak{b}_\psi (\Omega). \end{equation} In particular: \begin{itemize}
\item [(i)] If $\Omega \subset P_{\psi}$ has zero weighted Cheeger constant, and $\Sigma$ has constant mean curvature $H$, then $\Sigma$ is a minimal graph. \item [(ii)] If $\Omega \subset P$ has zero Cheeger constant, and $\Sigma \hookrightarrow M_{-\psi}$ has constant weighted mean curvature $H_{-\psi}$, then $\Sigma$ is a $(-\psi)$-minimal graph. \end{itemize}
\end{proposition*} Indeed, if $D \Subset P$ is a relatively compact domain in $P$ with boundary $\partial D = \Gamma$ and outward pointing unit normal $\nu_0\in TP$, integrating \eqref{capillary-3} and using the weighted version of the divergence theorem \[ \int_{D} \mathrm{div}_{\psi} Z\, \mathrm{d} P_{\psi} = \int_{\Gamma} \langle Z , \nu_{0} \rangle_{P}\, d\Gamma_{\psi} \] we obtain \[ \int_{D} nH \mathrm{d} P_{\psi} = \int_{\Gamma} \left\langle \frac{\nabla^P u}{W},\nu_0\right\rangle_{P} \, \mathrm{d} \Gamma_{\psi} \] If $H$ has constant sign, multiplying by $-1$ if necessary, we deduce that \begin{equation*}
\int_D n|H| \mathrm{d} P_{\psi} \leq \int_\Gamma \left\vert \left\langle \frac{\nabla^P u}{W},\nu_0 \right\rangle \right\vert \mathrm{d} \Gamma_{\psi} \end{equation*}
and recalling that $|\nabla^P u|/W\leq 1$, we conclude that \[
n\inf_D |H| \int_D \mathrm{d} P_{\psi} \leq \int_\Gamma \mathrm{d} \Gamma_{\psi}. \] that is \begin{equation}\label{Salavessa_weighted}
n \inf_D |H| \mathrm{vol}_{\psi}(D) \leq \mathrm{vol}_{\psi}(\partial D). \end{equation}
Note that \eqref{Salavessa_weighted} certainly holds even if $H$ has not constant sign, for then $\inf |H| =0$. In a completely similar way, starting from equation \eqref{capillary-2}, we obtain \begin{equation}\nonumber
n \inf_{D} |H_{\psi}| \mathrm{vol}(D) \leq \mathrm{vol}(\partial D). \end{equation} The desired conclusions now follow trivially.
This brief discussion should help to put in the appropriate perspective the following theorem which represents the main result of the paper.
\begin{theoremA} \label{th_fheightestimate}
Let $(M,\langle \cdot, \cdot \rangle_{M}) $ be a complete, $(n+1)$-dimensional Riemannian manifold endowed with a complete Killing vector field $Y$ whose orthogonal distribution has constant rank $n$ and it is integrable. Let $(P,\langle \cdot, \cdot \rangle_{P})$ be an integral leaf of that distribution and let $\Omega\subset P$ be a smooth domain. Set $\psi=-\log(|Y|)$ and assume that:
\begin{enumerate} \item[(a)] $-\infty<\inf_{{\Omega}} \psi \leq \sup_{{\Omega}} \psi<+\infty$;
\item[(b)] $\mathrm{vol}_{\psi}\left( B_{R}^{P}\cap\Omega\right) =\mathcal{O}(R^2) $, as $R\rightarrow+\infty$;
\item[(c)] $\mathrm{Ric}^{M}_\psi\geq 0$ in a neighborhood of $\Omega\times\mathbb{R} \subset M_{\psi}$.
\end{enumerate} Let $\Sigma=\mathrm{Graph}_{\bar \Omega}(u)$ be a Killing graph over $\bar \Omega$ with weighted mean curvature $H_{\psi} \equiv \mathrm{const} < 0$ with respect to the upward pointing Gauss map $\mathcal{N}$. Assume that:
\begin{enumerate} \item[(d)] The boundary $\partial\Sigma$ of $\Sigma$ lies in the slice $P\times\{0\}$;
\item[(e)] The weighted height function of $\Sigma$ is bounded: $\sup_{\Sigma} |u e^{-\psi}|< +\infty$. \end{enumerate}
\noindent Then \begin{equation} \label{hest}
0\leq u(x) e^{-\psi(x)} \leq\frac{C}{|H_{\psi}|}, \end{equation} where $C:=e^{2(\sup_\Omega \psi-\inf_\Omega\psi)}\geq 1$. \end{theoremA} It is worth pointing out that the constant $C$ in \eqref{hest} depends only on the variation of $\psi$ .
The strategy of proof follows the main steps in \cite{IPS-Crelle}. For instance, in order to to get the upper estimate, we will use potential theoretic properties of the weighted manifold with boundary $\Sigma_{\psi} = (\Sigma, \langle \cdot, \cdot \rangle_{\Sigma}, \mathrm{d} \Sigma_{\psi})$. The idea is to show that, thanks to the capillarity equation \eqref{capillary-2}, the moderate volume growth assumption (b) on $\Omega$ is inherited by $\Sigma_{\psi}$. Therefore, the weighted Laplacian, defined on a smooth function $w : \Sigma \to \mathbb{R}$ by \[ \Delta^{\Sigma}_{\psi} w = \mathrm{div}^{\Sigma}_{\psi}(\nabla^{\Sigma} w) = \Delta^{\Sigma} w - \langle \nabla^{\Sigma} \psi, \nabla^{\Sigma} w \rangle_{\Sigma}, \] satisfies a global maximum principle similar to that valid on a compact set. And it is precisely in the compact setting that, throughout the construction of explicit examples, we shall show that our height estimate is essentially sharp. Moreover, in the spirit of known results in the product case (see above), since we do not impose any restriction on the size of the mean curvature, the ambient space $M_{\psi}$ is assumed to have non-negative weighted (i.e. Bakry-\'Emery) Ricci curvature. Note however, see Remark~\ref{RmkHLR}, that the result extends to the case where $\mathrm{Ric}^{M}_\psi$ is bounded below by a negative constant, provided a suitable bound on $H^2$ is imposed.
On the other hand, to show that $u\geq 0$ one uses the parabolicity of $\Delta^\Sigma_{3\psi}$. This provides further instance of the interplay between different weight structures on $M.$
It is worth to point out that, as it often happens in submanifolds theory, the case $n=2$ is very special. In fact, for two dimensional Killing graphs, one can show that the curvature assumption (c) implies the boundedness condition (e). Therefore, the previous result takes the following striking form.
\begin{corollaryA} \label{coro_fheightestimate}
Let $(M,\langle \cdot, \cdot \rangle_{M}) $ be a complete, $3$-dimensional Riemannian manifold endowed with a complete Killing vector field $Y$ whose orthogonal distribution has constant rank $2$ and it is integrable. Let $(P,\langle \cdot, \cdot \rangle_{P})$ be an integral leaf of that distribution and let $\Omega\subset P$ be a smooth domain. Set $\psi=-\log(|Y|)$ and assume that: \begin{enumerate} \item[(a)] $-\infty<\inf_{{\Omega}} \psi \leq \sup_{{\Omega}} \psi<+\infty$;
\item[(b)] $\mathrm{vol}_{\psi}\left( B_{R}^{P}\cap\Omega\right) = \mathcal{O}(R^2) $, as $R\rightarrow+\infty$;
\item[(c)] $\mathrm{Ric}^{M}_\psi\geq 0$ in a neighborhood of $\Omega\times\mathbb{R} $.
\end{enumerate} Let $\Sigma=\mathrm{Graph}_{\Omega}(u)$ a be a $2$-dimensional Killing graph over $\Omega$ with constant weighted mean curvature $H_{\psi} \equiv \mathrm{const}<0$ with respect to the upward pointing Gauss map and with boundary $\partial \Sigma \subset P\times\{0\}$.
\noindent Then \[
0\leq u e^{-\psi} (x) \leq\frac{C}{|H_{\psi}|}, \] where $C:=e^{2(\sup_\Omega \psi-\inf_\Omega\psi)}\geq 1$. \end{corollaryA}
The organization of the paper is as follows:
\noindent In Section \ref{PotTheory} we prove some potential theoretic properties of weighted manifolds with boundary, related to global maximum principles for the weighted Laplacian. These results, using new direct arguments, extend to the weighted context previous investigation in \cite{IPS-Crelle}.
\noindent Section \ref{HeightEst} contains the proof of the quantitative height estimates for Killing graphs over possibly unbounded domains. To this end, we shall introduce: (i) some basic formulas for the weighted Laplacian of the height and the angle functions of the Killing graph and (ii) weighted volume growth estimate that will enable us to apply the global maximum principles obtained in Section \ref{PotTheory}.
\noindent In Section \ref{section-examples} we construct concrete examples of Killing graphs with constant weighted mean curvature which, in particular, show that the constant $C$ in our estimate has the correct functional dependence on the variation $\sup \psi-\inf \psi$ of $\psi$.\\
\noindent{\bf Acknowledgments.}
It is our pleasure to thank the anonymous referee for a very careful reading of the manuscript and for important suggestions that greatly improved the exposition.
\section{Some potential theory on weighted manifolds}\label{PotTheory} The aim of this section is to study potential theoretic properties, and, more precisely, parabolicity with respect to the weighted Laplacian, of a weighted Riemannian manifold with possibly empty boundary. This relies on the notion of weak sub (super) solution subject to Neumann boundary conditions that we are going to introduce. In order to avoid confusion, and since we are dealing with general results valid on any weighted manifold, throughout this Section we shall call $f:M \to \mathbb{R}$ the weight function. The symbol $\psi$ is thus reserved to the peculiar weight related to Killing graphs.
Let $M_f = (M,\langle \cdot, \cdot \rangle,e^{-f}\mathrm{d} M)$ be a smooth, $n$-dimensional, weighted manifold with smooth boundary $\partial M\neq\emptyset$ oriented by the exterior unit normal $\nu.$ The interior of $M$ is denoted by $\mathrm{int}M = M \setminus \partial M$. By a domain in $M$ we mean a non-necessarily connected open set $D\subseteq M$. We say that the domain $D$ is smooth if its topological boundary $\partial D$ is a smooth hypersurface $\Gamma$ with boundary $\partial\Gamma=\partial D\cap\partial M$. Adopting the notation in \cite{IPS-Crelle}, for any domain $D\subseteq M$ we define \begin{align*} \partial_{0}D&=\partial D\cap\mathrm{int}M,\\ \partial_{1}D&=\partial M\cap D \end{align*} We will refer to $\partial_0 D$ and to $\partial_1 D$ respectively as the \textit{Dirichlet boundary} and the \textit{Neumann boundary} of the domain $D$. Finally, the \textit{interior part} of $D$, in the sense of manifolds with boundary, is defined as \[
\mathrm{int}D=D \cap \mathrm{int}M, \] so that, in particular, \[ D= \mathrm{int}D \cup \partial_1 D. \]
We recall that the Sobolev space $W^{1,2}( \mathrm{int}D_f)$ is defined as the Banach space of functions $u \in L^2( \mathrm{int}D_f)$ whose distributional gradient satisfies $\nabla u \in L^2( \mathrm{int}D_f)$. Here we are using the notation \[ L^2( \mathrm{int}D_f):= \{w\, :\, \int_{D} w^2\mathrm{d} M_f<+\infty\}. \]
By the Meyers-Serrin density result, this space coincides with the closure of $C^{\infty}(\mathrm{int}D)$ with respect to the Sobolev norm $\|u\|_{W^{1,2}(M_f)}=\|u\|_{L^2(M_f)}+\|\nabla u\|_{L^2(M_f)}$. Moreover, when $D=M$ and $M$ is complete, $W^{1,2}(\mathrm{int}M_f)$ can be also realised as the $W^{1,2}(M_f)$-closure of $C^{\infty}_{c}(M)$.
Finally, the space $W^{1,2}_{\mathrm{loc}}(\mathrm{int}D_f)$ is defined by the condition that $u \cdot \chi \in W^{1,2}(\mathrm{int}D_f)$ for every cut-off function $\chi \in C^{\infty}_c(\mathrm{int}D)$. We extend this notion by including the Neumann boundary of the domain as follows \[ W^{1,2}_{\mathrm{loc}}(D_f)=\{u \in W^{1,2}(\mathrm{int}\Omega_f), \text{ }\forall \text{domain } \Omega \Subset D=\mathrm{int}D \cup \partial_1 D\}. \]
Now, suppose $D\subseteq M$ is any domain. We put the following Definition. Recall from the Introduction that the $f$-Laplacian of $M_{f}$ is the second order differential operator defined by \[ \Delta_{f} w = e^{f} \mathrm{div}(e^{-f} \nabla w) = \Delta w - \langle \nabla f , \nabla w \rangle, \] where $\Delta$ denotes the Laplace-Beltrami operator of $(M,\langle \cdot, \cdot \rangle)$. Clearly, as one can verify from the weighted divergence theorem, $-\Delta_{f}$ is a non-negative, symmetric operator on $L^{2}(M_{f})$. \begin{definition} By a weak Neumann sub-solution $u\in W_{loc}^{1,2}\left(D_f\right) $ of the $f$-Laplace equation, i.e., a weak solution of the problem \begin{equation} \left\{ \begin{array} [c]{ll} \Delta_f u\geq0 & \text{on }\mathrm{int}D\\ \dfrac{\partial u}{\partial\nu}\leq0 & \text{on }\partial_{1}D, \end{array} \right. \label{subneumannproblem} \end{equation} we mean that the following inequality \begin{equation} -\int_{D}\left\langle \nabla u,\nabla\varphi\right\rangle \mathrm{d} M_f\geq0 \label{fsubsol} \end{equation} holds for every $0\leq\varphi\in C_{c}^{\infty}\left( D\right) $. Similarly, by taking $D=M$, one defines the notion of weak Neumann subsolution of the $f$- Laplace equation on $M_f$ as a function $u\in W_{loc}^{1,2}\left( M_f\right) $ which satisfies (\ref{fsubsol}) for every $0\leq\varphi\in C_{c}^{\infty }\left( M\right) $. As usual, the notion of weak supersolution can be obtained by reversing the inequality and, finally, we speak of a weak solution when the equality holds in (\ref{fsubsol}) without any sign condition on $\varphi$. \end{definition}
\begin{remark} \rm{ Analogously to the classical case, in the above definition, it is equivalent to require that (\ref{fsubsol}) holds for every $0\leq\varphi\in \mathrm{Lip}_{c}\left( D\right) $. } \end{remark}
Following the terminology introduced in \cite{IPS-Crelle}, we are now ready to give the following \begin{definition} \label{def_fparab} A weighted manifold $M_f$ with boundary $\partial M\neq\emptyset$ oriented by the exterior unit normal $\nu$ is said to be $f$-parabolic if any bounded above, weak Neumann subsolution of the $f$-Laplace equation on $M_f$ must be constant. Namely, for every $u\in C^{0}\left( M\right) \cap W_{loc}^{1,2}\left( M_f\right) $, \begin{equation} \label{def_fpar} \begin{array} [c]{ccc} \left\{ \begin{array} [c]{ll} \Delta_f u\geq0 & \text{on }\mathrm{int}M\\ \dfrac{\partial u}{\partial\nu}\leq0 & \text{on }\partial M\\ \sup_{M}u<+\infty & \end{array} \right. & \Rightarrow & u\equiv\mathrm{const}. \end{array} \end{equation}
\end{definition}
In the boundary-less setting it is by now well-known that $f$--parabolicity is related to a wide class of equivalent properties involving the recurrence of the Brownian motion, $f$--capacities of condensers, the heat kernel associated to the drifted laplacian, weighted volume growth, function theoretic tests, global divergence theorems and many other geometric and potential-analytic properties. All these characterization can be proven to hold true also in case of weighted manifolds with non-empty boundaries. However, here we limit ourselves to pointing out the following characterization.
\begin{theorem}\label{thm_fAhlfors} A weighted manifold $M_f$ is $f$-parabolic if and only the following maximum principle holds. For every domain $D\subseteq M$ with $\partial_{0}D\neq\emptyset$ and for every $u\in C^{0}\left( \overline{D}\right) \cap W_{loc}^{1,2}\left( D_f\right) $ satisfying \[ \left\{ \begin{array} [c]{ll} \Delta_f u\geq0 & \text{on }\mathrm{int}D\\ \dfrac{\partial u}{\partial\nu}\leq0 & \text{on }\partial_{1}D\\ \sup\limits_{D}u<+\infty & \end{array} \right. \] in the weak sense, it holds \[ \sup_{D}u=\sup_{\partial_{0}D}u. \] Moreover, if $M_f$ is a $f$-parabolic manifold with boundary $\partial M\neq\emptyset$ and if $u\in C^{0}\left( M\right) \cap W_{loc} ^{1,2}\left( \mathrm{int}M_f\right) $ satisfies \[ \left\{ \begin{array} [c]{ll} \Delta_f u\geq0 & \text{on }\mathrm{int}M\\ \sup_{M}u<+\infty & \end{array} \right. \] then \[ \sup_{M}u=\sup_{\partial M}u. \] \end{theorem}
We refer the reader to \cite[Theorem 0.9]{IPS-Crelle} for a detailed proof of the previous result in the unweighted setting. Although the proof of this theorem can be deduced adapting to the weighted laplacian $\Delta_{f}$ the arguments in \cite{IPS-Crelle} (indeed, all results obtained there can be adapted to the weighted case with only minor modifications of the proofs), we provide here a shorter and more elegant argument. In order to do this we will need the following preliminary fact that will be proved without the use of any capacitary argument and, therefore, can be adapted to deal also with the $f$-parabolicity under Dirichlet boundary conditions; see \cite{PPS-L1Liouville} for old results and recent advances on the Dirichlet parabolicity of (unweighted) Riemannian manifolds.
\begin{proposition}\label{prop_equivfpar} Let $M_f$ be a weighted manifold with boundary $\partial M$ oriented by the exterior unit normal $\nu$ and consider the warped product manifold $\hat{M}=M\times_{e^{-f}}\mathbb{T}$, with boundary $\partial\hat{M}=\partial{M}\times\mathbb{T}$ oriented by the exterior unit normal $\hat \nu(x,\theta) = (\nu(x),0)$. Here we are setting $\mathbb{T}=\mathbb{R} /\mathbb{Z}$, normalized so that $\mathrm{vol}(\mathbb{T})=1$. Then $M_f$ is $f$-parabolic if and only if $\hat M$ is parabolic. \end{proposition} \begin{proof} To illustrate the argument we are going use pointwise computations for $C^{2}$ functions which however can be easily formulated in weak sense for functions in $C^{0}\cap W^{1,2}_{loc}$. We also recall that the $f$-Laplacian $\Delta_{f}$ on $M$ is related to the Laplace-Beltrami operator $\hat\Delta$ of $\hat M$ by the formula \[ \hat\Delta = \Delta_f + f^{-2} \Delta_{\mathbb{T}}. \] With this preparation, assume first that $\hat{M}$ is parabolic. We have to show that any (regular enough) solution of the problem \[ \left\{ \begin{array} [c]{ll} {\Delta}_{f} u\geq0 & \text{on }\mathrm{int}{M}\\ \dfrac{\partial u}{\partial {\nu}}\leq0 & \text{on }\partial {M}\\ \sup_{{M}} u<+\infty & \end{array} \right. \] must be constant. To this end, having selected $u$ we simply define $\hat u(x,t)=u(x)$, $(x,t)\in\hat{M}$, and we observe that $\hat u$ satisfies the analogous problem on $\hat M$. Since $\hat M$ is parabolic, $\hat u$ and hence $u$ must be constant, as required.
Conversely, assume that $M_f$ is $f$-parabolic and let $u$ be a (regular enough) solution of the problem \begin{equation} \label{subneumannproblem2} \begin{array} [c]{ccc} \left\{ \begin{array} [c]{ll} \hat \Delta u\geq0 & \text{on }\mathrm{int} \hat{M}\\ \dfrac{\partial u}{\partial \hat\nu}\leq0 & \text{on }\partial \hat{M}=\partial M\times \mathbb{T}\\ \sup_{\hat M}u<+\infty. & \end{array} \right. \end{array} \end{equation} By translating and scaling we may assume that $\sup u=1 $ and since $\max\{u,0\}$ is again a solution of \eqref{subneumannproblem2}, we may in fact assume that $0\leq u\leq 1$. Let \[ \bar{u}(x) = \int_{\mathbb{T}} u(x,t)dt. \] Recalling that rotations in $\mathbb{T}$ are isometries of $\hat{M}$, and therefore commute with the Laplacian $\hat \Delta$, we deduce that \[ \Delta_{f} \bar u (x) = \hat \Delta \bar u (x)= \int_{\mathbb{T}} \hat \Delta u (x,t) dt \geq 0 \] and \[ \frac{\partial \bar u}{\partial \nu}\leq 0, \] so, by the assumed $f$-parabolicity of $M_f$, $\bar u$ is constant, and \[ 0=\Delta_f \bar{u}=\int_{\mathbb{T}} \hat \Delta u (x,t). \] Since $\hat \Delta u \geq 0$ we conclude that $\hat \Delta u=0$ in $\mathrm{int} \,\hat{M}$ . Applying the above argument to $u^2$, which is again a solution of \eqref{subneumannproblem2}, we obtain that \[
0=\hat \Delta u^2 = 2 u \hat \Delta u + 2 |\hat \nabla u|^2 = 2 |\hat \nabla u|^2 \quad \text{ in } \mathrm{int} \, \hat{M}, \] and therefore $u$ is constant, as required. \end{proof}
We are now ready to present the
\begin{proof}[Proof of Theorem \ref{thm_fAhlfors}] Assume first that $M_f$ is $f$-parabolic. As a consequence of Proposition \ref{prop_equivfpar} this is equivalent to the parabolicity of $\hat{M}$ which, in turns, is equivalent to the validity of the following Ahlfors-type maximum principle (see \cite[Theorem 0.9]{IPS-Crelle}). For every domain $\hat{D}\subseteq \hat{M}$ with $\partial_{0}\hat{D}\neq\emptyset$ and for every $u\in C^{0}(\overline{\hat{D}}) \cap W_{loc}^{1,2}( \hat{D}) $ satisfying \begin{equation}\label{eq_ahlforsmhat} \left\{ \begin{array} [c]{ll} \hat{\Delta} u\geq0 & \text{on }\mathrm{int}\hat{D}\\ \dfrac{\partial u}{\partial\hat{\nu}}\leq0 & \text{on }\partial_{1}\hat{D}\\ \sup\limits_{\hat{D}}u<+\infty & \end{array} \right. \end{equation} in the weak sense, it holds \[ \sup_{\hat{D}}u=\sup_{\partial_{0}\hat{D}}u. \] Furthermore, in case $\hat{D}=\hat{M}$, for every $u\in C^{0}(\overline{\hat{D}}) \cap W_{loc}^{1,2}( \hat{D}) $ satisfying \[ \left\{ \begin{array} [c]{ll} \hat{\Delta} u\geq0 & \text{on }\mathrm{int}\hat{M}\\ \sup\limits_{\hat{M}}u<+\infty & \end{array} \right. \] in the weak sense, it holds \[ \sup_{\hat{M}}u=\sup_{\partial_{0}\hat{M}}u. \] In particular, if $D\subset M$ is any smooth domain with $\partial_0 D\neq\emptyset$ and if $u\in C^{0}\left( \overline{D}\right) \cap W_{loc}^{1,2}\left(D\right) $ satisfies \[ \left\{ \begin{array} [c]{ll} \Delta_f u\geq0 & \text{on }\mathrm{int}D\\ \dfrac{\partial u}{\partial\nu}\leq0 & \text{on }\partial_{1}D\\ \sup\limits_{D}u<+\infty & \end{array} \right. \] in the weak sense, then $\hat{u}(x,t)=u(x)$ is a solution of \eqref{eq_ahlforsmhat} on $\hat{D}\times\mathbb{T}$. Hence, the parabolicity $\hat{M}$ in the form of the Ahlfors-type maximum principle implies that \[ \sup_Du=\sup_{\hat{D}}\hat{u}=\sup_{\partial_{0}\hat{D}=\partial_0D\times\mathbb{T}}\hat{u}=\sup_{\partial_0D}u. \] The same reasoning applies in case $D=M$.
Conversely, assume that for every domain $D\subseteq M$ with $\partial_{0}D\neq\emptyset$ and for every $u\in C^{0}\left( \bar{D}\right) \cap W_{loc}^{1,2}( D_f) $ satisfying \[ \left\{ \begin{array} [c]{ll} \Delta_f u\geq0 & \text{on }\mathrm{int}D\\ \dfrac{\partial u}{\partial\nu}\leq0 & \text{on }\partial_{1}D\\ \sup\limits_{D}u<+\infty & \end{array} \right. \] in the weak sense, it holds \[ \sup_{D}u=\sup_{\partial_{0}D}u. \] Suppose by contradiction that $M_f$ is not $f$-parabolic. Then there exists a non-constant function $v\in C^{0}\left( M\right) \cap W_{loc}^{1,2}\left( M_f\right) $ satisfying \[ \left\{ \begin{array} [c]{ll} \Delta_f v\geq0 & \text{on }\mathrm{int}M\\ \dfrac{\partial v}{\partial\nu}\leq0 & \text{on }\partial_{1}M\\ \sup\limits_{M}v<+\infty & \end{array} \right. \] in the weak sense. Given $\eta<\sup_M v$ consider the domain $D_{\eta}=\{x\in M:v(x)>\eta \}\neq\emptyset$. We can choose $\eta$ sufficiently close to $\sup_M v$ in such a way that $\mathrm{int}M\not \subseteq D_{\eta}$. In particular, $\partial D_{\eta}\subseteq\left\{ v=\eta\right\} $ and $\partial _{0}D_{\eta}\neq\emptyset$. Now, $v\in C^{0}\left( \overline{D }_{\eta}\right) \cap W_{loc}^{1,2}\left( (D_{\eta})_f\right) $ is a bounded above weak Neumann subsolution of the $f$-Laplacian equation on $D_{\eta}$. Moreover, \[ \sup_{\partial_{0}D_{\eta}}{v}=\eta<\sup_{D_{\eta}}{v}, \] contradicting our assumptions. \end{proof}
From the geometric point of view, it can be proved that $f$--parabolicity is related to the growth rate of the weighted volume of intrinsic metric objects. Indeed, exploiting a result due to A. Grigor'yan, \cite{Gr1}, one can prove the following result; see also Theorem 0.7 and Remark 0.8 in \cite{IPS-Crelle}). For the sake of clarity, we recall from the Introduction that the weighted volume of the metric ball $B^{M}_{R}(o) = \{ x\in M : \mathrm{dist}_{M}(x,o)<R\}$ of $M_{f}$ is defined by $\mathrm{vol}_{f}(B^{M}_{R}\!(o)) = \int_{B^{M}_{R}\!(o)} \mathrm{d} M_{f}$. \begin{proposition}\label{prop-fparab-volume} Let $M_f$ be a complete weighted manifold with boundary $\partial M\neq\emptyset$. If, for some reference point $o\in M$, \[ \mathrm{vol}_f (B_{R}^{M}\!(o) ) =\mathcal{O}(R^2), \quad \mathrm{as}\quad R\rightarrow+\infty. \] then $M_f$ is $f$-parabolic. \end{proposition} \begin{proof} Set $\Omega_R:=B_{R}^{M}(o)\times \mathbb{T}$. Then, as a consequence of Fubini's Theorem, \[ \mathrm{vol}_f(B_{R}^{M} \!(o))=\int_{B_{R}^{M}\!(o)}\mathrm{d} M_f=\int_{\Omega_R}\mathrm{d} \hat{M}=\mathrm{vol}(\Omega_R). \] Denote by $B_R^{\hat{M}}\!(\hat{o})$ the geodesic ball in $\hat{M}$ with reference point $\hat{o}=(o,\hat{t})$. Given an arbitrary point $\hat{x}=(x,t)\in\hat{M}$, let $\hat{\alpha}=(\alpha,\beta):[0,1]\rightarrow \hat{M}$ be a curve in $\hat{M}$ such that $\hat{\alpha}(0)=\hat{o}$ and $\hat{\alpha}(1)=\hat{x}$. Then \begin{align*}
\ell(\hat{\alpha})=&\int_{0}^{1}\|\hat{\alpha}^{\prime}(s)\|\mathrm{d} s\\ =&\int_{0}^{1}\sqrt{({\alpha}^{\prime}(s))^2+e^{-2 f(\alpha(s))}({\beta}^{\prime}(s))^2}\mathrm{d} s\\
\geq &\int_{0}^{1}\|\alpha^{\prime}(s)\|\mathrm{d} s\\ \geq & \textrm{dist}_M(x,o). \end{align*} Hence, as a consequence of the previous chain of inequalities, if $\hat{x}\in B_R^{\hat{M}}\!(\hat{o})$, then $x\in B_R^{M}\!(o)$, which in turns implies that \[ B_R^{\hat{M}}\!(\hat{o})\subseteq \Omega_R. \] In particular, \[ \mathrm{vol}(B_R^{\hat{M}}\!(\hat{o}))\leq \mathrm{vol}(\Omega_R)\leq C R^2 \] for $R$ sufficiently large. The conclusion now follows from the above mentioned result by Grigor'yan and from the fact that, as proven in Proposition \ref{prop_equivfpar}, the parabolicity of $\hat{M}$ is equivalent to the $f$-parabolicity of $M_f$. \end{proof}
\section{The height estimates}\label{HeightEst} In this section we prove the main result of the paper, Theorem \ref{th_fheightestimate}, and show how the boundedness assumption can be dropped in dimension $2$; Corollary \ref{coro_fheightestimate}. To this end, we need some preparation: first we derive some basic formulas concerning the weighted Laplacian of the height function and the angle function of the Killing graph; see Proposition \ref{prop-formulas}. Next we extend to the weighted setting and for Killing graphs a crucial volume estimate obtained in \cite{IPS-Crelle, LW}; see Lemma \ref{lemma_volume}. This estimate will allow us to use the global maximum principles introduced in Section \ref{PotTheory}. The proofs of Theorem \ref{th_fheightestimate} and of its Corollary \ref{coro_fheightestimate} will be provided in Section \ref{subsection-proof}.
\subsection{Some basic formulas} This section aims to prove the following \begin{proposition}\label{prop-formulas}
Let $( M, \langle \cdot, \cdot \rangle_{M})$ be a complete $(n+1)$-dimensional Riemannian manifold endowed with a complete Killing vector field $Y$ whose orthogonal distribution has constant rank $n$ and is integrable. Let $(P,\langle \cdot, \cdot \rangle_{P}) $ be an integral leaf of that distribution and let $\Sigma=\mathrm{Graph_{\Omega}}(u)$ be a Killing graph over a smooth domain $\Omega\subset P$, with upward unit normal $\mathcal{N}$. Set $\psi=-\log|Y|$. Then, for any constant $C \in \mathbb{R}$ the following equations hold on $(\Sigma, \langle \cdot, \cdot \rangle)$: \begin{eqnarray} \label{eq-Deltafu} \Delta^{\Sigma}_{C\psi}u & = & \left( nH+(C-2)\langle {\nabla}^{M} \psi, \mathcal{N} \rangle_{M} \right) e^{2\psi}\langle Y,\mathcal{N} \rangle_{M},\\ \label{eq-DeltafAngle} \Delta^{\Sigma}_{C\psi}\langle Y, \mathcal{N} \rangle_{M} & = & -\langle Y, \mathcal{N}\rangle_{M}
\left( |A|^2+\mathrm{Ric}^{M}_{C\psi}(\mathcal{N},\mathcal{N}) \right)- nY^T(H_{C\psi}), \end{eqnarray}
where $\mathrm{Ric}^{M}_{C\psi}$ is the Bakry-\'Emery Ricci tensor of the weighted manifold $M_{C\psi}$, $|A|$ is the norm of the second fundamental form of $\Sigma$ and $H$, $H_{C\psi}$ denote respectively the mean curvature of $\Sigma$ and its $C\psi$-weighted modified version. \end{proposition} \begin{proof} Observe that \[
u=s|_{\Sigma}, \] where, we recall, $s$ is the flow parameter of $Y$. Using ${\nabla}s=e^{2\psi} Y$, we have \[ \nabla^{\Sigma} u = \nabla^{\Sigma} s = e^{2\psi} Y^T. \] Thus, letting $\{e_i\}$ be an orthonormal basis of $T\Sigma$, and recalling that, since $Y$ is Killing, $\langle \nabla^{M}_V Y, V\rangle_{M} = 0$ for every vector $V$, we compute \begin{align*} & \Delta^{\Sigma}u=\sum_{i=1}^{n}\langle\nabla_{e_{i}}^{\Sigma}\nabla^{\Sigma}u,e_{i}\rangle\\ &\,\, = \sum_{i=1}^{n}\langle\nabla^{\Sigma}_{e_{i}} e^{2\psi}Y^{T},e_{i}\rangle\\ &\,\, = e^{2\psi}\sum_{i=1}^{n}\langle{\nabla}^{M}_{e_{i}}(Y-\langle Y,\mathcal{N} \rangle_{M} \mathcal{N}),e_{i}\rangle_{M} +2\sum_{i=1}^{n}\langle e_{i},{\nabla^{\Sigma}}\psi\rangle\langle e_{i},{\nabla^{\Sigma}}s\rangle\\ &= e^{2\psi}\sum_{i=1}^{n}\langle{\nabla}^{M}_{e_{i}}Y,e_{i}\rangle_{M} -\langle Y,\mathcal{N}\rangle_{M} e^{2\psi}\sum_{i=1}^{n}\langle{\nabla}^{M}_{e_{i}} \mathcal{N},e_{i}\rangle_{M} +2\sum_{i=1}^{n}\langle e_{i},\nabla^{\Sigma}\psi \rangle\langle e_{i},\nabla^{\Sigma}u\rangle\\ & \,\,=nHe^{2\psi}\langle Y,\mathcal{N} \rangle_{M}+2\langle\nabla^{\Sigma}\psi,\nabla^{\Sigma}u\rangle. \end{align*} Equation \eqref{eq-Deltafu} follows since, by definition, \[ \Delta_{C\psi}^{\Sigma}u= \Delta^\Sigma u- C\langle\nabla^\Sigma \psi, \nabla^\Sigma u\rangle. \]
As for equation \eqref{eq-DeltafAngle}, note that, since $\mathcal{N}$ is a unit normal and $Y$ is a Killing vector field $\nabla^{M}_\mathcal{N} Y$ is tangent to $\Sigma$ and, for every vector $X$ tangent to $\Sigma$, \[ \langle \nabla^{M}_X \mathcal{N}, Y\rangle_{M} = \langle \nabla^{M}_X \mathcal{N}, Y^T + \langle Y, \mathcal{N} \rangle_{M} \mathcal{N} \rangle_{M} = \langle \nabla^{M}_X \mathcal{N}, Y^T\rangle_{M} = -\langle A_{\mathcal{N}}\,Y^T, X\rangle, \] so that \begin{equation*} \langle \nabla^\Sigma \langle Y,\mathcal{N} \rangle_{M}, X\rangle = X \langle Y,\mathcal{N}\rangle_{M} = -\langle \nabla^{M}_\mathcal{N} Y, X\rangle_{M} + \langle Y, \nabla^{M}_{X} \mathcal{N}\rangle_{M} = - \langle \nabla^{M}_\mathcal{N} Y + A_{\mathcal{N}} \, Y^T, X\rangle_{M} \end{equation*} and therefore \[ \nabla^{\Sigma}\langle Y, \mathcal{N} \rangle_{M}=-{\nabla}^{M}_\mathcal{N} Y- A_{\mathcal{N}} Y^T. \] Moreover, using the Codazzi equations, it is not difficult to prove that \[
\Delta^{\Sigma}\langle Y,\mathcal{N}\rangle_{M}=-\bigl(|A|^{2}+{\mathrm{Ric}^{M}}(\mathcal{N},\mathcal{N})\bigr)\langle Y, \mathcal{N} \rangle_{M}-nY^T(H), \] see, e.g., \cite[Prop. 1]{Fornari-Ripoll-Illinois}.
Using the definition of $H_{C\psi}= H+\frac 1n \langle \nabla^{M}( C\psi), \mathcal{N}\rangle_{M}$ we also have \begin{align*} -nY^T(H) &=-nY^T(H_{C\psi})+CY^T\langle {\nabla}^{M} {\psi}, \mathcal{N}\rangle_{M}\\ &=-nY^T(H_{C\psi})+C\langle{\nabla}^{M}_Y {\nabla}^{M} {\psi}, \mathcal{N} \rangle_{M} - C\langle Y,\mathcal{N}\rangle_{M} {\mathrm{Hess}^{M}}(\psi)(\mathcal{N},\mathcal{N})+ C\langle{\nabla}^{M} {\psi},{\nabla}^{M}_{Y^T} \mathcal{N}\rangle_{M}\\ &=-nY^T(H_{C\psi})+C\langle{\nabla}^{M}_{{\nabla} {\psi}} Y, \mathcal{N} \rangle_{M} -C\langle Y,\mathcal{N}\rangle_{M} {\mathrm{Hess}^{M}}(\psi)(\mathcal{N},\mathcal{N}) + C\langle{\nabla}^\Sigma {\psi},({\nabla}^{M}_{Y^T} \mathcal{N})^{T}\rangle\\ &=-nY^T(H_{C\psi})+C\langle\nabla^\Sigma {\psi},\nabla^\Sigma \langle Y, \mathcal{N} \rangle_{M} \rangle -C\langle Y,\mathcal{N}\rangle_{M} {\mathrm{Hess}}^{M}(\psi)(\mathcal{N},\mathcal{N}), \end{align*} where we have used that:\\ \begin{itemize}
\item [-] $\nabla^{M}_Y \nabla^{M}\psi = \nabla^{M}_{\nabla^{M} \psi}Y,$ since $\psi$ depends only on the $P$-variables and $Y=\partial_{s}$;
\item [-] $\langle \nabla^{M}_{\nabla^{M} \psi} Y, \mathcal{N} \rangle_{M} = -\langle \nabla^{M}_\mathcal{N} Y,\nabla^{M} \psi\rangle_{M}$ and $\langle \nabla^{M}_\mathcal{N} Y, \mathcal{N} \rangle_{M} =0$, since $Y$ is Killing;
;
\item [-] $\langle\nabla^{M} \psi, \nabla^{M}_{Y^T} \mathcal{N} \rangle_{M} = \langle \nabla^\Sigma \psi, (\nabla^{M}_{Y^T}\mathcal{N})^{T} \rangle$, since $\langle \mathcal{N}, \nabla^{M}_{Y^T}\mathcal{N} \rangle_{M}
= \frac 12 Y^T\langle \mathcal{N}, \mathcal{N} \rangle_{M} = 0$;
\item [-] the identity \begin{equation*} \begin{split} \nabla^\Sigma \psi\langle Y, \mathcal{N} \rangle_{M} &= \langle \nabla^{M}_{\nabla^\Sigma \psi} Y, \mathcal{N} \rangle_{M} + \langle Y, \nabla^{M}_{\nabla^\Sigma \psi} \mathcal{N} \rangle_{M}\\ &= \langle \nabla^{M}_{\nabla^{M} \psi} Y, \mathcal{N} \rangle_{M} - \langle \mathcal{N} , \nabla^{M} \psi \rangle_{M} \langle \nabla^{M}_\mathcal{N} Y, \mathcal{N}\rangle_{M}\\ &\,\,\,\,\,\, +\langle Y^T , (\nabla_{\nabla^\Sigma \psi} \mathcal{N})^{T}\rangle + \langle \mathcal{N}, Y \rangle_{M} \langle \mathcal{N}, \nabla^{M}_{\nabla^\Sigma \psi} \mathcal{N} \rangle_{M}\\ &= \langle \nabla^{M}_{\nabla^{M} \psi} Y, \mathcal{N} \rangle_{M} +\langle Y^T ,(\nabla^{M}_{\nabla^\Sigma \psi} \mathcal{N})^{T} \rangle\\ &=\langle \nabla^{M}_{\nabla^{M} \psi} Y, \mathcal{N} \rangle_{M} +\langle \nabla^\Sigma \psi ,(\nabla^{M}_{Y^T} \mathcal{N})^{T} \rangle. \end{split} \end{equation*}
\end{itemize} Inserting the above identities into \[ \Delta^\Sigma_{C\psi}\langle Y, \mathcal{N} \rangle_{M} = \Delta^\Sigma \langle Y,\mathcal{N} \rangle_{M} - C\langle \nabla^\Sigma \psi, \nabla^\Sigma \langle Y, \mathcal{N}\rangle\rangle \] and recalling that \[ \mathrm{Ric}^{M}_{C\psi}= \mathrm{Ric}^{M}+C\, \mathrm{Hess}^{M}(\psi) \] yield the validity of \eqref{eq-DeltafAngle}. \end{proof}
\subsection{Weighted volume estimates}
In this section we extend to the weighted setting of Killing graphs, with prescribed (weighted) mean curvature, an estimate of the extrinsic volume originally obtained in \cite{LW} and later extended in \cite{IPS-Crelle}.
\begin{lemma} \label{lemma_volume} Let $( M, \langle \cdot, \cdot \rangle_{M})$ be a complete $(n+1)$-dimensional Riemannian manifold endowed with a complete Killing vector field $Y$ whose orthogonal distribution has constant rank $n$ and is integrable. Let $(P,\langle \cdot, \cdot \rangle_{P}) $ be an integral leaf of that distribution so that $M$ can be identified with
$P \times_{e^{-\psi}}\mathbb{R}$, where $\psi=-\log|Y|$. Let $\Sigma=\mathrm{Graph_{ \Omega}}(u)$ be a Killing graph over a smooth domain $\Omega\subset P$, with mean curvature $H$ with respect to the upward unit normal $\mathcal{N}$ and let $\pi:\Sigma\to P$ be the projection map. Then, for any $y_{0}=( x_{0},u(x_0)) \in \Sigma$ and for every $R>0$, \[ \pi(B_R^\Sigma(y_0))
\subseteq \Omega_{R}\left(x_{0}\right)
\] where we have set \[ \Omega_{R}\left( x_{0}\right) =B_{R}^{P}(x_{0})\cap\Omega. \] Moreover, assume that given $D\in \mathbb{R}$, \begin{equation} A:=\sup_{\Omega}\left\vert u (x) e^{-\psi (x) }\right\vert +\sup_{\Omega}\left\vert H_{D\psi} (x) \right\vert <+\infty,\label{volume-u+Hf}. \end{equation} Then, there exists a constant $C>0$, depending on $n$ and $A$, such that, for every $\delta,R>0$, the corresponding $D\psi$-volume of the intrinsic ball of $\Sigma$ satisfies \begin{equation}\label{eq_volume} \mathrm{vol}_{D\psi}B_{R}^{\Sigma}\left( y_{0}\right) \leq C\left(1+\frac{1}{\delta R}\right) \mathrm{vol}_{D\psi}\left( \Omega_{\left( 1+\delta\right) R}\left( x_{0}\right) \right), \end{equation} where $y_{0}=(x_{0},u(x_{0}))\in\Sigma$ is a reference origin. \end{lemma}
\begin{proof} The Riemannian metric of $M$ writes as $\langle \cdot, \cdot \rangle_{M}=\langle \cdot, \cdot \rangle_{P}+e^{-2\psi} \mathrm{d} s \otimes \mathrm{d} s$. Let $y_{0}=(x_{0},u(x_{0}))$ and $y=(x,u(x))$ be points in $\Sigma$ connected by the curve $(\alpha(t), u(\alpha(t))$, where $\alpha(t)$, $t\in\lbrack0,1]$, is an arbitrary path connecting $x_{0}$ and $x$ in $\Omega\subseteq P$. Writing $s(t)=u(\alpha(t))$
we have \begin{align*}
\int_{0}^{1}\left\{|\alpha^{\prime}\left( t\right) ^{2}+e^{-2\psi(\alpha(t))}s^{\prime}(t)^{2}\right\}^{\frac{1}{2}} dt &
\geq \int_{0}^{1}|\alpha^{\prime}(t)|dt \geq d_{P}(x_{0},x).
\end{align*} Thus, if $y\in B_{R}^{\Sigma}(y_{0})$ we deduce that $x\in\Omega _{R}\left( x_{0}\right) $, proving the first half of the lemma.
Now we compute the volume of $B_{R}^{\Sigma}(y_{0})$. Since $\pi(B_{R}^{\Sigma}(y_{0}))\subset\Omega_{R}\left( x_{0}\right) $ we have \begin{align}
\mathrm{vol}_{D\psi}(B_{R}^{\Sigma}(y_{0})) & =\int_{\pi(B_{R}^{\Sigma}(y_{0}))}\sqrt{e^{2\psi}+|\nabla^{P}u|^{2}}e^{-(D+1)\psi}\,dP\label{volume-1}\\ & \leq\int_{\Omega_{R}\left( x_{0}\right) }
\sqrt{e^{2\psi}+|\nabla^{P}u|^{2}}e^{-(D+1)\psi}dP\nonumber\\ & =\int_{\Omega_{R}\left( x_{0}\right) }
\frac{|\nabla^{P}u|^{2}}{\sqrt{e^{2\psi}+|\nabla^{P}u|^{2}}}e^{-(D+1)\psi}\,dP\nonumber\\ & +\int_{\Omega_{R}\left( x_{0}\right) }\frac
{e^{2\psi}}{\sqrt{e^{2\psi(x)}+|\nabla^{P}u|^{2}}}e^{-(D+1)\psi}dP.\nonumber \end{align} We then consider the vector field \[ Z=\rho u\frac{e^{-\left(D+1\right) \psi}\nabla^{P}u}{\sqrt{e^{2\psi}
+|\nabla^{P}u|^{2}}}, \] where the function $\rho$ is given by \[ \rho(x)= \begin{cases} 1 & \mathrm{on}\quad B_{R}(x_{0})\\ \frac{\left( 1+\delta\right) R-r(x)}{\delta R} & \mathrm{on}\quad B_{\left( 1+\delta\right) R}(x_{0})\backslash B_{R}(x_{0})\\ 0 & \mathrm{elsewhere}, \end{cases} \] with $r(x) = \mathrm{dist}_{P}(x,x_{0})$. Recalling equation \eqref{capillary-2} in the Introduction, i.e., \[ \mathrm{div}^{P}\left\{ \frac{\nabla^{P}u}{\sqrt{e^{2\psi}+\left\vert \nabla^{P}u\right\vert ^{2}}}\right\}=nH+\frac{\left\langle \nabla^{P}u,\nabla^{P}\psi\right\rangle }{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}}} \] we compute \begin{align*} \mathrm{div} Z & =e^{-\left( D+1\right) \psi}\left\{ u\frac {\left\langle \nabla^{P}\rho,\nabla^{P}u\right\rangle }{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}}}+\rho\frac{\left\vert \nabla ^{P}u\right\vert ^{2}}{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}} }+n\rho uH_{D\psi}\right\} \\ & =e^{-D\psi}\left\{ u e^{-\psi}\frac{\left\langle \nabla^{P} \rho,\nabla^{P}u\right\rangle }{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}}}+\rho e^{-\psi}\frac{\left\vert \nabla^{P}u\right\vert ^{2} }{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}}}+n\rho u e^{-\psi}H_{D\psi}\right\} \\ & \geq e^{-D\psi}\left\{ -\left\vert u e^{-\psi}\right\vert \left\vert \nabla^{P}\rho\right\vert +\rho e^{-\psi} \frac{\left\vert \nabla^{P}u\right\vert ^{2}}{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}}}-n\rho\left\vert u e^{-\psi}\right\vert \left\vert H_{D\psi}\right\vert \right\} . \end{align*} Since $Z$ has compact support in $\Omega_{\left( 1+\delta\right) R}$, applying the divergence theorem and using the properties of $\rho$, from the above inequality we obtain \begin{align*} \int_{\Omega_{R}\left( x_{0}\right) }\frac{\left\vert \nabla^{P}u\right\vert ^{2}}{\sqrt{e^{2\psi}+\left\vert \nabla ^{P}u\right\vert ^{2}}}e^{-\psi}e^{-D\psi}\text{ }dP & \leq\frac{1}{\delta R}\int_{\Omega_{\left( 1+\delta\right) R}\left( x_{0}\right) }\left\vert u e^{-\psi}\right\vert e^{-D\psi}\text{ }dP\\ & +n\int_{\Omega_{\left( 1+\delta\right) R}\left( x_{0}\right) } \left\vert u e^{-\psi}\right\vert \left\vert H_{D\psi}\right\vert e^{-D\psi}\text{ }dP. \end{align*} Inserting this latter into (\ref{volume-1}) we get \begin{align*} \mathrm{vol}_{D\psi}(B_{R}^{\Sigma}(y_{0})) & \leq\frac{1}{\delta R} \int_{\Omega_{\left( 1+\delta\right) R}\left( x_{0}\right) } \left\vert u e^{-\psi}\right\vert e^{-D\psi}\text{ }dP\\ & +n\int_{\Omega_{\left( 1+\delta\right) R}\left( x_{0}\right) } \left\vert u e^{-\psi}\right\vert \left\vert H_{D\psi}\right\vert e^{-D\psi}\text{ }dP. \end{align*} To conclude the desired volume estimate, we now recall that, by assumption, \[ \sup_{\Omega}\left\vert u e^{-\psi}\right\vert +\sup_{\Omega }\left\vert H_{D\psi}\right\vert <+\infty. \] The proof of the Lemma is completed. \end{proof} \begin{remark}\label{rmk-fvolumes} \rm{ We note for further use that if, in the previous Lemma, we assume that $\inf_P\psi>-\infty$, then the following more general inequality holds: \[ \mathrm{vol}_{C\psi} B_{R}^{\Sigma} (y_0) \leq A \mathrm{vol}_{D\psi}(\Omega_R(x_0)) , \] for any constant $C>D$ and any $R \gg 1$. } \end{remark}
\subsection{Proofs of Theorem \ref{th_fheightestimate} and Corollary \ref{coro_fheightestimate}}\label{subsection-proof}
We are now in the position to give the \begin{proof}[Proof (of Theorem \ref{th_fheightestimate})] Since $H_{\psi} \equiv \mathrm{const}$, it follows by equation \eqref{eq-Deltafu} that \begin{align*} \Delta^{\Sigma}_{\psi}(H_{\psi} u) &= nH_{\psi}He^{2\psi}\langle Y,\mathcal{N} \rangle_{M}- H_{\psi}e^{2\psi}\langle \nabla^{M} {\psi}, \mathcal{N} \rangle_{M} \langle Y, \mathcal{N} \rangle_{M}\\ &= nH^2e^{2\psi}\langle Y, \mathcal{N} \rangle_{M} + He^{2\psi} \langle \nabla^{M} {\psi}, \mathcal{N} \rangle_{M} \langle Y, \mathcal{N} \rangle_{M}\\ &\,\,\,\,\, -He^{2\psi}\langle \nabla^{M} {\psi}, \mathcal{N} \rangle_{M} \langle Y,\mathcal{N} \rangle_{M} -\frac1n e^{2\psi} \langle \nabla^{M} {\psi}, \mathcal{N} \rangle_{M}^{2} \langle Y, \mathcal{N} \rangle_{M}\\ &\leq nH^2e^{2\psi}\langle Y, \mathcal{N} \rangle_{M}.
\end{align*} Combining this inequality with equation \eqref{eq-DeltafAngle}, it is straightforward to prove that, under our assumptions on $\mathrm{Ric}^{M}_{\psi}$ and $H_{\psi}$, the function \[ \varphi (x) =H_{\psi}u(x) e^{-2\sup_\Omega \psi}+\langle Y_{x}, \mathcal{N}_{x} \rangle_{M} \] satisfies \[ \Delta^{\Sigma}_{ \psi} \varphi \leq0\text{ on }\Sigma. \] On the other hand, using assumptions (b) and (e) we can apply Lemma \ref{lemma_volume} to deduce that \[ \mathrm{vol}_{\psi}(B_{R}^{\Sigma})=\mathcal{O}(R^2) \text{, as } R\rightarrow+\infty. \] In particular, by Proposition \ref{prop-fparab-volume}, $\Sigma$ is parabolic with respect to the weighted Laplacian $\Delta^{\Sigma}_{\psi}$. Since, again by (e), $\varphi$ is a bounded function and, according to (d), $u\equiv0$ on $\partial\Omega$, an application of the Ahlfors maximum principle stated in Theorem \ref{thm_fAhlfors} gives \[ \inf_{\Omega}\left( H_{\psi}u e^{-2\sup_{\Omega} \psi}+\langle Y, \mathcal{N} \rangle_{M}\right) =\inf_{\partial\Omega} \langle Y, \mathcal{N} \rangle_{M} \geq0. \] Combining this latter with the fact that $\langle Y, \mathcal{N} \rangle_{M} \leq e^{-\psi}$ we get the desired upper estimate on $u$.
Finally, note that equation \eqref{eq-Deltafu} can also be written in the form: \[ \Delta^{\Sigma}_{3\psi} u=H_{\psi} e^{2\psi}\langle Y, \mathcal{N} \rangle_{M}. \] Since, according to Remark \ref{rmk-fvolumes}, $\Sigma$ is also parabolic with respect to the weighted laplacian $\Delta^{\Sigma}_{3\psi}$ and $u$ is a bounded $3\psi$-superharmonic function, the desired lower estimate on $u$ follows again as an application of Theorem \ref{thm_fAhlfors}. \end{proof}
\begin{remark} \label{RmkHLR} { \rm It is clear from the proof that if we consider, as in \cite{HLR-TAMS}, the function $\varphi=cH_\psi e^{-2\psi} u +\langle Y,\mathcal{N}\rangle_M$, for some $0<c\leq 1$, it is possible to extend the height estimate to the case where $\mathrm{Ric}_\psi^M\geq -G^2(x)$ in a neighborhood of $\Omega\times \mathbb{R}$, provided $(1-c)H^2(x)/n\geq G^2(x)$. The resulting estimate becomes \[
0\leq ue^{-\psi}\leq \frac{e^{2(\sup_\Omega \psi-\inf_\Omega\psi)}}{c|H_\psi|}. \] } \end{remark}
To conclude this section, we show how the boundedness assumption (e) can be dropped in the case of $2$-dimensional graphs.
\begin{proof}[Proof (of Corollary \ref{coro_fheightestimate})] Let $\Sigma$ be a $2$-dimensional Killing graph over a domain $\Omega\subset P$ of constant weighted mean curvature $H_\psi<0$. Recall that the {\it Perelman scalar curvature} of $R_\psi$ of $M=P\times_{e^{-\psi}}\mathbb{R} $ is defined by \[
R^{M}_\psi=R+2 \Delta \psi-|\nabla \psi|^2=\mathrm{Tr}(\mathrm{Ric}^{M}_\psi)+\Delta_\psi\psi. \] Assume that $R^{M}_{\psi} \geq 0$. Then, as a consequence of a result due to Espinar \cite[Theorem 4.2]{Esp}, for every $y=(x,u(x))\in\Sigma$ it holds \[
|u(x)e^{-\psi}|\leq\mathrm{dist}(y,\partial \Sigma)\leq C(|H_\psi|)< +\infty. \] On the other hand, a straightforward calculation shows that \[ \Delta_\psi \psi=e^{2\psi}\mathrm{Ric}^{M}_\psi(Y,Y). \] Hence, $R^{M}_{\psi} \geq 0$ provided $\mathrm{Ric}^{M}_{\psi} \geq 0$. Putting everything together, it follows that condition (e) in Theorem \ref{th_fheightestimate} is automatically satisfied. \end{proof}
\section{Height estimates in model manifolds}\label{section-examples}
In this section we construct rotationally symmetric examples of Killing graphs with constant weighted mean curvature and exhibit explicit estimates on the maximum of their weighted height in terms of the weighted mean curvature. When the base space is $\mathbb{R} ^{n}$ and the Killing vector field has constant length $1$ (hence the ambient space is $\mathbb{R} ^{n}\times\mathbb{R} $) these graphs are standard half-spheres and the estimate on their maximal height is precisely the reciprocal of the mean curvature; \cite{He, Se}.
We shall assume that the induced metric $\langle \cdot, \cdot \rangle_{P}$ on $P$ is rotationally invariant. More precisely, we suppose that $P$ is a model space with pole at $o$ and Gaussian coordinates $(r,\theta)\in\left( 0,R\right) \times \mathbb{S}^{n-1}$, $R\in(0,+\infty],$ in terms of which $g_{P}$ is expressed by \[ g_{P}=dr^{2}+\xi^{2}(r)d\theta^{2}, \] for some $\xi\in C^{\infty}([0,R))$ satisfying \[ \begin{cases} & \xi>0 \text{ on }\left( 0,R\right) \\ & \xi^{(2k)}\left( 0\right) =0, \,\, k\in \mathbb{N} \\ & \xi^{\prime}\left( 0\right) =1 \end{cases} \] and where $d\theta^{2}$ denotes the usual metric in $\mathbb{S}^{n-1}$. We also assume that the norm of the Killing field does not depend on $\theta$, so that \[
|Y|^{2}=e^{-2\psi(r)}, \] In this case, the ambient metric $\langle \cdot, \cdot \rangle_{M}$ of $M=P\times_{e^{-\psi}}\mathbb{R} $ is written in terms of cylindrical coordinates $(s,r,\theta)$ as \[ \langle \cdot, \cdot \rangle_{M}=e^{-2\psi(r)}ds^{2}+dr^{2}+\xi^{2}(r)d\theta^{2}. \] and $M$ is a doubly-warped product with respect to warping functions of the coordinate $r$. The smoothness of $\psi$ implies that the pole $o$ is a critical point for $e^{-\psi}$, namely \[ \frac{d e^{-\psi(r)}}{dr}\left( 0\right)= -e^{-\psi(r)}\frac{d \psi(r)}{dr}=0. \] A rotationally invariant Killing graph $\Sigma_{0}\subset M$ is defined by a function $u$ that depends only on the radial coordinate $r$. In this case (\ref{capillary-3}) becomes \begin{equation} \label{H-ode} \bigg(\frac{u'(r)}{W}\bigg)' + \frac{u'}{W} \big(\Delta_P r - \langle \nabla^P \psi, \nabla^P r\rangle\big)= nH, \end{equation} where \[ W = \sqrt{e^{2\psi(r)}+u'^2(r)} \] and $'$ denotes derivatives with respect to $r$. Note that the weighted Laplacian of $r$ is given by \[
\Delta_P r- \langle \nabla^P \psi, \nabla^P r\rangle = -\psi'(r) + (n-1) \frac{\xi'(r)}{\xi(r)} = \frac{|Y|'(r)}{|Y|(r)} + (n-1) \frac{\xi'(r)}{\xi(r)}, \] which is, up to a factor $1/n$, the mean curvature $H_{{\rm cyl}}(r)$ of the cylinder over the geodesic sphere of radius $r$ centered at $o$ and ruled by the flow lines of $Y$ over that sphere. We also have \begin{equation} \label{H-Hpsi} nH_{\psi} = nH -\frac{u'(r)}{W}\langle\nabla^P \psi, \nabla^P r\rangle = nH -\psi'(r)\frac{u'(r)}{W} = {\rm div}^P_{2\psi}\bigg(\frac{u'(r)}{W}\nabla^P r\bigg)\cdot \end{equation} It follows from (\ref{H-ode}) that both $H$ and $H_{\psi}$ depend only on $r$. Integrating both extremes in (\ref{H-Hpsi}) against the weighted measure ${\rm d}P_\psi = e^{-\psi} {\rm d}P$ one obtains in this particular setting a first-order equation involving $u(r)$, namely \begin{equation} \label{H-rot-flux} \frac{u'(r)}{W} e^{-2\psi(r)}\xi^{n-1}(r) = \int_0^r nH_\psi e^{-2\psi(\tau)} \xi^{n-1}(\tau)\,{\rm d}\tau. \end{equation} Denoting the right hand side in (\ref{H-rot-flux}) by $I(r)$ and solving it for $u'(r)$ yields \begin{equation} \label{u-prima} u'^2= e^{2\psi} \frac{I^2 }{e^{-4\psi}\xi^{2(n-1)}-I^2} \end{equation} We assume that $u'(r)\le 0$ and denote \begin{equation}
V_\psi(r) = \frac{1}{|\mathbb{S}^{n-1}|}{\rm vol}_\psi (B_r(o)) =\int_0^r e^{-2\psi(\tau)}\xi^{n-1}(\tau) \, {\rm d}\tau \end{equation}
and \[
A_\psi(r) = \frac{1}{|\mathbb{S}^{n-1}|}{\rm vol}_\psi (\partial B_r(o)) = e^{-2\psi(r)}\xi^{n-1}(r). \]
Fixed this geometric setting, we prove the existence of compact rotationally symmetric Killing graphs with constant weighted mean curvature. \begin{theorem} \label{H-rot-existence} Suppose that the ratio $\frac{A_\psi(r)}{V_\psi(r)}$ is non-increasing for $r\in (0,R)$. Let $H_0$ be a constant with
\begin{equation} \label{isop-ineq}
n|H_0| = \frac{A_\psi(r_0)}{V_\psi(r_0)}
\end{equation} for some $r_0\in (0,R)$.
Then there exists a compact rotationally symmetric Killing graph $\Sigma_0\subset P\times_{e^{-\psi}}\mathbb{R}$ of a radial function $u(r)$, $r\in [0,r_0]$, given by \begin{equation} \label{u-solution} u(r) = \int^r_{r_0} e^{\psi(\tau)} \frac{I(\tau)}{\sqrt{e^{-4\psi(\tau)}\xi^{2(n-1)}(\tau)-I^2(\tau)}} \, {\rm d}\tau. \end{equation} with constant weighted mean curvature $H_\psi=H_0$ and boundary $\partial B_{r_0}(o)\subset P$. The weighted height function in this graph is bounded as follows \begin{equation} \label{u-estimate} e^{-\psi(r)} u(r) \le
e^{\sup_{B_r(o)} \psi-\inf_{B_r(o)}\psi}\int_{0}^{r_{0}}\frac{-nH_0}{\sqrt{\frac{A^2_\psi(\tau)}{V^2_\psi(\tau)}-n^2 H^2_0}}d\tau. \end{equation} \end{theorem}
\begin{proof} Since $A_\psi(r)/V_\psi(r)$ is non-increasing it follows from (\ref{isop-ineq}) that \[ e^{-4\psi(r)}\xi^{2(n-1)}(r)-I^2(r) \ge A_\psi^2(r) - n^2 H^2_\psi V_\psi^2(r)\ge 0 \] for $r\in (0,r_0]$ what guarantees that the expression \begin{equation} \label{u-rot} u(r) = \int_{r_0}^{r} e^{\psi(\tau)} \frac{I(\tau)}{\sqrt{e^{-4\psi(\tau)}\xi^{2(n-1)}(\tau)-I^2(\tau)}} \, {\rm d}\tau \end{equation} is well-defined for $r\in [0,r_0]$ with $u'(r)\le 0$. For further reference, we remark an application of L'H\^opital's rule shows that \begin{equation} \label{h-cyl}
\lim_{r\to 0} \frac{A_\psi(r)}{V_\psi(r)} =\lim_{r\to 0} \bigg(2\frac{|Y|'(r)}{|Y|(r)} + (n-1) \frac{\xi'(r)}{\xi(r)}\bigg)\cdot \end{equation}
In order to get a more detailed analysis at $r=r_0$ we consider a parametrization of $\Sigma_0$ in terms of cylindrical coordinates as \[ (t,\theta) \mapsto (r(t), s(t), \theta), \] where $t$ is the arc-lenght parameter defined by \[ \dot r^2 (t)+ e^{-2\psi(r(t))} \dot s^2(t) =1 \] and $\cdot$ denotes derivatives with respect to $s$. Since \[ u'(r(t)) = \frac{\dot s(t)}{\dot r(t)} \] we impose $\dot s\ge 0$ and $\dot r\le 0$. Hence $W=-e^{\psi}/\dot r$ whenever $\dot r(s)\neq 0$ and (\ref{H-rot-flux}) is written as \begin{equation} \label{flux-param} \dot s e^{-3\psi(r)}\xi^{n-1}(r) = -\int_0^r nH_\psi e^{-2\psi(\tau)} \xi^{n-1}(\tau)\,{\rm d}\tau \end{equation} Since $\xi(0)=0$, $\xi'(0)=1$ and $\psi'(0)=0$, applying L'H\^{o}pital's rule as above shows that \[ \lim_{r\to 0}\frac{\int_0^r nH_\psi e^{-2\psi(\tau)} \xi^{n-1}(\tau)\,{\rm d}\tau}{e^{-3\psi(r)}\xi^{n-1}(r)} = \lim_{r\to 0}\frac{nH_\psi e^{\psi(r)}}{-3\psi'(r) + (n-1)\frac{\xi'(r)}{\xi(r)}} =0
\] and we conclude that $e^{-\psi(r)}\dot s\to 0$ as $r\to 0$. Therefore $\dot s \to 0$ as $r\to 0$ what implies that $\Sigma_0$ is smooth at its intersection with the vertical axis of revolution.
Now, we have from (\ref{flux-param}) and (\ref{isop-ineq}) that $\dot r =0$ at $r=r_0$ since $e^{-\psi(r_0)}\dot s(r_0) =1$. Finally, let $\phi$ be the angle between a meridian $\theta={\rm cte.}$ in $\Sigma_0$ and and radial vector field $\partial_r$. We have \[ \frac{u'(r)}{W} = -\frac{\dot s}{e^{\psi(r)}} = -\sin\phi. \] Hence \begin{eqnarray*} \bigg(\frac{u'(r)}{W} \bigg)' = -\cos\phi \,\dot \phi (t)\frac{1}{\dot r(t)} = -\dot \phi(t). \end{eqnarray*} On the other hand \begin{eqnarray*}
\bigg(\frac{u'(r)}{W} \bigg)' &= & nH - \frac{u'(r)}{W} \bigg(\frac{|Y|'(r)}{|Y|(r)}+(n-1)\frac{\xi'(r)}{\xi(r)}\bigg)\\
& = & nH_\psi + \sin \phi \bigg(2\frac{|Y|'(r)}{|Y|(r)}+(n-1)\frac{\xi'(r)}{\xi(r)}\bigg) \end{eqnarray*}
In sum, $\Sigma_0$ is parameterized by the solution of the first order system \begin{equation} \begin{cases} \dot r & = \cos \phi \\ e^{-\psi (r)}\dot s & =\sin \phi\\
\dot \phi & = -nH_0 - \sin \phi \Big(2\frac{|Y|'(r)}{|Y|(r)}+(n-1)\frac{\xi'(r)}{\xi(r)}\Big), \end{cases} \end{equation} with initial conditions $r(0) = r_0, s(0)=0, \phi(0)= \frac{\pi}{2}$. The height estimate follows directly from (\ref{u-solution}). This finishes the proof. \end{proof}
It is worth to point out that, in the classical situation of the Euclidean space where $e^{-\psi}=|Y|\equiv1$ and $\xi\left( r\right) =r$, the Killing graph defined by $u(r)$ reduces to the standard sphere and (\ref{u-estimate}) gives rise to the expected sharp bound \[
u(r) \leq \frac{1}{|H|}. \] Actually, a similar conclusion can be achieved if we choose $\psi\left( r\right) $ in such a way that \[ e^{-2\psi\left( r\right)} \xi^{n-1}\left( r\right) =r^{n-1}. \] Note that this choice is possible and compatible with the request $d\psi/dr\left( 0\right) =0$ because $\xi\left( r\right) $ is odd at the origin. This choice corresponds to the case when \[ \frac{A_\psi(r)}{V_\psi(r)} = \frac{n}{r} \] as in the Euclidean space. We have thus obtained the following height estimate
\begin{corollary} \label{th_symm-example}Let $P=[0,R)\times_{\xi}\mathbb{S}^{n-1}$ be an $n$-dimensional model manifold with warping function $\xi$ and let $\psi:\left[0,R\right) \rightarrow\mathbb{R} _{>0}$ be the smooth, even function defined by \[ \psi(r) =c\cdot \begin{cases}
\frac{(n-1)}{2}\log\frac{\xi (r)}{r} & r\neq0\\ 1 & r=0, \end{cases} \] where $c>0$ is a given constant. Fix $0<r_{0}<R$, let $H_{0}=-1/r_{0}$ and define \[ u(r) =\int^{r_0}_{r}\frac{-H_{0}\tau}{\sqrt{1-H_0^2 \tau^{2}}}e^{\psi( \tau)} d\tau. \] Then, in the ambient manifold $M=P\times_{e^{-\psi}}\mathbb{R} $, the Killing graph of $u$ over $\Omega=B_{r_{0}}^{P}( o) \subset P$ has constant weighted mean curvature $H_\psi=H_0$ with respect to the upward pointing Gauss map. Moreover, \[ 0\leq e^{-\psi(r)}u(r) \leq\frac {\max_{\lbrack0,r_{0}]}e^{-\psi}}{\min_{[0,r_{0}]}e^{-\psi}}\cdot
\frac{1}{|H_0|}=e^{\sup_\Omega \psi-\inf_\Omega\psi}\frac{1}{|H_0|}. \] \end{corollary}
The counterpart of Theorem \ref{H-rot-existence} and Corollary \ref{th_symm-example} in the case of constant mean curvature can be obtained along the same lines by integrating both sides in (\ref{capillary-3}) instead of (\ref{H-Hpsi}). Denoting \[
V(r) = \frac{1}{|\mathbb{S}^{n-1}|}{\rm vol} (B_r(o)) =\int_0^r e^{-\psi(\tau)}\xi^{n-1}(\tau) \, {\rm d}\tau \] and \[
A(r) = \frac{1}{|\mathbb{S}^{n-1}|}{\rm vol} (\partial B_r(o)) = e^{-\psi(r)}\xi^{n-1}(r) \] one obtains \begin{theorem} \label{Hw-rot-existence} Suppose that the ratio $\frac{A(r)}{V(r)}$ is non-increasing for $r\in (0,R)$. Let $H_0$ be a non-positive constant with
\begin{equation} \label{isop-ineq}
n|H_{0}| = \frac{A(r_0)}{V(r_0)}
\end{equation} for some $r_0\in (0,R)$.
Then there exists a compact rotationally symmetric Killing graph $\Sigma_0\subset P\times_{e^{-\psi}} \mathbb{R}$ of a radial function $u(r)$, $r\in [0,r_0]$, given by \begin{equation} \label{u-solution} u(r) = \int^r_{r_0} \frac{nH_{0}V(\tau)}{\sqrt{A^2(\tau)-n^2 H_{0}^2 V^2(\tau)}} \, {\rm d}\tau. \end{equation} with constant mean curvature $H=H_{0}$ and boundary $\partial B_{r_0}(o)\subset P$. The height function in this graph is bounded as follows \begin{equation} \label{u-estimate}
e^{-\psi(r)}u(r) \le
e^{\sup_{B_r(o)} \psi-\inf_{B_r(o)}\psi} \int_{0}^{r_{0}}\frac{-nH_{0}}{\sqrt{\frac{A^2(\tau)}{V^2(\tau)}-n^2 H_{0}^2}}d\tau. \end{equation} \end{theorem}
The analog of Corollary \ref{th_symm-example} in the case of constant mean curvature is
\begin{corollary} \label{th_symm-example-w}Let $P=[0,R)\times_{\xi}\mathbb{S}^{n-1}$ be an $n$-dimensional model manifold with warping function $\xi$ and let $\psi:\left[0,R\right) \rightarrow\mathbb{R} _{>0}$ be the smooth, even function defined by \[ \psi(r) =c\cdot \begin{cases}
(n-1)\log\frac{\xi (r)}{r} & r\neq0\\ 1 & r=0, \end{cases} \] where $c>0$ is a given constant. Fix $0<r_{0}<R$, let $H_{0}=-1/r_{0}$ and define \[ u(r) =\int^{r}_{r_0}\frac{H_{0}\tau}{\sqrt{1-H_{0}^2 \tau^{2}}}d\tau. \] Then, in the ambient manifold $M=P\times_{e^{-\psi}}\mathbb{R} $, the Killing graph of $u$ over $\Omega=B_{r_{0}}^{P}( o) \subset P$ has constant mean curvature $H_{0}$ with respect to the upward pointing Gauss map. Moreover, \[ 0\leq e^{-\psi(r)} u(r) \leq \frac{\max_{\lbrack0,r_{0}]}e^{-\psi}}{\min_{[0,r_{0}]}e^{-\psi}}\cdot
\frac{1}{|H_{0}|}
=e^{\sup_\Omega \psi-\inf_\Omega\psi}\frac{1}{|H_0|}. \] \end{corollary}
\begin{remark} \rm{Up to the factor $2$ in the exponential, this is the estimate obtained in Theorem \ref{th_fheightestimate} above. We suspect that the high rotational symmetry considered in the example prevents to achieve the maximum height predicted by the theorem. On the other hand we conjecture that the rotationally symmetric graphs can be used as barriers to obtain sharp estimates in the case of general warped spaces $P\times_{e^{-\psi}}\mathbb{R}$ in the case when the radial sectional curvatures of $P$ are bounded from above by some radial function.}
\end{remark}
\end{document} | arXiv |
Ground expression
In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables.
In first-order logic with identity with constant symbols $a$ and $b$, the sentence $Q(a)\lor P(b)$ is a ground formula. A ground expression is a ground term or ground formula.
Examples
Consider the following expressions in first order logic over a signature containing the constant symbols $0$ and $1$ for the numbers 0 and 1, respectively, a unary function symbol $s$ for the successor function and a binary function symbol $+$ for addition.
• $s(0),s(s(0)),s(s(s(0))),\ldots $ are ground terms;
• $0+1,\;0+1+1,\ldots $ are ground terms;
• $0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0$ are ground terms;
• $x+s(1)$ and $s(x)$ are terms, but not ground terms;
• $s(0)=1$ and $0+0=0$ are ground formulae.
Formal definitions
What follows is a formal definition for first-order languages. Let a first-order language be given, with $C$ the set of constant symbols, $F$ the set of functional operators, and $P$ the set of predicate symbols.
Ground term
A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
1. Elements of $C$ are ground terms;
2. If $f\in F$ is an $n$-ary function symbol and $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ is a ground term.
3. Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms).
Roughly speaking, the Herbrand universe is the set of all ground terms.
Ground atom
A ground predicate, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.
If $p\in P$ is an $n$-ary predicate symbol and $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ is a ground predicate or ground atom.
Roughly speaking, the Herbrand base is the set of all ground atoms,[1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.
Ground formula
A ground formula or ground clause is a formula without variables.
Ground formulas may be defined by syntactic recursion as follows:
1. A ground atom is a ground formula.
2. If $\varphi $ and $\psi $ are ground formulas, then $\lnot \varphi $, $\varphi \lor \psi $, and $\varphi \land \psi $ are ground formulas.
Ground formulas are a particular kind of closed formulas.
See also
• Open formula – formula that contains at least one free variablePages displaying wikidata descriptions as a fallback
• Sentence (mathematical logic) – in mathematical logic, a well-formed formula with no free variablesPages displaying wikidata descriptions as a fallback
References
1. Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.
• Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.), Handbook of discrete and combinatorial mathematics, p. 68
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6
• First-Order Logic: Syntax and Semantics
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| Wikipedia |
\begin{definition}[Definition:Ordered Pair/Empty Set Formalization]
The concept of an ordered pair can be formalized by the definition:
:$\tuple {a, b} := \set {\set {\O, a}, \set {\set \O, b} }$
\end{definition} | ProofWiki |
ICIAM19
Modeling the Co-infection of Hepatitis E and Malaria in Uganda
By Lina Sorg
Overcrowding in Uganda from increased immigration activity is causing insufficient sanitation and poor nutrition in many regions of the country. Heightened movement in, out, and around Uganda also facilitates the outbreak and spread of multiple diseases, making epidemiological mathematics more important than ever. One such disease is hepatitis E, an inflammation of the liver caused by a virus of the same name.
A hepatitis E outbreak is typically self-limiting, unless it exists in tandem with another disease. Co-infection occurs when a population harbors two infections simultaneously. For this reason, the 2007-2009 outbreak of hepatitis E in Kitgum, a municipality in northern Uganda was particularly noteworthy. Malaria is endemic throughout Uganda, and its presence complicates the behavior and handling of hepatitis E. While no real treatment exists for hepatitis E other than improved hygiene, it does impact the effectiveness of the malaria drug.
During a scientific session at the 9th International Congress on Industrial and Applied Mathematics, currently taking place in Valencia, Spain, Betty Nannyonga of Makerere University described a simple model to depict the role of malaria in Kitgum's 2007-2009 hepatitis E outbreak. Her ultimate goal was to yield policy suggestions for the ministry's management of the disease—which was attributed to a dearth of both latrines and safe drinking water—and estimate the number of latrines and boreholes necessary to stop the outbreak. Malaria weakens the immune system, especially in the concurrent presence of other infections. "We wanted to see how these two infections affected this region of the country, and what the ministry has to do to combat it," Nannyonga said.
At ICIAM19, Betty Nannyonga described a simple model to depict the role of malaria in Kitgum's 2007 hepatitis E outbreak.
Approximately 9,449 cases of hepatitis—out of a population of 28,045 in 6,039 households—were reported in Kitgum during the outbreak, resulting in 160 deaths. To address the co-infection of malaria and hepatitis E, Nannyonga developed two models: one depicting only hepatitis E with a constant population, and one depicting the disease's co-infection with malaria. The latter model accounts for a changing population due to malaria's endemic nature. She fit her models to the existing data using two approaches: linear regression (under the assumption that \(\mu =0\)) and a nonlinear differential equations fitting tool that investigates \(\mu>0\). Based on the fitting, she estimated the transmission rate and basic reproduction number (\(R_0\)).
Without the presence of malaria, each person with hepatitis will likely affect two other individuals (with an \(R_0\) level of 2.11). Nannyonga then turned to a mathematical modeling tool called PottersWheel to investigate how a fluctuating population changes the \(R_0\). Because malaria is a continuous process, she used nonlinear ordinary differential equations to depict the varying ebb and flow of the population. "If you have malaria, what condition will facilitate hepatitis E, and the other way around?" she asked.
Using PottersWheel, Nannyonga set the fits in a sequence to 20, chose parameters, and ran the process 50s time for each value. The results of the fitting tool were substantial. "When you have malaria in your system, the chances of you ending up with hepatitis E is almost seven times greater than if you don't have malaria," Nannyonga said.
Next, she conducted cost-effective analysis of hepatitis E, beginning with an investigation of the disability-adjusted life year (DALY) — an indicator that quantifies a disease's burden as well as the associated functional limitation and premature mortality. In short, this metric investigates the present and future cost of a disease for the affected country. "The DALY can be used across cultures to measure health gaps as opposed to health expectancies, and the difference between a current and ideal situation where everyone lives up to the age of the standard life expectancy in perfect health," Nannyonga said. "It combines in one measure the time lived with the disability (YLD) and the time lost due to premature mortality (YLL)." Thus, \(DALY = YLD + YLL\). More specifically, the equation breaks down in the following way:
\(YLL=\) number of deaths \((P) \times\) standard life expectancy \((L)\) at the age at which death occurs. \(YLL= P \times L\).
\(YLD =\) number of disability cases \((I) \times\) the average duration of the disease \((D) \times\) a weight factor \((DW)\) that reflects the disease's severity on a scale of \(0\) (perfect health) to \(1\) (dead). \(YLD = I \times DW \times D\).
To estimate the net present value of years of life lost, Nannyonga applied a standard time discount rate to years of life lost in the future; this adjusts both the costs and health outcomes.
Finally, she turned to policymaking. There is currently one latrine per six households in Kitgum, and one borehole for every 263 households. To combat hepatitis E in the absence of malaria, one would have to increase latrine coverage to 16 percent from the current 3.7 percent, and borehole coverage to 17 percent from 0.38 percent. Malaria's presence in in the population demands latrine coverage of 17.5 percent, with borehole coverage of 18.1 percent.
Next, Nannyonga calculated the cost of constructing additional latrines. There are currently 1,038 latrines in Kitgum, meaning that 3,477 additional latrines would be necessary to avoid future hepatitis E outbreaks. The estimated cost of one basic pit latrine is $250. Thus, building the required 3,477 latrines would cost $869,250. In this case, the averted cost per DALY is $123.
While Nannyonga admits that her fitted models are fairly basic and may not fully reflect the complexity of either hepatitis E or malaria—in part due to a scarcity of credible data—fitting mathematical models to data is a great approach for exploring possible ways to eradicate diseases. "Our results highlight that protection against diseases is not only essential in terms of improving people's lives, but also an important means of bolstering economic and social development," she said.
Lina Sorg is the associate editor of SIAM News. | CommonCrawl |
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Tagged: basis of a vector space
by Yu · Published 10/02/2017
Every Basis of a Subspace Has the Same Number of Vectors
Problem 577
Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.
Prove that every basis of $V$ consists of $k$ vectors in $V$.
Read solution
Click here if solved 58
Add to solve later
by Yu · Published 09/27/2017 · Last modified 10/04/2017
Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.
(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.
(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.
Click here if solved 312
Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$
Let $V$ be a vector space over the field of real numbers $\R$.
Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.
The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization
Consider the $2\times 2$ real matrix
\[A=\begin{bmatrix}
1 & 1\\
1& 3
\end{bmatrix}.\]
(a) Prove that the matrix $A$ is positive definite.
(b) Since $A$ is positive definite by part (a), the formula
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans} A \mathbf{y}\] for $\mathbf{x}, \mathbf{y} \in \R^2$ defines an inner product on $\R^n$.
Consider $\R^2$ as an inner product space with this inner product.
Prove that the unit vectors
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
\end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix}
\end{bmatrix}\] are not orthogonal in the inner product space $\R^2$.
(c) Find an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ of $\R^2$ from the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ using the Gram-Schmidt orthogonalization process.
All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$
Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.
Dimension of the Sum of Two Subspaces
Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.
Then prove that
\[\dim(U+V) \leq \dim(U)+\dim(V).\]
Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices
Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ elements.
Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.
Coordinate Vectors and Dimension of Subspaces (Span)
Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.
Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.
\[[v_1]_B=\begin{bmatrix}
\end{bmatrix}, [v_2]_B=\begin{bmatrix}
The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.
Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix
Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.
Determine which matrices are in the subspace $W$ and find the dimension of $W$.
(a) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &b &0 \\
0 & 0 & c
\end{bmatrix},\] where $a, b, c$ are distinct real numbers.
(b) \[A=\begin{bmatrix}
0 &a &0 \\
0 & 0 & b
\end{bmatrix},\] where $a, b$ are distinct real numbers.
Prove a Given Subset is a Subspace and Find a Basis and Dimension
\end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.
\[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\]
(a) Prove that the subset $V$ is a subspace of $\R^2$.
(b) Find a basis for $V$ and determine the dimension of $V$.
Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions.
Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients.
Let $W$ be the subspace of $P_2$ by
\[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of $W$.
Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$
Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
If $\mathbf{e}_1=\mathbf{v}_1+2\mathbf{v}_2 \text{ and } \mathbf{e}_2=2\mathbf{v}_1-\mathbf{u}_2$, where $\mathbf{e}_1, \mathbf{e}_2$ are the standard unit vectors in $\R^2$, then find the matrix of $T$ with respect to the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$.
Linear Transformation and a Basis of the Vector Space $\R^3$
Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$.
Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$.
Show that the vectors $\mathbf{x}, T\mathbf{x}, T^2\mathbf{x}$ form a basis for $\R^3$.
(The Ohio State University Linear Algebra Exam Problem)
Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix
Suppose the following information is known about a $3\times 3$ matrix $A$.
\[A\begin{bmatrix}
\end{bmatrix}=6\begin{bmatrix}
\end{bmatrix},
\quad
A\begin{bmatrix}
-1 \\
\end{bmatrix}, \quad
(a) Find the eigenvalues of $A$.
(b) Find the corresponding eigenspaces.
(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?
(Johns Hopkins University Linear Algebra Exam)
A Matrix Representation of a Linear Transformation and Related Subspaces
Let $T:\R^4 \to \R^3$ be a linear transformation defined by
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.
(b) Find a basis for the null space of $T$.
(c) Find the rank of the linear transformation $T$.
Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known
Let $T: \R^2 \to \R^2$ be a linear transformation.
\mathbf{u}=\begin{bmatrix}
\end{bmatrix}, \mathbf{v}=\begin{bmatrix}
\end{bmatrix}\] be 2-dimensional vectors.
\begin{align*}
T(\mathbf{u})&=T\left( \begin{bmatrix}
\end{bmatrix} \right)=\begin{bmatrix}
\end{bmatrix},\\
T(\mathbf{v})&=T\left(\begin{bmatrix}
\end{bmatrix}\right)=\begin{bmatrix}
\end{bmatrix}.
\end{align*}
Let $\mathbf{w}=\begin{bmatrix}
x \\
\end{bmatrix}\in \R^2$.
Find the formula for $T(\mathbf{w})$ in terms of $x$ and $y$.
Vector Space of Polynomials and Coordinate Vectors
Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where
&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\
&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.
(a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.
(b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.
(c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.
Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$
Let $T: \R^3 \to \R^2$ be a linear transformation such that
\[T(\mathbf{e}_1)=\begin{bmatrix}
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
\end{bmatrix},\] where
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
\end{bmatrix}\] are the standard unit basis vectors of $\R^3$.
For any vector $\mathbf{x}=\begin{bmatrix}
\end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.
Any Vector is a Linear Combination of Basis Vectors Uniquely
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as
\[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\] where $c_1, c_2, c_3$ are scalars.
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\begin{document}
\begin{frontmatter} \title{Local H\"older regularity for nonlocal parabolic $p$-Laplace equations} \author[myaddress]{Karthik Adimurthi\tnoteref{thanksfirstauthor}} \ead{[email protected] and [email protected]} \author[myaddress]{Harsh Prasad\tnoteref{thankssecondauthor}} \ead{[email protected]} \author[myaddress]{Vivek Tewary\tnoteref{thankssecondauthor}} \ead{[email protected] and [email protected]}
\tnotetext[thanksfirstauthor]{Supported by the Department of Atomic Energy, Government of India, under project no. 12-R\&D-TFR-5.01-0520 and SERB grant SRG/2020/000081}
\tnotetext[thankssecondauthor]{Supported by the Department of Atomic Energy, Government of India, under project no. 12-R\&D-TFR-5.01-0520}
\address[myaddress]{Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, Karnataka, 560065, India} \begin{abstract} We prove local H\"older regularity for a nonlocal parabolic equations of the form \begin{align*}
\partial_t u + \text{P.V.}\oldint\limits_{\RR^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\,dy=0, \end{align*} for $p\in (1,\infty)$ and $s \in (0,1)$.
\end{abstract}
\begin{keyword} Nonlocal operators; Weak Solutions; H\"older regularity
\MSC[2010] 35K51, 35A01, 35A15, 35R11.
\end{keyword}
\end{frontmatter}
\tableofcontents
\section{Introduction}\label{sec1}
In this article, we prove local H\"older regularity for a nonlocal parabolic equation whose prototype equation is the parabolic fractional $p$-Laplacian of the form \begin{align*}
\partial_t u + \text{P.V.}\oldint\limits_{\RR^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\,dy=0, \end{align*} with $p\in (1,\infty)$ and $s \in (0,1)$. The main tools in our investigation are \begin{itemize} \item the exponential change of variable in time discovered by E.DiBenedetto, U.Gianazza and V.Vespri, which they used to prove Harnack's inequality for the parabolic $p$-Laplace equations. The standard references for these works are \cite{dibenedettoHarnackEstimatesQuasilinear2008,dibenedettoHarnackInequalityDegenerate2012}. The exponential change of variable in time leads to an expansion of positivity result which is suitable for proving H\"older regularity results. N.Liao in \cite{liaoUnifiedApproachHolder2020} has effected a direct proof of H\"older regularity for parabolic $p$-Laplace equation by using the expansion of positivity. It has also been used to give the first proofs of H\"older regularity for sign-changing solutions of doubly nonlinear equations of porous medium type in \cite{bogeleinHolderRegularitySigned2021,bogeleinOlderRegularitySigned2021,liaoHolderRegularitySigned2021}.
A recent work that employs the exponential change of variable to deduce regularity for elliptic anisotropic equations is \cite{liaoLocalRegularityAnisotropic2020}. It has also been used in the paper \cite{shangOlderRegularityMixed2021} for a mixed local and nonlocal parabolic equation. Our work differs from \cite{shangOlderRegularityMixed2021} in two important directions: the presence of the local term in \cite{shangOlderRegularityMixed2021} allows them to rely on the standard De Giorgi isoperimetric inequality (which we cannot use) and their estimates undergo a significant simplification due to their assumption of global boundedness on the solutions (we only assume local boundedness of solutions). \item the energy estimate for fractional parabolic $p$-Laplace equations containing the ``good term'' which substitutes the De Giorgi isoperimetric inequality in the so-called ``shrinking lemma''. The ``good term'' has been used to great effect in the works \cite{caffarelliDriftDiffusionEquations2010,caffarelliRegularityTheoryParabolic2011}. It has been used to define a novel De Giorgi class by M. Cozzi in \cite{cozziRegularityResultsHarnack2017} to prove H\"older regularity for fractional $p$-minimizers. Earlier, the absence of De Giorgi isoperimetric inequality was tackled by Di Castro et. al. through a logarithmic estimate\cite{dicastroLocalBehaviorFractional2016}. See \cite{cozziFractionalGiorgiClasses2019,adimurthiOlderRegularityFractional2022} for discussions on fractional versions of De Giorgi isoperimetric inequalities. \end{itemize}
We demonstrate two purely nonlocal phenomenon - one at a technical level which has implications for what the corresponding DeGiorgi conjectures related to the "best" Holder exponent in the nonlocal case should be (see \cref{rem:nonlocalA}) and another at the level of regularity which says that in certain cases, time regularity can beat space regularity when we are working with nonlocal equations (see \cref{rem:nonlocalB}).
Previously, local boundedness for fractional parabolic $p$-Laplace equations is proved in the papers \cite{stromqvistLocalBoundednessSolutions2019,dingLocalBoundednessHolder2021,prasadLocalBoundednessVariational2021}. Regarding H\"older regularity for fractional parabolic $p$-Laplace equations in the spirit of DeGiorgi-Nash-Moser, the only result that we are aware of is for the case $2 \leq p < \infty$ which was studied in \cite{dingLocalBoundednessHolder2021}, though we were unable to verify some of the calculations pertaining to the logarithmic estimates in their paper. Explicit exponents for H\"older regularity for fractional parabolic $p$-Laplace equations with no coefficients appears in \cite{brascoContinuitySolutionsNonlinear2021}. Existence has been studied in \cite{mazonFractionalPLaplacianEvolution2016,prasadExistenceVariationalSolutions2022}.
\subsection{A brief history of the problem}
Much of the early work on regularity of fractional elliptic equations in the case $p=2$ was carried out by Silvestre \cite{silvestreHolderEstimatesSolutions2006}, Caffarelli and Vasseur\cite{caffarelliDriftDiffusionEquations2010}, Caffarelli, Chan, Vasseur\cite{caffarelliRegularityTheoryParabolic2011} and also Bass-Kassmann \cite{bassHarnackInequalitiesNonlocal2005,bassHolderContinuityHarmonic2005, kassmannPrioriEstimatesIntegrodifferential2009}. An early formulation of the fractional $p$-Laplace operator was done by Ishii and Nakamura\cite{ishiiClassIntegralEquations2010} and existence of viscosity solutions was established. DiCastro, Kuusi and Palatucci extended the De Giorgi-Nash-Moser framework to study the regularity of the fractional $p$-Laplace equation in \cite{dicastroLocalBehaviorFractional2016}. The subsequent work of Cozzi \cite{cozziRegularityResultsHarnack2017} covered a stable (in the limit $s\to 1$) proof of H\"older regularity by defining a novel fractional DeGiorgi class. An alternate proof of H\"older regularity based on a differential inequality was given in \cite{adimurthiOlderRegularityFractional2022}. Explicit exponents for H\"older regularity were found in \cite{brascoHigherHolderRegularity2018,brascoContinuitySolutionsNonlinear2021}. Other works of interest are \cite{defilippisHolderRegularityNonlocal2019,byunOlderRegularityWeak2021,chakerRegularityNonlocalProblems2021,garainHigherOlderRegularity2022,de2022gradient} and in the parabolic context, some works of interest are \cite{dingLocalBoundednessHolder2021,prasadExistenceVariationalSolutions2022,prasadLocalBoundednessVariational2021}.
\subsection{On historical development of intrinsic scaling}
The method of intrinsic scaling was developed by E.DiBenedetto in \cite{dibenedettoLocalBehaviourSolutions1986} to prove H\"older regularity for degenerate quasilinear parabolic equations modelled on the $p$-Laplace operator. A technical requirement of the proof was a novel logarithmic estimate which aids in the expansion of positivity. Subsequently, the proof of H\"older regularity for the singular case was given in \cite{ya-zheLocalBehaviorSolutions1988} by switching the scaling from time variable to the space variable. These results were collected in E.DiBenedetto's treatise \cite{dibenedettoDegenerateParabolicEquations1993}. After a gap of several years, E.DiBenedetto, U.Gianazza and V.Vespri \cite{dibenedettoHarnackEstimatesQuasilinear2008, dibenedettoHarnackInequalityDegenerate2012} were able to prove Harnack's inequality for the parabolic $p$-Laplace equations by a new technique involving an exponential change of variable in time. This proof relies on expansion of positivity estimates and does not involve logarithmic test functions. Then, a new proof of H\"older regularity was given with a more geometric flavour in \cite{gianazzaNewProofHolder2010}. This theory was extended to generalized parabolic $p$-Laplace equations with Orlicz growth in \cite{hwangHolderContinuityBounded2015,hwangHolderContinuityBounded2015a}.
\begin{remark}
In this paper, we assume that solutions to the nonlocal equation are locally bounded. Local boundedness for the nonlocal elliptic equations is proved in \cite[Theorem 1.1]{dicastroLocalBehaviorFractional2016} and \cite[Theorem 6.2]{cozziRegularityResultsHarnack2017}. In the parabolic case, local boundedness is proved in \cite{stromqvistLocalBoundednessSolutions2019,dingLocalBoundednessHolder2021,prasadLocalBoundednessVariational2021}. \end{remark}
\section{Notations and Auxiliary Results}\label{sec2} In this section, we will fix the notation, provide definitions and state some standard auxiliary results that will be used in subsequent sections.
\subsection{Notations} We begin by collecting the standard notation that will be used throughout the paper: \begin{itemize} \item We shall denote $N$ to be the space dimension and by $z=(x,t)$ to be a point in $ \RR^N\times (0,T)$. \item We shall alternately use $\dfrac{\partial f}{\partial t},\partial_t f,f'$ to denote the time derivative of f.
\item Let $\Omega$ be an open bounded domain in $\mathbb{R}^N$ with boundary $\partial \Omega$ and for $0<T\leq\infty$, let $\Omega_T\coloneqq \Omega\times (0,T)$.
\item We shall use the notation \begin{equation*}
\begin{array}{ll}
B_{\rho}(x_0)=\{x\in\RR^N:|x-x_0|<\rho\}, &
\overline{B}_{\rho}(x_0)=\{x\in\RR^N:|x-x_0|\leq\rho\},\\
I_{\theta}(t_0)=\{t\in\RR:t_0-\theta<t<t_0\},
&Q_{\rho,\theta}(z_0)=B_{\rho}(x_0)\times I_\theta(t_0).
\end{array} \end{equation*} \item The maximum of two numbers $a$ and $b$ will be denoted by $a\wedge b\coloneqq \max(a,b)$. \item Integration with respect to either space or time only will be denoted by a single integral $\oldint\limits$ whereas integration on $\Omega\times\Omega$ or $\RR^N\times\RR^N$ will be denoted by a double integral $\oldiint\limits$. \item We will use $\oldiiint\limits$ to denote integral over $\RR^N \times \RR^N \times (0,T)$. More specifically, we will use the notation $\oldiiint\limits_{Q}$ and $\oldiiint\limits_{B \times I}$ to denote the integral over $\oldiiint\limits_{B \times B \times I}$. \item The notation $a \lesssim b$ is shorthand for $a\leq C b$ where $C$ is a universal constant which only depends on the dimension $N$, exponents $p$, $q$, and the numbers $M$, $s_1$ and $s_2$. \item For a function $u$ defined on the cylinder $Q_{\rho,\theta}(z_0)$ and any level $k \in \bb{R}$ we write $w_{\pm} = (u-k)_{\pm}$ \item For any fixed $t,k\in\RR$ and set $\Omega\subset\RR^N$, we denote $A_{\pm}(k,t) = \{x\in \Omega: (u-k)_{\pm}(\cdot,t) > 0\}$; for any ball $B_r$ we write $A_{\pm}(k,t) \cap (B_{r}\times I) = A_{\pm}(k,r,t)$. \item For any set $\Omega \subset \RR^N$, we denote $C_\Omega:=(\Omega^c\times\Omega^c)^c=\left(\Omega\times\Omega\right) \cup \left( \Omega\times(\RR^N\setminus\Omega)\right) \cup \left((\RR^N\setminus\Omega)\times\Omega\right)$. \end{itemize} Let $s \in (0,1)$ and $p >1$ be fixed, and $\Lambda \geq 1$ be a given constant. For almost every $x,y \in \RR^N$, let $K:\RR^N\times\RR^N\times \RR \to [0,\infty)$ be a symmetric measurable function satisfying \begin{equation}\label{boundsonKernel}
\frac{(1-s)}{\Lambda|x-y|^{N+ps}}\leq K(x,y,t)\leq \frac{(1-s)\Lambda}{|x-y|^{N+ps}}. \end{equation}
In this paper, we are interested in the regularity theory for the equation \begin{align}\label{maineq}
\partial_t u + \mathcal{L}u=0, \end{align} where $\mathcal{L}$ is the operator formally defined by \begin{equation*}
\mathcal{L}u=\text{P.V.}\oldint\limits_{\RR^N} K(x,y,t)|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy,\,x\in\RR^N. \end{equation*}
\subsection{Function spaces} Let $1<p<\infty$, we denote by $p'=p/(p-1)$ the conjugate exponent of $p$. Let $\Omega$ be an open subset of $\RR^N$, we define the {\it Sobolev-Slobodeki\u i} space, which is the fractional analogue of Sobolev spaces as follows: \begin{align*}
W^{s,p}(\Omega):=\left\{ \psi\in L^p(\Omega): [\psi]_{W^{s,p}(\Omega)}<\infty \right\},\qquad \text{for} \ s\in (0,1), \end{align*} where the seminorm $[\cdot]_{W^{s,p}(\Omega)}$ is defined by \begin{align*}
[\psi]_{W^{s,p}(\Omega)}:=\left( \oldiint\limits_{\Omega\times\Omega} \frac{|\psi(x)-\psi(y)|^p}{|x-y|^{N+ps}}\,dx\,dy \right)^{\frac 1p}. \end{align*} The space when endowed with the norm $\norm{\psi}_{W^{s,p}(\Omega)}=\norm{\psi}_{L^p(\Omega)}+[\psi]_{W^{s,p}(\Omega)}$ becomes a Banach space. The space $W^{s,p}_0(\Omega)$ is the subspace of $W^{s,p}(\RR^N)$ consisting of functions that vanish outside $\Omega$. We will use the notation $W^{s,p}_{(u_0)}(\Omega)$ to denote the space of functions in $W^{s,p}(\RR^N)$ such that $u-u_0\in W^{s,p}_0(\Omega)$.
Let $I$ be an interval and let $V$ be a separable, reflexive Banach space, endowed with a norm $\norm{\cdot}_V$. We denote by $V^*$ to be its topological dual space. Let $v$ be a mapping such that for a.e. $t\in I$, $v(t)\in V$. If the function $t\mapsto \norm{v(t)}_V$ is measurable on $I$, then $v$ is said to belong to the Banach space $L^p(I;V)$ provided $\oldint\limits_I\norm{v(t)}_V^p\,dt<\infty$. It is well known that the dual space $L^p(I;V)^*$ can be characterized as $L^{p'}(I;V^*)$.
Since the boundedness result requires some finiteness condition on the nonlocal tails, we define the tail space for some $m >0$ and $s >0$ as follows: \begin{equation*}
L^m_{s}(\RR^N):=\left\{ v\in L^m_{\text{loc}}(\RR^N):\oldint\limits_{\RR^N}\frac{|v(x)|^m}{1+|x|^{N+s}}\,dx<+\infty \right\}. \end{equation*} Then a nonlocal tail is defined by \begin{equation*}
\text{Tail}_{m,s,\infty}(v;x_0,R,I):=\text{Tail}_\infty(v;x_0,R,t_0-\theta,t_0):=\sup_{t\in (t_0-\theta, t_0)}\left( R^{sm}\oldint\limits_{\RR^N\setminus B_R(x_0)} \frac{|v(x,t)|^{m-1}}{|x-x_0|^{N+sm}}\,dx \right)^{\frac{1}{m-1}}, \end{equation*} where $(x_0,t_0)\in \RR^N\times (-T,T)$ and the interval $I=(t_0-\theta,t_0)\subseteq (-T,T)$. From this definition, it follows that for any $v\in L^\infty(-T,T;L^{m-1}_{sm}(\RR^N))$, there holds $\text{Tail}_{m,s,\infty}(v;x_0,R,I)<\infty$.
\subsection{Definitions}
Now, we are ready to state the definition of a weak sub(super)-solution.
\begin{definition}
A function $u\in L^p(I;W^{s,p}_{\text{loc}}(\Omega))\cap L^\infty(I;L^2_{\text{loc}}(\Omega))\cap L^\infty(I;L^{p-1}_{sp}(\RR^N))$ is said to be a local weak sub(super)-solution to \cref{maineq} if for any closed interval $[t_1,t_2]\subset I$, the following holds:
\begin{align*}
\oldint\limits_{\Omega}& u(x,t_2)\phi(x,t_2)\,dx - \oldint\limits_{\Omega} u(x,t_1)\phi(x,t_1)\,dx - \oldint\limits_{t_1}^{t_2}\oldint\limits_{\Omega} u(x,t)\partial_t\phi(x,t)\,dx\,dt\\
&+\oldiint\limits_{C_\Omega}\,K(x,y,t)|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))\,dy\,dx\,dt
\leq (\geq)0 ,
\end{align*} for all $\phi\in L^p(I,W^{s,p}(\Omega))\cap W^{1,2}(I,L^2(\Omega))$, where $C_\Omega:=(\Omega^c\times\Omega^c)^c=\left(\Omega\times\Omega\right) \cup \left( \Omega\times(\RR^N\setminus\Omega)\right) \cup \left((\RR^N\setminus\Omega)\times\Omega\right)$ and the spatial support of $\phi$ is contained in $\Omega$. \end{definition}
\subsection{Auxiliary Results} We collect the following standard results which will be used in the course of the paper. We begin with the Sobolev-type inequality~\cite[Lemma 2.3]{dingLocalBoundednessHolder2021}.
\begin{theorem}\label{fracpoin}
Let $t_2>t_1>0$ and suppose $s\in(0,1)$ and $1\leq p<\infty$. Then for any $f\in L^p(t_1,t_2;W^{s,p}(B_r))\cap L^\infty(t_1,t_2;L^2(B_r))$, we have
\begin{equation*}
\begin{array}{rcl}
\oldint\limits_{t_1}^{t_2}\oldfint\limits_{B_r}|f(x,t)|^{p\left(1+\frac{2s}{N}\right)}\,dx\,dt
& \leq & C(N,s,p)\left(r^{sp}\oldint\limits_{t_1}^{t_2}\oldint\limits_{B_r}\oldfint\limits_{B_r}\frac{|f(x,t)-f(y,t)|^p}{|x-y|^{N+sp}}\,dx\,dy\,dt+\oldint\limits_{t_1}^{t_2}\oldfint\limits_{B_r}|f(x,t)|^p\,dx\,dt\right) \\
&&\quad \times\left(\sup_{t_1<t<t_2}\oldfint\limits_{B_r}|f(x,t)|^2\,dx\right)^{\frac{sp}{N}}.
\end{array}
\end{equation*} \end{theorem}
We also list a number of algebraic inequalities that are customary in obtaining energy estimates for nonlinear nonlocal equations.
\begin{lemma}(\cite[Lemma 4.1]{cozziRegularityResultsHarnack2017})\label{pineq1}
Let $p\geq 1$ and $a,b\geq 0$, then for any $\theta\in (0,1)$, the following holds:
\begin{align*}
(a+b)^p-a^p\geq \theta p a^{p-1}b + (1-\theta)b^p.
\end{align*} \end{lemma}
\begin{lemma}(\cite[Lemma 4.3]{cozziRegularityResultsHarnack2017})\label{pineq3}
Let $p\geq 1$ and $a\geq b\geq 0$, then for any $\varepsilon>0$, the following holds:
\begin{align*}
a^p-b^p\leq \varepsilon a^p+\left(\frac{p-1}{\varepsilon}\right)^{p-1}(a-b)^p.
\end{align*} \end{lemma}
Finally, we recall the following well known lemma concerning the geometric convergence of sequence of numbers (see \cite[Lemma 4.1 from Section I]{dibenedettoDegenerateParabolicEquations1993} for the details): \begin{lemma}\label{geo_con}
Let $\{Y_n\}$, $n=0,1,2,\ldots$, be a sequence of positive number, satisfying the recursive inequalities
\[ Y_{n+1} \leq C b^n Y_{n}^{1+\alpha},\]
where $C > 1$, $b>1$, and $\alpha > 0$ are given numbers. If
\[ Y_0 \leq C^{-\frac{1}{\alpha}}b^{-\frac{1}{\alpha^2}},\]
then $\{Y_n\}$ converges to zero as $n\to \infty$. \end{lemma}
\subsection{Main results}
We prove the following main theorem.
\begin{theorem}\label{holderparabolic}
Let $p\in(1,\infty)$ and let $u\in L^p(I;W^{s,p}_{\text{loc}}(\Omega))\cap L^\infty(I;L^2_{\text{loc}}(\Omega))\cap L^\infty(I;L^{p-1}_{sp}(\RR^N))$ be a bounded local solution to \cref{maineq}. Then $u$ is locally H\"older continuous in $\Omega_T$, i.e., there exist constants $\gamma>1$, $C_0>1$, and $\alpha\in (0,1)$ depending only on the data, such that, for every cylinder
\[B_R(x_0)\times (t_0-L^{2-p}R^{sp},t_0)\subset B_{C_0\,R}(x_0)\times (t_0-L^{2-p}(C_0\,R)^{sp},t_0) \subset \Omega_T,\]
\begin{align*}
|u(x_1,t_1)-u(x_2,t_2)|\leq \gamma \left(\frac{|x_1-x_2|+L^{\frac{p-2}{sp}}|t_1-t_2|^{\frac{1}{sp}}}{R}\right)^{\alpha}\,L,
\end{align*} for every pair of points $(x_1,t_1), (x_2,t_2)\in B_R(x_0)\times (t_0-L^{2-p}R^{sp},t_0)$, where
\[L:=\|u\|_{L^\infty(Q_{C_0\,R})}+\text{Tail}_{\infty}(u;C_0\,R,x_0,(t_0-(C_0\,R)^{sp},t_0)).\] \end{theorem}
\section{Preliminary Results}\label{sec3}
\subsection{Energy estimates}
\begin{theorem}\label{energyest} Let $u$ be a local, weak sub(super)-solution. Let $B_{R}(x_0)\times (t_0-\theta,t_0)\Subset E_T$ and $k,\tau_1,\tau_2$ with $0<r< R, t_0-\theta<t_0-\tau_2<t_0-\tau_1\leq t_0-\frac{\theta}{2}$. Then there are positive universal constants $\gamma, C$ such that for every level $k \in \bb{R}$ and every piecewise smooth cutoff function $\xi(x,t)=\xi_1(x)\xi_2(t)$ vanishing on $\de K$ with $0 \leq \xi \leq 1$ we have \begin{align*}
\underset{t_0-\tau_1<t <t_0 }{\esssup}\oldint\limits_{B_r(x_0)}w_{\pm}^2&\xi^p(x,t)\,dx+ \oldint\limits_{t_0-\tau_1}^{t_0}\oldint\limits_{B_r(x_0)} w_{\pm}(x,t)\oldint\limits_{{B_R(x_0)}}\frac{w_{\mp}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\,dx\,dt \\
&+\underset{{B_r(x_0)}\times B_r(x_0) \times (t_0-\tau_1,t_0)}{\oldiiint\limits}|w_{\pm}(x,t)\xi_1(x)-w_{\pm}(y,t)\xi_1(y)|^p|\xi_2(t)|^p\,d\mu\,dt\\
&\leq \ \gamma \oldint\limits_{{B_R(x_0)}}w_{\pm}^2\xi^p(x,t_0-\tau_1)\,dx\\
&\quad + \gamma \oldint\limits_{t_0-\tau_2}^{t_0}\oldiint\limits_{{B_R(x_0)}\times {B_R(x_0)}} \max\{w_{\pm}(x,t),w_{\pm}(y,t)\}^{p}|\xi_1(x)-\xi_2(y)|^p|\xi_2(t)|^p\,d\mu\,dt \\
&\quad + \gamma \oldiint\limits_{(t_0-\tau_2,t_0)\times {B_R(x_0)}} w_{\pm}^2(x,t)|\de_t \xi(x,t)| \,dx\,dt \\
&\quad+ C\underset{\stackrel{t \in (t_0-\tau_2,t_0)}{x\in \spt\xi_1}}{\esssup}\,\oldint\limits_{{B_r(x_0)}^c}\frac{w_{\pm}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\lbr \oldiint\limits_{(t_0-\tau_2,t_0)\times {B_R(x_0)}} w_{\pm}(x,t)\,dx\,dt\rbr, \end{align*}
where $w_{\pm} = (u-k)_{\pm}$, $\xi_2 \in C^1(t_0-\theta,t_0)$ be a test function such that $\xi_2 = 0$ on $(t_0-\theta,t_0-\tau_2)$, $\xi_2 = 1$ on $(t_0-\tau_1,t_0)$ and $0\leq \xi_2 \leq 1$. Also, $\xi_1\in C_0^\infty(B_{R}(x_0))$ and $d\mu=\frac{dx\,dy}{|x-y|^{N+ps}}$. \end{theorem}
\begin{proof} All the heavy-lifting pertaining to time regularization has already been performed in \cite[Lemma 3.3]{dingLocalBoundednessHolder2021}. In fact, the proof of our theorem is the same as their proof except for their estimate for the terms $I_2$ and $I_3$. For this reason and in the interest of brevity, we only present below the part of their proof with the subheadings ``{\bf The estimate of $I_2$}'' and ``{\bf The estimate of $I_3$}'' with different estimates that produce the ``good term''. To this end, we follow \cite[Prop. 8.5]{cozziRegularityResultsHarnack2017}. Moreover, we only write the calculation for sub-solutions since those for super-solutions are similar and we assume without loss of generality that $(x_0,t_0)=(0,0)$. Let $\rho_1, \rho_2$ be positive numbers such that $0<r\leq \rho_1<\rho_2\leq R$. We take $\xi_1$ such that $0\leq \xi_1\leq 1$ and $\spt(\xi_1)=B_{\frac{\rho_1+\rho_2}{2}}:=B_{\frac{\rho_1+\rho_2}{2}}(0)$.
\begin{description}
\item[The estimate of $I_2$:]
Recall that \begin{align*}
I_2:=\frac{1}{2}\oldint\limits_{-\tau_2}^0 \oldiint\limits_{B_{\rho_2}\times B_{\rho_2}}|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))\xi_2^p(t)\,d\mu\,dt. \end{align*}
To estimate the nonlocal terms, we consider the following cases pointwise for $(x,t)$ and $(y,t)$ in $C_{B_{\rho_2}} \times I$ where $I=(-\tau_2,0)$. For any fixed $t\in I$, we claim that
\begin{itemize}
\item If $x \notin A_+(k,\rho_2,t)$ and $y \notin A_+(k,\rho_2,t)$ then
\begin{equation}\label{eq:A}
|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))=0.
\end{equation}
\item If $x \in A_+(k,\rho_2,t)$ and $ y \notin A_+(k,\rho_2,t)$ then
\begin{align}\label{eq:B}
|u(x,t)-u(y,t)|^{p-2}&(u(x,t)-u(y,t))(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))\nonumber\\
&\geq\min\{2^{p-2},1\}\left[|w_+(x,t)-w_+(y,t)|^p+w_-(y,t)^{p-1}w_+(x,t)\right]\xi_1(x)^p.
\end{align}
\item If $x, y \in A_+(k,\rho_2,t)$ then
\begin{align}\label{eq:C}
|u(x,t)-u(y,t)|^{p-2}&(u(x,t)-u(y,t))(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))\nonumber\\
& \geq \frac{1}{2}|w_+(x,t)-w_+(y,t)|^p\max\{\xi_1(x),\xi_1(y)\}^p\nonumber\\
& \qquad -C\max\{w_+(x,t),w_+(y,t)\}^p|\xi_1(x)-\xi(y)|^p,
\end{align} for some $C\geq 1$ depending only on $p$.
\end{itemize}
The estimate \cref{eq:A} follows from the fact that $(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))=0$ when $x \notin A_+(k,t)$ and $y \notin A_+(k,t)$.
To obtain estimate \cref{eq:B} we note that
\begin{align*}
|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))&(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))=\\
&(w_+(x,t)+w_+(y,t))^{p-1}w_+(x,t)\xi_1^p(x),
\end{align*} when $x \in A_+(k,\rho_2,t)$ and $y \notin A_+(k,\rho_2,t)$. Now, \cref{eq:B} follows by an application of \cref{pineq1}, with $\theta=0$ when $p\geq 2$ and Jensen's inequality when $p<2$.
Finally, to obtain estimate \cref{eq:C}, we begin by assuming without loss of generality that $u(x,t)\geq u(y,t)$. Since $x,y \in A_+(k,\rho_2,t)$, we have
\begin{align*}
|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))&(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))=\\
&(w_+(x,t)-w_+(y,t))^{p-1}(\xi_1^p(x)w_+(x,t)-\xi_1^p(y)w_+(y,t)).
\end{align*}
Now, if $\xi_1(x)\geq \xi_1(y)$, then
\begin{align*}
|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))&(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))=\\
&(w_+(x,t)-w_+(y,t))^{p}(\xi_1(x))^p,
\end{align*} and \cref{eq:C} is immediate.
However, if $\xi_1(x)<\xi_1(y)$, then
\begin{align}\label{caccest1}
|u(x,t)-u(y,t)|^{p-2}&(u(x,t)-u(y,t))(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))=\nonumber\\
&(w_+(x,t)-w_+(y,t))^{p}(\xi_1(y))^p-(w_+(x,t)-w_+(y,t))^{p-1}w_+(x,t)(\xi_1^p(x)-\xi_1^p(y)).
\end{align}
We apply \cref{pineq3} with $a=\xi_1(y)$ and $b=\xi_1(x)$ and
\begin{align*}
\varepsilon=\frac{1}{2}\frac{w_+(x,t)-w_+(y,t)}{w_+(x,t)},
\end{align*} to obtain
\begin{align}
\label{caccest2}
(w_+(x,t)-w_+(y,t))^{p-1}&w_+(x,t)(\xi_1^p(x)-\xi_1^p(y))\nonumber\\
& \frac{1}{2}(w_+(x,t)-w_+(y,t))^p\xi_1^p(y)+[2(p-1)]^{p-1}w_+(x,t)(\xi_1(y)-\xi(x))^p.
\end{align}
Combining \cref{caccest1} and \cref{caccest2}, we obtain \cref{eq:C} when $\xi_1(x)<\xi_1(y)$.
As a consequence of these cases and \cref{boundsonKernel}, we have
\begin{align*}
I_2\geq& C_{11}\Biggr[\oldint\limits_I\oldiint\limits_{B_{\rho_1}\times B_{\rho_1}} \frac{|w_+(x,t)-w_+(y,t)|^p}{|x-y|^{N+ps}}\max\{\xi_1(x),\xi_1(y)\}^p\xi_2^p(t)\,dx\,dy\,dt \nonumber\\
&\qquad+\oldint\limits_I\oldint\limits_{B_{\rho_1}}w_+(x,t)\left(\oldint\limits_{B_{\rho_2}}\frac{w_-(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right)\,dx\,dt\Biggr]\\
&\qquad\qquad-C_{22}\oldint\limits_I\oldiint\limits_{B_{\rho_2}\times B_{\rho_2}}\max\{w_+(x,t),w_+(y,t)\}^p\frac{|\xi_1(x)-\xi_1(y)|^p}{|x-y|^{N+sp}}\xi_2^p(t)\,dx\,dy\,dt. \end{align*}
\item[The estimate of $I_3$:] Recall that \begin{align}\label{caccest3}
I_3:&=\oldint\limits_{-\tau_2}^0 \oldiint\limits_{(\RR^N\setminus B_{\rho_2})\times B_{\rho_2}}|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_+(x,t)\xi_1^p(x)-w_+(y,t)\xi_1^p(y))\xi_2^p(t)\,d\mu\,dt\nonumber\\
&=\oldint\limits_{-\tau_2}^0\oldint\limits_{B_{\rho_2}}\xi_1^p(x)w_+(x,t)\left[\oldint\limits_{\RR^N\setminus B_{\rho_2}}{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}K(x,y,t)\,dy\right]\,dx\,dt\nonumber\\
&\geq C_{33}\underbrace{\oldint\limits_{-\tau_2}^0 \oldint\limits_{A^+(k,\rho_1,t)}w_+(x,t)\left[\oldint\limits_{(B_R(x)\cap \{u(x,t)\geq u(y,t)\})\setminus B_{\rho_2}} \frac{(u(x,t)-u(y,t))^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt}_{T_1}\nonumber\\
&\qquad - C_{44}\underbrace{\oldint\limits_{-\tau_2}^0 \oldint\limits_{A^+\left(k,\frac{\rho_1+\rho_2}{2},t\right)}w_+(x,t)\left[\oldint\limits_{\{u(y,t)> u(x,t)\}\setminus B_{\rho_2}} \frac{(u(x,t)-u(y,t))^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt}_{T_2}. \end{align} We estimate $T_1$ as \begin{align}\label{caccest4}
T_1&\geq \oldint\limits_{I}\oldint\limits_{B_{\rho_1}}w_+(x,t)\left[\oldint\limits_{(B_R(x)\cap A^-(k,t))\setminus B_{\rho_2}} \frac{(w_+(x,t)+w_+(y,t))^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt\nonumber\\
&\geq \oldint\limits_{I}\oldint\limits_{B_{\rho_1}}w_+(x,t)\left[\oldint\limits_{B_R(x)\setminus B_{\rho_2}} \frac{w_+(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt \end{align} We estimate $T_2$ as \begin{align}\label{caccest5}
T_2&\leq \oldint\limits_{-\tau_2}^0 \oldint\limits_{B_{\frac{\rho_1+\rho_2}{2}}}w_+(x,t)\left[\oldint\limits_{\stackrel{\RR^N\setminus B_{\rho_2}}{x\in\spt\xi_1}} \frac{w_+(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt\nonumber\\
&\leq \oldiint\limits_{I\times B_R}w_+(x,t)\,dx\,dt\lbr \esssup\limits_{t\in I}\left[\oldint\limits_{\stackrel{\RR^N\setminus B_{\rho_2}}{x\in\spt\xi_1}} \frac{w_+(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt\rbr. \end{align} Combining \cref{caccest4} and \cref{caccest5} in \cref{caccest3}, we obtain \begin{align*}
I_3\geq C_{33}&\oldint\limits_{I}\oldint\limits_{B_{\rho_1}}w_+(x,t)\left[\oldint\limits_{B_R(x)\setminus B_{\rho_2}} \frac{w_+(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt\\
&-C_{44}\oldiint\limits_{I\times B_R}w_+(x,t)\,dx\,dt \lbr \esssup\limits_{t\in I}\left[\oldint\limits_{\stackrel{\RR^N\setminus B_{\rho_2}}{x\in\spt\xi_1}} \frac{w_+(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt\rbr. \end{align*} \end{description} As mentioned earlier, the rest of the proof of \cref{energyest} is the same as in \cite[Lemma 3.3]{dingLocalBoundednessHolder2021}, however, we do not have the term $I_4$ since we are not considering source terms. \end{proof}
\begin{remark} This energy estimate first appears in \cite{prasadLocalBoundednessVariational2021} under the assumption that $\partial_t u\in L^2(E_T)$ (see Lemma 3.1 and Remark 3.2 in \cite{prasadLocalBoundednessVariational2021}). This assumption is generally dropped by working with regularizations in time, such as Steklov averages \cite{dibenedettoDegenerateParabolicEquations1993}. In fact, the estimate in \cite{prasadLocalBoundednessVariational2021} is proved for parabolic minimizers, however the same method works for solutions of equations. \end{remark} Henceforth, excepting the final section, we will let $u$ denote a nonnegative, locally bounded supersolution.
\subsection{Shrinking Lemma}
One of the main difficulties we face when dealing with regularity issues for nonlocal equations is the lack of a corresponding isoperimetric inequality for $W^{s,p}$ functions. Indeed, since such functions can have jumps, a generic isoperimetric inequality seems out of reach at this time (see \cite{cozziFractionalGiorgiClasses2019, adimurthiOlderRegularityFractional2022} ) One way around this issue is to note that since we are working with solutions of an equation, which we expect to be continuous and hence have no jumps, we could try and cook up an isoperimetric inequality for solutions. Such a strategy turns out to be feasible due to the presence of the ``good term'' or the isoperimetric term in the Caccioppoli inequality.
\begin{lemma}\label{lem:isop} Let $k<l<m$ be arbitrary levels and $A \geq 1$. Then, \[ (l-k)(m-l)^{p-1}\abs{[u>m]\cap K_{\rho}}\abs{[u<k]\cap K_{\rho}} \leq
C\rho^{N+sp}\oldint\limits_{K_{\rho}} (u-l)_{-}(x)\oldint\limits_{K_{A\rho}}\frac{(u-l)_{+}^{p-1}(y)}{|x-y|^{N+sp}}\,dy\,dx, \] where $C = C(N,s,p,A)>0$. \end{lemma}
\begin{proof} We estimate from below \begin{align*}
\oldint\limits_{K_{\rho}} (u-l)_{-}(x)\oldint\limits_{K_{A\rho}}\frac{(u-l)_{+}^{p-1}(y)}{|x-y|^{N+sp}}\,dy\,dx &\geq \frac{C}{\rho^{N+sp}}\oldint\limits_{K_{\rho}} (u-l)_{-}(x)\oldint\limits_{K_{\rho}}(u-l)_{+}^{p-1}(y)\,dy\,dx
\\
&\geq \frac{C}{\rho^{N+sp}}\oldint\limits_{K_{\rho}\cap \{u<k\}} (u-l)_{-}(x)\oldint\limits_{K_{\rho}\cap \{u>m\}}(u-l)_{+}^{p-1}(y)\,dy\,dx
\\
&\geq \frac{C}{\rho^{N+sp}}\oldint\limits_{K_{\rho}\cap \{u<k\}} (l-k)\oldint\limits_{K_{\rho}\cap \{u>m\}}(m-l)^{p-1}\,dy\,dx. \end{align*} \end{proof}
We now prove the shrinking lemma using \cref{lem:isop}. It is usually proved in the local case using the De Giorgi isoperimetric inequality.
\begin{lemma}\label{lem:shrinking} Let $u$ be a super-solution of \cref{maineq}. Suppose that for some level $m$, some constant $\nu \in (0,1)$ and all time levels $\tau$ in some interval $J$ we have \[
|[u(\cdot,\tau)>m]\cap K_{\rho}| \geq \nu|K_{\rho}|, \] and we can arrange that for some $A \geq 1$, the following is also satisfied: \begin{align}\label{smallness}
\oldint\limits_J\oldint\limits_{K_{\rho}} (u-l)_{-}(x,t)\oldint\limits_{K_{A\rho}}\frac{(u-l)_{+}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\,dx\,dt \leq C_1\frac{l^p}{\rho^{sp}}|Q|, \end{align} where \[ l = \frac{m}{2^{j}}, \qquad j\geq 1 \qquad \text{ and } \qquad Q = K_{\rho} \times J, \] then we have the following conclusion: \[
\left|\left[u<\frac{m}{2^{j+1}}\right]\cap Q\right| \leq \left(\frac{C}{2^{j}-1}\right)^{p-1}|Q|, \] where $C = (C_1,A,N,\nu) >0$. \end{lemma} \begin{proof} In the conclusion of \cref{lem:isop} we put $k = \frac{m}{2^{j+1}}$ and $l = \frac{m}{2^{j}}$, use the hypothesis and then integrate over the time interval $J$ to get \[
\frac{m}{2^{j+1}}m^{p-1}\left(\frac{2^j-1}{2^j}\right)^{p-1} \left|\left[u<\frac{m}{2^{j+1}}\right]\cap Q\right| \leq
\frac{C}{\nu}\left(\frac{m}{2^j}\right)^{p}|Q|, \] where $C$ depends on $C_1,A$ and $N$. The conclusion follows after a simple rearrangement. \end{proof}
\begin{remark} In applying the shrinking lemma in proving the expansion of positivity lemmas, in order to get the smallness condition \cref{smallness}, we shall need to impose the smallness condition on the Tail term - this is one of the ways in which the Tail alternatives enter the picture. \end{remark}
\begin{remark}\label{rem:nonlocalA} An intriguing aspect of our lemma is the fact that the dependence between the levels in the conclusion and how small we can make the corresponding level set is \textit{not} exponential; such a dependence is reflected in the best known lower bound for the H\"older exponent for solutions to second order linear nonlocal equations \cite{mosconiOptimalEllipticRegularity2018}, whereas, in the local case, this lower bound is much worse \cite{bombieriHarnackInequalityElliptic1972}. \end{remark}
\subsection{Tail Estimates}\label{sec:tail}
In this section, we shall outline how the estimate for the tail term is made and we shall refer to this section whenever a similar calculation is required in subsequent sections.
For any level $k$ we want to estimate \[
\underset{\stackrel{t \in J;}{x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy , \] where $J$ is some time interval and $\zeta$ is a cutoff function. We will typically choose $\zeta$ to be supported in $K_{\vartheta\rho}$ for some $\vartheta \in (0,1)$. In such cases, \[
|y-y_0| \leq |x-y|\left(1+\frac{|x-y_0|}{|x-y|}\right)\leq |x-y|\left(1+\frac{\vartheta}{(1-\vartheta)}\right), \] so we can make a first estimate \[
\underset{\stackrel{t \in J;}{ x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy
\leq \frac{1}{(1-\vartheta)^{N+sp}}\underset{t \in J}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{(u-k)_{-}^{p-1}(y,t)}{|y-y_0|^{N+sp}}\,dy. \] In the local case, we could always estimate $(u-k)_{-} \leq k$ because we take $u \geq 0$ locally. However, in the Tail term we are on the complement of a cube and so unless we make a global boundedness assumption (for e.g. $u \geq 0$ in full space), the best we can do is \[ (u-k)_{-} \leq u_{-} + k, \] which leads us to the next estimate \[
\underset{t \in J}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{(u-k)_{-}^{p-1}(y,t)}{|y-y_0|^{N+sp}}\,dy \leq C(p)\frac{k^{p-1}}{\rho^{sp}} + C(p)\,\underset{t \in J}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{u_{-}^{p-1}(y,t)}{|y-y_0|^{N+sp}}\,dy \] Putting together the above estimates yield \[
\underset{\stackrel{t \in J;}{ x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy \leq \frac{C(p)}{\rho^{sp}}\left[k^{p-1}+\text{Tail}_{\infty}^{p-1}(u_{-};y_0,\rho,J)\right]. \] Finally we usually want an estimate of the form \[
\underset{\stackrel{t \in J;}{ x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c(y_0)}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy \leq C\frac{k^{p-1}}{\rho^{sp}}, \] and so we impose the condition that \[ \text{Tail}_{\infty}^{p-1}(u_{-};y_0,\rho,J) \leq k^{p-1}. \] This is the origin of the various Tail alternatives.
\begin{remark} We will see that it is the Tail that captures the nonlocality of our equation in the sense that if it is small then the proofs become local. In fact, when dealing with nonnegative supersolutions, we only need the Tail of the negative part of the solution to be small and so if we assume that it is nonnegative in full space (as in \cite{shangOlderRegularityMixed2021}) then the Tail alternatives are automatically verified and our proofs become ``local proofs''. \end{remark}
\subsection{De Giorgi iteration Lemma} Let \[ \mu_{-} \leq \underset{(x_0,t_0)+Q_{2\rho}^{-}(\theta)}{\essinf} u \] and $M > 0$. Let $\xi \in (0,1]$ and $a\in (0,1)$ be fixed numbers. \begin{lemma}\label{lem:deG} Let $u$ be a locally bounded, local, weak supersolution of \cref{maineq} in $E_T$. Then there exists a number $\nu_{-}$ depending only on the data and the parameters $\theta,\xi,M$ and $a$ such that if the following is satisfied \[
|[u\leq \mu_{-} + \xi M]\cap (x_0,t_0)+Q_{2\rho}^{-}(\theta)| \leq \nu_{-}|Q_{2\rho}^{-}(\theta)|, \] then one of the following two conclusion holds: \[ \text{Tail}_{\infty}((u-\mu)_{-};x_0,2\rho,(t_0-\theta (2\rho)^{sp},t_0]) > \xi M, \] or \[ u \geq \mu_{-} + a\xi M \text{ a.e. in } [(x_0,t_0)+Q_{\rho}^{-}(\theta)]. \] \end{lemma} \begin{proof} Without loss of generality, we will assume that $(x_0,t_0) = (0,0)$ and for $n=0,1,2,\ldots$, we set \begin{equation*}
\rho_n = \rho + 2^{-n}\rho, \qquad
K_n = K_{\rho_n} \quad \text{and} \quad
Q_n = K_n\times(-\theta\rho_n^{sp},0]. \end{equation*} We apply the energy estimates over $K_n$ and $Q_n$ to $ (u-k_n)_{-}$ for the levels \begin{equation*}
k_n = \mu_{-} + \xi_nM \qquad \text{ where } \qquad \xi_n = a\xi + \frac{1-a}{2^n}\xi. \end{equation*} We take the following cutoff function $\zeta(x,t) = \zeta_1(x)\zeta_2(t)$ where $\zeta_1$ is supported in $K_{\tilde{n}}$, $\zeta_2(t) = 0$ for $t<-\theta\rho_n^{sp}$ and \begin{equation*}
\begin{array}{c}
\zeta_1 \equiv 1 \text{ on } K_{n+1}, \qquad |\de\zeta_1| \apprle \frac{1}{\rho}, \\
\zeta_2(t) = 1 \ \text{ for }\ t\geq -\theta\rho_{n+1}^{sp}, \qquad 0 \leq \de_t(\zeta_2) \leq \frac{2^{p(n+1)}}{\theta\rho^{sp}},
\end{array} \end{equation*} where $K_{\tilde{n}}$ is the cube with radius $(\rho_n+\rho_{n+1})/2$.
The energy inequality from \cref{energyest} gives \begin{align}
\underset{-\theta\rho_n^{sp} < t < 0}{\esssup}&\oldint\limits_{K_n}(u-k_n)_{-}^2\zeta^p(x,t)\,dx + {\oldiiint\limits_{Q_n}}\frac{|(u-k_n)_{-}(x,t)\zeta(x,t)-(u-k_n)_{-}\zeta(y,t)|^p}{|x-y|^{N+sp}}\,dx \,dy\,dt \nonumber
\\
&\leq \gamma \frac{2^{np}}{\rho^{sp}}\left((\xi M)^p+\frac{1}{\theta}(\xi M)^2\right)|[u<k_n]\cap Q_n| \nonumber
\\
&\quad + C\underset{\stackrel{-\theta\rho_n^{sp} < t < 0;}{ x\in \spt\zeta_1}}{\esssup}\oldint\limits_{K_n^c}\frac{(u-k_n)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\oldint\limits_{-\theta\rho_n^{sp}}^{0}\oldint\limits_{K_n} (u-k_n)_{-}(x,t)\zeta_1(x)\,dx\,dt. \label{energy_est1} \end{align}
To estimate the tail term, we first note that if $x \in K_{\tilde{n}}(0)$ and $y \in K_n^c(0)$ then \[
|y|\leq |x-y|\left(1+\frac{|x|}{|x-y|}\right) \leq c2^n|x-y| \ \Longrightarrow\ \frac{1}{|x-y|^{N+sp}} \leq c2^{n(N+sp)}\frac{1}{|y|^{N+sp}}. \]
Next, we note that the following \[ (u-k_n)_{-} \leq \xi_nM - (u - \mu_{-}) \leq \xi M + (u - \mu_{-})_{-}, \] holds globally. Therefore we have \begin{align*}
\underset{\stackrel{-\theta\rho_n^{sp} < t < 0;}{ x\in \spt\zeta_1}}{\esssup}\oldint\limits_{K_n^c}&\frac{(u-k_n)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\\
& \leq c2^{n(N+sp)}\underset{-\theta\rho_n^{sp} < t < 0}{\esssup}\oldint\limits_{K_n^c}\frac{(u - \mu_{-})_{-}^{p-1}(y,t)+(\xi M)^{p-1}}{|y|^{N+sp}}\,dy
\\
&\leq c2^{n(N+sp)}\frac{(\xi M)^{p-1}}{\rho^{sp}} +
c2^{n(N+sp)}\underset{-\theta\rho_n^{sp} < t < 0 }{\esssup}\oldint\limits_{K_{2\rho}^c}\frac{(u - \mu_{-})_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy
\\
&\leq c2^{n(N+sp)}\left(\frac{(\xi M)^{p-1}}{\rho^{sp}} + \frac{1}{(2\rho)^{sp}}\text{Tail}_{\infty}^{p-1}((u-\mu)_{-};0,2\rho,(-\theta (2\rho)^{sp},0])\right), \end{align*} where we used the fact that $u-\mu^- \geq 0$ in $Q_{2\rho}^{-}(\theta)$.
Thus, we can estimate the last term appearing on the right hand side of \cref{energy_est1} by \begin{align*}
C&\underset{\stackrel{-\theta\rho_n^{sp} < t < 0;}{ x\in \spt\zeta_1}}{\esssup}\oldint\limits_{K_n^c}\frac{(u-k_n)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\oldint\limits_{-\theta\rho_n^{sp}}^{0}\oldint\limits_{K_n} (u-k_n)_{-}(x,t)\zeta_1(x)\,dx\,dt
\\
&\leq cC2^{n(N+sp)}\left(\frac{(\xi M)^{p-1}}{\rho^{sp}} + \frac{1}{(2\rho)^{sp}}\text{Tail}_{\infty}^{p-1}((u-\mu)_{-};0,2\rho,(-\theta (2\rho)^{sp},0])\right)(\xi M)|[u<k_n]\cap Q_n|
\\
&\leq 2cC2^{n(N+sp)}\frac{1}{\rho^{sp}}\left((\xi M)^{p-1} + \text{Tail}_{\infty}^{p-1}((u-\mu)_{-};0,2\rho,(-\theta (2\rho)^{sp},0])\right)(\xi M)|[u<k_n]\cap Q_n|
\\
&\leq C2^{n(N+sp)}\frac{(\xi M)^{p}}{\rho^{sp}}|[u<k_n]\cap Q_n|, \end{align*} provided the following is satisfied: \[ \text{Tail}_{\infty}((u-\mu)_{-};0,2\rho,(-\theta (2\rho)^{sp},0]) \leq \xi M. \] Putting the above estimates together with \cref{energy_est1}, we get \begin{align*}
\underset{-\theta\rho_n^{sp} < t < 0}{\esssup}\oldint\limits_{K_n}(u-k_n)_{-}^2\zeta^p(x,t)\,dx + \oldiiint\limits_{Q_n}\frac{|(u-k_n)_{-}(x,t)\zeta(x,t)-(u-k_n)_{-}\zeta(y,t)|^p}{|x-y|^{N+sp}}\,dx \,dy\,dt
\\
\leq C\frac{2^{n(N+p)}}{\rho^{sp}}(\xi M)^{p}\left(2+\frac{1}{\theta}(\xi M)^{2-p}\right)|[u<k_n]\cap Q_n|. \end{align*}
We now apply the fractional parabolic Sobolev embedding from \cref{fracpoin} to get \begin{equation}\label{eq.1}
\begin{array}{rcl}
\oldiint\limits_{Q_n}[(u-k_n)_{-}\zeta]^{p\frac{N+2s}{N}} \,dx\,dt
&\apprle & \underset{Q_n}{\oldiiint\limits}\frac{|(u-k_n)_{-}(x,t)\zeta(x,t)-(u-k_n)_{-}\zeta(y,t)|^p}{|x-y|^{N+sp}}\,dx \,dy\,dt
\\&&\times \left(\underset{-\theta\rho_n^{sp} < t < 0}{\esssup}\oldint\limits_{K_n}[(u-k_n)_{-}\zeta(x,t)]^2\,dx\right)^{\frac{sp}{N}}
\\
&\apprle & \left[\frac{2^{n(N+p)}}{\rho^{sp}}(\xi M)^{p}\left(2+\frac{1}{\theta}(\xi M)^{2-p}\right)\right]^{\frac{N+sp}{N}}|[u<k_n]\cap Q_n|^{\frac{N+sp}{N}}.
\end{array} \end{equation}
Let us now estimate the term appearing on the left hand side of \cref{eq.1} from below \begin{equation}\label{eq.2}
\oldiint\limits_{Q_n}[(u-k_n)_{-}\zeta]^{p\frac{N+2s}{N}} \,dx\,dt \geq \left(\frac{(1-a)\xi M}{2^{n+1}}\right)^{p\frac{N+2s}{N}}|[u<k_{n+1}]\cap Q_{n+1}|. \end{equation} Finally, setting \[
Y_n = \frac{|[u<k_n]\cap Q_n|}{|Q_n|}, \] and combining \cref{eq.1} and \cref{eq.2}, we get \[ Y_{n+1}\leq \frac{Cb^n}{(1-a)^{(N+2s)\frac{p}{N}}}\left(\frac{\theta}{(\xi M)^{2-p}}\right)^{\frac{sp}{N}}\left(2+\frac{(\xi M)^{2-p}}{\theta}\right)^{\frac{N+sp}{N}}Y_n^{1+\frac{sp}{N}}, \] where $b = b(N,s,p) > 1$ and $C>0$ depends only on the data. By \cref{geo_con}, $Y_{\infty} = 0$ provided \[ Y_0 \leq C^{-\frac{N}{p}}b^{-\left(\frac{N}{p}\right)^2}(1-a)^{N+2s}\frac{\left(\frac{(\xi M)^{2-p}}{\theta}\right)^s}{\left(2+\frac{(\xi M)^{2-p}}{\theta}\right)^{\frac{N+sp}{p}}} =: \nu_{-}, \] which completes the proof of the lemma. \end{proof}
\begin{remark}
An analogous result holds for subsolutions. \end{remark}
\subsection{De Giorgi lemma: Forward in time version}
Let $u$ denote a nonnegative, local, weak supersolution of \cref{maineq} in $E_T$ and let $K_R\times I \subset E_T$ denote our reference cylinder. Suppose that we have the following information at a time level $t_0$: \begin{equation}\label{eq:deGleminitdata} u(x,t_0) \geq \xi M \qquad \text{ for a.e. } x \in K_{2\rho}(y), \end{equation} for some $M>0$ and $\xi \in (0,1]$. Then in the energy inequality from \cref{energyest}, for any level $k \leq \xi M$ over $[(x_0,t_0)+Q_{2\rho}^{+}(\theta)]$, the first term on the right hand side of \cref{energyest} over $K_{2\rho}\times\{t_0\}$ vanishes.
Moreover, taking a test function independent of time also kills the integral on the right involving the time derivative of the test function. Therefore we may repeat the same arguments as in \cref{lem:deG} for $(u-\xi_n M)_{-}$ over the cylinders $Q_n^+$ where \[ \xi_n = a\xi + \frac{1-a}{2^n}\xi \quad \text{and} \quad Q_n^+ = K_n \times (0,\theta(2\rho)^{sp}], \] to get \[ Y_{n+1}' \leq \frac{Cb^n}{(1-a)^{(N+2s)\frac{p}{N}}}\left(\frac{\theta}{(\xi M)^{2-p}}\right)^{\frac{sp}{N}}Y_n'^{1+\frac{sp}{N}}, \] provided \[ \text{Tail}_{\infty}(u_{-};x_0,2\rho,(t_0,t_0+\theta (2\rho)^{sp}]) \leq \xi M, \] where we have set \begin{equation}\label{Yprime}
Y_n' = \frac{|[u<\xi_nM]\cap Q_n^+|}{|Q_n^+|}. \end{equation}
Thus, by \cref{geo_con}, we have $Y_{\infty}' = 0$ if \begin{equation}\label{eq:initdatadeG}
Y_0' \leq \nu_0\left(\frac{(\xi M)^{2-p}}{\theta}\right)^s =: \tilde{\nu}. \end{equation} for a constant $\nu_0 \in (0,1)$ depending only on $a$ and data. The foregoing discussion leads to the following variant of the DeGiorgi lemma.
\begin{lemma}\label{lem:deGdata} Let $u$ denote a nonnegative, local, weak supersolution to \cref{maineq} in $E_T$. Let $M$ and $\xi$ be positive numbers and suppose \cref{eq:deGleminitdata} holds at time $t=t_0$, then there exists a constant $\tilde{\nu}$ (see \cref{eq:initdatadeG}) depending on data, $a$ and $\zeta, M, \theta$ such that if $Y_0' \leq \tilde{\nu}$ (see \cref{Yprime}), then either of the two alternatives hold: \[ \text{Tail}_{\infty}(u_{-};x_0,2\rho,(t_0,t_0+\theta (2\rho)^{sp}]) > \xi M, \] or \[ u \geq a\xi M \qquad \text{ a.e. in } K_{2\rho}(x_0)\times (t_0,t_0+\theta(2\rho)^{sp}]. \] \end{lemma}
\section{Qualitative expansion of positivity in time}\label{exptime} Let $u$ denote a nonnegative, local, weak supersolution to \cref{maineq} in $E_T$. In this section, we shall prove a general expansion of positivity estimate that will be used in \cref{sec6} and \cref{sec7}. \begin{lemma}\label{expandintime} Assume that for some $(x_0,t_0) \in E_T$ and some $\rho>0$, $M > 0$ and $\alpha \in (0,1)$, the following hypothesis is satisfied: \[
|[u(\cdot,t_0) \geq M]\cap K_{\rho}(x_0)| \geq \alpha|K_{\rho}(x_0)|, \] then there exist $\delta$ and $\epsilon$ in $(0,1)$ depending only on the data and $\alpha$ such that either \[ \text{Tail}_{\infty}(u_{-};x_0,\rho,(t_0,t_0+\delta\rho^{sp}M^{2-p})) > M, \] or \[
|[u(\cdot,t) \geq \epsilon M]\cap K_{\rho}(x_0)| \geq \frac{1}{2}\alpha|K_{\rho}(x_0)|, \] holds for all $t \in (t_0,t_0+\delta\rho^{sp}M^{2-p}]$. \end{lemma}
\begin{proof} Without loss of generality, we will assume that $(x_0,t_0)=(0,0)$. For $k>0$ and $t>0$, we set \[ A_{k,\rho}(t) = [u(\cdot,t) < k] \cap K_{\rho}, \] then the hypothesis of the lemma can be restated as: \[
|A_{M,\rho}(0)|\leq (1-\alpha)|K_{\rho}|. \]
We consider the energy estimate from \cref{energyest} for $(u-M)_{-}$ over the cylinder $K_{\rho}\times(0,\theta\rho^{sp}]$ where $\theta>0$ will be chosen later. Note that $(u-M)_{-} \leq M$ in $K_{\rho}$ because $u$ is nonnegative in $E_T$. For $\sigma \in (0,1/8]$ to be chosen later, we take a cutoff function $\zeta = \zeta(x)$, nonnegative such that it is supported in $K_{(1-\sigma/2)\rho}$, $\zeta = 1$ on $K_{(1-\sigma)\rho}$ and $|\de \zeta| \leq 2(\sigma\rho)^{-1}$ to get \begin{equation*}
\begin{array}{rcl}
\oldint\limits_{K_{(1-\sigma)\rho}}(u-M)_{-}^2(x,t) \,dx
&\leq & M^2|A_{M,\rho}(0)|+\frac{2\gamma M^p}{(\sigma \rho)^{sp}}\theta\rho^{sp}|K_{\rho}|
\\
&&+ C\underset{\stackrel{t \in (0,\theta\rho^{sp}];}{ x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c}\frac{(u-M)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\oldint\limits_{0}^{\theta\rho^{sp}}\oldint\limits_{K_{\rho}} (u-M)_{-}(x,t)\zeta(x,t)\,dx\,dt.
\end{array} \end{equation*} To estimate the tail term we first note that when $x \in K_{(1-\sigma/2)\rho}(0)$ and $y \in K_{\rho}^c(0)$ then \[
|y|\leq |x-y|\left(1+\frac{|x|}{|x-y|}\right) \leq \frac{2}{\sigma}|x-y|. \] In particular, this implies \[
\frac{1}{|x-y|^{N+sp}} \leq c\sigma^{-(N+sp)}\frac{1}{|y|^{N+sp}}. \] Next, we note that the following holds gobally: \[ (u-M)_{-} \leq M-u \leq M + u_{-}. \] Therefore, we have \begin{align*}
\underset{\stackrel{0<t<\theta\rho^{sp}; }{x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c}\frac{(u-M)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy &\leq c{\sigma}^{-(N+sp)}\underset{0<t<\theta\rho^{sp}}\oldint\limits_{K_{\rho}^c}\frac{(u )_{-}^{p-1}(y,t)+(M)^{p-1}}{|y|^{N+sp}}\,dy
\\
& \leq c{\sigma}^{-(N+sp)}\frac{M^{p-1}}{\rho^{sp}} +
c{\sigma}^{-(N+sp)}\underset{0<t<\theta\rho^{sp} }{\esssup}\oldint\limits_{K_{\rho}^c}\frac{(u )_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy
\\
& \leq c{\sigma}^{-(N+sp)}\left(\frac{M^{p-1}}{\rho^{sp}} + \frac{1}{\rho^{sp}}\text{Tail}_{\infty}^{p-1}(u_{-};0,\rho,(0,\theta\rho^{sp}))\right). \end{align*} Thus, the tail term can be estimated to get \begin{align*}
\underset{\stackrel{t \in (0,\theta\rho^{sp}];}{ x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{\rho}^c}&\frac{(u-M)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\oldint\limits_{0}^{\theta\rho^{sp}}\oldint\limits_{K_{\rho}} (u-M)_{-}(x,t)\zeta(x,t)\,dx\,dt
\\
&\leq c{\sigma}^{-(N+sp)}\left(\frac{M^{p-1}}{\rho^{sp}} + \frac{1}{\rho^{sp}}\text{Tail}_{\infty}^{p-1}(u_{-};0,\rho,(0,\theta\rho^{sp}))\right)(M)|K_{\rho}|\theta\rho^{sp}
\\
&\leq c{\sigma}^{-(N+sp)}M^{p}\theta|K_{\rho}|, \end{align*} provided \[ \text{Tail}_{\infty}(u_{-};0,\rho,(0,\theta\rho^{sp})) \leq M. \] Putting the above estimates together we get \begin{align*}
\oldint\limits_{K_{(1-\sigma)\rho}}(u-M)_{-}^2(x,t) \,dx
&\leq M^2|A_{M,\rho}(0)|+\frac{2\gamma M^p}{(\sigma \rho)^{sp}}\theta\rho^{sp}|K_{\rho}|
+ c{\sigma}^{-(N+sp)}M^{p}\theta|K_{\rho}|
\\
&\leq M^2\left((1-\alpha)+C\frac{\theta M^{p-2}}{\sigma^{N+sp}}\right)|K_{\rho}|, \end{align*} holds for all $t \in (0,\theta\rho^{sp}]$ and a universal constant $C>0$.
On the other hand, we have \begin{align*}
\oldint\limits_{K_{(1-\sigma)\rho}}(u-M)_{-}^2(x,t) \geq \oldint\limits_{K_{(1-\sigma)\rho}\cap [u<\epsilon M]}(u-M)_{-}^2(x,t)
\geq M^2(1-\epsilon)^2|A_{\epsilon M,(1-\sigma)\rho}(t)|, \end{align*} where $\epsilon \in (0,1)$ will be chosen below depending only on $\alpha$. Next, we note that \[
|A_{\epsilon M,\rho}(t)| \leq |A_{\epsilon M,(1-\sigma)\rho}(t)| + |K_{\rho}-K_{(1-\sigma)\rho}| \leq |A_{\epsilon M,(1-\sigma)\rho}(t)| + N\sigma|K_{\rho}|. \]
Thus, combining everything, we have \[
|A_{\epsilon M,\rho}(t)| \leq \frac{1}{(1-\epsilon)^2}\left((1-\alpha)+C\frac{\theta M^{p-2}}{\sigma^{N+sp}} + N\sigma \right)|K_{\rho}|. \] First, we set $\theta = \delta M^{2-p}$ and then choose \begin{equation*}
\sigma = \frac{\alpha}{8N}, \qquad \epsilon \leq 1 - \frac{\sqrt{1-\frac{3}{4}\alpha}}{\sqrt{1-\frac{1}{2}\alpha}}, \quad \txt{and} \quad \delta = \frac{\alpha^{N+sp+1}}{C8^{N+sp+1}N^{sp+N}}, \end{equation*} to get the desired conclusion. \end{proof}
\section{Expansion of positivity for Nonlocal Degenerate Equations}\label{sec6}
We will assume that $u$ is a nonnegative, local, weak supersolution to \cref{maineq} in $E_T$ and $p>2$. For $(x_0,t_0) \in E_T$ and some given positive number $M$ we consider the cylinder \[ K_{8\rho}(x_0) \times \left(t_0,t_0+\frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}\right] \subset K_{R}(x_0)\times [t_0-R^{sp},t_0+R^{sp}] \subset E_T, \] where $K_{R}(x_0)\times [t_0-R^{sp},t_0+R^{sp}] = K_R(x_0) \times I$ is our reference cylinder. The constants $b,\eta,\delta$ are constants given by \cref{prop:degexp} and $\rho > 0$ is chosen small enough.
\begin{proposition}\label{prop:degexp} Assume that for some $(x_0,t_0) \in E_T$, $\rho > 0$, $M>0$ and some $\alpha \in (0,1)$ the following assumption is satisfied: \begin{equation*}
|[u(\cdot,t_0) \geq M] \cap K_{\rho}(x_0)| \geq \alpha |K_{\rho}(x_0)|. \end{equation*}
Then there exist constants $\eta, \delta, \sigma \in (0,1)$ and $b>0$ depending only on the data and $\alpha$ such that either \[ \text{Tail}_{\infty}\left(u_{-};x_0,\rho,\left(t_0,t_0+\frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}\right)\right) > \eta M, \] or \begin{equation*}
u(\cdot,t) \geq \eta M \qquad \text{a.e. in } K_{2\rho}(x_0), \end{equation*} holds for all times \begin{equation*}
t_0 + \frac{b^{p-2}}{(\eta M)^{p-2}}\sigma\delta\rho^{sp} \leq t \leq t_0 + \frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}. \end{equation*} \end{proposition} \begin{proof} Without loss of generality, we will assume that $(x_0,t_0) = (0,0)$. The proof of the proposition is split into the following steps:
\begin{description}[leftmargin=*] \descitem{Step 1}{Step 1}\textit{Changing the time variable.} For $\tau \geq 0$, let us set \begin{equation*}
\sigma_{\tau} = \exp\left(-\frac{\tau}{p-2}\right) \leq 1, \end{equation*} then the hypothesis of the proposition implies the following holds \begin{equation*}
|[u(\cdot,0) \geq \sigma M] \cap K_{\rho}| \geq \alpha |K_{\rho}| \qquad \forall \ \sigma \leq 1. \end{equation*} Assuming \begin{equation}\label{tail_est_1} \text{Tail}_{\infty}(u_{-};0,\rho,J_1) \leq \sigma_{\tau} M \qquad \text{where} \quad J_1 := (0,\delta\rho^{sp}(\sigma_{\tau}M)^{2-p}), \end{equation} and making use of the calculations from \cref{exptime}, we get \begin{equation}\label{eq5.4}
\left|\left[u\lbr\cdot,\frac{\delta\rho^{sp}}{(\sigma_{\tau} M)^{p-2}}\rbr \geq \epsilon\sigma_{\tau} M\right] \cap K_{\rho}\right| \geq \frac{1}{2}\alpha |K_{\rho}|, \end{equation} for universal constants $\epsilon,\delta$ in $(0,1)$. Let us perform the change of variable \begin{equation}
w(x,\tau) := \frac{1}{\sigma_{\tau}M}(\delta\rho^{sp})^{\frac{1}{p-2}}u\left(x,\frac{e^{\tau}}{M^{p-2}}\delta\rho^{sp}\right), \end{equation} then for all $\tau \geq 0$, \cref{eq5.4} can be written as \[
\left|\left[u\left(\cdot,\frac{e^{\tau}}{ M^{p-2}}\delta\rho^{sp}\right) \geq \epsilon M\sigma_{\tau}\right] \cap K_{\rho}\right| \geq \frac12\alpha |K_{\rho}|, \] which in the new variable translates to \begin{equation}\label{eq5.5}
|[w(\cdot,\tau) \geq k_0] \cap K_{\rho}| \geq \frac12 \alpha |K_{\rho}| \quad \text{for all} \ \tau >0, \end{equation} where \begin{equation}\label{eq:k0}
k_0 := \epsilon(\delta\rho^{sp})^{\frac{1}{p-2}}. \end{equation} We can rewrite \cref{eq5.5} as \begin{equation}\label{timeexpdeg}
|K_{4\rho}- [w(\cdot,\tau) < k_0]| \geq \frac{1}{2}\alpha 4^{-N} |K_{4\rho}| \qquad \text{for all} \ \tau > 0. \end{equation}
\descitem{Step 2}{Step 2}{\it Relating $w$ to the evolution equation.} Since $u \geq 0$ in $E_T$, by formal calculations, we have \begin{align*}
w_{\tau} &= \left(\frac{1}{\sigma_{\tau}M}(\delta\rho^{sp})^{\frac{1}{p-2}}\right)^{p-1}u_t + \frac{1}{p-2}\frac{1}{M\sigma_{\tau}}(\delta\rho^{sp})^{\frac{1}{p-2}}u
\\
&\geq -\left(\frac{1}{\sigma_{\tau}M}(\delta\rho^{sp})^{\frac{1}{p-2}}\right)^{p-1}Lu
\\
&\geq - L_1w, \end{align*} in $E \times \bb{R}_{+}$ where \[ \left\{ \begin{array}{l} L_1\varphi(x,t) = P.V. \oldint\limits_{\bb{R}^N}K_1(x,y,t)J_p(\varphi(x,t)-\varphi(y,t)) dy, \\ K_1(x,y,t) = K\left(x,y,\frac{e^{\tau}}{M^{p-2}}\delta\rho^{sp}\right), \\
\frac{\Lambda^{-1}}{|x-y|^{N+sp}} \leq K_1(x,y,t) \leq
\frac{\Lambda}{|x-y|^{N+sp}}. \end{array} \right. \]
The formal calculation can be made rigorous by appealing to the weak formulation and the energy estimates for $u$ can be transferred to energy estimates for $w$ by change of variable.
For any level $k \in \bb{R}$, the energy estimate for $(w-k)_{-}$ in $Q_{8\rho}^{+}(\theta)$ yield \begin{align}\label{eq:wcaccio}
\oldint\limits_0^{\theta(8\rho)^{sp}}&\oldint\limits_{K_{8\rho}}(w-k)_{-}(x,t)\oldint\limits_{K_{8\rho}}\frac{|(w-k)_{+}(y,t)|^{p-1}}{|x-y|^{N+sp}}\,dx\,dy\,dt\ \nonumber \\ &\leq
\gamma \oldint\limits_0^{\theta(8\rho)^{sp}}\oldint\limits_{K_{8\rho}}\oldint\limits_{K_{8\rho}} \max\{(w-k)_{-}(x,t),(w-k)_{-}(y,t)\}^{p}|\xi(x,t)-\xi(y,t)|^p\,d\mu\,dt \nonumber
\\
& \quad+ \gamma \oldint\limits_0^{\theta(8\rho)^{sp}}\oldint\limits_{K_{8\rho}} (w-k)_{-}^2(x,t)|\de_t \xi(x,t)| \,dx\,dt \nonumber
\\
&\quad\quad +C\underset{\stackrel{t \in (0,\theta(8\rho)^{sp});}{x\in \spt\xi}}{\esssup}\oldint\limits_{K_{8\rho}^c}\frac{(w-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\lbr\oldint\limits_0^{\theta(8\rho)^{sp}}\oldint\limits_{K_{8\rho}} (w-k)_{-}(x,t)\xi(x,t)\,dx\,dt\rbr, \end{align} for any non-negative piecewise smooth cutoff vanishing on the parabolic boundary of $Q_{8\rho}^{+}(\theta)$. In particular, we choose $\zeta =1$ $ \mathcal{Q}_{4\rho}(\theta) = K_{4\rho} \times ((4\rho)^{sp}\theta,(8\rho)^{sp}\theta] $ with support in $K_{6\rho}\times(0,(8\rho)^{sp}\theta]$ and satisfying \[
|\nabla \zeta| \leq \frac{1}{2\rho} \qquad \text{and} \qquad |\zeta_{\tau}| \leq \frac{1}{\theta(4\rho)^{sp}}. \]
\descitem{Step 3}{Step 3}\textit{Shrinking lemma for $w$.} We claim that for every $\nu>0$ there exist $\epsilon_{\nu} \in (0,1)$ depending only on the data and $\alpha$, and $\theta = \theta(k_0,\epsilon_{\nu}) > 0$ depending only on $k_0$ and $\epsilon_{\nu}$ such that if \begin{equation}\label{tail_est_2}
\begin{array}{l} \text{Tail}_{\infty}(u_{-};0,8\rho,J_2) \leq 2\epsilon_{\nu}\epsilon M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),\\ J_2 = \left(M^{2-p}\delta\rho^{sp},\exp\left(\frac{8^{sp}}{(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)M^{2-p}\delta \rho^{sp}\right], \end{array} \end{equation}
then the following conclusion holds: \[
|[w < \epsilon_{\nu}k_0]\cap\mathcal{Q}_{4\rho}(\theta)| \leq \nu|\mathcal{Q}_{4\rho}(\theta)|. \]
\begin{proof}[Proof of Claim:] In \cref{eq:wcaccio} we work with the levels $k_j$ and the parameter $\theta$ as follows \[ k_j = \frac{1}{2^j}k_0 \quad \text{ for } \quad 1 \leq j \leq j_{\ast} \qquad \text{ and } \qquad \theta = k_{j_{\ast}}^{2-p}, \] where $k_0$ is given by \cref{eq:k0} and ${j_{\ast}}$ is to be chosen in \cref{eq:j*deg}. We first note that since $u \geq 0$ in $Q_{8\rho}^{+}(\theta)$, all the local integrals on the right in \cref{eq:wcaccio} can be estimated using \[ (w-k_j)_{-} \leq k_j. \] Thus, by our choice of the test function and \cref{eq:wcaccio}, we get \begin{align*}
\oldint\limits_0^{\theta(8\rho)^{sp}}\oldint\limits_{K_{8\rho}}(w-k_j)_{-}(x,t)&\oldint\limits_{K_{8\rho}}\frac{|(w-k_j)_{+}(y,t)|^{p-1}}{|x-y|^{N+sp}}\,dx\,dy\,dt\ \\
&\leq C\frac{k_{j}^p}{\rho^{sp}}|\mathcal{Q}_{4\rho}(\theta)|
+Ck_{j}|\mathcal{Q}_{4\rho}(\theta)|\underset{\stackrel{t \in (0,\theta(8\rho)^{sp});}{ x\in \spt\zeta}}{\esssup}\oldint\limits_{K_{8\rho}^c}\frac{(w-k_j)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy. \end{align*}
We recall from \cref{sec:tail} on Tail estimates to conclude that \[
\oldint\limits_0^{\theta(8\rho)^{sp}}\oldint\limits_{K_{8\rho}}(w-k_j)_{-}(x,t)\oldint\limits_{K_{8\rho}}\frac{|(w-k_j)_{+}(y,t)|^{p-1}}{|x-y|^{N+sp}}\,dx\,dy\,dt\ \leq C\frac{k_{j}^p}{\rho^{sp}}|\mathcal{Q}_{4\rho}(\theta)|, \] holds provided \[ \text{Tail}_{\infty}^{p-1}(w_{-};0,8\rho,(0,\theta(8\rho)^{sp})) \leq k_{j_{*}}^{p-1}. \]
We now invoke \cref{lem:shrinking} and \cref{timeexpdeg} to conclude that \[
|[w < k_{j+1}]\cap\mathcal{Q}_{4\rho}(\theta)| \leq \left(\frac{C}{2^{j}-1}\right)^{p-1}|\mathcal{Q}_{4\rho}(\theta)|, \] for a constant $C>0$ depending only on $\alpha$ and data. This proves the claim once we choose $j_*$ such that \begin{equation}\label{eq:j*deg}
\epsilon_{\nu} = \frac{1}{2^{j_*+1}} \qquad \text{and} \qquad \left(\frac{C}{2^{j_*}-1}\right)^{p-1}\leq \nu. \end{equation}
To conclude, we rewrite the Tail alternative in terms of $u_-$ as follows. \begin{equation*}
\begin{array}{rcl}
\underset{\tau \in (0,\theta(8\rho)^{sp})}{\esssup}\oldint\limits_{K_{8\rho}^c}\frac{w_{-}^{p-1}(y,\tau)}{|y|^{N+sp}}\,dy &=& (\delta\rho^{sp})^{\frac{p-1}{p-2}}\frac{1}{M^{p-1}}\underset{\tau \in (0,\theta(8\rho)^{sp})}{\esssup}\oldint\limits_{K_{8\rho}^c}\frac{u_{-}^{p-1}(y,e^{\tau}M^{2-p}\delta\rho^{sp})}{|y|^{N+sp}}e^{\tau\frac{p-1}{p-2}}\,dy
\\
&\leq & (\delta\rho^{sp})^{\frac{p-1}{p-2}}\frac{\exp\left(\theta(8\rho)^{sp}\frac{p-1}{p-2}\right)}{M^{p-1}}\underset{t \in J_2}{\esssup}\oldint\limits_{K_{8\rho}^c}\frac{u_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy,
\end{array} \end{equation*} where \[ J_2 = (M^{2-p}\delta\rho^{sp},\exp((8\rho)^{sp}\theta)M^{2-p}\delta \rho^{sp}]. \]
Thus, the Tail alternative is verified provided \[ \text{Tail}_{\infty}(u_{-};0,8\rho,J_2) \leq \frac{k_{j_{*}}}{(\delta\rho^{sp})^{\frac{1}{p-2}}}\frac{M}{\exp\left(\theta(8\rho)^{sp}\frac{1}{p-2}\right)}. \] Recalling that \[ k_{j_{*}} = \frac{k_0}{2^{j_{*}}} = \frac{\epsilon(\delta\rho^{sp})^{\frac{1}{p-2}}}{2^{j_{*}}} = 2\epsilon_{\nu}\epsilon(\delta\rho^{sp})^{\frac{1}{p-2}} = \theta^{\frac{1}{2-p}}, \] we get \[ \exp\left(\theta(8\rho)^{sp}\frac{1}{p-2}\right) = \exp\left(\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right), \] and \[ \frac{k_{j_{*}}}{(\delta\rho^{sp})^{\frac{1}{p-2}}}\frac{M}{\exp\left(\theta(8\rho)^{sp}\frac{1}{p-2}\right)} = 2\epsilon_{\nu}\epsilon M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right), \] which completes the proof of the claim. \end{proof}
\descitem{Step 4}{Step 4}\textit{Expansion of positivity for $w$.} We claim that there exists a $\nu \in (0,1)$ which depends only on the data and $\alpha$ such that either \[ \text{Tail}_{\infty}(u_{-};0,8\rho,J_2) > 2\epsilon_{\nu}\epsilon M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right) \] or \[ w(\cdot,\tau) \geq \frac{1}{2}\epsilon_{\nu}k_0 \qquad \text{ a.e. in } K_{2\rho}\times \left(\frac{(8\rho)^{sp}-(2\rho)^{sp}}{(2\epsilon_{\nu}k_0)^{p-2}},\frac{(8\rho)^{sp}}{(2\epsilon_{\nu}k_0)^{p-2}}\right] \] holds, where $\epsilon_{\nu}$ is the number corresponding to $\nu$ in \descref{Step 3}{Step 3}.
\begin{proof}[Proof of the claim] We apply \cref{lem:deG} to $w$ over the cylinder \[ \mathcal{Q}_{4\rho}(\theta) = (0,\tau^*)+Q_{4\rho}^{-}(\theta) \qquad \text{ for } \tau_{*} = \theta(8\rho)^{sp}. \] We work with $\epsilon_{\nu}k_0$ instead of $\xi\omega$, $a = 1/2$ and $\mu_{-} \geq 0$ is dropped to get that either \begin{equation}\label{eq:tail3deg}
\text{Tail}_{\infty}((w)_{-};0,4\rho,(\tau_{*}-\theta(4\rho)^{sp},\tau_{*}]) > \epsilon_{\nu}k_0 \end{equation} holds or \[ w(x,\tau) \geq \frac{1}{2}\epsilon_{\nu}k_0 \qquad \text{ for a.e. } \qquad (x,\tau) \in [(0,\tau^*)+Q_{2\rho}^{-}(\theta)], \] holds provided \[
\frac{\left|[w<\epsilon_{\nu}k_0]\cap \mathcal{Q}_{4\rho}(\theta)\right|}{|\mathcal{Q}_{4\rho}(\theta)|} \leq \frac{1}{C}\left(\frac{1}{2}\right)^{N+2s}\frac{[\theta(\epsilon_{\nu}k_0)^{p-2}]^{\frac{N}{p}}}{[1+2\theta(\epsilon_{\nu}k_0)^{p-2}]^{\frac{N+sp}{p}}} = \nu, \] for a universal constant $C>1$ and the Tail alternative as in \descref{Step 3}{Step 3} holds. Choosing $\nu$ from \descref{Step 3}{Step 3} (see \cref{eq:j*deg}) determines $\epsilon_{\nu}$ and therefore $\theta = (2\epsilon_\nu\epsilon)^{2-p}$ quantitatively. The tail alternative \cref{eq:tail3deg} can be reformulated in terms of $u$ as in \descref{Step 3}{Step 3} to get \begin{equation}\label{tail_est_3} \begin{array}{l} \text{Tail}_{\infty}(u_{-};0,8\rho,J_3) \leq 2\epsilon_{\nu}\epsilon M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),\\ J_3 = (\exp((8\rho)^{sp}\theta-(4\rho)^{sp}\theta)M^{2-p}\delta\rho^{sp},\exp((8\rho)^{sp}\theta)M^{2-p}\delta \rho^{sp}]. \end{array} \end{equation}
Since $J_2 \supset J_3$, the Tail alternative in \descref{Step 3}{Step 3} subsumes the Tail alternative from \cref{eq:tail3deg} as reflected in the claim. \end{proof}
\descitem{Step 5}{Step 5} {\it Expanding the positivity for $u$.} Assume that the Tail alternatives from \cref{tail_est_1},\cref{tail_est_2} and \cref{tail_est_3} hold . As $\tau$ ranges over the interval in \descref{Step 4}{Step 4}, $\exp(\tau/(p-2))$ ranges over the interval \[ b_1 := \exp\left(\frac{8^{sp}-2^{sp}}{(p-2)[2\epsilon_{\nu}\epsilon \delta^{\frac{1}{p-2}}]^{p-2}}\right) \leq f(\tau) \leq \exp\left(\frac{8^{sp}}{(p-2)[2\epsilon_{\nu}\epsilon \delta^{\frac{1}{p-2}}]^{p-2}}\right) =: b_2. \] Rewriting the conclusion of \descref{Step 4}{Step 4} in terms of $u$, we get that \begin{equation}\label{def_eta} u(x,t) \geq \frac{\epsilon_{\nu}\epsilon M}{2b_2} =: \eta M \qquad \text{ for a.e. } \quad x \in K_{2\rho}, \end{equation} holds for all times \[
\frac{b^{p-2}}{(\eta M)^{p-2}}\sigma\delta\rho^{sp} \leq t \leq \frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}. \] Here $b > 0$ is a constant depending only on $\alpha$ and data and $\sigma \in (0,1)$ is a fraction depending only on $\alpha$ and data. In fact, we have \[ b = \frac{\epsilon_{\nu}\epsilon}{2} \quad \text{and} \qquad \text{and} \quad \sigma = \left(\frac{b_1}{b_2}\right)^{p-2}. \]
\descitem{Step 6}{Step 6} {\it Pulling the Tail alternatives together.} We need the following Tail alternatives going from \descref{Step 1}{Step 1} and \descref{Step 3}{Step 3} noting that \cref{tail_est_2} implies \cref{tail_est_3}. In particular, we recall \[ \begin{array}{rcl} \text{Tail}_{\infty}(u_{-};0,\rho,J_1) &\leq&\sigma_{\tau} M,\\ \text{Tail}_{\infty}(u_{-};0,8\rho,J_2) &\leq & 2\epsilon_{\nu}\epsilon M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right), \end{array} \] where we have taken \[ \begin{array}{rcl} J_1 &=& (0,\delta\rho^{sp}(\sigma_{\tau}M)^{2-p}),\\ J_2 &=& \left(M^{2-p}\delta\rho^{sp},\exp\left(\frac{8^{sp}}{(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)M^{2-p}\delta \rho^{sp}\right]. \end{array} \]
Since we have $\tau\leq\theta(8\rho)^{sp}$ which implies the following inclusion holds: \[ J_1 \subset (0,\delta\rho^{sp}M^{2-p}\exp(\theta(8\rho)^{sp})) = (0,\delta\rho^{sp}M^{2-p}\exp(8^{sp}\delta^{-1}(2\epsilon_{\nu}\epsilon)^{2-p})). \] Furthermore, we have \[ 2\epsilon_{\nu}\epsilon M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right) = \frac{2\epsilon_{\nu}\epsilon M}{b_2} \overset{\cref{def_eta}}{=} 4\eta M. \] Hence, from the fact that $\eta \leq \sigma_{\tau}$, the final Tail alternative which subsumes all others is \[ \text{Tail}_{\infty}(u_{-};0,\rho,J) \leq \eta M \quad \text{where} \quad J := \left(0,\frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}\right]. \] \end{description} \end{proof} \begin{remark} The conclusion of \cref{prop:degexp} can also be written without Tail alternatives as \[ u(\cdot,t) \geq \eta M - \text{Tail}_{\infty}\left(u_{-};x_0,\rho,\left(t_0,t_0+\frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}\right)\right) \qquad \text{a.e. in } K_{2\rho}(x_0), \] holds for all times \[
t_0 + \frac{b^{p-2}}{(\eta M)^{p-2}}\sigma\delta\rho^{sp} \leq t \leq t_0 + \frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}, \] because we are working with nonnegative solutions. \end{remark}
\section{Expansion of positivity for Nonlocal Singular Equations}\label{sec7}
We will assume that $u$ is a nonnegative, local, weak supersolution to \cref{maineq} in $E_T$ for $1<p<2$. For $(x_0,t_0) \in E_T$ and some given positive numbers $M >0$ and $\delta \in (0,1)$, we consider the cylinder \[ (x_0,t_0) + Q_{16\rho}(\delta M^{2-p}) = K_{16\rho}(x_0) \times (t_0,t_0+\delta M^{2-p}\rho^{sp}] \subset K_{R}(x_0)\times [t_0-R^{sp},t_0+R^{sp}] \subset E_T, \] where $K_{R}(x_0)\times [t_0-R^{sp},t_0+R^{sp}] = K_R(x_0) \times I$ is our reference cylinder and $\rho > 0$ is chosen small enough.
\begin{proposition}\label{prop:sinexp} Assume that for some $(x_0,t_0) \in E_T$ and some $\rho > 0$, $M>0$ and $\alpha \in (0,1)$, the following hypothesis holds \begin{equation*}
|[u(\cdot,t_0) \geq M] \cap K_{\rho}(x_0)| \geq \alpha |K_{\rho}(x_0)|. \end{equation*}
Then there exist constants $\eta,\delta$ and $\varepsilon$ in $(0,1)$ depending only on the data and $\alpha$ such that either \[ \text{Tail}_{\infty}(u_{-};x_0,\rho,(t_0,t_0+\delta\rho^{sp}M^{2-p})) > \eta M, \] or \begin{equation*}
u(\cdot,t) \geq \eta M \qquad \text{a.e. in } K_{2\rho}(x_0), \end{equation*} holds for all times \begin{equation*}
t_0 + (1-\varepsilon)\delta M^{2-p}\rho^{sp} \leq t \leq t_0 + \delta M^{2-p}\rho^{sp}. \end{equation*} \end{proposition}
\begin{proof}Without loss of generality, we will assume that $(x_0,t_0) = (0,0)$.
\begin{description}[leftmargin=*] \descitem{Step 1}{step1} {\it Changing variables.} From \cref{exptime} we get that there exist $\delta$ and $\epsilon$ in $(0,1)$ depending only on the data and $\alpha$ such that one of the following two possibilities must hold: \[ \text{Tail}_{\infty}(u_{-};0,\rho,(0,\delta\rho^{sp}M^{2-p})) > M, \] or \begin{equation}\label{eq:timeexpsing}
|[u(\cdot,t) > \epsilon M] \cap K_{\rho}| \geq \frac{1}{2}\alpha|K_{\rho}| \qquad \text{ for all } \qquad t \in (0,\delta M^{2-p}\rho^{sp}]. \end{equation} Let us now define \begin{equation}\label{eq:covsing} z = \frac{x}{\rho}, \qquad -e^{-\tau} = \frac{t-\delta M^{2-p}\rho^{sp}}{\delta M^{2-p}\rho^{sp}}, \qquad \text{and} \quad v(z,\tau) = \frac{1}{M}u(x,t)e^{\frac{\tau}{2-p}}. \end{equation} Then the cylinder $Q_{16\rho}(\delta M^{2-p})$ in $(x,t)$ coordinates is mapped into $K_{16}\times (0,\infty)$ in the $(z,\tau)$ coordinates. By formal calculations (which can be made rigorous using the weak formulation), we have \begin{align}\label{eq:expeqsing}
v_{\tau} = \frac{1}{2-p}v + \frac{e^{\frac{p-1}{2-p}\tau}\delta\rho^{sp}}{M^{p-1}}u_t(z\rho,\delta M^{2-p}\rho^{sp}(1-e^{-\tau}))
= \frac{1}{2-p}v + \delta L_1v, \end{align} in $K_{16} \times (0,\infty)$, where \[\begin{array}{rcl} L_1\varphi(z,\tau) & = & P.V. \oldint\limits_{\bb{R}^N}K_1(z,z',\tau)J_p(\varphi(z,\tau)-\varphi(z',\tau)) \,dz', \\ K_1(z,z',\tau) &=& \rho^{N+sp}K\left(z\rho,z'\rho,\delta M^{2-p}\rho^{sp}(1-e^{-\tau})\right), \end{array} \] satisfying \[
\frac{\Lambda^{-1}}{|x-y|^{N+sp}} \leq K_1(x,y,t) \leq
\frac{\Lambda}{|x-y|^{N+sp}}. \]
We rewrite \cref{eq:timeexpsing} as \begin{equation}\label{eq:sing2}
|[v(\cdot,\tau) \geq \epsilon e^{\frac{\tau}{2-p}}]\cap K_1| \geq \frac{1}{2}\alpha |K_1| \qquad \text{ for all } \tau \in (0,\infty). \end{equation} For constants $\tau_0 > 0$ and $j_{*} \in \NN$ to be chosen, we take \begin{equation}\label{eq:k0sing} k_0 = \epsilon e^{\frac{\tau_0}{2-p}} \qquad \text{and} \quad k_j = k_02^{-j}\ \ \text{ for } j \geq 1. \end{equation} Then from \cref{eq:sing2}, we get \begin{equation*}
|[v(\cdot,\tau) \geq k_j]\cap K_{8}| \geq \frac{1}{2}\alpha 8^{-N}|K_8| \qquad \text{ for all } \tau \in (\tau_0,\infty) \ \text{and} \ \forall j \in \NN. \end{equation*}
Consider the cylinders \[ Q_{\tau_0} = K_8 \times (\tau_0+k_0^{2-p},\tau_0+2k_0^{2-p}) \quad \text{and} \quad Q_{\tau_0'} = K_{16} \times (\tau_0, \tau_0+2k_0^{2-p}), \] and a nonnegative, piecewise smooth cutoff in $Q_{\tau_0'}$ of the form $\zeta(z,\tau) = \zeta_1(z)\zeta_2(\tau)$ such that \[ \begin{array}{ll}
\zeta_1 = \left\{ \begin{array}{lcl} 1 & \text{in} & K_8 \\ 0 & \text{on} & \RR^N\setminus K_{12}\end{array}\right. & |\de \zeta_1| \leq \frac14,\\ \zeta_2 = \left\{ \begin{array}{lcl} 0 & \text{for} & \tau < \tau_0 \\ 1 & \text{for} & \tau \geq \tau_0 + k_0^{2-p} \end{array}\right. & 0 \leq \de_t \zeta_2 \leq \frac1{k_0^{2-p}}. \end{array} \]
From the energy estimates for $(v-k_j)_{-}$ over $Q_{\tau_0'}$ with the cutoff $\zeta$ we get \begin{align}\label{eq:vcaccio}
\oldint\limits_{\tau_0}^{\tau_0 +2k_0^{2-p}}&\oldint\limits_{K_{16}}(v-k_j)_{-}(x,t)\oldint\limits_{K_{16}}\frac{|(v-k_j)_{+}(y,t)|^{p-1}}{|x-y|^{N+sp}}\,dx\,dy\,dt\ \nonumber \\ &\leq
\gamma \oldint\limits_{\tau_0}^{\tau_0 +2k_0^{2-p}}\oldint\limits_{K_{16}}\oldint\limits_{K_{16}} \max\{(v-k_j)_{-}(x,t),(v-k_j)_{-}(y,t)\}^{p}|\xi(x,t)-\xi(y,t)|^p\,d\mu\,dt \nonumber
\\
&+ \gamma \oldint\limits_{\tau_0}^{\tau_0 +2k_0^{2-p}}\oldint\limits_{K_{16}} (v-k_j)_{-}^2(x,t)|\de_t \xi(x,t)| \,dx\,dt \nonumber
\\
&+C\gamma \underset{\stackrel{t \in (\tau_0, \tau_0+2k_0^{2-p});}{ x\in \spt\zeta_1}}{\esssup}\oldint\limits_{K_{16}^c}\frac{(v-k_j)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\oldint\limits_{\tau_0}^{\tau_0 +2k_0^{2-p}}\oldint\limits_{K_{16}} (v-k_j)_{-}(x,t)\xi(x,t)\,dx\,dt, \end{align} where $\gamma = \gamma/\delta$ (note the $\delta$ in \cref{eq:expeqsing}) is a constant depending only on data and $\delta$ (which came from \cref{exptime} and depended only on $\alpha$ and data.) To obtain the above estimate, we tested \cref{eq:expeqsing} with $-(v-k_j)_{-}\zeta^p$ and discarded the nonpositive contribution on the right-hand side from the nonnegative term $\frac{1}{2-p}v$.
\descitem{Step 2}{step2} {\it Shrinking lemma for $v$.} For any $j \geq 2$, we claim that either of the following two possibilities hold: \begin{equation}\label{tailest_sing_2} \begin{array}{l} \text{Tail}_{\infty}(u_{-};0,16\rho,J_2') > \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^j}, \\ J_2' = (M^{2-p}\delta\rho^{sp}(1-e^{-\tau_0}),M^{2-p}\delta\rho^{sp}(1-e^{-\tau_0-2k_0^{2-p}})], \end{array} \end{equation} or \[
|[v < k_{j}]\cap Q_{\tau_0}| \leq \nu|Q_{\tau_0}| \qquad \text{where} \quad \nu = \left(\frac{C}{2^{j-1}-1}\right)^{p-1}, \] for a universal constant $C>0$ depending only on $\alpha$ and data.
\begin{proof}[Proof of Claim] In \cref{eq:vcaccio}, we work with levels $k_j$ defined by \[ k_j = \frac{1}{2^j}k_0 \qquad \text{ for } \qquad j \geq 1, \] where $k_0$ is given by \cref{eq:k0sing}. We first note that since $v \geq 0$ in $Q_{\tau_0'}$ in all the local integrals on the right in \cref{eq:vcaccio}, we can estimate \[ (v-k_j)_{-} \leq k_j, \] and so by our choice of the test function we get \begin{align*}
\oldint\limits_{\tau_0}^{\tau_0 +2k_0^{2-p}}&\oldint\limits_{K_{16}}(v-k_j)_{-}(x,t)\oldint\limits_{K_{16}}\frac{|(v-k_j)_{+}(y,t)|^{p-1}}{|x-y|^{N+sp}}\,dy\,dx\,dt\
\leq C|Q_{\tau_0'}|\lbr k_{j}^p
+k_{j}\underset{\stackrel{t \in (\tau_0, \tau_0+2k_0^{2-p});}{ x\in \spt\zeta_1}}{\esssup}\oldint\limits_{K_{16}^c}\frac{(v-k_j)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\rbr \end{align*} where $C>0$ is a universal constant (note that we are on $K_{16}$ and so the $\rho^{sp}$ term is absorbed into the constant.) We recall the section on Tail estimates (\cref{sec:tail}) to get \[
\oldint\limits_{\tau_0}^{\tau_0 +2k_0^{2-p}}\oldint\limits_{K_{16}}(v-k_j)_{-}(x,t)\oldint\limits_{K_{16}}\frac{|(v-k_j)_{+}(y,t)|^{p-1}}{|x-y|^{N+sp}}\,dx\,dy\,dt\ \leq Ck_{j}^p|Q_{\tau_0'}|, \] holds provided \[ \text{Tail}_{\infty}^{p-1}(v_{-};0,16,(\tau_0, \tau_0+2k_0^{2-p})) \leq k_j^{p-1}. \]
We now invoke \cref{lem:shrinking} and \cref{eq:sing2} to conclude that \[
|[u < k_{j+1}]\cap\mathcal{Q}_{4\rho}(\theta)| \leq \left(\frac{C}{2^{j}-1}\right)^{p-1}|Q_{\tau_0'}|, \] for a constant $C>0$ depending only on $\alpha$ and data. This proves the claim for the choice \begin{equation*}
\nu = \left(\frac{C}{2^{j}-1}\right)^{p-1}. \end{equation*}
We now rewrite the Tail alternative in terms of $u_-$ as follows. \begin{align*}
\underset{\tau \in (\tau_0, \tau_0+2k_0^{2-p})}{\esssup}\oldint\limits_{K_{16}^c}\frac{v_{-}^{p-1}(z',\tau) }{|z'|^{N+sp}}\,dz'& \leq \frac{\exp((\tau_0+2k_0^{2-p})\frac{p-1}{2-p})}{M^{p-1}}\underset{t\in J_2'}{\esssup}\oldint\limits_{K_{16}^c}\frac{u(z'\rho,t)^{p-1}_{-}}{|z'|^{N+sp}}\,dy
\\
&\leq \frac{\exp((\tau_0+2k_0^{2-p})\frac{p-1}{2-p})}{M^{p-1}}\rho^{sp}\underset{t\in J_2'}{\esssup}\oldint\limits_{K_{16\rho}^c}\frac{u(y,t)^{p-1}_{-}}{|y|^{N+sp}}\,dy
\\
&\leq \frac{\exp((\tau_0+2k_0^{2-p})\frac{p-1}{2-p})}{M^{p-1}}\text{Tail}_{\infty}^{p-1}(u_{-};0,16\rho,J_2'), \end{align*} where \[ J_2' = (M^{2-p}\delta\rho^{sp}(1-e^{-\tau_0}),M^{2-p}\delta\rho^{sp}(1-e^{-\tau_0-2k_0^{2-p}})], \] and we used the change of variables from \cref{eq:covsing}. Thus, the Tail alternative is verified provided \[ \text{Tail}_{\infty}(u_{-};0,16\rho,J_2') \leq \exp\left(\lbr-\tau_0-2k_0^{2-p}\rbr\frac{1}{2-p}\right)Mk_j. \] Recalling \cref{eq:k0sing}, we get \[ \exp\left(\lbr-\tau_0-2k_0^{2-p}\rbr\frac{1}{2-p}\right)Mk_j = \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^j}, \] which completes the proof of the claim. \end{proof}
\descitem{Step 3}{step3} {\it Obtaining a good time slice for $v$.} We claim that there exists a $\sigma_0 \in (0,1)$ and $j_{*} \in (1,\infty)$ depending only on the data such that either \[ \text{Tail}_{\infty}(u_{-};0,8\rho,J_2') > \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^{j_{*}}}, \] holds (recall \cref{tailest_sing_2}) or there exists a time level \begin{equation*}
\tau_0 + k_0^{2-p} < \tau_1 < \tau_0 + 2k_0^{2-p}, \end{equation*} such that the following holds: \[ v(z,\tau_1) \geq \sigma_0e^{\frac{\tau_0}{2-p}}. \]
\begin{proof}[Proof of the claim] Suppose that $j_*$ and hence $\nu$ have been determined according to \descref{step2}{Step 2}. By increasing $j_*$ if required to be not necessarily an integer we can assume without loss of generality that $2^{j_{*}(2-p)}$ is an integer. Next, we subdivide $Q_{\tau_0}$ into $2^{j_{*}(2-p)}$ cylinders each of length $k_{j_*}^{2-p}$ given by \[ Q_n = K_8 \times (\tau_0 + k_0^{2-p} + nk_{j_*}^{2-p}, \tau_0 + k_0^{2-p} + (n+1)k_{j_*}^{2-p}) \qquad n = 0,1,\ldots,2^{j_{*}(2-p)}-1. \] Then either the tail alternative in \descref{step2}{Step 2} holds or for at least one of the sub-cylinders, we must have \[
|[v<k_{j_*}\cap Q_n]| \leq \nu |Q_n|. \] We now apply to \cref{lem:deG} to $v$ over $Q_n$ with \[ \nu_{-} = 0, \qquad \xi\omega = k_{j_*}, \qquad a = \frac{1}{2}, \qquad \text{and} \quad \theta = k_{j_*}^{2-p}, \] to get that if the following holds \begin{equation}\label{eq:tailstep3sing}
\begin{array}{rcl} \text{Tail}_{\infty}(v_{-};0,8,(\tau_0 + k_0^{2-p} + nk_{j_*}^{2-p}, \tau_0 + k_0^{2-p} + (n+1)k_{j_*}^{2-p})) &\leq& k_{j_{*}},\\
\frac{|[v<k_{j_*}\cap Q_n]|}{|Q_n|} &\leq& 3^{- \frac{N+sp}{p}}\overline{\gamma_0}(\text{data}) = \nu, \end{array} \end{equation} then the following conclusion follows \[ v(z,\tau_0 + k_0^{2-p} + (n+1)k_{j_*}^{2-p}) \geq \frac{1}{2}k_{j_*} \qquad \text{ a.e. in } K_4. \]
We choose $j_*$ and hence $\nu$ from this and \descref{step2}{Step 2} which implies the conclusion holds with $\sigma_0 = \epsilon2^{-(j_{*}+1)}$. We note that because $n+1 \leq 2^{j_*(2-p)}$ and \[ (\tau_0 + k_0^{2-p} + nk_{j_*}^{2-p}, \tau_0 + k_0^{2-p} + (n+1)k_{j_*}^{2-p}) \subset(\tau_0 + k_0^{2-p}, \tau_0 + 2k_0^{2-p}), \] the Tail alternative in \cref{eq:tailstep3sing} can be reformulated in terms of $u$ as in \descref{step2}{Step 2} to read \[ \text{Tail}_{\infty}(u_{-};0,8\rho,J_2') \leq \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^{j_{*}}}, \] which completes the proof of the claim. \end{proof}
\descitem{Step 4}{step4} {\it Expanding the positivity for $u$.} We switch back to the $(x,t)$ coordinates and assume that Tail alternatives in \descref{step}{Step 2} and \descref{step3}{Step 3} hold. Then, \[ u(\cdot,t_1) \geq \sigma_0Me^{-\frac{\tau_1-\tau_0}{2-p}} =: M_0 \qquad \text{ in } K_{4\rho}, \] where $t_1$ corresponds to $\tau_1$ via the change of variables \cref{eq:covsing}. We now apply \cref{lem:deGdata} with $M_0$ in place of $M$ and $\xi = 1$ over the cylinder \[ K_{4\rho} \times (t_1,t_1+\theta(4\rho)^{sp}]. \] By choosing \[ \theta = \nu_0^{\frac{1}{s}}M_0^{2-p} \qquad \text{ where }\ \nu_0 = \nu_0(\text{data}), \] the smallness assumption \cref{eq:initdatadeG} is satisfied and so we get that the following holds \begin{equation*}
u(\cdot,t) \geq \frac{1}{2}M_0 \geq \frac{1}{2}\sigma_0\exp\left(-\frac{2}{2-p}e^{\tau_0}\right)M \qquad \text{ in } K_{2\rho}, \end{equation*} for all times \begin{equation*}
t_1 \leq t \leq t_1 + \nu_0^{\frac{1}{s}}M_0^{2-p}(4\rho)^{sp}, \end{equation*} provided that the following Tail alternative is satisfied: \begin{equation*} \text{Tail}_{\infty}(u_{-};0,4\rho,(t_1,t_1+\theta (4\rho)^{sp}]) \leq M_0. \end{equation*} We now fix $\tau_0$ by stipulating \[ \delta M^{2-p}\rho^{sp}e^{-\tau_1} = \delta M^{2-p}\rho^{sp} - t_1 = \nu_0^{\frac{1}{s}} \sigma_0^{2-p} M^{2-p}(4\rho)^{sp}e^{-(\tau_1-\tau_0)}, \] which gives \[ \tau_0 = \ln\left(\frac{\delta}{4^{sp}\nu_0^{\frac{1}{s}}\sigma_0^{2-p}}\right). \] This determines $\tau_0$ in terms of $\alpha$ and data. Note that by increasing $j_{*}$ is necessary we can always ensure that $\tau_0 > 0$. Finally, taking \[ \eta := \frac{1}{2}\sigma_0\exp\left(-\frac{2}{2-p}e^{\tau_0}\right), \] to be a constant in $(0,1)$ depending only on $\alpha$ and data, we get that \[ u(\cdot,t) \geq \eta M \qquad \text{ a.e. in } K_{2\rho}, \] holds for all times \begin{equation}\label{def_ve} (1-\varepsilon)\delta M^{2-p}\rho^{sp} \leq t \leq M^{2-p}\delta\rho^{sp} \qquad \text{ where } \varepsilon = \exp(-\tau_0-2e^{\tau_0}). \end{equation}
\descitem{Step 5}{step5} {\it Pulling the Tail alternatives together.} We needed the following Tail assumptions going from \descref{step1}{Step 1} to \descref{step4}{Step 4} (recall $j \leq j_{*})$: \[ \begin{array}{rcl} \text{Tail}_{\infty}(u_{-};0,\rho,(0,\delta\rho^{sp}M^{2-p})) &\leq & M,\\ \text{Tail}_{\infty}(u_{-};0,16\rho,J_2') &\leq & \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^j},\\ \text{Tail}_{\infty}(u_{-};0,8\rho,J_2') &\leq & \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^{j_{*}}},\\ \text{Tail}_{\infty}(u_{-};0,4\rho,(t_1,t_1+\theta (4\rho)^{sp}]) &\leq & M_0. \end{array} \]
We note that in all the Tail alternatives, $\rho$ is the smallest radius and $(0,\delta\rho^{sp}M^{2-p})$ is the largest time interval. We estimate \[ \exp\left(-2k_0^{2-p}\frac{1}{2-p}\right)\frac{\epsilon M}{2^j} = \exp\left(-\frac{2}{2-p}\epsilon^{2-p}e^{\tau_0}\right)\frac{\epsilon M}{2^j} \geq \exp\left(-\frac{2}{2-p}e^{\tau_0}\right)\frac{\epsilon M}{2^j} = \frac{2\eta}{\sigma_0}\frac{\epsilon M}{2^j} \geq 4\eta M, \] and \[ M_0 \geq \sigma_0\exp\left(-\frac{2}{2-p}e^{\tau_0}\right)M = 2\eta M, \] where we recall $ \epsilon $ is from \cref{def_ve} and $\sigma_0 = \frac{\epsilon}{2^{j_{*}+1}}$.
Therefore all the Tail assumptions are satisfied if we require \[ \text{Tail}_{\infty}(u_{-};0,\rho,(0,\delta\rho^{sp}M^{2-p})) \leq \eta M, \] because $\eta \in (0,1)$. \end{description} This completes the proof of the proposition. \end{proof}
\begin{remark} The conclusion of \cref{prop:sinexp} can also be written without Tail alternatives as \[ u(\cdot,t) \geq \eta M - \text{Tail}_{\infty}(u_{-};x_0,\rho,(t_0,t_0+\delta\rho^{sp}M^{2-p})) \qquad \text{a.e. in } K_{2\rho}(x_0), \] for all times \[
t_0+(1-\varepsilon)\delta M^{2-p}\rho^{sp} \leq t \leq t_0+M^{2-p}\delta\rho^{sp}, \] because we are working with $u$ nonnegative. \end{remark} \begin{remark} In the degenerate case the final time level depends on the final lower level $\eta M$ that is achieved during the expansion whereas in the singular case, the final time level depends on the starting level $M$. \end{remark}
\hrule \hrule \hrule \hrule \hrule
\section{H\"older regularity for parabolic fractional $p$-Laplace equations}\label{sec9}
In this section, we present the proof of \cref{holderparabolic}. The induction argument is similar to the one in \cite{cozziRegularityResultsHarnack2017} and the covering argument is taken from \cite{hwangHolderContinuityBounded2015,hwangHolderContinuityBounded2015a}
\begin{proof}[Proof of \cref{holderparabolic}] We assume without loss of generality that $(x_0,t_0)=(0,0)$. Take \begin{align}\label{choiceofalpha}
0<\alpha<\min\left\{ \frac{sp}{p-1},\log_{C_0}\left(\frac{2}{2-\eta}\right)\right\}, \end{align} so that \begin{align}\label{cond2}
\oldint\limits_{C_0}^\infty \frac{(\rho^\alpha-1)^{p-1}}{\rho^{1+sp}}\,d\rho<\frac{\eta^{p-1}2^{-p}}{C_I\,C_0^{sp}}, \end{align} where $\eta$ is the constant that appears in \cref{prop:degexp} for $p>2$ and in \cref{prop:sinexp} for $p<2$ and $C_I$ appears in \cref{estim4}. The constant $C_0>0$ will be determined in a later step. The integral can be made small enough by taking small $\alpha$.
Also define \begin{align}\label{choiceofj0}
j_0:=\left\lceil\frac{1}{sp-\alpha(p-1)}\log_{C_0}\left(\frac{2(C_{III}+2^{p-1})}{\eta^{p-1}}\right)\right\rceil, \end{align}where $\eta$ is the constant that appears in \cref{prop:degexp} for $p>2$ and in \cref{prop:sinexp} for $p<2$ and $C_{III}$ appears in \cref{estim0}.
\begin{claim}
With these choices, we claim that there exist a non-decreasing sequence $\{m_i\}_{i=0}^\infty$ and a non-increasing sequence $\{M_i\}_{i=0}^\infty$ such that for any $i=0,1,2,\ldots$, we have
\begin{equation}\label{holdercond}
m_i\leq u \leq M_i \qquad \mbox{ in } \ Q_{R_i}(d_i):=B_{R_i}\times (-d_i\,R_i^{sp},0),
\end{equation} where we have taken
\begin{equation}\label{defofL}
\begin{array}{rcl}
M_i-m_i&=& C_0^{-\alpha i}L, \\
R_i &:= &C_0^{1-i}R,\\
L &:=& 2\cdot C_0^{\frac{sp}{p-1}\,j_0}\|u\|_{L^\infty(Q_{C_0\,R})}+\text{Tail}_{\infty}(u;C_0\,R,0,(-(C_0\,R)^{sp},0)),\\
d_i &:= &\begin{cases}
\frac{b^{p-2}}{(\eta\,C_0^{-\alpha i}L)^{p-2}}\delta , & p>2,\\
(C_0^{-\alpha i}L)^{2-p}\delta, & p<2.
\end{cases}
\end{array}
\end{equation}
\end{claim}
The proof of \cref{holdercond} is by induction on $i$. If we set $m_i:=-C_0^{-\alpha i}L/2$ and $M_i:=C_0^{-\alpha i}L/2$ for $i=0,1,2,\ldots, j_0$, then, by \cref{choiceofalpha} and \cref{defofL}, we see that \cref{holdercond} holds for this range of $i$.
Now, suppose that the sequences $m_i$ and $M_i$ have been defined for $i\leq j$ for some $j\geq j_0$. H\"older regularity will be proved once we prove the induction step, that is, we define $m_{j+1}$ and $M_{j+1}$ so that \cref{holdercond} is satisfied.
Define the function \begin{equation*}
v:=2u - (M_j+m_j). \end{equation*} We have by \cref{holdercond} and monotonicity properties of $M_j$ and $m_j$ that \begin{equation*} (M_j - M_0) + (m_0-m_j) \leq M_j + m_j \leq (M_0 - M_j) + (m_j - m_0), \end{equation*} which implies \begin{equation*}
|M_j+m_j|\leq (1-C_0^{-\alpha\,j})L. \end{equation*}
Using this, we have \begin{align}\label{estim1}
(C_0^{-\alpha j}L\pm v)_-^{p-1}\leq 2^{p-1}|u|^{p-1}+L^{p-1}\quad \mbox{ for a.e. }\ x\in \RR^N\setminus B_{C_0\,R}\ \mbox{ and }\ t\in (0,(C_0\,R)^{sp}). \end{align} Let $I_j:=(-d_j R_j^{sp},0)$ and $B_j:=B_{R_j}(0)$, then we wish to estimate \begin{align}\label{tailest1}
\tail((C_0^{-\alpha j}L\pm v)_{-}&;C_0^{1-j}R,I_j)^{p-1}= C_0^{(1-j)sp}R^{sp}\, \esssup\limits_{I_j}\oldint\limits_{\RR^N\setminus B_{j}}\frac{(C_0^{-\alpha j}L\pm v)_-^{p-1}}{|x|^{N+ps}}\,dx\nonumber\\
&= C_0^{(1-j)sp}R^{sp}\, \esssup\limits_{I_{j}}\Bigg[\oldint\limits_{B_0\setminus B_{j}}\frac{(C_0^{-\alpha j}L\pm v)_-^{p-1}}{|x|^{N+ps}}\,dx+\oldint\limits_{\RR^N\setminus B_0}\frac{(C_0^{-\alpha j}L\pm v)_-^{p-1}}{|x|^{N+ps}}\,dx\Bigg]\nonumber\\
&\stackrel{\cref{estim1}}{\leq} C_0^{(1-j)sp}R^{sp}\,\Bigg[\underbrace{\esssup\limits_{I_{j}}\oldint\limits_{B_0\setminus B_{j}}\frac{(C_0^{-\alpha j}L\pm v)_-^{p-1}}{|x|^{N+ps}}\,dx}_{I}\nonumber\\
&\qquad\qquad\qquad\qquad +\underbrace{2^{p-1}\esssup\limits_{I_{0}}\oldint\limits_{\RR^N\setminus B_{0}}\frac{|u|^{p-1}}{|x|^{N+ps}}\,dx}_{II}+\underbrace{L^{p-1}\oldint\limits_{\RR^N\setminus B_{0}}\frac{1}{|x|^{N+ps}}\,dx}_{III}\Bigg]. \end{align}
The estimate for $III$ is given by \begin{align}\label{estim0}
III\leq C_{III}\frac{L^{p-1}}{(C_0\,R)^{sp}}, \end{align}which is immediate after conversion to polar coordinates. The estimate for $II$ is given by \begin{align}\label{estim00}
II\leq 2^{p-1}\frac{L^{p-1}}{(C_0\,R)^{sp}}, \end{align} which follows from the definition of $L$ in \cref{defofL}.
To estimate $I$, for given $x \in B_0 \setminus B_j$, there is $l\in \{0,1,2,\ldots,j-1\}$ such that $x\in B_l\setminus B_{l-1}$ so that by the monotonicity of the sequence $m_i$ and by the induction hypothesis, for a.e. $t\in I_j$, we have: \begin{equation}\label{estim2}
\begin{array}{rcl}
\frac{v(x,t)}{2}&\leq& M_l - m_l + m_l - \frac{M_j+m_j}{2}\\
&\leq& M_l - m_l + m_j - \frac{M_j+m_j}{2}\\
&=&M_l-m_l-\frac{M_j-m_j}{2}
=\left(C_0^{-\alpha l}-\frac{C_0^{-\alpha j}}{2}\right)L\\
&\leq&\left(\left(\frac{|x|}{R}\right)^\alpha-\frac{C_0^{-\alpha j}}{2}\right)L.
\end{array} \end{equation}
Similarly, we obtain \begin{align}\label{estim3}
v(x,t)\geq -\left(2\left(\frac{|x|}{R}\right)^\alpha-C_0^{-\alpha j}\right)L. \end{align} Combining \cref{estim2} and \cref{estim3}, we get \begin{align*}
(C_0^{-\alpha j}L\pm v(x,t))_-^{p-1}\leq 2^{p-1}\left[ \left(\frac{|x|}{R}\right)^\alpha-C_0^{-\alpha j} \right]^{p-1}L^{p-1}. \end{align*}
We use this to estimate \begin{equation}\label{estim4}
\begin{array}{rcl}
I&\leq &C_0^{-\alpha j(p-1)}2^{p-1}L^{p-1}\oldint\limits_{\RR^N\setminus B_{C_0^{1-j}R}}\frac{\left(\left(\frac{C_0^j|x|}{R}\right)^\alpha-1\right)^{p-1}}{|x|^{N+ps}}\,dx\\
&=&\frac{C_0^{-\alpha j(p-1)}2^{p-1}L^{p-1}C_0^{jps}}{R^{sp}}\oldint\limits_{\RR^N\setminus B_{C_0}}\frac{\left(|y|^\alpha-1\right)^{p-1}}{|y|^{N+ps}}\,dy\\
&\leq& C_{I}\frac{C_0^{-\alpha j(p-1)}2^{p-1}L^{p-1}C_0^{jps}}{R^{sp}}\oldint\limits_{C_0}^\infty\frac{\left(\rho^\alpha-1\right)^{p-1}}{\rho^{1+ps}}\,d\rho.
\end{array} \end{equation}
Combining \cref{estim4}, \cref{estim00} and \cref{estim0} in \cref{tailest1}, we obtain \begin{align}\label{taildecay}
\tail(&(C_0^{-\alpha j}L\pm v)_{-};C_0^{1-j}R,I_j)^{p-1}\nonumber\\
&\leq C_0^{-\alpha j(p-1)}L^{p-1}\left( C_{I} 2^{p-1}C_0^{sp} \oldint\limits_{C_0}^\infty\frac{\left(\rho^\alpha-1\right)^{p-1}}{\rho^{1+ps}}\,d\rho+ ( C_{III}+2^{p-1})\,C_0^{-(sp-\alpha(p-1))j} \right)\nonumber\\
&\leq C_0^{-\alpha j(p-1)}L^{p-1} \eta^{p-1}, \end{align} where the last inequality follows from \cref{cond2} and \cref{choiceofj0} and the fact that $j\geq j_0$.
Now, one of the following two alternatives must hold: \begin{align}
\left|\{v(\cdot,s_j)\geq 0\}\cap B_{C_0^{-j}R/2}\right|\geq \frac 12 \left|B_{C_0^{-j}R/2}\right|, \label{alt1}\\
\left|\{v(\cdot,s_j)\geq 0\}\cap B_{C_0^{-j}R/2}\right|< \frac 12 \left|B_{C_0^{-j}R/2}\right|,\label{alt2} \end{align} where we have set \begin{align*}
s_j:=\begin{cases}
-\frac{b^{p-2}}{(\eta\,C_0^{-\alpha j}L)^{p-2}}\delta R_j^{sp} & p>2,\\
-(C_0^{-\alpha j}L)^{2-p}\delta R_j^{sp} & p<2.
\end{cases} \end{align*}
For $p>2$, the choice of $C_0$ is determined by the condition \begin{align*}
s_{j+1}>(-1+\sigma)d_jR_j^{sp}\quad \Longrightarrow \quad C_0 > \left(\frac{1}{1-\sigma}\right)^{\frac{1}{\alpha(2-p)+sp}}. \end{align*}
For $p<2$, the choice of $C_0$ is determined by the condition \begin{align*}
s_{j+1}>-\varepsilon d_jR_j^{sp} \quad \Longrightarrow \quad C_0 > \left(\frac{1}{\varepsilon}\right)^{\frac{1}{\alpha(2-p)+sp}}. \end{align*}
When \cref{alt1} holds, we define $w=C_0^{-\alpha j}L+v$, then \cref{alt1} becomes \begin{align*}
\left|\{w(\cdot,s_j)\geq C_0^{-\alpha j}L\}\cap B_{C_0^{-j}R/2}\right|\geq \frac 12 \left|B_{C_0^{-j}R/2}\right|. \end{align*} We have $w\geq 0$ in $Q_{R_j}(d_j)$ so due to \cref{taildecay} we are in a position to apply the expansion of positivity lemma, viz., \begin{align*}
w\geq \eta C_0^{-\alpha j}L \qquad \mbox{ a.e. }(x,t)\in Q_{R_{j+1}}(d_{j+1}). \end{align*} Therefore, \begin{align*} u(x,t)&=\frac{M_j+m_j}{2} + \frac{v}{2}\\
&=\frac{M_j+m_j}{2} + \frac{w-C_0^{-\alpha j}L}{2}\\
&\geq \frac{M_j+m_j}{2} - \frac{C_0^{-\alpha j}L}{2}(1-\eta)\\
&= M_j - \frac{M_j-m_j}{2} - \frac{C_0^{-\alpha j}L}{2}(1-\eta)\\
&= M_j - \frac{C_0^{-\alpha j}L}{2}(2-\eta)\\
&\geq M_j - C_0^{-\alpha(j+1)}L, \end{align*} where the last inequality in due to \cref{choiceofalpha}. So if we define $m_{j+1}:=M_j - C_0^{-\alpha(j+1)}L$, then the induction hypothesis will be satisfied. For the second alternative \cref{alt2}, we shall define $w=C_0^{-\alpha j}L-v$ and retrace the steps to obtain $M_{j+1}$.
A consequence of \cref{holdercond} is the following oscillation decay: \begin{align*}
\text{osc}_{Q_{R_i}(d_i)}\, u:=\sup\limits_{Q_{R_i}(d_i)}\,u - \inf\limits_{Q_{R_i}(d_i)}\, u \leq C_0^{-\alpha i}L. \end{align*}
Let $(x_1,t_1) ,(x_2,t_2)\in B_R(x_0)\times (t_0-L^{2-p}R^{sp},t_0)$ such that $x_1\neq x_2$ and $t_1\neq t_2$, then there exist non-negative integers $n$ and $m$ such that
\begin{align}\label{holdest1}
R_{n+1}<|x_1-x_2|\leq R_n, \qquad \text{and} \qquad d_{m+1}R_{m+1}^{sp}<|t_1-t_2|\leq d_{m}R_{m}^{sp}. \end{align}
As a result, we obtain \begin{align}\label{holdest0}
|u(x_1,t_1)-u(x_2,t_2)|\leq\max\{C_0^{-\alpha n}L,C_0^{-\alpha m}L\}. \end{align} From the first inequality in \cref{holdest1}, we deduce \begin{align}\label{holdest2.5}
\frac{|x_1-x_2|}{R}>C_0^{-n} \qquad \Longrightarrow \qquad \left(\frac{|x_1-x_2|}{R}\right)^\alpha\,L>C_0^{-\alpha n}L. \end{align} On the other hand, from the second inequality in \cref{holdest1}, we get \begin{align*}
C'\,L^{\frac{p-2}{sp}}\frac{|t_1-t_2|^{\frac{1}{sp}}}{R}>\left(C_0^{-\alpha(m+1)}\right)^{\frac{2-p}{sp}}\,C_{0}^{-m}, \end{align*} where $C'$ is a constant depending only on data. In order to further estimate the lower bound, we split into two cases: \begin{description}
\item[Case $p > 2$:] Using $C_0 > 1$ we get
\begin{align}\label{eq:holdestdeg}
C'\,L^{\frac{p-2}{sp}}\frac{|t_1-t_2|^{\frac{1}{sp}}}{R}>C_{0}^{-m} \qquad \Longrightarrow \qquad C_{0}^{-\alpha m} \leq \left(C'L^{\frac{p-2}{sp}}\frac{|t_1-t_2|^{\frac{1}{sp}}}{R}\right)^{\alpha}.
\end{align}
\item[Case $p<2$:] Using $p>1$ and $C_0 > 1$ we get
\begin{align*}
C'\,L^{\frac{p-2}{sp}}\frac{|t_1-t_2|^{\frac{1}{sp}}}{R}>\left(C_0^{-\alpha(m+1)}\right)^{\frac{1-p}{sp}}\,\left(C_0^{-\alpha(m+1)}\right)^{\frac{1}{sp}}\,C_{0}^{-m} > \left(C_0^{-\alpha(m+1)}\right)^{\frac{1}{sp}}\,C_{0}^{-m},
\end{align*}
which implies
\[
C''\,L^{\frac{p-2}{sp}}\frac{|t_1-t_2|^{\frac{1}{sp}}}{R} > C_0^{-m\left(1+\frac{\alpha}{sp}\right)},
\]
so that
\begin{equation}\label{eq:holdestsing}
C_{0}^{-\alpha m} = \left(C_0^{-m\left(1+\frac{\alpha}{sp}\right)}\right)^{\frac{\alpha}{1+\frac{\alpha}{sp}}}\leq \left(C''L^{\frac{p-2}{sp}}\frac{|t_1-t_2|^{\frac{1}{sp}}}{R}\right)^{\frac{\alpha}{1+\frac{\alpha}{sp}}},
\end{equation}
where $C'' = C'C_0^{\frac{\alpha}{sp}}$ is a constant depending only on data. \end{description}
The argument in the case when either $x_1=x_2$ or $t_1=t_2$ is similar. The conclusion follows from \cref{holdest0}, \cref{holdest2.5}, \cref{eq:holdestdeg} and \cref{eq:holdestsing}.
\end{proof}
\begin{remark}
We note that the proof provides the following relationship between the Holder exponents for space and time:
\begin{center}
\begin{tabular}{||c| c| c||}
\hline\hline
& x & t \\
\hline\hline
Degenerate & $\alpha$ & $\frac{\alpha}{sp}$\\
\hline
Singular & $\alpha$ & $\frac{\frac{\alpha}{sp}}{1+\frac{\alpha}{sp}}$\\
\hline\hline
\end{tabular}
\end{center}
In the degenerate case, the condition $\alpha(p-1) < sp$ forces $\alpha < sp$. \end{remark}
\begin{remark}\label{rem:nonlocalB}
We further note that in the degenerate case, the space Holder exponent is $\alpha$ while the time exponent is $\alpha/sp$. In particular, if we start with $sp<1$ time regularity beats space regularity - such a thing does not occur in the local case and is an instance of a purely nonlocal phenomena. \end{remark}
\end{document} | arXiv |
How many times does the digit 8 appear in the list of all integers from 1 to 1000?
The easiest approach is to consider how many times 8 can appear in the units place, how many times in the tens place, and how many times in the hundreds place. If we put a 8 in the units place, there are 10 choices for the tens place and 10 choices for the hundreds place (including having no hundreds digit) for a total of $10\times10=100$ options, which means that 8 will appear in the ones place 100 times. (If we choose the hundreds place to be equal to 0, we can just think of that as either a two digit or a one digit number.) Likewise, if we put an 8 in the tens place, there are 10 choices for the units place and 10 choices for the hundreds place for a total of 100 options and 100 appearances of 8 in the tens place. Finally, if we put a 8 in the hundreds place, we have 10 options for the units place and 10 options for the tens place for another 100 options and 100 appearances of 8. Since $100 + 100+100=300$, there will be a total of $\boxed{300}$ appearances of 8. | Math Dataset |
OSA Publishing > Optics Express > Volume 27 > Issue 24 > Page 35394
James Leger, Editor-in-Chief
Image-free real-time detection and tracking of fast moving object using a single-pixel detector
Zibang Zhang, Jiaquan Ye, Qiwen Deng, and Jingang Zhong
Zibang Zhang, Jiaquan Ye, Qiwen Deng, and Jingang Zhong*
Department of Optoelectronic Engineering, Jinan University, Guangzhou 510632, China
*Corresponding author: [email protected]
Zibang Zhang https://orcid.org/0000-0002-8241-0200
Z Zhang
J Ye
Q Deng
J Zhong
pp. 35394-35401
•https://doi.org/10.1364/OE.27.035394
Zibang Zhang, Jiaquan Ye, Qiwen Deng, and Jingang Zhong, "Image-free real-time detection and tracking of fast moving object using a single-pixel detector," Opt. Express 27, 35394-35401 (2019)
Image-free classification of fast-moving objects using "learned" structured illumination and single-pixel detection (OE)
Image-free real-time 3-D tracking of a fast-moving object using dual-pixel detection (OL)
Tracking and imaging of moving objects with temporal intensity difference correlation (OE)
Imaging Systems, Microscopy, and Displays
High speed photography
Object detection
Original Manuscript: October 9, 2019
Revised Manuscript: November 10, 2019
Manuscript Accepted: November 11, 2019
Suppl. Mat. (3)
Real-time detection and tracking for fast moving object has important applications in various fields. However, available methods, especially low-cost ones, can hardly achieve real-time and long-duration object detection and tracking. Here we report an image-free and cost-effective method for detecting and tracking a fast moving object in real time and for long duration. The method employs a spatial light modulator and a single-pixel detector for data acquisition. It uses Fourier basis patterns to illuminate the target moving object and collects the resulting light signal with a single-pixel detector. The proposed method is able to detect and track the object with the single-pixel measurements directly without image reconstruction. The detection and tracking algorithm of the proposed method is computationally efficient. We experimentally demonstrate that the method can achieve a temporal resolution of 1,666 frames per second by using a 10,000 Hz digital micro-mirror device. The latency time of the method is on the order of microseconds. Additionally, the method acquires only 600 bytes of data for each frame. The method therefore allows fast moving object detection and tracking in real time and for long duration. This image-free approach might open up a new avenue for spatial information acquisition in a highly efficient manner.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Object detection and tracking has found important applications in military, industry, and scientific research. Real-time object detection and tracking allows rapid response for emergencies. Various fast moving object detection and tracking approaches have been proposed [1–14], but few can achieve real-time and long-duration detection and tracking in a cost-effective manner. There are mainly two types of moving object detection and tracking approaches: image-free and image-based.
Radar [5] is an image-free technique for object detection and tracking. It sends out electromagnetic pulses in radio or microwaves range and collects the reflected pulses from the target object. To locate a moving object, radar measures the running time of the transmitted pulses (also known as, Time-of-Flight [6]). LiDAR [7–10] works on the principle of radar, but uses light waves instead of radiowaves. Radars are generally used for remote objects and large field-of-view. Radar systems with high spatial resolution are generally costly [11].
Image-based methods [12–14] rely on photography. Image sensors are used to capture images of the target object. Image post-processing and image analysis algorithms are further employed to detect and track the target object in the images. The accuracy of object detection and tracking depends on the quality of images captured and the performance of the algorithms utilized. However, the quality of images might be affected by two factors. The one is motion blur caused by high-speed moving objects. The other is exposure time which is short in order to reduce the undesired motion blur. Short exposure time might lead to a low signal-to-noise ratio. For detection and tracking of fast moving objects, high-speed cameras are preferable. It is because high-speed cameras have a high frame rate and are able to provide images with a relatively high signal-to-noise ratio even when the exposure time is short. However, high-speed cameras are expensive and their data throughput is generally huge. In practice, data storage and bandwidth are limited, resulting in long-duration object detection and tracking still a challenge. Moreover, computationally complex image analysis algorithms and limited data-processing capacity in actual also add challenge to real-time object detection and tracking.
Object detection is substantially to determine the presence of an object. It is a dual (yes-or-no) problem and consumes a little information in theory. On the other hand, object tracking is to acquire the two-dimensional (2-D) or three-dimensional (3-D) spatial coordinates of an object. Mathematically, there are only 2 or 3 unknowns to be determined. Thus, object tracking consumes a little information in theory as well. From this perspective, it turns out to be wasting resources that image-based methods (such as, high-speed photography) acquire a huge amount of image data for object detection and tracking. The redundancy in data acquisition is the main reason why real-time and long-duration object detection and tracking is challenging.
Recently, D. Shi et al. proposed an approach which is based on single-pixel imaging [15]. This method utilizes the 'slice theory' of Hadamard transform to obtain one-dimensional (1-D) images instead of 2-D images to locate the target object, so as to reduce the redundant data acquired. By using a 22,000 Hz digital micro-mirror device (DMD), this method achieves a temporal resolution of 1/177 seconds and a spatial resolution of 256×256 pixels. The method demonstrates reduction of redundancy in data acquisition is an effective way for real-time and long-duration object detection and tracking.
Here we propose an image-free and cost-effective method for fast moving object detection and tracking in real time and for long duration. The principle of the proposed image-free method is completely different from existing image-free approaches. Inspired by single-pixel imaging [16,17], the proposed method acquires the spatial information of target object by using spatial light modulation and single-pixel detection. The proposed method is based on Fourier single-pixel imaging [18–23], but acquires no images for object detection and tracking. Instead, it uses 6 Fourier basis patterns for structured light modulation to measure only 2 Fourier coefficients among the complete Fourier spectrum of object image. As Fourier transform is a global-to-point transformation, the 2 Fourier coefficients provide sufficient knowledge of presence or/and motion of object. Moreover, the property that, translation in spatial domain results in a linear phase shift in Fourier domain, allows us to estimate the displacement of a moving object with the 2 Fourier coefficients acquired. The algorithm for object detection and tracking of the proposed method is computationally efficient. Consequently, the proposed method enables real-time object detection and tracking.
2. Principle
Fourier single-pixel imaging was initially proposed for image acquisition. It is characterized by the use of Fourier basis (that is, sinusoidal intensity) patterns for spatial light modulation. As such, the spatial information of object is encoded into a 1-D temporal light signal. By measuring the intensity of the resulting light signal with a single-pixel detector, Fourier single-pixel imaging is able to recover the Fourier spectrum of the object image and the object image can be obtained by applying an inverse Fourier transform to the Fourier spectrum recovered.
Each Fourier basis pattern $P({x,\;y} )$ is characterized by its spatial frequency pair $({{f_x},{f_y}} )$ and initial phase ${\varphi _0}$:
(1)$$P({x,\;y|{{f_x},{f_y},{\varphi_0}} } )= A + B\cos [{2\pi ({{f_x}x + {f_y}y} )+ {\varphi_0}} ], $$
where $({x,\;y} )$ denotes 2-D coordinate in spatial domain, A is the average intensity of the pattern, and B denotes the contrast. Using the Fourier basis patterns for modulating the illumination light field or the detection light field, the resulting light intensity D is equivalent to an inner product of the object image I and the Fourier basis pattern $P$:
(2)$$\begin{aligned} D &= \left\langle {I({x,\;y} ),P({x,\;y} )} \right\rangle \\ &= \int\!\!\!\int {I({x,\;y} )\{{A + B\cos [{2\pi ({{f_x}x + {f_y}y} )+ {\varphi_0}} ]} \}\textrm{d}x\textrm{d}y} \end{aligned}. $$
As the response of single-pixel detector is linear to the light intensity input, we also use D to denote the single-pixel measurement for simplicity. The inner product is either the real or the imaginary part of a Fourier coefficient [18,20].
The integral in Eq. (2) implies that Fourier transform is a global-to-point transformation. Specifically, every single coefficient in Fourier domain is contributed by all points in spatial domain. Consequently, any changes in spatial domain will affect all coefficients in Fourier domain. Exploiting this property, we can detect presence or/and motion of objects in the scene by monitoring the change of one or a few Fourier coefficients in Fourier domain. It can be done by using Fourier single-pixel imaging, as Fourier single-pixel imaging allows Fourier coefficients acquisition by using Fourier basis patterns for spatial light modulation.
By modulating either the illumination light field or the detection light field with Fourier basis patterns, we can observe continuous and large variations in single-pixel measurements when an object enters the scene or starts moving in the scene. A single Fourier coefficient is sufficient for moving object detection by our method. Here we propose to use 2 Fourier coefficients. There are two reasons. The one is that object tracking by the proposed method, as will be introduced below, uses 2 Fourier coefficients. Object detection and object tracking can share the 2 Fourier coefficients, enabling efficient data acquisition. The second reason is that the more Fourier coefficients are used, the more reliable object detection will be. As only 2 Fourier coefficients instead of the complete Fourier spectrum are needed to be measured, our method is efficient in terms of data acquisition and advantageous for fast moving object detection.
For moving object tracking, we exploit the linear phase shift property of Fourier transform. The property illustrated by Eq. (3) indicates that an image with a displacement of $({{x_0},{y_0}} )$ in spatial domain results in a phase shift of $({ - 2\pi {f_x}{x_0}, - 2\pi {f_y}{y_0}} )$ in Fourier domain:
(3)$$I({x - {x_0},\;y - {y_0}} )= {F^{ - 1}}\{{\tilde{I}({{f_x},{f_y}} )\exp [{ - \textrm{j}2\pi ({{f_x}{x_0} + {f_y}{y_0}} )} ]} \}$$
where $({{f_x},{f_y}} )$ denotes the spatial frequency coordinate in Fourier domain, $\tilde{I}({{f_x},{f_y}} )$ denotes the Fourier spectrum of image $I({x,\;y} )$, and ${F^{ - 1}}$ denotes inverse Fourier transform operation. In order to obtain the displacement of the target object instead of the whole image, we apply background subtraction, which can be done by replacing $\tilde{I}$ with $\tilde{I} - {\tilde{I}_{\textrm{bg}}}$ in Eq. (3). The background, ${\tilde{I}_{\textrm{bg}}}$, is referred to the Fourier spectrum acquired before object enters the scene or starts moving. Please note that we don't need to acquire the complete Fourier spectrum of the background. Instead, as we'll discuss below, we only need to acquire a partial of the spectrum (specifically, 2 coefficients).
The displacement of object can be obtained from the phase term $\varphi ={-} 2\pi ({{f_x}{x_0} + {f_y}{y_0}} )$ in Eq. (3) where $\tilde{I}$ is replaced by $\tilde{I} - {\tilde{I}_{\textrm{bg}}}$. As there are 2 unknowns (${x_0}$ and ${y_0}$) in Eq. (3), we can establish a system of 2 equations to solve for the 2 unknowns. The linear equation system requires 2 Fourier coefficients. For simplicity, we propose to acquire $\tilde{I}({{f_x},0} )$ and $\tilde{I}({0,{f_y}} )$ for ${x_0}$ and ${y_0}$, respectively.
The acquisition of Fourier coefficients in Fourier domain can be referred to the N-step phase-shifting method. Specifically, 3-step phase shifting method [20] is chosen in this work, as it has demonstrated efficient in data acquisition and robust to noise and illumination fluctuation. To acquire a Fourier coefficient $\tilde{I}({{f_x},{f_y}} )$ by using the 3-step phase-shifting method, 3 Fourier basis patterns are needed. The 3 Fourier basis patterns have an identical spatial frequency pair $({{f_x},{f_y}} )$ but a different initial phase, that is, ${\varphi _0} = 0$, ${{2\pi } \mathord{\left/ {\vphantom {{2\pi } 3}} \right.} 3}$, or ${{4\pi } \mathord{\left/ {\vphantom {{4\pi } 3}} \right.} 3}$, respectively. The Fourier coefficient is obtained by the 3 corresponding single-pixel measurements:
(4)$$\tilde{I} = ({2{D_0} - {D_{{{2\pi } \mathord{\left/ {\vphantom {{2\pi } 3}} \right.} 3}}} - {D_{{{4\pi } \mathord{\left/ {\vphantom {{4\pi } 3}} \right.} 3}}}} )+ \sqrt 3 \textrm{j}({{D_{{{2\pi } \mathord{\left/ {\vphantom {{2\pi } 3}} \right.} 3}}} - {D_{{{4\pi } \mathord{\left/ {\vphantom {{4\pi } 3}} \right.} 3}}}} ), $$
where the coordinate of spatial frequency $({{f_x},{f_y}} )$ is omitted for simplicity, D denotes single-pixel measurement, and the subscription of D denotes the initial phase of Fourier basis pattern.
We define that one frame in the proposed method is to detect and to locate the target object for one time. As object detection and object tracking can use the same 2 Fourier coefficients and each coefficient uses 3 patterns, only 6 Fourier basis patterns are needed for one frame. With the corresponding 6 single-pixel measurements, object detection and tracking can be done simultaneously. As pattern displaying rate is commonly lower than data sampling rate, the frame rate of the proposed method is limited by pattern displaying rate. The frame rate is equal to the reciprocal of the time for displaying 6 patterns. Digital micro-mirror devices (DMDs) are a common choice for high-speed spatial light modulation. However, Fourier basis patterns are grayscale while DMDs can only generate binary patterns. Thus, we propose to use binarized Fourier basis patterns for spatial light modulation. The binarization strategy is detailed in [20]. The patterns are denoted by ${P_i}$ $({i = 1,2, \cdots ,6} )$ and the corresponding single-pixel measurements are denoted by ${D_i}$ $({i = 1,2, \cdots ,6} )$. The former 3 patterns are for the acquisition of $\tilde{I}({{f_x},0} )$, that is, ${P_1}({{f_x},{f_y},{\varphi_0}} )= P({{f_x},0,0} )$, ${P_2} = P({{f_x},0,{{2\pi } \mathord{\left/ {\vphantom {{2\pi } 3}} \right.} 3}} )$, and ${P_3} = P({{f_x},0,{{4\pi } \mathord{\left/ {\vphantom {{4\pi } 3}} \right.} 3}} )$. Similarly, the latter 3 patterns are for $\tilde{I}({0,{f_y}} )$.
Assuming the target moving object is not present at the beginning, the single-pixel measurements should be a constant in theory, but vary within a small range in actual. The variation might be caused by noise and/or ambient light. As soon as the moving object enters the scene or starts moving, it will cause large variation to the single-pixel measurements. We use the following criterion to determine the presence of moving object. If the difference between $\bar{D}$ (average of ${D_i}$, $i = 1,2, \cdots ,6$) of the current frame and the average of $\bar{D}$ in the latest 5 frames (current frame excluded) is larger than a threshold, we conclude that moving object is present. The threshold is adaptive and depends on the average and the standard deviation of $\bar{D}$ in the previous frames, which will be detailed in the experiments section. Similarly, we can detect if the object exits the scene or stops moving if the difference is smaller than the threshold.
As soon as moving object is detected, object tracking will be started. Specifically, the single-pixel measurements will be substituted into Eq. (4) to obtain the Fourier coefficients $\tilde{I}({{f_x},0} )$ and $\tilde{I}({0,{f_y}} )$, with which the displacement can be derived through:
(5)$${x_0} ={-} \frac{1}{{2\pi {f_x}}} \cdot \arg \{{[{\tilde{I}({{f_x},0} )- {{\tilde{I}}_{\textrm{bg}}}({{f_x},0} )} ]} \},$$
(6)$${y_0} ={-} \frac{1}{{2\pi {f_y}}} \cdot \arg \{{[{\tilde{I}({0,{f_y}} )- {{\tilde{I}}_{\textrm{bg}}}({0,{f_y}} )} ]} \},$$
where $\arg \{{} \}$ denotes argument operation, and ${\tilde{I}_{\textrm{bg}}}$ the coefficient obtained from a background frame. The background frame is referred to a frame which is acquired before the moving object enters the scene or starts moving.
3. Experiments
We demonstrate the proposed method with experiments. As Fig. 1(a) shows, our experimental set-up consists of an illumination system, a detection system, and a target moving object. The illumination system consists of a 10-watt white LED, a DMD (Taxes Instruments DLP Discovery 6500 development kit), and a projection lens.
Fig. 1. Experimental set up. (b) h-shaped track. (c) s-shaped track. Scale bar = 15 mm.
The illumination system generates Fourier basis patterns using the DMD which is 0.65 inch in size and has 1920×1080 micro mirrors. The mirror pitch is 7.6 µm. The DMD is under illumination by the LED. The DMD operates at its highest refreshing rate, that is, 10,000 Hz. Fourier basis patterns displayed on the DMD are projected onto the image plane through the projection lens. The field-of-view is determined by the size of patterns at the image plane. In our experiment, the field-of-view is 60 mm×34 mm.
We simulate a fast moving object using a hollow metallic ball. Initially, the ball is at the top of a 'track'. When the ball is released, it slides down along the track due to gravity (also watch Visualization 1 and Visualization 2). The track is prepared by bending a metallic wire. The track is flat and placed at the image plane of the DMD (illustrated by the red dash box in Fig. 1(a)). As such, the ball is restricted to motion in image plane of the DMD where the illumination patterns are in focus. We prepare two tracks in our experiments. The one is h-shaped (Fig. 1(b)) and the other is s-shaped (Fig. 1(c)).
The detection system consists of a collecting lens and a photodiode amplified (PDA) (Thorlabs PDA100A2). The target moving object is under Fourier basis patterns illumination. The resulting light is collected by the PDA through the collecting lens. The PDA converts the light intensities into analog electric signals which are fed to the computer after digitalization by the data acquisition board (National Instruments USB-6343 BNC). The data acquisition board operates at its maximum sampling rate, that is, 500,000 samples per second. The computer processes the input single-pixel measurements for object detection and tracking. In order to evaluate the accuracy of object tracking, we use a side camera (Point Grey CM3-U3-50S5M-CS) to capture images of the target object and track the object by image analysis. The camera operates at its highest frame rate 70 frames per second. The exposure time is set to be 0.2 ms.
We generate 6 binarized Fourier basis patterns (Fig. 2) for structured illumination. The size of the patterns is 1920×1080 pixels. The spatial frequency pair of ${P_1}\sim {P_3}$ is $({{f_x} = {2 \mathord{\left/ {\vphantom {2 {1\textrm{080}}}} \right.} {1\textrm{080}}},{f_y} = 0} )$ and the spatial frequency pair of ${P_4}\sim {P_6}$ is $({{f_x} = 0,{f_y} = {2 \mathord{\left/ {\vphantom {2 {\textrm{1920}}}} \right.} {\textrm{1920}}}} )$. The patterns are binarized using the Floyd-Steinberg dithering algorithm with the upsampling ratio $k = 1$ [20]. The patterns are repeatedly displayed on the DMD in sequence.
Fig. 2. Fourier basis patterns. (a)-(c) are used for the acquisition of $\tilde{I}({{f_x},0} )$ and (d)-(f) for $\tilde{I}({0,{f_y}} )$.
As the sampling rate of the utilized data acquisition board is 50 folds of the DMD refreshing rate, we therefore take 50 samples for each illumination pattern. Each single-pixel measurement ${D_i}$ is referred to the average of 50 samples. For each frame, 6 patterns and the corresponding 6 single-pixel measurements are required. Thus, each frame consumes 600 bytes data (16 bits (2 bytes)/sample × 50 samples/measurement × 6 measurements/frame). The data throughput of the proposed method in our experimental configuration is 1,000,000 bytes (0.95 mega bytes) per second, which is much less than that in high-speed photography systems. As the DMD switch 10,000 patterns per second and each frame requires 6 patterns, the temporal resolution we achieve is 600 µs. Figure 3 shows the acquired $\bar{D}$ of 600 frames for the two tracks.
Fig. 3. Single-pixel measurements $\bar{D}$: (a) for h-shaped track and (b) for s-shaped track.
As Fig. 3 shows, the single-pixel measurements are stationary before the target moving object enters the scene. As soon as the moving object enters, the single-pixel measurements drop rapidly as the ball block a portion of the light. The measurement varies continuously during the objects moves inside the scene. The measurements become stationary after the object leaves the scene.
In our experiments, the threshold used in the criterion for object detection is $\mu \pm ({{1 \mathord{\left/ {\vphantom {1 {40}}} \right.} {40}}} )\sigma $, where $\mu $ and $\sigma $ are the average and the standard deviation of $\bar{D}$ s for the latest 5 frames (current frame excluded), respectively. According to the criterion, we conclude that for the h-shaped track the hollow ball enters the scene at the 142nd frame and exits the scene at the 422nd frame. For the s-shaped track, the hollow ball enters at the 183rd frame and exits at 400th frame. The tracking results are shown in Fig. 4. As the figure shows, the results by the proposed image-free method coincide with the results obtained by high-speed photography. The lengths of h-shaped track and the s-shaped track are 0.135 m, and 0.158 m, respectively. The corresponding average moving speeds are 0.801 m/s and 1.214 m/s, respectively. Such speeds are high, because the ball crosses the scene within less than 0.2 seconds (also watch Visualization 1 and Visualization 2). Due to limited frame rate, the camera only captures 11 images for the h-shaped track and 9 images for the s-shaped track, respectively. With quite a few images captured by the camera, the path of the hollow ball is difficult to recover, as the positioning points are sparse in the space (also watch Visualization 3). On the contrary, the track results by the proposed method can well recover the path of the hollow ball. Particularly, the density of the positioning points varies with the speed of the ball accordingly. The positioning points are dense where the ball moves slowly (for instance, at the corners) and vice versa.
Fig. 4. Tracking results for (a) h-shaped and (b) s-shaped track, respectively. Red circles are results by proposed method and blue solid dots are the results by high-speed photography using the side camera.
The lowest detectable moving speed is determined by the pixel size of illumination pattern at the image plane, because the motion cannot be detected if the displacement of the object is less than one pixel size at the image plane within one frame interval. The highest detectable moving speed is determined by spatial frequency of the utilized Fourier basis patterns. Due to the periodicity of the Fourier basis patterns, it will cause 2-pi phase ambiguity if the displacement of object within one frame interval is larger than a fringe period. Thus, the higher spatial frequency of the used Fourier basis patterns is, the lower highest detectable speed will be.
In our experiment, the data acquisition time is limited by the utilized spatial light modulator. It is reasonable that one can use a high-speed spatial light modulator (for example, high-speed LED matrix [24,25]) so as to reduce the data acquisition time and achieve a higher frame rate.
The computational complexity of the proposed method is low. Averagely, the computational time is 67.3 µs for each frame (evaluated on a computer with an Intel 8700K 3.7 GHz CPU, 24GB RAM, and MATLAB 2014a). In other words, the latency time is on the order of microseconds. As the computational time is shorter than the data acquisition time, object detection and tracking can be conducted on-the-fly and data accumulation can be avoided, which allows for long-duration object detection and tracking.
The proposed method is potentially a cost-effective solution to real-time object detection and tracking. As only 6 binary patterns are needed, one can use a low-cost programmable projector, instead of the DMD development kit used in our experiment, for spatial light modulation.
We acknowledge that the proposed method at the current stage has two limitations. The one is that it can only detect and track only one moving object at a time. The other is the proposed method can only achieve 2-D tracking. Extending the method to multiple objects detection and 3-D tracking is our future work.
We propose an image-free and cost-effective method for object detection and tracking. The proposed method achieves a temporal resolution of 1/1666 seconds by using a 10,000 Hz DMD. The computationally efficient algorithm of the proposed method enables low latency time. Thus, the method is suitable for high-speed moving object detection and tracking in real time and for long duration. Given the advantages of wide spectral response by single-pixel detector, the proposed method potentially works at invisible wavebands, allowing hidden moving object detection and tracking. This image-free method also opens up a new avenue for spatial information acquisition in a highly efficient manner.
Fundamental Research Funds for the Central Universities (11618307); National Natural Science Foundation of China (61875074, 61905098).
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21. Z. Zhang, S. Liu, J. Peng, M. Yao, G. Zheng, and J. Zhong, "Simultaneous spatial, spectral, and 3D compressive imaging via efficient Fourier single-pixel measurements," Optica 5(3), 315–319 (2018). [CrossRef]
22. R. She, W. Liu, Y. Lu, Z. Zhou, and G. Li, "Fourier single-pixel imaging in the terahertz regime," Appl. Phys. Lett. 115(2), 021101 (2019). [CrossRef]
23. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, "Hadamard single-pixel imaging versus Fourier single-pixel imaging," Opt. Express 25(16), 19619–19639 (2017). [CrossRef]
24. Z. H. Xu, W. Chen, J. Penuelas, M. Padgett, and M. J. Sun, "1000 fps computational ghost imaging using LED-based structured illumination," Opt. Express 26(3), 2427–2434 (2018). [CrossRef]
25. E. Balaguer, P. Carmona, C. Chabert, F. Pla, J. Lancis, and E. Tajahuerce, "Low-cost single-pixel 3D imaging by using an LED array," Opt. Express 26(12), 15623–15631 (2018). [CrossRef]
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P. Li, D. Wang, L. Wang, and H. Lu, "Deep visual tracking: Review and experimental comparison," Pattern Recogn. 76, 323–338 (2018).
H. Yang, L. Shao, F. Zheng, L. Wang, and Z. Song, "Recent advances and trends in visual tracking: A review," Neurocomputing 74(18), 3823–3831 (2011).
P. Bahl, V. N. Padmanabhan, V. Bahl, and V. Padmanabhan, "RADAR: An in-building RF-based user location and tracking system," in Proceedings IEEE INFOCOM 2000 (IEEE, 2000), pp. 775.
A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, "Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging," Nat. Commun. 3(1), 745 (2012).
U. Wandinger, "Introduction to lidar," in Lidar (Springer, New York, 2005).
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G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, "Detection and tracking of moving objects hidden from view," Nat. Photonics 10(1), 23–26 (2016).
Y. Fang, M. Ichiro, and H. Berthold, "Depth-based target segmentation for intelligent vehicles: Fusion of radar and binocular stereo," IEEE Trans. Intell. Transport. Syst. 3(3), 196–202 (2002).
M. S. Wei, F. Xing, and Z. You, "A real-time detection and positioning method for small and weak targets using a 1D morphology-based approach in 2D images," Light: Sci. Appl. 7(5), 18006 (2018).
D. Comaniciu, V. Ramesh, and P. Meer, "Kernel-based object tracking," IEEE Trans. Pattern Anal. Machine Intell. 25(5), 564–577 (2003).
W. Zhong, H. Lu, and M. H. Yang, "Robust object tracking via sparsity-based collaborative model," IEEE Conference on Computer vision and pattern recognition (IEEE, 2015), pp. 1838–1845.
D. Shi, K. Yin, J. Huang, K. Yuan, W. Zhu, C. Xie, D. Liu, and Y. Wang, "Fast tracking of moving objects using single-pixel imaging," Opt. Commun. 440, 155–162 (2019).
M. P. Edgar, G. M. Gibson, and M. J. Padgett, "Principles and prospects for single-pixel imaging," Nat. Photonics 13(1), 13–20 (2019).
M. Sun and J. Zhang, "Single-pixel imaging and its application in three-dimensional reconstruction: a brief review," Sensors 19(3), 732 (2019).
Z. Zhang, X. Ma, and J. Zhong, "Single-pixel imaging by means of Fourier spectrum acquisition," Nat. Commun. 6(1), 6225 (2015).
Z. Zhang and J. Zhong, "Three-dimensional single-pixel imaging with far fewer measurements than effective image pixels," Opt. Lett. 41(11), 2497–2500 (2016).
Z. Zhang, X. Wang, G. Zheng, and J. Zhong, "Fast Fourier single-pixel imaging via binary illumination," Sci. Rep. 7(1), 12029 (2017).
Z. Zhang, S. Liu, J. Peng, M. Yao, G. Zheng, and J. Zhong, "Simultaneous spatial, spectral, and 3D compressive imaging via efficient Fourier single-pixel measurements," Optica 5(3), 315–319 (2018).
R. She, W. Liu, Y. Lu, Z. Zhou, and G. Li, "Fourier single-pixel imaging in the terahertz regime," Appl. Phys. Lett. 115(2), 021101 (2019).
Z. Zhang, X. Wang, G. Zheng, and J. Zhong, "Hadamard single-pixel imaging versus Fourier single-pixel imaging," Opt. Express 25(16), 19619–19639 (2017).
Z. H. Xu, W. Chen, J. Penuelas, M. Padgett, and M. J. Sun, "1000 fps computational ghost imaging using LED-based structured illumination," Opt. Express 26(3), 2427–2434 (2018).
E. Balaguer, P. Carmona, C. Chabert, F. Pla, J. Lancis, and E. Tajahuerce, "Low-cost single-pixel 3D imaging by using an LED array," Opt. Express 26(12), 15623–15631 (2018).
Albrecht, I.
Bahl, P.
Bahl, V.
Balaguer, E.
Bawendi, M. G.
Berthold, H.
Carmona, P.
Chabert, C.
Chen, W.
Comaniciu, D.
Edgar, M. P.
Faccio, D.
Fang, Y.
Fernald, F. G.
Gariepy, G.
Gibson, G. M.
Gupta, O.
Haber, J.
Hashimoto, K.
Hashimoto, M.
Henderson, R.
Huang, J.
Ichiro, M.
Ishikawa, M.
Lancis, J.
Leach, J.
Li, G.
Li, P.
Liu, D.
Liu, S.
Liu, W.
Lu, H.
Lu, Y.
Ma, X.
Magnor, M.
Meer, P.
Ogawa, N.
Ogawa, T.
Oku, H.
Padgett, M.
Padgett, M. J.
Padmanabhan, V.
Padmanabhan, V. N.
Peng, J.
Penuelas, J.
Pla, F.
Ramesh, V.
Raskar, R.
Sato, S.
Seidel, H. P.
Shao, L.
She, R.
Shi, D.
Song, Z.
Sun, M.
Sun, M. J.
Tajahuerce, E.
Takagi, K.
Takita, M.
Theobalt, C.
Tonolini, F.
Veeraraghavan, A.
Velten, A.
Wandinger, U.
Wang, D.
Wang, L.
Wei, M. S.
Willwacher, T.
Xie, C.
Xing, F.
Xu, Z. H.
Yang, H.
Yang, M. H.
Yao, M.
Yin, K.
You, Z.
Yuan, K.
Zhang, J.
Zhang, Z.
Zheng, F.
Zheng, G.
Zhong, J.
Zhong, W.
Zhou, Z.
Zhu, W.
ACM Trans. Graph. (1)
Appl. Opt. (1)
Appl. Phys. Lett. (1)
IEEE Trans. Intell. Transport. Syst. (1)
IEEE Trans. Pattern Anal. Machine Intell. (1)
IEEE Trans. Robot. (1)
Light: Sci. Appl. (1)
Nat. Commun. (2)
Nat. Photonics (2)
Neurocomputing (1)
Opt. Commun. (1)
Opt. Lett. (1)
Pattern Recogn. (1)
Sci. Rep. (1)
Supplementary Material (3)
» Visualization 1 Raw images captured by the side camera for the 'h-shaped track'.
» Visualization 2 Raw images captured by the side camera for the 's-shaped track'.
» Visualization 3 Tracking results comparison between the proposed image-free method and the image-based method.
Equations on this page are rendered with MathJax. Learn more.
(1) P(x,y|fx,fy,φ0)=A+Bcos[2π(fxx+fyy)+φ0],
(2) D=⟨I(x,y),P(x,y)⟩=∫∫I(x,y){A+Bcos[2π(fxx+fyy)+φ0]}dxdy.
(3) I(x−x0,y−y0)=F−1{I~(fx,fy)exp[−j2π(fxx0+fyy0)]}
(4) I~=(2D0−D2π/2π33−D4π/4π33)+3j(D2π/2π33−D4π/4π33),
(5) x0=−12πfx⋅arg{[I~(fx,0)−I~bg(fx,0)]},
(6) y0=−12πfy⋅arg{[I~(0,fy)−I~bg(0,fy)]}, | CommonCrawl |
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
"Kodaira map" redirects here. Not to be confused with Kodaira–Spencer map from cohomology theory.
These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space (X, OX).[1]
Linear system of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively.
A map determined by a linear system is sometimes called the Kodaira map.
Definitions
Given a general variety $X$, two divisors $D,E\in {\text{Div}}(X)$ are linearly equivalent if
$E=D+(f)\ $
for some non-zero rational function $f$ on $X$, or in other words a non-zero element $f$ of the function field $k(X)$. Here $(f)$ denotes the divisor of zeroes and poles of the function $f$.
Note that if $X$ has singular points, the notion of 'divisor' is inherently ambiguous (Cartier divisors, Weil divisors: see divisor (algebraic geometry)). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below.
A complete linear system on $X$ is defined as the set of all effective divisors linearly equivalent to some given divisor $D\in {\text{Div}}(X)$. It is denoted $|D|$. Let ${\mathcal {L}}$ be the line bundle associated to $D$. In the case that $X$ is a nonsingular projective variety, the set $|D|$ is in natural bijection with $(\Gamma (X,{\mathcal {L}})\smallsetminus \{0\})/k^{\ast },$[2] by associating the element $E=D+(f)$ of $|D|$ to the set of non-zero multiples of $f$ (this is well defined since two non-zero rational functions have the same divisor if and only if they are non-zero multiples of each other). A complete linear system $|D|$ is therefore a projective space.
A linear system ${\mathfrak {d}}$ is then a projective subspace of a complete linear system, so it corresponds to a vector subspace W of $\Gamma (X,{\mathcal {L}}).$ The dimension of the linear system ${\mathfrak {d}}$ is its dimension as a projective space. Hence $\dim {\mathfrak {d}}=\dim W-1$.
Linear systems can also be introduced by means of the line bundle or invertible sheaf language. In those terms, divisors $D$ (Cartier divisors, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.
Examples
Linear equivalence
Consider the line bundle ${\mathcal {O}}(2)$ on $\mathbb {P} ^{3}$ whose sections $s\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))$ define quadric surfaces. For the associated divisor $D_{s}=Z(s)$, it is linearly equivalent to any other divisor defined by the vanishing locus of some $t\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))$ using the rational function $\left(t/s\right)$[2] (Proposition 7.2). For example, the divisor $D$ associated to the vanishing locus of $x^{2}+y^{2}+z^{2}+w^{2}$ is linearly equivalent to the divisor $E$ associated to the vanishing locus of $xy$. Then, there is the equivalence of divisors
$D=E+\left({\frac {x^{2}+y^{2}+z^{2}+w^{2}}{xy}}\right)$
Linear systems on curves
One of the important complete linear systems on an algebraic curve $C$ of genus $g$ is given by the complete linear system associated with the canonical divisor $K$, denoted $|K|=\mathbb {P} (H^{0}(C,\omega _{C}))$. This definition follows from proposition II.7.7 of Hartshorne[2] since every effective divisor in the linear system comes from the zeros of some section of $\omega _{C}$.
Hyperelliptic curves
One application of linear systems is used in the classification of algebraic curves. A hyperelliptic curve is a curve $C$ with a degree $2$ morphism $f:C\to \mathbb {P} ^{1}$.[2] For the case $g=2$ all curves are hyperelliptic: the Riemann–Roch theorem then gives the degree of $K_{C}$ is $2g-2=2$ and $h^{0}(K_{C})=2$, hence there is a degree $2$ map to $\mathbb {P} ^{1}=\mathbb {P} (H^{0}(C,\omega _{C}))$.
grd
A $g_{r}^{d}$ is a linear system ${\mathfrak {d}}$ on a curve $C$ which is of degree $d$ and dimension $r$. For example, hyperelliptic curves have a $g_{2}^{1}$ since $|K_{C}|$ defines one. In fact, hyperelliptic curves have a unique $g_{2}^{1}$[2] from proposition 5.3. Another close set of examples are curves with a $g_{1}^{3}$ which are called trigonal curves. In fact, any curve has a $g_{1}^{d}$ for $d\geq (1/2)g+1$.[3]
Linear systems of hypersurfaces in a projective space
Consider the line bundle ${\mathcal {O}}(d)$ over $\mathbb {P} ^{n}$. If we take global sections $V=\Gamma ({\mathcal {O}}(d))$, then we can take its projectivization $\mathbb {P} (V)$. This is isomorphic to $\mathbb {P} ^{N}$ where
$N={\binom {n+d}{n}}-1$
Then, using any embedding $\mathbb {P} ^{k}\to \mathbb {P} ^{N}$ we can construct a linear system of dimension $k$.
Linear system of conics
Main article: Linear system of conics
Characteristic linear system of a family of curves
The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C.[4]
In modern terms, it is a subsystem of the linear system associated to the normal bundle to $C\hookrightarrow Y$. Note a characteristic system need not to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the Kodaira–Spencer theory can be used to answer the question of the completeness.
Other examples
The Cayley–Bacharach theorem is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.
Linear systems in birational geometry
In general linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions — the Riemann–Roch problem as it can be called — can be better phrased in terms of homological algebra. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible sheaves.
The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré's characteristic linear system of an algebraic family of curves on an algebraic surface.
Base locus
The base locus of a linear system of divisors on a variety refers to the subvariety of points 'common' to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines $x=a$ has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.
More precisely, suppose that $|D|$ is a complete linear system of divisors on some variety $X$. Consider the intersection
$\operatorname {Bl} (|D|):=\bigcap _{D_{\text{eff}}\in |D|}\operatorname {Supp} D_{\text{eff}}\ $
where $\operatorname {Supp} $ denotes the support of a divisor, and the intersection is taken over all effective divisors $D_{\text{eff}}$ in the linear system. This is the base locus of $|D|$ (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of $\operatorname {Bl} $ should be).
One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose $|D|$ is such a class on a variety $X$, and $C$ an irreducible curve on $X$. If $C$ is not contained in the base locus of $|D|$, then there exists some divisor ${\tilde {D}}$ in the class which does not contain $C$, and so intersects it properly. Basic facts from intersection theory then tell us that we must have $|D|\cdot C\geq 0$. The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef.
In the modern formulation of algebraic geometry, a complete linear system $|D|$ of (Cartier) divisors on a variety $X$ is viewed as a line bundle ${\mathcal {O}}(D)$ on $X$. From this viewpoint, the base locus $\operatorname {Bl} (|D|)$ is the set of common zeroes of all sections of ${\mathcal {O}}(D)$. A simple consequence is that the bundle is globally generated if and only if the base locus is empty.
The notion of the base locus still makes sense for a non-complete linear system as well: the base locus of it is still the intersection of the supports of all the effective divisors in the system.
See also: Theorem of Bertini
Example
Consider the Lefschetz pencil $p:{\mathfrak {X}}\to \mathbb {P} ^{1}$ given by two generic sections $f,g\in \Gamma (\mathbb {P} ^{n},{\mathcal {O}}(d))$, so ${\mathfrak {X}}$ given by the scheme
${\mathfrak {X}}={\text{Proj}}\left({\frac {k[s,t][x_{0},\ldots ,x_{n}]}{(sf+tg)}}\right)$
This has an associated linear system of divisors since each polynomial, $s_{0}f+t_{0}g$ for a fixed $[s_{0}:t_{0}]\in \mathbb {P} ^{1}$ is a divisor in $\mathbb {P} ^{n}$. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of $f,g$, so
${\text{Bl}}({\mathfrak {X}})={\text{Proj}}\left({\frac {k[s,t][x_{0},\ldots ,x_{n}]}{(f,g)}}\right)$
A map determined by a linear system
Each linear system on an algebraic variety determines a morphism from the complement of the base locus to a projective space of dimension of the system, as follows. (In a sense, the converse is also true; see the section below)
Let L be a line bundle on an algebraic variety X and $V\subset \Gamma (X,L)$ a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when V is base-point-free; in other words, the natural map $V\otimes _{k}{\mathcal {O}}_{X}\to L$ is surjective (here, k = the base field). Or equivalently, $\operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {O}}_{X}$ is surjective. Hence, writing $V_{X}=V\times X$ for the trivial vector bundle and passing the surjection to the relative Proj, there is a closed immersion:
$i:X\hookrightarrow \mathbb {P} (V_{X}^{*}\otimes L)\simeq \mathbb {P} (V_{X}^{*})=\mathbb {P} (V^{*})\times X$
where $\simeq $ on the right is the invariance of the projective bundle under a twist by a line bundle. Following i by a projection, there results in the map:[5]
$f:X\to \mathbb {P} (V^{*}).$
When the base locus of V is not empty, the above discussion still goes through with ${\mathcal {O}}_{X}$ in the direct sum replaced by an ideal sheaf defining the base locus and X replaced by the blow-up ${\widetilde {X}}$ of it along the (scheme-theoretic) base locus B. Precisely, as above, there is a surjection $\operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n}$ where ${\mathcal {I}}$ is the ideal sheaf of B and that gives rise to
$i:{\widetilde {X}}\hookrightarrow \mathbb {P} (V^{*})\times X.$
Since $X-B\simeq $ an open subset of ${\widetilde {X}}$, there results in the map:
$f:X-B\to \mathbb {P} (V^{*}).$
Finally, when a basis of V is chosen, the above discussion becomes more down-to-earth (and that is the style used in Hartshorne, Algebraic Geometry).
Linear system determined by a map to a projective space
Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably.
For a closed immersion $f:Y\hookrightarrow X$ of algebraic varieties there is a pullback of a linear system ${\mathfrak {d}}$ on $X$ to $Y$, defined as $f^{-1}({\mathfrak {d}})=\{f^{-1}(D)|D\in {\mathfrak {d}}\}$[2] (page 158).
O(1) on a projective variety
A projective variety $X$ embedded in $\mathbb {P} ^{r}$ has a natural linear system determining a map to projective space from ${\mathcal {O}}_{X}(1)={\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{\mathbb {P} ^{r}}}{\mathcal {O}}_{\mathbb {P} ^{r}}(1)$. This sends a point $x\in X$ to its corresponding point $[x_{0}:\cdots :x_{r}]\in \mathbb {P} ^{r}$.
See also
• Brill–Noether theory
• Lefschetz pencil
• bundle of principal parts
References
1. Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.
2. Hartshorne, R. 'Algebraic Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342
3. Kleiman, Steven L.; Laksov, Dan (1974). "Another proof of the existence of special divisors". Acta Mathematica. 132: 163–176. doi:10.1007/BF02392112. ISSN 0001-5962.
4. Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip (2011). Geometry of algebraic curves. Grundlehren der Mathematischen Wissenschaften. Vol. II, with a contribution by Joseph Daniel Harris. Heidelberg: Springer. p. 3. doi:10.1007/978-1-4757-5323-3. MR 2807457.
5. Fulton, § 4.4. harvnb error: no target: CITEREFFulton (help)
• P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 137. ISBN 0-471-05059-8.
• Hartshorne, R. Algebraic Geometry, Springer-Verlag, 1977; corrected 6th printing, 1993. ISBN 0-387-90244-9.
• Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. ISBN 3-540-22533-1.
| Wikipedia |
Central angle
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians).[1] The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.
The size of a central angle Θ is 0° < Θ < 360° or 0 < Θ < 2π (radians). When defining or drawing a central angle, in addition to specifying the points A and B, one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise.
Formulas
If the intersection points A and B of the legs of the angle with the circle form a diameter, then Θ = 180° is a straight angle. (In radians, Θ = π.)
Let L be the minor arc of the circle between points A and B, and let R be the radius of the circle.[2]
If the central angle Θ is subtended by L, then
$0^{\circ }<\Theta <180^{\circ }\,,\,\,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }={\frac {L}{R}}.$
Proof (for degrees)
The circumference of a circle with radius R is 2πR, and the minor arc L is the (Θ/360°) proportional part of the whole circumference (see arc). So:
$L={\frac {\Theta }{360^{\circ }}}\cdot 2\pi R\,\Rightarrow \,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }.$
Proof (for radians)
The circumference of a circle with radius R is 2πR, and the minor arc L is the (Θ/2π) proportional part of the whole circumference (see arc). So
$L={\frac {\Theta }{2\pi }}\cdot 2\pi R\,\Rightarrow \,\Theta ={\frac {L}{R}}.$
If the central angle Θ is not subtended by the minor arc L, then Θ is a reflex angle and
$180^{\circ }<\Theta <360^{\circ }\,,\,\,\Theta =\left(360-{\frac {180L}{\pi R}}\right)^{\circ }=2\pi -{\frac {L}{R}}.$
If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the center as O, the angles ∠BOA (convex) and ∠BPA are supplementary (sum to 180°).
Central angle of a regular polygon
A regular polygon with n sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is $2\pi /n.$
See also
• Inscribed angle
• Great-circle navigation
References
1. Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Central Angle" (PDF). Addison-Wesley. p. 122. Retrieved December 30, 2013.
2. "Central angle (of a circle)". Math Open Reference. 2009. Retrieved December 30, 2013. interactive
External links
• "Central angle (of a circle)". Math Open Reference. 2009. Retrieved December 30, 2013. interactive
• "Central Angle Theorem". Math Open Reference. 2009. Retrieved December 30, 2013. interactive
• Inscribed and Central Angles in a Circle
| Wikipedia |
GT Graphes et Optimisation
Vendredi 24 juin à 14h - Nathanaël Fijalkow
Universal graphs for playing games
Connexion: https://webconf.u-bordeaux.fr/b/mar-ef4-zed
I will introduce the notion of universal graphs for playing games: they are similar in spirit to the usual notion, but parameterised by an additional condition, such as (for weighted graphs) "all cycles have non negative total weights". I will briefly show what are their applications (algorithms for model checking games), and then focus on the combinatorial questions: understanding the size of minimal universal graphs.
This talk will survey a number of more or less recent results jointly obtained with Thomas Colcombet, Pierre Ohlmann, and Paweł Gawrychowski.
Vendredi 17 juin à 14h - František Kardoš
Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching
Let G be a bridgeless cubic graph. The Berge-Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan-Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that G admits two perfect matchings whose deletion yields a bipartite subgraph of G. It can be shown that given an arbitrary perfect matching of G, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan-Raspaud and the Berge-Fulkerson conjectures, respectively. In this talk, we show that given any 1+-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G, there always exists a perfect matching M of G containing e such that G∖(F∪M) is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in G, there exists a perfect matching of G containing at least one edge of each circuit in this collection.
This is a joint work with Edita Máčajová and Jean Paul Zerafa.
Vendredi 03 juin à 14h - Alp Muyesser
Rainbow matchings in groups
A rainbow matching in an edge-coloured graph is a matching whose edges all have different colours. Let G be a group of order n and consider an edge-coloured complete bipartite graph, whose parts are each a copy of the group G, and the edge (x, y) gets coloured by the group element xy. We call this graph the multiplication table of G. For which groups G does the multiplication table of G have a rainbow matching? This is an old question in combinatorial group theory due to Hall and Paige, with close connections to the study of Latin squares. The problem has been resolved in 2009 with a proof relying on the classification of finite simple groups. In 2021, a "simpler" proof for large groups appeared, this time using tools from analytic number theory. We present a third resolution of this problem, again only for large groups, and using techniques from probabilistic combinatorics. The main advantage of our approach is that we are able to find rainbow matchings in random subgraphs of the multiplication table of G. This flexibility allows us to settle numerous longstanding conjectures in this area. For example, Evans asked for a characterisation of groups whose elements can be ordered so that the product of each consecutive pair of elements is distinct. Using our results, we are able to answer this question for large groups. In this talk, we will give a gentle survey of this area. This is joint work with Alexey Pokrovskiy.
Vendredi 20 mai à 14h - Alexandra Wesolek
On asymptotic packing of geometric graphs
Only offline, no online version
A geometric graph G is a graph drawn in the Euclidean plane such that its vertices are points in general position and its edges are drawn as straight line segments. Given a complete geometric graph H_n on n vertices, we are interested in finding a large collection of plane copies of a graph G in H_n such that each edge of H_n appears in at most one copy of G. We say a graph G is geometric-packable if for every sequence of geometric complete graphs $(H_n)_{n \geq 1}$ all but o(n^2) edges of H_n can be packed by plane copies of G. In a joint work with Daniel W. Cranston, Jiaxi Nie and Jacques Verstraete, we study geometric-packability and show that if G is a triangle, plane 4-cycle or plane 4-cycle with a chord, the set of plane drawings of G is geometric-packable. In contrast, the analogous statement is false when G is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions).
Vendredi 13 mai à 11h - Frederik Garbe
Limits of Latin squares
We introduce a limit theory for Latin squares, paralleling the recent limit theories of dense graphs and permutations. We define a notion of density, an appropriate version of the cut distance, and a space of limit objects – so-called Latinons. Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence. Last, using Keevash's recent results on combinatorial designs, we prove that each Latinon can be approximated by a finite Latin square.
This is joint work with Robert Hancock, Jan Hladký and Maryam Sharifzadeh.
Vendredi 6 mai à 14h - Quentin Deschamps
Metric Dimension on sparse graph
The Metric Dimension problem consits in identifying vertices in a graph. The Metric Dimension of a graph G is the minimum cardinality of a subset S of vertices of G such that each vertex of G is uniquely determined by its distances to S. In a general case, add a vertex to a graph can drastically change its metric dimension, we prove we can bound this gap when the initial graph is the tree and one add several edges. To prove this result, we built a valid (but not minimal) set S which size can be bounded efficiently if the graphe is sparse enough.
Joint work with Nicolas Bousquet, Aline Parreau and Ignacio Pelayo
Vendredi 29 avril à 14h - Amedeo Sgueglia
Multistage Maker-Breaker Games
We consider a new procedure, which we call Multistage Maker-Breaker Game. Maker and Breaker start from $G_0:=K_n$ and play several stages of a usual Maker-Breaker game where, for $i \ge 1$, the $i$-th stage is played as follows. They claim edges of $G_{i-1}$ until all edges are distributed, and then they set $G_i$ to be the graph consisting only of Maker's edges. They will then play the next stage on $G_i$.
This creates a sequence of graphs $G_0 \supset G_1 \supset G_2 \supset \dots$ and, given a monotone graph property, the question is how long Maker can maintain it, i.e. what is the largest $k$ such that Maker has a strategy to guarantee that $G_k$ satisfies such property. We will answer this question for several graph properties and pose a number of interesting questions that remain open.
This is joint work with Juri Barkey, Dennis Clemens, Fabian Hamann, and Mirjana Mikalački.
Vendredi 22 avril à 14h - Tassio Naia Dos Santos
Seymour's second neighbourhood conjecture, for almost every oriented graph.
A famous conjecture of Seymour, known as Second Neighborhood Conjecture (SNC), says that every orientation of a graph contains a vertex whose second neighborhood is as large as its first neighborhood. I will present some recent results about the conjecture in the context of random graphs with either typical or arbitrary orientations.
Vendredi 8 avril à 14h - Jozef Skokan
Monochromatic partitions of graphs and hypergraphs.
As a variant on the traditional Ramsey-type questions, there has been a lot of research about the existence of spanning monochromatic subgraphs in complete edge-coloured graphs and hypergraphs. One of the central questions in this area was proposed by Lehel around 1979, who conjectured that the vertex set of every 2-edge-coloured complete graph can be partitioned into two monochromatic cycles of distinct colours. This was answered in the affirmative by Bessy and Thomassé in 2010. Similar partitioning problems have been considered for more colours and for other graphs and hypergraphs.
In this talk we will review some of these problems and offer some results, proofs, and open questions.
Vendredi 1 avril à 14h - Laurent Feuilloley
A tour of local certification and its connection to other fields.
A local certification is basically a labeling of a graph G, that can convince the vertices that G has some property (eg "G is planar"). It originates from the study of fault-tolerance in distributed computing, but it is also an interesting object from a graph theory perspective. For example, the minimum amount of information that is required to certify a graph class can be seen as an (inverse) measure of the locality of this class.
In this talk I will introduce the notion, and give a tour of some recent techniques and questions, highlighting the relations to other fields such as communication complexity, (simple) graph decompositions, regular automata, graph colorings etc.
Vendredi 25 mars à 14h - Daniel Cranston
Kempe Equivalent List Colorings
An $\alpha,\beta$-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors $\alpha$ and $\beta$. Two $k$-colorings of a graph are $k$-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than $k$ colors). Las Vergnas and Meyniel showed that if a graph is $(k-1)$-degenerate, then each pair of its $k$-colorings are $k$-Kempe equivalent. Mohar conjectured the same conclusion for connected $k$-regular graphs. This was proved for $k=3$ by Feghali, Johnson, and Paulusma (with a single exception $K_2\dbox K_3$, also called the 3-prism) and for $k\ge 4$ by Bonamy, Bousquet, Feghali, and Johnson.
In this paper we prove an analogous result for list-coloring. For a list-assignment $L$ and an $L$-coloring $\vph$, a Kempe swap is called $L$-valid for $\vph$ if performing the Kempe swap yields another $L$-coloring. Two $L$-colorings are called $L$-equivalent if we can form one from the other by a sequence of $L$-valid Kempe swaps. Let $G$ be a connected $k$-regular graph with $k\ge 3$. We prove that if $L$ is a $k$-assignment, then all $L$-colorings are $L$-equivalent (again excluding only $K_2\box K_3$). When $k\ge 4$, the proof is completely self-contained, implying an alternate proof of the result of Bonamy et al.
This is joint work with Reem Mahmoud.
Vendredi 18 mars à 14h - Colin Geniet
Twin-Width of groups and graphs of bounded degree
Twin-Width is a graph invariant introduced by Bonnet, Kim, Thomassé, and Watrigant, with applications in logic, FPT algorithms, etc. Although twin-width is designed for dense graphs, we study it in the context of graphs of bounded degree. It is known that the class of graphs of twin-width k is small: it contains n! c^n graphs on vertices 1,...,n for some constant c. This implies that almost all d-regular graphs have twin-width more than k for any fixed k and d≥3. However no explicit constructions of graphs with bounded degree and unbounded twin-width is known.
For infinite graphs of bounded degree, finiteness of twin-width is preserved by quasi-isometries, i.e. functions which preserve distances up to affine upper and lower bounds. This allows to define `finite twin-width' on finitely generated group groups: given a group Γ finitely generated by S, the Cayley graph of Γ has vertices Γ, with an edge x–x·s for all x in Γ, s in S. This graph depends on the choice of S, but all Cayley graphs of Γ are quasi-isometric, hence one can say that Γ has finite twin-width if (all) its Cayley graphs do. Abelian groups, nilpotent groups, groups with polynomial growth have finite twin-width. Using an embedding theorem of Osajda, we construct a finitely generated group with infinite twin-width. This implies the existence of a small class of finite graphs with unbounded twin-width, a question raised in previous work.
This is joint work with Édouard Bonnet, Romain Tessera, Stéphan Thomassé.
Vendredi 25 février à 14h - Claire Hilaire
Long induced paths in minor-closed graph classes and beyond
In this work we show that every graph of pathwidth less than k that has a path of order n also has an induced path of order at least 1/3 n1/ k. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound. This result is then used to prove the two following generalizations:
-every graph of treewidth less than k that has a path of order n contains an induced path of order at least 1/4 (log n)1/k;
- for every non-trivial graph class that is closed under topological minors there is a constant d∈(0, 1) such that every graph from this class that has a path of order n contains an induced path of order at least (log n)d.
We also describe consequences of these results beyond graph classes that are closed under topological minors.
This is joint work with Jean-Florent Raymond (available at https://arxiv.org/pdf/2201.03880).
Vendredi 18 février à 13h30 - Linda Cook
Detecting a long even hole
We call an induced cycle of even length in G an even hole. In 1991, Bienstock showed that it is NP-Hard to test whether a graph G has an even hole containing a specified vertex v in G. In 2002, Conforti, Cornuéjols, Kapoor and Vušković gave a polynomial-time algorithm to test whether a graph contains an even hole by applying a theorem about the structure of even-hole-free graphs from an earlier paper by the same group. In 2003, Chudnovsky, Kawarabayashi and Seymour provided a simpler polynomial time algorithm that searches for even holes directly. We extend this result by presenting a polynomial time algorithm to determine whether a graph has an even hole of length at least k for a given k ≥ 4. (Joint work with Paul Seymour)
Preprint at -- https://arxiv.org/abs/2009.05691
Vendredi 11 février à 14h - Marcin Briański
Separating polynomial χ-boundedness from χ-boundedness and thereabouts
If a graph contains no large complete subgraph but nonetheless has high chromatic number what can we say about the structure of such a graph? This question naturally leads to investigation of χ-bounded classes of graphs — graph classes where a graph needs to contain a large complete subgraph in order to have high chromatic number. This an active subfield of graph theory with many long standing open problems as well as interesting recent developments.
In this talk I will present a construction of a hereditary class of graphs which is χ-bounded but not polynomially χ-bounded. This construction provides a negative answer to a conjecture of Esperet that every χ-bounded hereditary class of graphs is polynomially χ-bounded. The construction is inspired by a recent paper of Carbonero, Hompe, Moore, and Spirkl which provided a counterexample to another conjecture of Esperet.
This is joint work with James Davies and Bartosz Walczak (available at https://arxiv.org/abs/2201.08814).
Vendredi 4 février à 14h - Florian Hörsch
Balancing spanning trees
Given a graph G, a spanning tree of G is a subgraph T of G such that T is a tree and V (T ) = V (G). We investigate the question whether every graph that admits a packing of a certain number of spanning trees also admits such a packing where the spanning trees are balanced meaning that for every vertex v of the graph the degree of v in each of the spanning trees is 'roughly' the same. We first show how to solve this problems for a packing of two spanning trees. More concretely, we show that every graph G that admits a packing of two spanning trees also admits a packing of two spanning trees T1, T2 such that |dT1 (v) − dT2 (v)| ≤ 5 for all v ∈ V (G). We further show that a similar statement also holds for spanning tree packings of arbitrary size, namely that every graph G that contains a packing of k spanning trees for some positive integer k also contains a packing of k spanning trees T1, . . . , Tk such that |dTi (v) − dTj (v)| ≤ ck for all v ∈ V (G) and i, j ∈ {1, . . . , k} where ck is a constant only depending on k. This solves a conjecture of Kriesell.
Vendredi 28 janvier à 14h - Alp Muyesser
Transversals in graph collections
Suppose we have a collection of graphs on a mutual vertex set, say V. A graph G on V is then called rainbow, if G uses at most one edge from each graph in the collection. Aharoni initiated the study of finding sufficient conditions for the existence of rainbow subgraphs in graph collections. Notably, he asked whether n graphs with minimum degree n/2 on a mutual vertex set of size n admit a rainbow Hamilton cycle. This is a far-reaching generalisation of Dirac's theorem, a cornerstone in graph theory.
We will talk about a general strategy for problems of this type that relies on a novel absorption argument. As an application, we will obtain rainbow versions of several classical theorems from extremal graph theory.
This is joint work with Richard Montgomery and Yani Pehova
Vendredi 21 janvier à 14h - Marek Sokołowski
Graphs of bounded twin-width are quasi-polynomially chi-bounded
We prove that for every $t \in \N$ there is a constant $c_t$ such that every graph with twin-width at most $t$ and clique number $\omega$ has chromatic number bounded by $2^{c_t \log^{O(t)} \omega}$. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially $\chi$-bounded. This provides a partial resolution of a conjecture of Bonnet et al. [SODA 2021] that they are polynomially $\chi$-bounded.
This is a joint work with Michal Pilipczuk.
Vendredi 14 janvier à 14h - Samuel Mohr
Uniform Turán density
In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K4(3)- and K4(3). The former question was solved only recently in [Israel J. Math. 211 (2016), 349--366] and [J. Eur. Math. Soc. 97 (2018), 77--97], while the latter still remains open for almost 40 years. In addition to K4(3)-, the only 3-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht~[J. London Math. Soc. 97 (2018), 77--97] and a specific family with uniform Turán density equal to 1/27.
In this talk, we give an introduction to the concept of uniform Turán densities, present a way to obtain lower bounds using color schemes, and give a glimpse of the proof for determining the uniform Turán density of the tight 3-uniform cycle Cℓ(3), ℓ ≥ 5$.
Vendredi 7 janvier à 14h - Tom Davot
Une approche gloutonne pour l'échafaudage du génome
L'échafaudage est un problème en bio-informatique dont le but est de compléter le processus d'assemblage de séquences génomiques (appelées contigs) en determinant leurs positions et orientations relatives. Ce problème peut être vu comme un problème de couverture par des cycles et des chemins d'un graphe particulier appelé "graphe d'échafaudage". Dans cette présentation, nous formulons quelques résultats sur la complexité de ce problème. Nous adaptons également un algorithme glouton, formulé originellement sur les graphes complets, afin qu'il fonctionne sur une classe particulière que nous espérons plus proche des instances réelles. Cet algorithme est le premier algorithme polynomial pour une classe différente des graphes complets.
Vendredi 17 décembre à 14h - Bruno Courcelle
The bounds of a class of graphs or hypergraphs
(Exposé en français. Transparents et questions en anglais)
A bound of a class of finite graphs C is a finite graph not in C whose proper induced subgraphs are all in C. In French «une borne de C», terminology by Fraïssé and Pouzet. The questions are: is Bounds(C) finite? If yes compute it. Which properties of Bounds(C) imply that C has bounded clique-width? I will present the following tools:
1. Graph Theory «pedestrian» arguments possibly using TRAG software (trag.labri.fr) for checking clique-width.
2. Monadic Second-Order descriptions + upper-bounds to clique-width
3. Well-quasi order arguments, give finiteness but no effective list.
- A working example will be the class of *probe cographs*, extending that of cographs and having finitely many bounds.
- Also: MSO tools do not work for ternary hypergraphs, but wqo arguments may work.
A full article is available.
Vendredi 10 décembre à 10h - Dimitri Lajou
PhD defense: On various graph coloring problems.
A29, amphitheater
In this thesis, we study some graph coloring problems. We are interested in two families of colorings. The first one consists in coloring graphs, called signed graphs, modeling social links. These signed graphs dispose of two types of edges: positive edges to represent friendship and negative edges for animosity. Coloring signed graphs is done through the notion of homomorphism: the chromatic number of a signed graph (G, σ) is the smallest order of a signed graph (H, π) to which (G, σ) admits a homomorphism. We study the complexity of homomorphisms of signed graphs when the target graph is fixed and when the input can be modified, giving P/NP-complete dichotomies and FPT/W[1]-hard dichotomies. We also present bounds on the chromatic number of signed graphs when the input graph has few cycles. Finally, we study the relationship between homomorphisms of signed graphs and the Cartesian product of signed graphs. The second family of colorings consists in coloring edges, instead of vertices, according to some constraints. We study four kinds of edge-colorings notions: packing edge-colorings, injective edge-colorings, AVD colorings and 1-2-3-labellings. Packing edge-coloring is a form of proper edge-coloring where each color has its own conflict rule, for example, color 1 may behave according to the rules of proper edge-colorings while color 2 behave according to the rules of strong edge-colorings. We study packing edge-coloring on subcubic graphs and provide bounds on the number of colors necessary to color the graphs. An injective edge-coloring is an edge-coloring where for any path of length 3, the two non-internal edges of the path receive different colors. We determine the complexity of injective edge-coloring for some classes of graphs. For AVD colorings, i.e. a proper edge-coloring where adjacent vertices are incident with different sets of colors, we obtain bounds on the number of colors required to color the graph when the graph has its maximum degree significantly greater than its maximum average degree and when the graph is planar and has maximum degree at least 12. Finally, we prove the Multiplicative 1-2-3 Conjecture, i.e. that every connected graph (which is not just an edge) can be edge-labelled with labels 1, 2 and 3 so that the coloring of G, obtained by associating with each vertex the product of the labels on edges incident with u, is proper.
Vendredi 10 décembre à 14h - Jacob Cooper
Uniform Turan density
In the early 1980s, Erdos and Sos initiated a study of the classical Turan problem with an additional uniformity condition: the uniform Turan density of a k-uniform hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that every linear-size subhypergraph has density at least d contains H. In particular, they raised the question of determining the uniform Turan densities of K_4^(3) (the complete 3-uniform hypergraph on four vertices) and K_4^{(3)-} (the complete 3-uniform hypergraph on four vertices minus an edge). The latter question was solved only recently in [Israel J. Math. 211 (2016), 349--366] and [J. Eur. Math. Soc. 97 (2018), 77--97], whilst the former continues to be open, now for almost 40 years. In addition to K_4^{(3)-}, the only 3-uniform hypergraphs whose uniform Turan density is known are those with uniform Turan density equal to zero, as classified by Reiher, Rodl and Schacht [J. London Math. Soc. 97 (2018), 77--97], and a specific family with uniform Turan density equal to 1/27.
In this talk, we will give a self-contained introduction to the concept of uniform Turan densities, present a way to obtain lower bounds using so-called colour schemes, and give a glimpse of the proof for determining the uniform Turan density of the tight 3-uniform cycle C_\ell^(3), for \ell\geq 5. Based on joint work with Matija Bucic, David Correia, Daniel Kral and Samuel Mohr.
Lundi 13 décembre à 14h - Alexandre Blanché
PhD defense: Décomposition en chemins de Gallai dans les graphes planaires
Labri, amphitheater
Cette thèse s'inscrit dans le domaine de la théorie des graphes, et traite d'une question posée en 1968 par Tibor Gallai, toujours sans réponse aujourd'hui. Gallai conjectura que les arêtes de tout graphe connexe à n sommets pouvaient être partitionnées en ⌈n/2⌉ chemins. Bien que cette conjecture fut attaquée et partiellement résolue au fil des ans, la propriété n'a été prouvée que pour des classes de graphes très spécifiques, comme les graphes dont les sommets de degré pair forment une forêt (Pyber, 1996), les graphes de degré maximum 5 (Bonamy, Perrett, 2016) ou les graphes de largeur arborescente au plus 3 (Botler, Sambinelli, Coelho, Lee, 2017). Les graphes planaires sont les graphes qui peuvent être plongés dans le plan, c'est-à-dire dessinés sans croisements d'arêtes. C'est une classe bien connue dans la théorie des graphes, et largement étudiée. Botler, Jiménez et Sambinelli ont récemment confirmé la conjecture dans le cas des graphes planaires sans triangles. Notre résultat consiste en une preuve de la conjecture sur la classe générale des graphes planaires. Cette classe est notablement plus générale que celles des précédents résultats, et de notre point de vue constitue une importante contribution à l'étude de la conjecture de Gallai. Plus précisément, nous travaillons sur une version plus forte de la conjecture, proposée par Bonamy et Perrett en 2016, et qui énonce que les graphes connexes à n sommets peuvent être décomposés en ⌊n/2⌋ chemins, à l'exception d'une famille de graphes denses. Nous confirmons cette conjecture dans le cas des graphes planaires, en démontrant que tout graphe planaire à n sommets, à l'exception de K3 et de K5 − (K5 moins une arête), peut être décomposé en ⌊n/2⌋ chemins. La preuve est divisée en trois parties : les deux premières montrent le lemme principal de la preuve, qui restreint la structure d'un contre-exemple hypothétique ayant un minimum de sommets, et la troisième partie utilise ce lemme pour montrer qu'un tel contre-exemple n'existe pas.
Vendredi 26 novembre à 14h - Sanjana Dey & Subhadeep Dev
Sanjana Dey: Discriminating Codes in Geometric Setups. 14h
We study geometric variations of the discriminating code problem. In the discrete version of the problem, a finite set of points P and a finite set of objects S are given in R^d. The objective is to choose a subset S^* \subseteq S of minimum cardinality such that for each point p_i in P the subset S_i^* \subseteq S^* covering p_i, satisfy S_i^*\neq \emptyset, and each pair p_i,p_j in P, i \neq j, satisfies S_i^* \neq S_j^*. In the continuous version of the problem, the solution set S^* can be chosen freely among a (potentially infinite) class of allowed geometric objects.
In the 1-dimensional case, d=1, the points in P are placed on a line L and the objects in S are finite-length line segments aligned with L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in one dimension. Still, for the 1-dimensional discrete version, we design a polynomial-time 2-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length.
We then study the 2-dimensional case, d=2, for axis-parallel unit square objects. We show that the continuous version is NP-complete, and design a polynomial-time approximation algorithm that produces (8+\epsilon)-approximate solutions, using rounding of suitably defined integer linear programming problems.
Subhadeep Dev: The k-Center Problem on Cactus Graphs. 14h55
The weighted k-center problem in graphs is a classical facility location problem where we place k centers on the graph which minimize the maximum weighted distance of a vertex to its nearest center. We study this problem when the underlying graph is a cactus with n vertices and present an O(n log^2 n) time algorithm for the same. This time complexity improves upon the O(n^2) time algorithm by Ben-Moshe et al. [TCS, 2007] which was the previous state of the art. We achieve this improvement by introducing methods to generalize Frederickson's [SODA, 1991] sorted matrix technique to cactus graphs. The existence of a subquadratic al-gorithm for this problem was open for more than a decade.
Lundi 14 au Vendredi 19 - JGA
Vendredi 12 novembre à 14h - Stijn Cambie
Maximising line subgraphs of diameter at most t
We consider an edge version of the famous (and hard) degree-diameter problem, where one is wondering about the maximum size of a graph given maximum degree and diameter of the line graph. This problem originated from 1988 and was proposed by Erdos and Nesetril. It was the inspiration for a series of research papers on variants of this problem. We will start discussing part of the history, which is related with e.g. the (still widely open) strong edge colouring conjecture. In the second part of the talk, we will look again to the initial inspirational question and the ideas behind some progress on this.
Vendredi 5 novembre à 14h - Maria Chudnovsky
Induced subgraphs and logarithmic tree width
Tree decompositions are a powerful tool in structural graph theory; they are traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs has until recently remained out of reach.
Tree decompositions are closely related to the existence of "laminar collections of separations" in a graph, which roughly means that the separations in the collection ``cooperate'' with each other, and the pieces that are obtained when the graph is simultaneously decomposed by all the separations in the collection ``line up'' to form a tree structure. Such collections of separations come up naturally in the context of forbidden minors.
In the case of families where induced subgraphs are excluded, while there are often natural separations, they are usually very far from forming a laminar collection. However, under certain circumstances, these collections of natural separations can be partitioned into a number of laminar collections, and the number of laminar collections needed is logarithmic in the number of vertices of the graph. This in turn allows us to obtain a wide variety of structural and algorithmic results, which we will discuss in this talk.
Vendredi 29 octobre à 14h - Hoang La
Further Extensions of the Grötzsch Theorem
The Grötzsch Theorem states that every triangle-free planar graph admits a proper 3-coloring. Among many of its generalizations, the one of Grünbaum and Aksenov, giving 3-colorability of planar graphs with at most three triangles, is perhaps the most known. A lot of attention was also given to extending 3-colorings of subgraphs to the whole graph. In this paper, we consider 3-colorings of planar graphs with at most one triangle. Particularly, we show that precoloring of any two non-adjacent vertices and precoloring of a face of length at most 4 can be extended to a 3-coloring of the graph. Additionally, we show that for every vertex of degree at most 3, a precoloring of its neighborhood with the same color extends to a 3-coloring of the graph. The latter result implies an affirmative answer to a conjecture on adynamic coloring. All the presented results are tight.
Vendredi 22 octobre à 14h - Bartosz Walczak
Distinguishing classes of intersection graphs of homothets or similarities of two convex disks
For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes Ghom(A) and Gsim(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that Ghom(A)=Ghom(B) if and only if A and B are affine equivalent, and Gsim(A)=Gsim(B) if and only if A and B are similar.
Joint work with Mikkel Abrahamsen.
Vendredi 15 octobre à 14h - Clément Legrand-Duchesne
On a recolouring version of Hadwiger's conjecture
We prove that for any epsilon > 0, for any large enough t, there is a graph G that admits no Kt-minor but admits a (3/2 - epsilon)t-colouring that is "frozen" with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.
Vendredi 8 octobre à 14h - Tassio Naia Dos Santos
Three questions about graphs of large chromatic number
Let G be an arbitrary graph with chromatic number k, where k is large. We discuss the state of the art of the following three open questions.
Erdős and Neumann-Lara (1979): does G admit an orientation with "high" dichromatic number? (The dichromatic number is the smallest size of a vertex partition in which each part induces an acyclic digraph.)
Burr (1980): is it true that every orientation of G contains all oriented trees of order k/2 +1 ?
Bukh (2015 or earlier): Is it true that typical subgraphs of G have chromatic number at least ck/log k for some positive constant c, independent of k ?
Vendredi 1 octobre à 15h - Nicole Wein
Token Swapping on Trees
Connexion: https://u-bordeaux-fr.zoom.us/j/4939941120
In the token swapping problem, we are given a graph with a labeled token on each vertex along with a final configuration of the tokens, and the goal is to find the minimum number of swaps of adjacent tokens to reach the final configuration. Token swapping is in the area of "Reconfiguration Problems" where the goal is generally to get from an initial configuration to a final configuration using a step-by-step series of changes. I will talk about both algorithms and hardness for the token swapping problem, mostly focusing on the case where the underlying graph is a tree.
Vendredi 24 septembre à 14h - Oliver Janzer
Counting H-free orientations of graphs.
In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. We resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős's question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.
Joint work with Matija Bucic and Benny Sudakov.
Vendredi 17 septembre à 14h - Julien Bensmail
A proof of the Multiplicative 1-2-3 Conjecture.
We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kaziów in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1,2,3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.
This is joint work with Hervé Hocquard, Dimitri Lajou and Éric Sopena.
Vendredi 3 septembre à 14h - Alexandra Wesolek
A tight local algorithm for the minimum dominating set problem in outerplanar graphs. / Limiting crossing numbers.
A tight local algorithm for the minimum dominating set problem in outerplanar graphs We present a deterministic local algorithm which computes a 5-approximation of a minimum dominating set on outerplanar graphs, and show that this is optimal. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.
This is joint work with Marthe Bonamy, Linda Cook and Carla Groenland.
Limiting crossing numbers In this talk we will explain a set up which shows that the theory of graph limits introduced by Lovász et al. can be applied to intersection graphs of graph drawings. We consider a model of random, geodesic drawings of the complete bipartite graph Kn,n on the unit sphere and show that the intersection graphs form a convergent series (for n going to infinity).
This talk is based on joint work with Marthe Bonamy and Bojan Mohar.
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2022-2023 (en ligne)
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\begin{document}
\title{\bf On the Second-Order Asymptotics for\\Entanglement-Assisted Communication}
\author{Nilanjana Datta\thanks{Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK} \and Marco Tomamichel\thanks{Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore, and School of Physics, The University of Sydney, Sydney, Australia} \and Mark M. Wilde\thanks{Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA}}
\date{\today}
\maketitle
\begin{abstract} The entanglement-assisted classical capacity of a quantum channel is known to provide the formal quantum generalization of Shannon's classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and thus completely characterize the convergence towards the entanglement-assisted classical capacity when the number of channel uses increases.
Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols. \end{abstract}
\section{Introduction}\label{intro} Let us consider the transmission of classical information through a memoryless quantum channel. If the sender and receiver initially share entangled states which they may use in their communication protocol, then the information transmission is said to be {\em{entanglement-assisted}}. The entanglement-assisted classical capacity $C_{\text{ea}}(\mathcal{N})$ of a quantum channel $\mathcal{N}$ is defined to be the maximum rate at which a sender and receiver can communicate classical information with vanishing error probability by using the channel $\mathcal{N}$ as many times as they wish and by using an arbitrary amount of shared entanglement of an arbitrary form. For a noiseless quantum channel, the entanglement-assisted classical capacity is twice as large as its unassisted one, an enhancement realized by the super-dense coding protocol~\cite{PhysRevLett.69.2881}. This is in stark contrast to the setting of classical channels where additional shared randomness or entanglement does not increase the capacity.
\MT{Similarly,} for a noisy quantum channel, the presence of entanglement as an auxiliary resource can also lead to an enhancement of its classical capacity \cite{PhysRevLett.83.3081,ieee2002bennett}. Somewhat surprisingly, entanglement assistance is advantageous even for some entanglement-breaking channels \cite{horodecki03}, such as depolarizing channels with sufficiently high error probability. Bennett, Shor, Smolin and Thapliyal~\cite{ieee2002bennett} proved that the entanglement-assisted classical capacity $C_{\rm ea}({\cal N})$ of a quantum channel ${\cal N}$ is given by a remarkably simple, single-letter formula in terms of the quantum mutual information \MT{(defined in the following section)}. This is in contrast to the unassisted classical capacity of a quantum channel \cite{{H02},{PhysRevA.56.131}}, for which the best known general expression involves a regularization of the Holevo formula over infinitely many instances of the channel \cite{H09}. The regularization renders the explicit evaluation of the capacity for a general quantum channel intractable. The formula for the entanglement-assisted capacity is formally analogous to Shannon's well-known formula~\cite{bell1948shannon} for the capacity of a discrete memoryless classical channel, which is given in terms of the mutual information between the channel's input and output. The entanglement-assisted capacity does not increase under the presence of a noiseless quantum feedback channel from receiver to sender \cite{B04}, much like the capacity of a classical channel does not increase in the presence of a noiseless classical feedback link \cite{S56}.
\MT{The formula for $C_{\rm ea}({\cal N})$ derived in~\cite{PhysRevLett.83.3081}, however, is only relevant if the channel is available for as many uses as the sender and receiver wish, with there being no correlations in the noise acting on its successive inputs.\footnote{In other words, the channel is assumed to be {\em{memoryless}.}} To see this, let us consider the practical scenario in which a memoryless channel is used a finite number $n$ times, and let ${\cal N}^n\equiv {\cal N}^{\otimes n}$. Let $\log M_{\rm ea}^*({\cal N}^n,\varepsilon)$ denote the maximum number of bits of information that can be transmitted through $n$ uses of the channel via an entanglement-assisted communication protocol, such that the average probability of failure is no larger than $\varepsilon \in (0,1)$. Then~\cite{PhysRevLett.83.3081} and the strong converse~\cite{BDHSW12,BCR09} imply that} \begin{align}
\lim_{n \to \infty} \frac1{n} \log M_{\rm ea}^*({\cal N}^n,\varepsilon) = C_{\rm ea}({\cal N}) . \end{align} The strong converse from \cite{GW13} implies that \begin{align} \log M_{\rm ea}^*({\cal N}^n,\varepsilon) = n C_{\rm ea}({\cal N}) + O(\sqrt{n}), \label{eq:cap-expand} \end{align} for all $\varepsilon \in (0,1)$. The results of \cite{CMW14} imply that this same expansion holds even when noiseless quantum feedback communication is allowed from receiver to sender.
\MT{We are interested in investigating the behavior of $M_{\rm ea}^*({\cal N}^n,\varepsilon)$ for large but finite $n$ as a function of~$\varepsilon$. In this paper, we derive a lower bound on $\log M_{\rm ea}^*({\cal N}^n,\varepsilon)$, for any fixed value of $\varepsilon \in (0,1)$ and~$n$ large enough, of the following form: \begin{equation}\label{asymp_expansion} \log M_{\rm ea}^*({\cal N}^n,\varepsilon) \ge n C_{\rm ea}({\cal N}) + \sqrt{n}\, b + {{O}}(\log n). \end{equation} The coefficient $b$ that we identify in this paper constitutes a second-order coding rate.} The second-order coding rate obtained here depends on the channel as well as on the allowed error threshold~$\varepsilon$, and we obtain an explicit expression for it in Theorem~\ref{th:main}. In addition, we conjecture that in fact $\log M_{\rm ea}^*({\cal N}^n,\varepsilon) = n C_{\rm ea}({\cal N}) + \sqrt{n}\,b + \MT{{{o}}(\sqrt{n})}$ for all quantum channels. We show that this conjecture is true for the class of {\em{covariant channels}} \cite{H02}.
Our lower bound on $\log M_{\rm ea}^*({\cal N}^n,\varepsilon)$ resembles the asymptotic expansion for the maximum number of bits of information which can be transmitted through $n$ uses of a generic discrete, memoryless {\em{classical}} channel ${\cal W}$, with an average probability of error no larger than $\varepsilon$, denoted $\log M^*({\cal W}^n, \varepsilon)$. The latter was first derived by Strassen in 1962~\cite{strassen62} and refined by Hayashi~\cite{Hay09} as well as Polyanskiy, Poor and Verd\'u~\cite{polyanskiy10}. It is given by \begin{equation} \log M^*({\cal W}^n,\varepsilon)= n C({\cal W}) + \sqrt{n V_\varepsilon({\cal W})}\Phi^{-1}(\varepsilon) + \MT{o(\sqrt{n})}, \label{eq:gauss} \end{equation} where ${\cal W}^n$ denotes $n$ uses of the channel, $C({\cal W})$ is its capacity given by Shannon's formula~\cite{bell1948shannon}, $\Phi^{-1}$ is the inverse of the cumulative distribution function of a standard normal random variable, and $V_\varepsilon({\cal W})$ is a property of the channel (which depends on $\varepsilon$) called its $\varepsilon$-{\em{dispersion}}~\cite{polyanskiy10}. The right hand side of \eqref{eq:gauss} is called the \emph{Gaussian approximation} of $\log M^*({\cal W}^n,\varepsilon)$.
This result has recently been generalized to classical coding for quantum channels~\cite{TT13} and it was shown that a formula reminiscent of~\eqref{eq:gauss} also holds for the classical capacity of quantum channels with product inputs. In fact, the Gaussian approximation is a common feature of the second-order asymptotics for optimal rates of many other quantum information processing tasks such as data compression, communication, entanglement manipulation and randomness extraction (see, e.g.,~\cite{TH12,KH13,TT13,DL14} and references therein).
Even though we focus our presentation throughout on entanglement-assisted classical communication, we would like to point out that all of the results established in this paper apply to entanglement-assisted quantum communication as well. This is because the protocols of teleportation \cite{BBCJPW93} and super-dense coding \cite{PhysRevLett.69.2881} establish an equivalence between entanglement-assisted classical and quantum communication. This equivalence was noted in early work on entanglement-assisted communication \cite{PhysRevLett.83.3081}. That this equivalence applies at the level of individual codes is a consequence of the development, e.g., in Appendix~B of \cite{LM14}, and as a result, the equivalence applies to second-order asymptotics as well. This point has also been noted in \cite{TB15}.
\MT{Finally, we note that a one-shot lower bound on $M_{\rm ea}^*({\cal N},\varepsilon)$ has already been derived in~\cite{DH13}.
Moreover, in \cite{MW12} a one-shot upper bound was obtained.
Even though these bounds converge in first order to the formula for the capacity obtained by Bennett {\em{et al.}}~\cite{ieee2002bennett}, neither of these works deals with characterizing second-order asymptotics.}
\MT{This paper is organized as follows. Section~\ref{sec_prelim}
introduces the necessary notation and definitions. Section~\ref{sec_results} presents our main theorem and our conjecture. The proof of the theorem is given in Section~\ref{sec_proofs}. In Section~\ref{sec_proofs}, we also provide a proof of our conjecture for the case of covariant channels.
We end with a discussion of open questions in Section~\ref{sec_discussion}, summarizing the problems encountered when trying to prove the converse for general channels.}
\section{Notations and Definitions}\label{sec_prelim}
Let ${\cal B}({\cal H})$ denote the algebra of linear operators acting on a finite-dimensional Hilbert space ${\cal H}$. Let ${{\cal P}}({{\cal H}})\subset {\cal B}({\cal H})$ be the set of positive semi-definite operators, and let ${{\cal D}}({{\cal H}})\subset {\cal P}({\cal H})$ denote the set of \emph{quantum states} (density matrices), ${{\cal D}}({{\cal H}}) :=\{\rho\in{\cal P}({\cal H}): \mathop{\rm Tr}\nolimits\rho=1\}$.
We denote the dimension of a Hilbert space~${{\cal H}}_A$ by $|A|$ and write ${\cal H}_A \simeq {\cal H}_{A'}$ when ${\cal H}_A$ and ${\cal H}_{A'}$ are isomorphic, i.e., if $|A| = |A'|$.
A quantum state $\psi$ is called pure if it is rank one; in this case, we associate with it an element $|\psi\rangle \in {{\cal H}}$ such that $\psi = |\psi\rangle \langle\psi|$. The set of \emph{pure quantum states} is denoted ${\cal D}_*({\cal H})$.
For a bipartite operator $\omega_{AB} \in {\cal B}({\cal H}_A \otimes {\cal H}_B)$, let $\omega_{A} := \mathop{\rm Tr}\nolimits_B ( \omega_{AB})$ denote its restriction to the subsystem $A$, where $\mathop{\rm Tr}\nolimits_B$ denotes the partial trace over $B$. Let ${I}_{A}$ denote the identity operator on ${\cal H}_A$, and let $\pi_A := {I}_A/|A|$ be the completely mixed state in ${\cal{D}}({\cal{H}}_A)$.
A \emph{positive operator-valued measure} (POVM) is a set $\{ \Lambda_{A}^x \}_{x \in {\cal X}} \subset {\cal P}({\cal H}_A)$ such that $\sum_{x \in {\cal X}} \Lambda_A^x = \identity_A$, where ${\cal X}$ denotes any index set. We use the convention that ${\cal E}_{A\to B}$ refers to a \emph{completely positive trace-preserving} (CPTP) map ${\cal E}_{A\to B}: {\cal B}({{\cal H}}_A) \to {\cal B}({{\cal H}}_B)$. We call such maps \emph{quantum channels} in the following.
The identity map on ${\cal B}({\cal H}_A)$ is denoted~${\rm id}_A$.
We employ the cumulative distribution function for a standard normal random variable: \begin{equation}
\Phi(a) := \frac{1}{\sqrt{2\pi}} \int_{-\infty}^a \rm{d} x \, \exp\left(-\frac{x^2}{2}\right).
\end{equation} and its inverse $\Phi^{-1}(\varepsilon) := \sup \left\{ a \in \mathbb{R} \,|\, \Phi(a) \le \varepsilon\right\}$.
\subsection{Entanglement-Assisted Codes}
We consider entanglement-assisted classical (EAC) communication through a noisy quantum channel, given by a CPTP map ${\cal N}_{A \to B}$. The sender (Alice) and the receiver (Bob) initially share an arbitrary pure state $\ket{\varphi_{A'B'}}$, where without loss of generality we assume that ${\cal H}_{A'} \simeq {\cal H}_{B'}$, the system $A'$ being with Alice and the system $B'$ with Bob. The goal is to transmit classical messages from Alice to Bob, labelled by the elements of an index set ${\cal M}$, through ${\cal N}_{A \to B}$.
Without loss of generality, any EAC communication protocol can be assumed to have the following form: Alice encodes her classical messages into states of the system $A'$ in her possession. Let the encoding CPTP map corresponding to message $m \in {\cal M}$ be denoted by ${\cal E}^m_{A'\to A}$. Alice then sends the system $A$ through ${\cal N}_{A \to B}$ to Bob. Subsequently, Bob performs a POVM $\{\Lambda^m_{BB'}\}_{m \in {\cal M}}$ on the system $BB'$ in his possession. This yields a classical register $\widehat{M}$ which contains his inference $\hat{m} \in {\cal M}$ of the message sent by Alice.
The above defines an \emph{EAC code} for the quantum channel ${\cal N}_{A \to B}$, which consists of a quadruple \begin{align} \label{eq:code}
{\cal C} = \Big\{{\cal M},\, |\varphi_{A'B'}\rangle,\, \{{\cal E}^m_{A'\to A}\}_{m\in{\cal M}},\, \{\Lambda^m_{BB'}\}_{m\in{\cal M}} \Big\} . \end{align}
The size of a code is denoted as $|{\cal C}| = |{\cal M}|$. The average probability of error for ${\cal C}$ on ${\cal N}_{A\to B}$ is
\begin{equation} p_{\text{err}}({\cal N}_{A \to B} ,{\cal C}) := \Pr[M \neq \widehat{M}] = 1 - \frac{1}{|{\cal M}|} \sum_m \mathop{\rm Tr}\nolimits \Big(\Lambda^m_{BB'}\, {\cal N}_{A \to B} \otimes {\rm id}_{B'} \big({\cal E}^m_{A'\to A} \otimes {\rm id}_{B'}(\varphi_{A'B'}) \big) \Big) . \end{equation}
The following quantity describes the maximum size of an EAC code for transmitting classical information through a single use of ${\cal N}_{A\to B}$ with average probability of error at most~$\varepsilon$.
\begin{definition} Let $\varepsilon \in (0,1)$ and ${\cal N} \equiv {\cal N}_{A\to B}$ be a quantum channel. We define \begin{equation}\label{pr}
M_{\rm ea}^*({\cal N},\varepsilon) := \max \big\{ m \in \mathbb{N} \,\big|\, \exists \, {\cal C} : |{\cal C}| = m \land p_{\text{err}}({\cal N},{\cal C}) \leq \varepsilon \big\} , \end{equation} where ${\cal C}$ is a code as prescribed in~\eqref{eq:code}. \end{definition}
We are particularly interested in the quantity $M_{\rm ea}^*({\cal N}^n, \varepsilon)$, where $n \in \mathbb{N}$ and ${\cal N}^n \equiv {\cal N}_{{A}^n\to B^n}^n = {\cal N}_{A_1 \to B_1} \otimes \ldots \otimes {\cal N}_{A_n \to B_n}$ is the $n$-fold memoryless repetition of ${\cal N}$.
\subsection{Information Quantities}
For a pair of positive semi-definite operators $\rho$ and $\sigma$ with ${\rm supp}\, \rho \subseteq {\rm supp} \,\sigma$, the {\em{quantum relative entropy}} and the {\em{relative entropy variance}}~\cite{li12,TH12} are respectively defined as follows:\footnote{All logarithms in this paper are taken to base two.} \begin{align}
D(\rho\|\sigma) &:= \mathop{\rm Tr}\nolimits \left[\rho\left(\log \rho - \log \sigma\right)\right], \qquad \textrm{and} \label{eq:rel-ent}\\
V(\rho\|\sigma) &:= \mathop{\rm Tr}\nolimits \left[\rho\left(\log \rho - \log \sigma - D(\rho\|\sigma)\right)^2\right]. \end{align}
For a bipartite state $\rho_{AB}$, let us define the \emph{mutual information} $I(A:B)_{\rho} := D(\rho_{AB} \| \rho_A \otimes \rho_B)$. Similarly, we define the \emph{mutual information variance} $V(A:B)_{\rho} := V(\rho_{AB} \| \rho_A \otimes \rho_B)$.
The EAC capacity of a quantum channel ${\cal N}$ is defined as \begin{align}
C_{\rm ea}({\cal N}) := \lim_{\varepsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} \log M_{\rm ea}^*({\cal N}^n, \varepsilon). \end{align}
Bennett, Shor, Smolin and Thapliyal~\cite{ieee2002bennett} established that the EAC capacity for a quantum channel ${\cal N}\equiv {\cal N}_{{A}\to B}$ satisfies
\begin{align}
C_{\rm ea}({\cal N}) = \max_{\psi_{AA'}} I(A':B)_{\omega}, \qquad \textrm{where} \quad \omega_{A'B} = {\cal N}_{A \to B} \otimes {\rm id}_{A'} (\psi_{AA'}), \label{eq:def-omega}
\end{align} and the maximum is taken over all $\psi_{AA'} \in {\cal D}_*({\cal H}_A \otimes {\cal H}_{A'})$ with ${\cal H}_{A'} \simeq {\cal H}_A$. Its proof was later simplified by Holevo \cite{Hol01a}, and an alternative proof was given in \cite{HDW05}.
In analogy with~\cite{TT13}, the following definitions will be used to characterize our lower bounds on the second-order asymptotic behavior of $M_{\rm ea}^*({\cal N}^n,\varepsilon)$.
\begin{definition}
Let ${\cal N} \equiv {\cal N}_{A \to B}$ be a quantum channel.
The set of \emph{capacity achieving resource states} on ${\cal N}$ is defined as
\begin{align}
\Pi({\cal N}) := \argmax_{\psi_{AA'}} I(A':B)_{\omega} \subseteq {\cal D}_*({\cal H}_A \otimes {\cal H}_{A'}),
\end{align}
where $\omega_{A'B}$ is given in~\eqref{eq:def-omega}.
The \emph{minimal mutual information variance} and the \emph{maximal mutual information variance} of ${\cal N}$ are respectively defined as
\begin{align}
V_{\rm ea,\min}({\cal N}) := \min_{\psi_{AA'}} V(A' : B)_{\omega} \qquad \textrm{and} \qquad
V_{\rm ea,\max}({\cal N}) := \max_{\psi_{AA'}} V(A' : B)_{\omega} ,
\end{align}
where the optimizations are over $\psi_{AA'} \in \Pi({\cal N})$ and $\omega_{A'B}$ is given in~\eqref{eq:def-omega}. \end{definition}
\section{Results} \label{sec_results} Our main result is stated in the following theorem, which provides a second-order lower bound on the maximum number of bits of classical message which can be transmitted through $n$ independent uses of a noisy channel via an entanglement-assisted protocol, for any given allowed error threshold.
\begin{theorem}
\label{th:main}
Let $\varepsilon \in (0,1)$ and let ${\cal N} \equiv {\cal N}_{A \to B}$ be a quantum channel. Then,
\begin{align}
\log M_{\rm ea}^*({\cal N}^n,\varepsilon) \geq
\begin{cases} n C_{\rm ea}({\cal N}) + \sqrt{n V_{\rm ea,\min}({\cal N})}\ \Phi^{-1}(\varepsilon) + K(n; {\cal N}, \varepsilon) & \text{if } \varepsilon < \frac12 \\
n C_{\rm ea}({\cal N}) + \sqrt{n V_{\rm ea,\max}({\cal N})}\ \Phi^{-1}(\varepsilon) + K(n; {\cal N}, \varepsilon) & \text{else} \end{cases} \label{eq:direct}
\end{align}
where $K(n; {\cal N},\varepsilon) = O(\log n)$. \end{theorem}
\MT{The proof of Theorem~\ref{th:main} is split into two parts, Proposition~\ref{thm-one-shot} in Section~\ref{SEC_MAIN} and Proposition~\ref{pr:direct-second} in Section~\ref{SEC_SECOND}. We first derive a one-shot lower bound on $\log M_{\rm ea}^*({\cal N},\varepsilon)$ using a coding scheme that is a one-shot version of the coding scheme given in \cite{HDW05} and reviewed in \cite[Sec.~20.4]{MW13}. Our one-shot bound is expressed in terms of an entropic quantity called the hypothesis testing relative entropy~\cite{WR12}, which has its roots in early work on the quantum Stein's lemma \cite{HP91} (see Section~\ref{sec_tech} for a definition). The relation between classical coding over a quantum channels and binary quantum hypothesis testing was first pointed out by Hayashi and Nagaoka~\cite{HN03}.
An asymptotic expansion for this quantity for product states was derived independently by Tomamichel and Hayashi~\cite{TH12} and Li \cite{li12}, and we make use of this to obtain our lower bound on $\log M_{\rm ea}^*({\cal N}^n,\varepsilon)$ in the second step.}
\begin{remark} In particular, Theorem~\ref{th:main} establishes that
\begin{align}
&\liminf_{n\to\infty} \frac{1}{\sqrt{n}} \Big( \log M_{\rm ea}^*({\cal N}^n,\varepsilon) - n C_{\rm ea}({\cal N}) \Big) \geq \begin{cases} \sqrt{V_{\rm ea,\min}({\cal N})}\ \Phi^{-1}(\varepsilon) & \text{if } \varepsilon < \frac12 \\
\sqrt{V_{\rm ea,\max}({\cal N})}\ \Phi^{-1}(\varepsilon) & \text{else} \end{cases} .
\end{align}
In analogy with~\cite[Eq.~(221)]{polyanskiy10} and~\cite[Rm.~4]{TT13}, we define the EAC $\varepsilon$-channel dispersion, for $\varepsilon \in (0,1)\setminus \{\tfrac12\}$ as
\begin{align}
V_{\rm ea,\varepsilon}({\cal N}) := \limsup_{n\to \infty} \frac{1}{n} \bigg( \frac{\log M_{\rm ea}^*({\cal N}^n,\varepsilon) - n C_{\rm ea}({\cal N})}{\Phi^{-1}(\varepsilon)} \bigg)^2 .
\end{align}
Theorem~\ref{th:main} shows that $V_{\rm ea,\varepsilon}({\cal N}) \leq V_{\rm ea,\min}({\cal N})$ if $\varepsilon < \frac12$ and $V_{\rm ea,\varepsilon}({\cal N}) \geq V_{\rm ea,\max}({\cal N})$ if $\varepsilon > \frac12$. \end{remark}
This leads us to the following conjecture:
\begin{conjecture}\label{conj:main}
We conjecture that~\eqref{eq:direct} is an equality with $K(n; {\cal N},\varepsilon) = o(\sqrt{n})$. In particular, we conjecture that the EAC $\varepsilon$-channel dispersion satisfies
\begin{align}
V_{\rm ea,\varepsilon}({\cal N}) = \begin{cases} V_{\rm ea,\min}({\cal N}) & \text{if } \varepsilon < \frac12 \\ V_{\rm ea,\max}({\cal N}) & \text{else} \end{cases}
\end{align}
and, thus, $\log M_{\rm ea}^*({\cal N}^n,\varepsilon) = n C_{\rm ea}({\cal N}) + \sqrt{n V_{\rm ea,\varepsilon}({\cal N})}\ \Phi^{-1}(\varepsilon) + o(\sqrt{n})$. \end{conjecture}
We show that Conjecture~\ref{conj:main} is true for the class of covariant quantum channels. This follows essentially from an analysis by Matthews and Wehner~\cite{MW12} which we recapitulate in Section~\ref{sec_converse} and the asymptotic expansion of the hypothesis testing relative entropy.
\section{Proofs} \label{sec_proofs}
\subsection{Technical Preliminaries} \label{sec_tech}
For given orthonormal bases $\{|i_{A}\rangle\}_{i=1}^d$ and $\{|i_{B}\rangle\}_{i=1}^d$ in isomorphic Hilbert spaces ${\cal{H}}_{A}\simeq{\cal{H}}_B\simeq{{\cal H}}$ of dimension~$d$, we define a maximally entangled state of Schmidt rank $d$ to be \begin{equation}\label{MES-m}
|\Phi_{AB}\rangle:= \frac{1}{\sqrt{d}} \sum_{i=1}^d |i_A\rangle\otimes |i_B\rangle. \end{equation}
Note that if $d = 1$ then $|\Phi_{AB}\rangle$ is a product state. We often make use of the following identity (``transpose trick''): for any operator~$M$, \begin{equation}\label{transpose}
(M_A \otimes I_B)|\Phi_{AB}\rangle = (I_A \otimes M_B^T)|\Phi_{AB}\rangle,
\end{equation} where $M_B^T := \sum_{i,j=1}^d |i\rangle_B \langle j|_A M_A |i\rangle_A \langle j|_B$ denotes the transpose.
\subsubsection{Distance Measures}
The \emph{trace distance} between two states $\rho$ and $\sigma$ is given by \begin{align}
\tfrac12 \Vert \rho - \sigma \Vert_1 = \max_{0 \leq Q \leq I} \mathop{\rm Tr}\nolimits \big( Q (\rho - \sigma) \big) =
\mathop{\rm Tr}\nolimits\bigl[\{\rho \ge \sigma\}(\rho-\sigma)\bigr] \label{eq:bd} \end{align} where $\{\rho\ge \sigma\}$ denotes the projector onto the subspace where the operator $\rho-\sigma$ is positive semi-definite. The fidelity of $\rho$ and $\sigma$ is defined as \begin{equation}\label{fidelity-aaa} F(\rho, \sigma):=\left \Vert{\sqrt{\rho}\sqrt{\sigma}}\right \Vert_1. \end{equation} For a pair of pure states $\phi$ and $\psi$, the trace distance and fidelity satisfy the following relation: \begin{equation} \tfrac{1}{2} \left \Vert\phi - \psi\right \Vert_1 = \sqrt{1 - F^2(\phi, \psi))} . \end{equation}
\subsubsection{Relative Entropies for One-Shot Analysis}
We will phrase our one-shot bounds in terms of the following relative entropy. \begin{definition} Let $\varepsilon \in (0,1)$, $\rho\in {\cal D}({\cal H})$ and $\sigma \in {\cal P}({\cal H})$. Then, the \emph{hypothesis testing relative entropy}~\cite{WR12} is defined as
\begin{equation} D_H^\varepsilon(\rho\|\sigma) := -\log \beta_\varepsilon (\rho\|\sigma), \end{equation} where \begin{equation}\label{beta}
\beta_\varepsilon (\rho\|\sigma) := \min \big\{ \mathop{\rm Tr}\nolimits (Q\sigma) : 0\le Q \le I \wedge \mathop{\rm Tr}\nolimits(Q\rho) \ge 1-\varepsilon \big\}. \end{equation} \end{definition}
Note that when $\sigma \in {\cal D}({\cal H})$, $\beta_\varepsilon (\rho\|\sigma)$
has an interpretation as the smallest type-II error of a hypothesis test between $\rho$ and $\sigma$, when the type-I error is at most~$\varepsilon$. The following lemma lists some properties of $D_H^\varepsilon(\rho\|\sigma)$.
\begin{lemma}\label{props-hypo} Let $\varepsilon \in (0,1)$. The hypothesis testing relative entropy has the following properties: \begin{enumerate}
\item For any $\rho \in {\cal D}({\cal H})$, $\sigma^\prime \ge \sigma \ge 0$ we have $D_H^\varepsilon(\rho\|\sigma)\ge D_H^\varepsilon(\rho\|\sigma^\prime)$.
\item For any $\rho \in {\cal D}({\cal H})$, $\sigma \geq 0$, $\alpha >0$, we have $D_H^\varepsilon(\rho\|\alpha \sigma)= D_H^\varepsilon(\rho\|\sigma) - \log \alpha$. \item For any classical-quantum state \begin{equation}
\rho_{XB} = \sum_{x \in {\cal X}} p(x) |x\rangle \langle x| \otimes \rho^x_B \in {\cal D}({\cal H}_X \otimes {\cal H}_B), \end{equation} for any
$\sigma_X= \sum_{x \in {\cal X}} q(x) |x\rangle \langle x|$ (with $\{p(x)\}_{x \in {\cal X}}$ and
$\{q(x)\}_{x \in {\cal X}}$ probability distributions on ${\cal X}$), and for any $\sigma_B \in {\cal D}({\cal H}_B)$, we have \begin{equation}
D_H^\varepsilon(\rho_{XB}\|\sigma_X \otimes \sigma_B) \ge \min_{x \in {\cal X}} D_H^\varepsilon(\rho_{B}^x\|\sigma_B), \end{equation}
\item For any $\delta \in (0,1-\varepsilon)$, $\rho, \rho' \in {\cal D}({\cal H})$ with $\frac12 \left\Vert\rho - \rho^\prime\right\Vert_1 \le \delta$, and $\sigma \in {\cal P}({\cal H})$, we have $D_H^\varepsilon(\rho'\|\sigma) \le D_H^{\varepsilon+\delta}(\rho\| \sigma)$. \end{enumerate} \end{lemma}
Properties 1--3 can be verified by close inspection and we omit their proofs.
\begin{proof}[Proof of Property 4] Consider $Q$ to be the operator achieving the minimum in the definition of
$ \beta_\varepsilon(\rho'\|\sigma)$, i.e.\ \begin{align}
D_H^\varepsilon(\rho'\|\sigma) = - \log \mathop{\rm Tr}\nolimits (Q \sigma) \qquad \textrm{and} \qquad \mathop{\rm Tr}\nolimits( Q \rho') \geq 1-\varepsilon. \end{align} From~\eqref{eq:bd}, we have $\mathop{\rm Tr}\nolimits\left[Q(\rho' -\rho)\right] \le \frac12 \left\Vert\rho - \rho'\right\Vert_1 \le \delta.$ Hence, $\mathop{\rm Tr}\nolimits (Q\rho) \ge \mathop{\rm Tr}\nolimits (Q\rho') - \frac12 \left\Vert\rho - \rho^\prime\right\Vert_1 \ge 1- \varepsilon - \delta$, and \begin{align}
D_H^\varepsilon(\rho'\|\sigma) & \le \max_{0\le Q' \le I\atop{ \mathop{\rm Tr}\nolimits (Q'\rho) \ge 1- \varepsilon -\delta}}\left[- \log \mathop{\rm Tr}\nolimits (Q' \sigma)\right] = D_H^{\varepsilon+\delta}(\rho\| \sigma) , \end{align} which concludes the proof. \end{proof} The following result, established independently in~\cite[Eq.~(34)]{TH12} and \cite{li12}, plays a central role in our analysis. \begin{lemma}[\cite{TH12,li12}]\label{second-order} Let $\varepsilon \in (0,1)$ and let $\rho, \sigma \in {\cal D}({\cal H})$. Then, \begin{equation}
D_H^\varepsilon(\rho^{\otimes n} \| \sigma^{\otimes n}) = nD(\rho\|\sigma) + \sqrt{n V(\rho\|\sigma)} \Phi^{-1}(\varepsilon) + O(\log n) . \end{equation} \end{lemma}
Two other generalized relative entropies which are relevant for our analysis are the {\em{collision relative entropy}} and the {\em{information-spectrum relative entropy}}~\cite[Def.~8]{TH12}. For any pair of positive semi-definite operators $\rho$ and $\sigma$ satisfying the condition ${\rm supp} \,\rho \subseteq {\rm supp} \,\sigma$, they are respectively defined as follows: \begin{equation}\label{collision}
D_2(\rho\|\sigma) := \log \left( \mathop{\rm Tr}\nolimits \left( \sigma^{-1/4} \rho \sigma^{-1/4}\right)^2 \right) , \end{equation} and, for any $\varepsilon \in (0, 1)$, \begin{equation}\label{infospec}
D_s^\varepsilon(\rho\|\sigma) := \sup\left\{R \,|\, \mathop{\rm Tr}\nolimits\left(\rho \big\{\rho \le 2^R \sigma\big\} \right) \le \varepsilon\right\} , \end{equation} where we write $A \geq B$ if $A - B$ is positive semidefinite. The following result, proved in~\cite[Thm.~4]{BG13}, relates these quantities. \begin{lemma}[\cite{BG13}] Let $\varepsilon, \lambda \in (0,1)$ and $\rho, \sigma \in {\cal D}({\cal H})$. Then, \begin{equation}
2^{D_2\left( \rho \| \lambda \rho +(1-\lambda)\sigma \right)} \ge (1-\varepsilon)
\left[\lambda + (1-\lambda) 2^{-D_s^\varepsilon(\rho\|\sigma)} \right]^{-1} . \end{equation} \label{lemBG} \end{lemma} Finally, the following lemma provides a useful relation between the hypothesis testing relative entropy and the information spectrum relative entropy~\cite[Lm.~12]{TH12}. \begin{lemma}[\cite{TH12}] Let $\varepsilon \in (0,1)$, $\delta \in (0, 1-\varepsilon)$, $\rho \in {\cal D}({\cal H})$, and $\sigma \in {{\cal P}}({\cal H})$. Then,
$D_H^{\varepsilon}(\rho\|\sigma) \ge D_s^{\varepsilon} (\rho\|\sigma) \ge D_H^{\varepsilon+\delta}(\rho\|\sigma) + \log \delta.$ \label{info-hypo} \end{lemma}
\subsection{One-Shot Achievability} \label{SEC_MAIN}
Our protocol is modeled after~\cite{ieee2002bennett}. Consider a quantum channel ${\cal N}_{A \to B}$ and introduce an auxiliary Hilbert space ${\cal H}_{A'} \simeq {\cal H}_A$.
Let \begin{equation}\label{directsum} {\cal H}_{A} \otimes {\cal H}_{A'} = \bigoplus_t {\cal H}_{A}^t \otimes {\cal H}_{A'}^t, \qquad \ {\cal H}_{A}^t \simeq {\cal H}_{A'}^t
\end{equation} be a decomposition of ${\cal H}_{A} \otimes {\cal H}_{A'}$, and set $d_t = |{\cal H}_{A}^t|$. We assume that $|\vartheta_{AA'}\rangle$ can be written as a superposition of maximally entangled states:
\begin{equation}\label{decomp} |\vartheta_{AA'}\rangle = \sum_t \sqrt{p(t)}\, |\Phi^t\rangle,
\end{equation} where $|\Phi^t\rangle$ denotes a maximally entangled state of Schmidt rank $d_t$ in ${\cal H}_{A}^t \otimes {\cal H}_{A'}^t$ and $p(t)$ is some probability distribution so that $\sum_t p(t) = 1$.
\begin{proposition}\label{thm-one-shot} Let $\varepsilon \in (0,1)$, $\delta \in (0, \frac{\varepsilon}2)$ and let ${\cal N} \equiv {\cal N}_{A\to B}$ be a quantum channel. Then for any
$\vartheta_{AA'}$ of the form~\eqref{decomp}, we have \begin{align}
\log M_{\rm ea}^*({\cal N},\varepsilon) \ge D_H^{\varepsilon - 2\delta}({\cal N}_{A \to B}( \vartheta_{AA'}) \,\|\, {\cal N}_{A \to B}(\kappa_{AA'} ) ) - f(\varepsilon, \delta), \end{align} where $f(\varepsilon, \delta) := \log \frac{1-\varepsilon}{\delta^2}$, $\kappa_{AA'} := \sum_t p(t)\, \pi_{A}^t \otimes \pi_{A'}^t$, and $\pi_A^t$ is the maximally mixed state on ${\cal H}_{A}^t$. \end{proposition}
\begin{remark} Note that the hypothesis testing relative entropy on the right hand side is not reminiscent of a mutual information type quantity since the second argument is not a product state.
\end{remark}
\begin{proof} Consider the set \begin{equation}
{\cal S}:= \big\{\left((x_t, z_t, b_t)\right)_t \,\big|\, x_t, z_t \in \{0,1,\cdots, d_t-1\}, b_t \in \{0,1\}\big\}, \end{equation} where the index $t$ labels the Hilbert spaces of the decomposition in~\reff{directsum}. For any $s \in {\cal S}$, consider the following unitary operator in ${\cal B}({\cal H}_{A})$: \begin{equation}\label{unitaryop} U_{A}(s) := \bigoplus_t (-1)^{b_t} X(x_t) Z(z_t), \end{equation} where $X(x_t)$ and $Z(z_t)$ are the Heisenberg-Weyl operators defined in~Appendix~\ref{app:hw}.
For any $M \in \mathbb{N}$, we now construct a random code as follows. Let ${\cal M} = \{1, 2, \ldots, M \}$. We set $A' \equiv A$ (i.e., we use the labels interchangeably), and ${\cal H}_{B'} \simeq {\cal H}_A$. We consider the resource state $\varphi_{AB'} = {\rm id}_{A' \to B'}(\vartheta_{AA'})$.
For each message $m\in {\cal M}$, choose a \emph{codeword}, $s_m$, uniformly at random from the set ${\cal S}$. The encoding operation, $\{\mathcal{E}_{A}^m \}_{m \in {\cal M}}$, is then given by the (random) unitary $U(s_m)$ as prescribed above. In particular, \begin{align}
{\cal E}^m_{A} \otimes {\rm id}_{B'}(\varphi_{AB'}) = \phi^{s_m}_{AB'}, \quad \textrm{where} \quad
|\phi^{s_m}_{AB'}\rangle := \left(U_{A}(s_m) \otimes \identity_{B'}\right)|\varphi_{AB'}\rangle.
\label{eq:enc} \end{align}
We denote the corresponding channel output state by $\rho^{s_m}_{BB'}:={\cal N}_{A \to B}\left(\phi^{s_m}_{AB'}\right)$ and use ``pretty good'' measurements for decoding. These are given by the POVM $\{\Lambda_{BB'}^m\}_{m \in {\cal M}}$, where \begin{align}
\Lambda_{BB'}^m :=\left( \sum_{m'\in {\cal M}}\rho^{s_{m'}}_{BB'}\right)^{-\frac{1}{2}}\rho^{s_m}_{BB'}\left(\sum_{m'\in {\cal M}}\rho^{s_{m'}}_{BB'}\right)^{-\frac{1}{2}} . \label{eq:dec} \end{align}
Let us now analyze the code ${\cal C} = \{{\cal M}, \varphi_{AB'}, \{{\cal E}^m_{A'}\}_{m\in{\cal M}}, \{\Lambda^m_{BB'}\}_{m\in{\cal M}} \}$ given by~\eqref{eq:enc} and~\eqref{eq:dec}, where we recall that $s_m$ is a random variable. For this purpose, consider the random state \begin{align}
\sigma_{MSBB'} := \frac{1}{M} \sum_{m \in {\cal M}} |m \rangle \langle m|_M \otimes |s_m\rangle \langle s_m|_S \otimes \rho^{s_m}_{BB'}. \end{align} Then, following Beigi and Gohari~\cite[Thm.~5]{BG13}, we find that the average probability of successfully inferring the sent message can be expressed as \begin{align}\label{succ} p_{{\rm succ}}({\cal C}, {\cal N}) &:= 1 - p_{\text{err}}({\cal C}, {\cal N}) = \frac{1}{M} \sum_{m \in {\cal M}} \mathop{\rm Tr}\nolimits (\Lambda_{BB'}^m \rho^{s_m}_{BB'}) \\
&= \frac{1}{M} 2^{D_2\left(\sigma_{MSBB'}\| \sigma_{MS} \,\otimes\, \sigma_{BB'} \right)}. \end{align} Moreover employing both the data-processing inequality and joint convexity of the collision relative entropy as in~\cite{BG13}, we establish the following lower bound on the expected value of ${p}_{{\rm succ}}$ with respect to the randomly chosen codewords: \begin{equation}
\mathbb{E}\left(p_{{\rm succ}}({\cal C}, {\cal N})\right)\ge \frac{1}{M} 2^{D_2\left( \mathbb{E}(\sigma_{SBB'})\| \mathbb{E}(\sigma_{S} \,\otimes\, \sigma_{BB}) \right)}. \end{equation} Note that
\begin{align}
\mathbb{E}(\sigma_{S} \otimes \sigma_{BB'}) &= \mathbb{E}\left(\frac{1}{M^2} \sum_{m \in {\cal M}} |s_m\rangle \langle s_m| \otimes \rho^{s_m}_{BB'} \right) + \mathbb{E}\left(\frac{1}{M^2} \sum_{m, m' \in {\cal M}\atop{m' \ne m}} |s_m\rangle \langle s_m| \otimes \rho^{s_{m'}}_{BB'} \right) \\ &= \frac{1}{M} \rho_{SBB'} + \left( 1- \frac{1}{M}\right) \rho_S \otimes \rho_{BB'}, \end{align} where \begin{align}
\rho_{SBB'} &:= \mathbb{E}(\sigma_{SBB'}) = {\cal N}_{A \to B} \left(\frac{1}{|{\cal S}|} \sum_{s \in {\cal S}} |s\rangle \langle s| \otimes U_{A}(s) \varphi_{AB'} U_{A}^\dagger(s)\right)\\
&= \frac{1}{|{\cal S}|} \sum_{s \in {\cal S}} |s\rangle \langle s| \otimes {\cal N}_{A \to B} \left(U_{A}(s) \varphi_{AB'} U_{A}^\dagger(s)\right),\label{c-q} \end{align} and $\rho_S$ and $\rho_{BB'}$ are the corresponding reduced states on the systems $S$ and $BB'$, respectively. In particular, defining $V(x_t, z_t) := X(x_t)Z(z_t)$, using the decomposition \reff{decomp} of the state $\ket{\varphi_{AB'}}$ and the definition \reff{unitaryop} of the unitary operators $U_{A}(s)$, we find that \begin{align} \rho_{BB'}&= {\cal N}_{A \to B}
\left(\frac{1}{|{\cal S}|}\sum_{s \in {\cal S}} U_{A}(s) \left(\sum_{t,t'} \sqrt{p(t)p(t')} |\Phi^t\rangle \langle \Phi^{t'}|\right)U_{A}^\dagger(s)\right)\\
&= {\cal N}_{A \to B} \left(\sum_{t}{p(t)} \frac{1}{d_t^2} \sum_{x_t, z_t =0}^{d_t - 1} V(x_t, z_t) |\Phi^t\rangle \langle \Phi^t|V^\dagger(x_t, z_t)\right) \\
& + {\cal N}_{A \to B} \left(\sum_{t,t'\atop{t'\ne t}} \sqrt{p(t)p(t')}\ \frac{1}{4} \sum_{b_t, b_{t'} \in \{0,1\}}(-1)^{b_t+ b_{t'}} \frac{1}{d_t^2d_{t'}^2} \sum_{x_t, z_t =0}^{d_t - 1} \sum_{x_{t'}, z_{t'} =0}^{d_{t'} - 1} V(x_{t},z_{t}) |\Phi^t\rangle \langle \Phi^{t'}| V^\dagger(x_{t'},z_{t'})\right) \end{align} can be written as the sum of a diagonal ($t = t'$) and an off-diagonal ($t \neq t'$) term. It can be verified (see, e.g.,~\cite[pp.~504--505]{MW13}) that the off-diagonal term vanishes and in fact \begin{equation}\label{decouple} \rho_{BB'} = \sum_t p(t) \, {\cal N}_{A \to B}(\pi_{A}^t) \otimes \pi_{B'}^t , \end{equation} where $\pi_{A}^t = \mathop{\rm Tr}\nolimits_{B'}(\Phi^t)$ and $\pi_{B'}^t = \mathop{\rm Tr}\nolimits_{A}(\Phi^t)$ are completely mixed states. The above identity follows from the fact that applying a Heisenberg-Weyl operator uniformly at random completely randomizes a quantum state, yielding a completely mixed state.
Hence, for any $0 < \delta < \varepsilon$, we have \begin{align}\label{last}
\mathbb{E}\left(p_{{\rm succ}}({\cal C}, {\cal N})\right)& \ge \frac{1}{M} 2^{D_2\left(\rho_{SBB'}\| \frac{1}{M} \rho_{SBB'} + (1- \frac{1}{M})(\rho_{S} \otimes \rho_{BB'} \right)}\\
& \ge \frac{1-(\varepsilon-\delta)}{1+(M-1)\,2^{-D_s^{\varepsilon-\delta}(\rho_{SBB'}\| \rho_S \otimes \rho_{BB'})}}, \end{align} where the last line follows from Lemma~\ref{lemBG}. Thus, provided that \begin{align} M &\leq
\frac{\delta}{1-\varepsilon}
2^{D_s^{\varepsilon - \delta}\left(\rho_{SBB'}\| \rho_S \otimes \rho_{BB'}\right)} + 1
\end{align} the random code satisfies $\mathbb{E}\left(p_{{\rm succ}}({\cal C}, {\cal N})\right) \ge 1- \varepsilon$. In particular, there exists a (deterministic) code which satisfies $p_{{\rm succ}}({\cal C}, {\cal N}) \ge 1- \varepsilon$. Hence, we conclude that \begin{align}
\log M_{\rm ea}^*({\cal N},\varepsilon) & \ge D_s^{\varepsilon - \delta}(\rho_{SBB'}\| \rho_S \otimes \rho_{BB'}) + \log \frac{\delta}{1-\varepsilon}\\ & \ge
D_H^{\varepsilon - 2\delta}(\rho_{SBB'}\| \rho_S \otimes \rho_{BB'}) - f(\varepsilon, \delta), \label{stp1} \end{align} where we require that $\varepsilon > 2 \delta$ and use \begin{equation}\label{fed} f(\varepsilon, \delta) = \log \frac{1-\varepsilon}{\delta^2}. \end{equation} The inequality in \reff{stp1} follows from Lemma~\ref{info-hypo}. Further, since $\rho_{SBB'}$ is a classical-quantum state as seen in \reff{c-q}, by item $3$ of Lemma~\ref{props-hypo} we have \begin{equation}
D_H^{\varepsilon - 2\delta}(\rho_{SBB'}\| \rho_S \otimes \rho_{BB'})
\ge \min_{s \in {\cal S}} D_H^{\varepsilon - 2\delta}(\rho_{BB'}^s\| \rho_{BB'}), \end{equation} where \begin{equation} \rho_{BB'}^s={\cal N}_{A \to B} \left(U_{A}(s) \varphi_{AB'} U_{A}^\dagger(s)\right). \end{equation} Using the decomposition \reff{decomp} of the state $\ket{\varphi_{AB'}}$ and the transpose trick \reff{transpose} we can write \begin{equation}\label{eqi} \rho_{BB'}^s= U_{B'}^T(s){\cal N}_{A \to B}\left( \varphi_{AB'}\right) U_{B'}^{T\dagger}(s). \end{equation} Further, from \reff{decouple} it follows that \begin{equation} U_{B'}^T(s) \rho_{BB'} U_{B'}^{T\dagger}(s) = \rho_{BB'}. \label{eqii} \end{equation} Hence, \reff{eqi}, \reff{eqii}, \reff{decouple}, and the invariance of the hypothesis testing relative entropy under the same unitary on both states imply that \begin{align}
D_H^{\varepsilon - 2\delta}(\rho_{BB'}^s\| \rho_{BB'})=
D_H^{\varepsilon - 2\delta}\Bigg({\cal N}_{A \to B}\left( \varphi_{AB'}\right) \Bigg\| \sum_t p(t)\left( {\cal N}_{A \to B}(\pi_{A}^t)\right) \otimes \pi_{B'}^t\Bigg), \label{eqiii} \end{align} From \reff{stp1} and \reff{eqiii} we obtain the statement of the proposition. \end{proof}
\begin{remark} Alternatively, one may also employ the one-shot achievability result of Hayashi and Nagaoka~\cite{HN03} (in the form of~\cite{WR12}), which leads to the following bound on the one-shot $\varepsilon$-error entanglement-assisted capacity. Let $\varepsilon \in (0,1)$. Then, for any $\delta \in (0,\varepsilon)$ and for any $\ket{\vartheta_{AA'}}$ with decomposition \reff{decomp},
we have \begin{equation}
\log M_{\rm ea}^*({\cal N},\varepsilon) \ge D_H^{\varepsilon - \delta}\Bigg({\cal N}_{A\to B}\left( \vartheta_{AA'}\right) \,\Bigg\|\, \sum_t p(t)\left( {\cal N}_{A \to B}(\pi_{A}^t)\right) \otimes \pi_{A'}^t\Bigg) - \log\frac{4\varepsilon}{\delta^2}. \end{equation} The proof of this lower bound uses the same coding scheme as given above while employing the error analysis and decoder given in \cite{HN03}. \end{remark}
\subsection{Second-Order Analysis for Achievability} \label{SEC_SECOND}
Theorem~\ref{th:main} is a direct corollary of the following result, for an appropriate choice of $\psi_{AA'}$.
\begin{proposition}
\label{pr:direct-second}
Let $\varepsilon \in (0,1)$, ${\cal N} \equiv {\cal N}_{A\to B}$ be a quantum channel, and $\psi_{AA'} \in {\cal D}_*({\cal H}_A \otimes {\cal H}_A')$, where ${\cal H}_{A'} \simeq {\cal H}_A$. Then, we have
\begin{align}
\log M_{\rm ea}^*({\cal N}^n,\varepsilon) \geq n I(A':B)_{\omega} + \sqrt{n V(A':B)_{\omega}}\,\Phi^{-1}(\varepsilon) + K(n; {\cal N},\varepsilon,\psi_{AA'}) ,
\end{align}
where $\omega_{A'B} = {\cal N}_{A \to B} \otimes {\rm id}_{A'} (\psi_{AA'})$ and $K(n; {\cal N},\varepsilon,\psi_{AA'}) = O(\log n)$. \end{proposition}
\begin{proof} We intend to apply Proposition~\ref{thm-one-shot} to the channel ${\cal N}^n:={\cal N}^{\otimes n}$ for a fixed $n$. For this purpose, let us first construct an appropriate resource state $\vartheta_{A^nA'^n}$. We write \begin{equation}
|\psi_{AA'}\rangle = \sum_{x \in {\cal X}} \sqrt{q(x)}\, |x\rangle_A \otimes |x\rangle_{A'}
\end{equation} in its Schmidt decomposition, where ${\cal X} = \{ 1,2, \cdots, d\}$ with $d = |{\cal H}_{A}| = |{\cal H}_{A'}|$ and define $\rho_{A'} = \mathop{\rm Tr}\nolimits_{A} (\psi_{AA'})$. For a sequence $x^n = (x_1, x_2, \ldots, x_n) \in {\cal X}^n$, we write $|x^n\rangle_{A^n} = \ket{x_{1}}_{A_1} \otimes \ket{x_{2}}_{A_2} \otimes \cdots \otimes \ket{x_{n}}_{A_n}$. We denote type classes (for sequences of length $n$) by ${\cal T}^t$, i.e.\ ${\cal T}^t = \{x^n \in {\cal X}^n \,:\, P_{x^n} = t\}$ where $P_{x^n}$ denotes the empirical distribution of the sequence $x^n \in {\cal X}^n$. The set of empirical distributions is denoted ${\cal P}_n$. (We refer to Appendix~\ref{sc:types} for a short overview of the method of types and relevant results.) We consider the decomposition \begin{equation} \left( {\cal H}_{A} \otimes {\cal H}_{A'}\right)^{\otimes n} = \bigoplus_{t \in {\cal P}_n} {{\cal H}_{A^n}^t} \otimes {\cal H}_{A'^n}^t,
\end{equation} where ${\cal H}_{A^n}^t = \textrm{span} \big\{ |x^n\rangle_{A^n} \,\big|\, x^n \in {\cal T}^t \big\}$ as in~\eqref{directsum}. Notably, since $\psi_{AA'}^{\otimes n}$ is a tensor-power state, we can write \begin{equation}
|\psi_{AA'}\rangle^{\otimes n} = \sum_{t \in {\cal P}_n} \sqrt{p'(t)}\, |\Phi^t\rangle,
\end{equation} where $|\Phi^t\rangle \in {\cal H}_{A^n}^t \otimes {\cal H}_{A'^n}^t$ denotes a maximally entangled state of Schmidt rank $d_t = |{\cal T}^t|$, and \begin{equation} p'(t) := \sum_{x^n \in {\cal T}^t} q^n(x^n), \quad \textrm{where} \quad q^n(x^n)=\prod_{i=1}^n q(x_i) \,. \end{equation}
Now, fix a small $\mu > 0$ and consider a restriction of $|\psi_{AA'}\rangle^{\otimes n}$ to types $\mu$-close to $q$. More precisely, we consider the set ${\cal P}_n^{q,\mu} := \{ t \in {\cal P}_n \,|\, D(t\|q) \leq \mu \}$ and define \begin{align} \label{aux}
&|\vartheta_{A^{n}A'^{n}}\rangle := \sum_{t \in {\cal P}_n^{q,\mu}} \sqrt{p(t)}\, |\Phi^t\rangle, \quad \textrm{where} \quad p(t) = \frac{p'(t)}{\alpha}, \quad \textrm{and} \\ &\alpha := \sum_{t \in {\cal P}_n^{q,\mu}} p'(t)
\ =\!\! \sum_{x^n \in {\cal X}^n \atop{ D(P_{x^n}\|q) \le \mu }}q^n(x^n) \ge 1 - 2^{-n\left( \mu - |{\cal X}| \frac{\log (n+1)}{n}\right)}, \label{alpha-bd} \end{align} where the last inequality follows from \reff{type6} in Appendix~\ref{sc:types}.
Note that \begin{align} \frac{1}{2} \left\Vert\vartheta_{A^{\prime n}B^{\prime n}} - \psi_{A'B'}^{\otimes n}\right\Vert_1 &= \sqrt{1 - F^2\!\left(\vartheta_{A^{\prime n}B^{\prime n}}, \psi_{A'B'}^{\otimes n} \right)} = \sqrt{ 1 - \alpha} \\
& \le 2^{-\frac{n}{2} \left( \mu - |{\cal X}|\frac{ \log (n+1)}{n}\right)} =: g(n,\mu), \label{gnmu} \end{align} where the last inequality follows from \reff{alpha-bd}.
Next, recall that ${{\cal N}}^n \equiv \left({\cal N}_{A \to B}\right)^{\otimes n}$. Then by Proposition~\ref{thm-one-shot}, for fixed $\varepsilon>0$ and $0< 2\delta < \varepsilon$ and $\vartheta_{A^{n}A'^{n}}$ given in~\reff{aux}, we establish that \begin{equation}
\log M_{\rm ea}^*({\cal N}^n,\varepsilon) \ge D_H^{\varepsilon - 2 \delta}\bigg({\cal N}^n\left( \vartheta_{A^{n}A'^{n}}\right) \, \bigg\| \, \sum_{t\in {\cal P}_n^{q,\mu}} p(t)\left( {\cal N}^n(\pi_{A^n}^t)\bigg) \otimes \pi_{A'^{n}}^t \right) - f(\varepsilon, \delta), \label{bd1shot} \end{equation}
where $f(\varepsilon, \delta)$ is given by \reff{fed}, and $\pi_{A^{n}}^t$ and $\pi_{A'^{n}}^t$ are completely mixed states. In particular, for any $t$ with $D(t\|q) \leq \mu$, we have \begin{align} \pi_{A'^{n}}^t = \frac{1}{d_t}\sum_{x^n \in
{\cal T}^t}|x^n \rangle \langle x^n| &\le (n+1)^{|{\cal X}|} 2^{n\mu} \sum_{x^n \in T_t } q^n({x^n}) |x^n \rangle \langle x^n| \\
& \le \underbrace{(n+1)^{|{\cal X}|} 2^{n\mu}}_{=:\, \gamma_{n,\mu}} \sum_{x^n \in {\cal X}^n} q^n({x^n}) |x^n \rangle \langle x^n| = \gamma_{n,\mu}\, \rho_{A'}^{\otimes n}\, . \label{bd2} \end{align}
The first inequality in \reff{bd2} follows from \reff{type4} in Appendix~\ref{sc:types}, which is a consequence of the fact that $D(t\|q) \le \mu$.
Next, we use \reff{bd1shot} and \reff{bd2} to obtain
\begin{align} & \log M_{\rm ea}^*({\cal N}^n,\varepsilon) \\
&\qquad \ge D_H^{\varepsilon - 2 \delta} \bigg({\cal N}^n\left( \vartheta_{A^{n}A'^{n}}\right) \,\bigg\|\ \sum_{t \in {\cal P}_n^{q,\mu}} p(t)\left( {\cal N}^n(\pi_{A^{ n}}^t)\right) \otimes \gamma_{n,\mu} \,\rho_{A'}^{\otimes n}\bigg) - f(\varepsilon, \delta) \\
&\qquad = D_H^{\varepsilon - 2 \delta}\bigg({\cal N}^n\left( \vartheta_{A^{ n}A'^n}\right) \,\bigg\|\, \sum_{t \in {\cal P}_n^{q,\mu}} p(t)\left( {\cal N}^n(\pi_{A^{ n}}^t)\right) \otimes \rho_{A'}^{\otimes n}\bigg) - f(\varepsilon, \delta) - \log \gamma_{n,\mu}\\ &\qquad \ge D_H^{\varepsilon - 2 \delta- g(n,\mu)}\bigg(
\big( {\cal N}(\psi_{AA'}) \big)^{\otimes n} \,\bigg\|\, \sum_{t \in {\cal P}_n^{q,\mu}} p(t)\, {\cal N}^n(\pi_{A^{ n}}^t) \otimes \rho_{A'}^{\otimes n}\bigg) - f(\varepsilon, \delta) - \log \gamma_{n,\mu},\\ & \qquad \ge D_H^{\varepsilon - 2 \delta- g(n,\mu)}\bigg(
\big( {\cal N}(\psi_{AA'}) \big)^{\otimes n} \,\bigg\|\, \sum_{t \in {\cal P}_n} p(t)\, {\cal N}^n(\pi_{A^{ n}}^t) \otimes \rho_{A'}^{\otimes n}\bigg) - f(\varepsilon, \delta) - \log \gamma_{n,\mu},\\ & \qquad = D_H^{\varepsilon - 2 \delta - g(n,\mu)}\left(
\big( {\cal N}(\psi_{AA'}) \big)^{\otimes n} \,\Big\|\, \Big({\cal N}(\rho_{A})\right)^{\otimes n} \otimes \rho_{A'}^{\otimes n}\Big) - f(\varepsilon, \delta) - \log \gamma_{n,\mu}. \label{long} \end{align} The first and second lines follow from items $1$ and $2$ of Lemma~\ref{props-hypo}, respectively. The third line follows from item $4$ of Lemma~\ref{props-hypo}. The fourth line also follows from item $2$ of Lemma~\ref{props-hypo}, since \begin{equation} \sum_{t \in {\cal P}_n} p(t) {\cal N}^n(\pi_{A^{ n}}^t) \otimes \rho_{A'}^{\otimes n} \ge \sum_{t \in {\cal P}_n^{q,\mu}} p(t) {\cal N}^n(\pi_{A^{ n}}^t) \otimes \rho_{A'}^{\otimes n}. \end{equation} The last line follows from the linearity of $ {\cal N}^n$ and the fact that \begin{equation}
\sum_{t \in {\cal P}_n} p(t)\pi_{A^{ n}}^t = \mathop{\rm Tr}\nolimits_{A'^{ n}} (\psi_{AA'}^{\otimes n}) = \rho_{A}^{\otimes n}. \end{equation}
Let us choose $\delta =1/{\sqrt{n}}$ and $\mu = \left((|{\cal X}| + 1) \log (n+1)\right)/n$. Then \begin{equation}
g(n,\mu) = \frac{1}{\sqrt{(n+1)}} \le \frac{1}{\sqrt{n}}, \quad \textrm{and} \quad \varepsilon - 2 \delta- g(n,\mu) \ge \varepsilon - {3}/{\sqrt{n}}.
\end{equation} Since $D_H^\varepsilon(\rho\|\sigma) \ge D_H^{\varepsilon'}(\rho\|\sigma)$ for $\varepsilon > \varepsilon'$, we obtain the following bound from \reff{long} \begin{equation}
\log M_{\rm ea}^*({\cal N}^n,\varepsilon) \ge D_H^{ \varepsilon - {3}/{\sqrt{n}}}\Big(
\big( {\cal N}(\psi_{AA'}) \big)^{\otimes n} \Big\| \left({\cal N}(\rho_{A})\right)^{\otimes n} \otimes \rho_{A'}^{\otimes n}\Big) - f(\varepsilon, \delta) - \log \gamma_{n,\mu}, \label{one} \end{equation} thus arriving at an expression involving the hypothesis testing relative entropy for product states. Substituting the above choices for $\delta$ and $\mu$ in the expressions \reff{fed} for $f(\varepsilon, \delta)$ and in $\gamma_{n,\mu}$, we find that \begin{equation}\label{two} f(\varepsilon, \delta) + \log \gamma_{n,\mu} = O(\log n). \end{equation}
Crucially, Lemma~\ref{second-order} applied to \eqref{one} now implies that \begin{align} \log M_{\rm ea}^*({\cal N}^n,\varepsilon) & \ge n D\big(
{\cal N}(\psi_{AA'}) \, \big\|\, {\cal N}(\rho_{A}) \otimes \rho_{A'}\big) \nonumber \\
& \qquad + \sqrt{nV\big({\cal N}(\psi_{AA'}) \,\big\|\, {\cal N}(\rho_{A}) \otimes \rho_{A'} \big)}\, \Phi^{-1}(\varepsilon - {3}/{\sqrt{n}}) + K'(n; {\cal N},\varepsilon,\psi_{AA'}) \\
&= n I(A':B)_{\omega} + \sqrt{n V(A':B)_{\omega}}\,\Phi^{-1}(\varepsilon - {3}/{\sqrt{n}}) + K'(n; {\cal N},\varepsilon,\psi_{AA'}) , \end{align} where $K'(n; {\cal N},\varepsilon,\psi_{AA'}) = O(\log n)$ due to \eqref{two}. To conclude the proof, note that $\Phi^{-1}$ is continuously differentiable around $\varepsilon > 0$, and thus $\Phi^{-1}(\varepsilon - {3}/{\sqrt{n}}) = \Phi^{-1}(\varepsilon) + O(1/\sqrt{n})$. \end{proof}
\subsection{Second-Order Converse for Covariant Quantum Channels} \label{sec_converse}
In this section, we observe that the Gaussian approximation is valid for the entanglement-assisted capacity of covariant quantum channels (i.e., Conjecture~\ref{conj:main} is true for this class of channels). Holevo first defined the class of covariant quantum channels \cite{H02}, and it is now known that many channels fall within this class, including depolarizing channels, transpose depolarizing channels \cite{WH02,FHMV04}, Pauli channels, cloning channels \cite{B11}, etc. Note that the following argument up to~\eqref{eq:matthews} has already essentially been proven in Section~III-E\ of Matthews and Wehner \cite{MW12}. However, we give a brief exposition in this section for completeness. We leave open the question of determining whether the Gaussian approximation is valid for the entanglement-assisted capacity of general discrete memoryless quantum channels.
Let $\mathcal{N}_{A\rightarrow B}$ be a quantum channel mapping density operators acting on an input Hilbert space~$\mathcal{H}_{A}$ to those acting on an output Hilbert space~$\mathcal{H}_{B}$. Let $G$ be a compact group, and for every $g\in G$, let $g\rightarrow U_{A}( g) $ and $g\rightarrow V_{B}( g) $ be continuous projective unitary representations of $G$ in $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, respectively. Then the channel $\mathcal{N}_{A\rightarrow B}$ is said to be covariant with respect to these representations if the following relation holds for all $g\in G$ and input density operators $\rho$: \begin{equation} \mathcal{N}_{A\rightarrow B}\!\left( U_{A}( g) \rho U_{A}^{\dag }( g) \right) =V_{B}( g) \mathcal{N}_{A\rightarrow B}( \rho) V_{B}^{\dag}( g) . \end{equation} We restrict our attention in this section to covariant channels for which the representation acting on the input space is irreducible.
In \cite[Thm.~14]{MW12}, Matthews and Wehner establish the following upper bound on the one-shot entanglement-assisted capacity of a channel $\mathcal{N}\equiv\mathcal{N}_{A\rightarrow B}$. \begin{equation} \log M_{\text{ea}}^{\ast}( \mathcal{N},\varepsilon) \leq \max_{\rho_{A}}\min_{\sigma_{B}}D_{H}^{\varepsilon}\big(\mathcal{N}(
\phi_{AA^{\prime}}^{\rho}) \big\|\,\rho_{A^{\prime}}\otimes\sigma _{B}\big),\label{eq:one-shot-converse} \end{equation} where $\phi_{AA^{\prime}}^{\rho}$ is a purification of $\rho_{A}$ and $\rho_{A^{\prime}}$ is the reduction of $\phi_{AA^{\prime}}^{\rho}$ to $A^{\prime}$. They also prove~\cite[Thm.~19]{MW12} that the quantity
$\beta_{\varepsilon}\left( \mathcal{N}_{A\to B}( \phi _{AA^{\prime}}^{\rho}) \|\rho_{A^{\prime}}\otimes\sigma_{B}\right) $\ defined through \eqref{beta} is convex in the input density operator $\rho_{A}$ for any $\sigma_{B}$, from which it follows that the quantity \begin{equation} \beta_{\varepsilon}( \mathcal{N}_{A\to B},\rho_{A})
:=\max_{\sigma_{B}}\beta_{\varepsilon}\big( \mathcal{N}_{A\to B}( \phi_{AA^{\prime}}^{\rho}) \big\|\,\rho_{A^{\prime}}\otimes \sigma_{B}\big) \end{equation} is convex in $\rho_{A}$ because it is the pointwise maximum of a set of convex functions.
We would now like to apply these results to the entanglement-assisted capacity of any discrete memoryless covariant channel $\mathcal{N}_{A^{n}\rightarrow B^{n}}\equiv\mathcal{N}^{\otimes n}$.
By definition, such channels have the following covariance: \begin{multline} \mathcal{N}_{A^{n}\rightarrow B^{n}}\left( \left[ U_{A_{1}}( g_{1}) \otimes\cdots\otimes U_{A_{n}}( g_{n}) \right] \rho_{A^{n}}\left[ U_{A_{1}}( g_{1}) \otimes\cdots\otimes U_{A_{n}}( g_{n}) \right] ^{\dag}\right) \\ =\left[ V_{B_{1}}( g_{1}) \otimes\cdots\otimes V_{B_{n}}( g_{n}) \right] \mathcal{N}_{A^{n}\rightarrow B^{n}}( \rho _{A^{n}}) \left[ V_{B_{1}}( g_{1}) \otimes\cdots\otimes V_{B_{n}}( g_{n}) \right] ^{\dag}. \end{multline} Let $T_{A^{n}}$ be a shorthand for
a sequence of local unitaries of the form $U_{A_{1}}( g_{1}) \otimes\cdots\otimes U_{A_{n}}( g_{n}) $. Let $\mathbb{E}$ denote the expectation over all such unitaries $T_{A^{n}}$, with the measure being the product Haar measure $\mu(g_1) \times \cdots \times \mu(g_n)$. Then following~\cite[Sec.~III-E]{MW12}, we can conclude the following chain of inequalities: \begin{align} \beta_{\varepsilon}( \mathcal{N}_{A^{n}\rightarrow B^{n}},\rho_{A^{n} }) & =\mathbb{E}\left\{ \beta_{\varepsilon}\left( \mathcal{N} _{A^{n}\rightarrow B^{n}},T_{A^{n}}\rho_{A^{n}}T_{A^{n}}^{\dag}\right) \right\} \\ & \geq\beta_{\varepsilon}\left( \mathcal{N}_{A^{n}\rightarrow B^{n} },\mathbb{E}\left\{ T_{A^{n}}\rho_{A^{n}}T_{A^{n}}^{\dag}\right\} \right) \\ & =\beta_{\varepsilon}\left( \mathcal{N}_{A^{n}\rightarrow B^{n}},\pi _{A_{1}}\otimes\cdots\otimes\pi_{A_{n}}\right) , \end{align} where $\pi$ is the maximally mixed state. The first equality is a result of \cite[Prop.~29]{MW12} (this follows directly from the assumption of channel covariance with respect to the operations $T_{A^{n}}$). The sole inequality exploits convexity as mentioned above. The last equality follows because the state $\mathbb{E}\big\{ T_{A^{n}}\rho_{A^{n}}T_{A^{n}}^{\dag}\big\}$ commutes with all local unitaries $U_{A_{1} }( g_{1}) \otimes\cdots\otimes U_{A_{n}}( g_{n}) $. As a consequence of Schur's lemma and the irreducibility of the representation on the input space, the only state which possesses such invariances is the tensor-power maximally mixed state. Note that we require irreducibility of the representation on only the input space in order for this argument to hold. So, by using the definition of $D_{H}^{\varepsilon}$, we can then conclude that \begin{align} \log M_{\text{ea}}^{\ast}( \mathcal{N}_{A^{n}\rightarrow B^{n} },\varepsilon) & \leq\max_{\rho_{A^{n}}}\min_{\sigma_{B^{n}}} D_{H}^{\varepsilon}( \mathcal{N}_{A^{n}\rightarrow B^{n}}(
\phi_{A^{n}A^{\prime n}}^{\rho}) \big\|\, \rho_{A^{\prime n}}\otimes \sigma_{B^{n}}) \\ & \leq\min_{\sigma_{B^{n}}}D_{H}^{\varepsilon}( ( \mathcal{N} _{A\rightarrow B}( \Phi_{AA^{\prime}}) )^{\otimes n}
\big\|\, \pi_{A^{\prime}}^{\otimes n}\otimes\sigma_{B^{n}}) \\ & \leq D_{H}^{\varepsilon}( ( \mathcal{N}_{A\rightarrow B}(
\Phi_{AA^{\prime}}) )^{\otimes n} \big\|\, \pi_{A}^{\otimes n} \otimes\left[ \mathcal{N}_{A\rightarrow B}\left( \pi_{A}\right) \right] ^{\otimes n}) \label{eq:matthews}\\ & =n I( A^{\prime}:B)_{\omega} +\sqrt{n V( A^{\prime}:B)_{\omega} }\, \Phi^{-1}( \varepsilon) +O( \log n) , \end{align} where the information quantities in the final line are with respect to the state $\omega_{A'B} := \mathcal{N}_{A\rightarrow B}( \Phi_{AA^{\prime}}) $. The final equality uses the asymptotic expansion in Lemma~\ref{second-order}.
\section{Discussion} \label{sec_discussion}
We have established the direct part of the Gaussian approximation in Theorem~\ref{th:main} and conjectured that the converse also holds in Conjecture~\ref{conj:main}. We again note that all of our results apply to entanglement-assisted quantum communication as well, due to the teleportation \cite{BBCJPW93} and super-dense coding \cite{PhysRevLett.69.2881} protocols and the results of \cite{LM14}. In the following we will discuss some of the approaches taken and difficulties encountered when trying to prove the converse for general channels.
\begin{description}
\item[Arimoto Converse:] Converse proofs using Arimoto's approach~\cite{arimoto73} and quantum generalizations of the R\'enyi divergence~\cite{lennert13,wilde13} as in~\cite{GW13} can be used to establish that the probability of successful decoding goes to zero exponentially fast for codes with $\frac{1}{n} \log |M| > C_{\textrm{ea}}$. However, they only yield trivial results when $\frac{1}{n} \log |M| = C_{\textrm{ea}} \pm O(1/\sqrt{n})$, as is the case in the Gaussian approximation.
\item[De Finetti Theorems:] Following Matthews and Wehner~\cite{MW12}, we find the following converse bound for $n$ uses of the channel employing the arguments presented in Section~\ref{sec_converse} and~\eqref{eq:one-shot-converse}.
\begin{align}
\log M_{\text{ea}}^*({\cal N}^{n}, \varepsilon) \leq \max_{\rho_{A^n}}\min_{\sigma_{B^n}} D_{H}^{\varepsilon}\big( {\cal N} (
\phi_{A^nA'^n}^{\rho}) \big\|\, \rho_{A'^n}\otimes\sigma_{B^n}\big) ,
\end{align}
where $\rho_{A^n}$ and $\sigma_B^n$ are invariant under permutations of the $n$ systems, and $\phi_{A^nA'^n}^{\rho}$ is chosen to have this property as well. One may now try to approximate the state $\phi_{A^nA'^n}$ by a convex combination of product states using the de Finetti theorem or the exponential de Finetti theorem~\cite{renner07}. However, the problem is that the number of systems that need to be sacrificed is at least of the order $\sqrt{n}$, and thus affects the second-order term significantly.
\item[Relation to Channel Simulation:] EAC coding is closely related to the classical communication cost in entanglement-assisted channel simulation~\cite{BDHSW12} and~\cite{BCR09}. In the latter paper, some bounds on the classical communication cost of entanglement-assisted channel simulation for a finite number of channels $n$ are given. However, these bounds turn out to be unsuitable for our purposes since the error is scaled by a factor polynomial in $n$ as a result of applying the
post-selection technique~\cite{christandl09}. It is not clear how the proof in~\cite{BCR09} can be adapted to yield a statement for fixed error. \end{description}
We believe that establishing Conjecture~\ref{conj:main} thus requires new techniques and that this constitutes an interesting open problem.
\paragraph*{Acknowledgements.} We are especially grateful to Milan Mosonyi for insightful discussions and for his help in establishing the proof of Propositions~\ref{thm-one-shot} and \ref{pr:direct-second}. We acknowledge discussions with Mario Berta, Ke~Li, Will Matthews, and Andreas Winter, \MT{and we thank the Isaac Newton Institute (Cambridge) for its hospitality while part of this work was completed.}
MT is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant ``Random numbers from quantum processes'' (MOE2012-T3-1-009). MMW acknowledges startup funds from the Department of Physics and Astronomy at LSU, support from the NSF through Award No.~CCF-1350397, and support from the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019.
\appendix
\section{Heisenberg-Weyl Operators} \label{app:hw}
For any $x,z \in \{0,1,\cdots, d\}$ the Heisenberg-Weyl Operators $X(x)$ and $Z(z)$ are defined through their actions on the vectors of the qudit computational basis $\{|j\rangle \}_{j \in \{0,1,\cdots, d-1\}}$ as follows:
\begin{align}\label{Weyl}
X(x) |j\rangle &= |j \oplus x\rangle,\\
Z(z)|j\rangle &= e^{2\pi i z j / d} |j\rangle , \end{align} where $j \oplus x = (j + x) \, {\rm mod }\, d $. Also note that if $d = 1$, then both $X(x)$ and $Z(z)$ are equal to the identity operator.
\section{The Method of Types} \label{sc:types}
In our proofs we employ the notion of {\em{types}}~\cite{csiszar98}, and hence we briefly recall certain relevant definitions and properties here.
Let ${\cal X}$ denote a discrete alphabet and fix $n \in \mathbb{N}$. The {\em{type}} (or empirical probability distribution) $P_{x^n}$ of a sequence $x^n \in {\cal X}^n$ is the empirical frequency of occurrences of each letter of ${\cal X}$, i.e., $P_{x^n}(a) := \frac1{n} \sum_{i=1}^n \delta_{x_i,a}$
for all $a \in {\cal X}$.
Let ${\cal P}_n$ denote the set of all types. The number of types, $|{\cal P}_n|$, satisfies the bound~\cite[Thm.~11.1.1]{CoverThomas} \begin{equation}\label{type1}
|{\cal P}_n| \le (n+1)^{|{\cal X}|}. \end{equation} For any type $t \in {\cal P}_n$, the {\em{type class}} ${\cal T}^t$ of $t$ is the set of sequences of type $t$, i.e.\ \begin{equation} {\cal T}^t := \{x^n \in {\cal X}^n \, : \, P_{x^n} = t \}. \end{equation} The number of types in a type class ${\cal T}^t$ satisfies the following lower bound~\cite[Lm.~II.2]{csiszar98}: \begin{equation}\label{type2}
|{\cal T}^t| \ge \frac{2^{nH(t)}}{(n+1)^{|{\cal X}|}}, \end{equation} where $H(t) := - \sum_{a \in {\cal X}} t(a) \log t(a)$, is the Shannon entropy of the type.
Let $q$ be any probability distribution on ${\cal X}$. For any sequence $x^n = (x_1, x_2, \ldots, x_n) \in {\cal X}^n$,
let $q^n(x^n) = \prod_{i=1}^n q(x_i)$. Then, we have \begin{equation}\label{type3}
q^n(x^n) = 2^{-n\left(H(t) + D(t\|q)\right)}, \qquad \textrm{where} \quad t = P_{x^n}
\end{equation} is the type of $x^n$ and $D(t\|q):= \sum_{a \in {\cal X}} t(a) \log \frac{t(a)}{q(a)}$ is the Kullback-Leibler divergence of the probability distributions $t$ and $q$.
From \reff{type1}, \reff{type2} and \reff{type3} it follows that for any sequence $x^n \in {\cal X}^n$ of type $t$, \begin{equation} \label{type4}
(n+1)^{|{\cal X}|} 2^{nD(t\|q)} q^n(x^n) = 2^{-nH(t)}(n+1)^{|{\cal X}|}\ge \frac{1}{|{\cal T}^t|}. \end{equation}
Finally, for any $\mu > 0$ we have~\cite[Eq.~(11.98)]{CoverThomas} \begin{equation}\label{type6}
\sum_{x^n \in {\cal X}^n \atop D(P_{x^n} \| q) > \mu} q^n(x^n) \le 2^{-n\left( \mu - |{\cal X}|\frac{ \log (n+1)}{n}\right)}. \end{equation}
\end{document} | arXiv |
\begin{document}
\title{Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng's F-B four-operator splitting method for solving monotone inclusions} \author{} \author{
M. Marques Alves \thanks{ Departamento de Matem\'atica, Universidade Federal de Santa Catarina, Florian\'opolis, Brazil, 88040-900 ({\tt [email protected]}). The work of this author was partially supported by CNPq grants no. 406250/2013-8, 306317/2014-1 and 405214/2016-2.} \and Marina Geremia \thanks{ Departamento de Matem\'atica, Universidade Federal de Santa Catarina, Florian\'opolis, Brazil, 88040-900 ({\tt [email protected]}).
} }
\maketitle
\begin{abstract}
In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng's forward-backward (F-B) splitting method for solving two-operator and four-operator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. Yao, in which an inexact DRS method is derived from a special instance of the hybrid proximal extragradient (HPE) method of Solodov and Svaiter, while the latter one combines the proposed inexact DRS method (used as an outer iteration) with a Tseng's F-B splitting type method (used as an inner iteration) for solving the corresponding subproblems. We prove iteration complexity bounds for both algorithms in the pointwise (non-ergodic) as well as in the ergodic sense by showing that they admit two different iterations: one that can be embedded into the HPE method, for which the iteration complexity is known since the work of Monteiro and Svaiter, and another one which demands a separate analysis.
Finally, we perform simple numerical experiments
to show the performance of the proposed methods when compared with other existing algorithms. \\ \\
2000 Mathematics Subject Classification: 47H05, 49M27, 90C25.
\\
\\
Key words: Inexact Douglas-Rachford method; splitting; monotone operators; HPE method; complexity;
Tseng's forward-backward method. \end{abstract}
\pagestyle{plain}
\section{Introduction}
Let $\mathcal{H}$ be a real Hilbert space. In this paper, we consider the \emph{two-operator monotone inclusion problem} (MIP) of finding $z$ such that
\begin{align}
\label{eq:mip.i}
0\in A(z)+B(z) \end{align}
as well as the \emph{four-operator} MIP
\begin{align}
\label{eq:drti}
0\in A(z)+C(z)+F_1(z)+F_2(z) \end{align}
where $A$, $B$ and $C$ are (set-valued) maximal monotone operators on $\mathcal{H}$, $F_1: D(F_1)\to \mathcal{H}$ is (point-to-point) \emph{Lipschitz continuous} and $F_2:\mathcal{H}\to \mathcal{H}$ is (point-to-point) \emph{cocoercive} (see Section \ref{sec:drt} for the precise statement).
Problems \eqref{eq:mip.i} and \eqref{eq:drti} appear in different fields of applied mathematics and optimization including convex optimization, signal processing, PDEs, inverse problems, among others~\cite{bau.com-book,glo.osh.yin-spl.spi16}.
Under mild conditions on the operators $C$, $F_1$ and $F_2$, problem \eqref{eq:drti} becomes a special instance of \eqref{eq:mip.i} with $B:=C+F_1+F_2$. This fact will be important later on in this paper.
In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting method (Algorithm \ref{inexact.dr}) and of a Douglas-Rachford-Tseng's forward-backward (F-B) four-operator splitting method (Algorithm \ref{drt}) for solving \eqref{eq:mip.i} and \eqref{eq:drti}, respectively.
The former method is inspired and motivated (although based on a slightly different mechanism of iteration) by the recent work of J. Eckstein and W. Yao~\cite{eck.yao-rel.mp17}, while the latter one, which, in particular, will be shown to be a special instance of the former one, is motivated by some variants of the standard Tseng's F-B splitting method~\cite{tse-mod.sjco00} recently proposed in the current literature~\cite{alv.mon.sva-reg.siam16,arias.davis-half,MonSva10-1}.
For more detailed information about the contributions of this paper in the light of reference \cite{eck.yao-rel.mp17}, we refer the reader to the first remark after Algorithm \ref{inexact.dr}. Moreover, we mention that Algorithm \ref{drt} is a purely primal splitting method for solving the \emph{four-operator} MIP \eqref{eq:drti}, and this seems to be new. The main contributions of this paper will be discussed in Subsection \ref{subsec:main.cont}.
\subsection{The Douglas-Rachford splitting (DRS) method}
\label{subsec:drsmi}
One of the most popular algorithms for finding approximate solutions of \eqref{eq:mip.i} is the \emph{Douglas-Rachford splitting} (DRS) \emph{method}. It consists of an iterative procedure in which at each iteration the resolvents $J_{\gamma A}=(\gamma A+I)^{-1}$ and $J_{\gamma B}=(\gamma B+I)^{-1}$ of $A$ and $B$, respectively, are employed separately instead of the resolvent $J_{\gamma(A+B)}$ of the full operator $A+B$, which may be expensive to compute numerically.
An iteration of the method can be described by
\begin{align}
\label{eq:dr.ite}
z_{k}= J_{\gamma A}(2J_{\gamma B}(z_{k-1})-z_{k-1})+z_{k-1}-J_{\gamma B}(z_{k-1})\qquad \forall k\geq 1, \end{align}
where $\gamma>0$ is a scaling parameter and $z_{k-1}$ is the current iterate. Originally proposed in \cite{dou.rac-num.tams56} for solving problems with linear operators, the DRS method was generalized in~\cite{lio.mer-spl.sjna79} for general nonlinear maximal monotone operators, where the formulation \eqref{eq:dr.ite} was first obtained.
It was proved in \cite{lio.mer-spl.sjna79} that $\{z_k\}$ converges (weakly, in infinite dimensional Hilbert spaces) to some $z^*$ such that $x^*:=J_{\gamma B}(z^*)$ is a solution of \eqref{eq:mip.i}.
Recently, \cite{sva-wea.sjco11} proved the (weak) convergence of the sequence $\{z_k\}$ generated in \eqref{eq:dr.ite} to a solution of \eqref{eq:mip.i}.
\subsection{The Rockafellar's proximal point (PP) method} \label{subsec:ppmi}
The \emph{proximal point} (PP) \emph{method} is an iterative method for seeking approximate solutions of the MIP
\begin{align}
\label{eq:mip.Ti}
0\in T(z) \end{align}
where $T$ is a maximal monotone operator on $\mathcal{H}$ for which the solution set of \eqref{eq:mip.Ti} is nonempty. In its exact formulation, an iteration of the PP method can be described by
\begin{align}
\label{eq:exact.prox}
z_k= (\lambda_k T+I)^{-1}z_{k-1}\qquad \forall k\geq 1, \end{align}
where $\lambda_k>0$ is a stepsize parameter and $z_{k-1}$ is the current iterate.
It is well-known that the practical applicability of numerical schemes based on the exact computation of resolvents of monotone operators strongly depends on strategies that allow for inexact computations. This is the case of the PP method \eqref{eq:exact.prox}. In his pioneering work~\cite{roc-mon.sjco76}, Rockafellar proved that if, at each iteration $k\geq 1$, $z_k$ is computed satisfying
\begin{align}
\label{eq:inexact.prox}
\norm{z_k-(\lambda_k T+I)^{-1}z_{k-1}}\leq e_k,\quad \sum_{k=1}^\infty\,e_k<\infty, \end{align}
and $\{\lambda_k\}$ is bounded away from zero, then $\{z_k\}$ converges (weakly, in infinite dimensions) to a solution of \eqref{eq:mip.Ti}. This result has found important applications in the design and analysis of many practical algorithms for solving challenging problems in optimization and related fields.
\subsection{The DRS method is an instance of the PP method (Eckstein and Bertsekas)}
\label{subsec:drsm.eh.ppm}
In~\cite{eck.ber-dou.mp92}, the DRS method \eqref{eq:dr.ite} was shown to be a special instance of the PP method \eqref{eq:exact.prox} with $\lambda_k\equiv 1$. More precisely, it was observed in~\cite{eck.ber-dou.mp92} (among other results) that the sequence $\{z_k\}$ in \eqref{eq:dr.ite} satisfies
\begin{align}
\label{eq:car.dr.s}
z_{k}=(S_{\gamma,\,A,\,B}+I)^{-1}z_{k-1}\qquad \forall k\geq 1, \end{align}
where $S_{\gamma,\,A,\,B}$ is the maximal monotone operator on $\mathcal{H}$ whose graph is
\begin{align}
\label{eq:def.s}
S_{\gamma,\,A,\,B}=\left\{(y+\gamma b,\gamma a+\gamma b)\in \mathcal{H}\times \mathcal{H}\;|\;b\in B(x),\,a\in A(y),\, \gamma a+y=x-\gamma b\right\}. \end{align}
It can be easily checked that $z^*$ is a solution of \eqref{eq:mip.i} if and only if $z^*=J_{\gamma B}(x^*)$ for some $x^*$ such that $0\in S_{\gamma,\,A\,B}(x^*)$.
The fact that \eqref{eq:dr.ite} is equivalent to \eqref{eq:car.dr.s} clarifies the proximal nature of the DRS method and allowed \cite{eck.ber-dou.mp92} to obtain inexact and relaxed versions of it by alternatively describing \eqref{eq:car.dr.s} according to the following procedure:
\begin{align}
\label{eq:dr.spp} &\mbox{compute}\;\; (x_k,b_k)\;\;\mbox{such that}\;\; b_k\in B(x_k)\;\;\mbox{and}\;\;\gamma b_k+x_k=z_{k-1};\\ \label{eq:dr.spp02}
&\mbox{compute}\;\; (y_k,a_k)\;\;\mbox{such that}\;\; a_k\in A(y_k)\;\;\mbox{and}\;\;\gamma a_k+y_k=x_k-\gamma b_k;\\ \label{eq:dr.spp03}
&\mbox{set}\;\; z_{k}=y_k+\gamma b_k.
\end{align}
\subsection{The hybrid proximal extragradient (HPE) method of Solodov and Svaiter}
\label{subsec:hpei}
Many modern inexact versions of the PP method, as opposed to the summable error criterion \eqref{eq:inexact.prox}, use \emph{relative error tolerances} for solving the associated subproblems. The first method of this type was proposed in \cite{sol.sva-hyb.jca99}, and subsequently studied, e.g., in~\cite{MonSva10-1,mon.sva-hpe.siam10,MonSva10-2,sol.sva-hyb.svva99,sol.sva-hyb.jca99,sol.sva-ine.mor00,Sol-Sv:hy.unif}. The key idea consists of decoupling \eqref{eq:exact.prox} in an inclusion-equation system:
\begin{align}
\label{eq:dec.prox}
v\in T(z_+),\quad \lambda v+z_{+}-z=0, \end{align}
where $(z,z_+,\lambda):=(z_{k-1},z_k,\lambda_k)$, and relaxing \eqref{eq:dec.prox} within relative error tolerance criteria.
Among these new methods, the \emph{hybrid proximal extragradient} (HPE) method of Solodov and Svaiter \cite{sol.sva-hyb.svva99}, which we discuss in details in Subsection \ref{sec:pp}, has been shown to be very effective as a framework for the design and analysis of many concrete algorithms (see, e.g.,~\cite{bot.cse-hyb.nfao15,cen.mor.yao-hyb.jota10,eck.sil-pra.mp13,he.mon-acc.siam16,ius.sos-pro.opt10,lol.par.sol-cla.jca09,mon.ort.sva-imp.coap14,mon.ort.sva-ada.coap16,MonSva10-2,sol.sva-hyb.svva99,sol.sva-ine.mor00,Sol-Sv:hy.unif}).
\subsection{The main contributions of this work}
\label{subsec:main.cont}
In \cite{eck.yao-rel.mp17}, J. Eckstein and W. Yao proposed and studied the (asymptotic) convergence of an inexact version of the DRS method \eqref{eq:dr.ite} by applying a special instance of HPE method to the maximal monotone operator given in \eqref{eq:def.s}.
The resulting algorithm (see \cite[Algorihm 3]{eck.yao-rel.mp17}) allows for inexact computations \emph{in the equation} in \eqref{eq:dr.spp} and, in particular, resulted in an inexact version of the ADMM which is suited for large-scale problems, in which fast inner solvers can be employed for solving the corresponding subproblems (see \cite[Section 6]{eck.yao-rel.mp17}).
In the present work, motivated by \cite{eck.yao-rel.mp17}, we first propose in Section \ref{sec:dr} an inexact version of the DRS method (Algorithm \ref{inexact.dr}) for solving \eqref{eq:mip.i} in which inexact computations are allowed \emph{in both the inclusion and the equation} in \eqref{eq:dr.spp}. At each iteration, instead of a point in the graph of $B$, Algorithm \ref{inexact.dr} computes a point in the graph of the $\varepsilon$-enlargement $B^\varepsilon$ of $B$ (it has the property that $B^{\varepsilon}(z)\supset B(z)$). Moreover, contrary to the reference \cite{eck.yao-rel.mp17}, we study the \emph{iteration complexity} of the proposed method for solving \eqref{eq:mip.i}.
We show that Algorithm \ref{inexact.dr} admits two type of iterations, one that can be embedded into the HPE method and, on the other hand, another one which demands a separate analysis. We emphasize again that, although motivated by the latter reference, the Douglas-Rachford type method proposed in this paper is based on a slightly different mechanism of iteration, specially designed for allowing its iteration complexity analysis (see Theorems \ref{th:idr.main} and \ref{th:idr.main.erg}).
Secondly, in Section \ref{sec:drt}, we consider the four-operator MIP \eqref{eq:drti} and propose and study the iteration complexity of a Douglas-Rachford-Tseng's F-B splitting type method (Algorithm \ref{drt}) which combines Algorithm \ref{inexact.dr} (as an outer iteration) and a Tseng's F-B splitting type method (Algorithm \ref{hff}) (as an inner iteration) for solving the corresponding subproblems. The resulting algorithm, namely Algorithm \ref{drt}, has a splitting nature and solves \eqref{eq:drti} without introducing extra variables.
Finally, in Section \ref{sec:num}, we perform simple numerical experiments
to show the performance of the proposed methods when compared with other existing algorithms.
\subsection{Most related works}
\label{subsec:mrw}
In \cite{ari-for.opt15}, the relaxed forward-Douglas-Rachford splitting (rFDRS) method was proposed and studied to solve \emph{three-operator MIPs} consisting of \eqref{eq:drti} with $C=N_V$, $V$ closed vector subspace, and $F_1=0$.
Subsequently, among other results, the iteration complexity of the latter method (specialized to variational problems) was analyzed in \cite{dav-con.sjo15}.
Problem \eqref{eq:drti} with $F_1=0$ was also considered in \cite{dav.yin-thr.svva17}, where a three-operator splitting (TOS) method was proposed and its iteration complexity studied.
On the other hand, problem \eqref{eq:drti} with $C=N_V$ and $F_2=0$ was studied in \cite{ari-for.jota15}, where the forward-partial inverse-forward splitting method was proposed and analyzed. In \cite{arias.davis-half}, a Tseng's F-B splitting type method was proposed and analyzed to solve the special instance of \eqref{eq:drti} in which $C=0$.
The iteration complexity of a relaxed Peaceman-Rachford splitting method for solving \eqref{eq:mip.i} was recently studied in \cite{mon.che-com.17}. The method of \cite{mon.che-com.17} was shown to be a special instance of a non-Euclidean HPE framework, for which the iteration complexity was also analyzed in the latter reference (see also \cite{gon.mel.mon-imp.pre16}).
Moreover, as we mentioned earlier, an inexact version of the DRS method for solving \eqref{eq:mip.i} was proposed and studied in \cite{eck.yao-rel.mp17}.
\section{Preliminaries and background materials}
\label{sec:pre}
\subsection{General notation and $\varepsilon$-enlargements} \label{sec:gn}
We denote by $\mathcal{H}$ a real Hilbert space with inner product $\inner{\cdot}{\cdot}$
and induced norm $\|\cdot\|:=\sqrt{\inner{\cdot}{\cdot}}$ and by $\mathcal{H}\times \mathcal{H}$ the product Cartesian endowed with usual inner product and norm.
A set-valued map $T:\mathcal{H}\rightrightarrows \mathcal{H}$ is said to be a \emph{monotone operator} on $\mathcal{H}$ if $\inner{z-z'}{v-v'}\geq 0$ for all $v\in T(z)$ and $v'\in T(z')$. Moreover, $T$ is a \emph{maximal monotone operator} if $T$ is monotone and $T=S$ whenever
$S$ is monotone on $\mathcal{H}$ and $T\subset S$. Here, we identify any monotone operator $T$ with its graph, i.e., we set $T=\{(z,v)\in \mathcal{H}\times \mathcal{H}\,|\,v\in T(z)\}$. The \emph{sum} $T+S$ of two set-valued maps $T,S$ is defined via
the usual Minkowski sum and for $\lambda\geq 0$ the operator $\lambda T$ is defined by $(\lambda T)(z)= \lambda T(z):=\{\lambda v\,|\,v\in T(z)\}$. The \emph{inverse} of $T:\mathcal{H}\rightrightarrows \mathcal{H}$ is $T^{-1}:\mathcal{H}\rightrightarrows \mathcal{H}$ defined by $v\in T^{-1}(z)$ if and only if $z\in T(v)$. In particular,
$\mbox{zer}(T):=T^{-1}(0)=\{z\in \mathcal{H}\,|\,0\in T(z)\}$. The \emph{resolvent} of a maximal monotone operator $T$ is $J_T:=(T+I)^{-1}$, where $I$ denotes the identity map on $\mathcal{H}$, and, in particular, the following holds: $x=J_{\lambda T}(z)$ if and only if $\lambda^{-1}(z-x)\in T(x)$ if and only if $0\in \lambda T(x)+x-z$. We denote by $\partial_\varepsilon f$ the usual $\varepsilon$-subdifferential of a proper closed convex function $f:\mathcal{H}\to (-\infty,+\infty]$ and by $\partial f:=\partial f_0$ the Fenchel-subdifferential of $f$ as well. The \emph{normal cone} of a closed convex set $X$ will be denoted by $N_X$ and by $P_X$ we denote the orthogonal projection onto $X$.
For $T:\mathcal{H}\rightrightarrows\mathcal{H}$ maximal monotone and $\varepsilon\geq 0$, the $\varepsilon$-enlargement~\cite{bur.ius.sva-enl.svva97,leg.the-sub.svva96} of $T$ is the operator $T^{\varepsilon}:\mathcal{H}\rightrightarrows\mathcal{H}$ defined by
\begin{align}
\label{eq:def.teps}
T^{\varepsilon}(z):=\{v\in \mathcal{H}\;|\;\inner{z-z'}{v-v'}\geq -\varepsilon\;\;\forall (z',v')\in T\}\quad \forall z\in \mathcal{H}. \end{align}
Note that $T(z)\subset T^{\varepsilon}(z)$ for all $z\in \mathcal{H}$.
The following summarizes some useful properties of $T^{\varepsilon}$ which will be useful in this paper.
\begin{proposition} \label{pr:teps} Let $T, S:\mathcal{H}\rightrightarrows \mathcal{H}$ be set-valued maps. Then, \begin{itemize}
\item[\emph{(a)}] if $\varepsilon \leq \varepsilon'$, then $T^{\varepsilon}(x)\subseteq T^{\varepsilon'}(x)$ for every $x \in \mathcal{H}$;
\item[\emph{(b)}] $T^{\varepsilon}(x)+S^{\,\varepsilon'}(x) \subseteq (T+S)^{\varepsilon+\varepsilon'}(x)$ for every $x \in \mathcal{H}$ and $\varepsilon, \varepsilon'\geq 0$; \item[\emph{(c)}] $T$ is monotone if, and only if, $T \subseteq T^{0}$; \item[\emph{(d)}] $T$ is maximal monotone if, and only if, $T = T^{0}$;
\end{itemize} \end{proposition}
Next we present the transportation formula for $\varepsilon$-enlargements.
\begin{theorem}\emph{(\cite[Theorem 2.3]{bur.sag.sva-enl.col99})}
\label{th:tf}
Suppose $T:\mathcal{H}\rightrightarrows \mathcal{H}$ is maximal monotone and
let $z_\ell, v_\ell\in \mathcal{H}$, $\varepsilon_\ell, \alpha_\ell\in \mathbb{R}_+$,
for $\ell=1,\dots, j$, be such that
\[
v_\ell\in T^{\varepsilon_\ell}(z_\ell),\quad \ell=1,\dots, j,\quad \sum_{\ell=1}^j\,\alpha_\ell=1,
\]
and define
\[
\overline{z}_j:=\sum_{\ell=1}^j\,\alpha_\ell\, z_\ell\,,\quad
\overline{v}_j:=\sum_{\ell=1}^j\,\alpha_\ell\; v_\ell\,,\quad
\overline{\varepsilon}_j:=\sum_{\ell=1}^j\,\alpha_\ell \left[\varepsilon_\ell+\inner{z_\ell-\overline{z}_j}
{v_\ell-\overline{v}_j}\right].
\]
Then, the following hold:
\begin{itemize}
\item[\emph{(a)}] $\overline{\varepsilon}_j\geq 0$ and
$\overline{v}_j\in T^{\overline{\varepsilon}_j}(\overline{z}_j)$.
\item[\emph{(b)}] If, in addition, $T=\partial f$ for some proper, convex and closed function
$f$ and $v_\ell\in \partial_{\varepsilon_\ell} f(z_{\ell})$ for $\ell=1,\dots, j$,
then $\overline{v}_j\in \partial_{\overline{\varepsilon}_j} f(\overline{z}_j)$.
\end{itemize} \end{theorem}
\subsection{The hybrid proximal extragradient (HPE) method} \label{sec:pp}
Consider the \emph{monotone inclusion problem} (MIP) \eqref{eq:mip.Ti}, i.e.,
\begin{align}
\label{eq:mip.T}
0\in T(z) \end{align}
where $T:\mathcal{H}\rightrightarrows \mathcal{H}$ is a maximal monotone operator for which the solution set $T^{-1}(0)$ of \eqref{eq:mip.T} is nonempty.
As we mentioned earlier, the proximal point (PP) method of Rockafellar~\cite{roc-mon.sjco76} is one of the most popular algorithms for finding approximate solutions of \eqref{eq:mip.T} and, among the modern inexact versions of the PP method, the \emph{hybrid proximal extragradient} (HPE) method of \cite{sol.sva-hyb.svva99}, which we present in what follows, has been shown to be very effective as a framework for the design and analysis of many concrete algorithms (see e.g.~\cite{bot.cse-hyb.nfao15,cen.mor.yao-hyb.jota10,eck.sil-pra.mp13, he.mon-acc.siam16,ius.sos-pro.opt10,lol.par.sol-cla.jca09,mon.ort.sva-imp.coap14,mon.ort.sva-ada.coap16,MonSva10-2,sol.sva-hyb.svva99,sol.sva-ine.mor00,Sol-Sv:hy.unif}).
\noindent \fbox{ \addtolength{\linewidth}{-2\fboxsep} \addtolength{\linewidth}{-2\fboxrule}
\begin{minipage}{\linewidth}
\begin{algorithm} \label{hpe} {\bf Hybrid proximal extragradient (HPE) method for \bf{(\ref{eq:mip.T})}} \end{algorithm} \begin{itemize}
\item[(0)] Let $z_0\in \mathcal{H}$ and $\sigma\in [0,1)$ be given and set $j\leftarrow 1$.
\item [(1)] Compute $(\widetilde z_j,v_j,\varepsilon_j)\in \mathcal{H}\times \mathcal{H}\times \mathbb{R}_+$ and $\lambda_j>0$ such that
\begin{align} \label{eq:hpe}
\begin{aligned}
v_j\in T^{\varepsilon_j}(\widetilde z_j),\quad \norm{\lambda_j v_j+\widetilde z_j-z_{j-1}}^2+
2\lambda_j\varepsilon_j \leq \sigma^2\norm{\widetilde z_j-z_{j-1}}^2. \end{aligned} \end{align}
\item[(2)] Define
\begin{align}
\label{eq:hpe2}
z_j=z_{j-1}-\lambda_j v_j,
\end{align}
set $j\leftarrow j+1$ and go to step 1.
\end{itemize} \noindent
\end{minipage} }
\noindent {\bf Remarks.}
\begin{enumerate}
\item If $\sigma=0$ in \eqref{eq:hpe}, then it follows from Proposition \ref{pr:teps}(d) and \eqref{eq:hpe2} that $(z_+,v):=(z_j,v_j)$ and $\lambda:=\lambda_j>0$ satisfy \eqref{eq:dec.prox}, which means that the HPE method generalizes the exact Rockafellar's PP method.
\item Condition \eqref{eq:hpe} clearly relaxes both the inclusion and the equation in \eqref{eq:dec.prox} within a relative error criterion. Recall that $T^{\varepsilon}(\cdot)$ denotes the $\varepsilon$-enlargement of $T$ and has the property that $T^{\varepsilon}(z)\supset T(z)$ (see Subsection \ref{sec:gn} for details). Moreover, in \eqref{eq:hpe2} an extragradient step from the current iterate $z_{j-1}$ gives the next iterate $z_{j}$.
\item We emphasize that specific strategies for computing the triple $(\widetilde z_j,v_j,\varepsilon_j)$ as well as the stepsize $\lambda_j>0$ satisfying \eqref{eq:hpe} will depend on the particular instance of the problem \eqref{eq:mip.T} under consideration. On the other hand, as mentioned before, the HPE method can also be used as a framework for the design and analysis of concrete algorithms for solving specific instances of \eqref{eq:mip.T} (see, e.g., \cite{eck.sil-pra.mp13,mon.ort.sva-imp.coap14,mon.ort.sva-ada.coap16,MonSva10-1,mon.sva-hpe.siam10,MonSva10-2}).
We also refer the reader to Sections \ref{sec:dr} and \ref{sec:drt}, in this work, for applications of the HPE method in the context of decomposition/splitting algorithms for monotone inclusions. \end{enumerate}
Since the appearance of the paper \cite{mon.sva-hpe.siam10}, we have seen an increasing interest in studding the \emph{iteration complexity} of the HPE method and its special instances (e.g., Tseng's forward-backward splitting method, Korpelevich extragradient method and ADMM~\cite{MonSva10-1,mon.sva-hpe.siam10,MonSva10-2}).
This depends on the following termination criterion~\cite{mon.sva-hpe.siam10}: given tolerances $\rho, \epsilon>0$, find $z, v\in \mathcal{H}$ and $\varepsilon>0$ such that
\begin{align}
\label{eq:tc}
v\in T^{\varepsilon}(z),\quad \norm{v}\leq \rho, \quad \varepsilon\leq \epsilon. \end{align}
Note that, by Proposition \ref{pr:teps}(d), if $\rho=\epsilon=0$ in \eqref{eq:tc} then $0\in T(z)$, i.e., $z\in T^{-1}(0)$.
We now summarize the main results on \emph{pointwise (non ergodic)} and \emph{ergodic} iteration complexity~\cite{mon.sva-hpe.siam10} of the HPE method that will be used in this paper.
The \emph{aggregate stepsize sequence} $\{\Lambda_j\}$ and the \emph{ergodic sequences} $\{\overline{\widetilde z}_j\}$, $\{\overline{v}_j\}$, $\{\overline{\varepsilon}_j\}$ associated to $\{\lambda_j\}$ and $\{\widetilde {z}_j\}$, $\{v_j\}$, and $\{\varepsilon_j\}$ are, respectively,
\begin{align} \label{eq:d.eg}
&\Lambda_j:=\sum_{\ell=1}^j\, \lambda_\ell\,,\\
\label{eq:d.eg2}
&\overline{\widetilde z}_j:= \frac{1}{\;\Lambda_j}\;
\sum_{\ell=1}^j\,\lambda_\ell\, \widetilde {z}_\ell, \quad
\overline{v}_j:= \frac{1}{\;\Lambda_j}\;\sum_{\ell=1}^j\, \lambda_\ell\, v_\ell,\\
\label{eq:d.eg3}
&\overline{\varepsilon}_j:=
\frac{1}{\;\Lambda_j}\;\sum_{\ell=1}^j\,\lambda_\ell \left[\varepsilon_\ell
+\inner{\widetilde {z}_\ell-
\overline{\widetilde z}_j}{v_\ell-\overline{v}_j}\right]=
\frac{1}{\;\Lambda_j}\;\sum_{\ell=1}^j\,\lambda_\ell \left[\varepsilon_\ell
+\inner{\widetilde {z}_\ell-
\overline{\widetilde z}_j}{v_\ell}\right].
\end{align}
\begin{theorem}[{\cite[Theorem 4.4(a) and 4.7]{mon.sva-hpe.siam10}}]
\label{lm:rhpe2}
Let $\{\widetilde z_j\}$, $\{v_j\}$, etc, be generated by the \emph{HPE method} \emph{(Algorithm \ref{hpe})}
and let $\{\overline{\widetilde z}_j\}$, $\{\overline{v}_j\}$, etc, be given in \eqref{eq:d.eg}--\eqref{eq:d.eg3}.
Let also $d_0$ denote the distance from $z_0$ to $T^{-1}(0)\neq\emptyset$
and assume that $\lambda_j\geq \underline{\lambda}>0$ for all $j\geq 1$. Then, the following hold:
\begin{enumerate}
\item[\emph{(a)}] For any $j\geq 1$,
there exists $i\in\{1,\dots,j\}$
such that
\begin{align*}
v_i\in T^{\varepsilon_i}(\widetilde z_i),\quad \norm{v_i}\leq \dfrac{d_0}{\underline{\lambda}\sqrt{j}}
\sqrt{\dfrac{1+\sigma}{1-\sigma}},\quad \varepsilon_i\leq \dfrac{\sigma^2 d_0^2}{2(1-\sigma^2)
\underline{\lambda}\;j}\,.
\end{align*}
\item[\emph{(b)}]
For any $j\geq 1$,
\begin{align*}
\overline{v}_j\in T^{\overline{\varepsilon}_j}(\overline{\widetilde z}_j),\quad
\norm{\overline{v}_j}\leq \dfrac{2d_0}{\underline{\lambda}\;j},\quad
\overline{\varepsilon}_j\leq \dfrac{2(1+\sigma/\sqrt{1-\sigma^2})d_0^2}{\underline{\lambda}\;j}\,.
\end{align*}
\end{enumerate} \end{theorem}
\noindent {\bf Remark.}
\begin{itemize}
\item[] The (\emph{pointwise} and \emph{ergodic}) bounds given in (a) and (b) of Theorem \ref{lm:rhpe2} guarantee, respectively, that
for given tolerances $\rho,\epsilon>0$, the termination criterion \eqref{eq:tc} is satisfied in at most
\begin{align*}
\mathcal{O}\left(\max\left\{\dfrac{d_0^2}{\underline{\lambda}^2\rho^2},
\dfrac{d_0^2}{\underline{\lambda}\epsilon}\right\}\right)
\;\;\mbox{and}\;\; \mathcal{O}\left(\max\left\{\dfrac{d_0}{\underline{\lambda}\rho},\dfrac{d_0^2}{\underline{\lambda}\epsilon} \right\}\right)
\end{align*}
iterations, respectively. We refer the reader to \cite{mon.sva-hpe.siam10} for a complete study of the iteration complexity of the HPE method and its special instances. \end{itemize}
\noindent The proposition below will be useful in the next sections.
\begin{proposition}[{\cite[Lemma 4.2 and Eq. (34)]{mon.sva-hpe.siam10}}] \label{pr:imp.fact}
Let $\{z_j\}$ be generated by the \emph{HPE method (Algorithm \ref{hpe})}. Then, for any $z^*\in T^{-1}(0)$, the sequence $\{\norm{z^*-z_j}\}$ is nonincreasing.
As a consequence, for every $j \ge 1$, we have
\begin{align} \label{eq:ineq.zk0}
\|z_j-z_0\| \le 2 d_0, \end{align} where $d_0$ denotes the distance of $z_0$ to $T^{-1}(0)$. \end{proposition}
\subsubsection{A HPE variant for strongly monotone sums}
\label{subsub:sms}
We now consider the MIP
\begin{align}
\label{eq:bmu}
0\in S(z)+B(z)=:T(z) \end{align}
where the following is assumed to hold:
\begin{itemize} \item[(C1)] $S$ and $B$ are maximal monotone operators on $\mathcal{H}$;
\item[(C2)] $S$ is (additionally) $\mu$--strongly monotone for some $\mu>0$, i.e., there exists $\mu>0$ such that
\begin{align}
\label{eq:def.strmon}
\inner{z-z'}{v-v'}\geq \mu\norm{z-z'}^2\qquad \forall v\in S(z),v'\in S(z'); \end{align}
\item[(C3)] the solution set $(S+B)^{-1}(0)$ of \eqref{eq:bmu} is nonempty. \end{itemize}
The main motivation to consider the above setting is Subsection \ref{subsec:solsub}, in which the monotone inclusion \eqref{eq:cff} is clearly a special instance of \eqref{eq:bmu} with $S(\cdot):=(1/\gamma)(\cdot-\bpt{z})$, which is obviously $(1/\gamma)$-strongly maximal monotone on $\mathcal{H}$.
The algorithm below was proposed and studied (with a different notation) in \cite[Algorithm 1]{alv.mon.sva-reg.siam16}.
\noindent \fbox{ \addtolength{\linewidth}{-2\fboxsep} \addtolength{\linewidth}{-2\fboxrule}
\begin{minipage}{\linewidth}
\begin{algorithm} \label{shpe} {\bf A specialized HPE method for solving strongly monotone inclusions} \end{algorithm} \begin{itemize}
\item[(0)] Let $z_0\in \mathcal{H}$ and $\sigma\in [0,1)$ be given and set $j\leftarrow 1$.
\item [(1)] Compute $(\widetilde z_j,v_j,\varepsilon_j)\in \mathcal{H}\times \mathcal{H}\times \mathbb{R}_+$ and $\lambda_j>0$ such that
\begin{align} \label{eq:shpe}
\begin{aligned}
v_j\in S(\widetilde z_j)+B^{\varepsilon_j}(\widetilde z_j),\quad
\norm{\lambda_j v_j+ \widetilde z_j-z_{j-1}}^2+
2\lambda_j \varepsilon_j \leq \sigma^2\norm{\widetilde z_j-z_{j-1}}^2. \end{aligned} \end{align}
\item[(2)] Define
\begin{align}
\label{eq:shpe2}
z_j=z_{j-1}-\lambda_j v_j,
\end{align}
set $j\leftarrow j+1$ and go to step 1.
\end{itemize} \noindent
\end{minipage} }
Next proposition will be useful in Subsection \ref{subsec:solsub}.
\begin{proposition}[{\cite[Proposition 2.2]{alv.mon.sva-reg.siam16}}] \label{pr:3m}
Let $\{\widetilde z_j\}$, $\{v_j\}$ and $\{\varepsilon_j\}$ be generated by \emph{Algorithm \ref{shpe}}, let
$z^*:=(S+B)^{-1}(0)$ and $d_0:=\norm{z_0-z^*}$. Assume that $\lambda_j\geq \underline{\lambda}>0$ for all $j\geq 1$ and define
\begin{align} \label{eq:def.alpha}
\alpha := \left( \frac{1}{2\underline{\lambda}\mu} +
\frac{1}{1-\sigma^2} \right)^{-1} \in (0,1). \end{align}
Then, for all $j\geq 1$,
\begin{align}
\label{eq:3m}
\begin{aligned}
&v_j\in S(\widetilde z_j)+B^{\varepsilon_j}(\widetilde z_j),\\
&\|v_j\| \le \sqrt{\dfrac{1+\sigma}{1-\sigma}}
\left(\frac{(1-\alpha)^{(j-1)/2}}{\underline{\lambda}} \right) d_0,\\ &\varepsilon_j \le \frac{\sigma^2}{2(1-\sigma^2)}
\left(\frac{(1-\alpha)^{j-1}}{\underline{\lambda}} \right) d_0^{\,2}. \end{aligned} \end{align}
\end{proposition}
Next section presents one of the main contributions of this paper, namely an inexact Douglas-Rachford type method for solving \eqref{eq:mip.i} and its iteration complexity analysis.
\section{An inexact Douglas-Rachford splitting (DRS) method and its iteration complexity} \label{sec:dr}
Consider problem \eqref{eq:mip.i}, i.e., the problem of finding $z\in \mathcal{H}$ such that
\begin{align}
\label{eq:mip}
0\in A(z)+B(z) \end{align}
where the following hold:
\begin{itemize} \item[(D1)] $A$ and $B$ are maximal monotone operators on $\mathcal{H}$; \item[(D2)] the solution set $(A+B)^{-1}(0)$ of \eqref{eq:mip} is nonempty. \end{itemize}
In this section, we propose and analyze the iteration complexity of an inexact version of the \emph{Douglas-Rachford splitting} (DRS) \emph{method}~\cite{lio.mer-spl.sjna79} for finding approximate solutions of \eqref{eq:mip} according to the following termination criterion: given tolerances $\rho, \epsilon>0$, find $a,b,x,y\in \mathcal{H}$ and $\varepsilon_a,\varepsilon_b\geq 0$ such that
\begin{align}
\label{eq:def.apsol}
a\in A^{\varepsilon_a}(y),\;b\in B^{\varepsilon_b}(x),\quad \gamma \norm{a+b}=\norm{x-y}\leq \rho,\; \varepsilon_a+\varepsilon_b\leq \epsilon, \end{align}
where $\gamma>0$ is a scaling parameter. Note that if $\rho=\epsilon=0$ in \eqref{eq:def.apsol}, then
$z^*:=x=y$ is a solution of \eqref{eq:mip}.
As we mentioned earlier, the algorithm below is motivated by \eqref{eq:dr.spp}--\eqref{eq:dr.spp03} as well as by the recent work of Eckstein and Yao~\cite{eck.yao-rel.mp17}.
\noindent \fbox{ \addtolength{\linewidth}{-2\fboxsep} \addtolength{\linewidth}{-2\fboxrule} \begin{minipage}{\linewidth}
\begin{algorithm} \label{inexact.dr} {\bf An inexact Douglas-Rachford splitting method for \bf{(\ref{eq:mip})}} \end{algorithm} \begin{itemize}
\item[(0)] Let $z_0\in \mathcal{H}$, $\gamma>0$, $\tau_0>0$ and $0<\sigma,\theta<1$ be given and set $k\leftarrow 1$.
\item [(1)] Compute $(x_k,b_k,\varepsilon_{b,\,k})\in \mathcal{H}\times \mathcal{H}\times \mathbb{R}_+$
such that
\begin{align}
\label{eq:err.b}
b_k\in B^{\varepsilon_{b,\,k}}(x_k),\quad \norm{\gamma b_k+x_k-z_{k-1}}^2+2\gamma\varepsilon_{b,\,k}\leq \tau_{k-1}. \end{align}
\item[(2)] Compute $(y_k,a_k)\in \mathcal{H}\times \mathcal{H}$ such that
\begin{align}
\label{eq:err.a}
\hspace{-3.0cm} a_k\in A(y_k),\quad \gamma a_k+y_k=x_k-\gamma b_k. \end{align}
\item[(3)] (3.a) If
\begin{align}
\label{eq:cond.err}
\hspace{-1.3cm}\norm{\gamma b_k+x_k-z_{k-1}}^2+2\gamma \varepsilon_{b,k}\leq \sigma^2\norm{\gamma b_k+y_k-z_{k-1}}^2, \end{align}
\hspace{0.9cm}then
\begin{align}
\label{eq:ext.step}
z_k=z_{k-1}-\gamma(a_k+b_k),\quad \tau_k=\tau_{k-1}\qquad \mbox{[extragradient step]}. \end{align}
(3.b) Else
\begin{align}
\label{eq:null.step}
\hspace{-3.6cm} z_k=z_{k-1},\quad \tau_k=\theta\,\tau_{k-1}\qquad \mbox{[null step]}. \end{align}
\item[(4)]
Set $k\leftarrow k+1$ and go to step 1.
\end{itemize} \noindent
\end{minipage} }
\noindent {\bf Remarks.} \begin{enumerate}
\item We emphasize that although it has been motivated by \cite[Algorithm 3]{eck.yao-rel.mp17}, Algorithm \ref{inexact.dr} is based on a slightly different mechanism of iteration. Moreover, it also allows for the computation of $(x_k,b_k)$ in \eqref{eq:err.b} in the
$\varepsilon_{b,k}$-- enlargement of $B$ (it has the property that $B^{\varepsilon_{b,k}}(x)\supset B(x)$ for all $x\in \mathcal{H}$); this will be crucial for the design and iteration complexity analysis of the four-operator splitting method of Section \ref{sec:drt}. We also mention that, contrary to this work, no iteration complexity analysis is performed in \cite{eck.yao-rel.mp17}.
\item Computation of $(x_k,b_k,\varepsilon_{b,\,k})$ satisfying \eqref{eq:err.b} will depend on the particular instance of the problem \eqref{eq:mip} under consideration. In Section \ref{sec:drt}, we will use Algorithm \ref{inexact.dr} for solving a four-operator splitting monotone inclusion. In this setting, at every iteration $k\geq 1$ of Algorithm \ref{inexact.dr}, called an outer iteration, a Tseng's forward-backward (F-B) splitting type method will be used, as an inner iteration, to solve the (prox) subproblem \eqref{eq:err.b}.
\item Whenever the resolvent $J_{\gamma B}=(\gamma B+I)^{-1}$ is computable, then it follows that
$(x_k,b_k):=(J_{\gamma B}(z_{k-1}),(z_{k-1}-x_k)/\gamma)$ and $\varepsilon_{b,\,k}:=0$ clearly solve \eqref{eq:err.b}. In this case, the left hand side of the inequality in \eqref{eq:err.b} is zero and, as a consequence, the inequality \eqref{eq:cond.err} is always satisfied. In particular, \eqref{eq:dr.spp}--\eqref{eq:dr.spp03} hold, i.e., in this case Algorithm \ref{inexact.dr} reduces to the (exact) DRS method.
\item In this paper, we assume that the resolvent $J_{\gamma A}=(\gamma A+I)^{-1}$ is computable, which implies that $(y_k,a_k):=(J_{\gamma A}(x_k-\gamma b_k),(x_k-\gamma b_k-y_k)/\gamma)$ is the demanded pair in \eqref{eq:err.a}.
\item Algorithm \ref{inexact.dr} potentially performs extragradient steps and null steps, depending on the condition \eqref{eq:cond.err}. It will be shown in Proposition \ref{pr:eHPE} that iterations corresponding to extragradient steps reduce to a special instance of the HPE method, in which case pointwise and ergodic iteration complexity results are available in the current literature (see Proposition \ref{pr:c.ihpe}). On the other hand, iterations corresponding to the null steps will demand a separate analysis (see Proposition \ref{pr:comp.b}).
\end{enumerate}
\noindent As we mentioned in the latter remark, each iteration of Algorithm \ref{inexact.dr} is either an extragradient step or a null step (see \eqref{eq:ext.step} and \eqref{eq:null.step}). This will be formally specified by considering the sets:
\begin{align}
\label{eq:def.ab}
\begin{aligned} &\mathcal{A}:=\mbox{indexes}\;k\geq 1\;\mbox{for which an extragradient step is executed at the iteration}\; k.\\ &\mathcal{B}:=\mbox{indexes}\;k\geq 1\;\mbox{for which a null step is executed at the iteration}\; k. \end{aligned} \end{align}
That said, we let
\begin{align}
\label{eq:car.a}
\mathcal{A}=\{k_j\}_{j\in J},\quad J:=\set{j\geq 1\;|\; j\leq \# \mathcal{A}}
\end{align}
where $k_0:=0$ and $k_0<k_j<k_{j+1}$ for all $j\in J$, and let $\beta_0:=0$ and
\begin{align}
\label{eq:def.betak}
\beta_k:=\mbox{the number of indexes for which a null step is executed until the iteration}\;k. \end{align}
Note that direct use of the above definition and \eqref{eq:null.step} yield
\begin{align}
\label{eq:tau.beta}
\tau_k=\theta^{\beta_k}\tau_0\quad \forall k\geq 0. \end{align}
In order to study the \emph{ergodic iteration complexity} of Algorithm \ref{inexact.dr} we also define the \emph{ergodic sequences} associated to the sequences $\{x_{k_j}\}_{j\in J}$, $\{y_{k_j}\}_{j\in J}$, $\{a_{k_j}\}_{j\in J}$, $\{b_{k_j}\}_{j\in J}$, and $\{\varepsilon_{b,\,{k_j}}\}_{j\in J}$, for all $j\in J$, as follows:
\begin{align} \label{eq:def.erg131} &\overline{x}_{k_j}:=\dfrac{1}{j}\sum_{\ell=1}^j x_{k_\ell},\quad
\overline{y}_{k_j}:=\dfrac{1}{j}\sum_{\ell=1}^j y_{k_\ell}\,,\\
\label{eq:def.erg130}
&\overline{a}_{k_j}:=\dfrac{1}{j}\sum_{\ell=1}^j a_{k_\ell}\,,\quad
\overline{b}_{k_j}:=\dfrac{1}{j}\sum_{\ell=1}^j b_{k_\ell}\,,\\ \label{eq:def.erg132} &\overline{\varepsilon}_{a,\,k_j}:=\dfrac{1}{j}\sum_{\ell=1}^j\, \inner{y_{k_\ell}-\overline{y}_{k_j}}{a_{k_\ell}-\overline{a}_{k_j}} =\dfrac{1}{j}\sum_{\ell=1}^j\, \inner{y_{k_\ell}-\overline{y}_{k_j}}{a_{k_\ell}},\,\\ \label{eq:def.erg133} &\overline{\varepsilon}_{b,\,k_j}:=\dfrac{1}{j}\sum_{\ell=1}^j \big[\varepsilon_{b,\,k_\ell}+\inner{x_{k_\ell}-\overline{x}_{k_j}}{b_{k_\ell}-\overline{b}_{k_j}}\big] =\dfrac{1}{j}\sum_{\ell=1}^j \big[\varepsilon_{b,\,k_\ell}+\inner{x_{k_\ell}-\overline{x}_{k_j}}{b_{k_\ell}} \big].
\end{align}
Moreover, the results on iteration complexity of Algorithm \ref{inexact.dr} (pointwise and ergodic) obtained in this paper will depend on the following quantity:
\begin{align}
\label{eq:def.d0}
d_{0,\,\gamma}:=\mbox{dist}\left(z_0,\mbox{zer}(S_{\gamma, A,B})\right)=
\min\,\{\norm{z_0-z}\;|\;z\in \mbox{zer}(S_{\gamma, A,B})\} \end{align}
which measures the quality of the initial guess $z_0$ in Algorithm \ref{inexact.dr} with respect to $\mbox{zer}(S_{\gamma, A,B})$, where the operator $S_{\gamma, A,B}$ is such that $J_{\gamma B}(\mbox{zer}(S_{\gamma,\,A,\,B}))=(A+B)^{-1}(0)$ (see \eqref{eq:def.s}).
In the next proposition, we show that the procedure resulting by selecting the extragradient steps in Algorithm \ref{inexact.dr} can be embedded into HPE method.
First, we need the following lemma.
\begin{lemma}
\label{lm:ixk} Let $\{z_k\}$ be generated by \emph{Algorithm \ref{inexact.dr}} and let the set $J$ be defined in \eqref{eq:car.a}. Then,
\begin{align} \label{eq:ixk}
z_{k_{j-1}}=z_{k_j-1}\qquad \forall j\in J. \end{align}
\end{lemma}
\begin{proof} Using \eqref{eq:def.ab} and \eqref{eq:car.a} we have
$\{k\geq 1\,|\, k_{j-1}<k<k_j\}\subset \mathcal{B}$, for all $j\in J$. Consequently, using the definition of $\mathcal{B}$ in \eqref{eq:def.ab} and \eqref{eq:null.step} we conclude that $z_k=z_{k_{j-1}}$ whenever $k_{j-1}\leq k< k_j$. As a consequence, we obtain that \eqref{eq:ixk} follows from the fact that $k_{j-1}\leq k_j-1<k_j$. \end{proof}
\begin{proposition}
\label{pr:eHPE}
Let $\{z_k\}$, $\{(x_k,b_k)\}$, $\{\varepsilon_{b,k}\}$ and $\{(y_k,a_k)\}$ be generated by \emph{Algorithm \ref{inexact.dr}} and let the operator $S_{\gamma,\,A,\,B}$ be defined in \eqref{eq:def.s}. Define, for all $j\in J$,
\begin{align}
\label{eq:pr:eHPE.02}
\widetilde z_{k_j}:= y_{k_j}+\gamma b_{k_j},\quad v_{k_j}:=\gamma(a_{k_j}+b_{k_j}),\quad
\varepsilon_{k_j}:=\gamma \varepsilon_{b,k_j}.
\end{align}
Then, for all $j\in J$,
\begin{align}
\begin{aligned}
\label{eq:pr.eHPE.01}
&v_{k_j}\in \left(S_{\gamma,\,A,\,B}\right)^{\varepsilon_{k_j}}(\widetilde z_{k_j}),\qquad
\norm{ v_{k_j}+\widetilde z_{k_j}-z_{k_{j-1}}}^2+2 \varepsilon_{k_j}\leq \sigma^2\norm{\widetilde z_{k_j}-z_{k_{j-1}}}^2,\\[2mm]
&z_{k_j}=z_{k_{j-1}} - v_{k_j}.
\end{aligned} \end{align}
As a consequence, the sequences $\{\widetilde z_{k_j}\}_{j\in J}$, $\{v_{k_j}\}_{j\in J}$, $\{\varepsilon_{k_j}\}_{j\in J}$ and $\{z_{k_j}\}_{j\in J}$ are generated by \emph{Algorithm \ref{hpe}} with $\lambda_j\equiv 1$ for solving \eqref{eq:mip.T} with $T:=S_{\gamma,\,A,\,B}$. \end{proposition}
\begin{proof} For any $(z',v'):=(y+\gamma b,\gamma a+\gamma b)\in S_{\gamma,\,A,\,B}$ we have, in particular, $b\in B(x)$ and $a\in A(y)$ (see \eqref{eq:def.s}). Using these inclusions, the inclusions in \eqref{eq:err.b} and \eqref{eq:err.a}, the monotonicity of the operator $A$ and \eqref{eq:def.teps} with $T=B$ we obtain
\begin{align}
\label{eq:11}
\inner{x_{k_j}-x}{b_{k_j}-b}\geq -\varepsilon_{b, k_j},\qquad
\inner{y_{k_j}-y}{a_{k_j}-a}\geq 0. \end{align}
Moreover, using the identity in \eqref{eq:err.a} and the corresponding one in \eqref{eq:def.s} we find
\begin{align}
\label{eq:10}
(y_{k_j}-y)+\gamma(b_{k_j}-b)=(x_{k_j}-x)-\gamma (a_{k_j}-a). \end{align}
Using \eqref{eq:pr:eHPE.02}, \eqref{eq:11} and \eqref{eq:10} we have
\begin{align}
\nonumber
\inner{\widetilde z_{k_j}-z'}{v_{k_j}-v'}&=\inner{(y_{k_j}+\gamma b_{k_j})-(y+\gamma b)}{(\gamma a_{k_j}+\gamma b_{k_j})-(\gamma a+\gamma b)}\\
\nonumber
&=\inner{y_{k_j}-y+\gamma (b_{k_j}-b)}
{\gamma(a_{k_j}-a)+\gamma(b_{k_j}-b)}\\ \nonumber &=\gamma\inner{y_{k_j}-y+\gamma (b_{k_j}-b)}
{a_{k_j}-a}+\gamma\inner{y_{k_j}-y+\gamma (b_{k_j}-b)}
{b_{k_j}-b}\\ \nonumber &=\gamma\inner{y_{k_j}-y+\gamma (b_{k_j}-b)}
{a_{k_j}-a}+\gamma\inner{x_{k_j}-x-\gamma(a_{k_j}-a)}
{b_{k_j}-b}\\ \nonumber &=\gamma\inner{y_{k_j}-y}{a_{k_j}-a}+\gamma\inner{x_{k_j}-x}{b_{k_j}-b}\\ \nonumber &\geq \gamma\inner{x_{k_j}-x}{b_{k_j}-b}\\ \nonumber &\geq -\varepsilon_{k_j}, \end{align}
which combined with definition \eqref{eq:def.teps} gives the inclusion in \eqref{eq:pr.eHPE.01}.
From \eqref{eq:pr:eHPE.02}, \eqref{eq:ixk}, the identity in \eqref{eq:err.a} and \eqref{eq:cond.err} we also obtain
\begin{align*} \norm{ v_{k_j}+\widetilde z_{k_j}-z_{k_{j-1}}}^2&= \norm{ \gamma(a_{k_j}+b_{k_j})+(y_{k_j}+\gamma b_{k_j})-z_{k_{j}-1}}^2\\ &=\norm{ (x_{k_j}-y_{k_j})+(y_{k_j}+\gamma b_{k_j})-z_{k_{j}-1}}^2\\ &=\norm{ \gamma b_{k_j}+x_{k_j}-z_{k_{j}-1}}^2\\ &\leq \sigma^2 \norm{ \gamma b_{k_j}+y_{k_j}-z_{k_{j}-1}}^2-2\gamma\varepsilon_{b,k_j}\\ &=\sigma^2 \norm{\widetilde z_{k_j}-z_{k_{j-1}}}^2-2\varepsilon_{k_j}, \end{align*}
which gives the inequality in \eqref{eq:pr.eHPE.01}. To finish the proof of \eqref{eq:pr.eHPE.01}, note that the desired identity in \eqref{eq:pr.eHPE.01} follows from the first one in \eqref{eq:ext.step}, the second one in \eqref{eq:pr:eHPE.02} and \eqref{eq:ixk}. The last statement of the proposition follows from \eqref{eq:pr:eHPE.02}, \eqref{eq:pr.eHPE.01} and Algorithm \ref{hpe}'s definition. \end{proof}
\begin{proposition}{\bf (rate of convergence for extragradient steps)}
\label{pr:c.ihpe} Let $\{(x_k,b_k)\}$, $\{(y_k,a_k)\}$ and $\{\varepsilon_{b,\,k}\}$ be generated by \emph{Algorithm~\ref{inexact.dr}} and consider the ergodic sequences defined in \eqref{eq:def.erg131}--\eqref{eq:def.erg133}. Let $d_{0,\gamma}$ and the set $J$ be defined in \eqref{eq:def.d0} and \eqref{eq:car.a}, respectively.
Then, \begin{enumerate}
\item[\emph{(a)}] For any $j\in J$,
there exists $i\in \{1,\dots,j\}$
such that
\begin{align}
\label{eq:706}
& a_{k_i}\in A(y_{k_i}),\quad b_{k_i}\in B^{\varepsilon_{b,\,k_i}}(x_{k_i}),\\
\label{eq:306}
&\gamma\norm{a_{k_i}+b_{k_i}}=\norm{x_{k_i}-y_{k_i}}\leq \dfrac{d_{0,\gamma}}{\sqrt{j}}
\sqrt{\dfrac{1+\sigma}{1-\sigma}},\\
\label{eq:206}
&\varepsilon_{b,\,k_i}\leq \dfrac{\sigma^2 d_{0,\gamma}^{\,2}}{2\gamma(1-\sigma^2) j}\;.
\end{align}
\item[\emph{(b)}]
For any $j\in J$,
\begin{align}
\label{eq:707}
&\overline{a}_{k_j}\in A^{\overline{\varepsilon}_{a,k_j}}(\overline{y}_{k_j}),
\quad \overline{b}_{k_j}\in B^{\overline{\varepsilon}_{b,\,k_j}}(\overline{x}_{k_j}),\\
\label{eq:618}
&\gamma\norm{\overline{a}_{k_j}+\overline{b}_{k_j}}
=\norm{\overline{x}_{k_j}-\overline{y}_{k_j}}\leq \dfrac{2d_{0,\gamma}}{j},\\
\label{eq:619}
&\overline{\varepsilon}_{a,\,k_j}+\overline{\varepsilon}_{b,\,k_j}\leq
\dfrac{2(1+\sigma/\sqrt{1-\sigma^2})d_{0,\gamma}^{\,2}}{\gamma j}\,.
\end{align}
\end{enumerate} \end{proposition}
\begin{proof} Note first that \eqref{eq:706} follow from the inclusions in \eqref{eq:err.b} and \eqref{eq:err.a}.
Using the last statement in Proposition \ref{pr:eHPE}, Theorem \ref{lm:rhpe2} (with $\underline{\lambda}=1$) and \eqref{eq:def.d0} we obtain that there exists $i\in \{1,\dots, j\}$ such that
\begin{align}
\norm{v_{k_i}}\leq \dfrac{d_{0,\gamma}}{\sqrt{j}}
\sqrt{\dfrac{1+\sigma}{1-\sigma}},\quad \varepsilon_{k_i}\leq \dfrac{\sigma^2 d_{0,\gamma}^2}{2(1-\sigma^2)j}\,, \end{align}
which, in turn, combined with the identity in \eqref{eq:err.a} and the definitions of $v_{k_i}$ and $\varepsilon_{k_i}$ in \eqref{eq:pr:eHPE.02} gives the desired inequalities in \eqref{eq:306} and \eqref{eq:206} (concluding the proof of (a)) and
\begin{align}
\label{eq:6001}
\norm{\overline{v}_j}\leq \dfrac{2d_{0,\gamma}}{j},\quad
\overline{\varepsilon}_j\leq \dfrac{2(1+\sigma/\sqrt{1-\sigma^2})d_{0,\gamma}^2}{j}\,, \end{align}
where $\overline{v}_j$ and $\overline{\varepsilon}_j$ are defined in \eqref{eq:d.eg2} and \eqref{eq:d.eg3}, respectively, with $\Lambda_j=j$ and
\begin{align}
\label{eq:2021}
\lambda_\ell:=1,\quad v_\ell:=v_{k_\ell},\quad \varepsilon_\ell:=\varepsilon_{k_\ell}, \quad \tilde z_\ell:=\widetilde z_{k_\ell}\qquad \forall \ell=1,\dots, j. \end{align}
Since the inclusions in \eqref{eq:707} are a direct consequence of the ones in \eqref{eq:err.b} and \eqref{eq:err.a}, Proposition \ref{pr:teps}(d), \eqref{eq:def.erg131}--\eqref{eq:def.erg133} and Theorem \ref{th:tf}, it follows from \eqref{eq:618}, \eqref{eq:619} and \eqref{eq:6001} that to finish the proof of (b), it suffices to prove that
\begin{align}
\label{eq:2022}
\overline{v_j}=\gamma(\overline{a}_{k_j}+\overline{b}_{k_j}),\quad \gamma(\overline{a}_{k_j}+\overline{b}_{k_j})=\overline{x}_{k_j}-\overline{y}_{k_j}, \qquad \overline{\varepsilon}_j= \gamma(\overline{\varepsilon}_{a,\,k_j}+\overline{\varepsilon}_{b,\,k_j}). \end{align}
The first identity in \eqref{eq:2022} follows from \eqref{eq:2021}, the second identities in \eqref{eq:d.eg2} and \eqref{eq:pr:eHPE.02}, and \eqref{eq:def.erg130}.
On the other hand, from \eqref{eq:err.a} we have $\gamma(a_{k_\ell}+b_{k_\ell})=x_{k_\ell}-y_{k_\ell}$, for all $\ell=1,\dots, j$, which combined with \eqref{eq:def.erg131} and \eqref{eq:def.erg130} gives the second identity in \eqref{eq:2022}. Using the latter identity and the second one in \eqref{eq:2022} we obtain
\begin{align}
\label{eq:2023}
(y_{k_\ell}-\overline{y}_{k_j})+\gamma(b_{k_\ell}-\overline{b}_{k_j})=(x_{k_\ell}-\overline{x}_{k_j})-\gamma(a_{k_\ell}-\overline{a}_{k_j}) \qquad \forall \ell=1,\dots, j. \end{align}
Moreover, it follows from \eqref{eq:d.eg2}, \eqref{eq:2021}, the first identity in \eqref{eq:pr:eHPE.02}, \eqref{eq:def.erg131} and \eqref{eq:def.erg130} that
\begin{align}
\label{eq:7001}
\overline{\widetilde z}_{j}=\overline{\widetilde z}_{k_j}=\dfrac{1}{j}\sum_{\ell=1}^j\,\left(y_{k_\ell}+\gamma b_{k_\ell}\right)=
\overline{y}_{k_j}+\gamma \overline{b}_{k_j}. \end{align}
Using \eqref{eq:7001}, \eqref{eq:2021}, \eqref{eq:pr:eHPE.02} and \eqref{eq:2023} we obtain, for all $\ell=1,\dots, j$,
\begin{align*}
\nonumber
\inner{\widetilde{z}_{\ell}-\overline{\widetilde z}_{j}}{v_{\ell}}&= \inner{(y_{k_\ell}+\gamma b_{k_\ell})-(\overline{y}_{k_j}+\gamma \overline{b}_{k_j})}{\gamma(a_{k_\ell}+b_{k_\ell})}\\
\nonumber
&=\gamma \inner{(y_{k_\ell}-\overline{y}_{k_j})+\gamma(b_{k_\ell}-\overline{b}_{k_j})}{a_{k_\ell}}+
\gamma \inner{(y_{k_\ell}-\overline{y}_{k_j})+\gamma(b_{k_\ell}-\overline{b}_{k_j})}{b_{k_\ell}}\\
\nonumber &=\gamma \inner{(y_{k_\ell}-\overline{y}_{k_j})+\gamma(b_{k_\ell}-\overline{b}_{k_j})}{a_{k_\ell}}+
\gamma \inner{(x_{k_\ell}-\overline{x}_{k_j})-\gamma(a_{k_\ell}-\overline{a}_{k_j})}{b_{k_\ell}}\\
&=\gamma\inner{y_{k_\ell}-\overline{y}_{k_j}}{a_{k_\ell}}+\gamma^2\inner{b_{k_\ell}-\overline{b}_{k_j}}{a_{k_\ell}}
+\gamma\inner{x_{k_\ell}-\overline{x}_{k_j}}{b_{k_\ell}}-\gamma^2\inner{a_{k_\ell}-\overline{a}_{k_j}}{b_{k_\ell}}, \end{align*}
which combined with \eqref{eq:d.eg3}, \eqref{eq:2021}, \eqref{eq:def.erg132} and \eqref{eq:def.erg133} yields
\begin{align*}
\overline{\varepsilon}_j=\dfrac{1}{j}\sum_{\ell=1}^j\,
\left[\varepsilon_\ell+\inner{\widetilde{z}_{\ell}-\overline{\widetilde z}_{j}}{v_{\ell}}\right]
&= \dfrac{1}{j}\sum_{\ell=1}^j\,
\gamma \left[\varepsilon_{b,\,k_\ell}+\inner{x_{k_\ell}-\overline{x}_{k_j}}{b_{k_\ell}}+\inner{y_{k_\ell}-\overline{y}_{k_j}}{a_{k_\ell}}\right]\\
&=\gamma(\overline{\varepsilon}_{a,\,k_j}+\overline{\varepsilon}_{b,\,k_j}),
\end{align*}
which is exactly the last identity in \eqref{eq:2022}. This finishes the proof. \end{proof}
\begin{proposition}{\bf (rate of convergence for null steps)}
\label{pr:comp.b}
Let $\{(x_k,b_k)\}$, $\{(y_k,a_k)\}$ and $\{\varepsilon_{b,k}\}$ be generated by \emph{Algorithm \ref{inexact.dr}}. Let $\{\beta_k\}$ and the set $\mathcal{B}$ be defined in \eqref{eq:def.betak} and \eqref{eq:def.ab}, respectively. Then, for $k\in \mathcal{B}$,
\begin{align}
\label{eq:com.bb}
& a_k\in A(y_k),\quad b_k\in B^{\varepsilon_{b,\,k}}(x_k),\\
\label{eq:com.bb2}
&\gamma\norm{a_k+b_k}=\norm{x_k-y_k}\leq \dfrac{2\sqrt{\tau_0}}{\sigma}\;\theta^{\frac{\beta_{k-1}}{2}},\\
\label{eq:com.bb3}
& \gamma \varepsilon_{b,\,k}\leq \dfrac{\tau_0}{2}\,\theta^{\beta_{k-1}}.
\end{align} \end{proposition}
\begin{proof} Note first that \eqref{eq:com.bb} follows from \eqref{eq:err.b} and \eqref{eq:err.a}.
Using \eqref{eq:def.ab}, \eqref{eq:err.b} and Step 3.b's definition (see Algorithm \ref{inexact.dr}) we obtain
\begin{align*}
\tau_{k-1}\geq \norm{\underbrace{\gamma b_k+x_k-z_{k-1}}_{p_k}}^2+2\gamma\varepsilon_{b,k}>
\sigma^2\norm{\underbrace{\gamma b_k+y_k-z_{k-1}}_{q_k}}^2, \end{align*}
which, in particular, gives
\begin{align}
\label{eq:1301}
\gamma \varepsilon_{b,\,k}\leq \dfrac{\tau_{k-1}}{2}, \end{align}
and combined with the identity in \eqref{eq:err.a} yields,
\begin{align}
\nonumber
\gamma\norm{a_k+b_k}=\norm{x_k-y_k}&=\norm{p_k-q_k}\\ \nonumber
&\leq \norm{p_k}+\norm{q_k}\\
\label{eq:ss34}
&\leq \left(1+\dfrac{1}{\sigma}\right)\sqrt{\tau_{k-1}}.
\end{align}
To finish the proof, use \eqref{eq:1301}, \eqref{eq:ss34} and \eqref{eq:tau.beta}.
\end{proof}
Next we present the main results regarding the pointwise and ergodic iteration complexity of Algorithm \ref{inexact.dr} for finding approximate solutions of \eqref{eq:mip} satisfying the termination criterion \eqref{eq:def.apsol}. While Theorem \ref{th:idr.main} is a consequence of Proposition \ref{pr:c.ihpe}(a) and Proposition \ref{pr:comp.b}, the ergodic iteration complexity of Algorithm \ref{inexact.dr}, namely Theorem \ref{th:idr.main.erg}, follows by combining the latter proposition and Proposition \ref{pr:c.ihpe}(b). Since the proof of Theorem \ref{th:idr.main.erg} follows the same outline of Theorem \ref{th:idr.main}'s proof, it will be omitted.
\begin{theorem}\emph{{\bf (pointwise iteration complexity of Algorithm \ref{inexact.dr})}}
\label{th:idr.main} Assume that $\max\{(1-\sigma)^{-1},\sigma^{-1}\}=\mathcal{O}(1)$ and let $d_{0,\gamma}$ be as in \eqref{eq:def.d0}. Then, for given tolerances $\rho,\epsilon>0$, \emph{Algorithm \ref{inexact.dr}} finds $a,b,x,y\in \mathcal{H}$ and $\varepsilon_b\geq 0$ such that
\begin{align}
\label{eq:main.02}
a\in A(y),\;b\in B^{\varepsilon_b}(x),\quad \gamma\norm{a+b}=\norm{x-y} \leq \rho,\quad \varepsilon_b\leq \epsilon \end{align}
after performing \emph{at most}
\begin{align}
\label{eq:ext.steps.alg3}
\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}^{\,2}}{\rho^2},
\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\} \right) \end{align}
\emph{extragradient steps} and
\begin{align}
\label{eq:null.steps.alg3}
\mathcal{O}\left(1+\max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon}\right)\right\}\right) \end{align}
\emph{null steps}. As a consequence, under the above assumptions, \emph{Algorithm \ref{inexact.dr}} terminates with $a,b,x,y\in \mathcal{H}$ and $\varepsilon_{b}\geq 0$ satisfying \eqref{eq:main.02} in at most
\begin{align}
\label{eq:main.01}
\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}^{\,2}}{\rho^2},
\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\}
+ \max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon} \right)\right\}\right) \end{align}
iterations. \end{theorem}
\begin{proof} Let $\mathcal{A}$ be as in \eqref{eq:def.ab} and consider the cases:
\begin{align}
\label{eq:def.m01}
\#\mathcal{A}\geq M_{\text{ext}}:=\left\lceil \max\left\{\dfrac{2\,d_{0,\gamma}^{\,2}}{(1-\sigma)\rho^2},
\dfrac{\sigma^2d_{0,\gamma}^{\,2}}{2\gamma(1-\sigma^2)\epsilon}\right\} \right\rceil\quad \mbox{and}\quad
\#\mathcal{A}<M_{\text{ext}}.
\end{align}
In the first case, the desired bound \eqref{eq:ext.steps.alg3} on the number of extragradient steps to find $a,b,x,y\in \mathcal{H}$ and $\varepsilon_b\geq 0$ satisfying \eqref{eq:main.02} follows from the definition of $J$ in \eqref{eq:car.a} and Proposition \ref{pr:c.ihpe}(a).
On the other hand, in the second case, i.e., $\#\mathcal{A}<M_{\text{ext}}$, the desired bound \eqref{eq:null.steps.alg3} is a direct consequence of Proposition \ref{pr:comp.b}. The last statement of the theorem follows from \eqref{eq:ext.steps.alg3} and \eqref{eq:null.steps.alg3}. \end{proof}
Next is the main result on the ergodic iteration complexity of Algorithm \ref{inexact.dr}. As mentioned before, its proof follows the same outline of Theorem \ref{th:idr.main}'s proof, now applying Proposition \ref{pr:c.ihpe}(b) instead of the item (a) of the latter proposition.
\begin{theorem}\emph{{\bf (ergodic iteration complexity of Algorithm \ref{inexact.dr})}}
\label{th:idr.main.erg} For given tolerances $\rho,\epsilon>0$, under the same assumptions of \emph{Theorem \ref{th:idr.main}}, \emph{Algorithm \ref{inexact.dr}} provides $a,b,x,y\in \mathcal{H}$ and $\varepsilon_a, \varepsilon_b\geq 0$ such that
\begin{align}
\label{eq:main.04}
a\in A^{\varepsilon_a}(y),\;b\in B^{\varepsilon_b}(x),\quad \gamma \norm{a+b}=\norm{x-y}\leq \rho,\quad \varepsilon_a+\varepsilon_b\leq \epsilon. \end{align}
after performing at most
\begin{align}
\label{eq:ext.steps.alg3e}
\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}}{\rho},
\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\} \right) \end{align}
\emph{extragradient steps} and
\begin{align}
\label{eq:null.steps.alg3e}
\mathcal{O}\left(1+\max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon}\right)\right\}\right) \end{align}
\emph{null steps}. As a consequence, under the above assumptions, \emph{Algorithm \ref{inexact.dr}} terminates with $a,b,x,y\in \mathcal{H}$ and $\varepsilon_{a},\varepsilon_{b}\geq 0$ satisfying \eqref{eq:main.04} in at most
\begin{align}
\label{eq:main.03}
\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}}{\rho},
\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\}
+ \max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon} \right)\right\}\right) \end{align}
iterations. \end{theorem} \begin{proof} The proof follows the same outline of Theorem \ref{th:idr.main}'s proof, now applying Proposition \ref{pr:c.ihpe}(b) instead of Proposition \ref{pr:c.ihpe}(a). \end{proof}
\noindent {\bf Remarks.}
\begin{enumerate}
\item Theorem \ref{th:idr.main.erg} ensures that for given tolerances $\rho, \epsilon>0$, up to an additive logarithmic factor, Algorithm \ref{inexact.dr} requires no more than
\[
\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}}{\rho},\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\}\right) \]
iterations to find an approximate solution of the monotone inclusion problem \eqref{eq:mip} according to the termination criterion \eqref{eq:def.apsol}.
\item While the (ergodic) upper bound on the number of iterations provided in \eqref{eq:main.03} is better than the corresponding one in \eqref{eq:main.01} (in terms of the dependence on the tolerance $\rho>0$) by a factor of $\mathcal{O}(1/\rho)$, the inclusion in \eqref{eq:main.04} is potentially weaker than the corresponding one in \eqref{eq:main.02}, since one may have $\varepsilon_a>0$ in \eqref{eq:main.04}, and the set $A^{\varepsilon_a}(y)$ is in general larger than $A(y)$.
\item Iteration complexity results similar to the ones in Proposition \ref{pr:c.ihpe} were recently obtained for a relaxed Peaceman-Rachford method in \cite{mon.che-com.17} . We emphasize that, in contrast to this work, the latter reference considers only the case where the resolvents $J_{\gamma A}$ and $J_{\gamma B}$ of $A$ and $B$, respectively, are both computable. \end{enumerate}
The proposition below will be important in the next section.
\begin{proposition} \label{pr:imp.fact2}
Let $\{z_k\}$ be generated by \emph{Algorithm \ref{inexact.dr}} and $d_{0,\gamma}$ be as in \eqref{eq:def.d0}. Then,
\begin{align} \label{eq:ineq.zk02}
\|z_k-z_0\| \le 2 d_{0,\gamma}\quad \forall k\geq 1. \end{align}
\end{proposition}
\begin{proof} Note that (i) if $k=k_j\in \mathcal{A}$, for some $j\in J$, see \eqref{eq:car.a}, then \eqref{eq:ineq.zk02} follows from the last statement in Proposition \ref{pr:eHPE} and Proposition \ref{pr:imp.fact}; (ii) if $k\in \mathcal{B}$, from the first identity in \eqref{eq:null.step}, see \eqref{eq:def.ab}, we find that
either $z_k=z_0$, in which case \eqref{eq:ineq.zk02} holds trivially, or $z_k=z_{k_j}$ for some $j\in J$, in which case the results follows from (i).
\end{proof}
\section{A Douglas-Rachford-Tseng's forward-backward (F-B) four-operator splitting method}
\label{sec:drt}
In this section, we consider problem \eqref{eq:drti}, i.e., the problem of finding $z\in \mathcal{H}$ such that
\begin{align}
\label{eq:drt}
0\in A(z)+C(z)+F_1(z)+F_2(z) \end{align}
where the following hold:
\begin{itemize}
\item[(E1)] $A$ and $C$ are (set-valued) maximal monotone operators on $\mathcal{H}$.
\item[(E2)] $F_1:D(F_1)\subset \mathcal{H}\to \mathcal{H}$ is monotone and $L$-Lipschitz continuous on a (nonempty) closed convex set $\Omega$ such that $D(C)\subset \Omega\subset D(F_1)$, i.e., $F_1$ is monotone on $\Omega$ and there exists $L\geq 0$ such that
\begin{align}
\label{eq:f.Lip}
\norm{F_1(z)-F_1(z')}\leq L\norm{z-z'}\qquad \forall z,z'\in \Omega.
\end{align}
\item[(E3)] $F_2:\mathcal{H}\to \mathcal{H}$ is $\eta-$cocoercivo, i.e., there exists $\eta>0$ such that
\begin{align}
\label{eq:f.coco}
\inner{F_2(z)-F_2(z')}{z-z'}\geq \eta\norm{F_2(z)-F_2(z')}^2\qquad \forall z,z'\in \mathcal{H}.
\end{align}
\item[(E4)] $B^{-1}(0)$ is nonempty, where
\begin{align}
\label{eq:def.bcff}
B:=C+F_1+F_2. \end{align}
\item[(E5)] The solution set of \eqref{eq:drt} is nonempty.
\end{itemize}
Aiming at solving the monotone inclusion \eqref{eq:drt}, we present and study the iteration complexity of a (four-operator) splitting method which combines Algorithm \ref{inexact.dr} (used as an outer iteration) and a Tseng's forward-backward (F-B) splitting type method (used as an inner iteration for solving, for each outer iteration, the prox subproblems in \eqref{eq:err.b}). We prove results on pointwise and ergodic iteration complexity of the proposed four-operator splitting algorithm by analyzing it in the framework of Algorithm \ref{inexact.dr} for solving \eqref{eq:mip} with $B$ as in \eqref{eq:def.bcff} and under assumptions (E1)--(E5). The (outer)
iteration complexities will follow from results on pointwise and ergodic iteration complexities of Algorithm \ref{inexact.dr}, obtained in Section \ref{sec:dr}, while the computation of an upper bound on the overall number of inner iterations required to achieve prescribed tolerances will require a separate analysis.
Still regarding the results on iteration complexity, we mention that we consider the following notion of approximate solution for \eqref{eq:drt}: given tolerances $\rho, \epsilon>0$, find $a,b,x,y\in \mathcal{H}$ and $\varepsilon_a,\varepsilon_b\geq 0$ such that
\begin{align}
\nonumber
&a\in A^{\varepsilon_a}(y),\\
\label{eq:app.sol.acff2}
&\mbox{either}\;\;b\in C(x)+F_1(x)+F_2^{\varepsilon_b}(x)\;\;\mbox{or}\;\;b\in \left(C+F_1+F_2\right)^{\varepsilon_b}(x),\\ \nonumber
&\gamma\norm{a+b}=\norm{x-y}\leq \rho,\;\; \varepsilon_a+\varepsilon_b\leq \epsilon,
\end{align}
where $\gamma>0$. Note that (i) for $\rho=\epsilon=0$, the above conditions
imply that $z^*:=x=y$ is a solution of the monotone inclusion \eqref{eq:drt}; (ii) the second inclusion in \eqref{eq:app.sol.acff2}, which will appear in the ergodic iteration complexity, is potentially weaker than the first one (see Proposition \ref{pr:teps}(b)), which will appear in the corresponding pointwise iteration complexity of the proposed method.
We also mention that problem \eqref{eq:drt} falls in the framework of the monotone inclusion \eqref{eq:mip} due to the facts that, in view of assumptions (E1), (E2) and (E3), the operator $A$ is maximal monotone, and the operator $F_1+F_2$ is monotone and $(L+1/\eta)$--Lipschitz continuous on the closed convex set $\Omega\supset D(C)$, which combined with the assumption on the operator $C$ in (E1) and with \cite[Proposition A.1]{MonSva10-1} implies that the operator $B$ defined in \eqref{eq:def.bcff} is maximal monotone as well. These facts combined with assumption (E5) give that conditions (D1) and (D2) of Section \ref{sec:dr} hold for $A$ and $B$ as in (E1) and \eqref{eq:def.bcff}, respectively. In particular, it gives that Algorithm \ref{inexact.dr} may be applied to solve the four-operator monotone inclusion \eqref{eq:drt}.
In this regard, we emphasize that any implementation of Algorithm \ref{inexact.dr} will heavily depend on specific strategies for solving each subproblem in \eqref{eq:err.b}, since $(y_k,a_k)$ required in \eqref{eq:err.a} can be computed by using the resolvent operator of $A$, available in closed form in many important cases.
In the next subsection, we show how the specific structure \eqref{eq:drt} allows for an application of a Tseng's F-B splitting type method for solving each subproblem in \eqref{eq:err.b}.
\subsection{Solving the subproblems in \eqref{eq:err.b} for $B$ as in \eqref{eq:def.bcff}}
\label{subsec:solsub}
In this subsection, we present and study
a Tseng's F-B splitting type method~\cite{bau.com-book,arias.davis-half,MonSva10-1,tse-mod.sjco00} for solving the corresponding proximal subproblem in \eqref{eq:err.b} at each (outer) iteration of Algorithm \ref{inexact.dr}, when used to solve \eqref{eq:drt}. To begin with, first consider the (strongly) monotone inclusion
\begin{align}
\label{eq:cff}
0\in B(z)+\dfrac{1}{\gamma}(z-\bpt{z}) \end{align}
where $B$ is as in \eqref{eq:def.bcff}, $\gamma>0$ and $\bpt{z}\in \mathcal{H}$, and note that the task of finding $(x_k,b_k,\varepsilon_{b,k})$ satisfying \eqref{eq:err.b} is related to the task of solving \eqref{eq:cff} with $\bpt{z}:=z_{k-1}$.
In the remaining part of this subsection, we present and study a Tseng's F-B splitting type method for solving \eqref{eq:cff}.
As we have mentioned before, the resulting algorithm will be used as an inner procedure for solving the subproblems \eqref{eq:err.b} at each iteration of Algorithm \ref{inexact.dr}, when applied to solve \eqref{eq:drt}.
\noindent \fbox{ \addtolength{\linewidth}{-2\fboxsep} \addtolength{\linewidth}{-2\fboxrule}
\begin{minipage}{\linewidth}
\begin{algorithm} \label{hff} {\bf A Tseng's F-B splitting type method for \bf{(\ref{eq:cff})}} \end{algorithm}
{\bf Input:} $C, F_1,\Omega,L, F_2$ and $\eta$ as in conditions (E1)--(E5), $\bpt{z}\in \mathcal{H}$, $\bpt{\tau}>0$, $\sigma\in (0,1)$ and $\gamma$ such that
\begin{align}
\label{eq:bound.gamma}
0<\gamma\leq \dfrac{4\eta \sigma^2}{1+\sqrt{1+16L^2\eta^2\sigma^2}}. \end{align}
\begin{itemize}
\item[(0)] Set $z_0\leftarrow \bpt{z}$ and $j\leftarrow 1$.
\item [(1)] Let $z'_{j-1}\leftarrow P_{\Omega}(z_{j-1})$ and compute
\begin{align} \label{eq:hff}
\begin{aligned}
&\widetilde z_j=
\left(\frac{\gamma}{2} C+I\right)^{-1}\left(\dfrac{\bpt{z}+z_{j-1}-\gamma(F_1+F_2)(z'_{j-1})}{2}\right),\\%\quad
[1mm]
&z_j=\widetilde z_j-\gamma\left(F_1(\widetilde z_j)-F_1(z'_{j-1})\right).
\end{aligned} \end{align}
\item[(2)] If
\begin{align}
\label{eq:hff2}
\hspace{-2.4cm}\norm{z_{j-1}-z_j}^2+\frac{\gamma\norm{z'_{j-1}-\widetilde z_j}^2}{2\eta}\leq \bpt{\tau},
\end{align}
then {\bf terminate}. Otherwise, set $j\leftarrow j+1$ and go to step 1.
\end{itemize} {\bf Output:} $(z_{j-1},z'_{j-1},z_j,\widetilde z_j)$. \noindent
\end{minipage} }
\noindent {\bf Remark.}
\begin{enumerate}
\item [] Algorithm \ref{hff} combines ideas from the standard Tseng's F-B splitting algorithm~\cite{tse-mod.sjco00} as well as from recent insights on the convergence and iteration complexity of some variants the latter method~\cite{alv.mon.sva-reg.siam16,arias.davis-half,MonSva10-1}. In this regard, evaluating the cocoercive component $F_2$ just once per iteration (see \cite[Theorem 1]{arias.davis-half}) is potentially important in many applications, where the evaluation of cocoercive operators is in general computationally expensive (see \cite{arias.davis-half} for a discussion). Nevertheless, we emphasize that the results obtained in this paper regarding the analysis of Algorithm \ref{hff} do not follow from any of the just mentioned references. \end{enumerate}
\noindent Next corollary ensures that Algorithm \ref{hff} always terminates with the desired output.
\begin{corollary}
\label{cor:cota.linear}
Assume that $(1-\sigma^2)^{-1}=\mathcal{O}(1)$ and let $d_{\bpt{z},b}$ denote
the distance of $\bpt{z}$ to $B^{-1}(0)\neq \emptyset$. Then, \emph{Algorithm \ref{hff}}
terminates with the desired output after performing no more than
\begin{align}
\label{eq:bound.alg4}
\mathcal{O}\left(1+\log^+\left(\dfrac{d_{\bpt{z},\,b}}{\sqrt{\bpt{\tau}}}\right)\right)
\end{align}
iterations. \end{corollary}
\begin{proof}
See Subsection \ref{sec:proof.drt}. \end{proof}
\subsection{A Douglas-Rachford-Tseng's F-B four-operator splitting method}
\label{subsec:drt}
In this subsection, we present and study the iteration complexity of the main algorithm in this work, for solving \eqref{eq:drt}, namely Algorithm \ref{drt}, which combines Algorithm \ref{inexact.dr}, used as an outer iteration, and Algorithm \ref{hff}, used as an inner iteration, for solving the corresponding subproblem in \eqref{eq:err.b}.
Algorithm \ref{drt} will be shown to be a special instance of Algorithm \ref{inexact.dr}, for which pointwise and ergodic iteration complexity results are available in Section \ref{sec:dr}. Corollary \ref{cor:cota.linear} will be specially important to compute a bound on the total number of inner iterations performed by Algorithm \ref{drt} to achieve prescribed tolerances.
\noindent \fbox{ \addtolength{\linewidth}{-2\fboxsep} \addtolength{\linewidth}{-2\fboxrule}
\begin{minipage}{\linewidth}
\begin{algorithm} \label{drt} {\bf A Douglas-Rachford-Tseng's F-B splitting type method for \bf{(\ref{eq:drt})}} \end{algorithm}
\begin{itemize}
\item[(0)] Let $z_0\in \mathcal{H}$, $\tau_0>0$ and $0<\sigma,\theta<1$ be given, let $C,F_1,\Omega, L, F_2$ and $\eta$ as in conditions (E1)--(E5) and $\gamma$ satisfying condition \eqref{eq:bound.gamma}, and set $k\leftarrow 1$.
\item [(1)] Call Algorithm \ref{hff} with inputs $C,F_1,\Omega, L, F_2$ and $\eta$, $(\bpt{z},\bpt{\tau}):=(z_{k-1},\tau_{k-1})$, $\sigma$ and $\gamma$ to obtain as output $(z_{j-1},z'_{j-1},z_j,\widetilde z_j)$, and set
\begin{align}
\label{eq:err.b2}
x_k=\widetilde z_j,\qquad b_k=\dfrac{z_{k-1}+z_{j-1}-(z_j+\widetilde z_j)}{\gamma},\qquad \varepsilon_{b,\,k}=\frac{\norm{z'_{j-1}-\widetilde z_j}^2}{4\eta}. \end{align}
\item[(2)] Compute $(y_k,a_k)\in \mathcal{H}\times \mathcal{H}$ such that
\begin{align}
\label{eq:err.a2}
\hspace{-3.0cm} a_k\in A(y_k),\quad \gamma a_k+y_k=x_k-\gamma b_k. \end{align}
\item[(3)] (3.a) If
\begin{align}
\label{eq:cond.err2}
\hspace{-1.3cm}\norm{\gamma b_k+x_k-z_{k-1}}^2+2\gamma \varepsilon_{b,k}\leq \sigma^2\norm{\gamma b_k+y_k-z_{k-1}}^2, \end{align}
\hspace{0.9cm}then
\begin{align}
\label{eq:ext.step2}
z_k=z_{k-1}-\gamma(a_k+b_k),\quad \tau_k=\tau_{k-1}\qquad \mbox{[extragradient step]}. \end{align}
(3.b) Else
\begin{align}
\label{eq:null.step2}
\hspace{-3.6cm} z_k=z_{k-1},\quad \tau_k=\theta\,\tau_{k-1}\qquad \mbox{[null step]}. \end{align}
\item[(4)]
Set $k\leftarrow k+1$ and go to step 1.
\end{itemize} \noindent
\end{minipage} }
In what follows we present the pointwise and ergodic iteration complexities of Algorithm \ref{drt} for solving the four-operator monotone inclusion problem \eqref{eq:drt}. The results will follow essentially from the corresponding ones for Algorithm \ref{inexact.dr} previously obtained in Section \ref{sec:dr}. On the other hand, bounds on the number of inner iterations executed before achieving prescribed tolerances will be proved by using Corollary \ref{cor:cota.linear}.
We start by showing that Algorithm \ref{drt} is a special instance of Algorithm \ref{inexact.dr}.
\begin{proposition}
\label{pr:5eh4}
The triple $(x_k,b_k,\varepsilon_{b,\,k})$ in \eqref{eq:err.b2} satisfies
condition \eqref{eq:err.b} in \emph{Step 1} of \emph{Algorithm \ref{inexact.dr}}, i.e.,
\begin{align}
\label{eq:err.b3}
b_k\in C(x_k)+F_1(x_k)+F_2^{\varepsilon_{b,k}}(x_k)\subset B^{\varepsilon_{b,\,k}}(x_k),\quad \norm{\gamma b_k+x_k-z_{k-1}}^2+2\gamma\varepsilon_{b,\,k}\leq \tau_{k-1},
\end{align}
where $B$ is as in \eqref{eq:def.bcff}. As a consequence, \emph{Algorithm \ref{drt}} is a special instance of \emph{Algorithm \ref{inexact.dr}} for solving \eqref{eq:mip} with $B$ as in \eqref{eq:def.bcff}. \end{proposition}
\begin{proof} Using the first identity in \eqref{eq:851}, the definition of $b_k$ in \eqref{eq:err.b2} as well as the fact that $\bpt{z}:=z_{k-1}$ in Step 1 of Algorithm \ref{drt} we find
\begin{align}
\label{eq:two.i}
b_k=v_j-\dfrac{1}{\gamma}(\widetilde z_j-z_{k-1})=v_j-\dfrac{1}{\gamma}(\widetilde z_j-\bpt{z}). \end{align}
Combining the latter identity with the second inclusion in \eqref{eq:pr:hff}, the second identity in \eqref{eq:851} and the definitions of $x_k$ and $\varepsilon_{b,\,k}$ in \eqref{eq:err.b2} we obtain the first inclusion in \eqref{eq:err.b3}. The second desired inclusion follows from \eqref{eq:def.bcff} and Proposition \ref{pr:teps}(b).
To finish the proof of \eqref{eq:err.b3}, note that from the first identity in \eqref{eq:two.i}, the definitions of $x_k$ and $\varepsilon_{b,\,k}$ in \eqref{eq:err.b2}, the definition of $v_j$ in \eqref{eq:851} and \eqref{eq:hff2} we have
\begin{align}
\norm{\gamma b_k+x_k-z_{k-1}}^2+2\gamma\varepsilon_{b,\,k}
=\norm{z_{j-1}-z_j}^2+\frac{\gamma\norm{z'_{j-1}-\widetilde z_j}^2}{2\eta}\leq \bpt{\tau}=\tau_{k-1}, \end{align}
which gives the inequality in \eqref{eq:err.b3}. The last statement of the proposition follows from \eqref{eq:err.b3}, \eqref{eq:err.b}--\eqref{eq:null.step} and \eqref{eq:err.a2}--\eqref{eq:null.step2}. \end{proof}
\begin{theorem}\emph{{\bf (pointwise iteration complexity of Algorithm \ref{drt})}}
\label{th:general} Let the operator $B$ and $d_{0,\gamma}$ be as in \eqref{eq:def.bcff} and \eqref{eq:def.d0}, respectively, and assume that $\max\{(1-\sigma)^{-1},\sigma^{-1}\}=\mathcal{O}(1)$. Let also $d_{0,b}$ be the distance of $z_0$ to $B^{-1}(0)\neq\emptyset$. Then, for given tolerances $\rho,\epsilon>0$, the following hold:
\begin{itemize}
\item[\emph{(a)}] \emph{Algorithm \ref{drt}} finds $a,b,x,y\in \mathcal{H}$ and $\varepsilon_b\geq 0$ such that
\begin{align}
\label{eq:opt.alg5}
a\in A(y),\;b\in C(x)+F_1(x)+F_2^{\varepsilon_b}(x),\quad \gamma\norm{a+b}=\norm{x-y}\leq \rho,\; \varepsilon_b\leq \epsilon
\end{align}
after performing no more than
\begin{align}
\label{eq:main.09}
k_{\emph{p;\,outer}}:=\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}^{\,2}}{\rho^2},
\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\}
+ \max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon} \right)\right\}\right) \end{align}
outer iterations.
\item[\emph{(b)}] Before achieving the desired tolerance $\rho,\epsilon>0$, each iteration of \emph{Algorithm \ref{drt}} performs at most
\begin{align}
\label{eq:inner.iter}
k_{\emph{inner}}:=\mathcal{O}\left(1+ \log^+\left(\dfrac{d_{0,\gamma}+d_{0,b}}{\sqrt{\tau_0}}\right)
+
\max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon}\right)\right\}
\right)
\end{align}
inner iterations; and hence evaluations of the $\eta$--cocoercive operator $F_2$.
\end{itemize}
As a consequence, \emph{Algorithm \ref{drt}} finds $a,b,x,y\in \mathcal{H}$ and $\varepsilon_b\geq 0$ satisfying \eqref{eq:opt.alg5} after performing no more than $k_{\emph{p;\,outer}}\times k_{\emph{inner}}$ inner iterations. \end{theorem}
\begin{proof} (a) The desired result is a direct consequence of the last statements in Proposition \ref{pr:5eh4} and Theorem \ref{th:idr.main}, and the inclusions in \eqref{eq:err.b3}.
(b) Using Step 1's definition and Corollary \ref{cor:cota.linear} we conclude that, at each iteration $k\geq 1$ of Algorithm \ref{drt}, the number of inner iterations
is bounded by
\begin{align}
\label{eq:bound.alg42}
\mathcal{O}\left(1+\log^+\left(\dfrac{d_{z_{k-1},\,b}}{\sqrt{\tau_{k-1}}}\right)\right)
\end{align}
where $d_{z_{k-1},\,b}$ denotes the distance of $z_{k-1}$ to $B^{-1}(0)$. Now, using the last statements in Propositions \ref{pr:5eh4} and \ref{pr:eHPE}, Proposition \ref{pr:imp.fact} and a simple argument based on the triangle inequality we obtain
\begin{align}
\label{eq:bound.alg43}
d_{z_{k-1},b}\leq 2d_{0,\gamma}+d_{0,b}\qquad \forall k\geq 1. \end{align}
By combining \eqref{eq:bound.alg42} and \eqref{eq:bound.alg43} and using \eqref{eq:tau.beta} we find that, at every iteration $k\geq 1$, the number of inner iterations is bounded by
\begin{align}
\mathcal{O}\left(1+\log^+\left(\dfrac{d_{0,\gamma}+d_{0,b}}{\sqrt{\theta^{\beta_{k-1}}\tau_0}}\right)\right)= \mathcal{O}\left(1+\log^+\left(\dfrac{d_{0,\gamma}+d_{0,b}}{\sqrt{\tau_0}}\right)+\beta_{k-1}\right). \end{align}
Using the latter bound, the last statement in Proposition \ref{pr:5eh4}, the bound on the number of null steps of Algorithm \ref{inexact.dr} given in Theorem \ref{th:idr.main}, and \eqref{eq:def.betak} we conclude that, before achieving the prescribed tolerance $\rho, \epsilon>0$, each iteration Algorithm \ref{drt} performs at most the number of iterations given in \eqref{eq:inner.iter}. This concludes the proof of (b).
To finish the proof, note that the last statement of the theorem follows directly from (a) and (b). \end{proof}
\begin{theorem}\emph{{\bf (ergodic iteration complexity of Algorithm \ref{drt})}}
\label{th:general.e} For given tolerances $\rho,\epsilon>0$, under the same assumptions of \emph{Theorem \ref{th:general}} the following hold:
\begin{itemize}
\item[\emph{(a)}] \emph{Algorithm \ref{drt}} provides \,$a,b,x,y\in \mathcal{H}$ and $\varepsilon_a,\varepsilon_b\geq 0$ such that
\begin{align}
\label{eq:eopt.alg5}
a\in A^{\varepsilon_a}(y),\;b\in \left(C+F_1+F_2\right)^{\varepsilon_b}(x),\quad \gamma\norm{a+b}=\norm{x-y}\leq \rho,\;
\varepsilon_a+\varepsilon_b\leq \epsilon
\end{align}
after performing no more than
\begin{align}
\label{eq:emain.09}
k_{\emph{e;\,outer}}:=\mathcal{O}\left(1+\max\left\{\dfrac{d_{0,\gamma}}{\rho},
\dfrac{d_{0,\gamma}^{\,2}}{\gamma\epsilon}\right\}
+ \max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon} \right)\right\}\right) \end{align}
outer iterations.
\item[\emph{(b)}] Before achieving the desired tolerance $\rho,\epsilon>0$, each iteration of \emph{Algorithm \ref{drt}} performs at most
\begin{align}
\label{eq:einner.iter}
k_{\emph{inner}}:=\mathcal{O}\left(1+ \log^+\left(\dfrac{d_{0,\gamma}+d_{0,b}}{\sqrt{\tau_0}}\right)
+
\max\left\{\log^+\left(\dfrac{\sqrt{\tau_0}}{\rho}\right),\log^+\left(\dfrac{\tau_0}{\gamma \epsilon}\right)\right\}
\right)
\end{align}
inner iterations; and hence evaluations of the $\eta$--cocoercive operator $F_2$.
\end{itemize}
As a consequence, \emph{Algorithm \ref{drt}} provides $a,b,x,y\in \mathcal{H}$ and $\varepsilon_b\geq 0$ satisfying \eqref{eq:eopt.alg5} after performing no more than $k_{\emph{e;\,outer}}\times k_{\emph{inner}}$ inner iterations. \end{theorem}
\begin{proof} The proof follows the same outline of Theorem \ref{th:general}'s proof. \end{proof}
\subsection{Proof of Corollary \ref{cor:cota.linear}}
\label{sec:proof.drt}
We start this subsection by showing that Algorithm \ref{hff} is a special instance of Algorithm \ref{shpe} for solving the strongly monotone inclusion \eqref{eq:cff}.
\begin{proposition}
\label{pr:hff}
Let $\{z_j\}$, $\{z'_j\}$ and $\{\widetilde z_j\}$ be generated by \emph{Algorithm \ref{hff}} and let the operator $B$ be as in \eqref{eq:def.bcff}. Define,
\begin{align}
\label{eq:851}
v_j:=\dfrac{z_{j-1}-z_j}{\gamma}, \qquad
\varepsilon_{j}:=\frac{\norm{z'_{j-1}-\widetilde z_j}^2}{4\eta},\quad \forall j\geq 1.
\end{align}
Then, for all $j\geq 1$,
\begin{align}
\label{eq:pr:hff}
&v_j\in (1/\gamma)(\widetilde z_j-\bpt{z})+C(\widetilde z_j)+F_1(\widetilde z_j)+F_2^{\,\varepsilon_{j}}(\widetilde z_j) \subset (1/\gamma)(\widetilde z_j-\bpt{z})+B^{\,\varepsilon_{j}}(\widetilde z_j),\\
\label{eq:pr:hff.2}
& \norm{\gamma v_j+\widetilde z_j-z_{j-1}}^2+
2\gamma \varepsilon_{j} \leq \sigma^2\norm{\widetilde z_j-z_{j-1}}^2,\\
\label{eq:pr:hff.3}
& z_j=z_{j-1}-\gamma v_j.
\end{align}
As a consequence, \emph{Algorithm \ref{hff}} is a special instance of \emph{Algorithm \ref{shpe}} with $\lambda_j\equiv \gamma$ for solving \eqref{eq:bmu} with $S(\cdot):=(1/\gamma)(\cdot-\bpt{z})$. \end{proposition}
\begin{proof}
Note that the first identity in \eqref{eq:hff} gives
\begin{align*}
\dfrac{z_{j-1}-\widetilde z_j}{\gamma}- F_1(z'_{j-1})\in
(1/\gamma)(\widetilde z_j-\bpt{z})+C(\widetilde z_j)+F_2(z'_{j-1}).
\end{align*}
Adding $F_1(\widetilde z_j)$ in both sides of the above identity and using the second and first identities in \eqref{eq:hff} and \eqref{eq:851}, respectively, we find
\begin{align}
v_j=\dfrac{z_{j-1}-z_j}{\gamma}\in
(1/\gamma)(\widetilde z_j-\bpt{z})+C(\widetilde z_j)+F_1(\widetilde z_j)+F_2(z'_{j-1}),
\end{align}
which, in turn, combined with Lemma \ref{lm:coco} and the definition of $\varepsilon_j$ in \eqref{eq:851} proves the first inclusion in \eqref{eq:pr:hff}.
Note now that the second inclusion in \eqref{eq:pr:hff} is a direct consequence of \eqref{eq:def.bcff} and Proposition \ref{pr:teps}(b). Moreover, \eqref{eq:pr:hff.3} is a direct consequence of the first identity in \eqref{eq:851}.
To prove \eqref{eq:pr:hff.2}, note that from \eqref{eq:851}, the second identity in \eqref{eq:hff}, \eqref{eq:bound.gamma} and \eqref{eq:f.Lip} we have
\begin{align*}
\norm{\gamma v_j+\widetilde z_j-z_{j-1}}^2+
2\gamma \varepsilon_{j}&=\gamma^2\norm{F_1(\widetilde z_j)-F_1(z'_{j-1})}^2+ \frac{\gamma\norm{z'_{j-1}-\widetilde z_j}^2}{2\eta}\\
&\leq \left(\gamma^2 L^2+\dfrac{\gamma}{2\eta}\right)\norm{z'_{j-1}-\widetilde z_j}^2\\
&\leq \sigma^2 \norm{z_{j-1}-\widetilde z_j}^2, \end{align*}
which is exactly the desired inequality, where we also used the facts that $z'_{j-1}=P_{\Omega}(z_{j-1})$, $\widetilde z_j\in D(C)\subset \Omega$ and that $P_\Omega$ is nonexpansive. The last statement of the proposition follows from \eqref{eq:pr:hff}--\eqref{eq:pr:hff.3}, \eqref{eq:cff}, \eqref{eq:shpe} and \eqref{eq:shpe2}. \end{proof}
\noindent {\bf Proof of Corollary \ref{cor:cota.linear}.}
Let, for all $j\geq 1$, $\{v_j\}$ and $\{\varepsilon_j\}$ be defined in \eqref{eq:851}.
Using the last statement in Proposition \ref{pr:hff} and Proposition \ref{pr:3m} with $\mu:=1/\gamma$ and $\underline{\lambda}:=\gamma$ we find
\begin{align}
\label{eq:cota.linear6}
\norm{\gamma v_j}^2+2\gamma\varepsilon_j\leq \dfrac{((1+\sigma)^2+\sigma^2)(1-\alpha)^{j-1}\norm{\bpt{z}-z_\gamma^*}^2}{1-\sigma^2}, \end{align}
where $z_\gamma^*:=(S+B)^{-1}(0)$ with $S(\cdot):=(1/\gamma)(\cdot-\bpt{z})$, i.e., $z^*_\gamma=(\gamma B+I)^{-1}(\bpt{z})$. Now, using \eqref{eq:cota.linear6}, \eqref{eq:851} and Lemma \ref{lm:rb} we obtain
\begin{align}
\label{eq:cota.linear2}
\norm{z_{j-1}-z_j}^2+\frac{\gamma\norm{z'_{j-1}-\widetilde z_j}^2}{2\eta}\leq \dfrac{((1+\sigma)^2+\sigma^2)(1-\alpha)^{j-1}d_{\bpt{z},\,b}^{\,2}}{1-\sigma^2}, \end{align}
which in turn combined with \eqref{eq:hff2}, after some direct calculations, gives \eqref{eq:bound.alg4}.
\section{Numerical experiments}
\label{sec:num}
In this section, we perform simple numerical experiments on the family of (convex) constrained quadratic programming problems
\begin{align}
\label{eq:cpro}
\begin{aligned}
& \mbox{minimize}\,\, \dfrac{1}{2} \inner{Qz}{z}+\inner{e}{z}\\
& \mbox{subject to }Kz=0, z\in X, \end{aligned} \end{align}
where $Q\in \mathbb{R}^{n\times n}$ is symmetric and either positive definite or positive semidefinite, $e=(1,\dots, 1)\in \mathbb{R}^n$, $K=(k_{j})\in \mathbb{R}^{1\times n}$, with $k_j\in \{-1,+1\}$ for all $j=1,\dots, n$, and $X=[0,10]^n$ is a box in $\mathbb{R}^n$. Problem \eqref{eq:cpro} appears, for instance, in support vector machine classifiers (see, e.g., \cite{cha.lin-lib.acm,dav-con.sjo15}). Here, $\inner{\cdot}{\cdot}$ denotes the usual inner product in $\mathbb{R}^n$.
A vector $z^*\in \mathbb{R}^n$ is a solution of \eqref{eq:cpro}, if and only if it solves the MIP
\begin{align}
\label{eq:cpro.foc}
0\in N_{\mathcal{M}}(z)+N_X(z)+Qz+e, \end{align}
where $\mathcal{M}:=\mathcal{N}(K):=\{z\in \mathbb{R}^n\,|\,Kz=0\}$. Problem \eqref{eq:cpro.foc} is clearly a special instance of \eqref{eq:drt}, in which
\begin{align}
\label{eq:eman}
A(\cdot):=N_{\mathcal{M}}(\cdot),\quad C(\cdot):=N_X(\cdot), \quad F_1(\cdot):=0\;\;\mbox{and}\;\;F_2(\cdot):=Q(\cdot)+e. \end{align}
Moreover, in this case, $J_{\gamma A}=P_{\mathcal{M}}$ and $J_{\gamma C}=P_X$.
In what follows, we analyze the numerical performance of the following three algorithms for solving the MIP \eqref{eq:cpro.foc}:
\begin{itemize}
\item The Douglas-Rachford-Tseng's F-B splitting method (Algorithm \ref{drt} (ALGO 5)) proposed in Section \ref{sec:drt}.
We set $\sigma=0.99$, $\theta=0.01$, the operators $A$, $C$, $F_1$ and $F_2$ as in \eqref{eq:eman}, and $\Omega=\mathbb{R}^n$, $L=0$ and $\eta=1/(\sup_{\norm{z}\leq 1}\norm{Qz})$ (which clearly satisfy the conditions (E1)--(E5) of Section \ref{sec:drt}). We also have set $\gamma=2\eta\sigma^2$ (see \eqref{eq:bound.gamma}) and $\tau_0=\norm{z_0-P_X(z_0)+Qz_0}^3+1$.
\item The relaxed forward-Douglas-Rachford splitting (rFDRS) from \cite[Algorithm 1]{dav-con.sjo15} (originally proposed in \cite{ari-for.opt15}). We set (in the notation of \cite{dav-con.sjo15}) $\beta_V=1/(\sup_{\norm{z}\leq 1}\norm{(P_{\mathcal{M}}\circ Q\circ P_{\mathcal{M}})z})$, $\gamma=1.99\beta_V$ and $\lambda_k\equiv 1$.
\item The three-operator splitting scheme (TOS) from \cite[Algorithm 1]{dav.yin-thr.svva17}. We set (in the notation of \cite{dav.yin-thr.svva17}) $\beta=1/(\sup_{\norm{z}\leq 1}\norm{Qz})$, $\gamma=1.99\beta$ and $\lambda_k\equiv 1$. \end{itemize}
For each dimension $n\in \{100,500,1000,2000, 6000\}$, we analyzed the performance of each the above mentioned algorithms on a set of 100 randomly generated instances of \eqref{eq:cpro}. All the experiments were performed on a laptop equipped with an Intel i7 7500U CPU, 8 GB DDR4 RAM and a nVidia GeForce 940MX.
In order to allow performance comparison of ALGO 5, rFDRS and TOS, we adopted the stopping criterion
\begin{align}
\label{eq:stc.impl}
\norm{z_{k}-z_{k-1}}\leq 10^{-6}, \end{align}
\textbf{for which we considered only extragradient steps when analyzing the performance of \mbox{ALGO 5}}.
The corresponding experiments are displayed in Tables 1 ($Q$ positive definite), 2 (Table 1 continued), 3 ($Q$ positive semidefinite) and 4 (Table 3 continued).
Now note that by using \eqref{eq:err.a2} and \eqref{eq:ext.step2}, we conclude that \eqref{eq:stc.impl} is equivalent to
\begin{align}
\label{eq:stc.impl2}
\gamma \norm{a_k+b_k}=\norm{x_k-y_k}\leq 10^{-6}. \end{align}
Motivated by the above observation, we analyzed the performance of ALGO 5 on solving \eqref{eq:cpro} while using the stopping criterion \eqref{eq:stc.impl2}, for which both extragradient and null steps are considered. The corresponding results are displayed on Tables 5 and 6.
Finally, we mention that \eqref{eq:cpro.foc} consists of a \emph{three-operator} MIP. For future research, we intend to study the numerical performance of Algorithm \ref{drt} in (true) \emph{four-operator} MIPs. One possibility would be to consider structured minimization problems of the form
\begin{align}
\label{eq:min4}
\min_{x\in \mathcal{H}}\,\left\{f(x)+g(x)+\varphi(\widetilde {K}x)+h(x)\right\} \end{align}
where $f,g,\varphi:\mathcal{H}\to (-\infty,+\infty]$ are proper closed convex functions, $h:\mathcal{H}\to \mathbb{R}$ is convex and differentiable and $\widetilde K:\mathcal{H}\to \mathcal{H}$ is a bounded linear operator. Under certain qualification conditions, \eqref{eq:min4} is equivalent to the MIP
\begin{align}
0\in \partial f(x)+\partial g(x)+\widetilde{K}^*\partial \varphi(\widetilde{K} x)+\nabla h(x) \end{align}
which, in turn, is clearly equivalent to
\begin{align}
\label{eq:mip.pd}
\begin{aligned}
& 0\in \partial f(x)+\partial g(x)+ \widetilde{K}^*y+\nabla h(x)\\ & 0\in \partial \varphi^*(y)-\widetilde{K}x,
\end{aligned} \end{align}
where $\varphi^*$ denotes the Fenchel-conjugate of $\varphi$. We now note that \eqref{eq:mip.pd} is a special instance of \eqref{eq:drt} where, by letting $z=(x,y)$,
\begin{align}
\begin{aligned}
&A(z):=\partial f(x)\times \partial\varphi^*(y),\quad C(z):=\partial g(x)\times \{0\},\\
& F_1(z):=(\widetilde K^*y,-\widetilde Kx),\quad F_2(z):=(\nabla h(x),0).
\end{aligned} \end{align}
Hence, under mild conditions on \eqref{eq:min4} (specially regarding conditions (E1)--(E5) on Section \ref{sec:drt}), Algorithm \ref{drt} is potentially applicable to solve \eqref{eq:mip.pd} (i.e., \eqref{eq:min4}).
We also mention that while the variational problem \eqref{eq:min4} appears in different applications in Imaging and related fields, the primal-dual formulation \eqref{eq:mip.pd} has been widely used in nowadays research in designing efficient primal-dual methods for, in particular, solving \eqref{eq:min4} (see, e.g., \cite{bot.cse.hei-con.mp15,bot.cse.hen-rec.mwb14,cha.poc-fir.jmiv11,com-sys.jco13,com.pes-pri.svva12}).
\begin{table}[!htpb]
\caption{Running time (in seconds) and number of iterations performed by ALGO 5, rFDRS and TOS to reach the stopping criterion \eqref{eq:stc.impl} on a set of 100 randomly generated instances of \eqref{eq:cpro} with the matrix $Q$ \textbf{positive definite}, with $n\in \{100,500,1000,2000,6000\}$. We can see that either ALGO 5 or TOS outperform the rFDRS in terms of (mean) running time, while ALGO 5 shows a slightly superior performance on large dimensions. Moreover, -- see Table 2 -- when compared to TOS, ALGO 5 provides a much more accurate approximate solution to the (unique) solution of \eqref{eq:cpro}. }
\begin{center}
\begin{tabular}{llcccp{0.25cm}rrr}
\hline \\
& \multicolumn {3}{r}{ Time } & & & \multicolumn {3}{c}{ Iterations} \\
\cline {3-5} \cline{7-9}\\
$n$ & Algorithm & Min & Max & Mean & & Min & Max & Mean \\ \hline \\
& ALGO 5 & 0.0014 & 0.0132 & 0.0017 & & 12 & 21 & 15.21 \\
$100$ & rFDRS & 0.0014 & 0.0170 & 0.0018 & & 6 & 15 & {\bf 8.31} \\
& TOS & 0.0010 & 0.0097 & \textbf{0.0012} & & 7 & 16 & 9.37 \\ \hline
& ALGO 5 & 0.0302 & 0.0451 & \textbf{0.0334} & & 15 & 22 & 17.24 \\
$500$ & rFDRS & 0.0517 & 0.0843 & 0.0626 & & 8 & 15 & {\bf 10.26} \\
& TOS & 0.0288 & 0.0434 & 0.0340 & & 9 & 16 & 11.28 \\ \hline
& ALGO 5 & 0.3177 & 0.4190 & \textbf{0.3540} & & 14 & 20 & 17.14 \\
$1000$ & rFDRS & 0.4768 & 0.7304 & 0.5771 & & 9 & 14 & {\bf 10.84} \\
& TOS & 0.3103 & 0.4504 & 0.3760 & & 10 & 15 & 11.86 \\ \hline
& ALGO 5 & 3.6411 & 3.9397 & \textbf{3.7648} & & 19 & 21 & 19.80 \\
$2000$ & rFDRS & 5.0062 & 5.5711 & 5.2795 & & 11 & 12 & {\bf 11.70} \\
& TOS & 3.6632 & 4.0798 & 3.7703 & & 12 & 13 & 12.70 \\ \hline
& ALGO 5 & 94.7551 & 121.1123 &\textbf{ 101.6311} & & 18 & 20 & 18.81 \\
$6000$ & rFDRS & 107.3812 & 125.9018 & 115.0631 & & 11 & 13 & {\bf12.20} \\
& TOS & 94.7152 & 123.6527 & 104.0517 & & 12 & 15 & 13.18 \\
\hline\\
\end{tabular}
\end{center}
\end{table}
\begin{table}[!htpb]
\caption{Table 1 continued. Here (1) we show the number of extragradient and null steps performed by ALGO 5 while reaching the stopping criterion \eqref{eq:stc.impl}; (2) we evaluate the absolute error between the provided iterate $z_k$ and the unique solution $z^*$ of \eqref{eq:cpro}. We can see that, when compared to TOS, both ALGO 5 and rFDRS provide a much more accurate approximate solution.}
\begin{center}
\begin{tabular}{lllllp{0.1cm}p{0.5cm}p{0.5cm}p{0.5cm}p{0.1cm}rrr}
\hline \\
& & \multicolumn {3}{c}{Extragradient steps} & & \multicolumn {3}{c}{Null steps} & & \multicolumn {3}{c}{$ || z_{k}-z_{*} || $} \\
\cline {3-5} \cline{7-9} \cline{11-13}\\
$n$ & Algorithm & Min & Max & Mean & & Min & Max & Mean & & Min & Max & Mean \\ \hline
& ALGO 5 & 8 & 13 & 10.24 & & 3 & 4 & 3.58 & & 0.0033 & 0.7829 & {\bf 0.2714} \\
$100$ & rFDRS & & & & & & & & & 0.0014 & 2.0768 & 0.3401 \\
& TOS & & & & & & & & & 2.8702 & 5.7959 & 4.2820 \\ \hline
& ALGO 5 & 9 & 14 & 11.64 & & 3 & 5 & 4.30 & & 0.0009 & 1.0619 & \textbf{0.2004} \\
$500$ & rFDRS & & & & & & & & & 0.0029 & 1.0275 & 0.3336 \\
& TOS & & & & & & & & & 5.7643 & 10.0704 & 7.8648 \\ \hline
& ALGO 5 & 10 & 16 & 12.55 & & 4 & 5 & 4.53 & & 0.0036 & 0.5649 & \textbf{0.1798} \\
$1000$ & rFDRS & & & & & & & & & 0.0008 & 0.9373 & 0.2763 \\
& TOS & & & & & & & & & 7.9120 & 13.3011 & 10.5356 \\ \hline
& ALGO 5 & 11 & 19 & 13.30 & & 4 & 6 & 4.70 & & 0.0766 & 0.5499 & 0.2795 \\
$2000$ & rFDRS & & & & & & & & & 0.1004 & 0.3433 & {\bf 0.2278} \\
& TOS & & & & & & & & & 13.3085 & 16.2610 & 14.4403\\ \hline
& ALGO 5 & 13 & 18 & 15.20 & & 4 & 7 & 5.20 & & 0.0437 & 0.6245 & \textbf{0.2333}\\
$6000$ & rFDRS & & & & & & & & & 0.1626 & 1.0021 & 0.4523 \\
& TOS & & & & & & & & & 19.9698 & 24.4657 & 22.9383 \\
\hline\\
\end{tabular}
\end{center}
\end{table}
\begin{table}[h!]
\caption{Running time (in seconds) and number of iterations performed by ALGO 5, rFDRS and TOS to reach the stopping criterion \eqref{eq:stc.impl} on a set of 100 randomly generated instances of \eqref{eq:cpro} with the matrix $Q$ \textbf{positive semidefinite}, with $n\in \{100,500,1000,2000,6000\}$.
Similarly to the case of $Q$ positive semidefinite, we can see that either ALGO 5 or TOS outperform the rFDRS in terms of (mean) running time, while ALGO 5 shows a slightly superior performance on large dimensions.}
\begin{center}
\begin{tabular}{llcccp{0.25cm}rrr}
\hline \\
& \multicolumn {3}{r}{ Time } & & & \multicolumn {3}{c}{ Iterations} \\
\cline {3-5} \cline{7-9}\\
$n$ & Algorithm & Min & Max & Mean & & Min & Max & Mean \\ \hline \\
& ALGO 5 & 0.0013 & 0.0130 & 0.0018 & & 11 & 20 & 15.31 \\
$100$ & rFDRS & 0.0013 & 0.0119 & 0.0017 & & 6 & 18 & {\bf 10.16} \\
& TOS & 0.0008 & 0.0056 & \textbf{0.0011} & & 7 & 19 & 11.19 \\ \hline
& ALGO 5 & 0.0249 & 0.0409 & \textbf{0.0283} & & 15 & 24 & 17.99 \\
$500$ & rFDRS & 0.0466 & 0.0895 & 0.0563 & & 9 & 18 & {\bf 12.13} \\
& TOS & 0.0236 & 0.0432 & 0.0291 & & 10 & 19 & 13.13 \\ \hline
& ALGO 5 & 0.3013 & 0.4061 & \textbf{0.3229} & & 17 & 24 & 19.88 \\
$1000$ & rFDRS & 0.4604 & 0.6780 & 0.5443 & & 10 & 16 & {\bf 13.18} \\
& TOS & 0.2910 & 0.4354 & 0.3348 & & 11 & 17 & 14.18 \\ \hline
& ALGO 5 & 3.6649 & 4.0284 & \textbf{3.7989} & & 18 & 25 & 21.75 \\
$2000$ & rFDRS & 5.0060 & 5.3364 & 5.1248 & & 13 & 15 & {\bf 14.20} \\
& TOS & 3.6933 & 4.0949 & 3.8607 & & 14 & 16 & 15.27 \\ \hline
& ALGO 5 & 101.0412 & 111.8105 & \textbf{107.9423} & & 20 & 23 & 21.80 \\
$6000$ & rFDRS & 115.9101 & 146.0409 & 130.2121 & & 13 & 16 & {\bf 15.01} \\
& TOS & 105.1307 & 116.9615 & 110.3801 & & 14 & 17 & 16.05 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h!]
\caption{Table 3 continued. Here we provide the number of extragradient and null steps performed by ALGO 5 while reaching the stopping criterion \eqref{eq:stc.impl}.}
\begin{center}
\begin{tabular}{llcccp{0.25cm}ccc}
\hline \\
& & \multicolumn {3}{c}{Extragradient steps} & & \multicolumn {3}{c}{Null steps} \\
\cline {3-5} \cline{7-9} \\
$n$ & Algorithm & Min & Max & Mean & & Min & Max & Mean \\ \hline \\
$100$ & ALGO 5 & 8 & 19 & 12.23 & & 3 & 5 & 3.77 \\
$500$ & ALGO 5 & 11 & 21 & 14.14 & & 4 & 5 & 4.31 \\
$1000$ & ALGO 5 & 12 & 20 & 15.42 & & 4 & 5 & 4.54 \\
$2000$ & ALGO 5 & 14 & 20 & 15.95 & & 4 & 6 & 4.70 \\
$6000$ & ALGO 5 & 14 & 21 & 16.55 & & 4 & 7 & 5.35 \\ \hline\\
\end{tabular}
\end{center}
\end{table}
\begin{table}[h!]
\begin{center}
\caption{Running time (in seconds) and number of iterations performed by ALGO 5 to reach the stopping criterion \eqref{eq:stc.impl2} on a set of 100 randomly generated instances of \eqref{eq:cpro} with the matrix $Q$ \textbf{positive definite}, with $n\in \{100,500,1000,2000,6000\}$. We can see a slight improvement when compared to the results obtained via the stopping criterion \eqref{eq:stc.impl} -- cf. Table 1.}
\begin{tabular}{lcccp{0.25cm}rrr}
\hline \\
& \multicolumn {3}{ c }{ Time } & \multicolumn {4}{c}{ Iterations} \\
\cline { 2-4 } \cline{6-8}\\
$n$ & Min & Max & Mean & & Min & Max & Mean \\ \hline \\
$100$ & 0.0012 & 0.0121 & 0.0016 & & 11 & 17 & 13.66 \\
$500$ & 0.0251 & 0.0492 & 0.0314 & & 14 & 19 & 16.10\\
$1000$ & 0.3000 & 0.3539 & 0.3201 & & 15 & 20 & 17.32\\
$2000$ & 3.5538 & 3.7583 & 3.5914 & & 16 & 20 & 17.72 \\
$6000$ & 98.8411 & 102.9118 & 99.7147& & 18 & 21 & 19.20 \\ \hline\\
\end{tabular}
\end{center}
\end{table}
\begin{table}[h!]
\begin{center}
\caption{Table 5 continued. Here, we show the number of extragradient and null steps performed by ALGO 5 while reaching the stopping criterion \eqref{eq:stc.impl2} and evaluate the absolute error between the provided iterate $z_k$ and the unique solution $z^*$ of \eqref{eq:cpro} -- cf. Table 2. }
\begin{tabular}{llccp{0.25cm}cccp{0.25cm}rrr}
\hline \\
& \multicolumn {3}{ c }{Extragradient steps} & & \multicolumn {3}{c}{Null steps} & & \multicolumn {3}{c}{Absolute Error} \\
\cline { 2-4 } \cline{6-8} \cline{10-12} \\
$n$ & Min & Max & Mean & & Min & Max & Mean & & Min & Max & Mean \\ \hline \\
$100$ & 8 & 16 & 10.49 & & 3 & 4 & 3.48 & & 0.0007 & 0.6949 & 0.2354\\
$500$ & 9 & 15 & 11.82 & & 3 & 5 & 4.22 & & 0.0007 & 0.7109 & 0.2134\\
$1000$ & 10 & 16 & 12.61 & & 4 & 5 & 4.42 & & 0.0011 & 0.6628 & 0.1989\\
$2000$ & 12 & 16 & 13.46 & & 4 & 6 & 4.64 & & 0.0061 & 0.4537 & 0.1596 \\
$6000$ & 13 & 17 & 14.50 & & 4 & 6 & 4.80 & & 0.0557 & 0.3666 & 0.2236 \\ \hline
\end{tabular}
\end{center} \end{table}
\appendix \section{Auxiliary results}
\begin{lemma}\emph{(\cite[Lemma 3.1]{alv.mon.sva-reg.siam16})}
\label{lm:rb}
Let $z_\gamma^*:=(\gamma B+I)^{-1}(\bpt{z})$ be the (unique) solution of \eqref{eq:cff}. Then,
\begin{align}
\norm{\bpt{z}-z_\gamma^*}\leq \norm{\bpt{z}-x^*}\qquad \forall x^*\in B^{-1}(0). \end{align} \end{lemma}
\begin{lemma}\emph{(\cite[Lemma 2.2]{sva-cla.jota14})}
\label{lm:coco}
Let $F:\mathcal{H}\to \mathcal{H}$ be $\eta$--cocoercive, for some $\eta>0$, and let $z',\widetilde z\in \mathcal{H}$. Then,
\[
F(z')\in F^{\varepsilon}(\widetilde z)\quad\mbox{where}\quad \varepsilon:=\dfrac{\norm{z'-\widetilde z}^2}{4\eta}.
\] \end{lemma}
\def$'${$'$}
\end{document} | arXiv |
Existence of periodic solution for a class of systems involving nonlinear wave equations
Claudianor O. Alves
In this paper, we show the existence of periodic solution for a class of systems involving nonlinear wave equations. The main tool used is the dual method.
Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4(3): 487-498. doi: 10.3934\/cpaa.2005.4.487.
Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates
D. Bartolucci and L. Orsina
We analyze the singular behavior of the Green's function for uniformly elliptic equations on smooth and bounded two dimensional domains. Then, we are able to generalize to the uniformly elliptic case some sharp estimates for Liouville type equations due to Brezis-Merle [7] and, in the same spirit of [3], a "mass" quantization result due to Y.Y. Li [21]. As a consequence, we obtain uniform a priori estimates for solutions of the corresponding Dirichlet problem. Then, we improve the standard existence theorem derived by direct minimization and, in the same spirit of [17] and [37], obtain the existence of Mountain Pass type solutions.
D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4(3): 499-522. doi: 10.3934\/cpaa.2005.4.499.
Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary
C. Brändle, F. Quirós and Julio D. Rossi
We study a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as non-simultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.
C. Br\u00E4ndle, F. Quir\u00F3s, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4(3): 523-536. doi: 10.3934\/cpaa.2005.4.523.
Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation
Qiang Du, Chun Liu, R. Ryham and X. Wang
2005, 4(3): 537-548 doi: 10.3934/cpaa.2005.4.537 +[Abstract](1831) +[PDF](1560.4KB)
In this paper, we study the effects of the spontaneous curvature on the static deformation of a vesicle membrane under the elastic bending energy, with prescribed bulk volume and surface area. Generalizing the phase field models developed in our previous works, we deduce a new energy formula involving the spontaneous curvature effects. Several axis-symmetric configurations are obtained through numerical simulations. Some analysis on the effects of the spontaneous curvature on the vesicle membrane shapes are also provided.
Qiang Du, Chun Liu, R. Ryham, X. Wang. Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation. Communications on Pure & Applied Analysis, 2005, 4(3): 537-548. doi: 10.3934\/cpaa.2005.4.537.
Existence and non-existence for a mean curvature equation in hyperbolic space
Elias M. Guio and Ricardo Sa Earp
There exists a well-known criterion for the solvability of the Dirichlet Problem for the constant mean curvature equation in bounded smooth domains in Euclidean space. This classical result was established by Serrin in 1969. Focusing the Dirichlet Problem for radial vertical graphs P.-A. Nitsche has established an existence and non-existence results on account of a criterion based on the notion of a hyperbolic cylinder. In this work we carry out a similar but distinct result in hyperbolic space considering a different Dirichlet Problem based on another system of coordinates. We consider a non standard cylinder generated by horocycles cutting orthogonally a geodesic plane $\mathcal P$ along the boundary of a domain $\Omega\subset \mathcal P.$ We prove that a non strict inequality between the mean curvature $\mathcal H'_{\mathcal C}(y)$ of this cylinder along $\partial \Omega$ and the prescribed mean curvature $\mathcal H(y),$ i.e $\mathcal H'_{\mathcal C}(y)\geq |\mathcal H(y)|, \forall y\in\partial\Omega$ yields existence of our Dirichlet Problem. Thus we obtain existence of surfaces whose graphs have prescribed mean curvature $\mathcal H(x)$ in hyperbolic space taking a smooth prescribed boundary data $\varphi.$ This result is sharp because if our condition fails at a point $y$ a non-existence result can be inferred.
Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4(3): 549-568. doi: 10.3934\/cpaa.2005.4.549.
Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals
Damiano Foschi
We discuss the asymptotic behaviour for the best constant in $L^p$-$L^q$ estimates for trigonometric polynomials and for an integral operator which is related to the solution of inhomogeneous Schrödinger equations. This gives us an opportunity to review some basic facts about oscillatory integrals and the method of stationary phase, and also to make some remarks in connection with Strichartz estimates.
Damiano Foschi. Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals. Communications on Pure & Applied Analysis, 2005, 4(3): 569-588. doi: 10.3934\/cpaa.2005.4.569.
On certain nonlinear parabolic equations with singular diffusion and data in $L^1$
C. García Vázquez and Francisco Ortegón Gallego
An existence result is established for a class of nonlinear parabolic equations having a coercive diffusion matrix blowing-up for a finite value of the unknown, a second hand $f\in L^1(Q)$, and an initial data $u_0\in L^1(\Omega)$. We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some uniqueness results are also shown under additional hypotheses.
C. Garc\u00EDa V\u00E1zquez, Francisco Orteg\u00F3n Gallego. On certain nonlinear parabolic equations with singular diffusion and data in $L^1$. Communications on Pure & Applied Analysis, 2005, 4(3): 589-612. doi: 10.3934\/cpaa.2005.4.589.
New dissipated energies for the thin fluid film equation
Richard S. Laugesen
The thin fluid film evolution $h_t = -(h^n h_{x x x})_x$ is known to conserve the fluid volume $\int h dx$ and to dissipate the "energies" $\int h^{1.5-n} dx$ and $\int h_x^2 dx$. We extend this last result by showing the energy $\int h^p h_x^2 dx$ is dissipated for some values of $p < 0$, when $\frac{1}{2} < n < 3$. For example when $n=1$, the Hele-Shaw equation $h_t = -(h h_{x x x})_x$ dissipates $\int h^{-1/2} h_x^2 dx$.
Richard S. Laugesen. New dissipated energies for the thin fluid film equation. Communications on Pure & Applied Analysis, 2005, 4(3): 613-634. doi: 10.3934\/cpaa.2005.4.613.
Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid
Yuming Qin, T. F. Ma, M. M. Cavalcanti and D. Andrade
This paper is concerned with the exponential stability of solutions in $H^4$ for the Navier--Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains $\Omega_n$ in $\mathbb{R}^n (n=2,3)$.
Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure & Applied Analysis, 2005, 4(3): 635-664. doi: 10.3934\/cpaa.2005.4.635.
A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions
Satoshi Kosugi, Yoshihisa Morita and Shoji Yotsutani
We consider the Ginzburg-Landau equation with a positive parameter, say lambda, and solve all equilibrium solutions with periodic boundary conditions. In particular we reveal a complete bifurcation diagram of the equilibrium solutions as lambda increases. Although it is known that this equation allows bifurcations from not only a trivial solution but also secondary bifurcations as lambda varies, the global structure of the secondary branches was open. We first classify all the equilibrium solutions by considering some configuration of the solutions. Then we formulate the problem to find a solution which bifurcates from a nontrivial solution and drive a reduced equation for the solution in terms of complete elliptic integrals involving useful parametrizations. Using some relations between the integrals, we investigate the reduced equation. In the sequel we obtain a global branch of the bifurcating solution.
Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4(3): 665-682. doi: 10.3934\/cpaa.2005.4.665.
Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions
Zhenhua Zhang
This paper is concerned with the asymptotic behavior of solutions to the phase-field equations subject to the Neumann boundary conditions where a Lojasiewicz-Simon type inequality plays an important role. In this paper, convergence of the solution of this problem to an equilibrium, as time goes to infinity, is proved.
Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4(3): 683-693. doi: 10.3934\/cpaa.2005.4.683. | CommonCrawl |
A novel systems biology approach to evaluate mouse models of late-onset Alzheimer's disease
Christoph Preuss1 na1,
Ravi Pandey1 na1,
Erin Piazza2,
Alexander Fine1,
Asli Uyar1,
Thanneer Perumal2,
Dylan Garceau1,
Kevin P. Kotredes1,
Harriet Williams1,
Lara M. Mangravite3,
Bruce T. Lamb4,
Adrian L. Oblak4,
Gareth R. Howell1,
Michael Sasner1,
Benjamin A. Logsdon3,
the MODEL-AD Consortium &
Gregory W. Carter ORCID: orcid.org/0000-0002-2834-81861
Late-onset Alzheimer's disease (LOAD) is the most common form of dementia worldwide. To date, animal models of Alzheimer's have focused on rare familial mutations, due to a lack of frank neuropathology from models based on common disease genes. Recent multi-cohort studies of postmortem human brain transcriptomes have identified a set of 30 gene co-expression modules associated with LOAD, providing a molecular catalog of relevant endophenotypes.
This resource enables precise gene-based alignment between new animal models and human molecular signatures of disease. Here, we describe a new resource to efficiently screen mouse models for LOAD relevance. A new NanoString nCounter® Mouse AD panel was designed to correlate key human disease processes and pathways with mRNA from mouse brains. Analysis of the 5xFAD mouse, a widely used amyloid pathology model, and three mouse models based on LOAD genetics carrying APOE4 and TREM2*R47H alleles demonstrated overlaps with distinct human AD modules that, in turn, were functionally enriched in key disease-associated pathways. Comprehensive comparison with full transcriptome data from same-sample RNA-Seq showed strong correlation between gene expression changes independent of experimental platform.
Taken together, we show that the nCounter Mouse AD panel offers a rapid, cost-effective and highly reproducible approach to assess disease relevance of potential LOAD mouse models.
Late-onset Alzheimer's disease (LOAD) is the most common cause of dementia worldwide [1]. LOAD presents as a heterogenous disease with highly variable outcomes. Recent efforts have been made to molecularly characterize LOAD using large cohorts of post-mortem human brain transcriptomic data [2]. Systems-level analysis of these large human data sets has revealed key drivers and molecular pathways that reflect specific changes resulting from disease [2, 3]. These studies have been primarily driven by gene co-expression analyses that reduce transcriptomes to modules representing specific disease processes or cell types across heterogenous tissue samples [2, 4,5,6]. Similar approaches have been used to characterize mouse models of neurodegenerative disease [7]. Detailed cross-species analysis reveals a translational gap between animal models and human disease, as no existing models fully recapitulate pathologies associated with LOAD [8,9,10,11]. New platforms to rapidly assess the translational relevance of new animal models of LOAD will allow efficient identification of the most promising preclinical models.
In this study, we describe a novel gene expression panel to assess LOAD-relevance of mouse models based on expression of key genes in the brain. We used a recent human molecular disease catalog based on harmonized co-expression data from three independent post mortem brain cohorts (ROSMAP, Mayo, Mount Sinai Brain bank) [12,13,14] and seven brain regions that define 30 human co-expression modules and five consensus clusters derived from the overlap of those modules [9]. These modules were used to design a mouse gene expression panel to assess the molecular overlap between human disease states and mouse models. This nCounter Mouse AD panel was piloted with samples from three novel mouse models of LOAD. Same-sample comparison between NanoString and RNA-Seq data demonstrated high per-gene correlation and overall concordance when separately compared to human disease co-expression modules. Taken together, the rapid screening of mouse models in the course of different life stages will allow better characterization of models based on alignment with specific human molecular pathologies.
Human-mouse co-expression module conservation and probe coverage across 30 LOAD associated modules
An overview of the Mouse AD panel design for translating the 30 human AMP-AD co-expression modules from three cohorts and seven brain regions is depicted in Fig. 1. Mouse to human gene prioritization resulted in the selection of 760 key mouse genes targeting a subset of highly co-expressed human genes plus 10 housekeeping genes, which explained a significant proportion of the observed variance across the 30 human AMP-AD modules (Methods). Co-expression modules were grouped into functionally distinct consensus clusters as previously described by Wan, et al. (see also Table S1) [9]. These consensus clusters contain expression modules from different brain regions and independent studies that share a high overlap in gene content and similar expression characteristics. Consensus clusters were annotated based on Reactome pathway enrichment analysis for the corresponding genes within each functionally distinct cluster (Methods, Table S1). Since consensus clusters showed an enrichment of multiple biological pathways, the highest rank and non-overlapping Reactome pathway was used to refer to each cluster (Table S2). In order to assess the conservation of sequence and gene expression levels between human and mouse genes for each of the 30 human co-expression modules, dN/dS values were correlated with the overall overlap in expression in brains from six-month-old C57BL/6 J (B6) mice (Fig. 2a). The fraction of orthologous genes expressed in the mouse brain, based on the presence or absence of transcripts at detectable levels, was very highly correlated with the overall module conservation (p < 2.2 × 10− 16, Pearson's correlation coefficient: − 0.96). Module conservation was based on the median dN/dS statistics measuring the rate of divergence in the coding sequence for all genes within a given module between both species (Figure S1). Notably, human co-expression modules of Consensus Cluster C, associated with the neuronal system and neurotransmission, showed the lowest degree of sequence divergence with a high proportion of human genes (64–72%) expressed in six-month-old B6 mice. In contrast to the highly conserved neuronal modules, immune modules of Consensus Cluster B contained genes that recently diverged on the sequence level and acquired a higher number of destabilizing missense variants. These modules showed the highest median dN/dS values and the lowest fraction of genes (27–46%) expressed in the mouse brain across all tested modules. The remaining human co-expression modules, associated with different functional categories (Fig. 2a, Table S1), had intermediate overlap in expression levels between human and mice. Each of the 30 human co-expression modules was covered with an average of 148 NanoString mouse probes (SD = 50 probes), where a single mouse probe can map to multiple human modules from different study cohorts and across several brain regions. Overall, mouse probe coverage for human co-expression modules ranged between 4 and 19%, depending on the size and level of conservation of the targeted human module (Fig. 2b and c, Tables S2 and S3). For three of the largest human co-expression modules harboring over 4000 transcripts, the probe coverage was slightly below the targeted 5% coverage threshold. However, these large modules are predominantly associated with neuronal function and show a high degree of expression and sequence conservation between human and mouse (Figs. 2a). Immune modules, containing genes that recently diverged on the coding sequence level, are well covered with a median coverage of 10% (Fig. 2c). A complete annotation of mouse probes to human transcripts for each human co-expression module is provided in Table S3. In addition, we compared our novel panel to the existing nCounter Mouse Neuropathology panel designed to assess expression changes in multiple neurodegenerative diseases. We observed an overlap of 105 probes (7%) between both panels, highlighting that most of our selected probe content is novel and specific to LOAD associated disease processes and pathways.
Overview of the nCounter Mouse AD panel design. The novel Mouse AD panel measures expression of genes from a set of 30 human co-expression modules from three human LOAD cohorts, including seven distinct brain regions. Human genes central to each of the human expression modules were prioritized for the Mouse AD panel to select conserved signatures of LOAD associated pathways
Human to mouse comparison and probe coverage summary statistics. a Human-mouse sequence divergence (median dN/dS values) is inversely correlated (Pearson's correlation coefficient: − 0.96) with the fraction of genes being expressed in B6 mouse brain for each of the human co-expression modules. b Coverage of the 770 selected mouse NanoString probes for the 30 human co-expression modules associated with five functional consensus clusters. The size and number of human co-expression modules differs for the three post-mortem brain cohorts (ROSMAP, Mayo, Mount Sinai Brain Bank) and across the seven included brain regions. c This results in a varying degree of probe coverage for each module with a number of disease associated consensus clusters (a-e), reflecting disease related pathways and processes
Prioritized subset of key genes shows a higher degree of sequence conservation and expression level across modules
In order to assess the level of sequence divergence and expression for the prioritized subset of genes on the novel panel, the selected subset of genes was compared to all genes across the 30 human co-expression modules. The 760 key genes, explaining a significant proportion of the observed variance in each human module, showed an overall lower level of sequence divergence (median dN/dS values) when compared to all other genes in the modules (Fig. 3, Figure S1). Furthermore, the selected key genes on the Mouse AD panel also displayed a higher average level of gene expression in brains of six-month-old B6 mice compared to the remaining genes for each of the 30 modules (Fig. 3). This highlights that our formal prioritization procedure resulted in the selection of a subset of highly expressed key genes, which are also more conserved between human and mouse facilitating the translation of co-expression profiles across species.
NanoString Mouse AD probe genes are strongly conserved and show high expression levels in the mouse brain. Comparison between gene-level sequence divergence and transcript abundances in 6 months old B6 mouse brains for all genes (red) in human co-expression modules and the subset of 770 genes covered by NanoString probes on the Mouse AD panel (green). Human transcripts within AMP-AD co-expression modules targeted by selected mouse NanoString probes highlighted in green showed higher levels of sequence conservation and transcript abundance across species when compared to unselected genes within modules
5xFAD mouse co-expression modules correlate with AMP-AD modules enriched for inflammatory and stress response pathways
To validate our novel Mouse AD panel, a time-course analysis was performed to correlate human AMP-AD co-expression modules with the 5xFAD mouse model carrying a transgenic insertion with five familial mutations in APP and PSEN1. The 5xFAD strain is a commonly used model of neurodegenerative disease for which neuropathology, histological as well as behavioral data are readily available (Fig. 4a). We analyzed mouse NanoString data from brain hemispheres from 1 to 12 months old mice in order to identify an overlap with human post-mortem co-expression modules in the course of amyloid deposition and aging. 5xFAD mice have been shown to accumulate high levels of intraneuronal Aβ42 at around 1.5 months of age and extracellular amyloid at around 2 months of age in the subiculum and cortex (Fig. 4a) [15]. Amyloid deposition in 5xFAD mice induced rising oxidative stress in surrounding cells and tissues and led to subsequent cell dysfunction and elevation of apoptosis markers. This is reflected in the transcriptomic response of one to 2 months old 5xFAD mice which showed a significant positive correlation (p < 0.05, Pearson's correlation coefficient > 0.25) with multiple human AMP-AD modules enriched for pathways linked to the cellular stress response in Consensus Cluster E (Fig. 4a). The 5XFAD strain also exhibited neuroinflammation after 2 months of age, as previously demonstrated by an increase in astrogliosis and microgliosis after initial plaque deposition [15]. This is in line with our 5xFAD data from the Mouse AD panel which showed an increased correlation between 3 and 4 months with several human modules in Consensus Cluster B enriched for immune related pathways. Furthermore, we observed an increased correlation with inflammatory AMP-AD co-expression modules after 4 months of age, where older mice at (6–12 months) showed a highly significant overlap with human immune modules in the cerebellum, superior temporal gyrus and inferior frontal gyrus (p < 0.05, Pearson's correlation coefficient > 0.35). This clear shift in inflammatory gene expression signatures after 3 months of age is a hallmark of the 5xFAD model in response to aggressive amyloid deposition in the brain and corresponded with previous findings from a transcriptomic survey of 5xFAD mice [16]. Reactome pathway analysis in 5xFAD mice compared to age-matched B6 controls supported the results from our correlation analysis (Fig. 4b). At a young age (3 months), several stress and immune related pathways were enriched in 5xFAD mice when compared to B6 mice. These pathways, including the activation of FOXO transcription factors pathway, are well known to mediate a cellular stress response to Aβ42 [17]. In older mice between the ages of 6 to 12 months, pathways linked to microglia and complement activation, such as the DAP12 signaling pathway, were enriched in 5xFAD mice which is in line with a previous study (Fig. 4b) [16]. Moreover, we observed a positive correlation with human AMP-AD modules enriched for neuronal pathways in Consensus Cluster C. This transcriptional response occurred after early amyloid deposition in male mice at 3 months (p < 0.05, Pearson's correlation coefficient > 0.15) and after the onset of neuronal loss in female mice at 12 months. Taken together, our novel approach identified several transcriptomic signatures in responses to amyloid deposition in the 5xFAD model that correlated with human post-mortem data from different brain regions. Despite the overlap with several key disease processes on the transcriptome level across species, the 5xFAD strain does not fully recapitulate LOAD pathologies. In addition, the highly penetrant nature of the early-onset familial variants expressed on the 5xFAD background made it difficult to identify disease related processes independent of amyloid pathology. For this reason, we used our approach to elucidate the role of additional AD risk variants in a set of novel mouse models .
Time-course correlation analysis between the 5xFAD mouse model and 30 human co-expression modules using the NanoString Mouse AD panel. a The 5xFAD mouse model shows a significant correlation with functionally distinct AMP-AD co-expression modules that correspond to previously reported phenotypes from by Oakley et al. [15] and Landel et al. [16]. Circles correspond to significant (p < 0.05) positive (blue) and negative (red) Pearson's correlation coefficients for gene expression changes in 5xFAD mice (log fold change of strain minus age matched B6 mice) and human disease (log fold change for cases minus controls). Correlations are based on the comparison of mouse NanoString data to human RNA-seq expression data from the three AMP-AD cohorts for seven brain regions. Human co-expression modules are ordered into Consensus Clusters A-E [9] describing major sources of AD-related alterations in transcriptional states across independent studies and brain regions. Consensus clusters are annotated based on the most significantly enriched and non-redundant Reactome pathway terms (Supplemental Tables S1, S2). b Reactome pathway enrichment analysis for multiple time points (1 month to 12 months) implicates multiple immune and stress-related processes in the response to amyloid deposition in the course of aging within the 5xFAD mouse model
Novel mouse models harboring LOAD associated risk variants correlate with distinct AMP-AD modules in a brain region- and pathway-specific manner
Three novel mouse models, harboring two LOAD risk alleles, (Table S4) were used to translate co-expression profiles between human and mouse brain transcriptome data using our novel Mouse AD panel. Transcriptome analysis was performed for the APOE4 KI mouse, carrying a humanized version of the strongest LOAD associated risk allele (APOE-ε4) and the Trem2*R47H mouse, which harbors a rare deleterious variant in TREM2. The rare TREM2 R47H missense variant (rs75932628) has been previously associated with LOAD in multiple independent studies [15]. In our RNA-Seq data, we observed that the variant impairs expression in the Trem2*R47H model and results in decreased expression of the major Trem2 isoform (ENSMUST00000024791) compared to B6 mice (adjusted FDR p = 4.26 × 10− 49, logFC = − 1.25). The decreased expression of Trem2 was also detected with the NanoString Mouse AD panel (adjusted FDR p = 0.03, logFC = − 0.29). The difference in log fold change reduction was potentially due to mismatched dynamic range between platforms or differences between aligned isoforms of RNA-Seq reads and probe design in the Mouse AD Panel. In addition, a mouse model harboring both, the common and rare AD risk variants (APOE4 KI/Trem2*R47H) was used to compare the transcriptional effects in mice carrying both variants to mice carrying only a single risk allele and B6 controls. Mouse transcriptome data for half brains was analyzed at different ages (2–14 months) to estimate the overlap with human post-mortem co-expression modules during aging. We observed specific overlaps with distinct disease processes and molecular pathways at different ages for the APOE4 KI and Trem2*R47H mouse models. At an early age (2–5 months), male APOE4 KI and Trem2*R47H mice showed strong negative correlations (p < 0.05, Pearson's correlation coefficient < − 0.3) with human co-expression modules in Consensus Cluster E that are enriched for transcripts associated with organelle biogenesis and cellular stress response pathways in multiple brain regions (Fig. 5a). Furthermore, Trem2*R47H male mice showed a significantly negative association (p < 0.05, Pearson's correlation coefficient < − 0.2) with immune related human modules in the superiortemporal gyrus, the inferiorfrontal gyrus, cerebellum and prefrontal cortex (Fig. 5a). This effect becomes more pronounced later in development, between six and 14 months, when the negative correlation with human immune modules is also observed in Trem2*R47H female mice. During mid-life, (6–9 months old age group), we observed an age-dependent effect for the APOE4 KI mouse in which human neuronal modules in Consensus Cluster C start to become positively correlated with the corresponding human expression modules (Fig. 5a). Interestingly, neuronal co-expression modules which are associated with synaptic signaling appear to be positively correlated with APOE4 KI, but not Trem2*R47H mice in an age dependent manner. This up-regulation of genes associated with synaptic signaling and a decrease of transcripts enriched for cell cycle, RNA non-mediated decay, myelination and glial development in aged mice was consistent for multiple brain regions and across three independent human AD cohorts. When compared to APOE4 KI mice, Trem2*R47H mice showed an age dependent decrease in genes associated with the immune response in several AMP-AD modules which is not observed for APOE4 KI mice (Fig. 5a). Notably, the APOE4 KI/Trem2*R47H mice showed characteristics of both single variant mouse models. At an early age, an overlap with both neuronal and immune associated human modules is observed and becomes more pronounced during aging.
Correlation analysis between three novel LOAD mouse models and human co-expression modules reveals age-related changes in immune function. a Correlation analysis highlights age-related changes in Trem2*R47H, APOE4 KI and APOE4 KI/Trem2*R47H mice. Circles correspond to significant (p < 0.05) positive (blue) and negative (red) Pearson's correlation coefficients for gene expression changes in mice associated with distinct human co-expression modules. This includes multiple modules linked to immune function (Consensus Cluster B) and stress response (Consensus Cluster E). b Pathway analysis for the Reactome and WikiPathway resources highlights a significant enrichment (FDR adjusted p < 0.05) of pathways involved in complement activation in both older (10–14 months) APOE4 KI and Trem2*R47H mice. c Genes encoding for complement component C1q show an antagonistic transcriptional effect between the Trem2*R47H and APOE4 KI/Trem2*R47H mice when compared to the humanized APOE4 knock-in model (*denotes FDR adjusted p < 0.05)
Differential expression and gene set enrichment analysis
In order to relate the human co-expression modules to disease associated genes and pathways, we performed differential expression (DE) analysis for the three novel mouse models and the 5xFAD mouse. Each mouse model was compared to the B6 control to assess the overall transcriptomic response and the differences in DE genes across models. Both the APOE4 KI and the APOE4 KI/Trem2*R47H models showed a moderate number of DE genes compared to B6 mice (< 100) at mid-life (6–9 months) while the number of DE is smaller (< 20) late in life. Early in life (2–5 months), only one gene was found to be DE in both the humanized APOE4 KI and the APOE4 KI/Trem2*R47H model (Tables S5, S6). We observed a significant decrease of the mouse Apoe gene (adjusted FDR p = 1.78 × 10− 69, logFC > − 3.5) reflecting that it was replaced by the human version. The Trem2*R47H model shows mostly down-regulated genes across all age groups, many of which are up-regulated in 5xFAD mice. While microglia related genes, including Tyrobp, Trem2, and complement components C1qa, C1qb, C1qc are highly up-regulated in 5xFAD mice, these genes are down-regulated in the Trem2*R47H model (Table S6). To elucidate the role of these immune related disease genes on the pathway level, gene set enrichment analysis (GSEA) was performed for the four mouse models and resulting pathways were compared to the human AMP-AD data. GSEA revealed multiple immune associated pathways up-regulated in 5xFAD mice when compared to B6 mice. The strongest association was observed in aged mice (10–14 months) where three immune related pathways (phagosome, Chagas disease, osteoclast differentiation) are significantly up-regulated (adjusted Benjamini-Hochberg p < 0.05) in the 5xFAD model. These pathways, which were also up-regulated in multiple brain regions from independent AMP-AD cohorts (cerebellum, superior temporal gyrus, temporal cortex), were down-regulated in the Trem2*R47H model (Figure S2). Notably, this neuro-protective effect of the Trem2*R47H allele was dampened in the presence of APOE4 on the APOE4 KI/Trem2*R47H background, which did not show any significant associations with immune pathways (Figure S2). To follow up on this antagonistic effect, pathway enrichment analysis for differentially expressed genes between APOE4 KI and Trem2*R47H mice was performed using both the Reactome [18] and WikiPathway knowledge pathway databases. Multiple pathways linked to immune function and specifically complement activation were significantly enriched (FDR adjusted p < 0.05) for genes showing opposite expression patterns in both mouse models late in life (10–14 months) (Fig. 5b). Among the genes that contribute significantly to this pathway enrichment were three members of the complement complex 1q (C1q), namely C1qa, C1qb, and C1qc that were also found to be up-regulated in the amyloidogenic 5xFAD model. These genes encode for the C1q complex and were significantly decreased in both the Trem2*R47H and APOE4 KI/Trem2*R47H model when compared to the APOE4 KI model (Fig. 5c, Table S7). This is in line with a recent study that linked the decrease in synaptic damage and vulnerability in a humanized Trem2*R47H tauopathy mouse model to reduced C1q expression [19]. Moreover, C1q accumulation had been shown to be drastically increased in synapses of APOE4 KI mice, when compared to APOE3 KI mice suggesting an important role of C1q in enhancing synaptic vulnerability to classical-complement-cascade mediated neurodegeneration [20]. Furthermore, expression of the C1q complex is critical for microglia function. Immune pathways up-regulated in 5xFAD mice and down-regulated in our Trem2*R47H model harbor multiple important genes (Tyrobp, Trem2, Ctss, and Apoe) linked to the activation of disease associated microglia (DAM). In order to further characterize the role of DAM genes, we compared DAM expression signatures based on recently published data single-cell transcriptomic data [21] (Figure S3). The expression signatures of 21 DAM associated genes on the NanoString panel supported an increased expression of DAM associated genes in the 5xFAD mice compared to B6 late in life (10-14 months) (Figure S3). This increased expression of DAM genes was absent in the APOE4 and Trem2 models when compared to age-matched B6 mice.
Comparison between nCounter mouse AD panel and RNA-Seq data
To assess the validity of the novel Mouse AD panel across transcriptomic platforms, we compared the results from the nCounter platform to RNA-Seq data for the same 137 mouse brain samples from three novel LOAD models carrying APOE4 and TREM2*R47H alleles for which both RNA-Seq and NanoString data was available. A correlation analysis was performed to compare the expression of the 770 NanoString probes across co-expression modules with RNA-Seq transcript expression for all ages, highlighting the different LOAD mouse models as independent variables (Fig. 6). For the direct comparison, between the 770 NanoString probes with corresponding RNA-Seq transcripts, a similar range of correlation coefficients between human data and the three mouse models was observed (Fig. 6a). Overall, the correlation between the RNA-Seq and NanoString platforms were high across all age groups (Pearson's correlation coefficients: 0.65–0.69) when comparing the subset of 760 key transcripts and 10 housekeeping transcripts across platforms. This demonstrates that the novel NanoString panel, despite the limited number of key custom probes, can achieve similar results when compared to high-throughput RNA-Seq data. Furthermore, the alignment of human and mouse modules based on the expression of all genes within each modules showed a weaker range of correlations when compared to transcripts covered by the 770 NanoString probes (Fig. 6b). Overall, we found strong and significant correlations between the results. Notably, these correlations generally increased with mouse age, suggesting that the human relevance of the models is increasing with age and that this relevance is captured well by both the NanoString and RNA-Seq platforms (Fig. 6b). A mild correlation at around 3 months of age (Pearson's correlation coefficient: 0.39) increased to a moderate correlation at 12 months of age (Pearson's correlation coefficient: 0.51). Furthermore, we observed a high correlation of log count values for the majority of NanoString probes when compared to log TPM transcript ratios from RNA-Seq data. The majority of the 770 measured NanoString probes (716/770 probes, 93%) were positively correlated with RNA-Seq transcripts (Figure S4). In order to test whether noise introduced by highly variable transcripts affects the correlation between NanoString probes and RNA-Seq transcripts, Pearson's correlation coefficients and variance in RNA-Seq expression across 137 samples were compared. There was no significant trend indicating an effect of highly variable transcripts on the overall correlation coefficients between transcripts measured by RNA-Seq and NanoString (Figure S4).
Platform comparison of how the Mouse AD Panel and RNA-Seq each correlate with the AMP-AD modules. Correlation coefficients for human AMP-AD co-expression modules and gene expression profiles derived from the RNA-Seq (x-axis), and the NanoString Mouse AD Panel for the same mouse samples (y-axis). Both data types were obtained from 137 samples, including three different ages and three mouse models carrying LOAD risk variants. a Strong positive correlations (p < 2.2 × 10− 16) were observed across all ages and samples combined when comparing expression of the 770 transcripts on the NanoString panel. b The correlation between NanoString and RNA-Seq expression analysis decreased overall when comparing all module transcripts measured by RNA-Seq to the subset of 770 probes on the NanoString panel. However, an age specific effect was observed for the mouse transcripts in which correlation with human co-expression modules increased with age (3–5 months p = 4.39 × 10− 8, 7–9 months p = 2.35 × 10− 8, 12–14 months p = 2.75 × 10− 13)
Here, we describe a novel systems biology approach to rapidly assess disease relevance for three novel mouse models carrying two human risk variants, strongly associated with LOAD. The nCounter Mouse AD gene expression panel was designed to align human brain transcriptome data covering 30 co-expression modules. Cross-species comparison of human and mouse revealed that immune associated co-expression modules which harbor genes that have recently diverged in sequence were more likely to be lowly expressed or absent at the transcript level in brains from 6 months old B6 mice. In contrast, neuronal modules containing genes with a lower degree of sequence divergence between both species were more likely to be highly and constitutively expressed in the mouse brain when compared to the remaining co-expression modules. This is in line with evidence from multiple studies highlighting that conserved neuronal process in the brain are under strong purifying selection while immune related genes are more likely to diverge in function and expression patterns across species [22, 23]. By using our prioritization approach, we selected for key mouse genes targeting a subset of highly co-expressed human genes. This subset of genes on the NanoString panel showed overall lower levels of sequence divergence compared to human genes and higher expression levels in the mouse brain, reducing potential noise introduced by lowly expressed transcripts across expression modules.
Cross-platform comparison between the novel Mouse AD panel and RNA-Seq data revealed a strong correlation between mouse gene expression changes independent of platform related effects. Notably, the correlation between nCounter probe and RNA-Seq transcript expression with human co-expression modules was highest in aged mice older than 12 months. This age-dependent overlap might be expected due to the late-onset nature of Alzheimer's disease resulting in an increased number of highly co-expressed genes in aged mice carrying human LOAD risk variants. In addition, the strongest correlation between human and mouse module signatures was observed when using the subset of 770 transcripts on the NanoString panel. This highlights that assessment of key genes in the brain, contributing highly to module expression, can improve the characterization of novel LOAD mouse models and their alignment with specific human co-expression modules.
The time-course analysis of the well-established 5xFAD mouse model using our novel panel revealed a significant overlap in transcriptional signatures with several human co-expression modules from distinct brain regions during aging. Several hallmark features of the 5xFAD model, such as the increased cellular stress response after early amyloid deposition, correlated well with human AMP-AD modules enriched for stress responsive transcripts. In addition, an age-related increase in several key inflammatory pathways and processes following more severe amyloid pathology was observed in 5xFAD mice. These findings highlighted that aged 5xFAD mice closely resemble pro-inflammatory transcriptional signatures of post-mortem brain samples from deceased LOAD patients. Interestingly, novel LOAD mouse models showed concordance with distinct human co-expression modules, reflecting a different transcriptional response driven by the human APOE and TREM2 associated LOAD risk variants. The strong negative correlation between the Trem2*R47H knock-in mice and immune related human co-expression highlights the important role of the LOAD associated TREM2 R47H variant in Alzheimer's related immune processes. This effect, which was reproduced across co-expression modules from multiple human brain regions (cerebellum, frontal cortex, temporal gyrus, frontal gyrus, frontal pole), was inverted in the presence of a high amyloid burden in the 5xFAD model. Immune related pathways containing genes linked to microglia activation were significantly increased in 5xFAD mice and decreased in Trem2*R47H mice. Interestingly, this inflammatory response was partially restored in the presence of APOE4 allele on the APOE4 KI/Trem2*R47H background suggesting an interaction between the two LOAD risk alleles. Pathway and differential gene expression analysis revealed antagonistic expression signatures between APOE4 KI and Trem2*R47H mice linked to the activation of the classical complement component through C1q members (C1qa, C1qb, C1qc). C1q protein accumulates at senescent synapses in the course of normal brain aging making them more vulnerable to complement mediated neurodegeneration. A recent study showed that expressing the R47H risk allele in a humanized Trem2 mouse model lowers the expression of C1q at synapses which in turn protects them from damage in the setting of a tauopathy mouse model [19]. This is in line with multiple studies which showed that the profound loss of synapses at the early stages of AD can be prevented by blocking activation of the complement cascade through C1q depletion in the mouse [24, 25]. In contrast, the APOE4 isoform increases C1q accumulation at synapses making them more vulnerable to degeneration when compared to the APOE2 and APOE3 isoforms in a set of APOE KI models [20]. However, C1q accumulation at synapses alone might not be sufficient to trigger synaptic loss in the aging brain. Other factors mediated by the Trem2 R47H and APOE4 risk alleles may activate the neuroinflammatory cascade that leads to age-related neurodegeneration which will require further studies in these novel LOAD models. Furthermore, a strong negative correlation between co-expression modules associated with cell cycle and DNA repair was observed for the mouse APOE4 KI model. This overlap with human late-onset co-expression signatures early in life was observed for a number of different brain regions and is absent in Trem2*R47H knock-in mice. Furthermore, aged APOE4 KI mice show a strong overlap with several human neuronal co-expression modules enriched for genes that play an important role in synaptic signaling and myelination. Although, APOE4 KI mice lack a clear neurodegenerative phenotype, this age dependent shift in co-expression patterns associated with core LOAD pathologies points to an increased susceptibility to cognitive decline in aged mice. This is in line with several studies which have shown that cognitive deficits in APOE4 transgenic mice develop late in life [26, 27]. Taken together, these results suggest that correlating gene expression signatures in LOAD and FAD mouse models to disease-associated AMP-AD modules can identify transcriptional disruptions relevant to human disease, even when the models are insufficiently advanced to exhibit full LOAD pathology. Assessing the effects of individual and combinations of LOAD variants in mouse models in this way can potentially separate the causal co-expression modules that drive LOAD pathology through genetic risk factors from modules that respond to established pathology. Furthermore, staging such in vivo models over a lifetime can determine the order of events, including microglia activation and, ultimately, neuronal loss observed in LOAD patients. The Mouse AD Panel described here provides an efficient platform to detect these events.
Limitations of the approach
Albeit being an excellent resource for characterizing molecular pathways and key drivers of disease, co-expression modules based on human post-mortem brain data have several limitations. As end stage measures, they might not reflect changes that occur early in disease pathogenesis. For this reason, we were unable to devise age-specific mouse panels that could be most informative at specific disease stages. However, as mouse models improve and are characterized at multiple ages, early transcriptomic indicators of LOAD might motivate additional panels corresponding to stages of pathogenesis. In addition, although a high concordance was observed across brain regions for the 30 modules, they might not cover individual or region-specific differences in patients in response to amyloid and tau pathology [9]. Furthermore, we used brain homogenates from our mouse models for the transcript comparison with different human brain regions in this study. Dissection of mouse brain regions to match the human studies might further improve the observed co-expression module correlations.
AMP-AD post-mortem brain cohorts and gene co-expression modules
Data on the 30 human AMP-AD co-expression modules was obtained from the Synapse data repository (https://www.synapse.org/#!Synapse:syn11932957/tables/; SynapseID: syn11932957). The modules derive from three independent LOAD cohorts, including 700 samples from the ROSMAP cohort, 300 samples from the Mount Sinai Brain bank and 270 samples from the Mayo cohort. Details on post-mortem brain sample collection, tissue and RNA preparation, sequencing, and sample QC can be found in previously published work related to each cohort [12,13,14]. A detailed description on how co-expression modules were identified can be found in the recent study that identified the harmonized human co-expression modules as part of transcriptome wide AD meta-analysis [9]. Briefly, Wan, et al. performed library normalization and covariate adjustments for each human study separately using fixed/mixed effects modeling to account for batch effects. Among the 2978 AMP-AD modules identified across all tissues (DOI:https://doi.org/10.7303/syn10309369.1), 660 modules were selected by Wan, et al. which showed an enrichment for at least one AD-specific differential expressed gene set from the meta-analysis (DOI:https://doi.org/10.7303/syn11914606) in cases compared to controls. Lastly, the edge betweenness graph clustering method was applied to identify 30 aggregate modules that are not only differentially expressed but are also replicated across multiple independent co-expression module algorithms [9]. Among the 30 aggregate co-expression modules, five consensus clusters have been described by Wan, et al. [9]. These consensus clusters consist of a subset of modules which are associated with similar AD related changes across the multiple studies and brain regions. Here, we used Reactome pathway (https://reactome.org/) enrichment analysis to identify specific biological themes across these five consensus clusters. A hypergeometric model, implemented in the clusterProfiler R package [28], was used to assess whether the number of selected genes associated within each set of AMP-AD modules defining a consensus cluster was larger than expected. All p-values were calculated based the hypergeometric model [29]. Pathways were ranked based on their Bonferroni corrected p-values to account for multiple testing. Finally, consensus clusters were annotated based on the highest ranked and non-overlapping term for each functionally distinct cluster.
Selection of NanoString probes for the nCounter mouse AD panel
Since NanoString gene expression panels are comprised of 770 probes with the option to customize 30 additional probes, we developed a formal prioritization procedure to identify the most representative genes and ensure broadest coverage across all modules (Fig. 1). Expression and transcript annotations for the 30 human co-expression modules were obtained via the AMP-AD knowledge portal (www.synapse.org/#!Synapse:syn11870970/tables/). To prioritize probe targets for the novel Mouse AD panel, human genes were ranked within each of the human AMP-AD co-expression modules based on their percentage of variation explaining the overall module behavior. First, we calculated a gene ranking score by multiplying correlations of transcripts with the percentage of variation explained by the first five principal components within each of the aggregated human AMP-AD modules. Secondly, the sums of the resulting gene scores for the first five principal components were calculated and converted to absolute values in order to rank highly positive or negative correlated transcripts within each human co-expression module. As a next step, only human transcripts with corresponding one-to-one mouse orthologous genes that are expressed in whole-brain tissue samples from six-month-old B6 mice were retained for downstream prioritization. While this filter risks excluding very few genes (6/760, < 1%) that may only be expressed at an advanced age, we maintained high representation of each human LOAD module. Disease-relevant effects are therefore robustly captured even if a few specific genes are omitted, as no module was determined to have more than five unexpressed genes in six-month-old mice. Furthermore, we included information on drug targets for LOAD from the AMP-AD Agora platform (agora.ampadportal.org), as nominated by members of the AMP-AD consortium (https://doi.org/10.7303/syn2580853). A total of 30 AMP-AD drug discovery targets that were highly ranked in our gene ranking approach and nominated by multiple AMP-AD groups were included on the panel (Table S3). Finally, ten housekeeping genes (AARS, ASB7, CCDC127, CNOT10, CSNK2A2, FAM104A, LARS, MTO1, SUPT7L, TADA2B) were included on the panel as internal standard references for probe normalization. This resulted in a total of 770 proposed NanoString probes, targeting the top 5% of ranked genes for each human AMP-AD expression module.
nCounter mouse AD panel probe design
The probe design process breaks a transcript's sequence down into 100 nucleotide (nt) windows to profile for probe characteristics, with the final goal of choosing the optimal pair of adjacent probes to profile any given target. Each window is profiled for intrinsic sequence makeup – non-canonical bases, G/C content, inverted and direct repeat regions, runs of poly-nucleotides, as well as the predicted melting temperature (Tm) for each potential probe-to-target interaction. The window is then divided in half to generate a probe pair, wherein each probe is thermodynamically tuned to determine the optimal probe length (ranging in size from 35 to 50 nt) within the 100 nt target region. Next, a cross-hybridization score is calculated for each probe region, using BLAST [30] to identify potential off-target interactions. In addition to a cross-hybridization score, a splice isoform coverage score was generated to identify transcripts that are isoforms of the gene intended to be targeted by the probe in question. Once all of this information is compiled, the final probe is then selected by identifying the candidate with the optimal splice form coverage, cross-hybridization score, and thermodynamic profile.
In-silico panel QC for intramolecular interactions
To ensure that there are no potential intramolecular probe-probe interactions that could cause elevated background for any individual probe pair, a stringent intermolecular screen is run on every collection of probes assembled into a panel. A sensitive algorithm was used that calculates both the Tm and the free energy potential of interactions between every possible pair of probes in the project. If two probes conflict in a way that would likely cause background based on this calculation, an alternative probe is selected for one of the targets and the screening is re-run until there are no known conflicts.
All experiments involving mice (Supplemental Table S5) were conducted in accordance with policies and procedures described in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health and were approved by the Institutional Animal Care and Use Committee at The Jackson Laboratory. All mice were bred and housed in a 12/12 h light/dark cycle. All experiments were performed on a unified genetic background (C57BL/6 J).
Whole-genome sequencing
Whole-genome sequencing was performed by Novogene in Bejing, China for three founders of the APOE4 KI/Trem2*R47H strain to exclude potential off-target effects in loci that were not targeted. Briefly, DNA was extracted from spleen and library preparation was performed using the KAPA HyperPrep sample preparation kit (KAPA Biosystems, Wilmington, MA, USA). Libraries were analyzed using a 2100 Bioanalyzer (Agilent Technologies, Santa Clara, CA, USA), with the DNA 2100 kit. Sequencing was performed on a HiSeq X sequencer according to the manufacturer's guidelines using 2x150bp paired-end reads. Reads were quality trimmed and filtered using the NGS QC toolkit. The resulting high-quality reads were aligned to the mm10 release of the mouse reference genome using BWA v0.5.10 [31]. SNPs and indels were called using the GATK tool suite v2.7 [32]. Finally, all variants were annotated using the SnpEff software [33]. All variants with a potential deleterious effect (missense, nonsense, splice-site, frame-shift) on the protein level are listed in Table S8 for three founders of the APOE4 KI/Trem2*R47H. Figure S5 gives an overview of deleterious variants shared across these three founders. Only three genes of which two are annotated as predicted genes (Gm11168, Gm10717) and one as an olfactory receptor (Vmn2r115) were identified that share private, deleterious variants in our models. Expression levels of these genes in the brain were below our cut-off criteria of 1 transcript per million reads.
Mouse brain sample collection
Upon arrival at the terminal endpoint for each aged mouse cohort, individual animals were weighed prior to intraperitoneal administration of ketamine (100 mg/kg) and xylazine (10 mg/kg). First confirming deep anesthetization via toe pinch, an incision was made along the midline to expose the thorax and abdomen followed by removal of the lateral borders of the diaphragm and ribcage revealed the heart. A small cut was placed in the right atrium to relieve pressure from the vascular system before transcardially perfusing the animal with 1XPBS via injection into the left ventricle. With the vascular system cleared, the entire brain was carefully removed and weighed before hemisecting along the midsagittal plane. Hemispheres were immediately placed in a cryovial and snap-frozen on dry ice. Brain samples were stored at − 80 °C until RNA extraction was performed.
RNA sample preparation
RNA was isolated from tissue using the MagMAX mirVana Total RNA Isolation Kit (ThermoFisher) and the KingFisher Flex purification system (ThermoFisher, Waltham, MA). Brain hemispheres were thawed to 0 °C and were lysed and homogenized in TRIzol Reagent (ThermoFisher). After the addition of chloroform, the RNA-containing aqueous layer was removed for RNA isolation according to the manufacturer's protocol, beginning with the RNA bead binding step. RNA concentration and quality were assessed using the Nanodrop 2000 spectrophotometer (Thermo Scientific) and the RNA Total RNA Nano assay (Agilent Technologies, Santa Clara, CA).
RNAseq library preparation and data collection
Sequencing libraries were constructed using TruSeq DNA V2 (Illumina, San Diego, CA) sample prep kits and quantified using qPCR (Kapa Biosystems, Wilmington, MA). The mRNA was fragmented, and double-stranded cDNA was generated by random priming. The ends of the fragmented DNA were converted into phosphorylated blunt ends. An 'A' base was added to the 3′ ends. Illumina®-specific adaptors were ligated to the DNA fragments. Using magnetic bead technology, the ligated fragments were size-selected and then a final PCR was performed to enrich the adapter-modified DNA fragments, since only the DNA fragments with adaptors at both ends will amplify. Libraries were pooled and sequenced by the Genome Technologies core facility at The Jackson Laboratory. Samples were sequenced on Illumina HiSeq 4000 using HiSeq 3000/4000 SBS Kit reagents (Illumina), targeting 30 million read pairs per sample. Samples were split across multiple lanes when being run on the Illumina HiSeq, once the data was received the samples were concatenated to have a single file for paired-end analysis.
NanoString gene expression panel and data collection
The NanoString Mouse AD gene expression panel was used for gene expression profiling on the nCounter platform (NanoString, Seattle, WA) as described by the manufacturer. nSolver software was used for analysis of NanoString gene expression values. Normalized log intensity and raw gene expression data can be accessed via NCBI GEO (Accession Number: GSE141509).
Normalization of NanoString data
Normalization was done by dividing counts within a lane by geometric mean of the housekeeping genes from the same lane. For the downstream analysis, counts were log-transformed from normalized count values.
Mouse-human expression comparison
First, we performed differential gene expression analysis for each mouse model and sex using the voom-limma [34] package in R. Secondly, we computed correlation between changes in expression (log fold change) for each gene in a given module with each mouse model, sex and age. Correlation coefficients were computed using cor.test function built in R as:
$$ \mathrm{cor}.\mathrm{test}\left(\ \mathrm{LogFC}\left(\mathrm{h}\right),\mathrm{LogFC}\left(\mathrm{m}\right)\ \right) $$
where LogFC(h) is the log fold change in transcript expression of human AD patients compared to control patients and LogFC(m) is the log fold change in expression of mouse transcripts compare to control mouse models. LogFC values for human transcripts were obtained via the AMP-AD knowledge portal (https://www.synapse.org/#!Synapse:syn11180450).
Differential expression, Gene set and pathway enrichment analysis
Differential gene expression analysis was performed using the limma package in the R software environment [34] for all analyzed mouse strains. Each model at each individual time point was compared to age-matched B6 wild type mice. In order to identify significantly enriched pathways across mouse models and human brain regions, gene set enrichment analysis was used based on the method proposed by Subramanian, et. al [35] as implanted in the clusterProfiler package for the KEGG pathway library. Briefly, human data with expression log fold changes for the seven AMP-AD brain regions were downloaded from Synapse (https://www.synapse.org/#!Synapse:syn14237651). We filtered to orthologous genes on the NanoString Mouse AD panel and KEGG pathway enrichment was performed for each brain region independently to identify significantly up and down-regulated gene sets. For the mouse data, differential expression analysis between each mouse model and B6 controls was performed to obtain a list of fold changes highlighting genes that are either up or down-regulated in the presence of the genetic risk variant. Enrichment scores for all significantly associated KEGG pathways were computed to compare relative expression on the pathway level between post-mortem brain samples and the four mouse models. Biological pathway enrichment analysis was performed using the clusterprofiler [28] package within the R software envirionment for the Reactome [18] and WikiPathways (wikipathways.org) knowledge bases. Pathways were determined to be significant after multiple testing correction (FDR adjusted p < 0.05).
Quality control of RNA-Seq data and read alignment
Sequence quality of reads was assessed using FastQC (v0.11.3, Babraham). Low-quality bases were trimmed from sequencing reads using Trimmomatic (v0.33) [36]. After trimming, reads of length longer than 36 bases were retained. The average quality score at each base position was greater than 30 and sequencing depth were in range of 60–120 million reads. All RNA-Seq samples were mapped to the mouse genome (mm10 reference, build 38, ENSEMBL) using ultrafast RNA-Seq aligner STAR [37] (v2.5.3). The genes annotated for mm10 (GRCm38) were quantified in two ways: Transcripts per million (TPM) using RSEM (v1.2.31) and raw read counts using HTSeq-count (v0.8.0).
Mouse-human co-expression module conservation
Genomic information on orthologous groups was obtained via the latest ENSEMBL build for human genome version GRCh38. All orthologous relationships were downloaded via BioMart [38] (biomart.org). dN/dS statistics were retrieved for all orthologous gene pairs with a one-to-one relationship between human and mouse. dN/dS values are calculated as the ratio of nonsynonymous substitutions to the number of synonymous substitutions in protein coding genes. The dN/dS values in ENSEMBL were calculated based on the latest version of the codeml (http://abacus.gene.ucl.ac.uk/software/paml.html) package using standard parameters (ensembl.org/info/genome/compara/homology_method.html) [39].
Taken together, we show that the novel nCounter Mouse AD gene expression panel offers a rapid and cost-effective approach to assess disease relevance of novel LOAD mouse models. Furthermore, this approach based on gene co-expression signatures offers a high level of reproducibility and will supplement methods solely based on differential expression analysis. Ultimately, this will help us to better understand the relevance of novel LOAD mouse models in regard to specific pathways and processes contributing to late-onset Alzheimer's disease.
The results published here are in whole or in part based on data obtained from the AMP-AD Knowledge Portal (doi:https://doi.org/10.7303/syn2580853). ROSMAP Study data were provided by the Rush Alzheimer's Disease Center, Rush University Medical Center, Chicago. Data collection was supported through funding by NIA grants P30AG10161, R01AG15819, R01AG17917, R01AG30146, R01AG36836, U01AG32984, U01AG46152, the Illinois Department of Public Health, and the Translational Genomics Research Institute. Mayo RNA-Seq Study data were provided by the following sources: The Mayo ClinicAlzheimer's Disease Genetic Studies, led by Dr. Nilufer Ertekin-Taner and Dr. Steven G. Younkin, Mayo Clinic, Jacksonville, FL using samples from the Mayo Clinic Study of Aging, the Mayo Clinic Alzheimer's Disease Research Center, and the Mayo Clinic Brain Bank. Data collection was supported through funding by NIA grants P50 AG016574, R01 AG032990, U01 AG046139, R01 AG018023, U01 AG006576, U01 AG006786, R01 AG025711, R01 AG017216, R01 AG003949, NINDS grant R01 NS080820, CurePSP Foundation, and support from Mayo Foundation. Study data includes samples collected through the Sun Health Research Institute Brain and Body Donation Program of Sun City, Arizona. The Brain and Body Donation Program is supported by the National Institute of Neurological Disorders and Stroke (U24 NS072026 National Brain and Tissue Resource for Parkinson's Disease and Related Disorders), the National Institute on Aging (P30 AG19610 Arizona Alzheimer's Disease CoreCenter), the Arizona Department of Health Services (contract 211002, Arizona Alzheimer's Research Center), the Arizona Biomedical Research Commission (contracts 4001, 0011, 05–901 and 1001 to the Arizona Parkinson's Disease Consortium) and the Michael J. Fox Foundation for Parkinson's Research. MSBB data were generated from postmortem brain tissue collected through the Mount Sinai VA MedicalCenter Brain Bank and were provided by Dr. Eric Schadt from Mount Sinai School of Medicine. Mouse RNA-Seq data from the MODEL-AD consortium is available through Synapse via the AMP-AD knowledge portal (www.synapse.org/#!Synapse:syn17095980).
B6:
C57BL/6 J mice
RNA-Seq:
AMP-AD:
Accelerating Medicines Partnership for Alzheimer's Disease
ROSMAP:
Religious Orders Study/Memory and Aging Project
Disease associated microglia (DAM)
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We thank the many institutions and their staff that provided support for this study and who were involved in this collaboration. We would like to acknowledge Jamie Kuhar for her critically reading of the manuscript.
This study was supported by the National Institutes of Health grant U54 AG 054345.
Christoph Preuss and Ravi Pandey contributed equally to this work.
The Jackson Laboratory, Bar Harbor, ME, 04609, USA
Christoph Preuss, Ravi Pandey, Alexander Fine, Asli Uyar, Dylan Garceau, Kevin P. Kotredes, Harriet Williams, Gareth R. Howell, Michael Sasner & Gregory W. Carter
NanoString Technologies, Seattle, WA, 98109, USA
Erin Piazza & Thanneer Perumal
Sage Bionetworks, Seattle, WA, 98121, USA
Lara M. Mangravite & Benjamin A. Logsdon
Stark Neurosciences Research Institute, Indiana University School of Medicine, Indianapolis, IN, 46202, USA
Bruce T. Lamb & Adrian L. Oblak
Christoph Preuss
Ravi Pandey
Erin Piazza
Alexander Fine
Asli Uyar
Thanneer Perumal
Dylan Garceau
Kevin P. Kotredes
Harriet Williams
Lara M. Mangravite
Bruce T. Lamb
Adrian L. Oblak
Gareth R. Howell
Michael Sasner
Benjamin A. Logsdon
Gregory W. Carter
the MODEL-AD Consortium
CP designed the novel transcriptome panel and performed bioinformatics analyses. RP, AF, AU, TP performed the gene-expression analyses in human and mouse brain tissue. EP designed the NanoString probes and guided the creation of the novel NanoString panel. BAL and LM curated human brain data. DG, GRH and MS performed mouse experiments. GWC and MS supervised and designed the project. CP, GWC and RP wrote the manuscript. All authors read and approved the final manuscript.
Correspondence to Gregory W. Carter.
All experiments involving mice were conducted in accordance with policies and procedures described in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health and were approved by the Institutional Animal Care and Use Committee at The Jackson Laboratory.
All authors have approved of the manuscript and agree with its submission.
Preuss, C., Pandey, R., Piazza, E. et al. A novel systems biology approach to evaluate mouse models of late-onset Alzheimer's disease. Mol Neurodegeneration 15, 67 (2020). https://doi.org/10.1186/s13024-020-00412-5
Received: 05 December 2019 | CommonCrawl |
Communications in Mathematical Physics
pp 1–54 | Cite as
Local Limit Theorem for Randomly Deforming Billiards
Mark F. Demers
Françoise Pène
Hong-Kun Zhang
We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a central limit theorem for the cell index of planar motion, as well as a mixing local limit theorem for piecewise Hölder continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.
Communicated by C. Liverani
This work was begun at the AIM workshop Stochastic Methods for Non-Equilibrium Dynamical Systems, in June 2015. Part of this work was carried out during visits by the authors to ESI, Vienna in 2016, to CIRM, Luminy in 2017 and 2018, and to BIRS, Canada in 2018, and by a visit of FP to the University of Massachusetts at Amherst in 2018. MD was supported in part by NSF Grant DMS 1800321. FP is grateful to the IUF for its important support.
Appendix A. Proof of Lemma 4.5
Here we prove the Lemma 4.5, which was used in Sect. 4.2, especially used in the proof of Theorem 2.4.
Let us prove that (36) holds true. By density, it suffices to perform the estimate for \(f \in \mathcal {C}^1({\bar{M}}_0)\). In the proof below, we use the fact that the invariant measure \({\bar{\mu }}_0\) is absolutely continuous with respect to the Lebesgure measure.
Choose \(\ell \ge 1\) and fix \({\underline{\omega }}_\ell := (\omega _1, \ldots , \omega _\ell )\). Let g be as in the statement of the lemma. For brevity, denote by \({\bar{T}}_{\underline{\omega }_\ell }^\ell = \bar{T}_{\omega _\ell } \circ \cdots \circ {\bar{T}}_{\omega _1}\) the composition of random maps and by \(\mathcal {L}_{\underline{\omega }_\ell }^\ell \) its associated transfer operator. Also, set \(H_\ell ^p(g) = |g|_\infty + \sup _{C\in \mathcal C_{\omega _1,\ldots ,\omega _\ell }}C_{g_{|C}}^{(p)}\). We must estimate
$$\begin{aligned} {\mathbb {E}}_{{\bar{\mu }}_0} [f \, g ] = \int _{{\bar{M}}_0} f \, g \, d{\bar{\mu }}_0 = \int _{{\bar{M}}_0} \mathcal {L}_{{\underline{\omega }}_\ell }^\ell f \cdot g \circ ({\bar{T}}_{{\underline{\omega }}_\ell }^{\ell })^{-1} \, d{\bar{\mu }}_0 . \end{aligned}$$
To do this, we decompose \({\bar{M}}_0\) into a countable collection of local rectangles, each foliated by a smooth collection of stable curves on which we may apply our norms. This technique follows closely the decomposition used in [16, Lemma 3.4].
We partition each connected component of \({\bar{M}}_0 {\setminus } (\cup _{|k| \ge k_0} {\mathbb {H}}_k)\), into finitely many boxes \(B_j\) whose boundary curves are elements of \(\mathcal {W}^s\) and \(\mathcal {W}^u\), as well as the horizontal boundaries of \({\mathbb {H}}_{\pm k_0}\). We construct the boxes \(B_j\) so that each has diameter in \((\delta /2, \delta )\), for some \(\delta >0\), and is foliated by a smooth foliation of stable curves \(\{ W_\xi \}_{\xi \in \Xi _j}\), such that each curve \(W_{\xi }\) is stretched completely between the two unstable boundaries of \(B_j\). Indeed, due to the continuity of the cones \(C^s(x)\) from (H1), we can choose \(\delta \) sufficiently small that the family \(\{ W_\xi \}_{\xi \in \Xi _j}\) is a family of parallel line segments.
We disintegrate the measure \({\bar{\mu }}_0\) on \(B_j\) into a family of conditional probability measures \(d\mu _{\xi } = c_\xi \cos \varphi \, dm_{W_\xi }\), \(\xi \in \Xi _j\), where \(c_\xi \) is a normalizing constant, and a factor measure \( \lambda _j(\xi )\) on the index set \(\Xi _j\). Since \({\bar{\mu }}_0\) is absolutely continuous with respect to Lebesgue measure on \({\bar{M}}_0\), we have \( \lambda _j(\Xi _j) = \bar{\mu }_0(B_j) = \mathcal {O}(\delta ^2)\).
Similarly, on each homogeneity strip \({\mathbb {H}}_t\), \(t \ge k_0\), we choose a smooth foliation of parallel line segments \(\{ W_\xi \}_{\xi \in \Xi _t} \subset {\mathbb {H}}_t\) which completely cross \({\mathbb {H}}_t\). Due to the uniform transversality of the stable cone with \(\partial {\mathbb {H}}_t\), we may choose a single index set \(\Xi _t\) for each homogeneity strip. We again disintegrate \({\bar{\mu }}_0\) into a family of conditional probability measures \(d\mu _\xi = c_\xi \cos \varphi \, dm_{W_\xi }\), \(\xi \in \Xi _t\), and a transverse measure \(\lambda _t(\xi )\) on the index set \(\Xi _t\). This implies that \(\lambda _t(\Xi _t) = {\bar{\mu }}_0({\mathbb {H}}_t) = \mathcal {O}(|t|^{-5})\) for each \(|t| \ge k_0\).
Notice that on each homogeneity strip \({\mathbb {H}}_k\), the function \(\cos \varphi \) satisfies,
$$\begin{aligned} |\log \cos \varphi (x)- \log \cos \varphi (y)| \le C d(x,y)^{1/3} \end{aligned}$$
for some uniform constant \(C>0\) (uniform in k).
We are ready to estimate the required integral. Let \(\mathcal {G}_\ell (W_\xi )\) denote the components of \((\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} W_{\xi }\), with long pieces subdivided to have length between \(\delta _0/2\) and \(\delta _0\), as in the proof of Lemma 3.14.
$$\begin{aligned} \begin{aligned}&\int \mathcal {L}_{ \underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} \, d{\bar{\mu }}_0 \\&\quad =\sum _j \int _{B_j} \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} d{\bar{\mu }}_0 + \sum _{|t| \ge k_0} \int _{{\mathbb {H}}_t} \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} d{\bar{\mu }}_0\\&\quad =\sum _j \!\! \int _{\Xi _j} \! \int _{W_\xi } \! \! \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} \, d\mu _\xi d\lambda _j(\xi )\\&\qquad + \sum _{|t| \ge k_0} \! \! \int _{\Xi _t} \! \int _{W_\xi } \! \! \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} \, d\mu _\xi d\lambda _t(\xi ) \\&\quad =\sum _j \int _{\Xi _j} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} \int _{W_{\xi ,i}} f \, g \, c_\xi \cos \varphi \circ \bar{T}_{{\underline{\omega }}_\ell }^\ell \, J_{W_{\xi ,i}}{\bar{T}}_{{\underline{\omega }}_\ell }^\ell \, dm_{W_{\xi ,i}} d\lambda _j(\xi ) \\&\qquad + \sum _{|t| \ge k_0} \int _{\Xi _t} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} \int _{W_{\xi ,i}} f \, g \, c_\xi \cos \varphi \circ {\bar{T}}_{{\underline{\omega }}_\ell }^\ell \, J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell \, dm_{W_{\xi ,i}} d\lambda _t(\xi ) \, . \end{aligned} \end{aligned}$$
Next we use the assumption that g is Hölder continuous on connected componts of \({\bar{M}}_0 {\setminus } (\cup _{k=1}^\ell \bar{T}_{\omega _1}^{-1} \circ \cdots \circ {\bar{T}}_{\omega _k}^{-1} (\mathcal {S}_{0,H}))\). Since elements of \(\mathcal {G}_\ell (W_\xi )\) are also subdivided according to these singularity sets, we have that g is Hölder continuous on each \(W_{\xi , i} \in \mathcal {G}_\ell (W_\xi )\). Thus,
$$\begin{aligned} \begin{aligned}&\int _{W_{\xi , i}} f \, g \, c_\xi \cos \varphi \circ \bar{T}_{{\underline{\omega }}_\ell }^\ell \, J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell \, dm_{W_{\xi ,i}}\\&\quad \le |f|_w |g|_{\mathcal {C}^p(W_{\xi ,i})} c_\xi |\cos \varphi \circ {\bar{T}}_{{\underline{\omega }}_\ell }^\ell | _{\mathcal {C}^p(W_{\xi ,i})} |J_{W_{\xi ,i}}{\bar{T}}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^p(W_{\xi ,i})} \\&\quad \le |f|_w H_\ell ^p(g) |J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})} \frac{C}{|W_{\xi }|}, \end{aligned} \end{aligned}$$
where we used (43) in the last estimate, as well as the fact that the normalizing constant \(c_\xi \) is proportional to \(|W_\xi |^{-1}\). This implies that
$$\begin{aligned} {\mathbb {E}}_{{\bar{\mu }}_0} [ f \, g ]&\le C |f|_w H_\ell ^p(g) \Big ( \sum _j \int _{\Xi _j} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} |J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})} |W_\xi |^{-1} \, d\lambda _j(\xi ) \\&\quad + \sum _{|t| \ge k_0} \int _{\Xi _t} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} |J_{W_{\xi ,i}}{\bar{T}}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})} |W_\xi |^{-1} \, d\lambda _t(\xi ) \Big ). \end{aligned}$$
Now \(\sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} |J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})}\) is bounded by a uniform constant independent of \(\xi \) and \({\underline{\omega }}_\ell \) by [16, Lemma 5.5(b)]. Moreover, \(\int _{\Xi _j} |W_{\xi }|^{-1} d \lambda _j(\xi ) \le C\delta _0\) for some constant \(C>0\) since we chose our foliation to be comprised of long cone-stable curves. We conclude that the first term to the right hand side of the last inequality is uniformly bounded by \(C_1|f|_w H_\ell ^p(g)\) since the sum over j is finite.
For the second term on the right hand side of the last inequality, we again use [16, Lemma 5.5(b)] as well as the fact that \(|W_{\xi }|^{-1} = \mathcal {O}(t^3)\) for \(\xi \in \Xi _t\), while \(\lambda _t(\Xi _t) = \mathcal {O}(t^{-5})\). Thus
$$\begin{aligned} \sum _{|t| \ge k_0} \int _{\Xi _t}|W_{\xi }|^{-1}d \lambda _t(\xi )\le \sum _{|t| \ge k_0} C t^{-2} \le C k_0^{-1} . \end{aligned}$$
We conclude that
$$\begin{aligned} \left| {\mathbb {E}}_{{\bar{\mu }}_0} [f \, g] \right| \le K_1 |f|_w H^p_\ell (g) , \end{aligned}$$
for some uniform constant \(K_1\) depending on \(\bar{\mathcal {F}}_{\vartheta _0}\), but not on f, \(\ell \) or \(\underline{\omega }_\ell \). This completes the proof of (36).
To prove (37), we follow the proof of Lemma 3.14. Note that for \(f \in \mathcal {C}^1({\bar{M}}_0)\), \(W \in \mathcal {W}^s\), and a test function \(\psi \), we have
$$\begin{aligned} \int _W \mathcal {L}_{u, \omega _\ell } \ldots \mathcal {L}_{u, \omega _1}(fg) \, \psi \, dm_W = \sum _{W_i} \int _{W_i} fg\, e^{iu \cdot S_\ell } \, \psi \circ {\bar{T}}^\ell _{{\underline{\omega }}_\ell } \, J_{W_i}\bar{T}^\ell _{{\underline{\omega }}_\ell } \, dm_{W_i} \, , \end{aligned}$$
where the sum is taken over \(W_i \in \mathcal {G}_\ell (W)\), the components of \(({\bar{T}}^\ell _{{\underline{\omega }}_\ell })^{-1}W\), subdivided as before. This is the same type of expression as in [16, eq. (5.24)] or [16, eq. (4.4)], but now the test function is
$$\begin{aligned} g\, e^{iu \cdot S_\ell } \, \psi \circ {\bar{T}}^\ell _{\underline{\omega }_\ell } \, J_{W_i}{\bar{T}}^\ell _{{\underline{\omega }}_\ell } \end{aligned}$$
rather than simply \(\psi \circ {\bar{T}}^\ell _{{\underline{\omega }}_\ell } \, J_{W_i}{\bar{T}}^\ell _{{\underline{\omega }}_\ell }\). Since \(S_\ell \) is constant on each \(W_i \in \mathcal {G}_\ell (W)\), and we have assumed that g is (uniformly in \(\ell \)) Hölder continuous on each \(W_i \in \mathcal {G}_\ell (W)\), the proof of the Lasota–Yorke inequalities follows as in the proof of [16, Proposition 5.6]. The bound (37) then follows as in the proof of Lemma 3.14.
Remark A.1
As a consequence of this lemma, if \(g:{\bar{M}} \rightarrow {\mathbb {R}}\) is a bounded measurable function such that, for every \(\underline{\omega }=(\omega _k)_{k\ge 0}\in E^{{\mathbb {N}}}\), there exists positive integer \(\ell _{\underline{\omega }}\) such that \(g(\cdot ,\underline{\omega })\) is p-Hölder on every connected component (uniformly on \(\underline{\omega }\)) of \({\bar{M}}_0{\setminus }\left( \cup _{k=0}^{\ell _{\underline{\omega }}-1} \bar{T}_{\omega _0}^{-1} \circ \cdots \circ \bar{T}_{\omega _{\ell (\underline{\omega })-1}}^{-1} (\mathcal {S}_{0,H})\right) \). Then, for every \(f\in \widetilde{{\mathcal {B}}}_w\), we have
$$\begin{aligned} \left| {\mathbb {E}}_{{\bar{\mu }}}[gf]\right|= & {} \left| \int _{E}{\mathbb {E}}_{{\bar{\mu }}_0}[g(\cdot ,\underline{\omega }) f(x,\underline{\omega })]\, d\eta (\underline{\omega })\right| \\= & {} K_1\Vert f\Vert _{{\widetilde{\mathcal {B}}}_w} \left( \Vert g\Vert _\infty +\sup _{\underline{\omega }\in E^{{\mathbb {N}}}}\sup _{C\in \mathcal C_{\omega _1,\ldots ,\omega _\ell (\omega )}}C_{(g(\cdot ,\underline{\omega }))_{|C}}^{(p)}\right) \, , \end{aligned}$$
with the same notations as in the previous lemma. Therefore, \({\mathbb {E}}_{{\bar{\mu }}} [ g \cdot ] \) is in \({\widetilde{\mathcal {B}}}_w'\).
Appendix B. Proof of Lemma 4.10
Note that \({\mathcal {V}}_n=n+2\sum _{1\le k<\ell \le n}\mathbf 1_{\{S_\ell =S_k,{\mathcal {I}}_\ell ={\mathcal {I}}_k\}}\). Hence
$$\begin{aligned} Var_{{\bar{\mu }}}({\mathcal {V}}_n)=4\sum _{1\le k_1<\ell _1\le n}\sum _{1\le k_2<\ell _2\le n} D_{k_1,\ell _1,k_2,\ell _2}, \end{aligned}$$
with \(D_{k_1,\ell _1,k_2,\ell _2}:={\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})-{\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})\). It follows that
$$\begin{aligned} \left| Var_{{\bar{\mu }}}({\mathcal {V}}_n)-8(A_2+A_3)\right| \le 8(A_1+A_4), \end{aligned}$$
$$\begin{aligned} A_1:= & {} \sum _{1\le k_1<\ell _1\le k_2<\ell _2\le n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| ,\quad A_2:=\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} D_{k_1,\ell _1,k_2,\ell _2}\, ,\\ A_3:= & {} \sum _{1< k_1< k_2<\ell _2<\ell _1\le n} D_{k_1,\ell _1,k_2,\ell _2},\quad A_4:=\sum _{(k_1,k_2,\ell _1,\ell _2)\in E_n\cup F_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| \, , \end{aligned}$$
$$\begin{aligned} E_n:= & {} \{(k_1,k_2,\ell _1,\ell _2)\in \{1,\ldots ,n\}\ :\ k_1=k_2<\min (\ell _1,\ell _2)\},\\ F_n:= & {} \{(k_1,k_2,\ell _1,\ell _2)\in \{1,\ldots ,n\}\ :\ \max (k_1, k_2)<\ell _1=\ell _2\}. \end{aligned}$$
We will start with the two easiest estimates: the estimates of the error terms \(A_1\) and \(A_4\). The method we will use to estimate the main terms \(A_2\) and \(A_3\) differs from [31].
Due to Lemma 4.9,
$$\begin{aligned} A_1\le \, I^2\sum _{1\le k_1<\ell _1\le k_2<\ell _2\le n}\frac{C_1\alpha ^{k_2-\ell _1}}{(\ell _1-k_1)(\ell _2-k_2)} =O(n(\log n)^2)=o(n^2). \end{aligned}$$
Let us now prove that \(A_4=o(n^2)\) by writing
$$\begin{aligned}&\sum _{(k_1,k_2,\ell _1,\ell _2)\in E_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| \\&\quad \le 2\sum _{1\le k<\ell _1\le \ell _2\le n} \left( {\bar{\mu }}(E_{k,\ell _1}\cap E_{k,\ell _2})+{\bar{\mu }}(E_{k,\ell _1}){\bar{\mu }}(E_{k,\ell _2})\right) \\&\quad \le 2\sum _{1\le k<\ell _1\le \ell _2\le n} \left( {\bar{\mu }}(S_{\ell _1}=S_{\ell _2}=S_k) +{\bar{\mu }}(S_{\ell _1}=S_k){\bar{\mu }}(S_{\ell _2}=S_k)\right) \\&\quad \le 2\sum _{1\le k<\ell _1\le \ell _2\le n} \left( {\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathcal {H}}_{0,\ell _2-\ell _1}{\mathcal {H}}_{0, \ell _1-k}({\mathbf {1}})\right] + {\mathbb {E}}_{{\bar{\mu }}} \left[ \mathcal H_{0, \ell _1-k}({\mathbf {1}}) \right] {\mathbb {E}}_{{\bar{\mu }}} \left[ {\mathcal {H}}_{0, \ell _2-k}({\mathbf {1}}) \right] \right) \\&\quad \le K'_0\sum _{1\le k<\ell _1\le \ell _2\le n} \left( \frac{1}{(\ell _1-k)(\ell _2-\ell _1+1)} +\frac{1}{(\ell _1-k)(\ell _2-k)}\right) \end{aligned}$$
for some \(K'_0>0\) due to Theorem 4.2, since \({\mathbb {E}}_{{\bar{\mu }}}[\cdot ]\) is a continuous linear operator on \({\widetilde{\mathcal {B}}}_1\) and since \({\mathbf {1}}\in {\widetilde{\mathcal {B}}}_1\). This leads to \(\sum _{(k_1,k_2,\ell _1,\ell _2)\in E_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| =O(n(\log n)^2)\). Analogously, we obtain \(\sum _{(k_1,k_2,\ell _1,\ell _2)\in F_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| =O(n(\log n)^2)\). Hence \(A_4=o(n^2)\).
For \(A_2\), we study separately the terms \({\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\) and the terms \({\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})\). First by Lemma 4.8,
$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} {\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}( E_{k_2,\ell _2})\nonumber \\&\quad =c_1^2\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}\left( (\ell _1-k_1)^{-1}+O((\ell _1-k_1)^{-3/2})\right) \nonumber \\&\qquad \left( (\ell _2-k_2)^{-1}+O((\ell _2-k_2)^{-3/2})\right) \nonumber \\&\quad =o(n^2)+c_1^2\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}\frac{1}{(\ell _1-k_1)(\ell _2-k_2)}\, , \end{aligned}$$
where we used the fact that
$$\begin{aligned}&\sum _{1\le k_1<k_2<\ell _1<\ell _2\le n}\frac{1}{\ell _1-k_1} \frac{1}{(\ell _2-k_2)^{\frac{3}{2}}}\\&\quad \le \sum _{m_1,m_2,m_3,m_4=1}^{n}\frac{1}{m_2+m_3}\frac{1}{(m_3+m_4)^{\frac{3}{2}}}\\&\quad \le n\sum _{m_3=1}^n\sum _{m_2=1}^n\frac{1}{m_2+m_3}\sum _{m_4=1}^{n}\frac{1}{(m_3+m_4)^{\frac{3}{2}}}\\&\quad = O\left( n\sum _{m_3=1}^n\log n \, m_3^{-\frac{1}{2}}\right) \\&\quad = O(n^{\frac{3}{2}}\log n)=o(n^2)\, . \end{aligned}$$
Therefore, due to the Lebesgue dominated convergence theorem, we obtain
$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} {\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}( E_{k_2,\ell _2}) \nonumber \\&\quad = o(n^2)+c_1^2 n^2\int _{\frac{1}{n}\le \frac{\lfloor nx\rfloor }{n}<\frac{\lfloor ny\rfloor }{n}<\frac{\lceil nz\rceil }{n}<\frac{\lceil nt\rceil }{n}\le 1}\frac{dxdydzdt}{\left( \frac{\lceil nz\rceil }{n}-\frac{\lfloor nx\rfloor }{n}\right) \left( \frac{\lceil nt\rceil }{n}-\frac{\lfloor ny\rfloor }{n}\right) } \nonumber \\&\quad \sim c_1^2 n^2\int _{0<x<y<z<t<1}\frac{dxdydzdt}{(z-x)(t-y)} \nonumber \\&\quad = c_1^2\frac{\pi ^2}{12}n^2 = \frac{n^2}{48\det \Sigma ^2}\left( \sum _{a=1}^I {\bar{\mu }}({\mathcal {I}}_0=a)^2\right) ^2. \end{aligned}$$
The rest of the estimate of \(A_2\) is new (it is different from [31]). Fix for the moment \(1\le k_1< k_2< \ell _1<\ell _2\le n\). Note that
$$\begin{aligned}&{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\\&\quad =\sum _{a,b=1}^I{\bar{\mu }}\left( {\bar{T}}^{-k_1}{\bar{O}}_a\cap \bar{T}^{-k_2}{\bar{O}}_b\cap {\bar{T}}^{-\ell _1}({\bar{O}}_a)\right. \\&\left. \qquad \cap {\bar{T}}^{-\ell _2}\bar{O}_b\cap \{S_{k_2}-S_{k_1}=-(S_{\ell _1}-S_{k_2})=S_{\ell _2}-S_{\ell _1}\}\right) \, . \end{aligned}$$
Using now (23) as for (24), we observe that \(\mathbf 1_{\{S_{k_2}-S_{k_1}=-(S_{\ell _1}-S_{k_2})=S_{\ell _2}-S_{\ell _1}\}}\) is equal to the following quantity
$$\begin{aligned} \frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2}e^{i u\cdot ((S_{k_2}-S_{k_1})+(S_{\ell _1}-S_{k_2}))} e^{i v\cdot ((S_{\ell _2}-S_{\ell _1})+(S_{\ell _1}-S_{k_2}))} \, du\, dv \, , \end{aligned}$$
which is also equal to
$$\begin{aligned} \begin{aligned}&\frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2}e^{i u\cdot (S_{k_2}-S_{k_1})}e^{i(u+v)\cdot (S_{\ell _1}-S_{k_2})} e^{i v\cdot (S_{\ell _2}-S_{\ell _1})} \, du\, dv \\&\quad = \frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2} e^{i u\cdot S_{k_2 - k_1} \circ {\bar{T}}^{k_1}} e^{i(u+v)\cdot S_{\ell _1 - k_2} \circ \bar{T}^{k_2}} e^{i v\cdot S_{\ell _2 - \ell _1} \circ {\bar{T}}^{\ell _1}} \, du\, dv \, . \end{aligned} \end{aligned}$$
Now using the P-invariance and \({\bar{T}}\)-invariance of \({\bar{\mu }}\) and several times the formula \(P^m(f.g\circ {\bar{T}}^m)=gP^m(f)\), we obtain
$$\begin{aligned}&{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\\&\quad = \sum _{a,b=1}^I\frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2}{\mathbb {E}}_{{\bar{\mu }}} \left[ \mathbf 1_{{\bar{O}}_b} P_v^{\ell _2-\ell _1}\left( {\mathbf {1}}_{{\bar{O}}_a} P_{u+v}^{\ell _1-k_2}\left( {\mathbf {1}}_{{\bar{O}}_b}P_u^{k_2-k_1}(\mathbf 1_{{\bar{O}}_a})\right) \right) \right] \, du\, dv\, . \end{aligned}$$
Due to our spectral assumptions, we observe that
$$\begin{aligned} P_u^n= \lambda _u^n\Pi _u + O(\alpha ^n)\, , \end{aligned}$$
up to defining \(\lambda _u=e^{-\frac{1}{2} \Sigma ^2 u\cdot u}\) for u outside \([-\beta ,\beta ]^2\) and so, proceding as in the proof of Theorem 4.2, we obtain that, for every \(n\ge 2\) and every \(u,v\in [-\pi ,\pi ]^2\),
$$\begin{aligned} P_u^n= & {} e^{-\frac{n}{2} \Sigma ^2u\cdot u}{\mathbb {E}}_{{\bar{\mu }}}[\cdot ]{\mathbf {1}} + O(\alpha ^n)+O(e^{-2na|u|^2}(|u|+n|u|^3))\\= & {} e^{-\frac{n}{2} \Sigma ^2u\cdot u}{\mathbb {E}}_{{\bar{\mu }}}[\cdot ]\mathbf 1 + O(e^{-n a|u|^2}|u|)\, , \end{aligned}$$
and \(|\lambda _u^n|\le e^{- 2a|u|^2}\) for some \(a>0\) (such that \(e^{-2a|\pi |^2}>\alpha ^n\), \(\max (\lambda _u^{n-1},e^{-\frac{n-1}{2}\Sigma ^2u\cdot u})\le e^{- 2an|u|^2}\)) since \(n|u|^2e^{-2n a|u|^2}= O(e^{-n a|u|^2})\). Therefore, we obtain
$$\begin{aligned}&{\mathbb {E}}_{{\bar{\mu }}} \left[ {\mathbf {1}}_{{\bar{O}}_b} P_v^{\ell _2-\ell _1}\left( {\mathbf {1}}_{\bar{O}_a} P_{u+v}^{\ell _1-k_2}\left( {\mathbf {1}}_{\bar{O}_b}P_u^{k_2-k_1}({\mathbf {1}}_{{\bar{O}}_a})\right) \right) \right] \nonumber \\&\quad =({\bar{\mu }}({\bar{O}}_a){\bar{\mu }}({\bar{O}}_b))^2 e^{-\frac{1}{2}Q(\Sigma u,\Sigma v)} +O\left( (|u|+|v|)e^{-naQ(u,v)}\right) \, , \end{aligned}$$
where we have set
$$\begin{aligned} Q(u,v):= & {} (\ell _2-\ell _1)|v|^2+(\ell _1-k_2)|u+v|^2+(k_2-k_1) |u|^2\\= & {} (\ell _2-k_2)|v|^2+2(\ell _1-k_2)u\cdot v+(\ell _1-k_1) |u|^2\\= & {} (A_Q (u,v))\cdot (A_Q(u,v))=|A_Q(u,v)|^2\, , \end{aligned}$$
with \(A^2_Q:=\left( \begin{array}{cccc} \ell _1-k_1&{}0&{}\ell _1-k_2&{}0\\ 0&{}\ell _1-k_1&{}0&{}\ell _1-k_2\\ \ell _1-k_2&{}0&{}\ell _2-k_2&{}0\\ 0&{}\ell _1-k_2&{}0&{}\ell _2-k_2\end{array}\right) \) which is symmetric with determinant
$$\begin{aligned} \det A^2_Q= & {} (\ell _1-k_1)^2(\ell _2-k_2)^2+(\ell _1-k_2)^4-2(\ell _1-k_2)^2(\ell _1-k_1)(\ell _2-k_2)\nonumber \\= & {} ((k_2-k_1)(\ell _1-k_2)+(k_2-k_1)(\ell _2-\ell _1)+(\ell _1-k_2)(\ell _2-\ell _1))^2.\nonumber \\ \end{aligned}$$
Due to the form of \(A^2_Q\), we observe that \(A^2_Q\) has eigenvectors of the forms \((*,0,*,0)\) and \((0,*,0,*)\), that it has two double eigenvalues of sum (without multiplicity) \(\ell _1-k_1+\ell _2-k_2\) and of product (without multiplicity) \(\sqrt{\det A_Q^2}\). Therefore its dominating eigenvalue is smaller than the sum and so is less than \(4\max (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)\) and so (using the fact that the product of the two eigenvalues is larger than the maximum times the median of these three values) the smallest eigenvalue of \(A^2_Q\) cannot be smaller than a quarter of the median of \(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1\), that we denote by \(med (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)\). So
$$\begin{aligned}&\int _{([-\pi ,\pi ]^2)^2} e^{-nQ(\Sigma u,\Sigma v)}\, dudv\, =(\det \Sigma )^{-2}\int _{(\Sigma [-\pi ,\pi ]^2)^2} e^{-nQ( u,v)}\, dudv\\&\quad =(\det A_Q)^{-1}(\det \Sigma )^{-2}\int _{A_Q(\Sigma ([-\pi ,\pi ]^2)^2)} e^{-|(x,y)|^2}\, dxdy\\&\quad =(\det A_Q)^{-1}(\det \Sigma )^{-2}\left( \int _{({\mathbb {R}}^2)^2} e^{-|(x,y)|^2}\, dxdy + O(e^{-a_1 {med (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^2}})\right) \\&\quad =(2\pi )^2(\det A_Q)^{-1}(\det \Sigma )^{-2}\left( 1 + O(e^{-a_1 {med (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^2}})\right) \, , \end{aligned}$$
for some \(a_1>0\). Moreover
$$\begin{aligned}&\int _{({\mathbb {R}}^2)^2} |(u,v)|e^{-naQ(u,v)}\, dudv =(\det A_Q)^{-1}\int _{({\mathbb {R}}^2)^2} |A_Q^{-1}(u,v)| e^{-a|(x,y)|^2}\, dxdy\\&\quad =O\left( (\det A_Q)^{-1}\, med(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^{-\frac{1}{2}} \right) \, . \end{aligned}$$
$$\begin{aligned}&{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\nonumber \\&=\frac{\left( \sum _{a=1}^I{\bar{\mu }}({\bar{O}}_a)^2\right) ^2}{(2\pi )^{2}\, \det A_Q\, \det \Sigma ^{2}}\left( 1+ O\left( med(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^{-\frac{1}{2}} \right) \right) \, . \end{aligned}$$
But using (48),
$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} (\det A_Q)^{-1}\\&\quad =\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}\frac{1}{(k_2-k_1)(\ell _1-k_2)+(k_2-k_1)(\ell _2-\ell _1)+(\ell _1-k_2)(\ell _2-\ell _1)}\\&\quad =\sum _{m_1,m_2,m_3,m_4\ge 1\ :\ m_1+m_2+m_3+m_4\le n}\frac{1}{m_2m_3+m_2m_4+m_3m_4}\\&\quad =n^2\int _{(0,+\infty )^4} \frac{{\mathbf {1}}_{\left\{ \frac{\lceil ny_1\rceil }{n}+\frac{\lceil ny_2\rceil }{n}+\frac{\lceil ny_3\rceil }{n}+\frac{\lceil ny_4\rceil }{n}\le 1\right\} }}{\frac{\lceil ny_2\rceil }{n} \frac{\lceil ny_3\rceil }{n} +\frac{\lceil ny_2\rceil }{n} \frac{\lceil ny_4\rceil }{n}+\frac{\lceil ny_3\rceil }{n}\frac{\lceil ny_4\rceil }{n}}\, dy_1\, dy_2\, dy_3\, dy_4\\&\quad \sim n^2\int _{(0,+\infty )^4} \frac{\mathbf 1_{\left\{ y_1+y_2+y_3+y_4\le 1\right\} }}{y_2y_3 +y_2y_4+y_3y_4}\, dy_1\, dy_2\, dy_3\, dy_4\, , \end{aligned}$$
due to the dominated convergence theorem. Therefore
$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} (\det A_Q)^{-1}\sim n^2\nonumber \\&\quad \int _{(0,+\infty )^3} \frac{(1-y_2-y_3-y_4)\mathbf 1_{\left\{ y_2+y_3+y_4\le 1\right\} }}{y_2y_3 +y_2y_4+y_3y_4}\, dy_2\, dy_3\, dy_4=n^2J\, . \end{aligned}$$
Analogously
$$\begin{aligned} \begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} (\det A_Q)^{-1}\, (med(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1))^{-\frac{1}{2}} \\&\quad =\sum _{m_1,m_2,m_3,m_4\ge 1\ :\ m_1+m_2+m_3+m_4\le n}\frac{1}{(m_2m_3+m_2m_4+m_3m_4)\, med(m_2,m_3,m_4)^{\frac{1}{2}}}\\&\quad \le n\sum _{1\le m_2\le m_3\le m_4\le n}\frac{1}{(m_2m_3+m_2m_4+m_3m_4)\, m_3^{\frac{1}{2}}}\\&\quad \le n\sum _{1\le m_2\le m_3\le m_4\le n}\frac{1}{ m_3^{\frac{3}{2}}m_4} \; \le \; n \log n \sum _{m_2=1}^n\sum _{m_3=m_2}^n m_3^{-\frac{3}{2}}\\&\quad \le n \log n \sum _{m_2=1}^n O( m_2^{-\frac{1}{2}})=O(n^{\frac{3}{2}}\log n)=o(n^2)\, . \end{aligned} \end{aligned}$$
Equations (49), (50) and (51) lead to
$$\begin{aligned} \sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})=\frac{\left( \sum _{a=1}^I{\bar{\mu }}(\bar{O}_a)^2\right) ^2}{((2\pi )^2\det \Sigma ^2)}J+o(n^2)\, . \end{aligned}$$
Combining this with (46), we conclude that
$$\begin{aligned} A_2\sim \frac{n^2}{\det \Sigma ^2}\left( \sum _{a=1}^I{\bar{\mu }}(\mathcal I_0=a)^2\right) ^2\left( \frac{-1}{48}+\frac{J}{4\pi ^2} \right) \, . \end{aligned}$$
The study of \(A_3\) is the most delicate. We can observe that both sums \(\sum _{1\le k_1< k_2<\ell _2<\ell _1\le n} {\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\) and \(\sum _{1\le k_1< k_2<\ell _2<\ell _1\le n}{\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})\) are in \(O(n^2\log n)\). However, we will see that their difference is in \(n^2\). Once again our proof differs from the one in [31] and is based on the same idea as the one used to prove \(A_2\). We set \(E_{k,\ell }(b):=E_{k,\ell }\cap \{{\mathcal {I}}_k=b\}\). Due to the first part of Lemma 4.8,
$$\begin{aligned} A_3&=\sum _{1\le k_1< k_2< \ell _2< \ell _1\le n} {\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2}) -{\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})\nonumber \\&=o(n^{2})+\sum _{1\le k_1< k_2< \ell _2< \ell _1\le n} \sum _{a,b=1}^I\left( -I_{k_1,k_1,l_1,l_2}+ {\bar{\mu }}(O_{k_1,k_2,l_1,l_2}\cap S_{k_1,k_2,l_1,l_2})\right) \nonumber \\&=o(n^{2})+\sum _{1\le k_1< k_2< \ell _2 < \ell _1\le n} \sum _{a,b=1}^I\left( -I_{k_1,k_1,l_1,l_2}\right) \end{aligned}$$
$$\begin{aligned}&\quad +\sum _{1\le k_1< k_2< \ell _2 < \ell _1\le n} \sum _{a,b=1}^I\left( \frac{1}{(2\pi )^2}\int _{[-\pi ,\pi ]^2}{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{\bar{O}_a}P_u^{\ell _1-\ell _2}\left( {\mathbf {1}}_{{\bar{O}}_b}\mathcal H_{0,\ell _2-k_2}\right. \right. \right. \nonumber \\&\left. \left. \left. \quad \left( {\mathbf {1}}_{{\bar{O}}_b}P_u^{k_2-k_1}\left( {\mathbf {1}}_{\bar{O}_a}\right) \right) \right) \right] \, du\right) \, , \end{aligned}$$
$$\begin{aligned} I_1(k_1,k_1,l_1,l_2)= & {} \frac{({\bar{\mu }}(\bar{O}_a))^2{\bar{\mu }}(E_{k_2,\ell _2}(b))}{2\pi \sqrt{\det \Sigma ^2}(\ell _1-k_1)},\\ O_{k_1,k_2,l_1,l_2}= & {} {\bar{O}}_a\cap {\bar{T}}^{-(k_2-k_1)}{\bar{O}}_b\cap \bar{T}^{-(\ell _2-k_1)}{\bar{O}}_b \cap {\bar{T}}^{-(\ell _1-k_1)}\bar{O}_a,\\ S_{k_1,k_2,l_1,l_2}= & {} \{S_{\ell _2-k_2}\circ {\bar{T}}^{k_2-k_1}=0\} \cap \{S_{\ell _1-\ell _2}\circ {\bar{T}}^{\ell _2-k_1}=-S_{k_2-k_1}\}. \end{aligned}$$
Now, as we did for (47) (and using Theorem 4.2), we get that
$$\begin{aligned}&{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{{\bar{O}}_a}P_u^{\ell _1-\ell _2}\left( {\mathbf {1}}_{\bar{O}_b}{\mathcal {H}}_{0,\ell _2-k_2}\left( {\mathbf {1}}_{\bar{O}_b}P_u^{k_2-k_1}\left( {\mathbf {1}}_{\bar{O}_a}\right) \right) \right) \right] \\&\quad =({\bar{\mu }}({\bar{O}}_a))^2 e^{-\frac{(\ell _1-\ell _2)+(k_2-k_1)}{2}|\Sigma u|^2}{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{{\bar{O}}_b}\mathcal H_{0,\ell _2-k_2}{\mathbf {1}}_{{\bar{O}}_b}\right] +O\left( \frac{|u|}{\ell _2-k_2}e^{-na|u|^2}\right) \, . \end{aligned}$$
$$\begin{aligned}&\frac{1}{(2\pi )^2}\int _{(-\pi ,\pi )^2}{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{{\bar{O}}_a}P_u^{\ell _1-\ell _2}\left( {\mathbf {1}}_{\bar{O}_b}{\mathcal {H}}_{0,\ell _2-k_2}\left( {\mathbf {1}}_{\bar{O}_b}P_u^{k_2-k_1}\left( {\mathbf {1}}_{\bar{O}_a}\right) \right) \right) \right] \, du\nonumber \\&\quad = \frac{({\bar{\mu }}({\bar{O}}_a))^2{{\bar{\mu }}}\left( E_{k_2,\ell _2}(b)\right) }{2\pi (\ell _1-\ell _2+k_2-k_1)\sqrt{\det \Sigma ^2}} +O\left( \frac{1}{(\ell _2-k_2)(\ell _1-\ell _2+k_2-k_1)^{\frac{3}{2}}}\right) \, . \end{aligned}$$
We will now prove that the term in O in this last formula is negligable. Indeed its sum over \(\{1\le k_1\le k_2\le \ell _2\le \ell _1\le n\}\) is in O of the following quantity:
$$\begin{aligned} \sum _{m_1+m_2+m_3+m_4\le n}\left( \frac{1}{m_3(m_4+m_2)^{\frac{3}{2}}}\right)\le & {} n\log n\sum _{m_2=1}^n\sum _{m_4=1}^n(m_4+m_2)^{-\frac{3}{2}}\\\le & {} O\left( n\log n\sum _{m_2=1}^nm_2^{-\frac{1}{2}}\right) =O(n^{\frac{3}{2}}\log n)=o(n^2)\, . \end{aligned}$$
This combined with (54) and (55) leads to
$$\begin{aligned} A_3= & {} o(n^{2})+\sum _{1\le k_1< k_2< \ell _2\le \ell _1\le n} \sum _{a,b=1}^I\frac{({\bar{\mu }}({\bar{O}}_a))^2{\bar{\mu }}\left( E_{k_2,\ell _2}(b)\right) }{2\pi \sqrt{\det \Sigma ^2}}\\&\left( \frac{1}{\ell _1-\ell _2+k_2-k_1}-\frac{1}{\ell _1-k_1}\right) \, , \end{aligned}$$
$$\begin{aligned} A_3= & {} o(n^{2})+\frac{\sum _{a}^I({\bar{\mu }}(\mathcal I_0=a))^2}{2\pi \sqrt{\det \Sigma ^2}}\sum _{m_1+m_2+m_3+m_4\le n}\left( \frac{c_1}{m_3}+O(m_3^{-\frac{3}{2}})\right) \\&\frac{m_3}{(m_2+m_4)(m_2+m_3+m_4)}\\= & {} o(n^{2})+c_1^2\sum _{m_1+m_2+m_3+m_4\le n}\frac{1}{(m_2+m_4)(m_2+m_3+m_4)}\, , \end{aligned}$$
$$\begin{aligned}&\sum _{m_1+m_2+m_3+m_4\le n}\frac{1}{m_3^{\frac{1}{2}}(m_2+m_4)(m_2+m_3+m_4)}\\&\quad =O\left( n\sum _{m_2,m_3,m_4=1}^nm_3^{-\frac{1}{2}}(m_2m_4)^{-1}\right) =o(n^2)\, . \end{aligned}$$
Therefore, due to the Lebesgue dominated convergence theorem,
$$\begin{aligned} A_3 \sim n^{2}c_1^2\int _{y_1,y_2,y_3,y_4>0:y_1+y_2+y_3+y_4<1} \frac{1}{(y_2+y_4)(y_2+y_3+y_4)}\, dy_1dy_2dy_3dy_4 \sim \frac{c_1^2}{2} n^{2} . \end{aligned}$$
To conclude the proof of the lemma, we use the estimate for \(A_3\) together with (44) and (52) to obtain,
$$\begin{aligned} 8A_2 + 8A_3&= 4c_1^2 n^2 + \frac{8n^2}{\det \Sigma ^2} \left( \sum _{a=1}^I {\bar{\mu }}({\bar{O}}_a)^2 \right) ^2 \left( \frac{-1}{48} + \frac{J}{4\pi ^2} \right) \\&= \frac{n^2}{\det \Sigma ^2} \left( \sum _{a=1}^I {\bar{\mu }}(\bar{O}_a)^2 \right) ^2 \left[ \frac{2J+1}{\pi ^2} - \frac{1}{6} \right] . \end{aligned}$$
This finished the proof.
Appendix C. Spectrum of \(\mathcal {P}_u\)
In this appendix, we are interested in the spectrum of the family of operators \(\mathcal {P}_u\). We start by stating a result for the unperturbed operators \({\mathcal {L}}_{u,0}\).
Lemma C.1
Let \(u\in {\mathbb {R}}^2\), \(h\in \mathcal {B}\) and \(\lambda \in {\mathbb {C}}\) be such that \({\mathcal {L}}_{u,0}h=\lambda h\) in \({{\mathcal {B}}}\) and \(|\lambda | \ge 1\). Then either \(h\equiv 0\) or \(u\in 2\pi {\mathbb {Z}}^2\), \(\lambda =1\) and h is \({\bar{\mu }}_0\)-almost surely constant.
Recall that for \(\psi \in \mathcal {C}^p({\bar{M}}_0)\), we have \(\psi \circ \bar{T}_0^n \in \mathcal {C}^p({\bar{T}}^{-n}\mathcal {W}^s)\). Note that
$$\begin{aligned} {\mathcal {L}}_{u,0} h(\psi )=h(e^{iu \cdot \Phi _0}\psi \circ {\bar{T}}_0). \end{aligned}$$
Thus for \(n\ge 1\),
$$\begin{aligned} {\mathcal {L}}_{u,0}^n h(\psi )= h(e^{iu \cdot S_n \Phi _0 }\psi \circ {\bar{T}}_0^n), \end{aligned}$$
where \(S_n \Phi _0 =\Phi _0+\Phi _0\circ {\bar{T}}_0+\cdots +\Phi _0\circ {\bar{T}}_0^{n-1}\) denotes the partial sum. By [16, Lemma 3.4], using the invariance of h,
$$\begin{aligned}&|h(\psi )| = |\lambda |^{-n} |h(e^{iu \cdot S_n\Phi _0}\psi \circ \bar{T}_0^n) | \nonumber \\&\quad \le C |\lambda |^{-n} | h |_w \big (|e^{i u \cdot S_n\Phi _0} \psi \circ {\bar{T}}_0^n|_\infty + C^{(p)}_{{\bar{T}}_0^{-n} \mathcal {W}^s}(e^{iu\cdot S_n\Phi _0}\cdot \psi \circ {\bar{T}}_0^n) \big ) , \end{aligned}$$
where \(C^{(p)}_{{\bar{T}}_0^{-n}\mathcal {W}^s}(\cdot )\) denotes the Hölder constant of exponent p measured along elements of \({\bar{T}}_0^{-n} \mathcal {W}^s\). Since \(|e^{i u \cdot S_n \Phi _0}| = 1\) and \(S_n \Phi _0\) is constant on each element of \({\bar{T}}_0^{-n} \mathcal {W}^s\), we have
$$\begin{aligned} \begin{aligned}&C^{(p)}_{{\bar{T}}_0^{-n} \mathcal {W}^s}(e^{iu\cdot S_n\Phi _0}\cdot \psi \circ {\bar{T}}_0^n) \\&\quad \le |e^{i u \cdot S_n \Phi _0}|_\infty C^{(p)}_{\bar{T}_0^{-n} \mathcal {W}^s}(\psi \circ {\bar{T}}_0^n) + |\psi \circ {\bar{T}}_0^n|_\infty C^{(p)}_{{\bar{T}}_0^{-n}\mathcal {W}^s}(e^{i u \cdot S_n \Phi _0}) \\&\quad \le C \Lambda ^{-pn} C^{(p)}_{\mathcal {W}^s}(\psi ) . \end{aligned} \end{aligned}$$
Using this estimate in (56) and taking the limit as \(n \rightarrow \infty \) yields \(|h(\psi )| = 0\) if \(|\lambda |>1\) and \(|h(\psi )| \le C| h |_w |\psi |_\infty \) for all \(\psi \in \mathcal {C}^p(\mathcal {W}^s)\) if \(|\lambda |=1\). From this we conclude that the spectrum of \(\mathcal {L}_{u,0}\) is always contained in the unit disk. Furthermore, when \(|\lambda |=1\), then h is a signed measure. For the remainder of the proof, we assume \(|\lambda |=1\).
Let \({\mathbb {V}}_{u,0}\) be the eigenspace of \({\mathcal {L}}_{u,0}\) corresponding to eigenvalue \(\lambda _{u,0}\), and \(\Pi _{u,0}\) the eigenprojection operator. Since we are assuming \({\mathbb {V}}_{u,0}\) is non-empty, Lemma 3.14 implies that \(\mathcal {L}_{u,0}\) is quasi-compact with essential spectral radius bounded by \(\tau < 1\). Moreover, Lemma 3.14 implies that \(\Vert \mathcal {L}_{u,0}^n \Vert _{L(\mathcal {B}, \mathcal {B})}\) remains bounded for all \(n \ge 0\), so using [15, Lemma 5.1], we conclude that \(\mathcal {L}_{u,0}\) has no Jordan blocks corresponding to its peripheral spectrum.
Using these facts, \(\Pi _{u,0}\) has the representation
$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n} \sum _{j=1}^n \lambda ^{-j} \mathcal L_{u,0}^j =\Pi _{u,0} . \end{aligned}$$
In addition, for \(f\in \mathcal {C}^1({\bar{M}}_0)\), \(\psi \in \mathcal {C}^p(\mathcal {W}^s)\),
$$\begin{aligned} \left| \Pi _{u,0} f(\psi ) \right| =\left| \lim _{n\rightarrow \infty }\frac{1}{n} \sum _{j=1}^n \lambda ^{-j} f((e^{iu \cdot S_j \Phi _0}\psi \circ {\bar{T}}_0^j) \right| \le |f|_{\infty }|\psi |_{\infty } . \end{aligned}$$
Since \(\Pi _{u,0} \mathcal {C}^1({\bar{M}}_0)\) is dense in the finite dimensional space \(\Pi _{u,0}\mathcal {B}\), therefore \(\Pi _{u,0} \mathcal {C}^1(\bar{M}_0)=\Pi _{u,0}\mathcal {B}={\mathbb {V}}_{u,0}\). So for \(h \in {\mathbb {V}}_{u,0}\), there exists \(f \in \mathcal {C}^1({\bar{M}}_0)\) such that \(\Pi _{u,0} f = h\). Now for each \(\psi \in \mathcal {C}^p({\bar{M}}_0)\),
$$\begin{aligned} |h(\psi )| = |\Pi _{u,0} f(\psi )| \le |f|_\infty \Pi _0 1(|\psi |) = |f|_\infty {\bar{\mu }}_0(|\psi |) . \end{aligned}$$
Thus h is absolutely continuous with respect to \({\bar{\mu }}_0\). For simplicity, we identify h and its density with respect to \({\bar{\mu }}_0\); then \(h \in L^\infty ({\bar{M}}_0, {\bar{\mu }}_0)\). Now for any \(\psi \in \mathcal {C}^p(\mathcal {W}^s)\), we have
$$\begin{aligned} \lambda \int _{{\bar{M}}_0} h \psi \, d\mu _0&=\int _{{\bar{M}}_0} \mathcal {L}_0(e^{iu \cdot \Phi _0} h)\cdot \psi \, d{\bar{\mu }}_0\\&=\int _{{\bar{M}}_0} (e^{iu\cdot \Phi _0} h)\circ {\bar{T}}_0^{-1}\cdot \psi \,d{\bar{\mu }}_0. \end{aligned}$$
Accordingly, \(\lambda \, h=(e^{iu \cdot \Phi _0} h)\circ \bar{T}_0^{-1}\), \({\bar{\mu }}_0\)-a.e. Or equivalently, we have \(\lambda \, h\circ {\bar{T}}_0=e^{iu \cdot \Phi _0}h\). Hence \(\lambda ^n\, h\circ {\bar{T}}_0^n=e^{iu\cdot S_n \Phi _0}h\).
Let \(G_\lambda \) be the closed multiplicative group generated by \(\lambda \) and let \(m_{\lambda }\) be the normalized Haar measure on \(G_\lambda \). (\(G_\lambda \) is finite if \(\lambda \) is a root of unity; it is \(\{z\in {\mathbb {C}}\, :\, |z|=1\}\) otherwise.) The dynamical system \((G_\lambda ,m_{\lambda },T_\lambda )\) is ergodic, where \(T_\lambda \) denotes multiplication by \(\lambda \) in \(G_\lambda \). Due to [28], the dynamical system \((M_0 \times G_\lambda ,\mu _0\otimes m_{\lambda }, T_0 \times T_\lambda )\) in infinite measure is conservative and ergodic. But the function \(H: M_0 \times G_\lambda \rightarrow {\mathbb {C}}\) defined as follows is \((T_0 \times T_\lambda )\)-invariant:
$$\begin{aligned} \forall ({\bar{x}},\ell ,y)\in {\bar{M}}_0\times {\mathbb {Z}}^2\times G_\lambda ,\quad H({\bar{x}}+\ell ,y):=y h({\bar{x}})e^{-i u\cdot \ell }. \end{aligned}$$
Indeed, for \(\mu _0\otimes m_{\lambda }\)-a.e. \(({\bar{x}}+\ell ,y)\in M_0 \times G_\lambda \),
$$\begin{aligned} H((T_0 \times T_\lambda )({\bar{x}}+\ell ,y))= & {} H({\bar{T}}_0(\bar{x})+\ell +\Phi _0({\bar{x}}),\lambda y) =\lambda y h({\bar{T}}_0({\bar{x}}))e^{-i u\cdot (\ell +\Phi _0({\bar{x}}))}\\= & {} ye^{-i u\cdot \ell }(\lambda h({\bar{T}}_0({\bar{x}})) e^{-i u\cdot \Phi _0({\bar{x}})})\\= & {} ye^{-i u\cdot \ell } h({\bar{x}})\, , \end{aligned}$$
due to our assumption on h. We conclude that H is a.e. equal to a constant, which implies that \(u\in 2\pi {\mathbb {Z}}^2\), \(\lambda =1\), and h is \({\bar{\mu }}_0\)-a.s. constant. \(\quad \square \)
Proposition C.2
Given \(\beta > 0\), there exists \(C>1\) and \(\alpha \in (0,1)\) such that
$$\begin{aligned} \forall n\in {\mathbb {N}}^*,\quad \sup _{\beta \le |u|\le \pi }\Vert {\mathcal {P}}_u^n\Vert _{L({\widetilde{\mathcal {B}}},{\widetilde{\mathcal {B}}})}\le C\alpha ^n\, . \end{aligned}$$
Fix \(\beta > 0\). Due to [1, Lemma 4.3], Lemma C.1, and the continuity in u provided by [17, Lemma 5.4] (see also Lemma 3.16 applied to \(\mathcal {L}_{u,0}\) rather than \(P_u\)), we know that there exists \(C>1\) and \(\alpha \in (0,1)\) such that
$$\begin{aligned} \forall n\in {\mathbb {N}}^*,\quad \sup _{\beta \le |u|\le \pi }\Vert {\mathcal {L}}_{u,0}^n\Vert _{L(\mathcal {B},\mathcal {B})}\le C\alpha ^n\, . \end{aligned}$$
Therefore, for every \(f\in {\widetilde{\mathcal {B}}}\), we have
$$\begin{aligned}&\sup _{\underline{\omega }\in E^{{\mathbb {N}}}}\left\| \mathcal P_u^nf(x,\underline{\omega })\right\| _{\mathcal {B}}\\&\quad =\sup _{\underline{\omega }\in E^{{\mathbb {N}}}}\left\| \int _{E^n}{\mathcal {L}}_{u,0}^nf(\cdot ,({{\tilde{\omega }}},\underline{\omega }))\, d\eta ^{\otimes n}({{\tilde{\omega }}}) \right\| _{\mathcal {B}}\\&\quad \le \sup _{\underline{\omega }\in E^{{\mathbb {N}}}} \int _{E^n}\left\| \mathcal L_{u,0}^nf(\cdot ,({{\tilde{\omega }}},\underline{\omega }))\right\| _{\mathcal {B}}\, d\eta ^{\otimes n}({{\tilde{\omega }}})\\&\quad \le \sup _{\underline{\omega }\in E^{{\mathbb {N}}}} C\alpha ^n\sup _{\underline{\omega '}}\left\| f(\cdot ,\underline{\omega '})\right\| _{\mathcal {B}}\, \end{aligned}$$
where we used Lemma 3.7 to obtain the second line. Analogously,
$$\begin{aligned}&\sup _{\underline{\omega }\ne \underline{\omega '}}\frac{\left\| \mathcal P_u^nf(x,\underline{\omega })-\mathcal P_u^nf(x,\underline{\omega '})\right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\\&\quad =\sup _{\underline{\omega }\ne \underline{\omega '}}\frac{\left\| \int _{E^n}\mathcal L_{u,0}^n\left( f(\cdot ,({{\tilde{\omega }}},\underline{\omega }))-f(\cdot ,({{\tilde{\omega }}},\underline{\omega '}))\right) \, d\eta ^{\otimes n}({{\tilde{\omega }}}) \right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\\&\quad \le \sup _{\underline{\omega }\ne \underline{\omega '}} \int _{E^n}\frac{\left\| \mathcal L_{u,0}^n\left( f(\cdot ,({{\tilde{\omega }}},\underline{\omega }))-f(\cdot ,({{\tilde{\omega }}},\underline{\omega '}))\right) \right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\, d\eta ^{\otimes n}({{\tilde{\omega }}})\\&\quad \le C\alpha ^n\varkappa ^n\sup _{\underline{\omega }\ne \underline{\omega '}} \frac{\left\| f(\cdot ,\underline{\omega '})-f(\cdot ,\underline{\omega '})\right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\, . \end{aligned}$$
We conclude by putting these two estimates together. \(\quad \square \)
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
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1.Department of MathematicsFairfield UniversityFairfieldUSA
2.Laboratoire de Mathématique de Bretagne Atlantique, LMBA, UMR CNRS 6205, Institut Universitaire de France, IUFUniv Brest, Université de BrestBrest CedexFrance
3.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA
Demers, M.F., Pène, F. & Zhang, HK. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-019-03670-7
Accepted 28 October 2019
First Online 09 January 2020
DOI https://doi.org/10.1007/s00220-019-03670-7
Publisher Name Springer Berlin Heidelberg
Cite article | CommonCrawl |
pp. 40878-40890
•https://doi.org/10.1364/OE.443788
Close packed random rectangular microlens array used for laser beam homogenization
Wei Yuan, Li Xue, Axiu Cao, Hui Pang, and Qiling Deng
Wei Yuan,1,2,3 Li Xue,1,2,3 Axiu Cao,1,* Hui Pang,1 and Qiling Deng1
1Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China
2School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
3These authors contributed equally to this work
*Corresponding author: [email protected]
Axiu Cao https://orcid.org/0000-0003-1943-184X
W Yuan
L Xue
A Cao
H Pang
Q Deng
Wei Yuan, Li Xue, Axiu Cao, Hui Pang, and Qiling Deng, "Close packed random rectangular microlens array used for laser beam homogenization," Opt. Express 29, 40878-40890 (2021)
Partially coherent beam smoothing using a microlens array
Jingjing Meng, et al.
Freeform microlens array homogenizer for excimer laser beam shaping
Yuhua Jin, et al.
Beam homogenizers based on chirped microlens arrays
Frank Wippermann, et al.
Opt. Express 15(10) 6218-6231 (2007)
Coherence, Statistical Optics, and Scattering
Diffractive optical elements
High power lasers
Laser arrays
Laser beam shaping
Laser beams
Stray light
Original Manuscript: September 19, 2021
Revised Manuscript: November 10, 2021
Design and simulations
Equations (15)
When the high coherence laser beam is homogenized by microlens array (MLA), interference fringes will be generated reducing the uniformity of homogenized spot. A novel close packed random rectangular microlens array (rRMLA) is proposed to solve this problem. By designing the MLA with random apertures and random focal lengths of sub-lenses, the phase regulation can be realized, so as to disturb the coherent superposition fringes for improving the uniformity. To realize the dense arrangement of a MLA with random rectangular aperture, an iterative segmentation method is proposed to design the structure of rRMLA with controllable divergence angle and high filling factor. Theoretical simulations and experimental results both demonstrate the improvement of uniformity of the homogenized spot based on the proposed rRMLA.
With the continuous development of laser technology, laser is widely used in industry, medical treatment, military, scientific research and many other fields. When it is applied to laser projection [1], laser background lighting [2,3] and other fields, the laser beam is always need to be shaped and homogenized. Diffractive optical element (DOE) and microlens array (MLA), as highly integrated beam control elements, have high practical value in laser beam homogenization and have been widely studied. The DOE can accurately control the laser beam with Gaussian distribution and realize the output of homogenized spot with arbitrary shape [4,5]. When the DOEs are used to homogenize the laser beam, the more the number of steps in the structure, the better the homogenization effect will, while the processing difficulty will increase as well. Meanwhile, the energy utilization rate, that is, diffraction efficiency will depend on the number of steps. For example, when the number of steps is two, four, eight, or sixteen, the energy utilization rate will be $40.5\%$, $81\%$, $94.9\%$, and $98.6\%$, respectively [6]. In addition, the DOEs operate in a very narrow wavelength band and are sensitive to the change of the wavelength, which limits the applicability of lasers with different wavelengths. The binary DOEs is always designed for a single wavelength. When the other wavelength is used to irradiate the DOEs, the central zero order strong intensity will be produced, which greatly reduces the uniformity of the homogenized spot. The method of MLA has the advantages of high energy utilization rate, small volume and high integration. In addition, it is not sensitive to the intensity distribution of incident light. The incident laser beam is divided into a series of sub-beams, which are superimposed on each other in the far field to eliminate the inhomogeneity between different sub-beams and form a homogenized spot [7,8].
Because the MLA is a refractive continuous surface structure with less stray light and it is suitable for beam homogenization of different lasers with a high energy utilization rate, researchers have carried out a lot of researches on beam homogenization by using MLA. M. Zimmermann et al. used diffraction theory to explain the diffraction effect of MLA with small aperture and found that periodic MLA is only suitable for laser beam homogenization with poor coherence [9]. For high coherence laser beams, due to the periodic structure of MLA, the divided sub-beams will interfere, resulting in interference fringes and periodic strong dots in the homogenized spot, which greatly reduces the uniformity. In order to disturb the interference fringes of periodic MLA during beam homogenization, researchers propose to change the polarization state or modulate the phase of the laser to disturb the spatial coherence. A. S. Victoria et al. proposed to use the random phase plates (RPP) of an optical birefringent material to smooth the homogenized spot [10]. It is proposed that the polarization direction of the incident beam is modulated by randomly etching structure at a specific depth on the surface, so as to disrupt the coherence condition of the beam. For the beam homogenization method based on MLA, the uniformity is usually improved by phase modulation. F. Wippermann et al. proposed that chirped MLA can eliminate the periodic lattice effect of homogenized spot [11]. However, due to the wedge-shaped distribution of sub-lenses of chirped MLA, it is difficult to process and prepare in practice, which is far from practical application.
In recent years, free-form surfaces MLA have also been widely studied in beam shaping [12–14]. In 2017, Y. Jin et al. proposed a free-form surface random MLA for beam homogenization of excimer lasers [12]. Taking advantage of the high degree of freedom in the design of free-form surface microlens, each free-form surface in the array is designed separately. By introducing an appropriate aberration to redistribute the irradiance of the beam, a higher beam homogenization effect can be obtained. This method can effectively reduce the diffraction effect caused by the small aperture of sub-lens. In 2019, Z. Liu et al. proposed random free-form surface MLA (rfMLA) for beam homogenization of digital micromirror device (DMD) lithography system [13]. The rfMLA with different structural parameters are designed to break the periodicity of MLA and achieve the goal of weakening the interference pattern in the homogenized spot to improving the uniformity of the spot projected onto the DMD. In 2020, W. Zhang et al. proposed a randomly distributed free-form surface cylindrical microlens for laser beam shaping and homogenization [14]. A linear uniform spot is generated by controlling the light field through a free-form surface cylindrical microlens. Although the scheme of free-form surface can improve the uniformity of coherent light homogenization, it is worth noting that the design of free-form surface lens depends on the irradiance distribution and beam profile of the incident light beam. Therefore, the designed free-form surface lens belongs to static optical system. If the energy distribution of the input beam changes or the beam shape is distorted, the homogenization effect will be adversely affected. Therefore, for different input laser beams, a new free-form surface lens needs to be redesigned to match it, which limits the application flexibility of this method.
In order to make the random MLA not only homogenize the laser beam with one type of energy distribution, it is considered that the free-form surface is not introduced to regulate the energy distribution of the input light field when designing the random MLA, under the condition of ensuring the homogenization effect. Our research group proposed to design and fabricate random MLA with small apertures to realize laser beam homogenization [15,16]. The designed MLA is fine enough to divide the input beam, so that the uniformity of the spot will be greatly improved. Meanwhile, the randomness of the MLA is used to reduce the interference fringes in the homogenized spot. However, the specific divergence angle needs to be controlled by adjusting the process parameters in the preparation process. The preparation results are easily affected by the process parameters and environmental factors, which reduces the robustness of the preparation results during the overall process, leading to the phenomenon that the intermediate energy is strong and the edge energy decreases gradually.
Based on the previous researches, this paper proposes to design a close packed random rectangular microlens array (rRMLA) for laser beam homogenization. The apertures and sag heights of the sub-lenses are manipulated and designed separately to make it a random distribution. While breaking the periodicity of the MLA, a high uniformity homogenized spot is obtained. Theoretical simulations and experiments are carried out to demonstrate the improvement of uniformity of the homogenized spot based on the proposed rRMLA. The main arrangement of this paper is as follows: Section 2 describes the beam homogenization principle of the proposed rRMLA. Section 3 introduces the design of rRMLA and simulations. Section 4 describes the experimental verification. Section 5 is the discussion, and Section 6 is the summary of the whole paper.
The principle of laser beam homogenization based on rRMLA is shown in Fig. 1. The collimated laser beam is incident on the surface of rRMLA1 and divided into several small sub-beams by the sub-lenses. These modulated sub-beams pass through the objective lens system composed of rRMLA2 and Fourier lens (FL), and finally get a uniform spot on the focal plane (FP). Due to the random aperture, radius of curvature and arrangement mode of each sub-lens in rRMLA, the beam is divided into several independent sub-beams after passing through the rRMLA, disrupting the interference conditions between the sub-beams and improving the uniformity of the output spot.
Fig. 1. Principle of laser beam homogenization.
Assuming that rRMLA1 and rRMLA2 are composed of $2N + 1$ sub-lenses, the transmittance function of MLA in y direction can be expressed by Eq. (1).
(1)$$t({y_1}) = [rect(\frac{{{y_1}}}{{{p_n}}})\cdot \exp ( - ik\frac{{y_1^2}}{{2{f_n}}})] \otimes \sum\limits_{n ={-} N}^N {\delta ({{y_1} - {p_n}n} )} .$$
where, ${p_n}$ is the aperture size of the $nth$ sub-lens in rRMLA, and the value range of n is $[{ - N,N} ]$. ${f_n}$ is the focal length of the $nth$ sub-lens. The apertures and focal lengths of different sub-lenses are different, which are randomly distributed. k is the wave number and its value is $2\pi /\lambda$. $rect$ is a rectangular function, which controls the aperture of sub-lens to be rectangular $\delta$ is an impulse function, which is used to produce an arrayed structure.
Fourier optics [17] is used to analyze the homogenized light field obtained by the beam passing through the whole homogenized structure. Assuming that the input laser is a collimated parallel light, so that the light field distribution close to the front surface of the rRMLA1 is constant 1, that is $E({y_1},{z_1}^ - ) = 1$, where z is the propagation direction of the light field. After the incident light passes through rRMLA1, the light field distribution function $E({y_1},{z_1}^ + )$ is obtained on the rear surface of rRMLA1, as shown in Eq. (2).
(2)$$E({y_1},{z_1}^ + ) = E({y_1},{z_1}^ - )\cdot t({y_1}).$$
After transmission through distance d, the field distribution $E({y_2},{z_2}^ - )$ on the front surface of rRMLA2 can be calculated by Fresnel diffraction formula, as shown in Eq. (3).
(3)$$\begin{aligned} E({y_2},{z_2}^ - ) &= \frac{1}{{i\lambda d}}\exp (ikd)\exp (ik\frac{{y_2^2}}{{2d}})\cdot \int {\left\{ {E({y_1},{z_1}^ + )\cdot \exp [i\frac{{ky_1^2}}{{2d}}]\exp ( - i2\pi \frac{{{y_2}}}{{\lambda d}}{y_1})} \right\}} d{y_1}\\ &= \frac{1}{{i\lambda d}}\exp (ikd)\exp (ik\frac{{y_2^2}}{{2d}})\cdot { {{{\cal F}}\{{E({y_1},{z_1}^ + )} \}} |_{f^{\prime} = \frac{{{y_2}}}{{\lambda d}}}} \otimes {\left. {{{\cal F}}\left\{ {\exp [i\frac{{ky_1^2}}{{2d}}]} \right\}} \right|_{f^{\prime} = \frac{{{y_2}}}{{\lambda d}}}}, \end{aligned}$$
where, ${{\cal F}}$ is Fourier transform function. The Fourier transform of the light field $E({y_1},{z_1}^ + )$ is shown in Eq. (4).
(4)$$\begin{aligned} {{\cal F}}\{{E({y_1},{z_1}^ + )} \}&= {{\cal F}}\{{t({y_1})} \}= {\left. {{{\cal F}}\left\{ {[rect(\frac{{{y_1}}}{{{p_n}}})\cdot \exp ( - ik\frac{{y_1^2}}{{2{f_n}}})] \otimes \sum\limits_{n ={-} N}^N {\delta ({{y_1} - {p_n}n} )} } \right\}} \right|_{f^{\prime} = \frac{{{y_2}}}{{\lambda d}}}}\\ &= [{p_n}\sin c({p_n}f^{\prime}) \otimes \sqrt { - i\lambda {f_n}} \exp ({i\pi \lambda {f_n} \cdot f{^{\prime}2}} )]\cdot \sum\limits_{n ={-} N}^N {\exp ({ - i2\pi {p_n}nf^{\prime}} )} \\ &= Diff({p_n},{f_n},f^{\prime})\cdot Interf(N,{p_n},f^{\prime}). \end{aligned}$$
where, the function $Diff({p_n},{f_n},f^{\prime}) = [{p_n}\sin c({p_n}f^{\prime}) \otimes \sqrt { - i\lambda {f_n}} \exp ({i\pi \lambda {f_n} \cdot f{^{\prime 2}}} )]$ describes the diffraction effect of the $nth$ sub-lens, and the function $Interf(N,{p_n},f^{\prime}) = \sum\limits_{n ={-} N}^N {\exp ({ - i2\pi {p_n}nf^{\prime}} )}$ describes the light field distribution of multi-slit interference with $2N + 1$ slits. Due to the random value of ${p_n}$ and ${f_n}$ in design, the phase and amplitude of the light field $E({y_2},{z_2}^ - )$ incident on the surface of rRMLA2 change randomly and are no longer periodic distribution. After passing through rRMLA2, the light field distribution is $E({y_2},{z_2}^ + )$, as shown in Eq. (5).
(5)$$\begin{aligned} E({y_2},{z_2}^ + ) &= E({y_2},{z_2}^ - )\cdot t({y_2}),\\ t({y_2}) &= [rect(\frac{{{y_2}}}{{{p_n}}})\cdot \exp ( - ik\frac{{y_2^2}}{{2{f_n}}})] \otimes \sum\limits_{n ={-} N}^N {\delta ({{y_1} - {p_n}n} )} . \end{aligned}$$
After passing through a Fourier lens with focal length ${f_F}$, the light field distribution $E({y_{{f_F}}},{z_{{f_F}}})$ on the FP is Fourier transform of $E({y_2},{z_2}^ + )$, as shown in Eq. (6).
(6)$$\begin{aligned} &E({y_{{f_F},}}{z_{{f_F}}}) = {{\cal F}}\{{E({{y_2},{z_2}^ + } )} \}\\ &= {{{\cal F}}\{{E({{y_2},{z_2}^ - } )} \}} |{\textrm{ }_{{f^{^{\prime\prime}}} = \frac{{{y_f}}}{{\lambda {f_F}}}}} \otimes {{{\cal F}}\{{t({{y_2}} )} \}} |{\textrm{ }_{{f^{^{\prime\prime}}} = \frac{{{y_f}}}{{\lambda {f_F}}}}}. \end{aligned}$$
According to Eqs. (3)–(5), the random ${p_n}$ and ${f_n}$ are introduced into the amplitude distribution in the final light field distribution $E({y_{{f_F}}},{z_{{f_F}}})$, so that the homogenized spot is no longer a periodic lattice distribution.
3. Design and simulations
When designing rRMLA, if the aperture and focal lengths of the sub-lenses change randomly and are not constrained, the divergence angle of the beam passing through rRMLA will not be controllable, so that the dimension and shape of the obtained homogenized spot will not be controllable. This section will introduce the design method of rRMLA and simulate its beam homogenization effect.
3.1 Control of the dimension of homogenized spot
In order to obtain a uniform spot with a target size of S, it is necessary to control the divergence angle (${\theta _n}$) of the beam after passing through rRMLA. Therefore, it is necessary to analyze the structural parameters of the sub-lens of the rRMLA in the optical path.
The overall optical path is shown in Fig. 2(a), and the value of the sub-lens aperture ${p_n}$ in rRMLA is shown in Eq. (7).
(7)$${p_n} = {p_0} \pm {C_{rand}}\cdot \Delta .$$
Fig. 2. (a) Optical path of beam homogenization. (b) Structural parameters of the sub-lens.
where, ${p_0}$ is the intermediate aperture value of the sub-lens. ${C_{rand}}$ is the random coefficient and varies randomly from $0$ to $1$. $\Delta$ is the maximum aperture variation of the sub-lens. That is, the aperture value of the sub-lens fluctuates up and down around the central value ${p_0}$, and the fluctuation value is ${\pm} {C_{rand}}\cdot \Delta$.
In the paraxial approximation, the divergence angle of the light emitted from the position ${x_i}$ of the sub-lens satisfies the equation $\tan {\theta _i} ={-} \frac{{{x_i}}}{{{f_n}}}$ (Fig. 2(b)). When ${x_i} = \frac{{{p_n}}}{2}$, it is the maximum divergence angle of the sub-lens, and its calculation formula is Eq. (8).
(8)$$\tan {\theta _n} = \frac{{{p_n}}}{{2{f_n}}}.$$
When using rRMLA1 and rRMLA2 for beam homogenization, the sub-lenses correspond one to one, so the size S of the homogenized spot is calculated by Eq. (9), which can be derived from matrix method in optics [18]. It should be noted that the distance z between rRMLA2 and FL (Fig. 2(a)) has no effect on the size S of the final homogenized spot, so it can be adjusted freely according to the needs of practical application.
(9)$$S = {p_n}\frac{{{f_F}({f_n} + {f_n} - d)}}{{{f_n}\cdot {f_n}}},(d \le 2{f_n}).$$
Equation (10) can be obtained by calculating and simplifying in combination with Eqs. (8) and (9).
(10)$$S = 4\tan {\theta _n}{f_F}(1 - \frac{{\tan {\theta _n}d}}{{{p_n}}}).$$
According to Eqs. (7) and (10), in order to obtain a uniform spot of a certain size, the fluctuation range ${\pm} {C_{rand}}\cdot \Delta$ in the aperture need to be controlled as a relatively small value. Besides, $\tan {\theta _n}$ needs to be controlled as a fixed value, that is ${\theta _n}$ is a fixed value such as ${\theta _0}$. Under this constraint, the surface profile of the sub-lens in rRMLA can be further designed.
From the knowledge of geometric optics [19], the focal length of the plano convex lens is shown in Eq. (11).
(11)$${f_n} = \frac{{{R_n}}}{{{n_\lambda } - 1}}.$$
where, ${R_n}$ is the curvature radius of the lens and ${n_\lambda }$ is the refractive index of the optical material at the wavelength of $\lambda$. Combining Eqs. (8) and (11), the curvature radius expression of the sub-lens can be obtained, as shown in Eq. (12).
(12)$${R_n} = \frac{{{p_n}({n_\lambda } - 1)}}{{2\tan {\theta _0}}}.$$
In the practical design of rRMLA, we control the diagonal divergence angle of each sub-lens to a unified fixed value. The curvature radius of the sub-lens can be calculated according to Eq. (12). At this time, the ${p_n}$ value is $\sqrt {p_{n - x}^2 + p_{n - y}^2}$, where ${p_{n - x}}$ is the aperture size in the x direction and ${p_{n - y}}$ is the aperture size in y the direction.
The surface profile function of the sub-lens can be calculated by Eq. (13).
(13)$$sa{g_n}({x,y} )= \frac{{\frac{1}{{{R_n}}}({x^2} + {y^2})}}{{1 + \sqrt {1 - \frac{1}{{{R_n}^2}}({x^2} + {y^2})} }},\textrm{ }\left\{ \begin{array}{l} - \frac{{{p_{n - x}}}}{2} \le x \le \frac{{{p_{n - x}}}}{2},\\ - \frac{{{p_{n - y}}}}{2} \le y \le \frac{{{p_{n - y}}}}{2}. \end{array} \right.$$
When the sub-lens surface function is known, the phase of the sub-lens can be calculated, as shown in Eq. (14).
(14)$${\Phi _n}(x,y) = \frac{{2\pi }}{\lambda }({n_\lambda } - 1)\cdot sa{g_n}(x,y).$$
It can be seen from Eq. (14) that the random change of the surface profile of the sub-lens leads to the random change of phase ${\Phi _n}$, which leads to the random phase difference between the divided sub-beams. It is further proved that the design method disrupts the interference conditions between the sub-beams divided by the sub-lenses, so that the homogenized spot pattern is no longer a periodic lattice.
3.2 Dense arrangement of rRMLA
Since the shape of the homogenization spot is determined by the aperture shape of the sub-lens in rRMLA. In order to obtain a rectangular homogenized spot, all of the apertures of the sub-lenses must be rectangular. Therefore, it is necessary to study how to realize the dense arrangement of rectangular sub-lenses with different aperture sizes to achieve high filling factor. In this paper, the iterative segmentation method is proposed to realize the design of rRMLA.
Assuming that the intermediate value of the aperture of each sub-lens in the direction x and direction y are both ${p_0}$. During iterative segmentation, the rectangular window with area $M{p_0} \times M{p_0}$ ($M$ is a positive integer) is cut into the array with the array number of $M \times M$. The aperture value of the sub-lens fluctuates around the intermediate value ${p_0}$ in the direction x and direction y, and the fluctuation range is ${\pm} {C_{rand}}\cdot \Delta$. The overall segmentation principle is shown in Fig. 3. Since the MLA with the number of $2 \times 2$ can be generated by each segmentation, the total number of iterations satisfy $num = {\log _2}M$. During segmentation, the median line of the rectangular window is taken as the benchmark and the random offset value for cutting meets $Offse{t_i} ={\pm} \frac{{{C_{rand}}\Delta }}{{{2^{num - k + 1}}}}$, where i is the number of offset corresponding to the number of cutting line, k is the iterative order, and the maximum value of k is $num$.
Fig. 3. Iterative segmentation principle: (a) First iteration; (b) Second iteration; (c) Third iteration; (d) $kth$ iteration.
In the first iteration, the rectangular window with area $M{p_0} \times M{p_0}$ is divided into rRMLA with array number of $2 \times 2$, as shown in Fig. 3(a). In this process, the rectangular window is divided for three times, and the cutting lines are cut1, cut2 and Cut3 respectively to generate sub-lenses with aperture shapes of A1, A2, A3 and A4. The values of the offset between the cutting line and the median lines of Mid1 and Mid1 equal to ${\pm} \frac{{{C_{rand}}\Delta }}{{{2^{num - 1 + 1}}}}$, so the apertures of the sub-lenses after the first iteration satisfy $p{x_1} = \frac{{M{p_0}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - 1 + 1}}}}$.
The second iteration is based on the completion of the first iteration, and the rectangular window with area $M{p_0} \times M{p_0}$ is further divided into rRMLA with array number of $4 \times 4$, as shown in Fig. 3(b). The apertures of the sub-lenses after the second iteration satisfy $p{x_2} = \frac{{{p_1}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - 2 + 1}}}}$.
By analogy, the third iteration is to further divide the rectangular window with area $M{p_0} \times M{p_0}$ into rRMLA with array number of $8 \times 8$ based on the completion of the second iteration, as shown in Fig. 3(c). The apertures of the sub-lenses satisfy $p{x_3} = \frac{{{p_2}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - 3 + 1}}}}$.
The $kth$ iteration is based on the completion of the $(k - 1)th$ iteration, and the rectangular window with area $M{p_0} \times M{p_0}$ is further divided into rRMLA with array number of ${2^k} \times {2^k}$, as shown in Fig. 3(d). The apertures of the sub-lenses satisfy $p{x_k} = \frac{{{p_{k - 1}}}}{2} \pm \frac{{{C_{rand}}\Delta }}{{{2^{num - k + 1}}}}$.
When $k = num$, the segmentation is completed, and the apertures of the sub-lenses in direction x and direction y are a series of random values around ${p_0}$. It can be seen that the iterative segmentation method can complete the dense arrangement between sub-lenses with different apertures on the basis of controlling the random variation range, so that the filling factor of the structure can reach $100\%$.
3.3 Simulations
Based on the above design method, the MLAs with a fixed divergence angle ${\theta _0}$ in the diagonal direction of $4^\circ$ and the number of arrays of $8 \times 8$ were designed. The sub-lens apertures of the periodic MLA are $500\mu m \times 500\mu m$, and the sub-lens apertures of the rRMLA are random values that fluctuate around ${p_0} = 500\mu m$ in the direction x and direction $y$ with fluctuation range of ${\pm} {C_{rand}}\cdot 50\mu m$. The design results are shown in Fig. 4. The surface profile and phase of the periodic MLA are periodically distributed (Fig. 4(a)-(c)), while the sub-lenses in rRMLA are randomly distributed (Fig. 4(d)-(f)). Due to the different apertures of the sub-lenses in rRMLA, the curvature radii of the sub-lenses are different under the condition of unchanged ${\theta _0}$ according to Eq. (12), resulting in different sag heights (Fig. 4(e)) and phase distributions (Fig. 4(f)).
Fig. 4. Design results: (a), (b) and (c) are the top view, three-dimensional view and phase view of the periodic MLA, respectively; (d), (e) and (f) are top view, three-dimensional view and phase view of rRMLA, respectively.
Furthermore, the numerical analysis software is used to simulate and compare the homogenization effect. During simulation, the focal length of Fourier lens was set as ${f_F} = 15cm$, and the distance between the two rRMLAs was selected as $d = 5.5mm$. After the beam was modulated by the two types of periodic MLA and rRMLA respectively, the homogenized spots on the focal plane of the Fourier lens were obtained with a size of $13.6mm \times 13.6mm$, as shown in Fig. 5. It can be seen that the light field distribution after periodic MLA homogenization is periodic strong dot distribution (Fig. 5(a) and 5(b)), while the light field distribution after rRMLA modulation is random distribution pattern (Fig. 5(c) and 5(d)), indicating that rRMLA can eliminate the phenomenon of periodic strong dot distribution to improve the uniformity of homogenized spot.
Fig. 5. Simulation results: (a) and (b) are the homogenized spot and energy profile generated by the periodic MLA, respectively; (c) and (d) are the homogenized spot and energy profile generated by the designed rRMLA, respectively.
4. Experiments
In order to verify the simulation results, the preparation of periodic MLA and rRMLA were carried out. The specific preparation principle can refer to the previous researches of our group [20]. The preparation results are shown in Fig. 6(a) and 6(b) respectively, and the beam homogenization experiments have been carried out with the prepared structure. The MLA in Fig. 6(a) is a periodic structure, the apertures and focal lengths of different sub-lenses are the same, as shown in Fig. 6(c). The apertures are the same as $500\mu m$ and the measured sag heights of sub-lenses are the same as $7.52\mu m$. According to Eqs. (11) and (13), the focal length can be calculated as $4.94mm$. The divergence angle in the diagonal direction can be calculated as $4.\textrm{09}^\circ$ according to Eq. (8), which is basically consistent with the design parameters. The apertures and focal lengths of different sub-lenses are the same, and the probability is $1$. The rRMLA in Fig. 6(b) is a random structure, the apertures and focal lengths of different sub-lenses are different. So, the sag heights of different sub-lenses are different as shown in Fig. 6(d). According to Eqs. (8), (11) and (13), the focal length can be calculated as $4.66mm$, $5.00mm$ and $4.64mm$, and the divergence angle in the diagonal direction can be calculated as $4.16^\circ$, $4.15^\circ$ and $4.29^\circ$. The divergence angle is approximate to the design value $4^\circ$ with a slight deviation, which may have a certain impact on the final size of the homogenized spot. The probability distribution of the apertures is determined by the mask. Therefore, when the mask is designed, the probability distribution of the apertures in the prepared rRMLA will be determined. Since the designed mask was converted according to the design parameters, the aperture distribution of the prepared rRMLA must be consistent with the design parameters. The probability distribution of focal lengths can be obtained by indirectly measuring the sag height of sub-lenses. It is calculated that the focal lengths of sub-lenses with different apertures are different. However, according to Eq. (8), it is found that the relationship between aperture and focal length follows the rule that the divergence angle in the diagonal direction is a fixed value close to $4^\circ$. Therefore, it can be considered that the probability distribution of focal length is consistent with that of aperture.
Fig. 6. Preparation results of periodic MLA and rRMLA: (a) and (c) are microscopic enlarged view and cross-section profile of periodic MLA, respectively; (b) and (d) are the microscopic enlarged view and cross-section profile of rRMLA, respectively.
In the experiment, a laser beam with wavelength of $650nm$ was used as the incident beam. The focal length of the Fourier lens was consistent with the simulation parameters, which was $15cm$. After the beam was phase modulated by the MLA, the homogenized spot was obtained on the focal plane of the Fourier lens and collected by CCD, as shown in Fig. 7(a) and 7(c), respectively. In the experiment, the misalignment was avoided by observing the homogenized spot in the CCD. Further, the collected spots were analyzed by using the numerical analysis software. When the periodic MLA is used for laser beam homogenization, the spot distribution is consistent with the simulation results, which is a typical periodic lattice distribution (Fig. 7(b)), and the spot size is $14.3mm$, which is calculated by multiplying the pixel number occupied by the homogenized spot by the CCD pixel size. The pixel number is $1934$ and the pixel size is $7.4\mu m$. When the designed rRMLA is used for laser beam homogenization, the lattice of homogenized spot is effectively eliminated (Fig. 7(d)), and the spot size is $14.1mm$ calculated by $1907 \times 7.4\mu m$. The results of homogenization experiment are consistent with the simulation results, which proves that the homogenized spot of rRMLA has better uniformity. However, there is a deviation in the size of the homogenized spot, which is mainly due to the fact that the distance between the two MLA cannot be accurately controlled and measured in the optical test experiments. Although there is a certain deviation in the size of the homogenized spot, it does not have a fundamental impact on the verification of the design principle of the whole article.
Fig. 7. Homogenization effect of periodic MLA and rRMLA: (a) and (b) are the homogenization spot and energy profile of periodic MLA, respectively; (c) and (d) are the homogenized spot and energy profile of rRMLA, respectively.
Next, the homogenized spot formed by this method is evaluated quantitatively. If the uniformity is evaluated by using the root mean square error (RMSE) of the traditional evaluation function, the uniformity of the homogenized spot of the periodic MLA and rRMLA captured in the experiment is $82.9\%$ and $80.6\%$, respectively. The calculation results are not very different, so the homogenized spots with these two different characteristics cannot be well identified and distinguished. Therefore, this paper proposes to use the effective speckle density $\rho$ to evaluate the uniformity of the homogenized spot. That is, the proportion of the area with normalized energy I greater than ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{e^2}}}} \right.}\!\lower0.7ex\hbox{${{e^2}}$}}$ in the overall homogenized spot is counted. The evaluation method is shown in Eq. (15). After beam homogenization by periodic MLA and rRMLA, the effective speckle density is $17.1\%$ and $96.9\%$, respectively. It can be seen that when the laser beam is homogenized by rRMLA, the effective speckle density and uniformity in the overall homogenized spot are higher.
(15)$$\rho = \frac{{{S_{(I > {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{e^2}}}} \right.}\!\lower0.7ex\hbox{${{e^2}}$}})}}}}{S} \times 100\%.$$
In Section 3.1, it has beam analyzed that, the divergence angle in the diagonal direction of each sub-lens is controlled to be a unified fixed value ${\theta _0}$ in the actual design of rRMLA. The influence of divergence angle constraints on the structural parameters of the overall rRMLA will be discussed below.
If the divergence angle in the diagonal direction of each sub-lens is controlled to be a fixed value, the random probability distribution of the focal length of the sub-lens will be consistent with the aperture in the diagonal direction, as shown in Fig. 8(a) and 8(b). The larger the aperture, the longer the focal length will, while the ratio of aperture to focal length is fixed. If the divergence angle is taken randomly, the focal length and aperture of the sub lens will obey different probability distribution functions, as shown in Fig. 8(c) and 8(d). The homogenized spot will be as shown in Fig. 8(e). In this case, the energy distribution of the homogenized spot is uneven. The middle energy of the spot is strong while the edge energy is weak, and the size of the homogenized spot is not controllable.
Fig. 8. (a) and (b) show the probability distribution of aperture and focal length when the divergence angle of the sub-lens in rRMLA is fixed. (c), (d) and (e) are the probability distribution of aperture and focal length and homogenized spot respectively when the divergence angle is a random value.
It can be seen from the above analysis that the divergence angle is a very important parameter in the design process of rRMLA. Next, the influence of the change of divergence angle on the homogenization effect will be analyzed. The divergence angle was changed and the homogenization effect was evaluated by using the effective speckle density. The results are shown in Fig. 9. From the overall trend, with the increase of divergence angle, the effective speckle density increases and the uniformity of spot improves. When the divergence angle satisfies $\tan {\theta _0} < 0.05$, the speckle density increases linearly with the divergence angle. When the divergence angle satisfies $\tan {\theta _0} \ge 0.05$, the speckle density tends to be stable. It can be seen that when using the design method in this paper to improve the beam homogenization effect, the divergence angle of rRMLA is designed to be preferably greater than $0.05$.
Fig. 9. Effect of variation of divergence angle on homogenization effect.
Meanwhile, different array numbers illuminated to yield sufficient smoothing have been discussed, as shown in Fig. 10. When the number of arrays is greater than $7 \times 7$, the periodicity of homogenization spot is completely broken, and a better homogenization effect can be obtained.
Fig. 10. Homogenized spot with different array numbers of rRMA illuminated: (a) $2 \times 2$ illuminated; (b) $3 \times 3$ illuminated; (c) $4 \times 4$ illuminated; (d) $5 \times 5$ illuminated; (e) $6 \times 6$ illuminated; (f) $7 \times 7$ illuminated; (g) $8 \times 8$ illuminated.
In this paper, a novel rRMLA is designed to homogenize the laser beam. By designing the MLA with random apertures and random focal lengths, the phase regulation of the laser beam is realized, so as to disturb the coherent superposition fringes between the sub-beams divided by MLA, improving the uniformity of the homogenized spot. By studying the dense arrangement method of random rectangular aperture MLA, the structure with controllable divergence angle is designed and verified by simulations. Finally, the prepared rRMLA has been used to test the homogenization effect. The effective speckle density of the homogenized spot is increased from $17.1\%$ to $96.9\%$.
National Natural Science Foundation of China (61905251).
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
1. A. Bewsher, I. Powell, and W. Boland, "Design of single-element laser-beam shape projectors," Appl. Opt. 35(10), 1654–1658 (1996). [CrossRef]
2. B. Kondász, B. Hopp, and T. Smausz, "Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern," J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018). [CrossRef]
3. J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27(16), 1463–1465 (2002). [CrossRef]
4. S. Tao and W. Yu, "Beam shaping of complex amplitude with separate constraints on the output beam," Opt. Express 23(2), 1052–1062 (2015). [CrossRef]
5. Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, "Shape the unstable laser beam using diffractive optical element array," Proc. SPIE 7848, 78481X (2010). [CrossRef]
6. W. Veldkamp and T McHugh, "Binary optics," Sci. Am. 266(5), 92–97 (1992). [CrossRef]
7. R. Voelkel and K. J. Weible, "Laser Beam Homogenizing: Limitations and Constraints," Proc. SPIE 7102, 71020 (2008). [CrossRef]
8. F. M. Dickey, Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC, 2014, Chap. 8).
9. M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, "Microlens laser beam homogenizer - from theory to application," Proc. SPIE 6663, 666302 (2007). [CrossRef]
10. A. S. Victoria, K. K. Galina, M. S. Maxim, A. Z Roman, and B. Y Evgeniy, "Speckle-free smoothing of coherence laser beams by a homogenizer on uniaxial high birefringent crystal," Opt. Mater. Express 9(5), 2392–2399 (2019). [CrossRef]
11. F. Wippermann, UD. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, "Beam homogenizers based on chirped microlens arrays," Opt. Express 15(10), 6218–6231 (2007). [CrossRef]
12. Y. Jin, A. Hassan, and Y. Jiang, "Freeform microlens array homogenizer for excimer laser beam shaping," Opt. Express 24(22), 24846–24858 (2016). [CrossRef]
13. Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, "A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses," Optics Communications 443, 211–215 (2019). [CrossRef]
14. W. Zhang, L. Xia, M. Gao, and C. Du, "Laser beam homogenization with randomly distributed freeform cylindrical microlens," Opt. Eng. 59(6), 065103 (2020). [CrossRef]
15. L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, "Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency," Micromachines 11(3), 338–342 (2020). [CrossRef]
16. W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, "Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization," Micromachines 12(6), 673–678 (2021). [CrossRef]
17. J. W. Goodman, Introduction To Fourier Optics (Roberts & Company Publishers, 2004, Chap. 3).
18. A. Gerrard and J. M. Burch, "Introduction to Matrix Methods in Optics," Am. J. Phys. 44(8), 338–341 (1976).
19. K Iizuka, Engineering Optics (Springer, 1985, pp. 105–144).
20. W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, "Fabrication of Multifocal Microlens Array by One Step Exposure Process," Micromachines 12(9), 1097–1102 (2021). [CrossRef]
A. Bewsher, I. Powell, and W. Boland, "Design of single-element laser-beam shape projectors," Appl. Opt. 35(10), 1654–1658 (1996).
B. Kondász, B. Hopp, and T. Smausz, "Homogenization with coherent light illuminated beam shaping diffusers for vision applications: spatial resolution limited by speckle pattern," J. Eur. Opt. Soc.-Rapid Publ. 14(1), 27 (2018).
J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27(16), 1463–1465 (2002).
S. Tao and W. Yu, "Beam shaping of complex amplitude with separate constraints on the output beam," Opt. Express 23(2), 1052–1062 (2015).
Y. Li, C. Qiu, P. Li, T. Xing, W. Lin, and C. Zhou, "Shape the unstable laser beam using diffractive optical element array," Proc. SPIE 7848, 78481X (2010).
W. Veldkamp and T McHugh, "Binary optics," Sci. Am. 266(5), 92–97 (1992).
R. Voelkel and K. J. Weible, "Laser Beam Homogenizing: Limitations and Constraints," Proc. SPIE 7102, 71020 (2008).
F. M. Dickey, Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC, 2014, Chap. 8).
M. Zimmermann, N. R. V. Lindlein, and K. J. Weible, "Microlens laser beam homogenizer - from theory to application," Proc. SPIE 6663, 666302 (2007).
A. S. Victoria, K. K. Galina, M. S. Maxim, A. Z Roman, and B. Y Evgeniy, "Speckle-free smoothing of coherence laser beams by a homogenizer on uniaxial high birefringent crystal," Opt. Mater. Express 9(5), 2392–2399 (2019).
F. Wippermann, UD. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, "Beam homogenizers based on chirped microlens arrays," Opt. Express 15(10), 6218–6231 (2007).
Y. Jin, A. Hassan, and Y. Jiang, "Freeform microlens array homogenizer for excimer laser beam shaping," Opt. Express 24(22), 24846–24858 (2016).
Z. Liu, H. Liu, L. Z. Lu, Q. Li, and J. Lu, "A beam homogenizer for digital micromirror device lithography system based on random freeform microlenses," Optics Communications 443, 211–215 (2019).
W. Zhang, L. Xia, M. Gao, and C. Du, "Laser beam homogenization with randomly distributed freeform cylindrical microlens," Opt. Eng. 59(6), 065103 (2020).
L. Xue, Y. Pang, W. Liu, L. Liu, H. Pang, A. Cao, L. Shi, Y. Fu, and Q. Deng, "Fabrication of Random Microlens Array for Laser Beam Homogenization with High Efficiency," Micromachines 11(3), 338–342 (2020).
W. Yuan, C. Xu, L. Xue, H. Pang, A. Cao, Y. Fu, and Q. Deng, "Integrated Double-Sided Random Microlens Array Used for Laser Beam Homogenization," Micromachines 12(6), 673–678 (2021).
J. W. Goodman, Introduction To Fourier Optics (Roberts & Company Publishers, 2004, Chap. 3).
A. Gerrard and J. M. Burch, "Introduction to Matrix Methods in Optics," Am. J. Phys. 44(8), 338–341 (1976).
K Iizuka, Engineering Optics (Springer, 1985, pp. 105–144).
W. Yuan, Y. Cai, C. Xu, H. Pang, A. Cao, Y. Fu, and Q. Deng, "Fabrication of Multifocal Microlens Array by One Step Exposure Process," Micromachines 12(9), 1097–1102 (2021).
Bewsher, A.
Boland, W.
Bräuer, A.
Burch, J. M.
Cai, Y.
Cao, A.
Dannberg, P.
Deng, Q.
Dickey, F. M.
Du, C.
Evgeniy, B. Y
Fu, Y.
Galina, K. K.
Gao, M.
Gerrard, A.
Goodman, J. W.
Hassan, A.
Hopp, B.
Iizuka, K
Jiang, Y.
Jin, Y.
Kondász, B.
Li, P.
Li, Q.
Li, Y.
Lindlein, N. R. V.
Liu, J. S.
Liu, L.
Liu, W.
Liu, Z.
Lu, J.
Lu, L. Z.
Maxim, M. S.
McHugh, T
Pang, H.
Pang, Y.
Powell, I.
Qiu, C.
Roman, A. Z
Shi, L.
Sinzinger, S.
Smausz, T.
Taghizadeh, M. R.
Tao, S.
Veldkamp, W.
Victoria, A. S.
Voelkel, R.
Weible, K. J.
Wippermann, F.
Xia, L.
Xing, T.
Xu, C.
Xue, L.
Yu, W.
Yuan, W.
Zeitner, UD.
Zhang, W.
Zhou, C.
Am. J. Phys. (1)
J. Eur. Opt. Soc.-Rapid Publ. (1)
Micromachines (3)
Opt. Eng. (1)
Opt. Lett. (1)
Opt. Mater. Express (1)
Optics Communications (1)
Proc. SPIE (3)
Sci. Am. (1)
Equations on this page are rendered with MathJax. Learn more.
(1) t ( y 1 ) = [ r e c t ( y 1 p n ) ⋅ exp ( − i k y 1 2 2 f n ) ] ⊗ ∑ n = − N N δ ( y 1 − p n n ) .
(2) E ( y 1 , z 1 + ) = E ( y 1 , z 1 − ) ⋅ t ( y 1 ) .
(3) E ( y 2 , z 2 − ) = 1 i λ d exp ( i k d ) exp ( i k y 2 2 2 d ) ⋅ ∫ { E ( y 1 , z 1 + ) ⋅ exp [ i k y 1 2 2 d ] exp ( − i 2 π y 2 λ d y 1 ) } d y 1 = 1 i λ d exp ( i k d ) exp ( i k y 2 2 2 d ) ⋅ F { E ( y 1 , z 1 + ) } | f ′ = y 2 λ d ⊗ F { exp [ i k y 1 2 2 d ] } | f ′ = y 2 λ d ,
(4) F { E ( y 1 , z 1 + ) } = F { t ( y 1 ) } = F { [ r e c t ( y 1 p n ) ⋅ exp ( − i k y 1 2 2 f n ) ] ⊗ ∑ n = − N N δ ( y 1 − p n n ) } | f ′ = y 2 λ d = [ p n sin c ( p n f ′ ) ⊗ − i λ f n exp ( i π λ f n ⋅ f ′ 2 ) ] ⋅ ∑ n = − N N exp ( − i 2 π p n n f ′ ) = D i f f ( p n , f n , f ′ ) ⋅ I n t e r f ( N , p n , f ′ ) .
(5) E ( y 2 , z 2 + ) = E ( y 2 , z 2 − ) ⋅ t ( y 2 ) , t ( y 2 ) = [ r e c t ( y 2 p n ) ⋅ exp ( − i k y 2 2 2 f n ) ] ⊗ ∑ n = − N N δ ( y 1 − p n n ) .
(6) E ( y f F , z f F ) = F { E ( y 2 , z 2 + ) } = F { E ( y 2 , z 2 − ) } | f ′ ′ = y f λ f F ⊗ F { t ( y 2 ) } | f ′ ′ = y f λ f F .
(7) p n = p 0 ± C r a n d ⋅ Δ .
(8) tan θ n = p n 2 f n .
(9) S = p n f F ( f n + f n − d ) f n ⋅ f n , ( d ≤ 2 f n ) .
(10) S = 4 tan θ n f F ( 1 − tan θ n d p n ) .
(11) f n = R n n λ − 1 .
(12) R n = p n ( n λ − 1 ) 2 tan θ 0 .
(13) s a g n ( x , y ) = 1 R n ( x 2 + y 2 ) 1 + 1 − 1 R n 2 ( x 2 + y 2 ) , { − p n − x 2 ≤ x ≤ p n − x 2 , − p n − y 2 ≤ y ≤ p n − y 2 .
(14) Φ n ( x , y ) = 2 π λ ( n λ − 1 ) ⋅ s a g n ( x , y ) .
(15) ρ = S ( I > 1 / 1 e 2 e 2 ) S × 100 % .
James Leger, Editor-in-Chief | CommonCrawl |
\begin{document}
\title{Almost reducibility of quasiperiodic $SL(2,\mathbb{R})$-cocycles in ultradifferentiable classes
}
\author{Maxime Chatal\footnote{ IMJ-PRG, Universit\'e Paris-Cit\'e, [email protected]}\\ and\\
Claire Chavaudret\footnote{IMJ-PRG, Universit\'e Paris-Cit\'e, [email protected]} }
\maketitle
\textbf{Abstract:} Given a quasiperiodic cocycle in $sl(2, \mathbb{R})$ sufficiently close to a constant, we prove that it is almost-reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. We also give a corollary on the H\" older regularity of the Lyapunov exponent.
\section{Introduction}
\subsection{Presentation of the result} Let $d\geq 1$ and $\omega = (\omega_1, \dots, \omega_d) \in \mathbb{R}^d$ a rationally independent vector (meaning that no non trivial integer combination of the $(\omega_i)_{ i=1,\dots , d}$ can vanish). We will assume that $\sup_i \vert \omega_i \vert \leq 1$. We will note $\mathbb{T} ^d := \mathbb{R}^d/\mathbb{Z}^d$ and $2\mathbb{T} ^d := \mathbb{R}^d / 2\mathbb{Z}^d$. Let $A : \mathbb{T} ^d \rightarrow sl(2, \mathbb{R})$ be in a certain class of continuous matrix-valued functions. We call quasi-periodic cocycle the solution $X : \mathbb{T} ^d \times \mathbb{R} \rightarrow SL(2, \mathbb{R})$ of the differential linear equation
\begin{equation} \left\{\begin{array}{l} \frac{\mathrm{d}}{\mathrm{d}t} X^t(\theta) = A(\theta+t\omega)X^t(\theta) \\X^0(\theta) = Id\end{array}\right. \tag{1} \label{eq:cocycle} \end{equation}
One of the main motivations for studying quasi-periodic cocycles is the study of quasi-periodic Schr\" odinger equations \[ -y''(t) + q(\theta +t \omega)y(t) = Ey(t) \] where $q:\mathbb{T}^d\rightarrow \mathbb{R}$ is called the potential, and $E\in\mathbb{R}$ the energy. It gives rise to a cocycle with values in $SL(2,\mathbb{R})$. The cocycle is said to be a constant cocycle if $A$ is a constant matrix. A quasi-periodic cocycle as in \eqref{eq:cocycle} is said reducible if it can be conjugated by a quasi-periodic change of variable $Z : \mathbb{T} ^d \rightarrow SL(2,\mathbb{R})$ to a constant cocycle, that is to say, if there exists $B \in sl(2,\mathbb{R})$ such that, for all $\theta \in 2 \mathbb{T} ^d$ : \[ \partial_\omega Z(\theta) = A(\theta)Z(\theta) - Z(\theta)B\]
In general, it is important to require the change of variables $Z$ to be regular enough. Is this paper, we will be interested in the perturbative setting, that is to say, in quasi-periodic cocycles close to a constant :
\begin{equation}\label{perturbative} \left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t} X^t(\theta) = (A + F(\theta + t\omega))X^t(\theta) \\X^0(\theta) = Id\end{array}\right. \tag{2}\end{equation}
\noindent where $A\in sl(2, \mathbb{R})$ and $F : \mathbb{T} ^d \rightarrow sl(2, \mathbb{R})$ is of ultra-differentiable class and small enough, with a smallness condition depending on $\omega$.
Reducibility is a strong property because it implies that the dynamics will be easily described by the constant equivalent of the system, in particular the Lyapunov exponents, the rotational properties of the solutions, the invariant subbundles etc. On the counterpart, reducibility results generally require many assumptions. Here we are interested in a weaker property which is almost reducibility. A cocycle like \eqref{perturbative} is said almost-reducible if it can be conjugated by a sequence of quasi-periodic changes of variables to a cocycle of the form \[ \left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t} X^t(\theta) = (\bar A_n(\theta + t\omega) + \bar F_n(\theta +t\omega)) X^t(\theta) \\X^0(\theta) = Id\end{array}\right. \] where $\bar{A}_n$ is reducible and $\bar{F}_n$ is arbitrarily small.
\noindent A quantitative version of almost reducibility, that is, almost reducibility together with estimates on the changes of variable, can have interesting corollaries such as approximate solutions, density of reducible cocycles, regularity of Lyapunov exponents.
\textbf{Ultra-differentiability :} To quantify the regularity of $F \in \mathcal C^{\infty} (\mathbb{T} ^d, sl(2, \mathbb{R}))$ and the size of the sequence $(\bar{F}_n)$ above, we introduce the weight function $\Lambda : [0, +\infty[ \rightarrow [0, + \infty[$ which we will assume to be increasing and differentiable. Expanding $F$ in Fourier series $F(\theta) = \sum_{k\in \mathbb{Z}^d} \hat F(k)e^{2i\pi \langle k, \theta \rangle}$, we will say that $F$ is $\Lambda$-ultra-differentiable if there exists $r > 0$ such that \[ \vert F \vert_{r} = \vert F \vert_{\Lambda, r} := \sum_{k \in \mathbb{Z}^d} \Vert \hat F(k) \Vert e^{2\pi \Lambda (\vert k \vert)r} < \infty \] where $\vert k\vert$ is the sum of the absolute values of the components of $k$, and we will denote $F\in U_{r}(\mathbb{T} ^d, sl(2, \mathbb{R})) = U_{\Lambda, r}(\mathbb{T} ^d, sl(2, \mathbb{R}))$. To make this space a Banach algebra, we will require $\Lambda$ to be subadditive : \[ \Lambda (x+y) \leq \Lambda (x) + \Lambda (y), \quad \forall x,y \geq 0\]
\noindent If $\Lambda \equiv id$, it is the analytic case.
\begin{remark} The standard definition of ultra-differentiable functions involves Denjoy-Carleman sequences, that is, real sequences satisfying certain conditions which act as bounds on the successive derivatives of a given function. However, the above definition, introduced by Braun-Meise-Taylor (\cite{BMT}), can be linked to Denjoy-Carleman classes (see \cite{Ra21}, Theorem 11.6). Since Fourier series appear naturally in the problem considered here, we chose to use Braun-Meise-Taylor classes as a starting point. \end{remark}
\textbf{Non-resonance condition on the frequency:} An often studied situation is the case where the frequency vector $\omega$ if Diophantine (which we denote by $\omega \in DC(\kappa, \tau)$), for some $0 < \kappa <1$ and $\tau \geq \max(1, d-1)$ : \[ \vert \langle k, \omega \rangle \vert \geq \frac{\kappa}{\vert k \vert^\tau}, \quad \forall k \in \mathbb{Z}^d \backslash \{0 \}\] where $\langle \cdot, \cdot\rangle$ is the standard Euclidean inner product. It was proved by Eliasson \cite{3} that in the analytic case, if $\omega \in DC(\kappa, \tau)$, and $F$ is sufficiently small, Equation \eqref{perturbative} is almost reducible. This result was improved by Chavaudret \cite{1} who proved that the convergence occurs on analyticity strips of fixed width (whereas Eliasson's theorem gave the convergence on strips of width going to zero).
One of the aims of the present paper is to weaken this arithmetic condition by introducing the approximating function \[ \Psi : [0, + \infty[ \rightarrow [0, +\infty[ \] with $\Psi \geq id$ (which is not restrictive since it is satisfied by the diophantine condition). We will assume $\Psi$ to be increasing, differentiable and satisfying, for all $x, y \in [1, +\infty[$, \[\Psi(x+y) \geq \Psi(x)+\Psi(y)\]
\noindent thus for all $n\in\mathbb{N}$, and for all $x\geq 1$, $\Psi(nx)\geq n\Psi(x)$. In our problem, $\omega$ will satisfy the following arithmetic condition for some $\kappa \in ]0, 1[$ : \[ \vert \langle k, \omega \rangle \vert \geq \frac{\kappa}{\Psi(\vert k \vert)}, \quad \forall k \in \mathbb{Z}^d \backslash \{0 \}\] (notice that the case $\Psi(.) = \vert . \vert^\tau$, is the Diophantine case).
We will require the following condition: \[\lim_{t \rightarrow +\infty}\frac{\log\Psi(t)}{\Lambda(t)} = 0.\] and $$\int_0^\infty \frac{\Lambda'(t)\ln \Psi(t)}{\Lambda(t)^2}dt<+\infty$$
This condition, known as the $\Lambda$-Brjuno-R\" ussmann condition, will be denoted by $\omega \in BR(\kappa)$. This coincides with the well-known Brjuno condition if $\Lambda$ is the identity.
If $A$ is elliptic, an almost reducibility theorem was given in \cite{2}.
The purpose of this article is to show the following theorem:
\begin{theorem}\label{maintheorem} Let $r_0 >0$, $A_0\in sl(2, \mathbb{R})$ and $F_0 \in U_{r_0}(\mathbb{T} ^d, sl(2, \mathbb{R}))$. Then, there exists $\varepsilon_0$ depending only on $A_0,\kappa,\Lambda,\Psi,r_0$
such that, if \[ \vert F_0 \vert_{r_0} \leq \varepsilon_0\] then for all $\varepsilon \leq \varepsilon_0$, there exist
\begin{itemize} \item $r_\varepsilon >0$, $\zeta \in ]0,\frac{1}{8}[$, \item $Z_\varepsilon \in U_{r_\varepsilon}( \mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $A_\varepsilon \in sl(2, \mathbb{R})$, \item $\bar A_\varepsilon, \bar F_\varepsilon \in U_{r_\varepsilon}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, \item $\psi_\varepsilon \in U_{r_{\varepsilon}}(2\mathbb{T} ^d, SL(2, \mathbb{R}))$ \end{itemize} such that \begin{enumerate} \item $\bar A_\varepsilon$ is reducible to $A_\varepsilon$ by $\psi_\varepsilon$, with $\vert \psi_\varepsilon \vert_{r_\varepsilon} \leq \varepsilon^{-\frac{1}{2}\zeta}$, \item $\vert \bar F_\varepsilon \vert_{r_\varepsilon} \leq \varepsilon$, \item $\lim_{\varepsilon \rightarrow 0} r_\varepsilon >0$, \item for all $\theta \in \mathbb{T} ^d$, \[ \partial_\omega Z_\varepsilon (\theta) = (A_0 + F_0(\theta)) Z_\varepsilon (\theta) - Z_\varepsilon (\theta)(\bar A_\varepsilon (\theta) + \bar F_\varepsilon (\theta))\] \item \[ \vert Z_\varepsilon ^{\pm 1} - Id \vert_{r_\varepsilon} \leq \varepsilon_0^{\frac{9}{10}} \] \end{enumerate}
Moreover, either $\Psi_\epsilon$ becomes constant as $\epsilon\rightarrow 0$, or there exist arbitrarily small $\varepsilon$ such that
$||A_\varepsilon||\leq \kappa \varepsilon^\zeta$. \end{theorem}
This theorem states almost reducibility in an ultradifferentiable class with the same weight function as that of the initial system, but with a smaller parameter $r_\varepsilon$. Notice however that the parameter $r_\varepsilon$ does not shrink to $0$. In order to achieve this, the resonance cancellation technique is similar to the one in \cite{1}. Notice that, for topological reasons, a period doubling is necessary in order to preserve the real structure. This phenomenon was already observed in \cite{C1}.
\subsection{Discussion}
Almost reducibility in itself is an interesting property of quasi-periodic cocycles, in particular the quantitative version. A perturbative almost reducibility result in arbitrary dimension of space was given in \cite{3}, in the analytic framework, under a diophantine condition on the frequency vector. A quantitative version of perturbative almost reducibility, in a similar framework, was then proved in \cite{1}. Using a technique developed in \cite{AFK}, in \cite{HY}, Hou and You managed to remove the diophantine assumption on the frequency vector, in case it is 2-dimensional ($d=2$); thus the result became non perturbative if not global (see also \cite{Puig} for a non perturbative reducibility result). It is yet unknown whether arithmetical conditions can be removed in case $d>2$. However in the analytic case, for any number of frequencies, it is known that the diophantine condition is not optimal for reducibility results and can be replaced by the Brjuno-R\"ussmann condition (see \cite{LD06}, \cite{CM12}). Here we give an almost reducibility result in which the arithmetical condition coincides with the Brjuno-R\"ussmann condition in the analytic case.
Concerning the functional framework, a few results were known in the Gevrey class. The reference \cite{1} contains almost reducibility in the Gevrey class as well, under a diophantine condition. The reference \cite{HP} gives a result on rigidity of reducibility in the Gevrey class, under a diophantine condition (see also \cite{LDG19} on Gevrey flows). But a simultaneous extension of Eliasson's reducibility result in \cite{E92} to more general ultradifferentiable classes and to a weaker arithmetical condition, which is linked to the considered class of functions, was given in \cite{2} (see also \cite{BF} for a result in a hamiltonian setting). Here, we obtain this generalization for almost reducibility, the proof of which is more technical. The link between the arithmetical condition and the functional setting is similar to the one in \cite{2}, and also coincides with the Brjuno-R\"ussmann condition in the analytic case.
\subsection{Comments on the proof}
The proof of the main result relies on the well-known KAM algorithm: a step of the algorithm will reduce the size of the perturbation to a power of it, by means of a change of variables which might be far from identity (if resonances have to be cancelled), but is still controlled by a small negative power of the size of the perturbation. The order at which one removes resonances to avoid small divisors has to be suitably chosen in order to decrease the perturbation sufficiently while having a sufficient control on the change of variables. One also has to shrink the parameter of the ultradifferentiable class at every step, and in order to have a strong almost reducibility result (i.e a sequence of parameters not shrinking to $0$), the $\Lambda$-Brjuno-R\"ussmann condition comes naturally.
\noindent If resonances are cancelled only finitely many times, then the change of variables remains close to identity at every step afterwards, which gives reducibility. Otherwise, the constant part of the system itself becomes small.
The main theorem, which is Theorem \ref{theoreme} below, is proved by iterating arbitrarily many times the Lemma \ref{352} below; Lemma \ref{352} gives a conjugation between to systems $\bar{A}+\bar{F}$ and $\bar{A}'+\bar{F}'$, where both $\bar{A}$ and $\bar{A}'$ are reducible maps and $\bar{F}'$ is smaller than $\bar{F}$, with a controlled loss of regularity.
The proof of Lemma \ref{352} can be sketched by the following diagram:
$$\bar{A}+\bar{F} \underset{Lemma\ \ref{351}}{\overset{\psi}{\longrightarrow} A+F\overset{\psi^{-1}\Phi^{-1}e^{X}\Phi}{\longrightarrow} }\bar{A}_1+\bar{F}_1 \overset{\Phi \psi}{\longrightarrow} A_1+F_1\underset{Lemma\ \ref{344}}{\overset{e^{X_1}}{\longrightarrow}}\dots \underset{Lemma\ \ref{344}}{\overset{e^{X_{l-1}}}{\longrightarrow}} A_l+F_l\overset{\psi^{-1}\Phi^{-1}}{\longrightarrow} \bar{A}_l+\bar{F}_l=\bar{A}'+\bar{F}'$$
where $A,A_1,\dots, A_l$ are constant matrices, $\bar{A},\bar{A}_1,\dots, \bar{A}_l$ are reducible, and $\bar{F},F,\bar{F}_i,F_i$ are small.
No non resonance condition is required on $A$, making it necessary to construct the change of variables $\Phi$ which will remove resonances, but may be far from the identity (Lemma \ref{renormalization}). However, once this is done, the matrices $A_1,\dots, A_{l-1}$ remain non resonant enough in order to reduce the perturbation a lot without having to remove resonances again.
The superscripts on the arrows refer to the changes of variables. The changes of variables with an exponential expression are close to the identity, therefore the total conjugation, from $\bar{A}+\bar{F}$ to $\bar{A}'+\bar{F}'$, is close to identity, which makes it possible to obtain the density of reducible systems in the neighbourhood of a constant.
\section{Notations}
The notation $E(x)$ will refer to the integer part of a number $x$.
If $F \in L^2(2\mathbb{T} ^d)$ and $N \in \mathbb{N}$, the truncation of $F$ at order $N$ (denoted $F^N$) is the function we obtain by cutting the Fourier series of $F$ : \[ F^N(\theta) = \sum_{\vert m \vert \leq N}\hat F(m)e^{2i\pi \langle k, \theta \rangle} \]
In order to simplify the notation throughout this paper, we will write $\Psi(\cdot)$ for $\Psi(\vert \cdot \vert)$, and $\Lambda(\cdot)$ for $\Lambda(\vert \cdot \vert)$.
We will denote by $||\cdot ||$ the norm of the greatest coefficient for matrices.
\section{Decompositions, triviality}
We take the following definitions from \cite{1}, describing decompositions of $\mathbb{R}^2$ and triviality, which will avoid to double the period more than once.
\begin{definition}[Decomposition] \begin{itemize}[label=$\bullet$] \item If $A \in sl(2,\mathbb{R})$ has distinct eigenvalues, we call \textit{$A$-decomposition} a decomposition of $\mathbb{R}^2$ as the direct sum of two eigenspaces of $A$. If $L$ is an eigenspace of $A$, we write $\sigma(A_{\vert L})$ the spectrum of the restriction of $A$ to the subspace $L$. We shall denote by $\mathcal{L}_{A}$ the decomposition of $\mathbb{R}^2$ into two distinct eigenspaces of $A$, if the related eigenvalues are distinct. \item If $\mathbb{R}^2=L_1\bigoplus L_2$, for all $u \in \mathbb{R}^2$, there exists a unique decomposition $u = u_1+u_2, u_1\in L_1,u_2\in L_2$. For $i=1,2$, we call \textit{projection on $L_i$ with respect to $\mathcal{L}=\{L_1,L_2\}$}, and we write $P_{L_i}^\mathcal{L}$ the map defined by $P_{L_i}^\mathcal{L} u = u_i$. \end{itemize} \end{definition}
Recall the following lemma on estimate of the projection (see \cite{3}) : \begin{lemma}[\cite{3}]\label{311} Let $\kappa ' >0$ and $A \in sl(2, \mathbb{R})$ with $\kappa'$-separated eigenvalues. There exists a constant $C_0 \geq 1$ such that, for any subspace $L \in \mathcal{L}=\mathcal{L}_{A}$, \[ \Vert P_L^\mathcal{L} \Vert \leq C_0\big( \frac{1}{\kappa '}\big)^{6}\] \end{lemma}
\begin{remark} The estimate given in \cite{3} is more general since it also concerns matrices $A$ with a nilpotent part. Here the setting in $sl(2,\mathbb{R} )$ makes the estimate a little better.
\end{remark}
\begin{definition}[Triviality] Let $\mathcal{L}=\{L_1,L_2\}$ such that $L_1\bigoplus L_2=\mathbb{R}^2$. We say that a function $\psi \in \mathcal{C}^0(2\mathbb{T} ^d,SL(2\mathbb{R}))$ is trivial with respect to $\mathcal{L}$ if there exists $m\in \frac{1}{2}\mathbb{Z}^d$ such that for all $\theta \in 2\mathbb{T} ^d$, \[ \psi(\theta) = e^{2i\pi \langle m, \theta \rangle}P_{L_1}^\mathcal{L} +e^{-2i\pi \langle m, \theta \rangle}P_{L_2}^\mathcal{L}\]
\noindent If $|m|\leq N$, we say that $\psi$ is trivial of order $N$. \end{definition}
\begin{remark}\label{period-properties} \begin{itemize} \item If $\psi_1, \psi_2 : 2\mathbb{T} ^d \rightarrow SL(2, \mathbb{R})$ are trivial with respect to $\mathcal{L}$, then the product $\psi_1 \psi_2$ is also trivial with respect to $\mathcal{L}$ since, for all $L \neq L', P_L^{\mathcal{L}}P_{L'}^{\mathcal{L}} = 0$. \item If $\psi$ is trivial with respect to a a decomposition $\mathcal{L}$ of $\mathbb{R}^2$, then for all $G \in \mathcal{C}^0(\mathbb{T} ^d, sl(2, \mathbb{R}))$, we have $\psi G \psi^{-1} \in \mathcal{C}^0(\mathbb{T} ^d, sl(2, \mathbb{R}))$. Indeed, notice that if $\psi = e^{2i\pi \langle m, \cdot \rangle}P_{L_1}^{\mathcal{L}} +e^{-2i\pi \langle m, \cdot \rangle}P_{L_2}^{\mathcal{L}}$ for some $m\in \frac{1}{2}\mathbb{Z}^d$, then \[ \psi^{-1} = e^{-2i\pi \langle m, \cdot \rangle}P_{L_1}^{\mathcal{L}} +e^{2i\pi \langle m, \cdot \rangle}P_{L_2}^{\mathcal{L}} \] (it's a simple calculus to check that with this expression, $\psi \psi^{-1} = \psi^{-1}\psi \equiv I$.) Then,
\begin{equation*}\begin{split} \psi G \psi^{-1} &= (e^{2i\pi \langle m, \cdot \rangle}P_{L_1}^{\mathcal{L}} +e^{-2i\pi \langle m, \cdot \rangle}P_{L_2}^{\mathcal{L}})G(e^{-2i\pi \langle m, \cdot \rangle}P_{L_1}^{\mathcal{L}} +e^{2i\pi \langle m, \cdot \rangle}P_{L_2}^{\mathcal{L}}) \\
&= P_{L_1}^{\mathcal{L}}GP_{L_1}^{\mathcal{L}} + P_{L_2}^{\mathcal{L}}GP_{L_2}^{\mathcal{L}} + e^{2i\pi\langle 2m, \cdot\rangle}P_{L_1}^{\mathcal{L}}GP_{L_2}^{\mathcal{L}}+e^{-2i\pi\langle 2m, \cdot\rangle}P_{L_2}^{\mathcal{L}}GP_{L_1}^{\mathcal{L}} \end{split}\end{equation*} which is well defined continuously on $\mathbb{T} ^d$. Hence the function $\psi$ will avoid a period doubling. \end{itemize} \end{remark}
\section{Choice of parameters} In this section, we define all the constants and parameters used in this paper. \[ \left\{\begin{array}{c} \delta= 100000\\ \zeta = \frac{1}{1728}\end{array}\right. \]
Let for all $r, \varepsilon >0$,
\[ N(r,\varepsilon) = \Lambda^{-1}\big(\frac{50 \vert \log \varepsilon \vert}{\pi r}\big) \] \[ R(r,\varepsilon) = \frac{1}{3N(r,\varepsilon)}\Psi^{-1}(\varepsilon^{-\zeta})\] \[ \kappa''(\varepsilon) = \kappa \varepsilon^{\zeta}\] \[ r'(r,\varepsilon) = r-\frac{50 \delta \vert \log \varepsilon \vert}{\pi \Lambda(R(r,\varepsilon)N(r,\varepsilon))}\]
\section{Smallness of the perturbation}\label{smallness-assumption}
Let $r_0>0,A_0\in sl(2,\mathbb{R}),F_0\in U_{r_0}(\mathbb{T}^d, sl(2,\mathbb{R}))$.
\begin{assumption}\label{assumption-2bis} The functions $\Lambda,\Psi$ satisfy
$$\lim_{t\rightarrow +\infty} \frac{\ln\Psi(t)}{\Lambda(t)} = 0$$
and $\varepsilon_0$ is small enough as to satisfy conditions of lemma \ref{smallness-epsilon} below and:
\[ \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} +\frac{150\delta}{\pi \zeta \log(2\delta)} \int_{\Psi^{-1}(\varepsilon_0^{-\zeta})}^{+\infty}\frac{\Lambda'(t)\ln \Psi(t)}{\Lambda(t)^2} dt <r_0 \]
\end{assumption}
These conditions depend on $r_0,\Lambda,\Psi$.
We shall define the following sequences of parameters used throughout the iteration: \[ \varepsilon_k := \varepsilon_0^{(2\delta)^k}\] \[ \Lambda(N_k) := \Lambda(N(r_k,\varepsilon_k)) = \frac{50 \vert \log \varepsilon_k \vert}{\pi r_k} \] \[ R_k := R(r_k,\varepsilon_k) = \frac{1}{3N(r_k,\varepsilon_k)}\Psi^{-1}(\varepsilon_k^{-\zeta}) \] and \[ r_k := r_0- \sum_{i=0}^{k-1}\frac{50 \delta \vert \log \varepsilon_i \vert}{\pi \Lambda(R(r_i,\varepsilon_i) N(r_i,\varepsilon_i))}\]
\begin{lemma}\label{rk-limite-positive} Under either the Assumption \ref{assumption-2bis}, the sequence $r_k$ converges to a positive limit. \end{lemma}
\begin{proof} Notice that, for all $k$, and since $\Lambda$ is subadditive, \[\Lambda(R_kN_k) \geq \frac{1}{3}\Lambda(3R_kN_k) \Rightarrow \frac{1}{\Lambda(R_kN_k)} \leq \frac{3}{\Lambda(3R_kN_k)}\] Then \begin{align*} \sum_{k\geq 0} \frac{50 \delta \vert \log \varepsilon_k \vert}{\pi \Lambda(R_k N_k)} & \leq \sum_{k\geq 0} \frac{150 \delta \vert \log \varepsilon_k \vert}{\pi \Lambda(3R_k N_k)} \\ & \leq \frac{150\delta}{\pi }\sum_{k \geq 0} \frac{(2\delta)^k\vert \log \varepsilon_0 \vert}{ \Lambda(3R_k N_k)} \\
& \leq \frac{150\delta}{\pi }\sum_{k \geq 0} \frac{(2\delta)^k\vert \log \varepsilon_0 \vert}{\Lambda (\Psi^{-1}(\varepsilon_k^{-\zeta}))} \\
& \leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})}
+\frac{150\delta\vert \log \varepsilon_0 \vert}{\pi }
\int_0^{+\infty} \frac{(2\delta)^x}{\Lambda (\Psi^{-1}(\varepsilon_0^{-\zeta(2\delta)^x}))}dx \\ \intertext{With the change of variable $t := (2\delta)^x$}
& \leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} +\frac{150\delta\vert \log \varepsilon_0 \vert}{\pi } \int_{1}^{+\infty} \frac{t}{\Lambda (\Psi^{-1}(\varepsilon_0^{-\zeta t}))} \frac{1}{t \log (2\delta)}dt \\
& \leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} + \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \log (2\delta)} \int_{1}^{+\infty} \frac{1}{\Lambda (\Psi^{-1}(\varepsilon_0^{-\zeta t}))} dt \\ \intertext{With the change of variable $v := \Psi^{-1}(\varepsilon_0^{-\zeta t})$}
& \leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} +\frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \log (2\delta)} \int_{\Psi^{-1}(\varepsilon_0^{-
\zeta})}^{+\infty} \frac{1}{\Lambda(v)}
\cdot \frac{ \Psi'(v)}{ -\zeta \log \varepsilon_0 \Psi(v)}dv \\
& \leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} +\frac{150\delta}{\pi \zeta \log(2\delta)}\int_{\Psi^{-1}(\varepsilon_0^{-
\zeta})}^{+\infty} \frac{\Psi'(v)}{\Lambda(v)\Psi(v)}dv \end{align*}
\noindent After integrating by parts,
\begin{equation}\begin{split} \sum_{k\geq 0} \frac{50 \delta \vert \log \varepsilon_k \vert}{\pi \Lambda( R_k N_k) }& \leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} + \frac{150\delta}{\pi \zeta \log(2\delta)} [ -\frac{\log(\varepsilon_0^{-\zeta })}{\Lambda(\Psi^{-1}(\varepsilon_0^{-\zeta}))}\\ &+\int_{\Psi^{-1}(\varepsilon_0^{-\zeta })}^{+\infty} \frac{\Lambda'(v)\log \Psi(v)}{\Lambda(v)^2}dv] \\ &\leq \frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} +\frac{150\delta}{\pi \zeta \log(2\delta)}\int_{\Psi^{-1}(\varepsilon_0^{-\zeta })}^{+\infty} \frac{\Lambda'(v)\log \Psi(v)}{\Lambda(v)^2}dv \end{split}\end{equation}
provided $\displaystyle \lim_{v \rightarrow + \infty}\frac{\log \Psi(v)}{\Lambda(v)} = 0$, thus the assumption \ref{assumption-2bis} implies that $(r_k)$ converges to a positive limit. \end{proof}
\textit{Remark :} We naturally find the Bruno-Rüssmann condition with respect to weight function $\Lambda$, which is the convergence of $\displaystyle \int \frac{\Lambda'(v)\log\Psi(v)}{\Lambda(v)^{2}}dv $.
\section{Elimination of resonances}
Given a matrix $A$, a useful technique in the KAM iteration will be to remove the resonances in the spectrum of the matrix $A$. To characterize the non-resonance of $z \in \mathbb{C}$ (depending on $\omega$, a constant $\kappa'>0$ and on an order $N \in \mathbb{N}$) we will write $z \in BR_{\omega}^N(\kappa')$ if and only if : \[ \forall k \in \mathbb{Z}^d \backslash \{0\}, \quad 0 < \vert k \vert \leq N \Rightarrow \vert z - 2i \pi \langle k, \omega \rangle \vert \geq \frac{\kappa'}{\Psi (k)} \]
\noindent \begin{definition} We will say that $A$ has $BR_\omega^N(\kappa')$ spectrum if \[ \sigma(A)=\{\alpha, \alpha'\} \Rightarrow \alpha - \alpha' \in BR_{\omega}^N(\kappa ') \] In particular, if the eigenvalues of $A$ are $i\alpha$ and $-i\alpha$ with $\alpha \in \mathbb{R}$, $A$ has if $BR_\omega^N(\kappa')$ spectrum if \[ 2i\alpha \in BR_{\omega}^N(\kappa ') \] \end{definition}
\begin{remark}
If $A$ has real eigenvalues $\alpha, \alpha'$, then for all $N\in \mathbb{N}$, $A$ has $BR_\omega^N(\kappa)$-spectrum because $|\alpha-\alpha'-2i\pi \langle m,\omega\rangle |\geq |2i\pi \langle m,\omega\rangle |\geq \frac{\kappa}{\Psi(m)}$.
\end{remark}
\begin{lemma} \label{resonances1} Let $\alpha \in \mathbb{R}$, $\tilde{N}\in\mathbb{N^*}$ and $\kappa' = \displaystyle \frac{\kappa}{\Psi(3\tilde{N})}$. There exists $m \in \frac{1}{2}\mathbb{Z}^d$, $\vert m \vert\leq \frac{1}{2}\tilde{N}$ such that, if we denote $\alpha' = \alpha - 2\pi \langle m,\omega\rangle$, then $2i\alpha' \in BR_{\omega}^{\tilde{N} }(\kappa')$ and if $m\neq 0$ then $\vert \alpha'\vert \leq \frac{\kappa'}{2}$.
\end{lemma}
\begin{proof} We want to remove the resonances between $i\alpha$ and $-i\alpha$. If $2i\alpha \in BR_{\omega}^{\tilde N}(\kappa ')$, let $m=0$ and we are done. Otherwise, if there exists $ m' \in \frac{1}{2}\mathbb{Z}^d$ with $\vert m' \vert \leq \tilde N$ such that \[ \vert 2\alpha - 2\pi \langle m', \omega \rangle \vert < \frac{\kappa}{\Psi(m')}\] then let $m = \frac{m'}{2}$ and $2\alpha ' = \alpha - 2\pi \langle m, \omega \rangle$. It's a simple calculus to check that, in this case, $\vert 2\alpha' \vert \leq \kappa'$, hence $\alpha'\leq \frac{\kappa'}{2}$. Now, for all $k\in \frac{1}{2}\mathbb{Z}^d$, $k\leq \tilde N$, \[ \vert 2 i \alpha' - 2i\pi \langle k, \omega \rangle \vert \geq \frac{\kappa}{\Psi(k)} - \kappa' \geq \frac{\kappa'}{\Psi(k)}\]
Then $2i\alpha' \in BR_{\omega}^{\tilde N}(\kappa')$. \end{proof}
\begin{lemma} \label{resonnances2} Let $\alpha\in\mathbb{R}$. For all $R\in\mathbb{R},N \in \mathbb{N}$, $N\geq 1, R \geq 2$, there exists $m \in \frac{1}{2}\mathbb{Z}^d$, $\vert m\vert \leq \frac{1}{2}N$ such that, if we denote $\displaystyle \kappa '' = \frac{\kappa}{ \Psi(3RN)}$ and $\alpha' = \alpha-2\pi\langle m,\omega\rangle$, then $2i\alpha' \in BR_{\omega}^{RN} (\kappa'')$ and if $m\neq 0$ then $\vert \alpha' \vert \leq \frac{\kappa''}{2}$.
\end{lemma}
\begin{proof} If $\alpha \in BR_{\omega}^{RN} (\kappa'')$, then $m=0$. Otherwise, apply the previous Lemma with $\tilde{N}=N$ and $\kappa'=\kappa''$ to obtain $\vert m\vert \leq \frac{1}{2}N$ such that $\vert \alpha - \langle m,\omega\rangle\vert \leq \frac{\kappa''}{2}$. Therefore for all $0\leq \vert k\vert \leq RN$,
$$\vert 2\alpha' - 2\pi\langle k,\omega\rangle\vert \geq \vert 2\pi\langle k,\omega\rangle \vert - \kappa''\geq \frac{\kappa}{\Psi( k )}-\kappa''\geq \frac{\kappa''}{\Psi(k)}$$ and then $2i\alpha' = 2i\alpha - 2\pi\langle m,\omega \rangle \in BR_{\omega}^{RN}(\kappa'')$. \end{proof}
\section{Renormalization} We want to define a map $\Phi$ which conjugates $A$ to a matrix with $BR_\omega^{R N}(\kappa'')$ spectrum.
\begin{lemma}\label{renormalization} Let $A \in sl(2, \mathbb{R}), R\geq 2, N \in \mathbb{N}\setminus\{0\}$. If $\kappa '' = \displaystyle \frac{\kappa}{\Psi(3RN)}$ and $A$ has $\kappa''$-separated eigenvalues, then there exists a map $\Phi \in \mathcal{C}^0(2\mathbb{T} ^d, SL(2, \mathbb{R}))$ which is trivial with respect to $\mathcal{L}_A$ (the decomposition into eigenspaces of $A$), and a constant $C_0\geq 1$ such that, \begin{enumerate} \item \label{item1} $\qquad$ For all $r' > 0$, \[ \vert \Phi ^{ \pm1} \vert_{r'} \leq 2 C_0e^{2\pi \Lambda(\frac{N}{2})r'} \big( \frac{1 }{\kappa '' }\big)^{6}\] \item \label{item2} $\qquad$ If $\tilde{A}$ is defined by the following condition: for all $\theta \in 2\mathbb{T} ^d$,
$$ \partial_\omega \Phi(\theta) = A \Phi(\theta) - \Phi(\theta)\tilde{A}$$
\noindent (note that $\tilde A$ actually does not depend on $\theta$), then $\Vert \tilde{A} - A \Vert \leq \pi N$ and $\tilde{A}$ has $BR_\omega ^{R N}(\kappa '')$ spectrum. \item \label{item3} For any function $G\in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, we have $\Phi G \Phi^{-1} \in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$. \item \label{item4} If $\tilde{A}\neq A$ then $\Vert \tilde{A}\Vert \leq \frac{1}{2}\kappa''$. \end{enumerate} \end{lemma}
\begin{proof} Let $m$ given by Lemma \ref{resonnances2} with $\alpha$ the imaginary part of an eigenvalue of matrix $A$. If the eigenvalues of $A$ are in $\mathbb{R}$, then $m=0$ and $\Phi\equiv I$.
Otherwise, let $L_1$ be the invariant subspace associated to $i\alpha$, $L_2$ associated to $-i\alpha$ and let for all $\theta \in 2 \mathbb{T} ^d$,
\[ \Phi(\theta) = e^{2i\pi \langle m, \theta \rangle}P_{L_1}^{\mathcal{L}_A} +e^{-2i\pi \langle m, \theta \rangle}P_{L_2}^{\mathcal{L}_A} \]
For all $\theta$, since the eigenvalues of $\Phi(\theta)$ are complex conjugate, we have $\Phi(\theta) \in SL(2, \mathbb{R})$, and from Lemma \ref{resonnances2}, $\tilde A$ has $BR_\omega^{R N}(\kappa '')$ spectrum (since the eigenvalues of $\tilde A$ are $\pm i \tilde \alpha$ obtained from lemma \ref{resonnances2} where $\pm i\alpha$ are the eigenvalues of $A$). Moreover, the spectrum of $\tilde A - A$ is $\{ \pm 2i\pi \langle m, \omega \rangle \}$ and $\vert 2i\pi \langle m, \omega \rangle \vert \leq \pi N$ (remind that $\vert m \vert \leq \frac{1}{2}N$, and that we supposed $\vert \omega \vert \leq 1$) whence \ref{item2}. Moreover, because $\vert m \vert \leq \frac{1}{2} N$, and from Lemma \ref{311}, \[ \vert \Phi \vert_{r'} \leq ( \Vert P_{L_1} \Vert + \Vert P_{L_2} \Vert) e^{2\pi \Lambda(\frac{N}{2})r'} \leq 2C_0\big( \frac{1 }{\kappa ''}\big)^{6} e^{2\pi \Lambda(\frac{N}{2})r'} \] whence \ref{item1}. The property \ref{item3} follows from the triviality of $\Phi$ (see the remark \ref{period-properties}).
For the estimate in \ref{item4}, notice that if $A \neq \tilde A$, that is to say if the spectrum of $A$ was resonant, $\Phi \not\equiv I$ conjugates $A$ to $\tilde A$. In particular, from lemma \ref{resonnances2}, the two eigenvalues $i \tilde \alpha$ and $- i\tilde \alpha$ of $\tilde A$ (which are the eigenvalues of $A$ translated by $2i\pi\langle m, \omega \rangle$) satisfy $\vert i\tilde \alpha - (-i \tilde \alpha) \vert \leq \kappa''$ and then $\Vert \tilde A \Vert \leq \frac{1}{2}\kappa''$. \end{proof} \begin{definition} A function $\Phi$ satisfying conclusions of lemma \ref{renormalization} will be called \textit{renormalization of $A$ of order $R,N$}. Here the resonance is removed up to order $RN$ whereas the estimate involves an exponential of $\Lambda(\frac{N}{2})$ and $\Psi(3RN)$. \end{definition}
\section{Cohomological equation}
In order to define a change of variables which will reduce the norm of the perturbation, we will first solve a linearized equation, which has the form:
\begin{equation} \forall \theta \in \mathbb{T} ^d, \partial_\omega \tilde X(\theta) = [\tilde A, \tilde X( \theta)] + \tilde F^N - \hat {\tilde F}(0), \quad \hat {\tilde{X}}(0) = 0 \end{equation}
Here $\tilde{A}\in sl(2,\mathbb{R})$, therefore either it has real non zero eigenvalues, or it is the zero matrix, or it is nilpotent, or it has two eigenvalues $i\alpha, -i\alpha,\alpha\in \mathbb{R}^*$. Only in the latter case can $\tilde{A}$ be resonant.
Assume the eigenvalues are different (so, either they are distinct reals or they are complex conjugates). Let $L_1,L_2$ be the eigenspaces. For all $L,L'\in \{L_1,L_2\}$, define the following operator:
$$\mathcal{A}_{L,L'} : gl(2,\mathbb{R}) \rightarrow gl(2, \mathbb{R}), M \mapsto \mathcal{A}_{L,L'} M := \tilde A P_LM - MP_{L'}\tilde A$$
It will be necessary to compute the spectrum of every $\mathcal{A}_{L,L'}$ to estimate the solution of the linearized equation. This is done in the following lemma:
\begin{lemma}\label{spectrum-ALL'} Let $L,L'\in \{L_1,L_2\}$, $\beta$ the eigenvalue associated to $L$ and $\gamma$ the eigenvalue associated to $L'$. The spectrum of $\mathcal{A}_{L,L'}$ is $\{ \beta,-\gamma, \beta-\gamma,0\}$. Moreover, the operator $\mathcal{A}_{L,L'}$ is diagonalizable. \end{lemma}
\begin{proof} Let $P\in GL(2,\mathbb{C})$ such that $P^{-1} \tilde{A} P=\left(\begin{array}{cc} \bar{\beta} & 0\\ 0 & \bar{\gamma}\\ \end{array}\right)$. Notice that $\{\beta,\gamma\}\subset \{\bar{\beta},\bar{\gamma}\}$. Denote by $E_{i,j}$ the elementary matrix which has $1$ as the coefficient situated on line $i$ and column $j$, and $0$ elsewhere.
Case 1: $L\neq L'$. Here $\{\beta,\gamma\}=\{\bar{\beta},\bar{\gamma}\}$. Without loss of generality, assume $\beta= \bar{\beta}, \gamma = \bar{\gamma}$, that is to say, $L=L_1,L'=L_2$.
Then $P_L=PE_{1,1}P^{-1}$ and $P_{L'}=PE_{2,2}P^{-1}$. Thus $PE_{1,1}P^{-1}$ is an eigenvector associated to $\bar{\beta}$ and $PE_{2,2}P^{-1}$ is an eigenvector associated to $-\bar{\gamma}$. The matrix $PE_{1,2}P^{-1}$ is an eigenvector associated to $\bar{\beta}-\bar{\gamma}$ and $PE_{2,1}P^{-1}$is in the kernel.
Case 2: $L=L'=L_1$. Here $\beta=\gamma=\bar{\beta}$.
Then $PE_{1,1}P^{-1}$ and $PE_{2,2}P^{-1}$ are in the kernel and $PE_{1,2}P^{-1},PE_{2,1}P^{-1}$ are eigenvectors associated to $\bar{\beta}$ and $-\bar{\beta}$ respectively.
Case 3: $L=L'=L_2$. This case is very similar to the previous one. \end{proof}
Now assume $0$ is the only eigenvalue of $\tilde A$. If $\tilde A$ is the zero matrix, then $ad_{\tilde A}=0$. Otherwise $\tilde A$ is nilpotent and in this case one has the following lemma:
\begin{lemma}\label{nilpotent-tildeA} Assume $\tilde A$ is nilpotent. Then the operator $ad_{\tilde A}$ has rank 2 and norm less than 1, and is nilpotent of order 3. \end{lemma}
\begin{proof} Let $P$ be such that $P^{-1}\tilde AP=\left(\begin{array}{cc} 0 & 1\\ 0 & 0\\ \end{array}\right)$. Then $PE_{12}P^{-1}$ and $P(E_{11}+E_{22})P^{-1}$ are in the kernel. Moreover $ad_{\tilde A} PE_{11}P^{-1}= -PE_{12}P^{-1} $ and $ad_{\tilde A} PE_{21}P^{-1}=P(E_{11}-E_{22})P^{-1}$, therefore $ad_{\tilde A}$ has norm less than 1.
This also implies that $ad_{\tilde A}^2(PE_{1,1}P^{-1})=0$ and $ad_{\tilde A}^3(PE_{2,1}P^{-1})=0$. Finally $ad_{\tilde A}$ is nilpotent of order 3. \end{proof}
\begin{proposition}\label{homologique} Let $N \in \mathbb{N}, \kappa' \in ]0, \kappa], r \in ]0, r_0[$. Let $\tilde{A} \in sl(2, \mathbb{R})$ with $BR_\omega ^N(\kappa')$ spectrum. Let $\tilde{F} \in U_r(\mathbb{T} ^d, sl(2,\mathbb{R}))$. Then there exists a solution $\tilde X \in$ $U_{r}( \mathbb{T} ^d, sl(2,\mathbb{R}))$ of the equation \begin{equation}\label{eq:homologique} \forall \theta \in \mathbb{T} ^d, \partial_\omega \tilde X(\theta) = [\tilde A, \tilde X( \theta)] + \tilde F^N - \hat {\tilde F}(0), \quad \hat {\tilde{X}}(0) = 0 \end{equation}
The truncation of $\tilde X$ at order $N$ is unique.
Moreover, \begin{enumerate} \item \label{cas1}
if $\tilde A$ is diagonalizable with distinct eigenvalues, let $\mathcal{L}_{\tilde A}=\{L_1,L_2\}$ (the decomposition into eigenspaces of $\tilde A$) and $\Phi=P^\mathcal{L}_{L_1}e^{2i\pi\langle m,\cdot\rangle }+P^\mathcal{L}_{L_2}e^{-2i\pi\langle m,\cdot\rangle }$ for some $m\in\frac{1}{2}\mathbb{Z}^d$, $|m|\leq N$, such that for $i\in \{1,2\}$, $\Vert P^\mathcal{L}_{L_i}\Vert \leq \frac{2C_0}{\kappa'^6}$, then
\[ \vert \Phi^{-1} \tilde X \Phi \vert _{r} \leq 4C_0^2 \big( \frac{1}{\kappa'}\big)^{13} \Psi(N) \vert \Phi^{-1} \tilde F \Phi \vert_{r}\]
\item \label{cas3} if $\tilde A$ is nilpotent,
\[ \vert \tilde X \vert _{r} \leq \frac{3}{\kappa^3}\Psi(N)^3 \vert \tilde F \vert_{r}\]
\item \label{cas4} if $ad_{\tilde{A}}=0$, then
\[ \vert \tilde X \vert _{r} \leq \frac{1}{\kappa} \Psi(N) \vert \tilde F \vert_{r}\]
\end{enumerate}
\end{proposition}
\begin{proof} About existence, uniqueness and continuity of $\tilde X \in sl(2, \mathbb{R})$ on $\mathbb{T} ^d$, the proof is the same as \cite{1}, proposition 3.2. We now have to show the estimate which also follows from \cite{1} and we will adapt the proof to ultra-differentiable setting.
\textbf{Case \ref{cas1}:} $\tilde{A}$ has two $\kappa'$-separated eigenvalues. Let $\Phi =P_{L_1} e^{2 i \pi \langle m_1, . \rangle} + P_{L_2} e^{-2 i \pi \langle m_1, . \rangle}$
where $L_1$ and $L_2$ are the eigenspaces of $\tilde A$, and $|m_1|\leq N$. For all $L,L'\in \mathcal{L}_{\tilde A}$, let the linear operator $\mathcal{A}_{L,L'} : gl(2,\mathbb{R}) \rightarrow gl(2, \mathbb{R}), M \mapsto \mathcal{A}_{L,L'} M := \tilde A P_LM - MP_{L'}\tilde A$. We decompose (\ref{eq:homologique}) into blocks, and we get for all $L, L' \in \mathcal{L}_{\tilde A}$, \[ \partial_\omega (P_L \tilde X(\theta) P_{L'}) =\mathcal{A}_{L,L'} P_L \tilde X (\theta) P_{L'} + P_L(\tilde F^N - \hat{\tilde F} (0))P_{L'} \]
Then for all $m \in \frac{1}{2} \mathbb{Z}^d$, $0<\vert m \vert \leq N$,
\[ 2i \pi \langle m, \omega \rangle(P_L \hat{\tilde X}(m) P_{L'}) = \mathcal{A}_{L,L'} (P_L \hat{\tilde X}(m) P_{L'} )+ P_L \hat{\tilde F}(m) P_{L'} \]
Let
\[ A_D:=(2i \pi \langle m, \omega \rangle I - \mathcal{A}_{L,L'}) \]
By Lemma \ref{spectrum-ALL'}, $\sigma(\mathcal{A}_{L,L'}) = \{ \alpha - \alpha',\alpha,-\alpha',0 ; \alpha \in \sigma(\tilde A_{\vert L}), \alpha' \in \sigma(\tilde A_{\vert L'}) \}$, therefore $\sigma(\mathcal{A}_{L,L'}-2i \pi \langle m, \omega \rangle I) = \{ \alpha - \alpha'-2i \pi \langle m, \omega \rangle ,\alpha-2i \pi \langle m, \omega \rangle,-\alpha'- 2i \pi \langle m, \omega \rangle,-2i \pi \langle m, \omega \rangle ; \alpha \in \sigma(\tilde A_{\vert L}), \alpha' \in \sigma(\tilde A_{\vert L'}) \}$. Moreover $\mathcal{A}_{L,L'}$ is diagonalizable, therefore $A_D$ as well, with non zero eigenvalues, and $\Vert A_D^{-1}\Vert =\max \{ \vert \beta\vert , \beta\in \sigma(A_D^{-1})\} =\max \{\vert \gamma\vert ^{-1}, \gamma\in \sigma(A_D)\} $.
Since $\forall \alpha \in \sigma(\tilde A_{\vert L}), \alpha' \in \sigma(\tilde A_{\vert L'})$, $\vert \alpha - \alpha' - 2i\pi \langle m, \omega \rangle \vert \geq \frac{\kappa'}{\Psi( m )} $ (for $m \in \mathbb{Z}^d$ if $L=L'$, $m \in\frac{1}{2}\mathbb{Z}^d$ if $L \neq L'$), then \[ \Vert (2i \pi \langle m, \omega \rangle - \mathcal{A}_{L,L'})^{-1} \Vert \leq \big( \frac{\Psi(m )}{\kappa '}\big) \]
Finally, for all $0<\vert m\vert \leq N$, \[ \Vert P_L \hat{\tilde X}(m) P_{L'} \Vert = \Vert (2i \pi \langle m , \omega \rangle - \mathcal{A}_{L,L'})^{-1}P_L \hat{\tilde F}(m ) P_{L'} \Vert \leq \big( \frac{\Psi(m )}{\kappa '}\big) \Vert P_L \hat{\tilde F}(m) P_{L'} \Vert \]
Denoting by $m_L$ the vector appearing in $\Phi$ along the projection onto $L$, this estimate implies:
\[ \vert P_L \tilde X e^{2i\pi \langle m_L - m_{L'}, . \rangle}P_{L'}\vert_{r'} = \sum_{\vert m-m_L+m_{L'}\vert \leq N} \Vert P_L \hat{\tilde X}(m-m_L+m_{L'}) P_{L'} \Vert e^{2\pi\Lambda( m ) r'} \] \[\leq \sum_{\vert m-m_L+m_{L'}\vert \leq N} \Vert P_L \hat{\tilde F}(m-m_L+m_{L'} ) P_{L'} \Vert e^{2\pi \Lambda( m ) r'} \frac{\Psi(\vert m-m_L+m_{L'} \vert)}{\kappa'} \]
\begin{equation}\label{estimate:homologique}\leq \frac{\Psi( N )}{\kappa'} \vert P_L \tilde Fe^{2i\pi \langle m_L - m_{L'}, . \rangle} P_{L'} \vert_{r'} \end{equation}
We finally estimate $\vert \Phi^{-1} \tilde X \Phi \vert _{r'}$.
\[ \vert \Phi^{-1} \tilde X \Phi \vert _{ r'} = \vert \sum_{L, L' \in \mathcal{L}} P_L \Phi^{-1} \tilde X \Phi P_{L'} \vert _{ r'} = \vert \sum_{L, L' \in \mathcal{L}} P_L \tilde X e^{2i\pi \langle m_L - m_{L'}, . \rangle} P_{L'} \vert _{ r'} \]
therefore, from (\ref{estimate:homologique}),
\[ \vert \Phi^{-1} \tilde X \Phi \vert _{r'} \leq \frac{\Psi( N ) }{\kappa'} \sum_{L, L' \in \mathcal{L}} \vert P_L \tilde F e^{2i\pi \langle m_L - m_{L'}, . \rangle} P_{L'} \vert_{r} =\frac{\Psi( N ) }{\kappa'} \sum_{L, L' \in \mathcal{L}} \vert P_L\Phi^{-1}\tilde{F}\Phi P_{L'}\vert _{r}\]
therefore, since $\Vert P_L \Vert\leq \frac{2C_0}{\kappa^{'6}}$, we get the result
\[ \vert \Phi^{-1} \tilde X \Phi \vert _{r'} \leq 4C_0^2 \big( \frac{1}{\kappa'}\big)^{13} \Psi( N ) \vert \Phi^{-1} \tilde F \Phi \vert_{ r}.\]
\textbf{Case \ref{cas3}:} $\tilde{A}$ is nilpotent. One has to estimate the inverse of the operator $ 2i\pi\langle m,\omega\rangle I-ad_{\tilde{A}}$. By Lemma \ref{nilpotent-tildeA},
\begin{equation}\begin{split}(2i\pi\langle m,\omega\rangle I-ad_{\tilde{A}})^{-1}& = (2i\pi\langle m,\omega\rangle)^{-1} [I+(2i\pi\langle m,\omega\rangle)^{-1}ad_{\tilde{A}}\\ &+ (2i\pi\langle m,\omega\rangle)^{-2}ad_{\tilde{A}}^2 ] \end{split}\end{equation}
\noindent Therefore
$$ \Vert (2i\pi\langle m,\omega\rangle I-ad_{\tilde{A}})^{-1}\Vert \leq 3\vert 2i\pi\langle m,\omega\rangle\vert ^{-3}$$
Finally, for all $0<\vert m\vert \leq N$,
\[ \Vert \hat{\tilde X}(m) \Vert = \Vert (2i \pi \langle m , \omega \rangle - \mathcal{A}_{L,L'})^{-1} \hat{\tilde F}(m ) \Vert \leq 3\big( \frac{\Psi(m )}{\kappa }\big)^3 \Vert \hat{\tilde F}(m) \Vert \]
Thus,
\[ \vert \tilde X \vert _{r'} \leq \frac{3}{\kappa^3} \Psi( N )^3 \vert \tilde F \vert_{ r}\]
\textbf{Case \ref{cas4}:} The operator to invert is just $2i\pi \langle m,\omega\rangle I$, which makes the estimate much simpler.
\end{proof}
\section{Inductive lemma without renormalization}
Before stating the inductive lemma, we will need this next result which will allow us to iterate the inductive lemma without needing a new renormalization map at each step.
\begin{lemma}\label{341} Let $\kappa' \in ]0,1[, \tilde F \in sl(2, \mathbb{R}), \tilde \varepsilon = \Vert \tilde F \Vert, \tilde N \in \mathbb{N}, \tilde A \in sl(2, \mathbb{R})$ with $BR_\omega ^{\tilde N}(\kappa ')$ spectrum.
If \[\tilde \varepsilon \leq \big(\frac{\kappa '}{32(1 +\Vert \tilde A \Vert)} \big)^{2}\frac{1}{\Psi(\tilde N)^2}, \] then $\tilde A + \tilde F$ has $BR_\omega^{\tilde{N}}(\frac{3\kappa '}{4})$ spectrum. \end{lemma} \begin{proof}
If $\tilde \alpha \in \sigma (\tilde A + \tilde F)$, there exists $\alpha \in \sigma(\tilde A)$ such that $\vert \alpha - \tilde \alpha \vert \leq 4(\Vert \tilde A \Vert +1)\tilde \varepsilon ^{\frac{1}{2 }}$ (see \cite{1}, lemma 4.1). Since $\tilde A$ has $BR_\omega ^{\tilde N}(\kappa ')$ spectrum, for all $\alpha, \alpha' \in \sigma (\tilde A + \tilde F)$, for all $m \in \frac{1}{2}\mathbb{Z}^d$, $ 0<\vert m \vert \leq \tilde N$,
\[ \vert \alpha - \alpha' - 2i\pi \langle m, \omega \rangle \vert \geq \frac{\kappa '}{\Psi( m )} - 8(\Vert \tilde A \Vert +1)\tilde \varepsilon ^{\frac{1}{2}} \] We have to check that $8(\Vert \tilde A \Vert +1)\tilde \varepsilon ^{\frac{1}{2}} \leq \frac{\kappa '}{4\Psi( m)}$, which is satisfied by assumption. \end{proof}
\begin{lemma}\label{NormeTroncation} Let $N\geq 1$. If $\mathcal{L}=\{L_1,L_2\}$ is a decomposition of $\mathbb{R}^2$ into supplementary subspaces, and $\Phi$ is trivial with respect to $\mathcal{L}$ of order $N$, then for all $0<r'<r$ and all $G\in U_r(\mathbb{T}^d,sl(2,\mathbb{R}))$, \[ \vert \Phi^{-1}(G - G^{3N}) \Phi\vert_{r'} \leq e^{-2 \pi \Lambda(N)(r-r')} \vert \Phi^{-1}G \Phi \vert_{ r} \] \end{lemma}
\begin{proof} Write $\Phi = P^\mathcal{L}_{L_1} e^{2i\pi \langle m_1, \cdot \rangle}+ P^\mathcal{L}_{L_2} e^{2i\pi \langle m_1, \cdot \rangle}$, $m_1\in\frac{1}{2}\mathbb{Z}^d$, then \[ \vert \Phi^{-1}(G-G^{3N})\Phi \vert_{r'} = \vert \sum_{L,L'\in\mathcal{L}}P^\mathcal{L}_L(G-G^{3N})e^{2i\pi \langle m_L - m_{L'}, \cdot \rangle} P^\mathcal{L}_{L'}\vert_{ r'} \] \[ = \sum_{k\in\mathbb{Z}^d} \Vert \sum_{L,L'} P^\mathcal{L}_L \widehat{(G-G^{3N})}(k-m_L +m_{L'}) P^\mathcal{L}_{L'}\Vert e^{2\pi \Lambda(k) r'} \] \[= \sum_{k\in\mathbb{Z}^d} \Vert \sum_{L,L'} P^\mathcal{L}_L \widehat{(G-G^{3N})}(k-m_L +m_{L'}) P^\mathcal{L}_{L'}\Vert e^{2\pi \Lambda(k)r} e^{2\pi \Lambda(k)(r'-r)}\]
\noindent Now if $\vert k\vert \leq N$, then for all $L,L'\in \mathcal{L}$, $\vert k-m_L+m_{L'}\vert \leq 3N$, and $\sum_{L,L'\in \mathcal{L}}P^\mathcal{L}_L\widehat{(G-G^{3N})}(k-m_L+m_{L'})P^\mathcal{L}_{L'}=0$, therefore
\[\vert \Phi^{-1}(G-G^{3N})\Phi \vert_{r'}\leq e^{2\pi \Lambda(N)(r'-r)} \sum_{\vert k\vert >N} \Vert \sum_{L,L'\in\mathcal{L}} P^\mathcal{L}_L \hat G(k-m_L +m_{L'}) P^\mathcal{L}_{L'}\Vert e^{2\pi \Lambda(k)r} \] \[\leq e^{2\pi \Lambda(N)(r'-r)} \vert \sum_{L,L'\in\mathcal{L}} P^\mathcal{L}_L G e^{2i\pi \langle m_L - m_{L'}, \cdot \rangle} P^\mathcal{L}_{L'}\vert_{ r} = e^{2\pi \Lambda(N)(r'-r)} \vert \Phi^{-1}G \Phi \vert_{r} \] \end{proof}
We can now state the first inductive lemma, which does not require a renormalization map.
\begin{lemma}\label{344} Let \begin{itemize} \item $\tilde \varepsilon >0, \tilde r>0, 0<\kappa' < 1, \tilde N \in \mathbb{N}^*, \tilde r' < \tilde r $, \item $\tilde F \in U_{\tilde r}(\mathbb{T} ^d, sl(2,\mathbb{R})), \tilde A \in sl(2,\mathbb{R})$. \end{itemize}
If \begin{enumerate} \item $\tilde A$ has $BR_\omega ^{\tilde N}(\kappa ')$ spectrum, \item \[ \Vert \hat{\tilde F}(0) \Vert \leq \tilde \varepsilon \leq \big(\frac{\kappa '}{32(1 + \Vert \tilde A \Vert)} \big)^{2}\frac{1}{\Psi(\tilde N)^2}\] \end{enumerate} then there exist \begin{itemize} \item $X \in U_{\tilde r'}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, \item $A' \in sl(2, \mathbb{R})$, \end{itemize} such that \begin{enumerate} \item $A'$ has $BR_\omega ^{\tilde N} (\frac{3\kappa '}{4})$ spectrum, \item $\Vert A'- \tilde A\Vert \leq \tilde \varepsilon$, \newline If $F' \in U_{ \tilde r '}(\mathbb{T} ^d, sl(2, \mathbb{R}))$ is defined by
\begin{equation}\label{defF'}\forall \theta \in \mathbb{T} ^d, \partial_\omega e^{X(\theta)} = (\tilde A + \tilde F(\theta))e^{X(\theta)} - e^{X(\theta)}(A' + F'(\theta)) , \end{equation}
\noindent then we have the following estimates :
If $\tilde A$ has two different eigenvalues, if $\Phi$ is of the form
$\Phi = P_{L_1}e^{2i\pi\langle m,\cdot\rangle}+P_{L_2}e^{-2i\pi\langle m,\cdot\rangle}$ where $L_1,L_2$ are the two eigenspaces of $\tilde A$, $|m|\leq \tilde N$ and $\Vert P_{L_i} \Vert \leq \frac{2C_0}{\kappa^{'6}}$, \item \label{estim3} \[ \vert \Phi^{-1}X\Phi\vert_{ \tilde r'}\leq 4C_0^2 \big( \frac{1}{\kappa'}\big)^{13} \Psi(3 \tilde N) \vert \Phi^{-1} \tilde F \Phi \vert_{ \tilde r},\] \item \label{estim4} \[\vert \Phi^{-1} F' \Phi \vert_{ \tilde r'} \leq 4C_0^2 e^{\vert \Phi^{-1}X\Phi \vert_{ \tilde r'}} \big( \frac{1}{\kappa'}\big)^{13} \vert \Phi^{-1}\tilde F \Phi \vert_{ \tilde r} \big[ e^{-2\pi \Lambda(\tilde N)(\tilde r-\tilde r')} \qquad \] \[ \qquad + \vert \Phi^{-1}\tilde F \Phi \vert_{ \tilde r} \Psi(3\tilde N)(2e+e^{\vert \Phi^{-1} X \Phi \vert_{ \tilde r'}}) \big].\]
If $\tilde A$ is nilpotent: \item \label{estim5} \[ \vert X\vert_{ \tilde r'}\leq \frac{3}{\kappa^3}\Psi( 3\tilde N)^3 \vert \tilde F \vert_{ \tilde r},\]
\item \label{estim7} \[\vert F' \vert_{ \tilde r'} \leq \frac{3}{\kappa^3}e^{\vert X \vert_{ \tilde r'}} \vert \tilde F \vert_{ \tilde r} \big[ e^{-2\pi \Lambda(\tilde N)(\tilde r-\tilde r')} \qquad \] \[ \qquad + \vert \tilde F \vert_{ \tilde r} \Psi(3\tilde N)^3(2e+e^{\vert X \vert_{ \tilde r'}}) \big].\]
If $ad_{\tilde A}=0$: \item \label{estim8} \[ \vert X\vert_{ \tilde r'}\leq \frac{1}{\kappa} \Psi(3\tilde N) \vert \tilde F \vert_{ \tilde r}, \] \item \label{estim9} \[\vert F' \vert_{ \tilde r'} \leq \frac{1}{\kappa}e^{\vert X \vert_{ \tilde r'}} \vert \tilde F \vert_{ \tilde r} \big[ e^{-2\pi \Lambda(\tilde N)(\tilde r-\tilde r')} \qquad \] \[ \qquad + \vert \tilde F \vert_{ \tilde r} \Psi(3\tilde N)(2e+e^{\vert X \vert_{\Lambda, \tilde r'}}) \big].\]
In any case, there is the estimate \item \label{estim6} \[\vert \partial_\omega X\vert _{\tilde r'}\leq 2\Vert \tilde A\Vert \ \vert X\vert _{\tilde r'}+\vert \tilde F\vert _{\tilde r'}. \]
\end{enumerate} \end{lemma}
\begin{proof} By assumption, $\tilde A$ has $BR_\omega^{\tilde N}(\kappa ')$ spectrum, so we apply Proposition \ref{homologique} with $N=3\tilde N$. Let $X \in U_{ \tilde r}(\mathbb{T} ^d, sl(2, \mathbb{R}))$ a solution of \[ \forall \theta \in 2\mathbb{T} ^d, \partial_\omega X(\theta) = [\tilde A, X(\theta)] + \tilde F^{3\tilde N}(\theta) - \hat{\tilde F}(0)\] satisfying the conclusion of Proposition \ref{homologique}. This obviously implies the property \ref{estim6}.
Let $A' := \tilde A + \hat{\tilde F}(0)$. We have $A' \in sl(2, \mathbb{R})$ and $\Vert \tilde A - A' \Vert = \Vert \hat{\tilde F}(0) \Vert$, and then property 2. With assumption (2) we can apply lemma \ref{341} to deduce that $A'$ has $BR_\omega^{\tilde N}(\frac{3\kappa'}{4})$ spectrum, and then property 1.
If $F'$ is defined in equation (\ref{defF'}),
\begin{equation}\label{expression-F'} F' = e^{-X}(\tilde F - \tilde F ^{3\tilde N}) + e^{-X}\tilde F (e^X-Id) + (e^{-X}-Id)\hat{\tilde F}(0) - e^{-X} \sum_{k \geq 2}\frac{1}{k!}\sum_{l=0}^{k-1}X^l(\tilde F ^{3\tilde N}-\hat{\tilde F}(0))X^{k-1-l} \end{equation}
$\bullet$ Case 1: $\tilde A$ has two different eigenvalues : Let $\Phi$ be as required, then \[ \vert \Phi^{-1} F' \Phi \vert_{ \tilde r'} \leq e^{\vert \Phi ^{-1}X \Phi \vert_{\tilde r'}} [\vert \Phi^{-1}(\tilde F - \tilde F^{3\tilde N})\Phi \vert_{\tilde r'} + \vert \Phi^{-1}\tilde F \Phi \vert_{ \tilde r}\vert \Phi^{-1}X \Phi \vert_{ \tilde r'}(2e+e^{\vert \Phi^{-1}X\Phi \vert_{\tilde r'}})] \]
From proposition \ref{homologique}, estimate \ref{cas1},
\[\vert \Phi^{-1} X \Phi \vert_{ \tilde r'} \leq \vert \Phi^{-1} X \Phi \vert_{ \tilde r} \leq 4C_0^2 \big( \frac{1}{\kappa'}\big)^{13} \Psi(3\tilde N) \vert \Phi^{-1} \tilde F \Phi \vert_{ \tilde r}\] whence \eqref{estim3}; and from lemma \ref{NormeTroncation}, since $\tilde r' < \tilde r$, \[ \vert \Phi^{-1}(\tilde F - \tilde F ^{3\tilde N}) \Phi \vert_{ \tilde r'} \leq e^{-2 \pi \Lambda(\tilde N)(\tilde r- \tilde r')} \vert\Phi^{-1} \tilde F \Phi\vert_{ \tilde r} \] which finally gives
\[ \vert \Phi^{-1} F' \Phi \vert_{ \tilde r'} \leq e^{\vert \Phi ^{-1}X \Phi \vert_{\tilde r'}} [e^{-2 \pi \Lambda(\tilde N)(\tilde r- \tilde r')} \vert\Phi^{-1} \tilde F \Phi\vert_{ \tilde r} \qquad \qquad\qquad\qquad \qquad \qquad \] \[ \qquad\qquad \qquad \qquad \qquad + \vert \Phi^{-1}\tilde F \Phi \vert_{ \tilde r}4C_0^2 \big( \frac{1}{\kappa'}\big)^{13} \Psi(3 \tilde N) \vert \Phi^{-1} \tilde F \Phi \vert_{ \tilde r}(2e+e^{\vert \Phi^{-1}X\Phi \vert_{ \tilde r'}})] \] \[ \leq 4C_0^2 e^{\vert \Phi^{-1}X\Phi \vert_{ \tilde r'}} \big( \frac{1}{\kappa'}\big)^{13} \vert \Phi^{-1}\tilde F \Phi \vert_{\tilde r} \big[ e^{-2\pi \Lambda(\tilde N)(\tilde r-\tilde r')} + \vert \Phi^{-1}\tilde F \Phi \vert_{ \tilde r} \Psi( 3\tilde N)(2e+e^{\vert \Phi^{-1} X \Phi \vert_{ \tilde r'}}) \big] \] hence \ref{estim4} holds.
$\bullet$ Case 2: $\tilde A$ is nilpotent : \eqref{expression-F'} implies
\[ \vert F' \vert_{ \tilde r'} \leq e^{\vert X \vert_{\tilde r'}} [\vert \tilde F - \tilde F^{3\tilde N} \vert_{\tilde r'} + \vert \tilde F \vert_{ \tilde r}\vert X \vert_{ \tilde r'}(2e+e^{\vert X \vert_{\tilde r'}})] \]
From proposition \ref{homologique}, estimate \ref{cas3},
\[\vert X \vert_{ \tilde r'} \leq \vert X \vert_{ \tilde r} \leq \frac{3}{\kappa^3} \Psi(3\tilde N)^3 \vert \tilde F \vert_{ \tilde r}\] which is estimate \ref{estim5}. Moreover, from Lemma \ref{NormeTroncation},
\[|\tilde F - \tilde F^{3\tilde N}|_{\tilde r'}\leq e^{-2\pi \Lambda (\tilde N)(\tilde r-\tilde r')}|\tilde F|_{\tilde r}\]
Therefore, similarly to the previous case, we get \[\vert F' \vert_{ \tilde r'} \leq \frac{3}{\kappa^3} e^{\vert X \vert_{ \tilde r'}} \vert \tilde F \vert_{ \tilde r} \big[ e^{-2\pi \Lambda(\tilde N)(\tilde r-\tilde r')} \qquad \] \[ \qquad + \vert \tilde F \vert_{ \tilde r} \Psi(3\tilde N)^3(2e+e^{\vert X \vert_{ \tilde r'}}) \big]\] which is estimate \ref{estim7}.
$\bullet$ Case 3: $ad_{\tilde A}=0$ : From proposition \ref{homologique}, estimate \ref{cas4} \[ \vert \tilde X \vert _{\tilde r} \leq \frac{1}{\kappa} \Psi(3\tilde N) \vert \tilde F \vert_{\tilde r} \] which is estimate \eqref{estim8}, and similarly to the two previous cases, we get the estimate \eqref{estim9}: \[\vert F' \vert_{ \tilde r'} \leq \frac{1}{\kappa} e^{\vert X \vert_{ \tilde r'}} \vert \tilde F \vert_{ \tilde r} \big[ e^{-2\pi \Lambda(\tilde N)(\tilde r-\tilde r')} \qquad \] \[ \qquad + \vert \tilde F \vert_{ \tilde r} \Psi(3\tilde N)(2e+e^{\vert X \vert_{ \tilde r'}}) \big].\] \end{proof}
\section{Inductive lemma with renormalization}
The following Lemma is used to define the smallness assumption on $\epsilon_0$ mentioned in section \ref{smallness-assumption}. This smallness assumption shall be sufficient for Lemmas \ref{351} and \ref{352}.
\begin{lemma}\label{smallness-epsilon}
Let $l=56$. There exists $\varepsilon_0 >0$ depending on $C_0$, $C'$ $\kappa$, $b_0$ and $D_5$, such that, for all $\varepsilon \in ]0, \varepsilon_0]$, the following inequalities hold for all $2 \leq j \leq l$ : \\ \textbf{In lemma \ref{351}} \begin{equation} \frac{1}{2}\kappa\varepsilon^{\frac{1}{1728}}+\varepsilon^{\frac{845}{864}} \leq \frac{3}{4}\kappa \varepsilon^{\frac{1}{1728}} \label{eq:cond1.1}\end{equation}
\begin{equation} \frac{4C_0^2}{\kappa^{13}}\varepsilon^{-13\zeta}\varepsilon^{-3\zeta}\varepsilon^{1-2\zeta} \leq \varepsilon^{\frac{7}{8}} \label{eq:cond1.2}\end{equation}
\begin{equation} 8C_0^2\varepsilon^{1-2\zeta-\frac{1}{96}}(\varepsilon^{100\delta}+3\varepsilon^{1-6\zeta}) \leq \varepsilon^{\frac{3}{2}-4\zeta-\frac{1}{96}} \label{eq:cond1.3} \end{equation}
\textbf{In lemma \ref{352}}
\begin{equation}\label{eq:cond2.0.0} \varepsilon^{1-576\zeta} \leq (2C_0)^{-96}\big(\frac{\kappa}{32(\varepsilon^{-\frac{\zeta}{2}} +1)}\big)^{576} \end{equation}
\begin{equation} \label{eq:cond2.0} \varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \big(\frac{\frac{3}{4}\frac{\kappa}{C_0}\varepsilon^{\zeta}}{32(1+(1+\pi)\varepsilon^{-\frac{\zeta}{2}}+\varepsilon^{\frac{23}{24}})} \big)^2\varepsilon^{2\zeta} \end{equation} \begin{equation}\varepsilon^{(\frac{5}{4})^j-\frac{1}{48}} \leq \big(\frac{(\frac{3}{4})^j\frac{\kappa }{C_0}\varepsilon^{\zeta}}{32(1+ \varepsilon^{\frac{23}{24}} + (1+\pi)\varepsilon^{-\frac{\zeta}{2}} + \sum_{i=1}^{j-1} \varepsilon^{(\frac{5}{4})^i-\frac{1}{96}})}\big)^2\varepsilon^{2\zeta} \label{eq:cond2.1}\end{equation}
\begin{equation} 256C_0^2 \varepsilon^{-14\zeta}\big( \frac{1}{(\frac{3}{4})^{j-1}\frac{\kappa}{C_0} }\big)^{13} \varepsilon^{(\frac{5}{4})^{j-1}} ( \varepsilon^{\frac{50\delta}{l}} + \varepsilon^{(\frac{5}{4})^{j-1}} ) \leq \varepsilon^{(\frac{5}{4})^j} \label{eq:cond2.2} \end{equation}
\begin{equation} \varepsilon^{\frac{23}{24}} + \pi \varepsilon^{-\frac{\zeta}{2}} + \sum_{i=1}^l \varepsilon^{(\frac{5}{4})^i-\frac{1}{48}} \leq \varepsilon^{-\zeta} \label{eq:cond2.3}\end{equation}
\begin{equation} \frac{1}{2}\kappa \varepsilon^{\zeta} + 2\varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \kappa \varepsilon^{\zeta} \label{eq:cond2.4}\end{equation}
\begin{equation}
\varepsilon^{-\frac{\zeta}{2}} + \varepsilon^{\frac{23}{24}} + \pi \varepsilon^{-\zeta} \leq \varepsilon^{-2\zeta} \label{eq:cond2.5.2} \end{equation} \begin{equation} 4 \varepsilon^{-2\zeta + \frac{59}{48}} + 2 \varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \varepsilon \label{eq:cond2.6} \end{equation}
\begin{equation}2 \varepsilon^{\frac{1}{2}} + 2 \varepsilon^{\frac{7}{8}} \leq \varepsilon^{\frac{1}{4}}\label{eq:cond2.7} \end{equation}
\end{lemma}
\begin{proof} \textbf{Equations in lemma \ref{351}} \\
Equation \eqref{eq:cond1.1} holds for \[\varepsilon \leq (\frac{1}{4}\kappa)^{\frac{1728}{2123}}.\]
Equation \eqref{eq:cond1.2} holds for \[\varepsilon \leq \big(\frac{4C_0^2}{\kappa^{13}} \big)^{-\frac{96}{11}}.\]
Equation \eqref{eq:cond1.3} holds if \[8C_0^2\varepsilon^{\frac{427}{432}}(\varepsilon^{100\delta}+3\varepsilon^{\frac{287}{288}}) \leq \varepsilon^{\frac{1285}{864}} \Leftrightarrow 8C_0^2(\varepsilon^{100\delta}+3\varepsilon^{\frac{287}{288}}) \leq \varepsilon^{\frac{431}{864}}\] therefore, if we have \[ \left\{\begin{array}{c} 8C_0^2\varepsilon^{100\delta}\leq \frac{1}{2}\varepsilon{\frac{431}{864}} \\ 24C_0^2 \varepsilon^{\frac{287}{288}}\leq \frac{1}{2}\varepsilon{\frac{431}{864}} \end{array}\right. \] which is satisfied if \[\varepsilon \leq (48C_0^2)^{-\frac{432}{215}} \] then inequality \eqref{eq:cond1.3} holds.
\textbf{Equations in lemma \ref{352}} \\
Equation \eqref{eq:cond2.0.0} holds if \[\varepsilon^{\frac{1}{1152}}+\varepsilon^{\frac{1}{864}} \leq (2C_0)^{-\frac{1}{6}}\frac{\kappa}{32} \] which is satisfied if \[ \left\{\begin{array}{c} \varepsilon^{\frac{1}{1152}} \leq \frac{\kappa}{64}(2C_0)^{-\frac{1}{6}}\\ \varepsilon^{\frac{1}{864}} \leq \frac{\kappa}{64}(2C_0)^{-\frac{1}{6}} \end{array}\right.
\Leftrightarrow
\left\{\begin{array}{c} \varepsilon \leq (\frac{\kappa}{64})^{1152}(2C_0)^{-192}\\ \varepsilon \leq (\frac{\kappa}{64})^{864}(2C_0)^{-144} \end{array}\right.
\]
Equation \eqref{eq:cond2.0} is satisfied if \[ \varepsilon^{\frac{5}{4}-\frac{1}{48}-4\zeta}(1+ \varepsilon^{\frac{23}{24}} + (1+\pi) \varepsilon^{-\frac{\zeta}{2}})^2 \leq (\frac{3\kappa}{128C_0})^2. \] For $\varepsilon \leq 1$, we have \[ 1+ \varepsilon^{\frac{23}{24}} + (1+\pi)\varepsilon^{-\frac{\zeta}{2}} \leq 4\pi \varepsilon^{-\frac{\zeta}{2}}\] then we need \[16\pi^2 \varepsilon^{\frac{5}{4}-\frac{1}{48}-5\zeta} \leq (\frac{3\kappa}{128C_0})^2 \] which is satisfied if \[ \varepsilon \leq (\frac{3\kappa}{512\pi C_0})^{\frac{2119}{1728}}\] Equation \eqref{eq:cond2.1} is satisfied if \[ \varepsilon^{(\frac{5}{4})^{j-1}-\frac{1}{96}-4\zeta}(1+ \varepsilon^{\frac{23}{24}} + (1+\pi) \varepsilon^{-\frac{\zeta}{2}}+ 2 \varepsilon^{\frac{5}{4}-\frac{1}{48}})^2 \leq (\frac{3}{4})^{2j}(\frac{\kappa}{32C_0})^2. \] If $\varepsilon \leq 1$, then \[1+\varepsilon^{\frac{23}{24}} + (1+\pi) \varepsilon^{-\frac{\zeta}{2}}+ 2 \varepsilon^{\frac{5}{4}-\frac{1}{96}} \leq 4\pi \varepsilon^{-\frac{\zeta}{2}}\] then it's enough to have \[16\pi^2\varepsilon^{\frac{5}{4}-\frac{1}{48}-4\zeta-\zeta} \leq (\frac{3}{4})^{2\cdot 104}(\frac{\kappa}{32C_0})^2 \] which is satisfied if \[\varepsilon \leq \big((\frac{3}{4})^{208}(\frac{\kappa}{128\pi C_0})^2) \big)^{\frac{3456}{2119}} \]
Equation \eqref{eq:cond2.2} holds if
\[ 256C_0^2(\frac{4}{3})^{13(j-1)}(\frac{C_0}{\kappa})^{13}( \varepsilon^{\frac{50\delta}{l}} + \varepsilon^{(\frac{5}{4})^{j-1}} ) \leq \varepsilon^{14\zeta+\frac{1}{4}(\frac{5}{4})^{j-1}}.\] We will first show that, for all $j \in \llbracket 2, l \rrbracket$, and for $\varepsilon$ small enough, \[\varepsilon^{\frac{50\delta}{l}} + \varepsilon^{(\frac{5}{4})^{j-1}}\leq \varepsilon^{\frac{1}{3}(\frac{5}{4})^{j-1}}. \] Since $l=56$, this condition is satisfied if for all $j\in \llbracket 2, l \rrbracket$ if \[ \left\{\begin{array}{c} 2\leq \varepsilon^{\frac{1}{3}(\frac{5}{4})^{j-1}-\frac{50\delta}{l}} \\ 2 \leq \varepsilon^{-\frac{1}{3}(\frac{5}{4})^{j-1}}
\end{array}\right. \]
\noindent which holds if
\[ \left\{\begin{array}{c} \varepsilon \leq 2^{\frac{1}{\frac{1}{2}(\frac{5}{4})^{55}-\frac{50\delta}{56}}} \\ \varepsilon \leq 2^{-\frac{12}{25}}
\end{array}\right.\] then equation \eqref{eq:cond2.2} is satisfied if \[ 256C_0^2(\frac{4}{3})^{13(j-1)}(\frac{C_0}{\kappa})^{13} \leq \varepsilon^{14\zeta - \frac{1}{12}(\frac{5}{4})^{j-1}}\Leftrightarrow \varepsilon \leq \big(256C_0^{15}(\frac{4}{3})^{13(j-1)}(\frac{1}{\kappa})^{13}\big)^{\frac{1}{14\zeta - \frac{1}{12}(\frac{5}{4})^{j-1}}}.\] Now, as $C_0 \geq 1$ and $0<\kappa<1$, since $\varepsilon \leq 1$, \[ \big(\frac{24C_0^{15}}{\kappa^{13}}\big)^{\frac{1}{5\zeta - \frac{1}{12}(\frac{5}{4})^{j-1}}} \geq \big(\frac{24C_0^{15}}{\kappa^{13}}\big)^{\frac{1}{14\zeta - \frac{1}{12}(\frac{5}{4})}} = \big(\frac{24C_0^{15}}{\kappa^{13}}\big)^{-\frac{864}{83}} \] and \[(\frac{4}{3})^{\frac{13(j-1)}{14\zeta - \frac{1}{12}(\frac{5}{4})^{j-1}}} \geq (\frac{4}{3})^{\frac{13\cdot 4}{14\zeta - \frac{1}{12}(\frac{5}{4})^{4}}} = (\frac{4}{3})^{-\frac{1437696}{5401}}\] Finally, equation \eqref{eq:cond2.2} is satisfied with
\[\varepsilon\leq \big(\frac{24C_0^{15}}{\kappa^{13}}\big)^{-\frac{1728}{178}}(\frac{4}{3})^{-\frac{1437696}{5545}}\]
Equation \eqref{eq:cond2.3} is satisfied if \[ \varepsilon^{\frac{23}{25}} + \pi \varepsilon^{-\frac{\zeta}{2}} + 2\varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \varepsilon^{-\zeta}.\] So if we have \[ \left\{\begin{array}{c} \varepsilon^{\frac{23}{24}} \leq \frac{1}{10}\varepsilon^{-\zeta} \\ \pi \varepsilon^{-\frac{\zeta}{2}} \leq \frac{4}{5}\varepsilon^{-\zeta} \\ 2\varepsilon^{\frac{59}{48}} \leq \frac{1}{10}\varepsilon^{-\zeta} \end{array}\right. \Leftrightarrow \left\{\begin{array}{c} \varepsilon \leq (\frac{1}{10})^{\frac{1728}{1628}} \\ \varepsilon \leq (\frac{4\pi}{5})^{3456} \\ \varepsilon \leq (\frac{1}{20})^{\frac{1728}{1837}} \end{array}\right. \] then equation \eqref{eq:cond2.3} holds.
Equation \eqref{eq:cond2.4} holds for \[\varepsilon \leq \big( \frac{\kappa}{4} \big)^{\frac{1728}{2141}}. \]
Equation \eqref{eq:cond2.5.2} holds if
\[ \left\{\begin{array}{c} \varepsilon^{-\frac{\zeta}{2}} \leq \frac{1}{3}\varepsilon^{-2\zeta} \\ \varepsilon^{\frac{23}{24}} \leq \frac{1}{3}\varepsilon^{-2\zeta} \\ \pi\varepsilon^{-\zeta} \leq \frac{1}{3}\varepsilon^{-2\zeta} \end{array}\right. \Leftrightarrow \left\{\begin{array}{c} \varepsilon \leq (\frac{1}{3})^{1152} \\ \varepsilon \leq (\frac{1}{3})^{\frac{864}{829}}\\ \varepsilon \leq (\frac{1}{3\pi})^{1728} \end{array}\right. \]
Equation \eqref{eq:cond2.6} holds if
\[ \left\{\begin{array}{c} 4\varepsilon^{\frac{1061}{864}} \leq \frac{1}{2}\varepsilon \\ 2\varepsilon^{\frac{59}{48}} \leq \frac{1}{2}\varepsilon \end{array}\right. \Leftrightarrow \left\{\begin{array}{c} \varepsilon \leq (\frac{1}{8})^{\frac{864}{197}} \\ \varepsilon \leq (\frac{1}{4})^{\frac{48}{11}}
\end{array}\right. \] and then equation \eqref{eq:cond2.6} holds.
Since, for $\varepsilon \leq 1$ we have $\varepsilon^{\frac{7}{8}}\leq \varepsilon^{\frac{1}{2}}$, equation \eqref{eq:cond2.7} holds if \[ 4\varepsilon^{\frac{1}{2}} \leq \varepsilon^{\frac{1}{4}} \] that's it to say, if \[ \varepsilon \leq \frac{1}{256}.\]
Now define $\varepsilon_0$ in order to satisfy conditions \eqref{eq:cond1.1} to \eqref{eq:cond2.7}.
\end{proof}
\begin{lemma}[Inductive lemma with renormalization]\label{351} Let
\begin{itemize} \item $A \in sl(2, \mathbb{R})$, \item $r>0$, \item $\bar A, \bar F \in U_{r}(\mathbb{T} ^d, sl(2, \mathbb{R})), \psi \in U_{r}(2\mathbb{T} ^d, SL(2,\mathbb{R}))$, \item $\vert \bar F \vert_r = \varepsilon$, \item \[ N= \Lambda^{-1}\big(\frac{50 \vert \log \varepsilon \vert}{\pi r }\big) \] \item \[ R = \frac{1}{3N}\Psi^{-1}(\varepsilon^{-\zeta}) \] \item \[ r' = r - \frac{50 \delta \vert \log \varepsilon \vert}{\pi \Lambda(R N)}\] \end{itemize}
Assume $r'>0$. Let $\displaystyle \kappa '' = \frac{\kappa }{\Psi(3RN)} = \kappa \varepsilon^{\zeta}$. Suppose that $\varepsilon\leq \varepsilon_0$ which was defined in Lemma \ref{351} and
\begin{enumerate} \item \label{351-epsilon-small} \[\varepsilon \leq (2C_0)^{-96} \big( \frac{ \kappa''}{32(\Vert A \Vert +1)} \big)^{576}\] \item $\bar A$ is reducible to $A$ by $\psi$, \item \label{normeA} $\Vert A \Vert \leq \varepsilon^{-\frac{\zeta}{2}}$, \item \label{period-psi}for all $G\in \mathcal{C}^0(\mathbb{T}^d, sl(2,\mathbb{R}))$, $\psi^{-1}G\psi \in \mathcal{C}^0(\mathbb{T}^d, sl(2,\mathbb{R}))$, \item \label{boundedPsi} $\vert \psi ^{\pm 1}\vert_{ r} \leq \varepsilon^{-\zeta }$, \end{enumerate} then there exist \begin{itemize} \item $Z' \in U_{ r'}(\mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $\bar A', \bar F' \in U_{ r'}(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $\psi ' \in U_{ r}(2\mathbb{T} ^d, SL(2,\mathbb{R}))$, \item $A'\in sl(2, \mathbb{R})$ \end{itemize} satisfying the following properties :
\begin{enumerate} \item \label{A'red} $\bar A'$ is reducible by $\psi '$ to $A'$, \item \label{period-psi'} for all $G\in \mathcal{C}^0(\mathbb{T}^d, sl(2,\mathbb{R}))$, $\psi'^{-1}G\psi' \in \mathcal{C}^0(\mathbb{T}^d, sl(2,\mathbb{R}))$, \item \label{spectrum-A'} $A'$ has $BR_\omega^{RN}(\frac{3}{4C_0}\kappa '')$ spectrum, where $C_0$ was defined in Lemma \ref{renormalization}, \item \label{conjug} \[ \partial _\omega Z' = (\bar A + \bar F) Z' - Z' (\bar A ' + \bar F')\] \item \label{norme-A'} $\Vert A' \Vert \leq \Vert A \Vert + \varepsilon^{\frac{23}{24}} + \pi N$, \item \label{estim-Z'} \[\vert Z'^{\pm 1} - Id \vert _{ r'} \leq \varepsilon^{\frac{8}{9}}\] \item \label{estim-renorm} for all $s' >0$, \[ \vert \psi'^{-1}\psi \vert_{ s'} \leq 2C_0 \big( \frac{1}{\kappa ''}\big)^{6}e^{2\pi \Lambda(\frac{N}{2})s'} \]\[ \vert \psi^{-1}\psi ' \vert_{s'} \leq 2 C_0 \big( \frac{1}{\kappa ''}\big)^{6}e^{2\pi \Lambda(\frac{N}{2})s'},\] \item \label{estim-psi'}$\vert \psi'^{\pm 1} \vert_{ r} \leq \varepsilon^{-\zeta-\frac{1}{96}}e^{2\pi \Lambda(\frac{N}{2})r}$, \item \label{estim-F'} $\vert \psi^{-1}\bar F' \psi \vert_{r'} \leq \varepsilon^{\frac{5}{4}}$. \item \label{norme2-A'} If moreover the spectrum of $A$ is not $BR_\omega^{RN}(\kappa'')$, then $\Vert A' \Vert \leq \frac{3}{4}\kappa''$, \item \label{estim-Z'-deriv} If the spectrum of $A$ is $BR_\omega^{RN}(\kappa'')$, we have $\Phi \equiv I$, then $\psi' = \psi$ and $\tilde A=A$, and then
\[\vert \psi^{-1}Z'^{\pm 1}\psi \vert_{r'} \leq e^{\varepsilon^{\frac{7}{8}}}, \] \[\vert \partial_\omega(\psi^{-1}Z'^{\pm 1}\psi) \vert_{r'} \leq \varepsilon^{\frac{1}{2}},\]
\end{enumerate}\end{lemma} \begin{proof} \textbf{Algebraic aspects} \\
If $A$ has a double eigenvalue or $\kappa''$-close eigenvalues, let $\Phi$ be defined on $2\mathbb{T}^d$ as constantly equal to $I$ and let $\tilde A=A$. Otherwise, let $\Phi$ a renormalization of $A$ of order $R, N$ given by lemma \ref{renormalization}. Let $\tilde A \in sl(2,\mathbb{R})$ such that \[ \forall \theta \in 2\mathbb{T} ^d, \partial_\omega \Phi(\theta) = A \Phi(\theta)-\Phi(\theta)\tilde A \]
\noindent so $||A-\tilde A||\leq \pi N$ and $\tilde A$ has $BR^{RN}_\omega(\kappa'')$ spectrum. Notice that in this case, $\tilde A$ is not nilpotent. Let $\psi' = \psi \Phi$, and \[ \tilde F:= \psi'^{-1} \bar F \psi ' \]
Moreover, $\Phi$ is trivial with respect to $\mathcal{L}_{A}$ :
\begin{equation}\label{trivialPhi-step} \Phi = P_{L_1}^{\mathcal{L}_{A}} e^{2i\pi \langle m, \cdot \rangle} + P_{L_2}^{\mathcal{L}_{A}} e^{-2i\pi \langle m, \cdot \rangle} \end{equation}
\noindent with $|m|\leq N$ and $||P_{L_i}||\leq \frac{C_0}{\kappa^{''6}}$. Since $\Phi$ is trivial with respect to $\mathcal{L}_{A}$, for all $s' \geq 0$, Lemma \ref{renormalization} implies \begin{equation}\label{estimPhi-prouvee} \vert \Phi ^{\pm1} \vert_{s'} \leq 2C_0\Big( \frac{1 }{\kappa ''} \Big)^{6}e^{2\pi \Lambda(\frac{N}{2})s'} \end{equation} which gives property \ref{estim-renorm}.
Let $\psi'=\psi\Phi$. Let $G\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, then by triviality of $\Phi$, $\Phi^{-1}G\Phi\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, and by the assumption \ref{period-psi}, $\psi^{'-1}G\psi'\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$. Therefore the property \ref{period-psi'} on $\psi'$ holds.
\noindent Moreover, \[ \Vert \hat {\tilde F}(0) \Vert \leq \vert \tilde F \vert_{0} \leq \vert \Phi \vert_{ 0} \vert \Phi ^{-1} \vert_{ 0} \vert \psi \vert_{ 0} \vert \psi^{-1} \vert_{ 0} \vert \bar F \vert_{ 0} \] Therefore by \eqref{estimPhi-prouvee} and by assumption \ref{boundedPsi},
\[ \Vert \hat{\tilde F}(0) \Vert \leq \varepsilon^{1-2\zeta}(2 C_0)^2 \big( \frac{1}{\kappa ''}\big)^{12}.\] Since $\varepsilon \leq (2C_0)^{-96} \big( \frac{ \kappa''}{32(\Vert A \Vert +1)} \big)^{576} \leq (2C_0)^{-96} \kappa''^{576 }$, we get \[ \Vert \hat{ \tilde F}(0) \Vert \leq \varepsilon^{1-2\zeta-\frac{1}{48}}. \] Since $\tilde A$ has a $BR_\omega ^{R N}(\kappa '')$ spectrum, we want to apply lemma \ref{344} with \[ \tilde \varepsilon = \varepsilon^{1-2\zeta-\frac{1}{48}}, \tilde r = r, \tilde r' = r', \kappa '= \frac{ \kappa ''}{C_0}, \tilde N = R N, \] then we need \[ \varepsilon^{1-2\zeta-\frac{1}{48}} \leq \left(\frac{1}{C_0}\cdot \frac{\kappa''}{32(1+\Vert \tilde A \Vert)}\varepsilon^{\zeta}\right)^2\] or sufficiently \[ \varepsilon^{1-2\zeta-\frac{1}{48}} \leq \left(\frac{1}{C_0}\cdot \frac{\kappa''}{32(1+\Vert A \Vert+\pi N)}\varepsilon^{\zeta}\right)^2\] which holds true if \[ \varepsilon^{1-2\zeta-\frac{1}{48}} \leq \left(\frac{1}{C_0}\cdot \frac{\kappa''}{32(2+\pi )}\varepsilon^{2\zeta}\right)^2\]
\noindent (where we have used the assumption that $\Psi\geq id$), which holds true by assumption \ref{351-epsilon-small}. Therefore we can apply lemma \ref{344} to get the maps $X \in U_{r'}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, $F' \in U_{ r'}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, and a matrix $A' \in sl(2, \mathbb{R})$ such that
\begin{itemize} \item $A'$ has $BR_\omega ^{R N} (\frac{3\kappa ''}{4C_0})$ spectrum, \item $\Vert A' - \tilde A \Vert \leq \tilde \varepsilon \leq \varepsilon^{\frac{23}{24}}$ (because $1-2\zeta - \frac{1}{48} \geq \frac{23}{24} $), which implies that \[ \Vert A' - A \Vert \leq \Vert A' - \tilde A \Vert + \Vert A - \tilde A \Vert \leq \varepsilon^{\frac{23}{24}}+\pi N \] and thus \[ \Vert A' \Vert \leq \Vert A \Vert + \varepsilon^{\frac{23}{24}} + \pi N \] which is property \ref{norme-A'}, \item $\partial_\omega e^X = (\tilde A + \tilde F)e^X - e^X(A' + F')$.
Let $\bar F' = \psi' F'(\psi' )^{-1}\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$ and $\bar A ' \in U_{r}(2 \mathbb{T} ^d, sl(2, \mathbb{R}))$ such that \[ \partial_\omega \psi' = \bar A' \psi' - \psi' A' \] (which means that $\bar A'$ is reducible to $A'$, hence Property \ref{A'red} with $\psi ' := \psi \Phi$). Then the function $Z' := \psi' e^X (\psi')^{-1}\in \mathcal{C}^0(\mathbb{T}^d,SL(2,\mathbb{R}))$ is solution of \[ \partial_\omega Z' = (\bar A + \bar F)Z' - Z'(\bar A' + \bar F') \] hence Property \ref{conjug}. This conjugation also implies that $\bar{A}'\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$.
\item if $\tilde A$ has two different eigenvalues, since $\Phi$ is trivial with respect to $\mathcal{L}_{A}$ which is identical to $\mathcal{L}_{\tilde A}$, by Lemma \ref{344} and the expression \eqref{trivialPhi-step}, \[ \vert \Phi X \Phi^{-1}\vert_{ r'}\leq \frac{4C_0^{15}}{\kappa^{''13}} \Psi( 3R N ) \vert \Phi \tilde F \Phi^{-1}\vert_{ r}\] and \[\vert \Phi F' \Phi^{-1} \vert_{ r'} \leq \frac{4C_0^{15}}{\kappa^{''13}} e^{\vert \Phi X\Phi^{-1} \vert_{ r'}} \big( \vert \Phi \tilde F \Phi^{-1} \vert_{ r} \big[ e^{-2\pi \Lambda(R N)(r-r')} + \vert \Phi \tilde F \Phi^{-1}\vert_{ r} \Psi(3RN)(2e+e^{\vert \Phi X \Phi^{-1} \vert_{ r'}}) \big]\]
otherwise if $\tilde A$ is nilpotent, \[ \vert X\vert _{r'}\leq \frac{3}{\kappa^3}\Psi( 3R N )^3\vert \tilde F \vert_{r}\] and \[ \vert F'\vert _{r'} \leq \frac{3}{\kappa^3} e^{\vert X \vert_{ r'}} \vert \tilde F \vert_{ r} \big[ e^{-2\pi \Lambda(R N)(r-r')} \qquad \] \[ \qquad + \vert \tilde F \vert_{ r} \Psi(3RN)^3(2e+e^{\vert X \vert_{ r'}}) \big]\] and if $ad_{\tilde{A}}=0$, \[ \vert X\vert _{r'}\leq \frac{1}{\kappa}\Psi( 3R N )\vert \tilde F \vert_{r}\] and \[ \vert F'\vert _{r'} \leq \frac{1}{\kappa} e^{\vert X \vert_{ r'}} \vert \tilde F \vert_{ r} \big[ e^{-2\pi \Lambda(R N)(r-r')} \qquad \] \[ \qquad + \vert \tilde F \vert_{ r} \Psi(3RN)(2e+e^{\vert X \vert_{ r'}}) \big].\]
Notice that in any case (since $\Phi\equiv I$ if $\tilde A$ is nilpotent or $\operatorname{ad}_{\tilde A}=0$), we have
\begin{equation}\label{estim-X-step} \vert \Phi X \Phi^{-1}\vert_{ r'}\leq \frac{4C_0^{15}}{\kappa^{''13}} \Psi( 3R N )^3 \vert \Phi \tilde F \Phi^{-1}\vert_{ r}\end{equation} and \begin{equation}\label{estim-F'-step} \vert \Phi F' \Phi^{-1} \vert_{ r'} \leq \frac{4C_0^{15}}{\kappa^{''13}} e^{\vert \Phi X\Phi^{-1} \vert_{ r'}} \vert \Phi \tilde F \Phi^{-1} \vert_{ r} \big[ e^{-2\pi \Lambda(R N)(r-r')} + \vert \Phi \tilde F \Phi^{-1}\vert_{ r} \Psi(3RN)^3(2e+e^{\vert \Phi X \Phi^{-1} \vert_{ r'}}) \big]\end{equation}
\end{itemize} \textbf{Estimates} \\
\textbf{Estimate of $\Psi',\Psi^{'-1}, A'$}
With the assumption $\varepsilon \leq (2C_0)^{-96} \big( \frac{ \kappa''}{\Vert A \Vert +1} \big)^{576}$, we have \[ \vert \Phi \vert_{ r} \leq \varepsilon^{-\frac{1}{96}}e^{2\pi \Lambda(\frac{N}{2}) r}\] and similarly for $\Phi^{-1}$. Moreover since $\vert \psi \vert_{ r} \leq \varepsilon^{-\zeta}$, we get property \ref{estim-psi'}: \[\vert \psi'\vert_{r} = \vert\psi \Phi\vert_{r} \leq \vert \psi \vert_{r} \vert \Phi \vert_{r} \leq \varepsilon^{-\zeta-\frac{1}{96}}e^{2\pi \Lambda(\frac{N}{2})r}\] and similarly for $\psi'^{-1}$. Notice that this inequality remains true if $\Phi \equiv id$.
Notice that if $\Phi \not\equiv I$ (that is to say if the spectrum of $A$ is resonant), then from lemma \ref{renormalization} we get $\Vert \tilde A \Vert \leq \frac{1}{2}\kappa''$ and then for $\varepsilon \leq \varepsilon_0$ defined in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond1.1}),
\begin{equation*}\begin{split} \Vert A' \Vert & \leq \Vert \tilde A \Vert + \Vert \hat{\tilde F}(0) \Vert \\
& \leq \frac{1}{2}\kappa \varepsilon^{\zeta} + \varepsilon^{1-2\zeta - \frac{1}{48}} \\
& \leq \frac{1}{2}\kappa \varepsilon^{\frac{1}{1728}} + \varepsilon^{\frac{845}{864}}\\
& \leq \frac{3}{4}\kappa \varepsilon^{\frac{1}{1728}} = \frac{3}{4}\kappa'' \end{split}\end{equation*} and property \ref{norme2-A'} is satisfied.
\textbf{Estimate of $Z^{'\pm 1}-I,\psi^{-1}(Z^{'\pm 1})\psi$ and its derivative}
Since $\tilde F = (\psi \Phi)^{-1} \bar F \psi \Phi$, then
\begin{equation}\label{estim-Ftilde} \vert \Phi\tilde F \Phi^{-1} \vert_{ r} = \vert \psi^{-1}\bar F \psi \vert_{ r'} \leq \vert \bar F \vert_{ r'} \varepsilon^{-2\zeta} = \varepsilon^{1-2\zeta} \end{equation}
Recall the estimate \eqref{estim-X-step}:
\begin{equation}\vert \Phi X \Phi^{-1}\vert_{ r'}\leq \frac{4C_0^{15}}{\kappa''^{13}} \Psi( 3R N )^3 \vert \Phi \tilde F \Phi^{-1}\vert_{ r}\end{equation} therefore by \eqref{estim-Ftilde}, and for $\varepsilon \leq \varepsilon_0$ defined in lemma \ref{smallness-epsilon} ((see equation \eqref{eq:cond1.2}), \begin{equation}\label{estim-X-bis} \vert \Phi X \Phi^{-1} \vert_{ r'} \leq \frac{4C_0^{15}}{\kappa^{13}}\varepsilon^{-16\zeta}\varepsilon^{1-2\zeta}\leq \varepsilon^{\frac{7}{8}} \end{equation} then \[ e^{ \vert \Phi X \Phi^{-1}\vert_{ r'} } \leq e^{\varepsilon^{\frac{7}{8}}} \leq 2 \]
\noindent We now estimate $\vert Z'-I\vert _{r'}=\vert \psi \Phi (e^X-I)(\psi \Phi)^{-1}\vert_{ r'}$. From \eqref{estim-X-bis}, \[ \vert \Phi e^X \Phi^{-1} -Id \vert_{r'}\leq e\vert \Phi X\Phi^{-1}\vert _{r'} \leq e \varepsilon^{\frac{7}{8}}\]
\noindent Then \[ \vert Z'-I\vert _{r'}=\vert \psi \Phi e^X (\psi \Phi)^{-1}-Id\vert_{ r'} \leq \vert \psi \vert_{ r'}\vert \Phi e^X \Phi^{-1}-Id\vert_{ r'} \vert \psi^{-1}\vert_{ r'}\leq e\varepsilon^{\frac{7}{8}-2\zeta}\]
\noindent hence property \ref{estim-Z'} is satisfied. If $\Phi \equiv I$, we have \[\psi^{-1}Z'\psi =\psi^{-1}\psi\Phi e^X (\psi \Phi)^{-1}\psi = e^X, \] therefore \[\vert \psi^{-1}Z'\psi \vert_{r'} \leq \vert e^X \vert_{r'}\leq e^{\varepsilon^{\frac{7}{8}}} \]
\noindent which is the first part of the property \ref{estim-Z'-deriv}. Now Lemma \ref{344} also states that if $\Phi\equiv I$ (that is, $\tilde A = A$), \[ \vert \partial_\omega X\vert_{r'}\leq 2\Vert A\Vert \ \vert X\vert_{r'}+\vert \tilde F\vert_{r'}\] which implies that
\[ \vert \partial_\omega X\vert_{r'}\leq 2\Vert A\Vert\varepsilon^{\frac{7}{8}}+\varepsilon^{1-2\zeta }\leq \varepsilon^{\frac{7}{8}}(2\Vert A \Vert+1)\]
\noindent and by the assumption \ref{351-epsilon-small}, \[\vert \partial_\omega X\vert_{r'}\leq\varepsilon^{\frac{4}{5}}.\] Therefore, \[\vert \partial_\omega(\psi^{-1}Z'\psi) \vert_{r'}= \vert \partial_\omega(X)e^X \vert_{r'}\leq e^{\varepsilon^{\frac{7}{8}}}\varepsilon^{\frac{4}{5}}\leq \varepsilon^{\frac{1}{2}}\] hence property \ref{estim-Z'-deriv}.
\textbf{Estimate of $\psi^{-1}\bar{F}'\psi=\Phi F'\Phi^{-1}$}
From Equation \eqref{estim-F'-step},
\[\vert \Phi F' \Phi^{-1} \vert_{ r'} \leq 4C_0^2 e^{\vert \Phi X\Phi^{-1} \vert_{ r'}} \big( \frac{C_0}{ \kappa''}\big)^{13} \vert \Phi \tilde F \Phi^{-1} \vert_{ r} \big[ e^{-2\pi \Lambda(R N)(r-r')} \qquad \] \[ \qquad + \vert \Phi \tilde F \Phi^{-1} \vert_{ r} \Psi(3R N)^3(2e+e^{\vert \Phi X \Phi^{-1} \vert_{ r'}}) \big]\]
Moreover, by definition, we have $\Lambda(RN) = \frac{50 \delta \vert \log \varepsilon \vert}{\pi (r-r')} $, thus \[e^{-2\pi \Lambda(R N)(r-r')} = \varepsilon^{100\delta} \] and then, because we assumed $\varepsilon \leq (2C_0)^{-96} \big( \frac{ \kappa''}{\Vert A \Vert +1} \big)^{576}$ and $\Psi(3 R N) = \varepsilon^{-\zeta} $, \[\vert \Phi F' \Phi^{-1} \vert_{ r'} \leq 8C_0^2\varepsilon^{-\frac{1}{96}}\varepsilon^{1-2\zeta}(\varepsilon^{100\delta}+ 8\Psi(3R N)^3 \varepsilon^{1-2\zeta}).\]
Thus \[\vert \Phi F' \Phi^{-1}\vert_{ r'} \leq 8C_0^2\varepsilon^{1-2\zeta-\frac{1}{96}}(\varepsilon^{100\delta}+\varepsilon^{1-6\zeta}) \leq \varepsilon^{\frac{3}{2}}. \] Hence property \ref{estim-F'} holds for $\varepsilon\leq \varepsilon_0$ as defined in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond1.3}). \newline
\end{proof}
\section{Inductive step}\label{inductivestep}
Let's define the following functions which will be used for the complete iterative step : \[ \tag{P}\label{parameters}
\left\{\begin{array}{c} \displaystyle \kappa''(\varepsilon) =\kappa \varepsilon^{\zeta}\\ \displaystyle N(r,\varepsilon) = \Lambda^{-1}\big(\frac{50 \vert \log \varepsilon \vert}{\pi r}\big) \\ \displaystyle R(r,\varepsilon) = \frac{1}{3N(r,\varepsilon)}\Psi^{-1}(\varepsilon^{-\zeta}) \\ \displaystyle r''(r,\varepsilon) = r-\frac{50 \delta \vert \log \varepsilon \vert}{\pi \Lambda(R(r,\varepsilon)N(r,\varepsilon))} \end{array}\right. \]
\noindent Note that these definitions match with lemma \ref{351}.
\begin{lemma}\label{352} Let \begin{itemize} \item $A \in sl(2, \mathbb{R})$, \item $r>0,$ \item $\bar A, \bar F \in U_{r}(\mathbb{T} ^d, sl(2, \mathbb{R})), \psi \in U_{r}(2\mathbb{T} ^d, SL(2,\mathbb{R}))$, \item $\vert \bar F \vert_r = \varepsilon$. \end{itemize} Suppose that
\begin{enumerate} \item $\varepsilon\leq \varepsilon_0$, where $\varepsilon_0$ is defined in Lemma \ref{smallness-epsilon}, \item $r''>0$, \item $\bar A$ is reducible to $A$ by $\psi$, \item for all $G\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, $\psi^{-1}G \psi \in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $\vert \psi ^{\pm 1}\vert_{r} \leq \varepsilon^{-\zeta}$, \item \label{estimationA} $\Vert A \Vert \leq \varepsilon^{-\frac{\zeta}{2}}$, \end{enumerate} then, there exist \begin{itemize} \item $Z' \in U_{ r''}(\mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $\bar A', \bar F' \in U_{ r''}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, \item $\psi' \in U_{ r}(2\mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $A' \in sl(2,\mathbb{R})$ \end{itemize} satisfying the following properties:
\begin{enumerate} \item \label{prop1} $\bar A'$ is reducible to $A'$ by $\psi '$, \item \label{prop2} for all $G\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, $\psi^{-1}G \psi \in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item \label{prop3} $\vert \bar{F'} \vert_{ r''} \leq \varepsilon^{2\delta}$, \item \label{prop4} $\vert \psi '^{\pm 1} \vert_{ r''} \leq \varepsilon^{-2\delta \zeta}$, \item \label{prop5} $\Vert A' \Vert \leq \Vert A \Vert +\varepsilon^{-\zeta}\leq \varepsilon^{-(2\delta)\frac{\zeta}{2}}$, \item \label{prop6} \[ \partial _\omega Z' = (\bar A + \bar F) Z' - Z' (\bar A ' + \bar F'),\] \item \label{prop7} \[ \vert Z '^{ \pm 1} - Id \vert _{ r''} \leq \varepsilon^{\frac{9}{10}}. \] \item \label{prop8} If moreover the spectrum of $A$ was not $BR_\omega^{R(r,\varepsilon)N(r, \varepsilon)}(\kappa''(\varepsilon))$, we actually have $\Vert A' \Vert \leq \kappa''(\varepsilon)$; \item \label{prop9} If the spectrum of $A$ was $BR_\omega^{R(r,\varepsilon)N(r, \varepsilon)}(\kappa''(\varepsilon))$, we actually have $\psi' = \psi$ and then \begin{equation}\label{prop9.1}\vert \psi^{-1}Z'^{\pm 1}\psi \vert_{r''}\leq (1+2\varepsilon)e^{2\varepsilon} \end{equation} and \[\vert \partial_\omega(\psi^{-1}Z'^{\pm 1}\psi )\vert_{r''}\leq \varepsilon^{\frac{1}{4}}. \] \end{enumerate} \end{lemma}
\begin{proof} \textbf{Removing the resonances and first step} \\
Let $R=R(r,\varepsilon)$, $N=N(r, \varepsilon)$, $ \kappa'' = \kappa''(r,\varepsilon)$, $r'' = r''(r, \varepsilon)$.
Since $\kappa''=\kappa\varepsilon^{\zeta}$, $\Vert A \Vert \leq \varepsilon^{-\frac{\zeta}{2}}$ and $\varepsilon \leq \varepsilon_0$ as defined in Lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.0.0}),
\[ \varepsilon^{1-576\zeta} \leq (2C_0)^{-96}\big(\frac{\kappa}{32(\Vert A \Vert +1)}\big)^{576} \] therefore \[ \varepsilon \leq (2C_0)^{-96}\Big(\frac{ \kappa '' }{32(\Vert A \Vert +1) }\Big)^{576}\] and the assumption of lemma \ref{351} is satisfied. We can apply lemma \ref{351} to get: \begin{itemize} \item $Z_1\in U_{ \frac{r+r''}{2}}(\mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $\psi' \in U_{ \frac{r+r''}{2}}(2\mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $A_1\in sl(2, \mathbb{R})$, \item $\bar A_1 \in U_{ \frac{r+r''}{2}}(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item and $F_1 = (\psi ')^{-1}\bar F_1 \psi' $, with $\bar{F}_1\in U_{ \frac{r+r''}{2}}(\mathbb{T}^d,sl(2,\mathbb{R}))$ \end{itemize} such that \begin{enumerate} \item $\bar A_1$ is reducible to $A_1$ by $\psi'$, \item for all $G\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, $\psi'^{-1}G\psi'\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, which implies that $F_1\in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $A_1$ has $BR_\omega^{R N}(\frac{3}{4}\frac{\kappa ''}{C_0})$ spectrum, \item \[ \partial _\omega Z_1 = (\bar A + \bar F) Z_1 - Z_1 (\bar A_1 + \bar F_1),\] \item
\begin{equation}\label{A_1-wrt-A}\Vert A_1\Vert \leq \Vert A \Vert + \varepsilon^{\frac{23}{24}} + \pi N,\end{equation} \item \begin{equation}\label{estim-Z_1-iteration}
\vert Z_1^{\pm 1} - Id \vert _{ r''} \leq \varepsilon^{\frac{8}{9}}, \end{equation} \item for all $s' >0$, \[ \vert \psi'^{-1}\psi \vert_{ s'} \leq 2C_0 \big( \frac{1}{\kappa ''}\big)^{6}e^{2\pi \Lambda(\frac{N}{2}) s'} \] \[\vert \psi^{-1}\psi ' \vert_{ s'} \leq 2 C_0 \big( \frac{1}{\kappa ''}\big)^{6}e^{2\pi \Lambda(\frac{N}{2}) s'}\] \\
\item $\vert \psi'^{\pm 1} \vert_{ r''} \leq \varepsilon^{-\zeta-\frac{1}{96}}e^{2\pi \Lambda(\frac{N}{2}) r}$,
\item \[\vert \psi^{-1} \bar F_1 \psi \vert_{r''} \leq \varepsilon^{\frac{5}{4}}, \] \item If the spectrum of $A$ was not $BR_\omega^{RN}(\kappa'')$, $\Vert A_1 \Vert \leq \frac{1}{2}\kappa''$; \item If the spectrum of $A$ was $BR_\omega^{RN}(\kappa'')$, we actually have $\psi' = \psi$ and then \[ \vert \psi^{-1}Z_1^{\pm 1}\psi \vert_{r''} \leq e^{\varepsilon^{\frac{7}{8}}}\] \begin{equation}\label{estim-derivZ1}\vert \partial_\omega(\psi^{-1}Z_1^{\pm 1}\psi)\vert_{r''} \leq \varepsilon^{\frac{1}{2}}. \end{equation} \end{enumerate} \textbf{Second step : iteration without resonances} \\
We will now iterate lemma \ref{344} a certain number of times, without renormalization. \\ Let $l = E(\frac{\log(100\delta)}{\log(\frac{4}{3})}) = 56$ which satisfies
\[ \varepsilon^{(\frac{4}{3})^{l+1}} \leq e^{-2\pi \Lambda(R N)(r-r'')} =\varepsilon^{100\delta} \leq \varepsilon^{(\frac{4}{3})^l}.\] Define for all $j\geq 0$, the sequences $\varepsilon'_j = \varepsilon^{(\frac{5}{4})^j}\varepsilon^{-\frac{1}{48}}$ and $r'_j= \frac{r+r''}{2} - j\frac{r-r''}{2l}$ . Thus $r'_0=\frac{r+r''}{2}$ and $r'_l=r''<r$.\\ We want to iterate $l-1$ times lemma \ref{344}, from $j=2$, with \begin{itemize} \item $\tilde \varepsilon = \varepsilon'_{j-1}$, \item $\tilde r = r'_{j-2} $, \item $\tilde r' = r'_{j-1} $, \item $\kappa ' = (\frac{3}{4})^{j-1}\frac{ \kappa ''}{C_0}$, \item $\tilde N = R N$, \item $\tilde F = F_{j-1}$, \item $\tilde A = A_{j-1}$, \item $\Phi = \psi^{-1}\psi'$, \end{itemize}
\underline{First iterate of lemma \ref{344}}: From \[\vert \psi^{-1}\bar F_1 \psi \vert_{0} \leq \varepsilon^{\frac{5}{4}}\] and
\[ \vert \psi'^{-1}\psi \vert_{0} \leq 2C_0 \big( \frac{1}{\kappa ''}\big)^{6}\leq \varepsilon^{-\frac{1}{96}},\ \vert \psi^{-1}\psi ' \vert_{0} \leq 2 C_0 \big( \frac{1}{\kappa ''}\big)^{6} \leq \varepsilon^{-\frac{1}{96}},\] then \[ \Vert \hat{F_1}(0) \Vert \leq \vert\psi'^{-1} \psi \vert_0 \vert\psi^{-1} \psi' \vert_0 \vert \psi^{-1}\bar{F_1}\psi \vert_0 \leq \varepsilon^{\frac{5}{4}-\frac{1}{48}}.\]
As $A_1$ has $BR_\omega^{RN}(\frac{3}{4}\frac{\kappa''}{C_0})$ spectrum, to apply lemma \ref{344} we need \[\varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \big(\frac{(\frac{3}{4})\frac{\kappa''}{C_0}}{32(1 + \Vert A_1 \Vert)} \big)^{2}\frac{1}{\Psi(RN)^2}\] and since \[\Vert A_1 \Vert \leq \Vert A \Vert + \varepsilon^{\frac{23}{24}} + \pi N \leq \varepsilon^{-\frac{\zeta}{2}} + \varepsilon^{\frac{23}{24}} + \pi \varepsilon^{-\zeta} \]
\noindent (this last inequality comes from the fact that $\Psi\geq id$, which implies that $N=\frac{1}{3R}\Psi^{-1}(\varepsilon^{-\zeta})\leq \varepsilon^{-\zeta}$), this remains true if \[\varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \big(\frac{(\frac{3}{4})\frac{\kappa''}{C_0}}{32(1 + (\pi +1)\varepsilon^{-\zeta}+\varepsilon^{\frac{23}{24}}) }\big)^{2}\frac{1}{\Psi(RN)^2}\] which holds for $\varepsilon \leq \varepsilon_0$ as in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.0}).
\underline{Iteration of lemma \ref{344}}
If for some $j\geq2$ \[ \varepsilon^{(\frac{5}{4})^j-\frac{1}{48}} \leq \Big( \frac{(\frac{3}{4})^j\frac{ \kappa''}{C_0}}{ {32}(1+\varepsilon^{\frac{23}{24}} +(1+\pi)\varepsilon^{- \zeta} + 2\varepsilon^{\frac{5}{4}-\frac{1}{48}})} \Big)^{2} \frac{1}{\Psi(R N)^2},\]
which holds true for $\varepsilon \leq \varepsilon_0$ as in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.1}), then \[\varepsilon'_j \leq \Big( \frac{(\frac{3}{4})^j\frac{ \kappa''}{C_0}}{{32}(1+\Vert A_1 \Vert + \sum_{i=1}^{j-1}\varepsilon _i)}\Big)^{2}\frac{1}{\Psi(R N)^2}. \]
Let $j\geq 2$ and assume that $A_{j-1}$ has $BR_\omega^{R N}((\frac{3}{4})^{j-1}\frac{\kappa''}{C_0})$ spectrum, $F_{j-1} \in U_{r'_{j-2}}(\mathbb{T} ^d, sl(2,\mathbb{R}))$, and
\[ \Vert \hat F_{j-1}(0) \Vert \leq \varepsilon'_{j-1};\ |\Psi^{-1}\Psi' F_{j-1}\Psi^{'-1}\Psi|_{r'_{j-2}}\leq \varepsilon^{(\frac{5}{4})^{j-1}}. \] We obtain via lemma \ref{344} functions $F_j$, $X_j\in U_{r_{j-1}}(\mathbb{T}^d,sl(2,\mathbb{R}))$ and a matrix $A_j \in sl(2,\mathbb{R})$ such that \begin{enumerate} \item $A_j$ has $BR_\omega ^{R N}((\frac{3}{4})^j \frac{\kappa ''}{C_0})$ spectrum, \item $\Vert A_{j} \Vert \leq \Vert A_{j-1}\Vert + \varepsilon'_{j-1}$, \item \[\partial_\omega e^{X_j} = (A_{j-1}+F_{j-1})e^{X_j} - e^{X_j}(A_j +F_j), \]
\item the following estimates hold: \begin{itemize}
\item if $A_{j-1}$ has two different eigenvalues:
\[\vert \psi^{-1}\psi' X_j \psi'^{-1}\psi \vert_{r'_{j-1}} \leq 4C_0^2 \big(\frac{1}{(\frac{3}{4})^{j-1}\frac{\kappa''}{C_0}} \big)^{13} \Psi(3RN) \vert \psi^{-1}\psi'F_j \psi'^{-1}\psi\vert_{r'_{j-1}}, \] \begin{equation*}\begin{split} \vert \psi^{-1} \psi ' F_j \psi'^{-1}\psi \vert_{ r'_{j-1}}& \leq 4C_0^2\big( \frac{1}{(\frac{3}{4})^{j-1}\frac{ \kappa ''}{C_0}}\big)^{13} e^{\vert \psi^{-1}\psi ' X_{j-1} \psi'^{-1}\psi\vert_{ r'_{j-1}}} \vert \psi^{-1} \psi ' F_{j-1} \psi'^{-1}\psi \vert_{ r'_{j-2}} \\ &\big[ e^{-2 \pi \Lambda(R N)(r'_{j-2}-r'_{j-1})} + \vert \psi^{-1} \psi ' F_{j-1} \psi'^{-1}\psi \vert_{ r'_{j-2}}\Psi(3R N)(2e+ e^{\vert \psi^{-1}\psi ' X_{j-1} \psi'^{-1}\psi\vert_{ r'_{j-1}}}) \big] ; \end{split}\end{equation*}
\item if $A_{j-1}$ is nilpotent:
\[ \vert X_j \vert_{r'_{j-1}} \leq \frac{3}{\kappa^3}\Psi(3RN)^3\vert F_{j-1}\vert_{r'_{j-1}}, \] \[ \vert F_j \vert_{r'_{j-1}} \leq \frac{3}{\kappa^3}e^{\vert X_{j-1}\vert_{r'_{j-1}}}\vert F_{j-1}\vert_{r'_{j-2}} \big[e^{-2\pi \Lambda(RN)(r'_{j-2}-r'_{j-1})}+\vert F_{j-1}\vert_{r'_{j-2}}\Psi(3RN)^3(2e+e^{\vert X_{j-1}\vert_{r'_{j-1}}}) \big].\]
\item if $ad_{A_{j-1}}=0$:
\[ \vert X_j \vert_{r'_{j-1}} \leq \frac{1}{\kappa}\Psi(3RN)\vert F_{j-1}\vert_{r'_{j-1}}, \] \[ \vert F_j \vert_{r'_{j-1}} \leq \frac{1}{\kappa}e^{\vert X_{j-1}\vert_{r'_{j-1}}}\vert F_{j-1}\vert_{r'_{j-2}} \big[e^{-2\pi \Lambda(RN)(r'_{j-2}-r'_{j-1})}+\vert F_{j-1}\vert_{r'_{j-2}}\Psi(3RN)(2e+e^{\vert X_{j-1}\vert_{r'_{j-1}}}) \big].\] \end{itemize} \end{enumerate}
Notice that in any case we have \begin{equation}\vert \psi^{-1}\psi' X_j \psi'^{-1}\psi \vert_{r'_{j-1}} \leq 4C_0^2 \big(\frac{1}{(\frac{3}{4})^{j-1}\frac{\kappa''}{C_0}} \big)^{13} \Psi(3RN) \vert \psi^{-1}\psi'F_j \psi'^{-1}\psi\vert_{r'_{j-1}} \label{Xj-}, \end{equation}
\begin{equation}\begin{split} \vert \psi^{-1} \psi ' F_j \psi'^{-1}\psi \vert_{ r'_{j-1}}& \leq 4C_0^2 \big( \frac{1}{(\frac{3}{4})^{j-1}\frac{ \kappa ''}{C_0}}\big)^{13} e^{\vert \psi^{-1}\psi ' X_{j-1} \psi'^{-1}\psi\vert_{ r'_{j-1}}} \vert \psi^{-1} \psi ' F_{j-1} \psi'^{-1}\psi \vert_{ r'_{j-2}} \\ &\big[ e^{-2 \pi \Lambda(R N)(r'_{j-2}-r'_{j-1})} +8 \vert \psi^{-1} \psi ' F_{j-1} \psi'^{-1}\psi \vert_{ r'_{j-2}}\Psi(3R N)\big] \label{estimFj}, \end{split}\end{equation} so we will use these estimates to iterate lemma \ref{344}.
Estimates $ \varepsilon \leq (2C_0)^{-96}\Big(\frac{ \kappa '' }{32(\Vert A \Vert +1) }\Big)^{576} $ and $\vert \psi^{-1}\psi' F_{j-1} \psi'^{-1} \psi \vert_{r_{j-2}} \leq \varepsilon^{(\frac{5}{4})^{j-1}}$ give \[ e^{\vert \psi^{-1}\psi'X_{j-1}\psi'^{-1}\psi \vert_{r'_{j-2}}} \leq 2 \] Hence from \eqref{estimFj}
\[ \vert \psi^{-1}\psi'F_j\psi'^{-1}\psi\vert_{ r'_{j-1}} \leq 64\cdot 4C_0^2 \Psi(3R N) \big( \frac{\Psi(3R N)}{(\frac{3}{4})^{j-1}\frac{\kappa}{C_0} }\big)^{13} \vert \psi^{-1}\psi' F_{j-1} \psi'^{-1}\psi \vert_{ r'_{j-2}} \big[ e^{-2\pi \Lambda(R N)(r'_{j-2}-r'_{j-1})} \] \[ \qquad + \vert \psi^{-1}\psi'F_{j-1} \psi'^{-1}\psi \vert_{ r'_{j-2}} \big] \] \[\leq 256C_0^2 \Psi(3R N)^{14}\big( \frac{1}{(\frac{3}{4})^{j-1}\frac{\kappa}{C_0} }\big)^{13} \varepsilon^{(\frac{5}{4})^{j-1}} ( \varepsilon^{\frac{50\delta}{l}} + \varepsilon^{(\frac{5}{4})^{j-1}} ) \] Since $\Psi(3RN) = \varepsilon^{-\zeta}$ and $\varepsilon\leq \varepsilon_0$ defined in lemma \ref{smallness-epsilon} ((see equation \eqref{eq:cond2.2}), \begin{equation}\label{Fj}\vert \psi^{-1}\psi'F_j\psi'^{-1}\psi\vert_{ r'_{j-1}} \leq \varepsilon^{(\frac{5}{4})^j} \end{equation}
We will now estimate $\Vert \hat F_j(0) \Vert$ to iterate lemma \ref{344}: \[ \Vert \hat{F}_j (0) \Vert \leq \vert F_j \vert_{ 0} = \vert ( \psi'^{-1}\psi )\psi^{-1}\psi'F_j \psi'^{-1} \psi(\psi^{-1} \psi') \vert_{ 0} \leq \vert \psi^{-1} \psi' \vert_{ 0} \vert \psi'^{-1} \psi \vert_{ 0} \vert \psi^{-1}\psi'F_j \psi'^{-1} \psi\vert_{ r_{j-1}} , \]
\noindent therefore \[ \Vert \hat F_j(0)\Vert \leq \vert \psi^{-1} \psi' \vert_{0} \vert \psi'^{-1} \psi \vert_{0} \varepsilon^{(\frac{5}{4})^j} \leq \varepsilon^{(\frac{5}{4})^j}\varepsilon^{-\frac{1}{48}}=\varepsilon'_j,\] and we can iterate lemma \ref{344}, $l-1$ times.
Equations \eqref{Fj} and \eqref{Xj-} imply that \begin{equation}\label{Xj}\vert \psi^{-1}\psi'X_j\psi'^{-1}\psi \vert_{r_{j-1}}\leq \varepsilon'_j\end{equation}
and
\[e^{\vert \psi^{-1}\psi'X_j\psi'^{-1}\psi \vert_{r_{j-1}}}\leq 2 \qquad .\]
\textbf{Conclusion :}
Let $Z = e^{X_2}...e^{X_{l+1}} \in U_{r''}(\mathbb{T} ^d,SL(2,\mathbb{R}))$.
Let $Z' = Z_1 \psi' Z \psi'^{-1}$, $A'=A_{l+1}$, $F'=F_{l+1}$, $\bar F' = \psi' F' \psi^{'-1}$ (hence Property \ref{prop2}) and $\bar A'$ such that \[ \partial_\omega \psi' = \bar A' \psi ' - \psi' A', \] then \[ \partial_\omega Z' = (\bar A + \bar F) Z' - Z' (\bar A' + \bar F'), \] hence the properties \ref{prop1} and \ref{prop6} hold.
\noindent We have \[ \partial_\omega Z = (A_1 + F_1)Z - Z(A_{l+1}+F_{l+1})\] and since for all $j\geq 2$, we have
\begin{equation}\label{major-Aj-iteration} \Vert A_j \Vert \leq \Vert A_{j-1}\Vert + \varepsilon^{(\frac{5}{4})^{j-1}}\varepsilon^{-\frac{1}{48}} \leq \Vert A_1 \Vert + \sum_{i=1}^{j-1}\varepsilon^{(\frac{5}{4})^{i}}\varepsilon^{-\frac{1}{48}}= \Vert A_1 \Vert + \sum_{i=1}^{j-1}\varepsilon'_i, \end{equation}
\noindent then
\[ \Vert A' \Vert \leq \Vert A_1 \Vert + \sum_{i=1}^l \varepsilon_i \leq \Vert A \Vert + \varepsilon^{\frac{23}{24}} + \pi N + \sum_{i=1}^l \varepsilon^{(\frac{5}{4})^i-\frac{1}{48}}. \] Remind that $\Psi\geq id$ implies \[N \leq \varepsilon^{-\frac{\zeta}{2}} \] and then, since $\Vert A \Vert \leq \varepsilon^{\frac{-\zeta}{2}}$,
\[ \Vert A' \Vert \leq \Vert A \Vert +\varepsilon^{-\zeta} \leq 2\varepsilon^{-\zeta} \leq \varepsilon^{-\delta\zeta} \] thus the property \ref{prop5} holds if $\varepsilon \leq \varepsilon_0$ as defined in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.3}).
Moreover,
\begin{equation}\label{estimFl+1} \vert \psi^{-1}\psi'F_{l+1}\psi'^{-1}\psi\vert_{ r'_{l}} \leq \varepsilon^{(\frac{5}{4})^{l+1}} \end{equation} and since $l=56$, one has
\[ \vert \psi'F_{l+1}\psi'^{-1}\vert_{ r'_{l}} \leq |\psi|_{r'_l}|\psi^{-1}|_{r'_l} \varepsilon^{(\frac{5}{4})^{l+1}}\leq \varepsilon^{(\frac{5}{4})^{57}-2\zeta} \leq \varepsilon^{2\delta} \]
\noindent thus the property \ref{prop3} holds. In the case the spectrum of $A$ was resonant, the function $\Phi$ used in lemma \ref{351} is not the identity and we have \[ \Vert A' \Vert \leq \Vert A_1 \Vert + \sum_{i=1}^l \varepsilon'_i \leq \frac{1}{2}\kappa''(r, \varepsilon) + 2 \varepsilon'_1 \leq \frac{1}{2}\kappa \varepsilon^{\zeta} + 2\varepsilon^{\frac{5}{4}-\frac{1}{48}} \leq \kappa \varepsilon^{\zeta} = \kappa''(r,\varepsilon) \] since $\varepsilon \leq \varepsilon_0$ as defined in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.4}), whence the property \ref{prop8}.
\textbf{Estimates}
Now we will show property \ref{prop4} : $\vert\psi'^{\pm 1}\vert_{r''} \leq \varepsilon^{- 2\delta\zeta}$ .\\ We know that $\displaystyle \vert \psi'^{\pm 1} \vert_{ r''} \leq \varepsilon^{-\zeta-\frac{1}{96}}e^{2\pi \Lambda(\frac{N}{2}) r}$. But, by definition of $\Lambda(N) = \frac{50\vert\log\varepsilon\vert}{\pi r}$,
\[e^{2\pi\Lambda(\frac{N}{2})r} \leq e^{2\pi\Lambda(N)r} \leq e^{100\vert \log \varepsilon \vert} = \varepsilon^{-100} \] therefore
\[ \vert\psi'^{\pm 1}\vert_{r''} \leq \varepsilon^{-\zeta - \frac{1}{96}-100} \leq \varepsilon^{-2\zeta \delta} .\]
\noindent which is Property \ref{prop4}. Now we will show the property \ref{prop3}. One has \[\vert \bar{F'}\vert_{r''} = \vert \psi'F'\psi'^{-1}\vert_{r''} = \vert \psi \psi^{-1}\psi' F' \psi'^{-1}\psi \psi^{-1}\vert_{r''} \leq \vert \psi \vert_r \vert \psi^{-1}\vert_r \vert \psi^{-1}\psi' F' \psi'^{-1}\psi \vert_{r''}\leq \varepsilon^{-2\zeta}\varepsilon^{(\frac{5}{4})^l} \leq \varepsilon^{2\delta} \] where the last inequality uses equation \eqref{estimFl+1}, which gives property \ref{prop3}.
According to the estimate \[ \vert Z_1^{\pm 1} - Id \vert _{ r''} \leq \varepsilon^{\frac{8}{9}}\]
\noindent obtained in \eqref{estim-Z_1-iteration}, we get \begin{equation*} \begin{split}
\vert Z' - Id \vert_{ r''} & \leq \vert Z_1 - Id \vert_{ r'_1} + \vert Z_1\vert_{r'_1} \vert \psi \vert_{ r}\vert \psi^{-1} \vert_{ r}\sum_{j=2}^{l+1} \vert \psi^{-1} \psi' X_j \psi'^{-1} \psi \vert_{ r'_j} \\
& \leq \varepsilon^{\frac{8}{9}}+(\varepsilon^{\frac{8}{9}}+1)\varepsilon^{-2\zeta}\sum_{j=2}^{l+1}\vert \psi^{-1}\psi'X_j\psi'^{-1}\psi \vert_{r'_j} \\ \end{split} \end{equation*}
Therefore, by the estimate \eqref{Xj} and by definition of $l=E(\frac{\log(100\delta)}{\log(\frac{7}{6})})$, \[ \vert Z' - Id \vert_{r''} \leq \varepsilon^{\frac{8}{9}} +(\varepsilon^{\frac{8}{9}}+1) 2\varepsilon'_1\varepsilon^{-2\zeta} \leq \varepsilon^{\frac{9}{10}} \] hence \ref{prop7}. \\
\textbf{Proof of the property \ref{prop9}}
We now have to estimate $\psi^{-1}Z'\psi$ and its directional derivative in the case $A$ has a $BR_\omega^{R(r,\varepsilon)N(r,\varepsilon)}(\kappa''(\varepsilon))$ spectrum. In this case $\Phi \equiv I$ and $\psi = \psi'$, therefore $\psi^{-1}Z'\psi = \psi^{-1}Z_1\psi Z$. Therefore \[\vert \psi^{-1}Z'\psi \vert_{r''} \leq \vert \psi^{-1}Z_1\psi\vert_{r''} \vert Z\vert_{r''} \leq (1+2\varepsilon) \vert Z \vert_{r''}. \]
\noindent (where the last inequality comes from \eqref{estim-Z_1-iteration}). Moreover, \[ \vert Z \vert_{r''} = \vert \Pi_{k=2}^{l+1} e^{X_k} \vert_{r''}. \]
Now for all $k \in \llbracket 2, l \rrbracket$, we have seen in \eqref{Xj} that $|X_k|_{r''}\leq \varepsilon'_k$.
\noindent Therefore \[ \vert Z \vert_{r''} \leq \vert \Pi_{k=2}^{l+1}e^{X_k} \vert_{r''} \leq e^{\sum_{k=2}^{l+1} \varepsilon'_k} \leq e^{2\varepsilon}\] and finally, \[ \vert \psi^{-1} Z' \psi \vert_{r''} \leq (1+2\varepsilon)e^{2\varepsilon}.\]
The estimate of $\psi^{-1}Z'^{-1}\psi$ is obtained in a similar way. This gives the property \eqref{prop9.1}.
Moreover, \[ \vert \partial_{\omega}(\psi^{-1}Z' \psi) \vert_{r''} \leq \vert \partial_\omega (\psi^{-1}Z_1\psi)Z \vert_{r''} +\vert \psi^{-1}Z_1\psi\partial_\omega(Z) \vert_{r''} \leq \varepsilon^{\frac{1}{2}} \vert Z \vert_{r''} + (1+2\varepsilon)\vert \partial_\omega(Z) \vert_{r''} \] where the last inequality comes from \eqref{estim-derivZ1}. Now \[\vert \partial_\omega(Z)\vert_{r''} \leq \sum_{k=2}^{l+1} \vert \partial_\omega X_k\vert_{r''} \prod_{j=2}^{l+1} e^{\vert X_j\vert _{r''}}. \] For all $k \in \llbracket 2, l+1 \rrbracket$, by construction of $X_k$,
\[ \vert \partial_\omega X_k \vert_{r''}\leq 2 \Vert A_k \Vert \vert X_k \vert_{r''} + \vert F_{k}\vert_{r''} \leq 2\Vert A_k \Vert \vert X_k \vert_{r''} + \varepsilon^{(\frac{5}{4})^{k}-\frac{1}{48}}. \] Now for all $k \in \llbracket 2, l+1 \rrbracket$, by the estimate \eqref{major-Aj-iteration}, the estimate \eqref{A_1-wrt-A} and condition \ref{normeA} of this lemma, for $\varepsilon \leq \varepsilon_0$ given by lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.5.2}),
\[ \Vert A_k \Vert \leq \varepsilon^{-\frac{\zeta}{2}} + \varepsilon^{\frac{23}{24}} + \pi \varepsilon^{- \zeta} \leq \varepsilon^{-2\zeta}\] therefore for $\varepsilon \leq \varepsilon_0$ given by lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.6}),
\begin{equation*}\begin{split} \sum_{k=2}^{l=1}\vert \partial_\omega X_k \vert_{r''} & \leq 2 \sum_{k=2}^{l+1}\Vert A_k \Vert \vert X_k \vert_{r''} + \sum_{k=2}^{l+1}\vert F_{k}\vert_{r''} \\ & \leq 2\sum_{k=2}^{l+1} \varepsilon^{-2\zeta} \varepsilon'_k+ \sum_{k=2}^{l+1}\varepsilon'_k \\ & \leq \varepsilon \end{split}\end{equation*} and finally, for $\varepsilon\leq \varepsilon_0$ like in lemma \ref{smallness-epsilon} (see equation \eqref{eq:cond2.7}),
\[ \vert \partial_{\omega}(\psi^{-1}Z' \psi) \vert_{r''} \leq 2\varepsilon^{\frac{1}{2}}+2\varepsilon^{\frac{7}{8}} \leq \varepsilon^{\frac{1}{4}}. \]
The estimate of $ \partial_{\omega}(\psi^{-1}Z^{'-1} \psi)$ is similar, which gives property \ref{prop9}. \end{proof}
\section{Almost reducibility}
Here we complete the proof of the main theorem.
\begin{theorem}\label{theoreme} Let $r_0 >0 $, $A\in sl(2, \mathbb{R})$ and $F \in U_{r_0}(\mathbb{T} ^d, sl(2, \mathbb{R}))$. Then, if \[ \vert F \vert_{r} \leq \varepsilon_0 \]
\noindent where $\varepsilon_0$ satisfies the assumptions above, and \[\Vert A \Vert \leq \varepsilon_0^{-\frac{\zeta}{2}}, \] then for all $\varepsilon \leq \varepsilon_0$, there exist
\begin{itemize} \item $r_\varepsilon >0$, $k_\varepsilon \in \mathbb{N}$, \item $Z_\varepsilon \in U_{r_\varepsilon}( \mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $A_\varepsilon \in sl(2, \mathbb{R})$, \item $\bar A_\varepsilon, \bar F_\varepsilon \in U_{r_\varepsilon}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, \item $\psi_\varepsilon \in U_{r_{\varepsilon}}(2\mathbb{T} ^d, SL(2, \mathbb{R}))$, \end{itemize}
such that \begin{enumerate} \item \label{thm-prop1} $\bar A_\varepsilon$ is reducible to $A_\varepsilon$ by $\psi_\varepsilon$, with $\vert \psi_{k_\varepsilon} \vert_{r_\varepsilon} \leq \varepsilon^{-\zeta}$, \item \label{thm-prop2} $\vert \bar F_\varepsilon \vert_{r_\varepsilon} \leq \varepsilon$, \item \label{thm-prop3} for all $\theta \in \mathbb{T} ^d$, \[ \partial_\omega Z_\varepsilon (\theta) = (A + F(\theta)) Z_\varepsilon (\theta) - Z_\varepsilon (\theta)(\bar A_\varepsilon (\theta) + \bar F_\varepsilon (\theta))\]
\item \label{thm-prop4} \[ \vert Z_\varepsilon ^{\pm 1} - Id \vert_{r_\varepsilon} \leq \varepsilon_0^{\frac{9}{10}}.\] \end{enumerate} Moreover, either $\vert \partial_\omega Z_\varepsilon \vert_{r_\varepsilon}$ is bounded as $\varepsilon\rightarrow 0$ and $A+F$ is a reducible cocycle in $U_{r_\infty}(\mathbb{T} ^d, sl(2,\mathbb{R}))$ for some $r_\infty > 0$, or for all $\varepsilon\leq \varepsilon_0$ there exists $\varepsilon'\leq \varepsilon$ such that \[ \Vert A_{\varepsilon'} \Vert \leq \kappa \varepsilon'^{\zeta}.\] \end{theorem}
\begin{proof} Remind parameters (\ref{parameters}) defined in section \ref{inductivestep} and define, for all $k \in \mathbb{N},k\geq 1$, \[ \varepsilon_k:=\varepsilon_0^{(2\delta)^k};\quad r_k := r_0- \displaystyle \sum_{i=0}^{k-1}\displaystyle \frac{50 \delta\vert \log \varepsilon_{i} \vert }{\pi \Lambda(R(r_{i},\varepsilon_{i})N(r_{i},\varepsilon_{i})) }.\] Notice that, by Lemma \ref{rk-limite-positive}, under assumption \ref{assumption-2bis}, for all $k\in \mathbb{N}$, $r_k >0$.
We can apply a first time lemma \ref{352}. There exist
\begin{itemize} \item $Z_1 \in U_{r_1}(\mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $\bar A_1$, $\bar F_1 \in U_{ r_1}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, \item $A_1 \in sl(2, \mathbb{R})$, \item $\psi_0 \in U_{r_1}(2\mathbb{T} ^d, SL(2, \mathbb{R}))$, \end{itemize} such that \begin{itemize} \item $\bar A_1$ is reducible to $A_1$ by $\psi_0$, \item for all $G\in \mathcal{C}^0(\mathbb{T}^d,sl(2,\mathbb{R}))$, $\psi_0^{-1}G\psi_0 \in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $\vert \bar F_1 \vert_{r_1} \leq \varepsilon_1$, \item $\vert \psi_0^{\pm 1} \vert_{r_1} \leq \varepsilon_1^{-\zeta}$, \item $\Vert A_1 \Vert \leq \varepsilon_1^{-\frac{\zeta}{2}} $, \item for all $\theta \in \mathbb{T} ^d$
\[ \partial_\omega Z_1(\theta) = (A + F(\theta))Z_1(\theta) - Z_1(\theta)(\bar A_1(\theta) + \bar F_1(\theta)), \] \item \[ \vert Z_1^{\pm 1} - Id \vert_{r_1} \leq \varepsilon_0^{\frac{9}{10}},\]
\item if moreover $A$ had a $BR_\omega^{R(r_0,\varepsilon_0)N(r_0,\varepsilon_0)}(\kappa''(\varepsilon_0))$ spectrum,
\begin{equation}\vert \psi_{0}^{-1}Z_{1}\psi_{0} \vert_{r_{1}} \leq (1+2\varepsilon_0)e^{2\varepsilon_0}\end{equation} and if not, \[\Vert A_{1} \Vert \leq \kappa''(\varepsilon_0)\] and \begin{equation} \vert \partial_\omega(\psi_{0}^{-1}Z_{1}\psi_{0}) \vert_{r_{1}} \leq \varepsilon_{0}^{\frac{1}{4}}. \end{equation} \end{itemize}
\underline{Iterative step :} let $k \geq 1$ and \begin{itemize} \item $ \bar A_k, \bar F_k \in U_{r_k}(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $A_k \in sl(2, \mathbb{R})$, \item $\psi_{k-1} \in U_{r_k}(2\mathbb{T} ^d, SL(2,\mathbb{R}))$, \end{itemize} such that \begin{itemize} \item $\bar A_k$ is reducible to $A_k$ by $\psi_{k-1}$, \item for all $G\in \mathcal{C}^0(\mathbb{T} ^d,sl(2,\mathbb{R}))$, $\psi_{k-1}^{-1}G\psi_{k-1} \in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $\vert \bar F_k \vert_{r_k} \leq \varepsilon_k$, \item $\vert \psi_{k-1}^{\pm 1} \vert_{r_k} \leq \varepsilon_k^{-\zeta}$, \item $\Vert A_k\Vert \leq \varepsilon_k^{-\frac{\zeta}{2}}$. \end{itemize}
We can one again apply lemma \ref{352} to get
\begin{itemize} \item $Z_{k+1} \in U_{r_{k+1}}( \mathbb{T} ^d, SL(2, \mathbb{R}))$, \item $\bar A_{k+1}$, $\bar F_{k+1} \in U_{ r_{k+1}}(\mathbb{T} ^d, sl(2, \mathbb{R}))$, \item $A_{k+1} \in sl(2, \mathbb{R})$, \item $\psi_k \in U_{r_{k+1}}(2\mathbb{T} ^d, SL(2, \mathbb{R}))$, \end{itemize} such that \begin{itemize} \item $\bar A_{k+1}$ is reducible to $A_{k+1}$ by $\psi_k$, \item for all $G\in \mathcal{C}^0(\mathbb{T} ^d,sl(2,\mathbb{R}))$, $\psi_k^{-1}G\psi_k \in \mathcal{C}^0(\mathbb{T} ^d, sl(2,\mathbb{R}))$, \item $\vert \bar F_{k+1}\vert_{r_{k+1}} \leq \varepsilon_{k+1}$, \item $\vert \psi_k^{\pm 1} \vert_{r_{k+1}} \leq \varepsilon_{k+1}^{-\zeta}$ , \item $\Vert A_{k+1} \Vert \leq \varepsilon_{k+1}^{-\frac{\zeta}{2}}$, \item for all $\theta \in \mathbb{T} ^d$, \[ \partial_\omega Z_{k+1}(\theta) = (\bar A_k(\theta) + \bar F_k(\theta))Z_1(\theta) - Z_{k+1}(\theta)(\bar A_{k+1}(\theta) + \bar F_{k+1}(\theta)), \] \item \[ \vert Z_{k+1}^{\pm 1} - Id \vert_{ r_{k+1}} \leq \varepsilon_k^{\frac{9}{10}}\]
\item if moreover $A_k$ had a $BR_\omega^{R(r_k,\varepsilon_k)N(r_k,\varepsilon_k)}(\kappa''(\varepsilon_k))$ spectrum,
\begin{equation}\label{bound-Z_k-nonres}\vert \psi_{k+1}^{-1}Z_{k+1}\psi_{k+1} \vert_{r_{k+1}} \leq (1+2\varepsilon_k)e^{2\varepsilon_k}\end{equation}
and if not,
$$||A_{k+1}||\leq \kappa''(\varepsilon_k)$$
and
\begin{equation} \label{bound-partial_omegaZ_k}\vert \partial_\omega(\psi_{k+1}^{-1}Z_{k+1}\psi_{k+1}) \vert_{r_{k+1}} \leq \varepsilon_{k}^{\frac{1}{4}}. \end{equation} \end{itemize}
\underline{Result:} Let $\varepsilon \leq \varepsilon_0$ and $k_\varepsilon \in \mathbb{N}$ such that $\vert F \vert_r^{(2\delta)^{k_{\varepsilon}}}\leq \varepsilon$. Let \[ \left\{\begin{array}{c}Z_\varepsilon = Z_1 \cdots Z_{k_\varepsilon} \\ \bar A_\varepsilon = \bar A_{k_\varepsilon} \\ \bar F_\varepsilon = \bar F_{k_\varepsilon} \\ \psi_\varepsilon = \psi_{k_\varepsilon} \\ r_\varepsilon = r_{k_\varepsilon} \end{array}\right. \] then the properties \ref{thm-prop1} and \ref{thm-prop2} hold. Moreover for all $\theta \in \mathbb{T} ^d$, \[ \partial_\omega Z_{\varepsilon}(\theta) = (A + F(\theta))Z_\varepsilon(\theta) - Z_\varepsilon(\theta)(\bar A_\varepsilon(\theta) + \bar F_\varepsilon(\theta))\] and the property \ref{thm-prop3} holds. Notice that, for all $k' > \kappa_{\varepsilon}$, if $\Vert A_{k_\varepsilon} \Vert \leq \kappa \varepsilon^{\zeta}$ (which is satisfied if, for example, the matrix $A_{k_{\varepsilon}-1}$ was resonant), then
\[ \Vert A_{k'} \Vert \leq \Vert A_{k_\varepsilon} \Vert + \sum_{i=k_\varepsilon+1}^{k'}\varepsilon_i\leq \kappa''(\varepsilon) + 2 \varepsilon \leq 2\kappa''(\varepsilon).\]
We also have \[\vert Z_1^{\pm 1} - Id \vert_{r_{\varepsilon}} \leq \varepsilon_0^{\frac{9}{10}} \Rightarrow \vert Z_1 \vert_{r_{\varepsilon}} \leq 1 + \varepsilon_0^{\frac{9}{10}}\] Let $k \in \mathbb{N}$ and suppose that for all $j \leq k-1$, \[ \vert Z_1 \dots Z_j \vert_{r_\varepsilon} \leq 2\] then \[ c_k := \vert Z_1 \dots Z_k - Id \vert_{r_\varepsilon} \leq \vert Z_{k-1} - Id \vert_{r_\varepsilon} \vert Z_1 \dots Z_{k-1} \vert_{r_\varepsilon} + \vert Z_1 \dots Z_{k-1} - Id \vert_{r_\varepsilon} \] \[\leq 2\varepsilon_{k-2}^{\frac{9}{10}}+c_{k-1}\]
\noindent which implies
\[c_k\leq 2\sum_{i=0}^{k-2}\varepsilon_i^{\frac{9}{10}}v\leq 4\varepsilon_0^{\frac{9}{10}}.\]
\noindent Finally \[ \vert (Z_1 \dots Z_k)^{\pm 1} -I\vert_{r_\varepsilon} \leq \varepsilon_0^{\frac{9}{10}}\] hence the property \ref{thm-prop4} holds.
\textbf{Reducible case}
Suppose that there exists $\bar{k}$ such that for all $k'\geq \bar{k}$, $\psi_{k'} \equiv \psi_{\bar{k}}$ (which means that for all $k'\geq \bar{k}$, $A_{k'}$ has a $BR_\omega^{N(r_{k'},\varepsilon_{k'})R(r_{k'},\varepsilon_{k'})}(\kappa''(\varepsilon_{k'}))$ spectrum). Then
\begin{equation}\begin{split} \partial_\omega Z_\varepsilon & = \partial_\omega (\prod_{i=1}^{k_\varepsilon}Z_i) \\
& = \partial_\omega(\prod_{i=1}^{\bar{k}-1} Z_i)(\prod_{j=\bar{k}}^{k_\varepsilon}Z_j) + (\prod_{i=1}^{\bar{k}-1}Z_i) \partial_\omega(\prod_{j=\bar{k}}^{k_\varepsilon}Z_j)\\
& = \partial_\omega(\prod_{i=1}^{\bar{k}-1} Z_i)(\prod_{j=\bar{k}}^{k_\varepsilon}Z_j) + (\prod_{i=1}^{\bar{k}-1}Z_i) \partial_\omega(\prod_{j=\bar{k}}^{k_\varepsilon}\psi_{\bar{k}}\psi^{-1}_j Z_j \psi_j \psi_{\bar{k}}^{-1}) \\
& = \partial_\omega(\prod_{i=1}^{\bar{k}-1} Z_i)(\prod_{j=\bar{k}}^{k_\varepsilon}Z_j) + (\prod_{i=1}^{\bar{k}-1}Z_i)[ \partial_\omega (\psi_{\bar{k}})\prod_{j=\bar{k}-1}^{k_\varepsilon}\psi_j^{-1}Z_j\psi_j \psi_{\bar{k}}^{-1} + \\
& \qquad \qquad \qquad \qquad \psi_{\bar{k}} \partial_\omega (\prod_{j=\bar{k}}^{k_\varepsilon}\psi_j^{-1}Z_j\psi_j) \psi_{\bar{k}}^{-1} + \psi_{\bar{k}} \prod_{j=\bar{k}}^{k_\varepsilon}(\psi_j^{-1}Z_j\psi_j) \partial_\omega (\psi_{\bar{k}}^{-1})], \end{split}\end{equation} thus \begin{equation*}\begin{split} \vert \partial_\omega Z_\varepsilon \vert_{r_\varepsilon} & \leq \vert \partial_\omega(\prod_{i=1}^{\bar{k}-1} Z_i )\vert_{r_\varepsilon} \vert \prod_{j=\bar{k}}^{k_\varepsilon}Z_j \vert_{r_\varepsilon} + \vert\prod_{i=1}^{\bar{k}-1}Z_i\vert_{r_\varepsilon} \vert\psi_{\bar{k}}\vert_{r_\varepsilon}\vert\partial_\omega(\prod_{j=\bar{k}}^{k_\varepsilon} \psi^{-1}_j Z_j\psi_j) \vert_{r_\varepsilon} \vert \psi_{\bar{k}}^{-1}\vert_{r_\varepsilon} \\ & + \vert\prod_{i=1}^{\bar{k}-1}Z_i\vert_{r_\varepsilon}\vert(\prod_{j=\bar{k}}^{k_\varepsilon}\psi^{-1}_j Z_j\psi_j) \vert_{r_\varepsilon}\vert \psi_{\bar{k}}^{-1}\vert_{r_\varepsilon}\vert\partial_\omega \psi_{\bar{k}}\vert_{r_\varepsilon} \\ &+\vert\prod_{i=1}^{\bar{k}-1}Z_i\vert_{r_\varepsilon}\vert(\prod_{j=\bar{k}}^{k_\varepsilon}\psi^{-1}_j Z_j \psi_j) \vert_{r_\varepsilon}\vert \partial_\omega \psi_{\bar{k}}^{-1}\vert_{r_\varepsilon} \vert\psi_{\bar{k}}\vert_{r_\varepsilon}. \end{split}\end{equation*}
\noindent Since the factors $\vert\prod_{i=1}^{\bar{k}-1}Z_i\vert_{r_\varepsilon},\vert\partial_\omega \prod_{i=1}^{\bar{k}-1}Z_i\vert_{r_\varepsilon},\vert(\prod_{j=\bar{k}}^{k_\varepsilon}Z_j) \vert_{r_\varepsilon},\vert\psi_{\bar{k}}\vert_{r_\varepsilon},\vert\psi_{\bar{k}}^{-1}\vert_{r_\varepsilon},\vert\partial_\omega \psi_{\bar{k}}\vert_{r_\varepsilon},\vert\partial_\omega \psi_{\bar{k}}^{-1}\vert_{r_\varepsilon}$ are bounded uniformly in $\varepsilon$ (here we use \eqref{bound-Z_k-nonres}), there exist $K_1,K_2\geq 0$ independent of $\varepsilon$ such that
\[\vert \partial_\omega Z_\varepsilon\vert _{r_\varepsilon}\leq K_1+K_2 \vert\partial_\omega(\prod_{j=\bar{k}}^{k_\varepsilon} \psi^{-1}_jZ_j\psi_j) \vert_{r_\varepsilon}.\]
Moreover, by \eqref{bound-partial_omegaZ_k} and \eqref{bound-Z_k-nonres}, \begin{equation*}\begin{split} \vert\partial_\omega(\prod_{j=\bar{k}}^{k_\varepsilon} \psi^{-1}_jZ_j\psi_j) \vert_{r_\varepsilon} & \leq \sum_{j=\bar{k}}^{k_\varepsilon} \vert \partial_\omega( \psi_j^{-1}Z_j\psi_j)\vert_{r_\varepsilon}
\prod_{\substack{\bar{k}\leq i
\leq k_\varepsilon \\ i \neq j}} \vert \psi_i^{-1}Z_i \psi_i \vert_{r_\varepsilon} \\
& \leq \sum_{j = \bar{k}}^{k_\varepsilon}\varepsilon_j^{\frac{1}{4}}\prod_{\substack{\bar{k}\leq i \leq k_\varepsilon \\ i \neq j}} (1 + 2e^{\varepsilon_i})e^{2\varepsilon_i} \end{split}\end{equation*}
therefore
\begin{equation*}\begin{split} \vert\partial_\omega(\prod_{j=\bar{k}}^{k_\varepsilon} \psi^{-1}_jZ_j\psi_j) \vert_{r_\varepsilon} & \leq 2 \sum_{j = \bar{k}}^{k_\varepsilon} \varepsilon_j^{\frac{1}{4}}e^{2\varepsilon_{\bar{k}}} \\
& \leq 8\varepsilon_{\bar{k}}^{\frac{1}{4}}e^{2\varepsilon_{\bar{k}}} \\
& \leq 16 \varepsilon_{\bar{k}}^{\frac{1}{4}} \end{split}\end{equation*} and finally, $\vert \partial_\omega Z_\varepsilon \vert_{r_\varepsilon}$ is bounded as $\varepsilon\rightarrow 0$. In this case, $Z_\varepsilon$ and $\partial_\omega Z_\varepsilon$ have adherent values; let $Z_\infty$ be an adherent value of $Z_\varepsilon$. Since
$$\partial_\omega (Z_\varepsilon \psi_{\bar{k}})=(A+F)Z_\varepsilon \psi_{\bar{k}} - Z_\varepsilon \psi_{\bar{k}}(A_\varepsilon + \psi_{\bar{k}}^{-1}\bar{F}_\varepsilon\psi_{\bar{k}}) $$
\noindent (where $A_\varepsilon\in sl(2,\mathbb{R})$), and since all factors except $A_\varepsilon$ are known to converge in a subsequence, then there exists a constant $A_\infty\in sl(2,\mathbb{R})$ such that
$$\partial_\omega (Z_\infty \psi_{\bar{k}})=(A+F)Z_\infty \psi_{\bar{k}} - Z_\infty \psi_{\bar{k}}A_\infty $$
and thus $A+F$ is actually is a reducible cocycle in $U_{r_\infty}(\mathbb{T} ^d, sl(2,\mathbb{R}))$ for some $r_\infty > 0$.
\textbf{Non reducible case}
If the system $A+F$ is not reducible, then for all $k\geq 1$, there exists $k'\geq k$ such that $A_{k'}$ does not have a $BR_\omega^{R_{k'}N_{k'}}(\kappa''(\varepsilon_{k'}))$ spectrum. In this case, $||A_{k'+1}||\leq \kappa''(\varepsilon_{k'})=\kappa \varepsilon_{k'}^\zeta$.
\end{proof}
We will now show a density corollary. \begin{corollary}[Density of reducible cocycles close to a constant cocycle]
Let $r_0>0$, $A\in sl(2, \mathbb{R})$, and $G\in U_{r_0}(2\mathbb{T} ^d,sl(2,\mathbb{R}))$ such that $\vert G - A \vert_{r_0} \leq \varepsilon_0$ and $\Vert A \Vert \leq \varepsilon_0^{-\frac{\zeta}{2}}$ with $\varepsilon_0$ as in \ref{smallness-epsilon}, and satisfying the assumption \ref{assumption-2bis}. Denote
\[\rho = r_0-\frac{150\delta\vert \log \varepsilon_0 \vert}{\pi \Lambda\circ \Psi^{-1}(\epsilon_0^{-\zeta})} -\frac{150\delta}{\pi \zeta \log(2\delta)} \int_{\Psi^{-1}(\varepsilon_0^{-\zeta})}^{+\infty}\frac{\Lambda'(t)\ln \Psi(t)}{\Lambda(t)^2} dt.\]
Then for all $\varepsilon > 0$ there exists $H \in U_{\rho}(2\mathbb{T} ^d, sl(2,\mathbb{R}))$ such that $\vert G - H \vert_{\rho} \leq \varepsilon$ and $H$ is reducible. \end{corollary} \begin{proof} Apply theorem \ref{theoreme} with $F = G-A$. Since $\rho \leq r_\varepsilon$, we in particular get matrices $Z_\varepsilon \in U_{\rho}(\mathbb{T} ^d, SL(2,\mathbb{R})), \bar A_\varepsilon, \bar F_\varepsilon \in U_{\rho}(\mathbb{T} ^d, sl(2,\mathbb{R}))$ and $A_\varepsilon \in sl(2,\mathbb{R})$ such that \begin{itemize} \item $\bar A_\varepsilon$ is reducible to $A_\varepsilon$, \item $\partial_\omega Z_\varepsilon =(A+(G-A))Z_\varepsilon - Z_\varepsilon(\bar A_\varepsilon + \bar F_\varepsilon) = G Z_\varepsilon - Z_\varepsilon(\bar A_\varepsilon + \bar F_\varepsilon)$, \item $\vert Z^{\pm 1}_\varepsilon \vert_{\rho} \leq 1 + \varepsilon_0^{\frac{9}{10}}\leq 2$, \item $\vert \bar F_\varepsilon \vert_{\rho} \leq \frac{\varepsilon}{4}$ \end{itemize} Let $H := G - Z_\varepsilon \bar F_\varepsilon Z_\varepsilon^{-1}$. We have \[\partial_\omega Z_\varepsilon = H Z_\varepsilon - Z_\varepsilon \bar A_\varepsilon \] and then $H$ is reducible to $A_\varepsilon$ (as $\bar A_\varepsilon$ is). Moreover, $H$ satisfies \[ \vert H - G \vert_{\rho} = \vert Z_\varepsilon^{-1}\bar F_\varepsilon Z_\varepsilon \vert_{\rho} \leq 4 \vert \bar F_\varepsilon\vert_{\rho} \leq \varepsilon. \] \end{proof}
\end{document} | arXiv |
Inclusion (Boolean algebra)
In Boolean algebra, the inclusion relation $a\leq b$ is defined as $ab'=0$ and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order.
The inclusion relation $a<b$ can be expressed in many ways:
• $a<b$
• $ab'=0$
• $a'+b=1$
• $b'<a'$
• $a+b=b$
• $ab=a$
The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }.
Some useful properties of the inclusion relation are:
• $a\leq a+b$
• $ab\leq a$
The inclusion relation may be used to define Boolean intervals such that $a\leq x\leq b$. A Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.
References
• Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 34, 52 ISBN 0486164594
| Wikipedia |
\begin{document}
\title {Self-similar spreading in a merging-splitting model of animal group size}
\author{Jian-Guo Liu$^{(1)}$, B. Niethammer$^{(2)}$, Robert L. Pego$^{(3)}$}
\maketitle
\begin{center} 1-Department of Physics and Department of Mathematics\\ Duke University, Durham, NC 27708, USA\\ email: [email protected] \end{center}
\begin{center} 2- Institut f\"ur Angewandte Mathematik\\ Universit\"at Bonn\\ Endenicher Allee 60\\ 53115 Bonn, Germany\\ email: [email protected] \end{center}
\begin{center} 3-Department of Mathematics and Center for Nonlinear Analysis\\ Carnegie Mellon University, Pittsburgh, Pennsylvania, PA 12513, USA\\ email: [email protected] \end{center}
\begin{abstract} In a recent study of certain merging-splitting models of animal-group size (Degond {\it et al.}, {\it J.~Nonl.~Sci.}~27 (2017) 379), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresponding to unbounded growth of group size. In the present paper we show that for any such initial distribution with a power-law tail, the solution approaches a self-similar spreading form. A one-parameter family of such self-similar solutions exists, with densities that are completely monotone, having power-law behavior in both small and large size regimes, with different exponents.
\end{abstract}
\noindent {\bf Key words: } Fish schools, Bernstein functions, complete monotonicity, heavy tails, convergence to equilibrium.
\noindent {\bf AMS Subject classification: }{45J05, 70F45, 92D50, 37L15, 44A10, 35Q99.}
\noindent {\bf Running head:} {Self-similar spreading in merging-splitting models} \vskip 0.4cm
\pagebreak
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\setcounter{equation}{0} \section{Introduction} \label{intro}
Coagulation-fragmentation equations can be used to describe a large variety of merging and splitting processes, including the evolution of animal group sizes \cite{Ma_etal_JTB11}.
We refer to \cite{DLP2017} for an extensive discussion of the relevant literature in this particular application area.
Here we consider a model with constant coagulation and overall fragmentation rate coefficients that lacks detailed balance and a corresponding $H$-theorem.
{This model is motivated by a compelling analysis of fisheries data that was carried out by H.-S. Niwa in \cite{Niwa-JTB2003}, and a first mathematical study of the behavior of its solutions was performed in \cite{DLP2017}.
As demonstrated in \cite{DLP2017}, the nature of} equilibria of this model as well as their domains of attractions can be rigorously studied using the theory of Bernstein functions. More precisely, it was shown that equilibria can be expressed by a single smooth scaling profile which is not explicit, but it has a convergent power-series representation and its behaviour for small and large cluster sizes can be completely characterized by different power laws with exponential cutoff \cite[eq. (1.5)-(1.7)]{DLP2017}. Furthermore, if the initial data have finite first moment, solutions converge to equilibrium in the large time limit.
In addition, it was also shown that if the initial data have infinite first moment, then solutions converge weakly to zero, which means that clusters grow without bound as time goes to infinity. Our goal in the present paper is to investigate whether this growth behaviour is described by self-similar solutions. Indeed, we are going to show that there exists a family of self-similar profiles with completely monotone densities, characterized by different power-law tail behaviours for small and large cluster sizes. Furthermore, if the cumulative mass distribution of the initial data has power law growth for large cluster sizes, the corresponding solution converges to the profile whose mass distribution diverges with the same power-law tail.
Self-similar solutions with fat tails have recently received quite some attention, in particular in the analysis of coagulation equations, starting with work on models with solvable kernels \cite{B_eternal,MP2004}. For coagulation equations with non-solvable kernels, existence of self-similar profiles with fat tails has been studied in \cite{NV2013,NTV2016,BNV2016}, but to our knowledge this is the first time that such solutions are found for a class of coagulation-fragmentation equations.
We describe both the discrete- and continuous-size versions of the model in section \ref{sec:CF_general}. Our proofs use and extend the methods of complex function theory and in particular Bernstein functions as developed in \cite{MP2004,MP2008,DLP2017} and we give a brief overview of the main definitions and results in section \ref{sec:prelim}. Our main results are stated in section \ref{sec:main}, while the remaining sections are devoted to their respective proofs.
\setcounter{equation}{0} \section{Coagulation-fragmentation Models D and C} \label{sec:CF_general}
In this section we describe both the discrete coagulation-fragmentation equations under study as well as their continuous-size analogue.
\subsection{Discrete-size distributions} The number density of clusters of size $i$ at time $t$ is denoted by $f_i(t)$. The size distribution $f(t)=(f_i(t))_{i\in\mathbb{N}}$ evolves according to discrete coagulation-fragmentation equations, written in strong form as follows: \begin{eqnarray} &&\hspace{-1.5cm} \frac{\partial f_i}{\partial t}(t) = Q_a(f)_i(t) + Q_b(f)_i(t) , \label{eq:CF3_disc}\\ &&\hspace{-1.5cm} Q_a(f)_i(t) = \frac{1}{2} \sum_{j=1}^{i-1} a_{j , i-j}\, f_j(t) \, f_{i-j}(t) - \sum_{j=1}^\infty a_{i,j} \, f_i(t) \, f_j(t) , \label{eq:CF4_disc} \\ &&\hspace{-1.5cm} Q_b(f)_i(t) = \sum_{j=1}^\infty b_{i,j} \, f_{i+j}(t) - \frac{1}{2} \sum_{j=1}^{i-1} b_{j , i-j} \, f_i(t) \ . \label{eq:CF5_disc} \end{eqnarray} The terms in $Q_a(f)_i(t)$ describe the gain and loss rate of clusters of size $i$ due to aggregation or coagulation, and correspondingly the terms in $Q_b(f)_i(t)$ describe the rate of breakup or frag\-mentation.
These equations can be written in the following weak form, suitable for comparing to the continuous-size analog: We require that for any bounded test sequence $(\varphi_i)$, \begin{eqnarray} &&\hspace{-1cm} \frac{d}{dt} \sum_{i=1}^\infty \varphi_i \, f_i(t) = \frac{1}{2} \sum_{i,j=1}^\infty \big( \varphi_{i+j} - \varphi_i - \varphi_j \big) \, a_{i,j} \, f_i(t) \, f_j(t) \nonumber \\ &&\hspace{1.5cm} - \frac{1}{2} \sum_{i=2}^\infty \Big( \sum_{j=1}^{i-1} \big( \varphi_i - \varphi_j - \varphi_{i-j} \big) \, b_{j , i-j} \, \Big) f_i(t) \,. \label{eq:CF2_disc} \end{eqnarray}
The present study deals with the particular case when the rate coefficients take the form \begin{eqnarray} a_{i,j} = \alpha\,, \qquad b_{i,j} = \frac{\beta}{i+j+1} \,, \qquad \alpha=\beta=2. \label{eq:rates_niwa_discD} \end{eqnarray} We refer to the coagulation-fragmentation equations \qref{eq:CF3_disc}-\qref{eq:CF5_disc} with the coefficients in \qref{eq:rates_niwa_discD} as \textbf{Model D} (D for discrete size). By a simple scaling we can achieve any values of $\alpha,\beta>0$ and so we keep $\alpha=\beta=2$ for simplicity. As discussed in \cite{DLP2017}, Model D arises as a modification of the time-discrete model written in \cite{Ma_etal_JTB11} which essentially corresponds to the choice of rate coefficients as
\begin{eqnarray} a_{i,j} = \alpha\,, \qquad b_{i,j} = \frac{\beta}{i+j-1}\,. \label{eq:rates_niwa_disc} \end{eqnarray} These choices correspond to taking the rate that pairs of individual clusters coalesce, and the rate that individual clusters fragment, to be constants independent of size.
The modification in \eqref{eq:rates_niwa_discD}, however, permits an analysis in terms of the {\sl Bernstein transform} of the size-distribution measure
$\sum_{j=1}^\infty f_j(t)\,\delta_j(dx)$. This Bernstein transform is given by \begin{equation}\label{eqD:BTdef} \breve f(\hat s,t) = \sum_{j=1}^\infty (1-e^{-j\hat s}) f_j(t) \ . \end{equation} Taking $\varphi_j = 1-e^{-j\hat s}$ in \qref{eq:CF2_disc}, it {can be shown (see \cite[Eq.(10.6)]{DLP2017})} that $\breve f(\hat s,t)$ satisfies the integro-differential equation \begin{equation}\label{eqD:BT1} \D_t \breve f(\hat s,t) = -\breve f^2 - \breve f +
\frac2{1-e^{-\hat s}}\int_0^{\hat s} \breve f(r,t) e^{-r}\,dr. \end{equation} for $\hat s, t>0$. By the simple change of variables \begin{equation}\label{eqD:change} s= 1-e^{-\hat s}\,, \qquad U(s,t)= \breve f(\hat s,t)\,, \end{equation} one finds that \qref{eqD:BT1} for $\hat s\in(0,\infty)$, $t>0$, is equivalent to \begin{equation}\label{eqD:BTueq} \D_t U(s,t) = -U^2-U + 2\int_0^1 U(sr,t)\,dr\,, \end{equation} for $s\in(0,1)$, $t>0$. This equation has the same form that arises in the continuous-size case, as we discuss next.
\subsection{Continuous-size distributions}
For clusters of any real size $x>0$, the size distribution at time $t$ is characterized by a measure $F_t$, whose distribution function we denote using the same symbol: \[ F_t(x) = \int_{(0,x]} F_t(dx). \] The measure $F_t$ evolves according to the following size-continuous coagulation-fragmentation equation, which we write in weak form. One requires that for any suitable test function $\varphi (x)$,
\begin{equation}\label{eq:CF2} \begin{split} &\frac{d}{dt} \int_{\rplus} \varphi(x) \, F_t(dx) = \frac{1}{2} \int_{{\mathbb R}_+^2} \big( \varphi (x+y) - \varphi(x) - \varphi(y) \big) a(x,y) \, F_t(dx) \, F_t(dy) \\ & \quad - \frac{1}{2} \int_{{\mathbb R}_+} \Big( \int_0^x \big( \varphi (x) - \varphi(y) - \varphi(x-y) \big) \, b(y,x-y) \, dy \, \Big) F_t(dx) . \end{split}\end{equation} The specific rate coefficients that we study correspond to constant coagulation rates and constant overall binary fragmentation rates with uniform distribution of fragments: \begin{eqnarray} &&\hspace{-1cm} a(x,y) = A\,, \qquad b(x,y) = \frac{B}{x+y}, \qquad A=B=2.\label{eq:rates_niwa_cont} \end{eqnarray} (Again, by scaling one can achieve any $A,B>0$.) We refer to the coagulation-fragmentation equations \qref{eq:CF2} with these coefficients as \textbf{Model C} (C for continuous size).
For size distributions with density, written as $F_t(dx)=f(x,t)\,dx$, Model C is written formally in strong form as follows: \begin{eqnarray} &&\hspace{-1.5cm} \D_t f(x,t) = Q_a(f)(x,t) + Q_b(f)(x,t) , \label{eq:CF3_Niwa_11}\\ &&\hspace{-1.5cm} Q_a(f)(x,t) = \int_0^x \, f(y,t) \, f(x-y,t) \, dy - 2 f(x,t) \, \int_0^\infty f(y,t) \, dy , \label{eq:CF4_Niwa_11} \\ &&\hspace{-1.5cm} Q_b(f)(x,t) = -f(x,t) + 2 \int_x^\infty \frac{f(y,t)}{y} \, dy . \label{eq:CF5_Niwa_11} \end{eqnarray} Importantly, Model C has a scaling invariance involving dilation of size. If $F_t(x)$ is any solution and $\lambda>0$, then \begin{equation}\label{scaleC:F} \hat F_t(x) := F_t(\lambda x) \end{equation} is also a solution.
When we take as test function $\varphi(x)=1-e^{-sx}$, we find that the Bernstein transform of $F_t$, defined by \begin{equation} \uu(s,t)=\breve F_t(s) = \int_{\rplus} (1-e^{-sx})\,F_t(dx)\,, \end{equation} satisfies \begin{equation} \D_t\uu(s,t) = -\uu^2 -\uu + 2 \int_0^1 \uu(sr,t)\,dr\,. \label{eqC:Bernstein} \end{equation} This equation has exactly the same form as \eqref{eqD:BTueq}.
According to the well-posedness result for Model C established in \cite[Thm.~6.1]{DLP2017}, given any initial $F_0\in \mathcal{M}_+(0,\infty)$ (the set of nonnegative finite measures on $(0,\infty)$), Model C admits a unique narrowly continuous map $t\mapsto F_t\in \mathcal{M}_+(0,\infty)$ that satisfies \eqref{eq:CF2} for all continuous $\varphi$ on $[0,\infty]$. In particular, \eqref{eqC:Bernstein} holds for all $s\in[0,\infty]$. For $s=\infty$ in particular this means that the zeroth moment $m_0(t)=U(\infty,t)$ satisfies the logistic equation \begin{equation}\label{e:m0} \D_t m_0(t) = -m_0(t)^2 + m_0(t) \,, \end{equation} whence $m_0(t)\to1$ as $t\to\infty$.
\section{Preliminaries}\label{sec:prelim}
All of our main results concern the behavior of solutions of Models C and D having power-law tails and infinite first moment, and the analysis involves the behavior of their Bernstein transforms. Hence, before we state our main results it is useful to recall some basic definitions and results on Bernstein functions and transforms.
A function $g\colon(0,\infty)\to\mathbb{R}$ is \textit{completely monotone} if it is infinitely differentiable and its derivatives satisfy $(-1)^ng^{(n)}(x)\ge0$ for all real $x>0$ and integer $n\ge0$. By Bernstein's theorem, $g$ is completely monotone if and only if it is the Laplace transform of some (Radon) measure on $[0,\infty)$.
\begin{definition} A function $U\colon (0,\infty)\to \mathbb{R}$ is a \textit{Bernstein function} if it is infinitely differentiable, nonnegative, and its derivative $U'$ is completely monotone. \end{definition}
The main representation theorem for these functions \cite[Thm. 3.2]{Schilling_etal_Bernstein} says that
a function $U\colon(0,\infty)\to\mathbb{R}$ is a Bernstein function if and only if it has the representation \begin{equation}\label{def:Btransform} U(s) = a_0s+a_\infty+\int_{(0,\infty)} (1-e^{-sx})\,F(dx)\,, \quad s\in(0,\infty), \end{equation} where $a_0$, $a_\infty\ge0$ and $F$ is a measure satisfying $\int_{(0,\infty)} (x\wedge1)F(dx)<\infty$. In particular, the triple $(a_0,a_\infty,F)$ uniquely determines $U$ and vice versa.
We point out that $U$ determines $a_0$ and $a_\infty$ via the relations \begin{equation}\label{eq:a0ainfty} a_0 = \lim_{s\to\infty} \frac{U(s)}s\,, \qquad a_\infty = U(0^+)=\lim_{s\to0}U(s)\, . \end{equation}
Whenever \qref{def:Btransform} holds, we call $U$ the {\sl Bernstein transform} of the L\'evy triple $(a_0,a_\infty,F)$. If $a_0=a_\infty=0$, we call $U$ the Bernstein transform of the L\'evy measure $F$, and write $U=\breve F$, in accordance with the definitions in section \ref{sec:CF_general}.
We will also make use of the theory of so-called \textit{complete} Bernstein functions, as developed in \cite[Chap. 6]{Schilling_etal_Bernstein}: \begin{theorem} \label{thm:CBF} The following are equivalent. \begin{itemize} \item[(i)] The L\'evy measure $F$ in \qref{def:Btransform} has a completely monotone density $g$, so that \begin{equation}\label{eq:CBF} U(s) = a_0s+a_\infty+\int_{(0,\infty)} (1-e^{-sx}) g(x)\,dx\,, \quad s\in(0,\infty). \end{equation} \item[(ii)] $U$ is a Bernstein function that admits a holomorphic extension to the cut plane $\mathbb{C}\setminus(-\infty,0]$ satisfying $(\im s)\im U(s) \ge0$. \end{itemize} \end{theorem} In complex function theory, a function holomorphic on the upper half of the complex plane that leaves it invariant is called a \textit{Pick function} (alternatively a \textit{Herglotz} or \textit{Nevalinna} function). Condition (ii) of the theorem above says simply that $U$ is a Pick function analytic and nonnegative on $(0,\infty)$. Such functions are called \textit{complete Bernstein functions} in \cite{Schilling_etal_Bernstein}.
The power-law tail behavior of size distributions is related to power-law behavior of Bernstein transforms near the origin through use of Karamata's Tauberian theorem \cite[Thm.~III.5.2]{Feller} and Lemma 3.3 of \cite{MP2004}. To explain, suppose a measure $F$ on $(0,\infty)$ has a density $f$ satisfying \begin{equation}\label{a:f} f(x)\sim Ax^{-\alpha-1}\,,\quad x\to\infty. \end{equation} Necessarily $\alpha\in(0,1]$ if $F$ is finite with infinite first moment. The derivative $\D_s\breve F$ of the Bernstein transform of $F$ is the Laplace transform of the measure with distribution function \begin{equation}\label{a:F} \int_0^x y\,F(dy) \sim \frac{A x^{1-\alpha} }{1-\alpha} \end{equation} for $\alpha\in(0,1)$. By Karamata's theorem, this is equivalent to \begin{equation}\label{a:L} \D_s\breve F(s) \sim \frac{A \Gamma(2-\alpha)}{1-\alpha} s^{\alpha-1}, \quad s\to0. \end{equation} Then by Lemma 3.3 of \cite{MP2004} this is equivalent to \begin{equation}\label{a:U} \breve F(s) \sim \frac{A \Gamma(2-\alpha)}{\alpha(1-\alpha)} s^\alpha, \quad s\to0. \end{equation}
\section{Main results}\label{sec:main}
The choice of coefficients in the asymptotic expressions below is made to simplify Bernstein transform calculations in the sequel. {In the following we denote by \[
F_t(x):=\int_{(0,x]} F_t(dx) \] the cumulative distribution function.}
\begin{theorem}\label{t:exist} (Self-similar solutions for Model C) For each $\alpha\in(0,1)$ and $\lambda>0$, Model C admits a unique self-similar solution having the form \begin{equation}\label{d:Fa} F_t(x) = F_{\star\alpha}(\lambda xe^{-\beta t}), \end{equation} where $F_{\star\alpha}$ is a probability measure having the tail behavior \begin{equation} \int_0^x y F_{\star\alpha}(dy) \sim \frac{ \alpha} {\Gamma(2-\alpha)} x^{1-\alpha}\,, \quad x\to\infty. \end{equation} For this solution, \begin{equation}\label{e:beta} \beta = \frac{1-\alpha}{\alpha(1+\alpha)}, \end{equation} and $F_{\star\alpha}$ has a completely monotone density $f_{\star\alpha}$ having the following asymptotics: \begin{equation}\label{e:faasym} f_{\star\alpha}(x) \sim \begin{cases} \displaystyle \frac{\alpha}{\Gamma(1-\alpha)} x^{-\alpha-1} & \quad x\to\infty\,,\\[8pt] \displaystyle \frac{\hat c}{\Gamma(-\hat\alpha)} x^{\hat\alpha-1} & \quad x\to 0^+\,, \end{cases} \end{equation} where the constants $\hat\alpha\in(0,1)$, $\hat c>0$ are as described in Lemma~\ref{lem:solbeta}. \end{theorem}
\begin{theorem}\label{t:mainC} (Large-time behavior for Model C with algebraic tails) Suppose that the initial data for Model C satisfies \begin{equation}\label{eqC:ics} \int_0^x yF_0(dy) \sim \int_0^x yF_{\star\alpha}(\lambda\,dy) \sim \frac{ \alpha\lambda^{-\alpha}} {\Gamma(2-\alpha)} x^{1-\alpha}\,, \quad x\to\infty, \end{equation} where $\alpha\in(0,1)$, $\lambda>0$. Then for every $x\in[0,\infty]$ we have \begin{equation}\label{te:limitC} F_t(x e^{\beta t}) \to F_{\star\alpha}(\lambda x) \,, \quad t\to\infty. \end{equation} \end{theorem}
\begin{theorem}\label{t:mainD} (Large-time behavior for Model D with algebraic tails) Suppose that the initial data for Model D satisfies \begin{equation}\label{eqD:ics} \sum_{1\le k\le x} k f_k(0) \sim \int_0^x yF_{\star\alpha}(\lambda\,dy) \sim \frac{ \alpha\lambda^{-\alpha}} {\Gamma(2-\alpha)} x^{1-\alpha}\,, \quad x\to\infty, \end{equation} where $\alpha\in(0,1)$, $\lambda>0$. Then for every $x\in[0,\infty]$ we have \begin{equation}\label{te:limitD} \sum_{1\le k\le xe^{\beta t}} f_k(t) \to F_{\star\alpha}(\lambda x) \,, \quad t\to\infty. \end{equation} \end{theorem}
These convergence results relate to the notion of weak convergence of measures on $(0,\infty)$ sometimes known as narrow convergence. Let $\mathcal{M}_+(0,\infty)$ be the space of nonnegative finite (Radon) measures on $(0,\infty)$. Given $F, F_n\in \mathcal{M}_+(0,\infty)$ for $n\in\mathbb{N}$, we say $F_n$ converges to $F$ {\it narrowly} and write $F_n \nto F$ if \[ \int_{(0,\infty)} g(x)\,F_n(dx) \to \int_{(0,\infty)} g(x)\,F(dx) \] for all functions $g\in C_b(0,\infty)$, the space of bounded continuous functions on $(0,\infty)$. The convergence statements \eqref{te:limitC} and \eqref{te:limitD} correspond to the statement that \[ \hat F_t(dx) \nto F_{\star\alpha}(\lambda\,dx),\quad t\to\infty \] where, respectively, \begin{equation} \hat F_t(dx) = \begin{cases} F_t(e^{\beta t}dx) & \mbox{for Model C,}\\ \sum_k f_k(t)\delta_{ke^{-\beta t}}(dx) & \mbox{for Model D.} \end{cases} \end{equation} The proofs of \eqref{te:limitC} and \eqref{te:limitD} make use of the following result from \cite{DLP2017} {(cf. \cite[Proposition 3.6]{DLP2017})} that characterizes narrow convergence in terms of the Bernstein transform. \begin{proposition} \label{p:narrowfat} Assume $F$, $F_n\in\mathcal{M}_+(0,\infty)$ for $n\in\mathbb{N}$. Then the following are equivalent as $n\to\infty$. \item[(i)] $F_n$ converges narrowly to $F$, i.e., $F_n\nto F$. \item[(ii)] The Bernstein transforms $\breve F_n(s) \to \breve F(s)$, for each $s\in[0,\infty]$. \item[(iii)] The Bernstein transforms $\breve F_n(s) \to \breve F(s)$, uniformly for $s\in(0,\infty)$. \end{proposition}
The proofs of our main results will proceed in stages as follows. In section~\ref{s:ss} we identify the family of relevant self-similar solutions of equation~\eqref{eqC:Bernstein}. The argument involves a phase plane analysis that does not yet establish that the profile function is actually a Bernstein function. In section~\ref{s:compare} we prove a comparison principle for the nonlocal evolution equation \eqref{eqC:Bernstein}, then use this in section~\ref{s:limits} to show that solutions of \eqref{eqC:Bernstein} with initial data $U_0(s)\sim s^\alpha$ approach the corresponding self-similar form found in section~\ref{s:ss}. From this we deduce the self-similar profiles are limits of complete Bernstein functions, hence they are Bernstein transforms themselves of measures $F_{\star\alpha}$ having completely monotone densities, and the results of Theorems~\ref{t:mainC} and \ref{t:mainD} follow. The remaining properties of the profiles stated in Theorem~\ref{t:exist}, including complete monotonicity of densities and asymptotics for small and large size, are established in sections~\ref{s:Pick} and \ref{s:Tauber}.
The results of Theorems~\ref{t:mainC} and \ref{t:mainD} show that the long-time behavior of solutions with algebraic tails depends upon the algebraic rate of decay. We recall that for the pure coagulation equation with constant rate kernel (corresponding to Model C without fragmentation), all domains of attraction for self-similiar solutions with algebraic tails were characterized in \cite{MP2004} by the condition that initial data are regularly varying. Here in Theorem~\ref{t:mainC}, for example, this would correspond to the condition that the initial data satisfy \[ \int_0^x y F_0(dy) \sim x^{1-\alpha} L(x) \] where $L$ is slowly varying at $\infty$. In the present context, however, we do not know whether this more general condition is either sufficient or necessary for convergence to self-similar form.
\section{Self-similar scaling---necessary conditions}\label{s:ss}
We begin our analysis by finding the necessary forms for any self-similar solution to equation \eqref{eqC:Bernstein} that governs the Bernstein transform of solutions to Model C.
We look for self-similar solutions to \eqref{eqC:Bernstein} of the form \[ \uu(s,t) = u(sX(t)), \] where $X(\cdot)$ is smooth with $X(t)\to\infty$ as $t\to\infty$. Because in general $\uu(\infty,t)=m_0(t)\to1$ as $t\to\infty$, we require $u(\infty)=1$. After substituting into \eqref{eqC:Bernstein}, we find that for nontrivial solutions we must have \[ \beta :=X'(t)/X(t) \] to be a positive constant independent of $t$, and $u(z)$ must satisfy \begin{equation} \beta z\D_z u + u^2+u = 2\int_0^1 u(zr)\,dr. \label{eq:SSu1} \end{equation} With \begin{equation} v(z) = \int_0^1 u(zr)\,dr = \frac1z\int_0^z u(r)\,dr, \end{equation} the variables $(v(z),u(z))$ satisfy the ODE system \begin{align} \label{e:ODEu} \beta z\D_z u &= - u - u^2 + 2v\,, \\ \label{e:ODEv} z\D_z v &= u-v \,. \end{align} Under the change of variables $\tau = \log z$ we have $\D_\tau = z\D_z$ and this system becomes autonomous. We seek a solution defined for $\tau\in\mathbb{R}$ satisfying \[ (u,v) \to \begin{cases}(0,0) & \tau\to-\infty,\cr (1,1) & \tau\to+\infty,\end{cases} \] with both components increasing in $\tau$. What is rather straightforward to check, is that the origin $(0,0)$ is a saddle point in the $(v,u)$ phase plane, and the region \[ R = \{(u,v)\mid 0<\frac12(u+u^2)<v<u\} \] is positively invariant and contained in the unit square $[0,1]^2$. Inside this region both $u$ and $v$ increase with $\tau$. The unstable manifold at $(0,0)$ enters this region and must approach the stable node $(1,1)$ as $\tau\to\infty$, satisfying $1\le dv/du \le \frac32$ asymptotically since the trajectory approaches from inside $R$.
This trajectory provides the following result.
\begin{lemma}\label{lem:solbeta} Let $\beta>0$. Then, up to a dilation in $z$, there is a unique solution of \eqref{eq:SSu1} which is positive and increasing for $z\in(0,\infty)$ with $u(0)=0$ and $u(\infty)=1$, satisfying \begin{align*} u(z)&\sim z^\alpha \qquad\mbox{as $z\to 0^+$},\\ 1-u(z)&\sim \hat c z^{-\hat\alpha} \quad\mbox{as $z\to \infty$}, \end{align*} where $\alpha\in(0,1)$, $\hat\alpha\in(0,\frac13)$ are determined by the relations \begin{equation}\label{e:eigsa} \beta = \frac{1-\alpha}{\alpha(1+\alpha)} = \frac{1-3\hat\alpha}{\hat\alpha(1-\hat\alpha)} \end{equation} \end{lemma} We note that the relations \eqref{e:eigsa} arise from the eigenvalue equations \begin{equation}\label{e:eigs}
\left|\begin{matrix} -1 -\beta\alpha & 2\cr 1 & -1-\alpha
\end{matrix}\right| = 0, \qquad
\left|\begin{matrix} -3 +\beta\hat\alpha & 2\cr 1 & -1+\hat\alpha
\end{matrix}\right| = 0\,. \end{equation}
In what follows we let $u_\alpha$ denote the solution described by this lemma, noting that the relation between $\beta$ and $\alpha$ is monotone and given by \eqref{e:beta}. The phase-plane argument above does not show that $u_\alpha$ is a Bernstein function, however. Our plan is to show that in fact $u_\alpha$ is a complete Bernstein function (a Pick function), by showing that it arises as the pointwise limit of rescaled solutions of \eqref{eqC:Bernstein} which are complete Bernstein functions. Thus, our proof of the existence theorem \ref{t:exist} will depend upon a proof of stability.
\section{Comparison principle}\label{s:compare}
Our next goal is to study the long-time dynamics of solutions of \eqref{eqC:Bernstein} with appropriate initial data. For this purpose we develop a comparison principle showing that solutions of \eqref{eqC:Bernstein} preserve the ordering of the initial data on any interval of the form $[0,S]$.
Given $S>0$ and $u\in C([0,S])$, {define an averaging operator $\mathcal{A}$ by} \begin{equation}\label{d:Av} (\mathcal{A} u)(s) = \int_0^1 u(sr)\,dr \,,\quad s\in[0,S]. \end{equation} Then clearly $\mathcal{A} $ is a linear contraction on $C([0,S])$, with \begin{equation}\label{e:A0} (\mathcal{A} u)(0)=u(0). \end{equation} We recall that by Hardy's inequality, \begin{equation}\label{e:hardy}
\left(\int_0^S |(\mathcal{A} u)(s)|^2\,ds \right)^{1/2}
\le 2 \left(\int_0^S |u(s)|^2\,ds\right)^{1/2}. \end{equation} { Indeed, due to Minkowski's inequality in integral form we have \[
\left(\int_0^S \left| \int_0^1 u(xr)\,dr\right|^2\,dx\right)^{1/2} \leq \int_0^1 \left(\int_0^S |u(sr)|^2\,ds \right)^{1/2}\,dr \] and thus \begin{align*}
\left(\int_0^S |(\mathcal{A} u)(s)|^2\,ds \right)^{1/2}&
\leq \int_0^1 \left(\int_0^S |u(sr)|^2\,ds \right)^{1/2}\,dr \\ &\le \int_0^1 \frac{dr}{r^{1/2}}
\left[\int_0^S |u(s)|^2\,ds \right]^{1/2}. \end{align*} }
\begin{proposition}\label{p:compare} Given $S, T>0$ suppose that $U, V\in C^1([0,T],C([0,S])$ have the following properties: \begin{itemize} \item[(i)] $U(s,0)\ge V(s,0)$ for all $s\in[0,S]$, \item[(ii)] for all $(s,t)\in[0,S]\times[0,T]$ the equations \begin{eqnarray} \D_t U + U^2 + U(s,t) = 2 \mathcal{A} U + F \,, \label{e:UF}\\ \D_t V + V^2 + V(s,t) = 2 \mathcal{A} V + G \,, \label{e:VG}\end{eqnarray} hold, where $F\ge G$. \end{itemize} Then $U\ge V$ everywhere in $[0,S]\times[0,T]$.
\end{proposition}
\begin{proof} We write \[ w=U-V = w_+-w_- \quad\mbox{ where $w_+$, $w_-\ge0$.} \]
Let $M\ge \max |U+V|$. Subtracting \eqref{e:VG} from \eqref{e:UF} we find \[
\D_t w + M |w| + w\ge 2\mathcal{A} w + F-G \,. \] Because $w_\pm$ is Lipschitz in $t$, $w_+w_-=0$, and $\mathcal{A} w_\pm\ge0$, we can multiply by $-2w_- \le0$ and invoke \cite[Lemma~7.6]{GT} to infer that the weak derivative \[ \D_t(w_-^2) -2M w_-^2 \le 4 w_- \mathcal{A} w_- \,. \] Integrating over $s\in[0,S]$ and using Hardy's inequality we find \[ \D_t \int_0^S w_-(s)^2\,ds \le (8+2M)\int_0^S w_-(s)^2\,ds. \] Because $w_-(s,0)=0$, integrating in $t$ and using Gronwall's lemma concludes the proof that $U\ge V$ in $[0,S]\times[0,T]$. \end{proof}
\section{Convergence to equilibrium for initial data with power-law tails} \label{s:limits}
We begin with a result for solutions of \eqref{eqC:Bernstein} that is suitable for use in treating both Model C and Model D. \begin{proposition}\label{p:compareU} Suppose $U(s,t)$ is any $C^1$ solution of \eqref{eqC:Bernstein} for $s\in[0,\bar s)$, $t\in[0,\infty)$, and assume that its initial data satisfies \begin{equation}\label{a:U0} U_0(s) \sim s^\alpha \quad\mbox{as $s\to0^+$,} \end{equation} where $\alpha\in(0,1)$. Then with $\beta$ given by \eqref{e:beta}, we have \begin{equation} U(se^{-\beta t},t) \to u_\alpha(s) \quad\mbox{as $t\to\infty$, for all $s\in(0,\infty)$,} \end{equation} with uniform convergence for $s$ in any bounded subset of $(0,\infty)$, where $u_\alpha$ is the self-similar profile $u$ described in Lemma~\ref{lem:solbeta}. \end{proposition}
The proof is rather different from the proof of convergence to equilibrium for initial data with finite first moment, in section 7 of \cite{DLP2017}. In the present case, the behavior of $U(s,t)$ globally in $t$ is determined by the local behavior of the initial data $U_0$ near $s=0$.
\begin{proof} First, let $u_\alpha$ be given by Lemma~\ref{lem:solbeta}, and note that for any $c>0$ the function given by \[ V(s,t)=u_\alpha(cse^{\beta t}) \] is a solution of \eqref{e:VG} with $G=0$. Second, it is not difficult to prove that \begin{equation}
u_\alpha(cz)\to u_\alpha(z) \quad\mbox{as $c\to1$, uniformly for $z\in(0,\infty)$}. \end{equation}
Now, let $S>0$ and let $\eps>0$. Choose $c<1<C$ such that \begin{equation} u_\alpha(cz)<u_\alpha(z)<u_\alpha(Cz)<u_\alpha(cz)+\eps \quad\mbox{for all $z\in(0,\infty)$}. \end{equation} Due to the hypothesis \eqref{a:U0}, there exists $S_0=S_0(c,C)>0$ such that \begin{equation}\label{a:cpm} u_\alpha(cs) \le U(s,0) \le u_\alpha(C s) \qquad\mbox{for all $s\in[0,S_0]$.} \end{equation} Invoking the comparison principle in Proposition~\ref{p:compare} we infer that \begin{equation}\label{e:comp1} u_\alpha(cse^{\beta t}) \le U(s,t) \le u_\alpha(C se^{\beta t}) \quad\mbox{for all $s\in[0,S_0]$, $t>0$}. \end{equation} Replacing $s\in[0,S_0]$ by $s e^{-\beta t}$ with $s\in[0,S_0e^{\beta t}]$, this gives \begin{equation}\label{e:comp2} u_\alpha(cs) \le U(se^{-\beta t},t) \le u_\alpha(C s) \quad\mbox{for all $s\in[0,S_0e^{\beta t}]$, $t>0$.} \end{equation} By consequence, whenever $S_0e^{\beta t}>S$ it follows that \[
|U(se^{-\beta t},t)-u_\alpha(s)|<\eps \quad\mbox{for all $s\in[0,S]$}. \] This finishes the proof. \end{proof}
\begin{proof}[Proof of Theorem~\ref{t:mainC}] Because of the dilation invariance of Model C, we may assume the initial data satisfies \eqref{eqC:ics} with $\lambda=1$. By the discussion of \eqref{a:F}--\eqref{a:U} we infer that \begin{equation} U_0(s) = \int_0^\infty (1-e^{-sx}) F_0(dx) \sim s^\alpha\,,\quad s\to0. \end{equation} Next, we invoke Proposition~\ref{p:compareU} to deduce that \begin{equation} U(se^{-\beta t},t) = \int_0^\infty (1-e^{-sx}) F_t(e^{\beta t}\,dx) \to u_\alpha(s) \end{equation} for all $s\in[0,\infty)$. The limit also holds for $s=\infty$ as a consequence of the logistic equation {\eqref{e:m0}} for $m_0(t)=U(\infty,t)$. At this point we use the fact that the pointwise limit $u_\alpha(s)$ of the Bernstein functions $s\mapsto U(se^{-\beta t},t)$ is necessarily Bernstein \cite[Cor.~3.7, p.~20]{Schilling_etal_Bernstein} and the facts that \[ \lim_{s\to0} u_\alpha(s) = 0, \qquad \lim_{s\to\infty} u_\alpha(s) = 1\,, \] to infer the following (cf.~\cite[Eq.~(3.3)]{DLP2017}).
\begin{lemma}\label{lem:ualphaBernstein} For any $\alpha\in(0,1)$, the function $u_\alpha$ described in Lemma~\ref{lem:solbeta} is the Bernstein transform of a probability measure $F_{\star\alpha}$ on $(0,\infty)$, satisfying \[ u_\alpha(s) = \int_0^\infty(1-e^{-sx})F_{\star\alpha}(dx) \,,\quad s\in[0,\infty]. \] \end{lemma} Finally, we use Proposition~\ref{p:narrowfat} to infer the narrow convergence result \begin{equation}\label{e:narrowC} F_t(e^{\beta t}\,dx) \nto F_{\star\alpha}(dx) \,,\quad t\to\infty, \end{equation} to conclude the proof of Theorem~\ref{t:mainC}, \end{proof}
\begin{proof}[Proof of Theorem~\ref{t:mainD}] For Model D, the discussion of \eqref{a:F}--\eqref{a:U} implies that the hypothesis \eqref{eqD:ics} on initial data is equivalent to the condition \begin{equation} \breve f(\hat s,0) \sim \lambda^{-\alpha}\hat s^\alpha\,, \quad\hat s\to0, \end{equation} on the Bernstein transform of the initial data. Under the change of variables $s=1-e^{-\hat s}$ in \eqref{eqD:change} this is evidently equivalent to \begin{equation} U(s,0)\sim \lambda^{-\alpha}s^\alpha\,,\quad s\to0. \end{equation} As $U(s,t)$ is a solution of the dilation-invariant equation \eqref{eqD:BTueq}, so is the function $\hat U(s,t)=U(\lambda s,t)$ which satisfies $\hat U(s,0)\sim s^{\alpha}$, $s\to0$. Invoking Proposition~\ref{p:compareU}, we deduce that for all $s\in[0,\infty)$, \begin{equation}\label{eqD:lim} U(se^{-\beta t},t) \to u_\alpha(s/\lambda) \qquad\mbox{as $t\to\infty$}. \end{equation} Note that the left-hand side is well-defined only for $e^{\beta t}>s$.
We can now write \begin{equation} \breve f(\hat se^{-\beta t},t) = U(\bar s(\hat s,t)e^{-\beta t},t), \end{equation} where $\bar s(\hat s,t) e^{-\beta t} = 1-\exp(-\hat s e^{-\beta t})$. Then for any fixed $\hat s\in(0,\infty)$, \begin{equation} \bar s(\hat s,t)=\hat s+O(e^{-\beta t}) \quad\mbox{as $t\to\infty$.} \end{equation} Because the convergence in \eqref{eqD:lim} is uniform for $s$ in bounded sets by Proposition~\ref{p:compareU}, it follows that for each $\hat s\in[0,\infty)$, \begin{equation}\label{eqD:flim} \breve f(\hat se^{-\beta t},t)\to u_\alpha(\hat s/\lambda). \end{equation}
Next we establish \eqref{eqD:flim} for $\hat s=\infty$, recalling $\breve f(\infty,t)=m_0(f(t))$. In the present case of Model D, the evolution equation for $m_0(f(t))$ is not closed, and we formulate our result as follows.
\begin{lemma}\label{l:m0limD} For any solution of Model D, $m_0(f(t)) \to 1$ as $t\to\infty$. \end{lemma}
\begin{proof} 1. According to \cite[Thm.~12.1]{DLP2017}, the zeroth moment $m_0(f(t))=\breve f(\infty,t)$ is a smooth function of $t\in[0,\infty)$ that satisfies the inequality \begin{equation} \D_t m_0(f(t)) \le -m_0(f(t))^2 + m_0(f(t))\,,\quad t\ge0. \end{equation} We infer that for all $t\ge0$, \begin{equation}\label{b:m0} m_0(f(t))\le \frac1{1-e^{-t}}\,, \end{equation} as the right-hand size solves the logistic equation $y'=-y^2+y$ on $(0,\infty)$. Thus we infer \begin{equation}\label{b:m0limsup} \limsup_{t\to\infty} m_0(f(t)) \le 1. \end{equation} 2. We claim $\liminf_{t\to\infty} m_0(f(t)) \ge 1$. For this we use the result of Proposition~\ref{p:compareU}, with $U(s,t)$ for $0<s<1$ determined from $\breve f(\hat s,t)$ by \eqref{eqD:change}. Choose $S>0$ such that $u_\alpha(S)>1-\eps$. Then for $t$ sufficiently large we have \[ m_0(f(t)) \ge U(S e^{-\beta t},t) >1-\eps. \] Hence $\liminf_{t\to\infty} m_0(f(t)) \ge 1$. This finishes the proof of the Lemma. \end{proof}
Now, because \eqref{eqD:flim} holds for all $s\in[0,\infty]$, the desired conclusion of narrow convergence in Theorem~\ref{t:mainD} follows by using Proposition~\ref{p:narrowfat}. \end{proof}
\section{Pick properties of self-similar profiles} \label{s:Pick}
\begin{lemma}\label{l:cmFa}
For any $\alpha\in(0,1)$ the measure $F_{\star\alpha}$ of {Lemma } \ref{lem:ualphaBernstein} has a completely monotone density $f_{\star\alpha}$, whose Bernstein transform is the function $u_\alpha$ described in Lemma~\ref{lem:solbeta}, i.e., \[ u_\alpha(s) = \int_0^\infty(1-e^{-sx})f_\alpha(x)\,dx, \quad s\in[0,\infty]. \] \end{lemma}
\begin{proof} By Theorem 6.1(ii) of \cite{DLP2017}, if the initial data $F_0$ for Model C has a completely monotone density, then the solution $F_t$ has a completely monotone density for every $t\ge0$, with $F_t(dx)=f_t(x)\,dx$ where $f_t$ is completely monotone. By the representation theorem for complete Bernstein functions, this property is equivalent to saying that the Bernstein transform $U(\cdot,t)=\breve F_t$ is a Pick function.
As dilates and pointwise limits of complete Bernstein functions are complete Bernstein functions \cite[Cor.~7.6]{Schilling_etal_Bernstein}, we infer directly from our Theorem \ref{t:mainC} that for any $\alpha\in(0,1)$, the self-similar profile $u_\alpha$ is a complete Bernstein function. Therefore, its L\'evy measure $F_{\star\alpha}$ has a completely monotone density $f_\alpha$. \end{proof}
\begin{remark} An example of Pick-function initial data which satisfy the hypotheses of the convergence theorem is the following: \begin{equation} U_0(s) = s^\alpha = \frac{\alpha}{\Gamma(1-\alpha)} \int_0^\infty (1-e^{-sx}) x^{-1-\alpha}\,dx\, \label{e:salphaBT} \end{equation} \end{remark}
\begin{remark} We have no argument establishing the monotonicity of densities for model C that avoids use of the representation theorem for complete Bernstein functions. It would be interesting to have such an argument. \end{remark}
{\it Decomposition.} A point which is interesting, but not essential to the main thrust of our analysis, is that we can sometimes `decompose' the Bernstein transforms $U(s,t)=\breve F_t(s)$ of solutions of Model C, writing \begin{equation}\label{d:Valp} U(s,t) = V(s^\alpha,t), \end{equation} where $V(\cdot,t)$ itself is a complete Bernstein function. By consider limits as $t\to\infty$, this can be used to say something more about the self-similar profiles $u_\alpha$. \begin{proposition}\label{p:VaCB} (a) Suppose $\alpha\in(0,1)$ and $U_0(s)=V_0(s^\alpha)$ where $V_0$ is completely Bernstein. Then for all $t\ge0$, \eqref{d:Valp} holds for the solution of \eqref{eqC:Bernstein} with initial data $U_0$, where $V(\cdot,t)$ is completely Bernstein.
(b) For each $\alpha\in(0,1)$, the Bernstein transform $u_\alpha$ of the self-similar profiles of Lemma~\ref{lem:solbeta} have the form \begin{equation}\label{e:uaVa} u_\alpha(s) = V_\alpha(s^\alpha) \end{equation} where $V_\alpha$ is completely Bernstein, having the representation \begin{equation}\label{r:Va} V_\alpha(s) = \int_0^\infty (1-e^{-sx})g_{\star\alpha}(x)\,dx \end{equation} for some completely monotone function $g_{\star\alpha}$. \end{proposition}
\begin{proof} To prove part (a), we define $V(\cdot,t)$ by \eqref{d:Valp} and compute that \begin{gather} \label{e:DtV} \D_t V(s,t) + V^2 + V = 2 A_\alpha V(s,t), \\ A_\alpha V(s,t) = \int_0^1 V(sr,t)\, d(r^{1/\alpha}). \label{e:Aa} \end{gather} The implicit-explicit difference scheme used in \cite[Sec.~6]{DLP2017} to solve \eqref{eqC:Bernstein} corresponds precisely here to the difference scheme \begin{equation} \hat V_n(s) = V_n(s) + 2\Delta t\, A_\alpha V_n(s)\,, \end{equation} \begin{equation} (1+\Delta t)V_{n+1}(s) + \Delta t\, V_{n+1}(s)^2 = \hat V_n(s) \end{equation} under the correspondence \begin{equation} U_n(s) = V_n(s^\alpha). \end{equation} Exactly as argued at the end of \cite[Sec.~6]{DLP2017}, if $V_n$ is completely Bernstein then so is $\hat V_n$ since complete Bernstein functions form a convex cone closed under dilations and taking pointwise limits. Then $V_{n+1}$ is completely Bernstein due to \cite[Prop.~3.4]{DLP2017} (i.e., for the same reason $U_{n+1}$ is). Because of the fact that $U_n(s)\to U(s,t)$ as $\Delta t\to0$ with $n\Delta t\to t$. which was shown in \cite{DLP2017}, we infer that similarly $V_n(s)\to V(s,t)$, and hence $V(\cdot,t)$ is completely Bernstein.
Next we prove part (b). From the convergence result of Proposition~\ref{p:compareU} it follows that if $V_0(s)=U_0(s^{1/\alpha})\sim s$ as $s\to0$, then for all $s>0$, \begin{equation} V(se^{-\alpha\beta t},t)=U(s^{1/\alpha}e^{-\beta t},t)\to V_\alpha(s) \quad\mbox{as $t\to\infty$,} \end{equation} where $V_\alpha$ is defined by \eqref{e:uaVa}. By taking $V_0$ to be completely Bernstein and applying part (a), we conclude $V_\alpha$ is completely Bernstein through taking the pointwise limit. \end{proof}
\begin{remark} Formulae such as \eqref{e:uaVa}, involving the composition of two Bernstein functions, are associated with the notion of subordination of probability measures, as is discussed by Feller \cite[XIII.7]{Feller}. See section~\ref{s:subord} below for further information. \end{remark}
\begin{remark} Equation \eqref{e:DtV} satisfied by $V(s,t)$ is close to one satisfied by the Bernstein transform of the solution of a system modeling coagulation with multiple-fragmentation \cite{Melzak1957,MLM1997,KKW2011}. This system takes the following strong form analogous to \eqref{eq:CF3_Niwa_11}--\eqref{eq:CF5_Niwa_11}: \begin{eqnarray} &&\hspace{-1.5cm} \D_t f(x,t) = Q_a(f)(x,t) + Q_b(f)(x,t) , \label{eq:CmF3_Niwa_11}\\ &&\hspace{-1.5cm} Q_a(f)(x,t) = \int_0^x \, f(y,t) \, f(x-y,t) \, dy - 2 f(x,t) \, \int_0^\infty f(y,t) \, dy , \label{eq:CmF4_Niwa_11} \\ &&\hspace{-1.5cm}
Q_b(f)(x,t) = -f(x,t) + \int_x^\infty b(x|y) {f(y,t)} \, dy , \label{eq:CmF5_Niwa_11} \end{eqnarray} where \begin{equation}\label{d:newb}
b(x| y) = (\gamma+2)\frac{x^\gamma}{y^{1+\gamma}} \,. \qquad \gamma = \frac{1-\alpha}{\alpha} \,. \end{equation} The coefficient $\gamma+2$ is determined by the requirement that mass is conserved: \[
1= \frac1y \int_0^y xb(x|y)\,dx = (\gamma+2)\int_0^1 r^{\gamma+1}\,dr\,. \] A key calculation is that with $\vp_s(x)=1-e^{-sx}$, \begin{align*}
\int_0^y \vp_s(x) b(x|y)\,dx
& = \int_0^y \vp_s(x) (\gamma+2) \left(\frac xy\right)^{\gamma}\,\frac{dx}y \\ & = \frac{\gamma+2}{\gamma+1}\int_0^1 \vp_s(ry)\,d(r^{\gamma+1}) \\ & = (\alpha+1)\int_0^1 \vp_s(ry)\,d(r^{1/\alpha}) \end{align*} As a consequence, the Bernstein transform of a solution of \eqref{eq:CmF3_Niwa_11}--\eqref{eq:CmF5_Niwa_11} should satisfy \begin{gather} \label{e:DtVm} \D_t V(s,t) + V^2 + V = (1+\alpha) A_\alpha V(s,t)\,. \end{gather} {The coefficient $(1+\alpha)$ here differs from the factor 2 in \eqref{e:DtV},} and we see no way to scale the $V$ in \eqref{d:Valp} to get exactly this coagulation--multiple-fragmentation model.
A last note is that the `number of clusters' produced from a cluster of size $y$ by this fragmentation mechanism is calculated to be \[
n(y) = \int_0^y b(x|y)\,dx = \frac{\gamma+2}{\gamma+1}=\alpha+1. \] \end{remark}
\section{Asymptotics of self-similar profiles} \label{s:Tauber}
Here we complete the proof of Theorem~\ref{t:exist}, characterizing self-similar solutions of Model C, by describing the asymptotic behavior of the self-similar size-distribution profiles $f_{\star\alpha}$ in the limits of large and small size. This involves a Tauberian analysis based on the behavior of the Bernstein transform $u_\alpha$ as described in Lemma~\ref{lem:solbeta}.
\begin{proof}[Proof of Theorem~\ref{t:exist}] Given $\alpha\in(0,1)$, recall we know that for any self-similar solution of Model C as in \eqref{d:Fa}, the measure $F_{\star\alpha}(dx)$ must have Bernstein transform $u_\alpha(s)$ as described by Lemma~\ref{lem:solbeta}. That indeed the function $u_\alpha$ is the Bernstein transform of a probability measure $F_{\star\alpha}$ follows from Lemma~\ref{lem:ualphaBernstein}, and the fact that $F_{\alpha\star}$ has a completely monotone density $f_{\star\alpha}$ was shown in Lemma~\ref{l:cmFa}. It remains only to establish that $f_{\alpha\star}$ enjoys the asymptotic properties stated in \eqref{e:faasym}.
From Lemma~\ref{lem:solbeta} we infer that as $z\to\infty$, \[ 1-u_\alpha(z) = \int_0^\infty e^{-zx} f_{\star\alpha}(x)\,dx \sim \hat c z^{-\hat\alpha} \quad\mbox{as $z\to\infty$}. \] Recalling $\hat\alpha\in(0,\frac13)$, invoking the Tauberian theorem \cite[Thm.~XIII.5.3]{Feller} and the fact that $f_{\star\alpha}$ is monotone, from \cite[Thm.~XIII.5.4]{Feller} we infer \begin{equation}\label{e:fa0} f_{\star\alpha}(x) \sim
\frac{\hat c }{\Gamma(\hat\alpha)} x^{\hat\alpha-1} \quad\mbox{as $x\to0$.} \end{equation}
Next, from Lemma~\ref{lem:solbeta}, {\eqref{e:ODEu} and \eqref{e:beta}} we infer that \[ \D_z u_\alpha(z) = \int_0^\infty e^{-zx} xf_{\star\alpha}(x)\,dx \sim \alpha z^{\alpha-1} \quad\mbox{as $z\to0$}. \] By Karamata's Tauberian theorem \cite[Thm.~XIII.5.2]{Feller} we deduce \[ \int_0^x y f_{\star\alpha}(y)\,dy \sim \frac{\alpha }{\Gamma(2-\alpha)} x^{1-\alpha} \quad\mbox{as $x\to\infty$}. \] Although we do not know $y\mapsto y f_\alpha(y)$ is eventually monotone, the selection argument used in the proof of \cite[Thm.~XIII.5.4]{Feller} works without change, allowing us to infer that \begin{equation} x f_{\star\alpha}(x) \sim \frac{\alpha }{\Gamma(1-\alpha)}x^{-\alpha} \quad\mbox{as $x\to\infty$}. \end{equation} This completes the proof of Theorem~\ref{t:exist}. \end{proof}
\begin{remark} We note that in the limit $\alpha\to1$ we have $\beta\to0$ and $\hat\alpha\to\frac13$, and the power-law exponent $\hat\alpha-1\to-\frac23$. This recovers the exponent governing the small-size behavior of the equilibrium distribution analyzed previously in \cite[Eq.~(1.6)]{DLP2017}. \end{remark}
\begin{remark} By \eqref{e:uaVa}, \[ 1-V_\alpha(z) = \int_0^\infty e^{-zx}g_{\star\alpha}(x)\,dx \sim \hat c z^{-\hat\alpha/\alpha}\,, \] hence by the same argument as that leading to \eqref{e:fa0} we find \begin{equation}\label{e:ga0} g_{\star\alpha}(x) \sim \frac{\hat c}{\Gamma(\hat\alpha/\alpha)} x^{-1+\hat\alpha/\alpha}\quad \mbox{as $x\to0.$} \end{equation} We note that $\hat\alpha/\alpha<1$ for all $\alpha\in(0,1)$, because the assumption $\hat\alpha=\alpha$ together with the relations \eqref{e:eigsa} lead to a contradiction. \end{remark}
\section{Series in fractional powers} \label{s:series}
In this section we show that the self-similar profile in Lemma~\ref{lem:solbeta} is expressed, for small $z>0$, in the form \begin{equation}\label{e:ustarseries} u_\alpha(z) = \sum_{n=1}^\infty (-1)^{n-1}c_n z^{\alpha n}\,, \end{equation} where the series converges for $z^\alpha\in(0,R_\alpha)$ for some positive but finite number $R_\alpha$, and the coefficient sequence $\{c_n\}$ is positive with a rather nice structure.
By substituting the series expansion \eqref{e:ustarseries} into \eqref{eq:SSu1} we find that $c_1=1$, and $c_n$ is necessarily determined recursively for $n\ge2$ by \begin{equation}\label{e:crecurs} c_n = \frac1{a_n} \sum_{k=1}^{n-1} c_k c_{n-k}\,, \end{equation} \begin{equation} a_n = \beta\alpha n + 1 - \frac2{\alpha n+1} = \frac{1-\alpha}{1+\alpha}n + \frac{\alpha n-1}{\alpha n+1}. \end{equation} Because the relation \eqref{e:beta} implies that indeed \begin{equation}\label{e:betaalpha2} \beta\alpha+1 = \frac2{1+\alpha}, \end{equation} plainly $a_1=0$ and $a_n$ increases with $n$, with $a_n>0$ for $n>1$.
Recall that we know from Proposition~\ref{p:VaCB} that $u_\alpha(s)=V_\alpha(s^\alpha)$ where $V_\alpha$ is completely Bernstein.
\begin{proposition} For each $\alpha\in(0,1)$, $V_\alpha$ is analytic in a neighborhood of $s=0$, given by the power series \[ V_\alpha(s) = \sum_{n=1}^\infty (-1)^{n-1}c_n s^{n}\,. \] This series has a positive radius of convergence $R_\alpha$ satisfying \begin{equation}\label{e:Rbds} \frac{1-\alpha}{1+\alpha} \le R_\alpha \le a_2 <1, \end{equation} and coefficients that take the form \begin{equation}\label{e:cng} c_n = \gamma^\star_{n-1} R_\alpha^{1-n}, \end{equation} where $(\gamma^\star_n)_{n\ge0}$ is a {\sl completely monotone sequence} with $\gamma^\star_0=1$. \end{proposition}
\begin{proof} It suffices to prove the bounds on the radius of convergence and the representation formula \eqref{e:cng}, as the validity of equation \eqref{eq:SSu1} then follows by substitution. By induction we will establish bounds on the radius of convergence of the power series \begin{equation}\label{e:vstarseries} v_\star(z) = \sum_{n=1}^\infty c_n z^n \,, \end{equation} which is evidently related to $V_\alpha$ by $V_\alpha(z)=-v_\star(-z)$. Observe that the inequality $c_k \le {m}/{r^k}$ for $1\le k<n$ implies \[ c_n \le \frac{n-1}{a_n} \frac{m^2}{r^n} \le \frac{m}{r^n} \,, \] provided that \[ m\le \frac{a_n}{n-1} = \frac{1-\alpha}{1+\alpha} + \frac{2\alpha}{(1+\alpha)(1+\alpha n)} \,. \] By choosing \[ m=r= \frac{1-\alpha}{1+\alpha}=\beta\alpha\,, \] we ensure $c_1=m/r$ and therefore $c_n\le r^{1-n}$ for all $n\ge1$, i.e., \begin{equation} c_n \le \left( \frac{1+\alpha}{1-\alpha}\right)^{n-1} \,, \quad n=1,2,\ldots, \end{equation} whence \[ v_\star(z) \le \frac{rz}{r-z}<\infty \quad\mbox{for $0<z<r$}. \] In a similar way, the choice \[ M=R = a_2 \ge \frac{a_n}{n-1} \] for all $n\ge2$ ensures $c_n \ge {M}/{R^n}$ for all $n\ge2$, whence \[ v_\star(z) \ge \frac{Rz}{R-z} \quad\mbox{for $0<z<R$} \] By consequence we infer the bounds in \eqref{e:Rbds} hold.
Now, because $V_\alpha$ is completely Bernstein, it is a Pick function analytic on the positive half-line. Hence, from what we have shown, the function $v_\star$ is a Pick function analytic on $(-\infty,R_\star)$. From this and Corollary 1 of \cite{LP2016}, it follows directly that the coefficients $c_n$ may be represented in the form \eqref{e:cng} where $\{\gamma_n\}_{n\ge0}$ is a completely monotone sequence with $\gamma^\star_0=1$. \end{proof}
\begin{remark} In the limiting case $\beta=0$, $\alpha=1$, the coefficients $c_n$ reduce to the explicit form appearing in eq. (5.19) of \cite{DLP2017}. Namely, \[ c_n = A_n(3,1) = \frac1{3n+1} \binom{3n+1}{n} \] in terms of the Fuss-Catalan numbers defined by \[ A_n(p,r) = \frac1{pn+r}\binom{pn+r}{n} \,. \] This can be verified directly from the recursion formulae in \eqref{e:crecurs} by using a known identity for the Fuss-Catalan numbers \cite[p.~148]{Riordan}. \end{remark}
\begin{remark} We are not aware of any combinatorial representation or interpretation of the coefficients $c_n(\alpha)$ for $\alpha\in(0,1)$, however. \end{remark}
\begin{remark} {\it (Nature of the singularity at $R_\alpha$)} Numerical evidence suggests that for $0<\alpha<1$, the singularity at $R_\alpha$ is a simple pole.
If true, this should imply that as $n\to\infty$, the coefficients $\gamma^\star_n \to \gamma^\star_\infty>0$, and the completely monotone L\'evy density $g_{\star\alpha}(x)$ for the complete Bernstein function $V_\alpha(z)$ has exponential decay at $\infty$, with \[ g_{\star\alpha}(x) \sim C_\star e^{-R_\alpha x} \quad \mbox{as $x\to\infty$}, \] where $C_\star>0$. \end{remark}
\section{A subordination formula} \label{s:subord} Here we use the subordination formulae from \cite[XIII.7(e)]{Feller} as linearized in \cite[Remark 3.10]{ILP2015}, to describe a relation between the completely monotone L\'evy densities for the Bernstein functions $u_\alpha$ and $V_\alpha$. Recall we have shown \[ u_\alpha(z) = \int_0^\infty (1-e^{-zx})f_{\star\alpha}(x)\,dx = V_\alpha(z^\alpha), \] where $V_\alpha$ is a complete Bernstein function, with \[ V_\alpha(z) = \int_0^\infty (1-e^{-zx})g_{\star\alpha}(x)\,dx\,, \] for some completely monotone function $g_{\star\alpha}$. The complete Bernstein function $z^\alpha$ has power-law L\'evy measure \[ \nu_0(dx)=c_\alpha x^{-1-\alpha} dx\,,\qquad c_\alpha = \frac{\alpha}{\Gamma(1-\alpha)}. \] This is the jump measure for an $\alpha$-stable L\'evy process $\{Y_\tau\}_{\tau\ge0}$ (increasing in $\tau$) whose time-$\tau$ transition kernel $Q_\tau(dy)$ has the Laplace transform \[ \mathbb{E}(e^{-qY_\tau}) = \int_0^\infty e^{-qy} Q_\tau(dy) = e^{-\tau q^\alpha}\,. \] Recalling the subordination formula in the linearized form (3.20) from \cite{ILP2015}, we infer that the self-similar profile $f_{\star\alpha}$ may be expressed as \[ f_{\star\alpha}(x) = \int_0^\infty Q_\tau(dx) g_{\star\alpha}(\tau)\,d\tau \,. \]
We know that $Q_1(dy)=p_\alpha(y)\,dy$ where $p_\alpha$ is the maximally skewed L\'evy-stable density from \cite[XVII.7]{Feller} given by \begin{equation}\label{e:palpha} p_\alpha(x) = p(x;\alpha,-\alpha) = \frac{-1}{\pi x} \sum_{k=1}^\infty \frac{\Gamma(k\alpha+1)}{k!} (-x^{-\alpha})^k \sin k\pi\alpha\,. \end{equation} Then by scaling dual to $\exp(-\tau q^\alpha) = \exp(-(\tau^{1/\alpha}q)^\alpha)$, we find \[ Q_\tau(dy) = p_\alpha\left(\frac{y}{\tau^{1/\alpha}}\right)\frac{dy}{\tau^{1/\alpha}}\,, \] and obtain the following. \begin{proposition} The self-similar profile $f_{\star\alpha}$ is related to the completely monotone L\'evy densitiy $g_{\star\alpha}$ of $V_\alpha$ by \[ f_{\star\alpha}(x) = \int_0^\infty g_{\star\alpha}(\tau) p_\alpha\left(\frac{x}{\tau^{1/\alpha}}\right) \frac{d\tau }{ \tau^{1/\alpha}} \] \end{proposition}
We note that in the limit $\alpha\to1$ one has $p_\alpha(y)\,dy\to \delta_1$, the delta mass at $1$, consistent with $g_{\star\alpha} \to f_{\star\alpha}$. Moreover, note that from \eqref{e:palpha} the large-$x$ behavior of the $\alpha$-stable density $p_\alpha$ is \begin{equation} p_\alpha(x) \sim {\Gamma(1+\alpha)}\frac{\sin \pi\alpha}{\pi} x^{-\alpha-1} \sim f_{\star\alpha}(x)\,,\quad x\to\infty\,, \end{equation} due to Euler's reflection formula for the $\Gamma$-function. This is consistent with the fact that the Bernstein transform of $p_\alpha$ is \[ \int_0^\infty(1-e^{-sx})p_\alpha(x)\,dx = 1-e^{-s^\alpha} \sim s^\alpha \sim u_\alpha(s)\,, \quad s\to0. \]
\input{fatniwa-arxiv.bbl}
\ifdraft {\bf Items to finish:}
\begin{itemize} \item [done] Edit acknowledgments \item [done] (remark added at end of sec 3) (Can we get necessary and/or sufficient conditions using regular variation?) \item [done] Describe test sequences better in \eqref{eq:CF2_disc}? [Bounded as in \cite{DLP2017}] \end{itemize} \fi
\end{document} | arXiv |
Home Journals MMEP Higher Order Chemical Reaction on MHD Nanofluid Flow with Slip Boundary Conditions: A Numerical Approach
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Higher Order Chemical Reaction on MHD Nanofluid Flow with Slip Boundary Conditions: A Numerical Approach
Kharabela Swain* | Sampada K. Parida | Gouranga C. Dash
Department of Mathematics, Radhakrishna Institute of Technology & Engineering, Biju Patnaik University of Technology, Bhubaneswar 752057, India
Department of Mathematics, Institute of Technical Education and Research, Siksha 'O' Anusandhan (Deemed to be University), Bhubaneswar 751030, India
[email protected]
https://doi.org/10.18280/mmep.060218
06.02_18.pdf
The present paper analyzes the MHD flow of nanofluid past a permeable stretched surface. The effect of non-linear radiative heat transfer, higher order chemical reaction and slip boundary conditions are also incorporated to the flow phenomena to enhance the heat transfer rate in the nanofluid. A suitable self-similar transformation is employed to convert PDEs into non-linear ODEs. The resulting set of differential systems is solved numerically by fourth order Runge-Kutta method with shooting technique. The impact of thermo-physical quantities on the flow field is shown via graphs. The numerical results for skin friction coefficient, local Nusselt and Sherwood numbers are calculated and demonstrated via table. It is found that heat generation is favorable to enhance the rate of shear stress as well as rate of heat transfer, further absorption retards mass transfer rate significantly. Also, the thickness of species distribution increases as the order of the chemical reaction n increases.
nanofluid, non-linear thermal radiation, chemical reaction, porous medium
A nanoparticle is a microscopic particle of diameter less than 100nm. The nanoparticles are made up metals, carbides, oxides or carbon nanotubes. By adding nanoparticles in the convectional fluids like ethylene glycol, water and engine oil, nanofluids are formed to enhance the heat transfer phenomena of the base fluids. There are several applications of nanofluids in the diverse field of engineering such as heat exchanger, refrigerator freezers, coolants, solar receiver, radiators etc. The pioneer experimental work on nanofluid is obtained Choi [1]. In his work, he takes water as a base fluid which enhances the heat transfer properties of the fluid. Further, development of model by incorporating Brownian motion and thermophoresis in the boundary layer flow of nanofluid was studied by Buongiorno [2].
The heat transfer radiation has many applications in the various fields such as design of engines and combustion chambers to operate at increased temperature to raise thermal efficiency, solar energy, semiconductor wafer processing, manufacturing of translucent crystals, energy transfer in furnaces etc. Akbar et al. [3] have studied the effects of the thermal radiation on nanofluid flow towards a stretching sheet with convective boundary condition. Second order slip MHD flow with the effects of radiation and chemical reaction has been studied by Zhu et al. [4]. Ibrahim [5] and Reddy [6] have considered the MHD nanofluid flow past a stretched surface with convective boundary condition and thermal radiation. Mabood et al. [7] projected their aim on Williamson nanofluid flow in presence of thermal radiation. Elbashbeshy et al. [8], Babu and Sandeep [9], Rahman and Eltayeb [10] and Mustafa et al. [11] studied numerically the results of nonlinear radiation mechanism for heat transfer of nanofluid past a vertical plate. Motsumi and Makinde [12] also extended their work by incorporating viscous dissipation and thermal radiation.
The application of heat and mass transfer is an important aspect now a day in industries such as the application of wet cooling water, drying, curing of plastics, food processing etc. Heat is generated when chemical reaction takes place between nanoparticles and base fluids. At that time the behavior of nanoparticles is observed which depending upon the sign of the chemical reaction. The rate of chemical reaction is controlled by the concentration of the species. Kandasamy and Devi [13] and Mahantesh et al. [14] have investigated the mixed convective nanofluid flow past a vertical plate with chemical reaction. Palani et al. [15] discussed unsteady nanofluid flow phenomena in presence of chemical of higher order.
The aim of the present study is to investigate the boundary layer flow of MHD nanofluid over stretching sheet under the influence of non-linear thermal radiation and higher order chemical reaction. Using similarity transformations, the governing PDEs are transferred into ODEs which were solved numerically by shooting technique. For the validation of the present result we have compared our result with that of earlier established result of Ibrahim [5] in a particular case.
2. Problem Formalism
Electrically conducting boundary layer flow of nanofluid over a permeable stretched surface immersed in a porous medium as shown in Figure 1 has been considered. Let us assume: (i) the flow is steady (ii) the sheet surface gets heated at temperature Tf and a heat transfer coefficient hf (iii) the flow is laminar (iv) B0 , magnetic field strength is applied normal to the flow (v) the induced magnetic field is neglected (vi) no slip boundary conditions are subjected to very thin moving fluid and stationary sheet (vii) the gravitational effect is negligible.
The governing boundary layer equations subjected to the conversation of mass, momentum and energy are respectively
Figure 1. Sketch of flow
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
$\rho_{f}\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=\mu_{f} \frac{\partial^{2} u}{\partial y^{2}}-\sigma B_{0}^{2} u-\frac{\mu_{f} u}{K p^{*}}$ (2)
$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\alpha_{f} \frac{\partial^{2} T}{\partial y^{2}}+\frac{(\rho C)_{p}}{(\rho C)_{f}}\left\{D_{B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_{T}}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^{2}\right\}$
$-\frac{1}{\left(\rho C_{p}\right)_{f}} \frac{\partial q_{r}}{\partial y}+\frac{\mu_{f}}{\left(\rho C_{p}\right)_{f}}\left(\frac{\partial u}{\partial y}\right)^{2}+\frac{Q_{0}}{\left(\rho C_{p}\right)_{f}}\left(T-T_{\infty}\right)$ (3)
$u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_{B} \frac{\partial^{2} C}{\partial y^{2}}+\frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}}-K_{n}\left(C-C_{\infty}\right)^{n}$ (4)
Using Rosseland approximation of radiation, the radiative heat flux is given by
$q_{r}=-\frac{4 \sigma^{*}}{3 k^{*}} \frac{\partial T^{4}}{\partial y}=-\frac{16 \sigma^{*}}{3 k^{*}} T^{3} \frac{\partial T}{\partial y}$ (5)
Substituting equation (5) in equation (4) we get
$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\frac{\partial}{\partial y}\left[\left\{\alpha_{f}+\frac{16 \sigma^{*} T^{3}}{3 k^{*}\left(\rho C_{p}\right)_{f}}\right\} \frac{\partial T}{\partial y}\right]$
$+\tau\left\{D_{B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_{T}}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^{2}\right\}+\frac{\mu_{f}}{\left(\rho C_{p}\right)_{f}}\left(\frac{\partial u}{\partial y}\right)^{2}$
$+\frac{Q_{0}}{\left(\rho C_{p}\right)_{f}}\left(T-T_{\infty}\right)$ (6)
The surface conditions are:
$u=u_{w}(x)=a x, v=v_{w},-k_{f} \frac{\partial T}{\partial y}=h_{f}\left(T_{f}-T\right)$
$-D_{m} \frac{\partial C}{\partial y}=h_{m}\left(C_{f}-C\right)$ at $y=0$
$u=0, T \rightarrow T_{s}, C \rightarrow C_{p}$ as $y \rightarrow \infty$ (7)
where uw and vw are the stretching velocity and wall mass transfer velocity respectively, a>0 for the stretching sheet.
The similarity transformations considered are:
$\eta=\sqrt{\frac{a}{v_{f}}} y, u=a x f^{\prime}(\eta), v=-\sqrt{v_{f} a} f(\eta)$
$\theta(\eta)=\frac{T-T_{\infty}}{T_{f}-T_{\infty}}, \phi(\eta)=\frac{C-C_{\infty}}{C_{f}-C_{\infty}}$$\}$ (8)
Using equation (8), equations (2), (6), and (4) can be written as
$f^{\prime \prime}-f^{\prime 2}+f f^{\prime \prime}-\left(M+\frac{1}{K p}\right) f^{\prime}=0$ (9)
$f^{\prime \prime \prime}-f^{\prime 2}+f f^{\prime \prime}-\left(M+\frac{1}{K p}\right) f^{\prime}=0$ (10)
$\phi^{\prime \prime}+L e \operatorname{Pr} f \phi^{\prime}+\frac{N t}{N b} \theta^{\prime \prime}-\gamma L e \operatorname{Pr} \phi^{n}=0$ (11)
The non-dimensional forms of equation (7) are
$f(0)=S, f^{\prime}(0)=1, \theta^{\prime}(0)=-B i_{t}[1-\theta(0)]$
$\phi^{\prime}(0)=-B i_{c}[1-\phi(0)]$ at $\eta=0$
$f^{\prime}(\infty)=0, \theta(\infty)=0, \phi(\infty)=0$ as $\eta \rightarrow \infty$ (12)
with non-dimensional parameters
$M=\frac{\sigma B_{0}^{2}}{a \rho_{j}}, K p=\frac{a K p^{*}}{v_{f}}, R=\frac{16 \sigma^{*} T_{\infty}^{3}}{3 k_{f} k^{*}}, \operatorname{Pr}=\frac{v_{f}}{\alpha_{f}}$
$N b=\frac{\tau D_{s}\left(C_{f}-C_{\alpha}\right)}{v_{f}}, \gamma=\frac{K_{n}\left(C_{f}-C_{\alpha}\right)^{n-1}}{a}, L e=\frac{\alpha_{f}}{D_{B}}$
$N t=\frac{\tau D_{T}\left(T_{f}-T_{\infty}\right)}{v_{f} T_{\infty}}, E c=\frac{\mu_{f} u_{w}^{2}}{\left(\rho C_{p}\right)_{f}\left(T_{f}-T_{\infty}\right)}, S=\frac{v_{w}}{\sqrt{a v_{f}}}$
$Q=\frac{v_{f} Q_{0}}{a \alpha\left(\rho C_{p}\right)_{f}}, B i_{t}=\frac{h_{f}}{k} \sqrt{\frac{v_{f}}{a}}, B i_{c}=\frac{h_{m}}{D_{m}} \sqrt{\frac{v_{f}}{a}}$
3. Physical Quantities of Engineering Interest
The important physical quantities used in these flow phenomena are shear stress coefficient Cf, local Nusselt number Nux
and local Sherwood number Shx are defined as
$\begin{aligned} C_{f} &=\frac{\tau_{w}}{\rho u_{w}^{2}} \\ N u_{x}=& \frac{x q_{w}}{k_{f}\left(T_{f}-T_{\infty}\right)} \\ S h_{x} &=\frac{x q_{m}}{D_{B}\left(C_{f}-C_{\infty}\right)} \end{aligned} \}$ (13)
Here, wall shear stress
$\tau_{w}=\mu_{f}\left(\frac{\partial u}{\partial y}\right)_{y=0} \Rightarrow C_{f} \sqrt{\operatorname{Re}_{x}}=-f^{\prime \prime}(0)$
The wall heat flux
$q_{w}=-k\left(\frac{\partial T}{\partial \dot{y}}\right)_{y=0}+\left(q_{r}\right)_{y=0} \Rightarrow \frac{N u_{x}}{\sqrt{\mathrm{Re}_{x}}}=-(1+R) \theta^{\prime}(0)$
The mass flux
$q_{m}=-D_{B}\left(\frac{\partial C}{\partial y}\right)_{y=0} \Rightarrow \frac{S h_{x}}{\sqrt{\mathrm{Re}_{x}}}=-\phi^{\prime}(0)$
where $R e_{x}=\frac{a x^{2}}{v}$ is the local Reynolds number.
The present study analyzes the boundary layer nanofluid flow past a permeable stretched sheet embedding with porous medium. Influence of thermal radiation and heat generation/absorption has been incorporated in the energy equation. The crux of the present investigation is the higher order solutal concentration. The governing transformed ODEs are nonlinear and coupled those are unable to solve analytically. The roust numerical technique such as Runge-Kutta fourth order method is employed accompanied with shooting to handle these set of ODEs. Initially, the unknown initial conditions for f"(0) , $\theta^{\prime}(0)$ and $\phi^{\prime}(0)$ are chosen and obtained by shooting technique and then solve numerically. The impact of various physical parameters on the flow phenomena are obtained and presented via Figures 2-14 and the computational results of engineering interest are shown in Table 1. The validation of the said results is obtained with that of established results of Ibrahim [5] in particular cases.
The numerical values of skin friction coefficient, Nusselt number and Sherwood number are calculated for distinct physical parameters such as M, Pr, R, Ec, γ, S, Q, n in Table 1. It is remarked that magnetic parameter (M) as well as mass transfer parameter (S) enhance the skin friction coefficient whereas chemical reaction parameter reduces it. Radiation parameter and chemical reaction parameter increase the heat flux at the surface but adverse effect is observed in case of mass flux. It is seen that Eckert number as well as heat generation/absorption parameter enhance both heat and mass fluxes at the wall. The increasing values of magnetic parameter and mass transfer parameter enhancing the skin friction coefficient and mass flux at the wall whereas opposite effect is noticed in heat flux. It is seen that higher order chemical reaction and Prandtl number has adverse effect on heat flux and mass flux at the wall.
4.1 Velocity distribution
Figure 2 shows the velocity profile for interaction of magnetic parameter (M) and porosity parameter (Kp). It is observed that the velocity reduces with the increase of M. The fact is the inclusion of magnetic field produces a resistive force called Lorentz force which resists the fluid motion throughout the boundary layer. Further, the effect of embedding parameter, the porous matrix is also exhibited. It is clear to note that the presence of porous matrix decelerates the fluid velocity as well. Similar to that of magnetic parameter, porosity also resists the velocity distribution resulted a thicker velocity boundary layer. Figure 3 exhibits the effect S on the velocity distributions. It is seen that increasing values of S decelerates the nanofluid velocity.
Figure 2. Velocity distributions for M and Kp
Figure 3. Velocity distributions for S
Table 1. Values of skin friction coefficients $f^{\prime \prime}(0),$ Nusselt number $\left\{-\theta^{\prime}(0)\right\}$ and Shervood number $\left\{-\phi^{\prime}(0)\right\}$
$\gamma $
$f^{\prime \prime}(0)$
$-\theta^{\prime}(0)$
$-\phi^{\prime}(0)$
4.2 Temperature distribution
Figure 4 illustrates the influences of Prandtl number (Pr) and Lewis number (Le) on the temperature profile. Increase values of Le resulting decrease in temperature profile. Since Pr has inverse relation with thermal conductivity, the larger Pr reverses weaker thermal diffusivity and hence decreases the thermal boundary layer thickness. Figure 5 presents the behavior of the heat transfer for distinct values of Eckert number (Ec). It is seen that an increase in Ec, the heat transfer rate increases. The positive values of Ec indicate cooling of the plate therefore heat loss occurs from the surface to the fluid so that the higher values of dissipative heat energy enhance the fluid temperature. The reason is in the thermal boundary layer certain amount of heat energy stored up which favors in to increase the nanofluid temperature. Figure 6 depicts the importance of thermal radiation on the temperature distribution. It is noteworthy that an increase in R increases the nanofluid temperature. It is the fact that the thermal radiation enhances the thermal diffusivity of the nanofluid resulting a thicker thermal boundary layer. Figure 7 describes the variation of temperature profile for distinct values of heat generation/absorption parameter. It is seen that the heat source parameter enhances the nanofluid temperature whereas the reverse effect is encountered for heat absorption. In the presence of magnetic parameter which retards the velocity profile due to the resistive force at the same time certain amount of heat energy stored up in the thermal boundary layer with an inclusion of heat generation parameter. However, in case of heat absorption due to loss of heat energy the profile becomes thinner and thinner. Figure 8 represents the variation of temperature profile with different values of thermophoresis parameter (Nt) and Brownian motion parameter (Nb). More importantly, the effect of increasing values of both Nt and Nb, increase the temperature profile. In Figure 9, the effects of thermal Biot number ( Bit ) and solutal Biot number ( Bic ) on the temperature distribution is shown. It is note that increasing values of both Bit and Bic , increase the temperature distribution.
Figure 4. Temperature distributions for Le and Pr
Figure 5. Temperature distributions for Ec
Figure 6. Temperature distributions for R
Figure 7. Temperature distributions for Q
Figure 8. Temperature distributions for Nb and Nt
Figure 9. Temperature distributions for and C
4.3 Concentration distribution
The influence of Lewis number (Le) on solutal transfer is demonstrated in Figure 10. As Le is the combined effect of both thermal diffusivity and mass diffusivity, both have retardation effects on the corresponding distribution resulted in the concentration profile decreases. Therefore, as Le increases the solutal boundary layer thickness decreases. Figure 11 displays the effect of $γ$ , chemical reaction parameter on mass transfer distribution. In the present study we consider destructive chemical reaction $γ>0$ , no chemical reaction $γ=0$ and generative chemical reaction $γ<0$ are taken care of. In the absence of chemical reaction present result validates with that of Ibrahim [5]. However, for $γ>0$ , the profiles decelerate and reverse effect is observed for $\gamma <0$ . Figure 12 describes the variation of species concentration for different orders of the chemical reaction. It is remarked that concentration profile enhances with higher order of the chemical reaction. From Figure 13, it is observed that the concentration profile increases as Bic increases. Figure 14 examines the effects of Brownian motion (Nb) and thermophoresis (Nt) parameters on concentration profile. As Nb increases the Brownian force increases for which the concentration gradient of nanoparticles increases at the surface. It is note that higher values of thermophoresis parameter increase the concentration uniformly at all the levels.
The opposite trend is observed in solutal boundary layer with increasing value of Nb and Nt.
Heat generation is favorable to enhance the rate of shear stress as well as rate of heat transfer, further absorption retards mass transfer rate significantly.
Thermal boundary layer thickness increases due to increase in both Eckert number Ec and Radiation parameter R.
The thickness of species distribution increases as the order of the chemical reaction n increases.
The magnitude of concentration $ϕ$ is greater for constructive chemical reaction parameter $γ<0$ in comparison to destructive chemical reaction parameter ( $γ>0$ ).
Figure 10. Concentration distributions for Le
Figure 11. Concentration distributions for
Figure 12. Concentration distributions for n
Figure 13. Concentration distributions for Bic
Figure 14. Concentration distributions for Nb and Nt
u, v
Velocity components along the x
and y directions respectively
Magnetic parameter
Porosity parameter
Prandtl number
Thermal radiation parameter
heat generation/absorption parameter
Eckert number
Mass transfer parameter
Lewis number
Brownian motion parameter
Thermophoresis parameter
Chemical reaction parameter
Thermal Biot number
Concentration Biot number
${{\rho }_{f}}$
Density of the fluid
${{\mu }_{f}}$
Dynamic viscosity
Thermal conductivity of the fluid
Convective heat transfer coefficient
Convective mass transfer coefficient
Molecular diffusivity
$\sigma$
$K{{p}^{*}}$
Permeability parameter
${{\alpha }_{f}}$
Thermal diffusivity
Brownian diffusion coefficient
Thermophoretic diffusion coefficient
Fluid temperature
${{T}_{\infty }}$
Ambient fluid temperature
Nanoparticle concentration
${{C}_{\infty }}$
Ambient nanoparticle concentration
Specific heat at constant pressure
Solutal concentration of order n
$\tau =\frac{{{\left( \rho C \right)}_{p}}}{{{\left( \rho C \right)}_{f}}}$
Ratio of effective heat capacity of
nanoparticles and heat capacity
of the fluid
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Variation and oscillation for harmonic operators in the inverse Gaussian setting
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A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition
February 2022, 21(2): 393-418. doi: 10.3934/cpaa.2021182
A convergent finite difference method for computing minimal Lagrangian graphs
Brittany Froese Hamfeldt , and Jacob Lesniewski
Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102
* Corresponding Author
Received February 2021 Revised June 2021 Published February 2022 Early access November 2021
Fund Project: The first author was partially supported by NSF DMS-1619807 and NSF DMS-1751996. The second author was partially supported by NSF DMS-1619807
We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an additive eigenvalue problem for a fully nonlinear elliptic partial differential equation. We introduce and implement a two-step generalized finite difference method, which we prove converges to the solution of the eigenvalue problem. Numerical experiments validate this approach in a range of challenging settings. We further discuss the generalization of this new framework to Monge-Ampère type equations arising in optimal transport. This approach holds great promise for applications where the data does not naturally satisfy the mass balance condition, and for the design of numerical methods with improved stability properties.
Keywords: finite difference methods, minimal Lagrangian graphs, second boundary value problem, fully nonlinear elliptic equations, eigenvalue problems.
Mathematics Subject Classification: Primary: 65N06, 65N12, 65N25; Secondary: 35J15, 35J25, 35J60, 35J66.
Citation: Brittany Froese Hamfeldt, Jacob Lesniewski. A convergent finite difference method for computing minimal Lagrangian graphs. Communications on Pure & Applied Analysis, 2022, 21 (2) : 393-418. doi: 10.3934/cpaa.2021182
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Figure 1. Discrete solution to Poisson's equation when viewed as an eigenvalue problem
19]">Figure 2. Examples of quadtree meshes. White squares are inside the domain, while gray squares intersect the boundary [19]
19]">Figure 3. Potential neighbors are circled in gray. Examples of selected neighbors are circled in black [19]
Figure 4. Examples of neighbors $ x_1, x_2 $ needed to construct a monotone approximation of the directional derivative in the direction $ n $ at the boundary point $ x_0 $
Figure 5. Domain and computed target ellipse
Figure 6. Computed maps from a square $ X $ to various targets $ Y $
Figure 7. Circular domain $ X $ and square target $ Y $
Figure 8. Circular domain $ X $ and degenerate target $ Y $
Table 1. Error in mapping an ellipse to an ellipse
$ h $ $ \|u^h- u_{\text{ex}}\|_\infty $ Ratio Observed Order
$ 2.625\times 10^{-1} $ $ 1.304 \times 10^{-1} $
$ 1.313\times 10^{-1} $ $ 5.703\times 10^{-2} $ 2.287 1.194
Table 2. Error in mapping a circle to a line segment
$ 6.875\times 10^{-2} $ $ 3.812 \times 10^{-2} $ 2.396 1.261
$ 8.59\times 10^{-3} $ $ 4.636\times 10^{-3} $ 2.333 1.222
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Brittany Froese Hamfeldt Jacob Lesniewski | CommonCrawl |
\begin{document}
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\begin{titlepage} \author{Pierre Berger~\footnote{[email protected], CNRS and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320} {} and Abed Bounemoura~\footnote{[email protected], IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320}} \title{\LARGE{\textbf{A geometrical proof of the persistence of normally hyperbolic submanifolds}}} \end{titlepage}
\maketitle
\begin{abstract} We present a simple, computation free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold which is invariant and normally hyperbolic persists under small perturbations of the diffeomorphism. The persistence of a Lipschitz invariant submanifold follows from an application of the Schauder fixed point theorem to a graph transform, while smoothness and uniqueness of the invariant submanifold are obtained through geometrical arguments. Moreover, our proof provides a new result on persistence and regularity of ``topologically" normally hyperbolic submanifolds, but without any uniqueness statement. \end{abstract}
\section{Introduction}
\paraga Let $M$ be smooth manifold, $f : M \rightarrow M$ a $C^1$-diffeomorphism and $N\subseteq M$ a $C^1$-submanifold invariant by $f$. Roughly speaking, $f$ is \emph{normally hyperbolic} at $N$ if the tangent map $Tf$, restricted to the normal direction to $N$, is hyperbolic (it expands and contracts complementary directions) and if it dominates the restriction of $Tf$ to the tangent direction $TN$ (that is, expansion and contraction in the tangent direction, if any, are weaker than those in the normal direction). A precise definition will be given below.
The importance of invariant normally hyperbolic submanifolds, both in theoretical and practical aspects of dynamical systems, is well-known and it does not need to be emphasised. Let us just point out that quite recently, they have acquired a major role in establishing instability properties for Hamiltonian systems which are close to integrable, a problem which goes back to the question of the stability of the solar system.
\paraga It has been known for a long time that compact invariant normally hyperbolic submanifolds are \emph{persistent}, in the following sense: any diffeomorphism $g : M \rightarrow M$, sufficiently close to $f$ in the $C^1$-topology, leaves invariant and is normally hyperbolic at a submanifold $N_g$ $C^1$-close to $N$. Classical references for this result are \cite{HPS77} and \cite{Fen71}.
In fact, normally hyperbolic submanifolds are persistent and \emph{uniformly locally maximal}: there exist neighbourhoods $U$ of $N$ in $M$ and $\mathcal{U}$ of $f$ in the space $\textrm{Diff}^1(M)$ of $C^1$-diffeomorphisms of $M$, such that for any $g\in\mathcal{U}$, $N_g=\bigcap_{k\in\mathbb Z}g^k(U)$ is a $C^1$-submanifold close to $N$, with $N_f=N$. The latter property implies uniqueness of the invariant submanifold.
The converse statement holds true: assuming $N$ is persistent and uniformly locally maximal, it was shown in \cite{Man78} that $N$ has to be normally hyperbolic.
In the case where $N$ is a point, then it is a hyperbolic fixed point and the persistence follows trivially from the implicit function theorem. In the general case, however, such a direct approach is not possible and it is customary to deduce the persistence of compact normally hyperbolic submanifolds from the existence and persistence of the associated local stable (respectively unstable) manifolds, which are located in a neighbourhood of $N$ and are tangent to the sum of the contracting (respectively expanding) and tangent direction to $N$. The existence and persistence of stable and unstable manifolds have a long history, that we shall go through only very briefly. In the case of a fixed point, this was first proved by Hadamard (\cite{Had01}) who introduced the so-called ``graph transform" method, which relies on the contraction principle. Another proof, based on the implicit function theorem, was later given by Perron (\cite{Per28}), which was subsequently greatly simplified by Irwin (\cite{Irw70}). For normally hyperbolic submanifolds which are not reduced to a point, the result was proved independently in \cite{HPS77} and \cite{Fen71}, using the graph transform method. Moreover, in \cite{HPS77}, results on persistence were obtained not only for normally hyperbolic submanifolds but also in the more general context of normally hyperbolic laminations. This result was then clarified and further generalised in \cite{Ber10} and \cite{Ber11}, where not only laminations but also certain stratifications of normally hyperbolic laminations were shown to be persistent. As a final remark, let us point out that the graph transform method has been successfully applied for semi-flows in infinite dimension (\cite{BLZ98}), with a view towards applications to partial differential equations.
\paraga The aim of this work is to give yet another proof of the persistence of compact normally hyperbolic submanifolds, a proof which we believe to be simpler. We will also use a graph transform, but at variance with all other proofs, we will not rely on the contraction principle. As a consequence, we will avoid any technical estimates that are usually required to show that the graph transform is indeed a contraction (on an appropriate Banach space). Also, we will be dispensed with giving the explicit, and usually rather cumbersome, expression of the graph transform acting on a suitable space of sections of a vector bundle. Instead, we will simply use the Schauder fixed point theorem to obtain the existence (but not the uniqueness) of the invariant submanifold. Such an invariant submanifold will be shown, at first, to be not more than Lipschitz regular. Then, to regain smoothness, we will use very simple geometrical properties of cone fields implied by the domination hypothesis. Compared to other proofs, and especially \cite{HPS77} where complicated techniques of ``Lipschitz jets" are used, our approach here is remarkably simple.
Moreover, our proof yields a new result since the existence and regularity works for a wider class of submanifolds which we call ``topologically" normally hyperbolic, where basically we will retain a suitable domination property but the normal contraction and expansion will be replaced by topological analogues (see below for a precise definition). Under this weaker assumption no result of uniqueness has to be expected. In the classical normally hyperbolic case, using some other simple geometrical arguments (where the contraction property of the graph transform is hidden behind), the uniqueness will be established.
The plan of the paper is the following. We state and explain the theorems for normally hyperbolic submanifolds in section~\ref{s2}, and for topologically normally submanifolds in section~\ref{s3}. The proofs of the results are given in section~\ref{s4}.
\section{Normally hyperbolic submanifolds}\label{s2}
\paraga Let us now detail the setting that we shall use in the formulation of the theorem below. The Riemannian manifold $M$ is $m$-dimensional and smooth ({\it i.e.} $C^\infty$). The diffeomorphism $f : M \rightarrow M$ is (at least) of class $C^1$. The submanifold $N\subseteq M$ is of class $C^1$, $n$-dimensional and \emph{closed}, that is without boundary, compact and connected. We suppose that $f$ leaves $N$ \emph{invariant}: $f(N)=N$.
\begin{definition} The diffeomorphism $f$ is normally hyperbolic at $N$ if there exist a splitting of the tangent bundle of $M$ over $N$ into three $Tf$-invariant subbundles:
\[ TM_{|N}=E^s \oplus E^u \oplus TN \]
and a constant $0<\lambda<1$ such that for all $x\in N$, with $||.||$ the operator norm induced by the Riemannian metric: \begin{equation}\label{hyp}
||T_xf_{|E_x^s}|| < \lambda, \quad ||T_xf_{|E_x^u}^{-1}|| < \lambda, \end{equation} and \begin{equation}\label{dom} \begin{cases}
||T_xf_{|E_x^s}||\cdot||T_{f(x)}f_{|T_{f(x)}N}^{-1}|| < \lambda \\
||T_xf_{|E_x^u}^{-1}||\cdot||T_{f^{-1}(x)}f_{|T_{f^{-1}(x)}N}|| < \lambda. \end{cases} \end{equation} \end{definition} One can check that the continuity of the splitting is then automatic. Condition~(\ref{hyp}) means that the normal behaviour of $Tf$ is hyperbolic while condition~(\ref{dom}) expresses the domination property with respect to the tangent behaviour of $Tf$ (in fact, condition~(\ref{dom}) can be expressed in many equivalent ways). If $E^u=\{0\}$ (respectively $E^s=\{0\}$), then $N$ is normally contracted (respectively normally expanded). Note that if the above conditions are satisfied not for $f$ but only for some iterate of $f$, then there exists another Riemannian metric (called \emph{adapted}) for which these two conditions hold true for $f$ (see \cite{Gou07} for the case of a general dominated splitting).
\paraga We endow the space of $C^1$-maps between $C^1$-manifolds with the compact-open topology (see \cite{Hir76}). This naturally defines a topology on the subset $\mathrm{Diff}^1(M)$ of $C^1$-diffeomorphisms of $M$. Moreover, two $C^1$-diffeomorphic submanifolds $N$ and $N'$ of $M$ are \emph{$C^1$-close} if there exists an embedding $i'$ of $N$ onto $N'$ which is $C^1$-close to the canonical inclusion $i : N \hookrightarrow M$. We are now ready to state the classical theorem.
\begin{theorem}\label{thmhps} Let $f$ be a $C^1$-diffeomorphism which leaves invariant and is normally hyperbolic at a closed $C^1$-submanifold $N$. Then there exists a neighbourhood $\mathcal{U}$ of $f$ in $\mathrm{Diff}^1(M)$ such that any $g\in\mathcal{U}$ leaves invariant and is normally hyperbolic at a $C^1$-submanifold $N_g$, diffeomorphic and $C^1$-close to $N$. Moreover, $N_g$ is unique and uniformly locally maximal. \end{theorem}
Let us point out that we will actually show the stable and unstable manifolds theorem: for any $g\in \mathcal{U}$, we will construct a local stable manifold $N^s_g$ (respectively a local unstable manifold $N^u_g$), $C^1$-close to $N^s_f$ (respectively $N^u_f$), the latter being the set of points whose forward (respectively backward) orbit lies in a small neighbourhood of $N$. Then the normally hyperbolic submanifold $N_g$ will be obtained as the (transverse) intersection between $N^s_g$ and $N^u_g$.
\paraga Given any $r\geq 1$, we can replace condition~(\ref{dom}) by the stronger condition \begin{equation}\label{domr} \begin{cases}
||T_xf_{|E_x^s}||\cdot||T_{f(x)}f_{|T_{f(x)}N}^{-1}||^k < \lambda \\
||T_xf_{|E_x^u}^{-1}||\cdot||T_{f^{-1}(x)}f_{|T_{f^{-1}(x)}N}||^k < \lambda \end{cases} \quad 1\leq k \leq r. \end{equation} If $f$ satisfies~(\ref{hyp}) and~(\ref{domr}), then it is \emph{$r$-normally hyperbolic} at $N$ (hence normally hyperbolic is just $1$-normally hyperbolic). Note that if condition~(\ref{hyp}) is satisfied, then it is sufficient to require condition~(\ref{domr}) only for $k=r$.
In this case, if $N$ and $f$ are $C^r$, using a trick from \cite{HPS77}, for $g$ in a $C^r$-neighbourhood of $f$, the invariant submanifold $N_g$ enjoys more regularity. Indeed, let $G_n(TM) \rightarrow M$ be the Grassmannian bundle with fibre at $x\in M$ the Grassmannian of $n$-planes of $T_x M$. The tangent map $Tf$ induces a canonical $C^{r-1}$-diffeomorphism $Gf$ of $G_n(TM)$. Moreover, the tangent bundle $TN$ can be considered as a closed $C^{r-1}$-submanifold of $G_n(TM)$ (although $TN$ is not compact as a submanifold of $TM$). As $f$ is $r$-normally hyperbolic at $N$, $Gf$ is $(r-1)$-normally hyperbolic at $TN$. By induction on $r\geq 1$, the existence and the uniqueness given by Theorem~\ref{thmhps} yields the following corollary.
\begin{corollary}\label{corhps} For any integer $r\geq 1$, let $f$ be a $C^r$-diffeomorphism which leaves invariant and is $r$-normally hyperbolic at a closed $C^r$-submanifold $N$. Then there exists a neighbourhood $\mathcal{U}$ of $f$ in $\mathrm{Diff}^r(M)$ such that any $g\in\mathcal{U}$ leaves invariant and is $r$-normally hyperbolic at a $C^r$-submanifold $N_g$, diffeomorphic and $C^r$-close to $N$. Moreover, $N_g$ is unique and uniformly locally maximal. \end{corollary}
Actually the assumption that $N$ is of class $C^r$ is automatic. In \cite{HPS77}, it is proved that if a $C^r$-diffeomorphism leaves invariant and is $r$-normally hyperbolic at a closed $C^1$-submanifold $N$, then $N$ is actually of class $C^r$.
\section{Topologically normally hyperbolic submanifolds}\label{s3}
To prove the persistence of normally hyperbolic submanifolds, we will use a geometric model which can be satisfied without being normally hyperbolic. Therefore this enables us to weaken the assumptions on normal hyperbolicity to obtain a new result of persistence. The submanifolds satisfying such a geometric model will be called topologically normally hyperbolic.
\paraga We first explain the geometric model in a simple case. We consider a closed manifold $N$, that we identify to the submanifold $N\times \{0\}$ of \[ V=N\times \mathbb R^s\times \mathbb R^u,\] with $s,u\geq 0$. Given two compact, convex neighbourhoods $B^s$ and $B^u$ of $0$ in respectively $\mathbb R^s$ and $\mathbb R^u$, we define the following compact neighbourhood of $N$ in $V$: \[ B = N\times B^s \times B^u, \] and the subsets \[ \partial^sB = N\times \partial B^s \times B^u, \quad \partial^u B = N\times B^s\times \partial B^u, \] where $\partial B^s$ (respectively $\partial B^u$) is the boundary of $B^s$ (respectively $B^u$).
We assume that $f$ is a $C^1$-embedding of $B$ into $V$, that is a $C^1$-diffeomorphism of $B$ onto its image in $V$, which leaves $N$ invariant. Then we also define the following cones for $z\in V$:
\[C_z^u=\{v=v_1+v_2\in T_zV \; | \; v_1\in T_zN\times \mathbb R^s, v_2\in \mathbb R^u,\|v_2\|_z\leq\|v_1\|_z\},\]
\[C_z^s=\{v=v_1+v_2\in T_zV \; | \; v_1\in T_zN\times \mathbb R^u, v_2\in \mathbb R^s, \|v_2\|_z\leq\|v_1\|_z\},\] with respect to a Riemannian metric on $V$.
\begin{definition} Under the above assumptions, $f$ is topologically normally hyperbolic at $N$ if it satisfies the following conditions: \begin{equation}\label{hyptop} f(B) \cap \partial^sB=\emptyset, \quad B \cap f(\partial^uB)=\emptyset, \tag{$1'$} \end{equation} \begin{equation}\label{domtop} T_zf(C_z^s)\subset \mathring C_{f(z)}^s\cup \{0\}, \quad C_{f(z)}^u\subset T_zf(\mathring C_z^u)\cup \{0\}, \tag{$2'$} \end{equation} for every $z\in B$. \end{definition}
Condition~(\ref{hyptop}) is equivalent to the requirement that $f(B)$ intersects the boundary of $B$ at most at $\partial^u B$, and that $f^{-1}(B)$ intersects the boundary of $B$ at most at $\partial^s B$. Moreover, by condition~(\ref{domtop}), we have the following transversality properties: for all $\xi_s\in \mathbb R^s$ and $\xi_u\in \mathbb R^u$, the image of the ``horizontal" $N\times\{\xi_s\}\times \mathbb R^u$ by $f$ intersects transversely each ``vertical" $N\times \mathbb R^s \times\{\xi_u\}$, and similarly, the image of $N\times \mathbb R^s \times\{\xi_u\}$ by $f^{-1}$ intersects transversely $N\times\{\xi_s\}\times \mathbb R^u$. The situation is depicted in figure~\ref{boite}. \begin{figure}
\caption{A topologically normally hyperbolic submanifold}
\label{boite}
\end{figure}
When $u=0$, then $N$ is topologically normally contracted, and the characterisation is simpler: $\partial B=\partial^sB$, the second half of condition~(\ref{hyptop}) is empty, the first half reads $f(B) \cap \partial B=\emptyset$ which can be seen to be equivalent to $f(B) \subseteq \mathring{B}$. A similar characterisation holds if $N$ is topologically normally expanded, that is when $s=0$.
Given a normally hyperbolic submanifold for which the stable and unstable bundles are trivial, we will construct a $C^1$-conjugacy of a neighbourhood of $N$ with such a geometric model. Basically, condition~(\ref{hyp}) will give condition~(\ref{hyptop}) and conditions~(\ref{hyp}) and~(\ref{dom}) will give condition~(\ref{domtop}).
\paraga Now we shall generalise the concept of topological normal hyperbolicity, to include in particular normally hyperbolic submanifolds for which the stable and unstable bundle are not necessarily trivial.
Let $V^s$ and $V^u$ be two vector bundles over a closed manifold $N$, whose fibres are of dimension $s$ and $u$, with $s,u\geq 0$. We denote by \[\pi:\; V=V^s\oplus V^u\rightarrow N\] the vector bundle whose fibre at $x\in N$ is the direct sum $V_x=V_x^s\oplus V_x^u$ of the fibres of $V^s$ and $V^u$ at $x\in N$. A \emph{horizontal distribution} for $\pi : V\rightarrow N$ is a smooth family of $n$-planes $(H_z)_{z\in V}$ such that $T_zV=H_z\oplus \ker T_z\pi$. Recall that a \emph{local trivialisation} of the vector bundle $\pi : V^s\oplus V^u\rightarrow N$ is an open set $W$ of $N$ and a diffeomorphism $\phi : \pi^{-1}(W)\rightarrow W\times \mathbb R^s\times \mathbb R^u$ such that its restriction to any fibre $\pi^{-1}(x)=V_x^s\oplus V_x^u$ is a linear automorphism onto $\{x\}\times \mathbb R^s\times \mathbb R^u$. A horizontal distribution $H$ is \emph{linear} if for any $z\in V_x^s\oplus V_x^u$, there exists a local trivialisation $\phi$ over a neighbourhood $W$ of $x$ such that $T_z\phi(H_z)= T_xW \times \{0\}$. Linear horizontal distribution exists on any vector bundle (the construction follows from the existence of a linear connection and parallel transport, see \cite{GHV73}).
Let $B^s\rightarrow N$ and $B^u\rightarrow N$ be two bundles whose fibres $B_x^s$ and $B_x^u$ at $x\in N$ are convex, compact neighbourhoods of $0$ in $V_x^s$ and $V_x^u$ respectively. Then we can define other bundles over $N$, namely $B= B^s \oplus B^u$, $\partial^sB = \partial B^s \oplus B^u$ and $\partial^u B = B^s\oplus \partial B^u$, where the fibre of $\partial B^s$ at $x \in N$ (respectively $\partial B^u$) is the boundary of $B_x^s$ (respectively $B_x^u$). Note that $B$, $\partial^sB$, and $\partial^u B$ are subbundles of $V$, and that $B$ is a compact neighbourhood of the graph of the zero section of $\pi : V\rightarrow N$.
Let $f$ be a $C^1$-embedding of $B$ into $V$. Identifying $N$ to the graph of the zero section of $\pi : V\rightarrow N$, we assume that $f$ leaves $N$ invariant.
Given a linear horizontal distribution $H$ for $\pi : V\rightarrow N$ and a Riemannian metric on $V$, we define the following cones for $z\in V$:
\[C_z^u=\{v=v_1+v_2\in T_zV \; | \; v_1\in H_z\oplus V_{\pi(z)}^{s}, v_2\in V_{\pi(z)}^{u},\|v_2\|_z\leq\|v_1\|_z\},\]
\[C_z^s=\{v=v_1+v_2\in T_zV \; | \; v_1\in H_z\oplus V_{\pi(z)}^{u}, v_2\in V_{\pi(z)}^{s}, \|v_2\|_z\leq\|v_1\|_z\},\] where we have identified the fibres $V_{\pi(z)}^{s}$ and $V_{\pi(z)}^{u}$ with their tangent spaces. Let us note that these cone fields are \emph{smooth} in the sense that $H_z, V_{\pi(z)}^{s}$ and $V_{\pi(z)}^{u}$ depend smoothly on $z\in B$.
\begin{definition}\label{defgen} Under the above assumptions, $f$ is topologically normally hyperbolic at $N$ if it satisfies conditions~(\ref{hyptop}) and~(\ref{domtop}). \end{definition}
This is clearly a generalisation of the simple geometric model, since the latter correspond to $V^s=N\times \mathbb R^s$, $V^u=N\times \mathbb R^s$ and there is a canonical linear horizontal distribution for the trivial vector bundle $\pi : V= N\times \mathbb R^s \times \mathbb R^u \rightarrow N$, given by $H_z=T_{\pi(z)}N \times \{0\}$ for $z\in V$.
\paraga Let us denote by $\mathrm{Emb}^1(B,V)$ the space of $C^1$-embeddings of $B$ into $V$, endowed with the $C^1$-topology. Here is the new result on persistence:
\begin{theorem}\label{thmhpstop} Let $f\in\mathrm{Emb}^1(B,V)$ which leaves invariant and is topologically normally hyperbolic at a closed $C^1$-submanifold $N$. Then there exists a neighbourhood $\mathcal{B}$ of $f$ in $\mathrm{Emb}^1(B,V)$ such that any $g\in\mathcal{B}$ leaves invariant and is topologically normally hyperbolic at a $C^1$-submanifold $N_g$, diffeomorphic to $N$. \end{theorem}
Under the assumptions of this theorem, for $g\in \mathcal{B}$, we will see that $N_g$ satisfies the same geometric model. In particular, $N_g$ is included in $B$ and its tangent space is in $C^s\cap C^u$. Thus, if $N$ is topologically normally hyperbolic for a certain geometric model such that $B$ and $C^s\cap C^u$ can be taken ``arbitrarily small" (when viewed as neighbourhoods of respectively $N$ and $TN$), then $N_g$ is $C^1$-close to $N$ when $g$ is $C^1$-close to $f$. In general, this stronger assumption is not any more satisfied by $N_g$.
The above result is analogous to Theorem~\ref{thmhps}, except that we do not obtain any uniqueness statement. One can only say that for all $g\in \mathcal{B}$, $g$ has at least one invariant submanifold $N_g$ contained in the maximal invariant subset of $B$. We will give below examples for which uniqueness fails, and as we already explained, the uniqueness property of Theorem~\ref{thmhps} is known to be characteristic of normal hyperbolicity.
\paraga Let us show that this generalisation is not free by giving some examples where Theorem~\ref{thmhps} fails, whereas Theorem~\ref{thmhpstop} applies.
The most simple example is when the submanifold has dimension zero, that is when it is a fixed point. Consider the map $f : \mathbb R^2 \rightarrow \mathbb R^2$ defined by \[ f(x,y)=(x-x^3,2y), \quad (x,y)\in \mathbb R^2 \] which is a diffeomorphism from a neighbourhood of the origin. The origin is a fixed point, which is not hyperbolic since the differential at this point has one eigenvalue equal to one. Hence we cannot apply Theorem~\ref{thmhps}. However, the fixed point is topologically hyperbolic: for $V=\mathbb R^2$, if $B^s= [-\delta,\delta]\times\{0\}$ and $B^u=\{0\}\times[-\delta,\delta]$ for $\delta>0$, then: \[B=[-\delta,\delta]^2, \quad f(B)=[-\delta+\delta^3,\delta-\delta^3]\times[-2\delta,2\delta].\] Hence condition~(\ref{hyptop}) is easily seen to be satisfied for any $0<\delta<1$. Moreover, the coordinates axes are invariant subspaces and, with respect to the Euclidean scalar product, the cone condition~(\ref{domtop}) is plainly satisfied. Then Theorem~\ref{thmhpstop} gives the existence of at least one topologically hyperbolic fixed point in $B$, for any small $C^1$-perturbation of $f$. It is very easy to see on examples that the fixed point is in general non-unique, and moreover it can be non-isolated. Let us also add that our result not only give the existence of a topologically hyperbolic fixed point, but also the existence of local stable and unstable manifolds.
Now a slightly less trivial example, when the submanifold is not reduced to a point, can be constructed as follows. Consider the circle diffeomorphism $b : \mathbb T \rightarrow \mathbb T$, $\mathbb T=\mathbb R/2\pi\mathbb Z$, defined by \[ b(\theta)=\theta-\alpha\sin\theta, \quad 0<\alpha<1. \] It has exactly two fixed points, $\theta=0$ and $\theta=\pi$ which are respectively attracting ($b'(0)=1-\alpha$) and repelling ($b'(\pi)=1+\alpha$). All other points in $\mathbb T$ are asymptotic to $0$ (respectively $\pi$) under positive (respectively negative) iterations. Consider a map $f$ which is a skew-product on $\mathbb R^2$ over $b$, of the form \[ f : \mathbb T \times \mathbb R^2 \rightarrow \mathbb T \times \mathbb R^2, \quad f(\theta,x,y)=(b(\theta),f^s_\theta(x), f^u_\theta(y)). \] We assume that for some $\beta$ with $\alpha<\beta<1$, \[ (f^s_0,f^u_0)(x,y)=((1-\beta)x,y+y^3), \quad (f^s_{\pi},f^u_{\pi})(x,y)=(x-x^3,(1+\beta)y), \] and we extend $f^s_\theta$ and $f^u_\theta$ for $\theta\in\mathbb T$ as follows: for every $0<\delta<1$ and $\theta\in\mathbb T$, $f^s_\theta$ and $(f^u_\theta)^{-1}$ fix $0$ and send $[-\delta,\delta]$ into $(-\delta,\delta)$ (in particular, $f$ leaves invariant the circle $\mathbb T\simeq\mathbb T\times\{(0,0)\}$), and we also ask the invariant splitting $T_{(\theta,0,0)}(\mathbb T \times \mathbb R^2)=T_\theta\mathbb T\oplus \mathbb R_x \oplus \mathbb R_y$ to be dominated. Put $V=\mathbb T \times \mathbb R^2$, $B^s=\mathbb T \times [-\delta,\delta]\times\{0\}$, $B^u=\mathbb T \times \{0\}\times[-\delta,\delta]$ and so $B=\mathbb T \times [-\delta,\delta]^2$, for some $\delta>0$. Then condition~(\ref{hyptop}) is clearly satisfied. Moreover, by the domination hypothesis, condition~(\ref{domtop}) is also satisfied with respect to the canonical Riemannian metric. Thus $\mathbb T$ is topologically normally hyperbolic but not normally hyperbolic. Therefore the invariant circle persists under small $C^1$-perturbations of $f$.
The examples we described above seem rather artificial. However, more complicated examples with a similar flavour do appear naturally in celestial mechanics (see for instance \cite{McG73} or \cite{Mos01}). We hope that our method will be useful in such situations.
\paraga To conclude, let us point out that one can easily give examples of persistent submanifolds which are not topologically normally hyperbolic. A simple example is given by the map \[ f(x,y)=(x-x^3,y+y^3), \quad (x,y)\in \mathbb R^2, \] which fixes the origin. Cone condition~(\ref{domtop}) is not satisfied, so our theorem cannot be applied. However, the origin is an isolated fixed point which has a non-zero index, hence by index theory (see \cite{KH97}, section $8.4$), this fixed point persists under $C^0$-perturbations.
In general, we ask the following question:
\begin{question} For $r\geq 1$, under which assumptions does an invariant closed $C^r$-submanifold persist under small $C^r$-perturbations ? \end{question}
For an isolated fixed point, index theory provides a good answer, but in general this seems quite difficult. We hope our work could be useful towards such a general answer.
\section{Proof of Theorem~\ref{thmhps} and Theorem~\ref{thmhpstop}}\label{s4}
This section is devoted to the proof of both Theorem~\ref{thmhps} and Theorem~\ref{thmhpstop}, but first we recall some useful facts.
\paraga To our knowledge, all the previous proofs of invariant manifolds theorems use the contraction principle (or the implicit function theorem) in a suitable Banach space. Here we shall only rely on Schauder's fixed point theorem (see \cite{GD03} for a proof).
\begin{theorem}[Schauder]\label{fix} Let $\Gamma$ be a Banach space, $K\subseteq \Gamma$ a compact convex subset and $F : K\rightarrow K$ a continuous map. Then $F$ has a fixed point. \end{theorem}
In fact, the hypotheses can be weakened as follows: $\Gamma$ can be replaced by a locally convex topological vector space, and $K$ needs not to be compact as long as its image by $F$ is relatively compact in $K$. However, such a greater generality will not be needed here.
\paraga Schauder's fixed point theorem will be used to prove the existence of an invariant submanifold, but \textit{a priori} the submanifold is not continuously differentiable. For these reasons, let us say that a subset of a smooth manifold is a Lipschitz submanifold of dimension $n$ if it is locally diffeomorphic to the graph of a Lipschitz map defined on an open set of $\mathbb R^n$.
Then, to decide whether a Lipschitz submanifold is differentiable, we will use the following notion of tangent cone. Given an arbitrary closed subset $Z \subseteq \mathbb R^m$ and $z\in Z$, the tangent cone $TC_zZ$ of $Z$ at $z$ is the set of vectors $v\in T_z\mathbb R^m$ which can be written as
\[ v=\alpha\lim_{n\rightarrow +\infty}\frac{z-z_n}{||z-z_n||}, \quad \alpha\in\mathbb R,\] for some sequence $z_n\in Z\setminus\{z\}$ converging to $z$. This definition is clearly independent of the choice of a norm. Also, it readily extends when $\mathbb R^m$ is replaced by the $m$-dimensional manifold $M$, choosing a local chart and then checking the definition is indeed independent of the local chart used. Note that for $z\in M$, $TC_zZ$ is always a subset of $T_zM$. Of course, if $Z$ is a differentiable $n$-dimensional submanifold, then for all $z\in Z$, $TC_zZ$ is a vector subspace as it coincides with the tangent space $T_zZ$. In the case where $Z$ is a Lipschitz submanifold, then some converse statement holds true.
Indeed, let $s : \mathbb R^n \rightarrow \mathbb R^d$ be a continuous function, $Z=\mathrm{Gr}(s)\subseteq \mathbb R^{n}\times\mathbb R^d$ and $z=(x,s(x))$ with $x\in\mathbb R^n$. Under those assumptions, if $TC_zZ$ is contained in a $n$-dimensional subspace $D_z$ of $\mathbb R^n\times\mathbb R^d$ which is transverse to $\{0\}\times\mathbb R^d$, then $D_z$ is the graph of a linear map $L_{z,s} : \mathbb R^n \rightarrow \mathbb R^d$ and we can easily prove (see \cite{Fle80} for instance) that $s$ is in fact differentiable at $x$, with $T_xs=L_{z,s}$. Moreover, if $s$ is Lipschitz, then the assumption that $D_z$ is transverse to $\{0\}\times\mathbb R^d$ is automatic, and this immediately gives the following:
\begin{lemma}\label{diff} Let $Z$ be a Lipschitz submanifold of dimension $n$. If for every $z\in Z$, the tangent cone $TC_zZ$ is contained in an $n$-dimensional space $D_z$, then $Z$ is a differentiable submanifold with $T_zZ=D_z$. If moreover $z\mapsto D_z$ is continuous, then $Z$ is of class $C^1$. \end{lemma}
\paraga \begin{proof}[Proof of Theorem~\ref{thmhpstop}] For a clearer exposition, we will first prove the theorem when $N$ satisfies a simple geometric model and is topologically normally contracted, that is $V=N \times \mathbb R^s$. The main ideas are already present in this simple situation. In the general case, where $N$ satisfies a general geometric model and is topologically normally hyperbolic, similar arguments will enable us to construct local stable and unstable manifolds and show their persistence. Then the invariant submanifold will be their transverse intersection. The proof is divided into three steps. The first two steps show the existence and the smoothness of the invariant submanifold in the simple case. The last step is devoted to the general case.
\noindent\emph{Step 1: existence.}
Recall that $B^s$ denotes a compact convex neighbourhood of $0$ in $\mathbb R^s$ and $f$ is a diffeomorphism from $B=N\times B^s$ onto its image in $V=N\times \mathbb R^s$. A Riemannian metric on $V$ is given such that conditions~(\ref{hyptop}) and~(\ref{domtop}) are satisfied. As $\partial^u B$ is empty, condition~(\ref{hyptop}) is equivalent to \begin{equation}\label{vois} f(B)\subseteq \mathring{B}. \end{equation} Indeed, an easy connectedness argument implies property~(\ref{vois}): since $f(B)\cap \partial B=\emptyset$, the set $f(B) \cap B=f(B) \cap \mathring{B}$ is both open and closed for the topology induced on $f(B)$, moreover it is non-empty (it contains $N$ which is invariant by $f$). As $f(B)$ is connected, it follows that $f(B) \cap \mathring{B}=f(B)$ and therefore $f(B)\subseteq \mathring{B}$.
Let us note that properties~(\ref{vois}) and~(\ref{domtop}) remain true if we replace $f$ by a $C^1$-embedding $g$ which is $C^1$-close to $f$. In other words, there exists a small neighbourhood $\mathcal{B}$ of $f$ in $\mathrm{Emb}^1(B,V)$ such that for any $g\in\mathcal{B}$, \begin{equation}\label{condg} g(B)\subseteq \mathring{B}, \quad T_zg(C_z^s)\subseteq \mathring{C}_{g(z)}^{s}\cup\{0\}, \quad z\in B. \end{equation} It sounds natural to consider the set of closed $n$-dimensional $C^1$-submanifolds $\tilde{N}$, contained in $B$ and with a tangent bundle $T\tilde{N}$ contained in $C^s$. Indeed, by~(\ref{condg}), any $C^1$-diffeomorphism $g \in \mathcal{B}$ sends this set into itself and the existence of a fixed point for this action of $g$ would give the desired invariant submanifold. Nevertheless, this set lacks compactness.
Therefore, we consider the set $\mathcal{S}$ of Lipschitz closed $n$-dimensional submanifolds $\tilde{N}$, contained in $B$ and with a tangent cone $TC\tilde{N}$ contained in $C^s$. The action of $g\in \mathcal{B}$ on this set is \[ \mathcal{G} : \mathcal{S} \rightarrow \mathcal{S}, \quad \mathcal{G}(\tilde{N})=g(\tilde{N}).\] If $\tilde{N}$ is a Lipschitz, closed submanifold of dimension $n$, then so is $g(\tilde{N})$. By the property~(\ref{condg}), the map $\mathcal{G}$ is well-defined.
To apply Schauder's fixed point theorem, we need to exhibit a linear structure and to do so we will restrict the map $\mathcal{G}$ to a proper $\mathcal{G}$-invariant subset of $\mathcal{S}$ which, roughly speaking, consists of Lipschitz submanifolds which are graphs over $N$.
Let $\Gamma$ be the space of continuous sections of the trivial vector bundle $N\times \mathbb R^s \rightarrow N$. Any continuous section $\sigma\in \Gamma$ is of the form $\sigma(x)=(x,s(x))$, for a continuous function $s: N \rightarrow \mathbb R^s$. Equipped with the $C^0$-norm, $\Gamma$ is a Banach space. Let us define the Lipschitz constant of a section $\sigma\in \Gamma$ at $x\in N$ by
\[ \mathrm{Lip}_x(\sigma)=\limsup_{y\rightarrow x, \; y\in N\setminus\{x\}}\frac{||s(y)-s(x)||_x}{d(y,x)}. \] It is not hard to check that for $\sigma\in \Gamma$, $\sigma(N)$ belongs to $\mathcal{S}$ if and only if $\sigma(x)$ belongs to $B^s$ and $\mathrm{Lip}_x(\sigma)\leq 1$ for every $x \in N$. So we consider the subset
\[ K=\{\sigma\in \Gamma \; | \; \forall x\in N, \; \sigma(x)\in B^s, \; \mathrm{Lip}_x(\sigma)\leq 1\}. \] This subset $K$ is convex, and it is compact by the Arzel\`a-Ascoli theorem.
Let us show that for any $\sigma \in K$, $\mathcal{G}(\sigma(N))=\tilde{\sigma}(N)$ for some other $\tilde{\sigma}\in K$. As we already know that $\sigma(N)\in \mathcal{S}$ for $\sigma \in K$, it remains to show that $\mathcal{G}$ preserves this graph property, and to do so we will restrict $\mathcal{B}$ to a connected neighbourhood of $f$. Indeed, by the cone condition, for every $x\in N$, the plane $F_x=\{x\}\times \mathbb R^s$ is a manifold which intersects transversally any $C^1$-manifold of $\mathcal{S}$. Thus it intersects transversally $\mathcal{G}(\sigma'(N))$, for every $g\in \mathcal{B}$ and $\sigma'\in K$ of class $C^1$. By connectedness and transversality, the intersection $\mathcal{G}(\sigma'(N)) \cap F_x$ is a unique point, since it is the case for $g=f$ and $\sigma'=0$. Now for any $\sigma \in K$, we can approximate $\sigma$ by a $C^1$-section $\sigma' \in K$ in the $C^0$-topology. This implies that, for any $g\in \mathcal{B}$, $\mathcal{G}(\sigma(N))$ is close to $\mathcal{G}(\sigma'(N))$ for the Hausdorff topology, and by the cone condition, one can check that $\mathcal{G}(\sigma(N)) \cap F_x$ remains close to $\mathcal{G}(\sigma'(N)) \cap F_x$, for every $x\in N$. Since we know that the latter set is reduced to a point, $\mathcal{G}(\sigma(N)) \cap F_x$ has to be reduced to a point too. This shows that $\mathcal{G}(\sigma(N))$ is still a graph, that is $\mathcal{G}(\sigma(N))=\tilde{\sigma}(N)$ for some other $\tilde{\sigma}\in K$.
Therefore $\mathcal{G}$ induces a map on $K$, which is obviously continuous, and as $K$ is compact and convex, by Theorem~\ref{fix} this induced map has a fixed point $\sigma_g$. Then $N_g=\sigma_g(N)$ is a $n$-dimensional Lipschitz submanifold, contained in $S$ and invariant by $g$.
\noindent\emph{Step 2: smoothness.}
So far we have shown the existence of a Lipschitz invariant submanifold $N_g$. Let us prove its differentiability. The cone condition implies that for any $z$ in the maximal invariant subset of $B$, \[ D_z=\bigcap_{k\geq 0} T_{g^{-k}(z)}g^k(C^s_{g^{-k}(z)}) \subseteq C_z^s \] is a $n$-dimensional subspace of $T_zM$ (see \cite{New04} for instance). The invariance of $N_g$ implies the invariance of its tangent cone under the tangent map $Tg$ and therefore \[ TC_zN_g= \bigcap_{k\geq 0} T_{g^{-k}(z)}g^k(TC_{g^{-k}(z)}N_g). \] By construction, $TCN_g \subseteq C^s$, and this implies $TCN_g \subseteq D$. By Lemma~\ref{diff}, the submanifold $N_g$ is differentiable, with $T_zN_g=D_z$ for $z\in N_g$.
Let us prove that $N_g$ is continuously differentiable. Given a convergent sequence $z_n \rightarrow z$ in $N_g$, we need to show that $D_{z_n}=T_{z_n}N_g$ converges to $D_z=T_zN_g$. By compactness of the Grassmannian, it is equivalent to prove that $D_z$ is the only accumulation point of $D_{z_n}$. So let $D'_z$ be an accumulation point of $D_{z_n}$. By continuity of $C_z^s$, $D'_z$ is included in $C_z^s$. For every $k\geq 0$, by continuity of $Tg^{-k}$, $T_zg^{-k} (D'_z)$ is also an accumulation point of $T_{z_n}g^{-k}(D_{z_n})=T_{g^{-k}(z_n)}N_g$ which is then included in $C_{g^{-k}(z)}^{s}$ for the same reasons. Thus $D'_z$ is included in $\bigcap_{k\geq 0} T_{g^{-k}(z)}g^k(C^s_{g^{-k}(z)})=D_z$, and since they have the same dimension, we conclude that $D'_z=D_z$. Therefore $N_g$ is continuously differentiable.
\noindent\emph{Step 3: general case.}
Now let us return to the general case where $N$ is topologically normally hyperbolic, and let us work with the general geometric model \[ B=B^u\oplus B^s\subset V=V^u\oplus V^s\rightarrow N.\] A horizontal linear distribution $H$ for $\pi : V \rightarrow N$ and a Riemannian metric on $V$ are fixed, and conditions~(\ref{hyptop}) and~(\ref{domtop}) are satisfied.
We recall that topological normal hyperbolicity is an open condition on the elements involved. Therefore, for every $g$ in a neighbourhood $\mathcal{B}$ of $f$, condition~(\ref{hyptop}) gives \begin{equation}\label{vois2} g(B) \cap \partial^sB=\emptyset, \quad B \cap g(\partial^u B)=\emptyset, \end{equation} and the cone condition~(\ref{domtop}) gives \begin{equation}\label{cone2} T_zg(C_z^{s})\subseteq \mathring{C}_{g(z)}^{s}\cup\{0\}, \quad C_{g(z)}^{u}\subseteq T_zg(\mathring{C}_z^{s})\cup\{0\}, \end{equation} for every $z\in B$.
As before, we will only use properties~(\ref{vois2}) and~(\ref{cone2}). Let $\mathcal{S}^u$ be the set of Lipschitz, compact $(n+u)$-dimensional submanifolds $\tilde{N}^u$, contained in $B$, with a topological boundary contained in $\partial^uB$, and with a tangent cone $TC\tilde{N}^u$ contained in $C^s$. For convenience, we add the empty set to $\mathcal{S}^u$. Now for $g\in\mathcal{B}$, we put \[ \mathcal{G}^u : \mathcal{S}^u \rightarrow \mathcal{S}^u, \quad \mathcal{G}^u(\tilde{N}^u)=g(\tilde{N}^u) \cap B.\] If $\mathcal{G}^u(\tilde{N}^u)$ is empty, then the map is trivially well-defined at $\tilde{N}^u$. Otherwise, $\mathcal{G}^u(\tilde{N}^u)$ is a Lipschitz and compact $(n+u)$-dimensional. By~(\ref{cone2}), if the tangent cone of $\tilde{N}^u$ is contained in $C^s$, then the same holds true for $\mathcal{G}^u(\tilde{N}^u)$. So to show that $\mathcal{G}^u(\tilde{N}^u)$ is well-defined we only need to check the boundary condition, and this will be a consequence of~(\ref{vois2}). Note that the boundary of $\mathcal{G}^u(\tilde{N}^u)$ is the union of $g(\partial\tilde{N}^u) \cap B$ and $g(\tilde{N}^u) \cap \partial B$. Since $\partial\tilde{N}^u \subseteq \partial^uB$ and $g(\partial^u B) \cap B =\emptyset$, then $g(\partial\tilde{N}^u) \cap B$ is empty so the boundary of $\mathcal{G}^u(\tilde{N}^u)$ actually reduces to $g(\tilde{N}^u) \cap \partial B$. As $g(B) \cap \partial^sB=\emptyset$, the boundary of $\mathcal{G}^u(\tilde{N}^u)$ is included in $\partial^uB$. Therefore $\mathcal{G}^u$ is a well-defined map.
Consider the bundle $\pi^u : V^s \oplus B^u \rightarrow B^u$, whose fibre at $\zeta\in B^u$ is the $s$-dimensional affine subspace $F_\zeta=\{v+\zeta \; | \; v\in V^s_{\pi(\zeta)}\}$ of $V^s_{\pi(\zeta)} \oplus V^u_{\pi(\zeta)}$. Let us denote by $\Gamma^u$ the associated space of continuous sections, which is a Banach space, endowed with the $C^0$-norm. Recall that the vector bundle $\pi : V^s \oplus V^u \rightarrow N$ is equipped with a linear horizontal distribution $H$, and any $x=\pi(z) \in N$ is contained in a local trivialisation $W \subseteq N$ such that \[\phi:\pi^{-1}(W)\rightarrow W\times \mathbb R^u\times \mathbb R^s\] is a diffeomorphism satisfying $T_z\phi(H_z)= T_xW \times \{0\}$. Any section $\sigma^u \in \Gamma^u$ satisfies, for $\zeta \in B^u$ with $\pi(\zeta)\in W$, $\phi\circ \sigma^u(\zeta)=(\zeta,s(\zeta))$ with $s : B^u \rightarrow \mathbb R^s$ a continuous function. Let us define the Lipschitz constant of an element $\sigma^u\in \Gamma^u$ at $\zeta\in B^u$ by
\[ \mathrm{Lip}_\zeta(\sigma^u)=\limsup_{\xi\rightarrow \zeta, \; \xi\in B^u\setminus\{\zeta\}}\frac{||s(\xi)-s(\zeta)||_\zeta}{d(\xi,\zeta)}. \]
It is not hard to check that for $\sigma^u\in \Gamma^u$, $\sigma^u(B^u)$ belongs to $\mathcal{S}^u$ if and only if $\sigma^u(\zeta)$ belongs to $B_\zeta=\{v+\zeta \; | \; v\in B_{\pi(\zeta)}^s\}$ and $\mathrm{Lip}_\zeta(\sigma^u)\leq 1$ for every $\zeta \in B^u$. So we consider the subset
\[ K^u=\{\sigma^u\in \Gamma^u \; | \; \forall \zeta\in B^u, \; \sigma^u(\zeta)\in B_\zeta, \; \mathrm{Lip}_\zeta(\sigma^u)\leq 1 \}. \] This subset is compact by the Arzel\`a-Ascoli theorem, and it is also convex (the linearity of the horizontal distribution, that we used here through the existence of a distinguished local trivialisation, is necessary to obtain the convexity).
By the boundary condition and the transversality given by cone condition~(\ref{cone2}), the cardinality of $f(B^u)\cap F_\zeta$ does not depend on $\zeta\in \mathring B^u$. When $\zeta$ lies in $N\subset \mathring B^u$, this cardinality is at least one since $f(B^u)$ contains $N=f(N)$. For every $\zeta\in V^u$, the intersection of $F_\zeta$ with $N$ is at most one point, so by transversality, it is also the case for the intersection with a small connected neighbourhood of $N$ in $f(B^u)$. By enlarging such a neighbourhood, transversality implies that $F_\zeta\cap f(B^u)$ is at most one point, for $\zeta\in V^u$. Therefore $F_\zeta\cap f(B^u)$ is exactly one point, for $\zeta\in B^u$. This proves that $\mathcal{G}^u(\sigma^u(B^u))$ is still a graph over $B^u$, for $g=f$ and $\sigma^u=0$. Now a similar argument as before shows that $\mathcal{G}^u$ induces a continuous map on $K^u$.
Hence we can repeat the first two steps we described above, and we find that $\mathcal{G}^u$ has a fixed point $N^u_g$, which is the graph of a $C^1$-section $\sigma_g^u : B^u \rightarrow B^u \oplus B^s$. The submanifold $N^u_g$ is a local unstable manifold, and since it is a fixed point of $\mathcal{G}^u$, it is locally invariant in the sense that $g(N^u_g)\cap B=N^u_g$.
Obviously, using $C^u$ and replacing $g$ by $g^{-1}$, we can define another set $\mathcal{S}^s$ and a map $\mathcal{G}^s : \mathcal{S}^s \rightarrow \mathcal{S}^s$ which has a fixed point $N^s_g$, given by the graph of a $C^1$-section $\sigma_g^s : B^s \rightarrow B^u \oplus B^s$. The submanifold $N^s_g$ is a local stable manifold, and it is locally invariant in the sense that $g^{-1}(N^s_g)\cap B=N^s_g$.
The transverse intersection $N_g=N^u_g \pitchfork N^s_g$ is an $n$-dimensional closed submanifold, invariant by $g$ and topologically normally hyperbolic with the same geometric model. This accomplishes the proof. \end{proof}
\paraga \begin{proof}[Proof of Theorem~\ref{thmhps}]
The proof is divided into two steps. In the first step, we will construct a geometric model for normally hyperbolic submanifolds to show that they are topologically normally hyperbolic. Using Theorem~\ref{thmhpstop}, this will immediately give us the existence and smoothness of the invariant submanifold. We shall also notice that an arbitrarily small geometric model can be constructed, so that in addition the invariant submanifold will be $C^1$-close to the unperturbed submanifold. Then, in the second step, we will fully use the hypotheses of normal hyperbolicity to prove that the invariant submanifold is uniformly locally maximal.
\noindent\emph{Step 1: existence and smoothness.}
First it is enough to consider the case where $N$ is a smooth submanifold of $M$. Indeed, there exists a $C^1$-diffeomorphism $\phi$ of $M$ such that $\phi(N)$ is a smooth submanifold of $M$ (see \cite{Hir76}, Theorem 3$.6$). The resulting metric on $M$ is then only $C^1$, but replacing it by a smooth approximation, the submanifold $\phi(N)$, which is invariant by $\phi f\phi^{-1}$, remains normally hyperbolic for this smooth metric (up to taking $\lambda$ slightly larger).
So from now $N$ is assumed to be smooth. By definition, there is a continuous, $Tf$-invariant splitting $ TN\oplus E^s \oplus E^u$ of $TM$ over $N$. The plane field $TN$ is smooth, but $E^s$ and $E^u$ are in general only continuous, so we regard smooth approximations $V^s$ and $V^u$ of them. In particular the sum $TN \oplus V^s\oplus V^u$ is direct and equal to the restriction of $TM$ to $N$. Note that $V^s$ and $V^u$ are no longer $Tf$-invariant. However, given $\gamma>0$, by taking $V^s$ and $V^u$ sufficiently close to $E^s$ and $E^u$, if we define
\[\chi_x^u=\{v=v_1+v_2\in T_xM \; | \; v_1\in T_xN\oplus V_x^s, v_2\in V_x^u,\|v_2\|_x\le \gamma\|v_1\|_x\},\]
\[\chi_x^s=\{v=v_1+v_2\in T_xM \; | \; v_1\in T_xN\oplus V_x^u, v_2\in V_x^s,\|v_2\|_x\le \gamma\|v_1\|_x\},\] then the following cone property is satisfied for $x\in N$: \begin{equation}\label{champchi} T_xf(\chi_x^s)\subset \mathring \chi_{f(x)}^s\cup \{0\}, \quad \chi_{f(x)}^u\subset T_xf(\mathring \chi_x^u)\cup \{0\}. \end{equation}
The plane fields $V^s$ and $V^u$ define a smooth vector bundle $V=V^s\oplus V^u\rightarrow N$ of dimension $m$ such that the fibre at $x\in N$ is the vector space $V_x^s\oplus V_x^u$. We identify the zero section of this bundle to $N$. Since $N$ is compact, by the tubular neighbourhood theorem, there exits a diffeomorphism $\Psi$ of $V$ onto an open neighbourhood $O$ of $N$ in $M$ which is the identity when restricted to the zero section. Let $B^s\rightarrow N$ and $B^u\rightarrow N$ be two bundles whose fibres $B_x^s$ and $B_x^u$ at $x\in N$, are convex, compact neighbourhoods of $0$ in $V_x^s$ and $V_x^u$. Let $B=B^s\oplus B^u$, and define $\Psi(B)=U$ which is a compact neighbourhood of $N$ in $M$, included in $O$. We consider the Riemannian metric on $V$ obtained by pulling-back the restriction to $O$ of the Riemannian metric on $M$.
Let us fix a linear horizontal distribution $(H_z)_{z\in V^s\oplus V^u}$, and consider the following cone fields over $V$:
\[ C_z^u=\{v=v_1+v_2\in T_zV \; | \; v_1\in H_z\oplus V_x^s, v_2\in V_x^u,\|v_2\|_z\le \gamma\|v_1\|_z\},\]
\[ C_z^s=\{v=v_1+v_2\in T_zV \; | \; v_1\in H_z\oplus V_x^u, v_2\in V_x^s,\|v_2\|_z\le \gamma\|v_1\|_z\},\] for any $z\in V$ and $x=\pi(z)\in N$. Restricting $B$ if necessary, cone property~(\ref{champchi}) implies that $f'=\Psi^{-1}f\Psi$ satisfies condition~(\ref{domtop}). Indeed, by rescaling the metric, we can define exactly the same cone fields with $\gamma=1$. Furthermore, we can choose this metric such that every vector in the complement of $\chi_x^s$ (respectively $\chi_x^u$) is contracted by $T_xf$ (respectively by $T_xf^{-1}$). Then it is easy to see that for $B$ small enough, condition~(\ref{hyptop}) is also satisfied.
Therefore a normally hyperbolic submanifold is topologically normally hyperbolic, and Theorem~\ref{thmhpstop} can be applied: there exists a neighbourhood $\mathcal{B}$ of $f'$ in $\mathrm{Emb}^1(B,V)$ such that any $g'\in\mathcal{B}$ leaves invariant and is topologically normally hyperbolic at a $C^1$-submanifold $N_{g'}$, diffeomorphic to $N$. This gives us a neighbourhood $\mathcal{U}$ of $f$ in $\mathrm{Diff}^1(M)$ such that any $g\in\mathcal{U}$ leaves invariant and is topologically normally hyperbolic at a $C^1$-submanifold $N_g$, diffeomorphic to $N$. Moreover, $N_g \subseteq U$ and $TN_g \subseteq T\Psi(C^s \cap C^u)$, and since here one has the freedom to choose $U$ and $C^s \cap C^u$ arbitrarily small (that is, $U$ and $\gamma>0$ can be taken arbitrarily small), $N_g$ is $C^1$-close to $N$. Moreover, it is easy to check that since $N$ is normally hyperbolic, then $N_g$ is not only topologically normally hyperbolic but also normally hyperbolic.
\noindent\emph{Step 2: uniqueness.}
Now it remains to show that $N_g=\bigcap_{k\in \mathbb Z}g^k(U)$. Recall that $N_g=N_g^u \cap N_g^s$, where $N_g^u$ and $N_g^s$ are respectively the local unstable and stable manifolds. It is enough to show that $N_g^u=\bigcap_{k\geq 0}g^k(U)$, since an analogous argument will show that $N_g^s=\bigcap_{k\leq 0}g^k(U)$ and so that $N_g=\bigcap_{k\in \mathbb Z}g^k(U)$.
So let us prove that $N_g^u=\bigcap_{k\geq 0}g^k(U)$ which is of course equivalent to $N_{g'}^u=\bigcap_{k\geq 0}g'^k(B)$, with $g'=\Psi^{-1}g\Psi$. The inclusion $\subseteq$ is obvious. For the other one, take $z\in\bigcap_{k\geq 0}g'^k(B)$. Then, for every $k\geq 0$, $g'^{-k}(z)$ belongs to $B$. The point $z$ belongs to $F_\zeta=\{v+\zeta \; | \; v\in V^s_{\pi(\zeta)}\}$, for a $\zeta \in B^u$. The disk $F_\zeta$ has its tangent space in the complement of $\mathring C^s$. By stability of $C^s$ under $Tg'$, the disk $g'^{-1}(F_\zeta)$ has also its tangent space in the complement of $\mathring C^s$. By condition~(\ref{hyptop}), this disk does not intersect $\partial^u B$, and by~(\ref{domtop}), it comes that $F_\zeta^1= B\cap g'^{-1}(F_\zeta)$ is connected. As $N_{g'}^u$ contains $g'^{-1}(N^u_{g'}$) and $F_\zeta$ intersects $N^u_{g'}$, the submanifold $F_\zeta^1$ contains both $g'^{-1}(z)$ and a point of $N_{g'}^u$. Applying the same argument $k$ times, it comes that the preimage $F_\zeta^k$ of $F_\zeta$ by $(g_{|B}')^k$ is a disk with tangent space in the complement of $\mathring C^s$, intersecting $N^u_{g'}$ and containing $g'^{-k}(z)$. Thus the diameter of $F_\zeta^k$ is bounded by a constant $c>0$ which depends only on $\gamma$ and the diameter of $B^s_x$, $x\in N$.
As $N_{g'}$ is normally hyperbolic, the vectors in the complement of $\mathring C^s$ are contracted by an iterate of $Tg'$. Therefore, for $B$ and $\mathcal{B}$ sufficiently small, $F^k_{\zeta}$ is contracted by $g'^k$. Letting $k$ goes to infinity, the distance between $z$ and $N_{g'}$ goes to zero and hence $z\in N_{g'}$. This ends the proof. \end{proof}
{\it Acknowledgments. The authors are grateful to the hospitality of IMPA. This work has been partially supported by the Balzan Research Project of J. Palis at IMPA.} \addcontentsline{toc}{section}{References}
\end{document} | arXiv |
Journal of the Egyptian Public Health Association
Non-steroidal anti-inflammatory drugs among chronic kidney disease patients: an epidemiological study
Samar Abd ElHafeez1,
Reem Hegazy2,
Yasmine Naga3,
Iman Wahdan1 &
Sunny Sallam1
Journal of the Egyptian Public Health Association volume 94, Article number: 8 (2019) Cite this article
Non-steroidal anti-inflammatory drugs (NSAIDs) should be avoided among chronic kidney disease (CKD) patients. Till now, limited data are available on NSAID use in Egypt, and we aimed to study the prevalence and pattern of NSAID use among CKD patients.
A cross-sectional study was done among 350 CKD adult patients presented to the Main Alexandria University Hospital. Those with end-stage renal disease and diagnosed with acute renal injury and pregnant women were excluded. Demographic and clinical data were collected by interviewing eligible patients. Data about the pattern, history of drug-drug interactions, and knowledge about the NSAID side effects were also gathered.
Of the enrolled patients, 57.1% were hypertensive, 46% were diabetics, 28% had osteoarthritis, and 18.3% had cardiovascular disease. CKD stages were 3.7%, 40.3%, and 56% in stages 2, 3, and 4, respectively. Almost two thirds (65.7%) were NSAID users. Among them, 82.6% were regular users. Headache was the most reported (68.7%) reason of use. The use of drugs which may have drug-drug interaction with the NSAIDs (as diuretics or renin-angiotensin-aldosterone system inhibitors) was reported in 36%. In multiple logistic regression, the odds of NSAID use decreased by 4% (odds ratio (OR) = 0.96, 95% confidence interval (CI) 0.93–0.99, p = 0.01) for every year increase in the patient's age and decreased by 3% (OR = 0.97, 95% CI 0.95–0.99, p = 0.01) for every 1 ml/min/1.73 m2 increase in glomerular filtration rate.
Despite the hazards of NSAID use on the kidney, still high proportion of CKD patients are using them for a long period and they are simultaneously using other drugs with possible drug-drug interactions. This study provided important information that would decrease the gap in knowledge about the use of NSAID in Egypt. It is recommended that NSAIDs should be used with caution among CKD patients and patients should be advised about its adverse health consequences.
Both over-the-counter and prescribed non-steroidal anti-inflammatory drugs (NSAIDs) are widely used all over the world. Although it is commonly used for the management of inflammation and pain, several guidelines including the Kidney Disease Initiative Global Outcome (KDIGO) guidelines recommended avoidance of NSAIDs (except aspirin and acetaminophen) for most patients with chronic kidney disease (CKD) [1].
The use of NSAIDs has been associated with renal function deterioration through variable mechanisms including alteration of the intraglomerular hemodynamic, nephrotic syndrome, glomerulonephritis, chronic interstitial nephritis, renal papillary necrosis, hyperkalemia, and podocyte injury [2,3,4]. This could lead to renal impairment and worsen the degree of renal dysfunction in CKD patients up to the development of end-stage renal disease (ESRD) [5]. Persons with CKD, however, are likely unaware of their disease and may also be unaware that NSAIDs should be avoided. Additionally, those with CKD are likely to be older and have multiple comorbid conditions or symptoms that lead to increased use of NSAIDs [6].
In addition, NSAIDs interact unfavorably with some commonly prescribed medications, including loop diuretics and renin-angiotensin-aldosterone system (RAAS) inhibitors. This is referred to as "triple whammy, leading to reduced effectiveness, along with increased risk of renal impairment. Although epidemiologic studies have linked NSAID use to progressive CKD, the risks of NSAIDs in patients with CKD, while supported by consensus and theoretical effect, remain less clearly established by evidence [7].
Despite the adverse effects of NSAIDs on renal functions, the available data on its pattern among CKD patients in Egypt is minimal. This study aimed to estimate the prevalence, to identify the pattern of NSAID use, and to assess the knowledge about their adverse effects in CKD patients. This will help in setting plans to reduce NSAID use in CKD patients, and spread awareness of their potential harms in this population.
The study was conducted in the outpatient clinics and the inpatient wards of the Internal Medicine Department, Alexandria Main University Hospital, which is the tertiary referral hospital for all patients from four governorates (Alexandria, El Beheira, Kafr-El Sheikh, and Marsa Matrouh).
A cross-sectional study was carried out.
Adult (18+ years) CKD patients diagnosed in pre-ESRD (i.e., before dialysis or transplantation) were included in the study. Patients < 18 years of age; those who were in stage 5 CKD (ESRD), on dialysis, with previous renal transplantation, and with acute renal injury; and pregnant women were excluded from the study.
Sample size and sampling techniques
The sample size was determined using Epi-info software 7.2.2.6 (CDC, 2018) based on power 80%, with confidence level of 95% and prevalence of using NSAIDs among CKD patients of 65.8% [8]. The minimum required sample size was 340 patients. The sample was rounded to 350 patients. Patients were consecutively included on daily basis from the outpatient clinics and the inpatient wards until the required sample was reached.
A predesigned interview questionnaire was used to collect data from the patients about their personal characteristics (sociodemographic characteristics and smoking), history of comorbid diseases, history of selected drugs interacting with NSAIDs (For patients who were illiterate, the researchers had to show them the boxes of the drugs to know which one is taken.), and NSAID use including the type, purpose, pattern, and source of advice. In addition, knowledge about the adverse effects of NSAIDs was determined.
Concerning smoking, patients were classified into never smokers (those who have not smoked 100 cigarettes during their lifetime), current smokers (those who report smoking at least 100 cigarettes in their lifetime and who smoke cigarettes every day or some days), and former smokers (those who has smoked at least 100 cigarettes in their lifetime but does not smoke cigarettes) [9].
Blood pressure, weight, and height were measured, and standardized serum creatinine was collected from the patients' records.
Regarding weight, patients were classified according to body mass index (BMI) [10] (weight in kilograms/height in meters2) into underweight (< 18.5 kg/m2), normal weight (18.5–24.9 kg/m2), overweight (25–29.9 kg/m2), and obese (≥ 30 kg/m2).
The estimated glomerular filtration rate (eGFR) was calculated by CKD-EPI equation [11]
$$ \mathrm{eGFR}=141\times \min {\left(\mathrm{Scr}/\kappa, 1\right)}^{\alpha}\times \max {\left(\mathrm{Scr}/\kappa, 1\right)}^{-1.209}\times {0.993}^{\mathrm{age}}\times 1.018\left[\mathrm{if}\ \mathrm{female}\right] $$
where Scr is serum creatinine (mg/dl), ĸ is 0.7 for females and 0.9 for males, α is − 0.329 for females and − 0.411 for males, min indicates the minimum of Scr/ĸ or 1, and max indicates the maximum of Scr/ĸ.
CKD was defined according to the 2012 KDIGO guidelines [1] according to the presence of the following criteria for more than 3 months:
Markers of kidney damage (one or more)
Albuminuria (AER ≥ 30 mg/24 h; ACR ≥ 30 mg/g [≥ 3 mg/mmol])
Abnormalities detected by histology
History of kidney transplantation
Decreased GFR (GFR < 60 ml/min/1.73 m2 (GFR categories G3a–G5))
CKD is classified into the following stages:
G1: normal or high GFR (> 90 ml/min/1.73 m2)
G2: mild decrease in GFR (60–89 ml/min/1.73 m2)
G3a: mild to moderate decrease in GFR (45–59 ml/min/1.73 m2)
G3b: moderate to severe decrease in GFR (30–44 ml/min/1.73 m2)
G4: severe decrease in GFR (15–29 ml/min/1.73 m2)
G5: kidney failure (GFR < 15)
We excluded patients in CKD stage 5 as those patients already reached ESRD and so prevention of NSAID use will not add any benefits to the disease progression.
Data was summarized using mean ± SD, medians and inter-quartile ranges or frequencies and percentage, as appropriate. Comparison between the variables was done using t test, Mann-Whitney or chi-square according to the data type. Correlation between the renal function and duration of NSAID use was done by Pearson's correlation. The factors associated with the use of NSAIDs were identified using multiple logistic regression analysis. The model included all variables which were significantly related to NSAID use in bivariate analysis. Results are considered significant when p < 0.05.
Table 1 summarizes the baseline characteristics of the study patients. The mean ± SD age of the study participants (N = 350) was 55 ± 13 years, 51% were males, 74.5% were married, 40.8% were smokers, 42% were obese, 57.1% were hypertensive, 46% were diabetics, 18.3% had cardiovascular disease, and 4.9% had autoimmune disease. More than half (56%) of the patients were in stage 4 CKD, and 40.3% were in stage 3 CKD. One fifth (20%) of the patients gave history of RAAS inhibitor use, 9.7% were using diuretics, and 6.3% reported the use of both.
Table 1 Characteristics of the study CKD patients, Alexandria Main University Hospital, Egypt, 2016
Prevalence and pattern of NSAID use
About two thirds (65.7%) of the study patients reported NSAID use (Fig. 1). Patients who use NSAIDs were younger, females, non-smokers, and hypertensive; had osteoarthritis; and were more users of RAAS inhibitors (Table 1).
Prevalence of NSAID use among the study CKD patients
Among NSAID users, ketoprofen was the most commonly used (50.3%), followed by diclofenac (33.7%), ibuprofen (23.1%), ketorolac (2.6%), and meloxicam (2.3%) (Table 2). More than two thirds (68.7%) of the patients reported headache as the main reason of NSAID use, followed by generalized pain (49.6%) and joint pain (43.9%). Forty percent of NSAID users used NSAIDs twice a week, 20.4% three times a week, and 22% every day within the last month. More than 80% (82.6%) of NSAID users gave history of regular use (at least twice a week for more than 2 months). More than 40% of the NSAID users used NSAIDs for a period exceeding 3 years or from 1 to 3 years (46.1% and 41.7%, respectively) (Table 2). There was an inverse correlation between the eGFR and the duration of NSAID use (r = − 0.25, p < 0.001) (Fig. 2).
Table 2 NSAID use by the study CKD patients (Alexandria Main University Hospital, Egypt, 2016)
Correlation between the duration of NSAID use and eGFR
The majority of NSAID users (76.5%) used NSAIDs by self-decision, while 25.2% used them after the advice of physicians. Those who mentioned relatives and friends constituted 13.5%. Pharmacists and previous prescription constituted 10.4% and 9.1%, respectively.
Knowledge about adverse effects of NSAIDs
More than half (53.2%) of CKD patients did not know whether NSAIDs have adverse effects or not. Those who mentioned that NSAIDs have adverse effects constituted 37.1%, and only 9.7% said that NSAIDs have no adverse effects. Among CKD patients who mentioned that NSAIDs have adverse effects, 55.4% thought that NSAIDs cause kidney problems. The patients who mentioned that NSAIDs lead to gastrointestinal tract, liver problems, and heart problems constituted 45.4%, 19.2%, and 2.3%, respectively (Table 3).
Table 3 Knowledge of CKD patients about adverse effects of NSAIDs (Alexandria Main University Hospital, Egypt, 2016)
Factors affecting NSAID use among CKD patients
Multiple logistic regression analysis of the factors affecting NSAID use among CKD patients revealed that NSAID use significantly decreased with increase in the age of the patients and increase in eGFR and those among hepatitis C patients. On the contrary, NSAID use was significantly higher among working, hypertensive, and osteoarthritis patients (Table 4).
Table 4 Multiple logistic regression analysis for the factors affecting NSAID use among CKD patients, Alexandria Main University Hospital, 2016
In this cross-sectional study of 350 pre-ESRD patients recruited from the Alexandria Main University hospital, the prevalence of NSAID usage was 65.7%. There was a gradient increase in NSAID use through CKD stages with a prevalence of 2.6% in stage 2, 9.1% in stage 3a, 27% in stage 3b, and 61.3% in stage 4. Ketoprofen was the most commonly used drug. In addition, prolonged use of NSAIDs was related to reduction of eGFR.
The use of NSAIDs has been reported to range between 8.9 and 69.2% [8,9,10,11,12,13]. Variations in the NSAID use in many studies could be explained by differences in the regulations on NSAID purchase and its availability in different countries, with the absence of restricted laws on the consumption of drugs, which encourages patients to self-treat their symptoms and signs, especially pain [14, 15]. Moreover, the silent nature of CKD and unawareness of the NSIAD complications predispose to the late diagnosis, with the inappropriate use of drugs [16, 17]. In addition, the high use of NSAIDs may indicate that clinicians tend to disregard evaluating the renal functions of patients when prescribing NSAIDs especially in high-risk group patients or they may want to achieve a better quality of life in some comorbid conditions warranting the use of NSAIDs despite the inherent risk.
Consistent with reports from other studies [9, 18, 19], headache was the most reported reason of NSAID use, followed by generalized pain and joint pain. Renal patients can experience pain as a result of several conditions and pathophysiological mechanisms. Osteodystrophy, osteoarthritis, calciphylaxis, and peripheral neuropathy may develop as consequence of the progressive course of CKD. Pain is also related to comorbidities such as peripheral vascular disease, cardiovascular disease, diabetic neuropathy, osteoporosis/osteopenia, and inflammatory/immunological diseases [20,21,22]. Their need for analgesia presents a serious challenge to clinicians and requires special considerations to achieve pain relief without causing toxicity [21].
Although it has been known that NSAIDs inhibit the therapeutic effect of RAAS inhibitors and diuretics (triple whammy), it was noted from the current study that many CKD patients who used NSAIDs were also receiving RAAS inhibitors (23%), diuretics (8.3%), or both (6.1%). A similar problem was reported by Plantinga et al. in the USA [23]. They found that among CKD patients who had prescriptions for NSAIDs, 16% had RAAS inhibitor prescriptions and 20% had diuretic prescriptions.
The pattern of NSAIDs in the current study showed that 40% of CKD patients used NSAIDs twice a week, 20.4% three times a week, and 22% every day. A considerable percentage used NSAIDs for a period exceeding 3 years or from 1 to 3 years (46.1% and 41.7% respectively). Only 12.2% of CKD patients used NSAIDs for less than 1 year. In other studies, the pattern of NSAID use was different. In Poland (2016) [18], 12.2% of CKD (stages 1–4) used NSAIDs few times a week, 14.8% few times a month, and 7.7% every day. In Southern Italy (2014) [13], more than one third (35.6%) of CKD patients were treated with NSAIDs for a period exceeding 90 days and almost 16.5% for more than 6 months. In USA (2011), Plantinga et al. [23] showed that 65% of CKD (stages 1 or 2) and 64.4% of CKD (stages 3 or 4) were long-term users of NSAIDs (1 year or longer).
A significant association between eGFR and NSAID use was found in the present study as the odds of NSAID use decreased by 3% for every 1 ml/min/1.73 m2 increase in eGFR. This association is consistent with what was reported in several studies. Senevirathna et al. [24] conducted a cohort study among 143 CKD patients of uncertain etiology to determine the possible factors associated with the progression and mortality of CKD. They reported that NSAIDs were a major individual factor for disease progression. Kuo and his colleagues [25] carried out a cohort study using a nationally representative sample of 19,163 newly diagnosed CKD patients to study the risk of various analgesics use on the progression of CKD. CKD patients using non-selective NSAIDs had an increased risk of CKD stage 5. The trends toward higher risk with increasing exposure dose were significant for all classes of analgesics.
Limitations of the study
Our study has limitations. First, it is a cross-sectional study, so we were not able to study the causal association between NSAID use and CKD progression. Second, our main definition did not include the most commonly used over-the-counter analgesics—acetaminophen and aspirin—because of their relatively low nephrotoxicity and indications for pain and cardiovascular prevention, respectively. Third, participants with chronic pain may over- or underestimate the duration and frequency of NSAID use. This could be due to high illiteracy rate. In addition, we did not have information on clinician recommendation, or detailed dosage and frequency. Fourth, there is the possibility of CKD misclassification, particularly for earlier stages. Finally, information on comorbid conditions was taken by clinical history.
As there are no available previous studies on the use of NSAID in kidney disease patients in Egypt, this study provided important information that would decrease the gap in knowledge about the use of NSAID in Egypt. It also has important implications. It reflects the importance of communicating the effects of NSAID use especially their nephrotoxicity and potential interactions with RAAS inhibitors and diuretics to physicians, other prescribers of medication, and the community through training courses and workshops.
Levin A, Stevens PE, Bilous RW, Coresh J, De Francisco AL, De Jong PE, et al. Kidney Disease: Improving Global Outcomes (KDIGO) CKD Work Group. KDIGO 2012 clinical practice guideline for the evaluation and management of chronic kidney disease. Kidney Int Suppl. 2013;3(1):1–150.
Harirforoosh S, Asghar W, Jamali F. Adverse effects of nonsteroidal antiinflammatory drugs: an update of gastrointestinal, cardiovascular and renal complications. J Pharm Pharm. 2014;16(5):821–47.
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Bilge U, Sahin G, Unluoglu I, Ipek M, Durdu M, Keskin A. Inappropriate use of nonsteroidal anti-inflammatory drugs and other drugs in chronic kidney disease patients without renal replacement therapy. Ren Fail. 2013;35(6):906–10.
Hull S, Mathur R, Dreyer G, Yaqoob MM. Evaluating ethnic differences in the prescription of NSAIDs for chronic kidney disease: a cross-sectional survey of patients in general practice. Br J Gen Pract. 2014;64(624):e448–e55.
Zhan M, Peter WLS, Doerfler RM, Woods CM, Blumenthal JB, Diamantidis CJ, et al. Patterns of NSAIDs use and their association with other analgesic use in CKD. Clin J Am Soc Nephrol. 2017;12:1778–86.
Adams RJ, Appleton SL, Gill TK, Taylor AW, Wilson DH, Hill CL. Cause for concern in the use of non-steroidal anti-inflammatory medications in the community--a population-based study. BMC Fam Pract. 2011;12:70.
Meuwesen WP, Du Plessis JM, Burger JR, Lubbe MS, Cockeran M. Prescribing patterns of non-steroidal anti-inflammatory drugs in chronic kidney disease patients in the South African private sector. Int J Clin Pharm. 2016;38(4):863–9.
Ingrasciotta Y, Sultana J, Giorgianni F, Caputi AP, Arcoraci V, Tari DU, et al. The burden of nephrotoxic drug prescriptions in patients with chronic kidney disease: a retrospective population-based study in Southern Italy. PLoS One. 2014;9(2):e89072.
Hawton K, Simkin S, Deeks J, Cooper J, Johnston A, Waters K, et al. UK legislation on analgesic packs: before and after study of long term effect on poisonings. BMJ. 2004;329(7474):1076.
Hughes CM, McElnay JC, Fleming GF. Benefits and risks of self medication. Drug Saf. 2001;24(14):1027–37.
Rothberg MB, Kehoe ED, Courtemanche AL, Kenosi T, Pekow PS, Brennan MJ, et al. Recognition and management of chronic kidney disease in an elderly ambulatory population. J Crit Care Med. 2008;23(8):1125.
Vassalotti JA, Stevens LA, Levey AS. Testing for chronic kidney disease: a position statement from the National Kidney Foundation. AJKD. 2007;50(2):169–80.
Heleniak Z, Cieplińska M, Szychliński T, Rychter D, Jagodzińska K, Kłos A, et al. Nonsteroidal anti-inflammatory drug use in patients with chronic kidney disease. J Nephrol. 2017;30(6):781–6.
Kaewput W, Disorn P, Satirapoj B. Selective cyclooxygenase-2 inhibitor use and progression of renal function in patients with chronic kidney disease: a single-center retrospective cohort study. Int J Nephrol Renovasc Dis. 2016;9:273–8.
Davison SN, Koncicki H, Brennan F. Pain in chronic kidney disease: a scoping review. Semin Dial. 2014;27(2):188–204.
Kafkia T, Chamney M, Drinkwater A, Pegoraro M, Sedgewick J. Pain in chronic kidney disease: prevalence, cause and management. J Ren Care. 2011;37(2):114–22.
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Plantinga L, Grubbs V, Sarkar U, Hsu C-y, Hedgeman E, Robinson B, et al. Nonsteroidal anti-inflammatory drug use among persons with chronic kidney disease in the United States. Ann Fam Med. 2011;9(5):423–30.
Senevirathna L, Abeysekera T, Nanayakkara S, Chandrajith R, Ratnatunga N, Harada KH, et al. Risk factors associated with disease progression and mortality in chronic kidney disease of uncertain etiology: a cohort study in Medawachchiya. Sri Lanka Environ Health Prev Med. 2012;17(3):191–8.
Kuo HW, Tsai SS, Tiao MM, Liu YC, Lee I-m, Yang CY. Analgesic use and the risk for progression of chronic kidney disease. Pharmacoepidemiol Drug Saf. 2010;19(7):745–51.
We would like to thank the patients for their participation and staff at the Internal Medicine Department, Alexandria Main University Hospital.
We would also like to acknowledge that part of this work was accepted as an abstract at the ERA-EDTA congress, 2018 in Copenhagen-Denmark.
Please contact the author for data requests.
Department of Epidemiology, High Institute of Public Health, Alexandria University, 165 ElHorreya Avenue, ElHadara, Alexandria, Egypt
Samar Abd ElHafeez, Iman Wahdan & Sunny Sallam
Ministry of Health, Alexandria, Egypt
Reem Hegazy
Department of Internal Medicine (Nephrology unit), Faculty of Medicine, Alexandria University, Alexandria, Egypt
Yasmine Naga
Samar Abd ElHafeez
Iman Wahdan
Sunny Sallam
SAE, RH, YN, and SS conceptualized and designed the study. SAE, RH, YN, and SS built the data collection tool and supervised the data collection. SAE, IW, and RH analyzed and interpreted the data. SAE, RH, IW, and YN wrote the first draft of the manuscript. SAE, RH, YN, IW, and SS revised and critically read the manuscript. All of the authors read and approved the final manuscript.
Correspondence to Samar Abd ElHafeez.
The study was approved by the Ethics Committee of the High Institute of Public Health. All methods were performed in accordance with the International Guidelines for Research Ethics. Verbal consent was obtained from each study participant after explanation of the purpose and benefits of the study. Verbal consent was used as the study was based on records without taking any biological samples or giving any investigational medicine, so it did not introduce more than minimal risk to the study patients. Anonymity and confidentiality of participants were maintained.
All authors agree on the manuscript and give consent for it to be published.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Abd ElHafeez, S., Hegazy, R., Naga, Y. et al. Non-steroidal anti-inflammatory drugs among chronic kidney disease patients: an epidemiological study. J. Egypt. Public. Health. Assoc. 94, 8 (2019). https://doi.org/10.1186/s42506-018-0005-2 | CommonCrawl |
Is it possible for me to derive Avogadro's number?
To my understanding the mole is the unit used to translate between mass on the atomic level and mass in the macro level, defined as the number of atoms in $12$ grams of Carbon-$12$, which apparently turns out to be $\pu{6.0221409e23}$ atoms, and this was defined as Avogadro's Number.
How was this number derived? Is it something I can derive myself? If I had $12$ grams of Carbon-$12$, how would I be able to deduce that it had $\pu{6.0221409e23}$ atoms?
stoichiometry mole units
mhchem
$\begingroup$ It would be hard, very hard. I'll point out that there is a gigantic different in stating that $N_A \approx 6.022 \times 10^{23}$ and stating that $N_A = 6.0221409\times 10^{23}$. Determining a value to eight significant figures requires extremely good experimental technique. $\endgroup$
– MaxW
$\begingroup$ @MaxW How did Avogadro do it in his time, to whatever degree of precision that he did? $\endgroup$
$\begingroup$ This information is readily available on the web in sources such as Wikipedia. en.wikipedia.org/wiki/Avogadro_constant, "By dividing the charge on a mole of electrons by the charge on a single electron..." Look up Millikan oild rop experiment for charge of e-, Faraday and electrolysis for atomic mass/charge ratio etc. $\endgroup$
$\begingroup$ @DrMoishePippik What is interesting to me is that apparently the oil drop experiment gave an answer that was technically wrong / inaccurate, but we still used it anyway to derive all these other numbers? $\endgroup$
$\begingroup$ It's been refined over time. As @MaxW states, original values had a very large margin of error. It's like Galileo using a chandelier as a clock... instrumentation has improved since then. $\endgroup$
To determine Avogadro's number you have to measure the same unit at the atomic and macroscopic scales.
This was first achieved by Millikan who measured the charge of an electron. The charge of one mole of electrons was already known and is a Faraday. Dividing both, you get Avogadro's number.
Before that, Josef Loschmidt was the first one to calculate the number of particles in a cubic meter of gas, using the kinetic theory of gases. This is, of course, also related to Avogadro's number. Avogadro himself did not give any number. He just stated that equal volumes of different gases at the same pressure and temperature have equal amount of particles.
A more modern way of doing it is determining the density of an ultra pure element and then determine the number of atoms and their distances in a unit cell with X-ray diffraction.
So yes, you could, in principle, derive Avogadro's number with pure monocrystalline carbon 12. You will, however, need very good instrumentation.
As per the answer to this question on Chemistry SE it might become the other way round. The Avogadro number will be absolute (defined, with 0 error) and then the kilogram will be redefined as a function of the number of atoms in a monocrystalline perfect $\ce{^{28}Si}$ sphere.
Raoul KesselsRaoul Kessels
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\begin{document}
\pagestyle{myheadings} \markright{IHARA-SELBERG...}
\title{The Ihara-Selberg zeta function for $\PGL_3$ and Hecke operators} \author{Anton Deitmar \& J. William Hoffman}
\date{} \maketitle
{\bf Abstract.} A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat-Tits building of $\PGL_3$. This formula expresses the zeta function in terms of Hecke-Operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.
\tableofcontents
\section*{Introduction} Y. Ihara \cite{Ihara} extended the theory of Selberg type zeta functions to $p$-adic settings. His work was later generalized by K. Hashimoto \cite{Hash1,Hash2,Hash3}, H. Bass \cite{Bass}, H. Stark and A. Terras \cite{ST}, and others. Ihara defined the zeta function in group theoretical terms first, but it can be described geometrically as follows. Let $\Ga\bs X$ be a finite quotient of the Bruhat-Tits tree of a rank one $p$-adic group modulo an arithmetic group $\Gamma$. Then define the zeta function by $$ Z(u)\=\prod_c (1-u^{l(c)}), $$ where the product runs over all primitive closed loops in $\Ga\bs X$. Ihara proved the remarkable formula $$ Z(u)\= (1-u^2)^{-\chi} \det( 1-Au + qu^2), $$ where $A$ is the adjacency operator on $\Ga\bs X$ which can be interpreted as the canonical generator of the unramified Hecke-algebra. Further, $\chi<0$ is the Euler-characteristic of $\Ga\bs X$, and $q$ is the order of the residue class field.
For $\Ga$ being the unit group of the maximal order in a quaternion algebra, this formula allowed Ihara to relate $Z(u)$ to the Hasse-Weil zeta function of the Shimura curve attached to $\Gamma$. This is the only proven link between Selberg-type zeta functions and arithmetical zeta functions.
In \cite{padgeom} the author gave a definition of an Ihara-type zeta function $Z(u)$ for a higher rank group. There is no Ihara-formula for higher rank up to date. In this paper we give an approximation to an Ihara formula in the case of the group $\PGL_3$. For this group the unramified Hecke-algebra has two generators $\pi_1,\pi_2$. The canonical replacement of the determinant factor in Ihara's formula is $$ \det(1-u\pi_1 +u^2 q \pi_2 -u^3 q^3). $$ The main result of the present paper is
\begin{theorem}\label{main} There are a natural number $n$ and a polynomial $P(u)$ such that $$ Z(u)\= \frac{\det(1-u\pi_1+ u^2 q \pi_2 -u^3 q^3)^n}{P(u)}. $$ \end{theorem}
I thank A. Setyadi for pointing out an error in an earlier version.
\section{The building} Let $F$ be a non-archimedean local field. Let $\CO$ be its valuation ring with maximal ideal $\m\subset\CO$. Fix a generator $\varpi$ of $\m$ and let $q$ be the cardinality of the residue class field $k=\CO/\m$.
Consider the locally compact group $G=\PGL_3(F)= \GL_3(F)/F^\times$. It is totally disconnected and every maximal compact subgroup is conjugate to $K=\PGL_3(\CO)=\GL_3(\CO)/\CO^\times$. Let $X$ be the Bruhat-Tits building \cite{Brown} of $G$. In this particular case the Bruhat-Tits building can be described rather nicely. The vertex set $X_0$ of $X$ is the set of homothety classes of $\CO$-lattices in $F^3$. Recall that an $\CO$-lattice in $F^3$ is a finitely generated $\CO$-submodule $\La$ of $F^3$ such that $F\La=F^3$. Two lattices $\Lambda, \Lambda'$ are \emph{homothetic}, if there exists $\alpha\in F^\times$ such that $\Lambda'=\alpha\Lambda$. Every lattice $\La$ is the image under some $g\in\GL_3(F)$ of the standard lattice $L_0=\CO e_1\oplus \CO e_2\oplus\CO e_3$, where $e_1,e_2,e_3$ is the standard basis of $F^3$. The set of all lattices thus can be identified with $\GL_3(F)/\GL_3(\CO)$ and the set $X_0$ of homothety classes of lattices with $G/K$. Let $G'$ denote the image of $\SL_3(F)$ in $G$. The set $X_0$ splits into three orbits under the action of $G'$. These orbits are given by $L_0$ as above, $L_1=\CO e_1\oplus\CO e_2\oplus\varpi\CO e_3$, and $L_2=\CO e_1\oplus\varpi\CO e_2\oplus\varpi\CO e_3$. For a given vertex $x\in X_0$ we say $x$ \emph{is of type} $j$ if $G' x=G' L_j$ for $j=0,1,2$. Two vertices $x\ne y$ are joined by an edge if and only if there are representatives $\La_1$ and $\La_2$ for $x$ and $y$ such that $\varpi\La_1\subset\La_2\subset\La_1$. It follows that $x$ and $y$ must be of different type. This describes the $1$-skeleton $X_1$ of $X$. The following Lemma gives further properties of the graph $X_1$.
\begin{lemma}\label{graph} \begin{enumerate} \item Every vertex in $X$ has $2(q^2+q+1)$ neighbours. \item Two neighboured vertices have $q+1$ common neighbours. \item Any three distinct vertices have at most one common neighbour. \end{enumerate} \end{lemma}
\prf For (a) it suffices to consider the vertex given by $L_0$. Every neighbour has a representative lattice $L$ with $$ \varpi L_0\ \subset\ L\ \subset\ L_0. $$ Now $L_0/\varpi L_0\cong \F_q^3$ as a vector space over $\F_q$, and $L$ defines a sub vector space. Thus the set of all neighbours of $[L_0]$ is in bijection with the set of all non-trivial sub vector spaces of $\F_q^3$ which are $2(1+q+q^2)$ in number. Part (b) and (c) are similar. \qed
Whenever three vertices $x,y,z$ are mutually connected by edges, then this triangle forms the boundary of a 2-cell of $X$, called a \emph{chamber}. This describes $X$ as a CW-complex. There is, however, more structure through the geometry of the apartments. For instance, whenever two edges meet in a vertex, there is an angle between them which can be $\pi/3, 2\pi/3,$ or $\pi$. A \emph{geodesic} $c$ in $X$ is a straight oriented line in one apartment. If $c$ happens to lie inside $X_1$, then it gives rise to a sequence of edges $(\dots,e_{-1},e_0,e_1,\dots)$ such that $e_k$ and $e_{k+1}$ have angle $\pi$ for every $k\in\Z$. In this case we say that $c$ is a \emph{rank-one geodesic}.
For each edge $e$ with vertices $\{x,y\}$ we fix an orientation, i.e., an ordering of the vertices $(x,y)$ such that if $x$ is of type $j$, then $y$ is of type $j+1\ \mod(3)$. An edge equipped with this orientation will be called \emph{positively oriented}. Likewise, for the chambers we fix a positive orientation by ordering the vertices by type.
\begin{lemma}\label{orientation} The action of $G$ on the edges and the chambers preserves the positive orientation. \end{lemma}
\prf Let $g\in G$. It suffices to show that if $gL_0$ is of type $j$, then $gL_1$ is of type $j+1\ \mod(3)$. First note that the double quotient $G'\bs G/K$ has three elements given by the class of $1$, $\diag(1,1,\varpi)$ and $\diag(1,\varpi,\varpi)$. Next note that the action of $K$ preserves the positive orientation on edges that contain the base point $L_0$. Thus it suffices to prove the claim for the three given elements which is easily done. \qed
\section{The zeta function} Let $\Ga\subset G$ be a discrete cocompact and torsion-free subgroup. Then $\Ga$ acts without fixed points on $X$ and thus $\Ga$ is the fundamental group of the quotient $\Ga\bs X$.
\subsection{Definition} A \emph{geodesic} $c$ in the quotient $\Ga\bs X$ is the image of a geodesic $\tilde c$ in $X$ under the projection map $X\to\Ga\bs X$. The geodesic $c$ is called \emph{rank-one} if $\tilde c$ is. For a geodesic $c$ we denote by $c^{-1}$ the geodesic with the reversed orientation. When speaking about \emph{closed geodesics} in $\Ga\bs X$ we adopt the convention that a closed geodesic comes with a multiplicity (going round more then once). A closed geodesic with multiplicity one is called a \emph{primitive} closed geodesic. To a given closed geodesic $c$ there is a unique primitive one $c_0$ such that $c$ is a power of $c_0$. For a closed geodesic $c$ in $\Ga\bs X$ let $l(c)$ denote its length. Here the length is normalized such that any edge gets the length 1. We define the \emph{zeta function} $$ Z(u)\=\prod_c\left( 1-u^{l(c)} \right) $$
as a formal power series at first, where $c$ ranges over the set of all primitive rank-one closed geodesics in $\Ga\bs X$ modulo homotopy and modulo change of orientation. It is easy to show that the Euler product defining $Z(u)$ actually converges for $u\in\C$ with $|u|$ small enough.
\subsection{A comparison} An element $g$ of $G$ is called \emph{neat} if for every rational representation $\rho\colon G\to \GL_n(F)$ over $F$ the matrix $\rho(g)$ has the following property: the subgroup of $\bar F^\times$ generated by all eigenvalues of $\rho(g)$ is torsion-free. Here $\bar F$ is an algebraic closure of $F$. The element $g$ is called \emph{weakly neat} if the adjoint $\Ad(g)\in\GL({\rm Lie}(G))$ has no non-trivial root of unity as eigenvalue. Obviously neat implies weakly neat. A subgroup $\Ga\subset G$ is called \emph{neat}/\emph{weakly neat} if every $\ga\in\Ga$ is neat/weakly neat in $G$. Every arithmetic group $\Ga$ has a subgroup of finite index which is neat \cite{Borel}.
An element $g$ of $G$ is called \emph{regular} if its centralizer is a torus. A subgroup $\Ga$ of $G$ is called regular if every $\ga\in\Ga$, $\ga\ne 1$ is regular in $G$. A regular group is weakly neat.
In \cite{padgeom}, the author defined for $\Ga$ being discrete, cocompact, and weakly neat a zeta function $Z_P(u)$ attached to a parabolic subgroup $P\subset G$ of splitrank one. It is shown that $Z_P(u)$ is a rational function and that its poles and zeros can be described in terms of certian cohomology groups.
\begin{proposition} Suppose the group $\Ga$ is discrete, cocompact and regular. Then $Z_P(u)=Z(u)$ for every parabolic $P$ of splitrank one. \end{proposition}
\prf We will recall the definition of $Z_P(u)$. Let $P=LN$ be a Levi decomposition of P and $A\subset L$ be a maximal split torus. The dimension of $A$ is one. Let $A^+$ be the set of all $a\in A$ that act on the Lie algebra of $N$ by eigenvalues $\mu$ with $|\mu|>1$. Fix an isomorphism $\ph\colon A\cong F^\times$ that maps $A^+$ to the set of $x\in F^\times$ with $v(x)>0$, where $v$ is the valuation of $F$. For $a\in A^+$ let $l(a)=v(\ph(a))$. Let $M$ be the derived group of $M$ and let $M_{\rm ell}$ be the set of all elliptic elements of $M$. Let $\CE_P(\Ga)$ denote the set of all conjugacy classes $[\ga]$ in $\Ga$ such that $\ga$ is in $G$ conjugate to an element $a_\ga m_\ga\in A^+ M_{\rm ell}$. An element $\ga\in\Ga$ is called \emph{primitive} if $\ga=\sigma^n$m $n\in\N$, $\sigma\in\Ga$ implies $n=1$. Let $\CE_P^p(\Ga)$ denote the set of primitive elements in $\CE_P(\Ga)$. The zeta function $Z_P$ is defined as $$ Z_P(u)\=\prod_{[\ga]\in\CE_P^p(\Ga)}(1-u^{l(a_\ga)})^{\chi_1(\Ga_\ga)}, $$ where $\Ga_\ga$ is the centralizer of $\ga$ in $\Ga$ and $$ \chi_1(\Ga_\ga)\=\sum_{p=0}^{\dim X} p(-1)^{p+1}\dim H^p(\Ga_\ga,\Q). $$ First we remark that since $\Ga$ is regular, we have that $\Ga_\ga\cong\Z$ for every $\ga\in\CE_P(\Ga)$ and thus the Euler numbers $\chi_1(\Ga_\ga)$ are all equal to $1$. Next let $[\ga]\in\CE_P^p(\Ga)$. The function $d_\ga(x)={\rm dist}(\ga x,x)$ on $X$ attains its minimum on a unique apartment of $X$. On this apartment, $\ga$ acts by a translation along a rank-one geodesic $\tilde c$ by the amount $l(a_\ga)$. So $\ga$ closes this geodesic and its image $c$ in $\Ga\bs X$ has length $l(c)=l(a_\ga)$. The other way round, every rank-one closed geodesic $c$ must be closed by one primitive element $\ga$ of $\Ga$. Then $\ga$ either lies in $\CE_P(\Ga)$ or has splitrank two in which case the geodesic it closes cannot be rank-one. This shows that the Euler products defining $Z(u)$ and $Z_P(u)$ coincide. \qed
\subsection{A factorization} Two rank-one geodesics in $X$ are called \emph{adjacent} if they lie in the same apartment, they are parallel, and there is only one row of chambers between them. Recall a \emph{gallery} \cite{Brown} in $X$ is a sequence $g=(C_0,\dots, C_n)$ of chambers such that $C_j$ and $C_{j+1}$ are adjacent for every $j$. We say that a gallery $g$ is \emph{rank-one} if $C_{j-1}\ne C_{j+1}$ for every $j$ and the gallery is located between two rank-one geodesics. The next picture shows an example of a rank-one gallery.
\begin{picture}(300,160)
\put(0,30){\line(1,0){350}} \put(0,70){\line(1,0){350}} \put(0,110){\line(1,0){350}} \put(0,150){\line(1,0){350}}
\put(0,80){\line(3,-5){48}} \put(0,160){\line(3,-5){96}} \put(48,160){\line(3,-5){96}} \put(96,160){\line(3,-5){96}} \put(144,160){\line(3,-5){96}} \put(192,160){\line(3,-5){96}} \put(240,160){\line(3,-5){96}} \put(288,160){\line(3,-5){62}} \put(336,160){\line(3,-5){14}}
\put(0,140){\line(3,5){12}} \put(0,60){\line(3,5){60}} \put(12,0){\line(3,5){96}} \put(60,0){\line(3,5){96}} \put(108,0){\line(3,5){96}} \put(156,0){\line(3,5){96}} \put(204,0){\line(3,5){96}} \put(252,0){\line(3,5){96}} \put(300,0){\line(3,5){50}}
\put(121,82){$C_0$} \put(145,97){$C_1$} \put(169,82){$C_2$} \put(192,97){$C_3$} \put(217,82){$C_4$} \put(241,97){$C_5$}
\end{picture}
A rank-one gallery in $\Ga\bs X$ is the image of a rank-one gallery in $X$ under the projection map. In $\Ga\bs X$ it may happen for a rank-one gallery $g=(C_0,\dots,C_n)$ that $C_0=C_n$ in which case we say that $g$ is \emph{closed}. In this case the number $n$ is even and we define the \emph{length} of $g$ to be $l(g)=n/2$. We say that $g$ is \emph{primitive} if furthermore $C_0\ne C_j$ for $0<j<n$. Two closed galleries $(C_0,\dots,C_n)$ and $(E_0,\dots,E_n)$ are \emph{equivalent} if there is $k\in\Z$ with $C_j=E_{j+k}$, where the indices run modulo $n$. An equivalence class of closed rank-one galleries is called a \emph{loop} of galleries.
Let $\CC_1$ denote the set of all primitive closed rank-one geodesics in $\Ga\bs X$ modulo reversal of orientation. Let $\CC_2$ denote the set of all primitive loops of galleries in $\Ga\bs X$ modulo reversal of orientation. Let $$ Z_j(u)\df \prod_{c\in\CC_j} (1-u^{l(c)}) $$ for $j=1,2$.
\begin{proposition}\label{2.3} For the zeta function $Z$ we have $\ds Z(u)=\frac{Z_1(u)}{Z_2(u)}$. Moreover, if $\Ga$ is regular, then $Z_2(u)=1$. \end{proposition}
\prf For any two topological spaces $X,Y$ let $[X,Y]$ be the set of homotopy classes of continuous maps from $X$ to $Y$. Let $S^1$ be the 1-sphere and consider the natural bijection $$ \Ga/{\rm conjugation} \ \to\ [S^1,\Ga\bs X] $$ given by the identification $\Ga\cong\pi_1(\Ga\bs X)$. If two closed geodesics $c_1,c_2$ are homotopic, then they are closed by conjugate elements of $\Ga$. So they have preimages $\tilde c_1,\tilde c_2$ in $X$ which are closed by the same element $\ga$. Hence $\tilde c_1$ and $\tilde c_2$ lie both in the apartment $\a$ where $d_\ga(x)$ is minimized. Since $\langle\ga\rangle\bs\a$ is a cylinder, $c_1$ and $c_2$ are homotopic through closed geodesics of the same length passing through loops of galleries or intermediate rank-one geodesics.
On the other hand, each closed loop of galleries in $\Ga\bs X$ induces a homotopy between two closed geodesics of the same length: the two boundary components of the gallery. Thus we see that the overcounting in $Z_1(u)$ is remedied by dividing by $Z_2(u)$ to result in $Z(u)$.
For the second part assume there is a closed loop $l$. Let $(C_0,\dots,C_n)$ be a gallery in $X$ being mapped to $l$. Then there is $\ga\in\Ga$ with $\ga C_0=C_n$ and $C_0\subset M_\ga$, where $$ M_\ga\=\{ x\in X : d_\ga(x)={\rm min}\}. $$ For any $\ga$ the set $M_\ga$ is either a geodesic line or an apartment. Since $C_0\subset M_\ga$ is our given case, it follows that $M_\ga$ is an apartment attached to a maximal split torus $A$ which contains $\ga$. But since $\ga$ translates along a rank-one geodesic it must lie in a one-dimensional standard subtorus of $A$, which means it is not regular. A contradiction. The claim follows. \qed
\section{The zeta function for the $1$-skeleton}
Let $E(X)$, resp. $E(\Ga\bs X)$ denote the set of positively oriented edges in $X$, resp. $\Ga\bs X$.
Consider the vector spaces $$ C_1(X) \= \prod_{e\in E(X)}\C e,\qquad C_1(\Ga\bs X) \= \prod_{e\in E(\Ga\bs X)}\C e. $$ The second space is finite dimensional. This notion actually makes sense due to Lemma \ref{orientation}. Define a linear operator $T$ on $C_1(\Ga\bs X)$ by $Te=\sum_{e': e\to e'} e'$, where the sum runs over all positively oriented edges $e'$ such that the endpoint of $e$ is the starting point of $e'$ and $e,e'$ lie on a rank-one geodesic, i.e., have angle $\pi$. By the same formula, we define an operator $\tilde T$ on $C_1(X)$. Note that $\Ga$ acts on $C_1(X)$ and that $\tilde T$ is $\Ga$-equivariant. One has a natural identification
$C_1(\Ga\bs X)\ \cong\ C_1(X)^\Ga$, and $T\cong \tilde T|_{C_1(X)^\Ga}$.
\begin{theorem} We have $Z_1(u)=\det(1-uT)$. In particular, $Z_1(u)$ is a polynomial of degree equal to the number of edges of $\Ga\bs X$, or, equivalently, $$ \deg Z_1(u)\= \frac{(q+1)N}2. $$ where $N$ is the number of vertices in $\Ga\bs X$. \end{theorem}
\prf One computes $$ \tr T^n\=\sum_e\sp{Te,e}\=\sum_{c: l(c)=n} l(c_0), $$ where the second sum runs over all closed geodesics of length $n$ and $c_0$ is the underlying primitive of $c$. In the next computation, we will use the letter $c$ for an arbitrary closed geodesic, $c_0$ for a primitive one, and if both occur, it will be understood that $c_0$ is the primitive underlying $c$. We compute \begin{eqnarray*} Z_1(u) &=& \exp\(-\sum_{c_0}\sum_{m=1}^\infty \frac{u^{l(c_0)m}}m\)\\ &=& \exp\(-\sum_c\frac{u^{l(c)}}{l(c)}l(c_0)\)\\ &=& \exp\(-\sum_{n=1}^\infty \frac{u^n}n\sum_{c:l(c)=n}l(c_0)\)\\ &=& \exp\(-\sum_{n=1}^\infty \frac{u^n}n\tr T^n\)\\ &=& \det (1-uT). \end{eqnarray*} For the last line we used the fact that for a matrix $A$ we have $\exp(\tr(A))=\det(\exp(A))$. To prove the final assertion of the Theorem it suffices to show that $T$ is invertible on $C_1(\Ga\bs X)$. For this in turn it suffices to show that $\tilde T$ has a right-inverse on $C_1(X)$. So let $e$ be a positively oriented edge with endpoint $[L_0]$, and let $e'$ be a positively oriented edge with start point $[L_0]$ such that $e,e'$ lie on a geodesic. Let $[\La_2]$ be the start point of $e$ and $[\La_1]$ the end point of $e'$. The situation is this: $$ \begin{diagram} \node{[\La_2]} \arrow{e,t}{e}
\node{[L_0]} \arrow{e,t}{e'}
\node{[\La_1],} \end{diagram} $$ where $[\La_j]$ is of type $j$ for $j=1,2$. We can choose representatives satisfying $$ \varpi L_0\ \subset\ \La_1,\La_2\ \subset\ L_0. $$ The condition on the types translates to the $\F_q$-vector space $\La_j/\varpi L_0$ being of dimension $j$. The condition that $e,e'$ lie on a geodesic is equivalent to $\La_1\subset\hspace{-10pt}/\hspace{5pt}\La_2$.
For $j=1,2$ let $W_j$ be the complex vector space formally spanned by the set of all $j$-dimensional sub vector spaces of $\F_q^3$. Let $$ T\colon W_2\ \to\ W_1 $$ be given by $$ T(\La_2)\=\sum_{\La_1\subset\hspace{-5pt}/\hspace{2pt}\La_2}\La_1. $$ Define $T'\colon W_1\to W_2$ by $$ T'(\La_1)\= \frac{-1}{q+1}\sum_{\La_2\supset\La_1}\La_2\ +\ \frac 1{q^2-q-1}\sum_{\La_2\supset\hspace{-5pt}/\hspace{2pt}\La_1}\La_2. $$ Then \begin{eqnarray*} \tilde T T'(\La_1) &=& \frac{-1}{q+1}\sum_{\La_2\supset\La_1}\sum_{\La_1'\subset\hspace{-5pt}/\hspace{2pt}\La_2}\La_1'\ +\ \frac 1{q^2-q-1}\sum_{\La_2\supset\hspace{-5pt}/\hspace{2pt}\La_1}\sum_{\La_1'\subset\hspace{-5pt}/\hspace{2pt}\La_2}\La_1'\\ &=& \sum_{\La_1'} c(\La_1')\La_1', \end{eqnarray*} where \begin{eqnarray*} c(\La_1') &=& \frac{-\#\{\La_2\supset\La_1, \La_2\supset\hspace{-11pt}/\hspace{5pt}\La_1'\}}{q+1}+\frac{\#\{\La_2\supset\hspace{-11pt}/\hspace{5pt}\La_1, \La_2\supset\hspace{-11pt}/\hspace{5pt}\La_1'\}}{q^2-q-1}\\ &=&\begin{cases} 1 & {\rm if}\ \La_1'=\La_1\\ 0 & {\rm if}\ \La_1'\ne\La_1.\end{cases} \end{eqnarray*} This calculation shows that the operator $T'$ on $C_1(X)$ given by $$ T'(e)\=\frac{-1}{q+1}\sum_{\stackrel{e'\to e}{\rm non\ geodesic}}e'+\frac 1{q^2-q-1}\sum_{\stackrel{e'\to e}{\rm geodesic}} e' $$ is a right-inverse to $\tilde T$. The claim follows. \qed
\subsection{A combinatorial computation} In the following, we will write $c$ for an arbitrary closed rank-one geodesic in $\Ga\bs X$ and $c_0$ for a primitive one. If $c$ and $c_0$ both occur, it will be understood that $c_0$ is the underlying primitive of $c$. We compute \begin{eqnarray*} \frac{Z_1'}{Z_1}(u) &=& \(\log Z_1(u)\)'\\ &=& -\sum_{c_0}\sum_{n=1}^\infty l(c_0)u^{l(c_0)n-1}\\ &=& -\sum_{n=1}^\infty u^{n-1} \sum_{c: l(c)=n}l(c_0). \end{eqnarray*} Note that the sums run modulo reversal of orientation.
There is a natural orientation on each rank-one geodesic in $X$ given as follows. We say that a rank-one geodesic $C$ in $X$ is \emph{positively oriented} if it runs through the vertices in the order of types: $0,1,2,0,1,2,\dots$. The image $c_\Ga$ of $c$ in $\Ga\bs X$ is isomorphic to the image in $\langle\ga\rangle\bs X$, where $\ga\in\Ga$ is the element in $\Ga$ that closes $c$. Since $\ga c=c$ and $\ga$ acts on $c$ by a translation it preserves the orientation of $c$ and so it does make sense to speak of positive or negative orientation for $c_\Ga$.
A \emph{line segment} in $X$ is a sequence of vertices $s=(x_0,\dots,x_k)$ such that they are consecutive vertices on a rank-one geodesic. A line segment in $\Ga\bs X$ is the image of one in $X$. The \emph{length} of a line segment $s=(x_0,\dots,x_n)$ is $l(s)=n$. On the vector spaces $$ C_0(X)\df \bigoplus_{x\ {\rm vertex\ in\ }X} \C x,\qquad C_0(\Ga\bs X)\df \bigoplus_{x\ {\rm vertex\ in\ }\Ga\bs X} \C x, $$ we define an operator $A_n$ for each $n\in\N$ by $$ A_nx\=\sum_{s:l(s)=n,\ o(s)=x} e(s), $$ where the sum runs over all positively oriented line segments $s$ in $\Ga\bs X$ with starting point $x$ and length $n$, and $e(s)$ denote the endpoint of $s$.
\begin{lemma} The operator $A_n$ has the trace $$ \tr A_n\= \sum_{c: l(c)=n} l(c_0), $$ where the sum runs over all closed rank-one geodesics in $\Ga\bs X$ modulo reversal of orientation. \end{lemma}
\prf Instead of summing modulo reversal of orientation one can as well sum over all positively oriented geodesics. Recall $\tr A_n=\sum_x \sp{A_n x,x}$, where the sum runs over all vertices of $\Ga\bs X$ and the pairing $\sp ,$ is the one given by $\sp{x,y}=\delta_{x,y}$ for vertices $x,y$. A vertex $x$ can only have a non-zero contribution $\sp{A_nx,x}$ if it lies on a close geodesic of length $n$. The contribution of each given geodesic $c$ equals $l(c_0)$. \qed
\subsection{The unramified Hecke algebra} Recall that $G$ is a unimodular group, so any Haar-measure will be left- and right-invariant. We normalize the Haar measure so that the compact open subgroup $K$ gets volume $1$. For a subset $A$ of $G$ we write $\1_A$ for its indicator function. Let $\CH_K$ denote the space of compactly supported functions $f\colon G\to\C$ with $f(k_1 xk_2)=f(x)$ for all $k_1,k_2\in K, x\in G$. This is an algebra under convolution, $$ f*g(x)\=\int_G f(y)g(y^{-1}x)\,dx. $$ Ii is known \cite{Cartier}, that $\CH_K$ is a commutative algebra. It has a unit element given by $\1_K$.
We will also write $KgK$ for the function $\1_{KgK}\in\CH_K$. So a typical element of $\CH_K$ is written as $$ f\=\sum_j c_j\, Kg_j K,\quad{\rm finite\ sum}, $$ and $$ I(f)\= \sum_j c_j\,\vol(Kg_jK). $$ The space $C_c(G/K)$ can be identified with $C_0(X)$ since $G/K$ can be identified with the set of vertices via $gK\mapsto gL_0$. Likewise, $C_c(\Ga\bs G/K)$ identifies with $C_0(\Ga\bs X)$. The Hecke algebra $\CH_K$ acts on $C_c(G/K)$ and $C_c(\Ga\bs G/K)$ via $g\mapsto g*f$, $g\in C_c(G/K)$, $f\in \CH_K$. This will be considered as a left action as is possible since $\CH_K$ is commutative. In $\CH_K$ we consider the elements $$ \pi_1\= K\,\diag(1,1,\varpi)\,K,\qquad \pi_2\= K\,\diag(1,\varpi,\varpi)\, K. $$
\begin{lemma}\label{pi_j} For $j=1,2$, $$ \pi_j L_0\=\sum_{\stackrel{x\ {\rm adjacent\ to\ }L_0}{x\ {\rm of\ type}\ j}}x. $$ \end{lemma}
\prf Clear. \qed
\begin{proposition}\label{relations} As operators on $C_0(X)$ or $C_0(\Ga\bs X)$ respectively, \begin{enumerate} \item $A_1=\pi_1$, \item $A_2=\pi_1^2-(q+1)\pi_2$, \item $A_3=\pi_1^3-(2q+1)\pi_1\pi_2+(1+q+q^2)q$, \item For $n\ge 3$, $$ A_{n+1}=A_n\pi_1-qA_{n-1}\pi_2+q^3 A_{n-2}. $$ \end{enumerate} \end{proposition}
\prf Part (a) follows from Lemma \ref{pi_j}. It is clear that $\pi_1^2=A_2+c\pi_2$ for some number $c$. From (b) in Lemma \ref{graph} it follows that $c=q+1$ which implies part (b). The rest follows similarly. \qed
Let $F(u)$ be the following formal powers series with values in the space $\End(C_0(\Ga\bs X))$, $$ F(u)\=\sum_{n=1}^\infty u^{n-1} A_n. $$ Then $\tr F(u)=\frac{Z_1'}{Z_1}(u)$. The relations in Proposition \ref{relations} imply the following Lemma.
\begin{lemma} We have $$ F(u)\= H(u)\( 1-u\pi_1+u^2 q\pi_2-u^3q^3\)^{-1}, $$ where $H(u)$ is the polynomial \begin{eqnarray*} H(u)&=& (\pi_2-\pi_1^2) +u(\pi_1^3-\pi_1\pi_2+\pi_1^2-(q+1)\pi_2)\\ &&+u^2(\pi_1^3-(2q+1)\pi_1\pi_2+(1+q+q^2)q). \end{eqnarray*} \end{lemma}
\prf This follows from Proposition \ref{relations} by a straightforward computation. \qed
\begin{theorem}\label{3.5} There is $m\in\N$ and a polynomial $Q(u)$ such that $$ Z_1(u)\=\frac{\det(1-u\pi_1+u^2 q\pi_2-u^3q^3)^m}{Q(u)}. $$ \end{theorem}
\prf We have $\frac{Z_1'}{Z_1}(u)=\tr F(u)$, so the poles of $\frac{Z_1'}{Z_1}(u)$ must be singularities of $F(u)$, which form a subset of the set of zeros of the polynomial $\det(1-u\pi_1+u^2 q\pi_2-u^3q^3)$. This implies the claim. \qed
\section{The zeta function on galleries} We now will show that the zeta function on galeries, $Z_2(u)$, also is a polynomial. Recall that every chamber $C$ of $X$ or $\Ga\bs X$ has three vertices, one of each type $0,1,2$. Accordingly, it has three edges of types $(0,1), (1,2)$, and $(2,0)$ respectively. So let $$ C_2(X)\=\prod_C \C C,\quad C_2(\Ga\bs X)\= \prod_{C\mod \Ga} \C C, $$ where the product runs over all chambers of the buildings $X$ and $\Ga\bs X$. On $C_1(\Ga\bs X)$ we define a linear operator $L_1$ mapping a chamber $C$ to the sum of all chambers $C'$ such that the (1,2)-edge of $C'$ is the direct geodesic prolongation of the (0,1)-edge of $C$ as in the following picture.
$ $
\begin{picture}(300,160)
\put(0,70){\line(1,0){350}}
\put(0,160){\line(3,-5){96}}
\put(96,160){\line(3,-5){96}}
\put(192,160){\line(3,-5){96}}
\put(12,0){\line(3,5){96}}
\put(108,0){\line(3,5){96}}
\put(204,0){\line(3,5){96}}
\put(300,0){\line(3,5){50}}
\put(95,100){$C$} \put(190,100){$L(C)$} \put(51,50){0} \put(147,50){1} \put(242,50){2} \put(99,130){2} \put(195,130){0}
\end{picture}
$ $
\noindent Similarly, define $L_2$ and $L_3$ by replacing $(0,1,2)$ by $(1,2,0)$ and $(2,0,1)$ respectively. Then let $L\df L_3L_2L_1$.
\begin{proposition}\label{gall} We have $$ Z_2(u)\= \det(1-u^3 L). $$ In particular, $Z_2(u)$ is a polynomial of degree at most $3$ times the number of chambers of $\Ga\bs X$. \end{proposition}
\prf It is easy to see that $$ \tr L^n \= \sum_{c: l(c)=3n} \frac{l(c_0)}{3}, $$ where the sum runs over all loops of galleries in $\Ga\bs X$. From this, the proposition follows by the same computation as before. \qed
Finally, Theorem \ref{main} follows from Proposition \ref{2.3} , Theorem \ref{3.5} and Proposition \ref{gall}.
\noindent {\small Mathematisches Institut\\ Auf der Morgenstelle 10\\ 72076 T\"ubingen\\ Germany\\ {\tt [email protected]}\\ \\ Department of Mathematics\\ Louisiana State University\\ Baton Rouge, LA 70803-4918\\ USA\\ \tt [email protected]}
\end{document} | arXiv |
If $j$ and $k$ are inversely proportional and $j = 16$ when $k = 21$, what is the value of $j$ when $k = 14$?
By the definition of inverse proportion, the product $jk=C$ for some constant $C$. Substituting the given values, we can see that $16\cdot 21=336=C$. Using this $C$ value, we can solve for $j$ when $k=14$: \begin{align*}
j\cdot 14&=336\\
\Rightarrow\qquad j&=\frac{336}{14}=\boxed{24}
\end{align*} | Math Dataset |
Emil Artin Junior Prize in Mathematics
Established in 2001, the Emil Artin Junior Prize in Mathematics is presented usually every year to a former student of an Armenian university, who is under the age of thirty-five,[1] for outstanding contributions in algebra, geometry, topology, and number theory. The award is announced in the Notices of the American Mathematical Society. The prize is named after Emil Artin, who was of Armenian descent. Although eligibility for the prize is not fully international, as the recipient has to have studied in Armenia, awards are made only for specific outstanding publications in leading international journals.
Recipient of the Emil Artin Junior Prize
• 2001 Vahagn Mikaelian[1]
• 2002 Artur Barkhudaryan[2]
• 2004 Gurgen R. Asatryan[3]
• 2005 Mihran Papikian[4]
• 2007 Ashot Minasyan[5]
• 2008 Nansen Petrosyan[6]
• 2009 Grigor Sargsyan[7]
• 2010 Hrant Hakobyan[8]
• 2011 Lilya Budaghyan[9]
• 2014 Sevak Mkrtchyan[10]
• 2015 Anush Tserunyan[11]
• 2016 Lilit Martirosyan[12]
• 2018 Davit Harutyunyan[13]
• 2019 Vahagn Aslanyan[14]
• 2020 Levon Haykazyan[15]
• 2021 Arman Darbinyan[16]
• 2022 Diana Davidova[17]
• 2023 Davit Karagulyan[18]
See also
• List of mathematics awards
References
1. "Mathematics People". ams.org. Retrieved 10 February 2023.
2. "Barkhudaryan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2003. Retrieved 10 February 2023.
3. "Asatryan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2004. Retrieved 10 February 2023.
4. "Papikian Awarded Emil Artin Junior Prize" (PDF). ams.org. 2005. Retrieved 10 February 2023.
5. "Minasyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2007. Retrieved 10 February 2023.
6. "Petrosyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2008. Retrieved 10 February 2023.
7. "Sargsyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2009. Retrieved 10 February 2023.
8. "Hakobyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2010. Retrieved 10 February 2023.
9. "Budaghyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2011. Retrieved 10 February 2023.
10. "Mkrtchyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2014. Retrieved 10 February 2023.
11. "Tserunyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2015. Retrieved 10 February 2023.
12. "Martirosyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2016. Retrieved 10 February 2023.
13. "Harutyunyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2018. Retrieved 10 February 2023.
14. "Aslanyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2019. Retrieved 10 February 2023.
15. "Haykazyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2020. Retrieved 10 February 2023.
16. "Darbinyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2021. Retrieved 10 February 2023.
17. "Davidova Awarded Emil Artin Junior Prize" (PDF). ams.org. 2022. Retrieved 10 February 2023.
18. "Karagulyan Awarded Emil Artin Junior Prize" (PDF). ams.org. 2023. Retrieved 10 February 2023.
| Wikipedia |
The combination of procalcitonin and C-reactive protein or presepsin alone improves the accuracy of diagnosis of neonatal sepsis: a meta-analysis and systematic review
Lin Ruan1Email author,
Guan-Yu Chen1,
Zhen Liu†1, 2,
Yun Zhao†1, 2,
Guang-Yu Xu†1, 2,
Shu-Fang Li†1, 2,
Chun-Ni Li†1, 2,
Lin-Shan Chen†1, 3 and
Zheng Tao†1, 3
Sepsis is an important cause of neonatal morbidity and mortality; therefore, the early diagnosis of neonatal sepsis is essential.
Our aim was to compare the diagnostic accuracy of procalcitonin (PCT), C-reactive protein (CRP), procalcitonin combined with C-reactive protein (PCT + CRP) and presepsin in the diagnosis of neonatal sepsis. We searched seven databases to identify studies that met the inclusion criteria. Two independent reviewers performed data extraction. The pooled sensitivity, specificity, positive likelihood ratio (PLR), negative likelihood ratio (NLR), diagnostic odds ratio (DOR), area under curve (AUC), and corresponding 95% credible interval (95% CI) were calculated by true positive (TP), false positive (FP), false negative (FN), and true negative (TN) classification using a bivariate regression model in STATA 14.0 software. The pooled sensitivity, specificity, PLR, NLR, DOR, AUC, and corresponding 95% CI were the primary outcomes. Secondary outcomes included the sensitivity and specificity in multiple subgroup analyses.
A total of 28 studies enrolling 2661 patients were included in our meta-analysis. The pooled sensitivity of CRP (0.71 (0.63, 0.78)) was weaker than that of PCT (0.85 (0.79, 0.89)), PCT + CRP (0.91 (0.84, 0.95)) and presepsin (0.94 (0.80, 0.99)) and the pooled NLR of presepsin (0.06 (0.02, 0.23)) and PCT + CRP (0.10 (0.05, 0.19)) were less than CRP (0.33 (0.26, 0.42)), and the AUC for presepsin (0.99 (0.98, 1.00)) was greater than PCT + CRP (0.96 (0.93, 0.97)), CRP (0.85 (0.82, 0.88)) and PCT (0.91 (0.89, 0.94)). The results of the subgroup analysis showed that 0.5–2 ng/mL may be the appropriate cutoff interval for PCT. A cut-off value > 10 mg/L for CRP had high sensitivity and specificity.
The combination of PCT and CRP or presepsin alone improves the accuracy of diagnosis of neonatal sepsis. However, further studies are required to confirm these findings.
C-reactive protein
Sepsis is a major cause of neonatal morbidity and mortality [1, 2, 3]. In routine clinical practice, the rapid and accurate diagnosis of neonatal sepsis is often difficult because the clinical presentation of neonatal sepsis may be confused with non-infectious disorders, the onset of sepsis may be acute, and the clinical process can quickly subside. Improving the accuracy of diagnostic testing may improve outcomes in those with true sepsis and decrease the indiscriminate use of antibiotics in those without sepsis [4]. Microbial cultures can help identify serious bacterial infections, but these often produce false negative results, especially after maternal use of antibiotics and may produce false positive results due to sample contamination. In addition, microbial cultures have a time delay (2–3 days) in obtaining results. Therefore, neonates with clinical manifestations of sepsis or risk factors for serious bacterial infections are usually treated with antibiotics while waiting for the results of microbiology testing [5]. This inevitably leads to the overuse of antibiotics, which in turn may lead to the emergence of multiple drug-resistant bacteria in the neonatal intensive care unit (NICU) [3, 6]. Therefore, to prevent microbial resistance due to unnecessary empirical treatment and to avoid unnecessary hospitalization, a definitive diagnosis should be ensured based on laboratory tests with higher diagnostic value [7]. Biomarkers can be important in the timely diagnosis of sepsis, helping in the differential diagnosis of non-infectious diseases and decision-making in initial treatment. C-reactive protein (CRP) is produced by the liver in response to inflammation and/or infectious stimuli, and thus it is considered to be an acute-phase protein [8, 9]. CRP may also be increased in some antenatal conditions, such as fetal distress, stress delivery, and maternal fever, in the absence of systemic infection [8]. Therefore, its specificity is low, and it is preferably used in combination with another serum biomarker. Procalcitonin (PCT) appears to be one of the most promising among the different molecules studied as biomarkers of sepsis. PCT is a procalcitonin precursor protein produced by monocytes and hepatocytes. After exposure to bacterial endotoxin, PCT levels within 2–4 h rise sharply, within 6–8 h they reach a plateau, and then they return to normal levels after 24 h [8, 10]. Serum PCT levels appear to correlate with the severity of the microbial attack and rapidly decrease after appropriate antibiotic treatment. In contrast to CRP, local bacterial infections, severe viral infections, and inflammatory reactions of non-infectious origin are either not associated with increased PCT or are only associated with a slight increase in PCT. In healthy and preterm neonates, there is a physiological increase in serum PCT after birth that peaks at 24 h of age [8, 10]. Presepsin is a nicked truncated form of soluble CD14 (sCD14), which is released by detachment from the surface of immune cells after stimulation by pathogens. Recently, presepsin has been described as a reliable diagnostic and prognostic marker for neonatal sepsis. Based on the above considerations, we performed a meta-analysis to compare the diagnostic accuracy of PCT, CRP, PCT combined with CRP, and presepsin in diagnosing neonatal sepsis.
This study was performed according to the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement. The protocol for this meta-analysis is available in PROSPERO (CRD 42018091339).
Search for trials
We searched PubMed, Web of Science, Cochrane Library, Embase, China National Knowledge Infrastructure (CNKI), Wanfang, and Weipu databases from their inception dates to 16 Aug. 2018 using the keywords "procalcitonin," "C-reactive protein," and "presepsin" to identify studies that met the inclusion criteria. There were no restrictions on language. The detailed search strategy is presented in Additional file 1: Table S1.
Studies were selected based on the following inclusion criteria: (1) neonatal patients with sepsis as the experimental group, whereas the participants with non-sepsis (the patient is suspected of having sepsis but has no sepsis) were regarded as the control group; (2) enough data to calculate the outcome data (true positive (TP), false positive (FP), true negative (TN), false negative (FN)); (3) the participants were diagnosed using the gold standard; (4) the gold standard for diagnosis of sepsis was defined in the study; and (5) blood measurement (of PCT, CRP, PCT + CRP, or presepsin) had to be performed at the time of clinical presentation with suspected sepsis before administration of antimicrobial therapy or in asymptomatic neonates at the time of inclusion in the study. The exclusion criteria were as follows: (1) the diagnostic method for sepsis was not serum PCT, CRP, PCT + CRP, or presepsin; (2) insufficient data to calculate the outcome data (TP, FP, TN, FN); (3) sepsis was diagnosed without a gold standard; (4) studies that used measurements that were made only on maternal or umbilical cord blood samples; (5) neonates treated with antibiotics within the first 72 h; (6) studies involving healthy neonates as controls; and (7) abstracts, reviews, and animal experiments.
Two researchers independently extracted the following information from each study: name of study, year; design country; region; assay method; test time; cutoff; study period; age (days); gestational age (weeks); weight (g); sepsis onset; characteristics and number of patients; and outcome data (TP, FP, FN, and TN). Discrepancies were resolved by consensus.
Risk-of-bias assessments
The analysis of risk of bias and applicability of diagnostic accuracy for the studies included was assessed independently by the two researchers based on the Quality Assessment of Diagnostic Accuracy Studies (QUADAS-2) by RevMan (version 5.2, Cochrane Collaboration, Oxford, UK). QUADAS-2 consists of four sections: patient selection, index test, reference standard, and flow and timing. The studies included were graded as low risk, high risk, or unclear bias based on the following criteria: (1) if the answers to all of the questions for a section were "yes," then the risk of bias was judged as "low;" (2) if any answer to a question in a section was "no," then risk of bias was judged as "high;" (3) the unclear bias was only to be used when insufficient information was provided. Applicability was judged as low, high, or unclear with the above criteria.
Threshold effects were calculated by testing Spearman correlation using Meta-DiSc (version 1.4) software, and P values <0.05 represent significant threshold effects. I2 and a bivariate boxplot were used to measure the heterogeneity caused by non-threshold effects. If the I2 value was ≥ 50% and the P value ≤0.05, or there were studies that fell outside the bivariate boxplot, indicating that the heterogeneity was significant due to the non-threshold effect, then meta-regression analysis to find sources of heterogeneity was performed. The pooled sensitivity, specificity, positive likelihood ratio (PLR), negative likelihood ratio (NLR), diagnostic odds ratio (DOR), AUC, and corresponding 95% credible interval (CI) were calculated by TP, FP, FN, and TN using a bivariate regression model using STATA 14.0 software. Deek's funnel plot was used to detect publication bias, with P < 0.05 indicating publication bias. The visual presentation of diagnostic performance was assessed by the Fagan plot. We performed two methods to evaluate if there was a significant difference in sensitivity, specificity, PLR, NLR, or AUC between any two diagnostic biomarkers. The qualitative method is to observe whether the 95% confidence intervals between different statistical indicators overlap. If there is overlap, there is no statistical significance. Quantitative tests are based on the z-test:
$$ \left({\mathrm{X}}_1-{\mathrm{X}}_2\right)/{\left({{\mathrm{SE}}_1}^2+{{\mathrm{SE}\mathrm{X}}_2}^2\right)}^{1/2}, $$
where X1 and X2 represent the AUC, and SE1 and SE2 are the corresponding standard errors, respectively. If the P value obtained from the z-test is less than p' (p' = 0.05/6), then it is considered there is a statistically significant difference between statistical indicators.
Studies retrieved and their characteristics
The database search identified 2525 records that potentially qualified for inclusion. The titles and abstracts of these records were then filtered. Full texts of 300 records were screened, and 40 met the inclusion criteria. Additional file 1: Table S2 lists the main characteristics of the 40 studies included in the meta-analysis. Of the 40 studies included, two studies included a control group that might have sepsis, and ten studies used healthy neonates as controls. We did not include these studies in the meta-analysis based on the inclusion criteria provided. Eventually, 28 studies (2661 participants) were included in the meta-analysis (Fig. 1), of which 9 studies were not written in English (Additional file 1: Table S1).
Literature search and screening process
Overall, 1281 participants were assigned to the sepsis group and 1380 to the non-sepsis group. In terms of region, 17 (60.7%) trials recruited patients from Asia, 8 (28.6%) from Europe, 1 (3.6%) from North America, and 2 (7.1%) from Africa.
In terms of sepsis onset, three studies included only patients with early-onset neonatal sepsis (diagnosed in the first 72 h of life), five studies included patients with late-onset neonatal sepsis (diagnosed after 72 h of life), and the remaining trials included early-onset and late-onset neonatal sepsis or did not provide relevant information.
In terms of trial design, 13 studies were prospective cohort studies, 12 studies were case-control studies, and 3 studies were cross-sectional studies.
Figures 2 and 3 show the results of assessment for risk of bias. In terms of the risk of bias, of the 28 studies included in our meta-analysis, 12 studies had unclear bias in patient selection. There were 18 studies that were judged as having low bias in the index tests, 27 studies were allocated as having low bias in terms of reference standards, and 26 studies were judged as having low bias in terms of flow and timing. In terms of applicability concerns, 8 studies had high bias in patient selection, 17 studies were judged as having low bias in relation to index tests, and 21 studies were classified as causing high concern about reference standards.
Risk of bias and applicability concerns
Risk of bias and applicability concerns - summary
Threshold effect and heterogeneity
The Spearman correlation coefficient and P value for PCT, CRP, PCT + CRP, and presepsin were 0.144 and 0.523, 0.301 and 0.174, 0.433 and 0.244, and 0.371 and 0.468, respectively, which indicated that there was no significant threshold effect, and thus we combined the sensitivity, specificity, PLR, NLR, DOR, and AUC. We used I2 and a bivariate boxplot (Additional file 2: Figure S1) to measure the heterogeneity caused by non-threshold effects. For PCT, CRP, PCT + CRP and presepsin, the I2 values were 96%, 98%, 0%, and 80%, respectively.
Forest plot and area under the summary ROC (SROC) curve
Forest plots of sensitivity and specificity are shown in Fig. 4. Additional file 1: Table S3 shows the pooled results of PCT, CRP, PCT + CRP and presepsin. Figure 5 shows the SROC curve for the diagnosis of neonatal sepsis. The pooled sensitivity, specificity, PLR, NLR, DOR, AUC and corresponding 95% CI (95% CI) of PCT, CRP, PCT + CRP, and presepsin were 0.85 (0.79, 0.89), 0.71 (0.63, 0.78), 0.91 (0.84, 0.95), 0.94 (0.80, 0.99); 0.84 (0.78, 0.89), 0.88 (0.80, 0.93), 0.89 (0.81, 0.93), 0.98 (0.87, 1.00); 5.4 (3.7, 7.9), 6.1 (3.6, 10.5), 8.0 (4.6, 14.0), 50.8 (6.5, 394.7); 0.18 (0.13, 0.25), 0.33 (0.26, 0.42), 0.10(0.05, 0.19), 0.06 (0.02, 0.23); 31 (17, 54), 19 (10, 35), 79 (26, 246), 864 (65, 11473); and 0.91(0.89–0.94), 0.85 (0.82–0.88), 0.96 (0.93–0.97), 0.99 (0.98–1.00), respectively.
Sensitivity and specificity. a C-reactive protein (CRP). b Procalcitonin (PCT). c PCT plus CRP. d Presepsin. Point estimates for sensitivity and 95% confidence intervals are shown with pooled estimates. Q = Cochran Q statistic
Summary receiver-operating characteristic (SROC) curves for the diagnosis of neonatal sepsis. a C-reactive protein (CRP). b Procalcitonin (PCT). c PCT plus CRP. d Presepsin. AUC = area under the curve
Pair-wise comparisons
Additional file 1: Table S3 shows the results of pair-wise comparisons between statistical indicators for sensitivity, specificity, PLR, NLR, and AUC. The pooled sensitivity of CRP (0.71 (0.63, 0.78)) was weaker than that of PCT (0.85 (0.79, 0.89)), PCT + CRP (0.91 (0.84, 0.95)) and presepsin (0.94 (0.80, 0.99)) and the pooled NLR of presepsin (0.06 (0.02, 0.23)) and PCT + CRP (0.10 (0.05, 0.19)) were less than for CRP (0.33 (0.26, 0.42)), and the AUC of presepsin (0.99 (0.98–1.00)) was greater than for PCT + CRP (0.96 (0.93–0.97)), CRP (0.85 (0.82–0.88)), and PCT (0.91 (0.89–0.94)).
Likelihood ratio scattergram
For PCT and CRP, the summary LRP and LRN for index testing were on the right lower quadrant (RLQ), indicating that PCT or CRP could not exclude or confirm neonatal sepsis (Additional file 3: Figure S2). For PCT + CRP, the summary LRP and LRN for index testing was between the left lower quadrant (LLQ) and right lower quadrant (RLQ), suggesting that PCT + CRP may exclude but not confirm neonatal sepsis (Additional file 3: Figure S2). For presepsin, the summary LRP and LRN for index testing was between the left upper quadrant (LUQ), indicating that presepsin could exclude and confirm neonatal sepsis (Additional file 3: Figure S2).
Fagan diagram and publication bias
Additional file 4: Figure S3 shows the assessment of publication bias. Based on the P values of PCT, CRP, PCT + CRP and presepsin (0.430, 0.735, 0.825, and 0.410, respectively) and the corresponding Deek's funnel plot, no significant publication bias was observed. Additional file 5: Figure S4 shows the Fagan diagrams. Based on the same pre-test probability of 20%, the post-test probability for presepsin (93%) was higher than for PCT + CRP (67%), PCT (58%), and CRP (60%).
Sensitivity analysis was performed with a method of reducing one article at a time, and the effect of a single study on the meta-analysis was evaluated. Additional file 1: Table S10 shows the combined DOR and 95% CI calculated after deleting a single study. We observed that regardless of the excluded study, the combined DOR after removal did not significantly change, suggesting that the results of this analysis were not excessively dependent on a certain study, and our findings were robust.
Meta-regression analysis
Meta-regression analysis of sensitivity, specificity, and joint models was performed to find potential sources of heterogeneity (Additional file 1: Tables S4, S5, and S6). According to the results of meta-regression analysis, we specified subgroups based on design, region, method, test time, and cutoff value.
Subgroup analysis
The results of the subgroup analysis are shown in Additional file 1: Tables S7, S8, and S9.
In terms of region, PCT had similar sensitivity in Asia and Europe, while its sensitivity in North America (0.98 (0.92–1.00)) was significantly higher than in Asia (0.85 (0.80–0.91)). In Asia, the sensitivity of PCT obtained at a cutoff level of 1.53 ng/mL (0.91 (0.77–1.00)) was higher than its sensitivity at a cutoff level of 1 ng/mL (0.59 (0.26–0.91)).
For CRP, in terms of region, the sensitivity of CRP in Africa (0.92 (0.80–1.00)) was significantly higher than in Asia (0.72 (0.63–0.80)) and Europe (0.63 (0.47–0.79)). In Europe, immunonephelometric assay (0.71 (0.54–0.83)) was significantly more sensitive than chemiluminescent immunoassay (0.28 (0.01–0.54)), but its specificity (0.85 (0.81–0.89)) was significantly lower than that of chemiluminescent immunoassay (0.98 (0.92–1.00)).
For PCT + CRP, in terms of region, the sensitivity of PCT + CRP in Asia (0.93 (0.90–0.97)) was significantly higher than in Europe (0.69 (0.51–0.87)).
For presepsin, in terms of region, the specificity of presepsin in Europe (1.00 (0.94–1.00)) was significantly higher than in Asia (0.90 (0.93–0.95)). In terms of study design, the sensitivity of case-control studies (0.92 (0.80–1.00)) was significantly higher than for cohort studies (0.80 (0.66–0.94)). The sensitivity and specificity of presepsin obtained at a cutoff level of 722 μg/L was higher than its sensitivity and specificity at a cutoff level of 539 μg/L.
In addition, we performed a study of the appropriate cutoff interval (Additional file 1: Table S9). For PCT, 0.5–2 ng/mL may be the appropriate cutoff interval. Moreover, the 0.5–1 ng/mL PCT range had high sensitivity (0.88 (0.82–0.95)), whereas the 1.5–2 ng/mL range had high specificity (0.90 (0.77–1.00)). For CRP, ae cutoff value > 10 mg/L had high sensitivity (0.85 (0.72–0.98)) and specificity (0.93 (0.82–1.00)).
The main finding of this meta-analysis was that presepsin or PCT plus CRP improves the accuracy of diagnosis of neonatal sepsis. In addition, our meta-analysis initially established suitable cutoff values and cutoff intervals. However, there is significant statistical heterogeneity in some of the analyses. This fact cannot be ignored in the interpretation of our findings.
Significant differences in the definition of neonatal sepsis were observed in the studies included in our meta-analysis. Although the concept of neonatal clinical sepsis is widely used, a standard definition for this common condition has not been established [11], thereby resulting in the variability in the criteria that are used for diagnoses [12]. Therefore, it is likely that the term neonatal sepsis encompasses diseases and disease severity that differ among the studies included in this meta-analysis, and may also explain the high levels of heterogeneity observed in our analysis. In addition, the potential source of heterogeneity may be the age of the patients, so we have removed studies in patients older than 28 days. Inclusion of preterm neonates in evaluation studies may also be a source of heterogeneity. However, because the percentage of neonates born prematurely is rarely reported in studies, we were unable to assess the impact of this particular factor on the diagnosis of neonatal sepsis using biomarkers. In infants with low birth weight, infection is more common than in those with normal birth weight. One of the important risk factors is early rupture of the membrane (PROM), which may pose a risk of upward infection to the fetus (microorganisms in the external genitalia cause infection of the internal genitalia by ascending) [13]. In addition, local microbiological characteristics may influence the value of PCT in predicting neonatal sepsis [14, 15]. However, we cannot explore this further because the studies included in this meta-analysis do not provide information on microbiological characteristics.
An important advantage of using biomarkers in screening for neonatal sepsis is the ability to correctly identify neonates with culture-negative sepsis, who require antibiotic therapy [12]. In addition, it is also important to exclude the diagnosis of sepsis so that the number of neonates treated with antibiotics can be minimized, hospital stays can be shortened, selection pressure for resistant strains may appear to be smaller, and medical and economic advantages may offset the financial costs of measuring PCT (the cost of measuring CRP is approximately 25% of the cost of measuring PCT) [16]. We observed that although PCT is more sensitive than CRP, the use of PCT or CRP alone cannot rule out a diagnosis of neonatal sepsis. The combination of CRP and PCT resulted in higher sensitivity and AUC and lower NLR, which helped in confirming and ruling out neonatal sepsis. Therefore, it is important to combine these two biomarkers for the diagnosis of neonatal sepsis. In addition, this meta-analysis shows that presepsin may be used alone to diagnose and rule out neonatal sepsis due to its high sensitivity and specificity. Many studies have found that the level of presepsin in patients with sepsis is significantly higher than in healthy infants, and over time, similar to CRP and PCT, it decreases with antibiotic treatment [17–22]. These findings indicate that presepsin can be used to monitor clinical response to therapeutic interventions prior to obtaining culture results. Poggi et al. reported that even on the first day of treatment, the level of presepsin decreased, and CRP and PCT did not differ from the baseline values, suggesting that presepsin may be able to detect neonatal sepsis earlier than PCT or CRP [17]. Moreover, some studies found no correlation between presepsin levels and gestational age in the control group, and it appears that presepsin is not affected by postnatal age [17, 18]. Therefore, a unique presepsin reference range can be used for preterm or term infants on any day after birth. However, the results of this meta-analysis are based on prospective and retrospective studies, which have relatively different methodological characteristics. Subgroup analysis showed that the sensitivity of presepsin in the diagnosis of neonatal sepsis was significantly different between prospective and retrospective studies, suggesting that different study designs may influence the accuracy of the diagnostic trial. In addition, these studies did not use a predetermined cutoff value, but used an optimal cutoff value instead, which could lead to overestimation of diagnostic accuracy [17–22]. In some studies, the inclusion of too many healthy newborns in the control group also improved the diagnostic accuracy, so we excluded these studies. Overall, there are too few high-quality studies on presepsin, and more research is needed to support its use in the diagnosis and exclusion of neonatal sepsis.
Our meta-analysis has several limitations. First, we observed heterogeneity between the studies included, in the selection of neonatal features and the broad definition of neonatal sepsis. Some studies only confirm septicemia by positive blood cultures, microscopy, or polymerase chain reaction, whereas others also consider a comprehensive assessment of the patient chart and assessment of clinical, radiological, and laboratory data [23, 24]. Second, some studies did not provide the exact time of blood sampling, only indicating "on admission" or "before antibiotic treatment" [25, 26]. Third, previous studies have shown that PCT has better diagnostic accuracy in the diagnosis of late-onset sepsis [27], but the current data on early-onset and late-onset sepsis in this meta-analysis is not sufficient to generate any definitive conclusions. Fourth, we consider that differences in diagnostic accuracy in different regions may be due to a variety of reasons, such as individual patient differences, diagnostic criteria for neonatal sepsis, methods for detecting samples, laboratory testing levels, and instruments used. Since these studies did not provide the aforementioned details, we could not further analyze the specific causes of differences in diagnostic accuracy in different regions.
The combination of PCT and CRP or presepsin alone improves the accuracy of the diagnosis of neonatal sepsis. However, further studies are required to confirm these findings.
Zhen Liu, Yun Zhao, Guang-Yu Xu, Shu-Fang Li, Chun-Ni Li, Lin-Shan Chen and Zheng Tao contributed equally to this work.
95% CI:
Corresponding 95% credible interval
AUC:
Area under the curve
CRP:
DOR:
Diagnostic odds ratio
FN:
False negative
FP:
NICU:
NLR:
Negative likelihood ratio
PCT:
PLR:
Positive likelihood ratio
QUADAS-2:
Quality Assessment of Diagnostic Accuracy Studies
SROC:
Summary receiver-operating characteristic
TN:
True negative
TP:
True positive
We thank LetPub for its linguistic assistance during the preparation of this manuscript.
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
The datasets supporting the conclusions of this article are included within the article and its additional files.
GYC had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. Acquisition, analysis, or interpretation of data: GYC, ZL. YZ, GYX, SFL, CNL, LSC, ZT. Drafting of the manuscript: GYC. Statistical analysis: GYC, GYX, SFL, CNL, ZT. Administrative, technical, or material support: GYC, LSC, ZT. Study supervision: LR. Role of the funding source: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. GYC, zhen liu had full access to all the data, and GYC was responsible for the decision to submit for publication. All authors read and approved the final manuscript.
This article is meta-analysis and does not require ethics committee approval or a consent statement.
Not applicable. The manuscript does not contain any personal data in any form (including personal details, pictures, or videos).
Additional file 1: Table S1. The characteristics of the studies included. Table S2. The characteristics of the studies included. Table S3. Pair-wise comparisons between modalities for sensitivity, specificity, PLR, NLR, and AUC. Table S4. The result of meta-regression and subgroup analysis for PCT. Table S5. The result of meta-regression and subgroup analysis for CRP. Table S6. The result of meta-regression and subgroup analysis for presepsin. Table S7. Subgroup analysis of region and detection method for PCT and CRP. Table S8. Subgroup analysis of region and cutoff level for PCT and CRP. Table S9. Subgroup analysis of cutoff level for PCT and CRP. Table S10. Sensitivity analyses of PCT, CRP, PCT + CRP, and presepsin. (ZIP 100 kb)
Additional file 2: Figure S1. Bivariate boxplots. Bivariate boxplots of CRP (A), PCT (B), PCT plus CRP (C), and presepsin (D). (TIF 2191 kb)
Additional file 3: Figure S2. Likelihood ratio scattergrams. Scattergrams evaluating the positive likelihood ratios in the diagnosis of neonatal sepsis for CRP (A), PCT (B), PCT plus CRP (C), and presepsin (D). (TIF 2591 kb)
Additional file 4: Figure S3. Deek's funnel plots. Funnel plots evaluating publication bias of CRP (A), PCT (B), PCT plus CRP (C), and presepsin (D). (TIF 1861 kb)
Additional file 5: Figure S4. Fagan diagram. A, Fagan diagram of CRP. B, Fagan diagram of PCT. C, Fagan diagram of PCT plus CRP. D, Fagan diagram of Presepsin. (TIF 3370 kb)
Departments of Anesthesiology, Guangxi Medical University Affiliated Tumor Hospital, Naning, Guangxi, China
Departments of Respiratory Oncology, Guangxi Medical University Affiliated Tumor Hospital, Naning, Guangxi, China
Departments of Urology, Guangxi Medical University Affiliated Tumor Hospital, Naning, Guangxi, China
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The Magnus Prediction Models
September 24, 2019, Micah Blake McCurdy, @IneffectiveMath
Estimating Shooter and Goalie Talent
This is the updated shooting and goaltending model, which is closely related to the model I introduced last year. Much of the exposition here is copied from that write-up.
I am interested in isolating which NHL players shoot the puck well, and which NHL goaltenders do a good job at preventing shots from becoming goals. To that end I have fit a regression model which replicates some of the simple features of shooting and saving. Throughout this article, when I say "shot" I will mean "unblocked shot", that is, goals, saves, and misses (including shots that hit the post or the crossbar). Furthermore, when I talk of shooting talent, I mean the ability to score more than one would expect given the shot location, so a player may well take a lot of shots from great scoring locations and still be "a bad shooter" in some sense. Generating many such shots is obviously desirable and surely can be done more often by talented players, but I do not consider any such talents to be part of shooting talent, which is (half of) the subject of this article.
In contrast to last year's model, which used only 5v5 shots, now I also use 5v4 shots. Although not the same, the fundemental mechanics of shooting and saving are similar in both cases.
Once a shot is being taken by a given player from a certain spot against a specific goaltender, I estimate the probability that such a shot will be a goal. This process is modelled with a linear model, and fit with a generalized ridge logistic regression. For a detailed exposition about how such models can be fit, please see Section 5. Briefly: I use a design matrix \(X\) for which every row is a shot with the following columns:
An indicator for shooter;
An indicator for the goaltender;
A set of indicators for shot type, where wrist and snap shots are (together, undistinguished) taken as the "base" shot type, and indicator variables are set to 1 for slap shots, backhands, wraparounds, and tips (including deflections);
An indicator for "rush shots", that is, shots for which the previous recorded play-by-play event is in a different zone and no more than four seconds prior;
An indicator for "rebound shots", that is, shots for which the previous recorded play-by-play event is another shot taken by the same team no more than three seconds prior;
The distance from the shot location to the net, divided by 89 ft; making the intersection of the red line and the split line distance "1" and shots from immediately in front of the net close to zero.
The "visible net", that is, the width of the net projected onto the plane which is square to the shooter, divided by six feet. For shots from the split line, the visible net has value 1, and for shots very close to the goal line, the visible net is close to 0; and
An indicator for teams which are leading and another for teams which are trailing; to be interpreted as representing change in configurations surrounding shots compared to when teams are tied.
An indicator for 5v4 shots, to be interpreted as the change compared to a similar shot taken at 5v5.
An intercept.
The items in bold are new in this year's model.
I make a slightly unusual modification to shot distances; namely, shots which are recorded as coming from closer than ten feet are assigned a distance of 10ft. This is to stop small variations in shot location from having outsize effects on the regression, and also because it is close to the threshold of minimum human reaction time for goaltenders given typical NHL wrist-shot speeds.
The observation vector \(Y\) is 1 for goals and 0 for saves or misses. The model itself is the usual linear one: $$ Y = X\beta $$ where \(\beta\) is the vector of covariate values.
The model is fit by maximizing the likelihood of the model, that is, for a given model, form the product of the predicted probabilities for all of the events that did happen (90% chance of a save here times 15% of that goal there, etc.). Large products are awkward, so we solve the mathematically equivalent problem of maximizing the logarithm of the likelihood, and before we do so we add a term of the form \(-\beta^T\Lambda\beta\), where we use \(\Lambda\) to encode our prior knowledge, as described below.
Simple formulas for the \(\beta\) which maximixes this likelihood to not seem to exist, but we can still find it by iteratively computing: $$ \beta_{n+1} = ( X^TX + \Lambda )^{-1} X^T ( X \beta_n + Y - f(X,\beta_n) ) $$ where \(f(X,\beta)\) is the vector function whose entry at position i is \((1 + \exp(-X_i\beta))^{-1}\) where \(X_i\) is the i'th row of \(X\) (this choice of \(f\) is what makes the regression logistic). By starting with \(\beta_0\) as the zero vector and iterating until convergence, I obtain estimates of shooter ability, goaltending ability, with suitable modifications for shot location and type, as well as the score and the skater strength.
This model is zero-biased, which is to say that we consider deviations from average ability to be on-their-face unlikely and bias our results towards average. Another way of saying the same thing is to say that we are beginning with an assumption (held with a certain strength) that all players are of league average ability and then letting the observed data slowly update our knowledge, instead of beginning with an assumption that we know nothing about the shooters and goaltenders at all. The bias controlled by the matrix \(\Lambda\), which must be positive definite for the above formula to be the well-defined solution which makes \(\beta\) the one which minimizes the total error. Similarly to my 5v5 shot rate model, I use a diagonal matrix, where the entries correspoding to goaltenders and shooters are \(\lambda = 100\) and those corresponding to all other columns are 0.001, that is, very close to zero. As for that model, the non-trivial \(\lambda\) values were chosen by varying \(\lambda\) and choosing a value where player estimates have stabilized.
(Incidentally, I do not yet see a way to convert this model, as the above 5v5 shot rate model has been, into a "chained" model where each year's results can be used as the prior for estimates formed after the following year. I would dearly like to do so, though.)
In the future, I will publish results for all seasons, but for now, I record the results of fitting this model on all of the 5v5 shots in the 2017-2019 regular seasons. First, the non-player covariates are:
Covariate
Constant -2.59
Slapshot -0.0971
Tip/Deflection +0.185
Backhand +0.0915
Wraparound -0.284
Rush +0.0969
Rebound +0.944
Distance -3.34
Visible Net +0.606
Leading +0.166
Trailing +0.0106
5v4 +0.413
Logistic regression coefficient values can be difficult to interpret, but negative values always mean "less likely to become a goal" and positive values mean "more likely to become a goal". To compute the probability that a shot with a given description will become a goal, add up all of the model covariates to obtain a number, and then apply the logistic function to it, that is, $$ x \mapsto \frac{1}{1 + \exp(-x)}$$ This function (after which the regression type is named) is very convenient for modelling probabilities, since it monotonically takes the midpoint of the number line (that is, zero) to 50% while taking large negative numbers to positive numbers close to zero and very large positive numbers to positive numbers close to one.
Thus, for instance, we might want to compute the goal probability of a 5v5 wrist shot from 30 feet out (just below the tops of the circles), in a tied game, on the split line, neither on the rush nor a rebound. To do this, begin with the constant value -2.59. We have encoded distance by dividing by 89, so we multiply 30/89 times the distance coefficient of -3.34 to obtain -1.13. From the split line, the visible net is 1, so we add +0.606. Wrist and snap shots are taken as the base category, so no shot type term needs to be added. Since the shot is neither a shot nor a rebound, taken while tied at 5v5, we have all the terms we need, adding them together gives -3.114. Applying the logistic function gives 4.2%, somewhat below the historical percentage of six to eight percent from this area, as we'd expect since most of the "special" things that could have described our shot would have increased it's chance of becoming a goal.
The overall features of the model are more or less as expected---shots from farther away are less likely to go in, seeing more of the net is good, rush shots are good, rebound shots are even better. Power-play shots are substantially more likely to result in goals from equivalently-described 5v5 shots. The shot type terms are somewhat surprising to me, especially the negative term for slapshots even after accounting for distance. Also mildly surprising is that both leading and trailing increases the chance of a shot becoming a goal, suggesting that games do "open up" when one team takes a lead, rather than the "losing teams dominate" pattern of score effects that are familiar from work on shot rates. Also interesting is that the effect of leading improves the goal odds of shots much more than that of trailing, suggesting perhaps that teams with the lead hold on to the puck a little more, preferring not to give up the puck unless they feel their chances of scoring are higher.
Player Results
As the above example shows, the model can already be used without specifying shooters or goaltenders. However, this is perhaps a little boring. Below are the values for all the goaltenders who faced at least one shot in the 2017-2019 regular seasons. I've inverted the scale so that the better performances are at the top.
The scale is the same units as for the non-player covariates above, so even the best or worst performances are smaller than the effect of a shot being a rush shot, for instance, consistent with goaltending performances being broadly similar across the league.
Similarly for forward and defender results, which I've put on separate pages for performance reasons.
Minimum Shots Faced: | CommonCrawl |
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4306 Periodontal disease and the oral microbiome in antiretroviral-treated patients with HIV
Medini K Annavajhala, Jayesh G. Shah, Jessica Weidler, Karolina Kister, Ryan T. Demmer, Sunil Wadhwa, Michael T. Yin, Anne-Catrin Uhlemann
Journal: Journal of Clinical and Translational Science / Volume 4 / Issue s1 / June 2020
Published online by Cambridge University Press: 29 July 2020, pp. 99-100
OBJECTIVES/GOALS: People living with HIV, despite antiretroviral therapy (ART), have increased burden of inflammatory and aging-related comorbidities such as periodontitis. Oral microbiota have been linked to periodontitis, but not in the context of HIV. We aim to compare relationships between the oral microbiome and periodontal disease in HIV+ vs healthy controls. METHODS/STUDY POPULATION: In an ongoing cohort study we have been recruiting pre- and post-menopausal women with HIV+ on ART for ≥6 months and HIV- controls matched by menopausal status (target n = 30 per arm; currently HIV+: n = 30 post- and 9 pre-M; HIV-: n = 15 post- and 6 pre-M). Patients age <18 or on antibiotics within 3 mos., except prophylaxis, are excluded. Patients provide saliva, then subgingival plaque collection during a dental examination through scaling from six index teeth. Standard CDC/AAP classifications of periodontitis are used. We will perform 16S rRNA and ITS sequencing to profile bacterial and fungal communities in saliva and plaque. Linear mixed effect regression and differential abundance analyses will be used to identify microbial and mycobial oral signatures of periodontal disease severity in HIV+ and HIV- populations. RESULTS/ANTICIPATED RESULTS: We found a markedly high prevalence of severe periodontal disease in HIV+ women despite ART (59%, compared to 11% in HIV- controls). In post-menopausal women with HIV, saliva bacterial α- and β-diversity in the saliva differed significantly with periodontal disease severity. Fungal α-diversity was also significantly lower in plaque from teeth with severe loss of tissue attachment (CAL ≥4 mm). We identified bacterial and fungal taxa significantly enriched in post-menopausal HIV+ women with severe compared to no or mild periodontitis. We hypothesize, similarly, associations between the oral microbiome and periodontitis in HIV- controls. However, we expect overall diversity metrics to be significantly altered in HIV+ compared to HIV- patients, indicating long-term dysbiosis despite treatment with ART. DISCUSSION/SIGNIFICANCE OF IMPACT: Contrasting associations between the oral microbiome and periodontal disease with respect to HIV will provide evidence for the role of microbiota in accelerated aging phenotype caused by HIV. Our results would also provide rationale for interventions targeting co-morbidities in people living with HIV to account for both inflammation and dysbiosis.
Evaluation of immunisation strategies for pertussis vaccines in Jinan, China – an interrupted time-series study
T. C. Liu, J. Zhang, S. Q. Liu, A. T. Yin, S. M. Ruan
Journal: Epidemiology & Infection / Volume 148 / 2020
Published online by Cambridge University Press: 12 February 2020, e26
Studies in countries with high immunisation coverage suggest that the re-emergence of pertussis may be caused by a decreased duration of protection resulting from the replacement of whole-cell pertussis vaccine (WPV) with the acellular pertussis vaccine (APV). In China, WPV was introduced in 1978. The pertussis vaccination schedule advanced from an all-WPV schedule (1978–2007), to a mixed WPV/APV schedule (2008–2009), then to an all-APV schedule (2010–2016). Increases in the incidence of pertussis have been reported in recent years in Jinan and other cities in China. However, there have been few Chinese-population-based studies focused on the impact of schedule changes. We obtained annual pertussis incidences from 1956 to 2016 from the Jinan Notifiable Conditions Database. We used interrupted time series and segmented regression analyses to assess changes in pertussis incidence at the beginning of each year, and average annual changes during the intervention. Pertussis incidence decreased by 1.11 cases per 100 000 population (P = 0.743) immediately following WPV introduction in 1978 and declined significantly by 1.21 cases per 100 000 population per year (P < 0.0001) between 1978 and 2001. Immediately after APV replaced the fourth dose of WPV in 2008, the second and third doses in 2009, then replaced all four doses in 2010, pertussis incidence declined by 1.98, 1.98 and 1.08 cases per 100 000 population, respectively. However, the results were not statistically significant. There were significant increasing trends in pertussis incidence after APV replacements: 1.63, 1.77 and 1.78 cases/year in 2008–2016, 2009–2016 and 2010–2016, respectively. Our study shows that the impact of an all-WPV schedule may be less than the impacts of the sequential WPV/APV schedules. The short-term impact of APV was better than that of WPV; however, the duration of APV-induced protection was not ideal. The impact and duration of protective immunity resulting from APVs produced in China need further evaluation. Further research on the effectiveness of pertussis vaccination programme in Jinan, China is also necessary.
P.053 Whole-genome sequencing identified a frameshift mutation at LMNB1 in a family with early-onset dystonia
RK Yuen, B Adhami-Mojarad, I Backstrom, A Yin, T Soman
Journal: Canadian Journal of Neurological Sciences / Volume 46 / Issue s1 / June 2019
Published online by Cambridge University Press: 05 June 2019, p. S28
Background: Dystonia is a hyperkinetic condition that produces abnormal movements or postures. Its diagnostic procedure is often challenging and time consuming. Genetic testing provides an effective approach for diagnosis, but currently only very few dystonia genes have been identified. We propose that studying early-onset forms of dystonia with the use of whole-genome sequencing (WGS) will improve the identification of dystonia-relevant genes and mutations. Methods: We performed deep WGS using the Illumina HiSeq X technology in a mother-proband pair with dystonia. The mother has generalized dystonia (age of onset: 15) and the proband has myoclonic dystonia (age of onset: 11). Results: No pathogenic mutation was identified in any of the known dystonia genes. However, we identified a rare heterozygous frameshift mutation (p.K342fs*7) at LMNB1 that was shared between the mother and the proband. Duplication of LMNB1 is known to cause Adult-onset Demyelinating Leukodystrophy. A heterozygous deletion of LMNB1 has been reported in a patient with microcephaly and global developmental disorder. Conclusions: Further characterization of phenotypes in the participants and their family members is needed to confirm the relationship between mutation in LMNB1 and dystonia. This work provides a proof-of-principle that novel disease-relevant genes can potentially be identified using the proposed approach.
Quantum electrodynamics experiments with colliding petawatt laser pulses
HPL_EP HEDP and High Power Laser 2018
I. C. E. Turcu, B. Shen, D. Neely, G. Sarri, K. A. Tanaka, P. McKenna, S. P. D. Mangles, T.-P. Yu, W. Luo, X.-L. Zhu, Y. Yin
Journal: High Power Laser Science and Engineering / Volume 7 / 2019
A new generation of high power laser facilities will provide laser pulses with extremely high powers of 10 petawatt (PW) and even 100 PW, capable of reaching intensities of $10^{23}~\text{W}/\text{cm}^{2}$ in the laser focus. These ultra-high intensities are nevertheless lower than the Schwinger intensity $I_{S}=2.3\times 10^{29}~\text{W}/\text{cm}^{2}$ at which the theory of quantum electrodynamics (QED) predicts that a large part of the energy of the laser photons will be transformed to hard Gamma-ray photons and even to matter, via electron–positron pair production. To enable the investigation of this physics at the intensities achievable with the next generation of high power laser facilities, an approach involving the interaction of two colliding PW laser pulses is being adopted. Theoretical simulations predict strong QED effects with colliding laser pulses of ${\geqslant}10~\text{PW}$ focused to intensities ${\geqslant}10^{22}~\text{W}/\text{cm}^{2}$ .
Management and population dynamics of diamondback moth (Plutella xylostella): planting regimes, crop hygiene, biological control and timing of interventions
Z. Li, M.J. Furlong, T. Yonow, D.J. Kriticos, H. Bao, F. Yin, Q. Lin, X. Feng, M.P. Zalucki
Journal: Bulletin of Entomological Research / Volume 109 / Issue 2 / April 2019
Using an age-structured process-based simulation model for diamondback moth (DBM), we model the population dynamics of this major Brassica pest using the cropping practices and climate of Guangdong, China. The model simulates two interacting sub-populations (demes), each representing a short season crop. The simulated DBM abundance, and hence pest problems, depend on planting regime, crop hygiene and biological control. A continuous supply of hosts, a low proportion of crop harvested and long residue times between harvest and replanting each exacerbate pest levels. Biological control provided by a larval parasitoid can reduce pest problems, but not eliminate them when climate is suitable for DBM and under certain planting practices. The classic Integrated Pest Management (IPM) method of insecticide application, when pest threshold is reached, proved effective and halved the number of insecticide sprays when compared with the typical practice of weekly insecticide application.
Adipose tissue uncoupling protein 1 levels and function are increased in a mouse model of developmental obesity induced by maternal exposure to high-fat diet
French DOHaD
E. Bytautiene Prewit, C. Porter, M. La Rosa, N. Bhattarai, H. Yin, P. Gamble, T. Kechichian, L. S. Sidossis
Journal: Journal of Developmental Origins of Health and Disease / Volume 9 / Issue 4 / August 2018
Published online by Cambridge University Press: 17 May 2018, pp. 401-408
With brown adipose tissue (BAT) becoming a possible therapeutic target to counteract obesity, the prenatal environment could represent a critical window to modify BAT function and browning of white AT. We investigated if levels of uncoupling protein 1 (UCP1) and UCP1-mediated thermogenesis are altered in offspring exposed to prenatal obesity. Female CD-1 mice were fed a high-fat (HF) or standard-fat (SF) diet for 3 months before breeding. After weaning, all pups were placed on SF. UCP1 mRNA and protein levels were quantified using quantitative real-time PCR and Western blot analysis, respectively, in brown (BAT), subcutaneous (SAT) and visceral (VAT) adipose tissues at 6 months of age. Total and UCP1-dependent mitochondrial respiration were determined by high-resolution respirometry. A Student's t-test and Mann–Whitney test were used (significance: P<0.05). UCP1 mRNA levels were not different between the HF and SF offspring. UCP1 protein levels, total mitochondrial respiration and UCP1-dependent respiration were significantly higher in BAT from HF males (P=0.02, P=0.04, P=0.005, respectively) and females (P=0.01, P=0.04, P=0.02, respectively). In SAT, the UCP1 protein was significantly lower in HF females (P=0.03), and the UCP1-dependent thermogenesis was significantly lower from HF males (P=0.04). In VAT, UCP1 protein levels and UCP1-dependent respiration were significantly lower only in HF females (P=0.03, P=0.04, respectively). There were no differences in total respiration in SAT and VAT. Prenatal exposure to maternal obesity leads to significant increases in UCP1 levels and function in BAT in offspring with little impact on UCP1 levels and function in SAT and VAT.
Seed treatment with glycine betaine enhances tolerance of cotton to chilling stress
C. Cheng, L. M. Pei, T. T. Yin, K. W. Zhang
Journal: The Journal of Agricultural Science / Volume 156 / Issue 3 / April 2018
Chilling injury is an important natural stress that can threaten cotton production, especially at the sowing and seedling stages in early spring. It is therefore important for cotton production to improve chilling tolerance at these stages. The current work examines the potential for glycine betaine (GB) treatment of seeds to increase the chilling tolerance of cotton at the seedling stage. Germination under cold stress was increased significantly by GB treatment. Under low temperature, the leaves of seedlings from treated seeds exhibited a higher net photosynthetic rate (PN), higher antioxidant enzyme activity including superoxide dismutase, ascorbate peroxidase and catalase, lower hydrogen peroxide (H2O2) content and less damage to the cell membrane. Enzyme activity was correlated negatively with H2O2 content and degree of damage to the cell membrane but correlated positively with GB content. The experimental results suggested that although GB was only used to treat cotton seed, the beneficial effect caused by the preliminary treatment of GB could play a significant role during germination that persisted to at least the four-leaf seedling stage. Therefore, it is crucial that this method is employed in agricultural production to improve chilling resistance in the seedling stage by soaking the seeds in GB.
Effects of nitrogen application and supplemental irrigation on canopy temperature and photosynthetic characteristics in winter wheat
D. Q. Yang, W. H. Dong, Y. L. Luo, W. T. Song, T. Cai, Y. Li, Y. P. Yin, Z. L. Wang
Journal: The Journal of Agricultural Science / Volume 156 / Issue 1 / January 2018
Print publication: January 2018
Nitrogen (N) application and irrigation to winter wheat may decrease leaf temperature and enhance photosynthesis: as a result, more photosynthates will be allocated to the grains, resulting in higher grain yields. To investigate this hypothesis, a 2-year field study was conducted with three levels of N fertilizer application (no fertilizer, N0; 240 kg N/ha, N1; 360 kg N/ha, N2) and two different water regimes (rainfed with no irrigation, R; irrigation at the over-wintering, stem elongation and grain filling stages, W). The results show that both N application and supplemental irrigation significantly increased grain yield with increases in both grain number/m2 and the 1000-grain weight, viz., WN2>WN1>WN0>RN2>RN1>RN0. In addition, application of N under both water regimes significantly increased flag leaf area, above-ground biomass and single stem productivity and decreased leaf temperature, which led to an increase in net photosynthesis rates and ribulose bisphosphate (RuBP) carboxylase activity. Moreover, analysis of the chlorophyll α fluorescence transient showed that N fertilizer application and supplemental irrigation significantly increased electron donor and acceptor performance of the photosystem II reaction centre.
Maternal dietary supplementation with ferrous N-carbamylglycinate chelate affects sow reproductive performance and iron status of neonatal piglets
D. Wan, Y. M. Zhang, X. Wu, X. Lin, X. G. Shu, X. H. Zhou, H. T. Du, W. G. Xing, H. N. Liu, L. Li, Y. Li, Y. L. Yin
Journal: animal / Volume 12 / Issue 7 / July 2018
Published online by Cambridge University Press: 27 November 2017, pp. 1372-1379
Iron-deficiency anemia is a public health concern that frequently occurs in pregnant mammals and neonatal offspring. Ferrous N-carbamylglycinate chelate (Fe-CGly) is a newly designed iron fortifier with proven effects in iron-deficient rats and weanling piglets. However, the effects of this new compound on pregnant mammals are unknown. Therefore, this experiment was conducted to evaluate the effects of Fe-CGly on sow reproductive performance and iron status of both sows and neonatal piglets. A total of 40 large-white sows after second parity were randomly assigned to two groups (n=20). They were receiving a diet including 80 mg Fe/kg as FeSO4 or Fe-CGly, respectively, from day 85 of gestation to parturition. The serum (day 110 of pregnancy) and placentas of sows were sampled. Litter size, mean weight of live born piglets, birth (live) litter weight, number of live born piglets, and the number of still-born piglets, mummies, and weak-born piglets were recorded. Once delivered, eight litters were randomly selected from the 20 litters per treatment, and one new-born male piglet (1.503±0.142 kg) from each selected litter was slaughtered within 3 h after birth from the selected litters, without colostrum ingestion. The serum, longissimus muscle, liver and kidneys of the piglets were collected. The iron status of the serum samples and the messenger RNA level of iron-related genes in the placenta, liver and kidney were analyzed. The results showed that litter weight of live born piglets was higher (P=0.030) in the Fe-CGly group (19.86 kg) than in the FeSO4 group (17.34 kg). Fe-CGly significantly increased placental iron concentration (P<0.05) of sows. It also significantly increased iron saturation and reduced the total iron-binding capacity of piglets (P<0.05) at birth. However, the results revealed that supplementation of Fe-CGly in sows reduced liver and kidney iron concentration of neonatal piglets (P<0.05), indicating decreased iron storage. In addition, the concentration of iron in the colostrum was not significantly changed. Therefore, the present results suggested that replacement of maternal FeSO4 supplement with Fe-CGly in the late-gestating period for sows could improve litter birth weight, probably via enhanced iron transportation in the placenta.
Formation of Single-atom-thick Copper Oxide Monolayers
Kuibo Yin, Yu-Yang Zhang, Yilong Zhou, Litao Sun, Matthew F. Chisholm, Sokrates T. Pantelides, Wu Zhou
Journal: Microscopy and Microanalysis / Volume 23 / Issue S1 / July 2017
Published online by Cambridge University Press: 04 August 2017, pp. 1684-1685
Impaired glucose tolerance in first-episode drug-naïve patients with schizophrenia: relationships with clinical phenotypes and cognitive deficits
D. C. Chen, X. D. Du, G. Z. Yin, K. B. Yang, Y. Nie, N. Wang, Y. L. Li, M. H. Xiu, S. C. He, F. D. Yang, R. Y. Cho, T. R. Kosten, J. C. Soares, J. P. Zhao, X. Y. Zhang
Journal: Psychological Medicine / Volume 46 / Issue 15 / November 2016
Schizophrenia patients have a higher prevalence of type 2 diabetes mellitus with impaired glucose tolerance (IGT) than normals. We examined the relationship between IGT and clinical phenotypes or cognitive deficits in first-episode, drug-naïve (FEDN) Han Chinese patients with schizophrenia.
A total of 175 in-patients were compared with 31 healthy controls on anthropometric measures and fasting plasma levels of glucose, insulin and lipids. They were also compared using a 75 g oral glucose tolerance test and the homeostasis model assessment of insulin resistance (HOMA-IR). Neurocognitive functioning was assessed using the MATRICS Consensus Cognitive Battery (MCCB). Patient psychopathology was assessed using the Positive and Negative Syndrome Scale (PANSS).
Of the patients, 24.5% had IGT compared with none of the controls, and they also had significantly higher levels of fasting blood glucose and 2-h glucose after an oral glucose load, and were more insulin resistant. Compared with those patients with normal glucose tolerance, the IGT patients were older, had a later age of onset, higher waist or hip circumference and body mass index, higher levels of low-density lipoprotein and triglycerides and higher insulin resistance. Furthermore, IGT patients had higher PANSS total and negative symptom subscale scores, but no greater cognitive impairment except on the emotional intelligence index of the MCCB.
IGT occurs with greater frequency in FEDN schizophrenia, and shows association with demographic and anthropometric parameters, as well as with clinical symptoms but minimally with cognitive impairment during the early course of the disorder.
Predominantly night-time feeding and maternal glycaemic levels during pregnancy
See Ling Loy, Tuck Seng Cheng, Marjorelee T. Colega, Yin Bun Cheung, Keith M. Godfrey, Peter D. Gluckman, Kenneth Kwek, Seang Mei Saw, Yap-Seng Chong, Natarajan Padmapriya, Falk Müller-Riemenschneider, Ngee Lek, Fabian Yap, Mary Foong-Fong Chong, Jerry Kok Yen Chan
Journal: British Journal of Nutrition / Volume 115 / Issue 9 / 14 May 2016
Published online by Cambridge University Press: 07 March 2016, pp. 1563-1570
Print publication: 14 May 2016
Little is known about the influence of meal timing and energy consumption patterns throughout the day on glucose regulation during pregnancy. We examined the association of maternal feeding patterns with glycaemic levels among lean and overweight pregnant women. In a prospective cohort study in Singapore, maternal 24-h dietary recalls, fasting glucose (FG) and 2-h postprandial glucose (2HPPG) concentrations were measured at 26–28 weeks of gestation. Women (n 985) were classified into lean (BMI<23 kg/m2) or overweight (BMI≥23 kg/m2) groups. They were further categorised as predominantly daytime (pDT) or predominantly night-time (pNT) feeders according to consumption of greater proportion of energy content from 07.00 to 18.59 hours or from 19.00 to 06.59 hours, respectively. On stratification by weight status, lean pNT feeders were found to have higher FG than lean pDT feeders (4·36 (sd 0·38) v. 4·22 (sd 0·35) mmol/l; P=0·002); however, such differences were not observed between overweight pDT and pNT feeders (4·49 (sd 0·60) v. 4·46 (sd 0·45) mmol/l; P=0·717). Using multiple linear regression with confounder adjustment, pNT feeding was associated with higher FG in the lean group (β=0·16 mmol/l; 95 % CI 0·05, 0·26; P=0·003) but not in the overweight group (β=0·02 mmol/l; 95 % CI −0·17, 0·20; P=0·879). No significant association was found between maternal feeding pattern and 2HPPG in both the lean and the overweight groups. In conclusion, pNT feeding was associated with higher FG concentration in lean but not in overweight pregnant women, suggesting that there may be an adiposity-dependent effect of maternal feeding patterns on glucose tolerance during pregnancy.
Developing one-dimensional implosions for inertial confinement fusion science
HEDP and HPL 2016
J. L. Kline, S. A. Yi, A. N. Simakov, R. E. Olson, D. C. Wilson, G. A. Kyrala, T. S. Perry, S. H. Batha, E. L. Dewald, J. E. Ralph, D. J. Strozzi, A. G. MacPhee, D. A. Callahan, D. Hinkel, O. A. Hurricane, R. J. Leeper, A. B. Zylstra, R. R. Peterson, B. M. Haines, L. Yin, P. A. Bradley, R. C. Shah, T. Braun, J. Biener, B. J. Kozioziemski, J. D. Sater, M. M. Biener, A. V. Hamza, A. Nikroo, L. F. Berzak Hopkins, D. Ho, S. LePape, N. B. Meezan, D. S. Montgomery, W. S. Daughton, E. C. Merritt, T. Cardenas, E. S. Dodd
Published online by Cambridge University Press: 12 December 2016, e44
Experiments on the National Ignition Facility show that multi-dimensional effects currently dominate the implosion performance. Low mode implosion symmetry and hydrodynamic instabilities seeded by capsule mounting features appear to be two key limiting factors for implosion performance. One reason these factors have a large impact on the performance of inertial confinement fusion implosions is the high convergence required to achieve high fusion gains. To tackle these problems, a predictable implosion platform is needed meaning experiments must trade-off high gain for performance. LANL has adopted three main approaches to develop a one-dimensional (1D) implosion platform where 1D means measured yield over the 1D clean calculation. A high adiabat, low convergence platform is being developed using beryllium capsules enabling larger case-to-capsule ratios to improve symmetry. The second approach is liquid fuel layers using wetted foam targets. With liquid fuel layers, the implosion convergence can be controlled via the initial vapor pressure set by the target fielding temperature. The last method is double shell targets. For double shells, the smaller inner shell houses the DT fuel and the convergence of this cavity is relatively small compared to hot spot ignition. However, double shell targets have a different set of trade-off versus advantages. Details for each of these approaches are described.
Scaling of ion energies in the relativistic-induced transparency regime
D. Jung, B.J. Albright, L. Yin, D.C. Gautier, B. Dromey, R. Shah, S. Palaniyappan, S. Letzring, H.-C. Wu, T. Shimada, R.P. Johnson, D. Habs, M. Roth, J.C. Fernández, B.M. Hegelich
Journal: Laser and Particle Beams / Volume 33 / Issue 4 / December 2015
Experimental data are presented showing maximum carbon C6+ ion energies obtained from nm-scaled targets in the relativistic transparent regime for laser intensities between 9 × 1019 and 2 × 1021 W/cm2. When combined with two-dimensional particle-in-cell simulations, these results show a steep linear scaling for carbon ions with the normalized laser amplitude a0 ( $a_0 \propto \sqrt ( I)$ ). The results are in good agreement with a semi-analytic model that allows one to calculate the optimum thickness and the maximum ion energies as functions of a0 and the laser pulse duration τλ for ion acceleration in the relativistic-induced transparency regime. Following our results, ion energies exceeding 100 MeV/amu may be accessible with currently available laser systems.
Spatiotemporal variation and social determinants of suicide in China, 2006–2012: findings from a nationally representative mortality surveillance system
S. Liu, A. Page, P. Yin, T. Astell-Burt, X. Feng, Y. Liu, J. Liu, L. Wang, M. Zhou
Suicide in China has declined since the 1990s. However, there has been limited investigation of the potential spatiotemporal variation and social determinants of suicide during subsequent periods.
Annual suicide counts from 2006 to 2012 stratified by county, 5-year age group (⩾15 years) and gender were obtained from the Chinese Disease Surveillance Points system. Trends and geographic differentials were examined using multilevel negative binomial regression models to explore spatiotemporal variation in suicide, and the role of key sociodemographic factors associated with suicide.
The suicide rate (per 100 000) in China decreased from 14.7 to 9.1, 2006–2012. Rates of suicide were higher in males than females and increased substantially with age. Suicide rates were higher in rural areas compared with urban areas; however, urban–rural disparities reduced over time with a faster decline for rural areas. Within both urban and rural areas, higher rates of suicide were evident in areas with lower socio-economic circumstances (SEC) [rate ratio (RR) 1.85, 95% confidence interval (CI) 1.31–2.62]. Suicide rates varied more than twofold (median RR 2.06) across counties, and were highest in central and southwest regions of China. A high proportion of the divorced population, especially for younger females, was associated with lower suicide rates (RR 0.60, 95% CI 0.46–0.79).
Geographic variations for suicide should be taken into account in policy making, particularly for older males living in rural areas and urban areas with low SEC. Measures to reduce disparities in socio-economic level and alleviate family relation stress are current priorities.
From phenotyping towards breeding strategies: using in vivo indicator traits and genetic markers to improve meat quality in an endangered pig breed
A. D. M. Biermann, T. Yin, U. U. König von Borstel, K. Rübesam, B. Kuhn, S. König
Journal: animal / Volume 9 / Issue 6 / June 2015
In endangered and local pig breeds of small population sizes, production has to focus on alternative niche markets with an emphasis on specific product and meat quality traits to achieve economic competiveness. For designing breeding strategies on meat quality, an adequate performance testing scheme focussing on phenotyped selection candidates is required. For the endangered German pig breed 'Bunte Bentheimer' (BB), no breeding program has been designed until now, and no performance testing scheme has been implemented. For local breeds, mainly reared in small-scale production systems, a performance test based on in vivo indicator traits might be a promising alternative in order to increase genetic gain for meat quality traits. Hence, the main objective of this study was to design and evaluate breeding strategies for the improvement of meat quality within the BB breed using in vivo indicator traits and genetic markers. The in vivo indicator trait was backfat thickness measured by ultrasound (BFiv), and genetic markers were allele variants at the ryanodine receptor 1 (RYR1) locus. In total, 1116 records of production and meat quality traits were collected, including 613 in vivo ultrasound measurements and 713 carcass and meat quality records. Additionally, 700 pigs were genotyped at the RYR1 locus. Data were used (1) to estimate genetic (co)variance components for production and meat quality traits, (2) to estimate allele substitution effects at the RYR1 locus using a selective genotyping approach and (3) to evaluate breeding strategies on meat quality by combining results from quantitative-genetic and molecular-genetic approaches. Heritability for the production trait BFiv was 0.27, and 0.48 for backfat thickness measured on carcass. Estimated heritabilities for meat quality traits ranged from 0.14 for meat brightness to 0.78 for the intramuscular fat content (IMF). Genetic correlations between BFiv and IMF were higher than estimates based on carcass backfat measurements (0.39 v. 0.25). The presence of the unfavorable n allele was associated with increased electric conductivity, paler meat and higher drip loss. The allele substitution effect on IMF was unfavorable, indicating lower IMF when the n allele is present. A breeding strategy including the phenotype (BFiv) combined with genetic marker information at the RYR1 locus from the selection candidate, resulted in a 20% increase in accuracy and selection response when compared with a breeding strategy without genetic marker information.
Detailed Characterization of Surface Ln-Doped Anatase TiO2 Nanoparticles by Hydrothermal Treatment for Photocatalysis and Gas Sensing Applications
Rezwanur Rahman, Sean T. Anderson, Sonal Dey, Robert A. Mayanovic
Journal: MRS Online Proceedings Library Archive / Volume 1806 / 2015
Nanostructured anatase TiO2 is a promising material for gas sensing and photocatalysis. In order to modify its catalytic properties, the lanthanide (Ln) ions Eu3+, Gd3+, Nd3+ and Yb3+ were precipitated on the surface of TiO2 nanoparticles (NPs) by hydrothermal treatment. Results from Raman spectroscopy and X-ray diffraction (XRD) measurements show that the anatase structure of the TiO2 nanoparticles was preserved after hydrothermal treatment. SEM and TEM show a heterogeneous distribution in size and a nanocrystallite morphology of the TiO2 NPs (∼ 14 nm in size) and EDX confirmed the presence of the Ln-ion surface doping after hydrothermal treatment. An increase in photoluminescence (PL) was observed for the Ln-surface-doped TiO2 NPs when measurements were made in forming gas (5% H2 + 95% Ar) at 520 °C. In contrast, the PL measurements made at room temperature did not show any noticeable difference in forming gas or in ambient air. Our temperature-dependent PL results obtained in different gas environments are consistent with modification of oxygen-vacancies and hole-defects due to a combination of hydrothermal treatment and surface Ln-doping.
Pedogenic alteration of illite in subtropical China
W. Han, H. L. Hong, K. Yin, G. J. Churchman, Z. H. Li, T. Chen
Journal: Clay Minerals / Volume 49 / Issue 3 / June 2014
Pedogenic alteration of illite from red earth sediments in Jiujiang in subtropical China was investigated using X-ray diffraction (XRD) and high-resolution transmission electron microscopy (HRTEM). Illite, hydroxy-interlayered vermiculite (HIV), kaolinite and mixed-layer illite-HIV (I-HIV) are present in the soils. The characteristic reflections of the clay phases were 14 Å, 10–14 Å, 10 Å, and 7 Å, respectively. After Mg-glycerol saturations, the 14 Å peak of the samples did not expand, and after heating at 350°C and 550°C it shifted to 13.8 Å and 12 Å respectively, with no residual 14 Å reflection, suggesting the occurrence of hydroxy-interlayered vermiculite. The randomly interstratified I-HIV clays were characterized by a broad peak at 10–14 Å, which did not change its position after Mg-glycerol saturation, but collapsed to 10 Å after heating at 350°C and 550°C. HRTEM analysis showed different lattice fringes of 12 Å, 10 Å and 7 Å . Mixed-layer I-HIV, HIV-K and illite-kaolinite (I-K) were observed in the HRTEM images which represented the intermediate phases during illite alteration. The merging of two 10 Å illite layers into a 12 Å HIV layer, lateral transformation of one HIV layer into one kaolinite layer and alteration of one illite layer into two kaolinite layers illustrated the mechanisms of illite-to-HIV, HIV-to-kaolinite and illite-tokaolinite transformation, respectively. The proposed pedogenic alteration of illite and the weathering sequence of the clay minerals in Jiujiang is illite → I-HIV → HIV → HIV-K → kaolinite. In addition, illite may transform directly to kaolinite.
Assessing the impact of natural service bulls and genotype by environment interactions on genetic gain and inbreeding in organic dairy cattle genomic breeding programs
T. Yin, M. Wensch-Dorendorf, H. Simianer, H. H. Swalve, S. König
The objective of the present study was to compare genetic gain and inbreeding coefficients of dairy cattle in organic breeding program designs by applying stochastic simulations. Evaluated breeding strategies were: (i) selecting bulls from conventional breeding programs, and taking into account genotype by environment (G×E) interactions, (ii) selecting genotyped bulls within the organic environment for artificial insemination (AI) programs and (iii) selecting genotyped natural service bulls within organic herds. The simulated conventional population comprised 148 800 cows from 2976 herds with an average herd size of 50 cows per herd, and 1200 cows were assigned to 60 organic herds. In a young bull program, selection criteria of young bulls in both production systems (conventional and organic) were either 'conventional' estimated breeding values (EBV) or genomic estimated breeding values (GEBV) for two traits with low (h 2=0.05) and moderate heritability (h 2=0.30). GEBV were calculated for different accuracies (r mg), and G×E interactions were considered by modifying originally simulated true breeding values in the range from r g=0.5 to 1.0. For both traits (h 2=0.05 and 0.30) and r mg⩾0.8, genomic selection of bulls directly in the organic population and using selected bulls via AI revealed higher genetic gain than selecting young bulls in the larger conventional population based on EBV; also without the existence of G×E interactions. Only for pronounced G×E interactions (r g=0.5), and for highly accurate GEBV for natural service bulls (r mg>0.9), results suggests the use of genotyped organic natural service bulls instead of implementing an AI program. Inbreeding coefficients of selected bulls and their offspring were generally lower when basing selection decisions for young bulls on GEBV compared with selection strategies based on pedigree indices.
By M. A. Allison, D. M. Alongi, N. Bi, T. S. Bianchi, G. Billen, N. Blair, D. Bombar, A. Borges, S. Bouillon, W. P. Broussard III, W.-J. Cai, J. Callens, S. Chakraborty, C. T. Arthur Chen, N. Chen, D. R. Corbett, M. Dai, J. W. Day, J. W. Dippner, S. Duan, C. Duarte, T. I. Eglinton, G. Erkens, C. France-Lanord, J. Gaillardet, V. Galy, J. Gan, J. Garnier, M. Goñi, S. L. Goodbred, K. Gundersen, L. Guo, D. Nhu Hai, A. Han, P. J. Harrison, C. Hein, P. J. Hernes, R. D. Hetland, R. M. Holmes, T. J. Hsu, G. Hunsinger, A. Kolker, S. A. Kuehl, H. S. Kung, Z. Lai, N. Ngoc Lam, E. L. Leithold, P. Liu, S. E. Lohrenz, N. Loick-Wilde, R. Macdonald, B. A. McKee, E. Meselhe, H. Middelkoop, S. Mitra, W. Moufaddal, M. C. Murrell, C. A. Nittrouer, A. S. Ogston, P. Passy, M. van der Perk, A. Ramanathan, P. A. Raymond, A. I. Robertson, B. E. Rosenheim, G. P. Shaffer, A. M. Shiller, M. Silvestre, R. G. M. Spencer, R. G. Striegl, A. Stubbins, S. E. Tank, V. Thieu, J. M. Visser, M. Voss, J. P. Walsh, H. Wang, W. R. Woerner, Y. Wu, J. Xu, Z. Yang, K. Yin, Z. Yin, G. L. Zhang, J. Zhang, Z. Y. Zhu, A. R. Zimmerman
Edited by Thomas S. Bianchi, Texas A & M University, Mead A. Allison, University of Texas, Austin, Wei-Jun Cai, University of Delaware
Book: Biogeochemical Dynamics at Major River-Coastal Interfaces
Print publication: 28 October 2013, pp ix-xii | CommonCrawl |
\begin{definition}[Definition:Arithmetic]
'''Arithmetic''' is the branch of mathematics which concerns the manipulation of numbers, using the operations addition, subtraction, multiplication and division, and the taking of powers.
\end{definition} | ProofWiki |
\begin{document}
\begin{abstract} We study the representability problem for torsion-free arithmetic matroids. By using a new operation called ``reduction'' and a ``signed Hermite normal form'', we provide and implement an algorithm to compute all the representations, up to equivalence. As an application, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement. \end{abstract}
\maketitle
\setcounter{tocdepth}{1} \tableofcontents
\section{Introduction} Arithmetic matroids are a generalization of matroids, inspired by the combinatorics of finite lists of vectors in $\mathbb{Z}^r$. Representations of arithmetic matroids come from many different contexts, such as:
arrangements of hypertori in an algebraic torus; vector partition functions; zonotopes \cite{deconcini2010topics, DAdderioMoci2013, BrandenMoci2014}.
However, not all arithmetic matroids admit a representation.
A natural question is to determine whether a given arithmetic matroid is representable, and characterize all possible representations. In this work, we give such a characterization in the case of torsion-free arithmetic matroids (i.e.\ when the multiplicity of the empty set is one). Our characterization is effective, and it yields an explicit algorithm to compute all representations. We implemented this algorithm as part of a new Sage library to work with arithmetic matroids, called Arithmat \cite{arithmat}.
After recalling some definitions (\Cref{sec:preliminaries}), we introduce two concepts that are used later: the \emph{strong gcd property} (\Cref{sec:strong-gcd}), and a new operation on \mbox{(quasi-)arithmetic} matroids that we call \emph{reduction} (\Cref{sec:reduction}). Roughly speaking, the strong gcd property requires the multiplicity function to be uniquely determined by the multiplicity of the bases.
The idea behind the reduction operation is the following: given a central toric arrangement, we can quotient the ambient torus by the subgroup of translations globally fixing the arrangement. In the quotient, the equations of the initial arrangement describe a new toric arrangement. The initial arrangement defines an arithmetic matroid and the final one defines its reduction. Indeed, the reduction operation consistently changes the multiplicity function, so that the resulting \mbox{(quasi-)arithmetic} matroid is torsion-free and surjective (i.e.\ the multiplicity of the empty set and of the full groundset are both equal to one).
In \Cref{sec:representations} we dive into the representation problem for torsion-free arithmetic matroids, which is the heart of our work. We start by considering the surjective case: following ideas of \cite{Lenz2017,Pagaria2017}, we show that there is at most one representation, and we describe how to compute it. Then we turn to the general case of torsion-free arithmetic matroids. Here the representation need not be unique, and we describe how to compute all representations. A consequence of our algorithm is that a torsion-free arithmetic matroid $(E, \rk, m)$ of rank $r$ has at most $m(E)^{r-1}$ essential representations up to equivalence.
The problem of recognizing equivalent representations reduces to the computation of a normal form of integer matrices up to left-multiplication by invertible matrices and change of sign of the columns. We tackle this problem in \Cref{sec:shnf}, where we describe a polynomial-time algorithm to compute such a normal form. We call this the \emph{signed Hermite normal form}, by analogy with the classical Hermite normal form (which is a normal form up to left-multiplication by invertible matrices). The signed Hermite normal form is also implemented in Arithmat \cite{arithmat}.
In \Cref{sec:decomposition} we tackle a related algorithmic problem, namely finding the decomposition of a represented arithmetic matroid as a sum of indecomposable ones.
Finally, in \Cref{sec:applications} we describe a few applications of our software library Arithmat. We disprove two known conjectures about the poset of layers and the arithmetic independence poset of a toric arrangement:
we exhibit an arithmetic matroid with $13$ non-equivalent representations (i.e.\ central toric arrangements), whose associated posets are not Cohen-Macaulay, and therefore not shellable. As already noted in \cite{Pagaria2018}, the toric arrangements associated with a fixed arithmetic matroid can have different posets of layers (in the previous example, the $13$ toric arrangements give rise to $3$ different posets of layers). We conclude with the following open question: is the arithmetic independence poset of a toric arrangement uniquely determined by the associated arithmetic matroid?
\subsection*{Acknowledgments} We thank Alessio d'Alì, Emanuele Delucchi, and Ivan Martino for the useful discussions. This work was supported by the Swiss National Science Foundation Professorship grant PP00P2\_179110/1.
\section{Preliminaries} \label{sec:preliminaries}
In this section, we recall the basic definitions and properties of arithmetic matroids. The main references are \cite{OxleyBook, DAdderioMoci2013, BrandenMoci2014}. We define a matroid in terms of its rank function.
\begin{definition}
A \emph{matroid} is a pair $\mathcal M = (E, \rk)$, where $E$ is a finite set and $\rk \colon \mathcal{P}(E) \to \mathbb{N}$ is a function satisfying the following properties:
\begin{enumerate}
\item $\rk(X) \leq \lvert X \rvert$ for every $X \subseteq E$;
\item $\rk(X\cup Y) + \rk(X \cap Y) \leq \rk(X) + \rk(Y)$ for every $X,Y \subseteq E$;
\item $\rk(X) \leq \rk(X \cup \{e\}) \leq \rk(X)+1$ for every $X \subseteq E$ and $e \in E$.
\end{enumerate} \end{definition}
For every matroid $\mathcal M=(E,\rk)$ and for every subset $X \subseteq E$, we denote by $\mathcal M/X$ the \textit{contraction} of $X$ and by $\mathcal M \setminus X$ the \textit{deletion} of $X$ (see \cite[Section 1.3]{OxleyBook}). For $X\subseteq E$, denote by $\cl(X)$ the maximal subset $Y\supseteq X$ of rank equal to $\rk(X)$.
Let us recall the definition of arithmetic matroids, introduced in \cite{DAdderioMoci2013,BrandenMoci2014}.
\begin{definition}
\label{def:molecule}
A \textit{molecule} $(X,Y)$ of a matroid $\mathcal M$ is a pair of sets $X\subset Y \subseteq E$ such that the matroid $(\mathcal{M}/X)\setminus Y^c$ has a unique basis.
Equivalently, it is possible to write $Y = X \sqcup T \sqcup F$ in such a way that, for every $Z$ with $X \subseteq Z \subseteq Y$, we have $\rk(Z) = \rk(X) + |Z \cap F|$.
Here $F = Y \setminus \cl(X)$ is the unique basis of $(M/X) \setminus Y^c$, and $T = \cl(X) \cap Y \setminus X$ is the set of loops of $(M/X) \setminus Y^c$. \end{definition}
\begin{definition}
\label{def:arithmetic-matroid}
An \textit{arithmetic matroid} is a triple $M = (E,\rk,m)$, such that $(E,\rk)$ is a matroid and $m \colon \mathcal{P}(E) \to \mathbb{N}_+=\set{ 1,2,\dots}$ is a function satisfying:
\begin{enumerate}
\item[(A1)] for every $X \subseteq E$ and $e \in E$, if $\rk(X\cup \{e\}) = \rk(X)$ then $m(X \cup \{e\}) \mid m(X)$, otherwise $m(X) \mid m(X \cup \{e\})$;
\item[(A2)] \label{eq:cond_on_molecule} if $(X,Y)$ is a molecule, with $Y = X \sqcup T \sqcup F$ as in \Cref{def:molecule}, then
\[ m(X) \, m(Y) = m(X \cup T) \, m(X \cup F); \]
\item[(P)] if $(X,Y)$ is a molecule, then
\[ \sum_{X \subseteq S \subseteq Y} (-1)^{\left|X \cup F \right|-|S|} \, m(S) \geq 0. \]
\end{enumerate}
We call $m$ the \textit{multiplicity function}.
If $M=(E,\rk,m)$ only satisfies axioms (A1) and (A2), we say that $M$ is a \emph{quasi-arithmetic matroid}.
If $M$ satisfies only axiom (P), we say that it is a \emph{pseudo-arithmetic matroid}. \end{definition}
If $m(\emptyset)=1$, we said that the arithmetic matroid $(E, \rk, m)$ is \textit{torsion-free}. If $m(E)=1$, the matroid is \textit{surjective}.
Recall that any finitely generated abelian group $G$ has a (finite) torsion subgroup, which we denote by $\Tor(G)$, and a well-defined rank $\rk(G) \in \mathbb{N}$.
\begin{definition} A \textit{representation} of an arithmetic matroid $M=(E,\rk,m)$ is a finitely generated abelian group $G$ together with elements $(v_e)_{e \in E}$ such that for all $X \subseteq E$ we have: \begin{itemize} \item $\rk (X)= \rk (\langle v_e\rangle_{e \in X})$, \item $m(X)= \lvert \Tor(G/\langle v_e\rangle_{e \in X}) \rvert$, \end{itemize} where $\langle v_e\rangle_{e \in X}$ is the subgroup generated by $v_e$ for $e \in X$.
A representation is \textit{essential} if $\rk(G)=\rk(E)$. \end{definition}
Notice that, if $M$ is torsion-free, every representation is a collection of integer vectors $(v_e)_{e \in E}$ in a lattice $\Lambda$, i.e.\ a free finitely generated abelian group. Once a basis $\mathcal{B}$ of $\Lambda \simeq \mathbb{Z}^r$ is fixed, the vectors $(v_e)_{e \in E}$ can be identified with the columns of a matrix $A \in \mathsf{M}(r,|E|;\mathbb{Z})$. A different choice for the basis gives a matrix $A'=UA$ for some $U\in \GL(r;\mathbb{Z})$.
\begin{definition} Let $(G, (v_e)_{e \in E})$ and $(H, (w_e)_{e \in E})$ be two representations of an arithmetic matroid $M$. The two representations are \textit{equivalent} if there exists a group isomorphism $\varphi \colon G \to H$ such that $\varphi(\langle v_e \rangle)= \langle w_e \rangle$ for all $e\in E$. \end{definition}
Notice that $\varphi(v_e) \in \set{w_e, -w_e}$, hence $\varphi(\langle v_e \rangle_{e \in X})= \langle w_e \rangle_{e \in X}$ for all $X \subseteq E$.
A topological motivation for the previous definitions comes from the fact that a representation of a torsion-free arithmetic matroid is a central toric arrangement.
\begin{definition}[Central toric arrangement]
A central toric arrangement is a finite collection $\mathcal{A}$
of hypertori in a torus $T \cong (\mathbb{C}^*)^r$, for some $r > 0$. \end{definition}
Two representations are equivalent if and only if they describe isomorphic toric arrangements.
\section{The strong gcd property} \label{sec:strong-gcd}
In this section, we introduce and study the strong gcd property. This is a variant of the gcd property, which was introduced in \cite[Section 3]{DAdderioMoci2013}. The gcd property is satisfied by all representable torsion-free arithmetic matroid, see \cite[Remark 3.1]{DAdderioMoci2013}. The strong version is satisfied by all representable, surjective, and torsion-free arithmetic matroid, see \Cref{cor:representable-strong-gcd} below.
\begin{definition} An arithmetic matroid $M = (E, \rk, m)$ satisfies the \emph{gcd property} if, for every subset $X \subseteq E$,
\[ m(X) = \gcd \set{m(I) \mid I \subseteq X, \, \left| I \right| = \rk(I) = \rk (X)}. \] \end{definition}
\begin{definition}
\label{def:strong-gcd}
An arithmetic matroid $M = (E, \rk, m)$ satisfies the \emph{strong gcd property} if, for every subset $X \subseteq E$,
\[ m(X) = \gcd \set{m(B) \mid \text{$B$ basis and } \left|B \cap X \right| = \rk(X)}. \] \end{definition}
\begin{lemma}
\label{lemma:strong-implies-gcd}
Let $M$ be an arithmetic matroid.
If $M$ satisfies the strong gcd property, then it also satisfies the gcd property. \end{lemma}
\begin{proof}
For every independent set $I \subseteq E$, we have that
\[ m(I) = \gcd \set{ m(B) \mid \text{$B$ basis and } I \subseteq B}. \]
Then, for a generic subset $X \subseteq E$,
\begin{align*}
m(X) &= \gcd \set{ m(B) \mid \text{$B$ basis and } \left|B \cap X \right| = \rk(X)} \\
&= \gcd \big\{ \gcd \set{ m(B) \mid \text{$B$ basis and } B \cap X = I } \mid I \subseteq X \text{ and } \\
& \;\;\quad\qquad \left|I\right| = \rk(I) = \rk(X) \big\} \\
&\smash{\stackrel{(*)}{=}} \gcd \big\{ \gcd \set{ m(B) \mid \text{$B$ basis and } I \subseteq B } \mid I \subseteq X \text{ and } \\
& \;\;\quad\qquad \left|I\right| = \rk(I) = \rk(X) \big\} \\
&= \gcd \set{ m(I) \mid I \subseteq X \text{ and } \left|I\right| = \rk(I) = \rk(X) }.
\end{align*}
The equality $(*)$ follows from $|I| = \rk(I) = \rk(X) \geq \rk(B \cap X) = |B \cap X|$. \end{proof}
\begin{lemma}
\label{lemma:strong-gcd-dual}
Let $M$ be an arithmetic matorid.
If $M$ satisfies the strong gcd property, then its dual $M^*$ also satisfies the strong gcd property. \end{lemma}
\begin{proof}
Let $M = (E, \rk, m)$ and $M^* = (E, \rk^*, m^*)$.
For every subset $X \subseteq E$, we have
\begin{align*}
m^*(X^c) &= m(X) = \gcd \set{ m(B) \mid \text{$B$ basis of $M$ and } \left|B \cap X\right| = \rk(X)} \\
&\smash{\stackrel{(*)}{=}} \gcd \set{ m^*(B^c) \mid \text{$B^c$ is a basis of $M^*$ and } \left|B^c\cap X^c\right| = \rk^*(X^c) }.
\end{align*}
The equality $(*)$ follows from $|B^c \cap X^c| = |(B \cup X)^c| = |E| - (|B| + |X| - |B \cap X|) = |X^c| - |B| + |B \cap X| = |X^c| - \rk(E) + \rk(X) = \rk^*(X^c)$. \end{proof}
\begin{theorem}
\label{thm:strong-iff-gcd}
Let $M$ be an arithmetic matroid.
Then $M$ satisfies the strong gcd property if and only if both $M$ and $M^*$ satisfy the gcd property. \end{theorem}
\begin{proof}
If $M$ satisfies the strong gcd property, then the same is true for $M^*$ by Lemma \ref{lemma:strong-gcd-dual}, and therefore both $M$ and $M^*$ satisfy the gcd property by Lemma \ref{lemma:strong-implies-gcd}.
Conversely, suppose that $M$ and $M^*$ both satisfy the gcd property.
By the gcd property for $M$, for every $X \subseteq E$, we have
\begin{equation}
\label{eq:M-gcd}
m(X) = \gcd \set{ m(I) \mid I \subseteq X \text{ and } \left|I\right| = \rk(I) = \rk(X) }.
\end{equation}
By the gcd property for $M^*$, for every independent set $I \subseteq E$ we have
\begin{align*}
m(I) &= m^*(I^c) = \gcd \set{m^*(B^c) \mid B^c \subseteq I^c \text{ and } \left|B^c\right| = \rk^*(B^c) = \rk^*(I^c) } \\
&= \gcd \set{m(B) \mid I \subseteq B \text{ and } \left|B^c\right| = \rk^*(B^c) = \rk^*(I^c) }.
\end{align*}
The condition $\left|B^c\right| = \rk^*(B^c) = \rk^*(I^c)$ can be rewritten as $\left|B^c\right| = |B^c| - \rk(E) + \rk(B) = |I^c| - \rk(E) + \rk(I)$.
The first equality implies that $\rk(B) = \rk(E)$.
By the second equality, we obtain $|B^c| = |I^c| - \rk(E) + |I| = |E| - \rk(E)$, thus $|B| = \rk(E)$.
Therefore $B$ is a basis.
Then
\begin{equation}
\label{eq:Mdual-gcd}
m(I) = \gcd \set{m(B) \mid I \subseteq B \text{ and $B$ is a basis}}.
\end{equation}
In particular, if $I \subseteq X \subseteq E$ and $|I| = \rk(I) = \rk(X)$, then $\rk(I) \leq \rk(B \cap X) \leq \rk(X)$ and therefore $|B \cap X| = \rk(B \cap X) = \rk(X)$.
Putting together \cref{eq:M-gcd,eq:Mdual-gcd}, we finally obtain
\[ m(X) = \gcd \set{m(B) \mid \text{$B$ basis and } \left|B \cap X \right| = \rk(X)}. \]
This proves the strong gcd property for $M$. \end{proof}
\begin{corollary}
\label{cor:representable-strong-gcd}
Let $M$ be a surjective, torsion-free, and representable arithmetic matroid.
Then $M$ satisfies the strong gcd property. \end{corollary}
\begin{proof}
By \cite[Remark 3.1]{DAdderioMoci2013}, a torsion-free representable arithmetic matroid satisfies the gcd property.
In particular, this applies to $M$.
Since $M$ is surjective and representable, its dual $M^* = (E, \rk^*, m^*)$ is torsion-free and representable, and thus it also satisfies the gcd property.
By Theorem \ref{thm:strong-iff-gcd}, we deduce that $M$ satisfies the strong gcd property. \end{proof}
As a final remark, notice that the strong gcd property is not preserved under deletion or contraction.
\section{Reduction of quasi-arithmetic matroids} \label{sec:reduction}
In this section, we introduce a new operation on quasi-arithmetic matroids, which we call \emph{reduction}. We will use this construction in the algorithm that computes the representations of a torsion-free arithmetic matroid.
\begin{definition}[Reduction]
Let $M = (E, \rk, m)$ be a quasi-arithmetic matroid.
Its reduction is the quasi-arithmetic matroid $\overline M = (E, \rk, \overline m)$ on the same groundset, with the same rank function, and with multiplicity function $\overline m$ is given by
\[ \overline m(X) = \frac{\gcd \set{ m(B) \mid \text{$B$ is a basis, and } \rk(X) = \left|X \cap B\right| }}{\gcd \set{ m(B) \mid \text{$B$ is a basis} }}. \] \end{definition}
Given a matroid $\mathcal{M} = (E, \rk)$ and two subsets $X,Y \subseteq E$, define \[ \begin{split}
\mathcal{B}_{(X,Y)} = \{\, (B_1, B_2) \mid \; & \text{$B_1$ and $B_2$ are bases of $\mathcal{M}$, } \rk(X) = |X \cap B_1|, \\
& \text{ and } \rk(Y) = |Y \cap B_2| \, \}. \end{split} \]
\begin{lemma}
\label{lemma:bijection}
Let $\mathcal{M} = (E, \rk)$ be a matroid, and let $(X,Y)$ be a molecule with $Y = X \sqcup T \sqcup F$ as in \Cref{def:molecule}.
Then there is a bijection $\varphi \colon \mathcal{B}_{(X,Y)} \to \mathcal{B}_{(X \sqcup T, \, X \sqcup F)}$ given by
\[ \varphi(B_1, B_2) = \big( (B_1 \setminus X) \cup (B_2 \cap (X \cup T)), \,
(B_2 \setminus (X \cup T)) \cup (B_1 \cap X) \big). \] \end{lemma}
\begin{proof}
Notice that $F \subseteq B_2$, because $\rk(Y) = \rk(Y \cap B_2) = \rk(X) + |B_2 \cap F|$ (the first equality is by definition of $\mathcal{B}_{(X,Y)}$, and the second equality is by definition of molecule).
We want to prove that $B_3 = (B_1 \setminus X) \cup (B_2 \cap (X \cup T))$ is a basis.
The set $B_1 \setminus X$ is independent, and its rank (or cardinality) is equal to $|B_1| - |X \cap B_1| = \rk(E) - \rk(X)$ by definition of $\mathcal{B}_{(X,Y)}$.
The set $B_2 \cap (X \cup T)$ is also independent, and since $F \subseteq B_2$ its rank (or cardinality) is equal to $|B_2 \cap Y| - |F| = \rk(X) + |F| - |F| = \rk(X)$.
Therefore $|B_3| \leq \rk(E)$.
Applying property (2) of the rank function to the pair $(B_3, X \cup T)$, we obtain
\[ \rk(B_3) + \rk(X \cup T) \geq \rk(B_3 \cup X \cup T) + \rk(B_3 \cap (X \cup T)). \]
Notice that $\rk(X \cup T) = \rk(X)$ (by definition of molecule), $B_1 \subseteq B_3 \cup X \cup T$, and $B_2 \cap (X \cup T) \subseteq B_3 \cap (X \cup T)$.
Then
\[ \rk(B_3) + \rk(X) \geq \rk(B_1) + \rk(B_2 \cap (X \cup T)) = \rk(E) + \rk(X). \]
Therefore $\rk(B_3) \geq \rk(E)$, and $B_3$ is a basis.
We want now to check that $|B_3 \cap (X \cup T)| = \rk(X \cup T)$.
We have $B_1 \cap T = \emptyset$, because
\begin{align*}
\rk(X) + |T \cap B_1| &= |X \cap B_1| + |T \cap B_1| = |(X \cap B_1) \sqcup (T \cap B_1)| \\
&= |(X \cup T) \cap B_1| = \rk((X \cup T) \cap B_1) \\
&\leq \rk(X \cup T) = \rk(X).
\end{align*}
Thus $B_3 \cap (X \cup T) = B_2 \cap (X \cup T)$, and this set has cardinality $\rk(X) = \rk(X \cup T)$.
Similarly, $B_4 = (B_2 \setminus (X \cup T)) \cup (B_1 \cap X)$ is a basis, and $B_4 \cap (X \cup F)| = \rk(X \cup F)$.
Therefore the map $\varphi$ is well-defined.
The map $\psi \colon \mathcal{B}_{(X \sqcup T, \, X \sqcup F)} \to \mathcal{B}_{(X,Y)}$ defined by
\[ \psi(B_3, B_4) = \big( (B_3 \setminus (X \cup T)) \cup (B_4 \cap X), \, (B_4 \setminus X) \cup (B_3 \cap (X \cup T)) \big) \]
can be verified to be the inverse of $\varphi$.
Therefore $\varphi$ is a bijection. \end{proof}
\begin{lemma}
\label{lemma:product}
Let $M = (E, \rk, m)$ be a quasi-arithmetic matroid, and let $(X,Y)$ be a molecule with $Y = X \sqcup T \sqcup F$ as in \Cref{def:molecule}.
If $\varphi \colon \mathcal{B}_{(X,Y)} \to \mathcal{B}_{(X \sqcup T, \, X \sqcup F)}$ is the bijection of \Cref{lemma:bijection}, and $(B_3, B_4) = \varphi(B_1, B_2)$, then
\[ m(B_1) \, m(B_2) = m(B_3) \, m(B_4). \] \end{lemma}
\begin{proof}
Consider the following four molecules:
\begin{align*}
& (B_1 \cap X, \, (B_2 \cap (X \cup T)) \cup B_1); \\
& (B_2 \cap (X \cup T), \, (B_1 \cap X) \cup B_2); \\
& (B_2 \cap (X \cup T), \, (B_2 \cap (X \cup T)) \cup B_1); \\
& (B_1 \cap X,\, (B_1 \cap X) \cup B_2).
\end{align*}
Applying axiom (A2) to these molecules, we get the following relations (we use the fact that $B_1 \cap T = \emptyset$, shown in the proof of \Cref{lemma:bijection}):
\begin{align}
& m(B_1 \cap X) \, m((B_2 \cap (X \cup T)) \cup B_1) = m((B_1 \cup B_2) \cap (X\cup T)) \, m(B_1); \\
& m(B_2 \cap (X \cup T)) \, m((B_1 \cap X) \cup B_2) = m((B_1 \cup B_2) \cap (X\cup T)) \, m(B_2); \\
& m(B_2 \cap (X \cup T)) \, m((B_2 \cap (X \cup T)) \cup B_1) = m((B_1 \cup B_2) \cap (X\cup T)) \, m(B_3); \\
& m(B_1 \cap X) \, m((B_1 \cap X) \cup B_2) = m((B_1 \cup B_2) \cap (X\cup T)) \, m(B_4).
\end{align}
Let $k = m((B_1 \cup B_2) \cap (X\cup T))$.
Multiplying the previous equations in pairs, we obtain $k^2 \, m(B_1) \, m(B_2) = k^2 \, m(B_3) \, m(B_4)$. Hence $m(B_1) \, m(B_2) = m(B_3) \, m(B_4)$. \end{proof}
\begin{theorem}
The reduction $\overline M$ of a quasi-arithmetic matroid $M = (E, \rk, m)$ is a torsion-free surjective quasi-arithmetic matroid, and it satisfies the strong gcd property. \end{theorem}
\begin{proof}
Let $d = \gcd \set{ m(B) \mid \text{$B$ is a basis} }$.
We start by checking axiom (A1) of \Cref{def:arithmetic-matroid}.
Consider a subset $X\subseteq E$ and an element $e \in E$.
\begin{itemize}
\item If $\rk(X \cup \{e\}) = \rk(X)$, then a basis $B$ such that $\rk(X) = \rk(X \cap B)$ also satisfies $\rk(X \cup \{e\}) = \rk((X \cup \{e\}) \cap B)$.
Therefore $d \cdot \overline m(X \cup \{e\}) \mid d \cdot \overline m(X)$.
\item Similarly, if $\rk(X \cup \{e\}) = \rk(X) + 1$, then a basis $B$ such that $\rk(X \cup \{e\}) = \rk((X \cup \{e\}) \cap B)$ also satisfies $\rk(X) = \rk(X \cap B)$.
Therefore $d \cdot \overline m(X) \mid d \cdot \overline m(X \cup \{e\})$.
\end{itemize}
We now check axiom (A2).
Let $(X,Y)$ be a molecule, with $Y = X \sqcup T \sqcup F$ as in \Cref{def:molecule}.
By definition of $\overline m$, we have that
\[ d^2 \, \overline m(X) \, \overline m(Y) = \gcd \set{ m(B_1) \, m(B_2) \mid (B_1, B_2) \in \mathcal{B}_{(X,Y)} }. \]
Similarly,
\[ d^2 \, \overline m(X \cup T) \, \overline m(X \cup F) = \gcd \set{ m(B_3) \, m(B_4) \mid (B_3, B_4) \in \mathcal{B}_{(X \cup T, \, X \cup F)} }. \]
By \Cref{lemma:bijection,lemma:product}, we obtain $d^2 \, \overline m(X) \, \overline m(Y) = d^2 \, \overline m(X \cup T) \, \overline m(X \cup F)$, hence $\overline m(X) \, \overline m(Y) = \overline m(X \cup T) \, \overline m(X \cup F)$.
Therefore $\overline M$ is a quasi-arithmetic matroid.
By definition of $\overline m$, we also have that $\overline m(\emptyset) = \overline m(E) = 1$, i.e.\ $M$ is torsion-free and surjective.
It is also immediate to check that $\overline M$ satisfies the strong gcd property. \end{proof}
It is not true in general that the reduction of an arithmetic matroid is an arithmetic matroid. We see this in the following example.
\begin{example}
Let $\mathcal{M} = (E,\rk)$ be the uniform matroid of rank $2$ on the groundset $E=\{1,2,\dots,6\}$.
Consider the multiplicity function $m \colon \mathcal{P}(E) \to \mathbb{N}_+$ defined as
\begin{align*}
& m(\emptyset)=1, & & \\
& m(\{1\})=m(\{2\})=2, & & \\
& m(\{ j \})=1 & & \text{if } j>2, \\
& m(\{ X\})=1 & & \text{if } |X \cap \{3, \ldots, 6\}|\geq 2, \\
& m(\{ i,j\})=2 & &\text{if } i=1,2 \text{ and } j>2, \\
& m(\{ 1,2 \})=4, & & \\
& m(\{ 1,2,3\})=1, & & \\
& m(\{ 1,2,j\})=2 & & \text{if } j>3.
\end{align*}
Then $M = (E, \rk, m)$ is an arithmetic matroid (this can be checked using the software library Arithmat \cite{arithmat}).
We have that $\overline m(X) = m(X)$ for every $X \subseteq E$, except that $\overline m(1,2,3) = 2$.
The quasi-arithmetic matroid $\overline M = (E, \rk, \overline m)$ does not satisfy axiom (P) for the molecule $(\{1,2\}, E)$. \end{example}
However, the reduction of a representable arithmetic matroid turns out to be a representable arithmetic matroid.
\begin{theorem}
\label{thm:reduction-rep}
If $M = (E, \rk, m)$ is a representable arithmetic matroid, then its reduction $\overline M$ is also a representable arithmetic matroid. \end{theorem}
\begin{proof}
Let $(v_e)_{e\in E} \subseteq G$ be a representation of $M$.
Denote by $K$ the quotient of $G$ by its torsion subgroup $T$.
Let $\overline G$ be the sublattice of $K$ generated by $\set{ \bar v_e \mid e \in E }$, where $\bar v_e$ is the class of $v_e$ in $K$.
We are going to show that $(\bar v_e)_{e \in E} \subseteq \overline G$ is a representation of $\overline M$.
Let $M' = (E, \rk, m')$ be the arithmetic matroid associated with the representation $(\bar v_e)_{e \in E} \subseteq \overline G$.
By construction, $M'$ is representable, torsion-free (because $\overline G$ is torsion-free), and surjective (because the vectors $\bar v_e$ generate $\overline G$).
Therefore, by \Cref{cor:representable-strong-gcd}, it satisfies the strong gcd property.
As a consequence,
\[ \gcd \set{ m'(B) \mid \text{$B$ basis} } = m(E) = 1. \]
Let $B$ be a basis of $M$.
Since $B$ is independent, we have that $T \cap \langle v_b \rangle_{b \in B} = \set{0}$.
Then,
\begin{align*}
m(B) &= \left| \faktor{G}{\langle v_b \rangle_{b \in B}} \right| = |T| \cdot \left| \faktor{K}{\langle \bar v_b \rangle_{b \in B}} \right| = |T| \cdot \left| \faktor{K}{\overline G} \right| \cdot \left| \faktor{\overline G}{\langle \bar v_b \rangle_{b \in B}} \right| \\
&= |T| \cdot \left| \faktor{K}{\overline G} \right| \cdot m'(B).
\end{align*}
If $B$ varies among all bases of $M$, taking the gcd of both sides we get
\[ \gcd \set{ m(B) \mid \text{$B$ basis} } = |T| \cdot \left| \faktor{K}{\overline G} \right|. \]
Therefore
\[ m'(B) = \frac{m(B)}{ \gcd \set{ m(B) \mid \text{$B$ basis} } } = \overline m(B). \]
Since both $M'$ and $\overline M$ satisfy the strong gcd property, $m'(X) = \overline m(X)$ for every subset $X \subseteq E$.
This means that $\overline M = M'$ is representable. \end{proof}
Finally, notice that the reduction does not commute with deletion and contraction. However, it commutes with taking the dual.
\section{Representations of arithmetic matroids} \label{sec:representations}
In this section, we prove that a torsion-free arithmetic matroid $M = (E, \rk, m)$ of rank $r$ has at most $m(E)^{r-1}$ essential representations, up to equivalence. At the same time, we describe an algorithm to list all such essential representations.
Callegaro and Delucchi showed that matroids with a unimodular basis admit at most one representation \cite{CallegaroDelucchi2017}. This result was later generalized by Lenz, in the case of weakly multiplicative matroids \cite{Lenz2017}. The first author proved the uniqueness of the representation for surjective matroids and showed that general torsion-free matroids admit at most $m(E)^r$ essential representations \cite{Pagaria2017}. In this work, we describe how to explicitly construct all representations, and improve the upper bound.
\subsection{Representation of torsion-free surjective matroids} Consider a torsion-free surjective arithmetic matroid $M = (E, \rk, m)$ of rank $r$. We want to describe how to choose $n = \vert E \vert$ vectors $(v_e)_{e \in E}$ in $\mathbb{Z}^r$ that form a representation of $M$ in the lattice $\Lambda = \langle v_e \mid e \in E \rangle_\mathbb{Z}$, if $M$ is representable.
Let $B \subseteq E$ be a basis of $M$. Relabel the groundset $E$ so that $E = \set{1, 2, \dots, n}$ and $B = \set{1, 2, \dots, r}$. For $i=1,\dots, r$, define $v_i = m(B) \, e_i$ where $(e_1, \dots, e_r)$ is the canonical basis of $\mathbb{Z}^r$. The absolute values of the coordinates of $v_{r+1}, \dots, v_n$ are uniquely determined by $M$, as described in \cite{Pagaria2017}. The entries $a_{ij}$ of the matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$ with columns $v_1, \dots, v_n$ satisfy
\[ |a_{ij}| =
\begin{cases}
m(B \setminus \{i\} \cup \{j\}) & \text{if $B \setminus \{i\} \cup \{j\}$ is a basis}; \\
0 & \text{otherwise}.
\end{cases} \]
To determine the signs of the entries $a_{ij}$, we follow the idea of Lenz \cite{Lenz2017}. Consider the bipartite graph $G$ on the vertex set $E = B \sqcup (E \setminus B)$, having an edge $(i,j)$ whenever $i \in B$, $j \in E \setminus B$, and $B \setminus \{i\} \cup \{j\}$ is a basis. Let $F$ be a spanning forest of $G$. Since reversing the sign of some vectors does not change the equivalence class of a representation, we can set $a_{ij}$ to be positive for $(i,j) \in F$ as shown by Lenz \cite[Lemma~6]{Lenz2017}. We determine the signs of the remaining entries $a_{ij}$ by iterating the following procedure.
\begin{enumerate}
\item Let $(i,j)$ be an edge of $G \setminus F$ such that the distance between $i$ and $j$ in $F$ is minimal.
\item Let $i_1, j_1, i_2, j_2, \dots, i_k, j_k$ be a minimal path from $i$ to $j$ in $F$, where $i_1=i$ and $j_k=j$.
Consider the $k\times k$ minor $A'$ of the matrix $A$ indexed by the rows $i_1,\dots, i_k$ and the columns $j_1, \dots, j_k$.
Notice that the signs of all entries of $A'$ have already been determined, except for $a_{ij}$.
The absolute value of the determinant of $A'$ must be equal to
\[ | \det A'| =
\begin{cases}
m(B)^{k-1} \cdot m(B') & \text{if $B'$ is a basis} \\
0 & \text{otherwise}
\end{cases}
\]
where $B' = B \setminus \set{i_1, \dots, i_k} \cup \set{j_1, \dots, j_k}$.
By minimality of the distance between $i$ and $j$, the only non-zero entries of $A'$ are $a_{i_\ell \, j_\ell}$ and $a_{i_\ell \, j_{\ell-1}}$ for $\ell=1,\dots, k$ (where $j_0=j_k$).
Then
\[ |\det A'| = \left| \prod_{\ell=1}^k a_{i_\ell \, j_\ell} - (-1)^k \prod_{\ell=1}^k a_{i_\ell \, j_{\ell-1}} \right|. \]
Comparing the two given expressions of $|\det A'|$, the sign of $a_{ij}$ can be uniquely determined.
\item Add the edge $(i,j)$ to $F$. \end{enumerate}
At some iteration of this procedure, the equation
\[ \left| \prod_{\ell=1}^k a_{i_\ell \, j_\ell} - (-1)^k \prod_{\ell=1}^k a_{i_\ell \, j_{\ell-1}} \right| \; = \; \begin{cases}
m(B)^{k-1} \cdot m(B') & \text{if $B'$ is a basis} \\
0 & \text{otherwise}
\end{cases} \] of the second step might have no solution. If this happens, we can conclude that the matroid $M$ is not representable.
\begin{remark}
\label{rmk:orientability}
If $M$ is orientable (in the sense of \cite{PagariaOAM}), then there exists a \emph{chirotope} $\chi\colon E^r \to \set{-1, 0, 1}$ such that $a_{ij} = \chi(B \setminus \{i\} \cup \{j\}) \cdot m(B \setminus \{i\} \cup \{j\})$.
This ensures that the equation of step (2) always has a solution.
Conversely, a failure of step (2) implies that $M$ is not orientable. \end{remark}
We have finally constructed a matrix $A$ whose columns $(v_e)_{e \in E}$ form a candidate representation of $M$ in the lattice $\Lambda = \langle v_e \mid e \in E \rangle_\mathbb{Z}$.
To recover the coordinates of the vectors $(v_e)_{e \in E}$ with respect to a basis of $\Lambda$, we use the Smith normal form as explained by the following lemma.
\begin{lemma}
Let $(v_e)_{e \in E}$ be a set of vectors in $\mathbb{Z}^r$, with coordinates described by a matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$ of rank $r$.
Let $D = UAV$ be the Smith normal form of $A$.
Then the $r \times n$ matrix consisting of the first $r$ rows of $V^{-1}$ gives the coordinates of the vectors $(v_e)_{e \in E}$ with respect to a basis of $\Lambda = \langle v_e \mid e \in E \rangle_\mathbb{Z}$. \end{lemma}
\begin{proof}
Recall that $U \in \GL(r;\mathbb{Z})$, $V \in \GL(n;\mathbb{Z})$, and $D \in \mathsf{M}(r, n; \mathbb{Z})$.
Let $D = D'I$, where $I$ is the block matrix $(\Id_{r \times r} \mid 0) \in \mathsf{M}(r,n;\mathbb{Z})$ and $D' \in \mathsf{M}(r,r;\mathbb{Z})$ is the matrix consisting of the first $r$ columns of $D$.
Consider the vectors $w_1, \dots, w_r$ of $\mathbb{Z}^r$ given by the columns of $U^{-1}D'$.
Then $U^{-1}D' \cdot IV^{-1} = A$, and therefore the columns of $IV^{-1}$ are the coordinates of $(v_e)_{e \in E}$ with respect to the $\mathbb{Q}$-basis $\mathcal{B} = (w_1, \dots, w_r)$.
Since the matrix $IV^{-1}$ has integer entries and Smith normal form equal to $I$, the basis $\mathcal{B}$ is also a lattice basis of $\Lambda$.
We conclude by noticing that $IV^{-1}$ is the matrix consisting of the first $r$ rows of $V^{-1}$. \end{proof}
At this point, we have a candidate representation of the matroid $M$. If $M$ is representable, this is the only possible representation of $M$ up to equivalence. We only need to verify if it is indeed a representation of $M$, checking the multiplicity $m(X)$ for every subset $X \subseteq E$.
\begin{remark}
Under our assumptions ($m(\emptyset) = m(E) = 1$), the matroid $M$ is representable if and only if it is orientable and satisfies the strong gcd property \cite[Proposition~8.3]{PagariaOAM}.
Before the final check of our algorithm, $M$ is known to be orientable by Remark \ref{rmk:orientability}.
Then the final check has a positive result if and only if $M$ satisfies the strong gcd property. \end{remark}
\subsection{Representations of general torsion-free matroids}
In this section, we describe how to construct all essential representations (up to equivalence) of a general torsion-free matroid $M = (E, \rk, m)$. Let $r = \rk(E)$.
Consider the reduction $\overline M$. If $M$ is representable, then $\overline M$ must also be a representable arithmetic matroid by \Cref{thm:reduction-rep}. Since $\overline M$ is torsion-free and surjective, using the algorithm of the previous section we can check if $\overline M$ is a representable arithmetic matroid. Assume from now on that this is the case. Then the previous algorithm also yields the unique essential representation of $\overline M$ (up to equivalence), which consists of some integer matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$.
\begin{theorem}
\label{thm:representations}
If $A \in \mathsf{M}(r,n;\mathbb{Z})$ is an essential representation of $\overline M$, then every essential representation of $M$ is equivalent to $HA$ for some matrix $H \in \mathsf{M}(r,r;\mathbb{Z})$ in Hermite normal form, with $\det(H) = m(E)$. \end{theorem}
\begin{proof}
Every essential representation $C \in \mathsf{M}(r,n;\mathbb{Z})$ of $M$ induces an essential representation $A' \in \mathsf{M}(r,n;\mathbb{Z})$ of $\overline M$, as shown in the proof of \Cref{thm:reduction-rep}.
These two representations are related as follows: $C' = H'A'$, where the matrix $H' \in \mathsf{M}(r,r;\mathbb{Z})$ describes (in the chosen coordinates) the inclusion $\overline G \hookrightarrow K$, and has rank $r$.
Since all representations of $\overline M$ are equivalent, we can write $A' = U' A S$ for some integer matrices $U' \in \GL(r;\mathbb{Z})$ and $S \in \mathbb{Z}_2^n \subseteq \GL(n,\mathbb{Z})$.
Then we have $CS = H'U'A$.
Let $U \in \GL(r, \mathbb{Z})$ be an integer matrix such that $UH'U'$ is in Hermite normal form.
We obtain that the representation $UCS$ of $M$ is equivalent to $C$ and can be written as $UCS = HA$, where $H=UH'U'$ is in Hermite normal form.
Notice that $m(E) = \det(H) \cdot \overline m(E) = \det(H)$. \end{proof}
Some of the representations given by \Cref{thm:representations} can be equivalent. To compute a list of representatives of the equivalence classes of representations, one needs to compute a normal form of matrices in $\mathsf{M}(r,n;\mathbb{Z})$ up to left-multiplication by $\GL(r;\mathbb{Z})$ and change of sign of the columns. We develop an algorithm to do this in the next section.
A direct consequence of \Cref{thm:representations} is a new upper bound on the number of non-equivalent representations of a torsion-free matroid.
\begin{corollary}
Every torsion-free arithmetic matroid $M = (E, \rk, m)$ of rank $r$ has at most $m(E)^{r-1}$ equivalence classes of essential representations. \end{corollary}
\begin{proof}
Reorder the groundset of $M$ so that the first $r$ elements form a basis.
Let $A \in \mathsf{M}(r,n;\mathbb{Z})$ be an essential representation of the reduction $\overline M = (E, \rk, \overline m)$.
Without loss of generality, we can assume that $A$ is in Hermite normal form.
Let $B_i$ be the $i\times i$ leading principal minor of a matrix $B$.
For every $i \in \set{1, \dots, r}$, we have that $A_i$ is upper triangular and $\det(A_i) = \overline m(\set{1,\dots, i})$.
If $H \in \mathsf{M}(r,r,\mathbb{Z})$ is an upper triangular matrix such that $HA$ is a representation of $M$, then $\det(H_i) \det(A_i) = \det((HA)_i) = m(\set{1,\dots,i})$.
By \Cref{thm:representations}, every essential representation of $M$ is equivalent to $HA$ for some matrix $H \in \mathsf{M}(r,r,\mathbb{Z})$ in Hermite normal form such that $\det(H) = m(E)$.
The diagonal entries $d_1, \dots, d_r$ of $H$ are uniquely determined by the previous relations.
The number of such matrices $H$ is $\prod_{i=1}^r d_i^{i-1} \leq \prod_{i=1}^r d_i^{r-1} = m(E)^{r-1}$.
\end{proof}
\begin{remark}
The orientability of $M$ is equivalent to the orientability of the reduction $\overline M$.
Then \Cref{rmk:orientability} yields an algorithm to check the orientability of $M$. \end{remark}
\section{Signed Hermite normal form} \label{sec:shnf}
In this section, we describe an algorithm that takes as input a matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$ and outputs a normal form with respect to the action of $\GL(r;\mathbb{Z}) \times \mathbb{Z}_2^n$. Here $\GL(r;\mathbb{Z})$ acts on $\mathsf{M}(r,n;\mathbb{Z})$ by left-multiplication, and the $j$-th standard generator of $\mathbb{Z}_2^n$ acts by changing the sign of the $j$-th column. It is convenient to view the elements of $\mathbb{Z}_2^n$ as the $n\times n$ diagonal matrices with diagonal entries equal to $\pm 1$. Then a pair $(U, S) \in \GL(r;\mathbb{Z}) \times \mathbb{Z}_2^n$ acts on $\mathsf{M}(r,n;\mathbb{Z})$ as $A \mapsto UAS$.
Recall that the (left) Hermite normal form is a canonical form for matrices in $\mathsf{M}(r,n;\mathbb{Z})$ with respect to the left action of $\GL(r;\mathbb{Z})$ (see for instance \cite{newman1972integral} and \cite{cohen1993course}). We write $\HNF(A)$ for the Hermite normal form of $A$. A matrix in Hermite normal form satisfies the following properties: \begin{itemize}
\item it is an upper triangular $r\times n$ matrix, and zero rows are located below non-zero rows;
\item the pivot (i.e.\ the first non-zero entry) of a non-zero row is positive, and is strictly to the right of the pivot of the row above it;
\item the elements below pivots are zero, and the elements above a pivot $q$ are non-negative and strictly smaller than $q$. \end{itemize} Our normal form with respect to the action of $\GL(r;\mathbb{Z}) \times \mathbb{Z}_2^n$ has a simple definition in terms of the Hermite normal form. We call it the \emph{signed Hermite normal form}.
\begin{definition}
The \emph{signed Hermite normal form} $\SHNF(A)$ of a matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$ is the lexicographically minimal matrix in the set $\set{ \HNF(AS) \mid S \in \mathbb{Z}_2^n }$.
To compare two matrices lexicographically, we look at the columns from left to right, and in each column, we look at the entries from bottom to top. \end{definition}
\begin{remark}
By definition, a matrix in signed Hermite normal form is also in Hermite normal form. \end{remark}
\begin{example}
Consider the following sequence of $2\times 2$ matrices:
\[
\begin{pmatrix}
4 & 2 \\
0 & 3
\end{pmatrix}
\longrightarrow
\begin{pmatrix}
4 & -2 \\
0 & -3
\end{pmatrix}
\longrightarrow
\begin{pmatrix}
4 & 1 \\
0 & 3
\end{pmatrix}.
\]
The leftmost matrix is in Hermite normal form, but it is not in signed Hermite normal form.
Indeed, if we change the sign of the second column, we obtain the matrix in the middle; its Hermite normal form is given by the rightmost matrix, which is lexicographically smaller than the leftmost one. \end{example}
A naive algorithm to compute the signed Hermite normal form could be: try all the $2^n$ elements $S \in \mathbb{Z}_2^n$; determine the left Hermite normal form of $AS$; choose the lexicographically minimal result. This algorithm runs in $2^n \cdot \text{poly}(n,r)$. In the rest of this section, we are going to describe an algorithm which is polynomial in $n$ and $r$.
Given a matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$, we indicate by $A_j \in \mathbb{Z}^r$ the $j$-th column of $A$, and by $A_{:j} \in \mathsf{M}(r,j;\mathbb{Z})$ the matrix consisting of the first $j$ columns of $A$. We write $\mathbb{Z}_2^j$ for the subgroup of $\mathbb{Z}_2^n$ generated by the first $j$ standard generators of $\mathbb{Z}_2^n$. Also, for every $m \leq r$, we regard the group $\GL(m;\mathbb{Z})$ as a subgroup of $\GL(r;\mathbb{Z})$ via the natural inclusion \[ U \mapsto \blockmatrix{U}{0}{0}{I_{(r-m)\times (r-m)}}. \] Define the stabilizer $\Stab(B)$ of a matrix $B \in \mathsf{M}(r,j;\mathbb{Z})$ as the subgroup \[ \Stab(B) = \set{ S \in \mathbb{Z}_2^{j} \mid BS = UB \text{ for some } U \in \GL(r;\mathbb{Z}) } \subseteq \mathbb{Z}_2^j. \] This is the stabilizer of the orbit $\set{ U B \mid U \in \GL(r;\mathbb{Z})}$ with respect to the right action of $\mathbb{Z}_2^j$. Notice that $\Stab(B) = \Stab(VBT)$ for every $(V,T) \in \GL(r;\mathbb{Z}) \times \mathbb{Z}_2^{j}$, because $\mathbb{Z}_2^j$ is abelian.
The pseudocode to compute the signed Hermite normal form is given in \Cref{alg:s-normal-form}. In the rest of this section, we are going to explain it with more details.
\begin{algorithm}
\caption{Signed Hermite normal form}
\label{alg:s-normal-form}
\textbf{Input}: a matrix $A \in \mathsf{M}(r,n; \mathbb{Z})$.\\
\textbf{Output}: the signed Hermite normal form of $A$.
\begin{algorithmic}[1]
\State $G \gets \set{0}$, as a subgroup of $\mathbb{Z}_2^n$
\State $A \gets$ Hermite normal form of $A$ \label{line:hnf-A}
\For{$j=1, 2, \dotsc, n$}\label{line:outer-loop}
\State $m \gets \rk(A_{: j-1})$\label{line:rank}
\State $q \gets A_{m+1, j}$\label{line:pivot}
\State $\varphi \gets$ the group homomorphism $G \to \GL(m;\mathbb{Z}) \subseteq \GL(r;\mathbb{Z})$ which maps $S \in G$ to the unique matrix $\varphi(S) \in \GL(m;\mathbb{Z})$ such that $\varphi(S)A_{:j-1}S=A_{:j-1}$\label{line:phi}
\State $G \gets G \times \mathbb{Z}_2$, where $\mathbb{Z}_2$ is the $j$-th factor of $\mathbb{Z}_2^n$\label{line:augment-G}
\State Extend $\varphi$ to a group homomorphism $G \to \GL(m;\mathbb{Z}) \subseteq \GL(r;\mathbb{Z})$, by sending the generator of the new $\mathbb{Z}_2$ factor to $-I_{m\times m}$\label{line:extend-phi}
\For{$i=m, m-1,\dotsc, 1$}\label{line:inner-loop}
\State $O \gets \set{ (\varphi(S) A_j)_{i} \!\mod q | S \in G }$\label{line:orbit}
\State $u \gets \min O$\label{line:min-orbit}
\State $\overline S \gets$ any element of $G$ such that $(\varphi(\overline S) A_j)_i \!\mod q = u$\label{line:S-min-orbit}
\State $A \gets$ Hermite normal form of $A \overline S$\label{line:HNF-inner-loop}
\State $G \gets \set{S \in G | (\varphi(S)A_j)_i \mmod q = u }$\label{line:restrict-G}
\State $\varphi \gets \varphi |_G$\label{line:restrict-phi}
\EndFor\label{line:end-inner-loop}
\EndFor
\State \Return $A$
\end{algorithmic} \end{algorithm}
Throughout the execution of \Cref{alg:s-normal-form}, $G$ is always a subgroup of $\mathbb{Z}_2^n$. It would require exponential time and space to compute and store the list of all its elements. For this reason, we rather describe it by giving one of its $\mathbb{Z}_2$-bases, i.e.\ a list of $k$ linearly independent vectors in $\mathbb{Z}_2^n$ (where $k$ is the dimension of $G$ as a $\mathbb{Z}_2$-vector space). Accordingly, the group homomorphism $\varphi \colon G \to \GL(r;\mathbb{Z})$ is always described by giving its values on the $\mathbb{Z}_2$-basis of $G$.
\Cref{alg:s-normal-form} adjusts the columns one at a time, from left to right. This is possible thanks to the following observation.
\begin{lemma}
For every $j$ we have $\SHNF(A)_{:j} = \SHNF(A_{:j})$. In particular, the $j$-th column of $\SHNF(A)$ only depends on the first $j$ columns of $A$. \end{lemma}
\begin{proof}
It is well known that $\HNF(A)_{:j} = \HNF(A_{:j})$.
Therefore
\begin{align*}
\SHNF(A)_{:j} &= \min \set{\HNF(AS) \mid S \in \mathbb{Z}_2^n}_{:j} \\
&= \min \set{\HNF(AS)_{:j} \mid S \in \mathbb{Z}_2^n} \\
&= \min \set{\HNF((AS)_{:j}) \mid S \in \mathbb{Z}_2^n} \\
&= \min \set{\HNF(A_{:j}T) \mid T \in \mathbb{Z}_2^j} \\
&= \SHNF(A_{:j}).
\end{align*}
In the second equality we used the fact that the lexicographic order privileges the first $j$ columns over the last $n-j$. \end{proof}
Let $j$ be the current column (\cref{line:outer-loop}). At the beginning of each iteration of the outer for loop, the following properties hold: \begin{enumerate}[(i)]
\item $A_{:j-1}$ is in signed Hermite normal form;
\item $A_{:j}$ is in Hermite normal form;
\item $G = \Stab(A_{:j-1}) \subseteq \mathbb{Z}_2^{j-1}$. \end{enumerate} The proof is by induction: for $j=1$ this is trivial; the induction step is given by \Cref{rmk:outer-loop}. Notice that $\Stab(A_{:j-1})$ describes the freedom we still have in changing the sign of the first $j-1$ columns, without affecting the Hermite normal form of $A_{:j-1}$.
Let $m = \rk(A_{:j-1})$ (\cref{line:rank}) and $q = A_{m+1, j}$ (\cref{line:pivot}). Here $q \geq 0$, and if $q \neq 0$ then $q$ is a pivot of the Hermite normal form $A_{:j}$. The matrix $A_{:j}$ looks like this:
\begin{equation}\label{eq:Aj} A_{:j} = \left(
\begin{array}{c|c}\\[-5pt] \;\;B\;\;\; & v \\[5pt] \hline & q \\ \cline{2-2} \multicolumn{1}{c}{\;\;0\;\;\;} & \multicolumn{1}{c}{} \\[5pt] \end{array} \right) \end{equation} where $B \in \mathsf{M}(m, j-1; \mathbb{Z})$, and $v$ is a column vector in $\mathbb{Z}^m$.
In \cref{line:phi}, we consider the group homomorphism $\varphi \colon G \to \GL(m;\mathbb{Z})$ defined as follows: $\varphi(S)$ is the unique matrix in $\GL(m;\mathbb{Z}) \subseteq \GL(r;\mathbb{Z})$ such that $\varphi(S) A_{:j-1}S = A_{:j-1}$. Since $A_{:j-1}$ has zeros in the last $r-m$ rows, this condition is equivalent to $\varphi(S)BS = B$. As we said before, we describe $\varphi$ by computing the image of each element $S$ of the $\mathbb{Z}_2$-basis of $G$. This is done by running the algorithm for the Hermite normal form of $BS$: this algorithm returns both the Hermite normal form (which we already know to be equal to $B$) and a matrix $\varphi(S) \in \GL(m;\mathbb{Z})$ such that $\varphi(S) B S = B$. The matrix $\varphi(S)$ is unique because $B$ has rank $m$. Notice that $\varphi$ is indeed a group homomorphism, because \begin{align*} \varphi(ST)BST &= B \\ & = \varphi(S)BS \\ & = \varphi(S)\varphi(T)BTS \\ & = \varphi(S)\varphi(T)BST. \end{align*}
In \cref{line:augment-G}, we replace $G$ with $G \times \mathbb{Z}_2$, where $\mathbb{Z}_2$ is the $j$-th factor of $\mathbb{Z}_2^n$. This is achieved by extending the basis of $G$ with the $j$-th element of the standard $\mathbb{Z}_2$-basis of $\mathbb{Z}_2^n$. In \cref{line:extend-phi} we extend $\varphi$ to the new basis element, by sending it to $-I_{m\times m}$. The definition of $\varphi$ is motivated by \Cref{lemma:phi} below.
Given $x \in \mathbb{Z}$, define $x \mmod q$ as the remainder of the division between $x$ and $q$ if $q > 0$, and define $x \mmod 0 = x$. Since the group $G$ is going to change throughout the inner loop, it is convenient to denote by $G_0$ the value of $G$ after \cref{line:extend-phi} is executed.
\begin{lemma}\label{lemma:phi}
Write $A_{:j}$ as in \cref{eq:Aj}.
Let $\mathcal{H} = \set{ \HNF(A_{:j}S) \mid S \in \mathbb{Z}_2^j} \subseteq \mathsf{M}(r, j, \mathbb{Z})$, and let $\mathcal{H}' = \set{A' \in \mathcal{H} \mid A'_{:j-1} = A_{:j-1}}$.
Then the matrices in $\mathcal{H}'$ are precisely those of the form
\[
A' =
\left(
\begin{array}{c|c}\\[-5pt]
\;\;B\;\;\; & w \\[5pt]
\hline
& q \\
\cline{2-2}
\multicolumn{1}{c}{\;\;0\;\;\;} & \multicolumn{1}{c}{} \\[5pt]
\end{array}
\right)
\]
where $w = \varphi(S)\cdot v$ mod $q$ for some $S \in G_0$. \end{lemma}
\begin{proof}
Denote by $S_j$ the $j$-th element of the standard $\mathbb{Z}_2$-basis of $\mathbb{Z}_2^n$.
Then $S_j$ acts on $\mathsf{M}(r,n;\mathbb{Z})$ by changing the sign of the $j$-th column.
Every element of $G_0$ is of the form $S_0S_j^\epsilon$ for some $S_0 \in \Stab (A_{:j-1})$ and $\epsilon \in \set{0,1}$.
We first show that a matrix $A'$ as above (for a given $S \in G_0$) belongs to $\mathcal{H}'$.
Notice that $A'$ is in Hermite normal form, and $A'_{:j-1} = A_{:j-1}$, so it is enough to show that $A'$ is in the same orbit as $A_{:j} S$ with respect to the left action of $\GL(r;\mathbb{Z})$.
Write $S = S_0S_j^\epsilon$.
Then, by definition of $\varphi$, we have
\[
\varphi(S_0) A_{:j} S_0 S_j^\epsilon =
\left(
\begin{array}{c|c}\\[-5pt]
\;\;B\;\;\; & \varphi(S_0) \cdot v \\[5pt]
\hline
& q \\
\cline{2-2}
\multicolumn{1}{c}{\;\;0\;\;\;} & \multicolumn{1}{c}{} \\[5pt]
\end{array}
\right) \cdot S_j^\epsilon
=
\left(
\begin{array}{c|c}\\[-5pt]
\;\;B\;\;\; & \varphi(S) \cdot v \\[5pt]
\hline
& (-1)^\epsilon q \\
\cline{2-2}
\multicolumn{1}{c}{\;\;0\;\;\;} & \multicolumn{1}{c}{} \\[5pt]
\end{array}
\right),
\]
and this matrix is in the same orbit as $A'$.
Conversely, let $A' \in \mathcal{H}'$. Since $A' \in \mathcal{H}$, we have $A' = \HNF(A_{:j}S)$ for some $S \in \mathbb{Z}_2^j$.
Write $S = S_0 S_j^\epsilon$ for some $S_0 \in \mathbb{Z}_2^{j-1}$ and $\epsilon \in \set{0,1}$.
We have $A'_{:j-1} = \HNF(A_{:j}S)_{:j-1} = \HNF(A_{:j-1} S_0)$.
Since $A' \in \mathcal{H}'$, we also have $A'_{:j-1} = A_{:j-1}$. Therefore $\HNF(A_{:j-1}S_0) = A_{:j-1}$, which implies that $S_0 \in \Stab (A_{:j-1})$.
Then $S = S_0 S_j^\epsilon \in G_0$.
We conclude by noticing that $A' = \HNF(A_{:j}S)$ is of the form given in the statement, with $w = \varphi(S)\cdot v \mmod q$. \end{proof}
\Cref{lemma:phi} gives an explicit characterization of the possible values of the $j$-th column of the Hermite normal form of $AS$, provided that $\HNF(AS)_{:j-1} = A_{:j-1}$. Then, to compute the $j$-th column of the signed Hermite normal form, we need to find the lexicographically minimal vector $w = \varphi(S) \cdot v$ mod $q$. This is done in the inner loop, starting from the $m$-th row and going up to the first row (lines \ref{line:inner-loop}-\ref{line:end-inner-loop}).
Let $i$ be the current row (\cref{line:inner-loop}).
At the beginning of each iteration of the inner loop, we have that \begin{equation} G = \set{S \in G_0 \mid (\varphi(S)A_j)_{i'} \mmod q = A_{i',j} \text{ for all } i'>i }. \label{eq:subgroup} \end{equation} This is proved by induction: it holds at the beginning of the first iteration ($i=m$) because $G=G_0$; the induction step is proved below.
In \cref{line:orbit} we explicitly compute the set $O$ of all possible values of the entry $(i,j)$. For ease of notation, write $A_{:j}$ as in \cref{eq:Aj}. Then \begin{align*} O &= \set{ (\varphi(S) A_j)_i \!\!\!\mod q \mid S \in G } \\ &= \set{ (\varphi(S) \cdot v)_i \!\!\!\mod q \mid S \in G }. \end{align*}
A key observation is that the set $O$ is very small, even if $G$ can be large.
\begin{lemma}
In \cref{line:orbit}, $\lvert O \rvert \in \set{1,2,4}$. \end{lemma}
\begin{proof}
By eq.\ \eqref{eq:subgroup} and since $\varphi(S)$ is upper triangular,
there is a well-defined action of $G$ on $\mathbb{Z}_q$: an element $S \in G$ acts as an affine automorphism $\rho(S) \in \Aff(\mathbb{Z}_q)$ given by
\[ x \;\mapsto\; \varphi(S)_{i,i} \, x + \sum_{k=i+1}^{r} \varphi(S)_{i,k} A_{k,j}. \]
This is how $G$ acts on the entry $(i,j)$ of $A$.
Notice that $\rho(S)$ has the form $x \mapsto \pm x + \beta$ for some $\beta \in \mathbb{Z}_q$, since $\varphi(S)_{i,i} = \pm 1$.
In addition, $\rho(S)$ is an involution, so it has one of the following forms:
\[ x \mapsto x, \quad x \mapsto x + q/2 \,\text{ (if $q$ is even)}, \quad x \mapsto -x + \beta \text{ for some } \beta \in \mathbb{Z}_q. \]
The maps $x \mapsto x$ and $x \mapsto x + q/2$ commute with each other and with any map of the form $x \mapsto -x+\beta$.
However, given two maps of the form $x \mapsto -x + \beta_1$ and $x \mapsto -x+\beta_2$, they commute if and only if $2(\beta_1 - \beta_2) = 0$, i.e.\ $\beta_1 = \beta_2$ or $\beta_1 = \beta_2 + q/2$.
Since $\rho(G)$ is abelian, there exists a $\beta \in \mathbb{Z}_q$ such that any element $S \in G$ acts as one of the following four maps:
\[ x \mapsto x, \quad x \mapsto x + q/2, \quad x \mapsto -x + \beta, \quad x \mapsto -x + \beta + q/2. \]
Therefore $\rho(G)$ is isomorphic to a subgroup of $\mathbb{Z}_2^2$.
By definition, $O$ is the orbit of $A_{i,j}$ in $\mathbb{Z}_q$, and so its cardinality divides $4$. \end{proof}
In \cref{line:min-orbit}, we select the smallest element $u \in O$. In \cref{line:S-min-orbit}, we choose any element $\overline S \in G$ such that $\rho(\overline S)(A_{i,j}) = u$. After \cref{line:HNF-inner-loop}, the entry $(i,j)$ of $A$ is equal to $u$. Finally, in \cref{line:restrict-G} we update the group $G$ in order to satisfy eq.\ \eqref{eq:subgroup}, and in \cref{line:restrict-phi} we restrict $\varphi$ to the new group $G$.
\begin{remark}
\label{rmk:outer-loop}
At the end of the inner loop (after \cref{line:end-inner-loop}), we have that: $G = \Stab(A_{:j})$, by eq.\ \eqref{eq:subgroup} for $i=0$;
$A$ is in Hermite normal form, by \cref{line:HNF-inner-loop}, so in particular $A_{:j+1}$ is in Hermite normal form;
$A_{:j}$ is in signed Hermite normal form, by construction. \end{remark}
\begin{example}
Consider the following $3\times 3$ matrix:
\[
A =
\begin{pmatrix}
1 & 1 & 4 \\
0 & 2 & 3 \\
0 & 0 & 6
\end{pmatrix}.
\]
The first two columns are already in signed Hermite normal form.
When \Cref{alg:s-normal-form} encounters the third column ($j=3$), the second entry ($i=2$) is already minimal.
At the beginning of the last iteration ($i=1$) of the inner loop, we have $G = \mathbb{Z}_2^3$.
The three standard generators of $G$ act on $\mathbb{Z}_6$ as $x \mapsto -x+3$, $x \mapsto x+3$, and $x \mapsto -x$.
Then $\rho(G)$ is a subgroup of $\Aff(\mathbb{Z}_6)$ isomorphic to $\mathbb{Z}_2^2$.
On \cref{line:orbit}, we have $O = \set{1,2,4,5}$.
By choosing $\overline S$ as the second standard generator of $\mathbb{Z}_2^3$, we obtain
\[
\varphi(\overline S) A \overline S =
\begin{pmatrix}
1 & 1 & 1 \\
0 & 2 & 3 \\
0 & 0 & 6
\end{pmatrix},
\]
which is the signed Hermite normal form. \end{example}
\begin{proposition}
\label{prop:complexity}
The running time of \Cref{alg:s-normal-form} is $O(r^{\theta-1}n^2(r^2+n))$, where $\theta$ is such that two $r\times r$ integer matrices can be multiplied in time $O(r^\theta)$. \end{proposition}
\begin{proof}
We assume that all operations between integers require time $O(1)$.
Then the algorithm of \cite{storjohann1996asymptotically} allows to compute the Hermite normal form $H$ of a matrix $A \in \mathsf{M}(r,n;\mathbb{Z})$ in time $O(r^{\theta-1}n)$.
This algorithm also returns a matrix $U$ such that $H = UA$.
Notice that the best exponent $\theta$ is between $2$ and $2.3728639$, by \cite{le2014powers}.
Throughout the execution of \Cref{alg:s-normal-form}, the group $G$ is always described by one of its $\mathbb{Z}_2$-bases.
Since the $\mathbb{Z}_2$-dimension of $G$ increases at most by $1$ at each iteration of the outer loop (\cref{line:augment-G}), it is always bounded by $n$.
The most expensive operations of \Cref{alg:s-normal-form} are the following.
\Cref{line:phi} requires $O(n)$ computations of Hermite normal forms, and is executed $n$ times (because it is inside the outer loop), so it takes $O(r^{\theta-1} n^3)$ time.
\Cref{line:orbit} requires to compute $O(n)$ elements of $\Aff(\mathbb{Z}_q)$, each of them being obtained as a dot product between two vectors of size $r$.
It is executed $rn$ times (because it is inside the inner loop), so it takes $O(r^2n^2)$ time.
\Cref{line:HNF-inner-loop} is executed $O(rn)$ times, and thus takes $O(r^\theta n^2)$ time.
\Cref{line:restrict-G,line:restrict-phi} require to compute $O(n)$ multiplications of $r \times r$ matrices.
They are executed $O(rn)$ times, so they take $O(r^{\theta+1}n^2)$ time.
The overall running time is $O(r^{\theta-1}n^3) + O(r^2n^2) + O(r^\theta n^2) + O(r^{\theta+1} n^2) = O(r^{\theta-1}n^2(r^2+n))$. \end{proof}
Since $\theta < 3$, assuming $r \leq n$ (otherwise some rows of $\HNF(A)$ are zero, and can be ignored), the time complexity of \Cref{alg:s-normal-form} is less than cubic in the input size $rn$.
\section{Decomposition of representable matroids} \label{sec:decomposition}
Given an arithmetic matroid $M = (E, \rk, m)$, a \emph{decomposition} of $M$ is a partition $E = E_1 \sqcup \dots \sqcup E_k$ of the groundset such that $\rk(X) = \rk(X \cap E_1) + \dots + \rk(X \cap E_k)$ and $m(X) = m(X \cap E_1) \dotsm m(X \cap E_k)$ for every $X \subseteq E$. An arithmetic matroid is \emph{indecomposable} if it has no non-trivial decompositions. Notice that, if $M$ is an indecomposable arithmetic matroid, the underlying matroid $\mathcal{M} = (E, \rk)$ can be decomposable.
The following lemma allows decomposing a represented arithmetic matroid into indecomposable ones, with a simple and fast algorithm.
\begin{lemma}
Let $M = (E, \rk, m)$ be a torsion-free arithmetic matroid of rank $r$.
Suppose that the first $r$ elements of the groundset $E$ form a basis $B$ of $M$.
Let $A \in \mathsf{M}(r,n;\mathbb{Z})$ be a representation of $M$, where $A$ is in Hermite normal form.
A partition $E = E_1 \sqcup E_2$ is a decomposition of $M$ if and only if $A_{ij} = 0$ for all $(i,j) \in (B \cap E_1) \times E_2 \cup (B \cap E_2) \times E_1$. \end{lemma}
\begin{proof}
If $A_{i,j} = 0$ for all $(i,j) \in (B \cap E_1) \times E_2 \cup (B \cap E_2) \times E_1$ then, up to a permutation of rows and columns, $A$ is a block diagonal matrix having the columns in $E_1$ in the first block and the columns in $E_2$ in the second block.
Then $E = E_1 \sqcup E_2$ is a decomposition of $M$.
Suppose now that $E = E_1 \sqcup E_2$ is a decomposition of $M$.
We prove the statement by induction on $j$.
Assume without loss of generality that $j \in E_2$.
We start with the case $j \in B$.
We have that
\[
\prod_{k=1}^j A_{k,k} = m(\set{1, \dots, j}) = m(\set{1,\dots,j} \cap E_1) \cdot m(\set{1,\dots,j} \cap E_2).
\]
By induction, $m(\set{1,\dots,j} \cap E_1) = \prod_{k \in \set{1,\dots,j} \cap E_1} A_{k,k}$.
In addition, we have $m(\set{1,\dotsc, j} \cap E_2) \mid \det(A')$, where $A'$ is the square submatrix of $A$ consisting of the rows $\set{i} \cup (\set{1,\dots,j-1} \cap E_2)$ and the columns $\set{1,\dots,j} \cap E_2$.
By induction, $\det(A') = A_{i,j} \cdot \prod_{k \in \set{1,\dots,j-1} \cap E_2} A_{k,k}$.
Putting everything together, we obtain that $A_{j,j} \mid A_{i,j}$.
Since $A$ is in Hermite normal form, $A_{i,j} = 0$.
Consider now the case $j \not\in B$.
The $j$-th column of $A$ is a linear combination of the columns in $B \cap E_2$. Therefore $A_{i,j} = 0$. \end{proof}
\section{Applications and examples} \label{sec:applications}
The software library Arithmat \cite{arithmat}, which is publicly available as a Sage package, implements arithmetic matroids, toric arrangements, and some of their most important operations. The algorithms of this paper are implemented, together with some additional ones such as Lenz's algorithm to compute the poset of layers of a toric arrangements \cite{lenz2017computing}.
As an application of our library and algorithms, we provide some examples of central toric arrangements with a non-shellable (nor Cohen-Macaulay) poset of layers, and a non-shellable (nor Cohen-Macaulay) arithmetic independence poset. This disproves two popular conjectures in the community of arrangements and matroids.
\begin{definition}[Poset of layers]
The poset of layers of a toric arrangement $\mathcal{A}$ is the set of connected components of intersections of elements of $\mathcal{A}$, ordered by reverse inclusion. \end{definition}
Posets of layers of toric arrangements associated with root systems were proved to be shellable \cite{delucchi2017shellability, paolini2018shellability}, and this led to the conjecture that posets of layers are always shellable.
A related poset is the \emph{(arithmetic) independence poset} of a toric arrangement, defined in \cite[Definition 5]{lenz2017stanley}, \cite[Section 2]{martino2018face} (under the name of \emph{poset of torsions}), and \cite[Section 7]{d2018stanley} (under the name of \emph{poset of independent sets}).
\begin{definition}[Arithmetic independence poset]
The arithmetic independence poset of a toric arrangement $\mathcal{A}$ is the set of pairs $(I, W)$ where $I \subseteq \mathcal{A}$ is an independent set and $W$ is a connected component of $\bigcap I$.
The order relation is defined as follows: $(I_1, W_1) \leq (I_2, W_2)$ if and only if $I_1 \subseteq I_2$ and $W_1 \supseteq W_2$. \end{definition}
D'Alì and Delucchi proved that both posets are homology Cohen-Macaulay over fields of all but a finite number of characteristics \cite{d2018stanley}. It was conjectured that the arithmetic independence poset is shellable. Notice that the non-arithmetic versions of these posets (the poset of flats and the independence poset of an ordinary matroid) are shellable, and therefore Cohen-Macaulay over fields of every characteristic.
Consider the example of \cite[Section 3]{Pagaria2018}: let $M$ be the arithmetic matroid associated with the matrix \[
A = \begin{pmatrix*}[c]
1 & 1 & 1 & -3 \\
0 & 5 & 0 & -5 \\
0 & 0 & 5 & -5
\end{pmatrix*}. \] Using the algorithm of \Cref{sec:representations}, we find that $M$ has $13$ non-equivalent essential representation. These $13$ representations give rise to $3$ non-isomorphic posets of layers. These $3$ posets are realized by the matrices $A$ and \[ A' = \begin{pmatrix*}[c] 1 & 1 & 1 & -1 \\ 0 & 5 & 0 & 5 \\ 0 & 0 & 5 & -5 \end{pmatrix*}, \quad A'' = \begin{pmatrix*}[c] 1 & 2 & 2 & 1 \\ 0 & 5 & 0 & 5 \\ 0 & 0 & 5 & -5 \end{pmatrix*}. \] The matrices $A, A', A''$ are given in signed Hermite normal form (see \Cref{sec:shnf}). The fact that $A$ and $A'$ give rise to non-isomorphic posets of layers was already proved by the first author in \cite{Pagaria2018}.
The homology of the order complex of the poset of layers (with the bottom element removed) is equal to $(0, \mathbb{Z}_5, \mathbb{Z}^{48})$ in all $3$ cases. In particular, these posets of layers are not Cohen-Macaulay over fields of characteristic $5$ and therefore are not shellable.
The arithmetic independence posets of the $13$ representations of $M$ are pairwise isomorphic. Their order complexes (with the bottom element removed) have homology $(0, \mathbb{Z}_5, \mathbb{Z}^{73})$. Therefore these posets are not Cohen-Macaulay in characteristic $5$ and are not shellable. Our computations settle some different conjectures about the posets associated with a toric arrangement, but also highlight the following problem.
\begin{question}
Let $M$ be an arithmetic matroid.
Are the arithmetic independence posets of the representations of $M$ always pairwise isomorphic? \end{question}
\nocite{*}
\end{document} | arXiv |
\begin{definition}[Definition:Cantor Set/Limit of Decreasing Sequence]
Let $\map {I_c} \R$ denote the set of all closed real intervals.
Define the mapping $t_1: \map {I_c} \R \to \map {I_c} \R$ by:
:$\map {t_1} {\closedint a b} := \closedint a {\dfrac 1 3 \paren {a + b} }$
and similarly $t_3: \map {I_c} \R \to \map {I_c} \R$ by:
:$\map {t_3} {\closedint a b} := \closedint {\dfrac 2 3 \paren {a + b} } b$
Note in particular how:
:$\map {t_1} {\closedint a b} \subseteq \closedint a b$
:$\map {t_3} {\closedint a b} \subseteq \closedint a b$
Subsequently, define inductively:
:$S_0 := \set {\closedint 0 1}$
:$S_{n + 1} := \map {t_1} {C_n} \cup \map {t_3} {C_n}$
and put, for all $n \in \N$:
:$C_n := \ds \bigcup S_n$
Note that $C_{n + 1} \subseteq C_n$ for all $n \in \N$, so that this forms a decreasing sequence of sets.
Then the '''Cantor set''' $\CC$ is defined as its limit, that is:
:$\ds \CC := \bigcap_{n \mathop \in \N} C_n$
Category:Definitions/Cantor Set
\end{definition} | ProofWiki |
\begin{definition}[Definition:Path Component/Equivalence Class]
Let $T$ be a topological space.
Let $\sim$ be the equivalence relation on $T$ defined as:
:$x \sim y \iff x$ and $y$ are path-connected.
The equivalence classes of $\sim$ are called the '''path components of $T$'''.
If $x \in T$, then the '''path component of $T$''' containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\map {\operatorname{PC}_x} T$.
From Path-Connectedness is Equivalence Relation, $\sim $ is an equivalence relation.
From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes.
\end{definition} | ProofWiki |
Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.
"Functoriality" redirects here. For the Langlands functoriality conjecture in number theory, see Langlands program § Functoriality.
The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively.[1] The latter used functor in a linguistic context;[2] see function word.
Definition
Let C and D be categories. A functor F from C to D is a mapping that[3]
• associates each object $X$ in C to an object $F(X)$ in D,
• associates each morphism $f\colon X\to Y$ in C to a morphism $F(f)\colon F(X)\to F(Y)$ in D such that the following two conditions hold:
• $F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!$ for every object $X$ in C,
• $F(g\circ f)=F(g)\circ F(f)$ for all morphisms $f\colon X\to Y\,\!$ and $g\colon Y\to Z$ in C.
That is, functors must preserve identity morphisms and composition of morphisms.
Covariance and contravariance
There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor F from C to D as a mapping that
• associates each object $X$ in C with an object $F(X)$ in D,
• associates each morphism $f\colon X\to Y$ in C with a morphism $F(f)\colon F(Y)\to F(X)$ in D such that the following two conditions hold:
• $F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!$ for every object $X$ in C,
• $F(g\circ f)=F(f)\circ F(g)$ for all morphisms $f\colon X\to Y$ and $g\colon Y\to Z$ in C.
Note that contravariant functors reverse the direction of composition.
Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the opposite category $C^{\mathrm {op} }$.[4] Some authors prefer to write all expressions covariantly. That is, instead of saying $F\colon C\to D$ is a contravariant functor, they simply write $F\colon C^{\mathrm {op} }\to D$ (or sometimes $F\colon C\to D^{\mathrm {op} }$) and call it a functor.
Contravariant functors are also occasionally called cofunctors.[5]
There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections $\Gamma (TM)$ of a tangent bundle $TM$—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections $\Gamma {\mathord {\left(T^{*}M\right)}}$ of a cotangent bundle $T^{*}M$—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as ${x'}^{\,i}=\Lambda _{j}^{i}x^{j}$ for $\mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} $ or $\omega '_{i}=\Lambda _{i}^{j}\omega _{j}$ for ${\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.$ In this formalism it is observed that the coordinate transformation symbol $\Lambda _{i}^{j}$ (representing the matrix ${\boldsymbol {\Lambda }}^{\textsf {T}}$) acts on the basis vectors "in the same way" as on the "covector coordinates": $\mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}$—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: $\mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}$). This terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant, whereas vectors in general are covariant since they can be pushed forward. See also Covariance and contravariance of vectors.
Opposite functor
Every functor $F\colon C\to D$ induces the opposite functor $F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }$, where $C^{\mathrm {op} }$ and $D^{\mathrm {op} }$ are the opposite categories to $C$ and $D$.[6] By definition, $F^{\mathrm {op} }$ maps objects and morphisms in the identical way as does $F$. Since $C^{\mathrm {op} }$ does not coincide with $C$ as a category, and similarly for $D$, $F^{\mathrm {op} }$ is distinguished from $F$. For example, when composing $F\colon C_{0}\to C_{1}$ with $G\colon C_{1}^{\mathrm {op} }\to C_{2}$, one should use either $G\circ F^{\mathrm {op} }$ or $G^{\mathrm {op} }\circ F$. Note that, following the property of opposite category, $\left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F$.
Bifunctors and multifunctors
A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type Cop × C → Set. It can be seen as a functor in two arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.
A multifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctor with n = 2.
Properties
Two important consequences of the functor axioms are:
• F transforms each commutative diagram in C into a commutative diagram in D;
• if f is an isomorphism in C, then F(f) is an isomorphism in D.
One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G ∘ F from A to C. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories.
A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
Examples
Diagram
For categories C and J, a diagram of type J in C is a covariant functor $D\colon J\to C$.
(Category theoretical) presheaf
For categories C and J, a J-presheaf on C is a contravariant functor $D\colon C\to J$.
In the special case when J is Set, the category of sets and functions, D is called a presheaf on C.
Presheaves (over a topological space)
If X is a topological space, then the open sets in X form a partially ordered set Open(X) under inclusion. Like every partially ordered set, Open(X) forms a small category by adding a single arrow U → V if and only if $U\subseteq V$. Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.
Constant functor
The functor C → D which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X. Such a functor is called a constant or selection functor.
Endofunctor
A functor that maps a category to that same category; e.g., polynomial functor.
Identity functor
In category C, written 1C or idC, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
Diagonal functor
The diagonal functor is defined as the functor from D to the functor category DC which sends each object in D to the constant functor at that object.
Limit functor
For a fixed index category J, if every functor J → C has a limit (for instance if C is complete), then the limit functor CJ → C assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitable version of the axiom of choice. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant).
Power sets functor
The power set functor P : Set → Set maps each set to its power set and each function $f\colon X\to Y$ to the map which sends $U\in {\mathcal {P}}(X)$ to its image $f(U)\in {\mathcal {P}}(Y)$. One can also consider the contravariant power set functor which sends $f\colon X\to Y$ to the map which sends $V\subseteq Y$ to its inverse image $f^{-1}(V)\subseteq X.$
For example, if $X=\{0,1\}$ then $F(X)={\mathcal {P}}(X)=\{\{\},\{0\},\{1\},X\}$. Suppose $f(0)=\{\}$ and $f(1)=X$. Then $F(f)$ is the function which sends any subset $U$ of $X$ to its image $f(U)$, which in this case means $\{\}\mapsto f(\{\})=\{\}$, where $\mapsto $ denotes the mapping under $F(f)$, so this could also be written as $(F(f))(\{\})=\{\}$. For the other values,$\{0\}\mapsto f(\{0\})=\{f(0)\}=\{\{\}\},\ $ $\{1\}\mapsto f(\{1\})=\{f(1)\}=\{X\},\ $ $\{0,1\}\mapsto f(\{0,1\})=\{f(0),f(1)\}=\{\{\},X\}.$ Note that $f(\{0,1\})$ consequently generates the trivial topology on $X$. Also note that although the function $f$ in this example mapped to the power set of $X$, that need not be the case in general.
Dual vector space
The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself.
Fundamental group
Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs (X, x0), where X is a topological space and x0 is a point in X. A morphism from (X, x0) to (Y, y0) is given by a continuous map f : X → Y with f(x0) = y0.
To every topological space X with distinguished point x0, one can define the fundamental group based at x0, denoted π1(X, x0). This is the group of homotopy classes of loops based at x0, with the group operation of concatenation. If f : X → Y is a morphism of pointed spaces, then every loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π(X, x0) to π(Y, y0). We thus obtain a functor from the category of pointed topological spaces to the category of groups.
In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial.
Algebra of continuous functions
A contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : X → Y induces an algebra homomorphism C(f) : C(Y) → C(X) by the rule C(f)(φ) = φ ∘ f for every φ in C(Y).
Tangent and cotangent bundles
The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles.
Doing this constructions pointwise gives the tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above.
Group actions/representations
Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i.e. a G-set. Likewise, a functor from G to the category of vector spaces, VectK, is a linear representation of G. In general, a functor G → C can be considered as an "action" of G on an object in the category C. If C is a group, then this action is a group homomorphism.
Lie algebras
Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor.
Tensor products
If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product $V\otimes W$ defines a functor C × C → C which is covariant in both arguments.[7]
Forgetful functors
The functor U : Grp → Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.[8] Functors like these, which "forget" some structure, are termed forgetful functors. Another example is the functor Rng → Ab which maps a ring to its underlying additive abelian group. Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).
Free functors
Going in the opposite direction of forgetful functors are free functors. The free functor F : Set → Grp sends every set X to the free group generated by X. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object.
Homomorphism groups
To every pair A, B of abelian groups one can assign the abelian group Hom(A, B) consisting of all group homomorphisms from A to B. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor Abop × Ab → Ab (where Ab denotes the category of abelian groups with group homomorphisms). If f : A1 → A2 and g : B1 → B2 are morphisms in Ab, then the group homomorphism Hom(f, g): Hom(A2, B1) → Hom(A1, B2) is given by φ ↦ g ∘ φ ∘ f. See Hom functor.
Representable functors
We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X, Y) of morphisms from X to Y. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor Cop × C → Set. If f : X1 → X2 and g : Y1 → Y2 are morphisms in C, then the map Hom(f, g) : Hom(X2, Y1) → Hom(X1, Y2) is given by φ ↦ g ∘ φ ∘ f.
Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable.
Relation to other categorical concepts
Let C and D be categories. The collection of all functors from C to D forms the objects of a category: the functor category. Morphisms in this category are natural transformations between functors.
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above.
Universal constructions often give rise to pairs of adjoint functors.
Computer implementations
Functors sometimes appear in functional programming. For instance, the programming language Haskell has a class Functor where fmap is a polytypic function used to map functions (morphisms on Hask, the category of Haskell types)[9] between existing types to functions between some new types.[10]
See also
• Functor category
• Kan extension
• Pseudofunctor
Notes
1. Mac Lane, Saunders (1971), Categories for the Working Mathematician, New York: Springer-Verlag, p. 30, ISBN 978-3-540-90035-1
2. Carnap, Rudolf (1937). The Logical Syntax of Language, Routledge & Kegan, pp. 13–14.
3. Jacobson (2009), p. 19, def. 1.2.
4. Jacobson (2009), pp. 19–20.
5. Popescu, Nicolae; Popescu, Liliana (1979). Theory of categories. Dordrecht: Springer. p. 12. ISBN 9789400995505. Retrieved 23 April 2016.
6. Mac Lane, Saunders; Moerdijk, Ieke (1992), Sheaves in geometry and logic: a first introduction to topos theory, Springer, ISBN 978-0-387-97710-2
7. Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004), Algebras, rings and modules, Springer, ISBN 978-1-4020-2690-4
8. Jacobson (2009), p. 20, ex. 2.
9. It's not entirely clear that Haskell datatypes truly form a category. See https://wiki.haskell.org/Hask for more details.
10. See https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell for more information.
References
• Jacobson, Nathan (2009), Basic algebra, vol. 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7.
External links
Look up functor in Wiktionary, the free dictionary.
• "Functor", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• see functor at the nLab and the variations discussed and linked to there.
• André Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematics
• Hillman, Chris (2001). "A Categorical Primer". CiteSeerX 10.1.1.24.3264: {{cite web}}: Missing or empty |url= (help) formal introduction to category theory.
• J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats Archived 2015-04-21 at the Wayback Machine
• Stanford Encyclopedia of Philosophy: "Category Theory" — by Jean-Pierre Marquis. Extensive bibliography.
• List of academic conferences on category theory
• Baez, John, 1996,"The Tale of n-categories." An informal introduction to higher order categories.
• WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors, natural transformations, universal properties.
• The catsters, a YouTube channel about category theory.
• Video archive of recorded talks relevant to categories, logic and the foundations of physics.
• Interactive Web page which generates examples of categorical constructions in the category of finite sets.
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
Functor types
• Additive
• Adjoint
• Conservative
• Derived
• Diagonal
• Enriched
• Essentially surjective
• Exact
• Forgetful
• Full and faithful
• Logical
• Monoidal
• Representable
• Smooth
| Wikipedia |
\begin{document}
\title{Ramanujan type congruences for modular forms of several variables}
\author{Toshiyuki Kikuta and Shoyu Nagaoka} \maketitle
\maketitle
\noindent {\bf Mathematics subject classification}: Primary 11F33 $\cdot$ Secondary 11F46\\ \noindent {\bf Key words}: Congruences for modular forms, Cusp forms
\begin{abstract} We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case. \end{abstract}
\section{Introduction} \label{intro} As is well known, Ramanujan's congruence asserts that \begin{align*} \sigma _{11}(n)\equiv \tau (n) \bmod{691}, \end{align*} where $\sigma _k(n)$ is the sum of $k$-th powers of the divisors of $n$ and $\tau (n)$ is the $n$-th Fourier coefficient of Ramanujan's $\Delta $ function. It should be noted that $\sigma _{11}(n)$ is the $n$-th Fourier coefficient of the elliptic Eisenstein series of weight $12$ for $n\ge 1$. We can regard this congruence in two ways:\\ ~~~~~~~~~~~~~~~~~(I) Congruences between Hecke eigenvalues of two eigenforms.\\ ~~~~~~~~~~~~~~~~~(II) Congruences between Fourier coefficients of two modular forms. \\ There are many studies on (I) related to congruence primes in the case of modular forms with several variables. For example, see \cite{Br,Kat1,Kat2,Kat-Miz,Kuro1,Kuro2,Kuro3}. On the other hand, it seems that studies on (II) are little known. However, Bringmann-Heim \cite{Bri-Hei} studied some congruences between Fourier coefficients of two Jacobi forms. These properties essentially give congruences for Fourier coefficients of two Siegel modular forms of degree $2$.
In the present paper, we study congruences for Siegel modular forms and for Hermitian modular forms in the sense of (II) in a way different from that of Bringmann-Heim. In particular, the results herein for the case of Hermitian modular forms are completely new. In addition, we should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$ (Theorem \ref{Thm2}). We will conclude by giving numerical examples for each case.
\section{Statement of results} \subsection{The case of Siegel modular forms} Let $M_k(\Gamma_n)$ denote the space of Siegel modular forms of weight $k$ for the Siegel modular group $\Gamma_n:=Sp_n(\mathbb{Z})$ and let $S_k(\Gamma_n)$ be the subspace of cusp forms. Any Siegel modular form $F(Z)$ in $M_k(\Gamma_n)$ has a Fourier expansion of the form \[ F(Z)=\sum_{0\leq T\in\Lambda_n}a_F(T)q^T,\quad q^T:=e^{2\pi i\text{tr}(TZ)}, \quad Z\in\mathbb{S}_n, \] where \[ \Lambda_n
:=\{ T=(t_{ij})\in Sym_n(\mathbb{Q})\;|\; t_{ii},\;2t_{ij}\in\mathbb{Z}\; \} \] (the lattice in $Sym_n(\mathbb{R})$ of half-integral, symmetric matrices), and $\mathbb{S}_n$ is the Siegel upper-half space of degree $n$.
For any subring $R\subset\mathbb{C}$, we define \begin{align*}
& M_k(\Gamma_n)_R:=\{ F=\sum_{T\in\Lambda_n}a_F(T)q^T\;|\; a_F(T)\in R\;(\forall\, T\in\Lambda_n)\;\},\\ & S_k(\Gamma_n)_R:=M_k(\Gamma_n)_R\cap S_k(\Gamma_n). \end{align*}
In this paper, we consider the degree 2 case. A typical example of Siegel modular forms is the Siegel-Eisenstein series \[ E_k(Z):=\sum_{M= \left( \begin{smallmatrix} * & * \\ C & D\end{smallmatrix} \right)}\det (CZ+D)^{-k},\quad Z\in \mathbb{S}_2, \] where $k>3$ is even, and $M$ runs over a set of representatives $\left\{\left( \begin{smallmatrix} * & * \\ 0_2 & * \end{smallmatrix} \right)\right\}\backslash\Gamma_2$. It is known that $E_k\in M_k(\Gamma_2)_{\mathbb{Q}}$. We set \[ G_k:=-\frac{B_{k}B_{2k-2}}{4k(k-1)}E_k, \] where $B_m$ is the $m$-th Bernoulli number.
The first main result is as follows: \begin{Thm} \label{ThmM} Let $(p,2k-2)$ be an irregular pair. Then there exists a cusp form $f\in M_k(\Gamma _2)_{\mathbb{Z}_{(p)}}$ such that \[ G_k\equiv f \bmod{p}. \] \end{Thm} \begin{Rem} Let $p$ be a prime number. A pair $(p,m)$ is called $irregular$
if $1<m<p$ and $p|B_{m}$.\\ \end{Rem}
\subsection{The case of Hermitian modular forms} The Hermitian upper half-space of degree $n$ is defined by \[
\mathbb{ H}_n:=\{ Z\in M_n(\mathbb{C})\;|\; \tfrac{1}{2i}(Z-{}^t\overline{Z})>0\; \}, \] where ${}^t\overline{Z}$ is the transpose, complex conjugate of $Z$. The Siegel upper-half space $\mathbb{S}_n$ is a subdomain of $\mathbb{H}_n$. Let $\boldsymbol{K}$ be an imaginary quadratic number field with discriminant $d_{\boldsymbol{K}}$ and ring of integers $\mathcal{O}_{\boldsymbol{K}}$. The Hermitian modular group $U_n(\mathcal{O}_{\boldsymbol{K}})$ is defined by \[ U_n(\mathcal{O}_{\boldsymbol{K}}):=
\{ M\in M_n(\mathcal{O}_{\boldsymbol{K}})\;|\; {}^t\overline{M}J_n M=J_n\},\; J_n=\begin{pmatrix}0_n & -1_n \\ 1_n & 0_n\end{pmatrix} \]
Define a character $\nu _k$ on $U_n(\mathcal{O}_{\boldsymbol{K}})$ as follows: \[ \nu_k:= \begin{cases} \text{det}^{\frac{k}{2}} & \text{if $d_{\boldsymbol{K}}=-4$},\\ \text{det}^{k} & \text{if $d_{\boldsymbol{K}}=-3$},\\ \text{trivial character} & \text{otherwise}. \end{cases} \]
We denote by $M_k(U_n(\mathcal{O}_{\boldsymbol{K}}), \nu _k)$ the space of Hermitian modular forms of weight $k$ and character $\nu_k$ with respect to $U_n(\mathcal{O}_{\boldsymbol{K}})$. Namely, it consists of holomorphic functions $F:\mathbb{H}_n\longrightarrow \mathbb{C}$ satisfying \[ F\mid_kM(Z):=\text{det}(CZ+D)^{-k}F((AZ+B)(CZ+D)^{-1})=\nu _k(M)F(Z), \] for all $M=\left( \begin{smallmatrix}A &B \\ C & D \end{smallmatrix} \right) \in U_n({\mathcal O}_{\boldsymbol K})$.
A cusp form is characterized by the condition \[ \Phi (F\mid_k\binom{\,\!{}^t\overline{U}\;\;0_n}{\;0_n\,\;U}) \equiv 0\quad \text{for}\;\text{all}\; U\in GL_n(\boldsymbol{K}) \] where $\Phi$ is the Siegel $\Phi$-operator. If the class number of ${\boldsymbol K}$ is $1$, this condition is equivalent to $\Phi (F)\equiv 0$. We denote by $S_k(U_n(\mathcal{O}_{\boldsymbol{K}}),\nu_k)$ the subspace consisting of cusp forms.
If $F\in M_k(U_n(\mathcal{O}_{\boldsymbol{K}}),\nu_k)$, then $F$ has a Fourier expansion of the form \[ F(Z)=\sum_{0\leq H\in\Lambda_n(\boldsymbol{K})}a_F(H)q^H,\quad q^H:=e^{2\pi i\text{tr}(HZ)},\quad Z\in\mathbb{H}_n, \] where \[ \Lambda_n(\boldsymbol{K}):=\{ H=(h_{ij})\in Her_n(\boldsymbol{K})
\;|\;h_{ii}\in\mathbb{Z},\sqrt{d_{\boldsymbol{K}}}h_{ij} \in\mathcal{O}_{\boldsymbol{K}} \}. \] Let $R$ be a subring of $\mathbb{C}$. As in the case of Siegel modular forms, we define \begin{align*} &M_k(U_n(\mathcal{O}_{\boldsymbol{K}}),\nu _k)_R\\
&:=\{ F=\sum a_F(H)q^H\in M_k(U_n(\mathcal{O}_{\boldsymbol{K}}),\nu _k)\,| \, a_F(H)\in R\ (\forall H\in \Lambda_n(\boldsymbol{K}))\}, \\ & S_k(U_n (\mathcal{O}_{\boldsymbol{K}}),\nu _k)_R:= M_k(U_n(\mathcal{O}_{\boldsymbol{K}}),\nu _k)_R\cap S_k(U_n (\mathcal{O}_{\boldsymbol{K}}),\nu _k). \end{align*}
We consider the Hermitian Eisenstein series of degree 2 \[ E_{k,\boldsymbol K}(Z) :=\sum_{M=\left( \begin{smallmatrix} * & * \\ C & D \end{smallmatrix} \right)} ({\det}M)^{\frac{k}{2}}{\det}(CZ+D)^{-k},\quad Z\in\mathbb{H}_2, \] where $k>4$ is even and $M$ runs over a set of representatives of $\left\{ \left( \begin{smallmatrix} * & * \\ 0_2 & *\end{smallmatrix} \right) \right\} \backslash U_2(\mathcal{O}_{\boldsymbol{K}})$. Then we have \[ E_{k,\boldsymbol K}\in M_k(U_2(\mathcal{O}_{\boldsymbol{K}}),\nu _k)_{\mathbb{Q}}. \] Moreover, $E_{4,\boldsymbol K}\in M_4(U_2(\mathcal{O}_{\boldsymbol{K}}),\nu _4)$ is constructed by the Maass lift (\cite{Kri}).
We set \[ G_{k,{\boldsymbol K}} :=\frac{B_k\cdot B_{k-1,\chi _{\boldsymbol K}}}{4k(k-1)}E_{k,{\boldsymbol K}}, \] where $B_{m,\chi_{\boldsymbol{K}}}$ is the generalized Bernoulli number with respect to the Kronecker character $\chi_{\boldsymbol{K}}$ of $\boldsymbol{K}$.
In this paper, we treat only the case that \[ \text{the class number of $\boldsymbol{K}$ is one}. \] \begin{Rem} It is well known that, if the class number of $\boldsymbol{K}$ is one, then $d_{\boldsymbol{K}}$ is one of \[ -3,-4,-7,-8,-11,-19,-43,-67,-163. \] \end{Rem} Our second main theorem is \begin{Thm} \label{Thm2} Assume that the class number of $\boldsymbol{K}$ is one and $k$ is even. Suppose that a prime number $p$ satisfies the following conditions (A) and (B).
\\ ~~~~~~~~~~~~~~~~(A) $p\nmid B_{3,\chi _{\boldsymbol K}}$ and $p\nmid B_{5,\chi _{\boldsymbol K}}$,
\\
~~~~~~~~~~~~~~~~(B) $k<p-1$ and $p|B_{k-1,\chi _{{\boldsymbol K}}}$,
\\ Then there exists a $non$-$trivial$ cusp form $f\in S_{k}(U_2({\mathcal O}),\nu _k)_{\mathbb{Z}_{(p)}}$ such that \[ G_{k,{\boldsymbol K}}\equiv f \bmod{p}. \] \end{Thm} \begin{Cor} \label{Cor1} (1)\; Assume that $d_{\boldsymbol{K}}$ is one of $-3,-4,-7$. If $p$ satisfies condition (B), then there exists a $non$-$trivial$ cusp form $f\in S_{k}(U_2({\mathcal O}_{\boldsymbol K}),\nu _k)_{\mathbb{Z}_{(p)}}$ such that \[ G_{k,{\boldsymbol K}}\equiv f \bmod{p}. \] (2j\; If $d_{\boldsymbol K}\neq -3$, then ${\rm dim}S_{8}(U_2({\mathcal O}_{\boldsymbol K}),\nu _8)\ge 1$. \end{Cor} \begin{Rem} (1)\; It is known that there are non-trivial cusp forms of weight $10$ and $12$. \\ (2)\; As introduced by Hao-Parry \cite{Ha-Pa,Ha-Pa2}, the prime $p$ satisfying condition (B) is related to the divisibility by $p$ of the class number of a certain algebraic number field. \end{Rem}
\section{Proof of theorems} \subsection{Proof of Theorem \ref{ThmM}} We start with proving that \begin{Lem} \label{Lem} Assume that $g=\sum a_g(T)q^T\in M_k(\Gamma _2)_{\mathbb{Z}_{(p)}}$ satisfies that $a_g(T)\equiv 0$ mod $p$ for all $T$ with $\det T=0$. Then there exists a cusp form $f\in S_k(\Gamma _2)_{\mathbb{Z}_{(p)}}$ such that \[ g \equiv f \bmod{p}. \] \end{Lem} \begin{proof} Since the proof is same as in the case of Hermitian modular forms, we omit it. See the proof of Lemma \ref{Lem3} in $\S$ \ref{prf2}. \end{proof}
\begin{proof}[Proof of Theorem \ref{ThmM}] We recall that the Fourier coefficients of the Eisenstein series $G_k$ are given as follows (e.g. cf. \cite{Ei-Za}): \[ a_{G_k}(T)= \begin{cases} \displaystyle -\frac{B_{k}\cdot B_{2k-2}}{4k(k-1)}\ & \text{if $T=0_2$},
\\ \displaystyle \frac{B_{2k-2}}{2k-2}\sigma _{k-1}(\varepsilon (T))\ & \text{if ${\rm rank}(T)=1$},
\\ \displaystyle \frac{B_{k-1,\chi _{D(T)}}}{k-1}
\sum _{d|\varepsilon (T)}d^{k-1}
\sum _{f|\frac{f(T)}{d}}\mu (f)\chi _{D(T)}(f)f^{k-2}\sigma _{2k-3} \left(\frac{f(T)}{fd}\right)\ & \text{if\ ${\rm rank}(T)=2$}, \end{cases} \] where \begin{align*}
& \text{$\varepsilon (T)$\,:\, the content of $T$, i.e. max$\{ l\in\mathbb{N}\,|\,l^{-1}T\in\Lambda_2\}$},\\ & \text{$\mu$\,:\, the M\"{o}bius function},\\ & \text{$D(T)$\,:\, the discriminant of the quadratic field $\mathbb{Q}(\sqrt{-4\det (T)})$},\\ & \text{$f(T)$\,:\, the natural number determined by $-4\det (T)=D(T)\cdot f(T)^2$},\\ & \text{$\chi_{D(T)}=\left(\frac{D(T)}{*} \right)$\,:\, the Kronecker character of $\mathbb{Q}(\sqrt{-4\det (T)})$},\\
& \text{$\sigma_m(N):=\sum_{0<d|N}d^m$}. \end{align*}
We shall show that Fourier coefficients corresponding $T$ with $\text{rank}(T)\leq 1$ vanish modulo $p$.
Note that $ p|\frac{B_{2k-2}}{2k-2}$ (by assumption), and $\frac{B_{k}}{2k}\in \mathbb{Z}_{(p)}$ (by $k<p-1$ and von Staudt-Clausen's theorem). Hence, $a_{G_k}(0_2)\equiv 0 \bmod{p}$ and $a_{G_k}(T)\equiv 0 \bmod{p}$ for $T$ with $\text{rank}(T)=1$.
In order to complete the proof, we need to show that $a_{G_k}(T)\in \mathbb{Z}_{(p)}$ for all $T$ with ${\rm rank}(T)=2$. By a result of Maass \cite{Ma} (cf. also B\"ocherer \cite{Bo}), $a_{G_k}(T)$ can be written in the form \[ a_{G_k}(T)=\frac{1}{2D^{**}_{2k-2}}c(T),\quad\ \text{for some $c(T)\in \mathbb{Z}$}, \] where $D^{**}_{2k-2}$ is the denominator of $B_{2k-2}$. By the irregularity of $p$, the prime $p$ does not divide $D^{**}_{2k-2}$. Therefore, we can take $g=G_k$ in Lemma \ref{Lem}. This completes the proof of Theorem \ref{ThmM}. \end{proof} \begin{Rem} In Theorem \ref{ThmM}, we do not refer to the non-triviality of the cusp form $f$ (compare with Theorem \ref{Thm2}). It is plausible that the non-triviality holds for any prime $p$ satisfying the assumption of Theorem \ref{ThmM}. Here we give a sufficient condition (see below $(*)$) for the non-triviality.
Under the same assumption as in Theorem \ref{ThmM}, we assume that there exists a fundamental discriminant $D_0<0$ such that \[ (*)\qquad\quad p\nmid B_{k-1,\chi_{D_0}}, \] then $f$ is non-trivial modulo $p$.
In \cite{Bru}, Bruinier proved that, if $p\nmid \frac{B_{2k-2}}{2k-2}$, then there exists such $D_0$. However we cannot apply his result because
$p$ satisfies $p| \frac{B_{2k-2}}{2k-2}$ in our case. \end{Rem}
\subsection{Proofs of Theorem \ref{Thm2} and Corollary \ref{Cor1}} \label{prf2} In this subsection, we prove Theorem \ref{Thm2} and Corollary \ref{Cor1}. Basically, the arguments are same as in the case of Siegel modular forms. First we prove \begin{Lem} \label{Lem5} Let $p$ be a prime satisfying condition (A) in Theorem \ref{Thm2}, namely, $p\nmid B_{3,\chi _{\boldsymbol K}}$ and $p\nmid B_{5,\chi _{\boldsymbol K}}$. Then we have \[ E_{4,\boldsymbol K}\in M_{4}(U_2({\mathcal O}_{\boldsymbol K}),\nu _4)_{\mathbb{Z}_{(p)}}\;\; \text{and}\;\; E_{6,{\boldsymbol K}}\in M_{6}(U_2({\mathcal O}_{\boldsymbol K}),\nu _6)_{\mathbb{Z}_{(p)}}. \] \end{Lem} \begin{proof}[Proof of Lemma \ref{Lem5}] The formula for the Fourier coefficient of $E_{k,\boldsymbol K}$ is given by Krieg \cite{Kri} and Dern \cite{Der,Der2} as follows: \begin{equation} \label{Coeff} a_{E_{k,{\boldsymbol K}}}(H)= \begin{cases} \displaystyle 1 \ & \text{if $H=0_2$},
\\ \displaystyle -\frac{2k}{B_k}\sigma _{k-1}(\varepsilon (H))\ & \text{if ${\rm rank}(H)=1$},
\\ \displaystyle \frac{4k(k-1)}{B_k\cdot B_{k-1,\chi _{\boldsymbol K}}}
\sum _{d|\varepsilon (H)}d^{k-1} G_{\boldsymbol K}
\!\left(k-2,\frac{|d_{\boldsymbol K}|\det (H)}{d^2}\right) & \text{if\ ${\rm rank}(H)=2$}, \end{cases} \end{equation}
where $\varepsilon (H)$ is the content of $H$, $\sigma _{m}(N):=\sum _{d|N}d^m$ and \begin{align*} & G_{\boldsymbol K}
\!(m,N):=\frac{1}{1+|\chi_{\boldsymbol{K}}(N)|} (\sigma _{m,\chi_{\boldsymbol{K}}}(N)-\sigma ^*_{m,\chi_{\boldsymbol{K}}}(N)),
\\
&\sigma _{m,\chi_{\boldsymbol{K}}}(N):=\sum _{0<d|N}\chi_{\boldsymbol{K}}(d)d^{m},
\quad \sigma ^*_{m,\chi_{\boldsymbol{K}}}(N):=\sum _{0<d|N}\chi_{\boldsymbol{K}} (N/d)d^{m}. \end{align*} Note that $G_{\boldsymbol K} \!(m,N)\in \mathbb{Z}$. If $k=4$ or $6$, then $\frac{2k}{B_k}\in \mathbb{Z}$. Moreover by the assumption, we have $\frac{1}{B_{3,\chi _{\boldsymbol K}}}$, $\frac{1}{B_{5,\chi _{\boldsymbol K}}}\in \mathbb{Z}_{(p)}$. Hence, the claim follows. \end{proof} \begin{Lem} \label{Lem3} Let $p$ be a prime satisfying condition (A) in Theorem \ref{Thm2}, namely, $p\nmid B_{3,\chi _{\boldsymbol K}}$ and $p\nmid B_{5,\chi _{\boldsymbol K}}$. Assume that $g=\sum a_g(H)q^H\in M_k(U_2({\mathcal O}_{\boldsymbol K}),\nu _k)_{\mathbb{Z}_{(p)}}$ satisfies that $a_g(H)\equiv 0 \bmod{p}$ for all $H$ with $\det H=0$. Then there exists a cusp form $f\in S_k(U_2({\mathcal O}_{\boldsymbol K}),\nu _k)_{\mathbb{Z}_{(p)}}$ such that $g \equiv f \bmod{p}$. \end{Lem} \begin{proof}[Proof of Lemma \ref{Lem3}] Take a polynomial $Q(X,Y)\in \mathbb{Z}_{(p)}[X,Y]$ such that $\Phi (g)=Q(E^{(1)}_4,E^{(1)}_6)$, where $E_4^{(1)}$ and $E_6^{(1)}$ are the normalized elliptic Eisenstein series of weight $4$ and $6$, respectively. Then the polynomial $Q$ must satisfy $\widetilde{Q}=0$ in $\mathbb{F}_p[X,Y]$ by the result of Swinnerton-Dyer \cite{Swi}. Now we consider $f:=g-Q(E_{4,\boldsymbol K},E_{6,{\boldsymbol K}})$. We see that $f\in S_k(U_2({\mathcal O}_{\boldsymbol K}),\nu _k)$ because of $\Phi (f)=0$. Moreover, by Lemma \ref{Lem5}, $E_{4,\boldsymbol K}\in M_{4}(U_2({\mathcal O}_{\boldsymbol K}),\nu _4)_{\mathbb{Z}_{(p)}}$ and $E_{6,{\boldsymbol K}}\in M_{6}(U_2({\mathcal O}_{\boldsymbol K}),\nu _6)_{\mathbb{Z}_{(p)}}$. Therefore, $Q(E_{4,\boldsymbol K},E_{6,{\boldsymbol K}})\equiv 0 \bmod{p}$, and hence $g \equiv f \bmod{p}$. \end{proof}
We return to the proof of Theorem \ref{Thm2}. \begin{proof}[Proof of Theorem \ref{Thm2}] The Fourier coefficient of $G_{k,\boldsymbol K}$ is given as follows: \[ a_{G_{k,{\boldsymbol K}}}(H)= \begin{cases} \displaystyle\frac{B_k\cdot B_{k-1,\chi _{\boldsymbol K}}}{4k(k-1)} & \text{if $H=0_2$},
\\ \displaystyle-\frac{B_{k-1,\chi _{\boldsymbol K}}}{2k-2}\sigma _{k-1}(\varepsilon (H)) & \text{if rank$(H)=1$},
\\
\displaystyle\sum _{d|\varepsilon (H)}d^{k-1} G_{\boldsymbol K}\!
\left(k-2,\frac{|d_{\boldsymbol K}|\det (H)}{d^2}\right) & \text{if rank$(H)=2$}, \end{cases} \] where the notation is as in (\ref{Coeff}).
From the assumption $k<p-1$ and von Staudt-Clausen's theorem, we have
$\frac{B_k}{2k}\in \mathbb{Z}_{(p)}$. Hence, $p\,|\,\frac{B_k\cdot B_{k-1,\chi _{\boldsymbol K}}}{4k(k-1)}$
and $p\,|\,\frac{B_{k-1,\chi _{\boldsymbol K}}}{2k-2}$. These facts imply that $a_{G_{k,{\boldsymbol K}}}(0_2)\equiv 0 \bmod{p}$ for all $H$ and $a_{G_{k,{\boldsymbol K}}}(H)\equiv 0 \bmod{p}$ with ${\rm rank}(H)=1$. For $H$ with ${\rm rank }(H)=2$, $a_{G_{k,\boldsymbol K}}(H)\in \mathbb{Z}$. Namely, $G_{k,{\boldsymbol K}}\in M_{k}(U_2({\mathcal O}_{{\boldsymbol K}}),\nu _k)_{\mathbb{Z}_{(p)}}$ and this implies that $a_{G_{k,\boldsymbol K}}(H)\equiv 0 \bmod{p}$ for all $H$ with $\det H=0$. Applying Lemma \ref{Lem3} to $G_{k,\boldsymbol K}$, there exists a cusp form such that $G_{k,\boldsymbol K}\equiv f \bmod{p}$.
In order to complete the proof of Theorem \ref{Thm2}, we need to show the non-triviality of $f$. The proof is reduced to proving that there exists $H$ with ${\rm rank}(H)=2$ which satisfies $a_{G_{k,{\boldsymbol K}}}(H)\not\equiv 0 \bmod{p}$. First, we consider the Fourier coefficient corresponding $H=1_2$ (the unit matrix of degree 2), which implies
$a_{G_{k,{\boldsymbol K}}}(1_2)=G_{\boldsymbol K}(k-2,|d_{\boldsymbol K}|)=1-|d_{\boldsymbol K}|^{k-2}$.
If $|d_{\boldsymbol K}|^{k-2}\not \equiv 1 \bmod{p}$, then $a_{G_{k,{\boldsymbol K}}}(1_2)\not \equiv 0 \bmod{p}$. This shows the non-triviality of
$G_{k,{\boldsymbol K}}$ in this case. Hence, it suffices to consider the case that $|d_{\boldsymbol K}|^{k-2}\equiv 1 \bmod{p}$.
Note that $(p,|d_{\boldsymbol K}|)=1$ in this case. Now we prove
\begin{Lem} \label{Lem2} Assume that $k-2<p-1$ and the prime $p$ does not divide $d_{\boldsymbol K}$. Then there exists a prime $q$ such that $\chi _{\boldsymbol K}(q)=-1$ and $q^{k-2}\not \equiv 1 \bmod{p}$. \end{Lem}
\begin{proof}[Proof of Lemma \ref{Lem2}] Let $\alpha $ be an integer such that $(\mathbb{Z}/p\mathbb{Z})^{\times }=\left< \overline{\alpha} \right>$. Applying the Chinese remainder theorem, we can find an integer $a$ such that
$a\equiv \alpha \bmod{p}$ and $a\equiv -1 \bmod{|d_{\boldsymbol K}|}$.
It is clear that $(a,p\cdot |d_{\boldsymbol K}|)=1$ for such $a$. By Dirichlet's theorem on arithmetic progressions, there exist infinitely
many primes in the sequence $\{a+p\cdot |d_{\boldsymbol K}|\cdot n\}_{n=1}^\infty$. We take a prime $q$ appearing in this sequence. Then $q$ satisfies both $\chi _{\boldsymbol K}(q)=-1$ and $q^{k-2}\not \equiv 1 \bmod{p}$. In fact, $q\equiv a \equiv \alpha \bmod{p}$, and hence $q^{k-2}\not\equiv 1 \bmod{p}$ because $k-2<p-1$.
On the other hand, by $q\equiv a\equiv -1$ mod $|d_{\boldsymbol K}|$, we have $\chi _{\boldsymbol K}(q)=-1$. \end{proof}
We return to the proof of the non-triviality of $f$. Note that $k-2<p-1$ by the assumption in Theorem \ref{Thm2}. By Lemma \ref{Lem2}, there exists a prime $q$ such that $\chi _{{\boldsymbol K}}(q)=-1$ and $q^{k-2}\not \equiv 1 \bmod{p}$. Considering the case $H=\left( \begin{smallmatrix} 1 & 0 \\ 0 & q \end{smallmatrix}\right)$, we have \begin{align*} a_{G_{k,{\boldsymbol K}}}(H)&=G_{\boldsymbol K}\!
(k-2,|d_{\boldsymbol K}|q)\\
&=1-q^{k-2}+|d_{\boldsymbol K}|^{k-2}-|d_{\boldsymbol K}|^{k-2}q^{k-2}\\
&=(1-q^{k-2})(1+|d_{\boldsymbol K}|^{k-2})\\ &\equiv 2(1-q^{k-2}) \bmod{p}. \end{align*} By the choice of $q$, this is not $0$ modulo $p$. This completes the proof of Theorem \ref{Thm2}. \end{proof}
\begin{proof}[Proof of Corollary \ref{Cor1}] See tables in $\S$ \ref{Table}. \\ (1)\; For the cases $d_{\boldsymbol{K}}=-3,-4,-7$, condition (A) is always satisfied. \\ (2)\; If $d_{\boldsymbol K}\neq -3$ and $k=8$, then there exists a prime $p$ satisfying conditions (A) and (B). Thus we the assertion of Corollary \ref{Cor1} (2) is obtained. \end{proof}
\section{Examples} \subsection{The case of Siegel modular forms} We set \begin{equation*} \begin{split} X_{10}:&=-\frac{43867}{2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 53}(E_{10}-E_4E_6), \\ X_{12}:&=\frac{131\cdot 593}{2^{11}\cdot 3^6\cdot 5^3\cdot 7^2\cdot 337} (3^2\cdot 7^2E_4^3+2\cdot 5^3E_6^2-691E_{12}). \end{split} \end{equation*} The following are cusp forms defined by Igusa.
\subsubsection*{The case of weight 10} \[ G_{10} \equiv 11313\cdot X_{10} \bmod{43867}. \] \begin{align*}
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 1 \end{pmatrix} \right) =-\frac{1618}{27} \equiv 11313= 11313\cdot a_{X_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 1 \end{pmatrix} \right) \bmod 43867,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) =-\frac{1385}{2} \equiv 11313\cdot (-2)= 11313\cdot a_{X_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) \bmod 43867,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 2 \end{pmatrix} \right) =-\frac{565184}{7} \equiv 11313\cdot (-16)= 11313\cdot a_{X_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 2 \end{pmatrix} \right) \bmod 43867,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) =-250737 \equiv 11313\cdot 36= 11313\cdot a_{X_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) \bmod 43867. \end{align*}
\subsubsection*{The case of weight 12} \[ G_{12} \equiv 53020\cdot X_{12} \bmod{131\cdot 593}. \] \begin{align*}
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 1 \end{pmatrix} \right) =\frac{3694}{3} \equiv 53020= 53020\cdot a_{X_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 1 \end{pmatrix} \right) \bmod 131\cdot 593,\\
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) =\frac{5052}{2} \equiv 53020\cdot 10= 53020\cdot a_{X_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) \bmod 131\cdot 593,\\
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 2 \end{pmatrix} \right) =9006448 \equiv 53020\cdot (-88)= 53020\cdot a_{X_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 2 \end{pmatrix} \right) \bmod 131\cdot 593,\\
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) =36581523 \equiv 53020\cdot (-132)= 53020\cdot a_{X_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) \bmod 131\cdot 593. \end{align*}
\subsection{The case of Hermitian modular forms} In this subsection, we introduce congruences for Hermitian modular forms. We use modular forms constructed in \cite{Kiku-Nag}.
\subsubsection*{Examples for $d_{\boldsymbol K}=-3$} We set \begin{equation*} \begin{split} F_{10}&= F_{10,\mathbb{Q}(\sqrt{-3})} :=-\frac{809}{21772800} (E_{10,\mathbb{Q}(\sqrt{-3})}-E_{4,\mathbb{Q}(\sqrt{-3})}\cdot E_{6,\mathbb{Q}(\sqrt{-3})}),\\ F_{12}&= F_{12,\mathbb{Q}(\sqrt{-3})} :=-\frac{1276277}{36578304000} \left(E_{12,\mathbb{Q}(\sqrt{-3})}-\frac{441}{691}E_{4,\mathbb{Q}(\sqrt{-3})}^3 -\frac{250}{691} E_{6,\mathbb{Q}(\sqrt{-3})}^2\right) \end{split} \end{equation*} (cf. \cite{Kiku-Nag}, Lemma 4.8 and Lemma 4.9).
\subsubsection*{The case of weight 10} \[ G_{10} \equiv 554\cdot F_{10} \bmod{809}, \] where $G_{10}=G_{10,\mathbb{Q}(\sqrt{-3})}$. \begin{align*}
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 1 \end{pmatrix} \right) =-255 \equiv 554= 554\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 1 \end{pmatrix} \right) \bmod 809,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) =-6560 \equiv 554\cdot (-6)= 554\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) \bmod 809,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 2 \end{pmatrix} \right) =-390624 \equiv 554\cdot (-10)= 554\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 2 \end{pmatrix} \right) \bmod 809,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) =-1673310 \equiv 554\cdot 90= 554\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) \bmod 809. \end{align*}
\subsubsection*{The case of weight 12} \[ G_{12} \equiv 824\cdot F_{12} \bmod{1847}, \] where $G_{12}=G_{12,\mathbb{Q}(\sqrt{-3})}$. \begin{align*}
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 1 \end{pmatrix} \right) =-1023 \equiv 824= 824\cdot a_{F_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 1 \end{pmatrix} \right) \bmod 1847,\\
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) =-59048 \equiv 824\cdot 18= 824\cdot a_{F_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) \bmod 1847,\\
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 2 \end{pmatrix} \right) =-9765624 \equiv 824\cdot (-106)= 824\cdot a_{F_{12}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{\sqrt{3}i}\\ -\tfrac{1}{\sqrt{3}i} & 2 \end{pmatrix} \right) \bmod 1847,\\
& a_{G_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) =-60408150 \equiv 824\cdot (-54)= 824\cdot a_{F_{12}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix} \right) \bmod 1847. \end{align*}
\subsubsection*{Examples for $d_{\boldsymbol K}=-4$} We set \begin{equation*} \begin{split} \chi_8& :=-\frac{61}{230400} (E_{8,\mathbb{Q}(\sqrt{-1})}-E_{4,\mathbb{Q}(\sqrt{-1})}^2), \\ F_{10}&= F_{10,\mathbb{Q}(\sqrt{-1})} :=-\frac{277}{2419200} (E_{10,\mathbb{Q}(\sqrt{-1})}-E_{4,\mathbb{Q}(\sqrt{-1})} E_{6,\mathbb{Q}(\sqrt{-1})}) \end{split} \end{equation*} (cf. \cite{Kiku-Nag}, Lemma 4.3 and Lemma 4.4).\\
\subsubsection*{The case of weight 8} \[ G_8 \equiv -2\cdot \chi_8 \bmod{61}, \] where $G_8=G_{8,\mathbb{Q}(\sqrt{-1})}$. \begin{align*}
& a_{G_8}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 1 \end{pmatrix} \right) =-63 \equiv -2= -2\cdot a_{\chi_8}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 1 \end{pmatrix} \right) \bmod 61,\\
& a_{G_8}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 1 \end{pmatrix} \right) =-728 \equiv -2\cdot (-2)= -2\cdot a_{\chi_8}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2}& 1 \end{pmatrix} \right) \bmod 61,\\
& a_{G_8}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) =-4095 \equiv -2\cdot 4= -2\cdot a_{\chi_8}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) \bmod 61,\\
& a_{G_8}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 2 \end{pmatrix} \right) =-47320 \equiv -2\cdot (-8)= -2\cdot a_{\chi_8}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 2 \end{pmatrix} \right) \bmod 61. \end{align*}
\subsubsection*{The case of weight 10} \[ G_{10} \equiv 22\cdot F_{10} \bmod{277}, \] where $G_{10}=G_{10,\mathbb{Q}(\sqrt{-1})}$.
\begin{align*}
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 1 \end{pmatrix} \right) =-255 \equiv 22= 22\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 1 \end{pmatrix} \right) \bmod 277,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2} & 1 \end{pmatrix} \right) =-6560 \equiv 22\cdot 4= 22\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1}{2}\\ \tfrac{1}{2}& 1 \end{pmatrix} \right) \bmod 277,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) =-65535 \equiv 22\cdot (-20)= 22\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \right) \bmod 277,\\
& a_{G_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 2 \end{pmatrix} \right) =-1685920 \equiv 22\cdot (-80)= 22\cdot a_{F_{10}}\!\!\left(\begin{pmatrix} 1 & \tfrac{1+i}{2}\\ \tfrac{1-i}{2} & 2 \end{pmatrix} \right) \bmod 277. \end{align*}
\section{Values of the generalized Bernoulli numbers} \subsection*{Values of $B_{k-1,\chi _{\boldsymbol K}}=\frac{N}{D}$ and $p$ satisfying (B) in Theorem \ref{Thm2}} \label{Table}
\begin{minipage}{0.5\hsize} \begin{center} $d_{\boldsymbol K}=-3$ \\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $3$ & -\\ $3$ & $2$ & $3$ & -\\ $5$ & $-2\cdot 5$ & $3$ & -\\ $7$ & $2\cdot 7^2$ & $3$ & -\\ $9$ & $-2\cdot 809$ & $3$ & $809$ \\ $11$ & $2\cdot 11\cdot 1847$ & $3$ & $1847$ \\ $13$ & $-2\cdot 7\cdot 13^3 \cdot 47$ & $3$ & $47$\\ $15$ & $2\cdot 5\cdot 419 \cdot 16519$ & $3$ & $419,\ 16519$ \end{tabular} \end{center} \end{minipage} \begin{minipage}{0.5\hsize} \begin{center} $d_{\boldsymbol K}=-4$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $2$ & -\\ $3$ & $3$ & $2$ & -\\ $5$ & $-5^2$ & $2$ & -\\ $7$ & $7\cdot 61$ & $2$ & $61$ \\ $9$ & $-3^2\cdot 5\cdot 277$ & $2$ & $277$ \\ $11$ & $11\cdot 19 \cdot 2659$ & $2$ & $19,\ 2659$ \\ $13$ & $-5\cdot 13^2\cdot 43 \cdot 967$ & $2$ & $43,\ 967$\\ $15$ & $3\cdot 5\cdot 47 \cdot 4241723$ & $2$ & $47,\ 4241723$ \end{tabular} \end{center} \end{minipage}
\begin{center} $d_{\boldsymbol K}=-7$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $2^4\cdot 3$ & $7$ & -\\ $5$ & $-2^5 \cdot 5$ & $1$ & -\\ $7$ & $2^4 \cdot 7 \cdot 73$ & $1$ & $73$\\ $9$ & $-2^6\cdot 3^2\cdot 8831$ & $7$ & $8831$ \\ $11$ & $2^4\cdot 11^2 \cdot 73\cdot 701$ & $1$ & $73,\ 701$ \\ $13$ & $-2^5\cdot 13\cdot 173 \cdot 266447$ & $1$ & $173,\ 266447$\\ $15$ & $2^4\cdot 3\cdot 5 \cdot 145764975331$ & $7$ & $145764975331$ \end{tabular} \end{center}
\begin{center} $d_{\boldsymbol K}=-8$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $3^2$ & $1$ & -\\ $5$ & $-3 \cdot 5 \cdot 19$ & $1$ & $19$ \\ $7$ & $3^2 \cdot 7 \cdot 307$ & $1$ & $307$\\ $9$ & $-3^3\cdot 83579$ & $1$ & $83579$ \\ $11$ & $3\cdot 11^2\cdot 23 \cdot 48197$ & $1$ & $23,\ 48197$ \\ $13$ & $-3^2\cdot 13\cdot 113 \cdot 811\cdot 9491$ & $1$ & $113,\ 811,\ 9491$\\ $15$ & $3^2\cdot 5\cdot 83\cdot 9275681267$ & $1$ & $83,\ 9275681267$ \end{tabular} \end{center}
\begin{center} $d_{\boldsymbol K}=-11$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $2\cdot 3^2$ & $1$ & -\\ $5$ & $- 2\cdot 3 \cdot 5^3 \cdot 17$ & $11$ & $17$ \\ $7$ & $2 \cdot 3^2 \cdot 7 \cdot 17 \cdot 71$ & $1$ & $17,\ 71$\\ $9$ & $-2\cdot 3^3 \cdot 5^3 \cdot 4999$ & $1$ & $4999$ \\ $11$ & $2\cdot 3\cdot 11 \cdot 43 \cdot 269\cdot 14923$ & $1$ & $43,\ 269,\ 14923$ \\ $13$ & $-2\cdot 3^2 \cdot 5^2 \cdot 13 \cdot 787 \cdot 1183579$ & $1$ & $787, \ 1183579$\\ $15$ & $2\cdot 3^2 \cdot 5\cdot 428708869630871$ & $11$ & $428708869630871$ \end{tabular} \end{center}
\begin{center} $d_{\boldsymbol K}=-19$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $2\cdot 3 \cdot 11$ & $1$ & $11$\\ $5$ & $- 2\cdot 5^2 \cdot 269$ & $1$ & $269$ \\ $7$ & $2 \cdot 7^2 \cdot 53 \cdot 1021$ & $1$ & $53,\ 1021$\\ $9$ & $-2\cdot 3^2 \cdot 5 \cdot 13 \cdot 67\cdot 851537$ & $19$ & $13,\ 67,\ 851537$ \\ $11$ & $2\cdot 11^3 \cdot 41 \cdot 32427511$ & $1$ & $41,\ 32427511$ \\ $13$ & $-2\cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 149 \cdot 3386245229$ & $1$ & $149,\ 3386245229$\\ $15$ & $2\cdot 3 \cdot 5 \cdot 829 \cdot 1249187 \cdot 312206737$ & $1$ & $829,\ 1249187,\ 312206737$ \end{tabular} \end{center}
\begin{center} $d_{\boldsymbol K}=-43$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $2\cdot 3 \cdot 83$ & $1$ & $83$\\ $5$ & $- 2\cdot 5 \cdot 29\cdot 31 \cdot 59$ & $1$ & $29,\ 31,\ 59$ \\ $7$ & $2 \cdot 7 \cdot 76565663$ & $1$ & $76565663$\\ $9$ & $-2\cdot 3^2 \cdot 202075601281$ & $1$ & $202075601281$ \\ $11$ & $2\cdot 11^2 \cdot 13^2 \cdot 509\cdot 901553753$ & $1$ & $509,\ 901553753$ \\ $13$ & $-2\cdot 13^2 \cdot 405842695582800517$ & $1$ & $405842695582800517$\\ $15$ & $2\cdot 3 \cdot 5 \cdot 223 \cdot 2791 \cdot 25889\cdot 113167\cdot 24665497$ & $1$ & $223,\ 2791,\ 25889\, 113167\, 24665497$ \end{tabular} \end{center}
\begin{center} $d_{\boldsymbol K}=-67$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $2\cdot 3 \cdot 251$ & $1$ & $251$\\ $5$ & $- 2\cdot 5 \cdot 19^2 \cdot 23 \cdot 47$ & $1$ & $19,\ 23,\ 47$ \\ $7$ & $2 \cdot 7 \cdot 1367650871$ & $1$ & $1367650871$\\ $9$ & $-2\cdot 3^2 \cdot 151 \cdot 58035119431$ & $1$ & $151,\ 58035119431$ \\ $11$ & $2\cdot 11 \cdot 3272681\cdot 27444275311$ & $1$ & $3272681,\ 27444275311$ \\ $13$ & $-2\cdot 13 \cdot 73\cdot 1439\cdot 56783\cdot 226088481721$ & $1$ & $73,\ 1439,\ 56783,\ 226088481721$\\ $15$ & $2\cdot 3 \cdot 5 \cdot 541355166251\cdot 51558395838661$ & $1$ & $541355166251,\ 51558395838661$ \end{tabular} \end{center}
\begin{center} $d_{\boldsymbol K}=-163$\\
\begin{tabular}{c|ccc} $k-1$ & $N$ & $D$ & $p$ \\ \hline $1$ & $-1$ & $1$ & -\\ $3$ & $2\cdot 3 \cdot 5\cdot 463$ & $1$ & $463$\\ $5$ & $- 2\cdot 5 \cdot 13^2 \cdot 281 \cdot 449$ & $1$ & $13,\ 281,\ 449$ \\ $7$ & $2 \cdot 5^3 \cdot 7 \cdot 3538330867$ & $1$ & $3538330867$\\ $9$ & $-2\cdot 3^2 \cdot 47 \cdot 1213\cdot 294217150811$ & $1$ & $47,\ 1213,\ 294217150811$ \\ $11$ & $2\cdot 5 \cdot 11 \cdot 29^2 \cdot 179\cdot 379\cdot 3566823499667$ & $1$ & $29,\ 179,\ 379,\ 3566823499667$ \\ $13$ & $-2\cdot 13 \cdot 103\cdot 172357\cdot 1097359\cdot 1883639\cdot 2464211$ & $1$ & $103,\ 172357,\ 1097359,\ 1883639,\ 2464211$\\ $15$ & $2\cdot 3 \cdot 5^2 \cdot 358181\cdot 6185071975972339006627199$ & $1$ & $ 358181,\ 6185071975972339006627199$ \end{tabular} \end{center}
\end{document} | arXiv |
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.[1] The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
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The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as $W=\mathbf {F} \cdot \mathbf {s} $, have natural continuous analogues in terms of line integrals, in this case $ W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} $, which computes the work done on an object moving through an electric or gravitational field F along a path $L$.
Vector calculus
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by z = f(x,y) and a curve C in the xy plane. The line integral of f would be the area of the "curtain" created—when the points of the surface that are directly over C are carved out.
Line integral of a scalar field
Definition
For some scalar field $f\colon U\to \mathbb {R} $ where $U\subseteq \mathbb {R} ^{n}$, the line integral along a piecewise smooth curve ${\mathcal {C}}\subset U$ is defined as
$\int _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)\left|\mathbf {r} '(t)\right|\,dt.$
where $\mathbf {r} \colon [a,b]\to {\mathcal {C}}$ is an arbitrary bijective parametrization of the curve ${\mathcal {C}}$ such that r(a) and r(b) give the endpoints of ${\mathcal {C}}$ and a < b. Here, and in the rest of the article, the absolute value bars denote the standard (Euclidean) norm of a vector.
The function f is called the integrand, the curve ${\mathcal {C}}$ is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length of the curve ${\mathcal {C}}$ (i.e., a differential length of ${\mathcal {C}}$). Line integrals of scalar fields over a curve ${\mathcal {C}}$ do not depend on the chosen parametrization r of ${\mathcal {C}}$.[2]
Geometrically, when the scalar field f is defined over a plane (n = 2), its graph is a surface z = f(x, y) in space, and the line integral gives the (signed) cross-sectional area bounded by the curve ${\mathcal {C}}$ and the graph of f. See the animation to the right.
Derivation
For a line integral over a scalar field, the integral can be constructed from a Riemann sum using the above definitions of f, C and a parametrization r of C. This can be done by partitioning the interval [a, b] into n sub-intervals [ti−1, ti] of length Δt = (b − a)/n, then r(ti) denotes some point, call it a sample point, on the curve C. We can use the set of sample points {r(ti): 1 ≤ i ≤ n} to approximate the curve C as a polygonal path by introducing the straight line piece between each of the sample points r(ti−1) and r(ti). (The approximation of a curve to a polygonal path is called rectification of a curve, see here for more details.) We then label the distance of the line segment between adjacent sample points on the curve as Δsi. The product of f(r(ti)) and Δsi can be associated with the signed area of a rectangle with a height and width of f(r(ti)) and Δsi, respectively. Taking the limit of the sum of the terms as the length of the partitions approaches zero gives us
$I=\lim _{\Delta s_{i}\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\,\Delta s_{i}.$
By the mean value theorem, the distance between subsequent points on the curve, is
$\Delta s_{i}=\left|\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\right|\approx \left|\mathbf {r} '(t_{i})\Delta t\right|$
Substituting this in the above Riemann sum yields
$I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}f(\mathbf {r} (t_{i}))\left|\mathbf {r} '(t_{i})\right|\Delta t$
which is the Riemann sum for the integral
$I=\int _{a}^{b}f(\mathbf {r} (t))\left|\mathbf {r} '(t)\right|dt.$
Definition
For a vector field F: U ⊆ Rn → Rn, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as
$\int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt$
where · is the dot product, and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.
A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line of the integration.
Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.[2]
From the viewpoint of differential geometry, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the musical isomorphism (which takes the vector field to the corresponding covector field), over the curve considered as an immersed 1-manifold.
Derivation
The line integral of a vector field can be derived in a manner very similar to the case of a scalar field, but this time with the inclusion of a dot product. Again using the above definitions of F, C and its parametrization r(t), we construct the integral from a Riemann sum. We partition the interval [a, b] (which is the range of the values of the parameter t) into n intervals of length Δt = (b − a)/n. Letting ti be the ith point on [a, b], then r(ti) gives us the position of the ith point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their displacement vectors, Δri. As before, evaluating F at all the points on the curve and taking the dot product with each displacement vector gives us the infinitesimal contribution of each partition of F on C. Letting the size of the partitions go to zero gives us a sum
$I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \Delta \mathbf {r} _{i}$
By the mean value theorem, we see that the displacement vector between adjacent points on the curve is
$\Delta \mathbf {r} _{i}=\mathbf {r} (t_{i}+\Delta t)-\mathbf {r} (t_{i})\approx \mathbf {r} '(t_{i})\,\Delta t.$
Substituting this in the above Riemann sum yields
$I=\lim _{\Delta t\to 0}\sum _{i=1}^{n}\mathbf {F} (\mathbf {r} (t_{i}))\cdot \mathbf {r} '(t_{i})\,\Delta t,$
which is the Riemann sum for the integral defined above.
Path independence
Main article: Gradient theorem
If a vector field F is the gradient of a scalar field G (i.e. if F is conservative), that is,
$\mathbf {F} =\nabla G,$
then by the multivariable chain rule the derivative of the composition of G and r(t) is
${\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)$
which happens to be the integrand for the line integral of F on r(t). It follows, given a path C, that
$\int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).$
In other words, the integral of F over C depends solely on the values of G at the points r(b) and r(a), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called path independent.
Applications
The line integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C.[3]
Flow across a curve
For a vector field $\mathbf {F} \colon U\subseteq \mathbb {R} ^{2}\to \mathbb {R} ^{2}$, F(x, y) = (P(x, y), Q(x, y)), the line integral across a curve C ⊂ U, also called the flux integral, is defined in terms of a piecewise smooth parametrization r: [a,b] → C, r(t) = (x(t), y(t)), as:
$\int _{C}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }=\int _{a}^{b}{\begin{bmatrix}P{\big (}x(t),y(t){\big )}\\Q{\big (}x(t),y(t){\big )}\end{bmatrix}}\cdot {\begin{bmatrix}y'(t)\\-x'(t)\end{bmatrix}}~dt=\int _{a}^{b}\left(-Q~dx+P~dy\right).$
Here ⋅ is the dot product, and $\mathbf {r} '(t)^{\perp }=(y'(t),-x'(t))$ is the clockwise perpendicular of the velocity vector $\mathbf {r} '(t)=(x'(t),y'(t))$.
The flow is computed in an oriented sense: the curve C has a specified forward direction from r(a) to r(b), and the flow is counted as positive when F(r(t)) is on the clockwise side of the forward velocity vector r'(t).
Complex line integral
In complex analysis, the line integral is defined in terms of multiplication and addition of complex numbers. Suppose U is an open subset of the complex plane C, f : U → C is a function, and $L\subset U$ is a curve of finite length, parametrized by γ: [a,b] → L, where γ(t) = x(t) + iy(t). The line integral
$\int _{L}f(z)\,dz$
may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression
$\sum _{k=1}^{n}f(\gamma (t_{k}))\,[\gamma (t_{k})-\gamma (t_{k-1})]=\sum _{k=1}^{n}f(\gamma _{k})\,\Delta \gamma _{k}.$
The integral is then the limit of this Riemann sum as the lengths of the subdivision intervals approach zero.
If the parametrization γ is continuously differentiable, the line integral can be evaluated as an integral of a function of a real variable:
$\int _{L}f(z)\,dz=\int _{a}^{b}f(\gamma (t))\gamma '(t)\,dt.$
When L is a closed curve (initial and final points coincide), the line integral is often denoted $ \oint _{L}f(z)\,dz,$ sometimes referred to in engineering as a cyclic integral.
The line integral with respect to the conjugate complex differential ${\overline {dz}}$ is defined[4] to be
$\int _{L}f(z){\overline {dz}}:={\overline {\int _{L}{\overline {f(z)}}\,dz}}=\int _{a}^{b}f(\gamma (t)){\overline {\gamma '(t)}}\,dt.$
The line integrals of complex functions can be evaluated using a number of techniques. The most direct is to split into real and imaginary parts, reducing the problem to evaluating two real-valued line integrals. The Cauchy integral theorem may be used to equate the line integral of an analytic function to the same integral over a more convenient curve. It also implies that over a closed curve enclosing a region where f(z) is analytic without singularities, the value of the integral is simply zero, or in case the region includes singularities, the residue theorem computes the integral in terms of the singularities. This also implies the path independence of complex line integral for analytic functions.
Example
Consider the function f(z) = 1/z, and let the contour L be the counterclockwise unit circle about 0, parametrized by z(t) = eit with t in [0, 2π] using the complex exponential. Substituting, we find:
${\begin{aligned}\oint _{L}{\frac {1}{z}}\,dz&=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }e^{-it}e^{it}\,dt\\&=i\int _{0}^{2\pi }dt=i(2\pi -0)=2\pi i.\end{aligned}}$
This is a typical result of Cauchy's integral formula and the residue theorem.
Relation of complex line integral and line integral of vector field
Viewing complex numbers as 2-dimensional vectors, the line integral of a complex-valued function $f(z)$ has real and complex parts equal to the line integral and the flux integral of the vector field corresponding to the conjugate function ${\overline {f(z)}}.$ Specifically, if $\mathbf {r} (t)=(x(t),y(t))$ parametrizes L, and $f(z)=u(z)+iv(z)$ corresponds to the vector field $\mathbf {F} (x,y)={\overline {f(x+iy)}}=(u(x+iy),-v(x+iy)),$ then:
${\begin{aligned}\int _{L}f(z)\,dz&=\int _{L}(u+iv)(dx+i\,dy)\\&=\int _{L}(u,-v)\cdot (dx,dy)+i\int _{L}(u,-v)\cdot (dy,-dx)\\&=\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} +i\int _{L}\mathbf {F} (\mathbf {r} )\cdot d\mathbf {r} ^{\perp }.\end{aligned}}$
By Cauchy's theorem, the left-hand integral is zero when $f(z)$ is analytic (satisfying the Cauchy–Riemann equations) for any smooth closed curve L. Correspondingly, by Green's theorem, the right-hand integrals are zero when $\mathbf {F} ={\overline {f(z)}}$ is irrotational (curl-free) and incompressible (divergence-free). In fact, the Cauchy-Riemann equations for $f(z)$ are identical to the vanishing of curl and divergence for F.
By Green's theorem, the area of a region enclosed by a smooth, closed, positively oriented curve $L$ is given by the integral $ {\frac {1}{2i}}\int _{L}{\overline {z}}\,dz.$ This fact is used, for example, in the proof of the area theorem.
Quantum mechanics
The path integral formulation of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function of a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.
See also
• Divergence theorem
• Gradient theorem
• Methods of contour integration
• Nachbin's theorem
• Surface integral
• Volume element
• Volume integral
References
1. Kwong-Tin Tang (30 November 2006). Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms. Springer Science & Business Media. ISBN 978-3-540-30268-1.
2. Nykamp, Duane. "Line integrals are independent of parametrization". Math Insight. Retrieved September 18, 2020.
3. "16.2 Line Integrals". www.whitman.edu. Retrieved 2020-09-18.
4. Ahlfors, Lars (1966). Complex Analysis (2nd ed.). New York: McGraw-Hill. p. 103.
External links
• "Integral over trajectories", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Khan Academy modules:
• "Introduction to the Line Integral"
• "Line Integral Example 1"
• "Line Integral Example 2 (part 1)"
• "Line Integral Example 2 (part 2)"
• Path integral at PlanetMath.
• Line integral of a vector field – Interactive
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| Wikipedia |
Sylvia Wiegand
Sylvia Margaret Wiegand (born March 8, 1945) is an American mathematician.[1]
Sylvia Margaret Wiegand
Born (1945-03-08) March 8, 1945
Cape Town, South Africa
Alma materUniversity of Wisconsin-Madison
Scientific career
FieldsCommutative algebra
math education, history of math
ThesisGalois Theory of Essential Expansions of Modules and Vanishing Tensor Powers (1972)
Doctoral advisorLawrence S. Levy
Doctoral studentsChristina Eubanks-Turner
Early life and education
Wiegand was born in Cape Town, South Africa. She is the daughter of mathematician Laurence Chisholm Young and through him the grand-daughter of mathematicians Grace Chisholm Young and William Henry Young.[2] Her family moved to Wisconsin in 1949, and she graduated from Bryn Mawr College in 1966 after three years of study.[1] In 1971 Wiegand earned her Ph.D. from the University of Wisconsin-Madison.[3] Her dissertation was titled Galois Theory of Essential Expansions of Modules and Vanishing Tensor Powers.[3]
Career
In 1987, she was named full professor at the University of Nebraska; at the time Wiegand was the only female professor in the department.[1] In 1988 Sylvia headed a search committee for two new jobs in the math department, for which two women were hired, although one stayed only a year and another left after four years.[4] In 1996 Sylvia and her husband, Roger Wiegand, established a fellowship for graduate student research at the university in honor of Sylvia's grandparents.[5]
From 1997 until 2000, Wiegand was president of the Association for Women in Mathematics.[6][7]
Wiegand has been an editor for Communications in Algebra and the Rocky Mountain Journal of Mathematics.[2] She was on the board of directors of the Canadian Mathematical Society from 1997 to 2000.[2]
Wiegand was an American Mathematical Society (AMS) Council member at large.[8]
Awards and recognition
Wiegand is featured in the book Notable Women in Mathematics: A Biographical Dictionary, edited by Charlene Morrow and Teri Perl, published in 1998.[1] For her work in improving the status of women in mathematics, she was awarded the University of Nebraska's Outstanding Contribution to the Status of Women Award in 2000.[4] In May 2005, the University of Nebraska hosted the Nebraska Commutative Algebra Conference: WiegandFest "in celebration of the many important contributions of Sylvia and her husband Roger Wiegand."[1]
In 2012 she became a fellow of the AMS.[9]
In 2017, she was selected as a fellow of the Association for Women in Mathematics in the inaugural class.[10]
References
1. "Sylvia Wiegand". Agnesscott.edu. 1945-03-08. Retrieved 2012-10-31.
2. "Sylvia Wiegand". www.agnesscott.edu. Retrieved 2018-10-06.
3. Sylvia Wiegand at the Mathematics Genealogy Project
4. "OCWW | Vol 32, Issue 3-4 | Features". Aacu.org. Archived from the original on 2003-11-10. Retrieved 2012-10-31.
5. PO BOX 880130 (2010-11-18). "UNL | Arts & Sciences | Math | Department | Awards | Graduate Student Awards". Math.unl.edu. Retrieved 2012-10-31.
6. "Sylvia Wiegand's Homepage". Math.unl.edu. Retrieved 2012-10-31.
7. "AWM Profile" (PDF). Ams.org. Retrieved 2012-10-31.
8. "AMS Committees". American Mathematical Society. Retrieved 2023-03-27.
9. List of Fellows of the American Mathematical Society, retrieved 2013-09-01.
10. "2018 Inaugural Class of AWM Fellows". Association for Women in Mathematics. Retrieved 9 January 2021.
External links
• Sylvia Wiegand's homepage
• Sylvia Wiegand's Author profile on MathSciNet
Presidents of the Association for Women in Mathematics
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| Wikipedia |
MRS Online Proceedings Library Archive (7)
British Journal of Nutrition (5)
Canadian Journal of Neurological Sciences (4)
Behavioral and Brain Sciences (1)
Bulletin of the Australian Mathematical Society (1)
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Cardiology in the Young (1)
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The Experience of Using Informational Systems to Improve the ACLS Process Optimization in the Emergency Department
Pei Fang Lai, Ying Fang Zhou, Pin Shou Chen
Journal: Prehospital and Disaster Medicine / Volume 34 / Issue s1 / May 2019
Published online by Cambridge University Press: 06 May 2019, p. s131
Print publication: May 2019
The best first-aid treatment for cardiac arrest patients is Advanced Cardiac Life Support (ACLS) to not only hope to save lives but to also leave minimal sequelae. The American Heart Association (AHA) published updated ACLS guidelines for care in 2015 emphasizing the concept of teamwork in resuscitation. However, the actual use of ACLS is not easy due to stress and unfamiliarity with the process.
Therefore, we want to use the information technology to assist the medical team to implement the ACLS process. This information system can help us to save time and labor, as well as increase precision. In addition to this, data analysis is more convenient, which facilitates the management and supervision of resuscitation quality.
An information system was developed using responsive web design (RWD) website. It can be used on a variety of devices, such as desktops, tablets, or mobile phones, and can be updated simultaneously. The system requires non-synchronous operation to be used in a wireless network environment. When the information system is in operation, the medical personnel can perform the resuscitation actions according to voice prompts, which can periodically remind staff to check rhythm, give correct medication dose, and identify whether defibrillation shock is needed. At the same time, the entire process can be recorded instantly. After the file is uploaded, the medical records are complete at the same time.
After 3 months, the satisfaction of medical staff reached 80.3%, the rate of return of spontaneous circulation (ROSC) of OHCA cases elevated to 45% from 15%, and discharge without neurological sequelae elevated to 33% from 27.4%.
All hospital staff can use this system to assist in the correct implementation of advanced CPR. It improves the quality of resuscitation and reduces the burden on clinical and writing medical records of medical staff.
The prevalence, temporal trends, and geographical distribution of HIV-1 subtypes among men who have sex with men in China: A systematic review and meta-analysis
Yueqi Yin, Yuxiang Liu, Jing Zhu, Xiang Hong, Rui Yuan, Gengfeng Fu, Ying Zhou, Bei Wang
Published online by Cambridge University Press: 19 February 2019, e83
The aim of this meta-analysis was to provide a comprehensive overview of human immunodeficiency virus (HIV)-1 subtypes and to investigate temporal and geographical trends of the HIV-1 epidemic among men who have sex with men (MSM) in China. Chinese and English articles published between January 2007 and December 2017 were systematically searched. Pooled HIV-1 prevalence was calculated, and its stability was analysed using sensitivity analysis. Subgroups were based on study time period, sampling area and prevalence. Publication bias was measured using Funnel plot and Egger's test. A total of 68 independent studies that included HIV-1 molecular investigations were eligible for meta-analysis. Circulating recombinant form (CRF) 01_AE (57.36%, 95% confidence interval (CI) 53.76–60.92) was confirmed as the most prevalent HIV-1 subtype among MSM in China. Subgroup analysis for time period found that CRF01_AE steadily increased prior to 2012 but decreased during 2012–2016. Further whereas CRF07_BC increased over time, B/B′ decreased over time. CRF55_01B has increased in recent years, with higher pooled estimated rate in Guangdong (12.22%, 95% CI 10.34–13.17) and Fujian (8.65%, 95% CI 4.98–13.17) provinces. The distribution of HIV-1 subtypes among MSM in China has changed across different regions and periods. HIV-1 strains in MSM are becoming more complex. Long-term molecular monitoring in this population remains necessary for HIV-1 epidemic control and prevention.
Breakfast replacement with a liquid formula improves glycaemic variability in patients with type 2 diabetes: a randomised clinical trial
Jiahui Peng, Jingyi Lu, Xiaojing Ma, Lingwen Ying, Wei Lu, Wei Zhu, Yuqian Bao, Jian Zhou
Journal: British Journal of Nutrition / Volume 121 / Issue 5 / 14 March 2019
Print publication: 14 March 2019
There is emerging evidence that glycaemic variability (GV) plays an important role in the development of diabetic complications. The current study aimed to compare the effects of lifestyle intervention (LI) with and without partial meal replacement (MR) on GV. A total of 123 patients with newly diagnosed and untreated type 2 diabetes (T2D) were randomised to receive either LI together with breakfast replacement with a liquid formula (LI+MR) (n 62) or LI alone (n 61) for 4 weeks and completed the study. Each participant was instructed to have three main meals per d and underwent 72-h continuous glucose monitoring (CGM) both before and after intervention. Measures of GV assessed by CGM included the incremental AUC of postprandial blood glucose (AUCpp), standard deviation of blood glucose (SDBG), glucose CV and mean amplitude of glycaemic excursions (MAGE). After a 4-week intervention, the improvements in systolic blood pressure (P=0·046) and time in range (P=0·033) were more pronounced in the LI+MR group than in the LI group. Furthermore, LI+MR caused significantly greater improvements in all GV metrics including SDBG (P=0·005), CV (P=0·002), MAGE (P=0·016) and AUCpp (P<0·001) than did LI. LI+MR (v. LI) was independently associated with improvements in GV after adjustment of covariates (all P<0·05). Our study showed that LI+MR led to significantly greater improvements in GV compared with LI, suggesting that LI+MR could be an effective treatment to alleviate glucose excursions.
Optimality is critical when it comes to testing computation-level hypotheses
Laura S. Geurts, Andrey Chetverikov, Ruben S. van Bergen, Ying J. Zhou, Andrea Bertana, Janneke F. M. Jehee
Journal: Behavioral and Brain Sciences / Volume 41 / 2018
Published online by Cambridge University Press: 10 January 2019, e231
We disagree with Rahnev & Denison (R&D) that optimality should be abandoned altogether. Rather, we argue that adopting a normative approach enables researchers to test hypotheses about the brain's computational goals, avoids just-so explanations, and offers insights into function that are simply inaccessible to the alternatives proposed by R&D.
Associations of plasma very-long-chain SFA and the metabolic syndrome in adults – ERRATUM
Jing Zhao, Xiaofan Li, Xiang Li, Qianqian Chu, Yunhua Zhou, Zi Li, Hong Zhang, Thomas J. Brenna, Yiqing Song, Ying Gao
Published online by Cambridge University Press: 07 December 2018, p. 120
Molecular investigation of the Pfmdr1 gene of Plasmodium falciparum isolates in Henan Province imported from Africa
Chengyun Yang, Ruimin Zhou, Ying Liu, Suhua Li, Dan Qian, Yuling Zhao, Hongwei Zhang, Bianli Xu
Journal: Parasitology / Volume 146 / Issue 3 / March 2019
Efficacious antimalarial drugs are important for malaria control and elimination, and continuous monitoring of their efficacy is essential. The prevalence and distribution of Pfmdr1 were evaluated in African migrant workers in Henan Province. Among 632 isolates, 13 haplotypes were identified, NYSND (39.87%, 252/632), YYSND (2.85%, 18/632), NFSND (31.01%, 196/632), NYSNY (0.47%, 3/632), YFSND (13.77%, 87/632), NFSNY (0.32%, 2/632), YYSNY (2.06%, 13/632), YFSNY (0.16%, 1/632), N/Y YSND (1.90%, 12/632), N Y/F SND (6.17%, 39/632), N/Y Y/F SND (0.47%, 3/632), YYSN D/Y (0.16%, 1/632) and N/Y FSND (0.79%, 5/632). The highest frequency of NYSND was observed in individuals from North Africa (63.64%, 7/11), followed by South Africa (61.33%, 111/181), Central Africa (33.33%, 56/168), West Africa (28.94%, 68/235) and East Africa (27.03%, 10/37) (χ2 = 54.605, P < 0.05). The highest frequency of NFSND was observed in East Africa (48.65%, 18/37), followed by West Africa (39.14%, 92/235), Central Africa (26.79%, 45/168), South Africa (22.65%, 41/181) and North Africa (9.09%, 1/11) (χ2 = 22.368 P < 0.05). The mutant prevalence of codons 86 and 184 decreased. These data may provide complementary information on antimalarial resistance that may be utilized in the development of a treatment regimen for Henan Province.
Associations of plasma very-long-chain SFA and the metabolic syndrome in adults
Journal: British Journal of Nutrition / Volume 120 / Issue 8 / 28 October 2018
Print publication: 28 October 2018
Plasma levels of very-long-chain SFA (VLCSFA) are associated with the metabolic syndrome (MetS). However, the associations may vary by different biological activities of individual VLCSFA or population characteristics. We aimed to examine the associations of VLCSFA and MetS risk in Chinese adults. Totally, 2008 Chinese population aged 35–59 years were recruited and followed up from 2010 to 2012. Baseline MetS status and plasma fatty acids data were available for 1729 individuals without serious diseases. Among 899 initially metabolically healthy individuals, we identified 212 incident MetS during the follow-up. Logistic regression analysis was used to estimate OR and 95 % CI. Cross-sectionally, each VLCSFA was inversely associated with MetS risk; comparing with the lowest quartile, the multivariate-adjusted OR for the highest quartile were 0·18 (95 % CI 0·13, 0·25) for C20 : 0, 0·26 (95 % CI 0·18, 0·35) for C22 : 0, 0·19 (95 % CI 0·13, 0·26) for C24 : 0 and 0·16 (0·11, 0·22) for total VLCSFA (all Pfor trend<0·001). The associations remained significant after further adjusting for C16 : 0, C18 : 0, C18 : 3n-3, C22 : 6n-3, n-6 PUFA and MUFA, respectively. Based on follow-up data, C20 : 0 or C22 : 0 was also inversely associated with incident MetS risk. Among the five individual MetS components, higher levels of VLCSFA were most strongly inversely associated with elevated TAG (≥1·7 mmol/l). Plasma levels of VLCSFA were significantly and inversely associated with MetS risk and individual MetS components, especially TAG. Further studies are warranted to confirm the findings and explore underlying mechanisms.
BDNF Val66Met in preclinical Alzheimer's disease is associated with short-term changes in episodic memory and hippocampal volume but not serum mBDNF
Yen Ying Lim, Stephanie Rainey-Smith, Yoon Lim, Simon M. Laws, Veer Gupta, Tenielle Porter, Pierrick Bourgeat, David Ames, Christopher Fowler, Olivier Salvado, Victor L. Villemagne, Christopher C. Rowe, Colin L. Masters, Xin Fu Zhou, Ralph N. Martins, Paul Maruff
Journal: International Psychogeriatrics / Volume 29 / Issue 11 / November 2017
Published online by Cambridge University Press: 19 July 2017, pp. 1825-1834
Print publication: November 2017
The brain-derived neurotrophic factor (BDNF) Val66Met polymorphism Met allele exacerbates amyloid (Aβ) related decline in episodic memory (EM) and hippocampal volume (HV) over 36–54 months in preclinical Alzheimer's disease (AD). However, the extent to which Aβ+ and BDNF Val66Met is related to circulating markers of BDNF (e.g. serum) is unknown. We aimed to determine the effect of Aβ and the BDNF Val66Met polymorphism on levels of serum mBDNF, EM, and HV at baseline and over 18-months.
Non-demented older adults (n = 446) underwent Aβ neuroimaging and BDNF Val66Met genotyping. EM and HV were assessed at baseline and 18 months later. Fasted blood samples were obtained from each participant at baseline and at 18-month follow-up. Aβ PET neuroimaging was used to classify participants as Aβ– or Aβ+.
At baseline, Aβ+ adults showed worse EM impairment and lower serum mBDNF levels relative to Aβ- adults. BDNF Val66Met polymorphism did not affect serum mBDNF, EM, or HV at baseline. When considered over 18-months, compared to Aβ– Val homozygotes, Aβ+ Val homozygotes showed significant decline in EM and HV but not serum mBDNF. Similarly, compared to Aβ+ Val homozygotes, Aβ+ Met carriers showed significant decline in EM and HV over 18-months but showed no change in serum mBDNF.
While allelic variation in BDNF Val66Met may influence Aβ+ related neurodegeneration and memory loss over the short term, this is not related to serum mBDNF. Longer follow-up intervals may be required to further determine any relationships between serum mBDNF, EM, and HV in preclinical AD.
Well dispersed Fe2N nanoparticles on surface of nitrogen-doped reduced graphite oxide for highly efficient electrochemical hydrogen evolution
Yi Zhang, Ying Xie, Yangtao Zhou, Xiuwen Wang, Kai Pan
Journal: Journal of Materials Research / Volume 32 / Issue 9 / 15 May 2017
Published online by Cambridge University Press: 20 April 2017, pp. 1770-1776
It is important to fabricate iron-based nitride/conductive material composite to obtain good catalytic performance. In this work, Fe2N nanoparticles with diameter of approximately 30 nm have been successfully dispersed on the surface of nitrogen-doped graphite oxide (NrGO) by a facile sol–gel method and further ammonia atmosphere treatment. XPS, XRD, Raman, and TEM proved that Fe2N nanoparticles are well monodispersed, and nitrogen atoms are doped in NrGO. The composite possessed two merits, that is, the more catalytic active site in Fe2N nanoparticles due to the well monodispersion, and fast electron transfer due to the nitrogen dope in rGO. With the proper ratio, the composite exhibited brilliant catalytic activity and durability in acidic media. It possesses overpotential of 94 mV to approach 10 mA/cm2, a small Tefel slope of 49 mV/dec, and maintains the good electrocatalytic activity for 10 h. Cyclic voltammetry and electrochemical impedance spectroscopy indicated that the electrocatalyst possessed high catalytic active area and fast electron transfer. Our work may provide a new avenue for the preparation of low-cost iron-based nitride/NrGO composite for highly efficient electrochemical hydrogen evolution.
Rings in which Every Element is a Sum of Two Tripotents
Zhiling Ying, Tamer Koşan, Yiqiang Zhou
Journal: Canadian Mathematical Bulletin / Volume 59 / Issue 3 / 01 September 2016
Print publication: 01 September 2016
Let $R$ be a ring. The following results are proved. $\left( 1 \right)$ Every element of $R$ is a sum of an idempotent and a tripotent that commute if and only if $R$ has the identity ${{x}^{6}}\,=\,{{x}^{4}}$ if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$ , where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$ . $\left( 2 \right)$ Every element of $R$ is either a sum or a difference of two commuting idempotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$ , where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $J\left( R \right)\,=\,0$ or $J\left( R \right)\,=\,\left\{ 0,\,2 \right\}$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$ . $\left( 3 \right)$ Every element of $R$ is a sum of two commuting tripotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}\,\times \,{{R}_{3}}$ , where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ , ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$ , and ${{R}_{3}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{5}}^{,}s$ .
Oxidative Stress and Environmental Exposures are Associated with Multiple System Atrophy in Chinese Patients
Linli Zhou, Ying Jiang, Cansheng Zhu, Lili Ma, Qiling Huang, Xiaohong Chen
Journal: Canadian Journal of Neurological Sciences / Volume 43 / Issue 5 / September 2016
Objective: Oxidative stress is involved in the pathogenesis of multiple system atrophy (MSA). The aim of this study is to examine oxidant biomarkers including homocysteine (Hcys), bilirubin, uric acid, lipids, and potential environmental risk factors and to ascertain whether these data correlate with MSA in a Chinese population. Methods: In this study, serum levels of Hcys, bilirubin, uric acid, and lipids were studied in 55 MSA patients and 76 healthy controls (HCs). Education, anti-parkinsonian agent usage, smoking, drinking, farming, and living area of the subjects also were analyzed. The Unified MSA Rating Scale (UMSARS), Hoehn & Yahr stage, International Cooperative Ataxia Rating Scale, and Mini-Mental State Examination were used to assess the disease severity, the parkinsonism, ataxia, and the cognitive ability of MSA, respectively. Results: The levels of Hcys were higher (p<0.001) and those of total bilirubin (p=0.007), indirect bilirubin (p=0.011), and total cholesterol (p=0.046) were lower in MSA patients than in healthy controls, whereas uric acid levels did not differ significantly between MSA and healthy controls. Moreover, Hcys levels in MSA patients had positive correlations with illness duration (r s =0.422, p=0.001) and UMSARS-I (r s =0.555, p<0.001), respectively. High-density lipoprotein cholesterol levels were negatively correlated with UMSARS-I (r s =−0.325, p=0.015). Farming was more frequent in MSA patients (1-20 years: odds ratio, 6.36; p<0.001; >20 years: odds ratio, 10.26; p=0.001), whereas current smoking was less frequent (odds ratio, 0.13, p=0.002). Conclusions: Elevated Hcys and decreased high-density lipoprotein cholesterol may be associated with the disease severity of MSA. Environmental exposures such as farming and smoking may contribute to the occurrence but not the progression of MSA.
Electrochemical corrosion behaviors of a stress-aged Al–Zn–Mg–Cu alloy
Y.C. Lin, Jin-Long Zhang, Ming-Song Chen, Ying Zhou, Xiang Ma
Journal: Journal of Materials Research / Volume 31 / Issue 16 / 29 August 2016
Published online by Cambridge University Press: 24 June 2016, pp. 2493-2505
The effects of stress-aging processing on corrosion resistance of an Al–Zn–Mg–Cu alloy were investigated. It is found that the one-stage stress-aged alloy is strongly sensitive to the electrochemical corrosion. The poor corrosion resistance of the one-stage stress-aged alloy can be attributed to fine intragranular aging precipitates and continuous distribution of grain boundary precipitates. Meanwhile, the incomplete precipitation of solute atoms results in high electrochemical activity of aluminum matrix. However, when the alloy is two-stage stress-aged, the corrosion resistance is greatly improved. Furthermore, the corrosion resistance decreases firstly and then increases with increasing the first stage stress-aging temperature. Increasing external stress can enhance the corrosion resistance of the two-stage stress-aged alloy. These phenomena are mainly related to aging precipitates within grains and along grain boundaries. The coarse and relatively low-density intragranular aging precipitates, as well as the discontinuously distributed grain boundary precipitates can enhance the corrosion resistance of the stress-aged alloy.
Effects of pre-treatments on precipitate microstructures and creep-rupture behavior of an Al–Zn–Mg–Cu alloy
Y.C. Lin, Zong-Wei Wang, Dao-Guang He, Ying Zhou, Ming-Song Chen, Ming-Hui Huang, Jin-Long Zhang
The effects of pre-treatments on the precipitate microstructures of an Al–Zn–Mg–Cu alloy are investigated. Meanwhile, the creep-rupture behavior of the under-aged and peak-aged alloys are comparatively analyzed. Additionally, the effects of pre-treatment on the fracture mechanisms are discussed. It is found that the precipitate microstructures are sensitive to pre-treatments. The intragranular precipitates of the peak-aged alloy are larger than those of the under-aged. The precipitate free zone of the peak-aged alloy is wider than that of the under-aged. Some large intergranular precipitates appear on the grain boundaries of the under-aged alloy, and induce the nucleation of microvoids. Eventually, the creep fracture of the under-aged alloy is accelerated. Therefore, the differences in microstructures lead to the shorter creep-rupture life of the under-aged alloy, compared to the peak-aged alloy.
Surface phase defects induced downstream laser intensity modulation in high-power laser facility
Xin Zhang, Wei Zhou, Wanjun Dai, Dongxia Hu, Xuewei Deng, Wanqing Huang, Lidan Zhou, Qiang Yuan, Xiaoxia Huang, De'en Wang, Ying Yang
Published online by Cambridge University Press: 02 March 2016, e6
Optics surface phase defects induced intensity modulation in high-power laser facility for inertial confinement fusion research is studied. Calculations and experiments reveal an exact mapping of the modulation patterns and the optics damage spot distributions from the surface phase defects. Origins are discussed during the processes of optics manufacturing and diagnostics, revealing potential improvements for future optics manufacturing techniques and diagnostic index, which is meaningful for fusion level laser facility construction and its operation safety.
A nonparametric method to test for associations between rare variants and multiple traits
YING ZHOU, YANGYANG CHENG, WENSHENG ZHU, QIAN ZHOU
Journal: Genetics Research / Volume 98 / 2016
Published online by Cambridge University Press: 13 January 2016, e1
More and more rare genetic variants are being detected in the human genome, and it is believed that besides common variants, some rare variants also explain part of the phenotypic variance for human diseases. Due to the importance of rare variants, many statistical methods have been proposed to test for associations between rare variants and human traits. However, in existing studies, most methods only test for associations between multiple loci and one trait; therefore, the joint information of multiple traits has not been considered simultaneously and sufficiently. In this article, we present a study of testing for associations between rare variants and multiple traits, where trait value can be binary, ordinal, quantitative and/or any mixture of them. Based on the method of generalized Kendall's τ, a nonparametric method called NM-RV is proposed. A new kernel function for U-statistic, which could incorporate the information of each rare variant itself, is also presented and is expected to enhance the power of rare variant analysis. We further consider the asymptotic distribution of the proposed association test statistic. Our simulation work suggests that the proposed method is more powerful and robust than existing methods in testing for associations between rare variants and multiple traits, especially for multivariate ordinal traits.
Ion size effects on thermoluminescence of terbium and europium doped magnesium orthosilicate
Ying Zhao, Yang Zhou, Yun Jiang, Weigong Zhou, Adrian A. Finch, Peter D. Townsend, Yafang Wang
Journal: Journal of Materials Research / Volume 30 / Issue 22 / 27 November 2015
Thermoluminescence (TL) and radioluminescence (RL) are reported over the temperature range 25–673 K from MgSiO4:Tb and MgSiO4:Eu. The dominant signals arise from the transitions within the Rare Earth (RE) dopants, with limited intensity from intrinsic or host defect sites. The Tb and Eu ions distort the lattice and alter the stability of the TL sites and the peak TL temperature scales with the Tb and Eu ion size. The larger Eu ions stabilize the trapped charges more than for the Tb, and so the Eu TL peak temperatures are ∼20% higher. There are further size effects linked to the TL driven by the volume of the upper state orbitals of the rare earth transitions. For Eu the temperatures of the TL peaks are wavelength dependent since higher excited states couple to distant traps via more extensive orbits. The same pattern of peak temperature data is encoded in RL during heating. The data imply that there are sites in which the rare earth and charge stabilizing defects are closely associated within the host lattice, and the stability of the entire complex is linked to the lattice distortions from inclusions of impurities.
Genotype-based clinical manifestation and treatment of Chinese long QT syndrome patients with KCNQ1 mutations – R380S and W305L
Hui Zhou, Wei Lai, Wengen Zhu, Jinyan Xie, Xin Liu, Yang Shen, Ping Yuan, Ying Liu, Qin Cao, Wenfeng He, Kui Hong
Journal: Cardiology in the Young / Volume 26 / Issue 4 / April 2016
Most long QT syndrome patients are associated with genetic mutations. We aimed to investigate the clinical and biochemical characteristics and look for genotype-based preventive implications in Chinese long QT syndrome patients.
Methods and results
We identified two missense mutations of the KCNQ1 gene in two independent Chinese families, including a previously reported mutation R380S in the C-terminus and a novel mutation W305L in the P-loop domain of the Kv7.1 channel, respectively. The proband with R380S was an 11-year-old girl who suffered a prolonged corrected QT interval of 660 ms, recurrent syncope, and sudden cardiac death, whose father was an asymptomatic carrier. The mutation W305L was detected in a 36-year-old woman with long QT syndrome and her immediate family members including the proband's younger sister with an unexplained syncope, her son, and her elder daughter without symptoms. Metoprolol appeared to be effective in preventing arrhythmias and syncope in long QT syndrome patients with mutation W305L. Both R380S and W305L mutations led to "loss-of-function" of the Kv7.1 channel accounting for the clinical phenotypes.
We first show two missense KCNQ1 mutations – R380S and W305L – in Chinese long QT syndrome patients, resulting in the loss of protein function. Mutation W305L in the P-loop domain of the Kv7.1 may derive a pronounced benefit from β-blocker therapy in symptomatic long QT syndrome patients, whereas mutation R380S located in the C-terminus may be associated with a high risk of sudden cardiac death.
Emergence of a Novel Binary Toxin–Positive Strain of Clostridium difficile Associated With Severe Diarrhea That Was Not Ribotype 027 and 078 in China
Chunhui Li, Sidi Liu, Pengcheng Zhou, Juping Duan, Qingya Dou, Rui Zhang, Hong Chen, Ying Cheng, Anhua Wu
Journal: Infection Control & Hospital Epidemiology / Volume 36 / Issue 9 / September 2015
Mortality from Parkinson's disease in China: Findings from a five-year follow up study in Shanghai
Gang Wang, Xin-Jian Li, Yi-Song Hu, Qi Cheng, Chun-Fang Wang, Qin Xiao, Jun Liu, Jian-Fang Ma, Hai-Yan Zhou, Jing Pan, Yu-Yan Tan, Ying Wang, Sheng-Di Chen
Journal: Canadian Journal of Neurological Sciences / Volume 42 / Issue 4 / July 2015
Introduction: The mortality of Parkinson's disease (PD) and its associated risk factors among clinically definite PD patients in China has been rarely investigated. Our study aimed to identify the mortality rates and predictors of death in PD patients in China. Methods: 157 consecutive, clinically definite PD patients from the urban area of Shanghai were recruited from a central hospital based movement disorder clinic in 2006. All patients were regularly followed up at the clinic until December 31, 2011, or death. Mortality and associations with baseline demographics, health and medical factors were then determined within the cohort. Results: After 5 years, 11(7%) patients had died. The standardised mortality ratio was 0.62 (95% CI 0.32 to 1.07, P=0.104). The main causes of death were pneumonia (54.5%, 6/11) and digestive disorders (18.2%, 2/11), respectively. Age at onset, independent living, the mini mental state examination score, the Parkinson's disease sleep scale score and the Epworth sleepiness scale score at baseline were statistically significantly different between the survival group and the deceased group (P<0.05). Across all participants, risk factors for death included low mini mental state examination score, and high Epworth sleepiness scale score according to a binary variable logistic regression analysis. Conclusions: This study confirms the similar survival of patients with PD to the control population up to a follow-up of 5 years. Interventions tailored to potential risk factors associated with death may offer further benefits.
Genome-wide interaction analysis of quantitative traits in outbred mice
WEIJUN MA, CHAOFENG YUAN, HAIDONG LIU, WEI ZHENG, YING ZHOU
Published online by Cambridge University Press: 20 April 2015, e9
With a large number of quantitative trait loci being identified in genome-wide association studies, researchers have become more interested in detecting interactions among genes or single nucleotide polymorphisms (SNPs). In this research, we carried out a two-stage model selection procedure to detect interacting gene pairs or SNP pairs associated with four important traits of outbred mice, including glucose, high-density lipoprotein cholesterol, diastolic blood pressure and triglyceride. In the first stage, a variance heterogeneity test was used to screen for candidate SNPs. In the second stage, the Lasso method and single pair analysis were used to select two-way interactions. Moreover, the shared Gene Ontology information about the selected interacting gene pairs was considered to study the interactions auxiliarily. Based on this method, we not only replicated the identification of important SNPs associated with each trait of outbred mice, but also found some SNP pairs and gene pairs with significant interaction effects on each trait. Simulation studies were also conducted to evaluate the performance of the two-stage method in different situations. | CommonCrawl |
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